--- a/Nominal/Ex/Let.thy	Tue Jun 28 14:30:30 2011 +0900
+++ b/Nominal/Ex/Let.thy	Tue Jun 28 14:45:30 2011 +0900
@@ -96,6 +96,90 @@
 (* TODO: should be provided by nominal *)
 lemmas [eqvt] = trm_assn.fv_bn_eqvt
 
+lemma Abs_lst_fcb2:
+  fixes as bs :: "'a :: fs"
+    and x y :: "'b :: fs"
+    and c::"'c::fs"
+  assumes eq: "[ba as]lst. x = [ba bs]lst. y"
+  and fcb1: "(set (ba as)) \<sharp>* f as x c"
+  and fresh1: "set (ba as) \<sharp>* c"
+  and fresh2: "set (ba bs) \<sharp>* c"
+  and perm1: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f as x c) = f (p \<bullet> as) (p \<bullet> x) c"
+  and perm2: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f bs y c) = f (p \<bullet> bs) (p \<bullet> y) c"
+  and props: "eqvt ba" "inj ba"
+  shows "f as x c = f bs y c"
+proof -
+  have "supp (as, x, c) supports (f as x c)"
+    unfolding  supports_def fresh_def[symmetric]
+    by (simp add: fresh_Pair perm1 fresh_star_def supp_swap swap_fresh_fresh)
+  then have fin1: "finite (supp (f as x c))"
+    by (auto intro: supports_finite simp add: finite_supp)
+  have "supp (bs, y, c) supports (f bs y c)"
+    unfolding  supports_def fresh_def[symmetric]
+    by (simp add: fresh_Pair perm2 fresh_star_def supp_swap swap_fresh_fresh)
+  then have fin2: "finite (supp (f bs y c))"
+    by (auto intro: supports_finite simp add: finite_supp)
+  obtain q::"perm" where 
+    fr1: "(q \<bullet> (set (ba as))) \<sharp>* (x, c, f as x c, f bs y c)" and 
+    fr2: "supp q \<sharp>* ([ba as]lst. x)" and 
+    inc: "supp q \<subseteq> (set (ba as)) \<union> q \<bullet> (set (ba as))"
+    using at_set_avoiding3[where xs="set (ba as)" and c="(x, c, f as x c, f bs y c)" and x="[ba as]lst. x"]  
+      fin1 fin2
+    by (auto simp add: supp_Pair finite_supp Abs_fresh_star dest: fresh_star_supp_conv)
+  have "[q \<bullet> (ba as)]lst. (q \<bullet> x) = q \<bullet> ([ba as]lst. x)" by simp
+  also have "\<dots> = [ba as]lst. x"
+    by (simp only: fr2 perm_supp_eq)
+  finally have "[q \<bullet> (ba as)]lst. (q \<bullet> x) = [ba bs]lst. y" using eq by simp
+  then obtain r::perm where 
+    qq1: "q \<bullet> x = r \<bullet> y" and 
+    qq2: "q \<bullet> (ba as) = r \<bullet> (ba bs)" and 
+    qq3: "supp r \<subseteq> (q \<bullet> (set (ba as))) \<union> set (ba bs)"
+    apply(drule_tac sym)
+    apply(simp only: Abs_eq_iff2 alphas)
+    apply(erule exE)
+    apply(erule conjE)+
+    apply(drule_tac x="p" in meta_spec)
+    apply(simp add: set_eqvt)
+    apply(blast)
+    done
+  have qq4: "q \<bullet> as = r \<bullet> bs" using qq2 props unfolding eqvt_def inj_on_def
+    apply(perm_simp)
+    apply(simp)
+    done
+  have "(set (ba as)) \<sharp>* f as x c" by (rule fcb1)
+  then have "q \<bullet> ((set (ba as)) \<sharp>* f as x c)"
+    by (simp add: permute_bool_def)
+  then have "set (q \<bullet> (ba as)) \<sharp>* f (q \<bullet> as) (q \<bullet> x) c"
+    apply(simp add: fresh_star_eqvt set_eqvt)
+    apply(subst (asm) perm1)
+    using inc fresh1 fr1
+    apply(auto simp add: fresh_star_def fresh_Pair)
+    done
+  then have "set (r \<bullet> (ba bs)) \<sharp>* f (r \<bullet> bs) (r \<bullet> y) c" using qq1 qq2 qq4
+    by simp
+  then have "r \<bullet> ((set (ba bs)) \<sharp>* f bs y c)"
+    apply(simp add: fresh_star_eqvt set_eqvt)
+    apply(subst (asm) perm2[symmetric])
+    using qq3 fresh2 fr1
+    apply(auto simp add: set_eqvt fresh_star_def fresh_Pair)
+    done
+  then have fcb2: "(set (ba bs)) \<sharp>* f bs y c" by (simp add: permute_bool_def)
+  have "f as x c = q \<bullet> (f as x c)"
+    apply(rule perm_supp_eq[symmetric])
+    using inc fcb1 fr1 by (auto simp add: fresh_star_def)
+  also have "\<dots> = f (q \<bullet> as) (q \<bullet> x) c" 
+    apply(rule perm1)
+    using inc fresh1 fr1 by (auto simp add: fresh_star_def)
+  also have "\<dots> = f (r \<bullet> bs) (r \<bullet> y) c" using qq1 qq4 by simp
+  also have "\<dots> = r \<bullet> (f bs y c)"
+    apply(rule perm2[symmetric])
+    using qq3 fresh2 fr1 by (auto simp add: fresh_star_def)
+  also have "... = f bs y c"
+    apply(rule perm_supp_eq)
+    using qq3 fr1 fcb2 by (auto simp add: fresh_star_def)
+  finally show ?thesis by simp
+qed
+
 (* PROBLEM: the proof needs induction on alpha_bn inside which is not possible... *)
 nominal_primrec
     height_trm :: "trm \<Rightarrow> nat"
@@ -154,12 +238,9 @@
   apply auto
   apply (simp_all add: meta_eq_to_obj_eq[OF subst_def, symmetric, unfolded fun_eq_iff])
   apply (simp_all add: meta_eq_to_obj_eq[OF substa_def, symmetric, unfolded fun_eq_iff])
-  (*apply (erule Abs_lst1_fcb)*)
-  prefer 3
-  apply (erule alpha_bn_inducts)
-  apply (simp add: alpha_bn_refl)
-  (* Needs an invariant *)
-  oops
+  prefer 2
+  apply (erule_tac Abs_lst_fcb2)
+ oops
 
 end