# HG changeset patch
# User haftmann
# Date 1312914288 -7200
# Node ID 0e018cbcc0de519b9422076e01c907ebe46b1f70
# Parent  04e51b7a342235d56790371c970904f397564a2a
tuned proofs

diff -r 04e51b7a3422 -r 0e018cbcc0de src/HOL/Algebra/Ideal.thy
--- a/src/HOL/Algebra/Ideal.thy	Tue Aug 09 18:52:18 2011 +0200
+++ b/src/HOL/Algebra/Ideal.thy	Tue Aug 09 20:24:48 2011 +0200
@@ -227,26 +227,14 @@
     and notempty: "S \<noteq> {}"
   shows "ideal (Inter S) R"
 apply (unfold_locales)
-apply (simp_all add: Inter_def INTER_def)
-      apply (rule, simp) defer 1
+apply (simp_all add: Inter_eq)
+      apply rule unfolding mem_Collect_eq defer 1
       apply rule defer 1
       apply rule defer 1
       apply (fold a_inv_def, rule) defer 1
       apply rule defer 1
       apply rule defer 1
 proof -
-  fix x
-  assume "\<forall>I\<in>S. x \<in> I"
-  hence xI: "\<And>I. I \<in> S \<Longrightarrow> x \<in> I" by simp
-
-  from notempty have "\<exists>I0. I0 \<in> S" by blast
-  from this obtain I0 where I0S: "I0 \<in> S" by auto
-
-  interpret ideal I0 R by (rule Sideals[OF I0S])
-
-  from xI[OF I0S] have "x \<in> I0" .
-  from this and a_subset show "x \<in> carrier R" by fast
-next
   fix x y
   assume "\<forall>I\<in>S. x \<in> I"
   hence xI: "\<And>I. I \<in> S \<Longrightarrow> x \<in> I" by simp
@@ -298,6 +286,20 @@
 
   from xI[OF JS] and ycarr
       show "x \<otimes> y \<in> J" by (rule I_r_closed)
+next
+  fix x
+  assume "\<forall>I\<in>S. x \<in> I"
+  hence xI: "\<And>I. I \<in> S \<Longrightarrow> x \<in> I" by simp
+
+  from notempty have "\<exists>I0. I0 \<in> S" by blast
+  from this obtain I0 where I0S: "I0 \<in> S" by auto
+
+  interpret ideal I0 R by (rule Sideals[OF I0S])
+
+  from xI[OF I0S] have "x \<in> I0" .
+  from this and a_subset show "x \<in> carrier R" by fast
+next
+
 qed
 
 
diff -r 04e51b7a3422 -r 0e018cbcc0de src/HOL/Import/HOLLight/hollight.imp
--- a/src/HOL/Import/HOLLight/hollight.imp	Tue Aug 09 18:52:18 2011 +0200
+++ b/src/HOL/Import/HOLLight/hollight.imp	Tue Aug 09 20:24:48 2011 +0200
@@ -590,7 +590,6 @@
   "UNIONS_INSERT" > "Complete_Lattice.Union_insert"
   "UNIONS_IMAGE" > "HOLLight.hollight.UNIONS_IMAGE"
   "UNIONS_GSPEC" > "HOLLight.hollight.UNIONS_GSPEC"
-  "UNIONS_2" > "Complete_Lattice.Un_eq_Union"
   "UNIONS_0" > "Complete_Lattice.Union_empty"
   "UNCURRY_def" > "HOLLight.hollight.UNCURRY_def"
   "TYDEF_recspace" > "HOLLight.hollight.TYDEF_recspace"
@@ -1596,7 +1595,6 @@
   "INTERS_INSERT" > "Complete_Lattice.Inter_insert"
   "INTERS_IMAGE" > "HOLLight.hollight.INTERS_IMAGE"
   "INTERS_GSPEC" > "HOLLight.hollight.INTERS_GSPEC"
-  "INTERS_2" > "Complete_Lattice.Int_eq_Inter"
   "INTERS_0" > "Complete_Lattice.Inter_empty"
   "INSERT_UNIV" > "HOLLight.hollight.INSERT_UNIV"
   "INSERT_UNION_EQ" > "Set.Un_insert_left"
diff -r 04e51b7a3422 -r 0e018cbcc0de src/HOL/Number_Theory/MiscAlgebra.thy
--- a/src/HOL/Number_Theory/MiscAlgebra.thy	Tue Aug 09 18:52:18 2011 +0200
+++ b/src/HOL/Number_Theory/MiscAlgebra.thy	Tue Aug 09 20:24:48 2011 +0200
@@ -8,7 +8,7 @@
 imports
   "~~/src/HOL/Algebra/Ring"
   "~~/src/HOL/Algebra/FiniteProduct"
-begin;
+begin
 
 (* finiteness stuff *)
 
@@ -34,7 +34,7 @@
 definition units_of :: "('a, 'b) monoid_scheme => 'a monoid" where
   "units_of G == (| carrier = Units G,
      Group.monoid.mult = Group.monoid.mult G,
-     one  = one G |)";
+     one  = one G |)"
 
 (*
 
@@ -264,7 +264,7 @@
       (ALL A:C. ALL B:C. A ~= B --> A Int B = {}) |] 
    ==> finprod G f (Union C) = finprod G (finprod G f) C" 
   apply (frule finprod_UN_disjoint [of C id f])
-  apply (unfold Union_def id_def, auto)
+  apply (auto simp add: SUP_def)
 done
 
 lemma (in comm_monoid) finprod_one [rule_format]: 
diff -r 04e51b7a3422 -r 0e018cbcc0de src/HOL/Probability/Caratheodory.thy
--- a/src/HOL/Probability/Caratheodory.thy	Tue Aug 09 18:52:18 2011 +0200
+++ b/src/HOL/Probability/Caratheodory.thy	Tue Aug 09 20:24:48 2011 +0200
@@ -6,7 +6,7 @@
 header {*Caratheodory Extension Theorem*}
 
 theory Caratheodory
-  imports Sigma_Algebra Extended_Real_Limits
+imports Sigma_Algebra "~~/src/HOL/Multivariate_Analysis/Extended_Real_Limits"
 begin
 
 lemma sums_def2:
@@ -433,8 +433,7 @@
             hence eq_fa: "f a = f (a \<inter> (\<Union>i\<in>{0..<n}. A i)) + f (a - (\<Union>i\<in>{0..<n}. A i))"
               by (simp add: lambda_system_eq UNION_in)
             have "f (a - (\<Union>i. A i)) \<le> f (a - (\<Union>i\<in>{0..<n}. A i))"
-              by (blast intro: increasingD [OF inc] UNION_eq_Union_image
-                               UNION_in U_in)
+              by (blast intro: increasingD [OF inc] UNION_in U_in)
             thus "(\<Sum>i<n. f (a \<inter> A i)) + f (a - (\<Union>i. A i)) \<le> f a"
               by (simp add: lambda_system_strong_sum pos A disj eq_fa add_left_mono atLeast0LessThan[symmetric])
           next
diff -r 04e51b7a3422 -r 0e018cbcc0de src/HOL/Probability/Sigma_Algebra.thy
--- a/src/HOL/Probability/Sigma_Algebra.thy	Tue Aug 09 18:52:18 2011 +0200
+++ b/src/HOL/Probability/Sigma_Algebra.thy	Tue Aug 09 20:24:48 2011 +0200
@@ -315,10 +315,10 @@
   by (auto simp add: binary_def)
 
 lemma Un_range_binary: "a \<union> b = (\<Union>i::nat. binary a b i)"
-  by (simp add: UNION_eq_Union_image range_binary_eq)
+  by (simp add: SUP_def range_binary_eq)
 
 lemma Int_range_binary: "a \<inter> b = (\<Inter>i::nat. binary a b i)"
-  by (simp add: INTER_eq_Inter_image range_binary_eq)
+  by (simp add: INF_def range_binary_eq)
 
 lemma sigma_algebra_iff2:
      "sigma_algebra M \<longleftrightarrow>
@@ -1109,7 +1109,7 @@
   done
 
 lemma UN_binaryset_eq: "(\<Union>i. binaryset A B i) = A \<union> B"
-  by (simp add: UNION_eq_Union_image range_binaryset_eq)
+  by (simp add: SUP_def range_binaryset_eq)
 
 section {* Closed CDI *}
 
diff -r 04e51b7a3422 -r 0e018cbcc0de src/HOL/UNITY/ELT.thy
--- a/src/HOL/UNITY/ELT.thy	Tue Aug 09 18:52:18 2011 +0200
+++ b/src/HOL/UNITY/ELT.thy	Tue Aug 09 20:24:48 2011 +0200
@@ -186,9 +186,7 @@
 lemma leadsETo_Un:
      "[| F : A leadsTo[CC] C; F : B leadsTo[CC] C |] 
       ==> F : (A Un B) leadsTo[CC] C"
-apply (subst Un_eq_Union)
-apply (blast intro: leadsETo_Union)
-done
+  using leadsETo_Union [of "{A, B}" F CC C] by auto
 
 lemma single_leadsETo_I:
      "(!!x. x : A ==> F : {x} leadsTo[CC] B) ==> F : A leadsTo[CC] B"
@@ -407,9 +405,7 @@
 lemma LeadsETo_Un:
      "[| F : A LeadsTo[CC] C; F : B LeadsTo[CC] C |]  
       ==> F : (A Un B) LeadsTo[CC] C"
-apply (subst Un_eq_Union)
-apply (blast intro: LeadsETo_Union)
-done
+  using LeadsETo_Union [of "{A, B}" F CC C] by auto
 
 (*Lets us look at the starting state*)
 lemma single_LeadsETo_I:
diff -r 04e51b7a3422 -r 0e018cbcc0de src/HOL/UNITY/ProgressSets.thy
--- a/src/HOL/UNITY/ProgressSets.thy	Tue Aug 09 18:52:18 2011 +0200
+++ b/src/HOL/UNITY/ProgressSets.thy	Tue Aug 09 20:24:48 2011 +0200
@@ -42,25 +42,21 @@
 
 lemma UN_in_lattice:
      "[|lattice L; !!i. i\<in>I ==> r i \<in> L|] ==> (\<Union>i\<in>I. r i) \<in> L"
-apply (simp add: UN_eq) 
+apply (unfold SUP_def)
 apply (blast intro: Union_in_lattice) 
 done
 
 lemma INT_in_lattice:
      "[|lattice L; !!i. i\<in>I ==> r i \<in> L|] ==> (\<Inter>i\<in>I. r i)  \<in> L"
-apply (simp add: INT_eq) 
+apply (unfold INF_def)
 apply (blast intro: Inter_in_lattice) 
 done
 
 lemma Un_in_lattice: "[|x\<in>L; y\<in>L; lattice L|] ==> x\<union>y \<in> L"
-apply (simp only: Un_eq_Union) 
-apply (blast intro: Union_in_lattice) 
-done
+  using Union_in_lattice [of "{x, y}" L] by simp
 
 lemma Int_in_lattice: "[|x\<in>L; y\<in>L; lattice L|] ==> x\<inter>y \<in> L"
-apply (simp only: Int_eq_Inter) 
-apply (blast intro: Inter_in_lattice) 
-done
+  using Inter_in_lattice [of "{x, y}" L] by simp
 
 lemma lattice_stable: "lattice {X. F \<in> stable X}"
 by (simp add: lattice_def stable_def constrains_def, blast)
diff -r 04e51b7a3422 -r 0e018cbcc0de src/HOL/UNITY/SubstAx.thy
--- a/src/HOL/UNITY/SubstAx.thy	Tue Aug 09 18:52:18 2011 +0200
+++ b/src/HOL/UNITY/SubstAx.thy	Tue Aug 09 20:24:48 2011 +0200
@@ -85,16 +85,14 @@
 
 lemma LeadsTo_UN: 
      "(!!i. i \<in> I ==> F \<in> (A i) LeadsTo B) ==> F \<in> (\<Union>i \<in> I. A i) LeadsTo B"
-apply (simp only: Union_image_eq [symmetric])
+apply (unfold SUP_def)
 apply (blast intro: LeadsTo_Union)
 done
 
 text{*Binary union introduction rule*}
 lemma LeadsTo_Un:
      "[| F \<in> A LeadsTo C; F \<in> B LeadsTo C |] ==> F \<in> (A \<union> B) LeadsTo C"
-apply (subst Un_eq_Union)
-apply (blast intro: LeadsTo_Union)
-done
+  using LeadsTo_UN [of "{A, B}" F id C] by auto
 
 text{*Lets us look at the starting state*}
 lemma single_LeadsTo_I:
diff -r 04e51b7a3422 -r 0e018cbcc0de src/HOL/UNITY/Transformers.thy
--- a/src/HOL/UNITY/Transformers.thy	Tue Aug 09 18:52:18 2011 +0200
+++ b/src/HOL/UNITY/Transformers.thy	Tue Aug 09 20:24:48 2011 +0200
@@ -467,7 +467,7 @@
       "single_valued act
        ==> insert (wens_single act B) (range (wens_single_finite act B)) \<subseteq> 
            wens_set (mk_program (init, {act}, allowed)) B"
-apply (simp add: wens_single_eq_Union UN_eq) 
+apply (simp add: SUP_def image_def wens_single_eq_Union) 
 apply (blast intro: wens_set.Union wens_single_finite_in_wens_set)
 done
 
diff -r 04e51b7a3422 -r 0e018cbcc0de src/HOL/UNITY/WFair.thy
--- a/src/HOL/UNITY/WFair.thy	Tue Aug 09 18:52:18 2011 +0200
+++ b/src/HOL/UNITY/WFair.thy	Tue Aug 09 20:24:48 2011 +0200
@@ -211,9 +211,7 @@
 text{*Binary union introduction rule*}
 lemma leadsTo_Un:
      "[| F \<in> A leadsTo C; F \<in> B leadsTo C |] ==> F \<in> (A \<union> B) leadsTo C"
-apply (subst Un_eq_Union)
-apply (blast intro: leadsTo_Union)
-done
+  using leadsTo_Union [of "{A, B}" F C] by auto
 
 lemma single_leadsTo_I: 
      "(!!x. x \<in> A ==> F \<in> {x} leadsTo B) ==> F \<in> A leadsTo B"