--- a/Nominal/Nominal2_Abs.thy	Mon Jul 04 23:56:19 2011 +0200
+++ b/Nominal/Nominal2_Abs.thy	Tue Jul 05 04:18:45 2011 +0200
@@ -1013,112 +1013,6 @@
   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_lst_fcb:
-  fixes xs ys :: "'a :: fs"
-    and S T :: "'b :: fs"
-  assumes e: "(Abs_lst (ba xs) T) = (Abs_lst (ba ys) S)"
-    and f1: "\<And>x. x \<in> set (ba xs) \<Longrightarrow> x \<sharp> f xs T"
-    and f2: "\<And>x. supp T - set (ba xs) = supp S - set (ba ys) \<Longrightarrow> x \<in> set (ba ys) \<Longrightarrow> x \<sharp> f xs T"
-    and eqv: "\<And>p. p \<bullet> T = S \<Longrightarrow> p \<bullet> ba xs = ba ys \<Longrightarrow> supp p \<subseteq> set (ba xs) \<union> set (ba ys) \<Longrightarrow> p \<bullet> (f xs T) = f ys S"
-  shows "f xs T = f ys S"
-  using e apply -
-  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 "(set (ba xs) \<union> set (ba ys)) \<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(simp add: fresh_star_def f2)
-  apply(simp add: eqv)
-  done
-
-lemma Abs_set_fcb:
-  fixes xs ys :: "'a :: fs"
-    and S T :: "'b :: fs"
-  assumes e: "(Abs_set (ba xs) T) = (Abs_set (ba ys) S)"
-    and f1: "\<And>x. x \<in> ba xs \<Longrightarrow> x \<sharp> f xs T"
-    and f2: "\<And>x. supp T - ba xs = supp S - ba ys \<Longrightarrow> x \<in> ba ys \<Longrightarrow> x \<sharp> f xs T"
-    and eqv: "\<And>p. p \<bullet> T = S \<Longrightarrow> p \<bullet> ba xs = ba ys \<Longrightarrow> supp p \<subseteq> ba xs \<union> ba ys \<Longrightarrow> p \<bullet> (f xs T) = f ys S"
-  shows "f xs T = f ys S"
-  using e apply -
-  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 "(ba xs \<union> ba ys) \<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(simp add: fresh_star_def f2)
-  apply(simp add: eqv)
-  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