(*  Title:      HOL/Nominal/nominal_primrec.ML

     ID:         $Id$

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

 Package for defining functions on nominal datatypes by primitive recursion.

 Taken from HOL/Tools/primrec_package.ML

 *)

 signature NOMINAL_PRIMREC =

 sig

   val add_primrec: string -> string list option -> string option ->

     ((Binding.T * string) * Attrib.src list) list -> theory -> Proof.state

   val add_primrec_unchecked: string -> string list option -> string option ->

     ((Binding.T * string) * Attrib.src list) list -> theory -> Proof.state

   val add_primrec_i: string -> term list option -> term option ->

     ((Binding.T * term) * attribute list) list -> theory -> Proof.state

   val add_primrec_unchecked_i: string -> term list option -> term option ->

     ((Binding.T * term) * attribute list) list -> theory -> Proof.state

 end;

 structure NominalPrimrec : NOMINAL_PRIMREC =

 struct

 open DatatypeAux;

 exception RecError of string;

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

 fun primrec_eq_err thy s eq =

   primrec_err (s ^ "\nin\n" ^ quote (Syntax.string_of_term_global thy eq));

 (* 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 (Logic.strip_imp_concl 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;

 val param_err = "Parameters must be the same for all recursive functions";

 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" 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, eqns) = 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 rs' = (case eqns of

                     (_, (ls', _, rs', _, _)) :: _ =>

                       let val (rs1, rs2) = chop (length rs') rs

                       in

                         if ls = map Free ls' andalso rs1 = map Free rs' then rs2

                         else raise RecError param_err

                       end

                   | _ => raise RecError ("no equations for " ^ quote fname'));

                 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 @ 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 (@{const_name 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, 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 SOME (_, _, eqns as (_, (ls, _, rs, _, _)) :: _) =

               AList.lookup (op =) rec_eqns fnameT;

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

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

           in

             (fnameTs', (i, (fname, ls, rs, 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 (@{const_name 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, rs, fs') => (fs' @ fs, (fname, ls, rs, rec_name, tname) :: defs);

 (* make definition *)

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

   let

     val used = map fst (fold Term.add_frees fs []);

     val x = (Name.variant used "x", dummyT);

     val frees = ls @ x :: rs;

     val rhs = list_abs_free (frees,

       list_comb (Const (rec_name, dummyT), fs @ [Free x]))

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

     val def_prop as _ $ _ $ t =

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

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

   in ((def_name, def_prop), subst_bounds (rev (map Free frees), strip_abs_body t)) end;

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

 fun find_dts (dt_info : NominalPackage.nominal_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 nominal 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 common_prefix eq ([], _) = []

   | common_prefix eq (_, []) = []

   | common_prefix eq (x :: xs, y :: ys) =

       if eq (x, y) then x :: common_prefix eq (xs, ys) else [];

 local

 fun gen_primrec_i note def alt_name invs fctxt eqns_atts thy =

   let

     val (raw_eqns, atts) = split_list eqns_atts;

     val eqns = map (apfst Binding.base_name) raw_eqns;

     val dt_info = NominalPackage.get_nominal_datatypes thy;

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

     val lsrs :: lsrss = maps (fn (_, (_, _, eqns)) =>

       map (fn (_, (ls, _, rs, _, _)) => ls @ rs) eqns) rec_eqns

     val _ =

       (if forall (curry eq_set lsrs) lsrss andalso forall

          (fn (_, (_, _, (_, (ls, _, rs, _, _)) :: eqns)) =>

                forall (fn (_, (ls', _, rs', _, _)) =>

                  ls = ls' andalso rs = rs') eqns

            | _ => true) rec_eqns

        then () else primrec_err param_err);

     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 descr = map (fn (i, (tname, args, constrs)) => (i, (tname, args,

       map (fn (cname, cargs) => (cname, fold (fn (dTs, dT) => fn dTs' =>

         dTs' @ dTs @ [dT]) cargs [])) constrs))) descr;

     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 cert = cterm_of thy';

     fun mk_idx eq =

       let

         val Const c = head_of (fst (HOLogic.dest_eq (HOLogic.dest_Trueprop

           (Logic.strip_imp_concl eq))));

         val SOME i = AList.lookup op = (map swap fnameTs) c;

         val SOME (_, _, constrs) = AList.lookup op = descr i;

         val SOME (_, _, eqns) = AList.lookup op = rec_eqns c;

         val SOME (cname, (_, cargs, _, _, _)) = find_first

           (fn (_, (_, _, _, _, eq')) => eq = eq') eqns

       in (i, find_index (fn (cname', _) => cname = cname') constrs, cargs) end;

     val rec_rewritess =

       unflat (map (fn (_, (_, _, constrs)) => constrs) descr) rec_rewrites;

     val fvars = rec_rewrites |> hd |> concl_of |> HOLogic.dest_Trueprop |>

       HOLogic.dest_eq |> fst |> strip_comb |> snd |> take_prefix is_Var |> fst;

     val (pvars, ctxtvars) = List.partition

       (equal HOLogic.boolT o body_type o snd)

       (fold_rev Term.add_vars (map Logic.strip_assums_concl

         (prems_of (hd rec_rewrites))) [] \\ map dest_Var fvars);

     val cfs = defs' |> hd |> snd |> strip_comb |> snd |>

       curry (List.take o swap) (length fvars) |> map cert;

     val invs' = (case invs of

         NONE => map (fn (i, _) =>

           let

             val SOME (_, T) = AList.lookup op = fnameTs i

             val (Ts, U) = strip_type T

           in

             Abs ("x", List.drop (Ts, length lsrs + 1) ---> U, HOLogic.true_const)

           end) descr

       | SOME invs' => invs');

     val inst = (map cert fvars ~~ cfs) @

       (map (cert o Var) pvars ~~ map cert invs') @

       (case ctxtvars of

          [ctxtvar] => [(cert (Var ctxtvar), cert (the_default HOLogic.unit fctxt))]

        | _ => []);

     val rec_rewrites' = map (fn (_, eq) =>

       let

         val (i, j, cargs) = mk_idx eq

         val th = nth (nth rec_rewritess i) j;

         val cargs' = th |> concl_of |> HOLogic.dest_Trueprop |>

           HOLogic.dest_eq |> fst |> strip_comb |> snd |> split_last |> snd |>

           strip_comb |> snd

       in (cargs, Logic.strip_imp_prems eq,

         Drule.cterm_instantiate (inst @

           (map (cterm_of thy') cargs' ~~ map (cterm_of thy' o Free) cargs)) th)

       end) eqns;

     val prems = foldr1 (common_prefix op aconv) (map (prems_of o #3) rec_rewrites');

     val cprems = map cert prems;

     val asms = map Thm.assume cprems;

     val premss = map (fn (cargs, eprems, eqn) =>

       map (fn t => list_all_free (cargs, Logic.list_implies (eprems, t)))

         (List.drop (prems_of eqn, length prems))) rec_rewrites';

     val cpremss = map (map cert) premss;

     val asmss = map (map Thm.assume) cpremss;

     fun mk_eqn ((cargs, eprems, eqn), asms') =

       let

         val ceprems = map cert eprems;

         val asms'' = map Thm.assume ceprems;

         val ccargs = map (cert o Free) cargs;

         val asms''' = map (fn th => implies_elim_list

           (forall_elim_list ccargs th) asms'') asms'

       in

         implies_elim_list eqn (asms @ asms''') |>

         implies_intr_list ceprems |>

         forall_intr_list ccargs

       end;

     val rule_prems = cprems @ flat cpremss;

     val rule = implies_intr_list rule_prems

       (Conjunction.intr_balanced (map mk_eqn (rec_rewrites' ~~ asmss)));

     val goals = map (fn ((cargs, _, _), (_, eqn)) =>

       (list_all_free (cargs, eqn), [])) (rec_rewrites' ~~ eqns);

   in

     thy' |>

     ProofContext.init |>

     Proof.theorem_i NONE

       (fn thss => ProofContext.theory (fn thy =>

          let

            val simps = map standard (flat thss);

            val (simps', thy') =

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

            val simps'' = maps snd simps'

          in

            thy'

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

            |> snd

            |> Sign.parent_path

          end))

       [goals] |>

     Proof.apply (Method.Basic (fn _ => Method.RAW_METHOD (fn _ =>

       rewrite_goals_tac (map snd defs_thms') THEN

       compose_tac (false, rule, length rule_prems) 1), Position.none)) |>

     Seq.hd

   end;

 fun gen_primrec note def alt_name invs fctxt 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 (Logic.strip_imp_concl eq))))

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

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

   in

     gen_primrec_i note def alt_name

       (Option.map (map (Syntax.read_term_global thy)) invs)

       (Option.map (Syntax.read_term_global thy) fctxt)

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

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

       PureThy.add_defs_unchecked 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);

 end; (*local*)

 (* outer syntax *)

 local structure P = OuterParse and K = OuterKeyword in

 val freshness_context = P.reserved "freshness_context";

 val invariant = P.reserved "invariant";

 fun unless_flag scan = Scan.unless ((freshness_context || invariant) -- P.$$$ ":") scan;

 val parser1 = (freshness_context -- P.$$$ ":") |-- unless_flag P.term >> SOME;

 val parser2 = (invariant -- P.$$$ ":") |--

     (Scan.repeat1 (unless_flag P.term) >> SOME) -- Scan.optional parser1 NONE ||

   (parser1 >> pair NONE);

 val parser3 =

   unless_flag P.name -- Scan.optional parser2 (NONE, NONE) ||

   (parser2 >> pair "");

 val parser4 =

   (P.$$$ "unchecked" >> K true) -- Scan.optional parser3 ("", (NONE, NONE)) ||

   (parser3 >> pair false);

 val options =

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

     (parser4 --| P.$$$ ")")) (false, ("", (NONE, NONE)));

 val primrec_decl =

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

 val _ =

   OuterSyntax.command "nominal_primrec" "define primitive recursive functions on nominal datatypes" K.thy_goal

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

       Toplevel.print o Toplevel.theory_to_proof

         ((if unchecked then add_primrec_unchecked else add_primrec) alt_name invs fctxt

           (map P.triple_swap eqns))));

 end;

 end;