--- a/src/HOL/Limits.thy Sun Apr 25 16:23:40 2010 -0700
+++ b/src/HOL/Limits.thy Sun Apr 25 20:48:19 2010 -0700
@@ -45,6 +45,10 @@
assumes "is_filter net" shows "eventually P (Abs_net net) = net P"
unfolding eventually_def using assms by (simp add: Abs_net_inverse)
+lemma expand_net_eq:
+ shows "net = net' \<longleftrightarrow> (\<forall>P. eventually P net = eventually P net')"
+unfolding Rep_net_inject [symmetric] expand_fun_eq eventually_def ..
+
lemma eventually_True [simp]: "eventually (\<lambda>x. True) net"
unfolding eventually_def
by (rule is_filter.True [OF is_filter_Rep_net])
@@ -95,6 +99,62 @@
using assms by (auto elim!: eventually_rev_mp)
+subsection {* Finer-than relation *}
+
+text {* @{term "net \<le> net'"} means that @{term net'} is finer than
+@{term net}. *}
+
+instantiation net :: (type) "{order,top}"
+begin
+
+definition
+ le_net_def [code del]:
+ "net \<le> net' \<longleftrightarrow> (\<forall>P. eventually P net \<longrightarrow> eventually P net')"
+
+definition
+ less_net_def [code del]:
+ "(net :: 'a net) < net' \<longleftrightarrow> net \<le> net' \<and> \<not> net' \<le> net"
+
+definition
+ top_net_def [code del]:
+ "top = Abs_net (\<lambda>P. True)"
+
+lemma eventually_top [simp]: "eventually P top"
+unfolding top_net_def
+by (subst eventually_Abs_net, rule is_filter.intro, auto)
+
+instance proof
+ fix x y :: "'a net" show "x < y \<longleftrightarrow> x \<le> y \<and> \<not> y \<le> x"
+ by (rule less_net_def)
+next
+ fix x :: "'a net" show "x \<le> x"
+ unfolding le_net_def by simp
+next
+ fix x y z :: "'a net" assume "x \<le> y" and "y \<le> z" thus "x \<le> z"
+ unfolding le_net_def by simp
+next
+ fix x y :: "'a net" assume "x \<le> y" and "y \<le> x" thus "x = y"
+ unfolding le_net_def expand_net_eq by fast
+next
+ fix x :: "'a net" show "x \<le> top"
+ unfolding le_net_def by simp
+qed
+
+end
+
+lemma net_leD:
+ "net \<le> net' \<Longrightarrow> eventually P net \<Longrightarrow> eventually P net'"
+unfolding le_net_def by simp
+
+lemma net_leI:
+ "(\<And>P. eventually P net \<Longrightarrow> eventually P net') \<Longrightarrow> net \<le> net'"
+unfolding le_net_def by simp
+
+lemma eventually_False:
+ "eventually (\<lambda>x. False) net \<longleftrightarrow> net = top"
+unfolding expand_net_eq by (auto elim: eventually_rev_mp)
+
+
subsection {* Standard Nets *}
definition
@@ -129,6 +189,9 @@
by (rule eventually_Abs_net, rule is_filter.intro)
(auto elim!: eventually_rev_mp)
+lemma within_UNIV: "net within UNIV = net"
+ unfolding expand_net_eq eventually_within by simp
+
lemma eventually_at_topological:
"eventually P (at a) \<longleftrightarrow> (\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. x \<noteq> a \<longrightarrow> P x))"
unfolding at_def