--- a/src/HOL/Analysis/Topology_Euclidean_Space.thy Tue Oct 10 14:03:51 2017 +0100
+++ b/src/HOL/Analysis/Topology_Euclidean_Space.thy Tue Oct 10 17:15:37 2017 +0100
@@ -7,7 +7,7 @@
section \<open>Elementary topology in Euclidean space.\<close>
theory Topology_Euclidean_Space
-imports
+imports
"HOL-Library.Indicator_Function"
"HOL-Library.Countable_Set"
"HOL-Library.FuncSet"
@@ -1208,6 +1208,38 @@
lemma ball_empty: "e \<le> 0 \<Longrightarrow> ball x e = {}" by simp
+lemma closed_cball [iff]: "closed (cball x e)"
+proof -
+ have "closed (dist x -` {..e})"
+ by (intro closed_vimage closed_atMost continuous_intros)
+ also have "dist x -` {..e} = cball x e"
+ by auto
+ finally show ?thesis .
+qed
+
+lemma open_contains_cball: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. cball x e \<subseteq> S)"
+proof -
+ {
+ fix x and e::real
+ assume "x\<in>S" "e>0" "ball x e \<subseteq> S"
+ then have "\<exists>d>0. cball x d \<subseteq> S" unfolding subset_eq by (rule_tac x="e/2" in exI, auto)
+ }
+ moreover
+ {
+ fix x and e::real
+ assume "x\<in>S" "e>0" "cball x e \<subseteq> S"
+ then have "\<exists>d>0. ball x d \<subseteq> S"
+ unfolding subset_eq
+ apply (rule_tac x="e/2" in exI, auto)
+ done
+ }
+ ultimately show ?thesis
+ unfolding open_contains_ball by auto
+qed
+
+lemma open_contains_cball_eq: "open S \<Longrightarrow> (\<forall>x. x \<in> S \<longleftrightarrow> (\<exists>e>0. cball x e \<subseteq> S))"
+ by (metis open_contains_cball subset_eq order_less_imp_le centre_in_cball)
+
lemma euclidean_dist_l2:
fixes x y :: "'a :: euclidean_space"
shows "dist x y = setL2 (\<lambda>i. dist (x \<bullet> i) (y \<bullet> i)) Basis"
@@ -1223,7 +1255,6 @@
lemma eventually_at_ball': "d > 0 \<Longrightarrow> eventually (\<lambda>t. t \<in> ball z d \<and> t \<noteq> z \<and> t \<in> A) (at z within A)"
unfolding eventually_at by (intro exI[of _ d]) (simp_all add: dist_commute)
-
subsection \<open>Boxes\<close>
abbreviation One :: "'a::euclidean_space"
@@ -1879,64 +1910,6 @@
qed
-subsection \<open>Connectedness\<close>
-
-lemma connected_local:
- "connected S \<longleftrightarrow>
- \<not> (\<exists>e1 e2.
- openin (subtopology euclidean S) e1 \<and>
- openin (subtopology euclidean S) e2 \<and>
- S \<subseteq> e1 \<union> e2 \<and>
- e1 \<inter> e2 = {} \<and>
- e1 \<noteq> {} \<and>
- e2 \<noteq> {})"
- unfolding connected_def openin_open
- by safe blast+
-
-lemma exists_diff:
- fixes P :: "'a set \<Rightarrow> bool"
- shows "(\<exists>S. P (- S)) \<longleftrightarrow> (\<exists>S. P S)"
- (is "?lhs \<longleftrightarrow> ?rhs")
-proof -
- have ?rhs if ?lhs
- using that by blast
- moreover have "P (- (- S))" if "P S" for S
- proof -
- have "S = - (- S)" by simp
- with that show ?thesis by metis
- qed
- ultimately show ?thesis by metis
-qed
-
-lemma connected_clopen: "connected S \<longleftrightarrow>
- (\<forall>T. openin (subtopology euclidean S) T \<and>
- closedin (subtopology euclidean S) T \<longrightarrow> T = {} \<or> T = S)" (is "?lhs \<longleftrightarrow> ?rhs")
-proof -
- have "\<not> connected S \<longleftrightarrow>
- (\<exists>e1 e2. open e1 \<and> open (- e2) \<and> S \<subseteq> e1 \<union> (- e2) \<and> e1 \<inter> (- e2) \<inter> S = {} \<and> e1 \<inter> S \<noteq> {} \<and> (- e2) \<inter> S \<noteq> {})"
- unfolding connected_def openin_open closedin_closed
- by (metis double_complement)
- then have th0: "connected S \<longleftrightarrow>
- \<not> (\<exists>e2 e1. closed e2 \<and> open e1 \<and> S \<subseteq> e1 \<union> (- e2) \<and> e1 \<inter> (- e2) \<inter> S = {} \<and> e1 \<inter> S \<noteq> {} \<and> (- e2) \<inter> S \<noteq> {})"
- (is " _ \<longleftrightarrow> \<not> (\<exists>e2 e1. ?P e2 e1)")
- by (simp add: closed_def) metis
- have th1: "?rhs \<longleftrightarrow> \<not> (\<exists>t' t. closed t'\<and>t = S\<inter>t' \<and> t\<noteq>{} \<and> t\<noteq>S \<and> (\<exists>t'. open t' \<and> t = S \<inter> t'))"
- (is "_ \<longleftrightarrow> \<not> (\<exists>t' t. ?Q t' t)")
- unfolding connected_def openin_open closedin_closed by auto
- have "(\<exists>e1. ?P e2 e1) \<longleftrightarrow> (\<exists>t. ?Q e2 t)" for e2
- proof -
- have "?P e2 e1 \<longleftrightarrow> (\<exists>t. closed e2 \<and> t = S\<inter>e2 \<and> open e1 \<and> t = S\<inter>e1 \<and> t\<noteq>{} \<and> t \<noteq> S)" for e1
- by auto
- then show ?thesis
- by metis
- qed
- then have "\<forall>e2. (\<exists>e1. ?P e2 e1) \<longleftrightarrow> (\<exists>t. ?Q e2 t)"
- by blast
- then show ?thesis
- by (simp add: th0 th1)
-qed
-
-
subsection \<open>Limit points\<close>
definition (in topological_space) islimpt:: "'a \<Rightarrow> 'a set \<Rightarrow> bool" (infixr "islimpt" 60)
@@ -2511,411 +2484,6 @@
qed
-subsection \<open>Connected components, considered as a connectedness relation or a set\<close>
-
-definition "connected_component s x y \<equiv> \<exists>t. connected t \<and> t \<subseteq> s \<and> x \<in> t \<and> y \<in> t"
-
-abbreviation "connected_component_set s x \<equiv> Collect (connected_component s x)"
-
-lemma connected_componentI:
- "connected t \<Longrightarrow> t \<subseteq> s \<Longrightarrow> x \<in> t \<Longrightarrow> y \<in> t \<Longrightarrow> connected_component s x y"
- by (auto simp: connected_component_def)
-
-lemma connected_component_in: "connected_component s x y \<Longrightarrow> x \<in> s \<and> y \<in> s"
- by (auto simp: connected_component_def)
-
-lemma connected_component_refl: "x \<in> s \<Longrightarrow> connected_component s x x"
- by (auto simp: connected_component_def) (use connected_sing in blast)
-
-lemma connected_component_refl_eq [simp]: "connected_component s x x \<longleftrightarrow> x \<in> s"
- by (auto simp: connected_component_refl) (auto simp: connected_component_def)
-
-lemma connected_component_sym: "connected_component s x y \<Longrightarrow> connected_component s y x"
- by (auto simp: connected_component_def)
-
-lemma connected_component_trans:
- "connected_component s x y \<Longrightarrow> connected_component s y z \<Longrightarrow> connected_component s x z"
- unfolding connected_component_def
- by (metis Int_iff Un_iff Un_subset_iff equals0D connected_Un)
-
-lemma connected_component_of_subset:
- "connected_component s x y \<Longrightarrow> s \<subseteq> t \<Longrightarrow> connected_component t x y"
- by (auto simp: connected_component_def)
-
-lemma connected_component_Union: "connected_component_set s x = \<Union>{t. connected t \<and> x \<in> t \<and> t \<subseteq> s}"
- by (auto simp: connected_component_def)
-
-lemma connected_connected_component [iff]: "connected (connected_component_set s x)"
- by (auto simp: connected_component_Union intro: connected_Union)
-
-lemma connected_iff_eq_connected_component_set:
- "connected s \<longleftrightarrow> (\<forall>x \<in> s. connected_component_set s x = s)"
-proof (cases "s = {}")
- case True
- then show ?thesis by simp
-next
- case False
- then obtain x where "x \<in> s" by auto
- show ?thesis
- proof
- assume "connected s"
- then show "\<forall>x \<in> s. connected_component_set s x = s"
- by (force simp: connected_component_def)
- next
- assume "\<forall>x \<in> s. connected_component_set s x = s"
- then show "connected s"
- by (metis \<open>x \<in> s\<close> connected_connected_component)
- qed
-qed
-
-lemma connected_component_subset: "connected_component_set s x \<subseteq> s"
- using connected_component_in by blast
-
-lemma connected_component_eq_self: "connected s \<Longrightarrow> x \<in> s \<Longrightarrow> connected_component_set s x = s"
- by (simp add: connected_iff_eq_connected_component_set)
-
-lemma connected_iff_connected_component:
- "connected s \<longleftrightarrow> (\<forall>x \<in> s. \<forall>y \<in> s. connected_component s x y)"
- using connected_component_in by (auto simp: connected_iff_eq_connected_component_set)
-
-lemma connected_component_maximal:
- "x \<in> t \<Longrightarrow> connected t \<Longrightarrow> t \<subseteq> s \<Longrightarrow> t \<subseteq> (connected_component_set s x)"
- using connected_component_eq_self connected_component_of_subset by blast
-
-lemma connected_component_mono:
- "s \<subseteq> t \<Longrightarrow> connected_component_set s x \<subseteq> connected_component_set t x"
- by (simp add: Collect_mono connected_component_of_subset)
-
-lemma connected_component_eq_empty [simp]: "connected_component_set s x = {} \<longleftrightarrow> x \<notin> s"
- using connected_component_refl by (fastforce simp: connected_component_in)
-
-lemma connected_component_set_empty [simp]: "connected_component_set {} x = {}"
- using connected_component_eq_empty by blast
-
-lemma connected_component_eq:
- "y \<in> connected_component_set s x \<Longrightarrow> (connected_component_set s y = connected_component_set s x)"
- by (metis (no_types, lifting)
- Collect_cong connected_component_sym connected_component_trans mem_Collect_eq)
-
-lemma closed_connected_component:
- assumes s: "closed s"
- shows "closed (connected_component_set s x)"
-proof (cases "x \<in> s")
- case False
- then show ?thesis
- by (metis connected_component_eq_empty closed_empty)
-next
- case True
- show ?thesis
- unfolding closure_eq [symmetric]
- proof
- show "closure (connected_component_set s x) \<subseteq> connected_component_set s x"
- apply (rule connected_component_maximal)
- apply (simp add: closure_def True)
- apply (simp add: connected_imp_connected_closure)
- apply (simp add: s closure_minimal connected_component_subset)
- done
- next
- show "connected_component_set s x \<subseteq> closure (connected_component_set s x)"
- by (simp add: closure_subset)
- qed
-qed
-
-lemma connected_component_disjoint:
- "connected_component_set s a \<inter> connected_component_set s b = {} \<longleftrightarrow>
- a \<notin> connected_component_set s b"
- apply (auto simp: connected_component_eq)
- using connected_component_eq connected_component_sym
- apply blast
- done
-
-lemma connected_component_nonoverlap:
- "connected_component_set s a \<inter> connected_component_set s b = {} \<longleftrightarrow>
- a \<notin> s \<or> b \<notin> s \<or> connected_component_set s a \<noteq> connected_component_set s b"
- apply (auto simp: connected_component_in)
- using connected_component_refl_eq
- apply blast
- apply (metis connected_component_eq mem_Collect_eq)
- apply (metis connected_component_eq mem_Collect_eq)
- done
-
-lemma connected_component_overlap:
- "connected_component_set s a \<inter> connected_component_set s b \<noteq> {} \<longleftrightarrow>
- a \<in> s \<and> b \<in> s \<and> connected_component_set s a = connected_component_set s b"
- by (auto simp: connected_component_nonoverlap)
-
-lemma connected_component_sym_eq: "connected_component s x y \<longleftrightarrow> connected_component s y x"
- using connected_component_sym by blast
-
-lemma connected_component_eq_eq:
- "connected_component_set s x = connected_component_set s y \<longleftrightarrow>
- x \<notin> s \<and> y \<notin> s \<or> x \<in> s \<and> y \<in> s \<and> connected_component s x y"
- apply (cases "y \<in> s", simp)
- apply (metis connected_component_eq connected_component_eq_empty connected_component_refl_eq mem_Collect_eq)
- apply (cases "x \<in> s", simp)
- apply (metis connected_component_eq_empty)
- using connected_component_eq_empty
- apply blast
- done
-
-lemma connected_iff_connected_component_eq:
- "connected s \<longleftrightarrow> (\<forall>x \<in> s. \<forall>y \<in> s. connected_component_set s x = connected_component_set s y)"
- by (simp add: connected_component_eq_eq connected_iff_connected_component)
-
-lemma connected_component_idemp:
- "connected_component_set (connected_component_set s x) x = connected_component_set s x"
- apply (rule subset_antisym)
- apply (simp add: connected_component_subset)
- apply (metis connected_component_eq_empty connected_component_maximal
- connected_component_refl_eq connected_connected_component mem_Collect_eq set_eq_subset)
- done
-
-lemma connected_component_unique:
- "\<lbrakk>x \<in> c; c \<subseteq> s; connected c;
- \<And>c'. x \<in> c' \<and> c' \<subseteq> s \<and> connected c'
- \<Longrightarrow> c' \<subseteq> c\<rbrakk>
- \<Longrightarrow> connected_component_set s x = c"
-apply (rule subset_antisym)
-apply (meson connected_component_maximal connected_component_subset connected_connected_component contra_subsetD)
-by (simp add: connected_component_maximal)
-
-lemma joinable_connected_component_eq:
- "\<lbrakk>connected t; t \<subseteq> s;
- connected_component_set s x \<inter> t \<noteq> {};
- connected_component_set s y \<inter> t \<noteq> {}\<rbrakk>
- \<Longrightarrow> connected_component_set s x = connected_component_set s y"
-apply (simp add: ex_in_conv [symmetric])
-apply (rule connected_component_eq)
-by (metis (no_types, hide_lams) connected_component_eq_eq connected_component_in connected_component_maximal subsetD mem_Collect_eq)
-
-
-lemma Union_connected_component: "\<Union>(connected_component_set s ` s) = s"
- apply (rule subset_antisym)
- apply (simp add: SUP_least connected_component_subset)
- using connected_component_refl_eq
- by force
-
-
-lemma complement_connected_component_unions:
- "s - connected_component_set s x =
- \<Union>(connected_component_set s ` s - {connected_component_set s x})"
- apply (subst Union_connected_component [symmetric], auto)
- apply (metis connected_component_eq_eq connected_component_in)
- by (metis connected_component_eq mem_Collect_eq)
-
-lemma connected_component_intermediate_subset:
- "\<lbrakk>connected_component_set u a \<subseteq> t; t \<subseteq> u\<rbrakk>
- \<Longrightarrow> connected_component_set t a = connected_component_set u a"
- apply (case_tac "a \<in> u")
- apply (simp add: connected_component_maximal connected_component_mono subset_antisym)
- using connected_component_eq_empty by blast
-
-proposition connected_Times:
- assumes S: "connected S" and T: "connected T"
- shows "connected (S \<times> T)"
-proof (clarsimp simp add: connected_iff_connected_component)
- fix x y x' y'
- assume xy: "x \<in> S" "y \<in> T" "x' \<in> S" "y' \<in> T"
- with xy obtain U V where U: "connected U" "U \<subseteq> S" "x \<in> U" "x' \<in> U"
- and V: "connected V" "V \<subseteq> T" "y \<in> V" "y' \<in> V"
- using S T \<open>x \<in> S\<close> \<open>x' \<in> S\<close> by blast+
- show "connected_component (S \<times> T) (x, y) (x', y')"
- unfolding connected_component_def
- proof (intro exI conjI)
- show "connected ((\<lambda>x. (x, y)) ` U \<union> Pair x' ` V)"
- proof (rule connected_Un)
- have "continuous_on U (\<lambda>x. (x, y))"
- by (intro continuous_intros)
- then show "connected ((\<lambda>x. (x, y)) ` U)"
- by (rule connected_continuous_image) (rule \<open>connected U\<close>)
- have "continuous_on V (Pair x')"
- by (intro continuous_intros)
- then show "connected (Pair x' ` V)"
- by (rule connected_continuous_image) (rule \<open>connected V\<close>)
- qed (use U V in auto)
- qed (use U V in auto)
-qed
-
-corollary connected_Times_eq [simp]:
- "connected (S \<times> T) \<longleftrightarrow> S = {} \<or> T = {} \<or> connected S \<and> connected T" (is "?lhs = ?rhs")
-proof
- assume L: ?lhs
- show ?rhs
- proof (cases "S = {} \<or> T = {}")
- case True
- then show ?thesis by auto
- next
- case False
- have "connected (fst ` (S \<times> T))" "connected (snd ` (S \<times> T))"
- using continuous_on_fst continuous_on_snd continuous_on_id
- by (blast intro: connected_continuous_image [OF _ L])+
- with False show ?thesis
- by auto
- qed
-next
- assume ?rhs
- then show ?lhs
- by (auto simp: connected_Times)
-qed
-
-
-subsection \<open>The set of connected components of a set\<close>
-
-definition components:: "'a::topological_space set \<Rightarrow> 'a set set"
- where "components s \<equiv> connected_component_set s ` s"
-
-lemma components_iff: "s \<in> components u \<longleftrightarrow> (\<exists>x. x \<in> u \<and> s = connected_component_set u x)"
- by (auto simp: components_def)
-
-lemma componentsI: "x \<in> u \<Longrightarrow> connected_component_set u x \<in> components u"
- by (auto simp: components_def)
-
-lemma componentsE:
- assumes "s \<in> components u"
- obtains x where "x \<in> u" "s = connected_component_set u x"
- using assms by (auto simp: components_def)
-
-lemma Union_components [simp]: "\<Union>(components u) = u"
- apply (rule subset_antisym)
- using Union_connected_component components_def apply fastforce
- apply (metis Union_connected_component components_def set_eq_subset)
- done
-
-lemma pairwise_disjoint_components: "pairwise (\<lambda>X Y. X \<inter> Y = {}) (components u)"
- apply (simp add: pairwise_def)
- apply (auto simp: components_iff)
- apply (metis connected_component_eq_eq connected_component_in)+
- done
-
-lemma in_components_nonempty: "c \<in> components s \<Longrightarrow> c \<noteq> {}"
- by (metis components_iff connected_component_eq_empty)
-
-lemma in_components_subset: "c \<in> components s \<Longrightarrow> c \<subseteq> s"
- using Union_components by blast
-
-lemma in_components_connected: "c \<in> components s \<Longrightarrow> connected c"
- by (metis components_iff connected_connected_component)
-
-lemma in_components_maximal:
- "c \<in> components s \<longleftrightarrow>
- c \<noteq> {} \<and> c \<subseteq> s \<and> connected c \<and> (\<forall>d. d \<noteq> {} \<and> c \<subseteq> d \<and> d \<subseteq> s \<and> connected d \<longrightarrow> d = c)"
- apply (rule iffI)
- apply (simp add: in_components_nonempty in_components_connected)
- apply (metis (full_types) components_iff connected_component_eq_self connected_component_intermediate_subset connected_component_refl in_components_subset mem_Collect_eq rev_subsetD)
- apply (metis bot.extremum_uniqueI components_iff connected_component_eq_empty connected_component_maximal connected_component_subset connected_connected_component subset_emptyI)
- done
-
-lemma joinable_components_eq:
- "connected t \<and> t \<subseteq> s \<and> c1 \<in> components s \<and> c2 \<in> components s \<and> c1 \<inter> t \<noteq> {} \<and> c2 \<inter> t \<noteq> {} \<Longrightarrow> c1 = c2"
- by (metis (full_types) components_iff joinable_connected_component_eq)
-
-lemma closed_components: "\<lbrakk>closed s; c \<in> components s\<rbrakk> \<Longrightarrow> closed c"
- by (metis closed_connected_component components_iff)
-
-lemma components_nonoverlap:
- "\<lbrakk>c \<in> components s; c' \<in> components s\<rbrakk> \<Longrightarrow> (c \<inter> c' = {}) \<longleftrightarrow> (c \<noteq> c')"
- apply (auto simp: in_components_nonempty components_iff)
- using connected_component_refl apply blast
- apply (metis connected_component_eq_eq connected_component_in)
- by (metis connected_component_eq mem_Collect_eq)
-
-lemma components_eq: "\<lbrakk>c \<in> components s; c' \<in> components s\<rbrakk> \<Longrightarrow> (c = c' \<longleftrightarrow> c \<inter> c' \<noteq> {})"
- by (metis components_nonoverlap)
-
-lemma components_eq_empty [simp]: "components s = {} \<longleftrightarrow> s = {}"
- by (simp add: components_def)
-
-lemma components_empty [simp]: "components {} = {}"
- by simp
-
-lemma connected_eq_connected_components_eq: "connected s \<longleftrightarrow> (\<forall>c \<in> components s. \<forall>c' \<in> components s. c = c')"
- by (metis (no_types, hide_lams) components_iff connected_component_eq_eq connected_iff_connected_component)
-
-lemma components_eq_sing_iff: "components s = {s} \<longleftrightarrow> connected s \<and> s \<noteq> {}"
- apply (rule iffI)
- using in_components_connected apply fastforce
- apply safe
- using Union_components apply fastforce
- apply (metis components_iff connected_component_eq_self)
- using in_components_maximal
- apply auto
- done
-
-lemma components_eq_sing_exists: "(\<exists>a. components s = {a}) \<longleftrightarrow> connected s \<and> s \<noteq> {}"
- apply (rule iffI)
- using connected_eq_connected_components_eq apply fastforce
- apply (metis components_eq_sing_iff)
- done
-
-lemma connected_eq_components_subset_sing: "connected s \<longleftrightarrow> components s \<subseteq> {s}"
- by (metis Union_components components_empty components_eq_sing_iff connected_empty insert_subset order_refl subset_singletonD)
-
-lemma connected_eq_components_subset_sing_exists: "connected s \<longleftrightarrow> (\<exists>a. components s \<subseteq> {a})"
- by (metis components_eq_sing_exists connected_eq_components_subset_sing empty_iff subset_iff subset_singletonD)
-
-lemma in_components_self: "s \<in> components s \<longleftrightarrow> connected s \<and> s \<noteq> {}"
- by (metis components_empty components_eq_sing_iff empty_iff in_components_connected insertI1)
-
-lemma components_maximal: "\<lbrakk>c \<in> components s; connected t; t \<subseteq> s; c \<inter> t \<noteq> {}\<rbrakk> \<Longrightarrow> t \<subseteq> c"
- apply (simp add: components_def ex_in_conv [symmetric], clarify)
- by (meson connected_component_def connected_component_trans)
-
-lemma exists_component_superset: "\<lbrakk>t \<subseteq> s; s \<noteq> {}; connected t\<rbrakk> \<Longrightarrow> \<exists>c. c \<in> components s \<and> t \<subseteq> c"
- apply (cases "t = {}", force)
- apply (metis components_def ex_in_conv connected_component_maximal contra_subsetD image_eqI)
- done
-
-lemma components_intermediate_subset: "\<lbrakk>s \<in> components u; s \<subseteq> t; t \<subseteq> u\<rbrakk> \<Longrightarrow> s \<in> components t"
- apply (auto simp: components_iff)
- apply (metis connected_component_eq_empty connected_component_intermediate_subset)
- done
-
-lemma in_components_unions_complement: "c \<in> components s \<Longrightarrow> s - c = \<Union>(components s - {c})"
- by (metis complement_connected_component_unions components_def components_iff)
-
-lemma connected_intermediate_closure:
- assumes cs: "connected s" and st: "s \<subseteq> t" and ts: "t \<subseteq> closure s"
- shows "connected t"
-proof (rule connectedI)
- fix A B
- assume A: "open A" and B: "open B" and Alap: "A \<inter> t \<noteq> {}" and Blap: "B \<inter> t \<noteq> {}"
- and disj: "A \<inter> B \<inter> t = {}" and cover: "t \<subseteq> A \<union> B"
- have disjs: "A \<inter> B \<inter> s = {}"
- using disj st by auto
- have "A \<inter> closure s \<noteq> {}"
- using Alap Int_absorb1 ts by blast
- then have Alaps: "A \<inter> s \<noteq> {}"
- by (simp add: A open_Int_closure_eq_empty)
- have "B \<inter> closure s \<noteq> {}"
- using Blap Int_absorb1 ts by blast
- then have Blaps: "B \<inter> s \<noteq> {}"
- by (simp add: B open_Int_closure_eq_empty)
- then show False
- using cs [unfolded connected_def] A B disjs Alaps Blaps cover st
- by blast
-qed
-
-lemma closedin_connected_component: "closedin (subtopology euclidean s) (connected_component_set s x)"
-proof (cases "connected_component_set s x = {}")
- case True
- then show ?thesis
- by (metis closedin_empty)
-next
- case False
- then obtain y where y: "connected_component s x y"
- by blast
- have *: "connected_component_set s x \<subseteq> s \<inter> closure (connected_component_set s x)"
- by (auto simp: closure_def connected_component_in)
- have "connected_component s x y \<Longrightarrow> s \<inter> closure (connected_component_set s x) \<subseteq> connected_component_set s x"
- apply (rule connected_component_maximal, simp)
- using closure_subset connected_component_in apply fastforce
- using * connected_intermediate_closure apply blast+
- done
- with y * show ?thesis
- by (auto simp: Topology_Euclidean_Space.closedin_closed)
-qed
-
-
subsection \<open>Frontier (also known as boundary)\<close>
definition "frontier S = closure S - interior S"
@@ -3629,477 +3197,6 @@
qed
-subsection \<open>Infimum Distance\<close>
-
-definition "infdist x A = (if A = {} then 0 else INF a:A. dist x a)"
-
-lemma bdd_below_infdist[intro, simp]: "bdd_below (dist x`A)"
- by (auto intro!: zero_le_dist)
-
-lemma infdist_notempty: "A \<noteq> {} \<Longrightarrow> infdist x A = (INF a:A. dist x a)"
- by (simp add: infdist_def)
-
-lemma infdist_nonneg: "0 \<le> infdist x A"
- by (auto simp: infdist_def intro: cINF_greatest)
-
-lemma infdist_le: "a \<in> A \<Longrightarrow> infdist x A \<le> dist x a"
- by (auto intro: cINF_lower simp add: infdist_def)
-
-lemma infdist_le2: "a \<in> A \<Longrightarrow> dist x a \<le> d \<Longrightarrow> infdist x A \<le> d"
- by (auto intro!: cINF_lower2 simp add: infdist_def)
-
-lemma infdist_zero[simp]: "a \<in> A \<Longrightarrow> infdist a A = 0"
- by (auto intro!: antisym infdist_nonneg infdist_le2)
-
-lemma infdist_triangle: "infdist x A \<le> infdist y A + dist x y"
-proof (cases "A = {}")
- case True
- then show ?thesis by (simp add: infdist_def)
-next
- case False
- then obtain a where "a \<in> A" by auto
- have "infdist x A \<le> Inf {dist x y + dist y a |a. a \<in> A}"
- proof (rule cInf_greatest)
- from \<open>A \<noteq> {}\<close> show "{dist x y + dist y a |a. a \<in> A} \<noteq> {}"
- by simp
- fix d
- assume "d \<in> {dist x y + dist y a |a. a \<in> A}"
- then obtain a where d: "d = dist x y + dist y a" "a \<in> A"
- by auto
- show "infdist x A \<le> d"
- unfolding infdist_notempty[OF \<open>A \<noteq> {}\<close>]
- proof (rule cINF_lower2)
- show "a \<in> A" by fact
- show "dist x a \<le> d"
- unfolding d by (rule dist_triangle)
- qed simp
- qed
- also have "\<dots> = dist x y + infdist y A"
- proof (rule cInf_eq, safe)
- fix a
- assume "a \<in> A"
- then show "dist x y + infdist y A \<le> dist x y + dist y a"
- by (auto intro: infdist_le)
- next
- fix i
- assume inf: "\<And>d. d \<in> {dist x y + dist y a |a. a \<in> A} \<Longrightarrow> i \<le> d"
- then have "i - dist x y \<le> infdist y A"
- unfolding infdist_notempty[OF \<open>A \<noteq> {}\<close>] using \<open>a \<in> A\<close>
- by (intro cINF_greatest) (auto simp: field_simps)
- then show "i \<le> dist x y + infdist y A"
- by simp
- qed
- finally show ?thesis by simp
-qed
-
-lemma in_closure_iff_infdist_zero:
- assumes "A \<noteq> {}"
- shows "x \<in> closure A \<longleftrightarrow> infdist x A = 0"
-proof
- assume "x \<in> closure A"
- show "infdist x A = 0"
- proof (rule ccontr)
- assume "infdist x A \<noteq> 0"
- with infdist_nonneg[of x A] have "infdist x A > 0"
- by auto
- then have "ball x (infdist x A) \<inter> closure A = {}"
- apply auto
- apply (metis \<open>x \<in> closure A\<close> closure_approachable dist_commute infdist_le not_less)
- done
- then have "x \<notin> closure A"
- by (metis \<open>0 < infdist x A\<close> centre_in_ball disjoint_iff_not_equal)
- then show False using \<open>x \<in> closure A\<close> by simp
- qed
-next
- assume x: "infdist x A = 0"
- then obtain a where "a \<in> A"
- by atomize_elim (metis all_not_in_conv assms)
- show "x \<in> closure A"
- unfolding closure_approachable
- apply safe
- proof (rule ccontr)
- fix e :: real
- assume "e > 0"
- assume "\<not> (\<exists>y\<in>A. dist y x < e)"
- then have "infdist x A \<ge> e" using \<open>a \<in> A\<close>
- unfolding infdist_def
- by (force simp: dist_commute intro: cINF_greatest)
- with x \<open>e > 0\<close> show False by auto
- qed
-qed
-
-lemma in_closed_iff_infdist_zero:
- assumes "closed A" "A \<noteq> {}"
- shows "x \<in> A \<longleftrightarrow> infdist x A = 0"
-proof -
- have "x \<in> closure A \<longleftrightarrow> infdist x A = 0"
- by (rule in_closure_iff_infdist_zero) fact
- with assms show ?thesis by simp
-qed
-
-lemma tendsto_infdist [tendsto_intros]:
- assumes f: "(f \<longlongrightarrow> l) F"
- shows "((\<lambda>x. infdist (f x) A) \<longlongrightarrow> infdist l A) F"
-proof (rule tendstoI)
- fix e ::real
- assume "e > 0"
- from tendstoD[OF f this]
- show "eventually (\<lambda>x. dist (infdist (f x) A) (infdist l A) < e) F"
- proof (eventually_elim)
- fix x
- from infdist_triangle[of l A "f x"] infdist_triangle[of "f x" A l]
- have "dist (infdist (f x) A) (infdist l A) \<le> dist (f x) l"
- by (simp add: dist_commute dist_real_def)
- also assume "dist (f x) l < e"
- finally show "dist (infdist (f x) A) (infdist l A) < e" .
- qed
-qed
-
-text\<open>Some other lemmas about sequences.\<close>
-
-lemma sequentially_offset: (* TODO: move to Topological_Spaces.thy *)
- assumes "eventually (\<lambda>i. P i) sequentially"
- shows "eventually (\<lambda>i. P (i + k)) sequentially"
- using assms by (rule eventually_sequentially_seg [THEN iffD2])
-
-lemma seq_offset_neg: (* TODO: move to Topological_Spaces.thy *)
- "(f \<longlongrightarrow> l) sequentially \<Longrightarrow> ((\<lambda>i. f(i - k)) \<longlongrightarrow> l) sequentially"
- apply (erule filterlim_compose)
- apply (simp add: filterlim_def le_sequentially eventually_filtermap eventually_sequentially, arith)
- done
-
-lemma seq_harmonic: "((\<lambda>n. inverse (real n)) \<longlongrightarrow> 0) sequentially"
- using LIMSEQ_inverse_real_of_nat by (rule LIMSEQ_imp_Suc) (* TODO: move to Limits.thy *)
-
-subsection \<open>More properties of closed balls\<close>
-
-lemma closed_cball [iff]: "closed (cball x e)"
-proof -
- have "closed (dist x -` {..e})"
- by (intro closed_vimage closed_atMost continuous_intros)
- also have "dist x -` {..e} = cball x e"
- by auto
- finally show ?thesis .
-qed
-
-lemma open_contains_cball: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. cball x e \<subseteq> S)"
-proof -
- {
- fix x and e::real
- assume "x\<in>S" "e>0" "ball x e \<subseteq> S"
- then have "\<exists>d>0. cball x d \<subseteq> S" unfolding subset_eq by (rule_tac x="e/2" in exI, auto)
- }
- moreover
- {
- fix x and e::real
- assume "x\<in>S" "e>0" "cball x e \<subseteq> S"
- then have "\<exists>d>0. ball x d \<subseteq> S"
- unfolding subset_eq
- apply (rule_tac x="e/2" in exI, auto)
- done
- }
- ultimately show ?thesis
- unfolding open_contains_ball by auto
-qed
-
-lemma open_contains_cball_eq: "open S \<Longrightarrow> (\<forall>x. x \<in> S \<longleftrightarrow> (\<exists>e>0. cball x e \<subseteq> S))"
- by (metis open_contains_cball subset_eq order_less_imp_le centre_in_cball)
-
-lemma mem_interior_cball: "x \<in> interior S \<longleftrightarrow> (\<exists>e>0. cball x e \<subseteq> S)"
- apply (simp add: interior_def, safe)
- apply (force simp: open_contains_cball)
- apply (rule_tac x="ball x e" in exI)
- apply (simp add: subset_trans [OF ball_subset_cball])
- done
-
-lemma islimpt_ball:
- fixes x y :: "'a::{real_normed_vector,perfect_space}"
- shows "y islimpt ball x e \<longleftrightarrow> 0 < e \<and> y \<in> cball x e"
- (is "?lhs \<longleftrightarrow> ?rhs")
-proof
- show ?rhs if ?lhs
- proof
- {
- assume "e \<le> 0"
- then have *: "ball x e = {}"
- using ball_eq_empty[of x e] by auto
- have False using \<open>?lhs\<close>
- unfolding * using islimpt_EMPTY[of y] by auto
- }
- then show "e > 0" by (metis not_less)
- show "y \<in> cball x e"
- using closed_cball[of x e] islimpt_subset[of y "ball x e" "cball x e"]
- ball_subset_cball[of x e] \<open>?lhs\<close>
- unfolding closed_limpt by auto
- qed
- show ?lhs if ?rhs
- proof -
- from that have "e > 0" by auto
- {
- fix d :: real
- assume "d > 0"
- have "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
- proof (cases "d \<le> dist x y")
- case True
- then show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
- proof (cases "x = y")
- case True
- then have False
- using \<open>d \<le> dist x y\<close> \<open>d>0\<close> by auto
- then show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
- by auto
- next
- case False
- have "dist x (y - (d / (2 * dist y x)) *\<^sub>R (y - x)) =
- norm (x - y + (d / (2 * norm (y - x))) *\<^sub>R (y - x))"
- unfolding mem_cball mem_ball dist_norm diff_diff_eq2 diff_add_eq[symmetric]
- by auto
- also have "\<dots> = \<bar>- 1 + d / (2 * norm (x - y))\<bar> * norm (x - y)"
- using scaleR_left_distrib[of "- 1" "d / (2 * norm (y - x))", symmetric, of "y - x"]
- unfolding scaleR_minus_left scaleR_one
- by (auto simp: norm_minus_commute)
- also have "\<dots> = \<bar>- norm (x - y) + d / 2\<bar>"
- unfolding abs_mult_pos[of "norm (x - y)", OF norm_ge_zero[of "x - y"]]
- unfolding distrib_right using \<open>x\<noteq>y\<close> by auto
- also have "\<dots> \<le> e - d/2" using \<open>d \<le> dist x y\<close> and \<open>d>0\<close> and \<open>?rhs\<close>
- by (auto simp: dist_norm)
- finally have "y - (d / (2 * dist y x)) *\<^sub>R (y - x) \<in> ball x e" using \<open>d>0\<close>
- by auto
- moreover
- have "(d / (2*dist y x)) *\<^sub>R (y - x) \<noteq> 0"
- using \<open>x\<noteq>y\<close>[unfolded dist_nz] \<open>d>0\<close> unfolding scaleR_eq_0_iff
- by (auto simp: dist_commute)
- moreover
- have "dist (y - (d / (2 * dist y x)) *\<^sub>R (y - x)) y < d"
- unfolding dist_norm
- apply simp
- unfolding norm_minus_cancel
- using \<open>d > 0\<close> \<open>x\<noteq>y\<close>[unfolded dist_nz] dist_commute[of x y]
- unfolding dist_norm
- apply auto
- done
- ultimately show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
- apply (rule_tac x = "y - (d / (2*dist y x)) *\<^sub>R (y - x)" in bexI)
- apply auto
- done
- qed
- next
- case False
- then have "d > dist x y" by auto
- show "\<exists>x' \<in> ball x e. x' \<noteq> y \<and> dist x' y < d"
- proof (cases "x = y")
- case True
- obtain z where **: "z \<noteq> y" "dist z y < min e d"
- using perfect_choose_dist[of "min e d" y]
- using \<open>d > 0\<close> \<open>e>0\<close> by auto
- show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
- unfolding \<open>x = y\<close>
- using \<open>z \<noteq> y\<close> **
- apply (rule_tac x=z in bexI)
- apply (auto simp: dist_commute)
- done
- next
- case False
- then show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
- using \<open>d>0\<close> \<open>d > dist x y\<close> \<open>?rhs\<close>
- apply (rule_tac x=x in bexI, auto)
- done
- qed
- qed
- }
- then show ?thesis
- unfolding mem_cball islimpt_approachable mem_ball by auto
- qed
-qed
-
-lemma closure_ball_lemma:
- fixes x y :: "'a::real_normed_vector"
- assumes "x \<noteq> y"
- shows "y islimpt ball x (dist x y)"
-proof (rule islimptI)
- fix T
- assume "y \<in> T" "open T"
- then obtain r where "0 < r" "\<forall>z. dist z y < r \<longrightarrow> z \<in> T"
- unfolding open_dist by fast
- (* choose point between x and y, within distance r of y. *)
- define k where "k = min 1 (r / (2 * dist x y))"
- define z where "z = y + scaleR k (x - y)"
- have z_def2: "z = x + scaleR (1 - k) (y - x)"
- unfolding z_def by (simp add: algebra_simps)
- have "dist z y < r"
- unfolding z_def k_def using \<open>0 < r\<close>
- by (simp add: dist_norm min_def)
- then have "z \<in> T"
- using \<open>\<forall>z. dist z y < r \<longrightarrow> z \<in> T\<close> by simp
- have "dist x z < dist x y"
- unfolding z_def2 dist_norm
- apply (simp add: norm_minus_commute)
- apply (simp only: dist_norm [symmetric])
- apply (subgoal_tac "\<bar>1 - k\<bar> * dist x y < 1 * dist x y", simp)
- apply (rule mult_strict_right_mono)
- apply (simp add: k_def \<open>0 < r\<close> \<open>x \<noteq> y\<close>)
- apply (simp add: \<open>x \<noteq> y\<close>)
- done
- then have "z \<in> ball x (dist x y)"
- by simp
- have "z \<noteq> y"
- unfolding z_def k_def using \<open>x \<noteq> y\<close> \<open>0 < r\<close>
- by (simp add: min_def)
- show "\<exists>z\<in>ball x (dist x y). z \<in> T \<and> z \<noteq> y"
- using \<open>z \<in> ball x (dist x y)\<close> \<open>z \<in> T\<close> \<open>z \<noteq> y\<close>
- by fast
-qed
-
-lemma closure_ball [simp]:
- fixes x :: "'a::real_normed_vector"
- shows "0 < e \<Longrightarrow> closure (ball x e) = cball x e"
- apply (rule equalityI)
- apply (rule closure_minimal)
- apply (rule ball_subset_cball)
- apply (rule closed_cball)
- apply (rule subsetI, rename_tac y)
- apply (simp add: le_less [where 'a=real])
- apply (erule disjE)
- apply (rule subsetD [OF closure_subset], simp)
- apply (simp add: closure_def, clarify)
- apply (rule closure_ball_lemma)
- apply (simp add: zero_less_dist_iff)
- done
-
-(* In a trivial vector space, this fails for e = 0. *)
-lemma interior_cball [simp]:
- fixes x :: "'a::{real_normed_vector, perfect_space}"
- shows "interior (cball x e) = ball x e"
-proof (cases "e \<ge> 0")
- case False note cs = this
- from cs have null: "ball x e = {}"
- using ball_empty[of e x] by auto
- moreover
- {
- fix y
- assume "y \<in> cball x e"
- then have False
- by (metis ball_eq_empty null cs dist_eq_0_iff dist_le_zero_iff empty_subsetI mem_cball subset_antisym subset_ball)
- }
- then have "cball x e = {}" by auto
- then have "interior (cball x e) = {}"
- using interior_empty by auto
- ultimately show ?thesis by blast
-next
- case True note cs = this
- have "ball x e \<subseteq> cball x e"
- using ball_subset_cball by auto
- moreover
- {
- fix S y
- assume as: "S \<subseteq> cball x e" "open S" "y\<in>S"
- then obtain d where "d>0" and d: "\<forall>x'. dist x' y < d \<longrightarrow> x' \<in> S"
- unfolding open_dist by blast
- then obtain xa where xa_y: "xa \<noteq> y" and xa: "dist xa y < d"
- using perfect_choose_dist [of d] by auto
- have "xa \<in> S"
- using d[THEN spec[where x = xa]]
- using xa by (auto simp: dist_commute)
- then have xa_cball: "xa \<in> cball x e"
- using as(1) by auto
- then have "y \<in> ball x e"
- proof (cases "x = y")
- case True
- then have "e > 0" using cs order.order_iff_strict xa_cball xa_y by fastforce
- then show "y \<in> ball x e"
- using \<open>x = y \<close> by simp
- next
- case False
- have "dist (y + (d / 2 / dist y x) *\<^sub>R (y - x)) y < d"
- unfolding dist_norm
- using \<open>d>0\<close> norm_ge_zero[of "y - x"] \<open>x \<noteq> y\<close> by auto
- then have *: "y + (d / 2 / dist y x) *\<^sub>R (y - x) \<in> cball x e"
- using d as(1)[unfolded subset_eq] by blast
- have "y - x \<noteq> 0" using \<open>x \<noteq> y\<close> by auto
- hence **:"d / (2 * norm (y - x)) > 0"
- unfolding zero_less_norm_iff[symmetric] using \<open>d>0\<close> by auto
- have "dist (y + (d / 2 / dist y x) *\<^sub>R (y - x)) x =
- norm (y + (d / (2 * norm (y - x))) *\<^sub>R y - (d / (2 * norm (y - x))) *\<^sub>R x - x)"
- by (auto simp: dist_norm algebra_simps)
- also have "\<dots> = norm ((1 + d / (2 * norm (y - x))) *\<^sub>R (y - x))"
- by (auto simp: algebra_simps)
- also have "\<dots> = \<bar>1 + d / (2 * norm (y - x))\<bar> * norm (y - x)"
- using ** by auto
- also have "\<dots> = (dist y x) + d/2"
- using ** by (auto simp: distrib_right dist_norm)
- finally have "e \<ge> dist x y +d/2"
- using *[unfolded mem_cball] by (auto simp: dist_commute)
- then show "y \<in> ball x e"
- unfolding mem_ball using \<open>d>0\<close> by auto
- qed
- }
- then have "\<forall>S \<subseteq> cball x e. open S \<longrightarrow> S \<subseteq> ball x e"
- by auto
- ultimately show ?thesis
- using interior_unique[of "ball x e" "cball x e"]
- using open_ball[of x e]
- by auto
-qed
-
-lemma interior_ball [simp]: "interior (ball x e) = ball x e"
- by (simp add: interior_open)
-
-lemma frontier_ball [simp]:
- fixes a :: "'a::real_normed_vector"
- shows "0 < e \<Longrightarrow> frontier (ball a e) = sphere a e"
- by (force simp: frontier_def)
-
-lemma frontier_cball [simp]:
- fixes a :: "'a::{real_normed_vector, perfect_space}"
- shows "frontier (cball a e) = sphere a e"
- by (force simp: frontier_def)
-
-lemma cball_eq_empty [simp]: "cball x e = {} \<longleftrightarrow> e < 0"
- apply (simp add: set_eq_iff not_le)
- apply (metis zero_le_dist dist_self order_less_le_trans)
- done
-
-lemma cball_empty [simp]: "e < 0 \<Longrightarrow> cball x e = {}"
- by (simp add: cball_eq_empty)
-
-lemma cball_eq_sing:
- fixes x :: "'a::{metric_space,perfect_space}"
- shows "cball x e = {x} \<longleftrightarrow> e = 0"
-proof (rule linorder_cases)
- assume e: "0 < e"
- obtain a where "a \<noteq> x" "dist a x < e"
- using perfect_choose_dist [OF e] by auto
- then have "a \<noteq> x" "dist x a \<le> e"
- by (auto simp: dist_commute)
- with e show ?thesis by (auto simp: set_eq_iff)
-qed auto
-
-lemma cball_sing:
- fixes x :: "'a::metric_space"
- shows "e = 0 \<Longrightarrow> cball x e = {x}"
- by (auto simp: set_eq_iff)
-
-lemma ball_divide_subset: "d \<ge> 1 \<Longrightarrow> ball x (e/d) \<subseteq> ball x e"
- apply (cases "e \<le> 0")
- apply (simp add: ball_empty divide_simps)
- apply (rule subset_ball)
- apply (simp add: divide_simps)
- done
-
-lemma ball_divide_subset_numeral: "ball x (e / numeral w) \<subseteq> ball x e"
- using ball_divide_subset one_le_numeral by blast
-
-lemma cball_divide_subset: "d \<ge> 1 \<Longrightarrow> cball x (e/d) \<subseteq> cball x e"
- apply (cases "e < 0")
- apply (simp add: divide_simps)
- apply (rule subset_cball)
- apply (metis div_by_1 frac_le not_le order_refl zero_less_one)
- done
-
-lemma cball_divide_subset_numeral: "cball x (e / numeral w) \<subseteq> cball x e"
- using cball_divide_subset one_le_numeral by blast
-
-
subsection \<open>Boundedness\<close>
(* FIXME: This has to be unified with BSEQ!! *)
@@ -4364,91 +3461,6 @@
by auto
-subsection\<open>Some theorems on sups and infs using the notion "bounded".\<close>
-
-lemma bounded_real: "bounded (S::real set) \<longleftrightarrow> (\<exists>a. \<forall>x\<in>S. \<bar>x\<bar> \<le> a)"
- by (simp add: bounded_iff)
-
-lemma bounded_imp_bdd_above: "bounded S \<Longrightarrow> bdd_above (S :: real set)"
- by (auto simp: bounded_def bdd_above_def dist_real_def)
- (metis abs_le_D1 abs_minus_commute diff_le_eq)
-
-lemma bounded_imp_bdd_below: "bounded S \<Longrightarrow> bdd_below (S :: real set)"
- by (auto simp: bounded_def bdd_below_def dist_real_def)
- (metis abs_le_D1 add.commute diff_le_eq)
-
-lemma bounded_inner_imp_bdd_above:
- assumes "bounded s"
- shows "bdd_above ((\<lambda>x. x \<bullet> a) ` s)"
-by (simp add: assms bounded_imp_bdd_above bounded_linear_image bounded_linear_inner_left)
-
-lemma bounded_inner_imp_bdd_below:
- assumes "bounded s"
- shows "bdd_below ((\<lambda>x. x \<bullet> a) ` s)"
-by (simp add: assms bounded_imp_bdd_below bounded_linear_image bounded_linear_inner_left)
-
-lemma bounded_has_Sup:
- fixes S :: "real set"
- assumes "bounded S"
- and "S \<noteq> {}"
- shows "\<forall>x\<in>S. x \<le> Sup S"
- and "\<forall>b. (\<forall>x\<in>S. x \<le> b) \<longrightarrow> Sup S \<le> b"
-proof
- show "\<forall>b. (\<forall>x\<in>S. x \<le> b) \<longrightarrow> Sup S \<le> b"
- using assms by (metis cSup_least)
-qed (metis cSup_upper assms(1) bounded_imp_bdd_above)
-
-lemma Sup_insert:
- fixes S :: "real set"
- shows "bounded S \<Longrightarrow> Sup (insert x S) = (if S = {} then x else max x (Sup S))"
- by (auto simp: bounded_imp_bdd_above sup_max cSup_insert_If)
-
-lemma Sup_insert_finite:
- fixes S :: "'a::conditionally_complete_linorder set"
- shows "finite S \<Longrightarrow> Sup (insert x S) = (if S = {} then x else max x (Sup S))"
-by (simp add: cSup_insert sup_max)
-
-lemma bounded_has_Inf:
- fixes S :: "real set"
- assumes "bounded S"
- and "S \<noteq> {}"
- shows "\<forall>x\<in>S. x \<ge> Inf S"
- and "\<forall>b. (\<forall>x\<in>S. x \<ge> b) \<longrightarrow> Inf S \<ge> b"
-proof
- show "\<forall>b. (\<forall>x\<in>S. x \<ge> b) \<longrightarrow> Inf S \<ge> b"
- using assms by (metis cInf_greatest)
-qed (metis cInf_lower assms(1) bounded_imp_bdd_below)
-
-lemma Inf_insert:
- fixes S :: "real set"
- shows "bounded S \<Longrightarrow> Inf (insert x S) = (if S = {} then x else min x (Inf S))"
- by (auto simp: bounded_imp_bdd_below inf_min cInf_insert_If)
-
-lemma Inf_insert_finite:
- fixes S :: "'a::conditionally_complete_linorder set"
- shows "finite S \<Longrightarrow> Inf (insert x S) = (if S = {} then x else min x (Inf S))"
-by (simp add: cInf_eq_Min)
-
-lemma finite_imp_less_Inf:
- fixes a :: "'a::conditionally_complete_linorder"
- shows "\<lbrakk>finite X; x \<in> X; \<And>x. x\<in>X \<Longrightarrow> a < x\<rbrakk> \<Longrightarrow> a < Inf X"
- by (induction X rule: finite_induct) (simp_all add: cInf_eq_Min Inf_insert_finite)
-
-lemma finite_less_Inf_iff:
- fixes a :: "'a :: conditionally_complete_linorder"
- shows "\<lbrakk>finite X; X \<noteq> {}\<rbrakk> \<Longrightarrow> a < Inf X \<longleftrightarrow> (\<forall>x \<in> X. a < x)"
- by (auto simp: cInf_eq_Min)
-
-lemma finite_imp_Sup_less:
- fixes a :: "'a::conditionally_complete_linorder"
- shows "\<lbrakk>finite X; x \<in> X; \<And>x. x\<in>X \<Longrightarrow> a > x\<rbrakk> \<Longrightarrow> a > Sup X"
- by (induction X rule: finite_induct) (simp_all add: cSup_eq_Max Sup_insert_finite)
-
-lemma finite_Sup_less_iff:
- fixes a :: "'a :: conditionally_complete_linorder"
- shows "\<lbrakk>finite X; X \<noteq> {}\<rbrakk> \<Longrightarrow> a > Sup X \<longleftrightarrow> (\<forall>x \<in> X. a > x)"
- by (auto simp: cSup_eq_Max)
-
subsection \<open>Compactness\<close>
subsubsection \<open>Bolzano-Weierstrass property\<close>
@@ -5428,11 +4440,6 @@
shows "compact(closure S) \<longleftrightarrow> bounded S"
by (meson bounded_closure bounded_subset closed_closure closure_subset compact_eq_bounded_closed)
-lemma compact_components:
- fixes s :: "'a::heine_borel set"
- shows "\<lbrakk>compact s; c \<in> components s\<rbrakk> \<Longrightarrow> compact c"
-by (meson bounded_subset closed_components in_components_subset compact_eq_bounded_closed)
-
lemma not_compact_UNIV[simp]:
fixes s :: "'a::{real_normed_vector,perfect_space,heine_borel} set"
shows "~ compact (UNIV::'a set)"
@@ -5591,193 +4598,6 @@
using l r by fast
qed
-subsubsection \<open>Intersecting chains of compact sets\<close>
-
-proposition bounded_closed_chain:
- fixes \<F> :: "'a::heine_borel set set"
- assumes "B \<in> \<F>" "bounded B" and \<F>: "\<And>S. S \<in> \<F> \<Longrightarrow> closed S" and "{} \<notin> \<F>"
- and chain: "\<And>S T. S \<in> \<F> \<and> T \<in> \<F> \<Longrightarrow> S \<subseteq> T \<or> T \<subseteq> S"
- shows "\<Inter>\<F> \<noteq> {}"
-proof -
- have "B \<inter> \<Inter>\<F> \<noteq> {}"
- proof (rule compact_imp_fip)
- show "compact B" "\<And>T. T \<in> \<F> \<Longrightarrow> closed T"
- by (simp_all add: assms compact_eq_bounded_closed)
- show "\<lbrakk>finite \<G>; \<G> \<subseteq> \<F>\<rbrakk> \<Longrightarrow> B \<inter> \<Inter>\<G> \<noteq> {}" for \<G>
- proof (induction \<G> rule: finite_induct)
- case empty
- with assms show ?case by force
- next
- case (insert U \<G>)
- then have "U \<in> \<F>" and ne: "B \<inter> \<Inter>\<G> \<noteq> {}" by auto
- then consider "B \<subseteq> U" | "U \<subseteq> B"
- using \<open>B \<in> \<F>\<close> chain by blast
- then show ?case
- proof cases
- case 1
- then show ?thesis
- using Int_left_commute ne by auto
- next
- case 2
- have "U \<noteq> {}"
- using \<open>U \<in> \<F>\<close> \<open>{} \<notin> \<F>\<close> by blast
- moreover
- have False if "\<And>x. x \<in> U \<Longrightarrow> \<exists>Y\<in>\<G>. x \<notin> Y"
- proof -
- have "\<And>x. x \<in> U \<Longrightarrow> \<exists>Y\<in>\<G>. Y \<subseteq> U"
- by (metis chain contra_subsetD insert.prems insert_subset that)
- then obtain Y where "Y \<in> \<G>" "Y \<subseteq> U"
- by (metis all_not_in_conv \<open>U \<noteq> {}\<close>)
- moreover obtain x where "x \<in> \<Inter>\<G>"
- by (metis Int_emptyI ne)
- ultimately show ?thesis
- by (metis Inf_lower subset_eq that)
- qed
- with 2 show ?thesis
- by blast
- qed
- qed
- qed
- then show ?thesis by blast
-qed
-
-corollary compact_chain:
- fixes \<F> :: "'a::heine_borel set set"
- assumes "\<And>S. S \<in> \<F> \<Longrightarrow> compact S" "{} \<notin> \<F>"
- "\<And>S T. S \<in> \<F> \<and> T \<in> \<F> \<Longrightarrow> S \<subseteq> T \<or> T \<subseteq> S"
- shows "\<Inter> \<F> \<noteq> {}"
-proof (cases "\<F> = {}")
- case True
- then show ?thesis by auto
-next
- case False
- show ?thesis
- by (metis False all_not_in_conv assms compact_imp_bounded compact_imp_closed bounded_closed_chain)
-qed
-
-lemma compact_nest:
- fixes F :: "'a::linorder \<Rightarrow> 'b::heine_borel set"
- assumes F: "\<And>n. compact(F n)" "\<And>n. F n \<noteq> {}" and mono: "\<And>m n. m \<le> n \<Longrightarrow> F n \<subseteq> F m"
- shows "\<Inter>range F \<noteq> {}"
-proof -
- have *: "\<And>S T. S \<in> range F \<and> T \<in> range F \<Longrightarrow> S \<subseteq> T \<or> T \<subseteq> S"
- by (metis mono image_iff le_cases)
- show ?thesis
- apply (rule compact_chain [OF _ _ *])
- using F apply (blast intro: dest: *)+
- done
-qed
-
-text\<open>The Baire property of dense sets\<close>
-theorem Baire:
- fixes S::"'a::{real_normed_vector,heine_borel} set"
- assumes "closed S" "countable \<G>"
- and ope: "\<And>T. T \<in> \<G> \<Longrightarrow> openin (subtopology euclidean S) T \<and> S \<subseteq> closure T"
- shows "S \<subseteq> closure(\<Inter>\<G>)"
-proof (cases "\<G> = {}")
- case True
- then show ?thesis
- using closure_subset by auto
-next
- let ?g = "from_nat_into \<G>"
- case False
- then have gin: "?g n \<in> \<G>" for n
- by (simp add: from_nat_into)
- show ?thesis
- proof (clarsimp simp: closure_approachable)
- fix x and e::real
- assume "x \<in> S" "0 < e"
- obtain TF where opeF: "\<And>n. openin (subtopology euclidean S) (TF n)"
- and ne: "\<And>n. TF n \<noteq> {}"
- and subg: "\<And>n. S \<inter> closure(TF n) \<subseteq> ?g n"
- and subball: "\<And>n. closure(TF n) \<subseteq> ball x e"
- and decr: "\<And>n. TF(Suc n) \<subseteq> TF n"
- proof -
- have *: "\<exists>Y. (openin (subtopology euclidean S) Y \<and> Y \<noteq> {} \<and>
- S \<inter> closure Y \<subseteq> ?g n \<and> closure Y \<subseteq> ball x e) \<and> Y \<subseteq> U"
- if opeU: "openin (subtopology euclidean S) U" and "U \<noteq> {}" and cloU: "closure U \<subseteq> ball x e" for U n
- proof -
- obtain T where T: "open T" "U = T \<inter> S"
- using \<open>openin (subtopology euclidean S) U\<close> by (auto simp: openin_subtopology)
- with \<open>U \<noteq> {}\<close> have "T \<inter> closure (?g n) \<noteq> {}"
- using gin ope by fastforce
- then have "T \<inter> ?g n \<noteq> {}"
- using \<open>open T\<close> open_Int_closure_eq_empty by blast
- then obtain y where "y \<in> U" "y \<in> ?g n"
- using T ope [of "?g n", OF gin] by (blast dest: openin_imp_subset)
- moreover have "openin (subtopology euclidean S) (U \<inter> ?g n)"
- using gin ope opeU by blast
- ultimately obtain d where U: "U \<inter> ?g n \<subseteq> S" and "d > 0" and d: "ball y d \<inter> S \<subseteq> U \<inter> ?g n"
- by (force simp: openin_contains_ball)
- show ?thesis
- proof (intro exI conjI)
- show "openin (subtopology euclidean S) (S \<inter> ball y (d/2))"
- by (simp add: openin_open_Int)
- show "S \<inter> ball y (d/2) \<noteq> {}"
- using \<open>0 < d\<close> \<open>y \<in> U\<close> opeU openin_imp_subset by fastforce
- have "S \<inter> closure (S \<inter> ball y (d/2)) \<subseteq> S \<inter> closure (ball y (d/2))"
- using closure_mono by blast
- also have "... \<subseteq> ?g n"
- using \<open>d > 0\<close> d by force
- finally show "S \<inter> closure (S \<inter> ball y (d/2)) \<subseteq> ?g n" .
- have "closure (S \<inter> ball y (d/2)) \<subseteq> S \<inter> ball y d"
- proof -
- have "closure (ball y (d/2)) \<subseteq> ball y d"
- using \<open>d > 0\<close> by auto
- then have "closure (S \<inter> ball y (d/2)) \<subseteq> ball y d"
- by (meson closure_mono inf.cobounded2 subset_trans)
- then show ?thesis
- by (simp add: \<open>closed S\<close> closure_minimal)
- qed
- also have "... \<subseteq> ball x e"
- using cloU closure_subset d by blast
- finally show "closure (S \<inter> ball y (d/2)) \<subseteq> ball x e" .
- show "S \<inter> ball y (d/2) \<subseteq> U"
- using ball_divide_subset_numeral d by blast
- qed
- qed
- let ?\<Phi> = "\<lambda>n X. openin (subtopology euclidean S) X \<and> X \<noteq> {} \<and>
- S \<inter> closure X \<subseteq> ?g n \<and> closure X \<subseteq> ball x e"
- have "closure (S \<inter> ball x (e / 2)) \<subseteq> closure(ball x (e/2))"
- by (simp add: closure_mono)
- also have "... \<subseteq> ball x e"
- using \<open>e > 0\<close> by auto
- finally have "closure (S \<inter> ball x (e / 2)) \<subseteq> ball x e" .
- moreover have"openin (subtopology euclidean S) (S \<inter> ball x (e / 2))" "S \<inter> ball x (e / 2) \<noteq> {}"
- using \<open>0 < e\<close> \<open>x \<in> S\<close> by auto
- ultimately obtain Y where Y: "?\<Phi> 0 Y \<and> Y \<subseteq> S \<inter> ball x (e / 2)"
- using * [of "S \<inter> ball x (e/2)" 0] by metis
- show thesis
- proof (rule exE [OF dependent_nat_choice [of ?\<Phi> "\<lambda>n X Y. Y \<subseteq> X"]])
- show "\<exists>x. ?\<Phi> 0 x"
- using Y by auto
- show "\<exists>Y. ?\<Phi> (Suc n) Y \<and> Y \<subseteq> X" if "?\<Phi> n X" for X n
- using that by (blast intro: *)
- qed (use that in metis)
- qed
- have "(\<Inter>n. S \<inter> closure (TF n)) \<noteq> {}"
- proof (rule compact_nest)
- show "\<And>n. compact (S \<inter> closure (TF n))"
- by (metis closed_closure subball bounded_subset_ballI compact_eq_bounded_closed closed_Int_compact [OF \<open>closed S\<close>])
- show "\<And>n. S \<inter> closure (TF n) \<noteq> {}"
- by (metis Int_absorb1 opeF \<open>closed S\<close> closure_eq_empty closure_minimal ne openin_imp_subset)
- show "\<And>m n. m \<le> n \<Longrightarrow> S \<inter> closure (TF n) \<subseteq> S \<inter> closure (TF m)"
- by (meson closure_mono decr dual_order.refl inf_mono lift_Suc_antimono_le)
- qed
- moreover have "(\<Inter>n. S \<inter> closure (TF n)) \<subseteq> {y \<in> \<Inter>\<G>. dist y x < e}"
- proof (clarsimp, intro conjI)
- fix y
- assume "y \<in> S" and y: "\<forall>n. y \<in> closure (TF n)"
- then show "\<forall>T\<in>\<G>. y \<in> T"
- by (metis Int_iff from_nat_into_surj [OF \<open>countable \<G>\<close>] set_mp subg)
- show "dist y x < e"
- by (metis y dist_commute mem_ball subball subsetCE)
- qed
- ultimately show "\<exists>y \<in> \<Inter>\<G>. dist y x < e"
- by auto
- qed
-qed
-
subsubsection \<open>Completeness\<close>
lemma (in metric_space) completeI:
@@ -6032,200 +4852,12 @@
shows "(S \<longlongrightarrow> l) sequentially \<Longrightarrow> bounded (range S)"
by (intro cauchy_imp_bounded LIMSEQ_imp_Cauchy)
-lemma compact_cball[simp]:
- fixes x :: "'a::heine_borel"
- shows "compact (cball x e)"
- using compact_eq_bounded_closed bounded_cball closed_cball
- by blast
-
-lemma compact_frontier_bounded[intro]:
- fixes S :: "'a::heine_borel set"
- shows "bounded S \<Longrightarrow> compact (frontier S)"
- unfolding frontier_def
- using compact_eq_bounded_closed
- by blast
-
-lemma compact_frontier[intro]:
- fixes S :: "'a::heine_borel set"
- shows "compact S \<Longrightarrow> compact (frontier S)"
- using compact_eq_bounded_closed compact_frontier_bounded
- by blast
-
-corollary compact_sphere [simp]:
- fixes a :: "'a::{real_normed_vector,perfect_space,heine_borel}"
- shows "compact (sphere a r)"
-using compact_frontier [of "cball a r"] by simp
-
-corollary bounded_sphere [simp]:
- fixes a :: "'a::{real_normed_vector,perfect_space,heine_borel}"
- shows "bounded (sphere a r)"
-by (simp add: compact_imp_bounded)
-
-corollary closed_sphere [simp]:
- fixes a :: "'a::{real_normed_vector,perfect_space,heine_borel}"
- shows "closed (sphere a r)"
-by (simp add: compact_imp_closed)
-
lemma frontier_subset_compact:
fixes S :: "'a::heine_borel set"
shows "compact S \<Longrightarrow> frontier S \<subseteq> S"
using frontier_subset_closed compact_eq_bounded_closed
by blast
-subsection\<open>Relations among convergence and absolute convergence for power series.\<close>
-
-lemma summable_imp_bounded:
- fixes f :: "nat \<Rightarrow> 'a::real_normed_vector"
- shows "summable f \<Longrightarrow> bounded (range f)"
-by (frule summable_LIMSEQ_zero) (simp add: convergent_imp_bounded)
-
-lemma summable_imp_sums_bounded:
- "summable f \<Longrightarrow> bounded (range (\<lambda>n. sum f {..<n}))"
-by (auto simp: summable_def sums_def dest: convergent_imp_bounded)
-
-lemma power_series_conv_imp_absconv_weak:
- fixes a:: "nat \<Rightarrow> 'a::{real_normed_div_algebra,banach}" and w :: 'a
- assumes sum: "summable (\<lambda>n. a n * z ^ n)" and no: "norm w < norm z"
- shows "summable (\<lambda>n. of_real(norm(a n)) * w ^ n)"
-proof -
- obtain M where M: "\<And>x. norm (a x * z ^ x) \<le> M"
- using summable_imp_bounded [OF sum] by (force simp: bounded_iff)
- then have *: "summable (\<lambda>n. norm (a n) * norm w ^ n)"
- by (rule_tac M=M in Abel_lemma) (auto simp: norm_mult norm_power intro: no)
- show ?thesis
- apply (rule series_comparison_complex [of "(\<lambda>n. of_real(norm(a n) * norm w ^ n))"])
- apply (simp only: summable_complex_of_real *)
- apply (auto simp: norm_mult norm_power)
- done
-qed
-
-subsection \<open>Bounded closed nest property (proof does not use Heine-Borel)\<close>
-
-lemma bounded_closed_nest:
- fixes s :: "nat \<Rightarrow> ('a::heine_borel) set"
- assumes "\<forall>n. closed (s n)"
- and "\<forall>n. s n \<noteq> {}"
- and "\<forall>m n. m \<le> n \<longrightarrow> s n \<subseteq> s m"
- and "bounded (s 0)"
- shows "\<exists>a. \<forall>n. a \<in> s n"
-proof -
- from assms(2) obtain x where x: "\<forall>n. x n \<in> s n"
- using choice[of "\<lambda>n x. x \<in> s n"] by auto
- from assms(4,1) have "seq_compact (s 0)"
- by (simp add: bounded_closed_imp_seq_compact)
- then obtain l r where lr: "l \<in> s 0" "strict_mono r" "(x \<circ> r) \<longlonglongrightarrow> l"
- using x and assms(3) unfolding seq_compact_def by blast
- have "\<forall>n. l \<in> s n"
- proof
- fix n :: nat
- have "closed (s n)"
- using assms(1) by simp
- moreover have "\<forall>i. (x \<circ> r) i \<in> s i"
- using x and assms(3) and lr(2) [THEN seq_suble] by auto
- then have "\<forall>i. (x \<circ> r) (i + n) \<in> s n"
- using assms(3) by (fast intro!: le_add2)
- moreover have "(\<lambda>i. (x \<circ> r) (i + n)) \<longlonglongrightarrow> l"
- using lr(3) by (rule LIMSEQ_ignore_initial_segment)
- ultimately show "l \<in> s n"
- by (rule closed_sequentially)
- qed
- then show ?thesis ..
-qed
-
-text \<open>Decreasing case does not even need compactness, just completeness.\<close>
-
-lemma decreasing_closed_nest:
- fixes s :: "nat \<Rightarrow> ('a::complete_space) set"
- assumes
- "\<forall>n. closed (s n)"
- "\<forall>n. s n \<noteq> {}"
- "\<forall>m n. m \<le> n \<longrightarrow> s n \<subseteq> s m"
- "\<forall>e>0. \<exists>n. \<forall>x\<in>s n. \<forall>y\<in>s n. dist x y < e"
- shows "\<exists>a. \<forall>n. a \<in> s n"
-proof -
- have "\<forall>n. \<exists>x. x \<in> s n"
- using assms(2) by auto
- then have "\<exists>t. \<forall>n. t n \<in> s n"
- using choice[of "\<lambda>n x. x \<in> s n"] by auto
- then obtain t where t: "\<forall>n. t n \<in> s n" by auto
- {
- fix e :: real
- assume "e > 0"
- then obtain N where N:"\<forall>x\<in>s N. \<forall>y\<in>s N. dist x y < e"
- using assms(4) by auto
- {
- fix m n :: nat
- assume "N \<le> m \<and> N \<le> n"
- then have "t m \<in> s N" "t n \<in> s N"
- using assms(3) t unfolding subset_eq t by blast+
- then have "dist (t m) (t n) < e"
- using N by auto
- }
- then have "\<exists>N. \<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (t m) (t n) < e"
- by auto
- }
- then have "Cauchy t"
- unfolding cauchy_def by auto
- then obtain l where l:"(t \<longlongrightarrow> l) sequentially"
- using complete_UNIV unfolding complete_def by auto
- {
- fix n :: nat
- {
- fix e :: real
- assume "e > 0"
- then obtain N :: nat where N: "\<forall>n\<ge>N. dist (t n) l < e"
- using l[unfolded lim_sequentially] by auto
- have "t (max n N) \<in> s n"
- using assms(3)
- unfolding subset_eq
- apply (erule_tac x=n in allE)
- apply (erule_tac x="max n N" in allE)
- using t
- apply auto
- done
- then have "\<exists>y\<in>s n. dist y l < e"
- apply (rule_tac x="t (max n N)" in bexI)
- using N
- apply auto
- done
- }
- then have "l \<in> s n"
- using closed_approachable[of "s n" l] assms(1) by auto
- }
- then show ?thesis by auto
-qed
-
-text \<open>Strengthen it to the intersection actually being a singleton.\<close>
-
-lemma decreasing_closed_nest_sing:
- fixes s :: "nat \<Rightarrow> 'a::complete_space set"
- assumes
- "\<forall>n. closed(s n)"
- "\<forall>n. s n \<noteq> {}"
- "\<forall>m n. m \<le> n \<longrightarrow> s n \<subseteq> s m"
- "\<forall>e>0. \<exists>n. \<forall>x \<in> (s n). \<forall> y\<in>(s n). dist x y < e"
- shows "\<exists>a. \<Inter>(range s) = {a}"
-proof -
- obtain a where a: "\<forall>n. a \<in> s n"
- using decreasing_closed_nest[of s] using assms by auto
- {
- fix b
- assume b: "b \<in> \<Inter>(range s)"
- {
- fix e :: real
- assume "e > 0"
- then have "dist a b < e"
- using assms(4) and b and a by blast
- }
- then have "dist a b = 0"
- by (metis dist_eq_0_iff dist_nz less_le)
- }
- with a have "\<Inter>(range s) = {a}"
- unfolding image_def by auto
- then show ?thesis ..
-qed
-
-
subsection \<open>Continuity\<close>
text\<open>Derive the epsilon-delta forms, which we often use as "definitions"\<close>
@@ -6546,11 +5178,6 @@
subsubsection \<open>Structural rules for pointwise continuity\<close>
-lemma continuous_infdist[continuous_intros]:
- assumes "continuous F f"
- shows "continuous F (\<lambda>x. infdist (f x) A)"
- using assms unfolding continuous_def by (rule tendsto_infdist)
-
lemma continuous_infnorm[continuous_intros]:
"continuous F f \<Longrightarrow> continuous F (\<lambda>x. infnorm (f x))"
unfolding continuous_def by (rule tendsto_infnorm)
@@ -6561,8 +5188,6 @@
shows "continuous F (\<lambda>x. inner (f x) (g x))"
using assms unfolding continuous_def by (rule tendsto_inner)
-lemmas continuous_at_inverse = isCont_inverse
-
subsubsection \<open>Structural rules for setwise continuity\<close>
lemma continuous_on_infnorm[continuous_intros]:
@@ -6655,8 +5280,6 @@
using assms uniformly_continuous_on_add [of s f "- g"]
by (simp add: fun_Compl_def uniformly_continuous_on_minus)
-lemmas continuous_at_compose = isCont_o
-
text \<open>Continuity in terms of open preimages.\<close>
lemma continuous_at_open:
@@ -6855,705 +5478,6 @@
with \<open>x = f y\<close> show "x \<in> f ` interior s" ..
qed
-subsection \<open>Equality of continuous functions on closure and related results.\<close>
-
-lemma continuous_closedin_preimage_constant:
- fixes f :: "_ \<Rightarrow> 'b::t1_space"
- shows "continuous_on s f \<Longrightarrow> closedin (subtopology euclidean s) {x \<in> s. f x = a}"
- using continuous_closedin_preimage[of s f "{a}"] by auto
-
-lemma continuous_closed_preimage_constant:
- fixes f :: "_ \<Rightarrow> 'b::t1_space"
- shows "continuous_on s f \<Longrightarrow> closed s \<Longrightarrow> closed {x \<in> s. f x = a}"
- using continuous_closed_preimage[of s f "{a}"] by auto
-
-lemma continuous_constant_on_closure:
- fixes f :: "_ \<Rightarrow> 'b::t1_space"
- assumes "continuous_on (closure S) f"
- and "\<And>x. x \<in> S \<Longrightarrow> f x = a"
- and "x \<in> closure S"
- shows "f x = a"
- using continuous_closed_preimage_constant[of "closure S" f a]
- assms closure_minimal[of S "{x \<in> closure S. f x = a}"] closure_subset
- unfolding subset_eq
- by auto
-
-lemma image_closure_subset:
- assumes "continuous_on (closure s) f"
- and "closed t"
- and "(f ` s) \<subseteq> t"
- shows "f ` (closure s) \<subseteq> t"
-proof -
- have "s \<subseteq> {x \<in> closure s. f x \<in> t}"
- using assms(3) closure_subset by auto
- moreover have "closed {x \<in> closure s. f x \<in> t}"
- using continuous_closed_preimage[OF assms(1)] and assms(2) by auto
- ultimately have "closure s = {x \<in> closure s . f x \<in> t}"
- using closure_minimal[of s "{x \<in> closure s. f x \<in> t}"] by auto
- then show ?thesis by auto
-qed
-
-lemma continuous_on_closure_norm_le:
- fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
- assumes "continuous_on (closure s) f"
- and "\<forall>y \<in> s. norm(f y) \<le> b"
- and "x \<in> (closure s)"
- shows "norm (f x) \<le> b"
-proof -
- have *: "f ` s \<subseteq> cball 0 b"
- using assms(2)[unfolded mem_cball_0[symmetric]] by auto
- show ?thesis
- using image_closure_subset[OF assms(1) closed_cball[of 0 b] *] assms(3)
- unfolding subset_eq
- apply (erule_tac x="f x" in ballE)
- apply (auto simp: dist_norm)
- done
-qed
-
-lemma isCont_indicator:
- fixes x :: "'a::t2_space"
- shows "isCont (indicator A :: 'a \<Rightarrow> real) x = (x \<notin> frontier A)"
-proof auto
- fix x
- assume cts_at: "isCont (indicator A :: 'a \<Rightarrow> real) x" and fr: "x \<in> frontier A"
- with continuous_at_open have 1: "\<forall>V::real set. open V \<and> indicator A x \<in> V \<longrightarrow>
- (\<exists>U::'a set. open U \<and> x \<in> U \<and> (\<forall>y\<in>U. indicator A y \<in> V))" by auto
- show False
- proof (cases "x \<in> A")
- assume x: "x \<in> A"
- hence "indicator A x \<in> ({0<..<2} :: real set)" by simp
- hence "\<exists>U. open U \<and> x \<in> U \<and> (\<forall>y\<in>U. indicator A y \<in> ({0<..<2} :: real set))"
- using 1 open_greaterThanLessThan by blast
- then guess U .. note U = this
- hence "\<forall>y\<in>U. indicator A y > (0::real)"
- unfolding greaterThanLessThan_def by auto
- hence "U \<subseteq> A" using indicator_eq_0_iff by force
- hence "x \<in> interior A" using U interiorI by auto
- thus ?thesis using fr unfolding frontier_def by simp
- next
- assume x: "x \<notin> A"
- hence "indicator A x \<in> ({-1<..<1} :: real set)" by simp
- hence "\<exists>U. open U \<and> x \<in> U \<and> (\<forall>y\<in>U. indicator A y \<in> ({-1<..<1} :: real set))"
- using 1 open_greaterThanLessThan by blast
- then guess U .. note U = this
- hence "\<forall>y\<in>U. indicator A y < (1::real)"
- unfolding greaterThanLessThan_def by auto
- hence "U \<subseteq> -A" by auto
- hence "x \<in> interior (-A)" using U interiorI by auto
- thus ?thesis using fr interior_complement unfolding frontier_def by auto
- qed
-next
- assume nfr: "x \<notin> frontier A"
- hence "x \<in> interior A \<or> x \<in> interior (-A)"
- by (auto simp: frontier_def closure_interior)
- thus "isCont ((indicator A)::'a \<Rightarrow> real) x"
- proof
- assume int: "x \<in> interior A"
- then obtain U where U: "open U" "x \<in> U" "U \<subseteq> A" unfolding interior_def by auto
- hence "\<forall>y\<in>U. indicator A y = (1::real)" unfolding indicator_def by auto
- hence "continuous_on U (indicator A)" by (simp add: continuous_on_const indicator_eq_1_iff)
- thus ?thesis using U continuous_on_eq_continuous_at by auto
- next
- assume ext: "x \<in> interior (-A)"
- then obtain U where U: "open U" "x \<in> U" "U \<subseteq> -A" unfolding interior_def by auto
- then have "continuous_on U (indicator A)"
- using continuous_on_topological by (auto simp: subset_iff)
- thus ?thesis using U continuous_on_eq_continuous_at by auto
- qed
-qed
-
-subsection\<open> Theorems relating continuity and uniform continuity to closures\<close>
-
-lemma continuous_on_closure:
- "continuous_on (closure S) f \<longleftrightarrow>
- (\<forall>x e. x \<in> closure S \<and> 0 < e
- \<longrightarrow> (\<exists>d. 0 < d \<and> (\<forall>y. y \<in> S \<and> dist y x < d \<longrightarrow> dist (f y) (f x) < e)))"
- (is "?lhs = ?rhs")
-proof
- assume ?lhs then show ?rhs
- unfolding continuous_on_iff by (metis Un_iff closure_def)
-next
- assume R [rule_format]: ?rhs
- show ?lhs
- proof
- fix x and e::real
- assume "0 < e" and x: "x \<in> closure S"
- obtain \<delta>::real where "\<delta> > 0"
- and \<delta>: "\<And>y. \<lbrakk>y \<in> S; dist y x < \<delta>\<rbrakk> \<Longrightarrow> dist (f y) (f x) < e/2"
- using R [of x "e/2"] \<open>0 < e\<close> x by auto
- have "dist (f y) (f x) \<le> e" if y: "y \<in> closure S" and dyx: "dist y x < \<delta>/2" for y
- proof -
- obtain \<delta>'::real where "\<delta>' > 0"
- and \<delta>': "\<And>z. \<lbrakk>z \<in> S; dist z y < \<delta>'\<rbrakk> \<Longrightarrow> dist (f z) (f y) < e/2"
- using R [of y "e/2"] \<open>0 < e\<close> y by auto
- obtain z where "z \<in> S" and z: "dist z y < min \<delta>' \<delta> / 2"
- using closure_approachable y
- by (metis \<open>0 < \<delta>'\<close> \<open>0 < \<delta>\<close> divide_pos_pos min_less_iff_conj zero_less_numeral)
- have "dist (f z) (f y) < e/2"
- apply (rule \<delta>' [OF \<open>z \<in> S\<close>])
- using z \<open>0 < \<delta>'\<close> by linarith
- moreover have "dist (f z) (f x) < e/2"
- apply (rule \<delta> [OF \<open>z \<in> S\<close>])
- using z \<open>0 < \<delta>\<close> dist_commute[of y z] dist_triangle_half_r [of y] dyx by auto
- ultimately show ?thesis
- by (metis dist_commute dist_triangle_half_l less_imp_le)
- qed
- then show "\<exists>d>0. \<forall>x'\<in>closure S. dist x' x < d \<longrightarrow> dist (f x') (f x) \<le> e"
- by (rule_tac x="\<delta>/2" in exI) (simp add: \<open>\<delta> > 0\<close>)
- qed
-qed
-
-lemma continuous_on_closure_sequentially:
- fixes f :: "'a::metric_space \<Rightarrow> 'b :: metric_space"
- shows
- "continuous_on (closure S) f \<longleftrightarrow>
- (\<forall>x a. a \<in> closure S \<and> (\<forall>n. x n \<in> S) \<and> x \<longlonglongrightarrow> a \<longrightarrow> (f \<circ> x) \<longlonglongrightarrow> f a)"
- (is "?lhs = ?rhs")
-proof -
- have "continuous_on (closure S) f \<longleftrightarrow>
- (\<forall>x \<in> closure S. continuous (at x within S) f)"
- by (force simp: continuous_on_closure Topology_Euclidean_Space.continuous_within_eps_delta)
- also have "... = ?rhs"
- by (force simp: continuous_within_sequentially)
- finally show ?thesis .
-qed
-
-lemma uniformly_continuous_on_closure:
- fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space"
- assumes ucont: "uniformly_continuous_on S f"
- and cont: "continuous_on (closure S) f"
- shows "uniformly_continuous_on (closure S) f"
-unfolding uniformly_continuous_on_def
-proof (intro allI impI)
- fix e::real
- assume "0 < e"
- then obtain d::real
- where "d>0"
- and d: "\<And>x x'. \<lbrakk>x\<in>S; x'\<in>S; dist x' x < d\<rbrakk> \<Longrightarrow> dist (f x') (f x) < e/3"
- using ucont [unfolded uniformly_continuous_on_def, rule_format, of "e/3"] by auto
- show "\<exists>d>0. \<forall>x\<in>closure S. \<forall>x'\<in>closure S. dist x' x < d \<longrightarrow> dist (f x') (f x) < e"
- proof (rule exI [where x="d/3"], clarsimp simp: \<open>d > 0\<close>)
- fix x y
- assume x: "x \<in> closure S" and y: "y \<in> closure S" and dyx: "dist y x * 3 < d"
- obtain d1::real where "d1 > 0"
- and d1: "\<And>w. \<lbrakk>w \<in> closure S; dist w x < d1\<rbrakk> \<Longrightarrow> dist (f w) (f x) < e/3"
- using cont [unfolded continuous_on_iff, rule_format, of "x" "e/3"] \<open>0 < e\<close> x by auto
- obtain x' where "x' \<in> S" and x': "dist x' x < min d1 (d / 3)"
- using closure_approachable [of x S]
- by (metis \<open>0 < d1\<close> \<open>0 < d\<close> divide_pos_pos min_less_iff_conj x zero_less_numeral)
- obtain d2::real where "d2 > 0"
- and d2: "\<forall>w \<in> closure S. dist w y < d2 \<longrightarrow> dist (f w) (f y) < e/3"
- using cont [unfolded continuous_on_iff, rule_format, of "y" "e/3"] \<open>0 < e\<close> y by auto
- obtain y' where "y' \<in> S" and y': "dist y' y < min d2 (d / 3)"
- using closure_approachable [of y S]
- by (metis \<open>0 < d2\<close> \<open>0 < d\<close> divide_pos_pos min_less_iff_conj y zero_less_numeral)
- have "dist x' x < d/3" using x' by auto
- moreover have "dist x y < d/3"
- by (metis dist_commute dyx less_divide_eq_numeral1(1))
- moreover have "dist y y' < d/3"
- by (metis (no_types) dist_commute min_less_iff_conj y')
- ultimately have "dist x' y' < d/3 + d/3 + d/3"
- by (meson dist_commute_lessI dist_triangle_lt add_strict_mono)
- then have "dist x' y' < d" by simp
- then have "dist (f x') (f y') < e/3"
- by (rule d [OF \<open>y' \<in> S\<close> \<open>x' \<in> S\<close>])
- moreover have "dist (f x') (f x) < e/3" using \<open>x' \<in> S\<close> closure_subset x' d1
- by (simp add: closure_def)
- moreover have "dist (f y') (f y) < e/3" using \<open>y' \<in> S\<close> closure_subset y' d2
- by (simp add: closure_def)
- ultimately have "dist (f y) (f x) < e/3 + e/3 + e/3"
- by (meson dist_commute_lessI dist_triangle_lt add_strict_mono)
- then show "dist (f y) (f x) < e" by simp
- qed
-qed
-
-lemma uniformly_continuous_on_extension_at_closure:
- fixes f::"'a::metric_space \<Rightarrow> 'b::complete_space"
- assumes uc: "uniformly_continuous_on X f"
- assumes "x \<in> closure X"
- obtains l where "(f \<longlongrightarrow> l) (at x within X)"
-proof -
- from assms obtain xs where xs: "xs \<longlonglongrightarrow> x" "\<And>n. xs n \<in> X"
- by (auto simp: closure_sequential)
-
- from uniformly_continuous_on_Cauchy[OF uc LIMSEQ_imp_Cauchy, OF xs]
- obtain l where l: "(\<lambda>n. f (xs n)) \<longlonglongrightarrow> l"
- by atomize_elim (simp only: convergent_eq_Cauchy)
-
- have "(f \<longlongrightarrow> l) (at x within X)"
- proof (safe intro!: Lim_within_LIMSEQ)
- fix xs'
- assume "\<forall>n. xs' n \<noteq> x \<and> xs' n \<in> X"
- and xs': "xs' \<longlonglongrightarrow> x"
- then have "xs' n \<noteq> x" "xs' n \<in> X" for n by auto
-
- from uniformly_continuous_on_Cauchy[OF uc LIMSEQ_imp_Cauchy, OF \<open>xs' \<longlonglongrightarrow> x\<close> \<open>xs' _ \<in> X\<close>]
- obtain l' where l': "(\<lambda>n. f (xs' n)) \<longlonglongrightarrow> l'"
- by atomize_elim (simp only: convergent_eq_Cauchy)
-
- show "(\<lambda>n. f (xs' n)) \<longlonglongrightarrow> l"
- proof (rule tendstoI)
- fix e::real assume "e > 0"
- define e' where "e' \<equiv> e / 2"
- have "e' > 0" using \<open>e > 0\<close> by (simp add: e'_def)
-
- have "\<forall>\<^sub>F n in sequentially. dist (f (xs n)) l < e'"
- by (simp add: \<open>0 < e'\<close> l tendstoD)
- moreover
- from uc[unfolded uniformly_continuous_on_def, rule_format, OF \<open>e' > 0\<close>]
- obtain d where d: "d > 0" "\<And>x x'. x \<in> X \<Longrightarrow> x' \<in> X \<Longrightarrow> dist x x' < d \<Longrightarrow> dist (f x) (f x') < e'"
- by auto
- have "\<forall>\<^sub>F n in sequentially. dist (xs n) (xs' n) < d"
- by (auto intro!: \<open>0 < d\<close> order_tendstoD tendsto_eq_intros xs xs')
- ultimately
- show "\<forall>\<^sub>F n in sequentially. dist (f (xs' n)) l < e"
- proof eventually_elim
- case (elim n)
- have "dist (f (xs' n)) l \<le> dist (f (xs n)) (f (xs' n)) + dist (f (xs n)) l"
- by (metis dist_triangle dist_commute)
- also have "dist (f (xs n)) (f (xs' n)) < e'"
- by (auto intro!: d xs \<open>xs' _ \<in> _\<close> elim)
- also note \<open>dist (f (xs n)) l < e'\<close>
- also have "e' + e' = e" by (simp add: e'_def)
- finally show ?case by simp
- qed
- qed
- qed
- thus ?thesis ..
-qed
-
-lemma uniformly_continuous_on_extension_on_closure:
- fixes f::"'a::metric_space \<Rightarrow> 'b::complete_space"
- assumes uc: "uniformly_continuous_on X f"
- obtains g where "uniformly_continuous_on (closure X) g" "\<And>x. x \<in> X \<Longrightarrow> f x = g x"
- "\<And>Y h x. X \<subseteq> Y \<Longrightarrow> Y \<subseteq> closure X \<Longrightarrow> continuous_on Y h \<Longrightarrow> (\<And>x. x \<in> X \<Longrightarrow> f x = h x) \<Longrightarrow> x \<in> Y \<Longrightarrow> h x = g x"
-proof -
- from uc have cont_f: "continuous_on X f"
- by (simp add: uniformly_continuous_imp_continuous)
- obtain y where y: "(f \<longlongrightarrow> y x) (at x within X)" if "x \<in> closure X" for x
- apply atomize_elim
- apply (rule choice)
- using uniformly_continuous_on_extension_at_closure[OF assms]
- by metis
- let ?g = "\<lambda>x. if x \<in> X then f x else y x"
-
- have "uniformly_continuous_on (closure X) ?g"
- unfolding uniformly_continuous_on_def
- proof safe
- fix e::real assume "e > 0"
- define e' where "e' \<equiv> e / 3"
- have "e' > 0" using \<open>e > 0\<close> by (simp add: e'_def)
- from uc[unfolded uniformly_continuous_on_def, rule_format, OF \<open>0 < e'\<close>]
- obtain d where "d > 0" and d: "\<And>x x'. x \<in> X \<Longrightarrow> x' \<in> X \<Longrightarrow> dist x' x < d \<Longrightarrow> dist (f x') (f x) < e'"
- by auto
- define d' where "d' = d / 3"
- have "d' > 0" using \<open>d > 0\<close> by (simp add: d'_def)
- show "\<exists>d>0. \<forall>x\<in>closure X. \<forall>x'\<in>closure X. dist x' x < d \<longrightarrow> dist (?g x') (?g x) < e"
- proof (safe intro!: exI[where x=d'] \<open>d' > 0\<close>)
- fix x x' assume x: "x \<in> closure X" and x': "x' \<in> closure X" and dist: "dist x' x < d'"
- then obtain xs xs' where xs: "xs \<longlonglongrightarrow> x" "\<And>n. xs n \<in> X"
- and xs': "xs' \<longlonglongrightarrow> x'" "\<And>n. xs' n \<in> X"
- by (auto simp: closure_sequential)
- have "\<forall>\<^sub>F n in sequentially. dist (xs' n) x' < d'"
- and "\<forall>\<^sub>F n in sequentially. dist (xs n) x < d'"
- by (auto intro!: \<open>0 < d'\<close> order_tendstoD tendsto_eq_intros xs xs')
- moreover
- have "(\<lambda>x. f (xs x)) \<longlonglongrightarrow> y x" if "x \<in> closure X" "x \<notin> X" "xs \<longlonglongrightarrow> x" "\<And>n. xs n \<in> X" for xs x
- using that not_eventuallyD
- by (force intro!: filterlim_compose[OF y[OF \<open>x \<in> closure X\<close>]] simp: filterlim_at)
- then have "(\<lambda>x. f (xs' x)) \<longlonglongrightarrow> ?g x'" "(\<lambda>x. f (xs x)) \<longlonglongrightarrow> ?g x"
- using x x'
- by (auto intro!: continuous_on_tendsto_compose[OF cont_f] simp: xs' xs)
- then have "\<forall>\<^sub>F n in sequentially. dist (f (xs' n)) (?g x') < e'"
- "\<forall>\<^sub>F n in sequentially. dist (f (xs n)) (?g x) < e'"
- by (auto intro!: \<open>0 < e'\<close> order_tendstoD tendsto_eq_intros)
- ultimately
- have "\<forall>\<^sub>F n in sequentially. dist (?g x') (?g x) < e"
- proof eventually_elim
- case (elim n)
- have "dist (?g x') (?g x) \<le>
- dist (f (xs' n)) (?g x') + dist (f (xs' n)) (f (xs n)) + dist (f (xs n)) (?g x)"
- by (metis add.commute add_le_cancel_left dist_commute dist_triangle dist_triangle_le)
- also
- {
- have "dist (xs' n) (xs n) \<le> dist (xs' n) x' + dist x' x + dist (xs n) x"
- by (metis add.commute add_le_cancel_left dist_triangle dist_triangle_le)
- also note \<open>dist (xs' n) x' < d'\<close>
- also note \<open>dist x' x < d'\<close>
- also note \<open>dist (xs n) x < d'\<close>
- finally have "dist (xs' n) (xs n) < d" by (simp add: d'_def)
- }
- with \<open>xs _ \<in> X\<close> \<open>xs' _ \<in> X\<close> have "dist (f (xs' n)) (f (xs n)) < e'"
- by (rule d)
- also note \<open>dist (f (xs' n)) (?g x') < e'\<close>
- also note \<open>dist (f (xs n)) (?g x) < e'\<close>
- finally show ?case by (simp add: e'_def)
- qed
- then show "dist (?g x') (?g x) < e" by simp
- qed
- qed
- moreover have "f x = ?g x" if "x \<in> X" for x using that by simp
- moreover
- {
- fix Y h x
- assume Y: "x \<in> Y" "X \<subseteq> Y" "Y \<subseteq> closure X" and cont_h: "continuous_on Y h"
- and extension: "(\<And>x. x \<in> X \<Longrightarrow> f x = h x)"
- {
- assume "x \<notin> X"
- have "x \<in> closure X" using Y by auto
- then obtain xs where xs: "xs \<longlonglongrightarrow> x" "\<And>n. xs n \<in> X"
- by (auto simp: closure_sequential)
- from continuous_on_tendsto_compose[OF cont_h xs(1)] xs(2) Y
- have hx: "(\<lambda>x. f (xs x)) \<longlonglongrightarrow> h x"
- by (auto simp: set_mp extension)
- then have "(\<lambda>x. f (xs x)) \<longlonglongrightarrow> y x"
- using \<open>x \<notin> X\<close> not_eventuallyD xs(2)
- by (force intro!: filterlim_compose[OF y[OF \<open>x \<in> closure X\<close>]] simp: filterlim_at xs)
- with hx have "h x = y x" by (rule LIMSEQ_unique)
- } then
- have "h x = ?g x"
- using extension by auto
- }
- ultimately show ?thesis ..
-qed
-
-lemma bounded_uniformly_continuous_image:
- fixes f :: "'a :: heine_borel \<Rightarrow> 'b :: heine_borel"
- assumes "uniformly_continuous_on S f" "bounded S"
- shows "bounded(image f S)"
- by (metis (no_types, lifting) assms bounded_closure_image compact_closure compact_continuous_image compact_eq_bounded_closed image_cong uniformly_continuous_imp_continuous uniformly_continuous_on_extension_on_closure)
-
-subsection\<open>Quotient maps\<close>
-
-lemma quotient_map_imp_continuous_open:
- assumes t: "f ` s \<subseteq> t"
- and ope: "\<And>u. u \<subseteq> t
- \<Longrightarrow> (openin (subtopology euclidean s) {x. x \<in> s \<and> f x \<in> u} \<longleftrightarrow>
- openin (subtopology euclidean t) u)"
- shows "continuous_on s f"
-proof -
- have [simp]: "{x \<in> s. f x \<in> f ` s} = s" by auto
- show ?thesis
- using ope [OF t]
- apply (simp add: continuous_on_open)
- by (metis (no_types, lifting) "ope" openin_imp_subset openin_trans)
-qed
-
-lemma quotient_map_imp_continuous_closed:
- assumes t: "f ` s \<subseteq> t"
- and ope: "\<And>u. u \<subseteq> t
- \<Longrightarrow> (closedin (subtopology euclidean s) {x. x \<in> s \<and> f x \<in> u} \<longleftrightarrow>
- closedin (subtopology euclidean t) u)"
- shows "continuous_on s f"
-proof -
- have [simp]: "{x \<in> s. f x \<in> f ` s} = s" by auto
- show ?thesis
- using ope [OF t]
- apply (simp add: continuous_on_closed)
- by (metis (no_types, lifting) "ope" closedin_imp_subset closedin_subtopology_refl closedin_trans openin_subtopology_refl openin_subtopology_self)
-qed
-
-lemma open_map_imp_quotient_map:
- assumes contf: "continuous_on s f"
- and t: "t \<subseteq> f ` s"
- and ope: "\<And>t. openin (subtopology euclidean s) t
- \<Longrightarrow> openin (subtopology euclidean (f ` s)) (f ` t)"
- shows "openin (subtopology euclidean s) {x \<in> s. f x \<in> t} =
- openin (subtopology euclidean (f ` s)) t"
-proof -
- have "t = image f {x. x \<in> s \<and> f x \<in> t}"
- using t by blast
- then show ?thesis
- using "ope" contf continuous_on_open by fastforce
-qed
-
-lemma closed_map_imp_quotient_map:
- assumes contf: "continuous_on s f"
- and t: "t \<subseteq> f ` s"
- and ope: "\<And>t. closedin (subtopology euclidean s) t
- \<Longrightarrow> closedin (subtopology euclidean (f ` s)) (f ` t)"
- shows "openin (subtopology euclidean s) {x \<in> s. f x \<in> t} \<longleftrightarrow>
- openin (subtopology euclidean (f ` s)) t"
- (is "?lhs = ?rhs")
-proof
- assume ?lhs
- then have *: "closedin (subtopology euclidean s) (s - {x \<in> s. f x \<in> t})"
- using closedin_diff by fastforce
- have [simp]: "(f ` s - f ` (s - {x \<in> s. f x \<in> t})) = t"
- using t by blast
- show ?rhs
- using ope [OF *, unfolded closedin_def] by auto
-next
- assume ?rhs
- with contf show ?lhs
- by (auto simp: continuous_on_open)
-qed
-
-lemma continuous_right_inverse_imp_quotient_map:
- assumes contf: "continuous_on s f" and imf: "f ` s \<subseteq> t"
- and contg: "continuous_on t g" and img: "g ` t \<subseteq> s"
- and fg [simp]: "\<And>y. y \<in> t \<Longrightarrow> f(g y) = y"
- and u: "u \<subseteq> t"
- shows "openin (subtopology euclidean s) {x. x \<in> s \<and> f x \<in> u} \<longleftrightarrow>
- openin (subtopology euclidean t) u"
- (is "?lhs = ?rhs")
-proof -
- have f: "\<And>z. openin (subtopology euclidean (f ` s)) z \<Longrightarrow>
- openin (subtopology euclidean s) {x \<in> s. f x \<in> z}"
- and g: "\<And>z. openin (subtopology euclidean (g ` t)) z \<Longrightarrow>
- openin (subtopology euclidean t) {x \<in> t. g x \<in> z}"
- using contf contg by (auto simp: continuous_on_open)
- show ?thesis
- proof
- have "{x \<in> t. g x \<in> g ` t \<and> g x \<in> s \<and> f (g x) \<in> u} = {x \<in> t. f (g x) \<in> u}"
- using imf img by blast
- also have "... = u"
- using u by auto
- finally have [simp]: "{x \<in> t. g x \<in> g ` t \<and> g x \<in> s \<and> f (g x) \<in> u} = u" .
- assume ?lhs
- then have *: "openin (subtopology euclidean (g ` t)) (g ` t \<inter> {x \<in> s. f x \<in> u})"
- by (meson img openin_Int openin_subtopology_Int_subset openin_subtopology_self)
- show ?rhs
- using g [OF *] by simp
- next
- assume rhs: ?rhs
- show ?lhs
- apply (rule f)
- by (metis fg image_eqI image_subset_iff imf img openin_subopen openin_subtopology_self openin_trans rhs)
- qed
-qed
-
-lemma continuous_left_inverse_imp_quotient_map:
- assumes "continuous_on s f"
- and "continuous_on (f ` s) g"
- and "\<And>x. x \<in> s \<Longrightarrow> g(f x) = x"
- and "u \<subseteq> f ` s"
- shows "openin (subtopology euclidean s) {x. x \<in> s \<and> f x \<in> u} \<longleftrightarrow>
- openin (subtopology euclidean (f ` s)) u"
-apply (rule continuous_right_inverse_imp_quotient_map)
-using assms
-apply force+
-done
-
-subsection \<open>A function constant on a set\<close>
-
-definition constant_on (infixl "(constant'_on)" 50)
- where "f constant_on A \<equiv> \<exists>y. \<forall>x\<in>A. f x = y"
-
-lemma constant_on_subset: "\<lbrakk>f constant_on A; B \<subseteq> A\<rbrakk> \<Longrightarrow> f constant_on B"
- unfolding constant_on_def by blast
-
-lemma injective_not_constant:
- fixes S :: "'a::{perfect_space} set"
- shows "\<lbrakk>open S; inj_on f S; f constant_on S\<rbrakk> \<Longrightarrow> S = {}"
-unfolding constant_on_def
-by (metis equals0I inj_on_contraD islimpt_UNIV islimpt_def)
-
-lemma constant_on_closureI:
- fixes f :: "_ \<Rightarrow> 'b::t1_space"
- assumes cof: "f constant_on S" and contf: "continuous_on (closure S) f"
- shows "f constant_on (closure S)"
-using continuous_constant_on_closure [OF contf] cof unfolding constant_on_def
-by metis
-
-text \<open>Making a continuous function avoid some value in a neighbourhood.\<close>
-
-lemma continuous_within_avoid:
- fixes f :: "'a::metric_space \<Rightarrow> 'b::t1_space"
- assumes "continuous (at x within s) f"
- and "f x \<noteq> a"
- shows "\<exists>e>0. \<forall>y \<in> s. dist x y < e --> f y \<noteq> a"
-proof -
- obtain U where "open U" and "f x \<in> U" and "a \<notin> U"
- using t1_space [OF \<open>f x \<noteq> a\<close>] by fast
- have "(f \<longlongrightarrow> f x) (at x within s)"
- using assms(1) by (simp add: continuous_within)
- then have "eventually (\<lambda>y. f y \<in> U) (at x within s)"
- using \<open>open U\<close> and \<open>f x \<in> U\<close>
- unfolding tendsto_def by fast
- then have "eventually (\<lambda>y. f y \<noteq> a) (at x within s)"
- using \<open>a \<notin> U\<close> by (fast elim: eventually_mono)
- then show ?thesis
- using \<open>f x \<noteq> a\<close> by (auto simp: dist_commute zero_less_dist_iff eventually_at)
-qed
-
-lemma continuous_at_avoid:
- fixes f :: "'a::metric_space \<Rightarrow> 'b::t1_space"
- assumes "continuous (at x) f"
- and "f x \<noteq> a"
- shows "\<exists>e>0. \<forall>y. dist x y < e \<longrightarrow> f y \<noteq> a"
- using assms continuous_within_avoid[of x UNIV f a] by simp
-
-lemma continuous_on_avoid:
- fixes f :: "'a::metric_space \<Rightarrow> 'b::t1_space"
- assumes "continuous_on s f"
- and "x \<in> s"
- and "f x \<noteq> a"
- shows "\<exists>e>0. \<forall>y \<in> s. dist x y < e \<longrightarrow> f y \<noteq> a"
- using assms(1)[unfolded continuous_on_eq_continuous_within, THEN bspec[where x=x],
- OF assms(2)] continuous_within_avoid[of x s f a]
- using assms(3)
- by auto
-
-lemma continuous_on_open_avoid:
- fixes f :: "'a::metric_space \<Rightarrow> 'b::t1_space"
- assumes "continuous_on s f"
- and "open s"
- and "x \<in> s"
- and "f x \<noteq> a"
- shows "\<exists>e>0. \<forall>y. dist x y < e \<longrightarrow> f y \<noteq> a"
- using assms(1)[unfolded continuous_on_eq_continuous_at[OF assms(2)], THEN bspec[where x=x], OF assms(3)]
- using continuous_at_avoid[of x f a] assms(4)
- by auto
-
-text \<open>Proving a function is constant by proving open-ness of level set.\<close>
-
-lemma continuous_levelset_openin_cases:
- fixes f :: "_ \<Rightarrow> 'b::t1_space"
- shows "connected s \<Longrightarrow> continuous_on s f \<Longrightarrow>
- openin (subtopology euclidean s) {x \<in> s. f x = a}
- \<Longrightarrow> (\<forall>x \<in> s. f x \<noteq> a) \<or> (\<forall>x \<in> s. f x = a)"
- unfolding connected_clopen
- using continuous_closedin_preimage_constant by auto
-
-lemma continuous_levelset_openin:
- fixes f :: "_ \<Rightarrow> 'b::t1_space"
- shows "connected s \<Longrightarrow> continuous_on s f \<Longrightarrow>
- openin (subtopology euclidean s) {x \<in> s. f x = a} \<Longrightarrow>
- (\<exists>x \<in> s. f x = a) \<Longrightarrow> (\<forall>x \<in> s. f x = a)"
- using continuous_levelset_openin_cases[of s f ]
- by meson
-
-lemma continuous_levelset_open:
- fixes f :: "_ \<Rightarrow> 'b::t1_space"
- assumes "connected s"
- and "continuous_on s f"
- and "open {x \<in> s. f x = a}"
- and "\<exists>x \<in> s. f x = a"
- shows "\<forall>x \<in> s. f x = a"
- using continuous_levelset_openin[OF assms(1,2), of a, unfolded openin_open]
- using assms (3,4)
- by fast
-
-text \<open>Some arithmetical combinations (more to prove).\<close>
-
-lemma open_scaling[intro]:
- fixes s :: "'a::real_normed_vector set"
- assumes "c \<noteq> 0"
- and "open s"
- shows "open((\<lambda>x. c *\<^sub>R x) ` s)"
-proof -
- {
- fix x
- assume "x \<in> s"
- then obtain e where "e>0"
- and e:"\<forall>x'. dist x' x < e \<longrightarrow> x' \<in> s" using assms(2)[unfolded open_dist, THEN bspec[where x=x]]
- by auto
- have "e * \<bar>c\<bar> > 0"
- using assms(1)[unfolded zero_less_abs_iff[symmetric]] \<open>e>0\<close> by auto
- moreover
- {
- fix y
- assume "dist y (c *\<^sub>R x) < e * \<bar>c\<bar>"
- then have "norm ((1 / c) *\<^sub>R y - x) < e"
- unfolding dist_norm
- using norm_scaleR[of c "(1 / c) *\<^sub>R y - x", unfolded scaleR_right_diff_distrib, unfolded scaleR_scaleR] assms(1)
- assms(1)[unfolded zero_less_abs_iff[symmetric]] by (simp del:zero_less_abs_iff)
- then have "y \<in> op *\<^sub>R c ` s"
- using rev_image_eqI[of "(1 / c) *\<^sub>R y" s y "op *\<^sub>R c"]
- using e[THEN spec[where x="(1 / c) *\<^sub>R y"]]
- using assms(1)
- unfolding dist_norm scaleR_scaleR
- by auto
- }
- ultimately have "\<exists>e>0. \<forall>x'. dist x' (c *\<^sub>R x) < e \<longrightarrow> x' \<in> op *\<^sub>R c ` s"
- apply (rule_tac x="e * \<bar>c\<bar>" in exI, auto)
- done
- }
- then show ?thesis unfolding open_dist by auto
-qed
-
-lemma minus_image_eq_vimage:
- fixes A :: "'a::ab_group_add set"
- shows "(\<lambda>x. - x) ` A = (\<lambda>x. - x) -` A"
- by (auto intro!: image_eqI [where f="\<lambda>x. - x"])
-
-lemma open_negations:
- fixes S :: "'a::real_normed_vector set"
- shows "open S \<Longrightarrow> open ((\<lambda>x. - x) ` S)"
- using open_scaling [of "- 1" S] by simp
-
-lemma open_translation:
- fixes S :: "'a::real_normed_vector set"
- assumes "open S"
- shows "open((\<lambda>x. a + x) ` S)"
-proof -
- {
- fix x
- have "continuous (at x) (\<lambda>x. x - a)"
- by (intro continuous_diff continuous_ident continuous_const)
- }
- moreover have "{x. x - a \<in> S} = op + a ` S"
- by force
- ultimately show ?thesis
- by (metis assms continuous_open_vimage vimage_def)
-qed
-
-lemma open_affinity:
- fixes S :: "'a::real_normed_vector set"
- assumes "open S" "c \<noteq> 0"
- shows "open ((\<lambda>x. a + c *\<^sub>R x) ` S)"
-proof -
- have *: "(\<lambda>x. a + c *\<^sub>R x) = (\<lambda>x. a + x) \<circ> (\<lambda>x. c *\<^sub>R x)"
- unfolding o_def ..
- have "op + a ` op *\<^sub>R c ` S = (op + a \<circ> op *\<^sub>R c) ` S"
- by auto
- then show ?thesis
- using assms open_translation[of "op *\<^sub>R c ` S" a]
- unfolding *
- by auto
-qed
-
-lemma interior_translation:
- fixes S :: "'a::real_normed_vector set"
- shows "interior ((\<lambda>x. a + x) ` S) = (\<lambda>x. a + x) ` (interior S)"
-proof (rule set_eqI, rule)
- fix x
- assume "x \<in> interior (op + a ` S)"
- then obtain e where "e > 0" and e: "ball x e \<subseteq> op + a ` S"
- unfolding mem_interior by auto
- then have "ball (x - a) e \<subseteq> S"
- unfolding subset_eq Ball_def mem_ball dist_norm
- by (auto simp: diff_diff_eq)
- then show "x \<in> op + a ` interior S"
- unfolding image_iff
- apply (rule_tac x="x - a" in bexI)
- unfolding mem_interior
- using \<open>e > 0\<close>
- apply auto
- done
-next
- fix x
- assume "x \<in> op + a ` interior S"
- then obtain y e where "e > 0" and e: "ball y e \<subseteq> S" and y: "x = a + y"
- unfolding image_iff Bex_def mem_interior by auto
- {
- fix z
- have *: "a + y - z = y + a - z" by auto
- assume "z \<in> ball x e"
- then have "z - a \<in> S"
- using e[unfolded subset_eq, THEN bspec[where x="z - a"]]
- unfolding mem_ball dist_norm y group_add_class.diff_diff_eq2 *
- by auto
- then have "z \<in> op + a ` S"
- unfolding image_iff by (auto intro!: bexI[where x="z - a"])
- }
- then have "ball x e \<subseteq> op + a ` S"
- unfolding subset_eq by auto
- then show "x \<in> interior (op + a ` S)"
- unfolding mem_interior using \<open>e > 0\<close> by auto
-qed
-
subsection \<open>Topological properties of linear functions.\<close>
lemma linear_lim_0:
@@ -7582,1106 +5506,7 @@
"bounded_linear f \<Longrightarrow> continuous_on s f"
using continuous_at_imp_continuous_on[of s f] using linear_continuous_at[of f] by auto
-subsubsection\<open>Relating linear images to open/closed/interior/closure.\<close>
-
-proposition open_surjective_linear_image:
- fixes f :: "'a::real_normed_vector \<Rightarrow> 'b::euclidean_space"
- assumes "open A" "linear f" "surj f"
- shows "open(f ` A)"
-unfolding open_dist
-proof clarify
- fix x
- assume "x \<in> A"
- have "bounded (inv f ` Basis)"
- by (simp add: finite_imp_bounded)
- with bounded_pos obtain B where "B > 0" and B: "\<And>x. x \<in> inv f ` Basis \<Longrightarrow> norm x \<le> B"
- by metis
- obtain e where "e > 0" and e: "\<And>z. dist z x < e \<Longrightarrow> z \<in> A"
- by (metis open_dist \<open>x \<in> A\<close> \<open>open A\<close>)
- define \<delta> where "\<delta> \<equiv> e / B / DIM('b)"
- show "\<exists>e>0. \<forall>y. dist y (f x) < e \<longrightarrow> y \<in> f ` A"
- proof (intro exI conjI)
- show "\<delta> > 0"
- using \<open>e > 0\<close> \<open>B > 0\<close> by (simp add: \<delta>_def divide_simps)
- have "y \<in> f ` A" if "dist y (f x) * (B * real DIM('b)) < e" for y
- proof -
- define u where "u \<equiv> y - f x"
- show ?thesis
- proof (rule image_eqI)
- show "y = f (x + (\<Sum>i\<in>Basis. (u \<bullet> i) *\<^sub>R inv f i))"
- apply (simp add: linear_add linear_sum linear.scaleR \<open>linear f\<close> surj_f_inv_f \<open>surj f\<close>)
- apply (simp add: euclidean_representation u_def)
- done
- have "dist (x + (\<Sum>i\<in>Basis. (u \<bullet> i) *\<^sub>R inv f i)) x \<le> (\<Sum>i\<in>Basis. norm ((u \<bullet> i) *\<^sub>R inv f i))"
- by (simp add: dist_norm sum_norm_le)
- also have "... = (\<Sum>i\<in>Basis. \<bar>u \<bullet> i\<bar> * norm (inv f i))"
- by simp
- also have "... \<le> (\<Sum>i\<in>Basis. \<bar>u \<bullet> i\<bar>) * B"
- by (simp add: B sum_distrib_right sum_mono mult_left_mono)
- also have "... \<le> DIM('b) * dist y (f x) * B"
- apply (rule mult_right_mono [OF sum_bounded_above])
- using \<open>0 < B\<close> by (auto simp: Basis_le_norm dist_norm u_def)
- also have "... < e"
- by (metis mult.commute mult.left_commute that)
- finally show "x + (\<Sum>i\<in>Basis. (u \<bullet> i) *\<^sub>R inv f i) \<in> A"
- by (rule e)
- qed
- qed
- then show "\<forall>y. dist y (f x) < \<delta> \<longrightarrow> y \<in> f ` A"
- using \<open>e > 0\<close> \<open>B > 0\<close>
- by (auto simp: \<delta>_def divide_simps mult_less_0_iff)
- qed
-qed
-
-corollary open_bijective_linear_image_eq:
- fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
- assumes "linear f" "bij f"
- shows "open(f ` A) \<longleftrightarrow> open A"
-proof
- assume "open(f ` A)"
- then have "open(f -` (f ` A))"
- using assms by (force simp: linear_continuous_at linear_conv_bounded_linear continuous_open_vimage)
- then show "open A"
- by (simp add: assms bij_is_inj inj_vimage_image_eq)
-next
- assume "open A"
- then show "open(f ` A)"
- by (simp add: assms bij_is_surj open_surjective_linear_image)
-qed
-
-corollary interior_bijective_linear_image:
- fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
- assumes "linear f" "bij f"
- shows "interior (f ` S) = f ` interior S" (is "?lhs = ?rhs")
-proof safe
- fix x
- assume x: "x \<in> ?lhs"
- then obtain T where "open T" and "x \<in> T" and "T \<subseteq> f ` S"
- by (metis interiorE)
- then show "x \<in> ?rhs"
- by (metis (no_types, hide_lams) assms subsetD interior_maximal open_bijective_linear_image_eq subset_image_iff)
-next
- fix x
- assume x: "x \<in> interior S"
- then show "f x \<in> interior (f ` S)"
- by (meson assms imageI image_mono interiorI interior_subset open_bijective_linear_image_eq open_interior)
-qed
-
-lemma interior_injective_linear_image:
- fixes f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"
- assumes "linear f" "inj f"
- shows "interior(f ` S) = f ` (interior S)"
- by (simp add: linear_injective_imp_surjective assms bijI interior_bijective_linear_image)
-
-lemma interior_surjective_linear_image:
- fixes f :: "'a::euclidean_space \<Rightarrow> 'a::euclidean_space"
- assumes "linear f" "surj f"
- shows "interior(f ` S) = f ` (interior S)"
- by (simp add: assms interior_injective_linear_image linear_surjective_imp_injective)
-
-lemma interior_negations:
- fixes S :: "'a::euclidean_space set"
- shows "interior(uminus ` S) = image uminus (interior S)"
- by (simp add: bij_uminus interior_bijective_linear_image linear_uminus)
-
-text \<open>Also bilinear functions, in composition form.\<close>
-
-lemma bilinear_continuous_at_compose:
- "continuous (at x) f \<Longrightarrow> continuous (at x) g \<Longrightarrow> bounded_bilinear h \<Longrightarrow>
- continuous (at x) (\<lambda>x. h (f x) (g x))"
- unfolding continuous_at
- using Lim_bilinear[of f "f x" "(at x)" g "g x" h]
- by auto
-
-lemma bilinear_continuous_within_compose:
- "continuous (at x within s) f \<Longrightarrow> continuous (at x within s) g \<Longrightarrow> bounded_bilinear h \<Longrightarrow>
- continuous (at x within s) (\<lambda>x. h (f x) (g x))"
- by (rule Limits.bounded_bilinear.continuous)
-
-lemma bilinear_continuous_on_compose:
- "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> bounded_bilinear h \<Longrightarrow>
- continuous_on s (\<lambda>x. h (f x) (g x))"
- by (rule Limits.bounded_bilinear.continuous_on)
-
-text \<open>Preservation of compactness and connectedness under continuous function.\<close>
-
-lemma compact_eq_openin_cover:
- "compact S \<longleftrightarrow>
- (\<forall>C. (\<forall>c\<in>C. openin (subtopology euclidean S) c) \<and> S \<subseteq> \<Union>C \<longrightarrow>
- (\<exists>D\<subseteq>C. finite D \<and> S \<subseteq> \<Union>D))"
-proof safe
- fix C
- assume "compact S" and "\<forall>c\<in>C. openin (subtopology euclidean S) c" and "S \<subseteq> \<Union>C"
- then have "\<forall>c\<in>{T. open T \<and> S \<inter> T \<in> C}. open c" and "S \<subseteq> \<Union>{T. open T \<and> S \<inter> T \<in> C}"
- unfolding openin_open by force+
- with \<open>compact S\<close> obtain D where "D \<subseteq> {T. open T \<and> S \<inter> T \<in> C}" and "finite D" and "S \<subseteq> \<Union>D"
- by (meson compactE)
- then have "image (\<lambda>T. S \<inter> T) D \<subseteq> C \<and> finite (image (\<lambda>T. S \<inter> T) D) \<and> S \<subseteq> \<Union>(image (\<lambda>T. S \<inter> T) D)"
- by auto
- then show "\<exists>D\<subseteq>C. finite D \<and> S \<subseteq> \<Union>D" ..
-next
- assume 1: "\<forall>C. (\<forall>c\<in>C. openin (subtopology euclidean S) c) \<and> S \<subseteq> \<Union>C \<longrightarrow>
- (\<exists>D\<subseteq>C. finite D \<and> S \<subseteq> \<Union>D)"
- show "compact S"
- proof (rule compactI)
- fix C
- let ?C = "image (\<lambda>T. S \<inter> T) C"
- assume "\<forall>t\<in>C. open t" and "S \<subseteq> \<Union>C"
- then have "(\<forall>c\<in>?C. openin (subtopology euclidean S) c) \<and> S \<subseteq> \<Union>?C"
- unfolding openin_open by auto
- with 1 obtain D where "D \<subseteq> ?C" and "finite D" and "S \<subseteq> \<Union>D"
- by metis
- let ?D = "inv_into C (\<lambda>T. S \<inter> T) ` D"
- have "?D \<subseteq> C \<and> finite ?D \<and> S \<subseteq> \<Union>?D"
- proof (intro conjI)
- from \<open>D \<subseteq> ?C\<close> show "?D \<subseteq> C"
- by (fast intro: inv_into_into)
- from \<open>finite D\<close> show "finite ?D"
- by (rule finite_imageI)
- from \<open>S \<subseteq> \<Union>D\<close> show "S \<subseteq> \<Union>?D"
- apply (rule subset_trans, clarsimp)
- apply (frule subsetD [OF \<open>D \<subseteq> ?C\<close>, THEN f_inv_into_f])
- apply (erule rev_bexI, fast)
- done
- qed
- then show "\<exists>D\<subseteq>C. finite D \<and> S \<subseteq> \<Union>D" ..
- qed
-qed
-
-lemma connected_continuous_image:
- assumes "continuous_on s f"
- and "connected s"
- shows "connected(f ` s)"
-proof -
- {
- fix T
- assume as:
- "T \<noteq> {}"
- "T \<noteq> f ` s"
- "openin (subtopology euclidean (f ` s)) T"
- "closedin (subtopology euclidean (f ` s)) T"
- have "{x \<in> s. f x \<in> T} = {} \<or> {x \<in> s. f x \<in> T} = s"
- using assms(1)[unfolded continuous_on_open, THEN spec[where x=T]]
- using assms(1)[unfolded continuous_on_closed, THEN spec[where x=T]]
- using assms(2)[unfolded connected_clopen, THEN spec[where x="{x \<in> s. f x \<in> T}"]] as(3,4) by auto
- then have False using as(1,2)
- using as(4)[unfolded closedin_def topspace_euclidean_subtopology] by auto
- }
- then show ?thesis
- unfolding connected_clopen by auto
-qed
-
-lemma connected_linear_image:
- fixes f :: "'a::euclidean_space \<Rightarrow> 'b::real_normed_vector"
- assumes "linear f" and "connected s"
- shows "connected (f ` s)"
-using connected_continuous_image assms linear_continuous_on linear_conv_bounded_linear by blast
-
-text \<open>Continuity implies uniform continuity on a compact domain.\<close>
-
-subsection \<open>Continuity implies uniform continuity on a compact domain.\<close>
-
-text\<open>From the proof of the Heine-Borel theorem: Lemma 2 in section 3.7, page 69 of
-J. C. Burkill and H. Burkill. A Second Course in Mathematical Analysis (CUP, 2002)\<close>
-
-lemma Heine_Borel_lemma:
- assumes "compact S" and Ssub: "S \<subseteq> \<Union>\<G>" and op: "\<And>G. G \<in> \<G> \<Longrightarrow> open G"
- obtains e where "0 < e" "\<And>x. x \<in> S \<Longrightarrow> \<exists>G \<in> \<G>. ball x e \<subseteq> G"
-proof -
- have False if neg: "\<And>e. 0 < e \<Longrightarrow> \<exists>x \<in> S. \<forall>G \<in> \<G>. \<not> ball x e \<subseteq> G"
- proof -
- have "\<exists>x \<in> S. \<forall>G \<in> \<G>. \<not> ball x (1 / Suc n) \<subseteq> G" for n
- using neg by simp
- then obtain f where "\<And>n. f n \<in> S" and fG: "\<And>G n. G \<in> \<G> \<Longrightarrow> \<not> ball (f n) (1 / Suc n) \<subseteq> G"
- by metis
- then obtain l r where "l \<in> S" "strict_mono r" and to_l: "(f \<circ> r) \<longlonglongrightarrow> l"
- using \<open>compact S\<close> compact_def that by metis
- then obtain G where "l \<in> G" "G \<in> \<G>"
- using Ssub by auto
- then obtain e where "0 < e" and e: "\<And>z. dist z l < e \<Longrightarrow> z \<in> G"
- using op open_dist by blast
- obtain N1 where N1: "\<And>n. n \<ge> N1 \<Longrightarrow> dist (f (r n)) l < e/2"
- using to_l apply (simp add: lim_sequentially)
- using \<open>0 < e\<close> half_gt_zero that by blast
- obtain N2 where N2: "of_nat N2 > 2/e"
- using reals_Archimedean2 by blast
- obtain x where "x \<in> ball (f (r (max N1 N2))) (1 / real (Suc (r (max N1 N2))))" and "x \<notin> G"
- using fG [OF \<open>G \<in> \<G>\<close>, of "r (max N1 N2)"] by blast
- then have "dist (f (r (max N1 N2))) x < 1 / real (Suc (r (max N1 N2)))"
- by simp
- also have "... \<le> 1 / real (Suc (max N1 N2))"
- apply (simp add: divide_simps del: max.bounded_iff)
- using \<open>strict_mono r\<close> seq_suble by blast
- also have "... \<le> 1 / real (Suc N2)"
- by (simp add: field_simps)
- also have "... < e/2"
- using N2 \<open>0 < e\<close> by (simp add: field_simps)
- finally have "dist (f (r (max N1 N2))) x < e / 2" .
- moreover have "dist (f (r (max N1 N2))) l < e/2"
- using N1 max.cobounded1 by blast
- ultimately have "dist x l < e"
- using dist_triangle_half_r by blast
- then show ?thesis
- using e \<open>x \<notin> G\<close> by blast
- qed
- then show ?thesis
- by (meson that)
-qed
-
-lemma compact_uniformly_equicontinuous:
- assumes "compact S"
- and cont: "\<And>x e. \<lbrakk>x \<in> S; 0 < e\<rbrakk>
- \<Longrightarrow> \<exists>d. 0 < d \<and>
- (\<forall>f \<in> \<F>. \<forall>x' \<in> S. dist x' x < d \<longrightarrow> dist (f x') (f x) < e)"
- and "0 < e"
- obtains d where "0 < d"
- "\<And>f x x'. \<lbrakk>f \<in> \<F>; x \<in> S; x' \<in> S; dist x' x < d\<rbrakk> \<Longrightarrow> dist (f x') (f x) < e"
-proof -
- obtain d where d_pos: "\<And>x e. \<lbrakk>x \<in> S; 0 < e\<rbrakk> \<Longrightarrow> 0 < d x e"
- and d_dist : "\<And>x x' e f. \<lbrakk>dist x' x < d x e; x \<in> S; x' \<in> S; 0 < e; f \<in> \<F>\<rbrakk> \<Longrightarrow> dist (f x') (f x) < e"
- using cont by metis
- let ?\<G> = "((\<lambda>x. ball x (d x (e / 2))) ` S)"
- have Ssub: "S \<subseteq> \<Union> ?\<G>"
- by clarsimp (metis d_pos \<open>0 < e\<close> dist_self half_gt_zero_iff)
- then obtain k where "0 < k" and k: "\<And>x. x \<in> S \<Longrightarrow> \<exists>G \<in> ?\<G>. ball x k \<subseteq> G"
- by (rule Heine_Borel_lemma [OF \<open>compact S\<close>]) auto
- moreover have "dist (f v) (f u) < e" if "f \<in> \<F>" "u \<in> S" "v \<in> S" "dist v u < k" for f u v
- proof -
- obtain G where "G \<in> ?\<G>" "u \<in> G" "v \<in> G"
- using k that
- by (metis \<open>dist v u < k\<close> \<open>u \<in> S\<close> \<open>0 < k\<close> centre_in_ball subsetD dist_commute mem_ball)
- then obtain w where w: "dist w u < d w (e / 2)" "dist w v < d w (e / 2)" "w \<in> S"
- by auto
- with that d_dist have "dist (f w) (f v) < e/2"
- by (metis \<open>0 < e\<close> dist_commute half_gt_zero)
- moreover
- have "dist (f w) (f u) < e/2"
- using that d_dist w by (metis \<open>0 < e\<close> dist_commute divide_pos_pos zero_less_numeral)
- ultimately show ?thesis
- using dist_triangle_half_r by blast
- qed
- ultimately show ?thesis using that by blast
-qed
-
-corollary compact_uniformly_continuous:
- fixes f :: "'a :: metric_space \<Rightarrow> 'b :: metric_space"
- assumes f: "continuous_on S f" and S: "compact S"
- shows "uniformly_continuous_on S f"
- using f
- unfolding continuous_on_iff uniformly_continuous_on_def
- by (force intro: compact_uniformly_equicontinuous [OF S, of "{f}"])
-
-subsection \<open>Topological stuff about the set of Reals\<close>
-
-lemma open_real:
- fixes s :: "real set"
- shows "open s \<longleftrightarrow> (\<forall>x \<in> s. \<exists>e>0. \<forall>x'. \<bar>x' - x\<bar> < e --> x' \<in> s)"
- unfolding open_dist dist_norm by simp
-
-lemma islimpt_approachable_real:
- fixes s :: "real set"
- shows "x islimpt s \<longleftrightarrow> (\<forall>e>0. \<exists>x'\<in> s. x' \<noteq> x \<and> \<bar>x' - x\<bar> < e)"
- unfolding islimpt_approachable dist_norm by simp
-
-lemma closed_real:
- fixes s :: "real set"
- shows "closed s \<longleftrightarrow> (\<forall>x. (\<forall>e>0. \<exists>x' \<in> s. x' \<noteq> x \<and> \<bar>x' - x\<bar> < e) \<longrightarrow> x \<in> s)"
- unfolding closed_limpt islimpt_approachable dist_norm by simp
-
-lemma continuous_at_real_range:
- fixes f :: "'a::real_normed_vector \<Rightarrow> real"
- shows "continuous (at x) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0. \<forall>x'. norm(x' - x) < d --> \<bar>f x' - f x\<bar> < e)"
- unfolding continuous_at
- unfolding Lim_at
- unfolding dist_norm
- apply auto
- apply (erule_tac x=e in allE, auto)
- apply (rule_tac x=d in exI, auto)
- apply (erule_tac x=x' in allE, auto)
- apply (erule_tac x=e in allE, auto)
- done
-
-lemma continuous_on_real_range:
- fixes f :: "'a::real_normed_vector \<Rightarrow> real"
- shows "continuous_on s f \<longleftrightarrow>
- (\<forall>x \<in> s. \<forall>e>0. \<exists>d>0. (\<forall>x' \<in> s. norm(x' - x) < d \<longrightarrow> \<bar>f x' - f x\<bar> < e))"
- unfolding continuous_on_iff dist_norm by simp
-
-text \<open>Hence some handy theorems on distance, diameter etc. of/from a set.\<close>
-
-lemma distance_attains_sup:
- assumes "compact s" "s \<noteq> {}"
- shows "\<exists>x\<in>s. \<forall>y\<in>s. dist a y \<le> dist a x"
-proof (rule continuous_attains_sup [OF assms])
- {
- fix x
- assume "x\<in>s"
- have "(dist a \<longlongrightarrow> dist a x) (at x within s)"
- by (intro tendsto_dist tendsto_const tendsto_ident_at)
- }
- then show "continuous_on s (dist a)"
- unfolding continuous_on ..
-qed
-
-text \<open>For \emph{minimal} distance, we only need closure, not compactness.\<close>
-
-lemma distance_attains_inf:
- fixes a :: "'a::heine_borel"
- assumes "closed s" and "s \<noteq> {}"
- obtains x where "x\<in>s" "\<And>y. y \<in> s \<Longrightarrow> dist a x \<le> dist a y"
-proof -
- from assms obtain b where "b \<in> s" by auto
- let ?B = "s \<inter> cball a (dist b a)"
- have "?B \<noteq> {}" using \<open>b \<in> s\<close>
- by (auto simp: dist_commute)
- moreover have "continuous_on ?B (dist a)"
- by (auto intro!: continuous_at_imp_continuous_on continuous_dist continuous_ident continuous_const)
- moreover have "compact ?B"
- by (intro closed_Int_compact \<open>closed s\<close> compact_cball)
- ultimately obtain x where "x \<in> ?B" "\<forall>y\<in>?B. dist a x \<le> dist a y"
- by (metis continuous_attains_inf)
- with that show ?thesis by fastforce
-qed
-
-
-subsection \<open>Cartesian products\<close>
-
-lemma bounded_Times:
- assumes "bounded s" "bounded t"
- shows "bounded (s \<times> t)"
-proof -
- obtain x y a b where "\<forall>z\<in>s. dist x z \<le> a" "\<forall>z\<in>t. dist y z \<le> b"
- using assms [unfolded bounded_def] by auto
- then have "\<forall>z\<in>s \<times> t. dist (x, y) z \<le> sqrt (a\<^sup>2 + b\<^sup>2)"
- by (auto simp: dist_Pair_Pair real_sqrt_le_mono add_mono power_mono)
- then show ?thesis unfolding bounded_any_center [where a="(x, y)"] by auto
-qed
-
-lemma mem_Times_iff: "x \<in> A \<times> B \<longleftrightarrow> fst x \<in> A \<and> snd x \<in> B"
- by (induct x) simp
-
-lemma seq_compact_Times: "seq_compact s \<Longrightarrow> seq_compact t \<Longrightarrow> seq_compact (s \<times> t)"
- unfolding seq_compact_def
- apply clarify
- apply (drule_tac x="fst \<circ> f" in spec)
- apply (drule mp, simp add: mem_Times_iff)
- apply (clarify, rename_tac l1 r1)
- apply (drule_tac x="snd \<circ> f \<circ> r1" in spec)
- apply (drule mp, simp add: mem_Times_iff)
- apply (clarify, rename_tac l2 r2)
- apply (rule_tac x="(l1, l2)" in rev_bexI, simp)
- apply (rule_tac x="r1 \<circ> r2" in exI)
- apply (rule conjI, simp add: strict_mono_def)
- apply (drule_tac f=r2 in LIMSEQ_subseq_LIMSEQ, assumption)
- apply (drule (1) tendsto_Pair) back
- apply (simp add: o_def)
- done
-
-lemma compact_Times:
- assumes "compact s" "compact t"
- shows "compact (s \<times> t)"
-proof (rule compactI)
- fix C
- assume C: "\<forall>t\<in>C. open t" "s \<times> t \<subseteq> \<Union>C"
- have "\<forall>x\<in>s. \<exists>a. open a \<and> x \<in> a \<and> (\<exists>d\<subseteq>C. finite d \<and> a \<times> t \<subseteq> \<Union>d)"
- proof
- fix x
- assume "x \<in> s"
- have "\<forall>y\<in>t. \<exists>a b c. c \<in> C \<and> open a \<and> open b \<and> x \<in> a \<and> y \<in> b \<and> a \<times> b \<subseteq> c" (is "\<forall>y\<in>t. ?P y")
- proof
- fix y
- assume "y \<in> t"
- with \<open>x \<in> s\<close> C obtain c where "c \<in> C" "(x, y) \<in> c" "open c" by auto
- then show "?P y" by (auto elim!: open_prod_elim)
- qed
- then obtain a b c where b: "\<And>y. y \<in> t \<Longrightarrow> open (b y)"
- and c: "\<And>y. y \<in> t \<Longrightarrow> c y \<in> C \<and> open (a y) \<and> open (b y) \<and> x \<in> a y \<and> y \<in> b y \<and> a y \<times> b y \<subseteq> c y"
- by metis
- then have "\<forall>y\<in>t. open (b y)" "t \<subseteq> (\<Union>y\<in>t. b y)" by auto
- with compactE_image[OF \<open>compact t\<close>] obtain D where D: "D \<subseteq> t" "finite D" "t \<subseteq> (\<Union>y\<in>D. b y)"
- by metis
- moreover from D c have "(\<Inter>y\<in>D. a y) \<times> t \<subseteq> (\<Union>y\<in>D. c y)"
- by (fastforce simp: subset_eq)
- ultimately show "\<exists>a. open a \<and> x \<in> a \<and> (\<exists>d\<subseteq>C. finite d \<and> a \<times> t \<subseteq> \<Union>d)"
- using c by (intro exI[of _ "c`D"] exI[of _ "\<Inter>(a`D)"] conjI) (auto intro!: open_INT)
- qed
- then obtain a d where a: "\<And>x. x\<in>s \<Longrightarrow> open (a x)" "s \<subseteq> (\<Union>x\<in>s. a x)"
- and d: "\<And>x. x \<in> s \<Longrightarrow> d x \<subseteq> C \<and> finite (d x) \<and> a x \<times> t \<subseteq> \<Union>d x"
- unfolding subset_eq UN_iff by metis
- moreover
- from compactE_image[OF \<open>compact s\<close> a]
- obtain e where e: "e \<subseteq> s" "finite e" and s: "s \<subseteq> (\<Union>x\<in>e. a x)"
- by auto
- moreover
- {
- from s have "s \<times> t \<subseteq> (\<Union>x\<in>e. a x \<times> t)"
- by auto
- also have "\<dots> \<subseteq> (\<Union>x\<in>e. \<Union>d x)"
- using d \<open>e \<subseteq> s\<close> by (intro UN_mono) auto
- finally have "s \<times> t \<subseteq> (\<Union>x\<in>e. \<Union>d x)" .
- }
- ultimately show "\<exists>C'\<subseteq>C. finite C' \<and> s \<times> t \<subseteq> \<Union>C'"
- by (intro exI[of _ "(\<Union>x\<in>e. d x)"]) (auto simp: subset_eq)
-qed
-
-text\<open>Hence some useful properties follow quite easily.\<close>
-
-lemma compact_scaling:
- fixes s :: "'a::real_normed_vector set"
- assumes "compact s"
- shows "compact ((\<lambda>x. c *\<^sub>R x) ` s)"
-proof -
- let ?f = "\<lambda>x. scaleR c x"
- have *: "bounded_linear ?f" by (rule bounded_linear_scaleR_right)
- show ?thesis
- using compact_continuous_image[of s ?f] continuous_at_imp_continuous_on[of s ?f]
- using linear_continuous_at[OF *] assms
- by auto
-qed
-
-lemma compact_negations:
- fixes s :: "'a::real_normed_vector set"
- assumes "compact s"
- shows "compact ((\<lambda>x. - x) ` s)"
- using compact_scaling [OF assms, of "- 1"] by auto
-
-lemma compact_sums:
- fixes s t :: "'a::real_normed_vector set"
- assumes "compact s"
- and "compact t"
- shows "compact {x + y | x y. x \<in> s \<and> y \<in> t}"
-proof -
- have *: "{x + y | x y. x \<in> s \<and> y \<in> t} = (\<lambda>z. fst z + snd z) ` (s \<times> t)"
- apply auto
- unfolding image_iff
- apply (rule_tac x="(xa, y)" in bexI)
- apply auto
- done
- have "continuous_on (s \<times> t) (\<lambda>z. fst z + snd z)"
- unfolding continuous_on by (rule ballI) (intro tendsto_intros)
- then show ?thesis
- unfolding * using compact_continuous_image compact_Times [OF assms] by auto
-qed
-
-lemma compact_differences:
- fixes s t :: "'a::real_normed_vector set"
- assumes "compact s"
- and "compact t"
- shows "compact {x - y | x y. x \<in> s \<and> y \<in> t}"
-proof-
- have "{x - y | x y. x\<in>s \<and> y \<in> t} = {x + y | x y. x \<in> s \<and> y \<in> (uminus ` t)}"
- apply auto
- apply (rule_tac x= xa in exI, auto)
- done
- then show ?thesis
- using compact_sums[OF assms(1) compact_negations[OF assms(2)]] by auto
-qed
-
-lemma compact_translation:
- fixes s :: "'a::real_normed_vector set"
- assumes "compact s"
- shows "compact ((\<lambda>x. a + x) ` s)"
-proof -
- have "{x + y |x y. x \<in> s \<and> y \<in> {a}} = (\<lambda>x. a + x) ` s"
- by auto
- then show ?thesis
- using compact_sums[OF assms compact_sing[of a]] by auto
-qed
-
-lemma compact_affinity:
- fixes s :: "'a::real_normed_vector set"
- assumes "compact s"
- shows "compact ((\<lambda>x. a + c *\<^sub>R x) ` s)"
-proof -
- have "op + a ` op *\<^sub>R c ` s = (\<lambda>x. a + c *\<^sub>R x) ` s"
- by auto
- then show ?thesis
- using compact_translation[OF compact_scaling[OF assms], of a c] by auto
-qed
-
-text \<open>Hence we get the following.\<close>
-
-lemma compact_sup_maxdistance:
- fixes s :: "'a::metric_space set"
- assumes "compact s"
- and "s \<noteq> {}"
- shows "\<exists>x\<in>s. \<exists>y\<in>s. \<forall>u\<in>s. \<forall>v\<in>s. dist u v \<le> dist x y"
-proof -
- have "compact (s \<times> s)"
- using \<open>compact s\<close> by (intro compact_Times)
- moreover have "s \<times> s \<noteq> {}"
- using \<open>s \<noteq> {}\<close> by auto
- moreover have "continuous_on (s \<times> s) (\<lambda>x. dist (fst x) (snd x))"
- by (intro continuous_at_imp_continuous_on ballI continuous_intros)
- ultimately show ?thesis
- using continuous_attains_sup[of "s \<times> s" "\<lambda>x. dist (fst x) (snd x)"] by auto
-qed
-
-subsection \<open>The diameter of a set.\<close>
-
-definition diameter :: "'a::metric_space set \<Rightarrow> real" where
- "diameter S = (if S = {} then 0 else SUP (x,y):S\<times>S. dist x y)"
-
-lemma diameter_empty [simp]: "diameter{} = 0"
- by (auto simp: diameter_def)
-
-lemma diameter_singleton [simp]: "diameter{x} = 0"
- by (auto simp: diameter_def)
-
-lemma diameter_le:
- assumes "S \<noteq> {} \<or> 0 \<le> d"
- and no: "\<And>x y. \<lbrakk>x \<in> S; y \<in> S\<rbrakk> \<Longrightarrow> norm(x - y) \<le> d"
- shows "diameter S \<le> d"
-using assms
- by (auto simp: dist_norm diameter_def intro: cSUP_least)
-
-lemma diameter_bounded_bound:
- fixes s :: "'a :: metric_space set"
- assumes s: "bounded s" "x \<in> s" "y \<in> s"
- shows "dist x y \<le> diameter s"
-proof -
- from s obtain z d where z: "\<And>x. x \<in> s \<Longrightarrow> dist z x \<le> d"
- unfolding bounded_def by auto
- have "bdd_above (case_prod dist ` (s\<times>s))"
- proof (intro bdd_aboveI, safe)
- fix a b
- assume "a \<in> s" "b \<in> s"
- with z[of a] z[of b] dist_triangle[of a b z]
- show "dist a b \<le> 2 * d"
- by (simp add: dist_commute)
- qed
- moreover have "(x,y) \<in> s\<times>s" using s by auto
- ultimately have "dist x y \<le> (SUP (x,y):s\<times>s. dist x y)"
- by (rule cSUP_upper2) simp
- with \<open>x \<in> s\<close> show ?thesis
- by (auto simp: diameter_def)
-qed
-
-lemma diameter_lower_bounded:
- fixes s :: "'a :: metric_space set"
- assumes s: "bounded s"
- and d: "0 < d" "d < diameter s"
- shows "\<exists>x\<in>s. \<exists>y\<in>s. d < dist x y"
-proof (rule ccontr)
- assume contr: "\<not> ?thesis"
- moreover have "s \<noteq> {}"
- using d by (auto simp: diameter_def)
- ultimately have "diameter s \<le> d"
- by (auto simp: not_less diameter_def intro!: cSUP_least)
- with \<open>d < diameter s\<close> show False by auto
-qed
-
-lemma diameter_bounded:
- assumes "bounded s"
- shows "\<forall>x\<in>s. \<forall>y\<in>s. dist x y \<le> diameter s"
- and "\<forall>d>0. d < diameter s \<longrightarrow> (\<exists>x\<in>s. \<exists>y\<in>s. dist x y > d)"
- using diameter_bounded_bound[of s] diameter_lower_bounded[of s] assms
- by auto
-
-lemma diameter_compact_attained:
- assumes "compact s"
- and "s \<noteq> {}"
- shows "\<exists>x\<in>s. \<exists>y\<in>s. dist x y = diameter s"
-proof -
- have b: "bounded s" using assms(1)
- by (rule compact_imp_bounded)
- then obtain x y where xys: "x\<in>s" "y\<in>s"
- and xy: "\<forall>u\<in>s. \<forall>v\<in>s. dist u v \<le> dist x y"
- using compact_sup_maxdistance[OF assms] by auto
- then have "diameter s \<le> dist x y"
- unfolding diameter_def
- apply clarsimp
- apply (rule cSUP_least, fast+)
- done
- then show ?thesis
- by (metis b diameter_bounded_bound order_antisym xys)
-qed
-
-lemma diameter_ge_0:
- assumes "bounded S" shows "0 \<le> diameter S"
- by (metis all_not_in_conv assms diameter_bounded_bound diameter_empty dist_self order_refl)
-
-lemma diameter_subset:
- assumes "S \<subseteq> T" "bounded T"
- shows "diameter S \<le> diameter T"
-proof (cases "S = {} \<or> T = {}")
- case True
- with assms show ?thesis
- by (force simp: diameter_ge_0)
-next
- case False
- then have "bdd_above ((\<lambda>x. case x of (x, xa) \<Rightarrow> dist x xa) ` (T \<times> T))"
- using \<open>bounded T\<close> diameter_bounded_bound by (force simp: bdd_above_def)
- with False \<open>S \<subseteq> T\<close> show ?thesis
- apply (simp add: diameter_def)
- apply (rule cSUP_subset_mono, auto)
- done
-qed
-
-lemma diameter_closure:
- assumes "bounded S"
- shows "diameter(closure S) = diameter S"
-proof (rule order_antisym)
- have "False" if "diameter S < diameter (closure S)"
- proof -
- define d where "d = diameter(closure S) - diameter(S)"
- have "d > 0"
- using that by (simp add: d_def)
- then have "diameter(closure(S)) - d / 2 < diameter(closure(S))"
- by simp
- have dd: "diameter (closure S) - d / 2 = (diameter(closure(S)) + diameter(S)) / 2"
- by (simp add: d_def divide_simps)
- have bocl: "bounded (closure S)"
- using assms by blast
- moreover have "0 \<le> diameter S"
- using assms diameter_ge_0 by blast
- ultimately obtain x y where "x \<in> closure S" "y \<in> closure S" and xy: "diameter(closure(S)) - d / 2 < dist x y"
- using diameter_bounded(2) [OF bocl, rule_format, of "diameter(closure(S)) - d / 2"] \<open>d > 0\<close> d_def by auto
- then obtain x' y' where x'y': "x' \<in> S" "dist x' x < d/4" "y' \<in> S" "dist y' y < d/4"
- using closure_approachable
- by (metis \<open>0 < d\<close> zero_less_divide_iff zero_less_numeral)
- then have "dist x' y' \<le> diameter S"
- using assms diameter_bounded_bound by blast
- with x'y' have "dist x y \<le> d / 4 + diameter S + d / 4"
- by (meson add_mono_thms_linordered_semiring(1) dist_triangle dist_triangle3 less_eq_real_def order_trans)
- then show ?thesis
- using xy d_def by linarith
- qed
- then show "diameter (closure S) \<le> diameter S"
- by fastforce
- next
- show "diameter S \<le> diameter (closure S)"
- by (simp add: assms bounded_closure closure_subset diameter_subset)
-qed
-
-lemma diameter_cball [simp]:
- fixes a :: "'a::euclidean_space"
- shows "diameter(cball a r) = (if r < 0 then 0 else 2*r)"
-proof -
- have "diameter(cball a r) = 2*r" if "r \<ge> 0"
- proof (rule order_antisym)
- show "diameter (cball a r) \<le> 2*r"
- proof (rule diameter_le)
- fix x y assume "x \<in> cball a r" "y \<in> cball a r"
- then have "norm (x - a) \<le> r" "norm (a - y) \<le> r"
- by (auto simp: dist_norm norm_minus_commute)
- then have "norm (x - y) \<le> r+r"
- using norm_diff_triangle_le by blast
- then show "norm (x - y) \<le> 2*r" by simp
- qed (simp add: that)
- have "2*r = dist (a + r *\<^sub>R (SOME i. i \<in> Basis)) (a - r *\<^sub>R (SOME i. i \<in> Basis))"
- apply (simp add: dist_norm)
- by (metis abs_of_nonneg mult.right_neutral norm_numeral norm_scaleR norm_some_Basis real_norm_def scaleR_2 that)
- also have "... \<le> diameter (cball a r)"
- apply (rule diameter_bounded_bound)
- using that by (auto simp: dist_norm)
- finally show "2*r \<le> diameter (cball a r)" .
- qed
- then show ?thesis by simp
-qed
-
-lemma diameter_ball [simp]:
- fixes a :: "'a::euclidean_space"
- shows "diameter(ball a r) = (if r < 0 then 0 else 2*r)"
-proof -
- have "diameter(ball a r) = 2*r" if "r > 0"
- by (metis bounded_ball diameter_closure closure_ball diameter_cball less_eq_real_def linorder_not_less that)
- then show ?thesis
- by (simp add: diameter_def)
-qed
-
-lemma diameter_closed_interval [simp]: "diameter {a..b} = (if b < a then 0 else b-a)"
-proof -
- have "{a .. b} = cball ((a+b)/2) ((b-a)/2)"
- by (auto simp: dist_norm abs_if divide_simps split: if_split_asm)
- then show ?thesis
- by simp
-qed
-
-lemma diameter_open_interval [simp]: "diameter {a<..<b} = (if b < a then 0 else b-a)"
-proof -
- have "{a <..< b} = ball ((a+b)/2) ((b-a)/2)"
- by (auto simp: dist_norm abs_if divide_simps split: if_split_asm)
- then show ?thesis
- by simp
-qed
-
-proposition Lebesgue_number_lemma:
- assumes "compact S" "\<C> \<noteq> {}" "S \<subseteq> \<Union>\<C>" and ope: "\<And>B. B \<in> \<C> \<Longrightarrow> open B"
- obtains \<delta> where "0 < \<delta>" "\<And>T. \<lbrakk>T \<subseteq> S; diameter T < \<delta>\<rbrakk> \<Longrightarrow> \<exists>B \<in> \<C>. T \<subseteq> B"
-proof (cases "S = {}")
- case True
- then show ?thesis
- by (metis \<open>\<C> \<noteq> {}\<close> zero_less_one empty_subsetI equals0I subset_trans that)
-next
- case False
- { fix x assume "x \<in> S"
- then obtain C where C: "x \<in> C" "C \<in> \<C>"
- using \<open>S \<subseteq> \<Union>\<C>\<close> by blast
- then obtain r where r: "r>0" "ball x (2*r) \<subseteq> C"
- by (metis mult.commute mult_2_right not_le ope openE real_sum_of_halves zero_le_numeral zero_less_mult_iff)
- then have "\<exists>r C. r > 0 \<and> ball x (2*r) \<subseteq> C \<and> C \<in> \<C>"
- using C by blast
- }
- then obtain r where r: "\<And>x. x \<in> S \<Longrightarrow> r x > 0 \<and> (\<exists>C \<in> \<C>. ball x (2*r x) \<subseteq> C)"
- by metis
- then have "S \<subseteq> (\<Union>x \<in> S. ball x (r x))"
- by auto
- then obtain \<T> where "finite \<T>" "S \<subseteq> \<Union>\<T>" and \<T>: "\<T> \<subseteq> (\<lambda>x. ball x (r x)) ` S"
- by (rule compactE [OF \<open>compact S\<close>]) auto
- then obtain S0 where "S0 \<subseteq> S" "finite S0" and S0: "\<T> = (\<lambda>x. ball x (r x)) ` S0"
- by (meson finite_subset_image)
- then have "S0 \<noteq> {}"
- using False \<open>S \<subseteq> \<Union>\<T>\<close> by auto
- define \<delta> where "\<delta> = Inf (r ` S0)"
- have "\<delta> > 0"
- using \<open>finite S0\<close> \<open>S0 \<subseteq> S\<close> \<open>S0 \<noteq> {}\<close> r by (auto simp: \<delta>_def finite_less_Inf_iff)
- show ?thesis
- proof
- show "0 < \<delta>"
- by (simp add: \<open>0 < \<delta>\<close>)
- show "\<exists>B \<in> \<C>. T \<subseteq> B" if "T \<subseteq> S" and dia: "diameter T < \<delta>" for T
- proof (cases "T = {}")
- case True
- then show ?thesis
- using \<open>\<C> \<noteq> {}\<close> by blast
- next
- case False
- then obtain y where "y \<in> T" by blast
- then have "y \<in> S"
- using \<open>T \<subseteq> S\<close> by auto
- then obtain x where "x \<in> S0" and x: "y \<in> ball x (r x)"
- using \<open>S \<subseteq> \<Union>\<T>\<close> S0 that by blast
- have "ball y \<delta> \<subseteq> ball y (r x)"
- by (metis \<delta>_def \<open>S0 \<noteq> {}\<close> \<open>finite S0\<close> \<open>x \<in> S0\<close> empty_is_image finite_imageI finite_less_Inf_iff imageI less_irrefl not_le subset_ball)
- also have "... \<subseteq> ball x (2*r x)"
- by clarsimp (metis dist_commute dist_triangle_less_add mem_ball mult_2 x)
- finally obtain C where "C \<in> \<C>" "ball y \<delta> \<subseteq> C"
- by (meson r \<open>S0 \<subseteq> S\<close> \<open>x \<in> S0\<close> dual_order.trans subsetCE)
- have "bounded T"
- using \<open>compact S\<close> bounded_subset compact_imp_bounded \<open>T \<subseteq> S\<close> by blast
- then have "T \<subseteq> ball y \<delta>"
- using \<open>y \<in> T\<close> dia diameter_bounded_bound by fastforce
- then show ?thesis
- apply (rule_tac x=C in bexI)
- using \<open>ball y \<delta> \<subseteq> C\<close> \<open>C \<in> \<C>\<close> by auto
- qed
- qed
-qed
-
-
-subsection \<open>Compact sets and the closure operation.\<close>
-
-lemma closed_scaling:
- fixes S :: "'a::real_normed_vector set"
- assumes "closed S"
- shows "closed ((\<lambda>x. c *\<^sub>R x) ` S)"
-proof (cases "c = 0")
- case True then show ?thesis
- by (auto simp: image_constant_conv)
-next
- case False
- from assms have "closed ((\<lambda>x. inverse c *\<^sub>R x) -` S)"
- by (simp add: continuous_closed_vimage)
- also have "(\<lambda>x. inverse c *\<^sub>R x) -` S = (\<lambda>x. c *\<^sub>R x) ` S"
- using \<open>c \<noteq> 0\<close> by (auto elim: image_eqI [rotated])
- finally show ?thesis .
-qed
-
-lemma closed_negations:
- fixes S :: "'a::real_normed_vector set"
- assumes "closed S"
- shows "closed ((\<lambda>x. -x) ` S)"
- using closed_scaling[OF assms, of "- 1"] by simp
-
-lemma compact_closed_sums:
- fixes S :: "'a::real_normed_vector set"
- assumes "compact S" and "closed T"
- shows "closed (\<Union>x\<in> S. \<Union>y \<in> T. {x + y})"
-proof -
- let ?S = "{x + y |x y. x \<in> S \<and> y \<in> T}"
- {
- fix x l
- assume as: "\<forall>n. x n \<in> ?S" "(x \<longlongrightarrow> l) sequentially"
- from as(1) obtain f where f: "\<forall>n. x n = fst (f n) + snd (f n)" "\<forall>n. fst (f n) \<in> S" "\<forall>n. snd (f n) \<in> T"
- using choice[of "\<lambda>n y. x n = (fst y) + (snd y) \<and> fst y \<in> S \<and> snd y \<in> T"] by auto
- obtain l' r where "l'\<in>S" and r: "strict_mono r" and lr: "(((\<lambda>n. fst (f n)) \<circ> r) \<longlongrightarrow> l') sequentially"
- using assms(1)[unfolded compact_def, THEN spec[where x="\<lambda> n. fst (f n)"]] using f(2) by auto
- have "((\<lambda>n. snd (f (r n))) \<longlongrightarrow> l - l') sequentially"
- using tendsto_diff[OF LIMSEQ_subseq_LIMSEQ[OF as(2) r] lr] and f(1)
- unfolding o_def
- by auto
- then have "l - l' \<in> T"
- using assms(2)[unfolded closed_sequential_limits,
- THEN spec[where x="\<lambda> n. snd (f (r n))"],
- THEN spec[where x="l - l'"]]
- using f(3)
- by auto
- then have "l \<in> ?S"
- using \<open>l' \<in> S\<close>
- apply auto
- apply (rule_tac x=l' in exI)
- apply (rule_tac x="l - l'" in exI, auto)
- done
- }
- moreover have "?S = (\<Union>x\<in> S. \<Union>y \<in> T. {x + y})"
- by force
- ultimately show ?thesis
- unfolding closed_sequential_limits
- by (metis (no_types, lifting))
-qed
-
-lemma closed_compact_sums:
- fixes S T :: "'a::real_normed_vector set"
- assumes "closed S" "compact T"
- shows "closed (\<Union>x\<in> S. \<Union>y \<in> T. {x + y})"
-proof -
- have "(\<Union>x\<in> T. \<Union>y \<in> S. {x + y}) = (\<Union>x\<in> S. \<Union>y \<in> T. {x + y})"
- by auto
- then show ?thesis
- using compact_closed_sums[OF assms(2,1)] by simp
-qed
-
-lemma compact_closed_differences:
- fixes S T :: "'a::real_normed_vector set"
- assumes "compact S" "closed T"
- shows "closed (\<Union>x\<in> S. \<Union>y \<in> T. {x - y})"
-proof -
- have "(\<Union>x\<in> S. \<Union>y \<in> uminus ` T. {x + y}) = (\<Union>x\<in> S. \<Union>y \<in> T. {x - y})"
- by force
- then show ?thesis
- using compact_closed_sums[OF assms(1) closed_negations[OF assms(2)]] by auto
-qed
-
-lemma closed_compact_differences:
- fixes S T :: "'a::real_normed_vector set"
- assumes "closed S" "compact T"
- shows "closed (\<Union>x\<in> S. \<Union>y \<in> T. {x - y})"
-proof -
- have "(\<Union>x\<in> S. \<Union>y \<in> uminus ` T. {x + y}) = {x - y |x y. x \<in> S \<and> y \<in> T}"
- by auto
- then show ?thesis
- using closed_compact_sums[OF assms(1) compact_negations[OF assms(2)]] by simp
-qed
-
-lemma closed_translation:
- fixes a :: "'a::real_normed_vector"
- assumes "closed S"
- shows "closed ((\<lambda>x. a + x) ` S)"
-proof -
- have "(\<Union>x\<in> {a}. \<Union>y \<in> S. {x + y}) = (op + a ` S)" by auto
- then show ?thesis
- using compact_closed_sums[OF compact_sing[of a] assms] by auto
-qed
-
-lemma translation_Compl:
- fixes a :: "'a::ab_group_add"
- shows "(\<lambda>x. a + x) ` (- t) = - ((\<lambda>x. a + x) ` t)"
- apply (auto simp: image_iff)
- apply (rule_tac x="x - a" in bexI, auto)
- done
-
-lemma translation_UNIV:
- fixes a :: "'a::ab_group_add"
- shows "range (\<lambda>x. a + x) = UNIV"
- apply (auto simp: image_iff)
- apply (rule_tac x="x - a" in exI, auto)
- done
-
-lemma translation_diff:
- fixes a :: "'a::ab_group_add"
- shows "(\<lambda>x. a + x) ` (s - t) = ((\<lambda>x. a + x) ` s) - ((\<lambda>x. a + x) ` t)"
- by auto
-
-lemma translation_Int:
- fixes a :: "'a::ab_group_add"
- shows "(\<lambda>x. a + x) ` (s \<inter> t) = ((\<lambda>x. a + x) ` s) \<inter> ((\<lambda>x. a + x) ` t)"
- by auto
-
-lemma closure_translation:
- fixes a :: "'a::real_normed_vector"
- shows "closure ((\<lambda>x. a + x) ` s) = (\<lambda>x. a + x) ` (closure s)"
-proof -
- have *: "op + a ` (- s) = - op + a ` s"
- apply auto
- unfolding image_iff
- apply (rule_tac x="x - a" in bexI, auto)
- done
- show ?thesis
- unfolding closure_interior translation_Compl
- using interior_translation[of a "- s"]
- unfolding *
- by auto
-qed
-
-lemma frontier_translation:
- fixes a :: "'a::real_normed_vector"
- shows "frontier((\<lambda>x. a + x) ` s) = (\<lambda>x. a + x) ` (frontier s)"
- unfolding frontier_def translation_diff interior_translation closure_translation
- by auto
-
-lemma sphere_translation:
- fixes a :: "'n::euclidean_space"
- shows "sphere (a+c) r = op+a ` sphere c r"
-apply safe
-apply (rule_tac x="x-a" in image_eqI)
-apply (auto simp: dist_norm algebra_simps)
-done
-
-lemma cball_translation:
- fixes a :: "'n::euclidean_space"
- shows "cball (a+c) r = op+a ` cball c r"
-apply safe
-apply (rule_tac x="x-a" in image_eqI)
-apply (auto simp: dist_norm algebra_simps)
-done
-
-lemma ball_translation:
- fixes a :: "'n::euclidean_space"
- shows "ball (a+c) r = op+a ` ball c r"
-apply safe
-apply (rule_tac x="x-a" in image_eqI)
-apply (auto simp: dist_norm algebra_simps)
-done
-
-
-subsection \<open>Separation between points and sets\<close>
-
-lemma separate_point_closed:
- fixes s :: "'a::heine_borel set"
- assumes "closed s" and "a \<notin> s"
- shows "\<exists>d>0. \<forall>x\<in>s. d \<le> dist a x"
-proof (cases "s = {}")
- case True
- then show ?thesis by(auto intro!: exI[where x=1])
-next
- case False
- from assms obtain x where "x\<in>s" "\<forall>y\<in>s. dist a x \<le> dist a y"
- using \<open>s \<noteq> {}\<close> by (blast intro: distance_attains_inf [of s a])
- with \<open>x\<in>s\<close> show ?thesis using dist_pos_lt[of a x] and\<open>a \<notin> s\<close>
- by blast
-qed
-
-lemma separate_compact_closed:
- fixes s t :: "'a::heine_borel set"
- assumes "compact s"
- and t: "closed t" "s \<inter> t = {}"
- shows "\<exists>d>0. \<forall>x\<in>s. \<forall>y\<in>t. d \<le> dist x y"
-proof cases
- assume "s \<noteq> {} \<and> t \<noteq> {}"
- then have "s \<noteq> {}" "t \<noteq> {}" by auto
- let ?inf = "\<lambda>x. infdist x t"
- have "continuous_on s ?inf"
- by (auto intro!: continuous_at_imp_continuous_on continuous_infdist continuous_ident)
- then obtain x where x: "x \<in> s" "\<forall>y\<in>s. ?inf x \<le> ?inf y"
- using continuous_attains_inf[OF \<open>compact s\<close> \<open>s \<noteq> {}\<close>] by auto
- then have "0 < ?inf x"
- using t \<open>t \<noteq> {}\<close> in_closed_iff_infdist_zero by (auto simp: less_le infdist_nonneg)
- moreover have "\<forall>x'\<in>s. \<forall>y\<in>t. ?inf x \<le> dist x' y"
- using x by (auto intro: order_trans infdist_le)
- ultimately show ?thesis by auto
-qed (auto intro!: exI[of _ 1])
-
-lemma separate_closed_compact:
- fixes s t :: "'a::heine_borel set"
- assumes "closed s"
- and "compact t"
- and "s \<inter> t = {}"
- shows "\<exists>d>0. \<forall>x\<in>s. \<forall>y\<in>t. d \<le> dist x y"
-proof -
- have *: "t \<inter> s = {}"
- using assms(3) by auto
- show ?thesis
- using separate_compact_closed[OF assms(2,1) *]
- apply auto
- apply (rule_tac x=d in exI, auto)
- apply (erule_tac x=y in ballE)
- apply (auto simp: dist_commute)
- done
-qed
-
-
-subsection \<open>Closure of halfspaces and hyperplanes\<close>
-
-lemma isCont_open_vimage:
- assumes "\<And>x. isCont f x"
- and "open s"
- shows "open (f -` s)"
-proof -
- from assms(1) have "continuous_on UNIV f"
- unfolding isCont_def continuous_on_def by simp
- then have "open {x \<in> UNIV. f x \<in> s}"
- using open_UNIV \<open>open s\<close> by (rule continuous_open_preimage)
- then show "open (f -` s)"
- by (simp add: vimage_def)
-qed
-
-lemma isCont_closed_vimage:
- assumes "\<And>x. isCont f x"
- and "closed s"
- shows "closed (f -` s)"
- using assms unfolding closed_def vimage_Compl [symmetric]
- by (rule isCont_open_vimage)
-
-lemma continuous_on_closed_Collect_le:
- fixes f g :: "'a::t2_space \<Rightarrow> real"
- assumes f: "continuous_on s f" and g: "continuous_on s g" and s: "closed s"
- shows "closed {x \<in> s. f x \<le> g x}"
-proof -
- have "closed ((\<lambda>x. g x - f x) -` {0..} \<inter> s)"
- using closed_real_atLeast continuous_on_diff [OF g f]
- by (simp add: continuous_on_closed_vimage [OF s])
- also have "((\<lambda>x. g x - f x) -` {0..} \<inter> s) = {x\<in>s. f x \<le> g x}"
- by auto
- finally show ?thesis .
-qed
-
-lemma continuous_at_inner: "continuous (at x) (inner a)"
- unfolding continuous_at by (intro tendsto_intros)
-
-lemma closed_halfspace_le: "closed {x. inner a x \<le> b}"
- by (simp add: closed_Collect_le continuous_on_inner continuous_on_const continuous_on_id)
-
-lemma closed_halfspace_ge: "closed {x. inner a x \<ge> b}"
- by (simp add: closed_Collect_le continuous_on_inner continuous_on_const continuous_on_id)
-
-lemma closed_hyperplane: "closed {x. inner a x = b}"
- by (simp add: closed_Collect_eq continuous_on_inner continuous_on_const continuous_on_id)
-
-lemma closed_halfspace_component_le: "closed {x::'a::euclidean_space. x\<bullet>i \<le> a}"
- by (simp add: closed_Collect_le continuous_on_inner continuous_on_const continuous_on_id)
-
-lemma closed_halfspace_component_ge: "closed {x::'a::euclidean_space. x\<bullet>i \<ge> a}"
- by (simp add: closed_Collect_le continuous_on_inner continuous_on_const continuous_on_id)
-
-lemma closed_interval_left:
- fixes b :: "'a::euclidean_space"
- shows "closed {x::'a. \<forall>i\<in>Basis. x\<bullet>i \<le> b\<bullet>i}"
- by (simp add: Collect_ball_eq closed_INT closed_Collect_le continuous_on_inner continuous_on_const continuous_on_id)
-
-lemma closed_interval_right:
- fixes a :: "'a::euclidean_space"
- shows "closed {x::'a. \<forall>i\<in>Basis. a\<bullet>i \<le> x\<bullet>i}"
- by (simp add: Collect_ball_eq closed_INT closed_Collect_le continuous_on_inner continuous_on_const continuous_on_id)
-
-lemma continuous_le_on_closure:
- fixes a::real
- assumes f: "continuous_on (closure s) f"
- and x: "x \<in> closure(s)"
- and xlo: "\<And>x. x \<in> s ==> f(x) \<le> a"
- shows "f(x) \<le> a"
- using image_closure_subset [OF f]
- using image_closure_subset [OF f] closed_halfspace_le [of "1::real" a] assms
- by force
-
-lemma continuous_ge_on_closure:
- fixes a::real
- assumes f: "continuous_on (closure s) f"
- and x: "x \<in> closure(s)"
- and xlo: "\<And>x. x \<in> s ==> f(x) \<ge> a"
- shows "f(x) \<ge> a"
- using image_closure_subset [OF f] closed_halfspace_ge [of a "1::real"] assms
- by force
+subsection \<open>Intervals\<close>
text \<open>Openness of halfspaces.\<close>
@@ -8699,136 +5524,6 @@
text \<open>This gives a simple derivation of limit component bounds.\<close>
-lemma Lim_component_le:
- fixes f :: "'a \<Rightarrow> 'b::euclidean_space"
- assumes "(f \<longlongrightarrow> l) net"
- and "\<not> (trivial_limit net)"
- and "eventually (\<lambda>x. f(x)\<bullet>i \<le> b) net"
- shows "l\<bullet>i \<le> b"
- by (rule tendsto_le[OF assms(2) tendsto_const tendsto_inner[OF assms(1) tendsto_const] assms(3)])
-
-lemma Lim_component_ge:
- fixes f :: "'a \<Rightarrow> 'b::euclidean_space"
- assumes "(f \<longlongrightarrow> l) net"
- and "\<not> (trivial_limit net)"
- and "eventually (\<lambda>x. b \<le> (f x)\<bullet>i) net"
- shows "b \<le> l\<bullet>i"
- by (rule tendsto_le[OF assms(2) tendsto_inner[OF assms(1) tendsto_const] tendsto_const assms(3)])
-
-lemma Lim_component_eq:
- fixes f :: "'a \<Rightarrow> 'b::euclidean_space"
- assumes net: "(f \<longlongrightarrow> l) net" "\<not> trivial_limit net"
- and ev:"eventually (\<lambda>x. f(x)\<bullet>i = b) net"
- shows "l\<bullet>i = b"
- using ev[unfolded order_eq_iff eventually_conj_iff]
- using Lim_component_ge[OF net, of b i]
- using Lim_component_le[OF net, of i b]
- by auto
-
-text \<open>Limits relative to a union.\<close>
-
-lemma eventually_within_Un:
- "eventually P (at x within (s \<union> t)) \<longleftrightarrow>
- eventually P (at x within s) \<and> eventually P (at x within t)"
- unfolding eventually_at_filter
- by (auto elim!: eventually_rev_mp)
-
-lemma Lim_within_union:
- "(f \<longlongrightarrow> l) (at x within (s \<union> t)) \<longleftrightarrow>
- (f \<longlongrightarrow> l) (at x within s) \<and> (f \<longlongrightarrow> l) (at x within t)"
- unfolding tendsto_def
- by (auto simp: eventually_within_Un)
-
-lemma Lim_topological:
- "(f \<longlongrightarrow> l) net \<longleftrightarrow>
- trivial_limit net \<or> (\<forall>S. open S \<longrightarrow> l \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) net)"
- unfolding tendsto_def trivial_limit_eq by auto
-
-text \<open>Continuity relative to a union.\<close>
-
-lemma continuous_on_Un_local:
- "\<lbrakk>closedin (subtopology euclidean (s \<union> t)) s; closedin (subtopology euclidean (s \<union> t)) t;
- continuous_on s f; continuous_on t f\<rbrakk>
- \<Longrightarrow> continuous_on (s \<union> t) f"
- unfolding continuous_on closedin_limpt
- by (metis Lim_trivial_limit Lim_within_union Un_iff trivial_limit_within)
-
-lemma continuous_on_cases_local:
- "\<lbrakk>closedin (subtopology euclidean (s \<union> t)) s; closedin (subtopology euclidean (s \<union> t)) t;
- continuous_on s f; continuous_on t g;
- \<And>x. \<lbrakk>x \<in> s \<and> ~P x \<or> x \<in> t \<and> P x\<rbrakk> \<Longrightarrow> f x = g x\<rbrakk>
- \<Longrightarrow> continuous_on (s \<union> t) (\<lambda>x. if P x then f x else g x)"
- by (rule continuous_on_Un_local) (auto intro: continuous_on_eq)
-
-lemma continuous_on_cases_le:
- fixes h :: "'a :: topological_space \<Rightarrow> real"
- assumes "continuous_on {t \<in> s. h t \<le> a} f"
- and "continuous_on {t \<in> s. a \<le> h t} g"
- and h: "continuous_on s h"
- and "\<And>t. \<lbrakk>t \<in> s; h t = a\<rbrakk> \<Longrightarrow> f t = g t"
- shows "continuous_on s (\<lambda>t. if h t \<le> a then f(t) else g(t))"
-proof -
- have s: "s = {t \<in> s. h t \<in> atMost a} \<union> {t \<in> s. h t \<in> atLeast a}"
- by force
- have 1: "closedin (subtopology euclidean s) {t \<in> s. h t \<in> atMost a}"
- by (rule continuous_closedin_preimage [OF h closed_atMost])
- have 2: "closedin (subtopology euclidean s) {t \<in> s. h t \<in> atLeast a}"
- by (rule continuous_closedin_preimage [OF h closed_atLeast])
- show ?thesis
- apply (rule continuous_on_subset [of s, OF _ order_refl])
- apply (subst s)
- apply (rule continuous_on_cases_local)
- using 1 2 s assms apply auto
- done
-qed
-
-lemma continuous_on_cases_1:
- fixes s :: "real set"
- assumes "continuous_on {t \<in> s. t \<le> a} f"
- and "continuous_on {t \<in> s. a \<le> t} g"
- and "a \<in> s \<Longrightarrow> f a = g a"
- shows "continuous_on s (\<lambda>t. if t \<le> a then f(t) else g(t))"
-using assms
-by (auto simp: continuous_on_id intro: continuous_on_cases_le [where h = id, simplified])
-
-text\<open>Some more convenient intermediate-value theorem formulations.\<close>
-
-lemma connected_ivt_hyperplane:
- assumes "connected s"
- and "x \<in> s"
- and "y \<in> s"
- and "inner a x \<le> b"
- and "b \<le> inner a y"
- shows "\<exists>z \<in> s. inner a z = b"
-proof (rule ccontr)
- assume as:"\<not> (\<exists>z\<in>s. inner a z = b)"
- let ?A = "{x. inner a x < b}"
- let ?B = "{x. inner a x > b}"
- have "open ?A" "open ?B"
- using open_halfspace_lt and open_halfspace_gt by auto
- moreover
- have "?A \<inter> ?B = {}" by auto
- moreover
- have "s \<subseteq> ?A \<union> ?B" using as by auto
- ultimately
- show False
- using assms(1)[unfolded connected_def not_ex,
- THEN spec[where x="?A"], THEN spec[where x="?B"]]
- using assms(2-5)
- by auto
-qed
-
-lemma connected_ivt_component:
- fixes x::"'a::euclidean_space"
- shows "connected s \<Longrightarrow>
- x \<in> s \<Longrightarrow> y \<in> s \<Longrightarrow>
- x\<bullet>k \<le> a \<Longrightarrow> a \<le> y\<bullet>k \<Longrightarrow> (\<exists>z\<in>s. z\<bullet>k = a)"
- using connected_ivt_hyperplane[of s x y "k::'a" a]
- by (auto simp: inner_commute)
-
-
-subsection \<open>Intervals\<close>
-
lemma open_box[intro]: "open (box a b)"
proof -
have "open (\<Inter>i\<in>Basis. (op \<bullet> i) -` {a \<bullet> i <..< b \<bullet> i})"
@@ -8973,63 +5668,6 @@
using bounded_closed_imp_seq_compact[of "cbox a b"] using bounded_cbox[of a b]
by (auto simp: compact_eq_seq_compact_metric)
-proposition is_interval_compact:
- "is_interval S \<and> compact S \<longleftrightarrow> (\<exists>a b. S = cbox a b)" (is "?lhs = ?rhs")
-proof (cases "S = {}")
- case True
- with empty_as_interval show ?thesis by auto
-next
- case False
- show ?thesis
- proof
- assume L: ?lhs
- then have "is_interval S" "compact S" by auto
- define a where "a \<equiv> \<Sum>i\<in>Basis. (INF x:S. x \<bullet> i) *\<^sub>R i"
- define b where "b \<equiv> \<Sum>i\<in>Basis. (SUP x:S. x \<bullet> i) *\<^sub>R i"
- have 1: "\<And>x i. \<lbrakk>x \<in> S; i \<in> Basis\<rbrakk> \<Longrightarrow> (INF x:S. x \<bullet> i) \<le> x \<bullet> i"
- by (simp add: cInf_lower bounded_inner_imp_bdd_below compact_imp_bounded L)
- have 2: "\<And>x i. \<lbrakk>x \<in> S; i \<in> Basis\<rbrakk> \<Longrightarrow> x \<bullet> i \<le> (SUP x:S. x \<bullet> i)"
- by (simp add: cSup_upper bounded_inner_imp_bdd_above compact_imp_bounded L)
- have 3: "x \<in> S" if inf: "\<And>i. i \<in> Basis \<Longrightarrow> (INF x:S. x \<bullet> i) \<le> x \<bullet> i"
- and sup: "\<And>i. i \<in> Basis \<Longrightarrow> x \<bullet> i \<le> (SUP x:S. x \<bullet> i)" for x
- proof (rule mem_box_componentwiseI [OF \<open>is_interval S\<close>])
- fix i::'a
- assume i: "i \<in> Basis"
- have cont: "continuous_on S (\<lambda>x. x \<bullet> i)"
- by (intro continuous_intros)
- obtain a where "a \<in> S" and a: "\<And>y. y\<in>S \<Longrightarrow> a \<bullet> i \<le> y \<bullet> i"
- using continuous_attains_inf [OF \<open>compact S\<close> False cont] by blast
- obtain b where "b \<in> S" and b: "\<And>y. y\<in>S \<Longrightarrow> y \<bullet> i \<le> b \<bullet> i"
- using continuous_attains_sup [OF \<open>compact S\<close> False cont] by blast
- have "a \<bullet> i \<le> (INF x:S. x \<bullet> i)"
- by (simp add: False a cINF_greatest)
- also have "\<dots> \<le> x \<bullet> i"
- by (simp add: i inf)
- finally have ai: "a \<bullet> i \<le> x \<bullet> i" .
- have "x \<bullet> i \<le> (SUP x:S. x \<bullet> i)"
- by (simp add: i sup)
- also have "(SUP x:S. x \<bullet> i) \<le> b \<bullet> i"
- by (simp add: False b cSUP_least)
- finally have bi: "x \<bullet> i \<le> b \<bullet> i" .
- show "x \<bullet> i \<in> (\<lambda>x. x \<bullet> i) ` S"
- apply (rule_tac x="\<Sum>j\<in>Basis. (if j = i then x \<bullet> i else a \<bullet> j) *\<^sub>R j" in image_eqI)
- apply (simp add: i)
- apply (rule mem_is_intervalI [OF \<open>is_interval S\<close> \<open>a \<in> S\<close> \<open>b \<in> S\<close>])
- using i ai bi apply force
- done
- qed
- have "S = cbox a b"
- by (auto simp: a_def b_def mem_box intro: 1 2 3)
- then show ?rhs
- by blast
- next
- assume R: ?rhs
- then show ?lhs
- using compact_cbox is_interval_cbox by blast
- qed
-qed
-
-
lemma box_midpoint:
fixes a :: "'a::euclidean_space"
assumes "box a b \<noteq> {}"
@@ -9072,18 +5710,12 @@
have "b \<bullet> i = e * (b\<bullet>i) + (1 - e) * (b\<bullet>i)"
unfolding left_diff_distrib by simp
also have "\<dots> > e * (x \<bullet> i) + (1 - e) * (y \<bullet> i)"
- apply (rule add_less_le_mono)
- using e unfolding mult_less_cancel_left and mult_le_cancel_left
- apply simp_all
- using x
- unfolding mem_box
- using i
- apply simp
- using y
- unfolding mem_box
- using i
- apply simp
- done
+ proof (rule add_less_le_mono)
+ show "e * (x \<bullet> i) < e * (b \<bullet> i)"
+ using e(1) i mem_box(1) x by auto
+ show "(1 - e) * (y \<bullet> i) \<le> (1 - e) * (b \<bullet> i)"
+ by (meson diff_ge_0_iff_ge e(2) i mem_box(2) ordered_comm_semiring_class.comm_mult_left_mono y)
+ qed
finally have "(e *\<^sub>R x + (1 - e) *\<^sub>R y) \<bullet> i < b \<bullet> i"
unfolding inner_simps by auto
}
@@ -9228,2118 +5860,18 @@
unfolding closure_box[OF assms, symmetric]
unfolding open_Int_closure_eq_empty[OF open_box] ..
-lemma diameter_cbox:
- fixes a b::"'a::euclidean_space"
- shows "(\<forall>i \<in> Basis. a \<bullet> i \<le> b \<bullet> i) \<Longrightarrow> diameter (cbox a b) = dist a b"
- by (force simp: diameter_def intro!: cSup_eq_maximum setL2_mono
- simp: euclidean_dist_l2[where 'a='a] cbox_def dist_norm)
-
lemma eucl_less_eq_halfspaces:
fixes a :: "'a::euclidean_space"
shows "{x. x <e a} = (\<Inter>i\<in>Basis. {x. x \<bullet> i < a \<bullet> i})"
"{x. a <e x} = (\<Inter>i\<in>Basis. {x. a \<bullet> i < x \<bullet> i})"
by (auto simp: eucl_less_def)
-lemma eucl_le_eq_halfspaces:
- fixes a :: "'a::euclidean_space"
- shows "{x. \<forall>i\<in>Basis. x \<bullet> i \<le> a \<bullet> i} = (\<Inter>i\<in>Basis. {x. x \<bullet> i \<le> a \<bullet> i})"
- "{x. \<forall>i\<in>Basis. a \<bullet> i \<le> x \<bullet> i} = (\<Inter>i\<in>Basis. {x. a \<bullet> i \<le> x \<bullet> i})"
- by auto
-
lemma open_Collect_eucl_less[simp, intro]:
fixes a :: "'a::euclidean_space"
shows "open {x. x <e a}"
"open {x. a <e x}"
by (auto simp: eucl_less_eq_halfspaces open_halfspace_component_lt open_halfspace_component_gt)
-lemma closed_Collect_eucl_le[simp, intro]:
- fixes a :: "'a::euclidean_space"
- shows "closed {x. \<forall>i\<in>Basis. a \<bullet> i \<le> x \<bullet> i}"
- "closed {x. \<forall>i\<in>Basis. x \<bullet> i \<le> a \<bullet> i}"
- unfolding eucl_le_eq_halfspaces
- by (simp_all add: closed_INT closed_Collect_le continuous_on_inner continuous_on_const continuous_on_id)
-
-lemma image_affinity_cbox: fixes m::real
- fixes a b c :: "'a::euclidean_space"
- shows "(\<lambda>x. m *\<^sub>R x + c) ` cbox a b =
- (if cbox a b = {} then {}
- else (if 0 \<le> m then cbox (m *\<^sub>R a + c) (m *\<^sub>R b + c)
- else cbox (m *\<^sub>R b + c) (m *\<^sub>R a + c)))"
-proof (cases "m = 0")
- case True
- {
- fix x
- assume "\<forall>i\<in>Basis. x \<bullet> i \<le> c \<bullet> i" "\<forall>i\<in>Basis. c \<bullet> i \<le> x \<bullet> i"
- then have "x = c"
- apply -
- apply (subst euclidean_eq_iff)
- apply (auto intro: order_antisym)
- done
- }
- moreover have "c \<in> cbox (m *\<^sub>R a + c) (m *\<^sub>R b + c)"
- unfolding True by (auto simp: cbox_sing)
- ultimately show ?thesis using True by (auto simp: cbox_def)
-next
- case False
- {
- fix y
- assume "\<forall>i\<in>Basis. a \<bullet> i \<le> y \<bullet> i" "\<forall>i\<in>Basis. y \<bullet> i \<le> b \<bullet> i" "m > 0"
- then have "\<forall>i\<in>Basis. (m *\<^sub>R a + c) \<bullet> i \<le> (m *\<^sub>R y + c) \<bullet> i" and "\<forall>i\<in>Basis. (m *\<^sub>R y + c) \<bullet> i \<le> (m *\<^sub>R b + c) \<bullet> i"
- by (auto simp: inner_distrib)
- }
- moreover
- {
- fix y
- assume "\<forall>i\<in>Basis. a \<bullet> i \<le> y \<bullet> i" "\<forall>i\<in>Basis. y \<bullet> i \<le> b \<bullet> i" "m < 0"
- then have "\<forall>i\<in>Basis. (m *\<^sub>R b + c) \<bullet> i \<le> (m *\<^sub>R y + c) \<bullet> i" and "\<forall>i\<in>Basis. (m *\<^sub>R y + c) \<bullet> i \<le> (m *\<^sub>R a + c) \<bullet> i"
- by (auto simp: mult_left_mono_neg inner_distrib)
- }
- moreover
- {
- fix y
- assume "m > 0" and "\<forall>i\<in>Basis. (m *\<^sub>R a + c) \<bullet> i \<le> y \<bullet> i" and "\<forall>i\<in>Basis. y \<bullet> i \<le> (m *\<^sub>R b + c) \<bullet> i"
- then have "y \<in> (\<lambda>x. m *\<^sub>R x + c) ` cbox a b"
- unfolding image_iff Bex_def mem_box
- apply (intro exI[where x="(1 / m) *\<^sub>R (y - c)"])
- apply (auto simp: pos_le_divide_eq pos_divide_le_eq mult.commute inner_distrib inner_diff_left)
- done
- }
- moreover
- {
- fix y
- assume "\<forall>i\<in>Basis. (m *\<^sub>R b + c) \<bullet> i \<le> y \<bullet> i" "\<forall>i\<in>Basis. y \<bullet> i \<le> (m *\<^sub>R a + c) \<bullet> i" "m < 0"
- then have "y \<in> (\<lambda>x. m *\<^sub>R x + c) ` cbox a b"
- unfolding image_iff Bex_def mem_box
- apply (intro exI[where x="(1 / m) *\<^sub>R (y - c)"])
- apply (auto simp: neg_le_divide_eq neg_divide_le_eq mult.commute inner_distrib inner_diff_left)
- done
- }
- ultimately show ?thesis using False by (auto simp: cbox_def)
-qed
-
-lemma image_smult_cbox:"(\<lambda>x. m *\<^sub>R (x::_::euclidean_space)) ` cbox a b =
- (if cbox a b = {} then {} else if 0 \<le> m then cbox (m *\<^sub>R a) (m *\<^sub>R b) else cbox (m *\<^sub>R b) (m *\<^sub>R a))"
- using image_affinity_cbox[of m 0 a b] by auto
-
-lemma islimpt_greaterThanLessThan1:
- fixes a b::"'a::{linorder_topology, dense_order}"
- assumes "a < b"
- shows "a islimpt {a<..<b}"
-proof (rule islimptI)
- fix T
- assume "open T" "a \<in> T"
- from open_right[OF this \<open>a < b\<close>]
- obtain c where c: "a < c" "{a..<c} \<subseteq> T" by auto
- with assms dense[of a "min c b"]
- show "\<exists>y\<in>{a<..<b}. y \<in> T \<and> y \<noteq> a"
- by (metis atLeastLessThan_iff greaterThanLessThan_iff min_less_iff_conj
- not_le order.strict_implies_order subset_eq)
-qed
-
-lemma islimpt_greaterThanLessThan2:
- fixes a b::"'a::{linorder_topology, dense_order}"
- assumes "a < b"
- shows "b islimpt {a<..<b}"
-proof (rule islimptI)
- fix T
- assume "open T" "b \<in> T"
- from open_left[OF this \<open>a < b\<close>]
- obtain c where c: "c < b" "{c<..b} \<subseteq> T" by auto
- with assms dense[of "max a c" b]
- show "\<exists>y\<in>{a<..<b}. y \<in> T \<and> y \<noteq> b"
- by (metis greaterThanAtMost_iff greaterThanLessThan_iff max_less_iff_conj
- not_le order.strict_implies_order subset_eq)
-qed
-
-lemma closure_greaterThanLessThan[simp]:
- fixes a b::"'a::{linorder_topology, dense_order}"
- shows "a < b \<Longrightarrow> closure {a <..< b} = {a .. b}" (is "_ \<Longrightarrow> ?l = ?r")
-proof
- have "?l \<subseteq> closure ?r"
- by (rule closure_mono) auto
- thus "closure {a<..<b} \<subseteq> {a..b}" by simp
-qed (auto simp: closure_def order.order_iff_strict islimpt_greaterThanLessThan1
- islimpt_greaterThanLessThan2)
-
-lemma closure_greaterThan[simp]:
- fixes a b::"'a::{no_top, linorder_topology, dense_order}"
- shows "closure {a<..} = {a..}"
-proof -
- from gt_ex obtain b where "a < b" by auto
- hence "{a<..} = {a<..<b} \<union> {b..}" by auto
- also have "closure \<dots> = {a..}" using \<open>a < b\<close> unfolding closure_Un
- by auto
- finally show ?thesis .
-qed
-
-lemma closure_lessThan[simp]:
- fixes b::"'a::{no_bot, linorder_topology, dense_order}"
- shows "closure {..<b} = {..b}"
-proof -
- from lt_ex obtain a where "a < b" by auto
- hence "{..<b} = {a<..<b} \<union> {..a}" by auto
- also have "closure \<dots> = {..b}" using \<open>a < b\<close> unfolding closure_Un
- by auto
- finally show ?thesis .
-qed
-
-lemma closure_atLeastLessThan[simp]:
- fixes a b::"'a::{linorder_topology, dense_order}"
- assumes "a < b"
- shows "closure {a ..< b} = {a .. b}"
-proof -
- from assms have "{a ..< b} = {a} \<union> {a <..< b}" by auto
- also have "closure \<dots> = {a .. b}" unfolding closure_Un
- by (auto simp: assms less_imp_le)
- finally show ?thesis .
-qed
-
-lemma closure_greaterThanAtMost[simp]:
- fixes a b::"'a::{linorder_topology, dense_order}"
- assumes "a < b"
- shows "closure {a <.. b} = {a .. b}"
-proof -
- from assms have "{a <.. b} = {b} \<union> {a <..< b}" by auto
- also have "closure \<dots> = {a .. b}" unfolding closure_Un
- by (auto simp: assms less_imp_le)
- finally show ?thesis .
-qed
-
-
-subsection \<open>Homeomorphisms\<close>
-
-definition "homeomorphism s t f g \<longleftrightarrow>
- (\<forall>x\<in>s. (g(f x) = x)) \<and> (f ` s = t) \<and> continuous_on s f \<and>
- (\<forall>y\<in>t. (f(g y) = y)) \<and> (g ` t = s) \<and> continuous_on t g"
-
-lemma homeomorphismI [intro?]:
- assumes "continuous_on S f" "continuous_on T g"
- "f ` S \<subseteq> T" "g ` T \<subseteq> S" "\<And>x. x \<in> S \<Longrightarrow> g(f x) = x" "\<And>y. y \<in> T \<Longrightarrow> f(g y) = y"
- shows "homeomorphism S T f g"
- using assms by (force simp: homeomorphism_def)
-
-lemma homeomorphism_translation:
- fixes a :: "'a :: real_normed_vector"
- shows "homeomorphism (op + a ` S) S (op + (- a)) (op + a)"
-unfolding homeomorphism_def by (auto simp: algebra_simps continuous_intros)
-
-lemma homeomorphism_ident: "homeomorphism T T (\<lambda>a. a) (\<lambda>a. a)"
- by (rule homeomorphismI) (auto simp: continuous_on_id)
-
-lemma homeomorphism_compose:
- assumes "homeomorphism S T f g" "homeomorphism T U h k"
- shows "homeomorphism S U (h o f) (g o k)"
- using assms
- unfolding homeomorphism_def
- by (intro conjI ballI continuous_on_compose) (auto simp: image_comp [symmetric])
-
-lemma homeomorphism_symD: "homeomorphism S t f g \<Longrightarrow> homeomorphism t S g f"
- by (simp add: homeomorphism_def)
-
-lemma homeomorphism_sym: "homeomorphism S t f g = homeomorphism t S g f"
- by (force simp: homeomorphism_def)
-
-definition homeomorphic :: "'a::topological_space set \<Rightarrow> 'b::topological_space set \<Rightarrow> bool"
- (infixr "homeomorphic" 60)
- where "s homeomorphic t \<equiv> (\<exists>f g. homeomorphism s t f g)"
-
-lemma homeomorphic_empty [iff]:
- "S homeomorphic {} \<longleftrightarrow> S = {}" "{} homeomorphic S \<longleftrightarrow> S = {}"
- by (auto simp: homeomorphic_def homeomorphism_def)
-
-lemma homeomorphic_refl: "s homeomorphic s"
- unfolding homeomorphic_def homeomorphism_def
- using continuous_on_id
- apply (rule_tac x = "(\<lambda>x. x)" in exI)
- apply (rule_tac x = "(\<lambda>x. x)" in exI)
- apply blast
- done
-
-lemma homeomorphic_sym: "s homeomorphic t \<longleftrightarrow> t homeomorphic s"
- unfolding homeomorphic_def homeomorphism_def
- by blast
-
-lemma homeomorphic_trans [trans]:
- assumes "S homeomorphic T"
- and "T homeomorphic U"
- shows "S homeomorphic U"
- using assms
- unfolding homeomorphic_def
-by (metis homeomorphism_compose)
-
-lemma homeomorphic_minimal:
- "s homeomorphic t \<longleftrightarrow>
- (\<exists>f g. (\<forall>x\<in>s. f(x) \<in> t \<and> (g(f(x)) = x)) \<and>
- (\<forall>y\<in>t. g(y) \<in> s \<and> (f(g(y)) = y)) \<and>
- continuous_on s f \<and> continuous_on t g)"
- (is "?lhs = ?rhs")
-proof
- assume ?lhs
- then show ?rhs
- by (fastforce simp: homeomorphic_def homeomorphism_def)
-next
- assume ?rhs
- then show ?lhs
- apply clarify
- unfolding homeomorphic_def homeomorphism_def
- by (metis equalityI image_subset_iff subsetI)
- qed
-
-lemma homeomorphicI [intro?]:
- "\<lbrakk>f ` S = T; g ` T = S;
- continuous_on S f; continuous_on T g;
- \<And>x. x \<in> S \<Longrightarrow> g(f(x)) = x;
- \<And>y. y \<in> T \<Longrightarrow> f(g(y)) = y\<rbrakk> \<Longrightarrow> S homeomorphic T"
-unfolding homeomorphic_def homeomorphism_def by metis
-
-lemma homeomorphism_of_subsets:
- "\<lbrakk>homeomorphism S T f g; S' \<subseteq> S; T'' \<subseteq> T; f ` S' = T'\<rbrakk>
- \<Longrightarrow> homeomorphism S' T' f g"
-apply (auto simp: homeomorphism_def elim!: continuous_on_subset)
-by (metis subsetD imageI)
-
-lemma homeomorphism_apply1: "\<lbrakk>homeomorphism S T f g; x \<in> S\<rbrakk> \<Longrightarrow> g(f x) = x"
- by (simp add: homeomorphism_def)
-
-lemma homeomorphism_apply2: "\<lbrakk>homeomorphism S T f g; x \<in> T\<rbrakk> \<Longrightarrow> f(g x) = x"
- by (simp add: homeomorphism_def)
-
-lemma homeomorphism_image1: "homeomorphism S T f g \<Longrightarrow> f ` S = T"
- by (simp add: homeomorphism_def)
-
-lemma homeomorphism_image2: "homeomorphism S T f g \<Longrightarrow> g ` T = S"
- by (simp add: homeomorphism_def)
-
-lemma homeomorphism_cont1: "homeomorphism S T f g \<Longrightarrow> continuous_on S f"
- by (simp add: homeomorphism_def)
-
-lemma homeomorphism_cont2: "homeomorphism S T f g \<Longrightarrow> continuous_on T g"
- by (simp add: homeomorphism_def)
-
-lemma continuous_on_no_limpt:
- "(\<And>x. \<not> x islimpt S) \<Longrightarrow> continuous_on S f"
- unfolding continuous_on_def
- by (metis UNIV_I empty_iff eventually_at_topological islimptE open_UNIV tendsto_def trivial_limit_within)
-
-lemma continuous_on_finite:
- fixes S :: "'a::t1_space set"
- shows "finite S \<Longrightarrow> continuous_on S f"
-by (metis continuous_on_no_limpt islimpt_finite)
-
-lemma homeomorphic_finite:
- fixes S :: "'a::t1_space set" and T :: "'b::t1_space set"
- assumes "finite T"
- shows "S homeomorphic T \<longleftrightarrow> finite S \<and> finite T \<and> card S = card T" (is "?lhs = ?rhs")
-proof
- assume "S homeomorphic T"
- with assms show ?rhs
- apply (auto simp: homeomorphic_def homeomorphism_def)
- apply (metis finite_imageI)
- by (metis card_image_le finite_imageI le_antisym)
-next
- assume R: ?rhs
- with finite_same_card_bij obtain h where "bij_betw h S T"
- by auto
- with R show ?lhs
- apply (auto simp: homeomorphic_def homeomorphism_def continuous_on_finite)
- apply (rule_tac x=h in exI)
- apply (rule_tac x="inv_into S h" in exI)
- apply (auto simp: bij_betw_inv_into_left bij_betw_inv_into_right bij_betw_imp_surj_on inv_into_into bij_betwE)
- apply (metis bij_betw_def bij_betw_inv_into)
- done
-qed
-
-text \<open>Relatively weak hypotheses if a set is compact.\<close>
-
-lemma homeomorphism_compact:
- fixes f :: "'a::topological_space \<Rightarrow> 'b::t2_space"
- assumes "compact s" "continuous_on s f" "f ` s = t" "inj_on f s"
- shows "\<exists>g. homeomorphism s t f g"
-proof -
- define g where "g x = (SOME y. y\<in>s \<and> f y = x)" for x
- have g: "\<forall>x\<in>s. g (f x) = x"
- using assms(3) assms(4)[unfolded inj_on_def] unfolding g_def by auto
- {
- fix y
- assume "y \<in> t"
- then obtain x where x:"f x = y" "x\<in>s"
- using assms(3) by auto
- then have "g (f x) = x" using g by auto
- then have "f (g y) = y" unfolding x(1)[symmetric] by auto
- }
- then have g':"\<forall>x\<in>t. f (g x) = x" by auto
- moreover
- {
- fix x
- have "x\<in>s \<Longrightarrow> x \<in> g ` t"
- using g[THEN bspec[where x=x]]
- unfolding image_iff
- using assms(3)
- by (auto intro!: bexI[where x="f x"])
- moreover
- {
- assume "x\<in>g ` t"
- then obtain y where y:"y\<in>t" "g y = x" by auto
- then obtain x' where x':"x'\<in>s" "f x' = y"
- using assms(3) by auto
- then have "x \<in> s"
- unfolding g_def
- using someI2[of "\<lambda>b. b\<in>s \<and> f b = y" x' "\<lambda>x. x\<in>s"]
- unfolding y(2)[symmetric] and g_def
- by auto
- }
- ultimately have "x\<in>s \<longleftrightarrow> x \<in> g ` t" ..
- }
- then have "g ` t = s" by auto
- ultimately show ?thesis
- unfolding homeomorphism_def homeomorphic_def
- apply (rule_tac x=g in exI)
- using g and assms(3) and continuous_on_inv[OF assms(2,1), of g, unfolded assms(3)] and assms(2)
- apply auto
- done
-qed
-
-lemma homeomorphic_compact:
- fixes f :: "'a::topological_space \<Rightarrow> 'b::t2_space"
- shows "compact s \<Longrightarrow> continuous_on s f \<Longrightarrow> (f ` s = t) \<Longrightarrow> inj_on f s \<Longrightarrow> s homeomorphic t"
- unfolding homeomorphic_def by (metis homeomorphism_compact)
-
-text\<open>Preservation of topological properties.\<close>
-
-lemma homeomorphic_compactness: "s homeomorphic t \<Longrightarrow> (compact s \<longleftrightarrow> compact t)"
- unfolding homeomorphic_def homeomorphism_def
- by (metis compact_continuous_image)
-
-text\<open>Results on translation, scaling etc.\<close>
-
-lemma homeomorphic_scaling:
- fixes s :: "'a::real_normed_vector set"
- assumes "c \<noteq> 0"
- shows "s homeomorphic ((\<lambda>x. c *\<^sub>R x) ` s)"
- unfolding homeomorphic_minimal
- apply (rule_tac x="\<lambda>x. c *\<^sub>R x" in exI)
- apply (rule_tac x="\<lambda>x. (1 / c) *\<^sub>R x" in exI)
- using assms
- apply (auto simp: continuous_intros)
- done
-
-lemma homeomorphic_translation:
- fixes s :: "'a::real_normed_vector set"
- shows "s homeomorphic ((\<lambda>x. a + x) ` s)"
- unfolding homeomorphic_minimal
- apply (rule_tac x="\<lambda>x. a + x" in exI)
- apply (rule_tac x="\<lambda>x. -a + x" in exI)
- using continuous_on_add [OF continuous_on_const continuous_on_id, of s a]
- continuous_on_add [OF continuous_on_const continuous_on_id, of "plus a ` s" "- a"]
- apply auto
- done
-
-lemma homeomorphic_affinity:
- fixes s :: "'a::real_normed_vector set"
- assumes "c \<noteq> 0"
- shows "s homeomorphic ((\<lambda>x. a + c *\<^sub>R x) ` s)"
-proof -
- have *: "op + a ` op *\<^sub>R c ` s = (\<lambda>x. a + c *\<^sub>R x) ` s" by auto
- show ?thesis
- using homeomorphic_trans
- using homeomorphic_scaling[OF assms, of s]
- using homeomorphic_translation[of "(\<lambda>x. c *\<^sub>R x) ` s" a]
- unfolding *
- by auto
-qed
-
-lemma homeomorphic_balls:
- fixes a b ::"'a::real_normed_vector"
- assumes "0 < d" "0 < e"
- shows "(ball a d) homeomorphic (ball b e)" (is ?th)
- and "(cball a d) homeomorphic (cball b e)" (is ?cth)
-proof -
- show ?th unfolding homeomorphic_minimal
- apply(rule_tac x="\<lambda>x. b + (e/d) *\<^sub>R (x - a)" in exI)
- apply(rule_tac x="\<lambda>x. a + (d/e) *\<^sub>R (x - b)" in exI)
- using assms
- apply (auto intro!: continuous_intros
- simp: dist_commute dist_norm pos_divide_less_eq mult_strict_left_mono)
- done
- show ?cth unfolding homeomorphic_minimal
- apply(rule_tac x="\<lambda>x. b + (e/d) *\<^sub>R (x - a)" in exI)
- apply(rule_tac x="\<lambda>x. a + (d/e) *\<^sub>R (x - b)" in exI)
- using assms
- apply (auto intro!: continuous_intros
- simp: dist_commute dist_norm pos_divide_le_eq mult_strict_left_mono)
- done
-qed
-
-lemma homeomorphic_spheres:
- fixes a b ::"'a::real_normed_vector"
- assumes "0 < d" "0 < e"
- shows "(sphere a d) homeomorphic (sphere b e)"
-unfolding homeomorphic_minimal
- apply(rule_tac x="\<lambda>x. b + (e/d) *\<^sub>R (x - a)" in exI)
- apply(rule_tac x="\<lambda>x. a + (d/e) *\<^sub>R (x - b)" in exI)
- using assms
- apply (auto intro!: continuous_intros
- simp: dist_commute dist_norm pos_divide_less_eq mult_strict_left_mono)
- done
-
-lemma homeomorphic_ball01_UNIV:
- "ball (0::'a::real_normed_vector) 1 homeomorphic (UNIV:: 'a set)"
- (is "?B homeomorphic ?U")
-proof
- have "x \<in> (\<lambda>z. z /\<^sub>R (1 - norm z)) ` ball 0 1" for x::'a
- apply (rule_tac x="x /\<^sub>R (1 + norm x)" in image_eqI)
- apply (auto simp: divide_simps)
- using norm_ge_zero [of x] apply linarith+
- done
- then show "(\<lambda>z::'a. z /\<^sub>R (1 - norm z)) ` ?B = ?U"
- by blast
- have "x \<in> range (\<lambda>z. (1 / (1 + norm z)) *\<^sub>R z)" if "norm x < 1" for x::'a
- apply (rule_tac x="x /\<^sub>R (1 - norm x)" in image_eqI)
- using that apply (auto simp: divide_simps)
- done
- then show "(\<lambda>z::'a. z /\<^sub>R (1 + norm z)) ` ?U = ?B"
- by (force simp: divide_simps dest: add_less_zeroD)
- show "continuous_on (ball 0 1) (\<lambda>z. z /\<^sub>R (1 - norm z))"
- by (rule continuous_intros | force)+
- show "continuous_on UNIV (\<lambda>z. z /\<^sub>R (1 + norm z))"
- apply (intro continuous_intros)
- apply (metis le_add_same_cancel1 norm_ge_zero not_le zero_less_one)
- done
- show "\<And>x. x \<in> ball 0 1 \<Longrightarrow>
- x /\<^sub>R (1 - norm x) /\<^sub>R (1 + norm (x /\<^sub>R (1 - norm x))) = x"
- by (auto simp: divide_simps)
- show "\<And>y. y /\<^sub>R (1 + norm y) /\<^sub>R (1 - norm (y /\<^sub>R (1 + norm y))) = y"
- apply (auto simp: divide_simps)
- apply (metis le_add_same_cancel1 norm_ge_zero not_le zero_less_one)
- done
-qed
-
-proposition homeomorphic_ball_UNIV:
- fixes a ::"'a::real_normed_vector"
- assumes "0 < r" shows "ball a r homeomorphic (UNIV:: 'a set)"
- using assms homeomorphic_ball01_UNIV homeomorphic_balls(1) homeomorphic_trans zero_less_one by blast
-
-subsection\<open>Inverse function property for open/closed maps\<close>
-
-lemma continuous_on_inverse_open_map:
- assumes contf: "continuous_on S f"
- and imf: "f ` S = T"
- and injf: "\<And>x. x \<in> S \<Longrightarrow> g (f x) = x"
- and oo: "\<And>U. openin (subtopology euclidean S) U \<Longrightarrow> openin (subtopology euclidean T) (f ` U)"
- shows "continuous_on T g"
-proof -
- from imf injf have gTS: "g ` T = S"
- by force
- from imf injf have fU: "U \<subseteq> S \<Longrightarrow> (f ` U) = {x \<in> T. g x \<in> U}" for U
- by force
- show ?thesis
- by (simp add: continuous_on_open [of T g] gTS) (metis openin_imp_subset fU oo)
-qed
-
-lemma continuous_on_inverse_closed_map:
- assumes contf: "continuous_on S f"
- and imf: "f ` S = T"
- and injf: "\<And>x. x \<in> S \<Longrightarrow> g(f x) = x"
- and oo: "\<And>U. closedin (subtopology euclidean S) U \<Longrightarrow> closedin (subtopology euclidean T) (f ` U)"
- shows "continuous_on T g"
-proof -
- from imf injf have gTS: "g ` T = S"
- by force
- from imf injf have fU: "U \<subseteq> S \<Longrightarrow> (f ` U) = {x \<in> T. g x \<in> U}" for U
- by force
- show ?thesis
- by (simp add: continuous_on_closed [of T g] gTS) (metis closedin_imp_subset fU oo)
-qed
-
-lemma homeomorphism_injective_open_map:
- assumes contf: "continuous_on S f"
- and imf: "f ` S = T"
- and injf: "inj_on f S"
- and oo: "\<And>U. openin (subtopology euclidean S) U \<Longrightarrow> openin (subtopology euclidean T) (f ` U)"
- obtains g where "homeomorphism S T f g"
-proof
- have "continuous_on T (inv_into S f)"
- by (metis contf continuous_on_inverse_open_map imf injf inv_into_f_f oo)
- with imf injf contf show "homeomorphism S T f (inv_into S f)"
- by (auto simp: homeomorphism_def)
-qed
-
-lemma homeomorphism_injective_closed_map:
- assumes contf: "continuous_on S f"
- and imf: "f ` S = T"
- and injf: "inj_on f S"
- and oo: "\<And>U. closedin (subtopology euclidean S) U \<Longrightarrow> closedin (subtopology euclidean T) (f ` U)"
- obtains g where "homeomorphism S T f g"
-proof
- have "continuous_on T (inv_into S f)"
- by (metis contf continuous_on_inverse_closed_map imf injf inv_into_f_f oo)
- with imf injf contf show "homeomorphism S T f (inv_into S f)"
- by (auto simp: homeomorphism_def)
-qed
-
-lemma homeomorphism_imp_open_map:
- assumes hom: "homeomorphism S T f g"
- and oo: "openin (subtopology euclidean S) U"
- shows "openin (subtopology euclidean T) (f ` U)"
-proof -
- from hom oo have [simp]: "f ` U = {y. y \<in> T \<and> g y \<in> U}"
- using openin_subset by (fastforce simp: homeomorphism_def rev_image_eqI)
- from hom have "continuous_on T g"
- unfolding homeomorphism_def by blast
- moreover have "g ` T = S"
- by (metis hom homeomorphism_def)
- ultimately show ?thesis
- by (simp add: continuous_on_open oo)
-qed
-
-lemma homeomorphism_imp_closed_map:
- assumes hom: "homeomorphism S T f g"
- and oo: "closedin (subtopology euclidean S) U"
- shows "closedin (subtopology euclidean T) (f ` U)"
-proof -
- from hom oo have [simp]: "f ` U = {y. y \<in> T \<and> g y \<in> U}"
- using closedin_subset by (fastforce simp: homeomorphism_def rev_image_eqI)
- from hom have "continuous_on T g"
- unfolding homeomorphism_def by blast
- moreover have "g ` T = S"
- by (metis hom homeomorphism_def)
- ultimately show ?thesis
- by (simp add: continuous_on_closed oo)
-qed
-
-
-subsection \<open>"Isometry" (up to constant bounds) of injective linear map etc.\<close>
-
-lemma cauchy_isometric:
- assumes e: "e > 0"
- and s: "subspace s"
- and f: "bounded_linear f"
- and normf: "\<forall>x\<in>s. norm (f x) \<ge> e * norm x"
- and xs: "\<forall>n. x n \<in> s"
- and cf: "Cauchy (f \<circ> x)"
- shows "Cauchy x"
-proof -
- interpret f: bounded_linear f by fact
- have "\<exists>N. \<forall>n\<ge>N. norm (x n - x N) < d" if "d > 0" for d :: real
- proof -
- from that obtain N where N: "\<forall>n\<ge>N. norm (f (x n) - f (x N)) < e * d"
- using cf[unfolded Cauchy_def o_def dist_norm, THEN spec[where x="e*d"]] e
- by auto
- have "norm (x n - x N) < d" if "n \<ge> N" for n
- proof -
- have "e * norm (x n - x N) \<le> norm (f (x n - x N))"
- using subspace_diff[OF s, of "x n" "x N"]
- using xs[THEN spec[where x=N]] and xs[THEN spec[where x=n]]
- using normf[THEN bspec[where x="x n - x N"]]
- by auto
- also have "norm (f (x n - x N)) < e * d"
- using \<open>N \<le> n\<close> N unfolding f.diff[symmetric] by auto
- finally show ?thesis
- using \<open>e>0\<close> by simp
- qed
- then show ?thesis by auto
- qed
- then show ?thesis
- by (simp add: Cauchy_altdef2 dist_norm)
-qed
-
-lemma complete_isometric_image:
- assumes "0 < e"
- and s: "subspace s"
- and f: "bounded_linear f"
- and normf: "\<forall>x\<in>s. norm(f x) \<ge> e * norm(x)"
- and cs: "complete s"
- shows "complete (f ` s)"
-proof -
- have "\<exists>l\<in>f ` s. (g \<longlongrightarrow> l) sequentially"
- if as:"\<forall>n::nat. g n \<in> f ` s" and cfg:"Cauchy g" for g
- proof -
- from that obtain x where "\<forall>n. x n \<in> s \<and> g n = f (x n)"
- using choice[of "\<lambda> n xa. xa \<in> s \<and> g n = f xa"] by auto
- then have x: "\<forall>n. x n \<in> s" "\<forall>n. g n = f (x n)" by auto
- then have "f \<circ> x = g" by (simp add: fun_eq_iff)
- then obtain l where "l\<in>s" and l:"(x \<longlongrightarrow> l) sequentially"
- using cs[unfolded complete_def, THEN spec[where x=x]]
- using cauchy_isometric[OF \<open>0 < e\<close> s f normf] and cfg and x(1)
- by auto
- then show ?thesis
- using linear_continuous_at[OF f, unfolded continuous_at_sequentially, THEN spec[where x=x], of l]
- by (auto simp: \<open>f \<circ> x = g\<close>)
- qed
- then show ?thesis
- unfolding complete_def by auto
-qed
-
-lemma injective_imp_isometric:
- fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
- assumes s: "closed s" "subspace s"
- and f: "bounded_linear f" "\<forall>x\<in>s. f x = 0 \<longrightarrow> x = 0"
- shows "\<exists>e>0. \<forall>x\<in>s. norm (f x) \<ge> e * norm x"
-proof (cases "s \<subseteq> {0::'a}")
- case True
- have "norm x \<le> norm (f x)" if "x \<in> s" for x
- proof -
- from True that have "x = 0" by auto
- then show ?thesis by simp
- qed
- then show ?thesis
- by (auto intro!: exI[where x=1])
-next
- case False
- interpret f: bounded_linear f by fact
- from False obtain a where a: "a \<noteq> 0" "a \<in> s"
- by auto
- from False have "s \<noteq> {}"
- by auto
- let ?S = "{f x| x. x \<in> s \<and> norm x = norm a}"
- let ?S' = "{x::'a. x\<in>s \<and> norm x = norm a}"
- let ?S'' = "{x::'a. norm x = norm a}"
-
- have "?S'' = frontier (cball 0 (norm a))"
- by (simp add: sphere_def dist_norm)
- then have "compact ?S''" by (metis compact_cball compact_frontier)
- moreover have "?S' = s \<inter> ?S''" by auto
- ultimately have "compact ?S'"
- using closed_Int_compact[of s ?S''] using s(1) by auto
- moreover have *:"f ` ?S' = ?S" by auto
- ultimately have "compact ?S"
- using compact_continuous_image[OF linear_continuous_on[OF f(1)], of ?S'] by auto
- then have "closed ?S"
- using compact_imp_closed by auto
- moreover from a have "?S \<noteq> {}" by auto
- ultimately obtain b' where "b'\<in>?S" "\<forall>y\<in>?S. norm b' \<le> norm y"
- using distance_attains_inf[of ?S 0] unfolding dist_0_norm by auto
- then obtain b where "b\<in>s"
- and ba: "norm b = norm a"
- and b: "\<forall>x\<in>{x \<in> s. norm x = norm a}. norm (f b) \<le> norm (f x)"
- unfolding *[symmetric] unfolding image_iff by auto
-
- let ?e = "norm (f b) / norm b"
- have "norm b > 0"
- using ba and a and norm_ge_zero by auto
- moreover have "norm (f b) > 0"
- using f(2)[THEN bspec[where x=b], OF \<open>b\<in>s\<close>]
- using \<open>norm b >0\<close> by simp
- ultimately have "0 < norm (f b) / norm b" by simp
- moreover
- have "norm (f b) / norm b * norm x \<le> norm (f x)" if "x\<in>s" for x
- proof (cases "x = 0")
- case True
- then show "norm (f b) / norm b * norm x \<le> norm (f x)"
- by auto
- next
- case False
- with \<open>a \<noteq> 0\<close> have *: "0 < norm a / norm x"
- unfolding zero_less_norm_iff[symmetric] by simp
- have "\<forall>x\<in>s. c *\<^sub>R x \<in> s" for c
- using s[unfolded subspace_def] by simp
- with \<open>x \<in> s\<close> \<open>x \<noteq> 0\<close> have "(norm a / norm x) *\<^sub>R x \<in> {x \<in> s. norm x = norm a}"
- by simp
- with \<open>x \<noteq> 0\<close> \<open>a \<noteq> 0\<close> show "norm (f b) / norm b * norm x \<le> norm (f x)"
- using b[THEN bspec[where x="(norm a / norm x) *\<^sub>R x"]]
- unfolding f.scaleR and ba
- by (auto simp: mult.commute pos_le_divide_eq pos_divide_le_eq)
- qed
- ultimately show ?thesis by auto
-qed
-
-lemma closed_injective_image_subspace:
- fixes f :: "'a::euclidean_space \<Rightarrow> 'b::euclidean_space"
- assumes "subspace s" "bounded_linear f" "\<forall>x\<in>s. f x = 0 \<longrightarrow> x = 0" "closed s"
- shows "closed(f ` s)"
-proof -
- obtain e where "e > 0" and e: "\<forall>x\<in>s. e * norm x \<le> norm (f x)"
- using injective_imp_isometric[OF assms(4,1,2,3)] by auto
- show ?thesis
- using complete_isometric_image[OF \<open>e>0\<close> assms(1,2) e] and assms(4)
- unfolding complete_eq_closed[symmetric] by auto
-qed
-
-
-subsection \<open>Some properties of a canonical subspace\<close>
-
-lemma subspace_substandard: "subspace {x::'a::euclidean_space. (\<forall>i\<in>Basis. P i \<longrightarrow> x\<bullet>i = 0)}"
- by (auto simp: subspace_def inner_add_left)
-
-lemma closed_substandard: "closed {x::'a::euclidean_space. \<forall>i\<in>Basis. P i \<longrightarrow> x\<bullet>i = 0}"
- (is "closed ?A")
-proof -
- let ?D = "{i\<in>Basis. P i}"
- have "closed (\<Inter>i\<in>?D. {x::'a. x\<bullet>i = 0})"
- by (simp add: closed_INT closed_Collect_eq continuous_on_inner
- continuous_on_const continuous_on_id)
- also have "(\<Inter>i\<in>?D. {x::'a. x\<bullet>i = 0}) = ?A"
- by auto
- finally show "closed ?A" .
-qed
-
-lemma dim_substandard:
- assumes d: "d \<subseteq> Basis"
- shows "dim {x::'a::euclidean_space. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x\<bullet>i = 0} = card d" (is "dim ?A = _")
-proof (rule dim_unique)
- from d show "d \<subseteq> ?A"
- by (auto simp: inner_Basis)
- from d show "independent d"
- by (rule independent_mono [OF independent_Basis])
- have "x \<in> span d" if "\<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0" for x
- proof -
- have "finite d"
- by (rule finite_subset [OF d finite_Basis])
- then have "(\<Sum>i\<in>d. (x \<bullet> i) *\<^sub>R i) \<in> span d"
- by (simp add: span_sum span_clauses)
- also have "(\<Sum>i\<in>d. (x \<bullet> i) *\<^sub>R i) = (\<Sum>i\<in>Basis. (x \<bullet> i) *\<^sub>R i)"
- by (rule sum.mono_neutral_cong_left [OF finite_Basis d]) (auto simp: that)
- finally show "x \<in> span d"
- by (simp only: euclidean_representation)
- qed
- then show "?A \<subseteq> span d" by auto
-qed simp
-
-text \<open>Hence closure and completeness of all subspaces.\<close>
-lemma ex_card:
- assumes "n \<le> card A"
- shows "\<exists>S\<subseteq>A. card S = n"
-proof (cases "finite A")
- case True
- from ex_bij_betw_nat_finite[OF this] obtain f where f: "bij_betw f {0..<card A} A" ..
- moreover from f \<open>n \<le> card A\<close> have "{..< n} \<subseteq> {..< card A}" "inj_on f {..< n}"
- by (auto simp: bij_betw_def intro: subset_inj_on)
- ultimately have "f ` {..< n} \<subseteq> A" "card (f ` {..< n}) = n"
- by (auto simp: bij_betw_def card_image)
- then show ?thesis by blast
-next
- case False
- with \<open>n \<le> card A\<close> show ?thesis by force
-qed
-
-lemma closed_subspace:
- fixes s :: "'a::euclidean_space set"
- assumes "subspace s"
- shows "closed s"
-proof -
- have "dim s \<le> card (Basis :: 'a set)"
- using dim_subset_UNIV by auto
- with ex_card[OF this] obtain d :: "'a set" where t: "card d = dim s" and d: "d \<subseteq> Basis"
- by auto
- let ?t = "{x::'a. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x\<bullet>i = 0}"
- have "\<exists>f. linear f \<and> f ` {x::'a. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0} = s \<and>
- inj_on f {x::'a. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0}"
- using dim_substandard[of d] t d assms
- by (intro subspace_isomorphism[OF subspace_substandard[of "\<lambda>i. i \<notin> d"]]) (auto simp: inner_Basis)
- then obtain f where f:
- "linear f"
- "f ` {x. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0} = s"
- "inj_on f {x. \<forall>i\<in>Basis. i \<notin> d \<longrightarrow> x \<bullet> i = 0}"
- by blast
- interpret f: bounded_linear f
- using f by (simp add: linear_conv_bounded_linear)
- have "x \<in> ?t \<Longrightarrow> f x = 0 \<Longrightarrow> x = 0" for x
- using f.zero d f(3)[THEN inj_onD, of x 0] by auto
- moreover have "closed ?t" by (rule closed_substandard)
- moreover have "subspace ?t" by (rule subspace_substandard)
- ultimately show ?thesis
- using closed_injective_image_subspace[of ?t f]
- unfolding f(2) using f(1) unfolding linear_conv_bounded_linear by auto
-qed
-
-lemma complete_subspace: "subspace s \<Longrightarrow> complete s"
- for s :: "'a::euclidean_space set"
- using complete_eq_closed closed_subspace by auto
-
-lemma closed_span [iff]: "closed (span s)"
- for s :: "'a::euclidean_space set"
- by (simp add: closed_subspace)
-
-lemma dim_closure [simp]: "dim (closure s) = dim s" (is "?dc = ?d")
- for s :: "'a::euclidean_space set"
-proof -
- have "?dc \<le> ?d"
- using closure_minimal[OF span_inc, of s]
- using closed_subspace[OF subspace_span, of s]
- using dim_subset[of "closure s" "span s"]
- by simp
- then show ?thesis
- using dim_subset[OF closure_subset, of s]
- by simp
-qed
-
-
-subsection \<open>Affine transformations of intervals\<close>
-
-lemma real_affinity_le: "0 < m \<Longrightarrow> m * x + c \<le> y \<longleftrightarrow> x \<le> inverse m * y + - (c / m)"
- for m :: "'a::linordered_field"
- by (simp add: field_simps)
-
-lemma real_le_affinity: "0 < m \<Longrightarrow> y \<le> m * x + c \<longleftrightarrow> inverse m * y + - (c / m) \<le> x"
- for m :: "'a::linordered_field"
- by (simp add: field_simps)
-
-lemma real_affinity_lt: "0 < m \<Longrightarrow> m * x + c < y \<longleftrightarrow> x < inverse m * y + - (c / m)"
- for m :: "'a::linordered_field"
- by (simp add: field_simps)
-
-lemma real_lt_affinity: "0 < m \<Longrightarrow> y < m * x + c \<longleftrightarrow> inverse m * y + - (c / m) < x"
- for m :: "'a::linordered_field"
- by (simp add: field_simps)
-
-lemma real_affinity_eq: "m \<noteq> 0 \<Longrightarrow> m * x + c = y \<longleftrightarrow> x = inverse m * y + - (c / m)"
- for m :: "'a::linordered_field"
- by (simp add: field_simps)
-
-lemma real_eq_affinity: "m \<noteq> 0 \<Longrightarrow> y = m * x + c \<longleftrightarrow> inverse m * y + - (c / m) = x"
- for m :: "'a::linordered_field"
- by (simp add: field_simps)
-
-
-subsection \<open>Banach fixed point theorem (not really topological ...)\<close>
-
-theorem banach_fix:
- assumes s: "complete s" "s \<noteq> {}"
- and c: "0 \<le> c" "c < 1"
- and f: "f ` s \<subseteq> s"
- and lipschitz: "\<forall>x\<in>s. \<forall>y\<in>s. dist (f x) (f y) \<le> c * dist x y"
- shows "\<exists>!x\<in>s. f x = x"
-proof -
- from c have "1 - c > 0" by simp
-
- from s(2) obtain z0 where z0: "z0 \<in> s" by blast
- define z where "z n = (f ^^ n) z0" for n
- with f z0 have z_in_s: "z n \<in> s" for n :: nat
- by (induct n) auto
- define d where "d = dist (z 0) (z 1)"
-
- have fzn: "f (z n) = z (Suc n)" for n
- by (simp add: z_def)
- have cf_z: "dist (z n) (z (Suc n)) \<le> (c ^ n) * d" for n :: nat
- proof (induct n)
- case 0
- then show ?case
- by (simp add: d_def)
- next
- case (Suc m)
- with \<open>0 \<le> c\<close> have "c * dist (z m) (z (Suc m)) \<le> c ^ Suc m * d"
- using mult_left_mono[of "dist (z m) (z (Suc m))" "c ^ m * d" c] by simp
- then show ?case
- using lipschitz[THEN bspec[where x="z m"], OF z_in_s, THEN bspec[where x="z (Suc m)"], OF z_in_s]
- by (simp add: fzn mult_le_cancel_left)
- qed
-
- have cf_z2: "(1 - c) * dist (z m) (z (m + n)) \<le> (c ^ m) * d * (1 - c ^ n)" for n m :: nat
- proof (induct n)
- case 0
- show ?case by simp
- next
- case (Suc k)
- from c have "(1 - c) * dist (z m) (z (m + Suc k)) \<le>
- (1 - c) * (dist (z m) (z (m + k)) + dist (z (m + k)) (z (Suc (m + k))))"
- by (simp add: dist_triangle)
- also from c cf_z[of "m + k"] have "\<dots> \<le> (1 - c) * (dist (z m) (z (m + k)) + c ^ (m + k) * d)"
- by simp
- also from Suc have "\<dots> \<le> c ^ m * d * (1 - c ^ k) + (1 - c) * c ^ (m + k) * d"
- by (simp add: field_simps)
- also have "\<dots> = (c ^ m) * (d * (1 - c ^ k) + (1 - c) * c ^ k * d)"
- by (simp add: power_add field_simps)
- also from c have "\<dots> \<le> (c ^ m) * d * (1 - c ^ Suc k)"
- by (simp add: field_simps)
- finally show ?case by simp
- qed
-
- have "\<exists>N. \<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (z m) (z n) < e" if "e > 0" for e
- proof (cases "d = 0")
- case True
- from \<open>1 - c > 0\<close> have "(1 - c) * x \<le> 0 \<longleftrightarrow> x \<le> 0" for x
- by (metis mult_zero_left mult.commute real_mult_le_cancel_iff1)
- with c cf_z2[of 0] True have "z n = z0" for n
- by (simp add: z_def)
- with \<open>e > 0\<close> show ?thesis by simp
- next
- case False
- with zero_le_dist[of "z 0" "z 1"] have "d > 0"
- by (metis d_def less_le)
- with \<open>1 - c > 0\<close> \<open>e > 0\<close> have "0 < e * (1 - c) / d"
- by simp
- with c obtain N where N: "c ^ N < e * (1 - c) / d"
- using real_arch_pow_inv[of "e * (1 - c) / d" c] by auto
- have *: "dist (z m) (z n) < e" if "m > n" and as: "m \<ge> N" "n \<ge> N" for m n :: nat
- proof -
- from c \<open>n \<ge> N\<close> have *: "c ^ n \<le> c ^ N"
- using power_decreasing[OF \<open>n\<ge>N\<close>, of c] by simp
- from c \<open>m > n\<close> have "1 - c ^ (m - n) > 0"
- using power_strict_mono[of c 1 "m - n"] by simp
- with \<open>d > 0\<close> \<open>0 < 1 - c\<close> have **: "d * (1 - c ^ (m - n)) / (1 - c) > 0"
- by simp
- from cf_z2[of n "m - n"] \<open>m > n\<close>
- have "dist (z m) (z n) \<le> c ^ n * d * (1 - c ^ (m - n)) / (1 - c)"
- by (simp add: pos_le_divide_eq[OF \<open>1 - c > 0\<close>] mult.commute dist_commute)
- also have "\<dots> \<le> c ^ N * d * (1 - c ^ (m - n)) / (1 - c)"
- using mult_right_mono[OF * order_less_imp_le[OF **]]
- by (simp add: mult.assoc)
- also have "\<dots> < (e * (1 - c) / d) * d * (1 - c ^ (m - n)) / (1 - c)"
- using mult_strict_right_mono[OF N **] by (auto simp: mult.assoc)
- also from c \<open>d > 0\<close> \<open>1 - c > 0\<close> have "\<dots> = e * (1 - c ^ (m - n))"
- by simp
- also from c \<open>1 - c ^ (m - n) > 0\<close> \<open>e > 0\<close> have "\<dots> \<le> e"
- using mult_right_le_one_le[of e "1 - c ^ (m - n)"] by auto
- finally show ?thesis by simp
- qed
- have "dist (z n) (z m) < e" if "N \<le> m" "N \<le> n" for m n :: nat
- proof (cases "n = m")
- case True
- with \<open>e > 0\<close> show ?thesis by simp
- next
- case False
- with *[of n m] *[of m n] and that show ?thesis
- by (auto simp: dist_commute nat_neq_iff)
- qed
- then show ?thesis by auto
- qed
- then have "Cauchy z"
- by (simp add: cauchy_def)
- then obtain x where "x\<in>s" and x:"(z \<longlongrightarrow> x) sequentially"
- using s(1)[unfolded compact_def complete_def, THEN spec[where x=z]] and z_in_s by auto
-
- define e where "e = dist (f x) x"
- have "e = 0"
- proof (rule ccontr)
- assume "e \<noteq> 0"
- then have "e > 0"
- unfolding e_def using zero_le_dist[of "f x" x]
- by (metis dist_eq_0_iff dist_nz e_def)
- then obtain N where N:"\<forall>n\<ge>N. dist (z n) x < e / 2"
- using x[unfolded lim_sequentially, THEN spec[where x="e/2"]] by auto
- then have N':"dist (z N) x < e / 2" by auto
- have *: "c * dist (z N) x \<le> dist (z N) x"
- unfolding mult_le_cancel_right2
- using zero_le_dist[of "z N" x] and c
- by (metis dist_eq_0_iff dist_nz order_less_asym less_le)
- have "dist (f (z N)) (f x) \<le> c * dist (z N) x"
- using lipschitz[THEN bspec[where x="z N"], THEN bspec[where x=x]]
- using z_in_s[of N] \<open>x\<in>s\<close>
- using c
- by auto
- also have "\<dots> < e / 2"
- using N' and c using * by auto
- finally show False
- unfolding fzn
- using N[THEN spec[where x="Suc N"]] and dist_triangle_half_r[of "z (Suc N)" "f x" e x]
- unfolding e_def
- by auto
- qed
- then have "f x = x" by (auto simp: e_def)
- moreover have "y = x" if "f y = y" "y \<in> s" for y
- proof -
- from \<open>x \<in> s\<close> \<open>f x = x\<close> that have "dist x y \<le> c * dist x y"
- using lipschitz[THEN bspec[where x=x], THEN bspec[where x=y]] by simp
- with c and zero_le_dist[of x y] have "dist x y = 0"
- by (simp add: mult_le_cancel_right1)
- then show ?thesis by simp
- qed
- ultimately show ?thesis
- using \<open>x\<in>s\<close> by blast
-qed
-
-
-subsection \<open>Edelstein fixed point theorem\<close>
-
-theorem edelstein_fix:
- fixes s :: "'a::metric_space set"
- assumes s: "compact s" "s \<noteq> {}"
- and gs: "(g ` s) \<subseteq> s"
- and dist: "\<forall>x\<in>s. \<forall>y\<in>s. x \<noteq> y \<longrightarrow> dist (g x) (g y) < dist x y"
- shows "\<exists>!x\<in>s. g x = x"
-proof -
- let ?D = "(\<lambda>x. (x, x)) ` s"
- have D: "compact ?D" "?D \<noteq> {}"
- by (rule compact_continuous_image)
- (auto intro!: s continuous_Pair continuous_ident simp: continuous_on_eq_continuous_within)
-
- have "\<And>x y e. x \<in> s \<Longrightarrow> y \<in> s \<Longrightarrow> 0 < e \<Longrightarrow> dist y x < e \<Longrightarrow> dist (g y) (g x) < e"
- using dist by fastforce
- then have "continuous_on s g"
- by (auto simp: continuous_on_iff)
- then have cont: "continuous_on ?D (\<lambda>x. dist ((g \<circ> fst) x) (snd x))"
- unfolding continuous_on_eq_continuous_within
- by (intro continuous_dist ballI continuous_within_compose)
- (auto intro!: continuous_fst continuous_snd continuous_ident simp: image_image)
-
- obtain a where "a \<in> s" and le: "\<And>x. x \<in> s \<Longrightarrow> dist (g a) a \<le> dist (g x) x"
- using continuous_attains_inf[OF D cont] by auto
-
- have "g a = a"
- proof (rule ccontr)
- assume "g a \<noteq> a"
- with \<open>a \<in> s\<close> gs have "dist (g (g a)) (g a) < dist (g a) a"
- by (intro dist[rule_format]) auto
- moreover have "dist (g a) a \<le> dist (g (g a)) (g a)"
- using \<open>a \<in> s\<close> gs by (intro le) auto
- ultimately show False by auto
- qed
- moreover have "\<And>x. x \<in> s \<Longrightarrow> g x = x \<Longrightarrow> x = a"
- using dist[THEN bspec[where x=a]] \<open>g a = a\<close> and \<open>a\<in>s\<close> by auto
- ultimately show "\<exists>!x\<in>s. g x = x"
- using \<open>a \<in> s\<close> by blast
-qed
-
-
-lemma cball_subset_cball_iff:
- fixes a :: "'a :: euclidean_space"
- shows "cball a r \<subseteq> cball a' r' \<longleftrightarrow> dist a a' + r \<le> r' \<or> r < 0"
- (is "?lhs \<longleftrightarrow> ?rhs")
-proof
- assume ?lhs
- then show ?rhs
- proof (cases "r < 0")
- case True
- then show ?rhs by simp
- next
- case False
- then have [simp]: "r \<ge> 0" by simp
- have "norm (a - a') + r \<le> r'"
- proof (cases "a = a'")
- case True
- then show ?thesis
- using subsetD [where c = "a + r *\<^sub>R (SOME i. i \<in> Basis)", OF \<open>?lhs\<close>] subsetD [where c = a, OF \<open>?lhs\<close>]
- by (force simp: SOME_Basis dist_norm)
- next
- case False
- have "norm (a' - (a + (r / norm (a - a')) *\<^sub>R (a - a'))) = norm (a' - a - (r / norm (a - a')) *\<^sub>R (a - a'))"
- by (simp add: algebra_simps)
- also have "... = norm ((-1 - (r / norm (a - a'))) *\<^sub>R (a - a'))"
- by (simp add: algebra_simps)
- also from \<open>a \<noteq> a'\<close> have "... = \<bar>- norm (a - a') - r\<bar>"
- by (simp add: abs_mult_pos field_simps)
- finally have [simp]: "norm (a' - (a + (r / norm (a - a')) *\<^sub>R (a - a'))) = \<bar>norm (a - a') + r\<bar>"
- by linarith
- from \<open>a \<noteq> a'\<close> show ?thesis
- using subsetD [where c = "a' + (1 + r / norm(a - a')) *\<^sub>R (a - a')", OF \<open>?lhs\<close>]
- by (simp add: dist_norm scaleR_add_left)
- qed
- then show ?rhs
- by (simp add: dist_norm)
- qed
-next
- assume ?rhs
- then show ?lhs
- by (auto simp: ball_def dist_norm)
- (metis add.commute add_le_cancel_right dist_norm dist_triangle3 order_trans)
-qed
-
-lemma cball_subset_ball_iff: "cball a r \<subseteq> ball a' r' \<longleftrightarrow> dist a a' + r < r' \<or> r < 0"
- (is "?lhs \<longleftrightarrow> ?rhs")
- for a :: "'a::euclidean_space"
-proof
- assume ?lhs
- then show ?rhs
- proof (cases "r < 0")
- case True then
- show ?rhs by simp
- next
- case False
- then have [simp]: "r \<ge> 0" by simp
- have "norm (a - a') + r < r'"
- proof (cases "a = a'")
- case True
- then show ?thesis
- using subsetD [where c = "a + r *\<^sub>R (SOME i. i \<in> Basis)", OF \<open>?lhs\<close>] subsetD [where c = a, OF \<open>?lhs\<close>]
- by (force simp: SOME_Basis dist_norm)
- next
- case False
- have False if "norm (a - a') + r \<ge> r'"
- proof -
- from that have "\<bar>r' - norm (a - a')\<bar> \<le> r"
- by (simp split: abs_split)
- (metis \<open>0 \<le> r\<close> \<open>?lhs\<close> centre_in_cball dist_commute dist_norm less_asym mem_ball subset_eq)
- then show ?thesis
- using subsetD [where c = "a + (r' / norm(a - a') - 1) *\<^sub>R (a - a')", OF \<open>?lhs\<close>] \<open>a \<noteq> a'\<close>
- by (simp add: dist_norm field_simps)
- (simp add: diff_divide_distrib scaleR_left_diff_distrib)
- qed
- then show ?thesis by force
- qed
- then show ?rhs by (simp add: dist_norm)
- qed
-next
- assume ?rhs
- then show ?lhs
- by (auto simp: ball_def dist_norm)
- (metis add.commute add_le_cancel_right dist_norm dist_triangle3 le_less_trans)
-qed
-
-lemma ball_subset_cball_iff: "ball a r \<subseteq> cball a' r' \<longleftrightarrow> dist a a' + r \<le> r' \<or> r \<le> 0"
- (is "?lhs = ?rhs")
- for a :: "'a::euclidean_space"
-proof (cases "r \<le> 0")
- case True
- then show ?thesis
- using dist_not_less_zero less_le_trans by force
-next
- case False
- show ?thesis
- proof
- assume ?lhs
- then have "(cball a r \<subseteq> cball a' r')"
- by (metis False closed_cball closure_ball closure_closed closure_mono not_less)
- with False show ?rhs
- by (fastforce iff: cball_subset_cball_iff)
- next
- assume ?rhs
- with False show ?lhs
- using ball_subset_cball cball_subset_cball_iff by blast
- qed
-qed
-
-lemma ball_subset_ball_iff:
- fixes a :: "'a :: euclidean_space"
- shows "ball a r \<subseteq> ball a' r' \<longleftrightarrow> dist a a' + r \<le> r' \<or> r \<le> 0"
- (is "?lhs = ?rhs")
-proof (cases "r \<le> 0")
- case True then show ?thesis
- using dist_not_less_zero less_le_trans by force
-next
- case False show ?thesis
- proof
- assume ?lhs
- then have "0 < r'"
- by (metis (no_types) False \<open>?lhs\<close> centre_in_ball dist_norm le_less_trans mem_ball norm_ge_zero not_less set_mp)
- then have "(cball a r \<subseteq> cball a' r')"
- by (metis False\<open>?lhs\<close> closure_ball closure_mono not_less)
- then show ?rhs
- using False cball_subset_cball_iff by fastforce
- next
- assume ?rhs then show ?lhs
- apply (auto simp: ball_def)
- apply (metis add.commute add_le_cancel_right dist_commute dist_triangle_lt not_le order_trans)
- using dist_not_less_zero order.strict_trans2 apply blast
- done
- qed
-qed
-
-
-lemma ball_eq_ball_iff:
- fixes x :: "'a :: euclidean_space"
- shows "ball x d = ball y e \<longleftrightarrow> d \<le> 0 \<and> e \<le> 0 \<or> x=y \<and> d=e"
- (is "?lhs = ?rhs")
-proof
- assume ?lhs
- then show ?rhs
- proof (cases "d \<le> 0 \<or> e \<le> 0")
- case True
- with \<open>?lhs\<close> show ?rhs
- by safe (simp_all only: ball_eq_empty [of y e, symmetric] ball_eq_empty [of x d, symmetric])
- next
- case False
- with \<open>?lhs\<close> show ?rhs
- apply (auto simp: set_eq_subset ball_subset_ball_iff dist_norm norm_minus_commute algebra_simps)
- apply (metis add_le_same_cancel1 le_add_same_cancel1 norm_ge_zero norm_pths(2) order_trans)
- apply (metis add_increasing2 add_le_imp_le_right eq_iff norm_ge_zero)
- done
- qed
-next
- assume ?rhs then show ?lhs
- by (auto simp: set_eq_subset ball_subset_ball_iff)
-qed
-
-lemma cball_eq_cball_iff:
- fixes x :: "'a :: euclidean_space"
- shows "cball x d = cball y e \<longleftrightarrow> d < 0 \<and> e < 0 \<or> x=y \<and> d=e"
- (is "?lhs = ?rhs")
-proof
- assume ?lhs
- then show ?rhs
- proof (cases "d < 0 \<or> e < 0")
- case True
- with \<open>?lhs\<close> show ?rhs
- by safe (simp_all only: cball_eq_empty [of y e, symmetric] cball_eq_empty [of x d, symmetric])
- next
- case False
- with \<open>?lhs\<close> show ?rhs
- apply (auto simp: set_eq_subset cball_subset_cball_iff dist_norm norm_minus_commute algebra_simps)
- apply (metis add_le_same_cancel1 le_add_same_cancel1 norm_ge_zero norm_pths(2) order_trans)
- apply (metis add_increasing2 add_le_imp_le_right eq_iff norm_ge_zero)
- done
- qed
-next
- assume ?rhs then show ?lhs
- by (auto simp: set_eq_subset cball_subset_cball_iff)
-qed
-
-lemma ball_eq_cball_iff:
- fixes x :: "'a :: euclidean_space"
- shows "ball x d = cball y e \<longleftrightarrow> d \<le> 0 \<and> e < 0" (is "?lhs = ?rhs")
-proof
- assume ?lhs
- then show ?rhs
- apply (auto simp: set_eq_subset ball_subset_cball_iff cball_subset_ball_iff algebra_simps)
- apply (metis add_increasing2 add_le_cancel_right add_less_same_cancel1 dist_not_less_zero less_le_trans zero_le_dist)
- apply (metis add_less_same_cancel1 dist_not_less_zero less_le_trans not_le)
- using \<open>?lhs\<close> ball_eq_empty cball_eq_empty apply blast+
- done
-next
- assume ?rhs then show ?lhs by auto
-qed
-
-lemma cball_eq_ball_iff:
- fixes x :: "'a :: euclidean_space"
- shows "cball x d = ball y e \<longleftrightarrow> d < 0 \<and> e \<le> 0"
- using ball_eq_cball_iff by blast
-
-lemma finite_ball_avoid:
- fixes S :: "'a :: euclidean_space set"
- assumes "open S" "finite X" "p \<in> S"
- shows "\<exists>e>0. \<forall>w\<in>ball p e. w\<in>S \<and> (w\<noteq>p \<longrightarrow> w\<notin>X)"
-proof -
- obtain e1 where "0 < e1" and e1_b:"ball p e1 \<subseteq> S"
- using open_contains_ball_eq[OF \<open>open S\<close>] assms by auto
- obtain e2 where "0 < e2" and "\<forall>x\<in>X. x \<noteq> p \<longrightarrow> e2 \<le> dist p x"
- using finite_set_avoid[OF \<open>finite X\<close>,of p] by auto
- hence "\<forall>w\<in>ball p (min e1 e2). w\<in>S \<and> (w\<noteq>p \<longrightarrow> w\<notin>X)" using e1_b by auto
- thus "\<exists>e>0. \<forall>w\<in>ball p e. w \<in> S \<and> (w \<noteq> p \<longrightarrow> w \<notin> X)" using \<open>e2>0\<close> \<open>e1>0\<close>
- apply (rule_tac x="min e1 e2" in exI)
- by auto
-qed
-
-lemma finite_cball_avoid:
- fixes S :: "'a :: euclidean_space set"
- assumes "open S" "finite X" "p \<in> S"
- shows "\<exists>e>0. \<forall>w\<in>cball p e. w\<in>S \<and> (w\<noteq>p \<longrightarrow> w\<notin>X)"
-proof -
- obtain e1 where "e1>0" and e1: "\<forall>w\<in>ball p e1. w\<in>S \<and> (w\<noteq>p \<longrightarrow> w\<notin>X)"
- using finite_ball_avoid[OF assms] by auto
- define e2 where "e2 \<equiv> e1/2"
- have "e2>0" and "e2 < e1" unfolding e2_def using \<open>e1>0\<close> by auto
- then have "cball p e2 \<subseteq> ball p e1" by (subst cball_subset_ball_iff,auto)
- then show "\<exists>e>0. \<forall>w\<in>cball p e. w \<in> S \<and> (w \<noteq> p \<longrightarrow> w \<notin> X)" using \<open>e2>0\<close> e1 by auto
-qed
-
-subsection\<open>Various separability-type properties\<close>
-
-lemma univ_second_countable:
- obtains \<B> :: "'a::euclidean_space set set"
- where "countable \<B>" "\<And>C. C \<in> \<B> \<Longrightarrow> open C"
- "\<And>S. open S \<Longrightarrow> \<exists>U. U \<subseteq> \<B> \<and> S = \<Union>U"
-by (metis ex_countable_basis topological_basis_def)
-
-lemma subset_second_countable:
- obtains \<B> :: "'a:: euclidean_space set set"
- where "countable \<B>"
- "{} \<notin> \<B>"
- "\<And>C. C \<in> \<B> \<Longrightarrow> openin(subtopology euclidean S) C"
- "\<And>T. openin(subtopology euclidean S) T \<Longrightarrow> \<exists>\<U>. \<U> \<subseteq> \<B> \<and> T = \<Union>\<U>"
-proof -
- obtain \<B> :: "'a set set"
- where "countable \<B>"
- and opeB: "\<And>C. C \<in> \<B> \<Longrightarrow> openin(subtopology euclidean S) C"
- and \<B>: "\<And>T. openin(subtopology euclidean S) T \<Longrightarrow> \<exists>\<U>. \<U> \<subseteq> \<B> \<and> T = \<Union>\<U>"
- proof -
- obtain \<C> :: "'a set set"
- where "countable \<C>" and ope: "\<And>C. C \<in> \<C> \<Longrightarrow> open C"
- and \<C>: "\<And>S. open S \<Longrightarrow> \<exists>U. U \<subseteq> \<C> \<and> S = \<Union>U"
- by (metis univ_second_countable that)
- show ?thesis
- proof
- show "countable ((\<lambda>C. S \<inter> C) ` \<C>)"
- by (simp add: \<open>countable \<C>\<close>)
- show "\<And>C. C \<in> op \<inter> S ` \<C> \<Longrightarrow> openin (subtopology euclidean S) C"
- using ope by auto
- show "\<And>T. openin (subtopology euclidean S) T \<Longrightarrow> \<exists>\<U>\<subseteq>op \<inter> S ` \<C>. T = \<Union>\<U>"
- by (metis \<C> image_mono inf_Sup openin_open)
- qed
- qed
- show ?thesis
- proof
- show "countable (\<B> - {{}})"
- using \<open>countable \<B>\<close> by blast
- show "\<And>C. \<lbrakk>C \<in> \<B> - {{}}\<rbrakk> \<Longrightarrow> openin (subtopology euclidean S) C"
- by (simp add: \<open>\<And>C. C \<in> \<B> \<Longrightarrow> openin (subtopology euclidean S) C\<close>)
- show "\<exists>\<U>\<subseteq>\<B> - {{}}. T = \<Union>\<U>" if "openin (subtopology euclidean S) T" for T
- using \<B> [OF that]
- apply clarify
- apply (rule_tac x="\<U> - {{}}" in exI, auto)
- done
- qed auto
-qed
-
-lemma univ_second_countable_sequence:
- obtains B :: "nat \<Rightarrow> 'a::euclidean_space set"
- where "inj B" "\<And>n. open(B n)" "\<And>S. open S \<Longrightarrow> \<exists>k. S = \<Union>{B n |n. n \<in> k}"
-proof -
- obtain \<B> :: "'a set set"
- where "countable \<B>"
- and op: "\<And>C. C \<in> \<B> \<Longrightarrow> open C"
- and Un: "\<And>S. open S \<Longrightarrow> \<exists>U. U \<subseteq> \<B> \<and> S = \<Union>U"
- using univ_second_countable by blast
- have *: "infinite (range (\<lambda>n. ball (0::'a) (inverse(Suc n))))"
- apply (rule Infinite_Set.range_inj_infinite)
- apply (simp add: inj_on_def ball_eq_ball_iff)
- done
- have "infinite \<B>"
- proof
- assume "finite \<B>"
- then have "finite (Union ` (Pow \<B>))"
- by simp
- then have "finite (range (\<lambda>n. ball (0::'a) (inverse(Suc n))))"
- apply (rule rev_finite_subset)
- by (metis (no_types, lifting) PowI image_eqI image_subset_iff Un [OF open_ball])
- with * show False by simp
- qed
- obtain f :: "nat \<Rightarrow> 'a set" where "\<B> = range f" "inj f"
- by (blast intro: countable_as_injective_image [OF \<open>countable \<B>\<close> \<open>infinite \<B>\<close>])
- have *: "\<exists>k. S = \<Union>{f n |n. n \<in> k}" if "open S" for S
- using Un [OF that]
- apply clarify
- apply (rule_tac x="f-`U" in exI)
- using \<open>inj f\<close> \<open>\<B> = range f\<close> apply force
- done
- show ?thesis
- apply (rule that [OF \<open>inj f\<close> _ *])
- apply (auto simp: \<open>\<B> = range f\<close> op)
- done
-qed
-
-proposition separable:
- fixes S :: "'a:: euclidean_space set"
- obtains T where "countable T" "T \<subseteq> S" "S \<subseteq> closure T"
-proof -
- obtain \<B> :: "'a:: euclidean_space set set"
- where "countable \<B>"
- and "{} \<notin> \<B>"
- and ope: "\<And>C. C \<in> \<B> \<Longrightarrow> openin(subtopology euclidean S) C"
- and if_ope: "\<And>T. openin(subtopology euclidean S) T \<Longrightarrow> \<exists>\<U>. \<U> \<subseteq> \<B> \<and> T = \<Union>\<U>"
- by (meson subset_second_countable)
- then obtain f where f: "\<And>C. C \<in> \<B> \<Longrightarrow> f C \<in> C"
- by (metis equals0I)
- show ?thesis
- proof
- show "countable (f ` \<B>)"
- by (simp add: \<open>countable \<B>\<close>)
- show "f ` \<B> \<subseteq> S"
- using ope f openin_imp_subset by blast
- show "S \<subseteq> closure (f ` \<B>)"
- proof (clarsimp simp: closure_approachable)
- fix x and e::real
- assume "x \<in> S" "0 < e"
- have "openin (subtopology euclidean S) (S \<inter> ball x e)"
- by (simp add: openin_Int_open)
- with if_ope obtain \<U> where \<U>: "\<U> \<subseteq> \<B>" "S \<inter> ball x e = \<Union>\<U>"
- by meson
- show "\<exists>C \<in> \<B>. dist (f C) x < e"
- proof (cases "\<U> = {}")
- case True
- then show ?thesis
- using \<open>0 < e\<close> \<U> \<open>x \<in> S\<close> by auto
- next
- case False
- then obtain C where "C \<in> \<U>" by blast
- show ?thesis
- proof
- show "dist (f C) x < e"
- by (metis Int_iff Union_iff \<U> \<open>C \<in> \<U>\<close> dist_commute f mem_ball subsetCE)
- show "C \<in> \<B>"
- using \<open>\<U> \<subseteq> \<B>\<close> \<open>C \<in> \<U>\<close> by blast
- qed
- qed
- qed
- qed
-qed
-
-proposition Lindelof:
- fixes \<F> :: "'a::euclidean_space set set"
- assumes \<F>: "\<And>S. S \<in> \<F> \<Longrightarrow> open S"
- obtains \<F>' where "\<F>' \<subseteq> \<F>" "countable \<F>'" "\<Union>\<F>' = \<Union>\<F>"
-proof -
- obtain \<B> :: "'a set set"
- where "countable \<B>" "\<And>C. C \<in> \<B> \<Longrightarrow> open C"
- and \<B>: "\<And>S. open S \<Longrightarrow> \<exists>U. U \<subseteq> \<B> \<and> S = \<Union>U"
- using univ_second_countable by blast
- define \<D> where "\<D> \<equiv> {S. S \<in> \<B> \<and> (\<exists>U. U \<in> \<F> \<and> S \<subseteq> U)}"
- have "countable \<D>"
- apply (rule countable_subset [OF _ \<open>countable \<B>\<close>])
- apply (force simp: \<D>_def)
- done
- have "\<And>S. \<exists>U. S \<in> \<D> \<longrightarrow> U \<in> \<F> \<and> S \<subseteq> U"
- by (simp add: \<D>_def)
- then obtain G where G: "\<And>S. S \<in> \<D> \<longrightarrow> G S \<in> \<F> \<and> S \<subseteq> G S"
- by metis
- have "\<Union>\<F> \<subseteq> \<Union>\<D>"
- unfolding \<D>_def by (blast dest: \<F> \<B>)
- moreover have "\<Union>\<D> \<subseteq> \<Union>\<F>"
- using \<D>_def by blast
- ultimately have eq1: "\<Union>\<F> = \<Union>\<D>" ..
- have eq2: "\<Union>\<D> = UNION \<D> G"
- using G eq1 by auto
- show ?thesis
- apply (rule_tac \<F>' = "G ` \<D>" in that)
- using G \<open>countable \<D>\<close> apply (auto simp: eq1 eq2)
- done
-qed
-
-lemma Lindelof_openin:
- fixes \<F> :: "'a::euclidean_space set set"
- assumes "\<And>S. S \<in> \<F> \<Longrightarrow> openin (subtopology euclidean U) S"
- obtains \<F>' where "\<F>' \<subseteq> \<F>" "countable \<F>'" "\<Union>\<F>' = \<Union>\<F>"
-proof -
- have "\<And>S. S \<in> \<F> \<Longrightarrow> \<exists>T. open T \<and> S = U \<inter> T"
- using assms by (simp add: openin_open)
- then obtain tf where tf: "\<And>S. S \<in> \<F> \<Longrightarrow> open (tf S) \<and> (S = U \<inter> tf S)"
- by metis
- have [simp]: "\<And>\<F>'. \<F>' \<subseteq> \<F> \<Longrightarrow> \<Union>\<F>' = U \<inter> \<Union>(tf ` \<F>')"
- using tf by fastforce
- obtain \<G> where "countable \<G> \<and> \<G> \<subseteq> tf ` \<F>" "\<Union>\<G> = UNION \<F> tf"
- using tf by (force intro: Lindelof [of "tf ` \<F>"])
- then obtain \<F>' where \<F>': "\<F>' \<subseteq> \<F>" "countable \<F>'" "\<Union>\<F>' = \<Union>\<F>"
- by (clarsimp simp add: countable_subset_image)
- then show ?thesis ..
-qed
-
-lemma countable_disjoint_open_subsets:
- fixes \<F> :: "'a::euclidean_space set set"
- assumes "\<And>S. S \<in> \<F> \<Longrightarrow> open S" and pw: "pairwise disjnt \<F>"
- shows "countable \<F>"
-proof -
- obtain \<F>' where "\<F>' \<subseteq> \<F>" "countable \<F>'" "\<Union>\<F>' = \<Union>\<F>"
- by (meson assms Lindelof)
- with pw have "\<F> \<subseteq> insert {} \<F>'"
- by (fastforce simp add: pairwise_def disjnt_iff)
- then show ?thesis
- by (simp add: \<open>countable \<F>'\<close> countable_subset)
-qed
-
-lemma closedin_compact:
- "\<lbrakk>compact S; closedin (subtopology euclidean S) T\<rbrakk> \<Longrightarrow> compact T"
-by (metis closedin_closed compact_Int_closed)
-
-lemma closedin_compact_eq:
- fixes S :: "'a::t2_space set"
- shows
- "compact S
- \<Longrightarrow> (closedin (subtopology euclidean S) T \<longleftrightarrow>
- compact T \<and> T \<subseteq> S)"
-by (metis closedin_imp_subset closedin_compact closed_subset compact_imp_closed)
-
-lemma continuous_imp_closed_map:
- fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space"
- assumes "closedin (subtopology euclidean S) U"
- "continuous_on S f" "image f S = T" "compact S"
- shows "closedin (subtopology euclidean T) (image f U)"
- by (metis assms closedin_compact_eq compact_continuous_image continuous_on_subset subset_image_iff)
-
-lemma continuous_imp_quotient_map:
- fixes f :: "'a::metric_space \<Rightarrow> 'b::metric_space"
- assumes "continuous_on S f" "image f S = T" "compact S" "U \<subseteq> T"
- shows "openin (subtopology euclidean S) {x. x \<in> S \<and> f x \<in> U} \<longleftrightarrow>
- openin (subtopology euclidean T) U"
- by (metis (no_types, lifting) Collect_cong assms closed_map_imp_quotient_map continuous_imp_closed_map)
-
-subsection\<open> Finite intersection property\<close>
-
-text\<open>Also developed in HOL's toplogical spaces theory, but the Heine-Borel type class isn't available there.\<close>
-
-lemma closed_imp_fip:
- fixes S :: "'a::heine_borel set"
- assumes "closed S"
- and T: "T \<in> \<F>" "bounded T"
- and clof: "\<And>T. T \<in> \<F> \<Longrightarrow> closed T"
- and none: "\<And>\<F>'. \<lbrakk>finite \<F>'; \<F>' \<subseteq> \<F>\<rbrakk> \<Longrightarrow> S \<inter> \<Inter>\<F>' \<noteq> {}"
- shows "S \<inter> \<Inter>\<F> \<noteq> {}"
-proof -
- have "compact (S \<inter> T)"
- using \<open>closed S\<close> clof compact_eq_bounded_closed T by blast
- then have "(S \<inter> T) \<inter> \<Inter>\<F> \<noteq> {}"
- apply (rule compact_imp_fip)
- apply (simp add: clof)
- by (metis Int_assoc complete_lattice_class.Inf_insert finite_insert insert_subset none \<open>T \<in> \<F>\<close>)
- then show ?thesis by blast
-qed
-
-lemma closed_imp_fip_compact:
- fixes S :: "'a::heine_borel set"
- shows
- "\<lbrakk>closed S; \<And>T. T \<in> \<F> \<Longrightarrow> compact T;
- \<And>\<F>'. \<lbrakk>finite \<F>'; \<F>' \<subseteq> \<F>\<rbrakk> \<Longrightarrow> S \<inter> \<Inter>\<F>' \<noteq> {}\<rbrakk>
- \<Longrightarrow> S \<inter> \<Inter>\<F> \<noteq> {}"
-by (metis Inf_greatest closed_imp_fip compact_eq_bounded_closed empty_subsetI finite.emptyI inf.orderE)
-
-lemma closed_fip_heine_borel:
- fixes \<F> :: "'a::heine_borel set set"
- assumes "closed S" "T \<in> \<F>" "bounded T"
- and "\<And>T. T \<in> \<F> \<Longrightarrow> closed T"
- and "\<And>\<F>'. \<lbrakk>finite \<F>'; \<F>' \<subseteq> \<F>\<rbrakk> \<Longrightarrow> \<Inter>\<F>' \<noteq> {}"
- shows "\<Inter>\<F> \<noteq> {}"
-proof -
- have "UNIV \<inter> \<Inter>\<F> \<noteq> {}"
- using assms closed_imp_fip [OF closed_UNIV] by auto
- then show ?thesis by simp
-qed
-
-lemma compact_fip_heine_borel:
- fixes \<F> :: "'a::heine_borel set set"
- assumes clof: "\<And>T. T \<in> \<F> \<Longrightarrow> compact T"
- and none: "\<And>\<F>'. \<lbrakk>finite \<F>'; \<F>' \<subseteq> \<F>\<rbrakk> \<Longrightarrow> \<Inter>\<F>' \<noteq> {}"
- shows "\<Inter>\<F> \<noteq> {}"
-by (metis InterI all_not_in_conv clof closed_fip_heine_borel compact_eq_bounded_closed none)
-
-lemma compact_sequence_with_limit:
- fixes f :: "nat \<Rightarrow> 'a::heine_borel"
- shows "(f \<longlongrightarrow> l) sequentially \<Longrightarrow> compact (insert l (range f))"
-apply (simp add: compact_eq_bounded_closed, auto)
-apply (simp add: convergent_imp_bounded)
-by (simp add: closed_limpt islimpt_insert sequence_unique_limpt)
-
-
-subsection\<open>Componentwise limits and continuity\<close>
-
-text\<open>But is the premise really necessary? Need to generalise @{thm euclidean_dist_l2}\<close>
-lemma Euclidean_dist_upper: "i \<in> Basis \<Longrightarrow> dist (x \<bullet> i) (y \<bullet> i) \<le> dist x y"
- by (metis (no_types) member_le_setL2 euclidean_dist_l2 finite_Basis)
-
-text\<open>But is the premise @{term \<open>i \<in> Basis\<close>} really necessary?\<close>
-lemma open_preimage_inner:
- assumes "open S" "i \<in> Basis"
- shows "open {x. x \<bullet> i \<in> S}"
-proof (rule openI, simp)
- fix x
- assume x: "x \<bullet> i \<in> S"
- with assms obtain e where "0 < e" and e: "ball (x \<bullet> i) e \<subseteq> S"
- by (auto simp: open_contains_ball_eq)
- have "\<exists>e>0. ball (y \<bullet> i) e \<subseteq> S" if dxy: "dist x y < e / 2" for y
- proof (intro exI conjI)
- have "dist (x \<bullet> i) (y \<bullet> i) < e / 2"
- by (meson \<open>i \<in> Basis\<close> dual_order.trans Euclidean_dist_upper not_le that)
- then have "dist (x \<bullet> i) z < e" if "dist (y \<bullet> i) z < e / 2" for z
- by (metis dist_commute dist_triangle_half_l that)
- then have "ball (y \<bullet> i) (e / 2) \<subseteq> ball (x \<bullet> i) e"
- using mem_ball by blast
- with e show "ball (y \<bullet> i) (e / 2) \<subseteq> S"
- by (metis order_trans)
- qed (simp add: \<open>0 < e\<close>)
- then show "\<exists>e>0. ball x e \<subseteq> {s. s \<bullet> i \<in> S}"
- by (metis (no_types, lifting) \<open>0 < e\<close> \<open>open S\<close> half_gt_zero_iff mem_Collect_eq mem_ball open_contains_ball_eq subsetI)
-qed
-
-proposition tendsto_componentwise_iff:
- fixes f :: "_ \<Rightarrow> 'b::euclidean_space"
- shows "(f \<longlongrightarrow> l) F \<longleftrightarrow> (\<forall>i \<in> Basis. ((\<lambda>x. (f x \<bullet> i)) \<longlongrightarrow> (l \<bullet> i)) F)"
- (is "?lhs = ?rhs")
-proof
- assume ?lhs
- then show ?rhs
- unfolding tendsto_def
- apply clarify
- apply (drule_tac x="{s. s \<bullet> i \<in> S}" in spec)
- apply (auto simp: open_preimage_inner)
- done
-next
- assume R: ?rhs
- then have "\<And>e. e > 0 \<Longrightarrow> \<forall>i\<in>Basis. \<forall>\<^sub>F x in F. dist (f x \<bullet> i) (l \<bullet> i) < e"
- unfolding tendsto_iff by blast
- then have R': "\<And>e. e > 0 \<Longrightarrow> \<forall>\<^sub>F x in F. \<forall>i\<in>Basis. dist (f x \<bullet> i) (l \<bullet> i) < e"
- by (simp add: eventually_ball_finite_distrib [symmetric])
- show ?lhs
- unfolding tendsto_iff
- proof clarify
- fix e::real
- assume "0 < e"
- have *: "setL2 (\<lambda>i. dist (f x \<bullet> i) (l \<bullet> i)) Basis < e"
- if "\<forall>i\<in>Basis. dist (f x \<bullet> i) (l \<bullet> i) < e / real DIM('b)" for x
- proof -
- have "setL2 (\<lambda>i. dist (f x \<bullet> i) (l \<bullet> i)) Basis \<le> sum (\<lambda>i. dist (f x \<bullet> i) (l \<bullet> i)) Basis"
- by (simp add: setL2_le_sum)
- also have "... < DIM('b) * (e / real DIM('b))"
- apply (rule sum_bounded_above_strict)
- using that by auto
- also have "... = e"
- by (simp add: field_simps)
- finally show "setL2 (\<lambda>i. dist (f x \<bullet> i) (l \<bullet> i)) Basis < e" .
- qed
- have "\<forall>\<^sub>F x in F. \<forall>i\<in>Basis. dist (f x \<bullet> i) (l \<bullet> i) < e / DIM('b)"
- apply (rule R')
- using \<open>0 < e\<close> by simp
- then show "\<forall>\<^sub>F x in F. dist (f x) l < e"
- apply (rule eventually_mono)
- apply (subst euclidean_dist_l2)
- using * by blast
- qed
-qed
-
-
-corollary continuous_componentwise:
- "continuous F f \<longleftrightarrow> (\<forall>i \<in> Basis. continuous F (\<lambda>x. (f x \<bullet> i)))"
-by (simp add: continuous_def tendsto_componentwise_iff [symmetric])
-
-corollary continuous_on_componentwise:
- fixes S :: "'a :: t2_space set"
- shows "continuous_on S f \<longleftrightarrow> (\<forall>i \<in> Basis. continuous_on S (\<lambda>x. (f x \<bullet> i)))"
- apply (simp add: continuous_on_eq_continuous_within)
- using continuous_componentwise by blast
-
-lemma linear_componentwise_iff:
- "(linear f') \<longleftrightarrow> (\<forall>i\<in>Basis. linear (\<lambda>x. f' x \<bullet> i))"
- apply (auto simp: linear_iff inner_left_distrib)
- apply (metis inner_left_distrib euclidean_eq_iff)
- by (metis euclidean_eqI inner_scaleR_left)
-
-lemma bounded_linear_componentwise_iff:
- "(bounded_linear f') \<longleftrightarrow> (\<forall>i\<in>Basis. bounded_linear (\<lambda>x. f' x \<bullet> i))"
- (is "?lhs = ?rhs")
-proof
- assume ?lhs then show ?rhs
- by (simp add: bounded_linear_inner_left_comp)
-next
- assume ?rhs
- then have "(\<forall>i\<in>Basis. \<exists>K. \<forall>x. \<bar>f' x \<bullet> i\<bar> \<le> norm x * K)" "linear f'"
- by (auto simp: bounded_linear_def bounded_linear_axioms_def linear_componentwise_iff [symmetric] ball_conj_distrib)
- then obtain F where F: "\<And>i x. i \<in> Basis \<Longrightarrow> \<bar>f' x \<bullet> i\<bar> \<le> norm x * F i"
- by metis
- have "norm (f' x) \<le> norm x * sum F Basis" for x
- proof -
- have "norm (f' x) \<le> (\<Sum>i\<in>Basis. \<bar>f' x \<bullet> i\<bar>)"
- by (rule norm_le_l1)
- also have "... \<le> (\<Sum>i\<in>Basis. norm x * F i)"
- by (metis F sum_mono)
- also have "... = norm x * sum F Basis"
- by (simp add: sum_distrib_left)
- finally show ?thesis .
- qed
- then show ?lhs
- by (force simp: bounded_linear_def bounded_linear_axioms_def \<open>linear f'\<close>)
-qed
-
-subsection\<open>Pasting functions together\<close>
-
-subsubsection\<open>on open sets\<close>
-
-lemma pasting_lemma:
- fixes f :: "'i \<Rightarrow> 'a::topological_space \<Rightarrow> 'b::topological_space"
- assumes clo: "\<And>i. i \<in> I \<Longrightarrow> openin (subtopology euclidean S) (T i)"
- and cont: "\<And>i. i \<in> I \<Longrightarrow> continuous_on (T i) (f i)"
- and f: "\<And>i j x. \<lbrakk>i \<in> I; j \<in> I; x \<in> S \<inter> T i \<inter> T j\<rbrakk> \<Longrightarrow> f i x = f j x"
- and g: "\<And>x. x \<in> S \<Longrightarrow> \<exists>j. j \<in> I \<and> x \<in> T j \<and> g x = f j x"
- shows "continuous_on S g"
-proof (clarsimp simp: continuous_openin_preimage_eq)
- fix U :: "'b set"
- assume "open U"
- have S: "\<And>i. i \<in> I \<Longrightarrow> (T i) \<subseteq> S"
- using clo openin_imp_subset by blast
- have *: "{x \<in> S. g x \<in> U} = \<Union>{{x. x \<in> (T i) \<and> (f i x) \<in> U} |i. i \<in> I}"
- apply (auto simp: dest: S)
- apply (metis (no_types, lifting) g mem_Collect_eq)
- using clo f g openin_imp_subset by fastforce
- show "openin (subtopology euclidean S) {x \<in> S. g x \<in> U}"
- apply (subst *)
- apply (rule openin_Union, clarify)
- apply (metis (full_types) \<open>open U\<close> cont clo openin_trans continuous_openin_preimage_gen)
- done
-qed
-
-lemma pasting_lemma_exists:
- fixes f :: "'i \<Rightarrow> 'a::topological_space \<Rightarrow> 'b::topological_space"
- assumes S: "S \<subseteq> (\<Union>i \<in> I. T i)"
- and clo: "\<And>i. i \<in> I \<Longrightarrow> openin (subtopology euclidean S) (T i)"
- and cont: "\<And>i. i \<in> I \<Longrightarrow> continuous_on (T i) (f i)"
- and f: "\<And>i j x. \<lbrakk>i \<in> I; j \<in> I; x \<in> S \<inter> T i \<inter> T j\<rbrakk> \<Longrightarrow> f i x = f j x"
- obtains g where "continuous_on S g" "\<And>x i. \<lbrakk>i \<in> I; x \<in> S \<inter> T i\<rbrakk> \<Longrightarrow> g x = f i x"
-proof
- show "continuous_on S (\<lambda>x. f (SOME i. i \<in> I \<and> x \<in> T i) x)"
- apply (rule pasting_lemma [OF clo cont])
- apply (blast intro: f)+
- apply (metis (mono_tags, lifting) S UN_iff subsetCE someI)
- done
-next
- fix x i
- assume "i \<in> I" "x \<in> S \<inter> T i"
- then show "f (SOME i. i \<in> I \<and> x \<in> T i) x = f i x"
- by (metis (no_types, lifting) IntD2 IntI f someI_ex)
-qed
-
-subsubsection\<open>Likewise on closed sets, with a finiteness assumption\<close>
-
-lemma pasting_lemma_closed:
- fixes f :: "'i \<Rightarrow> 'a::topological_space \<Rightarrow> 'b::topological_space"
- assumes "finite I"
- and clo: "\<And>i. i \<in> I \<Longrightarrow> closedin (subtopology euclidean S) (T i)"
- and cont: "\<And>i. i \<in> I \<Longrightarrow> continuous_on (T i) (f i)"
- and f: "\<And>i j x. \<lbrakk>i \<in> I; j \<in> I; x \<in> S \<inter> T i \<inter> T j\<rbrakk> \<Longrightarrow> f i x = f j x"
- and g: "\<And>x. x \<in> S \<Longrightarrow> \<exists>j. j \<in> I \<and> x \<in> T j \<and> g x = f j x"
- shows "continuous_on S g"
-proof (clarsimp simp: continuous_closedin_preimage_eq)
- fix U :: "'b set"
- assume "closed U"
- have *: "{x \<in> S. g x \<in> U} = \<Union>{{x. x \<in> (T i) \<and> (f i x) \<in> U} |i. i \<in> I}"
- apply auto
- apply (metis (no_types, lifting) g mem_Collect_eq)
- using clo closedin_closed apply blast
- apply (metis Int_iff f g clo closedin_limpt inf.absorb_iff2)
- done
- show "closedin (subtopology euclidean S) {x \<in> S. g x \<in> U}"
- apply (subst *)
- apply (rule closedin_Union)
- using \<open>finite I\<close> apply simp
- apply (blast intro: \<open>closed U\<close> continuous_closedin_preimage cont clo closedin_trans)
- done
-qed
-
-lemma pasting_lemma_exists_closed:
- fixes f :: "'i \<Rightarrow> 'a::topological_space \<Rightarrow> 'b::topological_space"
- assumes "finite I"
- and S: "S \<subseteq> (\<Union>i \<in> I. T i)"
- and clo: "\<And>i. i \<in> I \<Longrightarrow> closedin (subtopology euclidean S) (T i)"
- and cont: "\<And>i. i \<in> I \<Longrightarrow> continuous_on (T i) (f i)"
- and f: "\<And>i j x. \<lbrakk>i \<in> I; j \<in> I; x \<in> S \<inter> T i \<inter> T j\<rbrakk> \<Longrightarrow> f i x = f j x"
- obtains g where "continuous_on S g" "\<And>x i. \<lbrakk>i \<in> I; x \<in> S \<inter> T i\<rbrakk> \<Longrightarrow> g x = f i x"
-proof
- show "continuous_on S (\<lambda>x. f (SOME i. i \<in> I \<and> x \<in> T i) x)"
- apply (rule pasting_lemma_closed [OF \<open>finite I\<close> clo cont])
- apply (blast intro: f)+
- apply (metis (mono_tags, lifting) S UN_iff subsetCE someI)
- done
-next
- fix x i
- assume "i \<in> I" "x \<in> S \<inter> T i"
- then show "f (SOME i. i \<in> I \<and> x \<in> T i) x = f i x"
- by (metis (no_types, lifting) IntD2 IntI f someI_ex)
-qed
-
-lemma tube_lemma:
- assumes "compact K"
- assumes "open W"
- assumes "{x0} \<times> K \<subseteq> W"
- shows "\<exists>X0. x0 \<in> X0 \<and> open X0 \<and> X0 \<times> K \<subseteq> W"
-proof -
- {
- fix y assume "y \<in> K"
- then have "(x0, y) \<in> W" using assms by auto
- with \<open>open W\<close>
- have "\<exists>X0 Y. open X0 \<and> open Y \<and> x0 \<in> X0 \<and> y \<in> Y \<and> X0 \<times> Y \<subseteq> W"
- by (rule open_prod_elim) blast
- }
- then obtain X0 Y where
- *: "\<forall>y \<in> K. open (X0 y) \<and> open (Y y) \<and> x0 \<in> X0 y \<and> y \<in> Y y \<and> X0 y \<times> Y y \<subseteq> W"
- by metis
- from * have "\<forall>t\<in>Y ` K. open t" "K \<subseteq> \<Union>(Y ` K)" by auto
- with \<open>compact K\<close> obtain CC where CC: "CC \<subseteq> Y ` K" "finite CC" "K \<subseteq> \<Union>CC"
- by (meson compactE)
- then obtain c where c: "\<And>C. C \<in> CC \<Longrightarrow> c C \<in> K \<and> C = Y (c C)"
- by (force intro!: choice)
- with * CC show ?thesis
- by (force intro!: exI[where x="\<Inter>C\<in>CC. X0 (c C)"]) (* SLOW *)
-qed
-
-lemma continuous_on_prod_compactE:
- fixes fx::"'a::topological_space \<times> 'b::topological_space \<Rightarrow> 'c::metric_space"
- and e::real
- assumes cont_fx: "continuous_on (U \<times> C) fx"
- assumes "compact C"
- assumes [intro]: "x0 \<in> U"
- notes [continuous_intros] = continuous_on_compose2[OF cont_fx]
- assumes "e > 0"
- obtains X0 where "x0 \<in> X0" "open X0"
- "\<forall>x\<in>X0 \<inter> U. \<forall>t \<in> C. dist (fx (x, t)) (fx (x0, t)) \<le> e"
-proof -
- define psi where "psi = (\<lambda>(x, t). dist (fx (x, t)) (fx (x0, t)))"
- define W0 where "W0 = {(x, t) \<in> U \<times> C. psi (x, t) < e}"
- have W0_eq: "W0 = psi -` {..<e} \<inter> U \<times> C"
- by (auto simp: vimage_def W0_def)
- have "open {..<e}" by simp
- have "continuous_on (U \<times> C) psi"
- by (auto intro!: continuous_intros simp: psi_def split_beta')
- from this[unfolded continuous_on_open_invariant, rule_format, OF \<open>open {..<e}\<close>]
- obtain W where W: "open W" "W \<inter> U \<times> C = W0 \<inter> U \<times> C"
- unfolding W0_eq by blast
- have "{x0} \<times> C \<subseteq> W \<inter> U \<times> C"
- unfolding W
- by (auto simp: W0_def psi_def \<open>0 < e\<close>)
- then have "{x0} \<times> C \<subseteq> W" by blast
- from tube_lemma[OF \<open>compact C\<close> \<open>open W\<close> this]
- obtain X0 where X0: "x0 \<in> X0" "open X0" "X0 \<times> C \<subseteq> W"
- by blast
-
- have "\<forall>x\<in>X0 \<inter> U. \<forall>t \<in> C. dist (fx (x, t)) (fx (x0, t)) \<le> e"
- proof safe
- fix x assume x: "x \<in> X0" "x \<in> U"
- fix t assume t: "t \<in> C"
- have "dist (fx (x, t)) (fx (x0, t)) = psi (x, t)"
- by (auto simp: psi_def)
- also
- {
- have "(x, t) \<in> X0 \<times> C"
- using t x
- by auto
- also note \<open>\<dots> \<subseteq> W\<close>
- finally have "(x, t) \<in> W" .
- with t x have "(x, t) \<in> W \<inter> U \<times> C"
- by blast
- also note \<open>W \<inter> U \<times> C = W0 \<inter> U \<times> C\<close>
- finally have "psi (x, t) < e"
- by (auto simp: W0_def)
- }
- finally show "dist (fx (x, t)) (fx (x0, t)) \<le> e" by simp
- qed
- from X0(1,2) this show ?thesis ..
-qed
-
-
-subsection\<open>Constancy of a function from a connected set into a finite, disconnected or discrete set\<close>
-
-text\<open>Still missing: versions for a set that is smaller than R, or countable.\<close>
-
-lemma continuous_disconnected_range_constant:
- assumes S: "connected S"
- and conf: "continuous_on S f"
- and fim: "f ` S \<subseteq> t"
- and cct: "\<And>y. y \<in> t \<Longrightarrow> connected_component_set t y = {y}"
- shows "\<exists>a. \<forall>x \<in> S. f x = a"
-proof (cases "S = {}")
- case True then show ?thesis by force
-next
- case False
- { fix x assume "x \<in> S"
- then have "f ` S \<subseteq> {f x}"
- by (metis connected_continuous_image conf connected_component_maximal fim image_subset_iff rev_image_eqI S cct)
- }
- with False show ?thesis
- by blast
-qed
-
-lemma discrete_subset_disconnected:
- fixes S :: "'a::topological_space set"
- fixes t :: "'b::real_normed_vector set"
- assumes conf: "continuous_on S f"
- and no: "\<And>x. x \<in> S \<Longrightarrow> \<exists>e>0. \<forall>y. y \<in> S \<and> f y \<noteq> f x \<longrightarrow> e \<le> norm (f y - f x)"
- shows "f ` S \<subseteq> {y. connected_component_set (f ` S) y = {y}}"
-proof -
- { fix x assume x: "x \<in> S"
- then obtain e where "e>0" and ele: "\<And>y. \<lbrakk>y \<in> S; f y \<noteq> f x\<rbrakk> \<Longrightarrow> e \<le> norm (f y - f x)"
- using conf no [OF x] by auto
- then have e2: "0 \<le> e / 2"
- by simp
- have "f y = f x" if "y \<in> S" and ccs: "f y \<in> connected_component_set (f ` S) (f x)" for y
- apply (rule ccontr)
- using connected_closed [of "connected_component_set (f ` S) (f x)"] \<open>e>0\<close>
- apply (simp add: del: ex_simps)
- apply (drule spec [where x="cball (f x) (e / 2)"])
- apply (drule spec [where x="- ball(f x) e"])
- apply (auto simp: dist_norm open_closed [symmetric] simp del: le_divide_eq_numeral1 dest!: connected_component_in)
- apply (metis diff_self e2 ele norm_minus_commute norm_zero not_less)
- using centre_in_cball connected_component_refl_eq e2 x apply blast
- using ccs
- apply (force simp: cball_def dist_norm norm_minus_commute dest: ele [OF \<open>y \<in> S\<close>])
- done
- moreover have "connected_component_set (f ` S) (f x) \<subseteq> f ` S"
- by (auto simp: connected_component_in)
- ultimately have "connected_component_set (f ` S) (f x) = {f x}"
- by (auto simp: x)
- }
- with assms show ?thesis
- by blast
-qed
-
-lemma finite_implies_discrete:
- fixes S :: "'a::topological_space set"
- assumes "finite (f ` S)"
- shows "(\<forall>x \<in> S. \<exists>e>0. \<forall>y. y \<in> S \<and> f y \<noteq> f x \<longrightarrow> e \<le> norm (f y - f x))"
-proof -
- have "\<exists>e>0. \<forall>y. y \<in> S \<and> f y \<noteq> f x \<longrightarrow> e \<le> norm (f y - f x)" if "x \<in> S" for x
- proof (cases "f ` S - {f x} = {}")
- case True
- with zero_less_numeral show ?thesis
- by (fastforce simp add: Set.image_subset_iff cong: conj_cong)
- next
- case False
- then obtain z where z: "z \<in> S" "f z \<noteq> f x"
- by blast
- have finn: "finite {norm (z - f x) |z. z \<in> f ` S - {f x}}"
- using assms by simp
- then have *: "0 < Inf{norm(z - f x) | z. z \<in> f ` S - {f x}}"
- apply (rule finite_imp_less_Inf)
- using z apply force+
- done
- show ?thesis
- by (force intro!: * cInf_le_finite [OF finn])
- qed
- with assms show ?thesis
- by blast
-qed
-
-text\<open>This proof requires the existence of two separate values of the range type.\<close>
-lemma finite_range_constant_imp_connected:
- assumes "\<And>f::'a::topological_space \<Rightarrow> 'b::real_normed_algebra_1.
- \<lbrakk>continuous_on S f; finite(f ` S)\<rbrakk> \<Longrightarrow> \<exists>a. \<forall>x \<in> S. f x = a"
- shows "connected S"
-proof -
- { fix t u
- assume clt: "closedin (subtopology euclidean S) t"
- and clu: "closedin (subtopology euclidean S) u"
- and tue: "t \<inter> u = {}" and tus: "t \<union> u = S"
- have conif: "continuous_on S (\<lambda>x. if x \<in> t then 0 else 1)"
- apply (subst tus [symmetric])
- apply (rule continuous_on_cases_local)
- using clt clu tue
- apply (auto simp: tus continuous_on_const)
- done
- have fi: "finite ((\<lambda>x. if x \<in> t then 0 else 1) ` S)"
- by (rule finite_subset [of _ "{0,1}"]) auto
- have "t = {} \<or> u = {}"
- using assms [OF conif fi] tus [symmetric]
- by (auto simp: Ball_def) (metis IntI empty_iff one_neq_zero tue)
- }
- then show ?thesis
- by (simp add: connected_closedin_eq)
-qed
-
-lemma continuous_disconnected_range_constant_eq:
- "(connected S \<longleftrightarrow>
- (\<forall>f::'a::topological_space \<Rightarrow> 'b::real_normed_algebra_1.
- \<forall>t. continuous_on S f \<and> f ` S \<subseteq> t \<and> (\<forall>y \<in> t. connected_component_set t y = {y})
- \<longrightarrow> (\<exists>a::'b. \<forall>x \<in> S. f x = a)))" (is ?thesis1)
- and continuous_discrete_range_constant_eq:
- "(connected S \<longleftrightarrow>
- (\<forall>f::'a::topological_space \<Rightarrow> 'b::real_normed_algebra_1.
- continuous_on S f \<and>
- (\<forall>x \<in> S. \<exists>e. 0 < e \<and> (\<forall>y. y \<in> S \<and> (f y \<noteq> f x) \<longrightarrow> e \<le> norm(f y - f x)))
- \<longrightarrow> (\<exists>a::'b. \<forall>x \<in> S. f x = a)))" (is ?thesis2)
- and continuous_finite_range_constant_eq:
- "(connected S \<longleftrightarrow>
- (\<forall>f::'a::topological_space \<Rightarrow> 'b::real_normed_algebra_1.
- continuous_on S f \<and> finite (f ` S)
- \<longrightarrow> (\<exists>a::'b. \<forall>x \<in> S. f x = a)))" (is ?thesis3)
-proof -
- have *: "\<And>s t u v. \<lbrakk>s \<Longrightarrow> t; t \<Longrightarrow> u; u \<Longrightarrow> v; v \<Longrightarrow> s\<rbrakk>
- \<Longrightarrow> (s \<longleftrightarrow> t) \<and> (s \<longleftrightarrow> u) \<and> (s \<longleftrightarrow> v)"
- by blast
- have "?thesis1 \<and> ?thesis2 \<and> ?thesis3"
- apply (rule *)
- using continuous_disconnected_range_constant apply metis
- apply clarify
- apply (frule discrete_subset_disconnected; blast)
- apply (blast dest: finite_implies_discrete)
- apply (blast intro!: finite_range_constant_imp_connected)
- done
- then show ?thesis1 ?thesis2 ?thesis3
- by blast+
-qed
-
-lemma continuous_discrete_range_constant:
- fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_algebra_1"
- assumes S: "connected S"
- and "continuous_on S f"
- and "\<And>x. x \<in> S \<Longrightarrow> \<exists>e>0. \<forall>y. y \<in> S \<and> f y \<noteq> f x \<longrightarrow> e \<le> norm (f y - f x)"
- obtains a where "\<And>x. x \<in> S \<Longrightarrow> f x = a"
- using continuous_discrete_range_constant_eq [THEN iffD1, OF S] assms
- by blast
-
-lemma continuous_finite_range_constant:
- fixes f :: "'a::topological_space \<Rightarrow> 'b::real_normed_algebra_1"
- assumes "connected S"
- and "continuous_on S f"
- and "finite (f ` S)"
- obtains a where "\<And>x. x \<in> S \<Longrightarrow> f x = a"
- using assms continuous_finite_range_constant_eq
- by blast
-
-
-
-subsection \<open>Continuous Extension\<close>
-
-definition clamp :: "'a::euclidean_space \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a" where
- "clamp a b x = (if (\<forall>i\<in>Basis. a \<bullet> i \<le> b \<bullet> i)
- then (\<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)
- else a)"
-
-lemma clamp_in_interval[simp]:
- 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 clamp_cancel_cbox[simp]:
- fixes x a b :: "'a::euclidean_space"
- assumes x: "x \<in> cbox a b"
- shows "clamp a b x = x"
- using assms
- by (auto simp: clamp_def mem_box intro!: euclidean_eqI[where 'a='a])
-
-lemma clamp_empty_interval:
- assumes "i \<in> Basis" "a \<bullet> i > b \<bullet> i"
- shows "clamp a b = (\<lambda>_. a)"
- using assms
- by (force simp: clamp_def[abs_def] split: if_splits intro!: ext)
-
-lemma dist_clamps_le_dist_args:
- fixes x :: "'a::euclidean_space"
- shows "dist (clamp a b y) (clamp a b x) \<le> dist y x"
-proof cases
- assume le: "(\<forall>i\<in>Basis. a \<bullet> i \<le> b \<bullet> i)"
- 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!: sum_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 (auto simp: clamp_def)
-
-lemma clamp_continuous_at:
- fixes f :: "'a::euclidean_space \<Rightarrow> 'b::metric_space"
- and x :: 'a
- assumes f_cont: "continuous_on (cbox a b) f"
- shows "continuous (at x) (\<lambda>x. f (clamp a b x))"
-proof cases
- assume le: "(\<forall>i\<in>Basis. a \<bullet> i \<le> b \<bullet> i)"
- show ?thesis
- 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 le)
- 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"
- using le
- by (auto intro!: d clamp_in_interval dist_clamps_le_dist_args[THEN le_less_trans])
- qed
-qed (auto simp: clamp_empty_interval)
-
-lemma clamp_continuous_on:
- fixes f :: "'a::euclidean_space \<Rightarrow> 'b::metric_space"
- assumes f_cont: "continuous_on (cbox a b) f"
- shows "continuous_on S (\<lambda>x. f (clamp a b x))"
- using assms
- by (auto intro: continuous_at_imp_continuous_on clamp_continuous_at)
-
-lemma clamp_bounded:
- fixes f :: "'a::euclidean_space \<Rightarrow> 'b::metric_space"
- assumes bounded: "bounded (f ` (cbox a b))"
- shows "bounded (range (\<lambda>x. f (clamp a b x)))"
-proof cases
- assume le: "(\<forall>i\<in>Basis. a \<bullet> i \<le> b \<bullet> i)"
- from bounded obtain c where f_bound: "\<forall>x\<in>f ` cbox a b. dist undefined x \<le> c"
- by (auto simp: bounded_any_center[where a=undefined])
- then show ?thesis
- by (auto intro!: exI[where x=c] clamp_in_interval[OF le[rule_format]]
- simp: bounded_any_center[where a=undefined])
-qed (auto simp: clamp_empty_interval image_def)
-
-
-definition ext_cont :: "('a::euclidean_space \<Rightarrow> 'b::metric_space) \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> 'b"
- where "ext_cont f a b = (\<lambda>x. f (clamp a b x))"
-
-lemma ext_cont_cancel_cbox[simp]:
- fixes x a b :: "'a::euclidean_space"
- assumes x: "x \<in> cbox a b"
- shows "ext_cont f a b x = f x"
- using assms
- unfolding ext_cont_def
- by (auto simp: clamp_def mem_box intro!: euclidean_eqI[where 'a='a] arg_cong[where f=f])
-
-lemma continuous_on_ext_cont[continuous_intros]:
- "continuous_on (cbox a b) f \<Longrightarrow> continuous_on S (ext_cont f a b)"
- by (auto intro!: clamp_continuous_on simp: ext_cont_def)
-
no_notation
eucl_less (infix "<e" 50)