--- 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
 

--- a/Nominal/Ex/LetRecB.thy	Tue Jun 28 14:45:30 2011 +0900
+++ b/Nominal/Ex/LetRecB.thy	Wed Jun 29 00:48:50 2011 +0100
@@ -181,6 +181,11 @@
   apply (simp add: eqvt_at_def)
   apply (simp add: eqvt_at_def)
 --""
+  apply(simp_all add: eqvt_def inj_on_def)
+  apply(perm_simp)
+  apply(simp)
+  apply(perm_simp)
+  apply(simp)
   done
 
 termination by lexicographic_order

--- a/Nominal/Nominal2.thy	Tue Jun 28 14:45:30 2011 +0900
+++ b/Nominal/Nominal2.thy	Wed Jun 29 00:48:50 2011 +0100
@@ -218,7 +218,7 @@
   val (_, lthy3) = Local_Theory.note ((Binding.empty, [eqvt_attr]), raw_perm_simps) lthy2a
 
   val _ = trace_msg (K "Defining raw fv- and bn-functions...")
-  val (raw_bns, raw_bn_defs, raw_bn_info, raw_bn_induct, lthy3a) =
+  val (raw_bns, raw_bn_defs, raw_bn_info, raw_bn_inducts, lthy3a) =
     define_raw_bns raw_dt_names raw_dts raw_bn_funs raw_bn_eqs 
       (raw_inject_thms @ raw_distinct_thms) raw_size_thms lthy3
     
@@ -235,19 +235,22 @@
   val (alpha_trms, alpha_bn_trms, alpha_intros, alpha_cases, alpha_induct, lthy4) =
     define_raw_alpha raw_dt_names raw_tys raw_cns_info raw_bn_info raw_bclauses raw_fvs lthy3c
     
+  val _ = tracing ("alpha_induct\n" ^ Syntax.string_of_term lthy3c (prop_of alpha_induct))
+  val _ = tracing ("alpha_intros\n" ^ cat_lines (map (Syntax.string_of_term lthy4 o prop_of) alpha_intros))
+
   val alpha_tys = map (domain_type o fastype_of) alpha_trms  
 
   val _ = trace_msg (K "Proving distinct theorems...")
   val alpha_distincts = 
-    mk_alpha_distincts lthy4 alpha_cases raw_distinct_thms alpha_trms raw_dt_names
+    raw_prove_alpha_distincts lthy4 alpha_cases raw_distinct_thms alpha_trms raw_dt_names
 
   val _ = trace_msg (K "Proving eq-iff theorems...")
   val alpha_eq_iff = 
-    mk_alpha_eq_iff lthy4 alpha_intros raw_distinct_thms raw_inject_thms alpha_cases
+    raw_prove_alpha_eq_iff lthy4 alpha_intros raw_distinct_thms raw_inject_thms alpha_cases
     
   val _ = trace_msg (K "Proving equivariance of bns, fvs, size and alpha...")
   val raw_bn_eqvt = 
-    raw_prove_eqvt raw_bns raw_bn_induct (raw_bn_defs @ raw_perm_simps) lthy4
+    raw_prove_eqvt raw_bns raw_bn_inducts (raw_bn_defs @ raw_perm_simps) lthy4
     
   (* noting the raw_bn_eqvt lemmas in a temprorary theory *)
   val lthy_tmp = snd (Local_Theory.note ((Binding.empty, [eqvt_attr]), raw_bn_eqvt) lthy4)
@@ -288,6 +291,8 @@
   val alpha_bn_imp_thms = 
     raw_prove_bn_imp alpha_trms alpha_bn_trms alpha_intros alpha_induct lthy5 
 
+  val _ = tracing ("alpha_bn_imp_thms:\n" ^ cat_lines (map (Syntax.string_of_term lthy5 o prop_of) alpha_bn_imp_thms))
+
   val _ = trace_msg (K "Proving respectfulness...")
   val raw_funs_rsp_aux = 
     raw_fv_bn_rsp_aux alpha_trms alpha_bn_trms raw_fvs 
@@ -381,7 +386,8 @@
   val eq_iff_simps = @{thms alphas permute_prod.simps prod_fv.simps prod_alpha_def prod_rel_def
     prod.cases} 
 
-  val ((((((qdistincts, qeq_iffs), qfv_defs), qbn_defs), qperm_simps), qfv_qbn_eqvts), lthyA) = 
+  val (((((((qdistincts, qeq_iffs), qfv_defs), qbn_defs), qperm_simps), qfv_qbn_eqvts), qbn_inducts), 
+    lthyA) = 
     lthy9a    
     |> lift_thms qtys [] alpha_distincts  
     ||>> lift_thms qtys eq_iff_simps alpha_eq_iff       
@@ -389,6 +395,7 @@
     ||>> lift_thms qtys [] raw_bn_defs
     ||>> lift_thms qtys [] raw_perm_simps
     ||>> lift_thms qtys [] (raw_fv_eqvt @ raw_bn_eqvt)
+    ||>> lift_thms qtys [] raw_bn_inducts
 
   val ((((((qsize_eqvt, [qinduct]), qexhausts), qsize_simps), qperm_bn_simps), qalpha_refl_thms), lthyB) = 
     lthyA 
@@ -399,7 +406,7 @@
     ||>> lift_thms qtys [] raw_perm_bn_simps
     ||>> lift_thms qtys [] alpha_refl_thms
 
-  val qinducts = Project_Rule.projections lthyA qinduct
+  val qinducts = Project_Rule.projections lthyB qinduct
 
   val _ = trace_msg (K "Proving supp lemmas and fs-instances...")
   val qsupports_thms =
@@ -470,8 +477,9 @@
      ||>> Local_Theory.note ((thms_suffix "eq_iff", [induct_attr, simp_attr]), qeq_iffs')
      ||>> Local_Theory.note ((thms_suffix "fv_defs", []), qfv_defs) 
      ||>> Local_Theory.note ((thms_suffix "bn_defs", []), qbn_defs) 
+     ||>> Local_Theory.note ((thms_suffix "bn_inducts", []), qbn_inducts) 
      ||>> Local_Theory.note ((thms_suffix "perm_simps", [eqvt_attr, simp_attr]), qperm_simps) 
-     ||>> Local_Theory.note ((thms_suffix "fv_bn_eqvt", []), qfv_qbn_eqvts) 
+     ||>> Local_Theory.note ((thms_suffix "fv_bn_eqvt", [eqvt_attr]), qfv_qbn_eqvts) 
      ||>> Local_Theory.note ((thms_suffix "size", [simp_attr]), qsize_simps)
      ||>> Local_Theory.note ((thms_suffix "size_eqvt", []), qsize_eqvt)
      ||>> Local_Theory.note ((thms_suffix "induct", [case_names_attr]), [qinduct]) 

--- a/Nominal/nominal_dt_alpha.ML	Tue Jun 28 14:45:30 2011 +0900
+++ b/Nominal/nominal_dt_alpha.ML	Wed Jun 29 00:48:50 2011 +0100
@@ -13,10 +13,10 @@
     bclause list list list -> term list -> Proof.context -> 
     term list * term list * thm list * thm list * thm * local_theory
 
-  val mk_alpha_distincts: Proof.context -> thm list -> thm list -> 
+  val raw_prove_alpha_distincts: Proof.context -> thm list -> thm list -> 
     term list -> string list -> thm list
 
-  val mk_alpha_eq_iff: Proof.context -> thm list -> thm list -> thm list -> 
+  val raw_prove_alpha_eq_iff: Proof.context -> thm list -> thm list -> thm list -> 
     thm list -> thm list
 
   val induct_prove: typ list -> (typ * (term -> term)) list -> thm -> 
@@ -297,7 +297,7 @@
   THEN_ALL_NEW asm_full_simp_tac (HOL_ss addsimps distinct_thms)
 
 
-fun mk_alpha_distincts ctxt cases_thms distinct_thms alpha_trms alpha_str =
+fun raw_prove_alpha_distincts ctxt cases_thms distinct_thms alpha_trms alpha_str =
   let
     val ty_trm_assoc = alpha_str ~~ (map (fst o dest_Const) alpha_trms)
 
@@ -341,7 +341,7 @@
     else HOLogic.mk_Trueprop (HOLogic.mk_eq (concl, list_conj hyps))
   end;
 
-fun mk_alpha_eq_iff ctxt alpha_intros distinct_thms inject_thms alpha_elims =
+fun raw_prove_alpha_eq_iff ctxt alpha_intros distinct_thms inject_thms alpha_elims =
   let
     val ((_, thms_imp), ctxt') = Variable.import false alpha_intros ctxt;
     val goals = map mk_alpha_eq_iff_goal thms_imp;