--- a/Nominal/Ex/SingleLet.thy	Sat Aug 28 18:15:23 2010 +0800
+++ b/Nominal/Ex/SingleLet.thy	Sun Aug 29 00:09:45 2010 +0800
@@ -21,6 +21,8 @@
 where
   "bn (As x y t) = {atom x}"
 
+thm Ball_def Bex_def mem_def
+
 thm single_let.distinct
 thm single_let.induct
 thm single_let.exhaust
@@ -30,19 +32,88 @@
 thm single_let.eq_iff
 thm single_let.fv_bn_eqvt
 thm single_let.size_eqvt
-
+thm single_let.supports
 
-(*
+lemma test:
+  "finite (supp (t::trm)) \<and> finite (supp (a::assg))"
+apply(rule single_let.induct)
+apply(rule supports_finite)
+apply(rule single_let.supports)
+apply(simp only: finite_supp supp_Pair finite_Un)
+apply(rule supports_finite)
+apply(rule single_let.supports)
+apply(simp only: finite_supp supp_Pair finite_Un, simp)
+apply(rule supports_finite)
+apply(rule single_let.supports)
+apply(simp only: finite_supp supp_Pair finite_Un, simp)
+apply(rule supports_finite)
+apply(rule single_let.supports)
+apply(simp only: finite_supp supp_Pair finite_Un, simp)
+apply(rule supports_finite)
+apply(rule single_let.supports)
+apply(simp only: finite_supp supp_Pair finite_Un, simp)
+apply(rule supports_finite)
+apply(rule single_let.supports)
+apply(simp only: finite_supp supp_Pair finite_Un, simp)
+apply(rule supports_finite)
+apply(rule single_let.supports)
+apply(simp only: finite_supp supp_Pair finite_Un, simp)
+apply(rule supports_finite)
+apply(rule single_let.supports)
+apply(simp only: finite_supp supp_Pair finite_Un, simp)
+done
+
+instantiation trm and assg :: fs
+begin
+
+instance
+apply(default)
+apply(simp_all add: test)
+done
+
+end
 
 
+
+
+lemma test: 
+  "(\<exists>p. (bs, x) \<approx>lst (op=) f p (cs, y)) \<longleftrightarrow> (\<exists>p. (bs, x) \<approx>lst (op=) supp p (cs, y))"
+oops
+
+lemma Abs_eq_iff:
+  shows "Abs bs x = Abs cs y \<longleftrightarrow> (\<exists>p. (bs, x) \<approx>gen (op =) supp p (cs, y))"
+  and   "Abs_res bs x = Abs_res cs y \<longleftrightarrow> (\<exists>p. (bs, x) \<approx>res (op =) supp p (cs, y))"
+  and   "Abs_lst bsl x = Abs_lst csl y \<longleftrightarrow> (\<exists>p. (bsl, x) \<approx>lst (op =) supp p (csl, y))"
+  by (lifting alphas_abs)
+
+(*
 lemma supp_fv:
-  "supp t = fv_trm t"
-  "supp b = fv_bn b"
-apply(induct t and b rule: i1)
-apply(simp_all add: f1)
-apply(simp_all add: supp_def)
-apply(simp_all add: b1)
+  "supp t = fv_trm t \<and> supp b = fv_bn b"
+apply(rule single_let.induct)
+apply(simp_all only: single_let.fv_defs)[2]
+apply(simp_all only: supp_def)[2]
+apply(simp_all only: single_let.perm_simps)[2]
+apply(simp_all only: single_let.eq_iff)[2]
+apply(simp_all only: de_Morgan_conj)[2]
+apply(simp_all only: Collect_disj_eq)[2]
+apply(simp_all only: finite_Un)[2]
+apply(simp_all only: de_Morgan_conj)[2]
+apply(simp_all only: Collect_disj_eq)[2]
+apply(subgoal_tac "supp (Lam name trm) = supp (Abs_lst [atom name] trm)")
+apply(simp only: single_let.fv_defs)
+apply(simp only: supp_abs)
+apply(simp (no_asm) only: supp_def)
+apply(simp only: single_let.perm_simps)
+apply(simp only: single_let.eq_iff)
+apply(subst test)
+apply(simp only: Abs_eq_iff[symmetric])
+apply(simp only: alphas_abs[symmetric])
+apply(simp only: eqvts)
+thm Abs_eq_iff
+apply(simp only: alphas)
 sorry
+*)
+(*
 
 consts perm_bn_trm :: "perm \<Rightarrow> trm \<Rightarrow> trm"
 consts perm_bn_assg :: "perm \<Rightarrow> assg \<Rightarrow> assg"