tuned proofs -- clarified flow of facts wrt. calculation;
(* Title: HOL/Tools/inductive_realizer.ML
Author: Stefan Berghofer, TU Muenchen
Program 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: Local_Theory.note should return theorems with proper names! *) (* FIXME ?? *)
fun name_of_thm thm =
(case Proofterm.fold_proof_atoms false (fn PThm (_, ((name, _, _), _)) => cons name | _ => I)
[Thm.proof_of thm] [] of
[name] => name
| _ => error ("name_of_thm: bad proof of theorem\n" ^ Display.string_of_thm_without_context thm));
fun prf_of thm =
Reconstruct.proof_of thm
|> Reconstruct.expand_proof (Thm.theory_of_thm thm) [("", NONE)]; (* FIXME *)
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
[] => (singleton (Name.variant_list 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' (Term.add_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 = fold (fn ((a, i), T) => fn vs =>
(case strip_type T of
(_, Type (s, _)) => if s = @{type_name bool} then (a, T) :: vs else vs
| _ => vs)) (Term.add_vars prop []) [];
val attach_typeS = map_types (map_atyps
(fn TFree (s, []) => TFree (s, HOLogic.typeS)
| TVar (ixn, []) => TVar (ixn, HOLogic.typeS)
| T => T));
fun dt_of_intrs thy vs nparms intrs =
let
val iTs = rev (Term.add_tvars (prop_of (hd intrs)) []);
val (Const (s, _), ts) = strip_comb (HOLogic.dest_Trueprop
(Logic.strip_imp_concl (prop_of (hd intrs))));
val params = map dest_Var (take nparms ts);
val tname = Binding.name (space_implode "_" (Long_Name.base_name s ^ "T" :: vs));
fun constr_of_intr intr = (Binding.name (Long_Name.base_name (name_of_thm intr)),
map (Logic.unvarifyT_global o snd) (subtract (op =) params (rev (Term.add_vars (prop_of intr) []))) @
filter_out (equal Extraction.nullT) (map
(Logic.unvarifyT_global o Extraction.etype_of thy vs []) (prems_of intr)),
NoSyn);
in
((tname, map (rpair dummyS) (map (fn a => "'" ^ a) vs @ map (fst o fst) iTs), 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), [])
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 member (op =) vs a then
fold_rev Term.abs (("r", U) :: xs) (mk_rlz U $ Bound i $ u)
else
fold_rev Term.abs xs (mk_rlz Extraction.nullT $ Extraction.nullt $ u)
end
| gen_rvar _ t = t;
fun mk_realizes_eqn n vs nparms intrs =
let
val intr = map_types Type.strip_sorts (prop_of (hd intrs));
val concl = HOLogic.dest_Trueprop (Logic.strip_imp_concl intr);
val iTs = rev (Term.add_tvars intr []);
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), [])) vs @ Tvs);
val r = if n then Extraction.nullt else Var ((Long_Name.base_name s, 0), rT);
val S = list_comb (h, params @ xs);
val rvs = relevant_vars S;
val vs' = subtract (op =) vs (map fst rvs);
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), [])))
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 = Proof_Context.init_global thy
val args = map (Free o apfst fst o dest_Var) ivs;
val args' = map (Free o apfst fst)
(subtract (op =) params (Term.add_vars (prop_of intr) []));
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;
val is_rec = exists_Const (fn (c, _) => member (op =) rsets c);
fun is_meta (Const ("all", _) $ Abs (s, _, P)) = is_meta P
| is_meta (Const ("==>", _) $ _ $ Q) = is_meta Q
| is_meta (Const (@{const_name Trueprop}, _) $ t) =
(case head_of t of
Const (s, _) => can (Inductive.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 (@{type_name Product_Type.prod}, [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 =
fold_rev (Term.abs o 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 fold_rev (absfree o dest_Free) xs HOLogic.unit
else fold_rev (absfree o dest_Free) xs
(list_comb
(Free ("r" ^ Long_Name.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 = map_filter (fn (s, rs) => if member (op =) rsets s then
SOME (rs, map (fn (_, r) => nth (prems_of raw_induct)
(find_index (fn prp => prp = 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 (@{const_name default},
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" ^ Long_Name.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
(fold_rev absfree (("x", U) :: map_filter (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 (_, ((s, vs, mx), cs))) =
if Binding.eq_name (name, s)
then (true, ((s, vs, mx), (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 Datatype_Aux.Datatype_Empty name' =>
let
val name = Long_Name.base_name name';
val dname = singleton (Name.variant_list used) "Dummy";
in
thy
|> add_dummies f (map (add_dummy (Binding.name name) (Binding.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, _), _) => member (op =) rvs a) 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' = fold_rev Logic.all rs (prop_of rrule)
in (name, (vs,
if rt = Extraction.nullt then rt else fold_rev lambda vs1 rt,
Extraction.abs_corr_shyps thy rule vs vs2
(ProofRewriteRules.un_hhf_proof rlz' (attach_typeS rlz)
(fold_rev Proofterm.forall_intr_proof' rs (prf_of rrule)))))
end;
fun rename tab = map (fn x => the_default x (AList.lookup op = tab x));
fun add_ind_realizer rsets intrs induct raw_induct elims vs thy =
let
val qualifier = Long_Name.qualifier (name_of_thm induct);
val inducts = Global_Theory.get_thms thy (Long_Name.qualify qualifier "inducts");
val iTs = rev (Term.add_tvars (prop_of (hd intrs)) []);
val ar = length vs + length iTs;
val params = Inductive.params_of raw_induct;
val arities = Inductive.arities_of raw_induct;
val nparms = length params;
val params' = map dest_Var params;
val rss = Inductive.partition_rules raw_induct intrs;
val rss' = map (fn (((s, rs), (_, arity)), elim) =>
(s, (Inductive.infer_intro_vars elim arity rs ~~ rs)))
(rss ~~ arities ~~ elims);
val (prfx, _) = split_last (Long_Name.explode (fst (hd rss)));
val tnames = map (fn s => space_implode "_" (s ^ "T" :: vs)) rsets;
val thy1 = thy |>
Sign.root_path |>
Sign.add_path (Long_Name.implode prfx);
val (ty_eqs, rlz_eqs) = split_list
(map (fn (s, rs) => mk_realizes_eqn (not (member (op =) rsets s)) vs nparms rs) rss);
val thy1' = thy1 |>
Sign.add_types_global
(map (fn s => (Binding.name (Long_Name.base_name s), ar, NoSyn)) tnames) |>
Extraction.add_typeof_eqns_i ty_eqs;
val dts = map_filter (fn (s, rs) => if member (op =) rsets s then
SOME (dt_of_intrs thy1' vs nparms rs) else NONE) rss;
(** datatype representing computational content of inductive set **)
val ((dummies, some_dt_names), thy2) =
thy1
|> add_dummies (Datatype.add_datatype {strict = false, quiet = false})
(map (pair false) dts) []
||> Extraction.add_typeof_eqns_i ty_eqs
||> Extraction.add_realizes_eqns_i rlz_eqs;
val dt_names = these some_dt_names;
val case_thms = map (#case_rewrites o Datatype.the_info thy2) dt_names;
val rec_thms =
if null dt_names then []
else #rec_rewrites (Datatype.the_info thy2 (hd dt_names));
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) rec_thms);
val (constrss, _) = fold_map (fn (s, rs) => fn (recs, dummies) =>
if member (op =) rsets s then
let
val (d :: dummies') = dummies;
val (recs1, recs2) = chop (length rs) (if d then tl recs else recs)
in (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, (recs2, dummies'))
end
else (replicate (length rs) Extraction.nullt, (recs, dummies)))
rss (rec_thms, dummies);
val rintrs = map (fn (intr, c) => attach_typeS (Envir.eta_contract
(Extraction.realizes_of thy2 vs
(if c = Extraction.nullt then c else list_comb (c, map Var (rev
(subtract (op =) params' (Term.add_vars (prop_of intr) []))))) (prop_of intr))))
(maps snd rss ~~ flat constrss);
val (rlzpreds, rlzpreds') =
rintrs |> map (fn rintr =>
let
val Const (s, T) = head_of (HOLogic.dest_Trueprop (Logic.strip_assums_concl rintr));
val s' = Long_Name.base_name s;
val T' = Logic.unvarifyT_global T;
in (((s', T'), NoSyn), (Const (s, T'), Free (s', T'))) end)
|> distinct (op = o pairself (#1 o #1))
|> map (apfst (apfst (apfst Binding.name)))
|> split_list;
val rlzparams = map (fn Var ((s, _), T) => (s, Logic.unvarifyT_global T))
(List.take (snd (strip_comb
(HOLogic.dest_Trueprop (Logic.strip_assums_concl (hd rintrs)))), nparms));
(** realizability predicate **)
val (ind_info, thy3') = thy2 |>
Inductive.add_inductive_global
{quiet_mode = false, verbose = false, alt_name = Binding.empty, coind = false,
no_elim = false, no_ind = false, skip_mono = false}
rlzpreds rlzparams (map (fn (rintr, intr) =>
((Binding.name (Long_Name.base_name (name_of_thm intr)), []),
subst_atomic rlzpreds' (Logic.unvarify_global rintr)))
(rintrs ~~ maps snd rss)) [] ||>
Sign.root_path;
val thy3 = fold (Global_Theory.hide_fact false o name_of_thm) (#intrs ind_info) thy3';
(** realizer for induction rule **)
val Ps = map_filter (fn _ $ M $ P => if member (op =) rsets (pred_of M) 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 Ps thy =
let
val vs' = rename (map (pairself (fst o fst o dest_Var))
(params ~~ List.take (snd (strip_comb (HOLogic.dest_Trueprop
(hd (prems_of (hd inducts))))), nparms))) vs;
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 = fold Term.add_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_global 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 (@{thm fst_conv} :: @{thm snd_conv} :: rec_thms);
val thm = Goal.prove_global thy []
(map attach_typeS prems) (attach_typeS 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), Object_Logic.atomize_prems_tac,
DEPTH_SOLVE_1 o FIRST' [atac, etac allE, etac impE]]) 1)]);
val (thm', thy') = Global_Theory.store_thm (Binding.qualified_name (space_implode "_"
(Long_Name.qualify qualifier "induct" :: vs' @ Ps @ ["correctness"])), thm) thy;
val thms = map (fn th => zero_var_indexes (rotate_prems ~1 (th RS mp)))
(Datatype_Aux.split_conj_thm thm');
val ([thms'], thy'') = Global_Theory.add_thmss
[((Binding.qualified_name (space_implode "_"
(Long_Name.qualify 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.derivation_name ind, ind, corr, rlz, r))
realizers @ (case realizers of
[(((ind, corr), rlz), r)] =>
[mk_realizer thy'' (vs' @ Ps) (Long_Name.qualify 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) case_thms;
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)) =
fold (fn ((s, _), T) => Logic.all (Free (s, T)))
(subtract (op =) params' (Term.add_vars (prop_of intr) []))
(strip_all p);
fun reorder2 ((ivs, intr), i) =
let val fs = subtract (op =) params' (Term.add_vars (prop_of intr) [])
in fold (lambda o Var) fs (list_comb (Bound (i + length ivs), ivs)) 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
fold_rev (Term.abs o pair "x") Ts
(list_comb (Const (case_name, T),
(if dummy then
[Abs ("x", HOLogic.unitT, Const (@{const_name default}, 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' = attach_typeS (strip_all (Logic.unvarify_global 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 {context = ctxt, prems, ...} => EVERY
[cut_tac (hd prems) 1,
etac elimR 1,
ALLGOALS (asm_simp_tac (put_simpset HOL_basic_ss ctxt)),
rewrite_goals_tac rews,
REPEAT ((resolve_tac prems THEN_ALL_NEW (Object_Logic.atomize_prems_tac THEN'
DEPTH_SOLVE_1 o FIRST' [atac, etac allE, etac impE])) 1)]);
val (thm', thy') = Global_Theory.store_thm (Binding.qualified_name (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 = fold add_ind_realizer (subsets Ps) thy3;
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 (subtract (op =) params' (rev (Term.add_vars (prop_of rule) []))))))
(maps snd rss ~~ #intrs ind_info ~~ rintrs ~~ flat constrss))) thy4;
val elimps = map_filter (fn ((s, intrs), p) =>
if member (op =) rsets s then SOME (p, intrs) else NONE)
(rss' ~~ (elims ~~ #elims ind_info));
val thy6 =
fold (fn p as (((((elim, _), _), _), _), _) =>
add_elim_realizer [] p #>
add_elim_realizer [fst (fst (dest_Var (HOLogic.dest_Trueprop (concl_of elim))))] p)
(elimps ~~ case_thms ~~ case_names ~~ dummies) thy5;
in Sign.restore_naming thy thy6 end;
fun add_ind_realizers name rsets thy =
let
val (_, {intrs, induct, raw_induct, elims, ...}) =
Inductive.the_inductive (Proof_Context.init_global thy) name;
val vss = sort (int_ord o pairself length)
(subsets (map fst (relevant_vars (concl_of (hd intrs)))))
in
fold_rev (add_ind_realizer rsets intrs induct raw_induct elims) vss thy
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 () | List.Empty => err ();
in
add_ind_realizers (hd sets)
(case arg of
NONE => sets | SOME NONE => []
| SOME (SOME sets') => subtract (op =) sets' sets)
end I);
val setup =
Attrib.setup @{binding ind_realizer}
((Scan.option (Scan.lift (Args.$$$ "irrelevant") |--
Scan.option (Scan.lift (Args.colon) |-- Scan.repeat1 (Args.const true)))) >> rlz_attrib)
"add realizers for inductive set";
end;