use bdd_above and bdd_below for conditionally complete lattices
authorhoelzl
Tue Nov 05 09:44:58 2013 +0100 (2013-11-05)
changeset 54258adfc759263ab
parent 54257 5c7a3b6b05a9
child 54259 71c701dc5bf9
use bdd_above and bdd_below for conditionally complete lattices
src/HOL/Conditionally_Complete_Lattices.thy
src/HOL/Library/FSet.thy
src/HOL/Multivariate_Analysis/Convex_Euclidean_Space.thy
src/HOL/Multivariate_Analysis/Extended_Real_Limits.thy
src/HOL/Multivariate_Analysis/Integration.thy
src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy
src/HOL/Real.thy
src/HOL/Topological_Spaces.thy
     1.1 --- a/src/HOL/Conditionally_Complete_Lattices.thy	Tue Nov 05 09:44:57 2013 +0100
     1.2 +++ b/src/HOL/Conditionally_Complete_Lattices.thy	Tue Nov 05 09:44:58 2013 +0100
     1.3 @@ -1,6 +1,7 @@
     1.4  (*  Title:      HOL/Conditionally_Complete_Lattices.thy
     1.5      Author:     Amine Chaieb and L C Paulson, University of Cambridge
     1.6      Author:     Johannes Hölzl, TU München
     1.7 +    Author:     Luke S. Serafin, Carnegie Mellon University
     1.8  *)
     1.9  
    1.10  header {* Conditionally-complete Lattices *}
    1.11 @@ -15,6 +16,118 @@
    1.12  lemma Inf_fin_eq_Min: "finite X \<Longrightarrow> X \<noteq> {} \<Longrightarrow> Inf_fin X = Min X"
    1.13    by (induct X rule: finite_ne_induct) (simp_all add: inf_min)
    1.14  
    1.15 +context preorder
    1.16 +begin
    1.17 +
    1.18 +definition "bdd_above A \<longleftrightarrow> (\<exists>M. \<forall>x \<in> A. x \<le> M)"
    1.19 +definition "bdd_below A \<longleftrightarrow> (\<exists>m. \<forall>x \<in> A. m \<le> x)"
    1.20 +
    1.21 +lemma bdd_aboveI[intro]: "(\<And>x. x \<in> A \<Longrightarrow> x \<le> M) \<Longrightarrow> bdd_above A"
    1.22 +  by (auto simp: bdd_above_def)
    1.23 +
    1.24 +lemma bdd_belowI[intro]: "(\<And>x. x \<in> A \<Longrightarrow> m \<le> x) \<Longrightarrow> bdd_below A"
    1.25 +  by (auto simp: bdd_below_def)
    1.26 +
    1.27 +lemma bdd_above_empty [simp, intro]: "bdd_above {}"
    1.28 +  unfolding bdd_above_def by auto
    1.29 +
    1.30 +lemma bdd_below_empty [simp, intro]: "bdd_below {}"
    1.31 +  unfolding bdd_below_def by auto
    1.32 +
    1.33 +lemma bdd_above_mono: "bdd_above B \<Longrightarrow> A \<subseteq> B \<Longrightarrow> bdd_above A"
    1.34 +  by (metis (full_types) bdd_above_def order_class.le_neq_trans psubsetD)
    1.35 +
    1.36 +lemma bdd_below_mono: "bdd_below B \<Longrightarrow> A \<subseteq> B \<Longrightarrow> bdd_below A"
    1.37 +  by (metis bdd_below_def order_class.le_neq_trans psubsetD)
    1.38 +
    1.39 +lemma bdd_above_Int1 [simp]: "bdd_above A \<Longrightarrow> bdd_above (A \<inter> B)"
    1.40 +  using bdd_above_mono by auto
    1.41 +
    1.42 +lemma bdd_above_Int2 [simp]: "bdd_above B \<Longrightarrow> bdd_above (A \<inter> B)"
    1.43 +  using bdd_above_mono by auto
    1.44 +
    1.45 +lemma bdd_below_Int1 [simp]: "bdd_below A \<Longrightarrow> bdd_below (A \<inter> B)"
    1.46 +  using bdd_below_mono by auto
    1.47 +
    1.48 +lemma bdd_below_Int2 [simp]: "bdd_below B \<Longrightarrow> bdd_below (A \<inter> B)"
    1.49 +  using bdd_below_mono by auto
    1.50 +
    1.51 +lemma bdd_above_Ioo [simp, intro]: "bdd_above {a <..< b}"
    1.52 +  by (auto simp add: bdd_above_def intro!: exI[of _ b] less_imp_le)
    1.53 +
    1.54 +lemma bdd_above_Ico [simp, intro]: "bdd_above {a ..< b}"
    1.55 +  by (auto simp add: bdd_above_def intro!: exI[of _ b] less_imp_le)
    1.56 +
    1.57 +lemma bdd_above_Iio [simp, intro]: "bdd_above {..< b}"
    1.58 +  by (auto simp add: bdd_above_def intro: exI[of _ b] less_imp_le)
    1.59 +
    1.60 +lemma bdd_above_Ioc [simp, intro]: "bdd_above {a <.. b}"
    1.61 +  by (auto simp add: bdd_above_def intro: exI[of _ b] less_imp_le)
    1.62 +
    1.63 +lemma bdd_above_Icc [simp, intro]: "bdd_above {a .. b}"
    1.64 +  by (auto simp add: bdd_above_def intro: exI[of _ b] less_imp_le)
    1.65 +
    1.66 +lemma bdd_above_Iic [simp, intro]: "bdd_above {.. b}"
    1.67 +  by (auto simp add: bdd_above_def intro: exI[of _ b] less_imp_le)
    1.68 +
    1.69 +lemma bdd_below_Ioo [simp, intro]: "bdd_below {a <..< b}"
    1.70 +  by (auto simp add: bdd_below_def intro!: exI[of _ a] less_imp_le)
    1.71 +
    1.72 +lemma bdd_below_Ioc [simp, intro]: "bdd_below {a <.. b}"
    1.73 +  by (auto simp add: bdd_below_def intro!: exI[of _ a] less_imp_le)
    1.74 +
    1.75 +lemma bdd_below_Ioi [simp, intro]: "bdd_below {a <..}"
    1.76 +  by (auto simp add: bdd_below_def intro: exI[of _ a] less_imp_le)
    1.77 +
    1.78 +lemma bdd_below_Ico [simp, intro]: "bdd_below {a ..< b}"
    1.79 +  by (auto simp add: bdd_below_def intro: exI[of _ a] less_imp_le)
    1.80 +
    1.81 +lemma bdd_below_Icc [simp, intro]: "bdd_below {a .. b}"
    1.82 +  by (auto simp add: bdd_below_def intro: exI[of _ a] less_imp_le)
    1.83 +
    1.84 +lemma bdd_below_Ici [simp, intro]: "bdd_below {a ..}"
    1.85 +  by (auto simp add: bdd_below_def intro: exI[of _ a] less_imp_le)
    1.86 +
    1.87 +end
    1.88 +
    1.89 +context lattice
    1.90 +begin
    1.91 +
    1.92 +lemma bdd_above_insert [simp]: "bdd_above (insert a A) = bdd_above A"
    1.93 +  by (auto simp: bdd_above_def intro: le_supI2 sup_ge1)
    1.94 +
    1.95 +lemma bdd_below_insert [simp]: "bdd_below (insert a A) = bdd_below A"
    1.96 +  by (auto simp: bdd_below_def intro: le_infI2 inf_le1)
    1.97 +
    1.98 +lemma bdd_finite [simp]:
    1.99 +  assumes "finite A" shows bdd_above_finite: "bdd_above A" and bdd_below_finite: "bdd_below A"
   1.100 +  using assms by (induct rule: finite_induct, auto)
   1.101 +
   1.102 +lemma bdd_above_Un [simp]: "bdd_above (A \<union> B) = (bdd_above A \<and> bdd_above B)"
   1.103 +proof
   1.104 +  assume "bdd_above (A \<union> B)"
   1.105 +  thus "bdd_above A \<and> bdd_above B" unfolding bdd_above_def by auto
   1.106 +next
   1.107 +  assume "bdd_above A \<and> bdd_above B"
   1.108 +  then obtain a b where "\<forall>x\<in>A. x \<le> a" "\<forall>x\<in>B. x \<le> b" unfolding bdd_above_def by auto
   1.109 +  hence "\<forall>x \<in> A \<union> B. x \<le> sup a b" by (auto intro: Un_iff le_supI1 le_supI2)
   1.110 +  thus "bdd_above (A \<union> B)" unfolding bdd_above_def ..
   1.111 +qed
   1.112 +
   1.113 +lemma bdd_below_Un [simp]: "bdd_below (A \<union> B) = (bdd_below A \<and> bdd_below B)"
   1.114 +proof
   1.115 +  assume "bdd_below (A \<union> B)"
   1.116 +  thus "bdd_below A \<and> bdd_below B" unfolding bdd_below_def by auto
   1.117 +next
   1.118 +  assume "bdd_below A \<and> bdd_below B"
   1.119 +  then obtain a b where "\<forall>x\<in>A. a \<le> x" "\<forall>x\<in>B. b \<le> x" unfolding bdd_below_def by auto
   1.120 +  hence "\<forall>x \<in> A \<union> B. inf a b \<le> x" by (auto intro: Un_iff le_infI1 le_infI2)
   1.121 +  thus "bdd_below (A \<union> B)" unfolding bdd_below_def ..
   1.122 +qed
   1.123 +
   1.124 +end
   1.125 +
   1.126 +
   1.127  text {*
   1.128  
   1.129  To avoid name classes with the @{class complete_lattice}-class we prefix @{const Sup} and
   1.130 @@ -23,24 +136,22 @@
   1.131  *}
   1.132  
   1.133  class conditionally_complete_lattice = lattice + Sup + Inf +
   1.134 -  assumes cInf_lower: "x \<in> X \<Longrightarrow> (\<And>a. a \<in> X \<Longrightarrow> z \<le> a) \<Longrightarrow> Inf X \<le> x"
   1.135 +  assumes cInf_lower: "x \<in> X \<Longrightarrow> bdd_below X \<Longrightarrow> Inf X \<le> x"
   1.136      and cInf_greatest: "X \<noteq> {} \<Longrightarrow> (\<And>x. x \<in> X \<Longrightarrow> z \<le> x) \<Longrightarrow> z \<le> Inf X"
   1.137 -  assumes cSup_upper: "x \<in> X \<Longrightarrow> (\<And>a. a \<in> X \<Longrightarrow> a \<le> z) \<Longrightarrow> x \<le> Sup X"
   1.138 +  assumes cSup_upper: "x \<in> X \<Longrightarrow> bdd_above X \<Longrightarrow> x \<le> Sup X"
   1.139      and cSup_least: "X \<noteq> {} \<Longrightarrow> (\<And>x. x \<in> X \<Longrightarrow> x \<le> z) \<Longrightarrow> Sup X \<le> z"
   1.140  begin
   1.141  
   1.142 -lemma cSup_eq_maximum: (*REAL_SUP_MAX in HOL4*)
   1.143 -  "z \<in> X \<Longrightarrow> (\<And>x. x \<in> X \<Longrightarrow> x \<le> z) \<Longrightarrow> Sup X = z"
   1.144 -  by (blast intro: antisym cSup_upper cSup_least)
   1.145 +lemma cSup_eq_maximum: "z \<in> X \<Longrightarrow> (\<And>x. x \<in> X \<Longrightarrow> x \<le> z) \<Longrightarrow> Sup X = z"
   1.146 +  by (intro antisym cSup_upper[of z X] cSup_least[of X z]) auto
   1.147  
   1.148 -lemma cInf_eq_minimum: (*REAL_INF_MIN in HOL4*)
   1.149 -  "z \<in> X \<Longrightarrow> (\<And>x. x \<in> X \<Longrightarrow> z \<le> x) \<Longrightarrow> Inf X = z"
   1.150 -  by (intro antisym cInf_lower[of z X z] cInf_greatest[of X z]) auto
   1.151 +lemma cInf_eq_minimum: "z \<in> X \<Longrightarrow> (\<And>x. x \<in> X \<Longrightarrow> z \<le> x) \<Longrightarrow> Inf X = z"
   1.152 +  by (intro antisym cInf_lower[of z X] cInf_greatest[of X z]) auto
   1.153  
   1.154 -lemma cSup_le_iff: "S \<noteq> {} \<Longrightarrow> (\<And>a. a \<in> S \<Longrightarrow> a \<le> z) \<Longrightarrow> Sup S \<le> a \<longleftrightarrow> (\<forall>x\<in>S. x \<le> a)"
   1.155 +lemma cSup_le_iff: "S \<noteq> {} \<Longrightarrow> bdd_above S \<Longrightarrow> Sup S \<le> a \<longleftrightarrow> (\<forall>x\<in>S. x \<le> a)"
   1.156    by (metis order_trans cSup_upper cSup_least)
   1.157  
   1.158 -lemma le_cInf_iff: "S \<noteq> {} \<Longrightarrow> (\<And>a. a \<in> S \<Longrightarrow> z \<le> a) \<Longrightarrow> a \<le> Inf S \<longleftrightarrow> (\<forall>x\<in>S. a \<le> x)"
   1.159 +lemma le_cInf_iff: "S \<noteq> {} \<Longrightarrow> bdd_below S \<Longrightarrow> a \<le> Inf S \<longleftrightarrow> (\<forall>x\<in>S. a \<le> x)"
   1.160    by (metis order_trans cInf_lower cInf_greatest)
   1.161  
   1.162  lemma cSup_singleton [simp]: "Sup {x} = x"
   1.163 @@ -49,20 +160,12 @@
   1.164  lemma cInf_singleton [simp]: "Inf {x} = x"
   1.165    by (intro cInf_eq_minimum) auto
   1.166  
   1.167 -lemma cSup_upper2: (*REAL_IMP_LE_SUP in HOL4*)
   1.168 -  "x \<in> X \<Longrightarrow> y \<le> x \<Longrightarrow> (\<And>x. x \<in> X \<Longrightarrow> x \<le> z) \<Longrightarrow> y \<le> Sup X"
   1.169 +lemma cSup_upper2: "x \<in> X \<Longrightarrow> y \<le> x \<Longrightarrow> bdd_above X \<Longrightarrow> y \<le> Sup X"
   1.170    by (metis cSup_upper order_trans)
   1.171   
   1.172 -lemma cInf_lower2:
   1.173 -  "x \<in> X \<Longrightarrow> x \<le> y \<Longrightarrow> (\<And>x. x \<in> X \<Longrightarrow> z \<le> x) \<Longrightarrow> Inf X \<le> y"
   1.174 +lemma cInf_lower2: "x \<in> X \<Longrightarrow> x \<le> y \<Longrightarrow> bdd_below X \<Longrightarrow> Inf X \<le> y"
   1.175    by (metis cInf_lower order_trans)
   1.176  
   1.177 -lemma cSup_upper_EX: "x \<in> X \<Longrightarrow> \<exists>z. \<forall>x. x \<in> X \<longrightarrow> x \<le> z \<Longrightarrow> x \<le> Sup X"
   1.178 -  by (blast intro: cSup_upper)
   1.179 -
   1.180 -lemma cInf_lower_EX:  "x \<in> X \<Longrightarrow> \<exists>z. \<forall>x. x \<in> X \<longrightarrow> z \<le> x \<Longrightarrow> Inf X \<le> x"
   1.181 -  by (blast intro: cInf_lower)
   1.182 -
   1.183  lemma cSup_eq_non_empty:
   1.184    assumes 1: "X \<noteq> {}"
   1.185    assumes 2: "\<And>x. x \<in> X \<Longrightarrow> x \<le> a"
   1.186 @@ -77,67 +180,41 @@
   1.187    shows "Inf X = a"
   1.188    by (intro 3 1 antisym cInf_greatest) (auto intro: 2 1 cInf_lower)
   1.189  
   1.190 -lemma cInf_cSup: "S \<noteq> {} \<Longrightarrow> (\<And>x. x \<in> S \<Longrightarrow> z \<le> x) \<Longrightarrow> Inf S = Sup {x. \<forall>s\<in>S. x \<le> s}"
   1.191 -  by (rule cInf_eq_non_empty) (auto intro: cSup_upper cSup_least)
   1.192 -
   1.193 -lemma cSup_cInf: "S \<noteq> {} \<Longrightarrow> (\<And>x. x \<in> S \<Longrightarrow> x \<le> z) \<Longrightarrow> Sup S = Inf {x. \<forall>s\<in>S. s \<le> x}"
   1.194 -  by (rule cSup_eq_non_empty) (auto intro: cInf_lower cInf_greatest)
   1.195 +lemma cInf_cSup: "S \<noteq> {} \<Longrightarrow> bdd_below S \<Longrightarrow> Inf S = Sup {x. \<forall>s\<in>S. x \<le> s}"
   1.196 +  by (rule cInf_eq_non_empty) (auto intro!: cSup_upper cSup_least simp: bdd_below_def)
   1.197  
   1.198 -lemma cSup_insert: 
   1.199 -  assumes x: "X \<noteq> {}"
   1.200 -      and z: "\<And>x. x \<in> X \<Longrightarrow> x \<le> z"
   1.201 -  shows "Sup (insert a X) = sup a (Sup X)"
   1.202 -proof (intro cSup_eq_non_empty)
   1.203 -  fix y assume "\<And>x. x \<in> insert a X \<Longrightarrow> x \<le> y" with x show "sup a (Sup X) \<le> y" by (auto intro: cSup_least)
   1.204 -qed (auto intro: le_supI2 z cSup_upper)
   1.205 +lemma cSup_cInf: "S \<noteq> {} \<Longrightarrow> bdd_above S \<Longrightarrow> Sup S = Inf {x. \<forall>s\<in>S. s \<le> x}"
   1.206 +  by (rule cSup_eq_non_empty) (auto intro!: cInf_lower cInf_greatest simp: bdd_above_def)
   1.207 +
   1.208 +lemma cSup_insert: "X \<noteq> {} \<Longrightarrow> bdd_above X \<Longrightarrow> Sup (insert a X) = sup a (Sup X)"
   1.209 +  by (intro cSup_eq_non_empty) (auto intro: le_supI2 cSup_upper cSup_least)
   1.210  
   1.211 -lemma cInf_insert: 
   1.212 -  assumes x: "X \<noteq> {}"
   1.213 -      and z: "\<And>x. x \<in> X \<Longrightarrow> z \<le> x"
   1.214 -  shows "Inf (insert a X) = inf a (Inf X)"
   1.215 -proof (intro cInf_eq_non_empty)
   1.216 -  fix y assume "\<And>x. x \<in> insert a X \<Longrightarrow> y \<le> x" with x show "y \<le> inf a (Inf X)" by (auto intro: cInf_greatest)
   1.217 -qed (auto intro: le_infI2 z cInf_lower)
   1.218 +lemma cInf_insert: "X \<noteq> {} \<Longrightarrow> bdd_below X \<Longrightarrow> Inf (insert a X) = inf a (Inf X)"
   1.219 +  by (intro cInf_eq_non_empty) (auto intro: le_infI2 cInf_lower cInf_greatest)
   1.220  
   1.221 -lemma cSup_insert_If: 
   1.222 -  "(\<And>x. x \<in> X \<Longrightarrow> x \<le> z) \<Longrightarrow> Sup (insert a X) = (if X = {} then a else sup a (Sup X))"
   1.223 -  using cSup_insert[of X z] by simp
   1.224 +lemma cSup_insert_If:  "bdd_above X \<Longrightarrow> Sup (insert a X) = (if X = {} then a else sup a (Sup X))"
   1.225 +  using cSup_insert[of X] by simp
   1.226  
   1.227 -lemma cInf_insert_if: 
   1.228 -  "(\<And>x. x \<in> X \<Longrightarrow> z \<le> x) \<Longrightarrow> Inf (insert a X) = (if X = {} then a else inf a (Inf X))"
   1.229 -  using cInf_insert[of X z] by simp
   1.230 +lemma cInf_insert_if: "bdd_below X \<Longrightarrow> Inf (insert a X) = (if X = {} then a else inf a (Inf X))"
   1.231 +  using cInf_insert[of X] by simp
   1.232  
   1.233  lemma le_cSup_finite: "finite X \<Longrightarrow> x \<in> X \<Longrightarrow> x \<le> Sup X"
   1.234  proof (induct X arbitrary: x rule: finite_induct)
   1.235    case (insert x X y) then show ?case
   1.236 -    apply (cases "X = {}")
   1.237 -    apply simp
   1.238 -    apply (subst cSup_insert[of _ "Sup X"])
   1.239 -    apply (auto intro: le_supI2)
   1.240 -    done
   1.241 +    by (cases "X = {}") (auto simp: cSup_insert intro: le_supI2)
   1.242  qed simp
   1.243  
   1.244  lemma cInf_le_finite: "finite X \<Longrightarrow> x \<in> X \<Longrightarrow> Inf X \<le> x"
   1.245  proof (induct X arbitrary: x rule: finite_induct)
   1.246    case (insert x X y) then show ?case
   1.247 -    apply (cases "X = {}")
   1.248 -    apply simp
   1.249 -    apply (subst cInf_insert[of _ "Inf X"])
   1.250 -    apply (auto intro: le_infI2)
   1.251 -    done
   1.252 +    by (cases "X = {}") (auto simp: cInf_insert intro: le_infI2)
   1.253  qed simp
   1.254  
   1.255  lemma cSup_eq_Sup_fin: "finite X \<Longrightarrow> X \<noteq> {} \<Longrightarrow> Sup X = Sup_fin X"
   1.256 -proof (induct X rule: finite_ne_induct)
   1.257 -  case (insert x X) then show ?case
   1.258 -    using cSup_insert[of X "Sup_fin X" x] le_cSup_finite[of X] by simp
   1.259 -qed simp
   1.260 +  by (induct X rule: finite_ne_induct) (simp_all add: cSup_insert)
   1.261  
   1.262  lemma cInf_eq_Inf_fin: "finite X \<Longrightarrow> X \<noteq> {} \<Longrightarrow> Inf X = Inf_fin X"
   1.263 -proof (induct X rule: finite_ne_induct)
   1.264 -  case (insert x X) then show ?case
   1.265 -    using cInf_insert[of X "Inf_fin X" x] cInf_le_finite[of X] by simp
   1.266 -qed simp
   1.267 +  by (induct X rule: finite_ne_induct) (simp_all add: cInf_insert)
   1.268  
   1.269  lemma cSup_atMost[simp]: "Sup {..x} = x"
   1.270    by (auto intro!: cSup_eq_maximum)
   1.271 @@ -165,7 +242,7 @@
   1.272  lemma isLub_cSup: 
   1.273    "(S::'a :: conditionally_complete_lattice set) \<noteq> {} \<Longrightarrow> (\<exists>b. S *<= b) \<Longrightarrow> isLub UNIV S (Sup S)"
   1.274    by  (auto simp add: isLub_def setle_def leastP_def isUb_def
   1.275 -            intro!: setgeI intro: cSup_upper cSup_least)
   1.276 +            intro!: setgeI cSup_upper cSup_least)
   1.277  
   1.278  lemma cSup_eq:
   1.279    fixes a :: "'a :: {conditionally_complete_lattice, no_bot}"
   1.280 @@ -195,10 +272,10 @@
   1.281  begin
   1.282  
   1.283  lemma less_cSup_iff : (*REAL_SUP_LE in HOL4*)
   1.284 -  "X \<noteq> {} \<Longrightarrow> (\<And>x. x \<in> X \<Longrightarrow> x \<le> z) \<Longrightarrow> y < Sup X \<longleftrightarrow> (\<exists>x\<in>X. y < x)"
   1.285 +  "X \<noteq> {} \<Longrightarrow> bdd_above X \<Longrightarrow> y < Sup X \<longleftrightarrow> (\<exists>x\<in>X. y < x)"
   1.286    by (rule iffI) (metis cSup_least not_less, metis cSup_upper less_le_trans)
   1.287  
   1.288 -lemma cInf_less_iff: "X \<noteq> {} \<Longrightarrow> (\<And>x. x \<in> X \<Longrightarrow> z \<le> x) \<Longrightarrow> Inf X < y \<longleftrightarrow> (\<exists>x\<in>X. x < y)"
   1.289 +lemma cInf_less_iff: "X \<noteq> {} \<Longrightarrow> bdd_below X \<Longrightarrow> Inf X < y \<longleftrightarrow> (\<exists>x\<in>X. x < y)"
   1.290    by (rule iffI) (metis cInf_greatest not_less, metis cInf_lower le_less_trans)
   1.291  
   1.292  lemma less_cSupE:
   1.293 @@ -207,11 +284,11 @@
   1.294  
   1.295  lemma less_cSupD:
   1.296    "X \<noteq> {} \<Longrightarrow> z < Sup X \<Longrightarrow> \<exists>x\<in>X. z < x"
   1.297 -  by (metis less_cSup_iff not_leE)
   1.298 +  by (metis less_cSup_iff not_leE bdd_above_def)
   1.299  
   1.300  lemma cInf_lessD:
   1.301    "X \<noteq> {} \<Longrightarrow> Inf X < z \<Longrightarrow> \<exists>x\<in>X. x < z"
   1.302 -  by (metis cInf_less_iff not_leE)
   1.303 +  by (metis cInf_less_iff not_leE bdd_below_def)
   1.304  
   1.305  lemma complete_interval:
   1.306    assumes "a < b" and "P a" and "\<not> P b"
   1.307 @@ -219,7 +296,7 @@
   1.308               (\<forall>d. (\<forall>x. a \<le> x \<and> x < d \<longrightarrow> P x) \<longrightarrow> d \<le> c)"
   1.309  proof (rule exI [where x = "Sup {d. \<forall>x. a \<le> x & x < d --> P x}"], auto)
   1.310    show "a \<le> Sup {d. \<forall>c. a \<le> c \<and> c < d \<longrightarrow> P c}"
   1.311 -    by (rule cSup_upper [where z=b], auto)
   1.312 +    by (rule cSup_upper, auto simp: bdd_above_def)
   1.313         (metis `a < b` `\<not> P b` linear less_le)
   1.314  next
   1.315    show "Sup {d. \<forall>c. a \<le> c \<and> c < d \<longrightarrow> P c} \<le> b"
   1.316 @@ -240,7 +317,7 @@
   1.317    fix d
   1.318      assume 0: "\<forall>x. a \<le> x \<and> x < d \<longrightarrow> P x"
   1.319      thus "d \<le> Sup {d. \<forall>c. a \<le> c \<and> c < d \<longrightarrow> P c}"
   1.320 -      by (rule_tac z="b" in cSup_upper, auto) 
   1.321 +      by (rule_tac cSup_upper, auto simp: bdd_above_def)
   1.322           (metis `a<b` `~ P b` linear less_le)
   1.323  qed
   1.324  
     2.1 --- a/src/HOL/Library/FSet.thy	Tue Nov 05 09:44:57 2013 +0100
     2.2 +++ b/src/HOL/Library/FSet.thy	Tue Nov 05 09:44:58 2013 +0100
     2.3 @@ -101,19 +101,25 @@
     2.4  lemma finite_Sup: "\<exists>z. finite z \<and> (\<forall>a. a \<in> X \<longrightarrow> a \<le> z) \<Longrightarrow> finite (Sup X)"
     2.5  by (auto intro: finite_subset)
     2.6  
     2.7 +lemma transfer_bdd_below[transfer_rule]: "(set_rel (pcr_fset op =) ===> op =) bdd_below bdd_below"
     2.8 +  by auto
     2.9 +
    2.10  instance
    2.11  proof 
    2.12    fix x z :: "'a fset"
    2.13    fix X :: "'a fset set"
    2.14    {
    2.15 -    assume "x \<in> X" "(\<And>a. a \<in> X \<Longrightarrow> z |\<subseteq>| a)" 
    2.16 +    assume "x \<in> X" "bdd_below X" 
    2.17      then show "Inf X |\<subseteq>| x"  by transfer auto
    2.18    next
    2.19      assume "X \<noteq> {}" "(\<And>x. x \<in> X \<Longrightarrow> z |\<subseteq>| x)"
    2.20      then show "z |\<subseteq>| Inf X" by transfer (clarsimp, blast)
    2.21    next
    2.22 -    assume "x \<in> X" "(\<And>a. a \<in> X \<Longrightarrow> a |\<subseteq>| z)"
    2.23 -    then show "x |\<subseteq>| Sup X" by transfer (auto intro!: finite_Sup)
    2.24 +    assume "x \<in> X" "bdd_above X"
    2.25 +    then obtain z where "x \<in> X" "(\<And>x. x \<in> X \<Longrightarrow> x |\<subseteq>| z)"
    2.26 +      by (auto simp: bdd_above_def)
    2.27 +    then show "x |\<subseteq>| Sup X"
    2.28 +      by transfer (auto intro!: finite_Sup)
    2.29    next
    2.30      assume "X \<noteq> {}" "(\<And>x. x \<in> X \<Longrightarrow> x |\<subseteq>| z)"
    2.31      then show "Sup X |\<subseteq>| z" by transfer (clarsimp, blast)
     3.1 --- a/src/HOL/Multivariate_Analysis/Convex_Euclidean_Space.thy	Tue Nov 05 09:44:57 2013 +0100
     3.2 +++ b/src/HOL/Multivariate_Analysis/Convex_Euclidean_Space.thy	Tue Nov 05 09:44:58 2013 +0100
     3.3 @@ -8724,7 +8724,7 @@
     3.4      using interior_subset[of I] `x \<in> interior I` by auto
     3.5  
     3.6    have "Inf (?F x) \<le> (f x - f y) / (x - y)"
     3.7 -  proof (rule cInf_lower2)
     3.8 +  proof (intro bdd_belowI cInf_lower2)
     3.9      show "(f x - f t) / (x - t) \<in> ?F x"
    3.10        using `t \<in> I` `x < t` by auto
    3.11      show "(f x - f t) / (x - t) \<le> (f x - f y) / (x - y)"
     4.1 --- a/src/HOL/Multivariate_Analysis/Extended_Real_Limits.thy	Tue Nov 05 09:44:57 2013 +0100
     4.2 +++ b/src/HOL/Multivariate_Analysis/Extended_Real_Limits.thy	Tue Nov 05 09:44:58 2013 +0100
     4.3 @@ -660,7 +660,7 @@
     4.4      assume "S \<noteq> {}"
     4.5      { assume ex: "\<exists>B. \<forall>x\<in>S. B \<le> x"
     4.6        then have *: "\<forall>x\<in>S. Inf S \<le> x"
     4.7 -        using cInf_lower_EX[of _ S] ex by metis
     4.8 +        using cInf_lower[of _ S] ex by (metis bdd_below_def)
     4.9        then have "Inf S \<in> S"
    4.10          apply (subst closed_contains_Inf)
    4.11          using ex `S \<noteq> {}` `closed S`
     5.1 --- a/src/HOL/Multivariate_Analysis/Integration.thy	Tue Nov 05 09:44:57 2013 +0100
     5.2 +++ b/src/HOL/Multivariate_Analysis/Integration.thy	Tue Nov 05 09:44:58 2013 +0100
     5.3 @@ -13,7 +13,7 @@
     5.4  lemma cSup_abs_le: (* TODO: is this really needed? *)
     5.5    fixes S :: "real set"
     5.6    shows "S \<noteq> {} \<Longrightarrow> (\<forall>x\<in>S. \<bar>x\<bar> \<le> a) \<Longrightarrow> \<bar>Sup S\<bar> \<le> a"
     5.7 -  by (auto simp add: abs_le_interval_iff intro: cSup_least) (metis cSup_upper2)
     5.8 +  by (auto simp add: abs_le_interval_iff intro: cSup_least) (metis cSup_upper2 bdd_aboveI)
     5.9  
    5.10  lemma cInf_abs_ge: (* TODO: is this really needed? *)
    5.11    fixes S :: "real set"
    5.12 @@ -86,7 +86,7 @@
    5.13    apply (insert assms)
    5.14    apply (erule exE)
    5.15    apply (rule_tac x = b in exI)
    5.16 -  apply (auto simp: isLb_def setge_def intro: cInf_lower cInf_greatest)
    5.17 +  apply (auto simp: isLb_def setge_def intro!: cInf_lower cInf_greatest)
    5.18    done
    5.19  
    5.20  lemma real_ge_sup_subset:
    5.21 @@ -100,7 +100,7 @@
    5.22    apply (insert assms)
    5.23    apply (erule exE)
    5.24    apply (rule_tac x = b in exI)
    5.25 -  apply (auto simp: isUb_def setle_def intro: cSup_upper cSup_least)
    5.26 +  apply (auto simp: isUb_def setle_def intro!: cSup_upper cSup_least)
    5.27    done
    5.28  
    5.29  (*declare not_less[simp] not_le[simp]*)
    5.30 @@ -12728,8 +12728,8 @@
    5.31            assume x: "x \<in> s"
    5.32            have *: "\<And>x y::real. x \<ge> - y \<Longrightarrow> - x \<le> y" by auto
    5.33            show "Inf {f j x |j. n \<le> j} \<le> f n x"
    5.34 -            apply (rule cInf_lower[where z="- h x"])
    5.35 -            defer
    5.36 +            apply (intro cInf_lower bdd_belowI)
    5.37 +            apply auto []
    5.38              apply (rule *)
    5.39              using assms(3)[rule_format,OF x]
    5.40              unfolding real_norm_def abs_le_iff
    5.41 @@ -12741,8 +12741,7 @@
    5.42            fix x
    5.43            assume x: "x \<in> s"
    5.44            show "f n x \<le> Sup {f j x |j. n \<le> j}"
    5.45 -            apply (rule cSup_upper[where z="h x"])
    5.46 -            defer
    5.47 +            apply (rule cSup_upper)
    5.48              using assms(3)[rule_format,OF x]
    5.49              unfolding real_norm_def abs_le_iff
    5.50              apply auto
     6.1 --- a/src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy	Tue Nov 05 09:44:57 2013 +0100
     6.2 +++ b/src/HOL/Multivariate_Analysis/Topology_Euclidean_Space.thy	Tue Nov 05 09:44:58 2013 +0100
     6.3 @@ -1909,17 +1909,17 @@
     6.4  
     6.5  lemma closure_contains_Inf:
     6.6    fixes S :: "real set"
     6.7 -  assumes "S \<noteq> {}" "\<forall>x\<in>S. B \<le> x"
     6.8 +  assumes "S \<noteq> {}" "bdd_below S"
     6.9    shows "Inf S \<in> closure S"
    6.10  proof -
    6.11    have *: "\<forall>x\<in>S. Inf S \<le> x"
    6.12 -    using cInf_lower_EX[of _ S] assms by metis
    6.13 +    using cInf_lower[of _ S] assms by metis
    6.14    {
    6.15      fix e :: real
    6.16      assume "e > 0"
    6.17      then have "Inf S < Inf S + e" by simp
    6.18      with assms obtain x where "x \<in> S" "x < Inf S + e"
    6.19 -      by (subst (asm) cInf_less_iff[of _ B]) auto
    6.20 +      by (subst (asm) cInf_less_iff) auto
    6.21      with * have "\<exists>x\<in>S. dist x (Inf S) < e"
    6.22        by (intro bexI[of _ x]) (auto simp add: dist_real_def)
    6.23    }
    6.24 @@ -1928,12 +1928,9 @@
    6.25  
    6.26  lemma closed_contains_Inf:
    6.27    fixes S :: "real set"
    6.28 -  assumes "S \<noteq> {}" "\<forall>x\<in>S. B \<le> x"
    6.29 -    and "closed S"
    6.30 -  shows "Inf S \<in> S"
    6.31 +  shows "S \<noteq> {} \<Longrightarrow> bdd_below S \<Longrightarrow> closed S \<Longrightarrow> Inf S \<in> S"
    6.32    by (metis closure_contains_Inf closure_closed assms)
    6.33  
    6.34 -
    6.35  lemma not_trivial_limit_within_ball:
    6.36    "\<not> trivial_limit (at x within S) \<longleftrightarrow> (\<forall>e>0. S \<inter> ball x e - {x} \<noteq> {})"
    6.37    (is "?lhs = ?rhs")
    6.38 @@ -1977,27 +1974,20 @@
    6.39  
    6.40  definition "infdist x A = (if A = {} then 0 else Inf {dist x a|a. a \<in> A})"
    6.41  
    6.42 +lemma bdd_below_infdist[intro, simp]: "bdd_below {dist x a|a. a \<in> A}"
    6.43 +  by (auto intro!: zero_le_dist)
    6.44 +
    6.45  lemma infdist_notempty: "A \<noteq> {} \<Longrightarrow> infdist x A = Inf {dist x a|a. a \<in> A}"
    6.46    by (simp add: infdist_def)
    6.47  
    6.48  lemma infdist_nonneg: "0 \<le> infdist x A"
    6.49    by (auto simp add: infdist_def intro: cInf_greatest)
    6.50  
    6.51 -lemma infdist_le:
    6.52 -  assumes "a \<in> A"
    6.53 -    and "d = dist x a"
    6.54 -  shows "infdist x A \<le> d"
    6.55 -  using assms by (auto intro!: cInf_lower[where z=0] simp add: infdist_def)
    6.56 -
    6.57 -lemma infdist_zero[simp]:
    6.58 -  assumes "a \<in> A"
    6.59 -  shows "infdist a A = 0"
    6.60 -proof -
    6.61 -  from infdist_le[OF assms, of "dist a a"] have "infdist a A \<le> 0"
    6.62 -    by auto
    6.63 -  with infdist_nonneg[of a A] assms show "infdist a A = 0"
    6.64 -    by auto
    6.65 -qed
    6.66 +lemma infdist_le: "a \<in> A \<Longrightarrow> d = dist x a \<Longrightarrow> infdist x A \<le> d"
    6.67 +  using assms by (auto intro: cInf_lower simp add: infdist_def)
    6.68 +
    6.69 +lemma infdist_zero[simp]: "a \<in> A \<Longrightarrow> infdist a A = 0"
    6.70 +  by (auto intro!: antisym infdist_nonneg infdist_le)
    6.71  
    6.72  lemma infdist_triangle: "infdist x A \<le> infdist y A + dist x y"
    6.73  proof (cases "A = {}")
    6.74 @@ -2021,13 +2011,7 @@
    6.75          using `a \<in> A` by auto
    6.76        show "dist x a \<le> d"
    6.77          unfolding d by (rule dist_triangle)
    6.78 -      fix d
    6.79 -      assume "d \<in> {dist x a |a. a \<in> A}"
    6.80 -      then obtain a where "a \<in> A" "d = dist x a"
    6.81 -        by auto
    6.82 -      then show "infdist x A \<le> d"
    6.83 -        by (rule infdist_le)
    6.84 -    qed
    6.85 +    qed simp
    6.86    qed
    6.87    also have "\<dots> = dist x y + infdist y A"
    6.88    proof (rule cInf_eq, safe)
    6.89 @@ -2651,11 +2635,19 @@
    6.90  
    6.91  text{* Some theorems on sups and infs using the notion "bounded". *}
    6.92  
    6.93 -lemma bounded_real:
    6.94 -  fixes S :: "real set"
    6.95 -  shows "bounded S \<longleftrightarrow> (\<exists>a. \<forall>x\<in>S. abs x \<le> a)"
    6.96 +lemma bounded_real: "bounded (S::real set) \<longleftrightarrow> (\<exists>a. \<forall>x\<in>S. \<bar>x\<bar> \<le> a)"
    6.97    by (simp add: bounded_iff)
    6.98  
    6.99 +lemma bounded_imp_bdd_above: "bounded S \<Longrightarrow> bdd_above (S :: real set)"
   6.100 +  by (auto simp: bounded_def bdd_above_def dist_real_def)
   6.101 +     (metis abs_le_D1 abs_minus_commute diff_le_eq)
   6.102 +
   6.103 +lemma bounded_imp_bdd_below: "bounded S \<Longrightarrow> bdd_below (S :: real set)"
   6.104 +  by (auto simp: bounded_def bdd_below_def dist_real_def)
   6.105 +     (metis abs_le_D1 add_commute diff_le_eq)
   6.106 +
   6.107 +(* TODO: remove the following lemmas about Inf and Sup, is now in conditionally complete lattice *)
   6.108 +
   6.109  lemma bounded_has_Sup:
   6.110    fixes S :: "real set"
   6.111    assumes "bounded S"
   6.112 @@ -2663,22 +2655,14 @@
   6.113    shows "\<forall>x\<in>S. x \<le> Sup S"
   6.114      and "\<forall>b. (\<forall>x\<in>S. x \<le> b) \<longrightarrow> Sup S \<le> b"
   6.115  proof
   6.116 -  fix x
   6.117 -  assume "x\<in>S"
   6.118 -  then show "x \<le> Sup S"
   6.119 -    by (metis cSup_upper abs_le_D1 assms(1) bounded_real)
   6.120 -next
   6.121    show "\<forall>b. (\<forall>x\<in>S. x \<le> b) \<longrightarrow> Sup S \<le> b"
   6.122      using assms by (metis cSup_least)
   6.123 -qed
   6.124 +qed (metis cSup_upper assms(1) bounded_imp_bdd_above)
   6.125  
   6.126  lemma Sup_insert:
   6.127    fixes S :: "real set"
   6.128    shows "bounded S \<Longrightarrow> Sup (insert x S) = (if S = {} then x else max x (Sup S))"
   6.129 -  apply (subst cSup_insert_If)
   6.130 -  apply (rule bounded_has_Sup(1)[of S, rule_format])
   6.131 -  apply (auto simp: sup_max)
   6.132 -  done
   6.133 +  by (auto simp: bounded_imp_bdd_above sup_max cSup_insert_If)
   6.134  
   6.135  lemma Sup_insert_finite:
   6.136    fixes S :: "real set"
   6.137 @@ -2695,24 +2679,14 @@
   6.138    shows "\<forall>x\<in>S. x \<ge> Inf S"
   6.139      and "\<forall>b. (\<forall>x\<in>S. x \<ge> b) \<longrightarrow> Inf S \<ge> b"
   6.140  proof
   6.141 -  fix x
   6.142 -  assume "x \<in> S"
   6.143 -  from assms(1) obtain a where a: "\<forall>x\<in>S. \<bar>x\<bar> \<le> a"
   6.144 -    unfolding bounded_real by auto
   6.145 -  then show "x \<ge> Inf S" using `x \<in> S`
   6.146 -    by (metis cInf_lower_EX abs_le_D2 minus_le_iff)
   6.147 -next
   6.148    show "\<forall>b. (\<forall>x\<in>S. x \<ge> b) \<longrightarrow> Inf S \<ge> b"
   6.149      using assms by (metis cInf_greatest)
   6.150 -qed
   6.151 +qed (metis cInf_lower assms(1) bounded_imp_bdd_below)
   6.152  
   6.153  lemma Inf_insert:
   6.154    fixes S :: "real set"
   6.155    shows "bounded S \<Longrightarrow> Inf (insert x S) = (if S = {} then x else min x (Inf S))"
   6.156 -  apply (subst cInf_insert_if)
   6.157 -  apply (rule bounded_has_Inf(1)[of S, rule_format])
   6.158 -  apply (auto simp: inf_min)
   6.159 -  done
   6.160 +  by (auto simp: bounded_imp_bdd_below inf_min cInf_insert_if)
   6.161  
   6.162  lemma Inf_insert_finite:
   6.163    fixes S :: "real set"
   6.164 @@ -5738,12 +5712,16 @@
   6.165    from s obtain z d where z: "\<And>x. x \<in> s \<Longrightarrow> dist z x \<le> d"
   6.166      unfolding bounded_def by auto
   6.167    have "dist x y \<le> Sup ?D"
   6.168 -  proof (rule cSup_upper, safe)
   6.169 -    fix a b
   6.170 -    assume "a \<in> s" "b \<in> s"
   6.171 -    with z[of a] z[of b] dist_triangle[of a b z]
   6.172 -    show "dist a b \<le> 2 * d"
   6.173 -      by (simp add: dist_commute)
   6.174 +  proof (rule cSup_upper)
   6.175 +    show "bdd_above ?D"
   6.176 +      unfolding bdd_above_def
   6.177 +    proof (safe intro!: exI)
   6.178 +      fix a b
   6.179 +      assume "a \<in> s" "b \<in> s"
   6.180 +      with z[of a] z[of b] dist_triangle[of a b z]
   6.181 +      show "dist a b \<le> 2 * d"
   6.182 +        by (simp add: dist_commute)
   6.183 +    qed
   6.184    qed (insert s, auto)
   6.185    with `x \<in> s` show ?thesis
   6.186      by (auto simp add: diameter_def)
     7.1 --- a/src/HOL/Real.thy	Tue Nov 05 09:44:57 2013 +0100
     7.2 +++ b/src/HOL/Real.thy	Tue Nov 05 09:44:58 2013 +0100
     7.3 @@ -919,6 +919,12 @@
     7.4      using 1 2 3 by (rule_tac x="Real B" in exI, simp)
     7.5  qed
     7.6  
     7.7 +(* TODO: generalize to ordered group *)
     7.8 +lemma bdd_above_uminus[simp]: "bdd_above (uminus ` X) \<longleftrightarrow> bdd_below (X::real set)"
     7.9 +  by (auto simp: bdd_above_def bdd_below_def intro: le_imp_neg_le) (metis le_imp_neg_le minus_minus)
    7.10 +
    7.11 +lemma bdd_below_uminus[simp]: "bdd_below (uminus ` X) \<longleftrightarrow> bdd_above (X::real set)"
    7.12 +  by (auto simp: bdd_above_def bdd_below_def intro: le_imp_neg_le) (metis le_imp_neg_le minus_minus)
    7.13  
    7.14  instantiation real :: linear_continuum
    7.15  begin
    7.16 @@ -933,10 +939,10 @@
    7.17  
    7.18  instance
    7.19  proof
    7.20 -  { fix z x :: real and X :: "real set"
    7.21 -    assume x: "x \<in> X" and z: "\<And>x. x \<in> X \<Longrightarrow> x \<le> z"
    7.22 +  { fix x :: real and X :: "real set"
    7.23 +    assume x: "x \<in> X" "bdd_above X"
    7.24      then obtain s where s: "\<forall>y\<in>X. y \<le> s" "\<And>z. \<forall>y\<in>X. y \<le> z \<Longrightarrow> s \<le> z"
    7.25 -      using complete_real[of X] by blast
    7.26 +      using complete_real[of X] unfolding bdd_above_def by blast
    7.27      then show "x \<le> Sup X"
    7.28        unfolding Sup_real_def by (rule LeastI2_order) (auto simp: x) }
    7.29    note Sup_upper = this
    7.30 @@ -953,9 +959,9 @@
    7.31    note Sup_least = this
    7.32  
    7.33    { fix x z :: real and X :: "real set"
    7.34 -    assume x: "x \<in> X" and z: "\<And>x. x \<in> X \<Longrightarrow> z \<le> x"
    7.35 +    assume x: "x \<in> X" and [simp]: "bdd_below X"
    7.36      have "-x \<le> Sup (uminus ` X)"
    7.37 -      by (rule Sup_upper[of _ _ "- z"]) (auto simp add: image_iff x z)
    7.38 +      by (rule Sup_upper) (auto simp add: image_iff x)
    7.39      then show "Inf X \<le> x" 
    7.40        by (auto simp add: Inf_real_def) }
    7.41  
    7.42 @@ -976,7 +982,7 @@
    7.43  *}
    7.44  
    7.45  lemma reals_complete: "\<exists>X. X \<in> S \<Longrightarrow> \<exists>Y. isUb (UNIV::real set) S Y \<Longrightarrow> \<exists>t. isLub (UNIV :: real set) S t"
    7.46 -  by (intro exI[of _ "Sup S"] isLub_cSup) (auto simp: setle_def isUb_def intro: cSup_upper)
    7.47 +  by (intro exI[of _ "Sup S"] isLub_cSup) (auto simp: setle_def isUb_def intro!: cSup_upper)
    7.48  
    7.49  
    7.50  subsection {* Hiding implementation details *}
     8.1 --- a/src/HOL/Topological_Spaces.thy	Tue Nov 05 09:44:57 2013 +0100
     8.2 +++ b/src/HOL/Topological_Spaces.thy	Tue Nov 05 09:44:58 2013 +0100
     8.3 @@ -2112,7 +2112,7 @@
     8.4    with dense[of b "Inf A"] obtain c where "c < Inf A" "c \<in> A"
     8.5      by (auto simp: subset_eq)
     8.6    then show False
     8.7 -    using cInf_lower[OF `c \<in> A`, of x] bnd by (metis less_imp_le not_le)
     8.8 +    using cInf_lower[OF `c \<in> A`] bnd by (metis not_le less_imp_le bdd_belowI)
     8.9  qed
    8.10  
    8.11  lemma Sup_notin_open:
    8.12 @@ -2125,7 +2125,7 @@
    8.13    with dense[of "Sup A" b] obtain c where "Sup A < c" "c \<in> A"
    8.14      by (auto simp: subset_eq)
    8.15    then show False
    8.16 -    using cSup_upper[OF `c \<in> A`, of x] bnd by (metis less_imp_le not_le)
    8.17 +    using cSup_upper[OF `c \<in> A`] bnd by (metis less_imp_le not_le bdd_aboveI)
    8.18  qed
    8.19  
    8.20  end
    8.21 @@ -2151,7 +2151,7 @@
    8.22      let ?z = "Inf (B \<inter> {x <..})"
    8.23  
    8.24      have "x \<le> ?z" "?z \<le> y"
    8.25 -      using `y \<in> B` `x < y` by (auto intro: cInf_lower[where z=x] cInf_greatest)
    8.26 +      using `y \<in> B` `x < y` by (auto intro: cInf_lower cInf_greatest)
    8.27      with `x \<in> U` `y \<in> U` have "?z \<in> U"
    8.28        by (rule *)
    8.29      moreover have "?z \<notin> B \<inter> {x <..}"
    8.30 @@ -2163,11 +2163,11 @@
    8.31        obtain a where "?z < a" "{?z ..< a} \<subseteq> A"
    8.32          using open_right[OF `open A` `?z \<in> A` `?z < y`] by auto
    8.33        moreover obtain b where "b \<in> B" "x < b" "b < min a y"
    8.34 -        using cInf_less_iff[of "B \<inter> {x <..}" x "min a y"] `?z < a` `?z < y` `x < y` `y \<in> B`
    8.35 +        using cInf_less_iff[of "B \<inter> {x <..}" "min a y"] `?z < a` `?z < y` `x < y` `y \<in> B`
    8.36          by (auto intro: less_imp_le)
    8.37        moreover have "?z \<le> b"
    8.38          using `b \<in> B` `x < b`
    8.39 -        by (intro cInf_lower[where z=x]) auto
    8.40 +        by (intro cInf_lower) auto
    8.41        moreover have "b \<in> U"
    8.42          using `x \<le> ?z` `?z \<le> b` `b < min a y`
    8.43          by (intro *[OF `x \<in> U` `y \<in> U`]) (auto simp: less_imp_le)