src/HOL/Tools/inductive_package.ML
author wenzelm
Tue Feb 29 23:06:20 2000 +0100 (2000-02-29)
changeset 8316 74639e19eca0
parent 8312 b470bc28b59d
child 8336 fdf3ac335f77
permissions -rw-r--r--
add_cases_induct: project_rules accomodates mutual induction;
     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 (*Delete needless equality assumptions*)
   207 val refl_thin = prove_goal HOL.thy "!!P. [| a=a;  P |] ==> P"
   208      (fn _ => [assume_tac 1]);
   209 
   210 (*For simplifying the elimination rule*)
   211 val elim_rls = [asm_rl, FalseE, refl_thin, conjE, exE, Pair_inject];
   212 
   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";
   215 
   216 (* make injections needed in mutually recursive definitions *)
   217 
   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;
   232 
   233 (* make "vimage" terms for selecting out components of mutually rec.def. *)
   234 
   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;
   243 
   244 
   245 
   246 (** well-formedness checks **)
   247 
   248 fun err_in_rule sign t msg = error ("Ill-formed introduction rule\n" ^
   249   (Sign.string_of_term sign t) ^ "\n" ^ msg);
   250 
   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);
   254 
   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";
   259 
   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;
   264 
   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;
   275 
   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));
   279 
   280 
   281 
   282 (*** properties of (co)inductive sets ***)
   283 
   284 (** elimination rules **)
   285 
   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));
   291 
   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;
   296 
   297     val intrs = map dest_intr intr_ts;
   298 
   299     fun mk_elim (c, T) =
   300       let
   301         val a = Free (aname, T);
   302 
   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;
   313         
   314 
   315 
   316 (** premises and conclusions of induction rules **)
   317 
   318 fun mk_indrule cs cTs params intr_ts =
   319   let
   320     val used = foldr add_term_names (intr_ts, []);
   321 
   322     (* predicates for induction rule *)
   323 
   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);
   327 
   328     (* transform an introduction rule into a premise for induction rule *)
   329 
   330     fun mk_ind_prem r =
   331       let
   332         val frees = map dest_Free ((add_term_frees (r, [])) \\ params);
   333 
   334         val pred_of = curry (Library.gen_assoc (op aconv)) (cs ~~ preds);
   335 
   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)));
   349 
   350         fun mk_prem (s, prems) = (case subst s of
   351               (_, Some (t, u)) => t :: u :: prems
   352             | (t, _) => t :: prems);
   353           
   354         val Const ("op :", _) $ t $ u =
   355           HOLogic.dest_Trueprop (Logic.strip_imp_concl r)
   356 
   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;
   362 
   363     val ind_prems = map mk_ind_prem intr_ts;
   364 
   365     (* make conclusions for induction rules *)
   366 
   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;
   376 
   377     val mutual_ind_concl = HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
   378         (fst (foldr mk_ind_concl (cs ~~ preds, ([], "xa")))))
   379 
   380   in (preds, ind_prems, mutual_ind_concl)
   381   end;
   382 
   383 
   384 
   385 (** prepare cases and induct rules **)
   386 
   387 (*
   388   transform mutual rule:
   389     HH ==> (x1:A1 --> P1 x1) & ... & (xn:An --> Pn xn)
   390   into i-th projection:
   391     xi:Ai ==> HH ==> Pi xi
   392 *)
   393 
   394 fun project_rules [name] rule = [(name, rule)]
   395   | project_rules names mutual_rule =
   396       let
   397         val n = length names;
   398         fun proj i =
   399           (if i < n then (fn th => th RS conjunct1) else I)
   400             (Library.funpow (i - 1) (fn th => th RS conjunct2) mutual_rule)
   401             RS mp |> Thm.permute_prems 0 ~1 |> Drule.standard;
   402       in names ~~ map proj (1 upto n) end;
   403 
   404 fun add_cases_induct no_elim no_ind names elims induct =
   405   let
   406     val cases_specs =
   407       if no_elim then []
   408       else map2 (fn (name, elim) => (("", elim), [InductMethod.cases_set_global name]))
   409         (names, elims);
   410 
   411     val induct_specs =
   412       if no_ind then []
   413       else map (fn (name, th) => (("", th), [InductMethod.induct_set_global name]))
   414         (project_rules names induct);
   415   in PureThy.add_thms (cases_specs @ induct_specs) end;
   416 
   417 
   418 
   419 (*** proofs for (co)inductive sets ***)
   420 
   421 (** prove monotonicity **)
   422 
   423 fun prove_mono setT fp_fun monos thy =
   424   let
   425     val _ = message "  Proving monotonicity ...";
   426 
   427     val mono = prove_goalw_cterm [] (cterm_of (Theory.sign_of thy) (HOLogic.mk_Trueprop
   428       (Const (mono_name, (setT --> setT) --> HOLogic.boolT) $ fp_fun)))
   429         (fn _ => [rtac monoI 1, REPEAT (ares_tac (get_monos thy @ flat (map mk_mono monos)) 1)])
   430 
   431   in mono end;
   432 
   433 
   434 
   435 (** prove introduction rules **)
   436 
   437 fun prove_intrs coind mono fp_def intr_ts con_defs rec_sets_defs thy =
   438   let
   439     val _ = message "  Proving the introduction rules ...";
   440 
   441     val unfold = standard (mono RS (fp_def RS
   442       (if coind then def_gfp_Tarski else def_lfp_Tarski)));
   443 
   444     fun select_disj 1 1 = []
   445       | select_disj _ 1 = [rtac disjI1]
   446       | select_disj n i = (rtac disjI2)::(select_disj (n - 1) (i - 1));
   447 
   448     val intrs = map (fn (i, intr) => prove_goalw_cterm rec_sets_defs
   449       (cterm_of (Theory.sign_of thy) intr) (fn prems =>
   450        [(*insert prems and underlying sets*)
   451        cut_facts_tac prems 1,
   452        stac unfold 1,
   453        REPEAT (resolve_tac [vimageI2, CollectI] 1),
   454        (*Now 1-2 subgoals: the disjunction, perhaps equality.*)
   455        EVERY1 (select_disj (length intr_ts) i),
   456        (*Not ares_tac, since refl must be tried before any equality assumptions;
   457          backtracking may occur if the premises have extra variables!*)
   458        DEPTH_SOLVE_1 (resolve_tac [refl,exI,conjI] 1 APPEND assume_tac 1),
   459        (*Now solve the equations like Inl 0 = Inl ?b2*)
   460        rewrite_goals_tac con_defs,
   461        REPEAT (rtac refl 1)])) (1 upto (length intr_ts) ~~ intr_ts)
   462 
   463   in (intrs, unfold) end;
   464 
   465 
   466 
   467 (** prove elimination rules **)
   468 
   469 fun prove_elims cs cTs params intr_ts unfold rec_sets_defs thy =
   470   let
   471     val _ = message "  Proving the elimination rules ...";
   472 
   473     val rules1 = [CollectE, disjE, make_elim vimageD, exE];
   474     val rules2 = [conjE, Inl_neq_Inr, Inr_neq_Inl] @
   475       map make_elim [Inl_inject, Inr_inject];
   476 
   477     val elims = map (fn t => prove_goalw_cterm rec_sets_defs
   478       (cterm_of (Theory.sign_of thy) t) (fn prems =>
   479         [cut_facts_tac [hd prems] 1,
   480          dtac (unfold RS subst) 1,
   481          REPEAT (FIRSTGOAL (eresolve_tac rules1)),
   482          REPEAT (FIRSTGOAL (eresolve_tac rules2)),
   483          EVERY (map (fn prem =>
   484            DEPTH_SOLVE_1 (ares_tac [prem, conjI] 1)) (tl prems))]))
   485       (mk_elims cs cTs params intr_ts)
   486 
   487   in elims end;
   488 
   489 
   490 (** derivation of simplified elimination rules **)
   491 
   492 (*Applies freeness of the given constructors, which *must* be unfolded by
   493   the given defs.  Cannot simply use the local con_defs because con_defs=[] 
   494   for inference systems.
   495  *)
   496 fun con_elim_tac ss =
   497   let val elim_tac = REPEAT o (eresolve_tac elim_rls)
   498   in ALLGOALS(EVERY'[elim_tac,
   499 		     asm_full_simp_tac ss,
   500 		     elim_tac,
   501 		     REPEAT o bound_hyp_subst_tac])
   502      THEN prune_params_tac
   503   end;
   504 
   505 (*cprop should have the form t:Si where Si is an inductive set*)
   506 fun mk_cases_i elims ss cprop =
   507   let
   508     val prem = Thm.assume cprop;
   509     fun mk_elim rl = standard (rule_by_tactic (con_elim_tac ss) (prem RS rl));
   510   in
   511     (case get_first (try mk_elim) elims of
   512       Some r => r
   513     | None => error (Pretty.string_of (Pretty.block
   514         [Pretty.str "mk_cases: proposition not of form 't : S_i'", Pretty.fbrk,
   515           Display.pretty_cterm cprop])))
   516   end;
   517 
   518 fun mk_cases elims s =
   519   mk_cases_i elims (simpset()) (Thm.read_cterm (Thm.sign_of_thm (hd elims)) (s, propT));
   520 
   521 
   522 (* inductive_cases(_i) *)
   523 
   524 fun gen_inductive_cases prep_att prep_const prep_prop
   525     ((((name, raw_atts), raw_set), raw_props), comment) thy =
   526   let
   527     val sign = Theory.sign_of thy;
   528 
   529     val atts = map (prep_att thy) raw_atts;
   530     val (_, {elims, ...}) = get_inductive thy (prep_const sign raw_set);
   531     val cprops = map (Thm.cterm_of sign o prep_prop (ProofContext.init thy)) raw_props;
   532     val thms = map (mk_cases_i elims (Simplifier.simpset_of thy)) cprops;
   533   in
   534     thy
   535     |> IsarThy.have_theorems_i (((name, atts), map Thm.no_attributes thms), comment)
   536   end;
   537 
   538 val inductive_cases =
   539   gen_inductive_cases Attrib.global_attribute Sign.intern_const ProofContext.read_prop;
   540 
   541 val inductive_cases_i = gen_inductive_cases (K I) (K I) ProofContext.cert_prop;
   542 
   543 
   544 
   545 (** prove induction rule **)
   546 
   547 fun prove_indrule cs cTs sumT rec_const params intr_ts mono
   548     fp_def rec_sets_defs thy =
   549   let
   550     val _ = message "  Proving the induction rule ...";
   551 
   552     val sign = Theory.sign_of thy;
   553 
   554     val sum_case_rewrites = (case ThyInfo.lookup_theory "Datatype" of
   555         None => []
   556       | Some thy' => map mk_meta_eq (PureThy.get_thms thy' "sum.cases"));
   557 
   558     val (preds, ind_prems, mutual_ind_concl) = mk_indrule cs cTs params intr_ts;
   559 
   560     (* make predicate for instantiation of abstract induction rule *)
   561 
   562     fun mk_ind_pred _ [P] = P
   563       | mk_ind_pred T Ps =
   564          let val n = (length Ps) div 2;
   565              val Type (_, [T1, T2]) = T
   566          in Const ("Datatype.sum.sum_case",
   567            [T1 --> HOLogic.boolT, T2 --> HOLogic.boolT, T] ---> HOLogic.boolT) $
   568              mk_ind_pred T1 (take (n, Ps)) $ mk_ind_pred T2 (drop (n, Ps))
   569          end;
   570 
   571     val ind_pred = mk_ind_pred sumT preds;
   572 
   573     val ind_concl = HOLogic.mk_Trueprop
   574       (HOLogic.all_const sumT $ Abs ("x", sumT, HOLogic.mk_binop "op -->"
   575         (HOLogic.mk_mem (Bound 0, rec_const), ind_pred $ Bound 0)));
   576 
   577     (* simplification rules for vimage and Collect *)
   578 
   579     val vimage_simps = if length cs < 2 then [] else
   580       map (fn c => prove_goalw_cterm [] (cterm_of sign
   581         (HOLogic.mk_Trueprop (HOLogic.mk_eq
   582           (mk_vimage cs sumT (HOLogic.Collect_const sumT $ ind_pred) c,
   583            HOLogic.Collect_const (HOLogic.dest_setT (fastype_of c)) $
   584              nth_elem (find_index_eq c cs, preds)))))
   585         (fn _ => [rtac vimage_Collect 1, rewrite_goals_tac sum_case_rewrites,
   586           rtac refl 1])) cs;
   587 
   588     val induct = prove_goalw_cterm [] (cterm_of sign
   589       (Logic.list_implies (ind_prems, ind_concl))) (fn prems =>
   590         [rtac (impI RS allI) 1,
   591          DETERM (etac (mono RS (fp_def RS def_induct)) 1),
   592          rewrite_goals_tac (map mk_meta_eq (vimage_Int::Int_Collect::vimage_simps)),
   593          fold_goals_tac rec_sets_defs,
   594          (*This CollectE and disjE separates out the introduction rules*)
   595          REPEAT (FIRSTGOAL (eresolve_tac [CollectE, disjE, exE])),
   596          (*Now break down the individual cases.  No disjE here in case
   597            some premise involves disjunction.*)
   598          REPEAT (FIRSTGOAL (etac conjE ORELSE' hyp_subst_tac)),
   599          rewrite_goals_tac sum_case_rewrites,
   600          EVERY (map (fn prem =>
   601            DEPTH_SOLVE_1 (ares_tac [prem, conjI, refl] 1)) prems)]);
   602 
   603     val lemma = prove_goalw_cterm rec_sets_defs (cterm_of sign
   604       (Logic.mk_implies (ind_concl, mutual_ind_concl))) (fn prems =>
   605         [cut_facts_tac prems 1,
   606          REPEAT (EVERY
   607            [REPEAT (resolve_tac [conjI, impI] 1),
   608             TRY (dtac vimageD 1), etac allE 1, dtac mp 1, atac 1,
   609             rewrite_goals_tac sum_case_rewrites,
   610             atac 1])])
   611 
   612   in standard (split_rule (induct RS lemma))
   613   end;
   614 
   615 
   616 
   617 (*** specification of (co)inductive sets ****)
   618 
   619 (** definitional introduction of (co)inductive sets **)
   620 
   621 fun add_ind_def verbose declare_consts alt_name coind no_elim no_ind cs
   622     atts intros monos con_defs thy params paramTs cTs cnames =
   623   let
   624     val _ = if verbose then message ("Proofs for " ^ coind_prefix coind ^ "inductive set(s) " ^
   625       commas_quote cnames) else ();
   626 
   627     val sumT = fold_bal (fn (T, U) => Type ("+", [T, U])) cTs;
   628     val setT = HOLogic.mk_setT sumT;
   629 
   630     val fp_name = if coind then Sign.intern_const (Theory.sign_of Gfp.thy) "gfp"
   631       else Sign.intern_const (Theory.sign_of Lfp.thy) "lfp";
   632 
   633     val ((intr_names, intr_ts), intr_atts) = apfst split_list (split_list intros);
   634 
   635     val used = foldr add_term_names (intr_ts, []);
   636     val [sname, xname] = variantlist (["S", "x"], used);
   637 
   638     (* transform an introduction rule into a conjunction  *)
   639     (*   [| t : ... S_i ... ; ... |] ==> u : S_j          *)
   640     (* is transformed into                                *)
   641     (*   x = Inj_j u & t : ... Inj_i -`` S ... & ...      *)
   642 
   643     fun transform_rule r =
   644       let
   645         val frees = map dest_Free ((add_term_frees (r, [])) \\ params);
   646         val subst = subst_free
   647           (cs ~~ (map (mk_vimage cs sumT (Free (sname, setT))) cs));
   648         val Const ("op :", _) $ t $ u =
   649           HOLogic.dest_Trueprop (Logic.strip_imp_concl r)
   650 
   651       in foldr (fn ((x, T), P) => HOLogic.mk_exists (x, T, P))
   652         (frees, foldr1 HOLogic.mk_conj
   653           (((HOLogic.eq_const sumT) $ Free (xname, sumT) $ (mk_inj cs sumT u t))::
   654             (map (subst o HOLogic.dest_Trueprop)
   655               (Logic.strip_imp_prems r))))
   656       end
   657 
   658     (* make a disjunction of all introduction rules *)
   659 
   660     val fp_fun = absfree (sname, setT, (HOLogic.Collect_const sumT) $
   661       absfree (xname, sumT, foldr1 HOLogic.mk_disj (map transform_rule intr_ts)));
   662 
   663     (* add definiton of recursive sets to theory *)
   664 
   665     val rec_name = if alt_name = "" then space_implode "_" cnames else alt_name;
   666     val full_rec_name = Sign.full_name (Theory.sign_of thy) rec_name;
   667 
   668     val rec_const = list_comb
   669       (Const (full_rec_name, paramTs ---> setT), params);
   670 
   671     val fp_def_term = Logic.mk_equals (rec_const,
   672       Const (fp_name, (setT --> setT) --> setT) $ fp_fun)
   673 
   674     val def_terms = fp_def_term :: (if length cs < 2 then [] else
   675       map (fn c => Logic.mk_equals (c, mk_vimage cs sumT rec_const c)) cs);
   676 
   677     val thy' = thy |>
   678       (if declare_consts then
   679         Theory.add_consts_i (map (fn (c, n) =>
   680           (n, paramTs ---> fastype_of c, NoSyn)) (cs ~~ cnames))
   681        else I) |>
   682       (if length cs < 2 then I else
   683        Theory.add_consts_i [(rec_name, paramTs ---> setT, NoSyn)]) |>
   684       Theory.add_path rec_name |>
   685       PureThy.add_defss_i [(("defs", def_terms), [])];
   686 
   687     (* get definitions from theory *)
   688 
   689     val fp_def::rec_sets_defs = PureThy.get_thms thy' "defs";
   690 
   691     (* prove and store theorems *)
   692 
   693     val mono = prove_mono setT fp_fun monos thy';
   694     val (intrs, unfold) = prove_intrs coind mono fp_def intr_ts con_defs
   695       rec_sets_defs thy';
   696     val elims = if no_elim then [] else
   697       prove_elims cs cTs params intr_ts unfold rec_sets_defs thy';
   698     val raw_induct = if no_ind then Drule.asm_rl else
   699       if coind then standard (rule_by_tactic
   700         (rewrite_tac [mk_meta_eq vimage_Un] THEN
   701           fold_tac rec_sets_defs) (mono RS (fp_def RS def_Collect_coinduct)))
   702       else
   703         prove_indrule cs cTs sumT rec_const params intr_ts mono fp_def
   704           rec_sets_defs thy';
   705     val induct = if coind orelse no_ind orelse length cs > 1 then raw_induct
   706       else standard (raw_induct RSN (2, rev_mp));
   707 
   708     val thy'' = thy'
   709       |> PureThy.add_thmss [(("intrs", intrs), atts)]
   710       |> PureThy.add_thms ((intr_names ~~ intrs) ~~ intr_atts)
   711       |> (if no_elim then I else PureThy.add_thmss [(("elims", elims), [])])
   712       |> (if no_ind then I else PureThy.add_thms
   713         [((coind_prefix coind ^ "induct", induct), [])])
   714       |> Theory.parent_path;
   715     val intrs' = PureThy.get_thms thy'' "intrs";
   716     val elims' = if no_elim then elims else PureThy.get_thms thy'' "elims";  (* FIXME improve *)
   717     val induct' = if no_ind then induct else PureThy.get_thm thy'' (coind_prefix coind ^ "induct");  (* FIXME improve *)
   718   in (thy'',
   719     {defs = fp_def::rec_sets_defs,
   720      mono = mono,
   721      unfold = unfold,
   722      intrs = intrs',
   723      elims = elims',
   724      mk_cases = mk_cases elims',
   725      raw_induct = raw_induct,
   726      induct = induct'})
   727   end;
   728 
   729 
   730 
   731 (** axiomatic introduction of (co)inductive sets **)
   732 
   733 fun add_ind_axm verbose declare_consts alt_name coind no_elim no_ind cs
   734     atts intros monos con_defs thy params paramTs cTs cnames =
   735   let
   736     val rec_name = if alt_name = "" then space_implode "_" cnames else alt_name;
   737 
   738     val ((intr_names, intr_ts), intr_atts) = apfst split_list (split_list intros);
   739     val elim_ts = mk_elims cs cTs params intr_ts;
   740 
   741     val (_, ind_prems, mutual_ind_concl) = mk_indrule cs cTs params intr_ts;
   742     val ind_t = Logic.list_implies (ind_prems, mutual_ind_concl);
   743     
   744     val thy' = thy
   745       |> (if declare_consts then
   746             Theory.add_consts_i
   747               (map (fn (c, n) => (n, paramTs ---> fastype_of c, NoSyn)) (cs ~~ cnames))
   748          else I)
   749       |> Theory.add_path rec_name
   750       |> PureThy.add_axiomss_i [(("intrs", intr_ts), atts), (("elims", elim_ts), [])]
   751       |> (if coind then I else
   752             PureThy.add_axioms_i [(("raw_induct", ind_t), [apsnd (standard o split_rule)])]);
   753 
   754     val intrs = PureThy.get_thms thy' "intrs";
   755     val elims = PureThy.get_thms thy' "elims";
   756     val raw_induct = if coind then Drule.asm_rl else PureThy.get_thm thy' "raw_induct";
   757     val induct = if coind orelse length cs > 1 then raw_induct
   758       else standard (raw_induct RSN (2, rev_mp));
   759 
   760     val thy'' =
   761       thy'
   762       |> (if coind then I else PureThy.add_thms [(("induct", induct), [])])
   763       |> PureThy.add_thms ((intr_names ~~ intrs) ~~ intr_atts)
   764       |> Theory.parent_path;
   765     val induct' = if coind then raw_induct else PureThy.get_thm thy'' "induct";
   766   in (thy'',
   767     {defs = [],
   768      mono = Drule.asm_rl,
   769      unfold = Drule.asm_rl,
   770      intrs = intrs,
   771      elims = elims,
   772      mk_cases = mk_cases elims,
   773      raw_induct = raw_induct,
   774      induct = induct'})
   775   end;
   776 
   777 
   778 
   779 (** introduction of (co)inductive sets **)
   780 
   781 fun add_inductive_i verbose declare_consts alt_name coind no_elim no_ind cs
   782     atts intros monos con_defs thy =
   783   let
   784     val _ = Theory.requires thy "Inductive" (coind_prefix coind ^ "inductive definitions");
   785     val sign = Theory.sign_of thy;
   786 
   787     (*parameters should agree for all mutually recursive components*)
   788     val (_, params) = strip_comb (hd cs);
   789     val paramTs = map (try' (snd o dest_Free) "Parameter in recursive\
   790       \ component is not a free variable: " sign) params;
   791 
   792     val cTs = map (try' (HOLogic.dest_setT o fastype_of)
   793       "Recursive component not of type set: " sign) cs;
   794 
   795     val full_cnames = map (try' (fst o dest_Const o head_of)
   796       "Recursive set not previously declared as constant: " sign) cs;
   797     val cnames = map Sign.base_name full_cnames;
   798 
   799     val _ = seq (check_rule sign cs o snd o fst) intros;
   800 
   801     val (thy1, result) =
   802       (if ! quick_and_dirty then add_ind_axm else add_ind_def)
   803         verbose declare_consts alt_name coind no_elim no_ind cs atts intros monos
   804         con_defs thy params paramTs cTs cnames;
   805     val thy2 = thy1
   806       |> put_inductives full_cnames ({names = full_cnames, coind = coind}, result)
   807       |> add_cases_induct no_elim no_ind full_cnames (#elims result) (#induct result);
   808   in (thy2, result) end;
   809 
   810 
   811 
   812 (** external interface **)
   813 
   814 fun add_inductive verbose coind c_strings srcs intro_srcs raw_monos raw_con_defs thy =
   815   let
   816     val sign = Theory.sign_of thy;
   817     val cs = map (term_of o Thm.read_cterm sign o rpair HOLogic.termT) c_strings;
   818 
   819     val atts = map (Attrib.global_attribute thy) srcs;
   820     val intr_names = map (fst o fst) intro_srcs;
   821     val intr_ts = map (term_of o Thm.read_cterm sign o rpair propT o snd o fst) intro_srcs;
   822     val intr_atts = map (map (Attrib.global_attribute thy) o snd) intro_srcs;
   823     val (cs', intr_ts') = unify_consts sign cs intr_ts;
   824 
   825     val ((thy', con_defs), monos) = thy
   826       |> IsarThy.apply_theorems raw_monos
   827       |> apfst (IsarThy.apply_theorems raw_con_defs);
   828   in
   829     add_inductive_i verbose false "" coind false false cs'
   830       atts ((intr_names ~~ intr_ts') ~~ intr_atts) monos con_defs thy'
   831   end;
   832 
   833 
   834 
   835 (** package setup **)
   836 
   837 (* setup theory *)
   838 
   839 val setup = [InductiveData.init,
   840              Attrib.add_attributes [(monoN, mono_attr, "monotonicity rule")]];
   841 
   842 
   843 (* outer syntax *)
   844 
   845 local structure P = OuterParse and K = OuterSyntax.Keyword in
   846 
   847 fun mk_ind coind (((sets, (atts, intrs)), monos), con_defs) =
   848   #1 o add_inductive true coind sets atts (map P.triple_swap intrs) monos con_defs;
   849 
   850 fun ind_decl coind =
   851   (Scan.repeat1 P.term --| P.marg_comment) --
   852   (P.$$$ "intrs" |--
   853     P.!!! (P.opt_attribs -- Scan.repeat1 (P.opt_thm_name ":" -- P.prop --| P.marg_comment))) --
   854   Scan.optional (P.$$$ "monos" |-- P.!!! P.xthms1 --| P.marg_comment) [] --
   855   Scan.optional (P.$$$ "con_defs" |-- P.!!! P.xthms1 --| P.marg_comment) []
   856   >> (Toplevel.theory o mk_ind coind);
   857 
   858 val inductiveP =
   859   OuterSyntax.command "inductive" "define inductive sets" K.thy_decl (ind_decl false);
   860 
   861 val coinductiveP =
   862   OuterSyntax.command "coinductive" "define coinductive sets" K.thy_decl (ind_decl true);
   863 
   864 
   865 val ind_cases =
   866   P.opt_thm_name "=" -- P.xname --| P.$$$ ":" -- Scan.repeat1 P.prop -- P.marg_comment
   867   >> (Toplevel.theory o inductive_cases);
   868 
   869 val inductive_casesP =
   870   OuterSyntax.command "inductive_cases" "create simplified instances of elimination rules"
   871     K.thy_decl ind_cases;
   872 
   873 val _ = OuterSyntax.add_keywords ["intrs", "monos", "con_defs"];
   874 val _ = OuterSyntax.add_parsers [inductiveP, coinductiveP, inductive_casesP];
   875 
   876 end;
   877 
   878 
   879 end;