# File datatype_package.ML

    (*  Title:      HOL/Tools/datatype_package.ML
        ID:         $Id: datatype_package.ML,v 1.101 2005/09/20 14:17:34 haftmann Exp $
        Author:     Stefan Berghofer, TU Muenchen

    Datatype package for Isabelle/HOL.
    *)

    signature BASIC_DATATYPE_PACKAGE =
    sig
      val induct_tac : string -> int -> tactic
      val induct_thm_tac : thm -> string -> int -> tactic
      val case_tac : string -> int -> tactic
      val distinct_simproc : simproc
    end;

    signature DATATYPE_PACKAGE =
    sig
      include BASIC_DATATYPE_PACKAGE
      val quiet_mode : bool ref
      val add_datatype : bool -> string list -> (string list * bstring * mixfix *
        (bstring * string list * mixfix) list) list -> theory -> theory *
          {distinct : thm list list,
           inject : thm list list,
           exhaustion : thm list,
           rec_thms : thm list,
           case_thms : thm list list,
           split_thms : (thm * thm) list,
           induction : thm,
           size : thm list,
           simps : thm list}
      val add_datatype_i : bool -> bool -> string list -> (string list * bstring * mixfix *
        (bstring * typ list * mixfix) list) list -> theory -> theory *
          {distinct : thm list list,
           inject : thm list list,
           exhaustion : thm list,
           rec_thms : thm list,
           case_thms : thm list list,
           split_thms : (thm * thm) list,
           induction : thm,
           size : thm list,
           simps : thm list}
      val rep_datatype_i : string list option -> (thm list * theory attribute list) list list ->
        (thm list * theory attribute list) list list -> (thm list * theory attribute list) ->
        theory -> theory *
          {distinct : thm list list,
           inject : thm list list,
           exhaustion : thm list,
           rec_thms : thm list,
           case_thms : thm list list,
           split_thms : (thm * thm) list,
           induction : thm,
           size : thm list,
           simps : thm list}
      val rep_datatype : string list option -> (thmref * Attrib.src list) list list ->
        (thmref * Attrib.src list) list list -> thmref * Attrib.src list -> theory -> theory *
          {distinct : thm list list,
           inject : thm list list,
           exhaustion : thm list,
           rec_thms : thm list,
           case_thms : thm list list,
           split_thms : (thm * thm) list,
           induction : thm,
           size : thm list,
           simps : thm list}
      val get_datatypes : theory -> DatatypeAux.datatype_info Symtab.table
      val print_datatypes : theory -> unit
      val datatype_info : theory -> string -> DatatypeAux.datatype_info option
      val datatype_info_err : theory -> string -> DatatypeAux.datatype_info
      val constrs_of : theory -> string -> term list option
      val case_const_of : theory -> string -> term option
      val weak_case_congs_of : theory -> thm list
      val setup: (theory -> theory) list
    end;

    structure DatatypePackage : DATATYPE_PACKAGE =
    struct

    open DatatypeAux;

    val quiet_mode = quiet_mode;

    (* data kind 'HOL/datatypes' *)

    structure DatatypesData = TheoryDataFun
    (struct
      val name = "HOL/datatypes";
      type T = datatype_info Symtab.table;

      val empty = Symtab.empty;
      val copy = I;
      val extend = I;
      fun merge _ tabs : T = Symtab.merge (K true) tabs;

      fun print sg tab =
        Pretty.writeln (Pretty.strs ("datatypes:" ::
          map #1 (NameSpace.extern_table (Sign.type_space sg, tab))));
    end);

    val get_datatypes = DatatypesData.get;
    val put_datatypes = DatatypesData.put;
    val print_datatypes = DatatypesData.print;

    (** theory information about datatypes **)

    val datatype_info = Symtab.lookup o get_datatypes;

    fun datatype_info_err thy name = (case datatype_info thy name of
          SOME info => info
        | NONE => error ("Unknown datatype " ^ quote name));

    fun constrs_of thy tname = (case datatype_info thy tname of
       SOME {index, descr, ...} =>
         let val (_, _, constrs) = valOf (AList.lookup (op =) descr index)
         in SOME (map (fn (cname, _) => Const (cname, Sign.the_const_type thy cname)) constrs)
         end
     | _ => NONE);

    fun case_const_of thy tname = (case datatype_info thy tname of
       SOME {case_name, ...} => SOME (Const (case_name, Sign.the_const_type thy case_name))
     | _ => NONE);

    val weak_case_congs_of = map (#weak_case_cong o #2) o Symtab.dest o get_datatypes;

    fun find_tname var Bi =
      let val frees = map dest_Free (term_frees Bi)
          val params = rename_wrt_term Bi (Logic.strip_params Bi);
      in case AList.lookup (op =) (frees @ params) var of
           NONE => error ("No such variable in subgoal: " ^ quote var)
         | SOME(Type (tn, _)) => tn
         | _ => error ("Cannot determine type of " ^ quote var)
      end;

    fun infer_tname state i aterm =
      let
        val sign = Thm.sign_of_thm state;
        val (_, _, Bi, _) = Thm.dest_state (state, i)
        val params = Logic.strip_params Bi;   (*params of subgoal i*)
        val params = rev (rename_wrt_term Bi params);   (*as they are printed*)
        val (types, sorts) = types_sorts state;
        fun types' (a, ~1) = (case AList.lookup (op =) params a of NONE => types(a, ~1) | sm => sm)
          | types' ixn = types ixn;
        val (ct, _) = read_def_cterm (sign, types', sorts) [] false (aterm, TypeInfer.logicT);
      in case #T (rep_cterm ct) of
           Type (tn, _) => tn
         | _ => error ("Cannot determine type of " ^ quote aterm)
      end;

    (*Warn if the (induction) variable occurs Free among the premises, which
      usually signals a mistake.  But calls the tactic either way!*)
    fun occs_in_prems tacf vars =
      SUBGOAL (fn (Bi, i) =>
               (if  exists (fn Free (a, _) => a mem vars)
                          (foldr add_term_frees [] (#2 (strip_context Bi)))
                 then warning "Induction variable occurs also among premises!"
                 else ();
                tacf i));

    (* generic induction tactic for datatypes *)

    local

    fun prep_var (Var (ixn, _), SOME x) = SOME (ixn, x)
      | prep_var _ = NONE;

    fun prep_inst (concl, xs) =	(*exception UnequalLengths *)
      let val vs = InductAttrib.vars_of concl
      in List.mapPartial prep_var (Library.drop (length vs - length xs, vs) ~~ xs) end;

    in

    fun gen_induct_tac inst_tac (varss, opt_rule) i state =
      let
        val (_, _, Bi, _) = Thm.dest_state (state, i);
        val {sign, ...} = Thm.rep_thm state;
        val (rule, rule_name) =
          (case opt_rule of
            SOME r => (r, "Induction rule")
          | NONE =>
              let val tn = find_tname (hd (List.mapPartial I (List.concat varss))) Bi
              in (#induction (datatype_info_err sign tn), "Induction rule for type " ^ tn) end);

        val concls = HOLogic.dest_concls (Thm.concl_of rule);
        val insts = List.concat (map prep_inst (concls ~~ varss)) handle UnequalLengths =>
          error (rule_name ^ " has different numbers of variables");
      in occs_in_prems (inst_tac insts rule) (map #2 insts) i state end;

    fun induct_tac s =
      gen_induct_tac Tactic.res_inst_tac'
        (map (Library.single o SOME) (Syntax.read_idents s), NONE);

    fun induct_thm_tac th s =
      gen_induct_tac Tactic.res_inst_tac'
        ([map SOME (Syntax.read_idents s)], SOME th);

    end;

    (* generic case tactic for datatypes *)

    fun case_inst_tac inst_tac t rule i state =
      let
        val _ $ Var (ixn, _) $ _ = HOLogic.dest_Trueprop
          (hd (Logic.strip_assums_hyp (hd (Thm.prems_of rule))));
      in inst_tac [(ixn, t)] rule i state end;

    fun gen_case_tac inst_tac (t, SOME rule) i state =
          case_inst_tac inst_tac t rule i state
      | gen_case_tac inst_tac (t, NONE) i state =
          let val tn = infer_tname state i t in
            if tn = HOLogic.boolN then inst_tac [(("P", 0), t)] case_split_thm i state
            else case_inst_tac inst_tac t
                   (#exhaustion (datatype_info_err (Thm.sign_of_thm state) tn))
                   i state
          end handle THM _ => Seq.empty;

    fun case_tac t = gen_case_tac Tactic.res_inst_tac' (t, NONE);

    (** Isar tactic emulations **)

    local

    val rule_spec = Scan.lift (Args.$$$ "rule" -- Args.$$$ ":");
    val opt_rule = Scan.option (rule_spec |-- Attrib.local_thm);

    val varss =
      Args.and_list (Scan.repeat (Scan.unless rule_spec (Scan.lift (Args.maybe Args.name))));

    val inst_tac = Method.bires_inst_tac false;

    fun induct_meth ctxt (varss, opt_rule) =
      gen_induct_tac (inst_tac ctxt) (varss, opt_rule);
    fun case_meth ctxt (varss, opt_rule) =
      gen_case_tac (inst_tac ctxt) (varss, opt_rule);

    in

    val tactic_emulations =
     [("induct_tac", Method.goal_args_ctxt' (varss -- opt_rule) induct_meth,
        "induct_tac emulation (dynamic instantiation)"),
      ("case_tac", Method.goal_args_ctxt' (Scan.lift Args.name -- opt_rule) case_meth,
        "case_tac emulation (dynamic instantiation)")];

    end;

    (** induct method setup **)

    (* case names *)

    local

    fun dt_recs (DtTFree _) = []
      | dt_recs (DtType (_, dts)) = List.concat (map dt_recs dts)
      | dt_recs (DtRec i) = [i];

    fun dt_cases (descr: descr) (_, args, constrs) =
      let
        fun the_bname i = Sign.base_name (#1 (valOf (AList.lookup (op =) descr i)));
        val bnames = map the_bname (distinct (List.concat (map dt_recs args)));
      in map (fn (c, _) => space_implode "_" (Sign.base_name c :: bnames)) constrs end;

    fun induct_cases descr =
      DatatypeProp.indexify_names (List.concat (map (dt_cases descr) (map #2 descr)));

    fun exhaust_cases descr i = dt_cases descr (valOf (AList.lookup (op =) descr i));

    in

    fun mk_case_names_induct descr = RuleCases.case_names (induct_cases descr);

    fun mk_case_names_exhausts descr new =
      map (RuleCases.case_names o exhaust_cases descr o #1)
        (List.filter (fn ((_, (name, _, _))) => name mem_string new) descr);

    end;

    fun name_notE th =
        Thm.name_thm (Thm.name_of_thm th ^ "_E", th RS notE);

    fun add_rules simps case_thms size_thms rec_thms inject distinct
                      weak_case_congs cong_att =
      (#1 o PureThy.add_thmss [(("simps", simps), []),
        (("", List.concat case_thms @ size_thms @
              List.concat distinct  @ rec_thms), [Simplifier.simp_add_global]),
        (("", size_thms @ rec_thms),             [RecfunCodegen.add NONE]),
        (("", List.concat inject),               [iff_add_global]),
        (("", map name_notE (List.concat distinct)),  [Classical.safe_elim_global]),
        (("", weak_case_congs),                  [cong_att])]);

    (* add_cases_induct *)

    fun add_cases_induct infos induction =
      let
        val n = length (HOLogic.dest_concls (Thm.concl_of induction));
        fun proj i thm =
          if n = 1 then thm
          else (if i + 1 < n then (fn th => th RS conjunct1) else I)
            (Library.funpow i (fn th => th RS conjunct2) thm)
            |> Drule.zero_var_indexes
            |> RuleCases.save thm;

        fun named_rules (name, {index, exhaustion, ...}: datatype_info) =
          [(("", proj index induction), [InductAttrib.induct_type_global name]),
           (("", exhaustion), [InductAttrib.cases_type_global name])];
        fun unnamed_rule i =
          (("", proj i induction), [InductAttrib.induct_type_global ""]);
        val rules = List.concat (map named_rules infos) @ map unnamed_rule (length infos upto n - 1);
      in #1 o PureThy.add_thms rules end;

    (**** simplification procedure for showing distinctness of constructors ****)

    fun stripT (i, Type ("fun", [_, T])) = stripT (i + 1, T)
      | stripT p = p;

    fun stripC (i, f $ x) = stripC (i + 1, f)
      | stripC p = p;

    val distinctN = "constr_distinct";

    exception ConstrDistinct of term;

    fun distinct_proc sg ss (t as Const ("op =", _) $ t1 $ t2) =
      (case (stripC (0, t1), stripC (0, t2)) of
         ((i, Const (cname1, T1)), (j, Const (cname2, T2))) =>
             (case (stripT (0, T1), stripT (0, T2)) of
                ((i', Type (tname1, _)), (j', Type (tname2, _))) =>
                    if tname1 = tname2 andalso not (cname1 = cname2) andalso i = i' andalso j = j' then
                       (case (constrs_of sg tname1) of
                          SOME constrs => let val cnames = map (fst o dest_Const) constrs
                            in if cname1 mem cnames andalso cname2 mem cnames then
                                 let val eq_t = Logic.mk_equals (t, Const ("False", HOLogic.boolT));
                                     val eq_ct = cterm_of sg eq_t;
                                     val Datatype_thy = theory "Datatype";
                                     val [In0_inject, In1_inject, In0_not_In1, In1_not_In0] =
                                       map (get_thm Datatype_thy o Name)
                                         ["In0_inject", "In1_inject", "In0_not_In1", "In1_not_In0"]
                                 in (case (#distinct (datatype_info_err sg tname1)) of
                                     QuickAndDirty => SOME (Thm.invoke_oracle
                                       Datatype_thy distinctN (sg, ConstrDistinct eq_t))
                                   | FewConstrs thms => SOME (Tactic.prove sg [] [] eq_t (K
                                       (EVERY [rtac eq_reflection 1, rtac iffI 1, rtac notE 1,
                                        atac 2, resolve_tac thms 1, etac FalseE 1])))
                                   | ManyConstrs (thm, simpset) => SOME (Tactic.prove sg [] [] eq_t (K
                                       (EVERY [rtac eq_reflection 1, rtac iffI 1, dtac thm 1,
                                        full_simp_tac (Simplifier.inherit_bounds ss simpset) 1,
                                        REPEAT (dresolve_tac [In0_inject, In1_inject] 1),
                                        eresolve_tac [In0_not_In1 RS notE, In1_not_In0 RS notE] 1,
                                        etac FalseE 1]))))
                                 end
                               else NONE
                            end
                        | NONE => NONE)
                    else NONE
              | _ => NONE)
       | _ => NONE)
      | distinct_proc sg _ _ = NONE;

    val distinct_simproc =
      Simplifier.simproc HOL.thy distinctN ["s = t"] distinct_proc;

    val dist_ss = HOL_ss addsimprocs [distinct_simproc];

    val simproc_setup =
      [Theory.add_oracle (distinctN, fn (_, ConstrDistinct t) => t),
       fn thy => (simpset_ref_of thy := simpset_of thy addsimprocs [distinct_simproc]; thy)];

    (**** translation rules for case ****)

    fun case_tr sg [t, u] =
        let
          fun case_error s name ts = raise TERM ("Error in case expression" ^
            getOpt (Option.map (curry op ^ " for datatype ") name, "") ^ ":\n" ^ s, ts);
          fun dest_case1 (Const ("_case1", _) $ t $ u) = (case strip_comb t of
                (Const (s, _), ts) => (Sign.intern_const sg s, ts)
              | (Free (s, _), ts) => (Sign.intern_const sg s, ts)
              | _ => case_error "Head is not a constructor" NONE [t, u], u)
            | dest_case1 t = raise TERM ("dest_case1", [t]);
          fun dest_case2 (Const ("_case2", _) $ t $ u) = t :: dest_case2 u
            | dest_case2 t = [t];
          val cases as ((cname, _), _) :: _ = map dest_case1 (dest_case2 u);
          val tab = Symtab.dest (get_datatypes sg);
          val (cases', default) = (case split_last cases of
              (cases', (("dummy_pattern", []), t)) => (cases', SOME t)
            | _ => (cases, NONE))
          fun abstr (Free (x, T), body) = Term.absfree (x, T, body)
            | abstr (Const ("_constrain", _) $ Free (x, T) $ tT, body) =
                Syntax.const Syntax.constrainAbsC $ Term.absfree (x, T, body) $ tT
            | abstr (Const ("Pair", _) $ x $ y, body) =
                Syntax.const "split" $ abstr (x, abstr (y, body))
            | abstr (t, _) = case_error "Illegal pattern" NONE [t];
        in case find_first (fn (_, {descr, index, ...}) =>
          exists (equal cname o fst) (#3 (snd (List.nth (descr, index))))) tab of
            NONE => case_error ("Not a datatype constructor: " ^ cname) NONE [u]
          | SOME (tname, {descr, case_name, index, ...}) =>
            let
              val _ = if exists (equal "dummy_pattern" o fst o fst) cases' then
                case_error "Illegal occurrence of '_' dummy pattern" (SOME tname) [u] else ();
              val (_, (_, dts, constrs)) = List.nth (descr, index);
              val sorts = map (rpair [] o dest_DtTFree) dts;
              fun find_case (cases, (s, dt)) =
                (case find_first (equal s o fst o fst) cases' of
                   NONE => (case default of
                       NONE => case_error ("No clause for constructor " ^ s) (SOME tname) [u]
                     | SOME t => (cases, list_abs (map (rpair dummyT) (DatatypeProp.make_tnames
                         (map (typ_of_dtyp descr sorts) dt)), t)))
                 | SOME (c as ((_, vs), t)) =>
                     if length dt <> length vs then
                        case_error ("Wrong number of arguments for constructor " ^ s)
                          (SOME tname) vs
                     else (cases \ c, foldr abstr t vs))
              val (cases'', fs) = foldl_map find_case (cases', constrs)
            in case (cases'', length constrs = length cases', default) of
                ([], true, SOME _) =>
                  case_error "Extra '_' dummy pattern" (SOME tname) [u]
              | (_ :: _, _, _) =>
                  let val extra = distinct (map (fst o fst) cases'')
                  in case extra \\ map fst constrs of
                      [] => case_error ("More than one clause for constructor(s) " ^
                        commas extra) (SOME tname) [u]
                    | extra' => case_error ("Illegal constructor(s): " ^ commas extra')
                        (SOME tname) [u]
                  end
              | _ => list_comb (Syntax.const case_name, fs) $ t
            end
        end
      | case_tr sg ts = raise TERM ("case_tr", ts);

    fun case_tr' constrs sg ts =
      if length ts <> length constrs + 1 then raise Match else
      let
        val (fs, x) = split_last ts;
        fun strip_abs 0 t = ([], t)
          | strip_abs i (Abs p) =
            let val (x, u) = Syntax.atomic_abs_tr' p
            in apfst (cons x) (strip_abs (i-1) u) end
          | strip_abs i (Const ("split", _) $ t) = (case strip_abs (i+1) t of
              (v :: v' :: vs, u) => (Syntax.const "Pair" $ v $ v' :: vs, u));
        fun is_dependent i t =
          let val k = length (strip_abs_vars t) - i
          in k < 0 orelse exists (fn j => j >= k)
            (loose_bnos (strip_abs_body t))
          end;
        val cases = map (fn ((cname, dts), t) =>
          (Sign.extern_const sg cname,
           strip_abs (length dts) t, is_dependent (length dts) t))
          (constrs ~~ fs);
        fun count_cases (cs, (_, _, true)) = cs
          | count_cases (cs, (cname, (_, body), false)) = (case AList.lookup (op =) cs body of
              NONE => (body, [cname]) :: cs
            | SOME cnames => AList.update (op =) (body, cnames @ [cname]) cs);
        val cases' = sort (int_ord o Library.swap o pairself (length o snd))
          (Library.foldl count_cases ([], cases));
        fun mk_case1 (cname, (vs, body), _) = Syntax.const "_case1" $
          list_comb (Syntax.const cname, vs) $ body;
      in
        Syntax.const "_case_syntax" $ x $
          foldr1 (fn (t, u) => Syntax.const "_case2" $ t $ u) (map mk_case1
            (case cases' of
               [] => cases
             | (default, cnames) :: _ =>
               if length cnames = 1 then cases
               else if length cnames = length constrs then
                 [hd cases, ("dummy_pattern", ([], default), false)]
               else
                 filter_out (fn (cname, _, _) => cname mem cnames) cases @
                 [("dummy_pattern", ([], default), false)]))
      end;

    fun make_case_tr' case_names descr = List.concat (map
      (fn ((_, (_, _, constrs)), case_name) => map (rpair (case_tr' constrs))
        (NameSpace.accesses' case_name)) (descr ~~ case_names));

    val trfun_setup =
      [Theory.add_advanced_trfuns ([], [("_case_syntax", case_tr)], [], [])];

    (* prepare types *)

    fun read_typ sign ((Ts, sorts), str) =
      let
        val T = Type.no_tvars (Sign.read_typ (sign, AList.lookup (op =)
          (map (apfst (rpair ~1)) sorts)) str) handle TYPE (msg, _, _) => error msg
      in (Ts @ [T], add_typ_tfrees (T, sorts)) end;

    fun cert_typ sign ((Ts, sorts), raw_T) =
      let
        val T = Type.no_tvars (Sign.certify_typ sign raw_T) handle
          TYPE (msg, _, _) => error msg;
        val sorts' = add_typ_tfrees (T, sorts)
      in (Ts @ [T],
          case duplicates (map fst sorts') of
             [] => sorts'
           | dups => error ("Inconsistent sort constraints for " ^ commas dups))
      end;

    (**** make datatype info ****)

    fun make_dt_info descr induct reccomb_names rec_thms
        (((((((((i, (_, (tname, _, _))), case_name), case_thms),
          exhaustion_thm), distinct_thm), inject), nchotomy), case_cong), weak_case_cong) =
      (tname,
       {index = i,
        descr = descr,
        rec_names = reccomb_names,
        rec_rewrites = rec_thms,
        case_name = case_name,
        case_rewrites = case_thms,
        induction = induct,
        exhaustion = exhaustion_thm,
        distinct = distinct_thm,
        inject = inject,
        nchotomy = nchotomy,
        case_cong = case_cong,
        weak_case_cong = weak_case_cong});

    (********************* axiomatic introduction of datatypes ********************)

    fun add_and_get_axioms_atts label tnames attss ts thy =
      foldr (fn (((tname, atts), t), (thy', axs)) =>
        let
          val (thy'', [ax]) = thy' |>
            Theory.add_path tname |>
            PureThy.add_axioms_i [((label, t), atts)];
        in (Theory.parent_path thy'', ax::axs)
        end) (thy, []) (tnames ~~ attss ~~ ts);

    fun add_and_get_axioms label tnames =
      add_and_get_axioms_atts label tnames (replicate (length tnames) []);

    fun add_and_get_axiomss label tnames tss thy =
      foldr (fn ((tname, ts), (thy', axss)) =>
        let
          val (thy'', [axs]) = thy' |>
            Theory.add_path tname |>
            PureThy.add_axiomss_i [((label, ts), [])];
        in (Theory.parent_path thy'', axs::axss)
        end) (thy, []) (tnames ~~ tss);

    fun add_datatype_axm flat_names new_type_names descr sorts types_syntax constr_syntax dt_info
        case_names_induct case_names_exhausts thy =
      let
        val descr' = List.concat descr;
        val recTs = get_rec_types descr' sorts;
        val used = foldr add_typ_tfree_names [] recTs;
        val newTs = Library.take (length (hd descr), recTs);

        val no_size = exists (fn (_, (_, _, constrs)) => exists (fn (_, cargs) => exists
          (fn dt => is_rec_type dt andalso not (null (fst (strip_dtyp dt))))
            cargs) constrs) descr';

        (**** declare new types and constants ****)

        val tyvars = map (fn (_, (_, Ts, _)) => map dest_DtTFree Ts) (hd descr);

        val constr_decls = map (fn (((_, (_, _, constrs)), T), constr_syntax') =>
          map (fn ((_, cargs), (cname, mx)) =>
            (cname, map (typ_of_dtyp descr' sorts) cargs ---> T, mx))
              (constrs ~~ constr_syntax')) ((hd descr) ~~ newTs ~~ constr_syntax);

        val (rec_result_Ts, reccomb_fn_Ts) = DatatypeProp.make_primrec_Ts descr sorts used;

        val big_reccomb_name = (space_implode "_" new_type_names) ^ "_rec";
        val reccomb_names = if length descr' = 1 then [big_reccomb_name] else
          (map ((curry (op ^) (big_reccomb_name ^ "_")) o string_of_int)
            (1 upto (length descr')));

        val size_names = DatatypeProp.indexify_names
          (map (fn T => name_of_typ T ^ "_size") (Library.drop (length (hd descr), recTs)));

        val freeT = TFree (variant used "'t", HOLogic.typeS);
        val case_fn_Ts = map (fn (i, (_, _, constrs)) =>
          map (fn (_, cargs) =>
            let val Ts = map (typ_of_dtyp descr' sorts) cargs
            in Ts ---> freeT end) constrs) (hd descr);

        val case_names = map (fn s => (s ^ "_case")) new_type_names;

        val thy2' = thy |>

          (** new types **)

          curry (Library.foldr (fn (((name, mx), tvs), thy') => thy' |>
              TypedefPackage.add_typedecls [(name, tvs, mx)]))
            (types_syntax ~~ tyvars) |>
          add_path flat_names (space_implode "_" new_type_names) |>

          (** primrec combinators **)

          Theory.add_consts_i (map (fn ((name, T), T') =>
            (name, reccomb_fn_Ts @ [T] ---> T', NoSyn))
              (reccomb_names ~~ recTs ~~ rec_result_Ts)) |>

          (** case combinators **)

          Theory.add_consts_i (map (fn ((name, T), Ts) =>
            (name, Ts @ [T] ---> freeT, NoSyn))
              (case_names ~~ newTs ~~ case_fn_Ts));

        val reccomb_names' = map (Sign.intern_const thy2') reccomb_names;
        val case_names' = map (Sign.intern_const thy2') case_names;

        val thy2 = thy2' |>

          (** size functions **)

          (if no_size then I else Theory.add_consts_i (map (fn (s, T) =>
            (Sign.base_name s, T --> HOLogic.natT, NoSyn))
              (size_names ~~ Library.drop (length (hd descr), recTs)))) |>

          (** constructors **)

          parent_path flat_names |>
          curry (Library.foldr (fn (((((_, (_, _, constrs)), T), tname),
            constr_syntax'), thy') => thy' |>
              add_path flat_names tname |>
                Theory.add_consts_i (map (fn ((_, cargs), (cname, mx)) =>
                  (cname, map (typ_of_dtyp descr' sorts) cargs ---> T, mx))
                    (constrs ~~ constr_syntax')) |>
              parent_path flat_names))
                (hd descr ~~ newTs ~~ new_type_names ~~ constr_syntax);

        (**** introduction of axioms ****)

        val rec_axs = DatatypeProp.make_primrecs new_type_names descr sorts thy2;
        val size_axs = if no_size then [] else DatatypeProp.make_size descr sorts thy2;

        val (thy3, (([induct], [rec_thms]), inject)) =
          thy2 |>
          Theory.add_path (space_implode "_" new_type_names) |>
          PureThy.add_axioms_i [(("induct", DatatypeProp.make_ind descr sorts),
            [case_names_induct])] |>>>
          PureThy.add_axiomss_i [(("recs", rec_axs), [])] |>>
          (if no_size then I else #1 o PureThy.add_axiomss_i [(("size", size_axs), [])]) |>>
          Theory.parent_path |>>>
          add_and_get_axiomss "inject" new_type_names
            (DatatypeProp.make_injs descr sorts);
        val size_thms = if no_size then [] else get_thms thy3 (Name "size");
        val (thy4, distinct) = add_and_get_axiomss "distinct" new_type_names
          (DatatypeProp.make_distincts new_type_names descr sorts thy3) thy3;

        val exhaust_ts = DatatypeProp.make_casedists descr sorts;
        val (thy5, exhaustion) = add_and_get_axioms_atts "exhaust" new_type_names
          (map Library.single case_names_exhausts) exhaust_ts thy4;
        val (thy6, case_thms) = add_and_get_axiomss "cases" new_type_names
          (DatatypeProp.make_cases new_type_names descr sorts thy5) thy5;
        val (split_ts, split_asm_ts) = ListPair.unzip
          (DatatypeProp.make_splits new_type_names descr sorts thy6);
        val (thy7, split) = add_and_get_axioms "split" new_type_names split_ts thy6;
        val (thy8, split_asm) = add_and_get_axioms "split_asm" new_type_names
          split_asm_ts thy7;
        val (thy9, nchotomys) = add_and_get_axioms "nchotomy" new_type_names
          (DatatypeProp.make_nchotomys descr sorts) thy8;
        val (thy10, case_congs) = add_and_get_axioms "case_cong" new_type_names
          (DatatypeProp.make_case_congs new_type_names descr sorts thy9) thy9;
        val (thy11, weak_case_congs) = add_and_get_axioms "weak_case_cong" new_type_names
          (DatatypeProp.make_weak_case_congs new_type_names descr sorts thy10) thy10;

        val dt_infos = map (make_dt_info descr' induct reccomb_names' rec_thms)
          ((0 upto length (hd descr) - 1) ~~ (hd descr) ~~ case_names' ~~ case_thms ~~
            exhaustion ~~ replicate (length (hd descr)) QuickAndDirty ~~ inject ~~
              nchotomys ~~ case_congs ~~ weak_case_congs);

        val simps = List.concat (distinct @ inject @ case_thms) @ size_thms @ rec_thms;
        val split_thms = split ~~ split_asm;

        val thy12 = thy11 |>
          Theory.add_advanced_trfuns ([], [], make_case_tr' case_names' (hd descr), []) |>
          Theory.add_path (space_implode "_" new_type_names) |>
          add_rules simps case_thms size_thms rec_thms inject distinct
                    weak_case_congs Simplifier.cong_add_global |>
          put_datatypes (fold Symtab.update dt_infos dt_info) |>
          add_cases_induct dt_infos induct |>
          Theory.parent_path |>
          (#1 o store_thmss "splits" new_type_names (map (fn (x, y) => [x, y]) split_thms)) |>
          DatatypeRealizer.add_dt_realizers sorts (map snd dt_infos);
      in
        (thy12,
         {distinct = distinct,
          inject = inject,
          exhaustion = exhaustion,
          rec_thms = rec_thms,
          case_thms = case_thms,
          split_thms = split_thms,
          induction = induct,
          size = size_thms,
          simps = simps})
      end;

    (******************* definitional introduction of datatypes *******************)

    fun add_datatype_def flat_names new_type_names descr sorts types_syntax constr_syntax dt_info
        case_names_induct case_names_exhausts thy =
      let
        val _ = message ("Proofs for datatype(s) " ^ commas_quote new_type_names);

        val (thy2, inject, distinct, dist_rewrites, simproc_dists, induct) = thy |>
          DatatypeRepProofs.representation_proofs flat_names dt_info new_type_names descr sorts
            types_syntax constr_syntax case_names_induct;

        val (thy3, casedist_thms) = DatatypeAbsProofs.prove_casedist_thms new_type_names descr
          sorts induct case_names_exhausts thy2;
        val (thy4, (reccomb_names, rec_thms)) = DatatypeAbsProofs.prove_primrec_thms
          flat_names new_type_names descr sorts dt_info inject dist_rewrites dist_ss induct thy3;
        val (thy6, (case_thms, case_names)) = DatatypeAbsProofs.prove_case_thms
          flat_names new_type_names descr sorts reccomb_names rec_thms thy4;
        val (thy7, split_thms) = DatatypeAbsProofs.prove_split_thms new_type_names
          descr sorts inject dist_rewrites casedist_thms case_thms thy6;
        val (thy8, nchotomys) = DatatypeAbsProofs.prove_nchotomys new_type_names
          descr sorts casedist_thms thy7;
        val (thy9, case_congs) = DatatypeAbsProofs.prove_case_congs new_type_names
          descr sorts nchotomys case_thms thy8;
        val (thy10, weak_case_congs) = DatatypeAbsProofs.prove_weak_case_congs new_type_names
          descr sorts thy9;
        val (thy11, size_thms) = DatatypeAbsProofs.prove_size_thms flat_names new_type_names
          descr sorts reccomb_names rec_thms thy10;

        val dt_infos = map (make_dt_info (List.concat descr) induct reccomb_names rec_thms)
          ((0 upto length (hd descr) - 1) ~~ (hd descr) ~~ case_names ~~ case_thms ~~
            casedist_thms ~~ simproc_dists ~~ inject ~~ nchotomys ~~ case_congs ~~ weak_case_congs);

        val simps = List.concat (distinct @ inject @ case_thms) @ size_thms @ rec_thms;

        val thy12 = thy11 |>
          Theory.add_advanced_trfuns ([], [], make_case_tr' case_names (hd descr), []) |>
          Theory.add_path (space_implode "_" new_type_names) |>
          add_rules simps case_thms size_thms rec_thms inject distinct
                    weak_case_congs (Simplifier.change_global_ss (op addcongs)) |>
          put_datatypes (fold Symtab.update dt_infos dt_info) |>
          add_cases_induct dt_infos induct |>
          Theory.parent_path |>
          (#1 o store_thmss "splits" new_type_names (map (fn (x, y) => [x, y]) split_thms)) |>
          DatatypeRealizer.add_dt_realizers sorts (map snd dt_infos);
      in
        (thy12,
         {distinct = distinct,
          inject = inject,
          exhaustion = casedist_thms,
          rec_thms = rec_thms,
          case_thms = case_thms,
          split_thms = split_thms,
          induction = induct,
          size = size_thms,
          simps = simps})
      end;

    (*********************** declare existing type as datatype *********************)

    fun gen_rep_datatype apply_theorems alt_names raw_distinct raw_inject raw_induction thy0 =
      let
        val _ = Theory.requires thy0 "Inductive" "datatype representations";

        fun app_thmss srcs thy = foldl_map (fn (thy, x) => apply_theorems x thy) (thy, srcs);
        fun app_thm src thy = apsnd hd (apply_theorems [src] thy);

        val (((thy1, induction), inject), distinct) = thy0
          |> app_thmss raw_distinct
          |> apfst (app_thmss raw_inject)
          |> apfst (apfst (app_thm raw_induction));
        val sign = Theory.sign_of thy1;

        val induction' = freezeT induction;

        fun err t = error ("Ill-formed predicate in induction rule: " ^
          Sign.string_of_term sign t);

        fun get_typ (t as _ $ Var (_, Type (tname, Ts))) =
              ((tname, map dest_TFree Ts) handle TERM _ => err t)
          | get_typ t = err t;

        val dtnames = map get_typ (HOLogic.dest_conj (HOLogic.dest_Trueprop (Thm.concl_of induction')));
        val new_type_names = getOpt (alt_names, map fst dtnames);

        fun get_constr t = (case Logic.strip_assums_concl t of
            _ $ (_ $ t') => (case head_of t' of
                Const (cname, cT) => (case strip_type cT of
                    (Ts, Type (tname, _)) => (tname, (cname, map (dtyp_of_typ dtnames) Ts))
                  | _ => err t)
              | _ => err t)
          | _ => err t);

        fun make_dt_spec [] _ _ = []
          | make_dt_spec ((tname, tvs)::dtnames') i constrs =
              let val (constrs', constrs'') = take_prefix (equal tname o fst) constrs
              in (i, (tname, map DtTFree tvs, map snd constrs'))::
                (make_dt_spec dtnames' (i + 1) constrs'')
              end;

        val descr = make_dt_spec dtnames 0 (map get_constr (prems_of induction'));
        val sorts = add_term_tfrees (concl_of induction', []);
        val dt_info = get_datatypes thy1;

        val case_names_induct = mk_case_names_induct descr;
        val case_names_exhausts = mk_case_names_exhausts descr (map #1 dtnames);

        val _ = message ("Proofs for datatype(s) " ^ commas_quote new_type_names);

        val (thy2, casedist_thms) = thy1 |>
          DatatypeAbsProofs.prove_casedist_thms new_type_names [descr] sorts induction
            case_names_exhausts;
        val (thy3, (reccomb_names, rec_thms)) = DatatypeAbsProofs.prove_primrec_thms
          false new_type_names [descr] sorts dt_info inject distinct dist_ss induction thy2;
        val (thy4, (case_thms, case_names)) = DatatypeAbsProofs.prove_case_thms false
          new_type_names [descr] sorts reccomb_names rec_thms thy3;
        val (thy5, split_thms) = DatatypeAbsProofs.prove_split_thms
          new_type_names [descr] sorts inject distinct casedist_thms case_thms thy4;
        val (thy6, nchotomys) = DatatypeAbsProofs.prove_nchotomys new_type_names
          [descr] sorts casedist_thms thy5;
        val (thy7, case_congs) = DatatypeAbsProofs.prove_case_congs new_type_names
          [descr] sorts nchotomys case_thms thy6;
        val (thy8, weak_case_congs) = DatatypeAbsProofs.prove_weak_case_congs new_type_names
          [descr] sorts thy7;
        val (thy9, size_thms) =
          if Context.exists_name "NatArith" thy8 then
            DatatypeAbsProofs.prove_size_thms false new_type_names
              [descr] sorts reccomb_names rec_thms thy8
          else (thy8, []);

        val (thy10, [induction']) = thy9 |>
          (#1 o store_thmss "inject" new_type_names inject) |>
          (#1 o store_thmss "distinct" new_type_names distinct) |>
          Theory.add_path (space_implode "_" new_type_names) |>
          PureThy.add_thms [(("induct", induction), [case_names_induct])];

        val dt_infos = map (make_dt_info descr induction' reccomb_names rec_thms)
          ((0 upto length descr - 1) ~~ descr ~~ case_names ~~ case_thms ~~ casedist_thms ~~
            map FewConstrs distinct ~~ inject ~~ nchotomys ~~ case_congs ~~ weak_case_congs);

        val simps = List.concat (distinct @ inject @ case_thms) @ size_thms @ rec_thms;

        val thy11 = thy10 |>
          Theory.add_advanced_trfuns ([], [], make_case_tr' case_names descr, []) |>
          add_rules simps case_thms size_thms rec_thms inject distinct
                    weak_case_congs (Simplifier.change_global_ss (op addcongs)) |>
          put_datatypes (fold Symtab.update dt_infos dt_info) |>
          add_cases_induct dt_infos induction' |>
          Theory.parent_path |>
          (#1 o store_thmss "splits" new_type_names (map (fn (x, y) => [x, y]) split_thms)) |>
          DatatypeRealizer.add_dt_realizers sorts (map snd dt_infos);
      in
        (thy11,
         {distinct = distinct,
          inject = inject,
          exhaustion = casedist_thms,
          rec_thms = rec_thms,
          case_thms = case_thms,
          split_thms = split_thms,
          induction = induction',
          size = size_thms,
          simps = simps})
      end;

    val rep_datatype = gen_rep_datatype IsarThy.apply_theorems;
    val rep_datatype_i = gen_rep_datatype IsarThy.apply_theorems_i;

    (******************************** add datatype ********************************)

    fun gen_add_datatype prep_typ err flat_names new_type_names dts thy =
      let
        val _ = Theory.requires thy "Datatype_Universe" "datatype definitions";

        (* this theory is used just for parsing *)

        val tmp_thy = thy |>
          Theory.copy |>
          Theory.add_types (map (fn (tvs, tname, mx, _) =>
            (tname, length tvs, mx)) dts);

        val sign = Theory.sign_of tmp_thy;

        val (tyvars, _, _, _)::_ = dts;
        val (new_dts, types_syntax) = ListPair.unzip (map (fn (tvs, tname, mx, _) =>
          let val full_tname = Sign.full_name sign (Syntax.type_name tname mx)
          in (case duplicates tvs of
                [] => if eq_set (tyvars, tvs) then ((full_tname, tvs), (tname, mx))
                      else error ("Mutually recursive datatypes must have same type parameters")
              | dups => error ("Duplicate parameter(s) for datatype " ^ full_tname ^
                  " : " ^ commas dups))
          end) dts);

        val _ = (case duplicates (map fst new_dts) @ duplicates new_type_names of
          [] => () | dups => error ("Duplicate datatypes: " ^ commas dups));

        fun prep_dt_spec ((dts', constr_syntax, sorts, i), (tvs, tname, mx, constrs)) =
          let
            fun prep_constr ((constrs, constr_syntax', sorts'), (cname, cargs, mx')) =
              let
                val (cargs', sorts'') = Library.foldl (prep_typ sign) (([], sorts'), cargs);
                val _ = (case foldr add_typ_tfree_names [] cargs' \\ tvs of
                    [] => ()
                  | vs => error ("Extra type variables on rhs: " ^ commas vs))
              in (constrs @ [((if flat_names then Sign.full_name sign else
                    Sign.full_name_path sign tname) (Syntax.const_name cname mx'),
                       map (dtyp_of_typ new_dts) cargs')],
                  constr_syntax' @ [(cname, mx')], sorts'')
              end handle ERROR =>
                error ("The error above occured in constructor " ^ cname ^
                  " of datatype " ^ tname);

            val (constrs', constr_syntax', sorts') =
              Library.foldl prep_constr (([], [], sorts), constrs)

          in
            case duplicates (map fst constrs') of
               [] =>
                 (dts' @ [(i, (Sign.full_name sign (Syntax.type_name tname mx),
                    map DtTFree tvs, constrs'))],
                  constr_syntax @ [constr_syntax'], sorts', i + 1)
             | dups => error ("Duplicate constructors " ^ commas dups ^
                 " in datatype " ^ tname)
          end;

        val (dts', constr_syntax, sorts', i) = Library.foldl prep_dt_spec (([], [], [], 0), dts);
        val sorts = sorts' @ (map (rpair (Sign.defaultS sign)) (tyvars \\ map fst sorts'));
        val dt_info = get_datatypes thy;
        val (descr, _) = unfold_datatypes sign dts' sorts dt_info dts' i;
        val _ = check_nonempty descr handle (exn as Datatype_Empty s) =>
          if err then error ("Nonemptiness check failed for datatype " ^ s)
          else raise exn;

        val descr' = List.concat descr;
        val case_names_induct = mk_case_names_induct descr';
        val case_names_exhausts = mk_case_names_exhausts descr' (map #1 new_dts);
      in
        (if (!quick_and_dirty) then add_datatype_axm else add_datatype_def)
          flat_names new_type_names descr sorts types_syntax constr_syntax dt_info
          case_names_induct case_names_exhausts thy
      end;

    val add_datatype_i = gen_add_datatype cert_typ;
    val add_datatype = gen_add_datatype read_typ true;

    (** package setup **)

    (* setup theory *)

    val setup = [DatatypesData.init, Method.add_methods tactic_emulations] @ simproc_setup @ trfun_setup;

    (* outer syntax *)

    local structure P = OuterParse and K = OuterKeyword in

    val datatype_decl =
      Scan.option (P.$$$ "(" |-- P.name --| P.$$$ ")") -- P.type_args -- P.name -- P.opt_infix --
        (P.$$$ "=" |-- P.enum1 "|" (P.name -- Scan.repeat P.typ -- P.opt_mixfix));

    fun mk_datatype args =
      let
        val names = map (fn ((((NONE, _), t), _), _) => t | ((((SOME t, _), _), _), _) => t) args;
        val specs = map (fn ((((_, vs), t), mx), cons) =>
          (vs, t, mx, map (fn ((x, y), z) => (x, y, z)) cons)) args;
      in #1 o add_datatype false names specs end;

    val datatypeP =
      OuterSyntax.command "datatype" "define inductive datatypes" K.thy_decl
        (P.and_list1 datatype_decl >> (Toplevel.theory o mk_datatype));

    val rep_datatype_decl =
      Scan.option (Scan.repeat1 P.name) --
        Scan.optional (P.$$$ "distinct" |-- P.!!! (P.and_list1 P.xthms1)) [[]] --
        Scan.optional (P.$$$ "inject" |-- P.!!! (P.and_list1 P.xthms1)) [[]] --
        (P.$$$ "induction" |-- P.!!! P.xthm);

    fun mk_rep_datatype (((opt_ts, dss), iss), ind) = #1 o rep_datatype opt_ts dss iss ind;

    val rep_datatypeP =
      OuterSyntax.command "rep_datatype" "represent existing types inductively" K.thy_decl
        (rep_datatype_decl >> (Toplevel.theory o mk_rep_datatype));

    val _ = OuterSyntax.add_keywords ["distinct", "inject", "induction"];
    val _ = OuterSyntax.add_parsers [datatypeP, rep_datatypeP];

    end;

    end;

    structure BasicDatatypePackage: BASIC_DATATYPE_PACKAGE = DatatypePackage;
    open BasicDatatypePackage;