(*  Title:      HOL/Tools/primrec_package.ML

     ID:         $Id$

     Author:     Stefan Berghofer, TU Muenchen and Norbert Voelker, FernUni Hagen

 Package for defining functions on datatypes by primitive recursion.

 *)

 signature PRIMREC_PACKAGE =

 sig

   val quiet_mode: bool ref

   val unify_consts: theory -> term list -> term list -> term list * term list

   val add_primrec: string -> ((bstring * string) * Attrib.src list) list

     -> theory -> thm list * theory

   val add_primrec_unchecked: string -> ((bstring * string) * Attrib.src list) list

     -> theory -> thm list * theory

   val add_primrec_i: string -> ((bstring * term) * attribute list) list

     -> theory -> thm list * theory

   val add_primrec_unchecked_i: string -> ((bstring * term) * attribute list) list

     -> theory -> thm list * theory

   (* FIXME !? *)

   val gen_primrec: ((bstring * attribute list) * thm list -> theory -> (bstring * thm list) * theory)

     -> ((bstring * attribute list) * term -> theory -> (bstring * thm) * theory)

     -> string -> ((bstring * attribute list) * term) list

     -> theory -> thm list * theory;

 end;

 structure PrimrecPackage : PRIMREC_PACKAGE =

 struct

 open DatatypeAux;

 exception RecError of string;

 fun primrec_err s = error ("Primrec definition error:\n" ^ s);

 fun primrec_eq_err thy s eq =

   primrec_err (s ^ "\nin\n" ^ quote (Sign.string_of_term thy eq));

 (* messages *)

 val quiet_mode = ref false;

 fun message s = if ! quiet_mode then () else writeln s;

 (*the following code ensures that each recursive set always has the

   same type in all introduction rules*)

 fun unify_consts thy cs intr_ts =

   (let

     val add_term_consts_2 = fold_aterms (fn Const c => insert (op =) c | _ => I);

     fun varify (t, (i, ts)) =

       let val t' = map_types (Logic.incr_tvar (i + 1)) (snd (Type.varify [] t))

       in (maxidx_of_term t', t'::ts) end;

     val (i, cs') = foldr varify (~1, []) cs;

     val (i', intr_ts') = foldr varify (i, []) intr_ts;

     val rec_consts = fold add_term_consts_2 cs' [];

     val intr_consts = fold add_term_consts_2 intr_ts' [];

     fun unify (cname, cT) =

       let val consts = map snd (filter (fn (c, _) => c = cname) intr_consts)

       in fold (Sign.typ_unify thy) ((replicate (length consts) cT) ~~ consts) end;

     val (env, _) = fold unify rec_consts (Vartab.empty, i');

     val subst = Type.freeze o map_types (Envir.norm_type env)

   in (map subst cs', map subst intr_ts')

   end) handle Type.TUNIFY =>

     (warning "Occurrences of recursive constant have non-unifiable types"; (cs, intr_ts));

 (* preprocessing of equations *)

 fun process_eqn thy eq rec_fns =

   let

     val (lhs, rhs) =

       if null (term_vars eq) then

         HOLogic.dest_eq (HOLogic.dest_Trueprop eq)

         handle TERM _ => raise RecError "not a proper equation"

       else raise RecError "illegal schematic variable(s)";

     val (recfun, args) = strip_comb lhs;

     val fnameT = dest_Const recfun handle TERM _ =>

       raise RecError "function is not declared as constant in theory";

     val (ls', rest)  = take_prefix is_Free args;

     val (middle, rs') = take_suffix is_Free rest;

     val rpos = length ls';

     val (constr, cargs') = if null middle then raise RecError "constructor missing"

       else strip_comb (hd middle);

     val (cname, T) = dest_Const constr

       handle TERM _ => raise RecError "ill-formed constructor";

     val (tname, _) = dest_Type (body_type T) handle TYPE _ =>

       raise RecError "cannot determine datatype associated with function"

     val (ls, cargs, rs) =

       (map dest_Free ls', map dest_Free cargs', map dest_Free rs')

       handle TERM _ => raise RecError "illegal argument in pattern";

     val lfrees = ls @ rs @ cargs;

     fun check_vars _ [] = ()

       | check_vars s vars = raise RecError (s ^ commas_quote (map fst vars))

   in

     if length middle > 1 then

       raise RecError "more than one non-variable in pattern"

     else

      (check_vars "repeated variable names in pattern: " (duplicates (op =) lfrees);

       check_vars "extra variables on rhs: "

         (map dest_Free (term_frees rhs) \\ lfrees);

       case AList.lookup (op =) rec_fns fnameT of

         NONE =>

           (fnameT, (tname, rpos, [(cname, (ls, cargs, rs, rhs, eq))]))::rec_fns

       | SOME (_, rpos', eqns) =>

           if AList.defined (op =) eqns cname then

             raise RecError "constructor already occurred as pattern"

           else if rpos <> rpos' then

             raise RecError "position of recursive argument inconsistent"

           else

             AList.update (op =) (fnameT, (tname, rpos, (cname, (ls, cargs, rs, rhs, eq))::eqns))

               rec_fns)

   end

   handle RecError s => primrec_eq_err thy s eq;

 fun process_fun thy descr rec_eqns (i, fnameT as (fname, _)) (fnameTs, fnss) =

   let

     val (_, (tname, _, constrs)) = List.nth (descr, i);

     (* substitute "fname ls x rs" by "y ls rs" for (x, (_, y)) in subs *)

     fun subst [] t fs = (t, fs)

       | subst subs (Abs (a, T, t)) fs =

           fs

           |> subst subs t

           |-> (fn t' => pair (Abs (a, T, t')))

       | subst subs (t as (_ $ _)) fs =

           let

             val (f, ts) = strip_comb t;

           in

             if is_Const f andalso dest_Const f mem map fst rec_eqns then

               let

                 val fnameT' as (fname', _) = dest_Const f;

                 val (_, rpos, _) = the (AList.lookup (op =) rec_eqns fnameT');

                 val ls = Library.take (rpos, ts);

                 val rest = Library.drop (rpos, ts);

                 val (x', rs) = (hd rest, tl rest)

                   handle Empty => raise RecError ("not enough arguments\

                    \ in recursive application\nof function " ^ quote fname' ^ " on rhs");

                 val (x, xs) = strip_comb x'

               in case AList.lookup (op =) subs x

                of NONE =>

                     fs

                     |> fold_map (subst subs) ts

                     |-> (fn ts' => pair (list_comb (f, ts')))

                 | SOME (i', y) =>

                     fs

                     |> fold_map (subst subs) (xs @ ls @ rs)

                     ||> process_fun thy descr rec_eqns (i', fnameT')

                     |-> (fn ts' => pair (list_comb (y, ts')))

               end

             else

               fs

               |> fold_map (subst subs) (f :: ts)

               |-> (fn (f'::ts') => pair (list_comb (f', ts')))

           end

       | subst _ t fs = (t, fs);

     (* translate rec equations into function arguments suitable for rec comb *)

     fun trans eqns (cname, cargs) (fnameTs', fnss', fns) =

       (case AList.lookup (op =) eqns cname of

           NONE => (warning ("No equation for constructor " ^ quote cname ^

             "\nin definition of function " ^ quote fname);

               (fnameTs', fnss', (Const ("HOL.undefined", dummyT))::fns))

         | SOME (ls, cargs', rs, rhs, eq) =>

             let

               val recs = filter (is_rec_type o snd) (cargs' ~~ cargs);

               val rargs = map fst recs;

               val subs = map (rpair dummyT o fst)

                 (rev (rename_wrt_term rhs rargs));

               val (rhs', (fnameTs'', fnss'')) =

                   (subst (map (fn ((x, y), z) =>

                                (Free x, (body_index y, Free z)))

                           (recs ~~ subs)) rhs (fnameTs', fnss'))

                   handle RecError s => primrec_eq_err thy s eq

             in (fnameTs'', fnss'',

                 (list_abs_free (cargs' @ subs @ ls @ rs, rhs'))::fns)

             end)

   in (case AList.lookup (op =) fnameTs i of

       NONE =>

         if exists (equal fnameT o snd) fnameTs then

           raise RecError ("inconsistent functions for datatype " ^ quote tname)

         else

           let

             val (_, _, eqns) = the (AList.lookup (op =) rec_eqns fnameT);

             val (fnameTs', fnss', fns) = fold_rev (trans eqns) constrs

               ((i, fnameT)::fnameTs, fnss, [])

           in

             (fnameTs', (i, (fname, #1 (snd (hd eqns)), fns))::fnss')

           end

     | SOME fnameT' =>

         if fnameT = fnameT' then (fnameTs, fnss)

         else raise RecError ("inconsistent functions for datatype " ^ quote tname))

   end;

 (* prepare functions needed for definitions *)

 fun get_fns fns ((i : int, (tname, _, constrs)), rec_name) (fs, defs) =

   case AList.lookup (op =) fns i of

      NONE =>

        let

          val dummy_fns = map (fn (_, cargs) => Const ("HOL.undefined",

            replicate ((length cargs) + (length (List.filter is_rec_type cargs)))

              dummyT ---> HOLogic.unitT)) constrs;

          val _ = warning ("No function definition for datatype " ^ quote tname)

        in

          (dummy_fns @ fs, defs)

        end

    | SOME (fname, ls, fs') => (fs' @ fs, (fname, ls, rec_name, tname) :: defs);

 (* make definition *)

 fun make_def thy fs (fname, ls, rec_name, tname) =

   let

     val rhs = fold_rev (fn T => fn t => Abs ("", T, t))

                     ((map snd ls) @ [dummyT])

                     (list_comb (Const (rec_name, dummyT),

                                 fs @ map Bound (0 ::(length ls downto 1))))

     val def_name = Sign.base_name fname ^ "_" ^ Sign.base_name tname ^ "_def";

     val def_prop =

       singleton (Syntax.check_terms (ProofContext.init thy))

         (Logic.mk_equals (Const (fname, dummyT), rhs));

   in (def_name, def_prop) end;

 (* find datatypes which contain all datatypes in tnames' *)

 fun find_dts (dt_info : datatype_info Symtab.table) _ [] = []

   | find_dts dt_info tnames' (tname::tnames) =

       (case Symtab.lookup dt_info tname of

           NONE => primrec_err (quote tname ^ " is not a datatype")

         | SOME dt =>

             if tnames' subset (map (#1 o snd) (#descr dt)) then

               (tname, dt)::(find_dts dt_info tnames' tnames)

             else find_dts dt_info tnames' tnames);

 fun prepare_induct ({descr, induction, ...}: datatype_info) rec_eqns =

   let

     fun constrs_of (_, (_, _, cs)) =

       map (fn (cname:string, (_, cargs, _, _, _)) => (cname, map fst cargs)) cs;

     val params_of = these o AList.lookup (op =) (List.concat (map constrs_of rec_eqns));

   in

     induction

     |> RuleCases.rename_params (map params_of (List.concat (map (map #1 o #3 o #2) descr)))

     |> RuleCases.save induction

   end;

 local

 fun gen_primrec_i note def alt_name eqns_atts thy =

   let

     val (eqns, atts) = split_list eqns_atts;

     val dt_info = DatatypePackage.get_datatypes thy;

     val rec_eqns = fold_rev (process_eqn thy o snd) eqns [] ;

     val tnames = distinct (op =) (map (#1 o snd) rec_eqns);

     val dts = find_dts dt_info tnames tnames;

     val main_fns =

       map (fn (tname, {index, ...}) =>

         (index,

           (fst o the o find_first (fn f => (#1 o snd) f = tname)) rec_eqns))

       dts;

     val {descr, rec_names, rec_rewrites, ...} =

       if null dts then

         primrec_err ("datatypes " ^ commas_quote tnames ^ "\nare not mutually recursive")

       else snd (hd dts);

     val (fnameTs, fnss) =

       fold_rev (process_fun thy descr rec_eqns) main_fns ([], []);

     val (fs, defs) = fold_rev (get_fns fnss) (descr ~~ rec_names) ([], []);

     val defs' = map (make_def thy fs) defs;

     val nameTs1 = map snd fnameTs;

     val nameTs2 = map fst rec_eqns;

     val _ = if gen_eq_set (op =) (nameTs1, nameTs2) then ()

             else primrec_err ("functions " ^ commas_quote (map fst nameTs2) ^

               "\nare not mutually recursive");

     val primrec_name =

       if alt_name = "" then (space_implode "_" (map (Sign.base_name o #1) defs)) else alt_name;

     val (defs_thms', thy') =

       thy

       |> Sign.add_path primrec_name

       |> fold_map def (map (fn (name, t) => ((name, []), t)) defs');

     val rewrites = (map mk_meta_eq rec_rewrites) @ map snd defs_thms';

     val _ = message ("Proving equations for primrec function(s) " ^

       commas_quote (map fst nameTs1) ^ " ...");

     val simps = map (fn (_, t) => Goal.prove_global thy' [] [] t

         (fn _ => EVERY [rewrite_goals_tac rewrites, rtac refl 1])) eqns;

     val (simps', thy'') =

       thy'

       |> fold_map note ((map fst eqns ~~ atts) ~~ map single simps);

     val simps'' = maps snd simps';

   in

     thy''

     |> note (("simps", [Simplifier.simp_add, RecfunCodegen.add_default]), simps'')

     |> snd

     |> note (("induct", []), [prepare_induct (#2 (hd dts)) rec_eqns])

     |> snd

     |> Sign.parent_path

     |> pair simps''

   end;

 fun gen_primrec note def alt_name eqns thy =

   let

     val ((names, strings), srcss) = apfst split_list (split_list eqns);

     val atts = map (map (Attrib.attribute thy)) srcss;

     val eqn_ts = map (fn s => Syntax.read_prop_global thy s

       handle ERROR msg => cat_error msg ("The error(s) above occurred for " ^ s)) strings;

     val rec_ts = map (fn eq => head_of (fst (HOLogic.dest_eq (HOLogic.dest_Trueprop eq)))

       handle TERM _ => primrec_eq_err thy "not a proper equation" eq) eqn_ts;

     val (_, eqn_ts') = unify_consts thy rec_ts eqn_ts

   in

     gen_primrec_i note def alt_name (names ~~ eqn_ts' ~~ atts) thy

   end;

 fun thy_note ((name, atts), thms) =

   PureThy.add_thmss [((name, thms), atts)] #-> (fn [thms] => pair (name, thms));

 fun thy_def false ((name, atts), t) =

       PureThy.add_defs_i false [((name, t), atts)] #-> (fn [thm] => pair (name, thm))

   | thy_def true ((name, atts), t) =

       PureThy.add_defs_unchecked_i false [((name, t), atts)] #-> (fn [thm] => pair (name, thm));

 in

 val add_primrec = gen_primrec thy_note (thy_def false);

 val add_primrec_unchecked = gen_primrec thy_note (thy_def true);

 val add_primrec_i = gen_primrec_i thy_note (thy_def false);

 val add_primrec_unchecked_i = gen_primrec_i thy_note (thy_def true);

 fun gen_primrec note def alt_name specs =

   gen_primrec_i note def alt_name (map (fn ((name, t), atts) => ((name, atts), t)) specs);

 end;

 (* outer syntax *)

 local structure P = OuterParse and K = OuterKeyword in

 val opt_unchecked_name =

   Scan.optional (P.$$$ "(" |-- P.!!!

     (((P.$$$ "unchecked" >> K true) -- Scan.optional P.name "" ||

       P.name >> pair false) --| P.$$$ ")")) (false, "");

 val primrec_decl =

   opt_unchecked_name -- Scan.repeat1 (SpecParse.opt_thm_name ":" -- P.prop);

 val _ =

   OuterSyntax.command "primrec" "define primitive recursive functions on datatypes" K.thy_decl

     (primrec_decl >> (fn ((unchecked, alt_name), eqns) =>

       Toplevel.theory (snd o

         (if unchecked then add_primrec_unchecked else add_primrec) alt_name

           (map P.triple_swap eqns))));

 end;

 end;