(*  Title:      ZF/Tools/datatype_package.ML

    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory

    Copyright   1994  University of Cambridge

Datatype/Codatatype Definitions.

The functor will be instantiated for normal sums/products (datatype defs)

                         and non-standard sums/products (codatatype defs)

Sums are used only for mutual recursion;

Products are used only to derive "streamlined" induction rules for relations

*)

type datatype_result =

   {con_defs   : thm list,             (*definitions made in thy*)

    case_eqns  : thm list,             (*equations for case operator*)

    recursor_eqns : thm list,          (*equations for the recursor*)

    free_iffs  : thm list,             (*freeness rewrite rules*)

    free_SEs   : thm list,             (*freeness destruct rules*)

    mk_free    : string -> thm};       (*function to make freeness theorems*)

signature DATATYPE_ARG =

sig

  val intrs : thm list

  val elims : thm list

end;

signature DATATYPE_PACKAGE =

sig

  (*Insert definitions for the recursive sets, which

     must *already* be declared as constants in parent theory!*)

  val add_datatype_i: term * term list -> Ind_Syntax.constructor_spec list list ->

    thm list * thm list * thm list -> theory -> theory * inductive_result * datatype_result

  val add_datatype: string * string list -> (string * string list * mixfix) list list ->

    (Facts.ref * Attrib.src list) list * (Facts.ref * Attrib.src list) list *

    (Facts.ref * Attrib.src list) list -> theory -> theory * inductive_result * datatype_result

end;

functor Add_datatype_def_Fun

 (structure Fp: FP and Pr : PR and CP: CARTPROD and Su : SU

  and Ind_Package : INDUCTIVE_PACKAGE

  and Datatype_Arg : DATATYPE_ARG

  val coind : bool): DATATYPE_PACKAGE =

struct

(*con_ty_lists specifies the constructors in the form (name, prems, mixfix) *)

(*univ or quniv constitutes the sum domain for mutual recursion;

  it is applied to the datatype parameters and to Consts occurring in the

  definition other than Nat.nat and the datatype sets themselves.

  FIXME: could insert all constant set expressions, e.g. nat->nat.*)

fun data_domain co (rec_tms, con_ty_lists) =

    let val rec_hds = map head_of rec_tms

        val dummy = assert_all is_Const rec_hds

          (fn t => "Datatype set not previously declared as constant: " ^

                   Syntax.string_of_term_global @{theory IFOL} t);

        val rec_names = (*nat doesn't have to be added*)

            @{const_name nat} :: map (#1 o dest_Const) rec_hds

        val u = if co then @{const QUniv.quniv} else @{const Univ.univ}

        val cs = (fold o fold) (fn (_, _, _, prems) => prems |> (fold o fold_aterms)

          (fn t as Const (a, _) => if member (op =) rec_names a then I else insert (op =) t

            | _ => I)) con_ty_lists [];

    in  u $ Ind_Syntax.union_params (hd rec_tms, cs)  end;

fun add_datatype_i (dom_sum, rec_tms) con_ty_lists (monos, type_intrs, type_elims) thy =

 let

  val dummy = (*has essential ancestors?*)

    Theory.requires thy "Datatype_ZF" "(co)datatype definitions";

  val rec_hds = map head_of rec_tms;

  val dummy = assert_all is_Const rec_hds

          (fn t => "Datatype set not previously declared as constant: " ^

                   Syntax.string_of_term_global thy t);

  val rec_names = map (#1 o dest_Const) rec_hds

  val rec_base_names = map Long_Name.base_name rec_names

  val big_rec_base_name = space_implode "_" rec_base_names

  val thy_path = thy |> Sign.add_path big_rec_base_name

  val big_rec_name = Sign.intern_const thy_path big_rec_base_name;

  val intr_tms = Ind_Syntax.mk_all_intr_tms thy_path (rec_tms, con_ty_lists);

  val dummy =

    writeln ((if coind then "Codatatype" else "Datatype") ^ " definition " ^ quote big_rec_name);

  val case_varname = "f";                (*name for case variables*)

  (** Define the constructors **)

  (*The empty tuple is 0*)

  fun mk_tuple [] = @{const zero}

    | mk_tuple args = foldr1 (fn (t1, t2) => Pr.pair $ t1 $ t2) args;

  fun mk_inject n k u = Balanced_Tree.access

    {left = fn t => Su.inl $ t, right = fn t => Su.inr $ t, init = u} n k;

  val npart = length rec_names;  (*number of mutually recursive parts*)

  val full_name = Sign.full_bname thy_path;

  (*Make constructor definition;

    kpart is the number of this mutually recursive part*)

  fun mk_con_defs (kpart, con_ty_list) =

    let val ncon = length con_ty_list    (*number of constructors*)

        fun mk_def (((id,T,syn), name, args, prems), kcon) =

              (*kcon is index of constructor*)

            Misc_Legacy.mk_defpair (list_comb (Const (full_name name, T), args),

                        mk_inject npart kpart

                        (mk_inject ncon kcon (mk_tuple args)))

    in  ListPair.map mk_def (con_ty_list, 1 upto ncon)  end;

  (*** Define the case operator ***)

  (*Combine split terms using case; yields the case operator for one part*)

  fun call_case case_list =

    let fun call_f (free,[]) = Abs("null", @{typ i}, free)

          | call_f (free,args) =

                CP.ap_split (foldr1 CP.mk_prod (map (#2 o dest_Free) args))

                            @{typ i}

                            free

    in  Balanced_Tree.make (fn (t1, t2) => Su.elim $ t1 $ t2) (map call_f case_list)  end;

  (** Generating function variables for the case definition

      Non-identifiers (e.g. infixes) get a name of the form f_op_nnn. **)

  (*The function variable for a single constructor*)

  fun add_case ((_, T, _), name, args, _) (opno, cases) =

    if Lexicon.is_identifier name then

      (opno, (Free (case_varname ^ "_" ^ name, T), args) :: cases)

    else

      (opno + 1, (Free (case_varname ^ "_op_" ^ string_of_int opno, T), args)

       :: cases);

  (*Treatment of a list of constructors, for one part

    Result adds a list of terms, each a function variable with arguments*)

  fun add_case_list con_ty_list (opno, case_lists) =

    let val (opno', case_list) = fold_rev add_case con_ty_list (opno, [])

    in (opno', case_list :: case_lists) end;

  (*Treatment of all parts*)

  val (_, case_lists) = fold_rev add_case_list con_ty_lists (1, []);

  (*extract the types of all the variables*)

  val case_typ = maps (map (#2 o #1)) con_ty_lists ---> @{typ "i => i"};

  val case_base_name = big_rec_base_name ^ "_case";

  val case_name = full_name case_base_name;

  (*The list of all the function variables*)

  val case_args = maps (map #1) case_lists;

  val case_const = Const (case_name, case_typ);

  val case_tm = list_comb (case_const, case_args);

  val case_def = Misc_Legacy.mk_defpair

    (case_tm, Balanced_Tree.make (fn (t1, t2) => Su.elim $ t1 $ t2) (map call_case case_lists));

  (** Generating function variables for the recursor definition

      Non-identifiers (e.g. infixes) get a name of the form f_op_nnn. **)

  (*a recursive call for x is the application rec`x  *)

  val rec_call = @{const apply} $ Free ("rec", @{typ i});

  (*look back down the "case args" (which have been reversed) to

    determine the de Bruijn index*)

  fun make_rec_call ([], _) arg = error

          "Internal error in datatype (variable name mismatch)"

    | make_rec_call (a::args, i) arg =

           if a = arg then rec_call $ Bound i

           else make_rec_call (args, i+1) arg;

  (*creates one case of the "X_case" definition of the recursor*)

  fun call_recursor ((case_var, case_args), (recursor_var, recursor_args)) =

      let fun add_abs (Free(a,T), u) = Abs(a,T,u)

          val ncase_args = length case_args

          val bound_args = map Bound ((ncase_args - 1) downto 0)

          val rec_args = map (make_rec_call (rev case_args,0))

                         (List.drop(recursor_args, ncase_args))

      in

          List.foldr add_abs

            (list_comb (recursor_var,

                        bound_args @ rec_args)) case_args

      end

  (*Find each recursive argument and add a recursive call for it*)

  fun rec_args [] = []

    | rec_args ((Const(@{const_name mem},_)$arg$X)::prems) =

       (case head_of X of

            Const(a,_) => (*recursive occurrence?*)

                          if member (op =) rec_names a

                              then arg :: rec_args prems

                          else rec_args prems

          | _ => rec_args prems)

    | rec_args (_::prems) = rec_args prems;

  (*Add an argument position for each occurrence of a recursive set.

    Strictly speaking, the recursive arguments are the LAST of the function

    variable, but they all have type "i" anyway*)

  fun add_rec_args args' T = (map (fn _ => @{typ i}) args') ---> T

  (*Plug in the function variable type needed for the recursor

    as well as the new arguments (recursive calls)*)

  fun rec_ty_elem ((id, T, syn), name, args, prems) =

      let val args' = rec_args prems

      in ((id, add_rec_args args' T, syn),

          name, args @ args', prems)

      end;

  val rec_ty_lists = (map (map rec_ty_elem) con_ty_lists);

  (*Treatment of all parts*)

  val (_, recursor_lists) = fold_rev add_case_list rec_ty_lists (1, []);

  (*extract the types of all the variables*)

  val recursor_typ = maps (map (#2 o #1)) rec_ty_lists ---> @{typ "i => i"};

  val recursor_base_name = big_rec_base_name ^ "_rec";

  val recursor_name = full_name recursor_base_name;

  (*The list of all the function variables*)

  val recursor_args = maps (map #1) recursor_lists;

  val recursor_tm =

    list_comb (Const (recursor_name, recursor_typ), recursor_args);

  val recursor_cases = map call_recursor (flat case_lists ~~ flat recursor_lists);

  val recursor_def =

      Misc_Legacy.mk_defpair

        (recursor_tm,

         @{const Univ.Vrecursor} $

           absfree ("rec", @{typ i}) (list_comb (case_const, recursor_cases)));

  (* Build the new theory *)

  val need_recursor = (not coind andalso recursor_typ <> case_typ);

  fun add_recursor thy =

    if need_recursor then

      thy

      |> Sign.add_consts_i

        [(Binding.name recursor_base_name, recursor_typ, NoSyn)]

      |> (snd o Global_Theory.add_defs false [(Thm.no_attributes o apfst Binding.name) recursor_def])

    else thy;

  val (con_defs, thy0) = thy_path

             |> Sign.add_consts_i

                 (map (fn (c, T, mx) => (Binding.name c, T, mx))

                  ((case_base_name, case_typ, NoSyn) :: map #1 (flat con_ty_lists)))

             |> Global_Theory.add_defs false

                 (map (Thm.no_attributes o apfst Binding.name)

                  (case_def ::

                   flat (ListPair.map mk_con_defs (1 upto npart, con_ty_lists))))

             ||> add_recursor

             ||> Sign.parent_path

  val intr_names = map (Binding.name o #2) (flat con_ty_lists);

  val (thy1, ind_result) =

    thy0 |> Ind_Package.add_inductive_i

      false (rec_tms, dom_sum) (map Thm.no_attributes (intr_names ~~ intr_tms))

      (monos, con_defs, type_intrs @ Datatype_Arg.intrs, type_elims @ Datatype_Arg.elims);

  (**** Now prove the datatype theorems in this theory ****)

  (*** Prove the case theorems ***)

  (*Each equation has the form

    case(f_con1,...,f_conn)(coni(args)) = f_coni(args) *)

  fun mk_case_eqn (((_,T,_), name, args, _), case_free) =

    FOLogic.mk_Trueprop

      (FOLogic.mk_eq

       (case_tm $

         (list_comb (Const (Sign.intern_const thy1 name,T),

                     args)),

        list_comb (case_free, args)));

  val case_trans = hd con_defs RS @{thm def_trans}

  and split_trans = Pr.split_eq RS @{thm meta_eq_to_obj_eq} RS @{thm trans};

  fun prove_case_eqn (arg, con_def) =

    Goal.prove_global thy1 [] []

      (Ind_Syntax.traceIt "next case equation = " thy1 (mk_case_eqn arg))

      (*Proves a single case equation.  Could use simp_tac, but it's slower!*)

      (fn _ => EVERY

        [rewrite_goals_tac [con_def],

         rtac case_trans 1,

         REPEAT

           (resolve_tac [@{thm refl}, split_trans,

             Su.case_inl RS @{thm trans}, Su.case_inr RS @{thm trans}] 1)]);

  val free_iffs = map Drule.export_without_context (con_defs RL [@{thm def_swap_iff}]);

  val case_eqns = map prove_case_eqn (flat con_ty_lists ~~ case_args ~~ tl con_defs);

  (*** Prove the recursor theorems ***)

  val recursor_eqns = case try (Misc_Legacy.get_def thy1) recursor_base_name of

     NONE => (writeln "  [ No recursion operator ]";

              [])

   | SOME recursor_def =>

      let

        (*Replace subterms rec`x (where rec is a Free var) by recursor_tm(x) *)

        fun subst_rec (Const(@{const_name apply},_) $ Free _ $ arg) = recursor_tm $ arg

          | subst_rec tm =

              let val (head, args) = strip_comb tm

              in  list_comb (head, map subst_rec args)  end;

        (*Each equation has the form

          REC(coni(args)) = f_coni(args, REC(rec_arg), ...)

          where REC = recursor(f_con1,...,f_conn) and rec_arg is a recursive

          constructor argument.*)

        fun mk_recursor_eqn (((_,T,_), name, args, _), recursor_case) =

          FOLogic.mk_Trueprop

           (FOLogic.mk_eq

            (recursor_tm $

             (list_comb (Const (Sign.intern_const thy1 name,T),

                         args)),

             subst_rec (Term.betapplys (recursor_case, args))));

        val recursor_trans = recursor_def RS @{thm def_Vrecursor} RS @{thm trans};

        fun prove_recursor_eqn arg =

          Goal.prove_global thy1 [] []

            (Ind_Syntax.traceIt "next recursor equation = " thy1 (mk_recursor_eqn arg))

            (*Proves a single recursor equation.*)

            (fn _ => EVERY

              [rtac recursor_trans 1,

               simp_tac (rank_ss addsimps case_eqns) 1,

               IF_UNSOLVED (simp_tac (rank_ss addsimps tl con_defs) 1)]);

      in

         map prove_recursor_eqn (flat con_ty_lists ~~ recursor_cases)

      end

  val constructors =

      map (head_of o #1 o Logic.dest_equals o Thm.prop_of) (tl con_defs);

  val free_SEs = map Drule.export_without_context (Ind_Syntax.mk_free_SEs free_iffs);

  val {intrs, elim, induct, mutual_induct, ...} = ind_result

  (*Typical theorems have the form ~con1=con2, con1=con2==>False,

    con1(x)=con1(y) ==> x=y, con1(x)=con1(y) <-> x=y, etc.  *)

  fun mk_free s =

    let

      val thy = theory_of_thm elim;

      val ctxt = Proof_Context.init_global thy;

    in

      Goal.prove_global thy [] [] (Syntax.read_prop ctxt s)

        (fn _ => EVERY

         [rewrite_goals_tac con_defs,

          fast_tac (put_claset ZF_cs ctxt addSEs free_SEs @ Su.free_SEs) 1])

    end;

  val simps = case_eqns @ recursor_eqns;

  val dt_info =

        {inductive = true,

         constructors = constructors,

         rec_rewrites = recursor_eqns,

         case_rewrites = case_eqns,

         induct = induct,

         mutual_induct = mutual_induct,

         exhaustion = elim};

  val con_info =

        {big_rec_name = big_rec_name,

         constructors = constructors,

            (*let primrec handle definition by cases*)

         free_iffs = free_iffs,

         rec_rewrites = (case recursor_eqns of

                             [] => case_eqns | _ => recursor_eqns)};

  (*associate with each constructor the datatype name and rewrites*)

  val con_pairs = map (fn c => (#1 (dest_Const c), con_info)) constructors

 in

  (*Updating theory components: simprules and datatype info*)

  (thy1 |> Sign.add_path big_rec_base_name

        |> Global_Theory.add_thmss

         [((Binding.name "simps", simps), [Simplifier.simp_add]),

          ((Binding.empty, intrs), [Cla.safe_intro NONE]),

          ((Binding.name "con_defs", con_defs), []),

          ((Binding.name "case_eqns", case_eqns), []),

          ((Binding.name "recursor_eqns", recursor_eqns), []),

          ((Binding.name "free_iffs", free_iffs), []),

          ((Binding.name "free_elims", free_SEs), [])] |> snd

        |> DatatypesData.map (Symtab.update (big_rec_name, dt_info))

        |> ConstructorsData.map (fold Symtab.update con_pairs)

        |> Sign.parent_path,

   ind_result,

   {con_defs = con_defs,

    case_eqns = case_eqns,

    recursor_eqns = recursor_eqns,

    free_iffs = free_iffs,

    free_SEs = free_SEs,

    mk_free = mk_free})

  end;

fun add_datatype (sdom, srec_tms) scon_ty_lists (raw_monos, raw_type_intrs, raw_type_elims) thy =

  let

    val ctxt = Proof_Context.init_global thy;

    fun read_is strs =

      map (Syntax.parse_term ctxt #> Type.constraint @{typ i}) strs

      |> Syntax.check_terms ctxt;

    val rec_tms = read_is srec_tms;

    val con_ty_lists = Ind_Syntax.read_constructs ctxt scon_ty_lists;

    val dom_sum =

      if sdom = "" then data_domain coind (rec_tms, con_ty_lists)

      else singleton read_is sdom;

    val monos = Attrib.eval_thms ctxt raw_monos;

    val type_intrs = Attrib.eval_thms ctxt raw_type_intrs;

    val type_elims = Attrib.eval_thms ctxt raw_type_elims;

  in add_datatype_i (dom_sum, rec_tms) con_ty_lists (monos, type_intrs, type_elims) thy end;

(* outer syntax *)

fun mk_datatype ((((dom, dts), monos), type_intrs), type_elims) =

  #1 o add_datatype (dom, map fst dts) (map snd dts) (monos, type_intrs, type_elims);

val con_decl =

  Parse.name -- Scan.optional (@{keyword "("} |-- Parse.list1 Parse.term --| @{keyword ")"}) [] --

    Parse.opt_mixfix >> Parse.triple1;

val datatype_decl =

  (Scan.optional ((@{keyword "\<subseteq>"} || @{keyword "<="}) |-- Parse.!!! Parse.term) "") --

  Parse.and_list1 (Parse.term -- (@{keyword "="} |-- Parse.enum1 "|" con_decl)) --

  Scan.optional (@{keyword "monos"} |-- Parse.!!! Parse_Spec.xthms1) [] --

  Scan.optional (@{keyword "type_intros"} |-- Parse.!!! Parse_Spec.xthms1) [] --

  Scan.optional (@{keyword "type_elims"} |-- Parse.!!! Parse_Spec.xthms1) []

  >> (Toplevel.theory o mk_datatype);

val coind_prefix = if coind then "co" else "";

val _ =

  Outer_Syntax.command (coind_prefix ^ "datatype")

    ("define " ^ coind_prefix ^ "datatype") Keyword.thy_decl datatype_decl;

end;