# Theory Bounded_Continuous_Function

theory Bounded_Continuous_Function
imports Henstock_Kurzweil_Integration
```section ‹Bounded Continuous Functions›

theory Bounded_Continuous_Function
imports
Henstock_Kurzweil_Integration
begin

subsection ‹Definition›

definition "bcontfun = {f. continuous_on UNIV f ∧ bounded (range f)}"

typedef (overloaded) ('a, 'b) bcontfun ("(_ ⇒⇩C /_)" [22, 21] 21) =
"bcontfun::('a::topological_space ⇒ '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 ⇒⇩C'b::metric_space) ⇒ 'a ⇒ '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: "(⋀x. apply_bcontfun f x = apply_bcontfun g x) ⟹ f = g"
by transfer auto

lemma bcontfunE:
assumes "f ∈ bcontfun"
obtains g where "f = apply_bcontfun g"
by (blast intro: apply_bcontfun_cases assms )

lemma const_bcontfun: "(λx. b) ∈ bcontfun"
by (auto simp: bcontfun_def continuous_on_const image_def)

lift_definition const_bcontfun::"'b::metric_space ⇒ ('a::topological_space ⇒⇩C 'b)" is "λc _. c"
by (rule const_bcontfun)

(* TODO: Generalize to uniform spaces? *)
instantiation bcontfun :: (topological_space, metric_space) metric_space
begin

lift_definition dist_bcontfun :: "'a ⇒⇩C 'b ⇒ 'a ⇒⇩C 'b ⇒ real"
is "λf g. (SUP x. dist (f x) (g x))" .

definition uniformity_bcontfun :: "('a ⇒⇩C 'b × 'a ⇒⇩C 'b) filter"
where "uniformity_bcontfun = (INF e:{0 <..}. principal {(x, y). dist x y < e})"

definition open_bcontfun :: "('a ⇒⇩C 'b) set ⇒ bool"
where "open_bcontfun S = (∀x∈S. ∀⇩F (x', y) in uniformity. x' = x ⟶ y ∈ S)"

lemma bounded_dist_le_SUP_dist:
"bounded (range f) ⟹ bounded (range g) ⟹ dist (f x) (g x) ≤ (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 ⇒⇩C 'b"
shows "dist (f x) (g x) ≤ dist f g"
by transfer (auto intro!: bounded_dist_le_SUP_dist simp: bcontfun_def)

lemma dist_bound:
fixes f g :: "'a ⇒⇩C 'b"
assumes "⋀x. dist (f x) (g x) ≤ b"
shows "dist f g ≤ b"
using assms
by transfer (auto intro!: cSUP_least)

lemma dist_fun_lt_imp_dist_val_lt:
fixes f g :: "'a ⇒⇩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 ⇒⇩C 'b"
show "dist f g = 0 ⟷ f = g"
proof
have "⋀x. dist (f x) (g x) ≤ 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 ≤ dist f h + dist g h"
proof (rule dist_bound)
fix x
have "dist (f x) (g x) ≤ dist (f x) (h x) + dist (g x) (h x)"
by (rule dist_triangle2)
also have "dist (f x) (h x) ≤ dist f h"
by (rule dist_bounded)
also have "dist (g x) (h x) ≤ dist g h"
by (rule dist_bounded)
finally show "dist (f x) (g x) ≤ 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 ⇒ ('a ⇒ 'b set) ⇒ ('a ⇒⇩C 'b::metric_space) set"
is "λI X. Pi I X ∩ bcontfun"
by auto

lemma mem_PiC_iff: "x ∈ PiC I X ⟷ apply_bcontfun x ∈ 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 ⇒ 'a::topological_space ⇒⇩C 'b::metric_space"
assumes "(f ⤏ 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 "∀⇩F x in F. dist (f x) l < e" .
then show "∀⇩F n in F. ∀x∈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 ⇒ 'a::topological_space ⇒⇩C 'b::metric_space"
and l::"'a::topological_space ⇒⇩C 'b::metric_space"
assumes "uniform_limit UNIV f l F"
shows "(f ⤏ 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 "∀⇩F i in F. ∀x. dist (f i x) (l x) < e / 2" by simp
then have "∀⇩F x in F. dist (f x) l ≤ e / 2"
by eventually_elim (blast intro: dist_bound less_imp_le)
then show "∀⇩F x in F. dist (f x) l < e"
by eventually_elim (use ‹0 < e› in auto)
qed

lemma uniform_limit_bcontfunE:
fixes f::"'i ⇒ 'a::topological_space ⇒⇩C 'b::metric_space"
and l::"'a::topological_space ⇒ 'b::metric_space"
assumes "uniform_limit UNIV f l F" "F ≠ bot"
obtains l'::"'a::topological_space ⇒⇩C 'b::metric_space"
where "l = l'" "(f ⤏ 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 ⇒ 'b::complete_space set"
assumes "⋀i. i ∈ I ⟹ closed (X i)"
shows "closed (PiC I X)"
unfolding closed_sequential_limits
proof safe
fix f l
assume seq: "∀n. f n ∈ PiC I X" and lim: "f ⇢ l"
show "l ∈ PiC I X"
proof (safe intro!: mem_PiCI)
fix x assume "x ∈ I"
then have "closed (X x)"
using assms by simp
moreover have "eventually (λi. f i x ∈ X x) sequentially"
using seq ‹x ∈ I›
by (auto intro!: eventuallyI dest!: mem_PiCD simp: Pi_iff)
moreover note sequentially_bot
moreover have "(λn. (f n) x) ⇢ l x"
using tendsto_bcontfun_uniform_limit[OF lim]
by (rule tendsto_uniform_limitI) simp
ultimately show "l x ∈ X x"
by (rule Lim_in_closed_set)
qed
qed

subsection ‹Complete Space›

instance bcontfun :: (metric_space, complete_space) complete_space
proof
fix f :: "nat ⇒ ('a, 'b) bcontfun"
assume "Cauchy f"  ― ‹Cauchy equals uniform convergence›
then obtain g where "uniform_limit UNIV f g sequentially"
using uniformly_convergent_eq_cauchy[of "λ_. 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 ⇢ l')" by metis
then show "convergent f"
by (intro convergentI)
qed

subsection ‹Supremum norm for a normed vector space›

instantiation bcontfun :: (topological_space, real_normed_vector) real_vector
begin

lemma uminus_cont: "f ∈ bcontfun ⟹ (λx. - f x) ∈ bcontfun" for f::"'a ⇒ 'b"
by (auto simp: bcontfun_def intro!: continuous_intros)

lemma plus_cont: "f ∈ bcontfun ⟹ g ∈ bcontfun ⟹ (λx. f x + g x) ∈ bcontfun" for f g::"'a ⇒ 'b"
by (auto simp: bcontfun_def intro!: continuous_intros bounded_plus_comp)

lemma minus_cont: "f ∈ bcontfun ⟹ g ∈ bcontfun ⟹ (λx. f x - g x) ∈ bcontfun" for f g::"'a ⇒ 'b"
by (auto simp: bcontfun_def intro!: continuous_intros bounded_minus_comp)

lemma scaleR_cont: "f ∈ bcontfun ⟹ (λx. a *⇩R f x) ∈ bcontfun" for f :: "'a ⇒ 'b"
by (auto simp: bcontfun_def intro!: continuous_intros bounded_scaleR_comp)

lemma bcontfun_normI: "continuous_on UNIV f ⟹ (⋀x. norm (f x) ≤ b) ⟹ f ∈ bcontfun"
by (auto simp: bcontfun_def intro: boundedI)

lift_definition uminus_bcontfun::"('a ⇒⇩C 'b) ⇒ 'a ⇒⇩C 'b" is "λf x. - f x"
by (rule uminus_cont)

lift_definition plus_bcontfun::"('a ⇒⇩C 'b) ⇒ ('a ⇒⇩C 'b) ⇒ 'a ⇒⇩C 'b"  is "λf g x. f x + g x"
by (rule plus_cont)

lift_definition minus_bcontfun::"('a ⇒⇩C 'b) ⇒ ('a ⇒⇩C 'b) ⇒ 'a ⇒⇩C 'b"  is "λf g x. f x - g x"
by (rule minus_cont)

lift_definition zero_bcontfun::"'a ⇒⇩C 'b" is "λ_. 0"
by (rule const_bcontfun)

lemma const_bcontfun_0_eq_0[simp]: "const_bcontfun 0 = 0"
by transfer simp

lift_definition scaleR_bcontfun::"real ⇒ ('a ⇒⇩C 'b) ⇒ 'a ⇒⇩C 'b"  is "λr g x. r *⇩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) ⟹ norm (f x) ≤ (SUP x. norm (f x))"
by (auto intro!: cSUP_upper bounded_imp_bdd_above simp: bounded_norm_comp)

instantiation bcontfun :: (topological_space, real_normed_vector) real_normed_vector
begin

definition norm_bcontfun :: "('a, 'b) bcontfun ⇒ real"
where "norm_bcontfun f = dist f 0"

definition "sgn (f::('a,'b) bcontfun) = f /⇩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
show "norm (f + g) ≤ 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 *⇩R f) = ¦a¦ * 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)
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) ≤ norm f"
using dist_bounded[of f x 0]

lemma norm_bound:
fixes f :: "('a::topological_space, 'b::real_normed_vector) bcontfun"
assumes "⋀x. norm (apply_bcontfun f x) ≤ b"
shows "norm f ≤ b"
using dist_bound[of f 0 b] assms

subsection ‹(bounded) continuous extenstion›

lemma integral_clamp:
"integral {t0::real..clamp t0 t1 x} f =
(if x < t0 then 0 else if x ≤ 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 ⇒ 'b::metric_space"
assumes "continuous_on {a .. b} f"
obtains g::"'a ⇒⇩C 'b" where
"⋀x. a ≤ x ⟹ x ≤ b ⟹ g x = f x"
"⋀x. g x = f (clamp a b x)"
proof -
define g where "g ≡ ext_cont f a b"
have "g ∈ 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 ≤ x" "x ≤ 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
```