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
*)

header {* 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 < e2 / 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 (natceiling ¦x • b¦)"
      by (rule real_natceiling_ge)
    also have "… ≤ real (natceiling (Max ((λb. ¦x • b¦)`Basis)))"
      by (auto intro!: natceiling_mono)
    also have "… < real (natceiling (Max ((λb. ¦x • b¦)`Basis)) + 1)"
      by simp
    finally have "¦x • b¦ < real (natceiling (Max ((λb. ¦x • b¦)`Basis)) + 1)" . }
  then have "!!x::'a. ∃n::nat. ∀b∈Basis. ¦x • b¦ < real n"
    by auto
  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 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)

lemma frontier_straddle:
  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! *)

lemma Lim_cong_within(*[cong add]*):
  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

lemma Lim_cong_at(*[cong add]*):
  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 x`A)"
  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 have "eventually (λx. ∀b∈B. x ∈ b) F"
      by (auto intro!: eventually_Ball_finite)
    with F(2) have "eventually (λx. x ∈ U ∩ (\<Inter>B)) F"
      by eventually_elim auto
    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 _ "f`T"]) 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 tendsto_const 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{* Total boundedness *}

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

fun helper_1 :: "('a::metric_space set) => real => nat => 'a"
where
  "helper_1 s e n = (SOME y::'a. y ∈ s ∧ (∀m<n. ¬ (dist (helper_1 s e m) y < e)))"
declare helper_1.simps[simp del]

lemma seq_compact_imp_totally_bounded:
  assumes "seq_compact s"
  shows "∀e>0. ∃k. finite k ∧ k ⊆ s ∧ s ⊆ (\<Union>((λx. ball x e) ` k))"
proof (rule, rule, rule ccontr)
  fix e::real
  assume "e > 0"
  assume assm: "¬ (∃k. finite k ∧ k ⊆ s ∧ s ⊆ \<Union>((λx. ball x e) ` k))"
  def x  "helper_1 s e"
  {
    fix n
    have "x n ∈ s ∧ (∀m<n. ¬ dist (x m) (x n) < e)"
    proof (induct n rule: nat_less_induct)
      fix n
      def Q  "(λy. y ∈ s ∧ (∀m<n. ¬ dist (x m) y < e))"
      assume as: "∀m<n. x m ∈ s ∧ (∀ma<m. ¬ dist (x ma) (x m) < e)"
      have "¬ s ⊆ (\<Union>x∈x ` {0..<n}. ball x e)"
        using assm
        apply simp
        apply (erule_tac x="x ` {0 ..< n}" in allE)
        using as
        apply auto
        done
      then obtain z where z:"z∈s" "z ∉ (\<Union>x∈x ` {0..<n}. ball x e)"
        unfolding subset_eq by auto
      have "Q (x n)"
        unfolding x_def and helper_1.simps[of s e n]
        apply (rule someI2[where a=z])
        unfolding x_def[symmetric] and Q_def
        using z
        apply auto
        done
      then show "x n ∈ s ∧ (∀m<n. ¬ dist (x m) (x n) < e)"
        unfolding Q_def by auto
    qed
  }
  then have "∀n::nat. x n ∈ s" and x:"∀n. ∀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(1)[unfolded seq_compact_def, THEN spec[where x=x]] by auto
  from this(3) have "Cauchy (x o r)"
    using LIMSEQ_imp_Cauchy by auto
  then obtain N::nat where N:"∀m n. N ≤ m ∧ N ≤ n --> dist ((x o r) m) ((x o r) n) < e"
    unfolding cauchy_def using `e>0` by auto
  show False
    using N[THEN spec[where x=N], THEN spec[where x="N+1"]]
    using r[unfolded subseq_def, THEN spec[where x=N], THEN spec[where x="N+1"]]
    using x[THEN spec[where x="r (N+1)"], THEN spec[where x="r (N)"]]
    by auto
qed

subsubsection{* Heine-Borel theorem *}

lemma seq_compact_imp_heine_borel:
  fixes s :: "'a :: metric_space set"
  assumes "seq_compact s"
  shows "compact s"
proof -
  from seq_compact_imp_totally_bounded[OF `seq_compact s`]
  obtain f where f: "∀e>0. finite (f e) ∧ f e ⊆ s ∧ s ⊆ \<Union>((λx. ball x e) ` f 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 LIMSEQ_def, 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 LIMSEQ_def 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>((λx. ball x e) ` k)))"
    (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 LIMSEQ_def] 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 LIMSEQ_def] 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 x`and 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 LIMSEQ_def by blast
  }
  ultimately show ?thesis by auto
qed


subsection {* Continuity *}

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

lemma continuous_within_eps_delta:
  "continuous (at x within s) f <-> (∀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 LIMSEQ_def 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 LIMSEQ_def 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 LIMSEQ_def 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]],
        simp add: le)
  }
  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 .

lemma uniformly_continuous_on_add[continuous_intros]:
  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
  unfolding dist_norm tendsto_norm_zero_iff add_diff_add
  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
    apply auto
    apply (erule_tac x="a + xa" in allE)
    unfolding ab_group_add_class.diff_diff_eq[symmetric]
    apply auto
    done
  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 (snd`D) / 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 (a2 + b2)"
    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 _ "c`D"] exI[of _ "\<Inter>(a`D)"] 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] and`a ∉ 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)"
      apply (rule add_less_le_mono)
      using e unfolding mult_less_cancel_left and mult_le_cancel_left
      apply simp_all
      using x unfolding mem_box using i
      apply simp
      using y unfolding mem_box using i
      apply simp
      done
    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)"
        apply (rule add_less_le_mono)
        using e unfolding mult_less_cancel_left and mult_le_cancel_left
        apply simp_all
        using x
        unfolding mem_box
        using i
        apply simp
        using y
        unfolding mem_box
        using i
        apply simp
        done
      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 LIMSEQ_def 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 LIMSEQ_def, 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

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

no_notation
  eucl_less (infix "<e" 50)

end