src/HOL/Tools/inductive_package.ML
author wenzelm
Thu Jun 29 22:38:30 2000 +0200 (2000-06-29)
changeset 9201 435fef035d7f
parent 9116 9df44b5c610b
child 9235 1f734dc2e526
permissions -rw-r--r--
adapted args of IsarThy.have_theorems_i;
     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     
     7 
     8 (Co)Inductive Definition module for HOL.
     9 
    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
    16 
    17 The recursive sets must *already* be declared as constants in the
    18 current theory!
    19 
    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
    26 
    27 Sums are used only for mutual recursion.  Products are used only to
    28 derive "streamlined" induction rules for relations.
    29 *)
    30 
    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;
    58 
    59 structure InductivePackage: INDUCTIVE_PACKAGE =
    60 struct
    61 
    62 (*** theory data ***)
    63 
    64 (* data kind 'HOL/inductive' *)
    65 
    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};
    69 
    70 structure InductiveArgs =
    71 struct
    72   val name = "HOL/inductive";
    73   type T = inductive_info Symtab.table * thm list;
    74 
    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);
    80 
    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;
    86 
    87 structure InductiveData = TheoryDataFun(InductiveArgs);
    88 val print_inductives = InductiveData.print;
    89 
    90 
    91 (* get and put data *)
    92 
    93 fun get_inductive thy name = Symtab.lookup (fst (InductiveData.get thy), name);
    94 
    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;
   101 
   102 
   103 
   104 (** monotonicity rules **)
   105 
   106 val get_monos = snd o InductiveData.get;
   107 fun put_monos thms thy = InductiveData.put (fst (InductiveData.get thy), thms) thy;
   108 
   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;
   123 
   124 
   125 (* attributes *)
   126 
   127 local
   128 
   129 fun map_rules_global f thy = put_monos (f (get_monos thy)) thy;
   130 
   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);
   133 
   134 fun mk_att f g (x, thm) = (f (g thm) x, thm);
   135 
   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;
   140 
   141 val mono_attr =
   142  (Attrib.add_del_args mono_add_global mono_del_global,
   143   Attrib.add_del_args Attrib.undef_local_attribute Attrib.undef_local_attribute);
   144 
   145 
   146 
   147 (** utilities **)
   148 
   149 (* messages *)
   150 
   151 val quiet_mode = ref false;
   152 fun message s = if !quiet_mode then () else writeln s;
   153 
   154 fun coind_prefix true = "co"
   155   | coind_prefix false = "";
   156 
   157 
   158 (* the following code ensures that each recursive set *)
   159 (* always has the same type in all introduction rules *)
   160 
   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))
   183 
   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));
   187 
   188 
   189 (* misc *)
   190 
   191 val Const _ $ (vimage_f $ _) $ _ = HOLogic.dest_Trueprop (concl_of vimageD);
   192 
   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";
   195 
   196 (* make injections needed in mutually recursive definitions *)
   197 
   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;
   212 
   213 (* make "vimage" terms for selecting out components of mutually rec.def. *)
   214 
   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;
   223 
   224 
   225 
   226 (** well-formedness checks **)
   227 
   228 fun err_in_rule sign t msg = error ("Ill-formed introduction rule\n" ^
   229   (Sign.string_of_term sign t) ^ "\n" ^ msg);
   230 
   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);
   234 
   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";
   239 
   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;
   244 
   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;
   255 
   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));
   259 
   260 
   261 
   262 (*** properties of (co)inductive sets ***)
   263 
   264 (** elimination rules **)
   265 
   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));
   271 
   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;
   276 
   277     val intrs = map dest_intr intr_ts ~~ intr_names;
   278 
   279     fun mk_elim (c, T) =
   280       let
   281         val a = Free (aname, T);
   282 
   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;
   294         
   295 
   296 
   297 (** premises and conclusions of induction rules **)
   298 
   299 fun mk_indrule cs cTs params intr_ts =
   300   let
   301     val used = foldr add_term_names (intr_ts, []);
   302 
   303     (* predicates for induction rule *)
   304 
   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);
   308 
   309     (* transform an introduction rule into a premise for induction rule *)
   310 
   311     fun mk_ind_prem r =
   312       let
   313         val frees = map dest_Free ((add_term_frees (r, [])) \\ params);
   314 
   315         val pred_of = curry (Library.gen_assoc (op aconv)) (cs ~~ preds);
   316 
   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)));
   330 
   331         fun mk_prem (s, prems) = (case subst s of
   332               (_, Some (t, u)) => t :: u :: prems
   333             | (t, _) => t :: prems);
   334           
   335         val Const ("op :", _) $ t $ u =
   336           HOLogic.dest_Trueprop (Logic.strip_imp_concl r)
   337 
   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;
   343 
   344     val ind_prems = map mk_ind_prem intr_ts;
   345 
   346     (* make conclusions for induction rules *)
   347 
   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;
   357 
   358     val mutual_ind_concl = HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
   359         (fst (foldr mk_ind_concl (cs ~~ preds, ([], "xa")))))
   360 
   361   in (preds, ind_prems, mutual_ind_concl)
   362   end;
   363 
   364 
   365 
   366 (** prepare cases and induct rules **)
   367 
   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 *)
   374 
   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;
   384 
   385 fun add_cases_induct no_elim no_ind names elims induct induct_cases =
   386   let
   387     fun cases_spec (name, elim) = (("", elim), [InductMethod.cases_set_global name]);
   388     val cases_specs = if no_elim then [] else map2 cases_spec (names, elims);
   389 
   390     fun induct_spec (name, th) =
   391       (("", th), [RuleCases.case_names induct_cases, InductMethod.induct_set_global name]);
   392     val induct_specs = if no_ind then [] else map induct_spec (project_rules names induct);
   393   in #1 o PureThy.add_thms (cases_specs @ induct_specs) end;
   394 
   395 
   396 
   397 (*** proofs for (co)inductive sets ***)
   398 
   399 (** prove monotonicity **)
   400 
   401 fun prove_mono setT fp_fun monos thy =
   402   let
   403     val _ = message "  Proving monotonicity ...";
   404 
   405     val mono = prove_goalw_cterm [] (cterm_of (Theory.sign_of thy) (HOLogic.mk_Trueprop
   406       (Const (mono_name, (setT --> setT) --> HOLogic.boolT) $ fp_fun)))
   407         (fn _ => [rtac monoI 1, REPEAT (ares_tac (get_monos thy @ flat (map mk_mono monos)) 1)])
   408 
   409   in mono end;
   410 
   411 
   412 
   413 (** prove introduction rules **)
   414 
   415 fun prove_intrs coind mono fp_def intr_ts con_defs rec_sets_defs thy =
   416   let
   417     val _ = message "  Proving the introduction rules ...";
   418 
   419     val unfold = standard (mono RS (fp_def RS
   420       (if coind then def_gfp_Tarski else def_lfp_Tarski)));
   421 
   422     fun select_disj 1 1 = []
   423       | select_disj _ 1 = [rtac disjI1]
   424       | select_disj n i = (rtac disjI2)::(select_disj (n - 1) (i - 1));
   425 
   426     val intrs = map (fn (i, intr) => prove_goalw_cterm rec_sets_defs
   427       (cterm_of (Theory.sign_of thy) intr) (fn prems =>
   428        [(*insert prems and underlying sets*)
   429        cut_facts_tac prems 1,
   430        stac unfold 1,
   431        REPEAT (resolve_tac [vimageI2, CollectI] 1),
   432        (*Now 1-2 subgoals: the disjunction, perhaps equality.*)
   433        EVERY1 (select_disj (length intr_ts) i),
   434        (*Not ares_tac, since refl must be tried before any equality assumptions;
   435          backtracking may occur if the premises have extra variables!*)
   436        DEPTH_SOLVE_1 (resolve_tac [refl,exI,conjI] 1 APPEND assume_tac 1),
   437        (*Now solve the equations like Inl 0 = Inl ?b2*)
   438        rewrite_goals_tac con_defs,
   439        REPEAT (rtac refl 1)])) (1 upto (length intr_ts) ~~ intr_ts)
   440 
   441   in (intrs, unfold) end;
   442 
   443 
   444 
   445 (** prove elimination rules **)
   446 
   447 fun prove_elims cs cTs params intr_ts intr_names unfold rec_sets_defs thy =
   448   let
   449     val _ = message "  Proving the elimination rules ...";
   450 
   451     val rules1 = [CollectE, disjE, make_elim vimageD, exE];
   452     val rules2 = [conjE, Inl_neq_Inr, Inr_neq_Inl] @
   453       map make_elim [Inl_inject, Inr_inject];
   454   in
   455     map (fn (t, cases) => prove_goalw_cterm rec_sets_defs
   456       (cterm_of (Theory.sign_of thy) t) (fn prems =>
   457         [cut_facts_tac [hd prems] 1,
   458          dtac (unfold RS subst) 1,
   459          REPEAT (FIRSTGOAL (eresolve_tac rules1)),
   460          REPEAT (FIRSTGOAL (eresolve_tac rules2)),
   461          EVERY (map (fn prem =>
   462            DEPTH_SOLVE_1 (ares_tac [prem, conjI] 1)) (tl prems))])
   463       |> RuleCases.name cases)
   464       (mk_elims cs cTs params intr_ts intr_names)
   465   end;
   466 
   467 
   468 (** derivation of simplified elimination rules **)
   469 
   470 (*Applies freeness of the given constructors, which *must* be unfolded by
   471   the given defs.  Cannot simply use the local con_defs because con_defs=[] 
   472   for inference systems.
   473  *)
   474 
   475 (*cprop should have the form t:Si where Si is an inductive set*)
   476 fun mk_cases_i solved elims ss cprop =
   477   let
   478     val prem = Thm.assume cprop;
   479     val tac = if solved then InductMethod.con_elim_solved_tac else InductMethod.con_elim_tac;
   480     fun mk_elim rl = Drule.standard (Tactic.rule_by_tactic (tac ss) (prem RS rl));
   481   in
   482     (case get_first (try mk_elim) elims of
   483       Some r => r
   484     | None => error (Pretty.string_of (Pretty.block
   485         [Pretty.str "mk_cases: proposition not of form 't : S_i'", Pretty.fbrk,
   486           Display.pretty_cterm cprop])))
   487   end;
   488 
   489 fun mk_cases elims s =
   490   mk_cases_i false elims (simpset()) (Thm.read_cterm (Thm.sign_of_thm (hd elims)) (s, propT));
   491 
   492 
   493 (* inductive_cases(_i) *)
   494 
   495 fun gen_inductive_cases prep_att prep_const prep_prop
   496     ((((name, raw_atts), raw_set), raw_props), comment) thy =
   497   let val sign = Theory.sign_of thy;
   498   in (case get_inductive thy (prep_const sign raw_set) of
   499       None => error ("Unknown (co)inductive set " ^ quote name)
   500     | Some (_, {elims, ...}) =>
   501         let
   502           val atts = map (prep_att thy) raw_atts;
   503           val cprops = map
   504             (Thm.cterm_of sign o prep_prop (ProofContext.init thy)) raw_props;
   505           val thms = map
   506             (mk_cases_i true elims (Simplifier.simpset_of thy)) cprops;
   507         in
   508           thy |> IsarThy.have_theorems_i
   509             [(((name, atts), map Thm.no_attributes thms), comment)]
   510         end)
   511   end;
   512 
   513 val inductive_cases =
   514   gen_inductive_cases Attrib.global_attribute Sign.intern_const ProofContext.read_prop;
   515 
   516 val inductive_cases_i = gen_inductive_cases (K I) (K I) ProofContext.cert_prop;
   517 
   518 
   519 
   520 (** prove induction rule **)
   521 
   522 fun prove_indrule cs cTs sumT rec_const params intr_ts mono
   523     fp_def rec_sets_defs thy =
   524   let
   525     val _ = message "  Proving the induction rule ...";
   526 
   527     val sign = Theory.sign_of thy;
   528 
   529     val sum_case_rewrites = (case ThyInfo.lookup_theory "Datatype" of
   530         None => []
   531       | Some thy' => map mk_meta_eq (PureThy.get_thms thy' "sum.cases"));
   532 
   533     val (preds, ind_prems, mutual_ind_concl) = mk_indrule cs cTs params intr_ts;
   534 
   535     (* make predicate for instantiation of abstract induction rule *)
   536 
   537     fun mk_ind_pred _ [P] = P
   538       | mk_ind_pred T Ps =
   539          let val n = (length Ps) div 2;
   540              val Type (_, [T1, T2]) = T
   541          in Const ("Datatype.sum.sum_case",
   542            [T1 --> HOLogic.boolT, T2 --> HOLogic.boolT, T] ---> HOLogic.boolT) $
   543              mk_ind_pred T1 (take (n, Ps)) $ mk_ind_pred T2 (drop (n, Ps))
   544          end;
   545 
   546     val ind_pred = mk_ind_pred sumT preds;
   547 
   548     val ind_concl = HOLogic.mk_Trueprop
   549       (HOLogic.all_const sumT $ Abs ("x", sumT, HOLogic.mk_binop "op -->"
   550         (HOLogic.mk_mem (Bound 0, rec_const), ind_pred $ Bound 0)));
   551 
   552     (* simplification rules for vimage and Collect *)
   553 
   554     val vimage_simps = if length cs < 2 then [] else
   555       map (fn c => prove_goalw_cterm [] (cterm_of sign
   556         (HOLogic.mk_Trueprop (HOLogic.mk_eq
   557           (mk_vimage cs sumT (HOLogic.Collect_const sumT $ ind_pred) c,
   558            HOLogic.Collect_const (HOLogic.dest_setT (fastype_of c)) $
   559              nth_elem (find_index_eq c cs, preds)))))
   560         (fn _ => [rtac vimage_Collect 1, rewrite_goals_tac sum_case_rewrites,
   561           rtac refl 1])) cs;
   562 
   563     val induct = prove_goalw_cterm [] (cterm_of sign
   564       (Logic.list_implies (ind_prems, ind_concl))) (fn prems =>
   565         [rtac (impI RS allI) 1,
   566          DETERM (etac (mono RS (fp_def RS def_induct)) 1),
   567          rewrite_goals_tac (map mk_meta_eq (vimage_Int::Int_Collect::vimage_simps)),
   568          fold_goals_tac rec_sets_defs,
   569          (*This CollectE and disjE separates out the introduction rules*)
   570          REPEAT (FIRSTGOAL (eresolve_tac [CollectE, disjE, exE])),
   571          (*Now break down the individual cases.  No disjE here in case
   572            some premise involves disjunction.*)
   573          REPEAT (FIRSTGOAL (etac conjE ORELSE' hyp_subst_tac)),
   574          rewrite_goals_tac sum_case_rewrites,
   575          EVERY (map (fn prem =>
   576            DEPTH_SOLVE_1 (ares_tac [prem, conjI, refl] 1)) prems)]);
   577 
   578     val lemma = prove_goalw_cterm rec_sets_defs (cterm_of sign
   579       (Logic.mk_implies (ind_concl, mutual_ind_concl))) (fn prems =>
   580         [cut_facts_tac prems 1,
   581          REPEAT (EVERY
   582            [REPEAT (resolve_tac [conjI, impI] 1),
   583             TRY (dtac vimageD 1), etac allE 1, dtac mp 1, atac 1,
   584             rewrite_goals_tac sum_case_rewrites,
   585             atac 1])])
   586 
   587   in standard (split_rule (induct RS lemma))
   588   end;
   589 
   590 
   591 
   592 (*** specification of (co)inductive sets ****)
   593 
   594 (** definitional introduction of (co)inductive sets **)
   595 
   596 fun mk_ind_def declare_consts alt_name coind cs intr_ts monos con_defs thy
   597       params paramTs cTs cnames =
   598   let
   599     val sumT = fold_bal (fn (T, U) => Type ("+", [T, U])) cTs;
   600     val setT = HOLogic.mk_setT sumT;
   601 
   602     val fp_name = if coind then Sign.intern_const (Theory.sign_of Gfp.thy) "gfp"
   603       else Sign.intern_const (Theory.sign_of Lfp.thy) "lfp";
   604 
   605     val used = foldr add_term_names (intr_ts, []);
   606     val [sname, xname] = variantlist (["S", "x"], used);
   607 
   608     (* transform an introduction rule into a conjunction  *)
   609     (*   [| t : ... S_i ... ; ... |] ==> u : S_j          *)
   610     (* is transformed into                                *)
   611     (*   x = Inj_j u & t : ... Inj_i -`` S ... & ...      *)
   612 
   613     fun transform_rule r =
   614       let
   615         val frees = map dest_Free ((add_term_frees (r, [])) \\ params);
   616         val subst = subst_free
   617           (cs ~~ (map (mk_vimage cs sumT (Free (sname, setT))) cs));
   618         val Const ("op :", _) $ t $ u =
   619           HOLogic.dest_Trueprop (Logic.strip_imp_concl r)
   620 
   621       in foldr (fn ((x, T), P) => HOLogic.mk_exists (x, T, P))
   622         (frees, foldr1 HOLogic.mk_conj
   623           (((HOLogic.eq_const sumT) $ Free (xname, sumT) $ (mk_inj cs sumT u t))::
   624             (map (subst o HOLogic.dest_Trueprop)
   625               (Logic.strip_imp_prems r))))
   626       end
   627 
   628     (* make a disjunction of all introduction rules *)
   629 
   630     val fp_fun = absfree (sname, setT, (HOLogic.Collect_const sumT) $
   631       absfree (xname, sumT, foldr1 HOLogic.mk_disj (map transform_rule intr_ts)));
   632 
   633     (* add definiton of recursive sets to theory *)
   634 
   635     val rec_name = if alt_name = "" then space_implode "_" cnames else alt_name;
   636     val full_rec_name = Sign.full_name (Theory.sign_of thy) rec_name;
   637 
   638     val rec_const = list_comb
   639       (Const (full_rec_name, paramTs ---> setT), params);
   640 
   641     val fp_def_term = Logic.mk_equals (rec_const,
   642       Const (fp_name, (setT --> setT) --> setT) $ fp_fun)
   643 
   644     val def_terms = fp_def_term :: (if length cs < 2 then [] else
   645       map (fn c => Logic.mk_equals (c, mk_vimage cs sumT rec_const c)) cs);
   646 
   647     val (thy', [fp_def :: rec_sets_defs]) =
   648       thy
   649       |> (if declare_consts then
   650           Theory.add_consts_i (map (fn (c, n) =>
   651             (n, paramTs ---> fastype_of c, NoSyn)) (cs ~~ cnames))
   652           else I)
   653       |> (if length cs < 2 then I
   654           else Theory.add_consts_i [(rec_name, paramTs ---> setT, NoSyn)])
   655       |> Theory.add_path rec_name
   656       |> PureThy.add_defss_i [(("defs", def_terms), [])];
   657 
   658     val mono = prove_mono setT fp_fun monos thy'
   659 
   660   in
   661     (thy', mono, fp_def, rec_sets_defs, rec_const, sumT) 
   662   end;
   663 
   664 fun add_ind_def verbose declare_consts alt_name coind no_elim no_ind cs
   665     atts intros monos con_defs thy params paramTs cTs cnames induct_cases =
   666   let
   667     val _ = if verbose then message ("Proofs for " ^ coind_prefix coind ^ "inductive set(s) " ^
   668       commas_quote cnames) else ();
   669 
   670     val ((intr_names, intr_ts), intr_atts) = apfst split_list (split_list intros);
   671 
   672     val (thy', mono, fp_def, rec_sets_defs, rec_const, sumT) =
   673       mk_ind_def declare_consts alt_name coind cs intr_ts monos con_defs thy
   674         params paramTs cTs cnames;
   675 
   676     val (intrs, unfold) = prove_intrs coind mono fp_def intr_ts con_defs
   677       rec_sets_defs thy';
   678     val elims = if no_elim then [] else
   679       prove_elims cs cTs params intr_ts intr_names unfold rec_sets_defs thy';
   680     val raw_induct = if no_ind then Drule.asm_rl else
   681       if coind then standard (rule_by_tactic
   682         (rewrite_tac [mk_meta_eq vimage_Un] THEN
   683           fold_tac rec_sets_defs) (mono RS (fp_def RS def_Collect_coinduct)))
   684       else
   685         prove_indrule cs cTs sumT rec_const params intr_ts mono fp_def
   686           rec_sets_defs thy';
   687     val induct = if coind orelse no_ind orelse length cs > 1 then raw_induct
   688       else standard (raw_induct RSN (2, rev_mp));
   689 
   690     val (thy'', [intrs']) =
   691       thy'
   692       |> PureThy.add_thmss [(("intrs", intrs), atts)]
   693       |>> (#1 o PureThy.add_thms ((intr_names ~~ intrs) ~~ intr_atts))
   694       |>> (if no_elim then I else #1 o PureThy.add_thmss [(("elims", elims), [])])
   695       |>> (if no_ind then I else #1 o PureThy.add_thms
   696         [((coind_prefix coind ^ "induct", induct), [RuleCases.case_names induct_cases])])
   697       |>> Theory.parent_path;
   698     val elims' = if no_elim then elims else PureThy.get_thms thy'' "elims";  (* FIXME improve *)
   699     val induct' = if no_ind then induct else PureThy.get_thm thy'' (coind_prefix coind ^ "induct");  (* FIXME improve *)
   700   in (thy'',
   701     {defs = fp_def::rec_sets_defs,
   702      mono = mono,
   703      unfold = unfold,
   704      intrs = intrs',
   705      elims = elims',
   706      mk_cases = mk_cases elims',
   707      raw_induct = raw_induct,
   708      induct = induct'})
   709   end;
   710 
   711 
   712 
   713 (** axiomatic introduction of (co)inductive sets **)
   714 
   715 fun add_ind_axm verbose declare_consts alt_name coind no_elim no_ind cs
   716     atts intros monos con_defs thy params paramTs cTs cnames induct_cases =
   717   let
   718     val _ = message (coind_prefix coind ^ "inductive set(s) " ^ commas_quote cnames);
   719 
   720     val ((intr_names, intr_ts), intr_atts) = apfst split_list (split_list intros);
   721     val (thy', _, _, _, _, _) =
   722       mk_ind_def declare_consts alt_name coind cs intr_ts monos con_defs thy
   723         params paramTs cTs cnames;
   724     val (elim_ts, elim_cases) = Library.split_list (mk_elims cs cTs params intr_ts intr_names);
   725     val (_, ind_prems, mutual_ind_concl) = mk_indrule cs cTs params intr_ts;
   726     val ind_t = Logic.list_implies (ind_prems, mutual_ind_concl);
   727     
   728     val thy'' =
   729       thy'
   730       |> (#1 o PureThy.add_axiomss_i [(("intrs", intr_ts), atts), (("raw_elims", elim_ts), [])])
   731       |> (if coind then I else
   732             #1 o PureThy.add_axioms_i [(("raw_induct", ind_t), [apsnd (standard o split_rule)])]);
   733 
   734     val intrs = PureThy.get_thms thy'' "intrs";
   735     val elims = map2 (fn (th, cases) => RuleCases.name cases th)
   736       (PureThy.get_thms thy'' "raw_elims", elim_cases);
   737     val raw_induct = if coind then Drule.asm_rl else PureThy.get_thm thy'' "raw_induct";
   738     val induct = if coind orelse length cs > 1 then raw_induct
   739       else standard (raw_induct RSN (2, rev_mp));
   740 
   741     val (thy''', ([elims'], intrs')) =
   742       thy''
   743       |> PureThy.add_thmss [(("elims", elims), [])]
   744       |>> (if coind then I
   745           else #1 o PureThy.add_thms [(("induct", induct), [RuleCases.case_names induct_cases])])
   746       |>>> PureThy.add_thms ((intr_names ~~ intrs) ~~ intr_atts)
   747       |>> Theory.parent_path;
   748     val induct' = if coind then raw_induct else PureThy.get_thm thy''' "induct";
   749   in (thy''',
   750     {defs = [],
   751      mono = Drule.asm_rl,
   752      unfold = Drule.asm_rl,
   753      intrs = intrs',
   754      elims = elims',
   755      mk_cases = mk_cases elims',
   756      raw_induct = raw_induct,
   757      induct = induct'})
   758   end;
   759 
   760 
   761 
   762 (** introduction of (co)inductive sets **)
   763 
   764 fun add_inductive_i verbose declare_consts alt_name coind no_elim no_ind cs
   765     atts intros monos con_defs thy =
   766   let
   767     val _ = Theory.requires thy "Inductive" (coind_prefix coind ^ "inductive definitions");
   768     val sign = Theory.sign_of thy;
   769 
   770     (*parameters should agree for all mutually recursive components*)
   771     val (_, params) = strip_comb (hd cs);
   772     val paramTs = map (try' (snd o dest_Free) "Parameter in recursive\
   773       \ component is not a free variable: " sign) params;
   774 
   775     val cTs = map (try' (HOLogic.dest_setT o fastype_of)
   776       "Recursive component not of type set: " sign) cs;
   777 
   778     val full_cnames = map (try' (fst o dest_Const o head_of)
   779       "Recursive set not previously declared as constant: " sign) cs;
   780     val cnames = map Sign.base_name full_cnames;
   781 
   782     val _ = seq (check_rule sign cs o snd o fst) intros;
   783     val induct_cases = map (#1 o #1) intros;
   784 
   785     val (thy1, result) =
   786       (if ! quick_and_dirty then add_ind_axm else add_ind_def)
   787         verbose declare_consts alt_name coind no_elim no_ind cs atts intros monos
   788         con_defs thy params paramTs cTs cnames induct_cases;
   789     val thy2 = thy1
   790       |> put_inductives full_cnames ({names = full_cnames, coind = coind}, result)
   791       |> add_cases_induct no_elim (no_ind orelse coind) full_cnames
   792           (#elims result) (#induct result) induct_cases;
   793   in (thy2, result) end;
   794 
   795 
   796 
   797 (** external interface **)
   798 
   799 fun add_inductive verbose coind c_strings srcs intro_srcs raw_monos raw_con_defs thy =
   800   let
   801     val sign = Theory.sign_of thy;
   802     val cs = map (term_of o Thm.read_cterm sign o rpair HOLogic.termT) c_strings;
   803 
   804     val atts = map (Attrib.global_attribute thy) srcs;
   805     val intr_names = map (fst o fst) intro_srcs;
   806     val intr_ts = map (term_of o Thm.read_cterm sign o rpair propT o snd o fst) intro_srcs;
   807     val intr_atts = map (map (Attrib.global_attribute thy) o snd) intro_srcs;
   808     val (cs', intr_ts') = unify_consts sign cs intr_ts;
   809 
   810     val ((thy', con_defs), monos) = thy
   811       |> IsarThy.apply_theorems raw_monos
   812       |> apfst (IsarThy.apply_theorems raw_con_defs);
   813   in
   814     add_inductive_i verbose false "" coind false false cs'
   815       atts ((intr_names ~~ intr_ts') ~~ intr_atts) monos con_defs thy'
   816   end;
   817 
   818 
   819 
   820 (** package setup **)
   821 
   822 (* setup theory *)
   823 
   824 val setup =
   825  [InductiveData.init,
   826   Attrib.add_attributes [("mono", mono_attr, "monotonicity rule")]];
   827 
   828 
   829 (* outer syntax *)
   830 
   831 local structure P = OuterParse and K = OuterSyntax.Keyword in
   832 
   833 fun mk_ind coind (((sets, (atts, intrs)), monos), con_defs) =
   834   #1 o add_inductive true coind sets atts (map P.triple_swap intrs) monos con_defs;
   835 
   836 fun ind_decl coind =
   837   (Scan.repeat1 P.term --| P.marg_comment) --
   838   (P.$$$ "intrs" |--
   839     P.!!! (P.opt_attribs -- Scan.repeat1 (P.opt_thm_name ":" -- P.prop --| P.marg_comment))) --
   840   Scan.optional (P.$$$ "monos" |-- P.!!! P.xthms1 --| P.marg_comment) [] --
   841   Scan.optional (P.$$$ "con_defs" |-- P.!!! P.xthms1 --| P.marg_comment) []
   842   >> (Toplevel.theory o mk_ind coind);
   843 
   844 val inductiveP =
   845   OuterSyntax.command "inductive" "define inductive sets" K.thy_decl (ind_decl false);
   846 
   847 val coinductiveP =
   848   OuterSyntax.command "coinductive" "define coinductive sets" K.thy_decl (ind_decl true);
   849 
   850 
   851 val ind_cases =
   852   P.opt_thm_name "=" -- P.xname --| P.$$$ ":" -- Scan.repeat1 P.prop -- P.marg_comment
   853   >> (Toplevel.theory o inductive_cases);
   854 
   855 val inductive_casesP =
   856   OuterSyntax.command "inductive_cases" "create simplified instances of elimination rules"
   857     K.thy_decl ind_cases;
   858 
   859 val _ = OuterSyntax.add_keywords ["intrs", "monos", "con_defs"];
   860 val _ = OuterSyntax.add_parsers [inductiveP, coinductiveP, inductive_casesP];
   861 
   862 end;
   863 
   864 
   865 end;