(*  Title:      HOL/Tools/Datatype/datatype_data.ML

     Author:     Stefan Berghofer, TU Muenchen

 Datatype package: bookkeeping; interpretation of existing types as datatypes.

 *)

 signature DATATYPE_DATA =

 sig

   include DATATYPE_COMMON

   val derive_datatype_props : config -> string list -> string list option

     -> descr list -> (string * sort) list -> thm -> thm list list -> thm list list

     -> theory -> string list * 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 -> Datatype_Case.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 cert_typ: theory -> typ -> (string * sort) list -> typ * (string * sort) list

   val mk_case_names_induct: descr -> attribute

   val setup: theory -> theory

 end;

 structure Datatype_Data: DATATYPE_DATA =

 struct

 open Datatype_Aux;

 (** theory data **)

 (* data management *)

 structure DatatypesData = Theory_Data

 (

   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 extend = I;

   fun merge

     ({types = types1, constrs = constrs1, cases = cases1},

      {types = types2, constrs = constrs2, cases = cases2}) : T =

     {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 { 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 Datatype_Aux.dest_DtTFree) dtys;

     val cos = map

       (fn (co, tys) => (co, map (Datatype_Aux.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, _) = 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_global 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 =

   Datatype_Prop.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 = Rule_Cases.case_names (induct_cases descr);

 fun mk_case_names_exhausts descr new =

   map (Rule_Cases.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 = Datatype_Case.make_case

   (info_of_constr (ProofContext.theory_of ctxt)) ctxt;

 fun strip_case ctxt = Datatype_Case.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' = Syntax.mark_const case_name

       in (case_name', Datatype_Case.case_tr' info_of_case case_name')

       end) case_names, []) thy;

 val trfun_setup =

   Sign.add_advanced_trfuns ([],

     [(@{syntax_const "_case_syntax"}, Datatype_Case.case_tr true info_of_constr)],

     [], []);

 (** document antiquotation **)

 val _ = Thy_Output.antiquotation "datatype" (Args.type_name true)

   (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

       Thy_Output.output ctxt (Thy_Output.maybe_pretty_source (K (K pretty_datatype)) ctxt src [()])

     end);

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

 structure Datatype_Interpretation = Interpretation

   (type T = config * string list val eq: T * T -> bool = eq_snd op =);

 fun interpretation f = Datatype_Interpretation.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) = Datatype_Abs_Proofs.prove_casedist_thms config new_type_names

       descr sorts induct (mk_case_names_exhausts flat_descr dt_names) thy2;

     val (nchotomys, thy4) = Datatype_Abs_Proofs.prove_nchotomys config new_type_names

       descr sorts exhaust thy3;

     val ((rec_names, rec_rewrites), thy5) = Datatype_Abs_Proofs.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) = Datatype_Abs_Proofs.prove_case_thms

       config new_type_names descr sorts rec_names rec_rewrites thy5;

     val (case_congs, thy7) = Datatype_Abs_Proofs.prove_case_congs new_type_names

       descr sorts nchotomys case_rewrites thy6;

     val (weak_case_congs, thy8) = Datatype_Abs_Proofs.prove_weak_case_congs new_type_names

       descr sorts thy7;

     val (splits, thy9) = Datatype_Abs_Proofs.prove_split_thms

       config new_type_names descr sorts inject distinct exhaust case_rewrites thy8;

     val inducts = Project_Rule.projections (ProofContext.init_global 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]), [Rule_Cases.inner_rule, Induct.induct_type ""]))

         (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)]),

         ((Binding.empty, flat (distinct @ inject)), [Induct.induct_simp_add])] @

           named_rules @ unnamed_rules)

     |> snd

     |> add_case_tr' case_names

     |> register dt_infos

     |> Datatype_Interpretation.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 = Datatype_Prop.make_injs [descr] vs;

     val half_distincts = map snd (Datatype_Prop.make_distincts [descr] vs);

     val ind = Datatype_Prop.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.background_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_global

     |> Proof.theorem 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);

 (** package setup **)

 (* setup theory *)

 val setup =

   trfun_setup #>

   Datatype_Interpretation.init;

 (* outer syntax *)

 val _ =

   Outer_Syntax.command "rep_datatype" "represent existing types inductively" Keyword.thy_goal

     (Scan.option (Parse.$$$ "(" |-- Scan.repeat1 Parse.name --| Parse.$$$ ")") --

       Scan.repeat1 Parse.term

       >> (fn (alt_names, ts) =>

         Toplevel.print o Toplevel.theory_to_proof (rep_datatype_cmd alt_names ts)));

 end;