src/HOL/Library/Extended_Real.thy

 *)
 
 header {* Extended real number line *}
 
 theory Extended_Real
-imports Complex_Main Extended_Nat
+imports Complex_Main Extended_Nat Liminf_Limsup
 begin
 
 text {*
 
 For more lemmas about the extended real numbers go to
   @{file "~~/src/HOL/Multivariate_Analysis/Extended_Real_Limits.thy"}
 
 *}
-
-lemma SUPR_pair:
-  "(SUP i : A. SUP j : B. f i j) = (SUP p : A \<times> B. f (fst p) (snd p))"
-  by (rule antisym) (auto intro!: SUP_least SUP_upper2)
-
-lemma INFI_pair:
-  "(INF i : A. INF j : B. f i j) = (INF p : A \<times> B. f (fst p) (snd p))"
-  by (rule antisym) (auto intro!: INF_greatest INF_lower2)
-
-lemma le_Sup_iff_less:
-  fixes x :: "'a :: {complete_linorder, inner_dense_linorder}"
-  shows "x \<le> (SUP i:A. f i) \<longleftrightarrow> (\<forall>y<x. \<exists>i\<in>A. y \<le> f i)" (is "?lhs = ?rhs")
-  unfolding le_SUP_iff
-  by (blast intro: less_imp_le less_trans less_le_trans dest: dense)
-
-lemma Inf_le_iff_less:
-  fixes x :: "'a :: {complete_linorder, inner_dense_linorder}"
-  shows "(INF i:A. f i) \<le> x \<longleftrightarrow> (\<forall>y>x. \<exists>i\<in>A. f i \<le> y)"
-  unfolding INF_le_iff
-  by (blast intro: less_imp_le less_trans le_less_trans dest: dense)
 
 subsection {* Definition and basic properties *}
 
 datatype ereal = ereal real | PInfty | MInfty
 

   guess e using ereal_open_cont_interval[OF assms] .
   with that[of "x-e" "x+e"] ereal_between[OF x, of e]
   show thesis by auto
 qed
 
+instance ereal :: perfect_space
+proof (default, rule)
+  fix a :: ereal assume a: "open {a}"
+  show False
+  proof (cases a)
+    case MInf
+    then obtain y where "{..<ereal y} \<le> {a}" using a open_MInfty2[of "{a}"] by auto
+    then have "ereal (y - 1) \<in> {a}" apply (subst subsetD[of "{..<ereal y}"]) by auto
+    then show False using `a = -\<infinity>` by auto
+  next
+    case PInf
+    then obtain y where "{ereal y<..} \<le> {a}" using a open_PInfty2[of "{a}"] by auto
+    then have "ereal (y + 1) \<in> {a}" apply (subst subsetD[of "{ereal y<..}"]) by auto
+    then show False using `a = \<infinity>` by auto
+  next
+    case (real r)
+    then have fin: "\<bar>a\<bar> \<noteq> \<infinity>" by simp
+    from ereal_open_cont_interval[OF a singletonI this] guess e . note e = this
+    then obtain b where b_def: "a<b \<and> b<a+e"
+      using fin ereal_between dense[of a "a+e"] by auto
+    then have "b: {a-e <..< a+e}" using fin ereal_between[of a e] e by auto
+    then show False using b_def e by auto
+  qed
+qed
+
 subsubsection {* Convergent sequences *}
 
 lemma lim_ereal[simp]:
   "((\<lambda>n. ereal (f n)) ---> ereal x) net \<longleftrightarrow> (f ---> x) net" (is "?l = ?r")
 proof (intro iffI topological_tendstoI)

 next
   assume ?rhs then show "u ----> x"
     using ereal_LimI_finite[of x] assms by auto
 qed
 
-
-subsubsection {* @{text Liminf} and @{text Limsup} *}
-
-definition
-  "Liminf F f = (SUP P:{P. eventually P F}. INF x:{x. P x}. f x)"
-
-definition
-  "Limsup F f = (INF P:{P. eventually P F}. SUP x:{x. P x}. f x)"
-
-abbreviation "liminf \<equiv> Liminf sequentially"
-
-abbreviation "limsup \<equiv> Limsup sequentially"
-
-lemma Liminf_eqI:
-  "(\<And>P. eventually P F \<Longrightarrow> INFI (Collect P) f \<le> x) \<Longrightarrow>  
-    (\<And>y. (\<And>P. eventually P F \<Longrightarrow> INFI (Collect P) f \<le> y) \<Longrightarrow> x \<le> y) \<Longrightarrow> Liminf F f = x"
-  unfolding Liminf_def by (auto intro!: SUP_eqI)
-
-lemma Limsup_eqI:
-  "(\<And>P. eventually P F \<Longrightarrow> x \<le> SUPR (Collect P) f) \<Longrightarrow>  
-    (\<And>y. (\<And>P. eventually P F \<Longrightarrow> y \<le> SUPR (Collect P) f) \<Longrightarrow> y \<le> x) \<Longrightarrow> Limsup F f = x"
-  unfolding Limsup_def by (auto intro!: INF_eqI)
-
-lemma liminf_SUPR_INFI:
-  fixes f :: "nat \<Rightarrow> 'a :: complete_lattice"
-  shows "liminf f = (SUP n. INF m:{n..}. f m)"
-  unfolding Liminf_def eventually_sequentially
-  by (rule SUPR_eq) (auto simp: atLeast_def intro!: INF_mono)
-
-lemma limsup_INFI_SUPR:
-  fixes f :: "nat \<Rightarrow> 'a :: complete_lattice"
-  shows "limsup f = (INF n. SUP m:{n..}. f m)"
-  unfolding Limsup_def eventually_sequentially
-  by (rule INFI_eq) (auto simp: atLeast_def intro!: SUP_mono)
-
-lemma Limsup_const: 
-  assumes ntriv: "\<not> trivial_limit F"
-  shows "Limsup F (\<lambda>x. c) = (c::'a::complete_lattice)"
-proof -
-  have *: "\<And>P. Ex P \<longleftrightarrow> P \<noteq> (\<lambda>x. False)" by auto
-  have "\<And>P. eventually P F \<Longrightarrow> (SUP x : {x. P x}. c) = c"
-    using ntriv by (intro SUP_const) (auto simp: eventually_False *)
-  then show ?thesis
-    unfolding Limsup_def using eventually_True
-    by (subst INF_cong[where D="\<lambda>x. c"])
-       (auto intro!: INF_const simp del: eventually_True)
-qed
-
-lemma Liminf_const:
-  assumes ntriv: "\<not> trivial_limit F"
-  shows "Liminf F (\<lambda>x. c) = (c::'a::complete_lattice)"
-proof -
-  have *: "\<And>P. Ex P \<longleftrightarrow> P \<noteq> (\<lambda>x. False)" by auto
-  have "\<And>P. eventually P F \<Longrightarrow> (INF x : {x. P x}. c) = c"
-    using ntriv by (intro INF_const) (auto simp: eventually_False *)
-  then show ?thesis
-    unfolding Liminf_def using eventually_True
-    by (subst SUP_cong[where D="\<lambda>x. c"])
-       (auto intro!: SUP_const simp del: eventually_True)
-qed
-
-lemma Liminf_mono:
-  fixes f g :: "'a => 'b :: complete_lattice"
-  assumes ev: "eventually (\<lambda>x. f x \<le> g x) F"
-  shows "Liminf F f \<le> Liminf F g"
-  unfolding Liminf_def
-proof (safe intro!: SUP_mono)
-  fix P assume "eventually P F"
-  with ev have "eventually (\<lambda>x. f x \<le> g x \<and> P x) F" (is "eventually ?Q F") by (rule eventually_conj)
-  then show "\<exists>Q\<in>{P. eventually P F}. INFI (Collect P) f \<le> INFI (Collect Q) g"
-    by (intro bexI[of _ ?Q]) (auto intro!: INF_mono)
-qed
-
-lemma Liminf_eq:
-  fixes f g :: "'a \<Rightarrow> 'b :: complete_lattice"
-  assumes "eventually (\<lambda>x. f x = g x) F"
-  shows "Liminf F f = Liminf F g"
-  by (intro antisym Liminf_mono eventually_mono[OF _ assms]) auto
-
-lemma Limsup_mono:
-  fixes f g :: "'a \<Rightarrow> 'b :: complete_lattice"
-  assumes ev: "eventually (\<lambda>x. f x \<le> g x) F"
-  shows "Limsup F f \<le> Limsup F g"
-  unfolding Limsup_def
-proof (safe intro!: INF_mono)
-  fix P assume "eventually P F"
-  with ev have "eventually (\<lambda>x. f x \<le> g x \<and> P x) F" (is "eventually ?Q F") by (rule eventually_conj)
-  then show "\<exists>Q\<in>{P. eventually P F}. SUPR (Collect Q) f \<le> SUPR (Collect P) g"
-    by (intro bexI[of _ ?Q]) (auto intro!: SUP_mono)
-qed
-
-lemma Limsup_eq:
-  fixes f g :: "'a \<Rightarrow> 'b :: complete_lattice"
-  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
-
-lemma Liminf_le_Limsup:
-  fixes f :: "'a \<Rightarrow> 'b::complete_lattice"
-  assumes ntriv: "\<not> trivial_limit F"
-  shows "Liminf F f \<le> Limsup F f"
-  unfolding Limsup_def Liminf_def
-  apply (rule complete_lattice_class.SUP_least)
-  apply (rule complete_lattice_class.INF_greatest)
-proof safe
-  fix P Q assume "eventually P F" "eventually Q F"
-  then have "eventually (\<lambda>x. P x \<and> Q x) F" (is "eventually ?C F") by (rule eventually_conj)
-  then have not_False: "(\<lambda>x. P x \<and> Q x) \<noteq> (\<lambda>x. False)"
-    using ntriv by (auto simp add: eventually_False)
-  have "INFI (Collect P) f \<le> INFI (Collect ?C) f"
-    by (rule INF_mono) auto
-  also have "\<dots> \<le> SUPR (Collect ?C) f"
-    using not_False by (intro INF_le_SUP) auto
-  also have "\<dots> \<le> SUPR (Collect Q) f"
-    by (rule SUP_mono) auto
-  finally show "INFI (Collect P) f \<le> SUPR (Collect Q) f" .
-qed
+lemma ereal_Limsup_uminus:
+  fixes f :: "'a => ereal"
+  shows "Limsup net (\<lambda>x. - (f x)) = -(Liminf net f)"
+  unfolding Limsup_def Liminf_def ereal_SUPR_uminus ereal_INFI_uminus ..
+
+lemma liminf_bounded_iff:
+  fixes x :: "nat \<Rightarrow> ereal"
+  shows "C \<le> liminf x \<longleftrightarrow> (\<forall>B<C. \<exists>N. \<forall>n\<ge>N. B < x n)" (is "?lhs <-> ?rhs")
+  unfolding le_Liminf_iff eventually_sequentially ..
 
 lemma
   fixes X :: "nat \<Rightarrow> ereal"
   shows ereal_incseq_uminus[simp]: "incseq (\<lambda>i. - X i) = decseq X"
     and ereal_decseq_uminus[simp]: "decseq (\<lambda>i. - X i) = incseq X"
   unfolding incseq_def decseq_def by auto
 
-lemma Liminf_bounded:
-  fixes X Y :: "'a \<Rightarrow> 'b::complete_lattice"
-  assumes ntriv: "\<not> trivial_limit F"
-  assumes le: "eventually (\<lambda>n. C \<le> X n) F"
-  shows "C \<le> Liminf F X"
-  using Liminf_mono[OF le] Liminf_const[OF ntriv, of C] by simp
-
-lemma Limsup_bounded:
-  fixes X Y :: "'a \<Rightarrow> 'b::complete_lattice"
-  assumes ntriv: "\<not> trivial_limit F"
-  assumes le: "eventually (\<lambda>n. X n \<le> C) F"
-  shows "Limsup F X \<le> C"
-  using Limsup_mono[OF le] Limsup_const[OF ntriv, of C] by simp
-
-lemma le_Liminf_iff:
-  fixes X :: "_ \<Rightarrow> _ :: complete_linorder"
-  shows "C \<le> Liminf F X \<longleftrightarrow> (\<forall>y<C. eventually (\<lambda>x. y < X x) F)"
-proof -
-  { fix y P assume "eventually P F" "y < INFI (Collect P) X"
-    then have "eventually (\<lambda>x. y < X x) F"
-      by (auto elim!: eventually_elim1 dest: less_INF_D) }
-  moreover
-  { fix y P assume "y < C" and y: "\<forall>y<C. eventually (\<lambda>x. y < X x) F"
-    have "\<exists>P. eventually P F \<and> y < INFI (Collect P) X"
-    proof cases
-      assume "\<exists>z. y < z \<and> z < C"
-      then guess z ..
-      moreover then have "z \<le> INFI {x. z < X x} X"
-        by (auto intro!: INF_greatest)
-      ultimately show ?thesis
-        using y by (intro exI[of _ "\<lambda>x. z < X x"]) auto
-    next
-      assume "\<not> (\<exists>z. y < z \<and> z < C)"
-      then have "C \<le> INFI {x. y < X x} X"
-        by (intro INF_greatest) auto
-      with `y < C` show ?thesis
-        using y by (intro exI[of _ "\<lambda>x. y < X x"]) auto
-    qed }
-  ultimately show ?thesis
-    unfolding Liminf_def le_SUP_iff by auto
-qed
-
-lemma lim_imp_Liminf:
-  fixes f :: "'a \<Rightarrow> _ :: {complete_linorder, linorder_topology}"
-  assumes ntriv: "\<not> trivial_limit F"
-  assumes lim: "(f ---> f0) F"
-  shows "Liminf F f = f0"
-proof (intro Liminf_eqI)
-  fix P assume P: "eventually P F"
-  then have "eventually (\<lambda>x. INFI (Collect P) f \<le> f x) F"
-    by eventually_elim (auto intro!: INF_lower)
-  then show "INFI (Collect P) f \<le> f0"
-    by (rule tendsto_le[OF ntriv lim tendsto_const])
-next
-  fix y assume upper: "\<And>P. eventually P F \<Longrightarrow> INFI (Collect P) f \<le> y"
-  show "f0 \<le> y"
-  proof cases
-    assume "\<exists>z. y < z \<and> z < f0"
-    then guess z ..
-    moreover have "z \<le> INFI {x. z < f x} f"
-      by (rule INF_greatest) simp
-    ultimately show ?thesis
-      using lim[THEN topological_tendstoD, THEN upper, of "{z <..}"] by auto
-  next
-    assume discrete: "\<not> (\<exists>z. y < z \<and> z < f0)"
-    show ?thesis
-    proof (rule classical)
-      assume "\<not> f0 \<le> y"
-      then have "eventually (\<lambda>x. y < f x) F"
-        using lim[THEN topological_tendstoD, of "{y <..}"] by auto
-      then have "eventually (\<lambda>x. f0 \<le> f x) F"
-        using discrete by (auto elim!: eventually_elim1)
-      then have "INFI {x. f0 \<le> f x} f \<le> y"
-        by (rule upper)
-      moreover have "f0 \<le> INFI {x. f0 \<le> f x} f"
-        by (intro INF_greatest) simp
-      ultimately show "f0 \<le> y" by simp
-    qed
-  qed
-qed
-
-lemma lim_imp_Limsup:
-  fixes f :: "'a \<Rightarrow> _ :: {complete_linorder, linorder_topology}"
-  assumes ntriv: "\<not> trivial_limit F"
-  assumes lim: "(f ---> f0) F"
-  shows "Limsup F f = f0"
-proof (intro Limsup_eqI)
-  fix P assume P: "eventually P F"
-  then have "eventually (\<lambda>x. f x \<le> SUPR (Collect P) f) F"
-    by eventually_elim (auto intro!: SUP_upper)
-  then show "f0 \<le> SUPR (Collect P) f"
-    by (rule tendsto_le[OF ntriv tendsto_const lim])
-next
-  fix y assume lower: "\<And>P. eventually P F \<Longrightarrow> y \<le> SUPR (Collect P) f"
-  show "y \<le> f0"
-  proof cases
-    assume "\<exists>z. f0 < z \<and> z < y"
-    then guess z ..
-    moreover have "SUPR {x. f x < z} f \<le> z"
-      by (rule SUP_least) simp
-    ultimately show ?thesis
-      using lim[THEN topological_tendstoD, THEN lower, of "{..< z}"] by auto
-  next
-    assume discrete: "\<not> (\<exists>z. f0 < z \<and> z < y)"
-    show ?thesis
-    proof (rule classical)
-      assume "\<not> y \<le> f0"
-      then have "eventually (\<lambda>x. f x < y) F"
-        using lim[THEN topological_tendstoD, of "{..< y}"] by auto
-      then have "eventually (\<lambda>x. f x \<le> f0) F"
-        using discrete by (auto elim!: eventually_elim1 simp: not_less)
-      then have "y \<le> SUPR {x. f x \<le> f0} f"
-        by (rule lower)
-      moreover have "SUPR {x. f x \<le> f0} f \<le> f0"
-        by (intro SUP_least) simp
-      ultimately show "y \<le> f0" by simp
-    qed
-  qed
-qed
-
-lemma Liminf_eq_Limsup:
-  fixes f0 :: "'a :: {complete_linorder, linorder_topology}"
-  assumes ntriv: "\<not> trivial_limit F"
-    and lim: "Liminf F f = f0" "Limsup F f = f0"
-  shows "(f ---> f0) F"
-proof (rule order_tendstoI)
-  fix a assume "f0 < a"
-  with assms have "Limsup F f < a" by simp
-  then obtain P where "eventually P F" "SUPR (Collect P) f < a"
-    unfolding Limsup_def INF_less_iff by auto
-  then show "eventually (\<lambda>x. f x < a) F"
-    by (auto elim!: eventually_elim1 dest: SUP_lessD)
-next
-  fix a assume "a < f0"
-  with assms have "a < Liminf F f" by simp
-  then obtain P where "eventually P F" "a < INFI (Collect P) f"
-    unfolding Liminf_def less_SUP_iff by auto
-  then show "eventually (\<lambda>x. a < f x) F"
-    by (auto elim!: eventually_elim1 dest: less_INF_D)
-qed
-
-lemma tendsto_iff_Liminf_eq_Limsup:
-  fixes f0 :: "'a :: {complete_linorder, linorder_topology}"
-  shows "\<not> trivial_limit F \<Longrightarrow> (f ---> f0) F \<longleftrightarrow> (Liminf F f = f0 \<and> Limsup F f = f0)"
-  by (metis Liminf_eq_Limsup lim_imp_Limsup lim_imp_Liminf)
-
-lemma liminf_bounded_iff:
-  fixes x :: "nat \<Rightarrow> ereal"
-  shows "C \<le> liminf x \<longleftrightarrow> (\<forall>B<C. \<exists>N. \<forall>n\<ge>N. B < x n)" (is "?lhs <-> ?rhs")
-  unfolding le_Liminf_iff eventually_sequentially ..
-
-lemma liminf_subseq_mono:
-  fixes X :: "nat \<Rightarrow> 'a :: complete_linorder"
-  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> 'a :: complete_linorder"
-  assumes "subseq r"
-  shows "limsup (X \<circ> r) \<le> limsup X"
-proof-
-  have "\<And>n. (SUP m:{n..}. (X \<circ> r) m) \<le> (SUP m:{n..}. X m)"
-  proof (safe intro!: SUP_mono)
-    fix n m :: nat assume "n \<le> m" then show "\<exists>ma\<in>{n..}. (X \<circ> r) m \<le> X ma"
-      using seq_suble[OF `subseq r`, of m] by (intro bexI[of _ "r m"]) auto
-  qed
-  then show ?thesis by (auto intro!: INF_mono simp: limsup_INFI_SUPR comp_def)
-qed
-
-definition (in order) mono_set:
-  "mono_set S \<longleftrightarrow> (\<forall>x y. x \<le> y \<longrightarrow> x \<in> S \<longrightarrow> y \<in> S)"
-
-lemma (in order) mono_greaterThan [intro, simp]: "mono_set {B<..}" unfolding mono_set by auto
-lemma (in order) mono_atLeast [intro, simp]: "mono_set {B..}" unfolding mono_set by auto
-lemma (in order) mono_UNIV [intro, simp]: "mono_set UNIV" unfolding mono_set by auto
-lemma (in order) mono_empty [intro, simp]: "mono_set {}" unfolding mono_set by auto
-
-lemma (in complete_linorder) mono_set_iff:
-  fixes S :: "'a set"
-  defines "a \<equiv> Inf S"
-  shows "mono_set S \<longleftrightarrow> (S = {a <..} \<or> S = {a..})" (is "_ = ?c")
-proof
-  assume "mono_set 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: 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 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
-
 subsubsection {* Tests for code generator *}
 
 (* A small list of simple arithmetic expressions *)
 
 value [code] "- \<infinity> :: ereal"