(*  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]: "∀S∈K. 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]: "∀x∈A. open (B x) ==> open (∪x∈A. B x)"
  using open_Union [of "B ` A"] by simp

lemma open_Inter [continuous_intros, intro]: "finite S ==> ∀T∈S. open T ==> open (∩S)"
  by (induction set: finite) auto

lemma open_INT [continuous_intros, intro]: "finite A ==> ∀x∈A. open (B x) ==> open (∩x∈A. 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 ⟷ (∀x∈S. ∃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]: "∀x∈A. closed (B x) ==> closed (∩x∈A. B x)"
  unfolding closed_def by auto

lemma closed_Inter [continuous_intros, intro]: "∀S∈K. closed S ==> closed (∩K)"
  unfolding closed_def uminus_Inf by auto

lemma closed_Union [continuous_intros, intro]: "finite S ==> ∀T∈S. closed T ==> closed (∪S)"
  by (induct set: finite) auto

lemma closed_UN [continuous_intros, intro]:
  "finite A ==> ∀x∈A. closed (B x) ==> closed (∪x∈A. 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:
  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:
  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 topologies›

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 (∪k∈I. 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

lemma filterlim_atLeastAtMost_at_bot_at_top:
  fixes f g :: "'a → 'b :: linorder_topology"
  assumes "filterlim f at_bot F" "filterlim g at_top F"
  assumes [simp]: "∧a b. finite {a..b::'b}"
  shows   "filterlim (λx. {f x..g x}) finite_sets_at_top F"
  unfolding filterlim_finite_subsets_at_top
proof safe
  fix X :: "'b set"
  assume X: "finite X"
  from X obtain lb where lb: "∧x. x ∈ X ==> lb ≤ x"
    by (metis finite_has_minimal2 nle_le)
  from X obtain ub where ub: "∧x. x ∈ X ==> x ≤ ub"
    by (metis all_not_in_conv finite_has_maximal nle_le)
  have "eventually (λx. f x ≤ lb) F" "eventually (λx. g x ≥ ub) F"
    using assms by (simp_all add: filterlim_at_bot filterlim_at_top)
  thus "eventually (λx. finite {f x..g x} ∧ X ⊆ {f x..g x} ∧ {f x..g x} ⊆ UNIV) F"
  proof eventually_elim
    case (elim x)
    have "X ⊆ {f x..g x}"
    proof
      fix y assume "y ∈ X"
      thus "y ∈ {f x..g x}"
        using lb[of y] ub[of y] elim by auto
    qed
    thus ?case
      by auto
  qed
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 (∪a∈A. {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 (∪a∈A. {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∈{S∈T. 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 ∧ (∀x∈S. 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 ∧ (∀x∈S. x ≠ a ⟶ x ∈ s ⟶ P x))"
  by (simp add: eventually_nhds eventually_at_filter)

lemma eventually_nhds_conv_at:
  "eventually P (nhds x) ⟷ eventually P (at x) ∧ P x"
  unfolding eventually_at_topological eventually_nhds by fast

lemma eventually_at_in_open:
  assumes "open A" "x ∈ A"
  shows   "eventually (λy. y ∈ A - {x}) (at x)"
  using assms eventually_at_topological by blast

lemma eventually_at_in_open':
  assumes "open A" "x ∈ A"
  shows   "eventually (λy. y ∈ A) (at x)"
  using assms eventually_at_topological by blast

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
  by (metis UNIV_I empty_iff is_singletonE is_singletonI' singleton_iff)

lemma (in t1_space) eventually_neq_at_within:
  "eventually (λw. w ≠ x) (at z within A)"
  by (smt (verit, ccfv_threshold) eventually_True eventually_at_topological separation_t1)

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)

lemma filtermap_nhds_eq_imp_filtermap_at_eq: 
  assumes "filtermap f (nhds z) = nhds (f z)"
  assumes "eventually (λx. f x = f z ⟶ x = z) (at z)"
  shows   "filtermap f (at z) = at (f z)"
proof (rule filter_eqI)
  fix P :: "'a → bool"
  have "eventually P (filtermap f (at z)) ⟷ (∀_F x in nhds z. x ≠ z ⟶ P (f x))"
    by (simp add: eventually_filtermap eventually_at_filter)
  also have "… ⟷ (∀_F x in nhds z. f x ≠ f z ⟶ P (f x))"
    by (rule eventually_cong [OF assms(2)[unfolded eventually_at_filter]]) auto
  also have "… ⟷ (∀_F x in filtermap f (nhds z). x ≠ f z ⟶ P x)"
    by (simp add: eventually_filtermap)
  also have "filtermap f (nhds z) = nhds (f z)"
    by (rule assms)
  also have "(∀_F x in nhds (f z). x ≠ f z ⟶ P x) ⟷ (∀_F x in at (f z). P x)"
    by (simp add: eventually_at_filter)
  finally show "eventually P (filtermap f (at z)) = eventually P (at (f z))" .
qed

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 (🚫‹tendsto_eq_intros›,
 fn context =>
 Named_Theorems.get (Context.proof_of context) 🚫‹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)

(* Contributed by Dominique Unruh *)
lemma tendsto_principal_singleton[simp]:
  shows "(f ---> f x) (principal {x})"
  unfolding tendsto_def eventually_principal by simp

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.order_antisym filter_leI
      simp: eventually_at_filter less_le
      elim: eventually_elim2)

lemma (in order_topology)
  shows at_within_Ici_at_right: "at a within {a..} = at_right a"
    and at_within_Iic_at_left:  "at a within {..a} = at_left a"
  using order_tendstoD(2)[OF tendsto_ident_at [where s = "{a<..}"]]
  using order_tendstoD(1)[OF tendsto_ident_at[where s = "{..<a}"]]
  by (auto intro!: order_class.order_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 "∃_\<le>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 w∈s ∧ 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 x≠a" 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 (simp add: tendsto_Lim)

lemma Lim_cong:
  assumes "∀_F x in F. f x = g x" "F = G"
  shows "Lim F f = Lim F g"
  unfolding t2_space_class.Lim_def using tendsto_cong assms by fastforce

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 "∀s∈S. 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 ∧ (∀xa∈S. 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"  (‹(‹notation=‹infix LIMSEQ››(_)/ <---- (_))› [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. ∀n≥N. f n ∈ S))"
  unfolding tendsto_def eventually_sequentially by auto

lemma closed_sequentially:
  assumes "closed S" and "∧n. f n ∈ S" and "f <---- l"
  shows "l ∈ S"
  by (metis Lim_in_closed_set assms eventually_sequentially trivial_limit_sequentially)


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. ∀n≥m. X m ≤ X n) ∨ (∀m. ∀n≥m. X n ≤ X m)"

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

lemma incseq_def: "incseq X ⟷ (∀m. ∀n≥m. X n ≥ X m)"
  unfolding mono_def ..

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

lemma decseq_def: "decseq X ⟷ (∀m. ∀n≥m. 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"
  by (simp add: mono_iff_le_Suc)

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"
  by (simp add: antimono_iff_le_Suc)

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)›

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. ∀m≥p. s m ≤ s p")
  case True
  then have "∃f. ∀n. (∀m≥f 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::nat→nat. strict_mono r ∧ (∀n. ¬ P (r n))"
proof -
  from assms have "∀n. ∃m≥n. ¬ 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: "∃_\<le>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. ∀n≥N. 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. ∀n≥N. 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. ∀n≥N. 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 ==> ∀n≥M. 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: "∀n≥N. 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. ∀n≥m. a m ≤ a n) ∨
    (∀n. x ≤ a n) ∧ (∀m. ∀n≥m. 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 = (∩i≤n. 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. ∩i≤n. A i)"
      by (simp add: antimono_iff_le_Suc atMost_Suc)
    show "x ∈ (∩i≤n. A i)" "∧n. open (∩i≤n. A i)" for n
      using * by auto
    with * show "nhds x = (INF n. principal (∩i≤n. A i))"
      unfolding nhds_def
      apply (intro INF_eq)
       apply fastforce
      apply blast
      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" "∀x∈S. 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"
    (‹(‹notation=‹infix LIM››(_)/ ←-(_)/→ (_))› [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: "∃_\<le>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

lemma tendsto_nhds_iff: "(f ---> (c :: 'a :: t1_space)) (nhds x) ⟷ f ←-x→ c ∧ f x = c"
proof safe
  assume lim: "(f ---> c) (nhds x)"
  show "f x = c"
  proof (rule ccontr)
    assume "f x ≠ c"
    hence "c ≠ f x"
      by auto
    then obtain A where A: "open A" "c ∈ A" "f x ∉ A"
      by (subst (asm) separation_t1) auto
    with lim obtain B where "open B" "x ∈ B" "∧x. x ∈ B ==> f x ∈ A"
      unfolding tendsto_def eventually_nhds by metis 
    with ‹f x ∉ A› show False
      by blast
  qed
  show "(f ---> c) (at x)"
    using lim by (rule filterlim_mono) (auto simp: at_within_def)
next
  assume "f ←-x→ f x" "c = f x"
  thus "(f ---> f x) (nhds x)"
    unfolding tendsto_def eventually_at_filter by (fast elim: eventually_mono)
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_conv:
  "(∀S. (∀n. S n ≠ a) ∧ S <---- a ⟶ (λn. X (S n)) <---- L) ⟷ X ←-a→ L"  (is "?lhs=?rhs")
  for a :: "'a::first_countable_topology" and L :: "'b::topological_space"
proof
  assume ?lhs then show ?rhs
    by (simp add: sequentially_imp_eventually_within tendsto_def) 
next
  assume ?rhs then show ?lhs
    using tendsto_compose_eventually eventuallyI by blast
qed    

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 ⟷ (∀x∈s. (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 ⟷
    (∀x∈s. ∀B. open B ⟶ f x ∈ B ⟶ (∃A. open A ∧ x ∈ A ∧ (∀y∈s. 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 "∀x∈f -` 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 "∀x∈f -` 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 _ "∪x∈f -` 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 ∧ (∀y∈s. 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 (∪s∈S. 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 "∀x∈s. f x = ?h x" "∀x∈t. 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. (x∈s ∧ ¬ P x) ∨ (x ∈ t ∧ P x) ⟶ f x = g x ==>
    continuous_on (s ∪ t) (λx. if P x then f x else g x)"
  by (rule continuous_on_If) auto

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 _ "∪k∈K. 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∈{x∈A. 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∈{x∈A. 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)

lemma continuous_on_of_nat [continuous_intros]:
  assumes "continuous_on A f"
  shows   "continuous_on A (λn. of_nat (f n))"
  using continuous_on_compose[OF assms continuous_on_discrete[of _ of_nat]]
  by (simp add: o_def)

lemma continuous_on_of_int [continuous_intros]:
  assumes "continuous_on A f"
  shows   "continuous_on A (λn. of_int (f n))"
  using continuous_on_compose[OF assms continuous_on_discrete[of _ of_int]]
  by (simp add: o_def)

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 ∧ (∀y∈s. 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 ⟷ (∀x∈s. 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)

text ‹Continuity in terms of open preimages.›

lemma continuous_at_open:
  "continuous (at x) f ⟷ (∀t. open t ∧ f x ∈ t ⟶ (∃S. open S ∧ x ∈ S ∧ (∀x' ∈ S. (f x') ∈ t)))"
  by (metis UNIV_I continuous_within_topological)

lemma continuous_imp_tendsto:
  assumes "continuous (at x0) f" and "x <---- x0"
  shows "(f ∘ x) <---- (f x0)"
proof (rule topological_tendstoI)
  fix S
  assume "open S" "f x0 ∈ S"
  then obtain T where T_def: "open T" "x0 ∈ T" "∀x∈T. f x ∈ S"
     using assms continuous_at_open by metis
  then have "eventually (λn. x n ∈ T) sequentially"
    using assms T_def by (auto simp: tendsto_def)
  then show "eventually (λn. (f ∘ x) n ∈ S) sequentially"
    using T_def by (auto elim!: eventually_mono)
qed

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 ⟷ (∀x∈s. 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: "∀x∈s. 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 at_within_isCont_imp_nhds:
  fixes f:: "'a:: {t2_space,perfect_space} → 'b:: t2_space"
  assumes "∀_F w in at z. f w = g w" "isCont f z" "isCont g z"
  shows "∀_F w in nhds z. f w = g w"
proof -
  have "g ←-z→ f z"
    using assms isContD tendsto_cong by blast 
  moreover have "g ←-z→ g z" using ‹isCont g z› using isCont_def by blast
  ultimately have "f z=g z" using LIM_unique by auto
  moreover have "∀_F x in nhds z. x ≠ z ⟶ f x = g x"
    using assms unfolding eventually_at_filter by auto
  ultimately show ?thesis 
    by (auto elim:eventually_mono)
qed

lemma filtermap_nhds_open_map':
  assumes cont: "isCont f a"
    and "open A" "a ∈ A"
    and open_map: "∧S. open S ==> S ⊆ A ==> 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 S: "open S" "a ∈ S" "∀x∈S. P (f x)"
    by (auto simp: eventually_filtermap eventually_nhds)
  show "eventually P (nhds (f a))"
    unfolding eventually_nhds 
  proof (rule exI [of _ "f ` (A ∩ S)"], safe)
    show "open (f ` (A ∩ S))"
      using S by (intro open_Int assms) auto
    show "f a ∈ f ` (A ∩ S)"
      using assms S by auto
    show "P (f x)" if "x ∈ A" "x ∈ S" for x
      using S that by auto
  qed
qed (metis filterlim_iff tendsto_at_iff_tendsto_nhds isCont_def eventually_filtermap cont)

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)"
  using cont filtermap_nhds_open_map' open_map by blast

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. (∀c∈C. open c) ∧ S ⊆ ∪C ⟶ (∃D⊆C. finite D ∧ S ⊆ ∪D))"

lemma compactI:
  assumes "∧C. ∀t∈C. 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 ⊆ (∪c∈C. f c)"
  obtains C' where "C' ⊆ C" and "finite C'" and "S ⊆ (∪c∈C'. 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: "∀c∈C. open c"
  assume cover: "S ∩ T ⊆ ∪C"
  from C ‹closed T› have "∀c∈C ∪ {- T}. open c"
    by auto
  moreover from cover have "S ⊆ ∪(C ∪ {- T})"
    by auto
  ultimately have "∃D⊆C ∪ {- 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 "∃D⊆C. 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. (∀a∈A. 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: "∀a∈A. closed a" "U ∩ ∩A = {}"
  assume fin: "∀B ⊆ A. finite B ⟶ U ∩ ∩B ≠ {}"
  from A have "(∀a∈uminus`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 "∀a∈A. open a" "U ⊆ ∪A"
  then have "U ∩ ∩(uminus`A) = {}" "∀a∈uminus`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 "∃T⊆A. 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 ∩ (∩i∈I'. f i) ≠ {})"
  shows "s ∩ (∩i∈I. f i) ≠ {}"
proof -
  from P have "∀i ∈ f ` I. closed i"
    by blast
  moreover have "∀A. finite A ∧ A ⊆ f ` I ⟶ (s ∩ (∩A) ≠ {})"
    by (metis Q finite_subset_image)
  ultimately show "s ∩ (∩(f ` I)) ≠ {}"
    by (metis ‹compact s› compact_imp_fip)
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 = "∪x∈s. {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 "∀x∈D. 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 "∀c∈C. open c" and cover: "f`s ⊆ ∪C"
  with f have "∀c∈C. ∃A. open A ∧ A ∩ s = f -` c ∩ s"
    unfolding continuous_on_open_invariant by blast
  then obtain A where A: "∀c∈C. open (A c) ∧ A c ∩ s = f -` c ∩ s"
    unfolding bchoice_iff ..
  with cover have "∧c. c ∈ C ==> open (A c)" "s ⊆ (∪c∈C. 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 ⊆ (∪c∈D. 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 "∀x∈s. 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: "∀x∈s. 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 "∀y∈s. f y ∈ - f ` (s - B) ⟶ y ∈ B"
    by (simp add: 1)
  ultimately show "∃A. open A ∧ f x ∈ A ∧ (∀y∈s. 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 "∃s∈S. ∀t∈S. t ≤ s"
proof (rule classical)
  assume "¬ (∃s∈S. ∀t∈S. t ≤ s)"
  then obtain t where t: "∀s∈S. t s ∈ S" and "∀s∈S. s < t s"
    by (metis not_le)
  then have "∧s. s∈S ==> open {..< t s}" "S ⊆ (∪s∈S. {..< t s})"
    by auto
  with ‹compact S› obtain C where "C ⊆ S" "finite C" and C: "S ⊆ (∪s∈C. {..< t s})"
    by (metis compactE_image)
  with ‹S ≠ {}› have Max: "Max (t`C) ∈ t`C" and "∀s∈t`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 "∃s∈S. ∀t∈S. s ≤ t"
proof (rule classical)
  assume "¬ (∃s∈S. ∀t∈S. s ≤ t)"
  then obtain t where t: "∀s∈S. t s ∈ S" and "∀s∈S. t s < s"
    by (metis not_le)
  then have "∧s. s∈S ==> open {t s <..}" "S ⊆ (∪s∈S. {t s <..})"
    by auto
  with ‹compact S› obtain C where "C ⊆ S" "finite C" and C: "S ⊆ (∪s∈C. {t s <..})"
    by (metis compactE_image)
  with ‹S ≠ {}› have Min: "Min (t`C) ∈ t`C" and "∀s∈t`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 ==> (∃x∈s. ∀y∈s. 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 ==> (∃x∈s. ∀y∈s. 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. ∀s∈S. 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. ∀s∈S. 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. ∀s∈S. 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. ∀s∈S. 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. ∀s∈S. P s = c) ==> connected S"
  by (auto simp: connected_iff_const)

lemma connected_local_const:
  assumes "connected A" "a ∈ A" "b ∈ A"
    and *: "∀a∈A. 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∈{b∈A. f a = f b}. S b" and ?N = "∪b∈{b∈A. f a ≠ f b}. S b"
  have "?P ∩ A = {} ∨ ?N ∩ A = {}"
    using ‹connected A› S ‹a∈A›
    by (intro connectedD) (auto, metis)
  then show "f a = f b"
  proof
    assume "?N ∩ A = {}"
    then have "∀x∈A. f a = f x"
      using S(1) by auto
    with ‹b∈A› 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. ∀s∈f ` s. P s = c"
    by auto
qed

lemma connected_Un_UN:
  assumes "connected A" "∧X. X ∈ B ==> connected X" "∧X. X ∈ B ==> A ∩ X ≠ {}"
  shows   "connected (A ∪ ∪B)"
proof (rule connectedI_const)
  fix f :: "'a → bool"
  assume f: "continuous_on (A ∪ ∪B) f"
  have "connected A" "continuous_on A f"
    by (auto intro: assms continuous_on_subset[OF f(1)])
  from connectedD_const[OF this] obtain c where c: "∧x. x ∈ A ==> f x = c"
    by metis
  have "f x = c" if "x ∈ X" "X ∈ B" for x X
  proof -
    have "connected X" "continuous_on X f"
      using that by (auto intro: assms continuous_on_subset[OF f])
    from connectedD_const[OF this] obtain c' where c': "∧x. x ∈ X ==> f x = c'"
      by metis
    from assms(3) and that obtain y where "y ∈ A ∩ X"
      by auto
    with c[of y] c'[of y] c'[of x] that show ?thesis
      by auto
  qed
  with c show "∃c. ∀x∈A ∪ ∪ B. f x = c"
    by (intro exI[of _ c]) 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: "∀a∈A. 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: "∀a∈A. 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 "∃b∈B. 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 ⟷ (∀x∈U. ∀y∈U. ∀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 s∈S. 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 s∈S. 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 s∈S. f s"] obtain b where "b < Sup S"
        and *: "∧y. b < y ==> y < Sup S ==> (SUP s∈S. 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 s∈S. f s)"
proof (rule antisym)
  have f: "(f ---> f (Sup S)) (at_left (Sup S))"
    using cont unfolding continuous_within .
  show "(INF s∈S. 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 s∈S. f s"] obtain b where "b < Sup S"
        and *: "∧y. b < y ==> y < Sup S ==> f y < (INF s∈S. 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 s∈S. f s)"
proof (rule antisym)
  have f: "(f ---> f (Inf S)) (at_right (Inf S))"
    using cont unfolding continuous_within .
  show "(INF s∈S. 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 s∈S. f s"] obtain b where "Inf S < b"
        and *: "∧y. Inf S < y ==> y < b ==> f y < (INF s∈S. 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 s∈S. 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 s∈S. 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 s∈S. f s"] obtain b where "Inf S < b"
        and *: "∧y. Inf S < y ==> y < b ==> (SUP s∈S. 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 ⟷ (∀x∈U. 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 ∧ (∀x∈S. 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 ∧ (∀s∈S. ∃x∈X. 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 "∀S∈M. finite (X S) ∧ (∀s∈S. ∃x∈X S. E (x, s))"
    by (metis totally_bounded_def)
  with ‹finite M› show "∃X. finite X ∧ (∀s∈∪M. ∃x∈X. E (x, s))"
    by (intro exI[of _ "∪S∈M. X S"]) force
qed


subsubsection ‹Cauchy filter›

definition cauchy_filter :: "'a filter → bool"
  where "cauchy_filter F ⟷ F ×_F F ≤ uniformity"

definition Cauchy :: "(nat → 'a) → bool"
  where Cauchy_uniform: "Cauchy X = cauchy_filter (filtermap X sequentially)"

lemma Cauchy_uniform_iff:
  "Cauchy X ⟷ (∀P. eventually P uniformity ⟶ (∃N. ∀n≥N. ∀m≥N. P (X n, X m)))"
  unfolding Cauchy_uniform cauchy_filter_def le_filter_def eventually_prod_same
    eventually_filtermap eventually_sequentially
proof safe
  let ?U = "λP. eventually P uniformity"
  {
    fix P
    assume "?U P" "∀P. ?U P ⟶ (∃Q. (∃N. ∀n≥N. Q (X n)) ∧ (∀x y. Q x ⟶ Q y ⟶ P (x, y)))"
    then obtain Q N where "∧n. n ≥ N ==> Q (X n)" "∧x y. Q x ==> Q y ==> P (x, y)"
      by metis
    then show "∃N. ∀n≥N. ∀m≥N. P (X n, X m)"
      by blast
  next
    fix P
    assume "?U P" and P: "∀P. ?U P ⟶ (∃N. ∀n≥N. ∀m≥N. P (X n, X m))"
    then obtain Q where "?U Q" and Q: "∧x y z. Q (x, y) ==> Q (y, z) ==> P (x, z)"
      by (auto elim: uniformity_transE)
    then have "?U (λx. Q x ∧ (λ(x, y). Q (y, x)) x)"
      unfolding eventually_conj_iff by (simp add: uniformity_sym)
    from P[rule_format, OF this]
    obtain N where N: "∧n m. n ≥ N ==> m ≥ N ==> Q (X n, X m) ∧ Q (X m, X n)"
      by auto
    show "∃Q. (∃N. ∀n≥N. Q (X n)) ∧ (∀x y. Q x ⟶ Q y ⟶ P (x, y))"
    proof (safe intro!: exI[of _ "λx. ∀n≥N. Q (x, X n) ∧ Q (X n, x)"] exI[of _ N] N)
      fix x y
      assume "∀n≥N. Q (x, X n) ∧ Q (X n, x)" "∀n≥N. Q (y, X n) ∧ Q (X n, y)"
      then have "Q (x, X N)" "Q (X N, y)" by auto
      then show "P (x, y)"
        by (rule Q)
    qed
  }
qed

lemma nhds_imp_cauchy_filter:
  assumes *: "F ≤ nhds x"
  shows "cauchy_filter F"
proof -
  have "F ×_F F ≤ nhds x ×_F nhds x"
    by (intro prod_filter_mono *)
  also have "… ≤ uniformity"
    unfolding le_filter_def eventually_nhds_uniformity eventually_prod_same
  proof safe
    fix P
    assume "eventually P uniformity"
    then obtain Ql where ev: "eventually Ql uniformity"
      and "Ql (x, y) ==> Ql (y, z) ==> P (x, z)" for x y z
      by (rule uniformity_transE) simp
    with ev[THEN uniformity_sym]
    show "∃Q. eventually (λ(x', y). x' = x ⟶ Q y) uniformity ∧
        (∀x y. Q x ⟶ Q y ⟶ P (x, y))"
      by (rule_tac exI[of _ "λy. Ql (y, x) ∧ Ql (x, y)"]) (fastforce elim: eventually_elim2)
  qed
  finally show ?thesis
    by (simp add: cauchy_filter_def)
qed

lemma LIMSEQ_imp_Cauchy: "X <---- x ==> Cauchy X"
  unfolding Cauchy_uniform filterlim_def by (intro nhds_imp_cauchy_filter)

lemma Cauchy_subseq_Cauchy:
  assumes "Cauchy X" "strict_mono f"
  shows "Cauchy (X ∘ f)"
  unfolding Cauchy_uniform comp_def filtermap_filtermap[symmetric] cauchy_filter_def
  by (rule order_trans[OF _ ‹Cauchy X›[unfolded Cauchy_uniform cauchy_filter_def]])
     (intro prod_filter_mono filtermap_mono filterlim_subseq[OF ‹strict_mono f›, unfolded filterlim_def])

lemma convergent_Cauchy: "convergent X ==> Cauchy X"
  unfolding convergent_def by (erule exE, erule LIMSEQ_imp_Cauchy)

definition complete :: "'a set → bool"
  where complete_uniform: "complete S ⟷
    (∀F ≤ principal S. F ≠ bot ⟶ cauchy_filter F ⟶ (∃x∈S. F ≤ nhds x))"

lemma (in uniform_space) cauchy_filter_complete_converges:
  assumes "cauchy_filter F" "complete A" "F ≤ principal A" "F ≠ bot"
  shows   "∃c. F ≤ nhds c"
  using assms unfolding complete_uniform by blast

end

subsubsection ‹Uniformly continuous functions›

definition uniformly_continuous_on :: "'a set → ('a::uniform_space → 'b::uniform_space) → bool"
  where uniformly_continuous_on_uniformity: "uniformly_continuous_on s f ⟷
    (LIM (x, y) (uniformity_on s). (f x, f y) :> uniformity)"

lemma uniformly_continuous_onD:
  "uniformly_continuous_on s f ==> eventually E uniformity ==>
    eventually (λ(x, y). x ∈ s ⟶ y ∈ s ⟶ E (f x, f y)) uniformity"
  by (simp add: uniformly_continuous_on_uniformity filterlim_iff
      eventually_inf_principal split_beta' mem_Times_iff imp_conjL)

lemma uniformly_continuous_on_const[continuous_intros]: "uniformly_continuous_on s (λx. c)"
  by (auto simp: uniformly_continuous_on_uniformity filterlim_iff uniformity_refl)

lemma uniformly_continuous_on_id[continuous_intros]: "uniformly_continuous_on s (λx. x)"
  by (auto simp: uniformly_continuous_on_uniformity filterlim_def)

lemma uniformly_continuous_on_compose:
  "uniformly_continuous_on s g ==> uniformly_continuous_on (g`s) f ==>
    uniformly_continuous_on s (λx. f (g x))"
  using filterlim_compose[of "λ(x, y). (f x, f y)" uniformity
      "uniformity_on (g`s)"  "λ(x, y). (g x, g y)" "uniformity_on s"]
  by (simp add: split_beta' uniformly_continuous_on_uniformity
      filterlim_inf filterlim_principal eventually_inf_principal mem_Times_iff)

lemma uniformly_continuous_imp_continuous:
  assumes f: "uniformly_continuous_on s f"
  shows "continuous_on s f"
  by (auto simp: filterlim_iff eventually_at_filter eventually_nhds_uniformity continuous_on_def
           elim: eventually_mono dest!: uniformly_continuous_onD[OF f])


section ‹Product Topology›

subsection ‹Product is a topological space›

instantiation prod :: (topological_space, topological_space) topological_space
begin

definition open_prod_def[code del]:
  "open (S :: ('a × 'b) set) ⟷
    (∀x∈S. ∃A B. open A ∧ open B ∧ x ∈ A × B ∧ A × B ⊆ S)"

lemma open_prod_elim:
  assumes "open S" and "x ∈ S"
  obtains A B where "open A" and "open B" and "x ∈ A × B" and "A × B ⊆ S"
  using assms unfolding open_prod_def by fast

lemma open_prod_intro:
  assumes "∧x. x ∈ S ==> ∃A B. open A ∧ open B ∧ x ∈ A × B ∧ A × B ⊆ S"
  shows "open S"
  using assms unfolding open_prod_def by fast

instance
proof
  show "open (UNIV :: ('a × 'b) set)"
    unfolding open_prod_def by auto
next
  fix S T :: "('a × 'b) set"
  assume "open S" "open T"
  show "open (S ∩ T)"
  proof (rule open_prod_intro)
    fix x
    assume x: "x ∈ S ∩ T"
    from x have "x ∈ S" by simp
    obtain Sa Sb where A: "open Sa" "open Sb" "x ∈ Sa × Sb" "Sa × Sb ⊆ S"
      using ‹open S› and ‹x ∈ S› by (rule open_prod_elim)
    from x have "x ∈ T" by simp
    obtain Ta Tb where B: "open Ta" "open Tb" "x ∈ Ta × Tb" "Ta × Tb ⊆ T"
      using ‹open T› and ‹x ∈ T› by (rule open_prod_elim)
    let ?A = "Sa ∩ Ta" and ?B = "Sb ∩ Tb"
    have "open ?A ∧ open ?B ∧ x ∈ ?A × ?B ∧ ?A × ?B ⊆ S ∩ T"
      using A B by (auto simp add: open_Int)
    then show "∃A B. open A ∧ open B ∧ x ∈ A × B ∧ A × B ⊆ S ∩ T"
      by fast
  qed
next
  fix K :: "('a × 'b) set set"
  assume "∀S∈K. open S"
  then show "open (∪K)"
    unfolding open_prod_def by fast
qed

end

declare [[code abort: "open :: ('a::topological_space × 'b::topological_space) set → bool"]]

lemma open_Times: "open S ==> open T ==> open (S × T)"
  unfolding open_prod_def by auto

lemma fst_vimage_eq_Times: "fst -` S = S × UNIV"
  by auto

lemma snd_vimage_eq_Times: "snd -` S = UNIV × S"
  by auto

lemma open_vimage_fst: "open S ==> open (fst -` S)"
  by (simp add: fst_vimage_eq_Times open_Times)

lemma open_vimage_snd: "open S ==> open (snd -` S)"
  by (simp add: snd_vimage_eq_Times open_Times)

lemma closed_vimage_fst: "closed S ==> closed (fst -` S)"
  unfolding closed_open vimage_Compl [symmetric]
  by (rule open_vimage_fst)

lemma closed_vimage_snd: "closed S ==> closed (snd -` S)"
  unfolding closed_open vimage_Compl [symmetric]
  by (rule open_vimage_snd)

lemma closed_Times: "closed S ==> closed T ==> closed (S × T)"
proof -
  have "S × T = (fst -` S) ∩ (snd -` T)"
    by auto
  then show "closed S ==> closed T ==> closed (S × T)"
    by (simp add: closed_vimage_fst closed_vimage_snd closed_Int)
qed

lemma subset_fst_imageI: "A × B ⊆ S ==> y ∈ B ==> A ⊆ fst ` S"
  unfolding image_def subset_eq by force

lemma subset_snd_imageI: "A × B ⊆ S ==> x ∈ A ==> B ⊆ snd ` S"
  unfolding image_def subset_eq by force

lemma open_image_fst:
  assumes "open S"
  shows "open (fst ` S)"
proof (rule openI)
  fix x
  assume "x ∈ fst ` S"
  then obtain y where "(x, y) ∈ S"
    by auto
  then obtain A B where "open A" "open B" "x ∈ A" "y ∈ B" "A × B ⊆ S"
    using ‹open S› unfolding open_prod_def by auto
  from ‹A × B ⊆ S› ‹y ∈ B› have "A ⊆ fst ` S"
    by (rule subset_fst_imageI)
  with ‹open A› ‹x ∈ A› have "open A ∧ x ∈ A ∧ A ⊆ fst ` S"
    by simp
  then show "∃T. open T ∧ x ∈ T ∧ T ⊆ fst ` S" ..
qed

lemma open_image_snd:
  assumes "open S"
  shows "open (snd ` S)"
proof (rule openI)
  fix y
  assume "y ∈ snd ` S"
  then obtain x where "(x, y) ∈ S"
    by auto
  then obtain A B where "open A" "open B" "x ∈ A" "y ∈ B" "A × B ⊆ S"
    using ‹open S› unfolding open_prod_def by auto
  from ‹A × B ⊆ S› ‹x ∈ A› have "B ⊆ snd ` S"
    by (rule subset_snd_imageI)
  with ‹open B› ‹y ∈ B› have "open B ∧ y ∈ B ∧ B ⊆ snd ` S"
    by simp
  then show "∃T. open T ∧ y ∈ T ∧ T ⊆ snd ` S" ..
qed

lemma nhds_prod: "nhds (a, b) = nhds a ×_F nhds b"
  unfolding nhds_def
proof (subst prod_filter_INF, auto intro!: antisym INF_greatest simp: principal_prod_principal)
  fix S T
  assume "open S" "a ∈ S" "open T" "b ∈ T"
  then show "(INF x ∈ {S. open S ∧ (a, b) ∈ S}. principal x) ≤ principal (S × T)"
    by (intro INF_lower) (auto intro!: open_Times)
next
  fix S'
  assume "open S'" "(a, b) ∈ S'"
  then obtain S T where "open S" "a ∈ S" "open T" "b ∈ T" "S × T ⊆ S'"
    by (auto elim: open_prod_elim)
  then show "(INF x ∈ {S. open S ∧ a ∈ S}. INF y ∈ {S. open S ∧ b ∈ S}.
      principal (x × y)) ≤ principal S'"
    by (auto intro!: INF_lower2)
qed


subsubsection ‹Continuity of operations›

lemma tendsto_fst [tendsto_intros]:
  assumes "(f ---> a) F"
  shows "((λx. fst (f x)) ---> fst a) F"
proof (rule topological_tendstoI)
  fix S
  assume "open S" and "fst a ∈ S"
  then have "open (fst -` S)" and "a ∈ fst -` S"
    by (simp_all add: open_vimage_fst)
  with assms have "eventually (λx. f x ∈ fst -` S) F"
    by (rule topological_tendstoD)
  then show "eventually (λx. fst (f x) ∈ S) F"
    by simp
qed

lemma tendsto_snd [tendsto_intros]:
  assumes "(f ---> a) F"
  shows "((λx. snd (f x)) ---> snd a) F"
proof (rule topological_tendstoI)
  fix S
  assume "open S" and "snd a ∈ S"
  then have "open (snd -` S)" and "a ∈ snd -` S"
    by (simp_all add: open_vimage_snd)
  with assms have "eventually (λx. f x ∈ snd -` S) F"
    by (rule topological_tendstoD)
  then show "eventually (λx. snd (f x) ∈ S) F"
    by simp
qed

lemma tendsto_Pair [tendsto_intros]:
  assumes "(f ---> a) F" and "(g ---> b) F"
  shows "((λx. (f x, g x)) ---> (a, b)) F"
  unfolding nhds_prod using assms by (rule filterlim_Pair)

lemma continuous_fst[continuous_intros]: "continuous F f ==> continuous F (λx. fst (f x))"
  unfolding continuous_def by (rule tendsto_fst)

lemma continuous_snd[continuous_intros]: "continuous F f ==> continuous F (λx. snd (f x))"
  unfolding continuous_def by (rule tendsto_snd)

lemma continuous_Pair[continuous_intros]:
  "continuous F f ==> continuous F g ==> continuous F (λx. (f x, g x))"
  unfolding continuous_def by (rule tendsto_Pair)

lemma continuous_on_fst[continuous_intros]:
  "continuous_on s f ==> continuous_on s (λx. fst (f x))"
  unfolding continuous_on_def by (auto intro: tendsto_fst)

lemma continuous_on_snd[continuous_intros]:
  "continuous_on s f ==> continuous_on s (λx. snd (f x))"
  unfolding continuous_on_def by (auto intro: tendsto_snd)

lemma continuous_on_Pair[continuous_intros]:
  "continuous_on s f ==> continuous_on s g ==> continuous_on s (λx. (f x, g x))"
  unfolding continuous_on_def by (auto intro: tendsto_Pair)

lemma continuous_on_swap[continuous_intros]: "continuous_on A prod.swap"
  by (simp add: prod.swap_def continuous_on_fst continuous_on_snd
      continuous_on_Pair continuous_on_id)

lemma continuous_on_swap_args:
  assumes "continuous_on (A×B) (λ(x,y). d x y)"
    shows "continuous_on (B×A) (λ(x,y). d y x)"
proof -
  have "(λ(x,y). d y x) = (λ(x,y). d x y) ∘ prod.swap"
    by force
  then show ?thesis
    by (metis assms continuous_on_compose continuous_on_swap product_swap)
qed

lemma isCont_fst [simp]: "isCont f a ==> isCont (λx. fst (f x)) a"
  by (fact continuous_fst)

lemma isCont_snd [simp]: "isCont f a ==> isCont (λx. snd (f x)) a"
  by (fact continuous_snd)

lemma isCont_Pair [simp]: "[isCont f a; isCont g a] ==> isCont (λx. (f x, g x)) a"
  by (fact continuous_Pair)

lemma continuous_on_compose_Pair:
  assumes f: "continuous_on (Sigma A B) (λ(a, b). f a b)"
  assumes g: "continuous_on C g"
  assumes h: "continuous_on C h"
  assumes subset: "∧c. c ∈ C ==> g c ∈ A" "∧c. c ∈ C ==> h c ∈ B (g c)"
  shows "continuous_on C (λc. f (g c) (h c))"
  using continuous_on_compose2[OF f continuous_on_Pair[OF g h]] subset
  by auto


subsubsection ‹Connectedness of products›

proposition connected_Times:
  assumes S: "connected S" and T: "connected T"
  shows "connected (S × T)"
proof (rule connectedI_const)
  fix P::"'a × 'b → bool"
  assume P[THEN continuous_on_compose2, continuous_intros]: "continuous_on (S × T) P"
  have "continuous_on S (λs. P (s, t))" if "t ∈ T" for t
    by (auto intro!: continuous_intros that)
  from connectedD_const[OF S this]
  obtain c1 where c1: "∧s t. t ∈ T ==> s ∈ S ==> P (s, t) = c1 t"
    by metis
  moreover
  have "continuous_on T (λt. P (s, t))" if "s ∈ S" for s
    by (auto intro!: continuous_intros that)
  from connectedD_const[OF T this]
  obtain c2 where "∧s t. t ∈ T ==> s ∈ S ==> P (s, t) = c2 s"
    by metis
  ultimately show "∃c. ∀s∈S × T. P s = c"
    by auto
qed

corollary connected_Times_eq [simp]:
   "connected (S × T) ⟷ S = {} ∨ T = {} ∨ connected S ∧ connected T"  (is "?lhs = ?rhs")
proof
  assume L: ?lhs
  show ?rhs
  proof cases
    assume "S ≠ {} ∧ T ≠ {}"
    moreover
    have "connected (fst ` (S × T))" "connected (snd ` (S × T))"
      using continuous_on_fst continuous_on_snd continuous_on_id
      by (blast intro: connected_continuous_image [OF _ L])+
    ultimately show ?thesis
      by auto
  qed auto
qed (auto simp: connected_Times)


subsubsection ‹Separation axioms›

instance prod :: (t0_space, t0_space) t0_space
proof
  fix x y :: "'a × 'b"
  assume "x ≠ y"
  then have "fst x ≠ fst y ∨ snd x ≠ snd y"
    by (simp add: prod_eq_iff)
  then show "∃U. open U ∧ (x ∈ U) ≠ (y ∈ U)"
    by (fast dest: t0_space elim: open_vimage_fst open_vimage_snd)
qed

instance prod :: (t1_space, t1_space) t1_space
proof
  fix x y :: "'a × 'b"
  assume "x ≠ y"
  then have "fst x ≠ fst y ∨ snd x ≠ snd y"
    by (simp add: prod_eq_iff)
  then show "∃U. open U ∧ x ∈ U ∧ y ∉ U"
    by (fast dest: t1_space elim: open_vimage_fst open_vimage_snd)
qed

instance prod :: (t2_space, t2_space) t2_space
proof
  fix x y :: "'a × 'b"
  assume "x ≠ y"
  then have "fst x ≠ fst y ∨ snd x ≠ snd y"
    by (simp add: prod_eq_iff)
  then show "∃U V. open U ∧ open V ∧ x ∈ U ∧ y ∈ V ∧ U ∩ V = {}"
    by (fast dest: hausdorff elim: open_vimage_fst open_vimage_snd)
qed

lemma isCont_swap[continuous_intros]: "isCont prod.swap a"
  using continuous_on_eq_continuous_within continuous_on_swap by blast

lemma open_diagonal_complement:
  "open {(x,y) |x y. x ≠ (y::('a::t2_space))}"
proof -
  have "open {(x, y). x ≠ (y::'a)}"
    unfolding split_def by (intro open_Collect_neq continuous_intros)
  also have "{(x, y). x ≠ (y::'a)} = {(x, y) |x y. x ≠ (y::'a)}"
    by auto
  finally show ?thesis .
qed

lemma closed_diagonal:
  "closed {y. ∃ x::('a::t2_space). y = (x,x)}"
proof -
  have "{y. ∃ x::'a. y = (x,x)} = UNIV - {(x,y) | x y. x ≠ y}" by auto
  then show ?thesis using open_diagonal_complement closed_Diff by auto
qed

lemma open_superdiagonal:
  "open {(x,y) | x y. x > (y::'a::{linorder_topology})}"
proof -
  have "open {(x, y). x > (y::'a)}"
    unfolding split_def by (intro open_Collect_less continuous_intros)
  also have "{(x, y). x > (y::'a)} = {(x, y) |x y. x > (y::'a)}"
    by auto
  finally show ?thesis .
qed

lemma closed_subdiagonal:
  "closed {(x,y) | x y. x ≤ (y::'a::{linorder_topology})}"
proof -
  have "{(x,y) | x y. x ≤ (y::'a)} = UNIV - {(x,y) | x y. x > (y::'a)}" by auto
  then show ?thesis using open_superdiagonal closed_Diff by auto
qed

lemma open_subdiagonal:
  "open {(x,y) | x y. x < (y::'a::{linorder_topology})}"
proof -
  have "open {(x, y). x < (y::'a)}"
    unfolding split_def by (intro open_Collect_less continuous_intros)
  also have "{(x, y). x < (y::'a)} = {(x, y) |x y. x < (y::'a)}"
    by auto
  finally show ?thesis .
qed

lemma closed_superdiagonal:
  "closed {(x,y) | x y. x ≥ (y::('a::{linorder_topology}))}"
proof -
  have "{(x,y) | x y. x ≥ (y::'a)} = UNIV - {(x,y) | x y. x < y}" by auto
  then show ?thesis using open_subdiagonal closed_Diff by auto
qed

end