# HG changeset patch
# User huffman
# Date 1312856813 25200
# Node ID 730f7cced3a6ab5b0270f29f6f2419c3c3fdcd8c
# Parent  53d95b52954c5725827f9bdbf56cbb09c99d5cf0
rename type 'a net to 'a filter, following standard mathematical terminology

diff -r 53d95b52954c -r 730f7cced3a6 NEWS
--- a/NEWS	Mon Aug 08 18:36:32 2011 -0700
+++ b/NEWS	Mon Aug 08 19:26:53 2011 -0700
@@ -170,6 +170,9 @@
 Every theorem name containing "inat", "Fin", "Infty", or "iSuc" has
 been renamed accordingly.
 
+* Limits.thy: Type "'a net" has been renamed to "'a filter", in
+accordance with standard mathematical terminology. INCOMPATIBILITY.
+
 
 *** Document preparation ***
 
diff -r 53d95b52954c -r 730f7cced3a6 src/HOL/Limits.thy
--- a/src/HOL/Limits.thy	Mon Aug 08 18:36:32 2011 -0700
+++ b/src/HOL/Limits.thy	Mon Aug 08 19:26:53 2011 -0700
@@ -8,263 +8,262 @@
 imports RealVector
 begin
 
-subsection {* Nets *}
+subsection {* Filters *}
 
 text {*
-  A net is now defined simply as a filter on a set.
-  The definition also allows non-proper filters.
+  This definition also allows non-proper filters.
 *}
 
 locale is_filter =
-  fixes net :: "('a \<Rightarrow> bool) \<Rightarrow> bool"
-  assumes True: "net (\<lambda>x. True)"
-  assumes conj: "net (\<lambda>x. P x) \<Longrightarrow> net (\<lambda>x. Q x) \<Longrightarrow> net (\<lambda>x. P x \<and> Q x)"
-  assumes mono: "\<forall>x. P x \<longrightarrow> Q x \<Longrightarrow> net (\<lambda>x. P x) \<Longrightarrow> net (\<lambda>x. Q x)"
+  fixes F :: "('a \<Rightarrow> bool) \<Rightarrow> bool"
+  assumes True: "F (\<lambda>x. True)"
+  assumes conj: "F (\<lambda>x. P x) \<Longrightarrow> F (\<lambda>x. Q x) \<Longrightarrow> F (\<lambda>x. P x \<and> Q x)"
+  assumes mono: "\<forall>x. P x \<longrightarrow> Q x \<Longrightarrow> F (\<lambda>x. P x) \<Longrightarrow> F (\<lambda>x. Q x)"
 
-typedef (open) 'a net =
-  "{net :: ('a \<Rightarrow> bool) \<Rightarrow> bool. is_filter net}"
+typedef (open) 'a filter = "{F :: ('a \<Rightarrow> bool) \<Rightarrow> bool. is_filter F}"
 proof
-  show "(\<lambda>x. True) \<in> ?net" by (auto intro: is_filter.intro)
+  show "(\<lambda>x. True) \<in> ?filter" by (auto intro: is_filter.intro)
 qed
 
-lemma is_filter_Rep_net: "is_filter (Rep_net net)"
-using Rep_net [of net] by simp
+lemma is_filter_Rep_filter: "is_filter (Rep_filter A)"
+  using Rep_filter [of A] by simp
 
-lemma Abs_net_inverse':
-  assumes "is_filter net" shows "Rep_net (Abs_net net) = net"
-using assms by (simp add: Abs_net_inverse)
+lemma Abs_filter_inverse':
+  assumes "is_filter F" shows "Rep_filter (Abs_filter F) = F"
+  using assms by (simp add: Abs_filter_inverse)
 
 
 subsection {* Eventually *}
 
-definition eventually :: "('a \<Rightarrow> bool) \<Rightarrow> 'a net \<Rightarrow> bool" where
-  "eventually P net \<longleftrightarrow> Rep_net net P"
+definition eventually :: "('a \<Rightarrow> bool) \<Rightarrow> 'a filter \<Rightarrow> bool"
+  where "eventually P A \<longleftrightarrow> Rep_filter A P"
 
-lemma eventually_Abs_net:
-  assumes "is_filter net" shows "eventually P (Abs_net net) = net P"
-unfolding eventually_def using assms by (simp add: Abs_net_inverse)
+lemma eventually_Abs_filter:
+  assumes "is_filter F" shows "eventually P (Abs_filter F) = F P"
+  unfolding eventually_def using assms by (simp add: Abs_filter_inverse)
 
-lemma expand_net_eq:
-  shows "net = net' \<longleftrightarrow> (\<forall>P. eventually P net = eventually P net')"
-unfolding Rep_net_inject [symmetric] fun_eq_iff eventually_def ..
+lemma filter_eq_iff:
+  shows "A = B \<longleftrightarrow> (\<forall>P. eventually P A = eventually P B)"
+  unfolding Rep_filter_inject [symmetric] fun_eq_iff eventually_def ..
 
-lemma eventually_True [simp]: "eventually (\<lambda>x. True) net"
-unfolding eventually_def
-by (rule is_filter.True [OF is_filter_Rep_net])
+lemma eventually_True [simp]: "eventually (\<lambda>x. True) A"
+  unfolding eventually_def
+  by (rule is_filter.True [OF is_filter_Rep_filter])
 
-lemma always_eventually: "\<forall>x. P x \<Longrightarrow> eventually P net"
+lemma always_eventually: "\<forall>x. P x \<Longrightarrow> eventually P A"
 proof -
   assume "\<forall>x. P x" hence "P = (\<lambda>x. True)" by (simp add: ext)
-  thus "eventually P net" by simp
+  thus "eventually P A" by simp
 qed
 
 lemma eventually_mono:
-  "(\<forall>x. P x \<longrightarrow> Q x) \<Longrightarrow> eventually P net \<Longrightarrow> eventually Q net"
-unfolding eventually_def
-by (rule is_filter.mono [OF is_filter_Rep_net])
+  "(\<forall>x. P x \<longrightarrow> Q x) \<Longrightarrow> eventually P A \<Longrightarrow> eventually Q A"
+  unfolding eventually_def
+  by (rule is_filter.mono [OF is_filter_Rep_filter])
 
 lemma eventually_conj:
-  assumes P: "eventually (\<lambda>x. P x) net"
-  assumes Q: "eventually (\<lambda>x. Q x) net"
-  shows "eventually (\<lambda>x. P x \<and> Q x) net"
-using assms unfolding eventually_def
-by (rule is_filter.conj [OF is_filter_Rep_net])
+  assumes P: "eventually (\<lambda>x. P x) A"
+  assumes Q: "eventually (\<lambda>x. Q x) A"
+  shows "eventually (\<lambda>x. P x \<and> Q x) A"
+  using assms unfolding eventually_def
+  by (rule is_filter.conj [OF is_filter_Rep_filter])
 
 lemma eventually_mp:
-  assumes "eventually (\<lambda>x. P x \<longrightarrow> Q x) net"
-  assumes "eventually (\<lambda>x. P x) net"
-  shows "eventually (\<lambda>x. Q x) net"
+  assumes "eventually (\<lambda>x. P x \<longrightarrow> Q x) A"
+  assumes "eventually (\<lambda>x. P x) A"
+  shows "eventually (\<lambda>x. Q x) A"
 proof (rule eventually_mono)
   show "\<forall>x. (P x \<longrightarrow> Q x) \<and> P x \<longrightarrow> Q x" by simp
-  show "eventually (\<lambda>x. (P x \<longrightarrow> Q x) \<and> P x) net"
+  show "eventually (\<lambda>x. (P x \<longrightarrow> Q x) \<and> P x) A"
     using assms by (rule eventually_conj)
 qed
 
 lemma eventually_rev_mp:
-  assumes "eventually (\<lambda>x. P x) net"
-  assumes "eventually (\<lambda>x. P x \<longrightarrow> Q x) net"
-  shows "eventually (\<lambda>x. Q x) net"
+  assumes "eventually (\<lambda>x. P x) A"
+  assumes "eventually (\<lambda>x. P x \<longrightarrow> Q x) A"
+  shows "eventually (\<lambda>x. Q x) A"
 using assms(2) assms(1) by (rule eventually_mp)
 
 lemma eventually_conj_iff:
-  "eventually (\<lambda>x. P x \<and> Q x) net \<longleftrightarrow> eventually P net \<and> eventually Q net"
-by (auto intro: eventually_conj elim: eventually_rev_mp)
+  "eventually (\<lambda>x. P x \<and> Q x) A \<longleftrightarrow> eventually P A \<and> eventually Q A"
+  by (auto intro: eventually_conj elim: eventually_rev_mp)
 
 lemma eventually_elim1:
-  assumes "eventually (\<lambda>i. P i) net"
+  assumes "eventually (\<lambda>i. P i) A"
   assumes "\<And>i. P i \<Longrightarrow> Q i"
-  shows "eventually (\<lambda>i. Q i) net"
-using assms by (auto elim!: eventually_rev_mp)
+  shows "eventually (\<lambda>i. Q i) A"
+  using assms by (auto elim!: eventually_rev_mp)
 
 lemma eventually_elim2:
-  assumes "eventually (\<lambda>i. P i) net"
-  assumes "eventually (\<lambda>i. Q i) net"
+  assumes "eventually (\<lambda>i. P i) A"
+  assumes "eventually (\<lambda>i. Q i) A"
   assumes "\<And>i. P i \<Longrightarrow> Q i \<Longrightarrow> R i"
-  shows "eventually (\<lambda>i. R i) net"
-using assms by (auto elim!: eventually_rev_mp)
+  shows "eventually (\<lambda>i. R i) A"
+  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'}. *}
+text {* @{term "A \<le> B"} means that filter @{term A} is finer than
+filter @{term B}. *}
 
-instantiation net :: (type) complete_lattice
+instantiation filter :: (type) complete_lattice
 begin
 
-definition
-  le_net_def: "net \<le> net' \<longleftrightarrow> (\<forall>P. eventually P net' \<longrightarrow> eventually P net)"
+definition le_filter_def:
+  "A \<le> B \<longleftrightarrow> (\<forall>P. eventually P B \<longrightarrow> eventually P A)"
 
 definition
-  less_net_def: "(net :: 'a net) < net' \<longleftrightarrow> net \<le> net' \<and> \<not> net' \<le> net"
+  "(A :: 'a filter) < B \<longleftrightarrow> A \<le> B \<and> \<not> B \<le> A"
 
 definition
-  top_net_def: "top = Abs_net (\<lambda>P. \<forall>x. P x)"
+  "top = Abs_filter (\<lambda>P. \<forall>x. P x)"
 
 definition
-  bot_net_def: "bot = Abs_net (\<lambda>P. True)"
+  "bot = Abs_filter (\<lambda>P. True)"
 
 definition
-  sup_net_def: "sup net net' = Abs_net (\<lambda>P. eventually P net \<and> eventually P net')"
+  "sup A B = Abs_filter (\<lambda>P. eventually P A \<and> eventually P B)"
 
 definition
-  inf_net_def: "inf a b = Abs_net
-      (\<lambda>P. \<exists>Q R. eventually Q a \<and> eventually R b \<and> (\<forall>x. Q x \<and> R x \<longrightarrow> P x))"
+  "inf A B = Abs_filter
+      (\<lambda>P. \<exists>Q R. eventually Q A \<and> eventually R B \<and> (\<forall>x. Q x \<and> R x \<longrightarrow> P x))"
 
 definition
-  Sup_net_def: "Sup A = Abs_net (\<lambda>P. \<forall>net\<in>A. eventually P net)"
+  "Sup S = Abs_filter (\<lambda>P. \<forall>A\<in>S. eventually P A)"
 
 definition
-  Inf_net_def: "Inf A = Sup {x::'a net. \<forall>y\<in>A. x \<le> y}"
+  "Inf S = Sup {A::'a filter. \<forall>B\<in>S. A \<le> B}"
 
 lemma eventually_top [simp]: "eventually P top \<longleftrightarrow> (\<forall>x. P x)"
-unfolding top_net_def
-by (rule eventually_Abs_net, rule is_filter.intro, auto)
+  unfolding top_filter_def
+  by (rule eventually_Abs_filter, rule is_filter.intro, auto)
 
 lemma eventually_bot [simp]: "eventually P bot"
-unfolding bot_net_def
-by (subst eventually_Abs_net, rule is_filter.intro, auto)
+  unfolding bot_filter_def
+  by (subst eventually_Abs_filter, rule is_filter.intro, auto)
 
 lemma eventually_sup:
-  "eventually P (sup net net') \<longleftrightarrow> eventually P net \<and> eventually P net'"
-unfolding sup_net_def
-by (rule eventually_Abs_net, rule is_filter.intro)
-   (auto elim!: eventually_rev_mp)
+  "eventually P (sup A B) \<longleftrightarrow> eventually P A \<and> eventually P B"
+  unfolding sup_filter_def
+  by (rule eventually_Abs_filter, rule is_filter.intro)
+     (auto elim!: eventually_rev_mp)
 
 lemma eventually_inf:
-  "eventually P (inf a b) \<longleftrightarrow>
-   (\<exists>Q R. eventually Q a \<and> eventually R b \<and> (\<forall>x. Q x \<and> R x \<longrightarrow> P x))"
-unfolding inf_net_def
-apply (rule eventually_Abs_net, rule is_filter.intro)
-apply (fast intro: eventually_True)
-apply clarify
-apply (intro exI conjI)
-apply (erule (1) eventually_conj)
-apply (erule (1) eventually_conj)
-apply simp
-apply auto
-done
+  "eventually P (inf A B) \<longleftrightarrow>
+   (\<exists>Q R. eventually Q A \<and> eventually R B \<and> (\<forall>x. Q x \<and> R x \<longrightarrow> P x))"
+  unfolding inf_filter_def
+  apply (rule eventually_Abs_filter, rule is_filter.intro)
+  apply (fast intro: eventually_True)
+  apply clarify
+  apply (intro exI conjI)
+  apply (erule (1) eventually_conj)
+  apply (erule (1) eventually_conj)
+  apply simp
+  apply auto
+  done
 
 lemma eventually_Sup:
-  "eventually P (Sup A) \<longleftrightarrow> (\<forall>net\<in>A. eventually P net)"
-unfolding Sup_net_def
-apply (rule eventually_Abs_net, rule is_filter.intro)
-apply (auto intro: eventually_conj elim!: eventually_rev_mp)
-done
+  "eventually P (Sup S) \<longleftrightarrow> (\<forall>A\<in>S. eventually P A)"
+  unfolding Sup_filter_def
+  apply (rule eventually_Abs_filter, rule is_filter.intro)
+  apply (auto intro: eventually_conj elim!: eventually_rev_mp)
+  done
 
 instance proof
-  fix x y :: "'a net" show "x < y \<longleftrightarrow> x \<le> y \<and> \<not> y \<le> x"
-    by (rule less_net_def)
+  fix A B :: "'a filter" show "A < B \<longleftrightarrow> A \<le> B \<and> \<not> B \<le> A"
+    by (rule less_filter_def)
 next
-  fix x :: "'a net" show "x \<le> x"
-    unfolding le_net_def by simp
+  fix A :: "'a filter" show "A \<le> A"
+    unfolding le_filter_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
+  fix A B C :: "'a filter" assume "A \<le> B" and "B \<le> C" thus "A \<le> C"
+    unfolding le_filter_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
+  fix A B :: "'a filter" assume "A \<le> B" and "B \<le> A" thus "A = B"
+    unfolding le_filter_def filter_eq_iff by fast
 next
-  fix x :: "'a net" show "x \<le> top"
-    unfolding le_net_def eventually_top by (simp add: always_eventually)
+  fix A :: "'a filter" show "A \<le> top"
+    unfolding le_filter_def eventually_top by (simp add: always_eventually)
 next
-  fix x :: "'a net" show "bot \<le> x"
-    unfolding le_net_def by simp
+  fix A :: "'a filter" show "bot \<le> A"
+    unfolding le_filter_def by simp
 next
-  fix x y :: "'a net" show "x \<le> sup x y" and "y \<le> sup x y"
-    unfolding le_net_def eventually_sup by simp_all
+  fix A B :: "'a filter" show "A \<le> sup A B" and "B \<le> sup A B"
+    unfolding le_filter_def eventually_sup by simp_all
 next
-  fix x y z :: "'a net" assume "x \<le> z" and "y \<le> z" thus "sup x y \<le> z"
-    unfolding le_net_def eventually_sup by simp
+  fix A B C :: "'a filter" assume "A \<le> C" and "B \<le> C" thus "sup A B \<le> C"
+    unfolding le_filter_def eventually_sup by simp
 next
-  fix x y :: "'a net" show "inf x y \<le> x" and "inf x y \<le> y"
-    unfolding le_net_def eventually_inf by (auto intro: eventually_True)
+  fix A B :: "'a filter" show "inf A B \<le> A" and "inf A B \<le> B"
+    unfolding le_filter_def eventually_inf by (auto intro: eventually_True)
 next
-  fix x y z :: "'a net" assume "x \<le> y" and "x \<le> z" thus "x \<le> inf y z"
-    unfolding le_net_def eventually_inf
+  fix A B C :: "'a filter" assume "A \<le> B" and "A \<le> C" thus "A \<le> inf B C"
+    unfolding le_filter_def eventually_inf
     by (auto elim!: eventually_mono intro: eventually_conj)
 next
-  fix x :: "'a net" and A assume "x \<in> A" thus "x \<le> Sup A"
-    unfolding le_net_def eventually_Sup by simp
+  fix A :: "'a filter" and S assume "A \<in> S" thus "A \<le> Sup S"
+    unfolding le_filter_def eventually_Sup by simp
 next
-  fix A and y :: "'a net" assume "\<And>x. x \<in> A \<Longrightarrow> x \<le> y" thus "Sup A \<le> y"
-    unfolding le_net_def eventually_Sup by simp
+  fix S and B :: "'a filter" assume "\<And>A. A \<in> S \<Longrightarrow> A \<le> B" thus "Sup S \<le> B"
+    unfolding le_filter_def eventually_Sup by simp
 next
-  fix z :: "'a net" and A assume "z \<in> A" thus "Inf A \<le> z"
-    unfolding le_net_def Inf_net_def eventually_Sup Ball_def by simp
+  fix C :: "'a filter" and S assume "C \<in> S" thus "Inf S \<le> C"
+    unfolding le_filter_def Inf_filter_def eventually_Sup Ball_def by simp
 next
-  fix A and x :: "'a net" assume "\<And>y. y \<in> A \<Longrightarrow> x \<le> y" thus "x \<le> Inf A"
-    unfolding le_net_def Inf_net_def eventually_Sup Ball_def by simp
+  fix S and A :: "'a filter" assume "\<And>B. B \<in> S \<Longrightarrow> A \<le> B" thus "A \<le> Inf S"
+    unfolding le_filter_def Inf_filter_def eventually_Sup Ball_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 filter_leD:
+  "A \<le> B \<Longrightarrow> eventually P B \<Longrightarrow> eventually P A"
+  unfolding le_filter_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 filter_leI:
+  "(\<And>P. eventually P B \<Longrightarrow> eventually P A) \<Longrightarrow> A \<le> B"
+  unfolding le_filter_def by simp
 
 lemma eventually_False:
-  "eventually (\<lambda>x. False) net \<longleftrightarrow> net = bot"
-unfolding expand_net_eq by (auto elim: eventually_rev_mp)
+  "eventually (\<lambda>x. False) A \<longleftrightarrow> A = bot"
+  unfolding filter_eq_iff by (auto elim: eventually_rev_mp)
 
-subsection {* Map function for nets *}
+subsection {* Map function for filters *}
 
-definition netmap :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a net \<Rightarrow> 'b net" where
-  "netmap f net = Abs_net (\<lambda>P. eventually (\<lambda>x. P (f x)) net)"
+definition filtermap :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a filter \<Rightarrow> 'b filter"
+  where "filtermap f A = Abs_filter (\<lambda>P. eventually (\<lambda>x. P (f x)) A)"
 
-lemma eventually_netmap:
-  "eventually P (netmap f net) = eventually (\<lambda>x. P (f x)) net"
-unfolding netmap_def
-apply (rule eventually_Abs_net)
-apply (rule is_filter.intro)
-apply (auto elim!: eventually_rev_mp)
-done
+lemma eventually_filtermap:
+  "eventually P (filtermap f A) = eventually (\<lambda>x. P (f x)) A"
+  unfolding filtermap_def
+  apply (rule eventually_Abs_filter)
+  apply (rule is_filter.intro)
+  apply (auto elim!: eventually_rev_mp)
+  done
 
-lemma netmap_ident: "netmap (\<lambda>x. x) net = net"
-by (simp add: expand_net_eq eventually_netmap)
-
-lemma netmap_netmap: "netmap f (netmap g net) = netmap (\<lambda>x. f (g x)) net"
-by (simp add: expand_net_eq eventually_netmap)
+lemma filtermap_ident: "filtermap (\<lambda>x. x) A = A"
+  by (simp add: filter_eq_iff eventually_filtermap)
 
-lemma netmap_mono: "net \<le> net' \<Longrightarrow> netmap f net \<le> netmap f net'"
-unfolding le_net_def eventually_netmap by simp
+lemma filtermap_filtermap:
+  "filtermap f (filtermap g A) = filtermap (\<lambda>x. f (g x)) A"
+  by (simp add: filter_eq_iff eventually_filtermap)
 
-lemma netmap_bot [simp]: "netmap f bot = bot"
-by (simp add: expand_net_eq eventually_netmap)
+lemma filtermap_mono: "A \<le> B \<Longrightarrow> filtermap f A \<le> filtermap f B"
+  unfolding le_filter_def eventually_filtermap by simp
+
+lemma filtermap_bot [simp]: "filtermap f bot = bot"
+  by (simp add: filter_eq_iff eventually_filtermap)
 
 
 subsection {* Sequentially *}
 
-definition sequentially :: "nat net" where
-  "sequentially = Abs_net (\<lambda>P. \<exists>k. \<forall>n\<ge>k. P n)"
+definition sequentially :: "nat filter"
+  where "sequentially = Abs_filter (\<lambda>P. \<exists>k. \<forall>n\<ge>k. P n)"
 
 lemma eventually_sequentially:
   "eventually P sequentially \<longleftrightarrow> (\<exists>N. \<forall>n\<ge>N. P n)"
 unfolding sequentially_def
-proof (rule eventually_Abs_net, rule is_filter.intro)
+proof (rule eventually_Abs_filter, rule is_filter.intro)
   fix P Q :: "nat \<Rightarrow> bool"
   assume "\<exists>i. \<forall>n\<ge>i. P n" and "\<exists>j. \<forall>n\<ge>j. Q n"
   then obtain i j where "\<forall>n\<ge>i. P n" and "\<forall>n\<ge>j. Q n" by auto
@@ -273,49 +272,48 @@
 qed auto
 
 lemma sequentially_bot [simp]: "sequentially \<noteq> bot"
-unfolding expand_net_eq eventually_sequentially by auto
+  unfolding filter_eq_iff eventually_sequentially by auto
 
 lemma eventually_False_sequentially [simp]:
   "\<not> eventually (\<lambda>n. False) sequentially"
-by (simp add: eventually_False)
+  by (simp add: eventually_False)
 
 lemma le_sequentially:
-  "net \<le> sequentially \<longleftrightarrow> (\<forall>N. eventually (\<lambda>n. N \<le> n) net)"
-unfolding le_net_def eventually_sequentially
-by (safe, fast, drule_tac x=N in spec, auto elim: eventually_rev_mp)
+  "A \<le> sequentially \<longleftrightarrow> (\<forall>N. eventually (\<lambda>n. N \<le> n) A)"
+  unfolding le_filter_def eventually_sequentially
+  by (safe, fast, drule_tac x=N in spec, auto elim: eventually_rev_mp)
 
 
-definition
-  trivial_limit :: "'a net \<Rightarrow> bool" where
-  "trivial_limit net \<longleftrightarrow> eventually (\<lambda>x. False) net"
+definition trivial_limit :: "'a filter \<Rightarrow> bool"
+  where "trivial_limit A \<longleftrightarrow> eventually (\<lambda>x. False) A"
 
-lemma trivial_limit_sequentially[intro]: "\<not> trivial_limit sequentially"
+lemma trivial_limit_sequentially [intro]: "\<not> trivial_limit sequentially"
   by (auto simp add: trivial_limit_def eventually_sequentially)
 
-subsection {* Standard Nets *}
+subsection {* Standard filters *}
 
-definition within :: "'a net \<Rightarrow> 'a set \<Rightarrow> 'a net" (infixr "within" 70) where
-  "net within S = Abs_net (\<lambda>P. eventually (\<lambda>x. x \<in> S \<longrightarrow> P x) net)"
+definition within :: "'a filter \<Rightarrow> 'a set \<Rightarrow> 'a filter" (infixr "within" 70)
+  where "A within S = Abs_filter (\<lambda>P. eventually (\<lambda>x. x \<in> S \<longrightarrow> P x) A)"
 
-definition nhds :: "'a::topological_space \<Rightarrow> 'a net" where
-  "nhds a = Abs_net (\<lambda>P. \<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x))"
+definition nhds :: "'a::topological_space \<Rightarrow> 'a filter"
+  where "nhds a = Abs_filter (\<lambda>P. \<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x))"
 
-definition at :: "'a::topological_space \<Rightarrow> 'a net" where
-  "at a = nhds a within - {a}"
+definition at :: "'a::topological_space \<Rightarrow> 'a filter"
+  where "at a = nhds a within - {a}"
 
 lemma eventually_within:
-  "eventually P (net within S) = eventually (\<lambda>x. x \<in> S \<longrightarrow> P x) net"
-unfolding within_def
-by (rule eventually_Abs_net, rule is_filter.intro)
-   (auto elim!: eventually_rev_mp)
+  "eventually P (A within S) = eventually (\<lambda>x. x \<in> S \<longrightarrow> P x) A"
+  unfolding within_def
+  by (rule eventually_Abs_filter, 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 within_UNIV: "A within UNIV = A"
+  unfolding filter_eq_iff eventually_within by simp
 
 lemma eventually_nhds:
   "eventually P (nhds a) \<longleftrightarrow> (\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x))"
 unfolding nhds_def
-proof (rule eventually_Abs_net, rule is_filter.intro)
+proof (rule eventually_Abs_filter, rule is_filter.intro)
   have "open UNIV \<and> a \<in> UNIV \<and> (\<forall>x\<in>UNIV. True)" by simp
   thus "\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. True)" by - rule
 next
@@ -354,52 +352,52 @@
 
 subsection {* Boundedness *}
 
-definition Bfun :: "('a \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a net \<Rightarrow> bool" where
-  "Bfun f net = (\<exists>K>0. eventually (\<lambda>x. norm (f x) \<le> K) net)"
+definition Bfun :: "('a \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a filter \<Rightarrow> bool"
+  where "Bfun f A = (\<exists>K>0. eventually (\<lambda>x. norm (f x) \<le> K) A)"
 
 lemma BfunI:
-  assumes K: "eventually (\<lambda>x. norm (f x) \<le> K) net" shows "Bfun f net"
+  assumes K: "eventually (\<lambda>x. norm (f x) \<le> K) A" shows "Bfun f A"
 unfolding Bfun_def
 proof (intro exI conjI allI)
   show "0 < max K 1" by simp
 next
-  show "eventually (\<lambda>x. norm (f x) \<le> max K 1) net"
+  show "eventually (\<lambda>x. norm (f x) \<le> max K 1) A"
     using K by (rule eventually_elim1, simp)
 qed
 
 lemma BfunE:
-  assumes "Bfun f net"
-  obtains B where "0 < B" and "eventually (\<lambda>x. norm (f x) \<le> B) net"
+  assumes "Bfun f A"
+  obtains B where "0 < B" and "eventually (\<lambda>x. norm (f x) \<le> B) A"
 using assms unfolding Bfun_def by fast
 
 
 subsection {* Convergence to Zero *}
 
-definition Zfun :: "('a \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a net \<Rightarrow> bool" where
-  "Zfun f net = (\<forall>r>0. eventually (\<lambda>x. norm (f x) < r) net)"
+definition Zfun :: "('a \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a filter \<Rightarrow> bool"
+  where "Zfun f A = (\<forall>r>0. eventually (\<lambda>x. norm (f x) < r) A)"
 
 lemma ZfunI:
-  "(\<And>r. 0 < r \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) net) \<Longrightarrow> Zfun f net"
-unfolding Zfun_def by simp
+  "(\<And>r. 0 < r \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) A) \<Longrightarrow> Zfun f A"
+  unfolding Zfun_def by simp
 
 lemma ZfunD:
-  "\<lbrakk>Zfun f net; 0 < r\<rbrakk> \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) net"
-unfolding Zfun_def by simp
+  "\<lbrakk>Zfun f A; 0 < r\<rbrakk> \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) A"
+  unfolding Zfun_def by simp
 
 lemma Zfun_ssubst:
-  "eventually (\<lambda>x. f x = g x) net \<Longrightarrow> Zfun g net \<Longrightarrow> Zfun f net"
-unfolding Zfun_def by (auto elim!: eventually_rev_mp)
+  "eventually (\<lambda>x. f x = g x) A \<Longrightarrow> Zfun g A \<Longrightarrow> Zfun f A"
+  unfolding Zfun_def by (auto elim!: eventually_rev_mp)
 
-lemma Zfun_zero: "Zfun (\<lambda>x. 0) net"
-unfolding Zfun_def by simp
+lemma Zfun_zero: "Zfun (\<lambda>x. 0) A"
+  unfolding Zfun_def by simp
 
-lemma Zfun_norm_iff: "Zfun (\<lambda>x. norm (f x)) net = Zfun (\<lambda>x. f x) net"
-unfolding Zfun_def by simp
+lemma Zfun_norm_iff: "Zfun (\<lambda>x. norm (f x)) A = Zfun (\<lambda>x. f x) A"
+  unfolding Zfun_def by simp
 
 lemma Zfun_imp_Zfun:
-  assumes f: "Zfun f net"
-  assumes g: "eventually (\<lambda>x. norm (g x) \<le> norm (f x) * K) net"
-  shows "Zfun (\<lambda>x. g x) net"
+  assumes f: "Zfun f A"
+  assumes g: "eventually (\<lambda>x. norm (g x) \<le> norm (f x) * K) A"
+  shows "Zfun (\<lambda>x. g x) A"
 proof (cases)
   assume K: "0 < K"
   show ?thesis
@@ -407,9 +405,9 @@
     fix r::real assume "0 < r"
     hence "0 < r / K"
       using K by (rule divide_pos_pos)
-    then have "eventually (\<lambda>x. norm (f x) < r / K) net"
+    then have "eventually (\<lambda>x. norm (f x) < r / K) A"
       using ZfunD [OF f] by fast
-    with g show "eventually (\<lambda>x. norm (g x) < r) net"
+    with g show "eventually (\<lambda>x. norm (g x) < r) A"
     proof (rule eventually_elim2)
       fix x
       assume *: "norm (g x) \<le> norm (f x) * K"
@@ -427,7 +425,7 @@
   proof (rule ZfunI)
     fix r :: real
     assume "0 < r"
-    from g show "eventually (\<lambda>x. norm (g x) < r) net"
+    from g show "eventually (\<lambda>x. norm (g x) < r) A"
     proof (rule eventually_elim1)
       fix x
       assume "norm (g x) \<le> norm (f x) * K"
@@ -439,22 +437,22 @@
   qed
 qed
 
-lemma Zfun_le: "\<lbrakk>Zfun g net; \<forall>x. norm (f x) \<le> norm (g x)\<rbrakk> \<Longrightarrow> Zfun f net"
-by (erule_tac K="1" in Zfun_imp_Zfun, simp)
+lemma Zfun_le: "\<lbrakk>Zfun g A; \<forall>x. norm (f x) \<le> norm (g x)\<rbrakk> \<Longrightarrow> Zfun f A"
+  by (erule_tac K="1" in Zfun_imp_Zfun, simp)
 
 lemma Zfun_add:
-  assumes f: "Zfun f net" and g: "Zfun g net"
-  shows "Zfun (\<lambda>x. f x + g x) net"
+  assumes f: "Zfun f A" and g: "Zfun g A"
+  shows "Zfun (\<lambda>x. f x + g x) A"
 proof (rule ZfunI)
   fix r::real assume "0 < r"
   hence r: "0 < r / 2" by simp
-  have "eventually (\<lambda>x. norm (f x) < r/2) net"
+  have "eventually (\<lambda>x. norm (f x) < r/2) A"
     using f r by (rule ZfunD)
   moreover
-  have "eventually (\<lambda>x. norm (g x) < r/2) net"
+  have "eventually (\<lambda>x. norm (g x) < r/2) A"
     using g r by (rule ZfunD)
   ultimately
-  show "eventually (\<lambda>x. norm (f x + g x) < r) net"
+  show "eventually (\<lambda>x. norm (f x + g x) < r) A"
   proof (rule eventually_elim2)
     fix x
     assume *: "norm (f x) < r/2" "norm (g x) < r/2"
@@ -467,28 +465,28 @@
   qed
 qed
 
-lemma Zfun_minus: "Zfun f net \<Longrightarrow> Zfun (\<lambda>x. - f x) net"
-unfolding Zfun_def by simp
+lemma Zfun_minus: "Zfun f A \<Longrightarrow> Zfun (\<lambda>x. - f x) A"
+  unfolding Zfun_def by simp
 
-lemma Zfun_diff: "\<lbrakk>Zfun f net; Zfun g net\<rbrakk> \<Longrightarrow> Zfun (\<lambda>x. f x - g x) net"
-by (simp only: diff_minus Zfun_add Zfun_minus)
+lemma Zfun_diff: "\<lbrakk>Zfun f A; Zfun g A\<rbrakk> \<Longrightarrow> Zfun (\<lambda>x. f x - g x) A"
+  by (simp only: diff_minus Zfun_add Zfun_minus)
 
 lemma (in bounded_linear) Zfun:
-  assumes g: "Zfun g net"
-  shows "Zfun (\<lambda>x. f (g x)) net"
+  assumes g: "Zfun g A"
+  shows "Zfun (\<lambda>x. f (g x)) A"
 proof -
   obtain K where "\<And>x. norm (f x) \<le> norm x * K"
     using bounded by fast
-  then have "eventually (\<lambda>x. norm (f (g x)) \<le> norm (g x) * K) net"
+  then have "eventually (\<lambda>x. norm (f (g x)) \<le> norm (g x) * K) A"
     by simp
   with g show ?thesis
     by (rule Zfun_imp_Zfun)
 qed
 
 lemma (in bounded_bilinear) Zfun:
-  assumes f: "Zfun f net"
-  assumes g: "Zfun g net"
-  shows "Zfun (\<lambda>x. f x ** g x) net"
+  assumes f: "Zfun f A"
+  assumes g: "Zfun g A"
+  shows "Zfun (\<lambda>x. f x ** g x) A"
 proof (rule ZfunI)
   fix r::real assume r: "0 < r"
   obtain K where K: "0 < K"
@@ -496,13 +494,13 @@
     using pos_bounded by fast
   from K have K': "0 < inverse K"
     by (rule positive_imp_inverse_positive)
-  have "eventually (\<lambda>x. norm (f x) < r) net"
+  have "eventually (\<lambda>x. norm (f x) < r) A"
     using f r by (rule ZfunD)
   moreover
-  have "eventually (\<lambda>x. norm (g x) < inverse K) net"
+  have "eventually (\<lambda>x. norm (g x) < inverse K) A"
     using g K' by (rule ZfunD)
   ultimately
-  show "eventually (\<lambda>x. norm (f x ** g x) < r) net"
+  show "eventually (\<lambda>x. norm (f x ** g x) < r) A"
   proof (rule eventually_elim2)
     fix x
     assume *: "norm (f x) < r" "norm (g x) < inverse K"
@@ -517,12 +515,12 @@
 qed
 
 lemma (in bounded_bilinear) Zfun_left:
-  "Zfun f net \<Longrightarrow> Zfun (\<lambda>x. f x ** a) net"
-by (rule bounded_linear_left [THEN bounded_linear.Zfun])
+  "Zfun f A \<Longrightarrow> Zfun (\<lambda>x. f x ** a) A"
+  by (rule bounded_linear_left [THEN bounded_linear.Zfun])
 
 lemma (in bounded_bilinear) Zfun_right:
-  "Zfun f net \<Longrightarrow> Zfun (\<lambda>x. a ** f x) net"
-by (rule bounded_linear_right [THEN bounded_linear.Zfun])
+  "Zfun f A \<Longrightarrow> Zfun (\<lambda>x. a ** f x) A"
+  by (rule bounded_linear_right [THEN bounded_linear.Zfun])
 
 lemmas Zfun_mult = mult.Zfun
 lemmas Zfun_mult_right = mult.Zfun_right
@@ -531,9 +529,9 @@
 
 subsection {* Limits *}
 
-definition tendsto :: "('a \<Rightarrow> 'b::topological_space) \<Rightarrow> 'b \<Rightarrow> 'a net \<Rightarrow> bool"
+definition tendsto :: "('a \<Rightarrow> 'b::topological_space) \<Rightarrow> 'b \<Rightarrow> 'a filter \<Rightarrow> bool"
     (infixr "--->" 55) where
-  "(f ---> l) net \<longleftrightarrow> (\<forall>S. open S \<longrightarrow> l \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) net)"
+  "(f ---> l) A \<longleftrightarrow> (\<forall>S. open S \<longrightarrow> l \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) A)"
 
 ML {*
 structure Tendsto_Intros = Named_Thms
@@ -545,74 +543,74 @@
 
 setup Tendsto_Intros.setup
 
-lemma tendsto_mono: "net \<le> net' \<Longrightarrow> (f ---> l) net' \<Longrightarrow> (f ---> l) net"
-unfolding tendsto_def le_net_def by fast
+lemma tendsto_mono: "A \<le> A' \<Longrightarrow> (f ---> l) A' \<Longrightarrow> (f ---> l) A"
+  unfolding tendsto_def le_filter_def by fast
 
 lemma topological_tendstoI:
-  "(\<And>S. open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>x. f x \<in> S) net)
-    \<Longrightarrow> (f ---> l) net"
+  "(\<And>S. open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>x. f x \<in> S) A)
+    \<Longrightarrow> (f ---> l) A"
   unfolding tendsto_def by auto
 
 lemma topological_tendstoD:
-  "(f ---> l) net \<Longrightarrow> open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>x. f x \<in> S) net"
+  "(f ---> l) A \<Longrightarrow> open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>x. f x \<in> S) A"
   unfolding tendsto_def by auto
 
 lemma tendstoI:
-  assumes "\<And>e. 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) net"
-  shows "(f ---> l) net"
-apply (rule topological_tendstoI)
-apply (simp add: open_dist)
-apply (drule (1) bspec, clarify)
-apply (drule assms)
-apply (erule eventually_elim1, simp)
-done
+  assumes "\<And>e. 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) A"
+  shows "(f ---> l) A"
+  apply (rule topological_tendstoI)
+  apply (simp add: open_dist)
+  apply (drule (1) bspec, clarify)
+  apply (drule assms)
+  apply (erule eventually_elim1, simp)
+  done
 
 lemma tendstoD:
-  "(f ---> l) net \<Longrightarrow> 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) net"
-apply (drule_tac S="{x. dist x l < e}" in topological_tendstoD)
-apply (clarsimp simp add: open_dist)
-apply (rule_tac x="e - dist x l" in exI, clarsimp)
-apply (simp only: less_diff_eq)
-apply (erule le_less_trans [OF dist_triangle])
-apply simp
-apply simp
-done
+  "(f ---> l) A \<Longrightarrow> 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) A"
+  apply (drule_tac S="{x. dist x l < e}" in topological_tendstoD)
+  apply (clarsimp simp add: open_dist)
+  apply (rule_tac x="e - dist x l" in exI, clarsimp)
+  apply (simp only: less_diff_eq)
+  apply (erule le_less_trans [OF dist_triangle])
+  apply simp
+  apply simp
+  done
 
 lemma tendsto_iff:
-  "(f ---> l) net \<longleftrightarrow> (\<forall>e>0. eventually (\<lambda>x. dist (f x) l < e) net)"
-using tendstoI tendstoD by fast
+  "(f ---> l) A \<longleftrightarrow> (\<forall>e>0. eventually (\<lambda>x. dist (f x) l < e) A)"
+  using tendstoI tendstoD by fast
 
-lemma tendsto_Zfun_iff: "(f ---> a) net = Zfun (\<lambda>x. f x - a) net"
-by (simp only: tendsto_iff Zfun_def dist_norm)
+lemma tendsto_Zfun_iff: "(f ---> a) A = Zfun (\<lambda>x. f x - a) A"
+  by (simp only: tendsto_iff Zfun_def dist_norm)
 
 lemma tendsto_ident_at [tendsto_intros]: "((\<lambda>x. x) ---> a) (at a)"
-unfolding tendsto_def eventually_at_topological by auto
+  unfolding tendsto_def eventually_at_topological by auto
 
 lemma tendsto_ident_at_within [tendsto_intros]:
   "((\<lambda>x. x) ---> a) (at a within S)"
-unfolding tendsto_def eventually_within eventually_at_topological by auto
+  unfolding tendsto_def eventually_within eventually_at_topological by auto
 
-lemma tendsto_const [tendsto_intros]: "((\<lambda>x. k) ---> k) net"
-by (simp add: tendsto_def)
+lemma tendsto_const [tendsto_intros]: "((\<lambda>x. k) ---> k) A"
+  by (simp add: tendsto_def)
 
 lemma tendsto_const_iff:
   fixes k l :: "'a::metric_space"
-  assumes "net \<noteq> bot" shows "((\<lambda>n. k) ---> l) net \<longleftrightarrow> k = l"
-apply (safe intro!: tendsto_const)
-apply (rule ccontr)
-apply (drule_tac e="dist k l" in tendstoD)
-apply (simp add: zero_less_dist_iff)
-apply (simp add: eventually_False assms)
-done
+  assumes "A \<noteq> bot" shows "((\<lambda>n. k) ---> l) A \<longleftrightarrow> k = l"
+  apply (safe intro!: tendsto_const)
+  apply (rule ccontr)
+  apply (drule_tac e="dist k l" in tendstoD)
+  apply (simp add: zero_less_dist_iff)
+  apply (simp add: eventually_False assms)
+  done
 
 lemma tendsto_dist [tendsto_intros]:
-  assumes f: "(f ---> l) net" and g: "(g ---> m) net"
-  shows "((\<lambda>x. dist (f x) (g x)) ---> dist l m) net"
+  assumes f: "(f ---> l) A" and g: "(g ---> m) A"
+  shows "((\<lambda>x. dist (f x) (g x)) ---> dist l m) A"
 proof (rule tendstoI)
   fix e :: real assume "0 < e"
   hence e2: "0 < e/2" by simp
   from tendstoD [OF f e2] tendstoD [OF g e2]
-  show "eventually (\<lambda>x. dist (dist (f x) (g x)) (dist l m) < e) net"
+  show "eventually (\<lambda>x. dist (dist (f x) (g x)) (dist l m) < e) A"
   proof (rule eventually_elim2)
     fix x assume "dist (f x) l < e/2" "dist (g x) m < e/2"
     then show "dist (dist (f x) (g x)) (dist l m) < e"
@@ -626,48 +624,48 @@
 qed
 
 lemma norm_conv_dist: "norm x = dist x 0"
-unfolding dist_norm by simp
+  unfolding dist_norm by simp
 
 lemma tendsto_norm [tendsto_intros]:
-  "(f ---> a) net \<Longrightarrow> ((\<lambda>x. norm (f x)) ---> norm a) net"
-unfolding norm_conv_dist by (intro tendsto_intros)
+  "(f ---> a) A \<Longrightarrow> ((\<lambda>x. norm (f x)) ---> norm a) A"
+  unfolding norm_conv_dist by (intro tendsto_intros)
 
 lemma tendsto_norm_zero:
-  "(f ---> 0) net \<Longrightarrow> ((\<lambda>x. norm (f x)) ---> 0) net"
-by (drule tendsto_norm, simp)
+  "(f ---> 0) A \<Longrightarrow> ((\<lambda>x. norm (f x)) ---> 0) A"
+  by (drule tendsto_norm, simp)
 
 lemma tendsto_norm_zero_cancel:
-  "((\<lambda>x. norm (f x)) ---> 0) net \<Longrightarrow> (f ---> 0) net"
-unfolding tendsto_iff dist_norm by simp
+  "((\<lambda>x. norm (f x)) ---> 0) A \<Longrightarrow> (f ---> 0) A"
+  unfolding tendsto_iff dist_norm by simp
 
 lemma tendsto_norm_zero_iff:
-  "((\<lambda>x. norm (f x)) ---> 0) net \<longleftrightarrow> (f ---> 0) net"
-unfolding tendsto_iff dist_norm by simp
+  "((\<lambda>x. norm (f x)) ---> 0) A \<longleftrightarrow> (f ---> 0) A"
+  unfolding tendsto_iff dist_norm by simp
 
 lemma tendsto_add [tendsto_intros]:
   fixes a b :: "'a::real_normed_vector"
-  shows "\<lbrakk>(f ---> a) net; (g ---> b) net\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x + g x) ---> a + b) net"
-by (simp only: tendsto_Zfun_iff add_diff_add Zfun_add)
+  shows "\<lbrakk>(f ---> a) A; (g ---> b) A\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x + g x) ---> a + b) A"
+  by (simp only: tendsto_Zfun_iff add_diff_add Zfun_add)
 
 lemma tendsto_minus [tendsto_intros]:
   fixes a :: "'a::real_normed_vector"
-  shows "(f ---> a) net \<Longrightarrow> ((\<lambda>x. - f x) ---> - a) net"
-by (simp only: tendsto_Zfun_iff minus_diff_minus Zfun_minus)
+  shows "(f ---> a) A \<Longrightarrow> ((\<lambda>x. - f x) ---> - a) A"
+  by (simp only: tendsto_Zfun_iff minus_diff_minus Zfun_minus)
 
 lemma tendsto_minus_cancel:
   fixes a :: "'a::real_normed_vector"
-  shows "((\<lambda>x. - f x) ---> - a) net \<Longrightarrow> (f ---> a) net"
-by (drule tendsto_minus, simp)
+  shows "((\<lambda>x. - f x) ---> - a) A \<Longrightarrow> (f ---> a) A"
+  by (drule tendsto_minus, simp)
 
 lemma tendsto_diff [tendsto_intros]:
   fixes a b :: "'a::real_normed_vector"
-  shows "\<lbrakk>(f ---> a) net; (g ---> b) net\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x - g x) ---> a - b) net"
-by (simp add: diff_minus tendsto_add tendsto_minus)
+  shows "\<lbrakk>(f ---> a) A; (g ---> b) A\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x - g x) ---> a - b) A"
+  by (simp add: diff_minus tendsto_add tendsto_minus)
 
 lemma tendsto_setsum [tendsto_intros]:
   fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c::real_normed_vector"
-  assumes "\<And>i. i \<in> S \<Longrightarrow> (f i ---> a i) net"
-  shows "((\<lambda>x. \<Sum>i\<in>S. f i x) ---> (\<Sum>i\<in>S. a i)) net"
+  assumes "\<And>i. i \<in> S \<Longrightarrow> (f i ---> a i) A"
+  shows "((\<lambda>x. \<Sum>i\<in>S. f i x) ---> (\<Sum>i\<in>S. a i)) A"
 proof (cases "finite S")
   assume "finite S" thus ?thesis using assms
   proof (induct set: finite)
@@ -683,29 +681,29 @@
 qed
 
 lemma (in bounded_linear) tendsto [tendsto_intros]:
-  "(g ---> a) net \<Longrightarrow> ((\<lambda>x. f (g x)) ---> f a) net"
-by (simp only: tendsto_Zfun_iff diff [symmetric] Zfun)
+  "(g ---> a) A \<Longrightarrow> ((\<lambda>x. f (g x)) ---> f a) A"
+  by (simp only: tendsto_Zfun_iff diff [symmetric] Zfun)
 
 lemma (in bounded_bilinear) tendsto [tendsto_intros]:
-  "\<lbrakk>(f ---> a) net; (g ---> b) net\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x ** g x) ---> a ** b) net"
-by (simp only: tendsto_Zfun_iff prod_diff_prod
-               Zfun_add Zfun Zfun_left Zfun_right)
+  "\<lbrakk>(f ---> a) A; (g ---> b) A\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x ** g x) ---> a ** b) A"
+  by (simp only: tendsto_Zfun_iff prod_diff_prod
+                 Zfun_add Zfun Zfun_left Zfun_right)
 
 
 subsection {* Continuity of Inverse *}
 
 lemma (in bounded_bilinear) Zfun_prod_Bfun:
-  assumes f: "Zfun f net"
-  assumes g: "Bfun g net"
-  shows "Zfun (\<lambda>x. f x ** g x) net"
+  assumes f: "Zfun f A"
+  assumes g: "Bfun g A"
+  shows "Zfun (\<lambda>x. f x ** g x) A"
 proof -
   obtain K where K: "0 \<le> K"
     and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
     using nonneg_bounded by fast
   obtain B where B: "0 < B"
-    and norm_g: "eventually (\<lambda>x. norm (g x) \<le> B) net"
+    and norm_g: "eventually (\<lambda>x. norm (g x) \<le> B) A"
     using g by (rule BfunE)
-  have "eventually (\<lambda>x. norm (f x ** g x) \<le> norm (f x) * (B * K)) net"
+  have "eventually (\<lambda>x. norm (f x ** g x) \<le> norm (f x) * (B * K)) A"
   using norm_g proof (rule eventually_elim1)
     fix x
     assume *: "norm (g x) \<le> B"
@@ -724,39 +722,39 @@
 
 lemma (in bounded_bilinear) flip:
   "bounded_bilinear (\<lambda>x y. y ** x)"
-apply default
-apply (rule add_right)
-apply (rule add_left)
-apply (rule scaleR_right)
-apply (rule scaleR_left)
-apply (subst mult_commute)
-using bounded by fast
+  apply default
+  apply (rule add_right)
+  apply (rule add_left)
+  apply (rule scaleR_right)
+  apply (rule scaleR_left)
+  apply (subst mult_commute)
+  using bounded by fast
 
 lemma (in bounded_bilinear) Bfun_prod_Zfun:
-  assumes f: "Bfun f net"
-  assumes g: "Zfun g net"
-  shows "Zfun (\<lambda>x. f x ** g x) net"
-using flip g f by (rule bounded_bilinear.Zfun_prod_Bfun)
+  assumes f: "Bfun f A"
+  assumes g: "Zfun g A"
+  shows "Zfun (\<lambda>x. f x ** g x) A"
+  using flip g f by (rule bounded_bilinear.Zfun_prod_Bfun)
 
 lemma Bfun_inverse_lemma:
   fixes x :: "'a::real_normed_div_algebra"
   shows "\<lbrakk>r \<le> norm x; 0 < r\<rbrakk> \<Longrightarrow> norm (inverse x) \<le> inverse r"
-apply (subst nonzero_norm_inverse, clarsimp)
-apply (erule (1) le_imp_inverse_le)
-done
+  apply (subst nonzero_norm_inverse, clarsimp)
+  apply (erule (1) le_imp_inverse_le)
+  done
 
 lemma Bfun_inverse:
   fixes a :: "'a::real_normed_div_algebra"
-  assumes f: "(f ---> a) net"
+  assumes f: "(f ---> a) A"
   assumes a: "a \<noteq> 0"
-  shows "Bfun (\<lambda>x. inverse (f x)) net"
+  shows "Bfun (\<lambda>x. inverse (f x)) A"
 proof -
   from a have "0 < norm a" by simp
   hence "\<exists>r>0. r < norm a" by (rule dense)
   then obtain r where r1: "0 < r" and r2: "r < norm a" by fast
-  have "eventually (\<lambda>x. dist (f x) a < r) net"
+  have "eventually (\<lambda>x. dist (f x) a < r) A"
     using tendstoD [OF f r1] by fast
-  hence "eventually (\<lambda>x. norm (inverse (f x)) \<le> inverse (norm a - r)) net"
+  hence "eventually (\<lambda>x. norm (inverse (f x)) \<le> inverse (norm a - r)) A"
   proof (rule eventually_elim1)
     fix x
     assume "dist (f x) a < r"
@@ -783,29 +781,29 @@
 
 lemma tendsto_inverse_lemma:
   fixes a :: "'a::real_normed_div_algebra"
-  shows "\<lbrakk>(f ---> a) net; a \<noteq> 0; eventually (\<lambda>x. f x \<noteq> 0) net\<rbrakk>
-         \<Longrightarrow> ((\<lambda>x. inverse (f x)) ---> inverse a) net"
-apply (subst tendsto_Zfun_iff)
-apply (rule Zfun_ssubst)
-apply (erule eventually_elim1)
-apply (erule (1) inverse_diff_inverse)
-apply (rule Zfun_minus)
-apply (rule Zfun_mult_left)
-apply (rule mult.Bfun_prod_Zfun)
-apply (erule (1) Bfun_inverse)
-apply (simp add: tendsto_Zfun_iff)
-done
+  shows "\<lbrakk>(f ---> a) A; a \<noteq> 0; eventually (\<lambda>x. f x \<noteq> 0) A\<rbrakk>
+         \<Longrightarrow> ((\<lambda>x. inverse (f x)) ---> inverse a) A"
+  apply (subst tendsto_Zfun_iff)
+  apply (rule Zfun_ssubst)
+  apply (erule eventually_elim1)
+  apply (erule (1) inverse_diff_inverse)
+  apply (rule Zfun_minus)
+  apply (rule Zfun_mult_left)
+  apply (rule mult.Bfun_prod_Zfun)
+  apply (erule (1) Bfun_inverse)
+  apply (simp add: tendsto_Zfun_iff)
+  done
 
 lemma tendsto_inverse [tendsto_intros]:
   fixes a :: "'a::real_normed_div_algebra"
-  assumes f: "(f ---> a) net"
+  assumes f: "(f ---> a) A"
   assumes a: "a \<noteq> 0"
-  shows "((\<lambda>x. inverse (f x)) ---> inverse a) net"
+  shows "((\<lambda>x. inverse (f x)) ---> inverse a) A"
 proof -
   from a have "0 < norm a" by simp
-  with f have "eventually (\<lambda>x. dist (f x) a < norm a) net"
+  with f have "eventually (\<lambda>x. dist (f x) a < norm a) A"
     by (rule tendstoD)
-  then have "eventually (\<lambda>x. f x \<noteq> 0) net"
+  then have "eventually (\<lambda>x. f x \<noteq> 0) A"
     unfolding dist_norm by (auto elim!: eventually_elim1)
   with f a show ?thesis
     by (rule tendsto_inverse_lemma)
@@ -813,32 +811,32 @@
 
 lemma tendsto_divide [tendsto_intros]:
   fixes a b :: "'a::real_normed_field"
-  shows "\<lbrakk>(f ---> a) net; (g ---> b) net; b \<noteq> 0\<rbrakk>
-    \<Longrightarrow> ((\<lambda>x. f x / g x) ---> a / b) net"
-by (simp add: mult.tendsto tendsto_inverse divide_inverse)
+  shows "\<lbrakk>(f ---> a) A; (g ---> b) A; b \<noteq> 0\<rbrakk>
+    \<Longrightarrow> ((\<lambda>x. f x / g x) ---> a / b) A"
+  by (simp add: mult.tendsto tendsto_inverse divide_inverse)
 
 lemma tendsto_unique:
   fixes f :: "'a \<Rightarrow> 'b::t2_space"
-  assumes "\<not> trivial_limit net"  "(f ---> l) net"  "(f ---> l') net"
+  assumes "\<not> trivial_limit A"  "(f ---> l) A"  "(f ---> l') A"
   shows "l = l'"
 proof (rule ccontr)
   assume "l \<noteq> l'"
   obtain U V where "open U" "open V" "l \<in> U" "l' \<in> V" "U \<inter> V = {}"
     using hausdorff [OF `l \<noteq> l'`] by fast
-  have "eventually (\<lambda>x. f x \<in> U) net"
-    using `(f ---> l) net` `open U` `l \<in> U` by (rule topological_tendstoD)
+  have "eventually (\<lambda>x. f x \<in> U) A"
+    using `(f ---> l) A` `open U` `l \<in> U` by (rule topological_tendstoD)
   moreover
-  have "eventually (\<lambda>x. f x \<in> V) net"
-    using `(f ---> l') net` `open V` `l' \<in> V` by (rule topological_tendstoD)
+  have "eventually (\<lambda>x. f x \<in> V) A"
+    using `(f ---> l') A` `open V` `l' \<in> V` by (rule topological_tendstoD)
   ultimately
-  have "eventually (\<lambda>x. False) net"
+  have "eventually (\<lambda>x. False) A"
   proof (rule eventually_elim2)
     fix x
     assume "f x \<in> U" "f x \<in> V"
     hence "f x \<in> U \<inter> V" by simp
     with `U \<inter> V = {}` show "False" by simp
   qed
-  with `\<not> trivial_limit net` show "False"
+  with `\<not> trivial_limit A` show "False"
     by (simp add: trivial_limit_def)
 qed
 
diff -r 53d95b52954c -r 730f7cced3a6 src/HOL/Multivariate_Analysis/Derivative.thy
--- a/src/HOL/Multivariate_Analysis/Derivative.thy	Mon Aug 08 18:36:32 2011 -0700
+++ b/src/HOL/Multivariate_Analysis/Derivative.thy	Mon Aug 08 19:26:53 2011 -0700
@@ -18,7 +18,7 @@
   nets of a particular form. This lets us prove theorems generally and use 
   "at a" or "at a within s" for restriction to a set (1-sided on R etc.) *}
 
-definition has_derivative :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a net \<Rightarrow> bool)"
+definition has_derivative :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> ('a filter \<Rightarrow> bool)"
 (infixl "(has'_derivative)" 12) where
  "(f has_derivative f') net \<equiv> bounded_linear f' \<and> ((\<lambda>y. (1 / (norm (y - netlimit net))) *\<^sub>R
    (f y - (f (netlimit net) + f'(y - netlimit net)))) ---> 0) net"
@@ -291,7 +291,7 @@
 
 no_notation Deriv.differentiable (infixl "differentiable" 60)
 
-definition differentiable :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a net \<Rightarrow> bool" (infixr "differentiable" 30) where
+definition differentiable :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a filter \<Rightarrow> bool" (infixr "differentiable" 30) where
   "f differentiable net \<equiv> (\<exists>f'. (f has_derivative f') net)"
 
 definition differentiable_on :: "('a::real_normed_vector \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a set \<Rightarrow> bool" (infixr "differentiable'_on" 30) where
@@ -469,25 +469,25 @@
 
 subsection {* Composition rules stated just for differentiability. *}
 
-lemma differentiable_const[intro]: "(\<lambda>z. c) differentiable (net::'a::real_normed_vector net)"
+lemma differentiable_const[intro]: "(\<lambda>z. c) differentiable (net::'a::real_normed_vector filter)"
   unfolding differentiable_def using has_derivative_const by auto
 
-lemma differentiable_id[intro]: "(\<lambda>z. z) differentiable (net::'a::real_normed_vector net)"
+lemma differentiable_id[intro]: "(\<lambda>z. z) differentiable (net::'a::real_normed_vector filter)"
     unfolding differentiable_def using has_derivative_id by auto
 
-lemma differentiable_cmul[intro]: "f differentiable net \<Longrightarrow> (\<lambda>x. c *\<^sub>R f(x)) differentiable (net::'a::real_normed_vector net)"
+lemma differentiable_cmul[intro]: "f differentiable net \<Longrightarrow> (\<lambda>x. c *\<^sub>R f(x)) differentiable (net::'a::real_normed_vector filter)"
   unfolding differentiable_def apply(erule exE, drule has_derivative_cmul) by auto
 
-lemma differentiable_neg[intro]: "f differentiable net \<Longrightarrow> (\<lambda>z. -(f z)) differentiable (net::'a::real_normed_vector net)"
+lemma differentiable_neg[intro]: "f differentiable net \<Longrightarrow> (\<lambda>z. -(f z)) differentiable (net::'a::real_normed_vector filter)"
   unfolding differentiable_def apply(erule exE, drule has_derivative_neg) by auto
 
 lemma differentiable_add: "f differentiable net \<Longrightarrow> g differentiable net
-   \<Longrightarrow> (\<lambda>z. f z + g z) differentiable (net::'a::real_normed_vector net)"
+   \<Longrightarrow> (\<lambda>z. f z + g z) differentiable (net::'a::real_normed_vector filter)"
     unfolding differentiable_def apply(erule exE)+ apply(rule_tac x="\<lambda>z. f' z + f'a z" in exI)
     apply(rule has_derivative_add) by auto
 
 lemma differentiable_sub: "f differentiable net \<Longrightarrow> g differentiable net
-  \<Longrightarrow> (\<lambda>z. f z - g z) differentiable (net::'a::real_normed_vector net)"
+  \<Longrightarrow> (\<lambda>z. f z - g z) differentiable (net::'a::real_normed_vector filter)"
   unfolding differentiable_def apply(erule exE)+ apply(rule_tac x="\<lambda>z. f' z - f'a z" in exI)
     apply(rule has_derivative_sub) by auto 
 
@@ -1259,7 +1259,7 @@
 
 subsection {* Considering derivative @{typ "real \<Rightarrow> 'b\<Colon>real_normed_vector"} as a vector. *}
 
-definition has_vector_derivative :: "(real \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'b \<Rightarrow> (real net \<Rightarrow> bool)"
+definition has_vector_derivative :: "(real \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'b \<Rightarrow> (real filter \<Rightarrow> bool)"
 (infixl "has'_vector'_derivative" 12) where
  "(f has_vector_derivative f') net \<equiv> (f has_derivative (\<lambda>x. x *\<^sub>R f')) net"
 
diff -r 53d95b52954c -r 730f7cced3a6 src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy
--- a/src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy	Mon Aug 08 18:36:32 2011 -0700
+++ b/src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy	Mon Aug 08 19:26:53 2011 -0700
@@ -882,24 +882,25 @@
   using frontier_complement frontier_subset_eq[of "- S"]
   unfolding open_closed by auto
 
-subsection {* Nets and the ``eventually true'' quantifier *}
-
-text {* Common nets and The "within" modifier for nets. *}
+subsection {* Filters and the ``eventually true'' quantifier *}
+
+text {* Common filters and The "within" modifier for filters. *}
 
 definition
-  at_infinity :: "'a::real_normed_vector net" where
-  "at_infinity = Abs_net (\<lambda>P. \<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x)"
+  at_infinity :: "'a::real_normed_vector filter" where
+  "at_infinity = Abs_filter (\<lambda>P. \<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x)"
 
 definition
-  indirection :: "'a::real_normed_vector \<Rightarrow> 'a \<Rightarrow> 'a net" (infixr "indirection" 70) where
+  indirection :: "'a::real_normed_vector \<Rightarrow> 'a \<Rightarrow> 'a filter"
+    (infixr "indirection" 70) where
   "a indirection v = (at a) within {b. \<exists>c\<ge>0. b - a = scaleR c v}"
 
-text{* Prove That They are all nets. *}
+text{* Prove That They are all filters. *}
 
 lemma eventually_at_infinity:
   "eventually P at_infinity \<longleftrightarrow> (\<exists>b. \<forall>x. norm x >= b \<longrightarrow> P x)"
 unfolding at_infinity_def
-proof (rule eventually_Abs_net, rule is_filter.intro)
+proof (rule eventually_Abs_filter, rule is_filter.intro)
   fix P Q :: "'a \<Rightarrow> bool"
   assume "\<exists>r. \<forall>x. r \<le> norm x \<longrightarrow> P x" and "\<exists>s. \<forall>x. s \<le> norm x \<longrightarrow> Q x"
   then obtain r s where
@@ -944,7 +945,7 @@
   by (simp add: trivial_limit_at_iff)
 
 lemma trivial_limit_at_infinity:
-  "\<not> trivial_limit (at_infinity :: ('a::{real_normed_vector,perfect_space}) net)"
+  "\<not> trivial_limit (at_infinity :: ('a::{real_normed_vector,perfect_space}) filter)"
   unfolding trivial_limit_def eventually_at_infinity
   apply clarsimp
   apply (subgoal_tac "\<exists>x::'a. x \<noteq> 0", clarify)
@@ -972,12 +973,6 @@
   unfolding trivial_limit_def
   by (auto elim: eventually_rev_mp)
 
-lemma always_eventually: "(\<forall>x. P x) ==> eventually P net"
-proof -
-  assume "\<forall>x. P x" hence "P = (\<lambda>x. True)" by (simp add: ext)
-  thus "eventually P net" by simp
-qed
-
 lemma trivial_limit_eventually: "trivial_limit net \<Longrightarrow> eventually P net"
   unfolding trivial_limit_def by (auto elim: eventually_rev_mp)
 
@@ -1012,10 +1007,10 @@
 
 subsection {* Limits *}
 
-  text{* Notation Lim to avoid collition with lim defined in analysis *}
-definition
-  Lim :: "'a net \<Rightarrow> ('a \<Rightarrow> 'b::t2_space) \<Rightarrow> 'b" where
-  "Lim net f = (THE l. (f ---> l) net)"
+text{* Notation Lim to avoid collition with lim defined in analysis *}
+
+definition Lim :: "'a filter \<Rightarrow> ('a \<Rightarrow> 'b::t2_space) \<Rightarrow> 'b"
+  where "Lim A f = (THE l. (f ---> l) A)"
 
 lemma Lim:
  "(f ---> l) net \<longleftrightarrow>
@@ -1281,9 +1276,9 @@
   using assms by (rule scaleR.tendsto)
 
 lemma Lim_inv:
-  fixes f :: "'a \<Rightarrow> real"
-  assumes "(f ---> l) (net::'a net)"  "l \<noteq> 0"
-  shows "((inverse o f) ---> inverse l) net"
+  fixes f :: "'a \<Rightarrow> real" and A :: "'a filter"
+  assumes "(f ---> l) A" and "l \<noteq> 0"
+  shows "((inverse o f) ---> inverse l) A"
   unfolding o_def using assms by (rule tendsto_inverse)
 
 lemma Lim_vmul:
@@ -1485,10 +1480,10 @@
   thus "?lhs" by (rule topological_tendstoI)
 qed
 
-text{* It's also sometimes useful to extract the limit point from the net.  *}
+text{* It's also sometimes useful to extract the limit point from the filter. *}
 
 definition
-  netlimit :: "'a::t2_space net \<Rightarrow> 'a" where
+  netlimit :: "'a::t2_space filter \<Rightarrow> 'a" where
   "netlimit net = (SOME a. ((\<lambda>x. x) ---> a) net)"
 
 lemma netlimit_within:
@@ -1943,7 +1938,7 @@
 
 lemma at_within_interior:
   "x \<in> interior S \<Longrightarrow> at x within S = at x"
-  by (simp add: expand_net_eq eventually_within_interior)
+  by (simp add: filter_eq_iff eventually_within_interior)
 
 lemma lim_within_interior:
   "x \<in> interior S \<Longrightarrow> (f ---> l) (at x within S) \<longleftrightarrow> (f ---> l) (at x)"
@@ -3338,8 +3333,8 @@
 text {* Define continuity over a net to take in restrictions of the set. *}
 
 definition
-  continuous :: "'a::t2_space net \<Rightarrow> ('a \<Rightarrow> 'b::topological_space) \<Rightarrow> bool" where
-  "continuous net f \<longleftrightarrow> (f ---> f(netlimit net)) net"
+  continuous :: "'a::t2_space filter \<Rightarrow> ('a \<Rightarrow> 'b::topological_space) \<Rightarrow> bool"
+  where "continuous net f \<longleftrightarrow> (f ---> f(netlimit net)) net"
 
 lemma continuous_trivial_limit:
  "trivial_limit net ==> continuous net f"