src/HOL/Tools/inductive_package.ML
 author wenzelm Sun Feb 27 15:32:10 2000 +0100 (2000-02-27) changeset 8307 6600c6e53111 parent 8293 4a0e17cf8f70 child 8312 b470bc28b59d permissions -rw-r--r--
removed cases_of, all_cases, all_inducts;
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 (*Delete needless equality assumptions*)
207 val refl_thin = prove_goal HOL.thy "!!P. [| a=a;  P |] ==> P"
208      (fn _ => [assume_tac 1]);
210 (*For simplifying the elimination rule*)
211 val elim_rls = [asm_rl, FalseE, refl_thin, conjE, exE, Pair_inject];
213 val vimage_name = Sign.intern_const (Theory.sign_of Vimage.thy) "op -``";
214 val mono_name = Sign.intern_const (Theory.sign_of Ord.thy) "mono";
216 (* make injections needed in mutually recursive definitions *)
218 fun mk_inj cs sumT c x =
219   let
220     fun mk_inj' T n i =
221       if n = 1 then x else
222       let val n2 = n div 2;
223           val Type (_, [T1, T2]) = T
224       in
225         if i <= n2 then
226           Const ("Inl", T1 --> T) \$ (mk_inj' T1 n2 i)
227         else
228           Const ("Inr", T2 --> T) \$ (mk_inj' T2 (n - n2) (i - n2))
229       end
230   in mk_inj' sumT (length cs) (1 + find_index_eq c cs)
231   end;
233 (* make "vimage" terms for selecting out components of mutually rec.def. *)
235 fun mk_vimage cs sumT t c = if length cs < 2 then t else
236   let
237     val cT = HOLogic.dest_setT (fastype_of c);
238     val vimageT = [cT --> sumT, HOLogic.mk_setT sumT] ---> HOLogic.mk_setT cT
239   in
240     Const (vimage_name, vimageT) \$
241       Abs ("y", cT, mk_inj cs sumT c (Bound 0)) \$ t
242   end;
246 (** well-formedness checks **)
248 fun err_in_rule sign t msg = error ("Ill-formed introduction rule\n" ^
249   (Sign.string_of_term sign t) ^ "\n" ^ msg);
251 fun err_in_prem sign t p msg = error ("Ill-formed premise\n" ^
252   (Sign.string_of_term sign p) ^ "\nin introduction rule\n" ^
253   (Sign.string_of_term sign t) ^ "\n" ^ msg);
255 val msg1 = "Conclusion of introduction rule must have form\
256           \ ' t : S_i '";
257 val msg2 = "Non-atomic premise";
258 val msg3 = "Recursion term on left of member symbol";
260 fun check_rule sign cs r =
261   let
262     fun check_prem prem = if can HOLogic.dest_Trueprop prem then ()
263       else err_in_prem sign r prem msg2;
265   in (case HOLogic.dest_Trueprop (Logic.strip_imp_concl r) of
266         (Const ("op :", _) \$ t \$ u) =>
267           if u mem cs then
268             if exists (Logic.occs o (rpair t)) cs then
269               err_in_rule sign r msg3
270             else
271               seq check_prem (Logic.strip_imp_prems r)
272           else err_in_rule sign r msg1
273       | _ => err_in_rule sign r msg1)
274   end;
276 fun try' f msg sign t = (case (try f t) of
277       Some x => x
278     | None => error (msg ^ Sign.string_of_term sign t));
282 (*** properties of (co)inductive sets ***)
284 (** elimination rules **)
286 fun mk_elims cs cTs params intr_ts =
287   let
288     val used = foldr add_term_names (intr_ts, []);
289     val [aname, pname] = variantlist (["a", "P"], used);
290     val P = HOLogic.mk_Trueprop (Free (pname, HOLogic.boolT));
292     fun dest_intr r =
293       let val Const ("op :", _) \$ t \$ u =
294         HOLogic.dest_Trueprop (Logic.strip_imp_concl r)
295       in (u, t, Logic.strip_imp_prems r) end;
297     val intrs = map dest_intr intr_ts;
299     fun mk_elim (c, T) =
300       let
301         val a = Free (aname, T);
303         fun mk_elim_prem (_, t, ts) =
304           list_all_free (map dest_Free ((foldr add_term_frees (t::ts, [])) \\ params),
305             Logic.list_implies (HOLogic.mk_Trueprop (HOLogic.mk_eq (a, t)) :: ts, P));
306       in
307         Logic.list_implies (HOLogic.mk_Trueprop (HOLogic.mk_mem (a, c)) ::
308           map mk_elim_prem (filter (equal c o #1) intrs), P)
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 (*** proofs for (co)inductive sets ***)
387 (** prove monotonicity **)
389 fun prove_mono setT fp_fun monos thy =
390   let
391     val _ = message "  Proving monotonicity ...";
393     val mono = prove_goalw_cterm [] (cterm_of (Theory.sign_of thy) (HOLogic.mk_Trueprop
394       (Const (mono_name, (setT --> setT) --> HOLogic.boolT) \$ fp_fun)))
395         (fn _ => [rtac monoI 1, REPEAT (ares_tac (get_monos thy @ flat (map mk_mono monos)) 1)])
397   in mono end;
401 (** prove introduction rules **)
403 fun prove_intrs coind mono fp_def intr_ts con_defs rec_sets_defs thy =
404   let
405     val _ = message "  Proving the introduction rules ...";
407     val unfold = standard (mono RS (fp_def RS
408       (if coind then def_gfp_Tarski else def_lfp_Tarski)));
410     fun select_disj 1 1 = []
411       | select_disj _ 1 = [rtac disjI1]
412       | select_disj n i = (rtac disjI2)::(select_disj (n - 1) (i - 1));
414     val intrs = map (fn (i, intr) => prove_goalw_cterm rec_sets_defs
415       (cterm_of (Theory.sign_of thy) intr) (fn prems =>
416        [(*insert prems and underlying sets*)
417        cut_facts_tac prems 1,
418        stac unfold 1,
419        REPEAT (resolve_tac [vimageI2, CollectI] 1),
420        (*Now 1-2 subgoals: the disjunction, perhaps equality.*)
421        EVERY1 (select_disj (length intr_ts) i),
422        (*Not ares_tac, since refl must be tried before any equality assumptions;
423          backtracking may occur if the premises have extra variables!*)
424        DEPTH_SOLVE_1 (resolve_tac [refl,exI,conjI] 1 APPEND assume_tac 1),
425        (*Now solve the equations like Inl 0 = Inl ?b2*)
426        rewrite_goals_tac con_defs,
427        REPEAT (rtac refl 1)])) (1 upto (length intr_ts) ~~ intr_ts)
429   in (intrs, unfold) end;
433 (** prove elimination rules **)
435 fun prove_elims cs cTs params intr_ts unfold rec_sets_defs thy =
436   let
437     val _ = message "  Proving the elimination rules ...";
439     val rules1 = [CollectE, disjE, make_elim vimageD, exE];
440     val rules2 = [conjE, Inl_neq_Inr, Inr_neq_Inl] @
441       map make_elim [Inl_inject, Inr_inject];
443     val elims = map (fn t => prove_goalw_cterm rec_sets_defs
444       (cterm_of (Theory.sign_of thy) t) (fn prems =>
445         [cut_facts_tac [hd prems] 1,
446          dtac (unfold RS subst) 1,
447          REPEAT (FIRSTGOAL (eresolve_tac rules1)),
448          REPEAT (FIRSTGOAL (eresolve_tac rules2)),
449          EVERY (map (fn prem =>
450            DEPTH_SOLVE_1 (ares_tac [prem, conjI] 1)) (tl prems))]))
451       (mk_elims cs cTs params intr_ts)
453   in elims end;
456 (** derivation of simplified elimination rules **)
458 (*Applies freeness of the given constructors, which *must* be unfolded by
459   the given defs.  Cannot simply use the local con_defs because con_defs=[]
460   for inference systems.
461  *)
462 fun con_elim_tac ss =
463   let val elim_tac = REPEAT o (eresolve_tac elim_rls)
464   in ALLGOALS(EVERY'[elim_tac,
465 		     asm_full_simp_tac ss,
466 		     elim_tac,
467 		     REPEAT o bound_hyp_subst_tac])
468      THEN prune_params_tac
469   end;
471 (*cprop should have the form t:Si where Si is an inductive set*)
472 fun mk_cases_i elims ss cprop =
473   let
474     val prem = Thm.assume cprop;
475     fun mk_elim rl = standard (rule_by_tactic (con_elim_tac ss) (prem RS rl));
476   in
477     (case get_first (try mk_elim) elims of
478       Some r => r
479     | None => error (Pretty.string_of (Pretty.block
480         [Pretty.str "mk_cases: proposition not of form 't : S_i'", Pretty.fbrk,
481           Display.pretty_cterm cprop])))
482   end;
484 fun mk_cases elims s =
485   mk_cases_i elims (simpset()) (Thm.read_cterm (Thm.sign_of_thm (hd elims)) (s, propT));
488 (* inductive_cases(_i) *)
490 fun gen_inductive_cases prep_att prep_const prep_prop
491     ((((name, raw_atts), raw_set), raw_props), comment) thy =
492   let
493     val sign = Theory.sign_of thy;
495     val atts = map (prep_att thy) raw_atts;
496     val (_, {elims, ...}) = get_inductive thy (prep_const sign raw_set);
497     val cprops = map (Thm.cterm_of sign o prep_prop (ProofContext.init thy)) raw_props;
498     val thms = map (mk_cases_i elims (Simplifier.simpset_of thy)) cprops;
499   in
500     thy
501     |> IsarThy.have_theorems_i (((name, atts), map Thm.no_attributes thms), comment)
502   end;
504 val inductive_cases =
505   gen_inductive_cases Attrib.global_attribute Sign.intern_const ProofContext.read_prop;
507 val inductive_cases_i = gen_inductive_cases (K I) (K I) ProofContext.cert_prop;
511 (** prove induction rule **)
513 fun prove_indrule cs cTs sumT rec_const params intr_ts mono
514     fp_def rec_sets_defs thy =
515   let
516     val _ = message "  Proving the induction rule ...";
518     val sign = Theory.sign_of thy;
520     val sum_case_rewrites = (case ThyInfo.lookup_theory "Datatype" of
521         None => []
522       | Some thy' => map mk_meta_eq (PureThy.get_thms thy' "sum.cases"));
524     val (preds, ind_prems, mutual_ind_concl) = mk_indrule cs cTs params intr_ts;
526     (* make predicate for instantiation of abstract induction rule *)
528     fun mk_ind_pred _ [P] = P
529       | mk_ind_pred T Ps =
530          let val n = (length Ps) div 2;
531              val Type (_, [T1, T2]) = T
532          in Const ("Datatype.sum.sum_case",
533            [T1 --> HOLogic.boolT, T2 --> HOLogic.boolT, T] ---> HOLogic.boolT) \$
534              mk_ind_pred T1 (take (n, Ps)) \$ mk_ind_pred T2 (drop (n, Ps))
535          end;
537     val ind_pred = mk_ind_pred sumT preds;
539     val ind_concl = HOLogic.mk_Trueprop
540       (HOLogic.all_const sumT \$ Abs ("x", sumT, HOLogic.mk_binop "op -->"
541         (HOLogic.mk_mem (Bound 0, rec_const), ind_pred \$ Bound 0)));
543     (* simplification rules for vimage and Collect *)
545     val vimage_simps = if length cs < 2 then [] else
546       map (fn c => prove_goalw_cterm [] (cterm_of sign
547         (HOLogic.mk_Trueprop (HOLogic.mk_eq
548           (mk_vimage cs sumT (HOLogic.Collect_const sumT \$ ind_pred) c,
549            HOLogic.Collect_const (HOLogic.dest_setT (fastype_of c)) \$
550              nth_elem (find_index_eq c cs, preds)))))
551         (fn _ => [rtac vimage_Collect 1, rewrite_goals_tac sum_case_rewrites,
552           rtac refl 1])) cs;
554     val induct = prove_goalw_cterm [] (cterm_of sign
555       (Logic.list_implies (ind_prems, ind_concl))) (fn prems =>
556         [rtac (impI RS allI) 1,
557          DETERM (etac (mono RS (fp_def RS def_induct)) 1),
558          rewrite_goals_tac (map mk_meta_eq (vimage_Int::Int_Collect::vimage_simps)),
559          fold_goals_tac rec_sets_defs,
560          (*This CollectE and disjE separates out the introduction rules*)
561          REPEAT (FIRSTGOAL (eresolve_tac [CollectE, disjE, exE])),
562          (*Now break down the individual cases.  No disjE here in case
563            some premise involves disjunction.*)
564          REPEAT (FIRSTGOAL (etac conjE ORELSE' hyp_subst_tac)),
565          rewrite_goals_tac sum_case_rewrites,
566          EVERY (map (fn prem =>
567            DEPTH_SOLVE_1 (ares_tac [prem, conjI, refl] 1)) prems)]);
569     val lemma = prove_goalw_cterm rec_sets_defs (cterm_of sign
570       (Logic.mk_implies (ind_concl, mutual_ind_concl))) (fn prems =>
571         [cut_facts_tac prems 1,
572          REPEAT (EVERY
573            [REPEAT (resolve_tac [conjI, impI] 1),
574             TRY (dtac vimageD 1), etac allE 1, dtac mp 1, atac 1,
575             rewrite_goals_tac sum_case_rewrites,
576             atac 1])])
578   in standard (split_rule (induct RS lemma))
579   end;
583 (*** specification of (co)inductive sets ****)
585 (** definitional introduction of (co)inductive sets **)
587 fun add_ind_def verbose declare_consts alt_name coind no_elim no_ind cs
588     atts intros monos con_defs thy params paramTs cTs cnames =
589   let
590     val _ = if verbose then message ("Proofs for " ^ coind_prefix coind ^ "inductive set(s) " ^
591       commas_quote cnames) else ();
593     val sumT = fold_bal (fn (T, U) => Type ("+", [T, U])) cTs;
594     val setT = HOLogic.mk_setT sumT;
596     val fp_name = if coind then Sign.intern_const (Theory.sign_of Gfp.thy) "gfp"
597       else Sign.intern_const (Theory.sign_of Lfp.thy) "lfp";
599     val ((intr_names, intr_ts), intr_atts) = apfst split_list (split_list intros);
601     val used = foldr add_term_names (intr_ts, []);
602     val [sname, xname] = variantlist (["S", "x"], used);
604     (* transform an introduction rule into a conjunction  *)
605     (*   [| t : ... S_i ... ; ... |] ==> u : S_j          *)
606     (* is transformed into                                *)
607     (*   x = Inj_j u & t : ... Inj_i -`` S ... & ...      *)
609     fun transform_rule r =
610       let
611         val frees = map dest_Free ((add_term_frees (r, [])) \\ params);
612         val subst = subst_free
613           (cs ~~ (map (mk_vimage cs sumT (Free (sname, setT))) cs));
614         val Const ("op :", _) \$ t \$ u =
615           HOLogic.dest_Trueprop (Logic.strip_imp_concl r)
617       in foldr (fn ((x, T), P) => HOLogic.mk_exists (x, T, P))
618         (frees, foldr1 HOLogic.mk_conj
619           (((HOLogic.eq_const sumT) \$ Free (xname, sumT) \$ (mk_inj cs sumT u t))::
620             (map (subst o HOLogic.dest_Trueprop)
621               (Logic.strip_imp_prems r))))
622       end
624     (* make a disjunction of all introduction rules *)
626     val fp_fun = absfree (sname, setT, (HOLogic.Collect_const sumT) \$
627       absfree (xname, sumT, foldr1 HOLogic.mk_disj (map transform_rule intr_ts)));
629     (* add definiton of recursive sets to theory *)
631     val rec_name = if alt_name = "" then space_implode "_" cnames else alt_name;
632     val full_rec_name = Sign.full_name (Theory.sign_of thy) rec_name;
634     val rec_const = list_comb
635       (Const (full_rec_name, paramTs ---> setT), params);
637     val fp_def_term = Logic.mk_equals (rec_const,
638       Const (fp_name, (setT --> setT) --> setT) \$ fp_fun)
640     val def_terms = fp_def_term :: (if length cs < 2 then [] else
641       map (fn c => Logic.mk_equals (c, mk_vimage cs sumT rec_const c)) cs);
643     val thy' = thy |>
644       (if declare_consts then
645         Theory.add_consts_i (map (fn (c, n) =>
646           (n, paramTs ---> fastype_of c, NoSyn)) (cs ~~ cnames))
647        else I) |>
648       (if length cs < 2 then I else
649        Theory.add_consts_i [(rec_name, paramTs ---> setT, NoSyn)]) |>
650       Theory.add_path rec_name |>
651       PureThy.add_defss_i [(("defs", def_terms), [])];
653     (* get definitions from theory *)
655     val fp_def::rec_sets_defs = PureThy.get_thms thy' "defs";
657     (* prove and store theorems *)
659     val mono = prove_mono setT fp_fun monos thy';
660     val (intrs, unfold) = prove_intrs coind mono fp_def intr_ts con_defs
661       rec_sets_defs thy';
662     val elims = if no_elim then [] else
663       prove_elims cs cTs params intr_ts unfold rec_sets_defs thy';
664     val raw_induct = if no_ind then TrueI else
665       if coind then standard (rule_by_tactic
666         (rewrite_tac [mk_meta_eq vimage_Un] THEN
667           fold_tac rec_sets_defs) (mono RS (fp_def RS def_Collect_coinduct)))
668       else
669         prove_indrule cs cTs sumT rec_const params intr_ts mono fp_def
670           rec_sets_defs thy';
671     val induct = if coind orelse no_ind orelse length cs > 1 then raw_induct
672       else standard (raw_induct RSN (2, rev_mp));
674     val thy'' = thy'
675       |> PureThy.add_thmss [(("intrs", intrs), atts)]
676       |> PureThy.add_thms ((intr_names ~~ intrs) ~~ intr_atts)
677       |> (if no_elim then I else PureThy.add_thmss [(("elims", elims), [])])
678       |> (if no_ind then I else PureThy.add_thms
679         [((coind_prefix coind ^ "induct", induct), [])])
680       |> Theory.parent_path;
681     val intrs' = PureThy.get_thms thy'' "intrs";
682     val elims' = PureThy.get_thms thy'' "elims";
683     val induct' = PureThy.get_thm thy'' (coind_prefix coind ^ "induct");
684   in (thy'',
685     {defs = fp_def::rec_sets_defs,
686      mono = mono,
687      unfold = unfold,
688      intrs = intrs',
689      elims = elims',
690      mk_cases = mk_cases elims',
691      raw_induct = raw_induct,
692      induct = induct'})
693   end;
697 (** axiomatic introduction of (co)inductive sets **)
699 fun add_ind_axm verbose declare_consts alt_name coind no_elim no_ind cs
700     atts intros monos con_defs thy params paramTs cTs cnames =
701   let
702     val rec_name = if alt_name = "" then space_implode "_" cnames else alt_name;
704     val ((intr_names, intr_ts), intr_atts) = apfst split_list (split_list intros);
705     val elim_ts = mk_elims cs cTs params intr_ts;
707     val (_, ind_prems, mutual_ind_concl) = mk_indrule cs cTs params intr_ts;
708     val ind_t = Logic.list_implies (ind_prems, mutual_ind_concl);
710     val thy' = thy
711       |> (if declare_consts then
713               (map (fn (c, n) => (n, paramTs ---> fastype_of c, NoSyn)) (cs ~~ cnames))
714          else I)
715       |> Theory.add_path rec_name
716       |> PureThy.add_axiomss_i [(("intrs", intr_ts), atts), (("elims", elim_ts), [])]
717       |> (if coind then I else
718             PureThy.add_axioms_i [(("raw_induct", ind_t), [apsnd (standard o split_rule)])]);
720     val intrs = PureThy.get_thms thy' "intrs";
721     val elims = PureThy.get_thms thy' "elims";
722     val raw_induct = if coind then TrueI else PureThy.get_thm thy' "raw_induct";
723     val induct = if coind orelse length cs > 1 then raw_induct
724       else standard (raw_induct RSN (2, rev_mp));
726     val thy'' =
727       thy'
728       |> (if coind then I else PureThy.add_thms [(("induct", induct), [])])
729       |> PureThy.add_thms ((intr_names ~~ intrs) ~~ intr_atts)
730       |> Theory.parent_path;
731     val induct' = if coind then raw_induct else PureThy.get_thm thy'' "induct";
732   in (thy'',
733     {defs = [],
734      mono = TrueI,
735      unfold = TrueI,
736      intrs = intrs,
737      elims = elims,
738      mk_cases = mk_cases elims,
739      raw_induct = raw_induct,
740      induct = induct'})
741   end;
745 (** introduction of (co)inductive sets **)
747 fun add_cases_induct names elims induct =
749     (map (fn name => (("", induct), [InductMethod.induct_set_global name])) names @
750      map2 (fn (name, elim) => (("", elim), [InductMethod.cases_set_global name])) (names, elims));
753 fun add_inductive_i verbose declare_consts alt_name coind no_elim no_ind cs
754     atts intros monos con_defs thy =
755   let
756     val _ = Theory.requires thy "Inductive" (coind_prefix coind ^ "inductive definitions");
757     val sign = Theory.sign_of thy;
759     (*parameters should agree for all mutually recursive components*)
760     val (_, params) = strip_comb (hd cs);
761     val paramTs = map (try' (snd o dest_Free) "Parameter in recursive\
762       \ component is not a free variable: " sign) params;
764     val cTs = map (try' (HOLogic.dest_setT o fastype_of)
765       "Recursive component not of type set: " sign) cs;
767     val full_cnames = map (try' (fst o dest_Const o head_of)
768       "Recursive set not previously declared as constant: " sign) cs;
769     val cnames = map Sign.base_name full_cnames;
771     val _ = seq (check_rule sign cs o snd o fst) intros;
773     val (thy1, result) =
774       (if ! quick_and_dirty then add_ind_axm else add_ind_def)
775         verbose declare_consts alt_name coind no_elim no_ind cs atts intros monos
776         con_defs thy params paramTs cTs cnames;
777     val thy2 = thy1
778       |> put_inductives full_cnames ({names = full_cnames, coind = coind}, result)
779       |> add_cases_induct full_cnames (#elims result) (#induct result);
780   in (thy2, result) end;
784 (** external interface **)
786 fun add_inductive verbose coind c_strings srcs intro_srcs raw_monos raw_con_defs thy =
787   let
788     val sign = Theory.sign_of thy;
789     val cs = map (term_of o Thm.read_cterm sign o rpair HOLogic.termT) c_strings;
791     val atts = map (Attrib.global_attribute thy) srcs;
792     val intr_names = map (fst o fst) intro_srcs;
793     val intr_ts = map (term_of o Thm.read_cterm sign o rpair propT o snd o fst) intro_srcs;
794     val intr_atts = map (map (Attrib.global_attribute thy) o snd) intro_srcs;
795     val (cs', intr_ts') = unify_consts sign cs intr_ts;
797     val ((thy', con_defs), monos) = thy
798       |> IsarThy.apply_theorems raw_monos
799       |> apfst (IsarThy.apply_theorems raw_con_defs);
800   in
801     add_inductive_i verbose false "" coind false false cs'
802       atts ((intr_names ~~ intr_ts') ~~ intr_atts) monos con_defs thy'
803   end;
807 (** package setup **)
809 (* setup theory *)
811 val setup = [InductiveData.init,
812              Attrib.add_attributes [(monoN, mono_attr, "monotonicity rule")]];
815 (* outer syntax *)
817 local structure P = OuterParse and K = OuterSyntax.Keyword in
819 fun mk_ind coind (((sets, (atts, intrs)), monos), con_defs) =
820   #1 o add_inductive true coind sets atts (map P.triple_swap intrs) monos con_defs;
822 fun ind_decl coind =
823   (Scan.repeat1 P.term --| P.marg_comment) --
824   (P.\$\$\$ "intrs" |--
825     P.!!! (P.opt_attribs -- Scan.repeat1 (P.opt_thm_name ":" -- P.prop --| P.marg_comment))) --
826   Scan.optional (P.\$\$\$ "monos" |-- P.!!! P.xthms1 --| P.marg_comment) [] --
827   Scan.optional (P.\$\$\$ "con_defs" |-- P.!!! P.xthms1 --| P.marg_comment) []
828   >> (Toplevel.theory o mk_ind coind);
830 val inductiveP =
831   OuterSyntax.command "inductive" "define inductive sets" K.thy_decl (ind_decl false);
833 val coinductiveP =
834   OuterSyntax.command "coinductive" "define coinductive sets" K.thy_decl (ind_decl true);
837 val ind_cases =
838   P.opt_thm_name "=" -- P.xname --| P.\$\$\$ ":" -- Scan.repeat1 P.prop -- P.marg_comment
839   >> (Toplevel.theory o inductive_cases);
841 val inductive_casesP =
842   OuterSyntax.command "inductive_cases" "create simplified instances of elimination rules"
843     K.thy_decl ind_cases;
845 val _ = OuterSyntax.add_keywords ["intrs", "monos", "con_defs"];
846 val _ = OuterSyntax.add_parsers [inductiveP, coinductiveP, inductive_casesP];
848 end;
851 end;