(*  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: (binding * typ option * mixfix) list ->

     (Attrib.binding * term) list -> local_theory -> (term list * thm list list) * local_theory

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

     (Attrib.binding * string) list -> local_theory -> (term list * thm list list) * local_theory

   val add_primrec_global: (binding * typ option * mixfix) list ->

     (Attrib.binding * term) list -> theory -> (term list * thm list list) * theory

   val add_primrec_overloaded: (string * (string * typ) * bool) list ->

     (binding * typ option * mixfix) list ->

     (Attrib.binding * term) list -> theory -> (term list * thm list list) * theory

   val add_primrec_simple: ((binding * typ) * mixfix) list -> term list ->

     local_theory -> (string list * (term list * (int list list * thm list list))) * local_theory

   val add_primcorecursive_cmd: bool ->

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

     Proof.context -> Proof.state

   val add_primcorec_cmd: bool ->

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

     local_theory -> local_theory

 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

 val codeN = "code";

 val ctrN = "ctr";

 val discN = "disc";

 val selN = "sel";

 val nitpicksimp_attrs = @{attributes [nitpick_simp]};

 val simp_attrs = @{attributes [simp]};

 val code_nitpicksimp_attrs = Code.add_default_eqn_attrib :: nitpicksimp_attrs;

 val code_nitpicksimp_simp_attrs = Code.add_default_eqn_attrib :: nitpicksimp_attrs @ simp_attrs;

 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 free_name = try (fn Free (v, _) => v);

 val const_name = try (fn Const (v, _) => v);

 val undef_const = Const (@{const_name undefined}, dummyT);

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

   |> fold (K (Term.abs (Name.uu, dummyT))) (0 upto n);

 val abs_tuple = HOLogic.tupled_lambda o HOLogic.mk_tuple;

 fun drop_All t = subst_bounds (strip_qnt_vars @{const_name all} t |> map Free |> rev,

   strip_qnt_body @{const_name all} t)

 fun abstract vs =

   let fun a n (t $ u) = a n t $ a n u

         | a n (Abs (v, T, b)) = Abs (v, T, a (n + 1) b)

         | a n t = let val idx = find_index (equal t) vs in

             if idx < 0 then t else Bound (n + idx) end

   in a 0 end;

 fun mk_prod1 Ts (t, u) = HOLogic.pair_const (fastype_of1 (Ts, t)) (fastype_of1 (Ts, u)) $ t $ u;

 fun mk_tuple1 Ts = the_default HOLogic.unit o try (foldr1 (mk_prod1 Ts));

 fun get_indices fixes 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;

 (* 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 dissect_eqn lthy fun_names eqn' =

   let

     val eqn = drop_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 num_nonfrees = length nonfrees;

     val _ = num_nonfrees = 1 orelse if num_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;

 fun rewrite_map_arg get_ctr_pos rec_type res_type =

   let

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

     val maybe_suc = Option.map (fn x => x + 1);

     fun subst d (t as Bound d') = t |> d = SOME d' ? curry (op $) (fst_const pT)

       | subst d (Abs (v, T, b)) = Abs (v, if d = SOME ~1 then pT else T, subst (maybe_suc d) b)

       | subst d t =

         let

           val (u, vs) = strip_comb t;

           val ctr_pos = try (get_ctr_pos o the) (free_name u) |> the_default ~1;

         in

           if ctr_pos >= 0 then

             if d = SOME ~1 andalso length vs = ctr_pos then

               list_comb (permute_args ctr_pos (snd_const pT), vs)

             else if length vs > ctr_pos andalso is_some d

                 andalso d = try (fn Bound n => n) (nth vs ctr_pos) then

               list_comb (snd_const pT $ nth vs ctr_pos, map (subst d) (nth_drop ctr_pos vs))

             else

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

           else

             list_comb (u, map (subst (d |> d = SOME ~1 ? K NONE)) vs)

         end

   in

     subst (SOME ~1)

   end;

 fun subst_rec_calls lthy get_ctr_pos has_call ctr_args mutual_calls nested_calls =

   let

     fun try_nested_rec bound_Ts y t =

       AList.lookup (op =) nested_calls y

       |> Option.map (fn y' =>

         massage_nested_rec_call lthy has_call (rewrite_map_arg get_ctr_pos) bound_Ts y y' t);

     fun subst bound_Ts (t as g' $ y) =

         let

           fun subst_rec () = subst bound_Ts g' $ subst bound_Ts y;

           val y_head = head_of y;

         in

           if not (member (op =) ctr_args y_head) then

             subst_rec ()

           else

             (case try_nested_rec bound_Ts y_head t of

               SOME t' => t'

             | NONE =>

               let val (g, g_args) = strip_comb g' in

                 (case try (get_ctr_pos o the) (free_name g) of

                   SOME ctr_pos =>

                   (length g_args >= ctr_pos orelse

                    primrec_error_eqn "too few arguments in recursive call" t;

                    (case AList.lookup (op =) mutual_calls y of

                      SOME y' => list_comb (y', g_args)

                    | NONE => subst_rec ()))

                 | NONE => subst_rec ())

               end)

         end

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

       | subst _ t = t

     fun subst' t =

       if has_call t then

         (* FIXME detect this case earlier? *)

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

       else

         try_nested_rec [] (head_of t) t |> the_default t

   in

     subst' o subst []

   end;

 fun build_rec_arg lthy (funs_data : eqn_data list list) has_call (ctr_spec : rec_ctr_spec)

     (maybe_eqn_data : eqn_data option) =

   if is_none maybe_eqn_data then undef_const else

     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 (Integer.add o (fn Mutual_Rec _ => 2 | _ => 1)) calls 0;

       val no_calls' = tag_list 0 calls

         |> map_filter (try (apsnd (fn No_Rec p => p | Mutual_Rec (p, _) => p)));

       val mutual_calls' = tag_list 0 calls

         |> map_filter (try (apsnd (fn Mutual_Rec (_, p) => p)));

       val nested_calls' = tag_list 0 calls

         |> map_filter (try (apsnd (fn Nested_Rec p => p)));

       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, T)) =>

             nth_map arg_idx (K (retype_free T (nth ctr_args ctr_arg_idx))))

           mutual_calls'

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

             nth_map arg_idx (K (retype_free T (nth ctr_args ctr_arg_idx))))

           nested_calls';

       val fun_name_ctr_pos_list =

         map (fn (x :: _) => (#fun_name x, length (#left_args x))) funs_data;

       val get_ctr_pos = try (the o AList.lookup (op =) fun_name_ctr_pos_list) #> the_default ~1;

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

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

       val abstractions = args @ #left_args eqn_data @ #right_args eqn_data;

     in

       t

       |> subst_rec_calls lthy get_ctr_pos has_call ctr_args mutual_calls nested_calls

       |> fold_rev lambda abstractions

     end;

 fun build_defs lthy bs mxs (funs_data : eqn_data list list) (rec_specs : rec_spec list) has_call =

   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 funs_data has_call) 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

     |> map3 (fn b => fn mx => fn t => ((b, mx), ((Binding.conceal (Thm.def_binding b), []), t)))

       bs mxs

   end;

 fun massage_comp ctxt has_call bound_Ts t =

   massage_nested_corec_call ctxt has_call (K (K (K I))) bound_Ts (fastype_of1 (bound_Ts, t)) t;

 fun find_rec_calls ctxt has_call (eqn_data : eqn_data) =

   let

     fun find bound_Ts (Abs (_, T, b)) ctr_arg = find (T :: bound_Ts) b ctr_arg

       | find bound_Ts (t as _ $ _) ctr_arg =

         let

           val typof = curry fastype_of1 bound_Ts;

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

           val n = find_index (equal ctr_arg o head_of) args';

         in

           if n < 0 then

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

           else

             let

               val (f, args as arg :: _) = chop n args' |>> curry list_comb f'

               val (arg_head, arg_args) = Term.strip_comb arg;

             in

               if has_call f then

                 mk_partial_compN (length arg_args) (typof f) (typof arg_head) f ::

                 maps (fn x => find bound_Ts x ctr_arg) args

               else

                 find bound_Ts f ctr_arg @ maps (fn x => find bound_Ts 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 prepare_primrec fixes specs lthy =

   let

     val (bs, mxs) = map_split (apfst fst) fixes;

     val fun_names = map Binding.name_of bs;

     val eqns_data = map (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));

     val has_call = exists_subterm (map (fst #>> Binding.name_of #> Free) fixes |> member (op =));

     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 lthy has_call)));

     val ((n2m, rec_specs, _, induct_thm, induct_thms), lthy') =

       rec_specs_of bs arg_Ts res_Ts (get_indices fixes) 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 mxs funs_data rec_specs has_call;

     fun prove lthy def_thms' ({ctr_specs, nested_map_idents, nested_map_comps, ...} : rec_spec)

         (fun_data : eqn_data list) =

       let

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

         val simp_thmss = 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

             |> Thm.close_derivation);

         val poss = find_indices (fn (x, y) => #ctr x = #ctr y) fun_data eqns_data;

       in

         (poss, simp_thmss)

       end;

     val notes =

       (if n2m then map2 (fn name => fn thm =>

         (name, inductN, [thm], [])) fun_names (take actual_nn induct_thms) else [])

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

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

     val common_name = mk_common_name fun_names;

     val common_notes =

       (if n2m then [(inductN, [induct_thm], [])] else [])

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

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

   in

     (((fun_names, defs),

       fn lthy => fn defs =>

         split_list (map2 (prove lthy defs) (take actual_nn rec_specs) funs_data)),

       lthy' |> Local_Theory.notes (notes @ common_notes) |> snd)

   end;

 (* primrec definition *)

 fun add_primrec_simple fixes ts lthy =

   let

     val (((names, defs), prove), lthy) = prepare_primrec fixes ts lthy

       handle ERROR str => primrec_error str;

   in

     lthy

     |> fold_map Local_Theory.define defs

     |-> (fn defs => `(fn lthy => (names, (map fst defs, prove lthy defs))))

   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));

 local

 fun gen_primrec prep_spec (raw_fixes : (binding * 'a option * mixfix) list) raw_spec lthy =

   let

     val d = duplicates (op =) (map (Binding.name_of o #1) raw_fixes)

     val _ = null d orelse primrec_error ("duplicate function name(s): " ^ commas d);

     val (fixes, specs) = fst (prep_spec raw_fixes raw_spec lthy);

     val mk_notes =

       flat ooo map3 (fn poss => fn prefix => fn thms =>

         let

           val (bs, attrss) = map_split (fst o nth specs) poss;

           val notes =

             map3 (fn b => fn attrs => fn thm =>

               ((Binding.qualify false prefix b, code_nitpicksimp_simp_attrs @ attrs), [([thm], [])]))

             bs attrss thms;

         in

           ((Binding.qualify true prefix (Binding.name simpsN), []), [(thms, [])]) :: notes

         end);

   in

     lthy

     |> add_primrec_simple fixes (map snd specs)

     |-> (fn (names, (ts, (posss, simpss))) =>

       Spec_Rules.add Spec_Rules.Equational (ts, flat simpss)

       #> Local_Theory.notes (mk_notes posss names simpss)

       #>> pair ts o map snd)

   end;

 in

 val add_primrec = gen_primrec Specification.check_spec;

 val add_primrec_cmd = gen_primrec Specification.read_spec;

 end;

 fun add_primrec_global fixes specs thy =

   let

     val lthy = Named_Target.theory_init thy;

     val ((ts, simps), lthy') = add_primrec fixes specs lthy;

     val simps' = burrow (Proof_Context.export lthy' lthy) simps;

   in ((ts, simps'), Local_Theory.exit_global lthy') end;

 fun add_primrec_overloaded ops fixes specs thy =

   let

     val lthy = Overloading.overloading ops thy;

     val ((ts, simps), lthy') = add_primrec fixes specs lthy;

     val simps' = burrow (Proof_Context.export lthy' lthy) simps;

   in ((ts, simps'), Local_Theory.exit_global lthy') end;

 (* Primcorec *)

 type coeqn_data_disc = {

   fun_name: string,

   fun_T: typ,

   fun_args: term list,

   ctr: term,

   ctr_no: int, (*###*)

   disc: term,

   prems: term list,

   auto_gen: bool,

   maybe_ctr_rhs: term option,

   maybe_code_rhs: term option,

   user_eqn: term

 };

 type coeqn_data_sel = {

   fun_name: string,

   fun_T: typ,

   fun_args: term list,

   ctr: term,

   sel: term,

   rhs_term: term,

   user_eqn: term

 };

 datatype coeqn_data =

   Disc of coeqn_data_disc |

   Sel of coeqn_data_sel;

 fun dissect_coeqn_disc seq fun_names (basic_ctr_specss : basic_corec_ctr_spec list list)

     maybe_ctr_rhs maybe_code_rhs prems' concl matchedsss =

   let

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

       fun f (t as u $ v) = if p t then SOME t else merge_options (f u, f v)

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

       in f end;

     val applied_fun = concl

       |> find_subterm (member ((op =) o apsnd SOME) fun_names o try (fst o dest_Free o head_of))

       |> the

       handle Option.Option => primrec_error_eqn "malformed discriminator formula" concl;

     val ((fun_name, fun_T), fun_args) = strip_comb applied_fun |>> dest_Free;

     val basic_ctr_specs = the (AList.lookup (op =) (fun_names ~~ basic_ctr_specss) fun_name);

     val discs = map #disc basic_ctr_specs;

     val ctrs = map #ctr basic_ctr_specs;

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

     val _ = not_disc andalso length ctrs <> 2 andalso

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

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

     val eq_ctr0 = concl |> 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" concl;

     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 {ctr, disc, ...} = nth basic_ctr_specs ctr_no;

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

     val matchedss = AList.lookup (op =) matchedsss fun_name |> the_default [];

     val prems = map (abstract (List.rev fun_args)) prems';

     val real_prems =

       (if catch_all orelse seq then maps s_not_conj matchedss else []) @

       (if catch_all then [] else prems);

     val matchedsss' = AList.delete (op =) fun_name matchedsss

       |> cons (fun_name, if seq then matchedss @ [prems] else matchedss @ [real_prems]);

     val user_eqn =

       (real_prems, concl)

       |>> map HOLogic.mk_Trueprop ||> HOLogic.mk_Trueprop o abstract (List.rev fun_args)

       |> curry Logic.list_all (map dest_Free fun_args) o Logic.list_implies;

   in

     (Disc {

       fun_name = fun_name,

       fun_T = fun_T,

       fun_args = fun_args,

       ctr = ctr,

       ctr_no = ctr_no,

       disc = disc,

       prems = real_prems,

       auto_gen = catch_all,

       maybe_ctr_rhs = maybe_ctr_rhs,

       maybe_code_rhs = maybe_code_rhs,

       user_eqn = user_eqn

     }, matchedsss')

   end;

 fun dissect_coeqn_sel fun_names (basic_ctr_specss : basic_corec_ctr_spec list list) eqn' of_spec

     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_T), fun_args) = dest_comb lhs |> snd |> strip_comb |> apfst dest_Free

       handle TERM _ =>

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

     val basic_ctr_specs = the (AList.lookup (op =) (fun_names ~~ basic_ctr_specss) fun_name)

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

     val {ctr, ...} =

       if is_some of_spec

       then the (find_first (equal (the of_spec) o #ctr) basic_ctr_specs)

       else filter (exists (equal sel) o #sels) basic_ctr_specs |> the_single

         handle List.Empty => primrec_error_eqn "ambiguous selector - use \"of\"" eqn;

     val user_eqn = drop_All eqn';

   in

     Sel {

       fun_name = fun_name,

       fun_T = fun_T,

       fun_args = fun_args,

       ctr = ctr,

       sel = sel,

       rhs_term = rhs,

       user_eqn = user_eqn

     }

   end;

 fun dissect_coeqn_ctr seq fun_names (basic_ctr_specss : basic_corec_ctr_spec list list) eqn'

     maybe_code_rhs prems concl matchedsss =

   let

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

     val (fun_name, fun_args) = strip_comb lhs |>> fst o dest_Free;

     val basic_ctr_specs = the (AList.lookup (op =) (fun_names ~~ basic_ctr_specss) fun_name);

     val (ctr, ctr_args) = strip_comb (unfold_let rhs);

     val {disc, sels, ...} = the (find_first (equal ctr o #ctr) basic_ctr_specs)

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

     val disc_concl = betapply (disc, lhs);

     val (maybe_eqn_data_disc, matchedsss') = if length basic_ctr_specs = 1

       then (NONE, matchedsss)

       else apfst SOME (dissect_coeqn_disc seq fun_names basic_ctr_specss

           (SOME (abstract (List.rev fun_args) rhs)) maybe_code_rhs prems disc_concl matchedsss);

     val sel_concls = sels ~~ ctr_args

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

 (*

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

  (is_some maybe_eqn_data_disc ? K (Syntax.string_of_term @{context} disc_concl ^ "\n    \<cdot> ")) "" ^

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

  "\nfor premise(s)\n    \<cdot> " ^

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

 *)

     val eqns_data_sel =

       map (dissect_coeqn_sel fun_names basic_ctr_specss eqn' (SOME ctr)) sel_concls;

   in

     (the_list maybe_eqn_data_disc @ eqns_data_sel, matchedsss')

   end;

 fun dissect_coeqn_code lthy has_call fun_names basic_ctr_specss eqn' concl matchedsss =

   let

     val (lhs, (rhs', rhs)) = HOLogic.dest_eq concl ||> `(expand_corec_code_rhs lthy has_call []);

     val (fun_name, fun_args) = strip_comb lhs |>> fst o dest_Free;

     val basic_ctr_specs = the (AList.lookup (op =) (fun_names ~~ basic_ctr_specss) fun_name);

     val cond_ctrs = fold_rev_corec_code_rhs lthy (fn cs => fn ctr => fn _ =>

         if member ((op =) o apsnd #ctr) basic_ctr_specs ctr

         then cons (ctr, cs)

         else primrec_error_eqn "not a constructor" ctr) [] rhs' []

       |> AList.group (op =);

     val ctr_premss = (case cond_ctrs of [_] => [[]] | _ => map (s_dnf o snd) cond_ctrs);

     val ctr_concls = cond_ctrs |> map (fn (ctr, _) =>

         binder_types (fastype_of ctr)

         |> map_index (fn (n, T) => massage_corec_code_rhs lthy (fn _ => fn ctr' => fn args =>

           if ctr' = ctr then nth args n else Const (@{const_name undefined}, T)) [] rhs')

         |> curry list_comb ctr

         |> curry HOLogic.mk_eq lhs);

   in

     fold_map2 (dissect_coeqn_ctr false fun_names basic_ctr_specss eqn'

         (SOME (abstract (List.rev fun_args) rhs)))

       ctr_premss ctr_concls matchedsss

   end;

 fun dissect_coeqn lthy seq has_call fun_names (basic_ctr_specss : basic_corec_ctr_spec list list)

     eqn' of_spec matchedsss =

   let

     val eqn = drop_All eqn'

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

     val (prems, concl) = Logic.strip_horn eqn

       |> apfst (map HOLogic.dest_Trueprop) o apsnd HOLogic.dest_Trueprop;

     val head = concl

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

       |> head_of;

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

     val discs = maps (map #disc) basic_ctr_specss;

     val sels = maps (maps #sels) basic_ctr_specss;

     val ctrs = maps (map #ctr) basic_ctr_specss;

   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

       dissect_coeqn_disc seq fun_names basic_ctr_specss NONE NONE prems concl matchedsss

       |>> single

     else if member (op =) sels head then

       ([dissect_coeqn_sel fun_names basic_ctr_specss eqn' of_spec concl], matchedsss)

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

       member (op =) ctrs (head_of (unfold_let (the maybe_rhs))) then

       dissect_coeqn_ctr seq fun_names basic_ctr_specss eqn' NONE prems concl matchedsss

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

       null prems then

       dissect_coeqn_code lthy has_call fun_names basic_ctr_specss eqn' concl matchedsss

       |>> flat

     else

       primrec_error_eqn "malformed function equation" eqn

   end;

 fun build_corec_arg_disc (ctr_specs : corec_ctr_spec list)

     ({fun_args, ctr_no, prems, ...} : coeqn_data_disc) =

   if is_none (#pred (nth ctr_specs ctr_no)) then I else

     s_conjs prems

     |> curry subst_bounds (List.rev fun_args)

     |> HOLogic.tupled_lambda (HOLogic.mk_tuple fun_args)

     |> K |> nth_map (the (#pred (nth ctr_specs ctr_no)));

 fun build_corec_arg_no_call (sel_eqns : coeqn_data_sel list) sel =

   find_first (equal sel o #sel) sel_eqns

   |> try (fn SOME {fun_args, rhs_term, ...} => abs_tuple fun_args rhs_term)

   |> the_default undef_const

   |> K;

 fun build_corec_args_mutual_call lthy has_call (sel_eqns : coeqn_data_sel list) sel =

   let

     val maybe_sel_eqn = find_first (equal sel o #sel) sel_eqns;

   in

     if is_none maybe_sel_eqn then (I, I, I) else

     let

       val {fun_args, rhs_term, ... } = the maybe_sel_eqn;

       val bound_Ts = List.rev (map fastype_of fun_args);

       fun rewrite_stop _ t = if has_call t then @{term False} else @{term True};

       fun rewrite_end _ t = if has_call t then undef_const else t;

       fun rewrite_cont bound_Ts t =

         if has_call t then mk_tuple1 bound_Ts (snd (strip_comb t)) else undef_const;

       fun massage f _ = massage_mutual_corec_call lthy has_call f bound_Ts rhs_term

         |> abs_tuple fun_args;

     in

       (massage rewrite_stop, massage rewrite_end, massage rewrite_cont)

     end

   end;

 fun build_corec_arg_nested_call lthy has_call (sel_eqns : coeqn_data_sel list) sel =

   let

     val maybe_sel_eqn = find_first (equal sel o #sel) sel_eqns;

   in

     if is_none maybe_sel_eqn then I else

     let

       val {fun_args, rhs_term, ...} = the maybe_sel_eqn;

       val bound_Ts = List.rev (map fastype_of fun_args);

       fun rewrite bound_Ts U T (Abs (v, V, b)) = Abs (v, V, rewrite (V :: bound_Ts) U T b)

         | rewrite bound_Ts U T (t as _ $ _) =

           let val (u, vs) = strip_comb t in

             if is_Free u andalso has_call u then

               Inr_const U T $ mk_tuple1 bound_Ts vs

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

               map (rewrite bound_Ts U T) vs |> chop 1 |>> HOLogic.mk_split o the_single |> list_comb

             else

               list_comb (rewrite bound_Ts U T u, map (rewrite bound_Ts U T) vs)

           end

         | rewrite _ U T t =

           if is_Free t andalso has_call t then Inr_const U T $ HOLogic.unit else t;

       fun massage t =

         rhs_term

         |> massage_nested_corec_call lthy has_call rewrite bound_Ts (range_type (fastype_of t))

         |> abs_tuple fun_args;

     in

       massage

     end

   end;

 fun build_corec_args_sel lthy has_call (all_sel_eqns : coeqn_data_sel list)

     (ctr_spec : corec_ctr_spec) =

   let val sel_eqns = filter (equal (#ctr ctr_spec) o #ctr) all_sel_eqns in

     if null sel_eqns then I else

       let

         val sel_call_list = #sels ctr_spec ~~ #calls ctr_spec;

         val no_calls' = map_filter (try (apsnd (fn No_Corec n => n))) sel_call_list;

         val mutual_calls' = map_filter (try (apsnd (fn Mutual_Corec n => n))) sel_call_list;

         val nested_calls' = map_filter (try (apsnd (fn Nested_Corec n => n))) sel_call_list;

       in

         I

         #> fold (fn (sel, n) => nth_map n (build_corec_arg_no_call sel_eqns sel)) no_calls'

         #> fold (fn (sel, (q, g, h)) =>

           let val (fq, fg, fh) = build_corec_args_mutual_call lthy has_call sel_eqns sel in

             nth_map q fq o nth_map g fg o nth_map h fh end) mutual_calls'

         #> fold (fn (sel, n) => nth_map n

           (build_corec_arg_nested_call lthy has_call sel_eqns sel)) nested_calls'

       end

   end;

 fun build_codefs lthy bs mxs has_call arg_Tss (corec_specs : corec_spec list)

     (disc_eqnss : coeqn_data_disc list list) (sel_eqnss : coeqn_data_sel list list) =

   let

     val corecs = map #corec corec_specs;

     val ctr_specss = map #ctr_specs corec_specs;

     val corec_args = hd corecs

       |> fst o split_last o binder_types o fastype_of

       |> map (Const o pair @{const_name undefined})

       |> fold2 (fold o build_corec_arg_disc) ctr_specss disc_eqnss

       |> fold2 (fold o build_corec_args_sel lthy has_call) sel_eqnss ctr_specss;

     fun currys [] t = t

       | currys Ts t = t $ mk_tuple1 (List.rev Ts) (map Bound (length Ts - 1 downto 0))

           |> fold_rev (Term.abs o pair Name.uu) Ts;

 (*

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

  space_implode "\n    \<cdot> " (map (Syntax.string_of_term lthy) corec_args));

 *)

     val exclss' =

       disc_eqnss

       |> map (map (fn x => (#fun_args x, #ctr_no x, #prems x, #auto_gen x))

         #> fst o (fn xs => fold_map (fn x => fn ys => ((x, ys), ys @ [x])) xs [])

         #> maps (uncurry (map o pair)

           #> map (fn ((fun_args, c, x, a), (_, c', y, a')) =>

               ((c, c', a orelse a'), (x, s_not (s_conjs y)))

             ||> apfst (map HOLogic.mk_Trueprop) o apsnd HOLogic.mk_Trueprop

             ||> Logic.list_implies

             ||> curry Logic.list_all (map dest_Free fun_args))))

   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

     |> map3 (fn b => fn mx => fn t => ((b, mx), ((Binding.conceal (Thm.def_binding b), []), t)))

       bs mxs

     |> rpair exclss'

   end;

 fun mk_real_disc_eqns fun_binding arg_Ts ({ctr_specs, ...} : corec_spec)

     (sel_eqns : coeqn_data_sel list) (disc_eqns : coeqn_data_disc list) =

   if length disc_eqns <> length ctr_specs - 1 then disc_eqns else

     let

       val n = 0 upto length ctr_specs

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

       val fun_args = (try (#fun_args o hd) disc_eqns, try (#fun_args o hd) sel_eqns)

         |> the_default (map (curry Free Name.uu) arg_Ts) o merge_options;

       val extra_disc_eqn = {

         fun_name = Binding.name_of fun_binding,

         fun_T = arg_Ts ---> body_type (fastype_of (#ctr (hd ctr_specs))),

         fun_args = fun_args,

         ctr = #ctr (nth ctr_specs n),

         ctr_no = n,

         disc = #disc (nth ctr_specs n),

         prems = maps (s_not_conj o #prems) disc_eqns,

         auto_gen = true,

         maybe_ctr_rhs = NONE,

         maybe_code_rhs = NONE,

         user_eqn = undef_const};

     in

       chop n disc_eqns ||> cons extra_disc_eqn |> (op @)

     end;

 fun find_corec_calls has_call basic_ctr_specs {ctr, sel, rhs_term, ...} =

   let

     val sel_no = find_first (equal ctr o #ctr) basic_ctr_specs

       |> find_index (equal sel) o #sels o the;

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

       | find (t as _ $ _) = strip_comb t |>> find ||> maps find |> (op @)

       | find f = if is_Free f andalso has_call f then [f] else [];

   in

     find rhs_term

     |> K |> nth_map sel_no |> AList.map_entry (op =) ctr

   end;

 fun add_primcorec_ursive maybe_tac seq fixes specs of_specs lthy =

   let

     val (bs, mxs) = map_split (apfst fst) fixes;

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

     val fun_names = map Binding.name_of bs;

     val basic_ctr_specss = map (basic_corec_specs_of lthy) res_Ts;

     val has_call = exists_subterm (map (fst #>> Binding.name_of #> Free) fixes |> member (op =));

     val eqns_data =

       fold_map2 (dissect_coeqn lthy seq has_call fun_names basic_ctr_specss) (map snd specs)

         of_specs []

       |> flat o fst;

     val callssss =

       map_filter (try (fn Sel x => x)) eqns_data

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

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

       |> map (flat o snd)

       |> map2 (fold o find_corec_calls has_call) basic_ctr_specss

       |> map2 (curry (op |>)) (map (map (fn {ctr, sels, ...} =>

         (ctr, map (K []) sels))) basic_ctr_specss);

 (*

 val _ = tracing ("callssss = " ^ @{make_string} callssss);

 *)

     val ((n2m, corec_specs', _, coinduct_thm, strong_coinduct_thm, coinduct_thms,

           strong_coinduct_thms), lthy') =

       corec_specs_of bs arg_Ts res_Ts (get_indices fixes) callssss lthy;

     val actual_nn = length bs;

     val corec_specs = take actual_nn corec_specs'; (*###*)

     val ctr_specss = map #ctr_specs corec_specs;

     val disc_eqnss' = map_filter (try (fn Disc x => x)) eqns_data

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

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

       |> 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 formula in definition"

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

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

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

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

       |> map (flat o snd);

     val arg_Tss = map (binder_types o snd o fst) fixes;

     val disc_eqnss = map5 mk_real_disc_eqns bs arg_Tss corec_specs sel_eqnss disc_eqnss';

     val (defs, exclss') =

       build_codefs lthy' bs mxs has_call arg_Tss corec_specs disc_eqnss sel_eqnss;

     fun excl_tac (c, c', a) =

       if a orelse c = c' orelse seq then SOME (K (HEADGOAL (mk_primcorec_assumption_tac lthy [])))

       else maybe_tac;

 (*

 val _ = tracing ("exclusiveness properties:\n    \<cdot> " ^

  space_implode "\n    \<cdot> " (maps (map (Syntax.string_of_term lthy o snd)) exclss'));

 *)

     val exclss'' = exclss' |> map (map (fn (idx, t) =>

       (idx, (Option.map (Goal.prove lthy [] [] t #> Thm.close_derivation) (excl_tac idx), t))));

     val taut_thmss = map (map (apsnd (the o fst)) o filter (is_some o fst o snd)) exclss'';

     val (goal_idxss, goalss) = exclss''

       |> map (map (apsnd (rpair [] o snd)) o filter (is_none o fst o snd))

       |> split_list o map split_list;

     fun prove thmss' def_thms' lthy =

       let

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

         val exclss' = map (op ~~) (goal_idxss ~~ thmss');

         fun mk_exclsss excls n =

           (excls, map (fn k => replicate k [TrueI] @ replicate (n - k) []) (0 upto n - 1))

           |-> fold (fn ((c, c', _), thm) => nth_map c (nth_map c' (K [thm])));

         val exclssss = (exclss' ~~ taut_thmss |> map (op @), fun_names ~~ corec_specs)

           |-> map2 (fn excls => fn (_, {ctr_specs, ...}) => mk_exclsss excls (length ctr_specs));

         fun prove_disc ({ctr_specs, ...} : corec_spec) exclsss

             ({fun_name, fun_T, fun_args, ctr_no, prems, ...} : coeqn_data_disc) =

           if Term.aconv_untyped (#disc (nth ctr_specs ctr_no), @{term "\<lambda>x. x = x"}) then [] else

             let

               val {disc_corec, ...} = nth ctr_specs ctr_no;

               val k = 1 + ctr_no;

               val m = length prems;

               val t =

                 list_comb (Free (fun_name, fun_T), map Bound (length fun_args - 1 downto 0))

                 |> curry betapply (#disc (nth ctr_specs ctr_no)) (*###*)

                 |> HOLogic.mk_Trueprop

                 |> curry Logic.list_implies (map HOLogic.mk_Trueprop prems)

                 |> curry Logic.list_all (map dest_Free fun_args);

             in

               if prems = [@{term False}] then [] else

               mk_primcorec_disc_tac lthy def_thms disc_corec k m exclsss

               |> K |> Goal.prove lthy [] [] t

               |> Thm.close_derivation

               |> pair (#disc (nth ctr_specs ctr_no))

               |> single

             end;

         fun prove_sel ({nested_maps, nested_map_idents, nested_map_comps, ctr_specs, ...}

             : corec_spec) (disc_eqns : coeqn_data_disc list) exclsss

             ({fun_name, fun_T, fun_args, ctr, sel, rhs_term, ...} : coeqn_data_sel) =

           let

             val SOME ctr_spec = find_first (equal ctr o #ctr) ctr_specs;

             val ctr_no = find_index (equal ctr o #ctr) ctr_specs;

             val prems = the_default (maps (s_not_conj o #prems) disc_eqns)

                 (find_first (equal ctr_no o #ctr_no) disc_eqns |> Option.map #prems);

             val sel_corec = find_index (equal sel) (#sels ctr_spec)

               |> nth (#sel_corecs ctr_spec);

             val k = 1 + ctr_no;

             val m = length prems;

             val t =

               list_comb (Free (fun_name, fun_T), map Bound (length fun_args - 1 downto 0))

               |> curry betapply sel

               |> rpair (abstract (List.rev fun_args) rhs_term)

               |> HOLogic.mk_Trueprop o HOLogic.mk_eq

               |> curry Logic.list_implies (map HOLogic.mk_Trueprop prems)

               |> curry Logic.list_all (map dest_Free fun_args);

             val (distincts, _, sel_splits, sel_split_asms) = case_thms_of_term lthy [] rhs_term;

           in

             mk_primcorec_sel_tac lthy def_thms distincts sel_splits sel_split_asms nested_maps

               nested_map_idents nested_map_comps sel_corec k m exclsss

             |> K |> Goal.prove lthy [] [] t

             |> Thm.close_derivation

             |> pair sel

           end;

         fun prove_ctr disc_alist sel_alist (disc_eqns : coeqn_data_disc list)

             (sel_eqns : coeqn_data_sel list) ({ctr, disc, sels, collapse, ...} : corec_ctr_spec) =

           (* don't try to prove theorems when some sel_eqns are missing *)

           if not (exists (equal ctr o #ctr) disc_eqns)

               andalso not (exists (equal ctr o #ctr) sel_eqns)

             orelse

               filter (equal ctr o #ctr) sel_eqns

               |> fst o finds ((op =) o apsnd #sel) sels

               |> exists (null o snd)

           then [] else

             let

               val (fun_name, fun_T, fun_args, prems, maybe_rhs) =

                 (find_first (equal ctr o #ctr) disc_eqns, find_first (equal ctr o #ctr) sel_eqns)

                 |>> Option.map (fn x => (#fun_name x, #fun_T x, #fun_args x, #prems x,

                   #maybe_ctr_rhs x))

                 ||> Option.map (fn x => (#fun_name x, #fun_T x, #fun_args x, [], NONE))

                 |> the o merge_options;

               val m = length prems;

               val t = (if is_some maybe_rhs then the maybe_rhs else

                   filter (equal ctr o #ctr) sel_eqns

                   |> fst o finds ((op =) o apsnd #sel) sels

                   |> map (snd #> (fn [x] => (List.rev (#fun_args x), #rhs_term x)) #-> abstract)

                   |> curry list_comb ctr)

                 |> curry HOLogic.mk_eq (list_comb (Free (fun_name, fun_T),

                   map Bound (length fun_args - 1 downto 0)))

                 |> HOLogic.mk_Trueprop

                 |> curry Logic.list_implies (map HOLogic.mk_Trueprop prems)

                 |> curry Logic.list_all (map dest_Free fun_args);

               val maybe_disc_thm = AList.lookup (op =) disc_alist disc;

               val sel_thms = map snd (filter (member (op =) sels o fst) sel_alist);

             in

               if prems = [@{term False}] then [] else

                 mk_primcorec_ctr_of_dtr_tac lthy m collapse maybe_disc_thm sel_thms

                 |> K |> Goal.prove lthy [] [] t

                 |> Thm.close_derivation

                 |> pair ctr

                 |> single

             end;

         fun prove_code disc_eqns sel_eqns ctr_alist ctr_specs =

           let

             val (fun_name, fun_T, fun_args, maybe_rhs) =

               (find_first (member (op =) (map #ctr ctr_specs) o #ctr) disc_eqns,

                find_first (member (op =) (map #ctr ctr_specs) o #ctr) sel_eqns)

               |>> Option.map (fn x => (#fun_name x, #fun_T x, #fun_args x, #maybe_code_rhs x))

               ||> Option.map (fn x => (#fun_name x, #fun_T x, #fun_args x, NONE))

               |> the o merge_options;

             val bound_Ts = List.rev (map fastype_of fun_args);

             val lhs = list_comb (Free (fun_name, fun_T), map Bound (length fun_args - 1 downto 0));

             val maybe_rhs_info =

               (case maybe_rhs of

                 SOME rhs =>

                 let

                   val raw_rhs = expand_corec_code_rhs lthy has_call bound_Ts rhs;

                   val cond_ctrs =

                     fold_rev_corec_code_rhs lthy (K oo (cons oo pair)) bound_Ts raw_rhs [];

                   val ctr_thms = map (the o AList.lookup (op =) ctr_alist o snd) cond_ctrs;

                 in SOME (rhs, raw_rhs, ctr_thms) end

               | NONE =>

                 let

                   fun prove_code_ctr {ctr, sels, ...} =

                     if not (exists (equal ctr o fst) ctr_alist) then NONE else

                       let

                         val prems = find_first (equal ctr o #ctr) disc_eqns

                           |> Option.map #prems |> the_default [];

                         val t =

                           filter (equal ctr o #ctr) sel_eqns

                           |> fst o finds ((op =) o apsnd #sel) sels

                           |> map (snd #> (fn [x] => (List.rev (#fun_args x), #rhs_term x))

                             #-> abstract)

                           |> curry list_comb ctr;

                       in

                         SOME (prems, t)

                       end;

                   val maybe_ctr_conds_argss = map prove_code_ctr ctr_specs;

                 in

                   if exists is_none maybe_ctr_conds_argss then NONE else

                     let

                       val rhs = fold_rev (fn SOME (prems, u) => fn t => mk_If (s_conjs prems) u t)

                         maybe_ctr_conds_argss

                         (Const (@{const_name Code.abort}, @{typ String.literal} -->

                             (@{typ unit} --> body_type fun_T) --> body_type fun_T) $

                           HOLogic.mk_literal fun_name $

                           absdummy @{typ unit} (incr_boundvars 1 lhs));

                     in SOME (rhs, rhs, map snd ctr_alist) end

                 end);

           in

             (case maybe_rhs_info of

               NONE => []

             | SOME (rhs, raw_rhs, ctr_thms) =>

               let

                 val ms = map (Logic.count_prems o prop_of) ctr_thms;

                 val (raw_t, t) = (raw_rhs, rhs)

                   |> pairself

                     (curry HOLogic.mk_eq (list_comb (Free (fun_name, fun_T),

                       map Bound (length fun_args - 1 downto 0)))

                     #> HOLogic.mk_Trueprop

                     #> curry Logic.list_all (map dest_Free fun_args));

                 val (distincts, discIs, sel_splits, sel_split_asms) =

                   case_thms_of_term lthy bound_Ts raw_rhs;

                 val raw_code_thm = mk_primcorec_raw_code_of_ctr_tac lthy distincts discIs sel_splits

                     sel_split_asms ms ctr_thms

                   |> K |> Goal.prove lthy [] [] raw_t

                   |> Thm.close_derivation;

               in

                 mk_primcorec_code_of_raw_code_tac lthy distincts sel_splits raw_code_thm

                 |> K |> Goal.prove lthy [] [] t

                 |> Thm.close_derivation

                 |> single

               end)

           end;

         val disc_alists = map3 (maps oo prove_disc) corec_specs exclssss disc_eqnss;

         val sel_alists = map4 (map ooo prove_sel) corec_specs disc_eqnss exclssss sel_eqnss;

         val disc_thmss = map (map snd) disc_alists;

         val sel_thmss = map (map snd) sel_alists;

         val ctr_alists = map5 (maps oooo prove_ctr) disc_alists sel_alists disc_eqnss sel_eqnss

           ctr_specss;

         val ctr_thmss = map (map snd) ctr_alists;

         val code_thmss = map4 prove_code disc_eqnss sel_eqnss ctr_alists ctr_specss;

         val simp_thmss = map2 append disc_thmss sel_thmss

         val common_name = mk_common_name fun_names;

         val notes =

           [(coinductN, map (if n2m then single else K []) coinduct_thms, []),

            (codeN, code_thmss, code_nitpicksimp_attrs),

            (ctrN, ctr_thmss, []),

            (discN, disc_thmss, simp_attrs),

            (selN, sel_thmss, simp_attrs),

            (simpsN, simp_thmss, []),

            (strong_coinductN, map (if n2m then single else K []) strong_coinduct_thms, [])]

           |> maps (fn (thmN, thmss, attrs) =>

             map2 (fn fun_name => fn thms =>

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

               fun_names (take actual_nn thmss))

           |> filter_out (null o fst o hd o snd);

         val common_notes =

           [(coinductN, if n2m then [coinduct_thm] else [], []),

            (strong_coinductN, if n2m then [strong_coinduct_thm] else [], [])]

           |> filter_out (null o #2)

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

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

       in

         lthy |> Local_Theory.notes (notes @ common_notes) |> snd

       end;

     fun after_qed thmss' = fold_map Local_Theory.define defs #-> prove thmss';

   in

     (goalss, after_qed, lthy')

   end;

 fun add_primcorec_ursive_cmd maybe_tac seq (raw_fixes, raw_specs') lthy =

   let

     val (raw_specs, of_specs) = split_list raw_specs' ||> map (Option.map (Syntax.read_term lthy));

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

   in

     add_primcorec_ursive maybe_tac seq fixes specs of_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));

 val add_primcorecursive_cmd = (fn (goalss, after_qed, lthy) =>

   lthy

   |> Proof.theorem NONE after_qed goalss

   |> Proof.refine (Method.primitive_text I)

   |> Seq.hd) ooo add_primcorec_ursive_cmd NONE;

 val add_primcorec_cmd = (fn (goalss, after_qed, lthy) =>

   lthy

   |> after_qed (map (fn [] => []

       | _ => primrec_error "need exclusiveness proofs - use primcorecursive instead of primcorec")

     goalss)) ooo add_primcorec_ursive_cmd (SOME (fn {context = ctxt, ...} => auto_tac ctxt));

 end;