section \<open>Bounded Continuous Functions\<close>
theory Bounded_Continuous_Function
imports Integration
begin
subsection \<open>Definition\<close>
definition bcontfun :: "('a::topological_space \<Rightarrow> 'b::metric_space) set"
where "bcontfun = {f. continuous_on UNIV f \<and> bounded (range f)}"
typedef (overloaded) ('a, 'b) bcontfun =
"bcontfun :: ('a::topological_space \<Rightarrow> 'b::metric_space) set"
by (auto simp: bcontfun_def intro: continuous_intros simp: bounded_def)
lemma bcontfunE:
assumes "f \<in> bcontfun"
obtains y where "continuous_on UNIV f" "\<And>x. dist (f x) u \<le> y"
using assms unfolding bcontfun_def
by (metis (lifting) bounded_any_center dist_commute mem_Collect_eq rangeI)
lemma bcontfunE':
assumes "f \<in> bcontfun"
obtains y where "continuous_on UNIV f" "\<And>x. dist (f x) undefined \<le> y"
using assms bcontfunE
by metis
lemma bcontfunI: "continuous_on UNIV f \<Longrightarrow> (\<And>x. dist (f x) u \<le> b) \<Longrightarrow> f \<in> bcontfun"
unfolding bcontfun_def
by (metis (lifting, no_types) bounded_def dist_commute mem_Collect_eq rangeE)
lemma bcontfunI': "continuous_on UNIV f \<Longrightarrow> (\<And>x. dist (f x) undefined \<le> b) \<Longrightarrow> f \<in> bcontfun"
using bcontfunI by metis
lemma continuous_on_Rep_bcontfun[intro, simp]: "continuous_on T (Rep_bcontfun x)"
using Rep_bcontfun[of x]
by (auto simp: bcontfun_def intro: continuous_on_subset)
instantiation bcontfun :: (topological_space, metric_space) metric_space
begin
definition dist_bcontfun :: "('a, 'b) bcontfun \<Rightarrow> ('a, 'b) bcontfun \<Rightarrow> real"
where "dist_bcontfun f g = (SUP x. dist (Rep_bcontfun f x) (Rep_bcontfun g x))"
definition open_bcontfun :: "('a, 'b) bcontfun set \<Rightarrow> bool"
where "open_bcontfun S = (\<forall>x\<in>S. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> S)"
lemma dist_bounded:
fixes f :: "('a, 'b) bcontfun"
shows "dist (Rep_bcontfun f x) (Rep_bcontfun g x) \<le> dist f g"
proof -
have "Rep_bcontfun f \<in> bcontfun" by (rule Rep_bcontfun)
from bcontfunE'[OF this] obtain y where y:
"continuous_on UNIV (Rep_bcontfun f)"
"\<And>x. dist (Rep_bcontfun f x) undefined \<le> y"
by auto
have "Rep_bcontfun g \<in> bcontfun" by (rule Rep_bcontfun)
from bcontfunE'[OF this] obtain z where z:
"continuous_on UNIV (Rep_bcontfun g)"
"\<And>x. dist (Rep_bcontfun g x) undefined \<le> z"
by auto
show ?thesis
unfolding dist_bcontfun_def
proof (intro cSUP_upper bdd_aboveI2)
fix x
have "dist (Rep_bcontfun f x) (Rep_bcontfun g x) \<le>
dist (Rep_bcontfun f x) undefined + dist (Rep_bcontfun g x) undefined"
by (rule dist_triangle2)
also have "\<dots> \<le> y + z"
using y(2)[of x] z(2)[of x] by (rule add_mono)
finally show "dist (Rep_bcontfun f x) (Rep_bcontfun g x) \<le> y + z" .
qed simp
qed
lemma dist_bound:
fixes f :: "('a, 'b) bcontfun"
assumes "\<And>x. dist (Rep_bcontfun f x) (Rep_bcontfun g x) \<le> b"
shows "dist f g \<le> b"
using assms by (auto simp: dist_bcontfun_def intro: cSUP_least)
lemma dist_bounded_Abs:
fixes f g :: "'a \<Rightarrow> 'b"
assumes "f \<in> bcontfun" "g \<in> bcontfun"
shows "dist (f x) (g x) \<le> dist (Abs_bcontfun f) (Abs_bcontfun g)"
by (metis Abs_bcontfun_inverse assms dist_bounded)
lemma const_bcontfun: "(\<lambda>x::'a. b::'b) \<in> bcontfun"
by (auto intro: bcontfunI continuous_on_const)
lemma dist_fun_lt_imp_dist_val_lt:
assumes "dist f g < e"
shows "dist (Rep_bcontfun f x) (Rep_bcontfun g x) < e"
using dist_bounded assms by (rule le_less_trans)
lemma dist_val_lt_imp_dist_fun_le:
assumes "\<forall>x. dist (Rep_bcontfun f x) (Rep_bcontfun g x) < e"
shows "dist f g \<le> e"
unfolding dist_bcontfun_def
proof (intro cSUP_least)
fix x
show "dist (Rep_bcontfun f x) (Rep_bcontfun g x) \<le> e"
using assms[THEN spec[where x=x]] by (simp add: dist_norm)
qed simp
instance
proof
fix f g h :: "('a, 'b) bcontfun"
show "dist f g = 0 \<longleftrightarrow> f = g"
proof
have "\<And>x. dist (Rep_bcontfun f x) (Rep_bcontfun g x) \<le> dist f g"
by (rule dist_bounded)
also assume "dist f g = 0"
finally show "f = g"
by (auto simp: Rep_bcontfun_inject[symmetric] Abs_bcontfun_inverse)
qed (auto simp: dist_bcontfun_def SUP_def simp del: Sup_image_eq intro!: cSup_eq)
show "dist f g \<le> dist f h + dist g h"
proof (subst dist_bcontfun_def, safe intro!: cSUP_least)
fix x
have "dist (Rep_bcontfun f x) (Rep_bcontfun g x) \<le>
dist (Rep_bcontfun f x) (Rep_bcontfun h x) + dist (Rep_bcontfun g x) (Rep_bcontfun h x)"
by (rule dist_triangle2)
also have "dist (Rep_bcontfun f x) (Rep_bcontfun h x) \<le> dist f h"
by (rule dist_bounded)
also have "dist (Rep_bcontfun g x) (Rep_bcontfun h x) \<le> dist g h"
by (rule dist_bounded)
finally show "dist (Rep_bcontfun f x) (Rep_bcontfun g x) \<le> dist f h + dist g h"
by simp
qed
qed (simp add: open_bcontfun_def)
end
lemma closed_Pi_bcontfun:
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 (Abs_bcontfun ` (Pi I X \<inter> bcontfun))"
unfolding closed_sequential_limits
proof safe
fix f l
assume seq: "\<forall>n. f n \<in> Abs_bcontfun ` (Pi I X \<inter> bcontfun)" and lim: "f ----> l"
have lim_fun: "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<forall>x. dist (Rep_bcontfun (f n) x) (Rep_bcontfun l x) < e"
using LIMSEQ_imp_Cauchy[OF lim, simplified Cauchy_def] metric_LIMSEQ_D[OF lim]
by (intro uniformly_cauchy_imp_uniformly_convergent[where P="\<lambda>_. True", simplified])
(metis dist_fun_lt_imp_dist_val_lt)+
show "l \<in> Abs_bcontfun ` (Pi I X \<inter> bcontfun)"
proof (rule, safe)
fix x assume "x \<in> I"
then have "closed (X x)"
using assms by simp
moreover have "eventually (\<lambda>xa. Rep_bcontfun (f xa) x \<in> X x) sequentially"
proof (rule always_eventually, safe)
fix i
from seq[THEN spec, of i] \<open>x \<in> I\<close>
show "Rep_bcontfun (f i) x \<in> X x"
by (auto simp: Abs_bcontfun_inverse)
qed
moreover note sequentially_bot
moreover have "(\<lambda>n. Rep_bcontfun (f n) x) ----> Rep_bcontfun l x"
using lim_fun by (blast intro!: metric_LIMSEQ_I)
ultimately show "Rep_bcontfun l x \<in> X x"
by (rule Lim_in_closed_set)
qed (auto simp: Rep_bcontfun Rep_bcontfun_inverse)
qed
subsection \<open>Complete Space\<close>
instance bcontfun :: (metric_space, complete_space) complete_space
proof
fix f :: "nat \<Rightarrow> ('a, 'b) bcontfun"
assume "Cauchy f" \<comment> \<open>Cauchy equals uniform convergence\<close>
then obtain g where limit_function:
"\<forall>e>0. \<exists>N. \<forall>n\<ge>N. \<forall>x. dist (Rep_bcontfun (f n) x) (g x) < e"
using uniformly_convergent_eq_cauchy[of "\<lambda>_. True"
"\<lambda>n. Rep_bcontfun (f n)"]
unfolding Cauchy_def
by (metis dist_fun_lt_imp_dist_val_lt)
then obtain N where fg_dist: \<comment> \<open>for an upper bound on @{term g}\<close>
"\<forall>n\<ge>N. \<forall>x. dist (g x) ( Rep_bcontfun (f n) x) < 1"
by (force simp add: dist_commute)
from bcontfunE'[OF Rep_bcontfun, of "f N"] obtain b where
f_bound: "\<forall>x. dist (Rep_bcontfun (f N) x) undefined \<le> b"
by force
have "g \<in> bcontfun" \<comment> \<open>The limit function is bounded and continuous\<close>
proof (intro bcontfunI)
show "continuous_on UNIV g"
using bcontfunE[OF Rep_bcontfun] limit_function
by (intro continuous_uniform_limit[where f="\<lambda>n. Rep_bcontfun (f n)" and F="sequentially"])
(auto simp add: eventually_sequentially trivial_limit_def dist_norm)
next
fix x
from fg_dist have "dist (g x) (Rep_bcontfun (f N) x) < 1"
by (simp add: dist_norm norm_minus_commute)
with dist_triangle[of "g x" undefined "Rep_bcontfun (f N) x"]
show "dist (g x) undefined \<le> 1 + b" using f_bound[THEN spec, of x]
by simp
qed
show "convergent f"
proof (rule convergentI, subst lim_sequentially, safe)
\<comment> \<open>The limit function converges according to its norm\<close>
fix e :: real
assume "e > 0"
then have "e/2 > 0" by simp
with limit_function[THEN spec, of"e/2"]
have "\<exists>N. \<forall>n\<ge>N. \<forall>x. dist (Rep_bcontfun (f n) x) (g x) < e/2"
by simp
then obtain N where N: "\<forall>n\<ge>N. \<forall>x. dist (Rep_bcontfun (f n) x) (g x) < e / 2" by auto
show "\<exists>N. \<forall>n\<ge>N. dist (f n) (Abs_bcontfun g) < e"
proof (rule, safe)
fix n
assume "N \<le> n"
with N show "dist (f n) (Abs_bcontfun g) < e"
using dist_val_lt_imp_dist_fun_le[of
"f n" "Abs_bcontfun g" "e/2"]
Abs_bcontfun_inverse[OF \<open>g \<in> bcontfun\<close>] \<open>e > 0\<close> by simp
qed
qed
qed
subsection \<open>Supremum norm for a normed vector space\<close>
instantiation bcontfun :: (topological_space, real_normed_vector) real_vector
begin
definition "-f = Abs_bcontfun (\<lambda>x. -(Rep_bcontfun f x))"
definition "f + g = Abs_bcontfun (\<lambda>x. Rep_bcontfun f x + Rep_bcontfun g x)"
definition "f - g = Abs_bcontfun (\<lambda>x. Rep_bcontfun f x - Rep_bcontfun g x)"
definition "0 = Abs_bcontfun (\<lambda>x. 0)"
definition "scaleR r f = Abs_bcontfun (\<lambda>x. r *\<^sub>R Rep_bcontfun f x)"
lemma plus_cont:
fixes f g :: "'a \<Rightarrow> 'b"
assumes f: "f \<in> bcontfun"
and g: "g \<in> bcontfun"
shows "(\<lambda>x. f x + g x) \<in> bcontfun"
proof -
from bcontfunE'[OF f] obtain y where "continuous_on UNIV f" "\<And>x. dist (f x) undefined \<le> y"
by auto
moreover
from bcontfunE'[OF g] obtain z where "continuous_on UNIV g" "\<And>x. dist (g x) undefined \<le> z"
by auto
ultimately show ?thesis
proof (intro bcontfunI)
fix x
have "dist (f x + g x) 0 = dist (f x + g x) (0 + 0)"
by simp
also have "\<dots> \<le> dist (f x) 0 + dist (g x) 0"
by (rule dist_triangle_add)
also have "\<dots> \<le> dist (Abs_bcontfun f) 0 + dist (Abs_bcontfun g) 0"
unfolding zero_bcontfun_def using assms
by (auto intro!: add_mono dist_bounded_Abs const_bcontfun)
finally show "dist (f x + g x) 0 \<le> dist (Abs_bcontfun f) 0 + dist (Abs_bcontfun g) 0" .
qed (simp add: continuous_on_add)
qed
lemma Rep_bcontfun_plus[simp]: "Rep_bcontfun (f + g) x = Rep_bcontfun f x + Rep_bcontfun g x"
by (simp add: plus_bcontfun_def Abs_bcontfun_inverse plus_cont Rep_bcontfun)
lemma uminus_cont:
fixes f :: "'a \<Rightarrow> 'b"
assumes "f \<in> bcontfun"
shows "(\<lambda>x. - f x) \<in> bcontfun"
proof -
from bcontfunE[OF assms, of 0] obtain y
where "continuous_on UNIV f" "\<And>x. dist (f x) 0 \<le> y"
by auto
then show ?thesis
proof (intro bcontfunI)
fix x
assume "\<And>x. dist (f x) 0 \<le> y"
then show "dist (- f x) 0 \<le> y"
by (subst dist_minus[symmetric]) simp
qed (simp add: continuous_on_minus)
qed
lemma Rep_bcontfun_uminus[simp]: "Rep_bcontfun (- f) x = - Rep_bcontfun f x"
by (simp add: uminus_bcontfun_def Abs_bcontfun_inverse uminus_cont Rep_bcontfun)
lemma minus_cont:
fixes f g :: "'a \<Rightarrow> 'b"
assumes f: "f \<in> bcontfun"
and g: "g \<in> bcontfun"
shows "(\<lambda>x. f x - g x) \<in> bcontfun"
using plus_cont [of f "- g"] assms
by (simp add: uminus_cont fun_Compl_def)
lemma Rep_bcontfun_minus[simp]: "Rep_bcontfun (f - g) x = Rep_bcontfun f x - Rep_bcontfun g x"
by (simp add: minus_bcontfun_def Abs_bcontfun_inverse minus_cont Rep_bcontfun)
lemma scaleR_cont:
fixes a :: real
and f :: "'a \<Rightarrow> 'b"
assumes "f \<in> bcontfun"
shows " (\<lambda>x. a *\<^sub>R f x) \<in> bcontfun"
proof -
from bcontfunE[OF assms, of 0] obtain y
where "continuous_on UNIV f" "\<And>x. dist (f x) 0 \<le> y"
by auto
then show ?thesis
proof (intro bcontfunI)
fix x
assume "\<And>x. dist (f x) 0 \<le> y"
then show "dist (a *\<^sub>R f x) 0 \<le> abs a * y"
by (metis norm_cmul_rule_thm norm_conv_dist)
qed (simp add: continuous_intros)
qed
lemma Rep_bcontfun_scaleR[simp]: "Rep_bcontfun (a *\<^sub>R g) x = a *\<^sub>R Rep_bcontfun g x"
by (simp add: scaleR_bcontfun_def Abs_bcontfun_inverse scaleR_cont Rep_bcontfun)
instance
by standard
(simp_all add: plus_bcontfun_def zero_bcontfun_def minus_bcontfun_def scaleR_bcontfun_def
Abs_bcontfun_inverse Rep_bcontfun_inverse Rep_bcontfun algebra_simps
plus_cont const_bcontfun minus_cont scaleR_cont)
end
instantiation 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)"
by (simp add: norm_bcontfun_def dist_bcontfun_def zero_bcontfun_def
Abs_bcontfun_inverse const_bcontfun norm_conv_dist[symmetric] dist_norm)
show "norm (f + g) \<le> norm f + norm g"
unfolding norm_bcontfun_def
proof (subst dist_bcontfun_def, safe intro!: cSUP_least)
fix x
have "dist (Rep_bcontfun (f + g) x) (Rep_bcontfun 0 x) \<le>
dist (Rep_bcontfun f x) 0 + dist (Rep_bcontfun g x) 0"
by (metis (hide_lams, no_types) Rep_bcontfun_minus Rep_bcontfun_plus diff_0_right dist_norm
le_less_linear less_irrefl norm_triangle_lt)
also have "dist (Rep_bcontfun f x) 0 \<le> dist f 0"
using dist_bounded[of f x 0]
by (simp add: Abs_bcontfun_inverse const_bcontfun zero_bcontfun_def)
also have "dist (Rep_bcontfun g x) 0 \<le> dist g 0" using dist_bounded[of g x 0]
by (simp add: Abs_bcontfun_inverse const_bcontfun zero_bcontfun_def)
finally show "dist (Rep_bcontfun (f + g) x) (Rep_bcontfun 0 x) \<le> dist f 0 + dist g 0" by simp
qed
show "norm (a *\<^sub>R f) = \<bar>a\<bar> * norm f"
proof -
have "\<bar>a\<bar> * Sup (range (\<lambda>x. dist (Rep_bcontfun f x) 0)) =
(SUP i:range (\<lambda>x. dist (Rep_bcontfun f x) 0). \<bar>a\<bar> * i)"
proof (intro continuous_at_Sup_mono bdd_aboveI2)
fix x
show "dist (Rep_bcontfun f x) 0 \<le> norm f" using dist_bounded[of f x 0]
by (simp add: norm_bcontfun_def norm_conv_dist Abs_bcontfun_inverse zero_bcontfun_def
const_bcontfun)
qed (auto intro!: monoI mult_left_mono continuous_intros)
moreover
have "range (\<lambda>x. dist (Rep_bcontfun (a *\<^sub>R f) x) 0) =
(\<lambda>x. \<bar>a\<bar> * x) ` (range (\<lambda>x. dist (Rep_bcontfun f x) 0))"
by (auto simp: norm_conv_dist[symmetric])
ultimately
show "norm (a *\<^sub>R f) = \<bar>a\<bar> * norm f"
by (simp add: norm_bcontfun_def dist_bcontfun_def norm_conv_dist Abs_bcontfun_inverse
zero_bcontfun_def const_bcontfun SUP_def del: Sup_image_eq)
qed
qed (auto simp: norm_bcontfun_def sgn_bcontfun_def)
end
lemma bcontfun_normI: "continuous_on UNIV f \<Longrightarrow> (\<And>x. norm (f x) \<le> b) \<Longrightarrow> f \<in> bcontfun"
unfolding norm_conv_dist
by (auto intro: bcontfunI)
lemma norm_bounded:
fixes f :: "('a::topological_space, 'b::real_normed_vector) bcontfun"
shows "norm (Rep_bcontfun f x) \<le> norm f"
using dist_bounded[of f x 0]
by (simp add: norm_bcontfun_def norm_conv_dist Abs_bcontfun_inverse zero_bcontfun_def
const_bcontfun)
lemma norm_bound:
fixes f :: "('a::topological_space, 'b::real_normed_vector) bcontfun"
assumes "\<And>x. norm (Rep_bcontfun f x) \<le> b"
shows "norm f \<le> b"
using dist_bound[of f 0 b] assms
by (simp add: norm_bcontfun_def norm_conv_dist Abs_bcontfun_inverse zero_bcontfun_def
const_bcontfun)
subsection \<open>Continuously Extended Functions\<close>
definition clamp :: "'a::euclidean_space \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a" where
"clamp a b x = (\<Sum>i\<in>Basis. (if x\<bullet>i < a\<bullet>i then a\<bullet>i else if x\<bullet>i \<le> b\<bullet>i then x\<bullet>i else b\<bullet>i) *\<^sub>R i)"
definition ext_cont :: "('a::euclidean_space \<Rightarrow> 'b::real_normed_vector) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> ('a, 'b) bcontfun"
where "ext_cont f a b = Abs_bcontfun ((\<lambda>x. f (clamp a b x)))"
lemma ext_cont_def':
"ext_cont f a b = Abs_bcontfun (\<lambda>x.
f (\<Sum>i\<in>Basis. (if x\<bullet>i < a\<bullet>i then a\<bullet>i else if x\<bullet>i \<le> b\<bullet>i then x\<bullet>i else b\<bullet>i) *\<^sub>R i))"
unfolding ext_cont_def clamp_def ..
lemma clamp_in_interval:
assumes "\<And>i. i \<in> Basis \<Longrightarrow> a \<bullet> i \<le> b \<bullet> i"
shows "clamp a b x \<in> cbox a b"
unfolding clamp_def
using box_ne_empty(1)[of a b] assms by (auto simp: cbox_def)
lemma dist_clamps_le_dist_args:
fixes x :: "'a::euclidean_space"
assumes "\<And>i. i \<in> Basis \<Longrightarrow> a \<bullet> i \<le> b \<bullet> i"
shows "dist (clamp a b y) (clamp a b x) \<le> dist y x"
proof -
from box_ne_empty(1)[of a b] assms have "(\<forall>i\<in>Basis. a \<bullet> i \<le> b \<bullet> i)"
by (simp add: cbox_def)
then have "(\<Sum>i\<in>Basis. (dist (clamp a b y \<bullet> i) (clamp a b x \<bullet> i))\<^sup>2) \<le>
(\<Sum>i\<in>Basis. (dist (y \<bullet> i) (x \<bullet> i))\<^sup>2)"
by (auto intro!: setsum_mono simp: clamp_def dist_real_def abs_le_square_iff[symmetric])
then show ?thesis
by (auto intro: real_sqrt_le_mono
simp: euclidean_dist_l2[where y=x] euclidean_dist_l2[where y="clamp a b x"] setL2_def)
qed
lemma clamp_continuous_at:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::metric_space"
and x :: 'a
assumes "\<And>i. i \<in> Basis \<Longrightarrow> a \<bullet> i \<le> b \<bullet> i"
and f_cont: "continuous_on (cbox a b) f"
shows "continuous (at x) (\<lambda>x. f (clamp a b x))"
unfolding continuous_at_eps_delta
proof safe
fix x :: 'a
fix e :: real
assume "e > 0"
moreover have "clamp a b x \<in> cbox a b"
by (simp add: clamp_in_interval assms)
moreover note f_cont[simplified continuous_on_iff]
ultimately
obtain d where d: "0 < d"
"\<And>x'. x' \<in> cbox a b \<Longrightarrow> dist x' (clamp a b x) < d \<Longrightarrow> dist (f x') (f (clamp a b x)) < e"
by force
show "\<exists>d>0. \<forall>x'. dist x' x < d \<longrightarrow>
dist (f (clamp a b x')) (f (clamp a b x)) < e"
by (auto intro!: d clamp_in_interval assms dist_clamps_le_dist_args[THEN le_less_trans])
qed
lemma clamp_continuous_on:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::metric_space"
assumes "\<And>i. i \<in> Basis \<Longrightarrow> a \<bullet> i \<le> b \<bullet> i"
and f_cont: "continuous_on (cbox a b) f"
shows "continuous_on UNIV (\<lambda>x. f (clamp a b x))"
using assms
by (auto intro: continuous_at_imp_continuous_on clamp_continuous_at)
lemma clamp_bcontfun:
fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
assumes "\<And>i. i \<in> Basis \<Longrightarrow> a \<bullet> i \<le> b \<bullet> i"
and continuous: "continuous_on (cbox a b) f"
shows "(\<lambda>x. f (clamp a b x)) \<in> bcontfun"
proof -
have "bounded (f ` (cbox a b))"
by (rule compact_continuous_image[OF continuous compact_cbox[of a b], THEN compact_imp_bounded])
then obtain c where f_bound: "\<forall>x\<in>f ` cbox a b. norm x \<le> c"
by (auto simp add: bounded_pos)
show "(\<lambda>x. f (clamp a b x)) \<in> bcontfun"
proof (intro bcontfun_normI)
fix x
show "norm (f (clamp a b x)) \<le> c"
using clamp_in_interval[OF assms(1), of x] f_bound
by (simp add: ext_cont_def)
qed (simp add: clamp_continuous_on assms)
qed
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)
declare [[coercion Rep_bcontfun]]
lemma ext_cont_cancel[simp]:
fixes x a b :: "'a::euclidean_space"
assumes x: "x \<in> cbox a b"
and "continuous_on (cbox a b) f"
shows "ext_cont f a b x = f x"
using assms
unfolding ext_cont_def
proof (subst Abs_bcontfun_inverse[OF clamp_bcontfun])
show "f (clamp a b x) = f x"
using x unfolding clamp_def mem_box
by (intro arg_cong[where f=f] euclidean_eqI[where 'a='a]) (simp add: not_less)
qed (auto simp: cbox_def)
lemma ext_cont_cong:
assumes "t0 = s0"
and "t1 = s1"
and "\<And>t. t \<in> (cbox t0 t1) \<Longrightarrow> f t = g t"
and "continuous_on (cbox t0 t1) f"
and "continuous_on (cbox s0 s1) g"
and ord: "\<And>i. i \<in> Basis \<Longrightarrow> t0 \<bullet> i \<le> t1 \<bullet> i"
shows "ext_cont f t0 t1 = ext_cont g s0 s1"
unfolding assms ext_cont_def
using assms clamp_in_interval[OF ord]
by (subst Rep_bcontfun_inject[symmetric]) simp
end