Theory Extended_Real

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

section ‹Extended real number line›

theory Extended_Real
imports Complex_Main Extended_Nat Liminf_Limsup
begin

text ‹
  This should be part of theoryHOL-Library.Extended_Nat or theoryHOL-Library.Order_Continuity, but then the AFP-entry Jinja_Thread› fails, as it does overload
  certain named from theoryComplex_Main.
›

lemma incseq_sumI2:
  fixes f :: "'i  nat  'a::ordered_comm_monoid_add"
  shows "(n. n  A  mono (f n))  mono (λi. nA. f n i)"
  unfolding incseq_def by (auto intro: sum_mono)

lemma incseq_sumI:
  fixes f :: "nat  'a::ordered_comm_monoid_add"
  assumes "i. 0  f i"
  shows "incseq (λi. sum f {..< i})"
proof (intro incseq_SucI)
  fix n
  have "sum f {..< n} + 0  sum f {..<n} + f n"
    using assms by (rule add_left_mono)
  then show "sum f {..< n}  sum f {..< Suc n}"
    by auto
qed

lemma continuous_at_left_imp_sup_continuous:
  fixes f :: "'a::{complete_linorder, linorder_topology}  'b::{complete_linorder, linorder_topology}"
  assumes "mono f" "x. continuous (at_left x) f"
  shows "sup_continuous f"
  unfolding sup_continuous_def
proof safe
  fix M :: "nat  'a" assume "incseq M" then show "f (SUP i. M i) = (SUP i. f (M i))"
    using continuous_at_Sup_mono [OF assms, of "range M"] by (simp add: image_comp)
qed

lemma sup_continuous_at_left:
  fixes f :: "'a::{complete_linorder, linorder_topology, first_countable_topology} 
    'b::{complete_linorder, linorder_topology}"
  assumes f: "sup_continuous f"
  shows "continuous (at_left x) f"
proof cases
  assume "x = bot" then show ?thesis
    by (simp add: trivial_limit_at_left_bot)
next
  assume x: "x  bot"
  show ?thesis
    unfolding continuous_within
  proof (intro tendsto_at_left_sequentially[of bot])
    fix S :: "nat  'a" assume S: "incseq S" and S_x: "S  x"
    from S_x have x_eq: "x = (SUP i. S i)"
      by (rule LIMSEQ_unique) (intro LIMSEQ_SUP S)
    show "(λn. f (S n))  f x"
      unfolding x_eq sup_continuousD[OF f S]
      using S sup_continuous_mono[OF f] by (intro LIMSEQ_SUP) (auto simp: mono_def)
  qed (insert x, auto simp: bot_less)
qed

lemma sup_continuous_iff_at_left:
  fixes f :: "'a::{complete_linorder, linorder_topology, first_countable_topology} 
    'b::{complete_linorder, linorder_topology}"
  shows "sup_continuous f  (x. continuous (at_left x) f)  mono f"
  using sup_continuous_at_left[of f] continuous_at_left_imp_sup_continuous[of f]
    sup_continuous_mono[of f] by auto

lemma continuous_at_right_imp_inf_continuous:
  fixes f :: "'a::{complete_linorder, linorder_topology}  'b::{complete_linorder, linorder_topology}"
  assumes "mono f" "x. continuous (at_right x) f"
  shows "inf_continuous f"
  unfolding inf_continuous_def
proof safe
  fix M :: "nat  'a" assume "decseq M" then show "f (INF i. M i) = (INF i. f (M i))"
    using continuous_at_Inf_mono [OF assms, of "range M"] by (simp add: image_comp)
qed

lemma inf_continuous_at_right:
  fixes f :: "'a::{complete_linorder, linorder_topology, first_countable_topology} 
    'b::{complete_linorder, linorder_topology}"
  assumes f: "inf_continuous f"
  shows "continuous (at_right x) f"
proof cases
  assume "x = top" then show ?thesis
    by (simp add: trivial_limit_at_right_top)
next
  assume x: "x  top"
  show ?thesis
    unfolding continuous_within
  proof (intro tendsto_at_right_sequentially[of _ top])
    fix S :: "nat  'a" assume S: "decseq S" and S_x: "S  x"
    from S_x have x_eq: "x = (INF i. S i)"
      by (rule LIMSEQ_unique) (intro LIMSEQ_INF S)
    show "(λn. f (S n))  f x"
      unfolding x_eq inf_continuousD[OF f S]
      using S inf_continuous_mono[OF f] by (intro LIMSEQ_INF) (auto simp: mono_def antimono_def)
  qed (insert x, auto simp: less_top)
qed

lemma inf_continuous_iff_at_right:
  fixes f :: "'a::{complete_linorder, linorder_topology, first_countable_topology} 
    'b::{complete_linorder, linorder_topology}"
  shows "inf_continuous f  (x. continuous (at_right x) f)  mono f"
  using inf_continuous_at_right[of f] continuous_at_right_imp_inf_continuous[of f]
    inf_continuous_mono[of f] by auto

instantiation enat :: linorder_topology
begin

definition open_enat :: "enat set  bool" where
  "open_enat = generate_topology (range lessThan  range greaterThan)"

instance
  proof qed (rule open_enat_def)

end

lemma open_enat: "open {enat n}"
proof (cases n)
  case 0
  then have "{enat n} = {..< eSuc 0}"
    by (auto simp: enat_0)
  then show ?thesis
    by simp
next
  case (Suc n')
  then have "{enat n} = {enat n' <..< enat (Suc n)}"
    using enat_iless by (fastforce simp: set_eq_iff)
  then show ?thesis
    by simp
qed

lemma open_enat_iff:
  fixes A :: "enat set"
  shows "open A  (  A  (n::nat. {n <..}  A))"
proof safe
  assume "  A"
  then have "A = (n{n. enat n  A}. {enat n})"
    by (simp add: set_eq_iff) (metis not_enat_eq)
  moreover have "open "
    by (auto intro: open_enat)
  ultimately show "open A"
    by simp
next
  fix n assume "{enat n <..}  A"
  then have "A = (n{n. enat n  A}. {enat n})  {enat n <..}"
    using enat_ile leI by (simp add: set_eq_iff) blast
  moreover have "open "
    by (intro open_Un open_UN ballI open_enat open_greaterThan)
  ultimately show "open A"
    by simp
next
  assume "open A" "  A"
  then have "generate_topology (range lessThan  range greaterThan) A" "  A"
    unfolding open_enat_def by auto
  then show "n::nat. {n <..}  A"
  proof induction
    case (Int A B)
    then obtain n m where "{enat n<..}  A" "{enat m<..}  B"
      by auto
    then have "{enat (max n m) <..}  A  B"
      by (auto simp add: subset_eq Ball_def max_def simp flip: enat_ord_code(1))
    then show ?case
      by auto
  next
    case (UN K)
    then obtain k where "k  K" "  k"
      by auto
    with UN.IH[OF this] show ?case
      by auto
  qed auto
qed

lemma nhds_enat: "nhds x = (if x =  then INF i. principal {enat i..} else principal {x})"
proof auto
  show "nhds  = (INF i. principal {enat i..})"
  proof (rule antisym)
    show "nhds   (INF i. principal {enat i..})"
      unfolding nhds_def
      using Ioi_le_Ico by (intro INF_greatest INF_lower) (auto simp add: open_enat_iff)
    show "(INF i. principal {enat i..})  nhds "
      unfolding nhds_def
      by (intro INF_greatest) (force intro: INF_lower2[of "Suc _"] simp add: open_enat_iff Suc_ile_eq)
  qed
  show "nhds (enat i) = principal {enat i}" for i
    by (simp add: nhds_discrete_open open_enat)
qed

instance enat :: topological_comm_monoid_add
proof
  have [simp]: "enat i  aa  enat i  aa + ba" for aa ba i
    by (rule order_trans[OF _ add_mono[of aa aa 0 ba]]) auto
  then have [simp]: "enat i  ba  enat i  aa + ba" for aa ba i
    by (metis add.commute)
  fix a b :: enat show "((λx. fst x + snd x)  a + b) (nhds a ×F nhds b)"
    apply (auto simp: nhds_enat filterlim_INF prod_filter_INF1 prod_filter_INF2
                      filterlim_principal principal_prod_principal eventually_principal)
    subgoal for i
      by (auto intro!: eventually_INF1[of i] simp: eventually_principal)
    subgoal for j i
      by (auto intro!: eventually_INF1[of i] simp: eventually_principal)
    subgoal for j i
      by (auto intro!: eventually_INF1[of i] simp: eventually_principal)
    done
qed

text ‹
  For more lemmas about the extended real numbers see
  🗏‹~~/src/HOL/Analysis/Extended_Real_Limits.thy›.
›

subsection ‹Definition and basic properties›

datatype ereal = ereal real | PInfty | MInfty

lemma ereal_cong: "x = y  ereal x = ereal y" by simp

instantiation ereal :: uminus
begin

fun uminus_ereal where
  "- (ereal r) = ereal (- r)"
| "- PInfty = MInfty"
| "- MInfty = PInfty"

instance ..

end

instantiation ereal :: infinity
begin

definition "(::ereal) = PInfty"
instance ..

end

declare [[coercion "ereal :: real  ereal"]]

lemma ereal_uminus_uminus[simp]:
  fixes a :: ereal
  shows "- (- a) = a"
  by (cases a) simp_all

lemma
  shows PInfty_eq_infinity[simp]: "PInfty = "
    and MInfty_eq_minfinity[simp]: "MInfty = - "
    and MInfty_neq_PInfty[simp]: "  - (::ereal)" "-   (::ereal)"
    and MInfty_neq_ereal[simp]: "ereal r  - " "-   ereal r"
    and PInfty_neq_ereal[simp]: "ereal r  " "  ereal r"
    and PInfty_cases[simp]: "(case  of ereal r  f r | PInfty  y | MInfty  z) = y"
    and MInfty_cases[simp]: "(case -  of ereal r  f r | PInfty  y | MInfty  z) = z"
  by (simp_all add: infinity_ereal_def)

declare
  PInfty_eq_infinity[code_post]
  MInfty_eq_minfinity[code_post]

lemma [code_unfold]:
  " = PInfty"
  "- PInfty = MInfty"
  by simp_all

lemma inj_ereal[simp]: "inj_on ereal A"
  unfolding inj_on_def by auto

lemma ereal_cases[cases type: ereal]:
  obtains (real) r where "x = ereal r"
    | (PInf) "x = "
    | (MInf) "x = -"
  by (cases x) auto

lemmas ereal2_cases = ereal_cases[case_product ereal_cases]
lemmas ereal3_cases = ereal2_cases[case_product ereal_cases]

lemma ereal_all_split: "P. (x::ereal. P x)  P   (x. P (ereal x))  P (-)"
  by (metis ereal_cases)

lemma ereal_ex_split: "P. (x::ereal. P x)  P   (x. P (ereal x))  P (-)"
  by (metis ereal_cases)

lemma ereal_uminus_eq_iff[simp]:
  fixes a b :: ereal
  shows "-a = -b  a = b"
  by (cases rule: ereal2_cases[of a b]) simp_all

function real_of_ereal :: "ereal  real" where
  "real_of_ereal (ereal r) = r"
| "real_of_ereal  = 0"
| "real_of_ereal (-) = 0"
  by (auto intro: ereal_cases)
termination by standard (rule wf_empty)

lemma real_of_ereal[simp]:
  "real_of_ereal (- x :: ereal) = - (real_of_ereal x)"
  by (cases x) simp_all

lemma range_ereal[simp]: "range ereal = UNIV - {, -}"
proof safe
  fix x
  assume "x  range ereal" "x  "
  then show "x = -"
    by (cases x) auto
qed auto

lemma ereal_range_uminus[simp]: "range uminus = (UNIV::ereal set)"
proof safe
  fix x :: ereal
  show "x  range uminus"
    by (intro image_eqI[of _ _ "-x"]) auto
qed auto

instantiation ereal :: abs
begin

function abs_ereal where
  "¦ereal r¦ = ereal ¦r¦"
| "¦-¦ = (::ereal)"
| "¦¦ = (::ereal)"
by (auto intro: ereal_cases)
termination proof qed (rule wf_empty)

instance ..

end

lemma abs_eq_infinity_cases[elim!]:
  fixes x :: ereal
  assumes "¦x¦ = "
  obtains "x = " | "x = -"
  using assms by (cases x) auto

lemma abs_neq_infinity_cases[elim!]:
  fixes x :: ereal
  assumes "¦x¦  "
  obtains r where "x = ereal r"
  using assms by (cases x) auto

lemma abs_ereal_uminus[simp]:
  fixes x :: ereal
  shows "¦- x¦ = ¦x¦"
  by (cases x) auto

lemma ereal_infinity_cases:
  fixes a :: ereal
  shows "a    a  -  ¦a¦  "
  by auto

subsubsection "Addition"

instantiation ereal :: "{one,comm_monoid_add,zero_neq_one}"
begin

definition "0 = ereal 0"
definition "1 = ereal 1"

function plus_ereal where
  "ereal r + ereal p = ereal (r + p)"
| " + a = (::ereal)"
| "a +  = (::ereal)"
| "ereal r + - = - "
| "- + ereal p = -(::ereal)"
| "- + - = -(::ereal)"
proof goal_cases
  case prems: (1 P x)
  then obtain a b where "x = (a, b)"
    by (cases x) auto
  with prems show P
   by (cases rule: ereal2_cases[of a b]) auto
qed auto
termination by standard (rule wf_empty)

lemma Infty_neq_0[simp]:
  "(::ereal)  0" "0  (::ereal)"
  "-(::ereal)  0" "0  -(::ereal)"
  by (simp_all add: zero_ereal_def)

lemma ereal_eq_0[simp]:
  "ereal r = 0  r = 0"
  "0 = ereal r  r = 0"
  unfolding zero_ereal_def by simp_all

lemma ereal_eq_1[simp]:
  "ereal r = 1  r = 1"
  "1 = ereal r  r = 1"
  unfolding one_ereal_def by simp_all

instance
proof
  fix a b c :: ereal
  show "0 + a = a"
    by (cases a) (simp_all add: zero_ereal_def)
  show "a + b = b + a"
    by (cases rule: ereal2_cases[of a b]) simp_all
  show "a + b + c = a + (b + c)"
    by (cases rule: ereal3_cases[of a b c]) simp_all
  show "0  (1::ereal)"
    by (simp add: one_ereal_def zero_ereal_def)
qed

end

lemma ereal_0_plus [simp]: "ereal 0 + x = x"
  and plus_ereal_0 [simp]: "x + ereal 0 = x"
by(simp_all flip: zero_ereal_def)

instance ereal :: numeral ..

lemma real_of_ereal_0[simp]: "real_of_ereal (0::ereal) = 0"
  unfolding zero_ereal_def by simp

lemma abs_ereal_zero[simp]: "¦0¦ = (0::ereal)"
  unfolding zero_ereal_def abs_ereal.simps by simp

lemma ereal_uminus_zero[simp]: "- 0 = (0::ereal)"
  by (simp add: zero_ereal_def)

lemma ereal_uminus_zero_iff[simp]:
  fixes a :: ereal
  shows "-a = 0  a = 0"
  by (cases a) simp_all

lemma ereal_plus_eq_PInfty[simp]:
  fixes a b :: ereal
  shows "a + b =   a =   b = "
  by (cases rule: ereal2_cases[of a b]) auto

lemma ereal_plus_eq_MInfty[simp]:
  fixes a b :: ereal
  shows "a + b = -  (a = -  b = -)  a    b  "
  by (cases rule: ereal2_cases[of a b]) auto

lemma ereal_add_cancel_left:
  fixes a b :: ereal
  assumes "a  -"
  shows "a + b = a + c  a =   b = c"
  using assms by (cases rule: ereal3_cases[of a b c]) auto

lemma ereal_add_cancel_right:
  fixes a b :: ereal
  assumes "a  -"
  shows "b + a = c + a  a =   b = c"
  using assms by (cases rule: ereal3_cases[of a b c]) auto

lemma ereal_real: "ereal (real_of_ereal x) = (if ¦x¦ =  then 0 else x)"
  by (cases x) simp_all

lemma real_of_ereal_add:
  fixes a b :: ereal
  shows "real_of_ereal (a + b) =
    (if (¦a¦ = )  (¦b¦ = )  (¦a¦  )  (¦b¦  ) then real_of_ereal a + real_of_ereal b else 0)"
  by (cases rule: ereal2_cases[of a b]) auto


subsubsection "Linear order on typereal"

instantiation ereal :: linorder
begin

function less_ereal
where
  "   ereal x < ereal y      x < y"
| "(::ereal) < a            False"
| "         a < -(::ereal)  False"
| "ereal x    <             True"
| "        - < ereal r      True"
| "        - < (::ereal)  True"
proof goal_cases
  case prems: (1 P x)
  then obtain a b where "x = (a,b)" by (cases x) auto
  with prems show P by (cases rule: ereal2_cases[of a b]) auto
qed simp_all
termination by (relation "{}") simp

definition "x  (y::ereal)  x < y  x = y"

lemma ereal_infty_less[simp]:
  fixes x :: ereal
  shows "x <   (x  )"
    "- < x  (x  -)"
  by (cases x, simp_all) (cases x, simp_all)

lemma ereal_infty_less_eq[simp]:
  fixes x :: ereal
  shows "  x  x = "
    and "x  -  x = -"
  by (auto simp add: less_eq_ereal_def)

lemma ereal_less[simp]:
  "ereal r < 0  (r < 0)"
  "0 < ereal r  (0 < r)"
  "ereal r < 1  (r < 1)"
  "1 < ereal r  (1 < r)"
  "0 < (::ereal)"
  "-(::ereal) < 0"
  by (simp_all add: zero_ereal_def one_ereal_def)

lemma ereal_less_eq[simp]:
  "x  (::ereal)"
  "-(::ereal)  x"
  "ereal r  ereal p  r  p"
  "ereal r  0  r  0"
  "0  ereal r  0  r"
  "ereal r  1  r  1"
  "1  ereal r  1  r"
  by (auto simp add: less_eq_ereal_def zero_ereal_def one_ereal_def)

lemma ereal_infty_less_eq2:
  "a  b  a =   b = (::ereal)"
  "a  b  b = -  a = -(::ereal)"
  by simp_all

instance
proof
  fix x y z :: ereal
  show "x  x"
    by (cases x) simp_all
  show "x < y  x  y  ¬ y  x"
    by (cases rule: ereal2_cases[of x y]) auto
  show "x  y  y  x "
    by (cases rule: ereal2_cases[of x y]) auto
  {
    assume "x  y" "y  x"
    then show "x = y"
      by (cases rule: ereal2_cases[of x y]) auto
  }
  {
    assume "x  y" "y  z"
    then show "x  z"
      by (cases rule: ereal3_cases[of x y z]) auto
  }
qed

end

lemma ereal_dense2: "x < y  z. x < ereal z  ereal z < y"
  using lt_ex gt_ex dense by (cases x y rule: ereal2_cases) auto

instance ereal :: dense_linorder
  by standard (blast dest: ereal_dense2)

instance ereal :: ordered_comm_monoid_add
proof
  fix a b c :: ereal
  assume "a  b"
  then show "c + a  c + b"
    by (cases rule: ereal3_cases[of a b c]) auto
qed

lemma ereal_one_not_less_zero_ereal[simp]: "¬ 1 < (0::ereal)"
  by (simp add: zero_ereal_def)

lemma real_of_ereal_positive_mono:
  fixes x y :: ereal
  shows "0  x  x  y  y    real_of_ereal x  real_of_ereal y"
  by (cases rule: ereal2_cases[of x y]) auto

lemma ereal_MInfty_lessI[intro, simp]:
  fixes a :: ereal
  shows "a  -  - < a"
  by (cases a) auto

lemma ereal_less_PInfty[intro, simp]:
  fixes a :: ereal
  shows "a    a < "
  by (cases a) auto

lemma ereal_less_ereal_Ex:
  fixes a b :: ereal
  shows "x < ereal r  x = -  (p. p < r  x = ereal p)"
  by (cases x) auto

lemma less_PInf_Ex_of_nat: "x    (n::nat. x < ereal (real n))"
proof (cases x)
  case (real r)
  then show ?thesis
    using reals_Archimedean2[of r] by simp
qed simp_all

lemma ereal_add_strict_mono2:
  fixes a b c d :: ereal
  assumes "a < b" and "c < d"
  shows "a + c < b + d"
  using assms
  by (cases a; force simp add: elim: less_ereal.elims)

lemma ereal_minus_le_minus[simp]:
  fixes a b :: ereal
  shows "- a  - b  b  a"
  by (cases rule: ereal2_cases[of a b]) auto

lemma ereal_minus_less_minus[simp]:
  fixes a b :: ereal
  shows "- a < - b  b < a"
  by (cases rule: ereal2_cases[of a b]) auto

lemma ereal_le_real_iff:
  "x  real_of_ereal y  (¦y¦    ereal x  y)  (¦y¦ =   x  0)"
  by (cases y) auto

lemma real_le_ereal_iff:
  "real_of_ereal y  x  (¦y¦    y  ereal x)  (¦y¦ =   0  x)"
  by (cases y) auto

lemma ereal_less_real_iff:
  "x < real_of_ereal y  (¦y¦    ereal x < y)  (¦y¦ =   x < 0)"
  by (cases y) auto

lemma real_less_ereal_iff:
  "real_of_ereal y < x  (¦y¦    y < ereal x)  (¦y¦ =   0 < x)"
  by (cases y) auto

text ‹
  To help with inferences like propa < ereal x  x < y  a < ereal y,
  where x and y are real.
›

lemma le_ereal_le: "a  ereal x  x  y  a  ereal y"
  using ereal_less_eq(3) order.trans by blast

lemma le_ereal_less: "a  ereal x  x < y  a < ereal y"
  by (simp add: le_less_trans)

lemma less_ereal_le: "a < ereal x  x  y  a < ereal y"
  using ereal_less_ereal_Ex by auto

lemma ereal_le_le: "ereal y  a  x  y  ereal x  a"
  by (simp add: order_subst2)

lemma ereal_le_less: "ereal y  a  x < y  ereal x < a"
  by (simp add: dual_order.strict_trans1)

lemma ereal_less_le: "ereal y < a  x  y  ereal x < a"
  using ereal_less_eq(3) le_less_trans by blast

lemma real_of_ereal_pos:
  fixes x :: ereal
  shows "0  x  0  real_of_ereal x" by (cases x) auto

lemmas real_of_ereal_ord_simps =
  ereal_le_real_iff real_le_ereal_iff ereal_less_real_iff real_less_ereal_iff

lemma abs_ereal_ge0[simp]: "0  x  ¦x :: ereal¦ = x"
  by (cases x) auto

lemma abs_ereal_less0[simp]: "x < 0  ¦x :: ereal¦ = -x"
  by (cases x) auto

lemma abs_ereal_pos[simp]: "0  ¦x :: ereal¦"
  by (cases x) auto

lemma ereal_abs_leI:
  fixes x y :: ereal
  shows " x  y; -x  y   ¦x¦  y"
by(cases x y rule: ereal2_cases)(simp_all)

lemma ereal_abs_add:
  fixes a b::ereal
  shows "abs(a+b)  abs a + abs b"
by (cases rule: ereal2_cases[of a b]) (auto)

lemma real_of_ereal_le_0[simp]: "real_of_ereal (x :: ereal)  0  x  0  x = "
  by (cases x) auto

lemma abs_real_of_ereal[simp]: "¦real_of_ereal (x :: ereal)¦ = real_of_ereal ¦x¦"
  by (cases x) auto

lemma zero_less_real_of_ereal:
  fixes x :: ereal
  shows "0 < real_of_ereal x  0 < x  x  "
  by (cases x) auto

lemma ereal_0_le_uminus_iff[simp]:
  fixes a :: ereal
  shows "0  - a  a  0"
  by (cases rule: ereal2_cases[of a]) auto

lemma ereal_uminus_le_0_iff[simp]:
  fixes a :: ereal
  shows "- a  0  0  a"
  by (cases rule: ereal2_cases[of a]) auto

lemma ereal_add_strict_mono:
  fixes a b c d :: ereal
  assumes "a  b"
    and "0  a"
    and "a  "
    and "c < d"
  shows "a + c < b + d"
  using assms
  by (cases rule: ereal3_cases[case_product ereal_cases, of a b c d]) auto

lemma ereal_less_add:
  fixes a b c :: ereal
  shows "¦a¦    c < b  a + c < a + b"
  by (cases rule: ereal2_cases[of b c]) auto

lemma ereal_add_nonneg_eq_0_iff:
  fixes a b :: ereal
  shows "0  a  0  b  a + b = 0  a = 0  b = 0"
  by (cases a b rule: ereal2_cases) auto

lemma ereal_uminus_eq_reorder: "- a = b  a = (-b::ereal)"
  by auto

lemma ereal_uminus_less_reorder: "- a < b  -b < (a::ereal)"
  by (subst (3) ereal_uminus_uminus[symmetric]) (simp only: ereal_minus_less_minus)

lemma ereal_less_uminus_reorder: "a < - b  b < - (a::ereal)"
  by (subst (3) ereal_uminus_uminus[symmetric]) (simp only: ereal_minus_less_minus)

lemma ereal_uminus_le_reorder: "- a  b  -b  (a::ereal)"
  by (subst (3) ereal_uminus_uminus[symmetric]) (simp only: ereal_minus_le_minus)

lemmas ereal_uminus_reorder =
  ereal_uminus_eq_reorder ereal_uminus_less_reorder ereal_uminus_le_reorder

lemma ereal_bot:
  fixes x :: ereal
  assumes "B. x  ereal B"
  shows "x = - "
proof (cases x)
  case (real r)
  with assms[of "r - 1"] show ?thesis
    by auto
next
  case PInf
  with assms[of 0] show ?thesis
    by auto
next
  case MInf
  then show ?thesis
    by simp
qed

lemma ereal_top:
  fixes x :: ereal
  assumes "B. x  ereal B"
  shows "x = "
proof (cases x)
  case (real r)
  with assms[of "r + 1"] show ?thesis
    by auto
next
  case MInf
  with assms[of 0] show ?thesis
    by auto
next
  case PInf
  then show ?thesis
    by simp
qed

lemma
  shows ereal_max[simp]: "ereal (max x y) = max (ereal x) (ereal y)"
    and ereal_min[simp]: "ereal (min x y) = min (ereal x) (ereal y)"
  by (simp_all add: min_def max_def)

lemma ereal_max_0: "max 0 (ereal r) = ereal (max 0 r)"
  by (auto simp: zero_ereal_def)

lemma
  fixes f :: "nat  ereal"
  shows ereal_incseq_uminus[simp]: "incseq (λx. - f x)  decseq f"
    and ereal_decseq_uminus[simp]: "decseq (λx. - f x)  incseq f"
  unfolding decseq_def incseq_def by auto

lemma incseq_ereal: "incseq f  incseq (λx. ereal (f x))"
  unfolding incseq_def by auto

lemma sum_ereal[simp]: "(xA. ereal (f x)) = ereal (xA. f x)"
proof (cases "finite A")
  case True
  then show ?thesis by induct auto
next
  case False
  then show ?thesis by simp
qed

lemma sum_list_ereal [simp]: "sum_list (map (λx. ereal (f x)) xs) = ereal (sum_list (map f xs))"
  by (induction xs) simp_all

lemma sum_Pinfty:
  fixes f :: "'a  ereal"
  shows "(xP. f x) =   finite P  (iP. f i = )"
proof safe
  assume *: "sum f P = "
  show "finite P"
  proof (rule ccontr)
    assume "¬ finite P"
    with * show False
      by auto
  qed
  show "iP. f i = "
  proof (rule ccontr)
    assume "¬ ?thesis"
    then have "i. i  P  f i  "
      by auto
    with finite P have "sum f P  "
      by induct auto
    with * show False
      by auto
  qed
next
  fix i
  assume "finite P" and "i  P" and "f i = "
  then show "sum f P = "
  proof induct
    case (insert x A)
    show ?case using insert by (cases "x = i") auto
  qed simp
qed

lemma sum_Inf:
  fixes f :: "'a  ereal"
  shows "¦sum f A¦ =   finite A  (iA. ¦f i¦ = )"
proof
  assume *: "¦sum f A¦ = "
  have "finite A"
    by (rule ccontr) (insert *, auto)
  moreover have "iA. ¦f i¦ = "
  proof (rule ccontr)
    assume "¬ ?thesis"
    then have "iA. r. f i = ereal r"
      by auto
    from bchoice[OF this] obtain r where "xA. f x = ereal (r x)" ..
    with * show False
      by auto
  qed
  ultimately show "finite A  (iA. ¦f i¦ = )"
    by auto
next
  assume "finite A  (iA. ¦f i¦ = )"
  then obtain i where "finite A" "i  A" and "¦f i¦ = "
    by auto
  then show "¦sum f A¦ = "
  proof induct
    case (insert j A)
    then show ?case
      by (cases rule: ereal3_cases[of "f i" "f j" "sum f A"]) auto
  qed simp
qed

lemma sum_real_of_ereal:
  fixes f :: "'i  ereal"
  assumes "x. x  S  ¦f x¦  "
  shows "(xS. real_of_ereal (f x)) = real_of_ereal (sum f S)"
proof -
  have "xS. r. f x = ereal r"
  proof
    fix x
    assume "x  S"
    from assms[OF this] show "r. f x = ereal r"
      by (cases "f x") auto
  qed
  from bchoice[OF this] obtain r where "xS. f x = ereal (r x)" ..
  then show ?thesis
    by simp
qed

lemma sum_ereal_0:
  fixes f :: "'a  ereal"
  assumes "finite A"
    and "i. i  A  0  f i"
  shows "(xA. f x) = 0  (iA. f i = 0)"
proof
  assume "sum f A = 0" with assms show "iA. f i = 0"
  proof (induction A)
    case (insert a A)
    then have "f a = 0  (aA. f a) = 0"
      by (subst ereal_add_nonneg_eq_0_iff[symmetric]) (simp_all add: sum_nonneg)
    with insert show ?case
      by simp
  qed simp
qed auto

subsubsection "Multiplication"

instantiation ereal :: "{comm_monoid_mult,sgn}"
begin

function sgn_ereal :: "ereal  ereal" where
  "sgn (ereal r) = ereal (sgn r)"
| "sgn (::ereal) = 1"
| "sgn (-::ereal) = -1"
by (auto intro: ereal_cases)
termination by standard (rule wf_empty)

function times_ereal where
  "ereal r * ereal p = ereal (r * p)"
| "ereal r *  = (if r = 0 then 0 else if r > 0 then  else -)"
| " * ereal r = (if r = 0 then 0 else if r > 0 then  else -)"
| "ereal r * - = (if r = 0 then 0 else if r > 0 then - else )"
| "- * ereal r = (if r = 0 then 0 else if r > 0 then - else )"
| "(::ereal) *  = "
| "-(::ereal) *  = -"
| "(::ereal) * - = -"
| "-(::ereal) * - = "
proof goal_cases
  case prems: (1 P x)
  then obtain a b where "x = (a, b)"
    by (cases x) auto
  with prems show P
    by (cases rule: ereal2_cases[of a b]) auto
qed simp_all
termination by (relation "{}") simp

instance
proof
  fix a b c :: ereal
  show "1 * a = a"
    by (cases a) (simp_all add: one_ereal_def)
  show "a * b = b * a"
    by (cases rule: ereal2_cases[of a b]) simp_all
  show "a * b * c = a * (b * c)"
    by (cases rule: ereal3_cases[of a b c])
       (simp_all add: zero_ereal_def zero_less_mult_iff)
qed

end

lemma [simp]:
  shows ereal_1_times: "ereal 1 * x = x"
  and times_ereal_1: "x * ereal 1 = x"
by(simp_all flip: one_ereal_def)

lemma one_not_le_zero_ereal[simp]: "¬ (1  (0::ereal))"
  by (simp add: one_ereal_def zero_ereal_def)

lemma real_ereal_1[simp]: "real_of_ereal (1::ereal) = 1"
  unfolding one_ereal_def by simp

lemma real_of_ereal_le_1:
  fixes a :: ereal
  shows "a  1  real_of_ereal a  1"
  by (cases a) (auto simp: one_ereal_def)

lemma abs_ereal_one[simp]: "¦1¦ = (1::ereal)"
  unfolding one_ereal_def by simp

lemma ereal_mult_zero[simp]:
  fixes a :: ereal
  shows "a * 0 = 0"
  by (cases a) (simp_all add: zero_ereal_def)

lemma ereal_zero_mult[simp]:
  fixes a :: ereal
  shows "0 * a = 0"
  by (cases a) (simp_all add: zero_ereal_def)

lemma ereal_m1_less_0[simp]: "-(1::ereal) < 0"
  by (simp add: zero_ereal_def one_ereal_def)

lemma ereal_times[simp]:
  "1  (::ereal)" "(::ereal)  1"
  "1  -(::ereal)" "-(::ereal)  1"
  by (auto simp: one_ereal_def)

lemma ereal_plus_1[simp]:
  "1 + ereal r = ereal (r + 1)"
  "ereal r + 1 = ereal (r + 1)"
  "1 + -(::ereal) = -"
  "-(::ereal) + 1 = -"
  unfolding one_ereal_def by auto

lemma ereal_zero_times[simp]:
  fixes a b :: ereal
  shows "a * b = 0  a = 0  b = 0"
  by (cases rule: ereal2_cases[of a b]) auto

lemma ereal_mult_eq_PInfty[simp]:
  "a * b = (::ereal) 
    (a =   b > 0)  (a > 0  b = )  (a = -  b < 0)  (a < 0  b = -)"
  by (cases rule: ereal2_cases[of a b]) auto

lemma ereal_mult_eq_MInfty[simp]:
  "a * b = -(::ereal) 
    (a =   b < 0)  (a < 0  b = )  (a = -  b > 0)  (a > 0  b = -)"
  by (cases rule: ereal2_cases[of a b]) auto

lemma ereal_abs_mult: "¦x * y :: ereal¦ = ¦x¦ * ¦y¦"
  by (cases x y rule: ereal2_cases) (auto simp: abs_mult)

lemma ereal_0_less_1[simp]: "0 < (1::ereal)"
  by (simp_all add: zero_ereal_def one_ereal_def)

lemma ereal_mult_minus_left[simp]:
  fixes a b :: ereal
  shows "-a * b = - (a * b)"
  by (cases rule: ereal2_cases[of a b]) auto

lemma ereal_mult_minus_right[simp]:
  fixes a b :: ereal
  shows "a * -b = - (a * b)"
  by (cases rule: ereal2_cases[of a b]) auto

lemma ereal_mult_infty[simp]:
  "a * (::ereal) = (if a = 0 then 0 else if 0 < a then  else - )"
  by (cases a) auto

lemma ereal_infty_mult[simp]:
  "(::ereal) * a = (if a = 0 then 0 else if 0 < a then  else - )"
  by (cases a) auto

lemma ereal_mult_strict_right_mono:
  assumes "a < b"
    and "0 < c"
    and "c < (::ereal)"
  shows "a * c < b * c"
  using assms
  by (cases rule: ereal3_cases[of a b c]) (auto simp: zero_le_mult_iff)

lemma ereal_mult_strict_left_mono:
  "a < b  0 < c  c < (::ereal)  c * a < c * b"
  using ereal_mult_strict_right_mono
  by (simp add: mult.commute[of c])

lemma ereal_mult_right_mono:
  fixes a b c :: ereal
  assumes "a  b" "0  c"
  shows "a * c  b * c"
proof (cases "c = 0")
  case False
  with assms show ?thesis
    by (cases rule: ereal3_cases[of a b c]) auto
qed auto

lemma ereal_mult_left_mono:
  fixes a b c :: ereal
  shows "a  b  0  c  c * a  c * b"
  using ereal_mult_right_mono
  by (simp add: mult.commute[of c])

lemma ereal_mult_mono:
  fixes a b c d::ereal
  assumes "b  0" "c  0" "a  b" "c  d"
  shows "a * c  b * d"
by (metis ereal_mult_right_mono mult.commute order_trans assms)

lemma ereal_mult_mono':
  fixes a b c d::ereal
  assumes "a  0" "c  0" "a  b" "c  d"
  shows "a * c  b * d"
by (metis ereal_mult_right_mono mult.commute order_trans assms)

lemma ereal_mult_mono_strict:
  fixes a b c d::ereal
  assumes "b > 0" "c > 0" "a < b" "c < d"
  shows "a * c < b * d"
proof -
  have "c < " using c < d by auto
  then have "a * c < b * c" by (metis ereal_mult_strict_left_mono[OF assms(3) assms(2)] mult.commute)
  moreover have "b * c  b * d" using assms(2) assms(4) by (simp add: assms(1) ereal_mult_left_mono less_imp_le)
  ultimately show ?thesis by simp
qed

lemma ereal_mult_mono_strict':
  fixes a b c d::ereal
  assumes "a > 0" "c > 0" "a < b" "c < d"
  shows "a * c < b * d"
  using assms ereal_mult_mono_strict by auto

lemma zero_less_one_ereal[simp]: "0  (1::ereal)"
  by (simp add: one_ereal_def zero_ereal_def)

lemma ereal_0_le_mult[simp]: "0  a  0  b  0  a * (b :: ereal)"
  by (cases rule: ereal2_cases[of a b]) auto

lemma ereal_right_distrib:
  fixes r a b :: ereal
  shows "0  a  0  b  r * (a + b) = r * a + r * b"
  by (cases rule: ereal3_cases[of r a b]) (simp_all add: field_simps)

lemma ereal_left_distrib:
  fixes r a b :: ereal
  shows "0  a  0  b  (a + b) * r = a * r + b * r"
  by (cases rule: ereal3_cases[of r a b]) (simp_all add: field_simps)

lemma ereal_mult_le_0_iff:
  fixes a b :: ereal
  shows "a * b  0  (0  a  b  0)  (a  0  0  b)"
  by (cases rule: ereal2_cases[of a b]) (simp_all add: mult_le_0_iff)

lemma ereal_zero_le_0_iff:
  fixes a b :: ereal
  shows "0  a * b  (0  a  0  b)  (a  0  b  0)"
  by (cases rule: ereal2_cases[of a b]) (simp_all add: zero_le_mult_iff)

lemma ereal_mult_less_0_iff:
  fixes a b :: ereal
  shows "a * b < 0  (0 < a  b < 0)  (a < 0  0 < b)"
  by (cases rule: ereal2_cases[of a b]) (simp_all add: mult_less_0_iff)

lemma ereal_zero_less_0_iff:
  fixes a b :: ereal
  shows "0 < a * b  (0 < a  0 < b)  (a < 0  b < 0)"
  by (cases rule: ereal2_cases[of a b]) (simp_all add: zero_less_mult_iff)

lemma ereal_left_mult_cong:
  fixes a b c :: ereal
  shows  "c = d  (d  0  a = b)  a * c = b * d"
  by (cases "c = 0") simp_all

lemma ereal_right_mult_cong:
  fixes a b c :: ereal
  shows "c = d  (d  0  a = b)  c * a = d * b"
  by (cases "c = 0") simp_all

lemma ereal_distrib:
  fixes a b c :: ereal
  assumes "a    b  -"
    and "a  -  b  "
    and "¦c¦  "
  shows "(a + b) * c = a * c + b * c"
  using assms
  by (cases rule: ereal3_cases[of a b c]) (simp_all add: field_simps)

lemma numeral_eq_ereal [simp]: "numeral w = ereal (numeral w)"
proof (induct w rule: num_induct)
  case One
  then show ?case
    by simp
next
  case (inc x)
  then show ?case
    by (simp add: inc numeral_inc)
qed

lemma distrib_left_ereal_nn:
  "c  0  (x + y) * ereal c = x * ereal c + y * ereal c"
  by(cases x y rule: ereal2_cases)(simp_all add: ring_distribs)

lemma sum_ereal_right_distrib:
  fixes f :: "'a  ereal"
  shows "(i. i  A  0  f i)  r * sum f A = (nA. r * f n)"
  by (induct A rule: infinite_finite_induct)  (auto simp: ereal_right_distrib sum_nonneg)

lemma sum_ereal_left_distrib:
  "(i. i  A  0  f i)  sum f A * r = (nA. f n * r :: ereal)"
  using sum_ereal_right_distrib[of A f r] by (simp add: mult_ac)

lemma sum_distrib_right_ereal:
  "c  0  sum f A * ereal c = (xA. f x * c :: ereal)"
by(subst sum_comp_morphism[where h="λx. x * ereal c", symmetric])(simp_all add: distrib_left_ereal_nn)

lemma ereal_le_epsilon:
  fixes x y :: ereal
  assumes "e. 0 < e  x  y + e"
  shows "x  y"
proof (cases "x = -  x =   y = -  y = ")
  case True
  then show ?thesis
    using assms[of 1] by auto
next
  case False
  then obtain p q where "x = ereal p" "y = ereal q"
    by (metis MInfty_eq_minfinity ereal.distinct(3) uminus_ereal.elims)
  then show ?thesis 
    by (metis assms field_le_epsilon ereal_less(2) ereal_less_eq(3) plus_ereal.simps(1))
qed

lemma ereal_le_epsilon2:
  fixes x y :: ereal
  assumes "e::real. 0 < e  x  y + ereal e"
  shows "x  y"
proof (rule ereal_le_epsilon)
  show "ε::ereal. 0 < ε  x  y + ε"
  using assms less_ereal.elims(2) zero_less_real_of_ereal by fastforce
qed

lemma ereal_le_real:
  fixes x y :: ereal
  assumes "z. x  ereal z  y  ereal z"
  shows "y  x"
  by (metis assms ereal_bot ereal_cases ereal_infty_less_eq(2) ereal_less_eq(1) linorder_le_cases)

lemma prod_ereal_0:
  fixes f :: "'a  ereal"
  shows "(iA. f i) = 0  finite A  (iA. f i = 0)"
proof (cases "finite A")
  case True
  then show ?thesis by (induct A) auto
qed auto

lemma prod_ereal_pos:
  fixes f :: "'a  ereal"
  assumes pos: "i. i  I  0  f i"
  shows "0  (iI. f i)"
proof (cases "finite I")
  case True
  from this pos show ?thesis
    by induct auto
qed auto

lemma prod_PInf:
  fixes f :: "'a  ereal"
  assumes "i. i  I  0  f i"
  shows "(iI. f i) =   finite I  (iI. f i = )  (iI. f i  0)"
proof (cases "finite I")
  case True
  from this assms show ?thesis
  proof (induct I)
    case (insert i I)
    then have pos: "0  f i" "0  prod f I"
      by (auto intro!: prod_ereal_pos)
    from insert have "(jinsert i I. f j) =   prod f I * f i = "
      by auto
    also have "  (prod f I =   f i = )  f i  0  prod f I  0"
      using prod_ereal_pos[of I f] pos
      by (cases rule: ereal2_cases[of "f i" "prod f I"]) auto
    also have "  finite (insert i I)  (jinsert i I. f j = )  (jinsert i I. f j  0)"
      using insert by (auto simp: prod_ereal_0)
    finally show ?case .
  qed simp
qed auto

lemma prod_ereal: "(iA. ereal (f i)) = ereal (prod f A)"
proof (cases "finite A")
  case True
  then show ?thesis
    by induct (auto simp: one_ereal_def)
next
  case False
  then show ?thesis
    by (simp add: one_ereal_def)
qed


subsubsection ‹Power›

lemma ereal_power[simp]: "(ereal x) ^ n = ereal (x^n)"
  by (induct n) (auto simp: one_ereal_def)

lemma ereal_power_PInf[simp]: "(::ereal) ^ n = (if n = 0 then 1 else )"
  by (induct n) (auto simp: one_ereal_def)

lemma ereal_power_uminus[simp]:
  fixes x :: ereal
  shows "(- x) ^ n = (if even n then x ^ n else - (x^n))"
  by (induct n) (auto simp: one_ereal_def)

lemma ereal_power_numeral[simp]:
  "(numeral num :: ereal) ^ n = ereal (numeral num ^ n)"
  by (induct n) (auto simp: one_ereal_def)

lemma zero_le_power_ereal[simp]:
  fixes a :: ereal
  assumes "0  a"
  shows "0  a ^ n"
  using assms by (induct n) (auto simp: ereal_zero_le_0_iff)


subsubsection ‹Subtraction›

lemma ereal_minus_minus_image[simp]:
  fixes S :: "ereal set"
  shows "uminus ` uminus ` S = S"
  by (auto simp: image_iff)

lemma ereal_uminus_lessThan[simp]:
  fixes a :: ereal
  shows "uminus ` {..<a} = {-a<..}"
proof -
  {
    fix x
    assume "-a < x"
    then have "- x < - (- a)"
      by (simp del: ereal_uminus_uminus)
    then have "- x < a"
      by simp
  }
  then show ?thesis
    by force
qed

lemma ereal_uminus_greaterThan[simp]: "uminus ` {(a::ereal)<..} = {..<-a}"
  by (metis ereal_uminus_lessThan ereal_uminus_uminus ereal_minus_minus_image)

instantiation ereal :: minus
begin

definition "x - y = x + -(y::ereal)"
instance ..

end

lemma ereal_minus[simp]:
  "ereal r - ereal p = ereal (r - p)"
  "- - ereal r = -"
  "ereal r -  = -"
  "(::ereal) - x = "
  "-(::ereal) -  = -"
  "x - -y = x + y"
  "x - 0 = x"
  "0 - x = -x"
  by (simp_all add: minus_ereal_def)

lemma ereal_x_minus_x[simp]: "x - x = (if ¦x¦ =  then  else 0::ereal)"
  by (cases x) simp_all

lemma ereal_eq_minus_iff:
  fixes x y z :: ereal
  shows "x = z - y 
    (¦y¦    x + y = z) 
    (y = -  x = ) 
    (y =   z =   x = ) 
    (y =   z    x = -)"
  by (cases rule: ereal3_cases[of x y z]) auto

lemma ereal_eq_minus:
  fixes x y z :: ereal
  shows "¦y¦    x = z - y  x + y = z"
  by (auto simp: ereal_eq_minus_iff)

lemma ereal_less_minus_iff:
  fixes x y z :: ereal
  shows "x < z - y 
    (y =   z =   x  ) 
    (y = -  x  ) 
    (¦y¦   x + y < z)"
  by (cases rule: ereal3_cases[of x y z]) auto

lemma ereal_less_minus:
  fixes x y z :: ereal
  shows "¦y¦    x < z - y  x + y < z"
  by (auto simp: ereal_less_minus_iff)

lemma ereal_le_minus_iff:
  fixes x y z :: ereal
  shows "x  z - y  (y =   z    x = -)  (¦y¦    x + y  z)"
  by (cases rule: ereal3_cases[of x y z]) auto

lemma ereal_le_minus:
  fixes x y z :: ereal
  shows "¦y¦    x  z - y  x + y  z"
  by (auto simp: ereal_le_minus_iff)

lemma ereal_minus_less_iff:
  fixes x y z :: ereal
  shows "x - y < z  y  -  (y =   x    z  -)  (y    x < z + y)"
  by (cases rule: ereal3_cases[of x y z]) auto

lemma ereal_minus_less:
  fixes x y z :: ereal
  shows "¦y¦    x - y < z  x < z + y"
  by (auto simp: ereal_minus_less_iff)

lemma ereal_minus_le_iff:
  fixes x y z :: ereal
  shows "x - y  z 
    (y = -  z = ) 
    (y =   x =   z = ) 
    (¦y¦    x  z + y)"
  by (cases rule: ereal3_cases[of x y z]) auto

lemma ereal_minus_le:
  fixes x y z :: ereal
  shows "¦y¦    x - y  z  x  z + y"
  by (auto simp: ereal_minus_le_iff)

lemma ereal_minus_eq_minus_iff:
  fixes a b c :: ereal
  shows "a - b = a - c 
    b = c  a =   (a = -  b  -  c  -)"
  by (cases rule: ereal3_cases[of a b c]) auto

lemma ereal_add_le_add_iff:
  fixes a b c :: ereal
  shows "c + a  c + b 
    a  b  c =   (c = -  a    b  )"
  by (cases rule: ereal3_cases[of a b c]) (simp_all add: field_simps)

lemma ereal_add_le_add_iff2:
  fixes a b c :: ereal
  shows "a + c  b + c  a  b  c =   (c = -  a    b  )"
by(cases rule: ereal3_cases[of a b c])(simp_all add: field_simps)

lemma ereal_mult_le_mult_iff:
  fixes a b c :: ereal
  shows "¦c¦    c * a  c * b  (0 < c  a  b)  (c < 0  b  a)"
  by (cases rule: ereal3_cases[of a b c]) (simp_all add: mult_le_cancel_left)

lemma ereal_minus_mono:
  fixes A B C D :: ereal assumes "A  B" "D  C"
  shows "A - C  B - D"
  using assms
  by (cases rule: ereal3_cases[case_product ereal_cases, of A B C D]) simp_all

lemma ereal_mono_minus_cancel:
  fixes a b c :: ereal
  shows "c - a  c - b  0  c  c <   b  a"
  by (cases a b c rule: ereal3_cases) auto

lemma real_of_ereal_minus:
  fixes a b :: ereal
  shows "real_of_ereal (a - b) = (if ¦a¦ =   ¦b¦ =  then 0 else real_of_ereal a - real_of_ereal b)"
  by (cases rule: ereal2_cases[of a b]) auto

lemma real_of_ereal_minus': "¦x¦ =   ¦y¦ =   real_of_ereal x - real_of_ereal y = real_of_ereal (x - y :: ereal)"
by(subst real_of_ereal_minus) auto

lemma ereal_diff_positive:
  fixes a b :: ereal shows "a  b  0  b - a"
  by (cases rule: ereal2_cases[of a b]) auto

lemma ereal_between:
  fixes x e :: ereal
  assumes "¦x¦  "
    and "0 < e"
  shows "x - e < x"
    and "x < x + e"
  using assms  by (cases x, cases e, auto)+

lemma ereal_minus_eq_PInfty_iff:
  fixes x y :: ereal
  shows "x - y =   y = -  x = "
  by (cases x y rule: ereal2_cases) simp_all

lemma ereal_diff_add_eq_diff_diff_swap:
  fixes x y z :: ereal
  shows "¦y¦    x - (y + z) = x - y - z"
  by(cases x y z rule: ereal3_cases) simp_all

lemma ereal_diff_add_assoc2:
  fixes x y z :: ereal
  shows "x + y - z = x - z + y"
  by(cases x y z rule: ereal3_cases) simp_all

lemma ereal_add_uminus_conv_diff: fixes x y z :: ereal shows "- x + y = y - x"
  by(cases x y rule: ereal2_cases) simp_all

lemma ereal_minus_diff_eq:
  fixes x y :: ereal
  shows " x =   y  ; x = -  y  -    - (x - y) = y - x"
  by(cases x y rule: ereal2_cases) simp_all

lemma ediff_le_self [simp]: "x - y  (x :: enat)"
  by(cases x y rule: enat.exhaust[case_product enat.exhaust]) simp_all

lemma ereal_abs_diff:
  fixes a b::ereal
  shows "abs(a-b)  abs a + abs b"
  by (cases rule: ereal2_cases[of a b]) (auto)


subsubsection ‹Division›

instantiation ereal :: inverse
begin

function inverse_ereal where
  "inverse (ereal r) = (if r = 0 then  else ereal (inverse r))"
| "inverse (::ereal) = 0"
| "inverse (-::ereal) = 0"
  by (auto intro: ereal_cases)
termination by (relation "{}") simp

definition "x div y = x * inverse (y :: ereal)"

instance ..

end

lemma real_of_ereal_inverse[simp]:
  fixes a :: ereal
  shows "real_of_ereal (inverse a) = 1 / real_of_ereal a"
  by (cases a) (auto simp: inverse_eq_divide)

lemma ereal_inverse[simp]:
  "inverse (0::ereal) = "
  "inverse (1::ereal) = 1"
  by (simp_all add: one_ereal_def zero_ereal_def)

lemma ereal_divide[simp]:
  "ereal r / ereal p = (if p = 0 then ereal r *  else ereal (r / p))"
  unfolding divide_ereal_def by (auto simp: divide_real_def)

lemma ereal_divide_same[simp]:
  fixes x :: ereal
  shows "x / x = (if ¦x¦ =   x = 0 then 0 else 1)"
  by (cases x) (simp_all add: divide_real_def divide_ereal_def one_ereal_def)

lemma ereal_inv_inv[simp]:
  fixes x :: ereal
  shows "inverse (inverse x) = (if x  - then x else )"
  by (cases x) auto

lemma ereal_inverse_minus[simp]:
  fixes x :: ereal
  shows "inverse (- x) = (if x = 0 then  else -inverse x)"
  by (cases x) simp_all

lemma ereal_uminus_divide[simp]:
  fixes x y :: ereal
  shows "- x / y = - (x / y)"
  unfolding divide_ereal_def by simp

lemma ereal_divide_Infty[simp]:
  fixes x :: ereal
  shows "x /  = 0" "x / - = 0"
  unfolding divide_ereal_def by simp_all

lemma ereal_divide_one[simp]: "x / 1 = (x::ereal)"
  unfolding divide_ereal_def by simp

lemma ereal_divide_ereal[simp]: " / ereal r = (if 0  r then  else -)"
  unfolding divide_ereal_def by simp

lemma ereal_inverse_nonneg_iff: "0  inverse (x :: ereal)  0  x  x = -"
  by (cases x) auto

lemma inverse_ereal_ge0I: "0  (x :: ereal)  0  inverse x"
by(cases x) simp_all

lemma zero_le_divide_ereal[simp]:
  fixes a :: ereal
  assumes "0  a"
    and "0  b"
  shows "0  a / b"
  using assms by (cases rule: ereal2_cases[of a b]) (auto simp: zero_le_divide_iff)

lemma ereal_le_divide_pos:
  fixes x y z :: ereal
  shows "x > 0  x    y  z / x  x * y  z"
  by (cases rule: ereal3_cases[of x y z]) (auto simp: field_simps)

lemma ereal_divide_le_pos:
  fixes x y z :: ereal
  shows "x > 0  x    z / x  y  z  x * y"
  by (cases rule: ereal3_cases[of x y z]) (auto simp: field_simps)

lemma ereal_le_divide_neg:
  fixes x y z :: ereal
  shows "x < 0  x  -  y  z / x  z  x * y"
  by (cases rule: ereal3_cases[of x y z]) (auto simp: field_simps)

lemma ereal_divide_le_neg:
  fixes x y z :: ereal
  shows "x < 0  x  -  z / x  y  x * y  z"
  by (cases rule: ereal3_cases[of x y z]) (auto simp: field_simps)

lemma ereal_inverse_antimono_strict:
  fixes x y :: ereal
  shows "0  x  x < y  inverse y < inverse x"
  by (cases rule: ereal2_cases[of x y]) auto

lemma ereal_inverse_antimono:
  fixes x y :: ereal
  shows "0  x  x  y  inverse y  inverse x"
  by (cases rule: ereal2_cases[of x y]) auto

lemma inverse_inverse_Pinfty_iff[simp]:
  fixes x :: ereal
  shows "inverse x =   x = 0"
  by (cases x) auto

lemma ereal_inverse_eq_0:
  fixes x :: ereal
  shows "inverse x = 0  x =   x = -"
  by (cases x) auto

lemma ereal_0_gt_inverse:
  fixes x :: ereal
  shows "0 < inverse x  x    0  x"
  by (cases x) auto

lemma ereal_inverse_le_0_iff:
  fixes x :: ereal
  shows "inverse x  0  x < 0  x = "
  by(cases x) auto

lemma ereal_divide_eq_0_iff: "x / y = 0  x = 0  ¦y :: ereal¦ = "
by(cases x y rule: ereal2_cases) simp_all

lemma ereal_mult_less_right:
  fixes a b c :: ereal
  assumes "b * a < c * a"
    and "0 < a"
    and "a < "
  shows "b < c"
  using assms
  by (cases rule: ereal3_cases[of a b c])
     (auto split: if_split_asm simp: zero_less_mult_iff zero_le_mult_iff)

lemma ereal_mult_divide: fixes a b :: ereal shows "0 < b  b <   b * (a / b) = a"
  by (cases a b rule: ereal2_cases) auto

lemma ereal_power_divide:
  fixes x y :: ereal
  shows "y  0  (x / y) ^ n = x^n / y^n"
  by (cases rule: ereal2_cases [of x y])
     (auto simp: one_ereal_def zero_ereal_def power_divide zero_le_power_eq)

lemma ereal_le_mult_one_interval:
  fixes x y :: ereal
  assumes y: "y  -"
  assumes z: "z. 0 < z  z < 1  z * x  y"
  shows "x  y"
proof (cases x)
  case PInf
  with z[of "1 / 2"] show "x  y"
    by (simp add: one_ereal_def)
next
  case (real r)
  note r = this
  show "x  y"
  proof (cases y)
    case (real p)
    note p = this
    have "r  p"
    proof (rule field_le_mult_one_interval)
      fix z :: real
      assume "0 < z" and "z < 1"
      with z[of "ereal z"] show "z * r  p"
        using p r by (auto simp: zero_le_mult_iff one_ereal_def)
    qed
    then show "x  y"
      using p r by simp
  qed (insert y, simp_all)
qed simp

lemma ereal_divide_right_mono[simp]:
  fixes x y z :: ereal
  assumes "x  y"
    and "0 < z"
  shows "x / z  y / z"
  using assms by (cases x y z rule: ereal3_cases) (auto intro: divide_right_mono)

lemma ereal_divide_left_mono[simp]:
  fixes x y z :: ereal
  assumes "y  x"
    and "0 < z"
    and "0 < x * y"
  shows "z / x  z / y"
  using assms
  by (cases x y z rule: ereal3_cases)
     (auto intro: divide_left_mono simp: field_simps zero_less_mult_iff mult_less_0_iff split: if_split_asm)

lemma ereal_divide_zero_left[simp]:
  fixes a :: ereal
  shows "0 / a = 0"
  by (cases a) (auto simp: zero_ereal_def)

lemma ereal_times_divide_eq_left[simp]:
  fixes a b c :: ereal
  shows "b / c * a = b * a / c"
  by (cases a b c rule: ereal3_cases) (auto simp: field_simps zero_less_mult_iff mult_less_0_iff)

lemma ereal_times_divide_eq: "a * (b / c :: ereal) = a * b / c"
  by (cases a b c rule: ereal3_cases)
     (auto simp: field_simps zero_less_mult_iff)

lemma ereal_inverse_real [simp]: "¦z¦    z  0  ereal (inverse (real_of_ereal z)) = inverse z"
  by auto

lemma ereal_inverse_mult:
  "a  0  b  0  inverse (a * (b::ereal)) = inverse a * inverse b"
  by (cases a; cases b) auto

lemma inverse_eq_infinity_iff_eq_zero [simp]:
  "1/(x::ereal) =   x = 0"
by (simp add: divide_ereal_def)

lemma ereal_distrib_left:
  fixes a b c :: ereal
  assumes "a    b  -"
    and "a  -  b  "
    and "¦c¦  "
  shows "c * (a + b) = c * a + c * b"
using assms
by (cases rule: ereal3_cases[of a b c]) (simp_all add: field_simps)

lemma ereal_distrib_minus_left:
  fixes a b c :: ereal
  assumes "a    b  "
    and "a  -  b  -"
    and "¦c¦  "
  shows "c * (a - b) = c * a - c * b"
using assms
by (cases rule: ereal3_cases[of a b c]) (simp_all add: field_simps)

lemma ereal_distrib_minus_right:
  fixes a b c :: ereal
  assumes "a    b  "
    and "a  -  b  -"
    and "¦c¦  "
  shows "(a - b) * c = a * c - b * c"
using assms
by (cases rule: ereal3_cases[of a b c]) (simp_all add: field_simps)


subsection "Complete lattice"

instantiation ereal :: lattice
begin

definition [simp]: "sup x y = (max x y :: ereal)"
definition [simp]: "inf x y = (min x y :: ereal)"
instance by standard simp_all

end

instantiation ereal :: complete_lattice
begin

definition "bot = (-::ereal)"
definition "top = (::ereal)"

definition "Sup S = (SOME x :: ereal. (yS. y  x)  (z. (yS. y  z)  x  z))"
definition "Inf S = (SOME x :: ereal. (yS. x  y)  (z. (yS. z  y)  z  x))"

lemma ereal_complete_Sup:
  fixes S :: "ereal set"
  shows "x. (yS. y  x)  (z. (yS. y  z)  x  z)"
proof (cases "x. aS. a  ereal x")
  case True
  then obtain y where y: "a  ereal y" if "aS" for a
    by auto
  then have "  S"
    by force
  show ?thesis
  proof (cases "S  {-}  S  {}")
    case True
    with   S obtain x where x: "x  S" "¦x¦  "
      by auto
    obtain s where s: "xereal -` S. x  s" "(xereal -` S. x  z)  s  z" for z
    proof (atomize_elim, rule complete_real)
      show "x. x  ereal -` S"
        using x by auto
      show "z. xereal -` S. x  z"
        by (auto dest: y intro!: exI[of _ y])
    qed
    show ?thesis
    proof (safe intro!: exI[of _ "ereal s"])
      fix y
      assume "y  S"
      with s   S show "y  ereal s"
        by (cases y) auto
    next
      fix z
      assume "yS. y  z"
      with S  {-}  S  {} show "ereal s  z"
        by (cases z) (auto intro!: s)
    qed
  next
    case False
    then show ?thesis
      by (auto intro!: exI[of _ "-"])
  qed
next
  case False
  then show ?thesis
    by (fastforce intro!: exI[of _ ] ereal_top intro: order_trans dest: less_imp_le simp: not_le)
qed

lemma ereal_complete_uminus_eq:
  fixes S :: "ereal set"
  shows "(yuminus`S. y  x)  (z. (yuminus`S. y  z)  x  z)
      (yS. -x  y)  (z. (yS. z  y)  z  -x)"
  by simp (metis ereal_minus_le_minus ereal_uminus_uminus)

lemma ereal_complete_Inf:
  "x. (yS::ereal set. x  y)  (z. (yS. z  y)  z  x)"
  using ereal_complete_Sup[of "uminus ` S"]
  unfolding ereal_complete_uminus_eq
  by auto

instance
proof
  show "Sup {} = (bot::ereal)"
    using ereal_bot by (auto simp: bot_ereal_def Sup_ereal_def)
  show "Inf {} = (top::ereal)"
    unfolding top_ereal_def Inf_ereal_def
    using ereal_infty_less_eq(1) ereal_less_eq(1) by blast
qed (auto intro: someI2_ex ereal_complete_Sup ereal_complete_Inf
  simp: Sup_ereal_def Inf_ereal_def bot_ereal_def top_ereal_def)

end

instance ereal :: complete_linorder ..

instance ereal :: linear_continuum
proof
  show "a b::ereal. a  b"
    using zero_neq_one by blast
qed

lemma min_PInf [simp]: "min (::ereal) x = x"
  by (metis min_top top_ereal_def)

lemma min_PInf2 [simp]: "min x (::ereal) = x"
  by (metis min_top2 top_ereal_def)

lemma max_PInf [simp]: "max (::ereal) x = "
  by (metis max_top top_ereal_def)

lemma max_PInf2 [simp]: "max x (::ereal) = "
  by (metis max_top2 top_ereal_def)

lemma min_MInf [simp]: "min (-::ereal) x = -"
  by (metis min_bot bot_ereal_def)

lemma min_MInf2 [simp]: "min x (-::ereal) = -"
  by (metis min_bot2 bot_ereal_def)

lemma max_MInf [simp]: "max (-::ereal) x = x"
  by (metis max_bot bot_ereal_def)

lemma max_MInf2 [simp]: "max x (-::ereal) = x"
  by (metis max_bot2 bot_ereal_def)

subsection ‹Extended real intervals›

lemma real_greaterThanLessThan_infinity_eq:
  "real_of_ereal ` {N::ereal<..<} =
    (if N =  then {} else if N = - then UNIV else {real_of_ereal N<..})"
  by (force simp: real_less_ereal_iff intro!: image_eqI[where x="ereal _"] elim!: less_ereal.elims)

lemma real_greaterThanLessThan_minus_infinity_eq:
  "real_of_ereal ` {-<..<N::ereal} =
    (if N =  then UNIV else if N = - then {} else {..<real_of_ereal N})"
proof -
  have "real_of_ereal ` {-<..<N::ereal} = uminus ` real_of_ereal ` {-N<..<}"
    by (auto simp: ereal_uminus_less_reorder intro!: image_eqI[where x="-x" for x])
  also note real_greaterThanLessThan_infinity_eq
  finally show ?thesis by (auto intro!: image_eqI[where x="-x" for x])
qed

lemma real_greaterThanLessThan_inter:
  "real_of_ereal ` {N<..<M::ereal} = real_of_ereal ` {-<..<M}  real_of_ereal ` {N<..<}"
  by (force elim!: less_ereal.elims)

lemma real_atLeastGreaterThan_eq: "real_of_ereal ` {N<..<M::ereal} =
   (if N =  then {} else
   if N = - then
    (if M =  then UNIV
    else if M = - then {}
    else {..< real_of_ereal M})
  else if M = -  then {}
  else if M =  then {real_of_ereal N<..}
  else {real_of_ereal N <..< real_of_ereal M})"
proof (cases "M = -  M =   N = -  N = ")
  case True
  then show ?thesis
    by (auto simp: real_greaterThanLessThan_minus_infinity_eq real_greaterThanLessThan_infinity_eq )
next
  case False
  then obtain p q where "M = ereal p" "N = ereal q"
    by (metis MInfty_eq_minfinity ereal.distinct(3) uminus_ereal.elims)
  moreover have "x. q < x; x < p  x  real_of_ereal ` {ereal q<..<ereal p}"
    by (metis greaterThanLessThan_iff imageI less_ereal.simps(1) real_of_ereal.simps(1))
  ultimately show ?thesis 
    by (auto elim!: less_ereal.elims)
qed

lemma real_image_ereal_ivl:
  fixes a b::ereal
  shows
  "real_of_ereal ` {a<..<b} =
  (if a < b then (if a = -  then if b =  then UNIV else {..<real_of_ereal b}
  else if b =  then {real_of_ereal a<..} else {real_of_ereal a <..< real_of_ereal b}) else {})"
  by (cases a; cases b; simp add: real_atLeastGreaterThan_eq not_less)

lemma fixes a b c::ereal
  shows not_inftyI: "a < b  b < c  abs b  "
  by force

lemma
  interval_neqs:
  fixes r s t::real
  shows "{r<..<s}  {t<..}"
    and "{r<..<s}  {..<t}"
    and "{r<..<ra}  UNIV"
    and "{r<..}  {..<s}"
    and "{r<..}  UNIV"
    and "{..<r}  UNIV"
    and "{}  {r<..}"
    and "{}  {..<r}"
  subgoal
    by (metis dual_order.strict_trans greaterThanLessThan_iff greaterThan_iff gt_ex not_le order_refl)
  subgoal
    by (metis (no_types, opaque_lifting) greaterThanLessThan_empty_iff greaterThanLessThan_iff gt_ex
        lessThan_iff minus_minus neg_less_iff_less not_less order_less_irrefl)
  subgoal by force
  subgoal
    by (metis greaterThanLessThan_empty_iff greaterThanLessThan_eq greaterThan_iff inf.idem
        lessThan_iff lessThan_non_empty less_irrefl not_le)
  subgoal by force
  subgoal by force
  subgoal using greaterThan_non_empty by blast
  subgoal using lessThan_non_empty by blast
  done

lemma greaterThanLessThan_eq_iff:
  fixes r s t u::real
  shows "({r<..<s} = {t<..<u}) = (r  s  u  t  r = t  s = u)"
  by (metis cInf_greaterThanLessThan cSup_greaterThanLessThan greaterThanLessThan_empty_iff not_le)

lemma real_of_ereal_image_greaterThanLessThan_iff:
  "real_of_ereal ` {a <..< b} = real_of_ereal ` {c <..< d}  (a  b  c  d  a = c  b = d)"
  unfolding real_atLeastGreaterThan_eq
  by (cases a; cases b; cases c; cases d;
    simp add: greaterThanLessThan_eq_iff interval_neqs interval_neqs[symmetric])

lemma uminus_image_real_of_ereal_image_greaterThanLessThan:
  "uminus ` real_of_ereal ` {l <..< u} = real_of_ereal ` {-u <..< -l}"
  by (force simp: algebra_simps ereal_less_uminus_reorder
    ereal_uminus_less_reorder intro: image_eqI[where x="-x" for x])

lemma add_image_real_of_ereal_image_greaterThanLessThan:
  "(+) c ` real_of_ereal ` {l <..< u} = real_of_ereal ` {c + l <..< c + u}"
  apply safe
  subgoal for x
    using ereal_less_add[of c]
    by (force simp: real_of_ereal_add add.commute)
  subgoal for _ x
    by (force simp: add.commute real_of_ereal_minus ereal_minus_less ereal_less_minus
      intro: image_eqI[where x="x - c"])
  done

lemma add2_image_real_of_ereal_image_greaterThanLessThan:
  "(λx. x + c) ` real_of_ereal ` {l <..< u} = real_of_ereal ` {l + c <..< u + c}"
  using add_image_real_of_ereal_image_greaterThanLessThan[of c l u]
  by (metis add.commute image_cong)

lemma minus_image_real_of_ereal_image_greaterThanLessThan:
  "(-) c ` real_of_ereal ` {l <..< u} = real_of_ereal ` {c - u <..< c - l}"
  (is "?l = ?r")
proof -
  have "?l = (+) c ` uminus ` real_of_ereal ` {l <..< u}" by auto
  also note uminus_image_real_of_ereal_image_greaterThanLessThan
  also note add_image_real_of_ereal_image_greaterThanLessThan
  finally show ?thesis by (simp add: minus_ereal_def)
qed

lemma real_ereal_bound_lemma_up:
  assumes "s  real_of_ereal ` {a<..<b}"
  assumes "t  real_of_ereal ` {a<..<b}"
  assumes "s  t"
  shows "b  "
proof (cases b)
  case PInf
  then show ?thesis
    using assms
    apply clarsimp
    by (metis UNIV_I assms(1) ereal_less_PInfty greaterThan_iff less_eq_ereal_def less_le_trans real_image_ereal_ivl)
qed auto

lemma real_ereal_bound_lemma_down:
  assumes s: "s  real_of_ereal ` {a<..<b}"
  and t: "t  real_of_ereal ` {a<..<b}"
  and "t  s"
  shows "a  - "
proof (cases b)
  case (real r)
  then show ?thesis
    using assms real_greaterThanLessThan_minus_infinity_eq by force
next
  case PInf
  then show ?thesis
    using t real_greaterThanLessThan_infinity_eq by auto
next
  case MInf
  then show ?thesis
    using s by auto
qed


subsection "Topological space"

instantiation ereal :: linear_continuum_topology
begin

definition "open_ereal" :: "ereal set  bool" where
  open_ereal_generated: "open_ereal = generate_topology (range lessThan  range greaterThan)"

instance
  by standard (simp add: open_ereal_generated)

end

lemma continuous_on_ereal[continuous_intros]:
  assumes f: "continuous_on s f" shows "continuous_on s (λx. ereal (f x))"
  by (rule continuous_on_compose2 [OF continuous_onI_mono[of ereal UNIV] f]) auto

lemma tendsto_ereal[tendsto_intros, simp, intro]: "(f  x) F  ((λx. ereal (f x))  ereal x) F"
  using isCont_tendsto_compose[of x ereal f F] continuous_on_ereal[of UNIV "λx. x"]
  by (simp add: continuous_on_eq_continuous_at)

lemma tendsto_uminus_ereal[tendsto_intros, simp, intro]:
  assumes "(f  x) F"
  shows "((λx. - f x::ereal)  - x) F"
proof (rule tendsto_compose[OF order_tendstoI assms])
  show "a. a < - x  F x in at x. a < - x"
    by (metis ereal_less_uminus_reorder eventually_at_topological lessThan_iff open_lessThan)
  show "a. - x < a  F x in at x. - x < a"
    by (metis ereal_uminus_reorder(2) eventually_at_topological greaterThan_iff open_greaterThan)
qed

lemma at_infty_ereal_eq_at_top: "at  = filtermap ereal at_top"
  unfolding filter_eq_iff eventually_at_filter eventually_at_top_linorder eventually_filtermap
    top_ereal_def[symmetric]
  apply (subst eventually_nhds_top[of 0])
  apply (auto simp: top_ereal_def less_le ereal_all_split ereal_ex_split)
  apply (metis PInfty_neq_ereal(2) ereal_less_eq(3) ereal_top le_cases order_trans)
  done

lemma ereal_Lim_uminus: "(f  f0) net  ((λx. - f x::ereal)  - f0) net"
  using tendsto_uminus_ereal[of f f0 net] tendsto_uminus_ereal[of "λx. - f x" "- f0" net]
  by auto

lemma ereal_divide_less_iff: "0 < (c::ereal)  c <   a / c < b  a < b * c"
  by (cases a b c rule: ereal3_cases) (auto simp: field_simps)

lemma ereal_less_divide_iff: "0 < (c::ereal)  c <   a < b / c  a * c < b"
  by (cases a b c rule: ereal3_cases) (auto simp: field_simps)

lemma tendsto_cmult_ereal[tendsto_intros, simp, intro]:
  assumes c: "¦c¦  " and f: "(f  x) F" shows "((λx. c * f x::ereal)  c * x) F"
proof -
  { fix c :: ereal assume "0 < c" "c < "
    then have "((λx. c * f x::ereal)  c * x) F"
      apply (intro tendsto_compose[OF _ f])
      apply (auto intro!: order_tendstoI simp: eventually_at_topological)
      apply (rule_tac x="{a/c <..}" in exI)
      apply (auto split: ereal.split simp: ereal_divide_less_iff mult.commute) []
      apply (rule_tac x="{..< a/c}" in exI)
      apply (auto split: ereal.split simp: ereal_less_divide_iff mult.commute) []
      done }
  note * = this

  have "((0 < c  c < )  (- < c  c < 0)  c = 0)"
    using c by (cases c) auto
  then show ?thesis
  proof (elim disjE conjE)
    assume "-  < c" "c < 0"
    then have "0 < - c" "- c < "
      by (auto simp: ereal_uminus_reorder ereal_less_uminus_reorder[of 0])
    then have "((λx. (- c) * f x)  (- c) * x) F"
      by (rule *)
    from tendsto_uminus_ereal[OF this] show ?thesis
      by simp
  qed (auto intro!: *)
qed

lemma tendsto_cmult_ereal_not_0[tendsto_intros, simp, intro]:
  assumes "x  0" and f: "(f  x) F" shows "((λx. c * f x::ereal)  c * x) F"
proof cases
  assume "¦c¦ = "
  show ?thesis
  proof (rule filterlim_cong[THEN iffD1, OF refl refl _ tendsto_const])
    have "0 < x  x < 0"
      using x  0 by (auto simp add: neq_iff)
    then show "eventually (λx'. c * x = c * f x') F"
    proof
      assume "0 < x" from order_tendstoD(1)[OF f this] show ?thesis
        by eventually_elim (insert 0<x ¦c¦ = , auto)
    next
      assume "x < 0" from order_tendstoD(2)[OF f this] show ?thesis
        by eventually_elim (insert x<0 ¦c¦ = , auto)
    qed
  qed
qed (rule tendsto_cmult_ereal[OF _ f])

lemma tendsto_cadd_ereal[tendsto_intros, simp, intro]:
  assumes c: "y  - " "x  - " and f: "(f  x) F" shows "((λx. f x + y::ereal)  x + y) F"
  apply (intro tendsto_compose[OF _ f])
  apply (auto intro!: order_tendstoI simp: eventually_at_topological)
  apply (rule_tac x="{a - y <..}" in exI)
  apply (auto split: ereal.split simp: ereal_minus_less_iff c) []
  apply (rule_tac x="{..< a - y}" in exI)
  apply (auto split: ereal.split simp: ereal_less_minus_iff c) []
  done

lemma tendsto_add_left_ereal[tendsto_intros, simp, intro]:
  assumes c: "¦y¦  " and f: "(f  x) F" shows "((λx. f x + y::ereal)  x + y) F"
  apply (intro tendsto_compose[OF _ f])
  apply (auto intro!: order_tendstoI simp: eventually_at_topological)
  apply (rule_tac x="{a - y <..}" in exI)
  apply (insert c, auto split: ereal.split simp: ereal_minus_less_iff) []
  apply (rule_tac x="{..< a - y}" in exI)
  apply (auto split: ereal.split simp: ereal_less_minus_iff c) []
  done

lemma continuous_at_ereal[continuous_intros]: "continuous F f  continuous F (λx. ereal (f x))"
  unfolding continuous_def by auto

lemma ereal_Sup:
  assumes *: "¦SUP aA. ereal a¦  "
  shows "ereal (Sup A) = (SUP aA. ereal a)"
proof (rule continuous_at_Sup_mono)
  obtain r where r: "ereal r = (SUP aA. ereal a)" "A  {}"
    using * by (force simp: bot_ereal_def)
  then show "bdd_above A" "A  {}"
    by (auto intro!: SUP_upper bdd_aboveI[of _ r] simp flip: ereal_less_eq)
qed (auto simp: mono_def continuous_at_imp_continuous_at_within continuous_at_ereal)

lemma ereal_SUP: "¦SUP aA. ereal (f a)¦    ereal (SUP aA. f a) = (SUP aA. ereal (f a))"
  by (simp add: ereal_Sup image_comp)

lemma ereal_Inf:
  assumes *: "¦INF aA. ereal a¦  "
  shows "ereal (Inf A) = (INF aA. ereal a)"
proof (rule continuous_at_Inf_mono)
  obtain r where r: "ereal r = (INF aA. ereal a)" "A  {}"
    using * by (force simp: top_ereal_def)
  then show "bdd_below A" "A  {}"
    by (auto intro!: INF_lower bdd_belowI[of _ r] simp flip: ereal_less_eq)
qed (auto simp: mono_def continuous_at_imp_continuous_at_within continuous_at_ereal)

lemma ereal_Inf':
  assumes *: "bdd_below A" "A  {}"
  shows "ereal (Inf A) = (INF aA. ereal a)"
proof (rule ereal_Inf)
  from * obtain l u where "x  A  l  x" "u  A" for x
    by (auto simp: bdd_below_def)
  then have "l  (INF xA. ereal x)" "(INF xA. ereal x)  u"
    by (auto intro!: INF_greatest INF_lower)
  then show "¦INF aA. ereal a¦  "
    by auto
qed

lemma ereal_INF: "¦INF aA. ereal (f a)¦    ereal (INF aA. f a) = (INF aA. ereal (f a))"
  by (simp add: ereal_Inf image_comp)

lemma ereal_Sup_uminus_image_eq: "Sup (uminus ` S::ereal set) = - Inf S"
  by (auto intro!: SUP_eqI
           simp: Ball_def[symmetric] ereal_uminus_le_reorder le_Inf_iff
           intro!: complete_lattice_class.Inf_lower2)

lemma ereal_SUP_uminus_eq:
  fixes f :: "'a  ereal"
  shows "(SUP xS. uminus (f x)) = - (INF xS. f x)"
  using ereal_Sup_uminus_image_eq [of "f ` S"] by (simp add: image_comp)

lemma ereal_inj_on_uminus[intro, simp]: "inj_on uminus (A :: ereal set)"
  by (auto intro!: inj_onI)

lemma ereal_Inf_uminus_image_eq: "Inf (uminus ` S::ereal set) = - Sup S"
  using ereal_Sup_uminus_image_eq[of "uminus ` S"] by simp

lemma ereal_INF_uminus_eq:
  fixes f :: "'a  ereal"
  shows "(INF xS. - f x) = - (SUP xS. f x)"
  using ereal_Inf_uminus_image_eq [of "f ` S"] by (simp add: image_comp)

lemma ereal_SUP_uminus:
  fixes f :: "'a  ereal"
  shows "(SUP i  R. - f i) = - (INF i  R. f i)"
  using ereal_Sup_uminus_image_eq[of "f`R"]
  by (simp add: image_image)

lemma ereal_SUP_not_infty:
  fixes f :: "_  ereal"
  shows "A  {}  l  -  u    aA. l  f a  f a  u  ¦Sup (f ` A)¦  "
  using SUP_upper2[of _ A l f] SUP_least[of A f u]
  by (cases "Sup (f ` A)") auto

lemma ereal_INF_not_infty:
  fixes f :: "_  ereal"
  shows "A  {}  l  -  u    aA. l  f a  f a  u  ¦Inf (f ` A)¦  "
  using INF_lower2[of _ A f u] INF_greatest[of A l f]
  by (cases "Inf (f ` A)") auto

lemma ereal_image_uminus_shift:
  fixes X Y :: "ereal set"
  shows "uminus ` X = Y  X = uminus ` Y"
proof
  assume "uminus ` X = Y"
  then have "uminus ` uminus ` X = uminus ` Y"
    by (simp add: inj_image_eq_iff)
  then show "X = uminus ` Y"
    by (simp add: image_image)
qed (simp add: image_image)

lemma Sup_eq_MInfty:
  fixes S :: "ereal set"
  shows "Sup S = -  S = {}  S = {-}"
  unfolding bot_ereal_def[symmetric] by auto

lemma Inf_eq_PInfty:
  fixes S :: "ereal set"
  shows "Inf S =   S = {}  S = {}"
  using Sup_eq_MInfty[of "uminus`S"]
  unfolding ereal_Sup_uminus_image_eq ereal_image_uminus_shift by simp

lemma Inf_eq_MInfty:
  fixes S :: "ereal set"
  shows "-  S  Inf S = -"
  unfolding bot_ereal_def[symmetric] by auto

lemma Sup_eq_PInfty:
  fixes S :: "ereal set"
  shows "  S  Sup S = "
  unfolding top_ereal_def[symmetric] by auto

lemma not_MInfty_nonneg[simp]: "0  (x::ereal)  x  - "
  by auto

lemma Sup_ereal_close:
  fixes e :: ereal
  assumes "0 < e"
    and S: "¦Sup S¦  " "S  {}"
  shows "xS. Sup S - e < x"
  using assms by (cases e) (auto intro!: less_Sup_iff[THEN iffD1])

lemma Inf_ereal_close:
  fixes e :: ereal
  assumes "¦Inf X¦  "
    and "0 < e"
  shows "xX. x < Inf X + e"
proof (rule Inf_less_iff[THEN iffD1])
  show "Inf X < Inf X + e"
    using assms by (cases e) auto
qed

lemma SUP_PInfty:
  "(n::nat. iA. ereal (real n)  f i)  (SUP iA. f i :: ereal) = "
  unfolding top_ereal_def[symmetric] SUP_eq_top_iff
  by (metis MInfty_neq_PInfty(2) PInfty_neq_ereal(2) less_PInf_Ex_of_nat less_ereal.elims(2) less_le_trans)

lemma SUP_nat_Infty: "(SUP i. ereal (real i)) = "
  by (rule SUP_PInfty) auto

lemma SUP_ereal_add_left:
  assumes "I  {}" "c  -"
  shows "(SUP iI. f i + c :: ereal) = (SUP iI. f i) + c"
proof (cases "(SUP iI. f i) = - ")
  case True
  then have "i. i  I  f i = -"
    unfolding Sup_eq_MInfty by auto
  with True show ?thesis
    by (cases c) (auto simp: I  {})
next
  case False
  then show ?thesis
    by (subst continuous_at_Sup_mono[where f="λx. x + c"])
      (auto simp: continuous_at_imp_continuous_at_within continuous_at mono_def add_mono I  {}
      c  - image_comp)
qed

lemma SUP_ereal_add_right:
  fixes c :: ereal
  shows "I  {}  c  -  (SUP iI. c + f i) = c + (SUP iI. f i)"
  using SUP_ereal_add_left[of I c f] by (simp add: add.commute)

lemma SUP_ereal_minus_right:
  assumes "I  {}" "c  -"
  shows "(SUP iI. c - f i :: ereal) = c - (INF iI. f i)"
  using SUP_ereal_add_right[OF assms, of "λi. - f i"]
  by (simp add: ereal_SUP_uminus minus_ereal_def)

lemma SUP_ereal_minus_left:
  assumes "I  {}" "c  "
  shows "(SUP iI. f i - c:: ereal) = (SUP iI. f i) - c"
  using SUP_ereal_add_left[OF I  {}, of "-c" f] by (simp add: c   minus_ereal_def)

lemma INF_ereal_minus_right:
  assumes "I  {}" and "¦c¦  "
  shows "(INF iI. c - f i) = c - (SUP iI. f i::ereal)"
proof -
  { fix b have "(-c) + b = - (c - b)"
      using ¦c¦   by (cases c b rule: ereal2_cases) auto }
  note * = this
  show ?thesis
    using SUP_ereal_add_right[OF I  {}, of "-c" f] ¦c¦  
    by (auto simp add: * ereal_SUP_uminus_eq)
qed

lemma SUP_ereal_le_addI:
  fixes f :: "'i  ereal"
  assumes "i. f i + y  z" and "y  -"
  shows "Sup (f ` UNIV) + y  z"
  unfolding SUP_ereal_add_left[OF UNIV_not_empty y  -, symmetric]
  by (rule SUP_least assms)+

lemma SUP_combine:
  fixes f :: "'a::semilattice_sup  'a::semilattice_sup  'b::complete_lattice"
  assumes mono: "a b c d. a  b  c  d  f a c  f b d"
  shows "(SUP iUNIV. SUP jUNIV. f i j) = (SUP i. f i i)"
proof (rule antisym)
  show "(SUP i j. f i j)  (SUP i. f i i)"
    by (rule SUP_least SUP_upper2[where i="sup i j" for i j] UNIV_I mono sup_ge1 sup_ge2)+
  show "(SUP i. f i i)  (SUP i j. f i j)"
    by (rule SUP_least SUP_upper2 UNIV_I mono order_refl)+
qed

lemma SUP_ereal_add:
  fixes f g :: "nat  ereal"
  assumes inc: "incseq f" "incseq g"
    and pos: "i. f i  -" "i. g i  -"
  shows "(SUP i. f i + g i) = Sup (f ` UNIV) + Sup (g ` UNIV)"
  apply (subst SUP_ereal_add_left[symmetric, OF UNIV_not_empty])
   apply (metis SUP_upper UNIV_I assms(4) ereal_infty_less_eq(2))
  apply (subst (2) add.commute)
  apply (subst SUP_ereal_add_left[symmetric, OF UNIV_not_empty assms(3)])
  apply (subst (2) add.commute)
  apply (rule SUP_combine[symmetric] add_mono inc[THEN monoD] | assumption)+
  done

lemma INF_eq_minf: "(INF iI. f i::ereal)  -  (b>-. iI. b  f i)"
  unfolding bot_ereal_def[symmetric] INF_eq_bot_iff by (auto simp: not_less)

lemma INF_ereal_add_left:
  assumes "I  {}" "c  -" "x. x  I  0  f x"
  shows "(INF iI. f i + c :: ereal) = (INF iI. f i) + c"
proof -
  have "(INF iI. f i)  -"
    unfolding INF_eq_minf using assms by (intro exI[of _ 0]) auto
  then show ?thesis
    by (subst continuous_at_Inf_mono[where f="λx. x + c"])
       (auto simp: mono_def add_mono I  {} c  - continuous_at_imp_continuous_at_within
        continuous_at image_comp)
qed

lemma INF_ereal_add_right:
  assumes "I  {}" "c  -" "x. x  I  0  f x"
  shows "(INF iI. c + f i :: ereal) = c + (INF iI. f i)"
  using INF_ereal_add_left[OF assms] by (simp add: ac_simps)

lemma INF_ereal_add_directed:
  fixes f g :: "'a  ereal"
  assumes nonneg: "i. i  I  0  f i" "i. i  I  0  g i"
  assumes directed: "i j. i  I  j  I  kI. f i + g j  f k + g k"
  shows "(INF iI. f i + g i) = (INF iI. f i) + (INF iI. g i)"
proof cases
  assume "I = {}" then show ?thesis
    by (simp add: top_ereal_def)
next
  assume "I  {}"
  show ?thesis
  proof (rule antisym)
    show "(INF iI. f i) + (INF iI. g i)  (INF iI. f i + g i)"
      by (rule INF_greatest; intro add_mono INF_lower)
  next
    have "(INF iI. f i + g i)  (INF iI. (INF jI. f i + g j))"
      using directed by (intro INF_greatest) (blast intro: INF_lower2)
    also have " = (INF iI. f i + (INF iI. g i))"
      using nonneg I  {} by (auto simp: INF_ereal_add_right)
    also have " = (INF iI. f i) + (INF iI. g i)"
      using nonneg by (intro INF_ereal_add_left I  {}) (auto simp: INF_eq_minf intro!: exI[of _ 0])
    finally show "(INF iI. f i + g i)  (INF iI. f i) + (INF iI. g i)" .
  qed
qed

lemma INF_ereal_add:
  fixes f :: "nat  ereal"
  assumes "decseq f" "decseq g"
    and fin: "i. f i  " "i. g i  "
  shows "(INF i. f i + g i) = Inf (f ` UNIV) + Inf (g ` UNIV)"
proof -
  have INF_less: "(INF i. f i) < " "(INF i. g i) < "
    using assms unfolding INF_less_iff by auto
  { fix a b :: ereal assume "a  " "b  "
    then have "- ((- a) + (- b)) = a + b"
      by (cases a b rule: ereal2_cases) auto }
  note * = this
  have "(INF i. f i + g i) = (INF i. - ((- f i) + (- g i)))"
    by (simp add: fin *)
  also have " = Inf (f ` UNIV) + Inf (g ` UNIV)"
    unfolding ereal_INF_uminus_eq
    using assms INF_less
    by (subst SUP_ereal_add) (auto simp: ereal_SUP_uminus fin *)
  finally show ?thesis .
qed

lemma SUP_ereal_add_pos:
  fixes f g :: "nat  ereal"
  assumes inc: "incseq f" "incseq g"
    and pos: "i. 0  f i" "i. 0  g i"
  shows "(SUP i. f i + g i) = Sup (f ` UNIV) + Sup (g ` UNIV)"
proof (intro SUP_ereal_add inc)
  fix i
  show "f i  -" "g i  -"
    using pos[of i] by auto
qed

lemma SUP_ereal_sum:
  fixes f g :: "'a  nat  ereal"
  assumes "n. n  A  incseq (f n)"
    and pos: "n i. n  A  0  f n i"
  shows "(SUP i. nA. f n i) = (nA. Sup ((f n) ` UNIV))"
proof (cases "finite A")
  case True
  then show ?thesis using assms
    by induct (auto simp: incseq_sumI2 sum_nonneg SUP_ereal_add_pos)
next
  case False
  then show ?thesis by simp
qed

lemma SUP_ereal_mult_left:
  fixes f :: "'a  ereal"
  assumes "I  {}"
  assumes f: "i. i  I  0  f i" and c: "0  c"
  shows "(SUP iI. c * f i) = c * (SUP iI. f i)"
proof (cases "(SUP i  I. f i) = 0")
  case True
  then have "i. i  I  f i = 0"
    by (metis SUP_upper f antisym)
  with True show ?thesis
    by simp
next
  case False
  then show ?thesis
    by (subst continuous_at_Sup_mono[where f="λx. c * x"])
       (auto simp: mono_def continuous_at continuous_at_imp_continuous_at_within I  {} image_comp
             intro!: ereal_mult_left_mono c)
qed

lemma countable_approach:
  fixes x :: ereal
  assumes "x  -"
  shows "f. incseq f  (i::nat. f i < x)  (f  x)"
proof (cases x)
  case (real r)
  moreover have "(λn. r - inverse (real (Suc n)))  r - 0"
    by (intro tendsto_intros LIMSEQ_inverse_real_of_nat)
  ultimately show ?thesis
    by (intro exI[of _ "λn. x - inverse (Suc n)"]) (auto simp: incseq_def)
next
  case PInf with LIMSEQ_SUP[of "λn::nat. ereal (real n)"] show ?thesis
    by (intro exI[of _ "λn. ereal (real n)"]) (auto simp: incseq_def SUP_nat_Infty)
qed (simp add: assms)

lemma Sup_countable_SUP:
  assumes "A  {}"
  shows "f::nat  ereal. incseq f  range f  A  Sup A = (SUP i. f i)"
proof cases
  assume "Sup A = -"
  with A  {} have "A = {-}"
    by (auto simp: Sup_eq_MInfty)
  then show ?thesis
    by (auto intro!: exI[of _ "λ_. -"] simp: bot_ereal_def)
next
  assume "Sup A  -"
  then obtain l where "incseq l" and l: "l i < Sup A" and l_Sup: "l  Sup A" for i :: nat
    by (auto dest: countable_approach)

  have "f. n. (f n  A  l n  f n)  (f n  f (Suc n))" (is "f. ?P f")
  proof (rule dependent_nat_choice)
    show "x. x  A  l 0  x"
      using l[of 0] by (auto simp: less_Sup_iff)
  next
    fix x n assume "x  A  l n  x"
    moreover from l[of "Suc n"] obtain y where "y  A" "l (Suc n) < y"
      by (auto simp: less_Sup_iff)
    ultimately show "y. (y  A  l (Suc n)  y)  x  y"
      by (auto intro!: exI[of _ "max x y"] split: split_max)
  qed
  then obtain f where f: "?P f" ..
  then have "range f  A" "incseq f"
    by (auto simp: incseq_Suc_iff)
  moreover
  have "(SUP i. f i) = Sup A"
  proof (rule tendsto_unique)
    show "f  (SUP i. f i)"
      by (rule LIMSEQ_SUP incseq f)+
    show "f  Sup A"
      using l f
      by (intro tendsto_sandwich[OF _ _ l_Sup tendsto_const])
         (auto simp: Sup_upper)
  qed simp
  ultimately show ?thesis
    by auto
qed

lemma Inf_countable_INF:
  assumes "A  {}" shows "f::nat  ereal. decseq f  range f  A  Inf A = (INF i. f i)"
proof -
  obtain f where "incseq f" "range f  uminus`A" "Sup (uminus`A) = (SUP i. f i)"
    using Sup_countable_SUP[of "uminus ` A"] A  {} by auto
  then show ?thesis
    by (intro exI[of _ "λx. - f x"])
       (auto simp: ereal_Sup_uminus_image_eq ereal_INF_uminus_eq eq_commute[of "- _"])
qed

lemma SUP_countable_SUP:
  "A  {}  f::nat  ereal. range f  g`A  Sup (g ` A) = Sup (f ` UNIV)"
  using Sup_countable_SUP [of "g`A"] by auto

subsection "Relation to typenat"

definition "ereal_of_enat n = (case n of enat n  ereal (real n) |   )"

declare [[coercion "ereal_of_enat :: enat  ereal"]]
declare [[coercion "(λn. ereal (real n)) :: nat  ereal"]]

lemma ereal_of_enat_simps[simp]:
  "ereal_of_enat (enat n) = ereal n"
  "ereal_of_enat  = "
  by (simp_all add: ereal_of_enat_def)

lemma ereal_of_enat_le_iff[simp]: "ereal_of_enat m  ereal_of_enat n  m  n"
  by (cases m n rule: enat2_cases) auto

lemma ereal_of_enat_less_iff[simp]: "ereal_of_enat m < ereal_of_enat n  m < n"
  by (cases m n rule: enat2_cases) auto

lemma numeral_le_ereal_of_enat_iff[simp]: "numeral m  ereal_of_enat n  numeral m  n"
by (cases n) (auto)

lemma numeral_less_ereal_of_enat_iff[simp]: "numeral m < ereal_of_enat n  numeral m < n"
  by (cases n) auto

lemma ereal_of_enat_ge_zero_cancel_iff[simp]: "0  ereal_of_enat n  0  n"
  by (cases n) (auto simp flip: enat_0)

lemma ereal_of_enat_gt_zero_cancel_iff[simp]: "0 < ereal_of_enat n  0 < n"
  by (cases n) (auto simp flip: enat_0)

lemma ereal_of_enat_zero[simp]: "ereal_of_enat 0 = 0"
  by (auto simp flip: enat_0)

lemma ereal_of_enat_inf[simp]: "ereal_of_enat n =   n = "
  by (cases n) auto

lemma ereal_of_enat_add: "ereal_of_enat (m + n) = ereal_of_enat m + ereal_of_enat n"
  by (cases m n rule: enat2_cases) auto

lemma ereal_of_enat_sub:
  assumes "n  m"
  shows "ereal_of_enat (m - n) = ereal_of_enat m - ereal_of_enat n "
  using assms by (cases m n rule: enat2_cases) auto

lemma ereal_of_enat_mult:
  "ereal_of_enat (m * n) = ereal_of_enat m * ereal_of_enat n"
  by (cases m n rule: enat2_cases) auto

lemmas ereal_of_enat_pushin = ereal_of_enat_add ereal_of_enat_sub ereal_of_enat_mult
lemmas ereal_of_enat_pushout = ereal_of_enat_pushin[symmetric]

lemma ereal_of_enat_nonneg: "ereal_of_enat n  0"
by(cases n) simp_all

lemma ereal_of_enat_Sup:
  assumes "A  {}" shows "ereal_of_enat (Sup A) = (SUP a  A. ereal_of_enat a)"
proof (intro antisym mono_Sup)
  show "ereal_of_enat (Sup A)  (SUP a  A. ereal_of_enat a)"
  proof cases
    assume "finite A"
    with A  {} obtain a where "a  A" "ereal_of_enat (Sup A) = ereal_of_enat a"
      using Max_in[of A] by (auto simp: Sup_enat_def simp del: Max_in)
    then show ?thesis
      by (auto intro: SUP_upper)
  next
    assume "¬ finite A"
    have [simp]: "(SUP a  A. ereal_of_enat a) = top"
      unfolding SUP_eq_top_iff
    proof safe
      fix x :: ereal assume "x < top"
      then obtain n :: nat where "x < n"
        using less_PInf_Ex_of_nat top_ereal_def by auto
      obtain a where "a  A - enat ` {.. n}"
        by (metis ¬ finite A all_not_in_conv finite_Diff2 finite_atMost finite_imageI finite.emptyI)
      then have "a  A" "ereal n  ereal_of_enat a"
        by (auto simp: image_iff Ball_def)
           (metis enat_iless enat_ord_simps(1) ereal_of_enat_less_iff ereal_of_enat_simps(1) less_le not_less)
      with x < n show "iA. x < ereal_of_enat i"
        by (auto intro!: bexI[of _ a])
    qed
    show ?thesis
      by simp
  qed
qed (simp add: mono_def)

lemma ereal_of_enat_SUP:
  "A  {}  ereal_of_enat (SUP aA. f a) = (SUP a  A. ereal_of_enat (f a))"
  by (simp add: ereal_of_enat_Sup image_comp)


subsection "Limits on typereal"

lemma open_PInfty: "open A    A  (x. {ereal x<..}  A)"
  unfolding open_ereal_generated
proof (induct rule: generate_topology.induct)
  case (Int A B)
  then obtain x z where "  A  {ereal x <..}  A" "  B  {ereal z <..}  B"
    by auto
  with Int show ?case
    by (intro exI[of _ "max x z"]) fastforce
next
  case (Basis S)
  {
    fix x
    have "x    t. x  ereal t"
      by (cases x) auto
  }
  moreover note Basis
  ultimately show ?case
    by (auto split: ereal.split)
qed (fastforce simp add: vimage_Union)+

lemma open_MInfty: "open A  -  A  (x. {..<ereal x}  A)"
  unfolding open_ereal_generated
proof (induct rule: generate_topology.induct)
  case (Int A B)
  then obtain x z where "-  A  {..< ereal x}  A" "-  B  {..< ereal z}  B"
    by auto
  with Int show ?case
    by (intro exI[of _ "min x z"]) fastforce
next
  case (Basis S)
  {
    fix x
    have "x  -   t. ereal t  x"
      by (cases x) auto
  }
  moreover note Basis
  ultimately show ?case
    by (auto split: ereal.split)
qed (fastforce simp add: vimage_Union)+

lemma open_ereal_vimage: "open S  open (ereal -` S)"
  by (intro open_vimage continuous_intros)

lemma open_ereal: "open S  open (ereal ` S)"
  unfolding open_generated_order[where 'a=real]
proof (induct rule: generate_topology.induct)
  case (Basis S)
  moreover have "x. ereal ` {..< x} = { - <..< ereal x }"
    using ereal_less_ereal_Ex by auto
  moreover have "x. ereal ` {x <..} = { ereal x <..<  }"
    using less_ereal.elims(2) by fastforce
  ultimately show ?case
    by auto
qed (auto simp add: image_Union image_Int)

lemma open_image_real_of_ereal:
  fixes X::"ereal set"
  assumes "open X"
  assumes infty: "  X" "-  X"
  shows "open (real_of_ereal ` X)"
proof -
  have "real_of_ereal ` X = ereal -` X"
    using infty ereal_real by (force simp: set_eq_iff)
  thus ?thesis
    by (auto intro!: open_ereal_vimage assms)
qed

lemma eventually_finite:
  fixes x :: ereal
  assumes "¦x¦  " "(f  x) F"
  shows "eventually (λx. ¦f x¦  ) F"
proof -
  have "(f  ereal (real_of_ereal x)) F"
    using assms by (cases x) auto
  then have "eventually (λx. f x  ereal ` UNIV) F"
    by (rule topological_tendstoD) (auto intro: open_ereal)
  also have "(λx. f x  ereal ` UNIV) = (λx. ¦f x¦  )"
    by auto
  finally show ?thesis .
qed


lemma open_ereal_def:
  "open A  open (ereal -` A)  (  A  (x. {ereal x <..}  A))  (-  A  (x. {..<ereal x}  A))"
  (is "open A  ?rhs")
proof
  assume "open A"
  then show ?rhs
    using open_PInfty open_MInfty open_ereal_vimage by auto
next
  assume "?rhs"
  then obtain x y where A: "open (ereal -` A)" "  A  {ereal x<..}  A" "-  A  {..< ereal y}  A"
    by auto
  have *: "A = ereal ` (ereal -` A)  (if   A then {ereal x<..} else {})  (if -  A then {..< ereal y} else {})"
    using A(2,3) by auto
  from open_ereal[OF A(1)] show "open A"
    by (subst *) (auto simp: open_Un)
qed

lemma open_PInfty2:
  assumes "open A"
    and "  A"
  obtains x where "{ereal x<..}  A"
  using open_PInfty[OF assms] by auto

lemma open_MInfty2:
  assumes "open A"
    and "-  A"
  obtains x where "{..<ereal x}  A"
  using open_MInfty[OF assms] by auto

lemma ereal_openE:
  assumes "open A"
  obtains x y where "open (ereal -` A)"
    and "  A  {ereal x<..}  A"
    and "-  A  {..<ereal y}  A"
  using assms open_ereal_def by auto

lemmas open_ereal_lessThan = open_lessThan[where 'a=ereal]
lemmas open_ereal_greaterThan = open_greaterThan[where 'a=ereal]
lemmas ereal_open_greaterThanLessThan = open_greaterThanLessThan[where 'a=ereal]
lemmas closed_ereal_atLeast = closed_atLeast[where 'a=ereal]
lemmas closed_ereal_atMost = closed_atMost[where 'a=ereal]
lemmas closed_ereal_atLeastAtMost = closed_atLeastAtMost[where 'a=ereal]
lemmas closed_ereal_singleton = closed_singleton[where 'a=ereal]

lemma ereal_open_cont_interval:
  fixes S :: "ereal set"
  assumes "open S"
    and "x  S"
    and "¦x¦  "
  obtains e where "e > 0" and "{x-e <..< x+e}  S"
proof -
  from open S
  have "open (ereal -` S)"
    by (rule ereal_openE)
  then obtain e where "e > 0" and e: "dist y (real_of_ereal x) < e  ereal y  S" for y
    using assms unfolding open_dist by force
  show thesis
  proof (intro that subsetI)
    show "0 < ereal e"
      using 0 < e by auto
    fix y
    assume "y  {x - ereal e<..<x + ereal e}"
    with assms obtain t where "y = ereal t" "dist t (real_of_ereal x) < e"
      by (cases y) (auto simp: dist_real_def)
    then show "y  S"
      using e[of t] by auto
  qed
qed

lemma ereal_open_cont_interval2:
  fixes S :: "ereal set"
  assumes "open S"
    and "x  S"
    and x: "¦x¦  "
  obtains a b where "a < x" and "x < b" and "{a <..< b}  S"
proof -
  obtain e where "0 < e" "{x - e<..<x + e}  S"
    using assms by (rule ereal_open_cont_interval)
  with that[of "x - e" "x + e"] ereal_between[OF x, of e]
  show thesis
    by auto
qed

subsubsection ‹Convergent sequences›

lemma lim_real_of_ereal[simp]:
  assumes lim: "(f  ereal x) net"
  shows "((λx. real_of_ereal (f x))  x) net"
proof (intro topological_tendstoI)
  fix S
  assume "open S" and "x  S"
  then have S: "open S" "ereal x  ereal ` S"
    by (simp_all add: inj_image_mem_iff)
  show "eventually (λx. real_of_ereal (f x)  S) net"
    by (auto intro: eventually_mono [OF lim[THEN topological_tendstoD, OF open_ereal, OF S]])
qed

lemma lim_ereal[simp]: "((λn. ereal (f n))  ereal x) net  (f  x) net"
  by (auto dest!: lim_real_of_ereal)

lemma convergent_real_imp_convergent_ereal:
  assumes "convergent a"
  shows "convergent (λn. ereal (a n))" and "lim (λn. ereal (a n)) = ereal (lim a)"
proof -
  from assms obtain L where L: "a  L" unfolding convergent_def ..
  hence lim: "(λn. ereal (a n))  ereal L" using lim_ereal by auto
  thus "convergent (λn. ereal (a n))" unfolding convergent_def ..
  thus "lim (λn. ereal (a n)) = ereal (lim a)" using lim L limI by metis
qed

lemma tendsto_PInfty: "(f  ) F  (r. eventually (λx. ereal r < f x) F)"
proof -
  {
    fix l :: ereal
    assume "r. eventually (λx. ereal r < f x) F"
    from this[THEN spec, of "real_of_ereal l"] have "l    eventually (λx. l < f x) F"
      by (cases l) (auto elim: eventually_mono)
  }
  then show ?thesis
    by (auto simp: order_tendsto_iff)
qed

lemma tendsto_PInfty': "(f  ) F = (r>c. eventually (λx. ereal r < f x) F)"
proof (subst tendsto_PInfty, intro iffI allI impI)
  assume A: "r>c. eventually (λx. ereal r < f x) F"
  fix r :: real
  from A have A: "eventually (λx. ereal r < f x) F" if "r > c" for r using that by blast
  show "eventually (λx. ereal r < f x) F"
  proof (cases "r > c")
    case False
    hence B: "ereal r  ereal (c + 1)" by simp
    have "c < c + 1" by simp
    from A[OF this] show "eventually (λx. ereal r < f x) F"
      by eventually_elim (rule le_less_trans[OF B])
  qed (simp add: A)
qed simp

lemma tendsto_PInfty_eq_at_top:
  "((λz. ereal (f z))  ) F  (LIM z F. f z :> at_top)"
  unfolding tendsto_PInfty filterlim_at_top_dense by simp

lemma tendsto_MInfty: "(f  -) F  (r. eventually (λx. f x < ereal r) F)"
  unfolding tendsto_def
proof safe
  fix S :: "ereal set"
  assume "open S" "-  S"
  from open_MInfty[OF this] obtain B where "{..<ereal B}  S" ..
  moreover
  assume "r::real. eventually (λz. f z < r) F"
  then have "eventually (λz. f z  {..< B}) F"
    by auto
  ultimately show "eventually (λz. f z  S) F"
    by (auto elim!: eventually_mono)
next
  fix x
  assume "S. open S  -  S  eventually (λx. f x  S) F"
  from this[rule_format, of "{..< ereal x}"] show "eventually (λy. f y < ereal x) F"
    by auto
qed

lemma tendsto_MInfty': "(f  -) F = (r<c. eventually (λx. ereal r > f x) F)"
proof (subst tendsto_MInfty, intro iffI allI impI)
  assume A: "r<c. eventually (λx. ereal r > f x) F"
  fix r :: real
  from A have A: "eventually (λx. ereal r > f x) F" if "r < c" for r using that by blast
  show "eventually (λx. ereal r > f x) F"
  proof (cases "r < c")
    case False
    hence B: "ereal r  ereal (c - 1)" by simp
    have "c > c - 1" by simp
    from A[OF this] show "eventually (λx. ereal r > f x) F"
      by eventually_elim (erule less_le_trans[OF _ B])
  qed (simp add: A)
qed simp

lemma Lim_PInfty: "f    (B. N. nN. f n  ereal B)"
  unfolding tendsto_PInfty eventually_sequentially
proof safe
  fix r
  assume "r. N. nN. ereal r  f n"
  then obtain N where "nN. ereal (r + 1)  f n"
    by blast
  moreover have "ereal r < ereal (r + 1)"
    by auto
  ultimately show "N. nN. ereal r < f n"
    by (blast intro: less_le_trans)
qed (blast intro: less_imp_le)

lemma Lim_MInfty: "f  -  (B. N. nN. ereal B  f n)"
  unfolding tendsto_MInfty eventually_sequentially
proof safe
  fix r
  assume "r. N. nN. f n  ereal r"
  then obtain N where "nN. f n  ereal (r - 1)"
    by blast
  moreover have "ereal (r - 1) < ereal r"
    by auto
  ultimately show "N. nN. f n < ereal r"
    by (blast intro: le_less_trans)
qed (blast intro: less_imp_le)

lemma Lim_bounded_PInfty: "f  l  (n. f n  ereal B)  l  "
  using LIMSEQ_le_const2[of f l "ereal B"] by auto

lemma Lim_bounded_MInfty: "f  l  (n. ereal B  f n)  l  -"
  using LIMSEQ_le_const[of f l "ereal B"] by auto

lemma tendsto_zero_erealI:
  assumes "e. e > 0  eventually (λx. ¦f x¦ < ereal e) F"
  shows   "(f  0) F"
proof (subst filterlim_cong[OF refl refl])
  from assms[OF zero_less_one] show "eventually (λx. f x = ereal (real_of_ereal (f x))) F"
    by eventually_elim (auto simp: ereal_real)
  hence "eventually (λx. abs (real_of_ereal (f x)) < e) F" if "e > 0" for e using assms[OF that]
    by eventually_elim (simp add: real_less_ereal_iff that)
  hence "((λx. real_of_ereal (f x))  0) F" unfolding tendsto_iff
    by (auto simp: tendsto_iff dist_real_def)
  thus "((λx. ereal (real_of_ereal (f x)))  0) F" by (simp add: zero_ereal_def)
qed

lemma Lim_bounded_PInfty2: "f  l  nN. f n  ereal B  l  "
  using LIMSEQ_le_const2[of f l "ereal B"] by fastforce

lemma real_of_ereal_mult[simp]:
  fixes a b :: ereal
  shows "real_of_ereal (a * b) = real_of_ereal a * real_of_ereal b"
  by (cases rule: ereal2_cases[of a b]) auto

lemma real_of_ereal_eq_0:
  fixes x :: ereal
  shows "real_of_ereal x = 0  x =   x = -  x = 0"
  by (cases x) auto

lemma tendsto_ereal_realD:
  fixes f :: "'a  ereal"
  assumes "x  0"
    and tendsto: "((λx. ereal (real_of_ereal (f x)))  x) net"
  shows "(f  x) net"
proof (intro topological_tendstoI)
  fix S
  assume S: "open S" "x  S"
  with x  0 have "open (S - {0})" "x  S - {0}"
    by auto
  from tendsto[THEN topological_tendstoD, OF this]
  show "eventually (λx. f x  S) net"
    by (rule eventually_rev_mp) (auto simp: ereal_real)
qed

lemma tendsto_ereal_realI:
  fixes f :: "'a  ereal"
  assumes x: "¦x¦  " and tendsto: "(f  x) net"
  shows "((λx. ereal (real_of_ereal (f x)))  x) net"
proof (intro topological_tendstoI)
  fix S
  assume "open S" and "x  S"
  with x have "open (S - {, -})" "x  S - {, -}"
    by auto
  from tendsto[THEN topological_tendstoD, OF this]
  show "eventually (λx. ereal (real_of_ereal (f x))  S) net"
    by (elim eventually_mono) (auto simp: ereal_real)
qed

lemma ereal_mult_cancel_left:
  fixes a b c :: ereal
  shows "a * b = a * c  (¦a¦ =   0 < b * c)  a = 0  b = c"
  by (cases rule: ereal3_cases[of a b c]) (simp_all add: zero_less_mult_iff)

lemma tendsto_add_ereal:
  fixes x y :: ereal
  assumes x: "¦x¦  " and y: "¦y¦  "
  assumes f: "(f  x) F" and g: "(g  y) F"
  shows "((λx. f x + g x)  x + y) F"
proof -
  from x obtain r where x': "x = ereal r" by (cases x) auto
  with f have "((λi. real_of_ereal (f i))  r) F" by simp
  moreover
  from y obtain p where y': "y = ereal p" by (cases y) auto
  with g have "((λi. real_of_ereal (g i))  p) F" by simp
  ultimately have "((λi. real_of_ereal (f i) + real_of_ereal (g i))  r + p) F"
    by (rule tendsto_add)
  moreover
  from eventually_finite[OF x f] eventually_finite[OF y g]
  have "eventually (λx. f x + g x = ereal (real_of_ereal (f x) + real_of_ereal (g x))) F"
    by eventually_elim auto
  ultimately show ?thesis
    by (simp add: x' y' cong: filterlim_cong)
qed

lemma tendsto_add_ereal_nonneg:
  fixes x y :: "ereal"
  assumes "x  -" "y  -" "(f  x) F" "(g  y) F"
  shows "((λx. f x + g x)  x + y) F"
proof cases
  assume "x =   y = "
  moreover
  { fix y :: ereal and f g :: "'a  ereal" assume "y  -" "(f  ) F" "(g  y) F"
    then obtain y' where "- < y'" "y' < y"
      using dense[of "-" y] by auto
    have "((λx. f x + g x)  ) F"
    proof (rule tendsto_sandwich)
      have "F x in F. y' < g x"
        using order_tendstoD(1)[OF (g  y) F y' < y] by auto
      then show "F x in F. f x + y'  f x + g x"
        by eventually_elim (auto intro!: add_mono)
      show "F n in F. f n + g n  " "((λn. )  ) F"
        by auto
      show "((λx. f x + y')  ) F"
        using tendsto_cadd_ereal[of y'  f F] (f  ) F - < y' by auto
    qed }
  note this[of y f g] this[of x g f]
  ultimately show ?thesis
    using assms by (auto simp: add_ac)
next
  assume "¬ (x =   y = )"
  with assms tendsto_add_ereal[of x y f F g]
  show ?thesis
    by auto
qed

lemma ereal_inj_affinity:
  fixes m t :: ereal
  assumes "¦m¦  "
    and "m  0"
    and "¦t¦  "
  shows "inj_on (λx. m * x + t) A"
  using assms
  by (cases rule: ereal2_cases[of m t])
     (auto intro!: inj_onI simp: ereal_add_cancel_right ereal_mult_cancel_left)

lemma ereal_PInfty_eq_plus[simp]:
  fixes a b :: ereal
  shows " = a + b  a =   b = "
  by (cases rule: ereal2_cases[of a b]) auto

lemma ereal_MInfty_eq_plus[simp]:
  fixes a b :: ereal
  shows "- = a + b  (a = -  b  )  (b = -  a  )"
  by (cases rule: ereal2_cases[of a b]) auto

lemma ereal_less_divide_pos:
  fixes x y :: ereal
  shows "x > 0  x    y < z / x  x * y < z"
  by (cases rule: ereal3_cases[of x y z]) (auto simp: field_simps)

lemma ereal_divide_less_pos:
  fixes x y z :: ereal
  shows "x > 0  x    y / x < z  y < x * z"
  by (cases rule: ereal3_cases[of x y z]) (auto simp: field_simps)

lemma ereal_divide_eq:
  fixes a b c :: ereal
  shows "b  0  ¦b¦    a / b = c  a = b * c"
  by (cases rule: ereal3_cases[of a b c])
     (simp_all add: field_simps)

lemma ereal_inverse_not_MInfty[simp]: "inverse (a::ereal)  -"
  by (cases a) auto

lemma ereal_mult_m1[simp]: "x * ereal (-1) = -x"
  by (cases x) auto

lemma ereal_real':
  assumes "¦x¦  "
  shows "ereal (real_of_ereal x) = x"
  using assms by auto

lemma real_ereal_id: "real_of_ereal  ereal = id"
proof -
  {
    fix x
    have "(real_of_ereal  ereal) x = id x"
      by auto
  }
  then show ?thesis
    using ext by blast
qed

lemma open_image_ereal: "open(UNIV-{  , (- :: ereal)})"
  by (metis range_ereal open_ereal open_UNIV)

lemma ereal_le_distrib:
  fixes a b c :: ereal
  shows "c * (a + b)  c * a + c * b"
  by (cases rule: ereal3_cases[of a b c])
     (auto simp add: field_simps not_le mult_le_0_iff mult_less_0_iff)

lemma ereal_pos_distrib:
  fixes a b c :: ereal
  assumes "0  c"
    and "c  "
  shows "c * (a + b) = c * a + c * b"
  using assms
  by (cases rule: ereal3_cases[of a b c])
    (auto simp add: field_simps not_le mult_le_0_iff mult_less_0_iff)

lemma ereal_LimI_finite:
  fixes x :: ereal
  assumes "¦x¦  "
    and "r. 0 < r  N. nN. u n < x + r  x < u n + r"
  shows "u  x"
proof (rule topological_tendstoI, unfold eventually_sequentially)
  obtain rx where rx: "x = ereal rx"
    using assms by (cases x) auto
  fix S
  assume "open S" and "x  S"
  then have "open (ereal -` S)"
    unfolding open_ereal_def by auto
  with x  S obtain r where "0 < r" and dist: "dist y rx < r  ereal y  S" for y
    unfolding open_dist rx by auto
  then obtain n
    where upper: "u N < x + ereal r"
      and lower: "x < u N + ereal r"
      if "n  N" for N
    using assms(2)[of "ereal r"] by auto
  show "N. nN. u n  S"
  proof (safe intro!: exI[of _ n])
    fix N
    assume "n  N"
    from upper[OF this] lower[OF this] assms 0 < r
    have "u N  {,(-)}"
      by auto
    then obtain ra where ra_def: "(u N) = ereal ra"
      by (cases "u N") auto
    then have "rx < ra + r" and "ra < rx + r"
      using rx assms 0 < r lower[OF n  N] upper[OF n  N]
      by auto
    then have "dist (real_of_ereal (u N)) rx < r"
      using rx ra_def
      by (auto simp: dist_real_def abs_diff_less_iff field_simps)
    from dist[OF this] show "u N  S"
      using u N   {, -}
      by (auto simp: ereal_real split: if_split_asm)
  qed
qed

lemma tendsto_obtains_N:
  assumes "f  f0"
  assumes "open S"
    and "f0  S"
  obtains N where "nN. f n  S"
  using assms using tendsto_def
  using lim_explicit[of f f0] assms by auto

lemma ereal_LimI_finite_iff:
  fixes x :: ereal
  assumes "¦x¦  "
  shows "u  x  (r. 0 < r  (N. nN. u n < x + r  x < u n + r))"
  (is "?lhs  ?rhs")
proof
  assume lim: "u  x"
  {
    fix r :: ereal
    assume "r > 0"
    then obtain N where "nN. u n  {x - r <..< x + r}"
       apply (subst tendsto_obtains_N[of u x "{x - r <..< x + r}"])
       using lim ereal_between[of x r] assms r > 0
       apply auto
       done
    then have "N. nN. u n < x + r  x < u n + r"
      using ereal_minus_less[of r x]
      by (cases r) auto
  }
  then show ?rhs
    by auto
next
  assume ?rhs
  then show "u  x"
    using ereal_LimI_finite[of x] assms by auto
qed

lemma ereal_Limsup_uminus:
  fixes f :: "'a  ereal"
  shows "Limsup net (λx. - (f x)) = - Liminf net f"
  unfolding Limsup_def Liminf_def ereal_SUP_uminus ereal_INF_uminus_eq ..

lemma liminf_bounded_iff:
  fixes x :: "nat  ereal"
  shows "C  liminf x  (B<C. N. nN. B < x n)"
  (is "?lhs  ?rhs")
  unfolding le_Liminf_iff eventually_sequentially ..

lemma Liminf_add_le:
  fixes f g :: "_  ereal"
  assumes F: "F  bot"
  assumes ev: "eventually (λx. 0  f x) F" "eventually (λx. 0  g x) F"
  shows "Liminf F f + Liminf F g  Liminf F (λx. f x + g x)"
  unfolding Liminf_def
proof (subst SUP_ereal_add_left[symmetric])
  let ?F = "{P. eventually P F}"
  let ?INF = "λP g. Inf (g ` (Collect P))"
  show "?F  {}"
    by (auto intro: eventually_True)
  show "(SUP P?F. ?INF P g)  - "
    unfolding bot_ereal_def[symmetric] SUP_bot_conv INF_eq_bot_iff
    by (auto intro!: exI[of _ 0] ev simp: bot_ereal_def)
  have "(SUP P?F. ?INF P f + (SUP P?F. ?INF P g))  (SUP P?F. (SUP P'?F. ?INF P f + ?INF P' g))"
  proof (safe intro!: SUP_mono bexI[of _ "λx. P x  0  f x" for P])
    fix P let ?P' = "λx. P x  0  f x"
    assume "eventually P F"
    with ev show "eventually ?P' F"
      by eventually_elim auto
    have "?INF P f + (SUP P?F. ?INF P g)  ?INF ?P' f + (SUP P?F. ?INF P g)"
      by (intro add_mono INF_mono) auto
    also have " = (SUP P'?F. ?INF ?P' f + ?INF P' g)"
    proof (rule SUP_ereal_add_right[symmetric])
      show "Inf (f ` {x. P x  0  f x})  - "
        unfolding bot_ereal_def[symmetric] INF_eq_bot_iff
        by (auto intro!: exI[of _ 0] ev simp: bot_ereal_def)
    qed fact
    finally show "?INF P f + (SUP P?F. ?INF P g)  (SUP P'?F. ?INF ?P' f + ?INF P' g)" .
  qed
  also have "  (SUP P?F. INF xCollect P. f x + g x)"
  proof (safe intro!: SUP_least)
    fix P Q assume *: "eventually P F" "eventually Q F"
    show "?INF P f + ?INF Q g  (SUP P?F. INF xCollect P. f x + g x)"
    proof (rule SUP_upper2)
      show "(λx. P x  Q x)  ?F"
        using * by (auto simp: eventually_conj)
      show "?INF P f + ?INF Q g  (INF x{x. P x  Q x}. f x + g x)"
        by (intro INF_greatest add_mono) (auto intro: INF_lower)
    qed
  qed
  finally show "(SUP P?F. ?INF P f + (SUP P?F. ?INF P g))  (SUP P?F. INF xCollect P. f x + g x)" .
qed

lemma Sup_ereal_mult_right':
  assumes nonempty: "Y  {}"
  and x: "x  0"
  shows "(SUP iY. f i) * ereal x = (SUP iY. f i * ereal x)" (is "?lhs = ?rhs")
proof(cases "x = 0")
  case True thus ?thesis by(auto simp add: nonempty zero_ereal_def[symmetric])
next
  case False
  show ?thesis
  proof(rule antisym)
    show "?rhs  ?lhs"
      by(rule SUP_least)(simp add: ereal_mult_right_mono SUP_upper x)
  next
    have "?lhs / ereal x = (SUP iY. f i) * (ereal x / ereal x)" by(simp only: ereal_times_divide_eq)
    also have " = (SUP iY. f i)" using False by simp
    also have "  ?rhs / x"
    proof(rule SUP_least)
      fix i
      assume "i  Y"
      have "f i = f i * (ereal x / ereal x)" using False by simp
      also have " = f i * x / x" by(simp only: ereal_times_divide_eq)
      also from i  Y have "f i * x  ?rhs" by(rule SUP_upper)
      hence "f i * x / x  ?rhs / x" using x False by simp
      finally show "f i  ?rhs / x" .
    qed
    finally have "(?lhs / x) * x  (?rhs / x) * x"
      by(rule ereal_mult_right_mono)(simp add: x)
    also have " = ?rhs" using False ereal_divide_eq mult.commute by force
    also have "(?lhs / x) * x = ?lhs" using False ereal_divide_eq mult.commute by force
    finally show "?lhs  ?rhs" .
  qed
qed

lemma Sup_ereal_mult_left':
  " Y  {}; x  0   ereal x * (SUP iY. f i) = (SUP iY. ereal x * f i)"
by(subst (1 2) mult.commute)(rule Sup_ereal_mult_right')

lemma sup_continuous_add[order_continuous_intros]:
  fixes f g :: "'a::complete_lattice  ereal"
  assumes nn: "x. 0  f x" "x. 0  g x" and cont: "sup_continuous f" "sup_continuous g"
  shows "sup_continuous (λx. f x + g x)"
  unfolding sup_continuous_def
proof safe
  fix M :: "nat  'a" assume "incseq M"
  then show "f (SUP i. M i) + g (SUP i. M i) = (SUP i. f (M i) + g (M i))"
    using SUP_ereal_add_pos[of "λi. f (M i)" "λi. g (M i)"] nn
      cont[THEN sup_continuous_mono] cont[THEN sup_continuousD]
    by (auto simp: mono_def)
qed

lemma sup_continuous_mult_right[order_continuous_intros]:
  "0  c  c <   sup_continuous f  sup_continuous (λx. f x * c :: ereal)"
  by (cases c) (auto simp: sup_continuous_def fun_eq_iff Sup_ereal_mult_right')

lemma sup_continuous_mult_left[order_continuous_intros]:
  "0  c  c <   sup_continuous f  sup_continuous (λx. c * f x :: ereal)"
  using sup_continuous_mult_right[of c f] by (simp add: mult_ac)

lemma sup_continuous_ereal_of_enat[order_continuous_intros]:
  assumes f: "sup_continuous f" shows "sup_continuous (λx. ereal_of_enat (f x))"
  by (rule sup_continuous_compose[OF _ f])
     (auto simp: sup_continuous_def ereal_of_enat_SUP)

subsubsection ‹Sums›

lemma sums_ereal_positive:
  fixes f :: "nat  ereal"
  assumes "i. 0  f i"
  shows "f sums (SUP n. i<n. f i)"
proof -
  have "incseq (λi. j=0..<i. f j)"
    using add_mono[OF _ assms]
    by (auto intro!: incseq_SucI)
  from LIMSEQ_SUP[OF this]
  show ?thesis unfolding sums_def
    by (simp add: atLeast0LessThan)
qed

lemma summable_ereal_pos:
  fixes f :: "nat  ereal"
  assumes "i. 0  f i"
  shows "summable f"
  using sums_ereal_positive[of f, OF assms]
  unfolding summable_def
  by auto

lemma sums_ereal: "(λx. ereal (f x)) sums ereal x  f sums x"
  unfolding sums_def by simp

lemma suminf_ereal_eq_SUP:
  fixes f :: "nat  ereal"
  assumes "i. 0  f i"
  shows "(x. f x) = (SUP n. i<n. f i)"
  using sums_ereal_positive[of f, OF assms, THEN sums_unique]
  by simp

lemma suminf_bound:
  fixes f :: "nat  ereal"
  assumes "N. (n<N. f n)  x"
    and pos: "n. 0  f n"
  shows "suminf f  x"
proof (rule Lim_bounded)
  have "summable f" using pos[THEN summable_ereal_pos] .
  then show "(λN. n<N. f n)  suminf f"
    by (auto dest!: summable_sums simp: sums_def atLeast0LessThan)
  show "n0. sum f {..<n}  x"
    using assms by auto
qed

lemma suminf_bound_add:
  fixes f :: "nat  ereal"
  assumes "N. (n<N. f n) + y  x"
    and pos: "n. 0  f n"
    and "y  -"
  shows "suminf f + y  x"
proof (cases y)
  case (real r)
  then have "N. (n<N. f n)  x - y"
    using assms by (simp add: ereal_le_minus)
  then have "( n. f n)  x - y"
    using pos by (rule suminf_bound)
  then show "( n. f n) + y  x"
    using assms real by (simp add: ereal_le_minus)
qed (insert assms, auto)

lemma suminf_upper:
  fixes f :: "nat  ereal"
  assumes "n. 0  f n"
  shows "(n<N. f n)  (n. f n)"
  unfolding suminf_ereal_eq_SUP [OF assms]
  by (auto intro: complete_lattice_class.SUP_upper)

lemma suminf_0_le:
  fixes f :: "nat  ereal"
  assumes "n. 0  f n"
  shows "0  (n. f n)"
  using suminf_upper[of f 0, OF assms]
  by simp

lemma suminf_le_pos:
  fixes f g :: "nat  ereal"
  assumes "N. f N  g N"
    and "N. 0  f N"
  shows "suminf f  suminf g"
proof (safe intro!: suminf_bound)
  fix n
  {
    fix N
    have "0  g N"
      using assms(2,1)[of N] by auto
  }
  have "sum f {..<n}  sum g {..<n}"
    using assms by (auto intro: sum_mono)
  also have "  suminf g"
    using N. 0  g N
    by (rule suminf_upper)
  finally show "sum f {..<n}  suminf g" .
qed (rule assms(2))

lemma suminf_half_series_ereal: "(n. (1/2 :: ereal) ^ Suc n) = 1"
  using sums_ereal[THEN iffD2, OF power_half_series, THEN sums_unique, symmetric]
  by (simp add: one_ereal_def)

lemma suminf_add_ereal:
  fixes f g :: "nat  ereal"
  assumes "i. 0  f i" "i. 0  g i"
  shows "(i. f i + g i) = suminf f + suminf g"
proof -
  have "(SUP n. i<n. f i + g i) = (SUP n. sum f {..<n}) + (SUP n. sum g {..<n})"
    unfolding sum.distrib
    by (intro assms add_nonneg_nonneg SUP_ereal_add_pos incseq_sumI sum_nonneg ballI)
  with assms show ?thesis
    by (subst (1 2 3) suminf_ereal_eq_SUP) auto
qed

lemma suminf_cmult_ereal:
  fixes f g :: "nat  ereal"
  assumes "i. 0  f i"
    and "0  a"
  shows "(i. a * f i) = a * suminf f"
  by (auto simp: sum_ereal_right_distrib[symmetric] assms
       ereal_zero_le_0_iff sum_nonneg suminf_ereal_eq_SUP
       intro!: SUP_ereal_mult_left)

lemma suminf_PInfty:
  fixes f :: "nat  ereal"
  assumes "i. 0  f i"
    and "suminf f  "
  shows "f i  "
proof -
  from suminf_upper[of f "Suc i", OF assms(1)] assms(2)
  have "(i<Suc i. f i)  "
    by auto
  then show ?thesis
    unfolding sum_Pinfty by simp
qed

lemma suminf_PInfty_fun:
  assumes "i. 0  f i"
    and "suminf f  "
  shows "f'. f = (λx. ereal (f' x))"
proof -
  have "i. r. f i = ereal r"
  proof
    fix i
    show "r. f i = ereal r"
      using suminf_PInfty[OF assms] assms(1)[of i]
      by (cases "f i") auto
  qed
  from choice[OF this] show ?thesis
    by auto
qed

lemma summable_ereal:
  assumes "i. 0  f i"
    and "(i. ereal (f i))  "
  shows "summable f"
proof -
  have "0  (i. ereal (f i))"
    using assms by (intro suminf_0_le) auto
  with assms obtain r where r: "(i. ereal (f i)) = ereal r"
    by (cases "i. ereal (f i)") auto
  from summable_ereal_pos[of "λx. ereal (f x)"]
  have "summable (λx. ereal (f x))"
    using assms by auto
  from summable_sums[OF this]
  have "(λx. ereal (f x)) sums (x. ereal (f x))"
    by auto
  then show "summable f"
    unfolding r sums_ereal summable_def ..
qed

lemma suminf_ereal:
  assumes "i. 0  f i"
    and "(i. ereal (f i))  "
  shows "(i. ereal (f i)) = ereal (suminf f)"
proof (rule sums_unique[symmetric])
  from summable_ereal[OF assms]
  show "(λx. ereal (f x)) sums (ereal (suminf f))"
    unfolding sums_ereal
    using assms
    by (intro summable_sums summable_ereal)
qed

lemma suminf_ereal_minus:
  fixes f g :: "nat  ereal"
  assumes ord: "i. g i  f i" "i. 0  g i"
    and fin: "suminf f  " "suminf g  "
  shows "(i. f i - g i) = suminf f - suminf g"
proof -
  {
    fix i
    have "0  f i"
      using ord[of i] by auto
  }
  moreover
  from suminf_PInfty_fun[OF i. 0  f i fin(1)] obtain f' where [simp]: "f = (λx. ereal (f' x))" ..
  from suminf_PInfty_fun[OF i. 0  g i fin(2)] obtain g' where [simp]: "g = (λx. ereal (g' x))" ..
  {
    fix i
    have "0  f i - g i"
      using ord[of i] by (auto simp: ereal_le_minus_iff)
  }
  moreover
  have "suminf (λi. f i - g i)  suminf f"
    using assms by (auto intro!: suminf_le_pos simp: field_simps)
  then have "suminf (λi. f i - g i)  "
    using fin by auto
  ultimately show ?thesis
    using assms i. 0  f i
    apply simp
    apply (subst (1 2 3) suminf_ereal)
    apply (auto intro!: suminf_diff[symmetric] summable_ereal)
    done
qed

lemma suminf_ereal_PInf [simp]: "(x. ::ereal) = "
proof -
  have "(i<Suc 0. )  (x. ::ereal)"
    by (rule suminf_upper) auto
  then show ?thesis
    by simp
qed

lemma summable_real_of_ereal:
  fixes f :: "nat  ereal"
  assumes f: "i. 0  f i"
    and fin: "(i. f i)  "
  shows "summable (λi. real_of_ereal (f i))"
proof (rule summable_def[THEN iffD2])
  have "0  (i. f i)"
    using assms by (auto intro: suminf_0_le)
  with fin obtain r where r: "ereal r = (i. f i)"
    by (cases "(i. f i)") auto
  {
    fix i
    have "f i  "
      using f by (intro suminf_PInfty[OF _ fin]) auto
    then have "¦f i¦  "
      using f[of i] by auto
  }
  note fin = this
  have "(λi. ereal (real_of_ereal (f i))) sums (i. ereal (real_of_ereal (f i)))"
    using f
    by (auto intro!: summable_ereal_pos simp: ereal_le_real_iff zero_ereal_def)
  also have " = ereal r"
    using fin r by (auto simp: ereal_real)
  finally show "r. (λi. real_of_ereal (f i)) sums r"
    by (auto simp: sums_ereal)
qed

lemma suminf_SUP_eq:
  fixes f :: "nat  nat  ereal"
  assumes "i. incseq (λn. f n i)"
    and "n i. 0  f n i"
  shows "(i. SUP n. f n i) = (SUP n. i. f n i)"
proof -
  have *: "n. (i<n. SUP k. f k i) = (SUP k. i<n. f k i)"
    using assms
    by (auto intro!: SUP_ereal_sum [symmetric])
  show ?thesis
    using assms
    apply (subst (1 2) suminf_ereal_eq_SUP)
    apply (auto intro!: SUP_upper2 SUP_commute simp: *)
    done
qed

lemma suminf_sum_ereal:
  fixes f :: "_  _  ereal"
  assumes nonneg: "i a. a  A  0  f i a"
  shows "(i. aA. f i a) = (aA. i. f i a)"
proof (cases "finite A")
  case True
  then show ?thesis
    using nonneg
    by induct (simp_all add: suminf_add_ereal sum_nonneg)
next
  case False
  then show ?thesis by simp
qed

lemma suminf_ereal_eq_0:
  fixes f :: "nat  ereal"
  assumes nneg: "i. 0  f i"
  shows "(i. f i) = 0  (i. f i = 0)"
proof
  assume "(i. f i) = 0"
  {
    fix i
    assume "f i  0"
    with nneg have "0 < f i"
      by (auto simp: less_le)
    also have "f i = (j. if j = i then f i else 0)"
      by (subst suminf_finite[where N="{i}"]) auto
    also have "  (i. f i)"
      using nneg
      by (auto intro!: suminf_le_pos)
    finally have False
      using (i. f i) = 0 by auto
  }
  then show "i. f i = 0"
    by auto
qed simp

lemma suminf_ereal_offset_le:
  fixes f :: "nat  ereal"
  assumes f: "i. 0  f i"
  shows "(i. f (i + k))  suminf f"
proof -
  have "(λn. i<n. f (i + k))  (i. f (i + k))"
    using summable_sums[OF summable_ereal_pos]
    by (simp add: sums_def atLeast0LessThan f)
  moreover have "(λn. i<n. f i)  (i. f i)"
    using summable_sums[OF summable_ereal_pos]
    by (simp add: sums_def atLeast0LessThan f)
  then have "(λn. i<n + k. f i)  (i. f i)"
    by (rule LIMSEQ_ignore_initial_segment)
  ultimately show ?thesis
  proof (rule LIMSEQ_le, safe intro!: exI[of _ k])
    fix n assume "k  n"
    have "(i<n. f (i + k)) = (i<n. (f  plus k) i)"
      by (simp add: ac_simps)
    also have " = (i(plus k) ` {..<n}. f i)"
      by (rule sum.reindex [symmetric]) simp
    also have "  sum f {..<n + k}"
      by (intro sum_mono2) (auto simp: f)
    finally show "(i<n. f (i + k))  sum f {..<n + k}" .
  qed
qed

lemma sums_suminf_ereal: "f sums x  (i. ereal (f i)) = ereal x"
  by (metis sums_ereal sums_unique)

lemma suminf_ereal': "summable f  (i. ereal (f i)) = ereal (i. f i)"
  by (metis sums_ereal sums_unique summable_def)

lemma suminf_ereal_finite: "summable f  (i. ereal (f i))  "
  by (auto simp: summable_def simp flip: sums_ereal sums_unique)

lemma suminf_ereal_finite_neg:
  assumes "summable f"
  shows "(x. ereal (f x))  -"
proof-
  from assms obtain x where "f sums x" by blast
  hence "(λx. ereal (f x)) sums ereal x" by (simp add: sums_ereal)
  from sums_unique[OF this] have "(x. ereal (f x)) = ereal x" ..
  thus "(x. ereal (f x))  -" by simp_all
qed

lemma SUP_ereal_add_directed:
  fixes f g :: "'a  ereal"
  assumes nonneg: "i. i  I  0  f i" "i. i  I  0  g i"
  assumes directed: "i j. i  I  j  I  kI. f i + g j  f k + g k"
  shows "(SUP iI. f i + g i) = (SUP iI. f i) + (SUP iI. g i)"
proof cases
  assume "I = {}" then show ?thesis
    by (simp add: bot_ereal_def)
next
  assume "I  {}"
  show ?thesis
  proof (rule antisym)
    show "(SUP iI. f i + g i)  (SUP iI. f i) + (SUP iI. g i)"
      by (rule SUP_least; intro add_mono SUP_upper)
  next
    have "bot < (SUP iI. g i)"
      using I  {} nonneg(2) by (auto simp: bot_ereal_def less_SUP_iff)
    then have "(SUP iI. f i) + (SUP iI. g i) = (SUP iI. f i + (SUP iI. g i))"
      by (intro SUP_ereal_add_left[symmetric] I  {}) auto
    also have " = (SUP iI. (SUP jI. f i + g j))"
      using nonneg(1) I  {} by (simp add: SUP_ereal_add_right)
    also have "  (SUP iI. f i + g i)"
      using directed by (intro SUP_least) (blast intro: SUP_upper2)
    finally show "(SUP iI. f i) + (SUP iI. g i)  (SUP iI. f i + g i)" .
  qed
qed

lemma SUP_ereal_sum_directed:
  fixes f g :: "'a  'b  ereal"
  assumes "I  {}"
  assumes directed: "N i j. N  A  i  I  j  I  kI. nN. f n i  f n k  f n j  f n k"
  assumes nonneg: "n i. i  I  n  A  0  f n i"
  shows "(SUP iI. nA. f n i) = (nA. SUP iI. f n i)"
proof -
  have "N  A  (SUP iI. nN. f n i) = (nN. SUP iI. f n i)" for N
  proof (induction N rule: infinite_finite_induct)
    case (insert n N)
    have "(SUP iI. f n i + (lN. f l i)) = (SUP iI. f n i) + (SUP iI. lN. f l i)"
    proof (rule SUP_ereal_add_directed)
      fix i assume "i  I" then show "0  f n i" "0  (lN. f l i)"
        using insert by (auto intro!: sum_nonneg nonneg)
    next
      fix i j assume "i  I" "j  I"
      from directed[OF insert(4) this]
      show "kI. f n i + (lN. f l j)  f n k + (lN. f l k)"
        by (auto intro!: add_mono sum_mono)
    qed
    with insert show ?case
      by simp
  qed (simp_all add: SUP_constant I  {})
  from this[of A] show ?thesis by simp
qed

lemma suminf_SUP_eq_directed:
  fixes f :: "_  nat  ereal"
  assumes "I  {}"
  assumes directed: "N i j. i  I  j  I  finite N  kI. nN. f i n  f k n  f j n  f k n"
  assumes nonneg: "n i. 0  f n i"
  shows "(i. SUP nI. f n i) = (SUP nI. i. f n i)"
proof (subst (1 2) suminf_ereal_eq_SUP)
  show "n i. 0  f n i" "i. 0  (SUP nI. f n i)"
    using I  {} nonneg by (auto intro: SUP_upper2)
  show "(SUP n. i<n. SUP nI. f n i) = (SUP nI. SUP j. i<j. f n i)"
    by (auto simp: finite_subset SUP_commute SUP_ereal_sum_directed assms)
qed

lemma ereal_dense3:
  fixes x y :: ereal
  shows "x < y  r::rat. x < real_of_rat r  real_of_rat r < y"
proof (cases x y rule: ereal2_cases, simp_all)
  fix r q :: real
  assume "r < q"
  from Rats_dense_in_real[OF this] show "x. r < real_of_rat x  real_of_rat x < q"
    by (fastforce simp: Rats_def)
next
  fix r :: real
  show "x. r < real_of_rat x" "x. real_of_rat x < r"
    using gt_ex[of r] lt_ex[of r] Rats_dense_in_real
    by (auto simp: Rats_def)
qed

lemma continuous_within_ereal[intro, simp]: "x  A  continuous (at x within A) ereal"
  using continuous_on_eq_continuous_within[of A ereal]
  by (auto intro: continuous_on_ereal continuous_on_id)

lemma ereal_open_uminus:
  fixes S :: "ereal set"
  assumes "open S"
  shows "open (uminus ` S)"
  using open S[unfolded open_generated_order]
proof induct
  have "range uminus = (UNIV :: ereal set)"
    by (auto simp: image_iff ereal_uminus_eq_reorder)
  then show "open (range uminus :: ereal set)"
    by simp
qed (auto simp add: image_Union image_Int)

lemma ereal_uminus_complement:
  fixes S :: "ereal set"
  shows "uminus ` (- S) = - uminus ` S"
  by (auto intro!: bij_image_Compl_eq surjI[of _ uminus] simp: bij_betw_def)

lemma ereal_closed_uminus:
  fixes S :: "ereal set"
  assumes "closed S"
  shows "closed (uminus ` S)"
  using assms
  unfolding closed_def ereal_uminus_complement[symmetric]
  by (rule ereal_open_uminus)

lemma ereal_open_affinity_pos:
  fixes S :: "ereal set"
  assumes "open S"
    and m: "m  " "0 < m"
    and t: "¦t¦  "
  shows "open ((λx. m * x + t) ` S)"
proof -
  have "continuous_on UNIV (λx. inverse m * (x + - t))"
    using m t
    by (intro continuous_at_imp_continuous_on ballI continuous_at[THEN iffD2]; force)
  then have "open ((λx. inverse m * (x + -t)) -` S)"
    using open S open_vimage by blast
  also have "(λx. inverse m * (x + -t)) -` S = (λx. (x - t) / m) -` S"
    using m t by (auto simp: divide_ereal_def mult.commute minus_ereal_def
                       simp flip: uminus_ereal.simps)
  also have "(λx. (x - t) / m) -` S = (λx. m * x + t) ` S"
    using m t
    by (simp add: set_eq_iff image_iff)
       (metis abs_ereal_less0 abs_ereal_uminus ereal_divide_eq ereal_eq_minus ereal_minus(7,8)
              ereal_minus_less_minus ereal_mult_eq_PInfty ereal_uminus_uminus ereal_zero_mult)
  finally show ?thesis .
qed

lemma ereal_open_affinity:
  fixes S :: "ereal set"
  assumes "open S"
    and m: "¦m¦  " "m  0"
    and t: "¦t¦  "
  shows "open ((λx. m * x + t) ` S)"
proof cases
  assume "0 < m"
  then show ?thesis
    using ereal_open_affinity_pos[OF open S _ _ t, of m] m
    by auto
next
  assume "¬ 0 < m" then
  have "0 < -m"
    using m  0
    by (cases m) auto
  then have m: "-m  " "0 < -m"
    using ¦m¦  
    by (auto simp: ereal_uminus_eq_reorder)
  from ereal_open_affinity_pos[OF ereal_open_uminus[OF open S] m t] show ?thesis
    unfolding image_image by simp
qed

lemma open_uminus_iff:
  fixes S :: "ereal set"
  shows "open (uminus ` S)  open S"
  using ereal_open_uminus[of S] ereal_open_uminus[of "uminus ` S"]
  by auto

lemma ereal_Liminf_uminus:
  fixes f :: "'a  ereal"
  shows "Liminf net (λx. - (f x)) = - Limsup net f"
  using ereal_Limsup_uminus[of _ "(λx. - (f x))"] by auto

lemma Liminf_PInfty:
  fixes f :: "'a  ereal"
  assumes "¬ trivial_limit net"
  shows "(f  ) net  Liminf net f = "
  unfolding tendsto_iff_Liminf_eq_Limsup[OF assms]
  using Liminf_le_Limsup[OF assms, of f]
  by auto

lemma Limsup_MInfty:
  fixes f :: "'a  ereal"
  assumes "¬ trivial_limit net"
  shows "(f  -) net  Limsup net f = -"
  unfolding tendsto_iff_Liminf_eq_Limsup[OF assms]
  using Liminf_le_Limsup[OF assms, of f]
  by auto

lemma convergent_ereal: ― ‹RENAME›
  fixes X :: "nat  'a :: {complete_linorder,linorder_topology}"
  shows "convergent X  limsup X = liminf X"
  using tendsto_iff_Liminf_eq_Limsup[of sequentially]
  by (auto simp: convergent_def)

lemma limsup_le_liminf_real:
  fixes X :: "nat  real" and L :: real
  assumes 1: "limsup X  L" and 2: "L  liminf X"
  shows "X  L"
proof -
  from 1 2 have "limsup X  liminf X" by auto
  hence 3: "limsup X = liminf X"
    by (simp add: Liminf_le_Limsup order_class.order.antisym)
  hence 4: "convergent (λn. ereal (X n))"
    by (subst convergent_ereal)
  hence "limsup X = lim (λn. ereal(X n))"
    by (rule convergent_limsup_cl)
  also from 1 2 3 have "limsup X = L" by auto
  finally have "lim (λn. ereal(X n)) = L" ..
  hence "(λn. ereal (X n))  L"
    using "4" convergent_LIMSEQ_iff by force
  thus ?thesis by simp
qed

lemma liminf_PInfty:
  fixes X :: "nat  ereal"
  shows "X    liminf X = "
  by (metis Liminf_PInfty trivial_limit_sequentially)

lemma limsup_MInfty:
  fixes X :: "nat  ereal"
  shows "X  -  limsup X = -"
  by (metis Limsup_MInfty trivial_limit_sequentially)

lemma SUP_eq_LIMSEQ:
  assumes "mono f"
  shows "(SUP n. ereal (f n)) = ereal x  f  x"
proof
  have inc: "incseq (λi. ereal (f i))"
    using mono f unfolding mono_def incseq_def by auto
  {
    assume "f  x"
    then have "(λi. ereal (f i))  ereal x"
      by auto
    from SUP_Lim[OF inc this] show "(SUP n. ereal (f n)) = ereal x" .
  next
    assume "(SUP n. ereal (f n)) = ereal x"
    with LIMSEQ_SUP[OF inc] show "f  x" by auto
  }
qed

lemma liminf_ereal_cminus:
  fixes f :: "nat  ereal"
  assumes "c  -"
  shows "liminf (λx. c - f x) = c - limsup f"
proof (cases c)
  case PInf
  then show ?thesis
    by (simp add: Liminf_const)
next
  case (real r)
  then show ?thesis
    by (simp add: liminf_SUP_INF limsup_INF_SUP INF_ereal_minus_right SUP_ereal_minus_right)
qed (use c  - in simp)


subsubsection ‹Continuity›

lemma continuous_at_of_ereal:
  "¦x0 :: ereal¦    continuous (at x0) real_of_ereal"
  unfolding continuous_at
  by (rule lim_real_of_ereal) (simp add: ereal_real)

lemma nhds_ereal: "nhds (ereal r) = filtermap ereal (nhds r)"
  by (simp add: filtermap_nhds_open_map open_ereal continuous_at_of_ereal)

lemma at_ereal: "at (ereal r) = filtermap ereal (at r)"
  by (simp add: filter_eq_iff eventually_at_filter nhds_ereal eventually_filtermap)

lemma at_left_ereal: "at_left (ereal r) = filtermap ereal (at_left r)"
  by (simp add: filter_eq_iff eventually_at_filter nhds_ereal eventually_filtermap)

lemma at_right_ereal: "at_right (ereal r) = filtermap ereal (at_right r)"
  by (simp add: filter_eq_iff eventually_at_filter nhds_ereal eventually_filtermap)

lemma
  shows at_left_PInf: "at_left  = filtermap ereal at_top"
    and at_right_MInf: "at_right (-) = filtermap ereal at_bot"
  unfolding filter_eq_iff eventually_filtermap eventually_at_top_dense eventually_at_bot_dense
    eventually_at_left[OF ereal_less(5)] eventually_at_right[OF ereal_less(6)]
  by (auto simp add: ereal_all_split ereal_ex_split)

lemma ereal_tendsto_simps1:
  "((f  real_of_ereal)  y) (at_left (ereal x))  (f  y) (at_left x)"
  "((f  real_of_ereal)  y) (at_right (ereal x))  (f  y) (at_right x)"
  "((f  real_of_ereal)  y) (at_left (::ereal))  (f  y) at_top"
  "((f  real_of_ereal)  y) (at_right (-::ereal))  (f  y) at_bot"
  unfolding tendsto_compose_filtermap at_left_ereal at_right_ereal at_left_PInf at_right_MInf
  by (auto simp: filtermap_filtermap filtermap_ident)

lemma ereal_tendsto_simps2:
  "((ereal  f)  ereal a) F  (f  a) F"
  "((ereal  f)  ) F  (LIM x F. f x :> at_top)"
  "((ereal  f)  -) F  (LIM x F. f x :> at_bot)"
  unfolding tendsto_PInfty filterlim_at_top_dense tendsto_MInfty filterlim_at_bot_dense
  using lim_ereal by (simp_all add: comp_def)

lemma inverse_infty_ereal_tendsto_0: "inverse  (0::ereal)"
proof -
  have **: "((λx. ereal (inverse x))  ereal 0) at_infinity"
    by (intro tendsto_intros tendsto_inverse_0)
  then have "((λx. if x = 0 then  else ereal (inverse x))  0) at_top"
  proof (rule filterlim_mono_eventually)
    show "nhds (ereal 0)  nhds 0"
      by (simp add: zero_ereal_def)
    show "(at_top::real filter)  at_infinity"
      by (simp add: at_top_le_at_infinity)
  qed auto
  then show ?thesis
    unfolding at_infty_ereal_eq_at_top tendsto_compose_filtermap[symmetric] comp_def by auto
qed

lemma inverse_ereal_tendsto_pos:
  fixes x :: ereal assumes "0 < x"
  shows "inverse x inverse x"
proof (cases x)
  case (real r)
  with 0 < x have **: "(λx. ereal (inverse x)) r ereal (inverse r)"
    by (auto intro!: tendsto_inverse)
  from real 0 < x show ?thesis
    by (auto simp: at_ereal tendsto_compose_filtermap[symmetric] eventually_at_filter
             intro!: Lim_transform_eventually[OF **] t1_space_nhds)
qed (insert 0 < x, auto intro!: inverse_infty_ereal_tendsto_0)

lemma inverse_ereal_tendsto_at_right_0: "(inverse  ) (at_right (0::ereal))"
  unfolding tendsto_compose_filtermap[symmetric] at_right_ereal zero_ereal_def
  by (subst filterlim_cong[OF refl refl, where g="λx. ereal (inverse x)"])
     (auto simp: eventually_at_filter tendsto_PInfty_eq_at_top filterlim_inverse_at_top_right)

lemmas ereal_tendsto_simps = ereal_tendsto_simps1 ereal_tendsto_simps2

lemma continuous_at_iff_ereal:
  fixes f :: "'a::t2_space  real"
  shows "continuous (at x0 within s) f  continuous (at x0 within s) (ereal  f)"
  unfolding continuous_within comp_def lim_ereal ..

lemma continuous_on_iff_ereal:
  fixes f :: "'a::t2_space => real"
  assumes "open A"
  shows "continuous_on A f  continuous_on A (ereal  f)"
  unfolding continuous_on_def comp_def lim_ereal ..

lemma continuous_on_real: "continuous_on (UNIV - {, -::ereal}) real_of_ereal"
  using continuous_at_of_ereal continuous_on_eq_continuous_at open_image_ereal
  by auto

lemma continuous_on_iff_real:
  fixes f :: "'a::t2_space  ereal"
  assumes "x. x  A  ¦f x¦  "
  shows "continuous_on A f  continuous_on A (real_of_ereal  f)"
proof 
  assume L: "continuous_on A f"
  have "f ` A  UNIV - {, -}"
    using assms by force
  then show "continuous_on A (real_of_ereal  f)"
    by (meson L continuous_on_compose continuous_on_real continuous_on_subset)
next
  assume R: "continuous_on A (real_of_ereal  f)"
  then have "continuous_on A (ereal  (real_of_ereal  f))"
    by (meson continuous_at_iff_ereal continuous_on_eq_continuous_within)
  then show "continuous_on A f"
    using assms ereal_real' by auto 
qed

lemma continuous_uminus_ereal [continuous_intros]: "continuous_on (A :: ereal set) uminus"
  unfolding continuous_on_def
  by (intro ballI tendsto_uminus_ereal[of "λx. x::ereal"]) simp

lemma ereal_uminus_atMost [simp]: "uminus ` {..(a::ereal)} = {-a..}"
proof (intro equalityI subsetI)
  fix x :: ereal assume "x  {-a..}"
  hence "-(-x)  uminus ` {..a}" by (intro imageI) (simp add: ereal_uminus_le_reorder)
  thus "x  uminus ` {..a}" by simp
qed auto

lemma continuous_on_inverse_ereal [continuous_intros]:
  "continuous_on {0::ereal ..} inverse"
  unfolding continuous_on_def
proof clarsimp
  fix x :: ereal assume "0  x"
  moreover have "at 0 within {0 ..} = at_right (0::ereal)"
    by (auto simp: filter_eq_iff eventually_at_filter le_less)
  moreover have "at x within {0 ..} = at x" if "0 < x"
    using that by (intro at_within_nhd[of _ "{0<..}"]) auto
  ultimately show "(inverse  inverse x) (at x within {0..})"
    by (auto simp: le_less inverse_ereal_tendsto_at_right_0 inverse_ereal_tendsto_pos)
qed

lemma continuous_inverse_ereal_nonpos: "continuous_on ({..<0} :: ereal set) inverse"
proof (subst continuous_on_cong[OF refl])
  have "continuous_on {(0::ereal)<..} inverse"
    by (rule continuous_on_subset[OF continuous_on_inverse_ereal]) auto
  thus "continuous_on {..<(0::ereal)} (uminus  inverse  uminus)"
    by (intro continuous_intros) simp_all
qed simp

lemma tendsto_inverse_ereal:
  assumes "(f  (c :: ereal)) F"
  assumes "eventually (λx. f x  0) F"
  shows   "((λx. inverse (f x))  inverse c) F"
  by (cases "F = bot")
     (auto intro!: tendsto_lowerbound assms
                   continuous_on_tendsto_compose[OF continuous_on_inverse_ereal])


subsubsection ‹liminf and limsup›

lemma Limsup_ereal_mult_right:
  assumes "F  bot" "(c::real)  0"
  shows   "Limsup F (λn. f n * ereal c) = Limsup F f * ereal c"
proof (rule Limsup_compose_continuous_mono)
  from assms show "continuous_on UNIV (λa. a * ereal c)"
    using tendsto_cmult_ereal[of "ereal c" "λx. x" ]
    by (force simp: continuous_on_def mult_ac)
qed (insert assms, auto simp: mono_def ereal_mult_right_mono)

lemma Liminf_ereal_mult_right:
  assumes "F  bot" "(c::real)  0"
  shows   "Liminf F (λn. f n * ereal c) = Liminf F f * ereal c"
proof (rule Liminf_compose_continuous_mono)
  from assms show "continuous_on UNIV (λa. a * ereal c)"
    using tendsto_cmult_ereal[of "ereal c" "λx. x" ]
    by (force simp: continuous_on_def mult_ac)
qed (use assms in auto simp: mono_def ereal_mult_right_mono)

lemma Liminf_ereal_mult_left:
  assumes "F  bot" "(c::real)  0"
    shows "Liminf F (λn. ereal c * f n) = ereal c * Liminf F f"
using Liminf_ereal_mult_right[OF assms] by (subst (1 2) mult.commute)

lemma Limsup_ereal_mult_left:
  assumes "F  bot" "(c::real)  0"
  shows   "Limsup F (λn. ereal c * f n) = ereal c * Limsup F f"
  using Limsup_ereal_mult_right[OF assms] by (subst (1 2) mult.commute)

lemma limsup_ereal_mult_right:
  "(c::real)  0  limsup (λn. f n * ereal c) = limsup f * ereal c"
  by (rule Limsup_ereal_mult_right) simp_all

lemma limsup_ereal_mult_left:
  "(c::real)  0  limsup (λn. ereal c * f n) = ereal c * limsup f"
  by (subst (1 2) mult.commute, rule limsup_ereal_mult_right) simp_all

lemma Limsup_add_ereal_right:
  "F  bot  abs c    Limsup F (λn. g n + (c :: ereal)) = Limsup F g + c"
  by (rule Limsup_compose_continuous_mono) (auto simp: mono_def add_mono continuous_on_def)

lemma Limsup_add_ereal_left:
  "F  bot  abs c    Limsup F (λn. (c :: ereal) + g n) = c + Limsup F g"
  by (subst (1 2) add.commute) (rule Limsup_add_ereal_right)

lemma Liminf_add_ereal_right:
  "F  bot  abs c    Liminf F (λn. g n + (c :: ereal)) = Liminf F g + c"
  by (rule Liminf_compose_continuous_mono) (auto simp: mono_def add_mono continuous_on_def)

lemma Liminf_add_ereal_left:
  "F  bot  abs c    Liminf F (λn. (c :: ereal) + g n) = c + Liminf F g"
  by (subst (1 2) add.commute) (rule Liminf_add_ereal_right)

lemma
  assumes "F  bot"
  assumes nonneg: "eventually (λx. f x  (0::ereal)) F"
  shows   Liminf_inverse_ereal: "Liminf F (λx. inverse (f x)) = inverse (Limsup F f)"
  and     Limsup_inverse_ereal: "Limsup F (λx. inverse (f x)) = inverse (Liminf F f)"
proof -
  define inv where [abs_def]: "inv x = (if x  0 then  else inverse x)" for x :: ereal
  have "continuous_on ({..0}  {0..}) inv" unfolding inv_def
    by (intro continuous_on_If) (auto intro!: continuous_intros)
  also have "{..0}  {0..} = (UNIV :: ereal set)" by auto
  finally have cont: "continuous_on UNIV inv" .
  have antimono: "antimono inv" unfolding inv_def antimono_def
    by (auto intro!: ereal_inverse_antimono)

  have "Liminf F (λx. inverse (f x)) = Liminf F (λx. inv (f x))" using nonneg
    by (auto intro!: Liminf_eq elim!: eventually_mono simp: inv_def)
  also have "... = inv (Limsup F f)"
    by (simp add: assms(1) Liminf_compose_continuous_antimono[OF cont antimono])
  also from assms have "Limsup F f  0" by (intro le_Limsup) simp_all
  hence "inv (Limsup F f) = inverse (Limsup F f)" by (simp add: inv_def)
  finally show "Liminf F (λx. inverse (f x)) = inverse (Limsup F f)" .

  have "Limsup F (λx. inverse (f x)) = Limsup F (λx. inv (f x))" using nonneg
    by (auto intro!: Limsup_eq elim!: eventually_mono simp: inv_def)
  also have "... = inv (Liminf F f)"
    by (simp add: assms(1) Limsup_compose_continuous_antimono[OF cont antimono])
  also from assms have "Liminf F f  0" by (intro Liminf_bounded) simp_all
  hence "inv (Liminf F f) = inverse (Liminf F f)" by (simp add: inv_def)
  finally show "Limsup F (λx. inverse (f x)) = inverse (Liminf F f)" .
qed

lemma ereal_diff_le_mono_left: " x  z; 0  y   x - y  (z :: ereal)"
by(cases x y z rule: ereal3_cases) simp_all

lemma neg_0_less_iff_less_erea [simp]: "0 < - a  (a :: ereal) < 0"
by(cases a) simp_all

lemma not_infty_ereal: "¦x¦    (x'. x = ereal x')"
by(cases x) simp_all

lemma neq_PInf_trans: fixes x y :: ereal shows " y  ; x  y   x  "
by auto

lemma mult_2_ereal: "ereal 2 * x = x + x"
by(cases x) simp_all

lemma ereal_diff_le_self: "0  y  x - y  (x :: ereal)"
by(cases x y rule: ereal2_cases) simp_all

lemma ereal_le_add_self: "0  y  x  x + (y :: ereal)"
by(cases x y rule: ereal2_cases) simp_all

lemma ereal_le_add_self2: "0  y  x  y + (x :: ereal)"
by(cases x y rule: ereal2_cases) simp_all

lemma ereal_le_add_mono1: " x  y; 0  (z :: ereal)   x  y + z"
using add_mono by fastforce

lemma ereal_le_add_mono2: " x  z; 0  (y :: ereal)   x  y + z"
using add_mono by fastforce

lemma ereal_diff_nonpos:
  fixes a b :: ereal shows " a  b; a =   b  ; a = -  b  -   a - b  0"
  by (cases rule: ereal2_cases[of a b]) auto

lemma minus_ereal_0 [simp]: "x - ereal 0 = x"
by(simp flip: zero_ereal_def)

lemma ereal_diff_eq_0_iff: fixes a b :: ereal
  shows "(¦a¦ =   ¦b¦  )  a - b = 0  a = b"
by(cases a b rule: ereal2_cases) simp_all

lemma SUP_ereal_eq_0_iff_nonneg:
  fixes f :: "_  ereal" and A
  assumes nonneg: "xA. f x  0"
  and A:"A  {}"
  shows "(SUP xA. f x) = 0  (xA. f x = 0)" (is "?lhs  ?rhs")
proof(intro iffI ballI)
  fix x
  assume "?lhs" "x  A"
  from x  A have "f x  (SUP xA. f x)" by(rule SUP_upper)
  with ?lhs show "f x = 0" using nonneg x  A by auto
qed (simp add: A)

lemma ereal_divide_le_posI:
  fixes x y z :: ereal
  shows "x > 0  z  -   z  x * y  z / x  y"
by (cases rule: ereal3_cases[of x y z])(auto simp: field_simps split: if_split_asm)

lemma add_diff_eq_ereal: fixes x y z :: ereal
  shows "x + (y - z) = x + y - z"
by(cases x y z rule: ereal3_cases) simp_all

lemma ereal_diff_gr0:
  fixes a b :: ereal shows "a < b  0 < b - a"
  by (cases rule: ereal2_cases[of a b]) auto

lemma ereal_minus_minus: fixes x y z :: ereal shows
  "(¦y¦ =   ¦z¦  )  x - (y - z) = x + z - y"
by(cases x y z rule: ereal3_cases) simp_all

lemma diff_add_eq_ereal: fixes a b c :: ereal shows "a - b + c = a + c - b"
by(cases a b c rule: ereal3_cases) simp_all

lemma diff_diff_commute_ereal: fixes x y z :: ereal shows "x - y - z = x - z - y"
by(cases x y z rule: ereal3_cases) simp_all

lemma ereal_diff_eq_MInfty_iff: fixes x y :: ereal shows "x - y = -  x = -  y  -  y =   ¦x¦  "
by(cases x y rule: ereal2_cases) simp_all

lemma ereal_diff_add_inverse: fixes x y :: ereal shows "¦x¦    x + y - x = y"
by(cases x y rule: ereal2_cases) simp_all

lemma tendsto_diff_ereal:
  fixes x y :: ereal
  assumes x: "¦x¦  " and y: "¦y¦  "
  assumes f: "(f  x) F" and g: "(g  y) F"
  shows "((λx. f x - g x)  x - y) F"
proof -
  from x obtain r where x': "x = ereal r" by (cases x) auto
  with f have "((λi. real_of_ereal (f i))  r) F" by simp
  moreover
  from y obtain p where y': "y = ereal p" by (cases y) auto
  with g have "((λi. real_of_ereal (g i))  p) F" by simp
  ultimately have "((λi. real_of_ereal (f i) - real_of_ereal (g i))  r - p) F"
    by (rule tendsto_diff)
  moreover
  from eventually_finite[OF x f] eventually_finite[OF y g]
  have "eventually (λx. f x - g x = ereal (real_of_ereal (f x) - real_of_ereal (g x))) F"
    by eventually_elim auto
  ultimately show ?thesis
    by (simp add: x' y' cong: filterlim_cong)
qed

lemma continuous_on_diff_ereal:
  "continuous_on A f  continuous_on A g  (x. x  A  ¦f x¦  )  (x. x  A  ¦g x¦  )  continuous_on A (λz. f z - g z::ereal)"
  by (auto simp: tendsto_diff_ereal continuous_on_def)


subsubsection ‹Tests for code generator›

text ‹A small list of simple arithmetic expressions.›

value "-  :: ereal"
value "¦-¦ :: ereal"
value "4 + 5 / 4 - ereal 2 :: ereal"
value "ereal 3 < "
value "real_of_ereal (::ereal) = 0"

end