1.1 --- a/src/HOL/Library/Liminf_Limsup.thy	Tue Nov 05 09:44:59 2013 +0100
     1.2 +++ b/src/HOL/Library/Liminf_Limsup.thy	Tue Nov 05 09:45:00 2013 +0100
     1.3 @@ -32,10 +32,10 @@
     1.4  
     1.5  subsubsection {* @{text Liminf} and @{text Limsup} *}
     1.6  
     1.7 -definition
     1.8 +definition Liminf :: "'a filter \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b :: complete_lattice" where
     1.9    "Liminf F f = (SUP P:{P. eventually P F}. INF x:{x. P x}. f x)"
    1.10  
    1.11 -definition
    1.12 +definition Limsup :: "'a filter \<Rightarrow> ('a \<Rightarrow> 'b) \<Rightarrow> 'b :: complete_lattice" where
    1.13    "Limsup F f = (INF P:{P. eventually P F}. SUP x:{x. P x}. f x)"
    1.14  
    1.15  abbreviation "liminf \<equiv> Liminf sequentially"
    1.16 @@ -43,32 +43,26 @@
    1.17  abbreviation "limsup \<equiv> Limsup sequentially"
    1.18  
    1.19  lemma Liminf_eqI:
    1.20 -  fixes f :: "_ \<Rightarrow> _ :: complete_lattice"
    1.21 -  shows "(\<And>P. eventually P F \<Longrightarrow> INFI (Collect P) f \<le> x) \<Longrightarrow>  
    1.22 +  "(\<And>P. eventually P F \<Longrightarrow> INFI (Collect P) f \<le> x) \<Longrightarrow>  
    1.23      (\<And>y. (\<And>P. eventually P F \<Longrightarrow> INFI (Collect P) f \<le> y) \<Longrightarrow> x \<le> y) \<Longrightarrow> Liminf F f = x"
    1.24    unfolding Liminf_def by (auto intro!: SUP_eqI)
    1.25  
    1.26  lemma Limsup_eqI:
    1.27 -  fixes f :: "_ \<Rightarrow> _ :: complete_lattice"
    1.28 -  shows "(\<And>P. eventually P F \<Longrightarrow> x \<le> SUPR (Collect P) f) \<Longrightarrow>  
    1.29 +  "(\<And>P. eventually P F \<Longrightarrow> x \<le> SUPR (Collect P) f) \<Longrightarrow>  
    1.30      (\<And>y. (\<And>P. eventually P F \<Longrightarrow> y \<le> SUPR (Collect P) f) \<Longrightarrow> y \<le> x) \<Longrightarrow> Limsup F f = x"
    1.31    unfolding Limsup_def by (auto intro!: INF_eqI)
    1.32  
    1.33 -lemma liminf_SUPR_INFI:
    1.34 -  fixes f :: "nat \<Rightarrow> 'a :: complete_lattice"
    1.35 -  shows "liminf f = (SUP n. INF m:{n..}. f m)"
    1.36 +lemma liminf_SUPR_INFI: "liminf f = (SUP n. INF m:{n..}. f m)"
    1.37    unfolding Liminf_def eventually_sequentially
    1.38    by (rule SUPR_eq) (auto simp: atLeast_def intro!: INF_mono)
    1.39  
    1.40 -lemma limsup_INFI_SUPR:
    1.41 -  fixes f :: "nat \<Rightarrow> 'a :: complete_lattice"
    1.42 -  shows "limsup f = (INF n. SUP m:{n..}. f m)"
    1.43 +lemma limsup_INFI_SUPR: "limsup f = (INF n. SUP m:{n..}. f m)"
    1.44    unfolding Limsup_def eventually_sequentially
    1.45    by (rule INFI_eq) (auto simp: atLeast_def intro!: SUP_mono)
    1.46  
    1.47  lemma Limsup_const: 
    1.48    assumes ntriv: "\<not> trivial_limit F"
    1.49 -  shows "Limsup F (\<lambda>x. c) = (c::'a::complete_lattice)"
    1.50 +  shows "Limsup F (\<lambda>x. c) = c"
    1.51  proof -
    1.52    have *: "\<And>P. Ex P \<longleftrightarrow> P \<noteq> (\<lambda>x. False)" by auto
    1.53    have "\<And>P. eventually P F \<Longrightarrow> (SUP x : {x. P x}. c) = c"
    1.54 @@ -81,7 +75,7 @@
    1.55  
    1.56  lemma Liminf_const:
    1.57    assumes ntriv: "\<not> trivial_limit F"
    1.58 -  shows "Liminf F (\<lambda>x. c) = (c::'a::complete_lattice)"
    1.59 +  shows "Liminf F (\<lambda>x. c) = c"
    1.60  proof -
    1.61    have *: "\<And>P. Ex P \<longleftrightarrow> P \<noteq> (\<lambda>x. False)" by auto
    1.62    have "\<And>P. eventually P F \<Longrightarrow> (INF x : {x. P x}. c) = c"
    1.63 @@ -93,7 +87,6 @@
    1.64  qed
    1.65  
    1.66  lemma Liminf_mono:
    1.67 -  fixes f g :: "'a => 'b :: complete_lattice"
    1.68    assumes ev: "eventually (\<lambda>x. f x \<le> g x) F"
    1.69    shows "Liminf F f \<le> Liminf F g"
    1.70    unfolding Liminf_def
    1.71 @@ -105,13 +98,11 @@
    1.72  qed
    1.73  
    1.74  lemma Liminf_eq:
    1.75 -  fixes f g :: "'a \<Rightarrow> 'b :: complete_lattice"
    1.76    assumes "eventually (\<lambda>x. f x = g x) F"
    1.77    shows "Liminf F f = Liminf F g"
    1.78    by (intro antisym Liminf_mono eventually_mono[OF _ assms]) auto
    1.79  
    1.80  lemma Limsup_mono:
    1.81 -  fixes f g :: "'a \<Rightarrow> 'b :: complete_lattice"
    1.82    assumes ev: "eventually (\<lambda>x. f x \<le> g x) F"
    1.83    shows "Limsup F f \<le> Limsup F g"
    1.84    unfolding Limsup_def
    1.85 @@ -123,18 +114,16 @@
    1.86  qed
    1.87  
    1.88  lemma Limsup_eq:
    1.89 -  fixes f g :: "'a \<Rightarrow> 'b :: complete_lattice"
    1.90    assumes "eventually (\<lambda>x. f x = g x) net"
    1.91    shows "Limsup net f = Limsup net g"
    1.92    by (intro antisym Limsup_mono eventually_mono[OF _ assms]) auto
    1.93  
    1.94  lemma Liminf_le_Limsup:
    1.95 -  fixes f :: "'a \<Rightarrow> 'b::complete_lattice"
    1.96    assumes ntriv: "\<not> trivial_limit F"
    1.97    shows "Liminf F f \<le> Limsup F f"
    1.98    unfolding Limsup_def Liminf_def
    1.99 -  apply (rule complete_lattice_class.SUP_least)
   1.100 -  apply (rule complete_lattice_class.INF_greatest)
   1.101 +  apply (rule SUP_least)
   1.102 +  apply (rule INF_greatest)
   1.103  proof safe
   1.104    fix P Q assume "eventually P F" "eventually Q F"
   1.105    then have "eventually (\<lambda>x. P x \<and> Q x) F" (is "eventually ?C F") by (rule eventually_conj)
   1.106 @@ -150,14 +139,12 @@
   1.107  qed
   1.108  
   1.109  lemma Liminf_bounded:
   1.110 -  fixes X Y :: "'a \<Rightarrow> 'b::complete_lattice"
   1.111    assumes ntriv: "\<not> trivial_limit F"
   1.112    assumes le: "eventually (\<lambda>n. C \<le> X n) F"
   1.113    shows "C \<le> Liminf F X"
   1.114    using Liminf_mono[OF le] Liminf_const[OF ntriv, of C] by simp
   1.115  
   1.116  lemma Limsup_bounded:
   1.117 -  fixes X Y :: "'a \<Rightarrow> 'b::complete_lattice"
   1.118    assumes ntriv: "\<not> trivial_limit F"
   1.119    assumes le: "eventually (\<lambda>n. X n \<le> C) F"
   1.120    shows "Limsup F X \<le> C"