src/HOL/Tools/inductive_package.ML
 author wenzelm Wed Mar 08 18:06:12 2000 +0100 (2000-03-08) changeset 8375 0544749a5e8f parent 8336 fdf3ac335f77 child 8380 c96953faf0a4 permissions -rw-r--r--
mk_elims, add_cases_induct: name rule cases;
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 (([], i'), rec_consts);
192     fun typ_subst_TVars_2 env T = let val T' = typ_subst_TVars 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 tune_names raw_names =
280   let
281     fun tune ("", i) = Library.string_of_int i
282       | tune (s, _) = s;
283   in map2 tune (raw_names, 0 upto (length raw_names - 1)) end;
285 fun mk_elims cs cTs params intr_ts intr_names =
286   let
287     val used = foldr add_term_names (intr_ts, []);
288     val [aname, pname] = variantlist (["a", "P"], used);
289     val P = HOLogic.mk_Trueprop (Free (pname, HOLogic.boolT));
291     fun dest_intr r =
292       let val Const ("op :", _) \$ t \$ u =
293         HOLogic.dest_Trueprop (Logic.strip_imp_concl r)
294       in (u, t, Logic.strip_imp_prems r) end;
296     val intrs = map dest_intr intr_ts ~~ tune_names intr_names;
298     fun mk_elim (c, T) =
299       let
300         val a = Free (aname, T);
302         fun mk_elim_prem (_, t, ts) =
303           list_all_free (map dest_Free ((foldr add_term_frees (t::ts, [])) \\ params),
304             Logic.list_implies (HOLogic.mk_Trueprop (HOLogic.mk_eq (a, t)) :: ts, P));
305         val c_intrs = (filter (equal c o #1 o #1) intrs);
306       in
307         (Logic.list_implies (HOLogic.mk_Trueprop (HOLogic.mk_mem (a, c)) ::
308           map mk_elim_prem (map #1 c_intrs), P), map #2 c_intrs)
309       end
310   in
311     map mk_elim (cs ~~ cTs)
312   end;
316 (** premises and conclusions of induction rules **)
318 fun mk_indrule cs cTs params intr_ts =
319   let
320     val used = foldr add_term_names (intr_ts, []);
322     (* predicates for induction rule *)
324     val preds = map Free (variantlist (if length cs < 2 then ["P"] else
325       map (fn i => "P" ^ string_of_int i) (1 upto length cs), used) ~~
326         map (fn T => T --> HOLogic.boolT) cTs);
328     (* transform an introduction rule into a premise for induction rule *)
330     fun mk_ind_prem r =
331       let
332         val frees = map dest_Free ((add_term_frees (r, [])) \\ params);
334         val pred_of = curry (Library.gen_assoc (op aconv)) (cs ~~ preds);
336         fun subst (s as ((m as Const ("op :", T)) \$ t \$ u)) =
337               (case pred_of u of
338                   None => (m \$ fst (subst t) \$ fst (subst u), None)
339                 | Some P => (HOLogic.conj \$ s \$ (P \$ t), Some (s, P \$ t)))
340           | subst s =
341               (case pred_of s of
342                   Some P => (HOLogic.mk_binop "op Int"
343                     (s, HOLogic.Collect_const (HOLogic.dest_setT
344                       (fastype_of s)) \$ P), None)
345                 | None => (case s of
346                      (t \$ u) => (fst (subst t) \$ fst (subst u), None)
347                    | (Abs (a, T, t)) => (Abs (a, T, fst (subst t)), None)
348                    | _ => (s, None)));
350         fun mk_prem (s, prems) = (case subst s of
351               (_, Some (t, u)) => t :: u :: prems
352             | (t, _) => t :: prems);
354         val Const ("op :", _) \$ t \$ u =
355           HOLogic.dest_Trueprop (Logic.strip_imp_concl r)
357       in list_all_free (frees,
358            Logic.list_implies (map HOLogic.mk_Trueprop (foldr mk_prem
359              (map HOLogic.dest_Trueprop (Logic.strip_imp_prems r), [])),
360                HOLogic.mk_Trueprop (the (pred_of u) \$ t)))
361       end;
363     val ind_prems = map mk_ind_prem intr_ts;
365     (* make conclusions for induction rules *)
367     fun mk_ind_concl ((c, P), (ts, x)) =
368       let val T = HOLogic.dest_setT (fastype_of c);
369           val Ts = HOLogic.prodT_factors T;
370           val (frees, x') = foldr (fn (T', (fs, s)) =>
371             ((Free (s, T'))::fs, bump_string s)) (Ts, ([], x));
372           val tuple = HOLogic.mk_tuple T frees;
373       in ((HOLogic.mk_binop "op -->"
374         (HOLogic.mk_mem (tuple, c), P \$ tuple))::ts, x')
375       end;
377     val mutual_ind_concl = HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
378         (fst (foldr mk_ind_concl (cs ~~ preds, ([], "xa")))))
380   in (preds, ind_prems, mutual_ind_concl)
381   end;
385 (** prepare cases and induct rules **)
387 (*
388   transform mutual rule:
389     HH ==> (x1:A1 --> P1 x1) & ... & (xn:An --> Pn xn)
390   into i-th projection:
391     xi:Ai ==> HH ==> Pi xi
392 *)
394 fun project_rules [name] rule = [(name, rule)]
395   | project_rules names mutual_rule =
396       let
397         val n = length names;
398         fun proj i =
399           (if i < n then (fn th => th RS conjunct1) else I)
400             (Library.funpow (i - 1) (fn th => th RS conjunct2) mutual_rule)
401             RS mp |> Thm.permute_prems 0 ~1 |> Drule.standard;
402       in names ~~ map proj (1 upto n) end;
404 fun add_cases_induct no_elim no_ind names elims induct induct_cases =
405   let
406     fun cases_spec (name, elim) = (("", elim), [InductMethod.cases_set_global name]);
407     val cases_specs = if no_elim then [] else map2 cases_spec (names, elims);
409     fun induct_spec (name, th) =
410       (("", th), [RuleCases.case_names (tune_names induct_cases),
411         InductMethod.induct_set_global name]);
412     val induct_specs =
413       if no_ind then []
414       else map induct_spec (project_rules names induct);
415   in PureThy.add_thms (cases_specs @ induct_specs) end;
419 (*** proofs for (co)inductive sets ***)
421 (** prove monotonicity **)
423 fun prove_mono setT fp_fun monos thy =
424   let
425     val _ = message "  Proving monotonicity ...";
427     val mono = prove_goalw_cterm [] (cterm_of (Theory.sign_of thy) (HOLogic.mk_Trueprop
428       (Const (mono_name, (setT --> setT) --> HOLogic.boolT) \$ fp_fun)))
429         (fn _ => [rtac monoI 1, REPEAT (ares_tac (get_monos thy @ flat (map mk_mono monos)) 1)])
431   in mono end;
435 (** prove introduction rules **)
437 fun prove_intrs coind mono fp_def intr_ts con_defs rec_sets_defs thy =
438   let
439     val _ = message "  Proving the introduction rules ...";
441     val unfold = standard (mono RS (fp_def RS
442       (if coind then def_gfp_Tarski else def_lfp_Tarski)));
444     fun select_disj 1 1 = []
445       | select_disj _ 1 = [rtac disjI1]
446       | select_disj n i = (rtac disjI2)::(select_disj (n - 1) (i - 1));
448     val intrs = map (fn (i, intr) => prove_goalw_cterm rec_sets_defs
449       (cterm_of (Theory.sign_of thy) intr) (fn prems =>
450        [(*insert prems and underlying sets*)
451        cut_facts_tac prems 1,
452        stac unfold 1,
453        REPEAT (resolve_tac [vimageI2, CollectI] 1),
454        (*Now 1-2 subgoals: the disjunction, perhaps equality.*)
455        EVERY1 (select_disj (length intr_ts) i),
456        (*Not ares_tac, since refl must be tried before any equality assumptions;
457          backtracking may occur if the premises have extra variables!*)
458        DEPTH_SOLVE_1 (resolve_tac [refl,exI,conjI] 1 APPEND assume_tac 1),
459        (*Now solve the equations like Inl 0 = Inl ?b2*)
460        rewrite_goals_tac con_defs,
461        REPEAT (rtac refl 1)])) (1 upto (length intr_ts) ~~ intr_ts)
463   in (intrs, unfold) end;
467 (** prove elimination rules **)
469 fun prove_elims cs cTs params intr_ts intr_names unfold rec_sets_defs thy =
470   let
471     val _ = message "  Proving the elimination rules ...";
473     val rules1 = [CollectE, disjE, make_elim vimageD, exE];
474     val rules2 = [conjE, Inl_neq_Inr, Inr_neq_Inl] @
475       map make_elim [Inl_inject, Inr_inject];
476   in
477     map (fn (t, cases) => prove_goalw_cterm rec_sets_defs
478       (cterm_of (Theory.sign_of thy) t) (fn prems =>
479         [cut_facts_tac [hd prems] 1,
480          dtac (unfold RS subst) 1,
481          REPEAT (FIRSTGOAL (eresolve_tac rules1)),
482          REPEAT (FIRSTGOAL (eresolve_tac rules2)),
483          EVERY (map (fn prem =>
484            DEPTH_SOLVE_1 (ares_tac [prem, conjI] 1)) (tl prems))])
485       |> RuleCases.name cases)
486       (mk_elims cs cTs params intr_ts intr_names)
487   end;
490 (** derivation of simplified elimination rules **)
492 (*Applies freeness of the given constructors, which *must* be unfolded by
493   the given defs.  Cannot simply use the local con_defs because con_defs=[]
494   for inference systems.
495  *)
497 (*cprop should have the form t:Si where Si is an inductive set*)
498 fun mk_cases_i solved elims ss cprop =
499   let
500     val prem = Thm.assume cprop;
501     val tac = if solved then InductMethod.con_elim_solved_tac else InductMethod.con_elim_tac;
502     fun mk_elim rl = Drule.standard (Tactic.rule_by_tactic (tac ss) (prem RS rl));
503   in
504     (case get_first (try mk_elim) elims of
505       Some r => r
506     | None => error (Pretty.string_of (Pretty.block
507         [Pretty.str "mk_cases: proposition not of form 't : S_i'", Pretty.fbrk,
508           Display.pretty_cterm cprop])))
509   end;
511 fun mk_cases elims s =
512   mk_cases_i false elims (simpset()) (Thm.read_cterm (Thm.sign_of_thm (hd elims)) (s, propT));
515 (* inductive_cases(_i) *)
517 fun gen_inductive_cases prep_att prep_const prep_prop
518     ((((name, raw_atts), raw_set), raw_props), comment) thy =
519   let
520     val sign = Theory.sign_of thy;
522     val atts = map (prep_att thy) raw_atts;
523     val (_, {elims, ...}) = get_inductive thy (prep_const sign raw_set);
524     val cprops = map (Thm.cterm_of sign o prep_prop (ProofContext.init thy)) raw_props;
525     val thms = map (mk_cases_i true elims (Simplifier.simpset_of thy)) cprops;
526   in
527     thy
528     |> IsarThy.have_theorems_i (((name, atts), map Thm.no_attributes thms), comment)
529   end;
531 val inductive_cases =
532   gen_inductive_cases Attrib.global_attribute Sign.intern_const ProofContext.read_prop;
534 val inductive_cases_i = gen_inductive_cases (K I) (K I) ProofContext.cert_prop;
538 (** prove induction rule **)
540 fun prove_indrule cs cTs sumT rec_const params intr_ts mono
541     fp_def rec_sets_defs thy =
542   let
543     val _ = message "  Proving the induction rule ...";
545     val sign = Theory.sign_of thy;
547     val sum_case_rewrites = (case ThyInfo.lookup_theory "Datatype" of
548         None => []
549       | Some thy' => map mk_meta_eq (PureThy.get_thms thy' "sum.cases"));
551     val (preds, ind_prems, mutual_ind_concl) = mk_indrule cs cTs params intr_ts;
553     (* make predicate for instantiation of abstract induction rule *)
555     fun mk_ind_pred _ [P] = P
556       | mk_ind_pred T Ps =
557          let val n = (length Ps) div 2;
558              val Type (_, [T1, T2]) = T
559          in Const ("Datatype.sum.sum_case",
560            [T1 --> HOLogic.boolT, T2 --> HOLogic.boolT, T] ---> HOLogic.boolT) \$
561              mk_ind_pred T1 (take (n, Ps)) \$ mk_ind_pred T2 (drop (n, Ps))
562          end;
564     val ind_pred = mk_ind_pred sumT preds;
566     val ind_concl = HOLogic.mk_Trueprop
567       (HOLogic.all_const sumT \$ Abs ("x", sumT, HOLogic.mk_binop "op -->"
568         (HOLogic.mk_mem (Bound 0, rec_const), ind_pred \$ Bound 0)));
570     (* simplification rules for vimage and Collect *)
572     val vimage_simps = if length cs < 2 then [] else
573       map (fn c => prove_goalw_cterm [] (cterm_of sign
574         (HOLogic.mk_Trueprop (HOLogic.mk_eq
575           (mk_vimage cs sumT (HOLogic.Collect_const sumT \$ ind_pred) c,
576            HOLogic.Collect_const (HOLogic.dest_setT (fastype_of c)) \$
577              nth_elem (find_index_eq c cs, preds)))))
578         (fn _ => [rtac vimage_Collect 1, rewrite_goals_tac sum_case_rewrites,
579           rtac refl 1])) cs;
581     val induct = prove_goalw_cterm [] (cterm_of sign
582       (Logic.list_implies (ind_prems, ind_concl))) (fn prems =>
583         [rtac (impI RS allI) 1,
584          DETERM (etac (mono RS (fp_def RS def_induct)) 1),
585          rewrite_goals_tac (map mk_meta_eq (vimage_Int::Int_Collect::vimage_simps)),
586          fold_goals_tac rec_sets_defs,
587          (*This CollectE and disjE separates out the introduction rules*)
588          REPEAT (FIRSTGOAL (eresolve_tac [CollectE, disjE, exE])),
589          (*Now break down the individual cases.  No disjE here in case
590            some premise involves disjunction.*)
591          REPEAT (FIRSTGOAL (etac conjE ORELSE' hyp_subst_tac)),
592          rewrite_goals_tac sum_case_rewrites,
593          EVERY (map (fn prem =>
594            DEPTH_SOLVE_1 (ares_tac [prem, conjI, refl] 1)) prems)]);
596     val lemma = prove_goalw_cterm rec_sets_defs (cterm_of sign
597       (Logic.mk_implies (ind_concl, mutual_ind_concl))) (fn prems =>
598         [cut_facts_tac prems 1,
599          REPEAT (EVERY
600            [REPEAT (resolve_tac [conjI, impI] 1),
601             TRY (dtac vimageD 1), etac allE 1, dtac mp 1, atac 1,
602             rewrite_goals_tac sum_case_rewrites,
603             atac 1])])
605   in standard (split_rule (induct RS lemma))
606   end;
610 (*** specification of (co)inductive sets ****)
612 (** definitional introduction of (co)inductive sets **)
614 fun add_ind_def verbose declare_consts alt_name coind no_elim no_ind cs
615     atts intros monos con_defs thy params paramTs cTs cnames =
616   let
617     val _ = if verbose then message ("Proofs for " ^ coind_prefix coind ^ "inductive set(s) " ^
618       commas_quote cnames) else ();
620     val sumT = fold_bal (fn (T, U) => Type ("+", [T, U])) cTs;
621     val setT = HOLogic.mk_setT sumT;
623     val fp_name = if coind then Sign.intern_const (Theory.sign_of Gfp.thy) "gfp"
624       else Sign.intern_const (Theory.sign_of Lfp.thy) "lfp";
626     val ((intr_names, intr_ts), intr_atts) = apfst split_list (split_list intros);
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' = thy |>
671       (if declare_consts then
672         Theory.add_consts_i (map (fn (c, n) =>
673           (n, paramTs ---> fastype_of c, NoSyn)) (cs ~~ cnames))
674        else I) |>
675       (if length cs < 2 then I else
676        Theory.add_consts_i [(rec_name, paramTs ---> setT, NoSyn)]) |>
677       Theory.add_path rec_name |>
678       PureThy.add_defss_i [(("defs", def_terms), [])];
680     (* get definitions from theory *)
682     val fp_def::rec_sets_defs = PureThy.get_thms thy' "defs";
684     (* prove and store theorems *)
686     val mono = prove_mono setT fp_fun monos thy';
687     val (intrs, unfold) = prove_intrs coind mono fp_def intr_ts con_defs
688       rec_sets_defs thy';
689     val elims = if no_elim then [] else
690       prove_elims cs cTs params intr_ts intr_names unfold rec_sets_defs thy';
691     val raw_induct = if no_ind then Drule.asm_rl else
692       if coind then standard (rule_by_tactic
693         (rewrite_tac [mk_meta_eq vimage_Un] THEN
694           fold_tac rec_sets_defs) (mono RS (fp_def RS def_Collect_coinduct)))
695       else
696         prove_indrule cs cTs sumT rec_const params intr_ts mono fp_def
697           rec_sets_defs thy';
698     val induct = if coind orelse no_ind orelse length cs > 1 then raw_induct
699       else standard (raw_induct RSN (2, rev_mp));
701     val thy'' = thy'
702       |> PureThy.add_thmss [(("intrs", intrs), atts)]
703       |> PureThy.add_thms ((intr_names ~~ intrs) ~~ intr_atts)
704       |> (if no_elim then I else PureThy.add_thmss [(("elims", elims), [])])
705       |> (if no_ind then I else PureThy.add_thms
706         [((coind_prefix coind ^ "induct", induct), [])])
707       |> Theory.parent_path;
708     val intrs' = PureThy.get_thms thy'' "intrs";
709     val elims' = if no_elim then elims else PureThy.get_thms thy'' "elims";  (* FIXME improve *)
710     val induct' = if no_ind then induct else PureThy.get_thm thy'' (coind_prefix coind ^ "induct");  (* FIXME improve *)
711   in (thy'',
712     {defs = fp_def::rec_sets_defs,
713      mono = mono,
714      unfold = unfold,
715      intrs = intrs',
716      elims = elims',
717      mk_cases = mk_cases elims',
718      raw_induct = raw_induct,
719      induct = induct'})
720   end;
724 (** axiomatic introduction of (co)inductive sets **)
726 fun add_ind_axm verbose declare_consts alt_name coind no_elim no_ind cs
727     atts intros monos con_defs thy params paramTs cTs cnames =
728   let
729     val rec_name = if alt_name = "" then space_implode "_" cnames else alt_name;
731     val ((intr_names, intr_ts), intr_atts) = apfst split_list (split_list intros);
732     val (elim_ts, elim_cases) = Library.split_list (mk_elims cs cTs params intr_ts intr_names);
734     val (_, ind_prems, mutual_ind_concl) = mk_indrule cs cTs params intr_ts;
735     val ind_t = Logic.list_implies (ind_prems, mutual_ind_concl);
737     val thy' = thy
738       |> (if declare_consts then
740               (map (fn (c, n) => (n, paramTs ---> fastype_of c, NoSyn)) (cs ~~ cnames))
741          else I)
742       |> Theory.add_path rec_name
743       |> PureThy.add_axiomss_i [(("intrs", intr_ts), atts), (("raw_elims", elim_ts), [])]
744       |> (if coind then I else
745             PureThy.add_axioms_i [(("raw_induct", ind_t), [apsnd (standard o split_rule)])]);
747     val intrs = PureThy.get_thms thy' "intrs";
748     val elims = map2 (fn (th, cases) => RuleCases.name cases th)
749       (PureThy.get_thms thy' "raw_elims", elim_cases);
750     val raw_induct = if coind then Drule.asm_rl else PureThy.get_thm thy' "raw_induct";
751     val induct = if coind orelse length cs > 1 then raw_induct
752       else standard (raw_induct RSN (2, rev_mp));
754     val thy'' =
755       thy'
756       |> PureThy.add_thmss [(("elims", elims), [])]
757       |> (if coind then I else PureThy.add_thms [(("induct", induct), [])])
758       |> PureThy.add_thms ((intr_names ~~ intrs) ~~ intr_atts)
759       |> Theory.parent_path;
760     val induct' = if coind then raw_induct else PureThy.get_thm thy'' "induct";
761   in (thy'',
762     {defs = [],
763      mono = Drule.asm_rl,
764      unfold = Drule.asm_rl,
765      intrs = intrs,
766      elims = elims,
767      mk_cases = mk_cases elims,
768      raw_induct = raw_induct,
769      induct = induct'})
770   end;
774 (** introduction of (co)inductive sets **)
776 fun add_inductive_i verbose declare_consts alt_name coind no_elim no_ind cs
777     atts intros monos con_defs thy =
778   let
779     val _ = Theory.requires thy "Inductive" (coind_prefix coind ^ "inductive definitions");
780     val sign = Theory.sign_of thy;
782     (*parameters should agree for all mutually recursive components*)
783     val (_, params) = strip_comb (hd cs);
784     val paramTs = map (try' (snd o dest_Free) "Parameter in recursive\
785       \ component is not a free variable: " sign) params;
787     val cTs = map (try' (HOLogic.dest_setT o fastype_of)
788       "Recursive component not of type set: " sign) cs;
790     val full_cnames = map (try' (fst o dest_Const o head_of)
791       "Recursive set not previously declared as constant: " sign) cs;
792     val cnames = map Sign.base_name full_cnames;
794     val _ = seq (check_rule sign cs o snd o fst) intros;
796     val (thy1, result) =
797       (if ! quick_and_dirty then add_ind_axm else add_ind_def)
798         verbose declare_consts alt_name coind no_elim no_ind cs atts intros monos
799         con_defs thy params paramTs cTs cnames;
800     val thy2 = thy1
801       |> put_inductives full_cnames ({names = full_cnames, coind = coind}, result)
802       |> add_cases_induct no_elim no_ind full_cnames
803         (#elims result) (#induct result) (map (#1 o #1) intros);
804   in (thy2, result) end;
808 (** external interface **)
810 fun add_inductive verbose coind c_strings srcs intro_srcs raw_monos raw_con_defs thy =
811   let
812     val sign = Theory.sign_of thy;
813     val cs = map (term_of o Thm.read_cterm sign o rpair HOLogic.termT) c_strings;
815     val atts = map (Attrib.global_attribute thy) srcs;
816     val intr_names = map (fst o fst) intro_srcs;
817     val intr_ts = map (term_of o Thm.read_cterm sign o rpair propT o snd o fst) intro_srcs;
818     val intr_atts = map (map (Attrib.global_attribute thy) o snd) intro_srcs;
819     val (cs', intr_ts') = unify_consts sign cs intr_ts;
821     val ((thy', con_defs), monos) = thy
822       |> IsarThy.apply_theorems raw_monos
823       |> apfst (IsarThy.apply_theorems raw_con_defs);
824   in
825     add_inductive_i verbose false "" coind false false cs'
826       atts ((intr_names ~~ intr_ts') ~~ intr_atts) monos con_defs thy'
827   end;
831 (** package setup **)
833 (* setup theory *)
835 val setup = [InductiveData.init,
836              Attrib.add_attributes [(monoN, mono_attr, "monotonicity rule")]];
839 (* outer syntax *)
841 local structure P = OuterParse and K = OuterSyntax.Keyword in
843 fun mk_ind coind (((sets, (atts, intrs)), monos), con_defs) =
844   #1 o add_inductive true coind sets atts (map P.triple_swap intrs) monos con_defs;
846 fun ind_decl coind =
847   (Scan.repeat1 P.term --| P.marg_comment) --
848   (P.\$\$\$ "intrs" |--
849     P.!!! (P.opt_attribs -- Scan.repeat1 (P.opt_thm_name ":" -- P.prop --| P.marg_comment))) --
850   Scan.optional (P.\$\$\$ "monos" |-- P.!!! P.xthms1 --| P.marg_comment) [] --
851   Scan.optional (P.\$\$\$ "con_defs" |-- P.!!! P.xthms1 --| P.marg_comment) []
852   >> (Toplevel.theory o mk_ind coind);
854 val inductiveP =
855   OuterSyntax.command "inductive" "define inductive sets" K.thy_decl (ind_decl false);
857 val coinductiveP =
858   OuterSyntax.command "coinductive" "define coinductive sets" K.thy_decl (ind_decl true);
861 val ind_cases =
862   P.opt_thm_name "=" -- P.xname --| P.\$\$\$ ":" -- Scan.repeat1 P.prop -- P.marg_comment
863   >> (Toplevel.theory o inductive_cases);
865 val inductive_casesP =
866   OuterSyntax.command "inductive_cases" "create simplified instances of elimination rules"
867     K.thy_decl ind_cases;
869 val _ = OuterSyntax.add_keywords ["intrs", "monos", "con_defs"];
870 val _ = OuterSyntax.add_parsers [inductiveP, coinductiveP, inductive_casesP];
872 end;
875 end;