src/HOL/Tools/inductive_package.ML
 author berghofe Fri Mar 10 15:02:04 2000 +0100 (2000-03-10) changeset 8410 5902c02fa122 parent 8401 50d5f4402305 child 8433 8ae16c770fc8 permissions -rw-r--r--
Type.unify now uses Vartab instead of association lists.
1 (*  Title:      HOL/Tools/inductive_package.ML
2     ID:         \$Id\$
3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
4                 Stefan Berghofer,   TU Muenchen
5     Copyright   1994  University of Cambridge
6                 1998  TU Muenchen
8 (Co)Inductive Definition module for HOL.
10 Features:
11   * least or greatest fixedpoints
12   * user-specified product and sum constructions
13   * mutually recursive definitions
14   * definitions involving arbitrary monotone operators
15   * automatically proves introduction and elimination rules
17 The recursive sets must *already* be declared as constants in the
18 current theory!
20   Introduction rules have the form
21   [| ti:M(Sj), ..., P(x), ... |] ==> t: Sk
22   where M is some monotone operator (usually the identity)
23   P(x) is any side condition on the free variables
24   ti, t are any terms
25   Sj, Sk are two of the sets being defined in mutual recursion
27 Sums are used only for mutual recursion.  Products are used only to
28 derive "streamlined" induction rules for relations.
29 *)
31 signature INDUCTIVE_PACKAGE =
32 sig
33   val quiet_mode: bool ref
34   val unify_consts: Sign.sg -> term list -> term list -> term list * term list
35   val get_inductive: theory -> string ->
36     {names: string list, coind: bool} * {defs: thm list, elims: thm list, raw_induct: thm,
37       induct: thm, intrs: thm list, mk_cases: string -> thm, mono: thm, unfold: thm}
38   val print_inductives: theory -> unit
39   val mono_add_global: theory attribute
40   val mono_del_global: theory attribute
41   val get_monos: theory -> thm list
42   val add_inductive_i: bool -> bool -> bstring -> bool -> bool -> bool -> term list ->
43     theory attribute list -> ((bstring * term) * theory attribute list) list ->
44       thm list -> thm list -> theory -> theory *
45       {defs: thm list, elims: thm list, raw_induct: thm, induct: thm,
46        intrs: thm list, mk_cases: string -> thm, mono: thm, unfold: thm}
47   val add_inductive: bool -> bool -> string list -> Args.src list ->
48     ((bstring * string) * Args.src list) list -> (xstring * Args.src list) list ->
49       (xstring * Args.src list) list -> theory -> theory *
50       {defs: thm list, elims: thm list, raw_induct: thm, induct: thm,
51        intrs: thm list, mk_cases: string -> thm, mono: thm, unfold: thm}
52   val inductive_cases: (((bstring * Args.src list) * xstring) * string list) * Comment.text
53     -> theory -> theory
54   val inductive_cases_i: (((bstring * theory attribute list) * string) * term list) * Comment.text
55     -> theory -> theory
56   val setup: (theory -> theory) list
57 end;
59 structure InductivePackage: INDUCTIVE_PACKAGE =
60 struct
62 (*** theory data ***)
64 (* data kind 'HOL/inductive' *)
66 type inductive_info =
67   {names: string list, coind: bool} * {defs: thm list, elims: thm list, raw_induct: thm,
68     induct: thm, intrs: thm list, mk_cases: string -> thm, mono: thm, unfold: thm};
70 structure InductiveArgs =
71 struct
72   val name = "HOL/inductive";
73   type T = inductive_info Symtab.table * thm list;
75   val empty = (Symtab.empty, []);
76   val copy = I;
77   val prep_ext = I;
78   fun merge ((tab1, monos1), (tab2, monos2)) = (Symtab.merge (K true) (tab1, tab2),
79     Library.generic_merge Thm.eq_thm I I monos1 monos2);
81   fun print sg (tab, monos) =
82     (Pretty.writeln (Pretty.strs ("(co)inductives:" ::
83        map #1 (Sign.cond_extern_table sg Sign.constK tab)));
84      Pretty.writeln (Pretty.big_list "monotonicity rules:" (map Display.pretty_thm monos)));
85 end;
87 structure InductiveData = TheoryDataFun(InductiveArgs);
88 val print_inductives = InductiveData.print;
91 (* get and put data *)
93 fun get_inductive thy name =
94   (case Symtab.lookup (fst (InductiveData.get thy), name) of
95     Some info => info
96   | None => error ("Unknown (co)inductive set " ^ quote name));
98 fun put_inductives names info thy =
99   let
100     fun upd ((tab, monos), name) = (Symtab.update_new ((name, info), tab), monos);
101     val tab_monos = foldl upd (InductiveData.get thy, names)
102       handle Symtab.DUP name => error ("Duplicate definition of (co)inductive set " ^ quote name);
103   in InductiveData.put tab_monos thy end;
107 (** monotonicity rules **)
109 val get_monos = snd o InductiveData.get;
110 fun put_monos thms thy = InductiveData.put (fst (InductiveData.get thy), thms) thy;
112 fun mk_mono thm =
113   let
114     fun eq2mono thm' = [standard (thm' RS (thm' RS eq_to_mono))] @
115       (case concl_of thm of
116           (_ \$ (_ \$ (Const ("Not", _) \$ _) \$ _)) => []
117         | _ => [standard (thm' RS (thm' RS eq_to_mono2))]);
118     val concl = concl_of thm
119   in
120     if Logic.is_equals concl then
121       eq2mono (thm RS meta_eq_to_obj_eq)
122     else if can (HOLogic.dest_eq o HOLogic.dest_Trueprop) concl then
123       eq2mono thm
124     else [thm]
125   end;
127 (* mono add/del *)
129 local
131 fun map_rules_global f thy = put_monos (f (get_monos thy)) thy;
133 fun add_mono thm rules = Library.gen_union Thm.eq_thm (mk_mono thm, rules);
134 fun del_mono thm rules = Library.gen_rems Thm.eq_thm (rules, mk_mono thm);
136 fun mk_att f g (x, thm) = (f (g thm) x, thm);
138 in
140 val mono_add_global = mk_att map_rules_global add_mono;
141 val mono_del_global = mk_att map_rules_global del_mono;
143 end;
146 (* concrete syntax *)
148 val monoN = "mono";
150 val delN = "del";
152 fun mono_att add del =
153   Attrib.syntax (Scan.lift (Args.\$\$\$ addN >> K add || Args.\$\$\$ delN >> K del || Scan.succeed add));
155 val mono_attr =
156   (mono_att mono_add_global mono_del_global, mono_att Attrib.undef_local_attribute Attrib.undef_local_attribute);
160 (** utilities **)
162 (* messages *)
164 val quiet_mode = ref false;
165 fun message s = if !quiet_mode then () else writeln s;
167 fun coind_prefix true = "co"
168   | coind_prefix false = "";
171 (* the following code ensures that each recursive set *)
172 (* always has the same type in all introduction rules *)
174 fun unify_consts sign cs intr_ts =
175   (let
176     val {tsig, ...} = Sign.rep_sg sign;
177     val add_term_consts_2 =
178       foldl_aterms (fn (cs, Const c) => c ins cs | (cs, _) => cs);
179     fun varify (t, (i, ts)) =
180       let val t' = map_term_types (incr_tvar (i + 1)) (Type.varify (t, []))
181       in (maxidx_of_term t', t'::ts) end;
182     val (i, cs') = foldr varify (cs, (~1, []));
183     val (i', intr_ts') = foldr varify (intr_ts, (i, []));
184     val rec_consts = foldl add_term_consts_2 ([], cs');
185     val intr_consts = foldl add_term_consts_2 ([], intr_ts');
186     fun unify (env, (cname, cT)) =
187       let val consts = map snd (filter (fn c => fst c = cname) intr_consts)
188       in foldl (fn ((env', j'), Tp) => (Type.unify tsig j' env' Tp))
189           (env, (replicate (length consts) cT) ~~ consts)
190       end;
191     val (env, _) = foldl unify ((Vartab.empty, i'), rec_consts);
192     fun typ_subst_TVars_2 env T = let val T' = typ_subst_TVars_Vartab env T
193       in if T = T' then T else typ_subst_TVars_2 env T' end;
194     val subst = fst o Type.freeze_thaw o
195       (map_term_types (typ_subst_TVars_2 env))
197   in (map subst cs', map subst intr_ts')
198   end) handle Type.TUNIFY =>
199     (warning "Occurrences of recursive constant have non-unifiable types"; (cs, intr_ts));
202 (* misc *)
204 val Const _ \$ (vimage_f \$ _) \$ _ = HOLogic.dest_Trueprop (concl_of vimageD);
206 val vimage_name = Sign.intern_const (Theory.sign_of Vimage.thy) "op -``";
207 val mono_name = Sign.intern_const (Theory.sign_of Ord.thy) "mono";
209 (* make injections needed in mutually recursive definitions *)
211 fun mk_inj cs sumT c x =
212   let
213     fun mk_inj' T n i =
214       if n = 1 then x else
215       let val n2 = n div 2;
216           val Type (_, [T1, T2]) = T
217       in
218         if i <= n2 then
219           Const ("Inl", T1 --> T) \$ (mk_inj' T1 n2 i)
220         else
221           Const ("Inr", T2 --> T) \$ (mk_inj' T2 (n - n2) (i - n2))
222       end
223   in mk_inj' sumT (length cs) (1 + find_index_eq c cs)
224   end;
226 (* make "vimage" terms for selecting out components of mutually rec.def. *)
228 fun mk_vimage cs sumT t c = if length cs < 2 then t else
229   let
230     val cT = HOLogic.dest_setT (fastype_of c);
231     val vimageT = [cT --> sumT, HOLogic.mk_setT sumT] ---> HOLogic.mk_setT cT
232   in
233     Const (vimage_name, vimageT) \$
234       Abs ("y", cT, mk_inj cs sumT c (Bound 0)) \$ t
235   end;
239 (** well-formedness checks **)
241 fun err_in_rule sign t msg = error ("Ill-formed introduction rule\n" ^
242   (Sign.string_of_term sign t) ^ "\n" ^ msg);
244 fun err_in_prem sign t p msg = error ("Ill-formed premise\n" ^
245   (Sign.string_of_term sign p) ^ "\nin introduction rule\n" ^
246   (Sign.string_of_term sign t) ^ "\n" ^ msg);
248 val msg1 = "Conclusion of introduction rule must have form\
249           \ ' t : S_i '";
250 val msg2 = "Non-atomic premise";
251 val msg3 = "Recursion term on left of member symbol";
253 fun check_rule sign cs r =
254   let
255     fun check_prem prem = if can HOLogic.dest_Trueprop prem then ()
256       else err_in_prem sign r prem msg2;
258   in (case HOLogic.dest_Trueprop (Logic.strip_imp_concl r) of
259         (Const ("op :", _) \$ t \$ u) =>
260           if u mem cs then
261             if exists (Logic.occs o (rpair t)) cs then
262               err_in_rule sign r msg3
263             else
264               seq check_prem (Logic.strip_imp_prems r)
265           else err_in_rule sign r msg1
266       | _ => err_in_rule sign r msg1)
267   end;
269 fun try' f msg sign t = (case (try f t) of
270       Some x => x
271     | None => error (msg ^ Sign.string_of_term sign t));
275 (*** properties of (co)inductive sets ***)
277 (** elimination rules **)
279 fun mk_elims cs cTs params intr_ts intr_names =
280   let
281     val used = foldr add_term_names (intr_ts, []);
282     val [aname, pname] = variantlist (["a", "P"], used);
283     val P = HOLogic.mk_Trueprop (Free (pname, HOLogic.boolT));
285     fun dest_intr r =
286       let val Const ("op :", _) \$ t \$ u =
287         HOLogic.dest_Trueprop (Logic.strip_imp_concl r)
288       in (u, t, Logic.strip_imp_prems r) end;
290     val intrs = map dest_intr intr_ts ~~ intr_names;
292     fun mk_elim (c, T) =
293       let
294         val a = Free (aname, T);
296         fun mk_elim_prem (_, t, ts) =
297           list_all_free (map dest_Free ((foldr add_term_frees (t::ts, [])) \\ params),
298             Logic.list_implies (HOLogic.mk_Trueprop (HOLogic.mk_eq (a, t)) :: ts, P));
299         val c_intrs = (filter (equal c o #1 o #1) intrs);
300       in
301         (Logic.list_implies (HOLogic.mk_Trueprop (HOLogic.mk_mem (a, c)) ::
302           map mk_elim_prem (map #1 c_intrs), P), map #2 c_intrs)
303       end
304   in
305     map mk_elim (cs ~~ cTs)
306   end;
310 (** premises and conclusions of induction rules **)
312 fun mk_indrule cs cTs params intr_ts =
313   let
314     val used = foldr add_term_names (intr_ts, []);
316     (* predicates for induction rule *)
318     val preds = map Free (variantlist (if length cs < 2 then ["P"] else
319       map (fn i => "P" ^ string_of_int i) (1 upto length cs), used) ~~
320         map (fn T => T --> HOLogic.boolT) cTs);
322     (* transform an introduction rule into a premise for induction rule *)
324     fun mk_ind_prem r =
325       let
326         val frees = map dest_Free ((add_term_frees (r, [])) \\ params);
328         val pred_of = curry (Library.gen_assoc (op aconv)) (cs ~~ preds);
330         fun subst (s as ((m as Const ("op :", T)) \$ t \$ u)) =
331               (case pred_of u of
332                   None => (m \$ fst (subst t) \$ fst (subst u), None)
333                 | Some P => (HOLogic.conj \$ s \$ (P \$ t), Some (s, P \$ t)))
334           | subst s =
335               (case pred_of s of
336                   Some P => (HOLogic.mk_binop "op Int"
337                     (s, HOLogic.Collect_const (HOLogic.dest_setT
338                       (fastype_of s)) \$ P), None)
339                 | None => (case s of
340                      (t \$ u) => (fst (subst t) \$ fst (subst u), None)
341                    | (Abs (a, T, t)) => (Abs (a, T, fst (subst t)), None)
342                    | _ => (s, None)));
344         fun mk_prem (s, prems) = (case subst s of
345               (_, Some (t, u)) => t :: u :: prems
346             | (t, _) => t :: prems);
348         val Const ("op :", _) \$ t \$ u =
349           HOLogic.dest_Trueprop (Logic.strip_imp_concl r)
351       in list_all_free (frees,
352            Logic.list_implies (map HOLogic.mk_Trueprop (foldr mk_prem
353              (map HOLogic.dest_Trueprop (Logic.strip_imp_prems r), [])),
354                HOLogic.mk_Trueprop (the (pred_of u) \$ t)))
355       end;
357     val ind_prems = map mk_ind_prem intr_ts;
359     (* make conclusions for induction rules *)
361     fun mk_ind_concl ((c, P), (ts, x)) =
362       let val T = HOLogic.dest_setT (fastype_of c);
363           val Ts = HOLogic.prodT_factors T;
364           val (frees, x') = foldr (fn (T', (fs, s)) =>
365             ((Free (s, T'))::fs, bump_string s)) (Ts, ([], x));
366           val tuple = HOLogic.mk_tuple T frees;
367       in ((HOLogic.mk_binop "op -->"
368         (HOLogic.mk_mem (tuple, c), P \$ tuple))::ts, x')
369       end;
371     val mutual_ind_concl = HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
372         (fst (foldr mk_ind_concl (cs ~~ preds, ([], "xa")))))
374   in (preds, ind_prems, mutual_ind_concl)
375   end;
379 (** prepare cases and induct rules **)
381 (*
382   transform mutual rule:
383     HH ==> (x1:A1 --> P1 x1) & ... & (xn:An --> Pn xn)
384   into i-th projection:
385     xi:Ai ==> HH ==> Pi xi
386 *)
388 fun project_rules [name] rule = [(name, rule)]
389   | project_rules names mutual_rule =
390       let
391         val n = length names;
392         fun proj i =
393           (if i < n then (fn th => th RS conjunct1) else I)
394             (Library.funpow (i - 1) (fn th => th RS conjunct2) mutual_rule)
395             RS mp |> Thm.permute_prems 0 ~1 |> Drule.standard;
396       in names ~~ map proj (1 upto n) end;
398 fun add_cases_induct no_elim no_ind names elims induct induct_cases =
399   let
400     fun cases_spec (name, elim) = (("", elim), [InductMethod.cases_set_global name]);
401     val cases_specs = if no_elim then [] else map2 cases_spec (names, elims);
403     fun induct_spec (name, th) =
404       (("", th), [RuleCases.case_names induct_cases, InductMethod.induct_set_global name]);
405     val induct_specs = if no_ind then [] else map induct_spec (project_rules names induct);
406   in PureThy.add_thms (cases_specs @ induct_specs) end;
410 (*** proofs for (co)inductive sets ***)
412 (** prove monotonicity **)
414 fun prove_mono setT fp_fun monos thy =
415   let
416     val _ = message "  Proving monotonicity ...";
418     val mono = prove_goalw_cterm [] (cterm_of (Theory.sign_of thy) (HOLogic.mk_Trueprop
419       (Const (mono_name, (setT --> setT) --> HOLogic.boolT) \$ fp_fun)))
420         (fn _ => [rtac monoI 1, REPEAT (ares_tac (get_monos thy @ flat (map mk_mono monos)) 1)])
422   in mono end;
426 (** prove introduction rules **)
428 fun prove_intrs coind mono fp_def intr_ts con_defs rec_sets_defs thy =
429   let
430     val _ = message "  Proving the introduction rules ...";
432     val unfold = standard (mono RS (fp_def RS
433       (if coind then def_gfp_Tarski else def_lfp_Tarski)));
435     fun select_disj 1 1 = []
436       | select_disj _ 1 = [rtac disjI1]
437       | select_disj n i = (rtac disjI2)::(select_disj (n - 1) (i - 1));
439     val intrs = map (fn (i, intr) => prove_goalw_cterm rec_sets_defs
440       (cterm_of (Theory.sign_of thy) intr) (fn prems =>
441        [(*insert prems and underlying sets*)
442        cut_facts_tac prems 1,
443        stac unfold 1,
444        REPEAT (resolve_tac [vimageI2, CollectI] 1),
445        (*Now 1-2 subgoals: the disjunction, perhaps equality.*)
446        EVERY1 (select_disj (length intr_ts) i),
447        (*Not ares_tac, since refl must be tried before any equality assumptions;
448          backtracking may occur if the premises have extra variables!*)
449        DEPTH_SOLVE_1 (resolve_tac [refl,exI,conjI] 1 APPEND assume_tac 1),
450        (*Now solve the equations like Inl 0 = Inl ?b2*)
451        rewrite_goals_tac con_defs,
452        REPEAT (rtac refl 1)])) (1 upto (length intr_ts) ~~ intr_ts)
454   in (intrs, unfold) end;
458 (** prove elimination rules **)
460 fun prove_elims cs cTs params intr_ts intr_names unfold rec_sets_defs thy =
461   let
462     val _ = message "  Proving the elimination rules ...";
464     val rules1 = [CollectE, disjE, make_elim vimageD, exE];
465     val rules2 = [conjE, Inl_neq_Inr, Inr_neq_Inl] @
466       map make_elim [Inl_inject, Inr_inject];
467   in
468     map (fn (t, cases) => prove_goalw_cterm rec_sets_defs
469       (cterm_of (Theory.sign_of thy) t) (fn prems =>
470         [cut_facts_tac [hd prems] 1,
471          dtac (unfold RS subst) 1,
472          REPEAT (FIRSTGOAL (eresolve_tac rules1)),
473          REPEAT (FIRSTGOAL (eresolve_tac rules2)),
474          EVERY (map (fn prem =>
475            DEPTH_SOLVE_1 (ares_tac [prem, conjI] 1)) (tl prems))])
476       |> RuleCases.name cases)
477       (mk_elims cs cTs params intr_ts intr_names)
478   end;
481 (** derivation of simplified elimination rules **)
483 (*Applies freeness of the given constructors, which *must* be unfolded by
484   the given defs.  Cannot simply use the local con_defs because con_defs=[]
485   for inference systems.
486  *)
488 (*cprop should have the form t:Si where Si is an inductive set*)
489 fun mk_cases_i solved elims ss cprop =
490   let
491     val prem = Thm.assume cprop;
492     val tac = if solved then InductMethod.con_elim_solved_tac else InductMethod.con_elim_tac;
493     fun mk_elim rl = Drule.standard (Tactic.rule_by_tactic (tac ss) (prem RS rl));
494   in
495     (case get_first (try mk_elim) elims of
496       Some r => r
497     | None => error (Pretty.string_of (Pretty.block
498         [Pretty.str "mk_cases: proposition not of form 't : S_i'", Pretty.fbrk,
499           Display.pretty_cterm cprop])))
500   end;
502 fun mk_cases elims s =
503   mk_cases_i false elims (simpset()) (Thm.read_cterm (Thm.sign_of_thm (hd elims)) (s, propT));
506 (* inductive_cases(_i) *)
508 fun gen_inductive_cases prep_att prep_const prep_prop
509     ((((name, raw_atts), raw_set), raw_props), comment) thy =
510   let
511     val sign = Theory.sign_of thy;
513     val atts = map (prep_att thy) raw_atts;
514     val (_, {elims, ...}) = get_inductive thy (prep_const sign raw_set);
515     val cprops = map (Thm.cterm_of sign o prep_prop (ProofContext.init thy)) raw_props;
516     val thms = map (mk_cases_i true elims (Simplifier.simpset_of thy)) cprops;
517   in
518     thy
519     |> IsarThy.have_theorems_i (((name, atts), map Thm.no_attributes thms), comment)
520   end;
522 val inductive_cases =
523   gen_inductive_cases Attrib.global_attribute Sign.intern_const ProofContext.read_prop;
525 val inductive_cases_i = gen_inductive_cases (K I) (K I) ProofContext.cert_prop;
529 (** prove induction rule **)
531 fun prove_indrule cs cTs sumT rec_const params intr_ts mono
532     fp_def rec_sets_defs thy =
533   let
534     val _ = message "  Proving the induction rule ...";
536     val sign = Theory.sign_of thy;
538     val sum_case_rewrites = (case ThyInfo.lookup_theory "Datatype" of
539         None => []
540       | Some thy' => map mk_meta_eq (PureThy.get_thms thy' "sum.cases"));
542     val (preds, ind_prems, mutual_ind_concl) = mk_indrule cs cTs params intr_ts;
544     (* make predicate for instantiation of abstract induction rule *)
546     fun mk_ind_pred _ [P] = P
547       | mk_ind_pred T Ps =
548          let val n = (length Ps) div 2;
549              val Type (_, [T1, T2]) = T
550          in Const ("Datatype.sum.sum_case",
551            [T1 --> HOLogic.boolT, T2 --> HOLogic.boolT, T] ---> HOLogic.boolT) \$
552              mk_ind_pred T1 (take (n, Ps)) \$ mk_ind_pred T2 (drop (n, Ps))
553          end;
555     val ind_pred = mk_ind_pred sumT preds;
557     val ind_concl = HOLogic.mk_Trueprop
558       (HOLogic.all_const sumT \$ Abs ("x", sumT, HOLogic.mk_binop "op -->"
559         (HOLogic.mk_mem (Bound 0, rec_const), ind_pred \$ Bound 0)));
561     (* simplification rules for vimage and Collect *)
563     val vimage_simps = if length cs < 2 then [] else
564       map (fn c => prove_goalw_cterm [] (cterm_of sign
565         (HOLogic.mk_Trueprop (HOLogic.mk_eq
566           (mk_vimage cs sumT (HOLogic.Collect_const sumT \$ ind_pred) c,
567            HOLogic.Collect_const (HOLogic.dest_setT (fastype_of c)) \$
568              nth_elem (find_index_eq c cs, preds)))))
569         (fn _ => [rtac vimage_Collect 1, rewrite_goals_tac sum_case_rewrites,
570           rtac refl 1])) cs;
572     val induct = prove_goalw_cterm [] (cterm_of sign
573       (Logic.list_implies (ind_prems, ind_concl))) (fn prems =>
574         [rtac (impI RS allI) 1,
575          DETERM (etac (mono RS (fp_def RS def_induct)) 1),
576          rewrite_goals_tac (map mk_meta_eq (vimage_Int::Int_Collect::vimage_simps)),
577          fold_goals_tac rec_sets_defs,
578          (*This CollectE and disjE separates out the introduction rules*)
579          REPEAT (FIRSTGOAL (eresolve_tac [CollectE, disjE, exE])),
580          (*Now break down the individual cases.  No disjE here in case
581            some premise involves disjunction.*)
582          REPEAT (FIRSTGOAL (etac conjE ORELSE' hyp_subst_tac)),
583          rewrite_goals_tac sum_case_rewrites,
584          EVERY (map (fn prem =>
585            DEPTH_SOLVE_1 (ares_tac [prem, conjI, refl] 1)) prems)]);
587     val lemma = prove_goalw_cterm rec_sets_defs (cterm_of sign
588       (Logic.mk_implies (ind_concl, mutual_ind_concl))) (fn prems =>
589         [cut_facts_tac prems 1,
590          REPEAT (EVERY
591            [REPEAT (resolve_tac [conjI, impI] 1),
592             TRY (dtac vimageD 1), etac allE 1, dtac mp 1, atac 1,
593             rewrite_goals_tac sum_case_rewrites,
594             atac 1])])
596   in standard (split_rule (induct RS lemma))
597   end;
601 (*** specification of (co)inductive sets ****)
603 (** definitional introduction of (co)inductive sets **)
605 fun add_ind_def verbose declare_consts alt_name coind no_elim no_ind cs
606     atts intros monos con_defs thy params paramTs cTs cnames induct_cases =
607   let
608     val _ = if verbose then message ("Proofs for " ^ coind_prefix coind ^ "inductive set(s) " ^
609       commas_quote cnames) else ();
611     val sumT = fold_bal (fn (T, U) => Type ("+", [T, U])) cTs;
612     val setT = HOLogic.mk_setT sumT;
614     val fp_name = if coind then Sign.intern_const (Theory.sign_of Gfp.thy) "gfp"
615       else Sign.intern_const (Theory.sign_of Lfp.thy) "lfp";
617     val ((intr_names, intr_ts), intr_atts) = apfst split_list (split_list intros);
619     val used = foldr add_term_names (intr_ts, []);
620     val [sname, xname] = variantlist (["S", "x"], used);
622     (* transform an introduction rule into a conjunction  *)
623     (*   [| t : ... S_i ... ; ... |] ==> u : S_j          *)
624     (* is transformed into                                *)
625     (*   x = Inj_j u & t : ... Inj_i -`` S ... & ...      *)
627     fun transform_rule r =
628       let
629         val frees = map dest_Free ((add_term_frees (r, [])) \\ params);
630         val subst = subst_free
631           (cs ~~ (map (mk_vimage cs sumT (Free (sname, setT))) cs));
632         val Const ("op :", _) \$ t \$ u =
633           HOLogic.dest_Trueprop (Logic.strip_imp_concl r)
635       in foldr (fn ((x, T), P) => HOLogic.mk_exists (x, T, P))
636         (frees, foldr1 HOLogic.mk_conj
637           (((HOLogic.eq_const sumT) \$ Free (xname, sumT) \$ (mk_inj cs sumT u t))::
638             (map (subst o HOLogic.dest_Trueprop)
639               (Logic.strip_imp_prems r))))
640       end
642     (* make a disjunction of all introduction rules *)
644     val fp_fun = absfree (sname, setT, (HOLogic.Collect_const sumT) \$
645       absfree (xname, sumT, foldr1 HOLogic.mk_disj (map transform_rule intr_ts)));
647     (* add definiton of recursive sets to theory *)
649     val rec_name = if alt_name = "" then space_implode "_" cnames else alt_name;
650     val full_rec_name = Sign.full_name (Theory.sign_of thy) rec_name;
652     val rec_const = list_comb
653       (Const (full_rec_name, paramTs ---> setT), params);
655     val fp_def_term = Logic.mk_equals (rec_const,
656       Const (fp_name, (setT --> setT) --> setT) \$ fp_fun)
658     val def_terms = fp_def_term :: (if length cs < 2 then [] else
659       map (fn c => Logic.mk_equals (c, mk_vimage cs sumT rec_const c)) cs);
661     val thy' = thy |>
662       (if declare_consts then
663         Theory.add_consts_i (map (fn (c, n) =>
664           (n, paramTs ---> fastype_of c, NoSyn)) (cs ~~ cnames))
665        else I) |>
666       (if length cs < 2 then I else
667        Theory.add_consts_i [(rec_name, paramTs ---> setT, NoSyn)]) |>
668       Theory.add_path rec_name |>
669       PureThy.add_defss_i [(("defs", def_terms), [])];
671     (* get definitions from theory *)
673     val fp_def::rec_sets_defs = PureThy.get_thms thy' "defs";
675     (* prove and store theorems *)
677     val mono = prove_mono setT fp_fun monos thy';
678     val (intrs, unfold) = prove_intrs coind mono fp_def intr_ts con_defs
679       rec_sets_defs thy';
680     val elims = if no_elim then [] else
681       prove_elims cs cTs params intr_ts intr_names unfold rec_sets_defs thy';
682     val raw_induct = if no_ind then Drule.asm_rl else
683       if coind then standard (rule_by_tactic
684         (rewrite_tac [mk_meta_eq vimage_Un] THEN
685           fold_tac rec_sets_defs) (mono RS (fp_def RS def_Collect_coinduct)))
686       else
687         prove_indrule cs cTs sumT rec_const params intr_ts mono fp_def
688           rec_sets_defs thy';
689     val induct = if coind orelse no_ind orelse length cs > 1 then raw_induct
690       else standard (raw_induct RSN (2, rev_mp));
692     val thy'' = thy'
693       |> PureThy.add_thmss [(("intrs", intrs), atts)]
694       |> PureThy.add_thms ((intr_names ~~ intrs) ~~ intr_atts)
695       |> (if no_elim then I else PureThy.add_thmss [(("elims", elims), [])])
696       |> (if no_ind then I else PureThy.add_thms
697         [((coind_prefix coind ^ "induct", induct), [RuleCases.case_names induct_cases])])
698       |> Theory.parent_path;
699     val intrs' = PureThy.get_thms thy'' "intrs";
700     val elims' = if no_elim then elims else PureThy.get_thms thy'' "elims";  (* FIXME improve *)
701     val induct' = if no_ind then induct else PureThy.get_thm thy'' (coind_prefix coind ^ "induct");  (* FIXME improve *)
702   in (thy'',
703     {defs = fp_def::rec_sets_defs,
704      mono = mono,
705      unfold = unfold,
706      intrs = intrs',
707      elims = elims',
708      mk_cases = mk_cases elims',
709      raw_induct = raw_induct,
710      induct = induct'})
711   end;
715 (** axiomatic introduction of (co)inductive sets **)
717 fun add_ind_axm verbose declare_consts alt_name coind no_elim no_ind cs
718     atts intros monos con_defs thy params paramTs cTs cnames induct_cases =
719   let
720     val rec_name = if alt_name = "" then space_implode "_" cnames else alt_name;
722     val ((intr_names, intr_ts), intr_atts) = apfst split_list (split_list intros);
723     val (elim_ts, elim_cases) = Library.split_list (mk_elims cs cTs params intr_ts intr_names);
725     val (_, ind_prems, mutual_ind_concl) = mk_indrule cs cTs params intr_ts;
726     val ind_t = Logic.list_implies (ind_prems, mutual_ind_concl);
728     val thy' = thy
729       |> (if declare_consts then
731               (map (fn (c, n) => (n, paramTs ---> fastype_of c, NoSyn)) (cs ~~ cnames))
732          else I)
733       |> Theory.add_path rec_name
734       |> PureThy.add_axiomss_i [(("intrs", intr_ts), atts), (("raw_elims", elim_ts), [])]
735       |> (if coind then I else
736             PureThy.add_axioms_i [(("raw_induct", ind_t), [apsnd (standard o split_rule)])]);
738     val intrs = PureThy.get_thms thy' "intrs";
739     val elims = map2 (fn (th, cases) => RuleCases.name cases th)
740       (PureThy.get_thms thy' "raw_elims", elim_cases);
741     val raw_induct = if coind then Drule.asm_rl else PureThy.get_thm thy' "raw_induct";
742     val induct = if coind orelse length cs > 1 then raw_induct
743       else standard (raw_induct RSN (2, rev_mp));
745     val thy'' =
746       thy'
747       |> PureThy.add_thmss [(("elims", elims), [])]
748       |> (if coind then I else PureThy.add_thms [(("induct", induct),
749           [RuleCases.case_names induct_cases])])
750       |> PureThy.add_thms ((intr_names ~~ intrs) ~~ intr_atts)
751       |> Theory.parent_path;
752     val induct' = if coind then raw_induct else PureThy.get_thm thy'' "induct";
753   in (thy'',
754     {defs = [],
755      mono = Drule.asm_rl,
756      unfold = Drule.asm_rl,
757      intrs = intrs,
758      elims = elims,
759      mk_cases = mk_cases elims,
760      raw_induct = raw_induct,
761      induct = induct'})
762   end;
766 (** introduction of (co)inductive sets **)
768 fun add_inductive_i verbose declare_consts alt_name coind no_elim no_ind cs
769     atts intros monos con_defs thy =
770   let
771     val _ = Theory.requires thy "Inductive" (coind_prefix coind ^ "inductive definitions");
772     val sign = Theory.sign_of thy;
774     (*parameters should agree for all mutually recursive components*)
775     val (_, params) = strip_comb (hd cs);
776     val paramTs = map (try' (snd o dest_Free) "Parameter in recursive\
777       \ component is not a free variable: " sign) params;
779     val cTs = map (try' (HOLogic.dest_setT o fastype_of)
780       "Recursive component not of type set: " sign) cs;
782     val full_cnames = map (try' (fst o dest_Const o head_of)
783       "Recursive set not previously declared as constant: " sign) cs;
784     val cnames = map Sign.base_name full_cnames;
786     val _ = seq (check_rule sign cs o snd o fst) intros;
787     val induct_cases = map (#1 o #1) intros;
789     val (thy1, result) =
790       (if ! quick_and_dirty then add_ind_axm else add_ind_def)
791         verbose declare_consts alt_name coind no_elim no_ind cs atts intros monos
792         con_defs thy params paramTs cTs cnames induct_cases;
793     val thy2 = thy1
794       |> put_inductives full_cnames ({names = full_cnames, coind = coind}, result)
795       |> add_cases_induct no_elim (no_ind orelse coind) full_cnames
796           (#elims result) (#induct result) induct_cases;
797   in (thy2, result) end;
801 (** external interface **)
803 fun add_inductive verbose coind c_strings srcs intro_srcs raw_monos raw_con_defs thy =
804   let
805     val sign = Theory.sign_of thy;
806     val cs = map (term_of o Thm.read_cterm sign o rpair HOLogic.termT) c_strings;
808     val atts = map (Attrib.global_attribute thy) srcs;
809     val intr_names = map (fst o fst) intro_srcs;
810     val intr_ts = map (term_of o Thm.read_cterm sign o rpair propT o snd o fst) intro_srcs;
811     val intr_atts = map (map (Attrib.global_attribute thy) o snd) intro_srcs;
812     val (cs', intr_ts') = unify_consts sign cs intr_ts;
814     val ((thy', con_defs), monos) = thy
815       |> IsarThy.apply_theorems raw_monos
816       |> apfst (IsarThy.apply_theorems raw_con_defs);
817   in
818     add_inductive_i verbose false "" coind false false cs'
819       atts ((intr_names ~~ intr_ts') ~~ intr_atts) monos con_defs thy'
820   end;
824 (** package setup **)
826 (* setup theory *)
828 val setup = [InductiveData.init,
829              Attrib.add_attributes [(monoN, mono_attr, "monotonicity rule")]];
832 (* outer syntax *)
834 local structure P = OuterParse and K = OuterSyntax.Keyword in
836 fun mk_ind coind (((sets, (atts, intrs)), monos), con_defs) =
837   #1 o add_inductive true coind sets atts (map P.triple_swap intrs) monos con_defs;
839 fun ind_decl coind =
840   (Scan.repeat1 P.term --| P.marg_comment) --
841   (P.\$\$\$ "intrs" |--
842     P.!!! (P.opt_attribs -- Scan.repeat1 (P.opt_thm_name ":" -- P.prop --| P.marg_comment))) --
843   Scan.optional (P.\$\$\$ "monos" |-- P.!!! P.xthms1 --| P.marg_comment) [] --
844   Scan.optional (P.\$\$\$ "con_defs" |-- P.!!! P.xthms1 --| P.marg_comment) []
845   >> (Toplevel.theory o mk_ind coind);
847 val inductiveP =
848   OuterSyntax.command "inductive" "define inductive sets" K.thy_decl (ind_decl false);
850 val coinductiveP =
851   OuterSyntax.command "coinductive" "define coinductive sets" K.thy_decl (ind_decl true);
854 val ind_cases =
855   P.opt_thm_name "=" -- P.xname --| P.\$\$\$ ":" -- Scan.repeat1 P.prop -- P.marg_comment
856   >> (Toplevel.theory o inductive_cases);
858 val inductive_casesP =
859   OuterSyntax.command "inductive_cases" "create simplified instances of elimination rules"
860     K.thy_decl ind_cases;
862 val _ = OuterSyntax.add_keywords ["intrs", "monos", "con_defs"];
863 val _ = OuterSyntax.add_parsers [inductiveP, coinductiveP, inductive_casesP];
865 end;
868 end;