src/HOL/Tools/inductive_package.ML
author wenzelm
Wed Jun 18 18:55:00 2008 +0200 (2008-06-18)
changeset 27252 042aebff17e1
parent 26988 742e26213212
child 27353 71c4dd53d4cb
permissions -rw-r--r--
OldGoals.read_prop;
     1 (*  Title:      HOL/Tools/inductive_package.ML
     2     ID:         $Id$
     3     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
     4     Author:     Stefan Berghofer and Markus Wenzel, TU Muenchen
     5 
     6 (Co)Inductive Definition module for HOL.
     7 
     8 Features:
     9   * least or greatest fixedpoints
    10   * mutually recursive definitions
    11   * definitions involving arbitrary monotone operators
    12   * automatically proves introduction and elimination rules
    13 
    14   Introduction rules have the form
    15   [| M Pj ti, ..., Q x, ... |] ==> Pk t
    16   where M is some monotone operator (usually the identity)
    17   Q x is any side condition on the free variables
    18   ti, t are any terms
    19   Pj, Pk are two of the predicates being defined in mutual recursion
    20 *)
    21 
    22 signature BASIC_INDUCTIVE_PACKAGE =
    23 sig
    24   type inductive_result
    25   val morph_result: morphism -> inductive_result -> inductive_result
    26   type inductive_info
    27   val the_inductive: Proof.context -> string -> inductive_info
    28   val print_inductives: Proof.context -> unit
    29   val mono_add: attribute
    30   val mono_del: attribute
    31   val get_monos: Proof.context -> thm list
    32   val mk_cases: Proof.context -> term -> thm
    33   val inductive_forall_name: string
    34   val inductive_forall_def: thm
    35   val rulify: thm -> thm
    36   val inductive_cases: ((bstring * Attrib.src list) * string list) list ->
    37     Proof.context -> thm list list * local_theory
    38   val inductive_cases_i: ((bstring * Attrib.src list) * term list) list ->
    39     Proof.context -> thm list list * local_theory
    40   type inductive_flags
    41   val add_inductive_i:
    42     inductive_flags -> ((string * typ) * mixfix) list ->
    43     (string * typ) list -> ((bstring * Attrib.src list) * term) list -> thm list ->
    44       local_theory -> inductive_result * local_theory
    45   val add_inductive: bool -> bool -> (string * string option * mixfix) list ->
    46     (string * string option * mixfix) list ->
    47     ((bstring * Attrib.src list) * string) list -> (Facts.ref * Attrib.src list) list ->
    48     local_theory -> inductive_result * local_theory
    49   val add_inductive_global: string -> inductive_flags ->
    50     ((string * typ) * mixfix) list -> (string * typ) list ->
    51     ((bstring * Attrib.src list) * term) list -> thm list -> theory -> inductive_result * theory
    52   val arities_of: thm -> (string * int) list
    53   val params_of: thm -> term list
    54   val partition_rules: thm -> thm list -> (string * thm list) list
    55   val partition_rules': thm -> (thm * 'a) list -> (string * (thm * 'a) list) list
    56   val unpartition_rules: thm list -> (string * 'a list) list -> 'a list
    57   val infer_intro_vars: thm -> int -> thm list -> term list list
    58   val setup: theory -> theory
    59 end;
    60 
    61 signature INDUCTIVE_PACKAGE =
    62 sig
    63   include BASIC_INDUCTIVE_PACKAGE
    64   type add_ind_def
    65   val declare_rules: string -> bstring -> bool -> bool -> string list ->
    66     thm list -> bstring list -> Attrib.src list list -> (thm * string list) list ->
    67     thm -> local_theory -> thm list * thm list * thm * local_theory
    68   val add_ind_def: add_ind_def
    69   val gen_add_inductive_i: add_ind_def ->
    70     inductive_flags -> ((string * typ) * mixfix) list ->
    71     (string * typ) list -> ((bstring * Attrib.src list) * term) list -> thm list ->
    72       local_theory -> inductive_result * local_theory
    73   val gen_add_inductive: add_ind_def ->
    74     bool -> bool -> (string * string option * mixfix) list ->
    75     (string * string option * mixfix) list ->
    76     ((bstring * Attrib.src list) * string) list -> (Facts.ref * Attrib.src list) list ->
    77     local_theory -> inductive_result * local_theory
    78   val gen_ind_decl: add_ind_def -> bool ->
    79     OuterParse.token list -> (local_theory -> local_theory) * OuterParse.token list
    80 end;
    81 
    82 structure InductivePackage: INDUCTIVE_PACKAGE =
    83 struct
    84 
    85 
    86 (** theory context references **)
    87 
    88 val inductive_forall_name = "HOL.induct_forall";
    89 val inductive_forall_def = thm "induct_forall_def";
    90 val inductive_conj_name = "HOL.induct_conj";
    91 val inductive_conj_def = thm "induct_conj_def";
    92 val inductive_conj = thms "induct_conj";
    93 val inductive_atomize = thms "induct_atomize";
    94 val inductive_rulify = thms "induct_rulify";
    95 val inductive_rulify_fallback = thms "induct_rulify_fallback";
    96 
    97 val notTrueE = TrueI RSN (2, notE);
    98 val notFalseI = Seq.hd (atac 1 notI);
    99 val simp_thms' = map (fn s => mk_meta_eq (the (find_first
   100   (equal (OldGoals.read_prop HOL.thy s) o prop_of) simp_thms)))
   101   ["(~True) = False", "(~False) = True",
   102    "(True --> ?P) = ?P", "(False --> ?P) = True",
   103    "(?P & True) = ?P", "(True & ?P) = ?P"];
   104 
   105 
   106 
   107 (** context data **)
   108 
   109 type inductive_result =
   110   {preds: term list, elims: thm list, raw_induct: thm,
   111    induct: thm, intrs: thm list};
   112 
   113 fun morph_result phi {preds, elims, raw_induct: thm, induct, intrs} =
   114   let
   115     val term = Morphism.term phi;
   116     val thm = Morphism.thm phi;
   117     val fact = Morphism.fact phi;
   118   in
   119    {preds = map term preds, elims = fact elims, raw_induct = thm raw_induct,
   120     induct = thm induct, intrs = fact intrs}
   121   end;
   122 
   123 type inductive_info =
   124   {names: string list, coind: bool} * inductive_result;
   125 
   126 structure InductiveData = GenericDataFun
   127 (
   128   type T = inductive_info Symtab.table * thm list;
   129   val empty = (Symtab.empty, []);
   130   val extend = I;
   131   fun merge _ ((tab1, monos1), (tab2, monos2)) =
   132     (Symtab.merge (K true) (tab1, tab2), Thm.merge_thms (monos1, monos2));
   133 );
   134 
   135 val get_inductives = InductiveData.get o Context.Proof;
   136 
   137 fun print_inductives ctxt =
   138   let
   139     val (tab, monos) = get_inductives ctxt;
   140     val space = Consts.space_of (ProofContext.consts_of ctxt);
   141   in
   142     [Pretty.strs ("(co)inductives:" :: map #1 (NameSpace.extern_table (space, tab))),
   143      Pretty.big_list "monotonicity rules:" (map (ProofContext.pretty_thm ctxt) monos)]
   144     |> Pretty.chunks |> Pretty.writeln
   145   end;
   146 
   147 
   148 (* get and put data *)
   149 
   150 fun the_inductive ctxt name =
   151   (case Symtab.lookup (#1 (get_inductives ctxt)) name of
   152     NONE => error ("Unknown (co)inductive predicate " ^ quote name)
   153   | SOME info => info);
   154 
   155 fun put_inductives names info = InductiveData.map
   156   (apfst (fold (fn name => Symtab.update (name, info)) names));
   157 
   158 
   159 
   160 (** monotonicity rules **)
   161 
   162 val get_monos = #2 o get_inductives;
   163 val map_monos = InductiveData.map o apsnd;
   164 
   165 fun mk_mono thm =
   166   let
   167     val concl = concl_of thm;
   168     fun eq2mono thm' = [thm' RS (thm' RS eq_to_mono)] @
   169       (case concl of
   170           (_ $ (_ $ (Const ("Not", _) $ _) $ _)) => []
   171         | _ => [thm' RS (thm' RS eq_to_mono2)]);
   172     fun dest_less_concl thm = dest_less_concl (thm RS le_funD)
   173       handle THM _ => thm RS le_boolD
   174   in
   175     case concl of
   176       Const ("==", _) $ _ $ _ => eq2mono (thm RS meta_eq_to_obj_eq)
   177     | _ $ (Const ("op =", _) $ _ $ _) => eq2mono thm
   178     | _ $ (Const ("HOL.ord_class.less_eq", _) $ _ $ _) =>
   179       [dest_less_concl (Seq.hd (REPEAT (FIRSTGOAL
   180          (resolve_tac [le_funI, le_boolI'])) thm))]
   181     | _ => [thm]
   182   end handle THM _ => error ("Bad monotonicity theorem:\n" ^ Display.string_of_thm thm);
   183 
   184 val mono_add = Thm.declaration_attribute (map_monos o fold Thm.add_thm o mk_mono);
   185 val mono_del = Thm.declaration_attribute (map_monos o fold Thm.del_thm o mk_mono);
   186 
   187 
   188 
   189 (** misc utilities **)
   190 
   191 fun message quiet_mode s = if quiet_mode then () else writeln s;
   192 fun clean_message quiet_mode s = if ! quick_and_dirty then () else message quiet_mode s;
   193 
   194 fun coind_prefix true = "co"
   195   | coind_prefix false = "";
   196 
   197 fun log (b:int) m n = if m >= n then 0 else 1 + log b (b * m) n;
   198 
   199 fun make_bool_args f g [] i = []
   200   | make_bool_args f g (x :: xs) i =
   201       (if i mod 2 = 0 then f x else g x) :: make_bool_args f g xs (i div 2);
   202 
   203 fun make_bool_args' xs =
   204   make_bool_args (K HOLogic.false_const) (K HOLogic.true_const) xs;
   205 
   206 fun find_arg T x [] = sys_error "find_arg"
   207   | find_arg T x ((p as (_, (SOME _, _))) :: ps) =
   208       apsnd (cons p) (find_arg T x ps)
   209   | find_arg T x ((p as (U, (NONE, y))) :: ps) =
   210       if (T: typ) = U then (y, (U, (SOME x, y)) :: ps)
   211       else apsnd (cons p) (find_arg T x ps);
   212 
   213 fun make_args Ts xs =
   214   map (fn (T, (NONE, ())) => Const ("arbitrary", T) | (_, (SOME t, ())) => t)
   215     (fold (fn (t, T) => snd o find_arg T t) xs (map (rpair (NONE, ())) Ts));
   216 
   217 fun make_args' Ts xs Us =
   218   fst (fold_map (fn T => find_arg T ()) Us (Ts ~~ map (pair NONE) xs));
   219 
   220 fun dest_predicate cs params t =
   221   let
   222     val k = length params;
   223     val (c, ts) = strip_comb t;
   224     val (xs, ys) = chop k ts;
   225     val i = find_index_eq c cs;
   226   in
   227     if xs = params andalso i >= 0 then
   228       SOME (c, i, ys, chop (length ys)
   229         (List.drop (binder_types (fastype_of c), k)))
   230     else NONE
   231   end;
   232 
   233 fun mk_names a 0 = []
   234   | mk_names a 1 = [a]
   235   | mk_names a n = map (fn i => a ^ string_of_int i) (1 upto n);
   236 
   237 
   238 
   239 (** process rules **)
   240 
   241 local
   242 
   243 fun err_in_rule ctxt name t msg =
   244   error (cat_lines ["Ill-formed introduction rule " ^ quote name,
   245     Syntax.string_of_term ctxt t, msg]);
   246 
   247 fun err_in_prem ctxt name t p msg =
   248   error (cat_lines ["Ill-formed premise", Syntax.string_of_term ctxt p,
   249     "in introduction rule " ^ quote name, Syntax.string_of_term ctxt t, msg]);
   250 
   251 val bad_concl = "Conclusion of introduction rule must be an inductive predicate";
   252 
   253 val bad_ind_occ = "Inductive predicate occurs in argument of inductive predicate";
   254 
   255 val bad_app = "Inductive predicate must be applied to parameter(s) ";
   256 
   257 fun atomize_term thy = MetaSimplifier.rewrite_term thy inductive_atomize [];
   258 
   259 in
   260 
   261 fun check_rule ctxt cs params ((name, att), rule) =
   262   let
   263     val params' = Term.variant_frees rule (Logic.strip_params rule);
   264     val frees = rev (map Free params');
   265     val concl = subst_bounds (frees, Logic.strip_assums_concl rule);
   266     val prems = map (curry subst_bounds frees) (Logic.strip_assums_hyp rule);
   267     val rule' = Logic.list_implies (prems, concl);
   268     val aprems = map (atomize_term (ProofContext.theory_of ctxt)) prems;
   269     val arule = list_all_free (params', Logic.list_implies (aprems, concl));
   270 
   271     fun check_ind err t = case dest_predicate cs params t of
   272         NONE => err (bad_app ^
   273           commas (map (Syntax.string_of_term ctxt) params))
   274       | SOME (_, _, ys, _) =>
   275           if exists (fn c => exists (fn t => Logic.occs (c, t)) ys) cs
   276           then err bad_ind_occ else ();
   277 
   278     fun check_prem' prem t =
   279       if head_of t mem cs then
   280         check_ind (err_in_prem ctxt name rule prem) t
   281       else (case t of
   282           Abs (_, _, t) => check_prem' prem t
   283         | t $ u => (check_prem' prem t; check_prem' prem u)
   284         | _ => ());
   285 
   286     fun check_prem (prem, aprem) =
   287       if can HOLogic.dest_Trueprop aprem then check_prem' prem prem
   288       else err_in_prem ctxt name rule prem "Non-atomic premise";
   289   in
   290     (case concl of
   291        Const ("Trueprop", _) $ t =>
   292          if head_of t mem cs then
   293            (check_ind (err_in_rule ctxt name rule') t;
   294             List.app check_prem (prems ~~ aprems))
   295          else err_in_rule ctxt name rule' bad_concl
   296      | _ => err_in_rule ctxt name rule' bad_concl);
   297     ((name, att), arule)
   298   end;
   299 
   300 val rulify =
   301   hol_simplify inductive_conj
   302   #> hol_simplify inductive_rulify
   303   #> hol_simplify inductive_rulify_fallback
   304   #> MetaSimplifier.norm_hhf;
   305 
   306 end;
   307 
   308 
   309 
   310 (** proofs for (co)inductive predicates **)
   311 
   312 (* prove monotonicity *)
   313 
   314 fun prove_mono quiet_mode skip_mono predT fp_fun monos ctxt =
   315  (message (quiet_mode orelse skip_mono andalso !quick_and_dirty)
   316     "  Proving monotonicity ...";
   317   (if skip_mono then SkipProof.prove else Goal.prove) ctxt [] []
   318     (HOLogic.mk_Trueprop
   319       (Const (@{const_name Orderings.mono}, (predT --> predT) --> HOLogic.boolT) $ fp_fun))
   320     (fn _ => EVERY [rtac @{thm monoI} 1,
   321       REPEAT (resolve_tac [le_funI, le_boolI'] 1),
   322       REPEAT (FIRST
   323         [atac 1,
   324          resolve_tac (List.concat (map mk_mono monos) @ get_monos ctxt) 1,
   325          etac le_funE 1, dtac le_boolD 1])]));
   326 
   327 
   328 (* prove introduction rules *)
   329 
   330 fun prove_intrs quiet_mode coind mono fp_def k params intr_ts rec_preds_defs ctxt =
   331   let
   332     val _ = clean_message quiet_mode "  Proving the introduction rules ...";
   333 
   334     val unfold = funpow k (fn th => th RS fun_cong)
   335       (mono RS (fp_def RS
   336         (if coind then def_gfp_unfold else def_lfp_unfold)));
   337 
   338     fun select_disj 1 1 = []
   339       | select_disj _ 1 = [rtac disjI1]
   340       | select_disj n i = (rtac disjI2)::(select_disj (n - 1) (i - 1));
   341 
   342     val rules = [refl, TrueI, notFalseI, exI, conjI];
   343 
   344     val intrs = map_index (fn (i, intr) => rulify
   345       (SkipProof.prove ctxt (map (fst o dest_Free) params) [] intr (fn _ => EVERY
   346        [rewrite_goals_tac rec_preds_defs,
   347         rtac (unfold RS iffD2) 1,
   348         EVERY1 (select_disj (length intr_ts) (i + 1)),
   349         (*Not ares_tac, since refl must be tried before any equality assumptions;
   350           backtracking may occur if the premises have extra variables!*)
   351         DEPTH_SOLVE_1 (resolve_tac rules 1 APPEND assume_tac 1)]))) intr_ts
   352 
   353   in (intrs, unfold) end;
   354 
   355 
   356 (* prove elimination rules *)
   357 
   358 fun prove_elims quiet_mode cs params intr_ts intr_names unfold rec_preds_defs ctxt =
   359   let
   360     val _ = clean_message quiet_mode "  Proving the elimination rules ...";
   361 
   362     val ([pname], ctxt') = ctxt |>
   363       Variable.add_fixes (map (fst o dest_Free) params) |> snd |>
   364       Variable.variant_fixes ["P"];
   365     val P = HOLogic.mk_Trueprop (Free (pname, HOLogic.boolT));
   366 
   367     fun dest_intr r =
   368       (the (dest_predicate cs params (HOLogic.dest_Trueprop (Logic.strip_assums_concl r))),
   369        Logic.strip_assums_hyp r, Logic.strip_params r);
   370 
   371     val intrs = map dest_intr intr_ts ~~ intr_names;
   372 
   373     val rules1 = [disjE, exE, FalseE];
   374     val rules2 = [conjE, FalseE, notTrueE];
   375 
   376     fun prove_elim c =
   377       let
   378         val Ts = List.drop (binder_types (fastype_of c), length params);
   379         val (anames, ctxt'') = Variable.variant_fixes (mk_names "a" (length Ts)) ctxt';
   380         val frees = map Free (anames ~~ Ts);
   381 
   382         fun mk_elim_prem ((_, _, us, _), ts, params') =
   383           list_all (params',
   384             Logic.list_implies (map (HOLogic.mk_Trueprop o HOLogic.mk_eq)
   385               (frees ~~ us) @ ts, P));
   386         val c_intrs = (List.filter (equal c o #1 o #1 o #1) intrs);
   387         val prems = HOLogic.mk_Trueprop (list_comb (c, params @ frees)) ::
   388            map mk_elim_prem (map #1 c_intrs)
   389       in
   390         (SkipProof.prove ctxt'' [] prems P
   391           (fn {prems, ...} => EVERY
   392             [cut_facts_tac [hd prems] 1,
   393              rewrite_goals_tac rec_preds_defs,
   394              dtac (unfold RS iffD1) 1,
   395              REPEAT (FIRSTGOAL (eresolve_tac rules1)),
   396              REPEAT (FIRSTGOAL (eresolve_tac rules2)),
   397              EVERY (map (fn prem =>
   398                DEPTH_SOLVE_1 (ares_tac [rewrite_rule rec_preds_defs prem, conjI] 1)) (tl prems))])
   399           |> rulify
   400           |> singleton (ProofContext.export ctxt'' ctxt),
   401          map #2 c_intrs)
   402       end
   403 
   404    in map prove_elim cs end;
   405 
   406 
   407 (* derivation of simplified elimination rules *)
   408 
   409 local
   410 
   411 (*delete needless equality assumptions*)
   412 val refl_thin = Goal.prove_global HOL.thy [] [] @{prop "!!P. a = a ==> P ==> P"}
   413   (fn _ => assume_tac 1);
   414 val elim_rls = [asm_rl, FalseE, refl_thin, conjE, exE];
   415 val elim_tac = REPEAT o Tactic.eresolve_tac elim_rls;
   416 
   417 fun simp_case_tac ss i =
   418   EVERY' [elim_tac, asm_full_simp_tac ss, elim_tac, REPEAT o bound_hyp_subst_tac] i;
   419 
   420 in
   421 
   422 fun mk_cases ctxt prop =
   423   let
   424     val thy = ProofContext.theory_of ctxt;
   425     val ss = Simplifier.local_simpset_of ctxt;
   426 
   427     fun err msg =
   428       error (Pretty.string_of (Pretty.block
   429         [Pretty.str msg, Pretty.fbrk, Syntax.pretty_term ctxt prop]));
   430 
   431     val elims = Induct.find_casesP ctxt prop;
   432 
   433     val cprop = Thm.cterm_of thy prop;
   434     val tac = ALLGOALS (simp_case_tac ss) THEN prune_params_tac;
   435     fun mk_elim rl =
   436       Thm.implies_intr cprop (Tactic.rule_by_tactic tac (Thm.assume cprop RS rl))
   437       |> singleton (Variable.export (Variable.auto_fixes prop ctxt) ctxt);
   438   in
   439     (case get_first (try mk_elim) elims of
   440       SOME r => r
   441     | NONE => err "Proposition not an inductive predicate:")
   442   end;
   443 
   444 end;
   445 
   446 
   447 (* inductive_cases *)
   448 
   449 fun gen_inductive_cases prep_att prep_prop args lthy =
   450   let
   451     val thy = ProofContext.theory_of lthy;
   452     val facts = args |> map (fn ((a, atts), props) =>
   453       ((a, map (prep_att thy) atts),
   454         map (Thm.no_attributes o single o mk_cases lthy o prep_prop lthy) props));
   455   in lthy |> LocalTheory.notes Thm.theoremK facts |>> map snd end;
   456 
   457 val inductive_cases = gen_inductive_cases Attrib.intern_src Syntax.read_prop;
   458 val inductive_cases_i = gen_inductive_cases (K I) Syntax.check_prop;
   459 
   460 
   461 fun ind_cases src = Method.syntax (Scan.lift (Scan.repeat1 Args.name --
   462     Scan.optional (Args.$$$ "for" |-- Scan.repeat1 Args.name) [])) src
   463   #> (fn ((raw_props, fixes), ctxt) =>
   464     let
   465       val (_, ctxt') = Variable.add_fixes fixes ctxt;
   466       val props = Syntax.read_props ctxt' raw_props;
   467       val ctxt'' = fold Variable.declare_term props ctxt';
   468       val rules = ProofContext.export ctxt'' ctxt (map (mk_cases ctxt'') props)
   469     in Method.erule 0 rules end);
   470 
   471 
   472 
   473 (* prove induction rule *)
   474 
   475 fun prove_indrule quiet_mode cs argTs bs xs rec_const params intr_ts mono
   476     fp_def rec_preds_defs ctxt =
   477   let
   478     val _ = clean_message quiet_mode "  Proving the induction rule ...";
   479     val thy = ProofContext.theory_of ctxt;
   480 
   481     (* predicates for induction rule *)
   482 
   483     val (pnames, ctxt') = ctxt |>
   484       Variable.add_fixes (map (fst o dest_Free) params) |> snd |>
   485       Variable.variant_fixes (mk_names "P" (length cs));
   486     val preds = map Free (pnames ~~
   487       map (fn c => List.drop (binder_types (fastype_of c), length params) --->
   488         HOLogic.boolT) cs);
   489 
   490     (* transform an introduction rule into a premise for induction rule *)
   491 
   492     fun mk_ind_prem r =
   493       let
   494         fun subst s = (case dest_predicate cs params s of
   495             SOME (_, i, ys, (_, Ts)) =>
   496               let
   497                 val k = length Ts;
   498                 val bs = map Bound (k - 1 downto 0);
   499                 val P = list_comb (List.nth (preds, i),
   500                   map (incr_boundvars k) ys @ bs);
   501                 val Q = list_abs (mk_names "x" k ~~ Ts,
   502                   HOLogic.mk_binop inductive_conj_name
   503                     (list_comb (incr_boundvars k s, bs), P))
   504               in (Q, case Ts of [] => SOME (s, P) | _ => NONE) end
   505           | NONE => (case s of
   506               (t $ u) => (fst (subst t) $ fst (subst u), NONE)
   507             | (Abs (a, T, t)) => (Abs (a, T, fst (subst t)), NONE)
   508             | _ => (s, NONE)));
   509 
   510         fun mk_prem (s, prems) = (case subst s of
   511               (_, SOME (t, u)) => t :: u :: prems
   512             | (t, _) => t :: prems);
   513 
   514         val SOME (_, i, ys, _) = dest_predicate cs params
   515           (HOLogic.dest_Trueprop (Logic.strip_assums_concl r))
   516 
   517       in list_all_free (Logic.strip_params r,
   518         Logic.list_implies (map HOLogic.mk_Trueprop (foldr mk_prem
   519           [] (map HOLogic.dest_Trueprop (Logic.strip_assums_hyp r))),
   520             HOLogic.mk_Trueprop (list_comb (List.nth (preds, i), ys))))
   521       end;
   522 
   523     val ind_prems = map mk_ind_prem intr_ts;
   524 
   525 
   526     (* make conclusions for induction rules *)
   527 
   528     val Tss = map (binder_types o fastype_of) preds;
   529     val (xnames, ctxt'') =
   530       Variable.variant_fixes (mk_names "x" (length (flat Tss))) ctxt';
   531     val mutual_ind_concl = HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
   532         (map (fn (((xnames, Ts), c), P) =>
   533            let val frees = map Free (xnames ~~ Ts)
   534            in HOLogic.mk_imp
   535              (list_comb (c, params @ frees), list_comb (P, frees))
   536            end) (unflat Tss xnames ~~ Tss ~~ cs ~~ preds)));
   537 
   538 
   539     (* make predicate for instantiation of abstract induction rule *)
   540 
   541     val ind_pred = fold_rev lambda (bs @ xs) (foldr1 HOLogic.mk_conj
   542       (map_index (fn (i, P) => foldr HOLogic.mk_imp
   543          (list_comb (P, make_args' argTs xs (binder_types (fastype_of P))))
   544          (make_bool_args HOLogic.mk_not I bs i)) preds));
   545 
   546     val ind_concl = HOLogic.mk_Trueprop
   547       (HOLogic.mk_binrel "HOL.ord_class.less_eq" (rec_const, ind_pred));
   548 
   549     val raw_fp_induct = (mono RS (fp_def RS def_lfp_induct));
   550 
   551     val induct = SkipProof.prove ctxt'' [] ind_prems ind_concl
   552       (fn {prems, ...} => EVERY
   553         [rewrite_goals_tac [inductive_conj_def],
   554          DETERM (rtac raw_fp_induct 1),
   555          REPEAT (resolve_tac [le_funI, le_boolI] 1),
   556          rewrite_goals_tac (inf_fun_eq :: inf_bool_eq :: simp_thms'),
   557          (*This disjE separates out the introduction rules*)
   558          REPEAT (FIRSTGOAL (eresolve_tac [disjE, exE, FalseE])),
   559          (*Now break down the individual cases.  No disjE here in case
   560            some premise involves disjunction.*)
   561          REPEAT (FIRSTGOAL (etac conjE ORELSE' bound_hyp_subst_tac)),
   562          REPEAT (FIRSTGOAL
   563            (resolve_tac [conjI, impI] ORELSE' (etac notE THEN' atac))),
   564          EVERY (map (fn prem => DEPTH_SOLVE_1 (ares_tac [rewrite_rule
   565              (inductive_conj_def :: rec_preds_defs @ simp_thms') prem,
   566            conjI, refl] 1)) prems)]);
   567 
   568     val lemma = SkipProof.prove ctxt'' [] []
   569       (Logic.mk_implies (ind_concl, mutual_ind_concl)) (fn _ => EVERY
   570         [rewrite_goals_tac rec_preds_defs,
   571          REPEAT (EVERY
   572            [REPEAT (resolve_tac [conjI, impI] 1),
   573             REPEAT (eresolve_tac [le_funE, le_boolE] 1),
   574             atac 1,
   575             rewrite_goals_tac simp_thms',
   576             atac 1])])
   577 
   578   in singleton (ProofContext.export ctxt'' ctxt) (induct RS lemma) end;
   579 
   580 
   581 
   582 (** specification of (co)inductive predicates **)
   583 
   584 fun mk_ind_def quiet_mode skip_mono alt_name coind cs intr_ts monos params cnames_syn ctxt =
   585   let
   586     val fp_name = if coind then @{const_name Inductive.gfp} else @{const_name Inductive.lfp};
   587 
   588     val argTs = fold (fn c => fn Ts => Ts @
   589       (List.drop (binder_types (fastype_of c), length params) \\ Ts)) cs [];
   590     val k = log 2 1 (length cs);
   591     val predT = replicate k HOLogic.boolT ---> argTs ---> HOLogic.boolT;
   592     val p :: xs = map Free (Variable.variant_frees ctxt intr_ts
   593       (("p", predT) :: (mk_names "x" (length argTs) ~~ argTs)));
   594     val bs = map Free (Variable.variant_frees ctxt (p :: xs @ intr_ts)
   595       (map (rpair HOLogic.boolT) (mk_names "b" k)));
   596 
   597     fun subst t = (case dest_predicate cs params t of
   598         SOME (_, i, ts, (Ts, Us)) =>
   599           let
   600             val l = length Us;
   601             val zs = map Bound (l - 1 downto 0)
   602           in
   603             list_abs (map (pair "z") Us, list_comb (p,
   604               make_bool_args' bs i @ make_args argTs
   605                 ((map (incr_boundvars l) ts ~~ Ts) @ (zs ~~ Us))))
   606           end
   607       | NONE => (case t of
   608           t1 $ t2 => subst t1 $ subst t2
   609         | Abs (x, T, u) => Abs (x, T, subst u)
   610         | _ => t));
   611 
   612     (* transform an introduction rule into a conjunction  *)
   613     (*   [| p_i t; ... |] ==> p_j u                       *)
   614     (* is transformed into                                *)
   615     (*   b_j & x_j = u & p b_j t & ...                    *)
   616 
   617     fun transform_rule r =
   618       let
   619         val SOME (_, i, ts, (Ts, _)) = dest_predicate cs params
   620           (HOLogic.dest_Trueprop (Logic.strip_assums_concl r));
   621         val ps = make_bool_args HOLogic.mk_not I bs i @
   622           map HOLogic.mk_eq (make_args' argTs xs Ts ~~ ts) @
   623           map (subst o HOLogic.dest_Trueprop)
   624             (Logic.strip_assums_hyp r)
   625       in foldr (fn ((x, T), P) => HOLogic.exists_const T $ (Abs (x, T, P)))
   626         (if null ps then HOLogic.true_const else foldr1 HOLogic.mk_conj ps)
   627         (Logic.strip_params r)
   628       end
   629 
   630     (* make a disjunction of all introduction rules *)
   631 
   632     val fp_fun = fold_rev lambda (p :: bs @ xs)
   633       (if null intr_ts then HOLogic.false_const
   634        else foldr1 HOLogic.mk_disj (map transform_rule intr_ts));
   635 
   636     (* add definiton of recursive predicates to theory *)
   637 
   638     val rec_name = if alt_name = "" then
   639       space_implode "_" (map fst cnames_syn) else alt_name;
   640 
   641     val ((rec_const, (_, fp_def)), ctxt') = ctxt |>
   642       LocalTheory.define Thm.internalK
   643         ((rec_name, case cnames_syn of [(_, syn)] => syn | _ => NoSyn),
   644          (("", []), fold_rev lambda params
   645            (Const (fp_name, (predT --> predT) --> predT) $ fp_fun)));
   646     val fp_def' = Simplifier.rewrite (HOL_basic_ss addsimps [fp_def])
   647       (cterm_of (ProofContext.theory_of ctxt') (list_comb (rec_const, params)));
   648     val specs = if length cs < 2 then [] else
   649       map_index (fn (i, (name_mx, c)) =>
   650         let
   651           val Ts = List.drop (binder_types (fastype_of c), length params);
   652           val xs = map Free (Variable.variant_frees ctxt intr_ts
   653             (mk_names "x" (length Ts) ~~ Ts))
   654         in
   655           (name_mx, (("", []), fold_rev lambda (params @ xs)
   656             (list_comb (rec_const, params @ make_bool_args' bs i @
   657               make_args argTs (xs ~~ Ts)))))
   658         end) (cnames_syn ~~ cs);
   659     val (consts_defs, ctxt'') = fold_map (LocalTheory.define Thm.internalK) specs ctxt';
   660     val preds = (case cs of [_] => [rec_const] | _ => map #1 consts_defs);
   661 
   662     val mono = prove_mono quiet_mode skip_mono predT fp_fun monos ctxt''
   663 
   664   in (ctxt'', rec_name, mono, fp_def', map (#2 o #2) consts_defs,
   665     list_comb (rec_const, params), preds, argTs, bs, xs)
   666   end;
   667 
   668 fun declare_rules kind rec_name coind no_ind cnames intrs intr_names intr_atts
   669       elims raw_induct ctxt =
   670   let
   671     val ind_case_names = RuleCases.case_names intr_names;
   672     val induct =
   673       if coind then
   674         (raw_induct, [RuleCases.case_names [rec_name],
   675           RuleCases.case_conclusion (rec_name, intr_names),
   676           RuleCases.consumes 1, Induct.coinduct_pred (hd cnames)])
   677       else if no_ind orelse length cnames > 1 then
   678         (raw_induct, [ind_case_names, RuleCases.consumes 0])
   679       else (raw_induct RSN (2, rev_mp), [ind_case_names, RuleCases.consumes 1]);
   680 
   681     val (intrs', ctxt1) =
   682       ctxt |>
   683       LocalTheory.notes kind
   684         (map (NameSpace.qualified rec_name) intr_names ~~
   685          intr_atts ~~ map (fn th => [([th],
   686            [Attrib.internal (K (ContextRules.intro_query NONE))])]) intrs) |>>
   687       map (hd o snd);
   688     val (((_, elims'), (_, [induct'])), ctxt2) =
   689       ctxt1 |>
   690       LocalTheory.note kind ((NameSpace.qualified rec_name "intros", []), intrs') ||>>
   691       fold_map (fn (name, (elim, cases)) =>
   692         LocalTheory.note kind ((NameSpace.qualified (Sign.base_name name) "cases",
   693           [Attrib.internal (K (RuleCases.case_names cases)),
   694            Attrib.internal (K (RuleCases.consumes 1)),
   695            Attrib.internal (K (Induct.cases_pred name)),
   696            Attrib.internal (K (ContextRules.elim_query NONE))]), [elim]) #>
   697         apfst (hd o snd)) (if null elims then [] else cnames ~~ elims) ||>>
   698       LocalTheory.note kind ((NameSpace.qualified rec_name (coind_prefix coind ^ "induct"),
   699         map (Attrib.internal o K) (#2 induct)), [rulify (#1 induct)]);
   700 
   701     val ctxt3 = if no_ind orelse coind then ctxt2 else
   702       let val inducts = cnames ~~ ProjectRule.projects ctxt2 (1 upto length cnames) induct'
   703       in
   704         ctxt2 |>
   705         LocalTheory.notes kind [((NameSpace.qualified rec_name "inducts", []),
   706           inducts |> map (fn (name, th) => ([th],
   707             [Attrib.internal (K ind_case_names),
   708              Attrib.internal (K (RuleCases.consumes 1)),
   709              Attrib.internal (K (Induct.induct_pred name))])))] |> snd
   710       end
   711   in (intrs', elims', induct', ctxt3) end;
   712 
   713 type inductive_flags =
   714   {quiet_mode: bool, verbose: bool, kind: string, alt_name: bstring,
   715    coind: bool, no_elim: bool, no_ind: bool, skip_mono: bool}
   716 
   717 type add_ind_def =
   718   inductive_flags ->
   719   term list -> ((string * Attrib.src list) * term) list -> thm list ->
   720   term list -> (string * mixfix) list ->
   721   local_theory -> inductive_result * local_theory
   722 
   723 fun add_ind_def {quiet_mode, verbose, kind, alt_name, coind, no_elim, no_ind, skip_mono}
   724     cs intros monos params cnames_syn ctxt =
   725   let
   726     val _ = null cnames_syn andalso error "No inductive predicates given";
   727     val _ = message (quiet_mode andalso not verbose)
   728       ("Proofs for " ^ coind_prefix coind ^ "inductive predicate(s) " ^
   729         commas_quote (map fst cnames_syn));
   730 
   731     val cnames = map (Sign.full_name (ProofContext.theory_of ctxt) o #1) cnames_syn;  (* FIXME *)
   732     val ((intr_names, intr_atts), intr_ts) =
   733       apfst split_list (split_list (map (check_rule ctxt cs params) intros));
   734 
   735     val (ctxt1, rec_name, mono, fp_def, rec_preds_defs, rec_const, preds,
   736       argTs, bs, xs) = mk_ind_def quiet_mode skip_mono alt_name coind cs intr_ts
   737         monos params cnames_syn ctxt;
   738 
   739     val (intrs, unfold) = prove_intrs quiet_mode coind mono fp_def (length bs + length xs)
   740       params intr_ts rec_preds_defs ctxt1;
   741     val elims = if no_elim then [] else
   742       prove_elims quiet_mode cs params intr_ts intr_names unfold rec_preds_defs ctxt1;
   743     val raw_induct = zero_var_indexes
   744       (if no_ind then Drule.asm_rl else
   745        if coind then
   746          singleton (ProofContext.export
   747            (snd (Variable.add_fixes (map (fst o dest_Free) params) ctxt1)) ctxt1)
   748            (rotate_prems ~1 (ObjectLogic.rulify (rule_by_tactic
   749              (rewrite_tac [le_fun_def, le_bool_def, sup_fun_eq, sup_bool_eq] THEN
   750                fold_tac rec_preds_defs) (mono RS (fp_def RS def_coinduct)))))
   751        else
   752          prove_indrule quiet_mode cs argTs bs xs rec_const params intr_ts mono fp_def
   753            rec_preds_defs ctxt1);
   754 
   755     val (intrs', elims', induct, ctxt2) = declare_rules kind rec_name coind no_ind
   756       cnames intrs intr_names intr_atts elims raw_induct ctxt1;
   757 
   758     val names = map #1 cnames_syn;
   759     val result =
   760       {preds = preds,
   761        intrs = intrs',
   762        elims = elims',
   763        raw_induct = rulify raw_induct,
   764        induct = induct};
   765 
   766     val ctxt3 = ctxt2
   767       |> LocalTheory.declaration (fn phi =>
   768         let val result' = morph_result phi result;
   769         in put_inductives cnames (*global names!?*) ({names = cnames, coind = coind}, result') end);
   770   in (result, ctxt3) end;
   771 
   772 
   773 (* external interfaces *)
   774 
   775 fun gen_add_inductive_i mk_def
   776     (flags as {quiet_mode, verbose, kind, alt_name, coind, no_elim, no_ind, skip_mono})
   777     cnames_syn pnames spec monos lthy =
   778   let
   779     val thy = ProofContext.theory_of lthy;
   780     val _ = Theory.requires thy "Inductive" (coind_prefix coind ^ "inductive definitions");
   781 
   782 
   783     (* abbrevs *)
   784 
   785     val (_, ctxt1) = Variable.add_fixes (map (fst o fst) cnames_syn) lthy;
   786 
   787     fun get_abbrev ((name, atts), t) =
   788       if can (Logic.strip_assums_concl #> Logic.dest_equals) t then
   789         let
   790           val _ = name = "" andalso null atts orelse
   791             error "Abbreviations may not have names or attributes";
   792           val ((x, T), rhs) = LocalDefs.abs_def (snd (LocalDefs.cert_def ctxt1 t));
   793           val mx =
   794             (case find_first (fn ((c, _), _) => c = x) cnames_syn of
   795               NONE => error ("Undeclared head of abbreviation " ^ quote x)
   796             | SOME ((_, T'), mx) =>
   797                 if T <> T' then error ("Bad type specification for abbreviation " ^ quote x)
   798                 else mx);
   799         in SOME ((x, mx), rhs) end
   800       else NONE;
   801 
   802     val abbrevs = map_filter get_abbrev spec;
   803     val bs = map (fst o fst) abbrevs;
   804 
   805 
   806     (* predicates *)
   807 
   808     val pre_intros = filter_out (is_some o get_abbrev) spec;
   809     val cnames_syn' = filter_out (member (op =) bs o fst o fst) cnames_syn;
   810     val cs = map (Free o fst) cnames_syn';
   811     val ps = map Free pnames;
   812 
   813     val (_, ctxt2) = lthy |> Variable.add_fixes (map (fst o fst) cnames_syn');
   814     val _ = map (fn abbr => LocalDefs.fixed_abbrev abbr ctxt2) abbrevs;
   815     val ctxt3 = ctxt2 |> fold (snd oo LocalDefs.fixed_abbrev) abbrevs;
   816     val expand = Assumption.export_term ctxt3 lthy #> ProofContext.cert_term lthy;
   817 
   818     fun close_rule r = list_all_free (rev (fold_aterms
   819       (fn t as Free (v as (s, _)) =>
   820           if Variable.is_fixed ctxt1 s orelse
   821             member (op =) ps t then I else insert (op =) v
   822         | _ => I) r []), r);
   823 
   824     val intros = map (apsnd (Syntax.check_term lthy #> close_rule #> expand)) pre_intros;
   825     val preds = map (fn ((c, _), mx) => (c, mx)) cnames_syn';
   826   in
   827     lthy
   828     |> mk_def flags cs intros monos ps preds
   829     ||> fold (snd oo LocalTheory.abbrev Syntax.mode_default) abbrevs
   830   end;
   831 
   832 fun gen_add_inductive mk_def verbose coind cnames_syn pnames_syn intro_srcs raw_monos lthy =
   833   let
   834     val ((vars, specs), _) = lthy |> ProofContext.set_mode ProofContext.mode_abbrev
   835       |> Specification.read_specification
   836           (cnames_syn @ pnames_syn) (map (fn (a, s) => [(a, [s])]) intro_srcs);
   837     val (cs, ps) = chop (length cnames_syn) vars;
   838     val intrs = map (apsnd the_single) specs;
   839     val monos = Attrib.eval_thms lthy raw_monos;
   840     val flags = {quiet_mode = false, verbose = verbose, kind = Thm.theoremK, alt_name = "",
   841       coind = coind, no_elim = false, no_ind = false, skip_mono = false};
   842   in
   843     lthy
   844     |> LocalTheory.set_group (serial_string ())
   845     |> gen_add_inductive_i mk_def flags cs (map fst ps) intrs monos
   846   end;
   847 
   848 val add_inductive_i = gen_add_inductive_i add_ind_def;
   849 val add_inductive = gen_add_inductive add_ind_def;
   850 
   851 fun add_inductive_global group flags cnames_syn pnames pre_intros monos thy =
   852   let
   853     val name = Sign.full_name thy (fst (fst (hd cnames_syn)));
   854     val ctxt' = thy
   855       |> TheoryTarget.init NONE
   856       |> LocalTheory.set_group group
   857       |> add_inductive_i flags cnames_syn pnames pre_intros monos |> snd
   858       |> LocalTheory.exit;
   859     val info = #2 (the_inductive ctxt' name);
   860   in (info, ProofContext.theory_of ctxt') end;
   861 
   862 
   863 (* read off arities of inductive predicates from raw induction rule *)
   864 fun arities_of induct =
   865   map (fn (_ $ t $ u) =>
   866       (fst (dest_Const (head_of t)), length (snd (strip_comb u))))
   867     (HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of induct)));
   868 
   869 (* read off parameters of inductive predicate from raw induction rule *)
   870 fun params_of induct =
   871   let
   872     val (_ $ t $ u :: _) =
   873       HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of induct));
   874     val (_, ts) = strip_comb t;
   875     val (_, us) = strip_comb u
   876   in
   877     List.take (ts, length ts - length us)
   878   end;
   879 
   880 val pname_of_intr =
   881   concl_of #> HOLogic.dest_Trueprop #> head_of #> dest_Const #> fst;
   882 
   883 (* partition introduction rules according to predicate name *)
   884 fun gen_partition_rules f induct intros =
   885   fold_rev (fn r => AList.map_entry op = (pname_of_intr (f r)) (cons r)) intros
   886     (map (rpair [] o fst) (arities_of induct));
   887 
   888 val partition_rules = gen_partition_rules I;
   889 fun partition_rules' induct = gen_partition_rules fst induct;
   890 
   891 fun unpartition_rules intros xs =
   892   fold_map (fn r => AList.map_entry_yield op = (pname_of_intr r)
   893     (fn x :: xs => (x, xs)) #>> the) intros xs |> fst;
   894 
   895 (* infer order of variables in intro rules from order of quantifiers in elim rule *)
   896 fun infer_intro_vars elim arity intros =
   897   let
   898     val thy = theory_of_thm elim;
   899     val _ :: cases = prems_of elim;
   900     val used = map (fst o fst) (Term.add_vars (prop_of elim) []);
   901     fun mtch (t, u) =
   902       let
   903         val params = Logic.strip_params t;
   904         val vars = map (Var o apfst (rpair 0))
   905           (Name.variant_list used (map fst params) ~~ map snd params);
   906         val ts = map (curry subst_bounds (rev vars))
   907           (List.drop (Logic.strip_assums_hyp t, arity));
   908         val us = Logic.strip_imp_prems u;
   909         val tab = fold (Pattern.first_order_match thy) (ts ~~ us)
   910           (Vartab.empty, Vartab.empty);
   911       in
   912         map (Envir.subst_vars tab) vars
   913       end
   914   in
   915     map (mtch o apsnd prop_of) (cases ~~ intros)
   916   end;
   917 
   918 
   919 
   920 (** package setup **)
   921 
   922 (* setup theory *)
   923 
   924 val setup =
   925   Method.add_methods [("ind_cases", ind_cases,
   926     "dynamic case analysis on predicates")] #>
   927   Attrib.add_attributes [("mono", Attrib.add_del_args mono_add mono_del,
   928     "declaration of monotonicity rule")];
   929 
   930 
   931 (* outer syntax *)
   932 
   933 local structure P = OuterParse and K = OuterKeyword in
   934 
   935 val _ = OuterSyntax.keywords ["monos"];
   936 
   937 fun flatten_specification specs = specs |> maps
   938   (fn (a, (concl, [])) => concl |> map
   939         (fn ((b, atts), [B]) =>
   940               if a = "" then ((b, atts), B)
   941               else if b = "" then ((a, atts), B)
   942               else error ("Illegal nested case names " ^ quote (NameSpace.append a b))
   943           | ((b, _), _) => error ("Illegal simultaneous specification " ^ quote b))
   944     | (a, _) => error ("Illegal local specification parameters for " ^ quote a));
   945 
   946 fun gen_ind_decl mk_def coind =
   947   P.fixes -- P.for_fixes --
   948   Scan.optional (P.$$$ "where" |-- P.!!! SpecParse.specification) [] --
   949   Scan.optional (P.$$$ "monos" |-- P.!!! SpecParse.xthms1) []
   950   >> (fn (((preds, params), specs), monos) =>
   951       (snd o gen_add_inductive mk_def true coind preds params (flatten_specification specs) monos));
   952 
   953 val ind_decl = gen_ind_decl add_ind_def;
   954 
   955 val _ = OuterSyntax.local_theory "inductive" "define inductive predicates" K.thy_decl (ind_decl false);
   956 val _ = OuterSyntax.local_theory "coinductive" "define coinductive predicates" K.thy_decl (ind_decl true);
   957 
   958 val _ =
   959   OuterSyntax.local_theory "inductive_cases"
   960     "create simplified instances of elimination rules (improper)" K.thy_script
   961     (P.and_list1 SpecParse.spec >> (fn specs => snd o inductive_cases specs));
   962 
   963 end;
   964 
   965 end;