Lots of new material about matrices, etc.
(* Author: L C Paulson, University of Cambridge
Material split off from Topology_Euclidean_Space
*)
section \<open>Connected Components, Homeomorphisms, Baire property, etc.\<close>
theory Connected
imports Topology_Euclidean_Space
begin
subsection \<open>More properties of closed balls, spheres, etc.\<close>
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
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
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
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)
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
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
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
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 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 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: closedin_closed)
qed
lemma closedin_component:
"C \<in> components s \<Longrightarrow> closedin (subtopology euclidean s) C"
using closedin_connected_component componentsE by blast
subsection \<open>Intersecting chains of compact sets and the Baire property\<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
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)
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
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"
by (meson assms(3) contra_subsetD max.cobounded1 t)
then have "\<exists>y\<in>s n. dist y l < e"
using N max.cobounded2 by blast
}
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>Infimum Distance\<close>
definition "infdist x A = (if A = {} then 0 else INF a:A. dist x a)"
lemma bdd_below_image_dist[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 infdist_pos_not_in_closed:
assumes "closed S" "S \<noteq> {}" "x \<notin> S"
shows "infdist x S > 0"
using in_closed_iff_infdist_zero[OF assms(1) assms(2), of x] assms(3) infdist_nonneg le_less by fastforce
text \<open>Every metric space is a T4 space:\<close>
instance metric_space \<subseteq> t4_space
proof
fix S T::"'a set" assume H: "closed S" "closed T" "S \<inter> T = {}"
consider "S = {}" | "T = {}" | "S \<noteq> {} \<and> T \<noteq> {}" by auto
then show "\<exists>U V. open U \<and> open V \<and> S \<subseteq> U \<and> T \<subseteq> V \<and> U \<inter> V = {}"
proof (cases)
case 1
show ?thesis
apply (rule exI[of _ "{}"], rule exI[of _ UNIV]) using 1 by auto
next
case 2
show ?thesis
apply (rule exI[of _ UNIV], rule exI[of _ "{}"]) using 2 by auto
next
case 3
define U where "U = (\<Union>x\<in>S. ball x ((infdist x T)/2))"
have A: "open U" unfolding U_def by auto
have "infdist x T > 0" if "x \<in> S" for x
using H that 3 by (auto intro!: infdist_pos_not_in_closed)
then have B: "S \<subseteq> U" unfolding U_def by auto
define V where "V = (\<Union>x\<in>T. ball x ((infdist x S)/2))"
have C: "open V" unfolding V_def by auto
have "infdist x S > 0" if "x \<in> T" for x
using H that 3 by (auto intro!: infdist_pos_not_in_closed)
then have D: "T \<subseteq> V" unfolding V_def by auto
have "(ball x ((infdist x T)/2)) \<inter> (ball y ((infdist y S)/2)) = {}" if "x \<in> S" "y \<in> T" for x y
proof (auto)
fix z assume H: "dist x z * 2 < infdist x T" "dist y z * 2 < infdist y S"
have "2 * dist x y \<le> 2 * dist x z + 2 * dist y z"
using dist_triangle[of x y z] by (auto simp add: dist_commute)
also have "... < infdist x T + infdist y S"
using H by auto
finally have "dist x y < infdist x T \<or> dist x y < infdist y S"
by auto
then show False
using infdist_le[OF \<open>x \<in> S\<close>, of y] infdist_le[OF \<open>y \<in> T\<close>, of x] by (auto simp add: dist_commute)
qed
then have E: "U \<inter> V = {}"
unfolding U_def V_def by auto
show ?thesis
apply (rule exI[of _ U], rule exI[of _ V]) using A B C D E by auto
qed
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
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)
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 (simp add: vimage_def Collect_conj_eq)
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 (simp add: vimage_def Collect_conj_eq)
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 contf: "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 (closure S \<inter> f -` T)"
using continuous_closed_preimage[OF contf] \<open>closed T\<close> by auto
ultimately have "closure S = (closure S \<inter> f -` T)"
using closure_minimal[of S "(closure S \<inter> f -` 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
by (meson "*" assms(1) assms(3) closed_cball image_closure_subset image_subset_iff mem_cball_0)
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>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
subsection\<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>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
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 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(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>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 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
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) (S \<inter> f -` U) \<longleftrightarrow>
openin (subtopology euclidean T) U)"
shows "continuous_on S f"
proof -
have [simp]: "S \<inter> f -` f ` S = S" by auto
show ?thesis
using ope [OF T]
apply (simp add: continuous_on_open)
by (meson 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) (S \<inter> f -` U) \<longleftrightarrow>
closedin (subtopology euclidean T) U)"
shows "continuous_on S f"
proof -
have [simp]: "S \<inter> f -` 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_trans)
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) (S \<inter> f -` T) =
openin (subtopology euclidean (f ` S)) T"
proof -
have "T = f ` (S \<inter> f -` T)"
using T by blast
then show ?thesis
using "ope" contf continuous_on_open by metis
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) (S \<inter> f -` T) \<longleftrightarrow>
openin (subtopology euclidean (f ` S)) T"
(is "?lhs = ?rhs")
proof
assume ?lhs
then have *: "closedin (subtopology euclidean S) (S - (S \<inter> f -` T))"
using closedin_diff by fastforce
have [simp]: "(f ` S - f ` (S - (S \<inter> f -` 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) (S \<inter> f -` 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) (S \<inter> f -` Z)"
and g: "\<And>Z. openin (subtopology euclidean (g ` T)) Z \<Longrightarrow>
openin (subtopology euclidean T) (T \<inter> g -` Z)"
using contf contg by (auto simp: continuous_on_open)
show ?thesis
proof
have "T \<inter> g -` (g ` T \<inter> (S \<inter> f -` U)) = {x \<in> T. f (g x) \<in> U}"
using imf img by blast
also have "... = U"
using U by auto
finally have eq: "T \<inter> g -` (g ` T \<inter> (S \<inter> f -` U)) = U" .
assume ?lhs
then have *: "openin (subtopology euclidean (g ` T)) (g ` T \<inter> (S \<inter> f -` U))"
by (meson img openin_Int openin_subtopology_Int_subset openin_subtopology_self)
show ?rhs
using g [OF *] eq by auto
next
assume rhs: ?rhs
show ?lhs
by (metis f 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) (S \<inter> f -` U) \<longleftrightarrow>
openin (subtopology euclidean (f ` S)) U"
apply (rule continuous_right_inverse_imp_quotient_map)
using assms apply force+
done
text \<open>Proving a function is constant by proving that a level set is open\<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> ( *\<^sub>R) c ` s"
using rev_image_eqI[of "(1 / c) *\<^sub>R y" s y "( *\<^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> ( *\<^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} = (+) 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 "(+) a ` ( *\<^sub>R) c ` S = ((+) a \<circ> ( *\<^sub>R) c) ` S"
by auto
then show ?thesis
using assms open_translation[of "( *\<^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 ((+) a ` S)"
then obtain e where "e > 0" and e: "ball x e \<subseteq> (+) 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> (+) 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> (+) 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> (+) a ` S"
unfolding image_iff by (auto intro!: bexI[where x="z - a"])
}
then have "ball x e \<subseteq> (+) a ` S"
unfolding subset_eq by auto
then show "x \<in> interior ((+) a ` S)"
unfolding mem_interior using \<open>e > 0\<close> by auto
qed
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 opn: "\<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 opn 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 "(+) a ` ( *\<^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
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 L2_set_mono
simp: euclidean_dist_l2[where 'a='a] cbox_def dist_norm)
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) *] by (force simp: dist_commute)
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}) = ((+) a ` S)" by auto
then show ?thesis
using compact_closed_sums[OF compact_sing[of a] assms] by auto
qed
lemma closure_translation:
fixes a :: "'a::real_normed_vector"
shows "closure ((\<lambda>x. a + x) ` s) = (\<lambda>x. a + x) ` (closure s)"
proof -
have *: "(+) a ` (- s) = - (+) 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 = (+) 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 = (+) 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 = (+) 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>Closure of halfspaces and hyperplanes\<close>
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
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 = (s \<inter> h -` atMost a) \<union> (s \<inter> h -` atLeast a)"
by force
have 1: "closedin (subtopology euclidean s) (s \<inter> h -` atMost a)"
by (rule continuous_closedin_preimage [OF h closed_atMost])
have 2: "closedin (subtopology euclidean s) (s \<inter> h -` atLeast a)"
by (rule continuous_closedin_preimage [OF h closed_atLeast])
have eq: "s \<inter> h -` {..a} = {t \<in> s. h t \<le> a}" "s \<inter> h -` {a..} = {t \<in> s. a \<le> h t}"
by auto
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 simp: eq)
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])
subsubsection\<open>Some more convenient intermediate-value theorem formulations.\<close>
lemma connected_ivt_hyperplane:
assumes "connected S" and xy: "x \<in> S" "y \<in> S" and b: "inner a x \<le> b" "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 \<open>connected S\<close>[unfolded connected_def not_ex,
THEN spec[where x="?A"], THEN spec[where x="?B"]]
using xy b 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)
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"
by (simp add: dual_order.antisym euclidean_eqI)
}
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 ((+) a ` S) S ((+) (- a)) ((+) 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 *: "(+) a ` ( *\<^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) = T \<inter> g -` 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) = T \<inter> g -` 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 = T \<inter> g -` 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 = T \<inter> g -` 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> (\<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>(\<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 opn: "\<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> opn)
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 countable_disjoint_nonempty_interior_subsets:
fixes \<F> :: "'a::euclidean_space set set"
assumes pw: "pairwise disjnt \<F>" and int: "\<And>S. \<lbrakk>S \<in> \<F>; interior S = {}\<rbrakk> \<Longrightarrow> S = {}"
shows "countable \<F>"
proof (rule countable_image_inj_on)
have "disjoint (interior ` \<F>)"
using pw by (simp add: disjoint_image_subset interior_subset)
then show "countable (interior ` \<F>)"
by (auto intro: countable_disjoint_open_subsets)
show "inj_on interior \<F>"
using pw apply (clarsimp simp: inj_on_def pairwise_def)
apply (metis disjnt_def disjnt_subset1 inf.orderE int interior_subset)
done
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" "f ` S = T" "compact S"
shows "closedin (subtopology euclidean T) (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" "f ` S = T" "compact S" "U \<subseteq> T"
shows "openin (subtopology euclidean S) (S \<inter> f -` U) \<longleftrightarrow>
openin (subtopology euclidean T) U"
by (metis (no_types, lifting) assms closed_map_imp_quotient_map continuous_imp_closed_map)
lemma open_map_restrict:
assumes opeU: "openin (subtopology euclidean (S \<inter> f -` T')) U"
and oo: "\<And>U. openin (subtopology euclidean S) U \<Longrightarrow> openin (subtopology euclidean T) (f ` U)"
and "T' \<subseteq> T"
shows "openin (subtopology euclidean T') (f ` U)"
proof -
obtain V where "open V" "U = S \<inter> f -` T' \<inter> V"
using opeU by (auto simp: openin_open)
with oo [of "S \<inter> V"] \<open>T' \<subseteq> T\<close> show ?thesis
by (fastforce simp add: openin_open)
qed
lemma closed_map_restrict:
assumes cloU: "closedin (subtopology euclidean (S \<inter> f -` T')) U"
and cc: "\<And>U. closedin (subtopology euclidean S) U \<Longrightarrow> closedin (subtopology euclidean T) (f ` U)"
and "T' \<subseteq> T"
shows "closedin (subtopology euclidean T') (f ` U)"
proof -
obtain V where "closed V" "U = S \<inter> f -` T' \<inter> V"
using cloU by (auto simp: closedin_closed)
with cc [of "S \<inter> V"] \<open>T' \<subseteq> T\<close> show ?thesis
by (fastforce simp add: closedin_closed)
qed
lemma connected_monotone_quotient_preimage:
assumes "connected T"
and contf: "continuous_on S f" and fim: "f ` S = T"
and opT: "\<And>U. U \<subseteq> T
\<Longrightarrow> openin (subtopology euclidean S) (S \<inter> f -` U) \<longleftrightarrow>
openin (subtopology euclidean T) U"
and connT: "\<And>y. y \<in> T \<Longrightarrow> connected (S \<inter> f -` {y})"
shows "connected S"
proof (rule connectedI)
fix U V
assume "open U" and "open V" and "U \<inter> S \<noteq> {}" and "V \<inter> S \<noteq> {}"
and "U \<inter> V \<inter> S = {}" and "S \<subseteq> U \<union> V"
moreover
have disjoint: "f ` (S \<inter> U) \<inter> f ` (S \<inter> V) = {}"
proof -
have False if "y \<in> f ` (S \<inter> U) \<inter> f ` (S \<inter> V)" for y
proof -
have "y \<in> T"
using fim that by blast
show ?thesis
using connectedD [OF connT [OF \<open>y \<in> T\<close>] \<open>open U\<close> \<open>open V\<close>]
\<open>S \<subseteq> U \<union> V\<close> \<open>U \<inter> V \<inter> S = {}\<close> that by fastforce
qed
then show ?thesis by blast
qed
ultimately have UU: "(S \<inter> f -` f ` (S \<inter> U)) = S \<inter> U" and VV: "(S \<inter> f -` f ` (S \<inter> V)) = S \<inter> V"
by auto
have opeU: "openin (subtopology euclidean T) (f ` (S \<inter> U))"
by (metis UU \<open>open U\<close> fim image_Int_subset le_inf_iff opT openin_open_Int)
have opeV: "openin (subtopology euclidean T) (f ` (S \<inter> V))"
by (metis opT fim VV \<open>open V\<close> openin_open_Int image_Int_subset inf.bounded_iff)
have "T \<subseteq> f ` (S \<inter> U) \<union> f ` (S \<inter> V)"
using \<open>S \<subseteq> U \<union> V\<close> fim by auto
then show False
using \<open>connected T\<close> disjoint opeU opeV \<open>U \<inter> S \<noteq> {}\<close> \<open>V \<inter> S \<noteq> {}\<close>
by (auto simp: connected_openin)
qed
lemma connected_open_monotone_preimage:
assumes contf: "continuous_on S f" and fim: "f ` S = T"
and ST: "\<And>C. openin (subtopology euclidean S) C \<Longrightarrow> openin (subtopology euclidean T) (f ` C)"
and connT: "\<And>y. y \<in> T \<Longrightarrow> connected (S \<inter> f -` {y})"
and "connected C" "C \<subseteq> T"
shows "connected (S \<inter> f -` C)"
proof -
have contf': "continuous_on (S \<inter> f -` C) f"
by (meson contf continuous_on_subset inf_le1)
have eqC: "f ` (S \<inter> f -` C) = C"
using \<open>C \<subseteq> T\<close> fim by blast
show ?thesis
proof (rule connected_monotone_quotient_preimage [OF \<open>connected C\<close> contf' eqC])
show "connected (S \<inter> f -` C \<inter> f -` {y})" if "y \<in> C" for y
proof -
have "S \<inter> f -` C \<inter> f -` {y} = S \<inter> f -` {y}"
using that by blast
moreover have "connected (S \<inter> f -` {y})"
using \<open>C \<subseteq> T\<close> connT that by blast
ultimately show ?thesis
by metis
qed
have "\<And>U. openin (subtopology euclidean (S \<inter> f -` C)) U
\<Longrightarrow> openin (subtopology euclidean C) (f ` U)"
using open_map_restrict [OF _ ST \<open>C \<subseteq> T\<close>] by metis
then show "\<And>D. D \<subseteq> C
\<Longrightarrow> openin (subtopology euclidean (S \<inter> f -` C)) (S \<inter> f -` C \<inter> f -` D) =
openin (subtopology euclidean C) D"
using open_map_imp_quotient_map [of "(S \<inter> f -` C)" f] contf' by (simp add: eqC)
qed
qed
lemma connected_closed_monotone_preimage:
assumes contf: "continuous_on S f" and fim: "f ` S = T"
and ST: "\<And>C. closedin (subtopology euclidean S) C \<Longrightarrow> closedin (subtopology euclidean T) (f ` C)"
and connT: "\<And>y. y \<in> T \<Longrightarrow> connected (S \<inter> f -` {y})"
and "connected C" "C \<subseteq> T"
shows "connected (S \<inter> f -` C)"
proof -
have contf': "continuous_on (S \<inter> f -` C) f"
by (meson contf continuous_on_subset inf_le1)
have eqC: "f ` (S \<inter> f -` C) = C"
using \<open>C \<subseteq> T\<close> fim by blast
show ?thesis
proof (rule connected_monotone_quotient_preimage [OF \<open>connected C\<close> contf' eqC])
show "connected (S \<inter> f -` C \<inter> f -` {y})" if "y \<in> C" for y
proof -
have "S \<inter> f -` C \<inter> f -` {y} = S \<inter> f -` {y}"
using that by blast
moreover have "connected (S \<inter> f -` {y})"
using \<open>C \<subseteq> T\<close> connT that by blast
ultimately show ?thesis
by metis
qed
have "\<And>U. closedin (subtopology euclidean (S \<inter> f -` C)) U
\<Longrightarrow> closedin (subtopology euclidean C) (f ` U)"
using closed_map_restrict [OF _ ST \<open>C \<subseteq> T\<close>] by metis
then show "\<And>D. D \<subseteq> C
\<Longrightarrow> openin (subtopology euclidean (S \<inter> f -` C)) (S \<inter> f -` C \<inter> f -` D) =
openin (subtopology euclidean C) D"
using closed_map_imp_quotient_map [of "(S \<inter> f -` C)" f] contf' by (simp add: eqC)
qed
qed
subsection\<open>A couple of lemmas about components (see Newman IV, 3.3 and 3.4).\<close>
lemma connected_Un_clopen_in_complement:
fixes S U :: "'a::metric_space set"
assumes "connected S" "connected U" "S \<subseteq> U"
and opeT: "openin (subtopology euclidean (U - S)) T"
and cloT: "closedin (subtopology euclidean (U - S)) T"
shows "connected (S \<union> T)"
proof -
have *: "\<lbrakk>\<And>x y. P x y \<longleftrightarrow> P y x; \<And>x y. P x y \<Longrightarrow> S \<subseteq> x \<or> S \<subseteq> y;
\<And>x y. \<lbrakk>P x y; S \<subseteq> x\<rbrakk> \<Longrightarrow> False\<rbrakk> \<Longrightarrow> ~(\<exists>x y. (P x y))" for P
by metis
show ?thesis
unfolding connected_closedin_eq
proof (rule *)
fix H1 H2
assume H: "closedin (subtopology euclidean (S \<union> T)) H1 \<and>
closedin (subtopology euclidean (S \<union> T)) H2 \<and>
H1 \<union> H2 = S \<union> T \<and> H1 \<inter> H2 = {} \<and> H1 \<noteq> {} \<and> H2 \<noteq> {}"
then have clo: "closedin (subtopology euclidean S) (S \<inter> H1)"
"closedin (subtopology euclidean S) (S \<inter> H2)"
by (metis Un_upper1 closedin_closed_subset inf_commute)+
have Seq: "S \<inter> (H1 \<union> H2) = S"
by (simp add: H)
have "S \<inter> ((S \<union> T) \<inter> H1) \<union> S \<inter> ((S \<union> T) \<inter> H2) = S"
using Seq by auto
moreover have "H1 \<inter> (S \<inter> ((S \<union> T) \<inter> H2)) = {}"
using H by blast
ultimately have "S \<inter> H1 = {} \<or> S \<inter> H2 = {}"
by (metis (no_types) H Int_assoc \<open>S \<inter> (H1 \<union> H2) = S\<close> \<open>connected S\<close>
clo Seq connected_closedin inf_bot_right inf_le1)
then show "S \<subseteq> H1 \<or> S \<subseteq> H2"
using H \<open>connected S\<close> unfolding connected_closedin by blast
next
fix H1 H2
assume H: "closedin (subtopology euclidean (S \<union> T)) H1 \<and>
closedin (subtopology euclidean (S \<union> T)) H2 \<and>
H1 \<union> H2 = S \<union> T \<and> H1 \<inter> H2 = {} \<and> H1 \<noteq> {} \<and> H2 \<noteq> {}"
and "S \<subseteq> H1"
then have H2T: "H2 \<subseteq> T"
by auto
have "T \<subseteq> U"
using Diff_iff opeT openin_imp_subset by auto
with \<open>S \<subseteq> U\<close> have Ueq: "U = (U - S) \<union> (S \<union> T)"
by auto
have "openin (subtopology euclidean ((U - S) \<union> (S \<union> T))) H2"
proof (rule openin_subtopology_Un)
show "openin (subtopology euclidean (S \<union> T)) H2"
using \<open>H2 \<subseteq> T\<close> apply (auto simp: openin_closedin_eq)
by (metis Diff_Diff_Int Diff_disjoint Diff_partition Diff_subset H Int_absorb1 Un_Diff)
then show "openin (subtopology euclidean (U - S)) H2"
by (meson H2T Un_upper2 opeT openin_subset_trans openin_trans)
qed
moreover have "closedin (subtopology euclidean ((U - S) \<union> (S \<union> T))) H2"
proof (rule closedin_subtopology_Un)
show "closedin (subtopology euclidean (U - S)) H2"
using H H2T cloT closedin_subset_trans
by (blast intro: closedin_subtopology_Un closedin_trans)
qed (simp add: H)
ultimately
have H2: "H2 = {} \<or> H2 = U"
using Ueq \<open>connected U\<close> unfolding connected_clopen by metis
then have "H2 \<subseteq> S"
by (metis Diff_partition H Un_Diff_cancel Un_subset_iff \<open>H2 \<subseteq> T\<close> assms(3) inf.orderE opeT openin_imp_subset)
moreover have "T \<subseteq> H2 - S"
by (metis (no_types) H2 H opeT openin_closedin_eq topspace_euclidean_subtopology)
ultimately show False
using H \<open>S \<subseteq> H1\<close> by blast
qed blast
qed
proposition component_diff_connected:
fixes S :: "'a::metric_space set"
assumes "connected S" "connected U" "S \<subseteq> U" and C: "C \<in> components (U - S)"
shows "connected(U - C)"
using \<open>connected S\<close> unfolding connected_closedin_eq not_ex de_Morgan_conj
proof clarify
fix H3 H4
assume clo3: "closedin (subtopology euclidean (U - C)) H3"
and clo4: "closedin (subtopology euclidean (U - C)) H4"
and "H3 \<union> H4 = U - C" and "H3 \<inter> H4 = {}" and "H3 \<noteq> {}" and "H4 \<noteq> {}"
and * [rule_format]:
"\<forall>H1 H2. \<not> closedin (subtopology euclidean S) H1 \<or>
\<not> closedin (subtopology euclidean S) H2 \<or>
H1 \<union> H2 \<noteq> S \<or> H1 \<inter> H2 \<noteq> {} \<or> \<not> H1 \<noteq> {} \<or> \<not> H2 \<noteq> {}"
then have "H3 \<subseteq> U-C" and ope3: "openin (subtopology euclidean (U - C)) (U - C - H3)"
and "H4 \<subseteq> U-C" and ope4: "openin (subtopology euclidean (U - C)) (U - C - H4)"
by (auto simp: closedin_def)
have "C \<noteq> {}" "C \<subseteq> U-S" "connected C"
using C in_components_nonempty in_components_subset in_components_maximal by blast+
have cCH3: "connected (C \<union> H3)"
proof (rule connected_Un_clopen_in_complement [OF \<open>connected C\<close> \<open>connected U\<close> _ _ clo3])
show "openin (subtopology euclidean (U - C)) H3"
apply (simp add: openin_closedin_eq \<open>H3 \<subseteq> U - C\<close>)
apply (simp add: closedin_subtopology)
by (metis Diff_cancel Diff_triv Un_Diff clo4 \<open>H3 \<inter> H4 = {}\<close> \<open>H3 \<union> H4 = U - C\<close> closedin_closed inf_commute sup_bot.left_neutral)
qed (use clo3 \<open>C \<subseteq> U - S\<close> in auto)
have cCH4: "connected (C \<union> H4)"
proof (rule connected_Un_clopen_in_complement [OF \<open>connected C\<close> \<open>connected U\<close> _ _ clo4])
show "openin (subtopology euclidean (U - C)) H4"
apply (simp add: openin_closedin_eq \<open>H4 \<subseteq> U - C\<close>)
apply (simp add: closedin_subtopology)
by (metis Diff_cancel Int_commute Un_Diff Un_Diff_Int \<open>H3 \<inter> H4 = {}\<close> \<open>H3 \<union> H4 = U - C\<close> clo3 closedin_closed)
qed (use clo4 \<open>C \<subseteq> U - S\<close> in auto)
have "closedin (subtopology euclidean S) (S \<inter> H3)" "closedin (subtopology euclidean S) (S \<inter> H4)"
using clo3 clo4 \<open>S \<subseteq> U\<close> \<open>C \<subseteq> U - S\<close> by (auto simp: closedin_closed)
moreover have "S \<inter> H3 \<noteq> {}"
using components_maximal [OF C cCH3] \<open>C \<noteq> {}\<close> \<open>C \<subseteq> U - S\<close> \<open>H3 \<noteq> {}\<close> \<open>H3 \<subseteq> U - C\<close> by auto
moreover have "S \<inter> H4 \<noteq> {}"
using components_maximal [OF C cCH4] \<open>C \<noteq> {}\<close> \<open>C \<subseteq> U - S\<close> \<open>H4 \<noteq> {}\<close> \<open>H4 \<subseteq> U - C\<close> by auto
ultimately show False
using * [of "S \<inter> H3" "S \<inter> H4"] \<open>H3 \<inter> H4 = {}\<close> \<open>C \<subseteq> U - S\<close> \<open>H3 \<union> H4 = U - C\<close> \<open>S \<subseteq> U\<close>
by auto
qed
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_L2_set 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 *: "L2_set (\<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 "L2_set (\<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: L2_set_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 "L2_set (\<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 *: "(S \<inter> g -` U) = (\<Union>i \<in> I. T i \<inter> f i -` U)"
using S f g by fastforce
show "openin (subtopology euclidean S) (S \<inter> g -` U)"
apply (subst *)
apply (rule openin_Union, clarify)
using \<open>open U\<close> clo cont continuous_openin_preimage_gen openin_trans by blast
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 S: "\<And>i. i \<in> I \<Longrightarrow> (T i) \<subseteq> S"
using clo closedin_imp_subset by blast
have *: "(S \<inter> g -` U) = (\<Union>i \<in> I. T i \<inter> f i -` U)"
using S f g by fastforce
show "closedin (subtopology euclidean S) (S \<inter> g -` 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 "f constant_on S"
proof (cases "S = {}")
case True then show ?thesis
by (simp add: constant_on_def)
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
unfolding constant_on_def 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> f constant_on S"
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 constant_on_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> f constant_on S))" (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> f constant_on S))" (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> f constant_on S))" (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)"
shows "f constant_on S"
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)"
shows "f constant_on S"
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"] L2_set_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)
end