1 (* Title: ZF/Tools/inductive_package.ML
3 Author: Lawrence C Paulson, Cambridge University Computer Laboratory
4 Copyright 1994 University of Cambridge
6 Fixedpoint definition module -- for Inductive/Coinductive Definitions
8 The functor will be instantiated for normal sums/products (inductive defs)
9 and non-standard sums/products (coinductive defs)
11 Sums are used only for mutual recursion;
12 Products are used only to derive "streamlined" induction rules for relations
15 type inductive_result =
16 {defs : thm list, (*definitions made in thy*)
17 bnd_mono : thm, (*monotonicity for the lfp definition*)
18 dom_subset : thm, (*inclusion of recursive set in dom*)
19 intrs : thm list, (*introduction rules*)
20 elim : thm, (*case analysis theorem*)
21 mk_cases : string -> thm, (*generates case theorems*)
22 induct : thm, (*main induction rule*)
23 mutual_induct : thm}; (*mutual induction rule*)
26 (*Functor's result signature*)
27 signature INDUCTIVE_PACKAGE =
29 (*Insert definitions for the recursive sets, which
30 must *already* be declared as constants in parent theory!*)
31 val add_inductive_i: bool -> term list * term ->
32 ((bstring * term) * theory attribute list) list ->
33 thm list * thm list * thm list * thm list -> theory -> theory * inductive_result
34 val add_inductive: string list * string ->
35 ((bstring * string) * Attrib.src list) list ->
36 (thmref * Attrib.src list) list * (thmref * Attrib.src list) list *
37 (thmref * Attrib.src list) list * (thmref * Attrib.src list) list ->
38 theory -> theory * inductive_result
42 (*Declares functions to add fixedpoint/constructor defs to a theory.
43 Recursive sets must *already* be declared as constants.*)
44 functor Add_inductive_def_Fun
45 (structure Fp: FP and Pr : PR and CP: CARTPROD and Su : SU val coind: bool)
51 val co_prefix = if coind then "co" else "";
56 (*make distinct individual variables a1, a2, a3, ..., an. *)
57 fun mk_frees a [] = []
58 | mk_frees a (T::Ts) = Free(a,T) :: mk_frees (Symbol.bump_string a) Ts;
61 (* add_inductive(_i) *)
63 (*internal version, accepting terms*)
64 fun add_inductive_i verbose (rec_tms, dom_sum)
65 intr_specs (monos, con_defs, type_intrs, type_elims) thy =
67 val _ = Theory.requires thy "Inductive" "(co)inductive definitions";
68 val sign = sign_of thy;
70 val (intr_names, intr_tms) = split_list (map fst intr_specs);
71 val case_names = RuleCases.case_names intr_names;
73 (*recT and rec_params should agree for all mutually recursive components*)
74 val rec_hds = map head_of rec_tms;
76 val dummy = assert_all is_Const rec_hds
77 (fn t => "Recursive set not previously declared as constant: " ^
78 Sign.string_of_term sign t);
80 (*Now we know they are all Consts, so get their names, type and params*)
81 val rec_names = map (#1 o dest_Const) rec_hds
82 and (Const(_,recT),rec_params) = strip_comb (hd rec_tms);
84 val rec_base_names = map Sign.base_name rec_names;
85 val dummy = assert_all Syntax.is_identifier rec_base_names
86 (fn a => "Base name of recursive set not an identifier: " ^ a);
88 local (*Checking the introduction rules*)
89 val intr_sets = map (#2 o rule_concl_msg sign) intr_tms;
91 case head_of set of Const(a,recT) => a mem rec_names | _ => false;
93 val dummy = assert_all intr_ok intr_sets
94 (fn t => "Conclusion of rule does not name a recursive set: " ^
95 Sign.string_of_term sign t);
98 val dummy = assert_all is_Free rec_params
99 (fn t => "Param in recursion term not a free variable: " ^
100 Sign.string_of_term sign t);
102 (*** Construct the fixedpoint definition ***)
103 val mk_variant = variant (foldr add_term_names [] intr_tms);
105 val z' = mk_variant"z" and X' = mk_variant"X" and w' = mk_variant"w";
107 fun dest_tprop (Const("Trueprop",_) $ P) = P
108 | dest_tprop Q = error ("Ill-formed premise of introduction rule: " ^
109 Sign.string_of_term sign Q);
111 (*Makes a disjunct from an introduction rule*)
112 fun fp_part intr = (*quantify over rule's free vars except parameters*)
113 let val prems = map dest_tprop (Logic.strip_imp_prems intr)
114 val dummy = List.app (fn rec_hd => List.app (chk_prem rec_hd) prems) rec_hds
115 val exfrees = term_frees intr \\ rec_params
116 val zeq = FOLogic.mk_eq (Free(z',iT), #1 (rule_concl intr))
117 in foldr FOLogic.mk_exists
118 (fold_bal FOLogic.mk_conj (zeq::prems)) exfrees
121 (*The Part(A,h) terms -- compose injections to make h*)
122 fun mk_Part (Bound 0) = Free(X',iT) (*no mutual rec, no Part needed*)
123 | mk_Part h = Part_const $ Free(X',iT) $ Abs(w',iT,h);
125 (*Access to balanced disjoint sums via injections*)
127 map mk_Part (accesses_bal (fn t => Su.inl $ t, fn t => Su.inr $ t, Bound 0)
130 (*replace each set by the corresponding Part(A,h)*)
131 val part_intrs = map (subst_free (rec_tms ~~ parts) o fp_part) intr_tms;
133 val fp_abs = absfree(X', iT,
134 mk_Collect(z', dom_sum,
135 fold_bal FOLogic.mk_disj part_intrs));
137 val fp_rhs = Fp.oper $ dom_sum $ fp_abs
139 val dummy = List.app (fn rec_hd => deny (Logic.occs (rec_hd, fp_rhs))
140 "Illegal occurrence of recursion operator")
143 (*** Make the new theory ***)
146 If no mutual recursion then it equals the one recursive set.
147 If mutual recursion then it differs from all the recursive sets. *)
148 val big_rec_base_name = space_implode "_" rec_base_names;
149 val big_rec_name = Sign.intern_const sign big_rec_base_name;
152 val dummy = conditional verbose (fn () =>
153 writeln ((if coind then "Coind" else "Ind") ^ "uctive definition " ^ quote big_rec_name));
155 (*Forbid the inductive definition structure from clashing with a theory
156 name. This restriction may become obsolete as ML is de-emphasized.*)
157 val dummy = deny (big_rec_base_name mem (Context.names_of sign))
158 ("Definition " ^ big_rec_base_name ^
159 " would clash with the theory of the same name!");
161 (*Big_rec... is the union of the mutually recursive sets*)
162 val big_rec_tm = list_comb(Const(big_rec_name,recT), rec_params);
164 (*The individual sets must already be declared*)
165 val axpairs = map Logic.mk_defpair
166 ((big_rec_tm, fp_rhs) ::
168 [_] => [] (*no mutual recursion*)
169 | _ => rec_tms ~~ (*define the sets as Parts*)
170 map (subst_atomic [(Free(X',iT),big_rec_tm)]) parts));
172 (*tracing: print the fixedpoint definition*)
173 val dummy = if !Ind_Syntax.trace then
174 List.app (writeln o Sign.string_of_term sign o #2) axpairs
177 (*add definitions of the inductive sets*)
180 |> Theory.add_path big_rec_base_name
181 |> PureThy.add_defs_i false (map Thm.no_attributes axpairs)
184 (*fetch fp definitions from the theory*)
185 val big_rec_def::part_rec_defs =
187 (case rec_names of [_] => rec_names
188 | _ => big_rec_base_name::rec_names);
191 val sign1 = sign_of thy1;
194 val dummy = writeln " Proving monotonicity...";
197 standard (Goal.prove sign1 [] [] (FOLogic.mk_Trueprop (Fp.bnd_mono $ dom_sum $ fp_abs))
199 [rtac (Collect_subset RS bnd_monoI) 1,
200 REPEAT (ares_tac (basic_monos @ monos) 1)]));
202 val dom_subset = standard (big_rec_def RS Fp.subs);
204 val unfold = standard ([big_rec_def, bnd_mono] MRS Fp.Tarski);
207 val dummy = writeln " Proving the introduction rules...";
209 (*Mutual recursion? Helps to derive subset rules for the
214 | _ => standard (Part_subset RS subset_trans);
216 (*To type-check recursive occurrences of the inductive sets, possibly
217 enclosed in some monotonic operator M.*)
219 [dom_subset] RL (asm_rl :: ([Part_trans] RL monos))
222 (*Type-checking is hardest aspect of proof;
223 disjIn selects the correct disjunct after unfolding*)
224 fun intro_tacsf disjIn =
225 [DETERM (stac unfold 1),
226 REPEAT (resolve_tac [Part_eqI,CollectI] 1),
227 (*Now 2-3 subgoals: typechecking, the disjunction, perhaps equality.*)
229 (*Not ares_tac, since refl must be tried before equality assumptions;
230 backtracking may occur if the premises have extra variables!*)
231 DEPTH_SOLVE_1 (resolve_tac [refl,exI,conjI] 2 APPEND assume_tac 2),
232 (*Now solve the equations like Tcons(a,f) = Inl(?b4)*)
233 rewrite_goals_tac con_defs,
234 REPEAT (rtac refl 2),
235 (*Typechecking; this can fail*)
236 if !Ind_Syntax.trace then print_tac "The type-checking subgoal:"
238 REPEAT (FIRSTGOAL ( dresolve_tac rec_typechecks
239 ORELSE' eresolve_tac (asm_rl::PartE::SigmaE2::
241 ORELSE' hyp_subst_tac)),
242 if !Ind_Syntax.trace then print_tac "The subgoal after monos, type_elims:"
244 DEPTH_SOLVE (swap_res_tac (SigmaI::subsetI::type_intrs) 1)];
246 (*combines disjI1 and disjI2 to get the corresponding nested disjunct...*)
248 let fun f rl = rl RS disjI1
249 and g rl = rl RS disjI2
250 in accesses_bal(f, g, asm_rl) end;
253 (intr_tms, map intro_tacsf (mk_disj_rls (length intr_tms)))
254 |> ListPair.map (fn (t, tacs) =>
255 standard (Goal.prove sign1 [] [] t
256 (fn _ => EVERY (rewrite_goals_tac part_rec_defs :: tacs))))
257 handle MetaSimplifier.SIMPLIFIER (msg, thm) => (print_thm thm; error msg);
260 val dummy = writeln " Proving the elimination rule...";
262 (*Breaks down logical connectives in the monotonic function*)
264 REPEAT (SOMEGOAL (eresolve_tac (Ind_Syntax.elim_rls @ Su.free_SEs)
265 ORELSE' bound_hyp_subst_tac))
266 THEN prune_params_tac
267 (*Mutual recursion: collapse references to Part(D,h)*)
268 THEN fold_tac part_rec_defs;
271 val elim = rule_by_tactic basic_elim_tac
272 (unfold RS Ind_Syntax.equals_CollectD)
274 (*Applies freeness of the given constructors, which *must* be unfolded by
275 the given defs. Cannot simply use the local con_defs because
276 con_defs=[] for inference systems.
277 Proposition A should have the form t:Si where Si is an inductive set*)
278 fun make_cases ss A =
280 (basic_elim_tac THEN ALLGOALS (asm_full_simp_tac ss) THEN basic_elim_tac)
281 (Thm.assume A RS elim)
283 fun mk_cases a = make_cases (*delayed evaluation of body!*)
284 (simpset ()) (read_cterm (Thm.sign_of_thm elim) (a, propT));
286 fun induction_rules raw_induct thy =
288 val dummy = writeln " Proving the induction rule...";
290 (*** Prove the main induction rule ***)
292 val pred_name = "P"; (*name for predicate variables*)
294 (*Used to make induction rules;
295 ind_alist = [(rec_tm1,pred1),...] associates predicates with rec ops
296 prem is a premise of an intr rule*)
297 fun add_induct_prem ind_alist (prem as Const("Trueprop",_) $
298 (Const("op :",_)$t$X), iprems) =
299 (case AList.lookup (op aconv) ind_alist X of
300 SOME pred => prem :: FOLogic.mk_Trueprop (pred $ t) :: iprems
301 | NONE => (*possibly membership in M(rec_tm), for M monotone*)
302 let fun mk_sb (rec_tm,pred) =
303 (rec_tm, Ind_Syntax.Collect_const$rec_tm$pred)
304 in subst_free (map mk_sb ind_alist) prem :: iprems end)
305 | add_induct_prem ind_alist (prem,iprems) = prem :: iprems;
307 (*Make a premise of the induction rule.*)
308 fun induct_prem ind_alist intr =
309 let val quantfrees = map dest_Free (term_frees intr \\ rec_params)
310 val iprems = foldr (add_induct_prem ind_alist) []
311 (Logic.strip_imp_prems intr)
312 val (t,X) = Ind_Syntax.rule_concl intr
313 val (SOME pred) = AList.lookup (op aconv) ind_alist X
314 val concl = FOLogic.mk_Trueprop (pred $ t)
315 in list_all_free (quantfrees, Logic.list_implies (iprems,concl)) end
316 handle Bind => error"Recursion term not found in conclusion";
318 (*Minimizes backtracking by delivering the correct premise to each goal.
319 Intro rules with extra Vars in premises still cause some backtracking *)
320 fun ind_tac [] 0 = all_tac
321 | ind_tac(prem::prems) i =
322 DEPTH_SOLVE_1 (ares_tac [prem, refl] i) THEN ind_tac prems (i-1);
324 val pred = Free(pred_name, Ind_Syntax.iT --> FOLogic.oT);
326 val ind_prems = map (induct_prem (map (rpair pred) rec_tms))
329 val dummy = if !Ind_Syntax.trace then
330 (writeln "ind_prems = ";
331 List.app (writeln o Sign.string_of_term sign1) ind_prems;
332 writeln "raw_induct = "; print_thm raw_induct)
336 (*We use a MINIMAL simpset. Even FOL_ss contains too many simpules.
337 If the premises get simplified, then the proofs could fail.*)
338 val min_ss = Simplifier.theory_context thy empty_ss
339 setmksimps (map mk_eq o ZF_atomize o gen_all)
340 setSolver (mk_solver "minimal"
341 (fn prems => resolve_tac (triv_rls@prems)
343 ORELSE' etac FalseE));
346 standard (Goal.prove sign1 [] ind_prems
347 (FOLogic.mk_Trueprop (Ind_Syntax.mk_all_imp (big_rec_tm, pred)))
349 [rewrite_goals_tac part_rec_defs,
350 rtac (impI RS allI) 1,
351 DETERM (etac raw_induct 1),
352 (*Push Part inside Collect*)
353 full_simp_tac (min_ss addsimps [Part_Collect]) 1,
354 (*This CollectE and disjE separates out the introduction rules*)
355 REPEAT (FIRSTGOAL (eresolve_tac [CollectE, disjE])),
356 (*Now break down the individual cases. No disjE here in case
357 some premise involves disjunction.*)
358 REPEAT (FIRSTGOAL (eresolve_tac [CollectE, exE, conjE]
359 ORELSE' bound_hyp_subst_tac)),
360 ind_tac (rev (map (rewrite_rule part_rec_defs) prems)) (length prems)]));
362 val dummy = if !Ind_Syntax.trace then
363 (writeln "quant_induct = "; print_thm quant_induct)
367 (*** Prove the simultaneous induction rule ***)
369 (*Make distinct predicates for each inductive set*)
371 (*The components of the element type, several if it is a product*)
372 val elem_type = CP.pseudo_type dom_sum;
373 val elem_factors = CP.factors elem_type;
374 val elem_frees = mk_frees "za" elem_factors;
375 val elem_tuple = CP.mk_tuple Pr.pair elem_type elem_frees;
377 (*Given a recursive set and its domain, return the "fsplit" predicate
378 and a conclusion for the simultaneous induction rule.
379 NOTE. This will not work for mutually recursive predicates. Previously
380 a summand 'domt' was also an argument, but this required the domain of
381 mutual recursion to invariably be a disjoint sum.*)
382 fun mk_predpair rec_tm =
383 let val rec_name = (#1 o dest_Const o head_of) rec_tm
384 val pfree = Free(pred_name ^ "_" ^ Sign.base_name rec_name,
385 elem_factors ---> FOLogic.oT)
389 (Ind_Syntax.mem_const $ elem_tuple $ rec_tm)
390 $ (list_comb (pfree, elem_frees))) elem_frees
391 in (CP.ap_split elem_type FOLogic.oT pfree,
395 val (preds,qconcls) = split_list (map mk_predpair rec_tms);
397 (*Used to form simultaneous induction lemma*)
398 fun mk_rec_imp (rec_tm,pred) =
399 FOLogic.imp $ (Ind_Syntax.mem_const $ Bound 0 $ rec_tm) $
402 (*To instantiate the main induction rule*)
405 (Ind_Syntax.mk_all_imp
407 Abs("z", Ind_Syntax.iT,
408 fold_bal FOLogic.mk_conj
409 (ListPair.map mk_rec_imp (rec_tms, preds)))))
410 and mutual_induct_concl =
411 FOLogic.mk_Trueprop(fold_bal FOLogic.mk_conj qconcls);
413 val dummy = if !Ind_Syntax.trace then
414 (writeln ("induct_concl = " ^
415 Sign.string_of_term sign1 induct_concl);
416 writeln ("mutual_induct_concl = " ^
417 Sign.string_of_term sign1 mutual_induct_concl))
421 val lemma_tac = FIRST' [eresolve_tac [asm_rl, conjE, PartE, mp],
422 resolve_tac [allI, impI, conjI, Part_eqI],
423 dresolve_tac [spec, mp, Pr.fsplitD]];
425 val need_mutual = length rec_names > 1;
427 val lemma = (*makes the link between the two induction rules*)
429 (writeln " Proving the mutual induction rule...";
430 standard (Goal.prove sign1 [] []
431 (Logic.mk_implies (induct_concl, mutual_induct_concl))
433 [rewrite_goals_tac part_rec_defs,
434 REPEAT (rewrite_goals_tac [Pr.split_eq] THEN lemma_tac 1)])))
435 else (writeln " [ No mutual induction rule needed ]"; TrueI);
437 val dummy = if !Ind_Syntax.trace then
438 (writeln "lemma = "; print_thm lemma)
442 (*Mutual induction follows by freeness of Inl/Inr.*)
444 (*Simplification largely reduces the mutual induction rule to the
447 min_ss addsimps [Su.distinct, Su.distinct', Su.inl_iff, Su.inr_iff];
449 val all_defs = con_defs @ part_rec_defs;
451 (*Removes Collects caused by M-operators in the intro rules. It is very
453 list({v: tf. (v : t --> P_t(v)) & (v : f --> P_f(v))})
454 where t==Part(tf,Inl) and f==Part(tf,Inr) to list({v: tf. P_t(v)}).
455 Instead the following rules extract the relevant conjunct.
457 val cmonos = [subset_refl RS Collect_mono] RL monos
458 RLN (2,[rev_subsetD]);
460 (*Minimizes backtracking by delivering the correct premise to each goal*)
461 fun mutual_ind_tac [] 0 = all_tac
462 | mutual_ind_tac(prem::prems) i =
466 (*Simplify the assumptions and goal by unfolding Part and
467 using freeness of the Sum constructors; proves all but one
468 conjunct by contradiction*)
469 rewrite_goals_tac all_defs THEN
470 simp_tac (mut_ss addsimps [Part_iff]) 1 THEN
471 IF_UNSOLVED (*simp_tac may have finished it off!*)
472 ((*simplify assumptions*)
473 (*some risk of excessive simplification here -- might have
474 to identify the bare minimum set of rewrites*)
476 (mut_ss addsimps conj_simps @ imp_simps @ quant_simps) 1
478 (*unpackage and use "prem" in the corresponding place*)
479 REPEAT (rtac impI 1) THEN
480 rtac (rewrite_rule all_defs prem) 1 THEN
481 (*prem must not be REPEATed below: could loop!*)
482 DEPTH_SOLVE (FIRSTGOAL (ares_tac [impI] ORELSE'
483 eresolve_tac (conjE::mp::cmonos))))
485 THEN mutual_ind_tac prems (i-1);
487 val mutual_induct_fsplit =
489 standard (Goal.prove sign1 [] (map (induct_prem (rec_tms~~preds)) intr_tms)
492 [rtac (quant_induct RS lemma) 1,
493 mutual_ind_tac (rev prems) (length prems)]))
496 (** Uncurrying the predicate in the ordinary induction rule **)
498 (*instantiate the variable to a tuple, if it is non-trivial, in order to
499 allow the predicate to be "opened up".
500 The name "x.1" comes from the "RS spec" !*)
502 case elem_frees of [_] => I
503 | _ => instantiate ([], [(cterm_of sign1 (Var(("x",1), Ind_Syntax.iT)),
504 cterm_of sign1 elem_tuple)]);
506 (*strip quantifier and the implication*)
507 val induct0 = inst (quant_induct RS spec RSN (2,rev_mp));
509 val Const ("Trueprop", _) $ (pred_var $ _) = concl_of induct0
511 val induct = CP.split_rule_var(pred_var, elem_type-->FOLogic.oT, induct0)
513 and mutual_induct = CP.remove_split mutual_induct_fsplit
515 val ([induct', mutual_induct'], thy') =
517 |> PureThy.add_thms [((co_prefix ^ "induct", induct),
518 [case_names, Attrib.theory (InductAttrib.induct_set big_rec_name)]),
519 (("mutual_induct", mutual_induct), [case_names])];
520 in ((thy', induct'), mutual_induct')
521 end; (*of induction_rules*)
523 val raw_induct = standard ([big_rec_def, bnd_mono] MRS Fp.induct)
525 val ((thy2, induct), mutual_induct) =
526 if not coind then induction_rules raw_induct thy1
529 |> PureThy.add_thms [((co_prefix ^ "induct", raw_induct), [])]
530 |> apfst hd |> Library.swap, TrueI)
531 and defs = big_rec_def :: part_rec_defs
534 val (([bnd_mono', dom_subset', elim'], [defs', intrs']), thy3) =
536 |> IndCases.declare big_rec_name make_cases
538 [(("bnd_mono", bnd_mono), []),
539 (("dom_subset", dom_subset), []),
540 (("cases", elim), [case_names, Attrib.theory (InductAttrib.cases_set big_rec_name)])]
541 ||>> (PureThy.add_thmss o map Thm.no_attributes)
544 val (intrs'', thy4) =
546 |> PureThy.add_thms ((intr_names ~~ intrs') ~~ map #2 intr_specs)
547 ||> Theory.parent_path;
551 bnd_mono = bnd_mono',
552 dom_subset = dom_subset',
557 mutual_induct = mutual_induct})
561 fun add_inductive (srec_tms, sdom_sum) intr_srcs
562 (raw_monos, raw_con_defs, raw_type_intrs, raw_type_elims) thy =
564 val intr_atts = map (map (Attrib.global_attribute thy) o snd) intr_srcs;
565 val sintrs = map fst intr_srcs ~~ intr_atts;
566 val read = Sign.simple_read_term thy;
567 val rec_tms = map (read Ind_Syntax.iT) srec_tms;
568 val dom_sum = read Ind_Syntax.iT sdom_sum;
569 val intr_tms = map (read propT o snd o fst) sintrs;
570 val intr_specs = (map (fst o fst) sintrs ~~ intr_tms) ~~ map snd sintrs;
573 |> IsarThy.apply_theorems raw_monos
574 ||>> IsarThy.apply_theorems raw_con_defs
575 ||>> IsarThy.apply_theorems raw_type_intrs
576 ||>> IsarThy.apply_theorems raw_type_elims
577 |-> (fn (((monos, con_defs), type_intrs), type_elims) =>
578 add_inductive_i true (rec_tms, dom_sum) intr_specs
579 (monos, con_defs, type_intrs, type_elims))
585 local structure P = OuterParse and K = OuterKeyword in
587 fun mk_ind (((((doms, intrs), monos), con_defs), type_intrs), type_elims) =
588 #1 o add_inductive doms (map P.triple_swap intrs) (monos, con_defs, type_intrs, type_elims);
591 (P.$$$ "domains" |-- P.!!! (P.enum1 "+" P.term --
592 ((P.$$$ "\\<subseteq>" || P.$$$ "<=") |-- P.term))) --
594 P.!!! (Scan.repeat1 (P.opt_thm_name ":" -- P.prop))) --
595 Scan.optional (P.$$$ "monos" |-- P.!!! P.xthms1) [] --
596 Scan.optional (P.$$$ "con_defs" |-- P.!!! P.xthms1) [] --
597 Scan.optional (P.$$$ "type_intros" |-- P.!!! P.xthms1) [] --
598 Scan.optional (P.$$$ "type_elims" |-- P.!!! P.xthms1) []
599 >> (Toplevel.theory o mk_ind);
601 val inductiveP = OuterSyntax.command (co_prefix ^ "inductive")
602 ("define " ^ co_prefix ^ "inductive sets") K.thy_decl ind_decl;
604 val _ = OuterSyntax.add_keywords
605 ["domains", "intros", "monos", "con_defs", "type_intros", "type_elims"];
606 val _ = OuterSyntax.add_parsers [inductiveP];