theory Fv
imports "Nominal2_Atoms"
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)]]]

A SOME binding has to have a function returning an atom set,
and a NONE binding has to be on an argument that is an atom
or an atom set.

How the procedure works:
  For each of the defined datatypes,
  For each of the constructors,
  It creates a union of free variables for each argument.

  For an argument the free variables are the variables minus
  bound variables.

  The variables are:
    For an atom, a singleton set with the atom itself.
    For an atom set, the atom set itself.
    For a recursive argument, the appropriate fv function applied to it.
    (* TODO: This one is not implemented *)
    For other arguments it should be an appropriate fv function stored
      in the database.
  The bound variables are a union of results of all bindings that
  involve the given argument. For a paricular binding the result is:
    For a function applied to an argument this function with the argument.
    For an atom, a singleton set with the atom itself.
    For an atom set, the atom set itself.
    For a recursive argument, the appropriate fv function applied to it.
    (* TODO: This one is not implemented *)
    For other arguments it should be an appropriate fv function stored
      in the database.
*)

ML {*
  open Datatype_Aux; (* typ_of_dtyp, DtRec, ... *);
  (* 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
      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);
  fun mk_atoms 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;
  (* Copy from Term *)
  fun is_funtype (Type ("fun", [_, _])) = true
    | is_funtype _ = false;
*}


ML {*
(* Currently needs just one full_tname to access Datatype *)
fun define_raw_fv full_tname bindsall lthy =
let
  val thy = ProofContext.theory_of lthy;
  val {descr, ...} = Datatype.the_info thy full_tname;
  val sorts = []; (* TODO *)
  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 = nth fv_frees i;
      fun fv_bind (NONE, i) =
            if is_rec_type (nth dts i) then (nth fv_frees (body_index (nth dts i))) $ (nth args i) else
            (* TODO we assume that all can be 'atomized' *)
            if (is_funtype o fastype_of) (nth args i) then mk_atoms (nth args i) else
            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' *)
            if (is_funtype o fastype_of) x then mk_atoms x else
            HOLogic.mk_set @{typ atom} [mk_atom (fastype_of x) $ x]
          val sub = mk_union (map fv_bind bindxs)
        in
          mk_diff arg sub
        end;
        val _ = tracing ("d" ^ string_of_int (length dts));
        val _ = tracing (string_of_int (length args));
        val _ = tracing (string_of_int (length bindcs));
    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;
  fun fv_eq (i, (_, _, constrs)) binds = map2 (fv_eq_constr i) constrs binds;
  val fv_eqs = flat (map2 fv_eq descr bindsall)
in
  (* The snd will be removed later *)
  snd (Primrec.add_primrec
    (map (fn s => (Binding.name s, NONE, NoSyn)) fv_names) fv_eqs lthy)
end
*}

ML {*
fun define_alpha full_tname bindsall lthy =
let
  val thy = ProofContext.theory_of lthy;
  val {descr, ...} = Datatype.the_info thy full_tname;
  val sorts = []; (* TODO *)
  fun nth_dtyp i = typ_of_dtyp descr sorts (DtRec i);
  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);
  fun alpha_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 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 i));
      val alpha = nth alpha_frees i;
    in
      (Attrib.empty_binding, HOLogic.mk_Trueprop (alpha $ (list_comb (c, args)) $ (list_comb (c, args2))))
    end;
  fun alpha_eq (i, (_, _, constrs)) binds = map2 (alpha_eq_constr i) constrs binds;
  val alpha_eqs = flat (map2 alpha_eq descr bindsall)
in
  (* The snd will be removed later *)
  snd (Inductive.add_inductive_i
     {quiet_mode = false, verbose = true, 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_types) [] (alpha_eqs) [] lthy)
end
*}

atom_decl name

(*datatype ty =
  Var "name set"

ML {* Syntax.check_term @{context} (mk_atoms @{term "a :: name set"}) *}

local_setup {* define_raw_fv "Fv.ty" [[[[]]]] *}
print_theorems
*)

datatype rtrm1 =
  rVr1 "name"
| rAp1 "rtrm1" "rtrm1"
| rLm1 "name" "rtrm1"        --"name is bound in trm1"
| rLt1 "bp" "rtrm1" "rtrm1"   --"all variables in bp are bound in the 2nd trm1"
and bp =
  BUnit
| BVr "name"
| BPr "bp" "bp"

(* to be given by the user *)

primrec 
  bv1
where
  "bv1 (BUnit) = {}"
| "bv1 (BVr x) = {atom x}"
| "bv1 (BPr bp1 bp2) = (bv1 bp1) \<union> (bv1 bp1)"

local_setup {* define_raw_fv "Fv.rtrm1"
  [[[[]], [[], []], [[(NONE, 0)], [(NONE, 0)]], [[(NONE, 0)], [], [(SOME @{term bv1}, 0)]]],
   [[], [[]], [[], []]]] *}
print_theorems

local_setup {* define_alpha "Fv.rtrm1"
  [[[[]], [[], []], [[(NONE, 0)], [(NONE, 0)]], [[(NONE, 0)], [], [(SOME @{term bv1}, 0)]]],
   [[], [[]], [[], []]]] *}
print_theorems


end