Theory Topological_Spaces

(*  Title:      HOL/Topological_Spaces.thy
    Author:     Brian Huffman
    Author:     Johannes Hölzl
*)

section ‹Topological Spaces›

theory Topological_Spaces
  imports Main
begin

named_theorems continuous_intros "structural introduction rules for continuity"

subsection ‹Topological space›

class "open" =
  fixes "open" :: "'a set  bool"

class topological_space = "open" +
  assumes open_UNIV [simp, intro]: "open UNIV"
  assumes open_Int [intro]: "open S  open T  open (S  T)"
  assumes open_Union [intro]: "SK. open S  open (K)"
begin

definition closed :: "'a set  bool"
  where "closed S  open (- S)"

lemma open_empty [continuous_intros, intro, simp]: "open {}"
  using open_Union [of "{}"] by simp

lemma open_Un [continuous_intros, intro]: "open S  open T  open (S  T)"
  using open_Union [of "{S, T}"] by simp

lemma open_UN [continuous_intros, intro]: "xA. open (B x)  open (xA. B x)"
  using open_Union [of "B ` A"] by simp

lemma open_Inter [continuous_intros, intro]: "finite S  TS. open T  open (S)"
  by (induct set: finite) auto

lemma open_INT [continuous_intros, intro]: "finite A  xA. open (B x)  open (xA. B x)"
  using open_Inter [of "B ` A"] by simp

lemma openI:
  assumes "x. x  S  T. open T  x  T  T  S"
  shows "open S"
proof -
  have "open ({T. open T  T  S})" by auto
  moreover have "{T. open T  T  S} = S" by (auto dest!: assms)
  ultimately show "open S" by simp
qed

lemma open_subopen: "open S  (xS. T. open T  x  T  T  S)"
by (auto intro: openI)

lemma closed_empty [continuous_intros, intro, simp]: "closed {}"
  unfolding closed_def by simp

lemma closed_Un [continuous_intros, intro]: "closed S  closed T  closed (S  T)"
  unfolding closed_def by auto

lemma closed_UNIV [continuous_intros, intro, simp]: "closed UNIV"
  unfolding closed_def by simp

lemma closed_Int [continuous_intros, intro]: "closed S  closed T  closed (S  T)"
  unfolding closed_def by auto

lemma closed_INT [continuous_intros, intro]: "xA. closed (B x)  closed (xA. B x)"
  unfolding closed_def by auto

lemma closed_Inter [continuous_intros, intro]: "SK. closed S  closed (K)"
  unfolding closed_def uminus_Inf by auto

lemma closed_Union [continuous_intros, intro]: "finite S  TS. closed T  closed (S)"
  by (induct set: finite) auto

lemma closed_UN [continuous_intros, intro]:
  "finite A  xA. closed (B x)  closed (xA. B x)"
  using closed_Union [of "B ` A"] by simp

lemma open_closed: "open S  closed (- S)"
  by (simp add: closed_def)

lemma closed_open: "closed S  open (- S)"
  by (rule closed_def)

lemma open_Diff [continuous_intros, intro]: "open S  closed T  open (S - T)"
  by (simp add: closed_open Diff_eq open_Int)

lemma closed_Diff [continuous_intros, intro]: "closed S  open T  closed (S - T)"
  by (simp add: open_closed Diff_eq closed_Int)

lemma open_Compl [continuous_intros, intro]: "closed S  open (- S)"
  by (simp add: closed_open)

lemma closed_Compl [continuous_intros, intro]: "open S  closed (- S)"
  by (simp add: open_closed)

lemma open_Collect_neg: "closed {x. P x}  open {x. ¬ P x}"
  unfolding Collect_neg_eq by (rule open_Compl)

lemma open_Collect_conj:
  assumes "open {x. P x}" "open {x. Q x}"
  shows "open {x. P x  Q x}"
  using open_Int[OF assms] by (simp add: Int_def)

lemma open_Collect_disj:
  assumes "open {x. P x}" "open {x. Q x}"
  shows "open {x. P x  Q x}"
  using open_Un[OF assms] by (simp add: Un_def)

lemma open_Collect_ex: "(i. open {x. P i x})  open {x. i. P i x}"
  using open_UN[of UNIV "λi. {x. P i x}"] unfolding Collect_ex_eq by simp

lemma open_Collect_imp: "closed {x. P x}  open {x. Q x}  open {x. P x  Q x}"
  unfolding imp_conv_disj by (intro open_Collect_disj open_Collect_neg)

lemma open_Collect_const: "open {x. P}"
  by (cases P) auto

lemma closed_Collect_neg: "open {x. P x}  closed {x. ¬ P x}"
  unfolding Collect_neg_eq by (rule closed_Compl)

lemma closed_Collect_conj:
  assumes "closed {x. P x}" "closed {x. Q x}"
  shows "closed {x. P x  Q x}"
  using closed_Int[OF assms] by (simp add: Int_def)

lemma closed_Collect_disj:
  assumes "closed {x. P x}" "closed {x. Q x}"
  shows "closed {x. P x  Q x}"
  using closed_Un[OF assms] by (simp add: Un_def)

lemma closed_Collect_all: "(i. closed {x. P i x})  closed {x. i. P i x}"
  using closed_INT[of UNIV "λi. {x. P i x}"] by (simp add: Collect_all_eq)

lemma closed_Collect_imp: "open {x. P x}  closed {x. Q x}  closed {x. P x  Q x}"
  unfolding imp_conv_disj by (intro closed_Collect_disj closed_Collect_neg)

lemma closed_Collect_const: "closed {x. P}"
  by (cases P) auto

end


subsection ‹Hausdorff and other separation properties›

class t0_space = topological_space +
  assumes t0_space: "x  y  U. open U  ¬ (x  U  y  U)"

class t1_space = topological_space +
  assumes t1_space: "x  y  U. open U  x  U  y  U"

instance t1_space  t0_space
  by standard (fast dest: t1_space)

context t1_space begin

lemma separation_t1: "x  y  (U. open U  x  U  y  U)"
  using t1_space[of x y] by blast

lemma closed_singleton [iff]: "closed {a}"
proof -
  let ?T = "{S. open S  a  S}"
  have "open ?T"
    by (simp add: open_Union)
  also have "?T = - {a}"
    by (auto simp add: set_eq_iff separation_t1)
  finally show "closed {a}"
    by (simp only: closed_def)
qed

lemma closed_insert [continuous_intros, simp]:
  assumes "closed S"
  shows "closed (insert a S)"
proof -
  from closed_singleton assms have "closed ({a}  S)"
    by (rule closed_Un)
  then show "closed (insert a S)"
    by simp
qed

lemma finite_imp_closed: "finite S  closed S"
  by (induct pred: finite) simp_all

end

text ‹T2 spaces are also known as Hausdorff spaces.›

class t2_space = topological_space +
  assumes hausdorff: "x  y  U V. open U  open V  x  U  y  V  U  V = {}"

instance t2_space  t1_space
  by standard (fast dest: hausdorff)

lemma (in t2_space) separation_t2: "x  y  (U V. open U  open V  x  U  y  V  U  V = {})"
  using hausdorff [of x y] by blast

lemma (in t0_space) separation_t0: "x  y  (U. open U  ¬ (x  U  y  U))"
  using t0_space [of x y] by blast


text ‹A classical separation axiom for topological space, the T3 axiom -- also called regularity:
if a point is not in a closed set, then there are open sets separating them.›

class t3_space = t2_space +
  assumes t3_space: "closed S  y  S  U V. open U  open V  y  U  S  V  U  V = {}"

text ‹A classical separation axiom for topological space, the T4 axiom -- also called normality:
if two closed sets are disjoint, then there are open sets separating them.›

class t4_space = t2_space +
  assumes t4_space: "closed S  closed T  S  T = {}  U V. open U  open V  S  U  T  V  U  V = {}"

text ‹T4 is stronger than T3, and weaker than metric.›

instance t4_space  t3_space
proof
  fix S and y::'a assume "closed S" "y  S"
  then show "U V. open U  open V  y  U  S  V  U  V = {}"
    using t4_space[of "{y}" S] by auto
qed

text ‹A perfect space is a topological space with no isolated points.›

class perfect_space = topological_space +
  assumes not_open_singleton: "¬ open {x}"

lemma (in perfect_space) UNIV_not_singleton: "UNIV  {x}"
  for x::'a
  by (metis (no_types) open_UNIV not_open_singleton)


subsection ‹Generators for toplogies›

inductive generate_topology :: "'a set set  'a set  bool" for S :: "'a set set"
  where
    UNIV: "generate_topology S UNIV"
  | Int: "generate_topology S (a  b)" if "generate_topology S a" and "generate_topology S b"
  | UN: "generate_topology S (K)" if "(k. k  K  generate_topology S k)"
  | Basis: "generate_topology S s" if "s  S"

hide_fact (open) UNIV Int UN Basis

lemma generate_topology_Union:
  "(k. k  I  generate_topology S (K k))  generate_topology S (kI. K k)"
  using generate_topology.UN [of "K ` I"] by auto

lemma topological_space_generate_topology: "class.topological_space (generate_topology S)"
  by standard (auto intro: generate_topology.intros)


subsection ‹Order topologies›

class order_topology = order + "open" +
  assumes open_generated_order: "open = generate_topology (range (λa. {..< a})  range (λa. {a <..}))"
begin

subclass topological_space
  unfolding open_generated_order
  by (rule topological_space_generate_topology)

lemma open_greaterThan [continuous_intros, simp]: "open {a <..}"
  unfolding open_generated_order by (auto intro: generate_topology.Basis)

lemma open_lessThan [continuous_intros, simp]: "open {..< a}"
  unfolding open_generated_order by (auto intro: generate_topology.Basis)

lemma open_greaterThanLessThan [continuous_intros, simp]: "open {a <..< b}"
   unfolding greaterThanLessThan_eq by (simp add: open_Int)

end

class linorder_topology = linorder + order_topology

lemma closed_atMost [continuous_intros, simp]: "closed {..a}"
  for a :: "'a::linorder_topology"
  by (simp add: closed_open)

lemma closed_atLeast [continuous_intros, simp]: "closed {a..}"
  for a :: "'a::linorder_topology"
  by (simp add: closed_open)

lemma closed_atLeastAtMost [continuous_intros, simp]: "closed {a..b}"
  for a b :: "'a::linorder_topology"
proof -
  have "{a .. b} = {a ..}  {.. b}"
    by auto
  then show ?thesis
    by (simp add: closed_Int)
qed

lemma (in order) less_separate:
  assumes "x < y"
  shows "a b. x  {..< a}  y  {b <..}  {..< a}  {b <..} = {}"
proof (cases "z. x < z  z < y")
  case True
  then obtain z where "x < z  z < y" ..
  then have "x  {..< z}  y  {z <..}  {z <..}  {..< z} = {}"
    by auto
  then show ?thesis by blast
next
  case False
  with x < y have "x  {..< y}" "y  {x <..}" "{x <..}  {..< y} = {}"
    by auto
  then show ?thesis by blast
qed

instance linorder_topology  t2_space
proof
  fix x y :: 'a
  show "x  y  U V. open U  open V  x  U  y  V  U  V = {}"
    using less_separate [of x y] less_separate [of y x]
    by (elim neqE; metis open_lessThan open_greaterThan Int_commute)
qed

lemma (in linorder_topology) open_right:
  assumes "open S" "x  S"
    and gt_ex: "x < y"
  shows "b>x. {x ..< b}  S"
  using assms unfolding open_generated_order
proof induct
  case UNIV
  then show ?case by blast
next
  case (Int A B)
  then obtain a b where "a > x" "{x ..< a}  A"  "b > x" "{x ..< b}  B"
    by auto
  then show ?case
    by (auto intro!: exI[of _ "min a b"])
next
  case UN
  then show ?case by blast
next
  case Basis
  then show ?case
    by (fastforce intro: exI[of _ y] gt_ex)
qed

lemma (in linorder_topology) open_left:
  assumes "open S" "x  S"
    and lt_ex: "y < x"
  shows "b<x. {b <.. x}  S"
  using assms unfolding open_generated_order
proof induction
  case UNIV
  then show ?case by blast
next
  case (Int A B)
  then obtain a b where "a < x" "{a <.. x}  A"  "b < x" "{b <.. x}  B"
    by auto
  then show ?case
    by (auto intro!: exI[of _ "max a b"])
next
  case UN
  then show ?case by blast
next
  case Basis
  then show ?case
    by (fastforce intro: exI[of _ y] lt_ex)
qed


subsection ‹Setup some topologies›

subsubsection ‹Boolean is an order topology›

class discrete_topology = topological_space +
  assumes open_discrete: "A. open A"

instance discrete_topology < t2_space
proof
  fix x y :: 'a
  assume "x  y"
  then show "U V. open U  open V  x  U  y  V  U  V = {}"
    by (intro exI[of _ "{_}"]) (auto intro!: open_discrete)
qed

instantiation bool :: linorder_topology
begin

definition open_bool :: "bool set  bool"
  where "open_bool = generate_topology (range (λa. {..< a})  range (λa. {a <..}))"

instance
  by standard (rule open_bool_def)

end

instance bool :: discrete_topology
proof
  fix A :: "bool set"
  have *: "{False <..} = {True}" "{..< True} = {False}"
    by auto
  have "A = UNIV  A = {}  A = {False <..}  A = {..< True}"
    using subset_UNIV[of A] unfolding UNIV_bool * by blast
  then show "open A"
    by auto
qed

instantiation nat :: linorder_topology
begin

definition open_nat :: "nat set  bool"
  where "open_nat = generate_topology (range (λa. {..< a})  range (λa. {a <..}))"

instance
  by standard (rule open_nat_def)

end

instance nat :: discrete_topology
proof
  fix A :: "nat set"
  have "open {n}" for n :: nat
  proof (cases n)
    case 0
    moreover have "{0} = {..<1::nat}"
      by auto
    ultimately show ?thesis
       by auto
  next
    case (Suc n')
    then have "{n} = {..<Suc n}  {n' <..}"
      by auto
    with Suc show ?thesis
      by (auto intro: open_lessThan open_greaterThan)
  qed
  then have "open (aA. {a})"
    by (intro open_UN) auto
  then show "open A"
    by simp
qed

instantiation int :: linorder_topology
begin

definition open_int :: "int set  bool"
  where "open_int = generate_topology (range (λa. {..< a})  range (λa. {a <..}))"

instance
  by standard (rule open_int_def)

end

instance int :: discrete_topology
proof
  fix A :: "int set"
  have "{..<i + 1}  {i-1 <..} = {i}" for i :: int
    by auto
  then have "open {i}" for i :: int
    using open_Int[OF open_lessThan[of "i + 1"] open_greaterThan[of "i - 1"]] by auto
  then have "open (aA. {a})"
    by (intro open_UN) auto
  then show "open A"
    by simp
qed


subsubsection ‹Topological filters›

definition (in topological_space) nhds :: "'a  'a filter"
  where "nhds a = (INF S{S. open S  a  S}. principal S)"

definition (in topological_space) at_within :: "'a  'a set  'a filter"
    ("at (_)/ within (_)" [1000, 60] 60)
  where "at a within s = inf (nhds a) (principal (s - {a}))"

abbreviation (in topological_space) at :: "'a  'a filter"  ("at")
  where "at x  at x within (CONST UNIV)"

abbreviation (in order_topology) at_right :: "'a  'a filter"
  where "at_right x  at x within {x <..}"

abbreviation (in order_topology) at_left :: "'a  'a filter"
  where "at_left x  at x within {..< x}"

lemma (in topological_space) nhds_generated_topology:
  "open = generate_topology T  nhds x = (INF S{ST. x  S}. principal S)"
  unfolding nhds_def
proof (safe intro!: antisym INF_greatest)
  fix S
  assume "generate_topology T S" "x  S"
  then show "(INF S{S  T. x  S}. principal S)  principal S"
    by induct
      (auto intro: INF_lower order_trans simp: inf_principal[symmetric] simp del: inf_principal)
qed (auto intro!: INF_lower intro: generate_topology.intros)

lemma (in topological_space) eventually_nhds:
  "eventually P (nhds a)  (S. open S  a  S  (xS. P x))"
  unfolding nhds_def by (subst eventually_INF_base) (auto simp: eventually_principal)

lemma eventually_eventually:
  "eventually (λy. eventually P (nhds y)) (nhds x) = eventually P (nhds x)"
  by (auto simp: eventually_nhds)

lemma (in topological_space) eventually_nhds_in_open:
  "open s  x  s  eventually (λy. y  s) (nhds x)"
  by (subst eventually_nhds) blast

lemma (in topological_space) eventually_nhds_x_imp_x: "eventually P (nhds x)  P x"
  by (subst (asm) eventually_nhds) blast

lemma (in topological_space) nhds_neq_bot [simp]: "nhds a  bot"
  by (simp add: trivial_limit_def eventually_nhds)

lemma (in t1_space) t1_space_nhds: "x  y  (F x in nhds x. x  y)"
  by (drule t1_space) (auto simp: eventually_nhds)

lemma (in topological_space) nhds_discrete_open: "open {x}  nhds x = principal {x}"
  by (auto simp: nhds_def intro!: antisym INF_greatest INF_lower2[of "{x}"])

lemma (in discrete_topology) nhds_discrete: "nhds x = principal {x}"
  by (simp add: nhds_discrete_open open_discrete)

lemma (in discrete_topology) at_discrete: "at x within S = bot"
  unfolding at_within_def nhds_discrete by simp

lemma (in discrete_topology) tendsto_discrete:
  "filterlim (f :: 'b  'a) (nhds y) F  eventually (λx. f x = y) F"
  by (auto simp: nhds_discrete filterlim_principal)

lemma (in topological_space) at_within_eq:
  "at x within s = (INF S{S. open S  x  S}. principal (S  s - {x}))"
  unfolding nhds_def at_within_def
  by (subst INF_inf_const2[symmetric]) (auto simp: Diff_Int_distrib)

lemma (in topological_space) eventually_at_filter:
  "eventually P (at a within s)  eventually (λx. x  a  x  s  P x) (nhds a)"
  by (simp add: at_within_def eventually_inf_principal imp_conjL[symmetric] conj_commute)

lemma (in topological_space) at_le: "s  t  at x within s  at x within t"
  unfolding at_within_def by (intro inf_mono) auto

lemma (in topological_space) eventually_at_topological:
  "eventually P (at a within s)  (S. open S  a  S  (xS. x  a  x  s  P x))"
  by (simp add: eventually_nhds eventually_at_filter)

lemma (in topological_space) at_within_open: "a  S  open S  at a within S = at a"
  unfolding filter_eq_iff eventually_at_topological by (metis open_Int Int_iff UNIV_I)

lemma (in topological_space) at_within_open_NO_MATCH:
  "a  s  open s  NO_MATCH UNIV s  at a within s = at a"
  by (simp only: at_within_open)

lemma (in topological_space) at_within_open_subset:
  "a  S  open S  S  T  at a within T = at a"
  by (metis at_le at_within_open dual_order.antisym subset_UNIV)

lemma (in topological_space) at_within_nhd:
  assumes "x  S" "open S" "T  S - {x} = U  S - {x}"
  shows "at x within T = at x within U"
  unfolding filter_eq_iff eventually_at_filter
proof (intro allI eventually_subst)
  have "eventually (λx. x  S) (nhds x)"
    using x  S ‹open S by (auto simp: eventually_nhds)
  then show "F n in nhds x. (n  x  n  T  P n) = (n  x  n  U  P n)" for P
    by eventually_elim (insert T  S - {x} = U  S - {x}, blast)
qed

lemma (in topological_space) at_within_empty [simp]: "at a within {} = bot"
  unfolding at_within_def by simp

lemma (in topological_space) at_within_union:
  "at x within (S  T) = sup (at x within S) (at x within T)"
  unfolding filter_eq_iff eventually_sup eventually_at_filter
  by (auto elim!: eventually_rev_mp)

lemma (in topological_space) at_eq_bot_iff: "at a = bot  open {a}"
  unfolding trivial_limit_def eventually_at_topological
  apply safe
   apply (case_tac "S = {a}")
    apply simp
   apply fast
  apply fast
  done

lemma (in perfect_space) at_neq_bot [simp]: "at a  bot"
  by (simp add: at_eq_bot_iff not_open_singleton)

lemma (in order_topology) nhds_order:
  "nhds x = inf (INF a{x <..}. principal {..< a}) (INF a{..< x}. principal {a <..})"
proof -
  have 1: "{S  range lessThan  range greaterThan. x  S} =
      (λa. {..< a}) ` {x <..}  (λa. {a <..}) ` {..< x}"
    by auto
  show ?thesis
    by (simp only: nhds_generated_topology[OF open_generated_order] INF_union 1 INF_image comp_def)
qed

lemma (in topological_space) filterlim_at_within_If:
  assumes "filterlim f G (at x within (A  {x. P x}))"
    and "filterlim g G (at x within (A  {x. ¬P x}))"
  shows "filterlim (λx. if P x then f x else g x) G (at x within A)"
proof (rule filterlim_If)
  note assms(1)
  also have "at x within (A  {x. P x}) = inf (nhds x) (principal (A  Collect P - {x}))"
    by (simp add: at_within_def)
  also have "A  Collect P - {x} = (A - {x})  Collect P"
    by blast
  also have "inf (nhds x) (principal ) = inf (at x within A) (principal (Collect P))"
    by (simp add: at_within_def inf_assoc)
  finally show "filterlim f G (inf (at x within A) (principal (Collect P)))" .
next
  note assms(2)
  also have "at x within (A  {x. ¬ P x}) = inf (nhds x) (principal (A  {x. ¬ P x} - {x}))"
    by (simp add: at_within_def)
  also have "A  {x. ¬ P x} - {x} = (A - {x})  {x. ¬ P x}"
    by blast
  also have "inf (nhds x) (principal ) = inf (at x within A) (principal {x. ¬ P x})"
    by (simp add: at_within_def inf_assoc)
  finally show "filterlim g G (inf (at x within A) (principal {x. ¬ P x}))" .
qed

lemma (in topological_space) filterlim_at_If:
  assumes "filterlim f G (at x within {x. P x})"
    and "filterlim g G (at x within {x. ¬P x})"
  shows "filterlim (λx. if P x then f x else g x) G (at x)"
  using assms by (intro filterlim_at_within_If) simp_all
lemma (in linorder_topology) at_within_order:
  assumes "UNIV  {x}"
  shows "at x within s =
    inf (INF a{x <..}. principal ({..< a}  s - {x}))
        (INF a{..< x}. principal ({a <..}  s - {x}))"
proof (cases "{x <..} = {}" "{..< x} = {}" rule: case_split [case_product case_split])
  case True_True
  have "UNIV = {..< x}  {x}  {x <..}"
    by auto
  with assms True_True show ?thesis
    by auto
qed (auto simp del: inf_principal simp: at_within_def nhds_order Int_Diff
      inf_principal[symmetric] INF_inf_const2 inf_sup_aci[where 'a="'a filter"])

lemma (in linorder_topology) at_left_eq:
  "y < x  at_left x = (INF a{..< x}. principal {a <..< x})"
  by (subst at_within_order)
     (auto simp: greaterThan_Int_greaterThan greaterThanLessThan_eq[symmetric] min.absorb2 INF_constant
           intro!: INF_lower2 inf_absorb2)

lemma (in linorder_topology) eventually_at_left:
  "y < x  eventually P (at_left x)  (b<x. y>b. y < x  P y)"
  unfolding at_left_eq
  by (subst eventually_INF_base) (auto simp: eventually_principal Ball_def)

lemma (in linorder_topology) at_right_eq:
  "x < y  at_right x = (INF a{x <..}. principal {x <..< a})"
  by (subst at_within_order)
     (auto simp: lessThan_Int_lessThan greaterThanLessThan_eq[symmetric] max.absorb2 INF_constant Int_commute
           intro!: INF_lower2 inf_absorb1)

lemma (in linorder_topology) eventually_at_right:
  "x < y  eventually P (at_right x)  (b>x. y>x. y < b  P y)"
  unfolding at_right_eq
  by (subst eventually_INF_base) (auto simp: eventually_principal Ball_def)

lemma eventually_at_right_less: "F y in at_right (x::'a::{linorder_topology, no_top}). x < y"
  using gt_ex[of x] eventually_at_right[of x] by auto

lemma trivial_limit_at_right_top: "at_right (top::_::{order_top,linorder_topology}) = bot"
  by (auto simp: filter_eq_iff eventually_at_topological)

lemma trivial_limit_at_left_bot: "at_left (bot::_::{order_bot,linorder_topology}) = bot"
  by (auto simp: filter_eq_iff eventually_at_topological)

lemma trivial_limit_at_left_real [simp]: "¬ trivial_limit (at_left x)"
  for x :: "'a::{no_bot,dense_order,linorder_topology}"
  using lt_ex [of x]
  by safe (auto simp add: trivial_limit_def eventually_at_left dest: dense)

lemma trivial_limit_at_right_real [simp]: "¬ trivial_limit (at_right x)"
  for x :: "'a::{no_top,dense_order,linorder_topology}"
  using gt_ex[of x]
  by safe (auto simp add: trivial_limit_def eventually_at_right dest: dense)

lemma (in linorder_topology) at_eq_sup_left_right: "at x = sup (at_left x) (at_right x)"
  by (auto simp: eventually_at_filter filter_eq_iff eventually_sup
      elim: eventually_elim2 eventually_mono)

lemma (in linorder_topology) eventually_at_split:
  "eventually P (at x)  eventually P (at_left x)  eventually P (at_right x)"
  by (subst at_eq_sup_left_right) (simp add: eventually_sup)

lemma (in order_topology) eventually_at_leftI:
  assumes "x. x  {a<..<b}  P x" "a < b"
  shows   "eventually P (at_left b)"
  using assms unfolding eventually_at_topological by (intro exI[of _ "{a<..}"]) auto

lemma (in order_topology) eventually_at_rightI:
  assumes "x. x  {a<..<b}  P x" "a < b"
  shows   "eventually P (at_right a)"
  using assms unfolding eventually_at_topological by (intro exI[of _ "{..<b}"]) auto

lemma eventually_filtercomap_nhds:
  "eventually P (filtercomap f (nhds x))  (S. open S  x  S  (x. f x  S  P x))"
  unfolding eventually_filtercomap eventually_nhds by auto

lemma eventually_filtercomap_at_topological:
  "eventually P (filtercomap f (at A within B))  
     (S. open S  A  S  (x. f x  S  B - {A}  P x))" (is "?lhs = ?rhs")
  unfolding at_within_def filtercomap_inf eventually_inf_principal filtercomap_principal 
          eventually_filtercomap_nhds eventually_principal by blast

lemma eventually_at_right_field:
  "eventually P (at_right x)  (b>x. y>x. y < b  P y)"
  for x :: "'a::{linordered_field, linorder_topology}"
  using linordered_field_no_ub[rule_format, of x]
  by (auto simp: eventually_at_right)

lemma eventually_at_left_field:
  "eventually P (at_left x)  (b<x. y>b. y < x  P y)"
  for x :: "'a::{linordered_field, linorder_topology}"
  using linordered_field_no_lb[rule_format, of x]
  by (auto simp: eventually_at_left)


subsubsection ‹Tendsto›

abbreviation (in topological_space)
  tendsto :: "('b  'a)  'a  'b filter  bool"  (infixr "" 55)
  where "(f  l) F  filterlim f (nhds l) F"

definition (in t2_space) Lim :: "'f filter  ('f  'a)  'a"
  where "Lim A f = (THE l. (f  l) A)"

lemma (in topological_space) tendsto_eq_rhs: "(f  x) F  x = y  (f  y) F"
  by simp

named_theorems tendsto_intros "introduction rules for tendsto"
setup ‹
  Global_Theory.add_thms_dynamic (binding‹tendsto_eq_intros›,
    fn context =>
      Named_Theorems.get (Context.proof_of context) named_theorems‹tendsto_intros›
      |> map_filter (try (fn thm => @{thm tendsto_eq_rhs} OF [thm])))

context topological_space begin

lemma tendsto_def:
   "(f  l) F  (S. open S  l  S  eventually (λx. f x  S) F)"
   unfolding nhds_def filterlim_INF filterlim_principal by auto

lemma tendsto_cong: "(f  c) F  (g  c) F" if "eventually (λx. f x = g x) F"
  by (rule filterlim_cong [OF refl refl that])

lemma tendsto_mono: "F  F'  (f  l) F'  (f  l) F"
  unfolding tendsto_def le_filter_def by fast

lemma tendsto_ident_at [tendsto_intros, simp, intro]: "((λx. x)  a) (at a within s)"
  by (auto simp: tendsto_def eventually_at_topological)

lemma tendsto_const [tendsto_intros, simp, intro]: "((λx. k)  k) F"
  by (simp add: tendsto_def)

lemma filterlim_at:
  "(LIM x F. f x :> at b within s)  eventually (λx. f x  s  f x  b) F  (f  b) F"
  by (simp add: at_within_def filterlim_inf filterlim_principal conj_commute)

lemma (in -)
  assumes "filterlim f (nhds L) F"
  shows tendsto_imp_filterlim_at_right:
          "eventually (λx. f x > L) F  filterlim f (at_right L) F"
    and tendsto_imp_filterlim_at_left:
          "eventually (λx. f x < L) F  filterlim f (at_left L) F"
  using assms by (auto simp: filterlim_at elim: eventually_mono)

lemma  filterlim_at_withinI:
  assumes "filterlim f (nhds c) F"
  assumes "eventually (λx. f x  A - {c}) F"
  shows   "filterlim f (at c within A) F"
  using assms by (simp add: filterlim_at)

lemma filterlim_atI:
  assumes "filterlim f (nhds c) F"
  assumes "eventually (λx. f x  c) F"
  shows   "filterlim f (at c) F"
  using assms by (intro filterlim_at_withinI) simp_all

lemma topological_tendstoI:
  "(S. open S  l  S  eventually (λx. f x  S) F)  (f  l) F"
  by (auto simp: tendsto_def)

lemma topological_tendstoD:
  "(f  l) F  open S  l  S  eventually (λx. f x  S) F"
  by (auto simp: tendsto_def)

lemma tendsto_bot [simp]: "(f  a) bot"
  by (simp add: tendsto_def)

lemma tendsto_eventually: "eventually (λx. f x = l) net  ((λx. f x)  l) net"
  by (rule topological_tendstoI) (auto elim: eventually_mono)

end

lemma (in topological_space) filterlim_within_subset:
  "filterlim f l (at x within S)  T  S  filterlim f l (at x within T)"
  by (blast intro: filterlim_mono at_le)

lemmas tendsto_within_subset = filterlim_within_subset

lemma (in order_topology) order_tendsto_iff:
  "(f  x) F  (l<x. eventually (λx. l < f x) F)  (u>x. eventually (λx. f x < u) F)"
  by (auto simp: nhds_order filterlim_inf filterlim_INF filterlim_principal)

lemma (in order_topology) order_tendstoI:
  "(a. a < y  eventually (λx. a < f x) F)  (a. y < a  eventually (λx. f x < a) F) 
    (f  y) F"
  by (auto simp: order_tendsto_iff)

lemma (in order_topology) order_tendstoD:
  assumes "(f  y) F"
  shows "a < y  eventually (λx. a < f x) F"
    and "y < a  eventually (λx. f x < a) F"
  using assms by (auto simp: order_tendsto_iff)

lemma (in linorder_topology) tendsto_max[tendsto_intros]:
  assumes X: "(X  x) net"
    and Y: "(Y  y) net"
  shows "((λx. max (X x) (Y x))  max x y) net"
proof (rule order_tendstoI)
  fix a
  assume "a < max x y"
  then show "eventually (λx. a < max (X x) (Y x)) net"
    using order_tendstoD(1)[OF X, of a] order_tendstoD(1)[OF Y, of a]
    by (auto simp: less_max_iff_disj elim: eventually_mono)
next
  fix a
  assume "max x y < a"
  then show "eventually (λx. max (X x) (Y x) < a) net"
    using order_tendstoD(2)[OF X, of a] order_tendstoD(2)[OF Y, of a]
    by (auto simp: eventually_conj_iff)
qed

lemma (in linorder_topology) tendsto_min[tendsto_intros]:
  assumes X: "(X  x) net"
    and Y: "(Y  y) net"
  shows "((λx. min (X x) (Y x))  min x y) net"
proof (rule order_tendstoI)
  fix a
  assume "a < min x y"
  then show "eventually (λx. a < min (X x) (Y x)) net"
    using order_tendstoD(1)[OF X, of a] order_tendstoD(1)[OF Y, of a]
    by (auto simp: eventually_conj_iff)
next
  fix a
  assume "min x y < a"
  then show "eventually (λx. min (X x) (Y x) < a) net"
    using order_tendstoD(2)[OF X, of a] order_tendstoD(2)[OF Y, of a]
    by (auto simp: min_less_iff_disj elim: eventually_mono)
qed

lemma (in order_topology)
  assumes "a < b"
  shows at_within_Icc_at_right: "at a within {a..b} = at_right a"
    and at_within_Icc_at_left:  "at b within {a..b} = at_left b"
  using order_tendstoD(2)[OF tendsto_ident_at assms, of "{a<..}"]
  using order_tendstoD(1)[OF tendsto_ident_at assms, of "{..<b}"]
  by (auto intro!: order_class.antisym filter_leI
      simp: eventually_at_filter less_le
      elim: eventually_elim2)

lemma (in order_topology) at_within_Icc_at: "a < x  x < b  at x within {a..b} = at x"
  by (rule at_within_open_subset[where S="{a<..<b}"]) auto

lemma (in t2_space) tendsto_unique:
  assumes "F  bot"
    and "(f  a) F"
    and "(f  b) F"
  shows "a = b"
proof (rule ccontr)
  assume "a  b"
  obtain U V where "open U" "open V" "a  U" "b  V" "U  V = {}"
    using hausdorff [OF a  b] by fast
  have "eventually (λx. f x  U) F"
    using (f  a) F ‹open U a  U by (rule topological_tendstoD)
  moreover
  have "eventually (λx. f x  V) F"
    using (f  b) F ‹open V b  V by (rule topological_tendstoD)
  ultimately
  have "eventually (λx. False) F"
  proof eventually_elim
    case (elim x)
    then have "f x  U  V" by simp
    with U  V = {} show ?case by simp
  qed
  with ¬ trivial_limit F show "False"
    by (simp add: trivial_limit_def)
qed

lemma (in t2_space) tendsto_const_iff:
  fixes a b :: 'a
  assumes "¬ trivial_limit F"
  shows "((λx. a)  b) F  a = b"
  by (auto intro!: tendsto_unique [OF assms tendsto_const])

lemma (in t2_space) tendsto_unique':
 assumes "F  bot"
 shows "1l. (f  l) F"
 using Uniq_def assms local.tendsto_unique by fastforce

lemma Lim_in_closed_set:
  assumes "closed S" "eventually (λx. f(x)  S) F" "F  bot" "(f  l) F"
  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) F"
    by (rule topological_tendstoD)
  with assms(2) have "eventually (λx. False) F"
    by (rule eventually_elim2) simp
  with assms(3) show "False"
    by (simp add: eventually_False)
qed

lemma (in t3_space) nhds_closed:
  assumes "x  A" and "open A"
  shows   "A'. x  A'  closed A'  A'  A  eventually (λy. y  A') (nhds x)"
proof -
  from assms have "U V. open U  open V  x  U  - A  V  U  V = {}"
    by (intro t3_space) auto
  then obtain U V where UV: "open U" "open V" "x  U" "-A  V" "U  V = {}"
    by auto
  have "eventually (λy. y  U) (nhds x)"
    using ‹open U and x  U by (intro eventually_nhds_in_open)
  hence "eventually (λy. y  -V) (nhds x)"
    by eventually_elim (use UV in auto)
  with UV show ?thesis by (intro exI[of _ "-V"]) auto
qed

lemma (in order_topology) increasing_tendsto:
  assumes bdd: "eventually (λn. f n  l) F"
    and en: "x. x < l  eventually (λn. x < f n) F"
  shows "(f  l) F"
  using assms by (intro order_tendstoI) (auto elim!: eventually_mono)

lemma (in order_topology) decreasing_tendsto:
  assumes bdd: "eventually (λn. l  f n) F"
    and en: "x. l < x  eventually (λn. f n < x) F"
  shows "(f  l) F"
  using assms by (intro order_tendstoI) (auto elim!: eventually_mono)

lemma (in order_topology) tendsto_sandwich:
  assumes ev: "eventually (λn. f n  g n) net" "eventually (λn. g n  h n) net"
  assumes lim: "(f  c) net" "(h  c) net"
  shows "(g  c) net"
proof (rule order_tendstoI)
  fix a
  show "a < c  eventually (λx. a < g x) net"
    using order_tendstoD[OF lim(1), of a] ev by (auto elim: eventually_elim2)
next
  fix a
  show "c < a  eventually (λx. g x < a) net"
    using order_tendstoD[OF lim(2), of a] ev by (auto elim: eventually_elim2)
qed

lemma (in t1_space) limit_frequently_eq:
  assumes "F  bot"
    and "frequently (λx. f x = c) F"
    and "(f  d) F"
  shows "d = c"
proof (rule ccontr)
  assume "d  c"
  from t1_space[OF this] obtain U where "open U" "d  U" "c  U"
    by blast
  with assms have "eventually (λx. f x  U) F"
    unfolding tendsto_def by blast
  then have "eventually (λx. f x  c) F"
    by eventually_elim (insert c  U, blast)
  with assms(2) show False
    unfolding frequently_def by contradiction
qed

lemma (in t1_space) tendsto_imp_eventually_ne:
  assumes  "(f  c) F" "c  c'"
  shows "eventually (λz. f z  c') F"
proof (cases "F=bot")
  case True
  thus ?thesis by auto
next
  case False
  show ?thesis
  proof (rule ccontr)
    assume "¬ eventually (λz. f z  c') F"
    then have "frequently (λz. f z = c') F"
      by (simp add: frequently_def)
    from limit_frequently_eq[OF False this (f  c) F] and c  c' show False
      by contradiction
  qed
qed

lemma (in linorder_topology) tendsto_le:
  assumes F: "¬ trivial_limit F"
    and x: "(f  x) F"
    and y: "(g  y) F"
    and ev: "eventually (λx. g x  f x) F"
  shows "y  x"
proof (rule ccontr)
  assume "¬ y  x"
  with less_separate[of x y] obtain a b where xy: "x < a" "b < y" "{..<a}  {b<..} = {}"
    by (auto simp: not_le)
  then have "eventually (λx. f x < a) F" "eventually (λx. b < g x) F"
    using x y by (auto intro: order_tendstoD)
  with ev have "eventually (λx. False) F"
    by eventually_elim (insert xy, fastforce)
  with F show False
    by (simp add: eventually_False)
qed

lemma (in linorder_topology) tendsto_lowerbound:
  assumes x: "(f  x) F"
      and ev: "eventually (λi. a  f i) F"
      and F: "¬ trivial_limit F"
  shows "a  x"
  using F x tendsto_const ev by (rule tendsto_le)

lemma (in linorder_topology) tendsto_upperbound:
  assumes x: "(f  x) F"
      and ev: "eventually (λi. a  f i) F"
      and F: "¬ trivial_limit F"
  shows "a  x"
  by (rule tendsto_le [OF F tendsto_const x ev])

lemma filterlim_at_within_not_equal:
  fixes f::"'a  'b::t2_space"
  assumes "filterlim f (at a within s) F"
  shows "eventually (λw. f ws  f w b) F"
proof (cases "a=b")
  case True
  then show ?thesis using assms by (simp add: filterlim_at)
next
  case False
  from hausdorff[OF this] obtain U V where UV:"open U" "open V" "a  U" "b  V" "U  V = {}"
    by auto  
  have "(f  a) F" using assms filterlim_at by auto
  then have "F x in F. f x  U" using UV unfolding tendsto_def by auto
  moreover have  "F x in F. f x  s  f xa" using assms filterlim_at by auto
  ultimately show ?thesis 
    apply eventually_elim
    using UV by auto
qed

subsubsection ‹Rules about const‹Lim›

lemma tendsto_Lim: "¬ trivial_limit net  (f  l) net  Lim net f = l"
  unfolding Lim_def using tendsto_unique [of net f] by auto

lemma Lim_ident_at: "¬ trivial_limit (at x within s)  Lim (at x within s) (λx. x) = x"
  by (rule tendsto_Lim[OF _ tendsto_ident_at]) auto

lemma eventually_Lim_ident_at:
  "(F y in at x within X. P (Lim (at x within X) (λx. x)) y) 
    (F y in at x within X. P x y)" for x::"'a::t2_space"
  by (cases "at x within X = bot") (auto simp: Lim_ident_at)

lemma filterlim_at_bot_at_right:
  fixes f :: "'a::linorder_topology  'b::linorder"
  assumes mono: "x y. Q x  Q y  x  y  f x  f y"
    and bij: "x. P x  f (g x) = x" "x. P x  Q (g x)"
    and Q: "eventually Q (at_right a)"
    and bound: "b. Q b  a < b"
    and P: "eventually P at_bot"
  shows "filterlim f at_bot (at_right a)"
proof -
  from P obtain x where x: "y. y  x  P y"
    unfolding eventually_at_bot_linorder by auto
  show ?thesis
  proof (intro filterlim_at_bot_le[THEN iffD2] allI impI)
    fix z
    assume "z  x"
    with x have "P z" by auto
    have "eventually (λx. x  g z) (at_right a)"
      using bound[OF bij(2)[OF P z]]
      unfolding eventually_at_right[OF bound[OF bij(2)[OF P z]]]
      by (auto intro!: exI[of _ "g z"])
    with Q show "eventually (λx. f x  z) (at_right a)"
      by eventually_elim (metis bij P z mono)
  qed
qed

lemma filterlim_at_top_at_left:
  fixes f :: "'a::linorder_topology  'b::linorder"
  assumes mono: "x y. Q x  Q y  x  y  f x  f y"
    and bij: "x. P x  f (g x) = x" "x. P x  Q (g x)"
    and Q: "eventually Q (at_left a)"
    and bound: "b. Q b  b < a"
    and P: "eventually P at_top"
  shows "filterlim f at_top (at_left a)"
proof -
  from P obtain x where x: "y. x  y  P y"
    unfolding eventually_at_top_linorder by auto
  show ?thesis
  proof (intro filterlim_at_top_ge[THEN iffD2] allI impI)
    fix z
    assume "x  z"
    with x have "P z" by auto
    have "eventually (λx. g z  x) (at_left a)"
      using bound[OF bij(2)[OF P z]]
      unfolding eventually_at_left[OF bound[OF bij(2)[OF P z]]]
      by (auto intro!: exI[of _ "g z"])
    with Q show "eventually (λx. z  f x) (at_left a)"
      by eventually_elim (metis bij P z mono)
  qed
qed

lemma filterlim_split_at:
  "filterlim f F (at_left x)  filterlim f F (at_right x) 
    filterlim f F (at x)"
  for x :: "'a::linorder_topology"
  by (subst at_eq_sup_left_right) (rule filterlim_sup)

lemma filterlim_at_split:
  "filterlim f F (at x)  filterlim f F (at_left x)  filterlim f F (at_right x)"
  for x :: "'a::linorder_topology"
  by (subst at_eq_sup_left_right) (simp add: filterlim_def filtermap_sup)

lemma eventually_nhds_top:
  fixes P :: "'a :: {order_top,linorder_topology}  bool"
    and b :: 'a
  assumes "b < top"
  shows "eventually P (nhds top)  (b<top. (z. b < z  P z))"
  unfolding eventually_nhds
proof safe
  fix S :: "'a set"
  assume "open S" "top  S"
  note open_left[OF this b < top›]
  moreover assume "sS. P s"
  ultimately show "b<top. z>b. P z"
    by (auto simp: subset_eq Ball_def)
next
  fix b
  assume "b < top" "z>b. P z"
  then show "S. open S  top  S  (xaS. P xa)"
    by (intro exI[of _ "{b <..}"]) auto
qed

lemma tendsto_at_within_iff_tendsto_nhds:
  "(g  g l) (at l within S)  (g  g l) (inf (nhds l) (principal S))"
  unfolding tendsto_def eventually_at_filter eventually_inf_principal
  by (intro ext all_cong imp_cong) (auto elim!: eventually_mono)


subsection ‹Limits on sequences›

abbreviation (in topological_space)
  LIMSEQ :: "[nat  'a, 'a]  bool"  ("((_)/  (_))" [60, 60] 60)
  where "X  L  (X  L) sequentially"

abbreviation (in t2_space) lim :: "(nat  'a)  'a"
  where "lim X  Lim sequentially X"

definition (in topological_space) convergent :: "(nat  'a)  bool"
  where "convergent X = (L. X  L)"

lemma lim_def: "lim X = (THE L. X  L)"
  unfolding Lim_def ..

lemma lim_explicit:
  "f  f0  (S. open S  f0  S  (N. nN. f n  S))"
  unfolding tendsto_def eventually_sequentially by auto


subsection ‹Monotone sequences and subsequences›

text ‹
  Definition of monotonicity.
  The use of disjunction here complicates proofs considerably.
  One alternative is to add a Boolean argument to indicate the direction.
  Another is to develop the notions of increasing and decreasing first.
›
definition monoseq :: "(nat  'a::order)  bool"
  where "monoseq X  (m. nm. X m  X n)  (m. nm. X n  X m)"

abbreviation incseq :: "(nat  'a::order)  bool"
  where "incseq X  mono X"

lemma incseq_def: "incseq X  (m. nm. X n  X m)"
  unfolding mono_def ..

abbreviation decseq :: "(nat  'a::order)  bool"
  where "decseq X  antimono X"

lemma decseq_def: "decseq X  (m. nm. X n  X m)"
  unfolding antimono_def ..

subsubsection ‹Definition of subsequence.›

(* For compatibility with the old "subseq" *)
lemma strict_mono_leD: "strict_mono r  m  n  r m  r n"
  by (erule (1) monoD [OF strict_mono_mono])

lemma strict_mono_id: "strict_mono id"
  by (simp add: strict_mono_def)

lemma incseq_SucI: "(n. X n  X (Suc n))  incseq X"
  using lift_Suc_mono_le[of X] by (auto simp: incseq_def)

lemma incseqD: "incseq f  i  j  f i  f j"
  by (auto simp: incseq_def)

lemma incseq_SucD: "incseq A  A i  A (Suc i)"
  using incseqD[of A i "Suc i"] by auto

lemma incseq_Suc_iff: "incseq f  (n. f n  f (Suc n))"
  by (auto intro: incseq_SucI dest: incseq_SucD)

lemma incseq_const[simp, intro]: "incseq (λx. k)"
  unfolding incseq_def by auto

lemma decseq_SucI: "(n. X (Suc n)  X n)  decseq X"
  using order.lift_Suc_mono_le[OF dual_order, of X] by (auto simp: decseq_def)

lemma decseqD: "decseq f  i  j  f j  f i"
  by (auto simp: decseq_def)

lemma decseq_SucD: "decseq A  A (Suc i)  A i"
  using decseqD[of A i "Suc i"] by auto

lemma decseq_Suc_iff: "decseq f  (n. f (Suc n)  f n)"
  by (auto intro: decseq_SucI dest: decseq_SucD)

lemma decseq_const[simp, intro]: "decseq (λx. k)"
  unfolding decseq_def by auto

lemma monoseq_iff: "monoseq X  incseq X  decseq X"
  unfolding monoseq_def incseq_def decseq_def ..

lemma monoseq_Suc: "monoseq X  (n. X n  X (Suc n))  (n. X (Suc n)  X n)"
  unfolding monoseq_iff incseq_Suc_iff decseq_Suc_iff ..

lemma monoI1: "m. n  m. X m  X n  monoseq X"
  by (simp add: monoseq_def)

lemma monoI2: "m. n  m. X n  X m  monoseq X"
  by (simp add: monoseq_def)

lemma mono_SucI1: "n. X n  X (Suc n)  monoseq X"
  by (simp add: monoseq_Suc)

lemma mono_SucI2: "n. X (Suc n)  X n  monoseq X"
  by (simp add: monoseq_Suc)

lemma monoseq_minus:
  fixes a :: "nat  'a::ordered_ab_group_add"
  assumes "monoseq a"
  shows "monoseq (λ n. - a n)"
proof (cases "m. n  m. a m  a n")
  case True
  then have "m. n  m. - a n  - a m" by auto
  then show ?thesis by (rule monoI2)
next
  case False
  then have "m. n  m. - a m  - a n"
    using ‹monoseq a[unfolded monoseq_def] by auto
  then show ?thesis by (rule monoI1)
qed


subsubsection ‹Subsequence (alternative definition, (e.g. Hoskins)›

lemma strict_mono_Suc_iff: "strict_mono f  (n. f n < f (Suc n))"
proof (intro iffI strict_monoI)
  assume *: "n. f n < f (Suc n)"
  fix m n :: nat assume "m < n"
  thus "f m < f n"
    by (induction rule: less_Suc_induct) (use * in auto)
qed (auto simp: strict_mono_def)

lemma strict_mono_add: "strict_mono (λn::'a::linordered_semidom. n + k)"
  by (auto simp: strict_mono_def)

text ‹For any sequence, there is a monotonic subsequence.›
lemma seq_monosub:
  fixes s :: "nat  'a::linorder"
  shows "f. strict_mono f  monoseq (λn. (s (f n)))"
proof (cases "n. p>n. mp. s m  s p")
  case True
  then have "f. n. (mf n. s m  s (f n))  f n < f (Suc n)"
    by (intro dependent_nat_choice) (auto simp: conj_commute)
  then obtain f :: "nat  nat" 
    where f: "strict_mono f" and mono: "n m. f n  m  s m  s (f n)"
    by (auto simp: strict_mono_Suc_iff)
  then have "incseq f"
    unfolding strict_mono_Suc_iff incseq_Suc_iff by (auto intro: less_imp_le)
  then have "monoseq (λn. s (f n))"
    by (auto simp add: incseq_def intro!: mono monoI2)
  with f show ?thesis
    by auto
next
  case False
  then obtain N where N: "p > N  m>p. s p < s m" for p
    by (force simp: not_le le_less)
  have "f. n. N < f n  f n < f (Suc n)  s (f n)  s (f (Suc n))"
  proof (intro dependent_nat_choice)
    fix x
    assume "N < x" with N[of x]
    show "y>N. x < y  s x  s y"
      by (auto intro: less_trans)
  qed auto
  then show ?thesis
    by (auto simp: monoseq_iff incseq_Suc_iff strict_mono_Suc_iff)
qed

lemma seq_suble:
  assumes sf: "strict_mono (f :: nat  nat)"
  shows "n  f n"
proof (induct n)
  case 0
  show ?case by simp
next
  case (Suc n)
  with sf [unfolded strict_mono_Suc_iff, rule_format, of n] have "n < f (Suc n)"
     by arith
  then show ?case by arith
qed

lemma eventually_subseq:
  "strict_mono r  eventually P sequentially  eventually (λn. P (r n)) sequentially"
  unfolding eventually_sequentially by (metis seq_suble le_trans)

lemma not_eventually_sequentiallyD:
  assumes "¬ eventually P sequentially"
  shows "r::natnat. strict_mono r  (n. ¬ P (r n))"
proof -
  from assms have "n. mn. ¬ P m"
    unfolding eventually_sequentially by (simp add: not_less)
  then obtain r where "n. r n  n" "n. ¬ P (r n)"
    by (auto simp: choice_iff)
  then show ?thesis
    by (auto intro!: exI[of _ "λn. r (((Suc  r) ^^ Suc n) 0)"]
             simp: less_eq_Suc_le strict_mono_Suc_iff)
qed

lemma sequentially_offset: 
  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: 
  "(f  l) sequentially  ((λi. f(i - k))  l) sequentially"
  apply (erule filterlim_compose)
  apply (simp add: filterlim_def le_sequentially eventually_filtermap eventually_sequentially, arith)
  done

lemma filterlim_subseq: "strict_mono f  filterlim f sequentially sequentially"
  unfolding filterlim_iff by (metis eventually_subseq)

lemma strict_mono_o: "strict_mono r  strict_mono s  strict_mono (r  s)"
  unfolding strict_mono_def by simp

lemma strict_mono_compose: "strict_mono r  strict_mono s  strict_mono (λx. r (s x))"
  using strict_mono_o[of r s] by (simp add: o_def)

lemma incseq_imp_monoseq:  "incseq X  monoseq X"
  by (simp add: incseq_def monoseq_def)

lemma decseq_imp_monoseq:  "decseq X  monoseq X"
  by (simp add: decseq_def monoseq_def)

lemma decseq_eq_incseq: "decseq X = incseq (λn. - X n)"
  for X :: "nat  'a::ordered_ab_group_add"
  by (simp add: decseq_def incseq_def)

lemma INT_decseq_offset:
  assumes "decseq F"
  shows "(i. F i) = (i{n..}. F i)"
proof safe
  fix x i
  assume x: "x  (i{n..}. F i)"
  show "x  F i"
  proof cases
    from x have "x  F n" by auto
    also assume "i  n" with ‹decseq F have "F n  F i"
      unfolding decseq_def by simp
    finally show ?thesis .
  qed (insert x, simp)
qed auto

lemma LIMSEQ_const_iff: "(λn. k)  l  k = l"
  for k l :: "'a::t2_space"
  using trivial_limit_sequentially by (rule tendsto_const_iff)

lemma LIMSEQ_SUP: "incseq X  X  (SUP i. X i :: 'a::{complete_linorder,linorder_topology})"
  by (intro increasing_tendsto)
    (auto simp: SUP_upper less_SUP_iff incseq_def eventually_sequentially intro: less_le_trans)

lemma LIMSEQ_INF: "decseq X  X  (INF i. X i :: 'a::{complete_linorder,linorder_topology})"
  by (intro decreasing_tendsto)
    (auto simp: INF_lower INF_less_iff decseq_def eventually_sequentially intro: le_less_trans)

lemma LIMSEQ_ignore_initial_segment: "f  a  (λn. f (n + k))  a"
  unfolding tendsto_def by (subst eventually_sequentially_seg[where k=k])

lemma LIMSEQ_offset: "(λn. f (n + k))  a  f  a"
  unfolding tendsto_def
  by (subst (asm) eventually_sequentially_seg[where k=k])

lemma LIMSEQ_Suc: "f  l  (λn. f (Suc n))  l"
  by (drule LIMSEQ_ignore_initial_segment [where k="Suc 0"]) simp

lemma LIMSEQ_imp_Suc: "(λn. f (Suc n))  l  f  l"
  by (rule LIMSEQ_offset [where k="Suc 0"]) simp

lemma LIMSEQ_lessThan_iff_atMost:
  shows "(λn. f {..<n})  x  (λn. f {..n})  x"
  apply (subst filterlim_sequentially_Suc [symmetric])
  apply (simp only: lessThan_Suc_atMost)
  done

lemma (in t2_space) LIMSEQ_Uniq: "1l. X  l"
 by (simp add: tendsto_unique')

lemma (in t2_space) LIMSEQ_unique: "X  a  X  b  a = b"
  using trivial_limit_sequentially by (rule tendsto_unique)

lemma LIMSEQ_le_const: "X  x  N. nN. a  X n  a  x"
  for a x :: "'a::linorder_topology"
  by (simp add: eventually_at_top_linorder tendsto_lowerbound)

lemma LIMSEQ_le: "X  x  Y  y  N. nN. X n  Y n  x  y"
  for x y :: "'a::linorder_topology"
  using tendsto_le[of sequentially Y y X x] by (simp add: eventually_sequentially)

lemma LIMSEQ_le_const2: "X  x  N. nN. X n  a  x  a"
  for a x :: "'a::linorder_topology"
  by (rule LIMSEQ_le[of X x "λn. a"]) auto

lemma Lim_bounded: "f  l  nM. f n  C  l  C"
  for l :: "'a::linorder_topology"
  by (intro LIMSEQ_le_const2) auto

lemma Lim_bounded2:
  fixes f :: "nat  'a::linorder_topology"
  assumes lim:"f  l" and ge: "nN. f n  C"
  shows "l  C"
  using ge
  by (intro tendsto_le[OF trivial_limit_sequentially lim tendsto_const])
     (auto simp: eventually_sequentially)

lemma lim_mono:
  fixes X Y :: "nat  'a::linorder_topology"
  assumes "n. N  n  X n  Y n"
    and "X  x"
    and "Y  y"
  shows "x  y"
  using assms(1) by (intro LIMSEQ_le[OF assms(2,3)]) auto

lemma Sup_lim:
  fixes a :: "'a::{complete_linorder,linorder_topology}"
  assumes "n. b n  s"
    and "b  a"
  shows "a  Sup s"
  by (metis Lim_bounded assms complete_lattice_class.Sup_upper)

lemma Inf_lim:
  fixes a :: "'a::{complete_linorder,linorder_topology}"
  assumes "n. b n  s"
    and "b  a"
  shows "Inf s  a"
  by (metis Lim_bounded2 assms complete_lattice_class.Inf_lower)

lemma SUP_Lim:
  fixes X :: "nat  'a::{complete_linorder,linorder_topology}"
  assumes inc: "incseq X"
    and l: "X  l"
  shows "(SUP n. X n) = l"
  using LIMSEQ_SUP[OF inc] tendsto_unique[OF trivial_limit_sequentially l]
  by simp

lemma INF_Lim:
  fixes X :: "nat  'a::{complete_linorder,linorder_topology}"
  assumes dec: "decseq X"
    and l: "X  l"
  shows "(INF n. X n) = l"
  using LIMSEQ_INF[OF dec] tendsto_unique[OF trivial_limit_sequentially l]
  by simp

lemma convergentD: "convergent X  L. X  L"
  by (simp add: convergent_def)

lemma convergentI: "X  L  convergent X"
  by (auto simp add: convergent_def)

lemma convergent_LIMSEQ_iff: "convergent X  X  lim X"
  by (auto intro: theI LIMSEQ_unique simp add: convergent_def lim_def)

lemma convergent_const: "convergent (λn. c)"
  by (rule convergentI) (rule tendsto_const)

lemma monoseq_le:
  "monoseq a  a  x 
    (n. a n  x)  (m. nm. a m  a n) 
    (n. x  a n)  (m. nm. a n  a m)"
  for x :: "'a::linorder_topology"
  by (metis LIMSEQ_le_const LIMSEQ_le_const2 decseq_def incseq_def monoseq_iff)

lemma LIMSEQ_subseq_LIMSEQ: "X  L  strict_mono f  (X  f)  L"
  unfolding comp_def by (rule filterlim_compose [of X, OF _ filterlim_subseq])

lemma convergent_subseq_convergent: "convergent X  strict_mono f  convergent (X  f)"
  by (auto simp: convergent_def intro: LIMSEQ_subseq_LIMSEQ)

lemma limI: "X  L  lim X = L"
  by (rule tendsto_Lim) (rule trivial_limit_sequentially)

lemma lim_le: "convergent f  (n. f n  x)  lim f  x"
  for x :: "'a::linorder_topology"
  using LIMSEQ_le_const2[of f "lim f" x] by (simp add: convergent_LIMSEQ_iff)

lemma lim_const [simp]: "lim (λm. a) = a"
  by (simp add: limI)


subsubsection ‹Increasing and Decreasing Series›

lemma incseq_le: "incseq X  X  L  X n  L"
  for L :: "'a::linorder_topology"
  by (metis incseq_def LIMSEQ_le_const)

lemma decseq_ge: "decseq X  X  L  L  X n"
  for L :: "'a::linorder_topology"
  by (metis decseq_def LIMSEQ_le_const2)


subsection ‹First countable topologies›

class first_countable_topology = topological_space +
  assumes first_countable_basis:
    "A::nat  'a set. (i. x  A i  open (A i))  (S. open S  x  S  (i. A i  S))"

lemma (in first_countable_topology) countable_basis_at_decseq:
  obtains A :: "nat  'a set" where
    "i. open (A i)" "i. x  (A i)"
    "S. open S  x  S  eventually (λi. A i  S) sequentially"
proof atomize_elim
  from first_countable_basis[of x] obtain A :: "nat  'a set"
    where nhds: "i. open (A i)" "i. x  A i"
      and incl: "S. open S  x  S  i. A i  S"
    by auto
  define F where "F n = (in. A i)" for n
  show "A. (i. open (A i))  (i. x  A i) 
    (S. open S  x  S  eventually (λi. A i  S) sequentially)"
  proof (safe intro!: exI[of _ F])
    fix i
    show "open (F i)"
      using nhds(1) by (auto simp: F_def)
    show "x  F i"
      using nhds(2) by (auto simp: F_def)
  next
    fix S
    assume "open S" "x  S"
    from incl[OF this] obtain i where "F i  S"
      unfolding F_def by auto
    moreover have "j. i  j  F j  F i"
      by (simp add: Inf_superset_mono F_def image_mono)
    ultimately show "eventually (λi. F i  S) sequentially"
      by (auto simp: eventually_sequentially)
  qed
qed

lemma (in first_countable_topology) nhds_countable:
  obtains X :: "nat  'a set"
  where "decseq X" "n. open (X n)" "n. x  X n" "nhds x = (INF n. principal (X n))"
proof -
  from first_countable_basis obtain A :: "nat  'a set"
    where *: "n. x  A n" "n. open (A n)" "S. open S  x  S  i. A i  S"
    by metis
  show thesis
  proof
    show "decseq (λn. in. A i)"
      by (simp add: antimono_iff_le_Suc atMost_Suc)
    show "x  (in. A i)" "n. open (in. A i)" for n
      using * by auto
    show "nhds x = (INF n. principal (in. A i))"
      using *
      unfolding nhds_def
      apply -
      apply (rule INF_eq)
       apply simp_all
       apply fastforce
      apply (intro exI [of _ "in. A i" for n] conjI open_INT)
         apply auto
      done
  qed
qed

lemma (in first_countable_topology) countable_basis:
  obtains A :: "nat  'a set" where
    "i. open (A i)" "i. x  A i"
    "F. (n. F n  A n)  F  x"
proof atomize_elim
  obtain A :: "nat  'a set" where *:
    "i. open (A i)"
    "i. x  A i"
    "S. open S  x  S  eventually (λi. A i  S) sequentially"
    by (rule countable_basis_at_decseq) blast
  have "eventually (λn. F n  S) sequentially"
    if "n. F n  A n" "open S" "x  S" for F S
    using *(3)[of S] that by (auto elim: eventually_mono simp: subset_eq)
  with * show "A. (i. open (A i))  (i. x  A i)  (F. (n. F n  A n)  F  x)"
    by (intro exI[of _ A]) (auto simp: tendsto_def)
qed

lemma (in first_countable_topology) sequentially_imp_eventually_nhds_within:
  assumes "f. (n. f n  s)  f  a  eventually (λn. P (f n)) sequentially"
  shows "eventually P (inf (nhds a) (principal s))"
proof (rule ccontr)
  obtain A :: "nat  'a set" where *:
    "i. open (A i)"
    "i. a  A i"
    "F. n. F n  A n  F  a"
    by (rule countable_basis) blast
  assume "¬ ?thesis"
  with * have "F. n. F n  s  F n  A n  ¬ P (F n)"
    unfolding eventually_inf_principal eventually_nhds
    by (intro choice) fastforce
  then obtain F where F: "n. F n  s" and "n. F n  A n" and F': "n. ¬ P (F n)"
    by blast
  with * have "F  a"
    by auto
  then have "eventually (λn. P (F n)) sequentially"
    using assms F by simp
  then show False
    by (simp add: F')
qed

lemma (in first_countable_topology) eventually_nhds_within_iff_sequentially:
  "eventually P (inf (nhds a) (principal s)) 
    (f. (n. f n  s)  f  a  eventually (λn. P (f n)) sequentially)"
proof (safe intro!: sequentially_imp_eventually_nhds_within)
  assume "eventually P (inf (nhds a) (principal s))"
  then obtain S where "open S" "a  S" "xS. x  s  P x"
    by (auto simp: eventually_inf_principal eventually_nhds)
  moreover
  fix f
  assume "n. f n  s" "f  a"
  ultimately show "eventually (λn. P (f n)) sequentially"
    by (auto dest!: topological_tendstoD elim: eventually_mono)
qed

lemma (in first_countable_topology) eventually_nhds_iff_sequentially:
  "eventually P (nhds a)  (f. f  a  eventually (λn. P (f n)) sequentially)"
  using eventually_nhds_within_iff_sequentially[of P a UNIV] by simp

(*Thanks to Sébastien Gouëzel*)
lemma Inf_as_limit:
  fixes A::"'a::{linorder_topology, first_countable_topology, complete_linorder} set"
  assumes "A  {}"
  shows "u. (n. u n  A)  u  Inf A"
proof (cases "Inf A  A")
  case True
  show ?thesis
    by (rule exI[of _ "λn. Inf A"], auto simp add: True)
next
  case False
  obtain y where "y  A" using assms by auto
  then have "Inf A < y" using False Inf_lower less_le by auto
  obtain F :: "nat  'a set" where F: "i. open (F i)" "i. Inf A  F i"
                                       "u. (n. u n  F n)  u  Inf A"
    by (metis first_countable_topology_class.countable_basis)
  define u where "u = (λn. SOME z. z  F n  z  A)"
  have "z. z  U  z  A" if "Inf A  U" "open U" for U
  proof -
    obtain b where "b > Inf A" "{Inf A ..<b}  U"
      using open_right[OF ‹open U ‹Inf A  U ‹Inf A < y] by auto
    obtain z where "z < b" "z  A"
      using ‹Inf A < b Inf_less_iff by auto
    then have "z  {Inf A ..<b}"
      by (simp add: Inf_lower)
    then show ?thesis using z  A {Inf A ..<b}  U by auto
  qed
  then have *: "u n  F n  u n  A" for n
    using ‹Inf A  F n ‹open (F n) unfolding u_def by (metis (no_types, lifting) someI_ex)
  then have "u  Inf A" using F(3) by simp
  then show ?thesis using * by auto
qed

lemma tendsto_at_iff_sequentially:
  "(f  a) (at x within s)  (X. (i. X i  s - {x})  X  x  ((f  X)  a))"
  for f :: "'a::first_countable_topology  _"
  unfolding filterlim_def[of _ "nhds a"] le_filter_def eventually_filtermap
    at_within_def eventually_nhds_within_iff_sequentially comp_def
  by metis

lemma approx_from_above_dense_linorder:
  fixes x::"'a::{dense_linorder, linorder_topology, first_countable_topology}"
  assumes "x < y"
  shows "u. (n. u n > x)  (u  x)"
proof -
  obtain A :: "nat  'a set" where A: "i. open (A i)" "i. x  A i"
                                      "F. (n. F n  A n)  F  x"
    by (metis first_countable_topology_class.countable_basis)
  define u where "u = (λn. SOME z. z  A n  z > x)"
  have "z. z  U  x < z" if "x  U" "open U" for U
    using open_right[OF ‹open U x  U x < y]
    by (meson atLeastLessThan_iff dense less_imp_le subset_eq)
  then have *: "u n  A n  x < u n" for n
    using x  A n ‹open (A n) unfolding u_def by (metis (no_types, lifting) someI_ex)
  then have "u  x" using A(3) by simp
  then show ?thesis using * by auto
qed

lemma approx_from_below_dense_linorder:
  fixes x::"'a::{dense_linorder, linorder_topology, first_countable_topology}"
  assumes "x > y"
  shows "u. (n. u n < x)  (u  x)"
proof -
  obtain A :: "nat  'a set" where A: "i. open (A i)" "i. x  A i"
                                      "F. (n. F n  A n)  F  x"
    by (metis first_countable_topology_class.countable_basis)
  define u where "u = (λn. SOME z. z  A n  z < x)"
  have "z. z  U  z < x" if "x  U" "open U" for U
    using open_left[OF ‹open U x  U x > y]
    by (meson dense greaterThanAtMost_iff less_imp_le subset_eq)
  then have *: "u n  A n  u n < x" for n
    using x  A n ‹open (A n) unfolding u_def by (metis (no_types, lifting) someI_ex)
  then have "u  x" using A(3) by simp
  then show ?thesis using * by auto
qed


subsection ‹Function limit at a point›

abbreviation LIM :: "('a::topological_space  'b::topological_space)  'a  'b  bool"
    ("((_)/ (_)/ (_))" [60, 0, 60] 60)
  where "f a L  (f  L) (at a)"

lemma tendsto_within_open: "a  S  open S  (f  l) (at a within S)  (f a l)"
  by (simp add: tendsto_def at_within_open[where S = S])

lemma tendsto_within_open_NO_MATCH:
  "a  S  NO_MATCH UNIV S  open S  (f  l)(at a within S)  (f  l)(at a)"
  for f :: "'a::topological_space  'b::topological_space"
  using tendsto_within_open by blast

lemma LIM_const_not_eq[tendsto_intros]: "k  L  ¬ (λx. k) a L"
  for a :: "'a::perfect_space" and k L :: "'b::t2_space"
  by (simp add: tendsto_const_iff)

lemmas LIM_not_zero = LIM_const_not_eq [where L = 0]

lemma LIM_const_eq: "(λx. k) a L  k = L"
  for a :: "'a::perfect_space" and k L :: "'b::t2_space"
  by (simp add: tendsto_const_iff)

lemma LIM_unique: "f a L  f a M  L = M"
  for a :: "'a::perfect_space" and L M :: "'b::t2_space"
  using at_neq_bot by (rule tendsto_unique)

lemma LIM_Uniq: "1L::'b::t2_space. f a L"
  for a :: "'a::perfect_space"
 by (auto simp add: Uniq_def LIM_unique)


text ‹Limits are equal for functions equal except at limit point.›
lemma LIM_equal: "x. x  a  f x = g x  (f a l)  (g a l)"
  by (simp add: tendsto_def eventually_at_topological)

lemma LIM_cong: "a = b  (x. x  b  f x = g x)  l = m  (f a l)  (g b m)"
  by (simp add: LIM_equal)

lemma tendsto_cong_limit: "(f  l) F  k = l  (f  k) F"
  by simp

lemma tendsto_at_iff_tendsto_nhds: "g l g l  (g  g l) (nhds l)"
  unfolding tendsto_def eventually_at_filter
  by (intro ext all_cong imp_cong) (auto elim!: eventually_mono)

lemma tendsto_compose: "g l g l  (f  l) F  ((λx. g (f x))  g l) F"
  unfolding tendsto_at_iff_tendsto_nhds by (rule filterlim_compose[of g])

lemma tendsto_compose_eventually:
  "g l m  (f  l) F  eventually (λx. f x  l) F  ((λx. g (f x))  m) F"
  by (rule filterlim_compose[of g _ "at l"]) (auto simp add: filterlim_at)

lemma LIM_compose_eventually:
  assumes "f a b"
    and "g b c"
    and "eventually (λx. f x  b) (at a)"
  shows "(λx. g (f x)) a c"
  using assms(2,1,3) by (rule tendsto_compose_eventually)

lemma tendsto_compose_filtermap: "((g  f)  T) F  (g  T) (filtermap f F)"
  by (simp add: filterlim_def filtermap_filtermap comp_def)

lemma tendsto_compose_at:
  assumes f: "(f  y) F" and g: "(g  z) (at y)" and fg: "eventually (λw. f w = y  g y = z) F"
  shows "((g  f)  z) F"
proof -
  have "(F a in F. f a  y)  g y = z"
    using fg by force
  moreover have "(g  z) (filtermap f F)  ¬ (F a in F. f a  y)"
    by (metis (no_types) filterlim_atI filterlim_def tendsto_mono f g)
  ultimately show ?thesis
    by (metis (no_types) f filterlim_compose filterlim_filtermap g tendsto_at_iff_tendsto_nhds tendsto_compose_filtermap)
qed


subsubsection ‹Relation of LIM› and LIMSEQ›

lemma (in first_countable_topology) sequentially_imp_eventually_within:
  "(f. (n. f n  s  f n  a)  f  a  eventually (λn. P (f n)) sequentially) 
    eventually P (at a within s)"
  unfolding at_within_def
  by (intro sequentially_imp_eventually_nhds_within) auto

lemma (in first_countable_topology) sequentially_imp_eventually_at:
  "(f. (n. f n  a)  f  a  eventually (λn. P (f n)) sequentially)  eventually P (at a)"
  using sequentially_imp_eventually_within [where s=UNIV] by simp

lemma LIMSEQ_SEQ_conv1:
  fixes f :: "'a::topological_space  'b::topological_space"
  assumes f: "f a l"
  shows "S. (n. S n  a)  S  a  (λn. f (S n))  l"
  using tendsto_compose_eventually [OF f, where F=sequentially] by simp

lemma LIMSEQ_SEQ_conv2:
  fixes f :: "'a::first_countable_topology  'b::topological_space"
  assumes "S. (n. S n  a)  S  a  (λn. f (S n))  l"
  shows "f a l"
  using assms unfolding tendsto_def [where l=l] by (simp add: sequentially_imp_eventually_at)

lemma LIMSEQ_SEQ_conv: "(S. (n. S n  a)  S  a  (λn. X (S n))  L)  X a L"
  for a :: "'a::first_countable_topology" and L :: "'b::topological_space"
  using LIMSEQ_SEQ_conv2 LIMSEQ_SEQ_conv1 ..

lemma sequentially_imp_eventually_at_left:
  fixes a :: "'a::{linorder_topology,first_countable_topology}"
  assumes b[simp]: "b < a"
    and *: "f. (n. b < f n)  (n. f n < a)  incseq f  f  a 
      eventually (λn. P (f n)) sequentially"
  shows "eventually P (at_left a)"
proof (safe intro!: sequentially_imp_eventually_within)
  fix X
  assume X: "n. X n  {..< a}  X n  a" "X  a"
  show "eventually (λn. P (X n)) sequentially"
  proof (rule ccontr)
    assume neg: "¬ ?thesis"
    have "s. n. (¬ P (X (s n))  b < X (s n))  (X (s n)  X (s (Suc n))  Suc (s n)  s (Suc n))"
      (is "s. ?P s")
    proof (rule dependent_nat_choice)
      have "¬ eventually (λn. b < X n  P (X n)) sequentially"
        by (intro not_eventually_impI neg order_tendstoD(1) [OF X(2) b])
      then show "x. ¬ P (X x)  b < X x"
        by (auto dest!: not_eventuallyD)
    next
      fix x n
      have "¬ eventually (λn. Suc x  n  b < X n  X x < X n  P (X n)) sequentially"
        using X
        by (intro not_eventually_impI order_tendstoD(1)[OF X(2)] eventually_ge_at_top neg) auto
      then show "n. (¬ P (X n)  b < X n)  (X x  X n  Suc x  n)"
        by (auto dest!: not_eventuallyD)
    qed
    then obtain s where "?P s" ..
    with X have "b < X (s n)"
      and "X (s n) < a"
      and "incseq (λn. X (s n))"
      and "(λn. X (s n))  a"
      and "¬ P (X (s n))"
      for n
      by (auto simp: strict_mono_Suc_iff Suc_le_eq incseq_Suc_iff
          intro!: LIMSEQ_subseq_LIMSEQ[OF X  a, unfolded comp_def])
    from *[OF this(1,2,3,4)] this(5) show False
      by auto
  qed
qed

lemma tendsto_at_left_sequentially:
  fixes a b :: "'b::{linorder_topology,first_countable_topology}"
  assumes "b < a"
  assumes *: "S. (n. S n < a)  (n. b < S n)  incseq S  S  a 
    (λn. X (S n))  L"
  shows "(X  L) (at_left a)"
  using assms by (simp add: tendsto_def [where l=L] sequentially_imp_eventually_at_left)

lemma sequentially_imp_eventually_at_right:
  fixes a b :: "'a::{linorder_topology,first_countable_topology}"
  assumes b[simp]: "a < b"
  assumes *: "f. (n. a < f n)  (n. f n < b)  decseq f  f  a 
    eventually (λn. P (f n)) sequentially"
  shows "eventually P (at_right a)"
proof (safe intro!: sequentially_imp_eventually_within)
  fix X
  assume X: "n. X n  {a <..}  X n  a" "X  a"
  show "eventually (λn. P (X n)) sequentially"
  proof (rule ccontr)
    assume neg: "¬ ?thesis"
    have "s. n. (¬ P (X (s n))  X (s n) < b)  (X (s (Suc n))  X (s n)  Suc (s n)  s (Suc n))"
      (is "s. ?P s")
    proof (rule dependent_nat_choice)
      have "¬ eventually (λn. X n < b  P (X n)) sequentially"
        by (intro not_eventually_impI neg order_tendstoD(2) [OF X(2) b])
      then show "x. ¬ P (X x)  X x < b"
        by (auto dest!: not_eventuallyD)
    next
      fix x n
      have "¬ eventually (λn. Suc x  n  X n < b  X n < X x  P (X n)) sequentially"
        using X
        by (intro not_eventually_impI order_tendstoD(2)[OF X(2)] eventually_ge_at_top neg) auto
      then show "n. (¬ P (X n)  X n < b)  (X n  X x  Suc x  n)"
        by (auto dest!: not_eventuallyD)
    qed
    then obtain s where "?P s" ..
    with X have "a < X (s n)"
      and "X (s n) < b"
      and "decseq (λn. X (s n))"
      and "(λn. X (s n))  a"
      and "¬ P (X (s n))"
      for n
      by (auto simp: strict_mono_Suc_iff Suc_le_eq decseq_Suc_iff
          intro!: LIMSEQ_subseq_LIMSEQ[OF X  a, unfolded comp_def])
    from *[OF this(1,2,3,4)] this(5) show False
      by auto
  qed
qed

lemma tendsto_at_right_sequentially:
  fixes a :: "_ :: {linorder_topology, first_countable_topology}"
  assumes "a < b"
    and *: "S. (n. a < S n)  (n. S n < b)  decseq S  S  a 
      (λn. X (S n))  L"
  shows "(X  L) (at_right a)"
  using assms by (simp add: tendsto_def [where l=L] sequentially_imp_eventually_at_right)


subsection ‹Continuity›

subsubsection ‹Continuity on a set›

definition continuous_on :: "'a set  ('a::topological_space  'b::topological_space)  bool"
  where "continuous_on s f  (xs. (f  f x) (at x within s))"

lemma continuous_on_cong [cong]:
  "s = t  (x. x  t  f x = g x)  continuous_on s f  continuous_on t g"
  unfolding continuous_on_def
  by (intro ball_cong filterlim_cong) (auto simp: eventually_at_filter)

lemma continuous_on_cong_simp:
  "s = t  (x. x  t =simp=> f x = g x)  continuous_on s f  continuous_on t g"
  unfolding simp_implies_def by (rule continuous_on_cong)

lemma continuous_on_topological:
  "continuous_on s f 
    (xs. B. open B  f x  B  (A. open A  x  A  (ys. y  A  f y  B)))"
  unfolding continuous_on_def tendsto_def eventually_at_topological by metis

lemma continuous_on_open_invariant:
  "continuous_on s f  (B. open B  (A. open A  A  s = f -` B  s))"
proof safe
  fix B :: "'b set"
  assume "continuous_on s f" "open B"
  then have "xf -` B  s. (A. open A  x  A  s  A  f -` B)"
    by (auto simp: continuous_on_topological subset_eq Ball_def imp_conjL)
  then obtain A where "xf -` B  s. open (A x)  x  A x  s  A x  f -` B"
    unfolding bchoice_iff ..
  then show "A. open A  A  s = f -` B  s"
    by (intro exI[of _ "xf -` B  s. A x"]) auto
next
  assume B: "B. open B  (A. open A  A  s = f -` B  s)"
  show "continuous_on s f"
    unfolding continuous_on_topological
  proof safe
    fix x B
    assume "x  s" "open B" "f x  B"
    with B obtain A where A: "open A" "A  s = f -` B  s"
      by auto
    with x  s f x  B show "A. open A  x  A  (ys. y  A  f y  B)"
      by (intro exI[of _ A]) auto
  qed
qed

lemma continuous_on_open_vimage:
  "open s  continuous_on s f  (B. open B  open (f -` B  s))"
  unfolding continuous_on_open_invariant
  by (metis open_Int Int_absorb Int_commute[of s] Int_assoc[of _ _ s])

corollary continuous_imp_open_vimage:
  assumes "continuous_on s f" "open s" "open B" "f -` B  s"
  shows "open (f -` B)"
  by (metis assms continuous_on_open_vimage le_iff_inf)

corollary open_vimage[continuous_intros]:
  assumes "open s"
    and "continuous_on UNIV f"
  shows "open (f -` s)"
  using assms by (simp add: continuous_on_open_vimage [OF open_UNIV])

lemma continuous_on_closed_invariant:
  "continuous_on s f  (B. closed B  (A. closed A  A  s = f -` B  s))"
proof -
  have *: "(A. P A  Q (- A))  (A. P A)  (A. Q A)"
    for P Q :: "'b set  bool"
    by (metis double_compl)
  show ?thesis
    unfolding continuous_on_open_invariant
    by (intro *) (auto simp: open_closed[symmetric])
qed

lemma continuous_on_closed_vimage:
  "closed s  continuous_on s f  (B. closed B  closed (f -` B  s))"
  unfolding continuous_on_closed_invariant
  by (metis closed_Int Int_absorb Int_commute[of s] Int_assoc[of _ _ s])

corollary closed_vimage_Int[continuous_intros]:
  assumes "closed s"
    and "continuous_on t f"
    and t: "closed t"
  shows "closed (f -` s  t)"
  using assms by (simp add: continuous_on_closed_vimage [OF t])

corollary closed_vimage[continuous_intros]:
  assumes "closed s"
    and "continuous_on UNIV f"
  shows "closed (f -` s)"
  using closed_vimage_Int [OF assms] by simp

lemma continuous_on_empty [simp]: "continuous_on {} f"
  by (simp add: continuous_on_def)

lemma continuous_on_sing [simp]: "continuous_on {x} f"
  by (simp add: continuous_on_def at_within_def)

lemma continuous_on_open_Union:
  "(s. s  S  open s)  (s. s  S  continuous_on s f)  continuous_on (S) f"
  unfolding continuous_on_def
  by safe (metis open_Union at_within_open UnionI)

lemma continuous_on_open_UN:
  "(s. s  S  open (A s))  (s. s  S  continuous_on (A s) f) 
    continuous_on (sS. A s) f"
  by (rule continuous_on_open_Union) auto

lemma continuous_on_open_Un:
  "open s  open t  continuous_on s f  continuous_on t f  continuous_on (s  t) f"
  using continuous_on_open_Union [of "{s,t}"] by auto

lemma continuous_on_closed_Un:
  "closed s  closed t  continuous_on s f  continuous_on t f  continuous_on (s  t) f"
  by (auto simp add: continuous_on_closed_vimage closed_Un Int_Un_distrib)

lemma continuous_on_closed_Union:
  assumes "finite I"
    "i. i  I  closed (U i)"
    "i. i  I  continuous_on (U i) f"
  shows "continuous_on ( i  I. U i) f"
  using assms
  by (induction I) (auto intro!: continuous_on_closed_Un)

lemma continuous_on_If:
  assumes closed: "closed s" "closed t"
    and cont: "continuous_on s f" "continuous_on t g"
    and P: "x. x  s  ¬ P x  f x = g x" "x. x  t  P x  f x = g x"
  shows "continuous_on (s  t) (λx. if P x then f x else g x)"
    (is "continuous_on _ ?h")
proof-
  from P have "xs. f x = ?h x" "xt. g x = ?h x"
    by auto
  with cont have "continuous_on s ?h" "continuous_on t ?h"
    by simp_all
  with closed show ?thesis
    by (rule continuous_on_closed_Un)
qed

lemma continuous_on_cases:
  "closed s  closed t  continuous_on s f  continuous_on t g 
    x. (xs  ¬ 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

lemma continuous_on_id[continuous_intros,simp]: "continuous_on s (λx. x)"
  unfolding continuous_on_def by fast

lemma continuous_on_id'[continuous_intros,simp]: "continuous_on s id"
  unfolding continuous_on_def id_def by fast

lemma continuous_on_const[continuous_intros,simp]: "continuous_on s (λx. c)"
  unfolding continuous_on_def by auto

lemma continuous_on_subset: "continuous_on s f  t  s  continuous_on t f"
  unfolding continuous_on_def
  by (metis subset_eq tendsto_within_subset)

lemma continuous_on_compose[continuous_intros]:
  "continuous_on s f  continuous_on (f ` s) g  continuous_on s (g  f)"
  unfolding continuous_on_topological by simp metis

lemma continuous_on_compose2:
  "continuous_on t g  continuous_on s f  f ` s  t  continuous_on s (λx. g (f x))"
  using continuous_on_compose[of s f g] continuous_on_subset by (force simp add: comp_def)

lemma continuous_on_generate_topology:
  assumes *: "open = generate_topology X"
    and **: "B. B  X  C. open C  C  A = f -` B  A"
  shows "continuous_on A f"
  unfolding continuous_on_open_invariant
proof safe
  fix B :: "'a set"
  assume "open B"
  then show "C. open C  C  A = f -` B  A"
    unfolding *
  proof induct
    case (UN K)
    then obtain C where "k. k  K  open (C k)" "k. k  K  C k  A = f -` k  A"
      by metis
    then show ?case
      by (intro exI[of _ "kK. C k"]) blast
  qed (auto intro: **)
qed

lemma continuous_onI_mono:
  fixes f :: "'a::linorder_topology  'b::{dense_order,linorder_topology}"
  assumes "open (f`A)"
    and mono: "x y. x  A  y  A  x  y  f x  f y"
  shows "continuous_on A f"
proof (rule continuous_on_generate_topology[OF open_generated_order], safe)
  have monoD: "x y. x  A  y  A  f x < f y  x < y"
    by (auto simp: not_le[symmetric] mono)
  have "x. x  A  f x < b  a < x" if a: "a  A" and fa: "f a < b" for a b
  proof -
    obtain y where "f a < y" "{f a ..< y}  f`A"
      using open_right[OF ‹open (f`A), of "f a" b] a fa
      by auto
    obtain z where z: "f a < z" "z < min b y"
      using dense[of "f a" "min b y"] f a < y f a < b by auto
    then obtain c where "z = f c" "c  A"
      using {f a ..< y}  f`A[THEN subsetD, of z] by (auto simp: less_imp_le)
    with a z show ?thesis
      by (auto intro!: exI[of _ c] simp: monoD)
  qed
  then show "C. open C  C  A = f -` {..<b}  A" for b
    by (intro exI[of _ "(x{xA. f x < b}. {..< x})"])
       (auto intro: le_less_trans[OF mono] less_imp_le)

  have "x. x  A  b < f x  x < a" if a: "a  A" and fa: "b < f a" for a b
  proof -
    note a fa
    moreover
    obtain y where "y < f a" "{y <.. f a}  f`A"
      using open_left[OF ‹open (f`A), of "f a" b]  a fa
      by auto
    then obtain z where z: "max b y < z" "z < f a"
      using dense[of "max b y" "f a"] y < f a b < f a by auto
    then obtain c where "z = f c" "c  A"
      using {y <.. f a}  f`A[THEN subsetD, of z] by (auto simp: less_imp_le)
    with a z show ?thesis
      by (auto intro!: exI[of _ c] simp: monoD)
  qed
  then show "C. open C  C  A = f -` {b <..}  A" for b
    by (intro exI[of _ "(x{xA. b < f x}. {x <..})"])
       (auto intro: less_le_trans[OF _ mono] less_imp_le)
qed

lemma continuous_on_IccI:
  "(f  f a) (at_right a);
    (f  f b) (at_left b);
    (x. a < x  x < b  f x f x); a < b 
    continuous_on {a .. b} f"
  for a::"'a::linorder_topology"
  using at_within_open[of _ "{a<..<b}"]
  by (auto simp: continuous_on_def at_within_Icc_at_right at_within_Icc_at_left le_less
      at_within_Icc_at)

lemma
  fixes a b::"'a::linorder_topology"
  assumes "continuous_on {a .. b} f" "a < b"
  shows continuous_on_Icc_at_rightD: "(f  f a) (at_right a)"
    and continuous_on_Icc_at_leftD: "(f  f b) (at_left b)"
  using assms
  by (auto simp: at_within_Icc_at_right at_within_Icc_at_left continuous_on_def
      dest: bspec[where x=a] bspec[where x=b])

lemma continuous_on_discrete [simp]:
  "continuous_on A (f :: 'a :: discrete_topology  _)"
  by (auto simp: continuous_on_def at_discrete)

subsubsection ‹Continuity at a point›

definition continuous :: "'a::t2_space filter  ('a  'b::topological_space)  bool"
  where "continuous F f  (f  f (Lim F (λx. x))) F"

lemma continuous_bot[continuous_intros, simp]: "continuous bot f"
  unfolding continuous_def by auto

lemma continuous_trivial_limit: "trivial_limit net  continuous net f"
  by simp

lemma continuous_within: "continuous (at x within s) f  (f  f x) (at x within s)"
  by (cases "trivial_limit (at x within s)") (auto simp add: Lim_ident_at continuous_def)

lemma continuous_within_topological:
  "continuous (at x within s) f 
    (B. open B  f x  B  (A. open A  x  A  (ys. y  A  f y  B)))"
  unfolding continuous_within tendsto_def eventually_at_topological by metis

lemma continuous_within_compose[continuous_intros]:
  "continuous (at x within s) f  continuous (at (f x) within f ` s) g 
    continuous (at x within s) (g  f)"
  by (simp add: continuous_within_topological) metis

lemma continuous_within_compose2:
  "continuous (at x within s) f  continuous (at (f x) within f ` s) g 
    continuous (at x within s) (λx. g (f x))"
  using continuous_within_compose[of x s f g] by (simp add: comp_def)

lemma continuous_at: "continuous (at x) f  f x f x"
  using continuous_within[of x UNIV f] by simp

lemma continuous_ident[continuous_intros, simp]: "continuous (at x within S) (λx. x)"
  unfolding continuous_within by (rule tendsto_ident_at)

lemma continuous_id[continuous_intros, simp]: "continuous (at x within S) id"
  by (simp add: id_def)

lemma continuous_const[continuous_intros, simp]: "continuous F (λx. c)"
  unfolding continuous_def by (rule tendsto_const)

lemma continuous_on_eq_continuous_within:
  "continuous_on s f  (xs. continuous (at x within s) f)"
  unfolding continuous_on_def continuous_within ..

lemma continuous_discrete [simp]:
  "continuous (at x within A) (f :: 'a :: discrete_topology  _)"
  by (auto simp: continuous_def at_discrete)

abbreviation isCont :: "('a::t2_space  'b::topological_space)  'a  bool"
  where "isCont f a  continuous (at a) f"

lemma isCont_def: "isCont f a  f a f a"
  by (rule continuous_at)

lemma isContD: "isCont f x  f x f x"
  by (simp add: isCont_def)

lemma isCont_cong:
  assumes "eventually (λx. f x = g x) (nhds x)"
  shows "isCont f x  isCont g x"
proof -
  from assms have [simp]: "f x = g x"
    by (rule eventually_nhds_x_imp_x)
  from assms have "eventually (λx. f x = g x) (at x)"
    by (auto simp: eventually_at_filter elim!: eventually_mono)
  with assms have "isCont f x  isCont g x" unfolding isCont_def
    by (intro filterlim_cong) (auto elim!: eventually_mono)
  with assms show ?thesis by simp
qed

lemma continuous_at_imp_continuous_at_within: "isCont f x  continuous (at x within s) f"
  by (auto intro: tendsto_mono at_le simp: continuous_at continuous_within)

lemma continuous_on_eq_continuous_at: "open s  continuous_on s f  (xs. isCont f x)"
  by (simp add: continuous_on_def continuous_at at_within_open[of _ s])

lemma continuous_within_open: "a  A  open A  continuous (at a within A) f  isCont f a"
  by (simp add: at_within_open_NO_MATCH)

lemma continuous_at_imp_continuous_on: "xs. isCont f x  continuous_on s f"
  by (auto intro: continuous_at_imp_continuous_at_within simp: continuous_on_eq_continuous_within)

lemma isCont_o2: "isCont f a  isCont g (f a)  isCont (λx. g (f x)) a"
  unfolding isCont_def by (rule tendsto_compose)

lemma continuous_at_compose[continuous_intros]: "isCont f a  isCont g (f a)  isCont (g  f) a"
  unfolding o_def by (rule isCont_o2)

lemma isCont_tendsto_compose: "isCont g l  (f  l) F  ((λx. g (f x))  g l) F"
  unfolding isCont_def by (rule tendsto_compose)

lemma continuous_on_tendsto_compose:
  assumes f_cont: "continuous_on s f"
    and g: "(g  l) F"
    and l: "l  s"
    and ev: "Fx in F. g x  s"
  shows "((λx. f (g x))  f l) F"
proof -
  from f_cont l have f: "(f  f l) (at l within s)"
    by (simp add: continuous_on_def)
  have i: "((λx. if g x = l then f l else f (g x))  f l) F"
    by (rule filterlim_If)
       (auto intro!: filterlim_compose[OF f] eventually_conj tendsto_mono[OF _ g]
             simp: filterlim_at eventually_inf_principal eventually_mono[OF ev])
  show ?thesis
    by (rule filterlim_cong[THEN iffD1[OF _ i]]) auto
qed

lemma continuous_within_compose3:
  "isCont g (f x)  continuous (at x within s) f  continuous (at x within s) (λx. g (f x))"
  using continuous_at_imp_continuous_at_within continuous_within_compose2 by blast

lemma filtermap_nhds_open_map:
  assumes cont: "isCont f a"
    and open_map: "S. open S  open (f`S)"
  shows "filtermap f (nhds a) = nhds (f a)"
  unfolding filter_eq_iff
proof safe
  fix P
  assume "eventually P (filtermap f (nhds a))"
  then obtain S where "open S" "a  S" "xS. P (f x)"
    by (auto simp: eventually_filtermap eventually_nhds)
  then show "eventually P (nhds (f a))"
    unfolding eventually_nhds by (intro exI[of _ "f`S"]) (auto intro!: open_map)
qed (metis filterlim_iff tendsto_at_iff_tendsto_nhds isCont_def eventually_filtermap cont)

lemma continuous_at_split:
  "continuous (at x) f  continuous (at_left x) f  continuous (at_right x) f"
  for x :: "'a::linorder_topology"
  by (simp add: continuous_within filterlim_at_split)

lemma continuous_on_max [continuous_intros]:
  fixes f g :: "'a::topological_space  'b::linorder_topology"
  shows "continuous_on A f  continuous_on A g  continuous_on A (λx. max (f x) (g x))"
  by (auto simp: continuous_on_def intro!: tendsto_max)

lemma continuous_on_min [continuous_intros]:
  fixes f g :: "'a::topological_space  'b::linorder_topology"
  shows "continuous_on A f  continuous_on A g  continuous_on A (λx. min (f x) (g x))"
  by (auto simp: continuous_on_def intro!: tendsto_min)

lemma continuous_max [continuous_intros]:
  fixes f :: "'a::t2_space  'b::linorder_topology"
  shows "continuous F f; continuous F g  continuous F (λx. (max (f x) (g x)))"
  by (simp add: tendsto_max continuous_def)

lemma continuous_min [continuous_intros]:
  fixes f :: "'a::t2_space  'b::linorder_topology"
  shows "continuous F f; continuous F g  continuous F (λx. (min (f x) (g x)))"
  by (simp add: tendsto_min continuous_def)

text ‹
  The following open/closed Collect lemmas are ported from
  Sébastien Gouëzel's Ergodic_Theory›.
›
lemma open_Collect_neq:
  fixes f g :: "'a::topological_space  'b::t2_space"
  assumes f: "continuous_on UNIV f" and g: "continuous_on UNIV g"
  shows "open {x. f x  g x}"
proof (rule openI)
  fix t
  assume "t  {x. f x  g x}"
  then obtain U V where *: "open U" "open V" "f t  U" "g t  V" "U  V = {}"
    by (auto simp add: separation_t2)
  with open_vimage[OF ‹open U f] open_vimage[OF ‹open V g]
  show "T. open T  t  T  T  {x. f x  g x}"
    by (intro exI[of _ "f -` U  g -` V"]) auto
qed

lemma closed_Collect_eq:
  fixes f g :: "'a::topological_space  'b::t2_space"
  assumes f: "continuous_on UNIV f" and g: "continuous_on UNIV g"
  shows "closed {x. f x = g x}"
  using open_Collect_neq[OF f g] by (simp add: closed_def Collect_neg_eq)

lemma open_Collect_less:
  fixes f g :: "'a::topological_space  'b::linorder_topology"
  assumes f: "continuous_on UNIV f" and g: "continuous_on UNIV g"
  shows "open {x. f x < g x}"
proof (rule openI)
  fix t
  assume t: "t  {x. f x < g x}"
  show "T. open T  t  T  T  {x. f x < g x}"
  proof (cases "z. f t < z  z < g t")
    case True
    then obtain z where "f t < z  z < g t" by blast
    then show ?thesis
      using open_vimage[OF _ f, of "{..< z}"] open_vimage[OF _ g, of "{z <..}"]
      by (intro exI[of _ "f -` {..<z}  g -` {z<..}"]) auto
  next
    case False
    then have *: "{g t ..} = {f t <..}" "{..< g t} = {.. f t}"
      using t by (auto intro: leI)
    show ?thesis
      using open_vimage[OF _ f, of "{..< g t}"] open_vimage[OF _ g, of "{f t <..}"] t
      apply (intro exI[of _ "f -` {..< g t}  g -` {f t<..}"])
      apply (simp add: open_Int)
      apply (auto simp add: *)
      done
  qed
qed

lemma closed_Collect_le:
  fixes f g :: "'a :: topological_space  'b::linorder_topology"
  assumes f: "continuous_on UNIV f"
    and g: "continuous_on UNIV g"
  shows "closed {x. f x  g x}"
  using open_Collect_less [OF g f]
  by (simp add: closed_def Collect_neg_eq[symmetric] not_le)


subsubsection ‹Open-cover compactness›

context topological_space
begin

definition compact :: "'a set  bool" where
compact_eq_Heine_Borel:  (* This name is used for backwards compatibility *)
    "compact S  (C. (cC. open c)  S  C  (DC. finite D  S  D))"

lemma compactI:
  assumes "C. tC. open t  s  C  C'. C'  C  finite C'  s  C'"
  shows "compact s"
  unfolding compact_eq_Heine_Borel using assms by metis

lemma compact_empty[simp]: "compact {}"
  by (auto intro!: compactI)

lemma compactE: (*related to COMPACT_IMP_HEINE_BOREL in HOL Light*)
  assumes "compact S" "S  𝒯" "B. B  𝒯  open B"
  obtains 𝒯' where "𝒯'  𝒯" "finite 𝒯'" "S  𝒯'"
  by (meson assms compact_eq_Heine_Borel)

lemma compactE_image:
  assumes "compact S"
    and opn: "T. T  C  open (f T)"
    and S: "S  (cC. f c)"
  obtains C' where "C'  C" and "finite C'" and "S  (cC'. f c)"
    apply (rule compactE[OF ‹compact S S])
    using opn apply force
    by (metis finite_subset_image)

lemma compact_Int_closed [intro]:
  assumes "compact S"
    and "closed T"
  shows "compact (S  T)"
proof (rule compactI)
  fix C
  assume C: "cC. open c"
  assume cover: "S  T  C"
  from C ‹closed T have "cC  {- T}. open c"
    by auto
  moreover from cover have "S  (C  {- T})"
    by auto
  ultimately have "DC  {- T}. finite D  S  D"
    using ‹compact S unfolding compact_eq_Heine_Borel by auto
  then obtain D where "D  C  {- T}  finite D  S  D" ..
  then show "DC. finite D  S  T  D"
    by (intro exI[of _ "D - {-T}"]) auto
qed

lemma compact_diff: "compact S; open T  compact(S - T)"
  by (simp add: Diff_eq compact_Int_closed open_closed)

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


subsection ‹Finite intersection property›

lemma compact_fip:
  "compact U 
    (A. (aA. closed a)  (B  A. finite B  U  B  {})  U  A  {})"
  (is "_  ?R")
proof (safe intro!: compact_eq_Heine_Borel[THEN iffD2])
  fix A
  assume "compact U"
  assume A: "aA. closed a" "U  A = {}"
  assume fin: "B  A. finite B  U  B  {}"
  from A have "(auminus`A. open a)  U  (uminus`A)"
    by auto
  with ‹compact U obtain B where "B  A" "finite (uminus`B)" "U  (uminus`B)"
    unfolding compact_eq_Heine_Borel by (metis subset_image_iff)
  with fin[THEN spec, of B] show False
    by (auto dest: finite_imageD intro: inj_setminus)
next
  fix A
  assume ?R
  assume "aA. open a" "U  A"
  then have "U  (uminus`A) = {}" "auminus`A. closed a"
    by auto
  with ?R obtain B where "B  A" "finite (uminus`B)" "U  (uminus`B) = {}"
    by (metis subset_image_iff)
  then show "TA. finite T  U  T"
    by (auto intro!: exI[of _ B] inj_setminus dest: finite_imageD)
qed

lemma compact_imp_fip:
  assumes "compact S"
    and "T. T  F  closed T"
    and "F'. finite F'  F'  F  S  (F')  {}"
  shows "S  (F)  {}"
  using assms unfolding compact_fip by auto

lemma compact_imp_fip_image:
  assumes "compact s"
    and P: "i. i  I  closed (f i)"
    and Q: "I'. finite I'  I'  I  (s  (iI'. f i)  {})"
  shows "s  (iI. f i)  {}"
proof -
  note ‹compact s
  moreover from P have "i  f ` I. closed i"
    by blast
  moreover have "A. finite A  A  f ` I  (s  (A)  {})"
    apply rule
    apply rule
    apply (erule conjE)
  proof -
    fix A :: "'a set set"
    assume "finite A" and "A  f ` I"
    then obtain B where "B  I" and "finite B" and "A = f ` B"
      using finite_subset_image [of A f I] by blast
    with Q [of B] show "s  A  {}"
      by simp
  qed
  ultimately have "s  ((f ` I))  {}"
    by (metis compact_imp_fip)
  then show ?thesis by simp
qed

end

lemma (in t2_space) compact_imp_closed:
  assumes "compact s"
  shows "closed s"
  unfolding closed_def
proof (rule openI)
  fix y
  assume "y  - s"
  let ?C = "xs. {u. open u  x  u  eventually (λy. y  u) (nhds y)}"
  have "s  ?C"
  proof
    fix x
    assume "x  s"
    with y  - s have "x  y" by clarsimp
    then have "u v. open u  open v  x  u  y  v  u  v = {}"
      by (rule hausdorff)
    with x  s show "x  ?C"
      unfolding eventually_nhds by auto
  qed
  then obtain D where "D  ?C" and "finite D" and "s  D"
    by (rule compactE [OF ‹compact s]) auto
  from D  ?C have "xD. eventually (λy. y  x) (nhds y)"
    by auto
  with ‹finite D have "eventually (λy. y  D) (nhds y)"
    by (simp add: eventually_ball_finite)
  with s  D have "eventually (λy. y  s) (nhds y)"
    by (auto elim!: eventually_mono)
  then show "t. open t  y  t  t  - s"
    by (simp add: eventually_nhds subset_eq)
qed

lemma compact_continuous_image:
  assumes f: "continuous_on s f"
    and s: "compact s"
  shows "compact (f ` s)"
proof (rule compactI)
  fix C
  assume "cC. open c" and cover: "f`s  C"
  with f have "cC. A. open A  A  s = f -` c  s"
    unfolding continuous_on_open_invariant by blast
  then obtain A where A: "cC. open (A c)  A c  s = f -` c  s"
    unfolding bchoice_iff ..
  with cover have "c. c  C  open (A c)" "s  (cC. A c)"
    by (fastforce simp add: subset_eq set_eq_iff)+
  from compactE_image[OF s this] obtain D where "D  C" "finite D" "s  (cD. A c)" .
  with A show "D  C. finite D  f`s  D"
    by (intro exI[of _ D]) (fastforce simp add: subset_eq set_eq_iff)+
qed

lemma continuous_on_inv:
  fixes f :: "'a::topological_space  'b::t2_space"
  assumes "continuous_on s f"
    and "compact s"
    and "xs. g (f x) = x"
  shows "continuous_on (f ` s) g"
  unfolding continuous_on_topological
proof (clarsimp simp add: assms(3))
  fix x :: 'a and B :: "'a set"
  assume "x  s" and "open B" and "x  B"
  have 1: "xs. f x  f ` (s - B)  x  s - B"
    using assms(3) by (auto, metis)
  have "continuous_on (s - B) f"
    using ‹continuous_on s f Diff_subset
    by (rule continuous_on_subset)
  moreover have "compact (s - B)"
    using ‹open B and ‹compact s
    unfolding Diff_eq by (intro compact_Int_closed closed_Compl)
  ultimately have "compact (f ` (s - B))"
    by (rule compact_continuous_image)
  then have "closed (f ` (s - B))"
    by (rule compact_imp_closed)
  then have "open (- f ` (s - B))"
    by (rule open_Compl)
  moreover have "f x  - f ` (s - B)"
    using x  s and x  B by (simp add: 1)
  moreover have "ys. f y  - f ` (s - B)  y  B"
    by (simp add: 1)
  ultimately show "A. open A  f x  A  (ys. f y  A  y  B)"
    by fast
qed

lemma continuous_on_inv_into:
  fixes f :: "'a::topological_space  'b::t2_space"
  assumes s: "continuous_on s f" "compact s"
    and f: "inj_on f s"
  shows "continuous_on (f ` s) (the_inv_into s f)"
  by (rule continuous_on_inv[OF s]) (auto simp: the_inv_into_f_f[OF f])

lemma (in linorder_topology) compact_attains_sup:
  assumes "compact S" "S  {}"
  shows "sS. tS. t  s"
proof (rule classical)
  assume "¬ (sS. tS. t  s)"
  then obtain t where t: "sS. t s  S" and "sS. s < t s"
    by (metis not_le)
  then have "s. sS  open {..< t s}" "S  (sS. {..< t s})"
    by auto
  with ‹compact S obtain C where "C  S" "finite C" and C: "S  (sC. {..< t s})"
    by (metis compactE_image)
  with S  {} have Max: "Max (t`C)  t`C" and "st`C. s  Max (t`C)"
    by (auto intro!: Max_in)
  with C have "S  {..< Max (t`C)}"
    by (auto intro: less_le_trans simp: subset_eq)
  with t Max C  S show ?thesis
    by fastforce
qed

lemma (in linorder_topology) compact_attains_inf:
  assumes "compact S" "S  {}"
  shows "sS. tS. s  t"
proof (rule classical)
  assume "¬ (sS. tS. s  t)"
  then obtain t where t: "sS. t s  S" and "sS. t s < s"
    by (metis not_le)
  then have "s. sS  open {t s <..}" "S  (sS. {t s <..})"
    by auto
  with ‹compact S obtain C where "C  S" "finite C" and C: "S  (sC. {t s <..})"
    by (metis compactE_image)
  with S  {} have Min: "Min (t`C)  t`C" and "st`C. Min (t`C)  s"
    by (auto intro!: Min_in)
  with C have "S  {Min (t`C) <..}"
    by (auto intro: le_less_trans simp: subset_eq)
  with t Min C  S show ?thesis
    by fastforce
qed

lemma continuous_attains_sup:
  fixes f :: "'a::topological_space  'b::linorder_topology"
  shows "compact s  s  {}  continuous_on s f  (xs. ys.  f y  f x)"
  using compact_attains_sup[of "f ` s"] compact_continuous_image[of s f] by auto

lemma continuous_attains_inf:
  fixes f :: "'a::topological_space  'b::linorder_topology"
  shows "compact s  s  {}  continuous_on s f  (xs. ys. f x  f y)"
  using compact_attains_inf[of "f ` s"] compact_continuous_image[of s f] by auto


subsection ‹Connectedness›

context topological_space
begin

definition "connected S 
  ¬ (A B. open A  open B  S  A  B  A  B  S = {}  A  S  {}  B  S  {})"

lemma connectedI:
  "(A B. open A  open B  A  U  {}  B  U  {}  A  B  U = {}  U  A  B  False)
   connected U"
  by (auto simp: connected_def)

lemma connected_empty [simp]: "connected {}"
  by (auto intro!: connectedI)

lemma connected_sing [simp]: "connected {x}"
  by (auto intro!: connectedI)

lemma connectedD:
  "connected A  open U  open V  U  V  A = {}  A  U  V  U  A = {}  V  A = {}"
  by (auto simp: connected_def)

end

lemma connected_closed:
  "connected s 
    ¬ (A B. closed A  closed B  s  A  B  A  B  s = {}  A  s  {}  B  s  {})"
  apply (simp add: connected_def del: ex_simps, safe)
   apply (drule_tac x="-A" in spec)
   apply (drule_tac x="-B" in spec)
   apply (fastforce simp add: closed_def [symmetric])
  apply (drule_tac x="-A" in spec)
  apply (drule_tac x="-B" in spec)
  apply (fastforce simp add: open_closed [symmetric])
  done

lemma connected_closedD:
  "connected s; A  B  s = {}; s  A  B; closed A; closed B  A  s = {}  B  s = {}"
  by (simp add: connected_closed)

lemma connected_Union:
  assumes cs: "s. s  S  connected s"
    and ne: "S  {}"
  shows "connected(S)"
proof (rule connectedI)
  fix A B
  assume A: "open A" and B: "open B" and Alap: "A  S  {}" and Blap: "B  S  {}"
    and disj: "A  B  S = {}" and cover: "S  A  B"
  have disjs:"s. s  S  A  B  s = {}"
    using disj by auto
  obtain sa where sa: "sa  S" "A  sa  {}"
    using Alap by auto
  obtain sb where sb: "sb  S" "B  sb  {}"
    using Blap by auto
  obtain x where x: "s. s  S  x  s"
    using ne by auto
  then have "x  S"
    using sa  S by blast
  then have "x  A  x  B"
    using cover by auto
  then show False
    using cs [unfolded connected_def]
    by (metis A B IntI Sup_upper sa sb disjs x cover empty_iff subset_trans)
qed

lemma connected_Un: "connected s  connected t  s  t  {}  connected (s  t)"
  using connected_Union [of "{s,t}"] by auto

lemma connected_diff_open_from_closed:
  assumes st: "s  t"
    and tu: "t  u"
    and s: "open s"
    and t: "closed t"
    and u: "connected u"
    and ts: "connected (t - s)"
  shows "connected(u - s)"
proof (rule connectedI)
  fix A B
  assume AB: "open A" "open B" "A  (u - s)  {}" "B  (u - s)  {}"
    and disj: "A  B  (u - s) = {}"
    and cover: "u - s  A  B"
  then consider "A  (t - s) = {}" | "B  (t - s) = {}"
    using st ts tu connectedD [of "t-s" "A" "B"] by auto
  then show False
  proof cases
    case 1
    then have "(A - t)  (B  s)  u = {}"
      using disj st by auto
    moreover have "u  (A - t)  (B  s)"
      using 1 cover by auto
    ultimately show False
      using connectedD [of u "A - t" "B  s"] AB s t 1 u by auto
  next
    case 2
    then have "(A  s)  (B - t)  u = {}"
      using disj st by auto
    moreover have "u  (A  s)  (B - t)"
      using 2 cover by auto
    ultimately show False
      using connectedD [of u "A  s" "B - t"] AB s t 2 u by auto
  qed
qed

lemma connected_iff_const:
  fixes S :: "'a::topological_space set"
  shows "connected S  (P::'a  bool. continuous_on S P  (c. sS. P s = c))"
proof safe
  fix P :: "'a  bool"
  assume "connected S" "continuous_on S P"
  then have "b. A. open A  A  S = P -` {b}  S"
    unfolding continuous_on_open_invariant by (simp add: open_discrete)
  from this[of True] this[of False]
  obtain t f where "open t" "open f" and *: "f  S = P -` {False}  S" "t  S = P -` {True}  S"
    by meson
  then have "t  S = {}  f  S = {}"
    by (intro connectedD[OF ‹connected S])  auto
  then show "c. sS. P s = c"
  proof (rule disjE)
    assume "t  S = {}"
    then show ?thesis
      unfolding * by (intro exI[of _ False]) auto
  next
    assume "f  S = {}"
    then show ?thesis
      unfolding * by (intro exI[of _ True]) auto
  qed
next
  assume P: "P::'a  bool. continuous_on S P  (c. sS. P s = c)"
  show "connected S"
  proof (rule connectedI)
    fix A B
    assume *: "open A" "open B" "A  S  {}" "B  S  {}" "A  B  S = {}" "S  A  B"
    have "continuous_on S (λx. x  A)"
      unfolding continuous_on_open_invariant
    proof safe
      fix C :: "bool set"
      have "C = UNIV  C = {True}  C = {False}  C = {}"
        using subset_UNIV[of C] unfolding UNIV_bool by auto
      with * show "T. open T  T  S = (λx. x  A) -` C  S"
        by (intro exI[of _ "(if True  C then A else {})  (if False  C then B else {})"]) auto
    qed
    from P[rule_format, OF this] obtain c where "s. s  S  (s  A) = c"
      by blast
    with * show False
      by (cases c) auto
  qed
qed

lemma connectedD_const: "connected S  continuous_on S P  c. sS. P s = c"
  for P :: "'a::topological_space  bool"
  by (auto simp: connected_iff_const)

lemma connectedI_const:
  "(P::'a::topological_space  bool. continuous_on S P  c. sS. P s = c)  connected S"
  by (auto simp: connected_iff_const)

lemma connected_local_const:
  assumes "connected A" "a  A" "b  A"
    and *: "aA. eventually (λb. f a = f b) (at a within A)"
  shows "f a = f b"
proof -
  obtain S where S: "a. a  A  a  S a" "a. a  A  open (S a)"
    "a x. a  A  x  S a  x  A  f a = f x"
    using * unfolding eventually_at_topological by metis
  let ?P = "b{bA. f a = f b}. S b" and ?N = "b{bA. f a  f b}. S b"
  have "?P  A = {}  ?N  A = {}"
    using ‹connected A S aA
    by (intro connectedD) (auto, metis)
  then show "f a = f b"
  proof
    assume "?N  A = {}"
    then have "xA. f a = f x"
      using S(1) by auto
    with bA show ?thesis by auto
  next
    assume "?P  A = {}" then show ?thesis
      using a  A S(1)[of a] by auto
  qed
qed

lemma (in linorder_topology) connectedD_interval:
  assumes "connected U"
    and xy: "x  U" "y  U"
    and "x  z" "z  y"
  shows "z  U"
proof -
  have eq: "{..<z}  {z<..} = - {z}"
    by auto
  have "¬ connected U" if "z  U" "x < z" "z < y"
    using xy that
    apply (simp only: connected_def simp_thms)
    apply (rule_tac exI[of _ "{..< z}"])
    apply (rule_tac exI[of _ "{z <..}"])
    apply (auto simp add: eq)
    done
  with assms show "z  U"
    by (metis less_le)
qed

lemma (in linorder_topology) not_in_connected_cases:
  assumes conn: "connected S"
  assumes nbdd: "x  S"
  assumes ne: "S  {}"
  obtains "bdd_above S" "y. y  S  x  y" | "bdd_below S" "y. y  S  x  y"
proof -
  obtain s where "s  S" using ne by blast
  {
    assume "s  x"
    have "False" if "x  y" "y  S" for y
      using connectedD_interval[OF conn s  S y  S s  x x  y] x  S
      by simp
    then have wit: "y  S  x  y" for y
      using le_cases by blast
    then have "bdd_above S"
      by (rule local.bdd_aboveI)
    note this wit
  } moreover {
    assume "x  s"
    have "False" if "x  y" "y  S" for y
      using connectedD_interval[OF conn y  S s  S x  y s  x ] x  S
      by simp
    then have wit: "y  S  x  y" for y
      using le_cases by blast
    then have "bdd_below S"
      by (rule bdd_belowI)
    note this wit
  } ultimately show ?thesis
    by (meson le_cases that)
qed

lemma connected_continuous_image:
  assumes *: "continuous_on s f"
    and "connected s"
  shows "connected (f ` s)"
proof (rule connectedI_const)
  fix P :: "'b  bool"
  assume "continuous_on (f ` s) P"
  then have "continuous_on s (P  f)"
    by (rule continuous_on_compose[OF *])
  from connectedD_const[OF ‹connected s this] show "c. sf ` s. P s = c"
    by auto
qed


section ‹Linear Continuum Topologies›

class linear_continuum_topology = linorder_topology + linear_continuum
begin

lemma Inf_notin_open:
  assumes A: "open A"
    and bnd: "aA. x < a"
  shows "Inf A  A"
proof
  assume "Inf A  A"
  then obtain b where "b < Inf A" "{b <.. Inf A}  A"
    using open_left[of A "Inf A" x] assms by auto
  with dense[of b "Inf A"] obtain c where "c < Inf A" "c  A"
    by (auto simp: subset_eq)
  then show False
    using cInf_lower[OF c  A] bnd
    by (metis not_le less_imp_le bdd_belowI)
qed

lemma Sup_notin_open:
  assumes A: "open A"
    and bnd: "aA. a < x"
  shows "Sup A  A"
proof
  assume "Sup A  A"
  with assms obtain b where "Sup A < b" "{Sup A ..< b}  A"
    using open_right[of A "Sup A" x] by auto
  with dense[of "Sup A" b] obtain c where "Sup A < c" "c  A"
    by (auto simp: subset_eq)
  then show False
    using cSup_upper[OF c  A] bnd
    by (metis less_imp_le not_le bdd_aboveI)
qed

end

instance linear_continuum_topology  perfect_space
proof
  fix x :: 'a
  obtain y where "x < y  y < x"
    using ex_gt_or_lt [of x] ..
  with Inf_notin_open[of "{x}" y] Sup_notin_open[of "{x}" y] show "¬ open {x}"
    by auto
qed

lemma connectedI_interval:
  fixes U :: "'a :: linear_continuum_topology set"
  assumes *: "x y z. x  U  y  U  x  z  z  y  z  U"
  shows "connected U"
proof (rule connectedI)
  {
    fix A B
    assume "open A" "open B" "A  B  U = {}" "U  A  B"
    fix x y
    assume "x < y" "x  A" "y  B" "x  U" "y  U"

    let ?z = "Inf (B  {x <..})"

    have "x  ?z" "?z  y"
      using y  B x < y by (auto intro: cInf_lower cInf_greatest)
    with x  U y  U have "?z  U"
      by (rule *)
    moreover have "?z  B  {x <..}"
      using ‹open B by (intro Inf_notin_open) auto
    ultimately have "?z  A"
      using x  ?z A  B  U = {} x  A U  A  B by auto
    have "bB. b  A  b  U" if "?z < y"
    proof -
      obtain a where "?z < a" "{?z ..< a}  A"
        using open_right[OF ‹open A ?z  A ?z < y] by auto
      moreover obtain b where "b  B" "x < b" "b < min a y"
        using cInf_less_iff[of "B  {x <..}" "min a y"] ?z < a ?z < y x < y y  B
        by auto
      moreover have "?z  b"
        using b  B x < b
        by (intro cInf_lower) auto
      moreover have "b  U"
        using x  ?z ?z  b b < min a y
        by (intro *[OF x  U y  U]) (auto simp: less_imp_le)
      ultimately show ?thesis
        by (intro bexI[of _ b]) auto
    qed
    then have False
      using ?z  y ?z  A y  B y  U A  B  U = {}
      unfolding le_less by blast
  }
  note not_disjoint = this

  fix A B assume AB: "open A" "open B" "U  A  B" "A  B  U = {}"
  moreover assume "A  U  {}" then obtain x where x: "x  U" "x  A" by auto
  moreover assume "B  U  {}" then obtain y where y: "y  U" "y  B" by auto
  moreover note not_disjoint[of B A y x] not_disjoint[of A B x y]
  ultimately show False
    by (cases x y rule: linorder_cases) auto
qed

lemma connected_iff_interval: "connected U  (xU. yU. z. x  z  z  y  z  U)"
  for U :: "'a::linear_continuum_topology set"
  by (auto intro: connectedI_interval dest: connectedD_interval)

lemma connected_UNIV[simp]: "connected (UNIV::'a::linear_continuum_topology set)"
  by (simp add: connected_iff_interval)

lemma connected_Ioi[simp]: "connected {a<..}"
  for a :: "'a::linear_continuum_topology"
  by (auto simp: connected_iff_interval)

lemma connected_Ici[simp]: "connected {a..}"
  for a :: "'a::linear_continuum_topology"
  by (auto simp: connected_iff_interval)

lemma connected_Iio[simp]: "connected {..<a}"
  for a :: "'a::linear_continuum_topology"
  by (auto simp: connected_iff_interval)

lemma connected_Iic[simp]: "connected {..a}"
  for a :: "'a::linear_continuum_topology"
  by (auto simp: connected_iff_interval)

lemma connected_Ioo[simp]: "connected {a<..<b}"
  for a b :: "'a::linear_continuum_topology"
  unfolding connected_iff_interval by auto

lemma connected_Ioc[simp]: "connected {a<..b}"
  for a b :: "'a::linear_continuum_topology"
  by (auto simp: connected_iff_interval)

lemma connected_Ico[simp]: "connected {a..<b}"
  for a b :: "'a::linear_continuum_topology"
  by (auto simp: connected_iff_interval)

lemma connected_Icc[simp]: "connected {a..b}"
  for a b :: "'a::linear_continuum_topology"
  by (auto simp: connected_iff_interval)

lemma connected_contains_Ioo:
  fixes A :: "'a :: linorder_topology set"
  assumes "connected A" "a  A" "b  A" shows "{a <..< b}  A"
  using connectedD_interval[OF assms] by (simp add: subset_eq Ball_def less_imp_le)

lemma connected_contains_Icc:
  fixes A :: "'a::linorder_topology set"
  assumes "connected A" "a  A" "b  A"
  shows "{a..b}  A"
proof
  fix x assume "x  {a..b}"
  then have "x = a  x = b  x  {a<..<b}"
    by auto
  then show "x  A"
    using assms connected_contains_Ioo[of A a b] by auto
qed


subsection ‹Intermediate Value Theorem›

lemma IVT':
  fixes f :: "'a::linear_continuum_topology  'b::linorder_topology"
  assumes y: "f a  y" "y  f b" "a  b"
    and *: "continuous_on {a .. b} f"
  shows "x. a  x  x  b  f x = y"
proof -
  have "connected {a..b}"
    unfolding connected_iff_interval by auto
  from connected_continuous_image[OF * this, THEN connectedD_interval, of "f a" "f b" y] y
  show ?thesis
    by (auto simp add: atLeastAtMost_def atLeast_def atMost_def)
qed

lemma IVT2':
  fixes f :: "'a :: linear_continuum_topology  'b :: linorder_topology"
  assumes y: "f b  y" "y  f a" "a  b"
    and *: "continuous_on {a .. b} f"
  shows "x. a  x  x  b  f x = y"
proof -
  have "connected {a..b}"
    unfolding connected_iff_interval by auto
  from connected_continuous_image[OF * this, THEN connectedD_interval, of "f b" "f a" y] y
  show ?thesis
    by (auto simp add: atLeastAtMost_def atLeast_def atMost_def)
qed

lemma IVT:
  fixes f :: "'a::linear_continuum_topology  'b::linorder_topology"
  shows "f a  y  y  f b  a  b  (x. a  x  x  b  isCont f x) 
    x. a  x  x  b  f x = y"
  by (rule IVT') (auto intro: continuous_at_imp_continuous_on)

lemma IVT2:
  fixes f :: "'a::linear_continuum_topology  'b::linorder_topology"
  shows "f b  y  y  f a  a  b  (x. a  x  x  b  isCont f x) 
    x. a  x  x  b  f x = y"
  by (rule IVT2') (auto intro: continuous_at_imp_continuous_on)

lemma continuous_inj_imp_mono:
  fixes f :: "'a::linear_continuum_topology  'b::linorder_topology"
  assumes x: "a < x" "x < b"
    and cont: "continuous_on {a..b} f"
    and inj: "inj_on f {a..b}"
  shows "(f a < f x  f x < f b)  (f b < f x  f x < f a)"
proof -
  note I = inj_on_eq_iff[OF inj]
  {
    assume "f x < f a" "f x < f b"
    then obtain s t where "x  s" "s  b" "a  t" "t  x" "f s = f t" "f x < f s"
      using IVT'[of f x "min (f a) (f b)" b] IVT2'[of f x "min (f a) (f b)" a] x
      by (auto simp: continuous_on_subset[OF cont] less_imp_le)
    with x I have False by auto
  }
  moreover
  {
    assume "f a < f x" "f b < f x"
    then obtain s t where "x  s" "s  b" "a  t" "t  x" "f s = f t" "f s < f x"
      using IVT'[of f a "max (f a) (f b)" x] IVT2'[of f b "max (f a) (f b)" x] x
      by (auto simp: continuous_on_subset[OF cont] less_imp_le)
    with x I have False by auto
  }
  ultimately show ?thesis
    using I[of a x] I[of x b] x less_trans[OF x]
    by (auto simp add: le_less less_imp_neq neq_iff)
qed

lemma continuous_at_Sup_mono:
  fixes f :: "'a::{linorder_topology,conditionally_complete_linorder} 
    'b::{linorder_topology,conditionally_complete_linorder}"
  assumes "mono f"
    and cont: "continuous (at_left (Sup S)) f"
    and S: "S  {}" "bdd_above S"
  shows "f (Sup S) = (SUP sS. f s)"
proof (rule antisym)
  have f: "(f  f (Sup S)) (at_left (Sup S))"
    using cont unfolding continuous_within .
  show "f (Sup S)  (SUP sS. f s)"
  proof cases
    assume "Sup S  S"
    then show ?thesis
      by (rule cSUP_upper) (auto intro: bdd_above_image_mono S ‹mono f)
  next
    assume "Sup S  S"
    from S  {} obtain s where "s  S"
      by auto
    with ‹Sup S  S S have "s < Sup S"
      unfolding less_le by (blast intro: cSup_upper)
    show ?thesis
    proof (rule ccontr)
      assume "¬ ?thesis"
      with order_tendstoD(1)[OF f, of "SUP sS. f s"] obtain b where "b < Sup S"
        and *: "y. b < y  y < Sup S  (SUP sS. f s) < f y"
        by (auto simp: not_le eventually_at_left[OF s < Sup S])
      with S  {} obtain c where "c  S" "b < c"
        using less_cSupD[of S b] by auto
      with ‹Sup S  S S have "c < Sup S"
        unfolding less_le by (blast intro: cSup_upper)
      from *[OF b < c c < Sup S] cSUP_upper[OF c  S bdd_above_image_mono[of f]]
      show False
        by (auto simp: assms)
    qed
  qed
qed (intro cSUP_least ‹mono f[THEN monoD] cSup_upper S)

lemma continuous_at_Sup_antimono:
  fixes f :: "'a::{linorder_topology,conditionally_complete_linorder} 
    'b::{linorder_topology,conditionally_complete_linorder}"
  assumes "antimono f"
    and cont: "continuous (at_left (Sup S)) f"
    and S: "S  {}" "bdd_above S"
  shows "f (Sup S) = (INF sS. f s)"
proof (rule antisym)
  have f: "(f  f (Sup S)) (at_left (Sup S))"
    using cont unfolding continuous_within .
  show "(INF sS. f s)  f (Sup S)"
  proof cases
    assume "Sup S  S"
    then show ?thesis
      by (intro cINF_lower) (auto intro: bdd_below_image_antimono S ‹antimono f)
  next
    assume "Sup S  S"
    from S  {} obtain s where "s  S"
      by auto
    with ‹Sup S  S S have "s < Sup S"
      unfolding less_le by (blast intro: cSup_upper)
    show ?thesis
    proof (rule ccontr)
      assume "¬ ?thesis"
      with order_tendstoD(2)[OF f, of "INF sS. f s"] obtain b where "b < Sup S"
        and *: "y. b < y  y < Sup S  f y < (INF sS. f s)"
        by (auto simp: not_le eventually_at_left[OF s < Sup S])
      with S  {} obtain c where "c  S" "b < c"
        using less_cSupD[of S b] by auto
      with ‹Sup S  S S have "c < Sup S"
        unfolding less_le by (blast intro: cSup_upper)
      from *[OF b < c c < Sup S] cINF_lower[OF bdd_below_image_antimono, of f S c] c  S
      show False
        by (auto simp: assms)
    qed
  qed
qed (intro cINF_greatest ‹antimono f[THEN antimonoD] cSup_upper S)

lemma continuous_at_Inf_mono:
  fixes f :: "'a::{linorder_topology,conditionally_complete_linorder} 
    'b::{linorder_topology,conditionally_complete_linorder}"
  assumes "mono f"
    and cont: "continuous (at_right (Inf S)) f"
    and S: "S  {}" "bdd_below S"
  shows "f (Inf S) = (INF sS. f s)"
proof (rule antisym)
  have f: "(f  f (Inf S)) (at_right (Inf S))"
    using cont unfolding continuous_within .
  show "(INF sS. f s)  f (Inf S)"
  proof cases
    assume "Inf S  S"
    then show ?thesis
      by (rule cINF_lower[rotated]) (auto intro: bdd_below_image_mono S ‹mono f)
  next
    assume "Inf S  S"
    from S  {} obtain s where "s  S"
      by auto
    with ‹Inf S  S S have "Inf S < s"
      unfolding less_le by (blast intro: cInf_lower)
    show ?thesis
    proof (rule ccontr)
      assume "¬ ?thesis"
      with order_tendstoD(2)[OF f, of "INF sS. f s"] obtain b where "Inf S < b"
        and *: "y. Inf S < y  y < b  f y < (INF sS. f s)"
        by (auto simp: not_le eventually_at_right[OF ‹Inf S < s])
      with S  {} obtain c where "c  S" "c < b"
        using cInf_lessD[of S b] by auto
      with ‹Inf S  S S have "Inf S < c"
        unfolding less_le by (blast intro: cInf_lower)
      from *[OF ‹Inf S < c c < b] cINF_lower[OF bdd_below_image_mono[of f] c  S]
      show False
        by (auto simp: assms)
    qed
  qed
qed (intro cINF_greatest ‹mono f[THEN monoD] cInf_lower ‹bdd_below S S  {})

lemma continuous_at_Inf_antimono:
  fixes f :: "'a::{linorder_topology,conditionally_complete_linorder} 
    'b::{linorder_topology,conditionally_complete_linorder}"
  assumes "antimono f"
    and cont: "continuous (at_right (Inf S)) f"
    and S: "S  {}" "bdd_below S"
  shows "f (Inf S) = (SUP sS. f s)"
proof (rule antisym)
  have f: "(f  f (Inf S)) (at_right (Inf S))"
    using cont unfolding continuous_within .
  show "f (Inf S)  (SUP sS. f s)"
  proof cases
    assume "Inf S  S"
    then show ?thesis
      by (rule cSUP_upper) (auto intro: bdd_above_image_antimono S ‹antimono f)
  next
    assume "Inf S  S"
    from S  {} obtain s where "s  S"
      by auto
    with ‹Inf S  S S have "Inf S < s"
      unfolding less_le by (blast intro: cInf_lower)
    show ?thesis
    proof (rule ccontr)
      assume "¬ ?thesis"
      with order_tendstoD(1)[OF f, of "SUP sS. f s"] obtain b where "Inf S < b"
        and *: "y. Inf S < y  y < b  (SUP sS. f s) < f y"
        by (auto simp: not_le eventually_at_right[OF ‹Inf S < s])
      with S  {} obtain c where "c  S" "c < b"
        using cInf_lessD[of S b] by auto
      with ‹Inf S  S S have "Inf S < c"
        unfolding less_le by (blast intro: cInf_lower)
      from *[OF ‹Inf S < c c < b] cSUP_upper[OF c  S bdd_above_image_antimono[of f]]
      show False
        by (auto simp: assms)
    qed
  qed
qed (intro cSUP_least ‹antimono f[THEN antimonoD] cInf_lower S)


subsection ‹Uniform spaces›

class uniformity =
  fixes uniformity :: "('a × 'a) filter"
begin

abbreviation uniformity_on :: "'a set  ('a × 'a) filter"
  where "uniformity_on s  inf uniformity (principal (s×s))"

end

lemma uniformity_Abort:
  "uniformity =
    Filter.abstract_filter (λu. Code.abort (STR ''uniformity is not executable'') (λu. uniformity))"
  by simp

class open_uniformity = "open" + uniformity +
  assumes open_uniformity:
    "U. open U  (xU. eventually (λ(x', y). x' = x  y  U) uniformity)"
begin

subclass topological_space
  by standard (force elim: eventually_mono eventually_elim2 simp: split_beta' open_uniformity)+

end

class uniform_space = open_uniformity +
  assumes uniformity_refl: "eventually E uniformity  E (x, x)"
    and uniformity_sym: "eventually E uniformity  eventually (λ(x, y). E (y, x)) uniformity"
    and uniformity_trans:
      "eventually E uniformity 
        D. eventually D uniformity  (x y z. D (x, y)  D (y, z)  E (x, z))"
begin

lemma uniformity_bot: "uniformity  bot"
  using uniformity_refl by auto

lemma uniformity_trans':
  "eventually E uniformity 
    eventually (λ((x, y), (y', z)). y = y'  E (x, z)) (uniformity ×F uniformity)"
  by (drule uniformity_trans) (auto simp add: eventually_prod_same)

lemma uniformity_transE:
  assumes "eventually E uniformity"
  obtains D where "eventually D uniformity" "x y z. D (x, y)  D (y, z)  E (x, z)"
  using uniformity_trans [OF assms] by auto

lemma eventually_nhds_uniformity:
  "eventually P (nhds x)  eventually (λ(x', y). x' = x  P y) uniformity"
  (is "_  ?N P x")
  unfolding eventually_nhds
proof safe
  assume *: "?N P x"
  have "?N (?N P) x" if "?N P x" for x
  proof -
    from that obtain D where ev: "eventually D uniformity"
      and D: "D (a, b)  D (b, c)  case (a, c) of (x', y)  x' = x  P y" for a b c
      by (rule uniformity_transE) simp
    from ev show ?thesis
      by eventually_elim (insert ev D, force elim: eventually_mono split: prod.split)
  qed
  then have "open {x. ?N P x}"
    by (simp add: open_uniformity)
  then show "S. open S  x  S  (xS. P x)"
    by (intro exI[of _ "{x. ?N P x}"]) (auto dest: uniformity_refl simp: *)
qed (force simp add: open_uniformity elim: eventually_mono)


subsubsection ‹Totally bounded sets›

definition totally_bounded :: "'a set  bool"
  where "totally_bounded S 
    (E. eventually E uniformity  (X. finite X  (sS. xX. E (x, s))))"

lemma totally_bounded_empty[iff]: "totally_bounded {}"
  by (auto simp add: totally_bounded_def)

lemma totally_bounded_subset: "totally_bounded S  T  S  totally_bounded T"
  by (fastforce simp add: totally_bounded_def)

lemma totally_bounded_Union[intro]:
  assumes M: "finite M" "S. S  M  totally_bounded S"
  shows "totally_bounded (M)"
  unfolding totally_bounded_def
proof safe
  fix E
  assume "eventually E uniformity"
  with M obtain X where "SM. finite (X S)  (sS. xX S. E (x, s))"
    by (metis totally_bounded_def)
  with ‹finite M show "X. finite X  (sM. x