src/HOL/Tools/datatype_package.ML

Simplifier.cong_add_global;

(*  Title:      HOL/Tools/datatype_package.ML
    ID:         $Id$
    Author:     Stefan Berghofer
    Copyright   1998  TU Muenchen

Datatype package for Isabelle/HOL.
*)

signature BASIC_DATATYPE_PACKAGE =
sig
  val induct_tac : 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 -> 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 * theory attribute list) list list ->
    (thm * theory attribute list) list list -> (thm * 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 -> (xstring * Args.src list) list list ->
    (xstring * Args.src list) list list -> xstring * Args.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_sg : Sign.sg -> string -> DatatypeAux.datatype_info option
  val datatype_info : theory -> string -> DatatypeAux.datatype_info option
  val datatype_info_sg_err : Sign.sg -> string -> DatatypeAux.datatype_info
  val datatype_info_err : theory -> string -> DatatypeAux.datatype_info
  val constrs_of : theory -> string -> term list option
  val constrs_of_sg : Sign.sg -> string -> term list option
  val case_const_of : theory -> string -> term option
  val setup: (theory -> theory) list
end;

structure DatatypePackage : DATATYPE_PACKAGE =
struct

open DatatypeAux;

val quiet_mode = quiet_mode;


(* data kind 'HOL/datatypes' *)

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

  val empty = Symtab.empty;
  val copy = I;
  val prep_ext = I;
  val merge: T * T -> T = Symtab.merge (K true);

  fun print sg tab =
    Pretty.writeln (Pretty.strs ("datatypes:" ::
      map #1 (Sign.cond_extern_table sg Sign.typeK tab)));
end;

structure DatatypesData = TheoryDataFun(DatatypesArgs);
val get_datatypes_sg = DatatypesData.get_sg;
val get_datatypes = DatatypesData.get;
val put_datatypes = DatatypesData.put;
val print_datatypes = DatatypesData.print;


(** theory information about datatypes **)

fun datatype_info_sg sg name = Symtab.lookup (get_datatypes_sg sg, name);

fun datatype_info_sg_err sg name = (case datatype_info_sg sg name of
      Some info => info
    | None => error ("Unknown datatype " ^ quote name));

val datatype_info = datatype_info_sg o Theory.sign_of;

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

fun constrs_of_sg sg tname = (case datatype_info_sg sg tname of
   Some {index, descr, ...} =>
     let val (_, _, constrs) = the (assoc (descr, index))
     in Some (map (fn (cname, _) => Const (cname, the (Sign.const_type sg cname))) constrs)
     end
 | _ => None);

val constrs_of = constrs_of_sg o Theory.sign_of;

fun case_const_of thy tname = (case datatype_info thy tname of
   Some {case_name, ...} => Some (Const (case_name, the (Sign.const_type
     (Theory.sign_of thy) case_name)))
 | _ => None);

fun find_tname var Bi =
  let val frees = map dest_Free (term_frees Bi)
      val params = Logic.strip_params Bi;
  in case assoc (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 assoc (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 (implode (tl (explode (Syntax.string_of_vname ixn))), x)
  | prep_var _ = None;

fun prep_inst (concl, xs) =	(*exception LIST*)
  let val vs = InductMethod.vars_of concl
  in mapfilter prep_var (Library.drop (length vs - length xs, vs) ~~ xs) end;

in

fun gen_induct_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 (mapfilter I (flat varss))) Bi
          in (#induction (datatype_info_sg_err sign tn), "Induction rule for type " ^ tn) end);

    val concls = InductMethod.concls_of rule;
    val insts = flat (map prep_inst (concls ~~ varss)) handle LIST _ =>
      error (rule_name ^ " has different numbers of variables");
  in occs_in_prems (Tactic.res_inst_tac insts rule) (map #2 insts) i state end;

fun induct_tac s = gen_induct_tac (map (Library.single o Some) (Syntax.read_idents s), None);

end;


(* generic case tactic for datatypes *)

fun case_inst_tac t rule i state =
  let
    val _ $ Var (ixn, _) $ _ = HOLogic.dest_Trueprop
      (hd (Logic.strip_assums_hyp (hd (Thm.prems_of rule))));
    val exh_vname = implode (tl (explode (Syntax.string_of_vname ixn)));
  in Tactic.res_inst_tac [(exh_vname, t)] rule i state end;

fun gen_case_tac (t, Some rule) i state = case_inst_tac t rule i state
  | gen_case_tac (t, None) i state =
      let val tn = infer_tname state i t in
        if tn = HOLogic.boolN then Tactic.res_inst_tac [("P", t)] case_split_thm i state
        else case_inst_tac t (#exhaustion (datatype_info_sg_err (Thm.sign_of_thm state) tn)) i state
      end;

fun case_tac t = gen_case_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.name_dummy)));

in

val tactic_emulations =
 [("induct_tac", Method.goal_args' (varss -- opt_rule) gen_induct_tac,
    "induct_tac emulation (dynamic instantiation!)"),
  ("case_tac", Method.goal_args' (Scan.lift Args.name -- opt_rule) gen_case_tac,
    "case_tac emulation (dynamic instantiation!)")];

end;



(** induct method setup **)

(* case names *)

local

fun dt_recs (DtTFree _) = []
  | dt_recs (DtType (_, dts)) = flat (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 (the (assoc (descr, i))));
    val bnames = map the_bname (distinct (flat (map dt_recs args)));
  in map (fn (c, _) => space_implode "_" (Sign.base_name c :: bnames)) constrs end;


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

fun exhaust_cases descr i = dt_cases descr (the (assoc (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)
    (filter (fn ((_, (name, _, _))) => name mem_string new) descr);

end;


(* add_cases_induct *)

fun add_cases_induct infos =
  let
    fun proj _ 1 thm = thm
      | proj i n thm =
          (if i + 1 < n then (fn th => th RS conjunct1) else I)
            (Library.funpow i (fn th => th RS conjunct2) thm)
          |> Drule.standard
          |> RuleCases.name (RuleCases.get thm);

    fun add (ths, (name, {index, descr, induction, exhaustion, ...}: datatype_info)) =
      (("", proj index (length descr) induction), [InductMethod.induct_type_global name]) ::
      (("", exhaustion), [InductMethod.cases_type_global name]) :: ths;
  in #1 o PureThy.add_thms (foldl add ([], infos)) 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 _ (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 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)
                                     ["In0_inject", "In1_inject", "In0_not_In1", "In1_not_In0"]
                             in (case (#distinct (datatype_info_sg_err sg tname1)) of
                                 QuickAndDirty => Some (Thm.invoke_oracle
                                   Datatype_thy distinctN (sg, ConstrDistinct eq_t))
                               | FewConstrs thms => Some (prove_goalw_cterm [] eq_ct (K
                                   [rtac eq_reflection 1, rtac iffI 1, rtac notE 1,
                                    atac 2, resolve_tac thms 1, etac FalseE 1]))
                               | ManyConstrs (thm, ss) => Some (prove_goalw_cterm [] eq_ct (K
                                   [rtac eq_reflection 1, rtac iffI 1, dtac thm 1,
                                    full_simp_tac ss 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_pats = [Thm.read_cterm (Theory.sign_of HOL.thy) ("s = t", HOLogic.termT)];
val distinct_simproc = mk_simproc distinctN distinct_pats 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)];


(* prepare types *)

fun read_typ sign ((Ts, sorts), str) =
  let
    val T = Type.no_tvars (Sign.read_typ (sign, (curry assoc)
      (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) = (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});


(********************* 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) (tnames ~~ attss ~~ ts, (thy, []));

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) (tnames ~~ tss, (thy, []));

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' = flat descr;
    val recTs = get_rec_types descr' sorts;
    val used = foldr add_typ_tfree_names (recTs, []);
    val newTs = take (length (hd descr), recTs);

    val no_size = exists (fn (_, (_, _, constrs)) => exists (fn (_, cargs) => exists
      (fn (DtType ("fun", [_, DtRec _])) => true | _ => false) 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 = map TFree (variantlist (replicate (length descr') "'t", used) ~~
      replicate (length descr') HOLogic.termS);

    val reccomb_fn_Ts = flat (map (fn (i, (_, _, constrs)) =>
      map (fn (_, cargs) =>
        let
          val Ts = map (typ_of_dtyp descr' sorts) cargs;
          val recs = filter (is_rec_type o fst) (cargs ~~ Ts);

          fun mk_argT (DtRec k, _) = nth_elem (k, rec_result_Ts)
            | mk_argT (DtType ("fun", [_, DtRec k]), Type ("fun", [T, _])) =
               T --> nth_elem (k, rec_result_Ts);

          val argTs = Ts @ map mk_argT recs
        in argTs ---> nth_elem (i, rec_result_Ts)
        end) constrs) descr');

    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 big_size_name = space_implode "_" new_type_names ^ "_size";
    val size_names = if length (flat (tl descr)) = 1 then [big_size_name] else
      map (fn i => big_size_name ^ "_" ^ string_of_int i)
        (1 upto length (flat (tl descr)));

    val freeT = TFree (variant used "'t", HOLogic.termS);
    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 (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)) |>
      Theory.add_trrules_i (DatatypeProp.make_case_trrules new_type_names descr);

    val reccomb_names' = map (Sign.intern_const (Theory.sign_of thy2')) reccomb_names;
    val case_names' = map (Sign.intern_const (Theory.sign_of 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 ~~ drop (length (hd descr), recTs)))) |>

      (** constructors **)

      parent_path flat_names |>
      curry (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 new_type_names 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 "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);

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

    val thy12 = thy11 |>
      Theory.add_path (space_implode "_" new_type_names) |>
      (#1 o PureThy.add_thmss [(("simps", simps), []),
        (("", flat case_thms @ size_thms @ rec_thms), [Simplifier.simp_add_global]),
        (("", flat (inject @ distinct)), [iff_add_global]),
        (("", weak_case_congs), [Simplifier.cong_add_global])]) |>
      put_datatypes (foldr Symtab.update (dt_infos, dt_info)) |>
      add_cases_induct dt_infos |>
      Theory.parent_path |>
      (#1 o store_thmss "splits" new_type_names (map (fn (x, y) => [x, y]) split_thms));
  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 (flat 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);

    val simps = flat (distinct @ inject @ case_thms) @ size_thms @ rec_thms;

    val thy12 = thy11 |>
      Theory.add_path (space_implode "_" new_type_names) |>
      (#1 o PureThy.add_thmss [(("simps", simps), []),
        (("", flat case_thms @ size_thms @ rec_thms), [Simplifier.simp_add_global]),
        (("", flat (inject @ distinct)), [iff_add_global]),
        (("", weak_case_congs), [Simplifier.change_global_ss (op addcongs)])]) |>
      put_datatypes (foldr Symtab.update (dt_infos, dt_info)) |>
      add_cases_induct dt_infos |>
      Theory.parent_path |>
      (#1 o store_thmss "splits" new_type_names (map (fn (x, y) => [x, y]) split_thms));
  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
    fun app_thmss srcs thy = foldl_map (fn (thy, x) => apply_theorems x thy) (thy, srcs);
    fun app_thm src thy = apsnd Library.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 = if_none 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 exists (equal "Arith") (Sign.stamp_names_of (Theory.sign_of 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);

    val simps = flat (distinct @ inject @ case_thms) @ size_thms @ rec_thms;

    val thy11 = thy10 |>
      (#1 o PureThy.add_thmss [(("simps", simps), []),
        (("", flat case_thms @ size_thms @ rec_thms), [Simplifier.simp_add_global]),
        (("", flat (inject @ distinct)), [iff_add_global]),
        (("", weak_case_congs), [Simplifier.change_global_ss (op addcongs)])]) |>
      put_datatypes (foldr Symtab.update (dt_infos, dt_info)) |>
      add_cases_induct dt_infos |>
      Theory.parent_path |>
      (#1 o store_thmss "splits" new_type_names (map (fn (x, y) => [x, y]) split_thms));
  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 flat_names new_type_names dts thy =
  let
    val _ = Theory.requires thy "Datatype" "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'') = 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') =
          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) = 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;

    val descr' = flat 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;


(** package setup **)

(* setup theory *)

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


(* outer syntax *)

local structure P = OuterParse and K = OuterSyntax.Keyword 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 --| P.marg_comment));

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;