move first_countable_topology to the HOL image
authorhoelzl
Fri Mar 22 10:41:43 2013 +0100 (2013-03-22)
changeset 514731210309fddab
parent 51472 adb441e4b9e9
child 51474 1e9e68247ad1
move first_countable_topology to the HOL image
src/HOL/Lim.thy
src/HOL/Metric_Spaces.thy
src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy
src/HOL/Probability/Fin_Map.thy
src/HOL/Topological_Spaces.thy
     1.1 --- a/src/HOL/Lim.thy	Fri Mar 22 10:41:43 2013 +0100
     1.2 +++ b/src/HOL/Lim.thy	Fri Mar 22 10:41:43 2013 +0100
     1.3 @@ -226,58 +226,6 @@
     1.4  by (rule isUCont [THEN isUCont_Cauchy])
     1.5  
     1.6  
     1.7 -subsection {* Relation of LIM and LIMSEQ *}
     1.8 -
     1.9 -lemma sequentially_imp_eventually_within:
    1.10 -  fixes a :: "'a::metric_space"
    1.11 -  assumes "\<forall>f. (\<forall>n. f n \<in> s \<and> f n \<noteq> a) \<and> f ----> a \<longrightarrow>
    1.12 -    eventually (\<lambda>n. P (f n)) sequentially"
    1.13 -  shows "eventually P (at a within s)"
    1.14 -proof (rule ccontr)
    1.15 -  let ?I = "\<lambda>n. inverse (real (Suc n))"
    1.16 -  def F \<equiv> "\<lambda>n::nat. SOME x. x \<in> s \<and> x \<noteq> a \<and> dist x a < ?I n \<and> \<not> P x"
    1.17 -  assume "\<not> eventually P (at a within s)"
    1.18 -  hence P: "\<forall>d>0. \<exists>x. x \<in> s \<and> x \<noteq> a \<and> dist x a < d \<and> \<not> P x"
    1.19 -    unfolding eventually_within eventually_at by fast
    1.20 -  hence "\<And>n. \<exists>x. x \<in> s \<and> x \<noteq> a \<and> dist x a < ?I n \<and> \<not> P x" by simp
    1.21 -  hence F: "\<And>n. F n \<in> s \<and> F n \<noteq> a \<and> dist (F n) a < ?I n \<and> \<not> P (F n)"
    1.22 -    unfolding F_def by (rule someI_ex)
    1.23 -  hence F0: "\<forall>n. F n \<in> s" and F1: "\<forall>n. F n \<noteq> a"
    1.24 -    and F2: "\<forall>n. dist (F n) a < ?I n" and F3: "\<forall>n. \<not> P (F n)"
    1.25 -    by fast+
    1.26 -  from LIMSEQ_inverse_real_of_nat have "F ----> a"
    1.27 -    by (rule metric_tendsto_imp_tendsto,
    1.28 -      simp add: dist_norm F2 less_imp_le)
    1.29 -  hence "eventually (\<lambda>n. P (F n)) sequentially"
    1.30 -    using assms F0 F1 by simp
    1.31 -  thus "False" by (simp add: F3)
    1.32 -qed
    1.33 -
    1.34 -lemma sequentially_imp_eventually_at:
    1.35 -  fixes a :: "'a::metric_space"
    1.36 -  assumes "\<forall>f. (\<forall>n. f n \<noteq> a) \<and> f ----> a \<longrightarrow>
    1.37 -    eventually (\<lambda>n. P (f n)) sequentially"
    1.38 -  shows "eventually P (at a)"
    1.39 -  using assms sequentially_imp_eventually_within [where s=UNIV] by simp
    1.40 -
    1.41 -lemma LIMSEQ_SEQ_conv1:
    1.42 -  fixes f :: "'a::topological_space \<Rightarrow> 'b::topological_space"
    1.43 -  assumes f: "f -- a --> l"
    1.44 -  shows "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. f (S n)) ----> l"
    1.45 -  using tendsto_compose_eventually [OF f, where F=sequentially] by simp
    1.46 -
    1.47 -lemma LIMSEQ_SEQ_conv2:
    1.48 -  fixes f :: "'a::metric_space \<Rightarrow> 'b::topological_space"
    1.49 -  assumes "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. f (S n)) ----> l"
    1.50 -  shows "f -- a --> l"
    1.51 -  using assms unfolding tendsto_def [where l=l]
    1.52 -  by (simp add: sequentially_imp_eventually_at)
    1.53 -
    1.54 -lemma LIMSEQ_SEQ_conv:
    1.55 -  "(\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> (a::'a::metric_space) \<longrightarrow> (\<lambda>n. X (S n)) ----> L) =
    1.56 -   (X -- a --> (L::'b::topological_space))"
    1.57 -  using LIMSEQ_SEQ_conv2 LIMSEQ_SEQ_conv1 ..
    1.58 -
    1.59  lemma LIM_less_bound: 
    1.60    fixes f :: "real \<Rightarrow> real"
    1.61    assumes ev: "b < x" "\<forall> x' \<in> { b <..< x}. 0 \<le> f x'" and "isCont f x"
     2.1 --- a/src/HOL/Metric_Spaces.thy	Fri Mar 22 10:41:43 2013 +0100
     2.2 +++ b/src/HOL/Metric_Spaces.thy	Fri Mar 22 10:41:43 2013 +0100
     2.3 @@ -100,13 +100,31 @@
     2.4      unfolding open_dist by fast
     2.5  qed
     2.6  
     2.7 -lemma (in metric_space) open_ball: "open {y. dist x y < d}"
     2.8 +lemma open_ball: "open {y. dist x y < d}"
     2.9  proof (unfold open_dist, intro ballI)
    2.10    fix y assume *: "y \<in> {y. dist x y < d}"
    2.11    then show "\<exists>e>0. \<forall>z. dist z y < e \<longrightarrow> z \<in> {y. dist x y < d}"
    2.12      by (auto intro!: exI[of _ "d - dist x y"] simp: field_simps dist_triangle_lt)
    2.13  qed
    2.14  
    2.15 +subclass first_countable_topology
    2.16 +proof
    2.17 +  fix x 
    2.18 +  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))"
    2.19 +  proof (safe intro!: exI[of _ "\<lambda>n. {y. dist x y < inverse (Suc n)}"])
    2.20 +    fix S assume "open S" "x \<in> S"
    2.21 +    then obtain e where "0 < e" "{y. dist x y < e} \<subseteq> S"
    2.22 +      by (auto simp: open_dist subset_eq dist_commute)
    2.23 +    moreover
    2.24 +    then obtain i where "inverse (Suc i) < e"
    2.25 +      by (auto dest!: reals_Archimedean)
    2.26 +    then have "{y. dist x y < inverse (Suc i)} \<subseteq> {y. dist x y < e}"
    2.27 +      by auto
    2.28 +    ultimately show "\<exists>i. {y. dist x y < inverse (Suc i)} \<subseteq> S"
    2.29 +      by blast
    2.30 +  qed (auto intro: open_ball)
    2.31 +qed
    2.32 +
    2.33  end
    2.34  
    2.35  instance metric_space \<subseteq> t2_space
     3.1 --- a/src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy	Fri Mar 22 10:41:43 2013 +0100
     3.2 +++ b/src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy	Fri Mar 22 10:41:43 2013 +0100
     3.3 @@ -181,44 +181,15 @@
     3.4  
     3.5  end
     3.6  
     3.7 -class first_countable_topology = topological_space +
     3.8 -  assumes first_countable_basis:
     3.9 -    "\<exists>A. countable A \<and> (\<forall>a\<in>A. x \<in> a \<and> open a) \<and> (\<forall>S. open S \<and> x \<in> S \<longrightarrow> (\<exists>a\<in>A. a \<subseteq> S))"
    3.10 -
    3.11 -lemma (in first_countable_topology) countable_basis_at_decseq:
    3.12 -  obtains A :: "nat \<Rightarrow> 'a set" where
    3.13 -    "\<And>i. open (A i)" "\<And>i. x \<in> (A i)"
    3.14 -    "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> eventually (\<lambda>i. A i \<subseteq> S) sequentially"
    3.15 -proof atomize_elim
    3.16 -  from first_countable_basis[of x] obtain A
    3.17 -    where "countable A"
    3.18 -    and nhds: "\<And>a. a \<in> A \<Longrightarrow> open a" "\<And>a. a \<in> A \<Longrightarrow> x \<in> a"
    3.19 -    and incl: "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> \<exists>a\<in>A. a \<subseteq> S"  by auto
    3.20 -  then have "A \<noteq> {}" by auto
    3.21 -  with `countable A` have r: "A = range (from_nat_into A)" by auto
    3.22 -  def F \<equiv> "\<lambda>n. \<Inter>i\<le>n. from_nat_into A i"
    3.23 -  show "\<exists>A. (\<forall>i. open (A i)) \<and> (\<forall>i. x \<in> A i) \<and>
    3.24 -      (\<forall>S. open S \<longrightarrow> x \<in> S \<longrightarrow> eventually (\<lambda>i. A i \<subseteq> S) sequentially)"
    3.25 -  proof (safe intro!: exI[of _ F])
    3.26 -    fix i
    3.27 -    show "open (F i)" using nhds(1) r by (auto simp: F_def intro!: open_INT)
    3.28 -    show "x \<in> F i" using nhds(2) r by (auto simp: F_def)
    3.29 -  next
    3.30 -    fix S assume "open S" "x \<in> S"
    3.31 -    from incl[OF this] obtain i where "F i \<subseteq> S"
    3.32 -      by (subst (asm) r) (auto simp: F_def)
    3.33 -    moreover have "\<And>j. i \<le> j \<Longrightarrow> F j \<subseteq> F i"
    3.34 -      by (auto simp: F_def)
    3.35 -    ultimately show "eventually (\<lambda>i. F i \<subseteq> S) sequentially"
    3.36 -      by (auto simp: eventually_sequentially)
    3.37 -  qed
    3.38 -qed
    3.39 -
    3.40  lemma (in first_countable_topology) first_countable_basisE:
    3.41    obtains A where "countable A" "\<And>a. a \<in> A \<Longrightarrow> x \<in> a" "\<And>a. a \<in> A \<Longrightarrow> open a"
    3.42      "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> (\<exists>a\<in>A. a \<subseteq> S)"
    3.43    using first_countable_basis[of x]
    3.44 -  by atomize_elim auto
    3.45 +  apply atomize_elim
    3.46 +  apply (elim exE)
    3.47 +  apply (rule_tac x="range A" in exI)
    3.48 +  apply auto
    3.49 +  done
    3.50  
    3.51  lemma (in first_countable_topology) first_countable_basis_Int_stableE:
    3.52    obtains A where "countable A" "\<And>a. a \<in> A \<Longrightarrow> x \<in> a" "\<And>a. a \<in> A \<Longrightarrow> open a"
    3.53 @@ -245,77 +216,25 @@
    3.54    qed
    3.55  qed
    3.56  
    3.57 -
    3.58 -lemma countable_basis:
    3.59 -  obtains A :: "nat \<Rightarrow> 'a::first_countable_topology set" where
    3.60 -    "\<And>i. open (A i)" "\<And>i. x \<in> A i"
    3.61 -    "\<And>F. (\<forall>n. F n \<in> A n) \<Longrightarrow> F ----> x"
    3.62 -proof atomize_elim
    3.63 -  from countable_basis_at_decseq[of x] guess A . note A = this
    3.64 -  { fix F S assume "\<forall>n. F n \<in> A n" "open S" "x \<in> S"
    3.65 -    with A(3)[of S] have "eventually (\<lambda>n. F n \<in> S) sequentially"
    3.66 -      by (auto elim: eventually_elim1 simp: subset_eq) }
    3.67 -  with A show "\<exists>A. (\<forall>i. open (A i)) \<and> (\<forall>i. x \<in> A i) \<and> (\<forall>F. (\<forall>n. F n \<in> A n) \<longrightarrow> F ----> x)"
    3.68 -    by (intro exI[of _ A]) (auto simp: tendsto_def)
    3.69 -qed
    3.70 -
    3.71 -lemma sequentially_imp_eventually_nhds_within:
    3.72 -  fixes a :: "'a::first_countable_topology"
    3.73 -  assumes "\<forall>f. (\<forall>n. f n \<in> s) \<and> f ----> a \<longrightarrow> eventually (\<lambda>n. P (f n)) sequentially"
    3.74 -  shows "eventually P (nhds a within s)"
    3.75 -proof (rule ccontr)
    3.76 -  from countable_basis[of a] guess A . note A = this
    3.77 -  assume "\<not> eventually P (nhds a within s)"
    3.78 -  with A have P: "\<exists>F. \<forall>n. F n \<in> s \<and> F n \<in> A n \<and> \<not> P (F n)"
    3.79 -    unfolding Limits.eventually_within eventually_nhds by (intro choice) fastforce
    3.80 -  then guess F ..
    3.81 -  hence F0: "\<forall>n. F n \<in> s" and F2: "\<forall>n. F n \<in> A n" and F3: "\<forall>n. \<not> P (F n)"
    3.82 -    by fast+
    3.83 -  with A have "F ----> a" by auto
    3.84 -  hence "eventually (\<lambda>n. P (F n)) sequentially"
    3.85 -    using assms F0 by simp
    3.86 -  thus "False" by (simp add: F3)
    3.87 -qed
    3.88 -
    3.89 -lemma eventually_nhds_within_iff_sequentially:
    3.90 -  fixes a :: "'a::first_countable_topology"
    3.91 -  shows "eventually P (nhds a within s) \<longleftrightarrow> 
    3.92 -    (\<forall>f. (\<forall>n. f n \<in> s) \<and> f ----> a \<longrightarrow> eventually (\<lambda>n. P (f n)) sequentially)"
    3.93 -proof (safe intro!: sequentially_imp_eventually_nhds_within)
    3.94 -  assume "eventually P (nhds a within s)" 
    3.95 -  then obtain S where "open S" "a \<in> S" "\<forall>x\<in>S. x \<in> s \<longrightarrow> P x"
    3.96 -    by (auto simp: Limits.eventually_within eventually_nhds)
    3.97 -  moreover fix f assume "\<forall>n. f n \<in> s" "f ----> a"
    3.98 -  ultimately show "eventually (\<lambda>n. P (f n)) sequentially"
    3.99 -    by (auto dest!: topological_tendstoD elim: eventually_elim1)
   3.100 -qed
   3.101 -
   3.102 -lemma eventually_nhds_iff_sequentially:
   3.103 -  fixes a :: "'a::first_countable_topology"
   3.104 -  shows "eventually P (nhds a) \<longleftrightarrow> (\<forall>f. f ----> a \<longrightarrow> eventually (\<lambda>n. P (f n)) sequentially)"
   3.105 -  using eventually_nhds_within_iff_sequentially[of P a UNIV] by simp
   3.106 -
   3.107 -lemma not_eventually_sequentiallyD:
   3.108 -  assumes P: "\<not> eventually P sequentially"
   3.109 -  shows "\<exists>r. subseq r \<and> (\<forall>n. \<not> P (r n))"
   3.110 -proof -
   3.111 -  from P have "\<forall>n. \<exists>m\<ge>n. \<not> P m"
   3.112 -    unfolding eventually_sequentially by (simp add: not_less)
   3.113 -  then obtain r where "\<And>n. r n \<ge> n" "\<And>n. \<not> P (r n)"
   3.114 -    by (auto simp: choice_iff)
   3.115 -  then show ?thesis
   3.116 -    by (auto intro!: exI[of _ "\<lambda>n. r (((Suc \<circ> r) ^^ Suc n) 0)"]
   3.117 -             simp: less_eq_Suc_le subseq_Suc_iff)
   3.118 -qed
   3.119 -
   3.120 +lemma (in topological_space) first_countableI:
   3.121 +  assumes "countable A" and 1: "\<And>a. a \<in> A \<Longrightarrow> x \<in> a" "\<And>a. a \<in> A \<Longrightarrow> open a"
   3.122 +   and 2: "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> \<exists>a\<in>A. a \<subseteq> S"
   3.123 +  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))"
   3.124 +proof (safe intro!: exI[of _ "from_nat_into A"])
   3.125 +  have "A \<noteq> {}" using 2[of UNIV] by auto
   3.126 +  { fix i show "x \<in> from_nat_into A i" "open (from_nat_into A i)"
   3.127 +      using range_from_nat_into_subset[OF `A \<noteq> {}`] 1 by auto }
   3.128 +  { fix S assume "open S" "x\<in>S" from 2[OF this] show "\<exists>i. from_nat_into A i \<subseteq> S"
   3.129 +      using subset_range_from_nat_into[OF `countable A`] by auto }
   3.130 +qed
   3.131  
   3.132  instance prod :: (first_countable_topology, first_countable_topology) first_countable_topology
   3.133  proof
   3.134    fix x :: "'a \<times> 'b"
   3.135    from first_countable_basisE[of "fst x"] guess A :: "'a set set" . note A = this
   3.136    from first_countable_basisE[of "snd x"] guess B :: "'b set set" . note B = this
   3.137 -  show "\<exists>A::('a\<times>'b) set set. countable A \<and> (\<forall>a\<in>A. x \<in> a \<and> open a) \<and> (\<forall>S. open S \<and> x \<in> S \<longrightarrow> (\<exists>a\<in>A. a \<subseteq> S))"
   3.138 -  proof (intro exI[of _ "(\<lambda>(a, b). a \<times> b) ` (A \<times> B)"], safe)
   3.139 +  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))"
   3.140 +  proof (rule first_countableI[of "(\<lambda>(a, b). a \<times> b) ` (A \<times> B)"], safe)
   3.141      fix a b assume x: "a \<in> A" "b \<in> B"
   3.142      with A(2, 3)[of a] B(2, 3)[of b] show "x \<in> a \<times> b" "open (a \<times> b)"
   3.143        unfolding mem_Times_iff by (auto intro: open_Times)
   3.144 @@ -329,23 +248,6 @@
   3.145    qed (simp add: A B)
   3.146  qed
   3.147  
   3.148 -instance metric_space \<subseteq> first_countable_topology
   3.149 -proof
   3.150 -  fix x :: 'a
   3.151 -  show "\<exists>A. countable A \<and> (\<forall>a\<in>A. x \<in> a \<and> open a) \<and> (\<forall>S. open S \<and> x \<in> S \<longrightarrow> (\<exists>a\<in>A. a \<subseteq> S))"
   3.152 -  proof (intro exI, safe)
   3.153 -    fix S assume "open S" "x \<in> S"
   3.154 -    then obtain r where "0 < r" "{y. dist x y < r} \<subseteq> S"
   3.155 -      by (auto simp: open_dist dist_commute subset_eq)
   3.156 -    moreover from reals_Archimedean[OF `0 < r`] guess n ..
   3.157 -    moreover
   3.158 -    then have "{y. dist x y < inverse (Suc n)} \<subseteq> {y. dist x y < r}"
   3.159 -      by (auto simp: inverse_eq_divide)
   3.160 -    ultimately show "\<exists>a\<in>range (\<lambda>n. {y. dist x y < inverse (Suc n)}). a \<subseteq> S"
   3.161 -      by auto
   3.162 -  qed (auto intro: open_ball)
   3.163 -qed
   3.164 -
   3.165  class second_countable_topology = topological_space +
   3.166    assumes ex_countable_subbasis: "\<exists>B::'a::topological_space set set. countable B \<and> open = generate_topology B"
   3.167  begin
   3.168 @@ -417,9 +319,9 @@
   3.169    then have B: "countable B" "topological_basis B"
   3.170      using countable_basis is_basis
   3.171      by (auto simp: countable_basis is_basis)
   3.172 -  then show "\<exists>A. countable A \<and> (\<forall>a\<in>A. x \<in> a \<and> open a) \<and> (\<forall>S. open S \<and> x \<in> S \<longrightarrow> (\<exists>a\<in>A. a \<subseteq> S))"
   3.173 -    by (intro exI[of _ "{b\<in>B. x \<in> b}"])
   3.174 -       (fastforce simp: topological_space_class.topological_basis_def)
   3.175 +  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))"
   3.176 +    by (intro first_countableI[of "{b\<in>B. x \<in> b}"])
   3.177 +       (fastforce simp: topological_space_class.topological_basis_def)+
   3.178  qed
   3.179  
   3.180  subsection {* Polish spaces *}
   3.181 @@ -2353,7 +2255,7 @@
   3.182    apply (subgoal_tac "\<And>x (y::real). 0 < 1 + abs y \<and> (x <= y \<longrightarrow> x <= 1 + abs y)")
   3.183    by metis arith
   3.184  
   3.185 -lemma Bseq_eq_bounded: "Bseq f \<longleftrightarrow> bounded (range f)"
   3.186 +lemma Bseq_eq_bounded: "Bseq f \<longleftrightarrow> bounded (range f::_::real_normed_vector set)"
   3.187    unfolding Bseq_def bounded_pos by auto
   3.188  
   3.189  lemma bounded_Int[intro]: "bounded S \<or> bounded T \<Longrightarrow> bounded (S \<inter> T)"
     4.1 --- a/src/HOL/Probability/Fin_Map.thy	Fri Mar 22 10:41:43 2013 +0100
     4.2 +++ b/src/HOL/Probability/Fin_Map.thy	Fri Mar 22 10:41:43 2013 +0100
     4.3 @@ -222,8 +222,8 @@
     4.4    hence open_sub: "\<And>i S. i\<in>domain x \<Longrightarrow> open (S i) \<Longrightarrow> x i\<in>(S i) \<Longrightarrow> (\<exists>a\<in>A i. a\<subseteq>(S i))" by auto
     4.5    have A_notempty: "\<And>i. i \<in> domain x \<Longrightarrow> A i \<noteq> {}" using open_sub[of _ "\<lambda>_. UNIV"] by auto
     4.6    let ?A = "(\<lambda>f. Pi' (domain x) f) ` (Pi\<^isub>E (domain x) A)"
     4.7 -  show "\<exists>A. countable A \<and> (\<forall>a\<in>A. x \<in> a \<and> open a) \<and> (\<forall>S. open S \<and> x \<in> S \<longrightarrow> (\<exists>a\<in>A. a \<subseteq> S))"
     4.8 -  proof (rule exI[where x="?A"], safe)
     4.9 +  show "\<exists>A::nat \<Rightarrow> ('a\<Rightarrow>\<^isub>F'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))"
    4.10 +  proof (rule first_countableI[where A="?A"], safe)
    4.11      show "countable ?A" using A by (simp add: countable_PiE)
    4.12    next
    4.13      fix S::"('a \<Rightarrow>\<^isub>F 'b) set" assume "open S" "x \<in> S"
     5.1 --- a/src/HOL/Topological_Spaces.thy	Fri Mar 22 10:41:43 2013 +0100
     5.2 +++ b/src/HOL/Topological_Spaces.thy	Fri Mar 22 10:41:43 2013 +0100
     5.3 @@ -681,17 +681,17 @@
     5.4  definition (in topological_space) at :: "'a \<Rightarrow> 'a filter"
     5.5    where "at a = nhds a within - {a}"
     5.6  
     5.7 -abbreviation at_right :: "'a\<Colon>{topological_space, order} \<Rightarrow> 'a filter" where
     5.8 +abbreviation (in order_topology) at_right :: "'a \<Rightarrow> 'a filter" where
     5.9    "at_right x \<equiv> at x within {x <..}"
    5.10  
    5.11 -abbreviation at_left :: "'a\<Colon>{topological_space, order} \<Rightarrow> 'a filter" where
    5.12 +abbreviation (in order_topology) at_left :: "'a \<Rightarrow> 'a filter" where
    5.13    "at_left x \<equiv> at x within {..< x}"
    5.14  
    5.15 -lemma eventually_nhds:
    5.16 +lemma (in topological_space) eventually_nhds:
    5.17    "eventually P (nhds a) \<longleftrightarrow> (\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x))"
    5.18    unfolding nhds_def
    5.19  proof (rule eventually_Abs_filter, rule is_filter.intro)
    5.20 -  have "open (UNIV :: 'a :: topological_space set) \<and> a \<in> UNIV \<and> (\<forall>x\<in>UNIV. True)" by simp
    5.21 +  have "open UNIV \<and> a \<in> UNIV \<and> (\<forall>x\<in>UNIV. True)" by simp
    5.22    thus "\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. True)" ..
    5.23  next
    5.24    fix P Q
    5.25 @@ -843,7 +843,8 @@
    5.26      map (fn thm => @{thm tendsto_eq_rhs} OF [thm]) o Tendsto_Intros.get o Context.proof_of);
    5.27  *}
    5.28  
    5.29 -lemma tendsto_def: "(f ---> l) F \<longleftrightarrow> (\<forall>S. open S \<longrightarrow> l \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) F)"
    5.30 +lemma (in topological_space) tendsto_def:
    5.31 +   "(f ---> l) F \<longleftrightarrow> (\<forall>S. open S \<longrightarrow> l \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) F)"
    5.32    unfolding filterlim_def
    5.33  proof safe
    5.34    fix S assume "open S" "l \<in> S" "filtermap f F \<le> nhds l"
    5.35 @@ -859,12 +860,12 @@
    5.36  lemma tendsto_mono: "F \<le> F' \<Longrightarrow> (f ---> l) F' \<Longrightarrow> (f ---> l) F"
    5.37    unfolding tendsto_def le_filter_def by fast
    5.38  
    5.39 -lemma topological_tendstoI:
    5.40 +lemma (in topological_space) topological_tendstoI:
    5.41    "(\<And>S. open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>x. f x \<in> S) F)
    5.42      \<Longrightarrow> (f ---> l) F"
    5.43    unfolding tendsto_def by auto
    5.44  
    5.45 -lemma topological_tendstoD:
    5.46 +lemma (in topological_space) topological_tendstoD:
    5.47    "(f ---> l) F \<Longrightarrow> open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>x. f x \<in> S) F"
    5.48    unfolding tendsto_def by auto
    5.49  
    5.50 @@ -1290,6 +1291,19 @@
    5.51    "subseq r \<Longrightarrow> eventually P sequentially \<Longrightarrow> eventually (\<lambda>n. P (r n)) sequentially"
    5.52    unfolding eventually_sequentially by (metis seq_suble le_trans)
    5.53  
    5.54 +lemma not_eventually_sequentiallyD:
    5.55 +  assumes P: "\<not> eventually P sequentially"
    5.56 +  shows "\<exists>r. subseq r \<and> (\<forall>n. \<not> P (r n))"
    5.57 +proof -
    5.58 +  from P have "\<forall>n. \<exists>m\<ge>n. \<not> P m"
    5.59 +    unfolding eventually_sequentially by (simp add: not_less)
    5.60 +  then obtain r where "\<And>n. r n \<ge> n" "\<And>n. \<not> P (r n)"
    5.61 +    by (auto simp: choice_iff)
    5.62 +  then show ?thesis
    5.63 +    by (auto intro!: exI[of _ "\<lambda>n. r (((Suc \<circ> r) ^^ Suc n) 0)"]
    5.64 +             simp: less_eq_Suc_le subseq_Suc_iff)
    5.65 +qed
    5.66 +
    5.67  lemma filterlim_subseq: "subseq f \<Longrightarrow> filterlim f sequentially sequentially"
    5.68    unfolding filterlim_iff by (metis eventually_subseq)
    5.69  
    5.70 @@ -1427,6 +1441,83 @@
    5.71  lemma decseq_le: "decseq X \<Longrightarrow> X ----> L \<Longrightarrow> (L::'a::linorder_topology) \<le> X n"
    5.72    by (metis decseq_def LIMSEQ_le_const2)
    5.73  
    5.74 +subsection {* First countable topologies *}
    5.75 +
    5.76 +class first_countable_topology = topological_space +
    5.77 +  assumes first_countable_basis:
    5.78 +    "\<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))"
    5.79 +
    5.80 +lemma (in first_countable_topology) countable_basis_at_decseq:
    5.81 +  obtains A :: "nat \<Rightarrow> 'a set" where
    5.82 +    "\<And>i. open (A i)" "\<And>i. x \<in> (A i)"
    5.83 +    "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> eventually (\<lambda>i. A i \<subseteq> S) sequentially"
    5.84 +proof atomize_elim
    5.85 +  from first_countable_basis[of x] obtain A :: "nat \<Rightarrow> 'a set" where
    5.86 +    nhds: "\<And>i. open (A i)" "\<And>i. x \<in> A i"
    5.87 +    and incl: "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> \<exists>i. A i \<subseteq> S"  by auto
    5.88 +  def F \<equiv> "\<lambda>n. \<Inter>i\<le>n. A i"
    5.89 +  show "\<exists>A. (\<forall>i. open (A i)) \<and> (\<forall>i. x \<in> A i) \<and>
    5.90 +      (\<forall>S. open S \<longrightarrow> x \<in> S \<longrightarrow> eventually (\<lambda>i. A i \<subseteq> S) sequentially)"
    5.91 +  proof (safe intro!: exI[of _ F])
    5.92 +    fix i
    5.93 +    show "open (F i)" using nhds(1) by (auto simp: F_def intro!: open_INT)
    5.94 +    show "x \<in> F i" using nhds(2) by (auto simp: F_def)
    5.95 +  next
    5.96 +    fix S assume "open S" "x \<in> S"
    5.97 +    from incl[OF this] obtain i where "F i \<subseteq> S" unfolding F_def by auto
    5.98 +    moreover have "\<And>j. i \<le> j \<Longrightarrow> F j \<subseteq> F i"
    5.99 +      by (auto simp: F_def)
   5.100 +    ultimately show "eventually (\<lambda>i. F i \<subseteq> S) sequentially"
   5.101 +      by (auto simp: eventually_sequentially)
   5.102 +  qed
   5.103 +qed
   5.104 +
   5.105 +lemma (in first_countable_topology) countable_basis:
   5.106 +  obtains A :: "nat \<Rightarrow> 'a set" where
   5.107 +    "\<And>i. open (A i)" "\<And>i. x \<in> A i"
   5.108 +    "\<And>F. (\<forall>n. F n \<in> A n) \<Longrightarrow> F ----> x"
   5.109 +proof atomize_elim
   5.110 +  from countable_basis_at_decseq[of x] guess A . note A = this
   5.111 +  { fix F S assume "\<forall>n. F n \<in> A n" "open S" "x \<in> S"
   5.112 +    with A(3)[of S] have "eventually (\<lambda>n. F n \<in> S) sequentially"
   5.113 +      by (auto elim: eventually_elim1 simp: subset_eq) }
   5.114 +  with A show "\<exists>A. (\<forall>i. open (A i)) \<and> (\<forall>i. x \<in> A i) \<and> (\<forall>F. (\<forall>n. F n \<in> A n) \<longrightarrow> F ----> x)"
   5.115 +    by (intro exI[of _ A]) (auto simp: tendsto_def)
   5.116 +qed
   5.117 +
   5.118 +lemma (in first_countable_topology) sequentially_imp_eventually_nhds_within:
   5.119 +  assumes "\<forall>f. (\<forall>n. f n \<in> s) \<and> f ----> a \<longrightarrow> eventually (\<lambda>n. P (f n)) sequentially"
   5.120 +  shows "eventually P (nhds a within s)"
   5.121 +proof (rule ccontr)
   5.122 +  from countable_basis[of a] guess A . note A = this
   5.123 +  assume "\<not> eventually P (nhds a within s)"
   5.124 +  with A have P: "\<exists>F. \<forall>n. F n \<in> s \<and> F n \<in> A n \<and> \<not> P (F n)"
   5.125 +    unfolding eventually_within eventually_nhds by (intro choice) fastforce
   5.126 +  then guess F ..
   5.127 +  hence F0: "\<forall>n. F n \<in> s" and F2: "\<forall>n. F n \<in> A n" and F3: "\<forall>n. \<not> P (F n)"
   5.128 +    by fast+
   5.129 +  with A have "F ----> a" by auto
   5.130 +  hence "eventually (\<lambda>n. P (F n)) sequentially"
   5.131 +    using assms F0 by simp
   5.132 +  thus "False" by (simp add: F3)
   5.133 +qed
   5.134 +
   5.135 +lemma (in first_countable_topology) eventually_nhds_within_iff_sequentially:
   5.136 +  "eventually P (nhds a within s) \<longleftrightarrow> 
   5.137 +    (\<forall>f. (\<forall>n. f n \<in> s) \<and> f ----> a \<longrightarrow> eventually (\<lambda>n. P (f n)) sequentially)"
   5.138 +proof (safe intro!: sequentially_imp_eventually_nhds_within)
   5.139 +  assume "eventually P (nhds a within s)" 
   5.140 +  then obtain S where "open S" "a \<in> S" "\<forall>x\<in>S. x \<in> s \<longrightarrow> P x"
   5.141 +    by (auto simp: eventually_within eventually_nhds)
   5.142 +  moreover fix f assume "\<forall>n. f n \<in> s" "f ----> a"
   5.143 +  ultimately show "eventually (\<lambda>n. P (f n)) sequentially"
   5.144 +    by (auto dest!: topological_tendstoD elim: eventually_elim1)
   5.145 +qed
   5.146 +
   5.147 +lemma (in first_countable_topology) eventually_nhds_iff_sequentially:
   5.148 +  "eventually P (nhds a) \<longleftrightarrow> (\<forall>f. f ----> a \<longrightarrow> eventually (\<lambda>n. P (f n)) sequentially)"
   5.149 +  using eventually_nhds_within_iff_sequentially[of P a UNIV] by simp
   5.150 +
   5.151  subsection {* Function limit at a point *}
   5.152  
   5.153  abbreviation
   5.154 @@ -1487,6 +1578,35 @@
   5.155    shows "(\<lambda>x. g (f x)) -- a --> c"
   5.156    using g f inj by (rule tendsto_compose_eventually)
   5.157  
   5.158 +subsubsection {* Relation of LIM and LIMSEQ *}
   5.159 +
   5.160 +lemma (in first_countable_topology) sequentially_imp_eventually_within:
   5.161 +  "(\<forall>f. (\<forall>n. f n \<in> s \<and> f n \<noteq> a) \<and> f ----> a \<longrightarrow> eventually (\<lambda>n. P (f n)) sequentially) \<Longrightarrow>
   5.162 +    eventually P (at a within s)"
   5.163 +  unfolding at_def within_within_eq
   5.164 +  by (intro sequentially_imp_eventually_nhds_within) auto
   5.165 +
   5.166 +lemma (in first_countable_topology) sequentially_imp_eventually_at:
   5.167 +  "(\<forall>f. (\<forall>n. f n \<noteq> a) \<and> f ----> a \<longrightarrow> eventually (\<lambda>n. P (f n)) sequentially) \<Longrightarrow> eventually P (at a)"
   5.168 +  using assms sequentially_imp_eventually_within [where s=UNIV] by simp
   5.169 +
   5.170 +lemma LIMSEQ_SEQ_conv1:
   5.171 +  fixes f :: "'a::topological_space \<Rightarrow> 'b::topological_space"
   5.172 +  assumes f: "f -- a --> l"
   5.173 +  shows "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. f (S n)) ----> l"
   5.174 +  using tendsto_compose_eventually [OF f, where F=sequentially] by simp
   5.175 +
   5.176 +lemma LIMSEQ_SEQ_conv2:
   5.177 +  fixes f :: "'a::first_countable_topology \<Rightarrow> 'b::topological_space"
   5.178 +  assumes "\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> a \<longrightarrow> (\<lambda>n. f (S n)) ----> l"
   5.179 +  shows "f -- a --> l"
   5.180 +  using assms unfolding tendsto_def [where l=l] by (simp add: sequentially_imp_eventually_at)
   5.181 +
   5.182 +lemma LIMSEQ_SEQ_conv:
   5.183 +  "(\<forall>S. (\<forall>n. S n \<noteq> a) \<and> S ----> (a::'a::first_countable_topology) \<longrightarrow> (\<lambda>n. X (S n)) ----> L) =
   5.184 +   (X -- a --> (L::'b::topological_space))"
   5.185 +  using LIMSEQ_SEQ_conv2 LIMSEQ_SEQ_conv1 ..
   5.186 +
   5.187  subsection {* Continuity *}
   5.188  
   5.189  definition isCont :: "('a::topological_space \<Rightarrow> 'b::topological_space) \<Rightarrow> 'a \<Rightarrow> bool" where