src/HOL/Tools/inductive_realizer.ML

avoid polymorphic equality;

(*  Title:      HOL/Tools/inductive_realizer.ML
    ID:         $Id$
    Author:     Stefan Berghofer, TU Muenchen

Porgram extraction from proofs involving inductive predicates:
Realizers for induction and elimination rules
*)

signature INDUCTIVE_REALIZER =
sig
  val add_ind_realizers: string -> string list -> theory -> theory
  val setup: theory -> theory
end;

structure InductiveRealizer : INDUCTIVE_REALIZER =
struct

(* FIXME: LocalTheory.note should return theorems with proper names! *)
fun name_of_thm thm =
  (case Symtab.dest (Proofterm.thms_of_proof' (proof_of thm) Symtab.empty) of
     [(name, _)] => name
   | _ => error ("name_of_thm: bad proof of theorem\n" ^ string_of_thm thm));

val all_simps = map (symmetric o mk_meta_eq) (thms "HOL.all_simps");

fun prf_of thm =
  let val {thy, prop, der = (_, prf), ...} = rep_thm thm
  in Reconstruct.expand_proof thy [("", NONE)] (Reconstruct.reconstruct_proof thy prop prf) end; (* FIXME *)

fun forall_intr_prf (t, prf) =
  let val (a, T) = (case t of Var ((a, _), T) => (a, T) | Free p => p)
  in Abst (a, SOME T, Proofterm.prf_abstract_over t prf) end;

fun forall_intr_term (t, u) =
  let val (a, T) = (case t of Var ((a, _), T) => (a, T) | Free p => p)
  in all T $ Abs (a, T, abstract_over (t, u)) end;

fun subsets [] = [[]]
  | subsets (x::xs) =
      let val ys = subsets xs
      in ys @ map (cons x) ys end;

val pred_of = fst o dest_Const o head_of;

fun strip_all' used names (Const ("all", _) $ Abs (s, T, t)) =
      let val (s', names') = (case names of
          [] => (Name.variant used s, [])
        | name :: names' => (name, names'))
      in strip_all' (s'::used) names' (subst_bound (Free (s', T), t)) end
  | strip_all' used names ((t as Const ("==>", _) $ P) $ Q) =
      t $ strip_all' used names Q
  | strip_all' _ _ t = t;

fun strip_all t = strip_all' (add_term_free_names (t, [])) [] t;

fun strip_one name (Const ("all", _) $ Abs (s, T, Const ("==>", _) $ P $ Q)) =
      (subst_bound (Free (name, T), P), subst_bound (Free (name, T), Q))
  | strip_one _ (Const ("==>", _) $ P $ Q) = (P, Q);

fun relevant_vars prop = foldr (fn
      (Var ((a, i), T), vs) => (case strip_type T of
        (_, Type (s, _)) => if s mem ["bool"] then (a, T) :: vs else vs
      | _ => vs)
    | (_, vs) => vs) [] (term_vars prop);

fun dt_of_intrs thy vs nparms intrs =
  let
    val iTs = term_tvars (prop_of (hd intrs));
    val Tvs = map TVar iTs;
    val (Const (s, _), ts) = strip_comb (HOLogic.dest_Trueprop
      (Logic.strip_imp_concl (prop_of (hd intrs))));
    val params = map dest_Var (Library.take (nparms, ts));
    val tname = space_implode "_" (Sign.base_name s ^ "T" :: vs);
    fun constr_of_intr intr = (Sign.base_name (name_of_thm intr),
      map (Logic.unvarifyT o snd) (rev (Term.add_vars (prop_of intr) []) \\ params) @
        filter_out (equal Extraction.nullT) (map
          (Logic.unvarifyT o Extraction.etype_of thy vs []) (prems_of intr)),
            NoSyn);
  in (map (fn a => "'" ^ a) vs @ map (fst o fst) iTs, tname, NoSyn,
    map constr_of_intr intrs)
  end;

fun mk_rlz T = Const ("realizes", [T, HOLogic.boolT] ---> HOLogic.boolT);

(** turn "P" into "%r x. realizes r (P x)" **)

fun gen_rvar vs (t as Var ((a, 0), T)) =
      if body_type T <> HOLogic.boolT then t else
        let
          val U = TVar (("'" ^ a, 0), HOLogic.typeS)
          val Ts = binder_types T;
          val i = length Ts;
          val xs = map (pair "x") Ts;
          val u = list_comb (t, map Bound (i - 1 downto 0))
        in 
          if a mem vs then
            list_abs (("r", U) :: xs, mk_rlz U $ Bound i $ u)
          else list_abs (xs, mk_rlz Extraction.nullT $ Extraction.nullt $ u)
        end
  | gen_rvar _ t = t;

fun mk_realizes_eqn n vs nparms intrs =
  let
    val concl = HOLogic.dest_Trueprop (concl_of (hd intrs));
    val iTs = term_tvars concl;
    val Tvs = map TVar iTs;
    val (h as Const (s, T), us) = strip_comb concl;
    val params = List.take (us, nparms);
    val elTs = List.drop (binder_types T, nparms);
    val predT = elTs ---> HOLogic.boolT;
    val used = map (fst o fst o dest_Var) params;
    val xs = map (Var o apfst (rpair 0))
      (Name.variant_list used (replicate (length elTs) "x") ~~ elTs);
    val rT = if n then Extraction.nullT
      else Type (space_implode "_" (s ^ "T" :: vs),
        map (fn a => TVar (("'" ^ a, 0), HOLogic.typeS)) vs @ Tvs);
    val r = if n then Extraction.nullt else Var ((Sign.base_name s, 0), rT);
    val S = list_comb (h, params @ xs);
    val rvs = relevant_vars S;
    val vs' = map fst rvs \\ vs;
    val rname = space_implode "_" (s ^ "R" :: vs);

    fun mk_Tprem n v =
      let val T = (the o AList.lookup (op =) rvs) v
      in (Const ("typeof", T --> Type ("Type", [])) $ Var ((v, 0), T),
        Extraction.mk_typ (if n then Extraction.nullT
          else TVar (("'" ^ v, 0), HOLogic.typeS)))
      end;

    val prems = map (mk_Tprem true) vs' @ map (mk_Tprem false) vs;
    val ts = map (gen_rvar vs) params;
    val argTs = map fastype_of ts;

  in ((prems, (Const ("typeof", HOLogic.boolT --> Type ("Type", [])) $ S,
       Extraction.mk_typ rT)),
    (prems, (mk_rlz rT $ r $ S,
       if n then list_comb (Const (rname, argTs ---> predT), ts @ xs)
       else list_comb (Const (rname, argTs @ [rT] ---> predT), ts @ [r] @ xs))))
  end;

fun fun_of_prem thy rsets vs params rule ivs intr =
  let
    val ctxt = ProofContext.init thy
    val args = map (Free o apfst fst o dest_Var) ivs;
    val args' = map (Free o apfst fst)
      (Term.add_vars (prop_of intr) [] \\ params);
    val rule' = strip_all rule;
    val conclT = Extraction.etype_of thy vs [] (Logic.strip_imp_concl rule');
    val used = map (fst o dest_Free) args;

    fun is_rec t = not (null (term_consts t inter rsets));

    fun is_meta (Const ("all", _) $ Abs (s, _, P)) = is_meta P
      | is_meta (Const ("==>", _) $ _ $ Q) = is_meta Q
      | is_meta (Const ("Trueprop", _) $ t) = (case head_of t of
          Const (s, _) => can (InductivePackage.the_inductive ctxt) s
        | _ => true)
      | is_meta _ = false;

    fun fun_of ts rts args used (prem :: prems) =
          let
            val T = Extraction.etype_of thy vs [] prem;
            val [x, r] = Name.variant_list used ["x", "r"]
          in if T = Extraction.nullT
            then fun_of ts rts args used prems
            else if is_rec prem then
              if is_meta prem then
                let
                  val prem' :: prems' = prems;
                  val U = Extraction.etype_of thy vs [] prem';
                in if U = Extraction.nullT
                  then fun_of (Free (x, T) :: ts)
                    (Free (r, binder_types T ---> HOLogic.unitT) :: rts)
                    (Free (x, T) :: args) (x :: r :: used) prems'
                  else fun_of (Free (x, T) :: ts) (Free (r, U) :: rts)
                    (Free (r, U) :: Free (x, T) :: args) (x :: r :: used) prems'
                end
              else (case strip_type T of
                  (Ts, Type ("*", [T1, T2])) =>
                    let
                      val fx = Free (x, Ts ---> T1);
                      val fr = Free (r, Ts ---> T2);
                      val bs = map Bound (length Ts - 1 downto 0);
                      val t = list_abs (map (pair "z") Ts,
                        HOLogic.mk_prod (list_comb (fx, bs), list_comb (fr, bs)))
                    in fun_of (fx :: ts) (fr :: rts) (t::args)
                      (x :: r :: used) prems
                    end
                | (Ts, U) => fun_of (Free (x, T) :: ts)
                    (Free (r, binder_types T ---> HOLogic.unitT) :: rts)
                    (Free (x, T) :: args) (x :: r :: used) prems)
            else fun_of (Free (x, T) :: ts) rts (Free (x, T) :: args)
              (x :: used) prems
          end
      | fun_of ts rts args used [] =
          let val xs = rev (rts @ ts)
          in if conclT = Extraction.nullT
            then list_abs_free (map dest_Free xs, HOLogic.unit)
            else list_abs_free (map dest_Free xs, list_comb
              (Free ("r" ^ Sign.base_name (name_of_thm intr),
                map fastype_of (rev args) ---> conclT), rev args))
          end

  in fun_of args' [] (rev args) used (Logic.strip_imp_prems rule') end;

fun indrule_realizer thy induct raw_induct rsets params vs rec_names rss intrs dummies =
  let
    val concls = HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of raw_induct));
    val premss = List.mapPartial (fn (s, rs) => if s mem rsets then
      SOME (rs, map (fn (_, r) => List.nth (prems_of raw_induct,
        find_index_eq (prop_of r) (map prop_of intrs))) rs) else NONE) rss;
    val fs = maps (fn ((intrs, prems), dummy) =>
      let
        val fs = map (fn (rule, (ivs, intr)) =>
          fun_of_prem thy rsets vs params rule ivs intr) (prems ~~ intrs)
      in if dummy then Const ("arbitrary",
          HOLogic.unitT --> body_type (fastype_of (hd fs))) :: fs
        else fs
      end) (premss ~~ dummies);
    val frees = fold Term.add_frees fs [];
    val Ts = map fastype_of fs;
    fun name_of_fn intr = "r" ^ Sign.base_name (name_of_thm intr)
  in
    fst (fold_map (fn concl => fn names =>
      let val T = Extraction.etype_of thy vs [] concl
      in if T = Extraction.nullT then (Extraction.nullt, names) else
        let
          val Type ("fun", [U, _]) = T;
          val a :: names' = names
        in (list_abs_free (("x", U) :: List.mapPartial (fn intr =>
          Option.map (pair (name_of_fn intr))
            (AList.lookup (op =) frees (name_of_fn intr))) intrs,
          list_comb (Const (a, Ts ---> T), fs) $ Free ("x", U)), names')
        end
      end) concls rec_names)
  end;

fun add_dummy name dname (x as (_, (vs, s, mfx, cs))) =
  if (name: string) = s then (true, (vs, s, mfx, (dname, [HOLogic.unitT], NoSyn) :: cs))
  else x;

fun add_dummies f [] _ thy =
      (([], NONE), thy)
  | add_dummies f dts used thy =
      thy
      |> f (map snd dts)
      |-> (fn dtinfo => pair ((map fst dts), SOME dtinfo))
    handle DatatypeAux.Datatype_Empty name' =>
      let
        val name = Sign.base_name name';
        val dname = Name.variant used "Dummy"
      in
        thy
        |> add_dummies f (map (add_dummy name dname) dts) (dname :: used)
      end;

fun mk_realizer thy vs (name, rule, rrule, rlz, rt) =
  let
    val rvs = map fst (relevant_vars (prop_of rule));
    val xs = rev (Term.add_vars (prop_of rule) []);
    val vs1 = map Var (filter_out (fn ((a, _), _) => a mem rvs) xs);
    val rlzvs = rev (Term.add_vars (prop_of rrule) []);
    val vs2 = map (fn (ixn, _) => Var (ixn, (the o AList.lookup (op =) rlzvs) ixn)) xs;
    val rs = map Var (subtract (op = o pairself fst) xs rlzvs);
    val rlz' = foldr forall_intr_term (prop_of rrule) (vs2 @ rs);
    val rlz'' = foldr forall_intr_term rlz vs2
  in (name, (vs,
    if rt = Extraction.nullt then rt else
      foldr (uncurry lambda) rt vs1,
    ProofRewriteRules.un_hhf_proof rlz' rlz''
      (foldr forall_intr_prf (prf_of rrule) (vs2 @ rs))))
  end;

fun add_ind_realizer rsets intrs induct raw_induct elims (thy, vs) =
  let
    val qualifier = NameSpace.qualifier (name_of_thm induct);
    val inducts = PureThy.get_thms thy (Name
      (NameSpace.qualified qualifier "inducts"));
    val iTs = term_tvars (prop_of (hd intrs));
    val ar = length vs + length iTs;
    val params = InductivePackage.params_of raw_induct;
    val arities = InductivePackage.arities_of raw_induct;
    val nparms = length params;
    val params' = map dest_Var params;
    val rss = InductivePackage.partition_rules raw_induct intrs;
    val rss' = map (fn (((s, rs), (_, arity)), elim) =>
      (s, (InductivePackage.infer_intro_vars elim arity rs ~~ rs)))
        (rss ~~ arities ~~ elims);
    val (prfx, _) = split_last (NameSpace.explode (fst (hd rss)));
    val tnames = map (fn s => space_implode "_" (s ^ "T" :: vs)) rsets;

    val thy1 = thy |>
      Theory.root_path |>
      Theory.add_path (NameSpace.implode prfx);
    val (ty_eqs, rlz_eqs) = split_list
      (map (fn (s, rs) => mk_realizes_eqn (not (s mem rsets)) vs nparms rs) rss);

    val thy1' = thy1 |>
      Theory.copy |>
      Theory.add_types (map (fn s => (Sign.base_name s, ar, NoSyn)) tnames) |>
      fold (fn s => AxClass.axiomatize_arity_i
        (s, replicate ar HOLogic.typeS, HOLogic.typeS)) tnames |>
        Extraction.add_typeof_eqns_i ty_eqs;
    val dts = List.mapPartial (fn (s, rs) => if s mem rsets then
      SOME (dt_of_intrs thy1' vs nparms rs) else NONE) rss;

    (** datatype representing computational content of inductive set **)

    val ((dummies, dt_info), thy2) =
      thy1
      |> add_dummies
           (DatatypePackage.add_datatype_i false false (map #2 dts))
           (map (pair false) dts) []
      ||> Extraction.add_typeof_eqns_i ty_eqs
      ||> Extraction.add_realizes_eqns_i rlz_eqs;
    fun get f = (these oo Option.map) f;
    val rec_names = distinct (op =) (map (fst o dest_Const o head_of o fst o
      HOLogic.dest_eq o HOLogic.dest_Trueprop o prop_of) (get #rec_thms dt_info));
    val (_, constrss) = foldl_map (fn ((recs, dummies), (s, rs)) =>
      if s mem rsets then
        let
          val (d :: dummies') = dummies;
          val (recs1, recs2) = chop (length rs) (if d then tl recs else recs)
        in ((recs2, dummies'), map (head_of o hd o rev o snd o strip_comb o
          fst o HOLogic.dest_eq o HOLogic.dest_Trueprop o prop_of) recs1)
        end
      else ((recs, dummies), replicate (length rs) Extraction.nullt))
        ((get #rec_thms dt_info, dummies), rss);
    val rintrs = map (fn (intr, c) => Envir.eta_contract
      (Extraction.realizes_of thy2 vs
        (if c = Extraction.nullt then c else list_comb (c, map Var (rev
          (Term.add_vars (prop_of intr) []) \\ params'))) (prop_of intr)))
            (maps snd rss ~~ List.concat constrss);
    val (rlzpreds, rlzpreds') = split_list
      (distinct (op = o pairself (#1 o #1)) (map (fn rintr =>
        let
          val Const (s, T) = head_of (HOLogic.dest_Trueprop
            (Logic.strip_assums_concl rintr));
          val s' = Sign.base_name s;
          val T' = Logic.unvarifyT T
        in ((s', SOME T', NoSyn),
          (Const (s, T'), Free (s', T')))
        end) rintrs));
    val rlzparams = map (fn Var ((s, _), T) => (s, SOME (Logic.unvarifyT T)))
      (List.take (snd (strip_comb
        (HOLogic.dest_Trueprop (Logic.strip_assums_concl (hd rintrs)))), nparms));

    (** realizability predicate **)

    val (ind_info, thy3') = thy2 |>
      InductivePackage.add_inductive_global false "" false false false
        rlzpreds rlzparams (map (fn (rintr, intr) =>
          ((Sign.base_name (name_of_thm intr), []),
           subst_atomic rlzpreds' (Logic.unvarify rintr)))
             (rintrs ~~ maps snd rss)) [] ||>
      Theory.absolute_path;
    val thy3 = PureThy.hide_thms false
      (map name_of_thm (#intrs ind_info)) thy3';

    (** realizer for induction rule **)

    val Ps = List.mapPartial (fn _ $ M $ P => if pred_of M mem rsets then
      SOME (fst (fst (dest_Var (head_of P)))) else NONE)
        (HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of raw_induct)));

    fun add_ind_realizer (thy, Ps) =
      let
        val rs = indrule_realizer thy induct raw_induct rsets params'
          (vs @ Ps) rec_names rss' intrs dummies;
        val rlzs = map (fn (r, ind) => Extraction.realizes_of thy (vs @ Ps) r
          (prop_of ind)) (rs ~~ inducts);
        val used = foldr add_term_free_names [] rlzs;
        val rnames = Name.variant_list used (replicate (length inducts) "r");
        val rnames' = Name.variant_list
          (used @ rnames) (replicate (length intrs) "s");
        val rlzs' as (prems, _, _) :: _ = map (fn (rlz, name) =>
          let
            val (P, Q) = strip_one name (Logic.unvarify rlz);
            val Q' = strip_all' [] rnames' Q
          in
            (Logic.strip_imp_prems Q', P, Logic.strip_imp_concl Q')
          end) (rlzs ~~ rnames);
        val concl = HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj (map
          (fn (_, _ $ P, _ $ Q) => HOLogic.mk_imp (P, Q)) rlzs'));
        val rews = map mk_meta_eq
          (fst_conv :: snd_conv :: get #rec_thms dt_info);
        val thm = Goal.prove_global thy [] prems concl (fn prems => EVERY
          [rtac (#raw_induct ind_info) 1,
           rewrite_goals_tac rews,
           REPEAT ((resolve_tac prems THEN_ALL_NEW EVERY'
             [K (rewrite_goals_tac rews), ObjectLogic.atomize_tac,
              DEPTH_SOLVE_1 o FIRST' [atac, etac allE, etac impE]]) 1)]);
        val (thm', thy') = PureThy.store_thm ((space_implode "_"
          (NameSpace.qualified qualifier "induct" :: vs @ Ps @
             ["correctness"]), thm), []) thy;
        val thms = map (fn th => zero_var_indexes (rotate_prems ~1 (th RS mp)))
          (DatatypeAux.split_conj_thm thm');
        val ([thms'], thy'') = PureThy.add_thmss
          [((space_implode "_"
             (NameSpace.qualified qualifier "inducts" :: vs @ Ps @
               ["correctness"]), thms), [])] thy';
        val realizers = inducts ~~ thms' ~~ rlzs ~~ rs;
      in
        Extraction.add_realizers_i
          (map (fn (((ind, corr), rlz), r) =>
              mk_realizer thy' (vs @ Ps) (Thm.get_name ind, ind, corr, rlz, r))
            realizers @ (case realizers of
             [(((ind, corr), rlz), r)] =>
               [mk_realizer thy' (vs @ Ps) (NameSpace.qualified qualifier "induct",
                  ind, corr, rlz, r)]
           | _ => [])) thy''
      end;

    (** realizer for elimination rules **)

    val case_names = map (fst o dest_Const o head_of o fst o HOLogic.dest_eq o
      HOLogic.dest_Trueprop o prop_of o hd) (get #case_thms dt_info);

    fun add_elim_realizer Ps
      (((((elim, elimR), intrs), case_thms), case_name), dummy) thy =
      let
        val (prem :: prems) = prems_of elim;
        fun reorder1 (p, (_, intr)) =
          Library.foldl (fn (t, ((s, _), T)) => all T $ lambda (Free (s, T)) t)
            (strip_all p, Term.add_vars (prop_of intr) [] \\ params');
        fun reorder2 ((ivs, intr), i) =
          let val fs = Term.add_vars (prop_of intr) [] \\ params'
          in Library.foldl (fn (t, x) => lambda (Var x) t)
            (list_comb (Bound (i + length ivs), ivs), fs)
          end;
        val p = Logic.list_implies
          (map reorder1 (prems ~~ intrs) @ [prem], concl_of elim);
        val T' = Extraction.etype_of thy (vs @ Ps) [] p;
        val T = if dummy then (HOLogic.unitT --> body_type T') --> T' else T';
        val Ts = map (Extraction.etype_of thy (vs @ Ps) []) (prems_of elim);
        val r = if null Ps then Extraction.nullt
          else list_abs (map (pair "x") Ts, list_comb (Const (case_name, T),
            (if dummy then
               [Abs ("x", HOLogic.unitT, Const ("arbitrary", body_type T))]
             else []) @
            map reorder2 (intrs ~~ (length prems - 1 downto 0)) @
            [Bound (length prems)]));
        val rlz = Extraction.realizes_of thy (vs @ Ps) r (prop_of elim);
        val rlz' = strip_all (Logic.unvarify rlz);
        val rews = map mk_meta_eq case_thms;
        val thm = Goal.prove_global thy []
          (Logic.strip_imp_prems rlz') (Logic.strip_imp_concl rlz') (fn prems => EVERY
          [cut_facts_tac [hd prems] 1,
           etac elimR 1,
           ALLGOALS (asm_simp_tac HOL_basic_ss),
           rewrite_goals_tac rews,
           REPEAT ((resolve_tac prems THEN_ALL_NEW (ObjectLogic.atomize_tac THEN'
             DEPTH_SOLVE_1 o FIRST' [atac, etac allE, etac impE])) 1)]);
        val (thm', thy') = PureThy.store_thm ((space_implode "_"
          (name_of_thm elim :: vs @ Ps @ ["correctness"]), thm), []) thy
      in
        Extraction.add_realizers_i
          [mk_realizer thy' (vs @ Ps) (name_of_thm elim, elim, thm', rlz, r)] thy'
      end;

    (** add realizers to theory **)

    val thy4 = Library.foldl add_ind_realizer (thy3, subsets Ps);
    val thy5 = Extraction.add_realizers_i
      (map (mk_realizer thy4 vs) (map (fn (((rule, rrule), rlz), c) =>
         (name_of_thm rule, rule, rrule, rlz,
            list_comb (c, map Var (rev (Term.add_vars (prop_of rule) []) \\ params'))))
              (List.concat (map snd rss) ~~ #intrs ind_info ~~ rintrs ~~
                 List.concat constrss))) thy4;
    val elimps = List.mapPartial (fn ((s, intrs), p) =>
      if s mem rsets then SOME (p, intrs) else NONE)
        (rss' ~~ (elims ~~ #elims ind_info));
    val thy6 = Library.foldl (fn (thy, p as (((((elim, _), _), _), _), _)) => thy |>
      add_elim_realizer [] p |> add_elim_realizer [fst (fst (dest_Var
        (HOLogic.dest_Trueprop (concl_of elim))))] p) (thy5,
           elimps ~~ get #case_thms dt_info ~~ case_names ~~ dummies)

  in Theory.restore_naming thy thy6 end;

fun add_ind_realizers name rsets thy =
  let
    val (_, {intrs, induct, raw_induct, elims, ...}) =
      InductivePackage.the_inductive (ProofContext.init thy) name;
    val vss = sort (int_ord o pairself length)
      (subsets (map fst (relevant_vars (concl_of (hd intrs)))))
  in
    Library.foldl (add_ind_realizer rsets intrs induct raw_induct elims) (thy, vss)
  end

fun rlz_attrib arg = Thm.declaration_attribute (fn thm => Context.mapping
  let
    fun err () = error "ind_realizer: bad rule";
    val sets =
      (case HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of thm)) of
           [_] => [pred_of (HOLogic.dest_Trueprop (hd (prems_of thm)))]
         | xs => map (pred_of o fst o HOLogic.dest_imp) xs)
         handle TERM _ => err () | Empty => err ();
  in 
    add_ind_realizers (hd sets)
      (case arg of
        NONE => sets | SOME NONE => []
      | SOME (SOME sets') => sets \\ sets')
  end I);

val ind_realizer = Attrib.syntax
 ((Scan.option (Scan.lift (Args.$$$ "irrelevant") |--
    Scan.option (Scan.lift (Args.colon) |--
      Scan.repeat1 Args.const))) >> rlz_attrib);

val setup =
  Attrib.add_attributes [("ind_realizer", ind_realizer, "add realizers for inductive set")];

end;