src/HOL/Library/Extended_Real.thy

Simplification of some proofs. Also key lemmas using !! rather than ! in premises

(*  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 \<open>Extended real number line\<close>

theory Extended_Real
imports Complex_Main Extended_Nat Liminf_Limsup
begin

text \<open>This should be part of @{theory Extended_Nat} or @{theory Order_Continuity}, but then the
AFP-entry \<open>Jinja_Thread\<close> fails, as it does overload certain named from @{theory Complex_Main}.\<close>

lemma incseq_sumI2:
  fixes f :: "'i \<Rightarrow> nat \<Rightarrow> 'a::ordered_comm_monoid_add"
  shows "(\<And>n. n \<in> A \<Longrightarrow> mono (f n)) \<Longrightarrow> mono (\<lambda>i. \<Sum>n\<in>A. f n i)"
  unfolding incseq_def by (auto intro: sum_mono)

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

lemma continuous_at_left_imp_sup_continuous:
  fixes f :: "'a::{complete_linorder, linorder_topology} \<Rightarrow> 'b::{complete_linorder, linorder_topology}"
  assumes "mono f" "\<And>x. continuous (at_left x) f"
  shows "sup_continuous f"
  unfolding sup_continuous_def
proof safe
  fix M :: "nat \<Rightarrow> '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
qed

lemma sup_continuous_at_left:
  fixes f :: "'a::{complete_linorder, linorder_topology, first_countable_topology} \<Rightarrow>
    '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 \<noteq> bot"
  show ?thesis
    unfolding continuous_within
  proof (intro tendsto_at_left_sequentially[of bot])
    fix S :: "nat \<Rightarrow> 'a" assume S: "incseq S" and S_x: "S \<longlonglongrightarrow> x"
    from S_x have x_eq: "x = (SUP i. S i)"
      by (rule LIMSEQ_unique) (intro LIMSEQ_SUP S)
    show "(\<lambda>n. f (S n)) \<longlonglongrightarrow> 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} \<Rightarrow>
    'b::{complete_linorder, linorder_topology}"
  shows "sup_continuous f \<longleftrightarrow> (\<forall>x. continuous (at_left x) f) \<and> 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} \<Rightarrow> 'b::{complete_linorder, linorder_topology}"
  assumes "mono f" "\<And>x. continuous (at_right x) f"
  shows "inf_continuous f"
  unfolding inf_continuous_def
proof safe
  fix M :: "nat \<Rightarrow> '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
qed

lemma inf_continuous_at_right:
  fixes f :: "'a::{complete_linorder, linorder_topology, first_countable_topology} \<Rightarrow>
    '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 \<noteq> top"
  show ?thesis
    unfolding continuous_within
  proof (intro tendsto_at_right_sequentially[of _ top])
    fix S :: "nat \<Rightarrow> 'a" assume S: "decseq S" and S_x: "S \<longlonglongrightarrow> x"
    from S_x have x_eq: "x = (INF i. S i)"
      by (rule LIMSEQ_unique) (intro LIMSEQ_INF S)
    show "(\<lambda>n. f (S n)) \<longlonglongrightarrow> 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} \<Rightarrow>
    'b::{complete_linorder, linorder_topology}"
  shows "inf_continuous f \<longleftrightarrow> (\<forall>x. continuous (at_right x) f) \<and> 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 \<Rightarrow> bool" where
  "open_enat = generate_topology (range lessThan \<union> 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)}"
    apply auto
    apply (case_tac x)
    apply auto
    done
  then show ?thesis
    by simp
qed

lemma open_enat_iff:
  fixes A :: "enat set"
  shows "open A \<longleftrightarrow> (\<infinity> \<in> A \<longrightarrow> (\<exists>n::nat. {n <..} \<subseteq> A))"
proof safe
  assume "\<infinity> \<notin> A"
  then have "A = (\<Union>n\<in>{n. enat n \<in> A}. {enat n})"
    apply auto
    apply (case_tac x)
    apply auto
    done
  moreover have "open \<dots>"
    by (auto intro: open_enat)
  ultimately show "open A"
    by simp
next
  fix n assume "{enat n <..} \<subseteq> A"
  then have "A = (\<Union>n\<in>{n. enat n \<in> A}. {enat n}) \<union> {enat n <..}"
    apply auto
    apply (case_tac x)
    apply auto
    done
  moreover have "open \<dots>"
    by (intro open_Un open_UN ballI open_enat open_greaterThan)
  ultimately show "open A"
    by simp
next
  assume "open A" "\<infinity> \<in> A"
  then have "generate_topology (range lessThan \<union> range greaterThan) A" "\<infinity> \<in> A"
    unfolding open_enat_def by auto
  then show "\<exists>n::nat. {n <..} \<subseteq> A"
  proof induction
    case (Int A B)
    then obtain n m where "{enat n<..} \<subseteq> A" "{enat m<..} \<subseteq> B"
      by auto
    then have "{enat (max n m) <..} \<subseteq> A \<inter> B"
      by (auto simp add: subset_eq Ball_def max_def enat_ord_code(1)[symmetric] simp del: enat_ord_code(1))
    then show ?case
      by auto
  next
    case (UN K)
    then obtain k where "k \<in> K" "\<infinity> \<in> k"
      by auto
    with UN.IH[OF this] show ?case
      by auto
  qed auto
qed

lemma nhds_enat: "nhds x = (if x = \<infinity> then INF i. principal {enat i..} else principal {x})"
proof auto
  show "nhds \<infinity> = (INF i. principal {enat i..})"
    unfolding nhds_def
    apply (auto intro!: antisym INF_greatest simp add: open_enat_iff cong: rev_conj_cong)
    apply (auto intro!: INF_lower Ioi_le_Ico) []
    subgoal for x i
      by (auto intro!: INF_lower2[of "Suc i"] simp: subset_eq Ball_def eSuc_enat Suc_ile_eq)
    done
  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 \<le> aa \<Longrightarrow> enat i \<le> aa + ba" for aa ba i
    by (rule order_trans[OF _ add_mono[of aa aa 0 ba]]) auto
  then have [simp]: "enat i \<le> ba \<Longrightarrow> enat i \<le> aa + ba" for aa ba i
    by (metis add.commute)
  fix a b :: enat show "((\<lambda>x. fst x + snd x) \<longlongrightarrow> a + b) (nhds a \<times>\<^sub>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 \<open>
  For more lemmas about the extended real numbers see
  \<^file>\<open>~~/src/HOL/Analysis/Extended_Real_Limits.thy\<close>.
\<close>

subsection \<open>Definition and basic properties\<close>

datatype ereal = ereal real | PInfty | MInfty

lemma ereal_cong: "x = y \<Longrightarrow> 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 "(\<infinity>::ereal) = PInfty"
instance ..

end

declare [[coercion "ereal :: real \<Rightarrow> ereal"]]

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

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

declare
  PInfty_eq_infinity[code_post]
  MInfty_eq_minfinity[code_post]

lemma [code_unfold]:
  "\<infinity> = 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 = \<infinity>"
    | (MInf) "x = -\<infinity>"
  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: "\<And>P. (\<forall>x::ereal. P x) \<longleftrightarrow> P \<infinity> \<and> (\<forall>x. P (ereal x)) \<and> P (-\<infinity>)"
  by (metis ereal_cases)

lemma ereal_ex_split: "\<And>P. (\<exists>x::ereal. P x) \<longleftrightarrow> P \<infinity> \<or> (\<exists>x. P (ereal x)) \<or> P (-\<infinity>)"
  by (metis ereal_cases)

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

function real_of_ereal :: "ereal \<Rightarrow> real" where
  "real_of_ereal (ereal r) = r"
| "real_of_ereal \<infinity> = 0"
| "real_of_ereal (-\<infinity>) = 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 - {\<infinity>, -\<infinity>}"
proof safe
  fix x
  assume "x \<notin> range ereal" "x \<noteq> \<infinity>"
  then show "x = -\<infinity>"
    by (cases x) auto
qed auto

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

instantiation ereal :: abs
begin

function abs_ereal where
  "\<bar>ereal r\<bar> = ereal \<bar>r\<bar>"
| "\<bar>-\<infinity>\<bar> = (\<infinity>::ereal)"
| "\<bar>\<infinity>\<bar> = (\<infinity>::ereal)"
by (auto intro: ereal_cases)
termination proof qed (rule wf_empty)

instance ..

end

lemma abs_eq_infinity_cases[elim!]:
  fixes x :: ereal
  assumes "\<bar>x\<bar> = \<infinity>"
  obtains "x = \<infinity>" | "x = -\<infinity>"
  using assms by (cases x) auto

lemma abs_neq_infinity_cases[elim!]:
  fixes x :: ereal
  assumes "\<bar>x\<bar> \<noteq> \<infinity>"
  obtains r where "x = ereal r"
  using assms by (cases x) auto

lemma abs_ereal_uminus[simp]:
  fixes x :: ereal
  shows "\<bar>- x\<bar> = \<bar>x\<bar>"
  by (cases x) auto

lemma ereal_infinity_cases:
  fixes a :: ereal
  shows "a \<noteq> \<infinity> \<Longrightarrow> a \<noteq> -\<infinity> \<Longrightarrow> \<bar>a\<bar> \<noteq> \<infinity>"
  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)"
| "\<infinity> + a = (\<infinity>::ereal)"
| "a + \<infinity> = (\<infinity>::ereal)"
| "ereal r + -\<infinity> = - \<infinity>"
| "-\<infinity> + ereal p = -(\<infinity>::ereal)"
| "-\<infinity> + -\<infinity> = -(\<infinity>::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]:
  "(\<infinity>::ereal) \<noteq> 0" "0 \<noteq> (\<infinity>::ereal)"
  "-(\<infinity>::ereal) \<noteq> 0" "0 \<noteq> -(\<infinity>::ereal)"
  by (simp_all add: zero_ereal_def)

lemma ereal_eq_0[simp]:
  "ereal r = 0 \<longleftrightarrow> r = 0"
  "0 = ereal r \<longleftrightarrow> r = 0"
  unfolding zero_ereal_def by simp_all

lemma ereal_eq_1[simp]:
  "ereal r = 1 \<longleftrightarrow> r = 1"
  "1 = ereal r \<longleftrightarrow> 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 \<noteq> (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 add: zero_ereal_def[symmetric])

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]: "\<bar>0\<bar> = (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 \<longleftrightarrow> a = 0"
  by (cases a) simp_all

lemma ereal_plus_eq_PInfty[simp]:
  fixes a b :: ereal
  shows "a + b = \<infinity> \<longleftrightarrow> a = \<infinity> \<or> b = \<infinity>"
  by (cases rule: ereal2_cases[of a b]) auto

lemma ereal_plus_eq_MInfty[simp]:
  fixes a b :: ereal
  shows "a + b = -\<infinity> \<longleftrightarrow> (a = -\<infinity> \<or> b = -\<infinity>) \<and> a \<noteq> \<infinity> \<and> b \<noteq> \<infinity>"
  by (cases rule: ereal2_cases[of a b]) auto

lemma ereal_add_cancel_left:
  fixes a b :: ereal
  assumes "a \<noteq> -\<infinity>"
  shows "a + b = a + c \<longleftrightarrow> a = \<infinity> \<or> 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 \<noteq> -\<infinity>"
  shows "b + a = c + a \<longleftrightarrow> a = \<infinity> \<or> b = c"
  using assms by (cases rule: ereal3_cases[of a b c]) auto

lemma ereal_real: "ereal (real_of_ereal x) = (if \<bar>x\<bar> = \<infinity> 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 (\<bar>a\<bar> = \<infinity>) \<and> (\<bar>b\<bar> = \<infinity>) \<or> (\<bar>a\<bar> \<noteq> \<infinity>) \<and> (\<bar>b\<bar> \<noteq> \<infinity>) then real_of_ereal a + real_of_ereal b else 0)"
  by (cases rule: ereal2_cases[of a b]) auto


subsubsection "Linear order on @{typ ereal}"

instantiation ereal :: linorder
begin

function less_ereal
where
  "   ereal x < ereal y     \<longleftrightarrow> x < y"
| "(\<infinity>::ereal) < a           \<longleftrightarrow> False"
| "         a < -(\<infinity>::ereal) \<longleftrightarrow> False"
| "ereal x    < \<infinity>           \<longleftrightarrow> True"
| "        -\<infinity> < ereal r     \<longleftrightarrow> True"
| "        -\<infinity> < (\<infinity>::ereal) \<longleftrightarrow> 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 \<le> (y::ereal) \<longleftrightarrow> x < y \<or> x = y"

lemma ereal_infty_less[simp]:
  fixes x :: ereal
  shows "x < \<infinity> \<longleftrightarrow> (x \<noteq> \<infinity>)"
    "-\<infinity> < x \<longleftrightarrow> (x \<noteq> -\<infinity>)"
  by (cases x, simp_all) (cases x, simp_all)

lemma ereal_infty_less_eq[simp]:
  fixes x :: ereal
  shows "\<infinity> \<le> x \<longleftrightarrow> x = \<infinity>"
    and "x \<le> -\<infinity> \<longleftrightarrow> x = -\<infinity>"
  by (auto simp add: less_eq_ereal_def)

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

lemma ereal_less_eq[simp]:
  "x \<le> (\<infinity>::ereal)"
  "-(\<infinity>::ereal) \<le> x"
  "ereal r \<le> ereal p \<longleftrightarrow> r \<le> p"
  "ereal r \<le> 0 \<longleftrightarrow> r \<le> 0"
  "0 \<le> ereal r \<longleftrightarrow> 0 \<le> r"
  "ereal r \<le> 1 \<longleftrightarrow> r \<le> 1"
  "1 \<le> ereal r \<longleftrightarrow> 1 \<le> r"
  by (auto simp add: less_eq_ereal_def zero_ereal_def one_ereal_def)

lemma ereal_infty_less_eq2:
  "a \<le> b \<Longrightarrow> a = \<infinity> \<Longrightarrow> b = (\<infinity>::ereal)"
  "a \<le> b \<Longrightarrow> b = -\<infinity> \<Longrightarrow> a = -(\<infinity>::ereal)"
  by simp_all

instance
proof
  fix x y z :: ereal
  show "x \<le> x"
    by (cases x) simp_all
  show "x < y \<longleftrightarrow> x \<le> y \<and> \<not> y \<le> x"
    by (cases rule: ereal2_cases[of x y]) auto
  show "x \<le> y \<or> y \<le> x "
    by (cases rule: ereal2_cases[of x y]) auto
  {
    assume "x \<le> y" "y \<le> x"
    then show "x = y"
      by (cases rule: ereal2_cases[of x y]) auto
  }
  {
    assume "x \<le> y" "y \<le> z"
    then show "x \<le> z"
      by (cases rule: ereal3_cases[of x y z]) auto
  }
qed

end

lemma ereal_dense2: "x < y \<Longrightarrow> \<exists>z. x < ereal z \<and> 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 \<le> b"
  then show "c + a \<le> c + b"
    by (cases rule: ereal3_cases[of a b c]) auto
qed

lemma ereal_one_not_less_zero_ereal[simp]: "\<not> 1 < (0::ereal)"
  by (simp add: zero_ereal_def)

lemma real_of_ereal_positive_mono:
  fixes x y :: ereal
  shows "0 \<le> x \<Longrightarrow> x \<le> y \<Longrightarrow> y \<noteq> \<infinity> \<Longrightarrow> real_of_ereal x \<le> real_of_ereal y"
  by (cases rule: ereal2_cases[of x y]) auto

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

lemma ereal_less_PInfty[intro, simp]:
  fixes a :: ereal
  shows "a \<noteq> \<infinity> \<Longrightarrow> a < \<infinity>"
  by (cases a) auto

lemma ereal_less_ereal_Ex:
  fixes a b :: ereal
  shows "x < ereal r \<longleftrightarrow> x = -\<infinity> \<or> (\<exists>p. p < r \<and> x = ereal p)"
  by (cases x) auto

lemma less_PInf_Ex_of_nat: "x \<noteq> \<infinity> \<longleftrightarrow> (\<exists>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_mono:
  fixes a b c d :: ereal
  assumes "a \<le> b"
    and "c \<le> d"
  shows "a + c \<le> b + d"
  using assms
  apply (cases a)
  apply (cases rule: ereal3_cases[of b c d], auto)
  apply (cases rule: ereal3_cases[of b c d], auto)
  done

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

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

lemma ereal_le_real_iff:
  "x \<le> real_of_ereal y \<longleftrightarrow> (\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> ereal x \<le> y) \<and> (\<bar>y\<bar> = \<infinity> \<longrightarrow> x \<le> 0)"
  by (cases y) auto

lemma real_le_ereal_iff:
  "real_of_ereal y \<le> x \<longleftrightarrow> (\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> y \<le> ereal x) \<and> (\<bar>y\<bar> = \<infinity> \<longrightarrow> 0 \<le> x)"
  by (cases y) auto

lemma ereal_less_real_iff:
  "x < real_of_ereal y \<longleftrightarrow> (\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> ereal x < y) \<and> (\<bar>y\<bar> = \<infinity> \<longrightarrow> x < 0)"
  by (cases y) auto

lemma real_less_ereal_iff:
  "real_of_ereal y < x \<longleftrightarrow> (\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> y < ereal x) \<and> (\<bar>y\<bar> = \<infinity> \<longrightarrow> 0 < x)"
  by (cases y) auto

lemma real_of_ereal_pos:
  fixes x :: ereal
  shows "0 \<le> x \<Longrightarrow> 0 \<le> 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 \<le> x \<Longrightarrow> \<bar>x :: ereal\<bar> = x"
  by (cases x) auto

lemma abs_ereal_less0[simp]: "x < 0 \<Longrightarrow> \<bar>x :: ereal\<bar> = -x"
  by (cases x) auto

lemma abs_ereal_pos[simp]: "0 \<le> \<bar>x :: ereal\<bar>"
  by (cases x) auto

lemma ereal_abs_leI:
  fixes x y :: ereal
  shows "\<lbrakk> x \<le> y; -x \<le> y \<rbrakk> \<Longrightarrow> \<bar>x\<bar> \<le> y"
by(cases x y rule: ereal2_cases)(simp_all)

lemma real_of_ereal_le_0[simp]: "real_of_ereal (x :: ereal) \<le> 0 \<longleftrightarrow> x \<le> 0 \<or> x = \<infinity>"
  by (cases x) auto

lemma abs_real_of_ereal[simp]: "\<bar>real_of_ereal (x :: ereal)\<bar> = real_of_ereal \<bar>x\<bar>"
  by (cases x) auto

lemma zero_less_real_of_ereal:
  fixes x :: ereal
  shows "0 < real_of_ereal x \<longleftrightarrow> 0 < x \<and> x \<noteq> \<infinity>"
  by (cases x) auto

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

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

lemma ereal_add_strict_mono:
  fixes a b c d :: ereal
  assumes "a \<le> b"
    and "0 \<le> a"
    and "a \<noteq> \<infinity>"
    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 "\<bar>a\<bar> \<noteq> \<infinity> \<Longrightarrow> c < b \<Longrightarrow> 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 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> a + b = 0 \<longleftrightarrow> a = 0 \<and> b = 0"
  by (cases a b rule: ereal2_cases) auto

lemma ereal_uminus_eq_reorder: "- a = b \<longleftrightarrow> a = (-b::ereal)"
  by auto

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

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

lemma ereal_uminus_le_reorder: "- a \<le> b \<longleftrightarrow> -b \<le> (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 "\<And>B. x \<le> ereal B"
  shows "x = - \<infinity>"
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 "\<And>B. x \<ge> ereal B"
  shows "x = \<infinity>"
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 \<Rightarrow> ereal"
  shows ereal_incseq_uminus[simp]: "incseq (\<lambda>x. - f x) \<longleftrightarrow> decseq f"
    and ereal_decseq_uminus[simp]: "decseq (\<lambda>x. - f x) \<longleftrightarrow> incseq f"
  unfolding decseq_def incseq_def by auto

lemma incseq_ereal: "incseq f \<Longrightarrow> incseq (\<lambda>x. ereal (f x))"
  unfolding incseq_def by auto

lemma ereal_add_nonneg_nonneg[simp]:
  fixes a b :: ereal
  shows "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 0 \<le> a + b"
  using add_mono[of 0 a 0 b] by simp

lemma sum_ereal[simp]: "(\<Sum>x\<in>A. ereal (f x)) = ereal (\<Sum>x\<in>A. 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 (\<lambda>x. ereal (f x)) xs) = ereal (sum_list (map f xs))"
  by (induction xs) simp_all

lemma sum_Pinfty:
  fixes f :: "'a \<Rightarrow> ereal"
  shows "(\<Sum>x\<in>P. f x) = \<infinity> \<longleftrightarrow> finite P \<and> (\<exists>i\<in>P. f i = \<infinity>)"
proof safe
  assume *: "sum f P = \<infinity>"
  show "finite P"
  proof (rule ccontr)
    assume "\<not> finite P"
    with * show False
      by auto
  qed
  show "\<exists>i\<in>P. f i = \<infinity>"
  proof (rule ccontr)
    assume "\<not> ?thesis"
    then have "\<And>i. i \<in> P \<Longrightarrow> f i \<noteq> \<infinity>"
      by auto
    with \<open>finite P\<close> have "sum f P \<noteq> \<infinity>"
      by induct auto
    with * show False
      by auto
  qed
next
  fix i
  assume "finite P" and "i \<in> P" and "f i = \<infinity>"
  then show "sum f P = \<infinity>"
  proof induct
    case (insert x A)
    show ?case using insert by (cases "x = i") auto
  qed simp
qed

lemma sum_Inf:
  fixes f :: "'a \<Rightarrow> ereal"
  shows "\<bar>sum f A\<bar> = \<infinity> \<longleftrightarrow> finite A \<and> (\<exists>i\<in>A. \<bar>f i\<bar> = \<infinity>)"
proof
  assume *: "\<bar>sum f A\<bar> = \<infinity>"
  have "finite A"
    by (rule ccontr) (insert *, auto)
  moreover have "\<exists>i\<in>A. \<bar>f i\<bar> = \<infinity>"
  proof (rule ccontr)
    assume "\<not> ?thesis"
    then have "\<forall>i\<in>A. \<exists>r. f i = ereal r"
      by auto
    from bchoice[OF this] obtain r where "\<forall>x\<in>A. f x = ereal (r x)" ..
    with * show False
      by auto
  qed
  ultimately show "finite A \<and> (\<exists>i\<in>A. \<bar>f i\<bar> = \<infinity>)"
    by auto
next
  assume "finite A \<and> (\<exists>i\<in>A. \<bar>f i\<bar> = \<infinity>)"
  then obtain i where "finite A" "i \<in> A" and "\<bar>f i\<bar> = \<infinity>"
    by auto
  then show "\<bar>sum f A\<bar> = \<infinity>"
  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 \<Rightarrow> ereal"
  assumes "\<And>x. x \<in> S \<Longrightarrow> \<bar>f x\<bar> \<noteq> \<infinity>"
  shows "(\<Sum>x\<in>S. real_of_ereal (f x)) = real_of_ereal (sum f S)"
proof -
  have "\<forall>x\<in>S. \<exists>r. f x = ereal r"
  proof
    fix x
    assume "x \<in> S"
    from assms[OF this] show "\<exists>r. f x = ereal r"
      by (cases "f x") auto
  qed
  from bchoice[OF this] obtain r where "\<forall>x\<in>S. f x = ereal (r x)" ..
  then show ?thesis
    by simp
qed

lemma sum_ereal_0:
  fixes f :: "'a \<Rightarrow> ereal"
  assumes "finite A"
    and "\<And>i. i \<in> A \<Longrightarrow> 0 \<le> f i"
  shows "(\<Sum>x\<in>A. f x) = 0 \<longleftrightarrow> (\<forall>i\<in>A. f i = 0)"
proof
  assume "sum f A = 0" with assms show "\<forall>i\<in>A. f i = 0"
  proof (induction A)
    case (insert a A)
    then have "f a = 0 \<and> (\<Sum>a\<in>A. 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 \<Rightarrow> ereal" where
  "sgn (ereal r) = ereal (sgn r)"
| "sgn (\<infinity>::ereal) = 1"
| "sgn (-\<infinity>::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 * \<infinity> = (if r = 0 then 0 else if r > 0 then \<infinity> else -\<infinity>)"
| "\<infinity> * ereal r = (if r = 0 then 0 else if r > 0 then \<infinity> else -\<infinity>)"
| "ereal r * -\<infinity> = (if r = 0 then 0 else if r > 0 then -\<infinity> else \<infinity>)"
| "-\<infinity> * ereal r = (if r = 0 then 0 else if r > 0 then -\<infinity> else \<infinity>)"
| "(\<infinity>::ereal) * \<infinity> = \<infinity>"
| "-(\<infinity>::ereal) * \<infinity> = -\<infinity>"
| "(\<infinity>::ereal) * -\<infinity> = -\<infinity>"
| "-(\<infinity>::ereal) * -\<infinity> = \<infinity>"
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 add: one_ereal_def[symmetric])

lemma one_not_le_zero_ereal[simp]: "\<not> (1 \<le> (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 \<le> 1 \<Longrightarrow> real_of_ereal a \<le> 1"
  by (cases a) (auto simp: one_ereal_def)

lemma abs_ereal_one[simp]: "\<bar>1\<bar> = (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 \<noteq> (\<infinity>::ereal)" "(\<infinity>::ereal) \<noteq> 1"
  "1 \<noteq> -(\<infinity>::ereal)" "-(\<infinity>::ereal) \<noteq> 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 + -(\<infinity>::ereal) = -\<infinity>"
  "-(\<infinity>::ereal) + 1 = -\<infinity>"
  unfolding one_ereal_def by auto

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

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

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

lemma ereal_abs_mult: "\<bar>x * y :: ereal\<bar> = \<bar>x\<bar> * \<bar>y\<bar>"
  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 * (\<infinity>::ereal) = (if a = 0 then 0 else if 0 < a then \<infinity> else - \<infinity>)"
  by (cases a) auto

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

lemma ereal_mult_strict_right_mono:
  assumes "a < b"
    and "0 < c"
    and "c < (\<infinity>::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 \<Longrightarrow> 0 < c \<Longrightarrow> c < (\<infinity>::ereal) \<Longrightarrow> 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
  shows "a \<le> b \<Longrightarrow> 0 \<le> c \<Longrightarrow> a * c \<le> b * c"
  apply (cases "c = 0")
  apply simp
  apply (cases rule: ereal3_cases[of a b c])
  apply (auto simp: zero_le_mult_iff)
  done

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

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

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

lemma ereal_right_distrib:
  fixes r a b :: ereal
  shows "0 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> 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 \<le> a \<Longrightarrow> 0 \<le> b \<Longrightarrow> (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 \<le> 0 \<longleftrightarrow> (0 \<le> a \<and> b \<le> 0) \<or> (a \<le> 0 \<and> 0 \<le> 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 \<le> a * b \<longleftrightarrow> (0 \<le> a \<and> 0 \<le> b) \<or> (a \<le> 0 \<and> b \<le> 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 \<longleftrightarrow> (0 < a \<and> b < 0) \<or> (a < 0 \<and> 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 \<longleftrightarrow> (0 < a \<and> 0 < b) \<or> (a < 0 \<and> 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 \<Longrightarrow> (d \<noteq> 0 \<Longrightarrow> a = b) \<Longrightarrow> a * c = b * d"
  by (cases "c = 0") simp_all

lemma ereal_right_mult_cong:
  fixes a b c :: ereal
  shows "c = d \<Longrightarrow> (d \<noteq> 0 \<Longrightarrow> a = b) \<Longrightarrow> c * a = d * b"
  by (cases "c = 0") simp_all

lemma ereal_distrib:
  fixes a b c :: ereal
  assumes "a \<noteq> \<infinity> \<or> b \<noteq> -\<infinity>"
    and "a \<noteq> -\<infinity> \<or> b \<noteq> \<infinity>"
    and "\<bar>c\<bar> \<noteq> \<infinity>"
  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)"
  apply (induct w rule: num_induct)
  apply (simp only: numeral_One one_ereal_def)
  apply (simp only: numeral_inc ereal_plus_1)
  done

lemma distrib_left_ereal_nn:
  "c \<ge> 0 \<Longrightarrow> (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 \<Rightarrow> ereal"
  shows "(\<And>i. i \<in> A \<Longrightarrow> 0 \<le> f i) \<Longrightarrow> r * sum f A = (\<Sum>n\<in>A. r * f n)"
  by (induct A rule: infinite_finite_induct)  (auto simp: ereal_right_distrib sum_nonneg)

lemma sum_ereal_left_distrib:
  "(\<And>i. i \<in> A \<Longrightarrow> 0 \<le> f i) \<Longrightarrow> sum f A * r = (\<Sum>n\<in>A. f n * r :: ereal)"
  using sum_ereal_right_distrib[of A f r] by (simp add: mult_ac)

lemma sum_distrib_right_ereal:
  "c \<ge> 0 \<Longrightarrow> sum f A * ereal c = (\<Sum>x\<in>A. f x * c :: ereal)"
by(subst sum_comp_morphism[where h="\<lambda>x. x * ereal c", symmetric])(simp_all add: distrib_left_ereal_nn)

lemma ereal_le_epsilon:
  fixes x y :: ereal
  assumes "\<forall>e. 0 < e \<longrightarrow> x \<le> y + e"
  shows "x \<le> y"
proof -
  {
    assume a: "\<exists>r. y = ereal r"
    then obtain r where r_def: "y = ereal r"
      by auto
    {
      assume "x = -\<infinity>"
      then have ?thesis by auto
    }
    moreover
    {
      assume "x \<noteq> -\<infinity>"
      then obtain p where p_def: "x = ereal p"
      using a assms[rule_format, of 1]
        by (cases x) auto
      {
        fix e
        have "0 < e \<longrightarrow> p \<le> r + e"
          using assms[rule_format, of "ereal e"] p_def r_def by auto
      }
      then have "p \<le> r"
        apply (subst field_le_epsilon)
        apply auto
        done
      then have ?thesis
        using r_def p_def by auto
    }
    ultimately have ?thesis
      by blast
  }
  moreover
  {
    assume "y = -\<infinity> | y = \<infinity>"
    then have ?thesis
      using assms[rule_format, of 1] by (cases x) auto
  }
  ultimately show ?thesis
    by (cases y) auto
qed

lemma ereal_le_epsilon2:
  fixes x y :: ereal
  assumes "\<forall>e. 0 < e \<longrightarrow> x \<le> y + ereal e"
  shows "x \<le> y"
proof -
  {
    fix e :: ereal
    assume "e > 0"
    {
      assume "e = \<infinity>"
      then have "x \<le> y + e"
        by auto
    }
    moreover
    {
      assume "e \<noteq> \<infinity>"
      then obtain r where "e = ereal r"
        using \<open>e > 0\<close> by (cases e) auto
      then have "x \<le> y + e"
        using assms[rule_format, of r] \<open>e>0\<close> by auto
    }
    ultimately have "x \<le> y + e"
      by blast
  }
  then show ?thesis
    using ereal_le_epsilon by auto
qed

lemma ereal_le_real:
  fixes x y :: ereal
  assumes "\<forall>z. x \<le> ereal z \<longrightarrow> y \<le> ereal z"
  shows "y \<le> 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 \<Rightarrow> ereal"
  shows "(\<Prod>i\<in>A. f i) = 0 \<longleftrightarrow> finite A \<and> (\<exists>i\<in>A. f i = 0)"
proof (cases "finite A")
  case True
  then show ?thesis by (induct A) auto
next
  case False
  then show ?thesis by auto
qed

lemma prod_ereal_pos:
  fixes f :: "'a \<Rightarrow> ereal"
  assumes pos: "\<And>i. i \<in> I \<Longrightarrow> 0 \<le> f i"
  shows "0 \<le> (\<Prod>i\<in>I. f i)"
proof (cases "finite I")
  case True
  from this pos show ?thesis
    by induct auto
next
  case False
  then show ?thesis by simp
qed

lemma prod_PInf:
  fixes f :: "'a \<Rightarrow> ereal"
  assumes "\<And>i. i \<in> I \<Longrightarrow> 0 \<le> f i"
  shows "(\<Prod>i\<in>I. f i) = \<infinity> \<longleftrightarrow> finite I \<and> (\<exists>i\<in>I. f i = \<infinity>) \<and> (\<forall>i\<in>I. f i \<noteq> 0)"
proof (cases "finite I")
  case True
  from this assms show ?thesis
  proof (induct I)
    case (insert i I)
    then have pos: "0 \<le> f i" "0 \<le> prod f I"
      by (auto intro!: prod_ereal_pos)
    from insert have "(\<Prod>j\<in>insert i I. f j) = \<infinity> \<longleftrightarrow> prod f I * f i = \<infinity>"
      by auto
    also have "\<dots> \<longleftrightarrow> (prod f I = \<infinity> \<or> f i = \<infinity>) \<and> f i \<noteq> 0 \<and> prod f I \<noteq> 0"
      using prod_ereal_pos[of I f] pos
      by (cases rule: ereal2_cases[of "f i" "prod f I"]) auto
    also have "\<dots> \<longleftrightarrow> finite (insert i I) \<and> (\<exists>j\<in>insert i I. f j = \<infinity>) \<and> (\<forall>j\<in>insert i I. f j \<noteq> 0)"
      using insert by (auto simp: prod_ereal_0)
    finally show ?case .
  qed simp
next
  case False
  then show ?thesis by simp
qed

lemma prod_ereal: "(\<Prod>i\<in>A. 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 \<open>Power\<close>

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

lemma ereal_power_PInf[simp]: "(\<infinity>::ereal) ^ n = (if n = 0 then 1 else \<infinity>)"
  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 \<le> a"
  shows "0 \<le> a ^ n"
  using assms by (induct n) (auto simp: ereal_zero_le_0_iff)


subsubsection \<open>Subtraction\<close>

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)"
  "-\<infinity> - ereal r = -\<infinity>"
  "ereal r - \<infinity> = -\<infinity>"
  "(\<infinity>::ereal) - x = \<infinity>"
  "-(\<infinity>::ereal) - \<infinity> = -\<infinity>"
  "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 \<bar>x\<bar> = \<infinity> then \<infinity> else 0::ereal)"
  by (cases x) simp_all

lemma ereal_eq_minus_iff:
  fixes x y z :: ereal
  shows "x = z - y \<longleftrightarrow>
    (\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> x + y = z) \<and>
    (y = -\<infinity> \<longrightarrow> x = \<infinity>) \<and>
    (y = \<infinity> \<longrightarrow> z = \<infinity> \<longrightarrow> x = \<infinity>) \<and>
    (y = \<infinity> \<longrightarrow> z \<noteq> \<infinity> \<longrightarrow> x = -\<infinity>)"
  by (cases rule: ereal3_cases[of x y z]) auto

lemma ereal_eq_minus:
  fixes x y z :: ereal
  shows "\<bar>y\<bar> \<noteq> \<infinity> \<Longrightarrow> x = z - y \<longleftrightarrow> x + y = z"
  by (auto simp: ereal_eq_minus_iff)

lemma ereal_less_minus_iff:
  fixes x y z :: ereal
  shows "x < z - y \<longleftrightarrow>
    (y = \<infinity> \<longrightarrow> z = \<infinity> \<and> x \<noteq> \<infinity>) \<and>
    (y = -\<infinity> \<longrightarrow> x \<noteq> \<infinity>) \<and>
    (\<bar>y\<bar> \<noteq> \<infinity>\<longrightarrow> x + y < z)"
  by (cases rule: ereal3_cases[of x y z]) auto

lemma ereal_less_minus:
  fixes x y z :: ereal
  shows "\<bar>y\<bar> \<noteq> \<infinity> \<Longrightarrow> x < z - y \<longleftrightarrow> x + y < z"
  by (auto simp: ereal_less_minus_iff)

lemma ereal_le_minus_iff:
  fixes x y z :: ereal
  shows "x \<le> z - y \<longleftrightarrow> (y = \<infinity> \<longrightarrow> z \<noteq> \<infinity> \<longrightarrow> x = -\<infinity>) \<and> (\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> x + y \<le> z)"
  by (cases rule: ereal3_cases[of x y z]) auto

lemma ereal_le_minus:
  fixes x y z :: ereal
  shows "\<bar>y\<bar> \<noteq> \<infinity> \<Longrightarrow> x \<le> z - y \<longleftrightarrow> x + y \<le> z"
  by (auto simp: ereal_le_minus_iff)

lemma ereal_minus_less_iff:
  fixes x y z :: ereal
  shows "x - y < z \<longleftrightarrow> y \<noteq> -\<infinity> \<and> (y = \<infinity> \<longrightarrow> x \<noteq> \<infinity> \<and> z \<noteq> -\<infinity>) \<and> (y \<noteq> \<infinity> \<longrightarrow> x < z + y)"
  by (cases rule: ereal3_cases[of x y z]) auto

lemma ereal_minus_less:
  fixes x y z :: ereal
  shows "\<bar>y\<bar> \<noteq> \<infinity> \<Longrightarrow> x - y < z \<longleftrightarrow> x < z + y"
  by (auto simp: ereal_minus_less_iff)

lemma ereal_minus_le_iff:
  fixes x y z :: ereal
  shows "x - y \<le> z \<longleftrightarrow>
    (y = -\<infinity> \<longrightarrow> z = \<infinity>) \<and>
    (y = \<infinity> \<longrightarrow> x = \<infinity> \<longrightarrow> z = \<infinity>) \<and>
    (\<bar>y\<bar> \<noteq> \<infinity> \<longrightarrow> x \<le> z + y)"
  by (cases rule: ereal3_cases[of x y z]) auto

lemma ereal_minus_le:
  fixes x y z :: ereal
  shows "\<bar>y\<bar> \<noteq> \<infinity> \<Longrightarrow> x - y \<le> z \<longleftrightarrow> x \<le> 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 \<longleftrightarrow>
    b = c \<or> a = \<infinity> \<or> (a = -\<infinity> \<and> b \<noteq> -\<infinity> \<and> c \<noteq> -\<infinity>)"
  by (cases rule: ereal3_cases[of a b c]) auto

lemma ereal_add_le_add_iff:
  fixes a b c :: ereal
  shows "c + a \<le> c + b \<longleftrightarrow>
    a \<le> b \<or> c = \<infinity> \<or> (c = -\<infinity> \<and> a \<noteq> \<infinity> \<and> b \<noteq> \<infinity>)"
  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 \<le> b + c \<longleftrightarrow> a \<le> b \<or> c = \<infinity> \<or> (c = -\<infinity> \<and> a \<noteq> \<infinity> \<and> b \<noteq> \<infinity>)"
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 "\<bar>c\<bar> \<noteq> \<infinity> \<Longrightarrow> c * a \<le> c * b \<longleftrightarrow> (0 < c \<longrightarrow> a \<le> b) \<and> (c < 0 \<longrightarrow> b \<le> 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 \<le> B" "D \<le> C"
  shows "A - C \<le> 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 \<le> c - b \<Longrightarrow> 0 \<le> c \<Longrightarrow> c < \<infinity> \<Longrightarrow> b \<le> 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 \<bar>a\<bar> = \<infinity> \<or> \<bar>b\<bar> = \<infinity> 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': "\<bar>x\<bar> = \<infinity> \<longleftrightarrow> \<bar>y\<bar> = \<infinity> \<Longrightarrow> 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 \<le> b \<Longrightarrow> 0 \<le> b - a"
  by (cases rule: ereal2_cases[of a b]) auto

lemma ereal_between:
  fixes x e :: ereal
  assumes "\<bar>x\<bar> \<noteq> \<infinity>"
    and "0 < e"
  shows "x - e < x"
    and "x < x + e"
  using assms
  apply (cases x, cases e)
  apply auto
  using assms
  apply (cases x, cases e)
  apply auto
  done

lemma ereal_minus_eq_PInfty_iff:
  fixes x y :: ereal
  shows "x - y = \<infinity> \<longleftrightarrow> y = -\<infinity> \<or> x = \<infinity>"
  by (cases x y rule: ereal2_cases) simp_all

lemma ereal_diff_add_eq_diff_diff_swap:
  fixes x y z :: ereal
  shows "\<bar>y\<bar> \<noteq> \<infinity> \<Longrightarrow> 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 "\<lbrakk> x = \<infinity> \<longrightarrow> y \<noteq> \<infinity>; x = -\<infinity> \<longrightarrow> y \<noteq> - \<infinity> \<rbrakk> \<Longrightarrow> - (x - y) = y - x"
by(cases x y rule: ereal2_cases) simp_all

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

subsubsection \<open>Division\<close>

instantiation ereal :: inverse
begin

function inverse_ereal where
  "inverse (ereal r) = (if r = 0 then \<infinity> else ereal (inverse r))"
| "inverse (\<infinity>::ereal) = 0"
| "inverse (-\<infinity>::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) = \<infinity>"
  "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 * \<infinity> 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 \<bar>x\<bar> = \<infinity> \<or> 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 \<noteq> -\<infinity> then x else \<infinity>)"
  by (cases x) auto

lemma ereal_inverse_minus[simp]:
  fixes x :: ereal
  shows "inverse (- x) = (if x = 0 then \<infinity> 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 / \<infinity> = 0" "x / -\<infinity> = 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]: "\<infinity> / ereal r = (if 0 \<le> r then \<infinity> else -\<infinity>)"
  unfolding divide_ereal_def by simp

lemma ereal_inverse_nonneg_iff: "0 \<le> inverse (x :: ereal) \<longleftrightarrow> 0 \<le> x \<or> x = -\<infinity>"
  by (cases x) auto

lemma inverse_ereal_ge0I: "0 \<le> (x :: ereal) \<Longrightarrow> 0 \<le> inverse x"
by(cases x) simp_all

lemma zero_le_divide_ereal[simp]:
  fixes a :: ereal
  assumes "0 \<le> a"
    and "0 \<le> b"
  shows "0 \<le> 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 \<Longrightarrow> x \<noteq> \<infinity> \<Longrightarrow> y \<le> z / x \<longleftrightarrow> x * y \<le> 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 \<Longrightarrow> x \<noteq> \<infinity> \<Longrightarrow> z / x \<le> y \<longleftrightarrow> z \<le> 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 \<Longrightarrow> x \<noteq> -\<infinity> \<Longrightarrow> y \<le> z / x \<longleftrightarrow> z \<le> 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 \<Longrightarrow> x \<noteq> -\<infinity> \<Longrightarrow> z / x \<le> y \<longleftrightarrow> x * y \<le> 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 \<le> x \<Longrightarrow> x < y \<Longrightarrow> inverse y < inverse x"
  by (cases rule: ereal2_cases[of x y]) auto

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

lemma inverse_inverse_Pinfty_iff[simp]:
  fixes x :: ereal
  shows "inverse x = \<infinity> \<longleftrightarrow> x = 0"
  by (cases x) auto

lemma ereal_inverse_eq_0:
  fixes x :: ereal
  shows "inverse x = 0 \<longleftrightarrow> x = \<infinity> \<or> x = -\<infinity>"
  by (cases x) auto

lemma ereal_0_gt_inverse:
  fixes x :: ereal
  shows "0 < inverse x \<longleftrightarrow> x \<noteq> \<infinity> \<and> 0 \<le> x"
  by (cases x) auto

lemma ereal_inverse_le_0_iff:
  fixes x :: ereal
  shows "inverse x \<le> 0 \<longleftrightarrow> x < 0 \<or> x = \<infinity>"
  by(cases x) auto

lemma ereal_divide_eq_0_iff: "x / y = 0 \<longleftrightarrow> x = 0 \<or> \<bar>y :: ereal\<bar> = \<infinity>"
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 < \<infinity>"
  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 \<Longrightarrow> b < \<infinity> \<Longrightarrow> b * (a / b) = a"
  by (cases a b rule: ereal2_cases) auto

lemma ereal_power_divide:
  fixes x y :: ereal
  shows "y \<noteq> 0 \<Longrightarrow> (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 \<noteq> -\<infinity>"
  assumes z: "\<And>z. 0 < z \<Longrightarrow> z < 1 \<Longrightarrow> z * x \<le> y"
  shows "x \<le> y"
proof (cases x)
  case PInf
  with z[of "1 / 2"] show "x \<le> y"
    by (simp add: one_ereal_def)
next
  case (real r)
  note r = this
  show "x \<le> y"
  proof (cases y)
    case (real p)
    note p = this
    have "r \<le> 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 \<le> p"
        using p r by (auto simp: zero_le_mult_iff one_ereal_def)
    qed
    then show "x \<le> 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 \<le> y"
    and "0 < z"
  shows "x / z \<le> 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 \<le> x"
    and "0 < z"
    and "0 < x * y"
  shows "z / x \<le> 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: "\<bar>z\<bar> \<noteq> \<infinity> \<Longrightarrow> z \<noteq> 0 \<Longrightarrow> ereal (inverse (real_of_ereal z)) = inverse z"
  by (cases z) simp_all

lemma ereal_inverse_mult:
  "a \<noteq> 0 \<Longrightarrow> b \<noteq> 0 \<Longrightarrow> inverse (a * (b::ereal)) = inverse a * inverse b"
  by (cases a; cases b) auto


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 = (-\<infinity>::ereal)"
definition "top = (\<infinity>::ereal)"

definition "Sup S = (SOME x :: ereal. (\<forall>y\<in>S. y \<le> x) \<and> (\<forall>z. (\<forall>y\<in>S. y \<le> z) \<longrightarrow> x \<le> z))"
definition "Inf S = (SOME x :: ereal. (\<forall>y\<in>S. x \<le> y) \<and> (\<forall>z. (\<forall>y\<in>S. z \<le> y) \<longrightarrow> z \<le> x))"

lemma ereal_complete_Sup:
  fixes S :: "ereal set"
  shows "\<exists>x. (\<forall>y\<in>S. y \<le> x) \<and> (\<forall>z. (\<forall>y\<in>S. y \<le> z) \<longrightarrow> x \<le> z)"
proof (cases "\<exists>x. \<forall>a\<in>S. a \<le> ereal x")
  case True
  then obtain y where y: "a \<le> ereal y" if "a\<in>S" for a
    by auto
  then have "\<infinity> \<notin> S"
    by force
  show ?thesis
  proof (cases "S \<noteq> {-\<infinity>} \<and> S \<noteq> {}")
    case True
    with \<open>\<infinity> \<notin> S\<close> obtain x where x: "x \<in> S" "\<bar>x\<bar> \<noteq> \<infinity>"
      by auto
    obtain s where s: "\<forall>x\<in>ereal -` S. x \<le> s" "(\<forall>x\<in>ereal -` S. x \<le> z) \<Longrightarrow> s \<le> z" for z
    proof (atomize_elim, rule complete_real)
      show "\<exists>x. x \<in> ereal -` S"
        using x by auto
      show "\<exists>z. \<forall>x\<in>ereal -` S. x \<le> z"
        by (auto dest: y intro!: exI[of _ y])
    qed
    show ?thesis
    proof (safe intro!: exI[of _ "ereal s"])
      fix y
      assume "y \<in> S"
      with s \<open>\<infinity> \<notin> S\<close> show "y \<le> ereal s"
        by (cases y) auto
    next
      fix z
      assume "\<forall>y\<in>S. y \<le> z"
      with \<open>S \<noteq> {-\<infinity>} \<and> S \<noteq> {}\<close> show "ereal s \<le> z"
        by (cases z) (auto intro!: s)
    qed
  next
    case False
    then show ?thesis
      by (auto intro!: exI[of _ "-\<infinity>"])
  qed
next
  case False
  then show ?thesis
    by (fastforce intro!: exI[of _ \<infinity>] ereal_top intro: order_trans dest: less_imp_le simp: not_le)
qed

lemma ereal_complete_uminus_eq:
  fixes S :: "ereal set"
  shows "(\<forall>y\<in>uminus`S. y \<le> x) \<and> (\<forall>z. (\<forall>y\<in>uminus`S. y \<le> z) \<longrightarrow> x \<le> z)
     \<longleftrightarrow> (\<forall>y\<in>S. -x \<le> y) \<and> (\<forall>z. (\<forall>y\<in>S. z \<le> y) \<longrightarrow> z \<le> -x)"
  by simp (metis ereal_minus_le_minus ereal_uminus_uminus)

lemma ereal_complete_Inf:
  "\<exists>x. (\<forall>y\<in>S::ereal set. x \<le> y) \<and> (\<forall>z. (\<forall>y\<in>S. z \<le> y) \<longrightarrow> z \<le> x)"
  using ereal_complete_Sup[of "uminus ` S"]
  unfolding ereal_complete_uminus_eq
  by auto

instance
proof
  show "Sup {} = (bot::ereal)"
    apply (auto simp: bot_ereal_def Sup_ereal_def)
    apply (rule some1_equality)
    apply (metis ereal_bot ereal_less_eq(2))
    apply (metis ereal_less_eq(2))
    done
  show "Inf {} = (top::ereal)"
    apply (auto simp: top_ereal_def Inf_ereal_def)
    apply (rule some1_equality)
    apply (metis ereal_top ereal_less_eq(1))
    apply (metis ereal_less_eq(1))
    done
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 "\<exists>a b::ereal. a \<noteq> b"
    using zero_neq_one by blast
qed

subsubsection "Topological space"

instantiation ereal :: linear_continuum_topology
begin

definition "open_ereal" :: "ereal set \<Rightarrow> bool" where
  open_ereal_generated: "open_ereal = generate_topology (range lessThan \<union> 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 (\<lambda>x. 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 \<longlongrightarrow> x) F \<Longrightarrow> ((\<lambda>x. ereal (f x)) \<longlongrightarrow> ereal x) F"
  using isCont_tendsto_compose[of x ereal f F] continuous_on_ereal[of UNIV "\<lambda>x. x"]
  by (simp add: continuous_on_eq_continuous_at)

lemma tendsto_uminus_ereal[tendsto_intros, simp, intro]: "(f \<longlongrightarrow> x) F \<Longrightarrow> ((\<lambda>x. - f x::ereal) \<longlongrightarrow> - x) F"
  apply (rule tendsto_compose[where g=uminus])
  apply (auto intro!: order_tendstoI simp: eventually_at_topological)
  apply (rule_tac x="{..< -a}" in exI)
  apply (auto split: ereal.split simp: ereal_less_uminus_reorder) []
  apply (rule_tac x="{- a <..}" in exI)
  apply (auto split: ereal.split simp: ereal_uminus_reorder) []
  done

lemma at_infty_ereal_eq_at_top: "at \<infinity> = 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 \<longlongrightarrow> f0) net \<longleftrightarrow> ((\<lambda>x. - f x::ereal) \<longlongrightarrow> - f0) net"
  using tendsto_uminus_ereal[of f f0 net] tendsto_uminus_ereal[of "\<lambda>x. - f x" "- f0" net]
  by auto

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

lemma ereal_less_divide_iff: "0 < (c::ereal) \<Longrightarrow> c < \<infinity> \<Longrightarrow> a < b / c \<longleftrightarrow> 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: "\<bar>c\<bar> \<noteq> \<infinity>" and f: "(f \<longlongrightarrow> x) F" shows "((\<lambda>x. c * f x::ereal) \<longlongrightarrow> c * x) F"
proof -
  { fix c :: ereal assume "0 < c" "c < \<infinity>"
    then have "((\<lambda>x. c * f x::ereal) \<longlongrightarrow> 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 \<and> c < \<infinity>) \<or> (-\<infinity> < c \<and> c < 0) \<or> c = 0)"
    using c by (cases c) auto
  then show ?thesis
  proof (elim disjE conjE)
    assume "- \<infinity> < c" "c < 0"
    then have "0 < - c" "- c < \<infinity>"
      by (auto simp: ereal_uminus_reorder ereal_less_uminus_reorder[of 0])
    then have "((\<lambda>x. (- c) * f x) \<longlongrightarrow> (- 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 \<noteq> 0" and f: "(f \<longlongrightarrow> x) F" shows "((\<lambda>x. c * f x::ereal) \<longlongrightarrow> c * x) F"
proof cases
  assume "\<bar>c\<bar> = \<infinity>"
  show ?thesis
  proof (rule filterlim_cong[THEN iffD1, OF refl refl _ tendsto_const])
    have "0 < x \<or> x < 0"
      using \<open>x \<noteq> 0\<close> by (auto simp add: neq_iff)
    then show "eventually (\<lambda>x'. c * x = c * f x') F"
    proof
      assume "0 < x" from order_tendstoD(1)[OF f this] show ?thesis
        by eventually_elim (insert \<open>0<x\<close> \<open>\<bar>c\<bar> = \<infinity>\<close>, auto)
    next
      assume "x < 0" from order_tendstoD(2)[OF f this] show ?thesis
        by eventually_elim (insert \<open>x<0\<close> \<open>\<bar>c\<bar> = \<infinity>\<close>, auto)
    qed
  qed
qed (rule tendsto_cmult_ereal[OF _ f])

lemma tendsto_cadd_ereal[tendsto_intros, simp, intro]:
  assumes c: "y \<noteq> - \<infinity>" "x \<noteq> - \<infinity>" and f: "(f \<longlongrightarrow> x) F" shows "((\<lambda>x. f x + y::ereal) \<longlongrightarrow> 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: "\<bar>y\<bar> \<noteq> \<infinity>" and f: "(f \<longlongrightarrow> x) F" shows "((\<lambda>x. f x + y::ereal) \<longlongrightarrow> 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 \<Longrightarrow> continuous F (\<lambda>x. ereal (f x))"
  unfolding continuous_def by auto

lemma ereal_Sup:
  assumes *: "\<bar>SUP a:A. ereal a\<bar> \<noteq> \<infinity>"
  shows "ereal (Sup A) = (SUP a:A. ereal a)"
proof (rule continuous_at_Sup_mono)
  obtain r where r: "ereal r = (SUP a:A. ereal a)" "A \<noteq> {}"
    using * by (force simp: bot_ereal_def)
  then show "bdd_above A" "A \<noteq> {}"
    by (auto intro!: SUP_upper bdd_aboveI[of _ r] simp add: ereal_less_eq(3)[symmetric] simp del: ereal_less_eq)
qed (auto simp: mono_def continuous_at_imp_continuous_at_within continuous_at_ereal)

lemma ereal_SUP: "\<bar>SUP a:A. ereal (f a)\<bar> \<noteq> \<infinity> \<Longrightarrow> ereal (SUP a:A. f a) = (SUP a:A. ereal (f a))"
  using ereal_Sup[of "f`A"] by auto

lemma ereal_Inf:
  assumes *: "\<bar>INF a:A. ereal a\<bar> \<noteq> \<infinity>"
  shows "ereal (Inf A) = (INF a:A. ereal a)"
proof (rule continuous_at_Inf_mono)
  obtain r where r: "ereal r = (INF a:A. ereal a)" "A \<noteq> {}"
    using * by (force simp: top_ereal_def)
  then show "bdd_below A" "A \<noteq> {}"
    by (auto intro!: INF_lower bdd_belowI[of _ r] simp add: ereal_less_eq(3)[symmetric] simp del: 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 \<noteq> {}"
  shows "ereal (Inf A) = (INF a:A. ereal a)"
proof (rule ereal_Inf)
  from * obtain l u where "x \<in> A \<Longrightarrow> l \<le> x" "u \<in> A" for x
    by (auto simp: bdd_below_def)
  then have "l \<le> (INF x:A. ereal x)" "(INF x:A. ereal x) \<le> u"
    by (auto intro!: INF_greatest INF_lower)
  then show "\<bar>INF a:A. ereal a\<bar> \<noteq> \<infinity>"
    by auto
qed

lemma ereal_INF: "\<bar>INF a:A. ereal (f a)\<bar> \<noteq> \<infinity> \<Longrightarrow> ereal (INF a:A. f a) = (INF a:A. ereal (f a))"
  using ereal_Inf[of "f`A"] by auto

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 \<Rightarrow> ereal"
  shows "(SUP x:S. uminus (f x)) = - (INF x:S. f x)"
  using ereal_Sup_uminus_image_eq [of "f ` S"] by (simp add: comp_def)

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 \<Rightarrow> ereal"
  shows "(INF x:S. - f x) = - (SUP x:S. f x)"
  using ereal_Inf_uminus_image_eq [of "f ` S"] by (simp add: comp_def)

lemma ereal_SUP_uminus:
  fixes f :: "'a \<Rightarrow> 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 :: "_ \<Rightarrow> ereal"
  shows "A \<noteq> {} \<Longrightarrow> l \<noteq> -\<infinity> \<Longrightarrow> u \<noteq> \<infinity> \<Longrightarrow> \<forall>a\<in>A. l \<le> f a \<and> f a \<le> u \<Longrightarrow> \<bar>SUPREMUM A f\<bar> \<noteq> \<infinity>"
  using SUP_upper2[of _ A l f] SUP_least[of A f u]
  by (cases "SUPREMUM A f") auto

lemma ereal_INF_not_infty:
  fixes f :: "_ \<Rightarrow> ereal"
  shows "A \<noteq> {} \<Longrightarrow> l \<noteq> -\<infinity> \<Longrightarrow> u \<noteq> \<infinity> \<Longrightarrow> \<forall>a\<in>A. l \<le> f a \<and> f a \<le> u \<Longrightarrow> \<bar>INFIMUM A f\<bar> \<noteq> \<infinity>"
  using INF_lower2[of _ A f u] INF_greatest[of A l f]
  by (cases "INFIMUM A f") auto

lemma ereal_image_uminus_shift:
  fixes X Y :: "ereal set"
  shows "uminus ` X = Y \<longleftrightarrow> 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 = -\<infinity> \<longleftrightarrow> S = {} \<or> S = {-\<infinity>}"
  unfolding bot_ereal_def[symmetric] by auto

lemma Inf_eq_PInfty:
  fixes S :: "ereal set"
  shows "Inf S = \<infinity> \<longleftrightarrow> S = {} \<or> S = {\<infinity>}"
  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 "-\<infinity> \<in> S \<Longrightarrow> Inf S = -\<infinity>"
  unfolding bot_ereal_def[symmetric] by auto

lemma Sup_eq_PInfty:
  fixes S :: "ereal set"
  shows "\<infinity> \<in> S \<Longrightarrow> Sup S = \<infinity>"
  unfolding top_ereal_def[symmetric] by auto

lemma not_MInfty_nonneg[simp]: "0 \<le> (x::ereal) \<Longrightarrow> x \<noteq> - \<infinity>"
  by auto

lemma Sup_ereal_close:
  fixes e :: ereal
  assumes "0 < e"
    and S: "\<bar>Sup S\<bar> \<noteq> \<infinity>" "S \<noteq> {}"
  shows "\<exists>x\<in>S. Sup S - e < x"
  using assms by (cases e) (auto intro!: less_Sup_iff[THEN iffD1])

lemma Inf_ereal_close:
  fixes e :: ereal
  assumes "\<bar>Inf X\<bar> \<noteq> \<infinity>"
    and "0 < e"
  shows "\<exists>x\<in>X. 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:
  "(\<And>n::nat. \<exists>i\<in>A. ereal (real n) \<le> f i) \<Longrightarrow> (SUP i:A. f i :: ereal) = \<infinity>"
  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::nat. ereal (real i)) = \<infinity>"
  by (rule SUP_PInfty) auto

lemma SUP_ereal_add_left:
  assumes "I \<noteq> {}" "c \<noteq> -\<infinity>"
  shows "(SUP i:I. f i + c :: ereal) = (SUP i:I. f i) + c"
proof (cases "(SUP i:I. f i) = - \<infinity>")
  case True
  then have "\<And>i. i \<in> I \<Longrightarrow> f i = -\<infinity>"
    unfolding Sup_eq_MInfty by auto
  with True show ?thesis
    by (cases c) (auto simp: \<open>I \<noteq> {}\<close>)
next
  case False
  then show ?thesis
    by (subst continuous_at_Sup_mono[where f="\<lambda>x. x + c"])
       (auto simp: continuous_at_imp_continuous_at_within continuous_at mono_def ereal_add_mono \<open>I \<noteq> {}\<close> \<open>c \<noteq> -\<infinity>\<close>)
qed

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

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

lemma SUP_ereal_minus_left:
  assumes "I \<noteq> {}" "c \<noteq> \<infinity>"
  shows "(SUP i:I. f i - c:: ereal) = (SUP i:I. f i) - c"
  using SUP_ereal_add_left[OF \<open>I \<noteq> {}\<close>, of "-c" f] by (simp add: \<open>c \<noteq> \<infinity>\<close> minus_ereal_def)

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

lemma SUP_ereal_le_addI:
  fixes f :: "'i \<Rightarrow> ereal"
  assumes "\<And>i. f i + y \<le> z" and "y \<noteq> -\<infinity>"
  shows "SUPREMUM UNIV f + y \<le> z"
  unfolding SUP_ereal_add_left[OF UNIV_not_empty \<open>y \<noteq> -\<infinity>\<close>, symmetric]
  by (rule SUP_least assms)+

lemma SUP_combine:
  fixes f :: "'a::semilattice_sup \<Rightarrow> 'a::semilattice_sup \<Rightarrow> 'b::complete_lattice"
  assumes mono: "\<And>a b c d. a \<le> b \<Longrightarrow> c \<le> d \<Longrightarrow> f a c \<le> f b d"
  shows "(SUP i:UNIV. SUP j:UNIV. f i j) = (SUP i. f i i)"
proof (rule antisym)
  show "(SUP i j. f i j) \<le> (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) \<le> (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 \<Rightarrow> ereal"
  assumes inc: "incseq f" "incseq g"
    and pos: "\<And>i. f i \<noteq> -\<infinity>" "\<And>i. g i \<noteq> -\<infinity>"
  shows "(SUP i. f i + g i) = SUPREMUM UNIV f + SUPREMUM UNIV g"
  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] ereal_add_mono inc[THEN monoD] | assumption)+
  done

lemma INF_eq_minf: "(INF i:I. f i::ereal) \<noteq> -\<infinity> \<longleftrightarrow> (\<exists>b>-\<infinity>. \<forall>i\<in>I. b \<le> f i)"
  unfolding bot_ereal_def[symmetric] INF_eq_bot_iff by (auto simp: not_less)

lemma INF_ereal_add_left:
  assumes "I \<noteq> {}" "c \<noteq> -\<infinity>" "\<And>x. x \<in> I \<Longrightarrow> 0 \<le> f x"
  shows "(INF i:I. f i + c :: ereal) = (INF i:I. f i) + c"
proof -
  have "(INF i:I. f i) \<noteq> -\<infinity>"
    unfolding INF_eq_minf using assms by (intro exI[of _ 0]) auto
  then show ?thesis
    by (subst continuous_at_Inf_mono[where f="\<lambda>x. x + c"])
       (auto simp: mono_def ereal_add_mono \<open>I \<noteq> {}\<close> \<open>c \<noteq> -\<infinity>\<close> continuous_at_imp_continuous_at_within continuous_at)
qed

lemma INF_ereal_add_right:
  assumes "I \<noteq> {}" "c \<noteq> -\<infinity>" "\<And>x. x \<in> I \<Longrightarrow> 0 \<le> f x"
  shows "(INF i:I. c + f i :: ereal) = c + (INF i:I. f i)"
  using INF_ereal_add_left[OF assms] by (simp add: ac_simps)

lemma INF_ereal_add_directed:
  fixes f g :: "'a \<Rightarrow> ereal"
  assumes nonneg: "\<And>i. i \<in> I \<Longrightarrow> 0 \<le> f i" "\<And>i. i \<in> I \<Longrightarrow> 0 \<le> g i"
  assumes directed: "\<And>i j. i \<in> I \<Longrightarrow> j \<in> I \<Longrightarrow> \<exists>k\<in>I. f i + g j \<ge> f k + g k"
  shows "(INF i:I. f i + g i) = (INF i:I. f i) + (INF i:I. g i)"
proof cases
  assume "I = {}" then show ?thesis
    by (simp add: top_ereal_def)
next
  assume "I \<noteq> {}"
  show ?thesis
  proof (rule antisym)
    show "(INF i:I. f i) + (INF i:I. g i) \<le> (INF i:I. f i + g i)"
      by (rule INF_greatest; intro ereal_add_mono INF_lower)
  next
    have "(INF i:I. f i + g i) \<le> (INF i:I. (INF j:I. f i + g j))"
      using directed by (intro INF_greatest) (blast intro: INF_lower2)
    also have "\<dots> = (INF i:I. f i + (INF i:I. g i))"
      using nonneg by (intro INF_cong refl INF_ereal_add_right \<open>I \<noteq> {}\<close>) (auto simp: INF_eq_minf intro!: exI[of _ 0])
    also have "\<dots> = (INF i:I. f i) + (INF i:I. g i)"
      using nonneg by (intro INF_ereal_add_left \<open>I \<noteq> {}\<close>) (auto simp: INF_eq_minf intro!: exI[of _ 0])
    finally show "(INF i:I. f i + g i) \<le> (INF i:I. f i) + (INF i:I. g i)" .
  qed
qed

lemma INF_ereal_add:
  fixes f :: "nat \<Rightarrow> ereal"
  assumes "decseq f" "decseq g"
    and fin: "\<And>i. f i \<noteq> \<infinity>" "\<And>i. g i \<noteq> \<infinity>"
  shows "(INF i. f i + g i) = INFIMUM UNIV f + INFIMUM UNIV g"
proof -
  have INF_less: "(INF i. f i) < \<infinity>" "(INF i. g i) < \<infinity>"
    using assms unfolding INF_less_iff by auto
  { fix a b :: ereal assume "a \<noteq> \<infinity>" "b \<noteq> \<infinity>"
    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 "\<dots> = INFIMUM UNIV f + INFIMUM UNIV g"
    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 \<Rightarrow> ereal"
  assumes inc: "incseq f" "incseq g"
    and pos: "\<And>i. 0 \<le> f i" "\<And>i. 0 \<le> g i"
  shows "(SUP i. f i + g i) = SUPREMUM UNIV f + SUPREMUM UNIV g"
proof (intro SUP_ereal_add inc)
  fix i
  show "f i \<noteq> -\<infinity>" "g i \<noteq> -\<infinity>"
    using pos[of i] by auto
qed

lemma SUP_ereal_sum:
  fixes f g :: "'a \<Rightarrow> nat \<Rightarrow> ereal"
  assumes "\<And>n. n \<in> A \<Longrightarrow> incseq (f n)"
    and pos: "\<And>n i. n \<in> A \<Longrightarrow> 0 \<le> f n i"
  shows "(SUP i. \<Sum>n\<in>A. f n i) = (\<Sum>n\<in>A. SUPREMUM UNIV (f n))"
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 \<Rightarrow> ereal"
  assumes "I \<noteq> {}"
  assumes f: "\<And>i. i \<in> I \<Longrightarrow> 0 \<le> f i" and c: "0 \<le> c"
  shows "(SUP i:I. c * f i) = c * (SUP i:I. f i)"
proof (cases "(SUP i: I. f i) = 0")
  case True
  then have "\<And>i. i \<in> I \<Longrightarrow> 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="\<lambda>x. c * x"])
       (auto simp: mono_def continuous_at continuous_at_imp_continuous_at_within \<open>I \<noteq> {}\<close>
             intro!: ereal_mult_left_mono c)
qed

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

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

  have "\<exists>f. \<forall>n. (f n \<in> A \<and> l n \<le> f n) \<and> (f n \<le> f (Suc n))"
  proof (rule dependent_nat_choice)
    show "\<exists>x. x \<in> A \<and> l 0 \<le> x"
      using l[of 0] by (auto simp: less_Sup_iff)
  next
    fix x n assume "x \<in> A \<and> l n \<le> x"
    moreover from l[of "Suc n"] obtain y where "y \<in> A" "l (Suc n) < y"
      by (auto simp: less_Sup_iff)
    ultimately show "\<exists>y. (y \<in> A \<and> l (Suc n) \<le> y) \<and> x \<le> y"
      by (auto intro!: exI[of _ "max x y"] split: split_max)
  qed
  then guess f .. note f = this
  then have "range f \<subseteq> A" "incseq f"
    by (auto simp: incseq_Suc_iff)
  moreover
  have "(SUP i. f i) = Sup A"
  proof (rule tendsto_unique)
    show "f \<longlonglongrightarrow> (SUP i. f i)"
      by (rule LIMSEQ_SUP \<open>incseq f\<close>)+
    show "f \<longlonglongrightarrow> 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 \<noteq> {}" shows "\<exists>f::nat \<Rightarrow> ereal. decseq f \<and> range f \<subseteq> A \<and> Inf A = (INF i. f i)"
proof -
  obtain f where "incseq f" "range f \<subseteq> uminus`A" "Sup (uminus`A) = (SUP i. f i)"
    using Sup_countable_SUP[of "uminus ` A"] \<open>A \<noteq> {}\<close> by auto
  then show ?thesis
    by (intro exI[of _ "\<lambda>x. - f x"])
       (auto simp: ereal_Sup_uminus_image_eq ereal_INF_uminus_eq eq_commute[of "- _"])
qed

lemma SUP_countable_SUP:
  "A \<noteq> {} \<Longrightarrow> \<exists>f::nat \<Rightarrow> ereal. range f \<subseteq> g`A \<and> SUPREMUM A g = SUPREMUM UNIV f"
  using Sup_countable_SUP [of "g`A"] by auto

subsection "Relation to @{typ enat}"

definition "ereal_of_enat n = (case n of enat n \<Rightarrow> ereal (real n) | \<infinity> \<Rightarrow> \<infinity>)"

declare [[coercion "ereal_of_enat :: enat \<Rightarrow> ereal"]]
declare [[coercion "(\<lambda>n. ereal (real n)) :: nat \<Rightarrow> ereal"]]

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

lemma ereal_of_enat_le_iff[simp]: "ereal_of_enat m \<le> ereal_of_enat n \<longleftrightarrow> m \<le> n"
  by (cases m n rule: enat2_cases) auto

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

lemma numeral_le_ereal_of_enat_iff[simp]: "numeral m \<le> ereal_of_enat n \<longleftrightarrow> numeral m \<le> n"
by (cases n) (auto)

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

lemma ereal_of_enat_ge_zero_cancel_iff[simp]: "0 \<le> ereal_of_enat n \<longleftrightarrow> 0 \<le> n"
  by (cases n) (auto simp: enat_0[symmetric])

lemma ereal_of_enat_gt_zero_cancel_iff[simp]: "0 < ereal_of_enat n \<longleftrightarrow> 0 < n"
  by (cases n) (auto simp: enat_0[symmetric])

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

lemma ereal_of_enat_inf[simp]: "ereal_of_enat n = \<infinity> \<longleftrightarrow> n = \<infinity>"
  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 \<le> 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 \<ge> 0"
by(cases n) simp_all

lemma ereal_of_enat_Sup:
  assumes "A \<noteq> {}" shows "ereal_of_enat (Sup A) = (SUP a : A. ereal_of_enat a)"
proof (intro antisym mono_Sup)
  show "ereal_of_enat (Sup A) \<le> (SUP a : A. ereal_of_enat a)"
  proof cases
    assume "finite A"
    with \<open>A \<noteq> {}\<close> obtain a where "a \<in> 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 "\<not> 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 \<in> A - enat ` {.. n}"
        by (metis \<open>\<not> finite A\<close> all_not_in_conv finite_Diff2 finite_atMost finite_imageI finite.emptyI)
      then have "a \<in> A" "ereal n \<le> 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 \<open>x < n\<close> show "\<exists>i\<in>A. 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 \<noteq> {} \<Longrightarrow> ereal_of_enat (SUP a:A. f a) = (SUP a : A. ereal_of_enat (f a))"
  using ereal_of_enat_Sup[of "f`A"] by auto

subsection "Limits on @{typ ereal}"

lemma open_PInfty: "open A \<Longrightarrow> \<infinity> \<in> A \<Longrightarrow> (\<exists>x. {ereal x<..} \<subseteq> A)"
  unfolding open_ereal_generated
proof (induct rule: generate_topology.induct)
  case (Int A B)
  then obtain x z where "\<infinity> \<in> A \<Longrightarrow> {ereal x <..} \<subseteq> A" "\<infinity> \<in> B \<Longrightarrow> {ereal z <..} \<subseteq> B"
    by auto
  with Int show ?case
    by (intro exI[of _ "max x z"]) fastforce
next
  case (Basis S)
  {
    fix x
    have "x \<noteq> \<infinity> \<Longrightarrow> \<exists>t. x \<le> 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 \<Longrightarrow> -\<infinity> \<in> A \<Longrightarrow> (\<exists>x. {..<ereal x} \<subseteq> A)"
  unfolding open_ereal_generated
proof (induct rule: generate_topology.induct)
  case (Int A B)
  then obtain x z where "-\<infinity> \<in> A \<Longrightarrow> {..< ereal x} \<subseteq> A" "-\<infinity> \<in> B \<Longrightarrow> {..< ereal z} \<subseteq> B"
    by auto
  with Int show ?case
    by (intro exI[of _ "min x z"]) fastforce
next
  case (Basis S)
  {
    fix x
    have "x \<noteq> - \<infinity> \<Longrightarrow> \<exists>t. ereal t \<le> 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 \<Longrightarrow> open (ereal -` S)"
  by (intro open_vimage continuous_intros)

lemma open_ereal: "open S \<Longrightarrow> open (ereal ` S)"
  unfolding open_generated_order[where 'a=real]
proof (induct rule: generate_topology.induct)
  case (Basis S)
  moreover {
    fix x
    have "ereal ` {..< x} = { -\<infinity> <..< ereal x }"
      apply auto
      apply (case_tac xa)
      apply auto
      done
  }
  moreover {
    fix x
    have "ereal ` {x <..} = { ereal x <..< \<infinity> }"
      apply auto
      apply (case_tac xa)
      apply auto
      done
  }
  ultimately show ?case
     by auto
qed (auto simp add: image_Union image_Int)


lemma eventually_finite:
  fixes x :: ereal
  assumes "\<bar>x\<bar> \<noteq> \<infinity>" "(f \<longlongrightarrow> x) F"
  shows "eventually (\<lambda>x. \<bar>f x\<bar> \<noteq> \<infinity>) F"
proof -
  have "(f \<longlongrightarrow> ereal (real_of_ereal x)) F"
    using assms by (cases x) auto