diff -r fb201e383f1b -r da575186d492 Attic/Fv.thy
--- a/Attic/Fv.thy	Tue Feb 19 05:38:46 2013 +0000
+++ /dev/null	Thu Jan 01 00:00:00 1970 +0000
@@ -1,678 +0,0 @@
-theory Fv
-imports "../Nominal-General/Nominal2_Atoms" 
-        "Abs" "Perm" "Rsp" "Nominal2_FSet"
-begin
-
-(* The bindings data structure:
-
-  Bindings are a list of lists of lists of triples.
-
-   The first list represents the datatypes defined.
-   The second list represents the constructors.
-   The internal list is a list of all the bndings that
-   concern the constructor.
-
-   Every triple consists of a function, the binding and
-   the body.
-
-  Eg:
-nominal_datatype
-
-   C1
- | C2 x y z bind x in z
- | C3 x y z bind f x in z bind g y in z
-
-yields:
-[
- [],
- [(NONE, 0, 2)],
- [(SOME (Const f), 0, 2), (Some (Const g), 1, 2)]]
-
-A SOME binding has to have a function which takes an appropriate
-argument and returns an atom set. A NONE binding has to be on an
-argument that is an atom or an atom set.
-*)
-
-(*
-An overview of the generation of free variables:
-
-1) fv_bn functions are generated only for the non-recursive binds.
-
-   An fv_bn for a constructor is a union of values for the arguments:
-
-   For an argument x that is in the bn function
-   - if it is a recursive argument bn' we return: fv_bn' x
-   - otherwise empty
-
-   For an argument x that is not in the bn function
-   - for atom we return: {atom x}
-   - for atom set we return: atom ` x
-   - for a recursive call to type ty' we return: fv_ty' x
-     with fv of the appropriate type
-   - otherwise empty
-
-2) fv_ty functions generated for all types being defined:
-
-   fv_ty for a constructor is a union of values for the arguments.
-
-   For an argument that is bound in a shallow binding we return empty.
-
-   For an argument x that bound in a non-recursive deep binding
-   we return: fv_bn x.
-
-   Otherwise we return the free variables of the argument minus the
-   bound variables of the argument.
-
-   The free variables for an argument x are:
-   - for an atom: {atom x}
-   - for atom set: atom ` x
-   - for recursive call to type ty' return: fv_ty' x
-   - for nominal datatype ty' return: fv_ty' x
-
-   The bound variables are a union of results of all bindings that
-   involve the given argument. For a paricular binding:
-
-   - for a binding function bn: bn x
-   - for a recursive argument of type ty': fv_fy' x
-   - for nominal datatype ty' return: fv_ty' x
-*)
-
-(*
-An overview of the generation of alpha-equivalence:
-
-1) alpha_bn relations are generated for binding functions.
-
-   An alpha_bn for a constructor is true if a conjunction of
-   propositions for each argument holds.
-
-   For an argument a proposition is build as follows from
-   th:
-
-   - for a recursive argument in the bn function, we return: alpha_bn argl argr
-   - for a recursive argument for type ty not in bn, we return: alpha_ty argl argr
-   - for other arguments in the bn function we return: True
-   - for other arguments not in the bn function we return: argl = argr
-
-2) alpha_ty relations are generated for all the types being defined:
-
-   For each constructor we gather all the arguments that are bound,
-   and for each of those we add a permutation. We associate those
-   permutations with the bindings. Note that two bindings can have
-   the same permutation if the arguments being bound are the same.
-
-   An alpha_ty for a constructor is true if there exist permutations
-   as above such that a conjunction of propositions for all arguments holds.
-
-   For an argument we allow bindings where only one of the following
-   holds:
-
-   - Argument is bound in some shallow bindings: We return true
-   - Argument of type ty is bound recursively in some other
-     arguments [i1, .. in] with one binding function bn.
-     We return:
-
-     (bn argl, (argl, argl_i1, ..., argl_in)) \<approx>gen
-     \<lambda>(argl,argl1,..,argln) (argr,argr1,..,argrn). 
-         (alpha_ty argl argr) \<and> (alpha_i1 argl1 argr1) \<and> .. \<and> (alpha_in argln argrn)
-     \<lambda>(arg,arg1,..,argn). (fv_ty arg) \<union> (fv_i1 arg1) \<union> .. \<union> (fv_in argn)
-     pi
-     (bn argr, (argr, argr_i1, ..., argr_in))
-
-   - Argument is bound in some deep non-recursive bindings.
-     We return: alpha_bn argl argr
-   - Argument of type ty has some shallow bindings [b1..bn] and/or
-     non-recursive bindings [f1 a1, .., fm am], where the bindings
-     have the permutations p1..pl. We return:
-
-     (b1l \<union>..\<union> bnl \<union> f1 a1l \<union>..\<union> fn anl, argl) \<approx>gen
-     alpha_ty fv_ty (p1 +..+ pl)
-     (b1r \<union>..\<union> bnr \<union> f1 a1r \<union>..\<union> fn anr, argr)
-
-   - Argument has some recursive bindings. The bindings were
-     already treated in 2nd case so we return: True
-   - Argument has no bindings and is not bound.
-     If it is recursive for type ty, we return: alpha_ty argl argr
-     Otherwise we return: argl = argr
-
-*)
-
-
-ML {*
-datatype alpha_mode = AlphaGen | AlphaRes | AlphaLst;
-*}
-
-ML {*
-fun atyp_const AlphaGen = @{const_name alpha_gen}
-  | atyp_const AlphaRes = @{const_name alpha_res}
-  | atyp_const AlphaLst = @{const_name alpha_lst}
-*}
-
-(* TODO: make sure that parser checks that bindings are compatible *)
-ML {*
-fun alpha_const_for_binds [] = atyp_const AlphaGen
-  | alpha_const_for_binds ((NONE, _, _, at) :: t) = atyp_const at
-  | alpha_const_for_binds ((SOME (_, _), _, _, at) :: _) = atyp_const at
-*}
-
-ML {*
-fun is_atom thy typ =
-  Sign.of_sort thy (typ, @{sort at})
-
-fun is_atom_set thy (Type ("fun", [t, @{typ bool}])) = is_atom thy t
-  | is_atom_set _ _ = false;
-
-fun is_atom_fset thy (Type ("FSet.fset", [t])) = is_atom thy t
-  | is_atom_fset _ _ = false;
-*}
-
-
-(* Like map2, only if the second list is empty passes empty lists insted of error *)
-ML {*
-fun map2i _ [] [] = []
-  | map2i f (x :: xs) (y :: ys) = f x y :: map2i f xs ys
-  | map2i f (x :: xs) [] = f x [] :: map2i f xs []
-  | map2i _ _ _ = raise UnequalLengths;
-*}
-
-(* Finds bindings with the same function and binding, and gathers all
-   bodys for such pairs
- *)
-ML {*
-fun gather_binds binds =
-let
-  fun gather_binds_cons binds =
-    let
-      val common = map (fn (f, bi, _, aty) => (f, bi, aty)) binds
-      val nodups = distinct (op =) common
-      fun find_bodys (sf, sbi, sty) =
-        filter (fn (f, bi, _, aty) => f = sf andalso bi = sbi andalso aty = sty) binds
-      val bodys = map ((map (fn (_, _, bo, _) => bo)) o find_bodys) nodups
-    in
-      nodups ~~ bodys
-    end
-in
-  map (map gather_binds_cons) binds
-end
-*}
-
-ML {*
-fun un_gather_binds_cons binds =
-  flat (map (fn (((f, bi, aty), bos), pi) => map (fn bo => ((f, bi, bo, aty), pi)) bos) binds)
-*}
-
-ML {*
-  open Datatype_Aux; (* typ_of_dtyp, DtRec, ... *);
-*}
-ML {*
-  (* TODO: It is the same as one in 'nominal_atoms' *)
-  fun mk_atom ty = Const (@{const_name atom}, ty --> @{typ atom});
-  val noatoms = @{term "{} :: atom set"};
-  fun mk_single_atom x = HOLogic.mk_set @{typ atom} [mk_atom (type_of x) $ x];
-  fun mk_union sets =
-    fold (fn a => fn b =>
-      if a = noatoms then b else
-      if b = noatoms then a else
-      if a = b then a else
-      HOLogic.mk_binop @{const_name sup} (a, b)) (rev sets) noatoms;
-  val mk_inter = foldr1 (HOLogic.mk_binop @{const_name inf})
-  fun mk_diff a b =
-    if b = noatoms then a else
-    if b = a then noatoms else
-    HOLogic.mk_binop @{const_name minus} (a, b);
-  fun mk_atom_set t =
-    let
-      val ty = fastype_of t;
-      val atom_ty = HOLogic.dest_setT ty --> @{typ atom};
-      val img_ty = atom_ty --> ty --> @{typ "atom set"};
-    in
-      (Const (@{const_name image}, img_ty) $ Const (@{const_name atom}, atom_ty) $ t)
-    end;
-  fun mk_atom_fset t =
-    let
-      val ty = fastype_of t;
-      val atom_ty = dest_fsetT ty --> @{typ atom};
-      val fmap_ty = atom_ty --> ty --> @{typ "atom fset"};
-      val fset_to_set = @{term "fset_to_set :: atom fset \<Rightarrow> atom set"}
-    in
-      fset_to_set $ ((Const (@{const_name fmap}, fmap_ty) $ Const (@{const_name atom}, atom_ty) $ t))
-    end;
-  (* Similar to one in USyntax *)
-  fun mk_pair (fst, snd) =
-    let val ty1 = fastype_of fst
-      val ty2 = fastype_of snd
-      val c = HOLogic.pair_const ty1 ty2
-    in c $ fst $ snd
-    end;
-*}
-
-(* Given [fv1, fv2, fv3] creates %(x, y, z). fv1 x u fv2 y u fv3 z *)
-ML {*
-fun mk_compound_fv fvs =
-let
-  val nos = (length fvs - 1) downto 0;
-  val fvs_applied = map (fn (fv, no) => fv $ Bound no) (fvs ~~ nos);
-  val fvs_union = mk_union fvs_applied;
-  val (tyh :: tys) = rev (map (domain_type o fastype_of) fvs);
-  fun fold_fun ty t = HOLogic.mk_split (Abs ("", ty, t))
-in
-  fold fold_fun tys (Abs ("", tyh, fvs_union))
-end;
-*}
-
-(* Given [R1, R2, R3] creates %(x,x'). %(y,y'). %(z,z'). R x x' \<and> R y y' \<and> R z z' *)
-ML {*
-fun mk_compound_alpha Rs =
-let
-  val nos = (length Rs - 1) downto 0;
-  val nos2 = (2 * length Rs - 1) downto length Rs;
-  val Rs_applied = map (fn (R, (no2, no)) => R $ Bound no2 $ Bound no) (Rs ~~ (nos2 ~~ nos));
-  val Rs_conj = mk_conjl Rs_applied;
-  val (tyh :: tys) = rev (map (domain_type o fastype_of) Rs);
-  fun fold_fun ty t = HOLogic.mk_split (Abs ("", ty, t))
-  val abs_rhs = fold fold_fun tys (Abs ("", tyh, Rs_conj))
-in
-  fold fold_fun tys (Abs ("", tyh, abs_rhs))
-end;
-*}
-
-
-ML {*
-fun non_rec_binds l =
-let
-  fun is_non_rec (SOME (f, false), _, _, _) = SOME f
-    | is_non_rec _ = NONE
-in
-  distinct (op =) (map_filter is_non_rec (flat (flat l)))
-end
-*}
-
-(* We assume no bindings in the type on which bn is defined *)
-ML {*
-fun fv_bn thy (dt_info : Datatype_Aux.info) fv_frees bn_fvbn (fvbn, (bn, ith_dtyp, args_in_bns)) =
-let
-  val {descr, sorts, ...} = dt_info;
-  fun nth_dtyp i = typ_of_dtyp descr sorts (DtRec i);
-  fun fv_bn_constr (cname, dts) args_in_bn =
-  let
-    val Ts = map (typ_of_dtyp descr sorts) dts;
-    val names = Datatype_Prop.make_tnames Ts;
-    val args = map Free (names ~~ Ts);
-    val c = Const (cname, Ts ---> (nth_dtyp ith_dtyp));
-    fun fv_arg ((dt, x), arg_no) =
-      let
-        val ty = fastype_of x
-(*        val _ = tracing ("B 1" ^ PolyML.makestring args_in_bn);*)
-(*        val _ = tracing ("B 2" ^ PolyML.makestring bn_fvbn);*)
-      in
-        case AList.lookup (op=) args_in_bn arg_no of
-          SOME NONE => @{term "{} :: atom set"}
-        | SOME (SOME (f : term)) => (the (AList.lookup (op=) bn_fvbn f)) $ x
-        | NONE =>
-            if is_atom thy ty then mk_single_atom x else
-            if is_atom_set thy ty then mk_atom_set x else
-            if is_atom_fset thy ty then mk_atom_fset x else
-            if is_rec_type dt then nth fv_frees (body_index dt) $ x else
-            @{term "{} :: atom set"}
-      end;
-    val arg_nos = 0 upto (length dts - 1)
-  in
-    HOLogic.mk_Trueprop (HOLogic.mk_eq
-      (fvbn $ list_comb (c, args), mk_union (map fv_arg (dts ~~ args ~~ arg_nos))))
-  end;
-  val (_, (_, _, constrs)) = nth descr ith_dtyp;
-  val eqs = map2i fv_bn_constr constrs args_in_bns
-in
-  ((bn, fvbn), eqs)
-end
-*}
-
-ML {* print_depth 100 *}
-ML {*
-fun fv_bns thy dt_info fv_frees rel_bns =
-let
-  fun mk_fvbn_free (bn, ith, _) =
-    let
-      val fvbn_name = "fv_" ^ (Long_Name.base_name (fst (dest_Const bn)));
-    in
-      (fvbn_name, Free (fvbn_name, fastype_of (nth fv_frees ith)))
-    end;
-  val (fvbn_names, fvbn_frees) = split_list (map mk_fvbn_free rel_bns);
-  val bn_fvbn = (map (fn (bn, _, _) => bn) rel_bns) ~~ fvbn_frees
-  val (l1, l2) = split_list (map (fv_bn thy dt_info fv_frees bn_fvbn) (fvbn_frees ~~ rel_bns));
-in
-  (l1, (fvbn_names ~~ l2))
-end
-*}
-
-
-ML {*
-fun alpha_bn (dt_info : Datatype_Aux.info) alpha_frees bn_alphabn ((bn, ith_dtyp, args_in_bns), (alpha_bn_free, _ (*is_rec*) )) =
-let
-  val {descr, sorts, ...} = dt_info;
-  fun nth_dtyp i = typ_of_dtyp descr sorts (DtRec i);
-  fun alpha_bn_constr (cname, dts) args_in_bn =
-  let
-    val Ts = map (typ_of_dtyp descr sorts) dts;
-    val names = Name.variant_list ["pi"] (Datatype_Prop.make_tnames Ts);
-    val names2 = Name.variant_list ("pi" :: names) (Datatype_Prop.make_tnames Ts);
-    val args = map Free (names ~~ Ts);
-    val args2 = map Free (names2 ~~ Ts);
-    val c = Const (cname, Ts ---> (nth_dtyp ith_dtyp));
-    val rhs = HOLogic.mk_Trueprop
-      (alpha_bn_free $ (list_comb (c, args)) $ (list_comb (c, args2)));
-    fun lhs_arg ((dt, arg_no), (arg, arg2)) =
-      case AList.lookup (op=) args_in_bn arg_no of
-        SOME NONE => @{term True}
-      | SOME (SOME f) => (the (AList.lookup (op=) bn_alphabn f)) $ arg $ arg2
-      | NONE =>
-          if is_rec_type dt then (nth alpha_frees (body_index dt)) $ arg $ arg2
-          else HOLogic.mk_eq (arg, arg2)
-    val arg_nos = 0 upto (length dts - 1)
-    val lhss = mk_conjl (map lhs_arg (dts ~~ arg_nos ~~ (args ~~ args2)))
-    val eq = Logic.mk_implies (HOLogic.mk_Trueprop lhss, rhs)
-  in
-    eq
-  end
-  val (_, (_, _, constrs)) = nth descr ith_dtyp;
-  val eqs = map2i alpha_bn_constr constrs args_in_bns
-in
-  ((bn, alpha_bn_free), eqs)
-end
-*}
-
-ML {*
-fun alpha_bns dt_info alpha_frees rel_bns bns_rec =
-let
-  val {descr, sorts, ...} = dt_info;
-  fun nth_dtyp i = typ_of_dtyp descr sorts (DtRec i);
-  fun mk_alphabn_free (bn, ith, _) =
-    let
-      val alphabn_name = "alpha_" ^ (Long_Name.base_name (fst (dest_Const bn)));
-      val alphabn_type = nth_dtyp ith --> nth_dtyp ith --> @{typ bool};
-      val alphabn_free = Free(alphabn_name, alphabn_type);
-    in
-      (alphabn_name, alphabn_free)
-    end;
-  val (alphabn_names, alphabn_frees) = split_list (map mk_alphabn_free rel_bns);
-  val bn_alphabn = (map (fn (bn, _, _) => bn) rel_bns) ~~ alphabn_frees;
-  val pair = split_list (map (alpha_bn dt_info alpha_frees bn_alphabn)
-    (rel_bns ~~ (alphabn_frees ~~ bns_rec)))
-in
-  (alphabn_names, pair)
-end
-*}
-
-
-(* Checks that a list of bindings contains only compatible ones *)
-ML {*
-fun bns_same l =
-  length (distinct (op =) (map (fn ((b, _, _, atyp), _) => (b, atyp)) l)) = 1
-*}
-
-ML {*
-fun setify x =
-  if fastype_of x = @{typ "atom list"} then
-  Const (@{const_name set}, @{typ "atom list \<Rightarrow> atom set"}) $ x else x
-*}
-
-ML {*
-fun define_fv (dt_info : Datatype_Aux.info) bindsall bns lthy =
-let
-  val thy = ProofContext.theory_of lthy;
-  val {descr, sorts, ...} = dt_info;
-  fun nth_dtyp i = typ_of_dtyp descr sorts (DtRec i);
-  val fv_names = Datatype_Prop.indexify_names (map (fn (i, _) =>
-    "fv_" ^ name_of_typ (nth_dtyp i)) descr);
-  val fv_types = map (fn (i, _) => nth_dtyp i --> @{typ "atom set"}) descr;
-  val fv_frees = map Free (fv_names ~~ fv_types);
-(* TODO: We need a transitive closure, but instead we do this hack considering
-   all binding functions as recursive or not *)
-  val nr_bns =
-    if (non_rec_binds bindsall) = [] then []
-    else map (fn (bn, _, _) => bn) bns;
-  val rel_bns = filter (fn (bn, _, _) => bn mem nr_bns) bns;
-  val (bn_fv_bns, fv_bn_names_eqs) = fv_bns thy dt_info fv_frees rel_bns;
-  val fvbns = map snd bn_fv_bns;
-  val (fv_bn_names, fv_bn_eqs) = split_list fv_bn_names_eqs;
-
-  fun fv_constr ith_dtyp (cname, dts) bindcs =
-    let
-      val Ts = map (typ_of_dtyp descr sorts) dts;
-      val bindslen = length bindcs
-      val pi_strs_same = replicate bindslen "pi"
-      val pi_strs = Name.variant_list [] pi_strs_same;
-      val pis = map (fn ps => Free (ps, @{typ perm})) pi_strs;
-      val bind_pis_gath = bindcs ~~ pis;
-      val bind_pis = un_gather_binds_cons bind_pis_gath;
-      val bindcs = map fst bind_pis;
-      val names = Name.variant_list pi_strs (Datatype_Prop.make_tnames Ts);
-      val args = map Free (names ~~ Ts);
-      val c = Const (cname, Ts ---> (nth_dtyp ith_dtyp));
-      val fv_c = nth fv_frees ith_dtyp;
-      val arg_nos = 0 upto (length dts - 1)
-      fun fv_bind args (NONE, i, _, _) =
-            if is_rec_type (nth dts i) then (nth fv_frees (body_index (nth dts i))) $ (nth args i) else
-            if ((is_atom thy) o fastype_of) (nth args i) then mk_single_atom (nth args i) else
-            if ((is_atom_set thy) o fastype_of) (nth args i) then mk_atom_set (nth args i) else
-            if ((is_atom_fset thy) o fastype_of) (nth args i) then mk_atom_fset (nth args i) else
-            (* TODO goes the code for preiously defined nominal datatypes *)
-            @{term "{} :: atom set"}
-        | fv_bind args (SOME (f, _), i, _, _) = f $ (nth args i)
-      fun fv_binds_as_set args relevant = mk_union (map (setify o fv_bind args) relevant)
-      fun find_nonrec_binder j (SOME (f, false), i, _, _) = if i = j then SOME f else NONE
-        | find_nonrec_binder _ _ = NONE
-      fun fv_arg ((dt, x), arg_no) =
-        case get_first (find_nonrec_binder arg_no) bindcs of
-          SOME f =>
-            (case get_first (fn (x, y) => if x = f then SOME y else NONE) bn_fv_bns of
-                SOME fv_bn => fv_bn $ x
-              | NONE => error "bn specified in a non-rec binding but not in bn list")
-        | NONE =>
-            let
-              val arg =
-                if is_rec_type dt then nth fv_frees (body_index dt) $ x else
-                if ((is_atom thy) o fastype_of) x then mk_single_atom x else
-                if ((is_atom_set thy) o fastype_of) x then mk_atom_set x else
-                if ((is_atom_fset thy) o fastype_of) x then mk_atom_fset x else
-                (* TODO goes the code for preiously defined nominal datatypes *)
-                @{term "{} :: atom set"};
-              (* If i = j then we generate it only once *)
-              val relevant = filter (fn (_, i, j, _) => ((i = arg_no) orelse (j = arg_no))) bindcs;
-              val sub = fv_binds_as_set args relevant
-            in
-              mk_diff arg sub
-            end;
-      val fv_eq = HOLogic.mk_Trueprop (HOLogic.mk_eq
-        (fv_c $ list_comb (c, args), mk_union (map fv_arg  (dts ~~ args ~~ arg_nos))))
-    in
-      fv_eq
-    end;
-  fun fv_eq (i, (_, _, constrs)) binds = map2i (fv_constr i) constrs binds;
-  val fveqs = map2i fv_eq descr (gather_binds bindsall)
-  val fv_eqs_perfv = fveqs
-  val rel_bns_nos = map (fn (_, i, _) => i) rel_bns;
-  fun filter_fun (_, b) = b mem rel_bns_nos;
-  val all_fvs = (fv_names ~~ fv_eqs_perfv) ~~ (0 upto (length fv_names - 1))
-  val (fv_names_fst, fv_eqs_fst) = apsnd flat (split_list (map fst (filter_out filter_fun all_fvs)))
-  val (fv_names_snd, fv_eqs_snd) = apsnd flat (split_list (map fst (filter filter_fun all_fvs)))
-  val fv_eqs_all = fv_eqs_fst @ (flat fv_bn_eqs);
-  val fv_names_all = fv_names_fst @ fv_bn_names;
-  val add_binds = map (fn x => (Attrib.empty_binding, x))
-(* Function_Fun.add_fun Function_Common.default_config ... true *)
-  val (fvs, lthy') = (Primrec.add_primrec
-    (map (fn s => (Binding.name s, NONE, NoSyn)) fv_names_all) (add_binds fv_eqs_all) lthy)
-  val (fvs2, lthy'') =
-    if fv_eqs_snd = [] then (([], []), lthy') else
-   (Primrec.add_primrec
-    (map (fn s => (Binding.name s, NONE, NoSyn)) fv_names_snd) (add_binds fv_eqs_snd) lthy')
-  val ordered_fvs = fv_frees @ fvbns;
-  val all_fvs = (fst fvs @ fst fvs2, snd fvs @ snd fvs2)
-in
-  ((all_fvs, ordered_fvs), lthy'')
-end
-*}
-
-ML {*
-fun define_alpha (dt_info : Datatype_Aux.info) bindsall bns fv_frees lthy =
-let
-  val thy = ProofContext.theory_of lthy;
-  val {descr, sorts, ...} = dt_info;
-  fun nth_dtyp i = typ_of_dtyp descr sorts (DtRec i);
-(* TODO: We need a transitive closure, but instead we do this hack considering
-   all binding functions as recursive or not *)
-  val nr_bns =
-    if (non_rec_binds bindsall) = [] then []
-    else map (fn (bn, _, _) => bn) bns;
-  val alpha_names = Datatype_Prop.indexify_names (map (fn (i, _) =>
-    "alpha_" ^ name_of_typ (nth_dtyp i)) descr);
-  val alpha_types = map (fn (i, _) => nth_dtyp i --> nth_dtyp i --> @{typ bool}) descr;
-  val alpha_frees = map Free (alpha_names ~~ alpha_types);
-  (* We assume that a bn is either recursive or not *)
-  val bns_rec = map (fn (bn, _, _) => not (bn mem nr_bns)) bns;
-  val (alpha_bn_names, (bn_alpha_bns, alpha_bn_eqs)) =
-    alpha_bns dt_info alpha_frees bns bns_rec
-  val alpha_bn_frees = map snd bn_alpha_bns;
-  val alpha_bn_types = map fastype_of alpha_bn_frees;
-
-  fun alpha_constr ith_dtyp (cname, dts) bindcs =
-    let
-      val Ts = map (typ_of_dtyp descr sorts) dts;
-      val bindslen = length bindcs
-      val pi_strs_same = replicate bindslen "pi"
-      val pi_strs = Name.variant_list [] pi_strs_same;
-      val pis = map (fn ps => Free (ps, @{typ perm})) pi_strs;
-      val bind_pis_gath = bindcs ~~ pis;
-      val bind_pis = un_gather_binds_cons bind_pis_gath;
-      val names = Name.variant_list pi_strs (Datatype_Prop.make_tnames Ts);
-      val args = map Free (names ~~ Ts);
-      val names2 = Name.variant_list (pi_strs @ names) (Datatype_Prop.make_tnames Ts);
-      val args2 = map Free (names2 ~~ Ts);
-      val c = Const (cname, Ts ---> (nth_dtyp ith_dtyp));
-      val alpha = nth alpha_frees ith_dtyp;
-      val arg_nos = 0 upto (length dts - 1)
-      fun fv_bind args (NONE, i, _, _) =
-            if is_rec_type (nth dts i) then (nth fv_frees (body_index (nth dts i))) $ (nth args i) else
-            if ((is_atom thy) o fastype_of) (nth args i) then mk_single_atom (nth args i) else
-            if ((is_atom_set thy) o fastype_of) (nth args i) then mk_atom_set (nth args i) else
-            if ((is_atom_fset thy) o fastype_of) (nth args i) then mk_atom_fset (nth args i) else
-            (* TODO goes the code for preiously defined nominal datatypes *)
-            @{term "{} :: atom set"}
-        | fv_bind args (SOME (f, _), i, _, _) = f $ (nth args i)
-      fun fv_binds args relevant = mk_union (map (fv_bind args) relevant)
-      val alpha_rhs =
-        HOLogic.mk_Trueprop (alpha $ (list_comb (c, args)) $ (list_comb (c, args2)));
-      fun alpha_arg ((dt, arg_no), (arg, arg2)) =
-        let
-          val rel_in_simp_binds = filter (fn ((NONE, i, _, _), _) => i = arg_no | _ => false) bind_pis;
-          val rel_in_comp_binds = filter (fn ((SOME _, i, _, _), _) => i = arg_no | _ => false) bind_pis;
-          val rel_has_binds = filter (fn ((NONE, _, j, _), _) => j = arg_no
-                                       | ((SOME (_, false), _, j, _), _) => j = arg_no
-                                       | _ => false) bind_pis;
-          val rel_has_rec_binds = filter
-            (fn ((SOME (_, true), _, j, _), _) => j = arg_no | _ => false) bind_pis;
-        in
-          case (rel_in_simp_binds, rel_in_comp_binds, rel_has_binds, rel_has_rec_binds) of
-            ([], [], [], []) =>
-              if is_rec_type dt then (nth alpha_frees (body_index dt) $ arg $ arg2)
-              else (HOLogic.mk_eq (arg, arg2))
-          | (_, [], [], []) => @{term True}
-          | ([], [], [], _) => @{term True}
-          | ([], ((((SOME (bn, is_rec)), _, _, atyp), _) :: _), [], []) =>
-            if not (bns_same rel_in_comp_binds) then error "incompatible bindings for an argument" else
-            if is_rec then
-              let
-                val (rbinds, rpis) = split_list rel_in_comp_binds
-                val bound_in_nos = map (fn (_, _, i, _) => i) rbinds
-                val bound_in_ty_nos = map (fn i => body_index (nth dts i)) bound_in_nos;
-                val bound_args = arg :: map (nth args) bound_in_nos;
-                val bound_args2 = arg2 :: map (nth args2) bound_in_nos;
-                val lhs_binds = fv_binds args rbinds
-                val lhs_arg = foldr1 HOLogic.mk_prod bound_args
-                val lhs = mk_pair (lhs_binds, lhs_arg);
-                val rhs_binds = fv_binds args2 rbinds;
-                val rhs_arg = foldr1 HOLogic.mk_prod bound_args2;
-                val rhs = mk_pair (rhs_binds, rhs_arg);
-                val fvs = map (nth fv_frees) ((body_index dt) :: bound_in_ty_nos);
-                val fv = mk_compound_fv fvs;
-                val alphas = map (nth alpha_frees) ((body_index dt) :: bound_in_ty_nos);
-                val alpha = mk_compound_alpha alphas;
-                val pi = foldr1 (uncurry mk_plus) (distinct (op =) rpis);
-                val alpha_gen_pre = Const (atyp_const atyp, dummyT) $ lhs $ alpha $ fv $ pi $ rhs;
-                val alpha_gen = Syntax.check_term lthy alpha_gen_pre
-              in
-                alpha_gen
-              end
-            else
-              let
-                val alpha_bn_const =
-                  nth alpha_bn_frees (find_index (fn (b, _, _) => b = bn) bns)
-              in
-                alpha_bn_const $ arg $ arg2
-              end
-          | ([], [], relevant, []) =>
-            let
-              val (rbinds, rpis) = split_list relevant
-              val lhs_binds = fv_binds args rbinds
-              val lhs = mk_pair (lhs_binds, arg);
-              val rhs_binds = fv_binds args2 rbinds;
-              val rhs = mk_pair (rhs_binds, arg2);
-              val alpha = nth alpha_frees (body_index dt);
-              val fv = nth fv_frees (body_index dt);
-              val pi = foldr1 (uncurry mk_plus) (distinct (op =) rpis);
-              val alpha_const = alpha_const_for_binds rbinds;
-              val alpha_gen_pre = Const (alpha_const, dummyT) $ lhs $ alpha $ fv $ pi $ rhs;
-              val alpha_gen = Syntax.check_term lthy alpha_gen_pre
-            in
-              alpha_gen
-            end
-          | _ => error "Fv.alpha: not supported binding structure"
-        end
-      val alphas = map alpha_arg (dts ~~ arg_nos ~~ (args ~~ args2))
-      val alpha_lhss = mk_conjl alphas
-      val alpha_lhss_ex =
-        fold (fn pi_str => fn t => HOLogic.mk_exists (pi_str, @{typ perm}, t)) pi_strs alpha_lhss
-      val alpha_eq = Logic.mk_implies (HOLogic.mk_Trueprop alpha_lhss_ex, alpha_rhs)
-    in
-      alpha_eq
-    end;
-  fun alpha_eq (i, (_, _, constrs)) binds = map2i (alpha_constr i) constrs binds;
-  val alphaeqs = map2i alpha_eq descr (gather_binds bindsall)
-  val alpha_eqs = flat alphaeqs
-  val add_binds = map (fn x => (Attrib.empty_binding, x))
-  val (alphas, lthy') = (Inductive.add_inductive_i
-     {quiet_mode = true, verbose = false, alt_name = Binding.empty,
-      coind = false, no_elim = false, no_ind = false, skip_mono = true, fork_mono = false}
-     (map2 (fn x => fn y => ((Binding.name x, y), NoSyn)) (alpha_names @ alpha_bn_names)
-     (alpha_types @ alpha_bn_types)) []
-     (add_binds (alpha_eqs @ flat alpha_bn_eqs)) [] lthy)
-in
-  (alphas, lthy')
-end
-*}
-
-
-ML {*
-fun define_fv_alpha_export dt binds bns ctxt =
-let
-  val (((fv_ts_loc, fv_def_loc), ord_fv_ts_loc), ctxt') =
-    define_fv dt binds bns ctxt;
-  val (alpha, ctxt'') =
-    define_alpha dt binds bns fv_ts_loc ctxt';
-  val alpha_ts_loc = #preds alpha
-  val alpha_induct_loc = #induct alpha
-  val alpha_intros_loc = #intrs alpha;
-  val alpha_cases_loc = #elims alpha
-  val morphism = ProofContext.export_morphism ctxt'' ctxt;
-  val fv_ts = map (Morphism.term morphism) fv_ts_loc;
-  val ord_fv_ts = map (Morphism.term morphism) ord_fv_ts_loc;
-  val fv_def = Morphism.fact morphism fv_def_loc;
-  val alpha_ts = map (Morphism.term morphism) alpha_ts_loc;
-  val alpha_induct = Morphism.thm morphism alpha_induct_loc;
-  val alpha_intros = Morphism.fact morphism alpha_intros_loc
-  val alpha_cases = Morphism.fact morphism alpha_cases_loc
-in
-  ((((fv_ts, ord_fv_ts), fv_def), ((alpha_ts, alpha_intros), (alpha_cases, alpha_induct))), ctxt'')
-end;
-*}
-
-end