most of Topology_Euclidean_Space (now Elementary_Topology) requires fewer dependencies
--- a/src/HOL/Analysis/Conformal_Mappings.thy Thu Dec 27 22:54:29 2018 +0100
+++ b/src/HOL/Analysis/Conformal_Mappings.thy Thu Dec 27 23:38:55 2018 +0100
@@ -3260,7 +3260,7 @@
moreover have "isolated_singularity_at h z"
unfolding isolated_singularity_at_def h_def
apply (rule exI[where x=e])
- using e_holo e_nz \<open>e>0\<close> by (metis Topology_Euclidean_Space.open_ball analytic_on_open
+ using e_holo e_nz \<open>e>0\<close> by (metis open_ball analytic_on_open
holomorphic_on_inverse open_delete)
ultimately show ?thesis
using P_exist[of h] by auto
@@ -3482,7 +3482,7 @@
have [intro]: "fp \<midarrow>z\<rightarrow>fp z" "gp \<midarrow>z\<rightarrow>gp z"
using fr(1) \<open>fr>0\<close> gr(1) \<open>gr>0\<close>
- by (meson Topology_Euclidean_Space.open_ball ball_subset_cball centre_in_ball
+ by (meson open_ball ball_subset_cball centre_in_ball
continuous_on_eq_continuous_at continuous_within holomorphic_on_imp_continuous_on
holomorphic_on_subset)+
have ?thesis when "fn+gn>0"
@@ -3958,7 +3958,7 @@
apply (elim Lim_transform_within_open[where s="ball z r"])
using r by auto
moreover have "g \<midarrow>z\<rightarrow>g z"
- by (metis (mono_tags, lifting) Topology_Euclidean_Space.open_ball at_within_open_subset
+ by (metis (mono_tags, lifting) open_ball at_within_open_subset
ball_subset_cball centre_in_ball continuous_on holomorphic_on_imp_continuous_on r(1,3) subsetCE)
ultimately have "(\<lambda>w. (g w * (w - z) powr of_int n) / g w) \<midarrow>z\<rightarrow> f z/g z"
apply (rule_tac tendsto_divide)
@@ -4088,7 +4088,7 @@
assume " \<not> n < 0"
define c where "c=(if n=0 then g z else 0)"
have [simp]:"g \<midarrow>z\<rightarrow> g z"
- by (metis Topology_Euclidean_Space.open_ball at_within_open ball_subset_cball centre_in_ball
+ by (metis open_ball at_within_open ball_subset_cball centre_in_ball
continuous_on holomorphic_on_imp_continuous_on holomorphic_on_subset r(1) r(3) )
have "\<forall>\<^sub>F x in at z. f x = g x * (x - z) ^ nat n"
unfolding eventually_at_topological
@@ -4154,7 +4154,7 @@
then show "LIM w at z. w - z :> at 0"
unfolding filterlim_at by (auto intro:tendsto_eq_intros)
show "isolated_singularity_at g z"
- by (meson Diff_subset Topology_Euclidean_Space.open_ball analytic_on_holomorphic
+ by (meson Diff_subset open_ball analytic_on_holomorphic
assms(1,2,3) holomorphic_on_subset isolated_singularity_at_def openE)
qed
then show "not_essential f z"
@@ -4590,7 +4590,7 @@
assume "\<not> (\<exists>\<^sub>F w in at z. g w \<noteq> 0)"
then have "\<forall>\<^sub>F w in nhds z. g w = 0"
unfolding eventually_at eventually_nhds frequently_at using \<open>g z = 0\<close>
- by (metis Topology_Euclidean_Space.open_ball UNIV_I centre_in_ball dist_commute mem_ball)
+ by (metis open_ball UNIV_I centre_in_ball dist_commute mem_ball)
then have "deriv g z = deriv (\<lambda>_. 0) z"
by (intro deriv_cong_ev) auto
then have "deriv g z = 0" by auto
--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Analysis/Elementary_Topology.thy Thu Dec 27 23:38:55 2018 +0100
@@ -0,0 +1,5098 @@
+(* Author: L C Paulson, University of Cambridge
+ Author: Amine Chaieb, University of Cambridge
+ Author: Robert Himmelmann, TU Muenchen
+ Author: Brian Huffman, Portland State University
+*)
+
+section \<open>Elementary Topology\<close>
+
+theory Elementary_Topology
+imports
+ "HOL-Library.Indicator_Function"
+ "HOL-Library.Countable_Set"
+ "HOL-Library.FuncSet"
+ "HOL-Library.Set_Idioms"
+ "HOL-Library.Infinite_Set"
+ Product_Vector
+begin
+
+(* FIXME: move elsewhere *)
+
+lemma approachable_lt_le: "(\<exists>(d::real) > 0. \<forall>x. f x < d \<longrightarrow> P x) \<longleftrightarrow> (\<exists>d>0. \<forall>x. f x \<le> d \<longrightarrow> P x)"
+ apply auto
+ apply (rule_tac x="d/2" in exI)
+ apply auto
+ done
+
+lemma approachable_lt_le2: \<comment> \<open>like the above, but pushes aside an extra formula\<close>
+ "(\<exists>(d::real) > 0. \<forall>x. Q x \<longrightarrow> f x < d \<longrightarrow> P x) \<longleftrightarrow> (\<exists>d>0. \<forall>x. f x \<le> d \<longrightarrow> Q x \<longrightarrow> P x)"
+ apply auto
+ apply (rule_tac x="d/2" in exI, auto)
+ done
+
+lemma triangle_lemma:
+ fixes x y z :: real
+ assumes x: "0 \<le> x"
+ and y: "0 \<le> y"
+ and z: "0 \<le> z"
+ and xy: "x\<^sup>2 \<le> y\<^sup>2 + z\<^sup>2"
+ shows "x \<le> y + z"
+proof -
+ have "y\<^sup>2 + z\<^sup>2 \<le> y\<^sup>2 + 2 * y * z + z\<^sup>2"
+ using z y by simp
+ with xy have th: "x\<^sup>2 \<le> (y + z)\<^sup>2"
+ by (simp add: power2_eq_square field_simps)
+ from y z have yz: "y + z \<ge> 0"
+ by arith
+ from power2_le_imp_le[OF th yz] show ?thesis .
+qed
+
+definition (in monoid_add) support_on :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'b set"
+ where "support_on s f = {x\<in>s. f x \<noteq> 0}"
+
+lemma in_support_on: "x \<in> support_on s f \<longleftrightarrow> x \<in> s \<and> f x \<noteq> 0"
+ by (simp add: support_on_def)
+
+lemma support_on_simps[simp]:
+ "support_on {} f = {}"
+ "support_on (insert x s) f =
+ (if f x = 0 then support_on s f else insert x (support_on s f))"
+ "support_on (s \<union> t) f = support_on s f \<union> support_on t f"
+ "support_on (s \<inter> t) f = support_on s f \<inter> support_on t f"
+ "support_on (s - t) f = support_on s f - support_on t f"
+ "support_on (f ` s) g = f ` (support_on s (g \<circ> f))"
+ unfolding support_on_def by auto
+
+lemma support_on_cong:
+ "(\<And>x. x \<in> s \<Longrightarrow> f x = 0 \<longleftrightarrow> g x = 0) \<Longrightarrow> support_on s f = support_on s g"
+ by (auto simp: support_on_def)
+
+lemma support_on_if: "a \<noteq> 0 \<Longrightarrow> support_on A (\<lambda>x. if P x then a else 0) = {x\<in>A. P x}"
+ by (auto simp: support_on_def)
+
+lemma support_on_if_subset: "support_on A (\<lambda>x. if P x then a else 0) \<subseteq> {x \<in> A. P x}"
+ by (auto simp: support_on_def)
+
+lemma finite_support[intro]: "finite S \<Longrightarrow> finite (support_on S f)"
+ unfolding support_on_def by auto
+
+(* TODO: is supp_sum really needed? TODO: Generalize to Finite_Set.fold *)
+definition (in comm_monoid_add) supp_sum :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b set \<Rightarrow> 'a"
+ where "supp_sum f S = (\<Sum>x\<in>support_on S f. f x)"
+
+lemma supp_sum_empty[simp]: "supp_sum f {} = 0"
+ unfolding supp_sum_def by auto
+
+lemma supp_sum_insert[simp]:
+ "finite (support_on S f) \<Longrightarrow>
+ supp_sum f (insert x S) = (if x \<in> S then supp_sum f S else f x + supp_sum f S)"
+ by (simp add: supp_sum_def in_support_on insert_absorb)
+
+lemma supp_sum_divide_distrib: "supp_sum f A / (r::'a::field) = supp_sum (\<lambda>n. f n / r) A"
+ by (cases "r = 0")
+ (auto simp: supp_sum_def sum_divide_distrib intro!: sum.cong support_on_cong)
+
+(*END OF SUPPORT, ETC.*)
+
+lemma image_affinity_interval:
+ fixes c :: "'a::ordered_real_vector"
+ shows "((\<lambda>x. m *\<^sub>R x + c) ` {a..b}) =
+ (if {a..b}={} then {}
+ else if 0 \<le> m then {m *\<^sub>R a + c .. m *\<^sub>R b + c}
+ else {m *\<^sub>R b + c .. m *\<^sub>R a + c})"
+ (is "?lhs = ?rhs")
+proof (cases "m=0")
+ case True
+ then show ?thesis
+ by force
+next
+ case False
+ show ?thesis
+ proof
+ show "?lhs \<subseteq> ?rhs"
+ by (auto simp: scaleR_left_mono scaleR_left_mono_neg)
+ show "?rhs \<subseteq> ?lhs"
+ proof (clarsimp, intro conjI impI subsetI)
+ show "\<lbrakk>0 \<le> m; a \<le> b; x \<in> {m *\<^sub>R a + c..m *\<^sub>R b + c}\<rbrakk>
+ \<Longrightarrow> x \<in> (\<lambda>x. m *\<^sub>R x + c) ` {a..b}" for x
+ apply (rule_tac x="inverse m *\<^sub>R (x-c)" in rev_image_eqI)
+ using False apply (auto simp: le_diff_eq pos_le_divideRI)
+ using diff_le_eq pos_le_divideR_eq by force
+ show "\<lbrakk>\<not> 0 \<le> m; a \<le> b; x \<in> {m *\<^sub>R b + c..m *\<^sub>R a + c}\<rbrakk>
+ \<Longrightarrow> x \<in> (\<lambda>x. m *\<^sub>R x + c) ` {a..b}" for x
+ apply (rule_tac x="inverse m *\<^sub>R (x-c)" in rev_image_eqI)
+ apply (auto simp: diff_le_eq neg_le_divideR_eq)
+ using diff_eq_diff_less_eq linordered_field_class.sign_simps(11) minus_diff_eq not_less scaleR_le_cancel_left_neg by fastforce
+ qed
+ qed
+qed
+
+lemma countable_PiE:
+ "finite I \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> countable (F i)) \<Longrightarrow> countable (Pi\<^sub>E I F)"
+ by (induct I arbitrary: F rule: finite_induct) (auto simp: PiE_insert_eq)
+
+lemma open_sums:
+ fixes T :: "('b::real_normed_vector) set"
+ assumes "open S \<or> open T"
+ shows "open (\<Union>x\<in> S. \<Union>y \<in> T. {x + y})"
+ using assms
+proof
+ assume S: "open S"
+ show ?thesis
+ proof (clarsimp simp: open_dist)
+ fix x y
+ assume "x \<in> S" "y \<in> T"
+ with S obtain e where "e > 0" and e: "\<And>x'. dist x' x < e \<Longrightarrow> x' \<in> S"
+ by (auto simp: open_dist)
+ then have "\<And>z. dist z (x + y) < e \<Longrightarrow> \<exists>x\<in>S. \<exists>y\<in>T. z = x + y"
+ by (metis \<open>y \<in> T\<close> diff_add_cancel dist_add_cancel2)
+ then show "\<exists>e>0. \<forall>z. dist z (x + y) < e \<longrightarrow> (\<exists>x\<in>S. \<exists>y\<in>T. z = x + y)"
+ using \<open>0 < e\<close> \<open>x \<in> S\<close> by blast
+ qed
+next
+ assume T: "open T"
+ show ?thesis
+ proof (clarsimp simp: open_dist)
+ fix x y
+ assume "x \<in> S" "y \<in> T"
+ with T obtain e where "e > 0" and e: "\<And>x'. dist x' y < e \<Longrightarrow> x' \<in> T"
+ by (auto simp: open_dist)
+ then have "\<And>z. dist z (x + y) < e \<Longrightarrow> \<exists>x\<in>S. \<exists>y\<in>T. z = x + y"
+ by (metis \<open>x \<in> S\<close> add_diff_cancel_left' add_diff_eq diff_diff_add dist_norm)
+ then show "\<exists>e>0. \<forall>z. dist z (x + y) < e \<longrightarrow> (\<exists>x\<in>S. \<exists>y\<in>T. z = x + y)"
+ using \<open>0 < e\<close> \<open>y \<in> T\<close> by blast
+ qed
+qed
+
+
+subsection \<open>Topological Basis\<close>
+
+context topological_space
+begin
+
+definition%important "topological_basis B \<longleftrightarrow>
+ (\<forall>b\<in>B. open b) \<and> (\<forall>x. open x \<longrightarrow> (\<exists>B'. B' \<subseteq> B \<and> \<Union>B' = x))"
+
+lemma topological_basis:
+ "topological_basis B \<longleftrightarrow> (\<forall>x. open x \<longleftrightarrow> (\<exists>B'. B' \<subseteq> B \<and> \<Union>B' = x))"
+ unfolding topological_basis_def
+ apply safe
+ apply fastforce
+ apply fastforce
+ apply (erule_tac x=x in allE, simp)
+ apply (rule_tac x="{x}" in exI, auto)
+ done
+
+lemma topological_basis_iff:
+ assumes "\<And>B'. B' \<in> B \<Longrightarrow> open B'"
+ shows "topological_basis B \<longleftrightarrow> (\<forall>O'. open O' \<longrightarrow> (\<forall>x\<in>O'. \<exists>B'\<in>B. x \<in> B' \<and> B' \<subseteq> O'))"
+ (is "_ \<longleftrightarrow> ?rhs")
+proof safe
+ fix O' and x::'a
+ assume H: "topological_basis B" "open O'" "x \<in> O'"
+ then have "(\<exists>B'\<subseteq>B. \<Union>B' = O')" by (simp add: topological_basis_def)
+ then obtain B' where "B' \<subseteq> B" "O' = \<Union>B'" by auto
+ then show "\<exists>B'\<in>B. x \<in> B' \<and> B' \<subseteq> O'" using H by auto
+next
+ assume H: ?rhs
+ show "topological_basis B"
+ using assms unfolding topological_basis_def
+ proof safe
+ fix O' :: "'a set"
+ assume "open O'"
+ with H obtain f where "\<forall>x\<in>O'. f x \<in> B \<and> x \<in> f x \<and> f x \<subseteq> O'"
+ by (force intro: bchoice simp: Bex_def)
+ then show "\<exists>B'\<subseteq>B. \<Union>B' = O'"
+ by (auto intro: exI[where x="{f x |x. x \<in> O'}"])
+ qed
+qed
+
+lemma topological_basisI:
+ assumes "\<And>B'. B' \<in> B \<Longrightarrow> open B'"
+ and "\<And>O' x. open O' \<Longrightarrow> x \<in> O' \<Longrightarrow> \<exists>B'\<in>B. x \<in> B' \<and> B' \<subseteq> O'"
+ shows "topological_basis B"
+ using assms by (subst topological_basis_iff) auto
+
+lemma topological_basisE:
+ fixes O'
+ assumes "topological_basis B"
+ and "open O'"
+ and "x \<in> O'"
+ obtains B' where "B' \<in> B" "x \<in> B'" "B' \<subseteq> O'"
+proof atomize_elim
+ from assms have "\<And>B'. B'\<in>B \<Longrightarrow> open B'"
+ by (simp add: topological_basis_def)
+ with topological_basis_iff assms
+ show "\<exists>B'. B' \<in> B \<and> x \<in> B' \<and> B' \<subseteq> O'"
+ using assms by (simp add: Bex_def)
+qed
+
+lemma topological_basis_open:
+ assumes "topological_basis B"
+ and "X \<in> B"
+ shows "open X"
+ using assms by (simp add: topological_basis_def)
+
+lemma topological_basis_imp_subbasis:
+ assumes B: "topological_basis B"
+ shows "open = generate_topology B"
+proof (intro ext iffI)
+ fix S :: "'a set"
+ assume "open S"
+ with B obtain B' where "B' \<subseteq> B" "S = \<Union>B'"
+ unfolding topological_basis_def by blast
+ then show "generate_topology B S"
+ by (auto intro: generate_topology.intros dest: topological_basis_open)
+next
+ fix S :: "'a set"
+ assume "generate_topology B S"
+ then show "open S"
+ by induct (auto dest: topological_basis_open[OF B])
+qed
+
+lemma basis_dense:
+ fixes B :: "'a set set"
+ and f :: "'a set \<Rightarrow> 'a"
+ assumes "topological_basis B"
+ and choosefrom_basis: "\<And>B'. B' \<noteq> {} \<Longrightarrow> f B' \<in> B'"
+ shows "\<forall>X. open X \<longrightarrow> X \<noteq> {} \<longrightarrow> (\<exists>B' \<in> B. f B' \<in> X)"
+proof (intro allI impI)
+ fix X :: "'a set"
+ assume "open X" and "X \<noteq> {}"
+ from topological_basisE[OF \<open>topological_basis B\<close> \<open>open X\<close> choosefrom_basis[OF \<open>X \<noteq> {}\<close>]]
+ obtain B' where "B' \<in> B" "f X \<in> B'" "B' \<subseteq> X" .
+ then show "\<exists>B'\<in>B. f B' \<in> X"
+ by (auto intro!: choosefrom_basis)
+qed
+
+end
+
+lemma topological_basis_prod:
+ assumes A: "topological_basis A"
+ and B: "topological_basis B"
+ shows "topological_basis ((\<lambda>(a, b). a \<times> b) ` (A \<times> B))"
+ unfolding topological_basis_def
+proof (safe, simp_all del: ex_simps add: subset_image_iff ex_simps(1)[symmetric])
+ fix S :: "('a \<times> 'b) set"
+ assume "open S"
+ then show "\<exists>X\<subseteq>A \<times> B. (\<Union>(a,b)\<in>X. a \<times> b) = S"
+ proof (safe intro!: exI[of _ "{x\<in>A \<times> B. fst x \<times> snd x \<subseteq> S}"])
+ fix x y
+ assume "(x, y) \<in> S"
+ from open_prod_elim[OF \<open>open S\<close> this]
+ obtain a b where a: "open a""x \<in> a" and b: "open b" "y \<in> b" and "a \<times> b \<subseteq> S"
+ by (metis mem_Sigma_iff)
+ moreover
+ from A a obtain A0 where "A0 \<in> A" "x \<in> A0" "A0 \<subseteq> a"
+ by (rule topological_basisE)
+ moreover
+ from B b obtain B0 where "B0 \<in> B" "y \<in> B0" "B0 \<subseteq> b"
+ by (rule topological_basisE)
+ ultimately show "(x, y) \<in> (\<Union>(a, b)\<in>{X \<in> A \<times> B. fst X \<times> snd X \<subseteq> S}. a \<times> b)"
+ by (intro UN_I[of "(A0, B0)"]) auto
+ qed auto
+qed (metis A B topological_basis_open open_Times)
+
+
+subsection \<open>Countable Basis\<close>
+
+locale%important countable_basis =
+ fixes B :: "'a::topological_space set set"
+ assumes is_basis: "topological_basis B"
+ and countable_basis: "countable B"
+begin
+
+lemma open_countable_basis_ex:
+ assumes "open X"
+ shows "\<exists>B' \<subseteq> B. X = \<Union>B'"
+ using assms countable_basis is_basis
+ unfolding topological_basis_def by blast
+
+lemma open_countable_basisE:
+ assumes "open X"
+ obtains B' where "B' \<subseteq> B" "X = \<Union>B'"
+ using assms open_countable_basis_ex
+ by atomize_elim simp
+
+lemma countable_dense_exists:
+ "\<exists>D::'a set. countable D \<and> (\<forall>X. open X \<longrightarrow> X \<noteq> {} \<longrightarrow> (\<exists>d \<in> D. d \<in> X))"
+proof -
+ let ?f = "(\<lambda>B'. SOME x. x \<in> B')"
+ have "countable (?f ` B)" using countable_basis by simp
+ with basis_dense[OF is_basis, of ?f] show ?thesis
+ by (intro exI[where x="?f ` B"]) (metis (mono_tags) all_not_in_conv imageI someI)
+qed
+
+lemma countable_dense_setE:
+ obtains D :: "'a set"
+ where "countable D" "\<And>X. open X \<Longrightarrow> X \<noteq> {} \<Longrightarrow> \<exists>d \<in> D. d \<in> X"
+ using countable_dense_exists by blast
+
+end
+
+lemma (in first_countable_topology) first_countable_basisE:
+ fixes x :: 'a
+ obtains \<A> where "countable \<A>" "\<And>A. A \<in> \<A> \<Longrightarrow> x \<in> A" "\<And>A. A \<in> \<A> \<Longrightarrow> open A"
+ "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> (\<exists>A\<in>\<A>. A \<subseteq> S)"
+proof -
+ obtain \<A> where \<A>: "(\<forall>i::nat. x \<in> \<A> i \<and> open (\<A> i))" "(\<forall>S. open S \<and> x \<in> S \<longrightarrow> (\<exists>i. \<A> i \<subseteq> S))"
+ using first_countable_basis[of x] by metis
+ show thesis
+ proof
+ show "countable (range \<A>)"
+ by simp
+ qed (use \<A> in auto)
+qed
+
+lemma (in first_countable_topology) first_countable_basis_Int_stableE:
+ obtains \<A> where "countable \<A>" "\<And>A. A \<in> \<A> \<Longrightarrow> x \<in> A" "\<And>A. A \<in> \<A> \<Longrightarrow> open A"
+ "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> (\<exists>A\<in>\<A>. A \<subseteq> S)"
+ "\<And>A B. A \<in> \<A> \<Longrightarrow> B \<in> \<A> \<Longrightarrow> A \<inter> B \<in> \<A>"
+proof atomize_elim
+ obtain \<B> where \<B>:
+ "countable \<B>"
+ "\<And>B. B \<in> \<B> \<Longrightarrow> x \<in> B"
+ "\<And>B. B \<in> \<B> \<Longrightarrow> open B"
+ "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> \<exists>B\<in>\<B>. B \<subseteq> S"
+ by (rule first_countable_basisE) blast
+ define \<A> where [abs_def]:
+ "\<A> = (\<lambda>N. \<Inter>((\<lambda>n. from_nat_into \<B> n) ` N)) ` (Collect finite::nat set set)"
+ then show "\<exists>\<A>. countable \<A> \<and> (\<forall>A. A \<in> \<A> \<longrightarrow> x \<in> A) \<and> (\<forall>A. A \<in> \<A> \<longrightarrow> open A) \<and>
+ (\<forall>S. open S \<longrightarrow> x \<in> S \<longrightarrow> (\<exists>A\<in>\<A>. A \<subseteq> S)) \<and> (\<forall>A B. A \<in> \<A> \<longrightarrow> B \<in> \<A> \<longrightarrow> A \<inter> B \<in> \<A>)"
+ proof (safe intro!: exI[where x=\<A>])
+ show "countable \<A>"
+ unfolding \<A>_def by (intro countable_image countable_Collect_finite)
+ fix A
+ assume "A \<in> \<A>"
+ then show "x \<in> A" "open A"
+ using \<B>(4)[OF open_UNIV] by (auto simp: \<A>_def intro: \<B> from_nat_into)
+ next
+ let ?int = "\<lambda>N. \<Inter>(from_nat_into \<B> ` N)"
+ fix A B
+ assume "A \<in> \<A>" "B \<in> \<A>"
+ then obtain N M where "A = ?int N" "B = ?int M" "finite (N \<union> M)"
+ by (auto simp: \<A>_def)
+ then show "A \<inter> B \<in> \<A>"
+ by (auto simp: \<A>_def intro!: image_eqI[where x="N \<union> M"])
+ next
+ fix S
+ assume "open S" "x \<in> S"
+ then obtain a where a: "a\<in>\<B>" "a \<subseteq> S" using \<B> by blast
+ then show "\<exists>a\<in>\<A>. a \<subseteq> S" using a \<B>
+ by (intro bexI[where x=a]) (auto simp: \<A>_def intro: image_eqI[where x="{to_nat_on \<B> a}"])
+ qed
+qed
+
+lemma (in topological_space) first_countableI:
+ assumes "countable \<A>"
+ and 1: "\<And>A. A \<in> \<A> \<Longrightarrow> x \<in> A" "\<And>A. A \<in> \<A> \<Longrightarrow> open A"
+ and 2: "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> \<exists>A\<in>\<A>. A \<subseteq> S"
+ shows "\<exists>\<A>::nat \<Rightarrow> 'a set. (\<forall>i. x \<in> \<A> i \<and> open (\<A> i)) \<and> (\<forall>S. open S \<and> x \<in> S \<longrightarrow> (\<exists>i. \<A> i \<subseteq> S))"
+proof (safe intro!: exI[of _ "from_nat_into \<A>"])
+ fix i
+ have "\<A> \<noteq> {}" using 2[of UNIV] by auto
+ show "x \<in> from_nat_into \<A> i" "open (from_nat_into \<A> i)"
+ using range_from_nat_into_subset[OF \<open>\<A> \<noteq> {}\<close>] 1 by auto
+next
+ fix S
+ assume "open S" "x\<in>S" from 2[OF this]
+ show "\<exists>i. from_nat_into \<A> i \<subseteq> S"
+ using subset_range_from_nat_into[OF \<open>countable \<A>\<close>] by auto
+qed
+
+instance prod :: (first_countable_topology, first_countable_topology) first_countable_topology
+proof
+ fix x :: "'a \<times> 'b"
+ obtain \<A> where \<A>:
+ "countable \<A>"
+ "\<And>a. a \<in> \<A> \<Longrightarrow> fst x \<in> a"
+ "\<And>a. a \<in> \<A> \<Longrightarrow> open a"
+ "\<And>S. open S \<Longrightarrow> fst x \<in> S \<Longrightarrow> \<exists>a\<in>\<A>. a \<subseteq> S"
+ by (rule first_countable_basisE[of "fst x"]) blast
+ obtain B where B:
+ "countable B"
+ "\<And>a. a \<in> B \<Longrightarrow> snd x \<in> a"
+ "\<And>a. a \<in> B \<Longrightarrow> open a"
+ "\<And>S. open S \<Longrightarrow> snd x \<in> S \<Longrightarrow> \<exists>a\<in>B. a \<subseteq> S"
+ by (rule first_countable_basisE[of "snd x"]) blast
+ show "\<exists>\<A>::nat \<Rightarrow> ('a \<times> 'b) set.
+ (\<forall>i. x \<in> \<A> i \<and> open (\<A> i)) \<and> (\<forall>S. open S \<and> x \<in> S \<longrightarrow> (\<exists>i. \<A> i \<subseteq> S))"
+ proof (rule first_countableI[of "(\<lambda>(a, b). a \<times> b) ` (\<A> \<times> B)"], safe)
+ fix a b
+ assume x: "a \<in> \<A>" "b \<in> B"
+ show "x \<in> a \<times> b"
+ by (simp add: \<A>(2) B(2) mem_Times_iff x)
+ show "open (a \<times> b)"
+ by (simp add: \<A>(3) B(3) open_Times x)
+ next
+ fix S
+ assume "open S" "x \<in> S"
+ then obtain a' b' where a'b': "open a'" "open b'" "x \<in> a' \<times> b'" "a' \<times> b' \<subseteq> S"
+ by (rule open_prod_elim)
+ moreover
+ from a'b' \<A>(4)[of a'] B(4)[of b']
+ obtain a b where "a \<in> \<A>" "a \<subseteq> a'" "b \<in> B" "b \<subseteq> b'"
+ by auto
+ ultimately
+ show "\<exists>a\<in>(\<lambda>(a, b). a \<times> b) ` (\<A> \<times> B). a \<subseteq> S"
+ by (auto intro!: bexI[of _ "a \<times> b"] bexI[of _ a] bexI[of _ b])
+ qed (simp add: \<A> B)
+qed
+
+class second_countable_topology = topological_space +
+ assumes ex_countable_subbasis:
+ "\<exists>B::'a::topological_space set set. countable B \<and> open = generate_topology B"
+begin
+
+lemma ex_countable_basis: "\<exists>B::'a set set. countable B \<and> topological_basis B"
+proof -
+ from ex_countable_subbasis obtain B where B: "countable B" "open = generate_topology B"
+ by blast
+ let ?B = "Inter ` {b. finite b \<and> b \<subseteq> B }"
+
+ show ?thesis
+ proof (intro exI conjI)
+ show "countable ?B"
+ by (intro countable_image countable_Collect_finite_subset B)
+ {
+ fix S
+ assume "open S"
+ then have "\<exists>B'\<subseteq>{b. finite b \<and> b \<subseteq> B}. (\<Union>b\<in>B'. \<Inter>b) = S"
+ unfolding B
+ proof induct
+ case UNIV
+ show ?case by (intro exI[of _ "{{}}"]) simp
+ next
+ case (Int a b)
+ then obtain x y where x: "a = \<Union>(Inter ` x)" "\<And>i. i \<in> x \<Longrightarrow> finite i \<and> i \<subseteq> B"
+ and y: "b = \<Union>(Inter ` y)" "\<And>i. i \<in> y \<Longrightarrow> finite i \<and> i \<subseteq> B"
+ by blast
+ show ?case
+ unfolding x y Int_UN_distrib2
+ by (intro exI[of _ "{i \<union> j| i j. i \<in> x \<and> j \<in> y}"]) (auto dest: x(2) y(2))
+ next
+ case (UN K)
+ then have "\<forall>k\<in>K. \<exists>B'\<subseteq>{b. finite b \<and> b \<subseteq> B}. \<Union> (Inter ` B') = k" by auto
+ then obtain k where
+ "\<forall>ka\<in>K. k ka \<subseteq> {b. finite b \<and> b \<subseteq> B} \<and> \<Union>(Inter ` (k ka)) = ka"
+ unfolding bchoice_iff ..
+ then show "\<exists>B'\<subseteq>{b. finite b \<and> b \<subseteq> B}. \<Union> (Inter ` B') = \<Union>K"
+ by (intro exI[of _ "\<Union>(k ` K)"]) auto
+ next
+ case (Basis S)
+ then show ?case
+ by (intro exI[of _ "{{S}}"]) auto
+ qed
+ then have "(\<exists>B'\<subseteq>Inter ` {b. finite b \<and> b \<subseteq> B}. \<Union>B' = S)"
+ unfolding subset_image_iff by blast }
+ then show "topological_basis ?B"
+ unfolding topological_space_class.topological_basis_def
+ by (safe intro!: topological_space_class.open_Inter)
+ (simp_all add: B generate_topology.Basis subset_eq)
+ qed
+qed
+
+end
+
+sublocale second_countable_topology <
+ countable_basis "SOME B. countable B \<and> topological_basis B"
+ using someI_ex[OF ex_countable_basis]
+ by unfold_locales safe
+
+instance prod :: (second_countable_topology, second_countable_topology) second_countable_topology
+proof
+ obtain A :: "'a set set" where "countable A" "topological_basis A"
+ using ex_countable_basis by auto
+ moreover
+ obtain B :: "'b set set" where "countable B" "topological_basis B"
+ using ex_countable_basis by auto
+ ultimately show "\<exists>B::('a \<times> 'b) set set. countable B \<and> open = generate_topology B"
+ by (auto intro!: exI[of _ "(\<lambda>(a, b). a \<times> b) ` (A \<times> B)"] topological_basis_prod
+ topological_basis_imp_subbasis)
+qed
+
+instance second_countable_topology \<subseteq> first_countable_topology
+proof
+ fix x :: 'a
+ define B :: "'a set set" where "B = (SOME B. countable B \<and> topological_basis B)"
+ then have B: "countable B" "topological_basis B"
+ using countable_basis is_basis
+ by (auto simp: countable_basis is_basis)
+ then show "\<exists>A::nat \<Rightarrow> 'a set.
+ (\<forall>i. x \<in> A i \<and> open (A i)) \<and> (\<forall>S. open S \<and> x \<in> S \<longrightarrow> (\<exists>i. A i \<subseteq> S))"
+ by (intro first_countableI[of "{b\<in>B. x \<in> b}"])
+ (fastforce simp: topological_space_class.topological_basis_def)+
+qed
+
+instance nat :: second_countable_topology
+proof
+ show "\<exists>B::nat set set. countable B \<and> open = generate_topology B"
+ by (intro exI[of _ "range lessThan \<union> range greaterThan"]) (auto simp: open_nat_def)
+qed
+
+lemma countable_separating_set_linorder1:
+ shows "\<exists>B::('a::{linorder_topology, second_countable_topology} set). countable B \<and> (\<forall>x y. x < y \<longrightarrow> (\<exists>b \<in> B. x < b \<and> b \<le> y))"
+proof -
+ obtain A::"'a set set" where "countable A" "topological_basis A" using ex_countable_basis by auto
+ define B1 where "B1 = {(LEAST x. x \<in> U)| U. U \<in> A}"
+ then have "countable B1" using \<open>countable A\<close> by (simp add: Setcompr_eq_image)
+ define B2 where "B2 = {(SOME x. x \<in> U)| U. U \<in> A}"
+ then have "countable B2" using \<open>countable A\<close> by (simp add: Setcompr_eq_image)
+ have "\<exists>b \<in> B1 \<union> B2. x < b \<and> b \<le> y" if "x < y" for x y
+ proof (cases)
+ assume "\<exists>z. x < z \<and> z < y"
+ then obtain z where z: "x < z \<and> z < y" by auto
+ define U where "U = {x<..<y}"
+ then have "open U" by simp
+ moreover have "z \<in> U" using z U_def by simp
+ ultimately obtain V where "V \<in> A" "z \<in> V" "V \<subseteq> U" using topological_basisE[OF \<open>topological_basis A\<close>] by auto
+ define w where "w = (SOME x. x \<in> V)"
+ then have "w \<in> V" using \<open>z \<in> V\<close> by (metis someI2)
+ then have "x < w \<and> w \<le> y" using \<open>w \<in> V\<close> \<open>V \<subseteq> U\<close> U_def by fastforce
+ moreover have "w \<in> B1 \<union> B2" using w_def B2_def \<open>V \<in> A\<close> by auto
+ ultimately show ?thesis by auto
+ next
+ assume "\<not>(\<exists>z. x < z \<and> z < y)"
+ then have *: "\<And>z. z > x \<Longrightarrow> z \<ge> y" by auto
+ define U where "U = {x<..}"
+ then have "open U" by simp
+ moreover have "y \<in> U" using \<open>x < y\<close> U_def by simp
+ ultimately obtain "V" where "V \<in> A" "y \<in> V" "V \<subseteq> U" using topological_basisE[OF \<open>topological_basis A\<close>] by auto
+ have "U = {y..}" unfolding U_def using * \<open>x < y\<close> by auto
+ then have "V \<subseteq> {y..}" using \<open>V \<subseteq> U\<close> by simp
+ then have "(LEAST w. w \<in> V) = y" using \<open>y \<in> V\<close> by (meson Least_equality atLeast_iff subsetCE)
+ then have "y \<in> B1 \<union> B2" using \<open>V \<in> A\<close> B1_def by auto
+ moreover have "x < y \<and> y \<le> y" using \<open>x < y\<close> by simp
+ ultimately show ?thesis by auto
+ qed
+ moreover have "countable (B1 \<union> B2)" using \<open>countable B1\<close> \<open>countable B2\<close> by simp
+ ultimately show ?thesis by auto
+qed
+
+lemma countable_separating_set_linorder2:
+ shows "\<exists>B::('a::{linorder_topology, second_countable_topology} set). countable B \<and> (\<forall>x y. x < y \<longrightarrow> (\<exists>b \<in> B. x \<le> b \<and> b < y))"
+proof -
+ obtain A::"'a set set" where "countable A" "topological_basis A" using ex_countable_basis by auto
+ define B1 where "B1 = {(GREATEST x. x \<in> U) | U. U \<in> A}"
+ then have "countable B1" using \<open>countable A\<close> by (simp add: Setcompr_eq_image)
+ define B2 where "B2 = {(SOME x. x \<in> U)| U. U \<in> A}"
+ then have "countable B2" using \<open>countable A\<close> by (simp add: Setcompr_eq_image)
+ have "\<exists>b \<in> B1 \<union> B2. x \<le> b \<and> b < y" if "x < y" for x y
+ proof (cases)
+ assume "\<exists>z. x < z \<and> z < y"
+ then obtain z where z: "x < z \<and> z < y" by auto
+ define U where "U = {x<..<y}"
+ then have "open U" by simp
+ moreover have "z \<in> U" using z U_def by simp
+ ultimately obtain "V" where "V \<in> A" "z \<in> V" "V \<subseteq> U" using topological_basisE[OF \<open>topological_basis A\<close>] by auto
+ define w where "w = (SOME x. x \<in> V)"
+ then have "w \<in> V" using \<open>z \<in> V\<close> by (metis someI2)
+ then have "x \<le> w \<and> w < y" using \<open>w \<in> V\<close> \<open>V \<subseteq> U\<close> U_def by fastforce
+ moreover have "w \<in> B1 \<union> B2" using w_def B2_def \<open>V \<in> A\<close> by auto
+ ultimately show ?thesis by auto
+ next
+ assume "\<not>(\<exists>z. x < z \<and> z < y)"
+ then have *: "\<And>z. z < y \<Longrightarrow> z \<le> x" using leI by blast
+ define U where "U = {..<y}"
+ then have "open U" by simp
+ moreover have "x \<in> U" using \<open>x < y\<close> U_def by simp
+ ultimately obtain "V" where "V \<in> A" "x \<in> V" "V \<subseteq> U" using topological_basisE[OF \<open>topological_basis A\<close>] by auto
+ have "U = {..x}" unfolding U_def using * \<open>x < y\<close> by auto
+ then have "V \<subseteq> {..x}" using \<open>V \<subseteq> U\<close> by simp
+ then have "(GREATEST x. x \<in> V) = x" using \<open>x \<in> V\<close> by (meson Greatest_equality atMost_iff subsetCE)
+ then have "x \<in> B1 \<union> B2" using \<open>V \<in> A\<close> B1_def by auto
+ moreover have "x \<le> x \<and> x < y" using \<open>x < y\<close> by simp
+ ultimately show ?thesis by auto
+ qed
+ moreover have "countable (B1 \<union> B2)" using \<open>countable B1\<close> \<open>countable B2\<close> by simp
+ ultimately show ?thesis by auto
+qed
+
+lemma countable_separating_set_dense_linorder:
+ shows "\<exists>B::('a::{linorder_topology, dense_linorder, second_countable_topology} set). countable B \<and> (\<forall>x y. x < y \<longrightarrow> (\<exists>b \<in> B. x < b \<and> b < y))"
+proof -
+ obtain B::"'a set" where B: "countable B" "\<And>x y. x < y \<Longrightarrow> (\<exists>b \<in> B. x < b \<and> b \<le> y)"
+ using countable_separating_set_linorder1 by auto
+ have "\<exists>b \<in> B. x < b \<and> b < y" if "x < y" for x y
+ proof -
+ obtain z where "x < z" "z < y" using \<open>x < y\<close> dense by blast
+ then obtain b where "b \<in> B" "x < b \<and> b \<le> z" using B(2) by auto
+ then have "x < b \<and> b < y" using \<open>z < y\<close> by auto
+ then show ?thesis using \<open>b \<in> B\<close> by auto
+ qed
+ then show ?thesis using B(1) by auto
+qed
+
+subsection%important \<open>Polish spaces\<close>
+
+text \<open>Textbooks define Polish spaces as completely metrizable.
+ We assume the topology to be complete for a given metric.\<close>
+
+class polish_space = complete_space + second_countable_topology
+
+subsection \<open>General notion of a topology as a value\<close>
+
+definition%important "istopology L \<longleftrightarrow>
+ L {} \<and> (\<forall>S T. L S \<longrightarrow> L T \<longrightarrow> L (S \<inter> T)) \<and> (\<forall>K. Ball K L \<longrightarrow> L (\<Union>K))"
+
+typedef%important 'a topology = "{L::('a set) \<Rightarrow> bool. istopology L}"
+ morphisms "openin" "topology"
+ unfolding istopology_def by blast
+
+lemma istopology_openin[intro]: "istopology(openin U)"
+ using openin[of U] by blast
+
+lemma topology_inverse': "istopology U \<Longrightarrow> openin (topology U) = U"
+ using topology_inverse[unfolded mem_Collect_eq] .
+
+lemma topology_inverse_iff: "istopology U \<longleftrightarrow> openin (topology U) = U"
+ using topology_inverse[of U] istopology_openin[of "topology U"] by auto
+
+lemma topology_eq: "T1 = T2 \<longleftrightarrow> (\<forall>S. openin T1 S \<longleftrightarrow> openin T2 S)"
+proof
+ assume "T1 = T2"
+ then show "\<forall>S. openin T1 S \<longleftrightarrow> openin T2 S" by simp
+next
+ assume H: "\<forall>S. openin T1 S \<longleftrightarrow> openin T2 S"
+ then have "openin T1 = openin T2" by (simp add: fun_eq_iff)
+ then have "topology (openin T1) = topology (openin T2)" by simp
+ then show "T1 = T2" unfolding openin_inverse .
+qed
+
+
+text\<open>The "universe": the union of all sets in the topology.\<close>
+definition "topspace T = \<Union>{S. openin T S}"
+
+subsubsection \<open>Main properties of open sets\<close>
+
+proposition openin_clauses:
+ fixes U :: "'a topology"
+ shows
+ "openin U {}"
+ "\<And>S T. openin U S \<Longrightarrow> openin U T \<Longrightarrow> openin U (S\<inter>T)"
+ "\<And>K. (\<forall>S \<in> K. openin U S) \<Longrightarrow> openin U (\<Union>K)"
+ using openin[of U] unfolding istopology_def mem_Collect_eq by fast+
+
+lemma openin_subset[intro]: "openin U S \<Longrightarrow> S \<subseteq> topspace U"
+ unfolding topspace_def by blast
+
+lemma openin_empty[simp]: "openin U {}"
+ by (rule openin_clauses)
+
+lemma openin_Int[intro]: "openin U S \<Longrightarrow> openin U T \<Longrightarrow> openin U (S \<inter> T)"
+ by (rule openin_clauses)
+
+lemma openin_Union[intro]: "(\<And>S. S \<in> K \<Longrightarrow> openin U S) \<Longrightarrow> openin U (\<Union>K)"
+ using openin_clauses by blast
+
+lemma openin_Un[intro]: "openin U S \<Longrightarrow> openin U T \<Longrightarrow> openin U (S \<union> T)"
+ using openin_Union[of "{S,T}" U] by auto
+
+lemma openin_topspace[intro, simp]: "openin U (topspace U)"
+ by (force simp: openin_Union topspace_def)
+
+lemma openin_subopen: "openin U S \<longleftrightarrow> (\<forall>x \<in> S. \<exists>T. openin U T \<and> x \<in> T \<and> T \<subseteq> S)"
+ (is "?lhs \<longleftrightarrow> ?rhs")
+proof
+ assume ?lhs
+ then show ?rhs by auto
+next
+ assume H: ?rhs
+ let ?t = "\<Union>{T. openin U T \<and> T \<subseteq> S}"
+ have "openin U ?t" by (force simp: openin_Union)
+ also have "?t = S" using H by auto
+ finally show "openin U S" .
+qed
+
+lemma openin_INT [intro]:
+ assumes "finite I"
+ "\<And>i. i \<in> I \<Longrightarrow> openin T (U i)"
+ shows "openin T ((\<Inter>i \<in> I. U i) \<inter> topspace T)"
+using assms by (induct, auto simp: inf_sup_aci(2) openin_Int)
+
+lemma openin_INT2 [intro]:
+ assumes "finite I" "I \<noteq> {}"
+ "\<And>i. i \<in> I \<Longrightarrow> openin T (U i)"
+ shows "openin T (\<Inter>i \<in> I. U i)"
+proof -
+ have "(\<Inter>i \<in> I. U i) \<subseteq> topspace T"
+ using \<open>I \<noteq> {}\<close> openin_subset[OF assms(3)] by auto
+ then show ?thesis
+ using openin_INT[of _ _ U, OF assms(1) assms(3)] by (simp add: inf.absorb2 inf_commute)
+qed
+
+lemma openin_Inter [intro]:
+ assumes "finite \<F>" "\<F> \<noteq> {}" "\<And>X. X \<in> \<F> \<Longrightarrow> openin T X" shows "openin T (\<Inter>\<F>)"
+ by (metis (full_types) assms openin_INT2 image_ident)
+
+lemma openin_Int_Inter:
+ assumes "finite \<F>" "openin T U" "\<And>X. X \<in> \<F> \<Longrightarrow> openin T X" shows "openin T (U \<inter> \<Inter>\<F>)"
+ using openin_Inter [of "insert U \<F>"] assms by auto
+
+
+subsubsection \<open>Closed sets\<close>
+
+definition%important "closedin U S \<longleftrightarrow> S \<subseteq> topspace U \<and> openin U (topspace U - S)"
+
+lemma closedin_subset: "closedin U S \<Longrightarrow> S \<subseteq> topspace U"
+ by (metis closedin_def)
+
+lemma closedin_empty[simp]: "closedin U {}"
+ by (simp add: closedin_def)
+
+lemma closedin_topspace[intro, simp]: "closedin U (topspace U)"
+ by (simp add: closedin_def)
+
+lemma closedin_Un[intro]: "closedin U S \<Longrightarrow> closedin U T \<Longrightarrow> closedin U (S \<union> T)"
+ by (auto simp: Diff_Un closedin_def)
+
+lemma Diff_Inter[intro]: "A - \<Inter>S = \<Union>{A - s|s. s\<in>S}"
+ by auto
+
+lemma closedin_Union:
+ assumes "finite S" "\<And>T. T \<in> S \<Longrightarrow> closedin U T"
+ shows "closedin U (\<Union>S)"
+ using assms by induction auto
+
+lemma closedin_Inter[intro]:
+ assumes Ke: "K \<noteq> {}"
+ and Kc: "\<And>S. S \<in>K \<Longrightarrow> closedin U S"
+ shows "closedin U (\<Inter>K)"
+ using Ke Kc unfolding closedin_def Diff_Inter by auto
+
+lemma closedin_INT[intro]:
+ assumes "A \<noteq> {}" "\<And>x. x \<in> A \<Longrightarrow> closedin U (B x)"
+ shows "closedin U (\<Inter>x\<in>A. B x)"
+ apply (rule closedin_Inter)
+ using assms
+ apply auto
+ done
+
+lemma closedin_Int[intro]: "closedin U S \<Longrightarrow> closedin U T \<Longrightarrow> closedin U (S \<inter> T)"
+ using closedin_Inter[of "{S,T}" U] by auto
+
+lemma openin_closedin_eq: "openin U S \<longleftrightarrow> S \<subseteq> topspace U \<and> closedin U (topspace U - S)"
+ apply (auto simp: closedin_def Diff_Diff_Int inf_absorb2)
+ apply (metis openin_subset subset_eq)
+ done
+
+lemma openin_closedin: "S \<subseteq> topspace U \<Longrightarrow> (openin U S \<longleftrightarrow> closedin U (topspace U - S))"
+ by (simp add: openin_closedin_eq)
+
+lemma openin_diff[intro]:
+ assumes oS: "openin U S"
+ and cT: "closedin U T"
+ shows "openin U (S - T)"
+proof -
+ have "S - T = S \<inter> (topspace U - T)" using openin_subset[of U S] oS cT
+ by (auto simp: topspace_def openin_subset)
+ then show ?thesis using oS cT
+ by (auto simp: closedin_def)
+qed
+
+lemma closedin_diff[intro]:
+ assumes oS: "closedin U S"
+ and cT: "openin U T"
+ shows "closedin U (S - T)"
+proof -
+ have "S - T = S \<inter> (topspace U - T)"
+ using closedin_subset[of U S] oS cT by (auto simp: topspace_def)
+ then show ?thesis
+ using oS cT by (auto simp: openin_closedin_eq)
+qed
+
+
+subsection\<open>The discrete topology\<close>
+
+definition discrete_topology where "discrete_topology U \<equiv> topology (\<lambda>S. S \<subseteq> U)"
+
+lemma openin_discrete_topology [simp]: "openin (discrete_topology U) S \<longleftrightarrow> S \<subseteq> U"
+proof -
+ have "istopology (\<lambda>S. S \<subseteq> U)"
+ by (auto simp: istopology_def)
+ then show ?thesis
+ by (simp add: discrete_topology_def topology_inverse')
+qed
+
+lemma topspace_discrete_topology [simp]: "topspace(discrete_topology U) = U"
+ by (meson openin_discrete_topology openin_subset openin_topspace order_refl subset_antisym)
+
+lemma closedin_discrete_topology [simp]: "closedin (discrete_topology U) S \<longleftrightarrow> S \<subseteq> U"
+ by (simp add: closedin_def)
+
+lemma discrete_topology_unique:
+ "discrete_topology U = X \<longleftrightarrow> topspace X = U \<and> (\<forall>x \<in> U. openin X {x})" (is "?lhs = ?rhs")
+proof
+ assume R: ?rhs
+ then have "openin X S" if "S \<subseteq> U" for S
+ using openin_subopen subsetD that by fastforce
+ moreover have "x \<in> topspace X" if "openin X S" and "x \<in> S" for x S
+ using openin_subset that by blast
+ ultimately
+ show ?lhs
+ using R by (auto simp: topology_eq)
+qed auto
+
+lemma discrete_topology_unique_alt:
+ "discrete_topology U = X \<longleftrightarrow> topspace X \<subseteq> U \<and> (\<forall>x \<in> U. openin X {x})"
+ using openin_subset
+ by (auto simp: discrete_topology_unique)
+
+lemma subtopology_eq_discrete_topology_empty:
+ "X = discrete_topology {} \<longleftrightarrow> topspace X = {}"
+ using discrete_topology_unique [of "{}" X] by auto
+
+lemma subtopology_eq_discrete_topology_sing:
+ "X = discrete_topology {a} \<longleftrightarrow> topspace X = {a}"
+ by (metis discrete_topology_unique openin_topspace singletonD)
+
+
+subsection \<open>Subspace topology\<close>
+
+definition%important "subtopology U V = topology (\<lambda>T. \<exists>S. T = S \<inter> V \<and> openin U S)"
+
+lemma istopology_subtopology: "istopology (\<lambda>T. \<exists>S. T = S \<inter> V \<and> openin U S)"
+ (is "istopology ?L")
+proof -
+ have "?L {}" by blast
+ {
+ fix A B
+ assume A: "?L A" and B: "?L B"
+ from A B obtain Sa and Sb where Sa: "openin U Sa" "A = Sa \<inter> V" and Sb: "openin U Sb" "B = Sb \<inter> V"
+ by blast
+ have "A \<inter> B = (Sa \<inter> Sb) \<inter> V" "openin U (Sa \<inter> Sb)"
+ using Sa Sb by blast+
+ then have "?L (A \<inter> B)" by blast
+ }
+ moreover
+ {
+ fix K
+ assume K: "K \<subseteq> Collect ?L"
+ have th0: "Collect ?L = (\<lambda>S. S \<inter> V) ` Collect (openin U)"
+ by blast
+ from K[unfolded th0 subset_image_iff]
+ obtain Sk where Sk: "Sk \<subseteq> Collect (openin U)" "K = (\<lambda>S. S \<inter> V) ` Sk"
+ by blast
+ have "\<Union>K = (\<Union>Sk) \<inter> V"
+ using Sk by auto
+ moreover have "openin U (\<Union>Sk)"
+ using Sk by (auto simp: subset_eq)
+ ultimately have "?L (\<Union>K)" by blast
+ }
+ ultimately show ?thesis
+ unfolding subset_eq mem_Collect_eq istopology_def by auto
+qed
+
+lemma openin_subtopology: "openin (subtopology U V) S \<longleftrightarrow> (\<exists>T. openin U T \<and> S = T \<inter> V)"
+ unfolding subtopology_def topology_inverse'[OF istopology_subtopology]
+ by auto
+
+lemma openin_subtopology_Int:
+ "openin X S \<Longrightarrow> openin (subtopology X T) (S \<inter> T)"
+ using openin_subtopology by auto
+
+lemma openin_subtopology_Int2:
+ "openin X T \<Longrightarrow> openin (subtopology X S) (S \<inter> T)"
+ using openin_subtopology by auto
+
+lemma openin_subtopology_diff_closed:
+ "\<lbrakk>S \<subseteq> topspace X; closedin X T\<rbrakk> \<Longrightarrow> openin (subtopology X S) (S - T)"
+ unfolding closedin_def openin_subtopology
+ by (rule_tac x="topspace X - T" in exI) auto
+
+lemma openin_relative_to: "(openin X relative_to S) = openin (subtopology X S)"
+ by (force simp: relative_to_def openin_subtopology)
+
+lemma topspace_subtopology: "topspace (subtopology U V) = topspace U \<inter> V"
+ by (auto simp: topspace_def openin_subtopology)
+
+lemma closedin_subtopology: "closedin (subtopology U V) S \<longleftrightarrow> (\<exists>T. closedin U T \<and> S = T \<inter> V)"
+ unfolding closedin_def topspace_subtopology
+ by (auto simp: openin_subtopology)
+
+lemma openin_subtopology_refl: "openin (subtopology U V) V \<longleftrightarrow> V \<subseteq> topspace U"
+ unfolding openin_subtopology
+ by auto (metis IntD1 in_mono openin_subset)
+
+lemma subtopology_subtopology:
+ "subtopology (subtopology X S) T = subtopology X (S \<inter> T)"
+proof -
+ have eq: "\<And>T'. (\<exists>S'. T' = S' \<inter> T \<and> (\<exists>T. openin X T \<and> S' = T \<inter> S)) = (\<exists>Sa. T' = Sa \<inter> (S \<inter> T) \<and> openin X Sa)"
+ by (metis inf_assoc)
+ have "subtopology (subtopology X S) T = topology (\<lambda>Ta. \<exists>Sa. Ta = Sa \<inter> T \<and> openin (subtopology X S) Sa)"
+ by (simp add: subtopology_def)
+ also have "\<dots> = subtopology X (S \<inter> T)"
+ by (simp add: openin_subtopology eq) (simp add: subtopology_def)
+ finally show ?thesis .
+qed
+
+lemma openin_subtopology_alt:
+ "openin (subtopology X U) S \<longleftrightarrow> S \<in> (\<lambda>T. U \<inter> T) ` Collect (openin X)"
+ by (simp add: image_iff inf_commute openin_subtopology)
+
+lemma closedin_subtopology_alt:
+ "closedin (subtopology X U) S \<longleftrightarrow> S \<in> (\<lambda>T. U \<inter> T) ` Collect (closedin X)"
+ by (simp add: image_iff inf_commute closedin_subtopology)
+
+lemma subtopology_superset:
+ assumes UV: "topspace U \<subseteq> V"
+ shows "subtopology U V = U"
+proof -
+ {
+ fix S
+ {
+ fix T
+ assume T: "openin U T" "S = T \<inter> V"
+ from T openin_subset[OF T(1)] UV have eq: "S = T"
+ by blast
+ have "openin U S"
+ unfolding eq using T by blast
+ }
+ moreover
+ {
+ assume S: "openin U S"
+ then have "\<exists>T. openin U T \<and> S = T \<inter> V"
+ using openin_subset[OF S] UV by auto
+ }
+ ultimately have "(\<exists>T. openin U T \<and> S = T \<inter> V) \<longleftrightarrow> openin U S"
+ by blast
+ }
+ then show ?thesis
+ unfolding topology_eq openin_subtopology by blast
+qed
+
+lemma subtopology_topspace[simp]: "subtopology U (topspace U) = U"
+ by (simp add: subtopology_superset)
+
+lemma subtopology_UNIV[simp]: "subtopology U UNIV = U"
+ by (simp add: subtopology_superset)
+
+lemma openin_subtopology_empty:
+ "openin (subtopology U {}) S \<longleftrightarrow> S = {}"
+by (metis Int_empty_right openin_empty openin_subtopology)
+
+lemma closedin_subtopology_empty:
+ "closedin (subtopology U {}) S \<longleftrightarrow> S = {}"
+by (metis Int_empty_right closedin_empty closedin_subtopology)
+
+lemma closedin_subtopology_refl [simp]:
+ "closedin (subtopology U X) X \<longleftrightarrow> X \<subseteq> topspace U"
+by (metis closedin_def closedin_topspace inf.absorb_iff2 le_inf_iff topspace_subtopology)
+
+lemma closedin_topspace_empty: "topspace T = {} \<Longrightarrow> (closedin T S \<longleftrightarrow> S = {})"
+ by (simp add: closedin_def)
+
+lemma openin_imp_subset:
+ "openin (subtopology U S) T \<Longrightarrow> T \<subseteq> S"
+by (metis Int_iff openin_subtopology subsetI)
+
+lemma closedin_imp_subset:
+ "closedin (subtopology U S) T \<Longrightarrow> T \<subseteq> S"
+by (simp add: closedin_def topspace_subtopology)
+
+lemma openin_open_subtopology:
+ "openin X S \<Longrightarrow> openin (subtopology X S) T \<longleftrightarrow> openin X T \<and> T \<subseteq> S"
+ by (metis inf.orderE openin_Int openin_imp_subset openin_subtopology)
+
+lemma closedin_closed_subtopology:
+ "closedin X S \<Longrightarrow> (closedin (subtopology X S) T \<longleftrightarrow> closedin X T \<and> T \<subseteq> S)"
+ by (metis closedin_Int closedin_imp_subset closedin_subtopology inf.orderE)
+
+lemma openin_subtopology_Un:
+ "\<lbrakk>openin (subtopology X T) S; openin (subtopology X U) S\<rbrakk>
+ \<Longrightarrow> openin (subtopology X (T \<union> U)) S"
+by (simp add: openin_subtopology) blast
+
+lemma closedin_subtopology_Un:
+ "\<lbrakk>closedin (subtopology X T) S; closedin (subtopology X U) S\<rbrakk>
+ \<Longrightarrow> closedin (subtopology X (T \<union> U)) S"
+by (simp add: closedin_subtopology) blast
+
+
+subsection \<open>The standard Euclidean topology\<close>
+
+definition%important euclidean :: "'a::topological_space topology"
+ where "euclidean = topology open"
+
+lemma open_openin: "open S \<longleftrightarrow> openin euclidean S"
+ unfolding euclidean_def
+ apply (rule cong[where x=S and y=S])
+ apply (rule topology_inverse[symmetric])
+ apply (auto simp: istopology_def)
+ done
+
+declare open_openin [symmetric, simp]
+
+lemma topspace_euclidean [simp]: "topspace euclidean = UNIV"
+ by (force simp: topspace_def)
+
+lemma topspace_euclidean_subtopology[simp]: "topspace (subtopology euclidean S) = S"
+ by (simp add: topspace_subtopology)
+
+lemma closed_closedin: "closed S \<longleftrightarrow> closedin euclidean S"
+ by (simp add: closed_def closedin_def Compl_eq_Diff_UNIV)
+
+declare closed_closedin [symmetric, simp]
+
+lemma open_subopen: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> S)"
+ using openI by auto
+
+lemma openin_subtopology_self [simp]: "openin (subtopology euclidean S) S"
+ by (metis openin_topspace topspace_euclidean_subtopology)
+
+subsubsection\<open>The most basic facts about the usual topology and metric on R\<close>
+
+abbreviation euclideanreal :: "real topology"
+ where "euclideanreal \<equiv> topology open"
+
+lemma real_openin [simp]: "openin euclideanreal S = open S"
+ by (simp add: euclidean_def open_openin)
+
+lemma topspace_euclideanreal [simp]: "topspace euclideanreal = UNIV"
+ using openin_subset open_UNIV real_openin by blast
+
+lemma topspace_euclideanreal_subtopology [simp]:
+ "topspace (subtopology euclideanreal S) = S"
+ by (simp add: topspace_subtopology)
+
+lemma real_closedin [simp]: "closedin euclideanreal S = closed S"
+ by (simp add: closed_closedin euclidean_def)
+
+subsection \<open>Basic "localization" results are handy for connectedness.\<close>
+
+lemma openin_open: "openin (subtopology euclidean U) S \<longleftrightarrow> (\<exists>T. open T \<and> (S = U \<inter> T))"
+ by (auto simp: openin_subtopology)
+
+lemma openin_Int_open:
+ "\<lbrakk>openin (subtopology euclidean U) S; open T\<rbrakk>
+ \<Longrightarrow> openin (subtopology euclidean U) (S \<inter> T)"
+by (metis open_Int Int_assoc openin_open)
+
+lemma openin_open_Int[intro]: "open S \<Longrightarrow> openin (subtopology euclidean U) (U \<inter> S)"
+ by (auto simp: openin_open)
+
+lemma open_openin_trans[trans]:
+ "open S \<Longrightarrow> open T \<Longrightarrow> T \<subseteq> S \<Longrightarrow> openin (subtopology euclidean S) T"
+ by (metis Int_absorb1 openin_open_Int)
+
+lemma open_subset: "S \<subseteq> T \<Longrightarrow> open S \<Longrightarrow> openin (subtopology euclidean T) S"
+ by (auto simp: openin_open)
+
+lemma closedin_closed: "closedin (subtopology euclidean U) S \<longleftrightarrow> (\<exists>T. closed T \<and> S = U \<inter> T)"
+ by (simp add: closedin_subtopology Int_ac)
+
+lemma closedin_closed_Int: "closed S \<Longrightarrow> closedin (subtopology euclidean U) (U \<inter> S)"
+ by (metis closedin_closed)
+
+lemma closed_subset: "S \<subseteq> T \<Longrightarrow> closed S \<Longrightarrow> closedin (subtopology euclidean T) S"
+ by (auto simp: closedin_closed)
+
+lemma closedin_closed_subset:
+ "\<lbrakk>closedin (subtopology euclidean U) V; T \<subseteq> U; S = V \<inter> T\<rbrakk>
+ \<Longrightarrow> closedin (subtopology euclidean T) S"
+ by (metis (no_types, lifting) Int_assoc Int_commute closedin_closed inf.orderE)
+
+lemma finite_imp_closedin:
+ fixes S :: "'a::t1_space set"
+ shows "\<lbrakk>finite S; S \<subseteq> T\<rbrakk> \<Longrightarrow> closedin (subtopology euclidean T) S"
+ by (simp add: finite_imp_closed closed_subset)
+
+lemma closedin_singleton [simp]:
+ fixes a :: "'a::t1_space"
+ shows "closedin (subtopology euclidean U) {a} \<longleftrightarrow> a \<in> U"
+using closedin_subset by (force intro: closed_subset)
+
+lemma openin_euclidean_subtopology_iff:
+ fixes S U :: "'a::metric_space set"
+ shows "openin (subtopology euclidean U) S \<longleftrightarrow>
+ S \<subseteq> U \<and> (\<forall>x\<in>S. \<exists>e>0. \<forall>x'\<in>U. dist x' x < e \<longrightarrow> x'\<in> S)"
+ (is "?lhs \<longleftrightarrow> ?rhs")
+proof
+ assume ?lhs
+ then show ?rhs
+ unfolding openin_open open_dist by blast
+next
+ define T where "T = {x. \<exists>a\<in>S. \<exists>d>0. (\<forall>y\<in>U. dist y a < d \<longrightarrow> y \<in> S) \<and> dist x a < d}"
+ have 1: "\<forall>x\<in>T. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> T"
+ unfolding T_def
+ apply clarsimp
+ apply (rule_tac x="d - dist x a" in exI)
+ apply (clarsimp simp add: less_diff_eq)
+ by (metis dist_commute dist_triangle_lt)
+ assume ?rhs then have 2: "S = U \<inter> T"
+ unfolding T_def
+ by auto (metis dist_self)
+ from 1 2 show ?lhs
+ unfolding openin_open open_dist by fast
+qed
+
+lemma connected_openin:
+ "connected S \<longleftrightarrow>
+ \<not>(\<exists>E1 E2. openin (subtopology euclidean S) E1 \<and>
+ openin (subtopology euclidean S) E2 \<and>
+ S \<subseteq> E1 \<union> E2 \<and> E1 \<inter> E2 = {} \<and> E1 \<noteq> {} \<and> E2 \<noteq> {})"
+ apply (simp add: connected_def openin_open disjoint_iff_not_equal, safe)
+ apply (simp_all, blast+) (* SLOW *)
+ done
+
+lemma connected_openin_eq:
+ "connected S \<longleftrightarrow>
+ \<not>(\<exists>E1 E2. openin (subtopology euclidean S) E1 \<and>
+ openin (subtopology euclidean S) E2 \<and>
+ E1 \<union> E2 = S \<and> E1 \<inter> E2 = {} \<and>
+ E1 \<noteq> {} \<and> E2 \<noteq> {})"
+ apply (simp add: connected_openin, safe, blast)
+ by (metis Int_lower1 Un_subset_iff openin_open subset_antisym)
+
+lemma connected_closedin:
+ "connected S \<longleftrightarrow>
+ (\<nexists>E1 E2.
+ closedin (subtopology euclidean S) E1 \<and>
+ closedin (subtopology euclidean S) E2 \<and>
+ S \<subseteq> E1 \<union> E2 \<and> E1 \<inter> E2 = {} \<and> E1 \<noteq> {} \<and> E2 \<noteq> {})"
+ (is "?lhs = ?rhs")
+proof
+ assume ?lhs
+ then show ?rhs
+ by (auto simp add: connected_closed closedin_closed)
+next
+ assume R: ?rhs
+ then show ?lhs
+ proof (clarsimp simp add: connected_closed closedin_closed)
+ fix A B
+ assume s_sub: "S \<subseteq> A \<union> B" "B \<inter> S \<noteq> {}"
+ and disj: "A \<inter> B \<inter> S = {}"
+ and cl: "closed A" "closed B"
+ have "S \<inter> (A \<union> B) = S"
+ using s_sub(1) by auto
+ have "S - A = B \<inter> S"
+ using Diff_subset_conv Un_Diff_Int disj s_sub(1) by auto
+ then have "S \<inter> A = {}"
+ by (metis Diff_Diff_Int Diff_disjoint Un_Diff_Int R cl closedin_closed_Int inf_commute order_refl s_sub(2))
+ then show "A \<inter> S = {}"
+ by blast
+ qed
+qed
+
+lemma connected_closedin_eq:
+ "connected S \<longleftrightarrow>
+ \<not>(\<exists>E1 E2.
+ closedin (subtopology euclidean S) E1 \<and>
+ closedin (subtopology euclidean S) E2 \<and>
+ E1 \<union> E2 = S \<and> E1 \<inter> E2 = {} \<and>
+ E1 \<noteq> {} \<and> E2 \<noteq> {})"
+ apply (simp add: connected_closedin, safe, blast)
+ by (metis Int_lower1 Un_subset_iff closedin_closed subset_antisym)
+
+text \<open>These "transitivity" results are handy too\<close>
+
+lemma openin_trans[trans]:
+ "openin (subtopology euclidean T) S \<Longrightarrow> openin (subtopology euclidean U) T \<Longrightarrow>
+ openin (subtopology euclidean U) S"
+ unfolding open_openin openin_open by blast
+
+lemma openin_open_trans: "openin (subtopology euclidean T) S \<Longrightarrow> open T \<Longrightarrow> open S"
+ by (auto simp: openin_open intro: openin_trans)
+
+lemma closedin_trans[trans]:
+ "closedin (subtopology euclidean T) S \<Longrightarrow> closedin (subtopology euclidean U) T \<Longrightarrow>
+ closedin (subtopology euclidean U) S"
+ by (auto simp: closedin_closed closed_Inter Int_assoc)
+
+lemma closedin_closed_trans: "closedin (subtopology euclidean T) S \<Longrightarrow> closed T \<Longrightarrow> closed S"
+ by (auto simp: closedin_closed intro: closedin_trans)
+
+lemma openin_subtopology_Int_subset:
+ "\<lbrakk>openin (subtopology euclidean u) (u \<inter> S); v \<subseteq> u\<rbrakk> \<Longrightarrow> openin (subtopology euclidean v) (v \<inter> S)"
+ by (auto simp: openin_subtopology)
+
+lemma openin_open_eq: "open s \<Longrightarrow> (openin (subtopology euclidean s) t \<longleftrightarrow> open t \<and> t \<subseteq> s)"
+ using open_subset openin_open_trans openin_subset by fastforce
+
+
+subsection \<open>Open and closed balls\<close>
+
+definition%important ball :: "'a::metric_space \<Rightarrow> real \<Rightarrow> 'a set"
+ where "ball x e = {y. dist x y < e}"
+
+definition%important cball :: "'a::metric_space \<Rightarrow> real \<Rightarrow> 'a set"
+ where "cball x e = {y. dist x y \<le> e}"
+
+definition%important sphere :: "'a::metric_space \<Rightarrow> real \<Rightarrow> 'a set"
+ where "sphere x e = {y. dist x y = e}"
+
+lemma mem_ball [simp]: "y \<in> ball x e \<longleftrightarrow> dist x y < e"
+ by (simp add: ball_def)
+
+lemma mem_cball [simp]: "y \<in> cball x e \<longleftrightarrow> dist x y \<le> e"
+ by (simp add: cball_def)
+
+lemma mem_sphere [simp]: "y \<in> sphere x e \<longleftrightarrow> dist x y = e"
+ by (simp add: sphere_def)
+
+lemma ball_trivial [simp]: "ball x 0 = {}"
+ by (simp add: ball_def)
+
+lemma cball_trivial [simp]: "cball x 0 = {x}"
+ by (simp add: cball_def)
+
+lemma sphere_trivial [simp]: "sphere x 0 = {x}"
+ by (simp add: sphere_def)
+
+lemma mem_ball_0 [simp]: "x \<in> ball 0 e \<longleftrightarrow> norm x < e"
+ for x :: "'a::real_normed_vector"
+ by (simp add: dist_norm)
+
+lemma mem_cball_0 [simp]: "x \<in> cball 0 e \<longleftrightarrow> norm x \<le> e"
+ for x :: "'a::real_normed_vector"
+ by (simp add: dist_norm)
+
+lemma disjoint_ballI: "dist x y \<ge> r+s \<Longrightarrow> ball x r \<inter> ball y s = {}"
+ using dist_triangle_less_add not_le by fastforce
+
+lemma disjoint_cballI: "dist x y > r + s \<Longrightarrow> cball x r \<inter> cball y s = {}"
+ by (metis add_mono disjoint_iff_not_equal dist_triangle2 dual_order.trans leD mem_cball)
+
+lemma mem_sphere_0 [simp]: "x \<in> sphere 0 e \<longleftrightarrow> norm x = e"
+ for x :: "'a::real_normed_vector"
+ by (simp add: dist_norm)
+
+lemma sphere_empty [simp]: "r < 0 \<Longrightarrow> sphere a r = {}"
+ for a :: "'a::metric_space"
+ by auto
+
+lemma centre_in_ball [simp]: "x \<in> ball x e \<longleftrightarrow> 0 < e"
+ by simp
+
+lemma centre_in_cball [simp]: "x \<in> cball x e \<longleftrightarrow> 0 \<le> e"
+ by simp
+
+lemma ball_subset_cball [simp, intro]: "ball x e \<subseteq> cball x e"
+ by (simp add: subset_eq)
+
+lemma mem_ball_imp_mem_cball: "x \<in> ball y e \<Longrightarrow> x \<in> cball y e"
+ by (auto simp: mem_ball mem_cball)
+
+lemma sphere_cball [simp,intro]: "sphere z r \<subseteq> cball z r"
+ by force
+
+lemma cball_diff_sphere: "cball a r - sphere a r = ball a r"
+ by auto
+
+lemma subset_ball[intro]: "d \<le> e \<Longrightarrow> ball x d \<subseteq> ball x e"
+ by (simp add: subset_eq)
+
+lemma subset_cball[intro]: "d \<le> e \<Longrightarrow> cball x d \<subseteq> cball x e"
+ by (simp add: subset_eq)
+
+lemma mem_ball_leI: "x \<in> ball y e \<Longrightarrow> e \<le> f \<Longrightarrow> x \<in> ball y f"
+ by (auto simp: mem_ball mem_cball)
+
+lemma mem_cball_leI: "x \<in> cball y e \<Longrightarrow> e \<le> f \<Longrightarrow> x \<in> cball y f"
+ by (auto simp: mem_ball mem_cball)
+
+lemma cball_trans: "y \<in> cball z b \<Longrightarrow> x \<in> cball y a \<Longrightarrow> x \<in> cball z (b + a)"
+ unfolding mem_cball
+proof -
+ have "dist z x \<le> dist z y + dist y x"
+ by (rule dist_triangle)
+ also assume "dist z y \<le> b"
+ also assume "dist y x \<le> a"
+ finally show "dist z x \<le> b + a" by arith
+qed
+
+lemma ball_max_Un: "ball a (max r s) = ball a r \<union> ball a s"
+ by (simp add: set_eq_iff) arith
+
+lemma ball_min_Int: "ball a (min r s) = ball a r \<inter> ball a s"
+ by (simp add: set_eq_iff)
+
+lemma cball_max_Un: "cball a (max r s) = cball a r \<union> cball a s"
+ by (simp add: set_eq_iff) arith
+
+lemma cball_min_Int: "cball a (min r s) = cball a r \<inter> cball a s"
+ by (simp add: set_eq_iff)
+
+lemma cball_diff_eq_sphere: "cball a r - ball a r = sphere a r"
+ by (auto simp: cball_def ball_def dist_commute)
+
+lemma image_add_ball [simp]:
+ fixes a :: "'a::real_normed_vector"
+ shows "(+) b ` ball a r = ball (a+b) r"
+apply (intro equalityI subsetI)
+apply (force simp: dist_norm)
+apply (rule_tac x="x-b" in image_eqI)
+apply (auto simp: dist_norm algebra_simps)
+done
+
+lemma image_add_cball [simp]:
+ fixes a :: "'a::real_normed_vector"
+ shows "(+) b ` cball a r = cball (a+b) r"
+apply (intro equalityI subsetI)
+apply (force simp: dist_norm)
+apply (rule_tac x="x-b" in image_eqI)
+apply (auto simp: dist_norm algebra_simps)
+done
+
+lemma open_ball [intro, simp]: "open (ball x e)"
+proof -
+ have "open (dist x -` {..<e})"
+ by (intro open_vimage open_lessThan continuous_intros)
+ also have "dist x -` {..<e} = ball x e"
+ by auto
+ finally show ?thesis .
+qed
+
+lemma open_contains_ball: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. ball x e \<subseteq> S)"
+ by (simp add: open_dist subset_eq mem_ball Ball_def dist_commute)
+
+lemma openI [intro?]: "(\<And>x. x\<in>S \<Longrightarrow> \<exists>e>0. ball x e \<subseteq> S) \<Longrightarrow> open S"
+ by (auto simp: open_contains_ball)
+
+lemma openE[elim?]:
+ assumes "open S" "x\<in>S"
+ obtains e where "e>0" "ball x e \<subseteq> S"
+ using assms unfolding open_contains_ball by auto
+
+lemma open_contains_ball_eq: "open S \<Longrightarrow> x\<in>S \<longleftrightarrow> (\<exists>e>0. ball x e \<subseteq> S)"
+ by (metis open_contains_ball subset_eq centre_in_ball)
+
+lemma openin_contains_ball:
+ "openin (subtopology euclidean t) s \<longleftrightarrow>
+ s \<subseteq> t \<and> (\<forall>x \<in> s. \<exists>e. 0 < e \<and> ball x e \<inter> t \<subseteq> s)"
+ (is "?lhs = ?rhs")
+proof
+ assume ?lhs
+ then show ?rhs
+ apply (simp add: openin_open)
+ apply (metis Int_commute Int_mono inf.cobounded2 open_contains_ball order_refl subsetCE)
+ done
+next
+ assume ?rhs
+ then show ?lhs
+ apply (simp add: openin_euclidean_subtopology_iff)
+ by (metis (no_types) Int_iff dist_commute inf.absorb_iff2 mem_ball)
+qed
+
+lemma openin_contains_cball:
+ "openin (subtopology euclidean t) s \<longleftrightarrow>
+ s \<subseteq> t \<and>
+ (\<forall>x \<in> s. \<exists>e. 0 < e \<and> cball x e \<inter> t \<subseteq> s)"
+apply (simp add: openin_contains_ball)
+apply (rule iffI)
+apply (auto dest!: bspec)
+apply (rule_tac x="e/2" in exI, force+)
+done
+
+lemma ball_eq_empty[simp]: "ball x e = {} \<longleftrightarrow> e \<le> 0"
+ unfolding mem_ball set_eq_iff
+ apply (simp add: not_less)
+ apply (metis zero_le_dist order_trans dist_self)
+ done
+
+lemma ball_empty: "e \<le> 0 \<Longrightarrow> ball x e = {}" by simp
+
+lemma closed_cball [iff]: "closed (cball x e)"
+proof -
+ have "closed (dist x -` {..e})"
+ by (intro closed_vimage closed_atMost continuous_intros)
+ also have "dist x -` {..e} = cball x e"
+ by auto
+ finally show ?thesis .
+qed
+
+lemma open_contains_cball: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. cball x e \<subseteq> S)"
+proof -
+ {
+ fix x and e::real
+ assume "x\<in>S" "e>0" "ball x e \<subseteq> S"
+ then have "\<exists>d>0. cball x d \<subseteq> S" unfolding subset_eq by (rule_tac x="e/2" in exI, auto)
+ }
+ moreover
+ {
+ fix x and e::real
+ assume "x\<in>S" "e>0" "cball x e \<subseteq> S"
+ then have "\<exists>d>0. ball x d \<subseteq> S"
+ unfolding subset_eq
+ apply (rule_tac x="e/2" in exI, auto)
+ done
+ }
+ ultimately show ?thesis
+ unfolding open_contains_ball by auto
+qed
+
+lemma open_contains_cball_eq: "open S \<Longrightarrow> (\<forall>x. x \<in> S \<longleftrightarrow> (\<exists>e>0. cball x e \<subseteq> S))"
+ by (metis open_contains_cball subset_eq order_less_imp_le centre_in_cball)
+
+lemma eventually_nhds_ball: "d > 0 \<Longrightarrow> eventually (\<lambda>x. x \<in> ball z d) (nhds z)"
+ by (rule eventually_nhds_in_open) simp_all
+
+lemma eventually_at_ball: "d > 0 \<Longrightarrow> eventually (\<lambda>t. t \<in> ball z d \<and> t \<in> A) (at z within A)"
+ unfolding eventually_at by (intro exI[of _ d]) (simp_all add: dist_commute)
+
+lemma eventually_at_ball': "d > 0 \<Longrightarrow> eventually (\<lambda>t. t \<in> ball z d \<and> t \<noteq> z \<and> t \<in> A) (at z within A)"
+ unfolding eventually_at by (intro exI[of _ d]) (simp_all add: dist_commute)
+
+lemma at_within_ball: "e > 0 \<Longrightarrow> dist x y < e \<Longrightarrow> at y within ball x e = at y"
+ by (subst at_within_open) auto
+
+lemma atLeastAtMost_eq_cball:
+ fixes a b::real
+ shows "{a .. b} = cball ((a + b)/2) ((b - a)/2)"
+ by (auto simp: dist_real_def field_simps mem_cball)
+
+lemma greaterThanLessThan_eq_ball:
+ fixes a b::real
+ shows "{a <..< b} = ball ((a + b)/2) ((b - a)/2)"
+ by (auto simp: dist_real_def field_simps mem_ball)
+
+
+subsection \<open>Limit points\<close>
+
+definition%important (in topological_space) islimpt:: "'a \<Rightarrow> 'a set \<Rightarrow> bool" (infixr "islimpt" 60)
+ where "x islimpt S \<longleftrightarrow> (\<forall>T. x\<in>T \<longrightarrow> open T \<longrightarrow> (\<exists>y\<in>S. y\<in>T \<and> y\<noteq>x))"
+
+lemma islimptI:
+ assumes "\<And>T. x \<in> T \<Longrightarrow> open T \<Longrightarrow> \<exists>y\<in>S. y \<in> T \<and> y \<noteq> x"
+ shows "x islimpt S"
+ using assms unfolding islimpt_def by auto
+
+lemma islimptE:
+ assumes "x islimpt S" and "x \<in> T" and "open T"
+ obtains y where "y \<in> S" and "y \<in> T" and "y \<noteq> x"
+ using assms unfolding islimpt_def by auto
+
+lemma islimpt_iff_eventually: "x islimpt S \<longleftrightarrow> \<not> eventually (\<lambda>y. y \<notin> S) (at x)"
+ unfolding islimpt_def eventually_at_topological by auto
+
+lemma islimpt_subset: "x islimpt S \<Longrightarrow> S \<subseteq> T \<Longrightarrow> x islimpt T"
+ unfolding islimpt_def by fast
+
+lemma islimpt_approachable:
+ fixes x :: "'a::metric_space"
+ shows "x islimpt S \<longleftrightarrow> (\<forall>e>0. \<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e)"
+ unfolding islimpt_iff_eventually eventually_at by fast
+
+lemma islimpt_approachable_le: "x islimpt S \<longleftrightarrow> (\<forall>e>0. \<exists>x'\<in> S. x' \<noteq> x \<and> dist x' x \<le> e)"
+ for x :: "'a::metric_space"
+ unfolding islimpt_approachable
+ using approachable_lt_le [where f="\<lambda>y. dist y x" and P="\<lambda>y. y \<notin> S \<or> y = x",
+ THEN arg_cong [where f=Not]]
+ by (simp add: Bex_def conj_commute conj_left_commute)
+
+lemma islimpt_UNIV_iff: "x islimpt UNIV \<longleftrightarrow> \<not> open {x}"
+ unfolding islimpt_def by (safe, fast, case_tac "T = {x}", fast, fast)
+
+lemma islimpt_punctured: "x islimpt S = x islimpt (S-{x})"
+ unfolding islimpt_def by blast
+
+text \<open>A perfect space has no isolated points.\<close>
+
+lemma islimpt_UNIV [simp, intro]: "x islimpt UNIV"
+ for x :: "'a::perfect_space"
+ unfolding islimpt_UNIV_iff by (rule not_open_singleton)
+
+lemma perfect_choose_dist: "0 < r \<Longrightarrow> \<exists>a. a \<noteq> x \<and> dist a x < r"
+ for x :: "'a::{perfect_space,metric_space}"
+ using islimpt_UNIV [of x] by (simp add: islimpt_approachable)
+
+lemma closed_limpt: "closed S \<longleftrightarrow> (\<forall>x. x islimpt S \<longrightarrow> x \<in> S)"
+ unfolding closed_def
+ apply (subst open_subopen)
+ apply (simp add: islimpt_def subset_eq)
+ apply (metis ComplE ComplI)
+ done
+
+lemma islimpt_EMPTY[simp]: "\<not> x islimpt {}"
+ by (auto simp: islimpt_def)
+
+lemma finite_ball_include:
+ fixes a :: "'a::metric_space"
+ assumes "finite S"
+ shows "\<exists>e>0. S \<subseteq> ball a e"
+ using assms
+proof induction
+ case (insert x S)
+ then obtain e0 where "e0>0" and e0:"S \<subseteq> ball a e0" by auto
+ define e where "e = max e0 (2 * dist a x)"
+ have "e>0" unfolding e_def using \<open>e0>0\<close> by auto
+ moreover have "insert x S \<subseteq> ball a e"
+ using e0 \<open>e>0\<close> unfolding e_def by auto
+ ultimately show ?case by auto
+qed (auto intro: zero_less_one)
+
+lemma finite_set_avoid:
+ fixes a :: "'a::metric_space"
+ assumes "finite S"
+ shows "\<exists>d>0. \<forall>x\<in>S. x \<noteq> a \<longrightarrow> d \<le> dist a x"
+ using assms
+proof induction
+ case (insert x S)
+ then obtain d where "d > 0" and d: "\<forall>x\<in>S. x \<noteq> a \<longrightarrow> d \<le> dist a x"
+ by blast
+ show ?case
+ proof (cases "x = a")
+ case True
+ with \<open>d > 0 \<close>d show ?thesis by auto
+ next
+ case False
+ let ?d = "min d (dist a x)"
+ from False \<open>d > 0\<close> have dp: "?d > 0"
+ by auto
+ from d have d': "\<forall>x\<in>S. x \<noteq> a \<longrightarrow> ?d \<le> dist a x"
+ by auto
+ with dp False show ?thesis
+ by (metis insert_iff le_less min_less_iff_conj not_less)
+ qed
+qed (auto intro: zero_less_one)
+
+lemma islimpt_Un: "x islimpt (S \<union> T) \<longleftrightarrow> x islimpt S \<or> x islimpt T"
+ by (simp add: islimpt_iff_eventually eventually_conj_iff)
+
+lemma discrete_imp_closed:
+ fixes S :: "'a::metric_space set"
+ assumes e: "0 < e"
+ and d: "\<forall>x \<in> S. \<forall>y \<in> S. dist y x < e \<longrightarrow> y = x"
+ shows "closed S"
+proof -
+ have False if C: "\<And>e. e>0 \<Longrightarrow> \<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e" for x
+ proof -
+ from e have e2: "e/2 > 0" by arith
+ from C[rule_format, OF e2] obtain y where y: "y \<in> S" "y \<noteq> x" "dist y x < e/2"
+ by blast
+ let ?m = "min (e/2) (dist x y) "
+ from e2 y(2) have mp: "?m > 0"
+ by simp
+ from C[OF mp] obtain z where z: "z \<in> S" "z \<noteq> x" "dist z x < ?m"
+ by blast
+ from z y have "dist z y < e"
+ by (intro dist_triangle_lt [where z=x]) simp
+ from d[rule_format, OF y(1) z(1) this] y z show ?thesis
+ by (auto simp: dist_commute)
+ qed
+ then show ?thesis
+ by (metis islimpt_approachable closed_limpt [where 'a='a])
+qed
+
+lemma closed_of_nat_image: "closed (of_nat ` A :: 'a::real_normed_algebra_1 set)"
+ by (rule discrete_imp_closed[of 1]) (auto simp: dist_of_nat)
+
+lemma closed_of_int_image: "closed (of_int ` A :: 'a::real_normed_algebra_1 set)"
+ by (rule discrete_imp_closed[of 1]) (auto simp: dist_of_int)
+
+lemma closed_Nats [simp]: "closed (\<nat> :: 'a :: real_normed_algebra_1 set)"
+ unfolding Nats_def by (rule closed_of_nat_image)
+
+lemma closed_Ints [simp]: "closed (\<int> :: 'a :: real_normed_algebra_1 set)"
+ unfolding Ints_def by (rule closed_of_int_image)
+
+lemma closed_subset_Ints:
+ fixes A :: "'a :: real_normed_algebra_1 set"
+ assumes "A \<subseteq> \<int>"
+ shows "closed A"
+proof (intro discrete_imp_closed[OF zero_less_one] ballI impI, goal_cases)
+ case (1 x y)
+ with assms have "x \<in> \<int>" and "y \<in> \<int>" by auto
+ with \<open>dist y x < 1\<close> show "y = x"
+ by (auto elim!: Ints_cases simp: dist_of_int)
+qed
+
+
+subsection \<open>Interior of a Set\<close>
+
+definition%important "interior S = \<Union>{T. open T \<and> T \<subseteq> S}"
+
+lemma interiorI [intro?]:
+ assumes "open T" and "x \<in> T" and "T \<subseteq> S"
+ shows "x \<in> interior S"
+ using assms unfolding interior_def by fast
+
+lemma interiorE [elim?]:
+ assumes "x \<in> interior S"
+ obtains T where "open T" and "x \<in> T" and "T \<subseteq> S"
+ using assms unfolding interior_def by fast
+
+lemma open_interior [simp, intro]: "open (interior S)"
+ by (simp add: interior_def open_Union)
+
+lemma interior_subset: "interior S \<subseteq> S"
+ by (auto simp: interior_def)
+
+lemma interior_maximal: "T \<subseteq> S \<Longrightarrow> open T \<Longrightarrow> T \<subseteq> interior S"
+ by (auto simp: interior_def)
+
+lemma interior_open: "open S \<Longrightarrow> interior S = S"
+ by (intro equalityI interior_subset interior_maximal subset_refl)
+
+lemma interior_eq: "interior S = S \<longleftrightarrow> open S"
+ by (metis open_interior interior_open)
+
+lemma open_subset_interior: "open S \<Longrightarrow> S \<subseteq> interior T \<longleftrightarrow> S \<subseteq> T"
+ by (metis interior_maximal interior_subset subset_trans)
+
+lemma interior_empty [simp]: "interior {} = {}"
+ using open_empty by (rule interior_open)
+
+lemma interior_UNIV [simp]: "interior UNIV = UNIV"
+ using open_UNIV by (rule interior_open)
+
+lemma interior_interior [simp]: "interior (interior S) = interior S"
+ using open_interior by (rule interior_open)
+
+lemma interior_mono: "S \<subseteq> T \<Longrightarrow> interior S \<subseteq> interior T"
+ by (auto simp: interior_def)
+
+lemma interior_unique:
+ assumes "T \<subseteq> S" and "open T"
+ assumes "\<And>T'. T' \<subseteq> S \<Longrightarrow> open T' \<Longrightarrow> T' \<subseteq> T"
+ shows "interior S = T"
+ by (intro equalityI assms interior_subset open_interior interior_maximal)
+
+lemma interior_singleton [simp]: "interior {a} = {}"
+ for a :: "'a::perfect_space"
+ apply (rule interior_unique, simp_all)
+ using not_open_singleton subset_singletonD
+ apply fastforce
+ done
+
+lemma interior_Int [simp]: "interior (S \<inter> T) = interior S \<inter> interior T"
+ by (intro equalityI Int_mono Int_greatest interior_mono Int_lower1
+ Int_lower2 interior_maximal interior_subset open_Int open_interior)
+
+lemma mem_interior: "x \<in> interior S \<longleftrightarrow> (\<exists>e>0. ball x e \<subseteq> S)"
+ using open_contains_ball_eq [where S="interior S"]
+ by (simp add: open_subset_interior)
+
+lemma eventually_nhds_in_nhd: "x \<in> interior s \<Longrightarrow> eventually (\<lambda>y. y \<in> s) (nhds x)"
+ using interior_subset[of s] by (subst eventually_nhds) blast
+
+lemma interior_limit_point [intro]:
+ fixes x :: "'a::perfect_space"
+ assumes x: "x \<in> interior S"
+ shows "x islimpt S"
+ using x islimpt_UNIV [of x]
+ unfolding interior_def islimpt_def
+ apply (clarsimp, rename_tac T T')
+ apply (drule_tac x="T \<inter> T'" in spec)
+ apply (auto simp: open_Int)
+ done
+
+lemma interior_closed_Un_empty_interior:
+ assumes cS: "closed S"
+ and iT: "interior T = {}"
+ shows "interior (S \<union> T) = interior S"
+proof
+ show "interior S \<subseteq> interior (S \<union> T)"
+ by (rule interior_mono) (rule Un_upper1)
+ show "interior (S \<union> T) \<subseteq> interior S"
+ proof
+ fix x
+ assume "x \<in> interior (S \<union> T)"
+ then obtain R where "open R" "x \<in> R" "R \<subseteq> S \<union> T" ..
+ show "x \<in> interior S"
+ proof (rule ccontr)
+ assume "x \<notin> interior S"
+ with \<open>x \<in> R\<close> \<open>open R\<close> obtain y where "y \<in> R - S"
+ unfolding interior_def by fast
+ from \<open>open R\<close> \<open>closed S\<close> have "open (R - S)"
+ by (rule open_Diff)
+ from \<open>R \<subseteq> S \<union> T\<close> have "R - S \<subseteq> T"
+ by fast
+ from \<open>y \<in> R - S\<close> \<open>open (R - S)\<close> \<open>R - S \<subseteq> T\<close> \<open>interior T = {}\<close> show False
+ unfolding interior_def by fast
+ qed
+ qed
+qed
+
+lemma interior_Times: "interior (A \<times> B) = interior A \<times> interior B"
+proof (rule interior_unique)
+ show "interior A \<times> interior B \<subseteq> A \<times> B"
+ by (intro Sigma_mono interior_subset)
+ show "open (interior A \<times> interior B)"
+ by (intro open_Times open_interior)
+ fix T
+ assume "T \<subseteq> A \<times> B" and "open T"
+ then show "T \<subseteq> interior A \<times> interior B"
+ proof safe
+ fix x y
+ assume "(x, y) \<in> T"
+ then obtain C D where "open C" "open D" "C \<times> D \<subseteq> T" "x \<in> C" "y \<in> D"
+ using \<open>open T\<close> unfolding open_prod_def by fast
+ then have "open C" "open D" "C \<subseteq> A" "D \<subseteq> B" "x \<in> C" "y \<in> D"
+ using \<open>T \<subseteq> A \<times> B\<close> by auto
+ then show "x \<in> interior A" and "y \<in> interior B"
+ by (auto intro: interiorI)
+ qed
+qed
+
+lemma interior_Ici:
+ fixes x :: "'a :: {dense_linorder,linorder_topology}"
+ assumes "b < x"
+ shows "interior {x ..} = {x <..}"
+proof (rule interior_unique)
+ fix T
+ assume "T \<subseteq> {x ..}" "open T"
+ moreover have "x \<notin> T"
+ proof
+ assume "x \<in> T"
+ obtain y where "y < x" "{y <.. x} \<subseteq> T"
+ using open_left[OF \<open>open T\<close> \<open>x \<in> T\<close> \<open>b < x\<close>] by auto
+ with dense[OF \<open>y < x\<close>] obtain z where "z \<in> T" "z < x"
+ by (auto simp: subset_eq Ball_def)
+ with \<open>T \<subseteq> {x ..}\<close> show False by auto
+ qed
+ ultimately show "T \<subseteq> {x <..}"
+ by (auto simp: subset_eq less_le)
+qed auto
+
+lemma interior_Iic:
+ fixes x :: "'a ::{dense_linorder,linorder_topology}"
+ assumes "x < b"
+ shows "interior {.. x} = {..< x}"
+proof (rule interior_unique)
+ fix T
+ assume "T \<subseteq> {.. x}" "open T"
+ moreover have "x \<notin> T"
+ proof
+ assume "x \<in> T"
+ obtain y where "x < y" "{x ..< y} \<subseteq> T"
+ using open_right[OF \<open>open T\<close> \<open>x \<in> T\<close> \<open>x < b\<close>] by auto
+ with dense[OF \<open>x < y\<close>] obtain z where "z \<in> T" "x < z"
+ by (auto simp: subset_eq Ball_def less_le)
+ with \<open>T \<subseteq> {.. x}\<close> show False by auto
+ qed
+ ultimately show "T \<subseteq> {..< x}"
+ by (auto simp: subset_eq less_le)
+qed auto
+
+
+subsection \<open>Closure of a Set\<close>
+
+definition%important "closure S = S \<union> {x | x. x islimpt S}"
+
+lemma interior_closure: "interior S = - (closure (- S))"
+ by (auto simp: interior_def closure_def islimpt_def)
+
+lemma closure_interior: "closure S = - interior (- S)"
+ by (simp add: interior_closure)
+
+lemma closed_closure[simp, intro]: "closed (closure S)"
+ by (simp add: closure_interior closed_Compl)
+
+lemma closure_subset: "S \<subseteq> closure S"
+ by (simp add: closure_def)
+
+lemma closure_hull: "closure S = closed hull S"
+ by (auto simp: hull_def closure_interior interior_def)
+
+lemma closure_eq: "closure S = S \<longleftrightarrow> closed S"
+ unfolding closure_hull using closed_Inter by (rule hull_eq)
+
+lemma closure_closed [simp]: "closed S \<Longrightarrow> closure S = S"
+ by (simp only: closure_eq)
+
+lemma closure_closure [simp]: "closure (closure S) = closure S"
+ unfolding closure_hull by (rule hull_hull)
+
+lemma closure_mono: "S \<subseteq> T \<Longrightarrow> closure S \<subseteq> closure T"
+ unfolding closure_hull by (rule hull_mono)
+
+lemma closure_minimal: "S \<subseteq> T \<Longrightarrow> closed T \<Longrightarrow> closure S \<subseteq> T"
+ unfolding closure_hull by (rule hull_minimal)
+
+lemma closure_unique:
+ assumes "S \<subseteq> T"
+ and "closed T"
+ and "\<And>T'. S \<subseteq> T' \<Longrightarrow> closed T' \<Longrightarrow> T \<subseteq> T'"
+ shows "closure S = T"
+ using assms unfolding closure_hull by (rule hull_unique)
+
+lemma closure_empty [simp]: "closure {} = {}"
+ using closed_empty by (rule closure_closed)
+
+lemma closure_UNIV [simp]: "closure UNIV = UNIV"
+ using closed_UNIV by (rule closure_closed)
+
+lemma closure_Un [simp]: "closure (S \<union> T) = closure S \<union> closure T"
+ by (simp add: closure_interior)
+
+lemma closure_eq_empty [iff]: "closure S = {} \<longleftrightarrow> S = {}"
+ using closure_empty closure_subset[of S] by blast
+
+lemma closure_subset_eq: "closure S \<subseteq> S \<longleftrightarrow> closed S"
+ using closure_eq[of S] closure_subset[of S] by simp
+
+lemma open_Int_closure_eq_empty: "open S \<Longrightarrow> (S \<inter> closure T) = {} \<longleftrightarrow> S \<inter> T = {}"
+ using open_subset_interior[of S "- T"]
+ using interior_subset[of "- T"]
+ by (auto simp: closure_interior)
+
+lemma open_Int_closure_subset: "open S \<Longrightarrow> S \<inter> closure T \<subseteq> closure (S \<inter> T)"
+proof
+ fix x
+ assume *: "open S" "x \<in> S \<inter> closure T"
+ have "x islimpt (S \<inter> T)" if **: "x islimpt T"
+ proof (rule islimptI)
+ fix A
+ assume "x \<in> A" "open A"
+ with * have "x \<in> A \<inter> S" "open (A \<inter> S)"
+ by (simp_all add: open_Int)
+ with ** obtain y where "y \<in> T" "y \<in> A \<inter> S" "y \<noteq> x"
+ by (rule islimptE)
+ then have "y \<in> S \<inter> T" "y \<in> A \<and> y \<noteq> x"
+ by simp_all
+ then show "\<exists>y\<in>(S \<inter> T). y \<in> A \<and> y \<noteq> x" ..
+ qed
+ with * show "x \<in> closure (S \<inter> T)"
+ unfolding closure_def by blast
+qed
+
+lemma closure_complement: "closure (- S) = - interior S"
+ by (simp add: closure_interior)
+
+lemma interior_complement: "interior (- S) = - closure S"
+ by (simp add: closure_interior)
+
+lemma interior_diff: "interior(S - T) = interior S - closure T"
+ by (simp add: Diff_eq interior_complement)
+
+lemma closure_Times: "closure (A \<times> B) = closure A \<times> closure B"
+proof (rule closure_unique)
+ show "A \<times> B \<subseteq> closure A \<times> closure B"
+ by (intro Sigma_mono closure_subset)
+ show "closed (closure A \<times> closure B)"
+ by (intro closed_Times closed_closure)
+ fix T
+ assume "A \<times> B \<subseteq> T" and "closed T"
+ then show "closure A \<times> closure B \<subseteq> T"
+ apply (simp add: closed_def open_prod_def, clarify)
+ apply (rule ccontr)
+ apply (drule_tac x="(a, b)" in bspec, simp, clarify, rename_tac C D)
+ apply (simp add: closure_interior interior_def)
+ apply (drule_tac x=C in spec)
+ apply (drule_tac x=D in spec, auto)
+ done
+qed
+
+lemma closure_openin_Int_closure:
+ assumes ope: "openin (subtopology euclidean U) S" and "T \<subseteq> U"
+ shows "closure(S \<inter> closure T) = closure(S \<inter> T)"
+proof
+ obtain V where "open V" and S: "S = U \<inter> V"
+ using ope using openin_open by metis
+ show "closure (S \<inter> closure T) \<subseteq> closure (S \<inter> T)"
+ proof (clarsimp simp: S)
+ fix x
+ assume "x \<in> closure (U \<inter> V \<inter> closure T)"
+ then have "V \<inter> closure T \<subseteq> A \<Longrightarrow> x \<in> closure A" for A
+ by (metis closure_mono subsetD inf.coboundedI2 inf_assoc)
+ then have "x \<in> closure (T \<inter> V)"
+ by (metis \<open>open V\<close> closure_closure inf_commute open_Int_closure_subset)
+ then show "x \<in> closure (U \<inter> V \<inter> T)"
+ by (metis \<open>T \<subseteq> U\<close> inf.absorb_iff2 inf_assoc inf_commute)
+ qed
+next
+ show "closure (S \<inter> T) \<subseteq> closure (S \<inter> closure T)"
+ by (meson Int_mono closure_mono closure_subset order_refl)
+qed
+
+lemma islimpt_in_closure: "(x islimpt S) = (x\<in>closure(S-{x}))"
+ unfolding closure_def using islimpt_punctured by blast
+
+lemma connected_imp_connected_closure: "connected S \<Longrightarrow> connected (closure S)"
+ by (rule connectedI) (meson closure_subset open_Int open_Int_closure_eq_empty subset_trans connectedD)
+
+lemma limpt_of_limpts: "x islimpt {y. y islimpt S} \<Longrightarrow> x islimpt S"
+ for x :: "'a::metric_space"
+ apply (clarsimp simp add: islimpt_approachable)
+ apply (drule_tac x="e/2" in spec)
+ apply (auto simp: simp del: less_divide_eq_numeral1)
+ apply (drule_tac x="dist x' x" in spec)
+ apply (auto simp: zero_less_dist_iff simp del: less_divide_eq_numeral1)
+ apply (erule rev_bexI)
+ apply (metis dist_commute dist_triangle_half_r less_trans less_irrefl)
+ done
+
+lemma closed_limpts: "closed {x::'a::metric_space. x islimpt S}"
+ using closed_limpt limpt_of_limpts by blast
+
+lemma limpt_of_closure: "x islimpt closure S \<longleftrightarrow> x islimpt S"
+ for x :: "'a::metric_space"
+ by (auto simp: closure_def islimpt_Un dest: limpt_of_limpts)
+
+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)"
+ apply (simp add: closedin_closed, safe)
+ apply (simp add: closed_limpt islimpt_subset)
+ apply (rule_tac x="closure S" in exI, simp)
+ apply (force simp: closure_def)
+ done
+
+lemma closedin_closed_eq: "closed S \<Longrightarrow> closedin (subtopology euclidean S) T \<longleftrightarrow> closed T \<and> T \<subseteq> S"
+ by (meson closedin_limpt closed_subset closedin_closed_trans)
+
+lemma connected_closed_set:
+ "closed S
+ \<Longrightarrow> connected S \<longleftrightarrow> (\<nexists>A B. closed A \<and> closed B \<and> A \<noteq> {} \<and> B \<noteq> {} \<and> A \<union> B = S \<and> A \<inter> B = {})"
+ unfolding connected_closedin_eq closedin_closed_eq connected_closedin_eq by blast
+
+text \<open>If a connnected set is written as the union of two nonempty closed sets, then these sets
+have to intersect.\<close>
+
+lemma connected_as_closed_union:
+ assumes "connected C" "C = A \<union> B" "closed A" "closed B" "A \<noteq> {}" "B \<noteq> {}"
+ shows "A \<inter> B \<noteq> {}"
+by (metis assms closed_Un connected_closed_set)
+
+lemma closedin_subset_trans:
+ "closedin (subtopology euclidean U) S \<Longrightarrow> S \<subseteq> T \<Longrightarrow> T \<subseteq> U \<Longrightarrow>
+ closedin (subtopology euclidean T) S"
+ by (meson closedin_limpt subset_iff)
+
+lemma openin_subset_trans:
+ "openin (subtopology euclidean U) S \<Longrightarrow> S \<subseteq> T \<Longrightarrow> T \<subseteq> U \<Longrightarrow>
+ openin (subtopology euclidean T) S"
+ by (auto simp: openin_open)
+
+lemma openin_Times:
+ "openin (subtopology euclidean S) S' \<Longrightarrow> openin (subtopology euclidean T) T' \<Longrightarrow>
+ openin (subtopology euclidean (S \<times> T)) (S' \<times> T')"
+ unfolding openin_open using open_Times by blast
+
+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"
+ obtains V W where "openin (subtopology euclidean S) V" "x \<in> V"
+ "openin (subtopology euclidean T) W" "y \<in> W" "(V \<times> W) \<subseteq> U"
+proof -
+ from assms obtain e where "e > 0" and "U \<subseteq> S \<times> T"
+ and e: "\<And>x' y'. \<lbrakk>x'\<in>S; y'\<in>T; dist (x', y') (x, y) < e\<rbrakk> \<Longrightarrow> (x', y') \<in> U"
+ by (force simp: openin_euclidean_subtopology_iff)
+ with assms have "x \<in> S" "y \<in> T"
+ by auto
+ show ?thesis
+ proof
+ show "openin (subtopology euclidean S) (ball x (e/2) \<inter> S)"
+ by (simp add: Int_commute openin_open_Int)
+ show "x \<in> ball x (e / 2) \<inter> S"
+ by (simp add: \<open>0 < e\<close> \<open>x \<in> S\<close>)
+ show "openin (subtopology euclidean T) (ball y (e/2) \<inter> T)"
+ by (simp add: Int_commute openin_open_Int)
+ show "y \<in> ball y (e / 2) \<inter> T"
+ by (simp add: \<open>0 < e\<close> \<open>y \<in> T\<close>)
+ show "(ball x (e / 2) \<inter> S) \<times> (ball y (e / 2) \<inter> T) \<subseteq> U"
+ by clarify (simp add: e dist_Pair_Pair \<open>0 < e\<close> dist_commute sqrt_sum_squares_half_less)
+ qed
+qed
+
+lemma openin_Times_eq:
+ fixes S :: "'a::metric_space set" and T :: "'b::metric_space set"
+ shows
+ "openin (subtopology euclidean (S \<times> T)) (S' \<times> T') \<longleftrightarrow>
+ S' = {} \<or> T' = {} \<or> openin (subtopology euclidean S) S' \<and> openin (subtopology euclidean T) T'"
+ (is "?lhs = ?rhs")
+proof (cases "S' = {} \<or> T' = {}")
+ case True
+ then show ?thesis by auto
+next
+ case False
+ then obtain x y where "x \<in> S'" "y \<in> T'"
+ by blast
+ show ?thesis
+ proof
+ assume ?lhs
+ have "openin (subtopology euclidean S) S'"
+ apply (subst openin_subopen, clarify)
+ apply (rule Times_in_interior_subtopology [OF _ \<open>?lhs\<close>])
+ using \<open>y \<in> T'\<close>
+ apply auto
+ done
+ moreover have "openin (subtopology euclidean T) T'"
+ apply (subst openin_subopen, clarify)
+ apply (rule Times_in_interior_subtopology [OF _ \<open>?lhs\<close>])
+ using \<open>x \<in> S'\<close>
+ apply auto
+ done
+ ultimately show ?rhs
+ by simp
+ next
+ assume ?rhs
+ with False show ?lhs
+ by (simp add: openin_Times)
+ qed
+qed
+
+lemma closedin_Times:
+ "closedin (subtopology euclidean S) S' \<Longrightarrow> closedin (subtopology euclidean T) T' \<Longrightarrow>
+ closedin (subtopology euclidean (S \<times> T)) (S' \<times> T')"
+ unfolding closedin_closed using closed_Times by blast
+
+lemma bdd_below_closure:
+ fixes A :: "real set"
+ assumes "bdd_below A"
+ shows "bdd_below (closure A)"
+proof -
+ from assms obtain m where "\<And>x. x \<in> A \<Longrightarrow> m \<le> x"
+ by (auto simp: bdd_below_def)
+ then have "A \<subseteq> {m..}" by auto
+ then have "closure A \<subseteq> {m..}"
+ using closed_real_atLeast by (rule closure_minimal)
+ then show ?thesis
+ by (auto simp: bdd_below_def)
+qed
+
+
+subsection \<open>Frontier (also known as boundary)\<close>
+
+definition%important "frontier S = closure S - interior S"
+
+lemma frontier_closed [iff]: "closed (frontier S)"
+ by (simp add: frontier_def closed_Diff)
+
+lemma frontier_closures: "frontier S = closure S \<inter> closure (- S)"
+ by (auto simp: frontier_def interior_closure)
+
+lemma frontier_Int: "frontier(S \<inter> T) = closure(S \<inter> T) \<inter> (frontier S \<union> frontier T)"
+proof -
+ have "closure (S \<inter> T) \<subseteq> closure S" "closure (S \<inter> T) \<subseteq> closure T"
+ by (simp_all add: closure_mono)
+ then show ?thesis
+ by (auto simp: frontier_closures)
+qed
+
+lemma frontier_Int_subset: "frontier(S \<inter> T) \<subseteq> frontier S \<union> frontier T"
+ by (auto simp: frontier_Int)
+
+lemma frontier_Int_closed:
+ assumes "closed S" "closed T"
+ shows "frontier(S \<inter> T) = (frontier S \<inter> T) \<union> (S \<inter> frontier T)"
+proof -
+ have "closure (S \<inter> T) = T \<inter> S"
+ using assms by (simp add: Int_commute closed_Int)
+ moreover have "T \<inter> (closure S \<inter> closure (- S)) = frontier S \<inter> T"
+ by (simp add: Int_commute frontier_closures)
+ ultimately show ?thesis
+ by (simp add: Int_Un_distrib Int_assoc Int_left_commute assms frontier_closures)
+qed
+
+lemma frontier_straddle:
+ fixes a :: "'a::metric_space"
+ shows "a \<in> frontier S \<longleftrightarrow> (\<forall>e>0. (\<exists>x\<in>S. dist a x < e) \<and> (\<exists>x. x \<notin> S \<and> dist a x < e))"
+ unfolding frontier_def closure_interior
+ by (auto simp: mem_interior subset_eq ball_def)
+
+lemma frontier_subset_closed: "closed S \<Longrightarrow> frontier S \<subseteq> S"
+ by (metis frontier_def closure_closed Diff_subset)
+
+lemma frontier_empty [simp]: "frontier {} = {}"
+ by (simp add: frontier_def)
+
+lemma frontier_subset_eq: "frontier S \<subseteq> S \<longleftrightarrow> closed S"
+proof -
+ {
+ assume "frontier S \<subseteq> S"
+ then have "closure S \<subseteq> S"
+ using interior_subset unfolding frontier_def by auto
+ then have "closed S"
+ using closure_subset_eq by auto
+ }
+ then show ?thesis using frontier_subset_closed[of S] ..
+qed
+
+lemma frontier_complement [simp]: "frontier (- S) = frontier S"
+ by (auto simp: frontier_def closure_complement interior_complement)
+
+lemma frontier_Un_subset: "frontier(S \<union> T) \<subseteq> frontier S \<union> frontier T"
+ by (metis compl_sup frontier_Int_subset frontier_complement)
+
+lemma frontier_disjoint_eq: "frontier S \<inter> S = {} \<longleftrightarrow> open S"
+ using frontier_complement frontier_subset_eq[of "- S"]
+ unfolding open_closed by auto
+
+lemma frontier_UNIV [simp]: "frontier UNIV = {}"
+ using frontier_complement frontier_empty by fastforce
+
+lemma frontier_interiors: "frontier s = - interior(s) - interior(-s)"
+ by (simp add: Int_commute frontier_def interior_closure)
+
+lemma frontier_interior_subset: "frontier(interior S) \<subseteq> frontier S"
+ by (simp add: Diff_mono frontier_interiors interior_mono interior_subset)
+
+lemma connected_Int_frontier:
+ "\<lbrakk>connected s; s \<inter> t \<noteq> {}; s - t \<noteq> {}\<rbrakk> \<Longrightarrow> (s \<inter> frontier t \<noteq> {})"
+ apply (simp add: frontier_interiors connected_openin, safe)
+ apply (drule_tac x="s \<inter> interior t" in spec, safe)
+ apply (drule_tac [2] x="s \<inter> interior (-t)" in spec)
+ apply (auto simp: disjoint_eq_subset_Compl dest: interior_subset [THEN subsetD])
+ done
+
+lemma closure_Un_frontier: "closure S = S \<union> frontier S"
+proof -
+ have "S \<union> interior S = S"
+ using interior_subset by auto
+ then show ?thesis
+ using closure_subset by (auto simp: frontier_def)
+qed
+
+
+subsection%unimportant \<open>Filters and the ``eventually true'' quantifier\<close>
+
+definition indirection :: "'a::real_normed_vector \<Rightarrow> 'a \<Rightarrow> 'a filter" (infixr "indirection" 70)
+ where "a indirection v = at a within {b. \<exists>c\<ge>0. b - a = scaleR c v}"
+
+text \<open>Identify Trivial limits, where we can't approach arbitrarily closely.\<close>
+
+lemma trivial_limit_within: "trivial_limit (at a within S) \<longleftrightarrow> \<not> a islimpt S"
+proof
+ assume "trivial_limit (at a within S)"
+ then show "\<not> a islimpt S"
+ unfolding trivial_limit_def
+ unfolding eventually_at_topological
+ unfolding islimpt_def
+ apply (clarsimp simp add: set_eq_iff)
+ apply (rename_tac T, rule_tac x=T in exI)
+ apply (clarsimp, drule_tac x=y in bspec, simp_all)
+ done
+next
+ assume "\<not> a islimpt S"
+ then show "trivial_limit (at a within S)"
+ unfolding trivial_limit_def eventually_at_topological islimpt_def
+ by metis
+qed
+
+lemma trivial_limit_at_iff: "trivial_limit (at a) \<longleftrightarrow> \<not> a islimpt UNIV"
+ using trivial_limit_within [of a UNIV] by simp
+
+lemma trivial_limit_at: "\<not> trivial_limit (at a)"
+ for a :: "'a::perfect_space"
+ by (rule at_neq_bot)
+
+lemma trivial_limit_at_infinity:
+ "\<not> trivial_limit (at_infinity :: ('a::{real_normed_vector,perfect_space}) filter)"
+ unfolding trivial_limit_def eventually_at_infinity
+ apply clarsimp
+ apply (subgoal_tac "\<exists>x::'a. x \<noteq> 0", clarify)
+ apply (rule_tac x="scaleR (b / norm x) x" in exI, simp)
+ apply (cut_tac islimpt_UNIV [of "0::'a", unfolded islimpt_def])
+ apply (drule_tac x=UNIV in spec, simp)
+ done
+
+lemma not_trivial_limit_within: "\<not> trivial_limit (at x within S) = (x \<in> closure (S - {x}))"
+ using islimpt_in_closure by (metis trivial_limit_within)
+
+lemma not_in_closure_trivial_limitI:
+ "x \<notin> closure s \<Longrightarrow> trivial_limit (at x within s)"
+ using not_trivial_limit_within[of x s]
+ by safe (metis Diff_empty Diff_insert0 closure_subset contra_subsetD)
+
+lemma filterlim_at_within_closure_implies_filterlim: "filterlim f l (at x within s)"
+ if "x \<in> closure s \<Longrightarrow> filterlim f l (at x within s)"
+ by (metis bot.extremum filterlim_filtercomap filterlim_mono not_in_closure_trivial_limitI that)
+
+lemma at_within_eq_bot_iff: "at c within A = bot \<longleftrightarrow> c \<notin> closure (A - {c})"
+ using not_trivial_limit_within[of c A] by blast
+
+text \<open>Some property holds "sufficiently close" to the limit point.\<close>
+
+lemma trivial_limit_eventually: "trivial_limit net \<Longrightarrow> eventually P net"
+ by simp
+
+lemma trivial_limit_eq: "trivial_limit net \<longleftrightarrow> (\<forall>P. eventually P net)"
+ by (simp add: filter_eq_iff)
+
+
+subsection \<open>Limits\<close>
+
+proposition Lim: "(f \<longlongrightarrow> l) net \<longleftrightarrow> trivial_limit net \<or> (\<forall>e>0. eventually (\<lambda>x. dist (f x) l < e) net)"
+ by (auto simp: tendsto_iff trivial_limit_eq)
+
+text \<open>Show that they yield usual definitions in the various cases.\<close>
+
+proposition Lim_within_le: "(f \<longlongrightarrow> l)(at a within S) \<longleftrightarrow>
+ (\<forall>e>0. \<exists>d>0. \<forall>x\<in>S. 0 < dist x a \<and> dist x a \<le> d \<longrightarrow> dist (f x) l < e)"
+ by (auto simp: tendsto_iff eventually_at_le)
+
+proposition Lim_within: "(f \<longlongrightarrow> l) (at a within S) \<longleftrightarrow>
+ (\<forall>e >0. \<exists>d>0. \<forall>x \<in> S. 0 < dist x a \<and> dist x a < d \<longrightarrow> dist (f x) l < e)"
+ by (auto simp: tendsto_iff eventually_at)
+
+corollary Lim_withinI [intro?]:
+ assumes "\<And>e. e > 0 \<Longrightarrow> \<exists>d>0. \<forall>x \<in> S. 0 < dist x a \<and> dist x a < d \<longrightarrow> dist (f x) l \<le> e"
+ shows "(f \<longlongrightarrow> l) (at a within S)"
+ apply (simp add: Lim_within, clarify)
+ apply (rule ex_forward [OF assms [OF half_gt_zero]], auto)
+ done
+
+proposition Lim_at: "(f \<longlongrightarrow> l) (at a) \<longleftrightarrow>
+ (\<forall>e >0. \<exists>d>0. \<forall>x. 0 < dist x a \<and> dist x a < d \<longrightarrow> dist (f x) l < e)"
+ by (auto simp: tendsto_iff eventually_at)
+
+proposition Lim_at_infinity: "(f \<longlongrightarrow> l) at_infinity \<longleftrightarrow> (\<forall>e>0. \<exists>b. \<forall>x. norm x \<ge> b \<longrightarrow> dist (f x) l < e)"
+ by (auto simp: tendsto_iff eventually_at_infinity)
+
+corollary Lim_at_infinityI [intro?]:
+ assumes "\<And>e. e > 0 \<Longrightarrow> \<exists>B. \<forall>x. norm x \<ge> B \<longrightarrow> dist (f x) l \<le> e"
+ shows "(f \<longlongrightarrow> l) at_infinity"
+ apply (simp add: Lim_at_infinity, clarify)
+ apply (rule ex_forward [OF assms [OF half_gt_zero]], auto)
+ done
+
+lemma Lim_eventually: "eventually (\<lambda>x. f x = l) net \<Longrightarrow> (f \<longlongrightarrow> l) net"
+ by (rule topological_tendstoI) (auto elim: eventually_mono)
+
+lemma Lim_transform_within_set:
+ fixes a :: "'a::metric_space" and l :: "'b::metric_space"
+ shows "\<lbrakk>(f \<longlongrightarrow> l) (at a within S); eventually (\<lambda>x. x \<in> S \<longleftrightarrow> x \<in> T) (at a)\<rbrakk>
+ \<Longrightarrow> (f \<longlongrightarrow> l) (at a within T)"
+apply (clarsimp simp: eventually_at Lim_within)
+apply (drule_tac x=e in spec, clarify)
+apply (rename_tac k)
+apply (rule_tac x="min d k" in exI, simp)
+done
+
+lemma Lim_transform_within_set_eq:
+ fixes a l :: "'a::real_normed_vector"
+ shows "eventually (\<lambda>x. x \<in> s \<longleftrightarrow> x \<in> t) (at a)
+ \<Longrightarrow> ((f \<longlongrightarrow> l) (at a within s) \<longleftrightarrow> (f \<longlongrightarrow> l) (at a within t))"
+ by (force intro: Lim_transform_within_set elim: eventually_mono)
+
+lemma Lim_transform_within_openin:
+ fixes a :: "'a::metric_space"
+ assumes f: "(f \<longlongrightarrow> l) (at a within T)"
+ and "openin (subtopology euclidean T) S" "a \<in> S"
+ and eq: "\<And>x. \<lbrakk>x \<in> S; x \<noteq> a\<rbrakk> \<Longrightarrow> f x = g x"
+ shows "(g \<longlongrightarrow> l) (at a within T)"
+proof -
+ obtain \<epsilon> where "0 < \<epsilon>" and \<epsilon>: "ball a \<epsilon> \<inter> T \<subseteq> S"
+ using assms by (force simp: openin_contains_ball)
+ then have "a \<in> ball a \<epsilon>"
+ by simp
+ show ?thesis
+ by (rule Lim_transform_within [OF f \<open>0 < \<epsilon>\<close> eq]) (use \<epsilon> in \<open>auto simp: dist_commute subset_iff\<close>)
+qed
+
+lemma continuous_transform_within_openin:
+ fixes a :: "'a::metric_space"
+ assumes "continuous (at a within T) f"
+ and "openin (subtopology euclidean T) S" "a \<in> S"
+ and eq: "\<And>x. x \<in> S \<Longrightarrow> f x = g x"
+ shows "continuous (at a within T) g"
+ using assms by (simp add: Lim_transform_within_openin continuous_within)
+
+text \<open>The expected monotonicity property.\<close>
+
+lemma Lim_Un:
+ assumes "(f \<longlongrightarrow> l) (at x within S)" "(f \<longlongrightarrow> l) (at x within T)"
+ shows "(f \<longlongrightarrow> l) (at x within (S \<union> T))"
+ using assms unfolding at_within_union by (rule filterlim_sup)
+
+lemma Lim_Un_univ:
+ "(f \<longlongrightarrow> l) (at x within S) \<Longrightarrow> (f \<longlongrightarrow> l) (at x within T) \<Longrightarrow>
+ S \<union> T = UNIV \<Longrightarrow> (f \<longlongrightarrow> l) (at x)"
+ by (metis Lim_Un)
+
+text \<open>Interrelations between restricted and unrestricted limits.\<close>
+
+lemma Lim_at_imp_Lim_at_within: "(f \<longlongrightarrow> l) (at x) \<Longrightarrow> (f \<longlongrightarrow> l) (at x within S)"
+ by (metis order_refl filterlim_mono subset_UNIV at_le)
+
+lemma eventually_within_interior:
+ assumes "x \<in> interior S"
+ shows "eventually P (at x within S) \<longleftrightarrow> eventually P (at x)"
+ (is "?lhs = ?rhs")
+proof
+ from assms obtain T where T: "open T" "x \<in> T" "T \<subseteq> S" ..
+ {
+ assume ?lhs
+ then obtain A where "open A" and "x \<in> A" and "\<forall>y\<in>A. y \<noteq> x \<longrightarrow> y \<in> S \<longrightarrow> P y"
+ by (auto simp: eventually_at_topological)
+ with T have "open (A \<inter> T)" and "x \<in> A \<inter> T" and "\<forall>y \<in> A \<inter> T. y \<noteq> x \<longrightarrow> P y"
+ by auto
+ then show ?rhs
+ by (auto simp: eventually_at_topological)
+ next
+ assume ?rhs
+ then show ?lhs
+ by (auto elim: eventually_mono simp: eventually_at_filter)
+ }
+qed
+
+lemma at_within_interior: "x \<in> interior S \<Longrightarrow> at x within S = at x"
+ unfolding filter_eq_iff by (intro allI eventually_within_interior)
+
+lemma Lim_within_LIMSEQ:
+ fixes a :: "'a::first_countable_topology"
+ assumes "\<forall>S. (\<forall>n. S n \<noteq> a \<and> S n \<in> T) \<and> S \<longlonglongrightarrow> a \<longrightarrow> (\<lambda>n. X (S n)) \<longlonglongrightarrow> L"
+ shows "(X \<longlongrightarrow> L) (at a within T)"
+ using assms unfolding tendsto_def [where l=L]
+ by (simp add: sequentially_imp_eventually_within)
+
+lemma Lim_right_bound:
+ fixes f :: "'a :: {linorder_topology, conditionally_complete_linorder, no_top} \<Rightarrow>
+ 'b::{linorder_topology, conditionally_complete_linorder}"
+ assumes mono: "\<And>a b. a \<in> I \<Longrightarrow> b \<in> I \<Longrightarrow> x < a \<Longrightarrow> a \<le> b \<Longrightarrow> f a \<le> f b"
+ and bnd: "\<And>a. a \<in> I \<Longrightarrow> x < a \<Longrightarrow> K \<le> f a"
+ shows "(f \<longlongrightarrow> Inf (f ` ({x<..} \<inter> I))) (at x within ({x<..} \<inter> I))"
+proof (cases "{x<..} \<inter> I = {}")
+ case True
+ then show ?thesis by simp
+next
+ case False
+ show ?thesis
+ proof (rule order_tendstoI)
+ fix a
+ assume a: "a < Inf (f ` ({x<..} \<inter> I))"
+ {
+ fix y
+ assume "y \<in> {x<..} \<inter> I"
+ with False bnd have "Inf (f ` ({x<..} \<inter> I)) \<le> f y"
+ by (auto intro!: cInf_lower bdd_belowI2)
+ with a have "a < f y"
+ by (blast intro: less_le_trans)
+ }
+ then show "eventually (\<lambda>x. a < f x) (at x within ({x<..} \<inter> I))"
+ by (auto simp: eventually_at_filter intro: exI[of _ 1] zero_less_one)
+ next
+ fix a
+ assume "Inf (f ` ({x<..} \<inter> I)) < a"
+ from cInf_lessD[OF _ this] False obtain y where y: "x < y" "y \<in> I" "f y < a"
+ by auto
+ then have "eventually (\<lambda>x. x \<in> I \<longrightarrow> f x < a) (at_right x)"
+ unfolding eventually_at_right[OF \<open>x < y\<close>] by (metis less_imp_le le_less_trans mono)
+ then show "eventually (\<lambda>x. f x < a) (at x within ({x<..} \<inter> I))"
+ unfolding eventually_at_filter by eventually_elim simp
+ qed
+qed
+
+text \<open>Another limit point characterization.\<close>
+
+lemma limpt_sequential_inj:
+ fixes x :: "'a::metric_space"
+ shows "x islimpt S \<longleftrightarrow>
+ (\<exists>f. (\<forall>n::nat. f n \<in> S - {x}) \<and> inj f \<and> (f \<longlongrightarrow> x) sequentially)"
+ (is "?lhs = ?rhs")
+proof
+ assume ?lhs
+ then have "\<forall>e>0. \<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e"
+ by (force simp: islimpt_approachable)
+ then obtain y where y: "\<And>e. e>0 \<Longrightarrow> y e \<in> S \<and> y e \<noteq> x \<and> dist (y e) x < e"
+ by metis
+ define f where "f \<equiv> rec_nat (y 1) (\<lambda>n fn. y (min (inverse(2 ^ (Suc n))) (dist fn x)))"
+ have [simp]: "f 0 = y 1"
+ "f(Suc n) = y (min (inverse(2 ^ (Suc n))) (dist (f n) x))" for n
+ by (simp_all add: f_def)
+ have f: "f n \<in> S \<and> (f n \<noteq> x) \<and> dist (f n) x < inverse(2 ^ n)" for n
+ proof (induction n)
+ case 0 show ?case
+ by (simp add: y)
+ next
+ case (Suc n) then show ?case
+ apply (auto simp: y)
+ by (metis half_gt_zero_iff inverse_positive_iff_positive less_divide_eq_numeral1(1) min_less_iff_conj y zero_less_dist_iff zero_less_numeral zero_less_power)
+ qed
+ show ?rhs
+ proof (rule_tac x=f in exI, intro conjI allI)
+ show "\<And>n. f n \<in> S - {x}"
+ using f by blast
+ have "dist (f n) x < dist (f m) x" if "m < n" for m n
+ using that
+ proof (induction n)
+ case 0 then show ?case by simp
+ next
+ case (Suc n)
+ then consider "m < n" | "m = n" using less_Suc_eq by blast
+ then show ?case
+ proof cases
+ assume "m < n"
+ have "dist (f(Suc n)) x = dist (y (min (inverse(2 ^ (Suc n))) (dist (f n) x))) x"
+ by simp
+ also have "\<dots> < dist (f n) x"
+ by (metis dist_pos_lt f min.strict_order_iff min_less_iff_conj y)
+ also have "\<dots> < dist (f m) x"
+ using Suc.IH \<open>m < n\<close> by blast
+ finally show ?thesis .
+ next
+ assume "m = n" then show ?case
+ by simp (metis dist_pos_lt f half_gt_zero_iff inverse_positive_iff_positive min_less_iff_conj y zero_less_numeral zero_less_power)
+ qed
+ qed
+ then show "inj f"
+ by (metis less_irrefl linorder_injI)
+ show "f \<longlonglongrightarrow> x"
+ apply (rule tendstoI)
+ apply (rule_tac c="nat (ceiling(1/e))" in eventually_sequentiallyI)
+ apply (rule less_trans [OF f [THEN conjunct2, THEN conjunct2]])
+ apply (simp add: field_simps)
+ by (meson le_less_trans mult_less_cancel_left not_le of_nat_less_two_power)
+ qed
+next
+ assume ?rhs
+ then show ?lhs
+ by (fastforce simp add: islimpt_approachable lim_sequentially)
+qed
+
+(*could prove directly from islimpt_sequential_inj, but only for metric spaces*)
+lemma islimpt_sequential:
+ fixes x :: "'a::first_countable_topology"
+ shows "x islimpt S \<longleftrightarrow> (\<exists>f. (\<forall>n::nat. f n \<in> S - {x}) \<and> (f \<longlongrightarrow> x) sequentially)"
+ (is "?lhs = ?rhs")
+proof
+ assume ?lhs
+ from countable_basis_at_decseq[of x] obtain A where A:
+ "\<And>i. open (A i)"
+ "\<And>i. x \<in> A i"
+ "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> eventually (\<lambda>i. A i \<subseteq> S) sequentially"
+ by blast
+ define f where "f n = (SOME y. y \<in> S \<and> y \<in> A n \<and> x \<noteq> y)" for n
+ {
+ fix n
+ from \<open>?lhs\<close> have "\<exists>y. y \<in> S \<and> y \<in> A n \<and> x \<noteq> y"
+ unfolding islimpt_def using A(1,2)[of n] by auto
+ then have "f n \<in> S \<and> f n \<in> A n \<and> x \<noteq> f n"
+ unfolding f_def by (rule someI_ex)
+ then have "f n \<in> S" "f n \<in> A n" "x \<noteq> f n" by auto
+ }
+ then have "\<forall>n. f n \<in> S - {x}" by auto
+ moreover have "(\<lambda>n. f n) \<longlonglongrightarrow> x"
+ proof (rule topological_tendstoI)
+ fix S
+ assume "open S" "x \<in> S"
+ from A(3)[OF this] \<open>\<And>n. f n \<in> A n\<close>
+ show "eventually (\<lambda>x. f x \<in> S) sequentially"
+ by (auto elim!: eventually_mono)
+ qed
+ ultimately show ?rhs by fast
+next
+ assume ?rhs
+ then obtain f :: "nat \<Rightarrow> 'a" where f: "\<And>n. f n \<in> S - {x}" and lim: "f \<longlonglongrightarrow> x"
+ by auto
+ show ?lhs
+ unfolding islimpt_def
+ proof safe
+ fix T
+ assume "open T" "x \<in> T"
+ from lim[THEN topological_tendstoD, OF this] f
+ show "\<exists>y\<in>S. y \<in> T \<and> y \<noteq> x"
+ unfolding eventually_sequentially by auto
+ qed
+qed
+
+lemma Lim_null:
+ fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
+ shows "(f \<longlongrightarrow> l) net \<longleftrightarrow> ((\<lambda>x. f(x) - l) \<longlongrightarrow> 0) net"
+ by (simp add: Lim dist_norm)
+
+lemma Lim_null_comparison:
+ fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
+ assumes "eventually (\<lambda>x. norm (f x) \<le> g x) net" "(g \<longlongrightarrow> 0) net"
+ shows "(f \<longlongrightarrow> 0) net"
+ using assms(2)
+proof (rule metric_tendsto_imp_tendsto)
+ show "eventually (\<lambda>x. dist (f x) 0 \<le> dist (g x) 0) net"
+ using assms(1) by (rule eventually_mono) (simp add: dist_norm)
+qed
+
+lemma Lim_transform_bound:
+ fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
+ and g :: "'a \<Rightarrow> 'c::real_normed_vector"
+ assumes "eventually (\<lambda>n. norm (f n) \<le> norm (g n)) net"
+ and "(g \<longlongrightarrow> 0) net"
+ shows "(f \<longlongrightarrow> 0) net"
+ using assms(1) tendsto_norm_zero [OF assms(2)]
+ by (rule Lim_null_comparison)
+
+lemma lim_null_mult_right_bounded:
+ fixes f :: "'a \<Rightarrow> 'b::real_normed_div_algebra"
+ assumes f: "(f \<longlongrightarrow> 0) F" and g: "eventually (\<lambda>x. norm(g x) \<le> B) F"
+ shows "((\<lambda>z. f z * g z) \<longlongrightarrow> 0) F"
+proof -
+ have *: "((\<lambda>x. norm (f x) * B) \<longlongrightarrow> 0) F"
+ by (simp add: f tendsto_mult_left_zero tendsto_norm_zero)
+ have "((\<lambda>x. norm (f x) * norm (g x)) \<longlongrightarrow> 0) F"
+ apply (rule Lim_null_comparison [OF _ *])
+ apply (simp add: eventually_mono [OF g] mult_left_mono)
+ done
+ then show ?thesis
+ by (subst tendsto_norm_zero_iff [symmetric]) (simp add: norm_mult)
+qed
+
+lemma lim_null_mult_left_bounded:
+ fixes f :: "'a \<Rightarrow> 'b::real_normed_div_algebra"
+ assumes g: "eventually (\<lambda>x. norm(g x) \<le> B) F" and f: "(f \<longlongrightarrow> 0) F"
+ shows "((\<lambda>z. g z * f z) \<longlongrightarrow> 0) F"
+proof -
+ have *: "((\<lambda>x. B * norm (f x)) \<longlongrightarrow> 0) F"
+ by (simp add: f tendsto_mult_right_zero tendsto_norm_zero)
+ have "((\<lambda>x. norm (g x) * norm (f x)) \<longlongrightarrow> 0) F"
+ apply (rule Lim_null_comparison [OF _ *])
+ apply (simp add: eventually_mono [OF g] mult_right_mono)
+ done
+ then show ?thesis
+ by (subst tendsto_norm_zero_iff [symmetric]) (simp add: norm_mult)
+qed
+
+lemma lim_null_scaleR_bounded:
+ assumes f: "(f \<longlongrightarrow> 0) net" and gB: "eventually (\<lambda>a. f a = 0 \<or> norm(g a) \<le> B) net"
+ shows "((\<lambda>n. f n *\<^sub>R g n) \<longlongrightarrow> 0) net"
+proof
+ fix \<epsilon>::real
+ assume "0 < \<epsilon>"
+ then have B: "0 < \<epsilon> / (abs B + 1)" by simp
+ have *: "\<bar>f x\<bar> * norm (g x) < \<epsilon>" if f: "\<bar>f x\<bar> * (\<bar>B\<bar> + 1) < \<epsilon>" and g: "norm (g x) \<le> B" for x
+ proof -
+ have "\<bar>f x\<bar> * norm (g x) \<le> \<bar>f x\<bar> * B"
+ by (simp add: mult_left_mono g)
+ also have "\<dots> \<le> \<bar>f x\<bar> * (\<bar>B\<bar> + 1)"
+ by (simp add: mult_left_mono)
+ also have "\<dots> < \<epsilon>"
+ by (rule f)
+ finally show ?thesis .
+ qed
+ show "\<forall>\<^sub>F x in net. dist (f x *\<^sub>R g x) 0 < \<epsilon>"
+ apply (rule eventually_mono [OF eventually_conj [OF tendstoD [OF f B] gB] ])
+ apply (auto simp: \<open>0 < \<epsilon>\<close> divide_simps * split: if_split_asm)
+ done
+qed
+
+text\<open>Deducing things about the limit from the elements.\<close>
+
+lemma Lim_in_closed_set:
+ assumes "closed S"
+ and "eventually (\<lambda>x. f(x) \<in> S) net"
+ and "\<not> trivial_limit net" "(f \<longlongrightarrow> l) net"
+ shows "l \<in> S"
+proof (rule ccontr)
+ assume "l \<notin> S"
+ with \<open>closed S\<close> have "open (- S)" "l \<in> - S"
+ by (simp_all add: open_Compl)
+ with assms(4) have "eventually (\<lambda>x. f x \<in> - S) net"
+ by (rule topological_tendstoD)
+ with assms(2) have "eventually (\<lambda>x. False) net"
+ by (rule eventually_elim2) simp
+ with assms(3) show "False"
+ by (simp add: eventually_False)
+qed
+
+text\<open>Need to prove closed(cball(x,e)) before deducing this as a corollary.\<close>
+
+lemma Lim_dist_ubound:
+ assumes "\<not>(trivial_limit net)"
+ and "(f \<longlongrightarrow> l) net"
+ and "eventually (\<lambda>x. dist a (f x) \<le> e) net"
+ shows "dist a l \<le> e"
+ using assms by (fast intro: tendsto_le tendsto_intros)
+
+lemma Lim_norm_ubound:
+ fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
+ assumes "\<not>(trivial_limit net)" "(f \<longlongrightarrow> l) net" "eventually (\<lambda>x. norm(f x) \<le> e) net"
+ shows "norm(l) \<le> e"
+ using assms by (fast intro: tendsto_le tendsto_intros)
+
+lemma Lim_norm_lbound:
+ fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
+ assumes "\<not> trivial_limit net"
+ and "(f \<longlongrightarrow> l) net"
+ and "eventually (\<lambda>x. e \<le> norm (f x)) net"
+ shows "e \<le> norm l"
+ using assms by (fast intro: tendsto_le tendsto_intros)
+
+text\<open>Limit under bilinear function\<close>
+
+lemma Lim_bilinear:
+ assumes "(f \<longlongrightarrow> l) net"
+ and "(g \<longlongrightarrow> m) net"
+ and "bounded_bilinear h"
+ shows "((\<lambda>x. h (f x) (g x)) \<longlongrightarrow> (h l m)) net"
+ using \<open>bounded_bilinear h\<close> \<open>(f \<longlongrightarrow> l) net\<close> \<open>(g \<longlongrightarrow> m) net\<close>
+ by (rule bounded_bilinear.tendsto)
+
+text\<open>These are special for limits out of the same vector space.\<close>
+
+lemma Lim_within_id: "(id \<longlongrightarrow> a) (at a within s)"
+ unfolding id_def by (rule tendsto_ident_at)
+
+lemma Lim_at_id: "(id \<longlongrightarrow> a) (at a)"
+ unfolding id_def by (rule tendsto_ident_at)
+
+lemma Lim_at_zero:
+ fixes a :: "'a::real_normed_vector"
+ and l :: "'b::topological_space"
+ shows "(f \<longlongrightarrow> l) (at a) \<longleftrightarrow> ((\<lambda>x. f(a + x)) \<longlongrightarrow> l) (at 0)"
+ using LIM_offset_zero LIM_offset_zero_cancel ..
+
+text\<open>It's also sometimes useful to extract the limit point from the filter.\<close>
+
+abbreviation netlimit :: "'a::t2_space filter \<Rightarrow> 'a"
+ where "netlimit F \<equiv> Lim F (\<lambda>x. x)"
+
+lemma netlimit_within: "\<not> trivial_limit (at a within S) \<Longrightarrow> netlimit (at a within S) = a"
+ by (rule tendsto_Lim) (auto intro: tendsto_intros)
+
+lemma netlimit_at [simp]:
+ fixes a :: "'a::{perfect_space,t2_space}"
+ shows "netlimit (at a) = a"
+ using netlimit_within [of a UNIV] by simp
+
+lemma lim_within_interior:
+ "x \<in> interior S \<Longrightarrow> (f \<longlongrightarrow> l) (at x within S) \<longleftrightarrow> (f \<longlongrightarrow> l) (at x)"
+ by (metis at_within_interior)
+
+lemma netlimit_within_interior:
+ fixes x :: "'a::{t2_space,perfect_space}"
+ assumes "x \<in> interior S"
+ shows "netlimit (at x within S) = x"
+ using assms by (metis at_within_interior netlimit_at)
+
+lemma netlimit_at_vector:
+ fixes a :: "'a::real_normed_vector"
+ shows "netlimit (at a) = a"
+proof (cases "\<exists>x. x \<noteq> a")
+ case True then obtain x where x: "x \<noteq> a" ..
+ have "\<not> trivial_limit (at a)"
+ unfolding trivial_limit_def eventually_at dist_norm
+ apply clarsimp
+ apply (rule_tac x="a + scaleR (d / 2) (sgn (x - a))" in exI)
+ apply (simp add: norm_sgn sgn_zero_iff x)
+ done
+ then show ?thesis
+ by (rule netlimit_within [of a UNIV])
+qed simp
+
+
+text\<open>Useful lemmas on closure and set of possible sequential limits.\<close>
+
+lemma closure_sequential:
+ fixes l :: "'a::first_countable_topology"
+ shows "l \<in> closure S \<longleftrightarrow> (\<exists>x. (\<forall>n. x n \<in> S) \<and> (x \<longlongrightarrow> l) sequentially)"
+ (is "?lhs = ?rhs")
+proof
+ assume "?lhs"
+ moreover
+ {
+ assume "l \<in> S"
+ then have "?rhs" using tendsto_const[of l sequentially] by auto
+ }
+ moreover
+ {
+ assume "l islimpt S"
+ then have "?rhs" unfolding islimpt_sequential by auto
+ }
+ ultimately show "?rhs"
+ unfolding closure_def by auto
+next
+ assume "?rhs"
+ then show "?lhs" unfolding closure_def islimpt_sequential by auto
+qed
+
+lemma closed_sequential_limits:
+ fixes S :: "'a::first_countable_topology set"
+ shows "closed S \<longleftrightarrow> (\<forall>x l. (\<forall>n. x n \<in> S) \<and> (x \<longlongrightarrow> l) sequentially \<longrightarrow> l \<in> S)"
+by (metis closure_sequential closure_subset_eq subset_iff)
+
+lemma closure_approachable:
+ fixes S :: "'a::metric_space set"
+ shows "x \<in> closure S \<longleftrightarrow> (\<forall>e>0. \<exists>y\<in>S. dist y x < e)"
+ apply (auto simp: closure_def islimpt_approachable)
+ apply (metis dist_self)
+ done
+
+lemma closure_approachable_le:
+ fixes S :: "'a::metric_space set"
+ shows "x \<in> closure S \<longleftrightarrow> (\<forall>e>0. \<exists>y\<in>S. dist y x \<le> e)"
+ unfolding closure_approachable
+ using dense by force
+
+lemma closure_approachableD:
+ assumes "x \<in> closure S" "e>0"
+ shows "\<exists>y\<in>S. dist x y < e"
+ using assms unfolding closure_approachable by (auto simp: dist_commute)
+
+lemma closed_approachable:
+ fixes S :: "'a::metric_space set"
+ shows "closed S \<Longrightarrow> (\<forall>e>0. \<exists>y\<in>S. dist y x < e) \<longleftrightarrow> x \<in> S"
+ by (metis closure_closed closure_approachable)
+
+lemma closure_contains_Inf:
+ fixes S :: "real set"
+ assumes "S \<noteq> {}" "bdd_below S"
+ shows "Inf S \<in> closure S"
+proof -
+ have *: "\<forall>x\<in>S. Inf S \<le> x"
+ using cInf_lower[of _ S] assms by metis
+ {
+ fix e :: real
+ assume "e > 0"
+ then have "Inf S < Inf S + e" by simp
+ with assms obtain x where "x \<in> S" "x < Inf S + e"
+ by (subst (asm) cInf_less_iff) auto
+ with * have "\<exists>x\<in>S. dist x (Inf S) < e"
+ by (intro bexI[of _ x]) (auto simp: dist_real_def)
+ }
+ then show ?thesis unfolding closure_approachable by auto
+qed
+
+lemma closure_Int_ballI:
+ fixes S :: "'a :: metric_space set"
+ assumes "\<And>U. \<lbrakk>openin (subtopology euclidean S) U; U \<noteq> {}\<rbrakk> \<Longrightarrow> T \<inter> U \<noteq> {}"
+ shows "S \<subseteq> closure T"
+proof (clarsimp simp: closure_approachable dist_commute)
+ fix x and e::real
+ assume "x \<in> S" "0 < e"
+ with assms [of "S \<inter> ball x e"] show "\<exists>y\<in>T. dist x y < e"
+ by force
+qed
+
+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 not_trivial_limit_within_ball:
+ "\<not> trivial_limit (at x within S) \<longleftrightarrow> (\<forall>e>0. S \<inter> ball x e - {x} \<noteq> {})"
+ (is "?lhs \<longleftrightarrow> ?rhs")
+proof
+ show ?rhs if ?lhs
+ proof -
+ {
+ fix e :: real
+ assume "e > 0"
+ then obtain y where "y \<in> S - {x}" and "dist y x < e"
+ using \<open>?lhs\<close> not_trivial_limit_within[of x S] closure_approachable[of x "S - {x}"]
+ by auto
+ then have "y \<in> S \<inter> ball x e - {x}"
+ unfolding ball_def by (simp add: dist_commute)
+ then have "S \<inter> ball x e - {x} \<noteq> {}" by blast
+ }
+ then show ?thesis by auto
+ qed
+ show ?lhs if ?rhs
+ proof -
+ {
+ fix e :: real
+ assume "e > 0"
+ then obtain y where "y \<in> S \<inter> ball x e - {x}"
+ using \<open>?rhs\<close> by blast
+ then have "y \<in> S - {x}" and "dist y x < e"
+ unfolding ball_def by (simp_all add: dist_commute)
+ then have "\<exists>y \<in> S - {x}. dist y x < e"
+ by auto
+ }
+ then show ?thesis
+ using not_trivial_limit_within[of x S] closure_approachable[of x "S - {x}"]
+ by auto
+ qed
+qed
+
+lemma tendsto_If_within_closures:
+ assumes f: "x \<in> s \<union> (closure s \<inter> closure t) \<Longrightarrow>
+ (f \<longlongrightarrow> l x) (at x within s \<union> (closure s \<inter> closure t))"
+ assumes g: "x \<in> t \<union> (closure s \<inter> closure t) \<Longrightarrow>
+ (g \<longlongrightarrow> l x) (at x within t \<union> (closure s \<inter> closure t))"
+ assumes "x \<in> s \<union> t"
+ shows "((\<lambda>x. if x \<in> s then f x else g x) \<longlongrightarrow> l x) (at x within s \<union> t)"
+proof -
+ have *: "(s \<union> t) \<inter> {x. x \<in> s} = s" "(s \<union> t) \<inter> {x. x \<notin> s} = t - s"
+ by auto
+ have "(f \<longlongrightarrow> l x) (at x within s)"
+ by (rule filterlim_at_within_closure_implies_filterlim)
+ (use \<open>x \<in> _\<close> in \<open>auto simp: inf_commute closure_def intro: tendsto_within_subset[OF f]\<close>)
+ moreover
+ have "(g \<longlongrightarrow> l x) (at x within t - s)"
+ by (rule filterlim_at_within_closure_implies_filterlim)
+ (use \<open>x \<in> _\<close> in
+ \<open>auto intro!: tendsto_within_subset[OF g] simp: closure_def intro: islimpt_subset\<close>)
+ ultimately show ?thesis
+ by (intro filterlim_at_within_If) (simp_all only: *)
+qed
+
+
+subsection \<open>Boundedness\<close>
+
+ (* FIXME: This has to be unified with BSEQ!! *)
+definition%important (in metric_space) bounded :: "'a set \<Rightarrow> bool"
+ where "bounded S \<longleftrightarrow> (\<exists>x e. \<forall>y\<in>S. dist x y \<le> e)"
+
+lemma bounded_subset_cball: "bounded S \<longleftrightarrow> (\<exists>e x. S \<subseteq> cball x e \<and> 0 \<le> e)"
+ unfolding bounded_def subset_eq by auto (meson order_trans zero_le_dist)
+
+lemma bounded_any_center: "bounded S \<longleftrightarrow> (\<exists>e. \<forall>y\<in>S. dist a y \<le> e)"
+ unfolding bounded_def
+ by auto (metis add.commute add_le_cancel_right dist_commute dist_triangle_le)
+
+lemma bounded_iff: "bounded S \<longleftrightarrow> (\<exists>a. \<forall>x\<in>S. norm x \<le> a)"
+ unfolding bounded_any_center [where a=0]
+ by (simp add: dist_norm)
+
+lemma bdd_above_norm: "bdd_above (norm ` X) \<longleftrightarrow> bounded X"
+ by (simp add: bounded_iff bdd_above_def)
+
+lemma bounded_norm_comp: "bounded ((\<lambda>x. norm (f x)) ` S) = bounded (f ` S)"
+ by (simp add: bounded_iff)
+
+lemma boundedI:
+ assumes "\<And>x. x \<in> S \<Longrightarrow> norm x \<le> B"
+ shows "bounded S"
+ using assms bounded_iff by blast
+
+lemma bounded_empty [simp]: "bounded {}"
+ by (simp add: bounded_def)
+
+lemma bounded_subset: "bounded T \<Longrightarrow> S \<subseteq> T \<Longrightarrow> bounded S"
+ by (metis bounded_def subset_eq)
+
+lemma bounded_interior[intro]: "bounded S \<Longrightarrow> bounded(interior S)"
+ by (metis bounded_subset interior_subset)
+
+lemma bounded_closure[intro]:
+ assumes "bounded S"
+ shows "bounded (closure S)"
+proof -
+ from assms obtain x and a where a: "\<forall>y\<in>S. dist x y \<le> a"
+ unfolding bounded_def by auto
+ {
+ fix y
+ assume "y \<in> closure S"
+ then obtain f where f: "\<forall>n. f n \<in> S" "(f \<longlongrightarrow> y) sequentially"
+ unfolding closure_sequential by auto
+ have "\<forall>n. f n \<in> S \<longrightarrow> dist x (f n) \<le> a" using a by simp
+ then have "eventually (\<lambda>n. dist x (f n) \<le> a) sequentially"
+ by (simp add: f(1))
+ have "dist x y \<le> a"
+ apply (rule Lim_dist_ubound [of sequentially f])
+ apply (rule trivial_limit_sequentially)
+ apply (rule f(2))
+ apply fact
+ done
+ }
+ then show ?thesis
+ unfolding bounded_def by auto
+qed
+
+lemma bounded_closure_image: "bounded (f ` closure S) \<Longrightarrow> bounded (f ` S)"
+ by (simp add: bounded_subset closure_subset image_mono)
+
+lemma bounded_cball[simp,intro]: "bounded (cball x e)"
+ apply (simp add: bounded_def)
+ apply (rule_tac x=x in exI)
+ apply (rule_tac x=e in exI, auto)
+ done
+
+lemma bounded_ball[simp,intro]: "bounded (ball x e)"
+ by (metis ball_subset_cball bounded_cball bounded_subset)
+
+lemma bounded_Un[simp]: "bounded (S \<union> T) \<longleftrightarrow> bounded S \<and> bounded T"
+ by (auto simp: bounded_def) (metis Un_iff bounded_any_center le_max_iff_disj)
+
+lemma bounded_Union[intro]: "finite F \<Longrightarrow> \<forall>S\<in>F. bounded S \<Longrightarrow> bounded (\<Union>F)"
+ by (induct rule: finite_induct[of F]) auto
+
+lemma bounded_UN [intro]: "finite A \<Longrightarrow> \<forall>x\<in>A. bounded (B x) \<Longrightarrow> bounded (\<Union>x\<in>A. B x)"
+ by (induct set: finite) auto
+
+lemma bounded_insert [simp]: "bounded (insert x S) \<longleftrightarrow> bounded S"
+proof -
+ have "\<forall>y\<in>{x}. dist x y \<le> 0"
+ by simp
+ then have "bounded {x}"
+ unfolding bounded_def by fast
+ then show ?thesis
+ by (metis insert_is_Un bounded_Un)
+qed
+
+lemma bounded_subset_ballI: "S \<subseteq> ball x r \<Longrightarrow> bounded S"
+ by (meson bounded_ball bounded_subset)
+
+lemma bounded_subset_ballD:
+ assumes "bounded S" shows "\<exists>r. 0 < r \<and> S \<subseteq> ball x r"
+proof -
+ obtain e::real and y where "S \<subseteq> cball y e" "0 \<le> e"
+ using assms by (auto simp: bounded_subset_cball)
+ then show ?thesis
+ apply (rule_tac x="dist x y + e + 1" in exI)
+ apply (simp add: add.commute add_pos_nonneg)
+ apply (erule subset_trans)
+ apply (clarsimp simp add: cball_def)
+ by (metis add_le_cancel_right add_strict_increasing dist_commute dist_triangle_le zero_less_one)
+qed
+
+lemma finite_imp_bounded [intro]: "finite S \<Longrightarrow> bounded S"
+ by (induct set: finite) simp_all
+
+lemma bounded_pos: "bounded S \<longleftrightarrow> (\<exists>b>0. \<forall>x\<in> S. norm x \<le> b)"
+ apply (simp add: bounded_iff)
+ apply (subgoal_tac "\<And>x (y::real). 0 < 1 + \<bar>y\<bar> \<and> (x \<le> y \<longrightarrow> x \<le> 1 + \<bar>y\<bar>)")
+ apply metis
+ apply arith
+ done
+
+lemma bounded_pos_less: "bounded S \<longleftrightarrow> (\<exists>b>0. \<forall>x\<in> S. norm x < b)"
+ apply (simp add: bounded_pos)
+ apply (safe; rule_tac x="b+1" in exI; force)
+ done
+
+lemma Bseq_eq_bounded:
+ fixes f :: "nat \<Rightarrow> 'a::real_normed_vector"
+ shows "Bseq f \<longleftrightarrow> bounded (range f)"
+ unfolding Bseq_def bounded_pos by auto
+
+lemma bounded_Int[intro]: "bounded S \<or> bounded T \<Longrightarrow> bounded (S \<inter> T)"
+ by (metis Int_lower1 Int_lower2 bounded_subset)
+
+lemma bounded_diff[intro]: "bounded S \<Longrightarrow> bounded (S - T)"
+ by (metis Diff_subset bounded_subset)
+
+lemma not_bounded_UNIV[simp]:
+ "\<not> bounded (UNIV :: 'a::{real_normed_vector, perfect_space} set)"
+proof (auto simp: bounded_pos not_le)
+ obtain x :: 'a where "x \<noteq> 0"
+ using perfect_choose_dist [OF zero_less_one] by fast
+ fix b :: real
+ assume b: "b >0"
+ have b1: "b +1 \<ge> 0"
+ using b by simp
+ with \<open>x \<noteq> 0\<close> have "b < norm (scaleR (b + 1) (sgn x))"
+ by (simp add: norm_sgn)
+ then show "\<exists>x::'a. b < norm x" ..
+qed
+
+corollary cobounded_imp_unbounded:
+ fixes S :: "'a::{real_normed_vector, perfect_space} set"
+ shows "bounded (- S) \<Longrightarrow> \<not> bounded S"
+ using bounded_Un [of S "-S"] by (simp add: sup_compl_top)
+
+lemma bounded_dist_comp:
+ assumes "bounded (f ` S)" "bounded (g ` S)"
+ shows "bounded ((\<lambda>x. dist (f x) (g x)) ` S)"
+proof -
+ from assms obtain M1 M2 where *: "dist (f x) undefined \<le> M1" "dist undefined (g x) \<le> M2" if "x \<in> S" for x
+ by (auto simp: bounded_any_center[of _ undefined] dist_commute)
+ have "dist (f x) (g x) \<le> M1 + M2" if "x \<in> S" for x
+ using *[OF that]
+ by (rule order_trans[OF dist_triangle add_mono])
+ then show ?thesis
+ by (auto intro!: boundedI)
+qed
+
+lemma bounded_linear_image:
+ assumes "bounded S"
+ and "bounded_linear f"
+ shows "bounded (f ` S)"
+proof -
+ from assms(1) obtain b where "b > 0" and b: "\<forall>x\<in>S. norm x \<le> b"
+ unfolding bounded_pos by auto
+ from assms(2) obtain B where B: "B > 0" "\<forall>x. norm (f x) \<le> B * norm x"
+ using bounded_linear.pos_bounded by (auto simp: ac_simps)
+ show ?thesis
+ unfolding bounded_pos
+ proof (intro exI, safe)
+ show "norm (f x) \<le> B * b" if "x \<in> S" for x
+ by (meson B b less_imp_le mult_left_mono order_trans that)
+ qed (use \<open>b > 0\<close> \<open>B > 0\<close> in auto)
+qed
+
+lemma bounded_scaling:
+ fixes S :: "'a::real_normed_vector set"
+ shows "bounded S \<Longrightarrow> bounded ((\<lambda>x. c *\<^sub>R x) ` S)"
+ apply (rule bounded_linear_image, assumption)
+ apply (rule bounded_linear_scaleR_right)
+ done
+
+lemma bounded_scaleR_comp:
+ fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
+ assumes "bounded (f ` S)"
+ shows "bounded ((\<lambda>x. r *\<^sub>R f x) ` S)"
+ using bounded_scaling[of "f ` S" r] assms
+ by (auto simp: image_image)
+
+lemma bounded_translation:
+ fixes S :: "'a::real_normed_vector set"
+ assumes "bounded S"
+ shows "bounded ((\<lambda>x. a + x) ` S)"
+proof -
+ from assms obtain b where b: "b > 0" "\<forall>x\<in>S. norm x \<le> b"
+ unfolding bounded_pos by auto
+ {
+ fix x
+ assume "x \<in> S"
+ then have "norm (a + x) \<le> b + norm a"
+ using norm_triangle_ineq[of a x] b by auto
+ }
+ then show ?thesis
+ unfolding bounded_pos
+ using norm_ge_zero[of a] b(1) and add_strict_increasing[of b 0 "norm a"]
+ by (auto intro!: exI[of _ "b + norm a"])
+qed
+
+lemma bounded_translation_minus:
+ fixes S :: "'a::real_normed_vector set"
+ shows "bounded S \<Longrightarrow> bounded ((\<lambda>x. x - a) ` S)"
+using bounded_translation [of S "-a"] by simp
+
+lemma bounded_uminus [simp]:
+ fixes X :: "'a::real_normed_vector set"
+ shows "bounded (uminus ` X) \<longleftrightarrow> bounded X"
+by (auto simp: bounded_def dist_norm; rule_tac x="-x" in exI; force simp: add.commute norm_minus_commute)
+
+lemma uminus_bounded_comp [simp]:
+ fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
+ shows "bounded ((\<lambda>x. - f x) ` S) \<longleftrightarrow> bounded (f ` S)"
+ using bounded_uminus[of "f ` S"]
+ by (auto simp: image_image)
+
+lemma bounded_plus_comp:
+ fixes f g::"'a \<Rightarrow> 'b::real_normed_vector"
+ assumes "bounded (f ` S)"
+ assumes "bounded (g ` S)"
+ shows "bounded ((\<lambda>x. f x + g x) ` S)"
+proof -
+ {
+ fix B C
+ assume "\<And>x. x\<in>S \<Longrightarrow> norm (f x) \<le> B" "\<And>x. x\<in>S \<Longrightarrow> norm (g x) \<le> C"
+ then have "\<And>x. x \<in> S \<Longrightarrow> norm (f x + g x) \<le> B + C"
+ by (auto intro!: norm_triangle_le add_mono)
+ } then show ?thesis
+ using assms by (fastforce simp: bounded_iff)
+qed
+
+lemma bounded_plus:
+ fixes S ::"'a::real_normed_vector set"
+ assumes "bounded S" "bounded T"
+ shows "bounded ((\<lambda>(x,y). x + y) ` (S \<times> T))"
+ using bounded_plus_comp [of fst "S \<times> T" snd] assms
+ by (auto simp: split_def split: if_split_asm)
+
+lemma bounded_minus_comp:
+ "bounded (f ` S) \<Longrightarrow> bounded (g ` S) \<Longrightarrow> bounded ((\<lambda>x. f x - g x) ` S)"
+ for f g::"'a \<Rightarrow> 'b::real_normed_vector"
+ using bounded_plus_comp[of "f" S "\<lambda>x. - g x"]
+ by auto
+
+lemma bounded_minus:
+ fixes S ::"'a::real_normed_vector set"
+ assumes "bounded S" "bounded T"
+ shows "bounded ((\<lambda>(x,y). x - y) ` (S \<times> T))"
+ using bounded_minus_comp [of fst "S \<times> T" snd] assms
+ by (auto simp: split_def split: if_split_asm)
+
+
+subsection \<open>Compactness\<close>
+
+subsubsection \<open>Bolzano-Weierstrass property\<close>
+
+proposition heine_borel_imp_bolzano_weierstrass:
+ assumes "compact s"
+ and "infinite t"
+ and "t \<subseteq> s"
+ shows "\<exists>x \<in> s. x islimpt t"
+proof (rule ccontr)
+ assume "\<not> (\<exists>x \<in> s. x islimpt t)"
+ then obtain f where f: "\<forall>x\<in>s. x \<in> f x \<and> open (f x) \<and> (\<forall>y\<in>t. y \<in> f x \<longrightarrow> y = x)"
+ unfolding islimpt_def
+ using bchoice[of s "\<lambda> x T. x \<in> T \<and> open T \<and> (\<forall>y\<in>t. y \<in> T \<longrightarrow> y = x)"]
+ by auto
+ obtain g where g: "g \<subseteq> {t. \<exists>x. x \<in> s \<and> t = f x}" "finite g" "s \<subseteq> \<Union>g"
+ using assms(1)[unfolded compact_eq_heine_borel, THEN spec[where x="{t. \<exists>x. x\<in>s \<and> t = f x}"]]
+ using f by auto
+ from g(1,3) have g':"\<forall>x\<in>g. \<exists>xa \<in> s. x = f xa"
+ by auto
+ {
+ fix x y
+ assume "x \<in> t" "y \<in> t" "f x = f y"
+ then have "x \<in> f x" "y \<in> f x \<longrightarrow> y = x"
+ using f[THEN bspec[where x=x]] and \<open>t \<subseteq> s\<close> by auto
+ then have "x = y"
+ using \<open>f x = f y\<close> and f[THEN bspec[where x=y]] and \<open>y \<in> t\<close> and \<open>t \<subseteq> s\<close>
+ by auto
+ }
+ then have "inj_on f t"
+ unfolding inj_on_def by simp
+ then have "infinite (f ` t)"
+ using assms(2) using finite_imageD by auto
+ moreover
+ {
+ fix x
+ assume "x \<in> t" "f x \<notin> g"
+ from g(3) assms(3) \<open>x \<in> t\<close> obtain h where "h \<in> g" and "x \<in> h"
+ by auto
+ then obtain y where "y \<in> s" "h = f y"
+ using g'[THEN bspec[where x=h]] by auto
+ then have "y = x"
+ using f[THEN bspec[where x=y]] and \<open>x\<in>t\<close> and \<open>x\<in>h\<close>[unfolded \<open>h = f y\<close>]
+ by auto
+ then have False
+ using \<open>f x \<notin> g\<close> \<open>h \<in> g\<close> unfolding \<open>h = f y\<close>
+ by auto
+ }
+ then have "f ` t \<subseteq> g" by auto
+ ultimately show False
+ using g(2) using finite_subset by auto
+qed
+
+lemma acc_point_range_imp_convergent_subsequence:
+ fixes l :: "'a :: first_countable_topology"
+ assumes l: "\<forall>U. l\<in>U \<longrightarrow> open U \<longrightarrow> infinite (U \<inter> range f)"
+ shows "\<exists>r::nat\<Rightarrow>nat. strict_mono r \<and> (f \<circ> r) \<longlonglongrightarrow> l"
+proof -
+ from countable_basis_at_decseq[of l]
+ obtain A where A:
+ "\<And>i. open (A i)"
+ "\<And>i. l \<in> A i"
+ "\<And>S. open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>i. A i \<subseteq> S) sequentially"
+ by blast
+ define s where "s n i = (SOME j. i < j \<and> f j \<in> A (Suc n))" for n i
+ {
+ fix n i
+ have "infinite (A (Suc n) \<inter> range f - f`{.. i})"
+ using l A by auto
+ then have "\<exists>x. x \<in> A (Suc n) \<inter> range f - f`{.. i}"
+ unfolding ex_in_conv by (intro notI) simp
+ then have "\<exists>j. f j \<in> A (Suc n) \<and> j \<notin> {.. i}"
+ by auto
+ then have "\<exists>a. i < a \<and> f a \<in> A (Suc n)"
+ by (auto simp: not_le)
+ then have "i < s n i" "f (s n i) \<in> A (Suc n)"
+ unfolding s_def by (auto intro: someI2_ex)
+ }
+ note s = this
+ define r where "r = rec_nat (s 0 0) s"
+ have "strict_mono r"
+ by (auto simp: r_def s strict_mono_Suc_iff)
+ moreover
+ have "(\<lambda>n. f (r n)) \<longlonglongrightarrow> l"
+ proof (rule topological_tendstoI)
+ fix S
+ assume "open S" "l \<in> S"
+ with A(3) have "eventually (\<lambda>i. A i \<subseteq> S) sequentially"
+ by auto
+ moreover
+ {
+ fix i
+ assume "Suc 0 \<le> i"
+ then have "f (r i) \<in> A i"
+ by (cases i) (simp_all add: r_def s)
+ }
+ then have "eventually (\<lambda>i. f (r i) \<in> A i) sequentially"
+ by (auto simp: eventually_sequentially)
+ ultimately show "eventually (\<lambda>i. f (r i) \<in> S) sequentially"
+ by eventually_elim auto
+ qed
+ ultimately show "\<exists>r::nat\<Rightarrow>nat. strict_mono r \<and> (f \<circ> r) \<longlonglongrightarrow> l"
+ by (auto simp: convergent_def comp_def)
+qed
+
+lemma sequence_infinite_lemma:
+ fixes f :: "nat \<Rightarrow> 'a::t1_space"
+ assumes "\<forall>n. f n \<noteq> l"
+ and "(f \<longlongrightarrow> l) sequentially"
+ shows "infinite (range f)"
+proof
+ assume "finite (range f)"
+ then have "closed (range f)"
+ by (rule finite_imp_closed)
+ then have "open (- range f)"
+ by (rule open_Compl)
+ from assms(1) have "l \<in> - range f"
+ by auto
+ from assms(2) have "eventually (\<lambda>n. f n \<in> - range f) sequentially"
+ using \<open>open (- range f)\<close> \<open>l \<in> - range f\<close>
+ by (rule topological_tendstoD)
+ then show False
+ unfolding eventually_sequentially
+ by auto
+qed
+
+lemma closure_insert:
+ fixes x :: "'a::t1_space"
+ shows "closure (insert x s) = insert x (closure s)"
+ apply (rule closure_unique)
+ apply (rule insert_mono [OF closure_subset])
+ apply (rule closed_insert [OF closed_closure])
+ apply (simp add: closure_minimal)
+ done
+
+lemma islimpt_insert:
+ fixes x :: "'a::t1_space"
+ shows "x islimpt (insert a s) \<longleftrightarrow> x islimpt s"
+proof
+ assume *: "x islimpt (insert a s)"
+ show "x islimpt s"
+ proof (rule islimptI)
+ fix t
+ assume t: "x \<in> t" "open t"
+ show "\<exists>y\<in>s. y \<in> t \<and> y \<noteq> x"
+ proof (cases "x = a")
+ case True
+ obtain y where "y \<in> insert a s" "y \<in> t" "y \<noteq> x"
+ using * t by (rule islimptE)
+ with \<open>x = a\<close> show ?thesis by auto
+ next
+ case False
+ with t have t': "x \<in> t - {a}" "open (t - {a})"
+ by (simp_all add: open_Diff)
+ obtain y where "y \<in> insert a s" "y \<in> t - {a}" "y \<noteq> x"
+ using * t' by (rule islimptE)
+ then show ?thesis by auto
+ qed
+ qed
+next
+ assume "x islimpt s"
+ then show "x islimpt (insert a s)"
+ by (rule islimpt_subset) auto
+qed
+
+lemma islimpt_finite:
+ fixes x :: "'a::t1_space"
+ shows "finite s \<Longrightarrow> \<not> x islimpt s"
+ by (induct set: finite) (simp_all add: islimpt_insert)
+
+lemma islimpt_Un_finite:
+ fixes x :: "'a::t1_space"
+ shows "finite s \<Longrightarrow> x islimpt (s \<union> t) \<longleftrightarrow> x islimpt t"
+ by (simp add: islimpt_Un islimpt_finite)
+
+lemma islimpt_eq_acc_point:
+ fixes l :: "'a :: t1_space"
+ shows "l islimpt S \<longleftrightarrow> (\<forall>U. l\<in>U \<longrightarrow> open U \<longrightarrow> infinite (U \<inter> S))"
+proof (safe intro!: islimptI)
+ fix U
+ assume "l islimpt S" "l \<in> U" "open U" "finite (U \<inter> S)"
+ then have "l islimpt S" "l \<in> (U - (U \<inter> S - {l}))" "open (U - (U \<inter> S - {l}))"
+ by (auto intro: finite_imp_closed)
+ then show False
+ by (rule islimptE) auto
+next
+ fix T
+ assume *: "\<forall>U. l\<in>U \<longrightarrow> open U \<longrightarrow> infinite (U \<inter> S)" "l \<in> T" "open T"
+ then have "infinite (T \<inter> S - {l})"
+ by auto
+ then have "\<exists>x. x \<in> (T \<inter> S - {l})"
+ unfolding ex_in_conv by (intro notI) simp
+ then show "\<exists>y\<in>S. y \<in> T \<and> y \<noteq> l"
+ by auto
+qed
+
+corollary infinite_openin:
+ fixes S :: "'a :: t1_space set"
+ shows "\<lbrakk>openin (subtopology euclidean U) S; x \<in> S; x islimpt U\<rbrakk> \<Longrightarrow> infinite S"
+ by (clarsimp simp add: openin_open islimpt_eq_acc_point inf_commute)
+
+lemma islimpt_range_imp_convergent_subsequence:
+ fixes l :: "'a :: {t1_space, first_countable_topology}"
+ assumes l: "l islimpt (range f)"
+ shows "\<exists>r::nat\<Rightarrow>nat. strict_mono r \<and> (f \<circ> r) \<longlonglongrightarrow> l"
+ using l unfolding islimpt_eq_acc_point
+ by (rule acc_point_range_imp_convergent_subsequence)
+
+lemma islimpt_eq_infinite_ball: "x islimpt S \<longleftrightarrow> (\<forall>e>0. infinite(S \<inter> ball x e))"
+ apply (simp add: islimpt_eq_acc_point, safe)
+ apply (metis Int_commute open_ball centre_in_ball)
+ by (metis open_contains_ball Int_mono finite_subset inf_commute subset_refl)
+
+lemma islimpt_eq_infinite_cball: "x islimpt S \<longleftrightarrow> (\<forall>e>0. infinite(S \<inter> cball x e))"
+ apply (simp add: islimpt_eq_infinite_ball, safe)
+ apply (meson Int_mono ball_subset_cball finite_subset order_refl)
+ by (metis open_ball centre_in_ball finite_Int inf.absorb_iff2 inf_assoc open_contains_cball_eq)
+
+lemma sequence_unique_limpt:
+ fixes f :: "nat \<Rightarrow> 'a::t2_space"
+ assumes "(f \<longlongrightarrow> l) sequentially"
+ and "l' islimpt (range f)"
+ shows "l' = l"
+proof (rule ccontr)
+ assume "l' \<noteq> l"
+ obtain s t where "open s" "open t" "l' \<in> s" "l \<in> t" "s \<inter> t = {}"
+ using hausdorff [OF \<open>l' \<noteq> l\<close>] by auto
+ have "eventually (\<lambda>n. f n \<in> t) sequentially"
+ using assms(1) \<open>open t\<close> \<open>l \<in> t\<close> by (rule topological_tendstoD)
+ then obtain N where "\<forall>n\<ge>N. f n \<in> t"
+ unfolding eventually_sequentially by auto
+
+ have "UNIV = {..<N} \<union> {N..}"
+ by auto
+ then have "l' islimpt (f ` ({..<N} \<union> {N..}))"
+ using assms(2) by simp
+ then have "l' islimpt (f ` {..<N} \<union> f ` {N..})"
+ by (simp add: image_Un)
+ then have "l' islimpt (f ` {N..})"
+ by (simp add: islimpt_Un_finite)
+ then obtain y where "y \<in> f ` {N..}" "y \<in> s" "y \<noteq> l'"
+ using \<open>l' \<in> s\<close> \<open>open s\<close> by (rule islimptE)
+ then obtain n where "N \<le> n" "f n \<in> s" "f n \<noteq> l'"
+ by auto
+ with \<open>\<forall>n\<ge>N. f n \<in> t\<close> have "f n \<in> s \<inter> t"
+ by simp
+ with \<open>s \<inter> t = {}\<close> show False
+ by simp
+qed
+
+lemma bolzano_weierstrass_imp_closed:
+ fixes s :: "'a::{first_countable_topology,t2_space} set"
+ assumes "\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t)"
+ shows "closed s"
+proof -
+ {
+ fix x l
+ assume as: "\<forall>n::nat. x n \<in> s" "(x \<longlongrightarrow> l) sequentially"
+ then have "l \<in> s"
+ proof (cases "\<forall>n. x n \<noteq> l")
+ case False
+ then show "l\<in>s" using as(1) by auto
+ next
+ case True note cas = this
+ with as(2) have "infinite (range x)"
+ using sequence_infinite_lemma[of x l] by auto
+ then obtain l' where "l'\<in>s" "l' islimpt (range x)"
+ using assms[THEN spec[where x="range x"]] as(1) by auto
+ then show "l\<in>s" using sequence_unique_limpt[of x l l']
+ using as cas by auto
+ qed
+ }
+ then show ?thesis
+ unfolding closed_sequential_limits by fast
+qed
+
+lemma compact_imp_bounded:
+ assumes "compact U"
+ shows "bounded U"
+proof -
+ have "compact U" "\<forall>x\<in>U. open (ball x 1)" "U \<subseteq> (\<Union>x\<in>U. ball x 1)"
+ using assms by auto
+ then obtain D where D: "D \<subseteq> U" "finite D" "U \<subseteq> (\<Union>x\<in>D. ball x 1)"
+ by (metis compactE_image)
+ from \<open>finite D\<close> have "bounded (\<Union>x\<in>D. ball x 1)"
+ by (simp add: bounded_UN)
+ then show "bounded U" using \<open>U \<subseteq> (\<Union>x\<in>D. ball x 1)\<close>
+ by (rule bounded_subset)
+qed
+
+text\<open>In particular, some common special cases.\<close>
+
+lemma compact_Un [intro]:
+ assumes "compact s"
+ and "compact t"
+ shows " compact (s \<union> t)"
+proof (rule compactI)
+ fix f
+ assume *: "Ball f open" "s \<union> t \<subseteq> \<Union>f"
+ from * \<open>compact s\<close> obtain s' where "s' \<subseteq> f \<and> finite s' \<and> s \<subseteq> \<Union>s'"
+ unfolding compact_eq_heine_borel by (auto elim!: allE[of _ f])
+ moreover
+ from * \<open>compact t\<close> obtain t' where "t' \<subseteq> f \<and> finite t' \<and> t \<subseteq> \<Union>t'"
+ unfolding compact_eq_heine_borel by (auto elim!: allE[of _ f])
+ ultimately show "\<exists>f'\<subseteq>f. finite f' \<and> s \<union> t \<subseteq> \<Union>f'"
+ by (auto intro!: exI[of _ "s' \<union> t'"])
+qed
+
+lemma compact_Union [intro]: "finite S \<Longrightarrow> (\<And>T. T \<in> S \<Longrightarrow> compact T) \<Longrightarrow> compact (\<Union>S)"
+ by (induct set: finite) auto
+
+lemma compact_UN [intro]:
+ "finite A \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> compact (B x)) \<Longrightarrow> compact (\<Union>x\<in>A. B x)"
+ by (rule compact_Union) auto
+
+lemma closed_Int_compact [intro]:
+ assumes "closed s"
+ and "compact t"
+ shows "compact (s \<inter> t)"
+ using compact_Int_closed [of t s] assms
+ by (simp add: Int_commute)
+
+lemma compact_Int [intro]:
+ fixes s t :: "'a :: t2_space set"
+ assumes "compact s"
+ and "compact t"
+ shows "compact (s \<inter> t)"
+ using assms by (intro compact_Int_closed compact_imp_closed)
+
+lemma compact_sing [simp]: "compact {a}"
+ unfolding compact_eq_heine_borel by auto
+
+lemma compact_insert [simp]:
+ assumes "compact s"
+ shows "compact (insert x s)"
+proof -
+ have "compact ({x} \<union> s)"
+ using compact_sing assms by (rule compact_Un)
+ then show ?thesis by simp
+qed
+
+lemma finite_imp_compact: "finite s \<Longrightarrow> compact s"
+ by (induct set: finite) simp_all
+
+lemma open_delete:
+ fixes s :: "'a::t1_space set"
+ shows "open s \<Longrightarrow> open (s - {x})"
+ by (simp add: open_Diff)
+
+lemma openin_delete:
+ fixes a :: "'a :: t1_space"
+ shows "openin (subtopology euclidean u) s
+ \<Longrightarrow> openin (subtopology euclidean u) (s - {a})"
+by (metis Int_Diff open_delete openin_open)
+
+text\<open>Compactness expressed with filters\<close>
+
+lemma closure_iff_nhds_not_empty:
+ "x \<in> closure X \<longleftrightarrow> (\<forall>A. \<forall>S\<subseteq>A. open S \<longrightarrow> x \<in> S \<longrightarrow> X \<inter> A \<noteq> {})"
+proof safe
+ assume x: "x \<in> closure X"
+ fix S A
+ assume "open S" "x \<in> S" "X \<inter> A = {}" "S \<subseteq> A"
+ then have "x \<notin> closure (-S)"
+ by (auto simp: closure_complement subset_eq[symmetric] intro: interiorI)
+ with x have "x \<in> closure X - closure (-S)"
+ by auto
+ also have "\<dots> \<subseteq> closure (X \<inter> S)"
+ using \<open>open S\<close> open_Int_closure_subset[of S X] by (simp add: closed_Compl ac_simps)
+ finally have "X \<inter> S \<noteq> {}" by auto
+ then show False using \<open>X \<inter> A = {}\<close> \<open>S \<subseteq> A\<close> by auto
+next
+ assume "\<forall>A S. S \<subseteq> A \<longrightarrow> open S \<longrightarrow> x \<in> S \<longrightarrow> X \<inter> A \<noteq> {}"
+ from this[THEN spec, of "- X", THEN spec, of "- closure X"]
+ show "x \<in> closure X"
+ by (simp add: closure_subset open_Compl)
+qed
+
+corollary closure_Int_ball_not_empty:
+ assumes "S \<subseteq> closure T" "x \<in> S" "r > 0"
+ shows "T \<inter> ball x r \<noteq> {}"
+ using assms centre_in_ball closure_iff_nhds_not_empty by blast
+
+lemma compact_filter:
+ "compact U \<longleftrightarrow> (\<forall>F. F \<noteq> bot \<longrightarrow> eventually (\<lambda>x. x \<in> U) F \<longrightarrow> (\<exists>x\<in>U. inf (nhds x) F \<noteq> bot))"
+proof (intro allI iffI impI compact_fip[THEN iffD2] notI)
+ fix F
+ assume "compact U"
+ assume F: "F \<noteq> bot" "eventually (\<lambda>x. x \<in> U) F"
+ then have "U \<noteq> {}"
+ by (auto simp: eventually_False)
+
+ define Z where "Z = closure ` {A. eventually (\<lambda>x. x \<in> A) F}"
+ then have "\<forall>z\<in>Z. closed z"
+ by auto
+ moreover
+ have ev_Z: "\<And>z. z \<in> Z \<Longrightarrow> eventually (\<lambda>x. x \<in> z) F"
+ unfolding Z_def by (auto elim: eventually_mono intro: set_mp[OF closure_subset])
+ have "(\<forall>B \<subseteq> Z. finite B \<longrightarrow> U \<inter> \<Inter>B \<noteq> {})"
+ proof (intro allI impI)
+ fix B assume "finite B" "B \<subseteq> Z"
+ with \<open>finite B\<close> ev_Z F(2) have "eventually (\<lambda>x. x \<in> U \<inter> (\<Inter>B)) F"
+ by (auto simp: eventually_ball_finite_distrib eventually_conj_iff)
+ with F show "U \<inter> \<Inter>B \<noteq> {}"
+ by (intro notI) (simp add: eventually_False)
+ qed
+ ultimately have "U \<inter> \<Inter>Z \<noteq> {}"
+ using \<open>compact U\<close> unfolding compact_fip by blast
+ then obtain x where "x \<in> U" and x: "\<And>z. z \<in> Z \<Longrightarrow> x \<in> z"
+ by auto
+
+ have "\<And>P. eventually P (inf (nhds x) F) \<Longrightarrow> P \<noteq> bot"
+ unfolding eventually_inf eventually_nhds
+ proof safe
+ fix P Q R S
+ assume "eventually R F" "open S" "x \<in> S"
+ with open_Int_closure_eq_empty[of S "{x. R x}"] x[of "closure {x. R x}"]
+ have "S \<inter> {x. R x} \<noteq> {}" by (auto simp: Z_def)
+ moreover assume "Ball S Q" "\<forall>x. Q x \<and> R x \<longrightarrow> bot x"
+ ultimately show False by (auto simp: set_eq_iff)
+ qed
+ with \<open>x \<in> U\<close> show "\<exists>x\<in>U. inf (nhds x) F \<noteq> bot"
+ by (metis eventually_bot)
+next
+ fix A
+ assume A: "\<forall>a\<in>A. closed a" "\<forall>B\<subseteq>A. finite B \<longrightarrow> U \<inter> \<Inter>B \<noteq> {}" "U \<inter> \<Inter>A = {}"
+ define F where "F = (INF a\<in>insert U A. principal a)"
+ have "F \<noteq> bot"
+ unfolding F_def
+ proof (rule INF_filter_not_bot)
+ fix X
+ assume X: "X \<subseteq> insert U A" "finite X"
+ with A(2)[THEN spec, of "X - {U}"] have "U \<inter> \<Inter>(X - {U}) \<noteq> {}"
+ by auto
+ with X show "(INF a\<in>X. principal a) \<noteq> bot"
+ by (auto simp: INF_principal_finite principal_eq_bot_iff)
+ qed
+ moreover
+ have "F \<le> principal U"
+ unfolding F_def by auto
+ then have "eventually (\<lambda>x. x \<in> U) F"
+ by (auto simp: le_filter_def eventually_principal)
+ moreover
+ assume "\<forall>F. F \<noteq> bot \<longrightarrow> eventually (\<lambda>x. x \<in> U) F \<longrightarrow> (\<exists>x\<in>U. inf (nhds x) F \<noteq> bot)"
+ ultimately obtain x where "x \<in> U" and x: "inf (nhds x) F \<noteq> bot"
+ by auto
+
+ { fix V assume "V \<in> A"
+ then have "F \<le> principal V"
+ unfolding F_def by (intro INF_lower2[of V]) auto
+ then have V: "eventually (\<lambda>x. x \<in> V) F"
+ by (auto simp: le_filter_def eventually_principal)
+ have "x \<in> closure V"
+ unfolding closure_iff_nhds_not_empty
+ proof (intro impI allI)
+ fix S A
+ assume "open S" "x \<in> S" "S \<subseteq> A"
+ then have "eventually (\<lambda>x. x \<in> A) (nhds x)"
+ by (auto simp: eventually_nhds)
+ with V have "eventually (\<lambda>x. x \<in> V \<inter> A) (inf (nhds x) F)"
+ by (auto simp: eventually_inf)
+ with x show "V \<inter> A \<noteq> {}"
+ by (auto simp del: Int_iff simp add: trivial_limit_def)
+ qed
+ then have "x \<in> V"
+ using \<open>V \<in> A\<close> A(1) by simp
+ }
+ with \<open>x\<in>U\<close> have "x \<in> U \<inter> \<Inter>A" by auto
+ with \<open>U \<inter> \<Inter>A = {}\<close> show False by auto
+qed
+
+definition%important "countably_compact U \<longleftrightarrow>
+ (\<forall>A. countable A \<longrightarrow> (\<forall>a\<in>A. open a) \<longrightarrow> U \<subseteq> \<Union>A \<longrightarrow> (\<exists>T\<subseteq>A. finite T \<and> U \<subseteq> \<Union>T))"
+
+lemma countably_compactE:
+ assumes "countably_compact s" and "\<forall>t\<in>C. open t" and "s \<subseteq> \<Union>C" "countable C"
+ obtains C' where "C' \<subseteq> C" and "finite C'" and "s \<subseteq> \<Union>C'"
+ using assms unfolding countably_compact_def by metis
+
+lemma countably_compactI:
+ assumes "\<And>C. \<forall>t\<in>C. open t \<Longrightarrow> s \<subseteq> \<Union>C \<Longrightarrow> countable C \<Longrightarrow> (\<exists>C'\<subseteq>C. finite C' \<and> s \<subseteq> \<Union>C')"
+ shows "countably_compact s"
+ using assms unfolding countably_compact_def by metis
+
+lemma compact_imp_countably_compact: "compact U \<Longrightarrow> countably_compact U"
+ by (auto simp: compact_eq_heine_borel countably_compact_def)
+
+lemma countably_compact_imp_compact:
+ assumes "countably_compact U"
+ and ccover: "countable B" "\<forall>b\<in>B. open b"
+ and basis: "\<And>T x. open T \<Longrightarrow> x \<in> T \<Longrightarrow> x \<in> U \<Longrightarrow> \<exists>b\<in>B. x \<in> b \<and> b \<inter> U \<subseteq> T"
+ shows "compact U"
+ using \<open>countably_compact U\<close>
+ unfolding compact_eq_heine_borel countably_compact_def
+proof safe
+ fix A
+ assume A: "\<forall>a\<in>A. open a" "U \<subseteq> \<Union>A"
+ assume *: "\<forall>A. countable A \<longrightarrow> (\<forall>a\<in>A. open a) \<longrightarrow> U \<subseteq> \<Union>A \<longrightarrow> (\<exists>T\<subseteq>A. finite T \<and> U \<subseteq> \<Union>T)"
+ moreover define C where "C = {b\<in>B. \<exists>a\<in>A. b \<inter> U \<subseteq> a}"
+ ultimately have "countable C" "\<forall>a\<in>C. open a"
+ unfolding C_def using ccover by auto
+ moreover
+ have "\<Union>A \<inter> U \<subseteq> \<Union>C"
+ proof safe
+ fix x a
+ assume "x \<in> U" "x \<in> a" "a \<in> A"
+ with basis[of a x] A obtain b where "b \<in> B" "x \<in> b" "b \<inter> U \<subseteq> a"
+ by blast
+ with \<open>a \<in> A\<close> show "x \<in> \<Union>C"
+ unfolding C_def by auto
+ qed
+ then have "U \<subseteq> \<Union>C" using \<open>U \<subseteq> \<Union>A\<close> by auto
+ ultimately obtain T where T: "T\<subseteq>C" "finite T" "U \<subseteq> \<Union>T"
+ using * by metis
+ then have "\<forall>t\<in>T. \<exists>a\<in>A. t \<inter> U \<subseteq> a"
+ by (auto simp: C_def)
+ then obtain f where "\<forall>t\<in>T. f t \<in> A \<and> t \<inter> U \<subseteq> f t"
+ unfolding bchoice_iff Bex_def ..
+ with T show "\<exists>T\<subseteq>A. finite T \<and> U \<subseteq> \<Union>T"
+ unfolding C_def by (intro exI[of _ "f`T"]) fastforce
+qed
+
+proposition countably_compact_imp_compact_second_countable:
+ "countably_compact U \<Longrightarrow> compact (U :: 'a :: second_countable_topology set)"
+proof (rule countably_compact_imp_compact)
+ fix T and x :: 'a
+ assume "open T" "x \<in> T"
+ from topological_basisE[OF is_basis this] obtain b where
+ "b \<in> (SOME B. countable B \<and> topological_basis B)" "x \<in> b" "b \<subseteq> T" .
+ then show "\<exists>b\<in>SOME B. countable B \<and> topological_basis B. x \<in> b \<and> b \<inter> U \<subseteq> T"
+ by blast
+qed (insert countable_basis topological_basis_open[OF is_basis], auto)
+
+lemma countably_compact_eq_compact:
+ "countably_compact U \<longleftrightarrow> compact (U :: 'a :: second_countable_topology set)"
+ using countably_compact_imp_compact_second_countable compact_imp_countably_compact by blast
+
+subsubsection\<open>Sequential compactness\<close>
+
+definition%important seq_compact :: "'a::topological_space set \<Rightarrow> bool"
+ where "seq_compact S \<longleftrightarrow>
+ (\<forall>f. (\<forall>n. f n \<in> S) \<longrightarrow> (\<exists>l\<in>S. \<exists>r::nat\<Rightarrow>nat. strict_mono r \<and> ((f \<circ> r) \<longlongrightarrow> l) sequentially))"
+
+lemma seq_compactI:
+ assumes "\<And>f. \<forall>n. f n \<in> S \<Longrightarrow> \<exists>l\<in>S. \<exists>r::nat\<Rightarrow>nat. strict_mono r \<and> ((f \<circ> r) \<longlongrightarrow> l) sequentially"
+ shows "seq_compact S"
+ unfolding seq_compact_def using assms by fast
+
+lemma seq_compactE:
+ assumes "seq_compact S" "\<forall>n. f n \<in> S"
+ obtains l r where "l \<in> S" "strict_mono (r :: nat \<Rightarrow> nat)" "((f \<circ> r) \<longlongrightarrow> l) sequentially"
+ using assms unfolding seq_compact_def by fast
+
+lemma closed_sequentially: (* TODO: move upwards *)
+ assumes "closed s" and "\<forall>n. f n \<in> s" and "f \<longlonglongrightarrow> l"
+ shows "l \<in> s"
+proof (rule ccontr)
+ assume "l \<notin> s"
+ with \<open>closed s\<close> and \<open>f \<longlonglongrightarrow> l\<close> have "eventually (\<lambda>n. f n \<in> - s) sequentially"
+ by (fast intro: topological_tendstoD)
+ with \<open>\<forall>n. f n \<in> s\<close> show "False"
+ by simp
+qed
+
+lemma seq_compact_Int_closed:
+ assumes "seq_compact s" and "closed t"
+ shows "seq_compact (s \<inter> t)"
+proof (rule seq_compactI)
+ fix f assume "\<forall>n::nat. f n \<in> s \<inter> t"
+ hence "\<forall>n. f n \<in> s" and "\<forall>n. f n \<in> t"
+ by simp_all
+ from \<open>seq_compact s\<close> and \<open>\<forall>n. f n \<in> s\<close>
+ obtain l r where "l \<in> s" and r: "strict_mono r" and l: "(f \<circ> r) \<longlonglongrightarrow> l"
+ by (rule seq_compactE)
+ from \<open>\<forall>n. f n \<in> t\<close> have "\<forall>n. (f \<circ> r) n \<in> t"
+ by simp
+ from \<open>closed t\<close> and this and l have "l \<in> t"
+ by (rule closed_sequentially)
+ with \<open>l \<in> s\<close> and r and l show "\<exists>l\<in>s \<inter> t. \<exists>r. strict_mono r \<and> (f \<circ> r) \<longlonglongrightarrow> l"
+ by fast
+qed
+
+lemma seq_compact_closed_subset:
+ assumes "closed s" and "s \<subseteq> t" and "seq_compact t"
+ shows "seq_compact s"
+ using assms seq_compact_Int_closed [of t s] by (simp add: Int_absorb1)
+
+lemma seq_compact_imp_countably_compact:
+ fixes U :: "'a :: first_countable_topology set"
+ assumes "seq_compact U"
+ shows "countably_compact U"
+proof (safe intro!: countably_compactI)
+ fix A
+ assume A: "\<forall>a\<in>A. open a" "U \<subseteq> \<Union>A" "countable A"
+ have subseq: "\<And>X. range X \<subseteq> U \<Longrightarrow> \<exists>r x. x \<in> U \<and> strict_mono (r :: nat \<Rightarrow> nat) \<and> (X \<circ> r) \<longlonglongrightarrow> x"
+ using \<open>seq_compact U\<close> by (fastforce simp: seq_compact_def subset_eq)
+ show "\<exists>T\<subseteq>A. finite T \<and> U \<subseteq> \<Union>T"
+ proof cases
+ assume "finite A"
+ with A show ?thesis by auto
+ next
+ assume "infinite A"
+ then have "A \<noteq> {}" by auto
+ show ?thesis
+ proof (rule ccontr)
+ assume "\<not> (\<exists>T\<subseteq>A. finite T \<and> U \<subseteq> \<Union>T)"
+ then have "\<forall>T. \<exists>x. T \<subseteq> A \<and> finite T \<longrightarrow> (x \<in> U - \<Union>T)"
+ by auto
+ then obtain X' where T: "\<And>T. T \<subseteq> A \<Longrightarrow> finite T \<Longrightarrow> X' T \<in> U - \<Union>T"
+ by metis
+ define X where "X n = X' (from_nat_into A ` {.. n})" for n
+ have X: "\<And>n. X n \<in> U - (\<Union>i\<le>n. from_nat_into A i)"
+ using \<open>A \<noteq> {}\<close> unfolding X_def by (intro T) (auto intro: from_nat_into)
+ then have "range X \<subseteq> U"
+ by auto
+ with subseq[of X] obtain r x where "x \<in> U" and r: "strict_mono r" "(X \<circ> r) \<longlonglongrightarrow> x"
+ by auto
+ from \<open>x\<in>U\<close> \<open>U \<subseteq> \<Union>A\<close> from_nat_into_surj[OF \<open>countable A\<close>]
+ obtain n where "x \<in> from_nat_into A n" by auto
+ with r(2) A(1) from_nat_into[OF \<open>A \<noteq> {}\<close>, of n]
+ have "eventually (\<lambda>i. X (r i) \<in> from_nat_into A n) sequentially"
+ unfolding tendsto_def by (auto simp: comp_def)
+ then obtain N where "\<And>i. N \<le> i \<Longrightarrow> X (r i) \<in> from_nat_into A n"
+ by (auto simp: eventually_sequentially)
+ moreover from X have "\<And>i. n \<le> r i \<Longrightarrow> X (r i) \<notin> from_nat_into A n"
+ by auto
+ moreover from \<open>strict_mono r\<close>[THEN seq_suble, of "max n N"] have "\<exists>i. n \<le> r i \<and> N \<le> i"
+ by (auto intro!: exI[of _ "max n N"])
+ ultimately show False
+ by auto
+ qed
+ qed
+qed
+
+lemma compact_imp_seq_compact:
+ fixes U :: "'a :: first_countable_topology set"
+ assumes "compact U"
+ shows "seq_compact U"
+ unfolding seq_compact_def
+proof safe
+ fix X :: "nat \<Rightarrow> 'a"
+ assume "\<forall>n. X n \<in> U"
+ then have "eventually (\<lambda>x. x \<in> U) (filtermap X sequentially)"
+ by (auto simp: eventually_filtermap)
+ moreover
+ have "filtermap X sequentially \<noteq> bot"
+ by (simp add: trivial_limit_def eventually_filtermap)
+ ultimately
+ obtain x where "x \<in> U" and x: "inf (nhds x) (filtermap X sequentially) \<noteq> bot" (is "?F \<noteq> _")
+ using \<open>compact U\<close> by (auto simp: compact_filter)
+
+ from countable_basis_at_decseq[of x]
+ obtain A where A:
+ "\<And>i. open (A i)"
+ "\<And>i. x \<in> A i"
+ "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> eventually (\<lambda>i. A i \<subseteq> S) sequentially"
+ by blast
+ define s where "s n i = (SOME j. i < j \<and> X j \<in> A (Suc n))" for n i
+ {
+ fix n i
+ have "\<exists>a. i < a \<and> X a \<in> A (Suc n)"
+ proof (rule ccontr)
+ assume "\<not> (\<exists>a>i. X a \<in> A (Suc n))"
+ then have "\<And>a. Suc i \<le> a \<Longrightarrow> X a \<notin> A (Suc n)"
+ by auto
+ then have "eventually (\<lambda>x. x \<notin> A (Suc n)) (filtermap X sequentially)"
+ by (auto simp: eventually_filtermap eventually_sequentially)
+ moreover have "eventually (\<lambda>x. x \<in> A (Suc n)) (nhds x)"
+ using A(1,2)[of "Suc n"] by (auto simp: eventually_nhds)
+ ultimately have "eventually (\<lambda>x. False) ?F"
+ by (auto simp: eventually_inf)
+ with x show False
+ by (simp add: eventually_False)
+ qed
+ then have "i < s n i" "X (s n i) \<in> A (Suc n)"
+ unfolding s_def by (auto intro: someI2_ex)
+ }
+ note s = this
+ define r where "r = rec_nat (s 0 0) s"
+ have "strict_mono r"
+ by (auto simp: r_def s strict_mono_Suc_iff)
+ moreover
+ have "(\<lambda>n. X (r n)) \<longlonglongrightarrow> x"
+ proof (rule topological_tendstoI)
+ fix S
+ assume "open S" "x \<in> S"
+ with A(3) have "eventually (\<lambda>i. A i \<subseteq> S) sequentially"
+ by auto
+ moreover
+ {
+ fix i
+ assume "Suc 0 \<le> i"
+ then have "X (r i) \<in> A i"
+ by (cases i) (simp_all add: r_def s)
+ }
+ then have "eventually (\<lambda>i. X (r i) \<in> A i) sequentially"
+ by (auto simp: eventually_sequentially)
+ ultimately show "eventually (\<lambda>i. X (r i) \<in> S) sequentially"
+ by eventually_elim auto
+ qed
+ ultimately show "\<exists>x \<in> U. \<exists>r. strict_mono r \<and> (X \<circ> r) \<longlonglongrightarrow> x"
+ using \<open>x \<in> U\<close> by (auto simp: convergent_def comp_def)
+qed
+
+lemma countably_compact_imp_acc_point:
+ assumes "countably_compact s"
+ and "countable t"
+ and "infinite t"
+ and "t \<subseteq> s"
+ shows "\<exists>x\<in>s. \<forall>U. x\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> t)"
+proof (rule ccontr)
+ define C where "C = (\<lambda>F. interior (F \<union> (- t))) ` {F. finite F \<and> F \<subseteq> t }"
+ note \<open>countably_compact s\<close>
+ moreover have "\<forall>t\<in>C. open t"
+ by (auto simp: C_def)
+ moreover
+ assume "\<not> (\<exists>x\<in>s. \<forall>U. x\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> t))"
+ then have s: "\<And>x. x \<in> s \<Longrightarrow> \<exists>U. x\<in>U \<and> open U \<and> finite (U \<inter> t)" by metis
+ have "s \<subseteq> \<Union>C"
+ using \<open>t \<subseteq> s\<close>
+ unfolding C_def
+ apply (safe dest!: s)
+ apply (rule_tac a="U \<inter> t" in UN_I)
+ apply (auto intro!: interiorI simp add: finite_subset)
+ done
+ moreover
+ from \<open>countable t\<close> have "countable C"
+ unfolding C_def by (auto intro: countable_Collect_finite_subset)
+ ultimately
+ obtain D where "D \<subseteq> C" "finite D" "s \<subseteq> \<Union>D"
+ by (rule countably_compactE)
+ then obtain E where E: "E \<subseteq> {F. finite F \<and> F \<subseteq> t }" "finite E"
+ and s: "s \<subseteq> (\<Union>F\<in>E. interior (F \<union> (- t)))"
+ by (metis (lifting) finite_subset_image C_def)
+ from s \<open>t \<subseteq> s\<close> have "t \<subseteq> \<Union>E"
+ using interior_subset by blast
+ moreover have "finite (\<Union>E)"
+ using E by auto
+ ultimately show False using \<open>infinite t\<close>
+ by (auto simp: finite_subset)
+qed
+
+lemma countable_acc_point_imp_seq_compact:
+ fixes s :: "'a::first_countable_topology set"
+ assumes "\<forall>t. infinite t \<and> countable t \<and> t \<subseteq> s \<longrightarrow>
+ (\<exists>x\<in>s. \<forall>U. x\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> t))"
+ shows "seq_compact s"
+proof -
+ {
+ fix f :: "nat \<Rightarrow> 'a"
+ assume f: "\<forall>n. f n \<in> s"
+ have "\<exists>l\<in>s. \<exists>r. strict_mono r \<and> ((f \<circ> r) \<longlongrightarrow> l) sequentially"
+ proof (cases "finite (range f)")
+ case True
+ obtain l where "infinite {n. f n = f l}"
+ using pigeonhole_infinite[OF _ True] by auto
+ then obtain r :: "nat \<Rightarrow> nat" where "strict_mono r" and fr: "\<forall>n. f (r n) = f l"
+ using infinite_enumerate by blast
+ then have "strict_mono r \<and> (f \<circ> r) \<longlonglongrightarrow> f l"
+ by (simp add: fr o_def)
+ with f show "\<exists>l\<in>s. \<exists>r. strict_mono r \<and> (f \<circ> r) \<longlonglongrightarrow> l"
+ by auto
+ next
+ case False
+ with f assms have "\<exists>x\<in>s. \<forall>U. x\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> range f)"
+ by auto
+ then obtain l where "l \<in> s" "\<forall>U. l\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> range f)" ..
+ from this(2) have "\<exists>r. strict_mono r \<and> ((f \<circ> r) \<longlongrightarrow> l) sequentially"
+ using acc_point_range_imp_convergent_subsequence[of l f] by auto
+ with \<open>l \<in> s\<close> show "\<exists>l\<in>s. \<exists>r. strict_mono r \<and> ((f \<circ> r) \<longlongrightarrow> l) sequentially" ..
+ qed
+ }
+ then show ?thesis
+ unfolding seq_compact_def by auto
+qed
+
+lemma seq_compact_eq_countably_compact:
+ fixes U :: "'a :: first_countable_topology set"
+ shows "seq_compact U \<longleftrightarrow> countably_compact U"
+ using
+ countable_acc_point_imp_seq_compact
+ countably_compact_imp_acc_point
+ seq_compact_imp_countably_compact
+ by metis
+
+lemma seq_compact_eq_acc_point:
+ fixes s :: "'a :: first_countable_topology set"
+ shows "seq_compact s \<longleftrightarrow>
+ (\<forall>t. infinite t \<and> countable t \<and> t \<subseteq> s --> (\<exists>x\<in>s. \<forall>U. x\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> t)))"
+ using
+ countable_acc_point_imp_seq_compact[of s]
+ countably_compact_imp_acc_point[of s]
+ seq_compact_imp_countably_compact[of s]
+ by metis
+
+lemma seq_compact_eq_compact:
+ fixes U :: "'a :: second_countable_topology set"
+ shows "seq_compact U \<longleftrightarrow> compact U"
+ using seq_compact_eq_countably_compact countably_compact_eq_compact by blast
+
+proposition bolzano_weierstrass_imp_seq_compact:
+ fixes s :: "'a::{t1_space, first_countable_topology} set"
+ shows "\<forall>t. infinite t \<and> t \<subseteq> s \<longrightarrow> (\<exists>x \<in> s. x islimpt t) \<Longrightarrow> seq_compact s"
+ by (rule countable_acc_point_imp_seq_compact) (metis islimpt_eq_acc_point)
+
+
+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)"
+ unfolding Cauchy_def by metis
+
+proposition seq_compact_imp_totally_bounded:
+ assumes "seq_compact s"
+ shows "\<forall>e>0. \<exists>k. finite k \<and> k \<subseteq> s \<and> s \<subseteq> (\<Union>x\<in>k. ball x e)"
+proof -
+ { fix e::real assume "e > 0" assume *: "\<And>k. finite k \<Longrightarrow> k \<subseteq> s \<Longrightarrow> \<not> s \<subseteq> (\<Union>x\<in>k. ball x e)"
+ let ?Q = "\<lambda>x n r. r \<in> s \<and> (\<forall>m < (n::nat). \<not> (dist (x m) r < e))"
+ have "\<exists>x. \<forall>n::nat. ?Q x n (x n)"
+ proof (rule dependent_wellorder_choice)
+ fix n x assume "\<And>y. y < n \<Longrightarrow> ?Q x y (x y)"
+ then have "\<not> s \<subseteq> (\<Union>x\<in>x ` {0..<n}. ball x e)"
+ using *[of "x ` {0 ..< n}"] by (auto simp: subset_eq)
+ then obtain z where z:"z\<in>s" "z \<notin> (\<Union>x\<in>x ` {0..<n}. ball x e)"
+ unfolding subset_eq by auto
+ show "\<exists>r. ?Q x n r"
+ using z by auto
+ qed simp
+ then obtain x where "\<forall>n::nat. x n \<in> s" and x:"\<And>n m. m < n \<Longrightarrow> \<not> (dist (x m) (x n) < e)"
+ by blast
+ then obtain l r where "l \<in> s" and r:"strict_mono r" and "((x \<circ> r) \<longlongrightarrow> l) sequentially"
+ using assms by (metis seq_compact_def)
+ from this(3) have "Cauchy (x \<circ> r)"
+ using LIMSEQ_imp_Cauchy by auto
+ then obtain N::nat where "\<And>m n. N \<le> m \<Longrightarrow> N \<le> n \<Longrightarrow> dist ((x \<circ> r) m) ((x \<circ> r) n) < e"
+ unfolding cauchy_def using \<open>e > 0\<close> by blast
+ then have False
+ using x[of "r N" "r (N+1)"] r by (auto simp: strict_mono_def) }
+ then show ?thesis
+ by metis
+qed
+
+subsubsection\<open>Heine-Borel theorem\<close>
+
+proposition seq_compact_imp_heine_borel:
+ fixes s :: "'a :: metric_space set"
+ assumes "seq_compact s"
+ shows "compact s"
+proof -
+ from seq_compact_imp_totally_bounded[OF \<open>seq_compact s\<close>]
+ obtain f where f: "\<forall>e>0. finite (f e) \<and> f e \<subseteq> s \<and> s \<subseteq> (\<Union>x\<in>f e. ball x e)"
+ unfolding choice_iff' ..
+ define K where "K = (\<lambda>(x, r). ball x r) ` ((\<Union>e \<in> \<rat> \<inter> {0 <..}. f e) \<times> \<rat>)"
+ have "countably_compact s"
+ using \<open>seq_compact s\<close> by (rule seq_compact_imp_countably_compact)
+ then show "compact s"
+ proof (rule countably_compact_imp_compact)
+ show "countable K"
+ unfolding K_def using f
+ by (auto intro: countable_finite countable_subset countable_rat
+ intro!: countable_image countable_SIGMA countable_UN)
+ show "\<forall>b\<in>K. open b" by (auto simp: K_def)
+ next
+ fix T x
+ assume T: "open T" "x \<in> T" and x: "x \<in> s"
+ from openE[OF T] obtain e where "0 < e" "ball x e \<subseteq> T"
+ by auto
+ then have "0 < e / 2" "ball x (e / 2) \<subseteq> T"
+ by auto
+ from Rats_dense_in_real[OF \<open>0 < e / 2\<close>] obtain r where "r \<in> \<rat>" "0 < r" "r < e / 2"
+ by auto
+ from f[rule_format, of r] \<open>0 < r\<close> \<open>x \<in> s\<close> obtain k where "k \<in> f r" "x \<in> ball k r"
+ by auto
+ from \<open>r \<in> \<rat>\<close> \<open>0 < r\<close> \<open>k \<in> f r\<close> have "ball k r \<in> K"
+ by (auto simp: K_def)
+ then show "\<exists>b\<in>K. x \<in> b \<and> b \<inter> s \<subseteq> T"
+ proof (rule bexI[rotated], safe)
+ fix y
+ assume "y \<in> ball k r"
+ with \<open>r < e / 2\<close> \<open>x \<in> ball k r\<close> have "dist x y < e"
+ by (intro dist_triangle_half_r [of k _ e]) (auto simp: dist_commute)
+ with \<open>ball x e \<subseteq> T\<close> show "y \<in> T"
+ by auto
+ next
+ show "x \<in> ball k r" by fact
+ qed
+ qed
+qed
+
+proposition compact_eq_seq_compact_metric:
+ "compact (s :: 'a::metric_space set) \<longleftrightarrow> seq_compact s"
+ using compact_imp_seq_compact seq_compact_imp_heine_borel by blast
+
+proposition compact_def: \<comment> \<open>this is the definition of compactness in HOL Light\<close>
+ "compact (S :: 'a::metric_space set) \<longleftrightarrow>
+ (\<forall>f. (\<forall>n. f n \<in> S) \<longrightarrow> (\<exists>l\<in>S. \<exists>r::nat\<Rightarrow>nat. strict_mono r \<and> (f \<circ> r) \<longlonglongrightarrow> l))"
+ unfolding compact_eq_seq_compact_metric seq_compact_def by auto
+
+subsubsection \<open>Complete the chain of compactness variants\<close>
+
+proposition compact_eq_bolzano_weierstrass:
+ fixes s :: "'a::metric_space set"
+ shows "compact s \<longleftrightarrow> (\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t))"
+ (is "?lhs = ?rhs")
+proof
+ assume ?lhs
+ then show ?rhs
+ using heine_borel_imp_bolzano_weierstrass[of s] by auto
+next
+ assume ?rhs
+ then show ?lhs
+ unfolding compact_eq_seq_compact_metric by (rule bolzano_weierstrass_imp_seq_compact)
+qed
+
+proposition bolzano_weierstrass_imp_bounded:
+ "\<forall>t. infinite t \<and> t \<subseteq> s \<longrightarrow> (\<exists>x \<in> s. x islimpt t) \<Longrightarrow> bounded s"
+ using compact_imp_bounded unfolding compact_eq_bolzano_weierstrass .
+
+
+subsection \<open>Metric spaces with the Heine-Borel property\<close>
+
+text \<open>
+ A metric space (or topological vector space) is said to have the
+ Heine-Borel property if every closed and bounded subset is compact.
+\<close>
+
+class heine_borel = metric_space +
+ assumes bounded_imp_convergent_subsequence:
+ "bounded (range f) \<Longrightarrow> \<exists>l r. strict_mono (r::nat\<Rightarrow>nat) \<and> ((f \<circ> r) \<longlongrightarrow> l) sequentially"
+
+proposition bounded_closed_imp_seq_compact:
+ fixes s::"'a::heine_borel set"
+ assumes "bounded s"
+ and "closed s"
+ shows "seq_compact s"
+proof (unfold seq_compact_def, clarify)
+ fix f :: "nat \<Rightarrow> 'a"
+ assume f: "\<forall>n. f n \<in> s"
+ with \<open>bounded s\<close> have "bounded (range f)"
+ by (auto intro: bounded_subset)
+ obtain l r where r: "strict_mono (r :: nat \<Rightarrow> nat)" and l: "((f \<circ> r) \<longlongrightarrow> l) sequentially"
+ using bounded_imp_convergent_subsequence [OF \<open>bounded (range f)\<close>] by auto
+ from f have fr: "\<forall>n. (f \<circ> r) n \<in> s"
+ by simp
+ have "l \<in> s" using \<open>closed s\<close> fr l
+ by (rule closed_sequentially)
+ show "\<exists>l\<in>s. \<exists>r. strict_mono r \<and> ((f \<circ> r) \<longlongrightarrow> l) sequentially"
+ using \<open>l \<in> s\<close> r l by blast
+qed
+
+lemma compact_eq_bounded_closed:
+ fixes s :: "'a::heine_borel set"
+ shows "compact s \<longleftrightarrow> bounded s \<and> closed s"
+ (is "?lhs = ?rhs")
+proof
+ assume ?lhs
+ then show ?rhs
+ using compact_imp_closed compact_imp_bounded
+ by blast
+next
+ assume ?rhs
+ then show ?lhs
+ using bounded_closed_imp_seq_compact[of s]
+ unfolding compact_eq_seq_compact_metric
+ by auto
+qed
+
+lemma compact_Inter:
+ fixes \<F> :: "'a :: heine_borel set set"
+ assumes com: "\<And>S. S \<in> \<F> \<Longrightarrow> compact S" and "\<F> \<noteq> {}"
+ shows "compact(\<Inter> \<F>)"
+ using assms
+ by (meson Inf_lower all_not_in_conv bounded_subset closed_Inter compact_eq_bounded_closed)
+
+lemma compact_closure [simp]:
+ fixes S :: "'a::heine_borel set"
+ shows "compact(closure S) \<longleftrightarrow> bounded S"
+by (meson bounded_closure bounded_subset closed_closure closure_subset compact_eq_bounded_closed)
+
+lemma not_compact_UNIV[simp]:
+ fixes s :: "'a::{real_normed_vector,perfect_space,heine_borel} set"
+ shows "\<not> compact (UNIV::'a set)"
+ by (simp add: compact_eq_bounded_closed)
+
+text\<open>Representing sets as the union of a chain of compact sets.\<close>
+lemma closed_Union_compact_subsets:
+ fixes S :: "'a::{heine_borel,real_normed_vector} set"
+ assumes "closed S"
+ obtains F where "\<And>n. compact(F n)" "\<And>n. F n \<subseteq> S" "\<And>n. F n \<subseteq> F(Suc n)"
+ "(\<Union>n. F n) = S" "\<And>K. \<lbrakk>compact K; K \<subseteq> S\<rbrakk> \<Longrightarrow> \<exists>N. \<forall>n \<ge> N. K \<subseteq> F n"
+proof
+ show "compact (S \<inter> cball 0 (of_nat n))" for n
+ using assms compact_eq_bounded_closed by auto
+next
+ show "(\<Union>n. S \<inter> cball 0 (real n)) = S"
+ by (auto simp: real_arch_simple)
+next
+ fix K :: "'a set"
+ assume "compact K" "K \<subseteq> S"
+ then obtain N where "K \<subseteq> cball 0 N"
+ by (meson bounded_pos mem_cball_0 compact_imp_bounded subsetI)
+ then show "\<exists>N. \<forall>n\<ge>N. K \<subseteq> S \<inter> cball 0 (real n)"
+ by (metis of_nat_le_iff Int_subset_iff \<open>K \<subseteq> S\<close> real_arch_simple subset_cball subset_trans)
+qed auto
+
+instance%important real :: heine_borel
+proof%unimportant
+ fix f :: "nat \<Rightarrow> real"
+ assume f: "bounded (range f)"
+ obtain r :: "nat \<Rightarrow> nat" where r: "strict_mono r" "monoseq (f \<circ> r)"
+ unfolding comp_def by (metis seq_monosub)
+ then have "Bseq (f \<circ> r)"
+ unfolding Bseq_eq_bounded using f by (force intro: bounded_subset)
+ with r show "\<exists>l r. strict_mono r \<and> (f \<circ> r) \<longlonglongrightarrow> l"
+ using Bseq_monoseq_convergent[of "f \<circ> r"] by (auto simp: convergent_def)
+qed
+
+lemma compact_lemma_general:
+ fixes f :: "nat \<Rightarrow> 'a"
+ fixes proj::"'a \<Rightarrow> 'b \<Rightarrow> 'c::heine_borel" (infixl "proj" 60)
+ fixes unproj:: "('b \<Rightarrow> 'c) \<Rightarrow> 'a"
+ assumes finite_basis: "finite basis"
+ assumes bounded_proj: "\<And>k. k \<in> basis \<Longrightarrow> bounded ((\<lambda>x. x proj k) ` range f)"
+ assumes proj_unproj: "\<And>e k. k \<in> basis \<Longrightarrow> (unproj e) proj k = e k"
+ assumes unproj_proj: "\<And>x. unproj (\<lambda>k. x proj k) = x"
+ shows "\<forall>d\<subseteq>basis. \<exists>l::'a. \<exists> r::nat\<Rightarrow>nat.
+ strict_mono r \<and> (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) proj i) (l proj i) < e) sequentially)"
+proof safe
+ fix d :: "'b set"
+ assume d: "d \<subseteq> basis"
+ with finite_basis have "finite d"
+ by (blast intro: finite_subset)
+ from this d show "\<exists>l::'a. \<exists>r::nat\<Rightarrow>nat. strict_mono r \<and>
+ (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) proj i) (l proj i) < e) sequentially)"
+ proof (induct d)
+ case empty
+ then show ?case
+ unfolding strict_mono_def by auto
+ next
+ case (insert k d)
+ have k[intro]: "k \<in> basis"
+ using insert by auto
+ have s': "bounded ((\<lambda>x. x proj k) ` range f)"
+ using k
+ by (rule bounded_proj)
+ obtain l1::"'a" and r1 where r1: "strict_mono r1"
+ and lr1: "\<forall>e > 0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) proj i) (l1 proj i) < e) sequentially"
+ using insert(3) using insert(4) by auto
+ have f': "\<forall>n. f (r1 n) proj k \<in> (\<lambda>x. x proj k) ` range f"
+ by simp
+ have "bounded (range (\<lambda>i. f (r1 i) proj k))"
+ by (metis (lifting) bounded_subset f' image_subsetI s')
+ then obtain l2 r2 where r2:"strict_mono r2" and lr2:"((\<lambda>i. f (r1 (r2 i)) proj k) \<longlongrightarrow> l2) sequentially"
+ using bounded_imp_convergent_subsequence[of "\<lambda>i. f (r1 i) proj k"]
+ by (auto simp: o_def)
+ define r where "r = r1 \<circ> r2"
+ have r:"strict_mono r"
+ using r1 and r2 unfolding r_def o_def strict_mono_def by auto
+ moreover
+ define l where "l = unproj (\<lambda>i. if i = k then l2 else l1 proj i)"
+ {
+ fix e::real
+ assume "e > 0"
+ from lr1 \<open>e > 0\<close> have N1: "eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) proj i) (l1 proj i) < e) sequentially"
+ by blast
+ from lr2 \<open>e > 0\<close> have N2:"eventually (\<lambda>n. dist (f (r1 (r2 n)) proj k) l2 < e) sequentially"
+ by (rule tendstoD)
+ from r2 N1 have N1': "eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 (r2 n)) proj i) (l1 proj i) < e) sequentially"
+ by (rule eventually_subseq)
+ have "eventually (\<lambda>n. \<forall>i\<in>(insert k d). dist (f (r n) proj i) (l proj i) < e) sequentially"
+ using N1' N2
+ by eventually_elim (insert insert.prems, auto simp: l_def r_def o_def proj_unproj)
+ }
+ ultimately show ?case by auto
+ qed
+qed
+
+lemma bounded_fst: "bounded s \<Longrightarrow> bounded (fst ` s)"
+ unfolding bounded_def
+ by (metis (erased, hide_lams) dist_fst_le image_iff order_trans)
+
+lemma bounded_snd: "bounded s \<Longrightarrow> bounded (snd ` s)"
+ unfolding bounded_def
+ by (metis (no_types, hide_lams) dist_snd_le image_iff order.trans)
+
+instance%important prod :: (heine_borel, heine_borel) heine_borel
+proof%unimportant
+ fix f :: "nat \<Rightarrow> 'a \<times> 'b"
+ assume f: "bounded (range f)"
+ then have "bounded (fst ` range f)"
+ by (rule bounded_fst)
+ then have s1: "bounded (range (fst \<circ> f))"
+ by (simp add: image_comp)
+ obtain l1 r1 where r1: "strict_mono r1" and l1: "(\<lambda>n. fst (f (r1 n))) \<longlonglongrightarrow> l1"
+ using bounded_imp_convergent_subsequence [OF s1] unfolding o_def by fast
+ from f have s2: "bounded (range (snd \<circ> f \<circ> r1))"
+ by (auto simp: image_comp intro: bounded_snd bounded_subset)
+ obtain l2 r2 where r2: "strict_mono r2" and l2: "((\<lambda>n. snd (f (r1 (r2 n)))) \<longlongrightarrow> l2) sequentially"
+ using bounded_imp_convergent_subsequence [OF s2]
+ unfolding o_def by fast
+ have l1': "((\<lambda>n. fst (f (r1 (r2 n)))) \<longlongrightarrow> l1) sequentially"
+ using LIMSEQ_subseq_LIMSEQ [OF l1 r2] unfolding o_def .
+ have l: "((f \<circ> (r1 \<circ> r2)) \<longlongrightarrow> (l1, l2)) sequentially"
+ using tendsto_Pair [OF l1' l2] unfolding o_def by simp
+ have r: "strict_mono (r1 \<circ> r2)"
+ using r1 r2 unfolding strict_mono_def by simp
+ show "\<exists>l r. strict_mono r \<and> ((f \<circ> r) \<longlongrightarrow> l) sequentially"
+ using l r by fast
+qed
+
+subsubsection \<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"
+ shows "complete s"
+ using assms unfolding complete_def by fast
+
+proposition (in metric_space) completeE:
+ assumes "complete s" and "\<forall>n. f n \<in> s" and "Cauchy f"
+ obtains l where "l \<in> s" and "f \<longlonglongrightarrow> l"
+ using assms unfolding complete_def by fast
+
+(* TODO: generalize to uniform spaces *)
+lemma compact_imp_complete:
+ fixes s :: "'a::metric_space set"
+ assumes "compact s"
+ shows "complete s"
+proof -
+ {
+ fix f
+ assume as: "(\<forall>n::nat. f n \<in> s)" "Cauchy f"
+ from as(1) obtain l r where lr: "l\<in>s" "strict_mono r" "(f \<circ> r) \<longlonglongrightarrow> l"
+ using assms unfolding compact_def by blast
+
+ note lr' = seq_suble [OF lr(2)]
+ {
+ fix e :: real
+ assume "e > 0"
+ from as(2) obtain N where N:"\<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (f m) (f n) < e/2"
+ unfolding cauchy_def
+ using \<open>e > 0\<close>
+ apply (erule_tac x="e/2" in allE, auto)
+ done
+ from lr(3)[unfolded lim_sequentially, THEN spec[where x="e/2"]]
+ obtain M where M:"\<forall>n\<ge>M. dist ((f \<circ> r) n) l < e/2"
+ using \<open>e > 0\<close> by auto
+ {
+ fix n :: nat
+ assume n: "n \<ge> max N M"
+ have "dist ((f \<circ> r) n) l < e/2"
+ using n M by auto
+ moreover have "r n \<ge> N"
+ using lr'[of n] n by auto
+ then have "dist (f n) ((f \<circ> r) n) < e / 2"
+ using N and n by auto
+ ultimately have "dist (f n) l < e"
+ using dist_triangle_half_r[of "f (r n)" "f n" e l]
+ by (auto simp: dist_commute)
+ }
+ then have "\<exists>N. \<forall>n\<ge>N. dist (f n) l < e" by blast
+ }
+ then have "\<exists>l\<in>s. (f \<longlongrightarrow> l) sequentially" using \<open>l\<in>s\<close>
+ unfolding lim_sequentially by auto
+ }
+ then show ?thesis unfolding complete_def by auto
+qed
+
+proposition compact_eq_totally_bounded:
+ "compact s \<longleftrightarrow> complete s \<and> (\<forall>e>0. \<exists>k. finite k \<and> s \<subseteq> (\<Union>x\<in>k. ball x e))"
+ (is "_ \<longleftrightarrow> ?rhs")
+proof
+ assume assms: "?rhs"
+ then obtain k where k: "\<And>e. 0 < e \<Longrightarrow> finite (k e)" "\<And>e. 0 < e \<Longrightarrow> s \<subseteq> (\<Union>x\<in>k e. ball x e)"
+ by (auto simp: choice_iff')
+
+ show "compact s"
+ proof cases
+ assume "s = {}"
+ then show "compact s" by (simp add: compact_def)
+ next
+ assume "s \<noteq> {}"
+ show ?thesis
+ unfolding compact_def
+ proof safe
+ fix f :: "nat \<Rightarrow> 'a"
+ assume f: "\<forall>n. f n \<in> s"
+
+ define e where "e n = 1 / (2 * Suc n)" for n
+ then have [simp]: "\<And>n. 0 < e n" by auto
+ define B where "B n U = (SOME b. infinite {n. f n \<in> b} \<and> (\<exists>x. b \<subseteq> ball x (e n) \<inter> U))" for n U
+ {
+ fix n U
+ assume "infinite {n. f n \<in> U}"
+ then have "\<exists>b\<in>k (e n). infinite {i\<in>{n. f n \<in> U}. f i \<in> ball b (e n)}"
+ using k f by (intro pigeonhole_infinite_rel) (auto simp: subset_eq)
+ then obtain a where
+ "a \<in> k (e n)"
+ "infinite {i \<in> {n. f n \<in> U}. f i \<in> ball a (e n)}" ..
+ then have "\<exists>b. infinite {i. f i \<in> b} \<and> (\<exists>x. b \<subseteq> ball x (e n) \<inter> U)"
+ by (intro exI[of _ "ball a (e n) \<inter> U"] exI[of _ a]) (auto simp: ac_simps)
+ from someI_ex[OF this]
+ have "infinite {i. f i \<in> B n U}" "\<exists>x. B n U \<subseteq> ball x (e n) \<inter> U"
+ unfolding B_def by auto
+ }
+ note B = this
+
+ define F where "F = rec_nat (B 0 UNIV) B"
+ {
+ fix n
+ have "infinite {i. f i \<in> F n}"
+ by (induct n) (auto simp: F_def B)
+ }
+ then have F: "\<And>n. \<exists>x. F (Suc n) \<subseteq> ball x (e n) \<inter> F n"
+ using B by (simp add: F_def)
+ then have F_dec: "\<And>m n. m \<le> n \<Longrightarrow> F n \<subseteq> F m"
+ using decseq_SucI[of F] by (auto simp: decseq_def)
+
+ obtain sel where sel: "\<And>k i. i < sel k i" "\<And>k i. f (sel k i) \<in> F k"
+ proof (atomize_elim, unfold all_conj_distrib[symmetric], intro choice allI)
+ fix k i
+ have "infinite ({n. f n \<in> F k} - {.. i})"
+ using \<open>infinite {n. f n \<in> F k}\<close> by auto
+ from infinite_imp_nonempty[OF this]
+ show "\<exists>x>i. f x \<in> F k"
+ by (simp add: set_eq_iff not_le conj_commute)
+ qed
+
+ define t where "t = rec_nat (sel 0 0) (\<lambda>n i. sel (Suc n) i)"
+ have "strict_mono t"
+ unfolding strict_mono_Suc_iff by (simp add: t_def sel)
+ moreover have "\<forall>i. (f \<circ> t) i \<in> s"
+ using f by auto
+ moreover
+ {
+ fix n
+ have "(f \<circ> t) n \<in> F n"
+ by (cases n) (simp_all add: t_def sel)
+ }
+ note t = this
+
+ have "Cauchy (f \<circ> t)"
+ proof (safe intro!: metric_CauchyI exI elim!: nat_approx_posE)
+ fix r :: real and N n m
+ assume "1 / Suc N < r" "Suc N \<le> n" "Suc N \<le> m"
+ then have "(f \<circ> t) n \<in> F (Suc N)" "(f \<circ> t) m \<in> F (Suc N)" "2 * e N < r"
+ using F_dec t by (auto simp: e_def field_simps of_nat_Suc)
+ with F[of N] obtain x where "dist x ((f \<circ> t) n) < e N" "dist x ((f \<circ> t) m) < e N"
+ by (auto simp: subset_eq)
+ with dist_triangle[of "(f \<circ> t) m" "(f \<circ> t) n" x] \<open>2 * e N < r\<close>
+ show "dist ((f \<circ> t) m) ((f \<circ> t) n) < r"
+ by (simp add: dist_commute)
+ qed
+
+ ultimately show "\<exists>l\<in>s. \<exists>r. strict_mono r \<and> (f \<circ> r) \<longlonglongrightarrow> l"
+ using assms unfolding complete_def by blast
+ qed
+ qed
+qed (metis compact_imp_complete compact_imp_seq_compact seq_compact_imp_totally_bounded)
+
+lemma cauchy_imp_bounded:
+ assumes "Cauchy s"
+ shows "bounded (range s)"
+proof -
+ from assms obtain N :: nat where "\<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (s m) (s n) < 1"
+ unfolding cauchy_def by force
+ then have N:"\<forall>n. N \<le> n \<longrightarrow> dist (s N) (s n) < 1" by auto
+ moreover
+ have "bounded (s ` {0..N})"
+ using finite_imp_bounded[of "s ` {1..N}"] by auto
+ then obtain a where a:"\<forall>x\<in>s ` {0..N}. dist (s N) x \<le> a"
+ unfolding bounded_any_center [where a="s N"] by auto
+ ultimately show "?thesis"
+ unfolding bounded_any_center [where a="s N"]
+ apply (rule_tac x="max a 1" in exI, auto)
+ apply (erule_tac x=y in allE)
+ apply (erule_tac x=y in ballE, auto)
+ done
+qed
+
+instance heine_borel < complete_space
+proof
+ fix f :: "nat \<Rightarrow> 'a" assume "Cauchy f"
+ then have "bounded (range f)"
+ by (rule cauchy_imp_bounded)
+ then have "compact (closure (range f))"
+ unfolding compact_eq_bounded_closed by auto
+ then have "complete (closure (range f))"
+ by (rule compact_imp_complete)
+ moreover have "\<forall>n. f n \<in> closure (range f)"
+ using closure_subset [of "range f"] by auto
+ ultimately have "\<exists>l\<in>closure (range f). (f \<longlongrightarrow> l) sequentially"
+ using \<open>Cauchy f\<close> unfolding complete_def by auto
+ then show "convergent f"
+ unfolding convergent_def by auto
+qed
+
+lemma complete_UNIV: "complete (UNIV :: ('a::complete_space) set)"
+proof (rule completeI)
+ fix f :: "nat \<Rightarrow> 'a" assume "Cauchy f"
+ then have "convergent f" by (rule Cauchy_convergent)
+ then show "\<exists>l\<in>UNIV. f \<longlonglongrightarrow> l" unfolding convergent_def by simp
+qed
+
+lemma complete_imp_closed:
+ fixes S :: "'a::metric_space set"
+ assumes "complete S"
+ shows "closed S"
+proof (unfold closed_sequential_limits, clarify)
+ fix f x assume "\<forall>n. f n \<in> S" and "f \<longlonglongrightarrow> x"
+ from \<open>f \<longlonglongrightarrow> x\<close> have "Cauchy f"
+ by (rule LIMSEQ_imp_Cauchy)
+ with \<open>complete S\<close> and \<open>\<forall>n. f n \<in> S\<close> obtain l where "l \<in> S" and "f \<longlonglongrightarrow> l"
+ by (rule completeE)
+ from \<open>f \<longlonglongrightarrow> x\<close> and \<open>f \<longlonglongrightarrow> l\<close> have "x = l"
+ by (rule LIMSEQ_unique)
+ with \<open>l \<in> S\<close> show "x \<in> S"
+ by simp
+qed
+
+lemma complete_Int_closed:
+ fixes S :: "'a::metric_space set"
+ assumes "complete S" and "closed t"
+ shows "complete (S \<inter> t)"
+proof (rule completeI)
+ fix f assume "\<forall>n. f n \<in> S \<inter> t" and "Cauchy f"
+ then have "\<forall>n. f n \<in> S" and "\<forall>n. f n \<in> t"
+ by simp_all
+ from \<open>complete S\<close> obtain l where "l \<in> S" and "f \<longlonglongrightarrow> l"
+ using \<open>\<forall>n. f n \<in> S\<close> and \<open>Cauchy f\<close> by (rule completeE)
+ from \<open>closed t\<close> and \<open>\<forall>n. f n \<in> t\<close> and \<open>f \<longlonglongrightarrow> l\<close> have "l \<in> t"
+ by (rule closed_sequentially)
+ with \<open>l \<in> S\<close> and \<open>f \<longlonglongrightarrow> l\<close> show "\<exists>l\<in>S \<inter> t. f \<longlonglongrightarrow> l"
+ by fast
+qed
+
+lemma complete_closed_subset:
+ fixes S :: "'a::metric_space set"
+ assumes "closed S" and "S \<subseteq> t" and "complete t"
+ shows "complete S"
+ using assms complete_Int_closed [of t S] by (simp add: Int_absorb1)
+
+lemma complete_eq_closed:
+ fixes S :: "('a::complete_space) set"
+ shows "complete S \<longleftrightarrow> closed S"
+proof
+ assume "closed S" then show "complete S"
+ using subset_UNIV complete_UNIV by (rule complete_closed_subset)
+next
+ assume "complete S" then show "closed S"
+ by (rule complete_imp_closed)
+qed
+
+lemma convergent_eq_Cauchy:
+ fixes S :: "nat \<Rightarrow> 'a::complete_space"
+ shows "(\<exists>l. (S \<longlongrightarrow> l) sequentially) \<longleftrightarrow> Cauchy S"
+ unfolding Cauchy_convergent_iff convergent_def ..
+
+lemma convergent_imp_bounded:
+ fixes S :: "nat \<Rightarrow> 'a::metric_space"
+ shows "(S \<longlongrightarrow> l) sequentially \<Longrightarrow> bounded (range S)"
+ by (intro cauchy_imp_bounded LIMSEQ_imp_Cauchy)
+
+lemma frontier_subset_compact:
+ fixes S :: "'a::heine_borel set"
+ shows "compact S \<Longrightarrow> frontier S \<subseteq> S"
+ using frontier_subset_closed compact_eq_bounded_closed
+ by blast
+
+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
+
+lemma uniformly_continuous_onE:
+ assumes "uniformly_continuous_on s f" "0 < e"
+ obtains d where "d>0" "\<And>x x'. \<lbrakk>x\<in>s; x'\<in>s; dist x' x < d\<rbrakk> \<Longrightarrow> dist (f x') (f x) < e"
+using assms
+by (auto simp: uniformly_continuous_on_def)
+
+lemma continuous_at_imp_continuous_within:
+ "continuous (at x) f \<Longrightarrow> continuous (at x within s) f"
+ unfolding continuous_within continuous_at using Lim_at_imp_Lim_at_within by auto
+
+lemma Lim_trivial_limit: "trivial_limit net \<Longrightarrow> (f \<longlongrightarrow> l) net"
+ by simp
+
+lemmas continuous_on = continuous_on_def \<comment> \<open>legacy theorem name\<close>
+
+lemma continuous_within_subset:
+ "continuous (at x within s) f \<Longrightarrow> t \<subseteq> s \<Longrightarrow> continuous (at x within t) f"
+ unfolding continuous_within by(metis tendsto_within_subset)
+
+lemma continuous_on_interior:
+ "continuous_on s f \<Longrightarrow> x \<in> interior s \<Longrightarrow> continuous (at x) f"
+ by (metis continuous_on_eq_continuous_at continuous_on_subset interiorE)
+
+lemma continuous_on_eq:
+ "\<lbrakk>continuous_on s f; \<And>x. x \<in> s \<Longrightarrow> f x = g x\<rbrakk> \<Longrightarrow> continuous_on s g"
+ unfolding continuous_on_def tendsto_def eventually_at_topological
+ by simp
+
+text \<open>Characterization of various kinds of continuity in terms of sequences.\<close>
+
+lemma continuous_within_sequentiallyI:
+ fixes f :: "'a::{first_countable_topology, t2_space} \<Rightarrow> 'b::topological_space"
+ assumes "\<And>u::nat \<Rightarrow> 'a. u \<longlonglongrightarrow> a \<Longrightarrow> (\<forall>n. u n \<in> s) \<Longrightarrow> (\<lambda>n. f (u n)) \<longlonglongrightarrow> f a"
+ shows "continuous (at a within s) f"
+ using assms unfolding continuous_within tendsto_def[where l = "f a"]
+ by (auto intro!: sequentially_imp_eventually_within)
+
+lemma continuous_within_tendsto_compose:
+ fixes f::"'a::t2_space \<Rightarrow> 'b::topological_space"
+ assumes "continuous (at a within s) f"
+ "eventually (\<lambda>n. x n \<in> s) F"
+ "(x \<longlongrightarrow> a) F "
+ shows "((\<lambda>n. f (x n)) \<longlongrightarrow> f a) F"
+proof -
+ have *: "filterlim x (inf (nhds a) (principal s)) F"
+ using assms(2) assms(3) unfolding at_within_def filterlim_inf by (auto simp: filterlim_principal eventually_mono)
+ show ?thesis
+ by (auto simp: assms(1) continuous_within[symmetric] tendsto_at_within_iff_tendsto_nhds[symmetric] intro!: filterlim_compose[OF _ *])
+qed
+
+lemma continuous_within_tendsto_compose':
+ fixes f::"'a::t2_space \<Rightarrow> 'b::topological_space"
+ assumes "continuous (at a within s) f"
+ "\<And>n. x n \<in> s"
+ "(x \<longlongrightarrow> a) F "
+ shows "((\<lambda>n. f (x n)) \<longlongrightarrow> f a) F"
+ by (auto intro!: continuous_within_tendsto_compose[OF assms(1)] simp add: assms)
+
+lemma continuous_within_sequentially:
+ fixes f :: "'a::{first_countable_topology, t2_space} \<Rightarrow> 'b::topological_space"
+ shows "continuous (at a within s) f \<longleftrightarrow>
+ (\<forall>x. (\<forall>n::nat. x n \<in> s) \<and> (x \<longlongrightarrow> a) sequentially
+ \<longrightarrow> ((f \<circ> x) \<longlongrightarrow> f a) sequentially)"
+ using continuous_within_tendsto_compose'[of a s f _ sequentially]
+ continuous_within_sequentiallyI[of a s f]
+ by (auto simp: o_def)
+
+lemma continuous_at_sequentiallyI:
+ fixes f :: "'a::{first_countable_topology, t2_space} \<Rightarrow> 'b::topological_space"
+ assumes "\<And>u. u \<longlonglongrightarrow> a \<Longrightarrow> (\<lambda>n. f (u n)) \<longlonglongrightarrow> f a"
+ shows "continuous (at a) f"
+ using continuous_within_sequentiallyI[of a UNIV f] assms by auto
+
+lemma continuous_at_sequentially:
+ fixes f :: "'a::metric_space \<Rightarrow> 'b::topological_space"
+ shows "continuous (at a) f \<longleftrightarrow>
+ (\<forall>x. (x \<longlongrightarrow> a) sequentially --> ((f \<circ> x) \<longlongrightarrow> f a) sequentially)"
+ using continuous_within_sequentially[of a UNIV f] by simp
+
+lemma continuous_on_sequentiallyI:
+ fixes f :: "'a::{first_countable_topology, t2_space} \<Rightarrow> 'b::topological_space"
+ assumes "\<And>u a. (\<forall>n. u n \<in> s) \<Longrightarrow> a \<in> s \<Longrightarrow> u \<longlonglongrightarrow> a \<Longrightarrow> (\<lambda>n. f (u n)) \<longlonglongrightarrow> f a"
+ shows "continuous_on s f"
+ using assms unfolding continuous_on_eq_continuous_within
+ using continuous_within_sequentiallyI[of _ s f] by auto
+
+lemma continuous_on_sequentially:
+ fixes f :: "'a::metric_space \<Rightarrow> 'b::topological_space"
+ shows "continuous_on s f \<longleftrightarrow>
+ (\<forall>x. \<forall>a \<in> s. (\<forall>n. x(n) \<in> s) \<and> (x \<longlongrightarrow> a) sequentially
+ --> ((f \<circ> x) \<longlongrightarrow> f a) sequentially)"
+ (is "?lhs = ?rhs")
+proof
+ assume ?rhs
+ then show ?lhs
+ using continuous_within_sequentially[of _ s f]
+ unfolding continuous_on_eq_continuous_within
+ by auto
+next
+ assume ?lhs
+ then show ?rhs
+ unfolding continuous_on_eq_continuous_within
+ using continuous_within_sequentially[of _ s f]
+ by auto
+qed
+
+lemma uniformly_continuous_on_sequentially:
+ "uniformly_continuous_on s f \<longleftrightarrow> (\<forall>x y. (\<forall>n. x n \<in> s) \<and> (\<forall>n. y n \<in> s) \<and>
+ (\<lambda>n. dist (x n) (y n)) \<longlonglongrightarrow> 0 \<longrightarrow> (\<lambda>n. dist (f(x n)) (f(y n))) \<longlonglongrightarrow> 0)" (is "?lhs = ?rhs")
+proof
+ assume ?lhs
+ {
+ fix x y
+ assume x: "\<forall>n. x n \<in> s"
+ and y: "\<forall>n. y n \<in> s"
+ and xy: "((\<lambda>n. dist (x n) (y n)) \<longlongrightarrow> 0) sequentially"
+ {
+ fix e :: real
+ assume "e > 0"
+ then obtain d where "d > 0" and d: "\<forall>x\<in>s. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (f x') (f x) < e"
+ using \<open>?lhs\<close>[unfolded uniformly_continuous_on_def, THEN spec[where x=e]] by auto
+ obtain N where N: "\<forall>n\<ge>N. dist (x n) (y n) < d"
+ using xy[unfolded lim_sequentially dist_norm] and \<open>d>0\<close> by auto
+ {
+ fix n
+ assume "n\<ge>N"
+ then have "dist (f (x n)) (f (y n)) < e"
+ using N[THEN spec[where x=n]]
+ using d[THEN bspec[where x="x n"], THEN bspec[where x="y n"]]
+ using x and y
+ by (simp add: dist_commute)
+ }
+ then have "\<exists>N. \<forall>n\<ge>N. dist (f (x n)) (f (y n)) < e"
+ by auto
+ }
+ then have "((\<lambda>n. dist (f(x n)) (f(y n))) \<longlongrightarrow> 0) sequentially"
+ unfolding lim_sequentially and dist_real_def by auto
+ }
+ then show ?rhs by auto
+next
+ assume ?rhs
+ {
+ assume "\<not> ?lhs"
+ then obtain e where "e > 0" "\<forall>d>0. \<exists>x\<in>s. \<exists>x'\<in>s. dist x' x < d \<and> \<not> dist (f x') (f x) < e"
+ unfolding uniformly_continuous_on_def by auto
+ then obtain fa where fa:
+ "\<forall>x. 0 < x \<longrightarrow> fst (fa x) \<in> s \<and> snd (fa x) \<in> s \<and> dist (fst (fa x)) (snd (fa x)) < x \<and> \<not> dist (f (fst (fa x))) (f (snd (fa x))) < e"
+ using choice[of "\<lambda>d x. d>0 \<longrightarrow> fst x \<in> s \<and> snd x \<in> s \<and> dist (snd x) (fst x) < d \<and> \<not> dist (f (snd x)) (f (fst x)) < e"]
+ unfolding Bex_def
+ by (auto simp: dist_commute)
+ define x where "x n = fst (fa (inverse (real n + 1)))" for n
+ define y where "y n = snd (fa (inverse (real n + 1)))" for n
+ have xyn: "\<forall>n. x n \<in> s \<and> y n \<in> s"
+ and xy0: "\<forall>n. dist (x n) (y n) < inverse (real n + 1)"
+ and fxy:"\<forall>n. \<not> dist (f (x n)) (f (y n)) < e"
+ unfolding x_def and y_def using fa
+ by auto
+ {
+ fix e :: real
+ assume "e > 0"
+ then obtain N :: nat where "N \<noteq> 0" and N: "0 < inverse (real N) \<and> inverse (real N) < e"
+ unfolding real_arch_inverse[of e] by auto
+ {
+ fix n :: nat
+ assume "n \<ge> N"
+ then have "inverse (real n + 1) < inverse (real N)"
+ using of_nat_0_le_iff and \<open>N\<noteq>0\<close> by auto
+ also have "\<dots> < e" using N by auto
+ finally have "inverse (real n + 1) < e" by auto
+ then have "dist (x n) (y n) < e"
+ using xy0[THEN spec[where x=n]] by auto
+ }
+ then have "\<exists>N. \<forall>n\<ge>N. dist (x n) (y n) < e" by auto
+ }
+ then have "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. dist (f (x n)) (f (y n)) < e"
+ using \<open>?rhs\<close>[THEN spec[where x=x], THEN spec[where x=y]] and xyn
+ unfolding lim_sequentially dist_real_def by auto
+ then have False using fxy and \<open>e>0\<close> by auto
+ }
+ then show ?lhs
+ 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 uniformly_continuous_on_dist[continuous_intros]:
+ fixes f g :: "'a::metric_space \<Rightarrow> 'b::metric_space"
+ assumes "uniformly_continuous_on s f"
+ and "uniformly_continuous_on s g"
+ shows "uniformly_continuous_on s (\<lambda>x. dist (f x) (g x))"
+proof -
+ {
+ fix a b c d :: 'b
+ have "\<bar>dist a b - dist c d\<bar> \<le> dist a c + dist b d"
+ using dist_triangle2 [of a b c] dist_triangle2 [of b c d]
+ using dist_triangle3 [of c d a] dist_triangle [of a d b]
+ by arith
+ } note le = this
+ {
+ fix x y
+ assume f: "(\<lambda>n. dist (f (x n)) (f (y n))) \<longlonglongrightarrow> 0"
+ assume g: "(\<lambda>n. dist (g (x n)) (g (y n))) \<longlonglongrightarrow> 0"
+ have "(\<lambda>n. \<bar>dist (f (x n)) (g (x n)) - dist (f (y n)) (g (y n))\<bar>) \<longlonglongrightarrow> 0"
+ by (rule Lim_transform_bound [OF _ tendsto_add_zero [OF f g]],
+ simp add: le)
+ }
+ then show ?thesis
+ using assms unfolding uniformly_continuous_on_sequentially
+ unfolding dist_real_def by simp
+qed
+
+lemma uniformly_continuous_on_norm[continuous_intros]:
+ fixes f :: "'a :: metric_space \<Rightarrow> 'b :: real_normed_vector"
+ assumes "uniformly_continuous_on s f"
+ shows "uniformly_continuous_on s (\<lambda>x. norm (f x))"
+ unfolding norm_conv_dist using assms
+ by (intro uniformly_continuous_on_dist uniformly_continuous_on_const)
+
+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_cmul[continuous_intros]:
+ fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
+ assumes "uniformly_continuous_on s f"
+ shows "uniformly_continuous_on s (\<lambda>x. c *\<^sub>R f(x))"
+ using bounded_linear_scaleR_right assms
+ by (rule bounded_linear.uniformly_continuous_on)
+
+lemma dist_minus:
+ fixes x y :: "'a::real_normed_vector"
+ shows "dist (- x) (- y) = dist x y"
+ unfolding dist_norm minus_diff_minus norm_minus_cancel ..
+
+lemma uniformly_continuous_on_minus[continuous_intros]:
+ fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
+ shows "uniformly_continuous_on s f \<Longrightarrow> uniformly_continuous_on s (\<lambda>x. - f x)"
+ unfolding uniformly_continuous_on_def dist_minus .
+
+lemma uniformly_continuous_on_add[continuous_intros]:
+ fixes f g :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
+ assumes "uniformly_continuous_on s f"
+ and "uniformly_continuous_on s g"
+ shows "uniformly_continuous_on s (\<lambda>x. f x + g x)"
+ using assms
+ unfolding uniformly_continuous_on_sequentially
+ unfolding dist_norm tendsto_norm_zero_iff add_diff_add
+ by (auto intro: tendsto_add_zero)
+
+lemma uniformly_continuous_on_diff[continuous_intros]:
+ fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
+ assumes "uniformly_continuous_on s f"
+ and "uniformly_continuous_on s g"
+ shows "uniformly_continuous_on s (\<lambda>x. f x - g x)"
+ using assms uniformly_continuous_on_add [of s f "- g"]
+ by (simp add: fun_Compl_def uniformly_continuous_on_minus)
+
+text \<open>Continuity in terms of open preimages.\<close>
+
+lemma continuous_at_open:
+ "continuous (at x) f \<longleftrightarrow> (\<forall>t. open t \<and> f x \<in> t --> (\<exists>s. open s \<and> x \<in> s \<and> (\<forall>x' \<in> s. (f x') \<in> t)))"
+ unfolding continuous_within_topological [of x UNIV f]
+ unfolding imp_conjL
+ by (intro all_cong imp_cong ex_cong conj_cong refl) auto
+
+lemma continuous_imp_tendsto:
+ assumes "continuous (at x0) f"
+ and "x \<longlonglongrightarrow> x0"
+ shows "(f \<circ> x) \<longlonglongrightarrow> (f x0)"
+proof (rule topological_tendstoI)
+ fix S
+ assume "open S" "f x0 \<in> S"
+ then obtain T where T_def: "open T" "x0 \<in> T" "\<forall>x\<in>T. f x \<in> S"
+ using assms continuous_at_open by metis
+ then have "eventually (\<lambda>n. x n \<in> T) sequentially"
+ using assms T_def by (auto simp: tendsto_def)
+ then show "eventually (\<lambda>n. (f \<circ> x) n \<in> S) sequentially"
+ using T_def by (auto elim!: eventually_mono)
+qed
+
+lemma continuous_on_open:
+ "continuous_on S f \<longleftrightarrow>
+ (\<forall>T. openin (subtopology euclidean (f ` S)) T \<longrightarrow>
+ openin (subtopology euclidean S) (S \<inter> f -` T))"
+ unfolding continuous_on_open_invariant openin_open Int_def vimage_def Int_commute
+ by (simp add: imp_ex imageI conj_commute eq_commute cong: conj_cong)
+
+lemma continuous_on_open_gen:
+ fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space"
+ assumes "f ` S \<subseteq> T"
+ shows "continuous_on S f \<longleftrightarrow>
+ (\<forall>U. openin (subtopology euclidean T) U
+ \<longrightarrow> openin (subtopology euclidean S) (S \<inter> f -` U))"
+ (is "?lhs = ?rhs")
+proof
+ assume ?lhs
+ then show ?rhs
+ apply (clarsimp simp: openin_euclidean_subtopology_iff continuous_on_iff)
+ by (metis assms image_subset_iff)
+next
+ have ope: "openin (subtopology euclidean T) (ball y e \<inter> T)" for y e
+ by (simp add: Int_commute openin_open_Int)
+ assume R [rule_format]: ?rhs
+ show ?lhs
+ proof (clarsimp simp add: continuous_on_iff)
+ fix x and e::real
+ assume "x \<in> S" and "0 < e"
+ then have x: "x \<in> S \<inter> (f -` ball (f x) e \<inter> f -` T)"
+ using assms by auto
+ show "\<exists>d>0. \<forall>x'\<in>S. dist x' x < d \<longrightarrow> dist (f x') (f x) < e"
+ using R [of "ball (f x) e \<inter> T"] x
+ by (fastforce simp add: ope openin_euclidean_subtopology_iff [of S] dist_commute)
+ qed
+qed
+
+lemma continuous_openin_preimage:
+ fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space"
+ shows
+ "\<lbrakk>continuous_on S f; f ` S \<subseteq> T; openin (subtopology euclidean T) U\<rbrakk>
+ \<Longrightarrow> openin (subtopology euclidean S) (S \<inter> f -` U)"
+by (simp add: continuous_on_open_gen)
+
+text \<open>Similarly in terms of closed sets.\<close>
+
+lemma continuous_on_closed:
+ "continuous_on S f \<longleftrightarrow>
+ (\<forall>T. closedin (subtopology euclidean (f ` S)) T \<longrightarrow>
+ closedin (subtopology euclidean S) (S \<inter> f -` T))"
+ unfolding continuous_on_closed_invariant closedin_closed Int_def vimage_def Int_commute
+ by (simp add: imp_ex imageI conj_commute eq_commute cong: conj_cong)
+
+lemma continuous_on_closed_gen:
+ fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space"
+ assumes "f ` S \<subseteq> T"
+ shows "continuous_on S f \<longleftrightarrow>
+ (\<forall>U. closedin (subtopology euclidean T) U
+ \<longrightarrow> closedin (subtopology euclidean S) (S \<inter> f -` U))"
+ (is "?lhs = ?rhs")
+proof -
+ have *: "U \<subseteq> T \<Longrightarrow> S \<inter> f -` (T - U) = S - (S \<inter> f -` U)" for U
+ using assms by blast
+ show ?thesis
+ proof
+ assume L: ?lhs
+ show ?rhs
+ proof clarify
+ fix U
+ assume "closedin (subtopology euclidean T) U"
+ then show "closedin (subtopology euclidean S) (S \<inter> f -` U)"
+ using L unfolding continuous_on_open_gen [OF assms]
+ by (metis * closedin_def inf_le1 topspace_euclidean_subtopology)
+ qed
+ next
+ assume R [rule_format]: ?rhs
+ show ?lhs
+ unfolding continuous_on_open_gen [OF assms]
+ by (metis * R inf_le1 openin_closedin_eq topspace_euclidean_subtopology)
+ qed
+qed
+
+lemma continuous_closedin_preimage_gen:
+ fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space"
+ assumes "continuous_on S f" "f ` S \<subseteq> T" "closedin (subtopology euclidean T) U"
+ shows "closedin (subtopology euclidean S) (S \<inter> f -` U)"
+using assms continuous_on_closed_gen by blast
+
+lemma continuous_on_imp_closedin:
+ assumes "continuous_on S f" "closedin (subtopology euclidean (f ` S)) T"
+ shows "closedin (subtopology euclidean S) (S \<inter> f -` T)"
+using assms continuous_on_closed by blast
+
+subsection%unimportant \<open>Half-global and completely global cases\<close>
+
+lemma continuous_openin_preimage_gen:
+ assumes "continuous_on S f" "open T"
+ shows "openin (subtopology euclidean S) (S \<inter> f -` T)"
+proof -
+ have *: "(S \<inter> f -` T) = (S \<inter> f -` (T \<inter> f ` S))"
+ by auto
+ have "openin (subtopology euclidean (f ` S)) (T \<inter> f ` S)"
+ using openin_open_Int[of T "f ` S", OF assms(2)] unfolding openin_open by auto
+ then show ?thesis
+ using assms(1)[unfolded continuous_on_open, THEN spec[where x="T \<inter> f ` S"]]
+ using * by auto
+qed
+
+lemma continuous_closedin_preimage:
+ assumes "continuous_on S f" and "closed T"
+ shows "closedin (subtopology euclidean S) (S \<inter> f -` T)"
+proof -
+ have *: "(S \<inter> f -` T) = (S \<inter> f -` (T \<inter> f ` S))"
+ by auto
+ have "closedin (subtopology euclidean (f ` S)) (T \<inter> f ` S)"
+ using closedin_closed_Int[of T "f ` S", OF assms(2)]
+ by (simp add: Int_commute)
+ then show ?thesis
+ using assms(1)[unfolded continuous_on_closed, THEN spec[where x="T \<inter> f ` S"]]
+ using * by auto
+qed
+
+lemma continuous_openin_preimage_eq:
+ "continuous_on S f \<longleftrightarrow>
+ (\<forall>T. open T \<longrightarrow> openin (subtopology euclidean S) (S \<inter> f -` T))"
+apply safe
+apply (simp add: continuous_openin_preimage_gen)
+apply (fastforce simp add: continuous_on_open openin_open)
+done
+
+lemma continuous_closedin_preimage_eq:
+ "continuous_on S f \<longleftrightarrow>
+ (\<forall>T. closed T \<longrightarrow> closedin (subtopology euclidean S) (S \<inter> f -` T))"
+apply safe
+apply (simp add: continuous_closedin_preimage)
+apply (fastforce simp add: continuous_on_closed closedin_closed)
+done
+
+lemma continuous_open_preimage:
+ assumes contf: "continuous_on S f" and "open S" "open T"
+ shows "open (S \<inter> f -` T)"
+proof-
+ obtain U where "open U" "(S \<inter> f -` T) = S \<inter> U"
+ using continuous_openin_preimage_gen[OF contf \<open>open T\<close>]
+ unfolding openin_open by auto
+ then show ?thesis
+ using open_Int[of S U, OF \<open>open S\<close>] by auto
+qed
+
+lemma continuous_closed_preimage:
+ assumes contf: "continuous_on S f" and "closed S" "closed T"
+ shows "closed (S \<inter> f -` T)"
+proof-
+ obtain U where "closed U" "(S \<inter> f -` T) = S \<inter> U"
+ using continuous_closedin_preimage[OF contf \<open>closed T\<close>]
+ unfolding closedin_closed by auto
+ then show ?thesis using closed_Int[of S U, OF \<open>closed S\<close>] by auto
+qed
+
+lemma continuous_open_vimage: "open S \<Longrightarrow> (\<And>x. continuous (at x) f) \<Longrightarrow> open (f -` S)"
+ by (metis continuous_on_eq_continuous_within open_vimage)
+
+lemma continuous_closed_vimage: "closed S \<Longrightarrow> (\<And>x. continuous (at x) f) \<Longrightarrow> closed (f -` S)"
+ by (simp add: closed_vimage continuous_on_eq_continuous_within)
+
+lemma interior_image_subset:
+ assumes "inj f" "\<And>x. continuous (at x) f"
+ shows "interior (f ` S) \<subseteq> f ` (interior S)"
+proof
+ fix x assume "x \<in> interior (f ` S)"
+ then obtain T where as: "open T" "x \<in> T" "T \<subseteq> f ` S" ..
+ then have "x \<in> f ` S" by auto
+ then obtain y where y: "y \<in> S" "x = f y" by auto
+ have "open (f -` T)"
+ using assms \<open>open T\<close> by (simp add: continuous_at_imp_continuous_on open_vimage)
+ moreover have "y \<in> vimage f T"
+ using \<open>x = f y\<close> \<open>x \<in> T\<close> by simp
+ moreover have "vimage f T \<subseteq> S"
+ using \<open>T \<subseteq> image f S\<close> \<open>inj f\<close> unfolding inj_on_def subset_eq by auto
+ ultimately have "y \<in> interior S" ..
+ with \<open>x = f y\<close> show "x \<in> f ` interior S" ..
+qed
+
+subsection%unimportant \<open>Topological properties of linear functions\<close>
+
+lemma linear_lim_0:
+ assumes "bounded_linear f"
+ shows "(f \<longlongrightarrow> 0) (at (0))"
+proof -
+ interpret f: bounded_linear f by fact
+ have "(f \<longlongrightarrow> f 0) (at 0)"
+ using tendsto_ident_at by (rule f.tendsto)
+ then show ?thesis unfolding f.zero .
+qed
+
+lemma linear_continuous_at:
+ assumes "bounded_linear f"
+ shows "continuous (at a) f"
+ unfolding continuous_at using assms
+ apply (rule bounded_linear.tendsto)
+ apply (rule tendsto_ident_at)
+ done
+
+lemma linear_continuous_within:
+ "bounded_linear f \<Longrightarrow> continuous (at x within s) f"
+ using continuous_at_imp_continuous_within[of x f s] using linear_continuous_at[of f] by auto
+
+lemma linear_continuous_on:
+ "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
+
+end
\ No newline at end of file
--- a/src/HOL/Analysis/Euclidean_Space.thy Thu Dec 27 22:54:29 2018 +0100
+++ b/src/HOL/Analysis/Euclidean_Space.thy Thu Dec 27 23:38:55 2018 +0100
@@ -12,6 +12,16 @@
Product_Vector
begin
+
+subsection%unimportant \<open>Interlude: Some properties of real sets\<close>
+
+lemma seq_mono_lemma:
+ assumes "\<forall>(n::nat) \<ge> m. (d n :: real) < e n"
+ and "\<forall>n \<ge> m. e n \<le> e m"
+ shows "\<forall>n \<ge> m. d n < e m"
+ using assms by force
+
+
subsection \<open>Type class of Euclidean spaces\<close>
class euclidean_space = real_inner +
--- a/src/HOL/Analysis/Linear_Algebra.thy Thu Dec 27 22:54:29 2018 +0100
+++ b/src/HOL/Analysis/Linear_Algebra.thy Thu Dec 27 23:38:55 2018 +0100
@@ -310,51 +310,6 @@
by (rule adjoint_unique, simp add: adjoint_clauses [OF lf])
-subsection%unimportant \<open>Interlude: Some properties of real sets\<close>
-
-lemma seq_mono_lemma:
- assumes "\<forall>(n::nat) \<ge> m. (d n :: real) < e n"
- and "\<forall>n \<ge> m. e n \<le> e m"
- shows "\<forall>n \<ge> m. d n < e m"
- using assms by force
-
-lemma infinite_enumerate:
- assumes fS: "infinite S"
- shows "\<exists>r::nat\<Rightarrow>nat. strict_mono r \<and> (\<forall>n. r n \<in> S)"
- unfolding strict_mono_def
- using enumerate_in_set[OF fS] enumerate_mono[of _ _ S] fS by auto
-
-lemma approachable_lt_le: "(\<exists>(d::real) > 0. \<forall>x. f x < d \<longrightarrow> P x) \<longleftrightarrow> (\<exists>d>0. \<forall>x. f x \<le> d \<longrightarrow> P x)"
- apply auto
- apply (rule_tac x="d/2" in exI)
- apply auto
- done
-
-lemma approachable_lt_le2: \<comment> \<open>like the above, but pushes aside an extra formula\<close>
- "(\<exists>(d::real) > 0. \<forall>x. Q x \<longrightarrow> f x < d \<longrightarrow> P x) \<longleftrightarrow> (\<exists>d>0. \<forall>x. f x \<le> d \<longrightarrow> Q x \<longrightarrow> P x)"
- apply auto
- apply (rule_tac x="d/2" in exI, auto)
- done
-
-lemma triangle_lemma:
- fixes x y z :: real
- assumes x: "0 \<le> x"
- and y: "0 \<le> y"
- and z: "0 \<le> z"
- and xy: "x\<^sup>2 \<le> y\<^sup>2 + z\<^sup>2"
- shows "x \<le> y + z"
-proof -
- have "y\<^sup>2 + z\<^sup>2 \<le> y\<^sup>2 + 2 * y * z + z\<^sup>2"
- using z y by simp
- with xy have th: "x\<^sup>2 \<le> (y + z)\<^sup>2"
- by (simp add: power2_eq_square field_simps)
- from y z have yz: "y + z \<ge> 0"
- by arith
- from power2_le_imp_le[OF th yz] show ?thesis .
-qed
-
-
-
subsection \<open>Archimedean properties and useful consequences\<close>
text\<open>Bernoulli's inequality\<close>
--- a/src/HOL/Analysis/Path_Connected.thy Thu Dec 27 22:54:29 2018 +0100
+++ b/src/HOL/Analysis/Path_Connected.thy Thu Dec 27 23:38:55 2018 +0100
@@ -5,7 +5,7 @@
section \<open>Continuous paths and path-connected sets\<close>
theory Path_Connected
-imports Continuous_Extension Continuum_Not_Denumerable
+ imports Continuous_Extension Continuum_Not_Denumerable
begin
subsection \<open>Paths and Arcs\<close>
@@ -7421,7 +7421,7 @@
show "x \<in> ball x r \<inter> affine hull S"
using \<open>x \<in> S\<close> \<open>r > 0\<close> by (simp add: hull_inc)
have "openin (subtopology euclidean (affine hull S)) (ball x r \<inter> affine hull S)"
- by (simp add: inf.commute openin_Int_open)
+ by (subst inf.commute) (simp add: openin_Int_open)
then show "openin (subtopology euclidean S) (ball x r \<inter> affine hull S)"
by (rule openin_subset_trans [OF _ subS Ssub])
qed (use * path_component_trans in \<open>auto simp: path_connected_component path_component_of_subset [OF ST]\<close>)
--- a/src/HOL/Analysis/Starlike.thy Thu Dec 27 22:54:29 2018 +0100
+++ b/src/HOL/Analysis/Starlike.thy Thu Dec 27 23:38:55 2018 +0100
@@ -4921,6 +4921,11 @@
subsection%unimportant\<open>Basic lemmas about hyperplanes and halfspaces\<close>
+lemma halfspace_Int_eq:
+ "{x. a \<bullet> x \<le> b} \<inter> {x. b \<le> a \<bullet> x} = {x. a \<bullet> x = b}"
+ "{x. b \<le> a \<bullet> x} \<inter> {x. a \<bullet> x \<le> b} = {x. a \<bullet> x = b}"
+ by auto
+
lemma hyperplane_eq_Ex:
assumes "a \<noteq> 0" obtains x where "a \<bullet> x = b"
by (rule_tac x = "(b / (a \<bullet> a)) *\<^sub>R a" in that) (simp add: assms)
--- a/src/HOL/Analysis/Topology_Euclidean_Space.thy Thu Dec 27 22:54:29 2018 +0100
+++ b/src/HOL/Analysis/Topology_Euclidean_Space.thy Thu Dec 27 23:38:55 2018 +0100
@@ -4,1402 +4,15 @@
Author: Brian Huffman, Portland State University
*)
-section \<open>Elementary topology in Euclidean space\<close>
+section \<open>Elementary Topology in Euclidean Space\<close>
theory Topology_Euclidean_Space
-imports
- "HOL-Library.Indicator_Function"
- "HOL-Library.Countable_Set"
- "HOL-Library.FuncSet"
- "HOL-Library.Set_Idioms"
- Linear_Algebra
- Norm_Arith
-begin
-
-(* FIXME: move elsewhere *)
-
-lemma halfspace_Int_eq:
- "{x. a \<bullet> x \<le> b} \<inter> {x. b \<le> a \<bullet> x} = {x. a \<bullet> x = b}"
- "{x. b \<le> a \<bullet> x} \<inter> {x. a \<bullet> x \<le> b} = {x. a \<bullet> x = b}"
- by auto
-
-definition (in monoid_add) support_on :: "'b set \<Rightarrow> ('b \<Rightarrow> 'a) \<Rightarrow> 'b set"
- where "support_on s f = {x\<in>s. f x \<noteq> 0}"
-
-lemma in_support_on: "x \<in> support_on s f \<longleftrightarrow> x \<in> s \<and> f x \<noteq> 0"
- by (simp add: support_on_def)
-
-lemma support_on_simps[simp]:
- "support_on {} f = {}"
- "support_on (insert x s) f =
- (if f x = 0 then support_on s f else insert x (support_on s f))"
- "support_on (s \<union> t) f = support_on s f \<union> support_on t f"
- "support_on (s \<inter> t) f = support_on s f \<inter> support_on t f"
- "support_on (s - t) f = support_on s f - support_on t f"
- "support_on (f ` s) g = f ` (support_on s (g \<circ> f))"
- unfolding support_on_def by auto
-
-lemma support_on_cong:
- "(\<And>x. x \<in> s \<Longrightarrow> f x = 0 \<longleftrightarrow> g x = 0) \<Longrightarrow> support_on s f = support_on s g"
- by (auto simp: support_on_def)
-
-lemma support_on_if: "a \<noteq> 0 \<Longrightarrow> support_on A (\<lambda>x. if P x then a else 0) = {x\<in>A. P x}"
- by (auto simp: support_on_def)
-
-lemma support_on_if_subset: "support_on A (\<lambda>x. if P x then a else 0) \<subseteq> {x \<in> A. P x}"
- by (auto simp: support_on_def)
-
-lemma finite_support[intro]: "finite S \<Longrightarrow> finite (support_on S f)"
- unfolding support_on_def by auto
-
-(* TODO: is supp_sum really needed? TODO: Generalize to Finite_Set.fold *)
-definition (in comm_monoid_add) supp_sum :: "('b \<Rightarrow> 'a) \<Rightarrow> 'b set \<Rightarrow> 'a"
- where "supp_sum f S = (\<Sum>x\<in>support_on S f. f x)"
-
-lemma supp_sum_empty[simp]: "supp_sum f {} = 0"
- unfolding supp_sum_def by auto
-
-lemma supp_sum_insert[simp]:
- "finite (support_on S f) \<Longrightarrow>
- supp_sum f (insert x S) = (if x \<in> S then supp_sum f S else f x + supp_sum f S)"
- by (simp add: supp_sum_def in_support_on insert_absorb)
-
-lemma supp_sum_divide_distrib: "supp_sum f A / (r::'a::field) = supp_sum (\<lambda>n. f n / r) A"
- by (cases "r = 0")
- (auto simp: supp_sum_def sum_divide_distrib intro!: sum.cong support_on_cong)
-
-(*END OF SUPPORT, ETC.*)
-
-lemma image_affinity_interval:
- fixes c :: "'a::ordered_real_vector"
- shows "((\<lambda>x. m *\<^sub>R x + c) ` {a..b}) =
- (if {a..b}={} then {}
- else if 0 \<le> m then {m *\<^sub>R a + c .. m *\<^sub>R b + c}
- else {m *\<^sub>R b + c .. m *\<^sub>R a + c})"
- (is "?lhs = ?rhs")
-proof (cases "m=0")
- case True
- then show ?thesis
- by force
-next
- case False
- show ?thesis
- proof
- show "?lhs \<subseteq> ?rhs"
- by (auto simp: scaleR_left_mono scaleR_left_mono_neg)
- show "?rhs \<subseteq> ?lhs"
- proof (clarsimp, intro conjI impI subsetI)
- show "\<lbrakk>0 \<le> m; a \<le> b; x \<in> {m *\<^sub>R a + c..m *\<^sub>R b + c}\<rbrakk>
- \<Longrightarrow> x \<in> (\<lambda>x. m *\<^sub>R x + c) ` {a..b}" for x
- apply (rule_tac x="inverse m *\<^sub>R (x-c)" in rev_image_eqI)
- using False apply (auto simp: le_diff_eq pos_le_divideRI)
- using diff_le_eq pos_le_divideR_eq by force
- show "\<lbrakk>\<not> 0 \<le> m; a \<le> b; x \<in> {m *\<^sub>R b + c..m *\<^sub>R a + c}\<rbrakk>
- \<Longrightarrow> x \<in> (\<lambda>x. m *\<^sub>R x + c) ` {a..b}" for x
- apply (rule_tac x="inverse m *\<^sub>R (x-c)" in rev_image_eqI)
- apply (auto simp: diff_le_eq neg_le_divideR_eq)
- using diff_eq_diff_less_eq linordered_field_class.sign_simps(11) minus_diff_eq not_less scaleR_le_cancel_left_neg by fastforce
- qed
- qed
-qed
-
-lemma countable_PiE:
- "finite I \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> countable (F i)) \<Longrightarrow> countable (Pi\<^sub>E I F)"
- by (induct I arbitrary: F rule: finite_induct) (auto simp: PiE_insert_eq)
-
-lemma open_sums:
- fixes T :: "('b::real_normed_vector) set"
- assumes "open S \<or> open T"
- shows "open (\<Union>x\<in> S. \<Union>y \<in> T. {x + y})"
- using assms
-proof
- assume S: "open S"
- show ?thesis
- proof (clarsimp simp: open_dist)
- fix x y
- assume "x \<in> S" "y \<in> T"
- with S obtain e where "e > 0" and e: "\<And>x'. dist x' x < e \<Longrightarrow> x' \<in> S"
- by (auto simp: open_dist)
- then have "\<And>z. dist z (x + y) < e \<Longrightarrow> \<exists>x\<in>S. \<exists>y\<in>T. z = x + y"
- by (metis \<open>y \<in> T\<close> diff_add_cancel dist_add_cancel2)
- then show "\<exists>e>0. \<forall>z. dist z (x + y) < e \<longrightarrow> (\<exists>x\<in>S. \<exists>y\<in>T. z = x + y)"
- using \<open>0 < e\<close> \<open>x \<in> S\<close> by blast
- qed
-next
- assume T: "open T"
- show ?thesis
- proof (clarsimp simp: open_dist)
- fix x y
- assume "x \<in> S" "y \<in> T"
- with T obtain e where "e > 0" and e: "\<And>x'. dist x' y < e \<Longrightarrow> x' \<in> T"
- by (auto simp: open_dist)
- then have "\<And>z. dist z (x + y) < e \<Longrightarrow> \<exists>x\<in>S. \<exists>y\<in>T. z = x + y"
- by (metis \<open>x \<in> S\<close> add_diff_cancel_left' add_diff_eq diff_diff_add dist_norm)
- then show "\<exists>e>0. \<forall>z. dist z (x + y) < e \<longrightarrow> (\<exists>x\<in>S. \<exists>y\<in>T. z = x + y)"
- using \<open>0 < e\<close> \<open>y \<in> T\<close> by blast
- qed
-qed
-
-
-subsection \<open>Topological Basis\<close>
-
-context topological_space
-begin
-
-definition%important "topological_basis B \<longleftrightarrow>
- (\<forall>b\<in>B. open b) \<and> (\<forall>x. open x \<longrightarrow> (\<exists>B'. B' \<subseteq> B \<and> \<Union>B' = x))"
-
-lemma topological_basis:
- "topological_basis B \<longleftrightarrow> (\<forall>x. open x \<longleftrightarrow> (\<exists>B'. B' \<subseteq> B \<and> \<Union>B' = x))"
- unfolding topological_basis_def
- apply safe
- apply fastforce
- apply fastforce
- apply (erule_tac x=x in allE, simp)
- apply (rule_tac x="{x}" in exI, auto)
- done
-
-lemma topological_basis_iff:
- assumes "\<And>B'. B' \<in> B \<Longrightarrow> open B'"
- shows "topological_basis B \<longleftrightarrow> (\<forall>O'. open O' \<longrightarrow> (\<forall>x\<in>O'. \<exists>B'\<in>B. x \<in> B' \<and> B' \<subseteq> O'))"
- (is "_ \<longleftrightarrow> ?rhs")
-proof safe
- fix O' and x::'a
- assume H: "topological_basis B" "open O'" "x \<in> O'"
- then have "(\<exists>B'\<subseteq>B. \<Union>B' = O')" by (simp add: topological_basis_def)
- then obtain B' where "B' \<subseteq> B" "O' = \<Union>B'" by auto
- then show "\<exists>B'\<in>B. x \<in> B' \<and> B' \<subseteq> O'" using H by auto
-next
- assume H: ?rhs
- show "topological_basis B"
- using assms unfolding topological_basis_def
- proof safe
- fix O' :: "'a set"
- assume "open O'"
- with H obtain f where "\<forall>x\<in>O'. f x \<in> B \<and> x \<in> f x \<and> f x \<subseteq> O'"
- by (force intro: bchoice simp: Bex_def)
- then show "\<exists>B'\<subseteq>B. \<Union>B' = O'"
- by (auto intro: exI[where x="{f x |x. x \<in> O'}"])
- qed
-qed
-
-lemma topological_basisI:
- assumes "\<And>B'. B' \<in> B \<Longrightarrow> open B'"
- and "\<And>O' x. open O' \<Longrightarrow> x \<in> O' \<Longrightarrow> \<exists>B'\<in>B. x \<in> B' \<and> B' \<subseteq> O'"
- shows "topological_basis B"
- using assms by (subst topological_basis_iff) auto
-
-lemma topological_basisE:
- fixes O'
- assumes "topological_basis B"
- and "open O'"
- and "x \<in> O'"
- obtains B' where "B' \<in> B" "x \<in> B'" "B' \<subseteq> O'"
-proof atomize_elim
- from assms have "\<And>B'. B'\<in>B \<Longrightarrow> open B'"
- by (simp add: topological_basis_def)
- with topological_basis_iff assms
- show "\<exists>B'. B' \<in> B \<and> x \<in> B' \<and> B' \<subseteq> O'"
- using assms by (simp add: Bex_def)
-qed
-
-lemma topological_basis_open:
- assumes "topological_basis B"
- and "X \<in> B"
- shows "open X"
- using assms by (simp add: topological_basis_def)
-
-lemma topological_basis_imp_subbasis:
- assumes B: "topological_basis B"
- shows "open = generate_topology B"
-proof (intro ext iffI)
- fix S :: "'a set"
- assume "open S"
- with B obtain B' where "B' \<subseteq> B" "S = \<Union>B'"
- unfolding topological_basis_def by blast
- then show "generate_topology B S"
- by (auto intro: generate_topology.intros dest: topological_basis_open)
-next
- fix S :: "'a set"
- assume "generate_topology B S"
- then show "open S"
- by induct (auto dest: topological_basis_open[OF B])
-qed
-
-lemma basis_dense:
- fixes B :: "'a set set"
- and f :: "'a set \<Rightarrow> 'a"
- assumes "topological_basis B"
- and choosefrom_basis: "\<And>B'. B' \<noteq> {} \<Longrightarrow> f B' \<in> B'"
- shows "\<forall>X. open X \<longrightarrow> X \<noteq> {} \<longrightarrow> (\<exists>B' \<in> B. f B' \<in> X)"
-proof (intro allI impI)
- fix X :: "'a set"
- assume "open X" and "X \<noteq> {}"
- from topological_basisE[OF \<open>topological_basis B\<close> \<open>open X\<close> choosefrom_basis[OF \<open>X \<noteq> {}\<close>]]
- obtain B' where "B' \<in> B" "f X \<in> B'" "B' \<subseteq> X" .
- then show "\<exists>B'\<in>B. f B' \<in> X"
- by (auto intro!: choosefrom_basis)
-qed
-
-end
-
-lemma topological_basis_prod:
- assumes A: "topological_basis A"
- and B: "topological_basis B"
- shows "topological_basis ((\<lambda>(a, b). a \<times> b) ` (A \<times> B))"
- unfolding topological_basis_def
-proof (safe, simp_all del: ex_simps add: subset_image_iff ex_simps(1)[symmetric])
- fix S :: "('a \<times> 'b) set"
- assume "open S"
- then show "\<exists>X\<subseteq>A \<times> B. (\<Union>(a,b)\<in>X. a \<times> b) = S"
- proof (safe intro!: exI[of _ "{x\<in>A \<times> B. fst x \<times> snd x \<subseteq> S}"])
- fix x y
- assume "(x, y) \<in> S"
- from open_prod_elim[OF \<open>open S\<close> this]
- obtain a b where a: "open a""x \<in> a" and b: "open b" "y \<in> b" and "a \<times> b \<subseteq> S"
- by (metis mem_Sigma_iff)
- moreover
- from A a obtain A0 where "A0 \<in> A" "x \<in> A0" "A0 \<subseteq> a"
- by (rule topological_basisE)
- moreover
- from B b obtain B0 where "B0 \<in> B" "y \<in> B0" "B0 \<subseteq> b"
- by (rule topological_basisE)
- ultimately show "(x, y) \<in> (\<Union>(a, b)\<in>{X \<in> A \<times> B. fst X \<times> snd X \<subseteq> S}. a \<times> b)"
- by (intro UN_I[of "(A0, B0)"]) auto
- qed auto
-qed (metis A B topological_basis_open open_Times)
-
-
-subsection \<open>Countable Basis\<close>
-
-locale%important countable_basis =
- fixes B :: "'a::topological_space set set"
- assumes is_basis: "topological_basis B"
- and countable_basis: "countable B"
-begin
-
-lemma open_countable_basis_ex:
- assumes "open X"
- shows "\<exists>B' \<subseteq> B. X = \<Union>B'"
- using assms countable_basis is_basis
- unfolding topological_basis_def by blast
-
-lemma open_countable_basisE:
- assumes "open X"
- obtains B' where "B' \<subseteq> B" "X = \<Union>B'"
- using assms open_countable_basis_ex
- by atomize_elim simp
-
-lemma countable_dense_exists:
- "\<exists>D::'a set. countable D \<and> (\<forall>X. open X \<longrightarrow> X \<noteq> {} \<longrightarrow> (\<exists>d \<in> D. d \<in> X))"
-proof -
- let ?f = "(\<lambda>B'. SOME x. x \<in> B')"
- have "countable (?f ` B)" using countable_basis by simp
- with basis_dense[OF is_basis, of ?f] show ?thesis
- by (intro exI[where x="?f ` B"]) (metis (mono_tags) all_not_in_conv imageI someI)
-qed
-
-lemma countable_dense_setE:
- obtains D :: "'a set"
- where "countable D" "\<And>X. open X \<Longrightarrow> X \<noteq> {} \<Longrightarrow> \<exists>d \<in> D. d \<in> X"
- using countable_dense_exists by blast
-
-end
-
-lemma (in first_countable_topology) first_countable_basisE:
- fixes x :: 'a
- obtains \<A> where "countable \<A>" "\<And>A. A \<in> \<A> \<Longrightarrow> x \<in> A" "\<And>A. A \<in> \<A> \<Longrightarrow> open A"
- "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> (\<exists>A\<in>\<A>. A \<subseteq> S)"
-proof -
- obtain \<A> where \<A>: "(\<forall>i::nat. x \<in> \<A> i \<and> open (\<A> i))" "(\<forall>S. open S \<and> x \<in> S \<longrightarrow> (\<exists>i. \<A> i \<subseteq> S))"
- using first_countable_basis[of x] by metis
- show thesis
- proof
- show "countable (range \<A>)"
- by simp
- qed (use \<A> in auto)
-qed
-
-lemma (in first_countable_topology) first_countable_basis_Int_stableE:
- obtains \<A> where "countable \<A>" "\<And>A. A \<in> \<A> \<Longrightarrow> x \<in> A" "\<And>A. A \<in> \<A> \<Longrightarrow> open A"
- "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> (\<exists>A\<in>\<A>. A \<subseteq> S)"
- "\<And>A B. A \<in> \<A> \<Longrightarrow> B \<in> \<A> \<Longrightarrow> A \<inter> B \<in> \<A>"
-proof atomize_elim
- obtain \<B> where \<B>:
- "countable \<B>"
- "\<And>B. B \<in> \<B> \<Longrightarrow> x \<in> B"
- "\<And>B. B \<in> \<B> \<Longrightarrow> open B"
- "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> \<exists>B\<in>\<B>. B \<subseteq> S"
- by (rule first_countable_basisE) blast
- define \<A> where [abs_def]:
- "\<A> = (\<lambda>N. \<Inter>((\<lambda>n. from_nat_into \<B> n) ` N)) ` (Collect finite::nat set set)"
- then show "\<exists>\<A>. countable \<A> \<and> (\<forall>A. A \<in> \<A> \<longrightarrow> x \<in> A) \<and> (\<forall>A. A \<in> \<A> \<longrightarrow> open A) \<and>
- (\<forall>S. open S \<longrightarrow> x \<in> S \<longrightarrow> (\<exists>A\<in>\<A>. A \<subseteq> S)) \<and> (\<forall>A B. A \<in> \<A> \<longrightarrow> B \<in> \<A> \<longrightarrow> A \<inter> B \<in> \<A>)"
- proof (safe intro!: exI[where x=\<A>])
- show "countable \<A>"
- unfolding \<A>_def by (intro countable_image countable_Collect_finite)
- fix A
- assume "A \<in> \<A>"
- then show "x \<in> A" "open A"
- using \<B>(4)[OF open_UNIV] by (auto simp: \<A>_def intro: \<B> from_nat_into)
- next
- let ?int = "\<lambda>N. \<Inter>(from_nat_into \<B> ` N)"
- fix A B
- assume "A \<in> \<A>" "B \<in> \<A>"
- then obtain N M where "A = ?int N" "B = ?int M" "finite (N \<union> M)"
- by (auto simp: \<A>_def)
- then show "A \<inter> B \<in> \<A>"
- by (auto simp: \<A>_def intro!: image_eqI[where x="N \<union> M"])
- next
- fix S
- assume "open S" "x \<in> S"
- then obtain a where a: "a\<in>\<B>" "a \<subseteq> S" using \<B> by blast
- then show "\<exists>a\<in>\<A>. a \<subseteq> S" using a \<B>
- by (intro bexI[where x=a]) (auto simp: \<A>_def intro: image_eqI[where x="{to_nat_on \<B> a}"])
- qed
-qed
-
-lemma (in topological_space) first_countableI:
- assumes "countable \<A>"
- and 1: "\<And>A. A \<in> \<A> \<Longrightarrow> x \<in> A" "\<And>A. A \<in> \<A> \<Longrightarrow> open A"
- and 2: "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> \<exists>A\<in>\<A>. A \<subseteq> S"
- shows "\<exists>\<A>::nat \<Rightarrow> 'a set. (\<forall>i. x \<in> \<A> i \<and> open (\<A> i)) \<and> (\<forall>S. open S \<and> x \<in> S \<longrightarrow> (\<exists>i. \<A> i \<subseteq> S))"
-proof (safe intro!: exI[of _ "from_nat_into \<A>"])
- fix i
- have "\<A> \<noteq> {}" using 2[of UNIV] by auto
- show "x \<in> from_nat_into \<A> i" "open (from_nat_into \<A> i)"
- using range_from_nat_into_subset[OF \<open>\<A> \<noteq> {}\<close>] 1 by auto
-next
- fix S
- assume "open S" "x\<in>S" from 2[OF this]
- show "\<exists>i. from_nat_into \<A> i \<subseteq> S"
- using subset_range_from_nat_into[OF \<open>countable \<A>\<close>] by auto
-qed
-
-instance prod :: (first_countable_topology, first_countable_topology) first_countable_topology
-proof
- fix x :: "'a \<times> 'b"
- obtain \<A> where \<A>:
- "countable \<A>"
- "\<And>a. a \<in> \<A> \<Longrightarrow> fst x \<in> a"
- "\<And>a. a \<in> \<A> \<Longrightarrow> open a"
- "\<And>S. open S \<Longrightarrow> fst x \<in> S \<Longrightarrow> \<exists>a\<in>\<A>. a \<subseteq> S"
- by (rule first_countable_basisE[of "fst x"]) blast
- obtain B where B:
- "countable B"
- "\<And>a. a \<in> B \<Longrightarrow> snd x \<in> a"
- "\<And>a. a \<in> B \<Longrightarrow> open a"
- "\<And>S. open S \<Longrightarrow> snd x \<in> S \<Longrightarrow> \<exists>a\<in>B. a \<subseteq> S"
- by (rule first_countable_basisE[of "snd x"]) blast
- show "\<exists>\<A>::nat \<Rightarrow> ('a \<times> 'b) set.
- (\<forall>i. x \<in> \<A> i \<and> open (\<A> i)) \<and> (\<forall>S. open S \<and> x \<in> S \<longrightarrow> (\<exists>i. \<A> i \<subseteq> S))"
- proof (rule first_countableI[of "(\<lambda>(a, b). a \<times> b) ` (\<A> \<times> B)"], safe)
- fix a b
- assume x: "a \<in> \<A>" "b \<in> B"
- show "x \<in> a \<times> b"
- by (simp add: \<A>(2) B(2) mem_Times_iff x)
- show "open (a \<times> b)"
- by (simp add: \<A>(3) B(3) open_Times x)
- next
- fix S
- assume "open S" "x \<in> S"
- then obtain a' b' where a'b': "open a'" "open b'" "x \<in> a' \<times> b'" "a' \<times> b' \<subseteq> S"
- by (rule open_prod_elim)
- moreover
- from a'b' \<A>(4)[of a'] B(4)[of b']
- obtain a b where "a \<in> \<A>" "a \<subseteq> a'" "b \<in> B" "b \<subseteq> b'"
- by auto
- ultimately
- show "\<exists>a\<in>(\<lambda>(a, b). a \<times> b) ` (\<A> \<times> B). a \<subseteq> S"
- by (auto intro!: bexI[of _ "a \<times> b"] bexI[of _ a] bexI[of _ b])
- qed (simp add: \<A> B)
-qed
-
-class second_countable_topology = topological_space +
- assumes ex_countable_subbasis:
- "\<exists>B::'a::topological_space set set. countable B \<and> open = generate_topology B"
+ imports
+ Elementary_Topology
+ Linear_Algebra
+ Norm_Arith
begin
-lemma ex_countable_basis: "\<exists>B::'a set set. countable B \<and> topological_basis B"
-proof -
- from ex_countable_subbasis obtain B where B: "countable B" "open = generate_topology B"
- by blast
- let ?B = "Inter ` {b. finite b \<and> b \<subseteq> B }"
-
- show ?thesis
- proof (intro exI conjI)
- show "countable ?B"
- by (intro countable_image countable_Collect_finite_subset B)
- {
- fix S
- assume "open S"
- then have "\<exists>B'\<subseteq>{b. finite b \<and> b \<subseteq> B}. (\<Union>b\<in>B'. \<Inter>b) = S"
- unfolding B
- proof induct
- case UNIV
- show ?case by (intro exI[of _ "{{}}"]) simp
- next
- case (Int a b)
- then obtain x y where x: "a = \<Union>(Inter ` x)" "\<And>i. i \<in> x \<Longrightarrow> finite i \<and> i \<subseteq> B"
- and y: "b = \<Union>(Inter ` y)" "\<And>i. i \<in> y \<Longrightarrow> finite i \<and> i \<subseteq> B"
- by blast
- show ?case
- unfolding x y Int_UN_distrib2
- by (intro exI[of _ "{i \<union> j| i j. i \<in> x \<and> j \<in> y}"]) (auto dest: x(2) y(2))
- next
- case (UN K)
- then have "\<forall>k\<in>K. \<exists>B'\<subseteq>{b. finite b \<and> b \<subseteq> B}. \<Union> (Inter ` B') = k" by auto
- then obtain k where
- "\<forall>ka\<in>K. k ka \<subseteq> {b. finite b \<and> b \<subseteq> B} \<and> \<Union>(Inter ` (k ka)) = ka"
- unfolding bchoice_iff ..
- then show "\<exists>B'\<subseteq>{b. finite b \<and> b \<subseteq> B}. \<Union> (Inter ` B') = \<Union>K"
- by (intro exI[of _ "\<Union>(k ` K)"]) auto
- next
- case (Basis S)
- then show ?case
- by (intro exI[of _ "{{S}}"]) auto
- qed
- then have "(\<exists>B'\<subseteq>Inter ` {b. finite b \<and> b \<subseteq> B}. \<Union>B' = S)"
- unfolding subset_image_iff by blast }
- then show "topological_basis ?B"
- unfolding topological_space_class.topological_basis_def
- by (safe intro!: topological_space_class.open_Inter)
- (simp_all add: B generate_topology.Basis subset_eq)
- qed
-qed
-
-end
-
-sublocale second_countable_topology <
- countable_basis "SOME B. countable B \<and> topological_basis B"
- using someI_ex[OF ex_countable_basis]
- by unfold_locales safe
-
-instance prod :: (second_countable_topology, second_countable_topology) second_countable_topology
-proof
- obtain A :: "'a set set" where "countable A" "topological_basis A"
- using ex_countable_basis by auto
- moreover
- obtain B :: "'b set set" where "countable B" "topological_basis B"
- using ex_countable_basis by auto
- ultimately show "\<exists>B::('a \<times> 'b) set set. countable B \<and> open = generate_topology B"
- by (auto intro!: exI[of _ "(\<lambda>(a, b). a \<times> b) ` (A \<times> B)"] topological_basis_prod
- topological_basis_imp_subbasis)
-qed
-
-instance second_countable_topology \<subseteq> first_countable_topology
-proof
- fix x :: 'a
- define B :: "'a set set" where "B = (SOME B. countable B \<and> topological_basis B)"
- then have B: "countable B" "topological_basis B"
- using countable_basis is_basis
- by (auto simp: countable_basis is_basis)
- then show "\<exists>A::nat \<Rightarrow> 'a set.
- (\<forall>i. x \<in> A i \<and> open (A i)) \<and> (\<forall>S. open S \<and> x \<in> S \<longrightarrow> (\<exists>i. A i \<subseteq> S))"
- by (intro first_countableI[of "{b\<in>B. x \<in> b}"])
- (fastforce simp: topological_space_class.topological_basis_def)+
-qed
-
-instance nat :: second_countable_topology
-proof
- show "\<exists>B::nat set set. countable B \<and> open = generate_topology B"
- by (intro exI[of _ "range lessThan \<union> range greaterThan"]) (auto simp: open_nat_def)
-qed
-
-lemma countable_separating_set_linorder1:
- shows "\<exists>B::('a::{linorder_topology, second_countable_topology} set). countable B \<and> (\<forall>x y. x < y \<longrightarrow> (\<exists>b \<in> B. x < b \<and> b \<le> y))"
-proof -
- obtain A::"'a set set" where "countable A" "topological_basis A" using ex_countable_basis by auto
- define B1 where "B1 = {(LEAST x. x \<in> U)| U. U \<in> A}"
- then have "countable B1" using \<open>countable A\<close> by (simp add: Setcompr_eq_image)
- define B2 where "B2 = {(SOME x. x \<in> U)| U. U \<in> A}"
- then have "countable B2" using \<open>countable A\<close> by (simp add: Setcompr_eq_image)
- have "\<exists>b \<in> B1 \<union> B2. x < b \<and> b \<le> y" if "x < y" for x y
- proof (cases)
- assume "\<exists>z. x < z \<and> z < y"
- then obtain z where z: "x < z \<and> z < y" by auto
- define U where "U = {x<..<y}"
- then have "open U" by simp
- moreover have "z \<in> U" using z U_def by simp
- ultimately obtain V where "V \<in> A" "z \<in> V" "V \<subseteq> U" using topological_basisE[OF \<open>topological_basis A\<close>] by auto
- define w where "w = (SOME x. x \<in> V)"
- then have "w \<in> V" using \<open>z \<in> V\<close> by (metis someI2)
- then have "x < w \<and> w \<le> y" using \<open>w \<in> V\<close> \<open>V \<subseteq> U\<close> U_def by fastforce
- moreover have "w \<in> B1 \<union> B2" using w_def B2_def \<open>V \<in> A\<close> by auto
- ultimately show ?thesis by auto
- next
- assume "\<not>(\<exists>z. x < z \<and> z < y)"
- then have *: "\<And>z. z > x \<Longrightarrow> z \<ge> y" by auto
- define U where "U = {x<..}"
- then have "open U" by simp
- moreover have "y \<in> U" using \<open>x < y\<close> U_def by simp
- ultimately obtain "V" where "V \<in> A" "y \<in> V" "V \<subseteq> U" using topological_basisE[OF \<open>topological_basis A\<close>] by auto
- have "U = {y..}" unfolding U_def using * \<open>x < y\<close> by auto
- then have "V \<subseteq> {y..}" using \<open>V \<subseteq> U\<close> by simp
- then have "(LEAST w. w \<in> V) = y" using \<open>y \<in> V\<close> by (meson Least_equality atLeast_iff subsetCE)
- then have "y \<in> B1 \<union> B2" using \<open>V \<in> A\<close> B1_def by auto
- moreover have "x < y \<and> y \<le> y" using \<open>x < y\<close> by simp
- ultimately show ?thesis by auto
- qed
- moreover have "countable (B1 \<union> B2)" using \<open>countable B1\<close> \<open>countable B2\<close> by simp
- ultimately show ?thesis by auto
-qed
-
-lemma countable_separating_set_linorder2:
- shows "\<exists>B::('a::{linorder_topology, second_countable_topology} set). countable B \<and> (\<forall>x y. x < y \<longrightarrow> (\<exists>b \<in> B. x \<le> b \<and> b < y))"
-proof -
- obtain A::"'a set set" where "countable A" "topological_basis A" using ex_countable_basis by auto
- define B1 where "B1 = {(GREATEST x. x \<in> U) | U. U \<in> A}"
- then have "countable B1" using \<open>countable A\<close> by (simp add: Setcompr_eq_image)
- define B2 where "B2 = {(SOME x. x \<in> U)| U. U \<in> A}"
- then have "countable B2" using \<open>countable A\<close> by (simp add: Setcompr_eq_image)
- have "\<exists>b \<in> B1 \<union> B2. x \<le> b \<and> b < y" if "x < y" for x y
- proof (cases)
- assume "\<exists>z. x < z \<and> z < y"
- then obtain z where z: "x < z \<and> z < y" by auto
- define U where "U = {x<..<y}"
- then have "open U" by simp
- moreover have "z \<in> U" using z U_def by simp
- ultimately obtain "V" where "V \<in> A" "z \<in> V" "V \<subseteq> U" using topological_basisE[OF \<open>topological_basis A\<close>] by auto
- define w where "w = (SOME x. x \<in> V)"
- then have "w \<in> V" using \<open>z \<in> V\<close> by (metis someI2)
- then have "x \<le> w \<and> w < y" using \<open>w \<in> V\<close> \<open>V \<subseteq> U\<close> U_def by fastforce
- moreover have "w \<in> B1 \<union> B2" using w_def B2_def \<open>V \<in> A\<close> by auto
- ultimately show ?thesis by auto
- next
- assume "\<not>(\<exists>z. x < z \<and> z < y)"
- then have *: "\<And>z. z < y \<Longrightarrow> z \<le> x" using leI by blast
- define U where "U = {..<y}"
- then have "open U" by simp
- moreover have "x \<in> U" using \<open>x < y\<close> U_def by simp
- ultimately obtain "V" where "V \<in> A" "x \<in> V" "V \<subseteq> U" using topological_basisE[OF \<open>topological_basis A\<close>] by auto
- have "U = {..x}" unfolding U_def using * \<open>x < y\<close> by auto
- then have "V \<subseteq> {..x}" using \<open>V \<subseteq> U\<close> by simp
- then have "(GREATEST x. x \<in> V) = x" using \<open>x \<in> V\<close> by (meson Greatest_equality atMost_iff subsetCE)
- then have "x \<in> B1 \<union> B2" using \<open>V \<in> A\<close> B1_def by auto
- moreover have "x \<le> x \<and> x < y" using \<open>x < y\<close> by simp
- ultimately show ?thesis by auto
- qed
- moreover have "countable (B1 \<union> B2)" using \<open>countable B1\<close> \<open>countable B2\<close> by simp
- ultimately show ?thesis by auto
-qed
-
-lemma countable_separating_set_dense_linorder:
- shows "\<exists>B::('a::{linorder_topology, dense_linorder, second_countable_topology} set). countable B \<and> (\<forall>x y. x < y \<longrightarrow> (\<exists>b \<in> B. x < b \<and> b < y))"
-proof -
- obtain B::"'a set" where B: "countable B" "\<And>x y. x < y \<Longrightarrow> (\<exists>b \<in> B. x < b \<and> b \<le> y)"
- using countable_separating_set_linorder1 by auto
- have "\<exists>b \<in> B. x < b \<and> b < y" if "x < y" for x y
- proof -
- obtain z where "x < z" "z < y" using \<open>x < y\<close> dense by blast
- then obtain b where "b \<in> B" "x < b \<and> b \<le> z" using B(2) by auto
- then have "x < b \<and> b < y" using \<open>z < y\<close> by auto
- then show ?thesis using \<open>b \<in> B\<close> by auto
- qed
- then show ?thesis using B(1) by auto
-qed
-
-subsection%important \<open>Polish spaces\<close>
-
-text \<open>Textbooks define Polish spaces as completely metrizable.
- We assume the topology to be complete for a given metric.\<close>
-
-class polish_space = complete_space + second_countable_topology
-
-subsection \<open>General notion of a topology as a value\<close>
-
-definition%important "istopology L \<longleftrightarrow>
- L {} \<and> (\<forall>S T. L S \<longrightarrow> L T \<longrightarrow> L (S \<inter> T)) \<and> (\<forall>K. Ball K L \<longrightarrow> L (\<Union>K))"
-
-typedef%important 'a topology = "{L::('a set) \<Rightarrow> bool. istopology L}"
- morphisms "openin" "topology"
- unfolding istopology_def by blast
-
-lemma istopology_openin[intro]: "istopology(openin U)"
- using openin[of U] by blast
-
-lemma topology_inverse': "istopology U \<Longrightarrow> openin (topology U) = U"
- using topology_inverse[unfolded mem_Collect_eq] .
-
-lemma topology_inverse_iff: "istopology U \<longleftrightarrow> openin (topology U) = U"
- using topology_inverse[of U] istopology_openin[of "topology U"] by auto
-
-lemma topology_eq: "T1 = T2 \<longleftrightarrow> (\<forall>S. openin T1 S \<longleftrightarrow> openin T2 S)"
-proof
- assume "T1 = T2"
- then show "\<forall>S. openin T1 S \<longleftrightarrow> openin T2 S" by simp
-next
- assume H: "\<forall>S. openin T1 S \<longleftrightarrow> openin T2 S"
- then have "openin T1 = openin T2" by (simp add: fun_eq_iff)
- then have "topology (openin T1) = topology (openin T2)" by simp
- then show "T1 = T2" unfolding openin_inverse .
-qed
-
-
-text\<open>The "universe": the union of all sets in the topology.\<close>
-definition "topspace T = \<Union>{S. openin T S}"
-
-subsubsection \<open>Main properties of open sets\<close>
-
-proposition openin_clauses:
- fixes U :: "'a topology"
- shows
- "openin U {}"
- "\<And>S T. openin U S \<Longrightarrow> openin U T \<Longrightarrow> openin U (S\<inter>T)"
- "\<And>K. (\<forall>S \<in> K. openin U S) \<Longrightarrow> openin U (\<Union>K)"
- using openin[of U] unfolding istopology_def mem_Collect_eq by fast+
-
-lemma openin_subset[intro]: "openin U S \<Longrightarrow> S \<subseteq> topspace U"
- unfolding topspace_def by blast
-
-lemma openin_empty[simp]: "openin U {}"
- by (rule openin_clauses)
-
-lemma openin_Int[intro]: "openin U S \<Longrightarrow> openin U T \<Longrightarrow> openin U (S \<inter> T)"
- by (rule openin_clauses)
-
-lemma openin_Union[intro]: "(\<And>S. S \<in> K \<Longrightarrow> openin U S) \<Longrightarrow> openin U (\<Union>K)"
- using openin_clauses by blast
-
-lemma openin_Un[intro]: "openin U S \<Longrightarrow> openin U T \<Longrightarrow> openin U (S \<union> T)"
- using openin_Union[of "{S,T}" U] by auto
-
-lemma openin_topspace[intro, simp]: "openin U (topspace U)"
- by (force simp: openin_Union topspace_def)
-
-lemma openin_subopen: "openin U S \<longleftrightarrow> (\<forall>x \<in> S. \<exists>T. openin U T \<and> x \<in> T \<and> T \<subseteq> S)"
- (is "?lhs \<longleftrightarrow> ?rhs")
-proof
- assume ?lhs
- then show ?rhs by auto
-next
- assume H: ?rhs
- let ?t = "\<Union>{T. openin U T \<and> T \<subseteq> S}"
- have "openin U ?t" by (force simp: openin_Union)
- also have "?t = S" using H by auto
- finally show "openin U S" .
-qed
-
-lemma openin_INT [intro]:
- assumes "finite I"
- "\<And>i. i \<in> I \<Longrightarrow> openin T (U i)"
- shows "openin T ((\<Inter>i \<in> I. U i) \<inter> topspace T)"
-using assms by (induct, auto simp: inf_sup_aci(2) openin_Int)
-
-lemma openin_INT2 [intro]:
- assumes "finite I" "I \<noteq> {}"
- "\<And>i. i \<in> I \<Longrightarrow> openin T (U i)"
- shows "openin T (\<Inter>i \<in> I. U i)"
-proof -
- have "(\<Inter>i \<in> I. U i) \<subseteq> topspace T"
- using \<open>I \<noteq> {}\<close> openin_subset[OF assms(3)] by auto
- then show ?thesis
- using openin_INT[of _ _ U, OF assms(1) assms(3)] by (simp add: inf.absorb2 inf_commute)
-qed
-
-lemma openin_Inter [intro]:
- assumes "finite \<F>" "\<F> \<noteq> {}" "\<And>X. X \<in> \<F> \<Longrightarrow> openin T X" shows "openin T (\<Inter>\<F>)"
- by (metis (full_types) assms openin_INT2 image_ident)
-
-lemma openin_Int_Inter:
- assumes "finite \<F>" "openin T U" "\<And>X. X \<in> \<F> \<Longrightarrow> openin T X" shows "openin T (U \<inter> \<Inter>\<F>)"
- using openin_Inter [of "insert U \<F>"] assms by auto
-
-
-subsubsection \<open>Closed sets\<close>
-
-definition%important "closedin U S \<longleftrightarrow> S \<subseteq> topspace U \<and> openin U (topspace U - S)"
-
-lemma closedin_subset: "closedin U S \<Longrightarrow> S \<subseteq> topspace U"
- by (metis closedin_def)
-
-lemma closedin_empty[simp]: "closedin U {}"
- by (simp add: closedin_def)
-
-lemma closedin_topspace[intro, simp]: "closedin U (topspace U)"
- by (simp add: closedin_def)
-
-lemma closedin_Un[intro]: "closedin U S \<Longrightarrow> closedin U T \<Longrightarrow> closedin U (S \<union> T)"
- by (auto simp: Diff_Un closedin_def)
-
-lemma Diff_Inter[intro]: "A - \<Inter>S = \<Union>{A - s|s. s\<in>S}"
- by auto
-
-lemma closedin_Union:
- assumes "finite S" "\<And>T. T \<in> S \<Longrightarrow> closedin U T"
- shows "closedin U (\<Union>S)"
- using assms by induction auto
-
-lemma closedin_Inter[intro]:
- assumes Ke: "K \<noteq> {}"
- and Kc: "\<And>S. S \<in>K \<Longrightarrow> closedin U S"
- shows "closedin U (\<Inter>K)"
- using Ke Kc unfolding closedin_def Diff_Inter by auto
-
-lemma closedin_INT[intro]:
- assumes "A \<noteq> {}" "\<And>x. x \<in> A \<Longrightarrow> closedin U (B x)"
- shows "closedin U (\<Inter>x\<in>A. B x)"
- apply (rule closedin_Inter)
- using assms
- apply auto
- done
-
-lemma closedin_Int[intro]: "closedin U S \<Longrightarrow> closedin U T \<Longrightarrow> closedin U (S \<inter> T)"
- using closedin_Inter[of "{S,T}" U] by auto
-
-lemma openin_closedin_eq: "openin U S \<longleftrightarrow> S \<subseteq> topspace U \<and> closedin U (topspace U - S)"
- apply (auto simp: closedin_def Diff_Diff_Int inf_absorb2)
- apply (metis openin_subset subset_eq)
- done
-
-lemma openin_closedin: "S \<subseteq> topspace U \<Longrightarrow> (openin U S \<longleftrightarrow> closedin U (topspace U - S))"
- by (simp add: openin_closedin_eq)
-
-lemma openin_diff[intro]:
- assumes oS: "openin U S"
- and cT: "closedin U T"
- shows "openin U (S - T)"
-proof -
- have "S - T = S \<inter> (topspace U - T)" using openin_subset[of U S] oS cT
- by (auto simp: topspace_def openin_subset)
- then show ?thesis using oS cT
- by (auto simp: closedin_def)
-qed
-
-lemma closedin_diff[intro]:
- assumes oS: "closedin U S"
- and cT: "openin U T"
- shows "closedin U (S - T)"
-proof -
- have "S - T = S \<inter> (topspace U - T)"
- using closedin_subset[of U S] oS cT by (auto simp: topspace_def)
- then show ?thesis
- using oS cT by (auto simp: openin_closedin_eq)
-qed
-
-
-subsection\<open>The discrete topology\<close>
-
-definition discrete_topology where "discrete_topology U \<equiv> topology (\<lambda>S. S \<subseteq> U)"
-
-lemma openin_discrete_topology [simp]: "openin (discrete_topology U) S \<longleftrightarrow> S \<subseteq> U"
-proof -
- have "istopology (\<lambda>S. S \<subseteq> U)"
- by (auto simp: istopology_def)
- then show ?thesis
- by (simp add: discrete_topology_def topology_inverse')
-qed
-
-lemma topspace_discrete_topology [simp]: "topspace(discrete_topology U) = U"
- by (meson openin_discrete_topology openin_subset openin_topspace order_refl subset_antisym)
-
-lemma closedin_discrete_topology [simp]: "closedin (discrete_topology U) S \<longleftrightarrow> S \<subseteq> U"
- by (simp add: closedin_def)
-
-lemma discrete_topology_unique:
- "discrete_topology U = X \<longleftrightarrow> topspace X = U \<and> (\<forall>x \<in> U. openin X {x})" (is "?lhs = ?rhs")
-proof
- assume R: ?rhs
- then have "openin X S" if "S \<subseteq> U" for S
- using openin_subopen subsetD that by fastforce
- moreover have "x \<in> topspace X" if "openin X S" and "x \<in> S" for x S
- using openin_subset that by blast
- ultimately
- show ?lhs
- using R by (auto simp: topology_eq)
-qed auto
-
-lemma discrete_topology_unique_alt:
- "discrete_topology U = X \<longleftrightarrow> topspace X \<subseteq> U \<and> (\<forall>x \<in> U. openin X {x})"
- using openin_subset
- by (auto simp: discrete_topology_unique)
-
-lemma subtopology_eq_discrete_topology_empty:
- "X = discrete_topology {} \<longleftrightarrow> topspace X = {}"
- using discrete_topology_unique [of "{}" X] by auto
-
-lemma subtopology_eq_discrete_topology_sing:
- "X = discrete_topology {a} \<longleftrightarrow> topspace X = {a}"
- by (metis discrete_topology_unique openin_topspace singletonD)
-
-
-subsection \<open>Subspace topology\<close>
-
-definition%important "subtopology U V = topology (\<lambda>T. \<exists>S. T = S \<inter> V \<and> openin U S)"
-
-lemma istopology_subtopology: "istopology (\<lambda>T. \<exists>S. T = S \<inter> V \<and> openin U S)"
- (is "istopology ?L")
-proof -
- have "?L {}" by blast
- {
- fix A B
- assume A: "?L A" and B: "?L B"
- from A B obtain Sa and Sb where Sa: "openin U Sa" "A = Sa \<inter> V" and Sb: "openin U Sb" "B = Sb \<inter> V"
- by blast
- have "A \<inter> B = (Sa \<inter> Sb) \<inter> V" "openin U (Sa \<inter> Sb)"
- using Sa Sb by blast+
- then have "?L (A \<inter> B)" by blast
- }
- moreover
- {
- fix K
- assume K: "K \<subseteq> Collect ?L"
- have th0: "Collect ?L = (\<lambda>S. S \<inter> V) ` Collect (openin U)"
- by blast
- from K[unfolded th0 subset_image_iff]
- obtain Sk where Sk: "Sk \<subseteq> Collect (openin U)" "K = (\<lambda>S. S \<inter> V) ` Sk"
- by blast
- have "\<Union>K = (\<Union>Sk) \<inter> V"
- using Sk by auto
- moreover have "openin U (\<Union>Sk)"
- using Sk by (auto simp: subset_eq)
- ultimately have "?L (\<Union>K)" by blast
- }
- ultimately show ?thesis
- unfolding subset_eq mem_Collect_eq istopology_def by auto
-qed
-
-lemma openin_subtopology: "openin (subtopology U V) S \<longleftrightarrow> (\<exists>T. openin U T \<and> S = T \<inter> V)"
- unfolding subtopology_def topology_inverse'[OF istopology_subtopology]
- by auto
-
-lemma openin_subtopology_Int:
- "openin X S \<Longrightarrow> openin (subtopology X T) (S \<inter> T)"
- using openin_subtopology by auto
-
-lemma openin_subtopology_Int2:
- "openin X T \<Longrightarrow> openin (subtopology X S) (S \<inter> T)"
- using openin_subtopology by auto
-
-lemma openin_subtopology_diff_closed:
- "\<lbrakk>S \<subseteq> topspace X; closedin X T\<rbrakk> \<Longrightarrow> openin (subtopology X S) (S - T)"
- unfolding closedin_def openin_subtopology
- by (rule_tac x="topspace X - T" in exI) auto
-
-lemma openin_relative_to: "(openin X relative_to S) = openin (subtopology X S)"
- by (force simp: relative_to_def openin_subtopology)
-
-lemma topspace_subtopology: "topspace (subtopology U V) = topspace U \<inter> V"
- by (auto simp: topspace_def openin_subtopology)
-
-lemma closedin_subtopology: "closedin (subtopology U V) S \<longleftrightarrow> (\<exists>T. closedin U T \<and> S = T \<inter> V)"
- unfolding closedin_def topspace_subtopology
- by (auto simp: openin_subtopology)
-
-lemma openin_subtopology_refl: "openin (subtopology U V) V \<longleftrightarrow> V \<subseteq> topspace U"
- unfolding openin_subtopology
- by auto (metis IntD1 in_mono openin_subset)
-
-lemma subtopology_subtopology:
- "subtopology (subtopology X S) T = subtopology X (S \<inter> T)"
-proof -
- have eq: "\<And>T'. (\<exists>S'. T' = S' \<inter> T \<and> (\<exists>T. openin X T \<and> S' = T \<inter> S)) = (\<exists>Sa. T' = Sa \<inter> (S \<inter> T) \<and> openin X Sa)"
- by (metis inf_assoc)
- have "subtopology (subtopology X S) T = topology (\<lambda>Ta. \<exists>Sa. Ta = Sa \<inter> T \<and> openin (subtopology X S) Sa)"
- by (simp add: subtopology_def)
- also have "\<dots> = subtopology X (S \<inter> T)"
- by (simp add: openin_subtopology eq) (simp add: subtopology_def)
- finally show ?thesis .
-qed
-
-lemma openin_subtopology_alt:
- "openin (subtopology X U) S \<longleftrightarrow> S \<in> (\<lambda>T. U \<inter> T) ` Collect (openin X)"
- by (simp add: image_iff inf_commute openin_subtopology)
-
-lemma closedin_subtopology_alt:
- "closedin (subtopology X U) S \<longleftrightarrow> S \<in> (\<lambda>T. U \<inter> T) ` Collect (closedin X)"
- by (simp add: image_iff inf_commute closedin_subtopology)
-
-lemma subtopology_superset:
- assumes UV: "topspace U \<subseteq> V"
- shows "subtopology U V = U"
-proof -
- {
- fix S
- {
- fix T
- assume T: "openin U T" "S = T \<inter> V"
- from T openin_subset[OF T(1)] UV have eq: "S = T"
- by blast
- have "openin U S"
- unfolding eq using T by blast
- }
- moreover
- {
- assume S: "openin U S"
- then have "\<exists>T. openin U T \<and> S = T \<inter> V"
- using openin_subset[OF S] UV by auto
- }
- ultimately have "(\<exists>T. openin U T \<and> S = T \<inter> V) \<longleftrightarrow> openin U S"
- by blast
- }
- then show ?thesis
- unfolding topology_eq openin_subtopology by blast
-qed
-
-lemma subtopology_topspace[simp]: "subtopology U (topspace U) = U"
- by (simp add: subtopology_superset)
-
-lemma subtopology_UNIV[simp]: "subtopology U UNIV = U"
- by (simp add: subtopology_superset)
-
-lemma openin_subtopology_empty:
- "openin (subtopology U {}) S \<longleftrightarrow> S = {}"
-by (metis Int_empty_right openin_empty openin_subtopology)
-
-lemma closedin_subtopology_empty:
- "closedin (subtopology U {}) S \<longleftrightarrow> S = {}"
-by (metis Int_empty_right closedin_empty closedin_subtopology)
-
-lemma closedin_subtopology_refl [simp]:
- "closedin (subtopology U X) X \<longleftrightarrow> X \<subseteq> topspace U"
-by (metis closedin_def closedin_topspace inf.absorb_iff2 le_inf_iff topspace_subtopology)
-
-lemma closedin_topspace_empty: "topspace T = {} \<Longrightarrow> (closedin T S \<longleftrightarrow> S = {})"
- by (simp add: closedin_def)
-
-lemma openin_imp_subset:
- "openin (subtopology U S) T \<Longrightarrow> T \<subseteq> S"
-by (metis Int_iff openin_subtopology subsetI)
-
-lemma closedin_imp_subset:
- "closedin (subtopology U S) T \<Longrightarrow> T \<subseteq> S"
-by (simp add: closedin_def topspace_subtopology)
-
-lemma openin_open_subtopology:
- "openin X S \<Longrightarrow> openin (subtopology X S) T \<longleftrightarrow> openin X T \<and> T \<subseteq> S"
- by (metis inf.orderE openin_Int openin_imp_subset openin_subtopology)
-
-lemma closedin_closed_subtopology:
- "closedin X S \<Longrightarrow> (closedin (subtopology X S) T \<longleftrightarrow> closedin X T \<and> T \<subseteq> S)"
- by (metis closedin_Int closedin_imp_subset closedin_subtopology inf.orderE)
-
-lemma openin_subtopology_Un:
- "\<lbrakk>openin (subtopology X T) S; openin (subtopology X U) S\<rbrakk>
- \<Longrightarrow> openin (subtopology X (T \<union> U)) S"
-by (simp add: openin_subtopology) blast
-
-lemma closedin_subtopology_Un:
- "\<lbrakk>closedin (subtopology X T) S; closedin (subtopology X U) S\<rbrakk>
- \<Longrightarrow> closedin (subtopology X (T \<union> U)) S"
-by (simp add: closedin_subtopology) blast
-
-
-subsection \<open>The standard Euclidean topology\<close>
-
-definition%important euclidean :: "'a::topological_space topology"
- where "euclidean = topology open"
-
-lemma open_openin: "open S \<longleftrightarrow> openin euclidean S"
- unfolding euclidean_def
- apply (rule cong[where x=S and y=S])
- apply (rule topology_inverse[symmetric])
- apply (auto simp: istopology_def)
- done
-
-declare open_openin [symmetric, simp]
-
-lemma topspace_euclidean [simp]: "topspace euclidean = UNIV"
- by (force simp: topspace_def)
-
-lemma topspace_euclidean_subtopology[simp]: "topspace (subtopology euclidean S) = S"
- by (simp add: topspace_subtopology)
-
-lemma closed_closedin: "closed S \<longleftrightarrow> closedin euclidean S"
- by (simp add: closed_def closedin_def Compl_eq_Diff_UNIV)
-
-declare closed_closedin [symmetric, simp]
-
-lemma open_subopen: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> S)"
- using openI by auto
-
-lemma openin_subtopology_self [simp]: "openin (subtopology euclidean S) S"
- by (metis openin_topspace topspace_euclidean_subtopology)
-
-subsubsection\<open>The most basic facts about the usual topology and metric on R\<close>
-
-abbreviation euclideanreal :: "real topology"
- where "euclideanreal \<equiv> topology open"
-
-lemma real_openin [simp]: "openin euclideanreal S = open S"
- by (simp add: euclidean_def open_openin)
-
-lemma topspace_euclideanreal [simp]: "topspace euclideanreal = UNIV"
- using openin_subset open_UNIV real_openin by blast
-
-lemma topspace_euclideanreal_subtopology [simp]:
- "topspace (subtopology euclideanreal S) = S"
- by (simp add: topspace_subtopology)
-
-lemma real_closedin [simp]: "closedin euclideanreal S = closed S"
- by (simp add: closed_closedin euclidean_def)
-
-subsection \<open>Basic "localization" results are handy for connectedness.\<close>
-
-lemma openin_open: "openin (subtopology euclidean U) S \<longleftrightarrow> (\<exists>T. open T \<and> (S = U \<inter> T))"
- by (auto simp: openin_subtopology)
-
-lemma openin_Int_open:
- "\<lbrakk>openin (subtopology euclidean U) S; open T\<rbrakk>
- \<Longrightarrow> openin (subtopology euclidean U) (S \<inter> T)"
-by (metis open_Int Int_assoc openin_open)
-
-lemma openin_open_Int[intro]: "open S \<Longrightarrow> openin (subtopology euclidean U) (U \<inter> S)"
- by (auto simp: openin_open)
-
-lemma open_openin_trans[trans]:
- "open S \<Longrightarrow> open T \<Longrightarrow> T \<subseteq> S \<Longrightarrow> openin (subtopology euclidean S) T"
- by (metis Int_absorb1 openin_open_Int)
-
-lemma open_subset: "S \<subseteq> T \<Longrightarrow> open S \<Longrightarrow> openin (subtopology euclidean T) S"
- by (auto simp: openin_open)
-
-lemma closedin_closed: "closedin (subtopology euclidean U) S \<longleftrightarrow> (\<exists>T. closed T \<and> S = U \<inter> T)"
- by (simp add: closedin_subtopology Int_ac)
-
-lemma closedin_closed_Int: "closed S \<Longrightarrow> closedin (subtopology euclidean U) (U \<inter> S)"
- by (metis closedin_closed)
-
-lemma closed_subset: "S \<subseteq> T \<Longrightarrow> closed S \<Longrightarrow> closedin (subtopology euclidean T) S"
- by (auto simp: closedin_closed)
-
-lemma closedin_closed_subset:
- "\<lbrakk>closedin (subtopology euclidean U) V; T \<subseteq> U; S = V \<inter> T\<rbrakk>
- \<Longrightarrow> closedin (subtopology euclidean T) S"
- by (metis (no_types, lifting) Int_assoc Int_commute closedin_closed inf.orderE)
-
-lemma finite_imp_closedin:
- fixes S :: "'a::t1_space set"
- shows "\<lbrakk>finite S; S \<subseteq> T\<rbrakk> \<Longrightarrow> closedin (subtopology euclidean T) S"
- by (simp add: finite_imp_closed closed_subset)
-
-lemma closedin_singleton [simp]:
- fixes a :: "'a::t1_space"
- shows "closedin (subtopology euclidean U) {a} \<longleftrightarrow> a \<in> U"
-using closedin_subset by (force intro: closed_subset)
-
-lemma openin_euclidean_subtopology_iff:
- fixes S U :: "'a::metric_space set"
- shows "openin (subtopology euclidean U) S \<longleftrightarrow>
- S \<subseteq> U \<and> (\<forall>x\<in>S. \<exists>e>0. \<forall>x'\<in>U. dist x' x < e \<longrightarrow> x'\<in> S)"
- (is "?lhs \<longleftrightarrow> ?rhs")
-proof
- assume ?lhs
- then show ?rhs
- unfolding openin_open open_dist by blast
-next
- define T where "T = {x. \<exists>a\<in>S. \<exists>d>0. (\<forall>y\<in>U. dist y a < d \<longrightarrow> y \<in> S) \<and> dist x a < d}"
- have 1: "\<forall>x\<in>T. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> T"
- unfolding T_def
- apply clarsimp
- apply (rule_tac x="d - dist x a" in exI)
- apply (clarsimp simp add: less_diff_eq)
- by (metis dist_commute dist_triangle_lt)
- assume ?rhs then have 2: "S = U \<inter> T"
- unfolding T_def
- by auto (metis dist_self)
- from 1 2 show ?lhs
- unfolding openin_open open_dist by fast
-qed
-
-lemma connected_openin:
- "connected S \<longleftrightarrow>
- \<not>(\<exists>E1 E2. openin (subtopology euclidean S) E1 \<and>
- openin (subtopology euclidean S) E2 \<and>
- S \<subseteq> E1 \<union> E2 \<and> E1 \<inter> E2 = {} \<and> E1 \<noteq> {} \<and> E2 \<noteq> {})"
- apply (simp add: connected_def openin_open disjoint_iff_not_equal, safe)
- apply (simp_all, blast+) (* SLOW *)
- done
-
-lemma connected_openin_eq:
- "connected S \<longleftrightarrow>
- \<not>(\<exists>E1 E2. openin (subtopology euclidean S) E1 \<and>
- openin (subtopology euclidean S) E2 \<and>
- E1 \<union> E2 = S \<and> E1 \<inter> E2 = {} \<and>
- E1 \<noteq> {} \<and> E2 \<noteq> {})"
- apply (simp add: connected_openin, safe, blast)
- by (metis Int_lower1 Un_subset_iff openin_open subset_antisym)
-
-lemma connected_closedin:
- "connected S \<longleftrightarrow>
- (\<nexists>E1 E2.
- closedin (subtopology euclidean S) E1 \<and>
- closedin (subtopology euclidean S) E2 \<and>
- S \<subseteq> E1 \<union> E2 \<and> E1 \<inter> E2 = {} \<and> E1 \<noteq> {} \<and> E2 \<noteq> {})"
- (is "?lhs = ?rhs")
-proof
- assume ?lhs
- then show ?rhs
- by (auto simp add: connected_closed closedin_closed)
-next
- assume R: ?rhs
- then show ?lhs
- proof (clarsimp simp add: connected_closed closedin_closed)
- fix A B
- assume s_sub: "S \<subseteq> A \<union> B" "B \<inter> S \<noteq> {}"
- and disj: "A \<inter> B \<inter> S = {}"
- and cl: "closed A" "closed B"
- have "S \<inter> (A \<union> B) = S"
- using s_sub(1) by auto
- have "S - A = B \<inter> S"
- using Diff_subset_conv Un_Diff_Int disj s_sub(1) by auto
- then have "S \<inter> A = {}"
- by (metis Diff_Diff_Int Diff_disjoint Un_Diff_Int R cl closedin_closed_Int inf_commute order_refl s_sub(2))
- then show "A \<inter> S = {}"
- by blast
- qed
-qed
-
-lemma connected_closedin_eq:
- "connected S \<longleftrightarrow>
- \<not>(\<exists>E1 E2.
- closedin (subtopology euclidean S) E1 \<and>
- closedin (subtopology euclidean S) E2 \<and>
- E1 \<union> E2 = S \<and> E1 \<inter> E2 = {} \<and>
- E1 \<noteq> {} \<and> E2 \<noteq> {})"
- apply (simp add: connected_closedin, safe, blast)
- by (metis Int_lower1 Un_subset_iff closedin_closed subset_antisym)
-
-text \<open>These "transitivity" results are handy too\<close>
-
-lemma openin_trans[trans]:
- "openin (subtopology euclidean T) S \<Longrightarrow> openin (subtopology euclidean U) T \<Longrightarrow>
- openin (subtopology euclidean U) S"
- unfolding open_openin openin_open by blast
-
-lemma openin_open_trans: "openin (subtopology euclidean T) S \<Longrightarrow> open T \<Longrightarrow> open S"
- by (auto simp: openin_open intro: openin_trans)
-
-lemma closedin_trans[trans]:
- "closedin (subtopology euclidean T) S \<Longrightarrow> closedin (subtopology euclidean U) T \<Longrightarrow>
- closedin (subtopology euclidean U) S"
- by (auto simp: closedin_closed closed_Inter Int_assoc)
-
-lemma closedin_closed_trans: "closedin (subtopology euclidean T) S \<Longrightarrow> closed T \<Longrightarrow> closed S"
- by (auto simp: closedin_closed intro: closedin_trans)
-
-lemma openin_subtopology_Int_subset:
- "\<lbrakk>openin (subtopology euclidean u) (u \<inter> S); v \<subseteq> u\<rbrakk> \<Longrightarrow> openin (subtopology euclidean v) (v \<inter> S)"
- by (auto simp: openin_subtopology)
-
-lemma openin_open_eq: "open s \<Longrightarrow> (openin (subtopology euclidean s) t \<longleftrightarrow> open t \<and> t \<subseteq> s)"
- using open_subset openin_open_trans openin_subset by fastforce
-
-
-subsection \<open>Open and closed balls\<close>
-
-definition%important ball :: "'a::metric_space \<Rightarrow> real \<Rightarrow> 'a set"
- where "ball x e = {y. dist x y < e}"
-
-definition%important cball :: "'a::metric_space \<Rightarrow> real \<Rightarrow> 'a set"
- where "cball x e = {y. dist x y \<le> e}"
-
-definition%important sphere :: "'a::metric_space \<Rightarrow> real \<Rightarrow> 'a set"
- where "sphere x e = {y. dist x y = e}"
-
-lemma mem_ball [simp]: "y \<in> ball x e \<longleftrightarrow> dist x y < e"
- by (simp add: ball_def)
-
-lemma mem_cball [simp]: "y \<in> cball x e \<longleftrightarrow> dist x y \<le> e"
- by (simp add: cball_def)
-
-lemma mem_sphere [simp]: "y \<in> sphere x e \<longleftrightarrow> dist x y = e"
- by (simp add: sphere_def)
-
-lemma ball_trivial [simp]: "ball x 0 = {}"
- by (simp add: ball_def)
-
-lemma cball_trivial [simp]: "cball x 0 = {x}"
- by (simp add: cball_def)
-
-lemma sphere_trivial [simp]: "sphere x 0 = {x}"
- by (simp add: sphere_def)
-
-lemma mem_ball_0 [simp]: "x \<in> ball 0 e \<longleftrightarrow> norm x < e"
- for x :: "'a::real_normed_vector"
- by (simp add: dist_norm)
-
-lemma mem_cball_0 [simp]: "x \<in> cball 0 e \<longleftrightarrow> norm x \<le> e"
- for x :: "'a::real_normed_vector"
- by (simp add: dist_norm)
-
-lemma disjoint_ballI: "dist x y \<ge> r+s \<Longrightarrow> ball x r \<inter> ball y s = {}"
- using dist_triangle_less_add not_le by fastforce
-
-lemma disjoint_cballI: "dist x y > r + s \<Longrightarrow> cball x r \<inter> cball y s = {}"
- by (metis add_mono disjoint_iff_not_equal dist_triangle2 dual_order.trans leD mem_cball)
-
-lemma mem_sphere_0 [simp]: "x \<in> sphere 0 e \<longleftrightarrow> norm x = e"
- for x :: "'a::real_normed_vector"
- by (simp add: dist_norm)
-
-lemma sphere_empty [simp]: "r < 0 \<Longrightarrow> sphere a r = {}"
- for a :: "'a::metric_space"
- by auto
-
-lemma centre_in_ball [simp]: "x \<in> ball x e \<longleftrightarrow> 0 < e"
- by simp
-
-lemma centre_in_cball [simp]: "x \<in> cball x e \<longleftrightarrow> 0 \<le> e"
- by simp
-
-lemma ball_subset_cball [simp, intro]: "ball x e \<subseteq> cball x e"
- by (simp add: subset_eq)
-
-lemma mem_ball_imp_mem_cball: "x \<in> ball y e \<Longrightarrow> x \<in> cball y e"
- by (auto simp: mem_ball mem_cball)
-
-lemma sphere_cball [simp,intro]: "sphere z r \<subseteq> cball z r"
- by force
-
-lemma cball_diff_sphere: "cball a r - sphere a r = ball a r"
- by auto
-
-lemma subset_ball[intro]: "d \<le> e \<Longrightarrow> ball x d \<subseteq> ball x e"
- by (simp add: subset_eq)
-
-lemma subset_cball[intro]: "d \<le> e \<Longrightarrow> cball x d \<subseteq> cball x e"
- by (simp add: subset_eq)
-
-lemma mem_ball_leI: "x \<in> ball y e \<Longrightarrow> e \<le> f \<Longrightarrow> x \<in> ball y f"
- by (auto simp: mem_ball mem_cball)
-
-lemma mem_cball_leI: "x \<in> cball y e \<Longrightarrow> e \<le> f \<Longrightarrow> x \<in> cball y f"
- by (auto simp: mem_ball mem_cball)
-
-lemma cball_trans: "y \<in> cball z b \<Longrightarrow> x \<in> cball y a \<Longrightarrow> x \<in> cball z (b + a)"
- unfolding mem_cball
-proof -
- have "dist z x \<le> dist z y + dist y x"
- by (rule dist_triangle)
- also assume "dist z y \<le> b"
- also assume "dist y x \<le> a"
- finally show "dist z x \<le> b + a" by arith
-qed
-
-lemma ball_max_Un: "ball a (max r s) = ball a r \<union> ball a s"
- by (simp add: set_eq_iff) arith
-
-lemma ball_min_Int: "ball a (min r s) = ball a r \<inter> ball a s"
- by (simp add: set_eq_iff)
-
-lemma cball_max_Un: "cball a (max r s) = cball a r \<union> cball a s"
- by (simp add: set_eq_iff) arith
-
-lemma cball_min_Int: "cball a (min r s) = cball a r \<inter> cball a s"
- by (simp add: set_eq_iff)
-
-lemma cball_diff_eq_sphere: "cball a r - ball a r = sphere a r"
- by (auto simp: cball_def ball_def dist_commute)
-
-lemma image_add_ball [simp]:
- fixes a :: "'a::real_normed_vector"
- shows "(+) b ` ball a r = ball (a+b) r"
-apply (intro equalityI subsetI)
-apply (force simp: dist_norm)
-apply (rule_tac x="x-b" in image_eqI)
-apply (auto simp: dist_norm algebra_simps)
-done
-
-lemma image_add_cball [simp]:
- fixes a :: "'a::real_normed_vector"
- shows "(+) b ` cball a r = cball (a+b) r"
-apply (intro equalityI subsetI)
-apply (force simp: dist_norm)
-apply (rule_tac x="x-b" in image_eqI)
-apply (auto simp: dist_norm algebra_simps)
-done
-
-lemma open_ball [intro, simp]: "open (ball x e)"
-proof -
- have "open (dist x -` {..<e})"
- by (intro open_vimage open_lessThan continuous_intros)
- also have "dist x -` {..<e} = ball x e"
- by auto
- finally show ?thesis .
-qed
-
-lemma open_contains_ball: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. ball x e \<subseteq> S)"
- by (simp add: open_dist subset_eq mem_ball Ball_def dist_commute)
-
-lemma openI [intro?]: "(\<And>x. x\<in>S \<Longrightarrow> \<exists>e>0. ball x e \<subseteq> S) \<Longrightarrow> open S"
- by (auto simp: open_contains_ball)
-
-lemma openE[elim?]:
- assumes "open S" "x\<in>S"
- obtains e where "e>0" "ball x e \<subseteq> S"
- using assms unfolding open_contains_ball by auto
-
-lemma open_contains_ball_eq: "open S \<Longrightarrow> x\<in>S \<longleftrightarrow> (\<exists>e>0. ball x e \<subseteq> S)"
- by (metis open_contains_ball subset_eq centre_in_ball)
-
-lemma openin_contains_ball:
- "openin (subtopology euclidean t) s \<longleftrightarrow>
- s \<subseteq> t \<and> (\<forall>x \<in> s. \<exists>e. 0 < e \<and> ball x e \<inter> t \<subseteq> s)"
- (is "?lhs = ?rhs")
-proof
- assume ?lhs
- then show ?rhs
- apply (simp add: openin_open)
- apply (metis Int_commute Int_mono inf.cobounded2 open_contains_ball order_refl subsetCE)
- done
-next
- assume ?rhs
- then show ?lhs
- apply (simp add: openin_euclidean_subtopology_iff)
- by (metis (no_types) Int_iff dist_commute inf.absorb_iff2 mem_ball)
-qed
-
-lemma openin_contains_cball:
- "openin (subtopology euclidean t) s \<longleftrightarrow>
- s \<subseteq> t \<and>
- (\<forall>x \<in> s. \<exists>e. 0 < e \<and> cball x e \<inter> t \<subseteq> s)"
-apply (simp add: openin_contains_ball)
-apply (rule iffI)
-apply (auto dest!: bspec)
-apply (rule_tac x="e/2" in exI, force+)
-done
-
-lemma ball_eq_empty[simp]: "ball x e = {} \<longleftrightarrow> e \<le> 0"
- unfolding mem_ball set_eq_iff
- apply (simp add: not_less)
- apply (metis zero_le_dist order_trans dist_self)
- done
-
-lemma ball_empty: "e \<le> 0 \<Longrightarrow> ball x e = {}" by simp
-
-lemma closed_cball [iff]: "closed (cball x e)"
-proof -
- have "closed (dist x -` {..e})"
- by (intro closed_vimage closed_atMost continuous_intros)
- also have "dist x -` {..e} = cball x e"
- by auto
- finally show ?thesis .
-qed
-
-lemma open_contains_cball: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. cball x e \<subseteq> S)"
-proof -
- {
- fix x and e::real
- assume "x\<in>S" "e>0" "ball x e \<subseteq> S"
- then have "\<exists>d>0. cball x d \<subseteq> S" unfolding subset_eq by (rule_tac x="e/2" in exI, auto)
- }
- moreover
- {
- fix x and e::real
- assume "x\<in>S" "e>0" "cball x e \<subseteq> S"
- then have "\<exists>d>0. ball x d \<subseteq> S"
- unfolding subset_eq
- apply (rule_tac x="e/2" in exI, auto)
- done
- }
- ultimately show ?thesis
- unfolding open_contains_ball by auto
-qed
-
-lemma open_contains_cball_eq: "open S \<Longrightarrow> (\<forall>x. x \<in> S \<longleftrightarrow> (\<exists>e>0. cball x e \<subseteq> S))"
- by (metis open_contains_cball subset_eq order_less_imp_le centre_in_cball)
-
lemma euclidean_dist_l2:
fixes x y :: "'a :: euclidean_space"
shows "dist x y = L2_set (\<lambda>i. dist (x \<bullet> i) (y \<bullet> i)) Basis"
@@ -1417,28 +30,6 @@
by (auto intro!: real_le_rsqrt)
qed
-lemma eventually_nhds_ball: "d > 0 \<Longrightarrow> eventually (\<lambda>x. x \<in> ball z d) (nhds z)"
- by (rule eventually_nhds_in_open) simp_all
-
-lemma eventually_at_ball: "d > 0 \<Longrightarrow> eventually (\<lambda>t. t \<in> ball z d \<and> t \<in> A) (at z within A)"
- unfolding eventually_at by (intro exI[of _ d]) (simp_all add: dist_commute)
-
-lemma eventually_at_ball': "d > 0 \<Longrightarrow> eventually (\<lambda>t. t \<in> ball z d \<and> t \<noteq> z \<and> t \<in> A) (at z within A)"
- unfolding eventually_at by (intro exI[of _ d]) (simp_all add: dist_commute)
-
-lemma at_within_ball: "e > 0 \<Longrightarrow> dist x y < e \<Longrightarrow> at y within ball x e = at y"
- by (subst at_within_open) auto
-
-lemma atLeastAtMost_eq_cball:
- fixes a b::real
- shows "{a .. b} = cball ((a + b)/2) ((b - a)/2)"
- by (auto simp: dist_real_def field_simps mem_cball)
-
-lemma greaterThanLessThan_eq_ball:
- fixes a b::real
- shows "{a <..< b} = ball ((a + b)/2) ((b - a)/2)"
- by (auto simp: dist_real_def field_simps mem_ball)
-
subsection \<open>Boxes\<close>
@@ -1946,6 +537,7 @@
by (auto simp: box_def inner_sum_left inner_Basis sum.If_cases)
qed
+
subsection \<open>General Intervals\<close>
definition%important "is_interval (s::('a::euclidean_space) set) \<longleftrightarrow>
@@ -2123,2722 +715,6 @@
using is_interval_translation[of "-c" X]
by (metis image_cong uminus_add_conv_diff)
-
-subsection \<open>Limit points\<close>
-
-definition%important (in topological_space) islimpt:: "'a \<Rightarrow> 'a set \<Rightarrow> bool" (infixr "islimpt" 60)
- where "x islimpt S \<longleftrightarrow> (\<forall>T. x\<in>T \<longrightarrow> open T \<longrightarrow> (\<exists>y\<in>S. y\<in>T \<and> y\<noteq>x))"
-
-lemma islimptI:
- assumes "\<And>T. x \<in> T \<Longrightarrow> open T \<Longrightarrow> \<exists>y\<in>S. y \<in> T \<and> y \<noteq> x"
- shows "x islimpt S"
- using assms unfolding islimpt_def by auto
-
-lemma islimptE:
- assumes "x islimpt S" and "x \<in> T" and "open T"
- obtains y where "y \<in> S" and "y \<in> T" and "y \<noteq> x"
- using assms unfolding islimpt_def by auto
-
-lemma islimpt_iff_eventually: "x islimpt S \<longleftrightarrow> \<not> eventually (\<lambda>y. y \<notin> S) (at x)"
- unfolding islimpt_def eventually_at_topological by auto
-
-lemma islimpt_subset: "x islimpt S \<Longrightarrow> S \<subseteq> T \<Longrightarrow> x islimpt T"
- unfolding islimpt_def by fast
-
-lemma islimpt_approachable:
- fixes x :: "'a::metric_space"
- shows "x islimpt S \<longleftrightarrow> (\<forall>e>0. \<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e)"
- unfolding islimpt_iff_eventually eventually_at by fast
-
-lemma islimpt_approachable_le: "x islimpt S \<longleftrightarrow> (\<forall>e>0. \<exists>x'\<in> S. x' \<noteq> x \<and> dist x' x \<le> e)"
- for x :: "'a::metric_space"
- unfolding islimpt_approachable
- using approachable_lt_le [where f="\<lambda>y. dist y x" and P="\<lambda>y. y \<notin> S \<or> y = x",
- THEN arg_cong [where f=Not]]
- by (simp add: Bex_def conj_commute conj_left_commute)
-
-lemma islimpt_UNIV_iff: "x islimpt UNIV \<longleftrightarrow> \<not> open {x}"
- unfolding islimpt_def by (safe, fast, case_tac "T = {x}", fast, fast)
-
-lemma islimpt_punctured: "x islimpt S = x islimpt (S-{x})"
- unfolding islimpt_def by blast
-
-text \<open>A perfect space has no isolated points.\<close>
-
-lemma islimpt_UNIV [simp, intro]: "x islimpt UNIV"
- for x :: "'a::perfect_space"
- unfolding islimpt_UNIV_iff by (rule not_open_singleton)
-
-lemma perfect_choose_dist: "0 < r \<Longrightarrow> \<exists>a. a \<noteq> x \<and> dist a x < r"
- for x :: "'a::{perfect_space,metric_space}"
- using islimpt_UNIV [of x] by (simp add: islimpt_approachable)
-
-lemma closed_limpt: "closed S \<longleftrightarrow> (\<forall>x. x islimpt S \<longrightarrow> x \<in> S)"
- unfolding closed_def
- apply (subst open_subopen)
- apply (simp add: islimpt_def subset_eq)
- apply (metis ComplE ComplI)
- done
-
-lemma islimpt_EMPTY[simp]: "\<not> x islimpt {}"
- by (auto simp: islimpt_def)
-
-lemma finite_ball_include:
- fixes a :: "'a::metric_space"
- assumes "finite S"
- shows "\<exists>e>0. S \<subseteq> ball a e"
- using assms
-proof induction
- case (insert x S)
- then obtain e0 where "e0>0" and e0:"S \<subseteq> ball a e0" by auto
- define e where "e = max e0 (2 * dist a x)"
- have "e>0" unfolding e_def using \<open>e0>0\<close> by auto
- moreover have "insert x S \<subseteq> ball a e"
- using e0 \<open>e>0\<close> unfolding e_def by auto
- ultimately show ?case by auto
-qed (auto intro: zero_less_one)
-
-lemma finite_set_avoid:
- fixes a :: "'a::metric_space"
- assumes "finite S"
- shows "\<exists>d>0. \<forall>x\<in>S. x \<noteq> a \<longrightarrow> d \<le> dist a x"
- using assms
-proof induction
- case (insert x S)
- then obtain d where "d > 0" and d: "\<forall>x\<in>S. x \<noteq> a \<longrightarrow> d \<le> dist a x"
- by blast
- show ?case
- proof (cases "x = a")
- case True
- with \<open>d > 0 \<close>d show ?thesis by auto
- next
- case False
- let ?d = "min d (dist a x)"
- from False \<open>d > 0\<close> have dp: "?d > 0"
- by auto
- from d have d': "\<forall>x\<in>S. x \<noteq> a \<longrightarrow> ?d \<le> dist a x"
- by auto
- with dp False show ?thesis
- by (metis insert_iff le_less min_less_iff_conj not_less)
- qed
-qed (auto intro: zero_less_one)
-
-lemma islimpt_Un: "x islimpt (S \<union> T) \<longleftrightarrow> x islimpt S \<or> x islimpt T"
- by (simp add: islimpt_iff_eventually eventually_conj_iff)
-
-lemma discrete_imp_closed:
- fixes S :: "'a::metric_space set"
- assumes e: "0 < e"
- and d: "\<forall>x \<in> S. \<forall>y \<in> S. dist y x < e \<longrightarrow> y = x"
- shows "closed S"
-proof -
- have False if C: "\<And>e. e>0 \<Longrightarrow> \<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e" for x
- proof -
- from e have e2: "e/2 > 0" by arith
- from C[rule_format, OF e2] obtain y where y: "y \<in> S" "y \<noteq> x" "dist y x < e/2"
- by blast
- let ?m = "min (e/2) (dist x y) "
- from e2 y(2) have mp: "?m > 0"
- by simp
- from C[OF mp] obtain z where z: "z \<in> S" "z \<noteq> x" "dist z x < ?m"
- by blast
- from z y have "dist z y < e"
- by (intro dist_triangle_lt [where z=x]) simp
- from d[rule_format, OF y(1) z(1) this] y z show ?thesis
- by (auto simp: dist_commute)
- qed
- then show ?thesis
- by (metis islimpt_approachable closed_limpt [where 'a='a])
-qed
-
-lemma closed_of_nat_image: "closed (of_nat ` A :: 'a::real_normed_algebra_1 set)"
- by (rule discrete_imp_closed[of 1]) (auto simp: dist_of_nat)
-
-lemma closed_of_int_image: "closed (of_int ` A :: 'a::real_normed_algebra_1 set)"
- by (rule discrete_imp_closed[of 1]) (auto simp: dist_of_int)
-
-lemma closed_Nats [simp]: "closed (\<nat> :: 'a :: real_normed_algebra_1 set)"
- unfolding Nats_def by (rule closed_of_nat_image)
-
-lemma closed_Ints [simp]: "closed (\<int> :: 'a :: real_normed_algebra_1 set)"
- unfolding Ints_def by (rule closed_of_int_image)
-
-lemma closed_subset_Ints:
- fixes A :: "'a :: real_normed_algebra_1 set"
- assumes "A \<subseteq> \<int>"
- shows "closed A"
-proof (intro discrete_imp_closed[OF zero_less_one] ballI impI, goal_cases)
- case (1 x y)
- with assms have "x \<in> \<int>" and "y \<in> \<int>" by auto
- with \<open>dist y x < 1\<close> show "y = x"
- by (auto elim!: Ints_cases simp: dist_of_int)
-qed
-
-
-subsection \<open>Interior of a Set\<close>
-
-definition%important "interior S = \<Union>{T. open T \<and> T \<subseteq> S}"
-
-lemma interiorI [intro?]:
- assumes "open T" and "x \<in> T" and "T \<subseteq> S"
- shows "x \<in> interior S"
- using assms unfolding interior_def by fast
-
-lemma interiorE [elim?]:
- assumes "x \<in> interior S"
- obtains T where "open T" and "x \<in> T" and "T \<subseteq> S"
- using assms unfolding interior_def by fast
-
-lemma open_interior [simp, intro]: "open (interior S)"
- by (simp add: interior_def open_Union)
-
-lemma interior_subset: "interior S \<subseteq> S"
- by (auto simp: interior_def)
-
-lemma interior_maximal: "T \<subseteq> S \<Longrightarrow> open T \<Longrightarrow> T \<subseteq> interior S"
- by (auto simp: interior_def)
-
-lemma interior_open: "open S \<Longrightarrow> interior S = S"
- by (intro equalityI interior_subset interior_maximal subset_refl)
-
-lemma interior_eq: "interior S = S \<longleftrightarrow> open S"
- by (metis open_interior interior_open)
-
-lemma open_subset_interior: "open S \<Longrightarrow> S \<subseteq> interior T \<longleftrightarrow> S \<subseteq> T"
- by (metis interior_maximal interior_subset subset_trans)
-
-lemma interior_empty [simp]: "interior {} = {}"
- using open_empty by (rule interior_open)
-
-lemma interior_UNIV [simp]: "interior UNIV = UNIV"
- using open_UNIV by (rule interior_open)
-
-lemma interior_interior [simp]: "interior (interior S) = interior S"
- using open_interior by (rule interior_open)
-
-lemma interior_mono: "S \<subseteq> T \<Longrightarrow> interior S \<subseteq> interior T"
- by (auto simp: interior_def)
-
-lemma interior_unique:
- assumes "T \<subseteq> S" and "open T"
- assumes "\<And>T'. T' \<subseteq> S \<Longrightarrow> open T' \<Longrightarrow> T' \<subseteq> T"
- shows "interior S = T"
- by (intro equalityI assms interior_subset open_interior interior_maximal)
-
-lemma interior_singleton [simp]: "interior {a} = {}"
- for a :: "'a::perfect_space"
- apply (rule interior_unique, simp_all)
- using not_open_singleton subset_singletonD
- apply fastforce
- done
-
-lemma interior_Int [simp]: "interior (S \<inter> T) = interior S \<inter> interior T"
- by (intro equalityI Int_mono Int_greatest interior_mono Int_lower1
- Int_lower2 interior_maximal interior_subset open_Int open_interior)
-
-lemma mem_interior: "x \<in> interior S \<longleftrightarrow> (\<exists>e>0. ball x e \<subseteq> S)"
- using open_contains_ball_eq [where S="interior S"]
- by (simp add: open_subset_interior)
-
-lemma eventually_nhds_in_nhd: "x \<in> interior s \<Longrightarrow> eventually (\<lambda>y. y \<in> s) (nhds x)"
- using interior_subset[of s] by (subst eventually_nhds) blast
-
-lemma interior_limit_point [intro]:
- fixes x :: "'a::perfect_space"
- assumes x: "x \<in> interior S"
- shows "x islimpt S"
- using x islimpt_UNIV [of x]
- unfolding interior_def islimpt_def
- apply (clarsimp, rename_tac T T')
- apply (drule_tac x="T \<inter> T'" in spec)
- apply (auto simp: open_Int)
- done
-
-lemma interior_closed_Un_empty_interior:
- assumes cS: "closed S"
- and iT: "interior T = {}"
- shows "interior (S \<union> T) = interior S"
-proof
- show "interior S \<subseteq> interior (S \<union> T)"
- by (rule interior_mono) (rule Un_upper1)
- show "interior (S \<union> T) \<subseteq> interior S"
- proof
- fix x
- assume "x \<in> interior (S \<union> T)"
- then obtain R where "open R" "x \<in> R" "R \<subseteq> S \<union> T" ..
- show "x \<in> interior S"
- proof (rule ccontr)
- assume "x \<notin> interior S"
- with \<open>x \<in> R\<close> \<open>open R\<close> obtain y where "y \<in> R - S"
- unfolding interior_def by fast
- from \<open>open R\<close> \<open>closed S\<close> have "open (R - S)"
- by (rule open_Diff)
- from \<open>R \<subseteq> S \<union> T\<close> have "R - S \<subseteq> T"
- by fast
- from \<open>y \<in> R - S\<close> \<open>open (R - S)\<close> \<open>R - S \<subseteq> T\<close> \<open>interior T = {}\<close> show False
- unfolding interior_def by fast
- qed
- qed
-qed
-
-lemma interior_Times: "interior (A \<times> B) = interior A \<times> interior B"
-proof (rule interior_unique)
- show "interior A \<times> interior B \<subseteq> A \<times> B"
- by (intro Sigma_mono interior_subset)
- show "open (interior A \<times> interior B)"
- by (intro open_Times open_interior)
- fix T
- assume "T \<subseteq> A \<times> B" and "open T"
- then show "T \<subseteq> interior A \<times> interior B"
- proof safe
- fix x y
- assume "(x, y) \<in> T"
- then obtain C D where "open C" "open D" "C \<times> D \<subseteq> T" "x \<in> C" "y \<in> D"
- using \<open>open T\<close> unfolding open_prod_def by fast
- then have "open C" "open D" "C \<subseteq> A" "D \<subseteq> B" "x \<in> C" "y \<in> D"
- using \<open>T \<subseteq> A \<times> B\<close> by auto
- then show "x \<in> interior A" and "y \<in> interior B"
- by (auto intro: interiorI)
- qed
-qed
-
-lemma interior_Ici:
- fixes x :: "'a :: {dense_linorder,linorder_topology}"
- assumes "b < x"
- shows "interior {x ..} = {x <..}"
-proof (rule interior_unique)
- fix T
- assume "T \<subseteq> {x ..}" "open T"
- moreover have "x \<notin> T"
- proof
- assume "x \<in> T"
- obtain y where "y < x" "{y <.. x} \<subseteq> T"
- using open_left[OF \<open>open T\<close> \<open>x \<in> T\<close> \<open>b < x\<close>] by auto
- with dense[OF \<open>y < x\<close>] obtain z where "z \<in> T" "z < x"
- by (auto simp: subset_eq Ball_def)
- with \<open>T \<subseteq> {x ..}\<close> show False by auto
- qed
- ultimately show "T \<subseteq> {x <..}"
- by (auto simp: subset_eq less_le)
-qed auto
-
-lemma interior_Iic:
- fixes x :: "'a ::{dense_linorder,linorder_topology}"
- assumes "x < b"
- shows "interior {.. x} = {..< x}"
-proof (rule interior_unique)
- fix T
- assume "T \<subseteq> {.. x}" "open T"
- moreover have "x \<notin> T"
- proof
- assume "x \<in> T"
- obtain y where "x < y" "{x ..< y} \<subseteq> T"
- using open_right[OF \<open>open T\<close> \<open>x \<in> T\<close> \<open>x < b\<close>] by auto
- with dense[OF \<open>x < y\<close>] obtain z where "z \<in> T" "x < z"
- by (auto simp: subset_eq Ball_def less_le)
- with \<open>T \<subseteq> {.. x}\<close> show False by auto
- qed
- ultimately show "T \<subseteq> {..< x}"
- by (auto simp: subset_eq less_le)
-qed auto
-
-
-subsection \<open>Closure of a Set\<close>
-
-definition%important "closure S = S \<union> {x | x. x islimpt S}"
-
-lemma interior_closure: "interior S = - (closure (- S))"
- by (auto simp: interior_def closure_def islimpt_def)
-
-lemma closure_interior: "closure S = - interior (- S)"
- by (simp add: interior_closure)
-
-lemma closed_closure[simp, intro]: "closed (closure S)"
- by (simp add: closure_interior closed_Compl)
-
-lemma closure_subset: "S \<subseteq> closure S"
- by (simp add: closure_def)
-
-lemma closure_hull: "closure S = closed hull S"
- by (auto simp: hull_def closure_interior interior_def)
-
-lemma closure_eq: "closure S = S \<longleftrightarrow> closed S"
- unfolding closure_hull using closed_Inter by (rule hull_eq)
-
-lemma closure_closed [simp]: "closed S \<Longrightarrow> closure S = S"
- by (simp only: closure_eq)
-
-lemma closure_closure [simp]: "closure (closure S) = closure S"
- unfolding closure_hull by (rule hull_hull)
-
-lemma closure_mono: "S \<subseteq> T \<Longrightarrow> closure S \<subseteq> closure T"
- unfolding closure_hull by (rule hull_mono)
-
-lemma closure_minimal: "S \<subseteq> T \<Longrightarrow> closed T \<Longrightarrow> closure S \<subseteq> T"
- unfolding closure_hull by (rule hull_minimal)
-
-lemma closure_unique:
- assumes "S \<subseteq> T"
- and "closed T"
- and "\<And>T'. S \<subseteq> T' \<Longrightarrow> closed T' \<Longrightarrow> T \<subseteq> T'"
- shows "closure S = T"
- using assms unfolding closure_hull by (rule hull_unique)
-
-lemma closure_empty [simp]: "closure {} = {}"
- using closed_empty by (rule closure_closed)
-
-lemma closure_UNIV [simp]: "closure UNIV = UNIV"
- using closed_UNIV by (rule closure_closed)
-
-lemma closure_Un [simp]: "closure (S \<union> T) = closure S \<union> closure T"
- by (simp add: closure_interior)
-
-lemma closure_eq_empty [iff]: "closure S = {} \<longleftrightarrow> S = {}"
- using closure_empty closure_subset[of S] by blast
-
-lemma closure_subset_eq: "closure S \<subseteq> S \<longleftrightarrow> closed S"
- using closure_eq[of S] closure_subset[of S] by simp
-
-lemma open_Int_closure_eq_empty: "open S \<Longrightarrow> (S \<inter> closure T) = {} \<longleftrightarrow> S \<inter> T = {}"
- using open_subset_interior[of S "- T"]
- using interior_subset[of "- T"]
- by (auto simp: closure_interior)
-
-lemma open_Int_closure_subset: "open S \<Longrightarrow> S \<inter> closure T \<subseteq> closure (S \<inter> T)"
-proof
- fix x
- assume *: "open S" "x \<in> S \<inter> closure T"
- have "x islimpt (S \<inter> T)" if **: "x islimpt T"
- proof (rule islimptI)
- fix A
- assume "x \<in> A" "open A"
- with * have "x \<in> A \<inter> S" "open (A \<inter> S)"
- by (simp_all add: open_Int)
- with ** obtain y where "y \<in> T" "y \<in> A \<inter> S" "y \<noteq> x"
- by (rule islimptE)
- then have "y \<in> S \<inter> T" "y \<in> A \<and> y \<noteq> x"
- by simp_all
- then show "\<exists>y\<in>(S \<inter> T). y \<in> A \<and> y \<noteq> x" ..
- qed
- with * show "x \<in> closure (S \<inter> T)"
- unfolding closure_def by blast
-qed
-
-lemma closure_complement: "closure (- S) = - interior S"
- by (simp add: closure_interior)
-
-lemma interior_complement: "interior (- S) = - closure S"
- by (simp add: closure_interior)
-
-lemma interior_diff: "interior(S - T) = interior S - closure T"
- by (simp add: Diff_eq interior_complement)
-
-lemma closure_Times: "closure (A \<times> B) = closure A \<times> closure B"
-proof (rule closure_unique)
- show "A \<times> B \<subseteq> closure A \<times> closure B"
- by (intro Sigma_mono closure_subset)
- show "closed (closure A \<times> closure B)"
- by (intro closed_Times closed_closure)
- fix T
- assume "A \<times> B \<subseteq> T" and "closed T"
- then show "closure A \<times> closure B \<subseteq> T"
- apply (simp add: closed_def open_prod_def, clarify)
- apply (rule ccontr)
- apply (drule_tac x="(a, b)" in bspec, simp, clarify, rename_tac C D)
- apply (simp add: closure_interior interior_def)
- apply (drule_tac x=C in spec)
- apply (drule_tac x=D in spec, auto)
- done
-qed
-
-lemma closure_openin_Int_closure:
- assumes ope: "openin (subtopology euclidean U) S" and "T \<subseteq> U"
- shows "closure(S \<inter> closure T) = closure(S \<inter> T)"
-proof
- obtain V where "open V" and S: "S = U \<inter> V"
- using ope using openin_open by metis
- show "closure (S \<inter> closure T) \<subseteq> closure (S \<inter> T)"
- proof (clarsimp simp: S)
- fix x
- assume "x \<in> closure (U \<inter> V \<inter> closure T)"
- then have "V \<inter> closure T \<subseteq> A \<Longrightarrow> x \<in> closure A" for A
- by (metis closure_mono subsetD inf.coboundedI2 inf_assoc)
- then have "x \<in> closure (T \<inter> V)"
- by (metis \<open>open V\<close> closure_closure inf_commute open_Int_closure_subset)
- then show "x \<in> closure (U \<inter> V \<inter> T)"
- by (metis \<open>T \<subseteq> U\<close> inf.absorb_iff2 inf_assoc inf_commute)
- qed
-next
- show "closure (S \<inter> T) \<subseteq> closure (S \<inter> closure T)"
- by (meson Int_mono closure_mono closure_subset order_refl)
-qed
-
-lemma islimpt_in_closure: "(x islimpt S) = (x\<in>closure(S-{x}))"
- unfolding closure_def using islimpt_punctured by blast
-
-lemma connected_imp_connected_closure: "connected S \<Longrightarrow> connected (closure S)"
- by (rule connectedI) (meson closure_subset open_Int open_Int_closure_eq_empty subset_trans connectedD)
-
-lemma limpt_of_limpts: "x islimpt {y. y islimpt S} \<Longrightarrow> x islimpt S"
- for x :: "'a::metric_space"
- apply (clarsimp simp add: islimpt_approachable)
- apply (drule_tac x="e/2" in spec)
- apply (auto simp: simp del: less_divide_eq_numeral1)
- apply (drule_tac x="dist x' x" in spec)
- apply (auto simp: zero_less_dist_iff simp del: less_divide_eq_numeral1)
- apply (erule rev_bexI)
- apply (metis dist_commute dist_triangle_half_r less_trans less_irrefl)
- done
-
-lemma closed_limpts: "closed {x::'a::metric_space. x islimpt S}"
- using closed_limpt limpt_of_limpts by blast
-
-lemma limpt_of_closure: "x islimpt closure S \<longleftrightarrow> x islimpt S"
- for x :: "'a::metric_space"
- by (auto simp: closure_def islimpt_Un dest: limpt_of_limpts)
-
-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)"
- apply (simp add: closedin_closed, safe)
- apply (simp add: closed_limpt islimpt_subset)
- apply (rule_tac x="closure S" in exI, simp)
- apply (force simp: closure_def)
- done
-
-lemma closedin_closed_eq: "closed S \<Longrightarrow> closedin (subtopology euclidean S) T \<longleftrightarrow> closed T \<and> T \<subseteq> S"
- by (meson closedin_limpt closed_subset closedin_closed_trans)
-
-lemma connected_closed_set:
- "closed S
- \<Longrightarrow> connected S \<longleftrightarrow> (\<nexists>A B. closed A \<and> closed B \<and> A \<noteq> {} \<and> B \<noteq> {} \<and> A \<union> B = S \<and> A \<inter> B = {})"
- unfolding connected_closedin_eq closedin_closed_eq connected_closedin_eq by blast
-
-text \<open>If a connnected set is written as the union of two nonempty closed sets, then these sets
-have to intersect.\<close>
-
-lemma connected_as_closed_union:
- assumes "connected C" "C = A \<union> B" "closed A" "closed B" "A \<noteq> {}" "B \<noteq> {}"
- shows "A \<inter> B \<noteq> {}"
-by (metis assms closed_Un connected_closed_set)
-
-lemma closedin_subset_trans:
- "closedin (subtopology euclidean U) S \<Longrightarrow> S \<subseteq> T \<Longrightarrow> T \<subseteq> U \<Longrightarrow>
- closedin (subtopology euclidean T) S"
- by (meson closedin_limpt subset_iff)
-
-lemma openin_subset_trans:
- "openin (subtopology euclidean U) S \<Longrightarrow> S \<subseteq> T \<Longrightarrow> T \<subseteq> U \<Longrightarrow>
- openin (subtopology euclidean T) S"
- by (auto simp: openin_open)
-
-lemma openin_Times:
- "openin (subtopology euclidean S) S' \<Longrightarrow> openin (subtopology euclidean T) T' \<Longrightarrow>
- openin (subtopology euclidean (S \<times> T)) (S' \<times> T')"
- unfolding openin_open using open_Times by blast
-
-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"
- obtains V W where "openin (subtopology euclidean S) V" "x \<in> V"
- "openin (subtopology euclidean T) W" "y \<in> W" "(V \<times> W) \<subseteq> U"
-proof -
- from assms obtain e where "e > 0" and "U \<subseteq> S \<times> T"
- and e: "\<And>x' y'. \<lbrakk>x'\<in>S; y'\<in>T; dist (x', y') (x, y) < e\<rbrakk> \<Longrightarrow> (x', y') \<in> U"
- by (force simp: openin_euclidean_subtopology_iff)
- with assms have "x \<in> S" "y \<in> T"
- by auto
- show ?thesis
- proof
- show "openin (subtopology euclidean S) (ball x (e/2) \<inter> S)"
- by (simp add: Int_commute openin_open_Int)
- show "x \<in> ball x (e / 2) \<inter> S"
- by (simp add: \<open>0 < e\<close> \<open>x \<in> S\<close>)
- show "openin (subtopology euclidean T) (ball y (e/2) \<inter> T)"
- by (simp add: Int_commute openin_open_Int)
- show "y \<in> ball y (e / 2) \<inter> T"
- by (simp add: \<open>0 < e\<close> \<open>y \<in> T\<close>)
- show "(ball x (e / 2) \<inter> S) \<times> (ball y (e / 2) \<inter> T) \<subseteq> U"
- by clarify (simp add: e dist_Pair_Pair \<open>0 < e\<close> dist_commute sqrt_sum_squares_half_less)
- qed
-qed
-
-lemma openin_Times_eq:
- fixes S :: "'a::metric_space set" and T :: "'b::metric_space set"
- shows
- "openin (subtopology euclidean (S \<times> T)) (S' \<times> T') \<longleftrightarrow>
- S' = {} \<or> T' = {} \<or> openin (subtopology euclidean S) S' \<and> openin (subtopology euclidean T) T'"
- (is "?lhs = ?rhs")
-proof (cases "S' = {} \<or> T' = {}")
- case True
- then show ?thesis by auto
-next
- case False
- then obtain x y where "x \<in> S'" "y \<in> T'"
- by blast
- show ?thesis
- proof
- assume ?lhs
- have "openin (subtopology euclidean S) S'"
- apply (subst openin_subopen, clarify)
- apply (rule Times_in_interior_subtopology [OF _ \<open>?lhs\<close>])
- using \<open>y \<in> T'\<close>
- apply auto
- done
- moreover have "openin (subtopology euclidean T) T'"
- apply (subst openin_subopen, clarify)
- apply (rule Times_in_interior_subtopology [OF _ \<open>?lhs\<close>])
- using \<open>x \<in> S'\<close>
- apply auto
- done
- ultimately show ?rhs
- by simp
- next
- assume ?rhs
- with False show ?lhs
- by (simp add: openin_Times)
- qed
-qed
-
-lemma closedin_Times:
- "closedin (subtopology euclidean S) S' \<Longrightarrow> closedin (subtopology euclidean T) T' \<Longrightarrow>
- closedin (subtopology euclidean (S \<times> T)) (S' \<times> T')"
- unfolding closedin_closed using closed_Times by blast
-
-lemma bdd_below_closure:
- fixes A :: "real set"
- assumes "bdd_below A"
- shows "bdd_below (closure A)"
-proof -
- from assms obtain m where "\<And>x. x \<in> A \<Longrightarrow> m \<le> x"
- by (auto simp: bdd_below_def)
- then have "A \<subseteq> {m..}" by auto
- then have "closure A \<subseteq> {m..}"
- using closed_real_atLeast by (rule closure_minimal)
- then show ?thesis
- by (auto simp: bdd_below_def)
-qed
-
-
-subsection \<open>Frontier (also known as boundary)\<close>
-
-definition%important "frontier S = closure S - interior S"
-
-lemma frontier_closed [iff]: "closed (frontier S)"
- by (simp add: frontier_def closed_Diff)
-
-lemma frontier_closures: "frontier S = closure S \<inter> closure (- S)"
- by (auto simp: frontier_def interior_closure)
-
-lemma frontier_Int: "frontier(S \<inter> T) = closure(S \<inter> T) \<inter> (frontier S \<union> frontier T)"
-proof -
- have "closure (S \<inter> T) \<subseteq> closure S" "closure (S \<inter> T) \<subseteq> closure T"
- by (simp_all add: closure_mono)
- then show ?thesis
- by (auto simp: frontier_closures)
-qed
-
-lemma frontier_Int_subset: "frontier(S \<inter> T) \<subseteq> frontier S \<union> frontier T"
- by (auto simp: frontier_Int)
-
-lemma frontier_Int_closed:
- assumes "closed S" "closed T"
- shows "frontier(S \<inter> T) = (frontier S \<inter> T) \<union> (S \<inter> frontier T)"
-proof -
- have "closure (S \<inter> T) = T \<inter> S"
- using assms by (simp add: Int_commute closed_Int)
- moreover have "T \<inter> (closure S \<inter> closure (- S)) = frontier S \<inter> T"
- by (simp add: Int_commute frontier_closures)
- ultimately show ?thesis
- by (simp add: Int_Un_distrib Int_assoc Int_left_commute assms frontier_closures)
-qed
-
-lemma frontier_straddle:
- fixes a :: "'a::metric_space"
- shows "a \<in> frontier S \<longleftrightarrow> (\<forall>e>0. (\<exists>x\<in>S. dist a x < e) \<and> (\<exists>x. x \<notin> S \<and> dist a x < e))"
- unfolding frontier_def closure_interior
- by (auto simp: mem_interior subset_eq ball_def)
-
-lemma frontier_subset_closed: "closed S \<Longrightarrow> frontier S \<subseteq> S"
- by (metis frontier_def closure_closed Diff_subset)
-
-lemma frontier_empty [simp]: "frontier {} = {}"
- by (simp add: frontier_def)
-
-lemma frontier_subset_eq: "frontier S \<subseteq> S \<longleftrightarrow> closed S"
-proof -
- {
- assume "frontier S \<subseteq> S"
- then have "closure S \<subseteq> S"
- using interior_subset unfolding frontier_def by auto
- then have "closed S"
- using closure_subset_eq by auto
- }
- then show ?thesis using frontier_subset_closed[of S] ..
-qed
-
-lemma frontier_complement [simp]: "frontier (- S) = frontier S"
- by (auto simp: frontier_def closure_complement interior_complement)
-
-lemma frontier_Un_subset: "frontier(S \<union> T) \<subseteq> frontier S \<union> frontier T"
- by (metis compl_sup frontier_Int_subset frontier_complement)
-
-lemma frontier_disjoint_eq: "frontier S \<inter> S = {} \<longleftrightarrow> open S"
- using frontier_complement frontier_subset_eq[of "- S"]
- unfolding open_closed by auto
-
-lemma frontier_UNIV [simp]: "frontier UNIV = {}"
- using frontier_complement frontier_empty by fastforce
-
-lemma frontier_interiors: "frontier s = - interior(s) - interior(-s)"
- by (simp add: Int_commute frontier_def interior_closure)
-
-lemma frontier_interior_subset: "frontier(interior S) \<subseteq> frontier S"
- by (simp add: Diff_mono frontier_interiors interior_mono interior_subset)
-
-lemma connected_Int_frontier:
- "\<lbrakk>connected s; s \<inter> t \<noteq> {}; s - t \<noteq> {}\<rbrakk> \<Longrightarrow> (s \<inter> frontier t \<noteq> {})"
- apply (simp add: frontier_interiors connected_openin, safe)
- apply (drule_tac x="s \<inter> interior t" in spec, safe)
- apply (drule_tac [2] x="s \<inter> interior (-t)" in spec)
- apply (auto simp: disjoint_eq_subset_Compl dest: interior_subset [THEN subsetD])
- done
-
-lemma closure_Un_frontier: "closure S = S \<union> frontier S"
-proof -
- have "S \<union> interior S = S"
- using interior_subset by auto
- then show ?thesis
- using closure_subset by (auto simp: frontier_def)
-qed
-
-
-subsection%unimportant \<open>Filters and the ``eventually true'' quantifier\<close>
-
-definition indirection :: "'a::real_normed_vector \<Rightarrow> 'a \<Rightarrow> 'a filter" (infixr "indirection" 70)
- where "a indirection v = at a within {b. \<exists>c\<ge>0. b - a = scaleR c v}"
-
-text \<open>Identify Trivial limits, where we can't approach arbitrarily closely.\<close>
-
-lemma trivial_limit_within: "trivial_limit (at a within S) \<longleftrightarrow> \<not> a islimpt S"
-proof
- assume "trivial_limit (at a within S)"
- then show "\<not> a islimpt S"
- unfolding trivial_limit_def
- unfolding eventually_at_topological
- unfolding islimpt_def
- apply (clarsimp simp add: set_eq_iff)
- apply (rename_tac T, rule_tac x=T in exI)
- apply (clarsimp, drule_tac x=y in bspec, simp_all)
- done
-next
- assume "\<not> a islimpt S"
- then show "trivial_limit (at a within S)"
- unfolding trivial_limit_def eventually_at_topological islimpt_def
- by metis
-qed
-
-lemma trivial_limit_at_iff: "trivial_limit (at a) \<longleftrightarrow> \<not> a islimpt UNIV"
- using trivial_limit_within [of a UNIV] by simp
-
-lemma trivial_limit_at: "\<not> trivial_limit (at a)"
- for a :: "'a::perfect_space"
- by (rule at_neq_bot)
-
-lemma trivial_limit_at_infinity:
- "\<not> trivial_limit (at_infinity :: ('a::{real_normed_vector,perfect_space}) filter)"
- unfolding trivial_limit_def eventually_at_infinity
- apply clarsimp
- apply (subgoal_tac "\<exists>x::'a. x \<noteq> 0", clarify)
- apply (rule_tac x="scaleR (b / norm x) x" in exI, simp)
- apply (cut_tac islimpt_UNIV [of "0::'a", unfolded islimpt_def])
- apply (drule_tac x=UNIV in spec, simp)
- done
-
-lemma not_trivial_limit_within: "\<not> trivial_limit (at x within S) = (x \<in> closure (S - {x}))"
- using islimpt_in_closure by (metis trivial_limit_within)
-
-lemma not_in_closure_trivial_limitI:
- "x \<notin> closure s \<Longrightarrow> trivial_limit (at x within s)"
- using not_trivial_limit_within[of x s]
- by safe (metis Diff_empty Diff_insert0 closure_subset contra_subsetD)
-
-lemma filterlim_at_within_closure_implies_filterlim: "filterlim f l (at x within s)"
- if "x \<in> closure s \<Longrightarrow> filterlim f l (at x within s)"
- by (metis bot.extremum filterlim_filtercomap filterlim_mono not_in_closure_trivial_limitI that)
-
-lemma at_within_eq_bot_iff: "at c within A = bot \<longleftrightarrow> c \<notin> closure (A - {c})"
- using not_trivial_limit_within[of c A] by blast
-
-text \<open>Some property holds "sufficiently close" to the limit point.\<close>
-
-lemma trivial_limit_eventually: "trivial_limit net \<Longrightarrow> eventually P net"
- by simp
-
-lemma trivial_limit_eq: "trivial_limit net \<longleftrightarrow> (\<forall>P. eventually P net)"
- by (simp add: filter_eq_iff)
-
-
-subsection \<open>Limits\<close>
-
-proposition Lim: "(f \<longlongrightarrow> l) net \<longleftrightarrow> trivial_limit net \<or> (\<forall>e>0. eventually (\<lambda>x. dist (f x) l < e) net)"
- by (auto simp: tendsto_iff trivial_limit_eq)
-
-text \<open>Show that they yield usual definitions in the various cases.\<close>
-
-proposition Lim_within_le: "(f \<longlongrightarrow> l)(at a within S) \<longleftrightarrow>
- (\<forall>e>0. \<exists>d>0. \<forall>x\<in>S. 0 < dist x a \<and> dist x a \<le> d \<longrightarrow> dist (f x) l < e)"
- by (auto simp: tendsto_iff eventually_at_le)
-
-proposition Lim_within: "(f \<longlongrightarrow> l) (at a within S) \<longleftrightarrow>
- (\<forall>e >0. \<exists>d>0. \<forall>x \<in> S. 0 < dist x a \<and> dist x a < d \<longrightarrow> dist (f x) l < e)"
- by (auto simp: tendsto_iff eventually_at)
-
-corollary Lim_withinI [intro?]:
- assumes "\<And>e. e > 0 \<Longrightarrow> \<exists>d>0. \<forall>x \<in> S. 0 < dist x a \<and> dist x a < d \<longrightarrow> dist (f x) l \<le> e"
- shows "(f \<longlongrightarrow> l) (at a within S)"
- apply (simp add: Lim_within, clarify)
- apply (rule ex_forward [OF assms [OF half_gt_zero]], auto)
- done
-
-proposition Lim_at: "(f \<longlongrightarrow> l) (at a) \<longleftrightarrow>
- (\<forall>e >0. \<exists>d>0. \<forall>x. 0 < dist x a \<and> dist x a < d \<longrightarrow> dist (f x) l < e)"
- by (auto simp: tendsto_iff eventually_at)
-
-proposition Lim_at_infinity: "(f \<longlongrightarrow> l) at_infinity \<longleftrightarrow> (\<forall>e>0. \<exists>b. \<forall>x. norm x \<ge> b \<longrightarrow> dist (f x) l < e)"
- by (auto simp: tendsto_iff eventually_at_infinity)
-
-corollary Lim_at_infinityI [intro?]:
- assumes "\<And>e. e > 0 \<Longrightarrow> \<exists>B. \<forall>x. norm x \<ge> B \<longrightarrow> dist (f x) l \<le> e"
- shows "(f \<longlongrightarrow> l) at_infinity"
- apply (simp add: Lim_at_infinity, clarify)
- apply (rule ex_forward [OF assms [OF half_gt_zero]], auto)
- done
-
-lemma Lim_eventually: "eventually (\<lambda>x. f x = l) net \<Longrightarrow> (f \<longlongrightarrow> l) net"
- by (rule topological_tendstoI) (auto elim: eventually_mono)
-
-lemma Lim_transform_within_set:
- fixes a :: "'a::metric_space" and l :: "'b::metric_space"
- shows "\<lbrakk>(f \<longlongrightarrow> l) (at a within S); eventually (\<lambda>x. x \<in> S \<longleftrightarrow> x \<in> T) (at a)\<rbrakk>
- \<Longrightarrow> (f \<longlongrightarrow> l) (at a within T)"
-apply (clarsimp simp: eventually_at Lim_within)
-apply (drule_tac x=e in spec, clarify)
-apply (rename_tac k)
-apply (rule_tac x="min d k" in exI, simp)
-done
-
-lemma Lim_transform_within_set_eq:
- fixes a l :: "'a::real_normed_vector"
- shows "eventually (\<lambda>x. x \<in> s \<longleftrightarrow> x \<in> t) (at a)
- \<Longrightarrow> ((f \<longlongrightarrow> l) (at a within s) \<longleftrightarrow> (f \<longlongrightarrow> l) (at a within t))"
- by (force intro: Lim_transform_within_set elim: eventually_mono)
-
-lemma Lim_transform_within_openin:
- fixes a :: "'a::metric_space"
- assumes f: "(f \<longlongrightarrow> l) (at a within T)"
- and "openin (subtopology euclidean T) S" "a \<in> S"
- and eq: "\<And>x. \<lbrakk>x \<in> S; x \<noteq> a\<rbrakk> \<Longrightarrow> f x = g x"
- shows "(g \<longlongrightarrow> l) (at a within T)"
-proof -
- obtain \<epsilon> where "0 < \<epsilon>" and \<epsilon>: "ball a \<epsilon> \<inter> T \<subseteq> S"
- using assms by (force simp: openin_contains_ball)
- then have "a \<in> ball a \<epsilon>"
- by simp
- show ?thesis
- by (rule Lim_transform_within [OF f \<open>0 < \<epsilon>\<close> eq]) (use \<epsilon> in \<open>auto simp: dist_commute subset_iff\<close>)
-qed
-
-lemma continuous_transform_within_openin:
- fixes a :: "'a::metric_space"
- assumes "continuous (at a within T) f"
- and "openin (subtopology euclidean T) S" "a \<in> S"
- and eq: "\<And>x. x \<in> S \<Longrightarrow> f x = g x"
- shows "continuous (at a within T) g"
- using assms by (simp add: Lim_transform_within_openin continuous_within)
-
-text \<open>The expected monotonicity property.\<close>
-
-lemma Lim_Un:
- assumes "(f \<longlongrightarrow> l) (at x within S)" "(f \<longlongrightarrow> l) (at x within T)"
- shows "(f \<longlongrightarrow> l) (at x within (S \<union> T))"
- using assms unfolding at_within_union by (rule filterlim_sup)
-
-lemma Lim_Un_univ:
- "(f \<longlongrightarrow> l) (at x within S) \<Longrightarrow> (f \<longlongrightarrow> l) (at x within T) \<Longrightarrow>
- S \<union> T = UNIV \<Longrightarrow> (f \<longlongrightarrow> l) (at x)"
- by (metis Lim_Un)
-
-text \<open>Interrelations between restricted and unrestricted limits.\<close>
-
-lemma Lim_at_imp_Lim_at_within: "(f \<longlongrightarrow> l) (at x) \<Longrightarrow> (f \<longlongrightarrow> l) (at x within S)"
- by (metis order_refl filterlim_mono subset_UNIV at_le)
-
-lemma eventually_within_interior:
- assumes "x \<in> interior S"
- shows "eventually P (at x within S) \<longleftrightarrow> eventually P (at x)"
- (is "?lhs = ?rhs")
-proof
- from assms obtain T where T: "open T" "x \<in> T" "T \<subseteq> S" ..
- {
- assume ?lhs
- then obtain A where "open A" and "x \<in> A" and "\<forall>y\<in>A. y \<noteq> x \<longrightarrow> y \<in> S \<longrightarrow> P y"
- by (auto simp: eventually_at_topological)
- with T have "open (A \<inter> T)" and "x \<in> A \<inter> T" and "\<forall>y \<in> A \<inter> T. y \<noteq> x \<longrightarrow> P y"
- by auto
- then show ?rhs
- by (auto simp: eventually_at_topological)
- next
- assume ?rhs
- then show ?lhs
- by (auto elim: eventually_mono simp: eventually_at_filter)
- }
-qed
-
-lemma at_within_interior: "x \<in> interior S \<Longrightarrow> at x within S = at x"
- unfolding filter_eq_iff by (intro allI eventually_within_interior)
-
-lemma Lim_within_LIMSEQ:
- fixes a :: "'a::first_countable_topology"
- assumes "\<forall>S. (\<forall>n. S n \<noteq> a \<and> S n \<in> T) \<and> S \<longlonglongrightarrow> a \<longrightarrow> (\<lambda>n. X (S n)) \<longlonglongrightarrow> L"
- shows "(X \<longlongrightarrow> L) (at a within T)"
- using assms unfolding tendsto_def [where l=L]
- by (simp add: sequentially_imp_eventually_within)
-
-lemma Lim_right_bound:
- fixes f :: "'a :: {linorder_topology, conditionally_complete_linorder, no_top} \<Rightarrow>
- 'b::{linorder_topology, conditionally_complete_linorder}"
- assumes mono: "\<And>a b. a \<in> I \<Longrightarrow> b \<in> I \<Longrightarrow> x < a \<Longrightarrow> a \<le> b \<Longrightarrow> f a \<le> f b"
- and bnd: "\<And>a. a \<in> I \<Longrightarrow> x < a \<Longrightarrow> K \<le> f a"
- shows "(f \<longlongrightarrow> Inf (f ` ({x<..} \<inter> I))) (at x within ({x<..} \<inter> I))"
-proof (cases "{x<..} \<inter> I = {}")
- case True
- then show ?thesis by simp
-next
- case False
- show ?thesis
- proof (rule order_tendstoI)
- fix a
- assume a: "a < Inf (f ` ({x<..} \<inter> I))"
- {
- fix y
- assume "y \<in> {x<..} \<inter> I"
- with False bnd have "Inf (f ` ({x<..} \<inter> I)) \<le> f y"
- by (auto intro!: cInf_lower bdd_belowI2)
- with a have "a < f y"
- by (blast intro: less_le_trans)
- }
- then show "eventually (\<lambda>x. a < f x) (at x within ({x<..} \<inter> I))"
- by (auto simp: eventually_at_filter intro: exI[of _ 1] zero_less_one)
- next
- fix a
- assume "Inf (f ` ({x<..} \<inter> I)) < a"
- from cInf_lessD[OF _ this] False obtain y where y: "x < y" "y \<in> I" "f y < a"
- by auto
- then have "eventually (\<lambda>x. x \<in> I \<longrightarrow> f x < a) (at_right x)"
- unfolding eventually_at_right[OF \<open>x < y\<close>] by (metis less_imp_le le_less_trans mono)
- then show "eventually (\<lambda>x. f x < a) (at x within ({x<..} \<inter> I))"
- unfolding eventually_at_filter by eventually_elim simp
- qed
-qed
-
-text \<open>Another limit point characterization.\<close>
-
-lemma limpt_sequential_inj:
- fixes x :: "'a::metric_space"
- shows "x islimpt S \<longleftrightarrow>
- (\<exists>f. (\<forall>n::nat. f n \<in> S - {x}) \<and> inj f \<and> (f \<longlongrightarrow> x) sequentially)"
- (is "?lhs = ?rhs")
-proof
- assume ?lhs
- then have "\<forall>e>0. \<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e"
- by (force simp: islimpt_approachable)
- then obtain y where y: "\<And>e. e>0 \<Longrightarrow> y e \<in> S \<and> y e \<noteq> x \<and> dist (y e) x < e"
- by metis
- define f where "f \<equiv> rec_nat (y 1) (\<lambda>n fn. y (min (inverse(2 ^ (Suc n))) (dist fn x)))"
- have [simp]: "f 0 = y 1"
- "f(Suc n) = y (min (inverse(2 ^ (Suc n))) (dist (f n) x))" for n
- by (simp_all add: f_def)
- have f: "f n \<in> S \<and> (f n \<noteq> x) \<and> dist (f n) x < inverse(2 ^ n)" for n
- proof (induction n)
- case 0 show ?case
- by (simp add: y)
- next
- case (Suc n) then show ?case
- apply (auto simp: y)
- by (metis half_gt_zero_iff inverse_positive_iff_positive less_divide_eq_numeral1(1) min_less_iff_conj y zero_less_dist_iff zero_less_numeral zero_less_power)
- qed
- show ?rhs
- proof (rule_tac x=f in exI, intro conjI allI)
- show "\<And>n. f n \<in> S - {x}"
- using f by blast
- have "dist (f n) x < dist (f m) x" if "m < n" for m n
- using that
- proof (induction n)
- case 0 then show ?case by simp
- next
- case (Suc n)
- then consider "m < n" | "m = n" using less_Suc_eq by blast
- then show ?case
- proof cases
- assume "m < n"
- have "dist (f(Suc n)) x = dist (y (min (inverse(2 ^ (Suc n))) (dist (f n) x))) x"
- by simp
- also have "\<dots> < dist (f n) x"
- by (metis dist_pos_lt f min.strict_order_iff min_less_iff_conj y)
- also have "\<dots> < dist (f m) x"
- using Suc.IH \<open>m < n\<close> by blast
- finally show ?thesis .
- next
- assume "m = n" then show ?case
- by simp (metis dist_pos_lt f half_gt_zero_iff inverse_positive_iff_positive min_less_iff_conj y zero_less_numeral zero_less_power)
- qed
- qed
- then show "inj f"
- by (metis less_irrefl linorder_injI)
- show "f \<longlonglongrightarrow> x"
- apply (rule tendstoI)
- apply (rule_tac c="nat (ceiling(1/e))" in eventually_sequentiallyI)
- apply (rule less_trans [OF f [THEN conjunct2, THEN conjunct2]])
- apply (simp add: field_simps)
- by (meson le_less_trans mult_less_cancel_left not_le of_nat_less_two_power)
- qed
-next
- assume ?rhs
- then show ?lhs
- by (fastforce simp add: islimpt_approachable lim_sequentially)
-qed
-
-(*could prove directly from islimpt_sequential_inj, but only for metric spaces*)
-lemma islimpt_sequential:
- fixes x :: "'a::first_countable_topology"
- shows "x islimpt S \<longleftrightarrow> (\<exists>f. (\<forall>n::nat. f n \<in> S - {x}) \<and> (f \<longlongrightarrow> x) sequentially)"
- (is "?lhs = ?rhs")
-proof
- assume ?lhs
- from countable_basis_at_decseq[of x] obtain A where A:
- "\<And>i. open (A i)"
- "\<And>i. x \<in> A i"
- "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> eventually (\<lambda>i. A i \<subseteq> S) sequentially"
- by blast
- define f where "f n = (SOME y. y \<in> S \<and> y \<in> A n \<and> x \<noteq> y)" for n
- {
- fix n
- from \<open>?lhs\<close> have "\<exists>y. y \<in> S \<and> y \<in> A n \<and> x \<noteq> y"
- unfolding islimpt_def using A(1,2)[of n] by auto
- then have "f n \<in> S \<and> f n \<in> A n \<and> x \<noteq> f n"
- unfolding f_def by (rule someI_ex)
- then have "f n \<in> S" "f n \<in> A n" "x \<noteq> f n" by auto
- }
- then have "\<forall>n. f n \<in> S - {x}" by auto
- moreover have "(\<lambda>n. f n) \<longlonglongrightarrow> x"
- proof (rule topological_tendstoI)
- fix S
- assume "open S" "x \<in> S"
- from A(3)[OF this] \<open>\<And>n. f n \<in> A n\<close>
- show "eventually (\<lambda>x. f x \<in> S) sequentially"
- by (auto elim!: eventually_mono)
- qed
- ultimately show ?rhs by fast
-next
- assume ?rhs
- then obtain f :: "nat \<Rightarrow> 'a" where f: "\<And>n. f n \<in> S - {x}" and lim: "f \<longlonglongrightarrow> x"
- by auto
- show ?lhs
- unfolding islimpt_def
- proof safe
- fix T
- assume "open T" "x \<in> T"
- from lim[THEN topological_tendstoD, OF this] f
- show "\<exists>y\<in>S. y \<in> T \<and> y \<noteq> x"
- unfolding eventually_sequentially by auto
- qed
-qed
-
-lemma Lim_null:
- fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
- shows "(f \<longlongrightarrow> l) net \<longleftrightarrow> ((\<lambda>x. f(x) - l) \<longlongrightarrow> 0) net"
- by (simp add: Lim dist_norm)
-
-lemma Lim_null_comparison:
- fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
- assumes "eventually (\<lambda>x. norm (f x) \<le> g x) net" "(g \<longlongrightarrow> 0) net"
- shows "(f \<longlongrightarrow> 0) net"
- using assms(2)
-proof (rule metric_tendsto_imp_tendsto)
- show "eventually (\<lambda>x. dist (f x) 0 \<le> dist (g x) 0) net"
- using assms(1) by (rule eventually_mono) (simp add: dist_norm)
-qed
-
-lemma Lim_transform_bound:
- fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
- and g :: "'a \<Rightarrow> 'c::real_normed_vector"
- assumes "eventually (\<lambda>n. norm (f n) \<le> norm (g n)) net"
- and "(g \<longlongrightarrow> 0) net"
- shows "(f \<longlongrightarrow> 0) net"
- using assms(1) tendsto_norm_zero [OF assms(2)]
- by (rule Lim_null_comparison)
-
-lemma lim_null_mult_right_bounded:
- fixes f :: "'a \<Rightarrow> 'b::real_normed_div_algebra"
- assumes f: "(f \<longlongrightarrow> 0) F" and g: "eventually (\<lambda>x. norm(g x) \<le> B) F"
- shows "((\<lambda>z. f z * g z) \<longlongrightarrow> 0) F"
-proof -
- have *: "((\<lambda>x. norm (f x) * B) \<longlongrightarrow> 0) F"
- by (simp add: f tendsto_mult_left_zero tendsto_norm_zero)
- have "((\<lambda>x. norm (f x) * norm (g x)) \<longlongrightarrow> 0) F"
- apply (rule Lim_null_comparison [OF _ *])
- apply (simp add: eventually_mono [OF g] mult_left_mono)
- done
- then show ?thesis
- by (subst tendsto_norm_zero_iff [symmetric]) (simp add: norm_mult)
-qed
-
-lemma lim_null_mult_left_bounded:
- fixes f :: "'a \<Rightarrow> 'b::real_normed_div_algebra"
- assumes g: "eventually (\<lambda>x. norm(g x) \<le> B) F" and f: "(f \<longlongrightarrow> 0) F"
- shows "((\<lambda>z. g z * f z) \<longlongrightarrow> 0) F"
-proof -
- have *: "((\<lambda>x. B * norm (f x)) \<longlongrightarrow> 0) F"
- by (simp add: f tendsto_mult_right_zero tendsto_norm_zero)
- have "((\<lambda>x. norm (g x) * norm (f x)) \<longlongrightarrow> 0) F"
- apply (rule Lim_null_comparison [OF _ *])
- apply (simp add: eventually_mono [OF g] mult_right_mono)
- done
- then show ?thesis
- by (subst tendsto_norm_zero_iff [symmetric]) (simp add: norm_mult)
-qed
-
-lemma lim_null_scaleR_bounded:
- assumes f: "(f \<longlongrightarrow> 0) net" and gB: "eventually (\<lambda>a. f a = 0 \<or> norm(g a) \<le> B) net"
- shows "((\<lambda>n. f n *\<^sub>R g n) \<longlongrightarrow> 0) net"
-proof
- fix \<epsilon>::real
- assume "0 < \<epsilon>"
- then have B: "0 < \<epsilon> / (abs B + 1)" by simp
- have *: "\<bar>f x\<bar> * norm (g x) < \<epsilon>" if f: "\<bar>f x\<bar> * (\<bar>B\<bar> + 1) < \<epsilon>" and g: "norm (g x) \<le> B" for x
- proof -
- have "\<bar>f x\<bar> * norm (g x) \<le> \<bar>f x\<bar> * B"
- by (simp add: mult_left_mono g)
- also have "\<dots> \<le> \<bar>f x\<bar> * (\<bar>B\<bar> + 1)"
- by (simp add: mult_left_mono)
- also have "\<dots> < \<epsilon>"
- by (rule f)
- finally show ?thesis .
- qed
- show "\<forall>\<^sub>F x in net. dist (f x *\<^sub>R g x) 0 < \<epsilon>"
- apply (rule eventually_mono [OF eventually_conj [OF tendstoD [OF f B] gB] ])
- apply (auto simp: \<open>0 < \<epsilon>\<close> divide_simps * split: if_split_asm)
- done
-qed
-
-text\<open>Deducing things about the limit from the elements.\<close>
-
-lemma Lim_in_closed_set:
- assumes "closed S"
- and "eventually (\<lambda>x. f(x) \<in> S) net"
- and "\<not> trivial_limit net" "(f \<longlongrightarrow> l) net"
- shows "l \<in> S"
-proof (rule ccontr)
- assume "l \<notin> S"
- with \<open>closed S\<close> have "open (- S)" "l \<in> - S"
- by (simp_all add: open_Compl)
- with assms(4) have "eventually (\<lambda>x. f x \<in> - S) net"
- by (rule topological_tendstoD)
- with assms(2) have "eventually (\<lambda>x. False) net"
- by (rule eventually_elim2) simp
- with assms(3) show "False"
- by (simp add: eventually_False)
-qed
-
-text\<open>Need to prove closed(cball(x,e)) before deducing this as a corollary.\<close>
-
-lemma Lim_dist_ubound:
- assumes "\<not>(trivial_limit net)"
- and "(f \<longlongrightarrow> l) net"
- and "eventually (\<lambda>x. dist a (f x) \<le> e) net"
- shows "dist a l \<le> e"
- using assms by (fast intro: tendsto_le tendsto_intros)
-
-lemma Lim_norm_ubound:
- fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
- assumes "\<not>(trivial_limit net)" "(f \<longlongrightarrow> l) net" "eventually (\<lambda>x. norm(f x) \<le> e) net"
- shows "norm(l) \<le> e"
- using assms by (fast intro: tendsto_le tendsto_intros)
-
-lemma Lim_norm_lbound:
- fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
- assumes "\<not> trivial_limit net"
- and "(f \<longlongrightarrow> l) net"
- and "eventually (\<lambda>x. e \<le> norm (f x)) net"
- shows "e \<le> norm l"
- using assms by (fast intro: tendsto_le tendsto_intros)
-
-text\<open>Limit under bilinear function\<close>
-
-lemma Lim_bilinear:
- assumes "(f \<longlongrightarrow> l) net"
- and "(g \<longlongrightarrow> m) net"
- and "bounded_bilinear h"
- shows "((\<lambda>x. h (f x) (g x)) \<longlongrightarrow> (h l m)) net"
- using \<open>bounded_bilinear h\<close> \<open>(f \<longlongrightarrow> l) net\<close> \<open>(g \<longlongrightarrow> m) net\<close>
- by (rule bounded_bilinear.tendsto)
-
-text\<open>These are special for limits out of the same vector space.\<close>
-
-lemma Lim_within_id: "(id \<longlongrightarrow> a) (at a within s)"
- unfolding id_def by (rule tendsto_ident_at)
-
-lemma Lim_at_id: "(id \<longlongrightarrow> a) (at a)"
- unfolding id_def by (rule tendsto_ident_at)
-
-lemma Lim_at_zero:
- fixes a :: "'a::real_normed_vector"
- and l :: "'b::topological_space"
- shows "(f \<longlongrightarrow> l) (at a) \<longleftrightarrow> ((\<lambda>x. f(a + x)) \<longlongrightarrow> l) (at 0)"
- using LIM_offset_zero LIM_offset_zero_cancel ..
-
-text\<open>It's also sometimes useful to extract the limit point from the filter.\<close>
-
-abbreviation netlimit :: "'a::t2_space filter \<Rightarrow> 'a"
- where "netlimit F \<equiv> Lim F (\<lambda>x. x)"
-
-lemma netlimit_within: "\<not> trivial_limit (at a within S) \<Longrightarrow> netlimit (at a within S) = a"
- by (rule tendsto_Lim) (auto intro: tendsto_intros)
-
-lemma netlimit_at [simp]:
- fixes a :: "'a::{perfect_space,t2_space}"
- shows "netlimit (at a) = a"
- using netlimit_within [of a UNIV] by simp
-
-lemma lim_within_interior:
- "x \<in> interior S \<Longrightarrow> (f \<longlongrightarrow> l) (at x within S) \<longleftrightarrow> (f \<longlongrightarrow> l) (at x)"
- by (metis at_within_interior)
-
-lemma netlimit_within_interior:
- fixes x :: "'a::{t2_space,perfect_space}"
- assumes "x \<in> interior S"
- shows "netlimit (at x within S) = x"
- using assms by (metis at_within_interior netlimit_at)
-
-lemma netlimit_at_vector:
- fixes a :: "'a::real_normed_vector"
- shows "netlimit (at a) = a"
-proof (cases "\<exists>x. x \<noteq> a")
- case True then obtain x where x: "x \<noteq> a" ..
- have "\<not> trivial_limit (at a)"
- unfolding trivial_limit_def eventually_at dist_norm
- apply clarsimp
- apply (rule_tac x="a + scaleR (d / 2) (sgn (x - a))" in exI)
- apply (simp add: norm_sgn sgn_zero_iff x)
- done
- then show ?thesis
- by (rule netlimit_within [of a UNIV])
-qed simp
-
-
-text\<open>Useful lemmas on closure and set of possible sequential limits.\<close>
-
-lemma closure_sequential:
- fixes l :: "'a::first_countable_topology"
- shows "l \<in> closure S \<longleftrightarrow> (\<exists>x. (\<forall>n. x n \<in> S) \<and> (x \<longlongrightarrow> l) sequentially)"
- (is "?lhs = ?rhs")
-proof
- assume "?lhs"
- moreover
- {
- assume "l \<in> S"
- then have "?rhs" using tendsto_const[of l sequentially] by auto
- }
- moreover
- {
- assume "l islimpt S"
- then have "?rhs" unfolding islimpt_sequential by auto
- }
- ultimately show "?rhs"
- unfolding closure_def by auto
-next
- assume "?rhs"
- then show "?lhs" unfolding closure_def islimpt_sequential by auto
-qed
-
-lemma closed_sequential_limits:
- fixes S :: "'a::first_countable_topology set"
- shows "closed S \<longleftrightarrow> (\<forall>x l. (\<forall>n. x n \<in> S) \<and> (x \<longlongrightarrow> l) sequentially \<longrightarrow> l \<in> S)"
-by (metis closure_sequential closure_subset_eq subset_iff)
-
-lemma closure_approachable:
- fixes S :: "'a::metric_space set"
- shows "x \<in> closure S \<longleftrightarrow> (\<forall>e>0. \<exists>y\<in>S. dist y x < e)"
- apply (auto simp: closure_def islimpt_approachable)
- apply (metis dist_self)
- done
-
-lemma closure_approachable_le:
- fixes S :: "'a::metric_space set"
- shows "x \<in> closure S \<longleftrightarrow> (\<forall>e>0. \<exists>y\<in>S. dist y x \<le> e)"
- unfolding closure_approachable
- using dense by force
-
-lemma closure_approachableD:
- assumes "x \<in> closure S" "e>0"
- shows "\<exists>y\<in>S. dist x y < e"
- using assms unfolding closure_approachable by (auto simp: dist_commute)
-
-lemma closed_approachable:
- fixes S :: "'a::metric_space set"
- shows "closed S \<Longrightarrow> (\<forall>e>0. \<exists>y\<in>S. dist y x < e) \<longleftrightarrow> x \<in> S"
- by (metis closure_closed closure_approachable)
-
-lemma closure_contains_Inf:
- fixes S :: "real set"
- assumes "S \<noteq> {}" "bdd_below S"
- shows "Inf S \<in> closure S"
-proof -
- have *: "\<forall>x\<in>S. Inf S \<le> x"
- using cInf_lower[of _ S] assms by metis
- {
- fix e :: real
- assume "e > 0"
- then have "Inf S < Inf S + e" by simp
- with assms obtain x where "x \<in> S" "x < Inf S + e"
- by (subst (asm) cInf_less_iff) auto
- with * have "\<exists>x\<in>S. dist x (Inf S) < e"
- by (intro bexI[of _ x]) (auto simp: dist_real_def)
- }
- then show ?thesis unfolding closure_approachable by auto
-qed
-
-lemma closure_Int_ballI:
- fixes S :: "'a :: metric_space set"
- assumes "\<And>U. \<lbrakk>openin (subtopology euclidean S) U; U \<noteq> {}\<rbrakk> \<Longrightarrow> T \<inter> U \<noteq> {}"
- shows "S \<subseteq> closure T"
-proof (clarsimp simp: closure_approachable dist_commute)
- fix x and e::real
- assume "x \<in> S" "0 < e"
- with assms [of "S \<inter> ball x e"] show "\<exists>y\<in>T. dist x y < e"
- by force
-qed
-
-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 not_trivial_limit_within_ball:
- "\<not> trivial_limit (at x within S) \<longleftrightarrow> (\<forall>e>0. S \<inter> ball x e - {x} \<noteq> {})"
- (is "?lhs \<longleftrightarrow> ?rhs")
-proof
- show ?rhs if ?lhs
- proof -
- {
- fix e :: real
- assume "e > 0"
- then obtain y where "y \<in> S - {x}" and "dist y x < e"
- using \<open>?lhs\<close> not_trivial_limit_within[of x S] closure_approachable[of x "S - {x}"]
- by auto
- then have "y \<in> S \<inter> ball x e - {x}"
- unfolding ball_def by (simp add: dist_commute)
- then have "S \<inter> ball x e - {x} \<noteq> {}" by blast
- }
- then show ?thesis by auto
- qed
- show ?lhs if ?rhs
- proof -
- {
- fix e :: real
- assume "e > 0"
- then obtain y where "y \<in> S \<inter> ball x e - {x}"
- using \<open>?rhs\<close> by blast
- then have "y \<in> S - {x}" and "dist y x < e"
- unfolding ball_def by (simp_all add: dist_commute)
- then have "\<exists>y \<in> S - {x}. dist y x < e"
- by auto
- }
- then show ?thesis
- using not_trivial_limit_within[of x S] closure_approachable[of x "S - {x}"]
- by auto
- qed
-qed
-
-lemma tendsto_If_within_closures:
- assumes f: "x \<in> s \<union> (closure s \<inter> closure t) \<Longrightarrow>
- (f \<longlongrightarrow> l x) (at x within s \<union> (closure s \<inter> closure t))"
- assumes g: "x \<in> t \<union> (closure s \<inter> closure t) \<Longrightarrow>
- (g \<longlongrightarrow> l x) (at x within t \<union> (closure s \<inter> closure t))"
- assumes "x \<in> s \<union> t"
- shows "((\<lambda>x. if x \<in> s then f x else g x) \<longlongrightarrow> l x) (at x within s \<union> t)"
-proof -
- have *: "(s \<union> t) \<inter> {x. x \<in> s} = s" "(s \<union> t) \<inter> {x. x \<notin> s} = t - s"
- by auto
- have "(f \<longlongrightarrow> l x) (at x within s)"
- by (rule filterlim_at_within_closure_implies_filterlim)
- (use \<open>x \<in> _\<close> in \<open>auto simp: inf_commute closure_def intro: tendsto_within_subset[OF f]\<close>)
- moreover
- have "(g \<longlongrightarrow> l x) (at x within t - s)"
- by (rule filterlim_at_within_closure_implies_filterlim)
- (use \<open>x \<in> _\<close> in
- \<open>auto intro!: tendsto_within_subset[OF g] simp: closure_def intro: islimpt_subset\<close>)
- ultimately show ?thesis
- by (intro filterlim_at_within_If) (simp_all only: *)
-qed
-
-
-subsection \<open>Boundedness\<close>
-
- (* FIXME: This has to be unified with BSEQ!! *)
-definition%important (in metric_space) bounded :: "'a set \<Rightarrow> bool"
- where "bounded S \<longleftrightarrow> (\<exists>x e. \<forall>y\<in>S. dist x y \<le> e)"
-
-lemma bounded_subset_cball: "bounded S \<longleftrightarrow> (\<exists>e x. S \<subseteq> cball x e \<and> 0 \<le> e)"
- unfolding bounded_def subset_eq by auto (meson order_trans zero_le_dist)
-
-lemma bounded_any_center: "bounded S \<longleftrightarrow> (\<exists>e. \<forall>y\<in>S. dist a y \<le> e)"
- unfolding bounded_def
- by auto (metis add.commute add_le_cancel_right dist_commute dist_triangle_le)
-
-lemma bounded_iff: "bounded S \<longleftrightarrow> (\<exists>a. \<forall>x\<in>S. norm x \<le> a)"
- unfolding bounded_any_center [where a=0]
- by (simp add: dist_norm)
-
-lemma bdd_above_norm: "bdd_above (norm ` X) \<longleftrightarrow> bounded X"
- by (simp add: bounded_iff bdd_above_def)
-
-lemma bounded_norm_comp: "bounded ((\<lambda>x. norm (f x)) ` S) = bounded (f ` S)"
- by (simp add: bounded_iff)
-
-lemma boundedI:
- assumes "\<And>x. x \<in> S \<Longrightarrow> norm x \<le> B"
- shows "bounded S"
- using assms bounded_iff by blast
-
-lemma bounded_empty [simp]: "bounded {}"
- by (simp add: bounded_def)
-
-lemma bounded_subset: "bounded T \<Longrightarrow> S \<subseteq> T \<Longrightarrow> bounded S"
- by (metis bounded_def subset_eq)
-
-lemma bounded_interior[intro]: "bounded S \<Longrightarrow> bounded(interior S)"
- by (metis bounded_subset interior_subset)
-
-lemma bounded_closure[intro]:
- assumes "bounded S"
- shows "bounded (closure S)"
-proof -
- from assms obtain x and a where a: "\<forall>y\<in>S. dist x y \<le> a"
- unfolding bounded_def by auto
- {
- fix y
- assume "y \<in> closure S"
- then obtain f where f: "\<forall>n. f n \<in> S" "(f \<longlongrightarrow> y) sequentially"
- unfolding closure_sequential by auto
- have "\<forall>n. f n \<in> S \<longrightarrow> dist x (f n) \<le> a" using a by simp
- then have "eventually (\<lambda>n. dist x (f n) \<le> a) sequentially"
- by (simp add: f(1))
- have "dist x y \<le> a"
- apply (rule Lim_dist_ubound [of sequentially f])
- apply (rule trivial_limit_sequentially)
- apply (rule f(2))
- apply fact
- done
- }
- then show ?thesis
- unfolding bounded_def by auto
-qed
-
-lemma bounded_closure_image: "bounded (f ` closure S) \<Longrightarrow> bounded (f ` S)"
- by (simp add: bounded_subset closure_subset image_mono)
-
-lemma bounded_cball[simp,intro]: "bounded (cball x e)"
- apply (simp add: bounded_def)
- apply (rule_tac x=x in exI)
- apply (rule_tac x=e in exI, auto)
- done
-
-lemma bounded_ball[simp,intro]: "bounded (ball x e)"
- by (metis ball_subset_cball bounded_cball bounded_subset)
-
-lemma bounded_Un[simp]: "bounded (S \<union> T) \<longleftrightarrow> bounded S \<and> bounded T"
- by (auto simp: bounded_def) (metis Un_iff bounded_any_center le_max_iff_disj)
-
-lemma bounded_Union[intro]: "finite F \<Longrightarrow> \<forall>S\<in>F. bounded S \<Longrightarrow> bounded (\<Union>F)"
- by (induct rule: finite_induct[of F]) auto
-
-lemma bounded_UN [intro]: "finite A \<Longrightarrow> \<forall>x\<in>A. bounded (B x) \<Longrightarrow> bounded (\<Union>x\<in>A. B x)"
- by (induct set: finite) auto
-
-lemma bounded_insert [simp]: "bounded (insert x S) \<longleftrightarrow> bounded S"
-proof -
- have "\<forall>y\<in>{x}. dist x y \<le> 0"
- by simp
- then have "bounded {x}"
- unfolding bounded_def by fast
- then show ?thesis
- by (metis insert_is_Un bounded_Un)
-qed
-
-lemma bounded_subset_ballI: "S \<subseteq> ball x r \<Longrightarrow> bounded S"
- by (meson bounded_ball bounded_subset)
-
-lemma bounded_subset_ballD:
- assumes "bounded S" shows "\<exists>r. 0 < r \<and> S \<subseteq> ball x r"
-proof -
- obtain e::real and y where "S \<subseteq> cball y e" "0 \<le> e"
- using assms by (auto simp: bounded_subset_cball)
- then show ?thesis
- apply (rule_tac x="dist x y + e + 1" in exI)
- apply (simp add: add.commute add_pos_nonneg)
- apply (erule subset_trans)
- apply (clarsimp simp add: cball_def)
- by (metis add_le_cancel_right add_strict_increasing dist_commute dist_triangle_le zero_less_one)
-qed
-
-lemma finite_imp_bounded [intro]: "finite S \<Longrightarrow> bounded S"
- by (induct set: finite) simp_all
-
-lemma bounded_pos: "bounded S \<longleftrightarrow> (\<exists>b>0. \<forall>x\<in> S. norm x \<le> b)"
- apply (simp add: bounded_iff)
- apply (subgoal_tac "\<And>x (y::real). 0 < 1 + \<bar>y\<bar> \<and> (x \<le> y \<longrightarrow> x \<le> 1 + \<bar>y\<bar>)")
- apply metis
- apply arith
- done
-
-lemma bounded_pos_less: "bounded S \<longleftrightarrow> (\<exists>b>0. \<forall>x\<in> S. norm x < b)"
- apply (simp add: bounded_pos)
- apply (safe; rule_tac x="b+1" in exI; force)
- done
-
-lemma Bseq_eq_bounded:
- fixes f :: "nat \<Rightarrow> 'a::real_normed_vector"
- shows "Bseq f \<longleftrightarrow> bounded (range f)"
- unfolding Bseq_def bounded_pos by auto
-
-lemma bounded_Int[intro]: "bounded S \<or> bounded T \<Longrightarrow> bounded (S \<inter> T)"
- by (metis Int_lower1 Int_lower2 bounded_subset)
-
-lemma bounded_diff[intro]: "bounded S \<Longrightarrow> bounded (S - T)"
- by (metis Diff_subset bounded_subset)
-
-lemma not_bounded_UNIV[simp]:
- "\<not> bounded (UNIV :: 'a::{real_normed_vector, perfect_space} set)"
-proof (auto simp: bounded_pos not_le)
- obtain x :: 'a where "x \<noteq> 0"
- using perfect_choose_dist [OF zero_less_one] by fast
- fix b :: real
- assume b: "b >0"
- have b1: "b +1 \<ge> 0"
- using b by simp
- with \<open>x \<noteq> 0\<close> have "b < norm (scaleR (b + 1) (sgn x))"
- by (simp add: norm_sgn)
- then show "\<exists>x::'a. b < norm x" ..
-qed
-
-corollary cobounded_imp_unbounded:
- fixes S :: "'a::{real_normed_vector, perfect_space} set"
- shows "bounded (- S) \<Longrightarrow> \<not> bounded S"
- using bounded_Un [of S "-S"] by (simp add: sup_compl_top)
-
-lemma bounded_dist_comp:
- assumes "bounded (f ` S)" "bounded (g ` S)"
- shows "bounded ((\<lambda>x. dist (f x) (g x)) ` S)"
-proof -
- from assms obtain M1 M2 where *: "dist (f x) undefined \<le> M1" "dist undefined (g x) \<le> M2" if "x \<in> S" for x
- by (auto simp: bounded_any_center[of _ undefined] dist_commute)
- have "dist (f x) (g x) \<le> M1 + M2" if "x \<in> S" for x
- using *[OF that]
- by (rule order_trans[OF dist_triangle add_mono])
- then show ?thesis
- by (auto intro!: boundedI)
-qed
-
-lemma bounded_linear_image:
- assumes "bounded S"
- and "bounded_linear f"
- shows "bounded (f ` S)"
-proof -
- from assms(1) obtain b where "b > 0" and b: "\<forall>x\<in>S. norm x \<le> b"
- unfolding bounded_pos by auto
- from assms(2) obtain B where B: "B > 0" "\<forall>x. norm (f x) \<le> B * norm x"
- using bounded_linear.pos_bounded by (auto simp: ac_simps)
- show ?thesis
- unfolding bounded_pos
- proof (intro exI, safe)
- show "norm (f x) \<le> B * b" if "x \<in> S" for x
- by (meson B b less_imp_le mult_left_mono order_trans that)
- qed (use \<open>b > 0\<close> \<open>B > 0\<close> in auto)
-qed
-
-lemma bounded_scaling:
- fixes S :: "'a::real_normed_vector set"
- shows "bounded S \<Longrightarrow> bounded ((\<lambda>x. c *\<^sub>R x) ` S)"
- apply (rule bounded_linear_image, assumption)
- apply (rule bounded_linear_scaleR_right)
- done
-
-lemma bounded_scaleR_comp:
- fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
- assumes "bounded (f ` S)"
- shows "bounded ((\<lambda>x. r *\<^sub>R f x) ` S)"
- using bounded_scaling[of "f ` S" r] assms
- by (auto simp: image_image)
-
-lemma bounded_translation:
- fixes S :: "'a::real_normed_vector set"
- assumes "bounded S"
- shows "bounded ((\<lambda>x. a + x) ` S)"
-proof -
- from assms obtain b where b: "b > 0" "\<forall>x\<in>S. norm x \<le> b"
- unfolding bounded_pos by auto
- {
- fix x
- assume "x \<in> S"
- then have "norm (a + x) \<le> b + norm a"
- using norm_triangle_ineq[of a x] b by auto
- }
- then show ?thesis
- unfolding bounded_pos
- using norm_ge_zero[of a] b(1) and add_strict_increasing[of b 0 "norm a"]
- by (auto intro!: exI[of _ "b + norm a"])
-qed
-
-lemma bounded_translation_minus:
- fixes S :: "'a::real_normed_vector set"
- shows "bounded S \<Longrightarrow> bounded ((\<lambda>x. x - a) ` S)"
-using bounded_translation [of S "-a"] by simp
-
-lemma bounded_uminus [simp]:
- fixes X :: "'a::real_normed_vector set"
- shows "bounded (uminus ` X) \<longleftrightarrow> bounded X"
-by (auto simp: bounded_def dist_norm; rule_tac x="-x" in exI; force simp: add.commute norm_minus_commute)
-
-lemma uminus_bounded_comp [simp]:
- fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
- shows "bounded ((\<lambda>x. - f x) ` S) \<longleftrightarrow> bounded (f ` S)"
- using bounded_uminus[of "f ` S"]
- by (auto simp: image_image)
-
-lemma bounded_plus_comp:
- fixes f g::"'a \<Rightarrow> 'b::real_normed_vector"
- assumes "bounded (f ` S)"
- assumes "bounded (g ` S)"
- shows "bounded ((\<lambda>x. f x + g x) ` S)"
-proof -
- {
- fix B C
- assume "\<And>x. x\<in>S \<Longrightarrow> norm (f x) \<le> B" "\<And>x. x\<in>S \<Longrightarrow> norm (g x) \<le> C"
- then have "\<And>x. x \<in> S \<Longrightarrow> norm (f x + g x) \<le> B + C"
- by (auto intro!: norm_triangle_le add_mono)
- } then show ?thesis
- using assms by (fastforce simp: bounded_iff)
-qed
-
-lemma bounded_plus:
- fixes S ::"'a::real_normed_vector set"
- assumes "bounded S" "bounded T"
- shows "bounded ((\<lambda>(x,y). x + y) ` (S \<times> T))"
- using bounded_plus_comp [of fst "S \<times> T" snd] assms
- by (auto simp: split_def split: if_split_asm)
-
-lemma bounded_minus_comp:
- "bounded (f ` S) \<Longrightarrow> bounded (g ` S) \<Longrightarrow> bounded ((\<lambda>x. f x - g x) ` S)"
- for f g::"'a \<Rightarrow> 'b::real_normed_vector"
- using bounded_plus_comp[of "f" S "\<lambda>x. - g x"]
- by auto
-
-lemma bounded_minus:
- fixes S ::"'a::real_normed_vector set"
- assumes "bounded S" "bounded T"
- shows "bounded ((\<lambda>(x,y). x - y) ` (S \<times> T))"
- using bounded_minus_comp [of fst "S \<times> T" snd] assms
- by (auto simp: split_def split: if_split_asm)
-
-
-subsection \<open>Compactness\<close>
-
-subsubsection \<open>Bolzano-Weierstrass property\<close>
-
-proposition heine_borel_imp_bolzano_weierstrass:
- assumes "compact s"
- and "infinite t"
- and "t \<subseteq> s"
- shows "\<exists>x \<in> s. x islimpt t"
-proof (rule ccontr)
- assume "\<not> (\<exists>x \<in> s. x islimpt t)"
- then obtain f where f: "\<forall>x\<in>s. x \<in> f x \<and> open (f x) \<and> (\<forall>y\<in>t. y \<in> f x \<longrightarrow> y = x)"
- unfolding islimpt_def
- using bchoice[of s "\<lambda> x T. x \<in> T \<and> open T \<and> (\<forall>y\<in>t. y \<in> T \<longrightarrow> y = x)"]
- by auto
- obtain g where g: "g \<subseteq> {t. \<exists>x. x \<in> s \<and> t = f x}" "finite g" "s \<subseteq> \<Union>g"
- using assms(1)[unfolded compact_eq_heine_borel, THEN spec[where x="{t. \<exists>x. x\<in>s \<and> t = f x}"]]
- using f by auto
- from g(1,3) have g':"\<forall>x\<in>g. \<exists>xa \<in> s. x = f xa"
- by auto
- {
- fix x y
- assume "x \<in> t" "y \<in> t" "f x = f y"
- then have "x \<in> f x" "y \<in> f x \<longrightarrow> y = x"
- using f[THEN bspec[where x=x]] and \<open>t \<subseteq> s\<close> by auto
- then have "x = y"
- using \<open>f x = f y\<close> and f[THEN bspec[where x=y]] and \<open>y \<in> t\<close> and \<open>t \<subseteq> s\<close>
- by auto
- }
- then have "inj_on f t"
- unfolding inj_on_def by simp
- then have "infinite (f ` t)"
- using assms(2) using finite_imageD by auto
- moreover
- {
- fix x
- assume "x \<in> t" "f x \<notin> g"
- from g(3) assms(3) \<open>x \<in> t\<close> obtain h where "h \<in> g" and "x \<in> h"
- by auto
- then obtain y where "y \<in> s" "h = f y"
- using g'[THEN bspec[where x=h]] by auto
- then have "y = x"
- using f[THEN bspec[where x=y]] and \<open>x\<in>t\<close> and \<open>x\<in>h\<close>[unfolded \<open>h = f y\<close>]
- by auto
- then have False
- using \<open>f x \<notin> g\<close> \<open>h \<in> g\<close> unfolding \<open>h = f y\<close>
- by auto
- }
- then have "f ` t \<subseteq> g" by auto
- ultimately show False
- using g(2) using finite_subset by auto
-qed
-
-lemma acc_point_range_imp_convergent_subsequence:
- fixes l :: "'a :: first_countable_topology"
- assumes l: "\<forall>U. l\<in>U \<longrightarrow> open U \<longrightarrow> infinite (U \<inter> range f)"
- shows "\<exists>r::nat\<Rightarrow>nat. strict_mono r \<and> (f \<circ> r) \<longlonglongrightarrow> l"
-proof -
- from countable_basis_at_decseq[of l]
- obtain A where A:
- "\<And>i. open (A i)"
- "\<And>i. l \<in> A i"
- "\<And>S. open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>i. A i \<subseteq> S) sequentially"
- by blast
- define s where "s n i = (SOME j. i < j \<and> f j \<in> A (Suc n))" for n i
- {
- fix n i
- have "infinite (A (Suc n) \<inter> range f - f`{.. i})"
- using l A by auto
- then have "\<exists>x. x \<in> A (Suc n) \<inter> range f - f`{.. i}"
- unfolding ex_in_conv by (intro notI) simp
- then have "\<exists>j. f j \<in> A (Suc n) \<and> j \<notin> {.. i}"
- by auto
- then have "\<exists>a. i < a \<and> f a \<in> A (Suc n)"
- by (auto simp: not_le)
- then have "i < s n i" "f (s n i) \<in> A (Suc n)"
- unfolding s_def by (auto intro: someI2_ex)
- }
- note s = this
- define r where "r = rec_nat (s 0 0) s"
- have "strict_mono r"
- by (auto simp: r_def s strict_mono_Suc_iff)
- moreover
- have "(\<lambda>n. f (r n)) \<longlonglongrightarrow> l"
- proof (rule topological_tendstoI)
- fix S
- assume "open S" "l \<in> S"
- with A(3) have "eventually (\<lambda>i. A i \<subseteq> S) sequentially"
- by auto
- moreover
- {
- fix i
- assume "Suc 0 \<le> i"
- then have "f (r i) \<in> A i"
- by (cases i) (simp_all add: r_def s)
- }
- then have "eventually (\<lambda>i. f (r i) \<in> A i) sequentially"
- by (auto simp: eventually_sequentially)
- ultimately show "eventually (\<lambda>i. f (r i) \<in> S) sequentially"
- by eventually_elim auto
- qed
- ultimately show "\<exists>r::nat\<Rightarrow>nat. strict_mono r \<and> (f \<circ> r) \<longlonglongrightarrow> l"
- by (auto simp: convergent_def comp_def)
-qed
-
-lemma sequence_infinite_lemma:
- fixes f :: "nat \<Rightarrow> 'a::t1_space"
- assumes "\<forall>n. f n \<noteq> l"
- and "(f \<longlongrightarrow> l) sequentially"
- shows "infinite (range f)"
-proof
- assume "finite (range f)"
- then have "closed (range f)"
- by (rule finite_imp_closed)
- then have "open (- range f)"
- by (rule open_Compl)
- from assms(1) have "l \<in> - range f"
- by auto
- from assms(2) have "eventually (\<lambda>n. f n \<in> - range f) sequentially"
- using \<open>open (- range f)\<close> \<open>l \<in> - range f\<close>
- by (rule topological_tendstoD)
- then show False
- unfolding eventually_sequentially
- by auto
-qed
-
-lemma closure_insert:
- fixes x :: "'a::t1_space"
- shows "closure (insert x s) = insert x (closure s)"
- apply (rule closure_unique)
- apply (rule insert_mono [OF closure_subset])
- apply (rule closed_insert [OF closed_closure])
- apply (simp add: closure_minimal)
- done
-
-lemma islimpt_insert:
- fixes x :: "'a::t1_space"
- shows "x islimpt (insert a s) \<longleftrightarrow> x islimpt s"
-proof
- assume *: "x islimpt (insert a s)"
- show "x islimpt s"
- proof (rule islimptI)
- fix t
- assume t: "x \<in> t" "open t"
- show "\<exists>y\<in>s. y \<in> t \<and> y \<noteq> x"
- proof (cases "x = a")
- case True
- obtain y where "y \<in> insert a s" "y \<in> t" "y \<noteq> x"
- using * t by (rule islimptE)
- with \<open>x = a\<close> show ?thesis by auto
- next
- case False
- with t have t': "x \<in> t - {a}" "open (t - {a})"
- by (simp_all add: open_Diff)
- obtain y where "y \<in> insert a s" "y \<in> t - {a}" "y \<noteq> x"
- using * t' by (rule islimptE)
- then show ?thesis by auto
- qed
- qed
-next
- assume "x islimpt s"
- then show "x islimpt (insert a s)"
- by (rule islimpt_subset) auto
-qed
-
-lemma islimpt_finite:
- fixes x :: "'a::t1_space"
- shows "finite s \<Longrightarrow> \<not> x islimpt s"
- by (induct set: finite) (simp_all add: islimpt_insert)
-
-lemma islimpt_Un_finite:
- fixes x :: "'a::t1_space"
- shows "finite s \<Longrightarrow> x islimpt (s \<union> t) \<longleftrightarrow> x islimpt t"
- by (simp add: islimpt_Un islimpt_finite)
-
-lemma islimpt_eq_acc_point:
- fixes l :: "'a :: t1_space"
- shows "l islimpt S \<longleftrightarrow> (\<forall>U. l\<in>U \<longrightarrow> open U \<longrightarrow> infinite (U \<inter> S))"
-proof (safe intro!: islimptI)
- fix U
- assume "l islimpt S" "l \<in> U" "open U" "finite (U \<inter> S)"
- then have "l islimpt S" "l \<in> (U - (U \<inter> S - {l}))" "open (U - (U \<inter> S - {l}))"
- by (auto intro: finite_imp_closed)
- then show False
- by (rule islimptE) auto
-next
- fix T
- assume *: "\<forall>U. l\<in>U \<longrightarrow> open U \<longrightarrow> infinite (U \<inter> S)" "l \<in> T" "open T"
- then have "infinite (T \<inter> S - {l})"
- by auto
- then have "\<exists>x. x \<in> (T \<inter> S - {l})"
- unfolding ex_in_conv by (intro notI) simp
- then show "\<exists>y\<in>S. y \<in> T \<and> y \<noteq> l"
- by auto
-qed
-
-corollary infinite_openin:
- fixes S :: "'a :: t1_space set"
- shows "\<lbrakk>openin (subtopology euclidean U) S; x \<in> S; x islimpt U\<rbrakk> \<Longrightarrow> infinite S"
- by (clarsimp simp add: openin_open islimpt_eq_acc_point inf_commute)
-
-lemma islimpt_range_imp_convergent_subsequence:
- fixes l :: "'a :: {t1_space, first_countable_topology}"
- assumes l: "l islimpt (range f)"
- shows "\<exists>r::nat\<Rightarrow>nat. strict_mono r \<and> (f \<circ> r) \<longlonglongrightarrow> l"
- using l unfolding islimpt_eq_acc_point
- by (rule acc_point_range_imp_convergent_subsequence)
-
-lemma islimpt_eq_infinite_ball: "x islimpt S \<longleftrightarrow> (\<forall>e>0. infinite(S \<inter> ball x e))"
- apply (simp add: islimpt_eq_acc_point, safe)
- apply (metis Int_commute open_ball centre_in_ball)
- by (metis open_contains_ball Int_mono finite_subset inf_commute subset_refl)
-
-lemma islimpt_eq_infinite_cball: "x islimpt S \<longleftrightarrow> (\<forall>e>0. infinite(S \<inter> cball x e))"
- apply (simp add: islimpt_eq_infinite_ball, safe)
- apply (meson Int_mono ball_subset_cball finite_subset order_refl)
- by (metis open_ball centre_in_ball finite_Int inf.absorb_iff2 inf_assoc open_contains_cball_eq)
-
-lemma sequence_unique_limpt:
- fixes f :: "nat \<Rightarrow> 'a::t2_space"
- assumes "(f \<longlongrightarrow> l) sequentially"
- and "l' islimpt (range f)"
- shows "l' = l"
-proof (rule ccontr)
- assume "l' \<noteq> l"
- obtain s t where "open s" "open t" "l' \<in> s" "l \<in> t" "s \<inter> t = {}"
- using hausdorff [OF \<open>l' \<noteq> l\<close>] by auto
- have "eventually (\<lambda>n. f n \<in> t) sequentially"
- using assms(1) \<open>open t\<close> \<open>l \<in> t\<close> by (rule topological_tendstoD)
- then obtain N where "\<forall>n\<ge>N. f n \<in> t"
- unfolding eventually_sequentially by auto
-
- have "UNIV = {..<N} \<union> {N..}"
- by auto
- then have "l' islimpt (f ` ({..<N} \<union> {N..}))"
- using assms(2) by simp
- then have "l' islimpt (f ` {..<N} \<union> f ` {N..})"
- by (simp add: image_Un)
- then have "l' islimpt (f ` {N..})"
- by (simp add: islimpt_Un_finite)
- then obtain y where "y \<in> f ` {N..}" "y \<in> s" "y \<noteq> l'"
- using \<open>l' \<in> s\<close> \<open>open s\<close> by (rule islimptE)
- then obtain n where "N \<le> n" "f n \<in> s" "f n \<noteq> l'"
- by auto
- with \<open>\<forall>n\<ge>N. f n \<in> t\<close> have "f n \<in> s \<inter> t"
- by simp
- with \<open>s \<inter> t = {}\<close> show False
- by simp
-qed
-
-lemma bolzano_weierstrass_imp_closed:
- fixes s :: "'a::{first_countable_topology,t2_space} set"
- assumes "\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t)"
- shows "closed s"
-proof -
- {
- fix x l
- assume as: "\<forall>n::nat. x n \<in> s" "(x \<longlongrightarrow> l) sequentially"
- then have "l \<in> s"
- proof (cases "\<forall>n. x n \<noteq> l")
- case False
- then show "l\<in>s" using as(1) by auto
- next
- case True note cas = this
- with as(2) have "infinite (range x)"
- using sequence_infinite_lemma[of x l] by auto
- then obtain l' where "l'\<in>s" "l' islimpt (range x)"
- using assms[THEN spec[where x="range x"]] as(1) by auto
- then show "l\<in>s" using sequence_unique_limpt[of x l l']
- using as cas by auto
- qed
- }
- then show ?thesis
- unfolding closed_sequential_limits by fast
-qed
-
-lemma compact_imp_bounded:
- assumes "compact U"
- shows "bounded U"
-proof -
- have "compact U" "\<forall>x\<in>U. open (ball x 1)" "U \<subseteq> (\<Union>x\<in>U. ball x 1)"
- using assms by auto
- then obtain D where D: "D \<subseteq> U" "finite D" "U \<subseteq> (\<Union>x\<in>D. ball x 1)"
- by (metis compactE_image)
- from \<open>finite D\<close> have "bounded (\<Union>x\<in>D. ball x 1)"
- by (simp add: bounded_UN)
- then show "bounded U" using \<open>U \<subseteq> (\<Union>x\<in>D. ball x 1)\<close>
- by (rule bounded_subset)
-qed
-
-text\<open>In particular, some common special cases.\<close>
-
-lemma compact_Un [intro]:
- assumes "compact s"
- and "compact t"
- shows " compact (s \<union> t)"
-proof (rule compactI)
- fix f
- assume *: "Ball f open" "s \<union> t \<subseteq> \<Union>f"
- from * \<open>compact s\<close> obtain s' where "s' \<subseteq> f \<and> finite s' \<and> s \<subseteq> \<Union>s'"
- unfolding compact_eq_heine_borel by (auto elim!: allE[of _ f])
- moreover
- from * \<open>compact t\<close> obtain t' where "t' \<subseteq> f \<and> finite t' \<and> t \<subseteq> \<Union>t'"
- unfolding compact_eq_heine_borel by (auto elim!: allE[of _ f])
- ultimately show "\<exists>f'\<subseteq>f. finite f' \<and> s \<union> t \<subseteq> \<Union>f'"
- by (auto intro!: exI[of _ "s' \<union> t'"])
-qed
-
-lemma compact_Union [intro]: "finite S \<Longrightarrow> (\<And>T. T \<in> S \<Longrightarrow> compact T) \<Longrightarrow> compact (\<Union>S)"
- by (induct set: finite) auto
-
-lemma compact_UN [intro]:
- "finite A \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> compact (B x)) \<Longrightarrow> compact (\<Union>x\<in>A. B x)"
- by (rule compact_Union) auto
-
-lemma closed_Int_compact [intro]:
- assumes "closed s"
- and "compact t"
- shows "compact (s \<inter> t)"
- using compact_Int_closed [of t s] assms
- by (simp add: Int_commute)
-
-lemma compact_Int [intro]:
- fixes s t :: "'a :: t2_space set"
- assumes "compact s"
- and "compact t"
- shows "compact (s \<inter> t)"
- using assms by (intro compact_Int_closed compact_imp_closed)
-
-lemma compact_sing [simp]: "compact {a}"
- unfolding compact_eq_heine_borel by auto
-
-lemma compact_insert [simp]:
- assumes "compact s"
- shows "compact (insert x s)"
-proof -
- have "compact ({x} \<union> s)"
- using compact_sing assms by (rule compact_Un)
- then show ?thesis by simp
-qed
-
-lemma finite_imp_compact: "finite s \<Longrightarrow> compact s"
- by (induct set: finite) simp_all
-
-lemma open_delete:
- fixes s :: "'a::t1_space set"
- shows "open s \<Longrightarrow> open (s - {x})"
- by (simp add: open_Diff)
-
-lemma openin_delete:
- fixes a :: "'a :: t1_space"
- shows "openin (subtopology euclidean u) s
- \<Longrightarrow> openin (subtopology euclidean u) (s - {a})"
-by (metis Int_Diff open_delete openin_open)
-
-text\<open>Compactness expressed with filters\<close>
-
-lemma closure_iff_nhds_not_empty:
- "x \<in> closure X \<longleftrightarrow> (\<forall>A. \<forall>S\<subseteq>A. open S \<longrightarrow> x \<in> S \<longrightarrow> X \<inter> A \<noteq> {})"
-proof safe
- assume x: "x \<in> closure X"
- fix S A
- assume "open S" "x \<in> S" "X \<inter> A = {}" "S \<subseteq> A"
- then have "x \<notin> closure (-S)"
- by (auto simp: closure_complement subset_eq[symmetric] intro: interiorI)
- with x have "x \<in> closure X - closure (-S)"
- by auto
- also have "\<dots> \<subseteq> closure (X \<inter> S)"
- using \<open>open S\<close> open_Int_closure_subset[of S X] by (simp add: closed_Compl ac_simps)
- finally have "X \<inter> S \<noteq> {}" by auto
- then show False using \<open>X \<inter> A = {}\<close> \<open>S \<subseteq> A\<close> by auto
-next
- assume "\<forall>A S. S \<subseteq> A \<longrightarrow> open S \<longrightarrow> x \<in> S \<longrightarrow> X \<inter> A \<noteq> {}"
- from this[THEN spec, of "- X", THEN spec, of "- closure X"]
- show "x \<in> closure X"
- by (simp add: closure_subset open_Compl)
-qed
-
-corollary closure_Int_ball_not_empty:
- assumes "S \<subseteq> closure T" "x \<in> S" "r > 0"
- shows "T \<inter> ball x r \<noteq> {}"
- using assms centre_in_ball closure_iff_nhds_not_empty by blast
-
-lemma compact_filter:
- "compact U \<longleftrightarrow> (\<forall>F. F \<noteq> bot \<longrightarrow> eventually (\<lambda>x. x \<in> U) F \<longrightarrow> (\<exists>x\<in>U. inf (nhds x) F \<noteq> bot))"
-proof (intro allI iffI impI compact_fip[THEN iffD2] notI)
- fix F
- assume "compact U"
- assume F: "F \<noteq> bot" "eventually (\<lambda>x. x \<in> U) F"
- then have "U \<noteq> {}"
- by (auto simp: eventually_False)
-
- define Z where "Z = closure ` {A. eventually (\<lambda>x. x \<in> A) F}"
- then have "\<forall>z\<in>Z. closed z"
- by auto
- moreover
- have ev_Z: "\<And>z. z \<in> Z \<Longrightarrow> eventually (\<lambda>x. x \<in> z) F"
- unfolding Z_def by (auto elim: eventually_mono intro: set_mp[OF closure_subset])
- have "(\<forall>B \<subseteq> Z. finite B \<longrightarrow> U \<inter> \<Inter>B \<noteq> {})"
- proof (intro allI impI)
- fix B assume "finite B" "B \<subseteq> Z"
- with \<open>finite B\<close> ev_Z F(2) have "eventually (\<lambda>x. x \<in> U \<inter> (\<Inter>B)) F"
- by (auto simp: eventually_ball_finite_distrib eventually_conj_iff)
- with F show "U \<inter> \<Inter>B \<noteq> {}"
- by (intro notI) (simp add: eventually_False)
- qed
- ultimately have "U \<inter> \<Inter>Z \<noteq> {}"
- using \<open>compact U\<close> unfolding compact_fip by blast
- then obtain x where "x \<in> U" and x: "\<And>z. z \<in> Z \<Longrightarrow> x \<in> z"
- by auto
-
- have "\<And>P. eventually P (inf (nhds x) F) \<Longrightarrow> P \<noteq> bot"
- unfolding eventually_inf eventually_nhds
- proof safe
- fix P Q R S
- assume "eventually R F" "open S" "x \<in> S"
- with open_Int_closure_eq_empty[of S "{x. R x}"] x[of "closure {x. R x}"]
- have "S \<inter> {x. R x} \<noteq> {}" by (auto simp: Z_def)
- moreover assume "Ball S Q" "\<forall>x. Q x \<and> R x \<longrightarrow> bot x"
- ultimately show False by (auto simp: set_eq_iff)
- qed
- with \<open>x \<in> U\<close> show "\<exists>x\<in>U. inf (nhds x) F \<noteq> bot"
- by (metis eventually_bot)
-next
- fix A
- assume A: "\<forall>a\<in>A. closed a" "\<forall>B\<subseteq>A. finite B \<longrightarrow> U \<inter> \<Inter>B \<noteq> {}" "U \<inter> \<Inter>A = {}"
- define F where "F = (INF a\<in>insert U A. principal a)"
- have "F \<noteq> bot"
- unfolding F_def
- proof (rule INF_filter_not_bot)
- fix X
- assume X: "X \<subseteq> insert U A" "finite X"
- with A(2)[THEN spec, of "X - {U}"] have "U \<inter> \<Inter>(X - {U}) \<noteq> {}"
- by auto
- with X show "(INF a\<in>X. principal a) \<noteq> bot"
- by (auto simp: INF_principal_finite principal_eq_bot_iff)
- qed
- moreover
- have "F \<le> principal U"
- unfolding F_def by auto
- then have "eventually (\<lambda>x. x \<in> U) F"
- by (auto simp: le_filter_def eventually_principal)
- moreover
- assume "\<forall>F. F \<noteq> bot \<longrightarrow> eventually (\<lambda>x. x \<in> U) F \<longrightarrow> (\<exists>x\<in>U. inf (nhds x) F \<noteq> bot)"
- ultimately obtain x where "x \<in> U" and x: "inf (nhds x) F \<noteq> bot"
- by auto
-
- { fix V assume "V \<in> A"
- then have "F \<le> principal V"
- unfolding F_def by (intro INF_lower2[of V]) auto
- then have V: "eventually (\<lambda>x. x \<in> V) F"
- by (auto simp: le_filter_def eventually_principal)
- have "x \<in> closure V"
- unfolding closure_iff_nhds_not_empty
- proof (intro impI allI)
- fix S A
- assume "open S" "x \<in> S" "S \<subseteq> A"
- then have "eventually (\<lambda>x. x \<in> A) (nhds x)"
- by (auto simp: eventually_nhds)
- with V have "eventually (\<lambda>x. x \<in> V \<inter> A) (inf (nhds x) F)"
- by (auto simp: eventually_inf)
- with x show "V \<inter> A \<noteq> {}"
- by (auto simp del: Int_iff simp add: trivial_limit_def)
- qed
- then have "x \<in> V"
- using \<open>V \<in> A\<close> A(1) by simp
- }
- with \<open>x\<in>U\<close> have "x \<in> U \<inter> \<Inter>A" by auto
- with \<open>U \<inter> \<Inter>A = {}\<close> show False by auto
-qed
-
-definition%important "countably_compact U \<longleftrightarrow>
- (\<forall>A. countable A \<longrightarrow> (\<forall>a\<in>A. open a) \<longrightarrow> U \<subseteq> \<Union>A \<longrightarrow> (\<exists>T\<subseteq>A. finite T \<and> U \<subseteq> \<Union>T))"
-
-lemma countably_compactE:
- assumes "countably_compact s" and "\<forall>t\<in>C. open t" and "s \<subseteq> \<Union>C" "countable C"
- obtains C' where "C' \<subseteq> C" and "finite C'" and "s \<subseteq> \<Union>C'"
- using assms unfolding countably_compact_def by metis
-
-lemma countably_compactI:
- assumes "\<And>C. \<forall>t\<in>C. open t \<Longrightarrow> s \<subseteq> \<Union>C \<Longrightarrow> countable C \<Longrightarrow> (\<exists>C'\<subseteq>C. finite C' \<and> s \<subseteq> \<Union>C')"
- shows "countably_compact s"
- using assms unfolding countably_compact_def by metis
-
-lemma compact_imp_countably_compact: "compact U \<Longrightarrow> countably_compact U"
- by (auto simp: compact_eq_heine_borel countably_compact_def)
-
-lemma countably_compact_imp_compact:
- assumes "countably_compact U"
- and ccover: "countable B" "\<forall>b\<in>B. open b"
- and basis: "\<And>T x. open T \<Longrightarrow> x \<in> T \<Longrightarrow> x \<in> U \<Longrightarrow> \<exists>b\<in>B. x \<in> b \<and> b \<inter> U \<subseteq> T"
- shows "compact U"
- using \<open>countably_compact U\<close>
- unfolding compact_eq_heine_borel countably_compact_def
-proof safe
- fix A
- assume A: "\<forall>a\<in>A. open a" "U \<subseteq> \<Union>A"
- assume *: "\<forall>A. countable A \<longrightarrow> (\<forall>a\<in>A. open a) \<longrightarrow> U \<subseteq> \<Union>A \<longrightarrow> (\<exists>T\<subseteq>A. finite T \<and> U \<subseteq> \<Union>T)"
- moreover define C where "C = {b\<in>B. \<exists>a\<in>A. b \<inter> U \<subseteq> a}"
- ultimately have "countable C" "\<forall>a\<in>C. open a"
- unfolding C_def using ccover by auto
- moreover
- have "\<Union>A \<inter> U \<subseteq> \<Union>C"
- proof safe
- fix x a
- assume "x \<in> U" "x \<in> a" "a \<in> A"
- with basis[of a x] A obtain b where "b \<in> B" "x \<in> b" "b \<inter> U \<subseteq> a"
- by blast
- with \<open>a \<in> A\<close> show "x \<in> \<Union>C"
- unfolding C_def by auto
- qed
- then have "U \<subseteq> \<Union>C" using \<open>U \<subseteq> \<Union>A\<close> by auto
- ultimately obtain T where T: "T\<subseteq>C" "finite T" "U \<subseteq> \<Union>T"
- using * by metis
- then have "\<forall>t\<in>T. \<exists>a\<in>A. t \<inter> U \<subseteq> a"
- by (auto simp: C_def)
- then obtain f where "\<forall>t\<in>T. f t \<in> A \<and> t \<inter> U \<subseteq> f t"
- unfolding bchoice_iff Bex_def ..
- with T show "\<exists>T\<subseteq>A. finite T \<and> U \<subseteq> \<Union>T"
- unfolding C_def by (intro exI[of _ "f`T"]) fastforce
-qed
-
-proposition countably_compact_imp_compact_second_countable:
- "countably_compact U \<Longrightarrow> compact (U :: 'a :: second_countable_topology set)"
-proof (rule countably_compact_imp_compact)
- fix T and x :: 'a
- assume "open T" "x \<in> T"
- from topological_basisE[OF is_basis this] obtain b where
- "b \<in> (SOME B. countable B \<and> topological_basis B)" "x \<in> b" "b \<subseteq> T" .
- then show "\<exists>b\<in>SOME B. countable B \<and> topological_basis B. x \<in> b \<and> b \<inter> U \<subseteq> T"
- by blast
-qed (insert countable_basis topological_basis_open[OF is_basis], auto)
-
-lemma countably_compact_eq_compact:
- "countably_compact U \<longleftrightarrow> compact (U :: 'a :: second_countable_topology set)"
- using countably_compact_imp_compact_second_countable compact_imp_countably_compact by blast
-
-subsubsection\<open>Sequential compactness\<close>
-
-definition%important seq_compact :: "'a::topological_space set \<Rightarrow> bool"
- where "seq_compact S \<longleftrightarrow>
- (\<forall>f. (\<forall>n. f n \<in> S) \<longrightarrow> (\<exists>l\<in>S. \<exists>r::nat\<Rightarrow>nat. strict_mono r \<and> ((f \<circ> r) \<longlongrightarrow> l) sequentially))"
-
-lemma seq_compactI:
- assumes "\<And>f. \<forall>n. f n \<in> S \<Longrightarrow> \<exists>l\<in>S. \<exists>r::nat\<Rightarrow>nat. strict_mono r \<and> ((f \<circ> r) \<longlongrightarrow> l) sequentially"
- shows "seq_compact S"
- unfolding seq_compact_def using assms by fast
-
-lemma seq_compactE:
- assumes "seq_compact S" "\<forall>n. f n \<in> S"
- obtains l r where "l \<in> S" "strict_mono (r :: nat \<Rightarrow> nat)" "((f \<circ> r) \<longlongrightarrow> l) sequentially"
- using assms unfolding seq_compact_def by fast
-
-lemma closed_sequentially: (* TODO: move upwards *)
- assumes "closed s" and "\<forall>n. f n \<in> s" and "f \<longlonglongrightarrow> l"
- shows "l \<in> s"
-proof (rule ccontr)
- assume "l \<notin> s"
- with \<open>closed s\<close> and \<open>f \<longlonglongrightarrow> l\<close> have "eventually (\<lambda>n. f n \<in> - s) sequentially"
- by (fast intro: topological_tendstoD)
- with \<open>\<forall>n. f n \<in> s\<close> show "False"
- by simp
-qed
-
-lemma seq_compact_Int_closed:
- assumes "seq_compact s" and "closed t"
- shows "seq_compact (s \<inter> t)"
-proof (rule seq_compactI)
- fix f assume "\<forall>n::nat. f n \<in> s \<inter> t"
- hence "\<forall>n. f n \<in> s" and "\<forall>n. f n \<in> t"
- by simp_all
- from \<open>seq_compact s\<close> and \<open>\<forall>n. f n \<in> s\<close>
- obtain l r where "l \<in> s" and r: "strict_mono r" and l: "(f \<circ> r) \<longlonglongrightarrow> l"
- by (rule seq_compactE)
- from \<open>\<forall>n. f n \<in> t\<close> have "\<forall>n. (f \<circ> r) n \<in> t"
- by simp
- from \<open>closed t\<close> and this and l have "l \<in> t"
- by (rule closed_sequentially)
- with \<open>l \<in> s\<close> and r and l show "\<exists>l\<in>s \<inter> t. \<exists>r. strict_mono r \<and> (f \<circ> r) \<longlonglongrightarrow> l"
- by fast
-qed
-
-lemma seq_compact_closed_subset:
- assumes "closed s" and "s \<subseteq> t" and "seq_compact t"
- shows "seq_compact s"
- using assms seq_compact_Int_closed [of t s] by (simp add: Int_absorb1)
-
-lemma seq_compact_imp_countably_compact:
- fixes U :: "'a :: first_countable_topology set"
- assumes "seq_compact U"
- shows "countably_compact U"
-proof (safe intro!: countably_compactI)
- fix A
- assume A: "\<forall>a\<in>A. open a" "U \<subseteq> \<Union>A" "countable A"
- have subseq: "\<And>X. range X \<subseteq> U \<Longrightarrow> \<exists>r x. x \<in> U \<and> strict_mono (r :: nat \<Rightarrow> nat) \<and> (X \<circ> r) \<longlonglongrightarrow> x"
- using \<open>seq_compact U\<close> by (fastforce simp: seq_compact_def subset_eq)
- show "\<exists>T\<subseteq>A. finite T \<and> U \<subseteq> \<Union>T"
- proof cases
- assume "finite A"
- with A show ?thesis by auto
- next
- assume "infinite A"
- then have "A \<noteq> {}" by auto
- show ?thesis
- proof (rule ccontr)
- assume "\<not> (\<exists>T\<subseteq>A. finite T \<and> U \<subseteq> \<Union>T)"
- then have "\<forall>T. \<exists>x. T \<subseteq> A \<and> finite T \<longrightarrow> (x \<in> U - \<Union>T)"
- by auto
- then obtain X' where T: "\<And>T. T \<subseteq> A \<Longrightarrow> finite T \<Longrightarrow> X' T \<in> U - \<Union>T"
- by metis
- define X where "X n = X' (from_nat_into A ` {.. n})" for n
- have X: "\<And>n. X n \<in> U - (\<Union>i\<le>n. from_nat_into A i)"
- using \<open>A \<noteq> {}\<close> unfolding X_def by (intro T) (auto intro: from_nat_into)
- then have "range X \<subseteq> U"
- by auto
- with subseq[of X] obtain r x where "x \<in> U" and r: "strict_mono r" "(X \<circ> r) \<longlonglongrightarrow> x"
- by auto
- from \<open>x\<in>U\<close> \<open>U \<subseteq> \<Union>A\<close> from_nat_into_surj[OF \<open>countable A\<close>]
- obtain n where "x \<in> from_nat_into A n" by auto
- with r(2) A(1) from_nat_into[OF \<open>A \<noteq> {}\<close>, of n]
- have "eventually (\<lambda>i. X (r i) \<in> from_nat_into A n) sequentially"
- unfolding tendsto_def by (auto simp: comp_def)
- then obtain N where "\<And>i. N \<le> i \<Longrightarrow> X (r i) \<in> from_nat_into A n"
- by (auto simp: eventually_sequentially)
- moreover from X have "\<And>i. n \<le> r i \<Longrightarrow> X (r i) \<notin> from_nat_into A n"
- by auto
- moreover from \<open>strict_mono r\<close>[THEN seq_suble, of "max n N"] have "\<exists>i. n \<le> r i \<and> N \<le> i"
- by (auto intro!: exI[of _ "max n N"])
- ultimately show False
- by auto
- qed
- qed
-qed
-
-lemma compact_imp_seq_compact:
- fixes U :: "'a :: first_countable_topology set"
- assumes "compact U"
- shows "seq_compact U"
- unfolding seq_compact_def
-proof safe
- fix X :: "nat \<Rightarrow> 'a"
- assume "\<forall>n. X n \<in> U"
- then have "eventually (\<lambda>x. x \<in> U) (filtermap X sequentially)"
- by (auto simp: eventually_filtermap)
- moreover
- have "filtermap X sequentially \<noteq> bot"
- by (simp add: trivial_limit_def eventually_filtermap)
- ultimately
- obtain x where "x \<in> U" and x: "inf (nhds x) (filtermap X sequentially) \<noteq> bot" (is "?F \<noteq> _")
- using \<open>compact U\<close> by (auto simp: compact_filter)
-
- from countable_basis_at_decseq[of x]
- obtain A where A:
- "\<And>i. open (A i)"
- "\<And>i. x \<in> A i"
- "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> eventually (\<lambda>i. A i \<subseteq> S) sequentially"
- by blast
- define s where "s n i = (SOME j. i < j \<and> X j \<in> A (Suc n))" for n i
- {
- fix n i
- have "\<exists>a. i < a \<and> X a \<in> A (Suc n)"
- proof (rule ccontr)
- assume "\<not> (\<exists>a>i. X a \<in> A (Suc n))"
- then have "\<And>a. Suc i \<le> a \<Longrightarrow> X a \<notin> A (Suc n)"
- by auto
- then have "eventually (\<lambda>x. x \<notin> A (Suc n)) (filtermap X sequentially)"
- by (auto simp: eventually_filtermap eventually_sequentially)
- moreover have "eventually (\<lambda>x. x \<in> A (Suc n)) (nhds x)"
- using A(1,2)[of "Suc n"] by (auto simp: eventually_nhds)
- ultimately have "eventually (\<lambda>x. False) ?F"
- by (auto simp: eventually_inf)
- with x show False
- by (simp add: eventually_False)
- qed
- then have "i < s n i" "X (s n i) \<in> A (Suc n)"
- unfolding s_def by (auto intro: someI2_ex)
- }
- note s = this
- define r where "r = rec_nat (s 0 0) s"
- have "strict_mono r"
- by (auto simp: r_def s strict_mono_Suc_iff)
- moreover
- have "(\<lambda>n. X (r n)) \<longlonglongrightarrow> x"
- proof (rule topological_tendstoI)
- fix S
- assume "open S" "x \<in> S"
- with A(3) have "eventually (\<lambda>i. A i \<subseteq> S) sequentially"
- by auto
- moreover
- {
- fix i
- assume "Suc 0 \<le> i"
- then have "X (r i) \<in> A i"
- by (cases i) (simp_all add: r_def s)
- }
- then have "eventually (\<lambda>i. X (r i) \<in> A i) sequentially"
- by (auto simp: eventually_sequentially)
- ultimately show "eventually (\<lambda>i. X (r i) \<in> S) sequentially"
- by eventually_elim auto
- qed
- ultimately show "\<exists>x \<in> U. \<exists>r. strict_mono r \<and> (X \<circ> r) \<longlonglongrightarrow> x"
- using \<open>x \<in> U\<close> by (auto simp: convergent_def comp_def)
-qed
-
-lemma countably_compact_imp_acc_point:
- assumes "countably_compact s"
- and "countable t"
- and "infinite t"
- and "t \<subseteq> s"
- shows "\<exists>x\<in>s. \<forall>U. x\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> t)"
-proof (rule ccontr)
- define C where "C = (\<lambda>F. interior (F \<union> (- t))) ` {F. finite F \<and> F \<subseteq> t }"
- note \<open>countably_compact s\<close>
- moreover have "\<forall>t\<in>C. open t"
- by (auto simp: C_def)
- moreover
- assume "\<not> (\<exists>x\<in>s. \<forall>U. x\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> t))"
- then have s: "\<And>x. x \<in> s \<Longrightarrow> \<exists>U. x\<in>U \<and> open U \<and> finite (U \<inter> t)" by metis
- have "s \<subseteq> \<Union>C"
- using \<open>t \<subseteq> s\<close>
- unfolding C_def
- apply (safe dest!: s)
- apply (rule_tac a="U \<inter> t" in UN_I)
- apply (auto intro!: interiorI simp add: finite_subset)
- done
- moreover
- from \<open>countable t\<close> have "countable C"
- unfolding C_def by (auto intro: countable_Collect_finite_subset)
- ultimately
- obtain D where "D \<subseteq> C" "finite D" "s \<subseteq> \<Union>D"
- by (rule countably_compactE)
- then obtain E where E: "E \<subseteq> {F. finite F \<and> F \<subseteq> t }" "finite E"
- and s: "s \<subseteq> (\<Union>F\<in>E. interior (F \<union> (- t)))"
- by (metis (lifting) finite_subset_image C_def)
- from s \<open>t \<subseteq> s\<close> have "t \<subseteq> \<Union>E"
- using interior_subset by blast
- moreover have "finite (\<Union>E)"
- using E by auto
- ultimately show False using \<open>infinite t\<close>
- by (auto simp: finite_subset)
-qed
-
-lemma countable_acc_point_imp_seq_compact:
- fixes s :: "'a::first_countable_topology set"
- assumes "\<forall>t. infinite t \<and> countable t \<and> t \<subseteq> s \<longrightarrow>
- (\<exists>x\<in>s. \<forall>U. x\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> t))"
- shows "seq_compact s"
-proof -
- {
- fix f :: "nat \<Rightarrow> 'a"
- assume f: "\<forall>n. f n \<in> s"
- have "\<exists>l\<in>s. \<exists>r. strict_mono r \<and> ((f \<circ> r) \<longlongrightarrow> l) sequentially"
- proof (cases "finite (range f)")
- case True
- obtain l where "infinite {n. f n = f l}"
- using pigeonhole_infinite[OF _ True] by auto
- then obtain r :: "nat \<Rightarrow> nat" where "strict_mono r" and fr: "\<forall>n. f (r n) = f l"
- using infinite_enumerate by blast
- then have "strict_mono r \<and> (f \<circ> r) \<longlonglongrightarrow> f l"
- by (simp add: fr o_def)
- with f show "\<exists>l\<in>s. \<exists>r. strict_mono r \<and> (f \<circ> r) \<longlonglongrightarrow> l"
- by auto
- next
- case False
- with f assms have "\<exists>x\<in>s. \<forall>U. x\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> range f)"
- by auto
- then obtain l where "l \<in> s" "\<forall>U. l\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> range f)" ..
- from this(2) have "\<exists>r. strict_mono r \<and> ((f \<circ> r) \<longlongrightarrow> l) sequentially"
- using acc_point_range_imp_convergent_subsequence[of l f] by auto
- with \<open>l \<in> s\<close> show "\<exists>l\<in>s. \<exists>r. strict_mono r \<and> ((f \<circ> r) \<longlongrightarrow> l) sequentially" ..
- qed
- }
- then show ?thesis
- unfolding seq_compact_def by auto
-qed
-
-lemma seq_compact_eq_countably_compact:
- fixes U :: "'a :: first_countable_topology set"
- shows "seq_compact U \<longleftrightarrow> countably_compact U"
- using
- countable_acc_point_imp_seq_compact
- countably_compact_imp_acc_point
- seq_compact_imp_countably_compact
- by metis
-
-lemma seq_compact_eq_acc_point:
- fixes s :: "'a :: first_countable_topology set"
- shows "seq_compact s \<longleftrightarrow>
- (\<forall>t. infinite t \<and> countable t \<and> t \<subseteq> s --> (\<exists>x\<in>s. \<forall>U. x\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> t)))"
- using
- countable_acc_point_imp_seq_compact[of s]
- countably_compact_imp_acc_point[of s]
- seq_compact_imp_countably_compact[of s]
- by metis
-
-lemma seq_compact_eq_compact:
- fixes U :: "'a :: second_countable_topology set"
- shows "seq_compact U \<longleftrightarrow> compact U"
- using seq_compact_eq_countably_compact countably_compact_eq_compact by blast
-
-proposition bolzano_weierstrass_imp_seq_compact:
- fixes s :: "'a::{t1_space, first_countable_topology} set"
- shows "\<forall>t. infinite t \<and> t \<subseteq> s \<longrightarrow> (\<exists>x \<in> s. x islimpt t) \<Longrightarrow> seq_compact s"
- by (rule countable_acc_point_imp_seq_compact) (metis islimpt_eq_acc_point)
-
-
-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)"
- unfolding Cauchy_def by metis
-
-proposition seq_compact_imp_totally_bounded:
- assumes "seq_compact s"
- shows "\<forall>e>0. \<exists>k. finite k \<and> k \<subseteq> s \<and> s \<subseteq> (\<Union>x\<in>k. ball x e)"
-proof -
- { fix e::real assume "e > 0" assume *: "\<And>k. finite k \<Longrightarrow> k \<subseteq> s \<Longrightarrow> \<not> s \<subseteq> (\<Union>x\<in>k. ball x e)"
- let ?Q = "\<lambda>x n r. r \<in> s \<and> (\<forall>m < (n::nat). \<not> (dist (x m) r < e))"
- have "\<exists>x. \<forall>n::nat. ?Q x n (x n)"
- proof (rule dependent_wellorder_choice)
- fix n x assume "\<And>y. y < n \<Longrightarrow> ?Q x y (x y)"
- then have "\<not> s \<subseteq> (\<Union>x\<in>x ` {0..<n}. ball x e)"
- using *[of "x ` {0 ..< n}"] by (auto simp: subset_eq)
- then obtain z where z:"z\<in>s" "z \<notin> (\<Union>x\<in>x ` {0..<n}. ball x e)"
- unfolding subset_eq by auto
- show "\<exists>r. ?Q x n r"
- using z by auto
- qed simp
- then obtain x where "\<forall>n::nat. x n \<in> s" and x:"\<And>n m. m < n \<Longrightarrow> \<not> (dist (x m) (x n) < e)"
- by blast
- then obtain l r where "l \<in> s" and r:"strict_mono r" and "((x \<circ> r) \<longlongrightarrow> l) sequentially"
- using assms by (metis seq_compact_def)
- from this(3) have "Cauchy (x \<circ> r)"
- using LIMSEQ_imp_Cauchy by auto
- then obtain N::nat where "\<And>m n. N \<le> m \<Longrightarrow> N \<le> n \<Longrightarrow> dist ((x \<circ> r) m) ((x \<circ> r) n) < e"
- unfolding cauchy_def using \<open>e > 0\<close> by blast
- then have False
- using x[of "r N" "r (N+1)"] r by (auto simp: strict_mono_def) }
- then show ?thesis
- by metis
-qed
-
-subsubsection\<open>Heine-Borel theorem\<close>
-
-proposition seq_compact_imp_heine_borel:
- fixes s :: "'a :: metric_space set"
- assumes "seq_compact s"
- shows "compact s"
-proof -
- from seq_compact_imp_totally_bounded[OF \<open>seq_compact s\<close>]
- obtain f where f: "\<forall>e>0. finite (f e) \<and> f e \<subseteq> s \<and> s \<subseteq> (\<Union>x\<in>f e. ball x e)"
- unfolding choice_iff' ..
- define K where "K = (\<lambda>(x, r). ball x r) ` ((\<Union>e \<in> \<rat> \<inter> {0 <..}. f e) \<times> \<rat>)"
- have "countably_compact s"
- using \<open>seq_compact s\<close> by (rule seq_compact_imp_countably_compact)
- then show "compact s"
- proof (rule countably_compact_imp_compact)
- show "countable K"
- unfolding K_def using f
- by (auto intro: countable_finite countable_subset countable_rat
- intro!: countable_image countable_SIGMA countable_UN)
- show "\<forall>b\<in>K. open b" by (auto simp: K_def)
- next
- fix T x
- assume T: "open T" "x \<in> T" and x: "x \<in> s"
- from openE[OF T] obtain e where "0 < e" "ball x e \<subseteq> T"
- by auto
- then have "0 < e / 2" "ball x (e / 2) \<subseteq> T"
- by auto
- from Rats_dense_in_real[OF \<open>0 < e / 2\<close>] obtain r where "r \<in> \<rat>" "0 < r" "r < e / 2"
- by auto
- from f[rule_format, of r] \<open>0 < r\<close> \<open>x \<in> s\<close> obtain k where "k \<in> f r" "x \<in> ball k r"
- by auto
- from \<open>r \<in> \<rat>\<close> \<open>0 < r\<close> \<open>k \<in> f r\<close> have "ball k r \<in> K"
- by (auto simp: K_def)
- then show "\<exists>b\<in>K. x \<in> b \<and> b \<inter> s \<subseteq> T"
- proof (rule bexI[rotated], safe)
- fix y
- assume "y \<in> ball k r"
- with \<open>r < e / 2\<close> \<open>x \<in> ball k r\<close> have "dist x y < e"
- by (intro dist_triangle_half_r [of k _ e]) (auto simp: dist_commute)
- with \<open>ball x e \<subseteq> T\<close> show "y \<in> T"
- by auto
- next
- show "x \<in> ball k r" by fact
- qed
- qed
-qed
-
-proposition compact_eq_seq_compact_metric:
- "compact (s :: 'a::metric_space set) \<longleftrightarrow> seq_compact s"
- using compact_imp_seq_compact seq_compact_imp_heine_borel by blast
-
-proposition compact_def: \<comment> \<open>this is the definition of compactness in HOL Light\<close>
- "compact (S :: 'a::metric_space set) \<longleftrightarrow>
- (\<forall>f. (\<forall>n. f n \<in> S) \<longrightarrow> (\<exists>l\<in>S. \<exists>r::nat\<Rightarrow>nat. strict_mono r \<and> (f \<circ> r) \<longlonglongrightarrow> l))"
- unfolding compact_eq_seq_compact_metric seq_compact_def by auto
-
-subsubsection \<open>Complete the chain of compactness variants\<close>
-
-proposition compact_eq_bolzano_weierstrass:
- fixes s :: "'a::metric_space set"
- shows "compact s \<longleftrightarrow> (\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t))"
- (is "?lhs = ?rhs")
-proof
- assume ?lhs
- then show ?rhs
- using heine_borel_imp_bolzano_weierstrass[of s] by auto
-next
- assume ?rhs
- then show ?lhs
- unfolding compact_eq_seq_compact_metric by (rule bolzano_weierstrass_imp_seq_compact)
-qed
-
-proposition bolzano_weierstrass_imp_bounded:
- "\<forall>t. infinite t \<and> t \<subseteq> s \<longrightarrow> (\<exists>x \<in> s. x islimpt t) \<Longrightarrow> bounded s"
- using compact_imp_bounded unfolding compact_eq_bolzano_weierstrass .
-
-
-subsection \<open>Metric spaces with the Heine-Borel property\<close>
-
-text \<open>
- A metric space (or topological vector space) is said to have the
- Heine-Borel property if every closed and bounded subset is compact.
-\<close>
-
-class heine_borel = metric_space +
- assumes bounded_imp_convergent_subsequence:
- "bounded (range f) \<Longrightarrow> \<exists>l r. strict_mono (r::nat\<Rightarrow>nat) \<and> ((f \<circ> r) \<longlongrightarrow> l) sequentially"
-
-proposition bounded_closed_imp_seq_compact:
- fixes s::"'a::heine_borel set"
- assumes "bounded s"
- and "closed s"
- shows "seq_compact s"
-proof (unfold seq_compact_def, clarify)
- fix f :: "nat \<Rightarrow> 'a"
- assume f: "\<forall>n. f n \<in> s"
- with \<open>bounded s\<close> have "bounded (range f)"
- by (auto intro: bounded_subset)
- obtain l r where r: "strict_mono (r :: nat \<Rightarrow> nat)" and l: "((f \<circ> r) \<longlongrightarrow> l) sequentially"
- using bounded_imp_convergent_subsequence [OF \<open>bounded (range f)\<close>] by auto
- from f have fr: "\<forall>n. (f \<circ> r) n \<in> s"
- by simp
- have "l \<in> s" using \<open>closed s\<close> fr l
- by (rule closed_sequentially)
- show "\<exists>l\<in>s. \<exists>r. strict_mono r \<and> ((f \<circ> r) \<longlongrightarrow> l) sequentially"
- using \<open>l \<in> s\<close> r l by blast
-qed
-
-lemma compact_eq_bounded_closed:
- fixes s :: "'a::heine_borel set"
- shows "compact s \<longleftrightarrow> bounded s \<and> closed s"
- (is "?lhs = ?rhs")
-proof
- assume ?lhs
- then show ?rhs
- using compact_imp_closed compact_imp_bounded
- by blast
-next
- assume ?rhs
- then show ?lhs
- using bounded_closed_imp_seq_compact[of s]
- unfolding compact_eq_seq_compact_metric
- by auto
-qed
-
-lemma compact_Inter:
- fixes \<F> :: "'a :: heine_borel set set"
- assumes com: "\<And>S. S \<in> \<F> \<Longrightarrow> compact S" and "\<F> \<noteq> {}"
- shows "compact(\<Inter> \<F>)"
- using assms
- by (meson Inf_lower all_not_in_conv bounded_subset closed_Inter compact_eq_bounded_closed)
-
-lemma compact_closure [simp]:
- fixes S :: "'a::heine_borel set"
- shows "compact(closure S) \<longleftrightarrow> bounded S"
-by (meson bounded_closure bounded_subset closed_closure closure_subset compact_eq_bounded_closed)
-
-lemma not_compact_UNIV[simp]:
- fixes s :: "'a::{real_normed_vector,perfect_space,heine_borel} set"
- shows "\<not> compact (UNIV::'a set)"
- by (simp add: compact_eq_bounded_closed)
-
-text\<open>Representing sets as the union of a chain of compact sets.\<close>
-lemma closed_Union_compact_subsets:
- fixes S :: "'a::{heine_borel,real_normed_vector} set"
- assumes "closed S"
- obtains F where "\<And>n. compact(F n)" "\<And>n. F n \<subseteq> S" "\<And>n. F n \<subseteq> F(Suc n)"
- "(\<Union>n. F n) = S" "\<And>K. \<lbrakk>compact K; K \<subseteq> S\<rbrakk> \<Longrightarrow> \<exists>N. \<forall>n \<ge> N. K \<subseteq> F n"
-proof
- show "compact (S \<inter> cball 0 (of_nat n))" for n
- using assms compact_eq_bounded_closed by auto
-next
- show "(\<Union>n. S \<inter> cball 0 (real n)) = S"
- by (auto simp: real_arch_simple)
-next
- fix K :: "'a set"
- assume "compact K" "K \<subseteq> S"
- then obtain N where "K \<subseteq> cball 0 N"
- by (meson bounded_pos mem_cball_0 compact_imp_bounded subsetI)
- then show "\<exists>N. \<forall>n\<ge>N. K \<subseteq> S \<inter> cball 0 (real n)"
- by (metis of_nat_le_iff Int_subset_iff \<open>K \<subseteq> S\<close> real_arch_simple subset_cball subset_trans)
-qed auto
-
-instance%important real :: heine_borel
-proof%unimportant
- fix f :: "nat \<Rightarrow> real"
- assume f: "bounded (range f)"
- obtain r :: "nat \<Rightarrow> nat" where r: "strict_mono r" "monoseq (f \<circ> r)"
- unfolding comp_def by (metis seq_monosub)
- then have "Bseq (f \<circ> r)"
- unfolding Bseq_eq_bounded using f by (force intro: bounded_subset)
- with r show "\<exists>l r. strict_mono r \<and> (f \<circ> r) \<longlonglongrightarrow> l"
- using Bseq_monoseq_convergent[of "f \<circ> r"] by (auto simp: convergent_def)
-qed
-
-lemma compact_lemma_general:
- fixes f :: "nat \<Rightarrow> 'a"
- fixes proj::"'a \<Rightarrow> 'b \<Rightarrow> 'c::heine_borel" (infixl "proj" 60)
- fixes unproj:: "('b \<Rightarrow> 'c) \<Rightarrow> 'a"
- assumes finite_basis: "finite basis"
- assumes bounded_proj: "\<And>k. k \<in> basis \<Longrightarrow> bounded ((\<lambda>x. x proj k) ` range f)"
- assumes proj_unproj: "\<And>e k. k \<in> basis \<Longrightarrow> (unproj e) proj k = e k"
- assumes unproj_proj: "\<And>x. unproj (\<lambda>k. x proj k) = x"
- shows "\<forall>d\<subseteq>basis. \<exists>l::'a. \<exists> r::nat\<Rightarrow>nat.
- strict_mono r \<and> (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) proj i) (l proj i) < e) sequentially)"
-proof safe
- fix d :: "'b set"
- assume d: "d \<subseteq> basis"
- with finite_basis have "finite d"
- by (blast intro: finite_subset)
- from this d show "\<exists>l::'a. \<exists>r::nat\<Rightarrow>nat. strict_mono r \<and>
- (\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) proj i) (l proj i) < e) sequentially)"
- proof (induct d)
- case empty
- then show ?case
- unfolding strict_mono_def by auto
- next
- case (insert k d)
- have k[intro]: "k \<in> basis"
- using insert by auto
- have s': "bounded ((\<lambda>x. x proj k) ` range f)"
- using k
- by (rule bounded_proj)
- obtain l1::"'a" and r1 where r1: "strict_mono r1"
- and lr1: "\<forall>e > 0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) proj i) (l1 proj i) < e) sequentially"
- using insert(3) using insert(4) by auto
- have f': "\<forall>n. f (r1 n) proj k \<in> (\<lambda>x. x proj k) ` range f"
- by simp
- have "bounded (range (\<lambda>i. f (r1 i) proj k))"
- by (metis (lifting) bounded_subset f' image_subsetI s')
- then obtain l2 r2 where r2:"strict_mono r2" and lr2:"((\<lambda>i. f (r1 (r2 i)) proj k) \<longlongrightarrow> l2) sequentially"
- using bounded_imp_convergent_subsequence[of "\<lambda>i. f (r1 i) proj k"]
- by (auto simp: o_def)
- define r where "r = r1 \<circ> r2"
- have r:"strict_mono r"
- using r1 and r2 unfolding r_def o_def strict_mono_def by auto
- moreover
- define l where "l = unproj (\<lambda>i. if i = k then l2 else l1 proj i)"
- {
- fix e::real
- assume "e > 0"
- from lr1 \<open>e > 0\<close> have N1: "eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) proj i) (l1 proj i) < e) sequentially"
- by blast
- from lr2 \<open>e > 0\<close> have N2:"eventually (\<lambda>n. dist (f (r1 (r2 n)) proj k) l2 < e) sequentially"
- by (rule tendstoD)
- from r2 N1 have N1': "eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 (r2 n)) proj i) (l1 proj i) < e) sequentially"
- by (rule eventually_subseq)
- have "eventually (\<lambda>n. \<forall>i\<in>(insert k d). dist (f (r n) proj i) (l proj i) < e) sequentially"
- using N1' N2
- by eventually_elim (insert insert.prems, auto simp: l_def r_def o_def proj_unproj)
- }
- ultimately show ?case by auto
- qed
-qed
-
lemma compact_lemma:
fixes f :: "nat \<Rightarrow> 'a::euclidean_space"
assumes "bounded (range f)"
@@ -4887,630 +763,8 @@
by auto
qed
-lemma bounded_fst: "bounded s \<Longrightarrow> bounded (fst ` s)"
- unfolding bounded_def
- by (metis (erased, hide_lams) dist_fst_le image_iff order_trans)
-
-lemma bounded_snd: "bounded s \<Longrightarrow> bounded (snd ` s)"
- unfolding bounded_def
- by (metis (no_types, hide_lams) dist_snd_le image_iff order.trans)
-
-instance%important prod :: (heine_borel, heine_borel) heine_borel
-proof%unimportant
- fix f :: "nat \<Rightarrow> 'a \<times> 'b"
- assume f: "bounded (range f)"
- then have "bounded (fst ` range f)"
- by (rule bounded_fst)
- then have s1: "bounded (range (fst \<circ> f))"
- by (simp add: image_comp)
- obtain l1 r1 where r1: "strict_mono r1" and l1: "(\<lambda>n. fst (f (r1 n))) \<longlonglongrightarrow> l1"
- using bounded_imp_convergent_subsequence [OF s1] unfolding o_def by fast
- from f have s2: "bounded (range (snd \<circ> f \<circ> r1))"
- by (auto simp: image_comp intro: bounded_snd bounded_subset)
- obtain l2 r2 where r2: "strict_mono r2" and l2: "((\<lambda>n. snd (f (r1 (r2 n)))) \<longlongrightarrow> l2) sequentially"
- using bounded_imp_convergent_subsequence [OF s2]
- unfolding o_def by fast
- have l1': "((\<lambda>n. fst (f (r1 (r2 n)))) \<longlongrightarrow> l1) sequentially"
- using LIMSEQ_subseq_LIMSEQ [OF l1 r2] unfolding o_def .
- have l: "((f \<circ> (r1 \<circ> r2)) \<longlongrightarrow> (l1, l2)) sequentially"
- using tendsto_Pair [OF l1' l2] unfolding o_def by simp
- have r: "strict_mono (r1 \<circ> r2)"
- using r1 r2 unfolding strict_mono_def by simp
- show "\<exists>l r. strict_mono r \<and> ((f \<circ> r) \<longlongrightarrow> l) sequentially"
- using l r by fast
-qed
-
-subsubsection \<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"
- shows "complete s"
- using assms unfolding complete_def by fast
-
-proposition (in metric_space) completeE:
- assumes "complete s" and "\<forall>n. f n \<in> s" and "Cauchy f"
- obtains l where "l \<in> s" and "f \<longlonglongrightarrow> l"
- using assms unfolding complete_def by fast
-
-(* TODO: generalize to uniform spaces *)
-lemma compact_imp_complete:
- fixes s :: "'a::metric_space set"
- assumes "compact s"
- shows "complete s"
-proof -
- {
- fix f
- assume as: "(\<forall>n::nat. f n \<in> s)" "Cauchy f"
- from as(1) obtain l r where lr: "l\<in>s" "strict_mono r" "(f \<circ> r) \<longlonglongrightarrow> l"
- using assms unfolding compact_def by blast
-
- note lr' = seq_suble [OF lr(2)]
- {
- fix e :: real
- assume "e > 0"
- from as(2) obtain N where N:"\<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (f m) (f n) < e/2"
- unfolding cauchy_def
- using \<open>e > 0\<close>
- apply (erule_tac x="e/2" in allE, auto)
- done
- from lr(3)[unfolded lim_sequentially, THEN spec[where x="e/2"]]
- obtain M where M:"\<forall>n\<ge>M. dist ((f \<circ> r) n) l < e/2"
- using \<open>e > 0\<close> by auto
- {
- fix n :: nat
- assume n: "n \<ge> max N M"
- have "dist ((f \<circ> r) n) l < e/2"
- using n M by auto
- moreover have "r n \<ge> N"
- using lr'[of n] n by auto
- then have "dist (f n) ((f \<circ> r) n) < e / 2"
- using N and n by auto
- ultimately have "dist (f n) l < e"
- using dist_triangle_half_r[of "f (r n)" "f n" e l]
- by (auto simp: dist_commute)
- }
- then have "\<exists>N. \<forall>n\<ge>N. dist (f n) l < e" by blast
- }
- then have "\<exists>l\<in>s. (f \<longlongrightarrow> l) sequentially" using \<open>l\<in>s\<close>
- unfolding lim_sequentially by auto
- }
- then show ?thesis unfolding complete_def by auto
-qed
-
-proposition compact_eq_totally_bounded:
- "compact s \<longleftrightarrow> complete s \<and> (\<forall>e>0. \<exists>k. finite k \<and> s \<subseteq> (\<Union>x\<in>k. ball x e))"
- (is "_ \<longleftrightarrow> ?rhs")
-proof
- assume assms: "?rhs"
- then obtain k where k: "\<And>e. 0 < e \<Longrightarrow> finite (k e)" "\<And>e. 0 < e \<Longrightarrow> s \<subseteq> (\<Union>x\<in>k e. ball x e)"
- by (auto simp: choice_iff')
-
- show "compact s"
- proof cases
- assume "s = {}"
- then show "compact s" by (simp add: compact_def)
- next
- assume "s \<noteq> {}"
- show ?thesis
- unfolding compact_def
- proof safe
- fix f :: "nat \<Rightarrow> 'a"
- assume f: "\<forall>n. f n \<in> s"
-
- define e where "e n = 1 / (2 * Suc n)" for n
- then have [simp]: "\<And>n. 0 < e n" by auto
- define B where "B n U = (SOME b. infinite {n. f n \<in> b} \<and> (\<exists>x. b \<subseteq> ball x (e n) \<inter> U))" for n U
- {
- fix n U
- assume "infinite {n. f n \<in> U}"
- then have "\<exists>b\<in>k (e n). infinite {i\<in>{n. f n \<in> U}. f i \<in> ball b (e n)}"
- using k f by (intro pigeonhole_infinite_rel) (auto simp: subset_eq)
- then obtain a where
- "a \<in> k (e n)"
- "infinite {i \<in> {n. f n \<in> U}. f i \<in> ball a (e n)}" ..
- then have "\<exists>b. infinite {i. f i \<in> b} \<and> (\<exists>x. b \<subseteq> ball x (e n) \<inter> U)"
- by (intro exI[of _ "ball a (e n) \<inter> U"] exI[of _ a]) (auto simp: ac_simps)
- from someI_ex[OF this]
- have "infinite {i. f i \<in> B n U}" "\<exists>x. B n U \<subseteq> ball x (e n) \<inter> U"
- unfolding B_def by auto
- }
- note B = this
-
- define F where "F = rec_nat (B 0 UNIV) B"
- {
- fix n
- have "infinite {i. f i \<in> F n}"
- by (induct n) (auto simp: F_def B)
- }
- then have F: "\<And>n. \<exists>x. F (Suc n) \<subseteq> ball x (e n) \<inter> F n"
- using B by (simp add: F_def)
- then have F_dec: "\<And>m n. m \<le> n \<Longrightarrow> F n \<subseteq> F m"
- using decseq_SucI[of F] by (auto simp: decseq_def)
-
- obtain sel where sel: "\<And>k i. i < sel k i" "\<And>k i. f (sel k i) \<in> F k"
- proof (atomize_elim, unfold all_conj_distrib[symmetric], intro choice allI)
- fix k i
- have "infinite ({n. f n \<in> F k} - {.. i})"
- using \<open>infinite {n. f n \<in> F k}\<close> by auto
- from infinite_imp_nonempty[OF this]
- show "\<exists>x>i. f x \<in> F k"
- by (simp add: set_eq_iff not_le conj_commute)
- qed
-
- define t where "t = rec_nat (sel 0 0) (\<lambda>n i. sel (Suc n) i)"
- have "strict_mono t"
- unfolding strict_mono_Suc_iff by (simp add: t_def sel)
- moreover have "\<forall>i. (f \<circ> t) i \<in> s"
- using f by auto
- moreover
- {
- fix n
- have "(f \<circ> t) n \<in> F n"
- by (cases n) (simp_all add: t_def sel)
- }
- note t = this
-
- have "Cauchy (f \<circ> t)"
- proof (safe intro!: metric_CauchyI exI elim!: nat_approx_posE)
- fix r :: real and N n m
- assume "1 / Suc N < r" "Suc N \<le> n" "Suc N \<le> m"
- then have "(f \<circ> t) n \<in> F (Suc N)" "(f \<circ> t) m \<in> F (Suc N)" "2 * e N < r"
- using F_dec t by (auto simp: e_def field_simps of_nat_Suc)
- with F[of N] obtain x where "dist x ((f \<circ> t) n) < e N" "dist x ((f \<circ> t) m) < e N"
- by (auto simp: subset_eq)
- with dist_triangle[of "(f \<circ> t) m" "(f \<circ> t) n" x] \<open>2 * e N < r\<close>
- show "dist ((f \<circ> t) m) ((f \<circ> t) n) < r"
- by (simp add: dist_commute)
- qed
-
- ultimately show "\<exists>l\<in>s. \<exists>r. strict_mono r \<and> (f \<circ> r) \<longlonglongrightarrow> l"
- using assms unfolding complete_def by blast
- qed
- qed
-qed (metis compact_imp_complete compact_imp_seq_compact seq_compact_imp_totally_bounded)
-
-lemma cauchy_imp_bounded:
- assumes "Cauchy s"
- shows "bounded (range s)"
-proof -
- from assms obtain N :: nat where "\<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (s m) (s n) < 1"
- unfolding cauchy_def by force
- then have N:"\<forall>n. N \<le> n \<longrightarrow> dist (s N) (s n) < 1" by auto
- moreover
- have "bounded (s ` {0..N})"
- using finite_imp_bounded[of "s ` {1..N}"] by auto
- then obtain a where a:"\<forall>x\<in>s ` {0..N}. dist (s N) x \<le> a"
- unfolding bounded_any_center [where a="s N"] by auto
- ultimately show "?thesis"
- unfolding bounded_any_center [where a="s N"]
- apply (rule_tac x="max a 1" in exI, auto)
- apply (erule_tac x=y in allE)
- apply (erule_tac x=y in ballE, auto)
- done
-qed
-
-instance heine_borel < complete_space
-proof
- fix f :: "nat \<Rightarrow> 'a" assume "Cauchy f"
- then have "bounded (range f)"
- by (rule cauchy_imp_bounded)
- then have "compact (closure (range f))"
- unfolding compact_eq_bounded_closed by auto
- then have "complete (closure (range f))"
- by (rule compact_imp_complete)
- moreover have "\<forall>n. f n \<in> closure (range f)"
- using closure_subset [of "range f"] by auto
- ultimately have "\<exists>l\<in>closure (range f). (f \<longlongrightarrow> l) sequentially"
- using \<open>Cauchy f\<close> unfolding complete_def by auto
- then show "convergent f"
- unfolding convergent_def by auto
-qed
-
instance euclidean_space \<subseteq> banach ..
-lemma complete_UNIV: "complete (UNIV :: ('a::complete_space) set)"
-proof (rule completeI)
- fix f :: "nat \<Rightarrow> 'a" assume "Cauchy f"
- then have "convergent f" by (rule Cauchy_convergent)
- then show "\<exists>l\<in>UNIV. f \<longlonglongrightarrow> l" unfolding convergent_def by simp
-qed
-
-lemma complete_imp_closed:
- fixes S :: "'a::metric_space set"
- assumes "complete S"
- shows "closed S"
-proof (unfold closed_sequential_limits, clarify)
- fix f x assume "\<forall>n. f n \<in> S" and "f \<longlonglongrightarrow> x"
- from \<open>f \<longlonglongrightarrow> x\<close> have "Cauchy f"
- by (rule LIMSEQ_imp_Cauchy)
- with \<open>complete S\<close> and \<open>\<forall>n. f n \<in> S\<close> obtain l where "l \<in> S" and "f \<longlonglongrightarrow> l"
- by (rule completeE)
- from \<open>f \<longlonglongrightarrow> x\<close> and \<open>f \<longlonglongrightarrow> l\<close> have "x = l"
- by (rule LIMSEQ_unique)
- with \<open>l \<in> S\<close> show "x \<in> S"
- by simp
-qed
-
-lemma complete_Int_closed:
- fixes S :: "'a::metric_space set"
- assumes "complete S" and "closed t"
- shows "complete (S \<inter> t)"
-proof (rule completeI)
- fix f assume "\<forall>n. f n \<in> S \<inter> t" and "Cauchy f"
- then have "\<forall>n. f n \<in> S" and "\<forall>n. f n \<in> t"
- by simp_all
- from \<open>complete S\<close> obtain l where "l \<in> S" and "f \<longlonglongrightarrow> l"
- using \<open>\<forall>n. f n \<in> S\<close> and \<open>Cauchy f\<close> by (rule completeE)
- from \<open>closed t\<close> and \<open>\<forall>n. f n \<in> t\<close> and \<open>f \<longlonglongrightarrow> l\<close> have "l \<in> t"
- by (rule closed_sequentially)
- with \<open>l \<in> S\<close> and \<open>f \<longlonglongrightarrow> l\<close> show "\<exists>l\<in>S \<inter> t. f \<longlonglongrightarrow> l"
- by fast
-qed
-
-lemma complete_closed_subset:
- fixes S :: "'a::metric_space set"
- assumes "closed S" and "S \<subseteq> t" and "complete t"
- shows "complete S"
- using assms complete_Int_closed [of t S] by (simp add: Int_absorb1)
-
-lemma complete_eq_closed:
- fixes S :: "('a::complete_space) set"
- shows "complete S \<longleftrightarrow> closed S"
-proof
- assume "closed S" then show "complete S"
- using subset_UNIV complete_UNIV by (rule complete_closed_subset)
-next
- assume "complete S" then show "closed S"
- by (rule complete_imp_closed)
-qed
-
-lemma convergent_eq_Cauchy:
- fixes S :: "nat \<Rightarrow> 'a::complete_space"
- shows "(\<exists>l. (S \<longlongrightarrow> l) sequentially) \<longleftrightarrow> Cauchy S"
- unfolding Cauchy_convergent_iff convergent_def ..
-
-lemma convergent_imp_bounded:
- fixes S :: "nat \<Rightarrow> 'a::metric_space"
- shows "(S \<longlongrightarrow> l) sequentially \<Longrightarrow> bounded (range S)"
- by (intro cauchy_imp_bounded LIMSEQ_imp_Cauchy)
-
-lemma frontier_subset_compact:
- fixes S :: "'a::heine_borel set"
- shows "compact S \<Longrightarrow> frontier S \<subseteq> S"
- using frontier_subset_closed compact_eq_bounded_closed
- by blast
-
-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
-
-lemma uniformly_continuous_onE:
- assumes "uniformly_continuous_on s f" "0 < e"
- obtains d where "d>0" "\<And>x x'. \<lbrakk>x\<in>s; x'\<in>s; dist x' x < d\<rbrakk> \<Longrightarrow> dist (f x') (f x) < e"
-using assms
-by (auto simp: uniformly_continuous_on_def)
-
-lemma continuous_at_imp_continuous_within:
- "continuous (at x) f \<Longrightarrow> continuous (at x within s) f"
- unfolding continuous_within continuous_at using Lim_at_imp_Lim_at_within by auto
-
-lemma Lim_trivial_limit: "trivial_limit net \<Longrightarrow> (f \<longlongrightarrow> l) net"
- by simp
-
-lemmas continuous_on = continuous_on_def \<comment> \<open>legacy theorem name\<close>
-
-lemma continuous_within_subset:
- "continuous (at x within s) f \<Longrightarrow> t \<subseteq> s \<Longrightarrow> continuous (at x within t) f"
- unfolding continuous_within by(metis tendsto_within_subset)
-
-lemma continuous_on_interior:
- "continuous_on s f \<Longrightarrow> x \<in> interior s \<Longrightarrow> continuous (at x) f"
- by (metis continuous_on_eq_continuous_at continuous_on_subset interiorE)
-
-lemma continuous_on_eq:
- "\<lbrakk>continuous_on s f; \<And>x. x \<in> s \<Longrightarrow> f x = g x\<rbrakk> \<Longrightarrow> continuous_on s g"
- unfolding continuous_on_def tendsto_def eventually_at_topological
- by simp
-
-text \<open>Characterization of various kinds of continuity in terms of sequences.\<close>
-
-lemma continuous_within_sequentiallyI:
- fixes f :: "'a::{first_countable_topology, t2_space} \<Rightarrow> 'b::topological_space"
- assumes "\<And>u::nat \<Rightarrow> 'a. u \<longlonglongrightarrow> a \<Longrightarrow> (\<forall>n. u n \<in> s) \<Longrightarrow> (\<lambda>n. f (u n)) \<longlonglongrightarrow> f a"
- shows "continuous (at a within s) f"
- using assms unfolding continuous_within tendsto_def[where l = "f a"]
- by (auto intro!: sequentially_imp_eventually_within)
-
-lemma continuous_within_tendsto_compose:
- fixes f::"'a::t2_space \<Rightarrow> 'b::topological_space"
- assumes "continuous (at a within s) f"
- "eventually (\<lambda>n. x n \<in> s) F"
- "(x \<longlongrightarrow> a) F "
- shows "((\<lambda>n. f (x n)) \<longlongrightarrow> f a) F"
-proof -
- have *: "filterlim x (inf (nhds a) (principal s)) F"
- using assms(2) assms(3) unfolding at_within_def filterlim_inf by (auto simp: filterlim_principal eventually_mono)
- show ?thesis
- by (auto simp: assms(1) continuous_within[symmetric] tendsto_at_within_iff_tendsto_nhds[symmetric] intro!: filterlim_compose[OF _ *])
-qed
-
-lemma continuous_within_tendsto_compose':
- fixes f::"'a::t2_space \<Rightarrow> 'b::topological_space"
- assumes "continuous (at a within s) f"
- "\<And>n. x n \<in> s"
- "(x \<longlongrightarrow> a) F "
- shows "((\<lambda>n. f (x n)) \<longlongrightarrow> f a) F"
- by (auto intro!: continuous_within_tendsto_compose[OF assms(1)] simp add: assms)
-
-lemma continuous_within_sequentially:
- fixes f :: "'a::{first_countable_topology, t2_space} \<Rightarrow> 'b::topological_space"
- shows "continuous (at a within s) f \<longleftrightarrow>
- (\<forall>x. (\<forall>n::nat. x n \<in> s) \<and> (x \<longlongrightarrow> a) sequentially
- \<longrightarrow> ((f \<circ> x) \<longlongrightarrow> f a) sequentially)"
- using continuous_within_tendsto_compose'[of a s f _ sequentially]
- continuous_within_sequentiallyI[of a s f]
- by (auto simp: o_def)
-
-lemma continuous_at_sequentiallyI:
- fixes f :: "'a::{first_countable_topology, t2_space} \<Rightarrow> 'b::topological_space"
- assumes "\<And>u. u \<longlonglongrightarrow> a \<Longrightarrow> (\<lambda>n. f (u n)) \<longlonglongrightarrow> f a"
- shows "continuous (at a) f"
- using continuous_within_sequentiallyI[of a UNIV f] assms by auto
-
-lemma continuous_at_sequentially:
- fixes f :: "'a::metric_space \<Rightarrow> 'b::topological_space"
- shows "continuous (at a) f \<longleftrightarrow>
- (\<forall>x. (x \<longlongrightarrow> a) sequentially --> ((f \<circ> x) \<longlongrightarrow> f a) sequentially)"
- using continuous_within_sequentially[of a UNIV f] by simp
-
-lemma continuous_on_sequentiallyI:
- fixes f :: "'a::{first_countable_topology, t2_space} \<Rightarrow> 'b::topological_space"
- assumes "\<And>u a. (\<forall>n. u n \<in> s) \<Longrightarrow> a \<in> s \<Longrightarrow> u \<longlonglongrightarrow> a \<Longrightarrow> (\<lambda>n. f (u n)) \<longlonglongrightarrow> f a"
- shows "continuous_on s f"
- using assms unfolding continuous_on_eq_continuous_within
- using continuous_within_sequentiallyI[of _ s f] by auto
-
-lemma continuous_on_sequentially:
- fixes f :: "'a::metric_space \<Rightarrow> 'b::topological_space"
- shows "continuous_on s f \<longleftrightarrow>
- (\<forall>x. \<forall>a \<in> s. (\<forall>n. x(n) \<in> s) \<and> (x \<longlongrightarrow> a) sequentially
- --> ((f \<circ> x) \<longlongrightarrow> f a) sequentially)"
- (is "?lhs = ?rhs")
-proof
- assume ?rhs
- then show ?lhs
- using continuous_within_sequentially[of _ s f]
- unfolding continuous_on_eq_continuous_within
- by auto
-next
- assume ?lhs
- then show ?rhs
- unfolding continuous_on_eq_continuous_within
- using continuous_within_sequentially[of _ s f]
- by auto
-qed
-
-lemma uniformly_continuous_on_sequentially:
- "uniformly_continuous_on s f \<longleftrightarrow> (\<forall>x y. (\<forall>n. x n \<in> s) \<and> (\<forall>n. y n \<in> s) \<and>
- (\<lambda>n. dist (x n) (y n)) \<longlonglongrightarrow> 0 \<longrightarrow> (\<lambda>n. dist (f(x n)) (f(y n))) \<longlonglongrightarrow> 0)" (is "?lhs = ?rhs")
-proof
- assume ?lhs
- {
- fix x y
- assume x: "\<forall>n. x n \<in> s"
- and y: "\<forall>n. y n \<in> s"
- and xy: "((\<lambda>n. dist (x n) (y n)) \<longlongrightarrow> 0) sequentially"
- {
- fix e :: real
- assume "e > 0"
- then obtain d where "d > 0" and d: "\<forall>x\<in>s. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (f x') (f x) < e"
- using \<open>?lhs\<close>[unfolded uniformly_continuous_on_def, THEN spec[where x=e]] by auto
- obtain N where N: "\<forall>n\<ge>N. dist (x n) (y n) < d"
- using xy[unfolded lim_sequentially dist_norm] and \<open>d>0\<close> by auto
- {
- fix n
- assume "n\<ge>N"
- then have "dist (f (x n)) (f (y n)) < e"
- using N[THEN spec[where x=n]]
- using d[THEN bspec[where x="x n"], THEN bspec[where x="y n"]]
- using x and y
- by (simp add: dist_commute)
- }
- then have "\<exists>N. \<forall>n\<ge>N. dist (f (x n)) (f (y n)) < e"
- by auto
- }
- then have "((\<lambda>n. dist (f(x n)) (f(y n))) \<longlongrightarrow> 0) sequentially"
- unfolding lim_sequentially and dist_real_def by auto
- }
- then show ?rhs by auto
-next
- assume ?rhs
- {
- assume "\<not> ?lhs"
- then obtain e where "e > 0" "\<forall>d>0. \<exists>x\<in>s. \<exists>x'\<in>s. dist x' x < d \<and> \<not> dist (f x') (f x) < e"
- unfolding uniformly_continuous_on_def by auto
- then obtain fa where fa:
- "\<forall>x. 0 < x \<longrightarrow> fst (fa x) \<in> s \<and> snd (fa x) \<in> s \<and> dist (fst (fa x)) (snd (fa x)) < x \<and> \<not> dist (f (fst (fa x))) (f (snd (fa x))) < e"
- using choice[of "\<lambda>d x. d>0 \<longrightarrow> fst x \<in> s \<and> snd x \<in> s \<and> dist (snd x) (fst x) < d \<and> \<not> dist (f (snd x)) (f (fst x)) < e"]
- unfolding Bex_def
- by (auto simp: dist_commute)
- define x where "x n = fst (fa (inverse (real n + 1)))" for n
- define y where "y n = snd (fa (inverse (real n + 1)))" for n
- have xyn: "\<forall>n. x n \<in> s \<and> y n \<in> s"
- and xy0: "\<forall>n. dist (x n) (y n) < inverse (real n + 1)"
- and fxy:"\<forall>n. \<not> dist (f (x n)) (f (y n)) < e"
- unfolding x_def and y_def using fa
- by auto
- {
- fix e :: real
- assume "e > 0"
- then obtain N :: nat where "N \<noteq> 0" and N: "0 < inverse (real N) \<and> inverse (real N) < e"
- unfolding real_arch_inverse[of e] by auto
- {
- fix n :: nat
- assume "n \<ge> N"
- then have "inverse (real n + 1) < inverse (real N)"
- using of_nat_0_le_iff and \<open>N\<noteq>0\<close> by auto
- also have "\<dots> < e" using N by auto
- finally have "inverse (real n + 1) < e" by auto
- then have "dist (x n) (y n) < e"
- using xy0[THEN spec[where x=n]] by auto
- }
- then have "\<exists>N. \<forall>n\<ge>N. dist (x n) (y n) < e" by auto
- }
- then have "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. dist (f (x n)) (f (y n)) < e"
- using \<open>?rhs\<close>[THEN spec[where x=x], THEN spec[where x=y]] and xyn
- unfolding lim_sequentially dist_real_def by auto
- then have False using fxy and \<open>e>0\<close> by auto
- }
- then show ?lhs
- 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 pointwise continuity\<close>
@@ -5524,6 +778,7 @@
shows "continuous F (\<lambda>x. inner (f x) (g x))"
using assms unfolding continuous_def by (rule tendsto_inner)
+
subsubsection%unimportant \<open>Structural rules for setwise continuity\<close>
lemma continuous_on_infnorm[continuous_intros]:
@@ -5538,316 +793,6 @@
using bounded_bilinear_inner assms
by (rule bounded_bilinear.continuous_on)
-subsubsection%unimportant \<open>Structural rules for uniform continuity\<close>
-
-lemma uniformly_continuous_on_dist[continuous_intros]:
- fixes f g :: "'a::metric_space \<Rightarrow> 'b::metric_space"
- assumes "uniformly_continuous_on s f"
- and "uniformly_continuous_on s g"
- shows "uniformly_continuous_on s (\<lambda>x. dist (f x) (g x))"
-proof -
- {
- fix a b c d :: 'b
- have "\<bar>dist a b - dist c d\<bar> \<le> dist a c + dist b d"
- using dist_triangle2 [of a b c] dist_triangle2 [of b c d]
- using dist_triangle3 [of c d a] dist_triangle [of a d b]
- by arith
- } note le = this
- {
- fix x y
- assume f: "(\<lambda>n. dist (f (x n)) (f (y n))) \<longlonglongrightarrow> 0"
- assume g: "(\<lambda>n. dist (g (x n)) (g (y n))) \<longlonglongrightarrow> 0"
- have "(\<lambda>n. \<bar>dist (f (x n)) (g (x n)) - dist (f (y n)) (g (y n))\<bar>) \<longlonglongrightarrow> 0"
- by (rule Lim_transform_bound [OF _ tendsto_add_zero [OF f g]],
- simp add: le)
- }
- then show ?thesis
- using assms unfolding uniformly_continuous_on_sequentially
- unfolding dist_real_def by simp
-qed
-
-lemma uniformly_continuous_on_norm[continuous_intros]:
- fixes f :: "'a :: metric_space \<Rightarrow> 'b :: real_normed_vector"
- assumes "uniformly_continuous_on s f"
- shows "uniformly_continuous_on s (\<lambda>x. norm (f x))"
- unfolding norm_conv_dist using assms
- by (intro uniformly_continuous_on_dist uniformly_continuous_on_const)
-
-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_cmul[continuous_intros]:
- fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
- assumes "uniformly_continuous_on s f"
- shows "uniformly_continuous_on s (\<lambda>x. c *\<^sub>R f(x))"
- using bounded_linear_scaleR_right assms
- by (rule bounded_linear.uniformly_continuous_on)
-
-lemma dist_minus:
- fixes x y :: "'a::real_normed_vector"
- shows "dist (- x) (- y) = dist x y"
- unfolding dist_norm minus_diff_minus norm_minus_cancel ..
-
-lemma uniformly_continuous_on_minus[continuous_intros]:
- fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
- shows "uniformly_continuous_on s f \<Longrightarrow> uniformly_continuous_on s (\<lambda>x. - f x)"
- unfolding uniformly_continuous_on_def dist_minus .
-
-lemma uniformly_continuous_on_add[continuous_intros]:
- fixes f g :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
- assumes "uniformly_continuous_on s f"
- and "uniformly_continuous_on s g"
- shows "uniformly_continuous_on s (\<lambda>x. f x + g x)"
- using assms
- unfolding uniformly_continuous_on_sequentially
- unfolding dist_norm tendsto_norm_zero_iff add_diff_add
- by (auto intro: tendsto_add_zero)
-
-lemma uniformly_continuous_on_diff[continuous_intros]:
- fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
- assumes "uniformly_continuous_on s f"
- and "uniformly_continuous_on s g"
- shows "uniformly_continuous_on s (\<lambda>x. f x - g x)"
- using assms uniformly_continuous_on_add [of s f "- g"]
- by (simp add: fun_Compl_def uniformly_continuous_on_minus)
-
-text \<open>Continuity in terms of open preimages.\<close>
-
-lemma continuous_at_open:
- "continuous (at x) f \<longleftrightarrow> (\<forall>t. open t \<and> f x \<in> t --> (\<exists>s. open s \<and> x \<in> s \<and> (\<forall>x' \<in> s. (f x') \<in> t)))"
- unfolding continuous_within_topological [of x UNIV f]
- unfolding imp_conjL
- by (intro all_cong imp_cong ex_cong conj_cong refl) auto
-
-lemma continuous_imp_tendsto:
- assumes "continuous (at x0) f"
- and "x \<longlonglongrightarrow> x0"
- shows "(f \<circ> x) \<longlonglongrightarrow> (f x0)"
-proof (rule topological_tendstoI)
- fix S
- assume "open S" "f x0 \<in> S"
- then obtain T where T_def: "open T" "x0 \<in> T" "\<forall>x\<in>T. f x \<in> S"
- using assms continuous_at_open by metis
- then have "eventually (\<lambda>n. x n \<in> T) sequentially"
- using assms T_def by (auto simp: tendsto_def)
- then show "eventually (\<lambda>n. (f \<circ> x) n \<in> S) sequentially"
- using T_def by (auto elim!: eventually_mono)
-qed
-
-lemma continuous_on_open:
- "continuous_on S f \<longleftrightarrow>
- (\<forall>T. openin (subtopology euclidean (f ` S)) T \<longrightarrow>
- openin (subtopology euclidean S) (S \<inter> f -` T))"
- unfolding continuous_on_open_invariant openin_open Int_def vimage_def Int_commute
- by (simp add: imp_ex imageI conj_commute eq_commute cong: conj_cong)
-
-lemma continuous_on_open_gen:
- fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space"
- assumes "f ` S \<subseteq> T"
- shows "continuous_on S f \<longleftrightarrow>
- (\<forall>U. openin (subtopology euclidean T) U
- \<longrightarrow> openin (subtopology euclidean S) (S \<inter> f -` U))"
- (is "?lhs = ?rhs")
-proof
- assume ?lhs
- then show ?rhs
- apply (clarsimp simp: openin_euclidean_subtopology_iff continuous_on_iff)
- by (metis assms image_subset_iff)
-next
- have ope: "openin (subtopology euclidean T) (ball y e \<inter> T)" for y e
- by (simp add: Int_commute openin_open_Int)
- assume R [rule_format]: ?rhs
- show ?lhs
- proof (clarsimp simp add: continuous_on_iff)
- fix x and e::real
- assume "x \<in> S" and "0 < e"
- then have x: "x \<in> S \<inter> (f -` ball (f x) e \<inter> f -` T)"
- using assms by auto
- show "\<exists>d>0. \<forall>x'\<in>S. dist x' x < d \<longrightarrow> dist (f x') (f x) < e"
- using R [of "ball (f x) e \<inter> T"] x
- by (fastforce simp add: ope openin_euclidean_subtopology_iff [of S] dist_commute)
- qed
-qed
-
-lemma continuous_openin_preimage:
- fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space"
- shows
- "\<lbrakk>continuous_on S f; f ` S \<subseteq> T; openin (subtopology euclidean T) U\<rbrakk>
- \<Longrightarrow> openin (subtopology euclidean S) (S \<inter> f -` U)"
-by (simp add: continuous_on_open_gen)
-
-text \<open>Similarly in terms of closed sets.\<close>
-
-lemma continuous_on_closed:
- "continuous_on S f \<longleftrightarrow>
- (\<forall>T. closedin (subtopology euclidean (f ` S)) T \<longrightarrow>
- closedin (subtopology euclidean S) (S \<inter> f -` T))"
- unfolding continuous_on_closed_invariant closedin_closed Int_def vimage_def Int_commute
- by (simp add: imp_ex imageI conj_commute eq_commute cong: conj_cong)
-
-lemma continuous_on_closed_gen:
- fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space"
- assumes "f ` S \<subseteq> T"
- shows "continuous_on S f \<longleftrightarrow>
- (\<forall>U. closedin (subtopology euclidean T) U
- \<longrightarrow> closedin (subtopology euclidean S) (S \<inter> f -` U))"
- (is "?lhs = ?rhs")
-proof -
- have *: "U \<subseteq> T \<Longrightarrow> S \<inter> f -` (T - U) = S - (S \<inter> f -` U)" for U
- using assms by blast
- show ?thesis
- proof
- assume L: ?lhs
- show ?rhs
- proof clarify
- fix U
- assume "closedin (subtopology euclidean T) U"
- then show "closedin (subtopology euclidean S) (S \<inter> f -` U)"
- using L unfolding continuous_on_open_gen [OF assms]
- by (metis * closedin_def inf_le1 topspace_euclidean_subtopology)
- qed
- next
- assume R [rule_format]: ?rhs
- show ?lhs
- unfolding continuous_on_open_gen [OF assms]
- by (metis * R inf_le1 openin_closedin_eq topspace_euclidean_subtopology)
- qed
-qed
-
-lemma continuous_closedin_preimage_gen:
- fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space"
- assumes "continuous_on S f" "f ` S \<subseteq> T" "closedin (subtopology euclidean T) U"
- shows "closedin (subtopology euclidean S) (S \<inter> f -` U)"
-using assms continuous_on_closed_gen by blast
-
-lemma continuous_on_imp_closedin:
- assumes "continuous_on S f" "closedin (subtopology euclidean (f ` S)) T"
- shows "closedin (subtopology euclidean S) (S \<inter> f -` T)"
-using assms continuous_on_closed by blast
-
-subsection%unimportant \<open>Half-global and completely global cases\<close>
-
-lemma continuous_openin_preimage_gen:
- assumes "continuous_on S f" "open T"
- shows "openin (subtopology euclidean S) (S \<inter> f -` T)"
-proof -
- have *: "(S \<inter> f -` T) = (S \<inter> f -` (T \<inter> f ` S))"
- by auto
- have "openin (subtopology euclidean (f ` S)) (T \<inter> f ` S)"
- using openin_open_Int[of T "f ` S", OF assms(2)] unfolding openin_open by auto
- then show ?thesis
- using assms(1)[unfolded continuous_on_open, THEN spec[where x="T \<inter> f ` S"]]
- using * by auto
-qed
-
-lemma continuous_closedin_preimage:
- assumes "continuous_on S f" and "closed T"
- shows "closedin (subtopology euclidean S) (S \<inter> f -` T)"
-proof -
- have *: "(S \<inter> f -` T) = (S \<inter> f -` (T \<inter> f ` S))"
- by auto
- have "closedin (subtopology euclidean (f ` S)) (T \<inter> f ` S)"
- using closedin_closed_Int[of T "f ` S", OF assms(2)]
- by (simp add: Int_commute)
- then show ?thesis
- using assms(1)[unfolded continuous_on_closed, THEN spec[where x="T \<inter> f ` S"]]
- using * by auto
-qed
-
-lemma continuous_openin_preimage_eq:
- "continuous_on S f \<longleftrightarrow>
- (\<forall>T. open T \<longrightarrow> openin (subtopology euclidean S) (S \<inter> f -` T))"
-apply safe
-apply (simp add: continuous_openin_preimage_gen)
-apply (fastforce simp add: continuous_on_open openin_open)
-done
-
-lemma continuous_closedin_preimage_eq:
- "continuous_on S f \<longleftrightarrow>
- (\<forall>T. closed T \<longrightarrow> closedin (subtopology euclidean S) (S \<inter> f -` T))"
-apply safe
-apply (simp add: continuous_closedin_preimage)
-apply (fastforce simp add: continuous_on_closed closedin_closed)
-done
-
-lemma continuous_open_preimage:
- assumes contf: "continuous_on S f" and "open S" "open T"
- shows "open (S \<inter> f -` T)"
-proof-
- obtain U where "open U" "(S \<inter> f -` T) = S \<inter> U"
- using continuous_openin_preimage_gen[OF contf \<open>open T\<close>]
- unfolding openin_open by auto
- then show ?thesis
- using open_Int[of S U, OF \<open>open S\<close>] by auto
-qed
-
-lemma continuous_closed_preimage:
- assumes contf: "continuous_on S f" and "closed S" "closed T"
- shows "closed (S \<inter> f -` T)"
-proof-
- obtain U where "closed U" "(S \<inter> f -` T) = S \<inter> U"
- using continuous_closedin_preimage[OF contf \<open>closed T\<close>]
- unfolding closedin_closed by auto
- then show ?thesis using closed_Int[of S U, OF \<open>closed S\<close>] by auto
-qed
-
-lemma continuous_open_vimage: "open S \<Longrightarrow> (\<And>x. continuous (at x) f) \<Longrightarrow> open (f -` S)"
- by (metis continuous_on_eq_continuous_within open_vimage)
-
-lemma continuous_closed_vimage: "closed S \<Longrightarrow> (\<And>x. continuous (at x) f) \<Longrightarrow> closed (f -` S)"
- by (simp add: closed_vimage continuous_on_eq_continuous_within)
-
-lemma interior_image_subset:
- assumes "inj f" "\<And>x. continuous (at x) f"
- shows "interior (f ` S) \<subseteq> f ` (interior S)"
-proof
- fix x assume "x \<in> interior (f ` S)"
- then obtain T where as: "open T" "x \<in> T" "T \<subseteq> f ` S" ..
- then have "x \<in> f ` S" by auto
- then obtain y where y: "y \<in> S" "x = f y" by auto
- have "open (f -` T)"
- using assms \<open>open T\<close> by (simp add: continuous_at_imp_continuous_on open_vimage)
- moreover have "y \<in> vimage f T"
- using \<open>x = f y\<close> \<open>x \<in> T\<close> by simp
- moreover have "vimage f T \<subseteq> S"
- using \<open>T \<subseteq> image f S\<close> \<open>inj f\<close> unfolding inj_on_def subset_eq by auto
- ultimately have "y \<in> interior S" ..
- with \<open>x = f y\<close> show "x \<in> f ` interior S" ..
-qed
-
-subsection%unimportant \<open>Topological properties of linear functions\<close>
-
-lemma linear_lim_0:
- assumes "bounded_linear f"
- shows "(f \<longlongrightarrow> 0) (at (0))"
-proof -
- interpret f: bounded_linear f by fact
- have "(f \<longlongrightarrow> f 0) (at 0)"
- using tendsto_ident_at by (rule f.tendsto)
- then show ?thesis unfolding f.zero .
-qed
-
-lemma linear_continuous_at:
- assumes "bounded_linear f"
- shows "continuous (at a) f"
- unfolding continuous_at using assms
- apply (rule bounded_linear.tendsto)
- apply (rule tendsto_ident_at)
- done
-
-lemma linear_continuous_within:
- "bounded_linear f \<Longrightarrow> continuous (at x within s) f"
- using continuous_at_imp_continuous_within[of x f s] using linear_continuous_at[of f] by auto
-
-lemma linear_continuous_on:
- "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>Intervals\<close>
text \<open>Openness of halfspaces.\<close>
--- a/src/HOL/Library/Infinite_Set.thy Thu Dec 27 22:54:29 2018 +0100
+++ b/src/HOL/Library/Infinite_Set.thy Thu Dec 27 23:38:55 2018 +0100
@@ -266,6 +266,12 @@
finally show ?case by simp
qed
+lemma infinite_enumerate:
+ assumes fS: "infinite S"
+ shows "\<exists>r::nat\<Rightarrow>nat. strict_mono r \<and> (\<forall>n. r n \<in> S)"
+ unfolding strict_mono_def
+ using enumerate_in_set[OF fS] enumerate_mono[of _ _ S] fS by auto
+
lemma enumerate_Suc'':
fixes S :: "'a::wellorder set"
assumes "infinite S"