# Theory Topology_Euclidean_Space

theory Topology_Euclidean_Space
imports Countable_Set FuncSet Linear_Algebra Norm_Arith
(*  title:      HOL/Library/Topology_Euclidian_Space.thy
Author:     Amine Chaieb, University of Cambridge
Author:     Robert Himmelmann, TU Muenchen
Author:     Brian Huffman, Portland State University
*)

section {* Elementary topology in Euclidean space. *}

theory Topology_Euclidean_Space
imports
Complex_Main
"~~/src/HOL/Library/Countable_Set"
"~~/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 ==> dist x z < d / 2 ==> dist y z < d"
using dist_triangle[of y z x] by (simp add: dist_commute)

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

lemma countable_PiE:
"finite I ==> (!!i. i ∈ I ==> countable (F i)) ==> 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 => 'b::topological_space"
shows "a ∈ S ==> open S ==> (f ---> l)(at a within S) <-> (f ---> l)(at a)"
by (fact tendsto_within_open)

lemma continuous_on_union:
"closed s ==> closed t ==> continuous_on s f ==> continuous_on t f ==> continuous_on (s ∪ t) f"
by (fact continuous_on_closed_Un)

lemma continuous_on_cases:
"closed s ==> closed t ==> continuous_on s f ==> continuous_on t g ==>
∀x. (x∈s ∧ ¬ P x) ∨ (x ∈ t ∧ P x) --> f x = g x ==>
continuous_on (s ∪ t) (λx. 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 <->
(∀b∈B. open b) ∧ (∀x. open x --> (∃B'. B' ⊆ B ∧ \<Union>B' = x))"

lemma topological_basis:
"topological_basis B <-> (∀x. open x <-> (∃B'. B' ⊆ B ∧ \<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 "!!B'. B' ∈ B ==> open B'"
shows "topological_basis B <-> (∀O'. open O' --> (∀x∈O'. ∃B'∈B. x ∈ B' ∧ B' ⊆ O'))"
(is "_ <-> ?rhs")
proof safe
fix O' and x::'a
assume H: "topological_basis B" "open O'" "x ∈ O'"
then have "(∃B'⊆B. \<Union>B' = O')" by (simp add: topological_basis_def)
then obtain B' where "B' ⊆ B" "O' = \<Union>B'" by auto
then show "∃B'∈B. x ∈ B' ∧ B' ⊆ 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 "∀x∈O'. f x ∈ B ∧ x ∈ f x ∧ f x ⊆ O'"
by (force intro: bchoice simp: Bex_def)
then show "∃B'⊆B. \<Union>B' = O'"
by (auto intro: exI[where x="{f x |x. x ∈ O'}"])
qed
qed

lemma topological_basisI:
assumes "!!B'. B' ∈ B ==> open B'"
and "!!O' x. open O' ==> x ∈ O' ==> ∃B'∈B. x ∈ B' ∧ B' ⊆ O'"
shows "topological_basis B"
using assms by (subst topological_basis_iff) auto

lemma topological_basisE:
fixes O'
assumes "topological_basis B"
and "open O'"
and "x ∈ O'"
obtains B' where "B' ∈ B" "x ∈ B'" "B' ⊆ O'"
proof atomize_elim
from assms have "!!B'. B'∈B ==> open B'"
by (simp add: topological_basis_def)
with topological_basis_iff assms
show  "∃B'. B' ∈ B ∧ x ∈ B' ∧ B' ⊆ O'"
using assms by (simp add: Bex_def)
qed

lemma topological_basis_open:
assumes "topological_basis B"
and "X ∈ B"
shows "open X"
using assms by (simp add: topological_basis_def)

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' ⊆ 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 => 'a"
assumes "topological_basis B"
and choosefrom_basis: "!!B'. B' ≠ {} ==> f B' ∈ B'"
shows "∀X. open X --> X ≠ {} --> (∃B' ∈ B. f B' ∈ X)"
proof (intro allI impI)
fix X :: "'a set"
assume "open X" and "X ≠ {}"
from topological_basisE[OF topological_basis B open X choosefrom_basis[OF X ≠ {}]]
obtain B' where "B' ∈ B" "f X ∈ B'" "B' ⊆ X" .
then show "∃B'∈B. f B' ∈ 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 ((λ(a, b). a × b)  (A × B))"
unfolding topological_basis_def
proof (safe, simp_all del: ex_simps add: subset_image_iff ex_simps(1)[symmetric])
fix S :: "('a × 'b) set"
assume "open S"
then show "∃X⊆A × B. (\<Union>(a,b)∈X. a × b) = S"
proof (safe intro!: exI[of _ "{x∈A × B. fst x × snd x ⊆ S}"])
fix x y
assume "(x, y) ∈ S"
from open_prod_elim[OF open S this]
obtain a b where a: "open a""x ∈ a" and b: "open b" "y ∈ b" and "a × b ⊆ S"
by (metis mem_Sigma_iff)
moreover
from A a obtain A0 where "A0 ∈ A" "x ∈ A0" "A0 ⊆ a"
by (rule topological_basisE)
moreover
from B b obtain B0 where "B0 ∈ B" "y ∈ B0" "B0 ⊆ b"
by (rule topological_basisE)
ultimately show "(x, y) ∈ (\<Union>(a, b)∈{X ∈ A × B. fst X × snd X ⊆ S}. a × 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"
and countable_basis: "countable B"
begin

lemma open_countable_basis_ex:
assumes "open X"
shows "∃B' ⊆ 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' ⊆ B" "X = Union B'"
using assms open_countable_basis_ex
by (atomize_elim) simp

lemma countable_dense_exists:
"∃D::'a set. countable D ∧ (∀X. open X --> X ≠ {} --> (∃d ∈ D. d ∈ X))"
proof -
let ?f = "(λB'. SOME x. x ∈ 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" "!!X. open X ==> X ≠ {} ==> ∃d ∈ D. d ∈ X"
using countable_dense_exists by blast

end

lemma (in first_countable_topology) first_countable_basisE:
obtains A where "countable A" "!!a. a ∈ A ==> x ∈ a" "!!a. a ∈ A ==> open a"
"!!S. open S ==> x ∈ S ==> (∃a∈A. a ⊆ 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" "!!a. a ∈ A ==> x ∈ a" "!!a. a ∈ A ==> open a"
"!!S. open S ==> x ∈ S ==> (∃a∈A. a ⊆ S)"
"!!a b. a ∈ A ==> b ∈ A ==> a ∩ b ∈ A"
proof atomize_elim
obtain A' where A':
"countable A'"
"!!a. a ∈ A' ==> x ∈ a"
"!!a. a ∈ A' ==> open a"
"!!S. open S ==> x ∈ S ==> ∃a∈A'. a ⊆ S"
by (rule first_countable_basisE) blast
def A ≡ "(λN. \<Inter>((λn. from_nat_into A' n)  N))  (Collect finite::nat set set)"
then show "∃A. countable A ∧ (∀a. a ∈ A --> x ∈ a) ∧ (∀a. a ∈ A --> open a) ∧
(∀S. open S --> x ∈ S --> (∃a∈A. a ⊆ S)) ∧ (∀a b. a ∈ A --> b ∈ A --> a ∩ b ∈ 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 ∈ A"
then show "x ∈ a" "open a"
using A'(4)[OF open_UNIV] by (auto simp: A_def intro: A' from_nat_into)
next
let ?int = "λN. \<Inter>(from_nat_into A'  N)"
fix a b
assume "a ∈ A" "b ∈ A"
then obtain N M where "a = ?int N" "b = ?int M" "finite (N ∪ M)"
by (auto simp: A_def)
then show "a ∩ b ∈ A"
by (auto simp: A_def intro!: image_eqI[where x="N ∪ M"])
next
fix S
assume "open S" "x ∈ S"
then obtain a where a: "a∈A'" "a ⊆ S" using A' by blast
then show "∃a∈A. a ⊆ 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: "!!a. a ∈ A ==> x ∈ a" "!!a. a ∈ A ==> open a"
and 2: "!!S. open S ==> x ∈ S ==> ∃a∈A. a ⊆ S"
shows "∃A::nat => 'a set. (∀i. x ∈ A i ∧ open (A i)) ∧ (∀S. open S ∧ x ∈ S --> (∃i. A i ⊆ S))"
proof (safe intro!: exI[of _ "from_nat_into A"])
fix i
have "A ≠ {}" using 2[of UNIV] by auto
show "x ∈ from_nat_into A i" "open (from_nat_into A i)"
using range_from_nat_into_subset[OF A ≠ {}] 1 by auto
next
fix S
assume "open S" "x∈S" from 2[OF this]
show "∃i. from_nat_into A i ⊆ 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 × 'b"
obtain A where A:
"countable A"
"!!a. a ∈ A ==> fst x ∈ a"
"!!a. a ∈ A ==> open a"
"!!S. open S ==> fst x ∈ S ==> ∃a∈A. a ⊆ S"
by (rule first_countable_basisE[of "fst x"]) blast
obtain B where B:
"countable B"
"!!a. a ∈ B ==> snd x ∈ a"
"!!a. a ∈ B ==> open a"
"!!S. open S ==> snd x ∈ S ==> ∃a∈B. a ⊆ S"
by (rule first_countable_basisE[of "snd x"]) blast
show "∃A::nat => ('a × 'b) set.
(∀i. x ∈ A i ∧ open (A i)) ∧ (∀S. open S ∧ x ∈ S --> (∃i. A i ⊆ S))"
proof (rule first_countableI[of "(λ(a, b). a × b)  (A × B)"], safe)
fix a b
assume x: "a ∈ A" "b ∈ B"
with A(2, 3)[of a] B(2, 3)[of b] show "x ∈ a × b" and "open (a × b)"
unfolding mem_Times_iff
by (auto intro: open_Times)
next
fix S
assume "open S" "x ∈ S"
then obtain a' b' where a'b': "open a'" "open b'" "x ∈ a' × b'" "a' × b' ⊆ S"
by (rule open_prod_elim)
moreover
from a'b' A(4)[of a'] B(4)[of b']
obtain a b where "a ∈ A" "a ⊆ a'" "b ∈ B" "b ⊆ b'"
by auto
ultimately
show "∃a∈(λ(a, b). a × b)  (A × B). a ⊆ S"
by (auto intro!: bexI[of _ "a × b"] bexI[of _ a] bexI[of _ b])
qed (simp add: A B)
qed

class second_countable_topology = topological_space +
assumes ex_countable_subbasis:
"∃B::'a::topological_space set set. countable B ∧ open = generate_topology B"
begin

lemma ex_countable_basis: "∃B::'a set set. countable B ∧ 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 ∧ b ⊆ 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 "∃B'⊆{b. finite b ∧ b ⊆ B}. (\<Union>b∈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" "!!i. i ∈ x ==> finite i ∧ i ⊆ B"
and y: "b = UNION y Inter" "!!i. i ∈ y ==> finite i ∧ i ⊆ B"
by blast
show ?case
unfolding x y Int_UN_distrib2
by (intro exI[of _ "{i ∪ j| i j.  i ∈ x ∧ j ∈ y}"]) (auto dest: x(2) y(2))
next
case (UN K)
then have "∀k∈K. ∃B'⊆{b. finite b ∧ b ⊆ B}. UNION B' Inter = k" by auto
then obtain k where
"∀ka∈K. k ka ⊆ {b. finite b ∧ b ⊆ B} ∧ UNION (k ka) Inter = ka"
unfolding bchoice_iff ..
then show "∃B'⊆{b. finite b ∧ b ⊆ 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 "(∃B'⊆Inter  {b. finite b ∧ b ⊆ 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)
(simp_all add: B generate_topology.Basis subset_eq)
qed
qed

end

sublocale second_countable_topology <
countable_basis "SOME B. countable B ∧ 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 "∃B::('a × 'b) set set. countable B ∧ open = generate_topology B"
by (auto intro!: exI[of _ "(λ(a, b). a × b)  (A × B)"] topological_basis_prod
topological_basis_imp_subbasis)
qed

instance second_countable_topology ⊆ first_countable_topology
proof
fix x :: 'a
def B ≡ "SOME B::'a set set. countable B ∧ 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 "∃A::nat => 'a set.
(∀i. x ∈ A i ∧ open (A i)) ∧ (∀S. open S ∧ x ∈ S --> (∃i. A i ⊆ S))"
by (intro first_countableI[of "{b∈B. x ∈ 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 <->
L {} ∧ (∀S T. L S --> L T --> L (S ∩ T)) ∧ (∀K. Ball K L --> L (\<Union> K))"

typedef 'a topology = "{L::('a set) => 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 ==> openin (topology U) = U"
using topology_inverse[unfolded mem_Collect_eq] .

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

lemma topology_eq: "T1 = T2 <-> (∀S. openin T1 S <-> openin T2 S)"
proof
assume "T1 = T2"
then show "∀S. openin T1 S <-> openin T2 S" by simp
next
assume H: "∀S. openin T1 S <-> openin T2 S"
then have "openin T1 = openin T2" by (simp add: fun_eq_iff)
then have "topology (openin T1) = topology (openin T2)" by simp
then show "T1 = T2" unfolding openin_inverse .
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 {}"
"!!S T. openin U S ==> openin U T ==> openin U (S∩T)"
"!!K. (∀S ∈ K. openin U S) ==> openin U (\<Union>K)"
using openin[of U] unfolding istopology_def mem_Collect_eq by fast+

lemma openin_subset[intro]: "openin U S ==> S ⊆ 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 ==> openin U T ==> openin U (S ∩ T)"
using openin_clauses by simp

lemma openin_Union[intro]: "(∀S ∈K. openin U S) ==> openin U (\<Union> K)"
using openin_clauses by simp

lemma openin_Un[intro]: "openin U S ==> openin U T ==> openin U (S ∪ 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 <-> (∀x ∈ S. ∃T. openin U T ∧ x ∈ T ∧ T ⊆ S)"
(is "?lhs <-> ?rhs")
proof
assume ?lhs
then show ?rhs by auto
next
assume H: ?rhs
let ?t = "\<Union>{T. openin U T ∧ T ⊆ 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 <-> S ⊆ topspace U ∧ openin U (topspace U - S)"

lemma closedin_subset: "closedin U S ==> S ⊆ 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 ==> closedin U T ==> closedin U (S ∪ T)"
by (auto simp add: Diff_Un closedin_def)

lemma Diff_Inter[intro]: "A - \<Inter>S = \<Union> {A - s|s. s∈S}"
by auto

lemma closedin_Inter[intro]:
assumes Ke: "K ≠ {}"
and Kc: "∀S ∈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 ==> closedin U T ==> closedin U (S ∩ T)"
using closedin_Inter[of "{S,T}" U] by auto

lemma Diff_Diff_Int: "A - (A - B) = A ∩ B"
by blast

lemma openin_closedin_eq: "openin U S <-> S ⊆ topspace U ∧ 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 ⊆ topspace U ==> (openin U S <-> closedin U (topspace U - S))"
by (simp add: openin_closedin_eq)

lemma openin_diff[intro]:
assumes oS: "openin U S"
and cT: "closedin U T"
shows "openin U (S - T)"
proof -
have "S - T = S ∩ (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 ∩ (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 (λT. ∃S. T = S ∩ V ∧ openin U S)"

lemma istopology_subtopology: "istopology (λT. ∃S. T = S ∩ V ∧ 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 ∩ V" and Sb: "openin U Sb" "B = Sb ∩ V"
by blast
have "A ∩ B = (Sa ∩ Sb) ∩ V" "openin U (Sa ∩ Sb)"
using Sa Sb by blast+
then have "?L (A ∩ B)" by blast
}
moreover
{
fix K
assume K: "K ⊆ Collect ?L"
have th0: "Collect ?L = (λS. S ∩ V)  Collect (openin U)"
by blast
from K[unfolded th0 subset_image_iff]
obtain Sk where Sk: "Sk ⊆ Collect (openin U)" "K = (λS. S ∩ V)  Sk"
by blast
have "\<Union>K = (\<Union>Sk) ∩ 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 <-> (∃T. openin U T ∧ S = T ∩ V)"
unfolding subtopology_def topology_inverse'[OF istopology_subtopology]
by auto

lemma topspace_subtopology: "topspace (subtopology U V) = topspace U ∩ V"
by (auto simp add: topspace_def openin_subtopology)

lemma closedin_subtopology: "closedin (subtopology U V) S <-> (∃T. closedin U T ∧ S = T ∩ V)"
unfolding closedin_def topspace_subtopology
by (auto simp add: openin_subtopology)

lemma openin_subtopology_refl: "openin (subtopology U V) V <-> V ⊆ topspace U"
unfolding openin_subtopology
by auto (metis IntD1 in_mono openin_subset)

lemma subtopology_superset:
assumes UV: "topspace U ⊆ V"
shows "subtopology U V = U"
proof -
{
fix S
{
fix T
assume T: "openin U T" "S = T ∩ 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"
then have "∃T. openin U T ∧ S = T ∩ V"
using openin_subset[OF S] UV by auto
}
ultimately have "(∃T. openin U T ∧ S = T ∩ V) <-> 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"
by (simp add: subtopology_superset)

lemma subtopology_UNIV[simp]: "subtopology U UNIV = U"
by (simp add: subtopology_superset)

subsubsection {* The standard Euclidean topology *}

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

lemma open_openin: "open S <-> openin euclidean S"
unfolding euclidean_def
apply (rule cong[where x=S and y=S])
apply (rule topology_inverse[symmetric])
apply (auto simp add: istopology_def)
done

lemma topspace_euclidean: "topspace euclidean = UNIV"
apply (simp add: topspace_def)
apply (rule set_eqI)
apply (auto simp add: open_openin[symmetric])
done

lemma topspace_euclidean_subtopology[simp]: "topspace (subtopology euclidean S) = S"
by (simp add: topspace_euclidean topspace_subtopology)

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

lemma open_subopen: "open S <-> (∀x∈S. ∃T. open T ∧ x ∈ T ∧ T ⊆ S)"
by (simp add: open_openin openin_subopen[symmetric])

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

lemma openin_open: "openin (subtopology euclidean U) S <-> (∃T. open T ∧ (S = U ∩ T))"
by (auto simp add: openin_subtopology open_openin[symmetric])

lemma openin_open_Int[intro]: "open S ==> openin (subtopology euclidean U) (U ∩ S)"
by (auto simp add: openin_open)

lemma open_openin_trans[trans]:
"open S ==> open T ==> T ⊆ S ==> openin (subtopology euclidean S) T"
by (metis Int_absorb1  openin_open_Int)

lemma open_subset: "S ⊆ T ==> open S ==> openin (subtopology euclidean T) S"
by (auto simp add: openin_open)

lemma closedin_closed: "closedin (subtopology euclidean U) S <-> (∃T. closed T ∧ S = U ∩ T)"
by (simp add: closedin_subtopology closed_closedin Int_ac)

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

lemma closed_closedin_trans:
"closed S ==> closed T ==> T ⊆ S ==> closedin (subtopology euclidean S) T"
by (metis closedin_closed inf.absorb2)

lemma closed_subset: "S ⊆ T ==> closed S ==> closedin (subtopology euclidean T) S"
by (auto simp add: closedin_closed)

lemma openin_euclidean_subtopology_iff:
fixes S U :: "'a::metric_space set"
shows "openin (subtopology euclidean U) S <->
S ⊆ U ∧ (∀x∈S. ∃e>0. ∀x'∈U. dist x' x < e --> x'∈ S)"
(is "?lhs <-> ?rhs")
proof
assume ?lhs
then show ?rhs
unfolding openin_open open_dist by blast
next
def T ≡ "{x. ∃a∈S. ∃d>0. (∀y∈U. dist y a < d --> y ∈ S) ∧ dist x a < d}"
have 1: "∀x∈T. ∃e>0. ∀y. dist y x < e --> y ∈ T"
unfolding T_def
apply clarsimp
apply (rule_tac x="d - dist x a" in exI)
apply (clarsimp simp add: less_diff_eq)
by (metis dist_commute dist_triangle_lt)
assume ?rhs then have 2: "S = U ∩ T"
unfolding T_def
by auto (metis dist_self)
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 ==> openin (subtopology euclidean U) T ==>
openin (subtopology euclidean U) S"
unfolding open_openin openin_open by blast

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

lemma closedin_trans[trans]:
"closedin (subtopology euclidean T) S ==> 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 ==> closed T ==> closed S"
by (auto simp add: closedin_closed intro: closedin_trans)

subsection {* Open and closed balls *}

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

definition cball :: "'a::metric_space => real => 'a set"
where "cball x e = {y. dist x y ≤ e}"

lemma mem_ball [simp]: "y ∈ ball x e <-> dist x y < e"
by (simp add: ball_def)

lemma mem_cball [simp]: "y ∈ cball x e <-> dist x y ≤ e"
by (simp add: cball_def)

lemma mem_ball_0:
fixes x :: "'a::real_normed_vector"
shows "x ∈ ball 0 e <-> norm x < e"
by (simp add: dist_norm)

lemma mem_cball_0:
fixes x :: "'a::real_normed_vector"
shows "x ∈ cball 0 e <-> norm x ≤ e"
by (simp add: dist_norm)

lemma centre_in_ball: "x ∈ ball x e <-> 0 < e"
by simp

lemma centre_in_cball: "x ∈ cball x e <-> 0 ≤ e"
by simp

lemma ball_subset_cball[simp,intro]: "ball x e ⊆ cball x e"
by (simp add: subset_eq)

lemma subset_ball[intro]: "d ≤ e ==> ball x d ⊆ ball x e"
by (simp add: subset_eq)

lemma subset_cball[intro]: "d ≤ e ==> cball x d ⊆ cball x e"
by (simp add: subset_eq)

lemma ball_max_Un: "ball a (max r s) = ball a r ∪ ball a s"
by (simp add: set_eq_iff) arith

lemma ball_min_Int: "ball a (min r s) = ball a r ∩ ball a s"
by (simp add: set_eq_iff)

lemma diff_less_iff:
"(a::real) - b > 0 <-> a > b"
"(a::real) - b < 0 <-> a < b"
"a - b < c <-> a < c + b" "a - b > c <-> a > c + b"
by arith+

lemma diff_le_iff:
"(a::real) - b ≥ 0 <-> a ≥ b"
"(a::real) - b ≤ 0 <-> a ≤ b"
"a - b ≤ c <-> a ≤ c + b"
"a - b ≥ c <-> a ≥ c + b"
by arith+

lemma open_ball [intro, simp]: "open (ball x e)"
proof -
have "open (dist x - {..<e})"
by (intro open_vimage open_lessThan continuous_intros)
also have "dist x - {..<e} = ball x e"
by auto
finally show ?thesis .
qed

lemma open_contains_ball: "open S <-> (∀x∈S. ∃e>0. ball x e ⊆ S)"
unfolding open_dist subset_eq mem_ball Ball_def dist_commute ..

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

lemma open_contains_ball_eq: "open S ==> ∀x. x∈S <-> (∃e>0. ball x e ⊆ S)"
by (metis open_contains_ball subset_eq centre_in_ball)

lemma ball_eq_empty[simp]: "ball x e = {} <-> e ≤ 0"
unfolding mem_ball set_eq_iff
apply (simp add: not_less)
apply (metis zero_le_dist order_trans dist_self)
done

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

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

subsection {* Boxes *}

abbreviation One :: "'a::euclidean_space"
where "One ≡ ∑Basis"

definition (in euclidean_space) eucl_less (infix "<e" 50)
where "eucl_less a b <-> (∀i∈Basis. a • i < b • i)"

definition box_eucl_less: "box a b = {x. a <e x ∧ x <e b}"
definition "cbox a b = {x. ∀i∈Basis. a • i ≤ x • i ∧ x • i ≤ b • i}"

lemma box_def: "box a b = {x. ∀i∈Basis. a • i < x • i ∧ x • i < b • i}"
and in_box_eucl_less: "x ∈ box a b <-> a <e x ∧ x <e b"
and mem_box: "x ∈ box a b <-> (∀i∈Basis. a • i < x • i ∧ x • i < b • i)"
"x ∈ cbox a b <-> (∀i∈Basis. a • i ≤ x • i ∧ x • i ≤ b • i)"
by (auto simp: box_eucl_less eucl_less_def cbox_def)

lemma mem_box_real[simp]:
"(x::real) ∈ box a b <-> a < x ∧ x < b"
"(x::real) ∈ cbox a b <-> a ≤ x ∧ x ≤ b"
by (auto simp: mem_box)

lemma box_real[simp]:
fixes a b:: real
shows "box a b = {a <..< b}" "cbox a b = {a .. b}"
by auto

lemma box_Int_box:
fixes a :: "'a::euclidean_space"
shows "box a b ∩ box c d =
box (∑i∈Basis. max (a•i) (c•i) *⇩R i) (∑i∈Basis. min (b•i) (d•i) *⇩R i)"
unfolding set_eq_iff and Int_iff and mem_box by auto

lemma rational_boxes:
fixes x :: "'a::euclidean_space"
assumes "e > 0"
shows "∃a b. (∀i∈Basis. a • i ∈ \<rat> ∧ b • i ∈ \<rat> ) ∧ x ∈ box a b ∧ box a b ⊆ ball x e"
proof -
def e' ≡ "e / (2 * sqrt (real (DIM ('a))))"
then have e: "e' > 0"
using assms by (auto simp: DIM_positive)
have "∀i. ∃y. y ∈ \<rat> ∧ y < x • i ∧ x • i - y < e'" (is "∀i. ?th i")
proof
fix i
from Rats_dense_in_real[of "x • i - e'" "x • i"] e
show "?th i" by auto
qed
from choice[OF this] obtain a where
a: "∀xa. a xa ∈ \<rat> ∧ a xa < x • xa ∧ x • xa - a xa < e'" ..
have "∀i. ∃y. y ∈ \<rat> ∧ x • i < y ∧ y - x • i < e'" (is "∀i. ?th i")
proof
fix i
from Rats_dense_in_real[of "x • i" "x • i + e'"] e
show "?th i" by auto
qed
from choice[OF this] obtain b where
b: "∀xa. b xa ∈ \<rat> ∧ x • xa < b xa ∧ b xa - x • xa < e'" ..
let ?a = "∑i∈Basis. a i *⇩R i" and ?b = "∑i∈Basis. b i *⇩R i"
show ?thesis
proof (rule exI[of _ ?a], rule exI[of _ ?b], safe)
fix y :: 'a
assume *: "y ∈ box ?a ?b"
have "dist x y = sqrt (∑i∈Basis. (dist (x • i) (y • i))⇧2)"
unfolding setL2_def[symmetric] by (rule euclidean_dist_l2)
also have "… < sqrt (∑(i::'a)∈Basis. e^2 / real (DIM('a)))"
proof (rule real_sqrt_less_mono, rule setsum_strict_mono)
fix i :: "'a"
assume i: "i ∈ Basis"
have "a i < y•i ∧ y•i < b i"
using * i by (auto simp: box_def)
moreover have "a i < x•i" "x•i - a i < e'"
using a by auto
moreover have "x•i < b i" "b i - x•i < e'"
using b by auto
ultimately have "¦x•i - y•i¦ < 2 * e'"
by auto
then have "dist (x • i) (y • i) < e/sqrt (real (DIM('a)))"
unfolding e'_def by (auto simp: dist_real_def)
then have "(dist (x • i) (y • i))⇧2 < (e/sqrt (real (DIM('a))))⇧2"
by (rule power_strict_mono) auto
then show "(dist (x • i) (y • i))⇧2 < e⇧2 / real DIM('a)"
by (simp add: power_divide)
qed auto
also have "… = e"
using 0 < e by (simp add: real_eq_of_nat)
finally show "y ∈ ball x e"
by (auto simp: ball_def)
qed (insert a b, auto simp: box_def)
qed

lemma open_UNION_box:
fixes M :: "'a::euclidean_space set"
assumes "open M"
defines "a' ≡ λf :: 'a => real × real. (∑(i::'a)∈Basis. fst (f i) *⇩R i)"
defines "b' ≡ λf :: 'a => real × real. (∑(i::'a)∈Basis. snd (f i) *⇩R i)"
defines "I ≡ {f∈Basis ->⇩E \<rat> × \<rat>. box (a' f) (b' f) ⊆ M}"
shows "M = (\<Union>f∈I. box (a' f) (b' f))"
proof -
{
fix x assume "x ∈ M"
obtain e where e: "e > 0" "ball x e ⊆ M"
using openE[OF open M x ∈ M] by auto
moreover obtain a b where ab:
"x ∈ box a b"
"∀i ∈ Basis. a • i ∈ \<rat>"
"∀i∈Basis. b • i ∈ \<rat>"
"box a b ⊆ ball x e"
using rational_boxes[OF e(1)] by metis
ultimately have "x ∈ (\<Union>f∈I. box (a' f) (b' f))"
by (intro UN_I[of "λi∈Basis. (a • i, b • i)"])
(auto simp: euclidean_representation I_def a'_def b'_def)
}
then show ?thesis by (auto simp: I_def)
qed

lemma box_eq_empty:
fixes a :: "'a::euclidean_space"
shows "(box a b = {} <-> (∃i∈Basis. b•i ≤ a•i))" (is ?th1)
and "(cbox a b = {} <-> (∃i∈Basis. b•i < a•i))" (is ?th2)
proof -
{
fix i x
assume i: "i∈Basis" and as:"b•i ≤ a•i" and x:"x∈box a b"
then have "a • i < x • i ∧ x • i < b • i"
unfolding mem_box by (auto simp: box_def)
then have "a•i < b•i" by auto
then have False using as by auto
}
moreover
{
assume as: "∀i∈Basis. ¬ (b•i ≤ a•i)"
let ?x = "(1/2) *⇩R (a + b)"
{
fix i :: 'a
assume i: "i ∈ Basis"
have "a•i < b•i"
using as[THEN bspec[where x=i]] i by auto
then have "a•i < ((1/2) *⇩R (a+b)) • i" "((1/2) *⇩R (a+b)) • i < b•i"
by (auto simp: inner_add_left)
}
then have "box a b ≠ {}"
using mem_box(1)[of "?x" a b] by auto
}
ultimately show ?th1 by blast

{
fix i x
assume i: "i ∈ Basis" and as:"b•i < a•i" and x:"x∈cbox a b"
then have "a • i ≤ x • i ∧ x • i ≤ b • i"
unfolding mem_box by auto
then have "a•i ≤ b•i" by auto
then have False using as by auto
}
moreover
{
assume as:"∀i∈Basis. ¬ (b•i < a•i)"
let ?x = "(1/2) *⇩R (a + b)"
{
fix i :: 'a
assume i:"i ∈ Basis"
have "a•i ≤ b•i"
using as[THEN bspec[where x=i]] i by auto
then have "a•i ≤ ((1/2) *⇩R (a+b)) • i" "((1/2) *⇩R (a+b)) • i ≤ b•i"
by (auto simp: inner_add_left)
}
then have "cbox a b ≠ {}"
using mem_box(2)[of "?x" a b] by auto
}
ultimately show ?th2 by blast
qed

lemma box_ne_empty:
fixes a :: "'a::euclidean_space"
shows "cbox a b ≠ {} <-> (∀i∈Basis. a•i ≤ b•i)"
and "box a b ≠ {} <-> (∀i∈Basis. a•i < b•i)"
unfolding box_eq_empty[of a b] by fastforce+

lemma
fixes a :: "'a::euclidean_space"
shows cbox_sing: "cbox a a = {a}"
and box_sing: "box a a = {}"
unfolding set_eq_iff mem_box eq_iff [symmetric]
by (auto intro!: euclidean_eqI[where 'a='a])
(metis all_not_in_conv nonempty_Basis)

lemma subset_box_imp:
fixes a :: "'a::euclidean_space"
shows "(∀i∈Basis. a•i ≤ c•i ∧ d•i ≤ b•i) ==> cbox c d ⊆ cbox a b"
and "(∀i∈Basis. a•i < c•i ∧ d•i < b•i) ==> cbox c d ⊆ box a b"
and "(∀i∈Basis. a•i ≤ c•i ∧ d•i ≤ b•i) ==> box c d ⊆ cbox a b"
and "(∀i∈Basis. a•i ≤ c•i ∧ d•i ≤ b•i) ==> box c d ⊆ box a b"
unfolding subset_eq[unfolded Ball_def] unfolding mem_box
by (best intro: order_trans less_le_trans le_less_trans less_imp_le)+

lemma box_subset_cbox:
fixes a :: "'a::euclidean_space"
shows "box a b ⊆ cbox a b"
unfolding subset_eq [unfolded Ball_def] mem_box
by (fast intro: less_imp_le)

lemma subset_box:
fixes a :: "'a::euclidean_space"
shows "cbox c d ⊆ cbox a b <-> (∀i∈Basis. c•i ≤ d•i) --> (∀i∈Basis. a•i ≤ c•i ∧ d•i ≤ b•i)" (is ?th1)
and "cbox c d ⊆ box a b <-> (∀i∈Basis. c•i ≤ d•i) --> (∀i∈Basis. a•i < c•i ∧ d•i < b•i)" (is ?th2)
and "box c d ⊆ cbox a b <-> (∀i∈Basis. c•i < d•i) --> (∀i∈Basis. a•i ≤ c•i ∧ d•i ≤ b•i)" (is ?th3)
and "box c d ⊆ box a b <-> (∀i∈Basis. c•i < d•i) --> (∀i∈Basis. a•i ≤ c•i ∧ d•i ≤ b•i)" (is ?th4)
proof -
show ?th1
unfolding subset_eq and Ball_def and mem_box
by (auto intro: order_trans)
show ?th2
unfolding subset_eq and Ball_def and mem_box
by (auto intro: le_less_trans less_le_trans order_trans less_imp_le)
{
assume as: "box c d ⊆ cbox a b" "∀i∈Basis. c•i < d•i"
then have "box c d ≠ {}"
unfolding box_eq_empty by auto
fix i :: 'a
assume i: "i ∈ Basis"
(** TODO combine the following two parts as done in the HOL_light version. **)
{
let ?x = "(∑j∈Basis. (if j=i then ((min (a•j) (d•j))+c•j)/2 else (c•j+d•j)/2) *⇩R j)::'a"
assume as2: "a•i > c•i"
{
fix j :: 'a
assume j: "j ∈ Basis"
then have "c • j < ?x • j ∧ ?x • j < d • j"
apply (cases "j = i")
using as(2)[THEN bspec[where x=j]] i
apply (auto simp add: as2)
done
}
then have "?x∈box c d"
using i unfolding mem_box by auto
moreover
have "?x ∉ cbox a b"
unfolding mem_box
apply auto
apply (rule_tac x=i in bexI)
using as(2)[THEN bspec[where x=i]] and as2 i
apply auto
done
ultimately have False using as by auto
}
then have "a•i ≤ c•i" by (rule ccontr) auto
moreover
{
let ?x = "(∑j∈Basis. (if j=i then ((max (b•j) (c•j))+d•j)/2 else (c•j+d•j)/2) *⇩R j)::'a"
assume as2: "b•i < d•i"
{
fix j :: 'a
assume "j∈Basis"
then have "d • j > ?x • j ∧ ?x • j > c • j"
apply (cases "j = i")
using as(2)[THEN bspec[where x=j]]
apply (auto simp add: as2)
done
}
then have "?x∈box c d"
unfolding mem_box by auto
moreover
have "?x∉cbox a b"
unfolding mem_box
apply auto
apply (rule_tac x=i in bexI)
using as(2)[THEN bspec[where x=i]] and as2 using i
apply auto
done
ultimately have False using as by auto
}
then have "b•i ≥ d•i" by (rule ccontr) auto
ultimately
have "a•i ≤ c•i ∧ d•i ≤ b•i" by auto
} note part1 = this
show ?th3
unfolding subset_eq and Ball_def and mem_box
apply (rule, rule, rule, rule)
apply (rule part1)
unfolding subset_eq and Ball_def and mem_box
prefer 4
apply auto
apply (erule_tac x=xa in allE, erule_tac x=xa in allE, fastforce)+
done
{
assume as: "box c d ⊆ box a b" "∀i∈Basis. c•i < d•i"
fix i :: 'a
assume i:"i∈Basis"
from as(1) have "box c d ⊆ cbox a b"
using box_subset_cbox[of a b] by auto
then have "a•i ≤ c•i ∧ d•i ≤ b•i"
using part1 and as(2) using i by auto
} note * = this
show ?th4
unfolding subset_eq and Ball_def and mem_box
apply (rule, rule, rule, rule)
apply (rule *)
unfolding subset_eq and Ball_def and mem_box
prefer 4
apply auto
apply (erule_tac x=xa in allE, simp)+
done
qed

lemma inter_interval:
fixes a :: "'a::euclidean_space"
shows "cbox a b ∩ cbox c d =
cbox (∑i∈Basis. max (a•i) (c•i) *⇩R i) (∑i∈Basis. min (b•i) (d•i) *⇩R i)"
unfolding set_eq_iff and Int_iff and mem_box
by auto

lemma disjoint_interval:
fixes a::"'a::euclidean_space"
shows "cbox a b ∩ cbox c d = {} <-> (∃i∈Basis. (b•i < a•i ∨ d•i < c•i ∨ b•i < c•i ∨ d•i < a•i))" (is ?th1)
and "cbox a b ∩ box c d = {} <-> (∃i∈Basis. (b•i < a•i ∨ d•i ≤ c•i ∨ b•i ≤ c•i ∨ d•i ≤ a•i))" (is ?th2)
and "box a b ∩ cbox c d = {} <-> (∃i∈Basis. (b•i ≤ a•i ∨ d•i < c•i ∨ b•i ≤ c•i ∨ d•i ≤ a•i))" (is ?th3)
and "box a b ∩ box c d = {} <-> (∃i∈Basis. (b•i ≤ a•i ∨ d•i ≤ c•i ∨ b•i ≤ c•i ∨ d•i ≤ a•i))" (is ?th4)
proof -
let ?z = "(∑i∈Basis. (((max (a•i) (c•i)) + (min (b•i) (d•i))) / 2) *⇩R i)::'a"
have **: "!!P Q. (!!i :: 'a. i ∈ Basis ==> Q ?z i ==> P i) ==>
(!!i x :: 'a. i ∈ Basis ==> P i ==> Q x i) ==> (∀x. ∃i∈Basis. Q x i) <-> (∃i∈Basis. P i)"
by blast
note * = set_eq_iff Int_iff empty_iff mem_box 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

lemma UN_box_eq_UNIV: "(\<Union>i::nat. box (- (real i *⇩R One)) (real i *⇩R One)) = UNIV"
proof -
{
fix x b :: 'a
assume [simp]: "b ∈ Basis"
have "¦x • b¦ ≤ real (ceiling ¦x • b¦)"
by (rule real_of_int_ceiling_ge)
also have "… ≤ real (ceiling (Max ((λb. ¦x • b¦)Basis)))"
by (auto intro!: ceiling_mono)
also have "… < real (ceiling (Max ((λb. ¦x • b¦)Basis)) + 1)"
by simp
finally have "¦x • b¦ < real (ceiling (Max ((λb. ¦x • b¦)Basis)) + 1)" . }
then have "!!x::'a. ∃n::nat. ∀b∈Basis. ¦x • b¦ < real n"
by (metis order.strict_trans reals_Archimedean2)
moreover have "!!x b::'a. !!n::nat.  ¦x • b¦ < real n <-> - real n < x • b ∧ x • b < real n"
by auto
ultimately show ?thesis
by (auto simp: box_def inner_setsum_left inner_Basis setsum.If_cases)
qed

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

definition "is_interval (s::('a::euclidean_space) set) <->
(∀a∈s. ∀b∈s. ∀x. (∀i∈Basis. ((a•i ≤ x•i ∧ x•i ≤ b•i) ∨ (b•i ≤ x•i ∧ x•i ≤ a•i))) --> x ∈ s)"

lemma is_interval_cbox: "is_interval (cbox a (b::'a::euclidean_space))" (is ?th1)
and is_interval_box: "is_interval (box a b)" (is ?th2)
unfolding is_interval_def mem_box Ball_def atLeastAtMost_iff
by (meson order_trans le_less_trans less_le_trans less_trans)+

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

lemma mem_is_intervalI:
assumes "is_interval s"
assumes "a ∈ s" "b ∈ s"
assumes "!!i. i ∈ Basis ==> a • i ≤ x • i ∧ x • i ≤ b • i ∨ b • i ≤ x • i ∧ x • i ≤ a • i"
shows "x ∈ s"
by (rule assms(1)[simplified is_interval_def, rule_format, OF assms(2,3,4)])

lemma interval_subst:
fixes S::"'a::euclidean_space set"
assumes "is_interval S"
assumes "x ∈ S" "y j ∈ S"
assumes "j ∈ Basis"
shows "(∑i∈Basis. (if i = j then y i • i else x • i) *⇩R i) ∈ S"
by (rule mem_is_intervalI[OF assms(1,2)]) (auto simp: assms)

lemma mem_box_componentwiseI:
fixes S::"'a::euclidean_space set"
assumes "is_interval S"
assumes "!!i. i ∈ Basis ==> x • i ∈ ((λx. x • i)  S)"
shows "x ∈ S"
proof -
from assms have "∀i ∈ Basis. ∃s ∈ S. x • i = s • i"
by auto
with finite_Basis obtain s and bs::"'a list" where
s: "!!i. i ∈ Basis ==> x • i = s i • i" "!!i. i ∈ Basis ==> s i ∈ S" and
bs: "set bs = Basis" "distinct bs"
by (metis finite_distinct_list)
from nonempty_Basis s obtain j where j: "j ∈ Basis" "s j ∈ S" by blast
def y ≡ "rec_list
(s j)
(λj _ Y. (∑i∈Basis. (if i = j then s i • i else Y • i) *⇩R i))"
have "x = (∑i∈Basis. (if i ∈ set bs then s i • i else s j • i) *⇩R i)"
using bs by (auto simp add: s(1)[symmetric] euclidean_representation)
also have [symmetric]: "y bs = …"
using bs(2) bs(1)[THEN equalityD1]
by (induct bs) (auto simp: y_def euclidean_representation intro!: euclidean_eqI[where 'a='a])
also have "y bs ∈ S"
using bs(1)[THEN equalityD1]
apply (induct bs)
apply (auto simp: y_def j)
apply (rule interval_subst[OF assms(1)])
apply (auto simp: s)
done
finally show ?thesis .
qed

subsection{* Connectedness *}

lemma connected_local:
"connected S <->
¬ (∃e1 e2.
openin (subtopology euclidean S) e1 ∧
openin (subtopology euclidean S) e2 ∧
S ⊆ e1 ∪ e2 ∧
e1 ∩ e2 = {} ∧
e1 ≠ {} ∧
e2 ≠ {})"
unfolding connected_def openin_open
by safe blast+

lemma exists_diff:
fixes P :: "'a set => bool"
shows "(∃S. P(- S)) <-> (∃S. P S)" (is "?lhs <-> ?rhs")
proof -
{
assume "?lhs"
then have ?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 <->
(∀T. openin (subtopology euclidean S) T ∧
closedin (subtopology euclidean S) T --> T = {} ∨ T = S)" (is "?lhs <-> ?rhs")
proof -
have "¬ connected S <->
(∃e1 e2. open e1 ∧ open (- e2) ∧ S ⊆ e1 ∪ (- e2) ∧ e1 ∩ (- e2) ∩ S = {} ∧ e1 ∩ S ≠ {} ∧ (- e2) ∩ S ≠ {})"
unfolding connected_def openin_open closedin_closed
by (metis double_complement)
then have th0: "connected S <->
¬ (∃e2 e1. closed e2 ∧ open e1 ∧ S ⊆ e1 ∪ (- e2) ∧ e1 ∩ (- e2) ∩ S = {} ∧ e1 ∩ S ≠ {} ∧ (- e2) ∩ S ≠ {})"
(is " _ <-> ¬ (∃e2 e1. ?P e2 e1)")
apply (simp add: closed_def)
apply metis
done
have th1: "?rhs <-> ¬ (∃t' t. closed t'∧t = S∩t' ∧ t≠{} ∧ t≠S ∧ (∃t'. open t' ∧ t = S ∩ t'))"
(is "_ <-> ¬ (∃t' t. ?Q t' t)")
unfolding connected_def openin_open closedin_closed by auto
{
fix e2
{
fix e1
have "?P e2 e1 <-> (∃t. closed e2 ∧ t = S∩e2 ∧ open e1 ∧ t = S∩e1 ∧ t≠{} ∧ t ≠ S)"
by auto
}
then have "(∃e1. ?P e2 e1) <-> (∃t. ?Q e2 t)"
by metis
}
then have "∀e2. (∃e1. ?P e2 e1) <-> (∃t. ?Q e2 t)"
by blast
then show ?thesis
unfolding th0 th1 by simp
qed

subsection{* Limit points *}

definition (in topological_space) islimpt:: "'a => 'a set => bool"  (infixr "islimpt" 60)
where "x islimpt S <-> (∀T. x∈T --> open T --> (∃y∈S. y∈T ∧ y≠x))"

lemma islimptI:
assumes "!!T. x ∈ T ==> open T ==> ∃y∈S. y ∈ T ∧ y ≠ x"
shows "x islimpt S"
using assms unfolding islimpt_def by auto

lemma islimptE:
assumes "x islimpt S" and "x ∈ T" and "open T"
obtains y where "y ∈ S" and "y ∈ T" and "y ≠ x"
using assms unfolding islimpt_def by auto

lemma islimpt_iff_eventually: "x islimpt S <-> ¬ eventually (λy. y ∉ S) (at x)"
unfolding islimpt_def eventually_at_topological by auto

lemma islimpt_subset: "x islimpt S ==> S ⊆ T ==> x islimpt T"
unfolding islimpt_def by fast

lemma islimpt_approachable:
fixes x :: "'a::metric_space"
shows "x islimpt S <-> (∀e>0. ∃x'∈S. x' ≠ x ∧ 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 <-> (∀e>0. ∃x'∈ S. x' ≠ x ∧ dist x' x ≤ e)"
unfolding islimpt_approachable
using approachable_lt_le [where f="λy. dist y x" and P="λy. y ∉ S ∨ 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 <-> ¬ 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 ==> ∃a. a ≠ x ∧ dist a x < r"
using islimpt_UNIV [of x]
by (simp add: islimpt_approachable)

lemma closed_limpt: "closed S <-> (∀x. x islimpt S --> x ∈ S)"
unfolding closed_def
apply (subst open_subopen)
apply (simp add: islimpt_def subset_eq)
apply (metis ComplE ComplI)
done

lemma islimpt_EMPTY[simp]: "¬ x islimpt {}"
unfolding islimpt_def by auto

lemma finite_set_avoid:
fixes a :: "'a::metric_space"
assumes fS: "finite S"
shows  "∃d>0. ∀x∈S. x ≠ a --> d ≤ dist a x"
proof (induct rule: finite_induct[OF fS])
case 1
then show ?case by (auto intro: zero_less_one)
next
case (2 x F)
from 2 obtain d where d: "d >0" "∀x∈F. x≠a --> d ≤ dist a x"
by blast
show ?case
proof (cases "x = a")
case True
then show ?thesis using d by auto
next
case False
let ?d = "min d (dist a x)"
have dp: "?d > 0"
using False d(1) using dist_nz by auto
from d have d': "∀x∈F. x≠a --> ?d ≤ dist a x"
by auto
with dp False show ?thesis
by (auto intro!: exI[where x="?d"])
qed
qed

lemma islimpt_Un: "x islimpt (S ∪ T) <-> x islimpt S ∨ x islimpt T"
by (simp add: islimpt_iff_eventually eventually_conj_iff)

lemma discrete_imp_closed:
fixes S :: "'a::metric_space set"
assumes e: "0 < e"
and d: "∀x ∈ S. ∀y ∈ S. dist y x < e --> y = x"
shows "closed S"
proof -
{
fix x
assume C: "∀e>0. ∃x'∈S. x' ≠ x ∧ dist x' x < e"
from e have e2: "e/2 > 0" by arith
from C[rule_format, OF e2] obtain y where y: "y ∈ S" "y ≠ 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[symmetric])
from C[rule_format, OF mp] obtain z where z: "z ∈ S" "z ≠ 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 ∧ T ⊆ S}"

lemma interiorI [intro?]:
assumes "open T" and "x ∈ T" and "T ⊆ S"
shows "x ∈ interior S"
using assms unfolding interior_def by fast

lemma interiorE [elim?]:
assumes "x ∈ interior S"
obtains T where "open T" and "x ∈ T" and "T ⊆ S"
using assms unfolding interior_def by fast

lemma open_interior [simp, intro]: "open (interior S)"
by (simp add: interior_def open_Union)

lemma interior_subset: "interior S ⊆ S"
by (auto simp add: interior_def)

lemma interior_maximal: "T ⊆ S ==> open T ==> T ⊆ interior S"
by (auto simp add: interior_def)

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

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

lemma open_subset_interior: "open S ==> S ⊆ interior T <-> S ⊆ 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 ⊆ T ==> interior S ⊆ interior T"
by (auto simp add: interior_def)

lemma interior_unique:
assumes "T ⊆ S" and "open T"
assumes "!!T'. T' ⊆ S ==> open T' ==> T' ⊆ T"
shows "interior S = T"
by (intro equalityI assms interior_subset open_interior interior_maximal)

lemma interior_inter [simp]: "interior (S ∩ T) = interior S ∩ 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 ∈ interior S <-> (∃e>0. ball x e ⊆ S)"
using open_contains_ball_eq [where S="interior S"]
by (simp add: open_subset_interior)

lemma interior_limit_point [intro]:
fixes x :: "'a::perfect_space"
assumes x: "x ∈ 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 ∩ T'" in spec)
apply (auto simp add: open_Int)
done

lemma interior_closed_Un_empty_interior:
assumes cS: "closed S"
and iT: "interior T = {}"
shows "interior (S ∪ T) = interior S"
proof
show "interior S ⊆ interior (S ∪ T)"
by (rule interior_mono) (rule Un_upper1)
show "interior (S ∪ T) ⊆ interior S"
proof
fix x
assume "x ∈ interior (S ∪ T)"
then obtain R where "open R" "x ∈ R" "R ⊆ S ∪ T" ..
show "x ∈ interior S"
proof (rule ccontr)
assume "x ∉ interior S"
with x ∈ R open R obtain y where "y ∈ R - S"
unfolding interior_def by fast
from open R closed S have "open (R - S)"
by (rule open_Diff)
from R ⊆ S ∪ T have "R - S ⊆ T"
by fast
from y ∈ R - S open (R - S) R - S ⊆ T interior T = {} show False
unfolding interior_def by fast
qed
qed
qed

lemma interior_Times: "interior (A × B) = interior A × interior B"
proof (rule interior_unique)
show "interior A × interior B ⊆ A × B"
by (intro Sigma_mono interior_subset)
show "open (interior A × interior B)"
by (intro open_Times open_interior)
fix T
assume "T ⊆ A × B" and "open T"
then show "T ⊆ interior A × interior B"
proof safe
fix x y
assume "(x, y) ∈ T"
then obtain C D where "open C" "open D" "C × D ⊆ T" "x ∈ C" "y ∈ D"
using open T unfolding open_prod_def by fast
then have "open C" "open D" "C ⊆ A" "D ⊆ B" "x ∈ C" "y ∈ D"
using T ⊆ A × B by auto
then show "x ∈ interior A" and "y ∈ interior B"
by (auto intro: interiorI)
qed
qed

subsection {* Closure of a Set *}

definition "closure S = S ∪ {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 ⊆ 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 <-> closed S"
unfolding closure_hull using closed_Inter by (rule hull_eq)

lemma closure_closed [simp]: "closed S ==> 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 ⊆ T ==> closure S ⊆ closure T"
unfolding closure_hull by (rule hull_mono)

lemma closure_minimal: "S ⊆ T ==> closed T ==> closure S ⊆ T"
unfolding closure_hull by (rule hull_minimal)

lemma closure_unique:
assumes "S ⊆ T"
and "closed T"
and "!!T'. S ⊆ T' ==> closed T' ==> T ⊆ 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 ∪ T) = closure S ∪ closure T"
unfolding closure_interior by simp

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

lemma closure_subset_eq: "closure S ⊆ S <-> closed S"
using closure_eq[of S] closure_subset[of S]
by simp

lemma open_inter_closure_eq_empty:
"open S ==> (S ∩ closure T) = {} <-> S ∩ 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 ==> (S ∩ (closure T)) ⊆ closure(S ∩ T)"
proof
fix x
assume as: "open S" "x ∈ S ∩ closure T"
{
assume *: "x islimpt T"
have "x islimpt (S ∩ T)"
proof (rule islimptI)
fix A
assume "x ∈ A" "open A"
with as have "x ∈ A ∩ S" "open (A ∩ S)"
by (simp_all add: open_Int)
with * obtain y where "y ∈ T" "y ∈ A ∩ S" "y ≠ x"
by (rule islimptE)
then have "y ∈ S ∩ T" "y ∈ A ∧ y ≠ x"
by simp_all
then show "∃y∈(S ∩ T). y ∈ A ∧ y ≠ x" ..
qed
}
then show "x ∈ closure (S ∩ 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 × B) = closure A × closure B"
proof (rule closure_unique)
show "A × B ⊆ closure A × closure B"
by (intro Sigma_mono closure_subset)
show "closed (closure A × closure B)"
by (intro closed_Times closed_closure)
fix T
assume "A × B ⊆ T" and "closed T"
then show "closure A × closure B ⊆ 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 (simp add: closure_interior interior_def)
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)"
by (simp add: frontier_def closed_Diff)

lemma frontier_closures: "frontier S = (closure S) ∩ (closure(- S))"
by (auto simp add: frontier_def interior_closure)

fixes a :: "'a::metric_space"
shows "a ∈ frontier S <-> (∀e>0. (∃x∈S. dist a x < e) ∧ (∃x. x ∉ S ∧ 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 ==> frontier S ⊆ S"
by (metis frontier_def closure_closed Diff_subset)

lemma frontier_empty[simp]: "frontier {} = {}"
by (simp add: frontier_def)

lemma frontier_subset_eq: "frontier S ⊆ S <-> closed S"
proof -
{
assume "frontier S ⊆ S"
then have "closure S ⊆ S"
using interior_subset unfolding frontier_def by auto
then have "closed S"
using closure_subset_eq by auto
}
then show ?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 ∩ S = {} <-> 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 => 'a => 'a filter"
(infixr "indirection" 70)
where "a indirection v = at a within {b. ∃c≥0. b - a = scaleR c v}"

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

lemma trivial_limit_within: "trivial_limit (at a within S) <-> ¬ a islimpt S"
proof
assume "trivial_limit (at a within S)"
then show "¬ a islimpt S"
unfolding trivial_limit_def
unfolding eventually_at_topological
unfolding islimpt_def
apply (clarsimp simp add: set_eq_iff)
apply (rename_tac T, rule_tac x=T in exI)
apply (clarsimp, drule_tac x=y in bspec, simp_all)
done
next
assume "¬ a islimpt S"
then show "trivial_limit (at a within S)"
unfolding trivial_limit_def eventually_at_topological islimpt_def
by metis
qed

lemma trivial_limit_at_iff: "trivial_limit (at a) <-> ¬ a islimpt UNIV"
using trivial_limit_within [of a UNIV] by simp

lemma trivial_limit_at:
fixes a :: "'a::perfect_space"
shows "¬ trivial_limit (at a)"
by (rule at_neq_bot)

lemma trivial_limit_at_infinity:
"¬ trivial_limit (at_infinity :: ('a::{real_normed_vector,perfect_space}) filter)"
unfolding trivial_limit_def eventually_at_infinity
apply clarsimp
apply (subgoal_tac "∃x::'a. x ≠ 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) <-> (∃d>0. ∀x. 0 < dist x a ∧ dist x a < d --> P x)"
unfolding eventually_at dist_nz by auto

lemma eventually_happens: "eventually P net ==> trivial_limit net ∨ (∃x. P x)"
unfolding trivial_limit_def
by (auto elim: eventually_rev_mp)

lemma trivial_limit_eventually: "trivial_limit net ==> eventually P net"
by simp

lemma trivial_limit_eq: "trivial_limit net <-> (∀P. eventually P net)"
by (simp add: filter_eq_iff)

text{* Combining theorems for "eventually" *}

lemma eventually_rev_mono:
"eventually P net ==> (∀x. P x --> Q x) ==> eventually Q net"
using eventually_mono [of P Q] by fast

lemma not_eventually: "(∀x. ¬ P x ) ==> ¬ trivial_limit net ==> ¬ eventually (λx. P x) net"
by (simp add: eventually_False)

subsection {* Limits *}

lemma Lim:
"(f ---> l) net <->
trivial_limit net ∨
(∀e>0. eventually (λx. 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) <->
(∀e>0. ∃d>0. ∀x∈S. 0 < dist x a ∧ dist x a ≤ d --> 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) <->
(∀e >0. ∃d>0. ∀x ∈ S. 0 < dist x a ∧ dist x a  < d --> dist (f x) l < e)"
by (auto simp add: tendsto_iff eventually_at dist_nz)

lemma Lim_at: "(f ---> l) (at a) <->
(∀e >0. ∃d>0. ∀x. 0 < dist x a ∧ dist x a < d  --> dist (f x) l < e)"
by (auto simp add: tendsto_iff eventually_at2)

lemma Lim_at_infinity:
"(f ---> l) at_infinity <-> (∀e>0. ∃b. ∀x. norm x ≥ b --> dist (f x) l < e)"
by (auto simp add: tendsto_iff eventually_at_infinity)

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

text{* The expected monotonicity property. *}

lemma Lim_Un:
assumes "(f ---> l) (at x within S)" "(f ---> l) (at x within T)"
shows "(f ---> l) (at x within (S ∪ T))"
using assms unfolding at_within_union by (rule filterlim_sup)

lemma Lim_Un_univ:
"(f ---> l) (at x within S) ==> (f ---> l) (at x within T) ==>
S ∪ 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) ==> (f ---> l) (at x within S)"
by (metis order_refl filterlim_mono subset_UNIV at_le)

lemma eventually_within_interior:
assumes "x ∈ interior S"
shows "eventually P (at x within S) <-> eventually P (at x)"
(is "?lhs = ?rhs")
proof
from assms obtain T where T: "open T" "x ∈ T" "T ⊆ S" ..
{
assume "?lhs"
then obtain A where "open A" and "x ∈ A" and "∀y∈A. y ≠ x --> y ∈ S --> P y"
unfolding eventually_at_topological
by auto
with T have "open (A ∩ T)" and "x ∈ A ∩ T" and "∀y ∈ A ∩ T. y ≠ x --> P y"
by auto
then show "?rhs"
unfolding eventually_at_topological by auto
next
assume "?rhs"
then show "?lhs"
by (auto elim: eventually_elim1 simp: eventually_at_filter)
}
qed

lemma at_within_interior:
"x ∈ interior S ==> at x within S = at x"
unfolding filter_eq_iff by (intro allI eventually_within_interior)

lemma Lim_within_LIMSEQ:
fixes a :: "'a::first_countable_topology"
assumes "∀S. (∀n. S n ≠ a ∧ S n ∈ T) ∧ S ----> a --> (λn. X (S n)) ----> L"
shows "(X ---> L) (at a within T)"
using assms unfolding tendsto_def [where l=L]
by (simp add: sequentially_imp_eventually_within)

lemma Lim_right_bound:
fixes f :: "'a :: {linorder_topology, conditionally_complete_linorder, no_top} =>
'b::{linorder_topology, conditionally_complete_linorder}"
assumes mono: "!!a b. a ∈ I ==> b ∈ I ==> x < a ==> a ≤ b ==> f a ≤ f b"
and bnd: "!!a. a ∈ I ==> x < a ==> K ≤ f a"
shows "(f ---> Inf (f  ({x<..} ∩ I))) (at x within ({x<..} ∩ I))"
proof (cases "{x<..} ∩ I = {}")
case True
then show ?thesis by simp
next
case False
show ?thesis
proof (rule order_tendstoI)
fix a
assume a: "a < Inf (f  ({x<..} ∩ I))"
{
fix y
assume "y ∈ {x<..} ∩ I"
with False bnd have "Inf (f  ({x<..} ∩ I)) ≤ f y"
by (auto intro!: cInf_lower bdd_belowI2 simp del: Inf_image_eq)
with a have "a < f y"
by (blast intro: less_le_trans)
}
then show "eventually (λx. a < f x) (at x within ({x<..} ∩ I))"
by (auto simp: eventually_at_filter intro: exI[of _ 1] zero_less_one)
next
fix a
assume "Inf (f  ({x<..} ∩ I)) < a"
from cInf_lessD[OF _ this] False obtain y where y: "x < y" "y ∈ I" "f y < a"
by auto
then have "eventually (λx. x ∈ I --> f x < a) (at_right x)"
unfolding eventually_at_right[OF x < y] by (metis less_imp_le le_less_trans mono)
then show "eventually (λx. f x < a) (at x within ({x<..} ∩ 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 <-> (∃f. (∀n::nat. f n ∈ S - {x}) ∧ (f ---> x) sequentially)"
(is "?lhs = ?rhs")
proof
assume ?lhs
from countable_basis_at_decseq[of x] obtain A where A:
"!!i. open (A i)"
"!!i. x ∈ A i"
"!!S. open S ==> x ∈ S ==> eventually (λi. A i ⊆ S) sequentially"
by blast
def f ≡ "λn. SOME y. y ∈ S ∧ y ∈ A n ∧ x ≠ y"
{
fix n
from ?lhs have "∃y. y ∈ S ∧ y ∈ A n ∧ x ≠ y"
unfolding islimpt_def using A(1,2)[of n] by auto
then have "f n ∈ S ∧ f n ∈ A n ∧ x ≠ f n"
unfolding f_def by (rule someI_ex)
then have "f n ∈ S" "f n ∈ A n" "x ≠ f n" by auto
}
then have "∀n. f n ∈ S - {x}" by auto
moreover have "(λn. f n) ----> x"
proof (rule topological_tendstoI)
fix S
assume "open S" "x ∈ S"
from A(3)[OF this] !!n. f n ∈ A n
show "eventually (λx. f x ∈ S) sequentially"
by (auto elim!: eventually_elim1)
qed
ultimately show ?rhs by fast
next
assume ?rhs
then obtain f :: "nat => 'a" where f: "!!n. f n ∈ S - {x}" and lim: "f ----> x"
by auto
show ?lhs
unfolding islimpt_def
proof safe
fix T
assume "open T" "x ∈ T"
from lim[THEN topological_tendstoD, OF this] f
show "∃y∈S. y ∈ T ∧ y ≠ x"
unfolding eventually_sequentially by auto
qed
qed

lemma Lim_null:
fixes f :: "'a => 'b::real_normed_vector"
shows "(f ---> l) net <-> ((λx. f(x) - l) ---> 0) net"
by (simp add: Lim dist_norm)

lemma Lim_null_comparison:
fixes f :: "'a => 'b::real_normed_vector"
assumes "eventually (λx. norm (f x) ≤ g x) net" "(g ---> 0) net"
shows "(f ---> 0) net"
using assms(2)
proof (rule metric_tendsto_imp_tendsto)
show "eventually (λx. dist (f x) 0 ≤ dist (g x) 0) net"
using assms(1) by (rule eventually_elim1) (simp add: dist_norm)
qed

lemma Lim_transform_bound:
fixes f :: "'a => 'b::real_normed_vector"
and g :: "'a => 'c::real_normed_vector"
assumes "eventually (λn. norm (f n) ≤ norm (g n)) net"
and "(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"
and "eventually (λx. f(x) ∈ S) net"
and "¬ trivial_limit net" "(f ---> l) net"
shows "l ∈ S"
proof (rule ccontr)
assume "l ∉ S"
with closed S have "open (- S)" "l ∈ - S"
by (simp_all add: open_Compl)
with assms(4) have "eventually (λx. f x ∈ - S) net"
by (rule topological_tendstoD)
with assms(2) have "eventually (λx. False) net"
by (rule eventually_elim2) simp
with assms(3) show "False"
by (simp add: eventually_False)
qed

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

lemma Lim_dist_ubound:
assumes "¬(trivial_limit net)"
and "(f ---> l) net"
and "eventually (λx. dist a (f x) ≤ e) net"
shows "dist a l ≤ e"
using assms by (fast intro: tendsto_le tendsto_intros)

lemma Lim_norm_ubound:
fixes f :: "'a => 'b::real_normed_vector"
assumes "¬(trivial_limit net)" "(f ---> l) net" "eventually (λx. norm(f x) ≤ e) net"
shows "norm(l) ≤ e"
using assms by (fast intro: tendsto_le tendsto_intros)

lemma Lim_norm_lbound:
fixes f :: "'a => 'b::real_normed_vector"
assumes "¬ trivial_limit net"
and "(f ---> l) net"
and "eventually (λx. e ≤ norm (f x)) net"
shows "e ≤ norm l"
using assms by (fast intro: tendsto_le tendsto_intros)

text{* Limit under bilinear function *}

lemma Lim_bilinear:
assumes "(f ---> l) net"
and "(g ---> m) net"
and "bounded_bilinear h"
shows "((λx. 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"
and l :: "'b::topological_space"
shows "(f ---> l) (at a) <-> ((λx. f(a + x)) ---> l) (at 0)"
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 => 'a"
where "netlimit F ≡ Lim F (λx. x)"

lemma netlimit_within: "¬ trivial_limit (at a within S) ==> 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 ∈ interior S ==> (f ---> l) (at x within S) <-> (f ---> l) (at x)"
by (metis at_within_interior)

lemma netlimit_within_interior:
fixes x :: "'a::{t2_space,perfect_space}"
assumes "x ∈ 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 => 'b::real_normed_vector"
assumes "((λx. 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 (λx. f x = g x) net ==> (f ---> l) net ==> (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 "∀x'∈S. 0 < dist x' x ∧ dist x' x < d --> 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 (λx. 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 "∀x'. 0 < dist x' x ∧ dist x' x < d --> f x' = g x'"
and "(f ---> l) (at x)"
shows "(g ---> l) (at x)"
using _ assms(3)
proof (rule Lim_transform_eventually)
show "eventually (λx. f x = g x) (at x)"
unfolding eventually_at2
using assms(1,2) by auto
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 ≠ b"
and "∀x∈S. x ≠ a ∧ x ≠ b --> 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 (λx. 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≠b"
and fg: "∀x. x ≠ a ∧ x ≠ b --> 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 ∈ S"
and "∀x∈S. x ≠ a --> f x = g x"
and "(f ---> l) (at a)"
shows "(g ---> l) (at a)"
proof (rule Lim_transform_eventually)
show "eventually (λx. 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"
and "x = y"
and "S = T"
and "!!x. x ≠ b ==> x ∈ T ==> f x = g x"
shows "(f ---> x) (at a within S) <-> (g ---> y) (at b within T)"
unfolding tendsto_def eventually_at_topological
using assms by simp

assumes "a = b" "x = y"
and "!!x. x ≠ a ==> f x = g x"
shows "((λx. f x) ---> x) (at a) <-> ((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 ∈ closure S <-> (∃x. (∀n. x n ∈ S) ∧ (x ---> l) sequentially)"
(is "?lhs = ?rhs")
proof
assume "?lhs"
moreover
{
assume "l ∈ S"
then have "?rhs" using tendsto_const[of l sequentially] by auto
}
moreover
{
assume "l islimpt S"
then have "?rhs" unfolding islimpt_sequential by auto
}
ultimately show "?rhs"
unfolding closure_def by auto
next
assume "?rhs"
then show "?lhs" unfolding closure_def islimpt_sequential by auto
qed

lemma closed_sequential_limits:
fixes S :: "'a::first_countable_topology set"
shows "closed S <-> (∀x l. (∀n. x n ∈ S) ∧ (x ---> l) sequentially --> l ∈ S)"
by (metis closure_sequential closure_subset_eq subset_iff)

lemma closure_approachable:
fixes S :: "'a::metric_space set"
shows "x ∈ closure S <-> (∀e>0. ∃y∈S. dist y x < e)"
apply (auto simp add: closure_def islimpt_approachable)
apply (metis dist_self)
done

lemma closed_approachable:
fixes S :: "'a::metric_space set"
shows "closed S ==> (∀e>0. ∃y∈S. dist y x < e) <-> x ∈ S"
by (metis closure_closed closure_approachable)

lemma closure_contains_Inf:
fixes S :: "real set"
assumes "S ≠ {}" "bdd_below S"
shows "Inf S ∈ closure S"
proof -
have *: "∀x∈S. Inf S ≤ x"
using cInf_lower[of _ S] assms by metis
{
fix e :: real
assume "e > 0"
then have "Inf S < Inf S + e" by simp
with assms obtain x where "x ∈ S" "x < Inf S + e"
by (subst (asm) cInf_less_iff) auto
with * have "∃x∈S. dist x (Inf S) < e"
by (intro bexI[of _ x]) (auto simp add: dist_real_def)
}
then show ?thesis unfolding closure_approachable by auto
qed

lemma closed_contains_Inf:
fixes S :: "real set"
shows "S ≠ {} ==> bdd_below S ==> closed S ==> Inf S ∈ S"
by (metis closure_contains_Inf closure_closed assms)

lemma not_trivial_limit_within_ball:
"¬ trivial_limit (at x within S) <-> (∀e>0. S ∩ ball x e - {x} ≠ {})"
(is "?lhs = ?rhs")
proof -
{
assume "?lhs"
{
fix e :: real
assume "e > 0"
then obtain y where "y ∈ S - {x}" and "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 ∩ ball x e - {x}"
unfolding ball_def by (simp add: dist_commute)
then have "S ∩ 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 ∩ ball x e - {x}"
using ?rhs by blast
then have "y ∈ S - {x}" and "dist y x < e"
unfolding ball_def by (simp_all add: dist_commute)
then have "∃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 a:A. dist x a)"

lemma bdd_below_infdist[intro, simp]: "bdd_below (dist xA)"
by (auto intro!: zero_le_dist)

lemma infdist_notempty: "A ≠ {} ==> infdist x A = (INF a:A. dist x a)"
by (simp add: infdist_def)

lemma infdist_nonneg: "0 ≤ infdist x A"
by (auto simp add: infdist_def intro: cINF_greatest)

lemma infdist_le: "a ∈ A ==> infdist x A ≤ dist x a"
by (auto intro: cINF_lower simp add: infdist_def)

lemma infdist_le2: "a ∈ A ==> dist x a ≤ d ==> infdist x A ≤ d"
by (auto intro!: cINF_lower2 simp add: infdist_def)

lemma infdist_zero[simp]: "a ∈ A ==> infdist a A = 0"
by (auto intro!: antisym infdist_nonneg infdist_le2)

lemma infdist_triangle: "infdist x A ≤ infdist y A + dist x y"
proof (cases "A = {}")
case True
then show ?thesis by (simp add: infdist_def)
next
case False
then obtain a where "a ∈ A" by auto
have "infdist x A ≤ Inf {dist x y + dist y a |a. a ∈ A}"
proof (rule cInf_greatest)
from A ≠ {} show "{dist x y + dist y a |a. a ∈ A} ≠ {}"
by simp
fix d
assume "d ∈ {dist x y + dist y a |a. a ∈ A}"
then obtain a where d: "d = dist x y + dist y a" "a ∈ A"
by auto
show "infdist x A ≤ d"
unfolding infdist_notempty[OF A ≠ {}]
proof (rule cINF_lower2)
show "a ∈ A" by fact
show "dist x a ≤ d"
unfolding d by (rule dist_triangle)
qed simp
qed
also have "… = dist x y + infdist y A"
proof (rule cInf_eq, safe)
fix a
assume "a ∈ A"
then show "dist x y + infdist y A ≤ dist x y + dist y a"
by (auto intro: infdist_le)
next
fix i
assume inf: "!!d. d ∈ {dist x y + dist y a |a. a ∈ A} ==> i ≤ d"
then have "i - dist x y ≤ infdist y A"
unfolding infdist_notempty[OF A ≠ {}] using a ∈ A
by (intro cINF_greatest) (auto simp: field_simps)
then show "i ≤ dist x y + infdist y A"
by simp
qed
finally show ?thesis by simp
qed

lemma in_closure_iff_infdist_zero:
assumes "A ≠ {}"
shows "x ∈ closure A <-> infdist x A = 0"
proof
assume "x ∈ closure A"
show "infdist x A = 0"
proof (rule ccontr)
assume "infdist x A ≠ 0"
with infdist_nonneg[of x A] have "infdist x A > 0"
by auto
then have "ball x (infdist x A) ∩ closure A = {}"
apply auto
apply (metis x ∈ closure A closure_approachable dist_commute infdist_le not_less)
done
then have "x ∉ closure A"
by (metis 0 < infdist x A centre_in_ball disjoint_iff_not_equal)
then show False using x ∈ closure A by simp
qed
next
assume x: "infdist x A = 0"
then obtain a where "a ∈ A"
by atomize_elim (metis all_not_in_conv assms)
show "x ∈ closure A"
unfolding closure_approachable
apply safe
proof (rule ccontr)
fix e :: real
assume "e > 0"
assume "¬ (∃y∈A. dist y x < e)"
then have "infdist x A ≥ e" using a ∈ A
unfolding infdist_def
by (force simp: dist_commute intro: cINF_greatest)
with x e > 0 show False by auto
qed
qed

lemma in_closed_iff_infdist_zero:
assumes "closed A" "A ≠ {}"
shows "x ∈ A <-> infdist x A = 0"
proof -
have "x ∈ closure A <-> 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 "((λx. infdist (f x) A) ---> infdist l A) F"
proof (rule tendstoI)
fix e ::real
assume "e > 0"
from tendstoD[OF f this]
show "eventually (λx. 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) ≤ dist (f x) l"
by (simp add: dist_commute dist_real_def)
also assume "dist (f x) l < e"
finally show "dist (infdist (f x) A) (infdist l A) < e" .
qed
qed

text{* Some other lemmas about sequences. *}

lemma sequentially_offset: (* TODO: move to Topological_Spaces.thy *)
assumes "eventually (λi. P i) sequentially"
shows "eventually (λi. P (i + k)) sequentially"
using assms by (rule eventually_sequentially_seg [THEN iffD2])

lemma seq_offset_neg: (* TODO: move to Topological_Spaces.thy *)
"(f ---> l) sequentially ==> ((λi. f(i - k)) ---> l) sequentially"
apply (erule filterlim_compose)
apply (simp add: filterlim_def le_sequentially eventually_filtermap eventually_sequentially)
apply arith
done

lemma seq_harmonic: "((λn. inverse (real n)) ---> 0) sequentially"
using LIMSEQ_inverse_real_of_nat by (rule LIMSEQ_imp_Suc) (* TODO: move to Limits.thy *)

subsection {* More properties of closed balls *}

lemma closed_vimage: (* TODO: move to Topological_Spaces.thy *)
assumes "closed s" and "continuous_on UNIV f"
shows "closed (vimage f s)"
using assms unfolding continuous_on_closed_vimage [OF closed_UNIV]
by simp

lemma closed_cball: "closed (cball x e)"
proof -
have "closed (dist x - {..e})"
by (intro closed_vimage closed_atMost continuous_intros)
also have "dist x - {..e} = cball x e"
by auto
finally show ?thesis .
qed

lemma open_contains_cball: "open S <-> (∀x∈S. ∃e>0.  cball x e ⊆ S)"
proof -
{
fix x and e::real
assume "x∈S" "e>0" "ball x e ⊆ S"
then have "∃d>0. cball x d ⊆ S" unfolding subset_eq by (rule_tac x="e/2" in exI, auto)
}
moreover
{
fix x and e::real
assume "x∈S" "e>0" "cball x e ⊆ S"
then have "∃d>0. ball x d ⊆ S"
unfolding subset_eq
apply(rule_tac x="e/2" in exI)
apply auto
done
}
ultimately show ?thesis
unfolding open_contains_ball by auto
qed

lemma open_contains_cball_eq: "open S ==> (∀x. x ∈ S <-> (∃e>0. cball x e ⊆ S))"
by (metis open_contains_cball subset_eq order_less_imp_le centre_in_cball)

lemma mem_interior_cball: "x ∈ interior S <-> (∃e>0. cball x e ⊆ S)"
apply (simp add: interior_def, safe)
apply (force simp add: open_contains_cball)
apply (rule_tac x="ball x e" in exI)
apply (simp add: subset_trans [OF ball_subset_cball])
done

lemma islimpt_ball:
fixes x y :: "'a::{real_normed_vector,perfect_space}"
shows "y islimpt ball x e <-> 0 < e ∧ y ∈ cball x e"
(is "?lhs = ?rhs")
proof
assume "?lhs"
{
assume "e ≤ 0"
then have *:"ball x e = {}"
using ball_eq_empty[of x e] by auto
have False using ?lhs
unfolding * using islimpt_EMPTY[of y] by auto
}
then have "e > 0" by (metis not_less)
moreover
have "y ∈ 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"
then have "e > 0" by auto
{
fix d :: real
assume "d > 0"
have "∃x'∈ball x e. x' ≠ y ∧ dist x' y < d"
proof (cases "d ≤ dist x y")
case True
then show "∃x'∈ball x e. x' ≠ y ∧ dist x' y < d"
proof (cases "x = y")
case True
then have False
using d ≤ dist x y d>0 by auto
then show "∃x'∈ball x e. x' ≠ y ∧ dist x' y < d"
by auto
next
case False
have "dist x (y - (d / (2 * dist y x)) *⇩R (y - x)) =
norm (x - y + (d / (2 * norm (y - x))) *⇩R (y - x))"
unfolding mem_cball mem_ball dist_norm diff_diff_eq2 diff_add_eq[symmetric]
by auto
also have "… = ¦- 1 + d / (2 * norm (x - y))¦ * norm (x - y)"
using scaleR_left_distrib[of "- 1" "d / (2 * norm (y - x))", symmetric, of "y - x"]
unfolding scaleR_minus_left scaleR_one
by (auto simp add: norm_minus_commute)
also have "… = ¦- norm (x - y) + d / 2¦"
unfolding abs_mult_pos[of "norm (x - y)", OF norm_ge_zero[of "x - y"]]
unfolding distrib_right using x≠y[unfolded dist_nz, unfolded dist_norm]
by auto
also have "… ≤ e - d/2" using d ≤ dist x y and d>0 and ?rhs
by (auto simp add: dist_norm)
finally have "y - (d / (2 * dist y x)) *⇩R (y - x) ∈ ball x e" using d>0
by auto
moreover
have "(d / (2*dist y x)) *⇩R (y - x) ≠ 0"
using x≠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)) *⇩R (y - x)) y < d"
unfolding dist_norm
apply simp
unfolding norm_minus_cancel
using d > 0 x≠y[unfolded dist_nz] dist_commute[of x y]
unfolding dist_norm
apply auto
done
ultimately show "∃x'∈ball x e. x' ≠ y ∧ dist x' y < d"
apply (rule_tac x = "y - (d / (2*dist y x)) *⇩R (y - x)" in bexI)
apply auto
done
qed
next
case False
then have "d > dist x y" by auto
show "∃x' ∈ ball x e. x' ≠ y ∧ dist x' y < d"
proof (cases "x = y")
case True
obtain z where **: "z ≠ y" "dist z y < min e d"
using perfect_choose_dist[of "min e d" y]
using d > 0 e>0 by auto
show "∃x'∈ball x e. x' ≠ y ∧ dist x' y < d"
unfolding x = y
using z ≠ y **
apply (rule_tac x=z in bexI)
apply (auto simp add: dist_commute)
done
next
case False
then show "∃x'∈ball x e. x' ≠ y ∧ dist x' y < d"
using d>0 d > dist x y ?rhs
apply (rule_tac x=x in bexI)
apply auto
done
qed
qed
}
then show "?lhs"
unfolding mem_cball islimpt_approachable mem_ball by auto
qed

lemma closure_ball_lemma:
fixes x y :: "'a::real_normed_vector"
assumes "x ≠ y"
shows "y islimpt ball x (dist x y)"
proof (rule islimptI)
fix T
assume "y ∈ T" "open T"
then obtain r where "0 < r" "∀z. dist z y < r --> z ∈ T"
unfolding open_dist by fast
(* choose point between x and y, within distance r of y. *)
def k ≡ "min 1 (r / (2 * dist x y))"
def z ≡ "y + scaleR k (x - y)"
have z_def2: "z = x + scaleR (1 - k) (y - x)"
unfolding z_def by (simp add: algebra_simps)
have "dist z y < r"
unfolding z_def k_def using 0 < r
by (simp add: dist_norm min_def)
then have "z ∈ T"
using ∀z. dist z y < r --> z ∈ T by simp
have "dist x z < dist x y"
unfolding z_def2 dist_norm
apply (simp add: norm_minus_commute)
apply (simp only: dist_norm [symmetric])
apply (subgoal_tac "¦1 - k¦ * dist x y < 1 * dist x y", simp)
apply (rule mult_strict_right_mono)
apply (simp add: k_def zero_less_dist_iff 0 < r x ≠ y)
apply (simp add: zero_less_dist_iff x ≠ y)
done
then have "z ∈ ball x (dist x y)"
by simp
have "z ≠ y"
unfolding z_def k_def using x ≠ y 0 < r
by (simp add: min_def)
show "∃z∈ball x (dist x y). z ∈ T ∧ z ≠ y"
using z ∈ ball x (dist x y) z ∈ T z ≠ y
by fast
qed

lemma closure_ball:
fixes x :: "'a::real_normed_vector"
shows "0 < e ==> closure (ball x e) = cball x e"
apply (rule equalityI)
apply (rule closure_minimal)
apply (rule ball_subset_cball)
apply (rule closed_cball)
apply (rule subsetI, rename_tac y)
apply (simp add: le_less [where 'a=real])
apply (erule disjE)
apply (rule subsetD [OF closure_subset], simp)
apply (simp add: closure_def)
apply clarify
apply (rule closure_ball_lemma)
apply (simp add: zero_less_dist_iff)
done

(* In a trivial vector space, this fails for e = 0. *)
lemma interior_cball:
fixes x :: "'a::{real_normed_vector, perfect_space}"
shows "interior (cball x e) = ball x e"
proof (cases "e ≥ 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 ∈ cball x e"
then have False
unfolding mem_cball using dist_nz[of x y] cs by auto
}
then have "cball x e = {}" by auto
then have "interior (cball x e) = {}"
using interior_empty by auto
ultimately show ?thesis by blast
next
case True note cs = this
have "ball x e ⊆ cball x e"
using ball_subset_cball by auto
moreover
{
fix S y
assume as: "S ⊆ cball x e" "open S" "y∈S"
then obtain d where "d>0" and d: "∀x'. dist x' y < d --> x' ∈ S"
unfolding open_dist by blast
then obtain xa where xa_y: "xa ≠ y" and xa: "dist xa y < d"
using perfect_choose_dist [of d] by auto
have "xa ∈ S"
using d[THEN spec[where x = xa]]
using xa by (auto simp add: dist_commute)
then have xa_cball: "xa ∈ cball x e"
using as(1) by auto
then have "y ∈ ball x e"
proof (cases "x = y")
case True
then have "e > 0"
using xa_y[unfolded dist_nz] xa_cball[unfolded mem_cball]
by (auto simp add: dist_commute)
then show "y ∈ ball x e"
using x = y  by simp
next
case False
have "dist (y + (d / 2 / dist y x) *⇩R (y - x)) y < d"
unfolding dist_norm
using d>0 norm_ge_zero[of "y - x"] x ≠ y by auto
then have *: "y + (d / 2 / dist y x) *⇩R (y - x) ∈ cball x e"
using d as(1)[unfolded subset_eq] by blast
have "y - x ≠ 0" using x ≠ y by auto
hence **:"d / (2 * norm (y - x)) > 0"
unfolding zero_less_norm_iff[symmetric] using d>0 by auto
have "dist (y + (d / 2 / dist y x) *⇩R (y - x)) x =
norm (y + (d / (2 * norm (y - x))) *⇩R y - (d / (2 * norm (y - x))) *⇩R x - x)"
by (auto simp add: dist_norm algebra_simps)
also have "… = norm ((1 + d / (2 * norm (y - x))) *⇩R (y - x))"
by (auto simp add: algebra_simps)
also have "… = ¦1 + d / (2 * norm (y - x))¦ * norm (y - x)"
using ** by auto
also have "… = (dist y x) + d/2"
using ** by (auto simp add: distrib_right dist_norm)
finally have "e ≥ dist x y +d/2"
using *[unfolded mem_cball] by (auto simp add: dist_commute)
then show "y ∈ ball x e"
unfolding mem_ball using d>0 by auto
qed
}
then have "∀S ⊆ cball x e. open S --> S ⊆ 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)
apply (simp add: set_eq_iff)
apply arith
done

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)
apply (simp add: set_eq_iff)
apply arith
done

lemma cball_eq_empty: "cball x e = {} <-> e < 0"
apply (simp add: set_eq_iff not_le)
apply (metis zero_le_dist dist_self order_less_le_trans)
done

lemma cball_empty: "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} <-> e = 0"
proof (rule linorder_cases)
assume e: "0 < e"
obtain a where "a ≠ x" "dist a x < e"
using perfect_choose_dist [OF e] by auto
then have "a ≠ x" "dist x a ≤ 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}"
by (auto simp add: set_eq_iff)

subsection {* Boundedness *}

(* FIXME: This has to be unified with BSEQ!! *)
definition (in metric_space) bounded :: "'a set => bool"
where "bounded S <-> (∃x e. ∀y∈S. dist x y ≤ e)"

lemma bounded_subset_cball: "bounded S <-> (∃e x. S ⊆ cball x e)"
unfolding bounded_def subset_eq by auto

lemma bounded_any_center: "bounded S <-> (∃e. ∀y∈S. dist a y ≤ e)"
unfolding bounded_def
by auto (metis add.commute add_le_cancel_right dist_commute dist_triangle_le)

lemma bounded_iff: "bounded S <-> (∃a. ∀x∈S. norm x ≤ a)"
unfolding bounded_any_center [where a=0]
by (simp add: dist_norm)

lemma bounded_realI:
assumes "∀x∈s. abs (x::real) ≤ B"
shows "bounded s"
unfolding bounded_def dist_real_def
by (metis abs_minus_commute assms diff_0_right)

lemma bounded_empty [simp]: "bounded {}"
by (simp add: bounded_def)

lemma bounded_subset: "bounded T ==> S ⊆ 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: "∀y∈S. dist x y ≤ a"
unfolding bounded_def by auto
{
fix y
assume "y ∈ closure S"
then obtain f where f: "∀n. f n ∈ S"  "(f ---> y) sequentially"
unfolding closure_sequential by auto
have "∀n. f n ∈ S --> dist x (f n) ≤ a" using a by simp
then have "eventually (λn. dist x (f n) ≤ a) sequentially"
by (rule eventually_mono, simp add: f(1))
have "dist x y ≤ a"
apply (rule Lim_dist_ubound [of sequentially f])
apply (rule trivial_limit_sequentially)
apply (rule f(2))
apply fact
done
}
then show ?thesis
unfolding bounded_def by auto
qed

lemma bounded_cball[simp,intro]: "bounded (cball x e)"
apply (simp add: bounded_def)
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 ∪ T) <-> bounded S ∧ bounded T"
apply (auto simp add: bounded_def)
by (metis Un_iff add_le_cancel_left dist_triangle le_max_iff_disj max.order_iff)

lemma bounded_Union[intro]: "finite F ==> ∀S∈F. bounded S ==> bounded (\<Union>F)"
by (induct rule: finite_induct[of F]) auto

lemma bounded_UN [intro]: "finite A ==> ∀x∈A. bounded (B x) ==> bounded (\<Union>x∈A. B x)"
by (induct set: finite) auto

lemma bounded_insert [simp]: "bounded (insert x S) <-> bounded S"
proof -
have "∀y∈{x}. dist x y ≤ 0"
by simp
then have "bounded {x}"
unfolding bounded_def by fast
then show ?thesis
by (metis insert_is_Un bounded_Un)
qed

lemma finite_imp_bounded [intro]: "finite S ==> bounded S"
by (induct set: finite) simp_all

lemma bounded_pos: "bounded S <-> (∃b>0. ∀x∈ S. norm x ≤ b)"
apply (simp add: bounded_iff)
apply (subgoal_tac "!!x (y::real). 0 < 1 + abs y ∧ (x ≤ y --> x ≤ 1 + abs y)")
apply metis
apply arith
done

lemma Bseq_eq_bounded:
fixes f :: "nat => 'a::real_normed_vector"
shows "Bseq f <-> bounded (range f)"
unfolding Bseq_def bounded_pos by auto

lemma bounded_Int[intro]: "bounded S ∨ bounded T ==> bounded (S ∩ T)"
by (metis Int_lower1 Int_lower2 bounded_subset)

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

lemma not_bounded_UNIV[simp, intro]:
"¬ bounded (UNIV :: 'a::{real_normed_vector, perfect_space} set)"
proof (auto simp add: bounded_pos not_le)
obtain x :: 'a where "x ≠ 0"
using perfect_choose_dist [OF zero_less_one] by fast
fix b :: real
assume b: "b >0"
have b1: "b +1 ≥ 0"
using b by simp
with x ≠ 0 have "b < norm (scaleR (b + 1) (sgn x))"
by (simp add: norm_sgn)
then show "∃x::'a. b < norm x" ..
qed

lemma bounded_linear_image:
assumes "bounded S"
and "bounded_linear f"
shows "bounded (f  S)"
proof -
from assms(1) obtain b where b: "b > 0" "∀x∈S. norm x ≤ b"
unfolding bounded_pos by auto
from assms(2) obtain B where B: "B > 0" "∀x. norm (f x) ≤ B * norm x"
using bounded_linear.pos_bounded by (auto simp add: ac_simps)
{
fix x
assume "x ∈ S"
then have "norm x ≤ b"
using b by auto
then have "norm (f x) ≤ B * b"
using B(2)
apply (erule_tac x=x in allE)
apply (metis B(1) B(2) order_trans mult_le_cancel_left_pos)
done
}
then show ?thesis
unfolding bounded_pos
apply (rule_tac x="b*B" in exI)
using b B by (auto simp add: mult.commute)
qed

lemma bounded_scaling:
fixes S :: "'a::real_normed_vector set"
shows "bounded S ==> bounded ((λx. c *⇩R x)  S)"
apply (rule bounded_linear_image)
apply assumption
apply (rule bounded_linear_scaleR_right)
done

lemma bounded_translation:
fixes S :: "'a::real_normed_vector set"
assumes "bounded S"
shows "bounded ((λx. a + x)  S)"
proof -
from assms obtain b where b: "b > 0" "∀x∈S. norm x ≤ b"
unfolding bounded_pos by auto
{
fix x
assume "x ∈ S"
then have "norm (a + x) ≤ b + norm a"
using norm_triangle_ineq[of a x] b by auto
}
then show ?thesis
unfolding bounded_pos
using norm_ge_zero[of a] b(1) and 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: "bounded (S::real set) <-> (∃a. ∀x∈S. ¦x¦ ≤ a)"
by (simp add: bounded_iff)

lemma bounded_imp_bdd_above: "bounded S ==> bdd_above (S :: real set)"
by (auto simp: bounded_def bdd_above_def dist_real_def)
(metis abs_le_D1 abs_minus_commute diff_le_eq)

lemma bounded_imp_bdd_below: "bounded S ==> bdd_below (S :: real set)"
by (auto simp: bounded_def bdd_below_def dist_real_def)
(metis abs_le_D1 add.commute diff_le_eq)

(* TODO: remove the following lemmas about Inf and Sup, is now in conditionally complete lattice *)

lemma bounded_has_Sup:
fixes S :: "real set"
assumes "bounded S"
and "S ≠ {}"
shows "∀x∈S. x ≤ Sup S"
and "∀b. (∀x∈S. x ≤ b) --> Sup S ≤ b"
proof
show "∀b. (∀x∈S. x ≤ b) --> Sup S ≤ b"
using assms by (metis cSup_least)
qed (metis cSup_upper assms(1) bounded_imp_bdd_above)

lemma Sup_insert:
fixes S :: "real set"
shows "bounded S ==> Sup (insert x S) = (if S = {} then x else max x (Sup S))"
by (auto simp: bounded_imp_bdd_above sup_max cSup_insert_If)

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

lemma bounded_has_Inf:
fixes S :: "real set"
assumes "bounded S"
and "S ≠ {}"
shows "∀x∈S. x ≥ Inf S"
and "∀b. (∀x∈S. x ≥ b) --> Inf S ≥ b"
proof
show "∀b. (∀x∈S. x ≥ b) --> Inf S ≥ b"
using assms by (metis cInf_greatest)
qed (metis cInf_lower assms(1) bounded_imp_bdd_below)

lemma Inf_insert:
fixes S :: "real set"
shows "bounded S ==> Inf (insert x S) = (if S = {} then x else min x (Inf S))"
by (auto simp: bounded_imp_bdd_below inf_min cInf_insert_If)

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

subsection {* Compactness *}

subsubsection {* Bolzano-Weierstrass property *}

lemma heine_borel_imp_bolzano_weierstrass:
assumes "compact s"
and "infinite t"
and "t ⊆ s"
shows "∃x ∈ s. x islimpt t"
proof (rule ccontr)
assume "¬ (∃x ∈ s. x islimpt t)"
then obtain f where f: "∀x∈s. x ∈ f x ∧ open (f x) ∧ (∀y∈t. y ∈ f x --> y = x)"
unfolding islimpt_def
using bchoice[of s "λ x T. x ∈ T ∧ open T ∧ (∀y∈t. y ∈ T --> y = x)"]
by auto
obtain g where g: "g ⊆ {t. ∃x. x ∈ s ∧ t = f x}" "finite g" "s ⊆ \<Union>g"
using assms(1)[unfolded compact_eq_heine_borel, THEN spec[where x="{t. ∃x. x∈s ∧ t = f x}"]]
using f by auto
from g(1,3) have g':"∀x∈g. ∃xa ∈ s. x = f xa"
by auto
{
fix x y
assume "x ∈ t" "y ∈ t" "f x = f y"
then have "x ∈ f x"  "y ∈ f x --> y = x"
using f[THEN bspec[where x=x]] and t ⊆ s by auto
then have "x = y"
using f x = f y and f[THEN bspec[where x=y]] and y ∈ t and t ⊆ s
by auto
}
then have "inj_on f t"
unfolding inj_on_def by simp
then have "infinite (f  t)"
using assms(2) using finite_imageD by auto
moreover
{
fix x
assume "x ∈ t" "f x ∉ g"
from g(3) assms(3) x ∈ t obtain h where "h ∈ g" and "x ∈ h"
by auto
then obtain y where "y ∈ s" "h = f y"
using g'[THEN bspec[where x=h]] by auto
then have "y = x"
using f[THEN bspec[where x=y]] and x∈t and x∈h[unfolded h = f y]
by auto
then have False
using f x ∉ g h ∈ g unfolding h = f y
by auto
}
then have "f  t ⊆ 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: "∀U. l∈U --> open U --> infinite (U ∩ range f)"
shows "∃r. subseq r ∧ (f o r) ----> l"
proof -
from countable_basis_at_decseq[of l]
obtain A where A:
"!!i. open (A i)"
"!!i. l ∈ A i"
"!!S. open S ==> l ∈ S ==> eventually (λi. A i ⊆ S) sequentially"
by blast
def s ≡ "λn i. SOME j. i < j ∧ f j ∈ A (Suc n)"
{
fix n i
have "infinite (A (Suc n) ∩ range f - f{.. i})"
using l A by auto
then have "∃x. x ∈ A (Suc n) ∩ range f - f{.. i}"
unfolding ex_in_conv by (intro notI) simp
then have "∃j. f j ∈ A (Suc n) ∧ j ∉ {.. i}"
by auto
then have "∃a. i < a ∧ f a ∈ A (Suc n)"
by (auto simp: not_le)
then have "i < s n i" "f (s n i) ∈ A (Suc n)"
unfolding s_def by (auto intro: someI2_ex)
}
note s = this
def r ≡ "rec_nat (s 0 0) s"
have "subseq r"
by (auto simp: r_def s subseq_Suc_iff)
moreover
have "(λn. f (r n)) ----> l"
proof (rule topological_tendstoI)
fix S
assume "open S" "l ∈ S"
with A(3) have "eventually (λi. A i ⊆ S) sequentially"
by auto
moreover
{
fix i
assume "Suc 0 ≤ i"
then have "f (r i) ∈ A i"
by (cases i) (simp_all add: r_def s)
}
then have "eventually (λi. f (r i) ∈ A i) sequentially"
by (auto simp: eventually_sequentially)
ultimately show "eventually (λi. f (r i) ∈ S) sequentially"
by eventually_elim auto
qed
ultimately show "∃r. subseq r ∧ (f o r) ----> l"
by (auto simp: convergent_def comp_def)
qed

lemma sequence_infinite_lemma:
fixes f :: "nat => 'a::t1_space"
assumes "∀n. f n ≠ l"
and "(f ---> l) sequentially"
shows "infinite (range f)"
proof
assume "finite (range f)"
then have "closed (range f)"
by (rule finite_imp_closed)
then have "open (- range f)"
by (rule open_Compl)
from assms(1) have "l ∈ - range f"
by auto
from assms(2) have "eventually (λn. f n ∈ - range f) sequentially"
using open (- range f) l ∈ - range f
by (rule topological_tendstoD)
then show 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])
apply (simp add: closure_minimal)
done

lemma islimpt_insert:
fixes x :: "'a::t1_space"
shows "x islimpt (insert a s) <-> x islimpt s"
proof
assume *: "x islimpt (insert a s)"
show "x islimpt s"
proof (rule islimptI)
fix t
assume t: "x ∈ t" "open t"
show "∃y∈s. y ∈ t ∧ y ≠ x"
proof (cases "x = a")
case True
obtain y where "y ∈ insert a s" "y ∈ t" "y ≠ x"
using * t by (rule islimptE)
with x = a show ?thesis by auto
next
case False
with t have t': "x ∈ t - {a}" "open (t - {a})"
by (simp_all add: open_Diff)
obtain y where "y ∈ insert a s" "y ∈ t - {a}" "y ≠ x"
using * t' by (rule islimptE)
then show ?thesis by auto
qed
qed
next
assume "x islimpt s"
then show "x islimpt (insert a s)"
by (rule islimpt_subset) auto
qed

lemma islimpt_finite:
fixes x :: "'a::t1_space"
shows "finite s ==> ¬ x islimpt s"
by (induct set: finite) (simp_all add: islimpt_insert)

lemma islimpt_union_finite:
fixes x :: "'a::t1_space"
shows "finite s ==> x islimpt (s ∪ t) <-> x islimpt t"
by (simp add: islimpt_Un islimpt_finite)

lemma islimpt_eq_acc_point:
fixes l :: "'a :: t1_space"
shows "l islimpt S <-> (∀U. l∈U --> open U --> infinite (U ∩ S))"
proof (safe intro!: islimptI)
fix U
assume "l islimpt S" "l ∈ U" "open U" "finite (U ∩ S)"
then have "l islimpt S" "l ∈ (U - (U ∩ S - {l}))" "open (U - (U ∩ S - {l}))"
by (auto intro: finite_imp_closed)
then show False
by (rule islimptE) auto
next
fix T
assume *: "∀U. l∈U --> open U --> infinite (U ∩ S)" "l ∈ T" "open T"
then have "infinite (T ∩ S - {l})"
by auto
then have "∃x. x ∈ (T ∩ S - {l})"
unfolding ex_in_conv by (intro notI) simp
then show "∃y∈S. y ∈ T ∧ y ≠ 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 "∃r. subseq r ∧ (f o r) ----> l"
using l unfolding islimpt_eq_acc_point
by (rule acc_point_range_imp_convergent_subsequence)

lemma sequence_unique_limpt:
fixes f :: "nat => 'a::t2_space"
assumes "(f ---> l) sequentially"
and "l' islimpt (range f)"
shows "l' = l"
proof (rule ccontr)
assume "l' ≠ l"
obtain s t where "open s" "open t" "l' ∈ s" "l ∈ t" "s ∩ t = {}"
using hausdorff [OF l' ≠ l] by auto
have "eventually (λn. f n ∈ t) sequentially"
using assms(1) open t l ∈ t by (rule topological_tendstoD)
then obtain N where "∀n≥N. f n ∈ t"
unfolding eventually_sequentially by auto

have "UNIV = {..<N} ∪ {N..}"
by auto
then have "l' islimpt (f  ({..<N} ∪ {N..}))"
using assms(2) by simp
then have "l' islimpt (f  {..<N} ∪ f  {N..})"
by (simp add: image_Un)
then have "l' islimpt (f  {N..})"
by (simp add: islimpt_union_finite)
then obtain y where "y ∈ f  {N..}" "y ∈ s" "y ≠ l'"
using l' ∈ s open s by (rule islimptE)
then obtain n where "N ≤ n" "f n ∈ s" "f n ≠ l'"
by auto
with ∀n≥N. f n ∈ t have "f n ∈ s ∩ t"
by simp
with s ∩ t = {} show False
by simp
qed

lemma bolzano_weierstrass_imp_closed:
fixes s :: "'a::{first_countable_topology,t2_space} set"
assumes "∀t. infinite t ∧ t ⊆ s --> (∃x ∈ s. x islimpt t)"
shows "closed s"
proof -
{
fix x l
assume as: "∀n::nat. x n ∈ s" "(x ---> l) sequentially"
then have "l ∈ s"
proof (cases "∀n. x n ≠ l")
case False
then show "l∈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'∈s" "l' islimpt (range x)"
using assms[THEN spec[where x="range x"]] as(1) by auto
then show "l∈s" using sequence_unique_limpt[of x l l']
using as cas by auto
qed
}
then show ?thesis
unfolding closed_sequential_limits by fast
qed

lemma compact_imp_bounded:
assumes "compact U"
shows "bounded U"
proof -
have "compact U" "∀x∈U. open (ball x 1)" "U ⊆ (\<Union>x∈U. ball x 1)"
using assms by auto
then obtain D where D: "D ⊆ U" "finite D" "U ⊆ (\<Union>x∈D. ball x 1)"
by (rule compactE_image)
from finite D have "bounded (\<Union>x∈D. ball x 1)"
by (simp add: bounded_UN)
then show "bounded U" using U ⊆ (\<Union>x∈D. ball x 1)
by (rule bounded_subset)
qed

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

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

lemma compact_Union [intro]: "finite S ==> (!!T. T ∈ S ==> compact T) ==> compact (\<Union>S)"
by (induct set: finite) auto

lemma compact_UN [intro]:
"finite A ==> (!!x. x ∈ A ==> compact (B x)) ==> compact (\<Union>x∈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 ∩ t)"
using compact_inter_closed [of t s] assms
by (simp add: Int_commute)

lemma compact_inter [intro]:
fixes s t :: "'a :: t2_space set"
assumes "compact s"
and "compact t"
shows "compact (s ∩ 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} ∪ s)"
using compact_sing assms by (rule compact_union)
then show ?thesis by simp
qed

lemma finite_imp_compact: "finite s ==> compact s"
by (induct set: finite) simp_all

lemma open_delete:
fixes s :: "'a::t1_space set"
shows "open s ==> open (s - {x})"
by (simp add: open_Diff)

text{*Compactness expressed with filters*}

lemma closure_iff_nhds_not_empty:
"x ∈ closure X <-> (∀A. ∀S⊆A. open S --> x ∈ S --> X ∩ A ≠ {})"
proof safe
assume x: "x ∈ closure X"
fix S A
assume "open S" "x ∈ S" "X ∩ A = {}" "S ⊆ A"
then have "x ∉ closure (-S)"
by (auto simp: closure_complement subset_eq[symmetric] intro: interiorI)
with x have "x ∈ closure X - closure (-S)"
by auto
also have "… ⊆ closure (X ∩ S)"
using open S open_inter_closure_subset[of S X] by (simp add: closed_Compl ac_simps)
finally have "X ∩ S ≠ {}" by auto
then show False using X ∩ A = {} S ⊆ A by auto
next
assume "∀A S. S ⊆ A --> open S --> x ∈ S --> X ∩ A ≠ {}"
from this[THEN spec, of "- X", THEN spec, of "- closure X"]
show "x ∈ closure X"
by (simp add: closure_subset open_Compl)
qed

lemma compact_filter:
"compact U <-> (∀F. F ≠ bot --> eventually (λx. x ∈ U) F --> (∃x∈U. inf (nhds x) F ≠ bot))"
proof (intro allI iffI impI compact_fip[THEN iffD2] notI)
fix F
assume "compact U"
assume F: "F ≠ bot" "eventually (λx. x ∈ U) F"
then have "U ≠ {}"
by (auto simp: eventually_False)

def Z ≡ "closure  {A. eventually (λx. x ∈ A) F}"
then have "∀z∈Z. closed z"
by auto
moreover
have ev_Z: "!!z. z ∈ Z ==> eventually (λx. x ∈ z) F"
unfolding Z_def by (auto elim: eventually_elim1 intro: set_mp[OF closure_subset])
have "(∀B ⊆ Z. finite B --> U ∩ \<Inter>B ≠ {})"
proof (intro allI impI)
fix B assume "finite B" "B ⊆ Z"
with finite B ev_Z F(2) have "eventually (λx. x ∈ U ∩ (\<Inter>B)) F"
by (auto simp: eventually_ball_finite_distrib eventually_conj_iff)
with F show "U ∩ \<Inter>B ≠ {}"
by (intro notI) (simp add: eventually_False)
qed
ultimately have "U ∩ \<Inter>Z ≠ {}"
using compact U unfolding compact_fip by blast
then obtain x where "x ∈ U" and x: "!!z. z ∈ Z ==> x ∈ z"
by auto

have "!!P. eventually P (inf (nhds x) F) ==> P ≠ bot"
unfolding eventually_inf eventually_nhds
proof safe
fix P Q R S
assume "eventually R F" "open S" "x ∈ S"
with open_inter_closure_eq_empty[of S "{x. R x}"] x[of "closure {x. R x}"]
have "S ∩ {x. R x} ≠ {}" by (auto simp: Z_def)
moreover assume "Ball S Q" "∀x. Q x ∧ R x --> bot x"
ultimately show False by (auto simp: set_eq_iff)
qed
with x ∈ U show "∃x∈U. inf (nhds x) F ≠ bot"
by (metis eventually_bot)
next
fix A
assume A: "∀a∈A. closed a" "∀B⊆A. finite B --> U ∩ \<Inter>B ≠ {}" "U ∩ \<Inter>A = {}"
def F ≡ "INF a:insert U A. principal a"
have "F ≠ bot"
unfolding F_def
proof (rule INF_filter_not_bot)
fix X assume "X ⊆ insert U A" "finite X"
moreover with A(2)[THEN spec, of "X - {U}"] have "U ∩ \<Inter>(X - {U}) ≠ {}"
by auto
ultimately show "(INF a:X. principal a) ≠ bot"
by (auto simp add: INF_principal_finite principal_eq_bot_iff)
qed
moreover
have "F ≤ principal U"
unfolding F_def by auto
then have "eventually (λx. x ∈ U) F"
by (auto simp: le_filter_def eventually_principal)
moreover
assume "∀F. F ≠ bot --> eventually (λx. x ∈ U) F --> (∃x∈U. inf (nhds x) F ≠ bot)"
ultimately obtain x where "x ∈ U" and x: "inf (nhds x) F ≠ bot"
by auto

{ fix V assume "V ∈ A"
then have "F ≤ principal V"
unfolding F_def by (intro INF_lower2[of V]) auto
then have V: "eventually (λx. x ∈ V) F"
by (auto simp: le_filter_def eventually_principal)
have "x ∈ closure V"
unfolding closure_iff_nhds_not_empty
proof (intro impI allI)
fix S A
assume "open S" "x ∈ S" "S ⊆ A"
then have "eventually (λx. x ∈ A) (nhds x)"
by (auto simp: eventually_nhds)
with V have "eventually (λx. x ∈ V ∩ A) (inf (nhds x) F)"
by (auto simp: eventually_inf)
with x show "V ∩ A ≠ {}"
by (auto simp del: Int_iff simp add: trivial_limit_def)
qed
then have "x ∈ V"
using V ∈ A A(1) by simp
}
with x∈U have "x ∈ U ∩ \<Inter>A" by auto
with U ∩ \<Inter>A = {} show False by auto
qed

definition "countably_compact U <->
(∀A. countable A --> (∀a∈A. open a) --> U ⊆ \<Union>A --> (∃T⊆A. finite T ∧ U ⊆ \<Union>T))"

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

lemma countably_compactI:
assumes "!!C. ∀t∈C. open t ==> s ⊆ \<Union>C ==> countable C ==> (∃C'⊆C. finite C' ∧ s ⊆ \<Union>C')"
shows "countably_compact s"
using assms unfolding countably_compact_def by metis

lemma compact_imp_countably_compact: "compact U ==> countably_compact U"
by (auto simp: compact_eq_heine_borel countably_compact_def)

lemma countably_compact_imp_compact:
assumes "countably_compact U"
and ccover: "countable B" "∀b∈B. open b"
and basis: "!!T x. open T ==> x ∈ T ==> x ∈ U ==> ∃b∈B. x ∈ b ∧ b ∩ U ⊆ T"
shows "compact U"
using countably_compact U
unfolding compact_eq_heine_borel countably_compact_def
proof safe
fix A
assume A: "∀a∈A. open a" "U ⊆ \<Union>A"
assume *: "∀A. countable A --> (∀a∈A. open a) --> U ⊆ \<Union>A --> (∃T⊆A. finite T ∧ U ⊆ \<Union>T)"

moreover def C ≡ "{b∈B. ∃a∈A. b ∩ U ⊆ a}"
ultimately have "countable C" "∀a∈C. open a"
unfolding C_def using ccover by auto
moreover
have "\<Union>A ∩ U ⊆ \<Union>C"
proof safe
fix x a
assume "x ∈ U" "x ∈ a" "a ∈ A"
with basis[of a x] A obtain b where "b ∈ B" "x ∈ b" "b ∩ U ⊆ a"
by blast
with a ∈ A show "x ∈ \<Union>C"
unfolding C_def by auto
qed
then have "U ⊆ \<Union>C" using U ⊆ \<Union>A by auto
ultimately obtain T where T: "T⊆C" "finite T" "U ⊆ \<Union>T"
using * by metis
then have "∀t∈T. ∃a∈A. t ∩ U ⊆ a"
by (auto simp: C_def)
then obtain f where "∀t∈T. f t ∈ A ∧ t ∩ U ⊆ f t"
unfolding bchoice_iff Bex_def ..
with T show "∃T⊆A. finite T ∧ U ⊆ \<Union>T"
unfolding C_def by (intro exI[of _ "fT"]) fastforce
qed

lemma countably_compact_imp_compact_second_countable:
"countably_compact U ==> compact (U :: 'a :: second_countable_topology set)"
proof (rule countably_compact_imp_compact)
fix T and x :: 'a
assume "open T" "x ∈ T"
from topological_basisE[OF is_basis this] obtain b where
"b ∈ (SOME B. countable B ∧ topological_basis B)" "x ∈ b" "b ⊆ T" .
then show "∃b∈SOME B. countable B ∧ topological_basis B. x ∈ b ∧ b ∩ U ⊆ T"
by blast
qed (insert countable_basis topological_basis_open[OF is_basis], auto)

lemma countably_compact_eq_compact:
"countably_compact U <-> 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 => bool"
where "seq_compact S <->
(∀f. (∀n. f n ∈ S) --> (∃l∈S. ∃r. subseq r ∧ ((f o r) ---> l) sequentially))"

lemma seq_compactI:
assumes "!!f. ∀n. f n ∈ S ==> ∃l∈S. ∃r. subseq r ∧ ((f o r) ---> l) sequentially"
shows "seq_compact S"
unfolding seq_compact_def using assms by fast

lemma seq_compactE:
assumes "seq_compact S" "∀n. f n ∈ S"
obtains l r where "l ∈ S" "subseq r" "((f o r) ---> l) sequentially"
using assms unfolding seq_compact_def by fast

lemma closed_sequentially: (* TODO: move upwards *)
assumes "closed s" and "∀n. f n ∈ s" and "f ----> l"
shows "l ∈ s"
proof (rule ccontr)
assume "l ∉ s"
with closed s and f ----> l have "eventually (λn. f n ∈ - s) sequentially"
by (fast intro: topological_tendstoD)
with ∀n. f n ∈ s show "False"
by simp
qed

lemma seq_compact_inter_closed:
assumes "seq_compact s" and "closed t"
shows "seq_compact (s ∩ t)"
proof (rule seq_compactI)
fix f assume "∀n::nat. f n ∈ s ∩ t"
hence "∀n. f n ∈ s" and "∀n. f n ∈ t"
by simp_all
from seq_compact s and ∀n. f n ∈ s
obtain l r where "l ∈ s" and r: "subseq r" and l: "(f o r) ----> l"
by (rule seq_compactE)
from ∀n. f n ∈ t have "∀n. (f o r) n ∈ t"
by simp
from closed t and this and l have "l ∈ t"
by (rule closed_sequentially)
with l ∈ s and r and l show "∃l∈s ∩ t. ∃r. subseq r ∧ (f o r) ----> l"
by fast
qed

lemma seq_compact_closed_subset:
assumes "closed s" and "s ⊆ t" and "seq_compact t"
shows "seq_compact s"
using assms seq_compact_inter_closed [of t s] by (simp add: Int_absorb1)

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: "∀a∈A. open a" "U ⊆ \<Union>A" "countable A"
have subseq: "!!X. range X ⊆ U ==> ∃r x. x ∈ U ∧ subseq r ∧ (X o r) ----> x"
using seq_compact U by (fastforce simp: seq_compact_def subset_eq)
show "∃T⊆A. finite T ∧ U ⊆ \<Union>T"
proof cases
assume "finite A"
with A show ?thesis by auto
next
assume "infinite A"
then have "A ≠ {}" by auto
show ?thesis
proof (rule ccontr)
assume "¬ (∃T⊆A. finite T ∧ U ⊆ \<Union>T)"
then have "∀T. ∃x. T ⊆ A ∧ finite T --> (x ∈ U - \<Union>T)"
by auto
then obtain X' where T: "!!T. T ⊆ A ==> finite T ==> X' T ∈ U - \<Union>T"
by metis
def X ≡ "λn. X' (from_nat_into A  {.. n})"
have X: "!!n. X n ∈ U - (\<Union>i≤n. from_nat_into A i)"
using A ≠ {} unfolding X_def SUP_def by (intro T) (auto intro: from_nat_into)
then have "range X ⊆ U"
by auto
with subseq[of X] obtain r x where "x ∈ U" and r: "subseq r" "(X o r) ----> x"
by auto
from x∈U U ⊆ \<Union>A from_nat_into_surj[OF countable A]
obtain n where "x ∈ from_nat_into A n" by auto
with r(2) A(1) from_nat_into[OF A ≠ {}, of n]
have "eventually (λi. X (r i) ∈ from_nat_into A n) sequentially"
unfolding tendsto_def by (auto simp: comp_def)
then obtain N where "!!i. N ≤ i ==> X (r i) ∈ from_nat_into A n"
by (auto simp: eventually_sequentially)
moreover from X have "!!i. n ≤ r i ==> X (r i) ∉ from_nat_into A n"
by auto
moreover from subseq r[THEN seq_suble, of "max n N"] have "∃i. n ≤ r i ∧ N ≤ 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 => 'a"
assume "∀n. X n ∈ U"
then have "eventually (λx. x ∈ U) (filtermap X sequentially)"
by (auto simp: eventually_filtermap)
moreover
have "filtermap X sequentially ≠ bot"
by (simp add: trivial_limit_def eventually_filtermap)
ultimately
obtain x where "x ∈ U" and x: "inf (nhds x) (filtermap X sequentially) ≠ bot" (is "?F ≠ _")
using compact U by (auto simp: compact_filter)

from countable_basis_at_decseq[of x]
obtain A where A:
"!!i. open (A i)"
"!!i. x ∈ A i"
"!!S. open S ==> x ∈ S ==> eventually (λi. A i ⊆ S) sequentially"
by blast
def s ≡ "λn i. SOME j. i < j ∧ X j ∈ A (Suc n)"
{
fix n i
have "∃a. i < a ∧ X a ∈ A (Suc n)"
proof (rule ccontr)
assume "¬ (∃a>i. X a ∈ A (Suc n))"
then have "!!a. Suc i ≤ a ==> X a ∉ A (Suc n)"
by auto
then have "eventually (λx. x ∉ A (Suc n)) (filtermap X sequentially)"
by (auto simp: eventually_filtermap eventually_sequentially)
moreover have "eventually (λx. x ∈ A (Suc n)) (nhds x)"
using A(1,2)[of "Suc n"] by (auto simp: eventually_nhds)
ultimately have "eventually (λx. False) ?F"
by (auto simp add: eventually_inf)
with x show False
by (simp add: eventually_False)
qed
then have "i < s n i" "X (s n i) ∈ A (Suc n)"
unfolding s_def by (auto intro: someI2_ex)
}
note s = this
def r ≡ "rec_nat (s 0 0) s"
have "subseq r"
by (auto simp: r_def s subseq_Suc_iff)
moreover
have "(λn. X (r n)) ----> x"
proof (rule topological_tendstoI)
fix S
assume "open S" "x ∈ S"
with A(3) have "eventually (λi. A i ⊆ S) sequentially"
by auto
moreover
{
fix i
assume "Suc 0 ≤ i"
then have "X (r i) ∈ A i"
by (cases i) (simp_all add: r_def s)
}
then have "eventually (λi. X (r i) ∈ A i) sequentially"
by (auto simp: eventually_sequentially)
ultimately show "eventually (λi. X (r i) ∈ S) sequentially"
by eventually_elim auto
qed
ultimately show "∃x ∈ U. ∃r. subseq r ∧ (X o r) ----> x"
using x ∈ U by (auto simp: convergent_def comp_def)
qed

lemma countably_compact_imp_acc_point:
assumes "countably_compact s"
and "countable t"
and "infinite t"
and "t ⊆ s"
shows "∃x∈s. ∀U. x∈U ∧ open U --> infinite (U ∩ t)"
proof (rule ccontr)
def C ≡ "(λF. interior (F ∪ (- t)))  {F. finite F ∧ F ⊆ t }"
note countably_compact s
moreover have "∀t∈C. open t"
by (auto simp: C_def)
moreover
assume "¬ (∃x∈s. ∀U. x∈U ∧ open U --> infinite (U ∩ t))"
then have s: "!!x. x ∈ s ==> ∃U. x∈U ∧ open U ∧ finite (U ∩ t)" by metis
have "s ⊆ \<Union>C"
using t ⊆ s
unfolding C_def Union_image_eq
apply (safe dest!: s)
apply (rule_tac a="U ∩ 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
obtain D where "D ⊆ C" "finite D" "s ⊆ \<Union>D"
by (rule countably_compactE)
then obtain E where E: "E ⊆ {F. finite F ∧ F ⊆ t }" "finite E"
and s: "s ⊆ (\<Union>F∈E. interior (F ∪ (- t)))"
by (metis (lifting) Union_image_eq finite_subset_image C_def)
from s t ⊆ s have "t ⊆ \<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 "∀t. infinite t ∧ countable t ∧ t ⊆ s -->
(∃x∈s. ∀U. x∈U ∧ open U --> infinite (U ∩ t))"
shows "seq_compact s"
proof -
{
fix f :: "nat => 'a"
assume f: "∀n. f n ∈ s"
have "∃l∈s. ∃r. subseq r ∧ ((f o 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: "∀n. f (r n) = f l"
using infinite_enumerate by blast
then have "subseq r ∧ (f o r) ----> f l"
by (simp add: fr o_def)
with f show "∃l∈s. ∃r. subseq r ∧ (f o r) ----> l"
by auto
next
case False
with f assms have "∃x∈s. ∀U. x∈U ∧ open U --> infinite (U ∩ range f)"
by auto
then obtain l where "l ∈ s" "∀U. l∈U ∧ open U --> infinite (U ∩ range f)" ..
from this(2) have "∃r. subseq r ∧ ((f o r) ---> l) sequentially"
using acc_point_range_imp_convergent_subsequence[of l f] by auto
with l ∈ s show "∃l∈s. ∃r. subseq r ∧ ((f o r) ---> l) sequentially" ..
qed
}
then show ?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 <-> 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 <->
(∀t. infinite t ∧ countable t ∧ t ⊆ s --> (∃x∈s. ∀U. x∈U ∧ open U --> infinite (U ∩ 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 <-> 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 "∀t. infinite t ∧ t ⊆ s --> (∃x ∈ s. x islimpt t) ==> seq_compact s"
by (rule countable_acc_point_imp_seq_compact) (metis islimpt_eq_acc_point)

subsubsection{* Totally bounded *}

lemma cauchy_def: "Cauchy s <-> (∀e>0. ∃N. ∀m n. m ≥ N ∧ n ≥ N --> dist(s m)(s n) < e)"
unfolding Cauchy_def by metis

lemma seq_compact_imp_totally_bounded:
assumes "seq_compact s"
shows "∀e>0. ∃k. finite k ∧ k ⊆ s ∧ s ⊆ (\<Union>x∈k. ball x e)"
proof -
{ fix e::real assume "e > 0" assume *: "!!k. finite k ==> k ⊆ s ==> ¬ s ⊆ (\<Union>x∈k. ball x e)"
let ?Q = "λx n r. r ∈ s ∧ (∀m < (n::nat). ¬ (dist (x m) r < e))"
have "∃x. ∀n::nat. ?Q x n (x n)"
proof (rule dependent_wellorder_choice)
fix n x assume "!!y. y < n ==> ?Q x y (x y)"
then have "¬ s ⊆ (\<Union>x∈x  {0..<n}. ball x e)"
using *[of "x  {0 ..< n}"] by (auto simp: subset_eq)
then obtain z where z:"z∈s" "z ∉ (\<Union>x∈x  {0..<n}. ball x e)"
unfolding subset_eq by auto
show "∃r. ?Q x n r"
using z by auto
qed simp
then obtain x where "∀n::nat. x n ∈ s" and x:"!!n m. m < n ==> ¬ (dist (x m) (x n) < e)"
by blast
then obtain l r where "l ∈ s" and r:"subseq r" and "((x o r) ---> l) sequentially"
using assms by (metis seq_compact_def)
from this(3) have "Cauchy (x o r)"
using LIMSEQ_imp_Cauchy by auto
then obtain N::nat where "!!m n. N ≤ m ==> N ≤ n ==> dist ((x o r) m) ((x o r) n) < e"
unfolding cauchy_def using e > 0 by blast
then have False
using x[of "r N" "r (N+1)"] r by (auto simp: subseq_def) }
then show ?thesis
by metis
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]
obtain f where f: "∀e>0. finite (f e) ∧ f e ⊆ s ∧ s ⊆ (\<Union>x∈f e. ball x e)"
unfolding choice_iff' ..
def K ≡ "(λ(x, r). ball x r)  ((\<Union>e ∈ \<rat> ∩ {0 <..}. f e) × \<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 "∀b∈K. open b" by (auto simp: K_def)
next
fix T x
assume T: "open T" "x ∈ T" and x: "x ∈ s"
from openE[OF T] obtain e where "0 < e" "ball x e ⊆ T"
by auto
then have "0 < e / 2" "ball x (e / 2) ⊆ T"
by auto
from Rats_dense_in_real[OF 0 < e / 2] obtain r where "r ∈ \<rat>" "0 < r" "r < e / 2"
by auto
from f[rule_format, of r] 0 < r x ∈ s obtain k where "k ∈ f r" "x ∈ ball k r"
unfolding Union_image_eq by auto
from r ∈ \<rat> 0 < r k ∈ f r have "ball k r ∈ K"
by (auto simp: K_def)
then show "∃b∈K. x ∈ b ∧ b ∩ s ⊆ T"
proof (rule bexI[rotated], safe)
fix y
assume "y ∈ ball k r"
with r < e / 2 x ∈ 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 ⊆ T show "y ∈ T"
by auto
next
show "x ∈ ball k r" by fact
qed
qed
qed

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

lemma compact_def:
"compact (S :: 'a::metric_space set) <->
(∀f. (∀n. f n ∈ S) --> (∃l∈S. ∃r. subseq r ∧ (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 <-> (∀t. infinite t ∧ t ⊆ s --> (∃x ∈ s. x islimpt t))"
(is "?lhs = ?rhs")
proof
assume ?lhs
then show ?rhs
using heine_borel_imp_bolzano_weierstrass[of s] by auto
next
assume ?rhs
then show ?lhs
unfolding compact_eq_seq_compact_metric by (rule bolzano_weierstrass_imp_seq_compact)
qed

lemma bolzano_weierstrass_imp_bounded:
"∀t. infinite t ∧ t ⊆ s --> (∃x ∈ s. x islimpt t) ==> bounded s"
using compact_imp_bounded unfolding compact_eq_bolzano_weierstrass .

subsection {* Metric spaces with the Heine-Borel property *}

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) ==> ∃l r. subseq r ∧ ((f o 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 => 'a"
assume f: "∀n. f n ∈ s"
with bounded s have "bounded (range f)"
by (auto intro: bounded_subset)
obtain l r where r: "subseq r" and l: "((f o r) ---> l) sequentially"
using bounded_imp_convergent_subsequence [OF bounded (range f)] by auto
from f have fr: "∀n. (f o r) n ∈ s"
by simp
have "l ∈ s" using closed s fr l
by (rule closed_sequentially)
show "∃l∈s. ∃r. subseq r ∧ ((f o r) ---> l) sequentially"
using l ∈ s r l by blast
qed

lemma compact_eq_bounded_closed:
fixes s :: "'a::heine_borel set"
shows "compact s <-> bounded s ∧ closed s"
(is "?lhs = ?rhs")
proof
assume ?lhs
then show ?rhs
using compact_imp_closed compact_imp_bounded
by blast
next
assume ?rhs
then show ?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 => real"
shows "bounded {s n| n. True} ==> ∀n. s n ≤ s (Suc n) ==> ∃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 => real"
assume f: "bounded (range f)"
obtain r where r: "subseq r" "monoseq (f o r)"
unfolding comp_def by (metis seq_monosub)
then have "Bseq (f o r)"
unfolding Bseq_eq_bounded using f by (auto intro: bounded_subset)
with r show "∃l r. subseq r ∧ (f o r) ----> l"
using Bseq_monoseq_convergent[of "f o r"] by (auto simp: convergent_def)
qed

lemma compact_lemma:
fixes f :: "nat => 'a::euclidean_space"
assumes "bounded (range f)"
shows "∀d⊆Basis. ∃l::'a. ∃ r.
subseq r ∧ (∀e>0. eventually (λn. ∀i∈d. dist (f (r n) • i) (l • i) < e) sequentially)"
proof safe
fix d :: "'a set"
assume d: "d ⊆ Basis"
with finite_Basis have "finite d"
by (blast intro: finite_subset)
from this d show "∃l::'a. ∃r. subseq r ∧
(∀e>0. eventually (λn. ∀i∈d. dist (f (r n) • i) (l • i) < e) sequentially)"
proof (induct d)
case empty
then show ?case
unfolding subseq_def by auto
next
case (insert k d)
have k[intro]: "k ∈ Basis"
using insert by auto
have s': "bounded ((λx. x • 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: "∀e > 0. eventually (λn. ∀i∈d. dist (f (r1 n) • i) (l1 • i) < e) sequentially"
using insert(3) using insert(4) by auto
have f': "∀n. f (r1 n) • k ∈ (λx. x • k)  range f"
by simp
have "bounded (range (λi. f (r1 i) • k))"
by (metis (lifting) bounded_subset f' image_subsetI s')
then obtain l2 r2 where r2:"subseq r2" and lr2:"((λi. f (r1 (r2 i)) • k) ---> l2) sequentially"
using bounded_imp_convergent_subsequence[of "λi. f (r1 i) • k"]
by (auto simp: o_def)
def r ≡ "r1 o r2"
have r:"subseq r"
using r1 and r2 unfolding r_def o_def subseq_def by auto
moreover
def l ≡ "(∑i∈Basis. (if i = k then l2 else l1•i) *⇩R i)::'a"
{
fix e::real
assume "e > 0"
from lr1 e > 0 have N1: "eventually (λn. ∀i∈d. dist (f (r1 n) • i) (l1 • i) < e) sequentially"
by blast
from lr2 e > 0 have N2:"eventually (λn. dist (f (r1 (r2 n)) • k) l2 < e) sequentially"
by (rule tendstoD)
from r2 N1 have N1': "eventually (λn. ∀i∈d. dist (f (r1 (r2 n)) • i) (l1 • i) < e) sequentially"
by (rule eventually_subseq)
have "eventually (λn. ∀i∈(insert k d). dist (f (r n) • i) (l • 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 ⊆ heine_borel
proof
fix f :: "nat => 'a"
assume f: "bounded (range f)"
then obtain l::'a and r where r: "subseq r"
and l: "∀e>0. eventually (λn. ∀i∈Basis. dist (f (r n) • i) (l • i) < e) sequentially"
using compact_lemma [OF f] by blast
{
fix e::real
assume "e > 0"
hence "e / real_of_nat DIM('a) > 0" by (simp add: DIM_positive)
with l have "eventually (λn. ∀i∈Basis. dist (f (r n) • i) (l • i) < e / (real_of_nat DIM('a))) sequentially"
by simp
moreover
{
fix n
assume n: "∀i∈Basis. dist (f (r n) • i) (l • i) < e / (real_of_nat DIM('a))"
have "dist (f (r n)) l ≤ (∑i∈Basis. dist (f (r n) • i) (l • i))"
apply (subst euclidean_dist_l2)
using zero_le_dist
apply (rule setL2_le_setsum)
done
also have "… < (∑i∈(Basis::'a set). e / (real_of_nat DIM('a)))"
apply (rule setsum_strict_mono)
using n
apply auto
done
finally have "dist (f (r n)) l < e"
by auto
}
ultimately have "eventually (λn. dist (f (r n)) l < e) sequentially"
by (rule eventually_elim1)
}
then have *: "((f o r) ---> l) sequentially"
unfolding o_def tendsto_iff by simp
with r show "∃l r. subseq r ∧ ((f o r) ---> l) sequentially"
by auto
qed

lemma bounded_fst: "bounded s ==> bounded (fst  s)"
unfolding bounded_def
by (metis (erased, hide_lams) dist_fst_le image_iff order_trans)

lemma bounded_snd: "bounded s ==> bounded (snd  s)"
unfolding bounded_def
by (metis (no_types, hide_lams) dist_snd_le image_iff order.trans)

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

subsubsection {* Completeness *}

definition complete :: "'a::metric_space set => bool"
where "complete s <-> (∀f. (∀n. f n ∈ s) ∧ Cauchy f --> (∃l∈s. f ----> l))"

lemma completeI:
assumes "!!f. ∀n. f n ∈ s ==> Cauchy f ==> ∃l∈s. f ----> l"
shows "complete s"
using assms unfolding complete_def by fast

lemma completeE:
assumes "complete s" and "∀n. f n ∈ s" and "Cauchy f"
obtains l where "l ∈ s" and "f ----> l"
using assms unfolding complete_def by fast

lemma compact_imp_complete:
assumes "compact s"
shows "complete s"
proof -
{
fix f
assume as: "(∀n::nat. f n ∈ s)" "Cauchy f"
from as(1) obtain l r where lr: "l∈s" "subseq r" "(f o 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:"∀m n. N ≤ m ∧ N ≤ n --> dist (f m) (f n) < e/2"
unfolding cauchy_def
using e > 0
apply (erule_tac x="e/2" in allE)
apply auto
done
from lr(3)[unfolded lim_sequentially, THEN spec[where x="e/2"]]
obtain M where M:"∀n≥M. dist ((f o r) n) l < e/2"
using e > 0 by auto
{
fix n :: nat
assume n: "n ≥ max N M"
have "dist ((f o r) n) l < e/2"
using n M by auto
moreover have "r n ≥ N"
using lr'[of n] n by auto
then have "dist (f n) ((f o r) n) < e / 2"
using N and 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)
}
then have "∃N. ∀n≥N. dist (f n) l < e" by blast
}
then have "∃l∈s. (f ---> l) sequentially" using l∈s
unfolding lim_sequentially by auto
}
then show ?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 simp: 0 < e)
also have "1 / (ceiling (1/e)) ≤ 1 / (1/e)"
by (rule divide_left_mono) (auto simp: 0 < e)
also have "… = e" by simp
finally show  "∃n. 1 / real (Suc n) < e" ..
qed

lemma compact_eq_totally_bounded:
"compact s <-> complete s ∧ (∀e>0. ∃k. finite k ∧ s ⊆ (\<Union>x∈k. ball x e))"
(is "_ <-> ?rhs")
proof
assume assms: "?rhs"
then obtain k where k: "!!e. 0 < e ==> finite (k e)" "!!e. 0 < e ==> s ⊆ (\<Union>x∈k e. ball x e)"
by (auto simp: choice_iff')

show "compact s"
proof cases
assume "s = {}"
then show "compact s" by (simp add: compact_def)
next
assume "s ≠ {}"
show ?thesis
unfolding compact_def
proof safe
fix f :: "nat => 'a"
assume f: "∀n. f n ∈ s"

def e ≡ "λn. 1 / (2 * Suc n)"
then have [simp]: "!!n. 0 < e n" by auto
def B ≡ "λn U. SOME b. infinite {n. f n ∈ b} ∧ (∃x. b ⊆ ball x (e n) ∩ U)"
{
fix n U
assume "infinite {n. f n ∈ U}"
then have "∃b∈k (e n). infinite {i∈{n. f n ∈ U}. f i ∈ ball b (e n)}"
using k f by (intro pigeonhole_infinite_rel) (auto simp: subset_eq)
then obtain a where
"a ∈ k (e n)"
"infinite {i ∈ {n. f n ∈ U}. f i ∈ ball a (e n)}" ..
then have "∃b. infinite {i. f i ∈ b} ∧ (∃x. b ⊆ ball x (e n) ∩ U)"
by (intro exI[of _ "ball a (e n) ∩ U"] exI[of _ a]) (auto simp: ac_simps)
from someI_ex[OF this]
have "infinite {i. f i ∈ B n U}" "∃x. B n U ⊆ ball x (e n) ∩ U"
unfolding B_def by auto
}
note B = this

def F ≡ "rec_nat (B 0 UNIV) B"
{
fix n
have "infinite {i. f i ∈ F n}"
by (induct n) (auto simp: F_def B)
}
then have F: "!!n. ∃x. F (Suc n) ⊆ ball x (e n) ∩ F n"
using B by (simp add: F_def)
then have F_dec: "!!m n. m ≤ n ==> F n ⊆ F m"
using decseq_SucI[of F] by (auto simp: decseq_def)

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

def t ≡ "rec_nat (sel 0 0) (λn i. sel (Suc n) i)"
have "subseq t"
unfolding subseq_Suc_iff by (simp add: t_def sel)
moreover have "∀i. (f o t) i ∈ s"
using f by auto
moreover
{
fix n
have "(f o t) n ∈ F n"
by (cases n) (simp_all add: t_def sel)
}
note t = this

have "Cauchy (f o 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 ≤ n" "Suc N ≤ m"
then have "(f o t) n ∈ F (Suc N)" "(f o t) m ∈ 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 o t) n) < e N" "dist x ((f o t) m) < e N"
by (auto simp: subset_eq)
with dist_triangle[of "(f o t) m" "(f o t) n" x] 2 * e N < r
show "dist ((f o t) m) ((f o t) n) < r"
by (simp add: dist_commute)
qed

ultimately show "∃l∈s. ∃r. subseq r ∧ (f o 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 <-> (∀e>0.∃ N::nat. ∀n≥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:"∀n≥N. dist (s n) (s N) < e/2"
by (erule_tac x="e/2" in allE) auto
{
fix n m
assume nm:"N ≤ m ∧ N ≤ n"
then have "dist (s m) (s n) < e" using N
using dist_triangle_half_l[of "s m" "s N" "e" "s n"]
by blast
}
then have "∃N. ∀m n. N ≤ m ∧ N ≤ n --> dist (s m) (s n) < e"
by blast
}
then have ?lhs
unfolding cauchy_def
by blast
}
then show ?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 "∀m n. N ≤ m ∧ N ≤ n --> dist (s m) (s n) < 1"
unfolding cauchy_def
apply (erule_tac x= 1 in allE)
apply auto
done
then have N:"∀n. N ≤ n --> 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:"∀x∈s  {0..N}. dist (s N) x ≤ 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)
apply auto
done
qed

instance heine_borel < complete_space
proof
fix f :: "nat => 'a" assume "Cauchy f"
then have "bounded (range f)"
by (rule cauchy_imp_bounded)
then have "compact (closure (range f))"
unfolding compact_eq_bounded_closed by auto
then have "complete (closure (range f))"
by (rule compact_imp_complete)
moreover have "∀n. f n ∈ closure (range f)"
using closure_subset [of "range f"] by auto
ultimately have "∃l∈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 ⊆ banach ..

lemma complete_UNIV: "complete (UNIV :: ('a::complete_space) set)"
proof (rule completeI)
fix f :: "nat => 'a" assume "Cauchy f"
then have "convergent f" by (rule Cauchy_convergent)
then show "∃l∈UNIV. f ----> l" unfolding convergent_def by simp
qed

lemma complete_imp_closed:
assumes "complete s"
shows "closed s"
proof (unfold closed_sequential_limits, clarify)
fix f x assume "∀n. f n ∈ s" and "f ----> x"
from f ----> x have "Cauchy f"
by (rule LIMSEQ_imp_Cauchy)
with complete s and ∀n. f n ∈ s obtain l where "l ∈ s" and "f ----> l"
by (rule completeE)
from f ----> x and f ----> l have "x = l"
by (rule LIMSEQ_unique)
with l ∈ s show "x ∈ s"
by simp
qed

lemma complete_inter_closed:
assumes "complete s" and "closed t"
shows "complete (s ∩ t)"
proof (rule completeI)
fix f assume "∀n. f n ∈ s ∩ t" and "Cauchy f"
then have "∀n. f n ∈ s" and "∀n. f n ∈ t"
by simp_all
from complete s obtain l where "l ∈ s" and "f ----> l"
using ∀n. f n ∈ s and Cauchy f by (rule completeE)
from closed t and ∀n. f n ∈ t and f ----> l have "l ∈ t"
by (rule closed_sequentially)
with l ∈ s and f ----> l show "∃l∈s ∩ t. f ----> l"
by fast
qed

lemma complete_closed_subset:
assumes "closed s" and "s ⊆ t" and "complete t"
shows "complete s"
using assms complete_inter_closed [of t s] by (simp add: Int_absorb1)

lemma complete_eq_closed:
fixes s :: "('a::complete_space) set"
shows "complete s <-> closed s"
proof
assume "closed s" then show "complete s"
using subset_UNIV complete_UNIV by (rule complete_closed_subset)
next
assume "complete s" then show "closed s"
by (rule complete_imp_closed)
qed

lemma convergent_eq_cauchy:
fixes s :: "nat => 'a::complete_space"
shows "(∃l. (s ---> l) sequentially) <-> Cauchy s"
unfolding Cauchy_convergent_iff convergent_def ..

lemma convergent_imp_bounded:
fixes s :: "nat => 'a::metric_space"
shows "(s ---> l) sequentially ==> 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 ⊆ 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:
fixes s :: "nat => ('a::heine_borel) set"
assumes "∀n. closed (s n)"
and "∀n. s n ≠ {}"
and "∀m n. m ≤ n --> s n ⊆ s m"
and "bounded (s 0)"
shows "∃a. ∀n. a ∈ s n"
proof -
from assms(2) obtain x where x: "∀n. x n ∈ s n"
using choice[of "λn x. x ∈ s n"] by auto
from assms(4,1) have "seq_compact (s 0)"
by (simp add: bounded_closed_imp_seq_compact)
then obtain l r where lr: "l ∈ s 0" "subseq r" "(x o r) ----> l"
using x and assms(3) unfolding seq_compact_def by blast
have "∀n. l ∈ s n"
proof
fix n :: nat
have "closed (s n)"
using assms(1) by simp
moreover have "∀i. (x o r) i ∈ s i"
using x and assms(3) and lr(2) [THEN seq_suble] by auto
then have "∀i. (x o r) (i + n) ∈ s n"
using assms(3) by (fast intro!: le_add2)
moreover have "(λi. (x o r) (i + n)) ----> l"
using lr(3) by (rule LIMSEQ_ignore_initial_segment)
ultimately show "l ∈ s n"
by (rule closed_sequentially)
qed
then show ?thesis ..
qed

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

lemma decreasing_closed_nest:
fixes s :: "nat => ('a::complete_space) set"
assumes
"∀n. closed (s n)"
"∀n. s n ≠ {}"
"∀m n. m ≤ n --> s n ⊆ s m"
"∀e>0. ∃n. ∀x∈s n. ∀y∈s n. dist x y < e"
shows "∃a. ∀n. a ∈ s n"
proof -
have "∀n. ∃x. x ∈ s n"
using assms(2) by auto
then have "∃t. ∀n. t n ∈ s n"
using choice[of "λn x. x ∈ s n"] by auto
then obtain t where t: "∀n. t n ∈ s n" by auto
{
fix e :: real
assume "e > 0"
then obtain N where N:"∀x∈s N. ∀y∈s N. dist x y < e"
using assms(4) by auto
{
fix m n :: nat
assume "N ≤ m ∧ N ≤ n"
then have "t m ∈ s N" "t n ∈ s N"
using assms(3) t unfolding  subset_eq t by blast+
then have "dist (t m) (t n) < e"
using N by auto
}
then have "∃N. ∀m n. N ≤ m ∧ N ≤ n --> dist (t m) (t n) < e"
by auto
}
then have "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: "∀n≥N. dist (t n) l < e"
using l[unfolded lim_sequentially] by auto
have "t (max n N) ∈ s n"
using assms(3)
unfolding subset_eq
apply (erule_tac x=n in allE)
apply (erule_tac x="max n N" in allE)
using t
apply auto
done
then have "∃y∈s n. dist y l < e"
apply (rule_tac x="t (max n N)" in bexI)
using N
apply auto
done
}
then have "l ∈ 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 => 'a::complete_space set"
assumes
"∀n. closed(s n)"
"∀n. s n ≠ {}"
"∀m n. m ≤ n --> s n ⊆ s m"
"∀e>0. ∃n. ∀x ∈ (s n). ∀ y∈(s n). dist x y < e"
shows "∃a. \<Inter>(range s) = {a}"
proof -
obtain a where a: "∀n. a ∈ s n"
using decreasing_closed_nest[of s] using assms by auto
{
fix b
assume b: "b ∈ \<Inter>(range s)"
{
fix e :: real
assume "e > 0"
then have "dist a b < e"
using assms(4) and b and a by blast
}
then have "dist a b = 0"
by (metis dist_eq_0_iff dist_nz less_le)
}
with a have "\<Inter>(range s) = {a}"
unfolding image_def by auto
then show ?thesis ..
qed

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

lemma uniformly_convergent_eq_cauchy:
fixes s::"nat => 'b => 'a::complete_space"
shows
"(∃l. ∀e>0. ∃N. ∀n x. N ≤ n ∧ P x --> dist(s n x)(l x) < e) <->
(∀e>0. ∃N. ∀m n x. N ≤ m ∧ N ≤ n ∧ P x  --> dist (s m x) (s n x) < e)"
(is "?lhs = ?rhs")
proof
assume ?lhs
then obtain l where l:"∀e>0. ∃N. ∀n x. N ≤ n ∧ P x --> dist (s n x) (l x) < e"
by auto
{
fix e :: real
assume "e > 0"
then obtain N :: nat where N: "∀n x. N ≤ n ∧ P x --> 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 ≤ m ∧ N ≤ n ∧ P x"
then have "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
}
then have "∃N. ∀m n x. N ≤ m ∧ N ≤ n ∧ P x  --> dist (s m x) (s n x) < e"  by auto
}
then show ?rhs by auto
next
assume ?rhs
then have "∀x. P x --> Cauchy (λn. s n x)"
unfolding cauchy_def
apply auto
apply (erule_tac x=e in allE)
apply auto
done
then obtain l where l: "∀x. P x --> ((λn. s n x) ---> l x) sequentially"
unfolding convergent_eq_cauchy[symmetric]
using choice[of "λx l. P x --> ((λn. s n x) ---> l) sequentially"]
by auto
{
fix e :: real
assume "e > 0"
then obtain N where N:"∀m n x. N ≤ m ∧ N ≤ n ∧ P x --> 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:"∀n≥M. dist (s n x) (l x) < e/2"
using l[THEN spec[where x=x], unfolded lim_sequentially] and e > 0
by (auto elim!: allE[where x="e/2"])
fix n :: nat
assume "n ≥ N"
then have "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)
}
then have "∃N. ∀n x. N ≤ n ∧ P x --> dist(s n x)(l x) < e"
by auto
}
then show ?lhs by auto
qed

lemma uniformly_cauchy_imp_uniformly_convergent:
fixes s :: "nat => 'a => 'b::complete_space"
assumes "∀e>0.∃N. ∀m (n::nat) x. N ≤ m ∧ N ≤ n ∧ P x --> dist(s m x)(s n x) < e"
and "∀x. P x --> (∀e>0. ∃N. ∀n. N ≤ n --> dist(s n x)(l x) < e)"
shows "∀e>0. ∃N. ∀n x. N ≤ n ∧ P x --> dist(s n x)(l x) < e"
proof -
obtain l' where l:"∀e>0. ∃N. ∀n x. N ≤ n ∧ P x --> dist (s n x) (l' x) < e"
using assms(1) unfolding uniformly_convergent_eq_cauchy[symmetric] by auto
moreover
{
fix x
assume "P x"
then have "l x = l' x"
using tendsto_unique[OF trivial_limit_sequentially, of "λn. s n x" "l x" "l' x"]
using l and assms(2) unfolding lim_sequentially 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 <-> (∀e>0. ∃d>0. ∀x'∈ s.  dist x' x < d --> dist (f x') (f x) < e)"
unfolding continuous_within and Lim_within
apply auto
apply (metis dist_nz dist_self)
apply blast
done

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

lemma continuous_at_right_real_increasing:
fixes f :: "real => real"
assumes nondecF: "!!x y. x ≤ y ==> f x ≤ f y"
shows "continuous (at_right a) f <-> (∀e>0. ∃d>0. f (a + d) - f a < e)"
apply (simp add: greaterThan_def dist_real_def continuous_within Lim_within_le)
apply (intro all_cong ex_cong)
apply safe
apply (erule_tac x="a + d" in allE)
apply simp
apply (simp add: nondecF field_simps)
apply (drule nondecF)
apply simp
done

lemma continuous_at_left_real_increasing:
assumes nondecF: "!! x y. x ≤ y ==> f x ≤ ((f y) :: real)"
shows "(continuous (at_left (a :: real)) f) = (∀e > 0. ∃delta > 0. f a - f (a - delta) < e)"
apply (simp add: lessThan_def dist_real_def continuous_within Lim_within_le)
apply (intro all_cong ex_cong)
apply safe
apply (erule_tac x="a - d" in allE)
apply simp
apply (simp add: nondecF field_simps)
apply (cut_tac x="a - d" and y="x" in nondecF)
apply simp_all
done

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

lemma continuous_within_ball:
"continuous (at x within s) f <->
(∀e > 0. ∃d > 0. f  (ball x d ∩ s) ⊆ ball (f x) e)"
(is "?lhs = ?rhs")
proof
assume ?lhs
{
fix e :: real
assume "e > 0"
then obtain d where d: "d>0" "∀xa∈s. 0 < dist xa x ∧ dist xa x < d --> dist (f xa) (f x) < e"
using ?lhs[unfolded continuous_within Lim_within] by auto
{
fix y
assume "y ∈ f  (ball x d ∩ s)"
then have "y ∈ ball (f x) e"
using d(2)
unfolding dist_nz[symmetric]
apply (auto simp add: dist_commute)
apply (erule_tac x=xa in ballE)
apply auto
using e > 0
apply auto
done
}
then have "∃d>0. f  (ball x d ∩ s) ⊆ ball (f x) e"
using d > 0
unfolding subset_eq ball_def by (auto simp add: dist_commute)
}
then show ?rhs by auto
next
assume ?rhs
then show ?lhs
unfolding continuous_within Lim_within ball_def subset_eq
apply (auto simp add: dist_commute)
apply (erule_tac x=e in allE)
apply auto
done
qed

lemma continuous_at_ball:
"continuous (at x) f <-> (∀e>0. ∃d>0. f  (ball x d) ⊆ ball (f x) e)" (is "?lhs = ?rhs")
proof
assume ?lhs
then show ?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[symmetric]
apply auto
done
next
assume ?rhs
then show ?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)
apply (auto simp add: dist_commute dist_nz)
done
qed

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

lemma continuous_on_iff:
"continuous_on s f <->
(∀x∈s. ∀e>0. ∃d>0. ∀x'∈s. dist x' x < d --> dist (f x') (f x) < e)"
unfolding continuous_on_def Lim_within
by (metis dist_pos_lt dist_self)

definition uniformly_continuous_on :: "'a set => ('a::metric_space => 'b::metric_space) => bool"
where "uniformly_continuous_on s f <->
(∀e>0. ∃d>0. ∀x∈s. ∀x'∈s. dist x' x < d --> 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 ==> (f ---> l) net"
by simp

lemmas continuous_on = continuous_on_def -- "legacy theorem name"

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

lemma continuous_on_interior:
"continuous_on s f ==> x ∈ interior s ==> continuous (at x) f"
by (metis continuous_on_eq_continuous_at continuous_on_subset interiorE)

lemma continuous_on_eq:
"(∀x ∈ s. f x = g x) ==> continuous_on s f ==> 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 => 'b::topological_space"
shows "continuous (at a within s) f <->
(∀x. (∀n::nat. x n ∈ s) ∧ (x ---> a) sequentially
--> ((f o x) ---> f a) sequentially)"
(is "?lhs = ?rhs")
proof
assume ?lhs
{
fix x :: "nat => 'a"
assume x: "∀n. x n ∈ s" "∀e>0. eventually (λn. dist (x n) a < e) sequentially"
fix T :: "'b set"
assume "open T" and "f a ∈ T"
with ?lhs obtain d where "d>0" and d:"∀x∈s. 0 < dist x a ∧ dist x a < d --> f x ∈ T"
unfolding continuous_within tendsto_def eventually_at by (auto simp: dist_nz)
have "eventually (λn. dist (x n) a < d) sequentially"
using x(2) d>0 by simp
then have "eventually (λn. (f o x) n ∈ T) sequentially"
proof eventually_elim
case (elim n)
then show ?case
using d x(1) f a ∈ T unfolding dist_nz[symmetric] by auto
qed
}
then show ?rhs
unfolding tendsto_iff tendsto_def by simp
next
assume ?rhs
then show ?lhs
unfolding continuous_within tendsto_def [where l="f a"]
by (simp add: sequentially_imp_eventually_within)
qed

lemma continuous_at_sequentially:
fixes f :: "'a::metric_space => 'b::topological_space"
shows "continuous (at a) f <->
(∀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 => 'b::topological_space"
shows "continuous_on s f <->
(∀x. ∀a ∈ s. (∀n. x(n) ∈ s) ∧ (x ---> a) sequentially
--> ((f o x) ---> f a) sequentially)"
(is "?lhs = ?rhs")
proof
assume ?rhs
then show ?lhs
using continuous_within_sequentially[of _ s f]
unfolding continuous_on_eq_continuous_within
by auto
next
assume ?lhs
then show ?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 <-> (∀x y. (∀n. x n ∈ s) ∧ (∀n. y n ∈ s) ∧
((λn. dist (x n) (y n)) ---> 0) sequentially
--> ((λn. dist (f(x n)) (f(y n))) ---> 0) sequentially)" (is "?lhs = ?rhs")
proof
assume ?lhs
{
fix x y
assume x: "∀n. x n ∈ s"
and y: "∀n. y n ∈ s"
and xy: "((λn. dist (x n) (y n)) ---> 0) sequentially"
{
fix e :: real
assume "e > 0"
then obtain d where "d > 0" and d: "∀x∈s. ∀x'∈s. dist x' x < d --> dist (f x') (f x) < e"
using ?lhs[unfolded uniformly_continuous_on_def, THEN spec[where x=e]] by auto
obtain N where N: "∀n≥N. dist (x n) (y n) < d"
using xy[unfolded lim_sequentially dist_norm] and d>0 by auto
{
fix n
assume "n≥N"
then have "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
}
then have "∃N. ∀n≥N. dist (f (x n)) (f (y n)) < e"
by auto
}
then have "((λn. dist (f(x n)) (f(y n))) ---> 0) sequentially"
unfolding lim_sequentially and dist_real_def by auto
}
then show ?rhs by auto
next
assume ?rhs
{
assume "¬ ?lhs"
then obtain e where "e > 0" "∀d>0. ∃x∈s. ∃x'∈s. dist x' x < d ∧ ¬ dist (f x') (f x) < e"
unfolding uniformly_continuous_on_def by auto
then obtain fa where fa:
"∀x. 0 < x --> fst (fa x) ∈ s ∧ snd (fa x) ∈ s ∧ dist (fst (fa x)) (snd (fa x)) < x ∧ ¬ dist (f (fst (fa x))) (f (snd (fa x))) < e"
using choice[of "λd x. d>0 --> fst x ∈ s ∧ snd x ∈ s ∧ dist (snd x) (fst x) < d ∧ ¬ dist (f (snd x)) (f (fst x)) < e"]
unfolding Bex_def
by (auto simp add: dist_commute)
def x ≡ "λn::nat. fst (fa (inverse (real n + 1)))"
def y ≡ "λn::nat. snd (fa (inverse (real n + 1)))"
have xyn: "∀n. x n ∈ s ∧ y n ∈ s"
and xy0: "∀n. dist (x n) (y n) < inverse (real n + 1)"
and fxy:"∀n. ¬ 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 ≠ 0" and N: "0 < inverse (real N) ∧ inverse (real N) < e"
unfolding real_arch_inv[of e] by auto
{
fix n :: nat
assume "n ≥ N"
then have "inverse (real n + 1) < inverse (real N)"
using real_of_nat_ge_zero and N≠0 by auto
also have "… < e" using N by auto
finally have "inverse (real n + 1) < e" by auto
then have "dist (x n) (y n) < e"
using xy0[THEN spec[where x=n]] by auto
}
then have "∃N. ∀n≥N. dist (x n) (y n) < e" by auto
}
then have "∀e>0. ∃N. ∀n≥N. dist (f (x n)) (f (y n)) < e"
using ?rhs[THEN spec[where x=x], THEN spec[where x=y]] and xyn
unfolding lim_sequentially dist_real_def by auto
then have False using fxy and e>0 by auto
}
then show ?lhs
unfolding uniformly_continuous_on_def by blast
qed

text{* The usual transformation theorems. *}

lemma continuous_transform_within:
fixes f g :: "'a::metric_space => 'b::topological_space"
assumes "0 < d"
and "x ∈ s"
and "∀x' ∈ s. dist x' x < d --> f x' = g x'"
and "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 "∀x'∈s. 0 < dist x' x ∧ dist x' x < d --> f x' = g x'"
using assms(3) by auto
have "f x = g x"
using assms(1,2,3) by auto
then show "(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 => 'b::topological_space"
assumes "0 < d"
and "∀x'. dist x' x < d --> f x' = g x'"
and "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 (λx. infdist (f x) A)"
using assms unfolding continuous_def by (rule tendsto_infdist)

lemma continuous_infnorm[continuous_intros]:
"continuous F f ==> continuous F (λx. 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 (λx. 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_intros]:
"continuous_on s f ==> continuous_on s (λx. infnorm (f x))"
unfolding continuous_on by (fast intro: tendsto_infnorm)

lemma continuous_on_inner[continuous_intros]:
fixes g :: "'a::topological_space => 'b::real_inner"
assumes "continuous_on s f"
and "continuous_on s g"
shows "continuous_on s (λx. 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_intros]:
"uniformly_continuous_on s (λx. x)"
unfolding uniformly_continuous_on_def by auto

lemma uniformly_continuous_on_const[continuous_intros]:
"uniformly_continuous_on s (λx. c)"
unfolding uniformly_continuous_on_def by simp

lemma uniformly_continuous_on_dist[continuous_intros]:
fixes f g :: "'a::metric_space => 'b::metric_space"
assumes "uniformly_continuous_on s f"
and "uniformly_continuous_on s g"
shows "uniformly_continuous_on s (λx. dist (f x) (g x))"
proof -
{
fix a b c d :: 'b
have "¦dist a b - dist c d¦ ≤ 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: "(λn. dist (f (x n)) (f (y n))) ----> 0"
assume g: "(λn. dist (g (x n)) (g (y n))) ----> 0"
have "(λn. ¦dist (f (x n)) (g (x n)) - dist (f (y n)) (g (y n))¦) ----> 0"
by (rule Lim_transform_bound [OF _ tendsto_add_zero [OF f g]],
}
then show ?thesis
using assms unfolding uniformly_continuous_on_sequentially
unfolding dist_real_def by simp
qed

lemma uniformly_continuous_on_norm[continuous_intros]:
assumes "uniformly_continuous_on s f"
shows "uniformly_continuous_on s (λx. 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_intros]:
assumes "uniformly_continuous_on s g"
shows "uniformly_continuous_on s (λx. 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_intros]:
fixes f :: "'a::metric_space => 'b::real_normed_vector"
assumes "uniformly_continuous_on s f"
shows "uniformly_continuous_on s (λx. c *⇩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_intros]:
fixes f :: "'a::metric_space => 'b::real_normed_vector"
shows "uniformly_continuous_on s f ==> uniformly_continuous_on s (λx. - f x)"
unfolding uniformly_continuous_on_def dist_minus .

fixes f g :: "'a::metric_space => 'b::real_normed_vector"
assumes "uniformly_continuous_on s f"
and "uniformly_continuous_on s g"
shows "uniformly_continuous_on s (λx. f x + g x)"
using assms
unfolding uniformly_continuous_on_sequentially
by (auto intro: tendsto_add_zero)

lemma uniformly_continuous_on_diff[continuous_intros]:
fixes f :: "'a::metric_space => 'b::real_normed_vector"
assumes "uniformly_continuous_on s f"
and "uniformly_continuous_on s g"
shows "uniformly_continuous_on s (λx. f x - g x)"
using assms uniformly_continuous_on_add [of s f "- g"]
by (simp add: fun_Compl_def uniformly_continuous_on_minus)

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

lemmas continuous_at_compose = isCont_o

lemma uniformly_continuous_on_compose[continuous_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: "∀x∈f  s. ∀x'∈f  s. dist x' x < d --> dist (g x') (g x) < e"
using assms(2) unfolding uniformly_continuous_on_def by auto
obtain d' where "d'>0" "∀x∈s. ∀x'∈s. dist x' x < d' --> dist (f x') (f x) < d"
using d > 0 using assms(1) unfolding uniformly_continuous_on_def by auto
then have "∃d>0. ∀x∈s. ∀x'∈s. dist x' x < d --> dist ((g o f) x') ((g o f) x) < e"
using d>0 using d by auto
}
then show ?thesis
using assms unfolding uniformly_continuous_on_def by auto
qed

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

lemma continuous_at_open:
"continuous (at x) f <-> (∀t. open t ∧ f x ∈ t --> (∃s. open s ∧ x ∈ s ∧ (∀x' ∈ s. (f x') ∈ 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 o x) ----> (f x0)"
proof (rule topological_tendstoI)
fix S
assume "open S" "f x0 ∈ S"
then obtain T where T_def: "open T" "x0 ∈ T" "∀x∈T. f x ∈ S"
using assms continuous_at_open by metis
then have "eventually (λn. x n ∈ T) sequentially"
using assms T_def by (auto simp: tendsto_def)
then show "eventually (λn. (f o x) n ∈ S) sequentially"
using T_def by (auto elim!: eventually_elim1)
qed

lemma continuous_on_open:
"continuous_on s f <->
(∀t. openin (subtopology euclidean (f  s)) t -->
openin (subtopology euclidean s) {x ∈ s. f x ∈ t})"
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:
"continuous_on s f <->
(∀t. closedin (subtopology euclidean (f  s)) t -->
closedin (subtopology euclidean s) {x ∈ s. f x ∈ t})"
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 ∈ s. f x ∈ t}"
proof -
have *: "∀x. x ∈ s ∧ f x ∈ t <-> x ∈ s ∧ f x ∈ (t ∩ f  s)"
by auto
have "openin (subtopology euclidean (f  s)) (t ∩ f  s)"
using openin_open_Int[of t "f  s", OF assms(2)] unfolding openin_open by auto
then show ?thesis
using assms(1)[unfolded continuous_on_open, THEN spec[where x="t ∩ f  s"]]
using * by auto
qed

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

lemma continuous_open_preimage:
assumes "continuous_on s f"
and "open s"
and "open t"
shows "open {x ∈ s. f x ∈ t}"
proof-
obtain T where T: "open T" "{x ∈ s. f x ∈ t} = s ∩ T"
using continuous_open_in_preimage[OF assms(1,3)] unfolding openin_open by auto
then show ?thesis
using open_Int[of s T, OF assms(2)] by auto
qed

lemma continuous_closed_preimage:
assumes "continuous_on s f"
and "closed s"
and "closed t"
shows "closed {x ∈ s. f x ∈ t}"
proof-
obtain T where "closed T" "{x ∈ s. f x ∈ t} = s ∩ T"
using continuous_closed_in_preimage[OF assms(1,3)]
unfolding closedin_closed by auto
then show ?thesis using closed_Int[of s T, OF assms(2)] by auto
qed

lemma continuous_open_preimage_univ:
"∀x. continuous (at x) f ==> open s ==> open {x. f x ∈ s}"
using continuous_open_preimage[of UNIV f s] open_UNIV continuous_at_imp_continuous_on by auto

lemma continuous_closed_preimage_univ:
"(∀x. continuous (at x) f) ==> closed s ==> closed {x. f x ∈ s}"
using continuous_closed_preimage[of UNIV f s] closed_UNIV continuous_at_imp_continuous_on by auto

lemma continuous_open_vimage: "∀x. continuous (at x) f ==> open s ==> open (f - s)"
unfolding vimage_def by (rule continuous_open_preimage_univ)

lemma continuous_closed_vimage: "∀x. continuous (at x) f ==> closed s ==> closed (f - s)"
unfolding vimage_def by (rule continuous_closed_preimage_univ)

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

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

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

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

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

lemma image_closure_subset:
assumes "continuous_on (closure s) f"
and "closed t"
and "(f  s) ⊆ t"
shows "f  (closure s) ⊆ t"
proof -
have "s ⊆ {x ∈ closure s. f x ∈ t}"
using assms(3) closure_subset by auto
moreover have "closed {x ∈ closure s. f x ∈ t}"
using continuous_closed_preimage[OF assms(1)] and assms(2) by auto
ultimately have "closure s = {x ∈ closure s . f x ∈ t}"
using closure_minimal[of s "{x ∈ closure s. f x ∈ t}"] by auto
then show ?thesis by auto
qed

lemma continuous_on_closure_norm_le:
fixes f :: "'a::metric_space => 'b::real_normed_vector"
assumes "continuous_on (closure s) f"
and "∀y ∈ s. norm(f y) ≤ b"
and "x ∈ (closure s)"
shows "norm (f x) ≤ b"
proof -
have *: "f  s ⊆ cball 0 b"
using assms(2)[unfolded mem_cball_0[symmetric]] by auto
show ?thesis
using image_closure_subset[OF assms(1) closed_cball[of 0 b] *] assms(3)
unfolding subset_eq
apply (erule_tac x="f x" in ballE)
apply (auto simp add: dist_norm)
done
qed

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

lemma continuous_within_avoid:
fixes f :: "'a::metric_space => 'b::t1_space"
assumes "continuous (at x within s) f"
and "f x ≠ a"
shows "∃e>0. ∀y ∈ s. dist x y < e --> f y ≠ a"
proof -
obtain U where "open U" and "f x ∈ U" and "a ∉ U"
using t1_space [OF f x ≠ a] by fast
have "(f ---> f x) (at x within s)"
using assms(1) by (simp add: continuous_within)
then have "eventually (λy. f y ∈ U) (at x within s)"
using open U and f x ∈ U
unfolding tendsto_def by fast
then have "eventually (λy. f y ≠ a) (at x within s)"
using a ∉ U by (fast elim: eventually_mono [rotated])
then show ?thesis
using f x ≠ a by (auto simp: dist_commute zero_less_dist_iff eventually_at)
qed

lemma continuous_at_avoid:
fixes f :: "'a::metric_space => 'b::t1_space"
assumes "continuous (at x) f"
and "f x ≠ a"
shows "∃e>0. ∀y. dist x y < e --> f y ≠ a"
using assms continuous_within_avoid[of x UNIV f a] by simp

lemma continuous_on_avoid:
fixes f :: "'a::metric_space => 'b::t1_space"
assumes "continuous_on s f"
and "x ∈ s"
and "f x ≠ a"
shows "∃e>0. ∀y ∈ s. dist x y < e --> f y ≠ a"
using assms(1)[unfolded continuous_on_eq_continuous_within, THEN bspec[where x=x],
OF assms(2)] continuous_within_avoid[of x s f a]
using assms(3)
by auto

lemma continuous_on_open_avoid:
fixes f :: "'a::metric_space => 'b::t1_space"
assumes "continuous_on s f"
and "open s"
and "x ∈ s"
and "f x ≠ a"
shows "∃e>0. ∀y. dist x y < e --> f y ≠ a"
using assms(1)[unfolded continuous_on_eq_continuous_at[OF assms(2)], THEN bspec[where x=x], OF assms(3)]
using continuous_at_avoid[of x f a] assms(4)
by auto

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

lemma continuous_levelset_open_in_cases:
fixes f :: "_ => 'b::t1_space"
shows "connected s ==> continuous_on s f ==>
openin (subtopology euclidean s) {x ∈ s. f x = a}
==> (∀x ∈ s. f x ≠ a) ∨ (∀x ∈ s. f x = a)"
unfolding connected_clopen
using continuous_closed_in_preimage_constant by auto

lemma continuous_levelset_open_in:
fixes f :: "_ => 'b::t1_space"
shows "connected s ==> continuous_on s f ==>
openin (subtopology euclidean s) {x ∈ s. f x = a} ==>
(∃x ∈ s. f x = a)  ==> (∀x ∈ s. f x = a)"
using continuous_levelset_open_in_cases[of s f ]
by meson

lemma continuous_levelset_open:
fixes f :: "_ => 'b::t1_space"
assumes "connected s"
and "continuous_on s f"
and "open {x ∈ s. f x = a}"
and "∃x ∈ s.  f x = a"
shows "∀x ∈ 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 ≠ 0"
and "open s"
shows "open((λx. c *⇩R x)  s)"
proof -
{
fix x
assume "x ∈ s"
then obtain e where "e>0"
and e:"∀x'. dist x' x < e --> x' ∈ 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[symmetric]] e>0 by auto
moreover
{
fix y
assume "dist y (c *⇩R x) < e * ¦c¦"
then have "norm ((1 / c) *⇩R y - x) < e"
unfolding dist_norm
using norm_scaleR[of c "(1 / c) *⇩R y - x", unfolded scaleR_right_diff_distrib, unfolded scaleR_scaleR] assms(1)
assms(1)[unfolded zero_less_abs_iff[symmetric]] by (simp del:zero_less_abs_iff)
then have "y ∈ op *⇩R c  s"
using rev_image_eqI[of "(1 / c) *⇩R y" s y "op *⇩R c"]
using e[THEN spec[where x="(1 / c) *⇩R y"]]
using assms(1)
unfolding dist_norm scaleR_scaleR
by auto
}
ultimately have "∃e>0. ∀x'. dist x' (c *⇩R x) < e --> x' ∈ op *⇩R c  s"
apply (rule_tac x="e * abs c" in exI)
apply auto
done
}
then show ?thesis unfolding open_dist by auto
qed

lemma minus_image_eq_vimage:
fixes A :: "'a::ab_group_add set"
shows "(λx. - x)  A = (λx. - x) - A"
by (auto intro!: image_eqI [where f="λx. - x"])

lemma open_negations:
fixes s :: "'a::real_normed_vector set"
shows "open s ==> open ((λx. - x)  s)"
using open_scaling [of "- 1" s] by simp

lemma open_translation:
fixes s :: "'a::real_normed_vector set"
assumes "open s"
shows "open((λx. a + x)  s)"
proof -
{
fix x
have "continuous (at x) (λx. x - a)"
by (intro continuous_diff continuous_at_id continuous_const)
}
moreover have "{x. x - a ∈ s} = op + a  s"
by force
ultimately show ?thesis using continuous_open_preimage_univ[of "λx. x - a" s]
using assms by auto
qed

lemma open_affinity:
fixes s :: "'a::real_normed_vector set"
assumes "open s"  "c ≠ 0"
shows "open ((λx. a + c *⇩R x)  s)"
proof -
have *: "(λx. a + c *⇩R x) = (λx. a + x) o (λx. c *⇩R x)"
unfolding o_def ..
have "op + a  op *⇩R c  s = (op + a o op *⇩R c)  s"
by auto
then show ?thesis
using assms open_translation[of "op *⇩R c  s" a]
unfolding *
by auto
qed

lemma interior_translation:
fixes s :: "'a::real_normed_vector set"
shows "interior ((λx. a + x)  s) = (λx. a + x)  (interior s)"
proof (rule set_eqI, rule)
fix x
assume "x ∈ interior (op + a  s)"
then obtain e where "e > 0" and e: "ball x e ⊆ op + a  s"
unfolding mem_interior by auto
then have "ball (x - a) e ⊆ s"
unfolding subset_eq Ball_def mem_ball dist_norm
by (auto simp add: diff_diff_eq)
then show "x ∈ op + a  interior s"
unfolding image_iff
apply (rule_tac x="x - a" in bexI)
unfolding mem_interior
using e > 0
apply auto
done
next
fix x
assume "x ∈ op + a  interior s"
then obtain y e where "e > 0" and e: "ball y e ⊆ 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 ∈ ball x e"
then have "z - a ∈ s"
using e[unfolded subset_eq, THEN bspec[where x="z - a"]]
unfolding mem_ball dist_norm y group_add_class.diff_diff_eq2 *
by auto
then have "z ∈ op + a  s"
unfolding image_iff by (auto intro!: bexI[where x="z - a"])
}
then have "ball x e ⊆ op + a  s"
unfolding subset_eq by auto
then show "x ∈ 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)
then show ?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:
"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:
"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:
"continuous (at x) f ==> continuous (at x) g ==> bounded_bilinear h ==>
continuous (at x) (λx. h (f x) (g x))"
unfolding continuous_at
using Lim_bilinear[of f "f x" "(at x)" g "g x" h]
by auto

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

lemma bilinear_continuous_on_compose:
"continuous_on s f ==> continuous_on s g ==> bounded_bilinear h ==>
continuous_on s (λx. 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 <->
(∀C. (∀c∈C. openin (subtopology euclidean S) c) ∧ S ⊆ \<Union>C -->
(∃D⊆C. finite D ∧ S ⊆ \<Union>D))"
proof safe
fix C
assume "compact S" and "∀c∈C. openin (subtopology euclidean S) c" and "S ⊆ \<Union>C"
then have "∀c∈{T. open T ∧ S ∩ T ∈ C}. open c" and "S ⊆ \<Union>{T. open T ∧ S ∩ T ∈ C}"
unfolding openin_open by force+
with compact S obtain D where "D ⊆ {T. open T ∧ S ∩ T ∈ C}" and "finite D" and "S ⊆ \<Union>D"
by (rule compactE)
then have "image (λT. S ∩ T) D ⊆ C ∧ finite (image (λT. S ∩ T) D) ∧ S ⊆ \<Union>(image (λT. S ∩ T) D)"
by auto
then show "∃D⊆C. finite D ∧ S ⊆ \<Union>D" ..
next
assume 1: "∀C. (∀c∈C. openin (subtopology euclidean S) c) ∧ S ⊆ \<Union>C -->
(∃D⊆C. finite D ∧ S ⊆ \<Union>D)"
show "compact S"
proof (rule compactI)
fix C
let ?C = "image (λT. S ∩ T) C"
assume "∀t∈C. open t" and "S ⊆ \<Union>C"
then have "(∀c∈?C. openin (subtopology euclidean S) c) ∧ S ⊆ \<Union>?C"
unfolding openin_open by auto
with 1 obtain D where "D ⊆ ?C" and "finite D" and "S ⊆ \<Union>D"
by metis
let ?D = "inv_into C (λT. S ∩ T)  D"
have "?D ⊆ C ∧ finite ?D ∧ S ⊆ \<Union>?D"
proof (intro conjI)
from D ⊆ ?C show "?D ⊆ C"
by (fast intro: inv_into_into)
from finite D show "finite ?D"
by (rule finite_imageI)
from S ⊆ \<Union>D show "S ⊆ \<Union>?D"
apply (rule subset_trans)
apply clarsimp
apply (frule subsetD [OF D ⊆ ?C, THEN f_inv_into_f])
apply (erule rev_bexI, fast)
done
qed
then show "∃D⊆C. finite D ∧ S ⊆ \<Union>D" ..
qed
qed

lemma connected_continuous_image:
assumes "continuous_on s f"
and "connected s"
shows "connected(f  s)"
proof -
{
fix T
assume as:
"T ≠ {}"
"T ≠ f  s"
"openin (subtopology euclidean (f  s)) T"
"closedin (subtopology euclidean (f  s)) T"
have "{x ∈ s. f x ∈ T} = {} ∨ {x ∈ s. f x ∈ 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 ∈ s. f x ∈ T}"]] as(3,4) by auto
then have False using as(1,2)
using as(4)[unfolded closedin_def topspace_euclidean_subtopology] by auto
}
then show ?thesis
unfolding connected_clopen by auto
qed

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 ≠ {}"
def [simp]: R ≡ "{(y, d). y ∈ s ∧ 0 < d ∧ ball y d ∩ s ⊆ {x ∈ s. f x ∈ ball (f y) (e/2) } }"
let ?b = "(λ(y, d). ball y (d/2))"
have "(∀r∈R. open (?b r))" "s ⊆ (\<Union>r∈R. ?b r)"
proof safe
fix y
assume "y ∈ s"
from continuous_open_in_preimage[OF f open_ball]
obtain T where "open T" and T: "{x ∈ s. f x ∈ ball (f y) (e/2)} = T ∩ s"
unfolding openin_subtopology open_openin by metis
then obtain d where "ball y d ⊆ T" "0 < d"
using 0 < e y ∈ s by (auto elim!: openE)
with T y ∈ s show "y ∈ (\<Union>r∈R. ?b r)"
by (intro UN_I[of "(y, d)"]) auto
qed auto
with s obtain D where D: "finite D" "D ⊆ R" "s ⊆ (\<Union>(y, d)∈D. ball y (d/2))"
by (rule compactE_image)
with s ≠ {} have [simp]: "!!x. x < Min (snd  D) <-> (∀(y, d)∈D. x < d)"
by (subst Min_gr_iff) auto
show "∃d>0. ∀x∈s. ∀x'∈s. dist x' x < d --> dist (f x') (f x) < e"
proof (rule, safe)
fix x x'
assume in_s: "x' ∈ s" "x ∈ s"
with D obtain y d where x: "x ∈ ball y (d/2)" "(y, d) ∈ 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 => 'b::metric_space => 'c::metric_space"
assumes "¬ trivial_limit F"
and "eventually (λn. continuous_on s (f n)) F"
and "∀e>0. eventually (λn. ∀x∈s. dist (f n x) (g x) < e) F"
shows "continuous_on s g"
proof -
{
fix x and e :: real
assume "x∈s" "e>0"
have "eventually (λn. ∀x∈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:"∀x∈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:"∀x'∈s. dist x' x < d --> dist (f n x') (f n x) < e / 3"
using n(2)[unfolded continuous_on_iff, THEN bspec[where x=x], OF x∈s, THEN spec[where x="e/3"]] by blast
{
fix y
assume "y ∈ s" and "dist y x < d"
then have "dist (f n y) (f n x) < e / 3"
by (rule d [rule_format])
then have "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∈s]
by auto
then have "dist (g y) (g x) < e"
using n(1)[THEN bspec[where x=y], OF y∈s]
using dist_triangle3 [of "g y" "g x" "f n y"]
by auto
}
then have "∃d>0. ∀x'∈s. dist x' x < d --> dist (g x') (g x) < e"
using d>0 by auto
}
then show ?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 <-> (∀x ∈ s. ∃e>0. ∀x'. abs(x' - x) < e --> x' ∈ s)"
unfolding open_dist dist_norm by simp

lemma islimpt_approachable_real:
fixes s :: "real set"
shows "x islimpt s <-> (∀e>0.  ∃x'∈ s. x' ≠ x ∧ abs(x' - x) < e)"
unfolding islimpt_approachable dist_norm by simp

lemma closed_real:
fixes s :: "real set"
shows "closed s <-> (∀x. (∀e>0.  ∃x' ∈ s. x' ≠ x ∧ abs(x' - x) < e) --> x ∈ s)"
unfolding closed_limpt islimpt_approachable dist_norm by simp

lemma continuous_at_real_range:
fixes f :: "'a::real_normed_vector => real"
shows "continuous (at x) f <-> (∀e>0. ∃d>0. ∀x'. norm(x' - x) < d --> abs(f x' - f x) < e)"
unfolding continuous_at
unfolding Lim_at
unfolding dist_nz[symmetric]
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)
apply auto
done

lemma continuous_on_real_range:
fixes f :: "'a::real_normed_vector => real"
shows "continuous_on s f <->
(∀x ∈ s. ∀e>0. ∃d>0. (∀x' ∈ 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 ≠ {}"
shows "∃x∈s. ∀y∈s. dist a y ≤ dist a x"
proof (rule continuous_attains_sup [OF assms])
{
fix x
assume "x∈s"
have "(dist a ---> dist a x) (at x within s)"
by (intro tendsto_dist tendsto_const tendsto_ident_at)
}
then show "continuous_on s (dist a)"
unfolding continuous_on ..
qed

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

lemma distance_attains_inf:
fixes a :: "'a::heine_borel"
assumes "closed s"
and "s ≠ {}"
shows "∃x∈s. ∀y∈s. dist a x ≤ dist a y"
proof -
from assms(2) obtain b where "b ∈ s" by auto
let ?B = "s ∩ cball a (dist b a)"
have "?B ≠ {}" using b ∈ 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 ∈ ?B" "∀y∈?B. dist a x ≤ dist a y"
by (metis continuous_attains_inf)
then show ?thesis by fastforce
qed

subsection {* Pasted sets *}

lemma bounded_Times:
assumes "bounded s" "bounded t"
shows "bounded (s × t)"
proof -
obtain x y a b where "∀z∈s. dist x z ≤ a" "∀z∈t. dist y z ≤ b"
using assms [unfolded bounded_def] by auto
then have "∀z∈s × t. dist (x, y) z ≤ sqrt (a⇧2 + b⇧2)"
by (auto simp add: dist_Pair_Pair real_sqrt_le_mono add_mono power_mono)
then show ?thesis unfolding bounded_any_center [where a="(x, y)"] by auto
qed

lemma mem_Times_iff: "x ∈ A × B <-> fst x ∈ A ∧ snd x ∈ B"
by (induct x) simp

lemma seq_compact_Times: "seq_compact s ==> seq_compact t ==> seq_compact (s × t)"
unfolding seq_compact_def
apply clarify
apply (drule_tac x="fst o f" in spec)
apply (drule mp, simp add: mem_Times_iff)
apply (clarify, rename_tac l1 r1)
apply (drule_tac x="snd o f o 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 o 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
apply (simp add: o_def)
done

lemma compact_Times:
assumes "compact s" "compact t"
shows "compact (s × t)"
proof (rule compactI)
fix C
assume C: "∀t∈C. open t" "s × t ⊆ \<Union>C"
have "∀x∈s. ∃a. open a ∧ x ∈ a ∧ (∃d⊆C. finite d ∧ a × t ⊆ \<Union>d)"
proof
fix x
assume "x ∈ s"
have "∀y∈t. ∃a b c. c ∈ C ∧ open a ∧ open b ∧ x ∈ a ∧ y ∈ b ∧ a × b ⊆ c" (is "∀y∈t. ?P y")
proof
fix y
assume "y ∈ t"
with x ∈ s C obtain c where "c ∈ C" "(x, y) ∈ c" "open c" by auto
then show "?P y" by (auto elim!: open_prod_elim)
qed
then obtain a b c where b: "!!y. y ∈ t ==> open (b y)"
and c: "!!y. y ∈ t ==> c y ∈ C ∧ open (a y) ∧ open (b y) ∧ x ∈ a y ∧ y ∈ b y ∧ a y × b y ⊆ c y"
by metis
then have "∀y∈t. open (b y)" "t ⊆ (\<Union>y∈t. b y)" by auto
from compactE_image[OF compact t this] obtain D where D: "D ⊆ t" "finite D" "t ⊆ (\<Union>y∈D. b y)"
by auto
moreover from D c have "(\<Inter>y∈D. a y) × t ⊆ (\<Union>y∈D. c y)"
by (fastforce simp: subset_eq)
ultimately show "∃a. open a ∧ x ∈ a ∧ (∃d⊆C. finite d ∧ a × t ⊆ \<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: "∀x∈s. open (a x)" "s ⊆ (\<Union>x∈s. a x)"
and d: "!!x. x ∈ s ==> d x ⊆ C ∧ finite (d x) ∧ a x × t ⊆ \<Union>d x"
unfolding subset_eq UN_iff by metis
moreover
from compactE_image[OF compact s a]
obtain e where e: "e ⊆ s" "finite e" and s: "s ⊆ (\<Union>x∈e. a x)"
by auto
moreover
{
from s have "s × t ⊆ (\<Union>x∈e. a x × t)"
by auto
also have "… ⊆ (\<Union>x∈e. \<Union>d x)"
using d e ⊆ s by (intro UN_mono) auto
finally have "s × t ⊆ (\<Union>x∈e. \<Union>d x)" .
}
ultimately show "∃C'⊆C. finite C' ∧ s × t ⊆ \<Union>C'"
by (intro exI[of _ "(\<Union>x∈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 ((λx. c *⇩R x)  s)"
proof -
let ?f = "λx. 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 ((λx. - x)  s)"
using compact_scaling [OF assms, of "- 1"] by auto

lemma compact_sums:
fixes s t :: "'a::real_normed_vector set"
assumes "compact s"
and "compact t"
shows "compact {x + y | x y. x ∈ s ∧ y ∈ t}"
proof -
have *: "{x + y | x y. x ∈ s ∧ y ∈ t} = (λz. fst z + snd z)  (s × t)"
apply auto
unfolding image_iff
apply (rule_tac x="(xa, y)" in bexI)
apply auto
done
have "continuous_on (s × t) (λz. fst z + snd z)"
unfolding continuous_on by (rule ballI) (intro tendsto_intros)
then show ?thesis
unfolding * using compact_continuous_image compact_Times [OF assms] by auto
qed

lemma compact_differences:
fixes s t :: "'a::real_normed_vector set"
assumes "compact s"
and "compact t"
shows "compact {x - y | x y. x ∈ s ∧ y ∈ t}"
proof-
have "{x - y | x y. x∈s ∧ y ∈ t} =  {x + y | x y. x ∈ s ∧ y ∈ (uminus  t)}"
apply auto
apply (rule_tac x= xa in exI)
apply auto
done
then show ?thesis
using compact_sums[OF assms(1) compact_negations[OF assms(2)]] by auto
qed

lemma compact_translation:
fixes s :: "'a::real_normed_vector set"
assumes "compact s"
shows "compact ((λx. a + x)  s)"
proof -
have "{x + y |x y. x ∈ s ∧ y ∈ {a}} = (λx. a + x)  s"
by auto
then show ?thesis
using compact_sums[OF assms compact_sing[of a]] by auto
qed

lemma compact_affinity:
fixes s :: "'a::real_normed_vector set"
assumes "compact s"
shows "compact ((λx. a + c *⇩R x)  s)"
proof -
have "op + a  op *⇩R c  s = (λx. a + c *⇩R x)  s"
by auto
then show ?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"
and "s ≠ {}"
shows "∃x∈s. ∃y∈s. ∀u∈s. ∀v∈s. dist u v ≤ dist x y"
proof -
have "compact (s × s)"
using compact s by (intro compact_Times)
moreover have "s × s ≠ {}"
using s ≠ {} by auto
moreover have "continuous_on (s × s) (λx. dist (fst x) (snd x))"
by (intro continuous_at_imp_continuous_on ballI continuous_intros)
ultimately show ?thesis
using continuous_attains_sup[of "s × s" "λx. dist (fst x) (snd x)"] by auto
qed

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

definition diameter :: "'a::metric_space set => real" where
"diameter S = (if S = {} then 0 else SUP (x,y):S×S. dist x y)"

lemma diameter_bounded_bound:
fixes s :: "'a :: metric_space set"
assumes s: "bounded s" "x ∈ s" "y ∈ s"
shows "dist x y ≤ diameter s"
proof -
from s obtain z d where z: "!!x. x ∈ s ==> dist z x ≤ d"
unfolding bounded_def by auto
have "bdd_above (split dist  (s×s))"
proof (intro bdd_aboveI, safe)
fix a b
assume "a ∈ s" "b ∈ s"
with z[of a] z[of b] dist_triangle[of a b z]
show "dist a b ≤ 2 * d"
by (simp add: dist_commute)
qed
moreover have "(x,y) ∈ s×s" using s by auto
ultimately have "dist x y ≤ (SUP (x,y):s×s. dist x y)"
by (rule cSUP_upper2) simp
with x ∈ s show ?thesis
by (auto simp add: diameter_def)
qed

lemma diameter_lower_bounded:
fixes s :: "'a :: metric_space set"
assumes s: "bounded s"
and d: "0 < d" "d < diameter s"
shows "∃x∈s. ∃y∈s. d < dist x y"
proof (rule ccontr)
assume contr: "¬ ?thesis"
moreover have "s ≠ {}"
using d by (auto simp add: diameter_def)
ultimately have "diameter s ≤ d"
by (auto simp: not_less diameter_def intro!: cSUP_least)
with d < diameter s show False by auto
qed

lemma diameter_bounded:
assumes "bounded s"
shows "∀x∈s. ∀y∈s. dist x y ≤ diameter s"
and "∀d>0. d < diameter s --> (∃x∈s. ∃y∈s. dist x y > d)"
using diameter_bounded_bound[of s] diameter_lower_bounded[of s] assms
by auto

lemma diameter_compact_attained:
assumes "compact s"
and "s ≠ {}"
shows "∃x∈s. ∃y∈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∈s" "y∈s"
and xy: "∀u∈s. ∀v∈s. dist u v ≤ dist x y"
using compact_sup_maxdistance[OF assms] by auto
then have "diameter s ≤ dist x y"
unfolding diameter_def
apply clarsimp
apply (rule cSUP_least)
apply fast+
done
then show ?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 ((λx. c *⇩R x)  s)"
proof (cases "c = 0")
case True then show ?thesis
by (auto simp add: image_constant_conv)
next
case False
from assms have "closed ((λx. inverse c *⇩R x) - s)"
by (simp add: continuous_closed_vimage)
also have "(λx. inverse c *⇩R x) - s = (λx. c *⇩R x)  s"
using c ≠ 0 by (auto elim: image_eqI [rotated])
finally show ?thesis .
qed

lemma closed_negations:
fixes s :: "'a::real_normed_vector set"
assumes "closed s"
shows "closed ((λx. -x)  s)"
using closed_scaling[OF assms, of "- 1"] by simp

lemma compact_closed_sums:
fixes s :: "'a::real_normed_vector set"
assumes "compact s" and "closed t"
shows "closed {x + y | x y. x ∈ s ∧ y ∈ t}"
proof -
let ?S = "{x + y |x y. x ∈ s ∧ y ∈ t}"
{
fix x l
assume as: "∀n. x n ∈ ?S"  "(x ---> l) sequentially"
from as(1) obtain f where f: "∀n. x n = fst (f n) + snd (f n)"  "∀n. fst (f n) ∈ s"  "∀n. snd (f n) ∈ t"
using choice[of "λn y. x n = (fst y) + (snd y) ∧ fst y ∈ s ∧ snd y ∈ t"] by auto
obtain l' r where "l'∈s" and r: "subseq r" and lr: "(((λn. fst (f n)) o r) ---> l') sequentially"
using assms(1)[unfolded compact_def, THEN spec[where x="λ n. fst (f n)"]] using f(2) by auto
have "((λn. 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
then have "l - l' ∈ t"
using assms(2)[unfolded closed_sequential_limits,
THEN spec[where x="λ n. snd (f (r n))"],
THEN spec[where x="l - l'"]]
using f(3)
by auto
then have "l ∈ ?S"
using l' ∈ s
apply auto
apply (rule_tac x=l' in exI)
apply (rule_tac x="l - l'" in exI)
apply auto
done
}
then show ?thesis
unfolding closed_sequential_limits by fast
qed

lemma closed_compact_sums:
fixes s t :: "'a::real_normed_vector set"
assumes "closed s"
and "compact t"
shows "closed {x + y | x y. x ∈ s ∧ y ∈ t}"
proof -
have "{x + y |x y. x ∈ t ∧ y ∈ s} = {x + y |x y. x ∈ s ∧ y ∈ t}"
apply auto
apply (rule_tac x=y in exI)
apply auto
apply (rule_tac x=y in exI)
apply auto
done
then show ?thesis
using compact_closed_sums[OF assms(2,1)] by simp
qed

lemma compact_closed_differences:
fixes s t :: "'a::real_normed_vector set"
assumes "compact s"
and "closed t"
shows "closed {x - y | x y. x ∈ s ∧ y ∈ t}"
proof -
have "{x + y |x y. x ∈ s ∧ y ∈ uminus  t} =  {x - y |x y. x ∈ s ∧ y ∈ t}"
apply auto
apply (rule_tac x=xa in exI)
apply auto
apply (rule_tac x=xa in exI)
apply auto
done
then show ?thesis
using compact_closed_sums[OF assms(1) closed_negations[OF assms(2)]] by auto
qed

lemma closed_compact_differences:
fixes s t :: "'a::real_normed_vector set"
assumes "closed s"
and "compact t"
shows "closed {x - y | x y. x ∈ s ∧ y ∈ t}"
proof -
have "{x + y |x y. x ∈ s ∧ y ∈ uminus  t} = {x - y |x y. x ∈ s ∧ y ∈ t}"
apply auto
apply (rule_tac x=xa in exI)
apply auto
apply (rule_tac x=xa in exI)
apply auto
done
then show ?thesis
using closed_compact_sums[OF assms(1) compact_negations[OF assms(2)]] by simp
qed

lemma closed_translation:
fixes a :: "'a::real_normed_vector"
assumes "closed s"
shows "closed ((λx. a + x)  s)"
proof -
have "{a + y |y. y ∈ s} = (op + a  s)" by auto
then show ?thesis
using compact_closed_sums[OF compact_sing[of a] assms] by auto
qed

lemma translation_Compl:
fixes a :: "'a::ab_group_add"
shows "(λx. a + x)  (- t) = - ((λx. a + x)  t)"
apply (auto simp add: image_iff)
apply (rule_tac x="x - a" in bexI)
apply auto
done

lemma translation_UNIV:
fixes a :: "'a::ab_group_add"
shows "range (λx. a + x) = UNIV"
apply (auto simp add: image_iff)
apply (rule_tac x="x - a" in exI)
apply auto
done

lemma translation_diff:
fixes a :: "'a::ab_group_add"
shows "(λx. a + x)  (s - t) = ((λx. a + x)  s) - ((λx. a + x)  t)"
by auto

lemma closure_translation:
fixes a :: "'a::real_normed_vector"
shows "closure ((λx. a + x)  s) = (λx. 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)
apply auto
done
show ?thesis
unfolding closure_interior translation_Compl
using interior_translation[of a "- s"]
unfolding *
by auto
qed

lemma frontier_translation:
fixes a :: "'a::real_normed_vector"
shows "frontier((λx. a + x)  s) = (λx. 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"
assumes "closed s"
and "a ∉ s"
shows "∃d>0. ∀x∈s. d ≤ dist a x"
proof (cases "s = {}")
case True
then show ?thesis by(auto intro!: exI[where x=1])
next
case False
from assms obtain x where "x∈s" "∀y∈s. dist a x ≤ dist a y"
using s ≠ {} distance_attains_inf [of s a] by blast
with x∈s show ?thesis using dist_pos_lt[of a x] anda ∉ s
by blast
qed

lemma separate_compact_closed:
fixes s t :: "'a::heine_borel set"
assumes "compact s"
and t: "closed t" "s ∩ t = {}"
shows "∃d>0. ∀x∈s. ∀y∈t. d ≤ dist x y"
proof cases
assume "s ≠ {} ∧ t ≠ {}"
then have "s ≠ {}" "t ≠ {}" by auto
let ?inf = "λx. 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 ∈ s" "∀y∈s. ?inf x ≤ ?inf y"
using continuous_attains_inf[OF compact s s ≠ {}] by auto
then have "0 < ?inf x"
using t t ≠ {} in_closed_iff_infdist_zero by (auto simp: less_le infdist_nonneg)
moreover have "∀x'∈s. ∀y∈t. ?inf x ≤ 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 ∩ t = {}"
shows "∃d>0. ∀x∈s. ∀y∈t. d ≤ dist x y"
proof -
have *: "t ∩ 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)
apply (auto simp add: dist_commute)
done
qed

subsection {* Closure of halfspaces and hyperplanes *}

lemma isCont_open_vimage:
assumes "!!x. isCont f x"
and "open s"
shows "open (f - s)"
proof -
from assms(1) have "continuous_on UNIV f"
unfolding isCont_def continuous_on_def by simp
then have "open {x ∈ UNIV. f x ∈ s}"
using open_UNIV open s by (rule continuous_open_preimage)
then show "open (f - s)"
by (simp add: vimage_def)
qed

lemma isCont_closed_vimage:
assumes "!!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 => real"
assumes f: "!!x. isCont f x"
and g: "!!x. isCont g x"
shows "open {x. f x < g x}"
proof -
have "open ((λx. g x - f x) - {0<..})"
using isCont_diff [OF g f] open_real_greaterThan
by (rule isCont_open_vimage)
also have "((λx. 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 => real"
assumes f: "!!x. isCont f x"
and g: "!!x. isCont g x"
shows "closed {x. f x ≤ g x}"
proof -
have "closed ((λx. g x - f x) - {0..})"
using isCont_diff [OF g f] closed_real_atLeast
by (rule isCont_closed_vimage)
also have "((λx. g x - f x) - {0..}) = {x. f x ≤ g x}"
by auto
finally show ?thesis .
qed

lemma closed_Collect_eq:
fixes f g :: "'a::t2_space => 'b::t2_space"
assumes f: "!!x. isCont f x"
and g: "!!x. isCont g x"
shows "closed {x. f x = g x}"
proof -
have "open {(x::'b, y::'b). x ≠ y}"
unfolding open_prod_def by (auto dest!: hausdorff)
then have "closed {(x::'b, y::'b). x = y}"
unfolding closed_def split_def Collect_neg_eq .
with isCont_Pair [OF f g]
have "closed ((λx. (f x, g x)) - {(x, y). x = y})"
by (rule isCont_closed_vimage)
also have "… = {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 ≤ b}"
by (simp add: closed_Collect_le)

lemma closed_halfspace_ge: "closed {x. inner a x ≥ b}"
by (simp add: closed_Collect_le)

lemma closed_hyperplane: "closed {x. inner a x = b}"
by (simp add: closed_Collect_eq)

lemma closed_halfspace_component_le: "closed {x::'a::euclidean_space. x•i ≤ a}"
by (simp add: closed_Collect_le)

lemma closed_halfspace_component_ge: "closed {x::'a::euclidean_space. x•i ≥ a}"
by (simp add: closed_Collect_le)

lemma closed_interval_left:
fixes b :: "'a::euclidean_space"
shows "closed {x::'a. ∀i∈Basis. x•i ≤ b•i}"
by (simp add: Collect_ball_eq closed_INT closed_Collect_le)

lemma closed_interval_right:
fixes a :: "'a::euclidean_space"
shows "closed {x::'a. ∀i∈Basis. a•i ≤ x•i}"
by (simp add: Collect_ball_eq closed_INT closed_Collect_le)

text {* Openness of halfspaces. *}

lemma open_halfspace_lt: "open {x. inner a x < b}"
by (simp add: open_Collect_less)

lemma open_halfspace_gt: "open {x. inner a x > b}"
by (simp add: open_Collect_less)

lemma open_halfspace_component_lt: "open {x::'a::euclidean_space. x•i < a}"
by (simp add: open_Collect_less)

lemma open_halfspace_component_gt: "open {x::'a::euclidean_space. x•i > a}"
by (simp add: open_Collect_less)

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

lemma Lim_component_le:
fixes f :: "'a => 'b::euclidean_space"
assumes "(f ---> l) net"
and "¬ (trivial_limit net)"
and "eventually (λx. f(x)•i ≤ b) net"
shows "l•i ≤ 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 => 'b::euclidean_space"
assumes "(f ---> l) net"
and "¬ (trivial_limit net)"
and "eventually (λx. b ≤ (f x)•i) net"
shows "b ≤ l•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 => 'b::euclidean_space"
assumes net: "(f ---> l) net" "¬ trivial_limit net"
and ev:"eventually (λx. f(x)•i = b) net"
shows "l•i = b"
using ev[unfolded order_eq_iff eventually_conj_iff]
using Lim_component_ge[OF net, of b i]
using Lim_component_le[OF net, of i b]
by auto

text {* Limits relative to a union. *}

lemma eventually_within_Un:
"eventually P (at x within (s ∪ t)) <->
eventually P (at x within s) ∧ 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 ∪ t)) <->
(f ---> l) (at x within s) ∧ (f ---> l) (at x within t)"
unfolding tendsto_def
by (auto simp add: eventually_within_Un)

lemma Lim_topological:
"(f ---> l) net <->
trivial_limit net ∨ (∀S. open S --> l ∈ S --> eventually (λx. f x ∈ 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"
and "x ∈ s"
and "y ∈ s"
and "inner a x ≤ b"
and "b ≤ inner a y"
shows "∃z ∈ s. inner a z = b"
proof (rule ccontr)
assume as:"¬ (∃z∈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 ∩ ?B = {}" by auto
moreover
have "s ⊆ ?A ∪ ?B" using as by auto
ultimately
show False
using assms(1)[unfolded connected_def not_ex,
THEN spec[where x="?A"], THEN spec[where x="?B"]]
using assms(2-5)
by auto
qed

lemma connected_ivt_component:
fixes x::"'a::euclidean_space"
shows "connected s ==>
x ∈ s ==> y ∈ s ==>
x•k ≤ a ==> a ≤ y•k ==> (∃z∈s.  z•k = a)"
using connected_ivt_hyperplane[of s x y "k::'a" a]
by (auto simp: inner_commute)

subsection {* Intervals *}

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

instance euclidean_space ⊆ second_countable_topology
proof
def a ≡ "λf :: 'a => (real × real). ∑i∈Basis. fst (f i) *⇩R i"
then have a: "!!f. (∑i∈Basis. fst (f i) *⇩R i) = a f"
by simp
def b ≡ "λf :: 'a => (real × real). ∑i∈Basis. snd (f i) *⇩R i"
then have b: "!!f. (∑i∈Basis. snd (f i) *⇩R i) = b f"
by simp
def B ≡ "(λf. box (a f) (b f))  (Basis ->⇩E (\<rat> × \<rat>))"

have "Ball B open" by (simp add: B_def open_box)
moreover have "(∀A. open A --> (∃B'⊆B. \<Union>B' = A))"
proof safe
fix A::"'a set"
assume "open A"
show "∃B'⊆B. \<Union>B' = A"
apply (rule exI[of _ "{b∈B. b ⊆ 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 "∃B::'a set set. countable B ∧ open = generate_topology B"
by (blast intro: topological_basis_imp_subbasis)
qed

instance euclidean_space ⊆ polish_space ..

lemma closed_cbox[intro]:
fixes a b :: "'a::euclidean_space"
shows "closed (cbox a b)"
proof -
have "closed (\<Inter>i∈Basis. (λx. x•i) - {a•i .. b•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∈Basis. (λx. x•i) - {a•i .. b•i}) = cbox a b"
by (auto simp add: cbox_def)
finally show "closed (cbox a b)" .
qed

lemma interior_cbox [intro]:
fixes a b :: "'a::euclidean_space"
shows "interior (cbox a b) = box a b" (is "?L = ?R")
proof(rule subset_antisym)
show "?R ⊆ ?L"
using box_subset_cbox open_box
by (rule interior_maximal)
{
fix x
assume "x ∈ interior (cbox a b)"
then obtain s where s: "open s" "x ∈ s" "s ⊆ cbox a b" ..
then obtain e where "e>0" and e:"∀x'. dist x' x < e --> x' ∈ cbox a b"
unfolding open_dist and subset_eq by auto
{
fix i :: 'a
assume i: "i ∈ Basis"
have "dist (x - (e / 2) *⇩R i) x < e"
and "dist (x + (e / 2) *⇩R i) x < e"
unfolding dist_norm
apply auto
unfolding norm_minus_cancel
using norm_Basis[OF i] e>0
apply auto
done
then have "a • i ≤ (x - (e / 2) *⇩R i) • i" and "(x + (e / 2) *⇩R i) • i ≤ b • i"
using e[THEN spec[where x="x - (e/2) *⇩R i"]]
and e[THEN spec[where x="x + (e/2) *⇩R i"]]
unfolding mem_box
using i
by blast+
then have "a • i < x • i" and "x • i < b • i"
using e>0 i
by (auto simp: inner_diff_left inner_Basis inner_add_left)
}
then have "x ∈ box a b"
unfolding mem_box by auto
}
then show "?L ⊆ ?R" ..
qed

lemma bounded_cbox:
fixes a :: "'a::euclidean_space"
shows "bounded (cbox a b)"
proof -
let ?b = "∑i∈Basis. ¦a•i¦ + ¦b•i¦"
{
fix x :: "'a"
assume x: "∀i∈Basis. a • i ≤ x • i ∧ x • i ≤ b • i"
{
fix i :: 'a
assume "i ∈ Basis"
then have "¦x•i¦ ≤ ¦a•i¦ + ¦b•i¦"
using x[THEN bspec[where x=i]] by auto
}
then have "(∑i∈Basis. ¦x • i¦) ≤ ?b"
apply -
apply (rule setsum_mono)
apply auto
done
then have "norm x ≤ ?b"
using norm_le_l1[of x] by auto
}
then show ?thesis
unfolding cbox_def bounded_iff by auto
qed

lemma bounded_box:
fixes a :: "'a::euclidean_space"
shows "bounded (box a b)"
using bounded_cbox[of a b]
using box_subset_cbox[of a b]
using bounded_subset[of "cbox a b" "box a b"]
by simp

lemma not_interval_univ:
fixes a :: "'a::euclidean_space"
shows "cbox a b ≠ UNIV" "box a b ≠ UNIV"
using bounded_box[of a b] bounded_cbox[of a b] by auto

lemma compact_cbox:
fixes a :: "'a::euclidean_space"
shows "compact (cbox a b)"
using bounded_closed_imp_seq_compact[of "cbox a b"] using bounded_cbox[of a b]
by (auto simp: compact_eq_seq_compact_metric)

lemma box_midpoint:
fixes a :: "'a::euclidean_space"
assumes "box a b ≠ {}"
shows "((1/2) *⇩R (a + b)) ∈ box a b"
proof -
{
fix i :: 'a
assume "i ∈ Basis"
then have "a • i < ((1 / 2) *⇩R (a + b)) • i ∧ ((1 / 2) *⇩R (a + b)) • i < b • i"
using assms[unfolded box_ne_empty, THEN bspec[where x=i]] by (auto simp: inner_add_left)
}
then show ?thesis unfolding mem_box by auto
qed

lemma open_cbox_convex:
fixes x :: "'a::euclidean_space"
assumes x: "x ∈ box a b"
and y: "y ∈ cbox a b"
and e: "0 < e" "e ≤ 1"
shows "(e *⇩R x + (1 - e) *⇩R y) ∈ box a b"
proof -
{
fix i :: 'a
assume i: "i ∈ Basis"
have "a • i = e * (a • i) + (1 - e) * (a • i)"
unfolding left_diff_distrib by simp
also have "… < e * (x • i) + (1 - e) * (y • i)"
using e unfolding mult_less_cancel_left and mult_le_cancel_left
apply simp_all
using x unfolding mem_box using i
apply simp
using y unfolding mem_box using i
apply simp
done
finally have "a • i < (e *⇩R x + (1 - e) *⇩R y) • i"
unfolding inner_simps by auto
moreover
{
have "b • i = e * (b•i) + (1 - e) * (b•i)"
unfolding left_diff_distrib by simp
also have "… > e * (x • i) + (1 - e) * (y • i)"
using e unfolding mult_less_cancel_left and mult_le_cancel_left
apply simp_all
using x
unfolding mem_box
using i
apply simp
using y
unfolding mem_box
using i
apply simp
done
finally have "(e *⇩R x + (1 - e) *⇩R y) • i < b • i"
unfolding inner_simps by auto
}
ultimately have "a • i < (e *⇩R x + (1 - e) *⇩R y) • i ∧ (e *⇩R x + (1 - e) *⇩R y) • i < b • i"
by auto
}
then show ?thesis
unfolding mem_box by auto
qed

lemma closure_box:
fixes a :: "'a::euclidean_space"
assumes "box a b ≠ {}"
shows "closure (box a b) = cbox a b"
proof -
have ab: "a <e b"
using assms by (simp add: eucl_less_def box_ne_empty)
let ?c = "(1 / 2) *⇩R (a + b)"
{
fix x
assume as:"x ∈ cbox a b"
def f ≡ "λn::nat. x + (inverse (real n + 1)) *⇩R (?c - x)"
{
fix n
assume fn: "f n <e b --> a <e f n --> f n = x" and xc: "x ≠ ?c"
have *: "0 < inverse (real n + 1)" "inverse (real n + 1) ≤ 1"
unfolding inverse_le_1_iff by auto
have "(inverse (real n + 1)) *⇩R ((1 / 2) *⇩R (a + b)) + (1 - inverse (real n + 1)) *⇩R x =
x + (inverse (real n + 1)) *⇩R (((1 / 2) *⇩R (a + b)) - x)"
by (auto simp add: algebra_simps)
then have "f n <e b" and "a <e f n"
using open_cbox_convex[OF box_midpoint[OF assms] as *]
unfolding f_def by (auto simp: box_def eucl_less_def)
then have False
using fn unfolding f_def using xc by auto
}
moreover
{
assume "¬ (f ---> x) sequentially"
{
fix e :: real
assume "e > 0"
then have "∃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)
apply auto
done
then obtain N :: nat where "inverse (real (N + 1)) < e"
by auto
then have "∀n≥N. inverse (real n + 1) < e"
apply auto
apply (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)
done
then have "∃N::nat. ∀n≥N. inverse (real n + 1) < e" by auto
}
then have "((λn. inverse (real n + 1)) ---> 0) sequentially"
unfolding lim_sequentially by(auto simp add: dist_norm)
then have "(f ---> x) sequentially"
unfolding f_def
using tendsto_add[OF tendsto_const, of "λn::nat. (inverse (real n + 1)) *⇩R ((1 / 2) *⇩R (a + b) - x)" 0 sequentially x]
using tendsto_scaleR [OF _ tendsto_const, of "λn::nat. inverse (real n + 1)" 0 sequentially "((1 / 2) *⇩R (a + b) - x)"]
by auto
}
ultimately have "x ∈ closure (box a b)"
using as and box_midpoint[OF assms]
unfolding closure_def
unfolding islimpt_sequential
by (cases "x=?c") (auto simp: in_box_eucl_less)
}
then show ?thesis
using closure_minimal[OF box_subset_cbox, of a b] by blast
qed

lemma bounded_subset_box_symmetric:
fixes s::"('a::euclidean_space) set"
assumes "bounded s"
shows "∃a. s ⊆ box (-a) a"
proof -
obtain b where "b>0" and b: "∀x∈s. norm x ≤ b"
using assms[unfolded bounded_pos] by auto
def a ≡ "(∑i∈Basis. (b + 1) *⇩R i)::'a"
{
fix x
assume "x ∈ s"
fix i :: 'a
assume i: "i ∈ Basis"
then have "(-a)•i < x•i" and "x•i < a•i"
using b[THEN bspec[where x=x], OF x∈s]
using Basis_le_norm[OF i, of x]
unfolding inner_simps and a_def
by auto
}
then show ?thesis
by (auto intro: exI[where x=a] simp add: box_def)
qed

lemma bounded_subset_open_interval:
fixes s :: "('a::euclidean_space) set"
shows "bounded s ==> (∃a b. s ⊆ box a b)"
by (auto dest!: bounded_subset_box_symmetric)

lemma bounded_subset_cbox_symmetric:
fixes s :: "('a::euclidean_space) set"
assumes "bounded s"
shows "∃a. s ⊆ cbox (-a) a"
proof -
obtain a where "s ⊆ box (-a) a"
using bounded_subset_box_symmetric[OF assms] by auto
then show ?thesis
using box_subset_cbox[of "-a" a] by auto
qed

lemma bounded_subset_cbox:
fixes s :: "('a::euclidean_space) set"
shows "bounded s ==> ∃a b. s ⊆ cbox a b"
using bounded_subset_cbox_symmetric[of s] by auto

lemma frontier_cbox:
fixes a b :: "'a::euclidean_space"
shows "frontier (cbox a b) = cbox a b - box a b"
unfolding frontier_def unfolding interior_cbox and closure_closed[OF closed_cbox] ..

lemma frontier_box:
fixes a b :: "'a::euclidean_space"
shows "frontier (box a b) = (if box a b = {} then {} else cbox a b - box a b)"
proof (cases "box a b = {}")
case True
then show ?thesis
using frontier_empty by auto
next
case False
then show ?thesis
unfolding frontier_def and closure_box[OF False] and interior_open[OF open_box]
by auto
qed

lemma inter_interval_mixed_eq_empty:
fixes a :: "'a::euclidean_space"
assumes "box c d ≠ {}"
shows "box a b ∩ cbox c d = {} <-> box a b ∩ box c d = {}"
unfolding closure_box[OF assms, symmetric]
unfolding open_inter_closure_eq_empty[OF open_box] ..

lemma diameter_cbox:
fixes a b::"'a::euclidean_space"
shows "(∀i ∈ Basis. a • i ≤ b • i) ==> diameter (cbox a b) = dist a b"
by (force simp add: diameter_def SUP_def simp del: Sup_image_eq intro!: cSup_eq_maximum setL2_mono
simp: euclidean_dist_l2[where 'a='a] cbox_def dist_norm)

lemma eucl_less_eq_halfspaces:
fixes a :: "'a::euclidean_space"
shows "{x. x <e a} = (\<Inter>i∈Basis. {x. x • i < a • i})"
"{x. a <e x} = (\<Inter>i∈Basis. {x. a • i < x • i})"
by (auto simp: eucl_less_def)

lemma eucl_le_eq_halfspaces:
fixes a :: "'a::euclidean_space"
shows "{x. ∀i∈Basis. x • i ≤ a • i} = (\<Inter>i∈Basis. {x. x • i ≤ a • i})"
"{x. ∀i∈Basis. a • i ≤ x • i} = (\<Inter>i∈Basis. {x. a • i ≤ x • i})"
by auto

lemma open_Collect_eucl_less[simp, intro]:
fixes a :: "'a::euclidean_space"
shows "open {x. x <e a}"
"open {x. a <e x}"
by (auto simp: eucl_less_eq_halfspaces open_halfspace_component_lt open_halfspace_component_gt)

lemma closed_Collect_eucl_le[simp, intro]:
fixes a :: "'a::euclidean_space"
shows "closed {x. ∀i∈Basis. a • i ≤ x • i}"
"closed {x. ∀i∈Basis. x • i ≤ a • i}"
unfolding eucl_le_eq_halfspaces
by (simp_all add: closed_INT closed_Collect_le)

lemma image_affinity_cbox: fixes m::real
fixes a b c :: "'a::euclidean_space"
shows "(λx. m *⇩R x + c)  cbox a b =
(if cbox a b = {} then {}
else (if 0 ≤ m then cbox (m *⇩R a + c) (m *⇩R b + c)
else cbox (m *⇩R b + c) (m *⇩R a + c)))"
proof (cases "m = 0")
case True
{
fix x
assume "∀i∈Basis. x • i ≤ c • i" "∀i∈Basis. c • i ≤ x • i"
then have "x = c"
apply -
apply (subst euclidean_eq_iff)
apply (auto intro: order_antisym)
done
}
moreover have "c ∈ cbox (m *⇩R a + c) (m *⇩R b + c)"
unfolding True by (auto simp add: cbox_sing)
ultimately show ?thesis using True by (auto simp: cbox_def)
next
case False
{
fix y
assume "∀i∈Basis. a • i ≤ y • i" "∀i∈Basis. y • i ≤ b • i" "m > 0"
then have "∀i∈Basis. (m *⇩R a + c) • i ≤ (m *⇩R y + c) • i" and "∀i∈Basis. (m *⇩R y + c) • i ≤ (m *⇩R b + c) • i"
by (auto simp: inner_distrib)
}
moreover
{
fix y
assume "∀i∈Basis. a • i ≤ y • i" "∀i∈Basis. y • i ≤ b • i" "m < 0"
then have "∀i∈Basis. (m *⇩R b + c) • i ≤ (m *⇩R y + c) • i" and "∀i∈Basis. (m *⇩R y + c) • i ≤ (m *⇩R a + c) • i"
by (auto simp add: mult_left_mono_neg inner_distrib)
}
moreover
{
fix y
assume "m > 0" and "∀i∈Basis. (m *⇩R a + c) • i ≤ y • i" and "∀i∈Basis. y • i ≤ (m *⇩R b + c) • i"
then have "y ∈ (λx. m *⇩R x + c)  cbox a b"
unfolding image_iff Bex_def mem_box
apply (intro exI[where x="(1 / m) *⇩R (y - c)"])
apply (auto simp add: pos_le_divide_eq pos_divide_le_eq mult.commute diff_le_iff inner_distrib inner_diff_left)
done
}
moreover
{
fix y
assume "∀i∈Basis. (m *⇩R b + c) • i ≤ y • i" "∀i∈Basis. y • i ≤ (m *⇩R a + c) • i" "m < 0"
then have "y ∈ (λx. m *⇩R x + c)  cbox a b"
unfolding image_iff Bex_def mem_box
apply (intro exI[where x="(1 / m) *⇩R (y - c)"])
apply (auto simp add: neg_le_divide_eq neg_divide_le_eq mult.commute diff_le_iff inner_distrib inner_diff_left)
done
}
ultimately show ?thesis using False by (auto simp: cbox_def)
qed

lemma image_smult_cbox:"(λx. m *⇩R (x::_::euclidean_space))  cbox a b =
(if cbox a b = {} then {} else if 0 ≤ m then cbox (m *⇩R a) (m *⇩R b) else cbox (m *⇩R b) (m *⇩R a))"
using image_affinity_cbox[of m 0 a b] by auto

subsection {* Homeomorphisms *}

definition "homeomorphism s t f g <->
(∀x∈s. (g(f x) = x)) ∧ (f  s = t) ∧ continuous_on s f ∧
(∀y∈t. (f(g y) = y)) ∧ (g  t = s) ∧ continuous_on t g"

definition homeomorphic :: "'a::topological_space set => 'b::topological_space set => bool"
(infixr "homeomorphic" 60)
where "s homeomorphic t ≡ (∃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 = "(λx. x)" in exI)
apply (rule_tac x = "(λx. x)" in exI)
apply blast
done

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

lemma homeomorphic_trans:
assumes "s homeomorphic t"
and "t homeomorphic u"
shows "s homeomorphic u"
proof -
obtain f1 g1 where fg1: "∀x∈s. g1 (f1 x) = x"  "f1  s = t"
"continuous_on s f1" "∀y∈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: "∀x∈t. g2 (f2 x) = x"  "f2  t = u" "continuous_on t f2"
"∀y∈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∈s"
then have "(g1 o g2) ((f2 o 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 o f1)  s = u"
using fg1(2) fg2(2) by auto
moreover have "continuous_on s (f2 o f1)"
using continuous_on_compose[OF fg1(3)] and fg2(3) unfolding fg1(2) by auto
moreover
{
fix y
assume "y∈u"
then have "(f2 o f1) ((g1 o 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 o g2)  u = s" using fg1(5) fg2(5) by auto
moreover have "continuous_on u (g1 o 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 o f1" in exI)
apply (rule_tac x="g1 o g2" in exI)
apply auto
done
qed

lemma homeomorphic_minimal:
"s homeomorphic t <->
(∃f g. (∀x∈s. f(x) ∈ t ∧ (g(f(x)) = x)) ∧
(∀y∈t. g(y) ∈ s ∧ (f(g(y)) = y)) ∧
continuous_on s f ∧ 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)
apply auto
done

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

lemma homeomorphism_compact:
fixes f :: "'a::topological_space => 'b::t2_space"
assumes "compact s" "continuous_on s f"  "f  s = t"  "inj_on f s"
shows "∃g. homeomorphism s t f g"
proof -
def g ≡ "λx. SOME y. y∈s ∧ f y = x"
have g: "∀x∈s. g (f x) = x"
using assms(3) assms(4)[unfolded inj_on_def] unfolding g_def by auto
{
fix y
assume "y ∈ t"
then obtain x where x:"f x = y" "x∈s"
using assms(3) by auto
then have "g (f x) = x" using g by auto
then have "f (g y) = y" unfolding x(1)[symmetric] by auto
}
then have g':"∀x∈t. f (g x) = x" by auto
moreover
{
fix x
have "x∈s ==> x ∈ 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∈g  t"
then obtain y where y:"y∈t" "g y = x" by auto
then obtain x' where x':"x'∈s" "f x' = y"
using assms(3) by auto
then have "x ∈ s"
unfolding g_def
using someI2[of "λb. b∈s ∧ f b = y" x' "λx. x∈s"]
unfolding y(2)[symmetric] and g_def
by auto
}
ultimately have "x∈s <-> x ∈ g  t" ..
}
then have "g  t = s" by auto
ultimately show ?thesis
unfolding homeomorphism_def homeomorphic_def
apply (rule_tac x=g in exI)
using g and assms(3) and continuous_on_inv[OF assms(2,1), of g, unfolded assms(3)] and assms(2)
apply auto
done
qed

lemma homeomorphic_compact:
fixes f :: "'a::topological_space => 'b::t2_space"
shows "compact s ==> continuous_on s f ==> (f  s = t) ==> inj_on f s ==> s homeomorphic t"
unfolding homeomorphic_def by (metis homeomorphism_compact)

text{* Preservation of topological properties. *}

lemma homeomorphic_compactness: "s homeomorphic t ==> (compact s <-> 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 ≠ 0"
shows "s homeomorphic ((λx. c *⇩R x)  s)"
unfolding homeomorphic_minimal
apply (rule_tac x="λx. c *⇩R x" in exI)
apply (rule_tac x="λx. (1 / c) *⇩R x" in exI)
using assms
apply (auto simp add: continuous_intros)
done

lemma homeomorphic_translation:
fixes s :: "'a::real_normed_vector set"
shows "s homeomorphic ((λx. a + x)  s)"
unfolding homeomorphic_minimal
apply (rule_tac x="λx. a + x" in exI)
apply (rule_tac x="λx. -a + x" in exI)
using continuous_on_add [OF continuous_on_const continuous_on_id, of s a]
continuous_on_add [OF continuous_on_const continuous_on_id, of "plus a  s" "- a"]
apply auto
done

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

lemma homeomorphic_balls:
fixes a b ::"'a::real_normed_vector"
assumes "0 < d"  "0 < e"
shows "(ball a d) homeomorphic  (ball b e)" (is ?th)
and "(cball a d) homeomorphic (cball b e)" (is ?cth)
proof -
show ?th unfolding homeomorphic_minimal
apply(rule_tac x="λx. b + (e/d) *⇩R (x - a)" in exI)
apply(rule_tac x="λx. a + (d/e) *⇩R (x - b)" in exI)
using assms
apply (auto intro!: continuous_intros
simp: dist_commute dist_norm pos_divide_less_eq mult_strict_left_mono)
done
show ?cth unfolding homeomorphic_minimal
apply(rule_tac x="λx. b + (e/d) *⇩R (x - a)" in exI)
apply(rule_tac x="λx. a + (d/e) *⇩R (x - b)" in exI)
using assms
apply (auto intro!: continuous_intros
simp: dist_commute dist_norm pos_divide_le_eq mult_strict_left_mono)
done
qed

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

lemma cauchy_isometric:
assumes e: "e > 0"
and s: "subspace s"
and f: "bounded_linear f"
and normf: "∀x∈s. norm (f x) ≥ e * norm x"
and xs: "∀n. x n ∈ 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:"∀n≥N. norm (f (x n) - f (x N)) < e * d"
using cf[unfolded cauchy o_def dist_norm, THEN spec[where x="e*d"]] e
by auto
{
fix n
assume "n≥N"
have "e * norm (x n - x N) ≤ 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 ≤ n N unfolding f.diff[symmetric] by auto
finally have "norm (x n - x N) < d" using e>0 by simp
}
then have "∃N. ∀n≥N. norm (x n - x N) < d" by auto
}
then show ?thesis unfolding cauchy and dist_norm by auto
qed

lemma complete_isometric_image:
assumes "0 < e"
and s: "subspace s"
and f: "bounded_linear f"
and normf: "∀x∈s. norm(f x) ≥ e * norm(x)"
and cs: "complete s"
shows "complete (f  s)"
proof -
{
fix g
assume as:"∀n::nat. g n ∈ f  s" and cfg:"Cauchy g"
then obtain x where "∀n. x n ∈ s ∧ g n = f (x n)"
using choice[of "λ n xa. xa ∈ s ∧ g n = f xa"]
by auto
then have x:"∀n. x n ∈ s"  "∀n. g n = f (x n)"
by auto
then have "f o x = g"
unfolding fun_eq_iff
by auto
then obtain l where "l∈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
then have "∃l∈f  s. (g ---> l) sequentially"
using linear_continuous_at[OF f, unfolded continuous_at_sequentially, THEN spec[where x=x], of l]
unfolding f o x = g
by auto
}
then show ?thesis
unfolding complete_def by auto
qed

lemma injective_imp_isometric:
fixes f :: "'a::euclidean_space => 'b::euclidean_space"
assumes s: "closed s" "subspace s"
and f: "bounded_linear f" "∀x∈s. f x = 0 --> x = 0"
shows "∃e>0. ∀x∈s. norm (f x) ≥ e * norm x"
proof (cases "s ⊆ {0::'a}")
case True
{
fix x
assume "x ∈ s"
then have "x = 0" using True by auto
then have "norm x ≤ norm (f x)" by auto
}
then show ?thesis by (auto intro!: exI[where x=1])
next
interpret f: bounded_linear f by fact
case False
then obtain a where a: "a ≠ 0" "a ∈ s"
by auto
from False have "s ≠ {}"
by auto
let ?S = "{f x| x. (x ∈ s ∧ norm x = norm a)}"
let ?S' = "{x::'a. x∈s ∧ 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
then have "compact ?S''"
using compact_frontier[OF compact_cball, of 0 "norm a"] by auto
moreover have "?S' = s ∩ ?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
then have "closed ?S" using compact_imp_closed by auto
moreover have "?S ≠ {}" using a by auto
ultimately obtain b' where "b'∈?S" "∀y∈?S. norm b' ≤ norm y"
using distance_attains_inf[of ?S 0] unfolding dist_0_norm by auto
then obtain b where "b∈s"
and ba: "norm b = norm a"
and b: "∀x∈{x ∈ s. norm x = norm a}. norm (f b) ≤ norm (f x)"
unfolding *[symmetric] unfolding image_iff by auto

let ?e = "norm (f b) / norm b"
have "norm b > 0" using ba and a and norm_ge_zero by auto
moreover have "norm (f b) > 0"
using f(2)[THEN bspec[where x=b], OF b∈s]
using norm b >0
unfolding zero_less_norm_iff
by auto
ultimately have "0 < norm (f b) / norm b" by simp
moreover
{
fix x
assume "x∈s"
then have "norm (f b) / norm b * norm x ≤ norm (f x)"
proof (cases "x=0")
case True
then show "norm (f b) / norm b * norm x ≤ norm (f x)" by auto
next
case False
then have *: "0 < norm a / norm x"
using a≠0
unfolding zero_less_norm_iff[symmetric] by simp
have "∀c. ∀x∈s. c *⇩R x ∈ s"
using s[unfolded subspace_def] by auto
then have "(norm a / norm x) *⇩R x ∈ {x ∈ s. norm x = norm a}"
using x∈s and x≠0 by auto
then show "norm (f b) / norm b * norm x ≤ norm (f x)"
using b[THEN bspec[where x="(norm a / norm x) *⇩R x"]]
unfolding f.scaleR and ba using x≠0 a≠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 => 'b::euclidean_space"
assumes "subspace s" "bounded_linear f" "∀x∈s. f x = 0 --> x = 0" "closed s"
shows "closed(f  s)"
proof -
obtain e where "e > 0" and e: "∀x∈s. e * norm x ≤ 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[symmetric] by auto
qed

subsection {* Some properties of a canonical subspace *}

lemma subspace_substandard:
"subspace {x::'a::euclidean_space. (∀i∈Basis. P i --> x•i = 0)}"
unfolding subspace_def by (auto simp: inner_add_left)

lemma closed_substandard:
"closed {x::'a::euclidean_space. ∀i∈Basis. P i --> x•i = 0}" (is "closed ?A")
proof -
let ?D = "{i∈Basis. P i}"
have "closed (\<Inter>i∈?D. {x::'a. x•i = 0})"
by (simp add: closed_INT closed_Collect_eq)
also have "(\<Inter>i∈?D. {x::'a. x•i = 0}) = ?A"
by auto
finally show "closed ?A" .
qed

lemma dim_substandard:
assumes d: "d ⊆ Basis"
shows "dim {x::'a::euclidean_space. ∀i∈Basis. i ∉ d --> x•i = 0} = card d" (is "dim ?A = _")
proof (rule dim_unique)
show "d ⊆ ?A"
using d by (auto simp: inner_Basis)
show "independent d"
using independent_mono [OF independent_Basis d] .
show "?A ⊆ span d"
proof (clarify)
fix x assume x: "∀i∈Basis. i ∉ d --> x • i = 0"
have "finite d"
using finite_subset [OF d finite_Basis] .
then have "(∑i∈d. (x • i) *⇩R i) ∈ span d"
by (simp add: span_setsum span_clauses)
also have "(∑i∈d. (x • i) *⇩R i) = (∑i∈Basis. (x • i) *⇩R i)"
by (rule setsum.mono_neutral_cong_left [OF finite_Basis d]) (auto simp add: x)
finally show "x ∈ span d"
unfolding euclidean_representation .
qed
qed simp

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

lemma ex_card:
assumes "n ≤ card A"
shows "∃S⊆A. card S = n"
proof cases
assume "finite A"
from ex_bij_betw_nat_finite[OF this] obtain f where f: "bij_betw f {0..<card A} A" ..
moreover from f n ≤ card A have "{..< n} ⊆ {..< card A}" "inj_on f {..< n}"
by (auto simp: bij_betw_def intro: subset_inj_on)
ultimately have "f  {..< n} ⊆ A" "card (f  {..< n}) = n"
by (auto simp: bij_betw_def card_image)
then show ?thesis by blast
next
assume "¬ finite A"
with n ≤ 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 ≤ 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 ⊆ Basis"
by auto
let ?t = "{x::'a. ∀i∈Basis. i ∉ d --> x•i = 0}"
have "∃f. linear f ∧ f  {x::'a. ∀i∈Basis. i ∉ d --> x • i = 0} = s ∧
inj_on f {x::'a. ∀i∈Basis. i ∉ d --> x • i = 0}"
using dim_substandard[of d] t d assms
by (intro subspace_isomorphism[OF subspace_substandard[of "λi. i ∉ d"]]) (auto simp: inner_Basis)
then obtain f where f:
"linear f"
"f  {x. ∀i∈Basis. i ∉ d --> x • i = 0} = s"
"inj_on f {x. ∀i∈Basis. i ∉ d --> x • i = 0}"
by blast
interpret f: bounded_linear f
using f unfolding linear_conv_bounded_linear by auto
{
fix x
have "x∈?t ==> f x = 0 ==> 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 ≤ ?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
then show ?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 ≤ y <-> x ≤ inverse(m) * y + -(c / m))"
by (simp add: field_simps)

lemma real_le_affinity:
"0 < (m::'a::linordered_field) ==> (y ≤ m * x + c <-> inverse(m) * y + -(c / m) ≤ x)"
by (simp add: field_simps)

lemma real_affinity_lt:
"0 < (m::'a::linordered_field) ==> (m * x + c < y <-> x < inverse(m) * y + -(c / m))"
by (simp add: field_simps)

lemma real_lt_affinity:
"0 < (m::'a::linordered_field) ==> (y < m * x + c <-> inverse(m) * y + -(c / m) < x)"
by (simp add: field_simps)

lemma real_affinity_eq:
"(m::'a::linordered_field) ≠ 0 ==> (m * x + c = y <-> x = inverse(m) * y + -(c / m))"
by (simp add: field_simps)

lemma real_eq_affinity:
"(m::'a::linordered_field) ≠ 0 ==> (y = m * x + c  <-> inverse(m) * y + -(c / m) = x)"
by (simp add: field_simps)

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

lemma banach_fix:
assumes s: "complete s" "s ≠ {}"
and c: "0 ≤ c" "c < 1"
and f: "(f  s) ⊆ s"
and lipschitz: "∀x∈s. ∀y∈s. dist (f x) (f y) ≤ c * dist x y"
shows "∃!x∈s. f x = x"
proof -
have "1 - c > 0" using c by auto

from s(2) obtain z0 where "z0 ∈ s" by auto
def z ≡ "λn. (f ^^ n) z0"
{
fix n :: nat
have "z n ∈ s" unfolding z_def
proof (induct n)
case 0
then show ?case using z0 ∈ s by auto
next
case Suc
then show ?case using f by auto qed
} note z_in_s = this

def d ≡ "dist (z 0) (z 1)"

have fzn:"!!n. f (z n) = z (Suc n)" unfolding z_def by auto
{
fix n :: nat
have "dist (z n) (z (Suc n)) ≤ (c ^ n) * d"
proof (induct n)
case 0
then show ?case
unfolding d_def by auto
next
case (Suc m)
then have "c * dist (z m) (z (Suc m)) ≤ c ^ Suc m * d"
using 0 ≤ c
using mult_left_mono[of "dist (z m) (z (Suc m))" "c ^ m * d" c]
by auto
then show ?case
using lipschitz[THEN bspec[where x="z m"], OF z_in_s, THEN bspec[where x="z (Suc m)"], OF z_in_s]
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)) ≤ (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)) ≤
(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 "… ≤ (1 - c) * (dist (z m) (z (m + k)) + c ^ (m + k) * d)"
using cf_z[of "m + k"] and c by auto
also have "… ≤ c ^ m * d * (1 - c ^ k) + (1 - c) * c ^ (m + k) * d"
using Suc by (auto simp add: field_simps)
also have "… = (c ^ m) * (d * (1 - c ^ k) + (1 - c) * c ^ k * d)"
unfolding power_add by (auto simp add: field_simps)
also have "… ≤ (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"
then have "∃N. ∀m n. N ≤ m ∧ N ≤ n --> dist (z m) (z n) < e"
proof (cases "d = 0")
case True
have *: "!!x. ((1 - c) * x ≤ 0) = (x ≤ 0)" using 1 - c > 0
by (metis mult_zero_left mult.commute real_mult_le_cancel_iff1)
from True have "!!n. z n = z0" using cf_z2[of 0] and c unfolding z_def
by (simp add: *)
then show ?thesis using e>0 by auto
next
case False
then have "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 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≥N" "n≥N"
have *:"c ^ n ≤ c ^ N" using n≥N and c
using power_decreasing[OF n≥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 d>0 0 < 1 - c by auto

have "dist (z m) (z n) ≤ 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 "… ≤ 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 "… < (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 "… = e * (1 - c ^ (m - n))"
using c and d>0 and 1 - c > 0 by auto
also have "… ≤ 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 ≤ m" "N ≤ n"
then have "dist (z n) (z m) < e"
proof (cases "n = m")
case True
then show ?thesis using e>0 by auto
next
case False
then show ?thesis using as and *[of n m] *[of m n]
unfolding nat_neq_iff by (auto simp add: dist_commute)
qed
}
then show ?thesis by auto
qed
}
then have "Cauchy z"
unfolding cauchy_def by auto
then obtain x where "x∈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 ≡ "dist (f x) x"
have "e = 0"
proof (rule ccontr)
assume "e ≠ 0"
then have "e > 0"
unfolding e_def using zero_le_dist[of "f x" x]
by (metis dist_eq_0_iff dist_nz e_def)
then obtain N where N:"∀n≥N. dist (z n) x < e / 2"
using x[unfolded lim_sequentially, THEN spec[where x="e/2"]] by auto
then have N':"dist (z N) x < e / 2" by auto

have *: "c * dist (z N) x ≤ 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) ≤ 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∈s
using c
by auto
also have "… < e / 2"
using N' and c using * by auto
finally show False
unfolding fzn
using N[THEN spec[where x="Suc N"]] and dist_triangle_half_r[of "z (Suc N)" "f x" e x]
unfolding e_def
by auto
qed
then have "f x = x" unfolding e_def by auto
moreover
{
fix y
assume "f y = y" "y∈s"
then have "dist x y ≤ c * dist x y"
using lipschitz[THEN bspec[where x=x], THEN bspec[where x=y]]
using x∈s and f x = x
by auto
then have "dist x y = 0"
unfolding mult_le_cancel_right1
using c and zero_le_dist[of x y]
by auto
then have "y = x" by auto
}
ultimately show ?thesis using x∈s by blast+
qed

subsection {* Edelstein fixed point theorem *}

lemma edelstein_fix:
fixes s :: "'a::metric_space set"
assumes s: "compact s" "s ≠ {}"
and gs: "(g  s) ⊆ s"
and dist: "∀x∈s. ∀y∈s. x ≠ y --> dist (g x) (g y) < dist x y"
shows "∃!x∈s. g x = x"
proof -
let ?D = "(λx. (x, x))  s"
have D: "compact ?D" "?D ≠ {}"
by (rule compact_continuous_image)
(auto intro!: s continuous_Pair continuous_within_id simp: continuous_on_eq_continuous_within)

have "!!x y e. x ∈ s ==> y ∈ s ==> 0 < e ==> dist y x < e ==> 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 (λx. dist ((g o 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 ∈ s" and le: "!!x. x ∈ s ==> dist (g a) a ≤ dist (g x) x"
using continuous_attains_inf[OF D cont] by auto

have "g a = a"
proof (rule ccontr)
assume "g a ≠ a"
with a ∈ 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 ≤ dist (g (g a)) (g a)"
using a ∈ s gs by (intro le) auto
ultimately show False by auto
qed
moreover have "!!x. x ∈ s ==> g x = x ==> x = a"
using dist[THEN bspec[where x=a]] g a = a and a∈s by auto
ultimately show "∃!x∈s. g x = x" using a ∈ s by blast
qed

no_notation
eucl_less (infix "<e" 50)

end
`