merged

--- a/src/HOL/Finite_Set.thy	Sat Feb 14 01:23:38 2009 -0800
+++ b/src/HOL/Finite_Set.thy	Sat Feb 14 01:24:01 2009 -0800
@@ -144,8 +144,7 @@
 subsubsection{* Finiteness and set theoretic constructions *}
 
 lemma finite_UnI: "finite F ==> finite G ==> finite (F Un G)"
-  -- {* The union of two finite sets is finite. *}
-  by (induct set: finite) simp_all
+by (induct set: finite) simp_all
 
 lemma finite_subset: "A \<subseteq> B ==> finite B ==> finite A"
   -- {* Every subset of a finite set is finite. *}
@@ -174,15 +173,21 @@
   qed
 qed
 
-lemma finite_Collect_subset[simp]: "finite A \<Longrightarrow> finite{x \<in> A. P x}"
-using finite_subset[of "{x \<in> A. P x}" "A"] by blast
-
 lemma finite_Un [iff]: "finite (F Un G) = (finite F & finite G)"
-  by (blast intro: finite_subset [of _ "X Un Y", standard] finite_UnI)
+by (blast intro: finite_subset [of _ "X Un Y", standard] finite_UnI)
+
+lemma finite_disjI[simp]:
+  "finite{x. P x | Q x} = (finite{x. P x} & finite{x. Q x})"
+by(simp add:Collect_disj_eq)
 
 lemma finite_Int [simp, intro]: "finite F | finite G ==> finite (F Int G)"
   -- {* The converse obviously fails. *}
-  by (blast intro: finite_subset)
+by (blast intro: finite_subset)
+
+lemma finite_conjI [simp, intro]:
+  "finite{x. P x} | finite{x. Q x} ==> finite{x. P x & Q x}"
+  -- {* The converse obviously fails. *}
+by(simp add:Collect_conj_eq)
 
 lemma finite_insert [simp]: "finite (insert a A) = finite A"
   apply (subst insert_is_Un)
@@ -227,8 +232,24 @@
   then show ?thesis by simp
 qed
 
-lemma finite_Diff [simp]: "finite B ==> finite (B - Ba)"
-  by (rule Diff_subset [THEN finite_subset])
+lemma finite_Diff [simp]: "finite A ==> finite (A - B)"
+by (rule Diff_subset [THEN finite_subset])
+
+lemma finite_Diff2 [simp]:
+  assumes "finite B" shows "finite (A - B) = finite A"
+proof -
+  have "finite A \<longleftrightarrow> finite((A-B) Un (A Int B))" by(simp add: Un_Diff_Int)
+  also have "\<dots> \<longleftrightarrow> finite(A-B)" using `finite B` by(simp)
+  finally show ?thesis ..
+qed
+
+lemma finite_compl[simp]:
+  "finite(A::'a set) \<Longrightarrow> finite(-A) = finite(UNIV::'a set)"
+by(simp add:Compl_eq_Diff_UNIV)
+
+lemma finite_not[simp]:
+  "finite{x::'a. P x} \<Longrightarrow> finite{x. ~P x} = finite(UNIV::'a set)"
+by(simp add:Collect_neg_eq)
 
 lemma finite_Diff_insert [iff]: "finite (A - insert a B) = finite (A - B)"
   apply (subst Diff_insert)
@@ -237,9 +258,6 @@
    apply (subst insert_Diff, simp_all)
   done
 
-lemma finite_Diff_singleton [simp]: "finite (A - {a}) = finite A"
-  by simp
-
 
 text {* Image and Inverse Image over Finite Sets *}
 

--- a/src/HOL/Library/Infinite_Set.thy	Sat Feb 14 01:23:38 2009 -0800
+++ b/src/HOL/Library/Infinite_Set.thy	Sat Feb 14 01:24:01 2009 -0800
@@ -461,10 +461,11 @@
   by simp
 
 lemma enumerate_in_set: "infinite S \<Longrightarrow> enumerate S n : S"
-  apply (induct n arbitrary: S)
-   apply (fastsimp intro: LeastI dest!: infinite_imp_nonempty)
-  apply (fastsimp iff: finite_Diff_singleton)
-  done
+apply (induct n arbitrary: S)
+ apply (fastsimp intro: LeastI dest!: infinite_imp_nonempty)
+apply simp
+apply (metis Collect_def Collect_mem_eq DiffE infinite_remove)
+done
 
 declare enumerate_0 [simp del] enumerate_Suc [simp del]
 

--- a/src/HOL/Library/Multiset.thy	Sat Feb 14 01:23:38 2009 -0800
+++ b/src/HOL/Library/Multiset.thy	Sat Feb 14 01:24:01 2009 -0800
@@ -88,10 +88,8 @@
 
 lemma union_preserves_multiset:
   "M \<in> multiset ==> N \<in> multiset ==> (\<lambda>a. M a + N a) \<in> multiset"
-apply (simp add: multiset_def)
-apply (drule (1) finite_UnI)
-apply (simp del: finite_Un add: Un_def)
-done
+by (simp add: multiset_def)
+
 
 lemma diff_preserves_multiset:
   "M \<in> multiset ==> (\<lambda>a. M a - N a) \<in> multiset"

--- a/src/HOL/Nominal/Nominal.thy	Sat Feb 14 01:23:38 2009 -0800
+++ b/src/HOL/Nominal/Nominal.thy	Sat Feb 14 01:24:01 2009 -0800
@@ -558,12 +558,7 @@
   fixes x :: "'x"
   assumes at: "at TYPE('x)"
   shows "supp x = {x}"
-proof (simp add: supp_def Collect_conj_eq Collect_imp_eq at_calc[OF at], auto)
-  assume f: "finite {b::'x. b \<noteq> x}"
-  have a1: "{b::'x. b \<noteq> x} = UNIV-{x}" by force
-  have a2: "infinite (UNIV::'x set)" by (rule at4[OF at])
-  from f a1 a2 show False by force
-qed
+by(auto simp: supp_def Collect_conj_eq Collect_imp_eq at_calc[OF at] at4[OF at])
 
 lemma at_fresh:
   fixes a :: "'x"
@@ -1791,8 +1786,8 @@
     by (simp add: finite_infinite[OF f3,OF at4[OF at], simplified])
   hence "infinite ({c. [(a,c)]\<bullet>x = x \<and> [(b,c)]\<bullet>x = x}-{a,b})" 
     by (force dest: Diff_infinite_finite)
-  hence "({c. [(a,c)]\<bullet>x = x \<and> [(b,c)]\<bullet>x = x}-{a,b}) \<noteq> {}" 
-    by (auto iff del: finite_Diff_insert Diff_eq_empty_iff)
+  hence "({c. [(a,c)]\<bullet>x = x \<and> [(b,c)]\<bullet>x = x}-{a,b}) \<noteq> {}"
+    by (metis Collect_def finite_set set_empty2)
   hence "\<exists>c. c\<in>({c. [(a,c)]\<bullet>x = x \<and> [(b,c)]\<bullet>x = x}-{a,b})" by (force)
   then obtain c 
     where eq1: "[(a,c)]\<bullet>x = x" 

--- a/src/HOL/Set.thy	Sat Feb 14 01:23:38 2009 -0800
+++ b/src/HOL/Set.thy	Sat Feb 14 01:24:01 2009 -0800
@@ -787,6 +787,9 @@
 
 lemma set_diff_eq: "A - B = {x. x : A & ~ x : B}" by blast
 
+lemma Compl_eq_Diff_UNIV: "-A = (UNIV - A)"
+by blast
+
 
 subsubsection {* Augmenting a set -- insert *}