--- /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