src/HOL/Tools/inductive_package.ML
 author wenzelm Wed Aug 09 20:46:58 2000 +0200 (2000-08-09) changeset 9562 6b07b56aa3a8 parent 9405 3235873fdd90 child 9598 65ee72db0236 permissions -rw-r--r--
fixed mk_cases_i: TRYALL InductMethod.simp_case_tac;
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) * 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.strs ("(co)inductives:" :: map #1 (Sign.cond_extern_table sg Sign.constK tab)),
83      Pretty.big_list "monotonicity rules:" (map Display.pretty_thm monos)]
84     |> Pretty.chunks |> Pretty.writeln;
85 end;
87 structure InductiveData = TheoryDataFun(InductiveArgs);
88 val print_inductives = InductiveData.print;
91 (* get and put data *)
93 fun get_inductive thy name = Symtab.lookup (fst (InductiveData.get thy), name);
95 fun put_inductives names info thy =
96   let
97     fun upd ((tab, monos), name) = (Symtab.update_new ((name, info), tab), monos);
98     val tab_monos = foldl upd (InductiveData.get thy, names)
99       handle Symtab.DUP name => error ("Duplicate definition of (co)inductive set " ^ quote name);
100   in InductiveData.put tab_monos thy end;
104 (** monotonicity rules **)
106 val get_monos = snd o InductiveData.get;
107 fun put_monos thms thy = InductiveData.put (fst (InductiveData.get thy), thms) thy;
109 fun mk_mono thm =
110   let
111     fun eq2mono thm' = [standard (thm' RS (thm' RS eq_to_mono))] @
112       (case concl_of thm of
113           (_ \$ (_ \$ (Const ("Not", _) \$ _) \$ _)) => []
114         | _ => [standard (thm' RS (thm' RS eq_to_mono2))]);
115     val concl = concl_of thm
116   in
117     if Logic.is_equals concl then
118       eq2mono (thm RS meta_eq_to_obj_eq)
119     else if can (HOLogic.dest_eq o HOLogic.dest_Trueprop) concl then
120       eq2mono thm
121     else [thm]
122   end;
125 (* attributes *)
127 local
129 fun map_rules_global f thy = put_monos (f (get_monos thy)) thy;
131 fun add_mono thm rules = Library.gen_union Thm.eq_thm (mk_mono thm, rules);
132 fun del_mono thm rules = Library.gen_rems Thm.eq_thm (rules, mk_mono thm);
134 fun mk_att f g (x, thm) = (f (g thm) x, thm);
136 in
137   val mono_add_global = mk_att map_rules_global add_mono;
138   val mono_del_global = mk_att map_rules_global del_mono;
139 end;
141 val mono_attr =
143   Attrib.add_del_args Attrib.undef_local_attribute Attrib.undef_local_attribute);
147 (** utilities **)
149 (* messages *)
151 val quiet_mode = ref false;
152 fun message s = if !quiet_mode then () else writeln s;
154 fun coind_prefix true = "co"
155   | coind_prefix false = "";
158 (* the following code ensures that each recursive set *)
159 (* always has the same type in all introduction rules *)
161 fun unify_consts sign cs intr_ts =
162   (let
163     val {tsig, ...} = Sign.rep_sg sign;
164     val add_term_consts_2 =
165       foldl_aterms (fn (cs, Const c) => c ins cs | (cs, _) => cs);
166     fun varify (t, (i, ts)) =
167       let val t' = map_term_types (incr_tvar (i + 1)) (Type.varify (t, []))
168       in (maxidx_of_term t', t'::ts) end;
169     val (i, cs') = foldr varify (cs, (~1, []));
170     val (i', intr_ts') = foldr varify (intr_ts, (i, []));
171     val rec_consts = foldl add_term_consts_2 ([], cs');
172     val intr_consts = foldl add_term_consts_2 ([], intr_ts');
173     fun unify (env, (cname, cT)) =
174       let val consts = map snd (filter (fn c => fst c = cname) intr_consts)
175       in foldl (fn ((env', j'), Tp) => (Type.unify tsig j' env' Tp))
176           (env, (replicate (length consts) cT) ~~ consts)
177       end;
178     val (env, _) = foldl unify ((Vartab.empty, i'), rec_consts);
179     fun typ_subst_TVars_2 env T = let val T' = typ_subst_TVars_Vartab env T
180       in if T = T' then T else typ_subst_TVars_2 env T' end;
181     val subst = fst o Type.freeze_thaw o
182       (map_term_types (typ_subst_TVars_2 env))
184   in (map subst cs', map subst intr_ts')
185   end) handle Type.TUNIFY =>
186     (warning "Occurrences of recursive constant have non-unifiable types"; (cs, intr_ts));
189 (* misc *)
191 val Const _ \$ (vimage_f \$ _) \$ _ = HOLogic.dest_Trueprop (concl_of vimageD);
193 val vimage_name = Sign.intern_const (Theory.sign_of Vimage.thy) "op -``";
194 val mono_name = Sign.intern_const (Theory.sign_of Ord.thy) "mono";
196 (* make injections needed in mutually recursive definitions *)
198 fun mk_inj cs sumT c x =
199   let
200     fun mk_inj' T n i =
201       if n = 1 then x else
202       let val n2 = n div 2;
203           val Type (_, [T1, T2]) = T
204       in
205         if i <= n2 then
206           Const ("Inl", T1 --> T) \$ (mk_inj' T1 n2 i)
207         else
208           Const ("Inr", T2 --> T) \$ (mk_inj' T2 (n - n2) (i - n2))
209       end
210   in mk_inj' sumT (length cs) (1 + find_index_eq c cs)
211   end;
213 (* make "vimage" terms for selecting out components of mutually rec.def. *)
215 fun mk_vimage cs sumT t c = if length cs < 2 then t else
216   let
217     val cT = HOLogic.dest_setT (fastype_of c);
218     val vimageT = [cT --> sumT, HOLogic.mk_setT sumT] ---> HOLogic.mk_setT cT
219   in
220     Const (vimage_name, vimageT) \$
221       Abs ("y", cT, mk_inj cs sumT c (Bound 0)) \$ t
222   end;
226 (** well-formedness checks **)
228 fun err_in_rule sign t msg = error ("Ill-formed introduction rule\n" ^
229   (Sign.string_of_term sign t) ^ "\n" ^ msg);
231 fun err_in_prem sign t p msg = error ("Ill-formed premise\n" ^
232   (Sign.string_of_term sign p) ^ "\nin introduction rule\n" ^
233   (Sign.string_of_term sign t) ^ "\n" ^ msg);
235 val msg1 = "Conclusion of introduction rule must have form\
236           \ ' t : S_i '";
237 val msg2 = "Non-atomic premise";
238 val msg3 = "Recursion term on left of member symbol";
240 fun check_rule sign cs r =
241   let
242     fun check_prem prem = if can HOLogic.dest_Trueprop prem then ()
243       else err_in_prem sign r prem msg2;
245   in (case HOLogic.dest_Trueprop (Logic.strip_imp_concl r) of
246         (Const ("op :", _) \$ t \$ u) =>
247           if u mem cs then
248             if exists (Logic.occs o (rpair t)) cs then
249               err_in_rule sign r msg3
250             else
251               seq check_prem (Logic.strip_imp_prems r)
252           else err_in_rule sign r msg1
253       | _ => err_in_rule sign r msg1)
254   end;
256 fun try' f msg sign t = (case (try f t) of
257       Some x => x
258     | None => error (msg ^ Sign.string_of_term sign t));
262 (*** properties of (co)inductive sets ***)
264 (** elimination rules **)
266 fun mk_elims cs cTs params intr_ts intr_names =
267   let
268     val used = foldr add_term_names (intr_ts, []);
269     val [aname, pname] = variantlist (["a", "P"], used);
270     val P = HOLogic.mk_Trueprop (Free (pname, HOLogic.boolT));
272     fun dest_intr r =
273       let val Const ("op :", _) \$ t \$ u =
274         HOLogic.dest_Trueprop (Logic.strip_imp_concl r)
275       in (u, t, Logic.strip_imp_prems r) end;
277     val intrs = map dest_intr intr_ts ~~ intr_names;
279     fun mk_elim (c, T) =
280       let
281         val a = Free (aname, T);
283         fun mk_elim_prem (_, t, ts) =
284           list_all_free (map dest_Free ((foldr add_term_frees (t::ts, [])) \\ params),
285             Logic.list_implies (HOLogic.mk_Trueprop (HOLogic.mk_eq (a, t)) :: ts, P));
286         val c_intrs = (filter (equal c o #1 o #1) intrs);
287       in
288         (Logic.list_implies (HOLogic.mk_Trueprop (HOLogic.mk_mem (a, c)) ::
289           map mk_elim_prem (map #1 c_intrs), P), map #2 c_intrs)
290       end
291   in
292     map mk_elim (cs ~~ cTs)
293   end;
297 (** premises and conclusions of induction rules **)
299 fun mk_indrule cs cTs params intr_ts =
300   let
301     val used = foldr add_term_names (intr_ts, []);
303     (* predicates for induction rule *)
305     val preds = map Free (variantlist (if length cs < 2 then ["P"] else
306       map (fn i => "P" ^ string_of_int i) (1 upto length cs), used) ~~
307         map (fn T => T --> HOLogic.boolT) cTs);
309     (* transform an introduction rule into a premise for induction rule *)
311     fun mk_ind_prem r =
312       let
313         val frees = map dest_Free ((add_term_frees (r, [])) \\ params);
315         val pred_of = curry (Library.gen_assoc (op aconv)) (cs ~~ preds);
317         fun subst (s as ((m as Const ("op :", T)) \$ t \$ u)) =
318               (case pred_of u of
319                   None => (m \$ fst (subst t) \$ fst (subst u), None)
320                 | Some P => (HOLogic.conj \$ s \$ (P \$ t), Some (s, P \$ t)))
321           | subst s =
322               (case pred_of s of
323                   Some P => (HOLogic.mk_binop "op Int"
324                     (s, HOLogic.Collect_const (HOLogic.dest_setT
325                       (fastype_of s)) \$ P), None)
326                 | None => (case s of
327                      (t \$ u) => (fst (subst t) \$ fst (subst u), None)
328                    | (Abs (a, T, t)) => (Abs (a, T, fst (subst t)), None)
329                    | _ => (s, None)));
331         fun mk_prem (s, prems) = (case subst s of
332               (_, Some (t, u)) => t :: u :: prems
333             | (t, _) => t :: prems);
335         val Const ("op :", _) \$ t \$ u =
336           HOLogic.dest_Trueprop (Logic.strip_imp_concl r)
338       in list_all_free (frees,
339            Logic.list_implies (map HOLogic.mk_Trueprop (foldr mk_prem
340              (map HOLogic.dest_Trueprop (Logic.strip_imp_prems r), [])),
341                HOLogic.mk_Trueprop (the (pred_of u) \$ t)))
342       end;
344     val ind_prems = map mk_ind_prem intr_ts;
346     (* make conclusions for induction rules *)
348     fun mk_ind_concl ((c, P), (ts, x)) =
349       let val T = HOLogic.dest_setT (fastype_of c);
350           val Ts = HOLogic.prodT_factors T;
351           val (frees, x') = foldr (fn (T', (fs, s)) =>
352             ((Free (s, T'))::fs, bump_string s)) (Ts, ([], x));
353           val tuple = HOLogic.mk_tuple T frees;
354       in ((HOLogic.mk_binop "op -->"
355         (HOLogic.mk_mem (tuple, c), P \$ tuple))::ts, x')
356       end;
358     val mutual_ind_concl = HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
359         (fst (foldr mk_ind_concl (cs ~~ preds, ([], "xa")))))
361   in (preds, ind_prems, mutual_ind_concl)
362   end;
366 (** prepare cases and induct rules **)
368 (*
369   transform mutual rule:
370     HH ==> (x1:A1 --> P1 x1) & ... & (xn:An --> Pn xn)
371   into i-th projection:
372     xi:Ai ==> HH ==> Pi xi
373 *)
375 fun project_rules [name] rule = [(name, rule)]
376   | project_rules names mutual_rule =
377       let
378         val n = length names;
379         fun proj i =
380           (if i < n then (fn th => th RS conjunct1) else I)
381             (Library.funpow (i - 1) (fn th => th RS conjunct2) mutual_rule)
382             RS mp |> Thm.permute_prems 0 ~1 |> Drule.standard;
383       in names ~~ map proj (1 upto n) end;
385 fun add_cases_induct no_elim no_ind names elims induct induct_cases =
386   let
387     fun cases_spec (name, elim) thy =
388       thy
389       |> Theory.add_path (Sign.base_name name)
390       |> (#1 o PureThy.add_thms [(("cases", elim), [InductMethod.cases_set_global name])])
391       |> Theory.parent_path;
392     val cases_specs = if no_elim then [] else map2 cases_spec (names, elims);
394     fun induct_spec (name, th) = (#1 o PureThy.add_thms
395       [(("", th), [RuleCases.case_names induct_cases, InductMethod.induct_set_global name])]);
396     val induct_specs = if no_ind then [] else map induct_spec (project_rules names induct);
397   in Library.apply (cases_specs @ induct_specs) end;
401 (*** proofs for (co)inductive sets ***)
403 (** prove monotonicity **)
405 fun prove_mono setT fp_fun monos thy =
406   let
407     val _ = message "  Proving monotonicity ...";
409     val mono = prove_goalw_cterm [] (cterm_of (Theory.sign_of thy) (HOLogic.mk_Trueprop
410       (Const (mono_name, (setT --> setT) --> HOLogic.boolT) \$ fp_fun)))
411         (fn _ => [rtac monoI 1, REPEAT (ares_tac (get_monos thy @ flat (map mk_mono monos)) 1)])
413   in mono end;
417 (** prove introduction rules **)
419 fun prove_intrs coind mono fp_def intr_ts con_defs rec_sets_defs thy =
420   let
421     val _ = message "  Proving the introduction rules ...";
423     val unfold = standard (mono RS (fp_def RS
424       (if coind then def_gfp_Tarski else def_lfp_Tarski)));
426     fun select_disj 1 1 = []
427       | select_disj _ 1 = [rtac disjI1]
428       | select_disj n i = (rtac disjI2)::(select_disj (n - 1) (i - 1));
430     val intrs = map (fn (i, intr) => prove_goalw_cterm rec_sets_defs
431       (cterm_of (Theory.sign_of thy) intr) (fn prems =>
432        [(*insert prems and underlying sets*)
433        cut_facts_tac prems 1,
434        stac unfold 1,
435        REPEAT (resolve_tac [vimageI2, CollectI] 1),
436        (*Now 1-2 subgoals: the disjunction, perhaps equality.*)
437        EVERY1 (select_disj (length intr_ts) i),
438        (*Not ares_tac, since refl must be tried before any equality assumptions;
439          backtracking may occur if the premises have extra variables!*)
440        DEPTH_SOLVE_1 (resolve_tac [refl,exI,conjI] 1 APPEND assume_tac 1),
441        (*Now solve the equations like Inl 0 = Inl ?b2*)
442        rewrite_goals_tac con_defs,
443        REPEAT (rtac refl 1)])) (1 upto (length intr_ts) ~~ intr_ts)
445   in (intrs, unfold) end;
449 (** prove elimination rules **)
451 fun prove_elims cs cTs params intr_ts intr_names unfold rec_sets_defs thy =
452   let
453     val _ = message "  Proving the elimination rules ...";
455     val rules1 = [CollectE, disjE, make_elim vimageD, exE];
456     val rules2 = [conjE, Inl_neq_Inr, Inr_neq_Inl] @
457       map make_elim [Inl_inject, Inr_inject];
458   in
459     map (fn (t, cases) => prove_goalw_cterm rec_sets_defs
460       (cterm_of (Theory.sign_of thy) t) (fn prems =>
461         [cut_facts_tac [hd prems] 1,
462          dtac (unfold RS subst) 1,
463          REPEAT (FIRSTGOAL (eresolve_tac rules1)),
464          REPEAT (FIRSTGOAL (eresolve_tac rules2)),
465          EVERY (map (fn prem =>
466            DEPTH_SOLVE_1 (ares_tac [prem, conjI] 1)) (tl prems))])
467       |> RuleCases.name cases)
468       (mk_elims cs cTs params intr_ts intr_names)
469   end;
472 (** derivation of simplified elimination rules **)
474 (*Applies freeness of the given constructors, which *must* be unfolded by
475   the given defs.  Cannot simply use the local con_defs because con_defs=[]
476   for inference systems.
477  *)
479 (*cprop should have the form t:Si where Si is an inductive set*)
480 fun mk_cases_i solved elims ss cprop =
481   let
482     val prem = Thm.assume cprop;
483     val tac = TRYALL (InductMethod.simp_case_tac solved ss) THEN prune_params_tac;
484     fun mk_elim rl = Drule.standard (Tactic.rule_by_tactic tac (prem RS rl));
485   in
486     (case get_first (try mk_elim) elims of
487       Some r => r
488     | None => error (Pretty.string_of (Pretty.block
489         [Pretty.str "mk_cases: proposition not of form 't : S_i'", Pretty.fbrk,
490           Display.pretty_cterm cprop])))
491   end;
493 fun mk_cases elims s =
494   mk_cases_i false elims (simpset()) (Thm.read_cterm (Thm.sign_of_thm (hd elims)) (s, propT));
497 (* inductive_cases(_i) *)
499 fun gen_inductive_cases prep_att prep_const prep_prop
500     ((((name, raw_atts), raw_set), raw_props), comment) thy =
501   let val sign = Theory.sign_of thy;
502   in (case get_inductive thy (prep_const sign raw_set) of
503       None => error ("Unknown (co)inductive set " ^ quote name)
504     | Some (_, {elims, ...}) =>
505         let
506           val atts = map (prep_att thy) raw_atts;
507           val cprops = map
508             (Thm.cterm_of sign o prep_prop (ProofContext.init thy)) raw_props;
509           val thms = map
510             (mk_cases_i true elims (Simplifier.simpset_of thy)) cprops;
511         in
512           thy |> IsarThy.have_theorems_i
513             [(((name, atts), map Thm.no_attributes thms), comment)]
514         end)
515   end;
517 val inductive_cases =
518   gen_inductive_cases Attrib.global_attribute Sign.intern_const ProofContext.read_prop;
520 val inductive_cases_i = gen_inductive_cases (K I) (K I) ProofContext.cert_prop;
524 (** prove induction rule **)
526 fun prove_indrule cs cTs sumT rec_const params intr_ts mono
527     fp_def rec_sets_defs thy =
528   let
529     val _ = message "  Proving the induction rule ...";
531     val sign = Theory.sign_of thy;
533     val sum_case_rewrites = (case ThyInfo.lookup_theory "Datatype" of
534         None => []
535       | Some thy' => map mk_meta_eq (PureThy.get_thms thy' "sum.cases"));
537     val (preds, ind_prems, mutual_ind_concl) = mk_indrule cs cTs params intr_ts;
539     (* make predicate for instantiation of abstract induction rule *)
541     fun mk_ind_pred _ [P] = P
542       | mk_ind_pred T Ps =
543          let val n = (length Ps) div 2;
544              val Type (_, [T1, T2]) = T
545          in Const ("Datatype.sum.sum_case",
546            [T1 --> HOLogic.boolT, T2 --> HOLogic.boolT, T] ---> HOLogic.boolT) \$
547              mk_ind_pred T1 (take (n, Ps)) \$ mk_ind_pred T2 (drop (n, Ps))
548          end;
550     val ind_pred = mk_ind_pred sumT preds;
552     val ind_concl = HOLogic.mk_Trueprop
553       (HOLogic.all_const sumT \$ Abs ("x", sumT, HOLogic.mk_binop "op -->"
554         (HOLogic.mk_mem (Bound 0, rec_const), ind_pred \$ Bound 0)));
556     (* simplification rules for vimage and Collect *)
558     val vimage_simps = if length cs < 2 then [] else
559       map (fn c => prove_goalw_cterm [] (cterm_of sign
560         (HOLogic.mk_Trueprop (HOLogic.mk_eq
561           (mk_vimage cs sumT (HOLogic.Collect_const sumT \$ ind_pred) c,
562            HOLogic.Collect_const (HOLogic.dest_setT (fastype_of c)) \$
563              nth_elem (find_index_eq c cs, preds)))))
564         (fn _ => [rtac vimage_Collect 1, rewrite_goals_tac sum_case_rewrites,
565           rtac refl 1])) cs;
567     val induct = prove_goalw_cterm [] (cterm_of sign
568       (Logic.list_implies (ind_prems, ind_concl))) (fn prems =>
569         [rtac (impI RS allI) 1,
570          DETERM (etac (mono RS (fp_def RS def_induct)) 1),
571          rewrite_goals_tac (map mk_meta_eq (vimage_Int::Int_Collect::vimage_simps)),
572          fold_goals_tac rec_sets_defs,
573          (*This CollectE and disjE separates out the introduction rules*)
574          REPEAT (FIRSTGOAL (eresolve_tac [CollectE, disjE, exE])),
575          (*Now break down the individual cases.  No disjE here in case
576            some premise involves disjunction.*)
577          REPEAT (FIRSTGOAL (etac conjE ORELSE' hyp_subst_tac)),
578          rewrite_goals_tac sum_case_rewrites,
579          EVERY (map (fn prem =>
580            DEPTH_SOLVE_1 (ares_tac [prem, conjI, refl] 1)) prems)]);
582     val lemma = prove_goalw_cterm rec_sets_defs (cterm_of sign
583       (Logic.mk_implies (ind_concl, mutual_ind_concl))) (fn prems =>
584         [cut_facts_tac prems 1,
585          REPEAT (EVERY
586            [REPEAT (resolve_tac [conjI, impI] 1),
587             TRY (dtac vimageD 1), etac allE 1, dtac mp 1, atac 1,
588             rewrite_goals_tac sum_case_rewrites,
589             atac 1])])
591   in standard (split_rule (induct RS lemma))
592   end;
596 (*** specification of (co)inductive sets ****)
598 (** definitional introduction of (co)inductive sets **)
600 fun mk_ind_def declare_consts alt_name coind cs intr_ts monos con_defs thy
601       params paramTs cTs cnames =
602   let
603     val sumT = fold_bal (fn (T, U) => Type ("+", [T, U])) cTs;
604     val setT = HOLogic.mk_setT sumT;
606     val fp_name = if coind then Sign.intern_const (Theory.sign_of Gfp.thy) "gfp"
607       else Sign.intern_const (Theory.sign_of Lfp.thy) "lfp";
609     val used = foldr add_term_names (intr_ts, []);
610     val [sname, xname] = variantlist (["S", "x"], used);
612     (* transform an introduction rule into a conjunction  *)
613     (*   [| t : ... S_i ... ; ... |] ==> u : S_j          *)
614     (* is transformed into                                *)
615     (*   x = Inj_j u & t : ... Inj_i -`` S ... & ...      *)
617     fun transform_rule r =
618       let
619         val frees = map dest_Free ((add_term_frees (r, [])) \\ params);
620         val subst = subst_free
621           (cs ~~ (map (mk_vimage cs sumT (Free (sname, setT))) cs));
622         val Const ("op :", _) \$ t \$ u =
623           HOLogic.dest_Trueprop (Logic.strip_imp_concl r)
625       in foldr (fn ((x, T), P) => HOLogic.mk_exists (x, T, P))
626         (frees, foldr1 HOLogic.mk_conj
627           (((HOLogic.eq_const sumT) \$ Free (xname, sumT) \$ (mk_inj cs sumT u t))::
628             (map (subst o HOLogic.dest_Trueprop)
629               (Logic.strip_imp_prems r))))
630       end
632     (* make a disjunction of all introduction rules *)
634     val fp_fun = absfree (sname, setT, (HOLogic.Collect_const sumT) \$
635       absfree (xname, sumT, foldr1 HOLogic.mk_disj (map transform_rule intr_ts)));
637     (* add definiton of recursive sets to theory *)
639     val rec_name = if alt_name = "" then space_implode "_" cnames else alt_name;
640     val full_rec_name = Sign.full_name (Theory.sign_of thy) rec_name;
642     val rec_const = list_comb
643       (Const (full_rec_name, paramTs ---> setT), params);
645     val fp_def_term = Logic.mk_equals (rec_const,
646       Const (fp_name, (setT --> setT) --> setT) \$ fp_fun)
648     val def_terms = fp_def_term :: (if length cs < 2 then [] else
649       map (fn c => Logic.mk_equals (c, mk_vimage cs sumT rec_const c)) cs);
651     val (thy', [fp_def :: rec_sets_defs]) =
652       thy
653       |> (if declare_consts then
654           Theory.add_consts_i (map (fn (c, n) =>
655             (n, paramTs ---> fastype_of c, NoSyn)) (cs ~~ cnames))
656           else I)
657       |> (if length cs < 2 then I
658           else Theory.add_consts_i [(rec_name, paramTs ---> setT, NoSyn)])
659       |> Theory.add_path rec_name
660       |> PureThy.add_defss_i false [(("defs", def_terms), [])];
662     val mono = prove_mono setT fp_fun monos thy'
664   in
665     (thy', mono, fp_def, rec_sets_defs, rec_const, sumT)
666   end;
668 fun add_ind_def verbose declare_consts alt_name coind no_elim no_ind cs
669     atts intros monos con_defs thy params paramTs cTs cnames induct_cases =
670   let
671     val _ = if verbose then message ("Proofs for " ^ coind_prefix coind ^ "inductive set(s) " ^
672       commas_quote cnames) else ();
674     val ((intr_names, intr_ts), intr_atts) = apfst split_list (split_list intros);
676     val (thy', mono, fp_def, rec_sets_defs, rec_const, sumT) =
677       mk_ind_def declare_consts alt_name coind cs intr_ts monos con_defs thy
678         params paramTs cTs cnames;
680     val (intrs, unfold) = prove_intrs coind mono fp_def intr_ts con_defs
681       rec_sets_defs thy';
682     val elims = if no_elim then [] else
683       prove_elims cs cTs params intr_ts intr_names unfold rec_sets_defs thy';
684     val raw_induct = if no_ind then Drule.asm_rl else
685       if coind then standard (rule_by_tactic
686         (rewrite_tac [mk_meta_eq vimage_Un] THEN
687           fold_tac rec_sets_defs) (mono RS (fp_def RS def_Collect_coinduct)))
688       else
689         prove_indrule cs cTs sumT rec_const params intr_ts mono fp_def
690           rec_sets_defs thy';
691     val induct = if coind orelse no_ind orelse length cs > 1 then raw_induct
692       else standard (raw_induct RSN (2, rev_mp));
694     val (thy'', [intrs']) =
695       thy'
696       |> PureThy.add_thmss [(("intrs", intrs), atts)]
697       |>> (#1 o PureThy.add_thms ((intr_names ~~ intrs) ~~ intr_atts))
698       |>> (if no_elim then I else #1 o PureThy.add_thmss [(("elims", elims), [])])
699       |>> (if no_ind then I else #1 o PureThy.add_thms
700         [((coind_prefix coind ^ "induct", induct), [RuleCases.case_names induct_cases])])
701       |>> Theory.parent_path;
702     val elims' = if no_elim then elims else PureThy.get_thms thy'' "elims";  (* FIXME improve *)
703     val induct' = if no_ind then induct else PureThy.get_thm thy'' (coind_prefix coind ^ "induct");  (* FIXME improve *)
704   in (thy'',
705     {defs = fp_def::rec_sets_defs,
706      mono = mono,
707      unfold = unfold,
708      intrs = intrs',
709      elims = elims',
710      mk_cases = mk_cases elims',
711      raw_induct = raw_induct,
712      induct = induct'})
713   end;
717 (** axiomatic introduction of (co)inductive sets **)
719 fun add_ind_axm verbose declare_consts alt_name coind no_elim no_ind cs
720     atts intros monos con_defs thy params paramTs cTs cnames induct_cases =
721   let
722     val _ = message (coind_prefix coind ^ "inductive set(s) " ^ commas_quote cnames);
724     val ((intr_names, intr_ts), intr_atts) = apfst split_list (split_list intros);
725     val (thy', _, fp_def, rec_sets_defs, _, _) =
726       mk_ind_def declare_consts alt_name coind cs intr_ts monos con_defs thy
727         params paramTs cTs cnames;
728     val (elim_ts, elim_cases) = Library.split_list (mk_elims cs cTs params intr_ts intr_names);
729     val (_, ind_prems, mutual_ind_concl) = mk_indrule cs cTs params intr_ts;
730     val ind_t = Logic.list_implies (ind_prems, mutual_ind_concl);
732     val thy'' =
733       thy'
734       |> (#1 o PureThy.add_axiomss_i [(("intrs", intr_ts), atts), (("raw_elims", elim_ts), [])])
735       |> (if coind then I else
736             #1 o 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''', ([elims'], intrs')) =
746       thy''
747       |> PureThy.add_thmss [(("elims", elims), [])]
748       |>> (if coind then I
749           else #1 o PureThy.add_thms [(("induct", induct), [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 = fp_def :: rec_sets_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 as {elims, induct, ...}) =
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 elims induct induct_cases;
796   in (thy2, result) end;
800 (** external interface **)
802 fun add_inductive verbose coind c_strings srcs intro_srcs raw_monos raw_con_defs thy =
803   let
804     val sign = Theory.sign_of thy;
805     val cs = map (term_of o Thm.read_cterm sign o rpair HOLogic.termT) c_strings;
807     val atts = map (Attrib.global_attribute thy) srcs;
808     val intr_names = map (fst o fst) intro_srcs;
809     fun read_rule s = Thm.read_cterm sign (s, propT)
810       handle ERROR => error ("The error(s) above occurred for " ^ s);
811     val intr_ts = map (Thm.term_of o read_rule o snd o fst) intro_srcs;
812     val intr_atts = map (map (Attrib.global_attribute thy) o snd) intro_srcs;
813     val (cs', intr_ts') = unify_consts sign cs intr_ts;
815     val ((thy', con_defs), monos) = thy
816       |> IsarThy.apply_theorems raw_monos
817       |> apfst (IsarThy.apply_theorems raw_con_defs);
818   in
819     add_inductive_i verbose false "" coind false false cs'
820       atts ((intr_names ~~ intr_ts') ~~ intr_atts) monos con_defs thy'
821   end;
825 (** package setup **)
827 (* setup theory *)
829 val setup =
830  [InductiveData.init,
831   Attrib.add_attributes [("mono", mono_attr, "monotonicity rule")]];
834 (* outer syntax *)
836 local structure P = OuterParse and K = OuterSyntax.Keyword in
838 fun mk_ind coind (((sets, (atts, intrs)), monos), con_defs) =
839   #1 o add_inductive true coind sets atts (map P.triple_swap intrs) monos con_defs;
841 fun ind_decl coind =
842   (Scan.repeat1 P.term --| P.marg_comment) --
843   (P.\$\$\$ "intrs" |--
844     P.!!! (P.opt_attribs -- Scan.repeat1 (P.opt_thm_name ":" -- P.prop --| P.marg_comment))) --
845   Scan.optional (P.\$\$\$ "monos" |-- P.!!! P.xthms1 --| P.marg_comment) [] --
846   Scan.optional (P.\$\$\$ "con_defs" |-- P.!!! P.xthms1 --| P.marg_comment) []
847   >> (Toplevel.theory o mk_ind coind);
849 val inductiveP =
850   OuterSyntax.command "inductive" "define inductive sets" K.thy_decl (ind_decl false);
852 val coinductiveP =
853   OuterSyntax.command "coinductive" "define coinductive sets" K.thy_decl (ind_decl true);
856 val ind_cases =
857   P.opt_thm_name "=" -- P.xname --| P.\$\$\$ ":" -- Scan.repeat1 P.prop -- P.marg_comment
858   >> (Toplevel.theory o inductive_cases);
860 val inductive_casesP =
861   OuterSyntax.command "inductive_cases" "create simplified instances of elimination rules"
862     K.thy_decl ind_cases;
864 val _ = OuterSyntax.add_keywords ["intrs", "monos", "con_defs"];
865 val _ = OuterSyntax.add_parsers [inductiveP, coinductiveP, inductive_casesP];
867 end;
870 end;