author noschinl Tue Sep 13 16:21:48 2011 +0200 (2011-09-13) changeset 44918 6a80fbc4e72c parent 44917 8df4c332cb1c child 44919 482f1807976e
tune simpset for Complete_Lattices
 src/HOL/Big_Operators.thy file | annotate | diff | revisions src/HOL/Complete_Lattices.thy file | annotate | diff | revisions src/HOL/Induct/Sexp.thy file | annotate | diff | revisions src/HOL/Lattices.thy file | annotate | diff | revisions src/HOL/Library/Extended_Real.thy file | annotate | diff | revisions src/HOL/Library/Kleene_Algebra.thy file | annotate | diff | revisions src/HOL/Main.thy file | annotate | diff | revisions src/HOL/Multivariate_Analysis/Extended_Real_Limits.thy file | annotate | diff | revisions src/HOL/Probability/Caratheodory.thy file | annotate | diff | revisions src/HOL/Probability/Radon_Nikodym.thy file | annotate | diff | revisions src/HOL/UNITY/ProgressSets.thy file | annotate | diff | revisions
```     1.1 --- a/src/HOL/Big_Operators.thy	Tue Sep 13 13:17:52 2011 +0200
1.2 +++ b/src/HOL/Big_Operators.thy	Tue Sep 13 16:21:48 2011 +0200
1.3 @@ -1433,11 +1433,10 @@
1.4  proof -
1.5    interpret ab_semigroup_idem_mult inf
1.6      by (rule ab_semigroup_idem_mult_inf)
1.7 -  from `A \<noteq> {}` obtain b B where "A = insert b B" by auto
1.8 +  from `A \<noteq> {}` obtain b B where "A = {b} \<union> B" by auto
1.9    moreover with `finite A` have "finite B" by simp
1.10 -  ultimately show ?thesis
1.11 -  by (simp add: Inf_fin_def fold1_eq_fold_idem inf_Inf_fold_inf [symmetric])
1.12 -    (simp add: Inf_fold_inf)
1.13 +  ultimately show ?thesis
1.14 +    by (simp add: Inf_fin_def fold1_eq_fold_idem inf_Inf_fold_inf [symmetric])
1.15  qed
1.16
1.17  lemma Sup_fin_Sup:
1.18 @@ -1446,11 +1445,10 @@
1.19  proof -
1.20    interpret ab_semigroup_idem_mult sup
1.21      by (rule ab_semigroup_idem_mult_sup)
1.22 -  from `A \<noteq> {}` obtain b B where "A = insert b B" by auto
1.23 +  from `A \<noteq> {}` obtain b B where "A = {b} \<union> B" by auto
1.24    moreover with `finite A` have "finite B" by simp
1.25    ultimately show ?thesis
1.26    by (simp add: Sup_fin_def fold1_eq_fold_idem sup_Sup_fold_sup [symmetric])
1.27 -    (simp add: Sup_fold_sup)
1.28  qed
1.29
1.30  end
```
```     2.1 --- a/src/HOL/Complete_Lattices.thy	Tue Sep 13 13:17:52 2011 +0200
2.2 +++ b/src/HOL/Complete_Lattices.thy	Tue Sep 13 16:21:48 2011 +0200
2.3 @@ -126,16 +126,16 @@
2.4  lemma SUP_upper2: "i \<in> A \<Longrightarrow> u \<sqsubseteq> f i \<Longrightarrow> u \<sqsubseteq> (\<Squnion>i\<in>A. f i)"
2.5    using SUP_upper [of i A f] by auto
2.6
2.7 -lemma le_Inf_iff (*[simp]*): "b \<sqsubseteq> \<Sqinter>A \<longleftrightarrow> (\<forall>a\<in>A. b \<sqsubseteq> a)"
2.8 +lemma le_Inf_iff: "b \<sqsubseteq> \<Sqinter>A \<longleftrightarrow> (\<forall>a\<in>A. b \<sqsubseteq> a)"
2.9    by (auto intro: Inf_greatest dest: Inf_lower)
2.10
2.11 -lemma le_INF_iff (*[simp]*): "u \<sqsubseteq> (\<Sqinter>i\<in>A. f i) \<longleftrightarrow> (\<forall>i\<in>A. u \<sqsubseteq> f i)"
2.12 +lemma le_INF_iff: "u \<sqsubseteq> (\<Sqinter>i\<in>A. f i) \<longleftrightarrow> (\<forall>i\<in>A. u \<sqsubseteq> f i)"
2.13    by (auto simp add: INF_def le_Inf_iff)
2.14
2.15 -lemma Sup_le_iff (*[simp]*): "\<Squnion>A \<sqsubseteq> b \<longleftrightarrow> (\<forall>a\<in>A. a \<sqsubseteq> b)"
2.16 +lemma Sup_le_iff: "\<Squnion>A \<sqsubseteq> b \<longleftrightarrow> (\<forall>a\<in>A. a \<sqsubseteq> b)"
2.17    by (auto intro: Sup_least dest: Sup_upper)
2.18
2.19 -lemma SUP_le_iff (*[simp]*): "(\<Squnion>i\<in>A. f i) \<sqsubseteq> u \<longleftrightarrow> (\<forall>i\<in>A. f i \<sqsubseteq> u)"
2.20 +lemma SUP_le_iff: "(\<Squnion>i\<in>A. f i) \<sqsubseteq> u \<longleftrightarrow> (\<forall>i\<in>A. f i \<sqsubseteq> u)"
2.21    by (auto simp add: SUP_def Sup_le_iff)
2.22
2.23  lemma Inf_empty [simp]:
2.24 @@ -160,22 +160,22 @@
2.25    "\<Squnion>UNIV = \<top>"
2.26    by (auto intro!: antisym Sup_upper)
2.27
2.28 -lemma Inf_insert (*[simp]*): "\<Sqinter>insert a A = a \<sqinter> \<Sqinter>A"
2.29 +lemma Inf_insert [simp]: "\<Sqinter>insert a A = a \<sqinter> \<Sqinter>A"
2.30    by (auto intro: le_infI le_infI1 le_infI2 antisym Inf_greatest Inf_lower)
2.31
2.32  lemma INF_insert: "(\<Sqinter>x\<in>insert a A. f x) = f a \<sqinter> INFI A f"
2.33    by (simp add: INF_def Inf_insert)
2.34
2.35 -lemma Sup_insert (*[simp]*): "\<Squnion>insert a A = a \<squnion> \<Squnion>A"
2.36 +lemma Sup_insert [simp]: "\<Squnion>insert a A = a \<squnion> \<Squnion>A"
2.37    by (auto intro: le_supI le_supI1 le_supI2 antisym Sup_least Sup_upper)
2.38
2.39  lemma SUP_insert: "(\<Squnion>x\<in>insert a A. f x) = f a \<squnion> SUPR A f"
2.40    by (simp add: SUP_def Sup_insert)
2.41
2.42 -lemma INF_image (*[simp]*): "(\<Sqinter>x\<in>f`A. g x) = (\<Sqinter>x\<in>A. g (f x))"
2.43 +lemma INF_image [simp]: "(\<Sqinter>x\<in>f`A. g x) = (\<Sqinter>x\<in>A. g (f x))"
2.44    by (simp add: INF_def image_image)
2.45
2.46 -lemma SUP_image (*[simp]*): "(\<Squnion>x\<in>f`A. g x) = (\<Squnion>x\<in>A. g (f x))"
2.47 +lemma SUP_image [simp]: "(\<Squnion>x\<in>f`A. g x) = (\<Squnion>x\<in>A. g (f x))"
2.48    by (simp add: SUP_def image_image)
2.49
2.50  lemma Inf_Sup: "\<Sqinter>A = \<Squnion>{b. \<forall>a \<in> A. b \<sqsubseteq> a}"
2.51 @@ -210,7 +210,7 @@
2.52
2.53  lemma INF_mono:
2.54    "(\<And>m. m \<in> B \<Longrightarrow> \<exists>n\<in>A. f n \<sqsubseteq> g m) \<Longrightarrow> (\<Sqinter>n\<in>A. f n) \<sqsubseteq> (\<Sqinter>n\<in>B. g n)"
2.55 -  by (force intro!: Inf_mono simp: INF_def)
2.56 +  unfolding INF_def by (rule Inf_mono) fast
2.57
2.58  lemma Sup_mono:
2.59    assumes "\<And>a. a \<in> A \<Longrightarrow> \<exists>b\<in>B. a \<sqsubseteq> b"
2.60 @@ -224,7 +224,7 @@
2.61
2.62  lemma SUP_mono:
2.63    "(\<And>n. n \<in> A \<Longrightarrow> \<exists>m\<in>B. f n \<sqsubseteq> g m) \<Longrightarrow> (\<Squnion>n\<in>A. f n) \<sqsubseteq> (\<Squnion>n\<in>B. g n)"
2.64 -  by (force intro!: Sup_mono simp: SUP_def)
2.65 +  unfolding SUP_def by (rule Sup_mono) fast
2.66
2.67  lemma INF_superset_mono:
2.68    "B \<subseteq> A \<Longrightarrow> (\<And>x. x \<in> B \<Longrightarrow> f x \<sqsubseteq> g x) \<Longrightarrow> (\<Sqinter>x\<in>A. f x) \<sqsubseteq> (\<Sqinter>x\<in>B. g x)"
2.69 @@ -278,11 +278,14 @@
2.70  lemma INF_inf_distrib: "(\<Sqinter>a\<in>A. f a) \<sqinter> (\<Sqinter>a\<in>A. g a) = (\<Sqinter>a\<in>A. f a \<sqinter> g a)"
2.71    by (rule antisym) (rule INF_greatest, auto intro: le_infI1 le_infI2 INF_lower INF_mono)
2.72
2.73 -lemma SUP_sup_distrib: "(\<Squnion>a\<in>A. f a) \<squnion> (\<Squnion>a\<in>A. g a) = (\<Squnion>a\<in>A. f a \<squnion> g a)"
2.74 -  by (rule antisym) (auto intro: le_supI1 le_supI2 SUP_upper SUP_mono,
2.75 -    rule SUP_least, auto intro: le_supI1 le_supI2 SUP_upper SUP_mono)
2.76 +lemma SUP_sup_distrib: "(\<Squnion>a\<in>A. f a) \<squnion> (\<Squnion>a\<in>A. g a) = (\<Squnion>a\<in>A. f a \<squnion> g a)" (is "?L = ?R")
2.77 +proof (rule antisym)
2.78 +  show "?L \<le> ?R" by (auto intro: le_supI1 le_supI2 SUP_upper SUP_mono)
2.79 +next
2.80 +  show "?R \<le> ?L" by (rule SUP_least) (auto intro: le_supI1 le_supI2 SUP_upper)
2.81 +qed
2.82
2.83 -lemma Inf_top_conv (*[simp]*) [no_atp]:
2.84 +lemma Inf_top_conv [simp, no_atp]:
2.85    "\<Sqinter>A = \<top> \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)"
2.86    "\<top> = \<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)"
2.87  proof -
2.88 @@ -304,12 +307,12 @@
2.89    then show "\<top> = \<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x = \<top>)" by auto
2.90  qed
2.91
2.92 -lemma INF_top_conv (*[simp]*):
2.93 +lemma INF_top_conv [simp]:
2.94   "(\<Sqinter>x\<in>A. B x) = \<top> \<longleftrightarrow> (\<forall>x\<in>A. B x = \<top>)"
2.95   "\<top> = (\<Sqinter>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. B x = \<top>)"
2.96    by (auto simp add: INF_def Inf_top_conv)
2.97
2.98 -lemma Sup_bot_conv (*[simp]*) [no_atp]:
2.99 +lemma Sup_bot_conv [simp, no_atp]:
2.100    "\<Squnion>A = \<bottom> \<longleftrightarrow> (\<forall>x\<in>A. x = \<bottom>)" (is ?P)
2.101    "\<bottom> = \<Squnion>A \<longleftrightarrow> (\<forall>x\<in>A. x = \<bottom>)" (is ?Q)
2.102  proof -
2.103 @@ -318,7 +321,7 @@
2.104    from dual.Inf_top_conv show ?P and ?Q by simp_all
2.105  qed
2.106
2.107 -lemma SUP_bot_conv (*[simp]*):
2.108 +lemma SUP_bot_conv [simp]:
2.109   "(\<Squnion>x\<in>A. B x) = \<bottom> \<longleftrightarrow> (\<forall>x\<in>A. B x = \<bottom>)"
2.110   "\<bottom> = (\<Squnion>x\<in>A. B x) \<longleftrightarrow> (\<forall>x\<in>A. B x = \<bottom>)"
2.111    by (auto simp add: SUP_def Sup_bot_conv)
2.112 @@ -329,10 +332,10 @@
2.113  lemma SUP_const [simp]: "A \<noteq> {} \<Longrightarrow> (\<Squnion>i\<in>A. f) = f"
2.114    by (auto intro: antisym SUP_upper SUP_least)
2.115
2.116 -lemma INF_top (*[simp]*): "(\<Sqinter>x\<in>A. \<top>) = \<top>"
2.117 +lemma INF_top [simp]: "(\<Sqinter>x\<in>A. \<top>) = \<top>"
2.118    by (cases "A = {}") (simp_all add: INF_empty)
2.119
2.120 -lemma SUP_bot (*[simp]*): "(\<Squnion>x\<in>A. \<bottom>) = \<bottom>"
2.121 +lemma SUP_bot [simp]: "(\<Squnion>x\<in>A. \<bottom>) = \<bottom>"
2.122    by (cases "A = {}") (simp_all add: SUP_empty)
2.123
2.124  lemma INF_commute: "(\<Sqinter>i\<in>A. \<Sqinter>j\<in>B. f i j) = (\<Sqinter>j\<in>B. \<Sqinter>i\<in>A. f i j)"
2.125 @@ -492,23 +495,23 @@
2.126    "class.complete_linorder Sup Inf sup (op \<ge>) (op >) inf \<top> \<bottom>"
2.127    by (rule class.complete_linorder.intro, rule dual_complete_lattice, rule dual_linorder)
2.128
2.129 -lemma Inf_less_iff (*[simp]*):
2.130 +lemma Inf_less_iff:
2.131    "\<Sqinter>S \<sqsubset> a \<longleftrightarrow> (\<exists>x\<in>S. x \<sqsubset> a)"
2.132    unfolding not_le [symmetric] le_Inf_iff by auto
2.133
2.134 -lemma INF_less_iff (*[simp]*):
2.135 +lemma INF_less_iff:
2.136    "(\<Sqinter>i\<in>A. f i) \<sqsubset> a \<longleftrightarrow> (\<exists>x\<in>A. f x \<sqsubset> a)"
2.137    unfolding INF_def Inf_less_iff by auto
2.138
2.139 -lemma less_Sup_iff (*[simp]*):
2.140 +lemma less_Sup_iff:
2.141    "a \<sqsubset> \<Squnion>S \<longleftrightarrow> (\<exists>x\<in>S. a \<sqsubset> x)"
2.142    unfolding not_le [symmetric] Sup_le_iff by auto
2.143
2.144 -lemma less_SUP_iff (*[simp]*):
2.145 +lemma less_SUP_iff:
2.146    "a \<sqsubset> (\<Squnion>i\<in>A. f i) \<longleftrightarrow> (\<exists>x\<in>A. a \<sqsubset> f x)"
2.147    unfolding SUP_def less_Sup_iff by auto
2.148
2.149 -lemma Sup_eq_top_iff (*[simp]*):
2.150 +lemma Sup_eq_top_iff [simp]:
2.151    "\<Squnion>A = \<top> \<longleftrightarrow> (\<forall>x<\<top>. \<exists>i\<in>A. x < i)"
2.152  proof
2.153    assume *: "\<Squnion>A = \<top>"
2.154 @@ -530,11 +533,11 @@
2.155    qed
2.156  qed
2.157
2.158 -lemma SUP_eq_top_iff (*[simp]*):
2.159 +lemma SUP_eq_top_iff [simp]:
2.160    "(\<Squnion>i\<in>A. f i) = \<top> \<longleftrightarrow> (\<forall>x<\<top>. \<exists>i\<in>A. x < f i)"
2.161    unfolding SUP_def Sup_eq_top_iff by auto
2.162
2.163 -lemma Inf_eq_bot_iff (*[simp]*):
2.164 +lemma Inf_eq_bot_iff [simp]:
2.165    "\<Sqinter>A = \<bottom> \<longleftrightarrow> (\<forall>x>\<bottom>. \<exists>i\<in>A. i < x)"
2.166  proof -
2.167    interpret dual: complete_linorder Sup Inf sup "op \<ge>" "op >" inf \<top> \<bottom>
2.168 @@ -542,7 +545,7 @@
2.169    from dual.Sup_eq_top_iff show ?thesis .
2.170  qed
2.171
2.172 -lemma INF_eq_bot_iff (*[simp]*):
2.173 +lemma INF_eq_bot_iff [simp]:
2.174    "(\<Sqinter>i\<in>A. f i) = \<bottom> \<longleftrightarrow> (\<forall>x>\<bottom>. \<exists>i\<in>A. f i < x)"
2.175    unfolding INF_def Inf_eq_bot_iff by auto
2.176
```
```     3.1 --- a/src/HOL/Induct/Sexp.thy	Tue Sep 13 13:17:52 2011 +0200
3.2 +++ b/src/HOL/Induct/Sexp.thy	Tue Sep 13 16:21:48 2011 +0200
3.3 @@ -73,7 +73,7 @@
3.4  (** Introduction rules for 'pred_sexp' **)
3.5
3.6  lemma pred_sexp_subset_Sigma: "pred_sexp <= sexp <*> sexp"
3.7 -by (simp add: pred_sexp_def, blast)
3.8 +  by (simp add: pred_sexp_def) blast
3.9
3.10  (* (a,b) \<in> pred_sexp^+ ==> a \<in> sexp *)
3.11  lemmas trancl_pred_sexpD1 =
```
```     4.1 --- a/src/HOL/Lattices.thy	Tue Sep 13 13:17:52 2011 +0200
4.2 +++ b/src/HOL/Lattices.thy	Tue Sep 13 16:21:48 2011 +0200
4.3 @@ -180,10 +180,10 @@
4.4  lemma inf_left_commute: "x \<sqinter> (y \<sqinter> z) = y \<sqinter> (x \<sqinter> z)"
4.5    by (fact inf.left_commute)
4.6
4.7 -lemma inf_idem (*[simp]*): "x \<sqinter> x = x"
4.8 +lemma inf_idem [simp]: "x \<sqinter> x = x"
4.9    by (fact inf.idem)
4.10
4.11 -lemma inf_left_idem (*[simp]*): "x \<sqinter> (x \<sqinter> y) = x \<sqinter> y"
4.12 +lemma inf_left_idem [simp]: "x \<sqinter> (x \<sqinter> y) = x \<sqinter> y"
4.13    by (fact inf.left_idem)
4.14
4.15  lemma inf_absorb1: "x \<sqsubseteq> y \<Longrightarrow> x \<sqinter> y = x"
4.16 @@ -219,10 +219,10 @@
4.17  lemma sup_left_commute: "x \<squnion> (y \<squnion> z) = y \<squnion> (x \<squnion> z)"
4.18    by (fact sup.left_commute)
4.19
4.20 -lemma sup_idem (*[simp]*): "x \<squnion> x = x"
4.21 +lemma sup_idem [simp]: "x \<squnion> x = x"
4.22    by (fact sup.idem)
4.23
4.24 -lemma sup_left_idem (*[simp]*): "x \<squnion> (x \<squnion> y) = x \<squnion> y"
4.25 +lemma sup_left_idem [simp]: "x \<squnion> (x \<squnion> y) = x \<squnion> y"
4.26    by (fact sup.left_idem)
4.27
4.28  lemma sup_absorb1: "y \<sqsubseteq> x \<Longrightarrow> x \<squnion> y = x"
4.29 @@ -243,10 +243,10 @@
4.30    by (rule class.lattice.intro, rule dual_semilattice, rule class.semilattice_sup.intro, rule dual_order)
4.31      (unfold_locales, auto)
4.32
4.33 -lemma inf_sup_absorb (*[simp]*): "x \<sqinter> (x \<squnion> y) = x"
4.34 +lemma inf_sup_absorb [simp]: "x \<sqinter> (x \<squnion> y) = x"
4.35    by (blast intro: antisym inf_le1 inf_greatest sup_ge1)
4.36
4.37 -lemma sup_inf_absorb (*[simp]*): "x \<squnion> (x \<sqinter> y) = x"
4.38 +lemma sup_inf_absorb [simp]: "x \<squnion> (x \<sqinter> y) = x"
4.39    by (blast intro: antisym sup_ge1 sup_least inf_le1)
4.40
4.41  lemmas inf_sup_aci = inf_aci sup_aci
4.42 @@ -267,8 +267,9 @@
4.43  assumes D: "!!x y z. x \<sqinter> (y \<squnion> z) = (x \<sqinter> y) \<squnion> (x \<sqinter> z)"
4.44  shows "x \<squnion> (y \<sqinter> z) = (x \<squnion> y) \<sqinter> (x \<squnion> z)"
4.45  proof-
4.46 -  have "x \<squnion> (y \<sqinter> z) = (x \<squnion> (x \<sqinter> z)) \<squnion> (y \<sqinter> z)" by(simp add:sup_inf_absorb)
4.47 -  also have "\<dots> = x \<squnion> (z \<sqinter> (x \<squnion> y))" by(simp add:D inf_commute sup_assoc)
4.48 +  have "x \<squnion> (y \<sqinter> z) = (x \<squnion> (x \<sqinter> z)) \<squnion> (y \<sqinter> z)" by simp
4.49 +  also have "\<dots> = x \<squnion> (z \<sqinter> (x \<squnion> y))"
4.50 +    by (simp add: D inf_commute sup_assoc del: sup_inf_absorb)
4.51    also have "\<dots> = ((x \<squnion> y) \<sqinter> x) \<squnion> ((x \<squnion> y) \<sqinter> z)"
4.52      by(simp add:inf_sup_absorb inf_commute)
4.53    also have "\<dots> = (x \<squnion> y) \<sqinter> (x \<squnion> z)" by(simp add:D)
4.54 @@ -279,8 +280,9 @@
4.55  assumes D: "!!x y z. x \<squnion> (y \<sqinter> z) = (x \<squnion> y) \<sqinter> (x \<squnion> z)"
4.56  shows "x \<sqinter> (y \<squnion> z) = (x \<sqinter> y) \<squnion> (x \<sqinter> z)"
4.57  proof-
4.58 -  have "x \<sqinter> (y \<squnion> z) = (x \<sqinter> (x \<squnion> z)) \<sqinter> (y \<squnion> z)" by(simp add:inf_sup_absorb)
4.59 -  also have "\<dots> = x \<sqinter> (z \<squnion> (x \<sqinter> y))" by(simp add:D sup_commute inf_assoc)
4.60 +  have "x \<sqinter> (y \<squnion> z) = (x \<sqinter> (x \<squnion> z)) \<sqinter> (y \<squnion> z)" by simp
4.61 +  also have "\<dots> = x \<sqinter> (z \<squnion> (x \<sqinter> y))"
4.62 +    by (simp add: D sup_commute inf_assoc del: inf_sup_absorb)
4.63    also have "\<dots> = ((x \<sqinter> y) \<squnion> x) \<sqinter> ((x \<sqinter> y) \<squnion> z)"
4.64      by(simp add:sup_inf_absorb sup_commute)
4.65    also have "\<dots> = (x \<sqinter> y) \<squnion> (x \<sqinter> z)" by(simp add:D)
4.66 @@ -439,11 +441,11 @@
4.67    by (rule class.boolean_algebra.intro, rule dual_bounded_lattice, rule dual_distrib_lattice)
4.68      (unfold_locales, auto simp add: inf_compl_bot sup_compl_top diff_eq)
4.69
4.70 -lemma compl_inf_bot (*[simp]*):
4.71 +lemma compl_inf_bot [simp]:
4.72    "- x \<sqinter> x = \<bottom>"
4.73    by (simp add: inf_commute inf_compl_bot)
4.74
4.75 -lemma compl_sup_top (*[simp]*):
4.76 +lemma compl_sup_top [simp]:
4.77    "- x \<squnion> x = \<top>"
4.78    by (simp add: sup_commute sup_compl_top)
4.79
4.80 @@ -525,7 +527,7 @@
4.81    then show "- y \<sqsubseteq> - x" by (simp only: le_iff_inf)
4.82  qed
4.83
4.84 -lemma compl_le_compl_iff (*[simp]*):
4.85 +lemma compl_le_compl_iff [simp]:
4.86    "- x \<sqsubseteq> - y \<longleftrightarrow> y \<sqsubseteq> x"
4.87    by (auto dest: compl_mono)
4.88
```
```     5.1 --- a/src/HOL/Library/Extended_Real.thy	Tue Sep 13 13:17:52 2011 +0200
5.2 +++ b/src/HOL/Library/Extended_Real.thy	Tue Sep 13 16:21:48 2011 +0200
5.3 @@ -1506,7 +1506,8 @@
5.4        proof cases
5.5          assume "\<forall>i. f i = 0"
5.6          moreover then have "range f = {0}" by auto
5.7 -        ultimately show "c * SUPR UNIV f \<le> y" using * by (auto simp: SUPR_def)
5.8 +        ultimately show "c * SUPR UNIV f \<le> y" using *
5.9 +          by (auto simp: SUPR_def min_max.sup_absorb1)
5.10        next
5.11          assume "\<not> (\<forall>i. f i = 0)"
5.12          then obtain i where "f i \<noteq> 0" by auto
5.13 @@ -1568,7 +1569,8 @@
5.14          hence "0 < r" using `0 < e` by auto
5.15          then obtain n ::nat where *: "1 / real n < r" "0 < n"
5.16            using ex_inverse_of_nat_less by (auto simp: real_eq_of_nat inverse_eq_divide)
5.17 -        have "Sup A \<le> f n + 1 / ereal (real n)" using f[THEN spec, of n] by auto
5.18 +        have "Sup A \<le> f n + 1 / ereal (real n)" using f[THEN spec, of n]
5.19 +          by auto
5.20          also have "1 / ereal (real n) \<le> e" using real * by (auto simp: one_ereal_def )
5.21          with bound have "f n + 1 / ereal (real n) \<le> y + e" by (rule add_mono) simp
5.22          finally show "Sup A \<le> y + e" .
5.23 @@ -1625,7 +1627,7 @@
5.24    then show "Sup ((\<lambda>x. a + x) ` A) \<le> a + Sup A" .
5.25    show "a + Sup A \<le> Sup ((\<lambda>x. a + x) ` A)"
5.26    proof (cases a)
5.27 -    case PInf with `A \<noteq> {}` show ?thesis by (auto simp: image_constant)
5.28 +    case PInf with `A \<noteq> {}` show ?thesis by (auto simp: image_constant min_max.sup_absorb1)
5.29    next
5.30      case (real r)
5.31      then have **: "op + (- a) ` op + a ` A = A"
```
```     6.1 --- a/src/HOL/Library/Kleene_Algebra.thy	Tue Sep 13 13:17:52 2011 +0200
6.2 +++ b/src/HOL/Library/Kleene_Algebra.thy	Tue Sep 13 16:21:48 2011 +0200
6.3 @@ -377,19 +377,18 @@
6.4    have [simp]: "1 \<le> star a"
6.5      unfolding star_cont[of 1 a 1, simplified]
6.6      by (subst power_0[symmetric]) (rule le_SUPI [OF UNIV_I])
6.7 -
6.8 -  show "1 + a * star a \<le> star a"
6.9 -    apply (rule plus_leI, simp)
6.10 -    apply (simp add:star_cont[of a a 1, simplified])
6.11 -    apply (simp add:star_cont[of 1 a 1, simplified])
6.12 -    apply (subst power_Suc[symmetric])
6.13 -    by (intro SUP_leI le_SUPI UNIV_I)
6.14 +
6.15 +  have "a * star a \<le> star a"
6.16 +    using star_cont[of a a 1] star_cont[of 1 a 1]
6.17 +    by (auto simp add: power_Suc[symmetric] simp del: power_Suc
6.18 +      intro: SUP_leI le_SUPI)
6.19
6.20 -  show "1 + star a * a \<le> star a"
6.21 -    apply (rule plus_leI, simp)
6.22 -    apply (simp add:star_cont[of 1 a a, simplified])
6.23 -    apply (simp add:star_cont[of 1 a 1, simplified])
6.24 -    by (auto intro: SUP_leI le_SUPI simp add: power_Suc[symmetric] power_commutes simp del: power_Suc)
6.25 +  then show "1 + a * star a \<le> star a"
6.26 +    by simp
6.27 +
6.28 +  then show "1 + star a * a \<le> star a"
6.29 +    using star_cont[of a a 1] star_cont[of 1 a a]
6.30 +    by (simp add: power_commutes)
6.31
6.32    show "a * x \<le> x \<Longrightarrow> star a * x \<le> x"
6.33    proof -
```
```     7.1 --- a/src/HOL/Main.thy	Tue Sep 13 13:17:52 2011 +0200
7.2 +++ b/src/HOL/Main.thy	Tue Sep 13 16:21:48 2011 +0200
7.3 @@ -11,17 +11,4 @@
7.4
7.5  text {* See further \cite{Nipkow-et-al:2002:tutorial} *}
7.6
7.7 -text {* Compatibility layer -- to be dropped *}
7.8 -
7.9 -lemma Inf_bool_def:
7.10 -  "Inf A \<longleftrightarrow> (\<forall>x\<in>A. x)"
7.11 -  by (auto intro: bool_induct)
7.12 -
7.13 -lemma Sup_bool_def:
7.14 -  "Sup A \<longleftrightarrow> (\<exists>x\<in>A. x)"
7.15 -  by auto
7.16 -
7.17 -declare Complete_Lattices.Inf_bool_def [simp del]
7.18 -declare Complete_Lattices.Sup_bool_def [simp del]
7.19 -
7.20  end
```
```     8.1 --- a/src/HOL/Multivariate_Analysis/Extended_Real_Limits.thy	Tue Sep 13 13:17:52 2011 +0200
8.2 +++ b/src/HOL/Multivariate_Analysis/Extended_Real_Limits.thy	Tue Sep 13 16:21:48 2011 +0200
8.3 @@ -101,7 +101,7 @@
8.4      then show False using MInf by auto
8.5    next
8.6      case PInf then have "S={\<infinity>}" by (metis Inf_eq_PInfty assms(2))
8.7 -    then show False using `Inf S ~: S` by simp
8.8 +    then show False using `Inf S ~: S` by (simp add: top_ereal_def)
8.9    next
8.10      case (real r) then have fin: "\<bar>Inf S\<bar> \<noteq> \<infinity>" by simp
8.11      from ereal_open_cont_interval[OF a this] guess e . note e = this
8.12 @@ -143,7 +143,8 @@
8.13      from ereal_open_cont_interval[OF assms(1) * fin] guess e . note e = this
8.14      then obtain b where b_def: "Inf S-e<b & b<Inf S"
8.15        using fin ereal_between[of "Inf S" e] ereal_dense[of "Inf S-e"] by auto
8.16 -    hence "b: {Inf S-e <..< Inf S+e}" using e fin ereal_between[of "Inf S" e] by auto
8.17 +    hence "b: {Inf S-e <..< Inf S+e}" using e fin ereal_between[of "Inf S" e]
8.18 +      by auto
8.19      hence "b:S" using e by auto
8.20      hence False using b_def by (metis complete_lattice_class.Inf_lower leD)
8.21    } ultimately show False by auto
8.22 @@ -247,7 +248,7 @@
8.23      show "eventually (\<lambda>x. a * X x \<in> S) net"
8.24        by (rule eventually_mono[OF _ *]) auto
8.25    qed
8.26 -qed (auto intro: tendsto_const)
8.27 +qed auto
8.28
8.29  lemma ereal_lim_uminus:
8.30    fixes X :: "'a \<Rightarrow> ereal" shows "((\<lambda>i. - X i) ---> -L) net \<longleftrightarrow> (X ---> L) net"
8.31 @@ -306,7 +307,7 @@
8.32      assume S: "S = {Inf S<..}"
8.33      then have "Inf S < l" using `l \<in> S` by auto
8.34      then have "eventually (\<lambda>x. Inf S < f x) net" using ev by auto
8.35 -    then show "eventually (\<lambda>x. f x \<in> S) net"  by (subst S) auto
8.36 +    then show "eventually (\<lambda>x. f x \<in> S) net" by (subst S) auto
8.37    qed auto
8.38  next
8.39    fix l y assume S: "\<forall>S. open S \<longrightarrow> mono_set S \<longrightarrow> l \<in> S \<longrightarrow> eventually  (\<lambda>x. f x \<in> S) net" "y < l"
```
```     9.1 --- a/src/HOL/Probability/Caratheodory.thy	Tue Sep 13 13:17:52 2011 +0200
9.2 +++ b/src/HOL/Probability/Caratheodory.thy	Tue Sep 13 16:21:48 2011 +0200
9.3 @@ -613,7 +613,7 @@
9.4    assumes posf: "positive M f"
9.5    shows "increasing \<lparr> space = space M, sets = Pow (space M) \<rparr>
9.6                      (\<lambda>x. Inf (measure_set M f x))"
9.7 -apply (auto simp add: increasing_def)
9.8 +apply (clarsimp simp add: increasing_def)
9.9  apply (rule complete_lattice_class.Inf_greatest)
9.10  apply (rule complete_lattice_class.Inf_lower)
9.11  apply (clarsimp simp add: measure_set_def, rule_tac x=A in exI, blast)
```
```    10.1 --- a/src/HOL/Probability/Radon_Nikodym.thy	Tue Sep 13 13:17:52 2011 +0200
10.2 +++ b/src/HOL/Probability/Radon_Nikodym.thy	Tue Sep 13 16:21:48 2011 +0200
10.3 @@ -411,7 +411,9 @@
10.4    also have "\<dots> = ?y"
10.5    proof (rule antisym)
10.6      show "(SUP i. integral\<^isup>P M (?g i)) \<le> ?y"
10.7 -      using g_in_G by (auto intro!: exI Sup_mono simp: SUPR_def)
10.8 +      using g_in_G
10.9 +      using [[simp_trace]]
10.10 +      by (auto intro!: exI Sup_mono simp: SUPR_def)
10.11      show "?y \<le> (SUP i. integral\<^isup>P M (?g i))" unfolding y_eq
10.12        by (auto intro!: SUP_mono positive_integral_mono Max_ge)
10.13    qed
```
```    11.1 --- a/src/HOL/UNITY/ProgressSets.thy	Tue Sep 13 13:17:52 2011 +0200
11.2 +++ b/src/HOL/UNITY/ProgressSets.thy	Tue Sep 13 16:21:48 2011 +0200
11.3 @@ -77,7 +77,7 @@
11.4  by (simp add: cl_def, blast)
11.5
11.6  lemma subset_cl: "r \<subseteq> cl L r"
11.7 -by (simp add: cl_def, blast)
11.8 +by (simp add: cl_def le_Inf_iff)
11.9
11.10  text{*A reformulation of @{thm subset_cl}*}
11.11  lemma clI: "x \<in> r ==> x \<in> cl L r"
```