src/HOL/Tools/inductive_package.ML
 author wenzelm Sat Sep 02 21:50:38 2000 +0200 (2000-09-02) changeset 9804 ee0c337327cf parent 9643 c94db1a96f4e child 9831 9b883c416aef permissions -rw-r--r--
"inductive_cases": proper command;
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 -> ({names: string list, coind: bool} *
36     {defs: thm list, elims: thm list, raw_induct: thm, induct: thm,
37      intrs: thm list, mk_cases: string -> thm, mono: thm, unfold: thm}) option
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) * string list) * Comment.text
53     -> theory -> theory
54   val inductive_cases_i: ((bstring * theory attribute list) * term list) * Comment.text
55     -> theory -> theory
56   val setup: (theory -> theory) list
57 end;
59 structure InductivePackage: INDUCTIVE_PACKAGE =
60 struct
63 (*** theory data ***)
65 (* data kind 'HOL/inductive' *)
67 type inductive_info =
68   {names: string list, coind: bool} * {defs: thm list, elims: thm list, raw_induct: thm,
69     induct: thm, intrs: thm list, mk_cases: string -> thm, mono: thm, unfold: thm};
71 structure InductiveArgs =
72 struct
73   val name = "HOL/inductive";
74   type T = inductive_info Symtab.table * thm list;
76   val empty = (Symtab.empty, []);
77   val copy = I;
78   val prep_ext = I;
79   fun merge ((tab1, monos1), (tab2, monos2)) = (Symtab.merge (K true) (tab1, tab2),
80     Library.generic_merge Thm.eq_thm I I monos1 monos2);
82   fun print sg (tab, monos) =
83     [Pretty.strs ("(co)inductives:" :: map #1 (Sign.cond_extern_table sg Sign.constK tab)),
84      Pretty.big_list "monotonicity rules:" (map Display.pretty_thm monos)]
85     |> Pretty.chunks |> Pretty.writeln;
86 end;
88 structure InductiveData = TheoryDataFun(InductiveArgs);
89 val print_inductives = InductiveData.print;
92 (* get and put data *)
94 fun get_inductive thy name = Symtab.lookup (fst (InductiveData.get thy), name);
96 fun the_inductive thy name =
97   (case get_inductive thy name of
98     None => error ("Unknown (co)inductive set " ^ quote name)
99   | Some info => info);
101 fun put_inductives names info thy =
102   let
103     fun upd ((tab, monos), name) = (Symtab.update_new ((name, info), tab), monos);
104     val tab_monos = foldl upd (InductiveData.get thy, names)
105       handle Symtab.DUP name => error ("Duplicate definition of (co)inductive set " ^ quote name);
106   in InductiveData.put tab_monos thy end;
110 (** monotonicity rules **)
112 val get_monos = snd o InductiveData.get;
113 fun put_monos thms thy = InductiveData.put (fst (InductiveData.get thy), thms) thy;
115 fun mk_mono thm =
116   let
117     fun eq2mono thm' = [standard (thm' RS (thm' RS eq_to_mono))] @
118       (case concl_of thm of
119           (_ \$ (_ \$ (Const ("Not", _) \$ _) \$ _)) => []
120         | _ => [standard (thm' RS (thm' RS eq_to_mono2))]);
121     val concl = concl_of thm
122   in
123     if Logic.is_equals concl then
124       eq2mono (thm RS meta_eq_to_obj_eq)
125     else if can (HOLogic.dest_eq o HOLogic.dest_Trueprop) concl then
126       eq2mono thm
127     else [thm]
128   end;
131 (* attributes *)
133 local
135 fun map_rules_global f thy = put_monos (f (get_monos thy)) thy;
137 fun add_mono thm rules = Library.gen_union Thm.eq_thm (mk_mono thm, rules);
138 fun del_mono thm rules = Library.gen_rems Thm.eq_thm (rules, mk_mono thm);
140 fun mk_att f g (x, thm) = (f (g thm) x, thm);
142 in
143   val mono_add_global = mk_att map_rules_global add_mono;
144   val mono_del_global = mk_att map_rules_global del_mono;
145 end;
147 val mono_attr =
149   Attrib.add_del_args Attrib.undef_local_attribute Attrib.undef_local_attribute);
153 (** utilities **)
155 (* messages *)
157 val quiet_mode = ref false;
158 fun message s = if !quiet_mode then () else writeln s;
160 fun coind_prefix true = "co"
161   | coind_prefix false = "";
164 (* the following code ensures that each recursive set *)
165 (* always has the same type in all introduction rules *)
167 fun unify_consts sign cs intr_ts =
168   (let
169     val {tsig, ...} = Sign.rep_sg sign;
170     val add_term_consts_2 =
171       foldl_aterms (fn (cs, Const c) => c ins cs | (cs, _) => cs);
172     fun varify (t, (i, ts)) =
173       let val t' = map_term_types (incr_tvar (i + 1)) (Type.varify (t, []))
174       in (maxidx_of_term t', t'::ts) end;
175     val (i, cs') = foldr varify (cs, (~1, []));
176     val (i', intr_ts') = foldr varify (intr_ts, (i, []));
177     val rec_consts = foldl add_term_consts_2 ([], cs');
178     val intr_consts = foldl add_term_consts_2 ([], intr_ts');
179     fun unify (env, (cname, cT)) =
180       let val consts = map snd (filter (fn c => fst c = cname) intr_consts)
181       in foldl (fn ((env', j'), Tp) => (Type.unify tsig j' env' Tp))
182           (env, (replicate (length consts) cT) ~~ consts)
183       end;
184     val (env, _) = foldl unify ((Vartab.empty, i'), rec_consts);
185     fun typ_subst_TVars_2 env T = let val T' = typ_subst_TVars_Vartab env T
186       in if T = T' then T else typ_subst_TVars_2 env T' end;
187     val subst = fst o Type.freeze_thaw o
188       (map_term_types (typ_subst_TVars_2 env))
190   in (map subst cs', map subst intr_ts')
191   end) handle Type.TUNIFY =>
192     (warning "Occurrences of recursive constant have non-unifiable types"; (cs, intr_ts));
195 (* misc *)
197 val Const _ \$ (vimage_f \$ _) \$ _ = HOLogic.dest_Trueprop (concl_of vimageD);
199 val vimage_name = Sign.intern_const (Theory.sign_of Vimage.thy) "op -``";
200 val mono_name = Sign.intern_const (Theory.sign_of Ord.thy) "mono";
202 (* make injections needed in mutually recursive definitions *)
204 fun mk_inj cs sumT c x =
205   let
206     fun mk_inj' T n i =
207       if n = 1 then x else
208       let val n2 = n div 2;
209           val Type (_, [T1, T2]) = T
210       in
211         if i <= n2 then
212           Const ("Inl", T1 --> T) \$ (mk_inj' T1 n2 i)
213         else
214           Const ("Inr", T2 --> T) \$ (mk_inj' T2 (n - n2) (i - n2))
215       end
216   in mk_inj' sumT (length cs) (1 + find_index_eq c cs)
217   end;
219 (* make "vimage" terms for selecting out components of mutually rec.def. *)
221 fun mk_vimage cs sumT t c = if length cs < 2 then t else
222   let
223     val cT = HOLogic.dest_setT (fastype_of c);
224     val vimageT = [cT --> sumT, HOLogic.mk_setT sumT] ---> HOLogic.mk_setT cT
225   in
226     Const (vimage_name, vimageT) \$
227       Abs ("y", cT, mk_inj cs sumT c (Bound 0)) \$ t
228   end;
232 (** well-formedness checks **)
234 fun err_in_rule sign t msg = error ("Ill-formed introduction rule\n" ^
235   (Sign.string_of_term sign t) ^ "\n" ^ msg);
237 fun err_in_prem sign t p msg = error ("Ill-formed premise\n" ^
238   (Sign.string_of_term sign p) ^ "\nin introduction rule\n" ^
239   (Sign.string_of_term sign t) ^ "\n" ^ msg);
241 val msg1 = "Conclusion of introduction rule must have form\
242           \ ' t : S_i '";
243 val msg2 = "Non-atomic premise";
244 val msg3 = "Recursion term on left of member symbol";
246 fun check_rule sign cs r =
247   let
248     fun check_prem prem = if can HOLogic.dest_Trueprop prem then ()
249       else err_in_prem sign r prem msg2;
251   in (case HOLogic.dest_Trueprop (Logic.strip_imp_concl r) of
252         (Const ("op :", _) \$ t \$ u) =>
253           if u mem cs then
254             if exists (Logic.occs o (rpair t)) cs then
255               err_in_rule sign r msg3
256             else
257               seq check_prem (Logic.strip_imp_prems r)
258           else err_in_rule sign r msg1
259       | _ => err_in_rule sign r msg1)
260   end;
262 fun try' f msg sign t = (case (try f t) of
263       Some x => x
264     | None => error (msg ^ Sign.string_of_term sign t));
268 (*** properties of (co)inductive sets ***)
270 (** elimination rules **)
272 fun mk_elims cs cTs params intr_ts intr_names =
273   let
274     val used = foldr add_term_names (intr_ts, []);
275     val [aname, pname] = variantlist (["a", "P"], used);
276     val P = HOLogic.mk_Trueprop (Free (pname, HOLogic.boolT));
278     fun dest_intr r =
279       let val Const ("op :", _) \$ t \$ u =
280         HOLogic.dest_Trueprop (Logic.strip_imp_concl r)
281       in (u, t, Logic.strip_imp_prems r) end;
283     val intrs = map dest_intr intr_ts ~~ intr_names;
285     fun mk_elim (c, T) =
286       let
287         val a = Free (aname, T);
289         fun mk_elim_prem (_, t, ts) =
290           list_all_free (map dest_Free ((foldr add_term_frees (t::ts, [])) \\ params),
291             Logic.list_implies (HOLogic.mk_Trueprop (HOLogic.mk_eq (a, t)) :: ts, P));
292         val c_intrs = (filter (equal c o #1 o #1) intrs);
293       in
294         (Logic.list_implies (HOLogic.mk_Trueprop (HOLogic.mk_mem (a, c)) ::
295           map mk_elim_prem (map #1 c_intrs), P), map #2 c_intrs)
296       end
297   in
298     map mk_elim (cs ~~ cTs)
299   end;
303 (** premises and conclusions of induction rules **)
305 fun mk_indrule cs cTs params intr_ts =
306   let
307     val used = foldr add_term_names (intr_ts, []);
309     (* predicates for induction rule *)
311     val preds = map Free (variantlist (if length cs < 2 then ["P"] else
312       map (fn i => "P" ^ string_of_int i) (1 upto length cs), used) ~~
313         map (fn T => T --> HOLogic.boolT) cTs);
315     (* transform an introduction rule into a premise for induction rule *)
317     fun mk_ind_prem r =
318       let
319         val frees = map dest_Free ((add_term_frees (r, [])) \\ params);
321         val pred_of = curry (Library.gen_assoc (op aconv)) (cs ~~ preds);
323         fun subst (s as ((m as Const ("op :", T)) \$ t \$ u)) =
324               (case pred_of u of
325                   None => (m \$ fst (subst t) \$ fst (subst u), None)
326                 | Some P => (HOLogic.conj \$ s \$ (P \$ t), Some (s, P \$ t)))
327           | subst s =
328               (case pred_of s of
329                   Some P => (HOLogic.mk_binop "op Int"
330                     (s, HOLogic.Collect_const (HOLogic.dest_setT
331                       (fastype_of s)) \$ P), None)
332                 | None => (case s of
333                      (t \$ u) => (fst (subst t) \$ fst (subst u), None)
334                    | (Abs (a, T, t)) => (Abs (a, T, fst (subst t)), None)
335                    | _ => (s, None)));
337         fun mk_prem (s, prems) = (case subst s of
338               (_, Some (t, u)) => t :: u :: prems
339             | (t, _) => t :: prems);
341         val Const ("op :", _) \$ t \$ u =
342           HOLogic.dest_Trueprop (Logic.strip_imp_concl r)
344       in list_all_free (frees,
345            Logic.list_implies (map HOLogic.mk_Trueprop (foldr mk_prem
346              (map HOLogic.dest_Trueprop (Logic.strip_imp_prems r), [])),
347                HOLogic.mk_Trueprop (the (pred_of u) \$ t)))
348       end;
350     val ind_prems = map mk_ind_prem intr_ts;
352     (* make conclusions for induction rules *)
354     fun mk_ind_concl ((c, P), (ts, x)) =
355       let val T = HOLogic.dest_setT (fastype_of c);
356           val Ts = HOLogic.prodT_factors T;
357           val (frees, x') = foldr (fn (T', (fs, s)) =>
358             ((Free (s, T'))::fs, bump_string s)) (Ts, ([], x));
359           val tuple = HOLogic.mk_tuple T frees;
360       in ((HOLogic.mk_binop "op -->"
361         (HOLogic.mk_mem (tuple, c), P \$ tuple))::ts, x')
362       end;
364     val mutual_ind_concl = HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
365         (fst (foldr mk_ind_concl (cs ~~ preds, ([], "xa")))))
367   in (preds, ind_prems, mutual_ind_concl)
368   end;
372 (** prepare cases and induct rules **)
374 (*
375   transform mutual rule:
376     HH ==> (x1:A1 --> P1 x1) & ... & (xn:An --> Pn xn)
377   into i-th projection:
378     xi:Ai ==> HH ==> Pi xi
379 *)
381 fun project_rules [name] rule = [(name, rule)]
382   | project_rules names mutual_rule =
383       let
384         val n = length names;
385         fun proj i =
386           (if i < n then (fn th => th RS conjunct1) else I)
387             (Library.funpow (i - 1) (fn th => th RS conjunct2) mutual_rule)
388             RS mp |> Thm.permute_prems 0 ~1 |> Drule.standard;
389       in names ~~ map proj (1 upto n) end;
391 fun add_cases_induct no_elim no_ind names elims induct induct_cases =
392   let
393     fun cases_spec (name, elim) thy =
394       thy
395       |> Theory.add_path (Sign.base_name name)
396       |> (#1 o PureThy.add_thms [(("cases", elim), [InductMethod.cases_set_global name])])
397       |> Theory.parent_path;
398     val cases_specs = if no_elim then [] else map2 cases_spec (names, elims);
400     fun induct_spec (name, th) = (#1 o PureThy.add_thms
401       [(("", th), [RuleCases.case_names induct_cases, InductMethod.induct_set_global name])]);
402     val induct_specs = if no_ind then [] else map induct_spec (project_rules names induct);
403   in Library.apply (cases_specs @ induct_specs) end;
407 (*** proofs for (co)inductive sets ***)
409 (** prove monotonicity **)
411 fun prove_mono setT fp_fun monos thy =
412   let
413     val _ = message "  Proving monotonicity ...";
415     val mono = prove_goalw_cterm [] (cterm_of (Theory.sign_of thy) (HOLogic.mk_Trueprop
416       (Const (mono_name, (setT --> setT) --> HOLogic.boolT) \$ fp_fun)))
417         (fn _ => [rtac monoI 1, REPEAT (ares_tac (get_monos thy @ flat (map mk_mono monos)) 1)])
419   in mono end;
423 (** prove introduction rules **)
425 fun prove_intrs coind mono fp_def intr_ts con_defs rec_sets_defs thy =
426   let
427     val _ = message "  Proving the introduction rules ...";
429     val unfold = standard (mono RS (fp_def RS
430       (if coind then def_gfp_Tarski else def_lfp_Tarski)));
432     fun select_disj 1 1 = []
433       | select_disj _ 1 = [rtac disjI1]
434       | select_disj n i = (rtac disjI2)::(select_disj (n - 1) (i - 1));
436     val intrs = map (fn (i, intr) => prove_goalw_cterm rec_sets_defs
437       (cterm_of (Theory.sign_of thy) intr) (fn prems =>
438        [(*insert prems and underlying sets*)
439        cut_facts_tac prems 1,
440        stac unfold 1,
441        REPEAT (resolve_tac [vimageI2, CollectI] 1),
442        (*Now 1-2 subgoals: the disjunction, perhaps equality.*)
443        EVERY1 (select_disj (length intr_ts) i),
444        (*Not ares_tac, since refl must be tried before any equality assumptions;
445          backtracking may occur if the premises have extra variables!*)
446        DEPTH_SOLVE_1 (resolve_tac [refl,exI,conjI] 1 APPEND assume_tac 1),
447        (*Now solve the equations like Inl 0 = Inl ?b2*)
448        rewrite_goals_tac con_defs,
449        REPEAT (rtac refl 1)])) (1 upto (length intr_ts) ~~ intr_ts)
451   in (intrs, unfold) end;
455 (** prove elimination rules **)
457 fun prove_elims cs cTs params intr_ts intr_names unfold rec_sets_defs thy =
458   let
459     val _ = message "  Proving the elimination rules ...";
461     val rules1 = [CollectE, disjE, make_elim vimageD, exE];
462     val rules2 = [conjE, Inl_neq_Inr, Inr_neq_Inl] @
463       map make_elim [Inl_inject, Inr_inject];
464   in
465     map (fn (t, cases) => prove_goalw_cterm rec_sets_defs
466       (cterm_of (Theory.sign_of thy) t) (fn prems =>
467         [cut_facts_tac [hd prems] 1,
468          dtac (unfold RS subst) 1,
469          REPEAT (FIRSTGOAL (eresolve_tac rules1)),
470          REPEAT (FIRSTGOAL (eresolve_tac rules2)),
471          EVERY (map (fn prem =>
472            DEPTH_SOLVE_1 (ares_tac [prem, conjI] 1)) (tl prems))])
473       |> RuleCases.name cases)
474       (mk_elims cs cTs params intr_ts intr_names)
475   end;
478 (** derivation of simplified elimination rules **)
480 (*Applies freeness of the given constructors, which *must* be unfolded by
481   the given defs.  Cannot simply use the local con_defs because con_defs=[]
482   for inference systems.
483  *)
485 (*cprop should have the form t:Si where Si is an inductive set*)
487 val mk_cases_err = "mk_cases: proposition not of form 't : S_i'";
489 fun mk_cases_i elims ss cprop =
490   let
491     val prem = Thm.assume cprop;
492     val tac = ALLGOALS (InductMethod.simp_case_tac false ss) THEN prune_params_tac;
493     fun mk_elim rl = Drule.standard (Tactic.rule_by_tactic tac (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_err, Pretty.fbrk, Display.pretty_cterm cprop])))
499   end;
501 fun mk_cases elims s =
502   mk_cases_i elims (simpset()) (Thm.read_cterm (Thm.sign_of_thm (hd elims)) (s, propT));
504 fun smart_mk_cases thy ss cprop =
505   let
506     val c = #1 (Term.dest_Const (Term.head_of (#2 (HOLogic.dest_mem (HOLogic.dest_Trueprop
507       (Logic.strip_imp_concl (Thm.term_of cprop))))))) handle TERM _ => error mk_cases_err;
508     val (_, {elims, ...}) = the_inductive thy c;
509   in mk_cases_i elims ss cprop end;
512 (* inductive_cases(_i) *)
514 fun gen_inductive_cases prep_att prep_const prep_prop
515     (((name, raw_atts), raw_props), comment) thy =
516   let
517     val ss = Simplifier.simpset_of thy;
518     val sign = Theory.sign_of thy;
519     val cprops = map (Thm.cterm_of sign o prep_prop (ProofContext.init thy)) raw_props;
520     val atts = map (prep_att thy) raw_atts;
521     val thms = map (smart_mk_cases thy ss) cprops;
522   in thy |> IsarThy.have_theorems_i [(((name, atts), map Thm.no_attributes thms), comment)] end;
524 val inductive_cases =
525   gen_inductive_cases Attrib.global_attribute Sign.intern_const ProofContext.read_prop;
527 val inductive_cases_i = gen_inductive_cases (K I) (K I) ProofContext.cert_prop;
530 (* mk_cases_meth *)
532 fun mk_cases_meth (ctxt, raw_props) =
533   let
534     val thy = ProofContext.theory_of ctxt;
535     val ss = Simplifier.get_local_simpset ctxt;
536     val cprops = map (Thm.cterm_of (Theory.sign_of thy) o ProofContext.read_prop ctxt) raw_props;
537   in Method.erule (map (smart_mk_cases thy ss) cprops) end;
539 val mk_cases_args = Method.syntax (Scan.lift (Scan.repeat1 Args.name));
543 (** prove induction rule **)
545 fun prove_indrule cs cTs sumT rec_const params intr_ts mono
546     fp_def rec_sets_defs thy =
547   let
548     val _ = message "  Proving the induction rule ...";
550     val sign = Theory.sign_of thy;
552     val sum_case_rewrites = (case ThyInfo.lookup_theory "Datatype" of
553         None => []
554       | Some thy' => map mk_meta_eq (PureThy.get_thms thy' "sum.cases"));
556     val (preds, ind_prems, mutual_ind_concl) = mk_indrule cs cTs params intr_ts;
558     (* make predicate for instantiation of abstract induction rule *)
560     fun mk_ind_pred _ [P] = P
561       | mk_ind_pred T Ps =
562          let val n = (length Ps) div 2;
563              val Type (_, [T1, T2]) = T
564          in Const ("Datatype.sum.sum_case",
565            [T1 --> HOLogic.boolT, T2 --> HOLogic.boolT, T] ---> HOLogic.boolT) \$
566              mk_ind_pred T1 (take (n, Ps)) \$ mk_ind_pred T2 (drop (n, Ps))
567          end;
569     val ind_pred = mk_ind_pred sumT preds;
571     val ind_concl = HOLogic.mk_Trueprop
572       (HOLogic.all_const sumT \$ Abs ("x", sumT, HOLogic.mk_binop "op -->"
573         (HOLogic.mk_mem (Bound 0, rec_const), ind_pred \$ Bound 0)));
575     (* simplification rules for vimage and Collect *)
577     val vimage_simps = if length cs < 2 then [] else
578       map (fn c => prove_goalw_cterm [] (cterm_of sign
579         (HOLogic.mk_Trueprop (HOLogic.mk_eq
580           (mk_vimage cs sumT (HOLogic.Collect_const sumT \$ ind_pred) c,
581            HOLogic.Collect_const (HOLogic.dest_setT (fastype_of c)) \$
582              nth_elem (find_index_eq c cs, preds)))))
583         (fn _ => [rtac vimage_Collect 1, rewrite_goals_tac sum_case_rewrites,
584           rtac refl 1])) cs;
586     val induct = prove_goalw_cterm [] (cterm_of sign
587       (Logic.list_implies (ind_prems, ind_concl))) (fn prems =>
588         [rtac (impI RS allI) 1,
589          DETERM (etac (mono RS (fp_def RS def_induct)) 1),
590          rewrite_goals_tac (map mk_meta_eq (vimage_Int::Int_Collect::vimage_simps)),
591          fold_goals_tac rec_sets_defs,
592          (*This CollectE and disjE separates out the introduction rules*)
593          REPEAT (FIRSTGOAL (eresolve_tac [CollectE, disjE, exE])),
594          (*Now break down the individual cases.  No disjE here in case
595            some premise involves disjunction.*)
596          REPEAT (FIRSTGOAL (etac conjE ORELSE' hyp_subst_tac)),
597          rewrite_goals_tac sum_case_rewrites,
598          EVERY (map (fn prem =>
599            DEPTH_SOLVE_1 (ares_tac [prem, conjI, refl] 1)) prems)]);
601     val lemma = prove_goalw_cterm rec_sets_defs (cterm_of sign
602       (Logic.mk_implies (ind_concl, mutual_ind_concl))) (fn prems =>
603         [cut_facts_tac prems 1,
604          REPEAT (EVERY
605            [REPEAT (resolve_tac [conjI, impI] 1),
606             TRY (dtac vimageD 1), etac allE 1, dtac mp 1, atac 1,
607             rewrite_goals_tac sum_case_rewrites,
608             atac 1])])
610   in standard (split_rule (induct RS lemma))
611   end;
615 (*** specification of (co)inductive sets ****)
617 (** definitional introduction of (co)inductive sets **)
619 fun mk_ind_def declare_consts alt_name coind cs intr_ts monos con_defs thy
620       params paramTs cTs cnames =
621   let
622     val sumT = fold_bal (fn (T, U) => Type ("+", [T, U])) cTs;
623     val setT = HOLogic.mk_setT sumT;
625     val fp_name = if coind then Sign.intern_const (Theory.sign_of Gfp.thy) "gfp"
626       else Sign.intern_const (Theory.sign_of Lfp.thy) "lfp";
628     val used = foldr add_term_names (intr_ts, []);
629     val [sname, xname] = variantlist (["S", "x"], used);
631     (* transform an introduction rule into a conjunction  *)
632     (*   [| t : ... S_i ... ; ... |] ==> u : S_j          *)
633     (* is transformed into                                *)
634     (*   x = Inj_j u & t : ... Inj_i -`` S ... & ...      *)
636     fun transform_rule r =
637       let
638         val frees = map dest_Free ((add_term_frees (r, [])) \\ params);
639         val subst = subst_free
640           (cs ~~ (map (mk_vimage cs sumT (Free (sname, setT))) cs));
641         val Const ("op :", _) \$ t \$ u =
642           HOLogic.dest_Trueprop (Logic.strip_imp_concl r)
644       in foldr (fn ((x, T), P) => HOLogic.mk_exists (x, T, P))
645         (frees, foldr1 HOLogic.mk_conj
646           (((HOLogic.eq_const sumT) \$ Free (xname, sumT) \$ (mk_inj cs sumT u t))::
647             (map (subst o HOLogic.dest_Trueprop)
648               (Logic.strip_imp_prems r))))
649       end
651     (* make a disjunction of all introduction rules *)
653     val fp_fun = absfree (sname, setT, (HOLogic.Collect_const sumT) \$
654       absfree (xname, sumT, foldr1 HOLogic.mk_disj (map transform_rule intr_ts)));
656     (* add definiton of recursive sets to theory *)
658     val rec_name = if alt_name = "" then space_implode "_" cnames else alt_name;
659     val full_rec_name = Sign.full_name (Theory.sign_of thy) rec_name;
661     val rec_const = list_comb
662       (Const (full_rec_name, paramTs ---> setT), params);
664     val fp_def_term = Logic.mk_equals (rec_const,
665       Const (fp_name, (setT --> setT) --> setT) \$ fp_fun)
667     val def_terms = fp_def_term :: (if length cs < 2 then [] else
668       map (fn c => Logic.mk_equals (c, mk_vimage cs sumT rec_const c)) cs);
670     val (thy', [fp_def :: rec_sets_defs]) =
671       thy
672       |> (if declare_consts then
673           Theory.add_consts_i (map (fn (c, n) =>
674             (n, paramTs ---> fastype_of c, NoSyn)) (cs ~~ cnames))
675           else I)
676       |> (if length cs < 2 then I
677           else Theory.add_consts_i [(rec_name, paramTs ---> setT, NoSyn)])
678       |> Theory.add_path rec_name
679       |> PureThy.add_defss_i false [(("defs", def_terms), [])];
681     val mono = prove_mono setT fp_fun monos thy'
683   in
684     (thy', mono, fp_def, rec_sets_defs, rec_const, sumT)
685   end;
687 fun add_ind_def verbose declare_consts alt_name coind no_elim no_ind cs
688     atts intros monos con_defs thy params paramTs cTs cnames induct_cases =
689   let
690     val _ = if verbose then message ("Proofs for " ^ coind_prefix coind ^ "inductive set(s) " ^
691       commas_quote cnames) else ();
693     val ((intr_names, intr_ts), intr_atts) = apfst split_list (split_list intros);
695     val (thy', mono, fp_def, rec_sets_defs, rec_const, sumT) =
696       mk_ind_def declare_consts alt_name coind cs intr_ts monos con_defs thy
697         params paramTs cTs cnames;
699     val (intrs, unfold) = prove_intrs coind mono fp_def intr_ts con_defs
700       rec_sets_defs thy';
701     val elims = if no_elim then [] else
702       prove_elims cs cTs params intr_ts intr_names unfold rec_sets_defs thy';
703     val raw_induct = if no_ind then Drule.asm_rl else
704       if coind then standard (rule_by_tactic
705         (rewrite_tac [mk_meta_eq vimage_Un] THEN
706           fold_tac rec_sets_defs) (mono RS (fp_def RS def_Collect_coinduct)))
707       else
708         prove_indrule cs cTs sumT rec_const params intr_ts mono fp_def
709           rec_sets_defs thy';
710     val induct = if coind orelse no_ind orelse length cs > 1 then raw_induct
711       else standard (raw_induct RSN (2, rev_mp));
713     val (thy'', [intrs']) =
714       thy'
715       |> PureThy.add_thmss [(("intros", intrs), atts)]
716       |>> (#1 o PureThy.add_thms ((intr_names ~~ intrs) ~~ intr_atts))
717       |>> (if no_elim then I else #1 o PureThy.add_thmss [(("elims", elims), [])])
718       |>> (if no_ind then I else #1 o PureThy.add_thms
719         [((coind_prefix coind ^ "induct", induct), [RuleCases.case_names induct_cases])])
720       |>> Theory.parent_path;
721     val elims' = if no_elim then elims else PureThy.get_thms thy'' "elims";  (* FIXME improve *)
722     val induct' = if no_ind then induct else PureThy.get_thm thy'' (coind_prefix coind ^ "induct");  (* FIXME improve *)
723   in (thy'',
724     {defs = fp_def::rec_sets_defs,
725      mono = mono,
726      unfold = unfold,
727      intrs = intrs',
728      elims = elims',
729      mk_cases = mk_cases elims',
730      raw_induct = raw_induct,
731      induct = induct'})
732   end;
736 (** axiomatic introduction of (co)inductive sets **)
738 fun add_ind_axm verbose declare_consts alt_name coind no_elim no_ind cs
739     atts intros monos con_defs thy params paramTs cTs cnames induct_cases =
740   let
741     val _ = message (coind_prefix coind ^ "inductive set(s) " ^ commas_quote cnames);
743     val ((intr_names, intr_ts), intr_atts) = apfst split_list (split_list intros);
744     val (thy', _, fp_def, rec_sets_defs, _, _) =
745       mk_ind_def declare_consts alt_name coind cs intr_ts monos con_defs thy
746         params paramTs cTs cnames;
747     val (elim_ts, elim_cases) = Library.split_list (mk_elims cs cTs params intr_ts intr_names);
748     val (_, ind_prems, mutual_ind_concl) = mk_indrule cs cTs params intr_ts;
749     val ind_t = Logic.list_implies (ind_prems, mutual_ind_concl);
751     val (thy'', [intrs, raw_elims]) =
752       thy'
753       |> PureThy.add_axiomss_i [(("intros", intr_ts), atts), (("raw_elims", elim_ts), [])]
754       |>> (if coind then I else
755             #1 o PureThy.add_axioms_i [(("raw_induct", ind_t), [apsnd (standard o split_rule)])]);
757     val elims = map2 (fn (th, cases) => RuleCases.name cases th) (raw_elims, elim_cases);
758     val raw_induct = if coind then Drule.asm_rl else PureThy.get_thm thy'' "raw_induct";
759     val induct = if coind orelse length cs > 1 then raw_induct
760       else standard (raw_induct RSN (2, rev_mp));
762     val (thy''', ([elims'], intrs')) =
763       thy''
764       |> PureThy.add_thmss [(("elims", elims), [])]
765       |>> (if coind then I
766           else #1 o PureThy.add_thms [(("induct", induct), [RuleCases.case_names induct_cases])])
767       |>>> PureThy.add_thms ((intr_names ~~ intrs) ~~ intr_atts)
768       |>> Theory.parent_path;
769     val induct' = if coind then raw_induct else PureThy.get_thm thy''' "induct";
770   in (thy''',
771     {defs = fp_def :: rec_sets_defs,
772      mono = Drule.asm_rl,
773      unfold = Drule.asm_rl,
774      intrs = intrs',
775      elims = elims',
776      mk_cases = mk_cases elims',
777      raw_induct = raw_induct,
778      induct = induct'})
779   end;
783 (** introduction of (co)inductive sets **)
785 fun add_inductive_i verbose declare_consts alt_name coind no_elim no_ind cs
786     atts intros monos con_defs thy =
787   let
788     val _ = Theory.requires thy "Inductive" (coind_prefix coind ^ "inductive definitions");
789     val sign = Theory.sign_of thy;
791     (*parameters should agree for all mutually recursive components*)
792     val (_, params) = strip_comb (hd cs);
793     val paramTs = map (try' (snd o dest_Free) "Parameter in recursive\
794       \ component is not a free variable: " sign) params;
796     val cTs = map (try' (HOLogic.dest_setT o fastype_of)
797       "Recursive component not of type set: " sign) cs;
799     val full_cnames = map (try' (fst o dest_Const o head_of)
800       "Recursive set not previously declared as constant: " sign) cs;
801     val cnames = map Sign.base_name full_cnames;
803     val _ = seq (check_rule sign cs o snd o fst) intros;
804     val induct_cases = map (#1 o #1) intros;
806     val (thy1, result as {elims, induct, ...}) =
807       (if ! quick_and_dirty then add_ind_axm else add_ind_def)
808         verbose declare_consts alt_name coind no_elim no_ind cs atts intros monos
809         con_defs thy params paramTs cTs cnames induct_cases;
810     val thy2 = thy1
811       |> put_inductives full_cnames ({names = full_cnames, coind = coind}, result)
812       |> add_cases_induct no_elim (no_ind orelse coind) full_cnames elims induct induct_cases;
813   in (thy2, result) end;
817 (** external interface **)
819 fun add_inductive verbose coind c_strings srcs intro_srcs raw_monos raw_con_defs thy =
820   let
821     val sign = Theory.sign_of thy;
822     val cs = map (term_of o Thm.read_cterm sign o rpair HOLogic.termT) c_strings;
824     val atts = map (Attrib.global_attribute thy) srcs;
825     val intr_names = map (fst o fst) intro_srcs;
826     fun read_rule s = Thm.read_cterm sign (s, propT)
827       handle ERROR => error ("The error(s) above occurred for " ^ s);
828     val intr_ts = map (Thm.term_of o read_rule o snd o fst) intro_srcs;
829     val intr_atts = map (map (Attrib.global_attribute thy) o snd) intro_srcs;
830     val (cs', intr_ts') = unify_consts sign cs intr_ts;
832     val ((thy', con_defs), monos) = thy
833       |> IsarThy.apply_theorems raw_monos
834       |> apfst (IsarThy.apply_theorems raw_con_defs);
835   in
836     add_inductive_i verbose false "" coind false false cs'
837       atts ((intr_names ~~ intr_ts') ~~ intr_atts) monos con_defs thy'
838   end;
842 (** package setup **)
844 (* setup theory *)
846 val setup =
847  [InductiveData.init,
848   Method.add_methods [("ind_cases", mk_cases_meth oo mk_cases_args,
849     "dynamic case analysis on sets")],
850   Attrib.add_attributes [("mono", mono_attr, "monotonicity rule")]];
853 (* outer syntax *)
855 local structure P = OuterParse and K = OuterSyntax.Keyword in
857 fun mk_ind coind (((sets, (atts, intrs)), monos), con_defs) =
858   #1 o add_inductive true coind sets atts (map P.triple_swap intrs) monos con_defs;
860 fun ind_decl coind =
861   (Scan.repeat1 P.term --| P.marg_comment) --
862   (P.\$\$\$ "intros" |--
863     P.!!! (P.opt_attribs -- Scan.repeat1 (P.opt_thm_name ":" -- P.prop --| P.marg_comment))) --
864   Scan.optional (P.\$\$\$ "monos" |-- P.!!! P.xthms1 --| P.marg_comment) [] --
865   Scan.optional (P.\$\$\$ "con_defs" |-- P.!!! P.xthms1 --| P.marg_comment) []
866   >> (Toplevel.theory o mk_ind coind);
868 val inductiveP =
869   OuterSyntax.command "inductive" "define inductive sets" K.thy_decl (ind_decl false);
871 val coinductiveP =
872   OuterSyntax.command "coinductive" "define coinductive sets" K.thy_decl (ind_decl true);
875 val ind_cases =
876   P.opt_thm_name ":" -- Scan.repeat1 P.prop -- P.marg_comment
877   >> (Toplevel.theory o inductive_cases);
879 val inductive_casesP =
880   OuterSyntax.command "inductive_cases"
881     "create simplified instances of elimination rules (improper)" K.thy_script ind_cases;
883 val _ = OuterSyntax.add_keywords ["intros", "monos", "con_defs"];
884 val _ = OuterSyntax.add_parsers [inductiveP, coinductiveP, inductive_casesP];
886 end;
889 end;