src/HOL/Library/Product_Vector.thy
 changeset 31492 5400beeddb55 parent 31491 f7310185481d child 31562 10d0fb526643
```     1.1 --- a/src/HOL/Library/Product_Vector.thy	Sun Jun 07 15:18:52 2009 -0700
1.2 +++ b/src/HOL/Library/Product_Vector.thy	Sun Jun 07 17:59:54 2009 -0700
1.3 @@ -45,29 +45,29 @@
1.4    "*" :: (topological_space, topological_space) topological_space
1.5  begin
1.6
1.7 -definition topo_prod_def:
1.8 -  "topo = {S. \<forall>x\<in>S. \<exists>A\<in>topo. \<exists>B\<in>topo. x \<in> A \<times> B \<and> A \<times> B \<subseteq> S}"
1.9 +definition open_prod_def:
1.10 +  "open (S :: ('a \<times> 'b) set) \<longleftrightarrow>
1.11 +    (\<forall>x\<in>S. \<exists>A B. open A \<and> open B \<and> x \<in> A \<times> B \<and> A \<times> B \<subseteq> S)"
1.12
1.13  instance proof
1.14 -  show "(UNIV :: ('a \<times> 'b) set) \<in> topo"
1.15 -    unfolding topo_prod_def by (auto intro: topo_UNIV)
1.16 +  show "open (UNIV :: ('a \<times> 'b) set)"
1.17 +    unfolding open_prod_def by auto
1.18  next
1.19    fix S T :: "('a \<times> 'b) set"
1.20 -  assume "S \<in> topo" "T \<in> topo" thus "S \<inter> T \<in> topo"
1.21 -    unfolding topo_prod_def
1.22 +  assume "open S" "open T" thus "open (S \<inter> T)"
1.23 +    unfolding open_prod_def
1.24      apply clarify
1.25      apply (drule (1) bspec)+
1.26      apply (clarify, rename_tac Sa Ta Sb Tb)
1.27 -    apply (rule_tac x="Sa \<inter> Ta" in rev_bexI)
1.28 -    apply (simp add: topo_Int)
1.29 -    apply (rule_tac x="Sb \<inter> Tb" in rev_bexI)
1.30 -    apply (simp add: topo_Int)
1.31 +    apply (rule_tac x="Sa \<inter> Ta" in exI)
1.32 +    apply (rule_tac x="Sb \<inter> Tb" in exI)
1.33 +    apply (simp add: open_Int)
1.34      apply fast
1.35      done
1.36  next
1.37 -  fix T :: "('a \<times> 'b) set set"
1.38 -  assume "T \<subseteq> topo" thus "\<Union>T \<in> topo"
1.39 -    unfolding topo_prod_def Bex_def by fast
1.40 +  fix K :: "('a \<times> 'b) set set"
1.41 +  assume "\<forall>S\<in>K. open S" thus "open (\<Union>K)"
1.42 +    unfolding open_prod_def by fast
1.43  qed
1.44
1.45  end
1.46 @@ -104,9 +104,10 @@
1.47    (* FIXME: long proof! *)
1.48    (* Maybe it would be easier to define topological spaces *)
1.49    (* in terms of neighborhoods instead of open sets? *)
1.50 -  show "topo = {S::('a \<times> 'b) set. \<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S}"
1.51 -    unfolding topo_prod_def topo_dist
1.52 -    apply (safe, rename_tac S a b)
1.53 +  fix S :: "('a \<times> 'b) set"
1.54 +  show "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
1.55 +    unfolding open_prod_def open_dist
1.56 +    apply safe
1.57      apply (drule (1) bspec)
1.58      apply clarify
1.59      apply (drule (1) bspec)+
1.60 @@ -121,18 +122,19 @@
1.61      apply (drule spec, erule mp)
1.62      apply (erule le_less_trans [OF real_sqrt_sum_squares_ge2])
1.63
1.64 -    apply (rename_tac S a b)
1.65      apply (drule (1) bspec)
1.66      apply clarify
1.67      apply (subgoal_tac "\<exists>r>0. \<exists>s>0. e = sqrt (r\<twosuperior> + s\<twosuperior>)")
1.68      apply clarify
1.69 -    apply (rule_tac x="{y. dist y a < r}" in rev_bexI)
1.70 +    apply (rule_tac x="{y. dist y a < r}" in exI)
1.71 +    apply (rule_tac x="{y. dist y b < s}" in exI)
1.72 +    apply (rule conjI)
1.73      apply clarify
1.74      apply (rule_tac x="r - dist x a" in exI, rule conjI, simp)
1.75      apply clarify
1.76      apply (rule le_less_trans [OF dist_triangle])
1.77      apply (erule less_le_trans [OF add_strict_right_mono], simp)
1.78 -    apply (rule_tac x="{y. dist y b < s}" in rev_bexI)
1.79 +    apply (rule conjI)
1.80      apply clarify
1.81      apply (rule_tac x="s - dist x b" in exI, rule conjI, simp)
1.82      apply clarify
1.83 @@ -163,13 +165,13 @@
1.84    assumes "(f ---> a) net"
1.85    shows "((\<lambda>x. fst (f x)) ---> fst a) net"
1.86  proof (rule topological_tendstoI)
1.87 -  fix S assume "S \<in> topo" "fst a \<in> S"
1.88 -  then have "fst -` S \<in> topo" "a \<in> fst -` S"
1.89 -    unfolding topo_prod_def
1.90 +  fix S assume "open S" "fst a \<in> S"
1.91 +  then have "open (fst -` S)" "a \<in> fst -` S"
1.92 +    unfolding open_prod_def
1.93      apply simp_all
1.94      apply clarify
1.95 -    apply (erule rev_bexI, simp)
1.96 -    apply (rule rev_bexI [OF topo_UNIV])
1.97 +    apply (rule exI, erule conjI)
1.98 +    apply (rule exI, rule conjI [OF open_UNIV])
1.99      apply auto
1.100      done
1.101    with assms have "eventually (\<lambda>x. f x \<in> fst -` S) net"
1.102 @@ -182,13 +184,13 @@
1.103    assumes "(f ---> a) net"
1.104    shows "((\<lambda>x. snd (f x)) ---> snd a) net"
1.105  proof (rule topological_tendstoI)
1.106 -  fix S assume "S \<in> topo" "snd a \<in> S"
1.107 -  then have "snd -` S \<in> topo" "a \<in> snd -` S"
1.108 -    unfolding topo_prod_def
1.109 +  fix S assume "open S" "snd a \<in> S"
1.110 +  then have "open (snd -` S)" "a \<in> snd -` S"
1.111 +    unfolding open_prod_def
1.112      apply simp_all
1.113      apply clarify
1.114 -    apply (rule rev_bexI [OF topo_UNIV])
1.115 -    apply (erule rev_bexI)
1.116 +    apply (rule exI, rule conjI [OF open_UNIV])
1.117 +    apply (rule exI, erule conjI)
1.118      apply auto
1.119      done
1.120    with assms have "eventually (\<lambda>x. f x \<in> snd -` S) net"
1.121 @@ -201,15 +203,15 @@
1.122    assumes "(f ---> a) net" and "(g ---> b) net"
1.123    shows "((\<lambda>x. (f x, g x)) ---> (a, b)) net"
1.124  proof (rule topological_tendstoI)
1.125 -  fix S assume "S \<in> topo" "(a, b) \<in> S"
1.126 -  then obtain A B where "A \<in> topo" "B \<in> topo" "a \<in> A" "b \<in> B" "A \<times> B \<subseteq> S"
1.127 -    unfolding topo_prod_def by auto
1.128 +  fix S assume "open S" "(a, b) \<in> S"
1.129 +  then obtain A B where "open A" "open B" "a \<in> A" "b \<in> B" "A \<times> B \<subseteq> S"
1.130 +    unfolding open_prod_def by auto
1.131    have "eventually (\<lambda>x. f x \<in> A) net"
1.132 -    using `(f ---> a) net` `A \<in> topo` `a \<in> A`
1.133 +    using `(f ---> a) net` `open A` `a \<in> A`
1.134      by (rule topological_tendstoD)
1.135    moreover
1.136    have "eventually (\<lambda>x. g x \<in> B) net"
1.137 -    using `(g ---> b) net` `B \<in> topo` `b \<in> B`
1.138 +    using `(g ---> b) net` `open B` `b \<in> B`
1.139      by (rule topological_tendstoD)
1.140    ultimately
1.141    show "eventually (\<lambda>x. (f x, g x) \<in> S) net"
```