author huffman Sun Apr 25 20:48:19 2010 -0700 (2010-04-25) changeset 36360 9d8f7efd9289 parent 36359 e5c785c166b2 child 36361 1debc8e29f6a
define finer-than ordering on net type; move some theorems into Limits.thy
1.1 --- a/src/HOL/Limits.thy	Sun Apr 25 16:23:40 2010 -0700
1.2 +++ b/src/HOL/Limits.thy	Sun Apr 25 20:48:19 2010 -0700
1.3 @@ -45,6 +45,10 @@
1.4    assumes "is_filter net" shows "eventually P (Abs_net net) = net P"
1.5  unfolding eventually_def using assms by (simp add: Abs_net_inverse)
1.7 +lemma expand_net_eq:
1.8 +  shows "net = net' \<longleftrightarrow> (\<forall>P. eventually P net = eventually P net')"
1.9 +unfolding Rep_net_inject [symmetric] expand_fun_eq eventually_def ..
1.10 +
1.11  lemma eventually_True [simp]: "eventually (\<lambda>x. True) net"
1.12  unfolding eventually_def
1.13  by (rule is_filter.True [OF is_filter_Rep_net])
1.14 @@ -95,6 +99,62 @@
1.15  using assms by (auto elim!: eventually_rev_mp)
1.18 +subsection {* Finer-than relation *}
1.19 +
1.20 +text {* @{term "net \<le> net'"} means that @{term net'} is finer than
1.21 +@{term net}. *}
1.22 +
1.23 +instantiation net :: (type) "{order,top}"
1.24 +begin
1.25 +
1.26 +definition
1.27 +  le_net_def [code del]:
1.28 +    "net \<le> net' \<longleftrightarrow> (\<forall>P. eventually P net \<longrightarrow> eventually P net')"
1.29 +
1.30 +definition
1.31 +  less_net_def [code del]:
1.32 +    "(net :: 'a net) < net' \<longleftrightarrow> net \<le> net' \<and> \<not> net' \<le> net"
1.33 +
1.34 +definition
1.35 +  top_net_def [code del]:
1.36 +    "top = Abs_net (\<lambda>P. True)"
1.37 +
1.38 +lemma eventually_top [simp]: "eventually P top"
1.39 +unfolding top_net_def
1.40 +by (subst eventually_Abs_net, rule is_filter.intro, auto)
1.41 +
1.42 +instance proof
1.43 +  fix x y :: "'a net" show "x < y \<longleftrightarrow> x \<le> y \<and> \<not> y \<le> x"
1.44 +    by (rule less_net_def)
1.45 +next
1.46 +  fix x :: "'a net" show "x \<le> x"
1.47 +    unfolding le_net_def by simp
1.48 +next
1.49 +  fix x y z :: "'a net" assume "x \<le> y" and "y \<le> z" thus "x \<le> z"
1.50 +    unfolding le_net_def by simp
1.51 +next
1.52 +  fix x y :: "'a net" assume "x \<le> y" and "y \<le> x" thus "x = y"
1.53 +    unfolding le_net_def expand_net_eq by fast
1.54 +next
1.55 +  fix x :: "'a net" show "x \<le> top"
1.56 +    unfolding le_net_def by simp
1.57 +qed
1.58 +
1.59 +end
1.60 +
1.61 +lemma net_leD:
1.62 +  "net \<le> net' \<Longrightarrow> eventually P net \<Longrightarrow> eventually P net'"
1.63 +unfolding le_net_def by simp
1.64 +
1.65 +lemma net_leI:
1.66 +  "(\<And>P. eventually P net \<Longrightarrow> eventually P net') \<Longrightarrow> net \<le> net'"
1.67 +unfolding le_net_def by simp
1.68 +
1.69 +lemma eventually_False:
1.70 +  "eventually (\<lambda>x. False) net \<longleftrightarrow> net = top"
1.71 +unfolding expand_net_eq by (auto elim: eventually_rev_mp)
1.72 +
1.73 +
1.74  subsection {* Standard Nets *}
1.76  definition
1.77 @@ -129,6 +189,9 @@
1.78  by (rule eventually_Abs_net, rule is_filter.intro)
1.79     (auto elim!: eventually_rev_mp)
1.81 +lemma within_UNIV: "net within UNIV = net"
1.82 +  unfolding expand_net_eq eventually_within by simp
1.83 +
1.84  lemma eventually_at_topological:
1.85    "eventually P (at a) \<longleftrightarrow> (\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. x \<noteq> a \<longrightarrow> P x))"
1.86  unfolding at_def
2.1 --- a/src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy	Sun Apr 25 16:23:40 2010 -0700
2.2 +++ b/src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy	Sun Apr 25 20:48:19 2010 -0700
2.3 @@ -976,15 +976,6 @@
2.5  text{* Prove That They are all nets. *}
2.7 -(* TODO: move to HOL/Limits.thy *)
2.8 -lemma expand_net_eq:
2.9 -  "net = net' \<longleftrightarrow> (\<forall>P. eventually P net = eventually P net')"
2.10 -  unfolding Rep_net_inject [symmetric] expand_fun_eq eventually_def ..
2.11 -
2.12 -(* TODO: move to HOL/Limits.thy *)
2.13 -lemma within_UNIV: "net within UNIV = net"
2.14 -  unfolding expand_net_eq eventually_within by simp
2.15 -
2.16  lemma eventually_at_infinity:
2.17    "eventually P at_infinity \<longleftrightarrow> (\<exists>b. \<forall>x. norm x >= b \<longrightarrow> P x)"
2.18  unfolding at_infinity_def