# HG changeset patch
# User hoelzl
# Date 1422440241 -3600
# Node ID 4736ff5a41d8b2beabaab3c145bcd221822dd17d
# Parent  2538b2c51769f9e40c424644233f8f8c5cb6eac3
moved bcontfun from AFP/Ordinary_Differential_Equations

diff -r 2538b2c51769 -r 4736ff5a41d8 src/HOL/Multivariate_Analysis/Bounded_Continuous_Function.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Multivariate_Analysis/Bounded_Continuous_Function.thy	Wed Jan 28 11:17:21 2015 +0100
@@ -0,0 +1,498 @@
+section {* Bounded Continuous Functions *}
+theory Bounded_Continuous_Function
+imports Integration
+begin
+
+subsection{* Definition *}
+
+definition "bcontfun = {f :: 'a::topological_space \<Rightarrow> 'b::metric_space. continuous_on UNIV f \<and> bounded (range f)}"
+
+typedef ('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" using 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" using 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::"('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)
+next
+  fix f g h :: "('a, 'b) bcontfun"
+  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="%_. 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"
+    hence "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] `x \<in> I`
+      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 {* Complete Space *}
+
+instance bcontfun :: (metric_space, complete_space) complete_space
+proof
+  fix f::"nat \<Rightarrow> ('a,'b) bcontfun"
+  assume "Cauchy f" --{* Cauchy equals uniform convergence *}
+  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: --{* for an upper bound on g *}
+    "\<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" --{* The limit function is bounded and continuous *}
+  proof (intro bcontfunI)
+    show "continuous_on UNIV g"
+      using bcontfunE[OF Rep_bcontfun] limit_function
+      by (intro continuous_uniform_limit[where
+        f="%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 LIMSEQ_def, safe)
+    --{* The limit function converges according to its norm *}
+    fix e::real
+    assume "e > 0" hence "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 `g \<in> bcontfun`] `e > 0` by simp
+    qed
+  qed
+qed
+
+subsection{* Supremum norm for a normed vector space *}
+
+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 <= 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
+  thus ?thesis
+  proof (intro bcontfunI)
+    fix x
+    assume "\<And>x. dist (f x) 0 \<le> y"
+    thus "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 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
+  thus ?thesis
+  proof (intro bcontfunI)
+    fix x assume "\<And>x. dist (f x) 0 \<le> y"
+    thus "dist (a *\<^sub>R f x) 0 \<le> abs a * y"
+      by (metis abs_ge_zero comm_semiring_1_class.normalizing_semiring_rules(7) mult_right_mono
+        norm_conv_dist norm_scaleR)
+  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
+proof
+qed (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 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
+next
+  fix a and f g:: "('a, 'b) bcontfun"
+  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{* Continuously Extended Functions *}
+
+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)
+    hence "(\<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 add: clamp_def dist_real_def real_abs_le_square_iff[symmetric])
+    thus ?thesis
+      by (auto intro: real_sqrt_le_mono
+        simp add: 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"
+  fixes x
+  assumes "\<And>i. i \<in> Basis \<Longrightarrow> a \<bullet> i \<le> b \<bullet> i"
+  assumes 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 and e::real
+  assume "0 < e"
+  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"
+  assumes 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"
+  assumes continuous: "continuous_on (cbox a b) f"
+  shows "(\<lambda>x. f (clamp a b x)) \<in> bcontfun"
+proof -
+  from compact_continuous_image[OF continuous compact_cbox[of a b], THEN compact_imp_bounded]
+  have "bounded (f ` (cbox a b))" .
+  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
+    from clamp_in_interval[OF assms(1), of x] f_bound
+    show "norm (f (clamp a b x)) \<le> c" 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"
+  assumes "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"
+  assumes "t1 = s1"
+  assumes "\<And>t. t \<in> (cbox t0 t1) \<Longrightarrow> f t = g t"
+  assumes "continuous_on (cbox t0 t1) f"
+  assumes "continuous_on (cbox s0 s1) g"
+  assumes 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
diff -r 2538b2c51769 -r 4736ff5a41d8 src/HOL/Multivariate_Analysis/Multivariate_Analysis.thy
--- a/src/HOL/Multivariate_Analysis/Multivariate_Analysis.thy	Tue Jan 27 16:12:40 2015 +0100
+++ b/src/HOL/Multivariate_Analysis/Multivariate_Analysis.thy	Wed Jan 28 11:17:21 2015 +0100
@@ -1,5 +1,11 @@
 theory Multivariate_Analysis
-imports Fashoda Extended_Real_Limits Determinants Ordered_Euclidean_Space Complex_Analysis_Basics
+imports
+  Fashoda
+  Extended_Real_Limits
+  Determinants
+  Ordered_Euclidean_Space
+  Complex_Analysis_Basics
+  Bounded_Continuous_Function
 begin
 
 end