--- a/LamEx.thy	Sat Dec 05 23:35:09 2009 +0100
+++ b/LamEx.thy	Sun Dec 06 00:00:47 2009 +0100
@@ -190,14 +190,20 @@
 
 lemma fv_var: "fv (Var (a\<Colon>name)) = {a}"
 apply (tactic {* lift_tac_lam @{context} @{thm rfv_var} 1 *})
+apply (unfold fv_def[simplified id_simps])
+apply (tactic {* clean_tac @{context} 1 *})
 done
 
 lemma fv_app: "fv (App (x\<Colon>lam) (xa\<Colon>lam)) = fv x \<union> fv xa"
 apply (tactic {* lift_tac_lam @{context} @{thm rfv_app} 1 *})
+apply (unfold fv_def[simplified id_simps])
+apply (tactic {* clean_tac @{context} 1 *})
 done
 
 lemma fv_lam: "fv (Lam (a\<Colon>name) (x\<Colon>lam)) = fv x - {a}"
 apply (tactic {* lift_tac_lam @{context} @{thm rfv_lam} 1 *})
+apply (unfold fv_def[simplified id_simps])
+apply (tactic {* clean_tac @{context} 1 *})
 done
 
 lemma a1: "(a\<Colon>name) = (b\<Colon>name) \<Longrightarrow> Var a = Var b"
@@ -210,6 +216,8 @@
 
 lemma a3: "\<lbrakk>(x\<Colon>lam) = [(a\<Colon>name, b\<Colon>name)] \<bullet> (xa\<Colon>lam); a \<notin> fv (Lam b x)\<rbrakk> \<Longrightarrow> Lam a x = Lam b xa"
 apply (tactic {* lift_tac_lam @{context} @{thm a3} 1 *})
+apply (unfold fv_def[simplified id_simps])
+apply (tactic {* clean_tac @{context} 1 *})
 done
 
 lemma alpha_cases: "\<lbrakk>a1 = a2; \<And>a b. \<lbrakk>a1 = Var a; a2 = Var b; a = b\<rbrakk> \<Longrightarrow> P;
@@ -217,6 +225,8 @@
      \<And>x a b xa. \<lbrakk>a1 = Lam a x; a2 = Lam b xa; x = [(a, b)] \<bullet> xa; a \<notin> fv (Lam b x)\<rbrakk> \<Longrightarrow> P\<rbrakk>
     \<Longrightarrow> P"
 apply (tactic {* lift_tac_lam @{context} @{thm alpha.cases} 1 *})
+apply (unfold fv_def[simplified id_simps])
+apply (tactic {* clean_tac @{context} 1 *})
 done
 
 lemma alpha_induct: "\<lbrakk>(qx\<Colon>lam) = (qxa\<Colon>lam); \<And>(a\<Colon>name) b\<Colon>name. a = b \<Longrightarrow> (qxb\<Colon>lam \<Rightarrow> lam \<Rightarrow> bool) (Var a) (Var b);
@@ -225,6 +235,8 @@
         \<lbrakk>x = [(a, b)] \<bullet> xa; qxb x ([(a, b)] \<bullet> xa); a \<notin> fv (Lam b x)\<rbrakk> \<Longrightarrow> qxb (Lam a x) (Lam b xa)\<rbrakk>
     \<Longrightarrow> qxb qx qxa"
 apply (tactic {* lift_tac_lam @{context} @{thm alpha.induct} 1 *})
+apply (unfold fv_def[simplified id_simps])
+apply (tactic {* clean_tac @{context} 1 *})
 done
 
 lemma var_inject: "(Var a = Var b) = (a = b)"