(*  Title:      HOL/BNF/Tools/bnf_fp_rec_sugar.ML

     Author:     Lorenz Panny, TU Muenchen

     Copyright   2013

 Recursor and corecursor sugar.

 *)

 signature BNF_FP_REC_SUGAR =

 sig

   val add_primrec_cmd: (binding * string option * mixfix) list ->

     (Attrib.binding * string) list -> local_theory -> local_theory;

   val add_primcorec_cmd: bool ->

     (binding * string option * mixfix) list * (Attrib.binding * string) list -> Proof.context ->

     Proof.state

 end;

 structure BNF_FP_Rec_Sugar : BNF_FP_REC_SUGAR =

 struct

 open BNF_Util

 open BNF_FP_Util

 open BNF_FP_Rec_Sugar_Util

 open BNF_FP_Rec_Sugar_Tactics

 exception Primrec_Error of string * term list;

 fun primrec_error str = raise Primrec_Error (str, []);

 fun primrec_error_eqn str eqn = raise Primrec_Error (str, [eqn]);

 fun primrec_error_eqns str eqns = raise Primrec_Error (str, eqns);

 fun finds eq = fold_map (fn x => List.partition (curry eq x) #>> pair x);

 val simp_attrs = @{attributes [simp]};

 (* Primrec *)

 type eqn_data = {

   fun_name: string,

   rec_type: typ,

   ctr: term,

   ctr_args: term list,

   left_args: term list,

   right_args: term list,

   res_type: typ,

   rhs_term: term,

   user_eqn: term

 };

 fun permute_args n t = list_comb (t, map Bound (0 :: (n downto 1)))

   |> fold (K (fn u => Abs (Name.uu, dummyT, u))) (0 upto n);

 fun dissect_eqn lthy fun_names eqn' =

   let

     val eqn = subst_bounds (strip_qnt_vars @{const_name all} eqn' |> map Free |> rev,

         strip_qnt_body @{const_name all} eqn') |> HOLogic.dest_Trueprop

         handle TERM _ =>

           primrec_error_eqn "malformed function equation (expected \"lhs = rhs\")" eqn';

     val (lhs, rhs) = HOLogic.dest_eq eqn

         handle TERM _ =>

           primrec_error_eqn "malformed function equation (expected \"lhs = rhs\")" eqn';

     val (fun_name, args) = strip_comb lhs

       |>> (fn x => if is_Free x then fst (dest_Free x)

           else primrec_error_eqn "malformed function equation (does not start with free)" eqn);

     val (left_args, rest) = take_prefix is_Free args;

     val (nonfrees, right_args) = take_suffix is_Free rest;

     val _ = length nonfrees = 1 orelse if length nonfrees = 0 then

       primrec_error_eqn "constructor pattern missing in left-hand side" eqn else

       primrec_error_eqn "more than one non-variable argument in left-hand side" eqn;

     val _ = member (op =) fun_names fun_name orelse

       primrec_error_eqn "malformed function equation (does not start with function name)" eqn

     val (ctr, ctr_args) = strip_comb (the_single nonfrees);

     val _ = try (num_binder_types o fastype_of) ctr = SOME (length ctr_args) orelse

       primrec_error_eqn "partially applied constructor in pattern" eqn;

     val _ = let val d = duplicates (op =) (left_args @ ctr_args @ right_args) in null d orelse

       primrec_error_eqn ("duplicate variable \"" ^ Syntax.string_of_term lthy (hd d) ^

         "\" in left-hand side") eqn end;

     val _ = forall is_Free ctr_args orelse

       primrec_error_eqn "non-primitive pattern in left-hand side" eqn;

     val _ =

       let val b = fold_aterms (fn x as Free (v, _) =>

         if (not (member (op =) (left_args @ ctr_args @ right_args) x) andalso

         not (member (op =) fun_names v) andalso

         not (Variable.is_fixed lthy v)) then cons x else I | _ => I) rhs []

       in

         null b orelse

         primrec_error_eqn ("extra variable(s) in right-hand side: " ^

           commas (map (Syntax.string_of_term lthy) b)) eqn

       end;

   in

     {fun_name = fun_name,

      rec_type = body_type (type_of ctr),

      ctr = ctr,

      ctr_args = ctr_args,

      left_args = left_args,

      right_args = right_args,

      res_type = map fastype_of (left_args @ right_args) ---> fastype_of rhs,

      rhs_term = rhs,

      user_eqn = eqn'}

   end;

 (* substitutes (f ls x rs) by (y ls rs) for all f: get_idx f \<ge> 0, (x,y) \<in> substs *)

 fun subst_direct_calls get_idx get_ctr_pos substs = 

   let

     fun subst (Abs (v, T, b)) = Abs (v, T, subst b)

       | subst t =

         let

           val (f, args) = strip_comb t;

           val idx = get_idx f;

           val ctr_pos  = if idx >= 0 then get_ctr_pos idx else ~1;

         in

           if idx < 0 then

             list_comb (f, map subst args)

           else if ctr_pos >= length args then

             primrec_error_eqn "too few arguments in recursive call" t

           else

             let

               val (key, repl) = the (find_first (equal (nth args ctr_pos) o fst) substs)

                 handle Option.Option => primrec_error_eqn

                   "recursive call not directly applied to constructor argument" t;

             in

               remove (op =) key args |> map subst |> curry list_comb repl

             end

         end

   in subst end;

 (* FIXME get rid of funs_data or get_indices *)

 fun rewrite_map_arg funs_data get_indices y rec_type res_type =

   let

     val pT = HOLogic.mk_prodT (rec_type, res_type);

     val fstx = fst_const pT;

     val sndx = snd_const pT;

     val SOME ({fun_name, left_args, ...} :: _) =

       find_first (equal rec_type o #rec_type o hd) funs_data;

     val ctr_pos = length left_args;

     fun subst _ d (t as Bound d') = t |> d = d' ? curry (op $) fstx

       | subst l d (Abs (v, T, b)) = Abs (v, if d < 0 then pT else T, subst l (d + 1) b)

       | subst l d t =

         let val (u, vs) = strip_comb t in

           if try (fst o dest_Free) u = SOME fun_name then

             if l andalso length vs = ctr_pos then

               list_comb (sndx |> permute_args ctr_pos, vs)

             else if length vs <= ctr_pos then

               primrec_error_eqn "too few arguments in recursive call" t

             else if nth vs ctr_pos |> member (op =) [y, Bound d] then

               list_comb (sndx $ nth vs ctr_pos, nth_drop ctr_pos vs |> map (subst false d))

             else

               primrec_error_eqn "recursive call not directly applied to constructor argument" t

           else if try (fst o dest_Const) u = SOME @{const_name comp} then

             (hd vs |> get_indices |> null orelse

               primrec_error_eqn "recursive call not directly applied to constructor argument" t;

             list_comb

               (u |> map_types (strip_type #>> (fn Ts => Ts

                    |> nth_map (length Ts - 1) (K pT)

                    |> nth_map (length Ts - 2) (strip_type #>> nth_map 0 (K pT) #> (op --->)))

                  #> (op --->)),

               nth_map 1 (subst l d) vs))

           else

             list_comb (u, map (subst false d) vs)

         end

   in

     subst true ~1

   end;

 (* FIXME get rid of funs_data or get_indices *)

 fun subst_indirect_call lthy funs_data get_indices (y, y') =

   let

     fun massage massage_map_arg bound_Ts =

       massage_indirect_rec_call lthy (not o null o get_indices) massage_map_arg bound_Ts y y';

     fun subst bound_Ts (t as _ $ _) =

         let

           val ctr_args = fold_aterms (curry (op @) o get_indices) t []

             |> maps (maps #ctr_args o nth funs_data);

           val (f', args') = strip_comb t;

           val fun_arg_idx = find_index (exists_subterm (not o null o get_indices)) args';

           val arg_idx = find_index (exists_subterm (equal y)) args';

           val (f, args) = chop (arg_idx + 1) args' |>> curry list_comb f';

           val _ = fun_arg_idx < 0 orelse arg_idx >= 0 orelse

             exists (exists_subterm (member (op =) ctr_args)) args' orelse

             primrec_error_eqn "recursive call not applied to constructor argument" t;

         in

           if fun_arg_idx <> arg_idx andalso fun_arg_idx >= 0 andalso arg_idx >= 0 then

             if nth args' arg_idx = y then

               list_comb (massage (rewrite_map_arg funs_data get_indices y) bound_Ts f, args)

             else

               primrec_error_eqn "recursive call not directly applied to constructor argument" f

           else

             list_comb (f', map (subst bound_Ts) args')

         end

       | subst bound_Ts (Abs (v, T, b)) = Abs (v, T, subst (T :: bound_Ts) b)

       | subst bound_Ts t = t |> t = y ? massage (K I |> K) bound_Ts;

   in subst [] end;

 fun build_rec_arg lthy get_indices funs_data ctr_spec maybe_eqn_data =

   if is_some maybe_eqn_data then

     let

       val eqn_data = the maybe_eqn_data;

       val t = #rhs_term eqn_data;

       val ctr_args = #ctr_args eqn_data;

       val calls = #calls ctr_spec;

       val n_args = fold (curry (op +) o (fn Direct_Rec _ => 2 | _ => 1)) calls 0;

       val no_calls' = tag_list 0 calls

         |> map_filter (try (apsnd (fn No_Rec n => n | Direct_Rec (n, _) => n)));

       val direct_calls' = tag_list 0 calls

         |> map_filter (try (apsnd (fn Direct_Rec (_, n) => n)));

       val indirect_calls' = tag_list 0 calls

         |> map_filter (try (apsnd (fn Indirect_Rec n => n)));

       fun make_direct_type T = dummyT; (* FIXME? *)

       val rec_res_type_list = map (fn (x :: _) => (#rec_type x, #res_type x)) funs_data;

       fun make_indirect_type (Type (Tname, Ts)) = Type (Tname, Ts |> map (fn T =>

         let val maybe_res_type = AList.lookup (op =) rec_res_type_list T in

           if is_some maybe_res_type

           then HOLogic.mk_prodT (T, the maybe_res_type)

           else make_indirect_type T end))

         | make_indirect_type T = T;

       val args = replicate n_args ("", dummyT)

         |> Term.rename_wrt_term t

         |> map Free

         |> fold (fn (ctr_arg_idx, arg_idx) =>

             nth_map arg_idx (K (nth ctr_args ctr_arg_idx)))

           no_calls'

         |> fold (fn (ctr_arg_idx, arg_idx) =>

             nth_map arg_idx (K (nth ctr_args ctr_arg_idx |> map_types make_direct_type)))

           direct_calls'

         |> fold (fn (ctr_arg_idx, arg_idx) =>

             nth_map arg_idx (K (nth ctr_args ctr_arg_idx |> map_types make_indirect_type)))

           indirect_calls';

       val direct_calls = map (apfst (nth ctr_args) o apsnd (nth args)) direct_calls';

       val indirect_calls = map (apfst (nth ctr_args) o apsnd (nth args)) indirect_calls';

       val get_idx = (fn Free (v, _) => find_index (equal v o #fun_name o hd) funs_data | _ => ~1);

       val t' = t

         |> fold (subst_indirect_call lthy funs_data get_indices) indirect_calls

         |> subst_direct_calls get_idx (length o #left_args o hd o nth funs_data) direct_calls;

       val abstractions = map dest_Free (args @ #left_args eqn_data @ #right_args eqn_data);

     in

       t' |> fold_rev absfree abstractions

     end

   else Const (@{const_name undefined}, dummyT)

 fun build_defs lthy bs funs_data rec_specs get_indices =

   let

     val n_funs = length funs_data;

     val ctr_spec_eqn_data_list' =

       (take n_funs rec_specs |> map #ctr_specs) ~~ funs_data

       |> maps (uncurry (finds (fn (x, y) => #ctr x = #ctr y))

           ##> (fn x => null x orelse

             primrec_error_eqns "excess equations in definition" (map #rhs_term x)) #> fst);

     val _ = ctr_spec_eqn_data_list' |> map (fn (_, x) => length x <= 1 orelse

       primrec_error_eqns ("multiple equations for constructor") (map #user_eqn x));

     val ctr_spec_eqn_data_list =

       ctr_spec_eqn_data_list' @ (drop n_funs rec_specs |> maps #ctr_specs |> map (rpair []));

     val recs = take n_funs rec_specs |> map #recx;

     val rec_args = ctr_spec_eqn_data_list

       |> sort ((op <) o pairself (#offset o fst) |> make_ord)

       |> map (uncurry (build_rec_arg lthy get_indices funs_data) o apsnd (try the_single));

     val ctr_poss = map (fn x =>

       if length (distinct ((op =) o pairself (length o #left_args)) x) <> 1 then

         primrec_error ("inconstant constructor pattern position for function " ^

           quote (#fun_name (hd x)))

       else

         hd x |> #left_args |> length) funs_data;

   in

     (recs, ctr_poss)

     |-> map2 (fn recx => fn ctr_pos => list_comb (recx, rec_args) |> permute_args ctr_pos)

     |> Syntax.check_terms lthy

     |> map2 (fn b => fn t => ((b, NoSyn), ((Binding.map_name Thm.def_name b, []), t))) bs

   end;

 fun find_rec_calls get_indices eqn_data =

   let

     fun find (Abs (_, _, b)) ctr_arg = find b ctr_arg

       | find (t as _ $ _) ctr_arg =

         let

           val (f', args') = strip_comb t;

           val n = find_index (equal ctr_arg) args';

         in

           if n < 0 then

             find f' ctr_arg @ maps (fn x => find x ctr_arg) args'

           else

             let val (f, args) = chop n args' |>> curry list_comb f' in

               if exists_subterm (not o null o get_indices) f then

                 f :: maps (fn x => find x ctr_arg) args

               else

                 find f ctr_arg @ maps (fn x => find x ctr_arg) args

             end

         end

       | find _ _ = [];

   in

     map (find (#rhs_term eqn_data)) (#ctr_args eqn_data)

     |> (fn [] => NONE | callss => SOME (#ctr eqn_data, callss))

   end;

 fun add_primrec fixes specs lthy =

   let

     val bs = map (fst o fst) fixes;

     val fun_names = map Binding.name_of bs;

     val eqns_data = map (snd #> dissect_eqn lthy fun_names) specs;

     val funs_data = eqns_data

       |> partition_eq ((op =) o pairself #fun_name)

       |> finds (fn (x, y) => x = #fun_name (hd y)) fun_names |> fst

       |> map (fn (x, y) => the_single y handle List.Empty =>

           primrec_error ("missing equations for function " ^ quote x));

     fun get_indices t = map (fst #>> Binding.name_of #> Free) fixes

       |> map_index (fn (i, v) => if exists_subterm (equal v) t then SOME i else NONE)

       |> map_filter I;

     val arg_Ts = map (#rec_type o hd) funs_data;

     val res_Ts = map (#res_type o hd) funs_data;

     val callssss = funs_data

       |> map (partition_eq ((op =) o pairself #ctr))

       |> map (maps (map_filter (find_rec_calls get_indices)));

     val ((nontriv, rec_specs, _, induct_thm, induct_thms), lthy') =

       rec_specs_of bs arg_Ts res_Ts get_indices callssss lthy;

     val actual_nn = length funs_data;

     val _ = let val ctrs = (maps (map #ctr o #ctr_specs) rec_specs) in

       map (fn {ctr, user_eqn, ...} => member (op =) ctrs ctr orelse

         primrec_error_eqn ("argument " ^ quote (Syntax.string_of_term lthy' ctr) ^

           " is not a constructor in left-hand side") user_eqn) eqns_data end;

     val defs = build_defs lthy' bs funs_data rec_specs get_indices;

     fun prove def_thms' {ctr_specs, nested_map_idents, nested_map_comps, ...} induct_thm fun_data

         lthy =

       let

         val fun_name = #fun_name (hd fun_data);

         val def_thms = map (snd o snd) def_thms';

         val simp_thms = finds (fn (x, y) => #ctr x = #ctr y) fun_data ctr_specs

           |> fst

           |> map_filter (try (fn (x, [y]) =>

             (#user_eqn x, length (#left_args x) + length (#right_args x), #rec_thm y)))

           |> map (fn (user_eqn, num_extra_args, rec_thm) =>

             mk_primrec_tac lthy num_extra_args nested_map_idents nested_map_comps def_thms rec_thm

             |> K |> Goal.prove lthy [] [] user_eqn)

         val notes =

           [(inductN, if actual_nn > 1 then [induct_thm] else [], []),

            (simpsN, simp_thms, simp_attrs)]

           |> filter_out (null o #2)

           |> map (fn (thmN, thms, attrs) =>

             ((Binding.qualify true fun_name (Binding.name thmN), attrs), [(thms, [])]));

       in

         lthy |> Local_Theory.notes notes

       end;

     val common_name = mk_common_name fun_names;

     val common_notes =

       [(inductN, if nontriv andalso actual_nn > 1 then [induct_thm] else [], [])]

       |> filter_out (null o #2)

       |> map (fn (thmN, thms, attrs) =>

         ((Binding.qualify true common_name (Binding.name thmN), attrs), [(thms, [])]));

   in

     lthy'

     |> fold_map Local_Theory.define defs

     |-> (fn def_thms' => fold_map3 (prove def_thms') (take actual_nn rec_specs)

       (take actual_nn induct_thms) funs_data)

     |> snd

     |> Local_Theory.notes common_notes |> snd

   end;

 fun add_primrec_cmd raw_fixes raw_specs lthy =

   let

     val _ = let val d = duplicates (op =) (map (Binding.name_of o #1) raw_fixes) in null d orelse

       primrec_error ("duplicate function name(s): " ^ commas d) end;

     val (fixes, specs) = fst (Specification.read_spec raw_fixes raw_specs lthy);

   in

     add_primrec fixes specs lthy

       handle ERROR str => primrec_error str

   end

   handle Primrec_Error (str, eqns) =>

     if null eqns

     then error ("primrec_new error:\n  " ^ str)

     else error ("primrec_new error:\n  " ^ str ^ "\nin\n  " ^

       space_implode "\n  " (map (quote o Syntax.string_of_term lthy) eqns))

 (* Primcorec *)

 type co_eqn_data_dtr_disc = {

   fun_name: string,

   ctr_no: int,

   cond: term,

   user_eqn: term

 };

 type co_eqn_data_dtr_sel = {

   fun_name: string,

   ctr_no: int,

   sel_no: int,

   fun_args: term list,

   rhs_term: term,

   user_eqn: term

 };

 datatype co_eqn_data =

   Dtr_Disc of co_eqn_data_dtr_disc |

   Dtr_Sel of co_eqn_data_dtr_sel

 fun co_dissect_eqn_disc sequential fun_name_corec_spec_list eqn' imp_lhs' imp_rhs matched_conds_ps =

   let

     fun find_subterm p = let (* FIXME \<exists>? *)

       fun f (t as u $ v) =

         fold_rev (curry merge_options) [if p t then SOME t else NONE, f u, f v] NONE

         | f t = if p t then SOME t else NONE

       in f end;

     val fun_name = imp_rhs

       |> perhaps (try HOLogic.dest_not)

       |> `(try (fst o dest_Free o head_of o snd o dest_comb))

       ||> (try (fst o dest_Free o head_of o fst o HOLogic.dest_eq))

       |> the o merge_options;

     val corec_spec = the (AList.lookup (op =) fun_name_corec_spec_list fun_name)

       handle Option.Option => primrec_error_eqn "malformed discriminator equation" imp_rhs;

     val discs = #ctr_specs corec_spec |> map #disc;

     val ctrs = #ctr_specs corec_spec |> map #ctr;

     val n_ctrs = length ctrs;

     val not_disc = head_of imp_rhs = @{term Not};

     val _ = not_disc andalso n_ctrs <> 2 andalso

       primrec_error_eqn "\<not>ed discriminator for a type with \<noteq> 2 constructors" imp_rhs;

     val disc = find_subterm (member (op =) discs o head_of) imp_rhs;

     val eq_ctr0 = imp_rhs |> perhaps (try (HOLogic.dest_not)) |> try (HOLogic.dest_eq #> snd)

         |> (fn SOME t => let val n = find_index (equal t) ctrs in

           if n >= 0 then SOME n else NONE end | _ => NONE);

     val _ = is_some disc orelse is_some eq_ctr0 orelse

       primrec_error_eqn "no discriminator in equation" imp_rhs;

     val ctr_no' =

       if is_none disc then the eq_ctr0 else find_index (equal (head_of (the disc))) discs;

     val ctr_no = if not_disc then 1 - ctr_no' else ctr_no';

     val fun_args = if is_none disc

       then imp_rhs |> perhaps (try HOLogic.dest_not) |> HOLogic.dest_eq |> fst |> strip_comb |> snd

       else the disc |> the_single o snd o strip_comb

         |> (fn t => if try (fst o dest_Free o head_of) t = SOME fun_name

           then snd (strip_comb t) else []);

     val mk_conjs = try (foldr1 HOLogic.mk_conj) #> the_default @{const True};

     val mk_disjs = try (foldr1 HOLogic.mk_disj) #> the_default @{const False};

     val catch_all = try (fst o dest_Free o the_single) imp_lhs' = SOME Name.uu_;

     val matched_conds = filter (equal fun_name o fst) matched_conds_ps |> map snd;

     val imp_lhs = mk_conjs imp_lhs';

     val cond =

       if catch_all then

         if null matched_conds then fold_rev absfree (map dest_Free fun_args) @{const True} else

           (strip_abs_vars (hd matched_conds),

             mk_disjs (map strip_abs_body matched_conds) |> HOLogic.mk_not)

           |-> fold_rev (fn (v, T) => fn u => Abs (v, T, u))

       else if sequential then

         HOLogic.mk_conj (HOLogic.mk_not (mk_disjs (map strip_abs_body matched_conds)), imp_lhs)

         |> fold_rev absfree (map dest_Free fun_args)

       else

         imp_lhs |> fold_rev absfree (map dest_Free fun_args);

     val matched_cond =

       if sequential then fold_rev absfree (map dest_Free fun_args) imp_lhs else cond;

     val matched_conds_ps' = if catch_all

       then (fun_name, cond) :: filter (not_equal fun_name o fst) matched_conds_ps

       else (fun_name, matched_cond) :: matched_conds_ps;

   in

     (Dtr_Disc {

       fun_name = fun_name,

       ctr_no = ctr_no,

       cond = cond,

       user_eqn = eqn'

     }, matched_conds_ps')

   end;

 fun co_dissect_eqn_sel fun_name_corec_spec_list eqn' eqn =

   let

     val (lhs, rhs) = HOLogic.dest_eq eqn

       handle TERM _ =>

         primrec_error_eqn "malformed function equation (expected \"lhs = rhs\")" eqn;

     val sel = head_of lhs;

     val (fun_name, fun_args) = dest_comb lhs |> snd |> strip_comb |> apfst (fst o dest_Free)

       handle TERM _ =>

         primrec_error_eqn "malformed selector argument in left-hand side" eqn;

     val corec_spec = the (AList.lookup (op =) fun_name_corec_spec_list fun_name)

       handle Option.Option => primrec_error_eqn "malformed selector argument in left-hand side" eqn;

     val ((ctr_spec, ctr_no), sel) = #ctr_specs corec_spec

       |> the o get_index (try (the o find_first (equal sel) o #sels))

       |>> `(nth (#ctr_specs corec_spec));

     val sel_no = find_index (equal sel) (#sels ctr_spec);

   in

     Dtr_Sel {

       fun_name = fun_name,

       ctr_no = ctr_no,

       sel_no = sel_no,

       fun_args = fun_args,

       rhs_term = rhs,

       user_eqn = eqn'

     }

   end;

 fun co_dissect_eqn_ctr sequential fun_name_corec_spec_list eqn' imp_lhs' imp_rhs matched_conds_ps =

   let 

     val (lhs, rhs) = HOLogic.dest_eq imp_rhs;

     val fun_name = head_of lhs |> fst o dest_Free;

     val corec_spec = the (AList.lookup (op =) fun_name_corec_spec_list fun_name);

     val (ctr, ctr_args) = strip_comb rhs;

     val ctr_spec = the (find_first (equal ctr o #ctr) (#ctr_specs corec_spec))

       handle Option.Option => primrec_error_eqn "not a constructor" ctr;

     val disc_imp_rhs = betapply (#disc ctr_spec, lhs);

     val (eqn_data_disc, matched_conds_ps') = co_dissect_eqn_disc

         sequential fun_name_corec_spec_list eqn' imp_lhs' disc_imp_rhs matched_conds_ps;

     val sel_imp_rhss = (#sels ctr_spec ~~ ctr_args)

       |> map (fn (sel, ctr_arg) => HOLogic.mk_eq (betapply (sel, lhs), ctr_arg));

 val _ = warning ("reduced\n    " ^ Syntax.string_of_term @{context} imp_rhs ^ "\nto\n    \<cdot> " ^

  Syntax.string_of_term @{context} disc_imp_rhs ^ "\n    \<cdot> " ^

  space_implode "\n    \<cdot> " (map (Syntax.string_of_term @{context}) sel_imp_rhss));

     val eqns_data_sel =

       map (co_dissect_eqn_sel fun_name_corec_spec_list @{const True}(*###*)) sel_imp_rhss;

   in

     (eqn_data_disc :: eqns_data_sel, matched_conds_ps')

   end;

 fun co_dissect_eqn sequential fun_name_corec_spec_list eqn' matched_conds_ps =

   let

     val eqn = subst_bounds (strip_qnt_vars @{const_name all} eqn' |> map Free |> rev,

         strip_qnt_body @{const_name all} eqn')

         handle TERM _ => primrec_error_eqn "malformed function equation" eqn';

     val (imp_lhs', imp_rhs) =

       (map HOLogic.dest_Trueprop (Logic.strip_imp_prems eqn),

        HOLogic.dest_Trueprop (Logic.strip_imp_concl eqn));

     val head = imp_rhs

       |> perhaps (try HOLogic.dest_not) |> perhaps (try (fst o HOLogic.dest_eq))

       |> head_of;

     val maybe_rhs = imp_rhs |> perhaps (try (HOLogic.dest_not)) |> try (snd o HOLogic.dest_eq);

     val fun_names = map fst fun_name_corec_spec_list;

     val discs = maps (#ctr_specs o snd) fun_name_corec_spec_list |> map #disc;

     val sels = maps (#ctr_specs o snd) fun_name_corec_spec_list |> maps #sels;

     val ctrs = maps (#ctr_specs o snd) fun_name_corec_spec_list |> map #ctr;

   in

     if member (op =) discs head orelse

       is_some maybe_rhs andalso

         member (op =) (filter (null o binder_types o fastype_of) ctrs) (the maybe_rhs) then

       co_dissect_eqn_disc sequential fun_name_corec_spec_list eqn' imp_lhs' imp_rhs matched_conds_ps

       |>> single

     else if member (op =) sels head then

       ([co_dissect_eqn_sel fun_name_corec_spec_list eqn' imp_rhs], matched_conds_ps)

     else if is_Free head andalso member (op =) fun_names (fst (dest_Free head)) then

       co_dissect_eqn_ctr sequential fun_name_corec_spec_list eqn' imp_lhs' imp_rhs matched_conds_ps

     else

       primrec_error_eqn "malformed function equation" eqn

   end;

 fun build_corec_args_discs ctr_specs disc_eqns =

   let

     val conds = map #cond disc_eqns;

     val args =

       if length ctr_specs = 1 then []

       else if length disc_eqns = length ctr_specs then

         fst (split_last conds)

       else if length disc_eqns = length ctr_specs - 1 then

         let val n = 0 upto length ctr_specs - 1

             |> the o find_first (fn idx => not (exists (equal idx o #ctr_no) disc_eqns)) (*###*) in

           if n = length ctr_specs - 1 then

             conds

           else

             split_last conds

             ||> (fn t => fold_rev absfree (strip_abs_vars t) (strip_abs_body t |> HOLogic.mk_not))

             |>> chop n

             |> (fn ((l, r), x) => l @ (x :: r))

         end

       else

         0 upto length ctr_specs - 1

         |> map (fn idx => find_first (equal idx o #ctr_no) disc_eqns

           |> Option.map #cond

           |> the_default (Const (@{const_name undefined}, dummyT)))

         |> fst o split_last;

     fun finish t =

       let val n = length (fastype_of t |> binder_types) in

         if t = Const (@{const_name undefined}, dummyT) then t

         else if n = 0 then Abs (Name.uu_, @{typ unit}, t)

         else if n = 1 then t

         else Const (@{const_name prod_case}, dummyT) $ t

       end;

   in

     map finish args

   end;

 fun build_corec_args_sel sel_eqns ctr_spec =

   let

     (* FIXME *)

     val n_args = fold (curry (op +)) (map (fn Direct_Corec _ => 3 | _ => 1) (#calls ctr_spec)) 0;

   in

     replicate n_args (Const (@{const_name undefined}, dummyT))

   end;

 fun co_build_defs lthy sequential bs arg_Tss fun_name_corec_spec_list eqns_data =

   let

     val fun_names = map Binding.name_of bs;

 (*    fun group _ [] = [] (* FIXME \<exists>? *)

       | group eq (x :: xs) =

         let val (xs', ys) = List.partition (eq x) xs in (x :: xs') :: group eq ys end;*)

     val disc_eqnss = map_filter (try (fn Dtr_Disc x => x)) eqns_data

       |> partition_eq ((op =) o pairself #fun_name)

       |> finds (fn (x, ({fun_name, ...} :: _)) => x = fun_name) fun_names |> fst

       |> map (sort ((op <) o pairself #ctr_no |> make_ord) o flat o snd);

     val _ = disc_eqnss |> map (fn x =>

       let val d = duplicates ((op =) o pairself #ctr_no) x in null d orelse

         primrec_error_eqns "excess discriminator equations in definition"

           (maps (fn t => filter (equal (#ctr_no t) o #ctr_no) x) d |> map #user_eqn) end);

 val _ = warning ("disc_eqnss:\n    \<cdot> " ^ space_implode "\n    \<cdot> " (map @{make_string} disc_eqnss));

     val sel_eqnss = map_filter (try (fn Dtr_Sel x => x)) eqns_data

       |> partition_eq ((op =) o pairself #fun_name)

       |> finds (fn (x, ({fun_name, ...} :: _)) => x = fun_name) fun_names |> fst

       |> map (sort ((op <) o pairself #ctr_no |> make_ord) o flat o snd);

 val _ = warning ("sel_eqnss:\n    \<cdot> " ^ space_implode "\n    \<cdot> " (map @{make_string} sel_eqnss));

     fun splice (xs :: xss) (ys :: yss) = xs @ ys @ splice xss yss (* FIXME \<exists>? *)

       | splice xss yss = flat xss @ flat yss;

     val corecs = map (#corec o snd) fun_name_corec_spec_list;

     val corec_args = (map snd fun_name_corec_spec_list ~~ disc_eqnss ~~ sel_eqnss)

       |> maps (fn (({ctr_specs, ...}, disc_eqns), sel_eqns) =>

         splice (build_corec_args_discs ctr_specs disc_eqns |> map single)

           (map (build_corec_args_sel sel_eqns) ctr_specs));

 val _ = warning ("corecursor arguments:\n    \<cdot> " ^

  space_implode "\n    \<cdot> " (map (Syntax.string_of_term @{context}) corec_args));

     fun uneq_pairs_rev xs = xs (* FIXME \<exists>? *)

       |> these o try (split_last #> (fn (ys, y) => uneq_pairs_rev ys @ map (pair y) ys));

     val proof_obligations = if sequential then [] else

       maps (uneq_pairs_rev o map #cond) disc_eqnss

       |> map (fn (x, y) => ((strip_abs_body x, strip_abs_body y), strip_abs_vars x))

       |> map (apfst (apsnd HOLogic.mk_not #> pairself HOLogic.mk_Trueprop

         #> apfst (curry (op $) @{const ==>}) #> (op $)))

       |> map (fn (t, abs_vars) => fold_rev (fn (v, T) => fn u =>

           Const (@{const_name all}, (T --> @{typ prop}) --> @{typ prop}) $

             Abs (v, T, u)) abs_vars t);

     fun currys Ts t = if length Ts <= 1 then t else

       t $ foldr1 (fn (u, v) => HOLogic.pair_const dummyT dummyT $ u $ v)

         (length Ts - 1 downto 0 |> map Bound)

       |> fold_rev (fn T => fn u => Abs (Name.uu, T, u)) Ts;

   in

     map (list_comb o rpair corec_args) corecs

     |> map2 (fn Ts => fn t => if length Ts = 0 then t $ HOLogic.unit else t) arg_Tss

     |> map2 currys arg_Tss

     |> Syntax.check_terms lthy

     |> map2 (fn b => fn t => ((b, NoSyn), ((Binding.map_name Thm.def_name b, []), t))) bs

     |> rpair proof_obligations

   end;

 fun add_primcorec sequential fixes specs lthy =

   let

     val bs = map (fst o fst) fixes;

     val (arg_Ts, res_Ts) = map (strip_type o snd o fst #>> HOLogic.mk_tupleT) fixes |> split_list;

     (* copied from primrec_new *)

     fun get_indices t = map (fst #>> Binding.name_of #> Free) fixes

       |> map_index (fn (i, v) => if exists_subterm (equal v) t then SOME i else NONE)

       |> map_filter I;

     val callssss = []; (* FIXME *)

     val ((nontriv, corec_specs, _, coinduct_thm, strong_co_induct_thm, coinduct_thmss,

           strong_coinduct_thmss), lthy') =

       corec_specs_of bs arg_Ts res_Ts get_indices callssss lthy;

     val fun_names = map Binding.name_of bs;

     val fun_name_corec_spec_list = (fun_names ~~ res_Ts, corec_specs)

       |> uncurry (finds (fn ((v, T), {corec, ...}) => T = body_type (fastype_of corec))) |> fst

       |> map (apfst fst #> apsnd the_single); (*###*)

     val (eqns_data, _) =

       fold_map (co_dissect_eqn sequential fun_name_corec_spec_list) (map snd specs) []

       |>> flat;

     val (defs, proof_obligations) =

       co_build_defs lthy' sequential bs (map (binder_types o snd o fst) fixes)

         fun_name_corec_spec_list eqns_data;

   in

     lthy'

     |> fold_map Local_Theory.define defs |> snd

     |> Proof.theorem NONE (K I) [map (rpair []) proof_obligations]

     |> Proof.refine (Method.primitive_text I)

     |> Seq.hd

   end

 fun add_primcorec_cmd seq (raw_fixes, raw_specs) lthy =

   let

     val (fixes, specs) = fst (Specification.read_spec raw_fixes raw_specs lthy);

   in

     add_primcorec seq fixes specs lthy

     handle ERROR str => primrec_error str

   end

   handle Primrec_Error (str, eqns) =>

     if null eqns

     then error ("primcorec error:\n  " ^ str)

     else error ("primcorec error:\n  " ^ str ^ "\nin\n  " ^

       space_implode "\n  " (map (quote o Syntax.string_of_term lthy) eqns))

 end;