Theory HOL-Analysis.Extended_Real_Limits

(*  Title:      HOL/Analysis/Extended_Real_Limits.thy
    Author:     Johannes Hölzl, TU München
    Author:     Robert Himmelmann, TU München
    Author:     Armin Heller, TU München
    Author:     Bogdan Grechuk, University of Edinburgh
*)

section ‹Limits on the Extended Real Number Line› (* TO FIX: perhaps put all Nonstandard Analysis related
topics together? *)

theory Extended_Real_Limits
imports
  Topology_Euclidean_Space
  "HOL-Library.Extended_Real"
  "HOL-Library.Extended_Nonnegative_Real"
  "HOL-Library.Indicator_Function"
begin

lemma compact_UNIV:
  "compact (UNIV :: 'a::{complete_linorder,linorder_topology,second_countable_topology} set)"
  using compact_complete_linorder
  by (auto simp: seq_compact_eq_compact[symmetric] seq_compact_def)

lemma compact_eq_closed:
  fixes S :: "'a::{complete_linorder,linorder_topology,second_countable_topology} set"
  shows "compact S  closed S"
  using closed_Int_compact[of S, OF _ compact_UNIV] compact_imp_closed
  by auto

lemma closed_contains_Sup_cl:
  fixes S :: "'a::{complete_linorder,linorder_topology,second_countable_topology} set"
  assumes "closed S"
    and "S  {}"
  shows "Sup S  S"
proof -
  from compact_eq_closed[of S] compact_attains_sup[of S] assms
  obtain s where S: "s  S" "tS. t  s"
    by auto
  then have "Sup S = s"
    by (auto intro!: Sup_eqI)
  with S show ?thesis
    by simp
qed

lemma closed_contains_Inf_cl:
  fixes S :: "'a::{complete_linorder,linorder_topology,second_countable_topology} set"
  assumes "closed S"
    and "S  {}"
  shows "Inf S  S"
proof -
  from compact_eq_closed[of S] compact_attains_inf[of S] assms
  obtain s where S: "s  S" "tS. s  t"
    by auto
  then have "Inf S = s"
    by (auto intro!: Inf_eqI)
  with S show ?thesis
    by simp
qed

instancetag unimportant› enat :: second_countable_topology
proof
  show "B::enat set set. countable B  open = generate_topology B"
  proof (intro exI conjI)
    show "countable (range lessThan  range greaterThan::enat set set)"
      by auto
  qed (simp add: open_enat_def)
qed

instancetag unimportant› ereal :: second_countable_topology
proof (standard, intro exI conjI)
  let ?B = "(r. {{..< r}, {r <..}} :: ereal set set)"
  show "countable ?B"
    by (auto intro: countable_rat)
  show "open = generate_topology ?B"
  proof (intro ext iffI)
    fix S :: "ereal set"
    assume "open S"
    then show "generate_topology ?B S"
      unfolding open_generated_order
    proof induct
      case (Basis b)
      then obtain e where "b = {..<e}  b = {e<..}"
        by auto
      moreover have "{..<e} = {{..<x}|x. x    x < e}" "{e<..} = {{x<..}|x. x    e < x}"
        by (auto dest: ereal_dense3
                 simp del: ex_simps
                 simp add: ex_simps[symmetric] conj_commute Rats_def image_iff)
      ultimately show ?case
        by (auto intro: generate_topology.intros)
    qed (auto intro: generate_topology.intros)
  next
    fix S
    assume "generate_topology ?B S"
    then show "open S"
      by induct auto
  qed
qed

text ‹This is a copy from ereal :: second_countable_topology›. Maybe find a common super class of
topological spaces where the rational numbers are densely embedded ?›
instance ennreal :: second_countable_topology
proof (standard, intro exI conjI)
  let ?B = "(r. {{..< r}, {r <..}} :: ennreal set set)"
  show "countable ?B"
    by (auto intro: countable_rat)
  show "open = generate_topology ?B"
  proof (intro ext iffI)
    fix S :: "ennreal set"
    assume "open S"
    then show "generate_topology ?B S"
      unfolding open_generated_order
    proof induct
      case (Basis b)
      then obtain e where "b = {..<e}  b = {e<..}"
        by auto
      moreover have "{..<e} = {{..<x}|x. x    x < e}" "{e<..} = {{x<..}|x. x    e < x}"
        by (auto dest: ennreal_rat_dense
                 simp del: ex_simps
                 simp add: ex_simps[symmetric] conj_commute Rats_def image_iff)
      ultimately show ?case
        by (auto intro: generate_topology.intros)
    qed (auto intro: generate_topology.intros)
  next
    fix S
    assume "generate_topology ?B S"
    then show "open S"
      by induct auto
  qed
qed

lemma ereal_open_closed_aux:
  fixes S :: "ereal set"
  assumes "open S"
    and "closed S"
    and S: "(-)  S"
  shows "S = {}"
proof (rule ccontr)
  assume "¬ ?thesis"
  then have *: "Inf S  S"

    by (metis assms(2) closed_contains_Inf_cl)
  {
    assume "Inf S = -"
    then have False
      using * assms(3) by auto
  }
  moreover
  {
    assume "Inf S = "
    then have "S = {}"
      by (metis Inf_eq_PInfty S  {})
    then have False
      by (metis assms(1) not_open_singleton)
  }
  moreover
  {
    assume fin: "¦Inf S¦  "
    from ereal_open_cont_interval[OF assms(1) * fin]
    obtain e where e: "e > 0" "{Inf S - e<..<Inf S + e}  S" .
    then obtain b where b: "Inf S - e < b" "b < Inf S"
      using fin ereal_between[of "Inf S" e] dense[of "Inf S - e"]
      by auto
    then have "b  {Inf S - e <..< Inf S + e}"
      using e fin ereal_between[of "Inf S" e]
      by auto
    then have "b  S"
      using e by auto
    then have False
      using b by (metis complete_lattice_class.Inf_lower leD)
  }
  ultimately show False
    by auto
qed

lemma ereal_open_closed:
  fixes S :: "ereal set"
  shows "open S  closed S  S = {}  S = UNIV"
  using ereal_open_closed_aux open_closed by auto

lemma ereal_open_atLeast:
  fixes x :: ereal
  shows "open {x..}  x = -"
  by (metis atLeast_eq_UNIV_iff bot_ereal_def closed_atLeast ereal_open_closed not_Ici_eq_empty)

lemma mono_closed_real:
  fixes S :: "real set"
  assumes mono: "y z. y  S  y  z  z  S"
    and "closed S"
  shows "S = {}  S = UNIV  (a. S = {a..})"
proof -
  {
    assume "S  {}"
    { assume ex: "B. xS. B  x"
      then have *: "xS. Inf S  x"
        using cInf_lower[of _ S] ex by (metis bdd_below_def)
      then have "Inf S  S"
        by (meson S  {} assms(2) bdd_belowI closed_contains_Inf)
      then have "x. Inf S  x  x  S"
        using mono[rule_format, of "Inf S"] *
        by auto
      then have "S = {Inf S ..}"
        by auto
      then have "a. S = {a ..}"
        by auto
    }
    moreover
    {
      assume "¬ (B. xS. B  x)"
      then have nex: "B. xS. x < B"
        by (simp add: not_le)
      {
        fix y
        obtain x where "xS" and "x < y"
          using nex by auto
        then have "y  S"
          using mono[rule_format, of x y] by auto
      }
      then have "S = UNIV"
        by auto
    }
    ultimately have "S = UNIV  (a. S = {a ..})"
      by blast
  }
  then show ?thesis
    by blast
qed

lemma mono_closed_ereal:
  fixes S :: "real set"
  assumes mono: "y z. y  S  y  z  z  S"
    and "closed S"
  shows "a. S = {x. a  ereal x}"
proof -
  consider "S = {}  S = UNIV" | "(a. S = {a..})"
    using assms(2) mono mono_closed_real by blast
  then show ?thesis
  proof cases
    case 1
    then show ?thesis
      by (meson PInfty_neq_ereal(1) UNIV_eq_I bot.extremum empty_Collect_eq ereal_infty_less_eq(1) mem_Collect_eq)
  next
    case 2
    then show ?thesis
      by (metis atLeast_iff ereal_less_eq(3) mem_Collect_eq subsetI subset_antisym)
  qed
qed

lemma Liminf_within:
  fixes f :: "'a::metric_space  'b::complete_lattice"
  shows "Liminf (at x within S) f = (SUP e{0<..}. INF y(S  ball x e - {x}). f y)"
  unfolding Liminf_def eventually_at
proof (rule SUP_eq, simp_all add: Ball_def Bex_def, safe)
  fix P d
  assume "0 < d" and "y. y  S  y  x  dist y x < d  P y"
  then have "S  ball x d - {x}  {x. P x}"
    by (auto simp: dist_commute)
  then show "r>0. Inf (f ` (Collect P))  Inf (f ` (S  ball x r - {x}))"
    by (intro exI[of _ d] INF_mono conjI 0 < d) auto
next
  fix d :: real
  assume "0 < d"
  then show "P. (d>0. xa. xa  S  xa  x  dist xa x < d  P xa) 
    Inf (f ` (S  ball x d - {x}))  Inf (f ` (Collect P))"
    by (intro exI[of _ "λy. y  S  ball x d - {x}"])
       (auto intro!: INF_mono exI[of _ d] simp: dist_commute)
qed

lemma Limsup_within:
  fixes f :: "'a::metric_space  'b::complete_lattice"
  shows "Limsup (at x within S) f = (INF e{0<..}. SUP y(S  ball x e - {x}). f y)"
  unfolding Limsup_def eventually_at
proof (rule INF_eq, simp_all add: Ball_def Bex_def, safe)
  fix P d
  assume "0 < d" and "y. y  S  y  x  dist y x < d  P y"
  then have "S  ball x d - {x}  {x. P x}"
    by (auto simp: dist_commute)
  then show "r>0. Sup (f ` (S  ball x r - {x}))  Sup (f ` (Collect P))"
    by (intro exI[of _ d] SUP_mono conjI 0 < d) auto
next
  fix d :: real
  assume "0 < d"
  then show "P. (d>0. xa. xa  S  xa  x  dist xa x < d  P xa) 
    Sup (f ` (Collect P))  Sup (f ` (S  ball x d - {x}))"
    by (intro exI[of _ "λy. y  S  ball x d - {x}"])
       (auto intro!: SUP_mono exI[of _ d] simp: dist_commute)
qed

lemma Liminf_at:
  fixes f :: "'a::metric_space  'b::complete_lattice"
  shows "Liminf (at x) f = (SUP e{0<..}. INF y(ball x e - {x}). f y)"
  using Liminf_within[of x UNIV f] by simp

lemma Limsup_at:
  fixes f :: "'a::metric_space  'b::complete_lattice"
  shows "Limsup (at x) f = (INF e{0<..}. SUP y(ball x e - {x}). f y)"
  using Limsup_within[of x UNIV f] by simp

lemma min_Liminf_at:
  fixes f :: "'a::metric_space  'b::complete_linorder"
  shows "min (f x) (Liminf (at x) f) = (SUP e{0<..}. INF yball x e. f y)"
  apply (simp add: inf_min [symmetric] Liminf_at inf_commute [of "f x"] SUP_inf)
  apply (metis (no_types, lifting) INF_insert centre_in_ball greaterThan_iff image_cong inf_commute insert_Diff)
  done


subsection ‹Extended-Real.thy› (*FIX ME change title *)

lemma sum_constant_ereal:
  fixes a::ereal
  shows "(iI. a) = a * card I"
proof (induction I rule: infinite_finite_induct)
  case (insert i I)
  then show ?case
    by (simp add: ereal_right_distrib flip: plus_ereal.simps)
qed auto

lemma real_lim_then_eventually_real:
  assumes "(u  ereal l) F"
  shows "eventually (λn. u n = ereal(real_of_ereal(u n))) F"
proof -
  have "ereal l  {-<..<(::ereal)}" by simp
  moreover have "open {-<..<(::ereal)}" by simp
  ultimately have "eventually (λn. u n  {-<..<(::ereal)}) F" using assms tendsto_def by blast
  moreover have "x. x  {-<..<(::ereal)}  x = ereal(real_of_ereal x)" using ereal_real by auto
  ultimately show ?thesis by (metis (mono_tags, lifting) eventually_mono)
qed

lemma ereal_Inf_cmult:
  assumes "c>(0::real)"
  shows "Inf {ereal c * x |x. P x} = ereal c * Inf {x. P x}"
proof -
  have "bij ((*) (ereal c))"
    apply (rule bij_betw_byWitness[of _ "λx. (x::ereal) / c"], auto simp: assms ereal_mult_divide)
    using assms ereal_divide_eq by auto
  then have "ereal c * Inf {x. P x} = Inf ((*) (ereal c) ` {x. P x})"
    by (simp add: assms ereal_mult_left_mono less_imp_le mono_def mono_bij_Inf)
  then show ?thesis
    by (simp add: setcompr_eq_image)
qed


subsubsectiontag important› ‹Continuity of addition›

text ‹The next few lemmas remove an unnecessary assumption in tendsto_add_ereal›, culminating
in tendsto_add_ereal_general› which essentially says that the addition
is continuous on ereal times ereal, except at (-∞, ∞)› and (∞, -∞)›.
It is much more convenient in many situations, see for instance the proof of
tendsto_sum_ereal› below.›

lemma tendsto_add_ereal_PInf:
  fixes y :: ereal
  assumes y: "y  -"
  assumes f: "(f  ) F" and g: "(g  y) F"
  shows "((λx. f x + g x)  ) F"
proof -
  have "C. eventually (λx. g x > ereal C) F"
  proof (cases y)
    case (real r)
    have "y > y-1" using y real by (simp add: ereal_between(1))
    then have "eventually (λx. g x > y - 1) F" using g y order_tendsto_iff by auto
    moreover have "y-1 = ereal(real_of_ereal(y-1))"
      by (metis real ereal_eq_1(1) ereal_minus(1) real_of_ereal.simps(1))
    ultimately have "eventually (λx. g x > ereal(real_of_ereal(y - 1))) F" by simp
    then show ?thesis by auto
  next
    case (PInf)
    have "eventually (λx. g x > ereal 0) F" using g PInf by (simp add: tendsto_PInfty)
    then show ?thesis by auto
  qed (simp add: y)
  then obtain C::real where ge: "eventually (λx. g x > ereal C) F" by auto

  {
    fix M::real
    have "eventually (λx. f x > ereal(M - C)) F" using f by (simp add: tendsto_PInfty)
    then have "eventually (λx. (f x > ereal (M-C))  (g x > ereal C)) F"
      by (auto simp: ge eventually_conj_iff)
    moreover have "x. ((f x > ereal (M-C))  (g x > ereal C))  (f x + g x > ereal M)"
      using ereal_add_strict_mono2 by fastforce
    ultimately have "eventually (λx. f x + g x > ereal M) F" using eventually_mono by force
  }
  then show ?thesis by (simp add: tendsto_PInfty)
qed

text‹One would like to deduce the next lemma from the previous one, but the fact
that - (x + y)› is in general different from (- x) + (- y)› in ereal creates difficulties,
so it is more efficient to copy the previous proof.›

lemma tendsto_add_ereal_MInf:
  fixes y :: ereal
  assumes y: "y  "
  assumes f: "(f  -) F" and g: "(g  y) F"
  shows "((λx. f x + g x)  -) F"
proof -
  have "C. eventually (λx. g x < ereal C) F"
  proof (cases y)
    case (real r)
    have "y < y+1" using y real by (simp add: ereal_between(1))
    then have "eventually (λx. g x < y + 1) F" using g y order_tendsto_iff by force
    moreover have "y+1 = ereal(real_of_ereal (y+1))" by (simp add: real)
    ultimately have "eventually (λx. g x < ereal(real_of_ereal(y + 1))) F" by simp
    then show ?thesis by auto
  next
    case (MInf)
    have "eventually (λx. g x < ereal 0) F" using g MInf by (simp add: tendsto_MInfty)
    then show ?thesis by auto
  qed (simp add: y)
  then obtain C::real where ge: "eventually (λx. g x < ereal C) F" by auto

  {
    fix M::real
    have "eventually (λx. f x < ereal(M - C)) F" using f by (simp add: tendsto_MInfty)
    then have "eventually (λx. (f x < ereal (M- C))  (g x < ereal C)) F"
      by (auto simp: ge eventually_conj_iff)
    moreover have "x. ((f x < ereal (M-C))  (g x < ereal C))  (f x + g x < ereal M)"
      using ereal_add_strict_mono2 by fastforce
    ultimately have "eventually (λx. f x + g x < ereal M) F" using eventually_mono by force
  }
  then show ?thesis by (simp add: tendsto_MInfty)
qed

lemma tendsto_add_ereal_general1:
  fixes x y :: ereal
  assumes y: "¦y¦  "
  assumes f: "(f  x) F" and g: "(g  y) F"
  shows "((λx. f x + g x)  x + y) F"
proof (cases x)
  case (real r)
  have a: "¦x¦  " by (simp add: real)
  show ?thesis by (rule tendsto_add_ereal[OF a, OF y, OF f, OF g])
next
  case PInf
  then show ?thesis using tendsto_add_ereal_PInf assms by force
next
  case MInf
  then show ?thesis using tendsto_add_ereal_MInf assms
    by (metis abs_ereal.simps(3) ereal_MInfty_eq_plus)
qed

lemma tendsto_add_ereal_general2:
  fixes x y :: ereal
  assumes x: "¦x¦  "
      and f: "(f  x) F" and g: "(g  y) F"
  shows "((λx. f x + g x)  x + y) F"
proof -
  have "((λx. g x + f x)  x + y) F"
    by (metis (full_types) add.commute f g tendsto_add_ereal_general1 x)
  moreover have "x. g x + f x = f x + g x" using add.commute by auto
  ultimately show ?thesis by simp
qed

text ‹The next lemma says that the addition is continuous on ereal›, except at
the pairs (-∞, ∞)› and (∞, -∞)›.›

lemma tendsto_add_ereal_general [tendsto_intros]:
  fixes x y :: ereal
  assumes "¬((x=  y=-)  (x=-  y=))"
      and f: "(f  x) F" and g: "(g  y) F"
  shows "((λx. f x + g x)  x + y) F"
proof (cases x)
  case (real r)
  show ?thesis
    using real assms
    by (intro tendsto_add_ereal_general2; auto)
next
  case (PInf)
  then have "y  -" using assms by simp
  then show ?thesis using tendsto_add_ereal_PInf PInf assms by auto
next
  case (MInf)
  then have "y  " using assms by simp
  then show ?thesis 
    by (metis ereal_MInfty_eq_plus tendsto_add_ereal_MInf MInf f g)
qed

subsubsectiontag important› ‹Continuity of multiplication›

text ‹In the same way as for addition, we prove that the multiplication is continuous on
ereal times ereal, except at (∞, 0)› and (-∞, 0)› and (0, ∞)› and (0, -∞)›,
starting with specific situations.›

lemma tendsto_mult_real_ereal:
  assumes "(u  ereal l) F" "(v  ereal m) F"
  shows "((λn. u n * v n)  ereal l * ereal m) F"
proof -
  have ureal: "eventually (λn. u n = ereal(real_of_ereal(u n))) F" by (rule real_lim_then_eventually_real[OF assms(1)])
  then have "((λn. ereal(real_of_ereal(u n)))  ereal l) F" using assms by auto
  then have limu: "((λn. real_of_ereal(u n))  l) F" by auto
  have vreal: "eventually (λn. v n = ereal(real_of_ereal(v n))) F" by (rule real_lim_then_eventually_real[OF assms(2)])
  then have "((λn. ereal(real_of_ereal(v n)))  ereal m) F" using assms by auto
  then have limv: "((λn. real_of_ereal(v n))  m) F" by auto

  {
    fix n assume "u n = ereal(real_of_ereal(u n))" "v n = ereal(real_of_ereal(v n))"
    then have "ereal(real_of_ereal(u n) * real_of_ereal(v n)) = u n * v n" 
      by (metis times_ereal.simps(1))
  }
  then have *: "eventually (λn. ereal(real_of_ereal(u n) * real_of_ereal(v n)) = u n * v n) F"
    using eventually_elim2[OF ureal vreal] by auto

  have "((λn. real_of_ereal(u n) * real_of_ereal(v n))  l * m) F" 
    using tendsto_mult[OF limu limv] by auto
  then have "((λn. ereal(real_of_ereal(u n)) * real_of_ereal(v n))  ereal(l * m)) F" 
    by auto
  then show ?thesis using * filterlim_cong by fastforce
qed

lemma tendsto_mult_ereal_PInf:
  fixes f g::"_  ereal"
  assumes "(f  l) F" "l>0" "(g  ) F"
  shows "((λx. f x * g x)  ) F"
proof -
  obtain a::real where "0 < ereal a" "a < l" 
    using assms(2) using ereal_dense2 by blast
  have *: "eventually (λx. f x > a) F" 
    using a < l assms(1) by (simp add: order_tendsto_iff)
  {
    fix K::real
    define M where "M = max K 1"
    then have "M > 0" by simp
    then have "ereal(M/a) > 0" using ereal a > 0 by simp
    then have "x. ((f x > a)  (g x > M/a))  (f x * g x > ereal a * ereal(M/a))"
      using ereal_mult_mono_strict'[where ?c = "M/a", OF 0 < ereal a] by auto
    moreover have "ereal a * ereal(M/a) = M" using ereal a > 0 by simp
    ultimately have "x. ((f x > a)  (g x > M/a))  (f x * g x > M)" by simp
    moreover have "M  K" unfolding M_def by simp
    ultimately have Imp: "x. ((f x > a)  (g x > M/a))  (f x * g x > K)"
      using ereal_less_eq(3) le_less_trans by blast

    have "eventually (λx. g x > M/a) F" using assms(3) by (simp add: tendsto_PInfty)
    then have "eventually (λx. (f x > a)  (g x > M/a)) F"
      using * by (auto simp: eventually_conj_iff)
    then have "eventually (λx. f x * g x > K) F" using eventually_mono Imp by force
  }
  then show ?thesis by (auto simp: tendsto_PInfty)
qed

lemma tendsto_mult_ereal_pos:
  fixes f g::"_  ereal"
  assumes "(f  l) F" "(g  m) F" "l>0" "m>0"
  shows "((λx. f x * g x)  l * m) F"
proof (cases)
  assume *: "l =   m = "
  then show ?thesis
  proof (cases)
    assume "m = "
    then show ?thesis using tendsto_mult_ereal_PInf assms by auto
  next
    assume "¬(m = )"
    then have "l = " using * by simp
    then have "((λx. g x * f x)  l * m) F" using tendsto_mult_ereal_PInf assms by auto
    moreover have "x. g x * f x = f x * g x" using mult.commute by auto
    ultimately show ?thesis by simp
  qed
next
  assume "¬(l =   m = )"
  then have "l < " "m < " by auto
  then obtain lr mr where "l = ereal lr" "m = ereal mr"
    using l>0 m>0 by (metis ereal_cases ereal_less(6) not_less_iff_gr_or_eq)
  then show ?thesis using tendsto_mult_real_ereal assms by auto
qed

text ‹We reduce the general situation to the positive case by multiplying by suitable signs.
Unfortunately, as ereal is not a ring, all the neat sign lemmas are not available there. We
give the bare minimum we need.›

lemma ereal_sgn_abs:
  fixes l::ereal
  shows "sgn(l) * l = abs(l)"
    by (cases l, auto simp: sgn_if ereal_less_uminus_reorder)

lemma sgn_squared_ereal:
  assumes "l  (0::ereal)"
  shows "sgn(l) * sgn(l) = 1"
  using assms by (cases l, auto simp: one_ereal_def sgn_if)

lemma tendsto_mult_ereal [tendsto_intros]:
  fixes f g::"_  ereal"
  assumes "(f  l) F" "(g  m) F" "¬((l=0  abs(m) = )  (m=0  abs(l) = ))"
  shows "((λx. f x * g x)  l * m) F"
proof (cases)
  assume "l=0  m=0"
  then have "abs(l)  " "abs(m)  " using assms(3) by auto
  then obtain lr mr where "l = ereal lr" "m = ereal mr" by auto
  then show ?thesis using tendsto_mult_real_ereal assms by auto
next
  have sgn_finite: "a::ereal. abs(sgn a)  "
    by (metis MInfty_neq_ereal(2) PInfty_neq_ereal(2) abs_eq_infinity_cases ereal_times(1) ereal_times(3) ereal_uminus_eq_reorder sgn_ereal.elims)
  then have sgn_finite2: "a b::ereal. abs(sgn a * sgn b)  "
    by (metis abs_eq_infinity_cases abs_ereal.simps(2) abs_ereal.simps(3) ereal_mult_eq_MInfty ereal_mult_eq_PInfty)
  assume "¬(l=0  m=0)"
  then have "l  0" "m  0" by auto
  then have "abs(l) > 0" "abs(m) > 0"
    by (metis abs_ereal_ge0 abs_ereal_less0 abs_ereal_pos ereal_uminus_uminus ereal_uminus_zero less_le not_less)+
  then have "sgn(l) * l > 0" "sgn(m) * m > 0" using ereal_sgn_abs by auto
  moreover have "((λx. sgn(l) * f x)  (sgn(l) * l)) F"
    by (rule tendsto_cmult_ereal, auto simp: sgn_finite assms(1))
  moreover have "((λx. sgn(m) * g x)  (sgn(m) * m)) F"
    by (rule tendsto_cmult_ereal, auto simp: sgn_finite assms(2))
  ultimately have *: "((λx. (sgn(l) * f x) * (sgn(m) * g x))  (sgn(l) * l) * (sgn(m) * m)) F"
    using tendsto_mult_ereal_pos by force
  have "((λx. (sgn(l) * sgn(m)) * ((sgn(l) * f x) * (sgn(m) * g x)))  (sgn(l) * sgn(m)) * ((sgn(l) * l) * (sgn(m) * m))) F"
    by (rule tendsto_cmult_ereal, auto simp: sgn_finite2 *)
  moreover have "x. (sgn(l) * sgn(m)) * ((sgn(l) * f x) * (sgn(m) * g x)) = f x * g x"
    by (metis mult.left_neutral sgn_squared_ereal[OF l  0] sgn_squared_ereal[OF m  0] mult.assoc mult.commute)
  moreover have "(sgn(l) * sgn(m)) * ((sgn(l) * l) * (sgn(m) * m)) = l * m"
    by (metis mult.left_neutral sgn_squared_ereal[OF l  0] sgn_squared_ereal[OF m  0] mult.assoc mult.commute)
  ultimately show ?thesis by auto
qed

lemma tendsto_cmult_ereal_general [tendsto_intros]:
  fixes f::"_  ereal" and c::ereal
  assumes "(f  l) F" "¬ (l=0  abs(c) = )"
  shows "((λx. c * f x)  c * l) F"
by (cases "c = 0", auto simp: assms tendsto_mult_ereal)


subsubsectiontag important› ‹Continuity of division›

lemma tendsto_inverse_ereal_PInf:
  fixes u::"_  ereal"
  assumes "(u  ) F"
  shows "((λx. 1/ u x)  0) F"
proof -
  {
    fix e::real assume "e>0"
    have "1/e < " by auto
    then have "eventually (λn. u n > 1/e) F" using assms(1) by (simp add: tendsto_PInfty)
    moreover
    {
      fix z::ereal assume "z>1/e"
      then have "z>0" using e>0 using less_le_trans not_le by fastforce
      then have "1/z  0" by auto
      moreover have "1/z < e" 
      proof (cases z)
        case (real r)
        then show ?thesis
          using 0 < e 0 < z ereal (1 / e) < z divide_less_eq ereal_divide_less_pos by fastforce 
      qed (use 0 < e 0 < z in auto)
      ultimately have "1/z  0" "1/z < e" by auto
    }
    ultimately have "eventually (λn. 1/u n<e) F" "eventually (λn. 1/u n0) F" by (auto simp: eventually_mono)
  } note * = this
  show ?thesis
  proof (subst order_tendsto_iff, auto)
    fix a::ereal assume "a<0"
    then show "eventually (λn. 1/u n > a) F" using *(2) eventually_mono less_le_trans linordered_field_no_ub by fastforce
  next
    fix a::ereal assume "a>0"
    then obtain e::real where "e>0" "a>e" using ereal_dense2 ereal_less(2) by blast
    then have "eventually (λn. 1/u n < e) F" using *(1) by auto
    then show "eventually (λn. 1/u n < a) F" using a>e by (metis (mono_tags, lifting) eventually_mono less_trans)
  qed
qed

text ‹The next lemma deserves to exist by itself, as it is so common and useful.›

lemma tendsto_inverse_real [tendsto_intros]:
  fixes u::"_  real"
  shows "(u  l) F  l  0  ((λx. 1/ u x)  1/l) F"
  using tendsto_inverse unfolding inverse_eq_divide .

lemma tendsto_inverse_ereal [tendsto_intros]:
  fixes u::"_  ereal"
  assumes "(u  l) F" "l  0"
  shows "((λx. 1/ u x)  1/l) F"
proof (cases l)
  case (real r)
  then have "r  0" using assms(2) by auto
  then have "1/l = ereal(1/r)" using real by (simp add: one_ereal_def)
  define v where "v = (λn. real_of_ereal(u n))"
  have ureal: "eventually (λn. u n = ereal(v n)) F" unfolding v_def using real_lim_then_eventually_real assms(1) real by auto
  then have "((λn. ereal(v n))  ereal r) F" using assms real v_def by auto
  then have *: "((λn. v n)  r) F" by auto
  then have "((λn. 1/v n)  1/r) F" using r  0 tendsto_inverse_real by auto
  then have lim: "((λn. ereal(1/v n))  1/l) F" using 1/l = ereal(1/r) by auto

  have "r  -{0}" "open (-{(0::real)})" using r  0 by auto
  then have "eventually (λn. v n  -{0}) F" using * using topological_tendstoD by blast
  then have "eventually (λn. v n  0) F" by auto
  moreover
  {
    fix n assume H: "v n  0" "u n = ereal(v n)"
    then have "ereal(1/v n) = 1/ereal(v n)" by (simp add: one_ereal_def)
    then have "ereal(1/v n) = 1/u n" using H(2) by simp
  }
  ultimately have "eventually (λn. ereal(1/v n) = 1/u n) F" using ureal eventually_elim2 by force
  with Lim_transform_eventually[OF lim this] show ?thesis by simp
next
  case (PInf)
  then have "1/l = 0" by auto
  then show ?thesis using tendsto_inverse_ereal_PInf assms PInf by auto
next
  case (MInf)
  then have "1/l = 0" by auto
  have "1/z = -1/ -z" if "z < 0" for z::ereal
    apply (cases z) using divide_ereal_def z < 0 by auto
  moreover have "eventually (λn. u n < 0) F" by (metis (no_types) MInf assms(1) tendsto_MInfty zero_ereal_def)
  ultimately have *: "eventually (λn. -1/-u n = 1/u n) F" by (simp add: eventually_mono)

  define v where "v = (λn. - u n)"
  have "(v  ) F" unfolding v_def using MInf assms(1) tendsto_uminus_ereal by fastforce
  then have "((λn. 1/v n)  0) F" using tendsto_inverse_ereal_PInf by auto
  then have "((λn. -1/v n)  0) F" using tendsto_uminus_ereal by fastforce
  then show ?thesis unfolding v_def using Lim_transform_eventually[OF _ *] 1/l = 0 by auto
qed

lemma tendsto_divide_ereal [tendsto_intros]:
  fixes f g::"_  ereal"
  assumes "(f  l) F" "(g  m) F" "m  0" "¬(abs(l) =   abs(m) = )"
  shows "((λx. f x / g x)  l / m) F"
proof -
  define h where "h = (λx. 1/ g x)"
  have *: "(h  1/m) F" unfolding h_def using assms(2) assms(3) tendsto_inverse_ereal by auto
  have "((λx. f x * h x)  l * (1/m)) F"
    apply (rule tendsto_mult_ereal[OF assms(1) *]) using assms(3) assms(4) by (auto simp: divide_ereal_def)
  moreover have "f x * h x = f x / g x" for x unfolding h_def by (simp add: divide_ereal_def)
  moreover have "l * (1/m) = l/m" by (simp add: divide_ereal_def)
  ultimately show ?thesis unfolding h_def using Lim_transform_eventually by auto
qed


subsubsection ‹Further limits›

text ‹The assumptions of @{thm tendsto_diff_ereal} are too strong, we weaken them here.›

lemma tendsto_diff_ereal_general [tendsto_intros]:
  fixes u v::"'a  ereal"
  assumes "(u  l) F" "(v  m) F" "¬((l =   m = )  (l = -  m = -))"
  shows "((λn. u n - v n)  l - m) F"
proof -
  have " = l  ((λa. u a + - v a)  l + - m) F"
      by (metis (no_types) assms ereal_uminus_uminus tendsto_add_ereal_general tendsto_uminus_ereal)
  then have "((λn. u n + (-v n))  l + (-m)) F"
    by (simp add: assms ereal_uminus_eq_reorder tendsto_add_ereal_general)
  then show ?thesis 
    by (simp add: minus_ereal_def)
qed

lemma id_nat_ereal_tendsto_PInf [tendsto_intros]:
  "(λ n::nat. real n)  "
by (simp add: filterlim_real_sequentially tendsto_PInfty_eq_at_top)

lemma tendsto_at_top_pseudo_inverse [tendsto_intros]:
  fixes u::"nat  nat"
  assumes "LIM n sequentially. u n :> at_top"
  shows "LIM n sequentially. Inf {N. u N  n} :> at_top"
proof -
  {
    fix C::nat
    define M where "M = Max {u n| n. n  C}+1"
    {
      fix n assume "n  M"
      have "eventually (λN. u N  n) sequentially" using assms
        by (simp add: filterlim_at_top)
      then have *: "{N. u N  n}  {}" by force

      have "N > C" if "u N  n" for N
      proof (rule ccontr)
        assume "¬(N > C)"
        then have "u N  Max {u n| n. n  C}"
          using Max_ge by (simp add: setcompr_eq_image not_less)
        then show False using u N  n n  M unfolding M_def by auto
      qed
      then have **: "{N. u N  n}  {C..}" by fastforce
      have "Inf {N. u N  n}  C"
        by (metis "*" "**" Inf_nat_def1 atLeast_iff subset_eq)
    }
    then have "eventually (λn. Inf {N. u N  n}  C) sequentially"
      using eventually_sequentially by auto
  }
  then show ?thesis using filterlim_at_top by auto
qed

lemma pseudo_inverse_finite_set:
  fixes u::"nat  nat"
  assumes "LIM n sequentially. u n :> at_top"
  shows "finite {N. u N  n}"
proof -
  fix n
  have "eventually (λN. u N  n+1) sequentially" using assms
    by (simp add: filterlim_at_top)
  then obtain N1 where N1: "N. N  N1  u N  n + 1"
    using eventually_sequentially by auto
  have "{N. u N  n}  {..<N1}"
    by (metis (no_types, lifting) N1 Suc_eq_plus1 lessThan_iff less_le_not_le mem_Collect_eq nle_le not_less_eq_eq subset_eq)
  then show "finite {N. u N  n}" by (simp add: finite_subset)
qed

lemma tendsto_at_top_pseudo_inverse2 [tendsto_intros]:
  fixes u::"nat  nat"
  assumes "LIM n sequentially. u n :> at_top"
  shows "LIM n sequentially. Max {N. u N  n} :> at_top"
proof -
  {
    fix N0::nat
    have "N0  Max {N. u N  n}" if "n  u N0" for n
      by (simp add: assms pseudo_inverse_finite_set that)
    then have "eventually (λn. N0  Max {N. u N  n}) sequentially"
      using eventually_sequentially by blast
  }
  then show ?thesis using filterlim_at_top by auto
qed

lemma ereal_truncation_top [tendsto_intros]:
  fixes x::ereal
  shows "(λn::nat. min x n)  x"
proof (cases x)
  case (real r)
  then obtain K::nat where "K>0" "K > abs(r)" using reals_Archimedean2 gr0I by auto
  then have "min x n = x" if "n  K" for n apply (subst real, subst real, auto) using that eq_iff by fastforce
  then have "eventually (λn. min x n = x) sequentially" using eventually_at_top_linorder by blast
  then show ?thesis by (simp add: tendsto_eventually)
next
  case (PInf)
  then have "min x n = n" for n::nat by (auto simp: min_def)
  then show ?thesis using id_nat_ereal_tendsto_PInf PInf by auto
next
  case (MInf)
  then have "min x n = x" for n::nat by (auto simp: min_def)
  then show ?thesis by auto
qed

lemma ereal_truncation_real_top [tendsto_intros]:
  fixes x::ereal
  assumes "x  - "
  shows "(λn::nat. real_of_ereal(min x n))  x"
proof (cases x)
  case (real r)
  then obtain K::nat where "K>0" "K > abs(r)" using reals_Archimedean2 gr0I by auto
  then have "min x n = x" if "n  K" for n apply (subst real, subst real, auto) using that eq_iff by fastforce
  then have "real_of_ereal(min x n) = r" if "n  K" for n using real that by auto
  then have "eventually (λn. real_of_ereal(min x n) = r) sequentially" using eventually_at_top_linorder by blast
  then have "(λn. real_of_ereal(min x n))  r" by (simp add: tendsto_eventually)
  then show ?thesis using real by auto
next
  case (PInf)
  then have "real_of_ereal(min x n) = n" for n::nat by (auto simp: min_def)
  then show ?thesis using id_nat_ereal_tendsto_PInf PInf by auto
qed (simp add: assms)

lemma ereal_truncation_bottom [tendsto_intros]:
  fixes x::ereal
  shows "(λn::nat. max x (- real n))  x"
proof (cases x)
  case (real r)
  then obtain K::nat where "K>0" "K > abs(r)" using reals_Archimedean2 gr0I by auto
  then have "max x (-real n) = x" if "n  K" for n apply (subst real, subst real, auto) using that eq_iff by fastforce
  then have "eventually (λn. max x (-real n) = x) sequentially" using eventually_at_top_linorder by blast
  then show ?thesis by (simp add: tendsto_eventually)
next
  case (MInf)
  then have "max x (-real n) = (-1)* ereal(real n)" for n::nat by (auto simp: max_def)
  moreover have "(λn. (-1)* ereal(real n))  -"
    using tendsto_cmult_ereal[of "-1", OF _ id_nat_ereal_tendsto_PInf] by (simp add: one_ereal_def)
  ultimately show ?thesis using MInf by auto
next
  case (PInf)
  then have "max x (-real n) = x" for n::nat by (auto simp: max_def)
  then show ?thesis by auto
qed

lemma ereal_truncation_real_bottom [tendsto_intros]:
  fixes x::ereal
  assumes "x  "
  shows "(λn::nat. real_of_ereal(max x (- real n)))  x"
proof (cases x)
  case (real r)
  then obtain K::nat where "K>0" "K > abs(r)" using reals_Archimedean2 gr0I by auto
  then have "max x (-real n) = x" if "n  K" for n apply (subst real, subst real, auto) using that eq_iff by fastforce
  then have "real_of_ereal(max x (-real n)) = r" if "n  K" for n using real that by auto
  then have "eventually (λn. real_of_ereal(max x (-real n)) = r) sequentially" using eventually_at_top_linorder by blast
  then have "(λn. real_of_ereal(max x (-real n)))  r" by (simp add: tendsto_eventually)
  then show ?thesis using real by auto
next
  case (MInf)
  then have "real_of_ereal(max x (-real n)) = (-1)* ereal(real n)" for n::nat by (auto simp: max_def)
  moreover have "(λn. (-1)* ereal(real n))  -"
    using tendsto_cmult_ereal[of "-1", OF _ id_nat_ereal_tendsto_PInf] by (simp add: one_ereal_def)
  ultimately show ?thesis using MInf by auto
qed (simp add: assms)

text ‹the next one is copied from tendsto_sum›.›
lemma tendsto_sum_ereal [tendsto_intros]:
  fixes f :: "'a  'b  ereal"
  assumes "i. i  S  (f i  a i) F"
          "i. abs(a i)  "
  shows "((λx. iS. f i x)  (iS. a i)) F"
proof (cases "finite S")
  assume "finite S" then show ?thesis using assms
    by (induct, simp, simp add: tendsto_add_ereal_general2 assms)
qed(simp)


lemma continuous_ereal_abs:
  "continuous_on (UNIV::ereal set) abs"
proof -
  have "continuous_on ({..0}  {(0::ereal)..}) abs"
  proof (intro continuous_on_closed_Un continuous_intros)
    show "continuous_on {..0::ereal} abs"
      by (metis abs_ereal_ge0 abs_ereal_less0 continuous_on_eq antisym_conv1 atMost_iff continuous_uminus_ereal ereal_uminus_zero)
    show "continuous_on {0::ereal..} abs"
      by (metis abs_ereal_ge0 atLeast_iff continuous_on_eq continuous_on_id)
  qed
  moreover have "(UNIV::ereal set) = {..0}  {(0::ereal)..}" by auto
  ultimately show ?thesis by auto
qed

lemmas continuous_on_compose_ereal_abs[continuous_intros] =
  continuous_on_compose2[OF continuous_ereal_abs _ subset_UNIV]

lemma tendsto_abs_ereal [tendsto_intros]:
  assumes "(u  (l::ereal)) F"
  shows "((λn. abs(u n))  abs l) F"
using continuous_ereal_abs assms by (metis UNIV_I continuous_on tendsto_compose)

lemma ereal_minus_real_tendsto_MInf [tendsto_intros]:
  "(λx. ereal (- real x))  - "
by (subst uminus_ereal.simps(1)[symmetric], intro tendsto_intros)


subsection ‹Extended-Nonnegative-Real.thy› (*FIX title *)

lemma tendsto_diff_ennreal_general [tendsto_intros]:
  fixes u v::"'a  ennreal"
  assumes "(u  l) F" "(v  m) F" "¬(l =   m = )"
  shows "((λn. u n - v n)  l - m) F"
proof -
  have "((λn. e2ennreal(enn2ereal(u n) - enn2ereal(v n)))  e2ennreal(enn2ereal l - enn2ereal m)) F"
    by (intro tendsto_intros) (use assms in auto)
  then show ?thesis by auto
qed

lemma tendsto_mult_ennreal [tendsto_intros]:
  fixes l m::ennreal
  assumes "(u  l) F" "(v  m) F" "¬((l = 0  m = )  (l =   m = 0))"
  shows "((λn. u n * v n)  l * m) F"
proof -
  have "((λn. e2ennreal(enn2ereal (u n) * enn2ereal (v n)))  e2ennreal(enn2ereal l * enn2ereal m)) F"
    by (intro tendsto_intros) (use assms enn2ereal_inject zero_ennreal.rep_eq in fastforce)+
  moreover have "e2ennreal(enn2ereal (u n) * enn2ereal (v n)) = u n * v n" for n
    by (subst times_ennreal.abs_eq[symmetric], auto simp: eq_onp_same_args)
  moreover have "e2ennreal(enn2ereal l * enn2ereal m)  = l * m"
    by (subst times_ennreal.abs_eq[symmetric], auto simp: eq_onp_same_args)
  ultimately show ?thesis
    by auto
qed


subsection ‹monoset› (*FIX ME title *)

definition (in order) mono_set:
  "mono_set S  (x y. x  y  x  S  y  S)"

lemma (in order) mono_greaterThan [intro, simp]: "mono_set {B<..}" unfolding mono_set by auto
lemma (in order) mono_atLeast [intro, simp]: "mono_set {B..}" unfolding mono_set by auto
lemma (in order) mono_UNIV [intro, simp]: "mono_set UNIV" unfolding mono_set by auto
lemma (in order) mono_empty [intro, simp]: "mono_set {}" unfolding mono_set by auto

lemma (in complete_linorder) mono_set_iff:
  fixes S :: "'a set"
  defines "a  Inf S"
  shows "mono_set S  S = {a <..}  S = {a..}" (is "_ = ?c")
proof
  assume "mono_set S"
  then have mono: "x y. x  y  x  S  y  S"
    by (auto simp: mono_set)
  show ?c
  proof cases
    assume "a  S"
    show ?c
      using mono[OF _ a  S]
      by (auto intro: Inf_lower simp: a_def)
  next
    assume "a  S"
    have "S = {a <..}"
    proof safe
      fix x assume "x  S"
      then have "a  x"
        unfolding a_def by (rule Inf_lower)
      then show "a < x"
        using x  S a  S by (cases "a = x") auto
    next
      fix x assume "a < x"
      then obtain y where "y < x" "y  S"
        unfolding a_def Inf_less_iff ..
      with mono[of y x] show "x  S"
        by auto
    qed
    then show ?c ..
  qed
qed auto

lemma ereal_open_mono_set:
  fixes S :: "ereal set"
  shows "open S  mono_set S  S = UNIV  S = {Inf S <..}"
  by (metis Inf_UNIV atLeast_eq_UNIV_iff ereal_open_atLeast
    ereal_open_closed mono_set_iff open_ereal_greaterThan)

lemma ereal_closed_mono_set:
  fixes S :: "ereal set"
  shows "closed S  mono_set S  S = {}  S = {Inf S ..}"
  by (metis Inf_UNIV atLeast_eq_UNIV_iff closed_ereal_atLeast
    ereal_open_closed mono_empty mono_set_iff open_ereal_greaterThan)

lemma ereal_Liminf_Sup_monoset:
  fixes f :: "'a  ereal"
  shows "Liminf net f =
    Sup {l. S. open S  mono_set S  l  S  eventually (λx. f x  S) net}"
    (is "_ = Sup ?A")
proof (safe intro!: Liminf_eqI complete_lattice_class.Sup_upper complete_lattice_class.Sup_least)
  fix P
  assume P: "eventually P net"
  fix S
  assume S: "mono_set S" "Inf (f ` (Collect P))  S"
  {
    fix x
    assume "P x"
    then have "Inf (f ` (Collect P))  f x"
      by (intro complete_lattice_class.INF_lower) simp
    with S have "f x  S"
      by (simp add: mono_set)
  }
  with P show "eventually (λx. f x  S) net"
    by (auto elim: eventually_mono)
next
  fix y l
  assume S: "S. open S  mono_set S  l  S  eventually  (λx. f x  S) net"
  assume P: "P. eventually P net  Inf (f ` (Collect P))  y"
  show "l  y"
  proof (rule dense_le)
    fix B
    assume "B < l"
    then have "eventually (λx. f x  {B <..}) net"
      by (intro S[rule_format]) auto
    then have "Inf (f ` {x. B < f x})  y"
      using P by auto
    moreover have "B  Inf (f ` {x. B < f x})"
      by (intro INF_greatest) auto
    ultimately show "B  y"
      by simp
  qed
qed

lemma ereal_Limsup_Inf_monoset:
  fixes f :: "'a  ereal"
  shows "Limsup net f =
    Inf {l. S. open S  mono_set (uminus ` S)  l  S  eventually (λx. f x  S) net}"
    (is "_ = Inf ?A")
proof (safe intro!: Limsup_eqI complete_lattice_class.Inf_lower complete_lattice_class.Inf_greatest)
  fix P
  assume P: "eventually P net"
  fix S
  assume S: "mono_set (uminus`S)" "Sup (f ` (Collect P))  S"
  {
    fix x
    assume "P x"
    then have "f x  Sup (f ` (Collect P))"
      by (intro complete_lattice_class.SUP_upper) simp
    with S(1)[unfolded mono_set, rule_format, of "- Sup (f ` (Collect P))" "- f x"] S(2)
    have "f x  S"
      by (simp add: inj_image_mem_iff) }
  with P show "eventually (λx. f x  S) net"
    by (auto elim: eventually_mono)
next
  fix y l
  assume S: "S. open S  mono_set (uminus ` S)  l  S  eventually  (λx. f x  S) net"
  assume P: "P. eventually P net  y  Sup (f ` (Collect P))"
  show "y  l"
  proof (rule dense_ge)
    fix B
    assume "l < B"
    then have "eventually (λx. f x  {..< B}) net"
      by (intro S[rule_format]) auto
    then have "y  Sup (f ` {x. f x < B})"
      using P by auto
    moreover have "Sup (f ` {x. f x < B})  B"
      by (intro SUP_least) auto
    ultimately show "y  B"
      by simp
  qed
qed

lemma liminf_bounded_open:
  fixes x :: "nat  ereal"
  shows "x0  liminf x  (S. open S  mono_set S  x0  S  (N. nN. x n  S))"
  (is "_  ?P x0")
proof
  assume "?P x0"
  then show "x0  liminf x"
    unfolding ereal_Liminf_Sup_monoset eventually_sequentially
    by (intro complete_lattice_class.Sup_upper) auto
next
  assume "x0  liminf x"
  {
    fix S :: "ereal set"
    assume om: "open S" "mono_set S" "x0  S"
    then have "N. nN. x n  S"
        by (metis x0  liminf x ereal_open_mono_set greaterThan_iff liminf_bounded_iff om UNIV_I)
  }
  then show "?P x0"
    by auto
qed

lemma limsup_finite_then_bounded:
  fixes u::"nat  real"
  assumes "limsup u < "
  shows "C. n. u n  C"
proof -
  obtain C where C: "limsup u < C" "C < " using assms ereal_dense2 by blast
  then have "C = ereal(real_of_ereal C)" using ereal_real by force
  have "eventually (λn. u n < C) sequentially" 
    using SUP_lessD eventually_mono C(1)
    by (fastforce simp: INF_less_iff Limsup_def)
  then obtain N where N: "n. n  N  u n < C" using eventually_sequentially by auto
  define D where "D = max (real_of_ereal C) (Max {u n |n. n  N})"
  have "n. u n  D"
  proof -
    fix n show "u n  D"
    proof (cases)
      assume *: "n  N"
      have "u n  Max {u n |n. n  N}" by (rule Max_ge, auto simp: *)
      then show "u n  D" unfolding D_def by linarith
    next
      assume "¬(n  N)"
      then have "n  N" by simp
      then have "u n < C" using N by auto
      then have "u n < real_of_ereal C" using C = ereal(real_of_ereal C) less_ereal.simps(1) by fastforce
      then show "u n  D" unfolding D_def by linarith
    qed
  qed
  then show ?thesis by blast
qed

lemma liminf_finite_then_bounded_below:
  fixes u::"nat  real"
  assumes "liminf u > -"
  shows "C. n. u n  C"
proof -
  obtain C where C: "liminf u > C" "C > -" using assms using ereal_dense2 by blast
  then have "C = ereal(real_of_ereal C)" using ereal_real by force
  have "eventually (λn. u n > C) sequentially" 
    using eventually_elim2 less_INF_D C(1) 
    by (fastforce simp: less_SUP_iff Liminf_def)
  then obtain N where N: "n. n  N  u n > C" using eventually_sequentially by auto
  define D where "D = min (real_of_ereal C) (Min {u n |n. n  N})"
  have "n. u n  D"
  proof -
    fix n show "u n  D"
    proof (cases)
      assume *: "n  N"
      have "u n  Min {u n |n. n  N}" by (rule Min_le, auto simp: *)
      then show "u n  D" unfolding D_def by linarith
    next
      assume "¬(n  N)"
      then have "n  N" by simp
      then have "u n > C" using N by auto
      then have "u n > real_of_ereal C" 
        using C = ereal(real_of_ereal C) less_ereal.simps(1) by fastforce
      then show "u n  D" unfolding D_def by linarith
    qed
  qed
  then show ?thesis by blast
qed

lemma liminf_upper_bound:
  fixes u:: "nat  ereal"
  assumes "liminf u < l"
  shows "N>k. u N < l"
by (metis assms gt_ex less_le_trans liminf_bounded_iff not_less)

lemma limsup_shift:
  "limsup (λn. u (n+1)) = limsup u"
proof -
  have "(SUP m{n+1..}. u m) = (SUP m{n..}. u (m + 1))" for n
    by (rule SUP_eq) (use Suc_le_D in auto)
  then have a: "(INF n. SUP m{n..}. u (m + 1)) = (INF n. (SUP m{n+1..}. u m))" by auto
  have b: "(INF n. (SUP m{n+1..}. u m)) = (INF n{1..}. (SUP m{n..}. u m))"
    by (rule INF_eq) (use Suc_le_D in auto)
  have "(INF n{1..}. v n) = (INF n. v n)" if "decseq v" for v::"nat  'a"
    by (rule INF_eq) (use decseq v decseq_Suc_iff in auto)
  moreover have "decseq (λn. (SUP m{n..}. u m))" by (simp add: SUP_subset_mono decseq_def)
  ultimately have c: "(INF n{1..}. (SUP m{n..}. u m)) = (INF n. (SUP m{n..}. u m))" by simp
  have "(INF n. Sup (u ` {n..})) = (INF n. SUP m{n..}. u (m + 1))" using a b c by simp
  then show ?thesis by (auto cong: limsup_INF_SUP)
qed

lemma limsup_shift_k:
  "limsup (λn. u (n+k)) = limsup u"
proof (induction k)
  case (Suc k)
  have "limsup (λn. u (n+k+1)) = limsup (λn. u (n+k))" using limsup_shift[where ?u="λn. u(n+k)"] by simp
  then show ?case using Suc.IH by simp
qed (auto)

lemma liminf_shift:
  "liminf (λn. u (n+1)) = liminf u"
proof -
  have "(INF m{n+1..}. u m) = (INF m{n..}. u (m + 1))" for n
    by (rule INF_eq) (use Suc_le_D in auto)
  then have a: "(SUP n. INF m{n..}. u (m + 1)) = (SUP n. (INF m{n+1..}. u m))" by auto
  have b: "(SUP n. (INF m{n+1..}. u m)) = (SUP n{1..}. (INF m{n..}. u m))"
    by (rule SUP_eq) (use Suc_le_D in auto)
  have "(SUP n{1..}. v n) = (SUP n. v n)" if "incseq v" for v::"nat  'a"
    by (rule SUP_eq) (use incseq v incseq_Suc_iff in auto)
  moreover have "incseq (λn. (INF m{n..}. u m))" by (simp add: INF_superset_mono mono_def)
  ultimately have c: "(SUP n{1..}. (INF m{n..}. u m)) = (SUP n. (INF m{n..}. u m))" by simp
  have "(SUP n. Inf (u ` {n..})) = (SUP n. INF m{n..}. u (m + 1))" using a b c by simp
  then show ?thesis by (auto cong: liminf_SUP_INF)
qed

lemma liminf_shift_k:
  "liminf (λn. u (n+k)) = liminf u"
proof (induction k)
  case (Suc k)
  have "liminf (λn. u (n+k+1)) = liminf (λn. u (n+k))" 
    using liminf_shift[where ?u="λn. u(n+k)"] by simp
  then show ?case using Suc.IH by simp
qed (auto)

lemma Limsup_obtain:
  fixes u::"_  'a :: complete_linorder"
  assumes "Limsup F u > c"
  shows "i. u i > c"
proof -
  have "(INF P{P. eventually P F}. SUP x{x. P x}. u x) > c" using assms by (simp add: Limsup_def)
  then show ?thesis by (metis eventually_True mem_Collect_eq less_INF_D less_SUP_iff)
qed

text ‹The next lemma is extremely useful, as it often makes it possible to reduce statements
about limsups to statements about limits.›

lemma limsup_subseq_lim:
  fixes u::"nat  'a :: {complete_linorder, linorder_topology}"
  shows "r::natnat. strict_mono r  (u o r)  limsup u"
proof (cases)
  assume "n. p>n. mp. u m  u p"
  then have "r. n. (mr n. u m  u (r n))  r n < r (Suc n)"
    by (intro dependent_nat_choice) (auto simp: conj_commute)
  then obtain r :: "nat  nat" where "strict_mono r" and mono: "n m. r n  m  u m  u (r n)"
    by (auto simp: strict_mono_Suc_iff)
  define umax where "umax = (λn. (SUP m{n..}. u m))"
  have "decseq umax" unfolding umax_def by (simp add: SUP_subset_mono antimono_def)
  then have "umax  limsup u" unfolding umax_def by (metis LIMSEQ_INF limsup_INF_SUP)
  then have *: "(umax o r)  limsup u" by (simp add: LIMSEQ_subseq_LIMSEQ strict_mono r)
  have "n. umax(r n) = u(r n)" unfolding umax_def using mono
    by (metis SUP_le_iff antisym atLeast_def mem_Collect_eq order_refl)
  then have "umax o r = u o r" unfolding o_def by simp
  then have "(u o r)  limsup u" using * by simp
  then show ?thesis using strict_mono r by blast
next
  assume "¬ (n. p>n. (mp. u m  u p))"
  then obtain N where N: "p. p > N  m>p. u p < u m" by (force simp: not_le le_less)
  have "r. n. N < r n  r n < r (Suc n)  (i {N<..r (Suc n)}. u i  u (r (Suc n)))"
  proof (rule dependent_nat_choice)
    fix x assume "N < x"
    then have a: "finite {N<..x}" "{N<..x}  {}" by simp_all
    have "Max {u i |i. i  {N<..x}}  {u i |i. i  {N<..x}}" apply (rule Max_in) using a by (auto)
    then obtain p where "p  {N<..x}" and upmax: "u p = Max{u i |i. i  {N<..x}}" by auto
    define U where "U = {m. m > p  u p < u m}"
    have "U  {}" unfolding U_def using N[of p] p  {N<..x} by auto
    define y where "y = Inf U"
    then have "y  U" using U  {} by (simp add: Inf_nat_def1)
    have a: "i. i  {N<..x}  u i  u p"
    proof -
      fix i assume "i  {N<..x}"
      then have "u i  {u i |i. i  {N<..x}}" by blast
      then show "u i  u p" using upmax by simp
    qed
    moreover have "u p < u y" using y  U U_def by auto
    ultimately have "y  {N<..x}" using not_le by blast
    moreover have "y > N" using y  U U_def p  {N<..x} by auto
    ultimately have "y > x" by auto

    have "i. i  {N<..y}  u i  u y"
    proof -
      fix i assume "i  {N<..y}" show "u i  u y"
      proof (cases)
        assume "i = y"
        then show ?thesis by simp
      next
        assume "¬(i=y)"
        then have i:"i  {N<..<y}" using i  {N<..y} by simp
        have "u i  u p"
        proof (cases)
          assume "i  x"
          then have "i  {N<..x}" using i by simp
          then show ?thesis using a by simp
        next
          assume "¬(i  x)"
          then have "i > x" by simp
          then have *: "i > p" using p  {N<..x} by simp
          have "i < Inf U" using i y_def by simp
          then have "i  U" using Inf_nat_def not_less_Least by auto
          then show ?thesis using U_def * by auto
        qed
        then show "u i  u y" using u p < u y by auto
      qed
    qed
    then have "N < y  x < y  (i{N<..y}. u i  u y)" using y > x y > N by auto
    then show "y>N. x < y  (i{N<..y}. u i  u y)" by auto
  qed (auto)
  then obtain r where r: "n. N < r n  r n < r (Suc n)  (i {N<..r (Suc n)}. u i  u (r (Suc n)))" by auto
  have "strict_mono r" using r by (auto simp: strict_mono_Suc_iff)
  have "incseq (u o r)" unfolding o_def using r by (simp add: incseq_SucI order.strict_implies_order)
  then have "(u o r)  (SUP n. (u o r) n)" using LIMSEQ_SUP by blast
  then have "limsup (u o r) = (SUP n. (u o r) n)" by (simp add: lim_imp_Limsup)
  moreover have "limsup (u o r)  limsup u" using strict_mono r by (simp add: limsup_subseq_mono)
  ultimately have "(SUP n. (u o r) n)  limsup u" by simp

  {
    fix i assume i: "i  {N<..}"
    obtain n where "i < r (Suc n)" using strict_mono r using Suc_le_eq seq_suble by blast
    then have "i  {N<..r(Suc n)}" using i by simp
    then have "u i  u (r(Suc n))" using r by simp
    then have "u i  (SUP n. (u o r) n)" unfolding o_def by (meson SUP_upper2 UNIV_I)
  }
  then have "(SUP i{N<..}. u i)  (SUP n. (u o r) n)" using SUP_least by blast
  then have "limsup u  (SUP n. (u o r) n)" unfolding Limsup_def
    by (metis (mono_tags, lifting) INF_lower2 atLeast_Suc_greaterThan atLeast_def eventually_ge_at_top mem_Collect_eq)
  then have "limsup u = (SUP n. (u o r) n)" using (SUP n. (u o r) n)  limsup u by simp
  then have "(u o r)  limsup u" using (u o r)  (SUP n. (u o r) n) by simp
  then show ?thesis using strict_mono r by auto
qed

lemma liminf_subseq_lim:
  fixes u::"nat  'a :: {complete_linorder, linorder_topology}"
  shows "r::natnat. strict_mono r  (u o r)  liminf u"
proof (cases)
  assume "n. p>n. mp. u m  u p"
  then have "r. n. (mr n. u m  u (r n))  r n < r (Suc n)"
    by (intro dependent_nat_choice) (auto simp: conj_commute)
  then obtain r :: "nat  nat" where "strict_mono r" and mono: "n m. r n  m  u m  u (r n)"
    by (auto simp: strict_mono_Suc_iff)
  define umin where "umin = (λn. (INF m{n..}. u m))"
  have "incseq umin" unfolding umin_def by (simp add: INF_superset_mono incseq_def)
  then have "umin  liminf u" unfolding umin_def by (metis LIMSEQ_SUP liminf_SUP_INF)
  then have *: "(umin o r)  liminf u" by (simp add: LIMSEQ_subseq_LIMSEQ strict_mono r)
  have "n. umin(r n) = u(r n)" unfolding umin_def using mono
    by (metis le_INF_iff antisym atLeast_def mem_Collect_eq order_refl)
  then have "umin o r = u o r" unfolding o_def by simp
  then have "(u o r)  liminf u" using * by simp
  then show ?thesis using strict_mono r by blast
next
  assume "¬ (n. p>n. (mp. u m  u p))"
  then obtain N where N: "p. p > N  m>p. u p > u m" by (force simp: not_le le_less)
  have "r. n. N < r n  r n < r (Suc n)  (i {N<..r (Suc n)}. u i  u (r (Suc n)))"
  proof (rule dependent_nat_choice)
    fix x assume "N < x"
    then have a: "finite {N<..x}" "{N<..x}  {}" by simp_all
    have "Min {u i |i. i  {N<..x}}  {u i |i. i  {N<..x}}" apply (rule Min_in) using a by (auto)
    then obtain p where "p  {N<..x}" and upmin: "u p = Min{u i |i. i  {N<..x}}" by auto
    define U where "U = {m. m > p  u p > u m}"
    have "U  {}" unfolding U_def using N[of p] p  {N<..x} by auto
    define y where "y = Inf U"
    then have "y  U" using U  {} by (simp add: Inf_nat_def1)
    have a: "i. i  {N<..x}  u i  u p"
    proof -
      fix i assume "i  {N<..x}"
      then have "u i  {u i |i. i  {N<..x}}" by blast
      then show "u i  u p" using upmin by simp
    qed
    moreover have "u p > u y" using y  U U_def by auto
    ultimately have "y  {N<..x}" using not_le by blast
    moreover have "y > N" using y  U U_def p  {N<..x} by auto
    ultimately have "y > x" by auto

    have "i. i  {N<..y}  u i  u y"
    proof -
      fix i assume "i  {N<..y}" show "u i  u y"
      proof (cases)
        assume "i = y"
        then show ?thesis by simp
      next
        assume "¬(i=y)"
        then have i:"i  {N<..<y}" using i  {N<..y} by simp
        have "u i  u p"
        proof (cases)
          assume "i  x"
          then show ?thesis using a i  {N<..y} by force
        next
          assume "¬(i  x)"
          then have "i > x" by simp
          then have *: "i > p" using p  {N<..x} by simp
          have "i < Inf U" using i y_def by simp
          then have "i  U" using Inf_nat_def not_less_Least by auto
          then show ?thesis using U_def * by auto
        qed
        then show "u i  u y" using u p > u y by auto
      qed
    qed
    then have "N < y  x < y  (i{N<..y}. u i  u y)" using y > x y > N by auto
    then show "y>N. x < y  (i{N<..y}. u i  u y)" by auto
  qed (auto)
  then obtain r :: "nat  nat" 
    where r: "n. N < r n  r n < r (Suc n)  (i {N<..r (Suc n)}. u i  u (r (Suc n)))" by auto
  have "strict_mono r" using r by (auto simp: strict_mono_Suc_iff)
  have "decseq (u o r)" unfolding o_def using r by (simp add: decseq_SucI order.strict_implies_order)
  then have "(u o r)  (INF n. (u o r) n)" using LIMSEQ_INF by blast
  then have "liminf (u o r) = (INF n. (u o r) n)" by (simp add: lim_imp_Liminf)
  moreover have "liminf (u o r)  liminf u" using strict_mono r by (simp add: liminf_subseq_mono)
  ultimately have "(INF n. (u o r) n)  liminf u" by simp

  {
    fix i assume i: "i  {N<..}"
    obtain n where "i < r (Suc n)" using strict_mono r using Suc_le_eq seq_suble by blast
    then have "i  {N<..r(Suc n)}" using i by simp
    then have "u i  u (r(Suc n))" using r by simp
    then have "u i  (INF n. (u o r) n)" unfolding o_def by (meson INF_lower2 UNIV_I)
  }
  then have "(INF i{N<..}. u i)  (INF n. (u o r) n)" using INF_greatest by blast
  then have "liminf u  (INF n. (u o r) n)" unfolding Liminf_def
    by (metis (mono_tags, lifting) SUP_upper2 atLeast_Suc_greaterThan atLeast_def eventually_ge_at_top mem_Collect_eq)
  then have "liminf u = (INF n. (u o r) n)" using (INF n. (u o r) n)  liminf u by simp
  then have "(u o r)  liminf u" using (u o r)  (INF n. (u o r) n) by simp
  then show ?thesis using strict_mono r by auto
qed

text ‹The following statement about limsups is reduced to a statement about limits using
subsequences thanks to limsup_subseq_lim›. The statement for limits follows for instance from
tendsto_add_ereal_general›.›

lemma ereal_limsup_add_mono:
  fixes u v::"nat  ereal"
  shows "limsup (λn. u n + v n)  limsup u + limsup v"
proof (cases)
  assume "(limsup u = )  (limsup v = )"
  then have "limsup u + limsup v = " by simp
  then show ?thesis by auto
next
  assume "¬((limsup u = )  (limsup v = ))"
  then have "limsup u < " "limsup v < " by auto

  define w where "w = (λn. u n + v n)"
  obtain r where r: "strict_mono r" "(w o r)  limsup w" using limsup_subseq_lim by auto
  obtain s where s: "strict_mono s" "(u o r o s)  limsup (u o r)" using limsup_subseq_lim by auto
  obtain t where t: "strict_mono t" "(v o r o s o t)  limsup (v o r o s)" using limsup_subseq_lim by auto

  define a where "a = r o s o t"
  have "strict_mono a" using r s t by (simp add: a_def strict_mono_o)
  have l:"(w o a)  limsup w"
         "(u o a)  limsup (u o r)"
         "(v o a)  limsup (v o r o s)"
  apply (metis (no_types, lifting) r(2) s(1) t(1) LIMSEQ_subseq_LIMSEQ a_def comp_assoc)
  apply (metis (no_types, lifting) s(2) t(1) LIMSEQ_subseq_LIMSEQ a_def comp_assoc)
  apply (metis (no_types, lifting) t(2) a_def comp_assoc)
  done

  have "limsup (u o r)  limsup u" by (simp add: limsup_subseq_mono r(1))
  then have a: "limsup (u o r)  " using limsup u <  by auto
  have "limsup (v o r o s)  limsup v" 
    by (simp add: comp_assoc limsup_subseq_mono r(1) s(1) strict_mono_o)
  then have b: "limsup (v o r o s)  " using limsup v <  by auto

  have "(λn. (u o a) n + (v o a) n)  limsup (u o r) + limsup (v o r o s)"
    using l tendsto_add_ereal_general a b by fastforce
  moreover have "(λn. (u o a) n + (v o a) n) = (w o a)" unfolding w_def by auto
  ultimately have "(w o a)  limsup (u o r) + limsup (v o r o s)" by simp
  then have "limsup w = limsup (u o r) + limsup (v o r o s)" using l(1) LIMSEQ_unique by blast
  then have "limsup w  limsup u + limsup v"
    using limsup (u o r)  limsup u limsup (v o r o s)  limsup v add_mono by simp
  then show ?thesis unfolding w_def by simp
qed

text ‹There is an asymmetry between liminfs and limsups in ereal›, as ∞ + (-∞) = ∞›.
This explains why there are more assumptions in the next lemma dealing with liminfs that in the
previous one about limsups.›

lemma ereal_liminf_add_mono:
  fixes u v::"nat  ereal"
  assumes "¬((liminf u =   liminf v = -)  (liminf u = -  liminf v = ))"
  shows "liminf (λn. u n + v n)  liminf u + liminf v"
proof (cases)
  assume "(liminf u = -)  (liminf v = -)"
  then have *: "liminf u + liminf v = -" using assms by auto
  show ?thesis by (simp add: *)
next
  assume "¬((liminf u = -)  (liminf v = -))"
  then have "liminf u > -" "liminf v > -" by auto

  define w where "w = (λn. u n + v n)"
  obtain r where r: "strict_mono r" "(w o r)  liminf w" using liminf_subseq_lim by auto
  obtain s where s: "strict_mono s" "(u o r o s)  liminf (u o r)" using liminf_subseq_lim by auto
  obtain t where t: "strict_mono t" "(v o r o s o t)  liminf (v o r o s)" using liminf_subseq_lim by auto

  define a where "a = r o s o t"
  have "strict_mono a" using r s t by (simp add: a_def strict_mono_o)
  have l:"(w o a)  liminf w"
         "(u o a)  liminf (u o r)"
         "(v o a)  liminf (v o r o s)"
  apply (metis (no_types, lifting) r(2) s(1) t(1) LIMSEQ_subseq_LIMSEQ a_def comp_assoc)
  apply (metis (no_types, lifting) s(2) t(1) LIMSEQ_subseq_LIMSEQ a_def comp_assoc)
  apply (metis (no_types, lifting) t(2) a_def comp_assoc)
  done

  have "liminf (u o r)  liminf u" by (simp add: liminf_subseq_mono r(1))
  then have a: "liminf (u o r)  -" using liminf u > - by auto
  have "liminf (v o r o s)  liminf v" 
    by (simp add: comp_assoc liminf_subseq_mono r(1) s(1) strict_mono_o)
  then have b: "liminf (v o r o s)  -" using liminf v > - by auto

  have "(λn. (u o a) n + (v o a) n)  liminf (u o r) + liminf (v o r o s)"
    using l tendsto_add_ereal_general a b by fastforce
  moreover have "(λn. (u o a) n + (v o a) n) = (w o a)" unfolding w_def by auto
  ultimately have "(w o a)  liminf (u o r) + liminf (v o r o s)" by simp
  then have "liminf w = liminf (u o r) + liminf (v o r o s)" using l(1) LIMSEQ_unique by blast
  then have "liminf w  liminf u + liminf v"
    using liminf (u o r)  liminf u liminf (v o r o s)  liminf v add_mono by simp
  then show ?thesis unfolding w_def by simp
qed

lemma ereal_limsup_lim_add:
  fixes u v::"nat  ereal"
  assumes "u  a" "abs(a)  "
  shows "limsup (λn. u n + v n) = a + limsup v"
proof -
  have "limsup u = a" using assms(1) using tendsto_iff_Liminf_eq_Limsup trivial_limit_at_top_linorder by blast
  have "(λn. -u n)  -a" using assms(1) by auto
  then have "limsup (λn. -u n) = -a" using tendsto_iff_Liminf_eq_Limsup trivial_limit_at_top_linorder by blast

  have "limsup (λn. u n + v n)  limsup u + limsup v"
    by (rule ereal_limsup_add_mono)
  then have up: "limsup (λn. u n + v n)  a + limsup v" using limsup u = a by simp

  have a: "limsup (λn. (u n + v n) + (-u n))  limsup (λn. u n + v n) + limsup (λn. -u n)"
    by (rule ereal_limsup_add_mono)
  have "eventually (λn. u n = ereal(real_of_ereal(u n))) sequentially" using assms
    real_lim_then_eventually_real by auto
  moreover have "x. x = ereal(real_of_ereal(x))  x + (-x) = 0"
    by (metis plus_ereal.simps(1) right_minus uminus_ereal.simps(1) zero_ereal_def)
  ultimately have "eventually (λn. u n + (-u n) = 0) sequentially"
    by (metis (mono_tags, lifting) eventually_mono)
  moreover have "n. u n + (-u n) = 0  u n + v n + (-u n) = v n"
    by (metis add.commute add.left_commute add.left_neutral)
  ultimately have "eventually (λn. u n + v n + (-u n) = v n) sequentially"
    using eventually_mono by force
  then have "limsup v = limsup (λn. u n + v n + (-u n))" using Limsup_eq by force
  then have "limsup v  limsup (λn. u n + v n) -a" using a limsup (λn. -u n) = -a by (simp add: minus_ereal_def)
  then have "limsup (λn. u n + v n)  a + limsup v" using assms(2) by (metis add.commute ereal_le_minus)
  then show ?thesis using up by simp
qed

lemma ereal_limsup_lim_mult:
  fixes u v::"nat  ereal"
  assumes "u  a" "a>0" "a  "
  shows "limsup (λn. u n * v n) = a * limsup v"
proof -
  define w where "w = (λn. u n * v n)"
  obtain r where r: "strict_mono r" "(v o r)  limsup v" using limsup_subseq_lim by auto
  have "(u o r)  a" using assms(1) LIMSEQ_subseq_LIMSEQ r by auto
  with tendsto_mult_ereal[OF this r(2)] have "(λn. (u o r) n * (v o r) n)  a * limsup v" using assms(2) assms(3) by auto
  moreover have "n. (w o r) n = (u o r) n * (v o r) n" unfolding w_def by auto
  ultimately have "(w o r)  a * limsup v" unfolding w_def by presburger
  then have "limsup (w o r) = a * limsup v" by (simp add: tendsto_iff_Liminf_eq_Limsup)
  then have I: "limsup w  a * limsup v" by (metis limsup_subseq_mono r(1))

  obtain s where s: "strict_mono s" "(w o s)  limsup w" using limsup_subseq_lim by auto
  have *: "(u o s)  a" using assms(1) LIMSEQ_subseq_LIMSEQ s by auto
  have "eventually (λn. (u o s) n > 0) sequentially" using assms(2) * order_tendsto_iff by blast
  moreover have "eventually (λn. (u o s) n < ) sequentially" using assms(3) * order_tendsto_iff by blast
  moreover have "(w o s) n / (u o s) n = (v o s) n" if "(u o s) n > 0" "(u o s) n < " for n
    unfolding w_def using that by (auto simp: ereal_divide_eq)
  ultimately have "eventually (λn. (w o s) n / (u o s) n = (v o s) n) sequentially" using eventually_elim2 by force
  moreover have "(λn. (w o s) n / (u o s) n)  (limsup w) / a"
    apply (rule tendsto_divide_ereal[OF s(2) *]) using assms(2) assms(3) by auto
  ultimately have "(v o s)  (limsup w) / a" using Lim_transform_eventually by fastforce
  then have "limsup (v o s) = (limsup w) / a" by (simp add: tendsto_iff_Liminf_eq_Limsup)
  then have "limsup v  (limsup w) / a" by (metis limsup_subseq_mono s(1))
  then have "a * limsup v  limsup w" using assms(2) assms(3) by (simp add: ereal_divide_le_pos)
  then show ?thesis using I unfolding w_def by auto
qed

lemma ereal_liminf_lim_mult:
  fixes u v::"nat  ereal"
  assumes "u  a" "a>0" "a  "
  shows "liminf (λn. u n * v n) = a * liminf v"
proof -
  define w where "w = (λn. u n * v n)"
  obtain r where r: "strict_mono r" "(v o r)  liminf v" using liminf_subseq_lim by auto
  have "(u o r)  a" using assms(1) LIMSEQ_subseq_LIMSEQ r by auto
  with tendsto_mult_ereal[OF this r(2)] have "(λn. (u o r) n * (v o r) n)  a * liminf v" using assms(2) assms(3) by auto
  moreover have "n. (w o r) n = (u o r) n * (v o r) n" unfolding w_def by auto
  ultimately have "(w o r)  a * liminf v" unfolding w_def by presburger
  then have "liminf (w o r) = a * liminf v" by (simp add: tendsto_iff_Liminf_eq_Limsup)
  then have I: "liminf w  a * liminf v" by (metis liminf_subseq_mono r(1))

  obtain s where s: "strict_mono s" "(w o s)  liminf w" using liminf_subseq_lim by auto
  have *: "(u o s)  a" using assms(1) LIMSEQ_subseq_LIMSEQ s by auto
  have "eventually (λn. (u o s) n > 0) sequentially" using assms(2) * order_tendsto_iff by blast
  moreover have "eventually (λn. (u o s) n < ) sequentially" using assms(3) * order_tendsto_iff by blast
  moreover have "(w o s) n / (u o s) n = (v o s) n" if "(u o s) n > 0" "(u o s) n < " for n
    unfolding w_def using that by (auto simp: ereal_divide_eq)
  ultimately have "eventually (λn. (w o s) n / (u o s) n = (v o s) n) sequentially" using eventually_elim2 by force
  moreover have "(λn. (w o s) n / (u o s) n)  (liminf w) / a"
    using "*" assms s tendsto_divide_ereal by fastforce
  ultimately have "(v o s)  (liminf w) / a" using Lim_transform_eventually by fastforce
  then have "liminf (v o s) = (liminf w) / a" by (simp add: tendsto_iff_Liminf_eq_Limsup)
  then have "liminf v  (liminf w) / a" by (metis liminf_subseq_mono s(1))
  then have "a * liminf v  liminf w" using assms(2) assms(3) by (simp add: ereal_le_divide_pos)
  then show ?thesis using I unfolding w_def by auto
qed

lemma ereal_liminf_lim_add:
  fixes u v::"nat  ereal"
  assumes "u  a" "abs(a)  "
  shows "liminf (λn. u n + v n) = a + liminf v"
proof -
  have "liminf u = a" using assms(1) tendsto_iff_Liminf_eq_Limsup trivial_limit_at_top_linorder by blast
  then have *: "abs(liminf u)  " using assms(2) by auto
  have "(λn. -u n)  -a" using assms(1) by auto
  then have "liminf (λn. -u n) = -a" using tendsto_iff_Liminf_eq_Limsup trivial_limit_at_top_linorder by blast
  then have **: "abs(liminf (λn. -u n))  " using assms(2) by auto

  have "liminf (λn. u n + v n)  liminf u + liminf v"
    using abs_ereal.simps by (metis (full_types) "*" ereal_liminf_add_mono)
  then have up: "liminf (λn. u n + v n)  a + liminf v" using liminf u = a by simp

  have a: "liminf (λn. (u n + v n) + (-u n))  liminf (λn. u n + v n) + liminf (λn. -u n)"
    apply (rule ereal_liminf_add_mono) using ** by auto
  have "eventually (λn. u n = ereal(real_of_ereal(u n))) sequentially" using assms
    real_lim_then_eventually_real by auto
  moreover have "x. x = ereal(real_of_ereal(x))  x + (-x) = 0"
    by (metis plus_ereal.simps(1) right_minus uminus_ereal.simps(1) zero_ereal_def)
  ultimately have "eventually (λn. u n + (-u n) = 0) sequentially"
    by (metis (mono_tags, lifting) eventually_mono)
  moreover have "n. u n + (-u n) = 0  u n + v n + (-u n) = v n"
    by (metis add.commute add.left_commute add.left_neutral)
  ultimately have "eventually (λn. u n + v n + (-u n) = v n) sequentially"
    using eventually_mono by force
  then have "liminf v = liminf (λn. u n + v n + (-u n))" using Liminf_eq by force
  then have "liminf v  liminf (λn. u n + v n) -a" using a liminf (λn. -u n) = -a by (simp add: minus_ereal_def)
  then have "liminf (λn. u n + v n)  a + liminf v" using assms(2) by (metis add.commute ereal_minus_le)
  then show ?thesis using up by simp
qed

lemma ereal_liminf_limsup_add:
  fixes u v::"nat  ereal"
  shows "liminf (λn. u n + v n)  liminf u + limsup v"
proof (cases)
  assume "limsup v =   liminf u = "
  then show ?thesis by auto
next
  assume "¬(limsup v =   liminf u = )"
  then have "limsup v < " "liminf u < " by auto

  define w where "w = (λn. u n + v n)"
  obtain r where r: "strict_mono r" "(u o r)  liminf u" using liminf_subseq_lim by auto
  obtain s where s: "strict_mono s" "(w o r o s)  liminf (w o r)" using liminf_subseq_lim by auto
  obtain t where t: "strict_mono t" "(v o r o s o t)  limsup (v o r o s)" using limsup_subseq_lim by auto

  define a where "a = r o s o t"
  have "strict_mono a" using r s t by (simp add: a_def strict_mono_o)
  have l:"(u o a)  liminf u"
         "(w o a)  liminf (w o r)"
         "(v o a)  limsup (v o r o s)"
  apply (metis (no_types, lifting) r(2) s(1) t(1) LIMSEQ_subseq_LIMSEQ a_def comp_assoc)
  apply (metis (no_types, lifting) s(2) t(1) LIMSEQ_subseq_LIMSEQ a_def comp_assoc)
  apply (metis (no_types, lifting) t(2) a_def comp_assoc)
  done

  have "liminf (w o r)  liminf w" by (simp add: liminf_subseq_mono r(1))
  have "limsup (v o r o s)  limsup v" 
    by (simp add: comp_assoc limsup_subseq_mono r(1) s(1) strict_mono_o)
  then have b: "limsup (v o r o s) < " using limsup v <  by auto

  have "(λn. (u o a) n + (v o a) n)  liminf u + limsup (v o r o s)"
    apply (rule tendsto_add_ereal_general) using b liminf u <  l(1) l(3) by force+
  moreover have "(λn. (u o a) n + (v o a) n) = (w o a)" unfolding w_def by auto
  ultimately have "(w o a)  liminf u + limsup (v o r o s)" by simp
  then have "liminf (w o r) = liminf u + limsup (v o r o s)" using l(2) using LIMSEQ_unique by blast
  then have "liminf w  liminf u + limsup v"
    using liminf (w o r)  liminf w limsup (v o r o s)  limsup v
    by (metis add_mono_thms_linordered_semiring(2) le_less_trans not_less)
  then show ?thesis unfolding w_def by simp
qed

lemma ereal_liminf_limsup_minus:
  fixes u v::"nat  ereal"
  shows "liminf (λn. u n - v n)  limsup u - limsup v"
  unfolding minus_ereal_def
  apply (subst add.commute)
  apply (rule order_trans[OF ereal_liminf_limsup_add])
  using ereal_Limsup_uminus[of sequentially "λn. - v n"]
  apply (simp add: add.commute)
  done


lemma liminf_minus_ennreal:
  fixes u v::"nat  ennreal"
  shows "(n. v n  u n)  liminf (λn. u n - v n)  limsup u - limsup v"
  unfolding liminf_SUP_INF limsup_INF_SUP
  including ennreal.lifting
proof (transfer, clarsimp)
  fix v u :: "nat  ereal" assume *: "x. 0  v x" "x. 0  u x" "n. v n  u n"
  moreover have "0  limsup u - limsup v"
    using * by (intro ereal_diff_positive Limsup_mono always_eventually) simp
  moreover have "0  Sup (u ` {x..})" for x
    using * by (intro SUP_upper2[of x]) auto
  moreover have "0  Sup (v ` {x..})" for x
    using * by (intro SUP_upper2[of x]) auto
  ultimately show "(SUP n. INF n{n..}. max 0 (u n - v n))
             max 0 ((INF x. max 0 (Sup (u ` {x..}))) - (INF x. max 0 (Sup (v ` {x..}))))"
    by (auto simp: * ereal_diff_positive max.absorb2 liminf_SUP_INF[symmetric] limsup_INF_SUP[symmetric] ereal_liminf_limsup_minus)
qed

subsection "Relate extended reals and the indicator function"

lemma ereal_indicator_le_0: "(indicator S x::ereal)  0  x  S"
  by (auto split: split_indicator simp: one_ereal_def)

lemma ereal_indicator: "ereal (indicator A x) = indicator A x"
  by (auto simp: indicator_def one_ereal_def)

lemma ereal_mult_indicator: "ereal (x * indicator A y) = ereal x * indicator A y"
  by (simp split: split_indicator)

lemma ereal_indicator_mult: "ereal (indicator A y * x) = indicator A y * ereal x"
  by (simp split: split_indicator)

lemma ereal_indicator_nonneg[simp, intro]: "0  (indicator A x ::ereal)"
  unfolding indicator_def by auto

lemma indicator_inter_arith_ereal: "indicator A x * indicator B x = (indicator (A  B) x :: ereal)"
  by (simp split: split_indicator)

end