src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy
 author haftmann Sat, 25 May 2013 15:44:29 +0200 changeset 52141 eff000cab70f parent 51773 9328c6681f3c child 52624 8a7b59a81088 permissions -rw-r--r--
weaker precendence of syntax for big intersection and union on sets

(*  title:      HOL/Library/Topology_Euclidian_Space.thy
Author:     Amine Chaieb, University of Cambridge
Author:     Robert Himmelmann, TU Muenchen
Author:     Brian Huffman, Portland State University
*)

header {* Elementary topology in Euclidean space. *}

theory Topology_Euclidean_Space
imports
Complex_Main
"~~/src/HOL/Library/Countable_Set"
"~~/src/HOL/Library/Glbs"
"~~/src/HOL/Library/FuncSet"
Linear_Algebra
Norm_Arith
begin

lemma dist_0_norm:
fixes x :: "'a::real_normed_vector"
shows "dist 0 x = norm x"
unfolding dist_norm by simp

lemma dist_double: "dist x y < d / 2 \<Longrightarrow> dist x z < d / 2 \<Longrightarrow> dist y z < d"
using dist_triangle[of y z x] by (simp add: dist_commute)

(* LEGACY *)
lemma lim_subseq: "subseq r \<Longrightarrow> s ----> l \<Longrightarrow> (s o r) ----> l"
by (rule LIMSEQ_subseq_LIMSEQ)

lemmas real_isGlb_unique = isGlb_unique[where 'a=real]

lemma countable_PiE:
"finite I \<Longrightarrow> (\<And>i. i \<in> I \<Longrightarrow> countable (F i)) \<Longrightarrow> countable (PiE I F)"
by (induct I arbitrary: F rule: finite_induct) (auto simp: PiE_insert_eq)

lemma Lim_within_open:
fixes f :: "'a::topological_space \<Rightarrow> 'b::topological_space"
shows "a \<in> S \<Longrightarrow> open S \<Longrightarrow> (f ---> l)(at a within S) \<longleftrightarrow> (f ---> l)(at a)"
by (fact tendsto_within_open)

lemma continuous_on_union:
"closed s \<Longrightarrow> closed t \<Longrightarrow> continuous_on s f \<Longrightarrow> continuous_on t f \<Longrightarrow> continuous_on (s \<union> t) f"
by (fact continuous_on_closed_Un)

lemma continuous_on_cases:
"closed s \<Longrightarrow> closed t \<Longrightarrow> continuous_on s f \<Longrightarrow> continuous_on t g \<Longrightarrow>
\<forall>x. (x\<in>s \<and> \<not> P x) \<or> (x \<in> t \<and> P x) \<longrightarrow> f x = g x \<Longrightarrow>
continuous_on (s \<union> t) (\<lambda>x. if P x then f x else g x)"
by (rule continuous_on_If) auto

subsection {* Topological Basis *}

context topological_space
begin

definition "topological_basis B =
((\<forall>b\<in>B. open b) \<and> (\<forall>x. open x \<longrightarrow> (\<exists>B'. B' \<subseteq> B \<and> \<Union>B' = x)))"

lemma topological_basis:
"topological_basis B = (\<forall>x. open x \<longleftrightarrow> (\<exists>B'. B' \<subseteq> B \<and> \<Union>B' = x))"
unfolding topological_basis_def
apply safe
apply fastforce
apply fastforce
apply (erule_tac x="x" in allE)
apply simp
apply (rule_tac x="{x}" in exI)
apply auto
done

lemma topological_basis_iff:
assumes "\<And>B'. B' \<in> B \<Longrightarrow> open B'"
shows "topological_basis B \<longleftrightarrow> (\<forall>O'. open O' \<longrightarrow> (\<forall>x\<in>O'. \<exists>B'\<in>B. x \<in> B' \<and> B' \<subseteq> O'))"
(is "_ \<longleftrightarrow> ?rhs")
proof safe
fix O' and x::'a
assume H: "topological_basis B" "open O'" "x \<in> O'"
hence "(\<exists>B'\<subseteq>B. \<Union>B' = O')" by (simp add: topological_basis_def)
then obtain B' where "B' \<subseteq> B" "O' = \<Union>B'" by auto
thus "\<exists>B'\<in>B. x \<in> B' \<and> B' \<subseteq> O'" using H by auto
next
assume H: ?rhs
show "topological_basis B" using assms unfolding topological_basis_def
proof safe
fix O'::"'a set" assume "open O'"
with H obtain f where "\<forall>x\<in>O'. f x \<in> B \<and> x \<in> f x \<and> f x \<subseteq> O'"
by (force intro: bchoice simp: Bex_def)
thus "\<exists>B'\<subseteq>B. \<Union>B' = O'"
by (auto intro: exI[where x="{f x |x. x \<in> O'}"])
qed
qed

lemma topological_basisI:
assumes "\<And>B'. B' \<in> B \<Longrightarrow> open B'"
assumes "\<And>O' x. open O' \<Longrightarrow> x \<in> O' \<Longrightarrow> \<exists>B'\<in>B. x \<in> B' \<and> B' \<subseteq> O'"
shows "topological_basis B"
using assms by (subst topological_basis_iff) auto

lemma topological_basisE:
fixes O'
assumes "topological_basis B"
assumes "open O'"
assumes "x \<in> O'"
obtains B' where "B' \<in> B" "x \<in> B'" "B' \<subseteq> O'"
proof atomize_elim
from assms have "\<And>B'. B'\<in>B \<Longrightarrow> open B'" by (simp add: topological_basis_def)
with topological_basis_iff assms
show  "\<exists>B'. B' \<in> B \<and> x \<in> B' \<and> B' \<subseteq> O'" using assms by (simp add: Bex_def)
qed

lemma topological_basis_open:
assumes "topological_basis B"
assumes "X \<in> B"
shows "open X"
using assms

lemma topological_basis_imp_subbasis:
assumes B: "topological_basis B" shows "open = generate_topology B"
proof (intro ext iffI)
fix S :: "'a set" assume "open S"
with B obtain B' where "B' \<subseteq> B" "S = \<Union>B'"
unfolding topological_basis_def by blast
then show "generate_topology B S"
by (auto intro: generate_topology.intros dest: topological_basis_open)
next
fix S :: "'a set" assume "generate_topology B S" then show "open S"
by induct (auto dest: topological_basis_open[OF B])
qed

lemma basis_dense:
fixes B::"'a set set" and f::"'a set \<Rightarrow> 'a"
assumes "topological_basis B"
assumes choosefrom_basis: "\<And>B'. B' \<noteq> {} \<Longrightarrow> f B' \<in> B'"
shows "(\<forall>X. open X \<longrightarrow> X \<noteq> {} \<longrightarrow> (\<exists>B' \<in> B. f B' \<in> X))"
proof (intro allI impI)
fix X::"'a set" assume "open X" "X \<noteq> {}"
from topological_basisE[OF topological_basis B open X choosefrom_basis[OF X \<noteq> {}]]
guess B' . note B' = this
thus "\<exists>B'\<in>B. f B' \<in> X" by (auto intro!: choosefrom_basis)
qed

end

lemma topological_basis_prod:
assumes A: "topological_basis A" and B: "topological_basis B"
shows "topological_basis ((\<lambda>(a, b). a \<times> b)  (A \<times> B))"
unfolding topological_basis_def
proof (safe, simp_all del: ex_simps add: subset_image_iff ex_simps(1)[symmetric])
fix S :: "('a \<times> 'b) set" assume "open S"
then show "\<exists>X\<subseteq>A \<times> B. (\<Union>(a,b)\<in>X. a \<times> b) = S"
proof (safe intro!: exI[of _ "{x\<in>A \<times> B. fst x \<times> snd x \<subseteq> S}"])
fix x y assume "(x, y) \<in> S"
from open_prod_elim[OF open S this]
obtain a b where a: "open a""x \<in> a" and b: "open b" "y \<in> b" and "a \<times> b \<subseteq> S"
by (metis mem_Sigma_iff)
moreover from topological_basisE[OF A a] guess A0 .
moreover from topological_basisE[OF B b] guess B0 .
ultimately show "(x, y) \<in> (\<Union>(a, b)\<in>{X \<in> A \<times> B. fst X \<times> snd X \<subseteq> S}. a \<times> b)"
by (intro UN_I[of "(A0, B0)"]) auto
qed auto
qed (metis A B topological_basis_open open_Times)

subsection {* Countable Basis *}

locale countable_basis =
fixes B::"'a::topological_space set set"
assumes is_basis: "topological_basis B"
assumes countable_basis: "countable B"
begin

lemma open_countable_basis_ex:
assumes "open X"
shows "\<exists>B' \<subseteq> B. X = Union B'"
using assms countable_basis is_basis unfolding topological_basis_def by blast

lemma open_countable_basisE:
assumes "open X"
obtains B' where "B' \<subseteq> B" "X = Union B'"
using assms open_countable_basis_ex by (atomize_elim) simp

lemma countable_dense_exists:
shows "\<exists>D::'a set. countable D \<and> (\<forall>X. open X \<longrightarrow> X \<noteq> {} \<longrightarrow> (\<exists>d \<in> D. d \<in> X))"
proof -
let ?f = "(\<lambda>B'. SOME x. x \<in> B')"
have "countable (?f  B)" using countable_basis by simp
with basis_dense[OF is_basis, of ?f] show ?thesis
by (intro exI[where x="?f  B"]) (metis (mono_tags) all_not_in_conv imageI someI)
qed

lemma countable_dense_setE:
obtains D :: "'a set"
where "countable D" "\<And>X. open X \<Longrightarrow> X \<noteq> {} \<Longrightarrow> \<exists>d \<in> D. d \<in> X"
using countable_dense_exists by blast

end

lemma (in first_countable_topology) first_countable_basisE:
obtains A where "countable A" "\<And>a. a \<in> A \<Longrightarrow> x \<in> a" "\<And>a. a \<in> A \<Longrightarrow> open a"
"\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> (\<exists>a\<in>A. a \<subseteq> S)"
using first_countable_basis[of x]
apply atomize_elim
apply (elim exE)
apply (rule_tac x="range A" in exI)
apply auto
done

lemma (in first_countable_topology) first_countable_basis_Int_stableE:
obtains A where "countable A" "\<And>a. a \<in> A \<Longrightarrow> x \<in> a" "\<And>a. a \<in> A \<Longrightarrow> open a"
"\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> (\<exists>a\<in>A. a \<subseteq> S)"
"\<And>a b. a \<in> A \<Longrightarrow> b \<in> A \<Longrightarrow> a \<inter> b \<in> A"
proof atomize_elim
from first_countable_basisE[of x] guess A' . note A' = this
def A \<equiv> "(\<lambda>N. \<Inter>((\<lambda>n. from_nat_into A' n)  N))  (Collect finite::nat set set)"
thus "\<exists>A. countable A \<and> (\<forall>a. a \<in> A \<longrightarrow> x \<in> a) \<and> (\<forall>a. a \<in> A \<longrightarrow> open a) \<and>
(\<forall>S. open S \<longrightarrow> x \<in> S \<longrightarrow> (\<exists>a\<in>A. a \<subseteq> S)) \<and> (\<forall>a b. a \<in> A \<longrightarrow> b \<in> A \<longrightarrow> a \<inter> b \<in> A)"
proof (safe intro!: exI[where x=A])
show "countable A" unfolding A_def by (intro countable_image countable_Collect_finite)
fix a assume "a \<in> A"
thus "x \<in> a" "open a" using A'(4)[OF open_UNIV] by (auto simp: A_def intro: A' from_nat_into)
next
let ?int = "\<lambda>N. \<Inter>(from_nat_into A'  N)"
fix a b assume "a \<in> A" "b \<in> A"
then obtain N M where "a = ?int N" "b = ?int M" "finite (N \<union> M)" by (auto simp: A_def)
thus "a \<inter> b \<in> A" by (auto simp: A_def intro!: image_eqI[where x="N \<union> M"])
next
fix S assume "open S" "x \<in> S" then obtain a where a: "a\<in>A'" "a \<subseteq> S" using A' by blast
thus "\<exists>a\<in>A. a \<subseteq> S" using a A'
by (intro bexI[where x=a]) (auto simp: A_def intro: image_eqI[where x="{to_nat_on A' a}"])
qed
qed

lemma (in topological_space) first_countableI:
assumes "countable A" and 1: "\<And>a. a \<in> A \<Longrightarrow> x \<in> a" "\<And>a. a \<in> A \<Longrightarrow> open a"
and 2: "\<And>S. open S \<Longrightarrow> x \<in> S \<Longrightarrow> \<exists>a\<in>A. a \<subseteq> S"
shows "\<exists>A::nat \<Rightarrow> 'a set. (\<forall>i. x \<in> A i \<and> open (A i)) \<and> (\<forall>S. open S \<and> x \<in> S \<longrightarrow> (\<exists>i. A i \<subseteq> S))"
proof (safe intro!: exI[of _ "from_nat_into A"])
have "A \<noteq> {}" using 2[of UNIV] by auto
{ fix i show "x \<in> from_nat_into A i" "open (from_nat_into A i)"
using range_from_nat_into_subset[OF A \<noteq> {}] 1 by auto }
{ fix S assume "open S" "x\<in>S" from 2[OF this] show "\<exists>i. from_nat_into A i \<subseteq> S"
using subset_range_from_nat_into[OF countable A] by auto }
qed

instance prod :: (first_countable_topology, first_countable_topology) first_countable_topology
proof
fix x :: "'a \<times> 'b"
from first_countable_basisE[of "fst x"] guess A :: "'a set set" . note A = this
from first_countable_basisE[of "snd x"] guess B :: "'b set set" . note B = this
show "\<exists>A::nat \<Rightarrow> ('a \<times> 'b) set. (\<forall>i. x \<in> A i \<and> open (A i)) \<and> (\<forall>S. open S \<and> x \<in> S \<longrightarrow> (\<exists>i. A i \<subseteq> S))"
proof (rule first_countableI[of "(\<lambda>(a, b). a \<times> b)  (A \<times> B)"], safe)
fix a b assume x: "a \<in> A" "b \<in> B"
with A(2, 3)[of a] B(2, 3)[of b] show "x \<in> a \<times> b" "open (a \<times> b)"
unfolding mem_Times_iff by (auto intro: open_Times)
next
fix S assume "open S" "x \<in> S"
from open_prod_elim[OF this] guess a' b' .
moreover with A(4)[of a'] B(4)[of b']
obtain a b where "a \<in> A" "a \<subseteq> a'" "b \<in> B" "b \<subseteq> b'" by auto
ultimately show "\<exists>a\<in>(\<lambda>(a, b). a \<times> b)  (A \<times> B). a \<subseteq> S"
by (auto intro!: bexI[of _ "a \<times> b"] bexI[of _ a] bexI[of _ b])
qed

class second_countable_topology = topological_space +
assumes ex_countable_subbasis: "\<exists>B::'a::topological_space set set. countable B \<and> open = generate_topology B"
begin

lemma ex_countable_basis: "\<exists>B::'a set set. countable B \<and> topological_basis B"
proof -
from ex_countable_subbasis obtain B where B: "countable B" "open = generate_topology B" by blast
let ?B = "Inter  {b. finite b \<and> b \<subseteq> B }"

show ?thesis
proof (intro exI conjI)
show "countable ?B"
by (intro countable_image countable_Collect_finite_subset B)
{ fix S assume "open S"
then have "\<exists>B'\<subseteq>{b. finite b \<and> b \<subseteq> B}. (\<Union>b\<in>B'. \<Inter>b) = S"
unfolding B
proof induct
case UNIV show ?case by (intro exI[of _ "{{}}"]) simp
next
case (Int a b)
then obtain x y where x: "a = UNION x Inter" "\<And>i. i \<in> x \<Longrightarrow> finite i \<and> i \<subseteq> B"
and y: "b = UNION y Inter" "\<And>i. i \<in> y \<Longrightarrow> finite i \<and> i \<subseteq> B"
by blast
show ?case
unfolding x y Int_UN_distrib2
by (intro exI[of _ "{i \<union> j| i j.  i \<in> x \<and> j \<in> y}"]) (auto dest: x(2) y(2))
next
case (UN K)
then have "\<forall>k\<in>K. \<exists>B'\<subseteq>{b. finite b \<and> b \<subseteq> B}. UNION B' Inter = k" by auto
then guess k unfolding bchoice_iff ..
then show "\<exists>B'\<subseteq>{b. finite b \<and> b \<subseteq> B}. UNION B' Inter = \<Union>K"
by (intro exI[of _ "UNION K k"]) auto
next
case (Basis S) then show ?case
by (intro exI[of _ "{{S}}"]) auto
qed
then have "(\<exists>B'\<subseteq>Inter  {b. finite b \<and> b \<subseteq> B}. \<Union>B' = S)"
unfolding subset_image_iff by blast }
then show "topological_basis ?B"
unfolding topological_space_class.topological_basis_def
by (safe intro!: topological_space_class.open_Inter)
qed
qed

end

sublocale second_countable_topology <
countable_basis "SOME B. countable B \<and> topological_basis B"
using someI_ex[OF ex_countable_basis]
by unfold_locales safe

instance prod :: (second_countable_topology, second_countable_topology) second_countable_topology
proof
obtain A :: "'a set set" where "countable A" "topological_basis A"
using ex_countable_basis by auto
moreover
obtain B :: "'b set set" where "countable B" "topological_basis B"
using ex_countable_basis by auto
ultimately show "\<exists>B::('a \<times> 'b) set set. countable B \<and> open = generate_topology B"
by (auto intro!: exI[of _ "(\<lambda>(a, b). a \<times> b)  (A \<times> B)"] topological_basis_prod
topological_basis_imp_subbasis)
qed

instance second_countable_topology \<subseteq> first_countable_topology
proof
fix x :: 'a
def B \<equiv> "SOME B::'a set set. countable B \<and> topological_basis B"
then have B: "countable B" "topological_basis B"
using countable_basis is_basis
by (auto simp: countable_basis is_basis)
then show "\<exists>A::nat \<Rightarrow> 'a set. (\<forall>i. x \<in> A i \<and> open (A i)) \<and> (\<forall>S. open S \<and> x \<in> S \<longrightarrow> (\<exists>i. A i \<subseteq> S))"
by (intro first_countableI[of "{b\<in>B. x \<in> b}"])
(fastforce simp: topological_space_class.topological_basis_def)+
qed

subsection {* Polish spaces *}

text {* Textbooks define Polish spaces as completely metrizable.
We assume the topology to be complete for a given metric. *}

class polish_space = complete_space + second_countable_topology

subsection {* General notion of a topology as a value *}

definition "istopology L \<longleftrightarrow> L {} \<and> (\<forall>S T. L S \<longrightarrow> L T \<longrightarrow> L (S \<inter> T)) \<and> (\<forall>K. Ball K L \<longrightarrow> L (\<Union> K))"
typedef 'a topology = "{L::('a set) \<Rightarrow> bool. istopology L}"
morphisms "openin" "topology"
unfolding istopology_def by blast

lemma istopology_open_in[intro]: "istopology(openin U)"
using openin[of U] by blast

lemma topology_inverse': "istopology U \<Longrightarrow> openin (topology U) = U"
using topology_inverse[unfolded mem_Collect_eq] .

lemma topology_inverse_iff: "istopology U \<longleftrightarrow> openin (topology U) = U"
using topology_inverse[of U] istopology_open_in[of "topology U"] by auto

lemma topology_eq: "T1 = T2 \<longleftrightarrow> (\<forall>S. openin T1 S \<longleftrightarrow> openin T2 S)"
proof-
{ assume "T1=T2"
hence "\<forall>S. openin T1 S \<longleftrightarrow> openin T2 S" by simp }
moreover
{ assume H: "\<forall>S. openin T1 S \<longleftrightarrow> openin T2 S"
hence "openin T1 = openin T2" by (simp add: fun_eq_iff)
hence "topology (openin T1) = topology (openin T2)" by simp
hence "T1 = T2" unfolding openin_inverse .
}
ultimately show ?thesis by blast
qed

text{* Infer the "universe" from union of all sets in the topology. *}

definition "topspace T =  \<Union>{S. openin T S}"

subsubsection {* Main properties of open sets *}

lemma openin_clauses:
fixes U :: "'a topology"
shows "openin U {}"
"\<And>S T. openin U S \<Longrightarrow> openin U T \<Longrightarrow> openin U (S\<inter>T)"
"\<And>K. (\<forall>S \<in> K. openin U S) \<Longrightarrow> openin U (\<Union>K)"
using openin[of U] unfolding istopology_def mem_Collect_eq
by fast+

lemma openin_subset[intro]: "openin U S \<Longrightarrow> S \<subseteq> topspace U"
unfolding topspace_def by blast
lemma openin_empty[simp]: "openin U {}" by (simp add: openin_clauses)

lemma openin_Int[intro]: "openin U S \<Longrightarrow> openin U T \<Longrightarrow> openin U (S \<inter> T)"
using openin_clauses by simp

lemma openin_Union[intro]: "(\<forall>S \<in>K. openin U S) \<Longrightarrow> openin U (\<Union> K)"
using openin_clauses by simp

lemma openin_Un[intro]: "openin U S \<Longrightarrow> openin U T \<Longrightarrow> openin U (S \<union> T)"
using openin_Union[of "{S,T}" U] by auto

lemma openin_topspace[intro, simp]: "openin U (topspace U)" by (simp add: openin_Union topspace_def)

lemma openin_subopen: "openin U S \<longleftrightarrow> (\<forall>x \<in> S. \<exists>T. openin U T \<and> x \<in> T \<and> T \<subseteq> S)"
(is "?lhs \<longleftrightarrow> ?rhs")
proof
assume ?lhs
then show ?rhs by auto
next
assume H: ?rhs
let ?t = "\<Union>{T. openin U T \<and> T \<subseteq> S}"
have "openin U ?t" by (simp add: openin_Union)
also have "?t = S" using H by auto
finally show "openin U S" .
qed

subsubsection {* Closed sets *}

definition "closedin U S \<longleftrightarrow> S \<subseteq> topspace U \<and> openin U (topspace U - S)"

lemma closedin_subset: "closedin U S \<Longrightarrow> S \<subseteq> topspace U" by (metis closedin_def)
lemma closedin_empty[simp]: "closedin U {}" by (simp add: closedin_def)
lemma closedin_topspace[intro,simp]:
"closedin U (topspace U)" by (simp add: closedin_def)
lemma closedin_Un[intro]: "closedin U S \<Longrightarrow> closedin U T \<Longrightarrow> closedin U (S \<union> T)"
by (auto simp add: Diff_Un closedin_def)

lemma Diff_Inter[intro]: "A - \<Inter>S = \<Union> {A - s|s. s\<in>S}" by auto
lemma closedin_Inter[intro]: assumes Ke: "K \<noteq> {}" and Kc: "\<forall>S \<in>K. closedin U S"
shows "closedin U (\<Inter> K)"  using Ke Kc unfolding closedin_def Diff_Inter by auto

lemma closedin_Int[intro]: "closedin U S \<Longrightarrow> closedin U T \<Longrightarrow> closedin U (S \<inter> T)"
using closedin_Inter[of "{S,T}" U] by auto

lemma Diff_Diff_Int: "A - (A - B) = A \<inter> B" by blast
lemma openin_closedin_eq: "openin U S \<longleftrightarrow> S \<subseteq> topspace U \<and> closedin U (topspace U - S)"
apply (auto simp add: closedin_def Diff_Diff_Int inf_absorb2)
apply (metis openin_subset subset_eq)
done

lemma openin_closedin:  "S \<subseteq> topspace U \<Longrightarrow> (openin U S \<longleftrightarrow> closedin U (topspace U - S))"

lemma openin_diff[intro]: assumes oS: "openin U S" and cT: "closedin U T" shows "openin U (S - T)"
proof-
have "S - T = S \<inter> (topspace U - T)" using openin_subset[of U S]  oS cT
by (auto simp add: topspace_def openin_subset)
then show ?thesis using oS cT by (auto simp add: closedin_def)
qed

lemma closedin_diff[intro]: assumes oS: "closedin U S" and cT: "openin U T" shows "closedin U (S - T)"
proof-
have "S - T = S \<inter> (topspace U - T)" using closedin_subset[of U S]  oS cT
by (auto simp add: topspace_def )
then show ?thesis using oS cT by (auto simp add: openin_closedin_eq)
qed

subsubsection {* Subspace topology *}

definition "subtopology U V = topology (\<lambda>T. \<exists>S. T = S \<inter> V \<and> openin U S)"

lemma istopology_subtopology: "istopology (\<lambda>T. \<exists>S. T = S \<inter> V \<and> openin U S)"
(is "istopology ?L")
proof-
have "?L {}" by blast
{fix A B assume A: "?L A" and B: "?L B"
from A B obtain Sa and Sb where Sa: "openin U Sa" "A = Sa \<inter> V" and Sb: "openin U Sb" "B = Sb \<inter> V" by blast
have "A\<inter>B = (Sa \<inter> Sb) \<inter> V" "openin U (Sa \<inter> Sb)"  using Sa Sb by blast+
then have "?L (A \<inter> B)" by blast}
moreover
{fix K assume K: "K \<subseteq> Collect ?L"
have th0: "Collect ?L = (\<lambda>S. S \<inter> V)  Collect (openin U)"
apply (rule set_eqI)
by metis
from K[unfolded th0 subset_image_iff]
obtain Sk where Sk: "Sk \<subseteq> Collect (openin U)" "K = (\<lambda>S. S \<inter> V)  Sk" by blast
have "\<Union>K = (\<Union>Sk) \<inter> V" using Sk by auto
moreover have "openin U (\<Union> Sk)" using Sk by (auto simp add: subset_eq)
ultimately have "?L (\<Union>K)" by blast}
ultimately show ?thesis
unfolding subset_eq mem_Collect_eq istopology_def by blast
qed

lemma openin_subtopology:
"openin (subtopology U V) S \<longleftrightarrow> (\<exists> T. (openin U T) \<and> (S = T \<inter> V))"
unfolding subtopology_def topology_inverse'[OF istopology_subtopology]
by auto

lemma topspace_subtopology: "topspace(subtopology U V) = topspace U \<inter> V"
by (auto simp add: topspace_def openin_subtopology)

lemma closedin_subtopology:
"closedin (subtopology U V) S \<longleftrightarrow> (\<exists>T. closedin U T \<and> S = T \<inter> V)"
unfolding closedin_def topspace_subtopology
apply (rule iffI)
apply clarify
apply (rule_tac x="topspace U - T" in exI)
by auto

lemma openin_subtopology_refl: "openin (subtopology U V) V \<longleftrightarrow> V \<subseteq> topspace U"
unfolding openin_subtopology
apply (rule iffI, clarify)
apply (frule openin_subset[of U])  apply blast
apply (rule exI[where x="topspace U"])
apply auto
done

lemma subtopology_superset:
assumes UV: "topspace U \<subseteq> V"
shows "subtopology U V = U"
proof-
{fix S
{fix T assume T: "openin U T" "S = T \<inter> V"
from T openin_subset[OF T(1)] UV have eq: "S = T" by blast
have "openin U S" unfolding eq using T by blast}
moreover
{assume S: "openin U S"
hence "\<exists>T. openin U T \<and> S = T \<inter> V"
using openin_subset[OF S] UV by auto}
ultimately have "(\<exists>T. openin U T \<and> S = T \<inter> V) \<longleftrightarrow> openin U S" by blast}
then show ?thesis unfolding topology_eq openin_subtopology by blast
qed

lemma subtopology_topspace[simp]: "subtopology U (topspace U) = U"

lemma subtopology_UNIV[simp]: "subtopology U UNIV = U"

subsubsection {* The standard Euclidean topology *}

definition
euclidean :: "'a::topological_space topology" where
"euclidean = topology open"

lemma open_openin: "open S \<longleftrightarrow> openin euclidean S"
unfolding euclidean_def
apply (rule cong[where x=S and y=S])
apply (rule topology_inverse[symmetric])
done

lemma topspace_euclidean: "topspace euclidean = UNIV"
apply (rule set_eqI)

lemma topspace_euclidean_subtopology[simp]: "topspace (subtopology euclidean S) = S"

lemma closed_closedin: "closed S \<longleftrightarrow> closedin euclidean S"
by (simp add: closed_def closedin_def topspace_euclidean open_openin Compl_eq_Diff_UNIV)

lemma open_subopen: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>T. open T \<and> x \<in> T \<and> T \<subseteq> S)"

text {* Basic "localization" results are handy for connectedness. *}

lemma openin_open: "openin (subtopology euclidean U) S \<longleftrightarrow> (\<exists>T. open T \<and> (S = U \<inter> T))"
by (auto simp add: openin_subtopology open_openin[symmetric])

lemma openin_open_Int[intro]: "open S \<Longrightarrow> openin (subtopology euclidean U) (U \<inter> S)"

lemma open_openin_trans[trans]:
"open S \<Longrightarrow> open T \<Longrightarrow> T \<subseteq> S \<Longrightarrow> openin (subtopology euclidean S) T"
by (metis Int_absorb1  openin_open_Int)

lemma open_subset:  "S \<subseteq> T \<Longrightarrow> open S \<Longrightarrow> openin (subtopology euclidean T) S"

lemma closedin_closed: "closedin (subtopology euclidean U) S \<longleftrightarrow> (\<exists>T. closed T \<and> S = U \<inter> T)"
by (simp add: closedin_subtopology closed_closedin Int_ac)

lemma closedin_closed_Int: "closed S ==> closedin (subtopology euclidean U) (U \<inter> S)"
by (metis closedin_closed)

lemma closed_closedin_trans: "closed S \<Longrightarrow> closed T \<Longrightarrow> T \<subseteq> S \<Longrightarrow> closedin (subtopology euclidean S) T"
apply (subgoal_tac "S \<inter> T = T" )
apply auto
apply (frule closedin_closed_Int[of T S])
by simp

lemma closed_subset: "S \<subseteq> T \<Longrightarrow> closed S \<Longrightarrow> closedin (subtopology euclidean T) S"

lemma openin_euclidean_subtopology_iff:
fixes S U :: "'a::metric_space set"
shows "openin (subtopology euclidean U) S
\<longleftrightarrow> S \<subseteq> U \<and> (\<forall>x\<in>S. \<exists>e>0. \<forall>x'\<in>U. dist x' x < e \<longrightarrow> x'\<in> S)" (is "?lhs \<longleftrightarrow> ?rhs")
proof
assume ?lhs thus ?rhs unfolding openin_open open_dist by blast
next
def T \<equiv> "{x. \<exists>a\<in>S. \<exists>d>0. (\<forall>y\<in>U. dist y a < d \<longrightarrow> y \<in> S) \<and> dist x a < d}"
have 1: "\<forall>x\<in>T. \<exists>e>0. \<forall>y. dist y x < e \<longrightarrow> y \<in> T"
unfolding T_def
apply clarsimp
apply (rule_tac x="d - dist x a" in exI)
apply (erule rev_bexI)
apply (rule_tac x=d in exI, clarify)
apply (erule le_less_trans [OF dist_triangle])
done
assume ?rhs hence 2: "S = U \<inter> T"
unfolding T_def
apply auto
apply (drule (1) bspec, erule rev_bexI)
apply auto
done
from 1 2 show ?lhs
unfolding openin_open open_dist by fast
qed

text {* These "transitivity" results are handy too *}

lemma openin_trans[trans]: "openin (subtopology euclidean T) S \<Longrightarrow> openin (subtopology euclidean U) T
\<Longrightarrow> openin (subtopology euclidean U) S"
unfolding open_openin openin_open by blast

lemma openin_open_trans: "openin (subtopology euclidean T) S \<Longrightarrow> open T \<Longrightarrow> open S"
by (auto simp add: openin_open intro: openin_trans)

lemma closedin_trans[trans]:
"closedin (subtopology euclidean T) S \<Longrightarrow>
closedin (subtopology euclidean U) T
==> closedin (subtopology euclidean U) S"
by (auto simp add: closedin_closed closed_closedin closed_Inter Int_assoc)

lemma closedin_closed_trans: "closedin (subtopology euclidean T) S \<Longrightarrow> closed T \<Longrightarrow> closed S"
by (auto simp add: closedin_closed intro: closedin_trans)

subsection {* Open and closed balls *}

definition
ball :: "'a::metric_space \<Rightarrow> real \<Rightarrow> 'a set" where
"ball x e = {y. dist x y < e}"

definition
cball :: "'a::metric_space \<Rightarrow> real \<Rightarrow> 'a set" where
"cball x e = {y. dist x y \<le> e}"

lemma mem_ball [simp]: "y \<in> ball x e \<longleftrightarrow> dist x y < e"

lemma mem_cball [simp]: "y \<in> cball x e \<longleftrightarrow> dist x y \<le> e"

lemma mem_ball_0:
fixes x :: "'a::real_normed_vector"
shows "x \<in> ball 0 e \<longleftrightarrow> norm x < e"

lemma mem_cball_0:
fixes x :: "'a::real_normed_vector"
shows "x \<in> cball 0 e \<longleftrightarrow> norm x \<le> e"

lemma centre_in_ball: "x \<in> ball x e \<longleftrightarrow> 0 < e"
by simp

lemma centre_in_cball: "x \<in> cball x e \<longleftrightarrow> 0 \<le> e"
by simp

lemma ball_subset_cball[simp,intro]: "ball x e \<subseteq> cball x e" by (simp add: subset_eq)
lemma subset_ball[intro]: "d <= e ==> ball x d \<subseteq> ball x e" by (simp add: subset_eq)
lemma subset_cball[intro]: "d <= e ==> cball x d \<subseteq> cball x e" by (simp add: subset_eq)
lemma ball_max_Un: "ball a (max r s) = ball a r \<union> ball a s"

lemma ball_min_Int: "ball a (min r s) = ball a r \<inter> ball a s"

lemma diff_less_iff: "(a::real) - b > 0 \<longleftrightarrow> a > b"
"(a::real) - b < 0 \<longleftrightarrow> a < b"
"a - b < c \<longleftrightarrow> a < c +b" "a - b > c \<longleftrightarrow> a > c +b" by arith+
lemma diff_le_iff: "(a::real) - b \<ge> 0 \<longleftrightarrow> a \<ge> b" "(a::real) - b \<le> 0 \<longleftrightarrow> a \<le> b"
"a - b \<le> c \<longleftrightarrow> a \<le> c +b" "a - b \<ge> c \<longleftrightarrow> a \<ge> c +b"  by arith+

lemma open_ball[intro, simp]: "open (ball x e)"
unfolding open_dist ball_def mem_Collect_eq Ball_def
unfolding dist_commute
apply clarify
apply (rule_tac x="e - dist xa x" in exI)
using dist_triangle_alt[where z=x]
apply atomize
apply (erule_tac x="y" in allE)
apply (erule_tac x="xa" in allE)
by arith

lemma open_contains_ball: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0. ball x e \<subseteq> S)"
unfolding open_dist subset_eq mem_ball Ball_def dist_commute ..

lemma openE[elim?]:
assumes "open S" "x\<in>S"
obtains e where "e>0" "ball x e \<subseteq> S"
using assms unfolding open_contains_ball by auto

lemma open_contains_ball_eq: "open S \<Longrightarrow> \<forall>x. x\<in>S \<longleftrightarrow> (\<exists>e>0. ball x e \<subseteq> S)"
by (metis open_contains_ball subset_eq centre_in_ball)

lemma ball_eq_empty[simp]: "ball x e = {} \<longleftrightarrow> e \<le> 0"
unfolding mem_ball set_eq_iff
by (metis zero_le_dist order_trans dist_self)

lemma ball_empty[intro]: "e \<le> 0 ==> ball x e = {}" by simp

lemma euclidean_dist_l2:
fixes x y :: "'a :: euclidean_space"
shows "dist x y = setL2 (\<lambda>i. dist (x \<bullet> i) (y \<bullet> i)) Basis"
unfolding dist_norm norm_eq_sqrt_inner setL2_def
by (subst euclidean_inner) (simp add: power2_eq_square inner_diff_left)

definition "box a b = {x. \<forall>i\<in>Basis. a \<bullet> i < x \<bullet> i \<and> x \<bullet> i < b \<bullet> i}"

lemma rational_boxes:
fixes x :: "'a\<Colon>euclidean_space"
assumes "0 < e"
shows "\<exists>a b. (\<forall>i\<in>Basis. a \<bullet> i \<in> \<rat> \<and> b \<bullet> i \<in> \<rat> ) \<and> x \<in> box a b \<and> box a b \<subseteq> ball x e"
proof -
def e' \<equiv> "e / (2 * sqrt (real (DIM ('a))))"
then have e: "0 < e'" using assms by (auto intro!: divide_pos_pos simp: DIM_positive)
have "\<forall>i. \<exists>y. y \<in> \<rat> \<and> y < x \<bullet> i \<and> x \<bullet> i - y < e'" (is "\<forall>i. ?th i")
proof
fix i from Rats_dense_in_real[of "x \<bullet> i - e'" "x \<bullet> i"] e show "?th i" by auto
qed
from choice[OF this] guess a .. note a = this
have "\<forall>i. \<exists>y. y \<in> \<rat> \<and> x \<bullet> i < y \<and> y - x \<bullet> i < e'" (is "\<forall>i. ?th i")
proof
fix i from Rats_dense_in_real[of "x \<bullet> i" "x \<bullet> i + e'"] e show "?th i" by auto
qed
from choice[OF this] guess b .. note b = this
let ?a = "\<Sum>i\<in>Basis. a i *\<^sub>R i" and ?b = "\<Sum>i\<in>Basis. b i *\<^sub>R i"
show ?thesis
proof (rule exI[of _ ?a], rule exI[of _ ?b], safe)
fix y :: 'a assume *: "y \<in> box ?a ?b"
have "dist x y = sqrt (\<Sum>i\<in>Basis. (dist (x \<bullet> i) (y \<bullet> i))\<twosuperior>)"
unfolding setL2_def[symmetric] by (rule euclidean_dist_l2)
also have "\<dots> < sqrt (\<Sum>(i::'a)\<in>Basis. e^2 / real (DIM('a)))"
proof (rule real_sqrt_less_mono, rule setsum_strict_mono)
fix i :: "'a" assume i: "i \<in> Basis"
have "a i < y\<bullet>i \<and> y\<bullet>i < b i" using * i by (auto simp: box_def)
moreover have "a i < x\<bullet>i" "x\<bullet>i - a i < e'" using a by auto
moreover have "x\<bullet>i < b i" "b i - x\<bullet>i < e'" using b by auto
ultimately have "\<bar>x\<bullet>i - y\<bullet>i\<bar> < 2 * e'" by auto
then have "dist (x \<bullet> i) (y \<bullet> i) < e/sqrt (real (DIM('a)))"
unfolding e'_def by (auto simp: dist_real_def)
then have "(dist (x \<bullet> i) (y \<bullet> i))\<twosuperior> < (e/sqrt (real (DIM('a))))\<twosuperior>"
by (rule power_strict_mono) auto
then show "(dist (x \<bullet> i) (y \<bullet> i))\<twosuperior> < e\<twosuperior> / real DIM('a)"
qed auto
also have "\<dots> = e" using 0 < e by (simp add: real_eq_of_nat)
finally show "y \<in> ball x e" by (auto simp: ball_def)
qed (insert a b, auto simp: box_def)
qed

lemma open_UNION_box:
fixes M :: "'a\<Colon>euclidean_space set"
assumes "open M"
defines "a' \<equiv> \<lambda>f :: 'a \<Rightarrow> real \<times> real. (\<Sum>(i::'a)\<in>Basis. fst (f i) *\<^sub>R i)"
defines "b' \<equiv> \<lambda>f :: 'a \<Rightarrow> real \<times> real. (\<Sum>(i::'a)\<in>Basis. snd (f i) *\<^sub>R i)"
defines "I \<equiv> {f\<in>Basis \<rightarrow>\<^isub>E \<rat> \<times> \<rat>. box (a' f) (b' f) \<subseteq> M}"
shows "M = (\<Union>f\<in>I. box (a' f) (b' f))"
proof safe
fix x assume "x \<in> M"
obtain e where e: "e > 0" "ball x e \<subseteq> M"
using openE[OF open M x \<in> M] by auto
moreover then obtain a b where ab: "x \<in> box a b"
"\<forall>i \<in> Basis. a \<bullet> i \<in> \<rat>" "\<forall>i\<in>Basis. b \<bullet> i \<in> \<rat>" "box a b \<subseteq> ball x e"
using rational_boxes[OF e(1)] by metis
ultimately show "x \<in> (\<Union>f\<in>I. box (a' f) (b' f))"
by (intro UN_I[of "\<lambda>i\<in>Basis. (a \<bullet> i, b \<bullet> i)"])
(auto simp: euclidean_representation I_def a'_def b'_def)
qed (auto simp: I_def)

subsection{* Connectedness *}

lemma connected_local:
"connected S \<longleftrightarrow> ~(\<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 = {}) \<and>
~(e2 = {}))"
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-
{assume "?lhs" hence ?rhs by blast }
moreover
{fix S assume H: "P S"
have "S = - (- S)" by auto
with H have "P (- (- S))" by metis }
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
apply (subst exists_diff) by blast
hence 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)") apply (simp add: closed_def) by 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
{fix e2
{fix e1 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)"
by auto}
then have "(\<exists>e1. ?P e2 e1) \<longleftrightarrow> (\<exists>t. ?Q e2 t)" by metis}
then have "\<forall>e2. (\<exists>e1. ?P e2 e1) \<longleftrightarrow> (\<exists>t. ?Q e2 t)" by blast
then show ?thesis unfolding th0 th1 by simp
qed

lemma connected_empty[simp, intro]: "connected {}"

subsection{* Limit points *}

definition (in topological_space)
islimpt:: "'a \<Rightarrow> 'a set \<Rightarrow> bool" (infixr "islimpt" 60) where
"x islimpt S \<longleftrightarrow> (\<forall>T. x\<in>T \<longrightarrow> open T \<longrightarrow> (\<exists>y\<in>S. y\<in>T \<and> y\<noteq>x))"

lemma islimptI:
assumes "\<And>T. x \<in> T \<Longrightarrow> open T \<Longrightarrow> \<exists>y\<in>S. y \<in> T \<and> y \<noteq> x"
shows "x islimpt S"
using assms unfolding islimpt_def by auto

lemma islimptE:
assumes "x islimpt S" and "x \<in> T" and "open T"
obtains y where "y \<in> S" and "y \<in> T" and "y \<noteq> x"
using assms unfolding islimpt_def by auto

lemma islimpt_iff_eventually: "x islimpt S \<longleftrightarrow> \<not> eventually (\<lambda>y. y \<notin> S) (at x)"
unfolding islimpt_def eventually_at_topological by auto

lemma islimpt_subset: "\<lbrakk>x islimpt S; S \<subseteq> T\<rbrakk> \<Longrightarrow> x islimpt T"
unfolding islimpt_def by fast

lemma islimpt_approachable:
fixes x :: "'a::metric_space"
shows "x islimpt S \<longleftrightarrow> (\<forall>e>0. \<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e)"
unfolding islimpt_iff_eventually eventually_at by fast

lemma islimpt_approachable_le:
fixes x :: "'a::metric_space"
shows "x islimpt S \<longleftrightarrow> (\<forall>e>0. \<exists>x'\<in> S. x' \<noteq> x \<and> dist x' x <= e)"
unfolding islimpt_approachable
using approachable_lt_le [where f="\<lambda>y. dist y x" and P="\<lambda>y. y \<notin> S \<or> y = x",
THEN arg_cong [where f=Not]]
by (simp add: Bex_def conj_commute conj_left_commute)

lemma islimpt_UNIV_iff: "x islimpt UNIV \<longleftrightarrow> \<not> open {x}"
unfolding islimpt_def by (safe, fast, case_tac "T = {x}", fast, fast)

lemma islimpt_punctured: "x islimpt S = x islimpt (S-{x})"
unfolding islimpt_def by blast

text {* A perfect space has no isolated points. *}

lemma islimpt_UNIV [simp, intro]: "(x::'a::perfect_space) islimpt UNIV"
unfolding islimpt_UNIV_iff by (rule not_open_singleton)

lemma perfect_choose_dist:
fixes x :: "'a::{perfect_space, metric_space}"
shows "0 < r \<Longrightarrow> \<exists>a. a \<noteq> x \<and> dist a x < r"
using islimpt_UNIV [of x]

lemma closed_limpt: "closed S \<longleftrightarrow> (\<forall>x. x islimpt S \<longrightarrow> x \<in> S)"
unfolding closed_def
apply (subst open_subopen)
by (metis ComplE ComplI)

lemma islimpt_EMPTY[simp]: "\<not> x islimpt {}"
unfolding islimpt_def by auto

lemma finite_set_avoid:
fixes a :: "'a::metric_space"
assumes fS: "finite S" shows  "\<exists>d>0. \<forall>x\<in>S. x \<noteq> a \<longrightarrow> d <= dist a x"
proof(induct rule: finite_induct[OF fS])
case 1 thus ?case by (auto intro: zero_less_one)
next
case (2 x F)
from 2 obtain d where d: "d >0" "\<forall>x\<in>F. x\<noteq>a \<longrightarrow> d \<le> dist a x" by blast
{assume "x = a" hence ?case using d by auto  }
moreover
{assume xa: "x\<noteq>a"
let ?d = "min d (dist a x)"
have dp: "?d > 0" using xa d(1) using dist_nz by auto
from d have d': "\<forall>x\<in>F. x\<noteq>a \<longrightarrow> ?d \<le> dist a x" by auto
with dp xa have ?case by(auto intro!: exI[where x="?d"]) }
ultimately show ?case by blast
qed

lemma islimpt_Un: "x islimpt (S \<union> T) \<longleftrightarrow> x islimpt S \<or> x islimpt T"

lemma discrete_imp_closed:
fixes S :: "'a::metric_space set"
assumes e: "0 < e" and d: "\<forall>x \<in> S. \<forall>y \<in> S. dist y x < e \<longrightarrow> y = x"
shows "closed S"
proof-
{fix x assume C: "\<forall>e>0. \<exists>x'\<in>S. x' \<noteq> x \<and> dist x' x < e"
from e have e2: "e/2 > 0" by arith
from C[rule_format, OF e2] obtain y where y: "y \<in> S" "y\<noteq>x" "dist y x < e/2" by blast
let ?m = "min (e/2) (dist x y) "
from e2 y(2) have mp: "?m > 0" by (simp add: dist_nz[THEN sym])
from C[rule_format, OF mp] obtain z where z: "z \<in> S" "z\<noteq>x" "dist z x < ?m" by blast
have th: "dist z y < e" using z y
by (intro dist_triangle_lt [where z=x], simp)
from d[rule_format, OF y(1) z(1) th] y z
have False by (auto simp add: dist_commute)}
then show ?thesis by (metis islimpt_approachable closed_limpt [where 'a='a])
qed

subsection {* Interior of a Set *}

definition "interior S = \<Union>{T. open T \<and> T \<subseteq> S}"

lemma interiorI [intro?]:
assumes "open T" and "x \<in> T" and "T \<subseteq> S"
shows "x \<in> interior S"
using assms unfolding interior_def by fast

lemma interiorE [elim?]:
assumes "x \<in> interior S"
obtains T where "open T" and "x \<in> T" and "T \<subseteq> S"
using assms unfolding interior_def by fast

lemma open_interior [simp, intro]: "open (interior S)"

lemma interior_subset: "interior S \<subseteq> S"

lemma interior_maximal: "T \<subseteq> S \<Longrightarrow> open T \<Longrightarrow> T \<subseteq> interior S"

lemma interior_open: "open S \<Longrightarrow> interior S = S"
by (intro equalityI interior_subset interior_maximal subset_refl)

lemma interior_eq: "interior S = S \<longleftrightarrow> open S"
by (metis open_interior interior_open)

lemma open_subset_interior: "open S \<Longrightarrow> S \<subseteq> interior T \<longleftrightarrow> S \<subseteq> T"
by (metis interior_maximal interior_subset subset_trans)

lemma interior_empty [simp]: "interior {} = {}"
using open_empty by (rule interior_open)

lemma interior_UNIV [simp]: "interior UNIV = UNIV"
using open_UNIV by (rule interior_open)

lemma interior_interior [simp]: "interior (interior S) = interior S"
using open_interior by (rule interior_open)

lemma interior_mono: "S \<subseteq> T \<Longrightarrow> interior S \<subseteq> interior T"

lemma interior_unique:
assumes "T \<subseteq> S" and "open T"
assumes "\<And>T'. T' \<subseteq> S \<Longrightarrow> open T' \<Longrightarrow> T' \<subseteq> T"
shows "interior S = T"
by (intro equalityI assms interior_subset open_interior interior_maximal)

lemma interior_inter [simp]: "interior (S \<inter> T) = interior S \<inter> interior T"
by (intro equalityI Int_mono Int_greatest interior_mono Int_lower1
Int_lower2 interior_maximal interior_subset open_Int open_interior)

lemma mem_interior: "x \<in> interior S \<longleftrightarrow> (\<exists>e>0. ball x e \<subseteq> S)"
using open_contains_ball_eq [where S="interior S"]

lemma interior_limit_point [intro]:
fixes x :: "'a::perfect_space"
assumes x: "x \<in> interior S" shows "x islimpt S"
using x islimpt_UNIV [of x]
unfolding interior_def islimpt_def
apply (clarsimp, rename_tac T T')
apply (drule_tac x="T \<inter> T'" in spec)
done

lemma interior_closed_Un_empty_interior:
assumes cS: "closed S" and iT: "interior T = {}"
shows "interior (S \<union> T) = interior S"
proof
show "interior S \<subseteq> interior (S \<union> T)"
by (rule interior_mono, rule Un_upper1)
next
show "interior (S \<union> T) \<subseteq> interior S"
proof
fix x assume "x \<in> interior (S \<union> T)"
then obtain R where "open R" "x \<in> R" "R \<subseteq> S \<union> T" ..
show "x \<in> interior S"
proof (rule ccontr)
assume "x \<notin> interior S"
with x \<in> R open R obtain y where "y \<in> R - S"
unfolding interior_def by fast
from open R closed S have "open (R - S)" by (rule open_Diff)
from R \<subseteq> S \<union> T have "R - S \<subseteq> T" by fast
from y \<in> R - S open (R - S) R - S \<subseteq> T interior T = {}
show "False" unfolding interior_def by fast
qed
qed
qed

lemma interior_Times: "interior (A \<times> B) = interior A \<times> interior B"
proof (rule interior_unique)
show "interior A \<times> interior B \<subseteq> A \<times> B"
by (intro Sigma_mono interior_subset)
show "open (interior A \<times> interior B)"
by (intro open_Times open_interior)
fix T assume "T \<subseteq> A \<times> B" and "open T" thus "T \<subseteq> interior A \<times> interior B"
proof (safe)
fix x y assume "(x, y) \<in> T"
then obtain C D where "open C" "open D" "C \<times> D \<subseteq> T" "x \<in> C" "y \<in> D"
using open T unfolding open_prod_def by fast
hence "open C" "open D" "C \<subseteq> A" "D \<subseteq> B" "x \<in> C" "y \<in> D"
using T \<subseteq> A \<times> B by auto
thus "x \<in> interior A" and "y \<in> interior B"
by (auto intro: interiorI)
qed
qed

subsection {* Closure of a Set *}

definition "closure S = S \<union> {x | x. x islimpt S}"

lemma interior_closure: "interior S = - (closure (- S))"
unfolding interior_def closure_def islimpt_def by auto

lemma closure_interior: "closure S = - interior (- S)"
unfolding interior_closure by simp

lemma closed_closure[simp, intro]: "closed (closure S)"
unfolding closure_interior by (simp add: closed_Compl)

lemma closure_subset: "S \<subseteq> closure S"
unfolding closure_def by simp

lemma closure_hull: "closure S = closed hull S"
unfolding hull_def closure_interior interior_def by auto

lemma closure_eq: "closure S = S \<longleftrightarrow> closed S"
unfolding closure_hull using closed_Inter by (rule hull_eq)

lemma closure_closed [simp]: "closed S \<Longrightarrow> closure S = S"
unfolding closure_eq .

lemma closure_closure [simp]: "closure (closure S) = closure S"
unfolding closure_hull by (rule hull_hull)

lemma closure_mono: "S \<subseteq> T \<Longrightarrow> closure S \<subseteq> closure T"
unfolding closure_hull by (rule hull_mono)

lemma closure_minimal: "S \<subseteq> T \<Longrightarrow> closed T \<Longrightarrow> closure S \<subseteq> T"
unfolding closure_hull by (rule hull_minimal)

lemma closure_unique:
assumes "S \<subseteq> T" and "closed T"
assumes "\<And>T'. S \<subseteq> T' \<Longrightarrow> closed T' \<Longrightarrow> T \<subseteq> T'"
shows "closure S = T"
using assms unfolding closure_hull by (rule hull_unique)

lemma closure_empty [simp]: "closure {} = {}"
using closed_empty by (rule closure_closed)

lemma closure_UNIV [simp]: "closure UNIV = UNIV"
using closed_UNIV by (rule closure_closed)

lemma closure_union [simp]: "closure (S \<union> T) = closure S \<union> closure T"
unfolding closure_interior by simp

lemma closure_eq_empty: "closure S = {} \<longleftrightarrow> S = {}"
using closure_empty closure_subset[of S]
by blast

lemma closure_subset_eq: "closure S \<subseteq> S \<longleftrightarrow> closed S"
using closure_eq[of S] closure_subset[of S]
by simp

lemma open_inter_closure_eq_empty:
"open S \<Longrightarrow> (S \<inter> closure T) = {} \<longleftrightarrow> S \<inter> T = {}"
using open_subset_interior[of S "- T"]
using interior_subset[of "- T"]
unfolding closure_interior
by auto

lemma open_inter_closure_subset:
"open S \<Longrightarrow> (S \<inter> (closure T)) \<subseteq> closure(S \<inter> T)"
proof
fix x
assume as: "open S" "x \<in> S \<inter> closure T"
{ assume *:"x islimpt T"
have "x islimpt (S \<inter> T)"
proof (rule islimptI)
fix A
assume "x \<in> A" "open A"
with as have "x \<in> A \<inter> S" "open (A \<inter> S)"
with * obtain y where "y \<in> T" "y \<in> A \<inter> S" "y \<noteq> x"
by (rule islimptE)
hence "y \<in> S \<inter> T" "y \<in> A \<and> y \<noteq> x"
by simp_all
thus "\<exists>y\<in>(S \<inter> T). y \<in> A \<and> y \<noteq> x" ..
qed
}
then show "x \<in> closure (S \<inter> T)" using as
unfolding closure_def
by blast
qed

lemma closure_complement: "closure (- S) = - interior S"
unfolding closure_interior by simp

lemma interior_complement: "interior (- S) = - closure S"
unfolding closure_interior by simp

lemma closure_Times: "closure (A \<times> B) = closure A \<times> closure B"
proof (rule closure_unique)
show "A \<times> B \<subseteq> closure A \<times> closure B"
by (intro Sigma_mono closure_subset)
show "closed (closure A \<times> closure B)"
by (intro closed_Times closed_closure)
fix T assume "A \<times> B \<subseteq> T" and "closed T" thus "closure A \<times> closure B \<subseteq> T"
apply (simp add: closed_def open_prod_def, clarify)
apply (rule ccontr)
apply (drule_tac x="(a, b)" in bspec, simp, clarify, rename_tac C D)
apply (drule_tac x=C in spec)
apply (drule_tac x=D in spec)
apply auto
done
qed

lemma islimpt_in_closure: "(x islimpt S) = (x:closure(S-{x}))"
unfolding closure_def using islimpt_punctured by blast

subsection {* Frontier (aka boundary) *}

definition "frontier S = closure S - interior S"

lemma frontier_closed: "closed(frontier S)"

lemma frontier_closures: "frontier S = (closure S) \<inter> (closure(- S))"
by (auto simp add: frontier_def interior_closure)

fixes a :: "'a::metric_space"
shows "a \<in> frontier S \<longleftrightarrow> (\<forall>e>0. (\<exists>x\<in>S. dist a x < e) \<and> (\<exists>x. x \<notin> S \<and> dist a x < e))"
unfolding frontier_def closure_interior
by (auto simp add: mem_interior subset_eq ball_def)

lemma frontier_subset_closed: "closed S \<Longrightarrow> frontier S \<subseteq> S"
by (metis frontier_def closure_closed Diff_subset)

lemma frontier_empty[simp]: "frontier {} = {}"

lemma frontier_subset_eq: "frontier S \<subseteq> S \<longleftrightarrow> closed S"
proof-
{ assume "frontier S \<subseteq> S"
hence "closure S \<subseteq> S" using interior_subset unfolding frontier_def by auto
hence "closed S" using closure_subset_eq by auto
}
thus ?thesis using frontier_subset_closed[of S] ..
qed

lemma frontier_complement: "frontier(- S) = frontier S"
by (auto simp add: frontier_def closure_complement interior_complement)

lemma frontier_disjoint_eq: "frontier S \<inter> S = {} \<longleftrightarrow> open S"
using frontier_complement frontier_subset_eq[of "- S"]
unfolding open_closed by auto

subsection {* Filters and the eventually true'' quantifier *}

definition
indirection :: "'a::real_normed_vector \<Rightarrow> 'a \<Rightarrow> 'a filter"
(infixr "indirection" 70) where
"a indirection v = at a within {b. \<exists>c\<ge>0. b - a = scaleR c v}"

text {* Identify Trivial limits, where we can't approach arbitrarily closely. *}

lemma trivial_limit_within:
shows "trivial_limit (at a within S) \<longleftrightarrow> \<not> a islimpt S"
proof
assume "trivial_limit (at a within S)"
thus "\<not> a islimpt S"
unfolding trivial_limit_def
unfolding eventually_at_topological
unfolding islimpt_def
apply (rename_tac T, rule_tac x=T in exI)
apply (clarsimp, drule_tac x=y in bspec, simp_all)
done
next
assume "\<not> a islimpt S"
thus "trivial_limit (at a within S)"
unfolding trivial_limit_def
unfolding eventually_at_topological
unfolding islimpt_def
apply clarsimp
apply (rule_tac x=T in exI)
apply auto
done
qed

lemma trivial_limit_at_iff: "trivial_limit (at a) \<longleftrightarrow> \<not> a islimpt UNIV"
using trivial_limit_within [of a UNIV] by simp

lemma trivial_limit_at:
fixes a :: "'a::perfect_space"
shows "\<not> trivial_limit (at a)"
by (rule at_neq_bot)

lemma trivial_limit_at_infinity:
"\<not> trivial_limit (at_infinity :: ('a::{real_normed_vector,perfect_space}) filter)"
unfolding trivial_limit_def eventually_at_infinity
apply clarsimp
apply (subgoal_tac "\<exists>x::'a. x \<noteq> 0", clarify)
apply (rule_tac x="scaleR (b / norm x) x" in exI, simp)
apply (cut_tac islimpt_UNIV [of "0::'a", unfolded islimpt_def])
apply (drule_tac x=UNIV in spec, simp)
done

lemma not_trivial_limit_within: "~trivial_limit (at x within S) = (x:closure(S-{x}))"
using islimpt_in_closure by (metis trivial_limit_within)

text {* Some property holds "sufficiently close" to the limit point. *}

lemma eventually_at2:
"eventually P (at a) \<longleftrightarrow> (\<exists>d>0. \<forall>x. 0 < dist x a \<and> dist x a < d \<longrightarrow> P x)"
unfolding eventually_at dist_nz by auto

lemma eventually_happens: "eventually P net ==> trivial_limit net \<or> (\<exists>x. P x)"
unfolding trivial_limit_def
by (auto elim: eventually_rev_mp)

lemma trivial_limit_eventually: "trivial_limit net \<Longrightarrow> eventually P net"
by simp

lemma trivial_limit_eq: "trivial_limit net \<longleftrightarrow> (\<forall>P. eventually P net)"

text{* Combining theorems for "eventually" *}

lemma eventually_rev_mono:
"eventually P net \<Longrightarrow> (\<forall>x. P x \<longrightarrow> Q x) \<Longrightarrow> eventually Q net"
using eventually_mono [of P Q] by fast

lemma not_eventually: "(\<forall>x. \<not> P x ) \<Longrightarrow> ~(trivial_limit net) ==> ~(eventually (\<lambda>x. P x) net)"

subsection {* Limits *}

lemma Lim:
"(f ---> l) net \<longleftrightarrow>
trivial_limit net \<or>
(\<forall>e>0. eventually (\<lambda>x. dist (f x) l < e) net)"
unfolding tendsto_iff trivial_limit_eq by auto

text{* Show that they yield usual definitions in the various cases. *}

lemma Lim_within_le: "(f ---> l)(at a within S) \<longleftrightarrow>
(\<forall>e>0. \<exists>d>0. \<forall>x\<in>S. 0 < dist x a  \<and> dist x a  <= d \<longrightarrow> dist (f x) l < e)"
by (auto simp add: tendsto_iff eventually_at_le dist_nz)

lemma Lim_within: "(f ---> l) (at a within S) \<longleftrightarrow>
(\<forall>e >0. \<exists>d>0. \<forall>x \<in> S. 0 < dist x a  \<and> dist x a  < d  \<longrightarrow> dist (f x) l < e)"
by (auto simp add: tendsto_iff eventually_at dist_nz)

lemma Lim_at: "(f ---> l) (at a) \<longleftrightarrow>
(\<forall>e >0. \<exists>d>0. \<forall>x. 0 < dist x a  \<and> dist x a  < d  \<longrightarrow> dist (f x) l < e)"
by (auto simp add: tendsto_iff eventually_at2)

lemma Lim_at_infinity:
"(f ---> l) at_infinity \<longleftrightarrow> (\<forall>e>0. \<exists>b. \<forall>x. norm x >= b \<longrightarrow> dist (f x) l < e)"
by (auto simp add: tendsto_iff eventually_at_infinity)

lemma Lim_eventually: "eventually (\<lambda>x. f x = l) net \<Longrightarrow> (f ---> l) net"
by (rule topological_tendstoI, auto elim: eventually_rev_mono)

text{* The expected monotonicity property. *}

lemma Lim_within_empty: "(f ---> l) (at x within {})"
unfolding tendsto_def eventually_at_filter by simp

lemma Lim_Un: assumes "(f ---> l) (at x within S)" "(f ---> l) (at x within T)"
shows "(f ---> l) (at x within (S \<union> T))"
using assms unfolding tendsto_def eventually_at_filter
apply clarify
apply (drule spec, drule (1) mp, drule (1) mp)
apply (drule spec, drule (1) mp, drule (1) mp)
apply (auto elim: eventually_elim2)
done

lemma Lim_Un_univ:
"(f ---> l) (at x within S) \<Longrightarrow> (f ---> l) (at x within T) \<Longrightarrow>  S \<union> T = UNIV
==> (f ---> l) (at x)"
by (metis Lim_Un)

text{* Interrelations between restricted and unrestricted limits. *}

lemma Lim_at_within: (* FIXME: rename *)
"(f ---> l) (at x) \<Longrightarrow> (f ---> l) (at x within S)"
by (metis order_refl filterlim_mono subset_UNIV at_le)

lemma eventually_within_interior:
assumes "x \<in> interior S"
shows "eventually P (at x within S) \<longleftrightarrow> eventually P (at x)" (is "?lhs = ?rhs")
proof-
from assms obtain T where T: "open T" "x \<in> T" "T \<subseteq> S" ..
{ assume "?lhs"
then obtain A where "open A" "x \<in> A" "\<forall>y\<in>A. y \<noteq> x \<longrightarrow> y \<in> S \<longrightarrow> P y"
unfolding eventually_at_topological
by auto
with T have "open (A \<inter> T)" "x \<in> A \<inter> T" "\<forall>y\<in>(A \<inter> T). y \<noteq> x \<longrightarrow> P y"
by auto
then have "?rhs"
unfolding eventually_at_topological by auto
} moreover
{ assume "?rhs" hence "?lhs"
by (auto elim: eventually_elim1 simp: eventually_at_filter)
} ultimately
show "?thesis" ..
qed

lemma at_within_interior:
"x \<in> interior S \<Longrightarrow> at x within S = at x"
unfolding filter_eq_iff by (intro allI eventually_within_interior)

lemma Lim_within_LIMSEQ:
fixes a :: "'a::metric_space"
assumes "\<forall>S. (\<forall>n. S n \<noteq> a \<and> S n \<in> T) \<and> S ----> a \<longrightarrow> (\<lambda>n. X (S n)) ----> L"
shows "(X ---> L) (at a within T)"
using assms unfolding tendsto_def [where l=L]

lemma Lim_right_bound:
fixes f :: "'a :: {linorder_topology, conditionally_complete_linorder, no_top} \<Rightarrow>
'b::{linorder_topology, conditionally_complete_linorder}"
assumes mono: "\<And>a b. a \<in> I \<Longrightarrow> b \<in> I \<Longrightarrow> x < a \<Longrightarrow> a \<le> b \<Longrightarrow> f a \<le> f b"
assumes bnd: "\<And>a. a \<in> I \<Longrightarrow> x < a \<Longrightarrow> K \<le> f a"
shows "(f ---> Inf (f  ({x<..} \<inter> I))) (at x within ({x<..} \<inter> I))"
proof cases
assume "{x<..} \<inter> I = {}" then show ?thesis by (simp add: Lim_within_empty)
next
assume e: "{x<..} \<inter> I \<noteq> {}"
show ?thesis
proof (rule order_tendstoI)
fix a assume a: "a < Inf (f  ({x<..} \<inter> I))"
{ fix y assume "y \<in> {x<..} \<inter> I"
with e bnd have "Inf (f  ({x<..} \<inter> I)) \<le> f y"
by (auto intro: cInf_lower)
with a have "a < f y" by (blast intro: less_le_trans) }
then show "eventually (\<lambda>x. a < f x) (at x within ({x<..} \<inter> I))"
by (auto simp: eventually_at_filter intro: exI[of _ 1] zero_less_one)
next
fix a assume "Inf (f  ({x<..} \<inter> I)) < a"
from cInf_lessD[OF _ this] e obtain y where y: "x < y" "y \<in> I" "f y < a" by auto
then have "eventually (\<lambda>x. x \<in> I \<longrightarrow> f x < a) (at_right x)"
unfolding eventually_at_right by (metis less_imp_le le_less_trans mono)
then show "eventually (\<lambda>x. f x < a) (at x within ({x<..} \<inter> I))"
unfolding eventually_at_filter by eventually_elim simp
qed
qed

text{* Another limit point characterization. *}

lemma islimpt_sequential:
fixes x :: "'a::first_countable_topology"
shows "x islimpt S \<longleftrightarrow> (\<exists>f. (\<forall>n::nat. f n \<in> S - {x}) \<and> (f ---> x) sequentially)"
(is "?lhs = ?rhs")
proof
assume ?lhs
from countable_basis_at_decseq[of x] guess A . note A = this
def f \<equiv> "\<lambda>n. SOME y. y \<in> S \<and> y \<in> A n \<and> x \<noteq> y"
{ fix n
from ?lhs have "\<exists>y. y \<in> S \<and> y \<in> A n \<and> x \<noteq> y"
unfolding islimpt_def using A(1,2)[of n] by auto
then have "f n \<in> S \<and> f n \<in> A n \<and> x \<noteq> f n"
unfolding f_def by (rule someI_ex)
then have "f n \<in> S" "f n \<in> A n" "x \<noteq> f n" by auto }
then have "\<forall>n. f n \<in> S - {x}" by auto
moreover have "(\<lambda>n. f n) ----> x"
proof (rule topological_tendstoI)
fix S assume "open S" "x \<in> S"
from A(3)[OF this] \<And>n. f n \<in> A n
show "eventually (\<lambda>x. f x \<in> S) sequentially" by (auto elim!: eventually_elim1)
qed
ultimately show ?rhs by fast
next
assume ?rhs
then obtain f :: "nat \<Rightarrow> 'a" where f: "\<And>n. f n \<in> S - {x}" and lim: "f ----> x" by auto
show ?lhs
unfolding islimpt_def
proof safe
fix T assume "open T" "x \<in> T"
from lim[THEN topological_tendstoD, OF this] f
show "\<exists>y\<in>S. y \<in> T \<and> y \<noteq> x"
unfolding eventually_sequentially by auto
qed
qed

lemma Lim_inv: (* TODO: delete *)
fixes f :: "'a \<Rightarrow> real" and A :: "'a filter"
assumes "(f ---> l) A" and "l \<noteq> 0"
shows "((inverse o f) ---> inverse l) A"
unfolding o_def using assms by (rule tendsto_inverse)

lemma Lim_null:
fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
shows "(f ---> l) net \<longleftrightarrow> ((\<lambda>x. f(x) - l) ---> 0) net"

lemma Lim_null_comparison:
fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
assumes "eventually (\<lambda>x. norm (f x) \<le> g x) net" "(g ---> 0) net"
shows "(f ---> 0) net"
proof (rule metric_tendsto_imp_tendsto)
show "(g ---> 0) net" by fact
show "eventually (\<lambda>x. dist (f x) 0 \<le> dist (g x) 0) net"
using assms(1) by (rule eventually_elim1, simp add: dist_norm)
qed

lemma Lim_transform_bound:
fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
fixes g :: "'a \<Rightarrow> 'c::real_normed_vector"
assumes "eventually (\<lambda>n. norm(f n) <= norm(g n)) net"  "(g ---> 0) net"
shows "(f ---> 0) net"
using assms(1) tendsto_norm_zero [OF assms(2)]
by (rule Lim_null_comparison)

text{* Deducing things about the limit from the elements. *}

lemma Lim_in_closed_set:
assumes "closed S" "eventually (\<lambda>x. f(x) \<in> S) net" "\<not>(trivial_limit net)" "(f ---> l) net"
shows "l \<in> S"
proof (rule ccontr)
assume "l \<notin> S"
with closed S have "open (- S)" "l \<in> - S"
with assms(4) have "eventually (\<lambda>x. f x \<in> - S) net"
by (rule topological_tendstoD)
with assms(2) have "eventually (\<lambda>x. False) net"
by (rule eventually_elim2) simp
with assms(3) show "False"
qed

text{* Need to prove closed(cball(x,e)) before deducing this as a corollary. *}

lemma Lim_dist_ubound:
assumes "\<not>(trivial_limit net)" "(f ---> l) net" "eventually (\<lambda>x. dist a (f x) <= e) net"
shows "dist a l <= e"
proof-
have "dist a l \<in> {..e}"
proof (rule Lim_in_closed_set)
show "closed {..e}" by simp
show "eventually (\<lambda>x. dist a (f x) \<in> {..e}) net" by (simp add: assms)
show "\<not> trivial_limit net" by fact
show "((\<lambda>x. dist a (f x)) ---> dist a l) net" by (intro tendsto_intros assms)
qed
thus ?thesis by simp
qed

lemma Lim_norm_ubound:
fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
assumes "\<not>(trivial_limit net)" "(f ---> l) net" "eventually (\<lambda>x. norm(f x) <= e) net"
shows "norm(l) <= e"
proof-
have "norm l \<in> {..e}"
proof (rule Lim_in_closed_set)
show "closed {..e}" by simp
show "eventually (\<lambda>x. norm (f x) \<in> {..e}) net" by (simp add: assms)
show "\<not> trivial_limit net" by fact
show "((\<lambda>x. norm (f x)) ---> norm l) net" by (intro tendsto_intros assms)
qed
thus ?thesis by simp
qed

lemma Lim_norm_lbound:
fixes f :: "'a \<Rightarrow> 'b::real_normed_vector"
assumes "\<not> (trivial_limit net)"  "(f ---> l) net"  "eventually (\<lambda>x. e <= norm(f x)) net"
shows "e \<le> norm l"
proof-
have "norm l \<in> {e..}"
proof (rule Lim_in_closed_set)
show "closed {e..}" by simp
show "eventually (\<lambda>x. norm (f x) \<in> {e..}) net" by (simp add: assms)
show "\<not> trivial_limit net" by fact
show "((\<lambda>x. norm (f x)) ---> norm l) net" by (intro tendsto_intros assms)
qed
thus ?thesis by simp
qed

text{* Limit under bilinear function *}

lemma Lim_bilinear:
assumes "(f ---> l) net" and "(g ---> m) net" and "bounded_bilinear h"
shows "((\<lambda>x. h (f x) (g x)) ---> (h l m)) net"
using bounded_bilinear h (f ---> l) net (g ---> m) net
by (rule bounded_bilinear.tendsto)

text{* These are special for limits out of the same vector space. *}

lemma Lim_within_id: "(id ---> a) (at a within s)"
unfolding id_def by (rule tendsto_ident_at)

lemma Lim_at_id: "(id ---> a) (at a)"
unfolding id_def by (rule tendsto_ident_at)

lemma Lim_at_zero:
fixes a :: "'a::real_normed_vector"
fixes l :: "'b::topological_space"
shows "(f ---> l) (at a) \<longleftrightarrow> ((\<lambda>x. f(a + x)) ---> l) (at 0)" (is "?lhs = ?rhs")
using LIM_offset_zero LIM_offset_zero_cancel ..

text{* It's also sometimes useful to extract the limit point from the filter. *}

abbreviation netlimit :: "'a::t2_space filter \<Rightarrow> 'a" where
"netlimit F \<equiv> Lim F (\<lambda>x. x)"

lemma netlimit_within:
"\<not> trivial_limit (at a within S) \<Longrightarrow> netlimit (at a within S) = a"
by (rule tendsto_Lim) (auto intro: tendsto_intros)

lemma netlimit_at:
fixes a :: "'a::{perfect_space,t2_space}"
shows "netlimit (at a) = a"
using netlimit_within [of a UNIV] by simp

lemma lim_within_interior:
"x \<in> interior S \<Longrightarrow> (f ---> l) (at x within S) \<longleftrightarrow> (f ---> l) (at x)"
by (metis at_within_interior)

lemma netlimit_within_interior:
fixes x :: "'a::{t2_space,perfect_space}"
assumes "x \<in> interior S"
shows "netlimit (at x within S) = x"
using assms by (metis at_within_interior netlimit_at)

text{* Transformation of limit. *}

lemma Lim_transform:
fixes f g :: "'a::type \<Rightarrow> 'b::real_normed_vector"
assumes "((\<lambda>x. f x - g x) ---> 0) net" "(f ---> l) net"
shows "(g ---> l) net"
using tendsto_diff [OF assms(2) assms(1)] by simp

lemma Lim_transform_eventually:
"eventually (\<lambda>x. f x = g x) net \<Longrightarrow> (f ---> l) net \<Longrightarrow> (g ---> l) net"
apply (rule topological_tendstoI)
apply (drule (2) topological_tendstoD)
apply (erule (1) eventually_elim2, simp)
done

lemma Lim_transform_within:
assumes "0 < d" and "\<forall>x'\<in>S. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x'"
and "(f ---> l) (at x within S)"
shows "(g ---> l) (at x within S)"
proof (rule Lim_transform_eventually)
show "eventually (\<lambda>x. f x = g x) (at x within S)"
using assms(1,2) by (auto simp: dist_nz eventually_at)
show "(f ---> l) (at x within S)" by fact
qed

lemma Lim_transform_at:
assumes "0 < d" and "\<forall>x'. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x'"
and "(f ---> l) (at x)"
shows "(g ---> l) (at x)"
proof (rule Lim_transform_eventually)
show "eventually (\<lambda>x. f x = g x) (at x)"
unfolding eventually_at2
using assms(1,2) by auto
show "(f ---> l) (at x)" by fact
qed

text{* Common case assuming being away from some crucial point like 0. *}

lemma Lim_transform_away_within:
fixes a b :: "'a::t1_space"
assumes "a \<noteq> b" and "\<forall>x\<in>S. x \<noteq> a \<and> x \<noteq> b \<longrightarrow> f x = g x"
and "(f ---> l) (at a within S)"
shows "(g ---> l) (at a within S)"
proof (rule Lim_transform_eventually)
show "(f ---> l) (at a within S)" by fact
show "eventually (\<lambda>x. f x = g x) (at a within S)"
unfolding eventually_at_topological
by (rule exI [where x="- {b}"], simp add: open_Compl assms)
qed

lemma Lim_transform_away_at:
fixes a b :: "'a::t1_space"
assumes ab: "a\<noteq>b" and fg: "\<forall>x. x \<noteq> a \<and> x \<noteq> b \<longrightarrow> f x = g x"
and fl: "(f ---> l) (at a)"
shows "(g ---> l) (at a)"
using Lim_transform_away_within[OF ab, of UNIV f g l] fg fl
by simp

text{* Alternatively, within an open set. *}

lemma Lim_transform_within_open:
assumes "open S" and "a \<in> S" and "\<forall>x\<in>S. x \<noteq> a \<longrightarrow> f x = g x"
and "(f ---> l) (at a)"
shows "(g ---> l) (at a)"
proof (rule Lim_transform_eventually)
show "eventually (\<lambda>x. f x = g x) (at a)"
unfolding eventually_at_topological
using assms(1,2,3) by auto
show "(f ---> l) (at a)" by fact
qed

text{* A congruence rule allowing us to transform limits assuming not at point. *}

(* FIXME: Only one congruence rule for tendsto can be used at a time! *)

assumes "a = b" "x = y" "S = T"
assumes "\<And>x. x \<noteq> b \<Longrightarrow> x \<in> T \<Longrightarrow> f x = g x"
shows "(f ---> x) (at a within S) \<longleftrightarrow> (g ---> y) (at b within T)"
unfolding tendsto_def eventually_at_topological
using assms by simp

assumes "a = b" "x = y"
assumes "\<And>x. x \<noteq> a \<Longrightarrow> f x = g x"
shows "((\<lambda>x. f x) ---> x) (at a) \<longleftrightarrow> ((g ---> y) (at a))"
unfolding tendsto_def eventually_at_topological
using assms by simp

text{* Useful lemmas on closure and set of possible sequential limits.*}

lemma closure_sequential:
fixes l :: "'a::first_countable_topology"
shows "l \<in> closure S \<longleftrightarrow> (\<exists>x. (\<forall>n. x n \<in> S) \<and> (x ---> l) sequentially)" (is "?lhs = ?rhs")
proof
assume "?lhs" moreover
{ assume "l \<in> S"
hence "?rhs" using tendsto_const[of l sequentially] by auto
} moreover
{ assume "l islimpt S"
hence "?rhs" unfolding islimpt_sequential by auto
} ultimately
show "?rhs" unfolding closure_def by auto
next
assume "?rhs"
thus "?lhs" unfolding closure_def unfolding islimpt_sequential by auto
qed

lemma closed_sequential_limits:
fixes S :: "'a::first_countable_topology set"
shows "closed S \<longleftrightarrow> (\<forall>x l. (\<forall>n. x n \<in> S) \<and> (x ---> l) sequentially \<longrightarrow> l \<in> S)"
unfolding closed_limpt
using closure_sequential [where 'a='a] closure_closed [where 'a='a] closed_limpt [where 'a='a] islimpt_sequential [where 'a='a] mem_delete [where 'a='a]
by metis

lemma closure_approachable:
fixes S :: "'a::metric_space set"
shows "x \<in> closure S \<longleftrightarrow> (\<forall>e>0. \<exists>y\<in>S. dist y x < e)"
apply (auto simp add: closure_def islimpt_approachable)
by (metis dist_self)

lemma closed_approachable:
fixes S :: "'a::metric_space set"
shows "closed S ==> (\<forall>e>0. \<exists>y\<in>S. dist y x < e) \<longleftrightarrow> x \<in> S"
by (metis closure_closed closure_approachable)

lemma closure_contains_Inf:
fixes S :: "real set"
assumes "S \<noteq> {}" "\<forall>x\<in>S. B \<le> x"
shows "Inf S \<in> closure S"
unfolding closure_approachable
proof safe
have *: "\<forall>x\<in>S. Inf S \<le> x"
using cInf_lower_EX[of _ S] assms by metis

fix e :: real assume "0 < e"
then have "Inf S < Inf S + e" by simp
with assms obtain x where "x \<in> S" "x < Inf S + e"
by (subst (asm) cInf_less_iff[of _ B]) auto
with * show "\<exists>x\<in>S. dist x (Inf S) < e"
by (intro bexI[of _ x]) (auto simp add: dist_real_def)
qed

lemma closed_contains_Inf:
fixes S :: "real set"
assumes "S \<noteq> {}" "\<forall>x\<in>S. B \<le> x"
and "closed S"
shows "Inf S \<in> S"
by (metis closure_contains_Inf closure_closed assms)

lemma not_trivial_limit_within_ball:
"(\<not> trivial_limit (at x within S)) = (\<forall>e>0. S \<inter> ball x e - {x} \<noteq> {})"
(is "?lhs = ?rhs")
proof -
{ assume "?lhs"
{ fix e :: real
assume "e>0"
then obtain y where "y:(S-{x}) & dist y x < e"
using ?lhs not_trivial_limit_within[of x S] closure_approachable[of x "S - {x}"]
by auto
then have "y : (S Int ball x e - {x})"
unfolding ball_def by (simp add: dist_commute)
then have "S Int ball x e - {x} ~= {}" by blast
} then have "?rhs" by auto
}
moreover
{ assume "?rhs"
{ fix e :: real
assume "e>0"
then obtain y where "y : (S Int ball x e - {x})" using ?rhs by blast
then have "y:(S-{x}) & dist y x < e"
unfolding ball_def by (simp add: dist_commute)
then have "EX y:(S-{x}). dist y x < e" by auto
}
then have "?lhs"
using not_trivial_limit_within[of x S] closure_approachable[of x "S - {x}"] by auto
}
ultimately show ?thesis by auto
qed

subsection {* Infimum Distance *}

definition "infdist x A = (if A = {} then 0 else Inf {dist x a|a. a \<in> A})"

lemma infdist_notempty: "A \<noteq> {} \<Longrightarrow> infdist x A = Inf {dist x a|a. a \<in> A}"

lemma infdist_nonneg:
shows "0 \<le> infdist x A"
using assms by (auto simp add: infdist_def intro: cInf_greatest)

lemma infdist_le:
assumes "a \<in> A"
assumes "d = dist x a"
shows "infdist x A \<le> d"
using assms by (auto intro!: cInf_lower[where z=0] simp add: infdist_def)

lemma infdist_zero[simp]:
assumes "a \<in> A" shows "infdist a A = 0"
proof -
from infdist_le[OF assms, of "dist a a"] have "infdist a A \<le> 0" by auto
with infdist_nonneg[of a A] assms show "infdist a A = 0" by auto
qed

lemma infdist_triangle:
shows "infdist x A \<le> infdist y A + dist x y"
proof cases
assume "A = {}" thus ?thesis by (simp add: infdist_def)
next
assume "A \<noteq> {}" 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 A \<noteq> {} 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 A \<noteq> {}]
proof (rule cInf_lower2)
show "dist x a \<in> {dist x a |a. a \<in> A}" using a \<in> A by auto
show "dist x a \<le> d" unfolding d by (rule dist_triangle)
fix d assume "d \<in> {dist x a |a. a \<in> A}"
then obtain a where "a \<in> A" "d = dist x a" by auto
thus "infdist x A \<le> d" by (rule infdist_le)
qed
qed
also have "\<dots> = dist x y + infdist y A"
proof (rule cInf_eq, safe)
fix a assume "a \<in> A"
thus "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"
hence "i - dist x y \<le> infdist y A" unfolding infdist_notempty[OF A \<noteq> {}] using a \<in> A
by (intro cInf_greatest) (auto simp: field_simps)
thus "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
hence "ball x (infdist x A) \<inter> closure A = {}" apply auto
by (metis 0 < infdist x A x \<in> closure A closure_approachable dist_commute
eucl_less_not_refl euclidean_trans(2) infdist_le)
hence "x \<notin> closure A" by (metis 0 < infdist x A centre_in_ball disjoint_iff_not_equal)
thus False using x \<in> closure A 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
proof (safe, rule ccontr)
fix e::real assume "0 < e"
assume "\<not> (\<exists>y\<in>A. dist y x < e)"
hence "infdist x A \<ge> e" using a \<in> A
unfolding infdist_def
by (force simp: dist_commute intro: cInf_greatest)
with x 0 < e show False by auto
qed
qed

lemma in_closed_iff_infdist_zero:
assumes "closed A" "A \<noteq> {}"
shows "x \<in> A \<longleftrightarrow> infdist x A = 0"
proof -
have "x \<in> closure A \<longleftrightarrow> infdist x A = 0"
by (rule in_closure_iff_infdist_zero) fact
with assms show ?thesis by simp
qed

lemma tendsto_infdist [tendsto_intros]:
assumes f: "(f ---> l) F"
shows "((\<lambda>x. infdist (f x) A) ---> infdist l A) F"
proof (rule tendstoI)
fix e ::real assume "0 < e"
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"
also assume "dist (f x) l < e"
finally show "dist (infdist (f x) A) (infdist l A) < e" .
qed
qed

text{* Some other lemmas about sequences. *}

lemma sequentially_offset:
assumes "eventually (\<lambda>i. P i) sequentially"
shows "eventually (\<lambda>i. P (i + k)) sequentially"
using assms unfolding eventually_sequentially by (metis trans_le_add1)

lemma seq_offset:
assumes "(f ---> l) sequentially"
shows "((\<lambda>i. f (i + k)) ---> l) sequentially"
using assms by (rule LIMSEQ_ignore_initial_segment) (* FIXME: redundant *)

lemma seq_offset_neg:
"(f ---> l) sequentially ==> ((\<lambda>i. f(i - k)) ---> l) sequentially"
apply (rule topological_tendstoI)
apply (drule (2) topological_tendstoD)
apply (simp only: eventually_sequentially)
apply (subgoal_tac "\<And>N k (n::nat). N + k <= n ==> N <= n - k")
apply metis
by arith

lemma seq_offset_rev:
"((\<lambda>i. f(i + k)) ---> l) sequentially ==> (f ---> l) sequentially"
by (rule LIMSEQ_offset) (* FIXME: redundant *)

lemma seq_harmonic: "((\<lambda>n. inverse (real n)) ---> 0) sequentially"
using LIMSEQ_inverse_real_of_nat by (rule LIMSEQ_imp_Suc)

subsection {* More properties of closed balls *}

lemma closed_cball: "closed (cball x e)"
unfolding cball_def closed_def
unfolding Collect_neg_eq [symmetric] not_le
apply (clarsimp simp add: open_dist, rename_tac y)
apply (rule_tac x="dist x y - e" in exI, clarsimp)
apply (rename_tac x')
apply (cut_tac x=x and y=x' and z=y in dist_triangle)
apply simp
done

lemma open_contains_cball: "open S \<longleftrightarrow> (\<forall>x\<in>S. \<exists>e>0.  cball x e \<subseteq> S)"
proof-
{ fix x and e::real assume "x\<in>S" "e>0" "ball x e \<subseteq> S"
hence "\<exists>d>0. cball x d \<subseteq> S" unfolding subset_eq by (rule_tac x="e/2" in exI, auto)
} moreover
{ fix x and e::real assume "x\<in>S" "e>0" "cball x e \<subseteq> S"
hence "\<exists>d>0. ball x d \<subseteq> S" unfolding subset_eq apply(rule_tac x="e/2" in exI) by auto
} ultimately
show ?thesis unfolding open_contains_ball by auto
qed

lemma open_contains_cball_eq: "open S ==> (\<forall>x. x \<in> S \<longleftrightarrow> (\<exists>e>0. cball x e \<subseteq> S))"
by (metis open_contains_cball subset_eq order_less_imp_le centre_in_cball)

lemma mem_interior_cball: "x \<in> interior S \<longleftrightarrow> (\<exists>e>0. cball x e \<subseteq> S)"
apply (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 = ?rhs")
proof
assume "?lhs"
{ assume "e \<le> 0"
hence *:"ball x e = {}" using ball_eq_empty[of x e] by auto
have False using ?lhs unfolding * using islimpt_EMPTY[of y] by auto
}
hence "e > 0" by (metis not_less)
moreover
have "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] ?lhs unfolding closed_limpt by auto
ultimately show "?rhs" by auto
next
assume "?rhs" hence "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 thus "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
proof(cases "x=y")
case True hence False using d \<le> dist x y d>0 by auto
thus "\<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[THEN sym] 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))", THEN sym, of "y - x"]
unfolding scaleR_minus_left scaleR_one
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 x\<noteq>y[unfolded dist_nz, unfolded dist_norm] by auto
also have "\<dots> \<le> e - d/2" using d \<le> dist x y and d>0 and ?rhs by(auto simp add: dist_norm)
finally have "y - (d / (2 * dist y x)) *\<^sub>R (y - x) \<in> ball x e" using d>0 by auto

moreover

have "(d / (2*dist y x)) *\<^sub>R (y - x) \<noteq> 0"
using x\<noteq>y[unfolded dist_nz] d>0 unfolding scaleR_eq_0_iff by (auto simp add: 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 d>0 x\<noteq>y[unfolded dist_nz] dist_commute[of x y]
unfolding dist_norm by auto
ultimately show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d" by (rule_tac  x="y - (d / (2*dist y x)) *\<^sub>R (y - x)" in bexI) auto
qed
next
case False hence "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 d > 0 e>0 by auto
show "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
unfolding x = y
using z \<noteq> y **
by (rule_tac x=z in bexI, auto simp add: dist_commute)
next
case False thus "\<exists>x'\<in>ball x e. x' \<noteq> y \<and> dist x' y < d"
using d>0 d > dist x y ?rhs by(rule_tac x=x in bexI, auto)
qed
qed  }
thus "?lhs" unfolding mem_cball islimpt_approachable mem_ball by auto
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. *)
def k \<equiv> "min 1 (r / (2 * dist x y))"
def z \<equiv> "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 0 < r
hence "z \<in> T" using \<forall>z. dist z y < r \<longrightarrow> z \<in> T by simp
have "dist x z < dist x y"
unfolding z_def2 dist_norm
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 divide_pos_pos zero_less_dist_iff 0 < r x \<noteq> y)
apply (simp add: zero_less_dist_iff x \<noteq> y)
done
hence "z \<in> ball x (dist x y)" by simp
have "z \<noteq> y"
unfolding z_def k_def using x \<noteq> y 0 < r
show "\<exists>z\<in>ball x (dist x y). z \<in> T \<and> z \<noteq> y"
using z \<in> ball x (dist x y) z \<in> T z \<noteq> y
by fast
qed

lemma closure_ball:
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 clarify
apply (rule closure_ball_lemma)
done

(* In a trivial vector space, this fails for e = 0. *)
lemma interior_cball:
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 "ball x e = {}" using ball_empty[of e x] by auto moreover
{ fix y assume "y \<in> cball x e"
hence False unfolding mem_cball using dist_nz[of x y] cs by auto  }
hence "cball x e = {}" by auto
hence "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 add: dist_commute)
hence xa_cball:"xa \<in> cball x e" using as(1) by auto

hence "y \<in> ball x e" proof(cases "x = y")
case True
hence "e>0" using xa_y[unfolded dist_nz] xa_cball[unfolded mem_cball] by (auto simp add: dist_commute)
thus "y \<in> ball x e" using x = y  by simp
next
case False
have "dist (y + (d / 2 / dist y x) *\<^sub>R (y - x)) y < d" unfolding dist_norm
using d>0 norm_ge_zero[of "y - x"] x \<noteq> y by auto
hence *:"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 x \<noteq> y by auto
hence **:"d / (2 * norm (y - x)) > 0" unfolding zero_less_norm_iff[THEN sym]
using d>0 divide_pos_pos[of d "2*norm (y - x)"] 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 add: dist_norm algebra_simps)
also have "\<dots> = norm ((1 + d / (2 * norm (y - x))) *\<^sub>R (y - x))"
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 add: distrib_right dist_norm)
finally have "e \<ge> dist x y +d/2" using *[unfolded mem_cball] by (auto simp add: dist_commute)
thus "y \<in> ball x e" unfolding mem_ball using d>0 by auto
qed  }
hence "\<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 frontier_ball:
fixes a :: "'a::real_normed_vector"
shows "0 < e ==> frontier(ball a e) = {x. dist a x = e}"
apply (simp add: frontier_def closure_ball interior_open order_less_imp_le)
by arith

lemma frontier_cball:
fixes a :: "'a::{real_normed_vector, perfect_space}"
shows "frontier(cball a e) = {x. dist a x = e}"
apply (simp add: frontier_def interior_cball closed_cball order_less_imp_le)
by arith

lemma cball_eq_empty: "(cball x e = {}) \<longleftrightarrow> e < 0"
by (metis zero_le_dist dist_self order_less_le_trans)
lemma cball_empty: "e < 0 ==> cball x e = {}" by (simp add: cball_eq_empty)

lemma cball_eq_sing:
fixes x :: "'a::{metric_space,perfect_space}"
shows "(cball x e = {x}) \<longleftrightarrow> e = 0"
proof (rule linorder_cases)
assume e: "0 < e"
obtain a where "a \<noteq> x" "dist a x < e"
using perfect_choose_dist [OF e] by auto
hence "a \<noteq> x" "dist x a \<le> e" by (auto simp add: dist_commute)
with e show ?thesis by (auto simp add: set_eq_iff)
qed auto

lemma cball_sing:
fixes x :: "'a::metric_space"
shows "e = 0 ==> cball x e = {x}"

subsection {* Boundedness *}

(* FIXME: This has to be unified with BSEQ!! *)
definition (in metric_space)
bounded :: "'a set \<Rightarrow> bool" where
"bounded S \<longleftrightarrow> (\<exists>x e. \<forall>y\<in>S. dist x y \<le> e)"

lemma bounded_subset_cball: "bounded S \<longleftrightarrow> (\<exists>e x. S \<subseteq> cball x e)"
unfolding bounded_def subset_eq by auto

lemma bounded_any_center: "bounded S \<longleftrightarrow> (\<exists>e. \<forall>y\<in>S. dist a y \<le> e)"
unfolding bounded_def
apply safe
apply (rule_tac x="dist a x + e" in exI, clarify)
apply (drule (1) bspec)
apply (erule order_trans [OF dist_triangle add_left_mono])
apply auto
done

lemma bounded_iff: "bounded S \<longleftrightarrow> (\<exists>a. \<forall>x\<in>S. norm x \<le> a)"
unfolding bounded_any_center [where a=0]

lemma bounded_realI: assumes "\<forall>x\<in>s. abs (x::real) \<le> B" shows "bounded s"
unfolding bounded_def dist_real_def apply(rule_tac x=0 in exI)
using assms by auto

lemma bounded_empty [simp]: "bounded {}"

lemma bounded_subset: "bounded T \<Longrightarrow> S \<subseteq> T ==> bounded S"
by (metis bounded_def subset_eq)

lemma bounded_interior[intro]: "bounded S ==> bounded(interior S)"
by (metis bounded_subset interior_subset)

lemma bounded_closure[intro]: assumes "bounded S" shows "bounded(closure S)"
proof-
from assms obtain x and a where a: "\<forall>y\<in>S. dist x y \<le> a" unfolding bounded_def by auto
{ fix y assume "y \<in> closure S"
then obtain f where f: "\<forall>n. f n \<in> S"  "(f ---> y) sequentially"
unfolding closure_sequential by auto
have "\<forall>n. f n \<in> S \<longrightarrow> dist x (f n) \<le> a" using a by simp
hence "eventually (\<lambda>n. dist x (f n) \<le> a) sequentially"
by (rule eventually_mono, simp add: f(1))
have "dist x y \<le> a"
apply (rule Lim_dist_ubound [of sequentially f])
apply (rule trivial_limit_sequentially)
apply (rule f(2))
apply fact
done
}
thus ?thesis unfolding bounded_def by auto
qed

lemma bounded_cball[simp,intro]: "bounded (cball x e)"
apply (rule_tac x=x in exI)
apply (rule_tac x=e in exI)
apply auto
done

lemma bounded_ball[simp,intro]: "bounded(ball x e)"
by (metis ball_subset_cball bounded_cball bounded_subset)

lemma bounded_Un[simp]: "bounded (S \<union> T) \<longleftrightarrow> bounded S \<and> bounded T"
apply (rename_tac x y r s)
apply (rule_tac x=x in exI)
apply (rule_tac x="max r (dist x y + s)" in exI)
apply (rule ballI, rename_tac z, safe)
apply (drule (1) bspec, simp)
apply (drule (1) bspec)
apply (rule min_max.le_supI2)
apply (erule order_trans [OF dist_triangle add_left_mono])
done

lemma bounded_Union[intro]: "finite F \<Longrightarrow> (\<forall>S\<in>F. bounded S) \<Longrightarrow> bounded(\<Union>F)"
by (induct rule: finite_induct[of F], auto)

lemma bounded_UN [intro]: "finite A \<Longrightarrow> \<forall>x\<in>A. bounded (B x) \<Longrightarrow> bounded (\<Union>x\<in>A. B x)"
by (induct set: finite, auto)

lemma bounded_insert [simp]: "bounded (insert x S) \<longleftrightarrow> bounded S"
proof -
have "\<forall>y\<in>{x}. dist x y \<le> 0" by simp
hence "bounded {x}" unfolding bounded_def by fast
thus ?thesis by (metis insert_is_Un bounded_Un)
qed

lemma finite_imp_bounded [intro]: "finite S \<Longrightarrow> bounded S"
by (induct set: finite, simp_all)

lemma bounded_pos: "bounded S \<longleftrightarrow> (\<exists>b>0. \<forall>x\<in> S. norm x <= b)"
apply (subgoal_tac "\<And>x (y::real). 0 < 1 + abs y \<and> (x <= y \<longrightarrow> x <= 1 + abs y)")
by metis arith

lemma Bseq_eq_bounded: "Bseq f \<longleftrightarrow> bounded (range f::_::real_normed_vector set)"
unfolding Bseq_def bounded_pos by auto

lemma bounded_Int[intro]: "bounded S \<or> bounded T \<Longrightarrow> bounded (S \<inter> T)"
by (metis Int_lower1 Int_lower2 bounded_subset)

lemma bounded_diff[intro]: "bounded S ==> bounded (S - T)"
apply (metis Diff_subset bounded_subset)
done

lemma not_bounded_UNIV[simp, intro]:
"\<not> bounded (UNIV :: 'a::{real_normed_vector, perfect_space} set)"
obtain x :: 'a where "x \<noteq> 0"
using perfect_choose_dist [OF zero_less_one] by fast
fix b::real  assume b: "b >0"
have b1: "b +1 \<ge> 0" using b by simp
with x \<noteq> 0 have "b < norm (scaleR (b + 1) (sgn x))"
then show "\<exists>x::'a. b < norm x" ..
qed

lemma bounded_linear_image:
assumes "bounded S" "bounded_linear f"
shows "bounded(f  S)"
proof-
from assms(1) obtain b where b:"b>0" "\<forall>x\<in>S. norm x \<le> b" unfolding bounded_pos by auto
from assms(2) obtain B where B:"B>0" "\<forall>x. norm (f x) \<le> B * norm x" using bounded_linear.pos_bounded by (auto simp add: mult_ac)
{ fix x assume "x\<in>S"
hence "norm x \<le> b" using b by auto
hence "norm (f x) \<le> B * b" using B(2) apply(erule_tac x=x in allE)
by (metis B(1) B(2) order_trans mult_le_cancel_left_pos)
}
thus ?thesis unfolding bounded_pos apply(rule_tac x="b*B" in exI)
using b B mult_pos_pos [of b B] by (auto simp add: mult_commute)
qed

lemma bounded_scaling:
fixes S :: "'a::real_normed_vector set"
shows "bounded S \<Longrightarrow> bounded ((\<lambda>x. c *\<^sub>R x)  S)"
apply (rule bounded_linear_image, assumption)
apply (rule bounded_linear_scaleR_right)
done

lemma bounded_translation:
fixes S :: "'a::real_normed_vector set"
assumes "bounded S" shows "bounded ((\<lambda>x. a + x)  S)"
proof-
from assms obtain b where b:"b>0" "\<forall>x\<in>S. norm x \<le> b" unfolding bounded_pos by auto
{ fix x assume "x\<in>S"
hence "norm (a + x) \<le> b + norm a" using norm_triangle_ineq[of a x] b by auto
}
thus ?thesis unfolding bounded_pos using norm_ge_zero[of a] b(1) using add_strict_increasing[of b 0 "norm a"]
by (auto intro!: exI[of _ "b + norm a"])
qed

text{* Some theorems on sups and infs using the notion "bounded". *}

lemma bounded_real:
fixes S :: "real set"
shows "bounded S \<longleftrightarrow>  (\<exists>a. \<forall>x\<in>S. abs x <= a)"

lemma bounded_has_Sup:
fixes S :: "real set"
assumes "bounded S" "S \<noteq> {}"
shows "\<forall>x\<in>S. x <= Sup S" and "\<forall>b. (\<forall>x\<in>S. x <= b) \<longrightarrow> Sup S <= b"
proof
fix x assume "x\<in>S"
thus "x \<le> Sup S"
by (metis cSup_upper abs_le_D1 assms(1) bounded_real)
next
show "\<forall>b. (\<forall>x\<in>S. x \<le> b) \<longrightarrow> Sup S \<le> b" using assms
by (metis cSup_least)
qed

lemma Sup_insert:
fixes S :: "real set"
shows "bounded S ==> Sup(insert x S) = (if S = {} then x else max x (Sup S))"
apply (subst cSup_insert_If)
apply (rule bounded_has_Sup(1)[of S, rule_format])
apply (auto simp: sup_max)
done

lemma Sup_insert_finite:
fixes S :: "real set"
shows "finite S \<Longrightarrow> Sup(insert x S) = (if S = {} then x else max x (Sup S))"
apply (rule Sup_insert)
apply (rule finite_imp_bounded)
by simp

lemma bounded_has_Inf:
fixes S :: "real set"
assumes "bounded S"  "S \<noteq> {}"
shows "\<forall>x\<in>S. x >= Inf S" and "\<forall>b. (\<forall>x\<in>S. x >= b) \<longrightarrow> Inf S >= b"
proof
fix x assume "x\<in>S"
from assms(1) obtain a where a:"\<forall>x\<in>S. \<bar>x\<bar> \<le> a" unfolding bounded_real by auto
thus "x \<ge> Inf S" using x\<in>S
by (metis cInf_lower_EX abs_le_D2 minus_le_iff)
next
show "\<forall>b. (\<forall>x\<in>S. x >= b) \<longrightarrow> Inf S \<ge> b" using assms
by (metis cInf_greatest)
qed

lemma Inf_insert:
fixes S :: "real set"
shows "bounded S ==> Inf(insert x S) = (if S = {} then x else min x (Inf S))"
apply (subst cInf_insert_if)
apply (rule bounded_has_Inf(1)[of S, rule_format])
apply (auto simp: inf_min)
done

lemma Inf_insert_finite:
fixes S :: "real set"
shows "finite S ==> Inf(insert x S) = (if S = {} then x else min x (Inf S))"
by (rule Inf_insert, rule finite_imp_bounded, simp)

subsection {* Compactness *}

subsubsection {* Bolzano-Weierstrass property *}

lemma heine_borel_imp_bolzano_weierstrass:
assumes "compact s" "infinite t"  "t \<subseteq> s"
shows "\<exists>x \<in> s. x islimpt t"
proof(rule ccontr)
assume "\<not> (\<exists>x \<in> s. x islimpt t)"
then obtain f where f:"\<forall>x\<in>s. x \<in> f x \<and> open (f x) \<and> (\<forall>y\<in>t. y \<in> f x \<longrightarrow> y = x)" unfolding islimpt_def
using bchoice[of s "\<lambda> x T. x \<in> T \<and> open T \<and> (\<forall>y\<in>t. y \<in> T \<longrightarrow> y = x)"] by auto
obtain g where g:"g\<subseteq>{t. \<exists>x. x \<in> s \<and> t = f x}" "finite g" "s \<subseteq> \<Union>g"
using assms(1)[unfolded compact_eq_heine_borel, THEN spec[where x="{t. \<exists>x. x\<in>s \<and> t = f x}"]] using f by auto
from g(1,3) have g':"\<forall>x\<in>g. \<exists>xa \<in> s. x = f xa" by auto
{ fix x y assume "x\<in>t" "y\<in>t" "f x = f y"
hence "x \<in> f x"  "y \<in> f x \<longrightarrow> y = x" using f[THEN bspec[where x=x]] and t\<subseteq>s by auto
hence "x = y" using f x = f y and f[THEN bspec[where x=y]] and y\<in>t and t\<subseteq>s by auto  }
hence "inj_on f t" unfolding inj_on_def by simp
hence "infinite (f  t)" using assms(2) using finite_imageD by auto
moreover
{ fix x assume "x\<in>t" "f x \<notin> g"
from g(3) assms(3) x\<in>t obtain h where "h\<in>g" and "x\<in>h" by auto
then obtain y where "y\<in>s" "h = f y" using g'[THEN bspec[where x=h]] by auto
hence "y = x" using f[THEN bspec[where x=y]] and x\<in>t and x\<in>h[unfolded h = f y] by auto
hence False using f x \<notin> g h\<in>g unfolding h = f y by auto  }
hence "f  t \<subseteq> g" by auto
ultimately show False using g(2) using finite_subset by auto
qed

lemma acc_point_range_imp_convergent_subsequence:
fixes l :: "'a :: first_countable_topology"
assumes l: "\<forall>U. l\<in>U \<longrightarrow> open U \<longrightarrow> infinite (U \<inter> range f)"
shows "\<exists>r. subseq r \<and> (f \<circ> r) ----> l"
proof -
from countable_basis_at_decseq[of l] guess A . note A = this

def s \<equiv> "\<lambda>n i. SOME j. i < j \<and> f j \<in> A (Suc n)"
{ fix n i
have "infinite (A (Suc n) \<inter> range f - f{.. i})"
using l A by auto
then have "\<exists>x. x \<in> A (Suc n) \<inter> range f - f{.. i}"
unfolding ex_in_conv by (intro notI) simp
then have "\<exists>j. f j \<in> A (Suc n) \<and> j \<notin> {.. i}"
by auto
then have "\<exists>a. i < a \<and> f a \<in> A (Suc n)"
by (auto simp: not_le)
then have "i < s n i" "f (s n i) \<in> A (Suc n)"
unfolding s_def by (auto intro: someI2_ex) }
note s = this
def r \<equiv> "nat_rec (s 0 0) s"
have "subseq r"
by (auto simp: r_def s subseq_Suc_iff)
moreover
have "(\<lambda>n. f (r n)) ----> l"
proof (rule topological_tendstoI)
fix S assume "open S" "l \<in> S"
with A(3) have "eventually (\<lambda>i. A i \<subseteq> S) sequentially" by auto
moreover
{ fix i assume "Suc 0 \<le> i" then have "f (r i) \<in> A i"
by (cases i) (simp_all add: r_def s) }
then have "eventually (\<lambda>i. f (r i) \<in> A i) sequentially" by (auto simp: eventually_sequentially)
ultimately show "eventually (\<lambda>i. f (r i) \<in> S) sequentially"
by eventually_elim auto
qed
ultimately show "\<exists>r. subseq r \<and> (f \<circ> r) ----> l"
by (auto simp: convergent_def comp_def)
qed

lemma sequence_infinite_lemma:
fixes f :: "nat \<Rightarrow> 'a::t1_space"
assumes "\<forall>n. f n \<noteq> l" and "(f ---> l) sequentially"
shows "infinite (range f)"
proof
assume "finite (range f)"
hence "closed (range f)" by (rule finite_imp_closed)
hence "open (- range f)" by (rule open_Compl)
from assms(1) have "l \<in> - range f" by auto
from assms(2) have "eventually (\<lambda>n. f n \<in> - range f) sequentially"
using open (- range f) l \<in> - range f by (rule topological_tendstoD)
thus False unfolding eventually_sequentially by auto
qed

lemma closure_insert:
fixes x :: "'a::t1_space"
shows "closure (insert x s) = insert x (closure s)"
apply (rule closure_unique)
apply (rule insert_mono [OF closure_subset])
apply (rule closed_insert [OF closed_closure])
done

lemma islimpt_insert:
fixes x :: "'a::t1_space"
shows "x islimpt (insert a s) \<longleftrightarrow> x islimpt s"
proof
assume *: "x islimpt (insert a s)"
show "x islimpt s"
proof (rule islimptI)
fix t assume t: "x \<in> t" "open t"
show "\<exists>y\<in>s. y \<in> t \<and> y \<noteq> x"
proof (cases "x = a")
case True
obtain y where "y \<in> insert a s" "y \<in> t" "y \<noteq> x"
using * t by (rule islimptE)
with x = a show ?thesis by auto
next
case False
with t have t': "x \<in> t - {a}" "open (t - {a})"
obtain y where "y \<in> insert a s" "y \<in> t - {a}" "y \<noteq> x"
using * t' by (rule islimptE)
thus ?thesis by auto
qed
qed
next
assume "x islimpt s" thus "x islimpt (insert a s)"
by (rule islimpt_subset) auto
qed

lemma islimpt_finite:
fixes x :: "'a::t1_space"
shows "finite s \<Longrightarrow> \<not> x islimpt s"
by (induct set: finite, simp_all add: islimpt_insert)

lemma islimpt_union_finite:
fixes x :: "'a::t1_space"
shows "finite s \<Longrightarrow> x islimpt (s \<union> t) \<longleftrightarrow> x islimpt t"

lemma islimpt_eq_acc_point:
fixes l :: "'a :: t1_space"
shows "l islimpt S \<longleftrightarrow> (\<forall>U. l\<in>U \<longrightarrow> open U \<longrightarrow> infinite (U \<inter> S))"
proof (safe intro!: islimptI)
fix U assume "l islimpt S" "l \<in> U" "open U" "finite (U \<inter> S)"
then have "l islimpt S" "l \<in> (U - (U \<inter> S - {l}))" "open (U - (U \<inter> S - {l}))"
by (auto intro: finite_imp_closed)
then show False
by (rule islimptE) auto
next
fix T assume *: "\<forall>U. l\<in>U \<longrightarrow> open U \<longrightarrow> infinite (U \<inter> S)" "l \<in> T" "open T"
then have "infinite (T \<inter> S - {l})" by auto
then have "\<exists>x. x \<in> (T \<inter> S - {l})"
unfolding ex_in_conv by (intro notI) simp
then show "\<exists>y\<in>S. y \<in> T \<and> y \<noteq> l"
by auto
qed

lemma islimpt_range_imp_convergent_subsequence:
fixes l :: "'a :: {t1_space, first_countable_topology}"
assumes l: "l islimpt (range f)"
shows "\<exists>r. subseq r \<and> (f \<circ> r) ----> l"
using l unfolding islimpt_eq_acc_point
by (rule acc_point_range_imp_convergent_subsequence)

lemma sequence_unique_limpt:
fixes f :: "nat \<Rightarrow> 'a::t2_space"
assumes "(f ---> l) sequentially" and "l' islimpt (range f)"
shows "l' = l"
proof (rule ccontr)
assume "l' \<noteq> l"
obtain s t where "open s" "open t" "l' \<in> s" "l \<in> t" "s \<inter> t = {}"
using hausdorff [OF l' \<noteq> l] by auto
have "eventually (\<lambda>n. f n \<in> t) sequentially"
using assms(1) open t l \<in> t by (rule topological_tendstoD)
then obtain N where "\<forall>n\<ge>N. f n \<in> t"
unfolding eventually_sequentially by auto

have "UNIV = {..<N} \<union> {N..}" by auto
hence "l' islimpt (f  ({..<N} \<union> {N..}))" using assms(2) by simp
hence "l' islimpt (f  {..<N} \<union> f  {N..})" by (simp add: image_Un)
hence "l' islimpt (f  {N..})" by (simp add: islimpt_union_finite)
then obtain y where "y \<in> f  {N..}" "y \<in> s" "y \<noteq> l'"
using l' \<in> s open s by (rule islimptE)
then obtain n where "N \<le> n" "f n \<in> s" "f n \<noteq> l'" by auto
with \<forall>n\<ge>N. f n \<in> t have "f n \<in> s \<inter> t" by simp
with s \<inter> t = {} show False by simp
qed

lemma bolzano_weierstrass_imp_closed:
fixes s :: "'a::{first_countable_topology, t2_space} set"
assumes "\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t)"
shows "closed s"
proof-
{ fix x l assume as: "\<forall>n::nat. x n \<in> s" "(x ---> l) sequentially"
hence "l \<in> s"
proof(cases "\<forall>n. x n \<noteq> l")
case False thus "l\<in>s" using as(1) by auto
next
case True note cas = this
with as(2) have "infinite (range x)" using sequence_infinite_lemma[of x l] by auto
then obtain l' where "l'\<in>s" "l' islimpt (range x)" using assms[THEN spec[where x="range x"]] as(1) by auto
thus "l\<in>s" using sequence_unique_limpt[of x l l'] using as cas by auto
qed  }
thus ?thesis unfolding closed_sequential_limits by fast
qed

lemma compact_imp_bounded:
assumes "compact U" shows "bounded U"
proof -
have "compact U" "\<forall>x\<in>U. open (ball x 1)" "U \<subseteq> (\<Union>x\<in>U. ball x 1)" using assms by auto
then obtain D where D: "D \<subseteq> U" "finite D" "U \<subseteq> (\<Union>x\<in>D. ball x 1)"
by (elim compactE_image)
from finite D have "bounded (\<Union>x\<in>D. ball x 1)"
thus "bounded U" using U \<subseteq> (\<Union>x\<in>D. ball x 1)
by (rule bounded_subset)
qed

text{* In particular, some common special cases. *}

lemma compact_union [intro]:
assumes "compact s" "compact t" shows " compact (s \<union> t)"
proof (rule compactI)
fix f assume *: "Ball f open" "s \<union> t \<subseteq> \<Union>f"
from * compact s obtain s' where "s' \<subseteq> f \<and> finite s' \<and> s \<subseteq> \<Union>s'"
unfolding compact_eq_heine_borel by (auto elim!: allE[of _ f]) metis
moreover from * compact t obtain t' where "t' \<subseteq> f \<and> finite t' \<and> t \<subseteq> \<Union>t'"
unfolding compact_eq_heine_borel by (auto elim!: allE[of _ f]) metis
ultimately show "\<exists>f'\<subseteq>f. finite f' \<and> s \<union> t \<subseteq> \<Union>f'"
by (auto intro!: exI[of _ "s' \<union> t'"])
qed

lemma compact_Union [intro]: "finite S \<Longrightarrow> (\<And>T. T \<in> S \<Longrightarrow> compact T) \<Longrightarrow> compact (\<Union>S)"
by (induct set: finite) auto

lemma compact_UN [intro]:
"finite A \<Longrightarrow> (\<And>x. x \<in> A \<Longrightarrow> compact (B x)) \<Longrightarrow> compact (\<Union>x\<in>A. B x)"
unfolding SUP_def by (rule compact_Union) auto

lemma closed_inter_compact [intro]:
assumes "closed s" and "compact t"
shows "compact (s \<inter> t)"
using compact_inter_closed [of t s] assms

lemma compact_inter [intro]:
fixes s t :: "'a :: t2_space set"
assumes "compact s" and "compact t"
shows "compact (s \<inter> t)"
using assms by (intro compact_inter_closed compact_imp_closed)

lemma compact_sing [simp]: "compact {a}"
unfolding compact_eq_heine_borel by auto

lemma compact_insert [simp]:
assumes "compact s" shows "compact (insert x s)"
proof -
have "compact ({x} \<union> s)"
using compact_sing assms by (rule compact_union)
thus ?thesis by simp
qed

lemma finite_imp_compact:
shows "finite s \<Longrightarrow> compact s"
by (induct set: finite) simp_all

lemma open_delete:
fixes s :: "'a::t1_space set"
shows "open s \<Longrightarrow> open (s - {x})"

text{* Finite intersection property *}

lemma inj_setminus: "inj_on uminus (A::'a set set)"
by (auto simp: inj_on_def)

lemma compact_fip:
"compact U \<longleftrightarrow>
(\<forall>A. (\<forall>a\<in>A. closed a) \<longrightarrow> (\<forall>B \<subseteq> A. finite B \<longrightarrow> U \<inter> \<Inter>B \<noteq> {}) \<longrightarrow> U \<inter> \<Inter>A \<noteq> {})"
(is "_ \<longleftrightarrow> ?R")
proof (safe intro!: compact_eq_heine_borel[THEN iffD2])
fix A assume "compact U" and A: "\<forall>a\<in>A. closed a" "U \<inter> \<Inter>A = {}"
and fi: "\<forall>B \<subseteq> A. finite B \<longrightarrow> U \<inter> \<Inter>B \<noteq> {}"
from A have "(\<forall>a\<in>uminusA. open a) \<and> U \<subseteq> \<Union>(uminusA)"
by auto
with compact U obtain B where "B \<subseteq> A" "finite (uminusB)" "U \<subseteq> \<Union>(uminusB)"
unfolding compact_eq_heine_borel by (metis subset_image_iff)
with fi[THEN spec, of B] show False
by (auto dest: finite_imageD intro: inj_setminus)
next
fix A assume ?R and cover: "\<forall>a\<in>A. open a" "U \<subseteq> \<Union>A"
from cover have "U \<inter> \<Inter>(uminusA) = {}" "\<forall>a\<in>uminusA. closed a"
by auto
with ?R obtain B where "B \<subseteq> A" "finite (uminusB)" "U \<inter> \<Inter>(uminusB) = {}"
by (metis subset_image_iff)
then show "\<exists>T\<subseteq>A. finite T \<and> U \<subseteq> \<Union>T"
by  (auto intro!: exI[of _ B] inj_setminus dest: finite_imageD)
qed

lemma compact_imp_fip:
"compact s \<Longrightarrow> \<forall>t \<in> f. closed t \<Longrightarrow> \<forall>f'. finite f' \<and> f' \<subseteq> f \<longrightarrow> (s \<inter> (\<Inter> f') \<noteq> {}) \<Longrightarrow>
s \<inter> (\<Inter> f) \<noteq> {}"
unfolding compact_fip by auto

text{*Compactness expressed with filters*}

definition "filter_from_subbase B = Abs_filter (\<lambda>P. \<exists>X \<subseteq> B. finite X \<and> Inf X \<le> P)"

lemma eventually_filter_from_subbase:
"eventually P (filter_from_subbase B) \<longleftrightarrow> (\<exists>X \<subseteq> B. finite X \<and> Inf X \<le> P)"
(is "_ \<longleftrightarrow> ?R P")
unfolding filter_from_subbase_def
proof (rule eventually_Abs_filter is_filter.intro)+
show "?R (\<lambda>x. True)"
by (rule exI[of _ "{}"]) (simp add: le_fun_def)
next
fix P Q assume "?R P" then guess X ..
moreover assume "?R Q" then guess Y ..
ultimately show "?R (\<lambda>x. P x \<and> Q x)"
by (intro exI[of _ "X \<union> Y"]) auto
next
fix P Q
assume "?R P" then guess X ..
moreover assume "\<forall>x. P x \<longrightarrow> Q x"
ultimately show "?R Q"
by (intro exI[of _ X]) auto
qed

lemma eventually_filter_from_subbaseI: "P \<in> B \<Longrightarrow> eventually P (filter_from_subbase B)"
by (subst eventually_filter_from_subbase) (auto intro!: exI[of _ "{P}"])

lemma filter_from_subbase_not_bot:
"\<forall>X \<subseteq> B. finite X \<longrightarrow> Inf X \<noteq> bot \<Longrightarrow> filter_from_subbase B \<noteq> bot"
unfolding trivial_limit_def eventually_filter_from_subbase by auto

lemma closure_iff_nhds_not_empty:
"x \<in> closure X \<longleftrightarrow> (\<forall>A. \<forall>S\<subseteq>A. open S \<longrightarrow> x \<in> S \<longrightarrow> X \<inter> A \<noteq> {})"
proof safe
assume x: "x \<in> closure X"
fix S A assume "open S" "x \<in> S" "X \<inter> A = {}" "S \<subseteq> A"
then have "x \<notin> closure (-S)"
by (auto simp: closure_complement subset_eq[symmetric] intro: interiorI)
with x have "x \<in> closure X - closure (-S)"
by auto
also have "\<dots> \<subseteq> closure (X \<inter> S)"
using open S open_inter_closure_subset[of S X] by (simp add: closed_Compl ac_simps)
finally have "X \<inter> S \<noteq> {}" by auto
then show False using X \<inter> A = {} S \<subseteq> A by auto
next
assume "\<forall>A S. S \<subseteq> A \<longrightarrow> open S \<longrightarrow> x \<in> S \<longrightarrow> X \<inter> A \<noteq> {}"
from this[THEN spec, of "- X", THEN spec, of "- closure X"]
show "x \<in> closure X"
qed

lemma compact_filter:
"compact U \<longleftrightarrow> (\<forall>F. F \<noteq> bot \<longrightarrow> eventually (\<lambda>x. x \<in> U) F \<longrightarrow> (\<exists>x\<in>U. inf (nhds x) F \<noteq> bot))"
proof (intro allI iffI impI compact_fip[THEN iffD2] notI)
fix F assume "compact U" and F: "F \<noteq> bot" "eventually (\<lambda>x. x \<in> U) F"
from F have "U \<noteq> {}"
by (auto simp: eventually_False)

def Z \<equiv> "closure  {A. eventually (\<lambda>x. x \<in> A) F}"
then have "\<forall>z\<in>Z. closed z"
by auto
moreover
have ev_Z: "\<And>z. z \<in> Z \<Longrightarrow> eventually (\<lambda>x. x \<in> z) F"
unfolding Z_def by (auto elim: eventually_elim1 intro: set_mp[OF closure_subset])
have "(\<forall>B \<subseteq> Z. finite B \<longrightarrow> U \<inter> \<Inter>B \<noteq> {})"
proof (intro allI impI)
fix B assume "finite B" "B \<subseteq> Z"
with finite B ev_Z have "eventually (\<lambda>x. \<forall>b\<in>B. x \<in> b) F"
by (auto intro!: eventually_Ball_finite)
with F(2) have "eventually (\<lambda>x. x \<in> U \<inter> (\<Inter>B)) F"
by eventually_elim auto
with F show "U \<inter> \<Inter>B \<noteq> {}"
by (intro notI) (simp add: eventually_False)
qed
ultimately have "U \<inter> \<Inter>Z \<noteq> {}"
using compact U unfolding compact_fip by blast
then obtain x where "x \<in> U" and x: "\<And>z. z \<in> Z \<Longrightarrow> x \<in> z" by auto

have "\<And>P. eventually P (inf (nhds x) F) \<Longrightarrow> P \<noteq> bot"
unfolding eventually_inf eventually_nhds
proof safe
fix P Q R S
assume "eventually R F" "open S" "x \<in> S"
with open_inter_closure_eq_empty[of S "{x. R x}"] x[of "closure {x. R x}"]
have "S \<inter> {x. R x} \<noteq> {}" by (auto simp: Z_def)
moreover assume "Ball S Q" "\<forall>x. Q x \<and> R x \<longrightarrow> bot x"
ultimately show False by (auto simp: set_eq_iff)
qed
with x \<in> U show "\<exists>x\<in>U. inf (nhds x) F \<noteq> bot"
by (metis eventually_bot)
next
fix A assume A: "\<forall>a\<in>A. closed a" "\<forall>B\<subseteq>A. finite B \<longrightarrow> U \<inter> \<Inter>B \<noteq> {}" "U \<inter> \<Inter>A = {}"

def P' \<equiv> "(\<lambda>a (x::'a). x \<in> a)"
then have inj_P': "\<And>A. inj_on P' A"
by (auto intro!: inj_onI simp: fun_eq_iff)
def F \<equiv> "filter_from_subbase (P'  insert U A)"
have "F \<noteq> bot"
unfolding F_def
proof (safe intro!: filter_from_subbase_not_bot)
fix X assume "X \<subseteq> P'  insert U A" "finite X" "Inf X = bot"
then obtain B where "B \<subseteq> insert U A" "finite B" and B: "Inf (P'  B) = bot"
unfolding subset_image_iff by (auto intro: inj_P' finite_imageD)
with A(2)[THEN spec, of "B - {U}"] have "U \<inter> \<Inter>(B - {U}) \<noteq> {}" by auto
with B show False by (auto simp: P'_def fun_eq_iff)
qed
moreover have "eventually (\<lambda>x. x \<in> U) F"
unfolding F_def by (rule eventually_filter_from_subbaseI) (auto simp: P'_def)
moreover assume "\<forall>F. F \<noteq> bot \<longrightarrow> eventually (\<lambda>x. x \<in> U) F \<longrightarrow> (\<exists>x\<in>U. inf (nhds x) F \<noteq> bot)"
ultimately obtain x where "x \<in> U" and x: "inf (nhds x) F \<noteq> bot"
by auto

{ fix V assume "V \<in> A"
then have V: "eventually (\<lambda>x. x \<in> V) F"
by (auto simp add: F_def image_iff P'_def intro!: eventually_filter_from_subbaseI)
have "x \<in> closure V"
unfolding closure_iff_nhds_not_empty
proof (intro impI allI)
fix S A assume "open S" "x \<in> S" "S \<subseteq> A"
then have "eventually (\<lambda>x. x \<in> A) (nhds x)" by (auto simp: eventually_nhds)
with V have "eventually (\<lambda>x. x \<in> V \<inter> A) (inf (nhds x) F)"
by (auto simp: eventually_inf)
with x show "V \<inter> A \<noteq> {}"
by (auto simp del: Int_iff simp add: trivial_limit_def)
qed
then have "x \<in> V"
using V \<in> A A(1) by simp }
with x\<in>U have "x \<in> U \<inter> \<Inter>A" by auto
with U \<inter> \<Inter>A = {} show False by auto
qed

definition "countably_compact U \<longleftrightarrow>
(\<forall>A. countable A \<longrightarrow> (\<forall>a\<in>A. open a) \<longrightarrow> U \<subseteq> \<Union>A \<longrightarrow> (\<exists>T\<subseteq>A. finite T \<and> U \<subseteq> \<Union>T))"

lemma countably_compactE:
assumes "countably_compact s" and "\<forall>t\<in>C. open t" and "s \<subseteq> \<Union>C" "countable C"
obtains C' where "C' \<subseteq> C" and "finite C'" and "s \<subseteq> \<Union>C'"
using assms unfolding countably_compact_def by metis

lemma countably_compactI:
assumes "\<And>C. \<forall>t\<in>C. open t \<Longrightarrow> s \<subseteq> \<Union>C \<Longrightarrow> countable C \<Longrightarrow> (\<exists>C'\<subseteq>C. finite C' \<and> s \<subseteq> \<Union>C')"
shows "countably_compact s"
using assms unfolding countably_compact_def by metis

lemma compact_imp_countably_compact: "compact U \<Longrightarrow> countably_compact U"
by (auto simp: compact_eq_heine_borel countably_compact_def)

lemma countably_compact_imp_compact:
assumes "countably_compact U"
assumes ccover: "countable B" "\<forall>b\<in>B. open b"
assumes basis: "\<And>T x. open T \<Longrightarrow> x \<in> T \<Longrightarrow> x \<in> U \<Longrightarrow> \<exists>b\<in>B. x \<in> b \<and> b \<inter> U \<subseteq> T"
shows "compact U"
using countably_compact U unfolding compact_eq_heine_borel countably_compact_def
proof safe
fix A assume A: "\<forall>a\<in>A. open a" "U \<subseteq> \<Union>A"
assume *: "\<forall>A. countable A \<longrightarrow> (\<forall>a\<in>A. open a) \<longrightarrow> U \<subseteq> \<Union>A \<longrightarrow> (\<exists>T\<subseteq>A. finite T \<and> U \<subseteq> \<Union>T)"

moreover def C \<equiv> "{b\<in>B. \<exists>a\<in>A. b \<inter> U \<subseteq> a}"
ultimately have "countable C" "\<forall>a\<in>C. open a"
unfolding C_def using ccover by auto
moreover
have "\<Union>A \<inter> U \<subseteq> \<Union>C"
proof safe
fix x a assume "x \<in> U" "x \<in> a" "a \<in> A"
with basis[of a x] A obtain b where "b \<in> B" "x \<in> b" "b \<inter> U \<subseteq> a" by blast
with a \<in> A show "x \<in> \<Union>C" unfolding C_def
by auto
qed
then have "U \<subseteq> \<Union>C" using U \<subseteq> \<Union>A by auto
ultimately obtain T where "T\<subseteq>C" "finite T" "U \<subseteq> \<Union>T"
using * by metis
moreover then have "\<forall>t\<in>T. \<exists>a\<in>A. t \<inter> U \<subseteq> a"
by (auto simp: C_def)
then guess f unfolding bchoice_iff Bex_def ..
ultimately show "\<exists>T\<subseteq>A. finite T \<and> U \<subseteq> \<Union>T"
unfolding C_def by (intro exI[of _ "fT"]) fastforce
qed

lemma countably_compact_imp_compact_second_countable:
"countably_compact U \<Longrightarrow> compact (U :: 'a :: second_countable_topology set)"
proof (rule countably_compact_imp_compact)
fix T and x :: 'a assume "open T" "x \<in> T"
from topological_basisE[OF is_basis this] guess b .
then show "\<exists>b\<in>SOME B. countable B \<and> topological_basis B. x \<in> b \<and> b \<inter> U \<subseteq> T" by auto
qed (insert countable_basis topological_basis_open[OF is_basis], auto)

lemma countably_compact_eq_compact:
"countably_compact U \<longleftrightarrow> compact (U :: 'a :: second_countable_topology set)"
using countably_compact_imp_compact_second_countable compact_imp_countably_compact by blast

subsubsection{* Sequential compactness *}

definition
seq_compact :: "'a::topological_space set \<Rightarrow> bool" where
"seq_compact S \<longleftrightarrow>
(\<forall>f. (\<forall>n. f n \<in> S) \<longrightarrow>
(\<exists>l\<in>S. \<exists>r. subseq r \<and> ((f o r) ---> l) sequentially))"

lemma seq_compact_imp_countably_compact:
fixes U :: "'a :: first_countable_topology set"
assumes "seq_compact U"
shows "countably_compact U"
proof (safe intro!: countably_compactI)
fix A assume A: "\<forall>a\<in>A. open a" "U \<subseteq> \<Union>A" "countable A"
have subseq: "\<And>X. range X \<subseteq> U \<Longrightarrow> \<exists>r x. x \<in> U \<and> subseq r \<and> (X \<circ> r) ----> x"
using seq_compact U by (fastforce simp: seq_compact_def subset_eq)
show "\<exists>T\<subseteq>A. finite T \<and> U \<subseteq> \<Union>T"
proof cases
assume "finite A" with A show ?thesis by auto
next
assume "infinite A"
then have "A \<noteq> {}" by auto
show ?thesis
proof (rule ccontr)
assume "\<not> (\<exists>T\<subseteq>A. finite T \<and> U \<subseteq> \<Union>T)"
then have "\<forall>T. \<exists>x. T \<subseteq> A \<and> finite T \<longrightarrow> (x \<in> U - \<Union>T)" by auto
then obtain X' where T: "\<And>T. T \<subseteq> A \<Longrightarrow> finite T \<Longrightarrow> X' T \<in> U - \<Union>T" by metis
def X \<equiv> "\<lambda>n. X' (from_nat_into A  {.. n})"
have X: "\<And>n. X n \<in> U - (\<Union>i\<le>n. from_nat_into A i)"
using A \<noteq> {} unfolding X_def SUP_def by (intro T) (auto intro: from_nat_into)
then have "range X \<subseteq> U" by auto
with subseq[of X] obtain r x where "x \<in> U" and r: "subseq r" "(X \<circ> r) ----> x" by auto
from x\<in>U U \<subseteq> \<Union>A from_nat_into_surj[OF countable A]
obtain n where "x \<in> from_nat_into A n" by auto
with r(2) A(1) from_nat_into[OF A \<noteq> {}, of n]
have "eventually (\<lambda>i. X (r i) \<in> from_nat_into A n) sequentially"
unfolding tendsto_def by (auto simp: comp_def)
then obtain N where "\<And>i. N \<le> i \<Longrightarrow> X (r i) \<in> from_nat_into A n"
by (auto simp: eventually_sequentially)
moreover from X have "\<And>i. n \<le> r i \<Longrightarrow> X (r i) \<notin> from_nat_into A n"
by auto
moreover from subseq r[THEN seq_suble, of "max n N"] have "\<exists>i. n \<le> r i \<and> N \<le> i"
by (auto intro!: exI[of _ "max n N"])
ultimately show False
by auto
qed
qed
qed

lemma compact_imp_seq_compact:
fixes U :: "'a :: first_countable_topology set"
assumes "compact U" shows "seq_compact U"
unfolding seq_compact_def
proof safe
fix X :: "nat \<Rightarrow> 'a" assume "\<forall>n. X n \<in> U"
then have "eventually (\<lambda>x. x \<in> U) (filtermap X sequentially)"
by (auto simp: eventually_filtermap)
moreover have "filtermap X sequentially \<noteq> bot"
ultimately obtain x where "x \<in> U" and x: "inf (nhds x) (filtermap X sequentially) \<noteq> bot" (is "?F \<noteq> _")
using compact U by (auto simp: compact_filter)

from countable_basis_at_decseq[of x] guess A . note A = this
def s \<equiv> "\<lambda>n i. SOME j. i < j \<and> X j \<in> A (Suc n)"
{ fix n i
have "\<exists>a. i < a \<and> X a \<in> A (Suc n)"
proof (rule ccontr)
assume "\<not> (\<exists>a>i. X a \<in> A (Suc n))"
then have "\<And>a. Suc i \<le> a \<Longrightarrow> X a \<notin> A (Suc n)" by auto
then have "eventually (\<lambda>x. x \<notin> A (Suc n)) (filtermap X sequentially)"
by (auto simp: eventually_filtermap eventually_sequentially)
moreover have "eventually (\<lambda>x. x \<in> A (Suc n)) (nhds x)"
using A(1,2)[of "Suc n"] by (auto simp: eventually_nhds)
ultimately have "eventually (\<lambda>x. False) ?F"
with x show False
qed
then have "i < s n i" "X (s n i) \<in> A (Suc n)"
unfolding s_def by (auto intro: someI2_ex) }
note s = this
def r \<equiv> "nat_rec (s 0 0) s"
have "subseq r"
by (auto simp: r_def s subseq_Suc_iff)
moreover
have "(\<lambda>n. X (r n)) ----> x"
proof (rule topological_tendstoI)
fix S assume "open S" "x \<in> S"
with A(3) have "eventually (\<lambda>i. A i \<subseteq> S) sequentially" by auto
moreover
{ fix i assume "Suc 0 \<le> i" then have "X (r i) \<in> A i"
by (cases i) (simp_all add: r_def s) }
then have "eventually (\<lambda>i. X (r i) \<in> A i) sequentially" by (auto simp: eventually_sequentially)
ultimately show "eventually (\<lambda>i. X (r i) \<in> S) sequentially"
by eventually_elim auto
qed
ultimately show "\<exists>x \<in> U. \<exists>r. subseq r \<and> (X \<circ> r) ----> x"
using x \<in> U by (auto simp: convergent_def comp_def)
qed

lemma seq_compactI:
assumes "\<And>f. \<forall>n. f n \<in> S \<Longrightarrow> \<exists>l\<in>S. \<exists>r. subseq r \<and> ((f o r) ---> l) sequentially"
shows "seq_compact S"
unfolding seq_compact_def using assms by fast

lemma seq_compactE:
assumes "seq_compact S" "\<forall>n. f n \<in> S"
obtains l r where "l \<in> S" "subseq r" "((f \<circ> r) ---> l) sequentially"
using assms unfolding seq_compact_def by fast

lemma countably_compact_imp_acc_point:
assumes "countably_compact s" "countable t" "infinite t"  "t \<subseteq> s"
shows "\<exists>x\<in>s. \<forall>U. x\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> t)"
proof (rule ccontr)
def C \<equiv> "(\<lambda>F. interior (F \<union> (- t)))  {F. finite F \<and> F \<subseteq> t }"
note countably_compact s
moreover have "\<forall>t\<in>C. open t"
by (auto simp: C_def)
moreover
assume "\<not> (\<exists>x\<in>s. \<forall>U. x\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> t))"
then have s: "\<And>x. x \<in> s \<Longrightarrow> \<exists>U. x\<in>U \<and> open U \<and> finite (U \<inter> t)" by metis
have "s \<subseteq> \<Union>C"
using t \<subseteq> s
unfolding C_def Union_image_eq
apply (safe dest!: s)
apply (rule_tac a="U \<inter> t" in UN_I)
apply (auto intro!: interiorI simp add: finite_subset)
done
moreover
from countable t have "countable C"
unfolding C_def by (auto intro: countable_Collect_finite_subset)
ultimately guess D by (rule countably_compactE)
then obtain E where E: "E \<subseteq> {F. finite F \<and> F \<subseteq> t }" "finite E" and
s: "s \<subseteq> (\<Union>F\<in>E. interior (F \<union> (- t)))"
by (metis (lifting) Union_image_eq finite_subset_image C_def)
from s t \<subseteq> s have "t \<subseteq> \<Union>E"
using interior_subset by blast
moreover have "finite (\<Union>E)"
using E by auto
ultimately show False using infinite t by (auto simp: finite_subset)
qed

lemma countable_acc_point_imp_seq_compact:
fixes s :: "'a::first_countable_topology set"
assumes "\<forall>t. infinite t \<and> countable t \<and> t \<subseteq> s --> (\<exists>x\<in>s. \<forall>U. x\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> t))"
shows "seq_compact s"
proof -
{ fix f :: "nat \<Rightarrow> 'a" assume f: "\<forall>n. f n \<in> s"
have "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
proof (cases "finite (range f)")
case True
obtain l where "infinite {n. f n = f l}"
using pigeonhole_infinite[OF _ True] by auto
then obtain r where "subseq r" and fr: "\<forall>n. f (r n) = f l"
using infinite_enumerate by blast
hence "subseq r \<and> (f \<circ> r) ----> f l"
by (simp add: fr tendsto_const o_def)
with f show "\<exists>l\<in>s. \<exists>r. subseq r \<and> (f \<circ> r) ----> l"
by auto
next
case False
with f assms have "\<exists>x\<in>s. \<forall>U. x\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> range f)" by auto
then obtain l where "l \<in> s" "\<forall>U. l\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> range f)" ..
from this(2) have "\<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
using acc_point_range_imp_convergent_subsequence[of l f] by auto
with l \<in> s show "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially" ..
qed
}
thus ?thesis unfolding seq_compact_def by auto
qed

lemma seq_compact_eq_countably_compact:
fixes U :: "'a :: first_countable_topology set"
shows "seq_compact U \<longleftrightarrow> countably_compact U"
using
countable_acc_point_imp_seq_compact
countably_compact_imp_acc_point
seq_compact_imp_countably_compact
by metis

lemma seq_compact_eq_acc_point:
fixes s :: "'a :: first_countable_topology set"
shows "seq_compact s \<longleftrightarrow> (\<forall>t. infinite t \<and> countable t \<and> t \<subseteq> s --> (\<exists>x\<in>s. \<forall>U. x\<in>U \<and> open U \<longrightarrow> infinite (U \<inter> t)))"
using
countable_acc_point_imp_seq_compact[of s]
countably_compact_imp_acc_point[of s]
seq_compact_imp_countably_compact[of s]
by metis

lemma seq_compact_eq_compact:
fixes U :: "'a :: second_countable_topology set"
shows "seq_compact U \<longleftrightarrow> compact U"
using seq_compact_eq_countably_compact countably_compact_eq_compact by blast

lemma bolzano_weierstrass_imp_seq_compact:
fixes s :: "'a::{t1_space, first_countable_topology} set"
shows "\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t) \<Longrightarrow> seq_compact s"
by (rule countable_acc_point_imp_seq_compact) (metis islimpt_eq_acc_point)

subsubsection{* Total boundedness *}

lemma cauchy_def:
"Cauchy s \<longleftrightarrow> (\<forall>e>0. \<exists>N. \<forall>m n. m \<ge> N \<and> n \<ge> N --> dist(s m)(s n) < e)"
unfolding Cauchy_def by metis

fun helper_1::"('a::metric_space set) \<Rightarrow> real \<Rightarrow> nat \<Rightarrow> 'a" where
"helper_1 s e n = (SOME y::'a. y \<in> s \<and> (\<forall>m<n. \<not> (dist (helper_1 s e m) y < e)))"
declare helper_1.simps[simp del]

lemma seq_compact_imp_totally_bounded:
assumes "seq_compact s"
shows "\<forall>e>0. \<exists>k. finite k \<and> k \<subseteq> s \<and> s \<subseteq> (\<Union>((\<lambda>x. ball x e)  k))"
proof(rule, rule, rule ccontr)
fix e::real assume "e>0" and assm:"\<not> (\<exists>k. finite k \<and> k \<subseteq> s \<and> s \<subseteq> \<Union>((\<lambda>x. ball x e)  k))"
def x \<equiv> "helper_1 s e"
{ fix n
have "x n \<in> s \<and> (\<forall>m<n. \<not> dist (x m) (x n) < e)"
proof(induct_tac rule:nat_less_induct)
fix n  def Q \<equiv> "(\<lambda>y. y \<in> s \<and> (\<forall>m<n. \<not> dist (x m) y < e))"
assume as:"\<forall>m<n. x m \<in> s \<and> (\<forall>ma<m. \<not> dist (x ma) (x m) < e)"
have "\<not> s \<subseteq> (\<Union>x\<in>x  {0..<n}. ball x e)" using assm apply simp apply(erule_tac x="x  {0 ..< n}" in allE) using as by auto
then obtain z where z:"z\<in>s" "z \<notin> (\<Union>x\<in>x  {0..<n}. ball x e)" unfolding subset_eq by auto
have "Q (x n)" unfolding x_def and helper_1.simps[of s e n]
apply(rule someI2[where a=z]) unfolding x_def[symmetric] and Q_def using z by auto
thus "x n \<in> s \<and> (\<forall>m<n. \<not> dist (x m) (x n) < e)" unfolding Q_def by auto
qed }
hence "\<forall>n::nat. x n \<in> s" and x:"\<forall>n. \<forall>m < n. \<not> (dist (x m) (x n) < e)" by blast+
then obtain l r where "l\<in>s" and r:"subseq r" and "((x \<circ> r) ---> l) sequentially" using assms(1)[unfolded seq_compact_def, THEN spec[where x=x]] by auto
from this(3) have "Cauchy (x \<circ> r)" using LIMSEQ_imp_Cauchy by auto
then obtain N::nat where N:"\<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist ((x \<circ> r) m) ((x \<circ> r) n) < e" unfolding cauchy_def using e>0 by auto
show False
using N[THEN spec[where x=N], THEN spec[where x="N+1"]]
using r[unfolded subseq_def, THEN spec[where x=N], THEN spec[where x="N+1"]]
using x[THEN spec[where x="r (N+1)"], THEN spec[where x="r (N)"]] by auto
qed

subsubsection{* Heine-Borel theorem *}

lemma seq_compact_imp_heine_borel:
fixes s :: "'a :: metric_space set"
assumes "seq_compact s" shows "compact s"
proof -
from seq_compact_imp_totally_bounded[OF seq_compact s]
guess f unfolding choice_iff' .. note f = this
def K \<equiv> "(\<lambda>(x, r). ball x r)  ((\<Union>e \<in> \<rat> \<inter> {0 <..}. f e) \<times> \<rat>)"
have "countably_compact s"
using seq_compact s by (rule seq_compact_imp_countably_compact)
then show "compact s"
proof (rule countably_compact_imp_compact)
show "countable K"
unfolding K_def using f
by (auto intro: countable_finite countable_subset countable_rat
intro!: countable_image countable_SIGMA countable_UN)
show "\<forall>b\<in>K. open b" by (auto simp: K_def)
next
fix T x assume T: "open T" "x \<in> T" and x: "x \<in> s"
from openE[OF T] obtain e where "0 < e" "ball x e \<subseteq> T" by auto
then have "0 < e / 2" "ball x (e / 2) \<subseteq> T" by auto
from Rats_dense_in_real[OF 0 < e / 2] obtain r where "r \<in> \<rat>" "0 < r" "r < e / 2" by auto
from f[rule_format, of r] 0 < r x \<in> s obtain k where "k \<in> f r" "x \<in> ball k r"
unfolding Union_image_eq by auto
from r \<in> \<rat> 0 < r k \<in> f r have "ball k r \<in> K" by (auto simp: K_def)
then show "\<exists>b\<in>K. x \<in> b \<and> b \<inter> s \<subseteq> T"
proof (rule bexI[rotated], safe)
fix y assume "y \<in> ball k r"
with r < e / 2 x \<in> ball k r have "dist x y < e"
by (intro dist_double[where x = k and d=e]) (auto simp: dist_commute)
with ball x e \<subseteq> T show "y \<in> T" by auto
qed (rule x \<in> ball k r)
qed
qed

lemma compact_eq_seq_compact_metric:
"compact (s :: 'a::metric_space set) \<longleftrightarrow> seq_compact s"
using compact_imp_seq_compact seq_compact_imp_heine_borel by blast

lemma compact_def:
"compact (S :: 'a::metric_space set) \<longleftrightarrow>
(\<forall>f. (\<forall>n. f n \<in> S) \<longrightarrow> (\<exists>l\<in>S. \<exists>r. subseq r \<and> (f o r) ----> l))"
unfolding compact_eq_seq_compact_metric seq_compact_def by auto

subsubsection {* Complete the chain of compactness variants *}

lemma compact_eq_bolzano_weierstrass:
fixes s :: "'a::metric_space set"
shows "compact s \<longleftrightarrow> (\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t))" (is "?lhs = ?rhs")
proof
assume ?lhs thus ?rhs using heine_borel_imp_bolzano_weierstrass[of s] by auto
next
assume ?rhs thus ?lhs
unfolding compact_eq_seq_compact_metric by (rule bolzano_weierstrass_imp_seq_compact)
qed

lemma bolzano_weierstrass_imp_bounded:
"\<forall>t. infinite t \<and> t \<subseteq> s --> (\<exists>x \<in> s. x islimpt t) \<Longrightarrow> bounded s"
using compact_imp_bounded unfolding compact_eq_bolzano_weierstrass .

text {*
A metric space (or topological vector space) is said to have the
Heine-Borel property if every closed and bounded subset is compact.
*}

class heine_borel = metric_space +
assumes bounded_imp_convergent_subsequence:
"bounded (range f) \<Longrightarrow> \<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"

lemma bounded_closed_imp_seq_compact:
fixes s::"'a::heine_borel set"
assumes "bounded s" and "closed s" shows "seq_compact s"
proof (unfold seq_compact_def, clarify)
fix f :: "nat \<Rightarrow> 'a" assume f: "\<forall>n. f n \<in> s"
with bounded s have "bounded (range f)" by (auto intro: bounded_subset)
obtain l r where r: "subseq r" and l: "((f \<circ> r) ---> l) sequentially"
using bounded_imp_convergent_subsequence [OF bounded (range f)] by auto
from f have fr: "\<forall>n. (f \<circ> r) n \<in> s" by simp
have "l \<in> s" using closed s fr l
unfolding closed_sequential_limits by blast
show "\<exists>l\<in>s. \<exists>r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
using l \<in> s r l by blast
qed

lemma compact_eq_bounded_closed:
fixes s :: "'a::heine_borel set"
shows "compact s \<longleftrightarrow> bounded s \<and> closed s"  (is "?lhs = ?rhs")
proof
assume ?lhs thus ?rhs
using compact_imp_closed compact_imp_bounded by blast
next
assume ?rhs thus ?lhs
using bounded_closed_imp_seq_compact[of s] unfolding compact_eq_seq_compact_metric by auto
qed

(* TODO: is this lemma necessary? *)
lemma bounded_increasing_convergent:
fixes s :: "nat \<Rightarrow> real"
shows "bounded {s n| n. True} \<Longrightarrow> \<forall>n. s n \<le> s (Suc n) \<Longrightarrow> \<exists>l. s ----> l"
using Bseq_mono_convergent[of s] incseq_Suc_iff[of s]
by (auto simp: image_def Bseq_eq_bounded convergent_def incseq_def)

instance real :: heine_borel
proof
fix f :: "nat \<Rightarrow> real"
assume f: "bounded (range f)"
obtain r where r: "subseq r" "monoseq (f \<circ> r)"
unfolding comp_def by (metis seq_monosub)
moreover
then have "Bseq (f \<circ> r)"
unfolding Bseq_eq_bounded using f by (auto intro: bounded_subset)
ultimately show "\<exists>l r. subseq r \<and> (f \<circ> r) ----> l"
using Bseq_monoseq_convergent[of "f \<circ> r"] by (auto simp: convergent_def)
qed

lemma compact_lemma:
fixes f :: "nat \<Rightarrow> 'a::euclidean_space"
assumes "bounded (range f)"
shows "\<forall>d\<subseteq>Basis. \<exists>l::'a. \<exists> r. subseq r \<and>
(\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) \<bullet> i) (l \<bullet> i) < e) sequentially)"
proof safe
fix d :: "'a set" assume d: "d \<subseteq> Basis"
with finite_Basis have "finite d" by (blast intro: finite_subset)
from this d show "\<exists>l::'a. \<exists>r. subseq r \<and>
(\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r n) \<bullet> i) (l \<bullet> i) < e) sequentially)"
proof(induct d) case empty thus ?case unfolding subseq_def by auto
next case (insert k d) have k[intro]:"k\<in>Basis" using insert by auto
have s': "bounded ((\<lambda>x. x \<bullet> k)  range f)" using bounded (range f)
by (auto intro!: bounded_linear_image bounded_linear_inner_left)
obtain l1::"'a" and r1 where r1:"subseq r1" and
lr1:"\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) \<bullet> i) (l1 \<bullet> i) < e) sequentially"
using insert(3) using insert(4) by auto
have f': "\<forall>n. f (r1 n) \<bullet> k \<in> (\<lambda>x. x \<bullet> k)  range f" by simp
have "bounded (range (\<lambda>i. f (r1 i) \<bullet> k))"
by (metis (lifting) bounded_subset f' image_subsetI s')
then obtain l2 r2 where r2:"subseq r2" and lr2:"((\<lambda>i. f (r1 (r2 i)) \<bullet> k) ---> l2) sequentially"
using bounded_imp_convergent_subsequence[of "\<lambda>i. f (r1 i) \<bullet> k"] by (auto simp: o_def)
def r \<equiv> "r1 \<circ> r2" have r:"subseq r"
using r1 and r2 unfolding r_def o_def subseq_def by auto
moreover
def l \<equiv> "(\<Sum>i\<in>Basis. (if i = k then l2 else l1\<bullet>i) *\<^sub>R i)::'a"
{ fix e::real assume "e>0"
from lr1 e>0 have N1:"eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 n) \<bullet> i) (l1 \<bullet> i) < e) sequentially" by blast
from lr2 e>0 have N2:"eventually (\<lambda>n. dist (f (r1 (r2 n)) \<bullet> k) l2 < e) sequentially" by (rule tendstoD)
from r2 N1 have N1': "eventually (\<lambda>n. \<forall>i\<in>d. dist (f (r1 (r2 n)) \<bullet> i) (l1 \<bullet> i) < e) sequentially"
by (rule eventually_subseq)
have "eventually (\<lambda>n. \<forall>i\<in>(insert k d). dist (f (r n) \<bullet> i) (l \<bullet> i) < e) sequentially"
using N1' N2
by eventually_elim (insert insert.prems, auto simp: l_def r_def o_def)
}
ultimately show ?case by auto
qed
qed

instance euclidean_space \<subseteq> heine_borel
proof
fix f :: "nat \<Rightarrow> 'a"
assume f: "bounded (range f)"
then obtain l::'a and r where r: "subseq r"
and l: "\<forall>e>0. eventually (\<lambda>n. \<forall>i\<in>Basis. dist (f (r n) \<bullet> i) (l \<bullet> i) < e) sequentially"
using compact_lemma [OF f] by blast
{ fix e::real assume "e>0"
hence "0 < e / real_of_nat DIM('a)" by (auto intro!: divide_pos_pos DIM_positive)
with l have "eventually (\<lambda>n. \<forall>i\<in>Basis. dist (f (r n) \<bullet> i) (l \<bullet> i) < e / (real_of_nat DIM('a))) sequentially"
by simp
moreover
{ fix n assume n: "\<forall>i\<in>Basis. dist (f (r n) \<bullet> i) (l \<bullet> i) < e / (real_of_nat DIM('a))"
have "dist (f (r n)) l \<le> (\<Sum>i\<in>Basis. dist (f (r n) \<bullet> i) (l \<bullet> i))"
apply(subst euclidean_dist_l2) using zero_le_dist by (rule setL2_le_setsum)
also have "\<dots> < (\<Sum>i\<in>(Basis::'a set). e / (real_of_nat DIM('a)))"
apply(rule setsum_strict_mono) using n by auto
finally have "dist (f (r n)) l < e"
by auto
}
ultimately have "eventually (\<lambda>n. dist (f (r n)) l < e) sequentially"
by (rule eventually_elim1)
}
hence *:"((f \<circ> r) ---> l) sequentially" unfolding o_def tendsto_iff by simp
with r show "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially" by auto
qed

lemma bounded_fst: "bounded s \<Longrightarrow> bounded (fst  s)"
unfolding bounded_def
apply clarify
apply (rule_tac x="a" in exI)
apply (rule_tac x="e" in exI)
apply clarsimp
apply (drule (1) bspec)
apply (erule order_trans [OF real_sqrt_sum_squares_ge1])
done

lemma bounded_snd: "bounded s \<Longrightarrow> bounded (snd  s)"
unfolding bounded_def
apply clarify
apply (rule_tac x="b" in exI)
apply (rule_tac x="e" in exI)
apply clarsimp
apply (drule (1) bspec)
apply (erule order_trans [OF real_sqrt_sum_squares_ge2])
done

instance prod :: (heine_borel, heine_borel) heine_borel
proof
fix f :: "nat \<Rightarrow> 'a \<times> 'b"
assume f: "bounded (range f)"
from f have s1: "bounded (range (fst \<circ> f))" unfolding image_comp by (rule bounded_fst)
obtain l1 r1 where r1: "subseq r1" and l1: "(\<lambda>n. fst (f (r1 n))) ----> l1"
using bounded_imp_convergent_subsequence [OF s1] unfolding o_def by fast
from f have s2: "bounded (range (snd \<circ> f \<circ> r1))"
by (auto simp add: image_comp intro: bounded_snd bounded_subset)
obtain l2 r2 where r2: "subseq r2"
and l2: "((\<lambda>n. snd (f (r1 (r2 n)))) ---> l2) sequentially"
using bounded_imp_convergent_subsequence [OF s2]
unfolding o_def by fast
have l1': "((\<lambda>n. fst (f (r1 (r2 n)))) ---> l1) sequentially"
using LIMSEQ_subseq_LIMSEQ [OF l1 r2] unfolding o_def .
have l: "((f \<circ> (r1 \<circ> r2)) ---> (l1, l2)) sequentially"
using tendsto_Pair [OF l1' l2] unfolding o_def by simp
have r: "subseq (r1 \<circ> r2)"
using r1 r2 unfolding subseq_def by simp
show "\<exists>l r. subseq r \<and> ((f \<circ> r) ---> l) sequentially"
using l r by fast
qed

subsubsection{* Completeness *}

definition complete :: "'a::metric_space set \<Rightarrow> bool" where
"complete s \<longleftrightarrow> (\<forall>f. (\<forall>n. f n \<in> s) \<and> Cauchy f \<longrightarrow> (\<exists>l\<in>s. f ----> l))"

lemma compact_imp_complete: assumes "compact s" shows "complete s"
proof-
{ fix f assume as: "(\<forall>n::nat. f n \<in> s)" "Cauchy f"
from as(1) obtain l r where lr: "l\<in>s" "subseq r" "(f \<circ> r) ----> l"
using assms unfolding compact_def by blast

note lr' = seq_suble [OF lr(2)]

{ fix e::real assume "e>0"
from as(2) obtain N where N:"\<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (f m) (f n) < e/2" unfolding cauchy_def using e>0 apply (erule_tac x="e/2" in allE) by auto
from lr(3)[unfolded LIMSEQ_def, THEN spec[where x="e/2"]] obtain M where M:"\<forall>n\<ge>M. dist ((f \<circ> r) n) l < e/2" using e>0 by auto
{ fix n::nat assume n:"n \<ge> max N M"
have "dist ((f \<circ> r) n) l < e/2" using n M by auto
moreover have "r n \<ge> N" using lr'[of n] n by auto
hence "dist (f n) ((f \<circ> r) n) < e / 2" using N using n by auto
ultimately have "dist (f n) l < e" using dist_triangle_half_r[of "f (r n)" "f n" e l] by (auto simp add: dist_commute)  }
hence "\<exists>N. \<forall>n\<ge>N. dist (f n) l < e" by blast  }
hence "\<exists>l\<in>s. (f ---> l) sequentially" using l\<in>s unfolding LIMSEQ_def by auto  }
thus ?thesis unfolding complete_def by auto
qed

lemma nat_approx_posE:
fixes e::real
assumes "0 < e"
obtains n::nat where "1 / (Suc n) < e"
proof atomize_elim
have " 1 / real (Suc (nat (ceiling (1/e)))) < 1 / (ceiling (1/e))"
by (rule divide_strict_left_mono) (auto intro!: mult_pos_pos simp: 0 < e)
also have "1 / (ceiling (1/e)) \<le> 1 / (1/e)"
by (rule divide_left_mono) (auto intro!: divide_pos_pos simp: 0 < e)
also have "\<dots> = e" by simp
finally show  "\<exists>n. 1 / real (Suc n) < e" ..
qed

lemma compact_eq_totally_bounded:
"compact s \<longleftrightarrow> complete s \<and> (\<forall>e>0. \<exists>k. finite k \<and> s \<subseteq> (\<Union>((\<lambda>x. ball x e)  k)))"
(is "_ \<longleftrightarrow> ?rhs")
proof
assume assms: "?rhs"
then obtain k where k: "\<And>e. 0 < e \<Longrightarrow> finite (k e)" "\<And>e. 0 < e \<Longrightarrow> s \<subseteq> (\<Union>x\<in>k e. ball x e)"
by (auto simp: choice_iff')

show "compact s"
proof cases
assume "s = {}" thus "compact s" by (simp add: compact_def)
next
assume "s \<noteq> {}"
show ?thesis
unfolding compact_def
proof safe
fix f :: "nat \<Rightarrow> 'a" assume f: "\<forall>n. f n \<in> s"

def e \<equiv> "\<lambda>n. 1 / (2 * Suc n)"
then have [simp]: "\<And>n. 0 < e n" by auto
def B \<equiv> "\<lambda>n U. SOME b. infinite {n. f n \<in> b} \<and> (\<exists>x. b \<subseteq> ball x (e n) \<inter> U)"
{ fix n U assume "infinite {n. f n \<in> U}"
then have "\<exists>b\<in>k (e n). infinite {i\<in>{n. f n \<in> U}. f i \<in> ball b (e n)}"
using k f by (intro pigeonhole_infinite_rel) (auto simp: subset_eq)
then guess a ..
then have "\<exists>b. infinite {i. f i \<in> b} \<and> (\<exists>x. b \<subseteq> ball x (e n) \<inter> U)"
by (intro exI[of _ "ball a (e n) \<inter> U"] exI[of _ a]) (auto simp: ac_simps)
from someI_ex[OF this]
have "infinite {i. f i \<in> B n U}" "\<exists>x. B n U \<subseteq> ball x (e n) \<inter> U"
unfolding B_def by auto }
note B = this

def F \<equiv> "nat_rec (B 0 UNIV) B"
{ fix n have "infinite {i. f i \<in> F n}"
by (induct n) (auto simp: F_def B) }
then have F: "\<And>n. \<exists>x. F (Suc n) \<subseteq> ball x (e n) \<inter> F n"
using B by (simp add: F_def)
then have F_dec: "\<And>m n. m \<le> n \<Longrightarrow> F n \<subseteq> F m"
using decseq_SucI[of F] by (auto simp: decseq_def)

obtain sel where sel: "\<And>k i. i < sel k i" "\<And>k i. f (sel k i) \<in> F k"
proof (atomize_elim, unfold all_conj_distrib[symmetric], intro choice allI)
fix k i
have "infinite ({n. f n \<in> F k} - {.. i})"
using infinite {n. f n \<in> F k} by auto
from infinite_imp_nonempty[OF this]
show "\<exists>x>i. f x \<in> F k"
by (simp add: set_eq_iff not_le conj_commute)
qed

def t \<equiv> "nat_rec (sel 0 0) (\<lambda>n i. sel (Suc n) i)"
have "subseq t"
unfolding subseq_Suc_iff by (simp add: t_def sel)
moreover have "\<forall>i. (f \<circ> t) i \<in> s"
using f by auto
moreover
{ fix n have "(f \<circ> t) n \<in> F n"
by (cases n) (simp_all add: t_def sel) }
note t = this

have "Cauchy (f \<circ> t)"
proof (safe intro!: metric_CauchyI exI elim!: nat_approx_posE)
fix r :: real and N n m assume "1 / Suc N < r" "Suc N \<le> n" "Suc N \<le> m"
then have "(f \<circ> t) n \<in> F (Suc N)" "(f \<circ> t) m \<in> F (Suc N)" "2 * e N < r"
using F_dec t by (auto simp: e_def field_simps real_of_nat_Suc)
with F[of N] obtain x where "dist x ((f \<circ> t) n) < e N" "dist x ((f \<circ> t) m) < e N"
by (auto simp: subset_eq)
with dist_triangle[of "(f \<circ> t) m" "(f \<circ> t) n" x] 2 * e N < r
show "dist ((f \<circ> t) m) ((f \<circ> t) n) < r"
qed

ultimately show "\<exists>l\<in>s. \<exists>r. subseq r \<and> (f \<circ> r) ----> l"
using assms unfolding complete_def by blast
qed
qed
qed (metis compact_imp_complete compact_imp_seq_compact seq_compact_imp_totally_bounded)

lemma cauchy: "Cauchy s \<longleftrightarrow> (\<forall>e>0.\<exists> N::nat. \<forall>n\<ge>N. dist(s n)(s N) < e)" (is "?lhs = ?rhs")
proof-
{ assume ?rhs
{ fix e::real
assume "e>0"
with ?rhs obtain N where N:"\<forall>n\<ge>N. dist (s n) (s N) < e/2"
by (erule_tac x="e/2" in allE) auto
{ fix n m
assume nm:"N \<le> m \<and> N \<le> n"
hence "dist (s m) (s n) < e" using N
using dist_triangle_half_l[of "s m" "s N" "e" "s n"]
by blast
}
hence "\<exists>N. \<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (s m) (s n) < e"
by blast
}
hence ?lhs
unfolding cauchy_def
by blast
}
thus ?thesis
unfolding cauchy_def
using dist_triangle_half_l
by blast
qed

lemma cauchy_imp_bounded: assumes "Cauchy s" shows "bounded (range s)"
proof-
from assms obtain N::nat where "\<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (s m) (s n) < 1" unfolding cauchy_def apply(erule_tac x= 1 in allE) by auto
hence N:"\<forall>n. N \<le> n \<longrightarrow> dist (s N) (s n) < 1" by auto
moreover
have "bounded (s  {0..N})" using finite_imp_bounded[of "s  {1..N}"] by auto
then obtain a where a:"\<forall>x\<in>s  {0..N}. dist (s N) x \<le> a"
unfolding bounded_any_center [where a="s N"] by auto
ultimately show "?thesis"
unfolding bounded_any_center [where a="s N"]
apply(rule_tac x="max a 1" in exI) apply auto
apply(erule_tac x=y in allE) apply(erule_tac x=y in ballE) by auto
qed

instance heine_borel < complete_space
proof
fix f :: "nat \<Rightarrow> 'a" assume "Cauchy f"
hence "bounded (range f)"
by (rule cauchy_imp_bounded)
hence "compact (closure (range f))"
unfolding compact_eq_bounded_closed by auto
hence "complete (closure (range f))"
by (rule compact_imp_complete)
moreover have "\<forall>n. f n \<in> closure (range f)"
using closure_subset [of "range f"] by auto
ultimately have "\<exists>l\<in>closure (range f). (f ---> l) sequentially"
using Cauchy f unfolding complete_def by auto
then show "convergent f"
unfolding convergent_def by auto
qed

instance euclidean_space \<subseteq> banach ..

lemma complete_univ: "complete (UNIV :: 'a::complete_space set)"
fix f :: "nat \<Rightarrow> 'a" assume "Cauchy f"
hence "convergent f" by (rule Cauchy_convergent)
thus "\<exists>l. f ----> l" unfolding convergent_def .
qed

lemma complete_imp_closed: assumes "complete s" shows "closed s"
proof -
{ fix x assume "x islimpt s"
then obtain f where f: "\<forall>n. f n \<in> s - {x}" "(f ---> x) sequentially"
unfolding islimpt_sequential by auto
then obtain l where l: "l\<in>s" "(f ---> l) sequentially"
using complete s[unfolded complete_def] using LIMSEQ_imp_Cauchy[of f x] by auto
hence "x \<in> s"  using tendsto_unique[of sequentially f l x] trivial_limit_sequentially f(2) by auto
}
thus "closed s" unfolding closed_limpt by auto
qed

lemma complete_eq_closed:
fixes s :: "'a::complete_space set"
shows "complete s \<longleftrightarrow> closed s" (is "?lhs = ?rhs")
proof
assume ?lhs thus ?rhs by (rule complete_imp_closed)
next
assume ?rhs
{ fix f assume as:"\<forall>n::nat. f n \<in> s" "Cauchy f"
then obtain l where "(f ---> l) sequentially" using complete_univ[unfolded complete_def, THEN spec[where x=f]] by auto
hence "\<exists>l\<in>s. (f ---> l) sequentially" using ?rhs[unfolded closed_sequential_limits, THEN spec[where x=f], THEN spec[where x=l]] using as(1) by auto  }
thus ?lhs unfolding complete_def by auto
qed

lemma convergent_eq_cauchy:
fixes s :: "nat \<Rightarrow> 'a::complete_space"
shows "(\<exists>l. (s ---> l) sequentially) \<longleftrightarrow> Cauchy s"
unfolding Cauchy_convergent_iff convergent_def ..

lemma convergent_imp_bounded:
fixes s :: "nat \<Rightarrow> 'a::metric_space"
shows "(s ---> l) sequentially \<Longrightarrow> bounded (range s)"
by (intro cauchy_imp_bounded LIMSEQ_imp_Cauchy)

lemma compact_cball[simp]:
fixes x :: "'a::heine_borel"
shows "compact(cball x e)"
using compact_eq_bounded_closed bounded_cball closed_cball
by blast

lemma compact_frontier_bounded[intro]:
fixes s :: "'a::heine_borel set"
shows "bounded s ==> 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 ==> compact (frontier s)"
using compact_eq_bounded_closed compact_frontier_bounded
by blast

lemma frontier_subset_compact:
fixes s :: "'a::heine_borel set"
shows "compact s ==> frontier s \<subseteq> s"
using frontier_subset_closed compact_eq_bounded_closed
by blast

subsection {* Bounded closed nest property (proof does not use Heine-Borel) *}

lemma bounded_closed_nest:
assumes "\<forall>n. closed(s n)" "\<forall>n. (s n \<noteq> {})"
"(\<forall>m n. m \<le> n --> s n \<subseteq> s m)"  "bounded(s 0)"
shows "\<exists>a::'a::heine_borel. \<forall>n::nat. a \<in> s(n)"
proof-
from assms(2) obtain x where x:"\<forall>n::nat. 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)" using bounded_closed_imp_seq_compact[of "s 0"] by auto

then obtain l r where lr:"l\<in>s 0" "subseq r" "((x \<circ> r) ---> l) sequentially"
unfolding seq_compact_def apply(erule_tac x=x in allE)  using x using assms(3) by blast

{ fix n::nat
{ fix e::real assume "e>0"
with lr(3) obtain N where N:"\<forall>m\<ge>N. dist ((x \<circ> r) m) l < e" unfolding LIMSEQ_def by auto
hence "dist ((x \<circ> r) (max N n)) l < e" by auto
moreover
have "r (max N n) \<ge> n" using lr(2) using seq_suble[of r "max N n"] by auto
hence "(x \<circ> r) (max N n) \<in> s n"
using x apply(erule_tac x=n in allE)
using x apply(erule_tac x="r (max N n)" in allE)
using assms(3) apply(erule_tac x=n in allE) apply(erule_tac x="r (max N n)" in allE) by auto
ultimately have "\<exists>y\<in>s n. dist y l < e" by auto
}
hence "l \<in> s n" using closed_approachable[of "s n" l] assms(1) by blast
}
thus ?thesis by auto
qed

text {* Decreasing case does not even need compactness, just completeness. *}

lemma decreasing_closed_nest:
assumes "\<forall>n. closed(s n)"
"\<forall>n. (s n \<noteq> {})"
"\<forall>m n. m \<le> n --> 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::'a::complete_space. \<forall>n::nat. a \<in> s n"
proof-
have "\<forall>n. \<exists> x. x\<in>s n" using assms(2) by auto
hence "\<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"
hence "t m \<in> s N" "t n \<in> s N" using assms(3) t unfolding  subset_eq t by blast+
hence "dist (t m) (t n) < e" using N by auto
}
hence "\<exists>N. \<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (t m) (t n) < e" by auto
}
hence  "Cauchy t" unfolding cauchy_def by auto
then obtain l where l:"(t ---> 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 LIMSEQ_def] by auto
have "t (max n N) \<in> s n" using assms(3) unfolding subset_eq apply(erule_tac x=n in allE) apply (erule_tac x="max n N" in allE) using t by auto
hence "\<exists>y\<in>s n. dist y l < e" apply(rule_tac x="t (max n N)" in bexI) using N by auto
}
hence "l \<in> s n" using closed_approachable[of "s n" l] assms(1) by auto
}
then show ?thesis by auto
qed

text {* Strengthen it to the intersection actually being a singleton. *}

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 --> 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"
hence "dist a b < e" using assms(4 )using b using a by blast
}
hence "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
thus ?thesis ..
qed

text{* Cauchy-type criteria for uniform convergence. *}

lemma uniformly_convergent_eq_cauchy: fixes s::"nat \<Rightarrow> 'b \<Rightarrow> 'a::complete_space" shows
"(\<exists>l. \<forall>e>0. \<exists>N. \<forall>n x. N \<le> n \<and> P x --> dist(s n x)(l x) < e) \<longleftrightarrow>
(\<forall>e>0. \<exists>N. \<forall>m n x. N \<le> m \<and> N \<le> n \<and> P x  --> dist (s m x) (s n x) < e)" (is "?lhs = ?rhs")
proof(rule)
assume ?lhs
then obtain l where l:"\<forall>e>0. \<exists>N. \<forall>n x. N \<le> n \<and> P x \<longrightarrow> dist (s n x) (l x) < e" by auto
{ fix e::real assume "e>0"
then obtain N::nat where N:"\<forall>n x. N \<le> n \<and> P x \<longrightarrow> dist (s n x) (l x) < e / 2" using l[THEN spec[where x="e/2"]] by auto
{ fix n m::nat and x::"'b" assume "N \<le> m \<and> N \<le> n \<and> P x"
hence "dist (s m x) (s n x) < e"
using N[THEN spec[where x=m], THEN spec[where x=x]]
using N[THEN spec[where x=n], THEN spec[where x=x]]
using dist_triangle_half_l[of "s m x" "l x" e "s n x"] by auto  }
hence "\<exists>N. \<forall>m n x. N \<le> m \<and> N \<le> n \<and> P x  --> dist (s m x) (s n x) < e"  by auto  }
thus ?rhs by auto
next
assume ?rhs
hence "\<forall>x. P x \<longrightarrow> Cauchy (\<lambda>n. s n x)" unfolding cauchy_def apply auto by (erule_tac x=e in allE)auto
then obtain l where l:"\<forall>x. P x \<longrightarrow> ((\<lambda>n. s n x) ---> l x) sequentially" unfolding convergent_eq_cauchy[THEN sym]
using choice[of "\<lambda>x l. P x \<longrightarrow> ((\<lambda>n. s n x) ---> l) sequentially"] by auto
{ fix e::real assume "e>0"
then obtain N where N:"\<forall>m n x. N \<le> m \<and> N \<le> n \<and> P x \<longrightarrow> dist (s m x) (s n x) < e/2"
using ?rhs[THEN spec[where x="e/2"]] by auto
{ fix x assume "P x"
then obtain M where M:"\<forall>n\<ge>M. dist (s n x) (l x) < e/2"
using l[THEN spec[where x=x], unfolded LIMSEQ_def] using e>0 by(auto elim!: allE[where x="e/2"])
fix n::nat assume "n\<ge>N"
hence "dist(s n x)(l x) < e"  using P xand N[THEN spec[where x=n], THEN spec[where x="N+M"], THEN spec[where x=x]]
using M[THEN spec[where x="N+M"]] and dist_triangle_half_l[of "s n x" "s (N+M) x" e "l x"] by (auto simp add: dist_commute)  }
hence "\<exists>N. \<forall>n x. N \<le> n \<and> P x \<longrightarrow> dist(s n x)(l x) < e" by auto }
thus ?lhs by auto
qed

lemma uniformly_cauchy_imp_uniformly_convergent:
fixes s :: "nat \<Rightarrow> 'a \<Rightarrow> 'b::complete_space"
assumes "\<forall>e>0.\<exists>N. \<forall>m (n::nat) x. N \<le> m \<and> N \<le> n \<and> P x --> dist(s m x)(s n x) < e"
"\<forall>x. P x --> (\<forall>e>0. \<exists>N. \<forall>n. N \<le> n --> dist(s n x)(l x) < e)"
shows "\<forall>e>0. \<exists>N. \<forall>n x. N \<le> n \<and> P x --> dist(s n x)(l x) < e"
proof-
obtain l' where l:"\<forall>e>0. \<exists>N. \<forall>n x. N \<le> n \<and> P x \<longrightarrow> dist (s n x) (l' x) < e"
using assms(1) unfolding uniformly_convergent_eq_cauchy[THEN sym] by auto
moreover
{ fix x assume "P x"
hence "l x = l' x" using tendsto_unique[OF trivial_limit_sequentially, of "\<lambda>n. s n x" "l x" "l' x"]
using l and assms(2) unfolding LIMSEQ_def by blast  }
ultimately show ?thesis by auto
qed

subsection {* Continuity *}

text{* Derive the epsilon-delta forms, which we often use as "definitions" *}

lemma continuous_within_eps_delta:
"continuous (at x within s) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0. \<forall>x'\<in> s.  dist x' x < d --> dist (f x') (f x) < e)"
unfolding continuous_within and Lim_within
apply auto unfolding dist_nz[THEN sym] apply(auto del: allE elim!:allE) apply(rule_tac x=d in exI) by auto

lemma continuous_at_eps_delta: "continuous (at x) f \<longleftrightarrow>  (\<forall>e>0. \<exists>d>0.
\<forall>x'. dist x' x < d --> dist(f x')(f x) < e)"
using continuous_within_eps_delta [of x UNIV f] by simp

text{* Versions in terms of open balls. *}

lemma continuous_within_ball:
"continuous (at x within s) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0.
f  (ball x d \<inter> s) \<subseteq> ball (f x) e)" (is "?lhs = ?rhs")
proof
assume ?lhs
{ fix e::real assume "e>0"
then obtain d where d: "d>0" "\<forall>xa\<in>s. 0 < dist xa x \<and> dist xa x < d \<longrightarrow> dist (f xa) (f x) < e"
using ?lhs[unfolded continuous_within Lim_within] by auto
{ fix y assume "y\<in>f  (ball x d \<inter> s)"
hence "y \<in> ball (f x) e" using d(2) unfolding dist_nz[THEN sym]
apply (auto simp add: dist_commute) apply(erule_tac x=xa in ballE) apply auto using e>0 by auto
}
hence "\<exists>d>0. f  (ball x d \<inter> s) \<subseteq> ball (f x) e" using d>0 unfolding subset_eq ball_def by (auto simp add: dist_commute)  }
thus ?rhs by auto
next
assume ?rhs thus ?lhs unfolding continuous_within Lim_within ball_def subset_eq
apply (auto simp add: dist_commute) apply(erule_tac x=e in allE) by auto
qed

lemma continuous_at_ball:
"continuous (at x) f \<longleftrightarrow> (\<forall>e>0. \<exists>d>0. f  (ball x d) \<subseteq> ball (f x) e)" (is "?lhs = ?rhs")
proof
assume ?lhs thus ?rhs unfolding continuous_at Lim_at subset_eq Ball_def Bex_def image_iff mem_ball
apply auto apply(erule_tac x=e in allE) apply auto apply(rule_tac x=d in exI) apply auto apply(erule_tac x=xa in allE) apply (auto simp add: dist_commute dist_nz)
unfolding dist_nz[THEN sym] by auto
next
assume ?rhs thus ?lhs unfolding continuous_at Lim_at subset_eq Ball_def Bex_def image_iff mem_ball
apply auto apply(erule_tac x=e in allE) apply auto apply(rule_tac x=d in exI) apply auto apply(erule_tac x="f xa" in allE) by (auto simp add: dist_commute dist_nz)
qed

text{* Define setwise continuity in terms of limits within the set. *}

lemma continuous_on_iff:
"continuous_on s f \<longleftrightarrow>
(\<forall>x\<in>s. \<forall>e>0. \<exists>d>0. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (f x') (f x) < e)"
unfolding continuous_on_def Lim_within
apply (intro ball_cong [OF refl] all_cong ex_cong)
apply (rename_tac y, case_tac "y = x", simp)
done

definition
uniformly_continuous_on ::
"'a set \<Rightarrow> ('a::metric_space \<Rightarrow> 'b::metric_space) \<Rightarrow> bool"
where
"uniformly_continuous_on s f \<longleftrightarrow>
(\<forall>e>0. \<exists>d>0. \<forall>x\<in>s. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (f x') (f x) < e)"

text{* Some simple consequential lemmas. *}

lemma uniformly_continuous_imp_continuous:
" uniformly_continuous_on s f ==> continuous_on s f"
unfolding uniformly_continuous_on_def continuous_on_iff by blast

lemma continuous_at_imp_continuous_within:
"continuous (at x) f ==> continuous (at x within s) f"
unfolding continuous_within continuous_at using Lim_at_within by auto

lemma Lim_trivial_limit: "trivial_limit net \<Longrightarrow> (f ---> l) net"
by simp

lemmas continuous_on = continuous_on_def -- "legacy theorem name"

lemma continuous_within_subset:
"continuous (at x within s) f \<Longrightarrow> t \<subseteq> s
==> continuous (at x within t) f"
unfolding continuous_within by(metis tendsto_within_subset)

lemma continuous_on_interior:
shows "continuous_on s f \<Longrightarrow> x \<in> interior s \<Longrightarrow> continuous (at x) f"
by (erule interiorE, drule (1) continuous_on_subset,

lemma continuous_on_eq:
"(\<forall>x \<in> s. f x = g x) \<Longrightarrow> continuous_on s f \<Longrightarrow> continuous_on s g"
unfolding continuous_on_def tendsto_def eventually_at_topological
by simp

text {* Characterization of various kinds of continuity in terms of sequences. *}

lemma continuous_within_sequentially:
fixes f :: "'a::metric_space \<Rightarrow> 'b::topological_space"
shows "continuous (at a within s) f \<longleftrightarrow>
(\<forall>x. (\<forall>n::nat. x n \<in> s) \<and> (x ---> a) sequentially
--> ((f o x) ---> f a) sequentially)" (is "?lhs = ?rhs")
proof
assume ?lhs
{ fix x::"nat \<Rightarrow> 'a" assume x:"\<forall>n. x n \<in> s" "\<forall>e>0. eventually (\<lambda>n. dist (x n) a < e) sequentially"
fix T::"'b set" assume "open T" and "f a \<in> T"
with ?lhs obtain d where "d>0" and d:"\<forall>x\<in>s. 0 < dist x a \<and> dist x a < d \<longrightarrow> f x \<in> T"
unfolding continuous_within tendsto_def eventually_at by (auto simp: dist_nz)
have "eventually (\<lambda>n. dist (x n) a < d) sequentially"
using x(2) d>0 by simp
hence "eventually (\<lambda>n. (f \<circ> x) n \<in> T) sequentially"
proof eventually_elim
case (elim n) thus ?case
using d x(1) f a \<in> T unfolding dist_nz[THEN sym] by auto
qed
}
thus ?rhs unfolding tendsto_iff unfolding tendsto_def by simp
next
assume ?rhs thus ?lhs
unfolding continuous_within tendsto_def [where l="f a"]
qed

lemma continuous_at_sequentially:
fixes f :: "'a::metric_space \<Rightarrow> 'b::topological_space"
shows "continuous (at a) f \<longleftrightarrow> (\<forall>x. (x ---> a) sequentially
--> ((f o x) ---> f a) sequentially)"
using continuous_within_sequentially[of a UNIV f] by simp

lemma continuous_on_sequentially:
fixes f :: "'a::metric_space \<Rightarrow> 'b::topological_space"
shows "continuous_on s f \<longleftrightarrow>
(\<forall>x. \<forall>a \<in> s. (\<forall>n. x(n) \<in> s) \<and> (x ---> a) sequentially
--> ((f o x) ---> f(a)) sequentially)" (is "?lhs = ?rhs")
proof
assume ?rhs thus ?lhs using continuous_within_sequentially[of _ s f] unfolding continuous_on_eq_continuous_within by auto
next
assume ?lhs thus ?rhs unfolding continuous_on_eq_continuous_within using continuous_within_sequentially[of _ s f] by auto
qed

lemma uniformly_continuous_on_sequentially:
"uniformly_continuous_on s f \<longleftrightarrow> (\<forall>x y. (\<forall>n. x n \<in> s) \<and> (\<forall>n. y n \<in> s) \<and>
((\<lambda>n. dist (x n) (y n)) ---> 0) sequentially
\<longrightarrow> ((\<lambda>n. dist (f(x n)) (f(y n))) ---> 0) sequentially)" (is "?lhs = ?rhs")
proof
assume ?lhs
{ fix x y assume x:"\<forall>n. x n \<in> s" and y:"\<forall>n. y n \<in> s" and xy:"((\<lambda>n. dist (x n) (y n)) ---> 0) sequentially"
{ fix e::real assume "e>0"
then obtain d where "d>0" and d:"\<forall>x\<in>s. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (f x') (f x) < e"
using ?lhs[unfolded uniformly_continuous_on_def, THEN spec[where x=e]] by auto
obtain N where N:"\<forall>n\<ge>N. dist (x n) (y n) < d" using xy[unfolded LIMSEQ_def dist_norm] and d>0 by auto
{ fix n assume "n\<ge>N"
hence "dist (f (x n)) (f (y n)) < e"
using N[THEN spec[where x=n]] using d[THEN bspec[where x="x n"], THEN bspec[where x="y n"]] using x and y
unfolding dist_commute by simp  }
hence "\<exists>N. \<forall>n\<ge>N. dist (f (x n)) (f (y n)) < e"  by auto  }
hence "((\<lambda>n. dist (f(x n)) (f(y n))) ---> 0) sequentially" unfolding LIMSEQ_def and dist_real_def by auto  }
thus ?rhs by auto
next
assume ?rhs
{ assume "\<not> ?lhs"
then obtain e where "e>0" "\<forall>d>0. \<exists>x\<in>s. \<exists>x'\<in>s. dist x' x < d \<and> \<not> dist (f x') (f x) < e" unfolding uniformly_continuous_on_def by auto
then obtain fa where fa:"\<forall>x.  0 < x \<longrightarrow> fst (fa x) \<in> s \<and> snd (fa x) \<in> s \<and> dist (fst (fa x)) (snd (fa x)) < x \<and> \<not> dist (f (fst (fa x))) (f (snd (fa x))) < e"
using choice[of "\<lambda>d x. d>0 \<longrightarrow> fst x \<in> s \<and> snd x \<in> s \<and> dist (snd x) (fst x) < d \<and> \<not> dist (f (snd x)) (f (fst x)) < e"] unfolding Bex_def
def x \<equiv> "\<lambda>n::nat. fst (fa (inverse (real n + 1)))"
def y \<equiv> "\<lambda>n::nat. snd (fa (inverse (real n + 1)))"
have xyn:"\<forall>n. x n \<in> s \<and> y n \<in> s" and xy0:"\<forall>n. dist (x n) (y n) < inverse (real n + 1)" and fxy:"\<forall>n. \<not> dist (f (x n)) (f (y n)) < e"
unfolding x_def and y_def using fa by auto
{ fix e::real assume "e>0"
then obtain N::nat where "N \<noteq> 0" and N:"0 < inverse (real N) \<and> inverse (real N) < e" unfolding real_arch_inv[of e]   by auto
{ fix n::nat assume "n\<ge>N"
hence "inverse (real n + 1) < inverse (real N)" using real_of_nat_ge_zero and N\<noteq>0 by auto
also have "\<dots> < e" using N by auto
finally have "inverse (real n + 1) < e" by auto
hence "dist (x n) (y n) < e" using xy0[THEN spec[where x=n]] by auto  }
hence "\<exists>N. \<forall>n\<ge>N. dist (x n) (y n) < e" by auto  }
hence "\<forall>e>0. \<exists>N. \<forall>n\<ge>N. dist (f (x n)) (f (y n)) < e" using ?rhs[THEN spec[where x=x], THEN spec[where x=y]] and xyn unfolding LIMSEQ_def dist_real_def by auto
hence False using fxy and e>0 by auto  }
thus ?lhs unfolding uniformly_continuous_on_def by blast
qed

text{* The usual transformation theorems. *}

lemma continuous_transform_within:
fixes f g :: "'a::metric_space \<Rightarrow> 'b::topological_space"
assumes "0 < d" "x \<in> s" "\<forall>x' \<in> s. dist x' x < d --> f x' = g x'"
"continuous (at x within s) f"
shows "continuous (at x within s) g"
unfolding continuous_within
proof (rule Lim_transform_within)
show "0 < d" by fact
show "\<forall>x'\<in>s. 0 < dist x' x \<and> dist x' x < d \<longrightarrow> f x' = g x'"
using assms(3) by auto
have "f x = g x"
using assms(1,2,3) by auto
thus "(f ---> g x) (at x within s)"
using assms(4) unfolding continuous_within by simp
qed

lemma continuous_transform_at:
fixes f g :: "'a::metric_space \<Rightarrow> 'b::topological_space"
assumes "0 < d" "\<forall>x'. dist x' x < d --> f x' = g x'"
"continuous (at x) f"
shows "continuous (at x) g"
using continuous_transform_within [of d x UNIV f g] assms by simp

subsubsection {* Structural rules for pointwise continuity *}

lemmas continuous_within_id = continuous_ident

lemmas continuous_at_id = isCont_ident

lemma continuous_infdist[continuous_intros]:
assumes "continuous F f"
shows "continuous F (\<lambda>x. infdist (f x) A)"
using assms unfolding continuous_def by (rule tendsto_infdist)

lemma continuous_infnorm[continuous_intros]:
shows "continuous F f \<Longrightarrow> continuous F (\<lambda>x. infnorm (f x))"
unfolding continuous_def by (rule tendsto_infnorm)

lemma continuous_inner[continuous_intros]:
assumes "continuous F f" and "continuous F g"
shows "continuous F (\<lambda>x. inner (f x) (g x))"
using assms unfolding continuous_def by (rule tendsto_inner)

lemmas continuous_at_inverse = isCont_inverse

subsubsection {* Structural rules for setwise continuity *}

lemma continuous_on_infnorm[continuous_on_intros]:
shows "continuous_on s f \<Longrightarrow> continuous_on s (\<lambda>x. infnorm (f x))"
unfolding continuous_on by (fast intro: tendsto_infnorm)

lemma continuous_on_inner[continuous_on_intros]:
fixes g :: "'a::topological_space \<Rightarrow> 'b::real_inner"
assumes "continuous_on s f" and "continuous_on s g"
shows "continuous_on s (\<lambda>x. inner (f x) (g x))"
using bounded_bilinear_inner assms
by (rule bounded_bilinear.continuous_on)

subsubsection {* Structural rules for uniform continuity *}

lemma uniformly_continuous_on_id[continuous_on_intros]:
shows "uniformly_continuous_on s (\<lambda>x. x)"
unfolding uniformly_continuous_on_def by auto

lemma uniformly_continuous_on_const[continuous_on_intros]:
shows "uniformly_continuous_on s (\<lambda>x. c)"
unfolding uniformly_continuous_on_def by simp

lemma uniformly_continuous_on_dist[continuous_on_intros]:
fixes f g :: "'a::metric_space \<Rightarrow> 'b::metric_space"
assumes "uniformly_continuous_on s f"
assumes "uniformly_continuous_on s g"
shows "uniformly_continuous_on s (\<lambda>x. dist (f x) (g x))"
proof -
{ fix a b c d :: 'b have "\<bar>dist a b - dist c d\<bar> \<le> dist a c + dist b d"
using dist_triangle2 [of a b c] dist_triangle2 [of b c d]
using dist_triangle3 [of c d a] dist_triangle [of a d b]
by arith
} note le = this
{ fix x y
assume f: "(\<lambda>n. dist (f (x n)) (f (y n))) ----> 0"
assume g: "(\<lambda>n. dist (g (x n)) (g (y n))) ----> 0"
have "(\<lambda>n. \<bar>dist (f (x n)) (g (x n)) - dist (f (y n)) (g (y n))\<bar>) ----> 0"
by (rule Lim_transform_bound [OF _ tendsto_add_zero [OF f g]],
}
thus ?thesis using assms unfolding uniformly_continuous_on_sequentially
unfolding dist_real_def by simp
qed

lemma uniformly_continuous_on_norm[continuous_on_intros]:
assumes "uniformly_continuous_on s f"
shows "uniformly_continuous_on s (\<lambda>x. norm (f x))"
unfolding norm_conv_dist using assms
by (intro uniformly_continuous_on_dist uniformly_continuous_on_const)

lemma (in bounded_linear) uniformly_continuous_on[continuous_on_intros]:
assumes "uniformly_continuous_on s g"
shows "uniformly_continuous_on s (\<lambda>x. f (g x))"
using assms unfolding uniformly_continuous_on_sequentially
unfolding dist_norm tendsto_norm_zero_iff diff[symmetric]
by (auto intro: tendsto_zero)

lemma uniformly_continuous_on_cmul[continuous_on_intros]:
fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
assumes "uniformly_continuous_on s f"
shows "uniformly_continuous_on s (\<lambda>x. c *\<^sub>R f(x))"
using bounded_linear_scaleR_right assms
by (rule bounded_linear.uniformly_continuous_on)

lemma dist_minus:
fixes x y :: "'a::real_normed_vector"
shows "dist (- x) (- y) = dist x y"
unfolding dist_norm minus_diff_minus norm_minus_cancel ..

lemma uniformly_continuous_on_minus[continuous_on_intros]:
fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
shows "uniformly_continuous_on s f \<Longrightarrow> uniformly_continuous_on s (\<lambda>x. - f x)"
unfolding uniformly_continuous_on_def dist_minus .

fixes f g :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
assumes "uniformly_continuous_on s f"
assumes "uniformly_continuous_on s g"
shows "uniformly_continuous_on s (\<lambda>x. f x + g x)"
using assms unfolding uniformly_continuous_on_sequentially

lemma uniformly_continuous_on_diff[continuous_on_intros]:
fixes f :: "'a::metric_space \<Rightarrow> 'b::real_normed_vector"
assumes "uniformly_continuous_on s f" and "uniformly_continuous_on s g"
shows "uniformly_continuous_on s (\<lambda>x. f x - g x)"
unfolding ab_diff_minus using assms

text{* Continuity of all kinds is preserved under composition. *}

lemmas continuous_at_compose = isCont_o

lemma uniformly_continuous_on_compose[continuous_on_intros]:
assumes "uniformly_continuous_on s f"  "uniformly_continuous_on (f  s) g"
shows "uniformly_continuous_on s (g o f)"
proof-
{ fix e::real assume "e>0"
then obtain d where "d>0" and d:"\<forall>x\<in>f  s. \<forall>x'\<in>f  s. dist x' x < d \<longrightarrow> dist (g x') (g x) < e" using assms(2) unfolding uniformly_continuous_on_def by auto
obtain d' where "d'>0" "\<forall>x\<in>s. \<forall>x'\<in>s. dist x' x < d' \<longrightarrow> dist (f x') (f x) < d" using d>0 using assms(1) unfolding uniformly_continuous_on_def by auto
hence "\<exists>d>0. \<forall>x\<in>s. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist ((g \<circ> f) x') ((g \<circ> f) x) < e" using d>0 using d by auto  }
thus ?thesis using assms unfolding uniformly_continuous_on_def by auto
qed

text{* Continuity in terms of open preimages. *}

lemma continuous_at_open:
shows "continuous (at x) f \<longleftrightarrow> (\<forall>t. open t \<and> f x \<in> t --> (\<exists>s. open s \<and> x \<in> s \<and> (\<forall>x' \<in> s. (f x') \<in> t)))"
unfolding continuous_within_topological [of x UNIV f]
unfolding imp_conjL by (intro all_cong imp_cong ex_cong conj_cong refl) auto

lemma continuous_imp_tendsto:
assumes "continuous (at x0) f" and "x ----> x0"
shows "(f \<circ> x) ----> (f x0)"
proof (rule topological_tendstoI)
fix S
assume "open S" "f x0 \<in> S"
then obtain T where T_def: "open T" "x0 \<in> T" "\<forall>x\<in>T. f x \<in> S"
using assms continuous_at_open by metis
then have "eventually (\<lambda>n. x n \<in> T) sequentially"
using assms T_def by (auto simp: tendsto_def)
then show "eventually (\<lambda>n. (f \<circ> x) n \<in> S) sequentially"
using T_def by (auto elim!: eventually_elim1)
qed

lemma continuous_on_open:
"continuous_on s f \<longleftrightarrow>
(\<forall>t. openin (subtopology euclidean (f  s)) t
--> openin (subtopology euclidean s) {x \<in> s. f x \<in> t})" (is "?lhs = ?rhs")
unfolding continuous_on_open_invariant openin_open Int_def vimage_def Int_commute
by (simp add: imp_ex imageI conj_commute eq_commute cong: conj_cong)

text {* Similarly in terms of closed sets. *}

lemma continuous_on_closed:
shows "continuous_on s f \<longleftrightarrow>  (\<forall>t. closedin (subtopology euclidean (f  s)) t  --> closedin (subtopology euclidean s) {x \<in> s. f x \<in> t})" (is "?lhs = ?rhs")
unfolding continuous_on_closed_invariant closedin_closed Int_def vimage_def Int_commute
by (simp add: imp_ex imageI conj_commute eq_commute cong: conj_cong)

text {* Half-global and completely global cases. *}

lemma continuous_open_in_preimage:
assumes "continuous_on s f"  "open t"
shows "openin (subtopology euclidean s) {x \<in> s. f x \<in> t}"
proof-
have *:"\<forall>x. x \<in> s \<and> f x \<in> t \<longleftrightarrow> x \<in> s \<and> f x \<in> (t \<inter> f  s)" by auto
have "openin (subtopology euclidean (f  s)) (t \<inter> f  s)"
using openin_open_Int[of t "f  s", OF assms(2)] unfolding openin_open by auto
thus ?thesis using assms(1)[unfolded continuous_on_open, THEN spec[where x="t \<inter> f  s"]] using * by auto
qed

lemma continuous_closed_in_preimage:
assumes "continuous_on s f"  "closed t"
shows "closedin (subtopology euclidean s) {x \<in> s. f x \<in> t}"
proof-
have *:"\<forall>x. x \<in> s \<and> f x \<in> t \<longleftrightarrow> x \<in> s \<and> f x \<in> (t \<inter> f  s)" by auto
have "closedin (subtopology euclidean (f  s)) (t \<inter> f  s)"
using closedin_closed_Int[of t "f  s", OF assms(2)] unfolding Int_commute by auto
thus ?thesis
using assms(1)[unfolded continuous_on_closed, THEN spec[where x="t \<inter> f  s"]] using * by auto
qed

lemma continuous_open_preimage:
assumes "continuous_on s f" "open s" "open t"
shows "open {x \<in> s. f x \<in> t}"
proof-
obtain T where T: "open T" "{x \<in> s. f x \<in> t} = s \<inter> T"
using continuous_open_in_preimage[OF assms(1,3)] unfolding openin_open by auto
thus ?thesis using open_Int[of s T, OF assms(2)] by auto
qed

lemma continuous_closed_preimage:
assumes "continuous_on s f" "closed s" "closed t"
shows "closed {x \<in> s. f x \<in> t}"
proof-
obtain T where T: "closed T" "{x \<in> s. f x \<in> t} = s \<inter> T"
using continuous_closed_in_preimage[OF assms(1,3)] unfolding closedin_closed by auto
thus ?thesis using closed_Int[of s T, OF assms(2)] by auto
qed

lemma continuous_open_preimage_univ:
shows "\<forall>x. continuous (at x) f \<Longrightarrow> open s \<Longrightarrow> open {x. f x \<in> s}"
using continuous_open_preimage[of UNIV f s] open_UNIV continuous_at_imp_continuous_on by auto

lemma continuous_closed_preimage_univ:
shows "(\<forall>x. continuous (at x) f) \<Longrightarrow> closed s ==> closed {x. f x \<in> s}"
using continuous_closed_preimage[of UNIV f s] closed_UNIV continuous_at_imp_continuous_on by auto

lemma continuous_open_vimage:
shows "\<forall>x. continuous (at x) f \<Longrightarrow> open s \<Longrightarrow> open (f - s)"
unfolding vimage_def by (rule continuous_open_preimage_univ)

lemma continuous_closed_vimage:
shows "\<forall>x. continuous (at x) f \<Longrightarrow> closed s \<Longrightarrow> closed (f - s)"
unfolding vimage_def by (rule continuous_closed_preimage_univ)

lemma interior_image_subset:
assumes "\<forall>x. continuous (at x) f" "inj f"
shows "interior (f  s) \<subseteq> f  (interior s)"
proof
fix x assume "x \<in> interior (f  s)"
then obtain T where as: "open T" "x \<in> T" "T \<subseteq> f  s" ..
hence "x \<in> f  s" by auto
then obtain y where y: "y \<in> s" "x = f y" by auto
have "open (vimage f T)"
using assms(1) open T by (rule continuous_open_vimage)
moreover have "y \<in> vimage f T"
using x = f y x \<in> T by simp
moreover have "vimage f T \<subseteq> s"
using T \<subseteq> image f s inj f unfolding inj_on_def subset_eq by auto
ultimately have "y \<in> interior s" ..
with x = f y show "x \<in> f  interior s" ..
qed

text {* Equality of continuous functions on closure and related results. *}

lemma continuous_closed_in_preimage_constant:
fixes f :: "_ \<Rightarrow> 'b::t1_space"
shows "continuous_on s f ==> closedin (subtopology euclidean s) {x \<in> s. f x = a}"
using continuous_closed_in_preimage[of s f "{a}"] by auto

lemma continuous_closed_preimage_constant:
fixes f :: "_ \<Rightarrow> 'b::t1_space"
shows "continuous_on s f \<Longrightarrow> closed s ==> closed {x \<in> s. f x = a}"
using continuous_closed_preimage[of s f "{a}"] by auto

lemma continuous_constant_on_closure:
fixes f :: "_ \<Rightarrow> 'b::t1_space"
assumes "continuous_on (closure s) f"
"\<forall>x \<in> s. f x = a"
shows "\<forall>x \<in> (closure s). f x = a"
using continuous_closed_preimage_constant[of "closure s" f a]
assms closure_minimal[of s "{x \<in> closure s. f x = a}"] closure_subset unfolding subset_eq by auto

lemma image_closure_subset:
assumes "continuous_on (closure s) f"  "closed t"  "(f  s) \<subseteq> t"
shows "f  (closure s) \<subseteq> t"
proof-
have "s \<subseteq> {x \<in> closure s. f x \<in> t}" using assms(3) closure_subset by auto
moreover have "closed {x \<in> closure s. f x \<in> t}"
using continuous_closed_preimage[OF assms(1)] and assms(2) by auto
ultimately have "closure s = {x \<in> closure s . f x \<in> t}"
using closure_minimal[of s "{x \<in> closure s. f x \<in> t}"] by auto
thus ?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"  "\<forall>y \<in> s. norm(f y) \<le> b"  "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[THEN sym]] by auto
show ?thesis
using image_closure_subset[OF assms(1) closed_cball[of 0 b] *] assms(3)
unfolding subset_eq apply(erule_tac x="f x" in ballE) by (auto simp add: dist_norm)
qed

text {* Making a continuous function avoid some value in a neighbourhood. *}

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 f x \<noteq> a] by fast
have "(f ---> f x) (at x within s)"
using assms(1) by (simp add: continuous_within)
hence "eventually (\<lambda>y. f y \<in> U) (at x within s)"
using open U and f x \<in> U
unfolding tendsto_def by fast
hence "eventually (\<lambda>y. f y \<noteq> a) (at x within s)"
using a \<notin> U by (fast elim: eventually_mono [rotated])
thus ?thesis
using f x \<noteq> a by (auto simp: dist_commute zero_less_dist_iff eventually_at)
qed

lemma continuous_at_avoid:
fixes f :: "'a::metric_space \<Rightarrow> 'b::t1_space"
assumes "continuous (at x) f" and "f x \<noteq> a"
shows "\<exists>e>0. \<forall>y. dist x y < e \<longrightarrow> f y \<noteq> a"
using assms continuous_within_avoid[of x UNIV f a] by simp

lemma continuous_on_avoid:
fixes f :: "'a::metric_space \<Rightarrow> 'b::t1_space"
assumes "continuous_on s f"  "x \<in> s"  "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]  assms(3) by auto

lemma continuous_on_open_avoid:
fixes f :: "'a::metric_space \<Rightarrow> 'b::t1_space"
assumes "continuous_on s f"  "open s"  "x \<in> s"  "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)]  continuous_at_avoid[of x f a]  assms(4) by auto

text {* Proving a function is constant by proving open-ness of level set. *}

lemma continuous_levelset_open_in_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}
==> (\<forall>x \<in> s. f x \<noteq> a) \<or> (\<forall>x \<in> s. f x = a)"
unfolding connected_clopen using continuous_closed_in_preimage_constant by auto

lemma continuous_levelset_open_in:
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)  ==> (\<forall>x \<in> s. f x = a)"
using continuous_levelset_open_in_cases[of s f ]
by meson

lemma continuous_levelset_open:
fixes f :: "_ \<Rightarrow> 'b::t1_space"
assumes "connected s"  "continuous_on s f"  "open {x \<in> s. f x = a}"  "\<exists>x \<in> s.  f x = a"
shows "\<forall>x \<in> s. f x = a"
using continuous_levelset_open_in[OF assms(1,2), of a, unfolded openin_open] using assms (3,4) by fast

text {* Some arithmetical combinations (more to prove). *}

lemma open_scaling[intro]:
fixes s :: "'a::real_normed_vector set"
assumes "c \<noteq> 0"  "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 * abs c > 0" using assms(1)[unfolded zero_less_abs_iff[THEN sym]] using mult_pos_pos[OF e>0] by auto
moreover
{ fix y assume "dist y (c *\<^sub>R x) < e * \<bar>c\<bar>"
hence "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[THEN sym]] by (simp del:zero_less_abs_iff)
hence "y \<in> op *\<^sub>R c  s" using rev_image_eqI[of "(1 / c) *\<^sub>R y" s y "op *\<^sub>R c"]  e[THEN spec[where x="(1 / c) *\<^sub>R y"]]  assms(1) unfolding dist_norm scaleR_scaleR by auto  }
ultimately have "\<exists>e>0. \<forall>x'. dist x' (c *\<^sub>R x) < e \<longrightarrow> x' \<in> op *\<^sub>R c  s" apply(rule_tac x="e * abs c" in exI) by auto  }
thus ?thesis unfolding open_dist by auto
qed

lemma minus_image_eq_vimage:
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 ==> open ((\<lambda> x. -x)  s)"
unfolding scaleR_minus1_left [symmetric]
by (rule open_scaling, auto)

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_at_id continuous_const) }
moreover have "{x. x - a \<in> s} = op + a  s" by force
ultimately show ?thesis using continuous_open_preimage_univ[of "\<lambda>x. x - a" s] using assms by auto
qed

lemma open_affinity:
fixes s :: "'a::real_normed_vector set"
assumes "open s"  "c \<noteq> 0"
shows "open ((\<lambda>x. a + c *\<^sub>R x)  s)"
proof-
have *:"(\<lambda>x. a + c *\<^sub>R x) = (\<lambda>x. a + x) \<circ> (\<lambda>x. c *\<^sub>R x)" unfolding o_def ..
have "op + a  op *\<^sub>R c  s = (op + a \<circ> op *\<^sub>R c)  s" by auto
thus ?thesis using assms open_translation[of "op *\<^sub>R c  s" a] unfolding * by auto
qed

lemma interior_translation:
fixes s :: "'a::real_normed_vector set"
shows "interior ((\<lambda>x. a + x)  s) = (\<lambda>x. a + x)  (interior s)"
proof (rule set_eqI, rule)
fix x assume "x \<in> interior (op + a  s)"
then obtain e where "e>0" and e:"ball x e \<subseteq> op + a  s" unfolding mem_interior by auto
hence "ball (x - a) e \<subseteq> s" unfolding subset_eq Ball_def mem_ball dist_norm apply auto apply(erule_tac x="a + xa" in allE) unfolding ab_group_add_class.diff_diff_eq[THEN sym] by auto
thus "x \<in> op + a  interior s" unfolding image_iff apply(rule_tac x="x - a" in bexI) unfolding mem_interior using e > 0 by auto
next
fix x assume "x \<in> op + a  interior s"
then obtain y e where "e>0" and e:"ball y e \<subseteq> s" and y:"x = a + y" unfolding image_iff Bex_def mem_interior by auto
{ fix z have *:"a + y - z = y + a - z" by auto
assume "z\<in>ball x e"
hence "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
hence "z \<in> op + a  s" unfolding image_iff by(auto intro!: bexI[where x="z - a"])  }
hence "ball x e \<subseteq> op + a  s" unfolding subset_eq by auto
thus "x \<in> interior (op + a  s)" unfolding mem_interior using e>0 by auto
qed

text {* Topological properties of linear functions. *}

lemma linear_lim_0:
assumes "bounded_linear f" shows "(f ---> 0) (at (0))"
proof-
interpret f: bounded_linear f by fact
have "(f ---> f 0) (at 0)"
using tendsto_ident_at by (rule f.tendsto)
thus ?thesis unfolding f.zero .
qed

lemma linear_continuous_at:
assumes "bounded_linear f"  shows "continuous (at a) f"
unfolding continuous_at using assms
apply (rule bounded_linear.tendsto)
apply (rule tendsto_ident_at)
done

lemma linear_continuous_within:
shows "bounded_linear f ==> continuous (at x within s) f"
using continuous_at_imp_continuous_within[of x f s] using linear_continuous_at[of f] by auto

lemma linear_continuous_on:
shows "bounded_linear f ==> continuous_on s f"
using continuous_at_imp_continuous_on[of s f] using linear_continuous_at[of f] by auto

text {* Also bilinear functions, in composition form. *}

lemma bilinear_continuous_at_compose:
shows "continuous (at x) f \<Longrightarrow> continuous (at x) g \<Longrightarrow> bounded_bilinear h
==> continuous (at x) (\<lambda>x. h (f x) (g x))"
unfolding continuous_at using Lim_bilinear[of f "f x" "(at x)" g "g x" h] by auto

lemma bilinear_continuous_within_compose:
shows "continuous (at x within s) f \<Longrightarrow> continuous (at x within s) g \<Longrightarrow> bounded_bilinear h
==> continuous (at x within s) (\<lambda>x. h (f x) (g x))"
unfolding continuous_within using Lim_bilinear[of f "f x"] by auto

lemma bilinear_continuous_on_compose:
shows "continuous_on s f \<Longrightarrow> continuous_on s g \<Longrightarrow> bounded_bilinear h
==> continuous_on s (\<lambda>x. h (f x) (g x))"
unfolding continuous_on_def
by (fast elim: bounded_bilinear.tendsto)

text {* Preservation of compactness and connectedness under continuous function. *}

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"
hence "\<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 compact S obtain D where "D \<subseteq> {T. open T \<and> S \<inter> T \<in> C}" and "finite D" and "S \<subseteq> \<Union>D"
by (rule compactE)
hence "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
thus "\<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"
hence "(\<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 D \<subseteq> ?C show "?D \<subseteq> C"
by (fast intro: inv_into_into)
from finite D show "finite ?D"
by (rule finite_imageI)
from S \<subseteq> \<Union>D show "S \<subseteq> \<Union>?D"
apply (rule subset_trans)
apply clarsimp
apply (frule subsetD [OF D \<subseteq> ?C, THEN f_inv_into_f])
apply (erule rev_bexI, fast)
done
qed
thus "\<exists>D\<subseteq>C. finite D \<and> S \<subseteq> \<Union>D" ..
qed
qed

lemma connected_continuous_image:
assumes "continuous_on s f"  "connected s"
shows "connected(f  s)"
proof-
{ fix T assume as: "T \<noteq> {}"  "T \<noteq> f  s"  "openin (subtopology euclidean (f  s)) T"  "closedin (subtopology euclidean (f  s)) T"
have "{x \<in> s. f x \<in> T} = {} \<or> {x \<in> s. f x \<in> T} = s"
using assms(1)[unfolded continuous_on_open, THEN spec[where x=T]]
using assms(1)[unfolded continuous_on_closed, THEN spec[where x=T]]
using assms(2)[unfolded connected_clopen, THEN spec[where x="{x \<in> s. f x \<in> T}"]] as(3,4) by auto
hence False using as(1,2)
using as(4)[unfolded closedin_def topspace_euclidean_subtopology] by auto }
thus ?thesis unfolding connected_clopen by auto
qed

text {* Continuity implies uniform continuity on a compact domain. *}

lemma compact_uniformly_continuous:
assumes f: "continuous_on s f" and s: "compact s"
shows "uniformly_continuous_on s f"
unfolding uniformly_continuous_on_def
proof (cases, safe)
fix e :: real assume "0 < e" "s \<noteq> {}"
def [simp]: R \<equiv> "{(y, d). y \<in> s \<and> 0 < d \<and> ball y d \<inter> s \<subseteq> {x \<in> s. f x \<in> ball (f y) (e/2) } }"
let ?b = "(\<lambda>(y, d). ball y (d/2))"
have "(\<forall>r\<in>R. open (?b r))" "s \<subseteq> (\<Union>r\<in>R. ?b r)"
proof safe
fix y assume "y \<in> s"
from continuous_open_in_preimage[OF f open_ball]
obtain T where "open T" and T: "{x \<in> s. f x \<in> ball (f y) (e/2)} = T \<inter> s"
unfolding openin_subtopology open_openin by metis
then obtain d where "ball y d \<subseteq> T" "0 < d"
using 0 < e y \<in> s by (auto elim!: openE)
with T y \<in> s show "y \<in> (\<Union>r\<in>R. ?b r)"
by (intro UN_I[of "(y, d)"]) auto
qed auto
with s obtain D where D: "finite D" "D \<subseteq> R" "s \<subseteq> (\<Union>(y, d)\<in>D. ball y (d/2))"
by (rule compactE_image)
with s \<noteq> {} have [simp]: "\<And>x. x < Min (snd  D) \<longleftrightarrow> (\<forall>(y, d)\<in>D. x < d)"
by (subst Min_gr_iff) auto
show "\<exists>d>0. \<forall>x\<in>s. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (f x') (f x) < e"
proof (rule, safe)
fix x x' assume in_s: "x' \<in> s" "x \<in> s"
with D obtain y d where x: "x \<in> ball y (d/2)" "(y, d) \<in> D"
by blast
moreover assume "dist x x' < Min (sndD) / 2"
ultimately have "dist y x' < d"
by (intro dist_double[where x=x and d=d]) (auto simp: dist_commute)
with D x in_s show  "dist (f x) (f x') < e"
by (intro dist_double[where x="f y" and d=e]) (auto simp: dist_commute subset_eq)
qed (insert D, auto)
qed auto

text {* A uniformly convergent limit of continuous functions is continuous. *}

lemma continuous_uniform_limit:
fixes f :: "'a \<Rightarrow> 'b::metric_space \<Rightarrow> 'c::metric_space"
assumes "\<not> trivial_limit F"
assumes "eventually (\<lambda>n. continuous_on s (f n)) F"
assumes "\<forall>e>0. eventually (\<lambda>n. \<forall>x\<in>s. dist (f n x) (g x) < e) F"
shows "continuous_on s g"
proof-
{ fix x and e::real assume "x\<in>s" "e>0"
have "eventually (\<lambda>n. \<forall>x\<in>s. dist (f n x) (g x) < e / 3) F"
using e>0 assms(3)[THEN spec[where x="e/3"]] by auto
from eventually_happens [OF eventually_conj [OF this assms(2)]]
obtain n where n:"\<forall>x\<in>s. dist (f n x) (g x) < e / 3"  "continuous_on s (f n)"
using assms(1) by blast
have "e / 3 > 0" using e>0 by auto
then obtain d where "d>0" and d:"\<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (f n x') (f n x) < e / 3"
using n(2)[unfolded continuous_on_iff, THEN bspec[where x=x], OF x\<in>s, THEN spec[where x="e/3"]] by blast
{ fix y assume "y \<in> s" and "dist y x < d"
hence "dist (f n y) (f n x) < e / 3"
by (rule d [rule_format])
hence "dist (f n y) (g x) < 2 * e / 3"
using dist_triangle [of "f n y" "g x" "f n x"]
using n(1)[THEN bspec[where x=x], OF x\<in>s]
by auto
hence "dist (g y) (g x) < e"
using n(1)[THEN bspec[where x=y], OF y\<in>s]
using dist_triangle3 [of "g y" "g x" "f n y"]
by auto }
hence "\<exists>d>0. \<forall>x'\<in>s. dist x' x < d \<longrightarrow> dist (g x') (g x) < e"
using d>0 by auto }
thus ?thesis unfolding continuous_on_iff by auto
qed

subsection {* Topological stuff lifted from and dropped to R *}

lemma open_real:
fixes s :: "real set" shows
"open s \<longleftrightarrow>
(\<forall>x \<in> s. \<exists>e>0. \<forall>x'. abs(x' - x) < e --> x' \<in> s)" (is "?lhs = ?rhs")
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> abs(x' - x) < 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> abs(x' - x) < e)
--> 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 --> abs(f x' - f x) < e)"
unfolding continuous_at unfolding Lim_at
unfolding dist_nz[THEN sym] unfolding dist_norm apply auto
apply(erule_tac x=e in allE) apply auto apply (rule_tac x=d in exI) apply auto apply (erule_tac x=x' in allE) apply auto
apply(erule_tac x=e in allE) by auto

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 --> abs(f x' - f x) < e))"
unfolding continuous_on_iff dist_norm by simp

text {* Hence some handy theorems on distance, diameter etc. of/from a set. *}

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 ---> dist a x) (at x within s)"
by (intro tendsto_dist tendsto_const tendsto_ident_at)
}
thus "continuous_on s (dist a)"
unfolding continuous_on ..
qed

text {* For \emph{minimal} distance, we only need closure, not compactness. *}

lemma distance_attains_inf:
fixes a :: "'a::heine_borel"
assumes "closed s"  "s \<noteq> {}"
shows "\<exists>x\<in>s. \<forall>y\<in>s. dist a x \<le> dist a y"
proof-
from assms(2) obtain b where "b \<in> s" by auto
let ?B = "s \<inter> cball a (dist b a)"
have "?B \<noteq> {}" using b \<in> s by (auto simp add: dist_commute)
moreover have "continuous_on ?B (dist a)"
by (auto intro!: continuous_at_imp_continuous_on continuous_dist continuous_at_id continuous_const)
moreover have "compact ?B"
by (intro closed_inter_compact closed s 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)
thus ?thesis by fastforce
qed

subsection {* Pasted sets *}

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\<twosuperior> + b\<twosuperior>)"
thus ?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: subseq_def)
apply (drule_tac f=r2 in LIMSEQ_subseq_LIMSEQ, assumption)
apply (drule (1) tendsto_Pair) back
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 x \<in> s 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
from compactE_image[OF compact t this] obtain D where "D \<subseteq> t" "finite D" "t \<subseteq> (\<Union>y\<in>D. b y)"
by auto
moreover with 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 _ "cD"] exI[of _ "\<Inter>(aD)"] conjI) (auto intro!: open_INT)
qed
then obtain a d where a: "\<forall>x\<in>s. 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 compact s 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 e \<subseteq> s 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 add: subset_eq)
qed

text{* Hence some useful properties follow quite easily. *}

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"  "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) by auto
have "continuous_on (s \<times> t) (\<lambda>z. fst z + snd z)"
unfolding continuous_on by (rule ballI) (intro tendsto_intros)
thus ?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" "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) apply auto apply(rule_tac x=xa in exI) by auto
thus ?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
thus ?thesis using compact_sums[OF assms compact_sing[of a]] by auto
qed

lemma compact_affinity:
fixes s :: "'a::real_normed_vector set"
assumes "compact s"  shows "compact ((\<lambda>x. a + c *\<^sub>R x)  s)"
proof-
have "op + a  op *\<^sub>R c  s = (\<lambda>x. a + c *\<^sub>R x)  s" by auto
thus ?thesis using compact_translation[OF compact_scaling[OF assms], of a c] by auto
qed

text {* Hence we get the following. *}

lemma compact_sup_maxdistance:
fixes s :: "'a::metric_space set"
assumes "compact s"  "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 compact s by (intro compact_Times)
moreover have "s \<times> s \<noteq> {}" using s \<noteq> {} 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

text {* We can state this in terms of diameter of a set. *}

definition "diameter s = (if s = {} then 0::real else Sup {dist x y | x y. x \<in> s \<and> y \<in> s})"

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 -
let ?D = "{dist x y |x y. x \<in> s \<and> y \<in> s}"
from s obtain z d where z: "\<And>x. x \<in> s \<Longrightarrow> dist z x \<le> d"
unfolding bounded_def by auto
have "dist x y \<le> Sup ?D"
proof (rule cSup_upper, 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"
qed (insert s, auto)
with x \<in> s show ?thesis
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)
let ?D = "{dist x y |x y. x \<in> s \<and> y \<in> s}"
assume contr: "\<not> ?thesis"
moreover
from d have "s \<noteq> {}"
by (auto simp: diameter_def)
then have "?D \<noteq> {}" by auto
ultimately have "Sup ?D \<le> d"
by (intro cSup_least) (auto simp: not_less)
with d < diameter s s \<noteq> {} show False
by (auto simp: diameter_def)
qed

lemma diameter_bounded:
assumes "bounded s"
shows "\<forall>x\<in>s. \<forall>y\<in>s. dist x y \<le> diameter s"
"\<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"  "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
hence "diameter s \<le> dist x y"
unfolding diameter_def by clarsimp (rule cSup_least, fast+)
thus ?thesis
by (metis b diameter_bounded_bound order_antisym xys)
qed

text {* Related results with closure as the conclusion. *}

lemma closed_scaling:
fixes s :: "'a::real_normed_vector set"
assumes "closed s" shows "closed ((\<lambda>x. c *\<^sub>R x)  s)"
proof(cases "s={}")
case True thus ?thesis by auto
next
case False
show ?thesis
proof(cases "c=0")
have *:"(\<lambda>x. 0)  s = {0}" using s\<noteq>{} by auto
case True thus ?thesis apply auto unfolding * by auto
next
case False
{ fix x l assume as:"\<forall>n::nat. x n \<in> scaleR c  s"  "(x ---> l) sequentially"
{ fix n::nat have "scaleR (1 / c) (x n) \<in> s"
using as(1)[THEN spec[where x=n]]
using c\<noteq>0 by auto
}
moreover
{ fix e::real assume "e>0"
hence "0 < e *\<bar>c\<bar>"  using c\<noteq>0 mult_pos_pos[of e "abs c"] by auto
then obtain N where "\<forall>n\<ge>N. dist (x n) l < e * \<bar>c\<bar>"
using as(2)[unfolded LIMSEQ_def, THEN spec[where x="e * abs c"]] by auto
hence "\<exists>N. \<forall>n\<ge>N. dist (scaleR (1 / c) (x n)) (scaleR (1 / c) l) < e"
unfolding dist_norm unfolding scaleR_right_diff_distrib[THEN sym]
using mult_imp_div_pos_less[of "abs c" _ e] c\<noteq>0 by auto  }
hence "((\<lambda>n. scaleR (1 / c) (x n)) ---> scaleR (1 / c) l) sequentially" unfolding LIMSEQ_def by auto
ultimately have "l \<in> scaleR c  s"
using assms[unfolded closed_sequential_limits, THEN spec[where x="\<lambda>n. scaleR (1/c) (x n)"], THEN spec[where x="scaleR (1/c) l"]]
unfolding image_iff using c\<noteq>0 apply(rule_tac x="scaleR (1 / c) l" in bexI) by auto  }
thus ?thesis unfolding closed_sequential_limits by fast
qed
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"  "closed t"  shows "closed {x + y | x y. x \<in> s \<and> y \<in> t}"
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 ---> 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:"subseq r" and lr:"(((\<lambda>n. fst (f n)) \<circ> r) ---> 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))) ---> l - l') sequentially"
using tendsto_diff[OF LIMSEQ_subseq_LIMSEQ[OF as(2) r] lr] and f(1) unfolding o_def by auto
hence "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
hence "l \<in> ?S" using l' \<in> s apply auto apply(rule_tac x=l' in exI) apply(rule_tac x="l - l'" in exI) by auto
}
thus ?thesis unfolding closed_sequential_limits by fast
qed

lemma closed_compact_sums:
fixes s t :: "'a::real_normed_vector set"
assumes "closed s"  "compact t"
shows "closed {x + y | x y. x \<in> s \<and> y \<in> t}"
proof-
have "{x + y |x y. x \<in> t \<and> y \<in> s} = {x + y |x y. x \<in> s \<and> y \<in> t}" apply auto
apply(rule_tac x=y in exI) apply auto apply(rule_tac x=y in exI) by auto
thus ?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 {x - y | x y. x \<in> s \<and> y \<in> t}"
proof-
have "{x + y |x y. x \<in> s \<and> y \<in> uminus  t} =  {x - y |x y. x \<in> s \<and> y \<in> t}"
apply auto apply(rule_tac x=xa in exI) apply auto apply(rule_tac x=xa in exI) by auto
thus ?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 {x - y | x y. x \<in> s \<and> y \<in> t}"
proof-
have "{x + y |x y. x \<in> s \<and> y \<in> uminus  t} = {x - y |x y. x \<in> s \<and> y \<in> t}"
apply auto apply(rule_tac x=xa in exI) apply auto apply(rule_tac x=xa in exI) by auto
thus ?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 "{a + y |y. y \<in> s} = (op + a  s)" by auto
thus ?thesis using compact_closed_sums[OF compact_sing[of a] assms] by auto
qed

lemma translation_Compl:
shows "(\<lambda>x. a + x)  (- t) = - ((\<lambda>x. a + x)  t)"
apply (auto simp add: image_iff) apply(rule_tac x="x - a" in bexI) by auto

lemma translation_UNIV:
fixes a :: "'a::ab_group_add" shows "range (\<lambda>x. a + x) = UNIV"
apply (auto simp add: image_iff) apply(rule_tac x="x - a" in exI) by auto

lemma translation_diff:
shows "(\<lambda>x. a + x)  (s - t) = ((\<lambda>x. a + x)  s) - ((\<lambda>x. a + x)  t)"
by auto

lemma closure_translation:
fixes a :: "'a::real_normed_vector"
shows "closure ((\<lambda>x. a + x)  s) = (\<lambda>x. a + x)  (closure s)"
proof-
have *:"op + a  (- s) = - op + a  s"
apply auto unfolding image_iff apply(rule_tac x="x - a" in bexI) by auto
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

subsection {* Separation between points and sets *}

lemma separate_point_closed:
fixes s :: "'a::heine_borel set"
shows "closed s \<Longrightarrow> a \<notin> s  ==> (\<exists>d>0. \<forall>x\<in>s. d \<le> dist a x)"
proof(cases "s = {}")
case True
thus ?thesis by(auto intro!: exI[where x=1])
next
case False
assume "closed s" "a \<notin> s"
then obtain x where "x\<in>s" "\<forall>y\<in>s. dist a x \<le> dist a y" using s \<noteq> {} distance_attains_inf [of s a] by blast
with x\<in>s show ?thesis using dist_pos_lt[of a x] anda \<notin> s 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_at_id)
then obtain x where x: "x \<in> s" "\<forall>y\<in>s. ?inf x \<le> ?inf y"
using continuous_attains_inf[OF compact s s \<noteq> {}] by auto
then have "0 < ?inf x"
using t t \<noteq> {} in_closed_iff_infdist_zero by (auto simp: less_le infdist_nonneg)
moreover have "\<forall>x'\<in>s. \<forall>y\<in>t. ?inf x \<le> dist x' y"
using x by (auto intro: order_trans infdist_le)
ultimately show ?thesis
by auto
qed (auto intro!: exI[of _ 1])

lemma separate_closed_compact:
fixes s t :: "'a::heine_borel set"
assumes "closed s" and "compact t" and "s \<inter> t = {}"
shows "\<exists>d>0. \<forall>x\<in>s. \<forall>y\<in>t. d \<le> dist x y"
proof-
have *:"t \<inter> s = {}" using assms(3) by auto
show ?thesis using separate_compact_closed[OF assms(2,1) *]
apply auto apply(rule_tac x=d in exI) apply auto apply (erule_tac x=y in ballE)
qed

subsection {* Intervals *}

lemma interval: fixes a :: "'a::ordered_euclidean_space" shows
"{a <..< b} = {x::'a. \<forall>i\<in>Basis. a\<bullet>i < x\<bullet>i \<and> x\<bullet>i < b\<bullet>i}" and
"{a .. b} = {x::'a. \<forall>i\<in>Basis. a\<bullet>i \<le> x\<bullet>i \<and> x\<bullet>i \<le> b\<bullet>i}"
by(auto simp add:set_eq_iff eucl_le[where 'a='a] eucl_less[where 'a='a])

lemma mem_interval: fixes a :: "'a::ordered_euclidean_space" shows
"x \<in> {a<..<b} \<longleftrightarrow> (\<forall>i\<in>Basis. a\<bullet>i < x\<bullet>i \<and> x\<bullet>i < b\<bullet>i)"
"x \<in> {a .. b} \<longleftrightarrow> (\<forall>i\<in>Basis. a\<bullet>i \<le> x\<bullet>i \<and> x\<bullet>i \<le> b\<bullet>i)"
using interval[of a b] by(auto simp add: set_eq_iff eucl_le[where 'a='a] eucl_less[where 'a='a])

lemma interval_eq_empty: fixes a :: "'a::ordered_euclidean_space" shows
"({a <..< b} = {} \<longleftrightarrow> (\<exists>i\<in>Basis. b\<bullet>i \<le> a\<bullet>i))" (is ?th1) and
"({a  ..  b} = {} \<longleftrightarrow> (\<exists>i\<in>Basis. b\<bullet>i < a\<bullet>i))" (is ?th2)
proof-
{ fix i x assume i:"i\<in>Basis" and as:"b\<bullet>i \<le> a\<bullet>i" and x:"x\<in>{a <..< b}"
hence "a \<bullet> i < x \<bullet> i \<and> x \<bullet> i < b \<bullet> i" unfolding mem_interval by auto
hence "a\<bullet>i < b\<bullet>i" by auto
hence False using as by auto  }
moreover
{ assume as:"\<forall>i\<in>Basis. \<not> (b\<bullet>i \<le> a\<bullet>i)"
let ?x = "(1/2) *\<^sub>R (a + b)"
{ fix i :: 'a assume i:"i\<in>Basis"
have "a\<bullet>i < b\<bullet>i" using as[THEN bspec[where x=i]] i by auto
hence "a\<bullet>i < ((1/2) *\<^sub>R (a+b)) \<bullet> i" "((1/2) *\<^sub>R (a+b)) \<bullet> i < b\<bullet>i"
hence "{a <..< b} \<noteq> {}" using mem_interval(1)[of "?x" a b] by auto  }
ultimately show ?th1 by blast

{ fix i x assume i:"i\<in>Basis" and as:"b\<bullet>i < a\<bullet>i" and x:"x\<in>{a .. b}"
hence "a \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> b \<bullet> i" unfolding mem_interval by auto
hence "a\<bullet>i \<le> b\<bullet>i" by auto
hence False using as by auto  }
moreover
{ assume as:"\<forall>i\<in>Basis. \<not> (b\<bullet>i < a\<bullet>i)"
let ?x = "(1/2) *\<^sub>R (a + b)"
{ fix i :: 'a assume i:"i\<in>Basis"
have "a\<bullet>i \<le> b\<bullet>i" using as[THEN bspec[where x=i]] i by auto
hence "a\<bullet>i \<le> ((1/2) *\<^sub>R (a+b)) \<bullet> i" "((1/2) *\<^sub>R (a+b)) \<bullet> i \<le> b\<bullet>i"
hence "{a .. b} \<noteq> {}" using mem_interval(2)[of "?x" a b] by auto  }
ultimately show ?th2 by blast
qed

lemma interval_ne_empty: fixes a :: "'a::ordered_euclidean_space" shows
"{a  ..  b} \<noteq> {} \<longleftrightarrow> (\<forall>i\<in>Basis. a\<bullet>i \<le> b\<bullet>i)" and
"{a <..< b} \<noteq> {} \<longleftrightarrow> (\<forall>i\<in>Basis. a\<bullet>i < b\<bullet>i)"
unfolding interval_eq_empty[of a b] by fastforce+

lemma interval_sing:
fixes a :: "'a::ordered_euclidean_space"
shows "{a .. a} = {a}" and "{a<..<a} = {}"
unfolding set_eq_iff mem_interval eq_iff [symmetric]
by (auto intro: euclidean_eqI simp: ex_in_conv)

lemma subset_interval_imp: fixes a :: "'a::ordered_euclidean_space" shows
"(\<forall>i\<in>Basis. a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i) \<Longrightarrow> {c .. d} \<subseteq> {a .. b}" and
"(\<forall>i\<in>Basis. a\<bullet>i < c\<bullet>i \<and> d\<bullet>i < b\<bullet>i) \<Longrightarrow> {c .. d} \<subseteq> {a<..<b}" and
"(\<forall>i\<in>Basis. a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i) \<Longrightarrow> {c<..<d} \<subseteq> {a .. b}" and
"(\<forall>i\<in>Basis. a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i) \<Longrightarrow> {c<..<d} \<subseteq> {a<..<b}"
unfolding subset_eq[unfolded Ball_def] unfolding mem_interval
by (best intro: order_trans less_le_trans le_less_trans less_imp_le)+

lemma interval_open_subset_closed:
fixes a :: "'a::ordered_euclidean_space"
shows "{a<..<b} \<subseteq> {a .. b}"
unfolding subset_eq [unfolded Ball_def] mem_interval
by (fast intro: less_imp_le)

lemma subset_interval: fixes a :: "'a::ordered_euclidean_space" shows
"{c .. d} \<subseteq> {a .. b} \<longleftrightarrow> (\<forall>i\<in>Basis. c\<bullet>i \<le> d\<bullet>i) --> (\<forall>i\<in>Basis. a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i)" (is ?th1) and
"{c .. d} \<subseteq> {a<..<b} \<longleftrightarrow> (\<forall>i\<in>Basis. c\<bullet>i \<le> d\<bullet>i) --> (\<forall>i\<in>Basis. a\<bullet>i < c\<bullet>i \<and> d\<bullet>i < b\<bullet>i)" (is ?th2) and
"{c<..<d} \<subseteq> {a .. b} \<longleftrightarrow> (\<forall>i\<in>Basis. c\<bullet>i < d\<bullet>i) --> (\<forall>i\<in>Basis. a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i)" (is ?th3) and
"{c<..<d} \<subseteq> {a<..<b} \<longleftrightarrow> (\<forall>i\<in>Basis. c\<bullet>i < d\<bullet>i) --> (\<forall>i\<in>Basis. a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i)" (is ?th4)
proof-
show ?th1 unfolding subset_eq and Ball_def and mem_interval by (auto intro: order_trans)
show ?th2 unfolding subset_eq and Ball_def and mem_interval by (auto intro: le_less_trans less_le_trans order_trans less_imp_le)
{ assume as: "{c<..<d} \<subseteq> {a .. b}" "\<forall>i\<in>Basis. c\<bullet>i < d\<bullet>i"
hence "{c<..<d} \<noteq> {}" unfolding interval_eq_empty by auto
fix i :: 'a assume i:"i\<in>Basis"
(** TODO combine the following two parts as done in the HOL_light version. **)
{ let ?x = "(\<Sum>j\<in>Basis. (if j=i then ((min (a\<bullet>j) (d\<bullet>j))+c\<bullet>j)/2 else (c\<bullet>j+d\<bullet>j)/2) *\<^sub>R j)::'a"
assume as2: "a\<bullet>i > c\<bullet>i"
{ fix j :: 'a assume j:"j\<in>Basis"
hence "c \<bullet> j < ?x \<bullet> j \<and> ?x \<bullet> j < d \<bullet> j"
apply(cases "j=i") using as(2)[THEN bspec[where x=j]] i
by (auto simp add: as2)  }
hence "?x\<in>{c<..<d}" using i unfolding mem_interval by auto
moreover
have "?x\<notin>{a .. b}"
unfolding mem_interval apply auto apply(rule_tac x=i in bexI)
using as(2)[THEN bspec[where x=i]] and as2 i
by auto
ultimately have False using as by auto  }
hence "a\<bullet>i \<le> c\<bullet>i" by(rule ccontr)auto
moreover
{ let ?x = "(\<Sum>j\<in>Basis. (if j=i then ((max (b\<bullet>j) (c\<bullet>j))+d\<bullet>j)/2 else (c\<bullet>j+d\<bullet>j)/2) *\<^sub>R j)::'a"
assume as2: "b\<bullet>i < d\<bullet>i"
{ fix j :: 'a assume "j\<in>Basis"
hence "d \<bullet> j > ?x \<bullet> j \<and> ?x \<bullet> j > c \<bullet> j"
apply(cases "j=i") using as(2)[THEN bspec[where x=j]]
by (auto simp add: as2) }
hence "?x\<in>{c<..<d}" unfolding mem_interval by auto
moreover
have "?x\<notin>{a .. b}"
unfolding mem_interval apply auto apply(rule_tac x=i in bexI)
using as(2)[THEN bspec[where x=i]] and as2 using i
by auto
ultimately have False using as by auto  }
hence "b\<bullet>i \<ge> d\<bullet>i" by(rule ccontr)auto
ultimately
have "a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i" by auto
} note part1 = this
show ?th3
unfolding subset_eq and Ball_def and mem_interval
apply(rule,rule,rule,rule)
apply(rule part1)
unfolding subset_eq and Ball_def and mem_interval
prefer 4
apply auto
by(erule_tac x=xa in allE,erule_tac x=xa in allE,fastforce)+
{ assume as:"{c<..<d} \<subseteq> {a<..<b}" "\<forall>i\<in>Basis. c\<bullet>i < d\<bullet>i"
fix i :: 'a assume i:"i\<in>Basis"
from as(1) have "{c<..<d} \<subseteq> {a..b}" using interval_open_subset_closed[of a b] by auto
hence "a\<bullet>i \<le> c\<bullet>i \<and> d\<bullet>i \<le> b\<bullet>i" using part1 and as(2) using i by auto  } note * = this
show ?th4 unfolding subset_eq and Ball_def and mem_interval
apply(rule,rule,rule,rule) apply(rule *) unfolding subset_eq and Ball_def and mem_interval prefer 4
apply auto by(erule_tac x=xa in allE, simp)+
qed

lemma inter_interval: fixes a :: "'a::ordered_euclidean_space" shows
"{a .. b} \<inter> {c .. d} =  {(\<Sum>i\<in>Basis. max (a\<bullet>i) (c\<bullet>i) *\<^sub>R i) .. (\<Sum>i\<in>Basis. min (b\<bullet>i) (d\<bullet>i) *\<^sub>R i)}"
unfolding set_eq_iff and Int_iff and mem_interval by auto

lemma disjoint_interval: fixes a::"'a::ordered_euclidean_space" shows
"{a .. b} \<inter> {c .. d} = {} \<longleftrightarrow> (\<exists>i\<in>Basis. (b\<bullet>i < a\<bullet>i \<or> d\<bullet>i < c\<bullet>i \<or> b\<bullet>i < c\<bullet>i \<or> d\<bullet>i < a\<bullet>i))" (is ?th1) and
"{a .. b} \<inter> {c<..<d} = {} \<longleftrightarrow> (\<exists>i\<in>Basis. (b\<bullet>i < a\<bullet>i \<or> d\<bullet>i \<le> c\<bullet>i \<or> b\<bullet>i \<le> c\<bullet>i \<or> d\<bullet>i \<le> a\<bullet>i))" (is ?th2) and
"{a<..<b} \<inter> {c .. d} = {} \<longleftrightarrow> (\<exists>i\<in>Basis. (b\<bullet>i \<le> a\<bullet>i \<or> d\<bullet>i < c\<bullet>i \<or> b\<bullet>i \<le> c\<bullet>i \<or> d\<bullet>i \<le> a\<bullet>i))" (is ?th3) and
"{a<..<b} \<inter> {c<..<d} = {} \<longleftrightarrow> (\<exists>i\<in>Basis. (b\<bullet>i \<le> a\<bullet>i \<or> d\<bullet>i \<le> c\<bullet>i \<or> b\<bullet>i \<le> c\<bullet>i \<or> d\<bullet>i \<le> a\<bullet>i))" (is ?th4)
proof-
let ?z = "(\<Sum>i\<in>Basis. (((max (a\<bullet>i) (c\<bullet>i)) + (min (b\<bullet>i) (d\<bullet>i))) / 2) *\<^sub>R i)::'a"
have **: "\<And>P Q. (\<And>i :: 'a. i \<in> Basis \<Longrightarrow> Q ?z i \<Longrightarrow> P i) \<Longrightarrow>
(\<And>i x :: 'a. i \<in> Basis \<Longrightarrow> P i \<Longrightarrow> Q x i) \<Longrightarrow> (\<forall>x. \<exists>i\<in>Basis. Q x i) \<longleftrightarrow> (\<exists>i\<in>Basis. P i)"
by blast
note * = set_eq_iff Int_iff empty_iff mem_interval ball_conj_distrib[symmetric] eq_False ball_simps(10)
show ?th1 unfolding * by (intro **) auto
show ?th2 unfolding * by (intro **) auto
show ?th3 unfolding * by (intro **) auto
show ?th4 unfolding * by (intro **) auto
qed

(* Moved interval_open_subset_closed a bit upwards *)

lemma open_interval[intro]:
fixes a b :: "'a::ordered_euclidean_space" shows "open {a<..<b}"
proof-
have "open (\<Inter>i\<in>Basis. (\<lambda>x. x\<bullet>i) - {a\<bullet>i<..<b\<bullet>i})"
by (intro open_INT finite_lessThan ballI continuous_open_vimage allI
linear_continuous_at open_real_greaterThanLessThan finite_Basis bounded_linear_inner_left)
also have "(\<Inter>i\<in>Basis. (\<lambda>x. x\<bullet>i) - {a\<bullet>i<..<b\<bullet>i}) = {a<..<b}"
by (auto simp add: eucl_less [where 'a='a])
finally show "open {a<..<b}" .
qed

lemma closed_interval[intro]:
fixes a b :: "'a::ordered_euclidean_space" shows "closed {a .. b}"
proof-
have "closed (\<Inter>i\<in>Basis. (\<lambda>x. x\<bullet>i) - {a\<bullet>i .. b\<bullet>i})"
by (intro closed_INT ballI continuous_closed_vimage allI
linear_continuous_at closed_real_atLeastAtMost finite_Basis bounded_linear_inner_left)
also have "(\<Inter>i\<in>Basis. (\<lambda>x. x\<bullet>i) - {a\<bullet>i .. b\<bullet>i}) = {a .. b}"
by (auto simp add: eucl_le [where 'a='a])
finally show "closed {a .. b}" .
qed

lemma interior_closed_interval [intro]:
fixes a b :: "'a::ordered_euclidean_space"
shows "interior {a..b} = {a<..<b}" (is "?L = ?R")
proof(rule subset_antisym)
show "?R \<subseteq> ?L" using interval_open_subset_closed open_interval
by (rule interior_maximal)
next
{ fix x assume "x \<in> interior {a..b}"
then obtain s where s:"open s" "x \<in> s" "s \<subseteq> {a..b}" ..
then obtain e where "e>0" and e:"\<forall>x'. dist x' x < e \<longrightarrow> x' \<in> {a..b}" unfolding open_dist and subset_eq by auto
{ fix i :: 'a assume i:"i\<in>Basis"
have "dist (x - (e / 2) *\<^sub>R i) x < e"
"dist (x + (e / 2) *\<^sub>R i) x < e"
unfolding dist_norm apply auto
unfolding norm_minus_cancel using norm_Basis[OF i] e>0 by auto
hence "a \<bullet> i \<le> (x - (e / 2) *\<^sub>R i) \<bullet> i"
"(x + (e / 2) *\<^sub>R i) \<bullet> i \<le> b \<bullet> i"
using e[THEN spec[where x="x - (e/2) *\<^sub>R i"]]
and   e[THEN spec[where x="x + (e/2) *\<^sub>R i"]]
unfolding mem_interval using i by blast+
hence "a \<bullet> i < x \<bullet> i" and "x \<bullet> i < b \<bullet> i"
using e>0 i by (auto simp: inner_diff_left inner_Basis inner_add_left) }
hence "x \<in> {a<..<b}" unfolding mem_interval by auto  }
thus "?L \<subseteq> ?R" ..
qed

lemma bounded_closed_interval: fixes a :: "'a::ordered_euclidean_space" shows "bounded {a .. b}"
proof-
let ?b = "\<Sum>i\<in>Basis. \<bar>a\<bullet>i\<bar> + \<bar>b\<bullet>i\<bar>"
{ fix x::"'a" assume x:"\<forall>i\<in>Basis. a \<bullet> i \<le> x \<bullet> i \<and> x \<bullet> i \<le> b \<bullet> i"
{ fix i :: 'a assume "i\<in>Basis"
hence "\<bar>x\<bullet>i\<bar> \<le> \<bar>a\<bullet>i\<bar> + \<bar>b\<bullet>i\<bar>" using x[THEN bspec[where x=i]] by auto  }
hence "(\<Sum>i\<in>Basis. \<bar>x \<bullet> i\<bar>) \<le> ?b" apply-apply(rule setsum_mono) by auto
hence "norm x \<le> ?b" using norm_le_l1[of x] by auto  }
thus ?thesis unfolding interval and bounded_iff by auto
qed

lemma bounded_interval: fixes a :: "'a::ordered_euclidean_space" shows
"bounded {a .. b} \<and> bounded {a<..<b}"
using bounded_closed_interval[of a b]
using interval_open_subset_closed[of a b]
using bounded_subset[of "{a..b}" "{a<..<b}"]
by simp

lemma not_interval_univ: fixes a :: "'a::ordered_euclidean_space" shows
"({a .. b} \<noteq> UNIV) \<and> ({a<..<b} \<noteq> UNIV)"
using bounded_interval[of a b] by auto

lemma compact_interval: fixes a :: "'a::ordered_euclidean_space" shows "compact {a .. b}"
using bounded_closed_imp_seq_compact[of "{a..b}"] using bounded_interval[of a b]
by (auto simp: compact_eq_seq_compact_metric)

lemma open_interval_midpoint: fixes a :: "'a::ordered_euclidean_space"
assumes "{a<..<b} \<noteq> {}" shows "((1/2) *\<^sub>R (a + b)) \<in> {a<..<b}"
proof-
{ fix i :: 'a assume "i\<in>Basis"
hence "a \<bullet> i < ((1 / 2) *\<^sub>R (a + b)) \<bullet> i \<and> ((1 / 2) *\<^sub>R (a + b)) \<bullet> i < b \<bullet> i"
using assms[unfolded interval_ne_empty, THEN bspec[where x=i]] by (auto simp: inner_add_left)  }
thus ?thesis unfolding mem_interval by auto
qed

lemma open_closed_interval_convex: fixes x :: "'a::ordered_euclidean_space"
assumes x:"x \<in> {a<..<b}" and y:"y \<in> {a .. b}" and e:"0 < e" "e \<le> 1"
shows "(e *\<^sub>R x + (1 - e) *\<^sub>R y) \<in> {a<..<b}"
proof-
{ fix i :: 'a assume i:"i\<in>Basis"
have "a \<bullet> i = e * (a \<bullet> i) + (1 - e) * (a \<bullet> i)" unfolding left_diff_distrib by simp
also have "\<dots> < e * (x \<bullet> i) + (1 - e) * (y \<bullet> i)" apply(rule add_less_le_mono)
using e unfolding mult_less_cancel_left and mult_le_cancel_left apply simp_all
using x unfolding mem_interval using i apply simp
using y unfolding mem_interval using i apply simp
done
finally have "a \<bullet> i < (e *\<^sub>R x + (1 - e) *\<^sub>R y) \<bullet> i" unfolding inner_simps by auto
moreover {
have "b \<bullet> i = e * (b\<bullet>i) + (1 - e) * (b\<bullet>i)" unfolding left_diff_distrib by simp
also have "\<dots> > e * (x \<bullet> i) + (1 - e) * (y \<bullet> i)" apply(rule add_less_le_mono)
using e unfolding mult_less_cancel_left and mult_le_cancel_left apply simp_all
using x unfolding mem_interval using i apply simp
using y unfolding mem_interval using i apply simp
done
finally have "(e *\<^sub>R x + (1 - e) *\<^sub>R y) \<bullet> i < b \<bullet> i" unfolding inner_simps by auto
} ultimately have "a \<bullet> i < (e *\<^sub>R x + (1 - e) *\<^sub>R y) \<bullet> i \<and> (e *\<^sub>R x + (1 - e) *\<^sub>R y) \<bullet> i < b \<bullet> i" by auto }
thus ?thesis unfolding mem_interval by auto
qed

lemma closure_open_interval: fixes a :: "'a::ordered_euclidean_space"
assumes "{a<..<b} \<noteq> {}"
shows "closure {a<..<b} = {a .. b}"
proof-
have ab:"a < b" using assms[unfolded interval_ne_empty] apply(subst eucl_less) by auto
let ?c = "(1 / 2) *\<^sub>R (a + b)"
{ fix x assume as:"x \<in> {a .. b}"
def f == "\<lambda>n::nat. x + (inverse (real n + 1)) *\<^sub>R (?c - x)"
{ fix n assume fn:"f n < b \<longrightarrow> a < f n \<longrightarrow> f n = x" and xc:"x \<noteq> ?c"
have *:"0 < inverse (real n + 1)" "inverse (real n + 1) \<le> 1" unfolding inverse_le_1_iff by auto
have "(inverse (real n + 1)) *\<^sub>R ((1 / 2) *\<^sub>R (a + b)) + (1 - inverse (real n + 1)) *\<^sub>R x =
x + (inverse (real n + 1)) *\<^sub>R (((1 / 2) *\<^sub>R (a + b)) - x)"
hence "f n < b" and "a < f n" using open_closed_interval_convex[OF open_interval_midpoint[OF assms] as *] unfolding f_def by auto
hence False using fn unfolding f_def using xc by auto  }
moreover
{ assume "\<not> (f ---> x) sequentially"
{ fix e::real assume "e>0"
hence "\<exists>N::nat. inverse (real (N + 1)) < e" using real_arch_inv[of e] apply (auto simp add: Suc_pred') apply(rule_tac x="n - 1" in exI) by auto
then obtain N::nat where "inverse (real (N + 1)) < e" by auto
hence "\<forall>n\<ge>N. inverse (real n + 1) < e" by (auto, metis Suc_le_mono le_SucE less_imp_inverse_less nat_le_real_less order_less_trans real_of_nat_Suc real_of_nat_Suc_gt_zero)
hence "\<exists>N::nat. \<forall>n\<ge>N. inverse (real n + 1) < e" by auto  }
hence "((\<lambda>n. inverse (real n + 1)) ---> 0) sequentially"
unfolding LIMSEQ_def by(auto simp add: dist_norm)
hence "(f ---> x) sequentially" unfolding f_def
using tendsto_add[OF tendsto_const, of "\<lambda>n::nat. (inverse (real n + 1)) *\<^sub>R ((1 / 2) *\<^sub>R (a + b) - x)" 0 sequentially x]
using tendsto_scaleR [OF _ tendsto_const, of "\<lambda>n::nat. inverse (real n + 1)" 0 sequentially "((1 / 2) *\<^sub>R (a + b) - x)"] by auto  }
ultimately have "x \<in> closure {a<..<b}"
using as and open_interval_midpoint[OF assms] unfolding closure_def unfolding islimpt_sequential by(cases "x=?c")auto  }
thus ?thesis using closure_minimal[OF interval_open_subset_closed closed_interval, of a b] by blast
qed

lemma bounded_subset_open_interval_symmetric: fixes s::"('a::ordered_euclidean_space) set"
assumes "bounded s"  shows "\<exists>a. s \<subseteq> {-a<..<a}"
proof-
obtain b where "b>0" and b:"\<forall>x\<in>s. norm x \<le> b" using assms[unfolded bounded_pos] by auto
def a \<equiv> "(\<Sum>i\<in>Basis. (b + 1) *\<^sub>R i)::'a"
{ fix x assume "x\<in>s"
fix i :: 'a assume i:"i\<in>Basis"
hence "(-a)\<bullet>i < x\<bullet>i" and "x\<bullet>i < a\<bullet>i" using b[THEN bspec[where x=x], OF x\<in>s]
and Basis_le_norm[OF i, of x] unfolding inner_simps and a_def by auto }
thus ?thesis by(auto intro: exI[where x=a] simp add: eucl_less[where 'a='a])
qed

lemma bounded_subset_open_interval:
fixes s :: "('a::ordered_euclidean_space) set"
shows "bounded s ==> (\<exists>a b. s \<subseteq> {a<..<b})"
by (auto dest!: bounded_subset_open_interval_symmetric)

lemma bounded_subset_closed_interval_symmetric:
fixes s :: "('a::ordered_euclidean_space) set"
assumes "bounded s" shows "\<exists>a. s \<subseteq> {-a .. a}"
proof-
obtain a where "s \<subseteq> {- a<..<a}" using bounded_subset_open_interval_symmetric[OF assms] by auto
thus ?thesis using interval_open_subset_closed[of "-a" a] by auto
qed

lemma bounded_subset_closed_interval:
fixes s :: "('a::ordered_euclidean_space) set"
shows "bounded s ==> (\<exists>a b. s \<subseteq> {a .. b})"
using bounded_subset_closed_interval_symmetric[of s] by auto

lemma frontier_closed_interval:
fixes a b :: "'a::ordered_euclidean_space"
shows "frontier {a .. b} = {a .. b} - {a<..<b}"
unfolding frontier_def unfolding interior_closed_interval and closure_closed[OF closed_interval] ..

lemma frontier_open_interval:
fixes a b :: "'a::ordered_euclidean_space"
shows "frontier {a<..<b} = (if {a<..<b} = {} then {} else {a .. b} - {a<..<b})"
proof(cases "{a<..<b} = {}")
case True thus ?thesis using frontier_empty by auto
next
case False thus ?thesis unfolding frontier_def and closure_open_interval[OF False] and interior_open[OF open_interval] by auto
qed

lemma inter_interval_mixed_eq_empty: fixes a :: "'a::ordered_euclidean_space"
assumes "{c<..<d} \<noteq> {}"  shows "{a<..<b} \<inter> {c .. d} = {} \<longleftrightarrow> {a<..<b} \<inter> {c<..<d} = {}"
unfolding closure_open_interval[OF assms, THEN sym] unfolding open_inter_closure_eq_empty[OF open_interval] ..

(* Some stuff for half-infinite intervals too; FIXME: notation?  *)

lemma closed_interval_left: fixes b::"'a::euclidean_space"
shows "closed {x::'a. \<forall>i\<in>Basis. x\<bullet>i \<le> b\<bullet>i}"
proof-
{ fix i :: 'a assume i:"i\<in>Basis"
fix x::"'a" assume x:"\<forall>e>0. \<exists>x'\<in>{x. \<forall>i\<in>Basis. x \<bullet> i \<le> b \<bullet> i}. x' \<noteq> x \<and> dist x' x < e"
{ assume "x\<bullet>i > b\<bullet>i"
then obtain y where "y \<bullet> i \<le> b \<bullet> i"  "y \<noteq> x"  "dist y x < x\<bullet>i - b\<bullet>i"
using x[THEN spec[where x="x\<bullet>i - b\<bullet>i"]] using i by auto
hence False using Basis_le_norm[OF i, of "y - x"] unfolding dist_norm inner_simps using i
by auto }
hence "x\<bullet>i \<le> b\<bullet>i" by(rule ccontr)auto  }
thus ?thesis unfolding closed_limpt unfolding islimpt_approachable by blast
qed

lemma closed_interval_right: fixes a::"'a::euclidean_space"
shows "closed {x::'a. \<forall>i\<in>Basis. a\<bullet>i \<le> x\<bullet>i}"
proof-
{ fix i :: 'a assume i:"i\<in>Basis"
fix x::"'a" assume x:"\<forall>e>0. \<exists>x'\<in>{x. \<forall>i\<in>Basis. a \<bullet> i \<le> x \<bullet> i}. x' \<noteq> x \<and> dist x' x < e"
{ assume "a\<bullet>i > x\<bullet>i"
then obtain y where "a \<bullet> i \<le> y \<bullet> i"  "y \<noteq> x"  "dist y x < a\<bullet>i - x\<bullet>i"
using x[THEN spec[where x="a\<bullet>i - x\<bullet>i"]] i by auto
hence False using Basis_le_norm[OF i, of "y - x"] unfolding dist_norm inner_simps by auto }
hence "a\<bullet>i \<le> x\<bullet>i" by(rule ccontr)auto  }
thus ?thesis unfolding closed_limpt unfolding islimpt_approachable by blast
qed

lemma open_box: "open (box a b)"
proof -
have "open (\<Inter>i\<in>Basis. (op \<bullet> i) - {a \<bullet> i <..< b \<bullet> i})"
by (auto intro!: continuous_open_vimage continuous_inner continuous_at_id continuous_const)
also have "(\<Inter>i\<in>Basis. (op \<bullet> i) - {a \<bullet> i <..< b \<bullet> i}) = box a b"
by (auto simp add: box_def inner_commute)
finally show ?thesis .
qed

instance euclidean_space \<subseteq> second_countable_topology
proof
def a \<equiv> "\<lambda>f :: 'a \<Rightarrow> (real \<times> real). \<Sum>i\<in>Basis. fst (f i) *\<^sub>R i"
then have a: "\<And>f. (\<Sum>i\<in>Basis. fst (f i) *\<^sub>R i) = a f" by simp
def b \<equiv> "\<lambda>f :: 'a \<Rightarrow> (real \<times> real). \<Sum>i\<in>Basis. snd (f i) *\<^sub>R i"
then have b: "\<And>f. (\<Sum>i\<in>Basis. snd (f i) *\<^sub>R i) = b f" by simp
def B \<equiv> "(\<lambda>f. box (a f) (b f))  (Basis \<rightarrow>\<^isub>E (\<rat> \<times> \<rat>))"

have "Ball B open" by (simp add: B_def open_box)
moreover have "(\<forall>A. open A \<longrightarrow> (\<exists>B'\<subseteq>B. \<Union>B' = A))"
proof safe
fix A::"'a set" assume "open A"
show "\<exists>B'\<subseteq>B. \<Union>B' = A"
apply (rule exI[of _ "{b\<in>B. b \<subseteq> A}"])
apply (subst (3) open_UNION_box[OF open A])
apply (auto simp add: a b B_def)
done
qed
ultimately
have "topological_basis B" unfolding topological_basis_def by blast
moreover
have "countable B" unfolding B_def
by (intro countable_image countable_PiE finite_Basis countable_SIGMA countable_rat)
ultimately show "\<exists>B::'a set set. countable B \<and> open = generate_topology B"
by (blast intro: topological_basis_imp_subbasis)
qed

instance euclidean_space \<subseteq> polish_space ..

text {* Intervals in general, including infinite and mixtures of open and closed. *}

definition "is_interval (s::('a::euclidean_space) set) \<longleftrightarrow>
(\<forall>a\<in>s. \<forall>b\<in>s. \<forall>x. (\<forall>i\<in>Basis. ((a\<bullet>i \<le> x\<bullet>i \<and> x\<bullet>i \<le> b\<bullet>i) \<or> (b\<bullet>i \<le> x\<bullet>i \<and> x\<bullet>i \<le> a\<bullet>i))) \<longrightarrow> x \<in> s)"

lemma is_interval_interval: "is_interval {a .. b::'a::ordered_euclidean_space}" (is ?th1)
"is_interval {a<..<b}" (is ?th2) proof -
show ?th1 ?th2  unfolding is_interval_def mem_interval Ball_def atLeastAtMost_iff
by(meson order_trans le_less_trans less_le_trans less_trans)+ qed

lemma is_interval_empty:
"is_interval {}"
unfolding is_interval_def
by simp

lemma is_interval_univ:
"is_interval UNIV"
unfolding is_interval_def
by simp

subsection {* Closure of halfspaces and hyperplanes *}

lemma isCont_open_vimage:
assumes "\<And>x. isCont f x" and "open s" shows "open (f - s)"
proof -
from assms(1) have "continuous_on UNIV f"
unfolding isCont_def continuous_on_def by simp
hence "open {x \<in> UNIV. f x \<in> s}"
using open_UNIV open s by (rule continuous_open_preimage)
thus "open (f - s)"
qed

lemma isCont_closed_vimage:
assumes "\<And>x. isCont f x" and "closed s" shows "closed (f - s)"
using assms unfolding closed_def vimage_Compl [symmetric]
by (rule isCont_open_vimage)

lemma open_Collect_less:
fixes f g :: "'a::t2_space \<Rightarrow> real"
assumes f: "\<And>x. isCont f x"
assumes g: "\<And>x. isCont g x"
shows "open {x. f x < g x}"
proof -
have "open ((\<lambda>x. g x - f x) - {0<..})"
using isCont_diff [OF g f] open_real_greaterThan
by (rule isCont_open_vimage)
also have "((\<lambda>x. g x - f x) - {0<..}) = {x. f x < g x}"
by auto
finally show ?thesis .
qed

lemma closed_Collect_le:
fixes f g :: "'a::t2_space \<Rightarrow> real"
assumes f: "\<And>x. isCont f x"
assumes g: "\<And>x. isCont g x"
shows "closed {x. f x \<le> g x}"
proof -
have "closed ((\<lambda>x. g x - f x) - {0..})"
using isCont_diff [OF g f] closed_real_atLeast
by (rule isCont_closed_vimage)
also have "((\<lambda>x. g x - f x) - {0..}) = {x. f x \<le> g x}"
by auto
finally show ?thesis .
qed

lemma closed_Collect_eq:
fixes f g :: "'a::t2_space \<Rightarrow> 'b::t2_space"
assumes f: "\<And>x. isCont f x"
assumes g: "\<And>x. isCont g x"
shows "closed {x. f x = g x}"
proof -
have "open {(x::'b, y::'b). x \<noteq> y}"
unfolding open_prod_def by (auto dest!: hausdorff)
hence "closed {(x::'b, y::'b). x = y}"
unfolding closed_def split_def Collect_neg_eq .
with isCont_Pair [OF f g]
have "closed ((\<lambda>x. (f x, g x)) - {(x, y). x = y})"
by (rule isCont_closed_vimage)
also have "\<dots> = {x. f x = 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}"

lemma closed_halfspace_ge: "closed {x. inner a x \<ge> b}"

lemma closed_hyperplane: "closed {x. inner a x = b}"

lemma closed_halfspace_component_le:
shows "closed {x::'a::euclidean_space. x\<bullet>i \<le> a}"

lemma closed_halfspace_component_ge:
shows "closed {x::'a::euclidean_space. x\<bullet>i \<ge> a}"

text {* Openness of halfspaces. *}

lemma open_halfspace_lt: "open {x. inner a x < b}"

lemma open_halfspace_gt: "open {x. inner a x > b}"

lemma open_halfspace_component_lt:
shows "open {x::'a::euclidean_space. x\<bullet>i < a}"

lemma open_halfspace_component_gt:
shows "open {x::'a::euclidean_space. x\<bullet>i > a}"

text{* Instantiation for intervals on @{text ordered_euclidean_space} *}

lemma eucl_lessThan_eq_halfspaces:
fixes a :: "'a\<Colon>ordered_euclidean_space"
shows "{..<a} = (\<Inter>i\<in>Basis. {x. x \<bullet> i < a \<bullet> i})"
by (auto simp: eucl_less[where 'a='a])

lemma eucl_greaterThan_eq_halfspaces:
fixes a :: "'a\<Colon>ordered_euclidean_space"
shows "{a<..} = (\<Inter>i\<in>Basis. {x. a \<bullet> i < x \<bullet> i})"
by (auto simp: eucl_less[where 'a='a])

lemma eucl_atMost_eq_halfspaces:
fixes a :: "'a\<Colon>ordered_euclidean_space"
shows "{.. a} = (\<Inter>i\<in>Basis. {x. x \<bullet> i \<le> a \<bullet> i})"
by (auto simp: eucl_le[where 'a='a])

lemma eucl_atLeast_eq_halfspaces:
fixes a :: "'a\<Colon>ordered_euclidean_space"
shows "{a ..} = (\<Inter>i\<in>Basis. {x. a \<bullet> i \<le> x \<bullet> i})"
by (auto simp: eucl_le[where 'a='a])

lemma open_eucl_lessThan[simp, intro]:
fixes a :: "'a\<Colon>ordered_euclidean_space"
shows "open {..< a}"
by (auto simp: eucl_lessThan_eq_halfspaces open_halfspace_component_lt)

lemma open_eucl_greaterThan[simp, intro]:
fixes a :: "'a\<Colon>ordered_euclidean_space"
shows "open {a <..}"
by (auto simp: eucl_greaterThan_eq_halfspaces open_halfspace_component_gt)

lemma closed_eucl_atMost[simp, intro]:
fixes a :: "'a\<Colon>ordered_euclidean_space"
shows "closed {.. a}"
unfolding eucl_atMost_eq_halfspaces

lemma closed_eucl_atLeast[simp, intro]:
fixes a :: "'a\<Colon>ordered_euclidean_space"
shows "closed {a ..}"
unfolding eucl_atLeast_eq_halfspaces

text {* This gives a simple derivation of limit component bounds. *}

lemma Lim_component_le: fixes f :: "'a \<Rightarrow> 'b::euclidean_space"
assumes "(f ---> l) net" "\<not> (trivial_limit net)"  "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 ---> l) net"  "\<not> (trivial_limit net)"  "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 ---> l) net" "~(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] and Lim_component_le[OF net, of i b] by auto

text{* Limits relative to a union.                                               *}

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 ---> l) (at x within (s \<union> t)) \<longleftrightarrow>
(f ---> l) (at x within s) \<and> (f ---> l) (at x within t)"
unfolding tendsto_def

lemma Lim_topological:
"(f ---> 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{* Some more convenient intermediate-value theorem formulations.             *}

lemma connected_ivt_hyperplane:
assumes "connected s" "x \<in> s" "y \<in> s" "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 assms(1)[unfolded connected_def not_ex, THEN spec[where x="?A"], THEN spec[where x="?B"]] and assms(2-5) by auto
qed

lemma connected_ivt_component: fixes x::"'a::euclidean_space" shows
"connected s \<Longrightarrow> x \<in> s \<Longrightarrow> y \<in> s \<Longrightarrow> x\<bullet>k \<le> a \<Longrightarrow> a \<le> y\<bullet>k \<Longrightarrow> (\<exists>z\<in>s.  z\<bullet>k = a)"
using connected_ivt_hyperplane[of s x y "k::'a" a] by (auto simp: inner_commute)

subsection {* Homeomorphisms *}

definition "homeomorphism s t f g \<equiv>
(\<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"

definition
homeomorphic :: "'a::topological_space set \<Rightarrow> 'b::topological_space set \<Rightarrow> bool"
(infixr "homeomorphic" 60) where
homeomorphic_def: "s homeomorphic t \<equiv> (\<exists>f g. homeomorphism s t f g)"

lemma homeomorphic_refl: "s homeomorphic s"
unfolding homeomorphic_def
unfolding homeomorphism_def
using continuous_on_id
apply(rule_tac x = "(\<lambda>x. x)" in exI)
apply(rule_tac x = "(\<lambda>x. x)" in exI)
by blast

lemma homeomorphic_sym:
"s homeomorphic t \<longleftrightarrow> t homeomorphic s"
unfolding homeomorphic_def
unfolding homeomorphism_def
by blast

lemma homeomorphic_trans:
assumes "s homeomorphic t" "t homeomorphic u" shows "s homeomorphic u"
proof-
obtain f1 g1 where fg1:"\<forall>x\<in>s. g1 (f1 x) = x"  "f1  s = t" "continuous_on s f1" "\<forall>y\<in>t. f1 (g1 y) = y" "g1  t = s" "continuous_on t g1"
using assms(1) unfolding homeomorphic_def homeomorphism_def by auto
obtain f2 g2 where fg2:"\<forall>x\<in>t. g2 (f2 x) = x"  "f2  t = u" "continuous_on t f2" "\<forall>y\<in>u. f2 (g2 y) = y" "g2  u = t" "continuous_on u g2"
using assms(2) unfolding homeomorphic_def homeomorphism_def by auto

{ fix x assume "x\<in>s" hence "(g1 \<circ> g2) ((f2 \<circ> f1) x) = x" using fg1(1)[THEN bspec[where x=x]] and fg2(1)[THEN bspec[where x="f1 x"]] and fg1(2) by auto }
moreover have "(f2 \<circ> f1)  s = u" using fg1(2) fg2(2) by auto
moreover have "continuous_on s (f2 \<circ> f1)" using continuous_on_compose[OF fg1(3)] and fg2(3) unfolding fg1(2) by auto
moreover { fix y assume "y\<in>u" hence "(f2 \<circ> f1) ((g1 \<circ> g2) y) = y" using fg2(4)[THEN bspec[where x=y]] and fg1(4)[THEN bspec[where x="g2 y"]] and fg2(5) by auto }
moreover have "(g1 \<circ> g2)  u = s" using fg1(5) fg2(5) by auto
moreover have "continuous_on u (g1 \<circ> g2)" using continuous_on_compose[OF fg2(6)] and fg1(6)  unfolding fg2(5) by auto
ultimately show ?thesis unfolding homeomorphic_def homeomorphism_def apply(rule_tac x="f2 \<circ> f1" in exI) apply(rule_tac x="g1 \<circ> g2" in exI) by auto
qed

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)"
unfolding homeomorphic_def homeomorphism_def
apply auto apply (rule_tac x=f in exI) apply (rule_tac x=g in exI)
apply auto apply (rule_tac x=f in exI) apply (rule_tac x=g in exI) apply auto
unfolding image_iff
apply(erule_tac x="g x" in ballE) apply(erule_tac x="x" in ballE)
apply auto apply(rule_tac x="g x" in bexI) apply auto
apply(erule_tac x="f x" in ballE) apply(erule_tac x="x" in ballE)
apply auto apply(rule_tac x="f x" in bexI) by auto

text {* Relatively weak hypotheses if a set is compact. *}

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-
def g \<equiv> "\<lambda>x. SOME y. y\<in>s \<and> f y = 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
hence "g (f x) = x" using g by auto
hence "f (g y) = y" unfolding x(1)[THEN sym] by auto  }
hence 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
hence "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)[THEN sym] and g_def by auto }
ultimately have "x\<in>s \<longleftrightarrow> x \<in> g  t" ..  }
hence "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) by auto
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{* Preservation of topological properties.                                   *}

lemma homeomorphic_compactness:
"s homeomorphic t ==> (compact s \<longleftrightarrow> compact t)"
unfolding homeomorphic_def homeomorphism_def
by (metis compact_continuous_image)

text{* Results on translation, scaling etc.                                      *}

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 by (auto simp add: continuous_on_intros)

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] by auto

lemma homeomorphic_affinity:
fixes s :: "'a::real_normed_vector set"
assumes "c \<noteq> 0"  shows "s homeomorphic ((\<lambda>x. a + c *\<^sub>R x)  s)"
proof-
have *:"op + a  op *\<^sub>R c  s = (\<lambda>x. a + c *\<^sub>R x)  s" by auto
show ?thesis
using homeomorphic_trans
using homeomorphic_scaling[OF assms, of s]
using homeomorphic_translation[of "(\<lambda>x. c *\<^sub>R x)  s" a] unfolding * by auto
qed

lemma homeomorphic_balls:
fixes a b ::"'a::real_normed_vector"
assumes "0 < d"  "0 < e"
shows "(ball a d) homeomorphic  (ball b e)" (is ?th)
"(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_on_intros
simp: dist_commute dist_norm pos_divide_less_eq mult_strict_left_mono)
done
next
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_on_intros
simp: dist_commute dist_norm pos_divide_le_eq mult_strict_left_mono)
done
qed

text{* "Isometry" (up to constant bounds) of injective linear map etc.           *}

lemma cauchy_isometric:
fixes x :: "nat \<Rightarrow> 'a::euclidean_space"
assumes e:"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 xs:"\<forall>n::nat. x n \<in> s" and cf:"Cauchy(f o x)"
shows "Cauchy x"
proof-
interpret f: bounded_linear f by fact
{ fix d::real assume "d>0"
then obtain N where N:"\<forall>n\<ge>N. norm (f (x n) - f (x N)) < e * d"
using cf[unfolded cauchy o_def dist_norm, THEN spec[where x="e*d"]] and e and mult_pos_pos[of e d] by auto
{ fix n assume "n\<ge>N"
have "e * norm (x n - x N) \<le> norm (f (x n - x N))"
using subspace_sub[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 N \<le> n N unfolding f.diff[THEN sym] by auto
finally have "norm (x n - x N) < d" using e>0 by simp }
hence "\<exists>N. \<forall>n\<ge>N. norm (x n - x N) < d" by auto }
thus ?thesis unfolding cauchy and dist_norm by auto
qed

lemma complete_isometric_image:
fixes f :: "'a::euclidean_space => 'b::euclidean_space"
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-
{ fix g assume as:"\<forall>n::nat. g n \<in> f  s" and cfg:"Cauchy g"
then 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
hence x:"\<forall>n. x n \<in> s"  "\<forall>n. g n = f (x n)" by auto
hence "f \<circ> x = g" unfolding fun_eq_iff by auto
then obtain l where "l\<in>s" and l:"(x ---> l) sequentially"
using cs[unfolded complete_def, THEN spec[where x="x"]]
using cauchy_isometric[OF 0<e s f normf] and cfg and x(1) by auto
hence "\<exists>l\<in>f  s. (g ---> l) sequentially"
using linear_continuous_at[OF f, unfolded continuous_at_sequentially, THEN spec[where x=x], of l]
unfolding f \<circ> x = g by auto  }
thus ?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
{ fix x assume "x \<in> s"
hence "x = 0" using True by auto
hence "norm x \<le> norm (f x)" by auto  }
thus ?thesis by(auto intro!: exI[where x=1])
next
interpret f: bounded_linear f by fact
case False
then 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))" unfolding frontier_cball and dist_norm by auto
hence "compact ?S''" using compact_frontier[OF compact_cball, of 0 "norm a"] by auto
moreover have "?S' = s \<inter> ?S''" by auto
ultimately have "compact ?S'" using closed_inter_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
hence "closed ?S" using compact_imp_closed by auto
moreover have "?S \<noteq> {}" using a 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 *[THEN sym] 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 b\<in>s] using norm b >0 unfolding zero_less_norm_iff by auto
ultimately have "0 < norm (f b) / norm b" by(simp only: divide_pos_pos)
moreover
{ fix x assume "x\<in>s"
hence "norm (f b) / norm b * norm x \<le> norm (f x)"
proof(cases "x=0")
case True thus "norm (f b) / norm b * norm x \<le> norm (f x)" by auto
next
case False
hence *:"0 < norm a / norm x" using a\<noteq>0 unfolding zero_less_norm_iff[THEN sym] by(simp only: divide_pos_pos)
have "\<forall>c. \<forall>x\<in>s. c *\<^sub>R x \<in> s" using s[unfolded subspace_def] by auto
hence "(norm a / norm x) *\<^sub>R x \<in> {x \<in> s. norm x = norm a}" using x\<in>s and x\<noteq>0 by auto
thus "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 using x\<noteq>0 a\<noteq>0
by (auto simp add: 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 --> 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 e>0 assms(1,2) e] and assms(4)
unfolding complete_eq_closed[THEN sym] by auto
qed

subsection {* Some properties of a canonical subspace *}

lemma subspace_substandard:
"subspace {x::'a::euclidean_space. (\<forall>i\<in>Basis. P i \<longrightarrow> x\<bullet>i = 0)}"
unfolding subspace_def by (auto simp: inner_add_left)

lemma closed_substandard:
"closed {x::'a::euclidean_space. \<forall>i\<in>Basis. P i --> 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})"
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-
let ?D = "Basis :: 'a set"
have "d \<subseteq> ?A" using d by (auto simp: inner_Basis)
moreover
{ fix x::"'a" assume "x \<in> ?A"
hence "finite d" "x \<in> ?A" using assms by(auto intro: finite_subset[OF _ finite_Basis])
from this d have "x \<in> span d"
proof(induct d arbitrary: x)
case empty hence "x=0" apply(rule_tac euclidean_eqI) by auto
thus ?case using subspace_0[OF subspace_span[of "{}"]] by auto
next
case (insert k F)
hence *:"\<forall>i\<in>Basis. i \<notin> insert k F \<longrightarrow> x \<bullet> i = 0" by auto
have **:"F \<subseteq> insert k F" by auto
def y \<equiv> "x - (x\<bullet>k) *\<^sub>R k"
have y:"x = y + (x\<bullet>k) *\<^sub>R k" unfolding y_def by auto
{ fix i assume i': "i \<notin> F" "i \<in> Basis"
hence "y \<bullet> i = 0" unfolding y_def
using *[THEN bspec[where x=i]] insert by (auto simp: inner_simps inner_Basis) }
hence "y \<in> span F" using insert by auto
hence "y \<in> span (insert k F)"
using span_mono[of F "insert k F"] using assms by auto
moreover
have "k \<in> span (insert k F)" by(rule span_superset, auto)
hence "(x\<bullet>k) *\<^sub>R k \<in> span (insert k F)"
using span_mul by auto
ultimately
have "y + (x\<bullet>k) *\<^sub>R k \<in> span (insert k F)"
thus ?case using y by auto
qed
}
hence "?A \<subseteq> span d" by auto
moreover
{ fix x assume "x \<in> d" hence "x \<in> ?D" using assms by auto  }
hence "independent d" using independent_mono[OF independent_Basis, of d] and assms by auto
moreover
have "d \<subseteq> ?D" unfolding subset_eq using assms by auto
ultimately show ?thesis using dim_unique[of d ?A] by auto
qed

text{* Hence closure and completeness of all subspaces.                          *}

lemma ex_card: assumes "n \<le> card A" shows "\<exists>S\<subseteq>A. card S = n"
proof cases
assume "finite A"
from ex_bij_betw_nat_finite[OF this] guess f ..
moreover with n \<le> card A 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
assume "\<not> finite A" with n \<le> card A 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 guess f by (elim exE conjE) note f = this
interpret f: bounded_linear f using f unfolding linear_conv_bounded_linear by auto
{ fix x have "x\<in>?t \<Longrightarrow> f x = 0 \<Longrightarrow> x = 0" using f.zero d f(3)[THEN inj_onD, of x 0] by auto }
moreover have "closed ?t" using closed_substandard .
moreover have "subspace ?t" using 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:
fixes s :: "('a::euclidean_space) set" shows "subspace s ==> complete s"
using complete_eq_closed closed_subspace
by auto

lemma dim_closure:
fixes s :: "('a::euclidean_space) set"
shows "dim(closure s) = dim s" (is "?dc = ?d")
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"] unfolding dim_span by auto
thus ?thesis using dim_subset[OF closure_subset, of s] by auto
qed

subsection {* Affine transformations of intervals *}

lemma real_affinity_le:
"0 < (m::'a::linordered_field) ==> (m * x + c \<le> y \<longleftrightarrow> x \<le> inverse(m) * y + -(c / m))"

lemma real_le_affinity:
"0 < (m::'a::linordered_field) ==> (y \<le> m * x + c \<longleftrightarrow> inverse(m) * y + -(c / m) \<le> x)"

lemma real_affinity_lt:
"0 < (m::'a::linordered_field) ==> (m * x + c < y \<longleftrightarrow> x < inverse(m) * y + -(c / m))"

lemma real_lt_affinity:
"0 < (m::'a::linordered_field) ==> (y < m * x + c \<longleftrightarrow> inverse(m) * y + -(c / m) < x)"

lemma real_affinity_eq:
"(m::'a::linordered_field) \<noteq> 0 ==> (m * x + c = y \<longleftrightarrow> x = inverse(m) * y + -(c / m))"

lemma real_eq_affinity:
"(m::'a::linordered_field) \<noteq> 0 ==> (y = m * x + c  \<longleftrightarrow> inverse(m) * y + -(c / m) = x)"

lemma image_affinity_interval: fixes m::real
fixes a b c :: "'a::ordered_euclidean_space"
shows "(\<lambda>x. m *\<^sub>R x + c)  {a .. b} =
(if {a .. b} = {} then {}
else (if 0 \<le> m then {m *\<^sub>R a + c .. m *\<^sub>R b + c}
else {m *\<^sub>R b + c .. m *\<^sub>R a + c}))"
proof(cases "m=0")
{ fix x assume "x \<le> c" "c \<le> x"
hence "x=c" unfolding eucl_le[where 'a='a] apply-
apply(subst euclidean_eq_iff) by (auto intro: order_antisym) }
moreover case True
moreover have "c \<in> {m *\<^sub>R a + c..m *\<^sub>R b + c}" unfolding True by(auto simp add: eucl_le[where 'a='a])
ultimately show ?thesis by auto
next
case False
{ fix y assume "a \<le> y" "y \<le> b" "m > 0"
hence "m *\<^sub>R a + c \<le> m *\<^sub>R y + c"  "m *\<^sub>R y + c \<le> m *\<^sub>R b + c"
unfolding eucl_le[where 'a='a] by (auto simp: inner_simps)
} moreover
{ fix y assume "a \<le> y" "y \<le> b" "m < 0"
hence "m *\<^sub>R b + c \<le> m *\<^sub>R y + c"  "m *\<^sub>R y + c \<le> m *\<^sub>R a + c"
unfolding eucl_le[where 'a='a] by(auto simp add: mult_left_mono_neg inner_simps)
} moreover
{ fix y assume "m > 0"  "m *\<^sub>R a + c \<le> y"  "y \<le> m *\<^sub>R b + c"
hence "y \<in> (\<lambda>x. m *\<^sub>R x + c)  {a..b}"
unfolding image_iff Bex_def mem_interval eucl_le[where 'a='a]
apply (intro exI[where x="(1 / m) *\<^sub>R (y - c)"])
by(auto simp add: pos_le_divide_eq pos_divide_le_eq mult_commute diff_le_iff inner_simps)
} moreover
{ fix y assume "m *\<^sub>R b + c \<le> y" "y \<le> m *\<^sub>R a + c" "m < 0"
hence "y \<in> (\<lambda>x. m *\<^sub>R x + c)  {a..b}"
unfolding image_iff Bex_def mem_interval eucl_le[where 'a='a]
apply (intro exI[where x="(1 / m) *\<^sub>R (y - c)"])
by(auto simp add: neg_le_divide_eq neg_divide_le_eq mult_commute diff_le_iff inner_simps)
}
ultimately show ?thesis using False by auto
qed

lemma image_smult_interval:"(\<lambda>x. m *\<^sub>R (x::_::ordered_euclidean_space))  {a..b} =
(if {a..b} = {} then {} else if 0 \<le> m then {m *\<^sub>R a..m *\<^sub>R b} else {m *\<^sub>R b..m *\<^sub>R a})"
using image_affinity_interval[of m 0 a b] by auto

subsection {* Banach fixed point theorem (not really topological...) *}

lemma 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-
have "1 - c > 0" using c by auto

from s(2) obtain z0 where "z0 \<in> s" by auto
def z \<equiv> "\<lambda>n. (f ^^ n) z0"
{ fix n::nat
have "z n \<in> s" unfolding z_def
proof(induct n) case 0 thus ?case using z0 \<in>s by auto
next case Suc thus ?case using f by auto qed }
note z_in_s = this

def d \<equiv> "dist (z 0) (z 1)"

have fzn:"\<And>n. f (z n) = z (Suc n)" unfolding z_def by auto
{ fix n::nat
have "dist (z n) (z (Suc n)) \<le> (c ^ n) * d"
proof(induct n)
case 0 thus ?case unfolding d_def by auto
next
case (Suc m)
hence "c * dist (z m) (z (Suc m)) \<le> c ^ Suc m * d"
using 0 \<le> c using mult_left_mono[of "dist (z m) (z (Suc m))" "c ^ m * d" c] by auto
thus ?case using lipschitz[THEN bspec[where x="z m"], OF z_in_s, THEN bspec[where x="z (Suc m)"], OF z_in_s]
unfolding fzn and mult_le_cancel_left by auto
qed
} note cf_z = this

{ fix n m::nat
have "(1 - c) * dist (z m) (z (m+n)) \<le> (c ^ m) * d * (1 - c ^ n)"
proof(induct n)
case 0 show ?case by auto
next
case (Suc k)
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))))"
using dist_triangle and c by(auto simp add: dist_triangle)
also have "\<dots> \<le> (1 - c) * (dist (z m) (z (m + k)) + c ^ (m + k) * d)"
using cf_z[of "m + k"] and c by auto
also have "\<dots> \<le> c ^ m * d * (1 - c ^ k) + (1 - c) * c ^ (m + k) * d"
using Suc by (auto simp add: field_simps)
also have "\<dots> = (c ^ m) * (d * (1 - c ^ k) + (1 - c) * c ^ k * d)"
also have "\<dots> \<le> (c ^ m) * d * (1 - c ^ Suc k)"
using c by (auto simp add: field_simps)
finally show ?case by auto
qed
} note cf_z2 = this
{ fix e::real assume "e>0"
hence "\<exists>N. \<forall>m n. N \<le> m \<and> N \<le> n \<longrightarrow> dist (z m) (z n) < e"
proof(cases "d = 0")
case True
have *: "\<And>x. ((1 - c) * x \<le> 0) = (x \<le> 0)" using 1 - c > 0
by (metis mult_zero_left mult_commute real_mult_le_cancel_iff1)
from True have "\<And>n. z n = z0" using cf_z2[of 0] and c unfolding z_def
thus ?thesis using e>0 by auto
next
case False hence "d>0" unfolding d_def using zero_le_dist[of "z 0" "z 1"]
by (metis False d_def less_le)
hence "0 < e * (1 - c) / d" using e>0 and 1-c>0
using divide_pos_pos[of "e * (1 - c)" d] and mult_pos_pos[of e "1 - c"] by auto
then obtain N where N:"c ^ N < e * (1 - c) / d" using real_arch_pow_inv[of "e * (1 - c) / d" c] and c by auto
{ fix m n::nat assume "m>n" and as:"m\<ge>N" "n\<ge>N"
have *:"c ^ n \<le> c ^ N" using n\<ge>N and c using power_decreasing[OF n\<ge>N, of c] by auto
have "1 - c ^ (m - n) > 0" using c and power_strict_mono[of c 1 "m - n"] using m>n by auto
hence **:"d * (1 - c ^ (m - n)) / (1 - c) > 0"
using mult_pos_pos[OF d>0, of "1 - c ^ (m - n)"]
using divide_pos_pos[of "d * (1 - c ^ (m - n))" "1 - c"]
using 0 < 1 - c by auto

have "dist (z m) (z n) \<le> c ^ n * d * (1 - c ^ (m - n)) / (1 - c)"
using cf_z2[of n "m - n"] and m>n unfolding pos_le_divide_eq[OF 1-c>0]
by (auto simp add: 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 **]]
unfolding mult_assoc by auto
also have "\<dots> < (e * (1 - c) / d) * d * (1 - c ^ (m - n)) / (1 - c)"
using mult_strict_right_mono[OF N **] unfolding mult_assoc by auto
also have "\<dots> = e * (1 - c ^ (m - n))" using c and d>0 and 1 - c > 0 by auto
also have "\<dots> \<le> e" using c and 1 - c ^ (m - n) > 0 and e>0 using mult_right_le_one_le[of e "1 - c ^ (m - n)"] by auto
finally have  "dist (z m) (z n) < e" by auto
} note * = this
{ fix m n::nat assume as:"N\<le>m" "N\<le>n"
hence "dist (z n) (z m) < e"
proof(cases "n = m")
case True thus ?thesis using e>0 by auto
next
case False thus ?thesis using as and *[of n m] *[of m n] unfolding nat_neq_iff by (auto simp add: dist_commute)
qed }
thus ?thesis by auto
qed
}
hence "Cauchy z" unfolding cauchy_def by auto
then obtain x where "x\<in>s" and x:"(z ---> x) sequentially" using s(1)[unfolded compact_def complete_def, THEN spec[where x=z]] and z_in_s by auto

def e \<equiv> "dist (f x) x"
have "e = 0" proof(rule ccontr)
assume "e \<noteq> 0" hence "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 LIMSEQ_def, THEN spec[where x="e/2"]] by auto
hence 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] x\<in>s 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
hence "f x = x" unfolding e_def by auto
moreover
{ fix y assume "f y = y" "y\<in>s"
hence "dist x y \<le> c * dist x y" using lipschitz[THEN bspec[where x=x], THEN bspec[where x=y]]
using x\<in>s and f x = x by auto
hence "dist x y = 0" unfolding mult_le_cancel_right1
using c and zero_le_dist[of x y] by auto
hence "y = x" by auto
}
ultimately show ?thesis using x\<in>s by blast+
qed

subsection {* Edelstein fixed point theorem *}

lemma 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_within_id 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"
unfolding continuous_on_iff by auto
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_within_id 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 a \<in> s 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 a \<in> s 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]] g a = a and a\<in>s by auto
ultimately show "\<exists>!x\<in>s. g x = x" using a \<in> s by blast
qed

declare tendsto_const [intro] (* FIXME: move *)

end