1039 lemma Abs_lst_fcb:

  1040   fixes xs ys :: "'a :: fs"

  1041     and S T :: "'b :: fs"

  1042   assumes e: "(Abs_lst (ba xs) T) = (Abs_lst (ba ys) S)"

  1043     and f1: "\<And>x. x \<in> set (ba xs) \<Longrightarrow> x \<sharp> f xs T"

  1044     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"

  1045     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"

  1046   shows "f xs T = f ys S"

  1047   using e apply -

  1048   apply(subst (asm) Abs_eq_iff2)

  1049   apply(simp add: alphas)

  1050   apply(elim exE conjE)

  1051   apply(rule trans)

  1052   apply(rule_tac p="p" in supp_perm_eq[symmetric])

  1053   apply(rule fresh_star_supp_conv)

  1054   apply(drule fresh_star_perm_set_conv)

  1055   apply(rule finite_Diff)

  1056   apply(rule finite_supp)

  1057   apply(subgoal_tac "(set (ba xs) \<union> set (ba ys)) \<sharp>* f xs T")

  1058   apply(metis Un_absorb2 fresh_star_Un)

  1059   apply(subst fresh_star_Un)

  1060   apply(rule conjI)

  1061   apply(simp add: fresh_star_def f1)

  1062   apply(simp add: fresh_star_def f2)

  1063   apply(simp add: eqv)

  1064   done

  1065