--- a/Nominal/Ex/Let.thy	Tue Jun 28 14:45:30 2011 +0900
+++ b/Nominal/Ex/Let.thy	Wed Jun 29 00:48:50 2011 +0100
@@ -18,9 +18,12 @@
   "bn ANil = []"
 | "bn (ACons x t as) = (atom x) # (bn as)"
 
+print_theorems
+
 thm trm_assn.fv_defs
 thm trm_assn.eq_iff 
 thm trm_assn.bn_defs
+thm trm_assn.bn_inducts
 thm trm_assn.perm_simps
 thm trm_assn.induct
 thm trm_assn.inducts
@@ -30,6 +33,17 @@
 thm trm_assn.exhaust
 thm trm_assn.strong_exhaust
 
+lemma bn_inj:
+  assumes a: "alpha_bn_raw x y"
+  shows "bn_raw x = bn_raw y \<Longrightarrow> x = y"
+using a
+apply(induct)
+apply(auto)[6]
+apply(simp)
+apply(simp)
+oops
+  
+
 
 lemma lets_bla:
   "x \<noteq> z \<Longrightarrow> y \<noteq> z \<Longrightarrow> x \<noteq> y \<Longrightarrow>(Let (ACons x (Var y) ANil) (Var x)) \<noteq> (Let (ACons x (Var z) ANil) (Var x))"
@@ -96,17 +110,19 @@
 (* TODO: should be provided by nominal *)
 lemmas [eqvt] = trm_assn.fv_bn_eqvt
 
+thm Abs_lst_fcb
+
+(*
 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 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)"
@@ -123,8 +139,8 @@
     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
+    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"
@@ -142,10 +158,6 @@
     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)
@@ -155,32 +167,32 @@
     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
+  then have "set (r \<bullet> (ba bs)) \<sharp>* f (r \<bullet> bs) (r \<bullet> y) c" using qq1 qq2
     by simp
-  then have "r \<bullet> ((set (ba bs)) \<sharp>* f bs y c)"
+  then have "r \<bullet> ((set (ba bs)) \<sharp>* f (ba 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)"
+  then have fcb2: "(set (ba bs)) \<sharp>* f (ba bs) y c" by (simp add: permute_bool_def)
+  have "f (ba as) x c = q \<bullet> (f (ba 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" 
+  also have "\<dots> = f (q \<bullet> (ba 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)"
+  also have "\<dots> = f (r \<bullet> (ba bs)) (r \<bullet> y) c" using qq1 qq2 by simp
+  also have "\<dots> = r \<bullet> (f (ba bs) y c)"
     apply(rule perm2[symmetric])
     using qq3 fresh2 fr1 by (auto simp add: fresh_star_def)
-  also have "... = f bs y c"
+  also have "... = f (ba 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"
 and height_assn :: "assn \<Rightarrow> nat"
@@ -200,13 +212,42 @@
   apply (drule_tac x="trm" in meta_spec)
   apply (simp add: alpha_bn_refl)
   apply (case_tac b rule: trm_assn.exhaust(2))
-  apply (auto)
-  apply (erule Abs_lst1_fcb)
-  apply (simp_all add: pure_fresh)
+  apply (auto)[2]
+  apply(simp_all)
+  thm  trm_assn.perm_bn_alpha trm_assn.permute_bn
+  apply (erule_tac c="()" in Abs_lst_fcb2)
+  apply (simp_all add: pure_fresh fresh_star_def)[3]
+  apply (simp add: eqvt_at_def)
   apply (simp add: eqvt_at_def)
-  apply (erule Abs_lst_fcb)
-  apply (simp_all add: pure_fresh)
-  apply (simp_all add: eqvt_at_def eqvts)
+  apply(erule conjE)
+  apply (simp add: meta_eq_to_obj_eq[OF height_trm_def, symmetric, unfolded fun_eq_iff])
+  apply (simp add: meta_eq_to_obj_eq[OF height_assn_def, symmetric, unfolded fun_eq_iff])
+  apply (subgoal_tac "eqvt_at height_assn as")
+  apply (subgoal_tac "eqvt_at height_assn asa")
+  apply (subgoal_tac "eqvt_at height_trm b")
+  apply (subgoal_tac "eqvt_at height_trm ba")
+  apply (thin_tac "eqvt_at height_trm_height_assn_sumC (Inr as)")
+  apply (thin_tac "eqvt_at height_trm_height_assn_sumC (Inr asa)")
+  apply (thin_tac "eqvt_at height_trm_height_assn_sumC (Inl b)")
+  apply (thin_tac "eqvt_at height_trm_height_assn_sumC (Inl ba)")
+  defer
+  apply (simp add: eqvt_at_def height_trm_def)
+  apply (simp add: eqvt_at_def height_trm_def)
+  apply (simp add: eqvt_at_def height_assn_def)
+  apply (simp add: eqvt_at_def height_assn_def)
+  apply (subgoal_tac "height_assn as = height_assn asa")
+  apply (subgoal_tac "height_trm b = height_trm ba")
+  apply simp
+  apply (erule_tac c="()" in Abs_lst_fcb2)
+  apply (simp_all add: pure_fresh fresh_star_def)[3]
+  apply (simp_all add: eqvt_at_def)[2]
+  apply (drule_tac c="()" in Abs_lst_fcb2)
+  apply (simp_all add: pure_fresh fresh_star_def)[3]
+  apply (simp_all add: eqvt_at_def)[2]
+  apply(simp add: eqvt_def)
+  apply(perm_simp)
+  apply(simp)
+  apply(simp add: inj_on_def)
   apply (rule arg_cong) back
   oops