src/HOL/Conditionally_Complete_Lattices.thy
changeset 54258 adfc759263ab
parent 54257 5c7a3b6b05a9
child 54259 71c701dc5bf9
     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