src/HOL/Tools/inductive.ML
author wenzelm
Mon Apr 06 23:14:05 2015 +0200 (2015-04-06)
changeset 59940 087d81f5213e
parent 59936 b8ffc3dc9e24
child 60097 d20ca79d50e4
permissions -rw-r--r--
local setup of induction tools, with restricted access to auxiliary consts;
proper antiquotations for formerly inaccessible consts;
     1 (*  Title:      HOL/Tools/inductive.ML
     2     Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
     3     Author:     Stefan Berghofer and Markus Wenzel, TU Muenchen
     4 
     5 (Co)Inductive Definition module for HOL.
     6 
     7 Features:
     8   * least or greatest fixedpoints
     9   * mutually recursive definitions
    10   * definitions involving arbitrary monotone operators
    11   * automatically proves introduction and elimination rules
    12 
    13   Introduction rules have the form
    14   [| M Pj ti, ..., Q x, ... |] ==> Pk t
    15   where M is some monotone operator (usually the identity)
    16   Q x is any side condition on the free variables
    17   ti, t are any terms
    18   Pj, Pk are two of the predicates being defined in mutual recursion
    19 *)
    20 
    21 signature BASIC_INDUCTIVE =
    22 sig
    23   type inductive_result =
    24     {preds: term list, elims: thm list, raw_induct: thm,
    25      induct: thm, inducts: thm list, intrs: thm list, eqs: thm list}
    26   val transform_result: morphism -> inductive_result -> inductive_result
    27   type inductive_info = {names: string list, coind: bool} * inductive_result
    28   val the_inductive: Proof.context -> string -> inductive_info
    29   val print_inductives: bool -> Proof.context -> unit
    30   val get_monos: Proof.context -> thm list
    31   val mono_add: attribute
    32   val mono_del: attribute
    33   val mk_cases_tac: Proof.context -> tactic
    34   val mk_cases: Proof.context -> term -> thm
    35   val inductive_forall_def: thm
    36   val rulify: Proof.context -> thm -> thm
    37   val inductive_cases: (Attrib.binding * string list) list -> local_theory ->
    38     (string * thm list) list * local_theory
    39   val inductive_cases_i: (Attrib.binding * term list) list -> local_theory ->
    40     (string * thm list) list * local_theory
    41   val ind_cases_rules: Proof.context ->
    42     string list -> (binding * string option * mixfix) list -> thm list
    43   val inductive_simps: (Attrib.binding * string list) list -> local_theory ->
    44     (string * thm list) list * local_theory
    45   val inductive_simps_i: (Attrib.binding * term list) list -> local_theory ->
    46     (string * thm list) list * local_theory
    47   type inductive_flags =
    48     {quiet_mode: bool, verbose: bool, alt_name: binding, coind: bool,
    49       no_elim: bool, no_ind: bool, skip_mono: bool}
    50   val add_inductive_i:
    51     inductive_flags -> ((binding * typ) * mixfix) list ->
    52     (string * typ) list -> (Attrib.binding * term) list -> thm list -> local_theory ->
    53     inductive_result * local_theory
    54   val add_inductive: bool -> bool ->
    55     (binding * string option * mixfix) list ->
    56     (binding * string option * mixfix) list ->
    57     (Attrib.binding * string) list ->
    58     (Facts.ref * Token.src list) list ->
    59     local_theory -> inductive_result * local_theory
    60   val add_inductive_global: inductive_flags ->
    61     ((binding * typ) * mixfix) list -> (string * typ) list -> (Attrib.binding * term) list ->
    62     thm list -> theory -> inductive_result * theory
    63   val arities_of: thm -> (string * int) list
    64   val params_of: thm -> term list
    65   val partition_rules: thm -> thm list -> (string * thm list) list
    66   val partition_rules': thm -> (thm * 'a) list -> (string * (thm * 'a) list) list
    67   val unpartition_rules: thm list -> (string * 'a list) list -> 'a list
    68   val infer_intro_vars: thm -> int -> thm list -> term list list
    69 end;
    70 
    71 signature INDUCTIVE =
    72 sig
    73   include BASIC_INDUCTIVE
    74   val select_disj_tac: Proof.context -> int -> int -> int -> tactic
    75   type add_ind_def =
    76     inductive_flags ->
    77     term list -> (Attrib.binding * term) list -> thm list ->
    78     term list -> (binding * mixfix) list ->
    79     local_theory -> inductive_result * local_theory
    80   val declare_rules: binding -> bool -> bool -> string list -> term list ->
    81     thm list -> binding list -> Token.src list list -> (thm * string list * int) list ->
    82     thm list -> thm -> local_theory -> thm list * thm list * thm list * thm * thm list * local_theory
    83   val add_ind_def: add_ind_def
    84   val gen_add_inductive_i: add_ind_def -> inductive_flags ->
    85     ((binding * typ) * mixfix) list -> (string * typ) list -> (Attrib.binding * term) list ->
    86     thm list -> local_theory -> inductive_result * local_theory
    87   val gen_add_inductive: add_ind_def -> bool -> bool ->
    88     (binding * string option * mixfix) list ->
    89     (binding * string option * mixfix) list ->
    90     (Attrib.binding * string) list -> (Facts.ref * Token.src list) list ->
    91     local_theory -> inductive_result * local_theory
    92   val gen_ind_decl: add_ind_def -> bool -> (local_theory -> local_theory) parser
    93 end;
    94 
    95 structure Inductive: INDUCTIVE =
    96 struct
    97 
    98 (** theory context references **)
    99 
   100 val inductive_forall_def = @{thm HOL.induct_forall_def};
   101 val inductive_conj_def = @{thm HOL.induct_conj_def};
   102 val inductive_conj = @{thms induct_conj};
   103 val inductive_atomize = @{thms induct_atomize};
   104 val inductive_rulify = @{thms induct_rulify};
   105 val inductive_rulify_fallback = @{thms induct_rulify_fallback};
   106 
   107 val simp_thms1 =
   108   map mk_meta_eq
   109     @{lemma "(~ True) = False" "(~ False) = True"
   110         "(True --> P) = P" "(False --> P) = True"
   111         "(P & True) = P" "(True & P) = P"
   112       by (fact simp_thms)+};
   113 
   114 val simp_thms2 =
   115   map mk_meta_eq [@{thm inf_fun_def}, @{thm inf_bool_def}] @ simp_thms1;
   116 
   117 val simp_thms3 =
   118   map mk_meta_eq [@{thm le_fun_def}, @{thm le_bool_def}, @{thm sup_fun_def}, @{thm sup_bool_def}];
   119 
   120 
   121 
   122 (** misc utilities **)
   123 
   124 fun message quiet_mode s = if quiet_mode then () else writeln s;
   125 
   126 fun clean_message ctxt quiet_mode s =
   127   if Config.get ctxt quick_and_dirty then () else message quiet_mode s;
   128 
   129 fun coind_prefix true = "co"
   130   | coind_prefix false = "";
   131 
   132 fun log (b: int) m n = if m >= n then 0 else 1 + log b (b * m) n;
   133 
   134 fun make_bool_args f g [] i = []
   135   | make_bool_args f g (x :: xs) i =
   136       (if i mod 2 = 0 then f x else g x) :: make_bool_args f g xs (i div 2);
   137 
   138 fun make_bool_args' xs =
   139   make_bool_args (K @{term False}) (K @{term True}) xs;
   140 
   141 fun arg_types_of k c = drop k (binder_types (fastype_of c));
   142 
   143 fun find_arg T x [] = raise Fail "find_arg"
   144   | find_arg T x ((p as (_, (SOME _, _))) :: ps) =
   145       apsnd (cons p) (find_arg T x ps)
   146   | find_arg T x ((p as (U, (NONE, y))) :: ps) =
   147       if (T: typ) = U then (y, (U, (SOME x, y)) :: ps)
   148       else apsnd (cons p) (find_arg T x ps);
   149 
   150 fun make_args Ts xs =
   151   map (fn (T, (NONE, ())) => Const (@{const_name undefined}, T) | (_, (SOME t, ())) => t)
   152     (fold (fn (t, T) => snd o find_arg T t) xs (map (rpair (NONE, ())) Ts));
   153 
   154 fun make_args' Ts xs Us =
   155   fst (fold_map (fn T => find_arg T ()) Us (Ts ~~ map (pair NONE) xs));
   156 
   157 fun dest_predicate cs params t =
   158   let
   159     val k = length params;
   160     val (c, ts) = strip_comb t;
   161     val (xs, ys) = chop k ts;
   162     val i = find_index (fn c' => c' = c) cs;
   163   in
   164     if xs = params andalso i >= 0 then
   165       SOME (c, i, ys, chop (length ys) (arg_types_of k c))
   166     else NONE
   167   end;
   168 
   169 fun mk_names a 0 = []
   170   | mk_names a 1 = [a]
   171   | mk_names a n = map (fn i => a ^ string_of_int i) (1 upto n);
   172 
   173 fun select_disj_tac ctxt =
   174   let
   175     fun tacs 1 1 = []
   176       | tacs _ 1 = [resolve_tac ctxt @{thms disjI1}]
   177       | tacs n i = resolve_tac ctxt @{thms disjI2} :: tacs (n - 1) (i - 1);
   178   in fn n => fn i => EVERY' (tacs n i) end;
   179 
   180 
   181 
   182 (** context data **)
   183 
   184 type inductive_result =
   185   {preds: term list, elims: thm list, raw_induct: thm,
   186    induct: thm, inducts: thm list, intrs: thm list, eqs: thm list};
   187 
   188 fun transform_result phi {preds, elims, raw_induct: thm, induct, inducts, intrs, eqs} =
   189   let
   190     val term = Morphism.term phi;
   191     val thm = Morphism.thm phi;
   192     val fact = Morphism.fact phi;
   193   in
   194    {preds = map term preds, elims = fact elims, raw_induct = thm raw_induct,
   195     induct = thm induct, inducts = fact inducts, intrs = fact intrs, eqs = fact eqs}
   196   end;
   197 
   198 type inductive_info = {names: string list, coind: bool} * inductive_result;
   199 
   200 val empty_equations =
   201   Item_Net.init Thm.eq_thm_prop
   202     (single o fst o HOLogic.dest_eq o HOLogic.dest_Trueprop o Thm.prop_of);
   203 
   204 datatype data = Data of
   205  {infos: inductive_info Symtab.table,
   206   monos: thm list,
   207   equations: thm Item_Net.T};
   208 
   209 fun make_data (infos, monos, equations) =
   210   Data {infos = infos, monos = monos, equations = equations};
   211 
   212 structure Data = Generic_Data
   213 (
   214   type T = data;
   215   val empty = make_data (Symtab.empty, [], empty_equations);
   216   val extend = I;
   217   fun merge (Data {infos = infos1, monos = monos1, equations = equations1},
   218       Data {infos = infos2, monos = monos2, equations = equations2}) =
   219     make_data (Symtab.merge (K true) (infos1, infos2),
   220       Thm.merge_thms (monos1, monos2),
   221       Item_Net.merge (equations1, equations2));
   222 );
   223 
   224 fun map_data f =
   225   Data.map (fn Data {infos, monos, equations} => make_data (f (infos, monos, equations)));
   226 
   227 fun rep_data ctxt = Data.get (Context.Proof ctxt) |> (fn Data rep => rep);
   228 
   229 fun print_inductives verbose ctxt =
   230   let
   231     val {infos, monos, ...} = rep_data ctxt;
   232     val space = Consts.space_of (Proof_Context.consts_of ctxt);
   233   in
   234     [Pretty.block
   235       (Pretty.breaks
   236         (Pretty.str "(co)inductives:" ::
   237           map (Pretty.mark_str o #1)
   238             (Name_Space.markup_entries verbose ctxt space (Symtab.dest infos)))),
   239      Pretty.big_list "monotonicity rules:" (map (Display.pretty_thm_item ctxt) monos)]
   240   end |> Pretty.writeln_chunks;
   241 
   242 
   243 (* inductive info *)
   244 
   245 fun the_inductive ctxt name =
   246   (case Symtab.lookup (#infos (rep_data ctxt)) name of
   247     NONE => error ("Unknown (co)inductive predicate " ^ quote name)
   248   | SOME info => info);
   249 
   250 fun put_inductives names info =
   251   map_data (fn (infos, monos, equations) =>
   252     (fold (fn name => Symtab.update (name, info)) names infos, monos, equations));
   253 
   254 
   255 (* monotonicity rules *)
   256 
   257 val get_monos = #monos o rep_data;
   258 
   259 fun mk_mono ctxt thm =
   260   let
   261     fun eq_to_mono thm' = thm' RS (thm' RS @{thm eq_to_mono});
   262     fun dest_less_concl thm = dest_less_concl (thm RS @{thm le_funD})
   263       handle THM _ => thm RS @{thm le_boolD}
   264   in
   265     (case Thm.concl_of thm of
   266       Const (@{const_name Pure.eq}, _) $ _ $ _ => eq_to_mono (thm RS meta_eq_to_obj_eq)
   267     | _ $ (Const (@{const_name HOL.eq}, _) $ _ $ _) => eq_to_mono thm
   268     | _ $ (Const (@{const_name Orderings.less_eq}, _) $ _ $ _) =>
   269       dest_less_concl (Seq.hd (REPEAT (FIRSTGOAL
   270         (resolve_tac ctxt [@{thm le_funI}, @{thm le_boolI'}])) thm))
   271     | _ => thm)
   272   end handle THM _ => error ("Bad monotonicity theorem:\n" ^ Display.string_of_thm ctxt thm);
   273 
   274 val mono_add =
   275   Thm.declaration_attribute (fn thm => fn context =>
   276     map_data (fn (infos, monos, equations) =>
   277       (infos, Thm.add_thm (mk_mono (Context.proof_of context) thm) monos, equations)) context);
   278 
   279 val mono_del =
   280   Thm.declaration_attribute (fn thm => fn context =>
   281     map_data (fn (infos, monos, equations) =>
   282       (infos, Thm.del_thm (mk_mono (Context.proof_of context) thm) monos, equations)) context);
   283 
   284 val _ =
   285   Theory.setup
   286     (Attrib.setup @{binding mono} (Attrib.add_del mono_add mono_del)
   287       "declaration of monotonicity rule");
   288 
   289 
   290 (* equations *)
   291 
   292 val get_equations = #equations o rep_data;
   293 
   294 val equation_add_permissive =
   295   Thm.declaration_attribute (fn thm =>
   296     map_data (fn (infos, monos, equations) =>
   297       (infos, monos, perhaps (try (Item_Net.update thm)) equations)));
   298 
   299 
   300 
   301 (** process rules **)
   302 
   303 local
   304 
   305 fun err_in_rule ctxt name t msg =
   306   error (cat_lines ["Ill-formed introduction rule " ^ Binding.print name,
   307     Syntax.string_of_term ctxt t, msg]);
   308 
   309 fun err_in_prem ctxt name t p msg =
   310   error (cat_lines ["Ill-formed premise", Syntax.string_of_term ctxt p,
   311     "in introduction rule " ^ Binding.print name, Syntax.string_of_term ctxt t, msg]);
   312 
   313 val bad_concl = "Conclusion of introduction rule must be an inductive predicate";
   314 
   315 val bad_ind_occ = "Inductive predicate occurs in argument of inductive predicate";
   316 
   317 val bad_app = "Inductive predicate must be applied to parameter(s) ";
   318 
   319 fun atomize_term thy = Raw_Simplifier.rewrite_term thy inductive_atomize [];
   320 
   321 in
   322 
   323 fun check_rule ctxt cs params ((binding, att), rule) =
   324   let
   325     val params' = Term.variant_frees rule (Logic.strip_params rule);
   326     val frees = rev (map Free params');
   327     val concl = subst_bounds (frees, Logic.strip_assums_concl rule);
   328     val prems = map (curry subst_bounds frees) (Logic.strip_assums_hyp rule);
   329     val rule' = Logic.list_implies (prems, concl);
   330     val aprems = map (atomize_term (Proof_Context.theory_of ctxt)) prems;
   331     val arule = fold_rev (Logic.all o Free) params' (Logic.list_implies (aprems, concl));
   332 
   333     fun check_ind err t =
   334       (case dest_predicate cs params t of
   335         NONE => err (bad_app ^
   336           commas (map (Syntax.string_of_term ctxt) params))
   337       | SOME (_, _, ys, _) =>
   338           if exists (fn c => exists (fn t => Logic.occs (c, t)) ys) cs
   339           then err bad_ind_occ else ());
   340 
   341     fun check_prem' prem t =
   342       if member (op =) cs (head_of t) then
   343         check_ind (err_in_prem ctxt binding rule prem) t
   344       else
   345         (case t of
   346           Abs (_, _, t) => check_prem' prem t
   347         | t $ u => (check_prem' prem t; check_prem' prem u)
   348         | _ => ());
   349 
   350     fun check_prem (prem, aprem) =
   351       if can HOLogic.dest_Trueprop aprem then check_prem' prem prem
   352       else err_in_prem ctxt binding rule prem "Non-atomic premise";
   353 
   354     val _ =
   355       (case concl of
   356         Const (@{const_name Trueprop}, _) $ t =>
   357           if member (op =) cs (head_of t) then
   358            (check_ind (err_in_rule ctxt binding rule') t;
   359             List.app check_prem (prems ~~ aprems))
   360           else err_in_rule ctxt binding rule' bad_concl
   361        | _ => err_in_rule ctxt binding rule' bad_concl);
   362   in
   363     ((binding, att), arule)
   364   end;
   365 
   366 fun rulify ctxt =
   367   hol_simplify ctxt inductive_conj
   368   #> hol_simplify ctxt inductive_rulify
   369   #> hol_simplify ctxt inductive_rulify_fallback
   370   #> Simplifier.norm_hhf ctxt;
   371 
   372 end;
   373 
   374 
   375 
   376 (** proofs for (co)inductive predicates **)
   377 
   378 (* prove monotonicity *)
   379 
   380 fun prove_mono quiet_mode skip_mono predT fp_fun monos ctxt =
   381  (message (quiet_mode orelse skip_mono andalso Config.get ctxt quick_and_dirty)
   382     "  Proving monotonicity ...";
   383   (if skip_mono then Goal.prove_sorry else Goal.prove_future) ctxt
   384     [] []
   385     (HOLogic.mk_Trueprop
   386       (Const (@{const_name Orderings.mono}, (predT --> predT) --> HOLogic.boolT) $ fp_fun))
   387     (fn _ => EVERY [resolve_tac ctxt @{thms monoI} 1,
   388       REPEAT (resolve_tac ctxt [@{thm le_funI}, @{thm le_boolI'}] 1),
   389       REPEAT (FIRST
   390         [assume_tac ctxt 1,
   391          resolve_tac ctxt (map (mk_mono ctxt) monos @ get_monos ctxt) 1,
   392          eresolve_tac ctxt @{thms le_funE} 1,
   393          dresolve_tac ctxt @{thms le_boolD} 1])]));
   394 
   395 
   396 (* prove introduction rules *)
   397 
   398 fun prove_intrs quiet_mode coind mono fp_def k intr_ts rec_preds_defs ctxt ctxt' =
   399   let
   400     val _ = clean_message ctxt quiet_mode "  Proving the introduction rules ...";
   401 
   402     val unfold = funpow k (fn th => th RS fun_cong)
   403       (mono RS (fp_def RS
   404         (if coind then @{thm def_gfp_unfold} else @{thm def_lfp_unfold})));
   405 
   406     val rules = [refl, TrueI, @{lemma "~ False" by (rule notI)}, exI, conjI];
   407 
   408     val intrs = map_index (fn (i, intr) =>
   409       Goal.prove_sorry ctxt [] [] intr (fn _ => EVERY
   410        [rewrite_goals_tac ctxt rec_preds_defs,
   411         resolve_tac ctxt [unfold RS iffD2] 1,
   412         select_disj_tac ctxt (length intr_ts) (i + 1) 1,
   413         (*Not ares_tac, since refl must be tried before any equality assumptions;
   414           backtracking may occur if the premises have extra variables!*)
   415         DEPTH_SOLVE_1 (resolve_tac ctxt rules 1 APPEND assume_tac ctxt 1)])
   416        |> singleton (Proof_Context.export ctxt ctxt')) intr_ts
   417 
   418   in (intrs, unfold) end;
   419 
   420 
   421 (* prove elimination rules *)
   422 
   423 fun prove_elims quiet_mode cs params intr_ts intr_names unfold rec_preds_defs ctxt ctxt''' =
   424   let
   425     val _ = clean_message ctxt quiet_mode "  Proving the elimination rules ...";
   426 
   427     val ([pname], ctxt') = Variable.variant_fixes ["P"] ctxt;
   428     val P = HOLogic.mk_Trueprop (Free (pname, HOLogic.boolT));
   429 
   430     fun dest_intr r =
   431       (the (dest_predicate cs params (HOLogic.dest_Trueprop (Logic.strip_assums_concl r))),
   432        Logic.strip_assums_hyp r, Logic.strip_params r);
   433 
   434     val intrs = map dest_intr intr_ts ~~ intr_names;
   435 
   436     val rules1 = [disjE, exE, FalseE];
   437     val rules2 = [conjE, FalseE, @{lemma "~ True ==> R" by (rule notE [OF _ TrueI])}];
   438 
   439     fun prove_elim c =
   440       let
   441         val Ts = arg_types_of (length params) c;
   442         val (anames, ctxt'') = Variable.variant_fixes (mk_names "a" (length Ts)) ctxt';
   443         val frees = map Free (anames ~~ Ts);
   444 
   445         fun mk_elim_prem ((_, _, us, _), ts, params') =
   446           Logic.list_all (params',
   447             Logic.list_implies (map (HOLogic.mk_Trueprop o HOLogic.mk_eq)
   448               (frees ~~ us) @ ts, P));
   449         val c_intrs = filter (equal c o #1 o #1 o #1) intrs;
   450         val prems = HOLogic.mk_Trueprop (list_comb (c, params @ frees)) ::
   451            map mk_elim_prem (map #1 c_intrs)
   452       in
   453         (Goal.prove_sorry ctxt'' [] prems P
   454           (fn {context = ctxt4, prems} => EVERY
   455             [cut_tac (hd prems) 1,
   456              rewrite_goals_tac ctxt4 rec_preds_defs,
   457              dresolve_tac ctxt4 [unfold RS iffD1] 1,
   458              REPEAT (FIRSTGOAL (eresolve_tac ctxt4 rules1)),
   459              REPEAT (FIRSTGOAL (eresolve_tac ctxt4 rules2)),
   460              EVERY (map (fn prem =>
   461                DEPTH_SOLVE_1 (assume_tac ctxt4 1 ORELSE
   462                 resolve_tac ctxt [rewrite_rule ctxt4 rec_preds_defs prem, conjI] 1))
   463                 (tl prems))])
   464           |> singleton (Proof_Context.export ctxt'' ctxt'''),
   465          map #2 c_intrs, length Ts)
   466       end
   467 
   468    in map prove_elim cs end;
   469 
   470 
   471 (* prove simplification equations *)
   472 
   473 fun prove_eqs quiet_mode cs params intr_ts intrs
   474     (elims: (thm * bstring list * int) list) ctxt ctxt'' =  (* FIXME ctxt'' ?? *)
   475   let
   476     val _ = clean_message ctxt quiet_mode "  Proving the simplification rules ...";
   477 
   478     fun dest_intr r =
   479       (the (dest_predicate cs params (HOLogic.dest_Trueprop (Logic.strip_assums_concl r))),
   480        Logic.strip_assums_hyp r, Logic.strip_params r);
   481     val intr_ts' = map dest_intr intr_ts;
   482 
   483     fun prove_eq c (elim: thm * 'a * 'b) =
   484       let
   485         val Ts = arg_types_of (length params) c;
   486         val (anames, ctxt') = Variable.variant_fixes (mk_names "a" (length Ts)) ctxt;
   487         val frees = map Free (anames ~~ Ts);
   488         val c_intrs = filter (equal c o #1 o #1 o #1) (intr_ts' ~~ intrs);
   489         fun mk_intr_conj (((_, _, us, _), ts, params'), _) =
   490           let
   491             fun list_ex ([], t) = t
   492               | list_ex ((a, T) :: vars, t) =
   493                   HOLogic.exists_const T $ Abs (a, T, list_ex (vars, t));
   494             val conjs = map2 (curry HOLogic.mk_eq) frees us @ map HOLogic.dest_Trueprop ts;
   495           in
   496             list_ex (params', if null conjs then @{term True} else foldr1 HOLogic.mk_conj conjs)
   497           end;
   498         val lhs = list_comb (c, params @ frees);
   499         val rhs =
   500           if null c_intrs then @{term False}
   501           else foldr1 HOLogic.mk_disj (map mk_intr_conj c_intrs);
   502         val eq = HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, rhs));
   503         fun prove_intr1 (i, _) = Subgoal.FOCUS_PREMS (fn {context = ctxt'', params, prems, ...} =>
   504             select_disj_tac ctxt'' (length c_intrs) (i + 1) 1 THEN
   505             EVERY (replicate (length params) (resolve_tac ctxt'' @{thms exI} 1)) THEN
   506             (if null prems then resolve_tac ctxt'' @{thms TrueI} 1
   507              else
   508               let
   509                 val (prems', last_prem) = split_last prems;
   510               in
   511                 EVERY (map (fn prem =>
   512                   (resolve_tac ctxt'' @{thms conjI} 1 THEN resolve_tac ctxt'' [prem] 1)) prems')
   513                 THEN resolve_tac ctxt'' [last_prem] 1
   514               end)) ctxt' 1;
   515         fun prove_intr2 (((_, _, us, _), ts, params'), intr) =
   516           EVERY (replicate (length params') (eresolve_tac ctxt' @{thms exE} 1)) THEN
   517           (if null ts andalso null us then resolve_tac ctxt' [intr] 1
   518            else
   519             EVERY (replicate (length ts + length us - 1) (eresolve_tac ctxt' @{thms conjE} 1)) THEN
   520             Subgoal.FOCUS_PREMS (fn {context = ctxt'', prems, ...} =>
   521               let
   522                 val (eqs, prems') = chop (length us) prems;
   523                 val rew_thms = map (fn th => th RS @{thm eq_reflection}) eqs;
   524               in
   525                 rewrite_goal_tac ctxt'' rew_thms 1 THEN
   526                 resolve_tac ctxt'' [intr] 1 THEN
   527                 EVERY (map (fn p => resolve_tac ctxt'' [p] 1) prems')
   528               end) ctxt' 1);
   529       in
   530         Goal.prove_sorry ctxt' [] [] eq (fn _ =>
   531           resolve_tac ctxt' @{thms iffI} 1 THEN
   532           eresolve_tac ctxt' [#1 elim] 1 THEN
   533           EVERY (map_index prove_intr1 c_intrs) THEN
   534           (if null c_intrs then eresolve_tac ctxt' @{thms FalseE} 1
   535            else
   536             let val (c_intrs', last_c_intr) = split_last c_intrs in
   537               EVERY (map (fn ci => eresolve_tac ctxt' @{thms disjE} 1 THEN prove_intr2 ci) c_intrs')
   538               THEN prove_intr2 last_c_intr
   539             end))
   540         |> rulify ctxt'
   541         |> singleton (Proof_Context.export ctxt' ctxt'')
   542       end;
   543   in
   544     map2 prove_eq cs elims
   545   end;
   546 
   547 
   548 (* derivation of simplified elimination rules *)
   549 
   550 local
   551 
   552 (*delete needless equality assumptions*)
   553 val refl_thin = Goal.prove_global @{theory HOL} [] [] @{prop "!!P. a = a ==> P ==> P"}
   554   (fn {context = ctxt, ...} => assume_tac ctxt 1);
   555 val elim_rls = [asm_rl, FalseE, refl_thin, conjE, exE];
   556 fun elim_tac ctxt = REPEAT o eresolve_tac ctxt elim_rls;
   557 
   558 fun simp_case_tac ctxt i =
   559   EVERY' [elim_tac ctxt,
   560     asm_full_simp_tac ctxt,
   561     elim_tac ctxt,
   562     REPEAT o bound_hyp_subst_tac ctxt] i;
   563 
   564 in
   565 
   566 fun mk_cases_tac ctxt = ALLGOALS (simp_case_tac ctxt) THEN prune_params_tac ctxt;
   567 
   568 fun mk_cases ctxt prop =
   569   let
   570     fun err msg =
   571       error (Pretty.string_of (Pretty.block
   572         [Pretty.str msg, Pretty.fbrk, Syntax.pretty_term ctxt prop]));
   573 
   574     val elims = Induct.find_casesP ctxt prop;
   575 
   576     val cprop = Thm.cterm_of ctxt prop;
   577     fun mk_elim rl =
   578       Thm.implies_intr cprop
   579         (Tactic.rule_by_tactic ctxt (mk_cases_tac ctxt) (Thm.assume cprop RS rl))
   580       |> singleton (Variable.export (Variable.auto_fixes prop ctxt) ctxt);
   581   in
   582     (case get_first (try mk_elim) elims of
   583       SOME r => r
   584     | NONE => err "Proposition not an inductive predicate:")
   585   end;
   586 
   587 end;
   588 
   589 
   590 (* inductive_cases *)
   591 
   592 fun gen_inductive_cases prep_att prep_prop args lthy =
   593   let
   594     val thmss =
   595       map snd args
   596       |> burrow (grouped 10 Par_List.map_independent (mk_cases lthy o prep_prop lthy));
   597     val facts =
   598       map2 (fn ((a, atts), _) => fn thms => ((a, map (prep_att lthy) atts), [(thms, [])]))
   599         args thmss;
   600   in lthy |> Local_Theory.notes facts end;
   601 
   602 val inductive_cases = gen_inductive_cases Attrib.check_src Syntax.read_prop;
   603 val inductive_cases_i = gen_inductive_cases (K I) Syntax.check_prop;
   604 
   605 
   606 (* ind_cases *)
   607 
   608 fun ind_cases_rules ctxt raw_props raw_fixes =
   609   let
   610     val (props, ctxt' ) = Specification.read_props raw_props raw_fixes ctxt;
   611     val rules = Proof_Context.export ctxt' ctxt (map (mk_cases ctxt') props);
   612   in rules end;
   613 
   614 val _ =
   615   Theory.setup
   616     (Method.setup @{binding ind_cases}
   617       (Scan.lift (Scan.repeat1 Parse.prop -- Parse.for_fixes) >>
   618         (fn (props, fixes) => fn ctxt =>
   619           Method.erule ctxt 0 (ind_cases_rules ctxt props fixes)))
   620       "case analysis for inductive definitions, based on simplified elimination rule");
   621 
   622 
   623 (* derivation of simplified equation *)
   624 
   625 fun mk_simp_eq ctxt prop =
   626   let
   627     val thy = Proof_Context.theory_of ctxt;
   628     val ctxt' = Variable.auto_fixes prop ctxt;
   629     val lhs_of = fst o HOLogic.dest_eq o HOLogic.dest_Trueprop o Thm.prop_of;
   630     val substs =
   631       Item_Net.retrieve (get_equations ctxt) (HOLogic.dest_Trueprop prop)
   632       |> map_filter
   633         (fn eq => SOME (Pattern.match thy (lhs_of eq, HOLogic.dest_Trueprop prop)
   634             (Vartab.empty, Vartab.empty), eq)
   635           handle Pattern.MATCH => NONE);
   636     val (subst, eq) =
   637       (case substs of
   638         [s] => s
   639       | _ => error
   640         ("equations matching pattern " ^ Syntax.string_of_term ctxt prop ^ " is not unique"));
   641     val inst =
   642       map (fn v => apply2 (Thm.cterm_of ctxt') (Var v, Envir.subst_term subst (Var v)))
   643         (Term.add_vars (lhs_of eq) []);
   644   in
   645     Drule.cterm_instantiate inst eq
   646     |> Conv.fconv_rule (Conv.arg_conv (Conv.arg_conv (Simplifier.full_rewrite ctxt)))
   647     |> singleton (Variable.export ctxt' ctxt)
   648   end
   649 
   650 
   651 (* inductive simps *)
   652 
   653 fun gen_inductive_simps prep_att prep_prop args lthy =
   654   let
   655     val facts = args |> map (fn ((a, atts), props) =>
   656       ((a, map (prep_att lthy) atts),
   657         map (Thm.no_attributes o single o mk_simp_eq lthy o prep_prop lthy) props));
   658   in lthy |> Local_Theory.notes facts end;
   659 
   660 val inductive_simps = gen_inductive_simps Attrib.check_src Syntax.read_prop;
   661 val inductive_simps_i = gen_inductive_simps (K I) Syntax.check_prop;
   662 
   663 
   664 (* prove induction rule *)
   665 
   666 fun prove_indrule quiet_mode cs argTs bs xs rec_const params intr_ts mono
   667     fp_def rec_preds_defs ctxt ctxt''' =  (* FIXME ctxt''' ?? *)
   668   let
   669     val _ = clean_message ctxt quiet_mode "  Proving the induction rule ...";
   670 
   671     (* predicates for induction rule *)
   672 
   673     val (pnames, ctxt') = Variable.variant_fixes (mk_names "P" (length cs)) ctxt;
   674     val preds =
   675       map2 (curry Free) pnames
   676         (map (fn c => arg_types_of (length params) c ---> HOLogic.boolT) cs);
   677 
   678     (* transform an introduction rule into a premise for induction rule *)
   679 
   680     fun mk_ind_prem r =
   681       let
   682         fun subst s =
   683           (case dest_predicate cs params s of
   684             SOME (_, i, ys, (_, Ts)) =>
   685               let
   686                 val k = length Ts;
   687                 val bs = map Bound (k - 1 downto 0);
   688                 val P = list_comb (nth preds i, map (incr_boundvars k) ys @ bs);
   689                 val Q =
   690                   fold_rev Term.abs (mk_names "x" k ~~ Ts)
   691                     (HOLogic.mk_binop @{const_name HOL.induct_conj}
   692                       (list_comb (incr_boundvars k s, bs), P));
   693               in (Q, case Ts of [] => SOME (s, P) | _ => NONE) end
   694           | NONE =>
   695               (case s of
   696                 t $ u => (fst (subst t) $ fst (subst u), NONE)
   697               | Abs (a, T, t) => (Abs (a, T, fst (subst t)), NONE)
   698               | _ => (s, NONE)));
   699 
   700         fun mk_prem s prems =
   701           (case subst s of
   702             (_, SOME (t, u)) => t :: u :: prems
   703           | (t, _) => t :: prems);
   704 
   705         val SOME (_, i, ys, _) =
   706           dest_predicate cs params (HOLogic.dest_Trueprop (Logic.strip_assums_concl r));
   707       in
   708         fold_rev (Logic.all o Free) (Logic.strip_params r)
   709           (Logic.list_implies (map HOLogic.mk_Trueprop (fold_rev mk_prem
   710             (map HOLogic.dest_Trueprop (Logic.strip_assums_hyp r)) []),
   711               HOLogic.mk_Trueprop (list_comb (nth preds i, ys))))
   712       end;
   713 
   714     val ind_prems = map mk_ind_prem intr_ts;
   715 
   716 
   717     (* make conclusions for induction rules *)
   718 
   719     val Tss = map (binder_types o fastype_of) preds;
   720     val (xnames, ctxt'') = Variable.variant_fixes (mk_names "x" (length (flat Tss))) ctxt';
   721     val mutual_ind_concl =
   722       HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
   723         (map (fn (((xnames, Ts), c), P) =>
   724           let val frees = map Free (xnames ~~ Ts)
   725           in HOLogic.mk_imp (list_comb (c, params @ frees), list_comb (P, frees)) end)
   726         (unflat Tss xnames ~~ Tss ~~ cs ~~ preds)));
   727 
   728 
   729     (* make predicate for instantiation of abstract induction rule *)
   730 
   731     val ind_pred =
   732       fold_rev lambda (bs @ xs) (foldr1 HOLogic.mk_conj
   733         (map_index (fn (i, P) => fold_rev (curry HOLogic.mk_imp)
   734            (make_bool_args HOLogic.mk_not I bs i)
   735            (list_comb (P, make_args' argTs xs (binder_types (fastype_of P))))) preds));
   736 
   737     val ind_concl =
   738       HOLogic.mk_Trueprop
   739         (HOLogic.mk_binrel @{const_name Orderings.less_eq} (rec_const, ind_pred));
   740 
   741     val raw_fp_induct = mono RS (fp_def RS @{thm def_lfp_induct});
   742 
   743     val induct = Goal.prove_sorry ctxt'' [] ind_prems ind_concl
   744       (fn {context = ctxt3, prems} => EVERY
   745         [rewrite_goals_tac ctxt3 [inductive_conj_def],
   746          DETERM (resolve_tac ctxt3 [raw_fp_induct] 1),
   747          REPEAT (resolve_tac ctxt3 [@{thm le_funI}, @{thm le_boolI}] 1),
   748          rewrite_goals_tac ctxt3 simp_thms2,
   749          (*This disjE separates out the introduction rules*)
   750          REPEAT (FIRSTGOAL (eresolve_tac ctxt3 [disjE, exE, FalseE])),
   751          (*Now break down the individual cases.  No disjE here in case
   752            some premise involves disjunction.*)
   753          REPEAT (FIRSTGOAL (eresolve_tac ctxt3 [conjE] ORELSE' bound_hyp_subst_tac ctxt3)),
   754          REPEAT (FIRSTGOAL
   755            (resolve_tac ctxt3 [conjI, impI] ORELSE'
   756            (eresolve_tac ctxt3 [notE] THEN' assume_tac ctxt3))),
   757          EVERY (map (fn prem =>
   758             DEPTH_SOLVE_1 (assume_tac ctxt3 1 ORELSE
   759               resolve_tac ctxt3
   760                 [rewrite_rule ctxt3 (inductive_conj_def :: rec_preds_defs @ simp_thms2) prem,
   761                   conjI, refl] 1)) prems)]);
   762 
   763     val lemma = Goal.prove_sorry ctxt'' [] []
   764       (Logic.mk_implies (ind_concl, mutual_ind_concl)) (fn {context = ctxt3, ...} => EVERY
   765         [rewrite_goals_tac ctxt3 rec_preds_defs,
   766          REPEAT (EVERY
   767            [REPEAT (resolve_tac ctxt3 [conjI, impI] 1),
   768             REPEAT (eresolve_tac ctxt3 [@{thm le_funE}, @{thm le_boolE}] 1),
   769             assume_tac ctxt3 1,
   770             rewrite_goals_tac ctxt3 simp_thms1,
   771             assume_tac ctxt3 1])]);
   772 
   773   in singleton (Proof_Context.export ctxt'' ctxt''') (induct RS lemma) end;
   774 
   775 
   776 
   777 (** specification of (co)inductive predicates **)
   778 
   779 fun mk_ind_def quiet_mode skip_mono alt_name coind cs intr_ts monos params cnames_syn lthy =
   780   let
   781     val fp_name = if coind then @{const_name Inductive.gfp} else @{const_name Inductive.lfp};
   782 
   783     val argTs = fold (combine (op =) o arg_types_of (length params)) cs [];
   784     val k = log 2 1 (length cs);
   785     val predT = replicate k HOLogic.boolT ---> argTs ---> HOLogic.boolT;
   786     val p :: xs =
   787       map Free (Variable.variant_frees lthy intr_ts
   788         (("p", predT) :: (mk_names "x" (length argTs) ~~ argTs)));
   789     val bs =
   790       map Free (Variable.variant_frees lthy (p :: xs @ intr_ts)
   791         (map (rpair HOLogic.boolT) (mk_names "b" k)));
   792 
   793     fun subst t =
   794       (case dest_predicate cs params t of
   795         SOME (_, i, ts, (Ts, Us)) =>
   796           let
   797             val l = length Us;
   798             val zs = map Bound (l - 1 downto 0);
   799           in
   800             fold_rev (Term.abs o pair "z") Us
   801               (list_comb (p,
   802                 make_bool_args' bs i @ make_args argTs
   803                   ((map (incr_boundvars l) ts ~~ Ts) @ (zs ~~ Us))))
   804           end
   805       | NONE =>
   806           (case t of
   807             t1 $ t2 => subst t1 $ subst t2
   808           | Abs (x, T, u) => Abs (x, T, subst u)
   809           | _ => t));
   810 
   811     (* transform an introduction rule into a conjunction  *)
   812     (*   [| p_i t; ... |] ==> p_j u                       *)
   813     (* is transformed into                                *)
   814     (*   b_j & x_j = u & p b_j t & ...                    *)
   815 
   816     fun transform_rule r =
   817       let
   818         val SOME (_, i, ts, (Ts, _)) =
   819           dest_predicate cs params (HOLogic.dest_Trueprop (Logic.strip_assums_concl r));
   820         val ps =
   821           make_bool_args HOLogic.mk_not I bs i @
   822           map HOLogic.mk_eq (make_args' argTs xs Ts ~~ ts) @
   823           map (subst o HOLogic.dest_Trueprop) (Logic.strip_assums_hyp r);
   824       in
   825         fold_rev (fn (x, T) => fn P => HOLogic.exists_const T $ Abs (x, T, P))
   826           (Logic.strip_params r)
   827           (if null ps then @{term True} else foldr1 HOLogic.mk_conj ps)
   828       end;
   829 
   830     (* make a disjunction of all introduction rules *)
   831 
   832     val fp_fun =
   833       fold_rev lambda (p :: bs @ xs)
   834         (if null intr_ts then @{term False}
   835          else foldr1 HOLogic.mk_disj (map transform_rule intr_ts));
   836 
   837     (* add definiton of recursive predicates to theory *)
   838 
   839     val rec_name =
   840       if Binding.is_empty alt_name then
   841         Binding.name (space_implode "_" (map (Binding.name_of o fst) cnames_syn))
   842       else alt_name;
   843 
   844     val is_auxiliary = length cs >= 2; 
   845     val ((rec_const, (_, fp_def)), lthy') = lthy
   846       |> is_auxiliary ? Proof_Context.concealed
   847       |> Local_Theory.define
   848         ((rec_name, case cnames_syn of [(_, syn)] => syn | _ => NoSyn),
   849          ((Binding.concealed (Thm.def_binding rec_name), @{attributes [nitpick_unfold]}),
   850            fold_rev lambda params
   851              (Const (fp_name, (predT --> predT) --> predT) $ fp_fun)))
   852       ||> Proof_Context.restore_naming lthy;
   853     val fp_def' =
   854       Simplifier.rewrite (put_simpset HOL_basic_ss lthy' addsimps [fp_def])
   855         (Thm.cterm_of lthy' (list_comb (rec_const, params)));
   856     val specs =
   857       if length cs < 2 then []
   858       else
   859         map_index (fn (i, (name_mx, c)) =>
   860           let
   861             val Ts = arg_types_of (length params) c;
   862             val xs =
   863               map Free (Variable.variant_frees lthy intr_ts (mk_names "x" (length Ts) ~~ Ts));
   864           in
   865             (name_mx, (apfst Binding.concealed Attrib.empty_binding, fold_rev lambda (params @ xs)
   866               (list_comb (rec_const, params @ make_bool_args' bs i @
   867                 make_args argTs (xs ~~ Ts)))))
   868           end) (cnames_syn ~~ cs);
   869     val (consts_defs, lthy'') = lthy'
   870       |> fold_map Local_Theory.define specs;
   871     val preds = (case cs of [_] => [rec_const] | _ => map #1 consts_defs);
   872 
   873     val (_, lthy''') = Variable.add_fixes (map (fst o dest_Free) params) lthy'';
   874     val mono = prove_mono quiet_mode skip_mono predT fp_fun monos lthy''';
   875     val (_, lthy'''') =
   876       Local_Theory.note (apfst Binding.concealed Attrib.empty_binding,
   877         Proof_Context.export lthy''' lthy'' [mono]) lthy'';
   878 
   879   in (lthy'''', lthy''', rec_name, mono, fp_def', map (#2 o #2) consts_defs,
   880     list_comb (rec_const, params), preds, argTs, bs, xs)
   881   end;
   882 
   883 fun declare_rules rec_binding coind no_ind cnames
   884     preds intrs intr_bindings intr_atts elims eqs raw_induct lthy =
   885   let
   886     val rec_name = Binding.name_of rec_binding;
   887     fun rec_qualified qualified = Binding.qualify qualified rec_name;
   888     val intr_names = map Binding.name_of intr_bindings;
   889     val ind_case_names = Rule_Cases.case_names intr_names;
   890     val induct =
   891       if coind then
   892         (raw_induct,
   893          [Rule_Cases.case_names [rec_name],
   894           Rule_Cases.case_conclusion (rec_name, intr_names),
   895           Rule_Cases.consumes (1 - Thm.nprems_of raw_induct),
   896           Induct.coinduct_pred (hd cnames)])
   897       else if no_ind orelse length cnames > 1 then
   898         (raw_induct,
   899           [ind_case_names, Rule_Cases.consumes (~ (Thm.nprems_of raw_induct))])
   900       else
   901         (raw_induct RSN (2, rev_mp),
   902           [ind_case_names, Rule_Cases.consumes (~ (Thm.nprems_of raw_induct))]);
   903 
   904     val (intrs', lthy1) =
   905       lthy |>
   906       Spec_Rules.add
   907         (if coind then Spec_Rules.Co_Inductive else Spec_Rules.Inductive) (preds, intrs) |>
   908       Local_Theory.notes
   909         (map (rec_qualified false) intr_bindings ~~ intr_atts ~~
   910           map (fn th => [([th],
   911            [Attrib.internal (K (Context_Rules.intro_query NONE))])]) intrs) |>>
   912       map (hd o snd);
   913     val (((_, elims'), (_, [induct'])), lthy2) =
   914       lthy1 |>
   915       Local_Theory.note ((rec_qualified true (Binding.name "intros"), []), intrs') ||>>
   916       fold_map (fn (name, (elim, cases, k)) =>
   917         Local_Theory.note
   918           ((Binding.qualify true (Long_Name.base_name name) (Binding.name "cases"),
   919             [Attrib.internal (K (Rule_Cases.case_names cases)),
   920              Attrib.internal (K (Rule_Cases.consumes (1 - Thm.nprems_of elim))),
   921              Attrib.internal (K (Rule_Cases.constraints k)),
   922              Attrib.internal (K (Induct.cases_pred name)),
   923              Attrib.internal (K (Context_Rules.elim_query NONE))]), [elim]) #>
   924         apfst (hd o snd)) (if null elims then [] else cnames ~~ elims) ||>>
   925       Local_Theory.note
   926         ((rec_qualified true (Binding.name (coind_prefix coind ^ "induct")),
   927           map (Attrib.internal o K) (#2 induct)), [rulify lthy1 (#1 induct)]);
   928 
   929     val (eqs', lthy3) = lthy2 |>
   930       fold_map (fn (name, eq) => Local_Theory.note
   931           ((Binding.qualify true (Long_Name.base_name name) (Binding.name "simps"),
   932             [Attrib.internal (K equation_add_permissive)]), [eq])
   933           #> apfst (hd o snd))
   934         (if null eqs then [] else (cnames ~~ eqs))
   935     val (inducts, lthy4) =
   936       if no_ind orelse coind then ([], lthy3)
   937       else
   938         let val inducts = cnames ~~ Project_Rule.projects lthy3 (1 upto length cnames) induct' in
   939           lthy3 |>
   940           Local_Theory.notes [((rec_qualified true (Binding.name "inducts"), []),
   941             inducts |> map (fn (name, th) => ([th],
   942               [Attrib.internal (K ind_case_names),
   943                Attrib.internal (K (Rule_Cases.consumes (1 - Thm.nprems_of th))),
   944                Attrib.internal (K (Induct.induct_pred name))])))] |>> snd o hd
   945         end;
   946   in (intrs', elims', eqs', induct', inducts, lthy4) end;
   947 
   948 type inductive_flags =
   949   {quiet_mode: bool, verbose: bool, alt_name: binding, coind: bool,
   950     no_elim: bool, no_ind: bool, skip_mono: bool};
   951 
   952 type add_ind_def =
   953   inductive_flags ->
   954   term list -> (Attrib.binding * term) list -> thm list ->
   955   term list -> (binding * mixfix) list ->
   956   local_theory -> inductive_result * local_theory;
   957 
   958 fun add_ind_def {quiet_mode, verbose, alt_name, coind, no_elim, no_ind, skip_mono}
   959     cs intros monos params cnames_syn lthy =
   960   let
   961     val _ = null cnames_syn andalso error "No inductive predicates given";
   962     val names = map (Binding.name_of o fst) cnames_syn;
   963     val _ = message (quiet_mode andalso not verbose)
   964       ("Proofs for " ^ coind_prefix coind ^ "inductive predicate(s) " ^ commas_quote names);
   965 
   966     val cnames = map (Local_Theory.full_name lthy o #1) cnames_syn;  (* FIXME *)
   967     val ((intr_names, intr_atts), intr_ts) =
   968       apfst split_list (split_list (map (check_rule lthy cs params) intros));
   969 
   970     val (lthy1, lthy2, rec_name, mono, fp_def, rec_preds_defs, rec_const, preds,
   971       argTs, bs, xs) = mk_ind_def quiet_mode skip_mono alt_name coind cs intr_ts
   972         monos params cnames_syn lthy;
   973 
   974     val (intrs, unfold) = prove_intrs quiet_mode coind mono fp_def (length bs + length xs)
   975       intr_ts rec_preds_defs lthy2 lthy1;
   976     val elims =
   977       if no_elim then []
   978       else
   979         prove_elims quiet_mode cs params intr_ts (map Binding.name_of intr_names)
   980           unfold rec_preds_defs lthy2 lthy1;
   981     val raw_induct = zero_var_indexes
   982       (if no_ind then Drule.asm_rl
   983        else if coind then
   984          singleton (Proof_Context.export lthy2 lthy1)
   985            (rotate_prems ~1 (Object_Logic.rulify lthy2
   986              (fold_rule lthy2 rec_preds_defs
   987                (rewrite_rule lthy2 simp_thms3
   988                 (mono RS (fp_def RS @{thm def_coinduct}))))))
   989        else
   990          prove_indrule quiet_mode cs argTs bs xs rec_const params intr_ts mono fp_def
   991            rec_preds_defs lthy2 lthy1);
   992     val eqs =
   993       if no_elim then [] else prove_eqs quiet_mode cs params intr_ts intrs elims lthy2 lthy1;
   994 
   995     val elims' = map (fn (th, ns, i) => (rulify lthy1 th, ns, i)) elims;
   996     val intrs' = map (rulify lthy1) intrs;
   997 
   998     val (intrs'', elims'', eqs', induct, inducts, lthy3) =
   999       declare_rules rec_name coind no_ind
  1000         cnames preds intrs' intr_names intr_atts elims' eqs raw_induct lthy1;
  1001 
  1002     val result =
  1003       {preds = preds,
  1004        intrs = intrs'',
  1005        elims = elims'',
  1006        raw_induct = rulify lthy3 raw_induct,
  1007        induct = induct,
  1008        inducts = inducts,
  1009        eqs = eqs'};
  1010 
  1011     val lthy4 = lthy3
  1012       |> Local_Theory.declaration {syntax = false, pervasive = false} (fn phi =>
  1013         let val result' = transform_result phi result;
  1014         in put_inductives cnames (*global names!?*) ({names = cnames, coind = coind}, result') end);
  1015   in (result, lthy4) end;
  1016 
  1017 
  1018 (* external interfaces *)
  1019 
  1020 fun gen_add_inductive_i mk_def
  1021     flags cnames_syn pnames spec monos lthy =
  1022   let
  1023 
  1024     (* abbrevs *)
  1025 
  1026     val (_, ctxt1) = Variable.add_fixes (map (Binding.name_of o fst o fst) cnames_syn) lthy;
  1027 
  1028     fun get_abbrev ((name, atts), t) =
  1029       if can (Logic.strip_assums_concl #> Logic.dest_equals) t then
  1030         let
  1031           val _ = Binding.is_empty name andalso null atts orelse
  1032             error "Abbreviations may not have names or attributes";
  1033           val ((x, T), rhs) = Local_Defs.abs_def (snd (Local_Defs.cert_def ctxt1 t));
  1034           val var =
  1035             (case find_first (fn ((c, _), _) => Binding.name_of c = x) cnames_syn of
  1036               NONE => error ("Undeclared head of abbreviation " ^ quote x)
  1037             | SOME ((b, T'), mx) =>
  1038                 if T <> T' then error ("Bad type specification for abbreviation " ^ quote x)
  1039                 else (b, mx));
  1040         in SOME (var, rhs) end
  1041       else NONE;
  1042 
  1043     val abbrevs = map_filter get_abbrev spec;
  1044     val bs = map (Binding.name_of o fst o fst) abbrevs;
  1045 
  1046 
  1047     (* predicates *)
  1048 
  1049     val pre_intros = filter_out (is_some o get_abbrev) spec;
  1050     val cnames_syn' = filter_out (member (op =) bs o Binding.name_of o fst o fst) cnames_syn;
  1051     val cs = map (Free o apfst Binding.name_of o fst) cnames_syn';
  1052     val ps = map Free pnames;
  1053 
  1054     val (_, ctxt2) = lthy |> Variable.add_fixes (map (Binding.name_of o fst o fst) cnames_syn');
  1055     val _ = map (fn abbr => Local_Defs.fixed_abbrev abbr ctxt2) abbrevs;
  1056     val ctxt3 = ctxt2 |> fold (snd oo Local_Defs.fixed_abbrev) abbrevs;
  1057     val expand = Assumption.export_term ctxt3 lthy #> Proof_Context.cert_term lthy;
  1058 
  1059     fun close_rule r =
  1060       fold (Logic.all o Free) (fold_aterms
  1061         (fn t as Free (v as (s, _)) =>
  1062             if Variable.is_fixed ctxt1 s orelse
  1063               member (op =) ps t then I else insert (op =) v
  1064           | _ => I) r []) r;
  1065 
  1066     val intros = map (apsnd (Syntax.check_term lthy #> close_rule #> expand)) pre_intros;
  1067     val preds = map (fn ((c, _), mx) => (c, mx)) cnames_syn';
  1068   in
  1069     lthy
  1070     |> mk_def flags cs intros monos ps preds
  1071     ||> fold (snd oo Local_Theory.abbrev Syntax.mode_default) abbrevs
  1072   end;
  1073 
  1074 fun gen_add_inductive mk_def verbose coind cnames_syn pnames_syn intro_srcs raw_monos lthy =
  1075   let
  1076     val ((vars, intrs), _) = lthy
  1077       |> Proof_Context.set_mode Proof_Context.mode_abbrev
  1078       |> Specification.read_spec (cnames_syn @ pnames_syn) intro_srcs;
  1079     val (cs, ps) = chop (length cnames_syn) vars;
  1080     val monos = Attrib.eval_thms lthy raw_monos;
  1081     val flags =
  1082      {quiet_mode = false, verbose = verbose, alt_name = Binding.empty,
  1083       coind = coind, no_elim = false, no_ind = false, skip_mono = false};
  1084   in
  1085     lthy
  1086     |> gen_add_inductive_i mk_def flags cs (map (apfst Binding.name_of o fst) ps) intrs monos
  1087   end;
  1088 
  1089 val add_inductive_i = gen_add_inductive_i add_ind_def;
  1090 val add_inductive = gen_add_inductive add_ind_def;
  1091 
  1092 fun add_inductive_global flags cnames_syn pnames pre_intros monos thy =
  1093   let
  1094     val name = Sign.full_name thy (fst (fst (hd cnames_syn)));
  1095     val ctxt' = thy
  1096       |> Named_Target.theory_init
  1097       |> add_inductive_i flags cnames_syn pnames pre_intros monos |> snd
  1098       |> Local_Theory.exit;
  1099     val info = #2 (the_inductive ctxt' name);
  1100   in (info, Proof_Context.theory_of ctxt') end;
  1101 
  1102 
  1103 (* read off arities of inductive predicates from raw induction rule *)
  1104 fun arities_of induct =
  1105   map (fn (_ $ t $ u) =>
  1106       (fst (dest_Const (head_of t)), length (snd (strip_comb u))))
  1107     (HOLogic.dest_conj (HOLogic.dest_Trueprop (Thm.concl_of induct)));
  1108 
  1109 (* read off parameters of inductive predicate from raw induction rule *)
  1110 fun params_of induct =
  1111   let
  1112     val (_ $ t $ u :: _) = HOLogic.dest_conj (HOLogic.dest_Trueprop (Thm.concl_of induct));
  1113     val (_, ts) = strip_comb t;
  1114     val (_, us) = strip_comb u;
  1115   in
  1116     List.take (ts, length ts - length us)
  1117   end;
  1118 
  1119 val pname_of_intr =
  1120   Thm.concl_of #> HOLogic.dest_Trueprop #> head_of #> dest_Const #> fst;
  1121 
  1122 (* partition introduction rules according to predicate name *)
  1123 fun gen_partition_rules f induct intros =
  1124   fold_rev (fn r => AList.map_entry op = (pname_of_intr (f r)) (cons r)) intros
  1125     (map (rpair [] o fst) (arities_of induct));
  1126 
  1127 val partition_rules = gen_partition_rules I;
  1128 fun partition_rules' induct = gen_partition_rules fst induct;
  1129 
  1130 fun unpartition_rules intros xs =
  1131   fold_map (fn r => AList.map_entry_yield op = (pname_of_intr r)
  1132     (fn x :: xs => (x, xs)) #>> the) intros xs |> fst;
  1133 
  1134 (* infer order of variables in intro rules from order of quantifiers in elim rule *)
  1135 fun infer_intro_vars elim arity intros =
  1136   let
  1137     val thy = Thm.theory_of_thm elim;
  1138     val _ :: cases = Thm.prems_of elim;
  1139     val used = map (fst o fst) (Term.add_vars (Thm.prop_of elim) []);
  1140     fun mtch (t, u) =
  1141       let
  1142         val params = Logic.strip_params t;
  1143         val vars =
  1144           map (Var o apfst (rpair 0))
  1145             (Name.variant_list used (map fst params) ~~ map snd params);
  1146         val ts =
  1147           map (curry subst_bounds (rev vars))
  1148             (List.drop (Logic.strip_assums_hyp t, arity));
  1149         val us = Logic.strip_imp_prems u;
  1150         val tab =
  1151           fold (Pattern.first_order_match thy) (ts ~~ us) (Vartab.empty, Vartab.empty);
  1152       in
  1153         map (Envir.subst_term tab) vars
  1154       end
  1155   in
  1156     map (mtch o apsnd Thm.prop_of) (cases ~~ intros)
  1157   end;
  1158 
  1159 
  1160 
  1161 (** outer syntax **)
  1162 
  1163 fun gen_ind_decl mk_def coind =
  1164   Parse.fixes -- Parse.for_fixes --
  1165   Scan.optional Parse_Spec.where_alt_specs [] --
  1166   Scan.optional (@{keyword "monos"} |-- Parse.!!! Parse.xthms1) []
  1167   >> (fn (((preds, params), specs), monos) =>
  1168       (snd o gen_add_inductive mk_def true coind preds params specs monos));
  1169 
  1170 val ind_decl = gen_ind_decl add_ind_def;
  1171 
  1172 val _ =
  1173   Outer_Syntax.local_theory @{command_keyword inductive} "define inductive predicates"
  1174     (ind_decl false);
  1175 
  1176 val _ =
  1177   Outer_Syntax.local_theory @{command_keyword coinductive} "define coinductive predicates"
  1178     (ind_decl true);
  1179 
  1180 val _ =
  1181   Outer_Syntax.local_theory @{command_keyword inductive_cases}
  1182     "create simplified instances of elimination rules"
  1183     (Parse.and_list1 Parse_Spec.specs >> (snd oo inductive_cases));
  1184 
  1185 val _ =
  1186   Outer_Syntax.local_theory @{command_keyword inductive_simps}
  1187     "create simplification rules for inductive predicates"
  1188     (Parse.and_list1 Parse_Spec.specs >> (snd oo inductive_simps));
  1189 
  1190 val _ =
  1191   Outer_Syntax.command @{command_keyword print_inductives}
  1192     "print (co)inductive definitions and monotonicity rules"
  1193     (Parse.opt_bang >> (fn b => Toplevel.unknown_context o
  1194       Toplevel.keep (print_inductives b o Toplevel.context_of)));
  1195 
  1196 end;