1066 lemma Abs_set_fcb:

  1067   fixes xs ys :: "'a :: fs"

  1068     and S T :: "'b :: fs"

  1069   assumes e: "(Abs_set (ba xs) T) = (Abs_set (ba ys) S)"

  1070     and f1: "\<And>x. x \<in> ba xs \<Longrightarrow> x \<sharp> f xs T"

  1071     and f2: "\<And>x. supp T - ba xs = supp S - ba ys \<Longrightarrow> x \<in> ba ys \<Longrightarrow> x \<sharp> f xs T"

  1072     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"

  1073   shows "f xs T = f ys S"

  1074   using e apply -

  1075   apply(subst (asm) Abs_eq_iff2)

  1076   apply(simp add: alphas)

  1077   apply(elim exE conjE)

  1078   apply(rule trans)

  1079   apply(rule_tac p="p" in supp_perm_eq[symmetric])

  1080   apply(rule fresh_star_supp_conv)

  1081   apply(drule fresh_star_perm_set_conv)

  1082   apply(rule finite_Diff)

  1083   apply(rule finite_supp)

  1084   apply(subgoal_tac "(ba xs \<union> ba ys) \<sharp>* f xs T")

  1085   apply(metis Un_absorb2 fresh_star_Un)

  1086   apply(subst fresh_star_Un)

  1087   apply(rule conjI)

  1088   apply(simp add: fresh_star_def f1)

  1089   apply(simp add: fresh_star_def f2)

  1090   apply(simp add: eqv)

  1091   done

  1092