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_x: string list * string -> ((bstring * string) * theory attribute list) list
35 -> thm list * thm list * thm list * thm list -> theory -> theory * inductive_result
36 val add_inductive: string list * string ->
37 ((bstring * string) * Attrib.src list) list ->
38 (thmref * Attrib.src list) list * (thmref * Attrib.src list) list *
39 (thmref * Attrib.src list) list * (thmref * Attrib.src list) list ->
40 theory -> theory * inductive_result
44 (*Declares functions to add fixedpoint/constructor defs to a theory.
45 Recursive sets must *already* be declared as constants.*)
46 functor Add_inductive_def_Fun
47 (structure Fp: FP and Pr : PR and CP: CARTPROD and Su : SU val coind: bool)
51 open Logic Ind_Syntax;
53 val co_prefix = if coind then "co" else "";
58 (*make distinct individual variables a1, a2, a3, ..., an. *)
59 fun mk_frees a [] = []
60 | mk_frees a (T::Ts) = Free(a,T) :: mk_frees (Symbol.bump_string a) Ts;
63 (* add_inductive(_i) *)
65 (*internal version, accepting terms*)
66 fun add_inductive_i verbose (rec_tms, dom_sum)
67 intr_specs (monos, con_defs, type_intrs, type_elims) thy =
69 val _ = Theory.requires thy "Inductive" "(co)inductive definitions";
70 val sign = sign_of thy;
72 val (intr_names, intr_tms) = split_list (map fst intr_specs);
73 val case_names = RuleCases.case_names intr_names;
75 (*recT and rec_params should agree for all mutually recursive components*)
76 val rec_hds = map head_of rec_tms;
78 val dummy = assert_all is_Const rec_hds
79 (fn t => "Recursive set not previously declared as constant: " ^
80 Sign.string_of_term sign t);
82 (*Now we know they are all Consts, so get their names, type and params*)
83 val rec_names = map (#1 o dest_Const) rec_hds
84 and (Const(_,recT),rec_params) = strip_comb (hd rec_tms);
86 val rec_base_names = map Sign.base_name rec_names;
87 val dummy = assert_all Syntax.is_identifier rec_base_names
88 (fn a => "Base name of recursive set not an identifier: " ^ a);
90 local (*Checking the introduction rules*)
91 val intr_sets = map (#2 o rule_concl_msg sign) intr_tms;
93 case head_of set of Const(a,recT) => a mem rec_names | _ => false;
95 val dummy = assert_all intr_ok intr_sets
96 (fn t => "Conclusion of rule does not name a recursive set: " ^
97 Sign.string_of_term sign t);
100 val dummy = assert_all is_Free rec_params
101 (fn t => "Param in recursion term not a free variable: " ^
102 Sign.string_of_term sign t);
104 (*** Construct the fixedpoint definition ***)
105 val mk_variant = variant (foldr add_term_names [] intr_tms);
107 val z' = mk_variant"z" and X' = mk_variant"X" and w' = mk_variant"w";
109 fun dest_tprop (Const("Trueprop",_) $ P) = P
110 | dest_tprop Q = error ("Ill-formed premise of introduction rule: " ^
111 Sign.string_of_term sign Q);
113 (*Makes a disjunct from an introduction rule*)
114 fun fp_part intr = (*quantify over rule's free vars except parameters*)
115 let val prems = map dest_tprop (strip_imp_prems intr)
116 val dummy = List.app (fn rec_hd => List.app (chk_prem rec_hd) prems) rec_hds
117 val exfrees = term_frees intr \\ rec_params
118 val zeq = FOLogic.mk_eq (Free(z',iT), #1 (rule_concl intr))
119 in foldr FOLogic.mk_exists
120 (fold_bal FOLogic.mk_conj (zeq::prems)) exfrees
123 (*The Part(A,h) terms -- compose injections to make h*)
124 fun mk_Part (Bound 0) = Free(X',iT) (*no mutual rec, no Part needed*)
125 | mk_Part h = Part_const $ Free(X',iT) $ Abs(w',iT,h);
127 (*Access to balanced disjoint sums via injections*)
129 map mk_Part (accesses_bal (fn t => Su.inl $ t, fn t => Su.inr $ t, Bound 0)
132 (*replace each set by the corresponding Part(A,h)*)
133 val part_intrs = map (subst_free (rec_tms ~~ parts) o fp_part) intr_tms;
135 val fp_abs = absfree(X', iT,
136 mk_Collect(z', dom_sum,
137 fold_bal FOLogic.mk_disj part_intrs));
139 val fp_rhs = Fp.oper $ dom_sum $ fp_abs
141 val dummy = List.app (fn rec_hd => deny (rec_hd occs fp_rhs)
142 "Illegal occurrence of recursion operator")
145 (*** Make the new theory ***)
148 If no mutual recursion then it equals the one recursive set.
149 If mutual recursion then it differs from all the recursive sets. *)
150 val big_rec_base_name = space_implode "_" rec_base_names;
151 val big_rec_name = Sign.intern_const sign big_rec_base_name;
154 val dummy = conditional verbose (fn () =>
155 writeln ((if coind then "Coind" else "Ind") ^ "uctive definition " ^ quote big_rec_name));
157 (*Forbid the inductive definition structure from clashing with a theory
158 name. This restriction may become obsolete as ML is de-emphasized.*)
159 val dummy = deny (big_rec_base_name mem (Sign.stamp_names_of sign))
160 ("Definition " ^ big_rec_base_name ^
161 " would clash with the theory of the same name!");
163 (*Big_rec... is the union of the mutually recursive sets*)
164 val big_rec_tm = list_comb(Const(big_rec_name,recT), rec_params);
166 (*The individual sets must already be declared*)
167 val axpairs = map Logic.mk_defpair
168 ((big_rec_tm, fp_rhs) ::
170 [_] => [] (*no mutual recursion*)
171 | _ => rec_tms ~~ (*define the sets as Parts*)
172 map (subst_atomic [(Free(X',iT),big_rec_tm)]) parts));
174 (*tracing: print the fixedpoint definition*)
175 val dummy = if !Ind_Syntax.trace then
176 List.app (writeln o Sign.string_of_term sign o #2) axpairs
179 (*add definitions of the inductive sets*)
180 val thy1 = thy |> Theory.add_path big_rec_base_name
181 |> (#1 o 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...";
199 (FOLogic.mk_Trueprop (Fp.bnd_mono $ dom_sum $ fp_abs)))
201 [rtac (Collect_subset RS bnd_monoI) 1,
202 REPEAT (ares_tac (basic_monos @ monos) 1)]);
204 val dom_subset = standard (big_rec_def RS Fp.subs);
206 val unfold = standard ([big_rec_def, bnd_mono] MRS Fp.Tarski);
209 val dummy = writeln " Proving the introduction rules...";
211 (*Mutual recursion? Helps to derive subset rules for the
216 | _ => standard (Part_subset RS subset_trans);
218 (*To type-check recursive occurrences of the inductive sets, possibly
219 enclosed in some monotonic operator M.*)
221 [dom_subset] RL (asm_rl :: ([Part_trans] RL monos))
224 (*Type-checking is hardest aspect of proof;
225 disjIn selects the correct disjunct after unfolding*)
226 fun intro_tacsf disjIn prems =
227 [(*insert prems and underlying sets*)
228 cut_facts_tac prems 1,
229 DETERM (stac unfold 1),
230 REPEAT (resolve_tac [Part_eqI,CollectI] 1),
231 (*Now 2-3 subgoals: typechecking, the disjunction, perhaps equality.*)
233 (*Not ares_tac, since refl must be tried before equality assumptions;
234 backtracking may occur if the premises have extra variables!*)
235 DEPTH_SOLVE_1 (resolve_tac [refl,exI,conjI] 2 APPEND assume_tac 2),
236 (*Now solve the equations like Tcons(a,f) = Inl(?b4)*)
237 rewrite_goals_tac con_defs,
238 REPEAT (rtac refl 2),
239 (*Typechecking; this can fail*)
240 if !Ind_Syntax.trace then print_tac "The type-checking subgoal:"
242 REPEAT (FIRSTGOAL ( dresolve_tac rec_typechecks
243 ORELSE' eresolve_tac (asm_rl::PartE::SigmaE2::
245 ORELSE' hyp_subst_tac)),
246 if !Ind_Syntax.trace then print_tac "The subgoal after monos, type_elims:"
248 DEPTH_SOLVE (swap_res_tac (SigmaI::subsetI::type_intrs) 1)];
250 (*combines disjI1 and disjI2 to get the corresponding nested disjunct...*)
252 let fun f rl = rl RS disjI1
253 and g rl = rl RS disjI2
254 in accesses_bal(f, g, asm_rl) end;
256 fun prove_intr (ct, tacsf) = prove_goalw_cterm part_rec_defs ct tacsf;
258 val intrs = ListPair.map prove_intr
259 (map (cterm_of sign1) intr_tms,
260 map intro_tacsf (mk_disj_rls(length intr_tms)))
261 handle MetaSimplifier.SIMPLIFIER (msg,thm) => (print_thm thm; error msg);
264 val dummy = writeln " Proving the elimination rule...";
266 (*Breaks down logical connectives in the monotonic function*)
268 REPEAT (SOMEGOAL (eresolve_tac (Ind_Syntax.elim_rls @ Su.free_SEs)
269 ORELSE' bound_hyp_subst_tac))
270 THEN prune_params_tac
271 (*Mutual recursion: collapse references to Part(D,h)*)
272 THEN fold_tac part_rec_defs;
275 val elim = rule_by_tactic basic_elim_tac
276 (unfold RS Ind_Syntax.equals_CollectD)
278 (*Applies freeness of the given constructors, which *must* be unfolded by
279 the given defs. Cannot simply use the local con_defs because
280 con_defs=[] for inference systems.
281 Proposition A should have the form t:Si where Si is an inductive set*)
282 fun make_cases ss A =
284 (basic_elim_tac THEN ALLGOALS (asm_full_simp_tac ss) THEN basic_elim_tac)
285 (Thm.assume A RS elim)
287 fun mk_cases a = make_cases (*delayed evaluation of body!*)
288 (simpset ()) (read_cterm (Thm.sign_of_thm elim) (a, propT));
290 fun induction_rules raw_induct thy =
292 val dummy = writeln " Proving the induction rule...";
294 (*** Prove the main induction rule ***)
296 val pred_name = "P"; (*name for predicate variables*)
298 (*Used to make induction rules;
299 ind_alist = [(rec_tm1,pred1),...] associates predicates with rec ops
300 prem is a premise of an intr rule*)
301 fun add_induct_prem ind_alist (prem as Const("Trueprop",_) $
302 (Const("op :",_)$t$X), iprems) =
303 (case gen_assoc (op aconv) (ind_alist, X) of
304 SOME pred => prem :: FOLogic.mk_Trueprop (pred $ t) :: iprems
305 | NONE => (*possibly membership in M(rec_tm), for M monotone*)
306 let fun mk_sb (rec_tm,pred) =
307 (rec_tm, Ind_Syntax.Collect_const$rec_tm$pred)
308 in subst_free (map mk_sb ind_alist) prem :: iprems end)
309 | add_induct_prem ind_alist (prem,iprems) = prem :: iprems;
311 (*Make a premise of the induction rule.*)
312 fun induct_prem ind_alist intr =
313 let val quantfrees = map dest_Free (term_frees intr \\ rec_params)
314 val iprems = foldr (add_induct_prem ind_alist) []
315 (Logic.strip_imp_prems intr)
316 val (t,X) = Ind_Syntax.rule_concl intr
317 val (SOME pred) = gen_assoc (op aconv) (ind_alist, X)
318 val concl = FOLogic.mk_Trueprop (pred $ t)
319 in list_all_free (quantfrees, Logic.list_implies (iprems,concl)) end
320 handle Bind => error"Recursion term not found in conclusion";
322 (*Minimizes backtracking by delivering the correct premise to each goal.
323 Intro rules with extra Vars in premises still cause some backtracking *)
324 fun ind_tac [] 0 = all_tac
325 | ind_tac(prem::prems) i =
326 DEPTH_SOLVE_1 (ares_tac [prem, refl] i) THEN ind_tac prems (i-1);
328 val pred = Free(pred_name, Ind_Syntax.iT --> FOLogic.oT);
330 val ind_prems = map (induct_prem (map (rpair pred) rec_tms))
333 val dummy = if !Ind_Syntax.trace then
334 (writeln "ind_prems = ";
335 List.app (writeln o Sign.string_of_term sign1) ind_prems;
336 writeln "raw_induct = "; print_thm raw_induct)
340 (*We use a MINIMAL simpset. Even FOL_ss contains too many simpules.
341 If the premises get simplified, then the proofs could fail.*)
342 val min_ss = empty_ss
343 setmksimps (map mk_eq o ZF_atomize o gen_all)
344 setSolver (mk_solver "minimal"
345 (fn prems => resolve_tac (triv_rls@prems)
347 ORELSE' etac FalseE));
350 prove_goalw_cterm part_rec_defs
354 FOLogic.mk_Trueprop (Ind_Syntax.mk_all_imp(big_rec_tm,pred)))))
356 [rtac (impI RS allI) 1,
357 DETERM (etac raw_induct 1),
358 (*Push Part inside Collect*)
359 full_simp_tac (min_ss addsimps [Part_Collect]) 1,
360 (*This CollectE and disjE separates out the introduction rules*)
361 REPEAT (FIRSTGOAL (eresolve_tac [CollectE, disjE])),
362 (*Now break down the individual cases. No disjE here in case
363 some premise involves disjunction.*)
364 REPEAT (FIRSTGOAL (eresolve_tac [CollectE, exE, conjE]
365 ORELSE' bound_hyp_subst_tac)),
366 ind_tac (rev prems) (length prems) ]);
368 val dummy = if !Ind_Syntax.trace then
369 (writeln "quant_induct = "; print_thm quant_induct)
373 (*** Prove the simultaneous induction rule ***)
375 (*Make distinct predicates for each inductive set*)
377 (*The components of the element type, several if it is a product*)
378 val elem_type = CP.pseudo_type dom_sum;
379 val elem_factors = CP.factors elem_type;
380 val elem_frees = mk_frees "za" elem_factors;
381 val elem_tuple = CP.mk_tuple Pr.pair elem_type elem_frees;
383 (*Given a recursive set and its domain, return the "fsplit" predicate
384 and a conclusion for the simultaneous induction rule.
385 NOTE. This will not work for mutually recursive predicates. Previously
386 a summand 'domt' was also an argument, but this required the domain of
387 mutual recursion to invariably be a disjoint sum.*)
388 fun mk_predpair rec_tm =
389 let val rec_name = (#1 o dest_Const o head_of) rec_tm
390 val pfree = Free(pred_name ^ "_" ^ Sign.base_name rec_name,
391 elem_factors ---> FOLogic.oT)
395 (Ind_Syntax.mem_const $ elem_tuple $ rec_tm)
396 $ (list_comb (pfree, elem_frees))) elem_frees
397 in (CP.ap_split elem_type FOLogic.oT pfree,
401 val (preds,qconcls) = split_list (map mk_predpair rec_tms);
403 (*Used to form simultaneous induction lemma*)
404 fun mk_rec_imp (rec_tm,pred) =
405 FOLogic.imp $ (Ind_Syntax.mem_const $ Bound 0 $ rec_tm) $
408 (*To instantiate the main induction rule*)
411 (Ind_Syntax.mk_all_imp
413 Abs("z", Ind_Syntax.iT,
414 fold_bal FOLogic.mk_conj
415 (ListPair.map mk_rec_imp (rec_tms, preds)))))
416 and mutual_induct_concl =
417 FOLogic.mk_Trueprop(fold_bal FOLogic.mk_conj qconcls);
419 val dummy = if !Ind_Syntax.trace then
420 (writeln ("induct_concl = " ^
421 Sign.string_of_term sign1 induct_concl);
422 writeln ("mutual_induct_concl = " ^
423 Sign.string_of_term sign1 mutual_induct_concl))
427 val lemma_tac = FIRST' [eresolve_tac [asm_rl, conjE, PartE, mp],
428 resolve_tac [allI, impI, conjI, Part_eqI],
429 dresolve_tac [spec, mp, Pr.fsplitD]];
431 val need_mutual = length rec_names > 1;
433 val lemma = (*makes the link between the two induction rules*)
435 (writeln " Proving the mutual induction rule...";
436 prove_goalw_cterm part_rec_defs
437 (cterm_of sign1 (Logic.mk_implies (induct_concl,
438 mutual_induct_concl)))
440 [cut_facts_tac prems 1,
441 REPEAT (rewrite_goals_tac [Pr.split_eq] THEN
443 else (writeln " [ No mutual induction rule needed ]";
446 val dummy = if !Ind_Syntax.trace then
447 (writeln "lemma = "; print_thm lemma)
451 (*Mutual induction follows by freeness of Inl/Inr.*)
453 (*Simplification largely reduces the mutual induction rule to the
456 min_ss addsimps [Su.distinct, Su.distinct', Su.inl_iff, Su.inr_iff];
458 val all_defs = con_defs @ part_rec_defs;
460 (*Removes Collects caused by M-operators in the intro rules. It is very
462 list({v: tf. (v : t --> P_t(v)) & (v : f --> P_f(v))})
463 where t==Part(tf,Inl) and f==Part(tf,Inr) to list({v: tf. P_t(v)}).
464 Instead the following rules extract the relevant conjunct.
466 val cmonos = [subset_refl RS Collect_mono] RL monos
467 RLN (2,[rev_subsetD]);
469 (*Minimizes backtracking by delivering the correct premise to each goal*)
470 fun mutual_ind_tac [] 0 = all_tac
471 | mutual_ind_tac(prem::prems) i =
475 (*Simplify the assumptions and goal by unfolding Part and
476 using freeness of the Sum constructors; proves all but one
477 conjunct by contradiction*)
478 rewrite_goals_tac all_defs THEN
479 simp_tac (mut_ss addsimps [Part_iff]) 1 THEN
480 IF_UNSOLVED (*simp_tac may have finished it off!*)
481 ((*simplify assumptions*)
482 (*some risk of excessive simplification here -- might have
483 to identify the bare minimum set of rewrites*)
485 (mut_ss addsimps conj_simps @ imp_simps @ quant_simps) 1
487 (*unpackage and use "prem" in the corresponding place*)
488 REPEAT (rtac impI 1) THEN
489 rtac (rewrite_rule all_defs prem) 1 THEN
490 (*prem must not be REPEATed below: could loop!*)
491 DEPTH_SOLVE (FIRSTGOAL (ares_tac [impI] ORELSE'
492 eresolve_tac (conjE::mp::cmonos))))
494 THEN mutual_ind_tac prems (i-1);
496 val mutual_induct_fsplit =
501 (map (induct_prem (rec_tms~~preds)) intr_tms,
502 mutual_induct_concl)))
504 [rtac (quant_induct RS lemma) 1,
505 mutual_ind_tac (rev prems) (length prems)])
508 (** Uncurrying the predicate in the ordinary induction rule **)
510 (*instantiate the variable to a tuple, if it is non-trivial, in order to
511 allow the predicate to be "opened up".
512 The name "x.1" comes from the "RS spec" !*)
514 case elem_frees of [_] => I
515 | _ => instantiate ([], [(cterm_of sign1 (Var(("x",1), Ind_Syntax.iT)),
516 cterm_of sign1 elem_tuple)]);
518 (*strip quantifier and the implication*)
519 val induct0 = inst (quant_induct RS spec RSN (2,rev_mp));
521 val Const ("Trueprop", _) $ (pred_var $ _) = concl_of induct0
523 val induct = CP.split_rule_var(pred_var, elem_type-->FOLogic.oT, induct0)
525 and mutual_induct = CP.remove_split mutual_induct_fsplit
527 val (thy', [induct', mutual_induct']) = thy |> PureThy.add_thms
528 [((co_prefix ^ "induct", induct), [case_names, InductAttrib.induct_set_global big_rec_name]),
529 (("mutual_induct", mutual_induct), [case_names])];
530 in ((thy', induct'), mutual_induct')
531 end; (*of induction_rules*)
533 val raw_induct = standard ([big_rec_def, bnd_mono] MRS Fp.induct)
535 val ((thy2, induct), mutual_induct) =
536 if not coind then induction_rules raw_induct thy1
537 else (thy1 |> PureThy.add_thms [((co_prefix ^ "induct", raw_induct), [])] |> apsnd hd, TrueI)
538 and defs = big_rec_def :: part_rec_defs
541 val (thy3, ([bnd_mono', dom_subset', elim'], [defs', intrs'])) =
543 |> IndCases.declare big_rec_name make_cases
545 [(("bnd_mono", bnd_mono), []),
546 (("dom_subset", dom_subset), []),
547 (("cases", elim), [case_names, InductAttrib.cases_set_global big_rec_name])]
548 |>>> (PureThy.add_thmss o map Thm.no_attributes)
551 val (thy4, intrs'') =
552 thy3 |> PureThy.add_thms ((intr_names ~~ intrs') ~~ map #2 intr_specs)
553 |>> Theory.parent_path;
557 bnd_mono = bnd_mono',
558 dom_subset = dom_subset',
563 mutual_induct = mutual_induct})
567 (*external version, accepting strings*)
568 fun add_inductive_x (srec_tms, sdom_sum) sintrs (monos, con_defs, type_intrs, type_elims) thy =
570 val read = Sign.simple_read_term (Theory.sign_of thy);
571 val rec_tms = map (read Ind_Syntax.iT) srec_tms;
572 val dom_sum = read Ind_Syntax.iT sdom_sum;
573 val intr_tms = map (read propT o snd o fst) sintrs;
574 val intr_specs = (map (fst o fst) sintrs ~~ intr_tms) ~~ map snd sintrs;
576 add_inductive_i true (rec_tms, dom_sum) intr_specs
577 (monos, con_defs, type_intrs, type_elims) thy
582 fun add_inductive (srec_tms, sdom_sum) intr_srcs
583 (raw_monos, raw_con_defs, raw_type_intrs, raw_type_elims) thy =
585 val intr_atts = map (map (Attrib.global_attribute thy) o snd) intr_srcs;
586 val (thy', (((monos, con_defs), type_intrs), type_elims)) = thy
587 |> IsarThy.apply_theorems raw_monos
588 |>>> IsarThy.apply_theorems raw_con_defs
589 |>>> IsarThy.apply_theorems raw_type_intrs
590 |>>> IsarThy.apply_theorems raw_type_elims;
592 add_inductive_x (srec_tms, sdom_sum) (map fst intr_srcs ~~ intr_atts)
593 (monos, con_defs, type_intrs, type_elims) thy'
599 local structure P = OuterParse and K = OuterSyntax.Keyword in
601 fun mk_ind (((((doms, intrs), monos), con_defs), type_intrs), type_elims) =
602 #1 o add_inductive doms (map P.triple_swap intrs) (monos, con_defs, type_intrs, type_elims);
605 (P.$$$ "domains" |-- P.!!! (P.enum1 "+" P.term --
606 ((P.$$$ "\\<subseteq>" || P.$$$ "<=") |-- P.term))) --
608 P.!!! (Scan.repeat1 (P.opt_thm_name ":" -- P.prop))) --
609 Scan.optional (P.$$$ "monos" |-- P.!!! P.xthms1) [] --
610 Scan.optional (P.$$$ "con_defs" |-- P.!!! P.xthms1) [] --
611 Scan.optional (P.$$$ "type_intros" |-- P.!!! P.xthms1) [] --
612 Scan.optional (P.$$$ "type_elims" |-- P.!!! P.xthms1) []
613 >> (Toplevel.theory o mk_ind);
615 val inductiveP = OuterSyntax.command (co_prefix ^ "inductive")
616 ("define " ^ co_prefix ^ "inductive sets") K.thy_decl ind_decl;
618 val _ = OuterSyntax.add_keywords
619 ["domains", "intros", "monos", "con_defs", "type_intros", "type_elims"];
620 val _ = OuterSyntax.add_parsers [inductiveP];