(* Title: HOL/Tools/inductive_set.ML
Author: Stefan Berghofer, TU Muenchen
Wrapper for defining inductive sets using package for inductive predicates,
including infrastructure for converting between predicates and sets.
*)
signature INDUCTIVE_SET =
sig
val to_set_att: thm list -> attribute
val to_pred_att: thm list -> attribute
val to_pred : thm list -> Context.generic -> thm -> thm
val pred_set_conv_att: attribute
val add_inductive_i:
Inductive.inductive_flags ->
((binding * typ) * mixfix) list ->
(string * typ) list ->
(Attrib.binding * term) list -> thm list ->
local_theory -> Inductive.inductive_result * local_theory
val add_inductive: bool -> bool ->
(binding * string option * mixfix) list ->
(binding * string option * mixfix) list ->
(Attrib.binding * string) list -> (Facts.ref * Attrib.src list) list ->
bool -> local_theory -> Inductive.inductive_result * local_theory
val codegen_preproc: theory -> thm list -> thm list
val setup: theory -> theory
end;
structure Inductive_Set: INDUCTIVE_SET =
struct
(**** simplify {(x1, ..., xn). (x1, ..., xn) : S} to S ****)
val collect_mem_simproc =
Simplifier.simproc_global @{theory Set} "Collect_mem" ["Collect t"] (fn thy => fn ss =>
fn S as Const (@{const_name Collect}, Type ("fun", [_, T])) $ t =>
let val (u, _, ps) = HOLogic.strip_psplits t
in case u of
(c as Const (@{const_name Set.member}, _)) $ q $ S' =>
(case try (HOLogic.strip_ptuple ps) q of
NONE => NONE
| SOME ts =>
if not (loose_bvar (S', 0)) andalso
ts = map Bound (length ps downto 0)
then
let val simp = full_simp_tac (Simplifier.inherit_context ss
(HOL_basic_ss addsimps [@{thm split_paired_all}, @{thm split_conv}])) 1
in
SOME (Goal.prove (Simplifier.the_context ss) [] []
(Const ("==", T --> T --> propT) $ S $ S')
(K (EVERY
[rtac eq_reflection 1, rtac @{thm subset_antisym} 1,
rtac subsetI 1, dtac CollectD 1, simp,
rtac subsetI 1, rtac CollectI 1, simp])))
end
else NONE)
| _ => NONE
end
| _ => NONE);
(***********************************************************************************)
(* simplifies (%x y. (x, y) : S & P x y) to (%x y. (x, y) : S Int {(x, y). P x y}) *)
(* and (%x y. (x, y) : S | P x y) to (%x y. (x, y) : S Un {(x, y). P x y}) *)
(* used for converting "strong" (co)induction rules *)
(***********************************************************************************)
val anyt = Free ("t", TFree ("'t", []));
fun strong_ind_simproc tab =
Simplifier.simproc_global_i @{theory HOL} "strong_ind" [anyt] (fn thy => fn ss => fn t =>
let
fun close p t f =
let val vs = Term.add_vars t []
in Drule.instantiate' [] (rev (map (SOME o cterm_of thy o Var) vs))
(p (fold (Logic.all o Var) vs t) f)
end;
fun mkop @{const_name HOL.conj} T x =
SOME (Const (@{const_name Lattices.inf}, T --> T --> T), x)
| mkop @{const_name HOL.disj} T x =
SOME (Const (@{const_name Lattices.sup}, T --> T --> T), x)
| mkop _ _ _ = NONE;
fun mk_collect p T t =
let val U = HOLogic.dest_setT T
in HOLogic.Collect_const U $
HOLogic.mk_psplits (HOLogic.flat_tuple_paths p) U HOLogic.boolT t
end;
fun decomp (Const (s, _) $ ((m as Const (@{const_name Set.member},
Type (_, [_, Type (_, [T, _])]))) $ p $ S) $ u) =
mkop s T (m, p, S, mk_collect p T (head_of u))
| decomp (Const (s, _) $ u $ ((m as Const (@{const_name Set.member},
Type (_, [_, Type (_, [T, _])]))) $ p $ S)) =
mkop s T (m, p, mk_collect p T (head_of u), S)
| decomp _ = NONE;
val simp = full_simp_tac (Simplifier.inherit_context ss
(HOL_basic_ss addsimps [mem_Collect_eq, @{thm split_conv}])) 1;
fun mk_rew t = (case strip_abs_vars t of
[] => NONE
| xs => (case decomp (strip_abs_body t) of
NONE => NONE
| SOME (bop, (m, p, S, S')) =>
SOME (close (Goal.prove (Simplifier.the_context ss) [] [])
(Logic.mk_equals (t, list_abs (xs, m $ p $ (bop $ S $ S'))))
(K (EVERY
[rtac eq_reflection 1, REPEAT (rtac ext 1), rtac iffI 1,
EVERY [etac conjE 1, rtac IntI 1, simp, simp,
etac IntE 1, rtac conjI 1, simp, simp] ORELSE
EVERY [etac disjE 1, rtac UnI1 1, simp, rtac UnI2 1, simp,
etac UnE 1, rtac disjI1 1, simp, rtac disjI2 1, simp]])))
handle ERROR _ => NONE))
in
case strip_comb t of
(h as Const (name, _), ts) => (case Symtab.lookup tab name of
SOME _ =>
let val rews = map mk_rew ts
in
if forall is_none rews then NONE
else SOME (fold (fn th1 => fn th2 => Thm.combination th2 th1)
(map2 (fn SOME r => K r | NONE => Thm.reflexive o cterm_of thy)
rews ts) (Thm.reflexive (cterm_of thy h)))
end
| NONE => NONE)
| _ => NONE
end);
(* only eta contract terms occurring as arguments of functions satisfying p *)
fun eta_contract p =
let
fun eta b (Abs (a, T, body)) =
(case eta b body of
body' as (f $ Bound 0) =>
if loose_bvar1 (f, 0) orelse not b then Abs (a, T, body')
else incr_boundvars ~1 f
| body' => Abs (a, T, body'))
| eta b (t $ u) = eta b t $ eta (p (head_of t)) u
| eta b t = t
in eta false end;
fun eta_contract_thm p =
Conv.fconv_rule (Conv.then_conv (Thm.beta_conversion true, fn ct =>
Thm.transitive (Thm.eta_conversion ct)
(Thm.symmetric (Thm.eta_conversion
(cterm_of (theory_of_cterm ct) (eta_contract p (term_of ct)))))));
(***********************************************************)
(* rules for converting between predicate and set notation *)
(* *)
(* rules for converting predicates to sets have the form *)
(* P (%x y. (x, y) : s) = (%x y. (x, y) : S s) *)
(* *)
(* rules for converting sets to predicates have the form *)
(* S {(x, y). p x y} = {(x, y). P p x y} *)
(* *)
(* where s and p are parameters *)
(***********************************************************)
structure PredSetConvData = Generic_Data
(
type T =
{(* rules for converting predicates to sets *)
to_set_simps: thm list,
(* rules for converting sets to predicates *)
to_pred_simps: thm list,
(* arities of functions of type t set => ... => u set *)
set_arities: (typ * (int list list option list * int list list option)) list Symtab.table,
(* arities of functions of type (t => ... => bool) => u => ... => bool *)
pred_arities: (typ * (int list list option list * int list list option)) list Symtab.table};
val empty = {to_set_simps = [], to_pred_simps = [],
set_arities = Symtab.empty, pred_arities = Symtab.empty};
val extend = I;
fun merge
({to_set_simps = to_set_simps1, to_pred_simps = to_pred_simps1,
set_arities = set_arities1, pred_arities = pred_arities1},
{to_set_simps = to_set_simps2, to_pred_simps = to_pred_simps2,
set_arities = set_arities2, pred_arities = pred_arities2}) : T =
{to_set_simps = Thm.merge_thms (to_set_simps1, to_set_simps2),
to_pred_simps = Thm.merge_thms (to_pred_simps1, to_pred_simps2),
set_arities = Symtab.merge_list op = (set_arities1, set_arities2),
pred_arities = Symtab.merge_list op = (pred_arities1, pred_arities2)};
);
fun name_type_of (Free p) = SOME p
| name_type_of (Const p) = SOME p
| name_type_of _ = NONE;
fun map_type f (Free (s, T)) = Free (s, f T)
| map_type f (Var (ixn, T)) = Var (ixn, f T)
| map_type f _ = error "map_type";
fun find_most_specific is_inst f eq xs T =
find_first (fn U => is_inst (T, f U)
andalso forall (fn U' => eq (f U, f U') orelse not
(is_inst (T, f U') andalso is_inst (f U', f U)))
xs) xs;
fun lookup_arity thy arities (s, T) = case Symtab.lookup arities s of
NONE => NONE
| SOME xs => find_most_specific (Sign.typ_instance thy) fst (op =) xs T;
fun lookup_rule thy f rules = find_most_specific
(swap #> Pattern.matches thy) (f #> fst) (op aconv) rules;
fun infer_arities thy arities (optf, t) fs = case strip_comb t of
(Abs (s, T, u), []) => infer_arities thy arities (NONE, u) fs
| (Abs _, _) => infer_arities thy arities (NONE, Envir.beta_norm t) fs
| (u, ts) => (case Option.map (lookup_arity thy arities) (name_type_of u) of
SOME (SOME (_, (arity, _))) =>
(fold (infer_arities thy arities) (arity ~~ List.take (ts, length arity)) fs
handle Subscript => error "infer_arities: bad term")
| _ => fold (infer_arities thy arities) (map (pair NONE) ts)
(case optf of
NONE => fs
| SOME f => AList.update op = (u, the_default f
(Option.map (fn g => inter (op =) g f) (AList.lookup op = fs u))) fs));
(**************************************************************)
(* derive the to_pred equation from the to_set equation *)
(* *)
(* 1. instantiate each set parameter with {(x, y). p x y} *)
(* 2. apply %P. {(x, y). P x y} to both sides of the equation *)
(* 3. simplify *)
(**************************************************************)
fun mk_to_pred_inst thy fs =
map (fn (x, ps) =>
let
val U = HOLogic.dest_setT (fastype_of x);
val x' = map_type (K (HOLogic.strip_ptupleT ps U ---> HOLogic.boolT)) x;
in
(cterm_of thy x,
cterm_of thy (HOLogic.Collect_const U $
HOLogic.mk_psplits ps U HOLogic.boolT x'))
end) fs;
fun mk_to_pred_eq p fs optfs' T thm =
let
val thy = theory_of_thm thm;
val insts = mk_to_pred_inst thy fs;
val thm' = Thm.instantiate ([], insts) thm;
val thm'' = (case optfs' of
NONE => thm' RS sym
| SOME fs' =>
let
val (_, U) = split_last (binder_types T);
val Ts = HOLogic.strip_ptupleT fs' U;
(* FIXME: should cterm_instantiate increment indexes? *)
val arg_cong' = Thm.incr_indexes (Thm.maxidx_of thm + 1) arg_cong;
val (arg_cong_f, _) = arg_cong' |> cprop_of |> Drule.strip_imp_concl |>
Thm.dest_comb |> snd |> Drule.strip_comb |> snd |> hd |> Thm.dest_comb
in
thm' RS (Drule.cterm_instantiate [(arg_cong_f,
cterm_of thy (Abs ("P", Ts ---> HOLogic.boolT,
HOLogic.Collect_const U $ HOLogic.mk_psplits fs' U
HOLogic.boolT (Bound 0))))] arg_cong' RS sym)
end)
in
Simplifier.simplify (HOL_basic_ss addsimps [mem_Collect_eq, @{thm split_conv}]
addsimprocs [collect_mem_simproc]) thm'' |>
zero_var_indexes |> eta_contract_thm (equal p)
end;
(**** declare rules for converting predicates to sets ****)
fun add ctxt thm (tab as {to_set_simps, to_pred_simps, set_arities, pred_arities}) =
case prop_of thm of
Const (@{const_name Trueprop}, _) $ (Const (@{const_name HOL.eq}, Type (_, [T, _])) $ lhs $ rhs) =>
(case body_type T of
@{typ bool} =>
let
val thy = Context.theory_of ctxt;
fun factors_of t fs = case strip_abs_body t of
Const (@{const_name Set.member}, _) $ u $ S =>
if is_Free S orelse is_Var S then
let val ps = HOLogic.flat_tuple_paths u
in (SOME ps, (S, ps) :: fs) end
else (NONE, fs)
| _ => (NONE, fs);
val (h, ts) = strip_comb lhs
val (pfs, fs) = fold_map factors_of ts [];
val ((h', ts'), fs') = (case rhs of
Abs _ => (case strip_abs_body rhs of
Const (@{const_name Set.member}, _) $ u $ S =>
(strip_comb S, SOME (HOLogic.flat_tuple_paths u))
| _ => error "member symbol on right-hand side expected")
| _ => (strip_comb rhs, NONE))
in
case (name_type_of h, name_type_of h') of
(SOME (s, T), SOME (s', T')) =>
if exists (fn (U, _) =>
Sign.typ_instance thy (T', U) andalso
Sign.typ_instance thy (U, T'))
(Symtab.lookup_list set_arities s')
then
(warning ("Ignoring conversion rule for operator " ^ s'); tab)
else
{to_set_simps = thm :: to_set_simps,
to_pred_simps =
mk_to_pred_eq h fs fs' T' thm :: to_pred_simps,
set_arities = Symtab.insert_list op = (s',
(T', (map (AList.lookup op = fs) ts', fs'))) set_arities,
pred_arities = Symtab.insert_list op = (s,
(T, (pfs, fs'))) pred_arities}
| _ => error "set / predicate constant expected"
end
| _ => error "equation between predicates expected")
| _ => error "equation expected";
val pred_set_conv_att = Thm.declaration_attribute
(fn thm => fn ctxt => PredSetConvData.map (add ctxt thm) ctxt);
(**** convert theorem in set notation to predicate notation ****)
fun is_pred tab t =
case Option.map (Symtab.lookup tab o fst) (name_type_of t) of
SOME (SOME _) => true | _ => false;
fun to_pred_simproc rules =
let val rules' = map mk_meta_eq rules
in
Simplifier.simproc_global_i @{theory HOL} "to_pred" [anyt]
(fn thy => K (lookup_rule thy (prop_of #> Logic.dest_equals) rules'))
end;
fun to_pred_proc thy rules t = case lookup_rule thy I rules t of
NONE => NONE
| SOME (lhs, rhs) =>
SOME (Envir.subst_term
(Pattern.match thy (lhs, t) (Vartab.empty, Vartab.empty)) rhs);
fun to_pred thms ctxt thm =
let
val thy = Context.theory_of ctxt;
val {to_pred_simps, set_arities, pred_arities, ...} =
fold (add ctxt) thms (PredSetConvData.get ctxt);
val fs = filter (is_Var o fst)
(infer_arities thy set_arities (NONE, prop_of thm) []);
(* instantiate each set parameter with {(x, y). p x y} *)
val insts = mk_to_pred_inst thy fs
in
thm |>
Thm.instantiate ([], insts) |>
Simplifier.full_simplify (HOL_basic_ss addsimprocs
[to_pred_simproc (mem_Collect_eq :: @{thm split_conv} :: to_pred_simps)]) |>
eta_contract_thm (is_pred pred_arities) |>
Rule_Cases.save thm
end;
val to_pred_att = Thm.rule_attribute o to_pred;
(**** convert theorem in predicate notation to set notation ****)
fun to_set thms ctxt thm =
let
val thy = Context.theory_of ctxt;
val {to_set_simps, pred_arities, ...} =
fold (add ctxt) thms (PredSetConvData.get ctxt);
val fs = filter (is_Var o fst)
(infer_arities thy pred_arities (NONE, prop_of thm) []);
(* instantiate each predicate parameter with %x y. (x, y) : s *)
val insts = map (fn (x, ps) =>
let
val Ts = binder_types (fastype_of x);
val T = HOLogic.mk_ptupleT ps Ts;
val x' = map_type (K (HOLogic.mk_setT T)) x
in
(cterm_of thy x,
cterm_of thy (list_abs (map (pair "x") Ts, HOLogic.mk_mem
(HOLogic.mk_ptuple ps T (map Bound (length ps downto 0)), x'))))
end) fs
in
thm |>
Thm.instantiate ([], insts) |>
Simplifier.full_simplify (HOL_basic_ss addsimps to_set_simps
addsimprocs [strong_ind_simproc pred_arities, collect_mem_simproc]) |>
Rule_Cases.save thm
end;
val to_set_att = Thm.rule_attribute o to_set;
(**** preprocessor for code generator ****)
fun codegen_preproc thy =
let
val {to_pred_simps, set_arities, pred_arities, ...} =
PredSetConvData.get (Context.Theory thy);
fun preproc thm =
if exists_Const (fn (s, _) => case Symtab.lookup set_arities s of
NONE => false
| SOME arities => exists (fn (_, (xs, _)) =>
forall is_none xs) arities) (prop_of thm)
then
thm |>
Simplifier.full_simplify (HOL_basic_ss addsimprocs
[to_pred_simproc (mem_Collect_eq :: @{thm split_conv} :: to_pred_simps)]) |>
eta_contract_thm (is_pred pred_arities)
else thm
in map preproc end;
fun code_ind_att optmod = to_pred_att [] #> Inductive_Codegen.add optmod NONE;
(**** definition of inductive sets ****)
fun add_ind_set_def
{quiet_mode, verbose, alt_name, coind, no_elim, no_ind, skip_mono, fork_mono}
cs intros monos params cnames_syn lthy =
let
val thy = ProofContext.theory_of lthy;
val {set_arities, pred_arities, to_pred_simps, ...} =
PredSetConvData.get (Context.Proof lthy);
fun infer (Abs (_, _, t)) = infer t
| infer (Const (@{const_name Set.member}, _) $ t $ u) =
infer_arities thy set_arities (SOME (HOLogic.flat_tuple_paths t), u)
| infer (t $ u) = infer t #> infer u
| infer _ = I;
val new_arities = filter_out
(fn (x as Free (_, T), _) => member (op =) params x andalso length (binder_types T) > 1
| _ => false) (fold (snd #> infer) intros []);
val params' = map (fn x =>
(case AList.lookup op = new_arities x of
SOME fs =>
let
val T = HOLogic.dest_setT (fastype_of x);
val Ts = HOLogic.strip_ptupleT fs T;
val x' = map_type (K (Ts ---> HOLogic.boolT)) x
in
(x, (x',
(HOLogic.Collect_const T $
HOLogic.mk_psplits fs T HOLogic.boolT x',
list_abs (map (pair "x") Ts, HOLogic.mk_mem
(HOLogic.mk_ptuple fs T (map Bound (length fs downto 0)),
x)))))
end
| NONE => (x, (x, (x, x))))) params;
val (params1, (params2, params3)) =
params' |> map snd |> split_list ||> split_list;
val paramTs = map fastype_of params;
(* equations for converting sets to predicates *)
val ((cs', cs_info), eqns) = cs |> map (fn c as Free (s, T) =>
let
val fs = the_default [] (AList.lookup op = new_arities c);
val (Us, U) = split_last (binder_types T);
val _ = Us = paramTs orelse error (Pretty.string_of (Pretty.chunks
[Pretty.str "Argument types",
Pretty.block (Pretty.commas (map (Syntax.pretty_typ lthy) Us)),
Pretty.str ("of " ^ s ^ " do not agree with types"),
Pretty.block (Pretty.commas (map (Syntax.pretty_typ lthy) paramTs)),
Pretty.str "of declared parameters"]));
val Ts = HOLogic.strip_ptupleT fs U;
val c' = Free (s ^ "p",
map fastype_of params1 @ Ts ---> HOLogic.boolT)
in
((c', (fs, U, Ts)),
(list_comb (c, params2),
HOLogic.Collect_const U $ HOLogic.mk_psplits fs U HOLogic.boolT
(list_comb (c', params1))))
end) |> split_list |>> split_list;
val eqns' = eqns @
map (prop_of #> HOLogic.dest_Trueprop #> HOLogic.dest_eq)
(mem_Collect_eq :: @{thm split_conv} :: to_pred_simps);
(* predicate version of the introduction rules *)
val intros' =
map (fn (name_atts, t) => (name_atts,
t |>
map_aterms (fn u =>
(case AList.lookup op = params' u of
SOME (_, (u', _)) => u'
| NONE => u)) |>
Pattern.rewrite_term thy [] [to_pred_proc thy eqns'] |>
eta_contract (member op = cs' orf is_pred pred_arities))) intros;
val cnames_syn' = map (fn (b, _) => (Binding.suffix_name "p" b, NoSyn)) cnames_syn;
val monos' = map (to_pred [] (Context.Proof lthy)) monos;
val ({preds, intrs, elims, raw_induct, eqs, ...}, lthy1) =
Inductive.add_ind_def
{quiet_mode = quiet_mode, verbose = verbose, alt_name = Binding.empty,
coind = coind, no_elim = no_elim, no_ind = no_ind,
skip_mono = skip_mono, fork_mono = fork_mono}
cs' intros' monos' params1 cnames_syn' lthy;
(* define inductive sets using previously defined predicates *)
val (defs, lthy2) = lthy1
|> Local_Theory.conceal (* FIXME ?? *)
|> fold_map Local_Theory.define
(map (fn ((c_syn, (fs, U, _)), p) => (c_syn, (Attrib.empty_binding,
fold_rev lambda params (HOLogic.Collect_const U $
HOLogic.mk_psplits fs U HOLogic.boolT (list_comb (p, params3))))))
(cnames_syn ~~ cs_info ~~ preds))
||> Local_Theory.restore_naming lthy1;
(* prove theorems for converting predicate to set notation *)
val lthy3 = fold
(fn (((p, c as Free (s, _)), (fs, U, Ts)), (_, (_, def))) => fn lthy =>
let val conv_thm =
Goal.prove lthy (map (fst o dest_Free) params) []
(HOLogic.mk_Trueprop (HOLogic.mk_eq
(list_comb (p, params3),
list_abs (map (pair "x") Ts, HOLogic.mk_mem
(HOLogic.mk_ptuple fs U (map Bound (length fs downto 0)),
list_comb (c, params))))))
(K (REPEAT (rtac ext 1) THEN simp_tac (HOL_basic_ss addsimps
[def, mem_Collect_eq, @{thm split_conv}]) 1))
in
lthy |> Local_Theory.note ((Binding.name (s ^ "p_" ^ s ^ "_eq"),
[Attrib.internal (K pred_set_conv_att)]),
[conv_thm]) |> snd
end) (preds ~~ cs ~~ cs_info ~~ defs) lthy2;
(* convert theorems to set notation *)
val rec_name =
if Binding.is_empty alt_name then
Binding.name (space_implode "_" (map (Binding.name_of o fst) cnames_syn))
else alt_name;
val cnames = map (Local_Theory.full_name lthy3 o #1) cnames_syn; (* FIXME *)
val (intr_names, intr_atts) = split_list (map fst intros);
val raw_induct' = to_set [] (Context.Proof lthy3) raw_induct;
val (intrs', elims', eqs', induct, inducts, lthy4) =
Inductive.declare_rules rec_name coind no_ind cnames (map fst defs)
(map (to_set [] (Context.Proof lthy3)) intrs) intr_names intr_atts
(map (fn th => (to_set [] (Context.Proof lthy3) th,
map fst (fst (Rule_Cases.get th)),
Rule_Cases.get_constraints th)) elims)
(map (to_set [] (Context.Proof lthy3)) eqs) raw_induct' lthy3;
in
({intrs = intrs', elims = elims', induct = induct, inducts = inducts,
raw_induct = raw_induct', preds = map fst defs, eqs = eqs'},
lthy4)
end;
val add_inductive_i = Inductive.gen_add_inductive_i add_ind_set_def;
val add_inductive = Inductive.gen_add_inductive add_ind_set_def;
val mono_add_att = to_pred_att [] #> Inductive.mono_add;
val mono_del_att = to_pred_att [] #> Inductive.mono_del;
(** package setup **)
(* setup theory *)
val setup =
Attrib.setup @{binding pred_set_conv} (Scan.succeed pred_set_conv_att)
"declare rules for converting between predicate and set notation" #>
Attrib.setup @{binding to_set} (Attrib.thms >> to_set_att)
"convert rule to set notation" #>
Attrib.setup @{binding to_pred} (Attrib.thms >> to_pred_att)
"convert rule to predicate notation" #>
Attrib.setup @{binding code_ind_set}
(Scan.lift (Scan.option (Args.$$$ "target" |-- Args.colon |-- Args.name) >> code_ind_att))
"introduction rules for executable predicates" #>
Codegen.add_preprocessor codegen_preproc #>
Attrib.setup @{binding mono_set} (Attrib.add_del mono_add_att mono_del_att)
"declaration of monotonicity rule for set operators" #>
Context.theory_map (Simplifier.map_ss (fn ss => ss addsimprocs [collect_mem_simproc]));
(* outer syntax *)
val ind_set_decl = Inductive.gen_ind_decl add_ind_set_def;
val _ =
Outer_Syntax.local_theory' "inductive_set" "define inductive sets" Keyword.thy_decl
(ind_set_decl false);
val _ =
Outer_Syntax.local_theory' "coinductive_set" "define coinductive sets" Keyword.thy_decl
(ind_set_decl true);
end;