--- a/src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy Fri Mar 22 10:41:43 2013 +0100
+++ b/src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy Fri Mar 22 10:41:43 2013 +0100
@@ -181,44 +181,15 @@
end
-class first_countable_topology = topological_space +
- assumes first_countable_basis:
- "\<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))"
-
-lemma (in first_countable_topology) countable_basis_at_decseq:
- obtains A :: "nat \<Rightarrow> 'a set" where
- "\<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"
-proof atomize_elim
- from first_countable_basis[of x] obtain A
- where "countable A"
- and nhds: "\<And>a. a \<in> A \<Longrightarrow> open a" "\<And>a. a \<in> A \<Longrightarrow> x \<in> a"
- and incl: "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> \<exists>a\<in>A. a \<subseteq> S" by auto
- then have "A \<noteq> {}" by auto
- with `countable A` have r: "A = range (from_nat_into A)" by auto
- def F \<equiv> "\<lambda>n. \<Inter>i\<le>n. from_nat_into A i"
- show "\<exists>A. (\<forall>i. open (A i)) \<and> (\<forall>i. x \<in> A i) \<and>
- (\<forall>S. open S \<longrightarrow> x \<in> S \<longrightarrow> eventually (\<lambda>i. A i \<subseteq> S) sequentially)"
- proof (safe intro!: exI[of _ F])
- fix i
- show "open (F i)" using nhds(1) r by (auto simp: F_def intro!: open_INT)
- show "x \<in> F i" using nhds(2) r by (auto simp: F_def)
- next
- fix S assume "open S" "x \<in> S"
- from incl[OF this] obtain i where "F i \<subseteq> S"
- by (subst (asm) r) (auto simp: F_def)
- moreover have "\<And>j. i \<le> j \<Longrightarrow> F j \<subseteq> F i"
- by (auto simp: F_def)
- ultimately show "eventually (\<lambda>i. F i \<subseteq> S) sequentially"
- by (auto simp: eventually_sequentially)
- qed
-qed
-
lemma (in first_countable_topology) first_countable_basisE:
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)"
using first_countable_basis[of x]
- by atomize_elim auto
+ apply atomize_elim
+ apply (elim exE)
+ apply (rule_tac x="range A" in exI)
+ apply auto
+ done
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"
@@ -245,77 +216,25 @@
qed
qed
-
-lemma countable_basis:
- obtains A :: "nat \<Rightarrow> 'a::first_countable_topology set" where
- "\<And>i. open (A i)" "\<And>i. x \<in> A i"
- "\<And>F. (\<forall>n. F n \<in> A n) \<Longrightarrow> F ----> x"
-proof atomize_elim
- from countable_basis_at_decseq[of x] guess A . note A = this
- { fix F S assume "\<forall>n. F n \<in> A n" "open S" "x \<in> S"
- with A(3)[of S] have "eventually (\<lambda>n. F n \<in> S) sequentially"
- by (auto elim: eventually_elim1 simp: subset_eq) }
- 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)"
- by (intro exI[of _ A]) (auto simp: tendsto_def)
-qed
-
-lemma sequentially_imp_eventually_nhds_within:
- fixes a :: "'a::first_countable_topology"
- assumes "\<forall>f. (\<forall>n. f n \<in> s) \<and> f ----> a \<longrightarrow> eventually (\<lambda>n. P (f n)) sequentially"
- shows "eventually P (nhds a within s)"
-proof (rule ccontr)
- from countable_basis[of a] guess A . note A = this
- assume "\<not> eventually P (nhds a within s)"
- with A have P: "\<exists>F. \<forall>n. F n \<in> s \<and> F n \<in> A n \<and> \<not> P (F n)"
- unfolding Limits.eventually_within eventually_nhds by (intro choice) fastforce
- then guess F ..
- 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)"
- by fast+
- with A have "F ----> a" by auto
- hence "eventually (\<lambda>n. P (F n)) sequentially"
- using assms F0 by simp
- thus "False" by (simp add: F3)
-qed
-
-lemma eventually_nhds_within_iff_sequentially:
- fixes a :: "'a::first_countable_topology"
- shows "eventually P (nhds a within s) \<longleftrightarrow>
- (\<forall>f. (\<forall>n. f n \<in> s) \<and> f ----> a \<longrightarrow> eventually (\<lambda>n. P (f n)) sequentially)"
-proof (safe intro!: sequentially_imp_eventually_nhds_within)
- assume "eventually P (nhds a within s)"
- then obtain S where "open S" "a \<in> S" "\<forall>x\<in>S. x \<in> s \<longrightarrow> P x"
- by (auto simp: Limits.eventually_within eventually_nhds)
- moreover fix f assume "\<forall>n. f n \<in> s" "f ----> a"
- ultimately show "eventually (\<lambda>n. P (f n)) sequentially"
- by (auto dest!: topological_tendstoD elim: eventually_elim1)
-qed
-
-lemma eventually_nhds_iff_sequentially:
- fixes a :: "'a::first_countable_topology"
- shows "eventually P (nhds a) \<longleftrightarrow> (\<forall>f. f ----> a \<longrightarrow> eventually (\<lambda>n. P (f n)) sequentially)"
- using eventually_nhds_within_iff_sequentially[of P a UNIV] by simp
-
-lemma not_eventually_sequentiallyD:
- assumes P: "\<not> eventually P sequentially"
- shows "\<exists>r. subseq r \<and> (\<forall>n. \<not> P (r n))"
-proof -
- from P have "\<forall>n. \<exists>m\<ge>n. \<not> P m"
- unfolding eventually_sequentially by (simp add: not_less)
- then obtain r where "\<And>n. r n \<ge> n" "\<And>n. \<not> P (r n)"
- by (auto simp: choice_iff)
- then show ?thesis
- by (auto intro!: exI[of _ "\<lambda>n. r (((Suc \<circ> r) ^^ Suc n) 0)"]
- simp: less_eq_Suc_le subseq_Suc_iff)
-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"])
+ have "A \<noteq> {}" using 2[of UNIV] by auto
+ { fix i show "x \<in> from_nat_into A i" "open (from_nat_into A i)"
+ using range_from_nat_into_subset[OF `A \<noteq> {}`] 1 by auto }
+ { 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 `countable A`] by auto }
+qed
instance prod :: (first_countable_topology, first_countable_topology) first_countable_topology
proof
fix x :: "'a \<times> 'b"
from first_countable_basisE[of "fst x"] guess A :: "'a set set" . note A = this
from first_countable_basisE[of "snd x"] guess B :: "'b set set" . note B = this
- 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))"
- proof (intro exI[of _ "(\<lambda>(a, b). a \<times> b) ` (A \<times> B)"], safe)
+ 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"
with A(2, 3)[of a] B(2, 3)[of b] show "x \<in> a \<times> b" "open (a \<times> b)"
unfolding mem_Times_iff by (auto intro: open_Times)
@@ -329,23 +248,6 @@
qed (simp add: A B)
qed
-instance metric_space \<subseteq> first_countable_topology
-proof
- fix x :: 'a
- 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))"
- proof (intro exI, safe)
- fix S assume "open S" "x \<in> S"
- then obtain r where "0 < r" "{y. dist x y < r} \<subseteq> S"
- by (auto simp: open_dist dist_commute subset_eq)
- moreover from reals_Archimedean[OF `0 < r`] guess n ..
- moreover
- then have "{y. dist x y < inverse (Suc n)} \<subseteq> {y. dist x y < r}"
- by (auto simp: inverse_eq_divide)
- ultimately show "\<exists>a\<in>range (\<lambda>n. {y. dist x y < inverse (Suc n)}). a \<subseteq> S"
- by auto
- qed (auto intro: open_ball)
-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
@@ -417,9 +319,9 @@
then have B: "countable B" "topological_basis B"
using countable_basis is_basis
by (auto simp: countable_basis is_basis)
- 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))"
- by (intro exI[of _ "{b\<in>B. x \<in> b}"])
- (fastforce simp: topological_space_class.topological_basis_def)
+ 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
subsection {* Polish spaces *}
@@ -2353,7 +2255,7 @@
apply (subgoal_tac "\<And>x (y::real). 0 < 1 + abs y \<and> (x <= y \<longrightarrow> x <= 1 + abs y)")
by metis arith
-lemma Bseq_eq_bounded: "Bseq f \<longleftrightarrow> bounded (range f)"
+lemma Bseq_eq_bounded: "Bseq f \<longleftrightarrow> bounded (range f::_::real_normed_vector set)"
unfolding Bseq_def bounded_pos by auto
lemma bounded_Int[intro]: "bounded S \<or> bounded T \<Longrightarrow> bounded (S \<inter> T)"