# HG changeset patch
# User hoelzl
# Date 1300109859 -3600
# Node ID 15927c040731738042a4cfb59318a6e7477ba514
# Parent  8885ba6296927d1a947db67792439abe58172a35
add Extended_Reals from AFP/Lower_Semicontinuous

diff -r 8885ba629692 -r 15927c040731 src/HOL/IsaMakefile
--- a/src/HOL/IsaMakefile	Mon Mar 14 14:37:37 2011 +0100
+++ b/src/HOL/IsaMakefile	Mon Mar 14 14:37:39 2011 +0100
@@ -437,10 +437,10 @@
   Library/ContNotDenum.thy Library/Continuity.thy Library/Convex.thy	\
   Library/Countable.thy Library/Diagonalize.thy Library/Dlist.thy	\
   Library/Efficient_Nat.thy Library/Eval_Witness.thy 			\
-  Library/Executable_Set.thy Library/Float.thy				\
-  Library/Formal_Power_Series.thy Library/Fraction_Field.thy		\
-  Library/FrechetDeriv.thy Library/Cset.thy Library/FuncSet.thy		\
-  Library/Function_Algebras.thy						\
+  Library/Executable_Set.thy Library/Extended_Reals.thy			\
+  Library/Float.thy Library/Formal_Power_Series.thy			\
+  Library/Fraction_Field.thy Library/FrechetDeriv.thy Library/Cset.thy	\
+  Library/FuncSet.thy Library/Function_Algebras.thy			\
   Library/Fundamental_Theorem_Algebra.thy Library/Glbs.thy		\
   Library/Indicator_Function.thy Library/Infinite_Set.thy		\
   Library/Inner_Product.thy Library/Kleene_Algebra.thy			\
diff -r 8885ba629692 -r 15927c040731 src/HOL/Library/Extended_Reals.thy
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Library/Extended_Reals.thy	Mon Mar 14 14:37:39 2011 +0100
@@ -0,0 +1,3191 @@
+ (* Title: Extended_Reals.thy
+   Author: Johannes Hölzl, Robert Himmelmann, Armin Heller; TU München
+   Author: Bogdan Grechuk; University of Edinburgh *)
+
+header {* Extended real number line *}
+
+theory Extended_Reals
+  imports Topology_Euclidean_Space
+begin
+
+subsection {* Definition and basic properties *}
+
+datatype extreal = extreal real | PInfty | MInfty
+
+notation (xsymbols)
+  PInfty  ("\<infinity>")
+
+notation (HTML output)
+  PInfty  ("\<infinity>")
+
+instantiation extreal :: uminus
+begin
+  fun uminus_extreal where
+    "- (extreal r) = extreal (- r)"
+  | "- \<infinity> = MInfty"
+  | "- MInfty = \<infinity>"
+  instance ..
+end
+
+lemma MInfty_neq_PInfty[simp]:
+  "\<infinity> \<noteq> - \<infinity>" "- \<infinity> \<noteq> \<infinity>" by simp_all
+
+lemma MInfty_neq_extreal[simp]:
+  "extreal r \<noteq> - \<infinity>" "- \<infinity> \<noteq> extreal r" by simp_all
+
+lemma MInfinity_cases[simp]:
+  "(case - \<infinity> of extreal r \<Rightarrow> f r | \<infinity> \<Rightarrow> y | MInfinity \<Rightarrow> z) = z"
+  by simp
+
+lemma extreal_uminus_uminus[simp]:
+  fixes a :: extreal shows "- (- a) = a"
+  by (cases a) simp_all
+
+lemma MInfty_eq[simp]:
+  "MInfty = - \<infinity>" by simp
+
+declare uminus_extreal.simps(2)[simp del]
+
+lemma extreal_cases[case_names real PInf MInf, cases type: extreal]:
+  assumes "\<And>r. x = extreal r \<Longrightarrow> P"
+  assumes "x = \<infinity> \<Longrightarrow> P"
+  assumes "x = -\<infinity> \<Longrightarrow> P"
+  shows P
+  using assms by (cases x) auto
+
+lemma extreal2_cases[case_names
+  real_real real_PInf real_MInf
+  PInf_real PInf_PInf PInf_MInf
+  MInf_real MInf_PInf MInf_MInf]:
+  assumes "\<And>r p. y = extreal p \<Longrightarrow> x = extreal r \<Longrightarrow> P"
+  assumes "\<And>p. y = extreal p \<Longrightarrow> x = \<infinity> \<Longrightarrow> P"
+  assumes "\<And>p. y = extreal p \<Longrightarrow> x = -\<infinity> \<Longrightarrow> P"
+  assumes "\<And>r. y = \<infinity> \<Longrightarrow> x = extreal r \<Longrightarrow> P"
+  assumes "y = \<infinity> \<Longrightarrow> x = \<infinity> \<Longrightarrow> P"
+  assumes "y = \<infinity> \<Longrightarrow> x = -\<infinity> \<Longrightarrow> P"
+  assumes "\<And>r. y = -\<infinity> \<Longrightarrow> x = extreal r \<Longrightarrow> P"
+  assumes "y = -\<infinity> \<Longrightarrow> x = \<infinity> \<Longrightarrow> P"
+  assumes "y = -\<infinity> \<Longrightarrow> x = -\<infinity> \<Longrightarrow> P"
+  shows P
+  apply (cases x)
+  apply (cases y) using assms apply simp_all
+  apply (cases y) using assms apply simp_all
+  apply (cases y) using assms apply simp_all
+  done
+
+lemma extreal3_cases[case_names
+  real_real_real real_real_PInf real_real_MInf
+  real_PInf_real real_PInf_PInf real_PInf_MInf
+  real_MInf_real real_MInf_PInf real_MInf_MInf
+  PInf_real_real PInf_real_PInf PInf_real_MInf
+  PInf_PInf_real PInf_PInf_PInf PInf_PInf_MInf
+  PInf_MInf_real PInf_MInf_PInf PInf_MInf_MInf
+  MInf_real_real MInf_real_PInf MInf_real_MInf
+  MInf_PInf_real MInf_PInf_PInf MInf_PInf_MInf
+  MInf_MInf_real MInf_MInf_PInf MInf_MInf_MInf]:
+  assumes "\<And>r p q. z = extreal q \<Longrightarrow> y = extreal p \<Longrightarrow> x = extreal r \<Longrightarrow> P"
+  assumes "\<And>p q. z = extreal q \<Longrightarrow> y = extreal p \<Longrightarrow> x = \<infinity> \<Longrightarrow> P"
+  assumes "\<And>p q. z = extreal q \<Longrightarrow> y = extreal p \<Longrightarrow> x = -\<infinity> \<Longrightarrow> P"
+  assumes "\<And>r q. z = extreal q \<Longrightarrow> y = \<infinity> \<Longrightarrow> x = extreal r \<Longrightarrow> P"
+  assumes "\<And>q. z = extreal q \<Longrightarrow> y = \<infinity> \<Longrightarrow> x = \<infinity> \<Longrightarrow> P"
+  assumes "\<And>q. z = extreal q \<Longrightarrow> y = \<infinity> \<Longrightarrow> x = -\<infinity> \<Longrightarrow> P"
+  assumes "\<And>q r. z = extreal q \<Longrightarrow> y = -\<infinity> \<Longrightarrow> x = extreal r \<Longrightarrow> P"
+  assumes "\<And>q. z = extreal q \<Longrightarrow> y = -\<infinity> \<Longrightarrow> x = \<infinity> \<Longrightarrow> P"
+  assumes "\<And>q. z = extreal q \<Longrightarrow> y = -\<infinity> \<Longrightarrow> x = -\<infinity> \<Longrightarrow> P"
+  assumes "\<And>r p. z = \<infinity> \<Longrightarrow> y = extreal p \<Longrightarrow> x = extreal r \<Longrightarrow> P"
+  assumes "\<And>p. z = \<infinity> \<Longrightarrow> y = extreal p \<Longrightarrow> x = \<infinity> \<Longrightarrow> P"
+  assumes "\<And>p. z = \<infinity> \<Longrightarrow> y = extreal p \<Longrightarrow> x = -\<infinity> \<Longrightarrow> P"
+  assumes "\<And>r. z = \<infinity> \<Longrightarrow> y = \<infinity> \<Longrightarrow> x = extreal r \<Longrightarrow> P"
+  assumes "z = \<infinity> \<Longrightarrow> y = \<infinity> \<Longrightarrow> x = \<infinity> \<Longrightarrow> P"
+  assumes "z = \<infinity> \<Longrightarrow> y = \<infinity> \<Longrightarrow> x = -\<infinity> \<Longrightarrow> P"
+  assumes "\<And>r. z = \<infinity> \<Longrightarrow> y = -\<infinity> \<Longrightarrow> x = extreal r \<Longrightarrow> P"
+  assumes "z = \<infinity> \<Longrightarrow> y = -\<infinity> \<Longrightarrow> x = \<infinity> \<Longrightarrow> P"
+  assumes "z = \<infinity> \<Longrightarrow> y = -\<infinity> \<Longrightarrow> x = -\<infinity> \<Longrightarrow> P"
+  assumes "\<And>r p. z = -\<infinity> \<Longrightarrow> y = extreal p \<Longrightarrow> x = extreal r \<Longrightarrow> P"
+  assumes "\<And>p. z = -\<infinity> \<Longrightarrow> y = extreal p \<Longrightarrow> x = \<infinity> \<Longrightarrow> P"
+  assumes "\<And>p. z = -\<infinity> \<Longrightarrow> y = extreal p \<Longrightarrow> x = -\<infinity> \<Longrightarrow> P"
+  assumes "\<And>r. z = -\<infinity> \<Longrightarrow> y = \<infinity> \<Longrightarrow> x = extreal r \<Longrightarrow> P"
+  assumes "z = -\<infinity> \<Longrightarrow> y = \<infinity> \<Longrightarrow> x = \<infinity> \<Longrightarrow> P"
+  assumes "z = -\<infinity> \<Longrightarrow> y = \<infinity> \<Longrightarrow> x = -\<infinity> \<Longrightarrow> P"
+  assumes "\<And>r. z = -\<infinity> \<Longrightarrow> y = -\<infinity> \<Longrightarrow> x = extreal r \<Longrightarrow> P"
+  assumes "z = -\<infinity> \<Longrightarrow> y = -\<infinity> \<Longrightarrow> x = \<infinity> \<Longrightarrow> P"
+  assumes "z = -\<infinity> \<Longrightarrow> y = -\<infinity> \<Longrightarrow> x = -\<infinity> \<Longrightarrow> P"
+  shows P
+  apply (cases x)
+  apply (cases rule: extreal2_cases[of y z]) using assms apply simp_all
+  apply (cases  rule: extreal2_cases[of y z]) using assms apply simp_all
+  apply (cases  rule: extreal2_cases[of y z]) using assms apply simp_all
+  done
+
+lemma extreal_uminus_eq_iff[simp]:
+  fixes a b :: extreal shows "-a = -b \<longleftrightarrow> a = b"
+  by (cases rule: extreal2_cases[of a b]) simp_all
+
+function of_extreal :: "extreal \<Rightarrow> real" where
+"of_extreal (extreal r) = r" |
+"of_extreal \<infinity> = 0" |
+"of_extreal (-\<infinity>) = 0"
+  by (auto intro: extreal_cases)
+termination proof qed (rule wf_empty)
+
+defs (overloaded)
+  real_of_extreal_def [code_unfold]: "real \<equiv> of_extreal"
+
+lemma real_of_extreal[simp]:
+    "real (- x :: extreal) = - (real x)"
+    "real (extreal r) = r"
+    "real \<infinity> = 0"
+  by (cases x) (simp_all add: real_of_extreal_def)
+
+lemma range_extreal[simp]: "range extreal = UNIV - {\<infinity>, -\<infinity>}"
+proof safe
+  fix x assume "x \<notin> range extreal" "x \<noteq> \<infinity>"
+  then show "x = -\<infinity>" by (cases x) auto
+qed auto
+
+instantiation extreal :: number
+begin
+definition [simp]: "number_of x = extreal (number_of x)"
+instance proof qed
+end
+
+subsubsection "Addition"
+
+instantiation extreal :: comm_monoid_add
+begin
+
+definition "0 = extreal 0"
+
+function plus_extreal where
+"extreal r + extreal p = extreal (r + p)" |
+"\<infinity> + a = \<infinity>" |
+"a + \<infinity> = \<infinity>" |
+"extreal r + -\<infinity> = - \<infinity>" |
+"-\<infinity> + extreal p = -\<infinity>" |
+"-\<infinity> + -\<infinity> = -\<infinity>"
+proof -
+  case (goal1 P x)
+  moreover then obtain a b where "x = (a, b)" by (cases x) auto
+  ultimately show P
+   by (cases rule: extreal2_cases[of a b]) auto
+qed auto
+termination proof qed (rule wf_empty)
+
+lemma Infty_neq_0[simp]:
+  "\<infinity> \<noteq> 0" "0 \<noteq> \<infinity>"
+  "-\<infinity> \<noteq> 0" "0 \<noteq> -\<infinity>"
+  by (simp_all add: zero_extreal_def)
+
+lemma extreal_eq_0[simp]:
+  "extreal r = 0 \<longleftrightarrow> r = 0"
+  "0 = extreal r \<longleftrightarrow> r = 0"
+  unfolding zero_extreal_def by simp_all
+
+instance
+proof
+  fix a :: extreal show "0 + a = a"
+    by (cases a) (simp_all add: zero_extreal_def)
+  fix b :: extreal show "a + b = b + a"
+    by (cases rule: extreal2_cases[of a b]) simp_all
+  fix c :: extreal show "a + b + c = a + (b + c)"
+    by (cases rule: extreal3_cases[of a b c]) simp_all
+qed
+end
+
+lemma extreal_uminus_zero[simp]:
+  "- 0 = (0::extreal)"
+  by (simp add: zero_extreal_def)
+
+lemma extreal_uminus_zero_iff[simp]:
+  fixes a :: extreal shows "-a = 0 \<longleftrightarrow> a = 0"
+  by (cases a) simp_all
+
+lemma extreal_plus_eq_PInfty[simp]:
+  shows "a + b = \<infinity> \<longleftrightarrow> a = \<infinity> \<or> b = \<infinity>"
+  by (cases rule: extreal2_cases[of a b]) auto
+
+lemma extreal_plus_eq_MInfty[simp]:
+  shows "a + b = -\<infinity> \<longleftrightarrow>
+    (a = -\<infinity> \<or> b = -\<infinity>) \<and> a \<noteq> \<infinity> \<and> b \<noteq> \<infinity>"
+  by (cases rule: extreal2_cases[of a b]) auto
+
+lemma extreal_add_cancel_left:
+  assumes "a \<noteq> -\<infinity>"
+  shows "a + b = a + c \<longleftrightarrow> (a = \<infinity> \<or> b = c)"
+  using assms by (cases rule: extreal3_cases[of a b c]) auto
+
+lemma extreal_add_cancel_right:
+  assumes "a \<noteq> -\<infinity>"
+  shows "b + a = c + a \<longleftrightarrow> (a = \<infinity> \<or> b = c)"
+  using assms by (cases rule: extreal3_cases[of a b c]) auto
+
+lemma extreal_real:
+  "extreal (real x) = (if x = \<infinity> \<or> x = -\<infinity> then 0 else x)"
+  by (cases x) simp_all
+
+subsubsection "Linear order on @{typ extreal}"
+
+instantiation extreal :: linorder
+begin
+
+function less_extreal where
+"extreal x < extreal y \<longleftrightarrow> x < y" |
+"        \<infinity> < a         \<longleftrightarrow> False" |
+"        a < -\<infinity>        \<longleftrightarrow> False" |
+"extreal x < \<infinity>         \<longleftrightarrow> True" |
+"       -\<infinity> < extreal r \<longleftrightarrow> True" |
+"       -\<infinity> < \<infinity>         \<longleftrightarrow> True"
+proof -
+  case (goal1 P x)
+  moreover then obtain a b where "x = (a,b)" by (cases x) auto
+  ultimately show P by (cases rule: extreal2_cases[of a b]) auto
+qed simp_all
+termination by (relation "{}") simp
+
+definition "x \<le> (y::extreal) \<longleftrightarrow> x < y \<or> x = y"
+
+lemma extreal_infty_less[simp]:
+  "x < \<infinity> \<longleftrightarrow> (x \<noteq> \<infinity>)"
+  "-\<infinity> < x \<longleftrightarrow> (x \<noteq> -\<infinity>)"
+  by (cases x, simp_all) (cases x, simp_all)
+
+lemma extreal_infty_less_eq[simp]:
+  "\<infinity> \<le> x \<longleftrightarrow> x = \<infinity>"
+  "x \<le> -\<infinity> \<longleftrightarrow> x = -\<infinity>"
+  by (auto simp add: less_eq_extreal_def)
+
+lemma extreal_less[simp]:
+  "extreal r < 0 \<longleftrightarrow> (r < 0)"
+  "0 < extreal r \<longleftrightarrow> (0 < r)"
+  "0 < \<infinity>"
+  "-\<infinity> < 0"
+  by (simp_all add: zero_extreal_def)
+
+lemma extreal_less_eq[simp]:
+  "x \<le> \<infinity>"
+  "-\<infinity> \<le> x"
+  "extreal r \<le> extreal p \<longleftrightarrow> r \<le> p"
+  "extreal r \<le> 0 \<longleftrightarrow> r \<le> 0"
+  "0 \<le> extreal r \<longleftrightarrow> 0 \<le> r"
+  by (auto simp add: less_eq_extreal_def zero_extreal_def)
+
+lemma extreal_infty_less_eq2:
+  "a \<le> b \<Longrightarrow> a = \<infinity> \<Longrightarrow> b = \<infinity>"
+  "a \<le> b \<Longrightarrow> b = -\<infinity> \<Longrightarrow> a = -\<infinity>"
+  by simp_all
+
+instance
+proof
+  fix x :: extreal show "x \<le> x"
+    by (cases x) simp_all
+  fix y :: extreal show "x < y \<longleftrightarrow> x \<le> y \<and> \<not> y \<le> x"
+    by (cases rule: extreal2_cases[of x y]) auto
+  show "x \<le> y \<or> y \<le> x "
+    by (cases rule: extreal2_cases[of x y]) auto
+  { assume "x \<le> y" "y \<le> x" then show "x = y"
+    by (cases rule: extreal2_cases[of x y]) auto }
+  { fix z assume "x \<le> y" "y \<le> z" then show "x \<le> z"
+    by (cases rule: extreal3_cases[of x y z]) auto }
+qed
+end
+
+lemma extreal_MInfty_lessI[intro, simp]:
+  "a \<noteq> -\<infinity> \<Longrightarrow> -\<infinity> < a"
+  by (cases a) auto
+
+lemma extreal_less_PInfty[intro, simp]:
+  "a \<noteq> \<infinity> \<Longrightarrow> a < \<infinity>"
+  by (cases a) auto
+
+lemma extreal_less_extreal_Ex:
+  fixes a b :: extreal
+  shows "x < extreal r \<longleftrightarrow> x = -\<infinity> \<or> (\<exists>p. p < r \<and> x = extreal p)"
+  by (cases x) auto
+
+lemma extreal_add_mono:
+  fixes a b c d :: extreal assumes "a \<le> b" "c \<le> d" shows "a + c \<le> b + d"
+  using assms
+  apply (cases a)
+  apply (cases rule: extreal3_cases[of b c d], auto)
+  apply (cases rule: extreal3_cases[of b c d], auto)
+  done
+
+lemma extreal_minus_le_minus[simp]:
+  fixes a b :: extreal shows "- a \<le> - b \<longleftrightarrow> b \<le> a"
+  by (cases rule: extreal2_cases[of a b]) auto
+
+lemma extreal_minus_less_minus[simp]:
+  fixes a b :: extreal shows "- a < - b \<longleftrightarrow> b < a"
+  by (cases rule: extreal2_cases[of a b]) auto
+
+lemma extreal_le_real_iff:
+  "x \<le> real y \<longleftrightarrow> ((y \<noteq> -\<infinity> \<and> y \<noteq> \<infinity> \<longrightarrow> extreal x \<le> y) \<and> (y = -\<infinity> \<or> y = \<infinity> \<longrightarrow> x \<le> 0))"
+  by (cases y) auto
+
+lemma real_le_extreal_iff:
+  "real y \<le> x \<longleftrightarrow> ((y \<noteq> -\<infinity> \<and> y \<noteq> \<infinity> \<longrightarrow> y \<le> extreal x) \<and> (y = -\<infinity> \<or> y = \<infinity> \<longrightarrow> 0 \<le> x))"
+  by (cases y) auto
+
+lemma extreal_less_real_iff:
+  "x < real y \<longleftrightarrow> ((y \<noteq> -\<infinity> \<and> y \<noteq> \<infinity> \<longrightarrow> extreal x < y) \<and> (y = -\<infinity> \<or> y = \<infinity> \<longrightarrow> x < 0))"
+  by (cases y) auto
+
+lemma real_less_extreal_iff:
+  "real y < x \<longleftrightarrow> ((y \<noteq> -\<infinity> \<and> y \<noteq> \<infinity> \<longrightarrow> y < extreal x) \<and> (y = -\<infinity> \<or> y = \<infinity> \<longrightarrow> 0 < x))"
+  by (cases y) auto
+
+lemmas real_of_extreal_ord_simps =
+  extreal_le_real_iff real_le_extreal_iff extreal_less_real_iff real_less_extreal_iff
+
+lemma extreal_dense:
+  fixes x y :: extreal assumes "x < y"
+  shows "EX z. x < z & z < y"
+proof -
+{ assume a: "x = (-\<infinity>)"
+  { assume "y = \<infinity>" hence ?thesis using a by (auto intro!: exI[of _ "0"]) }
+  moreover
+  { assume "y ~= \<infinity>"
+    with `x < y` obtain r where r: "y = extreal r" by (cases y) auto
+    hence ?thesis using `x < y` a by (auto intro!: exI[of _ "extreal (r - 1)"])
+  } ultimately have ?thesis by auto
+}
+moreover
+{ assume "x ~= (-\<infinity>)"
+  with `x < y` obtain p where p: "x = extreal p" by (cases x) auto
+  { assume "y = \<infinity>" hence ?thesis using `x < y` p
+       by (auto intro!: exI[of _ "extreal (p + 1)"]) }
+  moreover
+  { assume "y ~= \<infinity>"
+    with `x < y` obtain r where r: "y = extreal r" by (cases y) auto
+    with p `x < y` have "p < r" by auto
+    with dense obtain z where "p < z" "z < r" by auto
+    hence ?thesis using r p by (auto intro!: exI[of _ "extreal z"])
+  } ultimately have ?thesis by auto
+} ultimately show ?thesis by auto
+qed
+
+lemma extreal_dense2:
+  fixes x y :: extreal assumes "x < y"
+  shows "EX z. x < extreal z & extreal z < y"
+  by (metis extreal_dense[OF `x < y`] extreal_cases less_extreal.simps(2,3))
+
+subsubsection "Multiplication"
+
+instantiation extreal :: comm_monoid_mult
+begin
+
+definition "1 = extreal 1"
+
+function times_extreal where
+"extreal r * extreal p = extreal (r * p)" |
+"extreal r * \<infinity> = (if r = 0 then 0 else if r > 0 then \<infinity> else -\<infinity>)" |
+"\<infinity> * extreal r = (if r = 0 then 0 else if r > 0 then \<infinity> else -\<infinity>)" |
+"extreal r * -\<infinity> = (if r = 0 then 0 else if r > 0 then -\<infinity> else \<infinity>)" |
+"-\<infinity> * extreal r = (if r = 0 then 0 else if r > 0 then -\<infinity> else \<infinity>)" |
+"\<infinity> * \<infinity> = \<infinity>" |
+"-\<infinity> * \<infinity> = -\<infinity>" |
+"\<infinity> * -\<infinity> = -\<infinity>" |
+"-\<infinity> * -\<infinity> = \<infinity>"
+proof -
+  case (goal1 P x)
+  moreover then obtain a b where "x = (a, b)" by (cases x) auto
+  ultimately show P by (cases rule: extreal2_cases[of a b]) auto
+qed simp_all
+termination by (relation "{}") simp
+
+instance
+proof
+  fix a :: extreal show "1 * a = a"
+    by (cases a) (simp_all add: one_extreal_def)
+  fix b :: extreal show "a * b = b * a"
+    by (cases rule: extreal2_cases[of a b]) simp_all
+  fix c :: extreal show "a * b * c = a * (b * c)"
+    by (cases rule: extreal3_cases[of a b c])
+       (simp_all add: zero_extreal_def zero_less_mult_iff)
+qed
+end
+
+lemma extreal_mult_zero[simp]:
+  fixes a :: extreal shows "a * 0 = 0"
+  by (cases a) (simp_all add: zero_extreal_def)
+
+lemma extreal_zero_mult[simp]:
+  fixes a :: extreal shows "0 * a = 0"
+  by (cases a) (simp_all add: zero_extreal_def)
+
+lemma extreal_m1_less_0[simp]:
+  "-(1::extreal) < 0"
+  by (simp add: zero_extreal_def one_extreal_def)
+
+lemma extreal_zero_m1[simp]:
+  "1 \<noteq> (0::extreal)"
+  by (simp add: zero_extreal_def one_extreal_def)
+
+lemma extreal_times_0[simp]:
+  fixes x :: extreal shows "0 * x = 0"
+  by (cases x) (auto simp: zero_extreal_def)
+
+lemma extreal_times[simp]:
+  "1 \<noteq> \<infinity>" "\<infinity> \<noteq> 1"
+  "1 \<noteq> -\<infinity>" "-\<infinity> \<noteq> 1"
+  by (auto simp add: times_extreal_def one_extreal_def)
+
+lemma extreal_plus_1[simp]:
+  "1 + extreal r = extreal (r + 1)" "extreal r + 1 = extreal (r + 1)"
+  "1 + -\<infinity> = -\<infinity>" "-\<infinity> + 1 = -\<infinity>"
+  unfolding one_extreal_def by auto
+
+lemma extreal_zero_times[simp]:
+  fixes a b :: extreal shows "a * b = 0 \<longleftrightarrow> a = 0 \<or> b = 0"
+  by (cases rule: extreal2_cases[of a b]) auto
+
+lemma extreal_mult_eq_PInfty[simp]:
+  shows "a * b = \<infinity> \<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: extreal2_cases[of a b]) auto
+
+lemma extreal_mult_eq_MInfty[simp]:
+  shows "a * b = -\<infinity> \<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: extreal2_cases[of a b]) auto
+
+lemma extreal_0_less_1[simp]: "0 < (1::extreal)"
+  by (simp_all add: zero_extreal_def one_extreal_def)
+
+lemma extreal_zero_one[simp]: "0 \<noteq> (1::extreal)"
+  by (simp_all add: zero_extreal_def one_extreal_def)
+
+lemma extreal_mult_minus_left[simp]:
+  fixes a b :: extreal shows "-a * b = - (a * b)"
+  by (cases rule: extreal2_cases[of a b]) auto
+
+lemma extreal_mult_minus_right[simp]:
+  fixes a b :: extreal shows "a * -b = - (a * b)"
+  by (cases rule: extreal2_cases[of a b]) auto
+
+lemma extreal_mult_infty[simp]:
+  "a * \<infinity> = (if a = 0 then 0 else if 0 < a then \<infinity> else - \<infinity>)"
+  by (cases a) auto
+
+lemma extreal_infty_mult[simp]:
+  "\<infinity> * a = (if a = 0 then 0 else if 0 < a then \<infinity> else - \<infinity>)"
+  by (cases a) auto
+
+lemma extreal_mult_strict_right_mono:
+  assumes "a < b" and "0 < c" "c < \<infinity>"
+  shows "a * c < b * c"
+  using assms
+  by (cases rule: extreal3_cases[of a b c])
+     (auto simp: zero_le_mult_iff extreal_less_PInfty)
+
+lemma extreal_mult_strict_left_mono:
+  "\<lbrakk> a < b ; 0 < c ; c < \<infinity>\<rbrakk> \<Longrightarrow> c * a < c * b"
+  using extreal_mult_strict_right_mono by (simp add: mult_commute[of c])
+
+lemma extreal_mult_right_mono:
+  fixes a b c :: extreal shows "\<lbrakk>a \<le> b; 0 \<le> c\<rbrakk> \<Longrightarrow> a*c \<le> b*c"
+  using assms
+  apply (cases "c = 0") apply simp
+  by (cases rule: extreal3_cases[of a b c])
+     (auto simp: zero_le_mult_iff extreal_less_PInfty)
+
+lemma extreal_mult_left_mono:
+  fixes a b c :: extreal shows "\<lbrakk>a \<le> b; 0 \<le> c\<rbrakk> \<Longrightarrow> c * a \<le> c * b"
+  using extreal_mult_right_mono by (simp add: mult_commute[of c])
+
+subsubsection {* Subtraction *}
+
+lemma extreal_minus_minus_image[simp]:
+  fixes S :: "extreal set"
+  shows "uminus ` uminus ` S = S"
+  by (auto simp: image_iff)
+
+lemma extreal_uminus_lessThan[simp]:
+  fixes a :: extreal shows "uminus ` {..<a} = {-a<..}"
+proof (safe intro!: image_eqI)
+  fix x assume "-a < x"
+  then have "- x < - (- a)" by (simp del: extreal_uminus_uminus)
+  then show "- x < a" by simp
+qed auto
+
+lemma extreal_uminus_greaterThan[simp]:
+  "uminus ` {(a::extreal)<..} = {..<-a}"
+  by (metis extreal_uminus_lessThan extreal_uminus_uminus
+            extreal_minus_minus_image)
+
+instantiation extreal :: minus
+begin
+definition "x - y = x + -(y::extreal)"
+instance ..
+end
+
+lemma extreal_minus[simp]:
+  "extreal r - extreal p = extreal (r - p)"
+  "-\<infinity> - extreal r = -\<infinity>"
+  "extreal r - \<infinity> = -\<infinity>"
+  "\<infinity> - x = \<infinity>"
+  "-\<infinity> - \<infinity> = -\<infinity>"
+  "x - -y = x + y"
+  "x - 0 = x"
+  "0 - x = -x"
+  by (simp_all add: minus_extreal_def)
+
+lemma extreal_x_minus_x[simp]:
+  "x - x = (if x = -\<infinity> \<or> x = \<infinity> then \<infinity> else 0)"
+  by (cases x) simp_all
+
+lemma extreal_eq_minus_iff:
+  fixes x y z :: extreal
+  shows "x = z - y \<longleftrightarrow>
+    (y \<noteq> \<infinity> \<longrightarrow> y \<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: extreal3_cases[of x y z]) auto
+
+lemma extreal_eq_minus:
+  fixes x y z :: extreal
+  shows "y \<noteq> \<infinity> \<Longrightarrow> y \<noteq> -\<infinity> \<Longrightarrow> x = z - y \<longleftrightarrow> x + y = z"
+  by (simp add: extreal_eq_minus_iff)
+
+lemma extreal_less_minus_iff:
+  fixes x y z :: extreal
+  shows "x < z - y \<longleftrightarrow>
+    (y = \<infinity> \<longrightarrow> z = \<infinity> \<and> x \<noteq> \<infinity>) \<and>
+    (y = -\<infinity> \<longrightarrow> x \<noteq> \<infinity>) \<and>
+    (y \<noteq> \<infinity> \<longrightarrow> y \<noteq> -\<infinity> \<longrightarrow> x + y < z)"
+  by (cases rule: extreal3_cases[of x y z]) auto
+
+lemma extreal_less_minus:
+  fixes x y z :: extreal
+  shows "y \<noteq> \<infinity> \<Longrightarrow> y \<noteq> -\<infinity> \<Longrightarrow> x < z - y \<longleftrightarrow> x + y < z"
+  by (simp add: extreal_less_minus_iff)
+
+lemma extreal_le_minus_iff:
+  fixes x y z :: extreal
+  shows "x \<le> z - y \<longleftrightarrow>
+    (y = \<infinity> \<longrightarrow> z \<noteq> \<infinity> \<longrightarrow> x = -\<infinity>) \<and>
+    (y \<noteq> \<infinity> \<longrightarrow> y \<noteq> -\<infinity> \<longrightarrow> x + y \<le> z)"
+  by (cases rule: extreal3_cases[of x y z]) auto
+
+lemma extreal_le_minus:
+  fixes x y z :: extreal
+  shows "y \<noteq> \<infinity> \<Longrightarrow> y \<noteq> -\<infinity> \<Longrightarrow> x \<le> z - y \<longleftrightarrow> x + y \<le> z"
+  by (simp add: extreal_le_minus_iff)
+
+lemma extreal_minus_less_iff:
+  fixes x y z :: extreal
+  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: extreal3_cases[of x y z]) auto
+
+lemma extreal_minus_less:
+  fixes x y z :: extreal
+  shows "y \<noteq> \<infinity> \<Longrightarrow> y \<noteq> -\<infinity> \<Longrightarrow> x - y < z \<longleftrightarrow> x < z + y"
+  by (simp add: extreal_minus_less_iff)
+
+lemma extreal_minus_le_iff:
+  fixes x y z :: extreal
+  shows "x - y \<le> z \<longleftrightarrow>
+    (y = -\<infinity> \<longrightarrow> z = \<infinity>) \<and>
+    (y = \<infinity> \<longrightarrow> x = \<infinity> \<longrightarrow> z = \<infinity>) \<and>
+    (y \<noteq> \<infinity> \<longrightarrow> y \<noteq> -\<infinity> \<longrightarrow> x \<le> z + y)"
+  by (cases rule: extreal3_cases[of x y z]) auto
+
+lemma extreal_minus_le:
+  fixes x y z :: extreal
+  shows "y \<noteq> \<infinity> \<Longrightarrow> y \<noteq> -\<infinity> \<Longrightarrow> x - y \<le> z \<longleftrightarrow> x \<le> z + y"
+  by (simp add: extreal_minus_le_iff)
+
+lemma extreal_minus_eq_minus_iff:
+  fixes a b c :: extreal
+  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: extreal3_cases[of a b c]) auto
+
+lemma extreal_add_le_add_iff:
+  "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: extreal3_cases[of a b c]) (simp_all add: field_simps)
+
+lemma extreal_mult_le_mult_iff:
+  "c \<noteq> \<infinity> \<Longrightarrow> c \<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: extreal3_cases[of a b c]) (simp_all add: mult_le_cancel_left)
+
+lemma extreal_between:
+  fixes x e :: extreal
+  assumes "x \<noteq> \<infinity>" "x \<noteq> -\<infinity>" "0 < e"
+  shows "x - e < x" "x < x + e"
+using assms apply (cases x, cases e) apply auto
+using assms by (cases x, cases e) auto
+
+lemma extreal_distrib:
+  fixes a b c :: extreal
+  assumes "a \<noteq> \<infinity> \<or> b \<noteq> -\<infinity>" "a \<noteq> -\<infinity> \<or> b \<noteq> \<infinity>" "c \<noteq> \<infinity>" "c \<noteq> -\<infinity>"
+  shows "(a + b) * c = a * c + b * c"
+  using assms
+  by (cases rule: extreal3_cases[of a b c]) (simp_all add: field_simps)
+
+subsubsection {* Division *}
+
+instantiation extreal :: inverse
+begin
+
+function inverse_extreal where
+"inverse (extreal r) = (if r = 0 then \<infinity> else extreal (inverse r))" |
+"inverse \<infinity> = 0" |
+"inverse (-\<infinity>) = 0"
+  by (auto intro: extreal_cases)
+termination by (relation "{}") simp
+
+definition "x / y = x * inverse (y :: extreal)"
+
+instance proof qed
+end
+
+lemma extreal_inverse[simp]:
+  "inverse 0 = \<infinity>"
+  "inverse (1::extreal) = 1"
+  by (simp_all add: one_extreal_def zero_extreal_def)
+
+lemma extreal_divide[simp]:
+  "extreal r / extreal p = (if p = 0 then extreal r * \<infinity> else extreal (r / p))"
+  unfolding divide_extreal_def by (auto simp: divide_real_def)
+
+lemma extreal_divide_same[simp]:
+  "x / x = (if x = \<infinity> \<or> x = -\<infinity> \<or> x = 0 then 0 else 1)"
+  by (cases x)
+     (simp_all add: divide_real_def divide_extreal_def one_extreal_def)
+
+lemma extreal_inv_inv[simp]:
+  "inverse (inverse x) = (if x \<noteq> -\<infinity> then x else \<infinity>)"
+  by (cases x) auto
+
+lemma extreal_inverse_minus[simp]:
+  "inverse (- x) = (if x = 0 then \<infinity> else -inverse x)"
+  by (cases x) simp_all
+
+lemma extreal_uminus_divide[simp]:
+  fixes x y :: extreal shows "- x / y = - (x / y)"
+  unfolding divide_extreal_def by simp
+
+lemma extreal_divide_Infty[simp]:
+  "x / \<infinity> = 0" "x / -\<infinity> = 0"
+  unfolding divide_extreal_def by simp_all
+
+lemma extreal_divide_one[simp]:
+  "x / 1 = (x::extreal)"
+  unfolding divide_extreal_def by simp
+
+lemma extreal_divide_extreal[simp]:
+  "\<infinity> / extreal r = (if 0 \<le> r then \<infinity> else -\<infinity>)"
+  unfolding divide_extreal_def by simp
+
+lemma extreal_mult_le_0_iff:
+  fixes a b :: extreal
+  shows "a * b \<le> 0 \<longleftrightarrow> (0 \<le> a \<and> b \<le> 0) \<or> (a \<le> 0 \<and> 0 \<le> b)"
+  by (cases rule: extreal2_cases[of a b]) (simp_all add: mult_le_0_iff)
+
+lemma extreal_zero_le_0_iff:
+  fixes a b :: extreal
+  shows "0 \<le> a * b \<longleftrightarrow> (0 \<le> a \<and> 0 \<le> b) \<or> (a \<le> 0 \<and> b \<le> 0)"
+  by (cases rule: extreal2_cases[of a b]) (simp_all add: zero_le_mult_iff)
+
+lemma extreal_mult_less_0_iff:
+  fixes a b :: extreal
+  shows "a * b < 0 \<longleftrightarrow> (0 < a \<and> b < 0) \<or> (a < 0 \<and> 0 < b)"
+  by (cases rule: extreal2_cases[of a b]) (simp_all add: mult_less_0_iff)
+
+lemma extreal_zero_less_0_iff:
+  fixes a b :: extreal
+  shows "0 < a * b \<longleftrightarrow> (0 < a \<and> 0 < b) \<or> (a < 0 \<and> b < 0)"
+  by (cases rule: extreal2_cases[of a b]) (simp_all add: zero_less_mult_iff)
+
+lemma extreal_le_divide_pos:
+  "x > 0 \<Longrightarrow> x \<noteq> \<infinity> \<Longrightarrow> y \<le> z / x \<longleftrightarrow> x * y \<le> z"
+  by (cases rule: extreal3_cases[of x y z]) (auto simp: field_simps)
+
+lemma extreal_divide_le_pos:
+  "x > 0 \<Longrightarrow> x \<noteq> \<infinity> \<Longrightarrow> z / x \<le> y \<longleftrightarrow> z \<le> x * y"
+  by (cases rule: extreal3_cases[of x y z]) (auto simp: field_simps)
+
+lemma extreal_le_divide_neg:
+  "x < 0 \<Longrightarrow> x \<noteq> -\<infinity> \<Longrightarrow> y \<le> z / x \<longleftrightarrow> z \<le> x * y"
+  by (cases rule: extreal3_cases[of x y z]) (auto simp: field_simps)
+
+lemma extreal_divide_le_neg:
+  "x < 0 \<Longrightarrow> x \<noteq> -\<infinity> \<Longrightarrow> z / x \<le> y \<longleftrightarrow> x * y \<le> z"
+  by (cases rule: extreal3_cases[of x y z]) (auto simp: field_simps)
+
+lemma extreal_inverse_antimono_strict:
+  fixes x y :: extreal
+  shows "0 \<le> x \<Longrightarrow> x < y \<Longrightarrow> inverse y < inverse x"
+  by (cases rule: extreal2_cases[of x y]) auto
+
+lemma extreal_inverse_antimono:
+  fixes x y :: extreal
+  shows "0 \<le> x \<Longrightarrow> x <= y \<Longrightarrow> inverse y <= inverse x"
+  by (cases rule: extreal2_cases[of x y]) auto
+
+lemma inverse_inverse_Pinfty_iff[simp]:
+  "inverse x = \<infinity> \<longleftrightarrow> x = 0"
+  by (cases x) auto
+
+lemma extreal_inverse_eq_0:
+  "inverse x = 0 \<longleftrightarrow> x = \<infinity> \<or> x = -\<infinity>"
+  by (cases x) auto
+
+lemma extreal_mult_less_right:
+  assumes "b * a < c * a" "0 < a" "a < \<infinity>"
+  shows "b < c"
+  using assms
+  by (cases rule: extreal3_cases[of a b c])
+     (auto split: split_if_asm simp: zero_less_mult_iff zero_le_mult_iff)
+
+subsection "Complete lattice"
+
+lemma extreal_bot:
+  fixes x :: extreal assumes "\<And>B. x \<le> extreal 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 extreal_top:
+  fixes x :: extreal assumes "\<And>B. x \<ge> extreal 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
+
+instantiation extreal :: lattice
+begin
+definition [simp]: "sup x y = (max x y :: extreal)"
+definition [simp]: "inf x y = (min x y :: extreal)"
+instance proof qed simp_all
+end
+
+instantiation extreal :: complete_lattice
+begin
+
+definition "bot = (-\<infinity>)"
+definition "top = \<infinity>"
+
+definition "Sup S = (LEAST z. ALL x:S. x <= z :: extreal)"
+definition "Inf S = (GREATEST z. ALL x:S. z <= x :: extreal)"
+
+lemma extreal_complete_Sup:
+  fixes S :: "extreal set" assumes "S \<noteq> {}"
+  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
+  assume "\<exists>x. \<forall>a\<in>S. a \<le> extreal x"
+  then obtain y where y: "\<And>a. a\<in>S \<Longrightarrow> a \<le> extreal y" by auto
+  then have "\<infinity> \<notin> S" by force
+  show ?thesis
+  proof cases
+    assume "S = {-\<infinity>}"
+    then show ?thesis by (auto intro!: exI[of _ "-\<infinity>"])
+  next
+    assume "S \<noteq> {-\<infinity>}"
+    with `S \<noteq> {}` `\<infinity> \<notin> S` obtain x where "x \<in> S - {-\<infinity>}" "x \<noteq> \<infinity>" by auto
+    with y `\<infinity> \<notin> S` have "\<forall>z\<in>real ` (S - {-\<infinity>}). z \<le> y"
+      by (auto simp: real_of_extreal_ord_simps)
+    with reals_complete2[of "real ` (S - {-\<infinity>})"] `x \<in> S - {-\<infinity>}`
+    obtain s where s:
+       "\<forall>y\<in>S - {-\<infinity>}. real y \<le> s" "\<And>z. (\<forall>y\<in>S - {-\<infinity>}. real y \<le> z) \<Longrightarrow> s \<le> z"
+       by auto
+    show ?thesis
+    proof (safe intro!: exI[of _ "extreal s"])
+      fix z assume "z \<in> S" with `\<infinity> \<notin> S` show "z \<le> extreal s"
+      proof (cases z)
+        case (real r)
+        then show ?thesis
+          using s(1)[rule_format, of z] `z \<in> S` `z = extreal r` by auto
+      qed auto
+    next
+      fix z assume *: "\<forall>y\<in>S. y \<le> z"
+      with `S \<noteq> {-\<infinity>}` `S \<noteq> {}` show "extreal s \<le> z"
+      proof (cases z)
+        case (real u)
+        with * have "s \<le> u"
+          by (intro s(2)[of u]) (auto simp: real_of_extreal_ord_simps)
+        then show ?thesis using real by simp
+      qed auto
+    qed
+  qed
+next
+  assume *: "\<not> (\<exists>x. \<forall>a\<in>S. a \<le> extreal x)"
+  show ?thesis
+  proof (safe intro!: exI[of _ \<infinity>])
+    fix y assume **: "\<forall>z\<in>S. z \<le> y"
+    with * show "\<infinity> \<le> y"
+    proof (cases y)
+      case MInf with * ** show ?thesis by (force simp: not_le)
+    qed auto
+  qed simp
+qed
+
+lemma extreal_complete_Inf:
+  fixes S :: "extreal set" assumes "S ~= {}"
+  shows "EX x. (ALL y:S. x <= y) & (ALL z. (ALL y:S. z <= y) --> z <= x)"
+proof-
+def S1 == "uminus ` S"
+hence "S1 ~= {}" using assms by auto
+from this obtain x where x_def: "(ALL y:S1. y <= x) & (ALL z. (ALL y:S1. y <= z) --> x <= z)"
+   using extreal_complete_Sup[of S1] by auto
+{ fix z assume "ALL y:S. z <= y"
+  hence "ALL y:S1. y <= -z" unfolding S1_def by auto
+  hence "x <= -z" using x_def by auto
+  hence "z <= -x"
+    apply (subst extreal_uminus_uminus[symmetric])
+    unfolding extreal_minus_le_minus . }
+moreover have "(ALL y:S. -x <= y)"
+   using x_def unfolding S1_def
+   apply simp
+   apply (subst (3) extreal_uminus_uminus[symmetric])
+   unfolding extreal_minus_le_minus by simp
+ultimately show ?thesis by auto
+qed
+
+lemma extreal_complete_uminus_eq:
+  fixes S :: "extreal 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 extreal_minus_le_minus extreal_uminus_uminus)
+
+lemma extreal_Sup_uminus_image_eq:
+  fixes S :: "extreal set"
+  shows "Sup (uminus ` S) = - Inf S"
+proof cases
+  assume "S = {}"
+  moreover have "(THE x. All (op \<le> x)) = (-\<infinity>::extreal)"
+    by (rule the_equality) (auto intro!: extreal_bot)
+  moreover have "(SOME x. \<forall>y. y \<le> x) = (\<infinity>::extreal)"
+    by (rule some_equality) (auto intro!: extreal_top)
+  ultimately show ?thesis unfolding Inf_extreal_def Sup_extreal_def
+    Least_def Greatest_def GreatestM_def by simp
+next
+  assume "S \<noteq> {}"
+  with extreal_complete_Sup[of "uminus`S"]
+  obtain x where x: "(\<forall>y\<in>S. -x \<le> y) \<and> (\<forall>z. (\<forall>y\<in>S. z \<le> y) \<longrightarrow> z \<le> -x)"
+    unfolding extreal_complete_uminus_eq by auto
+  show "Sup (uminus ` S) = - Inf S"
+    unfolding Inf_extreal_def Greatest_def GreatestM_def
+  proof (intro someI2[of _ _ "\<lambda>x. Sup (uminus`S) = - x"])
+    show "(\<forall>y\<in>S. -x \<le> y) \<and> (\<forall>y. (\<forall>z\<in>S. y \<le> z) \<longrightarrow> y \<le> -x)"
+      using x .
+    fix x' assume "(\<forall>y\<in>S. x' \<le> y) \<and> (\<forall>y. (\<forall>z\<in>S. y \<le> z) \<longrightarrow> y \<le> x')"
+    then have "(\<forall>y\<in>uminus`S. y \<le> - x') \<and> (\<forall>y. (\<forall>z\<in>uminus`S. z \<le> y) \<longrightarrow> - x' \<le> y)"
+      unfolding extreal_complete_uminus_eq by simp
+    then show "Sup (uminus ` S) = -x'"
+      unfolding Sup_extreal_def extreal_uminus_eq_iff
+      by (intro Least_equality) auto
+  qed
+qed
+
+instance
+proof
+  { fix x :: extreal and A
+    show "bot <= x" by (cases x) (simp_all add: bot_extreal_def)
+    show "x <= top" by (simp add: top_extreal_def) }
+
+  { fix x :: extreal and A assume "x : A"
+    with extreal_complete_Sup[of A]
+    obtain s where s: "\<forall>y\<in>A. y <= s" "\<forall>z. (\<forall>y\<in>A. y <= z) \<longrightarrow> s <= z" by auto
+    hence "x <= s" using `x : A` by auto
+    also have "... = Sup A" using s unfolding Sup_extreal_def
+      by (auto intro!: Least_equality[symmetric])
+    finally show "x <= Sup A" . }
+  note le_Sup = this
+
+  { fix x :: extreal and A assume *: "!!z. (z : A ==> z <= x)"
+    show "Sup A <= x"
+    proof (cases "A = {}")
+      case True
+      hence "Sup A = -\<infinity>" unfolding Sup_extreal_def
+        by (auto intro!: Least_equality)
+      thus "Sup A <= x" by simp
+    next
+      case False
+      with extreal_complete_Sup[of A]
+      obtain s where s: "\<forall>y\<in>A. y <= s" "\<forall>z. (\<forall>y\<in>A. y <= z) \<longrightarrow> s <= z" by auto
+      hence "Sup A = s"
+        unfolding Sup_extreal_def by (auto intro!: Least_equality)
+      also have "s <= x" using * s by auto
+      finally show "Sup A <= x" .
+    qed }
+  note Sup_le = this
+
+  { fix x :: extreal and A assume "x \<in> A"
+    with le_Sup[of "-x" "uminus`A"] show "Inf A \<le> x"
+      unfolding extreal_Sup_uminus_image_eq by simp }
+
+  { fix x :: extreal and A assume *: "!!z. (z : A ==> x <= z)"
+    with Sup_le[of "uminus`A" "-x"] show "x \<le> Inf A"
+      unfolding extreal_Sup_uminus_image_eq by force }
+qed
+end
+
+lemma extreal_SUPR_uminus:
+  fixes f :: "'a => extreal"
+  shows "(SUP i : R. -(f i)) = -(INF i : R. f i)"
+  unfolding SUPR_def INFI_def
+  using extreal_Sup_uminus_image_eq[of "f`R"]
+  by (simp add: image_image)
+
+lemma extreal_INFI_uminus:
+  fixes f :: "'a => extreal"
+  shows "(INF i : R. -(f i)) = -(SUP i : R. f i)"
+  using extreal_SUPR_uminus[of _ "\<lambda>x. - f x"] by simp
+
+lemma extreal_inj_on_uminus[intro, simp]: "inj_on uminus (A :: extreal set)"
+  by (auto intro!: inj_onI)
+
+lemma extreal_image_uminus_shift:
+  fixes X Y :: "extreal 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 Inf_extreal_iff:
+  fixes z :: extreal
+  shows "(!!x. x:X ==> z <= x) ==> (EX x:X. x<y) <-> Inf X < y"
+  by (metis complete_lattice_class.Inf_greatest complete_lattice_class.Inf_lower less_le_not_le linear
+            order_less_le_trans)
+
+lemma Sup_eq_MInfty:
+  fixes S :: "extreal set" shows "Sup S = -\<infinity> \<longleftrightarrow> S = {} \<or> S = {-\<infinity>}"
+proof
+  assume a: "Sup S = -\<infinity>"
+  with complete_lattice_class.Sup_upper[of _ S]
+  show "S={} \<or> S={-\<infinity>}" by auto
+next
+  assume "S={} \<or> S={-\<infinity>}" then show "Sup S = -\<infinity>"
+    unfolding Sup_extreal_def by (auto intro!: Least_equality)
+qed
+
+lemma Inf_eq_PInfty:
+  fixes S :: "extreal set" shows "Inf S = \<infinity> \<longleftrightarrow> S = {} \<or> S = {\<infinity>}"
+  using Sup_eq_MInfty[of "uminus`S"]
+  unfolding extreal_Sup_uminus_image_eq extreal_image_uminus_shift by simp
+
+lemma Inf_eq_MInfty: "-\<infinity> : S ==> Inf S = -\<infinity>"
+  unfolding Inf_extreal_def
+  by (auto intro!: Greatest_equality)
+
+lemma Sup_eq_PInfty: "\<infinity> : S ==> Sup S = \<infinity>"
+  unfolding Sup_extreal_def
+  by (auto intro!: Least_equality)
+
+lemma extreal_SUPI:
+  fixes x :: extreal
+  assumes "!!i. i : A ==> f i <= x"
+  assumes "!!y. (!!i. i : A ==> f i <= y) ==> x <= y"
+  shows "(SUP i:A. f i) = x"
+  unfolding SUPR_def Sup_extreal_def
+  using assms by (auto intro!: Least_equality)
+
+lemma extreal_INFI:
+  fixes x :: extreal
+  assumes "!!i. i : A ==> f i >= x"
+  assumes "!!y. (!!i. i : A ==> f i >= y) ==> x >= y"
+  shows "(INF i:A. f i) = x"
+  unfolding INFI_def Inf_extreal_def
+  using assms by (auto intro!: Greatest_equality)
+
+lemma Sup_extreal_close:
+  fixes e :: extreal
+  assumes "0 < e" and S: "Sup S \<noteq> \<infinity>" "Sup S \<noteq> -\<infinity>" "S \<noteq> {}"
+  shows "\<exists>x\<in>S. Sup S - e < x"
+proof (rule less_Sup_iff[THEN iffD1])
+  show "Sup S - e < Sup S " using assms
+    by (cases "Sup S", cases e) auto
+qed
+
+lemma Inf_extreal_close:
+  fixes e :: extreal assumes "Inf X \<noteq> \<infinity>" "Inf X \<noteq> -\<infinity>" "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 "Inf X", cases e) auto
+qed
+
+lemma (in complete_lattice) top_le:
+  "top \<le> x \<Longrightarrow> x = top"
+  by (rule antisym) auto
+
+lemma Sup_eq_top_iff:
+  fixes A :: "'a::{complete_lattice, linorder} set"
+  shows "Sup A = top \<longleftrightarrow> (\<forall>x<top. \<exists>i\<in>A. x < i)"
+proof
+  assume *: "Sup A = top"
+  show "(\<forall>x<top. \<exists>i\<in>A. x < i)" unfolding *[symmetric]
+  proof (intro allI impI)
+    fix x assume "x < Sup A" then show "\<exists>i\<in>A. x < i"
+      unfolding less_Sup_iff by auto
+  qed
+next
+  assume *: "\<forall>x<top. \<exists>i\<in>A. x < i"
+  show "Sup A = top"
+  proof (rule ccontr)
+    assume "Sup A \<noteq> top"
+    with top_greatest[of "Sup A"]
+    have "Sup A < top" unfolding le_less by auto
+    then have "Sup A < Sup A"
+      using * unfolding less_Sup_iff by auto
+    then show False by auto
+  qed
+qed
+
+lemma SUP_eq_top_iff:
+  fixes f :: "'a \<Rightarrow> 'b::{complete_lattice, linorder}"
+  shows "(SUP i:A. f i) = top \<longleftrightarrow> (\<forall>x<top. \<exists>i\<in>A. x < f i)"
+  unfolding SUPR_def Sup_eq_top_iff by auto
+
+lemma SUP_nat_Infty: "(SUP i::nat. extreal (real i)) = \<infinity>"
+proof -
+  { fix x assume "x \<noteq> \<infinity>"
+    then have "\<exists>k::nat. x < extreal (real k)"
+    proof (cases x)
+      case MInf then show ?thesis by (intro exI[of _ 0]) auto
+    next
+      case (real r)
+      moreover obtain k :: nat where "r < real k"
+        using ex_less_of_nat by (auto simp: real_eq_of_nat)
+      ultimately show ?thesis by auto
+    qed simp }
+  then show ?thesis
+    using SUP_eq_top_iff[of UNIV "\<lambda>n::nat. extreal (real n)"]
+    by (auto simp: top_extreal_def)
+qed
+
+lemma infeal_le_Sup:
+  fixes x :: extreal
+  shows "(x <= (SUP i:A. f i)) <-> (ALL y. y < x --> (EX i. i : A & y <= f i))"
+(is "?lhs <-> ?rhs")
+proof-
+{ assume "?rhs"
+  { assume "~(x <= (SUP i:A. f i))" hence "(SUP i:A. f i)<x" by (simp add: not_le)
+    from this obtain y where y_def: "(SUP i:A. f i)<y & y<x" using extreal_dense by auto
+    from this obtain i where "i : A & y <= f i" using `?rhs` by auto
+    hence "y <= (SUP i:A. f i)" using le_SUPI[of i A f] by auto
+    hence False using y_def by auto
+  } hence "?lhs" by auto
+}
+moreover
+{ assume "?lhs" hence "?rhs"
+  by (metis Collect_def Collect_mem_eq SUP_leI assms atLeastatMost_empty atLeastatMost_empty_iff
+      inf_sup_ord(4) linorder_le_cases sup_absorb1 xt1(8))
+} ultimately show ?thesis by auto
+qed
+
+lemma infeal_Inf_le:
+  fixes x :: extreal
+  shows "((INF i:A. f i) <= x) <-> (ALL y. x < y --> (EX i. i : A & f i <= y))"
+(is "?lhs <-> ?rhs")
+proof-
+{ assume "?rhs"
+  { assume "~((INF i:A. f i) <= x)" hence "x < (INF i:A. f i)" by (simp add: not_le)
+    from this obtain y where y_def: "x<y & y<(INF i:A. f i)" using extreal_dense by auto
+    from this obtain i where "i : A & f i <= y" using `?rhs` by auto
+    hence "(INF i:A. f i) <= y" using INF_leI[of i A f] by auto
+    hence False using y_def by auto
+  } hence "?lhs" by auto
+}
+moreover
+{ assume "?lhs" hence "?rhs"
+  by (metis Collect_def Collect_mem_eq le_INFI assms atLeastatMost_empty atLeastatMost_empty_iff
+      inf_sup_ord(4) linorder_le_cases sup_absorb1 xt1(8))
+} ultimately show ?thesis by auto
+qed
+
+lemma Inf_less:
+  fixes x :: extreal
+  assumes "(INF i:A. f i) < x"
+  shows "EX i. i : A & f i <= x"
+proof(rule ccontr)
+  assume "~ (EX i. i : A & f i <= x)"
+  hence "ALL i:A. f i > x" by auto
+  hence "(INF i:A. f i) >= x" apply (subst le_INFI) by auto
+  thus False using assms by auto
+qed
+
+lemma same_INF:
+  assumes "ALL e:A. f e = g e"
+  shows "(INF e:A. f e) = (INF e:A. g e)"
+proof-
+have "f ` A = g ` A" unfolding image_def using assms by auto
+thus ?thesis unfolding INFI_def by auto
+qed
+
+lemma same_SUP:
+  assumes "ALL e:A. f e = g e"
+  shows "(SUP e:A. f e) = (SUP e:A. g e)"
+proof-
+have "f ` A = g ` A" unfolding image_def using assms by auto
+thus ?thesis unfolding SUPR_def by auto
+qed
+
+subsection "Limits on @{typ extreal}"
+
+subsubsection "Topological space"
+
+instantiation extreal :: topological_space
+begin
+
+definition "open A \<longleftrightarrow>
+  (\<exists>T. open T \<and> extreal ` T = A - {\<infinity>, -\<infinity>})
+       \<and> (\<infinity> \<in> A \<longrightarrow> (\<exists>x. {extreal x <..} \<subseteq> A))
+       \<and> (-\<infinity> \<in> A \<longrightarrow> (\<exists>x. {..<extreal x} \<subseteq> A))"
+
+lemma open_PInfty: "open A ==> \<infinity> : A ==> (EX x. {extreal x<..} <= A)"
+  unfolding open_extreal_def by auto
+
+lemma open_MInfty: "open A ==> (-\<infinity>) : A ==> (EX x. {..<extreal x} <= A)"
+  unfolding open_extreal_def by auto
+
+lemma open_PInfty2: assumes "open A" "\<infinity> : A" obtains x where "{extreal x<..} <= A"
+  using open_PInfty[OF assms] by auto
+
+lemma open_MInfty2: assumes "open A" "(-\<infinity>) : A" obtains x where "{..<extreal x} <= A"
+  using open_MInfty[OF assms] by auto
+
+lemma extreal_openE: assumes "open A" obtains A' x y where
+  "open A'" "extreal ` A' = A - {\<infinity>, (-\<infinity>)}"
+  "\<infinity> : A ==> {extreal x<..} <= A"
+  "(-\<infinity>) : A ==> {..<extreal y} <= A"
+  using assms open_extreal_def by auto
+
+instance
+proof
+  let ?U = "UNIV::extreal set"
+  show "open ?U" unfolding open_extreal_def
+    by (auto intro!: exI[of _ "UNIV"] exI[of _ 0])
+next
+  fix S T::"extreal set" assume "open S" and "open T"
+  from `open S`[THEN extreal_openE] guess S' xS yS . note S' = this
+  from `open T`[THEN extreal_openE] guess T' xT yT . note T' = this
+
+  have "extreal ` (S' Int T') = (extreal ` S') Int (extreal ` T')" by auto
+  also have "... = S Int T - {\<infinity>, (-\<infinity>)}" using S' T' by auto
+  finally have "extreal ` (S' Int T') =  S Int T - {\<infinity>, (-\<infinity>)}" by auto
+  moreover have "open (S' Int T')" using S' T' by auto
+  moreover
+  { assume a: "\<infinity> : S Int T"
+    hence "EX x. {extreal x<..} <= S Int T"
+    apply(rule_tac x="max xS xT" in exI)
+    proof-
+    { fix x assume *: "extreal (max xS xT) < x"
+      hence "x : S Int T" apply (cases x, auto simp: max_def split: split_if_asm)
+      using a S' T' by auto
+    } thus "{extreal (max xS xT)<..} <= S Int T" by auto
+    qed }
+  moreover
+  { assume a: "(-\<infinity>) : S Int T"
+    hence "EX x. {..<extreal x} <= S Int T"
+    apply(rule_tac x="min yS yT" in exI)
+    proof-
+    { fix x assume *: "extreal (min yS yT) > x"
+      hence "x<extreal yS & x<extreal yT" by (cases x) auto
+      hence "x : S Int T" using a S' T' by auto
+    } thus "{..<extreal (min yS yT)} <= S Int T" by auto
+    qed }
+  ultimately show "open (S Int T)" unfolding open_extreal_def by auto
+next
+  fix K assume openK: "ALL S : K. open (S:: extreal set)"
+  hence "ALL S:K. EX T. open T & extreal ` T = S - {\<infinity>, (-\<infinity>)}" by (auto simp: open_extreal_def)
+  from bchoice[OF this] guess T .. note T = this[rule_format]
+
+  show "open (Union K)" unfolding open_extreal_def
+  proof (safe intro!: exI[of _ "Union (T ` K)"])
+    fix x S assume "x : T S" "S : K"
+    with T[OF `S : K`] show "extreal x : Union K" by auto
+  next
+    fix x S assume x: "x ~: extreal ` (Union (T ` K))" "S : K" "x : S" "x ~= \<infinity>"
+    hence "x ~: extreal ` (T S)"
+      by (auto simp: image_UN UN_simps[symmetric] simp del: UN_simps)
+    thus "x=(-\<infinity>)" using T[OF `S : K`] `x : S` `x ~= \<infinity>` by auto
+  next
+    fix S assume "\<infinity> : S" "S : K"
+    from openK[rule_format, OF `S : K`, THEN extreal_openE] guess S' x .
+    from this(3) `\<infinity> : S`
+    show "EX x. {extreal x<..} <= Union K"
+      by (auto intro!: exI[of _ x] bexI[OF _ `S : K`])
+  next
+    fix S assume "(-\<infinity>) : S" "S : K"
+    from openK[rule_format, OF `S : K`, THEN extreal_openE] guess S' x y .
+    from this(4) `(-\<infinity>) : S`
+    show "EX y. {..<extreal y} <= Union K"
+      by (auto intro!: exI[of _ y] bexI[OF _ `S : K`])
+  next
+    from T[THEN conjunct1] show "open (Union (T`K))" by auto
+  qed auto
+qed
+end
+
+lemma open_extreal_lessThan[simp]:
+  "open {..< a :: extreal}"
+proof (cases a)
+  case (real x)
+  then show ?thesis unfolding open_extreal_def
+  proof (safe intro!: exI[of _ "{..< x}"])
+    fix y assume "y < extreal x"
+    moreover assume "y ~: (extreal ` {..<x})"
+    ultimately have "y ~= extreal (real y)" using real by (cases y) auto
+    thus "y = (-\<infinity>)" apply (cases y) using `y < extreal x` by auto
+  qed auto
+qed (auto simp: open_extreal_def)
+
+lemma open_extreal_greaterThan[simp]:
+  "open {a :: extreal <..}"
+proof (cases a)
+  case (real x)
+  then show ?thesis unfolding open_extreal_def
+  proof (safe intro!: exI[of _ "{x<..}"])
+    fix y assume "extreal x < y"
+    moreover assume "y ~: (extreal ` {x<..})"
+    moreover assume "y ~= \<infinity>"
+    ultimately have "y ~= extreal (real y)" using real by (cases y) auto
+    hence False apply (cases y) using `extreal x < y` `y ~= \<infinity>` by auto
+    thus "y = (-\<infinity>)" by auto
+  qed auto
+qed (auto simp: open_extreal_def)
+
+lemma extreal_open_greaterThanLessThan[simp]: "open {a::extreal <..< b}"
+  unfolding greaterThanLessThan_def by auto
+
+lemma closed_extreal_atLeast[simp, intro]: "closed {a :: extreal ..}"
+proof -
+  have "- {a ..} = {..< a}" by auto
+  then show "closed {a ..}"
+    unfolding closed_def using open_extreal_lessThan by auto
+qed
+
+lemma closed_extreal_atMost[simp, intro]: "closed {.. b :: extreal}"
+proof -
+  have "- {.. b} = {b <..}" by auto
+  then show "closed {.. b}"
+    unfolding closed_def using open_extreal_greaterThan by auto
+qed
+
+lemma closed_extreal_atLeastAtMost[simp, intro]:
+  shows "closed {a :: extreal .. b}"
+  unfolding atLeastAtMost_def by auto
+
+lemma closed_extreal_singleton:
+  "closed {a :: extreal}"
+by (metis atLeastAtMost_singleton closed_extreal_atLeastAtMost)
+
+lemma extreal_open_cont_interval:
+  assumes "open S" "x \<in> S" and "x \<noteq> \<infinity>" "x \<noteq> - \<infinity>"
+  obtains e where "e>0" "{x-e <..< x+e} \<subseteq> S"
+proof-
+  obtain m where m_def: "x = extreal m" using assms by (cases x) auto
+  obtain A where "open A" and A_def: "extreal ` A = S - {\<infinity>,(-\<infinity>)}"
+    using assms by (auto elim!: extreal_openE)
+  hence "m : A" using m_def assms by auto
+  from this obtain e where e_def: "e>0 & ball m e <= A"
+    using open_contains_ball[of A] `open A` by auto
+  moreover have "ball m e = {m-e <..< m+e}" unfolding ball_def dist_norm by auto
+  ultimately have *: "{m-e <..< m+e} <= A" using e_def by auto
+  { fix y assume y_def: "y:{x-extreal e <..< x+extreal e}"
+    from this obtain z where z_def: "y = extreal z" by (cases y) auto
+    hence "z:A" using y_def m_def * by auto
+    hence "y:S" using z_def A_def by auto
+  } hence "{x-extreal e <..< x+extreal e} <= S" by auto
+  thus thesis apply- apply(rule that[of "extreal e"]) using e_def by auto
+qed
+
+lemma extreal_open_cont_interval2:
+  assumes "open S" "x \<in> S" and x: "x \<noteq> \<infinity>" "x \<noteq> -\<infinity>"
+  obtains a b where "a < x" "x < b" "{a <..< b} \<subseteq> S"
+proof-
+  guess e using extreal_open_cont_interval[OF assms] .
+  with that[of "x-e" "x+e"] extreal_between[OF x, of e]
+  show thesis by auto
+qed
+
+lemma extreal_uminus_eq_reorder: "- a = b \<longleftrightarrow> a = (-b::extreal)" by auto
+
+lemma extreal_uminus_less_reorder: "- a < b \<longleftrightarrow> -b < (a::extreal)"
+  by (subst (3) extreal_uminus_uminus[symmetric]) (simp only: extreal_minus_less_minus)
+
+lemma extreal_uminus_le_reorder: "- a \<le> b \<longleftrightarrow> -b \<le> (a::extreal)"
+  by (subst (3) extreal_uminus_uminus[symmetric]) (simp only: extreal_minus_le_minus)
+
+lemmas extreal_uminus_reorder =
+  extreal_uminus_eq_reorder extreal_uminus_less_reorder extreal_uminus_le_reorder
+
+lemma extreal_open_uminus:
+  fixes S :: "extreal set"
+  assumes "open S"
+  shows "open (uminus ` S)"
+proof-
+  obtain T x y where T_def: "open T & extreal ` T = S - {\<infinity>, (-\<infinity>)} &
+     (\<infinity> : S --> {extreal x<..} <= S) & ((-\<infinity>) : S --> {..<extreal y} <= S)"
+     using assms extreal_openE[of S] by metis
+  have "extreal ` uminus ` T = uminus ` extreal ` T" apply auto
+     by (metis imageI extreal_uminus_uminus uminus_extreal.simps)
+  also have "...=uminus ` (S - {\<infinity>, (-\<infinity>)})" using T_def by auto
+  finally have "extreal ` uminus ` T = uminus ` S - {\<infinity>, (-\<infinity>)}" by (auto simp: extreal_uminus_reorder)
+  moreover have "open (uminus ` T)" using T_def open_negations[of T] by auto
+  ultimately have "EX T. open T & extreal ` T = uminus ` S - {\<infinity>, (-\<infinity>)}" by auto
+  moreover
+  { assume "\<infinity>: uminus ` S"
+    hence "(-\<infinity>) : S" by (metis image_iff extreal_uminus_uminus)
+    hence "uminus ` {..<extreal y} <= uminus ` S" using T_def by (intro image_mono) auto
+    hence "EX x. {extreal x<..} <= uminus ` S" using extreal_uminus_lessThan by auto
+  } moreover
+  { assume "(-\<infinity>): uminus ` S"
+    hence "\<infinity> : S" by (metis image_iff extreal_uminus_uminus)
+    hence "uminus ` {extreal x<..} <= uminus ` S" using T_def by (intro image_mono) auto
+    hence "EX y. {..<extreal y} <= uminus ` S" using extreal_uminus_greaterThan by auto
+  }
+  ultimately show ?thesis unfolding open_extreal_def by auto
+qed
+
+lemma extreal_uminus_complement:
+  fixes S :: "extreal set"
+  shows "(uminus ` (- S)) = (- uminus ` S)"
+proof-
+{ fix x
+  have "x:uminus ` (- S) <-> -x:(- S)" by (metis image_iff extreal_uminus_uminus)
+  also have "... <-> x:(- uminus ` S)"
+     by (metis ComplI Compl_iff image_eqI extreal_uminus_uminus extreal_minus_minus_image)
+  finally have "x:uminus ` (- S) <-> x:(- uminus ` S)" by auto
+} thus ?thesis by auto
+qed
+
+lemma extreal_closed_uminus:
+  fixes S :: "extreal set"
+  assumes "closed S"
+  shows "closed (uminus ` S)"
+using assms unfolding closed_def
+using extreal_open_uminus[of "- S"] extreal_uminus_complement by auto
+
+
+lemma not_open_extreal_singleton:
+  "~(open {a :: extreal})"
+proof(rule ccontr)
+  assume "~ ~ open {a}" hence a: "open {a}" by auto
+  { assume "a=(-\<infinity>)"
+    then obtain y where "{..<extreal y} <= {a}" using a open_MInfty2[of "{a}"] by auto
+    hence "extreal(y - 1):{a}" apply (subst subsetD[of "{..<extreal y}"]) by auto
+    hence False using `a=(-\<infinity>)` by auto
+  } moreover
+  { assume "a=\<infinity>"
+    then obtain y where "{extreal y<..} <= {a}" using a open_PInfty2[of "{a}"] by auto
+    hence "extreal(y+1):{a}" apply (subst subsetD[of "{extreal y<..}"]) by auto
+    hence False using `a=\<infinity>` by auto
+  } moreover
+  { assume fin: "a~=(-\<infinity>)" "a~=\<infinity>"
+    from extreal_open_cont_interval[OF a singletonI this(2,1)] guess e . note e = this
+    then obtain b where b_def: "a<b & b<a+e"
+      using fin extreal_between extreal_dense[of a "a+e"] by auto
+    then have "b: {a-e <..< a+e}" using fin extreal_between[of a e] e by auto
+    then have False using b_def e by auto
+  } ultimately show False by auto
+qed
+
+lemma extreal_closed_contains_Inf:
+  fixes S :: "extreal set"
+  assumes "closed S" "S ~= {}"
+  shows "Inf S : S"
+proof(rule ccontr)
+assume "~(Inf S:S)" hence a: "open (-S)" "Inf S:(- S)" using assms by auto
+{ assume minf: "Inf S=(-\<infinity>)" hence "(-\<infinity>) : - S" using a by auto
+  then obtain y where "{..<extreal y} <= (-S)" using a open_MInfty2[of "- S"] by auto
+  hence "extreal y <= Inf S" by (metis Compl_anti_mono Compl_lessThan atLeast_iff
+     complete_lattice_class.Inf_greatest double_complement set_rev_mp)
+  hence False using minf by auto
+} moreover
+{ assume pinf: "Inf S=\<infinity>" hence "S={\<infinity>}" by (metis Inf_eq_PInfty assms(2))
+  hence False by (metis `Inf S ~: S` insert_code mem_def pinf)
+} moreover
+{ assume fin: "Inf S ~= \<infinity>" "Inf S ~= (-\<infinity>)"
+  from extreal_open_cont_interval[OF a this] guess e . note e = this
+  { fix x assume "x:S" hence "x>=Inf S" by (rule complete_lattice_class.Inf_lower)
+    hence *: "x>Inf S-e" using e by (metis fin extreal_between(1) order_less_le_trans)
+    { assume "x<Inf S+e" hence "x:{Inf S-e <..< Inf S+e}" using * by auto
+      hence False using e `x:S` by auto
+    } hence "x>=Inf S+e" by (metis linorder_le_less_linear)
+  } hence "Inf S + e <= Inf S" by (metis le_Inf_iff)
+  hence False by (metis calculation(1) calculation(2) e extreal_between(2) leD)
+} ultimately show False by auto
+qed
+
+lemma extreal_closed_contains_Sup:
+  fixes S :: "extreal set"
+  assumes "closed S" "S ~= {}"
+  shows "Sup S : S"
+proof-
+  have "closed (uminus ` S)" by (metis assms(1) extreal_closed_uminus)
+  hence "Inf (uminus ` S) : uminus ` S" using assms extreal_closed_contains_Inf[of "uminus ` S"] by auto
+  hence "- Sup S : uminus ` S" using extreal_Sup_uminus_image_eq[of "uminus ` S"] by (auto simp: image_image)
+  thus ?thesis by (metis imageI extreal_uminus_uminus extreal_minus_minus_image)
+qed
+
+lemma extreal_open_closed_aux:
+  fixes S :: "extreal set"
+  assumes "open S" "closed S"
+  assumes S: "(-\<infinity>) ~: S"
+  shows "S = {}"
+proof(rule ccontr)
+  assume "S ~= {}"
+  hence *: "(Inf S):S" by (metis assms(2) extreal_closed_contains_Inf)
+  { assume "Inf S=(-\<infinity>)" hence False using * assms(3) by auto }
+  moreover
+  { assume "Inf S=\<infinity>" hence "S={\<infinity>}" by (metis Inf_eq_PInfty `S ~= {}`)
+    hence False by (metis assms(1) not_open_extreal_singleton) }
+  moreover
+  { assume fin: "~(Inf S=\<infinity>)" "~(Inf S=(-\<infinity>))"
+    from extreal_open_cont_interval[OF assms(1) * fin] guess e . note e = this
+    then obtain b where b_def: "Inf S-e<b & b<Inf S"
+      using fin extreal_between[of "Inf S" e] extreal_dense[of "Inf S-e"] by auto
+    hence "b: {Inf S-e <..< Inf S+e}" using e fin extreal_between[of "Inf S" e] by auto
+    hence "b:S" using e by auto
+    hence False using b_def by (metis complete_lattice_class.Inf_lower leD)
+  } ultimately show False by auto
+qed
+
+
+lemma extreal_open_closed:
+  fixes S :: "extreal set"
+  shows "(open S & closed S) <-> (S = {} | S = UNIV)"
+proof-
+{ assume lhs: "open S & closed S"
+  { assume "(-\<infinity>) ~: S" hence "S={}" using lhs extreal_open_closed_aux by auto }
+  moreover
+  { assume "(-\<infinity>) : S" hence "(- S)={}" using lhs extreal_open_closed_aux[of "-S"] by auto }
+  ultimately have "S = {} | S = UNIV" by auto
+} thus ?thesis by auto
+qed
+
+
+lemma extreal_le_epsilon:
+  fixes x y :: extreal
+  assumes "ALL e. 0 < e --> x <= y + e"
+  shows "x <= y"
+proof-
+{ assume a: "EX r. y = extreal r"
+  from this obtain r where r_def: "y = extreal r" by auto
+  { assume "x=(-\<infinity>)" hence ?thesis by auto }
+  moreover
+  { assume "~(x=(-\<infinity>))"
+    from this obtain p where p_def: "x = extreal p"
+    using a assms[rule_format, of 1] by (cases x) auto
+    { fix e have "0 < e --> p <= r + e"
+      using assms[rule_format, of "extreal e"] p_def r_def by auto }
+    hence "p <= r" apply (subst field_le_epsilon) by auto
+    hence ?thesis using r_def p_def by auto
+  } ultimately have ?thesis by blast
+}
+moreover
+{ assume "y=(-\<infinity>) | y=\<infinity>" hence ?thesis
+    using assms[rule_format, of 1] by (cases x) auto
+} ultimately show ?thesis by (cases y) auto
+qed
+
+
+lemma extreal_le_epsilon2:
+  fixes x y :: extreal
+  assumes "ALL e. 0 < e --> x <= y + extreal e"
+  shows "x <= y"
+proof-
+{ fix e :: extreal assume "e>0"
+  { assume "e=\<infinity>" hence "x<=y+e" by auto }
+  moreover
+  { assume "e~=\<infinity>"
+    from this obtain r where "e = extreal r" using `e>0` apply (cases e) by auto
+    hence "x<=y+e" using assms[rule_format, of r] `e>0` by auto
+  } ultimately have "x<=y+e" by blast
+} from this show ?thesis using extreal_le_epsilon by auto
+qed
+
+lemma extreal_le_real:
+  fixes x y :: extreal
+  assumes "ALL z. x <= extreal z --> y <= extreal z"
+  shows "y <= x"
+by (metis assms extreal.exhaust extreal_bot extreal_less_eq(1)
+          extreal_less_eq(2) order_refl uminus_extreal.simps(2))
+
+lemma extreal_le_extreal:
+  fixes x y :: extreal
+  assumes "\<And>B. B < x \<Longrightarrow> B <= y"
+  shows "x <= y"
+by (metis assms extreal_dense leD linorder_le_less_linear)
+
+
+lemma extreal_ge_extreal:
+  fixes x y :: extreal
+  assumes "ALL B. B>x --> B >= y"
+  shows "x >= y"
+by (metis assms extreal_dense leD linorder_le_less_linear)
+
+
+instance extreal :: t2_space
+proof
+  fix x y :: extreal assume "x ~= y"
+  let "?P x (y::extreal)" = "EX U V. open U & open V & x : U & y : V & U Int V = {}"
+
+  { fix x y :: extreal assume "x < y"
+    from extreal_dense[OF this] obtain z where z: "x < z" "z < y" by auto
+    have "?P x y"
+      apply (rule exI[of _ "{..<z}"])
+      apply (rule exI[of _ "{z<..}"])
+      using z by auto }
+  note * = this
+
+  from `x ~= y`
+  show "EX U V. open U & open V & x : U & y : V & U Int V = {}"
+  proof (cases rule: linorder_cases)
+    assume "x = y" with `x ~= y` show ?thesis by simp
+  next assume "x < y" from *[OF this] show ?thesis by auto
+  next assume "y < x" from *[OF this] show ?thesis by auto
+  qed
+qed
+
+lemma open_extreal: assumes "open S" shows "open (extreal ` S)"
+  unfolding open_extreal_def apply(rule,rule,rule,rule assms) by auto
+
+lemma open_real_of_extreal:
+  fixes S :: "extreal set" assumes "open S"
+  shows "open (real ` (S - {\<infinity>, -\<infinity>}))"
+proof -
+  from `open S` obtain T where T: "open T" "S - {\<infinity>, -\<infinity>} = extreal ` T"
+    unfolding open_extreal_def by auto
+  show ?thesis using T by (simp add: image_image)
+qed
+
+subsubsection {* Convergent sequences *}
+
+lemma inj_extreal[simp, intro]: "inj_on extreal A" by (auto intro: inj_onI)
+
+lemma lim_extreal[simp]:
+  "((\<lambda>n. extreal (f n)) ---> extreal x) net \<longleftrightarrow> (f ---> x) net" (is "?l = ?r")
+proof (intro iffI topological_tendstoI)
+  fix S assume "?l" "open S" "x \<in> S"
+  then show "eventually (\<lambda>x. f x \<in> S) net"
+    using `?l`[THEN topological_tendstoD, OF open_extreal, OF `open S`]
+    by (simp add: inj_image_mem_iff)
+next
+  fix S assume "?r" "open S" "extreal x \<in> S"
+  have *: "\<And>x. x \<in> real ` (S - {\<infinity>, - \<infinity>}) \<longleftrightarrow> extreal x \<in> S"
+    apply (safe intro!: rev_image_eqI)
+    by (case_tac xa) auto
+  show "eventually (\<lambda>x. extreal (f x) \<in> S) net"
+    using `?r`[THEN topological_tendstoD, OF open_real_of_extreal, OF `open S`]
+    using `extreal x \<in> S` by (simp add: *)
+qed
+
+lemma lim_real_of_extreal[simp]:
+  assumes lim: "(f ---> extreal x) net"
+  shows "((\<lambda>x. real (f x)) ---> x) net"
+proof (intro topological_tendstoI)
+  fix S assume "open S" "x \<in> S"
+  then have S: "open S" "extreal x \<in> extreal ` S"
+    by (simp_all add: inj_image_mem_iff)
+  have "\<forall>x. f x \<in> extreal ` S \<longrightarrow> real (f x) \<in> S" by auto
+  from this lim[THEN topological_tendstoD, OF open_extreal, OF S]
+  show "eventually (\<lambda>x. real (f x) \<in> S) net"
+    by (rule eventually_mono)
+qed
+
+lemma Lim_PInfty: "f ----> \<infinity> <-> (ALL B. EX N. ALL n>=N. f n >= extreal B)" (is "?l = ?r")
+proof assume ?r show ?l apply(rule topological_tendstoI)
+    unfolding eventually_sequentially
+  proof- fix S assume "open S" "\<infinity> : S"
+    from open_PInfty[OF this] guess B .. note B=this
+    from `?r`[rule_format,of "B+1"] guess N .. note N=this
+    show "EX N. ALL n>=N. f n : S" apply(rule_tac x=N in exI)
+    proof safe case goal1
+      have "extreal B < extreal (B + 1)" by auto
+      also have "... <= f n" using goal1 N by auto
+      finally show ?case using B by fastsimp
+    qed
+  qed
+next assume ?l show ?r
+  proof fix B::real have "open {extreal B<..}" "\<infinity> : {extreal B<..}" by auto
+    from topological_tendstoD[OF `?l` this,unfolded eventually_sequentially]
+    guess N .. note N=this
+    show "EX N. ALL n>=N. extreal B <= f n" apply(rule_tac x=N in exI) using N by auto
+  qed
+qed
+
+
+lemma Lim_MInfty: "f ----> (-\<infinity>) <-> (ALL B. EX N. ALL n>=N. f n <= extreal B)" (is "?l = ?r")
+proof assume ?r show ?l apply(rule topological_tendstoI)
+    unfolding eventually_sequentially
+  proof- fix S assume "open S" "(-\<infinity>) : S"
+    from open_MInfty[OF this] guess B .. note B=this
+    from `?r`[rule_format,of "B-(1::real)"] guess N .. note N=this
+    show "EX N. ALL n>=N. f n : S" apply(rule_tac x=N in exI)
+    proof safe case goal1
+      have "extreal (B - 1) >= f n" using goal1 N by auto
+      also have "... < extreal B" by auto
+      finally show ?case using B by fastsimp
+    qed
+  qed
+next assume ?l show ?r
+  proof fix B::real have "open {..<extreal B}" "(-\<infinity>) : {..<extreal B}" by auto
+    from topological_tendstoD[OF `?l` this,unfolded eventually_sequentially]
+    guess N .. note N=this
+    show "EX N. ALL n>=N. extreal B >= f n" apply(rule_tac x=N in exI) using N by auto
+  qed
+qed
+
+
+lemma Lim_bounded_PInfty: assumes lim:"f ----> l" and "!!n. f n <= extreal B" shows "l ~= \<infinity>"
+proof(rule ccontr,unfold not_not) let ?B = "B + 1" assume as:"l=\<infinity>"
+  from lim[unfolded this Lim_PInfty,rule_format,of "?B"]
+  guess N .. note N=this[rule_format,OF le_refl]
+  hence "extreal ?B <= extreal B" using assms(2)[of N] by(rule order_trans)
+  hence "extreal ?B < extreal ?B" apply (rule le_less_trans) by auto
+  thus False by auto
+qed
+
+
+lemma Lim_bounded_MInfty: assumes lim:"f ----> l" and "!!n. f n >= extreal B" shows "l ~= (-\<infinity>)"
+proof(rule ccontr,unfold not_not) let ?B = "B - 1" assume as:"l=(-\<infinity>)"
+  from lim[unfolded this Lim_MInfty,rule_format,of "?B"]
+  guess N .. note N=this[rule_format,OF le_refl]
+  hence "extreal B <= extreal ?B" using assms(2)[of N] order_trans[of "extreal B" "f N" "extreal(B - 1)"] by blast
+  thus False by auto
+qed
+
+
+lemma tendsto_explicit:
+  "f ----> f0 <-> (ALL S. open S --> f0 : S --> (EX N. ALL n>=N. f n : S))"
+  unfolding tendsto_def eventually_sequentially by auto
+
+
+lemma tendsto_obtains_N:
+  assumes "f ----> f0"
+  assumes "open S" "f0 : S"
+  obtains N where "ALL n>=N. f n : S"
+  using tendsto_explicit[of f f0] assms by auto
+
+
+lemma tail_same_limit:
+  fixes X Y N
+  assumes "X ----> L" "ALL n>=N. X n = Y n"
+  shows "Y ----> L"
+proof-
+{ fix S assume "open S" and "L:S"
+  from this obtain N1 where "ALL n>=N1. X n : S"
+     using assms unfolding tendsto_def eventually_sequentially by auto
+  hence "ALL n>=max N N1. Y n : S" using assms by auto
+  hence "EX N. ALL n>=N. Y n : S" apply(rule_tac x="max N N1" in exI) by auto
+}
+thus ?thesis using tendsto_explicit by auto
+qed
+
+
+lemma Lim_bounded_PInfty2:
+assumes lim:"f ----> l" and "ALL n>=N. f n <= extreal B"
+shows "l ~= \<infinity>"
+proof-
+  def g == "(%n. if n>=N then f n else extreal B)"
+  hence "g ----> l" using tail_same_limit[of f l N g] lim by auto
+  moreover have "!!n. g n <= extreal B" using g_def assms by auto
+  ultimately show ?thesis using  Lim_bounded_PInfty by auto
+qed
+
+lemma Lim_bounded_extreal:
+  assumes lim:"f ----> (l :: extreal)"
+  and "ALL n>=M. f n <= C"
+  shows "l<=C"
+proof-
+{ assume "l=(-\<infinity>)" hence ?thesis by auto }
+moreover
+{ assume "~(l=(-\<infinity>))"
+  { assume "C=\<infinity>" hence ?thesis by auto }
+  moreover
+  { assume "C=(-\<infinity>)" hence "ALL n>=M. f n = (-\<infinity>)" using assms by auto
+    hence "l=(-\<infinity>)" using assms
+       Lim_unique[OF trivial_limit_sequentially] tail_same_limit[of "\<lambda>n. -\<infinity>" "-\<infinity>" M f, OF tendsto_const] by auto
+    hence ?thesis by auto }
+  moreover
+  { assume "EX B. C = extreal B"
+    from this obtain B where B_def: "C=extreal B" by auto
+    hence "~(l=\<infinity>)" using Lim_bounded_PInfty2 assms by auto
+    from this obtain m where m_def: "extreal m=l" using `~(l=(-\<infinity>))` by (cases l) auto
+    from this obtain N where N_def: "ALL n>=N. f n : {extreal(m - 1) <..< extreal(m+1)}"
+       apply (subst tendsto_obtains_N[of f l "{extreal(m - 1) <..< extreal(m+1)}"]) using assms by auto
+    { fix n assume "n>=N"
+      hence "EX r. extreal r = f n" using N_def by (cases "f n") auto
+    } from this obtain g where g_def: "ALL n>=N. extreal (g n) = f n" by metis
+    hence "(%n. extreal (g n)) ----> l" using tail_same_limit[of f l N] assms by auto
+    hence *: "(%n. g n) ----> m" using m_def by auto
+    { fix n assume "n>=max N M"
+      hence "extreal (g n) <= extreal B" using assms g_def B_def by auto
+      hence "g n <= B" by auto
+    } hence "EX N. ALL n>=N. g n <= B" by blast
+    hence "m<=B" using * LIMSEQ_le_const2[of g m B] by auto
+    hence ?thesis using m_def B_def by auto
+  } ultimately have ?thesis by (cases C) auto
+} ultimately show ?thesis by blast
+qed
+
+lemma real_of_extreal_0[simp]: "real (0::extreal) = 0"
+  unfolding real_of_extreal_def zero_extreal_def by simp
+
+lemma real_of_extreal_mult[simp]:
+  fixes a b :: extreal shows "real (a * b) = real a * real b"
+  by (cases rule: extreal2_cases[of a b]) auto
+
+lemma real_of_extreal_eq_0:
+  "real x = 0 \<longleftrightarrow> x = \<infinity> \<or> x = -\<infinity> \<or> x = 0"
+  by (cases x) auto
+
+lemma tendsto_extreal_realD:
+  fixes f :: "'a \<Rightarrow> extreal"
+  assumes "x \<noteq> 0" and tendsto: "((\<lambda>x. extreal (real (f x))) ---> x) net"
+  shows "(f ---> x) net"
+proof (intro topological_tendstoI)
+  fix S assume S: "open S" "x \<in> S"
+  with `x \<noteq> 0` have "open (S - {0})" "x \<in> S - {0}" by auto
+  from tendsto[THEN topological_tendstoD, OF this]
+  show "eventually (\<lambda>x. f x \<in> S) net"
+    by (rule eventually_rev_mp) (auto simp: extreal_real real_of_extreal_0)
+qed
+
+lemma tendsto_extreal_realI:
+  fixes f :: "'a \<Rightarrow> extreal"
+  assumes x: "x \<noteq> \<infinity>" "x \<noteq> -\<infinity>" and tendsto: "(f ---> x) net"
+  shows "((\<lambda>x. extreal (real (f x))) ---> x) net"
+proof (intro topological_tendstoI)
+  fix S assume "open S" "x \<in> S"
+  with x have "open (S - {\<infinity>, -\<infinity>})" "x \<in> S - {\<infinity>, -\<infinity>}" by auto
+  from tendsto[THEN topological_tendstoD, OF this]
+  show "eventually (\<lambda>x. extreal (real (f x)) \<in> S) net"
+    by (elim eventually_elim1) (auto simp: extreal_real)
+qed
+
+lemma extreal_mult_cancel_left:
+  fixes a b c :: extreal shows "a * b = a * c \<longleftrightarrow>
+    (((a = \<infinity> \<or> a = -\<infinity>) \<and> 0 < b * c) \<or> a = 0 \<or> b = c)"
+  by (cases rule: extreal3_cases[of a b c])
+     (simp_all add: zero_less_mult_iff)
+
+lemma extreal_inj_affinity:
+  assumes "m \<noteq> \<infinity>" "m \<noteq> -\<infinity>" "m \<noteq> 0" "t \<noteq> \<infinity>" "t \<noteq> -\<infinity>"
+  shows "inj_on (\<lambda>x. m * x + t) A"
+  using assms
+  by (cases rule: extreal2_cases[of m t])
+     (auto intro!: inj_onI simp: extreal_add_cancel_right extreal_mult_cancel_left)
+
+lemma extreal_PInfty_eq_plus[simp]:
+  shows "\<infinity> = a + b \<longleftrightarrow> a = \<infinity> \<or> b = \<infinity>"
+  by (cases rule: extreal2_cases[of a b]) auto
+
+lemma extreal_MInfty_eq_plus[simp]:
+  shows "-\<infinity> = a + b \<longleftrightarrow> (a = -\<infinity> \<and> b \<noteq> \<infinity>) \<or> (b = -\<infinity> \<and> a \<noteq> \<infinity>)"
+  by (cases rule: extreal2_cases[of a b]) auto
+
+lemma extreal_less_divide_pos:
+  "x > 0 \<Longrightarrow> x \<noteq> \<infinity> \<Longrightarrow> y < z / x \<longleftrightarrow> x * y < z"
+  by (cases rule: extreal3_cases[of x y z]) (auto simp: field_simps)
+
+lemma extreal_divide_less_pos:
+  "x > 0 \<Longrightarrow> x \<noteq> \<infinity> \<Longrightarrow> y / x < z \<longleftrightarrow> y < x * z"
+  by (cases rule: extreal3_cases[of x y z]) (auto simp: field_simps)
+
+lemma extreal_open_affinity_pos:
+  assumes "open S" and m: "m \<noteq> \<infinity>" "0 < m" and t: "t \<noteq> \<infinity>" "t \<noteq> -\<infinity>"
+  shows "open ((\<lambda>x. m * x + t) ` S)"
+proof -
+  obtain r where r[simp]: "m = extreal r" using m by (cases m) auto
+  obtain p where p[simp]: "t = extreal p" using t by (cases t) auto
+  have "r \<noteq> 0" "0 < r" and m': "m \<noteq> \<infinity>" "m \<noteq> -\<infinity>" "m \<noteq> 0" using m by auto
+  from `open S`[THEN extreal_openE] guess T l u . note T = this
+  let ?f = "(\<lambda>x. m * x + t)"
+  show ?thesis unfolding open_extreal_def
+  proof (intro conjI impI exI subsetI)
+    show "open ((\<lambda>x. r*x + p)`T)"
+      using open_affinity[OF `open T` `r \<noteq> 0`] by (auto simp: ac_simps)
+    have affine_infy: "?f ` {\<infinity>, - \<infinity>} = {\<infinity>, -\<infinity>}"
+      using `r \<noteq> 0` by auto
+    have "extreal ` (\<lambda>x. r * x + p) ` T = ?f ` (extreal ` T)"
+      by (simp add: image_image)
+    also have "\<dots> = ?f ` (S - {\<infinity>, -\<infinity>})"
+      using T(2) by simp
+    also have "\<dots> = ?f ` S - {\<infinity>, -\<infinity>}"
+      using extreal_inj_affinity[OF m' t] by (simp only: image_set_diff affine_infy)
+    finally show "extreal ` (\<lambda>x. r * x + p) ` T = ?f ` S - {\<infinity>, -\<infinity>}" .
+  next
+    assume "\<infinity> \<in> ?f`S" with `0 < r` have "\<infinity> \<in> S" by auto
+    fix x assume "x \<in> {extreal (r * l + p)<..}"
+    then have [simp]: "extreal (r * l + p) < x" by auto
+    show "x \<in> ?f`S"
+    proof (rule image_eqI)
+      show "x = m * ((x - t) / m) + t"
+        using m t by (cases rule: extreal3_cases[of m x t]) auto
+      have "extreal l < (x - t)/m"
+        using m t by (simp add: extreal_less_divide_pos extreal_less_minus)
+      then show "(x - t)/m \<in> S" using T(3)[OF `\<infinity> \<in> S`] by auto
+    qed
+  next
+    assume "-\<infinity> \<in> ?f`S" with `0 < r` have "-\<infinity> \<in> S" by auto
+    fix x assume "x \<in> {..<extreal (r * u + p)}"
+    then have [simp]: "x < extreal (r * u + p)" by auto
+    show "x \<in> ?f`S"
+    proof (rule image_eqI)
+      show "x = m * ((x - t) / m) + t"
+        using m t by (cases rule: extreal3_cases[of m x t]) auto
+      have "(x - t)/m < extreal u"
+        using m t by (simp add: extreal_divide_less_pos extreal_minus_less)
+      then show "(x - t)/m \<in> S" using T(4)[OF `-\<infinity> \<in> S`] by auto
+    qed
+  qed
+qed
+
+lemma extreal_open_affinity:
+  assumes "open S" and m: "m \<noteq> \<infinity>" "m \<noteq> -\<infinity>" "m \<noteq> 0" and t: "t \<noteq> \<infinity>" "t \<noteq> -\<infinity>"
+  shows "open ((\<lambda>x. m * x + t) ` S)"
+proof cases
+  assume "0 < m" then show ?thesis
+    using extreal_open_affinity_pos[OF `open S` `m \<noteq> \<infinity>` _ t] by auto
+next
+  assume "\<not> 0 < m" then
+  have "0 < -m" using `m \<noteq> 0` by (cases m) auto
+  then have m: "-m \<noteq> \<infinity>" "0 < -m" using `m \<noteq> -\<infinity>`
+    by (simp_all add: extreal_uminus_eq_reorder)
+  from extreal_open_affinity_pos[OF extreal_open_uminus[OF `open S`] m t]
+  show ?thesis unfolding image_image by simp
+qed
+
+lemma extreal_divide_eq:
+  "b \<noteq> 0 \<Longrightarrow> b \<noteq> \<infinity> \<Longrightarrow> b \<noteq> -\<infinity> \<Longrightarrow> a / b = c \<longleftrightarrow> a = b * c"
+  by (cases rule: extreal3_cases[of a b c])
+     (simp_all add: field_simps)
+
+lemma extreal_inverse_not_MInfty[simp]: "inverse a \<noteq> -\<infinity>"
+  by (cases a) auto
+
+lemma extreal_lim_mult:
+  fixes X :: "'a \<Rightarrow> extreal"
+  assumes lim: "(X ---> L) net" and a: "a \<noteq> \<infinity>" "a \<noteq> -\<infinity>"
+  shows "((\<lambda>i. a * X i) ---> a * L) net"
+proof cases
+  assume "a \<noteq> 0"
+  show ?thesis
+  proof (rule topological_tendstoI)
+    fix S assume "open S" "a * L \<in> S"
+    have "a * L / a = L"
+      using `a \<noteq> 0` a by (cases rule: extreal2_cases[of a L]) auto
+    then have L: "L \<in> ((\<lambda>x. x / a) ` S)"
+      using `a * L \<in> S` by (force simp: image_iff)
+    moreover have "open ((\<lambda>x. x / a) ` S)"
+      using extreal_open_affinity[OF `open S`, of "inverse a" 0] `a \<noteq> 0` a
+      by (simp add: extreal_divide_eq extreal_inverse_eq_0 divide_extreal_def ac_simps)
+    note * = lim[THEN topological_tendstoD, OF this L]
+    { fix x from a `a \<noteq> 0` have "a * (x / a) = x"
+        by (cases rule: extreal2_cases[of a x]) auto }
+    note this[simp]
+    show "eventually (\<lambda>x. a * X x \<in> S) net"
+      by (rule eventually_mono[OF _ *]) auto
+  qed
+qed auto
+
+lemma extreal_mult_m1[simp]: "x * extreal (-1) = -x"
+  by (cases x) auto
+
+lemma extreal_lim_uminus:
+  fixes X :: "'a \<Rightarrow> extreal" shows "((\<lambda>i. - X i) ---> -L) net \<longleftrightarrow> (X ---> L) net"
+  using extreal_lim_mult[of X L net "extreal (-1)"]
+        extreal_lim_mult[of "(\<lambda>i. - X i)" "-L" net "extreal (-1)"]
+  by (auto simp add: algebra_simps)
+
+lemma Lim_bounded2_extreal:
+  assumes lim:"f ----> (l :: extreal)"
+  and ge: "ALL n>=N. f n >= C"
+  shows "l>=C"
+proof-
+def g == "(%i. -(f i))"
+{ fix n assume "n>=N" hence "g n <= -C" using assms extreal_minus_le_minus g_def by auto }
+hence "ALL n>=N. g n <= -C" by auto
+moreover have limg: "g ----> (-l)" using g_def extreal_lim_uminus lim by auto
+ultimately have "-l <= -C" using Lim_bounded_extreal[of g "-l" _ "-C"] by auto
+from this show ?thesis using extreal_minus_le_minus by auto
+qed
+
+
+lemma extreal_LimI_finite:
+  assumes "x ~= \<infinity>" "x ~= (-\<infinity>)"
+  assumes "!!r. 0 < r ==> EX N. ALL n>=N. u n < x + r & x < u n + r"
+  shows "u ----> x"
+proof (rule topological_tendstoI, unfold eventually_sequentially)
+  obtain rx where rx_def: "x=extreal rx" using assms by (cases x) auto
+  fix S assume "open S" "x : S"
+  then obtain A where "open A" and A_eq: "extreal ` A = S - {\<infinity>,(-\<infinity>)}"
+     by (auto elim!: extreal_openE)
+  then have "x : extreal ` A" using `x : S` assms by auto
+  then have "rx : A" using rx_def by auto
+  then obtain r where "0 < r" and dist: "!!y. dist y (real x) < r ==> y : A"
+    using `open A` unfolding open_real_def rx_def by auto
+  then obtain n where
+    upper: "!!N. n <= N ==> u N < x + extreal r" and
+    lower: "!!N. n <= N ==> x < u N + extreal r" using assms(3)[of "extreal r"] by auto
+  show "EX N. ALL n>=N. 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 ~: {\<infinity>,(-\<infinity>)}" by auto
+    from this obtain ra where ra_def: "(u N) = extreal ra" by (cases "u N") auto
+    hence "rx < ra + r" and "ra < rx + r"
+       using rx_def assms `0 < r` lower[OF `n <= N`] upper[OF `n <= N`] by auto
+    hence "dist (real (u N)) (real x) < r"
+      using rx_def ra_def
+      by (auto simp: dist_real_def abs_diff_less_iff field_simps)
+    from dist[OF this]
+    have "u N : extreal ` A" using `u N  ~: {\<infinity>,(-\<infinity>)}`
+      by (auto intro!: image_eqI[of _ _ "real (u N)"] simp: extreal_real)
+    thus "u N : S" using A_eq by simp
+  qed
+qed
+
+lemma extreal_LimI_finite_iff:
+  assumes "x ~= \<infinity>" "x ~= (-\<infinity>)"
+  shows "u ----> x <-> (ALL r. 0 < r --> (EX N. ALL n>=N. u n < x + r & x < u n + r))"
+  (is "?lhs <-> ?rhs")
+proof-
+{ assume lim: "u ----> x"
+  { fix r assume "(r::extreal)>0"
+    from this obtain N where N_def: "ALL n>=N. u n : {x - r <..< x + r}"
+       apply (subst tendsto_obtains_N[of u x "{x - r <..< x + r}"])
+       using lim extreal_between[of x r] assms `r>0` by auto
+    hence "EX N. ALL n>=N. u n < x + r & x < u n + r"
+      using extreal_minus_less[of r x] by (cases r) auto
+  } hence "?rhs" by auto
+} from this show ?thesis using extreal_LimI_finite assms by blast
+qed
+
+
+subsubsection {* @{text Liminf} and @{text Limsup} *}
+
+definition
+  "Liminf net f = (GREATEST l. \<forall>y<l. eventually (\<lambda>x. y < f x) net)"
+
+definition
+  "Limsup net f = (LEAST l. \<forall>y>l. eventually (\<lambda>x. f x < y) net)"
+
+lemma Liminf_Sup:
+  fixes f :: "'a => 'b::{complete_lattice, linorder}"
+  shows "Liminf net f = Sup {l. \<forall>y<l. eventually (\<lambda>x. y < f x) net}"
+  by (auto intro!: Greatest_equality complete_lattice_class.Sup_upper simp: less_Sup_iff Liminf_def)
+
+lemma Limsup_Inf:
+  fixes f :: "'a => 'b::{complete_lattice, linorder}"
+  shows "Limsup net f = Inf {l. \<forall>y>l. eventually (\<lambda>x. f x < y) net}"
+  by (auto intro!: Least_equality complete_lattice_class.Inf_lower simp: Inf_less_iff Limsup_def)
+
+lemma extreal_SupI:
+  fixes x :: extreal
+  assumes "\<And>y. y \<in> A \<Longrightarrow> y \<le> x"
+  assumes "\<And>y. (\<And>z. z \<in> A \<Longrightarrow> z \<le> y) \<Longrightarrow> x \<le> y"
+  shows "Sup A = x"
+  unfolding Sup_extreal_def
+  using assms by (auto intro!: Least_equality)
+
+lemma extreal_InfI:
+  fixes x :: extreal
+  assumes "\<And>i. i \<in> A \<Longrightarrow> x \<le> i"
+  assumes "\<And>y. (\<And>i. i \<in> A \<Longrightarrow> y \<le> i) \<Longrightarrow> y \<le> x"
+  shows "Inf A = x"
+  unfolding Inf_extreal_def
+  using assms by (auto intro!: Greatest_equality)
+
+lemma Limsup_const:
+  fixes c :: "'a::{complete_lattice, linorder}"
+  assumes ntriv: "\<not> trivial_limit net"
+  shows "Limsup net (\<lambda>x. c) = c"
+  unfolding Limsup_Inf
+proof (safe intro!: antisym complete_lattice_class.Inf_greatest complete_lattice_class.Inf_lower)
+  fix x assume *: "\<forall>y>x. eventually (\<lambda>_. c < y) net"
+  show "c \<le> x"
+  proof (rule ccontr)
+    assume "\<not> c \<le> x" then have "x < c" by auto
+    then show False using ntriv * by (auto simp: trivial_limit_def)
+  qed
+qed auto
+
+lemma Liminf_const:
+  fixes c :: "'a::{complete_lattice, linorder}"
+  assumes ntriv: "\<not> trivial_limit net"
+  shows "Liminf net (\<lambda>x. c) = c"
+  unfolding Liminf_Sup
+proof (safe intro!: antisym complete_lattice_class.Sup_least complete_lattice_class.Sup_upper)
+  fix x assume *: "\<forall>y<x. eventually (\<lambda>_. y < c) net"
+  show "x \<le> c"
+  proof (rule ccontr)
+    assume "\<not> x \<le> c" then have "c < x" by auto
+    then show False using ntriv * by (auto simp: trivial_limit_def)
+  qed
+qed auto
+
+lemma mono_set:
+  fixes S :: "('a::order) set"
+  shows "mono S \<longleftrightarrow> (\<forall>x y. x \<le> y \<longrightarrow> x \<in> S \<longrightarrow> y \<in> S)"
+  by (auto simp: mono_def mem_def)
+
+lemma mono_greaterThan[intro, simp]: "mono {B<..}" unfolding mono_set by auto
+lemma mono_atLeast[intro, simp]: "mono {B..}" unfolding mono_set by auto
+lemma mono_UNIV[intro, simp]: "mono UNIV" unfolding mono_set by auto
+lemma mono_empty[intro, simp]: "mono {}" unfolding mono_set by auto
+
+lemma mono_set_iff:
+  fixes S :: "'a::{linorder,complete_lattice} set"
+  defines "a \<equiv> Inf S"
+  shows "mono S \<longleftrightarrow> (S = {a <..} \<or> S = {a..})" (is "_ = ?c")
+proof
+  assume "mono S"
+  then have mono: "\<And>x y. x \<le> y \<Longrightarrow> x \<in> S \<Longrightarrow> y \<in> S" by (auto simp: mono_set)
+  show ?c
+  proof cases
+    assume "a \<in> S"
+    show ?c
+      using mono[OF _ `a \<in> S`]
+      by (auto intro: complete_lattice_class.Inf_lower simp: a_def)
+  next
+    assume "a \<notin> S"
+    have "S = {a <..}"
+    proof safe
+      fix x assume "x \<in> S"
+      then have "a \<le> x" unfolding a_def by (rule complete_lattice_class.Inf_lower)
+      then show "a < x" using `x \<in> S` `a \<notin> S` by (cases "a = x") auto
+    next
+      fix x assume "a < x"
+      then obtain y where "y < x" "y \<in> S" unfolding a_def Inf_less_iff ..
+      with mono[of y x] show "x \<in> S" by auto
+    qed
+    then show ?c ..
+  qed
+qed auto
+
+lemma (in complete_lattice) not_less_bot[simp]: "\<not> (x < bot)"
+proof
+  assume "x < bot"
+  with bot_least[of x] show False by (auto simp: le_less)
+qed
+
+lemma (in complete_lattice) atLeast_eq_UNIV_iff: "{x..} = UNIV \<longleftrightarrow> x = bot"
+proof
+  assume "{x..} = UNIV"
+  show "x = bot"
+  proof (rule ccontr)
+    assume "x \<noteq> bot" then have "bot \<notin> {x..}" by (simp add: le_less)
+    then show False using `{x..} = UNIV` by simp
+  qed
+qed auto
+
+
+lemma extreal_open_atLeast: "open {x..} \<longleftrightarrow> x = -\<infinity>"
+proof
+  assume "x = -\<infinity>" then have "{x..} = UNIV" by auto
+  then show "open {x..}" by auto
+next
+  assume "open {x..}"
+  then have "open {x..} \<and> closed {x..}" by auto
+  then have "{x..} = UNIV" unfolding extreal_open_closed by auto
+  then show "x = -\<infinity>" by (simp add: bot_extreal_def atLeast_eq_UNIV_iff)
+qed
+
+lemma extreal_open_mono_set:
+  fixes S :: "extreal set"
+  defines "a \<equiv> Inf S"
+  shows "(open S \<and> mono S) \<longleftrightarrow> (S = UNIV \<or> S = {a <..})"
+  by (metis Inf_UNIV a_def atLeast_eq_UNIV_iff extreal_open_atLeast
+            extreal_open_closed mono_set_iff open_extreal_greaterThan)
+
+lemma extreal_closed_mono_set:
+  fixes S :: "extreal set"
+  shows "(closed S \<and> mono S) \<longleftrightarrow> (S = {} \<or> S = {Inf S ..})"
+  by (metis Inf_UNIV atLeast_eq_UNIV_iff closed_extreal_atLeast
+            extreal_open_closed mono_empty mono_set_iff open_extreal_greaterThan)
+
+lemma extreal_Liminf_Sup_monoset:
+  fixes f :: "'a => extreal"
+  shows "Liminf net f = Sup {l. \<forall>S. open S \<longrightarrow> mono S \<longrightarrow> l \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) net}"
+  unfolding Liminf_Sup
+proof (intro arg_cong[where f="\<lambda>P. Sup (Collect P)"] ext iffI allI impI)
+  fix l S assume ev: "\<forall>y<l. eventually (\<lambda>x. y < f x) net" and "open S" "mono S" "l \<in> S"
+  then have "S = UNIV \<or> S = {Inf S <..}"
+    using extreal_open_mono_set[of S] by auto
+  then show "eventually (\<lambda>x. f x \<in> S) net"
+  proof
+    assume S: "S = {Inf S<..}"
+    then have "Inf S < l" using `l \<in> S` by auto
+    then have "eventually (\<lambda>x. Inf S < f x) net" using ev by auto
+    then show "eventually (\<lambda>x. f x \<in> S) net"  by (subst S) auto
+  qed auto
+next
+  fix l y assume S: "\<forall>S. open S \<longrightarrow> mono S \<longrightarrow> l \<in> S \<longrightarrow> eventually  (\<lambda>x. f x \<in> S) net" "y < l"
+  have "eventually  (\<lambda>x. f x \<in> {y <..}) net"
+    using `y < l` by (intro S[rule_format]) auto
+  then show "eventually (\<lambda>x. y < f x) net" by auto
+qed
+
+lemma extreal_Inf_uminus_image_eq: "Inf (uminus ` S) = - Sup (S::extreal set)"
+  using extreal_Sup_uminus_image_eq[of "uminus ` S"] by (simp add: image_image)
+
+lemma extreal_range_uminus[simp]: "range uminus = (UNIV::extreal set)"
+proof safe
+  fix x :: extreal show "x \<in> range uminus" by (intro image_eqI[of _ _ "-x"]) auto
+qed auto
+
+lemma extreal_Limsup_Inf_monoset:
+  fixes f :: "'a => extreal"
+  shows "Limsup net f = Inf {l. \<forall>S. open S \<longrightarrow> mono (uminus ` S) \<longrightarrow> l \<in> S \<longrightarrow> eventually (\<lambda>x. f x \<in> S) net}"
+  unfolding Limsup_Inf
+proof (intro arg_cong[where f="\<lambda>P. Inf (Collect P)"] ext iffI allI impI)
+  fix l S assume ev: "\<forall>y>l. eventually (\<lambda>x. f x < y) net" and "open S" "mono (uminus`S)" "l \<in> S"
+  then have "open (uminus`S) \<and> mono (uminus`S)" by (simp add: extreal_open_uminus)
+  then have "S = UNIV \<or> S = {..< Sup S}"
+    unfolding extreal_open_mono_set extreal_Inf_uminus_image_eq extreal_image_uminus_shift by simp
+  then show "eventually (\<lambda>x. f x \<in> S) net"
+  proof
+    assume S: "S = {..< Sup S}"
+    then have "l < Sup S" using `l \<in> S` by auto
+    then have "eventually (\<lambda>x. f x < Sup S) net" using ev by auto
+    then show "eventually (\<lambda>x. f x \<in> S) net"  by (subst S) auto
+  qed auto
+next
+  fix l y assume S: "\<forall>S. open S \<longrightarrow> mono (uminus`S) \<longrightarrow> l \<in> S \<longrightarrow> eventually  (\<lambda>x. f x \<in> S) net" "l < y"
+  have "eventually  (\<lambda>x. f x \<in> {..< y}) net"
+    using `l < y` by (intro S[rule_format]) auto
+  then show "eventually (\<lambda>x. f x < y) net" by auto
+qed
+
+lemma open_uminus_iff: "open (uminus ` S) \<longleftrightarrow> open (S::extreal set)"
+  using extreal_open_uminus[of S] extreal_open_uminus[of "uminus`S"] by auto
+
+lemma extreal_Limsup_uminus:
+  fixes f :: "'a => extreal"
+  shows "Limsup net (\<lambda>x. - (f x)) = -(Liminf net f)"
+proof -
+  { fix P l have "(\<exists>x. (l::extreal) = -x \<and> P x) \<longleftrightarrow> P (-l)" by (auto intro!: exI[of _ "-l"]) }
+  note Ex_cancel = this
+  { fix P :: "extreal set \<Rightarrow> bool" have "(\<forall>S. P S) \<longleftrightarrow> (\<forall>S. P (uminus`S))"
+      apply auto by (erule_tac x="uminus`S" in allE) (auto simp: image_image) }
+  note add_uminus_image = this
+  { fix x S have "(x::extreal) \<in> uminus`S \<longleftrightarrow> -x\<in>S" by (auto intro!: image_eqI[of _ _ "-x"]) }
+  note remove_uminus_image = this
+  show ?thesis
+    unfolding extreal_Limsup_Inf_monoset extreal_Liminf_Sup_monoset
+    unfolding extreal_Inf_uminus_image_eq[symmetric] image_Collect Ex_cancel
+    by (subst add_uminus_image) (simp add: open_uminus_iff remove_uminus_image)
+qed
+
+lemma extreal_Liminf_uminus:
+  fixes f :: "'a => extreal"
+  shows "Liminf net (\<lambda>x. - (f x)) = -(Limsup net f)"
+  using extreal_Limsup_uminus[of _ "(\<lambda>x. - (f x))"] by auto
+
+lemma extreal_Lim_uminus:
+  fixes f :: "'a \<Rightarrow> extreal" shows "(f ---> f0) net \<longleftrightarrow> ((\<lambda>x. - f x) ---> - f0) net"
+  using
+    extreal_lim_mult[of f f0 net "- 1"]
+    extreal_lim_mult[of "\<lambda>x. - (f x)" "-f0" net "- 1"]
+  by (auto simp: extreal_uminus_reorder)
+
+lemma lim_imp_Liminf:
+  fixes f :: "'a \<Rightarrow> extreal"
+  assumes ntriv: "\<not> trivial_limit net"
+  assumes lim: "(f ---> f0) net"
+  shows "Liminf net f = f0"
+  unfolding Liminf_Sup
+proof (safe intro!: extreal_SupI)
+  fix y assume *: "\<forall>y'<y. eventually (\<lambda>x. y' < f x) net"
+  show "y \<le> f0"
+  proof (rule extreal_le_extreal)
+    fix B assume "B < y"
+    { assume "f0 < B"
+      then have "eventually (\<lambda>x. f x < B \<and> B < f x) net"
+         using topological_tendstoD[OF lim, of "{..<B}"] *[rule_format, OF `B < y`]
+         by (auto intro: eventually_conj)
+      also have "(\<lambda>x. f x < B \<and> B < f x) = (\<lambda>x. False)" by (auto simp: fun_eq_iff)
+      finally have False using ntriv[unfolded trivial_limit_def] by auto
+    } then show "B \<le> f0" by (metis linorder_le_less_linear)
+  qed
+next
+  fix y assume *: "\<forall>z. z \<in> {l. \<forall>y<l. eventually (\<lambda>x. y < f x) net} \<longrightarrow> z \<le> y"
+  show "f0 \<le> y"
+  proof (safe intro!: *[rule_format])
+    fix y assume "y < f0" then show "eventually (\<lambda>x. y < f x) net"
+      using lim[THEN topological_tendstoD, of "{y <..}"] by auto
+  qed
+qed
+
+lemma lim_imp_Limsup:
+  fixes f :: "'a => extreal"
+  assumes "\<not> trivial_limit net"
+  assumes lim: "(f ---> f0) net"
+  shows "Limsup net f = f0"
+  using extreal_Lim_uminus[of f f0] lim_imp_Liminf[of net "(%x. -(f x))" "-f0"]
+     extreal_Liminf_uminus[of net f] assms by simp
+
+lemma extreal_Liminf_le_Limsup:
+  fixes f :: "'a \<Rightarrow> extreal"
+  assumes ntriv: "\<not> trivial_limit net"
+  shows "Liminf net f \<le> Limsup net f"
+  unfolding Limsup_Inf Liminf_Sup
+proof (safe intro!: complete_lattice_class.Inf_greatest  complete_lattice_class.Sup_least)
+  fix u v assume *: "\<forall>y<u. eventually (\<lambda>x. y < f x) net" "\<forall>y>v. eventually (\<lambda>x. f x < y) net"
+  show "u \<le> v"
+  proof (rule ccontr)
+    assume "\<not> u \<le> v"
+    then obtain t where "t < u" "v < t"
+      using extreal_dense[of v u] by (auto simp: not_le)
+    then have "eventually (\<lambda>x. t < f x \<and> f x < t) net"
+      using * by (auto intro: eventually_conj)
+    also have "(\<lambda>x. t < f x \<and> f x < t) = (\<lambda>x. False)" by (auto simp: fun_eq_iff)
+    finally show False using ntriv by (auto simp: trivial_limit_def)
+  qed
+qed
+
+lemma Liminf_PInfty:
+  fixes f :: "'a \<Rightarrow> extreal"
+  assumes "\<not> trivial_limit net"
+  shows "(f ---> \<infinity>) net \<longleftrightarrow> Liminf net f = \<infinity>"
+proof (intro lim_imp_Liminf iffI assms)
+  assume rhs: "Liminf net f = \<infinity>"
+  { fix S assume "open S & \<infinity> : S"
+    then obtain m where "{extreal m<..} <= S" using open_PInfty2 by auto
+    moreover have "eventually (\<lambda>x. f x \<in> {extreal m<..}) net"
+      using rhs unfolding Liminf_Sup top_extreal_def[symmetric] Sup_eq_top_iff
+      by (auto elim!: allE[where x="extreal m"] simp: top_extreal_def)
+    ultimately have "eventually (%x. f x : S) net" apply (subst eventually_mono) by auto
+  } then show "(f ---> \<infinity>) net" unfolding tendsto_def by auto
+qed
+
+lemma Limsup_MInfty:
+  fixes f :: "'a \<Rightarrow> extreal"
+  assumes "\<not> trivial_limit net"
+  shows "(f ---> -\<infinity>) net \<longleftrightarrow> Limsup net f = -\<infinity>"
+  using assms extreal_Lim_uminus[of f "-\<infinity>"] Liminf_PInfty[of _ "\<lambda>x. - (f x)"]
+        extreal_Liminf_uminus[of _ f] by (auto simp: extreal_uminus_eq_reorder)
+
+lemma extreal_Liminf_eq_Limsup:
+  fixes f :: "'a \<Rightarrow> extreal"
+  assumes ntriv: "\<not> trivial_limit net"
+  assumes lim: "Liminf net f = f0" "Limsup net f = f0"
+  shows "(f ---> f0) net"
+proof (cases f0)
+  case PInf then show ?thesis using Liminf_PInfty[OF ntriv] lim by auto
+next
+  case MInf then show ?thesis using Limsup_MInfty[OF ntriv] lim by auto
+next
+  case (real r)
+  show "(f ---> f0) net"
+  proof (rule topological_tendstoI)
+    fix S assume "open S""f0 \<in> S"
+    then obtain a b where "a < Liminf net f" "Limsup net f < b" "{a<..<b} \<subseteq> S"
+      using extreal_open_cont_interval2[of S f0] real lim by auto
+    then have "eventually (\<lambda>x. f x \<in> {a<..<b}) net"
+      unfolding Liminf_Sup Limsup_Inf less_Sup_iff Inf_less_iff
+      by (auto intro!: eventually_conj simp add: greaterThanLessThan_iff)
+    with `{a<..<b} \<subseteq> S` show "eventually (%x. f x : S) net"
+      by (rule_tac eventually_mono) auto
+  qed
+qed
+
+lemma extreal_Liminf_eq_Limsup_iff:
+  fixes f :: "'a \<Rightarrow> extreal"
+  assumes "\<not> trivial_limit net"
+  shows "(f ---> f0) net \<longleftrightarrow> Liminf net f = f0 \<and> Limsup net f = f0"
+  by (metis assms extreal_Liminf_eq_Limsup lim_imp_Liminf lim_imp_Limsup)
+
+
+lemma Liminf_mono:
+  fixes f g :: "'a => extreal"
+  assumes ev: "eventually (\<lambda>x. f x \<le> g x) net"
+  shows "Liminf net f \<le> Liminf net g"
+  unfolding Liminf_Sup
+proof (safe intro!: Sup_mono bexI)
+  fix a y assume "\<forall>y<a. eventually (\<lambda>x. y < f x) net" and "y < a"
+  then have "eventually (\<lambda>x. y < f x) net" by auto
+  then show "eventually (\<lambda>x. y < g x) net"
+    by (rule eventually_rev_mp) (rule eventually_mono[OF _ ev], auto)
+qed simp
+
+lemma Liminf_eq:
+  fixes f g :: "'a \<Rightarrow> extreal"
+  assumes "eventually (\<lambda>x. f x = g x) net"
+  shows "Liminf net f = Liminf net g"
+  by (intro antisym Liminf_mono eventually_mono[OF _ assms]) auto
+
+lemma Liminf_mono_all:
+  fixes f g :: "'a \<Rightarrow> extreal"
+  assumes "\<And>x. f x \<le> g x"
+  shows "Liminf net f \<le> Liminf net g"
+  using assms by (intro Liminf_mono always_eventually) auto
+
+lemma Limsup_mono:
+  fixes f g :: "'a \<Rightarrow> extreal"
+  assumes ev: "eventually (\<lambda>x. f x \<le> g x) net"
+  shows "Limsup net f \<le> Limsup net g"
+  unfolding Limsup_Inf
+proof (safe intro!: Inf_mono bexI)
+  fix a y assume "\<forall>y>a. eventually (\<lambda>x. g x < y) net" and "a < y"
+  then have "eventually (\<lambda>x. g x < y) net" by auto
+  then show "eventually (\<lambda>x. f x < y) net"
+    by (rule eventually_rev_mp) (rule eventually_mono[OF _ ev], auto)
+qed simp
+
+lemma Limsup_mono_all:
+  fixes f g :: "'a \<Rightarrow> extreal"
+  assumes "\<And>x. f x \<le> g x"
+  shows "Limsup net f \<le> Limsup net g"
+  using assms by (intro Limsup_mono always_eventually) auto
+
+lemma Limsup_eq:
+  fixes f g :: "'a \<Rightarrow> extreal"
+  assumes "eventually (\<lambda>x. f x = g x) net"
+  shows "Limsup net f = Limsup net g"
+  by (intro antisym Limsup_mono eventually_mono[OF _ assms]) auto
+
+abbreviation "liminf \<equiv> Liminf sequentially"
+
+abbreviation "limsup \<equiv> Limsup sequentially"
+
+lemma (in complete_lattice) less_INFD:
+  assumes "y < INFI A f"" i \<in> A" shows "y < f i"
+proof -
+  note `y < INFI A f`
+  also have "INFI A f \<le> f i" using `i \<in> A` by (rule INF_leI)
+  finally show "y < f i" .
+qed
+
+lemma liminf_SUPR_INFI:
+  fixes f :: "nat \<Rightarrow> extreal"
+  shows "liminf f = (SUP n. INF m:{n..}. f m)"
+  unfolding Liminf_Sup eventually_sequentially
+proof (safe intro!: antisym complete_lattice_class.Sup_least)
+  fix x assume *: "\<forall>y<x. \<exists>N. \<forall>n\<ge>N. y < f n" show "x \<le> (SUP n. INF m:{n..}. f m)"
+  proof (rule extreal_le_extreal)
+    fix y assume "y < x"
+    with * obtain N where "\<And>n. N \<le> n \<Longrightarrow> y < f n" by auto
+    then have "y \<le> (INF m:{N..}. f m)" by (force simp: le_INF_iff)
+    also have "\<dots> \<le> (SUP n. INF m:{n..}. f m)" by (intro le_SUPI) auto
+    finally show "y \<le> (SUP n. INF m:{n..}. f m)" .
+  qed
+next
+  show "(SUP n. INF m:{n..}. f m) \<le> Sup {l. \<forall>y<l. \<exists>N. \<forall>n\<ge>N. y < f n}"
+  proof (unfold SUPR_def, safe intro!: Sup_mono bexI)
+    fix y n assume "y < INFI {n..} f"
+    from less_INFD[OF this] show "\<exists>N. \<forall>n\<ge>N. y < f n" by (intro exI[of _ n]) auto
+  qed (rule order_refl)
+qed
+
+lemma limsup_INFI_SUPR:
+  fixes f :: "nat \<Rightarrow> extreal"
+  shows "limsup f = (INF n. SUP m:{n..}. f m)"
+  using extreal_Limsup_uminus[of sequentially "\<lambda>x. - f x"]
+  by (simp add: liminf_SUPR_INFI extreal_INFI_uminus extreal_SUPR_uminus)
+
+lemma liminf_PInfty:
+  fixes X :: "nat => extreal"
+  shows "X ----> \<infinity> <-> liminf X = \<infinity>"
+by (metis Liminf_PInfty trivial_limit_sequentially)
+
+lemma limsup_MInfty:
+  fixes X :: "nat => extreal"
+  shows "X ----> (-\<infinity>) <-> limsup X = (-\<infinity>)"
+by (metis Limsup_MInfty trivial_limit_sequentially)
+
+lemma tail_same_limsup:
+  fixes X Y :: "nat => extreal"
+  assumes "\<And>n. N \<le> n \<Longrightarrow> X n = Y n"
+  shows "limsup X = limsup Y"
+  using Limsup_eq[of X Y sequentially] eventually_sequentially assms by auto
+
+lemma tail_same_liminf:
+  fixes X Y :: "nat => extreal"
+  assumes "\<And>n. N \<le> n \<Longrightarrow> X n = Y n"
+  shows "liminf X = liminf Y"
+  using Liminf_eq[of X Y sequentially] eventually_sequentially assms by auto
+
+lemma liminf_mono:
+  fixes X Y :: "nat \<Rightarrow> extreal"
+  assumes "\<And>n. N \<le> n \<Longrightarrow> X n <= Y n"
+  shows "liminf X \<le> liminf Y"
+  using Liminf_mono[of X Y sequentially] eventually_sequentially assms by auto
+
+lemma limsup_mono:
+  fixes X Y :: "nat => extreal"
+  assumes "\<And>n. N \<le> n \<Longrightarrow> X n <= Y n"
+  shows "limsup X \<le> limsup Y"
+  using Limsup_mono[of X Y sequentially] eventually_sequentially assms by auto
+
+declare trivial_limit_sequentially[simp]
+
+lemma liminf_bounded:
+  fixes X Y :: "nat \<Rightarrow> extreal"
+  assumes "\<And>n. N \<le> n \<Longrightarrow> C \<le> X n"
+  shows "C \<le> liminf X"
+  using liminf_mono[of N "\<lambda>n. C" X] assms Liminf_const[of sequentially C] by simp
+
+lemma limsup_bounded:
+  fixes X Y :: "nat => extreal"
+  assumes "\<And>n. N \<le> n \<Longrightarrow> X n <= C"
+  shows "limsup X \<le> C"
+  using limsup_mono[of N X "\<lambda>n. C"] assms Limsup_const[of sequentially C] by simp
+
+lemma liminf_bounded_iff:
+  fixes x :: "nat \<Rightarrow> extreal"
+  shows "C \<le> liminf x \<longleftrightarrow> (\<forall>B<C. \<exists>N. \<forall>n\<ge>N. B < x n)" (is "?lhs <-> ?rhs")
+proof safe
+  fix B assume "B < C" "C \<le> liminf x"
+  then have "B < liminf x" by auto
+  then obtain N where "B < (INF m:{N..}. x m)"
+    unfolding liminf_SUPR_INFI SUPR_def less_Sup_iff by auto
+  from less_INFD[OF this] show "\<exists>N. \<forall>n\<ge>N. B < x n" by auto
+next
+  assume *: "\<forall>B<C. \<exists>N. \<forall>n\<ge>N. B < x n"
+  { fix B assume "B<C"
+    then obtain N where "\<forall>n\<ge>N. B < x n" using `?rhs` by auto
+    hence "B \<le> (INF m:{N..}. x m)" by (intro le_INFI) auto
+    also have "... \<le> liminf x" unfolding liminf_SUPR_INFI by (intro le_SUPI) simp
+    finally have "B \<le> liminf x" .
+  } then show "?lhs" by (metis * leD liminf_bounded linorder_le_less_linear)
+qed
+
+lemma liminf_bounded_open:
+  fixes x :: "nat \<Rightarrow> extreal"
+  shows "x0 \<le> liminf x \<longleftrightarrow> (\<forall>S. open S \<longrightarrow> mono S \<longrightarrow> x0 \<in> S \<longrightarrow> (\<exists>N. \<forall>n\<ge>N. x n \<in> S))" 
+  (is "_ \<longleftrightarrow> ?P x0")
+proof
+  assume "?P x0" then show "x0 \<le> liminf x"
+    unfolding extreal_Liminf_Sup_monoset eventually_sequentially
+    by (intro complete_lattice_class.Sup_upper) auto
+next
+  assume "x0 \<le> liminf x"
+  { fix S :: "extreal set" assume om: "open S & mono S & x0:S"
+    { assume "S = UNIV" hence "EX N. (ALL n>=N. x n : S)" by auto }
+    moreover
+    { assume "~(S=UNIV)"
+      then obtain B where B_def: "S = {B<..}" using om extreal_open_mono_set by auto
+      hence "B<x0" using om by auto
+      hence "EX N. ALL n>=N. x n : S" unfolding B_def using `x0 \<le> liminf x` liminf_bounded_iff by auto
+    } ultimately have "EX N. (ALL n>=N. x n : S)" by auto
+  } then show "?P x0" by auto
+qed
+
+
+lemma extreal_lim_mono:
+  fixes X Y :: "nat => extreal"
+  assumes "\<And>n. N \<le> n \<Longrightarrow> X n <= Y n"
+  assumes "X ----> x" "Y ----> y"
+  shows "x <= y"
+  by (metis extreal_Liminf_eq_Limsup_iff[OF trivial_limit_sequentially] assms liminf_mono)
+
+lemma liminf_subseq_mono:
+  fixes X :: "nat \<Rightarrow> extreal"
+  assumes "subseq r"
+  shows "liminf X \<le> liminf (X \<circ> r) "
+proof-
+  have "\<And>n. (INF m:{n..}. X m) \<le> (INF m:{n..}. (X \<circ> r) m)"
+  proof (safe intro!: INF_mono)
+    fix n m :: nat assume "n \<le> m" then show "\<exists>ma\<in>{n..}. X ma \<le> (X \<circ> r) m"
+      using seq_suble[OF `subseq r`, of m] by (intro bexI[of _ "r m"]) auto
+  qed
+  then show ?thesis by (auto intro!: SUP_mono simp: liminf_SUPR_INFI comp_def)
+qed
+
+lemma limsup_subseq_mono:
+  fixes X :: "nat \<Rightarrow> extreal"
+  assumes "subseq r"
+  shows "limsup (X \<circ> r) \<le> limsup X"
+proof-
+  have "(\<lambda>n. - X n) \<circ> r = (\<lambda>n. - (X \<circ> r) n)" by (simp add: fun_eq_iff)
+  then have "- limsup X \<le> - limsup (X \<circ> r)"
+     using liminf_subseq_mono[of r "(%n. - X n)"]
+       extreal_Liminf_uminus[of sequentially X]
+       extreal_Liminf_uminus[of sequentially "X o r"] assms by auto
+  then show ?thesis by auto
+qed
+
+lemma bounded_abs:
+  assumes "(a::real)<=x" "x<=b"
+  shows "abs x <= max (abs a) (abs b)"
+by (metis abs_less_iff assms leI le_max_iff_disj less_eq_real_def less_le_not_le less_minus_iff minus_minus)
+
+
+lemma bounded_increasing_convergent2: fixes f::"nat => real"
+  assumes "ALL n. f n <= B"  "ALL n m. n>=m --> f n >= f m"
+  shows "EX l. (f ---> l) sequentially"
+proof-
+def N == "max (abs (f 0)) (abs B)"
+{ fix n have "abs (f n) <= N" unfolding N_def apply (subst bounded_abs) using assms by auto }
+hence "bounded {f n| n::nat. True}" unfolding bounded_real by auto
+from this show ?thesis apply(rule Topology_Euclidean_Space.bounded_increasing_convergent)
+   using assms by auto
+qed
+
+
+lemma extreal_real': assumes "x~=\<infinity>" and "x~=(-\<infinity>)" shows "extreal (real x) = x"
+   using assms extreal_real by auto
+
+
+lemma lim_extreal_increasing: assumes "\<And>n m. n >= m \<Longrightarrow> f n >= f m"
+  obtains l where "f ----> (l::extreal)"
+proof(cases "f = (\<lambda>x. - \<infinity>)")
+  case True then show thesis using Lim_const[of "- \<infinity>" sequentially] by (intro that[of "-\<infinity>"]) auto
+next
+  case False
+  from this obtain N where N_def: "f N > (-\<infinity>)" by (auto simp: fun_eq_iff)
+  have "ALL n>=N. f n >= f N" using assms by auto
+  hence minf: "ALL n>=N. f n > (-\<infinity>)" using N_def by auto
+  def Y == "(%n. (if n>=N then f n else f N))"
+  hence incy: "!!n m. n>=m ==> Y n >= Y m" using assms by auto
+  from minf have minfy: "ALL n. Y n ~= (-\<infinity>)" using Y_def by auto
+  show thesis
+  proof(cases "EX B. ALL n. f n < extreal B")
+    case False thus thesis apply- apply(rule that[of \<infinity>]) unfolding Lim_PInfty not_ex not_all
+    apply safe apply(erule_tac x=B in allE,safe) apply(rule_tac x=x in exI,safe)
+    apply(rule order_trans[OF _ assms[rule_format]]) by auto
+  next case True then guess B ..
+    hence "ALL n. Y n < extreal B" using Y_def by auto note B = this[rule_format]
+    { fix n have "Y n < \<infinity>" using B[of n] apply (subst less_le_trans) by auto
+      hence "Y n ~= \<infinity> & Y n ~= (-\<infinity>)" using minfy by auto
+    } hence *: "ALL n. Y n ~= \<infinity> & Y n ~= (-\<infinity>)" by auto
+    { fix n have "real (Y n) < B" proof- case goal1 thus ?case
+        using B[of n] apply-apply(subst(asm) extreal_real'[THEN sym]) defer defer
+        unfolding extreal_less using * by auto
+      qed
+    }
+    hence B': "ALL n. (real (Y n) <= B)" using less_imp_le by auto
+    have "EX l. (%n. real (Y n)) ----> l"
+      apply(rule bounded_increasing_convergent2)
+    proof safe show "!!n. real (Y n) <= B" using B' by auto
+      fix n m::nat assume "n<=m"
+      hence "extreal (real (Y n)) <= extreal (real (Y m))"
+        using incy[rule_format,of n m] apply(subst extreal_real)+
+        using *[rule_format, of n] *[rule_format, of m] by auto
+      thus "real (Y n) <= real (Y m)" by auto
+    qed then guess l .. note l=this
+    have "Y ----> extreal l" using l apply-apply(subst(asm) lim_extreal[THEN sym])
+    unfolding extreal_real using * by auto
+    thus thesis apply-apply(rule that[of "extreal l"])
+       apply (subst tail_same_limit[of Y _ N]) using Y_def by auto
+  qed
+qed
+
+
+lemma lim_extreal_decreasing: assumes "\<And>n m. n >= m \<Longrightarrow> f n <= f m"
+  obtains l where "f ----> (l::extreal)"
+proof -
+  from lim_extreal_increasing[of "\<lambda>x. - f x"] assms
+  obtain l where "(\<lambda>x. - f x) ----> l" by auto
+  from extreal_lim_mult[OF this, of "- 1"] show thesis 
+    by (intro that[of "-l"]) (simp add: extreal_uminus_eq_reorder)
+qed
+
+lemma compact_extreal:
+  fixes X :: "nat \<Rightarrow> extreal"
+  shows "\<exists>l r. subseq r \<and> (X \<circ> r) ----> l"
+proof -
+  obtain r where "subseq r" and mono: "monoseq (X \<circ> r)"
+    using seq_monosub[of X] unfolding comp_def by auto
+  then have "(\<forall>n m. m \<le> n \<longrightarrow> (X \<circ> r) m \<le> (X \<circ> r) n) \<or> (\<forall>n m. m \<le> n \<longrightarrow> (X \<circ> r) n \<le> (X \<circ> r) m)"
+    by (auto simp add: monoseq_def)
+  then obtain l where "(X\<circ>r) ----> l"
+     using lim_extreal_increasing[of "X \<circ> r"] lim_extreal_decreasing[of "X \<circ> r"] by auto
+  then show ?thesis using `subseq r` by auto
+qed
+
+lemma extreal_Sup_lim:
+  assumes "\<And>n. b n \<in> s" "b ----> (a::extreal)"
+  shows "a \<le> Sup s"
+by (metis Lim_bounded_extreal assms complete_lattice_class.Sup_upper)
+
+lemma extreal_Inf_lim:
+  assumes "\<And>n. b n \<in> s" "b ----> (a::extreal)"
+  shows "Inf s \<le> a"
+by (metis Lim_bounded2_extreal assms complete_lattice_class.Inf_lower)
+
+lemma incseq_le_extreal: assumes inc: "!!n m. n>=m ==> X n >= X m"
+  and lim: "X ----> (L::extreal)" shows "X N <= L"
+proof(cases "X N = (-\<infinity>)")
+case True thus ?thesis by auto
+next
+case False
+   have "ALL n>=N. X n >= X N" using inc by auto
+   hence minf: "ALL n>=N. X n > (-\<infinity>)" using False by auto
+   def Y == "(%n. (if n>=N then X n else X N))"
+   hence incy: "!!n m. n>=m ==> Y n >= Y m" using inc by auto
+   from minf have minfy: "ALL n. Y n ~= (-\<infinity>)" using Y_def by auto
+   from lim have limy: "Y ----> L"
+      apply (subst tail_same_limit[of X _ N]) using Y_def by auto
+show ?thesis
+proof(cases "L = \<infinity> | L=(-\<infinity>)")
+  case False have "ALL n. Y n ~= \<infinity>"
+  proof(rule ccontr,unfold not_all not_not,safe)
+    case goal1 hence "ALL n>=x. Y n = \<infinity>" using incy[of x] by auto
+    hence "Y ----> \<infinity>" unfolding tendsto_def eventually_sequentially
+      apply safe apply(rule_tac x=x in exI) by auto
+    note Lim_unique[OF trivial_limit_sequentially this limy]
+    with False show False by auto
+  qed note * =this[rule_format]
+
+  have **:"ALL m n. m <= n --> extreal (real (Y m)) <= extreal (real (Y n))"
+    unfolding extreal_real using minfy * incy apply (cases "Y m", cases "Y n") by auto
+  have "real (Y N) <= real L" apply-apply(rule incseq_le) defer
+  apply(subst lim_extreal[THEN sym])
+    unfolding extreal_real
+    unfolding incseq_def using minfy * ** limy False by auto
+  hence "extreal (real (Y N)) <= extreal (real L)" by auto
+  hence ***: "Y N <= L" unfolding extreal_real using minfy * False by auto
+  thus ?thesis using Y_def by auto
+next
+case True
+show ?thesis proof(cases "L=(-\<infinity>)")
+  case True
+  have "open {..<X N}" by auto
+  moreover have "(-\<infinity>) : {..<X N}" using False by auto
+  ultimately obtain N1 where "ALL n>=N1. X n : {..<X N}" using lim True
+     unfolding tendsto_def eventually_sequentially by metis
+  hence "X (max N N1) : {..<X N}" by auto
+  with inc[of N "max N N1"] show ?thesis by auto
+next
+case False thus ?thesis using True by auto qed
+qed
+qed
+
+
+lemma decseq_ge_extreal: assumes dec: "!!n m. n>=m ==> X n <= X m"
+  and lim: "X ----> (L::extreal)" shows "X N >= L"
+proof-
+def Y == "(%i. -(X i))"
+hence inc: "!!n m. n>=m ==> Y n >= Y m" using dec extreal_minus_le_minus by auto
+moreover have limy: "Y ----> (-L)" using Y_def extreal_lim_uminus lim by auto
+ultimately have "Y N <= -L" using incseq_le_extreal[of Y "-L"] by auto
+from this show ?thesis using Y_def extreal_minus_le_minus by auto
+qed
+
+
+lemma real_interm:
+ assumes "(a::real)<b"
+ shows "a + (b-a)/2 < b"
+by (metis Bit0_def assms comm_semiring_1_class.normalizing_semiring_rules(24) diff_minus_eq_add number_of_is_id one_is_num_one pth_2 real_average_minus_second real_gt_half_sum succ_def)
+
+
+lemma SUP_Lim_extreal: assumes "!!n m. n>=m ==> f n >= f m" "f ----> l"
+  shows "(SUP n. f n) = (l::extreal)" unfolding SUPR_def Sup_extreal_def
+proof (safe intro!: Least_equality)
+  fix n::nat show "f n <= l" apply(rule incseq_le_extreal)
+    using assms by auto
+next fix y assume y:"ALL x:range f. x <= y" show "l <= y"
+  proof-
+    { assume ym: "y ~= (-\<infinity>)" and yp: "y ~= \<infinity>"
+      { assume as:"y < l"
+        hence lm: "l ~= (-\<infinity>)" by auto
+        have lp:"l ~= \<infinity>" apply(rule Lim_bounded_PInfty[OF assms(2), of "real y"])
+          using y yp unfolding extreal_real by auto
+        have [simp]: "extreal (1 / 2) = 1 / 2" by (auto simp: divide_extreal_def)
+        have yl:"real y < real l" using as apply-
+          apply(subst(asm) extreal_real'[THEN sym,OF `y~=\<infinity>` `y~=(-\<infinity>)`])
+          apply(subst(asm) extreal_real'[THEN sym,OF `l~=\<infinity>` `l~=(-\<infinity>)`])
+          unfolding extreal_less by auto
+        hence "y + (l - y) * 1 / 2 < l" apply-
+          apply(subst extreal_real'[THEN sym,OF `y~=\<infinity>` `y~=(-\<infinity>)`])
+          apply(subst(2) extreal_real'[THEN sym,OF `y~=\<infinity>` `y~=(-\<infinity>)`])
+          apply(subst extreal_real'[THEN sym,OF `l~=\<infinity>` `l~=(-\<infinity>)`])
+          apply(subst(2) extreal_real'[THEN sym,OF `l~=\<infinity>` `l~=(-\<infinity>)`])
+          using real_interm by auto
+        hence *:"l : {y + (l - y) / 2<..}" by auto
+        have "open {y + (l-y)/2 <..}" by auto
+        note topological_tendstoD[OF assms(2) this *]
+        from this[unfolded eventually_sequentially] guess N .. note this[rule_format, of N]
+        hence "y + (l - y)  / 2 < y" using y[rule_format,of "f N"] by auto
+        hence "extreal (real y) + (extreal (real l) - extreal (real y)) / 2 < extreal (real y)"
+          unfolding extreal_real using `y~=\<infinity>` `y~=(-\<infinity>)` `l~=\<infinity>` `l~=(-\<infinity>)` by auto
+        hence False using yl by auto
+      } hence ?thesis using not_le by auto
+    }
+    moreover
+    { assume "y=(-\<infinity>)" hence "f = (\<lambda>_. -\<infinity>)" using y by (auto simp: fun_eq_iff)
+      hence "l=(-\<infinity>)" using `f ----> l` using tendsto_const[of "-\<infinity>"]
+         Lim_unique[OF trivial_limit_sequentially] by auto
+      hence ?thesis by auto
+    }
+    moreover have "y=\<infinity> --> l <= y" by auto
+    ultimately show ?thesis by blast
+  qed
+qed
+
+lemma INF_Lim_extreal: assumes "!!n m. n>=m ==> f n <= f m" "f ----> l"
+  shows "(INF n. f n) = (l::extreal)"
+proof-
+def Y == "(%i. -(f i))"
+hence inc: "!!n m. n>=m ==> Y n >= Y m" using assms extreal_minus_le_minus by auto
+moreover have limy: "Y ----> (-l)" using Y_def extreal_lim_uminus assms by auto
+ultimately have "(SUP n. Y n) = -l" using SUP_Lim_extreal[of Y "-l"] by auto
+hence "- (INF n. f n) = - l" using Y_def extreal_SUPR_uminus[of "UNIV" f] by auto
+from this show ?thesis by simp
+qed
+
+
+lemma incseq_mono: "mono f <-> incseq f"
+  unfolding mono_def incseq_def by auto
+
+
+lemma SUP_eq_LIMSEQ:
+  assumes "mono f"
+  shows "(SUP n. extreal (f n)) = extreal x <-> f ----> x"
+proof
+  assume x: "(SUP n. extreal (f n)) = extreal x"
+  { fix n
+    have "extreal (f n) <= extreal x" using x[symmetric] by (auto intro: le_SUPI)
+    hence "f n <= x" using assms by simp }
+  show "f ----> x"
+  proof (rule LIMSEQ_I)
+    fix r :: real assume "0 < r"
+    show "EX no. ALL n>=no. norm (f n - x) < r"
+    proof (rule ccontr)
+      assume *: "~ ?thesis"
+      { fix N
+        from * obtain n where "N <= n" "r <= x - f n"
+          using `!!n. f n <= x` by (auto simp: not_less)
+        hence "f N <= f n" using `mono f` by (auto dest: monoD)
+        hence "f N <= x - r" using `r <= x - f n` by auto
+        hence "extreal (f N) <= extreal (x - r)" by auto }
+      hence "(SUP n. extreal (f n)) <= extreal (x - r)"
+        and "extreal (x - r) < extreal x" using `0 < r` by (auto intro: SUP_leI)
+      hence "(SUP n. extreal (f n)) < extreal x" by (rule le_less_trans)
+      thus False using x by auto
+    qed
+  qed
+next
+  assume "f ----> x"
+  show "(SUP n. extreal (f n)) = extreal x"
+  proof (rule extreal_SUPI)
+    fix n
+    from incseq_le[of f x] `mono f` `f ----> x`
+    show "extreal (f n) <= extreal x" using assms incseq_mono by auto
+  next
+    fix y assume *: "!!n. n:UNIV ==> extreal (f n) <= y"
+    show "extreal x <= y"
+    proof-
+    { assume "EX r. y = extreal r"
+      from this obtain r where r_def: "y = extreal r" by auto
+      with * have "EX N. ALL n>=N. f n <= r" using assms by fastsimp
+      from LIMSEQ_le_const2[OF `f ----> x` this]
+      have "extreal x <= y" using r_def by auto
+    }
+    moreover
+    { assume "y=\<infinity> | y=(-\<infinity>)"
+      hence ?thesis using * by auto
+    } ultimately show ?thesis by (cases y) auto
+    qed
+  qed
+qed
+
+
+lemma Liminf_within:
+  fixes f :: "'a::metric_space => extreal"
+  shows "Liminf (at x within S) f = (SUP e:{0<..}. INF y:(S Int ball x e - {x}). f y)"
+proof-
+let ?l="(SUP e:{0<..}. INF y:(S Int ball x e - {x}). f y)"
+{ fix T assume T_def: "open T & mono T & ?l:T"
+  have "EX d>0. ALL y:S. 0 < dist y x & dist y x < d --> f y : T"
+  proof-
+  { assume "T=UNIV" hence ?thesis by (simp add: gt_ex) }
+  moreover
+  { assume "~(T=UNIV)"
+    then obtain B where "T={B<..}" using T_def extreal_open_mono_set[of T] by auto
+    hence "B<?l" using T_def by auto
+    then obtain d where d_def: "0<d & B<(INF y:(S Int ball x d - {x}). f y)"
+      unfolding less_SUP_iff by auto
+    { fix y assume "y:S & 0 < dist y x & dist y x < d"
+      hence "y:(S Int ball x d - {x})" unfolding ball_def by (auto simp add: dist_commute)
+      hence "f y:T" using d_def INF_leI[of y "S Int ball x d - {x}" f] `T={B<..}` by auto
+    } hence ?thesis apply(rule_tac x="d" in exI) using d_def by auto
+  } ultimately show ?thesis by auto
+  qed
+}
+moreover
+{ fix z
+  assume a: "ALL T. open T --> mono T --> z : T -->
+     (EX d>0. ALL y:S. 0 < dist y x & dist y x < d --> f y : T)"
+  { fix B assume "B<z"
+    then obtain d where d_def: "d>0 & (ALL y:S. 0 < dist y x & dist y x < d --> B < f y)"
+       using a[rule_format, of "{B<..}"] mono_greaterThan by auto
+    { fix y assume "y:(S Int ball x d - {x})"
+      hence "y:S & 0 < dist y x & dist y x < d" unfolding ball_def apply (simp add: dist_commute)
+         by (metis dist_eq_0_iff real_less_def zero_le_dist)
+      hence "B <= f y" using d_def by auto
+    } hence "B <= INFI (S Int ball x d - {x}) f" apply (subst le_INFI) by auto
+    also have "...<=?l" apply (subst le_SUPI) using d_def by auto
+    finally have "B<=?l" by auto
+  } hence "z <= ?l" using extreal_le_extreal[of z "?l"] by auto
+}
+ultimately show ?thesis unfolding extreal_Liminf_Sup_monoset eventually_within
+   apply (subst extreal_SupI[of _ "(SUP e:{0<..}. INFI (S Int ball x e - {x}) f)"]) by auto
+qed
+
+lemma Limsup_within:
+  fixes f :: "'a::metric_space => extreal"
+  shows "Limsup (at x within S) f = (INF e:{0<..}. SUP y:(S Int ball x e - {x}). f y)"
+proof-
+let ?l="(INF e:{0<..}. SUP y:(S Int ball x e - {x}). f y)"
+{ fix T assume T_def: "open T & mono (uminus ` T) & ?l:T"
+  have "EX d>0. ALL y:S. 0 < dist y x & dist y x < d --> f y : T"
+  proof-
+  { assume "T=UNIV" hence ?thesis by (simp add: gt_ex) }
+  moreover
+  { assume "~(T=UNIV)" hence "~(uminus ` T = UNIV)"
+       by (metis Int_UNIV_right Int_absorb1 image_mono extreal_minus_minus_image subset_UNIV)
+    hence "uminus ` T = {Inf (uminus ` T)<..}" using T_def extreal_open_mono_set[of "uminus ` T"]
+       extreal_open_uminus[of T] by auto
+    then obtain B where "T={..<B}"
+      unfolding extreal_Inf_uminus_image_eq extreal_uminus_lessThan[symmetric]
+      unfolding inj_image_eq_iff[OF extreal_inj_on_uminus] by simp
+    hence "?l<B" using T_def by auto
+    then obtain d where d_def: "0<d & (SUP y:(S Int ball x d - {x}). f y)<B"
+      unfolding INF_less_iff by auto
+    { fix y assume "y:S & 0 < dist y x & dist y x < d"
+      hence "y:(S Int ball x d - {x})" unfolding ball_def by (auto simp add: dist_commute)
+      hence "f y:T" using d_def le_SUPI[of y "S Int ball x d - {x}" f] `T={..<B}` by auto
+    } hence ?thesis apply(rule_tac x="d" in exI) using d_def by auto
+  } ultimately show ?thesis by auto
+  qed
+}
+moreover
+{ fix z
+  assume a: "ALL T. open T --> mono (uminus ` T) --> z : T -->
+     (EX d>0. ALL y:S. 0 < dist y x & dist y x < d --> f y : T)"
+  { fix B assume "z<B"
+    then obtain d where d_def: "d>0 & (ALL y:S. 0 < dist y x & dist y x < d --> f y<B)"
+       using a[rule_format, of "{..<B}"] by auto
+    { fix y assume "y:(S Int ball x d - {x})"
+      hence "y:S & 0 < dist y x & dist y x < d" unfolding ball_def apply (simp add: dist_commute)
+         by (metis dist_eq_0_iff real_less_def zero_le_dist)
+      hence "f y <= B" using d_def by auto
+    } hence "SUPR (S Int ball x d - {x}) f <= B" apply (subst SUP_leI) by auto
+    moreover have "?l<=SUPR (S Int ball x d - {x}) f" apply (subst INF_leI) using d_def by auto
+    ultimately have "?l<=B" by auto
+  } hence "?l <= z" using extreal_ge_extreal[of z "?l"] by auto
+}
+ultimately show ?thesis unfolding extreal_Limsup_Inf_monoset eventually_within
+   apply (subst extreal_InfI) by auto
+qed
+
+
+lemma Liminf_within_UNIV:
+  fixes f :: "'a::metric_space => extreal"
+  shows "Liminf (at x) f = Liminf (at x within UNIV) f"
+by (metis within_UNIV)
+
+
+lemma Liminf_at:
+  fixes f :: "'a::metric_space => extreal"
+  shows "Liminf (at x) f = (SUP e:{0<..}. INF y:(ball x e - {x}). f y)"
+using Liminf_within[of x UNIV f] Liminf_within_UNIV[of x f] by auto
+
+
+lemma Limsup_within_UNIV:
+  fixes f :: "'a::metric_space => extreal"
+  shows "Limsup (at x) f = Limsup (at x within UNIV) f"
+by (metis within_UNIV)
+
+
+lemma Limsup_at:
+  fixes f :: "'a::metric_space => extreal"
+  shows "Limsup (at x) f = (INF e:{0<..}. SUP y:(ball x e - {x}). f y)"
+using Limsup_within[of x UNIV f] Limsup_within_UNIV[of x f] by auto
+
+lemma Lim_within_constant:
+  fixes f :: "'a::metric_space => 'b::topological_space"
+  assumes "ALL y:S. f y = C"
+  shows "(f ---> C) (at x within S)"
+unfolding tendsto_def eventually_within
+by (metis assms(1) linorder_le_less_linear n_not_Suc_n real_of_nat_le_zero_cancel_iff)
+
+lemma Liminf_within_constant:
+  fixes f :: "'a::metric_space => extreal"
+  assumes "ALL y:S. f y = C"
+  assumes "~trivial_limit (at x within S)"
+  shows "Liminf (at x within S) f = C"
+by (metis Lim_within_constant assms lim_imp_Liminf)
+
+lemma Limsup_within_constant:
+  fixes f :: "'a::metric_space => extreal"
+  assumes "ALL y:S. f y = C"
+  assumes "~trivial_limit (at x within S)"
+  shows "Limsup (at x within S) f = C"
+by (metis Lim_within_constant assms lim_imp_Limsup)
+
+lemma islimpt_punctured:
+"x islimpt S = x islimpt (S-{x})"
+unfolding islimpt_def by blast
+
+
+lemma islimpt_in_closure:
+"(x islimpt S) = (x:closure(S-{x}))"
+unfolding closure_def using islimpt_punctured by blast
+
+
+lemma not_trivial_limit_within:
+  "~trivial_limit (at x within S) = (x:closure(S-{x}))"
+using islimpt_in_closure by (metis trivial_limit_within)
+
+
+lemma not_trivial_limit_within_ball:
+  "(~trivial_limit (at x within S)) = (ALL e>0. S Int ball x e - {x} ~= {})"
+  (is "?lhs = ?rhs")
+proof-
+{ assume "?lhs"
+  { fix e :: real assume "e>0"
+    then obtain y where "y:(S-{x}) & dist y x < e"
+       using `?lhs` not_trivial_limit_within[of x S] closure_approachable[of x "S - {x}"] by auto
+    hence "y : (S Int ball x e - {x})" unfolding ball_def by (simp add: dist_commute)
+    hence "S Int ball x e - {x} ~= {}" by blast
+  } hence "?rhs" by auto
+}
+moreover
+{ assume "?rhs"
+  { fix e :: real assume "e>0"
+    then obtain y where "y : (S Int ball x e - {x})" using `?rhs` by blast
+    hence "y:(S-{x}) & dist y x < e" unfolding ball_def by (simp add: dist_commute)
+    hence "EX y:(S-{x}). dist y x < e" by auto
+  } hence "?lhs" using not_trivial_limit_within[of x S] closure_approachable[of x "S - {x}"] by auto
+} ultimately show ?thesis by auto
+qed
+
+subsubsection {* Continuity *}
+
+lemma continuous_imp_tendsto:
+  assumes "continuous (at x0) f"
+  assumes "x ----> x0"
+  shows "(f o x) ----> (f x0)"
+proof-
+{ fix S assume "open S & (f x0):S"
+  from this obtain T where T_def: "open T & x0 : T & (ALL x:T. f x : S)"
+     using assms continuous_at_open by metis
+  hence "(EX N. ALL n>=N. x n : T)" using assms tendsto_explicit T_def by auto
+  hence "(EX N. ALL n>=N. f(x n) : S)" using T_def by auto
+} from this show ?thesis using tendsto_explicit[of "f o x" "f x0"] by auto
+qed
+
+
+lemma continuous_at_sequentially2:
+fixes f :: "'a::metric_space => 'b:: topological_space"
+shows "continuous (at x0) f <-> (ALL x. (x ----> x0) --> (f o x) ----> (f x0))"
+proof-
+{ assume "~(continuous (at x0) f)"
+  from this obtain T where T_def:
+     "open T & f x0 : T & (ALL S. (open S & x0 : S) --> (EX x':S. f x' ~: T))"
+     using continuous_at_open[of x0 f] by metis
+  def X == "{x'. f x' ~: T}" hence "x0 islimpt X" unfolding islimpt_def using T_def by auto
+  from this obtain x where x_def: "(ALL n. x n : X) & x ----> x0"
+     using islimpt_sequential[of x0 X] by auto
+  hence "~(f o x) ----> (f x0)" unfolding tendsto_explicit using X_def T_def by auto
+  hence "EX x. x ----> x0 & (~(f o x) ----> (f x0))" using x_def by auto
+}
+from this show ?thesis using continuous_imp_tendsto by auto
+qed
+
+
+lemma continuous_at_extreal:
+fixes x0 :: real
+shows "continuous (at x0) extreal"
+proof-
+{ fix T assume T_def: "open T & extreal x0 : T"
+  from this obtain S where S_def: "open S & extreal ` S = T - {\<infinity>, (-\<infinity>)}"
+     using extreal_openE[of T] by metis
+  moreover hence "x0 : S" using T_def by auto
+  moreover have "ALL y:S. extreal y : T" using S_def by auto
+  ultimately have "EX S. x0 : S & open S & (ALL y:S. extreal y : T)" by auto
+} from this show ?thesis unfolding continuous_at_open by blast
+qed
+
+
+lemma continuous_at_of_extreal:
+fixes x0 :: extreal
+assumes "x0 ~: {\<infinity>, (-\<infinity>)}"
+shows "continuous (at x0) real"
+proof-
+{ fix T assume T_def: "open T & real x0 : T"
+  def S == "extreal ` T"
+  hence "extreal (real x0) : S" using T_def by auto
+  hence "x0 : S" using assms extreal_real by auto
+  moreover have "open S" using open_extreal S_def T_def by auto
+  moreover have "ALL y:S. real y : T" using S_def T_def by auto
+  ultimately have "EX S. x0 : S & open S & (ALL y:S. real y : T)" by auto
+} from this show ?thesis unfolding continuous_at_open by blast
+qed
+
+
+lemma real_extreal_id: "real o extreal = id"
+proof-
+{ fix x have "(real o extreal) x = id x" by auto }
+from this show ?thesis using ext by blast
+qed
+
+
+lemma continuous_at_iff_extreal:
+fixes f :: "'a::t2_space => real"
+shows "continuous (at x0) f <-> continuous (at x0) (extreal o f)"
+proof-
+{ assume "continuous (at x0) f" hence "continuous (at x0) (extreal o f)"
+     using continuous_at_extreal continuous_at_compose[of x0 f extreal] by auto
+}
+moreover
+{ assume "continuous (at x0) (extreal o f)"
+  hence "continuous (at x0) (real o (extreal o f))"
+     using continuous_at_of_extreal by (intro continuous_at_compose[of x0 "extreal o f"]) auto
+  moreover have "real o (extreal o f) = f" using real_extreal_id by (simp add: o_assoc)
+  ultimately have "continuous (at x0) f" by auto
+} ultimately show ?thesis by auto
+qed
+
+
+lemma continuous_on_iff_extreal:
+fixes f :: "'a::t2_space => real"
+fixes A assumes "open A"
+shows "continuous_on A f <-> continuous_on A (extreal o f)"
+   using continuous_at_iff_extreal assms by (auto simp add: continuous_on_eq_continuous_at)
+
+
+lemma continuous_on_extreal: "continuous_on UNIV extreal"
+   using continuous_at_extreal continuous_on_eq_continuous_at by auto
+
+lemma open_image_extreal: "open(UNIV-{\<infinity>,(-\<infinity>)})"
+by (metis range_extreal open_extreal open_UNIV)
+
+lemma continuous_on_real: "continuous_on (UNIV-{\<infinity>,(-\<infinity>)}) real"
+   using continuous_at_of_extreal continuous_on_eq_continuous_at open_image_extreal by auto
+
+
+lemma continuous_on_iff_real:
+fixes f :: "'a::t2_space => extreal"
+assumes "ALL x. x:A --> (f x ~: {\<infinity>,(-\<infinity>)})"
+shows "continuous_on A f <-> continuous_on A (real o f)"
+proof-
+have "f ` A <= UNIV-{\<infinity>,(-\<infinity>)}" using assms by auto
+hence *: "continuous_on (f ` A) real"
+   using continuous_on_real by (simp add: continuous_on_subset)
+have **: "continuous_on ((real o f) ` A) extreal"
+   using continuous_on_extreal continuous_on_subset[of "UNIV" "extreal" "(real o f) ` A"] by blast
+{ assume "continuous_on A f" hence "continuous_on A (real o f)"
+  apply (subst continuous_on_compose) using * by auto
+}
+moreover
+{ assume "continuous_on A (real o f)"
+  hence "continuous_on A (extreal o (real o f))"
+     apply (subst continuous_on_compose) using ** by auto
+  hence "continuous_on A f"
+     apply (subst continuous_on_eq[of A "extreal o (real o f)" f])
+     using assms extreal_real by auto
+}
+ultimately show ?thesis by auto
+qed
+
+
+lemma continuous_at_const:
+  fixes f :: "'a::t2_space => extreal"
+  assumes "ALL x. (f x = C)"
+  shows "ALL x. continuous (at x) f"
+unfolding continuous_at_open using assms t1_space by auto
+
+
+lemma closure_contains_Inf:
+  fixes S :: "real set"
+  assumes "S ~= {}" "EX B. ALL x:S. B<=x"
+  shows "Inf S : closure S"
+proof-
+have *: "ALL x:S. Inf S <= x" using Inf_lower_EX[of _ S] assms by metis
+{ fix e assume "e>(0 :: real)"
+  from this obtain x where x_def: "x:S & x < Inf S + e" using Inf_close `S ~= {}` by auto
+  moreover hence "x > Inf S - e" using * by auto
+  ultimately have "abs (x - Inf S) < e" by (simp add: abs_diff_less_iff)
+  hence "EX x:S. abs (x - Inf S) < e" using x_def by auto
+} from this show ?thesis apply (subst closure_approachable) unfolding dist_norm by auto
+qed
+
+
+lemma closed_contains_Inf:
+  fixes S :: "real set"
+  assumes "S ~= {}" "EX B. ALL x:S. B<=x"
+  assumes "closed S"
+  shows "Inf S : S"
+by (metis closure_contains_Inf closure_closed assms)
+
+
+lemma mono_closed_real:
+  fixes S :: "real set"
+  assumes mono: "ALL y z. y:S & y<=z --> z:S"
+  assumes "closed S"
+  shows "S = {} | S = UNIV | (EX a. S = {a ..})"
+proof-
+{ assume "S ~= {}"
+  { assume ex: "EX B. ALL x:S. B<=x"
+    hence *: "ALL x:S. Inf S <= x" using Inf_lower_EX[of _ S] ex by metis
+    hence "Inf S : S" apply (subst closed_contains_Inf) using ex `S ~= {}` `closed S` by auto
+    hence "ALL x. (Inf S <= x <-> x:S)" using mono[rule_format, of "Inf S"] * by auto
+    hence "S = {Inf S ..}" by auto
+    hence "EX a. S = {a ..}" by auto
+  }
+  moreover
+  { assume "~(EX B. ALL x:S. B<=x)"
+    hence nex: "ALL B. EX x:S. x<B" by (simp add: not_le)
+    { fix y obtain x where "x:S & x < y" using nex by auto
+      hence "y:S" using mono[rule_format, of x y] by auto
+    } hence "S = UNIV" by auto
+  } ultimately have "S = UNIV | (EX a. S = {a ..})" by blast
+} from this show ?thesis by blast
+qed
+
+
+lemma mono_closed_extreal:
+  fixes S :: "real set"
+  assumes mono: "ALL y z. y:S & y<=z --> z:S"
+  assumes "closed S"
+  shows "EX a. S = {x. a <= extreal x}"
+proof-
+{ assume "S = {}" hence ?thesis apply(rule_tac x=PInfty in exI) by auto }
+moreover
+{ assume "S = UNIV" hence ?thesis apply(rule_tac x="-\<infinity>" in exI) by auto }
+moreover
+{ assume "EX a. S = {a ..}"
+  from this obtain a where "S={a ..}" by auto
+  hence ?thesis apply(rule_tac x="extreal a" in exI) by auto
+} ultimately show ?thesis using mono_closed_real[of S] assms by auto
+qed
+
+lemma extreal_le_distrib:
+  fixes a b c :: extreal shows "c * (a + b) \<le> c * a + c * b"
+  by (cases rule: extreal3_cases[of a b c])
+     (auto simp add: field_simps not_le mult_le_0_iff mult_less_0_iff)
+
+lemma extreal_pos_distrib:
+  fixes a b c :: extreal assumes "0 \<le> c" "c \<noteq> \<infinity>" shows "c * (a + b) = c * a + c * b"
+  using assms by (cases rule: extreal3_cases[of a b c])
+                 (auto simp add: field_simps not_le mult_le_0_iff mult_less_0_iff)
+
+lemma extreal_pos_le_distrib:
+fixes a b c :: extreal
+assumes "c>=0"
+shows "c * (a + b) <= c * a + c * b"
+  using assms by (cases rule: extreal3_cases[of a b c])
+                 (auto simp add: field_simps)
+
+lemma extreal_max_mono:
+  "[| (a::extreal) <= b; c <= d |] ==> max a c <= max b d"
+  by (metis sup_extreal_def sup_mono)
+
+
+lemma extreal_max_least:
+  "[| (a::extreal) <= x; c <= x |] ==> max a c <= x"
+  by (metis sup_extreal_def sup_least)
+
+end