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));

      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

  snd (Primrec.add_primrec

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

end

*}

atom_decl name

  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

end