src/HOL/Tools/Datatype/datatype.ML

separate "inner_rule" tag indicates parts of induction rules -- avoids unclear overlap with "internal" tag;

(*  Title:      HOL/Tools/datatype.ML
    Author:     Stefan Berghofer, TU Muenchen

Datatype package for Isabelle/HOL.
*)

signature DATATYPE =
sig
  include DATATYPE_COMMON
  val add_datatype : config -> string list -> (string list * binding * mixfix *
    (binding * typ list * mixfix) list) list -> theory -> string list * theory
  val datatype_cmd : string list -> (string list * binding * mixfix *
    (binding * string list * mixfix) list) list -> theory -> theory
  val rep_datatype : config -> (string list -> Proof.context -> Proof.context)
    -> string list option -> term list -> theory -> Proof.state
  val rep_datatype_cmd : string list option -> string list -> theory -> Proof.state
  val get_info : theory -> string -> info option
  val the_info : theory -> string -> info
  val the_descr : theory -> string list
    -> descr * (string * sort) list * string list
      * string * (string list * string list) * (typ list * typ list)
  val the_spec : theory -> string -> (string * sort) list * (string * typ list) list
  val all_distincts : theory -> typ list -> thm list list
  val get_constrs : theory -> string -> (string * typ) list option
  val get_all : theory -> info Symtab.table
  val info_of_constr : theory -> string * typ -> info option
  val info_of_case : theory -> string -> info option
  val interpretation : (config -> string list -> theory -> theory) -> theory -> theory
  val make_case :  Proof.context -> DatatypeCase.config -> string list -> term ->
    (term * term) list -> term * (term * (int * bool)) list
  val strip_case : Proof.context -> bool -> term -> (term * (term * term) list) option
  val read_typ: theory -> string -> (string * sort) list -> typ * (string * sort) list
  val setup: theory -> theory
end;

structure Datatype : DATATYPE =
struct

open DatatypeAux;

(** theory data **)

(* data management *)

structure DatatypesData = TheoryDataFun
(
  type T =
    {types: info Symtab.table,
     constrs: (string * info) list Symtab.table,
     cases: info Symtab.table};

  val empty =
    {types = Symtab.empty, constrs = Symtab.empty, cases = Symtab.empty};
  val copy = I;
  val extend = I;
  fun merge _
    ({types = types1, constrs = constrs1, cases = cases1},
     {types = types2, constrs = constrs2, cases = cases2}) =
    {types = Symtab.merge (K true) (types1, types2),
     constrs = Symtab.join (K (AList.merge (op =) (K true))) (constrs1, constrs2),
     cases = Symtab.merge (K true) (cases1, cases2)};
);

val get_all = #types o DatatypesData.get;
val get_info = Symtab.lookup o get_all;

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

fun info_of_constr thy (c, T) =
  let
    val tab = Symtab.lookup_list ((#constrs o DatatypesData.get) thy) c;
    val hint = case strip_type T of (_, Type (tyco, _)) => SOME tyco | _ => NONE;
    val default = if null tab then NONE
      else SOME (snd (Library.last_elem tab))
        (*conservative wrt. overloaded constructors*);
  in case hint
   of NONE => default
    | SOME tyco => case AList.lookup (op =) tab tyco
       of NONE => default (*permissive*)
        | SOME info => SOME info
  end;

val info_of_case = Symtab.lookup o #cases o DatatypesData.get;

fun register (dt_infos : (string * info) list) =
  DatatypesData.map (fn {types, constrs, cases} =>
    {types = types |> fold Symtab.update dt_infos,
     constrs = constrs |> fold (fn (constr, dtname_info) =>
         Symtab.map_default (constr, []) (cons dtname_info))
       (maps (fn (dtname, info as {descr, index, ...}) =>
          map (rpair (dtname, info) o fst)
            (#3 (the (AList.lookup op = descr index)))) dt_infos),
     cases = cases |> fold Symtab.update
       (map (fn (_, info as {case_name, ...}) => (case_name, info)) dt_infos)});

(* complex queries *)

fun the_spec thy dtco =
  let
    val info as { descr, index, sorts = raw_sorts, ... } = the_info thy dtco;
    val SOME (_, dtys, raw_cos) = AList.lookup (op =) descr index;
    val sorts = map ((fn v => (v, (the o AList.lookup (op =) raw_sorts) v))
      o DatatypeAux.dest_DtTFree) dtys;
    val cos = map
      (fn (co, tys) => (co, map (DatatypeAux.typ_of_dtyp descr sorts) tys)) raw_cos;
  in (sorts, cos) end;

fun the_descr thy (raw_tycos as raw_tyco :: _) =
  let
    val info = the_info thy raw_tyco;
    val descr = #descr info;

    val SOME (_, dtys, raw_cos) = AList.lookup (op =) descr (#index info);
    val vs = map ((fn v => (v, (the o AList.lookup (op =) (#sorts info)) v))
      o dest_DtTFree) dtys;

    fun is_DtTFree (DtTFree _) = true
      | is_DtTFree _ = false
    val k = find_index (fn (_, (_, dTs, _)) => not (forall is_DtTFree dTs)) descr;
    val protoTs as (dataTs, _) = chop k descr
      |> (pairself o map) (fn (_, (tyco, dTs, _)) => (tyco, map (typ_of_dtyp descr vs) dTs));
    
    val tycos = map fst dataTs;
    val _ = if eq_set (op =) (tycos, raw_tycos) then ()
      else error ("Type constructors " ^ commas (map quote raw_tycos)
        ^ " do not belong exhaustively to one mutual recursive datatype");

    val (Ts, Us) = (pairself o map) Type protoTs;

    val names = map Long_Name.base_name (the_default tycos (#alt_names info));
    val (auxnames, _) = Name.make_context names
      |> fold_map (yield_singleton Name.variants o name_of_typ) Us;
    val prefix = space_implode "_" names;

  in (descr, vs, tycos, prefix, (names, auxnames), (Ts, Us)) end;

fun all_distincts thy Ts =
  let
    fun add_tycos (Type (tyco, Ts)) = insert (op =) tyco #> fold add_tycos Ts
      | add_tycos _ = I;
    val tycos = fold add_tycos Ts [];
  in map_filter (Option.map #distinct o get_info thy) tycos end;

fun get_constrs thy dtco =
  case try (the_spec thy) dtco
   of SOME (sorts, cos) =>
        let
          fun subst (v, sort) = TVar ((v, 0), sort);
          fun subst_ty (TFree v) = subst v
            | subst_ty ty = ty;
          val dty = Type (dtco, map subst sorts);
          fun mk_co (co, tys) = (co, map (Term.map_atyps subst_ty) tys ---> dty);
        in SOME (map mk_co cos) end
    | NONE => NONE;


(** various auxiliary **)

(* prepare datatype specifications *)

fun read_typ thy str sorts =
  let
    val ctxt = ProofContext.init thy
      |> fold (Variable.declare_typ o TFree) sorts;
    val T = Syntax.read_typ ctxt str;
  in (T, Term.add_tfreesT T sorts) end;

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


(* case names *)

local

fun dt_recs (DtTFree _) = []
  | dt_recs (DtType (_, dts)) = maps dt_recs dts
  | dt_recs (DtRec i) = [i];

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

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

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

end;

(* translation rules for case *)

fun make_case ctxt = DatatypeCase.make_case
  (info_of_constr (ProofContext.theory_of ctxt)) ctxt;

fun strip_case ctxt = DatatypeCase.strip_case
  (info_of_case (ProofContext.theory_of ctxt));

fun add_case_tr' case_names thy =
  Sign.add_advanced_trfuns ([], [],
    map (fn case_name =>
      let val case_name' = Sign.const_syntax_name thy case_name
      in (case_name', DatatypeCase.case_tr' info_of_case case_name')
      end) case_names, []) thy;

val trfun_setup =
  Sign.add_advanced_trfuns ([],
    [("_case_syntax", DatatypeCase.case_tr true info_of_constr)],
    [], []);


(** document antiquotation **)

val _ = ThyOutput.antiquotation "datatype" Args.tyname
  (fn {source = src, context = ctxt, ...} => fn dtco =>
    let
      val thy = ProofContext.theory_of ctxt;
      val (vs, cos) = the_spec thy dtco;
      val ty = Type (dtco, map TFree vs);
      fun pretty_typ_bracket (ty as Type (_, _ :: _)) =
            Pretty.enclose "(" ")" [Syntax.pretty_typ ctxt ty]
        | pretty_typ_bracket ty =
            Syntax.pretty_typ ctxt ty;
      fun pretty_constr (co, tys) =
        (Pretty.block o Pretty.breaks)
          (Syntax.pretty_term ctxt (Const (co, tys ---> ty)) ::
            map pretty_typ_bracket tys);
      val pretty_datatype =
        Pretty.block
          (Pretty.command "datatype" :: Pretty.brk 1 ::
           Syntax.pretty_typ ctxt ty ::
           Pretty.str " =" :: Pretty.brk 1 ::
           flat (separate [Pretty.brk 1, Pretty.str "| "]
             (map (single o pretty_constr) cos)));
    in ThyOutput.output (ThyOutput.maybe_pretty_source (K pretty_datatype) src [()]) end);


(** abstract theory extensions relative to a datatype characterisation **)

structure DatatypeInterpretation = InterpretationFun
  (type T = config * string list val eq: T * T -> bool = eq_snd op =);
fun interpretation f = DatatypeInterpretation.interpretation (uncurry f);

fun make_dt_info alt_names descr sorts induct inducts rec_names rec_rewrites
    (index, (((((((((((_, (tname, _, _))), inject), distinct),
      exhaust), nchotomy), case_name), case_rewrites), case_cong), weak_case_cong), (split, split_asm))) =
  (tname,
   {index = index,
    alt_names = alt_names,
    descr = descr,
    sorts = sorts,
    inject = inject,
    distinct = distinct,
    induct = induct,
    inducts = inducts,
    exhaust = exhaust,
    nchotomy = nchotomy,
    rec_names = rec_names,
    rec_rewrites = rec_rewrites,
    case_name = case_name,
    case_rewrites = case_rewrites,
    case_cong = case_cong,
    weak_case_cong = weak_case_cong,
    split = split,
    split_asm = split_asm});

fun derive_datatype_props config dt_names alt_names descr sorts
    induct inject distinct thy1 =
  let
    val thy2 = thy1 |> Theory.checkpoint;
    val flat_descr = flat descr;
    val new_type_names = map Long_Name.base_name (the_default dt_names alt_names);
    val _ = message config ("Deriving properties for datatype(s) " ^ commas_quote new_type_names);

    val (exhaust, thy3) = DatatypeAbsProofs.prove_casedist_thms config new_type_names
      descr sorts induct (mk_case_names_exhausts flat_descr dt_names) thy2;
    val (nchotomys, thy4) = DatatypeAbsProofs.prove_nchotomys config new_type_names
      descr sorts exhaust thy3;
    val ((rec_names, rec_rewrites), thy5) = DatatypeAbsProofs.prove_primrec_thms
      config new_type_names descr sorts (#inject o the o Symtab.lookup (get_all thy4))
      inject (distinct, all_distincts thy2 (get_rec_types flat_descr sorts))
      induct thy4;
    val ((case_rewrites, case_names), thy6) = DatatypeAbsProofs.prove_case_thms
      config new_type_names descr sorts rec_names rec_rewrites thy5;
    val (case_congs, thy7) = DatatypeAbsProofs.prove_case_congs new_type_names
      descr sorts nchotomys case_rewrites thy6;
    val (weak_case_congs, thy8) = DatatypeAbsProofs.prove_weak_case_congs new_type_names
      descr sorts thy7;
    val (splits, thy9) = DatatypeAbsProofs.prove_split_thms
      config new_type_names descr sorts inject distinct exhaust case_rewrites thy8;

    val inducts = Project_Rule.projections (ProofContext.init thy2) induct;
    val dt_infos = map_index (make_dt_info alt_names flat_descr sorts induct inducts rec_names rec_rewrites)
      (hd descr ~~ inject ~~ distinct ~~ exhaust ~~ nchotomys ~~
        case_names ~~ case_rewrites ~~ case_congs ~~ weak_case_congs ~~ splits);
    val dt_names = map fst dt_infos;
    val prfx = Binding.qualify true (space_implode "_" new_type_names);
    val simps = flat (inject @ distinct @ case_rewrites) @ rec_rewrites;
    val named_rules = flat (map_index (fn (index, tname) =>
      [((Binding.empty, [nth inducts index]), [Induct.induct_type tname]),
       ((Binding.empty, [nth exhaust index]), [Induct.cases_type tname])]) dt_names);
    val unnamed_rules = map (fn induct =>
      ((Binding.empty, [induct]), [RuleCases.inner_rule, Induct.induct_type ""]))
        (Library.drop (length dt_names, inducts));
  in
    thy9
    |> PureThy.add_thmss ([((prfx (Binding.name "simps"), simps), []),
        ((prfx (Binding.name "inducts"), inducts), []),
        ((prfx (Binding.name "splits"), maps (fn (x, y) => [x, y]) splits), []),
        ((Binding.empty, flat case_rewrites @ flat distinct @ rec_rewrites),
          [Simplifier.simp_add]),
        ((Binding.empty, rec_rewrites), [Code.add_default_eqn_attribute]),
        ((Binding.empty, flat inject), [iff_add]),
        ((Binding.empty, map (fn th => th RS notE) (flat distinct)),
          [Classical.safe_elim NONE]),
        ((Binding.empty, weak_case_congs), [Simplifier.attrib (op addcongs)])]
        @ named_rules @ unnamed_rules)
    |> snd
    |> add_case_tr' case_names
    |> register dt_infos
    |> DatatypeInterpretation.data (config, dt_names)
    |> pair dt_names
  end;


(** declare existing type as datatype **)

fun prove_rep_datatype config dt_names alt_names descr sorts raw_inject half_distinct raw_induct thy1 =
  let
    val raw_distinct = (map o maps) (fn thm => [thm, thm RS not_sym]) half_distinct;
    val new_type_names = map Long_Name.base_name (the_default dt_names alt_names);
    val (((inject, distinct), [induct]), thy2) =
      thy1
      |> store_thmss "inject" new_type_names raw_inject
      ||>> store_thmss "distinct" new_type_names raw_distinct
      ||> Sign.add_path (space_implode "_" new_type_names)
      ||>> PureThy.add_thms [((Binding.name "induct", raw_induct), [mk_case_names_induct descr])]
      ||> Sign.restore_naming thy1;
  in
    thy2
    |> derive_datatype_props config dt_names alt_names [descr] sorts
         induct inject distinct
 end;

fun gen_rep_datatype prep_term config after_qed alt_names raw_ts thy =
  let
    fun constr_of_term (Const (c, T)) = (c, T)
      | constr_of_term t =
          error ("Not a constant: " ^ Syntax.string_of_term_global thy t);
    fun no_constr (c, T) = error ("Bad constructor: "
      ^ Sign.extern_const thy c ^ "::"
      ^ Syntax.string_of_typ_global thy T);
    fun type_of_constr (cT as (_, T)) =
      let
        val frees = OldTerm.typ_tfrees T;
        val (tyco, vs) = ((apsnd o map) (dest_TFree) o dest_Type o snd o strip_type) T
          handle TYPE _ => no_constr cT
        val _ = if has_duplicates (eq_fst (op =)) vs then no_constr cT else ();
        val _ = if length frees <> length vs then no_constr cT else ();
      in (tyco, (vs, cT)) end;

    val raw_cs = AList.group (op =) (map (type_of_constr o constr_of_term o prep_term thy) raw_ts);
    val _ = case map_filter (fn (tyco, _) =>
        if Symtab.defined (get_all thy) tyco then SOME tyco else NONE) raw_cs
     of [] => ()
      | tycos => error ("Type(s) " ^ commas (map quote tycos)
          ^ " already represented inductivly");
    val raw_vss = maps (map (map snd o fst) o snd) raw_cs;
    val ms = case distinct (op =) (map length raw_vss)
     of [n] => 0 upto n - 1
      | _ => error ("Different types in given constructors");
    fun inter_sort m = map (fn xs => nth xs m) raw_vss
      |> Library.foldr1 (Sorts.inter_sort (Sign.classes_of thy))
    val sorts = map inter_sort ms;
    val vs = Name.names Name.context Name.aT sorts;

    fun norm_constr (raw_vs, (c, T)) = (c, map_atyps
      (TFree o (the o AList.lookup (op =) (map fst raw_vs ~~ vs)) o fst o dest_TFree) T);

    val cs = map (apsnd (map norm_constr)) raw_cs;
    val dtyps_of_typ = map (dtyp_of_typ (map (rpair (map fst vs) o fst) cs))
      o fst o strip_type;
    val dt_names = map fst cs;

    fun mk_spec (i, (tyco, constr)) = (i, (tyco,
      map (DtTFree o fst) vs,
      (map o apsnd) dtyps_of_typ constr))
    val descr = map_index mk_spec cs;
    val injs = DatatypeProp.make_injs [descr] vs;
    val half_distincts = map snd (DatatypeProp.make_distincts [descr] vs);
    val ind = DatatypeProp.make_ind [descr] vs;
    val rules = (map o map o map) Logic.close_form [[[ind]], injs, half_distincts];

    fun after_qed' raw_thms =
      let
        val [[[raw_induct]], raw_inject, half_distinct] =
          unflat rules (map Drule.zero_var_indexes_list raw_thms);
            (*FIXME somehow dubious*)
      in
        ProofContext.theory_result
          (prove_rep_datatype config dt_names alt_names descr vs raw_inject half_distinct raw_induct)
        #-> after_qed
      end;
  in
    thy
    |> ProofContext.init
    |> Proof.theorem_i NONE after_qed' ((map o map) (rpair []) (flat rules))
  end;

val rep_datatype = gen_rep_datatype Sign.cert_term;
val rep_datatype_cmd = gen_rep_datatype Syntax.read_term_global default_config (K I);


(** definitional introduction of datatypes **)

fun gen_add_datatype prep_typ config 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 |>
      Sign.add_types (map (fn (tvs, tname, mx, _) =>
        (tname, length tvs, mx)) dts);

    val (tyvars, _, _, _)::_ = dts;
    val (new_dts, types_syntax) = ListPair.unzip (map (fn (tvs, tname, mx, _) =>
      let val full_tname = Sign.full_name tmp_thy (Binding.map_name (Syntax.type_name mx) tname)
      in (case duplicates (op =) tvs of
            [] => if eq_set (op =) (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 " ^ quote (Binding.str_of tname) ^
              " : " ^ commas dups))
      end) dts);
    val dt_names = map fst new_dts;

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

    fun prep_dt_spec ((tvs, tname, mx, constrs), tname') (dts', constr_syntax, sorts, i) =
      let
        fun prep_constr (cname, cargs, mx') (constrs, constr_syntax', sorts') =
          let
            val (cargs', sorts'') = fold_map (prep_typ tmp_thy) cargs sorts';
            val _ =
              (case subtract (op =) tvs (fold (curry OldTerm.add_typ_tfree_names) cargs' []) of
                [] => ()
              | vs => error ("Extra type variables on rhs: " ^ commas vs))
          in (constrs @ [(Sign.full_name_path tmp_thy tname'
                  (Binding.map_name (Syntax.const_name mx') cname),
                   map (dtyp_of_typ new_dts) cargs')],
              constr_syntax' @ [(cname, mx')], sorts'')
          end handle ERROR msg => cat_error msg
           ("The error above occured in constructor " ^ quote (Binding.str_of cname) ^
            " of datatype " ^ quote (Binding.str_of tname));

        val (constrs', constr_syntax', sorts') =
          fold prep_constr constrs ([], [], sorts)

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

    val (dts', constr_syntax, sorts', i) =
      fold prep_dt_spec (dts ~~ new_type_names) ([], [], [], 0);
    val sorts = sorts' @ (map (rpair (Sign.defaultS tmp_thy)) (subtract (op =) (map fst sorts') tyvars));
    val dt_info = get_all thy;
    val (descr, _) = unfold_datatypes tmp_thy dts' sorts dt_info dts' i;
    val _ = check_nonempty descr handle (exn as Datatype_Empty s) =>
      if #strict config then error ("Nonemptiness check failed for datatype " ^ s)
      else raise exn;

    val _ = message config ("Constructing datatype(s) " ^ commas_quote new_type_names);
    val ((inject, distinct, induct), thy') = thy |>
      DatatypeRepProofs.representation_proofs config dt_info new_type_names descr sorts
        types_syntax constr_syntax (mk_case_names_induct (flat descr));
  in
    derive_datatype_props config dt_names (SOME new_type_names) descr sorts
      induct inject distinct thy'
  end;

val add_datatype = gen_add_datatype cert_typ;
val datatype_cmd = snd ooo gen_add_datatype read_typ default_config;


(** package setup **)

(* setup theory *)

val setup =
  trfun_setup #>
  DatatypeInterpretation.init;

(* outer syntax *)

local

structure P = OuterParse and K = OuterKeyword

fun prep_datatype_decls args =
  let
    val names = map
      (fn ((((NONE, _), t), _), _) => Binding.name_of 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 (names, specs) end;

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

val parse_datatype_decls = P.and_list1 parse_datatype_decl >> prep_datatype_decls;

in

val _ =
  OuterSyntax.command "datatype" "define inductive datatypes" K.thy_decl
    (parse_datatype_decls >> (fn (names, specs) => Toplevel.theory (datatype_cmd names specs)));

val _ =
  OuterSyntax.command "rep_datatype" "represent existing types inductively" K.thy_goal
    (Scan.option (P.$$$ "(" |-- Scan.repeat1 P.name --| P.$$$ ")") -- Scan.repeat1 P.term
      >> (fn (alt_names, ts) => Toplevel.print
           o Toplevel.theory_to_proof (rep_datatype_cmd alt_names ts)));

end;

end;