--- a/Nominal/Nominal2_Abs.thy	Thu Jun 09 15:34:51 2011 +0900
+++ b/Nominal/Nominal2_Abs.thy	Fri Jun 10 15:30:09 2011 +0900
@@ -1011,9 +1011,61 @@
 
 lemma prod_alpha_eq:
   shows "prod_alpha (op=) (op=) = (op=)"
-unfolding prod_alpha_def
-by (auto intro!: ext)
+  unfolding prod_alpha_def
+  by (auto intro!: ext)
+
+lemma Abs_lst1_fcb:
+  fixes x y :: "'a :: at_base"
+    and S T :: "'b :: fs"
+  assumes e: "(Abs_lst [atom x] T) = (Abs_lst [atom y] S)"
+  and f1: "x \<noteq> y \<Longrightarrow> atom y \<sharp> T \<Longrightarrow> atom x \<sharp> (atom y \<rightleftharpoons> atom x) \<bullet> T \<Longrightarrow> atom x \<sharp> f x T"
+  and f2: "x \<noteq> y \<Longrightarrow> atom y \<sharp> T \<Longrightarrow> atom x \<sharp> (atom y \<rightleftharpoons> atom x) \<bullet> T \<Longrightarrow> atom y \<sharp> f x T"
+  and p: "S = (atom x \<rightleftharpoons> atom y) \<bullet> T \<Longrightarrow> x \<noteq> y \<Longrightarrow> atom y \<sharp> T \<Longrightarrow> atom x \<sharp> S \<Longrightarrow> (atom x \<rightleftharpoons> atom y) \<bullet> (f x T) = f y S"
+  and s: "sort_of (atom x) = sort_of (atom y)"
+  shows "f x T = f y S"
+  using e
+  apply(case_tac "atom x \<sharp> S")
+  apply(simp add: Abs1_eq_iff'[OF s s])
+  apply(elim conjE disjE)
+  apply(simp)
+  apply(rule trans)
+  apply(rule_tac p="(atom x \<rightleftharpoons> atom y)" in supp_perm_eq[symmetric])
+  apply(rule fresh_star_supp_conv)
+  apply(simp add: supp_swap fresh_star_def s f1 f2)
+  apply(simp add: swap_commute p)
+  apply(simp add: Abs1_eq_iff[OF s s])
+  done
 
+lemma Abs_res_fcb:
+  fixes xs ys :: "('a :: at_base) set"
+    and S T :: "'b :: fs"
+  assumes e: "(Abs_res (atom ` xs) T) = (Abs_res (atom ` ys) S)"
+    and f1: "\<And>x. x \<in> atom ` xs \<Longrightarrow> x \<in> supp T \<Longrightarrow> x \<sharp> f xs T"
+    and f2: "\<And>x. supp T - atom ` xs = supp S - atom ` ys \<Longrightarrow> x \<in> atom ` ys \<Longrightarrow> x \<in> supp S \<Longrightarrow> x \<sharp> f xs T"
+    and eqv: "\<And>p. p \<bullet> T = S \<Longrightarrow> supp p \<subseteq> atom ` xs \<inter> supp T \<union> atom ` ys \<inter> supp S
+               \<Longrightarrow> p \<bullet> (atom ` xs \<inter> supp T) = atom ` ys \<inter> supp S \<Longrightarrow> p \<bullet> (f xs T) = f ys S"
+  shows "f xs T = f ys S"
+  using e apply -
+  apply(subst (asm) Abs_eq_res_set)
+  apply(subst (asm) Abs_eq_iff2)
+  apply(simp add: alphas)
+  apply(elim exE conjE)
+  apply(rule trans)
+  apply(rule_tac p="p" in supp_perm_eq[symmetric])
+  apply(rule fresh_star_supp_conv)
+  apply(drule fresh_star_perm_set_conv)
+  apply(rule finite_Diff)
+  apply(rule finite_supp)
+  apply(subgoal_tac "(atom ` xs \<inter> supp T \<union> atom ` ys \<inter> supp S) \<sharp>* f xs T")
+  apply(metis Un_absorb2 fresh_star_Un)
+  apply(subst fresh_star_Un)
+  apply(rule conjI)
+  apply(simp add: fresh_star_def f1)
+  apply(subgoal_tac "supp T - atom ` xs = supp S - atom ` ys")
+  apply(simp add: fresh_star_def f2)
+  apply(blast)
+  apply(simp add: eqv)
+  done
 
 end