new material on path_component_sets, inside, outside, etc. And more default simprules
(* Title: HOL/Multivariate_Analysis/Uniform_Limit.thy
Author: Christoph Traut, TU München
Author: Fabian Immler, TU München
*)
section \<open>Uniform Limit and Uniform Convergence\<close>
theory Uniform_Limit
imports Topology_Euclidean_Space
begin
definition uniformly_on :: "'a set \<Rightarrow> ('a \<Rightarrow> 'b::metric_space) \<Rightarrow> ('a \<Rightarrow> 'b) filter"
where "uniformly_on S l = (INF e:{0 <..}. principal {f. \<forall>x\<in>S. dist (f x) (l x) < e})"
abbreviation
"uniform_limit S f l \<equiv> filterlim f (uniformly_on S l)"
lemma uniform_limit_iff:
"uniform_limit S f l F \<longleftrightarrow> (\<forall>e>0. \<forall>\<^sub>F n in F. \<forall>x\<in>S. dist (f n x) (l x) < e)"
unfolding filterlim_iff uniformly_on_def
by (subst eventually_INF_base)
(fastforce
simp: eventually_principal uniformly_on_def
intro: bexI[where x="min a b" for a b]
elim: eventually_elim1)+
lemma uniform_limitD:
"uniform_limit S f l F \<Longrightarrow> e > 0 \<Longrightarrow> \<forall>\<^sub>F n in F. \<forall>x\<in>S. dist (f n x) (l x) < e"
by (simp add: uniform_limit_iff)
lemma uniform_limitI:
"(\<And>e. e > 0 \<Longrightarrow> \<forall>\<^sub>F n in F. \<forall>x\<in>S. dist (f n x) (l x) < e) \<Longrightarrow> uniform_limit S f l F"
by (simp add: uniform_limit_iff)
lemma uniform_limit_sequentially_iff:
"uniform_limit S f l sequentially \<longleftrightarrow> (\<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<forall>x \<in> S. dist (f n x) (l x) < e)"
unfolding uniform_limit_iff eventually_sequentially ..
lemma uniform_limit_at_iff:
"uniform_limit S f l (at x) \<longleftrightarrow>
(\<forall>e>0. \<exists>d>0. \<forall>z. 0 < dist z x \<and> dist z x < d \<longrightarrow> (\<forall>x\<in>S. dist (f z x) (l x) < e))"
unfolding uniform_limit_iff eventually_at2 ..
lemma uniform_limit_at_le_iff:
"uniform_limit S f l (at x) \<longleftrightarrow>
(\<forall>e>0. \<exists>d>0. \<forall>z. 0 < dist z x \<and> dist z x < d \<longrightarrow> (\<forall>x\<in>S. dist (f z x) (l x) \<le> e))"
unfolding uniform_limit_iff eventually_at2
by (fastforce dest: spec[where x = "e / 2" for e])
lemma swap_uniform_limit:
assumes f: "\<forall>\<^sub>F n in F. (f n ---> g n) (at x within S)"
assumes g: "(g ---> l) F"
assumes uc: "uniform_limit S f h F"
assumes "\<not>trivial_limit F"
shows "(h ---> l) (at x within S)"
proof (rule tendstoI)
fix e :: real
def e' \<equiv> "e/3"
assume "0 < e"
then have "0 < e'" by (simp add: e'_def)
from uniform_limitD[OF uc \<open>0 < e'\<close>]
have "\<forall>\<^sub>F n in F. \<forall>x\<in>S. dist (h x) (f n x) < e'"
by (simp add: dist_commute)
moreover
from f
have "\<forall>\<^sub>F n in F. \<forall>\<^sub>F x in at x within S. dist (g n) (f n x) < e'"
by eventually_elim (auto dest!: tendstoD[OF _ \<open>0 < e'\<close>] simp: dist_commute)
moreover
from tendstoD[OF g \<open>0 < e'\<close>] have "\<forall>\<^sub>F x in F. dist l (g x) < e'"
by (simp add: dist_commute)
ultimately
have "\<forall>\<^sub>F _ in F. \<forall>\<^sub>F x in at x within S. dist (h x) l < e"
proof eventually_elim
case (elim n)
note fh = elim(1)
note gl = elim(3)
have "\<forall>\<^sub>F x in at x within S. x \<in> S"
by (auto simp: eventually_at_filter)
with elim(2)
show ?case
proof eventually_elim
case (elim x)
from fh[rule_format, OF \<open>x \<in> S\<close>] elim(1)
have "dist (h x) (g n) < e' + e'"
by (rule dist_triangle_lt[OF add_strict_mono])
from dist_triangle_lt[OF add_strict_mono, OF this gl]
show ?case by (simp add: e'_def)
qed
qed
thus "\<forall>\<^sub>F x in at x within S. dist (h x) l < e"
using eventually_happens by (metis \<open>\<not>trivial_limit F\<close>)
qed
lemma
tendsto_uniform_limitI:
assumes "uniform_limit S f l F"
assumes "x \<in> S"
shows "((\<lambda>y. f y x) ---> l x) F"
using assms
by (auto intro!: tendstoI simp: eventually_elim1 dest!: uniform_limitD)
lemma uniform_limit_theorem:
assumes c: "\<forall>\<^sub>F n in F. continuous_on A (f n)"
assumes ul: "uniform_limit A f l F"
assumes "\<not> trivial_limit F"
shows "continuous_on A l"
unfolding continuous_on_def
proof safe
fix x assume "x \<in> A"
then have "\<forall>\<^sub>F n in F. (f n ---> f n x) (at x within A)" "((\<lambda>n. f n x) ---> l x) F"
using c ul
by (auto simp: continuous_on_def eventually_elim1 tendsto_uniform_limitI)
then show "(l ---> l x) (at x within A)"
by (rule swap_uniform_limit) fact+
qed
lemma weierstrass_m_test:
fixes f :: "_ \<Rightarrow> _ \<Rightarrow> _ :: banach"
assumes "\<And>n x. x \<in> A \<Longrightarrow> norm (f n x) \<le> M n"
assumes "summable M"
shows "uniform_limit A (\<lambda>n x. \<Sum>i<n. f i x) (\<lambda>x. suminf (\<lambda>i. f i x)) sequentially"
proof (rule uniform_limitI)
fix e :: real
assume "0 < e"
from suminf_exist_split[OF \<open>0 < e\<close> \<open>summable M\<close>]
have "\<forall>\<^sub>F k in sequentially. norm (\<Sum>i. M (i + k)) < e"
by (auto simp: eventually_sequentially)
thus "\<forall>\<^sub>F n in sequentially. \<forall>x\<in>A. dist (\<Sum>i<n. f i x) (\<Sum>i. f i x) < e"
proof eventually_elim
case (elim k)
show ?case
proof safe
fix x assume "x \<in> A"
have "\<exists>N. \<forall>n\<ge>N. norm (f n x) \<le> M n"
using assms(1)[OF \<open>x \<in> A\<close>] by simp
hence summable_norm_f: "summable (\<lambda>n. norm (f n x))"
by(rule summable_norm_comparison_test[OF _ \<open>summable M\<close>])
have summable_f: "summable (\<lambda>n. f n x)"
using summable_norm_cancel[OF summable_norm_f] .
have summable_norm_f_plus_k: "summable (\<lambda>i. norm (f (i + k) x))"
using summable_ignore_initial_segment[OF summable_norm_f]
by auto
have summable_M_plus_k: "summable (\<lambda>i. M (i + k))"
using summable_ignore_initial_segment[OF \<open>summable M\<close>]
by auto
have "dist (\<Sum>i<k. f i x) (\<Sum>i. f i x) = norm ((\<Sum>i. f i x) - (\<Sum>i<k. f i x))"
using dist_norm dist_commute by (subst dist_commute)
also have "... = norm (\<Sum>i. f (i + k) x)"
using suminf_minus_initial_segment[OF summable_f, where k=k] by simp
also have "... \<le> (\<Sum>i. norm (f (i + k) x))"
using summable_norm[OF summable_norm_f_plus_k] .
also have "... \<le> (\<Sum>i. M (i + k))"
by (rule suminf_le[OF _ summable_norm_f_plus_k summable_M_plus_k])
(simp add: assms(1)[OF \<open>x \<in> A\<close>])
finally show "dist (\<Sum>i<k. f i x) (\<Sum>i. f i x) < e"
using elim by auto
qed
qed
qed
lemma uniform_limit_eq_rhs: "uniform_limit X f l F \<Longrightarrow> l = m \<Longrightarrow> uniform_limit X f m F"
by simp
named_theorems uniform_limit_intros "introduction rules for uniform_limit"
setup \<open>
Global_Theory.add_thms_dynamic (@{binding uniform_limit_eq_intros},
fn context =>
Named_Theorems.get (Context.proof_of context) @{named_theorems uniform_limit_intros}
|> map_filter (try (fn thm => @{thm uniform_limit_eq_rhs} OF [thm])))
\<close>
lemma (in bounded_linear) uniform_limit[uniform_limit_intros]:
assumes "uniform_limit X g l F"
shows "uniform_limit X (\<lambda>a b. f (g a b)) (\<lambda>a. f (l a)) F"
proof (rule uniform_limitI)
fix e::real
from pos_bounded obtain K
where K: "\<And>x y. dist (f x) (f y) \<le> K * dist x y" "K > 0"
by (auto simp: ac_simps dist_norm diff[symmetric])
assume "0 < e" with \<open>K > 0\<close> have "e / K > 0" by simp
from uniform_limitD[OF assms this]
show "\<forall>\<^sub>F n in F. \<forall>x\<in>X. dist (f (g n x)) (f (l x)) < e"
by eventually_elim (metis le_less_trans mult.commute pos_less_divide_eq K)
qed
lemmas bounded_linear_uniform_limit_intros[uniform_limit_intros] =
bounded_linear.uniform_limit[OF bounded_linear_Im]
bounded_linear.uniform_limit[OF bounded_linear_Re]
bounded_linear.uniform_limit[OF bounded_linear_cnj]
bounded_linear.uniform_limit[OF bounded_linear_fst]
bounded_linear.uniform_limit[OF bounded_linear_snd]
bounded_linear.uniform_limit[OF bounded_linear_zero]
bounded_linear.uniform_limit[OF bounded_linear_of_real]
bounded_linear.uniform_limit[OF bounded_linear_inner_left]
bounded_linear.uniform_limit[OF bounded_linear_inner_right]
bounded_linear.uniform_limit[OF bounded_linear_divide]
bounded_linear.uniform_limit[OF bounded_linear_scaleR_right]
bounded_linear.uniform_limit[OF bounded_linear_mult_left]
bounded_linear.uniform_limit[OF bounded_linear_mult_right]
bounded_linear.uniform_limit[OF bounded_linear_scaleR_left]
lemmas uniform_limit_uminus[uniform_limit_intros] =
bounded_linear.uniform_limit[OF bounded_linear_minus[OF bounded_linear_ident]]
lemma uniform_limit_add[uniform_limit_intros]:
fixes f g::"'a \<Rightarrow> 'b \<Rightarrow> 'c::real_normed_vector"
assumes "uniform_limit X f l F"
assumes "uniform_limit X g m F"
shows "uniform_limit X (\<lambda>a b. f a b + g a b) (\<lambda>a. l a + m a) F"
proof (rule uniform_limitI)
fix e::real
assume "0 < e"
hence "0 < e / 2" by simp
from
uniform_limitD[OF assms(1) this]
uniform_limitD[OF assms(2) this]
show "\<forall>\<^sub>F n in F. \<forall>x\<in>X. dist (f n x + g n x) (l x + m x) < e"
by eventually_elim (simp add: dist_triangle_add_half)
qed
lemma uniform_limit_minus[uniform_limit_intros]:
fixes f g::"'a \<Rightarrow> 'b \<Rightarrow> 'c::real_normed_vector"
assumes "uniform_limit X f l F"
assumes "uniform_limit X g m F"
shows "uniform_limit X (\<lambda>a b. f a b - g a b) (\<lambda>a. l a - m a) F"
unfolding diff_conv_add_uminus
by (rule uniform_limit_intros assms)+
lemma (in bounded_bilinear) bounded_uniform_limit[uniform_limit_intros]:
assumes "uniform_limit X f l F"
assumes "uniform_limit X g m F"
assumes "bounded (m ` X)"
assumes "bounded (l ` X)"
shows "uniform_limit X (\<lambda>a b. prod (f a b) (g a b)) (\<lambda>a. prod (l a) (m a)) F"
proof (rule uniform_limitI)
fix e::real
from pos_bounded obtain K where K:
"0 < K" "\<And>a b. norm (prod a b) \<le> norm a * norm b * K"
by auto
hence "sqrt (K*4) > 0" by simp
from assms obtain Km Kl
where Km: "Km > 0" "\<And>x. x \<in> X \<Longrightarrow> norm (m x) \<le> Km"
and Kl: "Kl > 0" "\<And>x. x \<in> X \<Longrightarrow> norm (l x) \<le> Kl"
by (auto simp: bounded_pos)
hence "K * Km * 4 > 0" "K * Kl * 4 > 0"
using \<open>K > 0\<close>
by simp_all
assume "0 < e"
hence "sqrt e > 0" by simp
from uniform_limitD[OF assms(1) divide_pos_pos[OF this \<open>sqrt (K*4) > 0\<close>]]
uniform_limitD[OF assms(2) divide_pos_pos[OF this \<open>sqrt (K*4) > 0\<close>]]
uniform_limitD[OF assms(1) divide_pos_pos[OF \<open>e > 0\<close> \<open>K * Km * 4 > 0\<close>]]
uniform_limitD[OF assms(2) divide_pos_pos[OF \<open>e > 0\<close> \<open>K * Kl * 4 > 0\<close>]]
show "\<forall>\<^sub>F n in F. \<forall>x\<in>X. dist (prod (f n x) (g n x)) (prod (l x) (m x)) < e"
proof eventually_elim
case (elim n)
show ?case
proof safe
fix x assume "x \<in> X"
have "dist (prod (f n x) (g n x)) (prod (l x) (m x)) \<le>
norm (prod (f n x - l x) (g n x - m x)) +
norm (prod (f n x - l x) (m x)) +
norm (prod (l x) (g n x - m x))"
by (auto simp: dist_norm prod_diff_prod intro: order_trans norm_triangle_ineq add_mono)
also note K(2)[of "f n x - l x" "g n x - m x"]
also from elim(1)[THEN bspec, OF \<open>_ \<in> X\<close>, unfolded dist_norm]
have "norm (f n x - l x) \<le> sqrt e / sqrt (K * 4)"
by simp
also from elim(2)[THEN bspec, OF \<open>_ \<in> X\<close>, unfolded dist_norm]
have "norm (g n x - m x) \<le> sqrt e / sqrt (K * 4)"
by simp
also have "sqrt e / sqrt (K * 4) * (sqrt e / sqrt (K * 4)) * K = e / 4"
using \<open>K > 0\<close> \<open>e > 0\<close> by auto
also note K(2)[of "f n x - l x" "m x"]
also note K(2)[of "l x" "g n x - m x"]
also from elim(3)[THEN bspec, OF \<open>_ \<in> X\<close>, unfolded dist_norm]
have "norm (f n x - l x) \<le> e / (K * Km * 4)"
by simp
also from elim(4)[THEN bspec, OF \<open>_ \<in> X\<close>, unfolded dist_norm]
have "norm (g n x - m x) \<le> e / (K * Kl * 4)"
by simp
also note Kl(2)[OF \<open>_ \<in> X\<close>]
also note Km(2)[OF \<open>_ \<in> X\<close>]
also have "e / (K * Km * 4) * Km * K = e / 4"
using \<open>K > 0\<close> \<open>Km > 0\<close> by simp
also have " Kl * (e / (K * Kl * 4)) * K = e / 4"
using \<open>K > 0\<close> \<open>Kl > 0\<close> by simp
also have "e / 4 + e / 4 + e / 4 < e" using \<open>e > 0\<close> by simp
finally show "dist (prod (f n x) (g n x)) (prod (l x) (m x)) < e"
using \<open>K > 0\<close> \<open>Kl > 0\<close> \<open>Km > 0\<close> \<open>e > 0\<close>
by (simp add: algebra_simps mult_right_mono divide_right_mono)
qed
qed
qed
lemmas bounded_bilinear_bounded_uniform_limit_intros[uniform_limit_intros] =
bounded_bilinear.bounded_uniform_limit[OF Inner_Product.bounded_bilinear_inner]
bounded_bilinear.bounded_uniform_limit[OF Real_Vector_Spaces.bounded_bilinear_mult]
bounded_bilinear.bounded_uniform_limit[OF Real_Vector_Spaces.bounded_bilinear_scaleR]
lemma metric_uniform_limit_imp_uniform_limit:
assumes f: "uniform_limit S f a F"
assumes le: "eventually (\<lambda>x. \<forall>y\<in>S. dist (g x y) (b y) \<le> dist (f x y) (a y)) F"
shows "uniform_limit S g b F"
proof (rule uniform_limitI)
fix e :: real assume "0 < e"
from uniform_limitD[OF f this] le
show "\<forall>\<^sub>F x in F. \<forall>y\<in>S. dist (g x y) (b y) < e"
by eventually_elim force
qed
lemma uniform_limit_null_comparison:
assumes "\<forall>\<^sub>F x in F. \<forall>a\<in>S. norm (f x a) \<le> g x a"
assumes "uniform_limit S g (\<lambda>_. 0) F"
shows "uniform_limit S f (\<lambda>_. 0) F"
using assms(2)
proof (rule metric_uniform_limit_imp_uniform_limit)
show "\<forall>\<^sub>F x in F. \<forall>y\<in>S. dist (f x y) 0 \<le> dist (g x y) 0"
using assms(1) by (rule eventually_elim1) (force simp add: dist_norm)
qed
lemma uniform_limit_on_union:
"uniform_limit I f g F \<Longrightarrow> uniform_limit J f g F \<Longrightarrow> uniform_limit (I \<union> J) f g F"
by (auto intro!: uniform_limitI dest!: uniform_limitD elim: eventually_elim2)
lemma uniform_limit_on_empty:
"uniform_limit {} f g F"
by (auto intro!: uniform_limitI)
lemma uniform_limit_on_UNION:
assumes "finite S"
assumes "\<And>s. s \<in> S \<Longrightarrow> uniform_limit (h s) f g F"
shows "uniform_limit (UNION S h) f g F"
using assms
by induct (auto intro: uniform_limit_on_empty uniform_limit_on_union)
lemma uniform_limit_on_Union:
assumes "finite I"
assumes "\<And>J. J \<in> I \<Longrightarrow> uniform_limit J f g F"
shows "uniform_limit (Union I) f g F"
by (metis SUP_identity_eq assms uniform_limit_on_UNION)
lemma uniform_limit_on_subset:
"uniform_limit J f g F \<Longrightarrow> I \<subseteq> J \<Longrightarrow> uniform_limit I f g F"
by (auto intro!: uniform_limitI dest!: uniform_limitD intro: eventually_rev_mono)
end