--- a/src/HOL/Lim.thy Tue Jun 02 15:37:59 2009 -0700
+++ b/src/HOL/Lim.thy Tue Jun 02 17:03:22 2009 -0700
@@ -13,10 +13,6 @@
text{*Standard Definitions*}
definition
- at :: "'a::metric_space \<Rightarrow> 'a filter" where
- [code del]: "at a = Abs_filter (\<lambda>P. \<exists>r>0. \<forall>x. x \<noteq> a \<and> dist x a < r \<longrightarrow> P x)"
-
-definition
LIM :: "['a::metric_space \<Rightarrow> 'b::metric_space, 'a, 'b] \<Rightarrow> bool"
("((_)/ -- (_)/ --> (_))" [60, 0, 60] 60) where
[code del]: "f -- a --> L =
@@ -31,23 +27,11 @@
isUCont :: "['a::metric_space \<Rightarrow> 'b::metric_space] \<Rightarrow> bool" where
[code del]: "isUCont f = (\<forall>r>0. \<exists>s>0. \<forall>x y. dist x y < s \<longrightarrow> dist (f x) (f y) < r)"
-subsection {* Neighborhood Filter *}
-
-lemma eventually_at:
- "eventually P (at a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> P x)"
-unfolding at_def
-apply (rule eventually_Abs_filter)
-apply (rule_tac x=1 in exI, simp)
-apply (clarify, rule_tac x=r in exI, simp)
-apply (clarify, rename_tac r s)
-apply (rule_tac x="min r s" in exI, simp)
-done
+subsection {* Limits of Functions *}
lemma LIM_conv_tendsto: "(f -- a --> L) \<longleftrightarrow> tendsto f L (at a)"
unfolding LIM_def tendsto_def eventually_at ..
-subsection {* Limits of Functions *}
-
lemma metric_LIM_I:
"(\<And>r. 0 < r \<Longrightarrow> \<exists>s>0. \<forall>x. x \<noteq> a \<and> dist x a < s \<longrightarrow> dist (f x) L < r)
\<Longrightarrow> f -- a --> L"
--- a/src/HOL/Limits.thy Tue Jun 02 15:37:59 2009 -0700
+++ b/src/HOL/Limits.thy Tue Jun 02 17:03:22 2009 -0700
@@ -8,128 +8,204 @@
imports RealVector RComplete
begin
-subsection {* Filters *}
+subsection {* Nets *}
+
+text {*
+ A net is now defined as a filter base.
+ The definition also allows non-proper filter bases.
+*}
+
+typedef (open) 'a net =
+ "{net :: 'a set set. (\<exists>A. A \<in> net)
+ \<and> (\<forall>A\<in>net. \<forall>B\<in>net. \<exists>C\<in>net. C \<subseteq> A \<and> C \<subseteq> B)}"
+proof
+ show "UNIV \<in> ?net" by auto
+qed
-typedef (open) 'a filter =
- "{f :: ('a \<Rightarrow> bool) \<Rightarrow> bool. f (\<lambda>x. True)
- \<and> (\<forall>P Q. (\<forall>x. P x \<longrightarrow> Q x) \<longrightarrow> f P \<longrightarrow> f Q)
- \<and> (\<forall>P Q. f P \<longrightarrow> f Q \<longrightarrow> f (\<lambda>x. P x \<and> Q x))}"
-proof
- show "(\<lambda>P. True) \<in> ?filter" by simp
-qed
+lemma Rep_net_nonempty: "\<exists>A. A \<in> Rep_net net"
+using Rep_net [of net] by simp
+
+lemma Rep_net_directed:
+ "A \<in> Rep_net net \<Longrightarrow> B \<in> Rep_net net \<Longrightarrow> \<exists>C\<in>Rep_net net. C \<subseteq> A \<and> C \<subseteq> B"
+using Rep_net [of net] by simp
+
+lemma Abs_net_inverse':
+ assumes "\<exists>A. A \<in> net"
+ assumes "\<And>A B. A \<in> net \<Longrightarrow> B \<in> net \<Longrightarrow> \<exists>C\<in>net. C \<subseteq> A \<and> C \<subseteq> B"
+ shows "Rep_net (Abs_net net) = net"
+using assms by (simp add: Abs_net_inverse)
+
+lemma image_nonempty: "\<exists>x. x \<in> A \<Longrightarrow> \<exists>x. x \<in> f ` A"
+by auto
+
+
+subsection {* Eventually *}
definition
- eventually :: "('a \<Rightarrow> bool) \<Rightarrow> 'a filter \<Rightarrow> bool" where
- [code del]: "eventually P F \<longleftrightarrow> Rep_filter F P"
+ eventually :: "('a \<Rightarrow> bool) \<Rightarrow> 'a net \<Rightarrow> bool" where
+ [code del]: "eventually P net \<longleftrightarrow> (\<exists>A\<in>Rep_net net. \<forall>x\<in>A. P x)"
-lemma eventually_True [simp]: "eventually (\<lambda>x. True) F"
-unfolding eventually_def using Rep_filter [of F] by blast
+lemma eventually_True [simp]: "eventually (\<lambda>x. True) net"
+unfolding eventually_def using Rep_net_nonempty [of net] by fast
lemma eventually_mono:
- "(\<forall>x. P x \<longrightarrow> Q x) \<Longrightarrow> eventually P F \<Longrightarrow> eventually Q F"
-unfolding eventually_def using Rep_filter [of F] by blast
+ "(\<forall>x. P x \<longrightarrow> Q x) \<Longrightarrow> eventually P net \<Longrightarrow> eventually Q net"
+unfolding eventually_def by blast
lemma eventually_conj:
- "\<lbrakk>eventually (\<lambda>x. P x) F; eventually (\<lambda>x. Q x) F\<rbrakk>
- \<Longrightarrow> eventually (\<lambda>x. P x \<and> Q x) F"
-unfolding eventually_def using Rep_filter [of F] by blast
+ 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"
+proof -
+ obtain A where A: "A \<in> Rep_net net" "\<forall>x\<in>A. P x"
+ using P unfolding eventually_def by fast
+ obtain B where B: "B \<in> Rep_net net" "\<forall>x\<in>B. Q x"
+ using Q unfolding eventually_def by fast
+ obtain C where C: "C \<in> Rep_net net" "C \<subseteq> A" "C \<subseteq> B"
+ using Rep_net_directed [OF A(1) B(1)] by fast
+ then have "\<forall>x\<in>C. P x \<and> Q x" "C \<in> Rep_net net"
+ using A(2) B(2) by auto
+ then show ?thesis unfolding eventually_def ..
+qed
lemma eventually_mp:
- assumes "eventually (\<lambda>x. P x \<longrightarrow> Q x) F"
- assumes "eventually (\<lambda>x. P x) F"
- shows "eventually (\<lambda>x. Q x) F"
+ assumes "eventually (\<lambda>x. P x \<longrightarrow> Q x) net"
+ assumes "eventually (\<lambda>x. P x) net"
+ shows "eventually (\<lambda>x. Q x) net"
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) F"
+ show "eventually (\<lambda>x. (P x \<longrightarrow> Q x) \<and> P x) net"
using assms by (rule eventually_conj)
qed
lemma eventually_rev_mp:
- assumes "eventually (\<lambda>x. P x) F"
- assumes "eventually (\<lambda>x. P x \<longrightarrow> Q x) F"
- shows "eventually (\<lambda>x. Q x) F"
+ assumes "eventually (\<lambda>x. P x) net"
+ assumes "eventually (\<lambda>x. P x \<longrightarrow> Q x) net"
+ shows "eventually (\<lambda>x. Q x) net"
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)
-lemma eventually_Abs_filter:
- assumes "f (\<lambda>x. True)"
- assumes "\<And>P Q. (\<forall>x. P x \<longrightarrow> Q x) \<Longrightarrow> f P \<Longrightarrow> f Q"
- assumes "\<And>P Q. f P \<Longrightarrow> f Q \<Longrightarrow> f (\<lambda>x. P x \<and> Q x)"
- shows "eventually P (Abs_filter f) \<longleftrightarrow> f P"
-unfolding eventually_def using assms
-by (subst Abs_filter_inverse, auto)
-
-lemma filter_ext:
- "(\<And>P. eventually P F \<longleftrightarrow> eventually P F') \<Longrightarrow> F = F'"
-unfolding eventually_def
-by (simp add: Rep_filter_inject [THEN iffD1] ext)
-
lemma eventually_elim1:
- assumes "eventually (\<lambda>i. P i) F"
+ assumes "eventually (\<lambda>i. P i) net"
assumes "\<And>i. P i \<Longrightarrow> Q i"
- shows "eventually (\<lambda>i. Q i) F"
+ shows "eventually (\<lambda>i. Q i) net"
using assms by (auto elim!: eventually_rev_mp)
lemma eventually_elim2:
- assumes "eventually (\<lambda>i. P i) F"
- assumes "eventually (\<lambda>i. Q i) F"
+ assumes "eventually (\<lambda>i. P i) net"
+ assumes "eventually (\<lambda>i. Q i) net"
assumes "\<And>i. P i \<Longrightarrow> Q i \<Longrightarrow> R i"
- shows "eventually (\<lambda>i. R i) F"
+ shows "eventually (\<lambda>i. R i) net"
using assms by (auto elim!: eventually_rev_mp)
+subsection {* Standard Nets *}
+
+definition
+ sequentially :: "nat net" where
+ [code del]: "sequentially = Abs_net (range (\<lambda>n. {n..}))"
+
+definition
+ at :: "'a::metric_space \<Rightarrow> 'a net" where
+ [code del]: "at a = Abs_net ((\<lambda>r. {x. x \<noteq> a \<and> dist x a < r}) ` {r. 0 < r})"
+
+definition
+ within :: "'a net \<Rightarrow> 'a set \<Rightarrow> 'a net" (infixr "within" 70) where
+ [code del]: "net within S = Abs_net ((\<lambda>A. A \<inter> S) ` Rep_net net)"
+
+lemma Rep_net_sequentially:
+ "Rep_net sequentially = range (\<lambda>n. {n..})"
+unfolding sequentially_def
+apply (rule Abs_net_inverse')
+apply (rule image_nonempty, simp)
+apply (clarsimp, rename_tac m n)
+apply (rule_tac x="max m n" in exI, auto)
+done
+
+lemma Rep_net_at:
+ "Rep_net (at a) = ((\<lambda>r. {x. x \<noteq> a \<and> dist x a < r}) ` {r. 0 < r})"
+unfolding at_def
+apply (rule Abs_net_inverse')
+apply (rule image_nonempty, rule_tac x=1 in exI, simp)
+apply (clarsimp, rename_tac r s)
+apply (rule_tac x="min r s" in exI, auto)
+done
+
+lemma Rep_net_within:
+ "Rep_net (net within S) = (\<lambda>A. A \<inter> S) ` Rep_net net"
+unfolding within_def
+apply (rule Abs_net_inverse')
+apply (rule image_nonempty, rule Rep_net_nonempty)
+apply (clarsimp, rename_tac A B)
+apply (drule (1) Rep_net_directed)
+apply (clarify, rule_tac x=C in bexI, auto)
+done
+
+lemma eventually_sequentially:
+ "eventually P sequentially \<longleftrightarrow> (\<exists>N. \<forall>n\<ge>N. P n)"
+unfolding eventually_def Rep_net_sequentially by auto
+
+lemma eventually_at:
+ "eventually P (at a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> P x)"
+unfolding eventually_def Rep_net_at by auto
+
+lemma eventually_within:
+ "eventually P (net within S) = eventually (\<lambda>x. x \<in> S \<longrightarrow> P x) net"
+unfolding eventually_def Rep_net_within by auto
+
+
subsection {* Boundedness *}
definition
- Bfun :: "('a \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a filter \<Rightarrow> bool" where
- [code del]: "Bfun S F = (\<exists>K>0. eventually (\<lambda>i. norm (S i) \<le> K) F)"
+ Bfun :: "('a \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a net \<Rightarrow> bool" where
+ [code del]: "Bfun S net = (\<exists>K>0. eventually (\<lambda>i. norm (S i) \<le> K) net)"
-lemma BfunI: assumes K: "eventually (\<lambda>i. norm (X i) \<le> K) F" shows "Bfun X F"
+lemma BfunI: assumes K: "eventually (\<lambda>i. norm (X i) \<le> K) net" shows "Bfun X net"
unfolding Bfun_def
proof (intro exI conjI allI)
show "0 < max K 1" by simp
next
- show "eventually (\<lambda>i. norm (X i) \<le> max K 1) F"
+ show "eventually (\<lambda>i. norm (X i) \<le> max K 1) net"
using K by (rule eventually_elim1, simp)
qed
lemma BfunE:
- assumes "Bfun S F"
- obtains B where "0 < B" and "eventually (\<lambda>i. norm (S i) \<le> B) F"
+ assumes "Bfun S net"
+ obtains B where "0 < B" and "eventually (\<lambda>i. norm (S i) \<le> B) net"
using assms unfolding Bfun_def by fast
subsection {* Convergence to Zero *}
definition
- Zfun :: "('a \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a filter \<Rightarrow> bool" where
- [code del]: "Zfun S F = (\<forall>r>0. eventually (\<lambda>i. norm (S i) < r) F)"
+ Zfun :: "('a \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a net \<Rightarrow> bool" where
+ [code del]: "Zfun S net = (\<forall>r>0. eventually (\<lambda>i. norm (S i) < r) net)"
lemma ZfunI:
- "(\<And>r. 0 < r \<Longrightarrow> eventually (\<lambda>i. norm (S i) < r) F) \<Longrightarrow> Zfun S F"
+ "(\<And>r. 0 < r \<Longrightarrow> eventually (\<lambda>i. norm (S i) < r) net) \<Longrightarrow> Zfun S net"
unfolding Zfun_def by simp
lemma ZfunD:
- "\<lbrakk>Zfun S F; 0 < r\<rbrakk> \<Longrightarrow> eventually (\<lambda>i. norm (S i) < r) F"
+ "\<lbrakk>Zfun S net; 0 < r\<rbrakk> \<Longrightarrow> eventually (\<lambda>i. norm (S i) < r) net"
unfolding Zfun_def by simp
lemma Zfun_ssubst:
- "eventually (\<lambda>i. X i = Y i) F \<Longrightarrow> Zfun Y F \<Longrightarrow> Zfun X F"
+ "eventually (\<lambda>i. X i = Y i) net \<Longrightarrow> Zfun Y net \<Longrightarrow> Zfun X net"
unfolding Zfun_def by (auto elim!: eventually_rev_mp)
-lemma Zfun_zero: "Zfun (\<lambda>i. 0) F"
+lemma Zfun_zero: "Zfun (\<lambda>i. 0) net"
unfolding Zfun_def by simp
-lemma Zfun_norm_iff: "Zfun (\<lambda>i. norm (S i)) F = Zfun (\<lambda>i. S i) F"
+lemma Zfun_norm_iff: "Zfun (\<lambda>i. norm (S i)) net = Zfun (\<lambda>i. S i) net"
unfolding Zfun_def by simp
lemma Zfun_imp_Zfun:
- assumes X: "Zfun X F"
- assumes Y: "eventually (\<lambda>i. norm (Y i) \<le> norm (X i) * K) F"
- shows "Zfun (\<lambda>n. Y n) F"
+ assumes X: "Zfun X net"
+ assumes Y: "eventually (\<lambda>i. norm (Y i) \<le> norm (X i) * K) net"
+ shows "Zfun (\<lambda>n. Y n) net"
proof (cases)
assume K: "0 < K"
show ?thesis
@@ -137,9 +213,9 @@
fix r::real assume "0 < r"
hence "0 < r / K"
using K by (rule divide_pos_pos)
- then have "eventually (\<lambda>i. norm (X i) < r / K) F"
+ then have "eventually (\<lambda>i. norm (X i) < r / K) net"
using ZfunD [OF X] by fast
- with Y show "eventually (\<lambda>i. norm (Y i) < r) F"
+ with Y show "eventually (\<lambda>i. norm (Y i) < r) net"
proof (rule eventually_elim2)
fix i
assume *: "norm (Y i) \<le> norm (X i) * K"
@@ -157,7 +233,7 @@
proof (rule ZfunI)
fix r :: real
assume "0 < r"
- from Y show "eventually (\<lambda>i. norm (Y i) < r) F"
+ from Y show "eventually (\<lambda>i. norm (Y i) < r) net"
proof (rule eventually_elim1)
fix i
assume "norm (Y i) \<le> norm (X i) * K"
@@ -169,22 +245,22 @@
qed
qed
-lemma Zfun_le: "\<lbrakk>Zfun Y F; \<forall>n. norm (X n) \<le> norm (Y n)\<rbrakk> \<Longrightarrow> Zfun X F"
+lemma Zfun_le: "\<lbrakk>Zfun Y net; \<forall>n. norm (X n) \<le> norm (Y n)\<rbrakk> \<Longrightarrow> Zfun X net"
by (erule_tac K="1" in Zfun_imp_Zfun, simp)
lemma Zfun_add:
- assumes X: "Zfun X F" and Y: "Zfun Y F"
- shows "Zfun (\<lambda>n. X n + Y n) F"
+ assumes X: "Zfun X net" and Y: "Zfun Y net"
+ shows "Zfun (\<lambda>n. X n + Y n) net"
proof (rule ZfunI)
fix r::real assume "0 < r"
hence r: "0 < r / 2" by simp
- have "eventually (\<lambda>i. norm (X i) < r/2) F"
+ have "eventually (\<lambda>i. norm (X i) < r/2) net"
using X r by (rule ZfunD)
moreover
- have "eventually (\<lambda>i. norm (Y i) < r/2) F"
+ have "eventually (\<lambda>i. norm (Y i) < r/2) net"
using Y r by (rule ZfunD)
ultimately
- show "eventually (\<lambda>i. norm (X i + Y i) < r) F"
+ show "eventually (\<lambda>i. norm (X i + Y i) < r) net"
proof (rule eventually_elim2)
fix i
assume *: "norm (X i) < r/2" "norm (Y i) < r/2"
@@ -197,28 +273,28 @@
qed
qed
-lemma Zfun_minus: "Zfun X F \<Longrightarrow> Zfun (\<lambda>i. - X i) F"
+lemma Zfun_minus: "Zfun X net \<Longrightarrow> Zfun (\<lambda>i. - X i) net"
unfolding Zfun_def by simp
-lemma Zfun_diff: "\<lbrakk>Zfun X F; Zfun Y F\<rbrakk> \<Longrightarrow> Zfun (\<lambda>i. X i - Y i) F"
+lemma Zfun_diff: "\<lbrakk>Zfun X net; Zfun Y net\<rbrakk> \<Longrightarrow> Zfun (\<lambda>i. X i - Y i) net"
by (simp only: diff_minus Zfun_add Zfun_minus)
lemma (in bounded_linear) Zfun:
- assumes X: "Zfun X F"
- shows "Zfun (\<lambda>n. f (X n)) F"
+ assumes X: "Zfun X net"
+ shows "Zfun (\<lambda>n. f (X n)) net"
proof -
obtain K where "\<And>x. norm (f x) \<le> norm x * K"
using bounded by fast
- then have "eventually (\<lambda>i. norm (f (X i)) \<le> norm (X i) * K) F"
+ then have "eventually (\<lambda>i. norm (f (X i)) \<le> norm (X i) * K) net"
by simp
with X show ?thesis
by (rule Zfun_imp_Zfun)
qed
lemma (in bounded_bilinear) Zfun:
- assumes X: "Zfun X F"
- assumes Y: "Zfun Y F"
- shows "Zfun (\<lambda>n. X n ** Y n) F"
+ assumes X: "Zfun X net"
+ assumes Y: "Zfun Y net"
+ shows "Zfun (\<lambda>n. X n ** Y n) net"
proof (rule ZfunI)
fix r::real assume r: "0 < r"
obtain K where K: "0 < K"
@@ -226,13 +302,13 @@
using pos_bounded by fast
from K have K': "0 < inverse K"
by (rule positive_imp_inverse_positive)
- have "eventually (\<lambda>i. norm (X i) < r) F"
+ have "eventually (\<lambda>i. norm (X i) < r) net"
using X r by (rule ZfunD)
moreover
- have "eventually (\<lambda>i. norm (Y i) < inverse K) F"
+ have "eventually (\<lambda>i. norm (Y i) < inverse K) net"
using Y K' by (rule ZfunD)
ultimately
- show "eventually (\<lambda>i. norm (X i ** Y i) < r) F"
+ show "eventually (\<lambda>i. norm (X i ** Y i) < r) net"
proof (rule eventually_elim2)
fix i
assume *: "norm (X i) < r" "norm (Y i) < inverse K"
@@ -247,11 +323,11 @@
qed
lemma (in bounded_bilinear) Zfun_left:
- "Zfun X F \<Longrightarrow> Zfun (\<lambda>n. X n ** a) F"
+ "Zfun X net \<Longrightarrow> Zfun (\<lambda>n. X n ** a) net"
by (rule bounded_linear_left [THEN bounded_linear.Zfun])
lemma (in bounded_bilinear) Zfun_right:
- "Zfun X F \<Longrightarrow> Zfun (\<lambda>n. a ** X n) F"
+ "Zfun X net \<Longrightarrow> Zfun (\<lambda>n. a ** X n) net"
by (rule bounded_linear_right [THEN bounded_linear.Zfun])
lemmas Zfun_mult = mult.Zfun
@@ -262,7 +338,7 @@
subsection{* Limits *}
definition
- tendsto :: "('a \<Rightarrow> 'b::metric_space) \<Rightarrow> 'b \<Rightarrow> 'a filter \<Rightarrow> bool" where
+ tendsto :: "('a \<Rightarrow> 'b::metric_space) \<Rightarrow> 'b \<Rightarrow> 'a net \<Rightarrow> bool" where
[code del]: "tendsto f l net \<longleftrightarrow> (\<forall>e>0. eventually (\<lambda>x. dist (f x) l < e) net)"
lemma tendstoI:
@@ -274,15 +350,15 @@
"tendsto f l net \<Longrightarrow> 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) net"
unfolding tendsto_def by auto
-lemma tendsto_Zfun_iff: "tendsto (\<lambda>n. X n) L F = Zfun (\<lambda>n. X n - L) F"
+lemma tendsto_Zfun_iff: "tendsto (\<lambda>n. X n) L net = Zfun (\<lambda>n. X n - L) net"
by (simp only: tendsto_def Zfun_def dist_norm)
-lemma tendsto_const: "tendsto (\<lambda>n. k) k F"
+lemma tendsto_const: "tendsto (\<lambda>n. k) k net"
by (simp add: tendsto_def)
lemma tendsto_norm:
fixes a :: "'a::real_normed_vector"
- shows "tendsto X a F \<Longrightarrow> tendsto (\<lambda>n. norm (X n)) (norm a) F"
+ shows "tendsto X a net \<Longrightarrow> tendsto (\<lambda>n. norm (X n)) (norm a) net"
apply (simp add: tendsto_def dist_norm, safe)
apply (drule_tac x="e" in spec, safe)
apply (erule eventually_elim1)
@@ -301,30 +377,30 @@
lemma tendsto_add:
fixes a b :: "'a::real_normed_vector"
- shows "\<lbrakk>tendsto X a F; tendsto Y b F\<rbrakk> \<Longrightarrow> tendsto (\<lambda>n. X n + Y n) (a + b) F"
+ shows "\<lbrakk>tendsto X a net; tendsto Y b net\<rbrakk> \<Longrightarrow> tendsto (\<lambda>n. X n + Y n) (a + b) net"
by (simp only: tendsto_Zfun_iff add_diff_add Zfun_add)
lemma tendsto_minus:
fixes a :: "'a::real_normed_vector"
- shows "tendsto X a F \<Longrightarrow> tendsto (\<lambda>n. - X n) (- a) F"
+ shows "tendsto X a net \<Longrightarrow> tendsto (\<lambda>n. - X n) (- a) net"
by (simp only: tendsto_Zfun_iff minus_diff_minus Zfun_minus)
lemma tendsto_minus_cancel:
fixes a :: "'a::real_normed_vector"
- shows "tendsto (\<lambda>n. - X n) (- a) F \<Longrightarrow> tendsto X a F"
+ shows "tendsto (\<lambda>n. - X n) (- a) net \<Longrightarrow> tendsto X a net"
by (drule tendsto_minus, simp)
lemma tendsto_diff:
fixes a b :: "'a::real_normed_vector"
- shows "\<lbrakk>tendsto X a F; tendsto Y b F\<rbrakk> \<Longrightarrow> tendsto (\<lambda>n. X n - Y n) (a - b) F"
+ shows "\<lbrakk>tendsto X a net; tendsto Y b net\<rbrakk> \<Longrightarrow> tendsto (\<lambda>n. X n - Y n) (a - b) net"
by (simp add: diff_minus tendsto_add tendsto_minus)
lemma (in bounded_linear) tendsto:
- "tendsto X a F \<Longrightarrow> tendsto (\<lambda>n. f (X n)) (f a) F"
+ "tendsto X a net \<Longrightarrow> tendsto (\<lambda>n. f (X n)) (f a) net"
by (simp only: tendsto_Zfun_iff diff [symmetric] Zfun)
lemma (in bounded_bilinear) tendsto:
- "\<lbrakk>tendsto X a F; tendsto Y b F\<rbrakk> \<Longrightarrow> tendsto (\<lambda>n. X n ** Y n) (a ** b) F"
+ "\<lbrakk>tendsto X a net; tendsto Y b net\<rbrakk> \<Longrightarrow> tendsto (\<lambda>n. X n ** Y n) (a ** b) net"
by (simp only: tendsto_Zfun_iff prod_diff_prod
Zfun_add Zfun Zfun_left Zfun_right)
@@ -332,17 +408,17 @@
subsection {* Continuity of Inverse *}
lemma (in bounded_bilinear) Zfun_prod_Bfun:
- assumes X: "Zfun X F"
- assumes Y: "Bfun Y F"
- shows "Zfun (\<lambda>n. X n ** Y n) F"
+ assumes X: "Zfun X net"
+ assumes Y: "Bfun Y net"
+ shows "Zfun (\<lambda>n. X n ** Y n) net"
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_Y: "eventually (\<lambda>i. norm (Y i) \<le> B) F"
+ and norm_Y: "eventually (\<lambda>i. norm (Y i) \<le> B) net"
using Y by (rule BfunE)
- have "eventually (\<lambda>i. norm (X i ** Y i) \<le> norm (X i) * (B * K)) F"
+ have "eventually (\<lambda>i. norm (X i ** Y i) \<le> norm (X i) * (B * K)) net"
using norm_Y proof (rule eventually_elim1)
fix i
assume *: "norm (Y i) \<le> B"
@@ -370,9 +446,9 @@
using bounded by fast
lemma (in bounded_bilinear) Bfun_prod_Zfun:
- assumes X: "Bfun X F"
- assumes Y: "Zfun Y F"
- shows "Zfun (\<lambda>n. X n ** Y n) F"
+ assumes X: "Bfun X net"
+ assumes Y: "Zfun Y net"
+ shows "Zfun (\<lambda>n. X n ** Y n) net"
using flip Y X by (rule bounded_bilinear.Zfun_prod_Bfun)
lemma inverse_diff_inverse:
@@ -389,16 +465,16 @@
lemma Bfun_inverse:
fixes a :: "'a::real_normed_div_algebra"
- assumes X: "tendsto X a F"
+ assumes X: "tendsto X a net"
assumes a: "a \<noteq> 0"
- shows "Bfun (\<lambda>n. inverse (X n)) F"
+ shows "Bfun (\<lambda>n. inverse (X n)) net"
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>i. dist (X i) a < r) F"
+ have "eventually (\<lambda>i. dist (X i) a < r) net"
using tendstoD [OF X r1] by fast
- hence "eventually (\<lambda>i. norm (inverse (X i)) \<le> inverse (norm a - r)) F"
+ hence "eventually (\<lambda>i. norm (inverse (X i)) \<le> inverse (norm a - r)) net"
proof (rule eventually_elim1)
fix i
assume "dist (X i) a < r"
@@ -425,8 +501,8 @@
lemma tendsto_inverse_lemma:
fixes a :: "'a::real_normed_div_algebra"
- shows "\<lbrakk>tendsto X a F; a \<noteq> 0; eventually (\<lambda>i. X i \<noteq> 0) F\<rbrakk>
- \<Longrightarrow> tendsto (\<lambda>i. inverse (X i)) (inverse a) F"
+ shows "\<lbrakk>tendsto X a net; a \<noteq> 0; eventually (\<lambda>i. X i \<noteq> 0) net\<rbrakk>
+ \<Longrightarrow> tendsto (\<lambda>i. inverse (X i)) (inverse a) net"
apply (subst tendsto_Zfun_iff)
apply (rule Zfun_ssubst)
apply (erule eventually_elim1)
@@ -440,14 +516,14 @@
lemma tendsto_inverse:
fixes a :: "'a::real_normed_div_algebra"
- assumes X: "tendsto X a F"
+ assumes X: "tendsto X a net"
assumes a: "a \<noteq> 0"
- shows "tendsto (\<lambda>i. inverse (X i)) (inverse a) F"
+ shows "tendsto (\<lambda>i. inverse (X i)) (inverse a) net"
proof -
from a have "0 < norm a" by simp
- with X have "eventually (\<lambda>i. dist (X i) a < norm a) F"
+ with X have "eventually (\<lambda>i. dist (X i) a < norm a) net"
by (rule tendstoD)
- then have "eventually (\<lambda>i. X i \<noteq> 0) F"
+ then have "eventually (\<lambda>i. X i \<noteq> 0) net"
unfolding dist_norm by (auto elim!: eventually_elim1)
with X a show ?thesis
by (rule tendsto_inverse_lemma)
@@ -455,8 +531,8 @@
lemma tendsto_divide:
fixes a b :: "'a::real_normed_field"
- shows "\<lbrakk>tendsto X a F; tendsto Y b F; b \<noteq> 0\<rbrakk>
- \<Longrightarrow> tendsto (\<lambda>n. X n / Y n) (a / b) F"
+ shows "\<lbrakk>tendsto X a net; tendsto Y b net; b \<noteq> 0\<rbrakk>
+ \<Longrightarrow> tendsto (\<lambda>n. X n / Y n) (a / b) net"
by (simp add: mult.tendsto tendsto_inverse divide_inverse)
end
--- a/src/HOL/SEQ.thy Tue Jun 02 15:37:59 2009 -0700
+++ b/src/HOL/SEQ.thy Tue Jun 02 17:03:22 2009 -0700
@@ -13,10 +13,6 @@
begin
definition
- sequentially :: "nat filter" where
- [code del]: "sequentially = Abs_filter (\<lambda>P. \<exists>N. \<forall>n\<ge>N. P n)"
-
-definition
Zseq :: "[nat \<Rightarrow> 'a::real_normed_vector] \<Rightarrow> bool" where
--{*Standard definition of sequence converging to zero*}
[code del]: "Zseq X = (\<forall>r>0. \<exists>no. \<forall>n\<ge>no. norm (X n) < r)"
@@ -71,24 +67,6 @@
[code del]: "Cauchy X = (\<forall>e>0. \<exists>M. \<forall>m \<ge> M. \<forall>n \<ge> M. dist (X m) (X n) < e)"
-subsection {* Sequentially *}
-
-lemma eventually_sequentially:
- "eventually P sequentially \<longleftrightarrow> (\<exists>N. \<forall>n\<ge>N. P n)"
-unfolding sequentially_def
-apply (rule eventually_Abs_filter)
-apply simp
-apply (clarify, rule_tac x=N in exI, simp)
-apply (clarify, rename_tac M N)
-apply (rule_tac x="max M N" in exI, simp)
-done
-
-lemma Zseq_conv_Zfun: "Zseq X \<longleftrightarrow> Zfun X sequentially"
-unfolding Zseq_def Zfun_def eventually_sequentially ..
-
-lemma LIMSEQ_conv_tendsto: "(X ----> L) \<longleftrightarrow> tendsto X L sequentially"
-unfolding LIMSEQ_def tendsto_def eventually_sequentially ..
-
subsection {* Bounded Sequences *}
lemma BseqI': assumes K: "\<And>n. norm (X n) \<le> K" shows "Bseq X"
@@ -150,6 +128,9 @@
"\<lbrakk>Zseq X; 0 < r\<rbrakk> \<Longrightarrow> \<exists>no. \<forall>n\<ge>no. norm (X n) < r"
unfolding Zseq_def by simp
+lemma Zseq_conv_Zfun: "Zseq X \<longleftrightarrow> Zfun X sequentially"
+unfolding Zseq_def Zfun_def eventually_sequentially ..
+
lemma Zseq_zero: "Zseq (\<lambda>n. 0)"
unfolding Zseq_def by simp
@@ -212,6 +193,9 @@
subsection {* Limits of Sequences *}
+lemma LIMSEQ_conv_tendsto: "(X ----> L) \<longleftrightarrow> tendsto X L sequentially"
+unfolding LIMSEQ_def tendsto_def eventually_sequentially ..
+
lemma LIMSEQ_iff:
fixes L :: "'a::real_normed_vector"
shows "(X ----> L) = (\<forall>r>0. \<exists>no. \<forall>n \<ge> no. norm (X n - L) < r)"