theory Fv

imports "Nominal2_Atoms" "Nominal2_Eqvt" "Nominal2_Supp" "Abs"

begin

(* Bindings are given as a list which has a length being equal

   to the length of the number of constructors.

   Each element is a list whose length is equal to the number

   of arguents.

   Every element specifies bindings of this argument given as

   a tuple: function, bound argument.

  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)]],

 [[], [], [(SOME (Const f), 0), (Some (Const g), 1)]]]

*)

ML {*

  open Datatype_Aux; (* typ_of_dtyp, DtRec, ... *);

  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

      HOLogic.mk_binop @{const_name union} (a, b)) (rev sets) noatoms;

  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);

*}

atom_decl name

ML {*

  val sorts = [];

  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);

  fun fv_eq_constr i (cname, dts) bindcs =

    let

      val Ts = map (typ_of_dtyp descr sorts) dts;

      val names = Name.variant_list ["pi"] (Datatype_Prop.make_tnames Ts);

      val args = map Free (names ~~ Ts);

      val c = Const (cname, Ts ---> (nth_dtyp i));

      val fv_c = Free (nth fv_names i, (nth_dtyp i) --> @{typ "atom set"});

      (* TODO we assume that all can be 'atomized' *)

      fun fv_bind (NONE, i) = mk_single_atom (nth args i)

        | fv_bind (SOME f, i) = f $ (nth args i);

      fun fv_arg ((dt, x), bindxs) =

        let

          val arg =

            if is_rec_type dt then nth fv_frees (body_index dt) $ x else

            (* TODO: we just assume everything can be 'atomized' *)

            HOLogic.mk_set @{typ atom} [mk_atom (type_of x) $ x]

          val sub = mk_union (map fv_bind bindxs)

        in

          mk_diff arg sub

        end;

    in

      (Attrib.empty_binding, HOLogic.mk_Trueprop (HOLogic.mk_eq

        (fv_c $ list_comb (c, args), mk_union (map fv_arg (dts ~~ args ~~ bindcs)))))

    end;

*}

local_setup {*

snd o (Primrec.add_primrec

  (map (fn s => (Binding.name s, NONE, NoSyn)) fv_names) fv_eqs)

*}

print_theorems

end