diff -r bbcb05504fdc -r bd218a9320b5 src/HOL/Multivariate_Analysis/Extension.thy
--- a/src/HOL/Multivariate_Analysis/Extension.thy	Thu Aug 04 18:45:28 2016 +0200
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,547 +0,0 @@
-(*  Title:      HOL/Multivariate_Analysis/Extension.thy
-    Authors:    LC Paulson, based on material from HOL Light
-*)
-
-section \<open>Continuous extensions of functions: Urysohn's lemma, Dugundji extension theorem, Tietze\<close>
-
-theory Extension
-imports Convex_Euclidean_Space
-begin
-
-subsection\<open>Partitions of unity subordinate to locally finite open coverings\<close>
-
-text\<open>A difference from HOL Light: all summations over infinite sets equal zero,
-   so the "support" must be made explicit in the summation below!\<close>
-
-proposition subordinate_partition_of_unity:
-  fixes S :: "'a :: euclidean_space set"
-  assumes "S \<subseteq> \<Union>\<C>" and opC: "\<And>T. T \<in> \<C> \<Longrightarrow> open T"
-      and fin: "\<And>x. x \<in> S \<Longrightarrow> \<exists>V. open V \<and> x \<in> V \<and> finite {U \<in> \<C>. U \<inter> V \<noteq> {}}"
-  obtains F :: "['a set, 'a] \<Rightarrow> real"
-    where "\<And>U. U \<in> \<C> \<Longrightarrow> continuous_on S (F U) \<and> (\<forall>x \<in> S. 0 \<le> F U x)"
-      and "\<And>x U. \<lbrakk>U \<in> \<C>; x \<in> S; x \<notin> U\<rbrakk> \<Longrightarrow> F U x = 0"
-      and "\<And>x. x \<in> S \<Longrightarrow> supp_setsum (\<lambda>W. F W x) \<C> = 1"
-      and "\<And>x. x \<in> S \<Longrightarrow> \<exists>V. open V \<and> x \<in> V \<and> finite {U \<in> \<C>. \<exists>x\<in>V. F U x \<noteq> 0}"
-proof (cases "\<exists>W. W \<in> \<C> \<and> S \<subseteq> W")
-  case True
-    then obtain W where "W \<in> \<C>" "S \<subseteq> W" by metis
-    then show ?thesis
-      apply (rule_tac F = "\<lambda>V x. if V = W then 1 else 0" in that)
-      apply (auto simp: continuous_on_const supp_setsum_def support_on_def)
-      done
-next
-  case False
-    have nonneg: "0 \<le> supp_setsum (\<lambda>V. setdist {x} (S - V)) \<C>" for x
-      by (simp add: supp_setsum_def setsum_nonneg)
-    have sd_pos: "0 < setdist {x} (S - V)" if "V \<in> \<C>" "x \<in> S" "x \<in> V" for V x
-    proof -
-      have "closedin (subtopology euclidean S) (S - V)"
-        by (simp add: Diff_Diff_Int Diff_subset closedin_def opC openin_open_Int \<open>V \<in> \<C>\<close>)
-      with that False setdist_eq_0_closedin [of S "S-V" x] setdist_pos_le [of "{x}" "S - V"]
-        show ?thesis
-          by (simp add: order_class.order.order_iff_strict)
-    qed
-    have ss_pos: "0 < supp_setsum (\<lambda>V. setdist {x} (S - V)) \<C>" if "x \<in> S" for x
-    proof -
-      obtain U where "U \<in> \<C>" "x \<in> U" using \<open>x \<in> S\<close> \<open>S \<subseteq> \<Union>\<C>\<close>
-        by blast
-      obtain V where "open V" "x \<in> V" "finite {U \<in> \<C>. U \<inter> V \<noteq> {}}"
-        using \<open>x \<in> S\<close> fin by blast
-      then have *: "finite {A \<in> \<C>. \<not> S \<subseteq> A \<and> x \<notin> closure (S - A)}"
-        using closure_def that by (blast intro: rev_finite_subset)
-      have "x \<notin> closure (S - U)"
-        by (metis \<open>U \<in> \<C>\<close> \<open>x \<in> U\<close> less_irrefl sd_pos setdist_eq_0_sing_1 that)
-      then show ?thesis
-        apply (simp add: setdist_eq_0_sing_1 supp_setsum_def support_on_def)
-        apply (rule ordered_comm_monoid_add_class.setsum_pos2 [OF *, of U])
-        using \<open>U \<in> \<C>\<close> \<open>x \<in> U\<close> False
-        apply (auto simp: setdist_pos_le sd_pos that)
-        done
-    qed
-    define F where
-      "F \<equiv> \<lambda>W x. if x \<in> S then setdist {x} (S - W) / supp_setsum (\<lambda>V. setdist {x} (S - V)) \<C>
-                 else 0"
-    show ?thesis
-    proof (rule_tac F = F in that)
-      have "continuous_on S (F U)" if "U \<in> \<C>" for U
-      proof -
-        have *: "continuous_on S (\<lambda>x. supp_setsum (\<lambda>V. setdist {x} (S - V)) \<C>)"
-        proof (clarsimp simp add: continuous_on_eq_continuous_within)
-          fix x assume "x \<in> S"
-          then obtain X where "open X" and x: "x \<in> S \<inter> X" and finX: "finite {U \<in> \<C>. U \<inter> X \<noteq> {}}"
-            using assms by blast
-          then have OSX: "openin (subtopology euclidean S) (S \<inter> X)" by blast
-          have sumeq: "\<And>x. x \<in> S \<inter> X \<Longrightarrow>
-                     (\<Sum>V | V \<in> \<C> \<and> V \<inter> X \<noteq> {}. setdist {x} (S - V))
-                     = supp_setsum (\<lambda>V. setdist {x} (S - V)) \<C>"
-            apply (simp add: supp_setsum_def)
-            apply (rule setsum.mono_neutral_right [OF finX])
-            apply (auto simp: setdist_eq_0_sing_1 support_on_def subset_iff)
-            apply (meson DiffI closure_subset disjoint_iff_not_equal subsetCE)
-            done
-          show "continuous (at x within S) (\<lambda>x. supp_setsum (\<lambda>V. setdist {x} (S - V)) \<C>)"
-            apply (rule continuous_transform_within_openin
-                     [where f = "\<lambda>x. (setsum (\<lambda>V. setdist {x} (S - V)) {V \<in> \<C>. V \<inter> X \<noteq> {}})"
-                        and S ="S \<inter> X"])
-            apply (rule continuous_intros continuous_at_setdist continuous_at_imp_continuous_at_within OSX x)+
-            apply (simp add: sumeq)
-            done
-        qed
-        show ?thesis
-          apply (simp add: F_def)
-          apply (rule continuous_intros *)+
-          using ss_pos apply force
-          done
-      qed
-      moreover have "\<lbrakk>U \<in> \<C>; x \<in> S\<rbrakk> \<Longrightarrow> 0 \<le> F U x" for U x
-        using nonneg [of x] by (simp add: F_def divide_simps setdist_pos_le)
-      ultimately show "\<And>U. U \<in> \<C> \<Longrightarrow> continuous_on S (F U) \<and> (\<forall>x\<in>S. 0 \<le> F U x)"
-        by metis
-    next
-      show "\<And>x U. \<lbrakk>U \<in> \<C>; x \<in> S; x \<notin> U\<rbrakk> \<Longrightarrow> F U x = 0"
-        by (simp add: setdist_eq_0_sing_1 closure_def F_def)
-    next
-      show "supp_setsum (\<lambda>W. F W x) \<C> = 1" if "x \<in> S" for x
-        using that ss_pos [OF that]
-        by (simp add: F_def divide_simps supp_setsum_divide_distrib [symmetric])
-    next
-      show "\<exists>V. open V \<and> x \<in> V \<and> finite {U \<in> \<C>. \<exists>x\<in>V. F U x \<noteq> 0}" if "x \<in> S" for x
-        using fin [OF that] that
-        by (fastforce simp: setdist_eq_0_sing_1 closure_def F_def elim!: rev_finite_subset)
-    qed
-qed
-
-
-subsection\<open>Urysohn's lemma (for Euclidean spaces, where the proof is easy using distances)\<close>
-
-lemma Urysohn_both_ne:
-  assumes US: "closedin (subtopology euclidean U) S"
-      and UT: "closedin (subtopology euclidean U) T"
-      and "S \<inter> T = {}" "S \<noteq> {}" "T \<noteq> {}" "a \<noteq> b"
-  obtains f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
-    where "continuous_on U f"
-          "\<And>x. x \<in> U \<Longrightarrow> f x \<in> closed_segment a b"
-          "\<And>x. x \<in> U \<Longrightarrow> (f x = a \<longleftrightarrow> x \<in> S)"
-          "\<And>x. x \<in> U \<Longrightarrow> (f x = b \<longleftrightarrow> x \<in> T)"
-proof -
-  have S0: "\<And>x. x \<in> U \<Longrightarrow> setdist {x} S = 0 \<longleftrightarrow> x \<in> S"
-    using \<open>S \<noteq> {}\<close>  US setdist_eq_0_closedin  by auto
-  have T0: "\<And>x. x \<in> U \<Longrightarrow> setdist {x} T = 0 \<longleftrightarrow> x \<in> T"
-    using \<open>T \<noteq> {}\<close>  UT setdist_eq_0_closedin  by auto
-  have sdpos: "0 < setdist {x} S + setdist {x} T" if "x \<in> U" for x
-  proof -
-    have "~ (setdist {x} S = 0 \<and> setdist {x} T = 0)"
-      using assms by (metis IntI empty_iff setdist_eq_0_closedin that)
-    then show ?thesis
-      by (metis add.left_neutral add.right_neutral add_pos_pos linorder_neqE_linordered_idom not_le setdist_pos_le)
-  qed
-  define f where "f \<equiv> \<lambda>x. a + (setdist {x} S / (setdist {x} S + setdist {x} T)) *\<^sub>R (b - a)"
-  show ?thesis
-  proof (rule_tac f = f in that)
-    show "continuous_on U f"
-      using sdpos unfolding f_def
-      by (intro continuous_intros | force)+
-    show "f x \<in> closed_segment a b" if "x \<in> U" for x
-        unfolding f_def
-      apply (simp add: closed_segment_def)
-      apply (rule_tac x="(setdist {x} S / (setdist {x} S + setdist {x} T))" in exI)
-      using sdpos that apply (simp add: algebra_simps)
-      done
-    show "\<And>x. x \<in> U \<Longrightarrow> (f x = a \<longleftrightarrow> x \<in> S)"
-      using S0 \<open>a \<noteq> b\<close> f_def sdpos by force
-    show "(f x = b \<longleftrightarrow> x \<in> T)" if "x \<in> U" for x
-    proof -
-      have "f x = b \<longleftrightarrow> (setdist {x} S / (setdist {x} S + setdist {x} T)) = 1"
-        unfolding f_def
-        apply (rule iffI)
-         apply (metis  \<open>a \<noteq> b\<close> add_diff_cancel_left' eq_iff_diff_eq_0 pth_1 real_vector.scale_right_imp_eq, force)
-        done
-      also have "...  \<longleftrightarrow> setdist {x} T = 0 \<and> setdist {x} S \<noteq> 0"
-        using sdpos that
-        by (simp add: divide_simps) linarith
-      also have "...  \<longleftrightarrow> x \<in> T"
-        using \<open>S \<noteq> {}\<close> \<open>T \<noteq> {}\<close> \<open>S \<inter> T = {}\<close> that
-        by (force simp: S0 T0)
-      finally show ?thesis .
-    qed
-  qed
-qed
-
-proposition Urysohn_local_strong:
-  assumes US: "closedin (subtopology euclidean U) S"
-      and UT: "closedin (subtopology euclidean U) T"
-      and "S \<inter> T = {}" "a \<noteq> b"
-  obtains f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
-    where "continuous_on U f"
-          "\<And>x. x \<in> U \<Longrightarrow> f x \<in> closed_segment a b"
-          "\<And>x. x \<in> U \<Longrightarrow> (f x = a \<longleftrightarrow> x \<in> S)"
-          "\<And>x. x \<in> U \<Longrightarrow> (f x = b \<longleftrightarrow> x \<in> T)"
-proof (cases "S = {}")
-  case True show ?thesis
-  proof (cases "T = {}")
-    case True show ?thesis
-    proof (rule_tac f = "\<lambda>x. midpoint a b" in that)
-      show "continuous_on U (\<lambda>x. midpoint a b)"
-        by (intro continuous_intros)
-      show "midpoint a b \<in> closed_segment a b"
-        using csegment_midpoint_subset by blast
-      show "(midpoint a b = a) = (x \<in> S)" for x
-        using \<open>S = {}\<close> \<open>a \<noteq> b\<close> by simp
-      show "(midpoint a b = b) = (x \<in> T)" for x
-        using \<open>T = {}\<close> \<open>a \<noteq> b\<close> by simp
-    qed
-  next
-    case False
-    show ?thesis
-    proof (cases "T = U")
-      case True with \<open>S = {}\<close> \<open>a \<noteq> b\<close> show ?thesis
-        by (rule_tac f = "\<lambda>x. b" in that) (auto simp: continuous_on_const)
-    next
-      case False
-      with UT closedin_subset obtain c where c: "c \<in> U" "c \<notin> T"
-        by fastforce
-      obtain f where f: "continuous_on U f"
-                "\<And>x. x \<in> U \<Longrightarrow> f x \<in> closed_segment (midpoint a b) b"
-                "\<And>x. x \<in> U \<Longrightarrow> (f x = midpoint a b \<longleftrightarrow> x = c)"
-                "\<And>x. x \<in> U \<Longrightarrow> (f x = b \<longleftrightarrow> x \<in> T)"
-        apply (rule Urysohn_both_ne [of U "{c}" T "midpoint a b" "b"])
-        using c \<open>T \<noteq> {}\<close> assms apply simp_all
-        done
-      show ?thesis
-        apply (rule_tac f=f in that)
-        using \<open>S = {}\<close> \<open>T \<noteq> {}\<close> f csegment_midpoint_subset notin_segment_midpoint [OF \<open>a \<noteq> b\<close>]
-        apply force+
-        done
-    qed
-  qed
-next
-  case False
-  show ?thesis
-  proof (cases "T = {}")
-    case True show ?thesis
-    proof (cases "S = U")
-      case True with \<open>T = {}\<close> \<open>a \<noteq> b\<close> show ?thesis
-        by (rule_tac f = "\<lambda>x. a" in that) (auto simp: continuous_on_const)
-    next
-      case False
-      with US closedin_subset obtain c where c: "c \<in> U" "c \<notin> S"
-        by fastforce
-      obtain f where f: "continuous_on U f"
-                "\<And>x. x \<in> U \<Longrightarrow> f x \<in> closed_segment a (midpoint a b)"
-                "\<And>x. x \<in> U \<Longrightarrow> (f x = midpoint a b \<longleftrightarrow> x = c)"
-                "\<And>x. x \<in> U \<Longrightarrow> (f x = a \<longleftrightarrow> x \<in> S)"
-        apply (rule Urysohn_both_ne [of U S "{c}" a "midpoint a b"])
-        using c \<open>S \<noteq> {}\<close> assms apply simp_all
-        apply (metis midpoint_eq_endpoint)
-        done
-      show ?thesis
-        apply (rule_tac f=f in that)
-        using \<open>S \<noteq> {}\<close> \<open>T = {}\<close> f  \<open>a \<noteq> b\<close>
-        apply simp_all
-        apply (metis (no_types) closed_segment_commute csegment_midpoint_subset midpoint_sym subset_iff)
-        apply (metis closed_segment_commute midpoint_sym notin_segment_midpoint)
-        done
-    qed
-  next
-    case False
-    show ?thesis
-      using Urysohn_both_ne [OF US UT \<open>S \<inter> T = {}\<close> \<open>S \<noteq> {}\<close> \<open>T \<noteq> {}\<close> \<open>a \<noteq> b\<close>] that
-      by blast
-  qed
-qed
-
-lemma Urysohn_local:
-  assumes US: "closedin (subtopology euclidean U) S"
-      and UT: "closedin (subtopology euclidean U) T"
-      and "S \<inter> T = {}"
-  obtains f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
-    where "continuous_on U f"
-          "\<And>x. x \<in> U \<Longrightarrow> f x \<in> closed_segment a b"
-          "\<And>x. x \<in> S \<Longrightarrow> f x = a"
-          "\<And>x. x \<in> T \<Longrightarrow> f x = b"
-proof (cases "a = b")
-  case True then show ?thesis
-    by (rule_tac f = "\<lambda>x. b" in that) (auto simp: continuous_on_const)
-next
-  case False
-  then show ?thesis
-    apply (rule Urysohn_local_strong [OF assms])
-    apply (erule that, assumption)
-    apply (meson US closedin_singleton closedin_trans)
-    apply (meson UT closedin_singleton closedin_trans)
-    done
-qed
-
-lemma Urysohn_strong:
-  assumes US: "closed S"
-      and UT: "closed T"
-      and "S \<inter> T = {}" "a \<noteq> b"
-  obtains f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
-    where "continuous_on UNIV f"
-          "\<And>x. f x \<in> closed_segment a b"
-          "\<And>x. f x = a \<longleftrightarrow> x \<in> S"
-          "\<And>x. f x = b \<longleftrightarrow> x \<in> T"
-apply (rule Urysohn_local_strong [of UNIV S T])
-using assms
-apply (auto simp: closed_closedin)
-done
-
-proposition Urysohn:
-  assumes US: "closed S"
-      and UT: "closed T"
-      and "S \<inter> T = {}"
-  obtains f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
-    where "continuous_on UNIV f"
-          "\<And>x. f x \<in> closed_segment a b"
-          "\<And>x. x \<in> S \<Longrightarrow> f x = a"
-          "\<And>x. x \<in> T \<Longrightarrow> f x = b"
-apply (rule Urysohn_local [of UNIV S T a b])
-using assms
-apply (auto simp: closed_closedin)
-done
-
-
-subsection\<open> The Dugundji extension theorem, and Tietze variants as corollaries.\<close>
-
-text\<open>J. Dugundji. An extension of Tietze's theorem. Pacific J. Math. Volume 1, Number 3 (1951), 353-367.
-http://projecteuclid.org/euclid.pjm/1103052106\<close>
-
-theorem Dugundji:
-  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_inner"
-  assumes "convex C" "C \<noteq> {}"
-      and cloin: "closedin (subtopology euclidean U) S"
-      and contf: "continuous_on S f" and "f ` S \<subseteq> C"
-  obtains g where "continuous_on U g" "g ` U \<subseteq> C"
-                  "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
-proof (cases "S = {}")
-  case True then show thesis
-    apply (rule_tac g="\<lambda>x. @y. y \<in> C" in that)
-      apply (rule continuous_intros)
-     apply (meson all_not_in_conv \<open>C \<noteq> {}\<close> image_subsetI someI_ex, simp)
-    done
-next
-  case False
-  then have sd_pos: "\<And>x. \<lbrakk>x \<in> U; x \<notin> S\<rbrakk> \<Longrightarrow> 0 < setdist {x} S"
-    using setdist_eq_0_closedin [OF cloin] le_less setdist_pos_le by fastforce
-  define \<B> where "\<B> = {ball x (setdist {x} S / 2) |x. x \<in> U - S}"
-  have [simp]: "\<And>T. T \<in> \<B> \<Longrightarrow> open T"
-    by (auto simp: \<B>_def)
-  have USS: "U - S \<subseteq> \<Union>\<B>"
-    by (auto simp: sd_pos \<B>_def)
-  obtain \<C> where USsub: "U - S \<subseteq> \<Union>\<C>"
-       and nbrhd: "\<And>U. U \<in> \<C> \<Longrightarrow> open U \<and> (\<exists>T. T \<in> \<B> \<and> U \<subseteq> T)"
-       and fin: "\<And>x. x \<in> U - S
-                     \<Longrightarrow> \<exists>V. open V \<and> x \<in> V \<and> finite {U. U \<in> \<C> \<and> U \<inter> V \<noteq> {}}"
-    using paracompact [OF USS] by auto
-  have "\<exists>v a. v \<in> U \<and> v \<notin> S \<and> a \<in> S \<and>
-              T \<subseteq> ball v (setdist {v} S / 2) \<and>
-              dist v a \<le> 2 * setdist {v} S" if "T \<in> \<C>" for T
-  proof -
-    obtain v where v: "T \<subseteq> ball v (setdist {v} S / 2)" "v \<in> U" "v \<notin> S"
-      using \<open>T \<in> \<C>\<close> nbrhd by (force simp: \<B>_def)
-    then obtain a where "a \<in> S" "dist v a < 2 * setdist {v} S"
-      using setdist_ltE [of "{v}" S "2 * setdist {v} S"]
-      using False sd_pos by force
-    with v show ?thesis
-      apply (rule_tac x=v in exI)
-      apply (rule_tac x=a in exI, auto)
-      done
-  qed
-  then obtain \<V> \<A> where
-    VA: "\<And>T. T \<in> \<C> \<Longrightarrow> \<V> T \<in> U \<and> \<V> T \<notin> S \<and> \<A> T \<in> S \<and>
-              T \<subseteq> ball (\<V> T) (setdist {\<V> T} S / 2) \<and>
-              dist (\<V> T) (\<A> T) \<le> 2 * setdist {\<V> T} S"
-    by metis
-  have sdle: "setdist {\<V> T} S \<le> 2 * setdist {v} S" if "T \<in> \<C>" "v \<in> T" for T v
-    using setdist_Lipschitz [of "\<V> T" S v] VA [OF \<open>T \<in> \<C>\<close>] \<open>v \<in> T\<close> by auto
-  have d6: "dist a (\<A> T) \<le> 6 * dist a v" if "T \<in> \<C>" "v \<in> T" "a \<in> S" for T v a
-  proof -
-    have "dist (\<V> T) v < setdist {\<V> T} S / 2"
-      using that VA mem_ball by blast
-    also have "... \<le> setdist {v} S"
-      using sdle [OF \<open>T \<in> \<C>\<close> \<open>v \<in> T\<close>] by simp
-    also have vS: "setdist {v} S \<le> dist a v"
-      by (simp add: setdist_le_dist setdist_sym \<open>a \<in> S\<close>)
-    finally have VTV: "dist (\<V> T) v < dist a v" .
-    have VTS: "setdist {\<V> T} S \<le> 2 * dist a v"
-      using sdle that vS by force
-    have "dist a (\<A> T) \<le> dist a v + dist v (\<V> T) + dist (\<V> T) (\<A> T)"
-      by (metis add.commute add_le_cancel_left dist_commute dist_triangle2 dist_triangle_le)
-    also have "... \<le> dist a v + dist a v + dist (\<V> T) (\<A> T)"
-      using VTV by (simp add: dist_commute)
-    also have "... \<le> 2 * dist a v + 2 * setdist {\<V> T} S"
-      using VA [OF \<open>T \<in> \<C>\<close>] by auto
-    finally show ?thesis
-      using VTS by linarith
-  qed
-  obtain H :: "['a set, 'a] \<Rightarrow> real"
-    where Hcont: "\<And>Z. Z \<in> \<C> \<Longrightarrow> continuous_on (U-S) (H Z)"
-      and Hge0: "\<And>Z x. \<lbrakk>Z \<in> \<C>; x \<in> U-S\<rbrakk> \<Longrightarrow> 0 \<le> H Z x"
-      and Heq0: "\<And>x Z. \<lbrakk>Z \<in> \<C>; x \<in> U-S; x \<notin> Z\<rbrakk> \<Longrightarrow> H Z x = 0"
-      and H1: "\<And>x. x \<in> U-S \<Longrightarrow> supp_setsum (\<lambda>W. H W x) \<C> = 1"
-      and Hfin: "\<And>x. x \<in> U-S \<Longrightarrow> \<exists>V. open V \<and> x \<in> V \<and> finite {U \<in> \<C>. \<exists>x\<in>V. H U x \<noteq> 0}"
-    apply (rule subordinate_partition_of_unity [OF USsub _ fin])
-    using nbrhd by auto
-  define g where "g \<equiv> \<lambda>x. if x \<in> S then f x else supp_setsum (\<lambda>T. H T x *\<^sub>R f(\<A> T)) \<C>"
-  show ?thesis
-  proof (rule that)
-    show "continuous_on U g"
-    proof (clarsimp simp: continuous_on_eq_continuous_within)
-      fix a assume "a \<in> U"
-      show "continuous (at a within U) g"
-      proof (cases "a \<in> S")
-        case True show ?thesis
-        proof (clarsimp simp add: continuous_within_topological)
-          fix W
-          assume "open W" "g a \<in> W"
-          then obtain e where "0 < e" and e: "ball (f a) e \<subseteq> W"
-            using openE True g_def by auto
-          have "continuous (at a within S) f"
-            using True contf continuous_on_eq_continuous_within by blast
-          then obtain d where "0 < d"
-                        and d: "\<And>x. \<lbrakk>x \<in> S; dist x a < d\<rbrakk> \<Longrightarrow> dist (f x) (f a) < e"
-            using continuous_within_eps_delta \<open>0 < e\<close> by force
-          have "g y \<in> ball (f a) e" if "y \<in> U" and y: "y \<in> ball a (d / 6)" for y
-          proof (cases "y \<in> S")
-            case True
-            then have "dist (f a) (f y) < e"
-              by (metis ball_divide_subset_numeral dist_commute in_mono mem_ball y d)
-            then show ?thesis
-              by (simp add: True g_def)
-          next
-            case False
-            have *: "dist (f (\<A> T)) (f a) < e" if "T \<in> \<C>" "H T y \<noteq> 0" for T
-            proof -
-              have "y \<in> T"
-                using Heq0 that False \<open>y \<in> U\<close> by blast
-              have "dist (\<A> T) a < d"
-                using d6 [OF \<open>T \<in> \<C>\<close> \<open>y \<in> T\<close> \<open>a \<in> S\<close>] y
-                by (simp add: dist_commute mult.commute)
-              then show ?thesis
-                using VA [OF \<open>T \<in> \<C>\<close>] by (auto simp: d)
-            qed
-            have "supp_setsum (\<lambda>T. H T y *\<^sub>R f (\<A> T)) \<C> \<in> ball (f a) e"
-              apply (rule convex_supp_setsum [OF convex_ball])
-              apply (simp_all add: False H1 Hge0 \<open>y \<in> U\<close>)
-              by (metis dist_commute *)
-            then show ?thesis
-              by (simp add: False g_def)
-          qed
-          then show "\<exists>A. open A \<and> a \<in> A \<and> (\<forall>y\<in>U. y \<in> A \<longrightarrow> g y \<in> W)"
-            apply (rule_tac x = "ball a (d / 6)" in exI)
-            using e \<open>0 < d\<close> by fastforce
-        qed
-      next
-        case False
-        obtain N where N: "open N" "a \<in> N"
-                   and finN: "finite {U \<in> \<C>. \<exists>a\<in>N. H U a \<noteq> 0}"
-          using Hfin False \<open>a \<in> U\<close> by auto
-        have oUS: "openin (subtopology euclidean U) (U - S)"
-          using cloin by (simp add: openin_diff)
-        have HcontU: "continuous (at a within U) (H T)" if "T \<in> \<C>" for T
-          using Hcont [OF \<open>T \<in> \<C>\<close>] False  \<open>a \<in> U\<close> \<open>T \<in> \<C>\<close>
-          apply (simp add: continuous_on_eq_continuous_within continuous_within)
-          apply (rule Lim_transform_within_set)
-          using oUS
-            apply (force simp: eventually_at openin_contains_ball dist_commute dest!: bspec)+
-          done
-        show ?thesis
-        proof (rule continuous_transform_within_openin [OF _ oUS])
-          show "continuous (at a within U) (\<lambda>x. supp_setsum (\<lambda>T. H T x *\<^sub>R f (\<A> T)) \<C>)"
-          proof (rule continuous_transform_within_openin)
-            show "continuous (at a within U)
-                    (\<lambda>x. \<Sum>T\<in>{U \<in> \<C>. \<exists>x\<in>N. H U x \<noteq> 0}. H T x *\<^sub>R f (\<A> T))"
-              by (force intro: continuous_intros HcontU)+
-          next
-            show "openin (subtopology euclidean U) ((U - S) \<inter> N)"
-              using N oUS openin_trans by blast
-          next
-            show "a \<in> (U - S) \<inter> N" using False \<open>a \<in> U\<close> N by blast
-          next
-            show "\<And>x. x \<in> (U - S) \<inter> N \<Longrightarrow>
-                         (\<Sum>T \<in> {U \<in> \<C>. \<exists>x\<in>N. H U x \<noteq> 0}. H T x *\<^sub>R f (\<A> T))
-                         = supp_setsum (\<lambda>T. H T x *\<^sub>R f (\<A> T)) \<C>"
-              by (auto simp: supp_setsum_def support_on_def
-                       intro: setsum.mono_neutral_right [OF finN])
-          qed
-        next
-          show "a \<in> U - S" using False \<open>a \<in> U\<close> by blast
-        next
-          show "\<And>x. x \<in> U - S \<Longrightarrow> supp_setsum (\<lambda>T. H T x *\<^sub>R f (\<A> T)) \<C> = g x"
-            by (simp add: g_def)
-        qed
-      qed
-    qed
-    show "g ` U \<subseteq> C"
-      using \<open>f ` S \<subseteq> C\<close> VA
-      by (fastforce simp: g_def Hge0 intro!: convex_supp_setsum [OF \<open>convex C\<close>] H1)
-    show "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
-      by (simp add: g_def)
-  qed
-qed
-
-
-corollary Tietze:
-  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_inner"
-  assumes "continuous_on S f"
-      and "closedin (subtopology euclidean U) S"
-      and "0 \<le> B"
-      and "\<And>x. x \<in> S \<Longrightarrow> norm(f x) \<le> B"
-  obtains g where "continuous_on U g" "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
-                  "\<And>x. x \<in> U \<Longrightarrow> norm(g x) \<le> B"
-using assms
-by (auto simp: image_subset_iff intro: Dugundji [of "cball 0 B" U S f])
-
-corollary Tietze_closed_interval:
-  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
-  assumes "continuous_on S f"
-      and "closedin (subtopology euclidean U) S"
-      and "cbox a b \<noteq> {}"
-      and "\<And>x. x \<in> S \<Longrightarrow> f x \<in> cbox a b"
-  obtains g where "continuous_on U g" "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
-                  "\<And>x. x \<in> U \<Longrightarrow> g x \<in> cbox a b"
-apply (rule Dugundji [of "cbox a b" U S f])
-using assms by auto
-
-corollary Tietze_closed_interval_1:
-  fixes f :: "'a::euclidean_space \<Rightarrow> real"
-  assumes "continuous_on S f"
-      and "closedin (subtopology euclidean U) S"
-      and "a \<le> b"
-      and "\<And>x. x \<in> S \<Longrightarrow> f x \<in> cbox a b"
-  obtains g where "continuous_on U g" "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
-                  "\<And>x. x \<in> U \<Longrightarrow> g x \<in> cbox a b"
-apply (rule Dugundji [of "cbox a b" U S f])
-using assms by (auto simp: image_subset_iff)
-
-corollary Tietze_open_interval:
-  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
-  assumes "continuous_on S f"
-      and "closedin (subtopology euclidean U) S"
-      and "box a b \<noteq> {}"
-      and "\<And>x. x \<in> S \<Longrightarrow> f x \<in> box a b"
-  obtains g where "continuous_on U g" "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
-                  "\<And>x. x \<in> U \<Longrightarrow> g x \<in> box a b"
-apply (rule Dugundji [of "box a b" U S f])
-using assms by auto
-
-corollary Tietze_open_interval_1:
-  fixes f :: "'a::euclidean_space \<Rightarrow> real"
-  assumes "continuous_on S f"
-      and "closedin (subtopology euclidean U) S"
-      and "a < b"
-      and no: "\<And>x. x \<in> S \<Longrightarrow> f x \<in> box a b"
-  obtains g where "continuous_on U g" "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
-                  "\<And>x. x \<in> U \<Longrightarrow> g x \<in> box a b"
-apply (rule Dugundji [of "box a b" U S f])
-using assms by (auto simp: image_subset_iff)
-
-corollary Tietze_unbounded:
-  fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_inner"
-  assumes "continuous_on S f"
-      and "closedin (subtopology euclidean U) S"
-  obtains g where "continuous_on U g" "\<And>x. x \<in> S \<Longrightarrow> g x = f x"
-apply (rule Dugundji [of UNIV U S f])
-using assms by auto
-
-end