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 ((Name.binding * term) * attribute list) list ->
33 thm list * thm list * thm list * thm list -> theory -> theory * inductive_result
34 val add_inductive: string list * string ->
35 ((Name.binding * string) * Attrib.src list) list ->
36 (Facts.ref * Attrib.src list) list * (Facts.ref * Attrib.src list) list *
37 (Facts.ref * Attrib.src list) list * (Facts.ref * 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 raw_intr_specs (monos, con_defs, type_intrs, type_elims) thy =
67 val _ = Theory.requires thy "Inductive_ZF" "(co)inductive definitions";
68 val ctxt = ProofContext.init thy;
70 val intr_specs = map (apfst (apfst Name.name_of)) raw_intr_specs;
71 val (intr_names, intr_tms) = split_list (map fst intr_specs);
72 val case_names = RuleCases.case_names intr_names;
74 (*recT and rec_params should agree for all mutually recursive components*)
75 val rec_hds = map head_of rec_tms;
77 val dummy = assert_all is_Const rec_hds
78 (fn t => "Recursive set not previously declared as constant: " ^
79 Syntax.string_of_term ctxt t);
81 (*Now we know they are all Consts, so get their names, type and params*)
82 val rec_names = map (#1 o dest_Const) rec_hds
83 and (Const(_,recT),rec_params) = strip_comb (hd rec_tms);
85 val rec_base_names = map Sign.base_name rec_names;
86 val dummy = assert_all Syntax.is_identifier rec_base_names
87 (fn a => "Base name of recursive set not an identifier: " ^ a);
89 local (*Checking the introduction rules*)
90 val intr_sets = map (#2 o rule_concl_msg thy) intr_tms;
92 case head_of set of Const(a,recT) => a mem rec_names | _ => false;
94 val dummy = assert_all intr_ok intr_sets
95 (fn t => "Conclusion of rule does not name a recursive set: " ^
96 Syntax.string_of_term ctxt t);
99 val dummy = assert_all is_Free rec_params
100 (fn t => "Param in recursion term not a free variable: " ^
101 Syntax.string_of_term ctxt t);
103 (*** Construct the fixedpoint definition ***)
104 val mk_variant = Name.variant (foldr add_term_names [] intr_tms);
106 val z' = mk_variant"z" and X' = mk_variant"X" and w' = mk_variant"w";
108 fun dest_tprop (Const("Trueprop",_) $ P) = P
109 | dest_tprop Q = error ("Ill-formed premise of introduction rule: " ^
110 Syntax.string_of_term ctxt Q);
112 (*Makes a disjunct from an introduction rule*)
113 fun fp_part intr = (*quantify over rule's free vars except parameters*)
114 let val prems = map dest_tprop (Logic.strip_imp_prems intr)
115 val dummy = List.app (fn rec_hd => List.app (chk_prem rec_hd) prems) rec_hds
116 val exfrees = term_frees intr \\ rec_params
117 val zeq = FOLogic.mk_eq (Free(z',iT), #1 (rule_concl intr))
118 in foldr FOLogic.mk_exists
119 (BalancedTree.make FOLogic.mk_conj (zeq::prems)) exfrees
122 (*The Part(A,h) terms -- compose injections to make h*)
123 fun mk_Part (Bound 0) = Free(X',iT) (*no mutual rec, no Part needed*)
124 | mk_Part h = @{const Part} $ Free(X',iT) $ Abs(w',iT,h);
126 (*Access to balanced disjoint sums via injections*)
127 val parts = map mk_Part
128 (BalancedTree.accesses {left = fn t => Su.inl $ t, right = fn t => Su.inr $ t, init = Bound 0}
131 (*replace each set by the corresponding Part(A,h)*)
132 val part_intrs = map (subst_free (rec_tms ~~ parts) o fp_part) intr_tms;
134 val fp_abs = absfree(X', iT,
135 mk_Collect(z', dom_sum,
136 BalancedTree.make FOLogic.mk_disj part_intrs));
138 val fp_rhs = Fp.oper $ dom_sum $ fp_abs
140 val dummy = List.app (fn rec_hd => (Logic.occs (rec_hd, fp_rhs) andalso
141 error "Illegal occurrence of recursion operator"; ()))
144 (*** Make the new theory ***)
147 If no mutual recursion then it equals the one recursive set.
148 If mutual recursion then it differs from all the recursive sets. *)
149 val big_rec_base_name = space_implode "_" rec_base_names;
150 val big_rec_name = Sign.intern_const thy big_rec_base_name;
155 writeln ((if coind then "Coind" else "Ind") ^ "uctive definition " ^ quote big_rec_name)
158 (*Big_rec... is the union of the mutually recursive sets*)
159 val big_rec_tm = list_comb(Const(big_rec_name,recT), rec_params);
161 (*The individual sets must already be declared*)
162 val axpairs = map PrimitiveDefs.mk_defpair
163 ((big_rec_tm, fp_rhs) ::
165 [_] => [] (*no mutual recursion*)
166 | _ => rec_tms ~~ (*define the sets as Parts*)
167 map (subst_atomic [(Free(X',iT),big_rec_tm)]) parts));
169 (*tracing: print the fixedpoint definition*)
170 val dummy = if !Ind_Syntax.trace then
171 writeln (cat_lines (map (Syntax.string_of_term ctxt o #2) axpairs))
174 (*add definitions of the inductive sets*)
177 |> Sign.add_path big_rec_base_name
178 |> PureThy.add_defs false (map Thm.no_attributes axpairs);
180 val ctxt1 = ProofContext.init thy1;
183 (*fetch fp definitions from the theory*)
184 val big_rec_def::part_rec_defs =
185 map (Thm.get_def thy1)
186 (case rec_names of [_] => rec_names
187 | _ => big_rec_base_name::rec_names);
191 val dummy = writeln " Proving monotonicity...";
194 Goal.prove_global thy1 [] [] (FOLogic.mk_Trueprop (Fp.bnd_mono $ dom_sum $ fp_abs))
196 [rtac (@{thm Collect_subset} RS @{thm bnd_monoI}) 1,
197 REPEAT (ares_tac (@{thms basic_monos} @ monos) 1)]);
199 val dom_subset = standard (big_rec_def RS Fp.subs);
201 val unfold = standard ([big_rec_def, bnd_mono] MRS Fp.Tarski);
204 val dummy = writeln " Proving the introduction rules...";
206 (*Mutual recursion? Helps to derive subset rules for the
211 | _ => standard (@{thm Part_subset} RS @{thm subset_trans});
213 (*To type-check recursive occurrences of the inductive sets, possibly
214 enclosed in some monotonic operator M.*)
216 [dom_subset] RL (asm_rl :: ([Part_trans] RL monos))
219 (*Type-checking is hardest aspect of proof;
220 disjIn selects the correct disjunct after unfolding*)
221 fun intro_tacsf disjIn =
222 [DETERM (stac unfold 1),
223 REPEAT (resolve_tac [@{thm Part_eqI}, @{thm CollectI}] 1),
224 (*Now 2-3 subgoals: typechecking, the disjunction, perhaps equality.*)
226 (*Not ares_tac, since refl must be tried before equality assumptions;
227 backtracking may occur if the premises have extra variables!*)
228 DEPTH_SOLVE_1 (resolve_tac [refl,exI,conjI] 2 APPEND assume_tac 2),
229 (*Now solve the equations like Tcons(a,f) = Inl(?b4)*)
230 rewrite_goals_tac con_defs,
231 REPEAT (rtac @{thm refl} 2),
232 (*Typechecking; this can fail*)
233 if !Ind_Syntax.trace then print_tac "The type-checking subgoal:"
235 REPEAT (FIRSTGOAL ( dresolve_tac rec_typechecks
236 ORELSE' eresolve_tac (asm_rl::@{thm PartE}::@{thm SigmaE2}::
238 ORELSE' hyp_subst_tac)),
239 if !Ind_Syntax.trace then print_tac "The subgoal after monos, type_elims:"
241 DEPTH_SOLVE (swap_res_tac (@{thm SigmaI}::@{thm subsetI}::type_intrs) 1)];
243 (*combines disjI1 and disjI2 to get the corresponding nested disjunct...*)
244 val mk_disj_rls = BalancedTree.accesses
245 {left = fn rl => rl RS @{thm disjI1},
246 right = fn rl => rl RS @{thm disjI2},
247 init = @{thm asm_rl}};
250 (intr_tms, map intro_tacsf (mk_disj_rls (length intr_tms)))
251 |> ListPair.map (fn (t, tacs) =>
252 Goal.prove_global thy1 [] [] t
253 (fn _ => EVERY (rewrite_goals_tac part_rec_defs :: tacs)))
254 handle MetaSimplifier.SIMPLIFIER (msg, thm) => (Display.print_thm thm; error msg);
257 val dummy = writeln " Proving the elimination rule...";
259 (*Breaks down logical connectives in the monotonic function*)
261 REPEAT (SOMEGOAL (eresolve_tac (Ind_Syntax.elim_rls @ Su.free_SEs)
262 ORELSE' bound_hyp_subst_tac))
263 THEN prune_params_tac
264 (*Mutual recursion: collapse references to Part(D,h)*)
265 THEN fold_tac part_rec_defs;
268 val elim = rule_by_tactic basic_elim_tac
269 (unfold RS Ind_Syntax.equals_CollectD)
271 (*Applies freeness of the given constructors, which *must* be unfolded by
272 the given defs. Cannot simply use the local con_defs because
273 con_defs=[] for inference systems.
274 Proposition A should have the form t:Si where Si is an inductive set*)
275 fun make_cases ss A =
277 (basic_elim_tac THEN ALLGOALS (asm_full_simp_tac ss) THEN basic_elim_tac)
278 (Thm.assume A RS elim)
280 fun mk_cases a = make_cases (*delayed evaluation of body!*)
282 let val thy = Thm.theory_of_thm elim in cterm_of thy (Syntax.read_prop_global thy a) end;
284 fun induction_rules raw_induct thy =
286 val dummy = writeln " Proving the induction rule...";
288 (*** Prove the main induction rule ***)
290 val pred_name = "P"; (*name for predicate variables*)
292 (*Used to make induction rules;
293 ind_alist = [(rec_tm1,pred1),...] associates predicates with rec ops
294 prem is a premise of an intr rule*)
295 fun add_induct_prem ind_alist (prem as Const (@{const_name Trueprop}, _) $
296 (Const (@{const_name mem}, _) $ t $ X), iprems) =
297 (case AList.lookup (op aconv) ind_alist X of
298 SOME pred => prem :: FOLogic.mk_Trueprop (pred $ t) :: iprems
299 | NONE => (*possibly membership in M(rec_tm), for M monotone*)
300 let fun mk_sb (rec_tm,pred) =
301 (rec_tm, @{const Collect} $ rec_tm $ pred)
302 in subst_free (map mk_sb ind_alist) prem :: iprems end)
303 | add_induct_prem ind_alist (prem,iprems) = prem :: iprems;
305 (*Make a premise of the induction rule.*)
306 fun induct_prem ind_alist intr =
307 let val quantfrees = map dest_Free (term_frees intr \\ rec_params)
308 val iprems = foldr (add_induct_prem ind_alist) []
309 (Logic.strip_imp_prems intr)
310 val (t,X) = Ind_Syntax.rule_concl intr
311 val (SOME pred) = AList.lookup (op aconv) ind_alist X
312 val concl = FOLogic.mk_Trueprop (pred $ t)
313 in list_all_free (quantfrees, Logic.list_implies (iprems,concl)) end
314 handle Bind => error"Recursion term not found in conclusion";
316 (*Minimizes backtracking by delivering the correct premise to each goal.
317 Intro rules with extra Vars in premises still cause some backtracking *)
318 fun ind_tac [] 0 = all_tac
319 | ind_tac(prem::prems) i =
320 DEPTH_SOLVE_1 (ares_tac [prem, refl] i) THEN ind_tac prems (i-1);
322 val pred = Free(pred_name, Ind_Syntax.iT --> FOLogic.oT);
324 val ind_prems = map (induct_prem (map (rpair pred) rec_tms))
327 val dummy = if !Ind_Syntax.trace then
328 (writeln "ind_prems = ";
329 List.app (writeln o Syntax.string_of_term ctxt1) ind_prems;
330 writeln "raw_induct = "; Display.print_thm raw_induct)
334 (*We use a MINIMAL simpset. Even FOL_ss contains too many simpules.
335 If the premises get simplified, then the proofs could fail.*)
336 val min_ss = Simplifier.theory_context thy empty_ss
337 setmksimps (map mk_eq o ZF_atomize o gen_all)
338 setSolver (mk_solver "minimal"
339 (fn prems => resolve_tac (triv_rls@prems)
341 ORELSE' etac FalseE));
344 Goal.prove_global thy1 [] ind_prems
345 (FOLogic.mk_Trueprop (Ind_Syntax.mk_all_imp (big_rec_tm, pred)))
346 (fn {prems, ...} => EVERY
347 [rewrite_goals_tac part_rec_defs,
348 rtac (@{thm impI} RS @{thm allI}) 1,
349 DETERM (etac raw_induct 1),
350 (*Push Part inside Collect*)
351 full_simp_tac (min_ss addsimps [@{thm Part_Collect}]) 1,
352 (*This CollectE and disjE separates out the introduction rules*)
353 REPEAT (FIRSTGOAL (eresolve_tac [@{thm CollectE}, @{thm disjE}])),
354 (*Now break down the individual cases. No disjE here in case
355 some premise involves disjunction.*)
356 REPEAT (FIRSTGOAL (eresolve_tac [@{thm CollectE}, @{thm exE}, @{thm conjE}]
357 ORELSE' bound_hyp_subst_tac)),
358 ind_tac (rev (map (rewrite_rule part_rec_defs) prems)) (length prems)]);
360 val dummy = if !Ind_Syntax.trace then
361 (writeln "quant_induct = "; Display.print_thm quant_induct)
365 (*** Prove the simultaneous induction rule ***)
367 (*Make distinct predicates for each inductive set*)
369 (*The components of the element type, several if it is a product*)
370 val elem_type = CP.pseudo_type dom_sum;
371 val elem_factors = CP.factors elem_type;
372 val elem_frees = mk_frees "za" elem_factors;
373 val elem_tuple = CP.mk_tuple Pr.pair elem_type elem_frees;
375 (*Given a recursive set and its domain, return the "fsplit" predicate
376 and a conclusion for the simultaneous induction rule.
377 NOTE. This will not work for mutually recursive predicates. Previously
378 a summand 'domt' was also an argument, but this required the domain of
379 mutual recursion to invariably be a disjoint sum.*)
380 fun mk_predpair rec_tm =
381 let val rec_name = (#1 o dest_Const o head_of) rec_tm
382 val pfree = Free(pred_name ^ "_" ^ Sign.base_name rec_name,
383 elem_factors ---> FOLogic.oT)
387 (@{const mem} $ elem_tuple $ rec_tm)
388 $ (list_comb (pfree, elem_frees))) elem_frees
389 in (CP.ap_split elem_type FOLogic.oT pfree,
393 val (preds,qconcls) = split_list (map mk_predpair rec_tms);
395 (*Used to form simultaneous induction lemma*)
396 fun mk_rec_imp (rec_tm,pred) =
397 FOLogic.imp $ (@{const mem} $ Bound 0 $ rec_tm) $
400 (*To instantiate the main induction rule*)
403 (Ind_Syntax.mk_all_imp
405 Abs("z", Ind_Syntax.iT,
406 BalancedTree.make FOLogic.mk_conj
407 (ListPair.map mk_rec_imp (rec_tms, preds)))))
408 and mutual_induct_concl =
409 FOLogic.mk_Trueprop(BalancedTree.make FOLogic.mk_conj qconcls);
411 val dummy = if !Ind_Syntax.trace then
412 (writeln ("induct_concl = " ^
413 Syntax.string_of_term ctxt1 induct_concl);
414 writeln ("mutual_induct_concl = " ^
415 Syntax.string_of_term ctxt1 mutual_induct_concl))
419 val lemma_tac = FIRST' [eresolve_tac [@{thm asm_rl}, @{thm conjE}, @{thm PartE}, @{thm mp}],
420 resolve_tac [@{thm allI}, @{thm impI}, @{thm conjI}, @{thm Part_eqI}],
421 dresolve_tac [@{thm spec}, @{thm mp}, Pr.fsplitD]];
423 val need_mutual = length rec_names > 1;
425 val lemma = (*makes the link between the two induction rules*)
427 (writeln " Proving the mutual induction rule...";
428 Goal.prove_global thy1 [] []
429 (Logic.mk_implies (induct_concl, mutual_induct_concl))
431 [rewrite_goals_tac part_rec_defs,
432 REPEAT (rewrite_goals_tac [Pr.split_eq] THEN lemma_tac 1)]))
433 else (writeln " [ No mutual induction rule needed ]"; @{thm TrueI});
435 val dummy = if !Ind_Syntax.trace then
436 (writeln "lemma = "; Display.print_thm lemma)
440 (*Mutual induction follows by freeness of Inl/Inr.*)
442 (*Simplification largely reduces the mutual induction rule to the
445 min_ss addsimps [Su.distinct, Su.distinct', Su.inl_iff, Su.inr_iff];
447 val all_defs = con_defs @ part_rec_defs;
449 (*Removes Collects caused by M-operators in the intro rules. It is very
451 list({v: tf. (v : t --> P_t(v)) & (v : f --> P_f(v))})
452 where t==Part(tf,Inl) and f==Part(tf,Inr) to list({v: tf. P_t(v)}).
453 Instead the following rules extract the relevant conjunct.
455 val cmonos = [@{thm subset_refl} RS @{thm Collect_mono}] RL monos
456 RLN (2,[@{thm rev_subsetD}]);
458 (*Minimizes backtracking by delivering the correct premise to each goal*)
459 fun mutual_ind_tac [] 0 = all_tac
460 | mutual_ind_tac(prem::prems) i =
464 (*Simplify the assumptions and goal by unfolding Part and
465 using freeness of the Sum constructors; proves all but one
466 conjunct by contradiction*)
467 rewrite_goals_tac all_defs THEN
468 simp_tac (mut_ss addsimps [@{thm Part_iff}]) 1 THEN
469 IF_UNSOLVED (*simp_tac may have finished it off!*)
470 ((*simplify assumptions*)
471 (*some risk of excessive simplification here -- might have
472 to identify the bare minimum set of rewrites*)
474 (mut_ss addsimps @{thms conj_simps} @ @{thms imp_simps} @ @{thms quant_simps}) 1
476 (*unpackage and use "prem" in the corresponding place*)
477 REPEAT (rtac impI 1) THEN
478 rtac (rewrite_rule all_defs prem) 1 THEN
479 (*prem must not be REPEATed below: could loop!*)
480 DEPTH_SOLVE (FIRSTGOAL (ares_tac [impI] ORELSE'
481 eresolve_tac (conjE::mp::cmonos))))
483 THEN mutual_ind_tac prems (i-1);
485 val mutual_induct_fsplit =
487 Goal.prove_global thy1 [] (map (induct_prem (rec_tms~~preds)) intr_tms)
489 (fn {prems, ...} => EVERY
490 [rtac (quant_induct RS lemma) 1,
491 mutual_ind_tac (rev prems) (length prems)])
494 (** Uncurrying the predicate in the ordinary induction rule **)
496 (*instantiate the variable to a tuple, if it is non-trivial, in order to
497 allow the predicate to be "opened up".
498 The name "x.1" comes from the "RS spec" !*)
500 case elem_frees of [_] => I
501 | _ => instantiate ([], [(cterm_of thy1 (Var(("x",1), Ind_Syntax.iT)),
502 cterm_of thy1 elem_tuple)]);
504 (*strip quantifier and the implication*)
505 val induct0 = inst (quant_induct RS spec RSN (2, @{thm rev_mp}));
507 val Const (@{const_name Trueprop}, _) $ (pred_var $ _) = concl_of induct0
509 val induct = CP.split_rule_var(pred_var, elem_type-->FOLogic.oT, induct0)
511 and mutual_induct = CP.remove_split mutual_induct_fsplit
513 val ([induct', mutual_induct'], thy') =
515 |> PureThy.add_thms [((co_prefix ^ "induct", induct),
516 [case_names, Induct.induct_pred big_rec_name]),
517 (("mutual_induct", mutual_induct), [case_names])];
518 in ((thy', induct'), mutual_induct')
519 end; (*of induction_rules*)
521 val raw_induct = standard ([big_rec_def, bnd_mono] MRS Fp.induct)
523 val ((thy2, induct), mutual_induct) =
524 if not coind then induction_rules raw_induct thy1
527 |> PureThy.add_thms [((co_prefix ^ "induct", raw_induct), [])]
528 |> apfst hd |> Library.swap, TrueI)
529 and defs = big_rec_def :: part_rec_defs
532 val (([bnd_mono', dom_subset', elim'], [defs', intrs']), thy3) =
534 |> IndCases.declare big_rec_name make_cases
536 [(("bnd_mono", bnd_mono), []),
537 (("dom_subset", dom_subset), []),
538 (("cases", elim), [case_names, Induct.cases_pred big_rec_name])]
539 ||>> (PureThy.add_thmss o map Thm.no_attributes)
542 val (intrs'', thy4) =
544 |> PureThy.add_thms ((intr_names ~~ intrs') ~~ map #2 intr_specs)
545 ||> Sign.parent_path;
549 bnd_mono = bnd_mono',
550 dom_subset = dom_subset',
555 mutual_induct = mutual_induct})
559 fun add_inductive (srec_tms, sdom_sum) intr_srcs
560 (raw_monos, raw_con_defs, raw_type_intrs, raw_type_elims) thy =
562 val ctxt = ProofContext.init thy;
563 val read_terms = map (Syntax.parse_term ctxt #> TypeInfer.constrain Ind_Syntax.iT)
564 #> Syntax.check_terms ctxt;
566 val intr_atts = map (map (Attrib.attribute thy) o snd) intr_srcs;
567 val sintrs = map fst intr_srcs ~~ intr_atts;
568 val rec_tms = read_terms srec_tms;
569 val dom_sum = singleton read_terms sdom_sum;
570 val intr_tms = Syntax.read_props ctxt (map (snd o fst) sintrs);
571 val intr_specs = (map (fst o fst) sintrs ~~ intr_tms) ~~ map snd sintrs;
572 val monos = Attrib.eval_thms ctxt raw_monos;
573 val con_defs = Attrib.eval_thms ctxt raw_con_defs;
574 val type_intrs = Attrib.eval_thms ctxt raw_type_intrs;
575 val type_elims = Attrib.eval_thms ctxt raw_type_elims;
578 |> add_inductive_i true (rec_tms, dom_sum) intr_specs (monos, con_defs, type_intrs, type_elims)
584 local structure P = OuterParse and K = OuterKeyword in
586 val _ = List.app OuterKeyword.keyword
587 ["domains", "intros", "monos", "con_defs", "type_intros", "type_elims"];
589 fun mk_ind (((((doms, intrs), monos), con_defs), type_intrs), type_elims) =
590 #1 o add_inductive doms (map P.triple_swap intrs) (monos, con_defs, type_intrs, type_elims);
593 (P.$$$ "domains" |-- P.!!! (P.enum1 "+" P.term --
594 ((P.$$$ "\<subseteq>" || P.$$$ "<=") |-- P.term))) --
596 P.!!! (Scan.repeat1 (SpecParse.opt_thm_name ":" -- P.prop))) --
597 Scan.optional (P.$$$ "monos" |-- P.!!! SpecParse.xthms1) [] --
598 Scan.optional (P.$$$ "con_defs" |-- P.!!! SpecParse.xthms1) [] --
599 Scan.optional (P.$$$ "type_intros" |-- P.!!! SpecParse.xthms1) [] --
600 Scan.optional (P.$$$ "type_elims" |-- P.!!! SpecParse.xthms1) []
601 >> (Toplevel.theory o mk_ind);
603 val _ = OuterSyntax.command (co_prefix ^ "inductive")
604 ("define " ^ co_prefix ^ "inductive sets") K.thy_decl ind_decl;