move class perfect_space into RealVector.thy;
use not_open_singleton as perfect_space class axiom;
generalize some lemmas to class perfect_space;
(* Title : Limits.thy
Author : Brian Huffman
*)
header {* Filters and Limits *}
theory Limits
imports RealVector
begin
subsection {* Filters *}
text {*
This definition also allows non-proper filters.
*}
locale is_filter =
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 filter = "{F :: ('a \<Rightarrow> bool) \<Rightarrow> bool. is_filter F}"
proof
show "(\<lambda>x. True) \<in> ?filter" by (auto intro: is_filter.intro)
qed
lemma is_filter_Rep_filter: "is_filter (Rep_filter F)"
using Rep_filter [of F] by simp
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 filter \<Rightarrow> bool"
where "eventually P F \<longleftrightarrow> Rep_filter F P"
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 filter_eq_iff:
shows "F = F' \<longleftrightarrow> (\<forall>P. eventually P F = eventually P F')"
unfolding Rep_filter_inject [symmetric] fun_eq_iff eventually_def ..
lemma eventually_True [simp]: "eventually (\<lambda>x. True) F"
unfolding eventually_def
by (rule is_filter.True [OF is_filter_Rep_filter])
lemma always_eventually: "\<forall>x. P x \<Longrightarrow> eventually P F"
proof -
assume "\<forall>x. P x" hence "P = (\<lambda>x. True)" by (simp add: ext)
thus "eventually P F" by simp
qed
lemma eventually_mono:
"(\<forall>x. P x \<longrightarrow> Q x) \<Longrightarrow> eventually P F \<Longrightarrow> eventually Q F"
unfolding eventually_def
by (rule is_filter.mono [OF is_filter_Rep_filter])
lemma eventually_conj:
assumes P: "eventually (\<lambda>x. P x) F"
assumes Q: "eventually (\<lambda>x. Q x) F"
shows "eventually (\<lambda>x. P x \<and> Q x) F"
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) F"
assumes "eventually (\<lambda>x. P x) F"
shows "eventually (\<lambda>x. Q x) F"
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"
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"
using assms(2) assms(1) by (rule eventually_mp)
lemma eventually_conj_iff:
"eventually (\<lambda>x. P x \<and> Q x) F \<longleftrightarrow> eventually P F \<and> eventually Q F"
by (auto intro: eventually_conj elim: eventually_rev_mp)
lemma eventually_elim1:
assumes "eventually (\<lambda>i. P i) F"
assumes "\<And>i. P i \<Longrightarrow> Q i"
shows "eventually (\<lambda>i. Q i) F"
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 "\<And>i. P i \<Longrightarrow> Q i \<Longrightarrow> R i"
shows "eventually (\<lambda>i. R i) F"
using assms by (auto elim!: eventually_rev_mp)
subsection {* Finer-than relation *}
text {* @{term "F \<le> F'"} means that filter @{term F} is finer than
filter @{term F'}. *}
instantiation filter :: (type) complete_lattice
begin
definition le_filter_def:
"F \<le> F' \<longleftrightarrow> (\<forall>P. eventually P F' \<longrightarrow> eventually P F)"
definition
"(F :: 'a filter) < F' \<longleftrightarrow> F \<le> F' \<and> \<not> F' \<le> F"
definition
"top = Abs_filter (\<lambda>P. \<forall>x. P x)"
definition
"bot = Abs_filter (\<lambda>P. True)"
definition
"sup F F' = Abs_filter (\<lambda>P. eventually P F \<and> eventually P F')"
definition
"inf F F' = Abs_filter
(\<lambda>P. \<exists>Q R. eventually Q F \<and> eventually R F' \<and> (\<forall>x. Q x \<and> R x \<longrightarrow> P x))"
definition
"Sup S = Abs_filter (\<lambda>P. \<forall>F\<in>S. eventually P F)"
definition
"Inf S = Sup {F::'a filter. \<forall>F'\<in>S. F \<le> F'}"
lemma eventually_top [simp]: "eventually P top \<longleftrightarrow> (\<forall>x. P x)"
unfolding top_filter_def
by (rule eventually_Abs_filter, rule is_filter.intro, auto)
lemma eventually_bot [simp]: "eventually P bot"
unfolding bot_filter_def
by (subst eventually_Abs_filter, rule is_filter.intro, auto)
lemma eventually_sup:
"eventually P (sup F F') \<longleftrightarrow> eventually P F \<and> eventually P F'"
unfolding sup_filter_def
by (rule eventually_Abs_filter, rule is_filter.intro)
(auto elim!: eventually_rev_mp)
lemma eventually_inf:
"eventually P (inf F F') \<longleftrightarrow>
(\<exists>Q R. eventually Q F \<and> eventually R F' \<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 S) \<longleftrightarrow> (\<forall>F\<in>S. eventually P F)"
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 F F' F'' :: "'a filter" and S :: "'a filter set"
{ show "F < F' \<longleftrightarrow> F \<le> F' \<and> \<not> F' \<le> F"
by (rule less_filter_def) }
{ show "F \<le> F"
unfolding le_filter_def by simp }
{ assume "F \<le> F'" and "F' \<le> F''" thus "F \<le> F''"
unfolding le_filter_def by simp }
{ assume "F \<le> F'" and "F' \<le> F" thus "F = F'"
unfolding le_filter_def filter_eq_iff by fast }
{ show "F \<le> top"
unfolding le_filter_def eventually_top by (simp add: always_eventually) }
{ show "bot \<le> F"
unfolding le_filter_def by simp }
{ show "F \<le> sup F F'" and "F' \<le> sup F F'"
unfolding le_filter_def eventually_sup by simp_all }
{ assume "F \<le> F''" and "F' \<le> F''" thus "sup F F' \<le> F''"
unfolding le_filter_def eventually_sup by simp }
{ show "inf F F' \<le> F" and "inf F F' \<le> F'"
unfolding le_filter_def eventually_inf by (auto intro: eventually_True) }
{ assume "F \<le> F'" and "F \<le> F''" thus "F \<le> inf F' F''"
unfolding le_filter_def eventually_inf
by (auto elim!: eventually_mono intro: eventually_conj) }
{ assume "F \<in> S" thus "F \<le> Sup S"
unfolding le_filter_def eventually_Sup by simp }
{ assume "\<And>F. F \<in> S \<Longrightarrow> F \<le> F'" thus "Sup S \<le> F'"
unfolding le_filter_def eventually_Sup by simp }
{ assume "F'' \<in> S" thus "Inf S \<le> F''"
unfolding le_filter_def Inf_filter_def eventually_Sup Ball_def by simp }
{ assume "\<And>F'. F' \<in> S \<Longrightarrow> F \<le> F'" thus "F \<le> Inf S"
unfolding le_filter_def Inf_filter_def eventually_Sup Ball_def by simp }
qed
end
lemma filter_leD:
"F \<le> F' \<Longrightarrow> eventually P F' \<Longrightarrow> eventually P F"
unfolding le_filter_def by simp
lemma filter_leI:
"(\<And>P. eventually P F' \<Longrightarrow> eventually P F) \<Longrightarrow> F \<le> F'"
unfolding le_filter_def by simp
lemma eventually_False:
"eventually (\<lambda>x. False) F \<longleftrightarrow> F = bot"
unfolding filter_eq_iff by (auto elim: eventually_rev_mp)
abbreviation (input) trivial_limit :: "'a filter \<Rightarrow> bool"
where "trivial_limit F \<equiv> F = bot"
lemma trivial_limit_def: "trivial_limit F \<longleftrightarrow> eventually (\<lambda>x. False) F"
by (rule eventually_False [symmetric])
subsection {* Map function for filters *}
definition filtermap :: "('a \<Rightarrow> 'b) \<Rightarrow> 'a filter \<Rightarrow> 'b filter"
where "filtermap f F = Abs_filter (\<lambda>P. eventually (\<lambda>x. P (f x)) F)"
lemma eventually_filtermap:
"eventually P (filtermap f F) = eventually (\<lambda>x. P (f x)) F"
unfolding filtermap_def
apply (rule eventually_Abs_filter)
apply (rule is_filter.intro)
apply (auto elim!: eventually_rev_mp)
done
lemma filtermap_ident: "filtermap (\<lambda>x. x) F = F"
by (simp add: filter_eq_iff eventually_filtermap)
lemma filtermap_filtermap:
"filtermap f (filtermap g F) = filtermap (\<lambda>x. f (g x)) F"
by (simp add: filter_eq_iff eventually_filtermap)
lemma filtermap_mono: "F \<le> F' \<Longrightarrow> filtermap f F \<le> filtermap f F'"
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 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_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
then have "\<forall>n\<ge>max i j. P n \<and> Q n" by simp
then show "\<exists>k. \<forall>n\<ge>k. P n \<and> Q n" ..
qed auto
lemma sequentially_bot [simp, intro]: "sequentially \<noteq> bot"
unfolding filter_eq_iff eventually_sequentially by auto
lemmas trivial_limit_sequentially = sequentially_bot
lemma eventually_False_sequentially [simp]:
"\<not> eventually (\<lambda>n. False) sequentially"
by (simp add: eventually_False)
lemma le_sequentially:
"F \<le> sequentially \<longleftrightarrow> (\<forall>N. eventually (\<lambda>n. N \<le> n) F)"
unfolding le_filter_def eventually_sequentially
by (safe, fast, drule_tac x=N in spec, auto elim: eventually_rev_mp)
subsection {* Standard filters *}
definition within :: "'a filter \<Rightarrow> 'a set \<Rightarrow> 'a filter" (infixr "within" 70)
where "F within S = Abs_filter (\<lambda>P. eventually (\<lambda>x. x \<in> S \<longrightarrow> P x) F)"
definition (in topological_space) nhds :: "'a \<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 (in topological_space) at :: "'a \<Rightarrow> 'a filter"
where "at a = nhds a within - {a}"
lemma eventually_within:
"eventually P (F within S) = eventually (\<lambda>x. x \<in> S \<longrightarrow> P x) F"
unfolding within_def
by (rule eventually_Abs_filter, rule is_filter.intro)
(auto elim!: eventually_rev_mp)
lemma within_UNIV: "F within UNIV = F"
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_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
fix P Q
assume "\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x)"
and "\<exists>T. open T \<and> a \<in> T \<and> (\<forall>x\<in>T. Q x)"
then obtain S T where
"open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x)"
"open T \<and> a \<in> T \<and> (\<forall>x\<in>T. Q x)" by auto
hence "open (S \<inter> T) \<and> a \<in> S \<inter> T \<and> (\<forall>x\<in>(S \<inter> T). P x \<and> Q x)"
by (simp add: open_Int)
thus "\<exists>S. open S \<and> a \<in> S \<and> (\<forall>x\<in>S. P x \<and> Q x)" by - rule
qed auto
lemma eventually_nhds_metric:
"eventually P (nhds a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. dist x a < d \<longrightarrow> P x)"
unfolding eventually_nhds open_dist
apply safe
apply fast
apply (rule_tac x="{x. dist x a < d}" in exI, simp)
apply clarsimp
apply (rule_tac x="d - dist x a" in exI, clarsimp)
apply (simp only: less_diff_eq)
apply (erule le_less_trans [OF dist_triangle])
done
lemma nhds_neq_bot [simp]: "nhds a \<noteq> bot"
unfolding trivial_limit_def eventually_nhds 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 eventually_within eventually_nhds by simp
lemma eventually_at:
fixes a :: "'a::metric_space"
shows "eventually P (at a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. x \<noteq> a \<and> dist x a < d \<longrightarrow> P x)"
unfolding at_def eventually_within eventually_nhds_metric by auto
lemma at_eq_bot_iff: "at a = bot \<longleftrightarrow> open {a}"
unfolding trivial_limit_def eventually_at_topological
by (safe, case_tac "S = {a}", simp, fast, fast)
lemma at_neq_bot [simp]: "at (a::'a::perfect_space) \<noteq> bot"
by (simp add: at_eq_bot_iff not_open_singleton)
subsection {* Boundedness *}
definition Bfun :: "('a \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a filter \<Rightarrow> bool"
where "Bfun f F = (\<exists>K>0. eventually (\<lambda>x. norm (f x) \<le> K) F)"
lemma BfunI:
assumes K: "eventually (\<lambda>x. norm (f x) \<le> K) F" shows "Bfun f F"
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) F"
using K by (rule eventually_elim1, simp)
qed
lemma BfunE:
assumes "Bfun f F"
obtains B where "0 < B" and "eventually (\<lambda>x. norm (f x) \<le> B) F"
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 "Zfun f F = (\<forall>r>0. eventually (\<lambda>x. norm (f x) < r) F)"
lemma ZfunI:
"(\<And>r. 0 < r \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) F) \<Longrightarrow> Zfun f F"
unfolding Zfun_def by simp
lemma ZfunD:
"\<lbrakk>Zfun f F; 0 < r\<rbrakk> \<Longrightarrow> eventually (\<lambda>x. norm (f x) < r) F"
unfolding Zfun_def by simp
lemma Zfun_ssubst:
"eventually (\<lambda>x. f x = g x) F \<Longrightarrow> Zfun g F \<Longrightarrow> Zfun f F"
unfolding Zfun_def by (auto elim!: eventually_rev_mp)
lemma Zfun_zero: "Zfun (\<lambda>x. 0) F"
unfolding Zfun_def by simp
lemma Zfun_norm_iff: "Zfun (\<lambda>x. norm (f x)) F = Zfun (\<lambda>x. f x) F"
unfolding Zfun_def by simp
lemma Zfun_imp_Zfun:
assumes f: "Zfun f F"
assumes g: "eventually (\<lambda>x. norm (g x) \<le> norm (f x) * K) F"
shows "Zfun (\<lambda>x. g x) F"
proof (cases)
assume K: "0 < K"
show ?thesis
proof (rule ZfunI)
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) F"
using ZfunD [OF f] by fast
with g show "eventually (\<lambda>x. norm (g x) < r) F"
proof (rule eventually_elim2)
fix x
assume *: "norm (g x) \<le> norm (f x) * K"
assume "norm (f x) < r / K"
hence "norm (f x) * K < r"
by (simp add: pos_less_divide_eq K)
thus "norm (g x) < r"
by (simp add: order_le_less_trans [OF *])
qed
qed
next
assume "\<not> 0 < K"
hence K: "K \<le> 0" by (simp only: not_less)
show ?thesis
proof (rule ZfunI)
fix r :: real
assume "0 < r"
from g show "eventually (\<lambda>x. norm (g x) < r) F"
proof (rule eventually_elim1)
fix x
assume "norm (g x) \<le> norm (f x) * K"
also have "\<dots> \<le> norm (f x) * 0"
using K norm_ge_zero by (rule mult_left_mono)
finally show "norm (g x) < r"
using `0 < r` by simp
qed
qed
qed
lemma Zfun_le: "\<lbrakk>Zfun g F; \<forall>x. norm (f x) \<le> norm (g x)\<rbrakk> \<Longrightarrow> Zfun f F"
by (erule_tac K="1" in Zfun_imp_Zfun, simp)
lemma Zfun_add:
assumes f: "Zfun f F" and g: "Zfun g F"
shows "Zfun (\<lambda>x. f x + g x) F"
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) F"
using f r by (rule ZfunD)
moreover
have "eventually (\<lambda>x. norm (g x) < r/2) F"
using g r by (rule ZfunD)
ultimately
show "eventually (\<lambda>x. norm (f x + g x) < r) F"
proof (rule eventually_elim2)
fix x
assume *: "norm (f x) < r/2" "norm (g x) < r/2"
have "norm (f x + g x) \<le> norm (f x) + norm (g x)"
by (rule norm_triangle_ineq)
also have "\<dots> < r/2 + r/2"
using * by (rule add_strict_mono)
finally show "norm (f x + g x) < r"
by simp
qed
qed
lemma Zfun_minus: "Zfun f F \<Longrightarrow> Zfun (\<lambda>x. - f x) F"
unfolding Zfun_def by simp
lemma Zfun_diff: "\<lbrakk>Zfun f F; Zfun g F\<rbrakk> \<Longrightarrow> Zfun (\<lambda>x. f x - g x) F"
by (simp only: diff_minus Zfun_add Zfun_minus)
lemma (in bounded_linear) Zfun:
assumes g: "Zfun g F"
shows "Zfun (\<lambda>x. f (g x)) F"
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) F"
by simp
with g show ?thesis
by (rule Zfun_imp_Zfun)
qed
lemma (in bounded_bilinear) Zfun:
assumes f: "Zfun f F"
assumes g: "Zfun g F"
shows "Zfun (\<lambda>x. f x ** g x) F"
proof (rule ZfunI)
fix r::real assume r: "0 < r"
obtain K where K: "0 < K"
and norm_le: "\<And>x y. norm (x ** y) \<le> norm x * norm y * K"
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) F"
using f r by (rule ZfunD)
moreover
have "eventually (\<lambda>x. norm (g x) < inverse K) F"
using g K' by (rule ZfunD)
ultimately
show "eventually (\<lambda>x. norm (f x ** g x) < r) F"
proof (rule eventually_elim2)
fix x
assume *: "norm (f x) < r" "norm (g x) < inverse K"
have "norm (f x ** g x) \<le> norm (f x) * norm (g x) * K"
by (rule norm_le)
also have "norm (f x) * norm (g x) * K < r * inverse K * K"
by (intro mult_strict_right_mono mult_strict_mono' norm_ge_zero * K)
also from K have "r * inverse K * K = r"
by simp
finally show "norm (f x ** g x) < r" .
qed
qed
lemma (in bounded_bilinear) Zfun_left:
"Zfun f F \<Longrightarrow> Zfun (\<lambda>x. f x ** a) F"
by (rule bounded_linear_left [THEN bounded_linear.Zfun])
lemma (in bounded_bilinear) Zfun_right:
"Zfun f F \<Longrightarrow> Zfun (\<lambda>x. a ** f x) F"
by (rule bounded_linear_right [THEN bounded_linear.Zfun])
lemmas Zfun_mult = bounded_bilinear.Zfun [OF bounded_bilinear_mult]
lemmas Zfun_mult_right = bounded_bilinear.Zfun_right [OF bounded_bilinear_mult]
lemmas Zfun_mult_left = bounded_bilinear.Zfun_left [OF bounded_bilinear_mult]
subsection {* Limits *}
definition (in topological_space)
tendsto :: "('b \<Rightarrow> 'a) \<Rightarrow> 'a \<Rightarrow> 'b filter \<Rightarrow> bool" (infixr "--->" 55) where
"(f ---> l) F \<longleftrightarrow> (\<forall>S. open S \<longrightarrow> l \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) F)"
ML {*
structure Tendsto_Intros = Named_Thms
(
val name = "tendsto_intros"
val description = "introduction rules for tendsto"
)
*}
setup Tendsto_Intros.setup
lemma tendsto_mono: "F \<le> F' \<Longrightarrow> (f ---> l) F' \<Longrightarrow> (f ---> l) F"
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) F)
\<Longrightarrow> (f ---> l) F"
unfolding tendsto_def by auto
lemma topological_tendstoD:
"(f ---> l) F \<Longrightarrow> open S \<Longrightarrow> l \<in> S \<Longrightarrow> eventually (\<lambda>x. f x \<in> S) F"
unfolding tendsto_def by auto
lemma tendstoI:
assumes "\<And>e. 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) F"
shows "(f ---> l) F"
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) F \<Longrightarrow> 0 < e \<Longrightarrow> eventually (\<lambda>x. dist (f x) l < e) F"
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) F \<longleftrightarrow> (\<forall>e>0. eventually (\<lambda>x. dist (f x) l < e) F)"
using tendstoI tendstoD by fast
lemma tendsto_Zfun_iff: "(f ---> a) F = Zfun (\<lambda>x. f x - a) F"
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
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
lemma tendsto_const [tendsto_intros]: "((\<lambda>x. k) ---> k) F"
by (simp add: tendsto_def)
lemma tendsto_unique:
fixes f :: "'a \<Rightarrow> 'b::t2_space"
assumes "\<not> trivial_limit F" and "(f ---> a) F" and "(f ---> b) F"
shows "a = b"
proof (rule ccontr)
assume "a \<noteq> b"
obtain U V where "open U" "open V" "a \<in> U" "b \<in> V" "U \<inter> V = {}"
using hausdorff [OF `a \<noteq> b`] by fast
have "eventually (\<lambda>x. f x \<in> U) F"
using `(f ---> a) F` `open U` `a \<in> U` by (rule topological_tendstoD)
moreover
have "eventually (\<lambda>x. f x \<in> V) F"
using `(f ---> b) F` `open V` `b \<in> V` by (rule topological_tendstoD)
ultimately
have "eventually (\<lambda>x. False) F"
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 F` show "False"
by (simp add: trivial_limit_def)
qed
lemma tendsto_const_iff:
fixes a b :: "'a::t2_space"
assumes "\<not> trivial_limit F" shows "((\<lambda>x. a) ---> b) F \<longleftrightarrow> a = b"
by (safe intro!: tendsto_const tendsto_unique [OF assms tendsto_const])
lemma tendsto_compose:
assumes g: "(g ---> g l) (at l)"
assumes f: "(f ---> l) F"
shows "((\<lambda>x. g (f x)) ---> g l) F"
proof (rule topological_tendstoI)
fix B assume B: "open B" "g l \<in> B"
obtain A where A: "open A" "l \<in> A"
and gB: "\<forall>y. y \<in> A \<longrightarrow> g y \<in> B"
using topological_tendstoD [OF g B] B(2)
unfolding eventually_at_topological by fast
hence "\<forall>x. f x \<in> A \<longrightarrow> g (f x) \<in> B" by simp
from this topological_tendstoD [OF f A]
show "eventually (\<lambda>x. g (f x) \<in> B) F"
by (rule eventually_mono)
qed
lemma tendsto_compose_eventually:
assumes g: "(g ---> m) (at l)"
assumes f: "(f ---> l) F"
assumes inj: "eventually (\<lambda>x. f x \<noteq> l) F"
shows "((\<lambda>x. g (f x)) ---> m) F"
proof (rule topological_tendstoI)
fix B assume B: "open B" "m \<in> B"
obtain A where A: "open A" "l \<in> A"
and gB: "\<And>y. y \<in> A \<Longrightarrow> y \<noteq> l \<Longrightarrow> g y \<in> B"
using topological_tendstoD [OF g B]
unfolding eventually_at_topological by fast
show "eventually (\<lambda>x. g (f x) \<in> B) F"
using topological_tendstoD [OF f A] inj
by (rule eventually_elim2) (simp add: gB)
qed
lemma metric_tendsto_imp_tendsto:
assumes f: "(f ---> a) F"
assumes le: "eventually (\<lambda>x. dist (g x) b \<le> dist (f x) a) F"
shows "(g ---> b) F"
proof (rule tendstoI)
fix e :: real assume "0 < e"
with f have "eventually (\<lambda>x. dist (f x) a < e) F" by (rule tendstoD)
with le show "eventually (\<lambda>x. dist (g x) b < e) F"
using le_less_trans by (rule eventually_elim2)
qed
subsubsection {* Distance and norms *}
lemma tendsto_dist [tendsto_intros]:
assumes f: "(f ---> l) F" and g: "(g ---> m) F"
shows "((\<lambda>x. dist (f x) (g x)) ---> dist l m) F"
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) F"
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"
unfolding dist_real_def
using dist_triangle2 [of "f x" "g x" "l"]
using dist_triangle2 [of "g x" "l" "m"]
using dist_triangle3 [of "l" "m" "f x"]
using dist_triangle [of "f x" "m" "g x"]
by arith
qed
qed
lemma norm_conv_dist: "norm x = dist x 0"
unfolding dist_norm by simp
lemma tendsto_norm [tendsto_intros]:
"(f ---> a) F \<Longrightarrow> ((\<lambda>x. norm (f x)) ---> norm a) F"
unfolding norm_conv_dist by (intro tendsto_intros)
lemma tendsto_norm_zero:
"(f ---> 0) F \<Longrightarrow> ((\<lambda>x. norm (f x)) ---> 0) F"
by (drule tendsto_norm, simp)
lemma tendsto_norm_zero_cancel:
"((\<lambda>x. norm (f x)) ---> 0) F \<Longrightarrow> (f ---> 0) F"
unfolding tendsto_iff dist_norm by simp
lemma tendsto_norm_zero_iff:
"((\<lambda>x. norm (f x)) ---> 0) F \<longleftrightarrow> (f ---> 0) F"
unfolding tendsto_iff dist_norm by simp
lemma tendsto_rabs [tendsto_intros]:
"(f ---> (l::real)) F \<Longrightarrow> ((\<lambda>x. \<bar>f x\<bar>) ---> \<bar>l\<bar>) F"
by (fold real_norm_def, rule tendsto_norm)
lemma tendsto_rabs_zero:
"(f ---> (0::real)) F \<Longrightarrow> ((\<lambda>x. \<bar>f x\<bar>) ---> 0) F"
by (fold real_norm_def, rule tendsto_norm_zero)
lemma tendsto_rabs_zero_cancel:
"((\<lambda>x. \<bar>f x\<bar>) ---> (0::real)) F \<Longrightarrow> (f ---> 0) F"
by (fold real_norm_def, rule tendsto_norm_zero_cancel)
lemma tendsto_rabs_zero_iff:
"((\<lambda>x. \<bar>f x\<bar>) ---> (0::real)) F \<longleftrightarrow> (f ---> 0) F"
by (fold real_norm_def, rule tendsto_norm_zero_iff)
subsubsection {* Addition and subtraction *}
lemma tendsto_add [tendsto_intros]:
fixes a b :: "'a::real_normed_vector"
shows "\<lbrakk>(f ---> a) F; (g ---> b) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x + g x) ---> a + b) F"
by (simp only: tendsto_Zfun_iff add_diff_add Zfun_add)
lemma tendsto_add_zero:
fixes f g :: "'a::type \<Rightarrow> 'b::real_normed_vector"
shows "\<lbrakk>(f ---> 0) F; (g ---> 0) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x + g x) ---> 0) F"
by (drule (1) tendsto_add, simp)
lemma tendsto_minus [tendsto_intros]:
fixes a :: "'a::real_normed_vector"
shows "(f ---> a) F \<Longrightarrow> ((\<lambda>x. - f x) ---> - a) F"
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) F \<Longrightarrow> (f ---> a) F"
by (drule tendsto_minus, simp)
lemma tendsto_diff [tendsto_intros]:
fixes a b :: "'a::real_normed_vector"
shows "\<lbrakk>(f ---> a) F; (g ---> b) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x - g x) ---> a - b) F"
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) F"
shows "((\<lambda>x. \<Sum>i\<in>S. f i x) ---> (\<Sum>i\<in>S. a i)) F"
proof (cases "finite S")
assume "finite S" thus ?thesis using assms
by (induct, simp add: tendsto_const, simp add: tendsto_add)
next
assume "\<not> finite S" thus ?thesis
by (simp add: tendsto_const)
qed
subsubsection {* Linear operators and multiplication *}
lemma (in bounded_linear) tendsto:
"(g ---> a) F \<Longrightarrow> ((\<lambda>x. f (g x)) ---> f a) F"
by (simp only: tendsto_Zfun_iff diff [symmetric] Zfun)
lemma (in bounded_linear) tendsto_zero:
"(g ---> 0) F \<Longrightarrow> ((\<lambda>x. f (g x)) ---> 0) F"
by (drule tendsto, simp only: zero)
lemma (in bounded_bilinear) tendsto:
"\<lbrakk>(f ---> a) F; (g ---> b) F\<rbrakk> \<Longrightarrow> ((\<lambda>x. f x ** g x) ---> a ** b) F"
by (simp only: tendsto_Zfun_iff prod_diff_prod
Zfun_add Zfun Zfun_left Zfun_right)
lemma (in bounded_bilinear) tendsto_zero:
assumes f: "(f ---> 0) F"
assumes g: "(g ---> 0) F"
shows "((\<lambda>x. f x ** g x) ---> 0) F"
using tendsto [OF f g] by (simp add: zero_left)
lemma (in bounded_bilinear) tendsto_left_zero:
"(f ---> 0) F \<Longrightarrow> ((\<lambda>x. f x ** c) ---> 0) F"
by (rule bounded_linear.tendsto_zero [OF bounded_linear_left])
lemma (in bounded_bilinear) tendsto_right_zero:
"(f ---> 0) F \<Longrightarrow> ((\<lambda>x. c ** f x) ---> 0) F"
by (rule bounded_linear.tendsto_zero [OF bounded_linear_right])
lemmas tendsto_of_real [tendsto_intros] =
bounded_linear.tendsto [OF bounded_linear_of_real]
lemmas tendsto_scaleR [tendsto_intros] =
bounded_bilinear.tendsto [OF bounded_bilinear_scaleR]
lemmas tendsto_mult [tendsto_intros] =
bounded_bilinear.tendsto [OF bounded_bilinear_mult]
lemmas tendsto_mult_zero =
bounded_bilinear.tendsto_zero [OF bounded_bilinear_mult]
lemmas tendsto_mult_left_zero =
bounded_bilinear.tendsto_left_zero [OF bounded_bilinear_mult]
lemmas tendsto_mult_right_zero =
bounded_bilinear.tendsto_right_zero [OF bounded_bilinear_mult]
lemma tendsto_power [tendsto_intros]:
fixes f :: "'a \<Rightarrow> 'b::{power,real_normed_algebra}"
shows "(f ---> a) F \<Longrightarrow> ((\<lambda>x. f x ^ n) ---> a ^ n) F"
by (induct n) (simp_all add: tendsto_const tendsto_mult)
lemma tendsto_setprod [tendsto_intros]:
fixes f :: "'a \<Rightarrow> 'b \<Rightarrow> 'c::{real_normed_algebra,comm_ring_1}"
assumes "\<And>i. i \<in> S \<Longrightarrow> (f i ---> L i) F"
shows "((\<lambda>x. \<Prod>i\<in>S. f i x) ---> (\<Prod>i\<in>S. L i)) F"
proof (cases "finite S")
assume "finite S" thus ?thesis using assms
by (induct, simp add: tendsto_const, simp add: tendsto_mult)
next
assume "\<not> finite S" thus ?thesis
by (simp add: tendsto_const)
qed
subsubsection {* Inverse and division *}
lemma (in bounded_bilinear) Zfun_prod_Bfun:
assumes f: "Zfun f F"
assumes g: "Bfun g F"
shows "Zfun (\<lambda>x. f x ** g x) F"
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) F"
using g by (rule BfunE)
have "eventually (\<lambda>x. norm (f x ** g x) \<le> norm (f x) * (B * K)) F"
using norm_g proof (rule eventually_elim1)
fix x
assume *: "norm (g x) \<le> B"
have "norm (f x ** g x) \<le> norm (f x) * norm (g x) * K"
by (rule norm_le)
also have "\<dots> \<le> norm (f x) * B * K"
by (intro mult_mono' order_refl norm_g norm_ge_zero
mult_nonneg_nonneg K *)
also have "\<dots> = norm (f x) * (B * K)"
by (rule mult_assoc)
finally show "norm (f x ** g x) \<le> norm (f x) * (B * K)" .
qed
with f show ?thesis
by (rule Zfun_imp_Zfun)
qed
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
lemma (in bounded_bilinear) Bfun_prod_Zfun:
assumes f: "Bfun f F"
assumes g: "Zfun g F"
shows "Zfun (\<lambda>x. f x ** g x) F"
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
lemma Bfun_inverse:
fixes a :: "'a::real_normed_div_algebra"
assumes f: "(f ---> a) F"
assumes a: "a \<noteq> 0"
shows "Bfun (\<lambda>x. inverse (f x)) F"
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) F"
using tendstoD [OF f r1] by fast
hence "eventually (\<lambda>x. norm (inverse (f x)) \<le> inverse (norm a - r)) F"
proof (rule eventually_elim1)
fix x
assume "dist (f x) a < r"
hence 1: "norm (f x - a) < r"
by (simp add: dist_norm)
hence 2: "f x \<noteq> 0" using r2 by auto
hence "norm (inverse (f x)) = inverse (norm (f x))"
by (rule nonzero_norm_inverse)
also have "\<dots> \<le> inverse (norm a - r)"
proof (rule le_imp_inverse_le)
show "0 < norm a - r" using r2 by simp
next
have "norm a - norm (f x) \<le> norm (a - f x)"
by (rule norm_triangle_ineq2)
also have "\<dots> = norm (f x - a)"
by (rule norm_minus_commute)
also have "\<dots> < r" using 1 .
finally show "norm a - r \<le> norm (f x)" by simp
qed
finally show "norm (inverse (f x)) \<le> inverse (norm a - r)" .
qed
thus ?thesis by (rule BfunI)
qed
lemma tendsto_inverse_lemma:
fixes a :: "'a::real_normed_div_algebra"
shows "\<lbrakk>(f ---> a) F; a \<noteq> 0; eventually (\<lambda>x. f x \<noteq> 0) F\<rbrakk>
\<Longrightarrow> ((\<lambda>x. inverse (f x)) ---> inverse a) F"
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 bounded_bilinear.Bfun_prod_Zfun [OF bounded_bilinear_mult])
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) F"
assumes a: "a \<noteq> 0"
shows "((\<lambda>x. inverse (f x)) ---> inverse a) F"
proof -
from a have "0 < norm a" by simp
with f have "eventually (\<lambda>x. dist (f x) a < norm a) F"
by (rule tendstoD)
then have "eventually (\<lambda>x. f x \<noteq> 0) F"
unfolding dist_norm by (auto elim!: eventually_elim1)
with f a show ?thesis
by (rule tendsto_inverse_lemma)
qed
lemma tendsto_divide [tendsto_intros]:
fixes a b :: "'a::real_normed_field"
shows "\<lbrakk>(f ---> a) F; (g ---> b) F; b \<noteq> 0\<rbrakk>
\<Longrightarrow> ((\<lambda>x. f x / g x) ---> a / b) F"
by (simp add: tendsto_mult tendsto_inverse divide_inverse)
lemma tendsto_sgn [tendsto_intros]:
fixes l :: "'a::real_normed_vector"
shows "\<lbrakk>(f ---> l) F; l \<noteq> 0\<rbrakk> \<Longrightarrow> ((\<lambda>x. sgn (f x)) ---> sgn l) F"
unfolding sgn_div_norm by (simp add: tendsto_intros)
end