(*  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 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 mk_not @{const True} = @{const False}

   | mk_not @{const False} = @{const True}

   | mk_not (@{const Not} $ t) = t

   | mk_not (@{const Trueprop} $ t) = @{const Trueprop} $ mk_not t

   | mk_not t = HOLogic.mk_not t

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

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

 fun invert_prems [t] = map mk_not (HOLogic.disjuncts t)

   | invert_prems ts = [mk_disjs (map mk_not ts)];

 fun invert_prems_disj [t] = map mk_not (HOLogic.disjuncts t)

   | invert_prems_disj ts = [mk_disjs (map (mk_conjs o map mk_not o HOLogic.disjuncts) ts)];

 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;

 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 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 _ = 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;

 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 if d = SOME ~1 andalso const_name u = SOME @{const_name comp} then

             list_comb (map_types (K dummyT) u, map2 subst [NONE, d] vs)

           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 direct_calls indirect_calls t =

   let

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

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

         let

           val maybe_direct_y' = AList.lookup (op =) direct_calls y;

           val maybe_indirect_y' = AList.lookup (op =) indirect_calls y;

           val (g, g_args) = strip_comb g';

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

           val _ = ctr_pos < 0 orelse length g_args >= ctr_pos orelse

             primrec_error_eqn "too few arguments in recursive call" t;

         in

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

             pairself (subst bound_Ts) (g', y) |> (op $)

           else if ctr_pos >= 0 then

             list_comb (the maybe_direct_y', g_args)

           else if is_some maybe_indirect_y' then

             (if has_call g' then t else y)

             |> massage_indirect_rec_call lthy has_call

               (rewrite_map_arg get_ctr_pos) bound_Ts y (the maybe_indirect_y')

             |> (if has_call g' then I else curry (op $) g')

           else

             t

         end

       | subst _ t = t

   in

     subst [] t

     |> tap (fn u => has_call u andalso (* FIXME detect this case earlier *)

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

   end;

 fun build_rec_arg lthy funs_data has_call ctr_spec maybe_eqn_data =

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

     in

       t

       |> subst_rec_calls lthy get_ctr_pos has_call ctr_args direct_calls indirect_calls

       |> fold_rev lambda abstractions

     end;

 fun build_defs lthy bs mxs funs_data rec_specs 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.map_name Thm.def_name b, []), t))) bs mxs

   end;

 fun find_rec_calls has_call 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 has_call 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, mxs) = map_split (apfst 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));

     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 has_call)));

     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 ((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 mxs funs_data rec_specs has_call;

     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

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

       (take actual_nn induct_thms) funs_data)

     |> snd o Local_Theory.notes common_notes

   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_disc = {

   fun_name: string,

   fun_T: typ,

   fun_args: term list,

   ctr: term,

   ctr_no: int, (*###*)

   disc: term,

   prems: term list,

   user_eqn: term

 };

 type co_eqn_data_sel = {

   fun_name: string,

   fun_T: typ,

   fun_args: term list,

   ctr: term,

   sel: term,

   rhs_term: term,

   user_eqn: term

 };

 datatype co_eqn_data =

   Disc of co_eqn_data_disc |

   Sel of co_eqn_data_sel;

 fun co_dissect_eqn_disc sequential fun_names corec_specs 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 equation" concl;

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

     val {ctr_specs, ...} = the (AList.lookup (op =) (fun_names ~~ corec_specs) fun_name);

     val discs = map #disc ctr_specs;

     val ctrs = map #ctr 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 = #ctr (nth 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 sequential then maps invert_prems_disj matchedss else []) @

       (if catch_all then [] else prems);

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

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

     val user_eqn =

       (real_prems, betapply (#disc (nth ctr_specs ctr_no), applied_fun))

       |>> map HOLogic.mk_Trueprop ||> HOLogic.mk_Trueprop

       |> 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 (nth ctr_specs ctr_no),

       prems = real_prems,

       user_eqn = user_eqn

     }, matchedsss')

   end;

 fun co_dissect_eqn_sel fun_names corec_specs 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_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 corec_spec = the (AList.lookup (op =) (fun_names ~~ corec_specs) fun_name)

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

     val (ctr_spec, sel) = #ctr_specs corec_spec

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

       |>> nth (#ctr_specs corec_spec);

     val user_eqn = drop_All eqn';

   in

     Sel {

       fun_name = fun_name,

       fun_T = fun_T,

       fun_args = fun_args,

       ctr = #ctr ctr_spec,

       sel = sel,

       rhs_term = rhs,

       user_eqn = user_eqn

     }

   end;

 fun co_dissect_eqn_ctr sequential fun_names corec_specs eqn' imp_prems imp_rhs matchedsss =

   let 

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

     val fun_name = head_of lhs |> fst o dest_Free;

     val {ctr_specs, ...} = the (AList.lookup (op =) (fun_names ~~ corec_specs) fun_name);

     val (ctr, ctr_args) = strip_comb rhs;

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

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

     val disc_imp_rhs = betapply (disc, lhs);

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

       then (NONE, matchedsss)

       else apfst SOME (co_dissect_eqn_disc

           sequential fun_names corec_specs imp_prems disc_imp_rhs matchedsss);

     val sel_imp_rhss = (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} imp_rhs ^ "\nto\n    \<cdot> " ^

  (is_some maybe_eqn_data_disc ? K (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_names corec_specs eqn') sel_imp_rhss;

   in

     (the_list maybe_eqn_data_disc @ eqns_data_sel, matchedsss')

   end;

 fun co_dissect_eqn sequential fun_names corec_specs eqn' matchedsss =

   let

     val eqn = drop_All eqn'

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

     val (imp_prems, imp_rhs) = Logic.strip_horn eqn

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

     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 discs = maps #ctr_specs corec_specs |> map #disc;

     val sels = maps #ctr_specs corec_specs |> maps #sels;

     val ctrs = maps #ctr_specs corec_specs |> 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_names corec_specs imp_prems imp_rhs matchedsss

       |>> single

     else if member (op =) sels head then

       ([co_dissect_eqn_sel fun_names corec_specs eqn' imp_rhs], matchedsss)

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

       co_dissect_eqn_ctr sequential fun_names corec_specs eqn' imp_prems imp_rhs matchedsss

     else

       primrec_error_eqn "malformed function equation" eqn

   end;

 fun build_corec_arg_disc ctr_specs {fun_args, ctr_no, prems, ...} =

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

     mk_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 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_arg_direct_call lthy has_call sel_eqns sel =

   let

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

     fun rewrite is_end U _ t =

       if U = @{typ bool} then @{term True} |> has_call t ? K @{term False} (* stop? *)

       else if is_end = has_call t then undef_const

       else if is_end then t (* end *)

       else HOLogic.mk_tuple (snd (strip_comb t)); (* continue *)

     fun massage rhs_term is_end t = massage_direct_corec_call

       lthy has_call (rewrite is_end) [] (range_type (fastype_of t)) rhs_term;

   in

     if is_none maybe_sel_eqn then K I else

       abs_tuple (#fun_args (the maybe_sel_eqn)) oo massage (#rhs_term (the maybe_sel_eqn))

   end;

 fun build_corec_arg_indirect_call lthy has_call sel_eqns sel =

   let

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

     fun rewrite _ _ =

       let

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

           | subst (t as _ $ _) =

             let val (u, vs) = strip_comb t in

               if is_Free u andalso has_call u then

                 Const (@{const_name Inr}, dummyT) $

                   (try (foldr1 (fn (x, y) => Const (@{const_name Pair}, dummyT) $ x $ y)) vs

                     |> the_default (hd vs))

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

                 list_comb (u |> map_types (K dummyT), map subst vs)

               else

                 list_comb (subst u, map subst vs)

             end

           | subst t = t;

       in

         subst

       end;

     fun massage rhs_term t = massage_indirect_corec_call

       lthy has_call rewrite [] (range_type (fastype_of t)) rhs_term;

   in

     if is_none maybe_sel_eqn then I else

       abs_tuple (#fun_args (the maybe_sel_eqn)) o massage (#rhs_term (the maybe_sel_eqn))

   end;

 fun build_corec_args_sel lthy has_call all_sel_eqns 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 direct_calls' = map_filter (try (apsnd (fn Direct_Corec n => n))) sel_call_list;

         val indirect_calls' = map_filter (try (apsnd (fn Indirect_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 f = build_corec_arg_direct_call lthy has_call sel_eqns sel in

             nth_map h (f false) o nth_map g (f true) o nth_map q (f true) end) direct_calls'

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

           (build_corec_arg_indirect_call lthy has_call sel_eqns sel)) indirect_calls'

       end

   end;

 fun co_build_defs lthy bs mxs has_call arg_Tss corec_specs disc_eqnss sel_eqnss =

   let

     val corec_specs' = take (length bs) corec_specs;

     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 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 (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 {fun_args, ctr_no, prems, ...} => (fun_args, ctr_no, prems))

         #> 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), (_, c', y)) => ((c, c'), (x, mk_not (mk_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.map_name Thm.def_name b, []), t))) bs mxs

     |> rpair exclss'

   end;

 fun mk_real_disc_eqns fun_binding arg_Ts {ctr_specs, ...} sel_eqns disc_eqns =

   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 (invert_prems o #prems) disc_eqns,

         user_eqn = undef_const};

     in

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

     end;

 fun add_primcorec sequential fixes 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;

     (* 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 corec_specs = take (length fun_names) corec_specs'; (*###*)

     val (eqns_data, _) =

       fold_map (co_dissect_eqn sequential fun_names corec_specs) (map snd specs) []

       |>> flat;

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

     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') =

       co_build_defs lthy' bs mxs has_call arg_Tss corec_specs disc_eqnss sel_eqnss;

     (* try to prove (automatically generated) tautologies by ourselves *)

     val exclss'' = exclss'

       |> map (map (apsnd

         (`(try (fn t => Goal.prove lthy [] [] t (mk_primcorec_assumption_tac lthy |> K))))));

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

     val (obligation_idxss, obligationss) = 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 ~~) (obligation_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, ...} exclsss

             {fun_name, fun_T, fun_args, ctr_no, prems, user_eqn, ...} =

           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

               mk_primcorec_disc_tac lthy def_thms disc_corec k m exclsss

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

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

               |> single

             end;

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

             disc_eqns exclsss {fun_name, fun_T, fun_args, ctr, sel, rhs_term, ...} =

           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 (invert_prems 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);

           in

             mk_primcorec_ctr_or_sel_tac lthy def_thms sel_corec k m exclsss

               nested_maps nested_map_idents nested_map_comps

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

             |> pair sel

           end;

         fun prove_ctr (_, disc_thms) (_, sel_thms') disc_eqns sel_eqns

             {ctr, disc, sels, collapse, ...} =

 let val _ = tracing ("disc = " ^ @{make_string} disc); in

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

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

 andalso (warning ("no eqns for ctr " ^ Syntax.string_of_term lthy ctr); true)

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

               filter (equal ctr o #ctr) sel_eqns

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

               |> exists (null o snd)

 andalso (warning ("sel_eqn(s) missing for ctr " ^ Syntax.string_of_term lthy ctr); true)

           then [] else

             let

 val _ = tracing ("ctr = " ^ Syntax.string_of_term lthy ctr);

 val _ = tracing (the_default "NO disc_eqn" (Option.map (curry (op ^) "disc = " o Syntax.string_of_term lthy o #disc) (find_first (equal ctr o #ctr) disc_eqns)));

               val (fun_name, fun_T, fun_args, prems) =

                 (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))

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

                 |> the o merge_options;

               val m = length prems;

               val t = 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_thms disc;

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

 val _ = tracing ("t = " ^ Syntax.string_of_term lthy t);

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

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

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

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

             in

               mk_primcorec_ctr_of_dtr_tac lthy m collapse maybe_disc_thm sel_thms

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

               |> single

 (*handle ERROR x => (warning x; []))*)

 end

           end;

         val (disc_notes, disc_thmss) =

           fun_names ~~ map3 (maps oo prove_disc) corec_specs exclssss disc_eqnss

           |> `(map (fn (fun_name, thms) =>

             ((Binding.qualify true fun_name (@{binding disc}), simp_attrs), [(map snd thms, [])])));

         val (sel_notes, sel_thmss) =

           fun_names ~~ map4 (map ooo prove_sel) corec_specs disc_eqnss exclssss sel_eqnss

           |> `(map (fn (fun_name, thms) =>

             ((Binding.qualify true fun_name (@{binding sel}), simp_attrs), [(map snd thms, [])])));

         val ctr_notes =

           fun_names ~~ map5 (maps oooo prove_ctr) disc_thmss sel_thmss

             disc_eqnss sel_eqnss (map #ctr_specs corec_specs)

           |> map (fn (fun_name, thms) =>

             ((Binding.qualify true fun_name (@{binding ctr}), simp_attrs), [(thms, [])]));

       in

         lthy |> snd o Local_Theory.notes

           (filter (not o null o fst o the_single o snd) (disc_notes @ sel_notes @ ctr_notes))

       end;

   in

     lthy'

     |> Proof.theorem NONE (curry (op #->) (fold_map Local_Theory.define defs) o prove) obligationss

     |> 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;