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

     Author:     Jasmin Blanchette, TU Muenchen

     Copyright   2012, 2013

 Sugared datatype and codatatype constructions.

 *)

 signature BNF_FP_DEF_SUGAR =

 sig

   type fp_sugar =

     {T: typ,

      fp: BNF_FP_Util.fp_kind,

      index: int,

      pre_bnfs: BNF_Def.bnf list,

      nested_bnfs: BNF_Def.bnf list,

      nesting_bnfs: BNF_Def.bnf list,

      fp_res: BNF_FP_Util.fp_result,

      ctr_defss: thm list list,

      ctr_sugars: Ctr_Sugar.ctr_sugar list,

      co_iterss: term list list,

      mapss: thm list list,

      co_inducts: thm list,

      co_iter_thmsss: thm list list list,

      disc_co_itersss: thm list list list,

      sel_co_iterssss: thm list list list list};

   val of_fp_sugar: (fp_sugar -> 'a list) -> fp_sugar -> 'a

   val eq_fp_sugar: fp_sugar * fp_sugar -> bool

   val morph_fp_sugar: morphism -> fp_sugar -> fp_sugar

   val transfer_fp_sugar: Proof.context -> fp_sugar -> fp_sugar

   val fp_sugar_of: Proof.context -> string -> fp_sugar option

   val fp_sugars_of: Proof.context -> fp_sugar list

   val co_induct_of: 'a list -> 'a

   val strong_co_induct_of: 'a list -> 'a

   val tvar_subst: theory -> typ list -> typ list -> ((string * int) * typ) list

   val exists_subtype_in: typ list -> typ -> bool

   val flat_rec_arg_args: 'a list list -> 'a list

   val flat_corec_preds_predsss_gettersss: 'a list -> 'a list list list -> 'a list list list ->

     'a list

   val mk_co_iter: theory -> BNF_FP_Util.fp_kind -> typ -> typ list -> term -> term

   val nesty_bnfs: Proof.context -> typ list list list -> typ list -> BNF_Def.bnf list

   type lfp_sugar_thms =

     (thm list * thm * Args.src list)

     * (thm list list * thm list list * Args.src list)

   val morph_lfp_sugar_thms: morphism -> lfp_sugar_thms -> lfp_sugar_thms

   val transfer_lfp_sugar_thms: Proof.context -> lfp_sugar_thms -> lfp_sugar_thms

   type gfp_sugar_thms =

     ((thm list * thm) list * Args.src list)

     * (thm list list * thm list list * Args.src list)

     * (thm list list * thm list list * Args.src list)

     * (thm list list * thm list list * Args.src list)

     * (thm list list list * thm list list list * Args.src list)

   val morph_gfp_sugar_thms: morphism -> gfp_sugar_thms -> gfp_sugar_thms

   val transfer_gfp_sugar_thms: Proof.context -> gfp_sugar_thms -> gfp_sugar_thms

   val mk_co_iters_prelims: BNF_FP_Util.fp_kind -> typ list list list -> typ list -> typ list ->

     int list -> int list list -> term list list -> Proof.context ->

     (term list list

      * (typ list list * typ list list list list * term list list

         * term list list list list) list option

      * (string * term list * term list list

         * ((term list list * term list list list) * (typ list * typ list list)) list) option)

     * Proof.context

   val mk_iter_fun_arg_types: typ list list list -> int list -> int list list -> term ->

     typ list list list list

   val mk_coiter_fun_arg_types: typ list list list -> typ list -> int list -> term ->

     typ list list

     * (typ list list list list * typ list list list * typ list list list list * typ list)

   val define_iters: string list ->

     (typ list list * typ list list list list * term list list * term list list list list) list ->

     (string -> binding) -> typ list -> typ list -> term list -> Proof.context ->

     (term list * thm list) * Proof.context

   val define_coiters: string list -> string * term list * term list list

     * ((term list list * term list list list) * (typ list * typ list list)) list ->

     (string -> binding) -> typ list -> typ list -> term list -> Proof.context ->

     (term list * thm list) * Proof.context

   val derive_induct_iters_thms_for_types: BNF_Def.bnf list ->

     (typ list list * typ list list list list * term list list * term list list list list) list ->

     thm -> thm list list -> BNF_Def.bnf list -> BNF_Def.bnf list -> typ list -> typ list ->

     typ list -> typ list list list -> term list list -> thm list list -> term list list ->

     thm list list -> local_theory -> lfp_sugar_thms

   val derive_coinduct_coiters_thms_for_types: BNF_Def.bnf list ->

     string * term list * term list list * ((term list list * term list list list)

       * (typ list * typ list list)) list ->

     thm -> thm list -> thm list -> thm list list -> BNF_Def.bnf list -> typ list -> typ list ->

     typ list -> typ list list list -> int list list -> int list list -> int list -> thm list list ->

     Ctr_Sugar.ctr_sugar list -> term list list -> thm list list -> (thm list -> thm list) ->

     local_theory -> gfp_sugar_thms

   val co_datatypes: BNF_FP_Util.fp_kind -> (mixfix list -> binding list -> binding list ->

       binding list list -> binding list -> (string * sort) list -> typ list * typ list list ->

       BNF_Def.bnf list -> local_theory -> BNF_FP_Util.fp_result * local_theory) ->

     (bool * bool) * (((((binding * (typ * sort)) list * binding) * (binding * binding)) * mixfix) *

       ((((binding * binding) * (binding * typ) list) * (binding * term) list) *

         mixfix) list) list ->

     local_theory -> local_theory

   val parse_co_datatype_cmd: BNF_FP_Util.fp_kind -> (mixfix list -> binding list -> binding list ->

       binding list list -> binding list -> (string * sort) list -> typ list * typ list list ->

       BNF_Def.bnf list -> local_theory -> BNF_FP_Util.fp_result * local_theory) ->

     (local_theory -> local_theory) parser

 end;

 structure BNF_FP_Def_Sugar : BNF_FP_DEF_SUGAR =

 struct

 open Ctr_Sugar

 open BNF_Util

 open BNF_Comp

 open BNF_Def

 open BNF_FP_Util

 open BNF_FP_Def_Sugar_Tactics

 val EqN = "Eq_";

 type fp_sugar =

   {T: typ,

    fp: fp_kind,

    index: int,

    pre_bnfs: bnf list,

    nested_bnfs: bnf list,

    nesting_bnfs: bnf list,

    fp_res: fp_result,

    ctr_defss: thm list list,

    ctr_sugars: ctr_sugar list,

    co_iterss: term list list,

    mapss: thm list list,

    co_inducts: thm list,

    co_iter_thmsss: thm list list list,

    disc_co_itersss: thm list list list,

    sel_co_iterssss: thm list list list list};

 fun of_fp_sugar f (fp_sugar as ({index, ...}: fp_sugar)) = nth (f fp_sugar) index;

 fun eq_fp_sugar ({T = T1, fp = fp1, index = index1, fp_res = fp_res1, ...} : fp_sugar,

     {T = T2, fp = fp2, index = index2, fp_res = fp_res2, ...} : fp_sugar) =

   T1 = T2 andalso fp1 = fp2 andalso index1 = index2 andalso eq_fp_result (fp_res1, fp_res2);

 fun morph_fp_sugar phi ({T, fp, index, pre_bnfs, nested_bnfs, nesting_bnfs, fp_res, ctr_defss,

     ctr_sugars, co_iterss, mapss, co_inducts, co_iter_thmsss, disc_co_itersss, sel_co_iterssss}

     : fp_sugar) =

   {T = Morphism.typ phi T, fp = fp, index = index, pre_bnfs = map (morph_bnf phi) pre_bnfs,

     nested_bnfs = map (morph_bnf phi) nested_bnfs, nesting_bnfs = map (morph_bnf phi) nesting_bnfs,

    fp_res = morph_fp_result phi fp_res,

    ctr_defss = map (map (Morphism.thm phi)) ctr_defss,

    ctr_sugars = map (morph_ctr_sugar phi) ctr_sugars,

    co_iterss = map (map (Morphism.term phi)) co_iterss,

    mapss = map (map (Morphism.thm phi)) mapss,

    co_inducts = map (Morphism.thm phi) co_inducts,

    co_iter_thmsss = map (map (map (Morphism.thm phi))) co_iter_thmsss,

    disc_co_itersss = map (map (map (Morphism.thm phi))) disc_co_itersss,

    sel_co_iterssss = map (map (map (map (Morphism.thm phi)))) sel_co_iterssss};

 val transfer_fp_sugar =

   morph_fp_sugar o Morphism.thm_morphism o Thm.transfer o Proof_Context.theory_of;

 structure Data = Generic_Data

 (

   type T = fp_sugar Symtab.table;

   val empty = Symtab.empty;

   val extend = I;

   val merge = Symtab.merge eq_fp_sugar;

 );

 fun fp_sugar_of ctxt =

   Symtab.lookup (Data.get (Context.Proof ctxt))

   #> Option.map (transfer_fp_sugar ctxt);

 fun fp_sugars_of ctxt =

   Symtab.fold (cons o transfer_fp_sugar ctxt o snd) (Data.get (Context.Proof ctxt)) [];

 fun co_induct_of (i :: _) = i;

 fun strong_co_induct_of [_, s] = s;

 (* TODO: register "sum" and "prod" as datatypes to enable N2M reduction for them *)

 fun register_fp_sugar key fp_sugar =

   Local_Theory.declaration {syntax = false, pervasive = true}

     (fn phi => Data.map (Symtab.default (key, morph_fp_sugar phi fp_sugar)));

 fun register_fp_sugars fp pre_bnfs nested_bnfs nesting_bnfs (fp_res as {Ts, ...}) ctr_defss

     ctr_sugars co_iterss mapss co_inducts co_iter_thmsss disc_co_itersss sel_co_iterssss lthy =

   (0, lthy)

   |> fold (fn T as Type (s, _) => fn (kk, lthy) => (kk + 1,

     register_fp_sugar s {T = T, fp = fp, index = kk, pre_bnfs = pre_bnfs,

         nested_bnfs = nested_bnfs, nesting_bnfs = nesting_bnfs, fp_res = fp_res,

         ctr_defss = ctr_defss, ctr_sugars = ctr_sugars, co_iterss = co_iterss, mapss = mapss,

         co_inducts = co_inducts, co_iter_thmsss = co_iter_thmsss, disc_co_itersss = disc_co_itersss,

         sel_co_iterssss = sel_co_iterssss}

       lthy)) Ts

   |> snd;

 (* This function could produce clashes in contrived examples (e.g., "x.A", "x.x_A", "y.A"). *)

 fun quasi_unambiguous_case_names names =

   let

     val ps = map (`Long_Name.base_name) names;

     val dups = Library.duplicates (op =) (map fst ps);

     fun underscore s =

       let val ss = space_explode Long_Name.separator s in

         space_implode "_" (drop (length ss - 2) ss)

       end;

   in

     map (fn (base, full) => if member (op =) dups base then underscore full else base) ps

   end;

 val id_def = @{thm id_def};

 val mp_conj = @{thm mp_conj};

 val nitpicksimp_attrs = @{attributes [nitpick_simp]};

 val code_nitpicksimp_attrs = Code.add_default_eqn_attrib :: nitpicksimp_attrs;

 val simp_attrs = @{attributes [simp]};

 fun tvar_subst thy Ts Us =

   Vartab.fold (cons o apsnd snd) (fold (Sign.typ_match thy) (Ts ~~ Us) Vartab.empty) [];

 val exists_subtype_in = Term.exists_subtype o member (op =);

 val lists_bmoc = fold (fn xs => fn t => Term.list_comb (t, xs));

 fun flat_rec_arg_args xss =

   (* FIXME (once the old datatype package is phased out): The first line below gives the preferred

      order. The second line is for compatibility with the old datatype package. *)

 (*

   flat xss

 *)

   map hd xss @ maps tl xss;

 fun flat_corec_predss_getterss qss fss = maps (op @) (qss ~~ fss);

 fun flat_corec_preds_predsss_gettersss [] [qss] [fss] = flat_corec_predss_getterss qss fss

   | flat_corec_preds_predsss_gettersss (p :: ps) (qss :: qsss) (fss :: fsss) =

     p :: flat_corec_predss_getterss qss fss @ flat_corec_preds_predsss_gettersss ps qsss fsss;

 fun mk_tupled_fun x f xs =

   if xs = [x] then f else HOLogic.tupled_lambda x (Term.list_comb (f, xs));

 fun mk_uncurried2_fun f xss =

   mk_tupled_fun (HOLogic.mk_tuple (map HOLogic.mk_tuple xss)) f (flat_rec_arg_args xss);

 fun mk_flip (x, Type (_, [T1, Type (_, [T2, T3])])) =

   Abs ("x", T1, Abs ("y", T2, Var (x, T2 --> T1 --> T3) $ Bound 0 $ Bound 1));

 fun flip_rels lthy n thm =

   let

     val Rs = Term.add_vars (prop_of thm) [];

     val Rs' = rev (drop (length Rs - n) Rs);

     val cRs = map (fn f => (certify lthy (Var f), certify lthy (mk_flip f))) Rs';

   in

     Drule.cterm_instantiate cRs thm

   end;

 fun mk_ctor_or_dtor get_T Ts t =

   let val Type (_, Ts0) = get_T (fastype_of t) in

     Term.subst_atomic_types (Ts0 ~~ Ts) t

   end;

 val mk_ctor = mk_ctor_or_dtor range_type;

 val mk_dtor = mk_ctor_or_dtor domain_type;

 fun mk_co_iter thy fp fpT Cs t =

   let

     val (f_Cs, Type (_, [prebody, body])) = strip_fun_type (fastype_of t);

     val fpT0 = fp_case fp prebody body;

     val Cs0 = distinct (op =) (map (fp_case fp body_type domain_type) f_Cs);

     val rho = tvar_subst thy (fpT0 :: Cs0) (fpT :: Cs);

   in

     Term.subst_TVars rho t

   end;

 fun mk_co_iters thy fp fpTs Cs ts0 =

   let

     val nn = length fpTs;

     val (fpTs0, Cs0) =

       map ((fp = Greatest_FP ? swap) o dest_funT o snd o strip_typeN nn o fastype_of) ts0

       |> split_list;

     val rho = tvar_subst thy (fpTs0 @ Cs0) (fpTs @ Cs);

   in

     map (Term.subst_TVars rho) ts0

   end;

 val mk_fp_iter_fun_types = binder_fun_types o fastype_of;

 fun unzip_recT (Type (@{type_name prod}, _)) T = [T]

   | unzip_recT _ (Type (@{type_name prod}, Ts)) = Ts

   | unzip_recT _ T = [T];

 fun unzip_corecT (Type (@{type_name sum}, _)) T = [T]

   | unzip_corecT _ (Type (@{type_name sum}, Ts)) = Ts

   | unzip_corecT _ T = [T];

 fun liveness_of_fp_bnf n bnf =

   (case T_of_bnf bnf of

     Type (_, Ts) => map (not o member (op =) (deads_of_bnf bnf)) Ts

   | _ => replicate n false);

 fun cannot_merge_types () = error "Mutually recursive types must have the same type parameters";

 fun merge_type_arg T T' = if T = T' then T else cannot_merge_types ();

 fun merge_type_args (As, As') =

   if length As = length As' then map2 merge_type_arg As As' else cannot_merge_types ();

 fun reassoc_conjs thm =

   reassoc_conjs (thm RS @{thm conj_assoc[THEN iffD1]})

   handle THM _ => thm;

 fun type_args_named_constrained_of ((((ncAs, _), _), _), _) = ncAs;

 fun type_binding_of ((((_, b), _), _), _) = b;

 fun map_binding_of (((_, (b, _)), _), _) = b;

 fun rel_binding_of (((_, (_, b)), _), _) = b;

 fun mixfix_of ((_, mx), _) = mx;

 fun ctr_specs_of (_, ctr_specs) = ctr_specs;

 fun disc_of ((((disc, _), _), _), _) = disc;

 fun ctr_of ((((_, ctr), _), _), _) = ctr;

 fun args_of (((_, args), _), _) = args;

 fun defaults_of ((_, ds), _) = ds;

 fun ctr_mixfix_of (_, mx) = mx;

 fun add_nesty_bnf_names Us =

   let

     fun add (Type (s, Ts)) ss =

         let val (needs, ss') = fold_map add Ts ss in

           if exists I needs then (true, insert (op =) s ss') else (false, ss')

         end

       | add T ss = (member (op =) Us T, ss);

   in snd oo add end;

 fun nesty_bnfs ctxt ctr_Tsss Us =

   map_filter (bnf_of ctxt) (fold (fold (fold (add_nesty_bnf_names Us))) ctr_Tsss []);

 fun indexify proj xs f p = f (find_index (curry (op =) (proj p)) xs) p;

 type lfp_sugar_thms =

   (thm list * thm * Args.src list)

   * (thm list list * thm list list * Args.src list)

 fun morph_lfp_sugar_thms phi ((inducts, induct, induct_attrs), (foldss, recss, iter_attrs)) =

   ((map (Morphism.thm phi) inducts, Morphism.thm phi induct, induct_attrs),

    (map (map (Morphism.thm phi)) foldss, map (map (Morphism.thm phi)) recss, iter_attrs));

 val transfer_lfp_sugar_thms =

   morph_lfp_sugar_thms o Morphism.thm_morphism o Thm.transfer o Proof_Context.theory_of;

 type gfp_sugar_thms =

   ((thm list * thm) list * Args.src list)

   * (thm list list * thm list list * Args.src list)

   * (thm list list * thm list list * Args.src list)

   * (thm list list * thm list list * Args.src list)

   * (thm list list list * thm list list list * Args.src list);

 fun morph_gfp_sugar_thms phi ((coinducts_pairs, coinduct_attrs),

     (unfoldss, corecss, coiter_attrs), (disc_unfoldss, disc_corecss, disc_iter_attrs),

     (disc_unfold_iffss, disc_corec_iffss, disc_iter_iff_attrs),

     (sel_unfoldsss, sel_corecsss, sel_iter_attrs)) =

   ((map (apfst (map (Morphism.thm phi)) o apsnd (Morphism.thm phi)) coinducts_pairs,

     coinduct_attrs),

    (map (map (Morphism.thm phi)) unfoldss, map (map (Morphism.thm phi)) corecss, coiter_attrs),

    (map (map (Morphism.thm phi)) disc_unfoldss, map (map (Morphism.thm phi)) disc_corecss,

     disc_iter_attrs),

    (map (map (Morphism.thm phi)) disc_unfold_iffss, map (map (Morphism.thm phi)) disc_corec_iffss,

     disc_iter_iff_attrs),

    (map (map (map (Morphism.thm phi))) sel_unfoldsss,

     map (map (map (Morphism.thm phi))) sel_corecsss, sel_iter_attrs));

 val transfer_gfp_sugar_thms =

   morph_gfp_sugar_thms o Morphism.thm_morphism o Thm.transfer o Proof_Context.theory_of;

 fun mk_iter_fun_arg_types0 n ms = map2 dest_tupleT ms o dest_sumTN_balanced n o domain_type;

 fun mk_iter_fun_arg_types ctr_Tsss ns mss =

   mk_fp_iter_fun_types

   #> map3 mk_iter_fun_arg_types0 ns mss

   #> map2 (map2 (map2 unzip_recT)) ctr_Tsss;

 fun mk_iters_args_types ctr_Tsss Cs ns mss ctor_iter_fun_Tss lthy =

   let

     val Css = map2 replicate ns Cs;

     val y_Tsss = map3 mk_iter_fun_arg_types0 ns mss (map un_fold_of ctor_iter_fun_Tss);

     val g_Tss = map2 (fn C => map (fn y_Ts => y_Ts ---> C)) Cs y_Tsss;

     val ((gss, ysss), lthy) =

       lthy

       |> mk_Freess "f" g_Tss

       ||>> mk_Freesss "x" y_Tsss;

     val y_Tssss = map (map (map single)) y_Tsss;

     val yssss = map (map (map single)) ysss;

     val z_Tssss =

       map4 (fn n => fn ms => fn ctr_Tss => fn ctor_iter_fun_Ts =>

           map3 (fn m => fn ctr_Ts => fn ctor_iter_fun_T =>

               map2 unzip_recT ctr_Ts (dest_tupleT m ctor_iter_fun_T))

             ms ctr_Tss (dest_sumTN_balanced n (domain_type (co_rec_of ctor_iter_fun_Ts))))

         ns mss ctr_Tsss ctor_iter_fun_Tss;

     val z_Tsss' = map (map flat_rec_arg_args) z_Tssss;

     val h_Tss = map2 (map2 (curry (op --->))) z_Tsss' Css;

     val hss = map2 (map2 retype_free) h_Tss gss;

     val zssss_hd = map2 (map2 (map2 (retype_free o hd))) z_Tssss ysss;

     val (zssss_tl, lthy) =

       lthy

       |> mk_Freessss "y" (map (map (map tl)) z_Tssss);

     val zssss = map2 (map2 (map2 cons)) zssss_hd zssss_tl;

   in

     ([(g_Tss, y_Tssss, gss, yssss), (h_Tss, z_Tssss, hss, zssss)], lthy)

   end;

 fun mk_coiter_fun_arg_types0 ctr_Tsss Cs ns fun_Ts =

   let

     (*avoid "'a itself" arguments in coiterators*)

     fun repair_arity [[]] = [[@{typ unit}]]

       | repair_arity Tss = Tss;

     val ctr_Tsss' = map repair_arity ctr_Tsss;

     val f_sum_prod_Ts = map range_type fun_Ts;

     val f_prod_Tss = map2 dest_sumTN_balanced ns f_sum_prod_Ts;

     val f_Tsss = map2 (map2 (dest_tupleT o length)) ctr_Tsss' f_prod_Tss;

     val f_Tssss = map3 (fn C => map2 (map2 (map (curry (op -->) C) oo unzip_corecT)))

       Cs ctr_Tsss' f_Tsss;

     val q_Tssss = map (map (map (fn [_] => [] | [_, T] => [mk_pred1T (domain_type T)]))) f_Tssss;

   in

     (q_Tssss, f_Tsss, f_Tssss, f_sum_prod_Ts)

   end;

 fun mk_coiter_p_pred_types Cs ns = map2 (fn n => replicate (Int.max (0, n - 1)) o mk_pred1T) ns Cs;

 fun mk_coiter_fun_arg_types ctr_Tsss Cs ns dtor_coiter =

   (mk_coiter_p_pred_types Cs ns,

    mk_fp_iter_fun_types dtor_coiter |> mk_coiter_fun_arg_types0 ctr_Tsss Cs ns);

 fun mk_coiters_args_types ctr_Tsss Cs ns dtor_coiter_fun_Tss lthy =

   let

     val p_Tss = mk_coiter_p_pred_types Cs ns;

     fun mk_types get_Ts =

       let

         val fun_Ts = map get_Ts dtor_coiter_fun_Tss;

         val (q_Tssss, f_Tsss, f_Tssss, f_sum_prod_Ts) = mk_coiter_fun_arg_types0 ctr_Tsss Cs ns fun_Ts;

         val pf_Tss = map3 flat_corec_preds_predsss_gettersss p_Tss q_Tssss f_Tssss;

       in

         (q_Tssss, f_Tsss, f_Tssss, (f_sum_prod_Ts, pf_Tss))

       end;

     val (r_Tssss, g_Tsss, g_Tssss, unfold_types) = mk_types un_fold_of;

     val (s_Tssss, h_Tsss, h_Tssss, corec_types) = mk_types co_rec_of;

     val ((((Free (z, _), cs), pss), gssss), lthy) =

       lthy

       |> yield_singleton (mk_Frees "z") dummyT

       ||>> mk_Frees "a" Cs

       ||>> mk_Freess "p" p_Tss

       ||>> mk_Freessss "g" g_Tssss;

     val rssss = map (map (map (fn [] => []))) r_Tssss;

     val hssss_hd = map2 (map2 (map2 (fn T :: _ => fn [g] => retype_free T g))) h_Tssss gssss;

     val ((sssss, hssss_tl), lthy) =

       lthy

       |> mk_Freessss "q" s_Tssss

       ||>> mk_Freessss "h" (map (map (map tl)) h_Tssss);

     val hssss = map2 (map2 (map2 cons)) hssss_hd hssss_tl;

     val cpss = map2 (map o rapp) cs pss;

     fun build_sum_inj mk_inj = build_map lthy (uncurry mk_inj o dest_sumT o snd);

     fun build_dtor_coiter_arg _ [] [cf] = cf

       | build_dtor_coiter_arg T [cq] [cf, cf'] =

         mk_If cq (build_sum_inj Inl_const (fastype_of cf, T) $ cf)

           (build_sum_inj Inr_const (fastype_of cf', T) $ cf');

     fun mk_args qssss fssss f_Tsss =

       let

         val pfss = map3 flat_corec_preds_predsss_gettersss pss qssss fssss;

         val cqssss = map2 (map o map o map o rapp) cs qssss;

         val cfssss = map2 (map o map o map o rapp) cs fssss;

         val cqfsss = map3 (map3 (map3 build_dtor_coiter_arg)) f_Tsss cqssss cfssss;

       in (pfss, cqfsss) end;

     val unfold_args = mk_args rssss gssss g_Tsss;

     val corec_args = mk_args sssss hssss h_Tsss;

   in

     ((z, cs, cpss, [(unfold_args, unfold_types), (corec_args, corec_types)]), lthy)

   end;

 fun mk_co_iters_prelims fp ctr_Tsss fpTs Cs ns mss xtor_co_iterss0 lthy =

   let

     val thy = Proof_Context.theory_of lthy;

     val (xtor_co_iter_fun_Tss, xtor_co_iterss) =

       map (mk_co_iters thy fp fpTs Cs #> `(mk_fp_iter_fun_types o hd)) (transpose xtor_co_iterss0)

       |> apsnd transpose o apfst transpose o split_list;

     val ((iters_args_types, coiters_args_types), lthy') =

       if fp = Least_FP then

         mk_iters_args_types ctr_Tsss Cs ns mss xtor_co_iter_fun_Tss lthy |>> (rpair NONE o SOME)

       else

         mk_coiters_args_types ctr_Tsss Cs ns xtor_co_iter_fun_Tss lthy |>> (pair NONE o SOME)

   in

     ((xtor_co_iterss, iters_args_types, coiters_args_types), lthy')

   end;

 fun mk_iter_body ctor_iter fss xssss =

   Term.list_comb (ctor_iter, map2 (mk_sum_caseN_balanced oo map2 mk_uncurried2_fun) fss xssss);

 fun mk_preds_getterss_join c cps sum_prod_T cqfss =

   let val n = length cqfss in

     Term.lambda c (mk_IfN sum_prod_T cps

       (map2 (mk_InN_balanced sum_prod_T n) (map HOLogic.mk_tuple cqfss) (1 upto n)))

   end;

 fun mk_coiter_body cs cpss f_sum_prod_Ts cqfsss dtor_coiter =

   Term.list_comb (dtor_coiter, map4 mk_preds_getterss_join cs cpss f_sum_prod_Ts cqfsss);

 fun define_co_iters fp fpT Cs binding_specs lthy0 =

   let

     val thy = Proof_Context.theory_of lthy0;

     val maybe_conceal_def_binding = Thm.def_binding

       #> Config.get lthy0 bnf_note_all = false ? Binding.conceal;

     val ((csts, defs), (lthy', lthy)) = lthy0

       |> apfst split_list o fold_map (fn (b, spec) =>

         Specification.definition (SOME (b, NONE, NoSyn), ((maybe_conceal_def_binding b, []), spec))

         #>> apsnd snd) binding_specs

       ||> `Local_Theory.restore;

     val phi = Proof_Context.export_morphism lthy lthy';

     val csts' = map (mk_co_iter thy fp fpT Cs o Morphism.term phi) csts;

     val defs' = map (Morphism.thm phi) defs;

   in

     ((csts', defs'), lthy')

   end;

 fun define_iters iterNs iter_args_typess' mk_binding fpTs Cs ctor_iters lthy =

   let

     val nn = length fpTs;

     val fpT_to_C as Type (_, [fpT, _]) = snd (strip_typeN nn (fastype_of (hd ctor_iters)));

     fun generate_iter suf (f_Tss, _, fss, xssss) ctor_iter =

       let

         val res_T = fold_rev (curry (op --->)) f_Tss fpT_to_C;

         val b = mk_binding suf;

         val spec =

           mk_Trueprop_eq (lists_bmoc fss (Free (Binding.name_of b, res_T)),

             mk_iter_body ctor_iter fss xssss);

       in (b, spec) end;

   in

     define_co_iters Least_FP fpT Cs (map3 generate_iter iterNs iter_args_typess' ctor_iters) lthy

   end;

 fun define_coiters coiterNs (_, cs, cpss, coiter_args_typess') mk_binding fpTs Cs dtor_coiters

     lthy =

   let

     val nn = length fpTs;

     val C_to_fpT as Type (_, [_, fpT]) = snd (strip_typeN nn (fastype_of (hd dtor_coiters)));

     fun generate_coiter suf ((pfss, cqfsss), (f_sum_prod_Ts, pf_Tss)) dtor_coiter =

       let

         val res_T = fold_rev (curry (op --->)) pf_Tss C_to_fpT;

         val b = mk_binding suf;

         val spec =

           mk_Trueprop_eq (lists_bmoc pfss (Free (Binding.name_of b, res_T)),

             mk_coiter_body cs cpss f_sum_prod_Ts cqfsss dtor_coiter);

       in (b, spec) end;

   in

     define_co_iters Greatest_FP fpT Cs

       (map3 generate_coiter coiterNs coiter_args_typess' dtor_coiters) lthy

   end;

 fun derive_induct_iters_thms_for_types pre_bnfs [fold_args_types, rec_args_types] ctor_induct

     ctor_iter_thmss nesting_bnfs nested_bnfs fpTs Cs Xs ctrXs_Tsss ctrss ctr_defss iterss iter_defss

     lthy =

   let

     val iterss' = transpose iterss;

     val iter_defss' = transpose iter_defss;

     val [folds, recs] = iterss';

     val [fold_defs, rec_defs] = iter_defss';

     val ctr_Tsss = map (map (binder_types o fastype_of)) ctrss;

     val nn = length pre_bnfs;

     val ns = map length ctr_Tsss;

     val mss = map (map length) ctr_Tsss;

     val pre_map_defs = map map_def_of_bnf pre_bnfs;

     val pre_set_defss = map set_defs_of_bnf pre_bnfs;

     val nesting_map_idents = map (unfold_thms lthy [id_def] o map_id0_of_bnf) nesting_bnfs;

     val nested_map_idents = map (unfold_thms lthy [id_def] o map_id0_of_bnf) nested_bnfs;

     val nested_set_maps = maps set_map_of_bnf nested_bnfs;

     val fp_b_names = map base_name_of_typ fpTs;

     val ((((ps, ps'), xsss), us'), names_lthy) =

       lthy

       |> mk_Frees' "P" (map mk_pred1T fpTs)

       ||>> mk_Freesss "x" ctr_Tsss

       ||>> Variable.variant_fixes fp_b_names;

     val us = map2 (curry Free) us' fpTs;

     fun mk_sets_nested bnf =

       let

         val Type (T_name, Us) = T_of_bnf bnf;

         val lives = lives_of_bnf bnf;

         val sets = sets_of_bnf bnf;

         fun mk_set U =

           (case find_index (curry (op =) U) lives of

             ~1 => Term.dummy

           | i => nth sets i);

       in

         (T_name, map mk_set Us)

       end;

     val setss_nested = map mk_sets_nested nested_bnfs;

     val (induct_thms, induct_thm) =

       let

         fun mk_set Ts t =

           let val Type (_, Ts0) = domain_type (fastype_of t) in

             Term.subst_atomic_types (Ts0 ~~ Ts) t

           end;

         fun mk_raw_prem_prems _ (x as Free (_, Type _)) (X as TFree _) =

             [([], (find_index (curry (op =) X) Xs + 1, x))]

           | mk_raw_prem_prems names_lthy (x as Free (s, Type (T_name, Ts0))) (Type (_, Xs_Ts0)) =

             (case AList.lookup (op =) setss_nested T_name of

               NONE => []

             | SOME raw_sets0 =>

               let

                 val (Xs_Ts, (Ts, raw_sets)) =

                   filter (exists_subtype_in Xs o fst) (Xs_Ts0 ~~ (Ts0 ~~ raw_sets0))

                   |> split_list ||> split_list;

                 val sets = map (mk_set Ts0) raw_sets;

                 val (ys, names_lthy') = names_lthy |> mk_Frees s Ts;

                 val xysets = map (pair x) (ys ~~ sets);

                 val ppremss = map2 (mk_raw_prem_prems names_lthy') ys Xs_Ts;

               in

                 flat (map2 (map o apfst o cons) xysets ppremss)

               end)

           | mk_raw_prem_prems _ _ _ = [];

         fun close_prem_prem xs t =

           fold_rev Logic.all (map Free (drop (nn + length xs)

             (rev (Term.add_frees t (map dest_Free xs @ ps'))))) t;

         fun mk_prem_prem xs (xysets, (j, x)) =

           close_prem_prem xs (Logic.list_implies (map (fn (x', (y, set)) =>

               HOLogic.mk_Trueprop (HOLogic.mk_mem (y, set $ x'))) xysets,

             HOLogic.mk_Trueprop (nth ps (j - 1) $ x)));

         fun mk_raw_prem phi ctr ctr_Ts ctrXs_Ts =

           let

             val (xs, names_lthy') = names_lthy |> mk_Frees "x" ctr_Ts;

             val pprems = flat (map2 (mk_raw_prem_prems names_lthy') xs ctrXs_Ts);

           in (xs, pprems, HOLogic.mk_Trueprop (phi $ Term.list_comb (ctr, xs))) end;

         fun mk_prem (xs, raw_pprems, concl) =

           fold_rev Logic.all xs (Logic.list_implies (map (mk_prem_prem xs) raw_pprems, concl));

         val raw_premss = map4 (map3 o mk_raw_prem) ps ctrss ctr_Tsss ctrXs_Tsss;

         val goal =

           Library.foldr (Logic.list_implies o apfst (map mk_prem)) (raw_premss,

             HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj (map2 (curry (op $)) ps us)));

         val kksss = map (map (map (fst o snd) o #2)) raw_premss;

         val ctor_induct' = ctor_induct OF (map mk_sumEN_tupled_balanced mss);

         val thm =

           Goal.prove_sorry lthy [] [] goal (fn {context = ctxt, ...} =>

             mk_induct_tac ctxt nn ns mss kksss (flat ctr_defss) ctor_induct' nested_set_maps

               pre_set_defss)

           |> singleton (Proof_Context.export names_lthy lthy)

           |> Thm.close_derivation;

       in

         `(conj_dests nn) thm

       end;

     val induct_cases = quasi_unambiguous_case_names (maps (map name_of_ctr) ctrss);

     val induct_case_names_attr = Attrib.internal (K (Rule_Cases.case_names induct_cases));

     val xctrss = map2 (map2 (curry Term.list_comb)) ctrss xsss;

     fun mk_iter_thmss (_, x_Tssss, fss, _) iters iter_defs ctor_iter_thms =

       let

         val fiters = map (lists_bmoc fss) iters;

         fun mk_goal fss fiter xctr f xs fxs =

           fold_rev (fold_rev Logic.all) (xs :: fss)

             (mk_Trueprop_eq (fiter $ xctr, Term.list_comb (f, fxs)));

         fun maybe_tick (T, U) u f =

           if try (fst o HOLogic.dest_prodT) U = SOME T then

             Term.lambda u (HOLogic.mk_prod (u, f $ u))

           else

             f;

         fun build_iter (x as Free (_, T)) U =

           if T = U then

             x

           else

             build_map lthy (indexify (perhaps (try (snd o HOLogic.dest_prodT)) o snd) Cs

               (fn kk => fn TU => maybe_tick TU (nth us kk) (nth fiters kk))) (T, U) $ x;

         val fxsss = map2 (map2 (flat_rec_arg_args oo map2 (map o build_iter))) xsss x_Tssss;

         val goalss = map5 (map4 o mk_goal fss) fiters xctrss fss xsss fxsss;

         val tacss =

           map2 (map o mk_iter_tac pre_map_defs (nested_map_idents @ nesting_map_idents) iter_defs)

             ctor_iter_thms ctr_defss;

         fun prove goal tac =

           Goal.prove_sorry lthy [] [] goal (tac o #context)

           |> Thm.close_derivation;

       in

         map2 (map2 prove) goalss tacss

       end;

     val fold_thmss = mk_iter_thmss fold_args_types folds fold_defs (map un_fold_of ctor_iter_thmss);

     val rec_thmss = mk_iter_thmss rec_args_types recs rec_defs (map co_rec_of ctor_iter_thmss);

   in

     ((induct_thms, induct_thm, [induct_case_names_attr]),

      (fold_thmss, rec_thmss, code_nitpicksimp_attrs @ simp_attrs))

   end;

 fun derive_coinduct_coiters_thms_for_types pre_bnfs (z, cs, cpss,

       coiters_args_types as [((pgss, crgsss), _), ((phss, cshsss), _)])

     dtor_coinduct dtor_injects dtor_ctors dtor_coiter_thmss nesting_bnfs fpTs Cs Xs ctrXs_Tsss kss

     mss ns ctr_defss (ctr_sugars : ctr_sugar list) coiterss coiter_defss export_args lthy =

   let

     fun mk_ctor_dtor_coiter_thm dtor_inject dtor_ctor coiter =

       iffD1 OF [dtor_inject, trans OF [coiter, dtor_ctor RS sym]];

     val ctor_dtor_coiter_thmss =

       map3 (map oo mk_ctor_dtor_coiter_thm) dtor_injects dtor_ctors dtor_coiter_thmss;

     val coiterss' = transpose coiterss;

     val coiter_defss' = transpose coiter_defss;

     val [unfold_defs, corec_defs] = coiter_defss';

     val nn = length pre_bnfs;

     val pre_map_defs = map map_def_of_bnf pre_bnfs;

     val pre_rel_defs = map rel_def_of_bnf pre_bnfs;

     val nesting_map_idents = map (unfold_thms lthy [id_def] o map_id0_of_bnf) nesting_bnfs;

     val nesting_rel_eqs = map rel_eq_of_bnf nesting_bnfs;

     val fp_b_names = map base_name_of_typ fpTs;

     val ctrss = map #ctrs ctr_sugars;

     val discss = map #discs ctr_sugars;

     val selsss = map #selss ctr_sugars;

     val exhausts = map #exhaust ctr_sugars;

     val disc_thmsss = map #disc_thmss ctr_sugars;

     val discIss = map #discIs ctr_sugars;

     val sel_thmsss = map #sel_thmss ctr_sugars;

     val (((rs, us'), vs'), names_lthy) =

       lthy

       |> mk_Frees "R" (map (fn T => mk_pred2T T T) fpTs)

       ||>> Variable.variant_fixes fp_b_names

       ||>> Variable.variant_fixes (map (suffix "'") fp_b_names);

     val us = map2 (curry Free) us' fpTs;

     val udiscss = map2 (map o rapp) us discss;

     val uselsss = map2 (map o map o rapp) us selsss;

     val vs = map2 (curry Free) vs' fpTs;

     val vdiscss = map2 (map o rapp) vs discss;

     val vselsss = map2 (map o map o rapp) vs selsss;

     val coinduct_thms_pairs =

       let

         val uvrs = map3 (fn r => fn u => fn v => r $ u $ v) rs us vs;

         val uv_eqs = map2 (curry HOLogic.mk_eq) us vs;

         val strong_rs =

           map4 (fn u => fn v => fn uvr => fn uv_eq =>

             fold_rev Term.lambda [u, v] (HOLogic.mk_disj (uvr, uv_eq))) us vs uvrs uv_eqs;

         fun build_the_rel rs' T Xs_T =

           build_rel lthy (fn (_, X) => nth rs' (find_index (curry (op =) X) Xs)) (T, Xs_T)

           |> Term.subst_atomic_types (Xs ~~ fpTs);

         fun build_rel_app rs' usel vsel Xs_T =

           fold rapp [usel, vsel] (build_the_rel rs' (fastype_of usel) Xs_T);

         fun mk_prem_ctr_concls rs' n k udisc usels vdisc vsels ctrXs_Ts =

           (if k = n then [] else [HOLogic.mk_eq (udisc, vdisc)]) @

           (if null usels then

              []

            else

              [Library.foldr HOLogic.mk_imp (if n = 1 then [] else [udisc, vdisc],

                 Library.foldr1 HOLogic.mk_conj (map3 (build_rel_app rs') usels vsels ctrXs_Ts))]);

         fun mk_prem_concl rs' n udiscs uselss vdiscs vselss ctrXs_Tss =

           Library.foldr1 HOLogic.mk_conj (flat (map6 (mk_prem_ctr_concls rs' n)

             (1 upto n) udiscs uselss vdiscs vselss ctrXs_Tss))

           handle List.Empty => @{term True};

         fun mk_prem rs' uvr u v n udiscs uselss vdiscs vselss ctrXs_Tss =

           fold_rev Logic.all [u, v] (Logic.mk_implies (HOLogic.mk_Trueprop uvr,

             HOLogic.mk_Trueprop (mk_prem_concl rs' n udiscs uselss vdiscs vselss ctrXs_Tss)));

         val concl =

           HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj

             (map3 (fn uvr => fn u => fn v => HOLogic.mk_imp (uvr, HOLogic.mk_eq (u, v)))

                uvrs us vs));

         fun mk_goal rs' =

           Logic.list_implies (map9 (mk_prem rs') uvrs us vs ns udiscss uselsss vdiscss vselsss

             ctrXs_Tsss, concl);

         val goals = map mk_goal [rs, strong_rs];

         fun prove dtor_coinduct' goal =

           Goal.prove_sorry lthy [] [] goal (fn {context = ctxt, ...} =>

             mk_coinduct_tac ctxt nesting_rel_eqs nn ns dtor_coinduct' pre_rel_defs dtor_ctors

               exhausts ctr_defss disc_thmsss sel_thmsss)

           |> singleton (Proof_Context.export names_lthy lthy)

           |> Thm.close_derivation;

         fun postproc nn thm =

           Thm.permute_prems 0 nn

             (if nn = 1 then thm RS mp else funpow nn (fn thm => reassoc_conjs (thm RS mp_conj)) thm)

           |> Drule.zero_var_indexes

           |> `(conj_dests nn);

         val rel_eqs = map rel_eq_of_bnf pre_bnfs;

         val rel_monos = map rel_mono_of_bnf pre_bnfs;

         val dtor_coinducts =

           [dtor_coinduct, mk_strong_coinduct_thm dtor_coinduct rel_eqs rel_monos lthy];

       in

         map2 (postproc nn oo prove) dtor_coinducts goals

       end;

     fun mk_coinduct_concls ms discs ctrs =

       let

         fun mk_disc_concl disc = [name_of_disc disc];

         fun mk_ctr_concl 0 _ = []

           | mk_ctr_concl _ ctor = [name_of_ctr ctor];

         val disc_concls = map mk_disc_concl (fst (split_last discs)) @ [[]];

         val ctr_concls = map2 mk_ctr_concl ms ctrs;

       in

         flat (map2 append disc_concls ctr_concls)

       end;

     val coinduct_cases = quasi_unambiguous_case_names (map (prefix EqN) fp_b_names);

     val coinduct_conclss =

       map3 (quasi_unambiguous_case_names ooo mk_coinduct_concls) mss discss ctrss;

     fun mk_maybe_not pos = not pos ? HOLogic.mk_not;

     val fcoiterss' as [gunfolds, hcorecs] =

       map2 (fn (pfss, _) => map (lists_bmoc pfss)) (map fst coiters_args_types) coiterss';

     val (unfold_thmss, corec_thmss) =

       let

         fun mk_goal pfss c cps fcoiter n k ctr m cfs' =

           fold_rev (fold_rev Logic.all) ([c] :: pfss)

             (Logic.list_implies (seq_conds (HOLogic.mk_Trueprop oo mk_maybe_not) n k cps,

                mk_Trueprop_eq (fcoiter $ c, Term.list_comb (ctr, take m cfs'))));

         fun mk_U maybe_mk_sumT =

           typ_subst_nonatomic (map2 (fn C => fn fpT => (maybe_mk_sumT fpT C, fpT)) Cs fpTs);

         fun tack z_name (c, u) f =

           let val z = Free (z_name, mk_sumT (fastype_of u, fastype_of c)) in

             Term.lambda z (mk_sum_case (Term.lambda u u, Term.lambda c (f $ c)) $ z)

           end;

         fun build_coiter fcoiters maybe_mk_sumT maybe_tack cqf =

           let val T = fastype_of cqf in

             if exists_subtype_in Cs T then

               let val U = mk_U maybe_mk_sumT T in

                 build_map lthy (indexify snd fpTs (fn kk => fn _ =>

                   maybe_tack (nth cs kk, nth us kk) (nth fcoiters kk))) (T, U) $ cqf

               end

             else

               cqf

           end;

         val crgsss' = map (map (map (build_coiter (un_fold_of fcoiterss') (K I) (K I)))) crgsss;

         val cshsss' = map (map (map (build_coiter (co_rec_of fcoiterss') (curry mk_sumT) (tack z))))

           cshsss;

         val unfold_goalss = map8 (map4 oooo mk_goal pgss) cs cpss gunfolds ns kss ctrss mss crgsss';

         val corec_goalss = map8 (map4 oooo mk_goal phss) cs cpss hcorecs ns kss ctrss mss cshsss';

         val unfold_tacss =

           map3 (map oo mk_coiter_tac unfold_defs nesting_map_idents)

             (map un_fold_of ctor_dtor_coiter_thmss) pre_map_defs ctr_defss;

         val corec_tacss =

           map3 (map oo mk_coiter_tac corec_defs nesting_map_idents)

             (map co_rec_of ctor_dtor_coiter_thmss) pre_map_defs ctr_defss;

         fun prove goal tac =

           Goal.prove_sorry lthy [] [] goal (tac o #context)

           |> Thm.close_derivation;

         val unfold_thmss = map2 (map2 prove) unfold_goalss unfold_tacss;

         val corec_thmss =

           map2 (map2 prove) corec_goalss corec_tacss

           |> map (map (unfold_thms lthy @{thms sum_case_if}));

       in

         (unfold_thmss, corec_thmss)

       end;

     val (disc_unfold_iff_thmss, disc_corec_iff_thmss) =

       let

         fun mk_goal c cps fcoiter n k disc =

           mk_Trueprop_eq (disc $ (fcoiter $ c),

             if n = 1 then @{const True}

             else Library.foldr1 HOLogic.mk_conj (seq_conds mk_maybe_not n k cps));

         val unfold_goalss = map6 (map2 oooo mk_goal) cs cpss gunfolds ns kss discss;

         val corec_goalss = map6 (map2 oooo mk_goal) cs cpss hcorecs ns kss discss;

         fun mk_case_split' cp = Drule.instantiate' [] [SOME (certify lthy cp)] @{thm case_split};

         val case_splitss' = map (map mk_case_split') cpss;

         val unfold_tacss =

           map3 (map oo mk_disc_coiter_iff_tac) case_splitss' unfold_thmss disc_thmsss;

         val corec_tacss =

           map3 (map oo mk_disc_coiter_iff_tac) case_splitss' corec_thmss disc_thmsss;

         fun prove goal tac =

           Goal.prove_sorry lthy [] [] goal (tac o #context)

           |> singleton export_args

           |> singleton (Proof_Context.export names_lthy lthy)

           |> Thm.close_derivation;

         fun proves [_] [_] = []

           | proves goals tacs = map2 prove goals tacs;

       in

         (map2 proves unfold_goalss unfold_tacss, map2 proves corec_goalss corec_tacss)

       end;

     fun mk_disc_coiter_thms coiters discIs = map (op RS) (coiters ~~ discIs);

     val disc_unfold_thmss = map2 mk_disc_coiter_thms unfold_thmss discIss;

     val disc_corec_thmss = map2 mk_disc_coiter_thms corec_thmss discIss;

     fun mk_sel_coiter_thm coiter_thm sel sel_thm =

       let

         val (domT, ranT) = dest_funT (fastype_of sel);

         val arg_cong' =

           Drule.instantiate' (map (SOME o certifyT lthy) [domT, ranT])

             [NONE, NONE, SOME (certify lthy sel)] arg_cong

           |> Thm.varifyT_global;

         val sel_thm' = sel_thm RSN (2, trans);

       in

         coiter_thm RS arg_cong' RS sel_thm'

       end;

     fun mk_sel_coiter_thms coiter_thmss =

       map3 (map3 (map2 o mk_sel_coiter_thm)) coiter_thmss selsss sel_thmsss;

     val sel_unfold_thmsss = mk_sel_coiter_thms unfold_thmss;

     val sel_corec_thmsss = mk_sel_coiter_thms corec_thmss;

     val coinduct_consumes_attr = Attrib.internal (K (Rule_Cases.consumes nn));

     val coinduct_case_names_attr = Attrib.internal (K (Rule_Cases.case_names coinduct_cases));

     val coinduct_case_concl_attrs =

       map2 (fn casex => fn concls =>

           Attrib.internal (K (Rule_Cases.case_conclusion (casex, concls))))

         coinduct_cases coinduct_conclss;

     val coinduct_case_attrs =

       coinduct_consumes_attr :: coinduct_case_names_attr :: coinduct_case_concl_attrs;

   in

     ((coinduct_thms_pairs, coinduct_case_attrs),

      (unfold_thmss, corec_thmss, code_nitpicksimp_attrs),

      (disc_unfold_thmss, disc_corec_thmss, []),

      (disc_unfold_iff_thmss, disc_corec_iff_thmss, simp_attrs),

      (sel_unfold_thmsss, sel_corec_thmsss, simp_attrs))

   end;

 fun define_co_datatypes prepare_constraint prepare_typ prepare_term fp construct_fp

     (wrap_opts as (no_discs_sels, rep_compat), specs) no_defs_lthy0 =

   let

     (* TODO: sanity checks on arguments *)

     val _ = if fp = Greatest_FP andalso no_discs_sels then

         error "Cannot define codatatypes without discriminators and selectors"

       else

         ();

     fun qualify mandatory fp_b_name =

       Binding.qualify mandatory fp_b_name o (rep_compat ? Binding.qualify false rep_compat_prefix);

     val nn = length specs;

     val fp_bs = map type_binding_of specs;

     val fp_b_names = map Binding.name_of fp_bs;

     val fp_common_name = mk_common_name fp_b_names;

     val map_bs = map map_binding_of specs;

     val rel_bs = map rel_binding_of specs;

     fun prepare_type_arg (_, (ty, c)) =

       let val TFree (s, _) = prepare_typ no_defs_lthy0 ty in

         TFree (s, prepare_constraint no_defs_lthy0 c)

       end;

     val Ass0 = map (map prepare_type_arg o type_args_named_constrained_of) specs;

     val unsorted_Ass0 = map (map (resort_tfree HOLogic.typeS)) Ass0;

     val unsorted_As = Library.foldr1 merge_type_args unsorted_Ass0;

     val num_As = length unsorted_As;

     val set_bss = map (map fst o type_args_named_constrained_of) specs;

     val (((Bs0, Cs), Xs), no_defs_lthy) =

       no_defs_lthy0

       |> fold (Variable.declare_typ o resort_tfree dummyS) unsorted_As

       |> mk_TFrees num_As

       ||>> mk_TFrees nn

       ||>> variant_tfrees fp_b_names;

     fun add_fake_type spec = Typedecl.basic_typedecl (type_binding_of spec, num_As, mixfix_of spec);

     val (fake_T_names, fake_lthy) = fold_map add_fake_type specs no_defs_lthy0;

     val qsoty = quote o Syntax.string_of_typ fake_lthy;

     val _ = (case Library.duplicates (op =) unsorted_As of [] => ()

       | A :: _ => error ("Duplicate type parameter " ^ qsoty A ^ " in " ^ co_prefix fp ^

           "datatype specification"));

     val bad_args =

       map (Logic.type_map (singleton (Variable.polymorphic no_defs_lthy0))) unsorted_As

       |> filter_out Term.is_TVar;

     val _ = null bad_args orelse

       error ("Locally fixed type argument " ^ qsoty (hd bad_args) ^ " in " ^ co_prefix fp ^

         "datatype specification");

     val mixfixes = map mixfix_of specs;

     val _ = (case Library.duplicates Binding.eq_name fp_bs of [] => ()

       | b :: _ => error ("Duplicate type name declaration " ^ quote (Binding.name_of b)));

     val ctr_specss = map ctr_specs_of specs;

     val disc_bindingss = map (map disc_of) ctr_specss;

     val ctr_bindingss =

       map2 (fn fp_b_name => map (qualify false fp_b_name o ctr_of)) fp_b_names ctr_specss;

     val ctr_argsss = map (map args_of) ctr_specss;

     val ctr_mixfixess = map (map ctr_mixfix_of) ctr_specss;

     val sel_bindingsss = map (map (map fst)) ctr_argsss;

     val fake_ctr_Tsss0 = map (map (map (prepare_typ fake_lthy o snd))) ctr_argsss;

     val raw_sel_defaultsss = map (map defaults_of) ctr_specss;

     val (As :: _) :: fake_ctr_Tsss =

       burrow (burrow (Syntax.check_typs fake_lthy)) (Ass0 :: fake_ctr_Tsss0);

     val As' = map dest_TFree As;

     val rhs_As' = fold (fold (fold Term.add_tfreesT)) fake_ctr_Tsss [];

     val _ = (case subtract (op =) As' rhs_As' of [] => ()

       | extras => error ("Extra type variables on right-hand side: " ^

           commas (map (qsoty o TFree) extras)));

     val fake_Ts = map (fn s => Type (s, As)) fake_T_names;

     fun eq_fpT_check (T as Type (s, Ts)) (T' as Type (s', Ts')) =

         s = s' andalso (Ts = Ts' orelse

           error ("Wrong type arguments in " ^ co_prefix fp ^ "recursive type " ^ qsoty T ^

             " (expected " ^ qsoty T' ^ ")"))

       | eq_fpT_check _ _ = false;

     fun freeze_fp (T as Type (s, Ts)) =

         (case find_index (eq_fpT_check T) fake_Ts of

           ~1 => Type (s, map freeze_fp Ts)

         | kk => nth Xs kk)

       | freeze_fp T = T;

     val unfreeze_fp = Term.typ_subst_atomic (Xs ~~ fake_Ts);

     val ctrXs_Tsss = map (map (map freeze_fp)) fake_ctr_Tsss;

     val ctrXs_sum_prod_Ts = map (mk_sumTN_balanced o map HOLogic.mk_tupleT) ctrXs_Tsss;

     val fp_eqs =

       map dest_TFree Xs ~~ map (Term.typ_subst_atomic (As ~~ unsorted_As)) ctrXs_sum_prod_Ts;

     val rhsXs_As' = fold (fold (fold Term.add_tfreesT)) ctrXs_Tsss [];

     val _ = (case subtract (op =) rhsXs_As' As' of [] => ()

       | extras => List.app (fn extra => warning ("Unused type variable on right-hand side of " ^

           co_prefix fp ^ "datatype definition: " ^ qsoty (TFree extra))) extras);

     val (pre_bnfs, (fp_res as {bnfs = fp_bnfs as any_fp_bnf :: _, ctors = ctors0, dtors = dtors0,

            xtor_co_iterss = xtor_co_iterss0, xtor_co_induct, dtor_ctors, ctor_dtors, ctor_injects,

            dtor_injects, xtor_map_thms, xtor_set_thmss, xtor_rel_thms, xtor_co_iter_thmss, ...},

            lthy)) =

       fp_bnf (construct_fp mixfixes map_bs rel_bs set_bss) fp_bs (map dest_TFree unsorted_As) fp_eqs

         no_defs_lthy0

       handle BAD_DEAD (X, X_backdrop) =>

         (case X_backdrop of

           Type (bad_tc, _) =>

           let

             val fake_T = qsoty (unfreeze_fp X);

             val fake_T_backdrop = qsoty (unfreeze_fp X_backdrop);

             fun register_hint () =

               "\nUse the " ^ quote (fst (fst @{command_spec "bnf"})) ^ " command to register " ^

               quote bad_tc ^ " as a bounded natural functor to allow nested (co)recursion through \

               \it";

           in

             if is_some (bnf_of no_defs_lthy bad_tc) orelse

                is_some (fp_sugar_of no_defs_lthy bad_tc) then

               error ("Inadmissible " ^ co_prefix fp ^ "recursive occurrence of type " ^ fake_T ^

                 " in type expression " ^ fake_T_backdrop)

             else if is_some (Datatype_Data.get_info (Proof_Context.theory_of no_defs_lthy)

                 bad_tc) then

               error ("Unsupported " ^ co_prefix fp ^ "recursive occurrence of type " ^ fake_T ^

                 " via the old-style datatype " ^ quote bad_tc ^ " in type expression " ^

                 fake_T_backdrop ^ register_hint ())

             else

               error ("Unsupported " ^ co_prefix fp ^ "recursive occurrence of type " ^ fake_T ^

                 " via type constructor " ^ quote bad_tc ^ " in type expression " ^ fake_T_backdrop ^

                 register_hint ())

           end);

     val time = time lthy;

     val timer = time (Timer.startRealTimer ());

     val nesting_bnfs = nesty_bnfs lthy ctrXs_Tsss As;

     val nested_bnfs = nesty_bnfs lthy ctrXs_Tsss Xs;

     val pre_map_defs = map map_def_of_bnf pre_bnfs;

     val pre_set_defss = map set_defs_of_bnf pre_bnfs;

     val pre_rel_defs = map rel_def_of_bnf pre_bnfs;

     val nesting_set_maps = maps set_map_of_bnf nesting_bnfs;

     val nested_set_maps = maps set_map_of_bnf nested_bnfs;

     val live = live_of_bnf any_fp_bnf;

     val _ =

       if live = 0 andalso exists (not o Binding.is_empty) (map_bs @ rel_bs) then

         warning "Map function and relator names ignored"

       else

         ();

     val Bs =

       map3 (fn alive => fn A as TFree (_, S) => fn B => if alive then resort_tfree S B else A)

         (liveness_of_fp_bnf num_As any_fp_bnf) As Bs0;

     val B_ify = Term.typ_subst_atomic (As ~~ Bs);

     val ctors = map (mk_ctor As) ctors0;

     val dtors = map (mk_dtor As) dtors0;

     val fpTs = map (domain_type o fastype_of) dtors;

     fun massage_simple_notes base =

       filter_out (null o #2)

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

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

     val massage_multi_notes =

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

         map3 (fn fp_b_name => fn Type (T_name, _) => fn thms =>

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

           fp_b_names fpTs thmss)

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

     val ctr_Tsss = map (map (map (Term.typ_subst_atomic (Xs ~~ fpTs)))) ctrXs_Tsss;

     val ns = map length ctr_Tsss;

     val kss = map (fn n => 1 upto n) ns;

     val mss = map (map length) ctr_Tsss;

     val ((xtor_co_iterss, iters_args_types, coiters_args_types), lthy') =

       mk_co_iters_prelims fp ctr_Tsss fpTs Cs ns mss xtor_co_iterss0 lthy;

     fun define_ctrs_dtrs_for_type (((((((((((((((((((((((fp_bnf, fp_b), fpT), ctor), dtor),

             xtor_co_iters), ctor_dtor), dtor_ctor), ctor_inject), pre_map_def), pre_set_defs),

           pre_rel_def), fp_map_thm), fp_set_thms), fp_rel_thm), n), ks), ms), ctr_bindings),

         ctr_mixfixes), ctr_Tss), disc_bindings), sel_bindingss), raw_sel_defaultss) no_defs_lthy =

       let

         val fp_b_name = Binding.name_of fp_b;

         val dtorT = domain_type (fastype_of ctor);

         val ctr_prod_Ts = map HOLogic.mk_tupleT ctr_Tss;

         val ctr_sum_prod_T = mk_sumTN_balanced ctr_prod_Ts;

         val ((((w, xss), yss), u'), names_lthy) =

           no_defs_lthy

           |> yield_singleton (mk_Frees "w") dtorT

           ||>> mk_Freess "x" ctr_Tss

           ||>> mk_Freess "y" (map (map B_ify) ctr_Tss)

           ||>> yield_singleton Variable.variant_fixes fp_b_name;

         val u = Free (u', fpT);

         val tuple_xs = map HOLogic.mk_tuple xss;

         val tuple_ys = map HOLogic.mk_tuple yss;

         val ctr_rhss =

           map3 (fn k => fn xs => fn tuple_x => fold_rev Term.lambda xs (ctor $

             mk_InN_balanced ctr_sum_prod_T n tuple_x k)) ks xss tuple_xs;

         val maybe_conceal_def_binding = Thm.def_binding

           #> Config.get no_defs_lthy bnf_note_all = false ? Binding.conceal;

         val ((raw_ctrs, raw_ctr_defs), (lthy', lthy)) = no_defs_lthy

           |> apfst split_list o fold_map3 (fn b => fn mx => fn rhs =>

               Local_Theory.define ((b, mx), ((maybe_conceal_def_binding b, []), rhs)) #>> apsnd snd)

             ctr_bindings ctr_mixfixes ctr_rhss

           ||> `Local_Theory.restore;

         val phi = Proof_Context.export_morphism lthy lthy';

         val ctr_defs = map (Morphism.thm phi) raw_ctr_defs;

         val ctr_defs' =

           map2 (fn m => fn def => mk_unabs_def m (def RS meta_eq_to_obj_eq)) ms ctr_defs;

         val ctrs0 = map (Morphism.term phi) raw_ctrs;

         val ctrs = map (mk_ctr As) ctrs0;

         fun wrap_ctrs lthy =

           let

             fun exhaust_tac {context = ctxt, prems = _} =

               let

                 val ctor_iff_dtor_thm =

                   let

                     val goal =

                       fold_rev Logic.all [w, u]

                         (mk_Trueprop_eq (HOLogic.mk_eq (u, ctor $ w), HOLogic.mk_eq (dtor $ u, w)));

                   in

                     Goal.prove_sorry lthy [] [] goal (fn {context = ctxt, ...} =>

                       mk_ctor_iff_dtor_tac ctxt (map (SOME o certifyT lthy) [dtorT, fpT])

                         (certify lthy ctor) (certify lthy dtor) ctor_dtor dtor_ctor)

                     |> Thm.close_derivation

                     |> Morphism.thm phi

                   end;

                 val sumEN_thm' =

                   unfold_thms lthy @{thms unit_all_eq1}

                     (Drule.instantiate' (map (SOME o certifyT lthy) ctr_prod_Ts) []

                        (mk_sumEN_balanced n))

                   |> Morphism.thm phi;

               in

                 mk_exhaust_tac ctxt n ctr_defs ctor_iff_dtor_thm sumEN_thm'

               end;

             val inject_tacss =

               map2 (fn 0 => K [] | _ => fn ctr_def => [fn {context = ctxt, ...} =>

                 mk_inject_tac ctxt ctr_def ctor_inject]) ms ctr_defs;

             val half_distinct_tacss =

               map (map (fn (def, def') => fn {context = ctxt, ...} =>

                 mk_half_distinct_tac ctxt ctor_inject [def, def'])) (mk_half_pairss (`I ctr_defs));

             val tacss = [exhaust_tac] :: inject_tacss @ half_distinct_tacss;

             val sel_defaultss = map (map (apsnd (prepare_term lthy))) raw_sel_defaultss

           in

             wrap_free_constructors tacss (((wrap_opts, ctrs0), standard_binding), (disc_bindings,

               (sel_bindingss, sel_defaultss))) lthy

           end;

         fun derive_maps_sets_rels (ctr_sugar, lthy) =

           if live = 0 then

             ((([], [], [], []), ctr_sugar), lthy)

           else

             let

               val rel_flip = rel_flip_of_bnf fp_bnf;

               val nones = replicate live NONE;

               val ctor_cong =

                 if fp = Least_FP then

                   Drule.dummy_thm

                 else

                   let val ctor' = mk_ctor Bs ctor in

                     cterm_instantiate_pos [NONE, NONE, SOME (certify lthy ctor')] arg_cong

                   end;

               fun mk_cIn ify =

                 certify lthy o (fp = Greatest_FP ? curry (op $) (map_types ify ctor)) oo

                 mk_InN_balanced (ify ctr_sum_prod_T) n;

               val cxIns = map2 (mk_cIn I) tuple_xs ks;

               val cyIns = map2 (mk_cIn B_ify) tuple_ys ks;

               fun mk_map_thm ctr_def' cxIn =

                 fold_thms lthy [ctr_def']

                   (unfold_thms lthy (pre_map_def ::

                        (if fp = Least_FP then [] else [ctor_dtor, dtor_ctor]) @ sum_prod_thms_map)

                      (cterm_instantiate_pos (nones @ [SOME cxIn])

                         (if fp = Least_FP then fp_map_thm else fp_map_thm RS ctor_cong)))

                 |> singleton (Proof_Context.export names_lthy no_defs_lthy);

               fun mk_set_thm fp_set_thm ctr_def' cxIn =

                 fold_thms lthy [ctr_def']

                   (unfold_thms lthy (pre_set_defs @ nested_set_maps @ nesting_set_maps @

                        (if fp = Least_FP then [] else [dtor_ctor]) @ sum_prod_thms_set)

                      (cterm_instantiate_pos [SOME cxIn] fp_set_thm))

                 |> singleton (Proof_Context.export names_lthy no_defs_lthy);

               fun mk_set_thms fp_set_thm = map2 (mk_set_thm fp_set_thm) ctr_defs' cxIns;

               val map_thms = map2 mk_map_thm ctr_defs' cxIns;

               val set_thmss = map mk_set_thms fp_set_thms;

               val rel_infos = (ctr_defs' ~~ cxIns, ctr_defs' ~~ cyIns);

               fun mk_rel_thm postproc ctr_defs' cxIn cyIn =

                 fold_thms lthy ctr_defs'

                   (unfold_thms lthy (@{thm Inl_Inr_False} :: pre_rel_def ::

                        (if fp = Least_FP then [] else [dtor_ctor]) @ sum_prod_thms_rel)

                      (cterm_instantiate_pos (nones @ [SOME cxIn, SOME cyIn]) fp_rel_thm))

                 |> postproc

                 |> singleton (Proof_Context.export names_lthy no_defs_lthy);

               fun mk_rel_inject_thm ((ctr_def', cxIn), (_, cyIn)) =

                 mk_rel_thm (unfold_thms lthy @{thms eq_sym_Unity_conv}) [ctr_def'] cxIn cyIn;

               val rel_inject_thms = map mk_rel_inject_thm (op ~~ rel_infos);

               fun mk_half_rel_distinct_thm ((xctr_def', cxIn), (yctr_def', cyIn)) =

                 mk_rel_thm (fn thm => thm RS @{thm eq_False[THEN iffD1]}) [xctr_def', yctr_def']

                   cxIn cyIn;

               fun mk_other_half_rel_distinct_thm thm =

                 flip_rels lthy live thm

                 RS (rel_flip RS sym RS @{thm arg_cong[of _ _ Not]} RS iffD2);

               val half_rel_distinct_thmss =

                 map (map mk_half_rel_distinct_thm) (mk_half_pairss rel_infos);

               val other_half_rel_distinct_thmss =

                 map (map mk_other_half_rel_distinct_thm) half_rel_distinct_thmss;

               val (rel_distinct_thms, _) =

                 join_halves n half_rel_distinct_thmss other_half_rel_distinct_thmss;

               val anonymous_notes =

                 [(map (fn th => th RS @{thm eq_False[THEN iffD2]}) rel_distinct_thms,

                   code_nitpicksimp_attrs),

                  (map2 (fn th => fn 0 => th RS @{thm eq_True[THEN iffD2]} | _ => th)

                     rel_inject_thms ms, code_nitpicksimp_attrs)]

                 |> map (fn (thms, attrs) => ((Binding.empty, attrs), [(thms, [])]));

               val notes =

                 [(mapN, map_thms, code_nitpicksimp_attrs @ simp_attrs),

                  (rel_distinctN, rel_distinct_thms, simp_attrs),

                  (rel_injectN, rel_inject_thms, simp_attrs),

                  (setN, flat set_thmss, code_nitpicksimp_attrs @ simp_attrs)]

                 |> massage_simple_notes fp_b_name;

             in

               (((map_thms, rel_inject_thms, rel_distinct_thms, set_thmss), ctr_sugar),

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

             end;

         fun mk_binding suf = qualify false fp_b_name (Binding.suffix_name ("_" ^ suf) fp_b);

         fun massage_res (((maps_sets_rels, ctr_sugar), co_iter_res), lthy) =

           (((maps_sets_rels, (ctrs, xss, ctr_defs, ctr_sugar)), co_iter_res), lthy);

       in

         (wrap_ctrs

          #> derive_maps_sets_rels

          ##>>

            (if fp = Least_FP then define_iters [foldN, recN] (the iters_args_types)

             else define_coiters [unfoldN, corecN] (the coiters_args_types))

              mk_binding fpTs Cs xtor_co_iters

          #> massage_res, lthy')

       end;

     fun wrap_types_etc (wrap_types_etcs, lthy) =

       fold_map I wrap_types_etcs lthy

       |>> apsnd split_list o apfst (apsnd split_list4 o apfst split_list4 o split_list)

         o split_list;

     fun mk_simp_thms ({injects, distincts, case_thms, ...} : ctr_sugar) un_folds co_recs

         mapsx rel_injects rel_distincts setss =

       injects @ distincts @ case_thms @ co_recs @ un_folds @ mapsx @ rel_injects @ rel_distincts

       @ flat setss;

     fun derive_note_induct_iters_thms_for_types

         ((((mapss, rel_injects, rel_distincts, setss), (ctrss, _, ctr_defss, ctr_sugars)),

           (iterss, iter_defss)), lthy) =

       let

         val ((induct_thms, induct_thm, induct_attrs), (fold_thmss, rec_thmss, iter_attrs)) =

           derive_induct_iters_thms_for_types pre_bnfs (the iters_args_types) xtor_co_induct

             xtor_co_iter_thmss nesting_bnfs nested_bnfs fpTs Cs Xs ctrXs_Tsss ctrss ctr_defss iterss

             iter_defss lthy;

         val induct_type_attr = Attrib.internal o K o Induct.induct_type;

         val simp_thmss =

           map7 mk_simp_thms ctr_sugars fold_thmss rec_thmss mapss rel_injects rel_distincts setss;

         val common_notes =

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

           |> massage_simple_notes fp_common_name;

         val notes =

           [(foldN, fold_thmss, K iter_attrs),

            (inductN, map single induct_thms, fn T_name => induct_attrs @ [induct_type_attr T_name]),

            (recN, rec_thmss, K iter_attrs),

            (simpsN, simp_thmss, K [])]

           |> massage_multi_notes;

       in

         lthy

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

         |> register_fp_sugars Least_FP pre_bnfs nested_bnfs nesting_bnfs fp_res ctr_defss ctr_sugars

           iterss mapss [induct_thm] (transpose [fold_thmss, rec_thmss]) [] []

       end;

     fun derive_note_coinduct_coiters_thms_for_types

         ((((mapss, rel_injects, rel_distincts, setss), (_, _, ctr_defss, ctr_sugars)),

           (coiterss, coiter_defss)), lthy) =

       let

         val (([(coinduct_thms, coinduct_thm), (strong_coinduct_thms, strong_coinduct_thm)],

               coinduct_attrs),

              (unfold_thmss, corec_thmss, coiter_attrs),

              (disc_unfold_thmss, disc_corec_thmss, disc_coiter_attrs),

              (disc_unfold_iff_thmss, disc_corec_iff_thmss, disc_coiter_iff_attrs),

              (sel_unfold_thmsss, sel_corec_thmsss, sel_coiter_attrs)) =

           derive_coinduct_coiters_thms_for_types pre_bnfs (the coiters_args_types) xtor_co_induct

             dtor_injects dtor_ctors xtor_co_iter_thmss nesting_bnfs fpTs Cs Xs ctrXs_Tsss kss mss ns

             ctr_defss ctr_sugars coiterss coiter_defss (Proof_Context.export lthy' no_defs_lthy)

             lthy;

         val sel_unfold_thmss = map flat sel_unfold_thmsss;

         val sel_corec_thmss = map flat sel_corec_thmsss;

         val coinduct_type_attr = Attrib.internal o K o Induct.coinduct_type;

         val flat_coiter_thms = append oo append;

         val simp_thmss =

           map7 mk_simp_thms ctr_sugars

             (map3 flat_coiter_thms disc_unfold_thmss disc_unfold_iff_thmss sel_unfold_thmss)

             (map3 flat_coiter_thms disc_corec_thmss disc_corec_iff_thmss sel_corec_thmss)

             mapss rel_injects rel_distincts setss;

         val common_notes =

           (if nn > 1 then

              [(coinductN, [coinduct_thm], coinduct_attrs),

               (strong_coinductN, [strong_coinduct_thm], coinduct_attrs)]

            else

              [])

           |> massage_simple_notes fp_common_name;

         val notes =

           [(coinductN, map single coinduct_thms,

             fn T_name => coinduct_attrs @ [coinduct_type_attr T_name]),

            (corecN, corec_thmss, K coiter_attrs),

            (disc_corecN, disc_corec_thmss, K disc_coiter_attrs),

            (disc_corec_iffN, disc_corec_iff_thmss, K disc_coiter_iff_attrs),

            (disc_unfoldN, disc_unfold_thmss, K disc_coiter_attrs),

            (disc_unfold_iffN, disc_unfold_iff_thmss, K disc_coiter_iff_attrs),

            (sel_corecN, sel_corec_thmss, K sel_coiter_attrs),

            (sel_unfoldN, sel_unfold_thmss, K sel_coiter_attrs),

            (simpsN, simp_thmss, K []),

            (strong_coinductN, map single strong_coinduct_thms, K coinduct_attrs),

            (unfoldN, unfold_thmss, K coiter_attrs)]

           |> massage_multi_notes;

         fun is_codatatype (Type (s, _)) =

             (case fp_sugar_of lthy s of SOME {fp = Greatest_FP, ...} => true | _ => false)

           | is_codatatype _ = false;

         val nitpick_supported = forall (is_codatatype o T_of_bnf) nested_bnfs;

         fun register_nitpick fpT ({ctrs, casex, ...} : ctr_sugar) =

           Nitpick_HOL.register_codatatype fpT (fst (dest_Const casex))

             (map (dest_Const o mk_ctr As) ctrs)

           |> Generic_Target.theory_declaration;

       in

         lthy

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

         |> register_fp_sugars Greatest_FP pre_bnfs nested_bnfs nesting_bnfs fp_res ctr_defss

           ctr_sugars coiterss mapss [coinduct_thm, strong_coinduct_thm]

           (transpose [unfold_thmss, corec_thmss]) (transpose [disc_unfold_thmss, disc_corec_thmss])

           (transpose [sel_unfold_thmsss, sel_corec_thmsss])

         |> nitpick_supported ? fold2 register_nitpick fpTs ctr_sugars

       end;

     val lthy'' = lthy'

       |> fold_map define_ctrs_dtrs_for_type (fp_bnfs ~~ fp_bs ~~ fpTs ~~ ctors ~~ dtors ~~

         xtor_co_iterss ~~ ctor_dtors ~~ dtor_ctors ~~ ctor_injects ~~ pre_map_defs ~~

         pre_set_defss ~~ pre_rel_defs ~~ xtor_map_thms ~~ xtor_set_thmss ~~ xtor_rel_thms ~~ ns ~~

         kss ~~ mss ~~ ctr_bindingss ~~ ctr_mixfixess ~~ ctr_Tsss ~~ disc_bindingss ~~

         sel_bindingsss ~~ raw_sel_defaultsss)

       |> wrap_types_etc

       |> fp_case fp derive_note_induct_iters_thms_for_types

            derive_note_coinduct_coiters_thms_for_types;

     val timer = time (timer ("Constructors, discriminators, selectors, etc., for the new " ^

       co_prefix fp ^ "datatype"));

   in

     timer; lthy''

   end;

 fun co_datatypes x = define_co_datatypes (K I) (K I) (K I) x;

 fun co_datatype_cmd x =

   define_co_datatypes Typedecl.read_constraint Syntax.parse_typ Syntax.parse_term x;

 val parse_ctr_arg =

   @{keyword "("} |-- parse_binding_colon -- Parse.typ --| @{keyword ")"} ||

   (Parse.typ >> pair Binding.empty);

 val parse_defaults =

   @{keyword "("} |-- Parse.reserved "defaults" |-- Scan.repeat parse_bound_term --| @{keyword ")"};

 val parse_type_arg_constrained =

   Parse.type_ident -- Scan.option (@{keyword "::"} |-- Parse.!!! Parse.sort);

 val parse_type_arg_named_constrained = parse_opt_binding_colon -- parse_type_arg_constrained;

 val parse_type_args_named_constrained =

   parse_type_arg_constrained >> (single o pair Binding.empty) ||

   @{keyword "("} |-- Parse.!!! (Parse.list1 parse_type_arg_named_constrained --| @{keyword ")"}) ||

   Scan.succeed [];

 val parse_map_rel_binding = Parse.short_ident --| @{keyword ":"} -- parse_binding;

 val no_map_rel = (Binding.empty, Binding.empty);

 fun extract_map_rel ("map", b) = apfst (K b)

   | extract_map_rel ("rel", b) = apsnd (K b)

   | extract_map_rel (s, _) = error ("Unknown label " ^ quote s ^ " (expected \"map\" or \"rel\")");

 val parse_map_rel_bindings =

   @{keyword "("} |-- Scan.repeat parse_map_rel_binding --| @{keyword ")"}

     >> (fn ps => fold extract_map_rel ps no_map_rel) ||

   Scan.succeed no_map_rel;

 val parse_ctr_spec =

   parse_opt_binding_colon -- parse_binding -- Scan.repeat parse_ctr_arg --

   Scan.optional parse_defaults [] -- Parse.opt_mixfix;

 val parse_spec =

   parse_type_args_named_constrained -- parse_binding -- parse_map_rel_bindings --

   Parse.opt_mixfix -- (@{keyword "="} |-- Parse.enum1 "|" parse_ctr_spec);

 val parse_co_datatype = parse_wrap_free_constructors_options -- Parse.and_list1 parse_spec;

 fun parse_co_datatype_cmd fp construct_fp = parse_co_datatype >> co_datatype_cmd fp construct_fp;

 end;