more concise definition for Inf, Sup on bool

--- a/NEWS	Fri Aug 19 17:05:10 2011 +0900
+++ b/NEWS	Fri Aug 19 19:33:31 2011 +0200
@@ -70,7 +70,8 @@
 generalized theorems INF_cong and SUP_cong.  New type classes for complete
 boolean algebras and complete linear orders.  Lemmas Inf_less_iff,
 less_Sup_iff, INF_less_iff, less_SUP_iff now reside in class complete_linorder.
-Changed proposition of lemmas Inf_fun_def, Sup_fun_def, Inf_apply, Sup_apply.
+Changed proposition of lemmas Inf_bool_def, Sup_bool_def, Inf_fun_def, Sup_fun_def,
+Inf_apply, Sup_apply.
 Redundant lemmas Inf_singleton, Sup_singleton, Inf_binary, Sup_binary,
 INF_eq, SUP_eq, INF_UNIV_range, SUP_UNIV_range, Int_eq_Inter,
 INTER_eq_Inter_image, Inter_def, INT_eq, Un_eq_Union, UNION_eq_Union_image,

--- a/src/HOL/Complete_Lattice.thy	Fri Aug 19 17:05:10 2011 +0900
+++ b/src/HOL/Complete_Lattice.thy	Fri Aug 19 19:33:31 2011 +0200
@@ -414,8 +414,7 @@
   apply (simp_all only: INF_foundation_dual SUP_foundation_dual inf_Sup sup_Inf)
   done
 
-subclass distrib_lattice proof -- {* Question: is it sufficient to include @{class distrib_lattice}
-  and prove @{text inf_Sup} and @{text sup_Inf} from that? *}
+subclass distrib_lattice proof
   fix a b c
   from sup_Inf have "a \<squnion> \<Sqinter>{b, c} = (\<Sqinter>d\<in>{b, c}. a \<squnion> d)" .
   then show "a \<squnion> b \<sqinter> c = (a \<squnion> b) \<sqinter> (a \<squnion> c)" by (simp add: INF_def Inf_insert)
@@ -556,13 +555,13 @@
 begin
 
 definition
-  "\<Sqinter>A \<longleftrightarrow> (\<forall>x\<in>A. x)"
+  [simp]: "\<Sqinter>A \<longleftrightarrow> False \<notin> A"
 
 definition
-  "\<Squnion>A \<longleftrightarrow> (\<exists>x\<in>A. x)"
+  [simp]: "\<Squnion>A \<longleftrightarrow> True \<in> A"
 
 instance proof
-qed (auto simp add: Inf_bool_def Sup_bool_def)
+qed (auto intro: bool_induct)
 
 end
 
@@ -572,7 +571,7 @@
   fix A :: "'a set"
   fix P :: "'a \<Rightarrow> bool"
   show "(\<Sqinter>x\<in>A. P x) \<longleftrightarrow> (\<forall>x\<in>A. P x)"
-    by (auto simp add: Ball_def INF_def Inf_bool_def)
+    by (auto simp add: INF_def)
 qed
 
 lemma SUP_bool_eq [simp]:
@@ -581,11 +580,11 @@
   fix A :: "'a set"
   fix P :: "'a \<Rightarrow> bool"
   show "(\<Squnion>x\<in>A. P x) \<longleftrightarrow> (\<exists>x\<in>A. P x)"
-    by (auto simp add: Bex_def SUP_def Sup_bool_def)
+    by (auto simp add: SUP_def)
 qed
 
 instance bool :: complete_boolean_algebra proof
-qed (auto simp add: Inf_bool_def Sup_bool_def)
+qed (auto intro: bool_induct)
 
 instantiation "fun" :: (type, complete_lattice) complete_lattice
 begin
@@ -638,7 +637,7 @@
   have "(\<forall>Q\<in>{P. \<exists>B\<in>A. P \<longleftrightarrow> x \<in> B}. Q) \<longleftrightarrow> (\<forall>B\<in>A. x \<in> B)"
     by auto
   then show "x \<in> \<Inter>A \<longleftrightarrow> x \<in> {x. \<forall>B \<in> A. x \<in> B}"
-    by (simp add: Inf_fun_def Inf_bool_def) (simp add: mem_def)
+    by (simp add: Inf_fun_def) (simp add: mem_def)
 qed
 
 lemma Inter_iff [simp,no_atp]: "A \<in> \<Inter>C \<longleftrightarrow> (\<forall>X\<in>C. A \<in> X)"
@@ -821,7 +820,7 @@
   have "(\<exists>Q\<in>{P. \<exists>B\<in>A. P \<longleftrightarrow> x \<in> B}. Q) \<longleftrightarrow> (\<exists>B\<in>A. x \<in> B)"
     by auto
   then show "x \<in> \<Union>A \<longleftrightarrow> x \<in> {x. \<exists>B\<in>A. x \<in> B}"
-    by (simp add: Sup_fun_def Sup_bool_def) (simp add: mem_def)
+    by (simp add: Sup_fun_def) (simp add: mem_def)
 qed
 
 lemma Union_iff [simp, no_atp]: