author nipkow Sun, 08 Dec 2019 17:42:53 +0100 changeset 71255 4258ee13f5d4 parent 71244 38457af660bc child 71256 dd74e0558fd1
moved lemmas
```--- a/src/HOL/Analysis/Continuous_Extension.thy	Fri Dec 06 17:03:58 2019 +0100
+++ b/src/HOL/Analysis/Continuous_Extension.thy	Sun Dec 08 17:42:53 2019 +0100
@@ -298,6 +298,20 @@

text \<open>See \cite{dugundji}.\<close>

+lemma convex_supp_sum:
+  assumes "convex S" and 1: "supp_sum u I = 1"
+      and "\<And>i. i \<in> I \<Longrightarrow> 0 \<le> u i \<and> (u i = 0 \<or> f i \<in> S)"
+    shows "supp_sum (\<lambda>i. u i *\<^sub>R f i) I \<in> S"
+proof -
+  have fin: "finite {i \<in> I. u i \<noteq> 0}"
+    using 1 sum.infinite by (force simp: supp_sum_def support_on_def)
+  then have "supp_sum (\<lambda>i. u i *\<^sub>R f i) I = sum (\<lambda>i. u i *\<^sub>R f i) {i \<in> I. u i \<noteq> 0}"
+    by (force intro: sum.mono_neutral_left simp: supp_sum_def support_on_def)
+  also have "... \<in> S"
+    using 1 assms by (force simp: supp_sum_def support_on_def intro: convex_sum [OF fin \<open>convex S\<close>])
+  finally show ?thesis .
+qed
+
theorem Dugundji:
fixes f :: "'a::{metric_space,second_countable_topology} \<Rightarrow> 'b::real_inner"
assumes "convex C" "C \<noteq> {}"```
```--- a/src/HOL/Analysis/Line_Segment.thy	Fri Dec 06 17:03:58 2019 +0100
+++ b/src/HOL/Analysis/Line_Segment.thy	Sun Dec 08 17:42:53 2019 +0100
@@ -30,76 +30,6 @@
finally show ?thesis .
qed

-lemma closure_bounded_linear_image_subset:
-  assumes f: "bounded_linear f"
-  shows "f ` closure S \<subseteq> closure (f ` S)"
-  using linear_continuous_on [OF f] closed_closure closure_subset
-  by (rule image_closure_subset)
-
-lemma closure_linear_image_subset:
-  fixes f :: "'m::euclidean_space \<Rightarrow> 'n::real_normed_vector"
-  assumes "linear f"
-  shows "f ` (closure S) \<subseteq> closure (f ` S)"
-  using assms unfolding linear_conv_bounded_linear
-  by (rule closure_bounded_linear_image_subset)
-
-lemma closed_injective_linear_image:
-    fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
-    assumes S: "closed S" and f: "linear f" "inj f"
-    shows "closed (f ` S)"
-proof -
-  obtain g where g: "linear g" "g \<circ> f = id"
-    using linear_injective_left_inverse [OF f] by blast
-  then have confg: "continuous_on (range f) g"
-    using linear_continuous_on linear_conv_bounded_linear by blast
-  have [simp]: "g ` f ` S = S"
-    using g by (simp add: image_comp)
-  have cgf: "closed (g ` f ` S)"
-    by (simp add: \<open>g \<circ> f = id\<close> S image_comp)
-  have [simp]: "(range f \<inter> g -` S) = f ` S"
-    using g unfolding o_def id_def image_def by auto metis+
-  show ?thesis
-  proof (rule closedin_closed_trans [of "range f"])
-    show "closedin (top_of_set (range f)) (f ` S)"
-      using continuous_closedin_preimage [OF confg cgf] by simp
-    show "closed (range f)"
-      apply (rule closed_injective_image_subspace)
-      using f apply (auto simp: linear_linear linear_injective_0)
-      done
-  qed
-qed
-
-lemma closed_injective_linear_image_eq:
-    fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
-    assumes f: "linear f" "inj f"
-      shows "(closed(image f s) \<longleftrightarrow> closed s)"
-  by (metis closed_injective_linear_image closure_eq closure_linear_image_subset closure_subset_eq f(1) f(2) inj_image_subset_iff)
-
-lemma closure_injective_linear_image:
-    fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
-    shows "\<lbrakk>linear f; inj f\<rbrakk> \<Longrightarrow> f ` (closure S) = closure (f ` S)"
-  apply (rule subset_antisym)
-  by (simp add: closure_minimal closed_injective_linear_image closure_subset image_mono)
-
-lemma closure_bounded_linear_image:
-    fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
-    shows "\<lbrakk>linear f; bounded S\<rbrakk> \<Longrightarrow> f ` (closure S) = closure (f ` S)"
-  apply (rule subset_antisym, simp add: closure_linear_image_subset)
-  apply (rule closure_minimal, simp add: closure_subset image_mono)
-  by (meson bounded_closure closed_closure compact_continuous_image compact_eq_bounded_closed linear_continuous_on linear_conv_bounded_linear)
-
-lemma closure_scaleR:
-  fixes S :: "'a::real_normed_vector set"
-  shows "((*\<^sub>R) c) ` (closure S) = closure (((*\<^sub>R) c) ` S)"
-proof
-  show "((*\<^sub>R) c) ` (closure S) \<subseteq> closure (((*\<^sub>R) c) ` S)"
-    using bounded_linear_scaleR_right
-    by (rule closure_bounded_linear_image_subset)
-  show "closure (((*\<^sub>R) c) ` S) \<subseteq> ((*\<^sub>R) c) ` (closure S)"
-    by (intro closure_minimal image_mono closure_subset closed_scaling closed_closure)
-qed
-
lemma sphere_eq_empty [simp]:
fixes a :: "'a::{real_normed_vector, perfect_space}"
shows "sphere a r = {} \<longleftrightarrow> r < 0"```
```--- a/src/HOL/Analysis/Topology_Euclidean_Space.thy	Fri Dec 06 17:03:58 2019 +0100
+++ b/src/HOL/Analysis/Topology_Euclidean_Space.thy	Sun Dec 08 17:42:53 2019 +0100
@@ -2340,6 +2340,77 @@
using complete_isometric_image[OF \<open>e>0\<close> assms(1,2) e] and assms(4)
unfolding complete_eq_closed[symmetric] by auto
qed
+
+
+lemma closure_bounded_linear_image_subset:
+  assumes f: "bounded_linear f"
+  shows "f ` closure S \<subseteq> closure (f ` S)"
+  using linear_continuous_on [OF f] closed_closure closure_subset
+  by (rule image_closure_subset)
+
+lemma closure_linear_image_subset:
+  fixes f :: "'m::euclidean_space \<Rightarrow> 'n::real_normed_vector"
+  assumes "linear f"
+  shows "f ` (closure S) \<subseteq> closure (f ` S)"
+  using assms unfolding linear_conv_bounded_linear
+  by (rule closure_bounded_linear_image_subset)
+
+lemma closed_injective_linear_image:
+    fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
+    assumes S: "closed S" and f: "linear f" "inj f"
+    shows "closed (f ` S)"
+proof -
+  obtain g where g: "linear g" "g \<circ> f = id"
+    using linear_injective_left_inverse [OF f] by blast
+  then have confg: "continuous_on (range f) g"
+    using linear_continuous_on linear_conv_bounded_linear by blast
+  have [simp]: "g ` f ` S = S"
+    using g by (simp add: image_comp)
+  have cgf: "closed (g ` f ` S)"
+    by (simp add: \<open>g \<circ> f = id\<close> S image_comp)
+  have [simp]: "(range f \<inter> g -` S) = f ` S"
+    using g unfolding o_def id_def image_def by auto metis+
+  show ?thesis
+  proof (rule closedin_closed_trans [of "range f"])
+    show "closedin (top_of_set (range f)) (f ` S)"
+      using continuous_closedin_preimage [OF confg cgf] by simp
+    show "closed (range f)"
+      apply (rule closed_injective_image_subspace)
+      using f apply (auto simp: linear_linear linear_injective_0)
+      done
+  qed
+qed
+
+lemma closed_injective_linear_image_eq:
+    fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
+    assumes f: "linear f" "inj f"
+      shows "(closed(image f s) \<longleftrightarrow> closed s)"
+  by (metis closed_injective_linear_image closure_eq closure_linear_image_subset closure_subset_eq f(1) f(2) inj_image_subset_iff)
+
+lemma closure_injective_linear_image:
+    fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
+    shows "\<lbrakk>linear f; inj f\<rbrakk> \<Longrightarrow> f ` (closure S) = closure (f ` S)"
+  apply (rule subset_antisym)
+  by (simp add: closure_minimal closed_injective_linear_image closure_subset image_mono)
+
+lemma closure_bounded_linear_image:
+    fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
+    shows "\<lbrakk>linear f; bounded S\<rbrakk> \<Longrightarrow> f ` (closure S) = closure (f ` S)"
+  apply (rule subset_antisym, simp add: closure_linear_image_subset)
+  apply (rule closure_minimal, simp add: closure_subset image_mono)
+  by (meson bounded_closure closed_closure compact_continuous_image compact_eq_bounded_closed linear_continuous_on linear_conv_bounded_linear)
+
+lemma closure_scaleR:
+  fixes S :: "'a::real_normed_vector set"
+  shows "((*\<^sub>R) c) ` (closure S) = closure (((*\<^sub>R) c) ` S)"
+proof
+  show "((*\<^sub>R) c) ` (closure S) \<subseteq> closure (((*\<^sub>R) c) ` S)"
+    using bounded_linear_scaleR_right
+    by (rule closure_bounded_linear_image_subset)
+  show "closure (((*\<^sub>R) c) ` S) \<subseteq> ((*\<^sub>R) c) ` (closure S)"
+    by (intro closure_minimal image_mono closure_subset closed_scaling closed_closure)
+qed

subsection\<^marker>\<open>tag unimportant\<close> \<open>Some properties of a canonical subspace\<close>```