avoid context-sensitive simp rules whose context-free form (image_comp) is not simp by default
section \<open>Bounded Continuous Functions\<close>
theory Bounded_Continuous_Function
imports
Henstock_Kurzweil_Integration
begin
subsection \<open>Definition\<close>
definition%important "bcontfun = {f. continuous_on UNIV f \<and> bounded (range f)}"
typedef (overloaded) ('a, 'b) bcontfun ("(_ \<Rightarrow>\<^sub>C /_)" [22, 21] 21) =
"bcontfun::('a::topological_space \<Rightarrow> 'b::metric_space) set"
morphisms apply_bcontfun Bcontfun
by (auto intro: continuous_intros simp: bounded_def bcontfun_def)
declare [[coercion "apply_bcontfun :: ('a::topological_space \<Rightarrow>\<^sub>C'b::metric_space) \<Rightarrow> 'a \<Rightarrow> 'b"]]
setup_lifting type_definition_bcontfun
lemma continuous_on_apply_bcontfun[intro, simp]: "continuous_on T (apply_bcontfun x)"
and bounded_apply_bcontfun[intro, simp]: "bounded (range (apply_bcontfun x))"
using apply_bcontfun[of x]
by (auto simp: bcontfun_def intro: continuous_on_subset)
lemma bcontfun_eqI: "(\<And>x. apply_bcontfun f x = apply_bcontfun g x) \<Longrightarrow> f = g"
by transfer auto
lemma bcontfunE:
assumes "f \<in> bcontfun"
obtains g where "f = apply_bcontfun g"
by (blast intro: apply_bcontfun_cases assms )
lemma const_bcontfun: "(\<lambda>x. b) \<in> bcontfun"
by (auto simp: bcontfun_def continuous_on_const image_def)
lift_definition const_bcontfun::"'b::metric_space \<Rightarrow> ('a::topological_space \<Rightarrow>\<^sub>C 'b)" is "\<lambda>c _. c"
by (rule const_bcontfun)
(* TODO: Generalize to uniform spaces? *)
instantiation bcontfun :: (topological_space, metric_space) metric_space
begin
lift_definition%important dist_bcontfun :: "'a \<Rightarrow>\<^sub>C 'b \<Rightarrow> 'a \<Rightarrow>\<^sub>C 'b \<Rightarrow> real"
is "\<lambda>f g. (SUP x. dist (f x) (g x))" .
definition uniformity_bcontfun :: "('a \<Rightarrow>\<^sub>C 'b \<times> 'a \<Rightarrow>\<^sub>C 'b) filter"
where "uniformity_bcontfun = (INF e\<in>{0 <..}. principal {(x, y). dist x y < e})"
definition open_bcontfun :: "('a \<Rightarrow>\<^sub>C 'b) set \<Rightarrow> bool"
where "open_bcontfun S = (\<forall>x\<in>S. \<forall>\<^sub>F (x', y) in uniformity. x' = x \<longrightarrow> y \<in> S)"
lemma bounded_dist_le_SUP_dist:
"bounded (range f) \<Longrightarrow> bounded (range g) \<Longrightarrow> dist (f x) (g x) \<le> (SUP x. dist (f x) (g x))"
by (auto intro!: cSUP_upper bounded_imp_bdd_above bounded_dist_comp)
lemma dist_bounded:
fixes f g :: "'a \<Rightarrow>\<^sub>C 'b"
shows "dist (f x) (g x) \<le> dist f g"
by transfer (auto intro!: bounded_dist_le_SUP_dist simp: bcontfun_def)
lemma dist_bound:
fixes f g :: "'a \<Rightarrow>\<^sub>C 'b"
assumes "\<And>x. dist (f x) (g x) \<le> b"
shows "dist f g \<le> b"
using assms
by transfer (auto intro!: cSUP_least)
lemma dist_fun_lt_imp_dist_val_lt:
fixes f g :: "'a \<Rightarrow>\<^sub>C 'b"
assumes "dist f g < e"
shows "dist (f x) (g x) < e"
using dist_bounded assms by (rule le_less_trans)
instance
proof
fix f g h :: "'a \<Rightarrow>\<^sub>C 'b"
show "dist f g = 0 \<longleftrightarrow> f = g"
proof
have "\<And>x. dist (f x) (g x) \<le> dist f g"
by (rule dist_bounded)
also assume "dist f g = 0"
finally show "f = g"
by (auto simp: apply_bcontfun_inject[symmetric])
qed (auto simp: dist_bcontfun_def intro!: cSup_eq)
show "dist f g \<le> dist f h + dist g h"
proof (rule dist_bound)
fix x
have "dist (f x) (g x) \<le> dist (f x) (h x) + dist (g x) (h x)"
by (rule dist_triangle2)
also have "dist (f x) (h x) \<le> dist f h"
by (rule dist_bounded)
also have "dist (g x) (h x) \<le> dist g h"
by (rule dist_bounded)
finally show "dist (f x) (g x) \<le> dist f h + dist g h"
by simp
qed
qed (rule open_bcontfun_def uniformity_bcontfun_def)+
end
lift_definition PiC::"'a::topological_space set \<Rightarrow> ('a \<Rightarrow> 'b set) \<Rightarrow> ('a \<Rightarrow>\<^sub>C 'b::metric_space) set"
is "\<lambda>I X. Pi I X \<inter> bcontfun"
by auto
lemma mem_PiC_iff: "x \<in> PiC I X \<longleftrightarrow> apply_bcontfun x \<in> Pi I X"
by transfer simp
lemmas mem_PiCD = mem_PiC_iff[THEN iffD1]
and mem_PiCI = mem_PiC_iff[THEN iffD2]
lemma tendsto_bcontfun_uniform_limit:
fixes f::"'i \<Rightarrow> 'a::topological_space \<Rightarrow>\<^sub>C 'b::metric_space"
assumes "(f \<longlongrightarrow> l) F"
shows "uniform_limit UNIV f l F"
proof (rule uniform_limitI)
fix e::real assume "e > 0"
from tendstoD[OF assms this] have "\<forall>\<^sub>F x in F. dist (f x) l < e" .
then show "\<forall>\<^sub>F n in F. \<forall>x\<in>UNIV. dist ((f n) x) (l x) < e"
by eventually_elim (auto simp: dist_fun_lt_imp_dist_val_lt)
qed
lemma uniform_limit_tendsto_bcontfun:
fixes f::"'i \<Rightarrow> 'a::topological_space \<Rightarrow>\<^sub>C 'b::metric_space"
and l::"'a::topological_space \<Rightarrow>\<^sub>C 'b::metric_space"
assumes "uniform_limit UNIV f l F"
shows "(f \<longlongrightarrow> l) F"
proof (rule tendstoI)
fix e::real assume "e > 0"
then have "e / 2 > 0" by simp
from uniform_limitD[OF assms this]
have "\<forall>\<^sub>F i in F. \<forall>x. dist (f i x) (l x) < e / 2" by simp
then have "\<forall>\<^sub>F x in F. dist (f x) l \<le> e / 2"
by eventually_elim (blast intro: dist_bound less_imp_le)
then show "\<forall>\<^sub>F x in F. dist (f x) l < e"
by eventually_elim (use \<open>0 < e\<close> in auto)
qed
lemma uniform_limit_bcontfunE:
fixes f::"'i \<Rightarrow> 'a::topological_space \<Rightarrow>\<^sub>C 'b::metric_space"
and l::"'a::topological_space \<Rightarrow> 'b::metric_space"
assumes "uniform_limit UNIV f l F" "F \<noteq> bot"
obtains l'::"'a::topological_space \<Rightarrow>\<^sub>C 'b::metric_space"
where "l = l'" "(f \<longlongrightarrow> l') F"
by (metis (mono_tags, lifting) always_eventually apply_bcontfun apply_bcontfun_cases assms
bcontfun_def mem_Collect_eq uniform_limit_bounded uniform_limit_tendsto_bcontfun
uniform_limit_theorem)
lemma closed_PiC:
fixes I :: "'a::metric_space set"
and X :: "'a \<Rightarrow> 'b::complete_space set"
assumes "\<And>i. i \<in> I \<Longrightarrow> closed (X i)"
shows "closed (PiC I X)"
unfolding closed_sequential_limits
proof safe
fix f l
assume seq: "\<forall>n. f n \<in> PiC I X" and lim: "f \<longlonglongrightarrow> l"
show "l \<in> PiC I X"
proof (safe intro!: mem_PiCI)
fix x assume "x \<in> I"
then have "closed (X x)"
using assms by simp
moreover have "eventually (\<lambda>i. f i x \<in> X x) sequentially"
using seq \<open>x \<in> I\<close>
by (auto intro!: eventuallyI dest!: mem_PiCD simp: Pi_iff)
moreover note sequentially_bot
moreover have "(\<lambda>n. (f n) x) \<longlonglongrightarrow> l x"
using tendsto_bcontfun_uniform_limit[OF lim]
by (rule tendsto_uniform_limitI) simp
ultimately show "l x \<in> X x"
by (rule Lim_in_closed_set)
qed
qed
subsection \<open>Complete Space\<close>
instance%important bcontfun :: (metric_space, complete_space) complete_space
proof%unimportant
fix f :: "nat \<Rightarrow> ('a, 'b) bcontfun"
assume "Cauchy f" \<comment> \<open>Cauchy equals uniform convergence\<close>
then obtain g where "uniform_limit UNIV f g sequentially"
using uniformly_convergent_eq_cauchy[of "\<lambda>_. True" f]
unfolding Cauchy_def uniform_limit_sequentially_iff
by (metis dist_fun_lt_imp_dist_val_lt)
from uniform_limit_bcontfunE[OF this sequentially_bot]
obtain l' where "g = apply_bcontfun l'" "(f \<longlonglongrightarrow> l')" by metis
then show "convergent f"
by (intro convergentI)
qed
subsection%unimportant \<open>Supremum norm for a normed vector space\<close>
instantiation%unimportant bcontfun :: (topological_space, real_normed_vector) real_vector
begin
lemma uminus_cont: "f \<in> bcontfun \<Longrightarrow> (\<lambda>x. - f x) \<in> bcontfun" for f::"'a \<Rightarrow> 'b"
by (auto simp: bcontfun_def intro!: continuous_intros)
lemma plus_cont: "f \<in> bcontfun \<Longrightarrow> g \<in> bcontfun \<Longrightarrow> (\<lambda>x. f x + g x) \<in> bcontfun" for f g::"'a \<Rightarrow> 'b"
by (auto simp: bcontfun_def intro!: continuous_intros bounded_plus_comp)
lemma minus_cont: "f \<in> bcontfun \<Longrightarrow> g \<in> bcontfun \<Longrightarrow> (\<lambda>x. f x - g x) \<in> bcontfun" for f g::"'a \<Rightarrow> 'b"
by (auto simp: bcontfun_def intro!: continuous_intros bounded_minus_comp)
lemma scaleR_cont: "f \<in> bcontfun \<Longrightarrow> (\<lambda>x. a *\<^sub>R f x) \<in> bcontfun" for f :: "'a \<Rightarrow> 'b"
by (auto simp: bcontfun_def intro!: continuous_intros bounded_scaleR_comp)
lemma bcontfun_normI: "continuous_on UNIV f \<Longrightarrow> (\<And>x. norm (f x) \<le> b) \<Longrightarrow> f \<in> bcontfun"
by (auto simp: bcontfun_def intro: boundedI)
lift_definition uminus_bcontfun::"('a \<Rightarrow>\<^sub>C 'b) \<Rightarrow> 'a \<Rightarrow>\<^sub>C 'b" is "\<lambda>f x. - f x"
by (rule uminus_cont)
lift_definition plus_bcontfun::"('a \<Rightarrow>\<^sub>C 'b) \<Rightarrow> ('a \<Rightarrow>\<^sub>C 'b) \<Rightarrow> 'a \<Rightarrow>\<^sub>C 'b" is "\<lambda>f g x. f x + g x"
by (rule plus_cont)
lift_definition minus_bcontfun::"('a \<Rightarrow>\<^sub>C 'b) \<Rightarrow> ('a \<Rightarrow>\<^sub>C 'b) \<Rightarrow> 'a \<Rightarrow>\<^sub>C 'b" is "\<lambda>f g x. f x - g x"
by (rule minus_cont)
lift_definition zero_bcontfun::"'a \<Rightarrow>\<^sub>C 'b" is "\<lambda>_. 0"
by (rule const_bcontfun)
lemma const_bcontfun_0_eq_0[simp]: "const_bcontfun 0 = 0"
by transfer simp
lift_definition scaleR_bcontfun::"real \<Rightarrow> ('a \<Rightarrow>\<^sub>C 'b) \<Rightarrow> 'a \<Rightarrow>\<^sub>C 'b" is "\<lambda>r g x. r *\<^sub>R g x"
by (rule scaleR_cont)
lemmas [simp] =
const_bcontfun.rep_eq
uminus_bcontfun.rep_eq
plus_bcontfun.rep_eq
minus_bcontfun.rep_eq
zero_bcontfun.rep_eq
scaleR_bcontfun.rep_eq
instance
by standard (auto intro!: bcontfun_eqI simp: algebra_simps)
end
lemma bounded_norm_le_SUP_norm:
"bounded (range f) \<Longrightarrow> norm (f x) \<le> (SUP x. norm (f x))"
by (auto intro!: cSUP_upper bounded_imp_bdd_above simp: bounded_norm_comp)
instantiation%unimportant bcontfun :: (topological_space, real_normed_vector) real_normed_vector
begin
definition norm_bcontfun :: "('a, 'b) bcontfun \<Rightarrow> real"
where "norm_bcontfun f = dist f 0"
definition "sgn (f::('a,'b) bcontfun) = f /\<^sub>R norm f"
instance
proof
fix a :: real
fix f g :: "('a, 'b) bcontfun"
show "dist f g = norm (f - g)"
unfolding norm_bcontfun_def
by transfer (simp add: dist_norm)
show "norm (f + g) \<le> norm f + norm g"
unfolding norm_bcontfun_def
by transfer
(auto intro!: cSUP_least norm_triangle_le add_mono bounded_norm_le_SUP_norm
simp: dist_norm bcontfun_def)
show "norm (a *\<^sub>R f) = \<bar>a\<bar> * norm f"
unfolding norm_bcontfun_def
apply transfer
by (rule trans[OF _ continuous_at_Sup_mono[symmetric]])
(auto intro!: monoI mult_left_mono continuous_intros bounded_imp_bdd_above
simp: bounded_norm_comp bcontfun_def image_comp)
qed (auto simp: norm_bcontfun_def sgn_bcontfun_def)
end
lemma norm_bounded:
fixes f :: "('a::topological_space, 'b::real_normed_vector) bcontfun"
shows "norm (apply_bcontfun f x) \<le> norm f"
using dist_bounded[of f x 0]
by (simp add: dist_norm)
lemma norm_bound:
fixes f :: "('a::topological_space, 'b::real_normed_vector) bcontfun"
assumes "\<And>x. norm (apply_bcontfun f x) \<le> b"
shows "norm f \<le> b"
using dist_bound[of f 0 b] assms
by (simp add: dist_norm)
subsection%unimportant \<open>(bounded) continuous extenstion\<close>
lemma integral_clamp:
"integral {t0::real..clamp t0 t1 x} f =
(if x < t0 then 0 else if x \<le> t1 then integral {t0..x} f else integral {t0..t1} f)"
by (auto simp: clamp_def)
lemma continuous_on_interval_bcontfunE:
fixes f::"'a::ordered_euclidean_space \<Rightarrow> 'b::metric_space"
assumes "continuous_on {a .. b} f"
obtains g::"'a \<Rightarrow>\<^sub>C 'b" where
"\<And>x. a \<le> x \<Longrightarrow> x \<le> b \<Longrightarrow> g x = f x"
"\<And>x. g x = f (clamp a b x)"
proof -
define g where "g \<equiv> ext_cont f a b"
have "g \<in> bcontfun"
using assms
by (auto intro!: continuous_on_ext_cont simp: g_def cbox_interval bcontfun_def)
(auto simp: g_def ext_cont_def cbox_interval
intro!: compact_interval clamp_bounded compact_imp_bounded[OF compact_continuous_image] assms)
then obtain h where h: "g = apply_bcontfun h" by (rule bcontfunE)
then have "h x = f x" if "a \<le> x" "x \<le> b" for x
by (auto simp: h[symmetric] g_def cbox_interval that)
moreover
have "h x = f (clamp a b x)" for x
by (auto simp: h[symmetric] g_def ext_cont_def)
ultimately show ?thesis ..
qed
lifting_update bcontfun.lifting
lifting_forget bcontfun.lifting
end