src/HOL/Tools/inductive_package.ML
author wenzelm
Fri Mar 03 21:01:57 2000 +0100 (2000-03-03)
changeset 8336 fdf3ac335f77
parent 8316 74639e19eca0
child 8375 0544749a5e8f
permissions -rw-r--r--
mk_cases / inductive_cases: use InductMethod.con_elim_(solved_)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     
     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 ->
    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;
    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.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;
    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 =
    94   (case Symtab.lookup (fst (InductiveData.get thy), name) of
    95     Some info => info
    96   | None => error ("Unknown (co)inductive set " ^ quote name));
    97 
    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;
   104 
   105 
   106 
   107 (** monotonicity rules **)
   108 
   109 val get_monos = snd o InductiveData.get;
   110 fun put_monos thms thy = InductiveData.put (fst (InductiveData.get thy), thms) thy;
   111 
   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;
   126 
   127 (* mono add/del *)
   128 
   129 local
   130 
   131 fun map_rules_global f thy = put_monos (f (get_monos thy)) thy;
   132 
   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);
   135 
   136 fun mk_att f g (x, thm) = (f (g thm) x, thm);
   137 
   138 in
   139 
   140 val mono_add_global = mk_att map_rules_global add_mono;
   141 val mono_del_global = mk_att map_rules_global del_mono;
   142 
   143 end;
   144 
   145 
   146 (* concrete syntax *)
   147 
   148 val monoN = "mono";
   149 val addN = "add";
   150 val delN = "del";
   151 
   152 fun mono_att add del =
   153   Attrib.syntax (Scan.lift (Args.$$$ addN >> K add || Args.$$$ delN >> K del || Scan.succeed add));
   154 
   155 val mono_attr =
   156   (mono_att mono_add_global mono_del_global, mono_att Attrib.undef_local_attribute Attrib.undef_local_attribute);
   157 
   158 
   159 
   160 (** utilities **)
   161 
   162 (* messages *)
   163 
   164 val quiet_mode = ref false;
   165 fun message s = if !quiet_mode then () else writeln s;
   166 
   167 fun coind_prefix true = "co"
   168   | coind_prefix false = "";
   169 
   170 
   171 (* the following code ensures that each recursive set *)
   172 (* always has the same type in all introduction rules *)
   173 
   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))
   196 
   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));
   200 
   201 
   202 (* misc *)
   203 
   204 val Const _ $ (vimage_f $ _) $ _ = HOLogic.dest_Trueprop (concl_of vimageD);
   205 
   206 val vimage_name = Sign.intern_const (Theory.sign_of Vimage.thy) "op -``";
   207 val mono_name = Sign.intern_const (Theory.sign_of Ord.thy) "mono";
   208 
   209 (* make injections needed in mutually recursive definitions *)
   210 
   211 fun mk_inj cs sumT c x =
   212   let
   213     fun mk_inj' T n i =
   214       if n = 1 then x else
   215       let val n2 = n div 2;
   216           val Type (_, [T1, T2]) = T
   217       in
   218         if i <= n2 then
   219           Const ("Inl", T1 --> T) $ (mk_inj' T1 n2 i)
   220         else
   221           Const ("Inr", T2 --> T) $ (mk_inj' T2 (n - n2) (i - n2))
   222       end
   223   in mk_inj' sumT (length cs) (1 + find_index_eq c cs)
   224   end;
   225 
   226 (* make "vimage" terms for selecting out components of mutually rec.def. *)
   227 
   228 fun mk_vimage cs sumT t c = if length cs < 2 then t else
   229   let
   230     val cT = HOLogic.dest_setT (fastype_of c);
   231     val vimageT = [cT --> sumT, HOLogic.mk_setT sumT] ---> HOLogic.mk_setT cT
   232   in
   233     Const (vimage_name, vimageT) $
   234       Abs ("y", cT, mk_inj cs sumT c (Bound 0)) $ t
   235   end;
   236 
   237 
   238 
   239 (** well-formedness checks **)
   240 
   241 fun err_in_rule sign t msg = error ("Ill-formed introduction rule\n" ^
   242   (Sign.string_of_term sign t) ^ "\n" ^ msg);
   243 
   244 fun err_in_prem sign t p msg = error ("Ill-formed premise\n" ^
   245   (Sign.string_of_term sign p) ^ "\nin introduction rule\n" ^
   246   (Sign.string_of_term sign t) ^ "\n" ^ msg);
   247 
   248 val msg1 = "Conclusion of introduction rule must have form\
   249           \ ' t : S_i '";
   250 val msg2 = "Non-atomic premise";
   251 val msg3 = "Recursion term on left of member symbol";
   252 
   253 fun check_rule sign cs r =
   254   let
   255     fun check_prem prem = if can HOLogic.dest_Trueprop prem then ()
   256       else err_in_prem sign r prem msg2;
   257 
   258   in (case HOLogic.dest_Trueprop (Logic.strip_imp_concl r) of
   259         (Const ("op :", _) $ t $ u) =>
   260           if u mem cs then
   261             if exists (Logic.occs o (rpair t)) cs then
   262               err_in_rule sign r msg3
   263             else
   264               seq check_prem (Logic.strip_imp_prems r)
   265           else err_in_rule sign r msg1
   266       | _ => err_in_rule sign r msg1)
   267   end;
   268 
   269 fun try' f msg sign t = (case (try f t) of
   270       Some x => x
   271     | None => error (msg ^ Sign.string_of_term sign t));
   272 
   273 
   274 
   275 (*** properties of (co)inductive sets ***)
   276 
   277 (** elimination rules **)
   278 
   279 fun mk_elims cs cTs params intr_ts =
   280   let
   281     val used = foldr add_term_names (intr_ts, []);
   282     val [aname, pname] = variantlist (["a", "P"], used);
   283     val P = HOLogic.mk_Trueprop (Free (pname, HOLogic.boolT));
   284 
   285     fun dest_intr r =
   286       let val Const ("op :", _) $ t $ u =
   287         HOLogic.dest_Trueprop (Logic.strip_imp_concl r)
   288       in (u, t, Logic.strip_imp_prems r) end;
   289 
   290     val intrs = map dest_intr intr_ts;
   291 
   292     fun mk_elim (c, T) =
   293       let
   294         val a = Free (aname, T);
   295 
   296         fun mk_elim_prem (_, t, ts) =
   297           list_all_free (map dest_Free ((foldr add_term_frees (t::ts, [])) \\ params),
   298             Logic.list_implies (HOLogic.mk_Trueprop (HOLogic.mk_eq (a, t)) :: ts, P));
   299       in
   300         Logic.list_implies (HOLogic.mk_Trueprop (HOLogic.mk_mem (a, c)) ::
   301           map mk_elim_prem (filter (equal c o #1) intrs), P)
   302       end
   303   in
   304     map mk_elim (cs ~~ cTs)
   305   end;
   306         
   307 
   308 
   309 (** premises and conclusions of induction rules **)
   310 
   311 fun mk_indrule cs cTs params intr_ts =
   312   let
   313     val used = foldr add_term_names (intr_ts, []);
   314 
   315     (* predicates for induction rule *)
   316 
   317     val preds = map Free (variantlist (if length cs < 2 then ["P"] else
   318       map (fn i => "P" ^ string_of_int i) (1 upto length cs), used) ~~
   319         map (fn T => T --> HOLogic.boolT) cTs);
   320 
   321     (* transform an introduction rule into a premise for induction rule *)
   322 
   323     fun mk_ind_prem r =
   324       let
   325         val frees = map dest_Free ((add_term_frees (r, [])) \\ params);
   326 
   327         val pred_of = curry (Library.gen_assoc (op aconv)) (cs ~~ preds);
   328 
   329         fun subst (s as ((m as Const ("op :", T)) $ t $ u)) =
   330               (case pred_of u of
   331                   None => (m $ fst (subst t) $ fst (subst u), None)
   332                 | Some P => (HOLogic.conj $ s $ (P $ t), Some (s, P $ t)))
   333           | subst s =
   334               (case pred_of s of
   335                   Some P => (HOLogic.mk_binop "op Int"
   336                     (s, HOLogic.Collect_const (HOLogic.dest_setT
   337                       (fastype_of s)) $ P), None)
   338                 | None => (case s of
   339                      (t $ u) => (fst (subst t) $ fst (subst u), None)
   340                    | (Abs (a, T, t)) => (Abs (a, T, fst (subst t)), None)
   341                    | _ => (s, None)));
   342 
   343         fun mk_prem (s, prems) = (case subst s of
   344               (_, Some (t, u)) => t :: u :: prems
   345             | (t, _) => t :: prems);
   346           
   347         val Const ("op :", _) $ t $ u =
   348           HOLogic.dest_Trueprop (Logic.strip_imp_concl r)
   349 
   350       in list_all_free (frees,
   351            Logic.list_implies (map HOLogic.mk_Trueprop (foldr mk_prem
   352              (map HOLogic.dest_Trueprop (Logic.strip_imp_prems r), [])),
   353                HOLogic.mk_Trueprop (the (pred_of u) $ t)))
   354       end;
   355 
   356     val ind_prems = map mk_ind_prem intr_ts;
   357 
   358     (* make conclusions for induction rules *)
   359 
   360     fun mk_ind_concl ((c, P), (ts, x)) =
   361       let val T = HOLogic.dest_setT (fastype_of c);
   362           val Ts = HOLogic.prodT_factors T;
   363           val (frees, x') = foldr (fn (T', (fs, s)) =>
   364             ((Free (s, T'))::fs, bump_string s)) (Ts, ([], x));
   365           val tuple = HOLogic.mk_tuple T frees;
   366       in ((HOLogic.mk_binop "op -->"
   367         (HOLogic.mk_mem (tuple, c), P $ tuple))::ts, x')
   368       end;
   369 
   370     val mutual_ind_concl = HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
   371         (fst (foldr mk_ind_concl (cs ~~ preds, ([], "xa")))))
   372 
   373   in (preds, ind_prems, mutual_ind_concl)
   374   end;
   375 
   376 
   377 
   378 (** prepare cases and induct rules **)
   379 
   380 (*
   381   transform mutual rule:
   382     HH ==> (x1:A1 --> P1 x1) & ... & (xn:An --> Pn xn)
   383   into i-th projection:
   384     xi:Ai ==> HH ==> Pi xi
   385 *)
   386 
   387 fun project_rules [name] rule = [(name, rule)]
   388   | project_rules names mutual_rule =
   389       let
   390         val n = length names;
   391         fun proj i =
   392           (if i < n then (fn th => th RS conjunct1) else I)
   393             (Library.funpow (i - 1) (fn th => th RS conjunct2) mutual_rule)
   394             RS mp |> Thm.permute_prems 0 ~1 |> Drule.standard;
   395       in names ~~ map proj (1 upto n) end;
   396 
   397 fun add_cases_induct no_elim no_ind names elims induct =
   398   let
   399     val cases_specs =
   400       if no_elim then []
   401       else map2 (fn (name, elim) => (("", elim), [InductMethod.cases_set_global name]))
   402         (names, elims);
   403 
   404     val induct_specs =
   405       if no_ind then []
   406       else map (fn (name, th) => (("", th), [InductMethod.induct_set_global name]))
   407         (project_rules names induct);
   408   in PureThy.add_thms (cases_specs @ induct_specs) end;
   409 
   410 
   411 
   412 (*** proofs for (co)inductive sets ***)
   413 
   414 (** prove monotonicity **)
   415 
   416 fun prove_mono setT fp_fun monos thy =
   417   let
   418     val _ = message "  Proving monotonicity ...";
   419 
   420     val mono = prove_goalw_cterm [] (cterm_of (Theory.sign_of thy) (HOLogic.mk_Trueprop
   421       (Const (mono_name, (setT --> setT) --> HOLogic.boolT) $ fp_fun)))
   422         (fn _ => [rtac monoI 1, REPEAT (ares_tac (get_monos thy @ flat (map mk_mono monos)) 1)])
   423 
   424   in mono end;
   425 
   426 
   427 
   428 (** prove introduction rules **)
   429 
   430 fun prove_intrs coind mono fp_def intr_ts con_defs rec_sets_defs thy =
   431   let
   432     val _ = message "  Proving the introduction rules ...";
   433 
   434     val unfold = standard (mono RS (fp_def RS
   435       (if coind then def_gfp_Tarski else def_lfp_Tarski)));
   436 
   437     fun select_disj 1 1 = []
   438       | select_disj _ 1 = [rtac disjI1]
   439       | select_disj n i = (rtac disjI2)::(select_disj (n - 1) (i - 1));
   440 
   441     val intrs = map (fn (i, intr) => prove_goalw_cterm rec_sets_defs
   442       (cterm_of (Theory.sign_of thy) intr) (fn prems =>
   443        [(*insert prems and underlying sets*)
   444        cut_facts_tac prems 1,
   445        stac unfold 1,
   446        REPEAT (resolve_tac [vimageI2, CollectI] 1),
   447        (*Now 1-2 subgoals: the disjunction, perhaps equality.*)
   448        EVERY1 (select_disj (length intr_ts) i),
   449        (*Not ares_tac, since refl must be tried before any equality assumptions;
   450          backtracking may occur if the premises have extra variables!*)
   451        DEPTH_SOLVE_1 (resolve_tac [refl,exI,conjI] 1 APPEND assume_tac 1),
   452        (*Now solve the equations like Inl 0 = Inl ?b2*)
   453        rewrite_goals_tac con_defs,
   454        REPEAT (rtac refl 1)])) (1 upto (length intr_ts) ~~ intr_ts)
   455 
   456   in (intrs, unfold) end;
   457 
   458 
   459 
   460 (** prove elimination rules **)
   461 
   462 fun prove_elims cs cTs params intr_ts unfold rec_sets_defs thy =
   463   let
   464     val _ = message "  Proving the elimination rules ...";
   465 
   466     val rules1 = [CollectE, disjE, make_elim vimageD, exE];
   467     val rules2 = [conjE, Inl_neq_Inr, Inr_neq_Inl] @
   468       map make_elim [Inl_inject, Inr_inject];
   469 
   470     val elims = map (fn t => prove_goalw_cterm rec_sets_defs
   471       (cterm_of (Theory.sign_of thy) t) (fn prems =>
   472         [cut_facts_tac [hd prems] 1,
   473          dtac (unfold RS subst) 1,
   474          REPEAT (FIRSTGOAL (eresolve_tac rules1)),
   475          REPEAT (FIRSTGOAL (eresolve_tac rules2)),
   476          EVERY (map (fn prem =>
   477            DEPTH_SOLVE_1 (ares_tac [prem, conjI] 1)) (tl prems))]))
   478       (mk_elims cs cTs params intr_ts)
   479 
   480   in elims end;
   481 
   482 
   483 (** derivation of simplified elimination rules **)
   484 
   485 (*Applies freeness of the given constructors, which *must* be unfolded by
   486   the given defs.  Cannot simply use the local con_defs because con_defs=[] 
   487   for inference systems.
   488  *)
   489 
   490 (*cprop should have the form t:Si where Si is an inductive set*)
   491 fun mk_cases_i solved elims ss cprop =
   492   let
   493     val prem = Thm.assume cprop;
   494     val tac = if solved then InductMethod.con_elim_solved_tac else InductMethod.con_elim_tac;
   495     fun mk_elim rl = Drule.standard (Tactic.rule_by_tactic (tac ss) (prem RS rl));
   496   in
   497     (case get_first (try mk_elim) elims of
   498       Some r => r
   499     | None => error (Pretty.string_of (Pretty.block
   500         [Pretty.str "mk_cases: proposition not of form 't : S_i'", Pretty.fbrk,
   501           Display.pretty_cterm cprop])))
   502   end;
   503 
   504 fun mk_cases elims s =
   505   mk_cases_i false elims (simpset()) (Thm.read_cterm (Thm.sign_of_thm (hd elims)) (s, propT));
   506 
   507 
   508 (* inductive_cases(_i) *)
   509 
   510 fun gen_inductive_cases prep_att prep_const prep_prop
   511     ((((name, raw_atts), raw_set), raw_props), comment) thy =
   512   let
   513     val sign = Theory.sign_of thy;
   514 
   515     val atts = map (prep_att thy) raw_atts;
   516     val (_, {elims, ...}) = get_inductive thy (prep_const sign raw_set);
   517     val cprops = map (Thm.cterm_of sign o prep_prop (ProofContext.init thy)) raw_props;
   518     val thms = map (mk_cases_i true elims (Simplifier.simpset_of thy)) cprops;
   519   in
   520     thy
   521     |> IsarThy.have_theorems_i (((name, atts), map Thm.no_attributes thms), comment)
   522   end;
   523 
   524 val inductive_cases =
   525   gen_inductive_cases Attrib.global_attribute Sign.intern_const ProofContext.read_prop;
   526 
   527 val inductive_cases_i = gen_inductive_cases (K I) (K I) ProofContext.cert_prop;
   528 
   529 
   530 
   531 (** prove induction rule **)
   532 
   533 fun prove_indrule cs cTs sumT rec_const params intr_ts mono
   534     fp_def rec_sets_defs thy =
   535   let
   536     val _ = message "  Proving the induction rule ...";
   537 
   538     val sign = Theory.sign_of thy;
   539 
   540     val sum_case_rewrites = (case ThyInfo.lookup_theory "Datatype" of
   541         None => []
   542       | Some thy' => map mk_meta_eq (PureThy.get_thms thy' "sum.cases"));
   543 
   544     val (preds, ind_prems, mutual_ind_concl) = mk_indrule cs cTs params intr_ts;
   545 
   546     (* make predicate for instantiation of abstract induction rule *)
   547 
   548     fun mk_ind_pred _ [P] = P
   549       | mk_ind_pred T Ps =
   550          let val n = (length Ps) div 2;
   551              val Type (_, [T1, T2]) = T
   552          in Const ("Datatype.sum.sum_case",
   553            [T1 --> HOLogic.boolT, T2 --> HOLogic.boolT, T] ---> HOLogic.boolT) $
   554              mk_ind_pred T1 (take (n, Ps)) $ mk_ind_pred T2 (drop (n, Ps))
   555          end;
   556 
   557     val ind_pred = mk_ind_pred sumT preds;
   558 
   559     val ind_concl = HOLogic.mk_Trueprop
   560       (HOLogic.all_const sumT $ Abs ("x", sumT, HOLogic.mk_binop "op -->"
   561         (HOLogic.mk_mem (Bound 0, rec_const), ind_pred $ Bound 0)));
   562 
   563     (* simplification rules for vimage and Collect *)
   564 
   565     val vimage_simps = if length cs < 2 then [] else
   566       map (fn c => prove_goalw_cterm [] (cterm_of sign
   567         (HOLogic.mk_Trueprop (HOLogic.mk_eq
   568           (mk_vimage cs sumT (HOLogic.Collect_const sumT $ ind_pred) c,
   569            HOLogic.Collect_const (HOLogic.dest_setT (fastype_of c)) $
   570              nth_elem (find_index_eq c cs, preds)))))
   571         (fn _ => [rtac vimage_Collect 1, rewrite_goals_tac sum_case_rewrites,
   572           rtac refl 1])) cs;
   573 
   574     val induct = prove_goalw_cterm [] (cterm_of sign
   575       (Logic.list_implies (ind_prems, ind_concl))) (fn prems =>
   576         [rtac (impI RS allI) 1,
   577          DETERM (etac (mono RS (fp_def RS def_induct)) 1),
   578          rewrite_goals_tac (map mk_meta_eq (vimage_Int::Int_Collect::vimage_simps)),
   579          fold_goals_tac rec_sets_defs,
   580          (*This CollectE and disjE separates out the introduction rules*)
   581          REPEAT (FIRSTGOAL (eresolve_tac [CollectE, disjE, exE])),
   582          (*Now break down the individual cases.  No disjE here in case
   583            some premise involves disjunction.*)
   584          REPEAT (FIRSTGOAL (etac conjE ORELSE' hyp_subst_tac)),
   585          rewrite_goals_tac sum_case_rewrites,
   586          EVERY (map (fn prem =>
   587            DEPTH_SOLVE_1 (ares_tac [prem, conjI, refl] 1)) prems)]);
   588 
   589     val lemma = prove_goalw_cterm rec_sets_defs (cterm_of sign
   590       (Logic.mk_implies (ind_concl, mutual_ind_concl))) (fn prems =>
   591         [cut_facts_tac prems 1,
   592          REPEAT (EVERY
   593            [REPEAT (resolve_tac [conjI, impI] 1),
   594             TRY (dtac vimageD 1), etac allE 1, dtac mp 1, atac 1,
   595             rewrite_goals_tac sum_case_rewrites,
   596             atac 1])])
   597 
   598   in standard (split_rule (induct RS lemma))
   599   end;
   600 
   601 
   602 
   603 (*** specification of (co)inductive sets ****)
   604 
   605 (** definitional introduction of (co)inductive sets **)
   606 
   607 fun add_ind_def verbose declare_consts alt_name coind no_elim no_ind cs
   608     atts intros monos con_defs thy params paramTs cTs cnames =
   609   let
   610     val _ = if verbose then message ("Proofs for " ^ coind_prefix coind ^ "inductive set(s) " ^
   611       commas_quote cnames) else ();
   612 
   613     val sumT = fold_bal (fn (T, U) => Type ("+", [T, U])) cTs;
   614     val setT = HOLogic.mk_setT sumT;
   615 
   616     val fp_name = if coind then Sign.intern_const (Theory.sign_of Gfp.thy) "gfp"
   617       else Sign.intern_const (Theory.sign_of Lfp.thy) "lfp";
   618 
   619     val ((intr_names, intr_ts), intr_atts) = apfst split_list (split_list intros);
   620 
   621     val used = foldr add_term_names (intr_ts, []);
   622     val [sname, xname] = variantlist (["S", "x"], used);
   623 
   624     (* transform an introduction rule into a conjunction  *)
   625     (*   [| t : ... S_i ... ; ... |] ==> u : S_j          *)
   626     (* is transformed into                                *)
   627     (*   x = Inj_j u & t : ... Inj_i -`` S ... & ...      *)
   628 
   629     fun transform_rule r =
   630       let
   631         val frees = map dest_Free ((add_term_frees (r, [])) \\ params);
   632         val subst = subst_free
   633           (cs ~~ (map (mk_vimage cs sumT (Free (sname, setT))) cs));
   634         val Const ("op :", _) $ t $ u =
   635           HOLogic.dest_Trueprop (Logic.strip_imp_concl r)
   636 
   637       in foldr (fn ((x, T), P) => HOLogic.mk_exists (x, T, P))
   638         (frees, foldr1 HOLogic.mk_conj
   639           (((HOLogic.eq_const sumT) $ Free (xname, sumT) $ (mk_inj cs sumT u t))::
   640             (map (subst o HOLogic.dest_Trueprop)
   641               (Logic.strip_imp_prems r))))
   642       end
   643 
   644     (* make a disjunction of all introduction rules *)
   645 
   646     val fp_fun = absfree (sname, setT, (HOLogic.Collect_const sumT) $
   647       absfree (xname, sumT, foldr1 HOLogic.mk_disj (map transform_rule intr_ts)));
   648 
   649     (* add definiton of recursive sets to theory *)
   650 
   651     val rec_name = if alt_name = "" then space_implode "_" cnames else alt_name;
   652     val full_rec_name = Sign.full_name (Theory.sign_of thy) rec_name;
   653 
   654     val rec_const = list_comb
   655       (Const (full_rec_name, paramTs ---> setT), params);
   656 
   657     val fp_def_term = Logic.mk_equals (rec_const,
   658       Const (fp_name, (setT --> setT) --> setT) $ fp_fun)
   659 
   660     val def_terms = fp_def_term :: (if length cs < 2 then [] else
   661       map (fn c => Logic.mk_equals (c, mk_vimage cs sumT rec_const c)) cs);
   662 
   663     val thy' = thy |>
   664       (if declare_consts then
   665         Theory.add_consts_i (map (fn (c, n) =>
   666           (n, paramTs ---> fastype_of c, NoSyn)) (cs ~~ cnames))
   667        else I) |>
   668       (if length cs < 2 then I else
   669        Theory.add_consts_i [(rec_name, paramTs ---> setT, NoSyn)]) |>
   670       Theory.add_path rec_name |>
   671       PureThy.add_defss_i [(("defs", def_terms), [])];
   672 
   673     (* get definitions from theory *)
   674 
   675     val fp_def::rec_sets_defs = PureThy.get_thms thy' "defs";
   676 
   677     (* prove and store theorems *)
   678 
   679     val mono = prove_mono setT fp_fun monos thy';
   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 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));
   693 
   694     val thy'' = thy'
   695       |> PureThy.add_thmss [(("intrs", intrs), atts)]
   696       |> PureThy.add_thms ((intr_names ~~ intrs) ~~ intr_atts)
   697       |> (if no_elim then I else PureThy.add_thmss [(("elims", elims), [])])
   698       |> (if no_ind then I else PureThy.add_thms
   699         [((coind_prefix coind ^ "induct", induct), [])])
   700       |> Theory.parent_path;
   701     val intrs' = PureThy.get_thms thy'' "intrs";
   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;
   714 
   715 
   716 
   717 (** axiomatic introduction of (co)inductive sets **)
   718 
   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 =
   721   let
   722     val rec_name = if alt_name = "" then space_implode "_" cnames else alt_name;
   723 
   724     val ((intr_names, intr_ts), intr_atts) = apfst split_list (split_list intros);
   725     val elim_ts = mk_elims cs cTs params intr_ts;
   726 
   727     val (_, ind_prems, mutual_ind_concl) = mk_indrule cs cTs params intr_ts;
   728     val ind_t = Logic.list_implies (ind_prems, mutual_ind_concl);
   729     
   730     val thy' = thy
   731       |> (if declare_consts then
   732             Theory.add_consts_i
   733               (map (fn (c, n) => (n, paramTs ---> fastype_of c, NoSyn)) (cs ~~ cnames))
   734          else I)
   735       |> Theory.add_path rec_name
   736       |> PureThy.add_axiomss_i [(("intrs", intr_ts), atts), (("elims", elim_ts), [])]
   737       |> (if coind then I else
   738             PureThy.add_axioms_i [(("raw_induct", ind_t), [apsnd (standard o split_rule)])]);
   739 
   740     val intrs = PureThy.get_thms thy' "intrs";
   741     val elims = PureThy.get_thms thy' "elims";
   742     val raw_induct = if coind then Drule.asm_rl else PureThy.get_thm thy' "raw_induct";
   743     val induct = if coind orelse length cs > 1 then raw_induct
   744       else standard (raw_induct RSN (2, rev_mp));
   745 
   746     val thy'' =
   747       thy'
   748       |> (if coind then I else PureThy.add_thms [(("induct", induct), [])])
   749       |> PureThy.add_thms ((intr_names ~~ intrs) ~~ intr_atts)
   750       |> Theory.parent_path;
   751     val induct' = if coind then raw_induct else PureThy.get_thm thy'' "induct";
   752   in (thy'',
   753     {defs = [],
   754      mono = Drule.asm_rl,
   755      unfold = Drule.asm_rl,
   756      intrs = intrs,
   757      elims = elims,
   758      mk_cases = mk_cases elims,
   759      raw_induct = raw_induct,
   760      induct = induct'})
   761   end;
   762 
   763 
   764 
   765 (** introduction of (co)inductive sets **)
   766 
   767 fun add_inductive_i verbose declare_consts alt_name coind no_elim no_ind cs
   768     atts intros monos con_defs thy =
   769   let
   770     val _ = Theory.requires thy "Inductive" (coind_prefix coind ^ "inductive definitions");
   771     val sign = Theory.sign_of thy;
   772 
   773     (*parameters should agree for all mutually recursive components*)
   774     val (_, params) = strip_comb (hd cs);
   775     val paramTs = map (try' (snd o dest_Free) "Parameter in recursive\
   776       \ component is not a free variable: " sign) params;
   777 
   778     val cTs = map (try' (HOLogic.dest_setT o fastype_of)
   779       "Recursive component not of type set: " sign) cs;
   780 
   781     val full_cnames = map (try' (fst o dest_Const o head_of)
   782       "Recursive set not previously declared as constant: " sign) cs;
   783     val cnames = map Sign.base_name full_cnames;
   784 
   785     val _ = seq (check_rule sign cs o snd o fst) intros;
   786 
   787     val (thy1, result) =
   788       (if ! quick_and_dirty then add_ind_axm else add_ind_def)
   789         verbose declare_consts alt_name coind no_elim no_ind cs atts intros monos
   790         con_defs thy params paramTs cTs cnames;
   791     val thy2 = thy1
   792       |> put_inductives full_cnames ({names = full_cnames, coind = coind}, result)
   793       |> add_cases_induct no_elim no_ind full_cnames (#elims result) (#induct result);
   794   in (thy2, result) end;
   795 
   796 
   797 
   798 (** external interface **)
   799 
   800 fun add_inductive verbose coind c_strings srcs intro_srcs raw_monos raw_con_defs thy =
   801   let
   802     val sign = Theory.sign_of thy;
   803     val cs = map (term_of o Thm.read_cterm sign o rpair HOLogic.termT) c_strings;
   804 
   805     val atts = map (Attrib.global_attribute thy) srcs;
   806     val intr_names = map (fst o fst) intro_srcs;
   807     val intr_ts = map (term_of o Thm.read_cterm sign o rpair propT o snd o fst) intro_srcs;
   808     val intr_atts = map (map (Attrib.global_attribute thy) o snd) intro_srcs;
   809     val (cs', intr_ts') = unify_consts sign cs intr_ts;
   810 
   811     val ((thy', con_defs), monos) = thy
   812       |> IsarThy.apply_theorems raw_monos
   813       |> apfst (IsarThy.apply_theorems raw_con_defs);
   814   in
   815     add_inductive_i verbose false "" coind false false cs'
   816       atts ((intr_names ~~ intr_ts') ~~ intr_atts) monos con_defs thy'
   817   end;
   818 
   819 
   820 
   821 (** package setup **)
   822 
   823 (* setup theory *)
   824 
   825 val setup = [InductiveData.init,
   826              Attrib.add_attributes [(monoN, 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;