src/HOL/BNF/Tools/bnf_fp_def_sugar.ML

tuning

(*  Title:      HOL/BNF/Tools/bnf_fp_def_sugar.ML
    Author:     Jasmin Blanchette, TU Muenchen
    Copyright   2012

Sugared datatype and codatatype constructions.
*)

signature BNF_FP_DEF_SUGAR =
sig
  type fp_sugar =
    {T: typ,
     lfp: bool,
     index: int,
     pre_bnfs: BNF_Def.bnf list,
     fp_res: BNF_FP_Util.fp_result,
     ctr_sugars: BNF_Ctr_Sugar.ctr_sugar list,
     un_folds: term list,
     co_recs: term list,
     co_induct: thm,
     strong_co_induct: thm,
     un_fold_thmss: thm list list,
     co_rec_thmss: thm list list};

  val fp_sugar_of: local_theory -> string -> fp_sugar option

  val exists_subtype_in: typ list -> typ -> bool
  val indexify_fst: ''a list -> (int -> ''a * 'b -> 'c) -> ''a * 'b -> 'c
  val build_map: local_theory -> (typ * typ -> term) -> typ * typ -> term

  val mk_fp_iter_fun_types: term -> typ list
  val mk_fun_arg_typess: int -> int list -> typ -> typ list list
  val unzip_recT: typ list -> typ -> typ list * typ list
  val mk_fold_fun_typess: typ list list list -> typ list list -> typ list list
  val mk_rec_fun_typess: typ list -> typ list list list -> typ list list -> typ list list

  val derive_induct_fold_rec_thms_for_types: BNF_Def.bnf list -> term list -> term list -> thm ->
    thm list -> thm list -> BNF_Def.bnf list -> BNF_Def.bnf list -> typ list -> typ list ->
    typ list -> term list list -> thm list list -> term list -> term list -> thm list -> thm list ->
    local_theory ->
    (thm * thm list * Args.src list) * (thm list list * Args.src list)
    * (thm list list * Args.src list)
  val derive_coinduct_unfold_corec_thms_for_types: BNF_Def.bnf list -> term list -> term list ->
    thm -> thm -> thm list -> thm list -> thm list -> BNF_Def.bnf list -> BNF_Def.bnf list ->
    typ list -> typ list -> typ list -> int list list -> int list list -> int list ->
    thm list list -> BNF_Ctr_Sugar.ctr_sugar list -> term list -> term list -> thm list ->
    thm list -> local_theory ->
    (thm * thm list * thm * thm list * Args.src list)
    * (thm list list * thm list list * Args.src list)
    * (thm list list * thm list 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)

  val co_datatypes: bool -> (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: bool -> (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 BNF_Util
open BNF_Ctr_Sugar
open BNF_Def
open BNF_FP_Util
open BNF_FP_Def_Sugar_Tactics

val EqN = "Eq_";

type fp_sugar =
  {T: typ,
   lfp: bool,
   index: int,
   pre_bnfs: bnf list,
   fp_res: fp_result,
   ctr_sugars: ctr_sugar list,
   un_folds: term list,
   co_recs: term list,
   co_induct: thm,
   strong_co_induct: thm,
   un_fold_thmss: thm list list,
   co_rec_thmss: thm list list};

fun eq_fp_sugar ({T = T1, lfp = lfp1, index = index1, fp_res = fp_res1, ...} : fp_sugar,
    {T = T2, lfp = lfp2, index = index2, fp_res = fp_res2, ...} : fp_sugar) =
  T1 = T2 andalso lfp1 = lfp2 andalso index1 = index2 andalso eq_fp_result (fp_res1, fp_res2);

fun morph_fp_sugar phi {T, lfp, index, pre_bnfs, fp_res, ctr_sugars, un_folds, co_recs, co_induct,
    strong_co_induct, un_fold_thmss, co_rec_thmss} =
  {T = Morphism.typ phi T, lfp = lfp, index = index, pre_bnfs = map (morph_bnf phi) pre_bnfs,
   fp_res = morph_fp_result phi fp_res, ctr_sugars = map (morph_ctr_sugar phi) ctr_sugars,
   un_folds = map (Morphism.term phi) un_folds, co_recs = map (Morphism.term phi) co_recs,
   co_induct = Morphism.thm phi co_induct, strong_co_induct = Morphism.thm phi strong_co_induct,
   un_fold_thmss = map (map (Morphism.thm phi)) un_fold_thmss,
   co_rec_thmss = map (map (Morphism.thm phi)) co_rec_thmss};

structure Data = Generic_Data
(
  type T = fp_sugar Symtab.table;
  val empty = Symtab.empty;
  val extend = I;
  val merge = Symtab.merge eq_fp_sugar;
);

val fp_sugar_of = Symtab.lookup o Data.get o Context.Proof;

fun register_fp_sugar key fp_sugar =
  Local_Theory.declaration {syntax = false, pervasive = true}
    (fn phi => Data.map (Symtab.update_new (key, morph_fp_sugar phi fp_sugar)));

fun register_fp_sugars lfp pre_bnfs (fp_res as {Ts, ...}) ctr_sugars un_folds co_recs co_induct
    strong_co_induct un_fold_thmss co_rec_thmss lthy =
  (0, lthy)
  |> fold (fn T as Type (s, _) => fn (kk, lthy) => (kk + 1,
    register_fp_sugar s {T = T, lfp = lfp, index = kk, pre_bnfs = pre_bnfs, fp_res = fp_res,
      ctr_sugars = ctr_sugars, un_folds = un_folds, co_recs = co_recs, co_induct = co_induct,
      strong_co_induct = strong_co_induct, un_fold_thmss = un_fold_thmss,
      co_rec_thmss = co_rec_thmss} 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 mp_conj = @{thm mp_conj};

val simp_attrs = @{attributes [simp]};
val code_simp_attrs = Code.add_default_eqn_attrib :: simp_attrs;

fun split_list4 [] = ([], [], [], [])
  | split_list4 ((x1, x2, x3, x4) :: xs) =
    let val (xs1, xs2, xs3, xs4) = split_list4 xs;
    in (x1 :: xs1, x2 :: xs2, x3 :: xs3, x4 :: xs4) end;

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

fun resort_tfree S (TFree (s, _)) = TFree (s, S);

fun typ_subst_nonatomic inst (T as Type (s, Ts)) =
    (case AList.lookup (op =) inst T of
      NONE => Type (s, map (typ_subst_nonatomic inst) Ts)
    | SOME T' => T')
  | typ_subst_nonatomic inst T = the_default T (AList.lookup (op =) inst T);

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

fun mk_tupled_fun x f xs = HOLogic.tupled_lambda x (Term.list_comb (f, xs));
fun mk_uncurried_fun f xs = mk_tupled_fun (HOLogic.mk_tuple xs) f xs;

fun flat_rec unzipf xs =
  let val ps = map unzipf xs in
    (* The first line below gives the preferred order. The second line is for compatibility with the
       old datatype package: *)
(*
    maps (op @) ps
*)
    maps fst ps @ maps snd ps
  end;

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

fun flat_preds_predsss_gettersss [] [qss] [fss] = flat_predss_getterss qss fss
  | flat_preds_predsss_gettersss (p :: ps) (qss :: qsss) (fss :: fsss) =
    p :: flat_predss_getterss qss fss @ flat_preds_predsss_gettersss ps qsss fsss;

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 lfp Ts Us t =
  let
    val (bindings, body) = strip_type (fastype_of t);
    val (f_Us, prebody) = split_last bindings;
    val Type (_, Ts0) = if lfp then prebody else body;
    val Us0 = distinct (op =) (map (if lfp then body_type else domain_type) f_Us);
  in
    Term.subst_atomic_types (Ts0 @ Us0 ~~ Ts @ Us) t
  end;

val mk_fp_iter_fun_types = fst o split_last o binder_types o fastype_of;

fun mk_fp_iter lfp As Cs fp_iters0 =
  map (mk_co_iter lfp As Cs) fp_iters0
  |> (fn ts => (ts, mk_fp_iter_fun_types (hd ts)));

fun unzip_recT fpTs T =
  let
    fun project_recT proj =
      let
        fun project (Type (s as @{type_name prod}, Ts as [T, U])) =
            if member (op =) fpTs T then proj (T, U) else Type (s, map project Ts)
          | project (Type (s, Ts)) = Type (s, map project Ts)
          | project T = T;
      in project end;
  in
    if exists_subtype_in fpTs T then ([project_recT fst T], [project_recT snd T]) else ([T], [])
  end;

val massage_rec_fun_arg_typesss = map o map o flat_rec o unzip_recT;

val mk_fold_fun_typess = map2 (map2 (curry (op --->)));
val mk_rec_fun_typess = mk_fold_fun_typess oo massage_rec_fun_arg_typesss;

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

fun mk_fold_rec_terms_types fpTs Css ns mss ctor_fold_fun_Ts ctor_rec_fun_Ts lthy =
  let
    val y_Tsss = map3 mk_fun_arg_typess ns mss ctor_fold_fun_Ts;
    val g_Tss = mk_fold_fun_typess y_Tsss Css;

    val ((gss, ysss), lthy) =
      lthy
      |> mk_Freess "f" g_Tss
      ||>> mk_Freesss "x" y_Tsss;

    val z_Tsss = map3 mk_fun_arg_typess ns mss ctor_rec_fun_Ts;
    val h_Tss = mk_rec_fun_typess fpTs z_Tsss Css;

    val hss = map2 (map2 retype_free) h_Tss gss;
    val zsss = map2 (map2 (map2 retype_free)) z_Tsss ysss;
  in
    (((gss, g_Tss, ysss), (hss, h_Tss, zsss)), lthy)
  end;

fun mk_unfold_corec_terms_types fpTs Cs ns mss dtor_unfold_fun_Ts dtor_corec_fun_Ts lthy =
  let
    (*avoid "'a itself" arguments in coiterators and corecursors*)
    fun repair_arity [0] = [1]
      | repair_arity ms = ms;

    fun project_corecT proj =
      let
        fun project (Type (s as @{type_name sum}, Ts as [T, U])) =
            if member (op =) fpTs T then proj (T, U) else Type (s, map project Ts)
          | project (Type (s, Ts)) = Type (s, map project Ts)
          | project T = T;
      in project end;

    fun unzip_corecT T =
      if exists_subtype_in fpTs T then [project_corecT fst T, project_corecT snd T] else [T];

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

    fun mk_types maybe_unzipT fun_Ts =
      let
        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 repair_arity) mss f_prod_Tss;
        val f_Tssss =
          map2 (fn C => map (map (map (curry (op -->) C) o maybe_unzipT))) Cs f_Tsss;
        val q_Tssss =
          map (map (map (fn [_] => [] | [_, C] => [mk_pred1T (domain_type C)]))) f_Tssss;
        val pf_Tss = map3 flat_preds_predsss_gettersss p_Tss q_Tssss f_Tssss;
      in (q_Tssss, f_sum_prod_Ts, f_Tsss, f_Tssss, pf_Tss) end;

    val (r_Tssss, g_sum_prod_Ts, g_Tsss, g_Tssss, pg_Tss) = mk_types single dtor_unfold_fun_Ts;
    val (s_Tssss, h_sum_prod_Ts, h_Tsss, h_Tssss, ph_Tss) = mk_types unzip_corecT dtor_corec_fun_Ts;

    val (((cs, pss), gssss), lthy) =
      lthy
      |> 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 mk_terms qssss fssss =
      let
        val pfss = map3 flat_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;
      in (pfss, cqssss, cfssss) end;

    val unfold_terms = mk_terms rssss gssss;
    val corec_terms = mk_terms sssss hssss;
  in
    ((cs, cpss, (unfold_terms, (g_sum_prod_Ts, g_Tsss, pg_Tss)),
      (corec_terms, (h_sum_prod_Ts, h_Tsss, ph_Tss))), lthy)
  end;

fun mk_map live Ts Us t =
  let val (Type (_, Ts0), Type (_, Us0)) = strip_typeN (live + 1) (fastype_of t) |>> List.last in
    Term.subst_atomic_types (Ts0 @ Us0 ~~ Ts @ Us) t
  end;

fun mk_rel live Ts Us t =
  let val [Type (_, Ts0), Type (_, Us0)] = binder_types (snd (strip_typeN live (fastype_of t))) in
    Term.subst_atomic_types (Ts0 @ Us0 ~~ Ts @ Us) t
  end;

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 indexify_fst xs f (x, y) = f (find_index (curry (op =) x) xs) (x, y);

fun build_map lthy build_simple =
  let
    fun build (TU as (T, U)) =
      if T = U then
        id_const T
      else
        (case TU of
          (Type (s, Ts), Type (s', Us)) =>
          if s = s' then
            let
              val bnf = the (bnf_of lthy s);
              val live = live_of_bnf bnf;
              val mapx = mk_map live Ts Us (map_of_bnf bnf);
              val TUs' = map dest_funT (fst (strip_typeN live (fastype_of mapx)));
            in Term.list_comb (mapx, map build TUs') end
          else
            build_simple TU
        | _ => build_simple TU);
  in build end;

fun derive_induct_fold_rec_thms_for_types pre_bnfs ctor_folds0 ctor_recs0 ctor_induct ctor_fold_thms
    ctor_rec_thms nesting_bnfs nested_bnfs fpTs Cs As ctrss ctr_defss folds recs fold_defs rec_defs
    lthy =
  let
    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 Css = map2 replicate ns Cs;

    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_ids'' = map (unfold_thms lthy @{thms id_def} o map_id_of_bnf) nesting_bnfs;
    val nested_map_comp's = map map_comp'_of_bnf nested_bnfs;
    val nested_map_ids'' = map (unfold_thms lthy @{thms id_def} o map_id_of_bnf) nested_bnfs;
    val nested_set_map's = maps set_map'_of_bnf nested_bnfs;

    val fp_b_names = map base_name_of_typ fpTs;

    val (_, ctor_fold_fun_Ts) = mk_fp_iter true As Cs ctor_folds0;
    val (_, ctor_rec_fun_Ts) = mk_fp_iter true As Cs ctor_recs0;

    val (((gss, _, _), (hss, _, _)), names_lthy0) =
      mk_fold_rec_terms_types fpTs Css ns mss ctor_fold_fun_Ts ctor_rec_fun_Ts lthy;

    val ((((ps, ps'), xsss), us'), names_lthy) =
      names_lthy0
      |> 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 names_lthy (x as Free (s, T as Type (T_name, Ts0))) =
            (case find_index (curry (op =) T) fpTs of
              ~1 =>
              (case AList.lookup (op =) setss_nested T_name of
                NONE => []
              | SOME raw_sets0 =>
                let
                  val (Ts, raw_sets) =
                    split_list (filter (exists_subtype_in fpTs o fst) (Ts0 ~~ raw_sets0));
                  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 = map (mk_raw_prem_prems names_lthy') ys;
                in
                  flat (map2 (map o apfst o cons) xysets ppremss)
                end)
            | kk => [([], (kk + 1, x))])
          | 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 =
          let
            val (xs, names_lthy') = names_lthy |> mk_Frees "x" ctr_Ts;
            val pprems = maps (mk_raw_prem_prems names_lthy') xs;
          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 = map3 (map2 o mk_raw_prem) ps ctrss ctr_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_map's
              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 (fold_thmss, rec_thmss) =
      let
        val xctrss = map2 (map2 (curry Term.list_comb)) ctrss xsss;
        val gfolds = map (lists_bmoc gss) folds;
        val hrecs = map (lists_bmoc hss) recs;

        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)));

        val mk_U = typ_subst_nonatomic (map2 pair fpTs Cs);

        fun unzip_iters fiters combine (x as Free (_, T)) =
          if exists_subtype_in fpTs T then
            combine (x, build_map lthy (indexify_fst fpTs (K o nth fiters)) (T, mk_U T) $ x)
          else
            ([x], []);

        val gxsss = map (map (flat_rec (unzip_iters gfolds (fn (_, t) => ([t], []))))) xsss;
        val hxsss = map (map (flat_rec (unzip_iters hrecs (pairself single)))) xsss;

        val fold_goalss = map5 (map4 o mk_goal gss) gfolds xctrss gss xsss gxsss;
        val rec_goalss = map5 (map4 o mk_goal hss) hrecs xctrss hss xsss hxsss;

        val fold_tacss =
          map2 (map o mk_iter_tac pre_map_defs [] nesting_map_ids'' fold_defs) ctor_fold_thms
            ctr_defss;
        val rec_tacss =
          map2 (map o mk_iter_tac pre_map_defs nested_map_comp's
            (nested_map_ids'' @ nesting_map_ids'') rec_defs) ctor_rec_thms ctr_defss;

        fun prove goal tac =
          Goal.prove_sorry lthy [] [] goal (tac o #context)
          |> Thm.close_derivation;
      in
        (map2 (map2 prove) fold_goalss fold_tacss, map2 (map2 prove) rec_goalss rec_tacss)
      end;

    val induct_case_names_attr = Attrib.internal (K (Rule_Cases.case_names induct_cases));
  in
    ((induct_thm, induct_thms, [induct_case_names_attr]),
     (fold_thmss, code_simp_attrs), (rec_thmss, code_simp_attrs))
  end;

fun derive_coinduct_unfold_corec_thms_for_types pre_bnfs dtor_unfolds0 dtor_corecs0 dtor_coinduct
    dtor_strong_induct dtor_ctors dtor_unfold_thms dtor_corec_thms nesting_bnfs nested_bnfs fpTs Cs
    As kss mss ns ctr_defss ctr_sugars unfolds corecs unfold_defs corec_defs lthy =
  let
    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_ids'' = map (unfold_thms lthy @{thms id_def} o map_id_of_bnf) nesting_bnfs;
    val nesting_rel_eqs = map rel_eq_of_bnf nesting_bnfs;
    val nested_map_comp's = map map_comp'_of_bnf nested_bnfs;
    val nested_map_comps'' = map ((fn thm => thm RS sym) o map_comp_of_bnf) nested_bnfs;
    val nested_map_ids'' = map (unfold_thms lthy @{thms id_def} o map_id_of_bnf) nested_bnfs;

    val fp_b_names = map base_name_of_typ fpTs;

    val (_, dtor_unfold_fun_Ts) = mk_fp_iter false As Cs dtor_unfolds0;
    val (_, dtor_corec_fun_Ts) = mk_fp_iter false As Cs dtor_corecs0;

    val ctrss = map (map (mk_ctr As) o #ctrs) ctr_sugars;
    val discss = map (map (mk_disc_or_sel As) o #discs) ctr_sugars;
    val selsss = map (map (map (mk_disc_or_sel As)) o #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 ((cs, cpss, ((pgss, crssss, cgssss), _), ((phss, csssss, chssss), _)), names_lthy0) =
      mk_unfold_corec_terms_types fpTs Cs ns mss dtor_unfold_fun_Ts dtor_corec_fun_Ts lthy;

    val (((rs, us'), vs'), names_lthy) =
      names_lthy0
      |> 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, coinduct_thm), (strong_coinduct_thms, strong_coinduct_thm)) =
      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_rel rs' T =
          (case find_index (curry (op =) T) fpTs of
            ~1 =>
            if exists_subtype_in fpTs T then
              let
                val Type (s, Ts) = T
                val bnf = the (bnf_of lthy s);
                val live = live_of_bnf bnf;
                val rel = mk_rel live Ts Ts (rel_of_bnf bnf);
                val Ts' = map domain_type (fst (strip_typeN live (fastype_of rel)));
              in Term.list_comb (rel, map (build_rel rs') Ts') end
            else
              HOLogic.eq_const T
          | kk => nth rs' kk);

        fun build_rel_app rs' usel vsel = fold rapp [usel, vsel] (build_rel rs' (fastype_of usel));

        fun mk_prem_ctr_concls rs' n k udisc usels vdisc vsels =
          (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 (map2 (build_rel_app rs') usels vsels))]);

        fun mk_prem_concl rs' n udiscs uselss vdiscs vselss =
          Library.foldr1 HOLogic.mk_conj
            (flat (map5 (mk_prem_ctr_concls rs' n) (1 upto n) udiscs uselss vdiscs vselss))
          handle List.Empty => @{term True};

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

        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 (map8 (mk_prem rs') uvrs us vs ns udiscss uselsss vdiscss vselsss,
            concl);

        val goal = mk_goal rs;
        val strong_goal = mk_goal 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);
      in
        (postproc nn (prove dtor_coinduct goal), postproc nn (prove dtor_strong_induct strong_goal))
      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 gunfolds = map (lists_bmoc pgss) unfolds;
    val hcorecs = map (lists_bmoc phss) corecs;

    val (unfold_thmss, corec_thmss, safe_unfold_thmss, safe_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'))));

        val mk_U = typ_subst_nonatomic (map2 pair Cs fpTs);

        fun intr_coiters fcoiters [] [cf] =
            let val T = fastype_of cf in
              if exists_subtype_in Cs T then
                build_map lthy (indexify_fst Cs (K o nth fcoiters)) (T, mk_U T) $ cf
              else
                cf
            end
          | intr_coiters fcoiters [cq] [cf, cf'] =
            mk_If cq (intr_coiters fcoiters [] [cf]) (intr_coiters fcoiters [] [cf']);

        val crgsss = map2 (map2 (map2 (intr_coiters gunfolds))) crssss cgssss;
        val cshsss = map2 (map2 (map2 (intr_coiters hcorecs))) csssss chssss;

        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;

        fun mk_map_if_distrib bnf =
          let
            val mapx = map_of_bnf bnf;
            val live = live_of_bnf bnf;
            val ((Ts, T), U) = strip_typeN (live + 1) (fastype_of mapx) |>> split_last;
            val fs = Variable.variant_frees lthy [mapx] (map (pair "f") Ts);
            val t = Term.list_comb (mapx, map (Var o apfst (rpair 0)) fs);
          in
            Drule.instantiate' (map (SOME o certifyT lthy) [U, T]) [SOME (certify lthy t)]
              @{thm if_distrib}
          end;

        val nested_map_if_distribs = map mk_map_if_distrib nested_bnfs;

        val unfold_tacss =
          map3 (map oo mk_coiter_tac unfold_defs [] [] nesting_map_ids'' [])
            dtor_unfold_thms pre_map_defs ctr_defss;
        val corec_tacss =
          map3 (map oo mk_coiter_tac corec_defs nested_map_comps'' nested_map_comp's
              (nested_map_ids'' @ nesting_map_ids'') nested_map_if_distribs)
            dtor_corec_thms 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;

        val filter_safesss =
          map2 (map_filter (fn (safes, thm) => if forall I safes then SOME thm else NONE) oo
            curry (op ~~)) (map2 (map2 (map2 (member (op =)))) cgssss crgsss);

        val safe_unfold_thmss = filter_safesss unfold_thmss;
        val safe_corec_thmss = filter_safesss corec_thmss;
      in
        (unfold_thmss, corec_thmss, safe_unfold_thmss, safe_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 (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;

    val is_triv_discI = is_triv_implies orf is_concl_refl;

    fun mk_disc_coiter_thms coiters discIs =
      map (op RS) (filter_out (is_triv_discI o snd) (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 coiterss =
      map3 (map3 (map2 o mk_sel_coiter_thm)) coiterss selsss sel_thmsss |> map flat;

    val sel_unfold_thmss = mk_sel_coiter_thms unfold_thmss;
    val sel_corec_thmss = 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_thm, coinduct_thms, strong_coinduct_thm, strong_coinduct_thms, coinduct_case_attrs),
     (unfold_thmss, corec_thmss, []),
     (safe_unfold_thmss, safe_corec_thmss),
     (disc_unfold_thmss, disc_corec_thmss, simp_attrs),
     (disc_unfold_iff_thmss, disc_corec_iff_thmss, simp_attrs),
     (sel_unfold_thmss, sel_corec_thmss, simp_attrs))
  end;

fun define_co_datatypes prepare_constraint prepare_typ prepare_term lfp construct_fp
    (wrap_opts as (no_dests, rep_compat), specs) no_defs_lthy0 =
  let
    (* TODO: sanity checks on arguments *)

    val _ = if not lfp andalso no_dests then error "Cannot define destructor-less codatatypes"
      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 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 (length unsorted_As)
      ||>> mk_TFrees nn
      ||>> variant_tfrees fp_b_names;

    (* TODO: cleaner handling of fake contexts, without "background_theory" *)
    (*the "perhaps o try" below helps gracefully handles the case where the new type is defined in a
      locale and shadows an existing global type*)

    fun add_fake_type spec =
      Sign.add_type no_defs_lthy (type_binding_of spec,
        length (type_args_named_constrained_of spec), mixfix_of spec);

    val fake_thy = Theory.copy #> fold add_fake_type specs;
    val fake_lthy = Proof_Context.background_theory fake_thy no_defs_lthy;

    fun mk_fake_T b =
      Type (fst (Term.dest_Type (Proof_Context.read_type_name fake_lthy true (Binding.name_of b))),
        unsorted_As);

    val fake_Ts = map mk_fake_T fp_bs;

    val mixfixes = map mixfix_of specs;

    val _ = (case 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 _ = (case duplicates (op =) unsorted_As of [] => ()
      | A :: _ => error ("Duplicate type parameter " ^
          quote (Syntax.string_of_typ no_defs_lthy A)));

    val rhs_As' = fold (fold (fold Term.add_tfreesT)) fake_ctr_Tsss [];
    val _ = (case subtract (op =) (map dest_TFree As) rhs_As' of
        [] => ()
      | A' :: _ => error ("Extra type variable on right-hand side: " ^
          quote (Syntax.string_of_typ no_defs_lthy (TFree A'))));

    fun eq_fpT_check (T as Type (s, Us)) (Type (s', Us')) =
        s = s' andalso (Us = Us' orelse error ("Illegal occurrence of recursive type " ^
          quote (Syntax.string_of_typ fake_lthy T)))
      | eq_fpT_check _ _ = false;

    fun freeze_fp (T as Type (s, Us)) =
        (case find_index (eq_fpT_check T) fake_Ts of
          ~1 => Type (s, map freeze_fp Us)
        | kk => nth Xs kk)
      | freeze_fp T = T;

    val ctr_TsssXs = map (map (map freeze_fp)) fake_ctr_Tsss;
    val ctr_sum_prod_TsXs = map (mk_sumTN_balanced o map HOLogic.mk_tupleT) ctr_TsssXs;

    val fp_eqs =
      map dest_TFree Xs ~~ map (Term.typ_subst_atomic (As ~~ unsorted_As)) ctr_sum_prod_TsXs;

    val (pre_bnfs, (fp_res as {bnfs = fp_bnfs as any_fp_bnf :: _, ctors = ctors0, dtors = dtors0,
           un_folds = fp_folds0, co_recs = fp_recs0, co_induct = fp_induct,
           strong_co_induct = fp_strong_induct, dtor_ctors, ctor_dtors, ctor_injects,
           map_thms = fp_map_thms, set_thmss = fp_set_thmss, rel_thms = fp_rel_thms,
           un_fold_thms = fp_fold_thms, co_rec_thms = fp_rec_thms, ...}, lthy)) =
      fp_bnf (construct_fp mixfixes map_bs rel_bs set_bss) fp_bs (map dest_TFree unsorted_As) fp_eqs
        no_defs_lthy0;

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

    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 Us =
      map_filter (bnf_of lthy) (fold (fold (fold (add_nesty_bnf_names Us))) ctr_TsssXs []);

    val nesting_bnfs = nesty_bnfs As;
    val nested_bnfs = nesty_bnfs 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_map's = maps set_map'_of_bnf nesting_bnfs;
    val nested_set_map's = maps set_map'_of_bnf nested_bnfs;

    val live = live_of_bnf any_fp_bnf;

    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 (length 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) =>
        if forall null thmss then
          []
        else
          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);

    val ctr_Tsss = map (map (map (Term.typ_subst_atomic (Xs ~~ fpTs)))) ctr_TsssXs;
    val ns = map length ctr_Tsss;
    val kss = map (fn n => 1 upto n) ns;
    val mss = map (map length) ctr_Tsss;
    val Css = map2 replicate ns Cs;

    val (fp_folds, fp_fold_fun_Ts) = mk_fp_iter lfp As Cs fp_folds0;
    val (fp_recs, fp_rec_fun_Ts) = mk_fp_iter lfp As Cs fp_recs0;

    val (((fold_only, rec_only),
          (cs, cpss, unfold_only as ((_, crssss, cgssss), (_, g_Tsss, _)),
           corec_only as ((_, csssss, chssss), (_, h_Tsss, _)))), _) =
      if lfp then
        mk_fold_rec_terms_types fpTs Css ns mss fp_fold_fun_Ts fp_rec_fun_Ts lthy
        |>> rpair ([], [], (([], [], []), ([], [], [])), (([], [], []), ([], [], [])))
      else
        mk_unfold_corec_terms_types fpTs Cs ns mss fp_fold_fun_Ts fp_rec_fun_Ts lthy
        |>> pair (([], [], []), ([], [], []));

    fun define_ctrs_case_for_type (((((((((((((((((((((((((fp_bnf, fp_b), fpT), C), ctor), dtor),
            fp_fold), fp_rec), 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 case_Ts = map (fn Ts => Ts ---> C) ctr_Tss;

        val (((((w, fs), xss), yss), u'), names_lthy) =
          no_defs_lthy
          |> yield_singleton (mk_Frees "w") dtorT
          ||>> mk_Frees "f" case_Ts
          ||>> 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 case_binding = qualify false fp_b_name (Binding.suffix_name ("_" ^ caseN) fp_b);

        val case_rhs =
          fold_rev Term.lambda (fs @ [u])
            (mk_sum_caseN_balanced (map2 mk_uncurried_fun fs xss) $ (dtor $ u));

        val ((raw_case :: raw_ctrs, raw_case_def :: 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), ((Thm.def_binding b, []), rhs)) #>> apsnd snd)
            (case_binding :: ctr_bindings) (NoSyn :: ctr_mixfixes) (case_rhs :: 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 case_def = Morphism.thm phi raw_case_def;

        val ctrs0 = map (Morphism.term phi) raw_ctrs;
        val casex0 = Morphism.term phi raw_case;

        val ctrs = map (mk_ctr As) ctrs0;

        fun wrap 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 all_unit_eq}
                    (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 case_tacs =
              map3 (fn k => fn m => fn ctr_def => fn {context = ctxt, ...} =>
                mk_case_tac ctxt n k m case_def ctr_def dtor_ctor) ks ms ctr_defs;

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

            val sel_defaultss = map (map (apsnd (prepare_term lthy))) raw_sel_defaultss
          in
            wrap_free_constructors tacss (((wrap_opts, ctrs0), casex0), (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 lfp 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 (not lfp ? 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 lfp then [] else [ctor_dtor, dtor_ctor]) @ sum_prod_thms_map)
                     (cterm_instantiate_pos (nones @ [SOME cxIn])
                        (if lfp 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_map's @ nesting_set_map's @
                       (if lfp 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 lfp 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 notes =
                [(mapN, map_thms, code_simp_attrs),
                 (rel_distinctN, rel_distinct_thms, code_simp_attrs),
                 (rel_injectN, rel_inject_thms, code_simp_attrs),
                 (setsN, flat set_thmss, code_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 notes |> snd)
            end;

        fun define_fold_rec no_defs_lthy =
          let
            val fpT_to_C = fpT --> C;

            fun build_prod_proj mk_proj = build_map lthy (mk_proj o fst);

            (* TODO: Avoid these complications; cf. corec case *)
            fun mk_U proj (Type (s as @{type_name prod}, Ts as [T', U])) =
                if member (op =) fpTs T' then proj (T', U) else Type (s, map (mk_U proj) Ts)
              | mk_U proj (Type (s, Ts)) = Type (s, map (mk_U proj) Ts)
              | mk_U _ T = T;

            fun unzip_rec (x as Free (_, T)) =
              if exists_subtype_in fpTs T then
                ([build_prod_proj fst_const (T, mk_U fst T) $ x],
                 [build_prod_proj snd_const (T, mk_U snd T) $ x])
              else
                ([x], []);

            fun mk_iter_arg f xs =
              mk_tupled_fun (HOLogic.mk_tuple xs) f (flat_rec unzip_rec xs);

            fun generate_iter (suf, ctor_iter, (fss, f_Tss, xsss)) =
              let
                val res_T = fold_rev (curry (op --->)) f_Tss fpT_to_C;
                val binding = qualify false fp_b_name (Binding.suffix_name ("_" ^ suf) fp_b);
                val spec =
                  mk_Trueprop_eq (lists_bmoc fss (Free (Binding.name_of binding, res_T)),
                    Term.list_comb (ctor_iter,
                      map2 (mk_sum_caseN_balanced oo map2 mk_iter_arg) fss xsss));
              in (binding, spec) end;

            val iter_infos =
              [(foldN, fp_fold, fold_only),
               (recN, fp_rec, rec_only)];

            val (bindings, specs) = map generate_iter iter_infos |> split_list;

            val ((csts, defs), (lthy', lthy)) = no_defs_lthy
              |> apfst split_list o fold_map2 (fn b => fn spec =>
                Specification.definition (SOME (b, NONE, NoSyn), ((Thm.def_binding b, []), spec))
                #>> apsnd snd) bindings specs
              ||> `Local_Theory.restore;

            val phi = Proof_Context.export_morphism lthy lthy';

            val [fold_def, rec_def] = map (Morphism.thm phi) defs;

            val [foldx, recx] = map (mk_co_iter lfp As Cs o Morphism.term phi) csts;
          in
            ((foldx, recx, fold_def, rec_def), lthy')
          end;

        fun define_unfold_corec no_defs_lthy =
          let
            val B_to_fpT = C --> fpT;

            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')

            val crgsss = map3 (map3 (map3 build_dtor_coiter_arg)) g_Tsss crssss cgssss;
            val cshsss = map3 (map3 (map3 build_dtor_coiter_arg)) h_Tsss csssss chssss;

            fun mk_preds_getterss_join c n cps sum_prod_T cqfss =
              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)));

            fun generate_coiter (suf, dtor_coiter, (cqfsss, ((pfss, _, _),
                (f_sum_prod_Ts, _, pf_Tss)))) =
              let
                val res_T = fold_rev (curry (op --->)) pf_Tss B_to_fpT;
                val binding = qualify false fp_b_name (Binding.suffix_name ("_" ^ suf) fp_b);
                val spec =
                  mk_Trueprop_eq (lists_bmoc pfss (Free (Binding.name_of binding, res_T)),
                    Term.list_comb (dtor_coiter,
                      map5 mk_preds_getterss_join cs ns cpss f_sum_prod_Ts cqfsss));
              in (binding, spec) end;

            val coiter_infos =
              [(unfoldN, fp_fold, (crgsss, unfold_only)),
               (corecN, fp_rec, (cshsss, corec_only))];

            val (bindings, specs) = map generate_coiter coiter_infos |> split_list;

            val ((csts, defs), (lthy', lthy)) = no_defs_lthy
              |> apfst split_list o fold_map2 (fn b => fn spec =>
                Specification.definition (SOME (b, NONE, NoSyn), ((Thm.def_binding b, []), spec))
                #>> apsnd snd) bindings specs
              ||> `Local_Theory.restore;

            val phi = Proof_Context.export_morphism lthy lthy';

            val [unfold_def, corec_def] = map (Morphism.thm phi) defs;

            val [unfold, corec] = map (mk_co_iter lfp As Cs o Morphism.term phi) csts;
          in
            ((unfold, corec, unfold_def, corec_def), lthy')
          end;

        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
         #> derive_maps_sets_rels
         ##>> (if lfp then define_fold_rec else define_unfold_corec)
         #> massage_res, lthy')
      end;

    fun wrap_types_etc (wrap_types_etcs, lthy) =
      fold_map I wrap_types_etcs lthy
      |>> apsnd split_list4 o apfst (apsnd split_list4 o apfst split_list4 o split_list)
        o split_list;

    val mk_simp_thmss =
      map7 (fn {injects, distincts, case_thms, ...} => fn un_folds => fn co_recs =>
        fn mapsx => fn rel_injects => fn rel_distincts => fn setss =>
          injects @ distincts @ case_thms @ co_recs @ un_folds @ mapsx @ rel_injects
          @ rel_distincts @ flat setss);

    fun derive_and_note_induct_fold_rec_thms_for_types
        ((((mapsx, rel_injects, rel_distincts, setss), (ctrss, _, ctr_defss, ctr_sugars)),
          (folds, recs, fold_defs, rec_defs)), lthy) =
      let
        val ((induct_thm, induct_thms, induct_attrs), (fold_thmss, fold_attrs),
             (rec_thmss, rec_attrs)) =
          derive_induct_fold_rec_thms_for_types pre_bnfs fp_folds0 fp_recs0 fp_induct fp_fold_thms
            fp_rec_thms nesting_bnfs nested_bnfs fpTs Cs As ctrss ctr_defss folds recs fold_defs
            rec_defs lthy;

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

        val simp_thmss =
          mk_simp_thmss ctr_sugars fold_thmss rec_thmss mapsx 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 fold_attrs),
           (inductN, map single induct_thms, fn T_name => induct_attrs @ [induct_type_attr T_name]),
           (recN, rec_thmss, K rec_attrs),
           (simpsN, simp_thmss, K [])]
          |> massage_multi_notes;
      in
        lthy
        |> Local_Theory.notes (common_notes @ notes) |> snd
        |> register_fp_sugars true pre_bnfs fp_res ctr_sugars folds recs induct_thm induct_thm
          fold_thmss rec_thmss
      end;

    fun derive_and_note_coinduct_unfold_corec_thms_for_types
        ((((mapsx, rel_injects, rel_distincts, setss), (_, _, ctr_defss, ctr_sugars)),
          (unfolds, corecs, unfold_defs, corec_defs)), lthy) =
      let
        val ((coinduct_thm, coinduct_thms, strong_coinduct_thm, strong_coinduct_thms,
              coinduct_attrs),
             (unfold_thmss, corec_thmss, coiter_attrs),
             (safe_unfold_thmss, safe_corec_thmss),
             (disc_unfold_thmss, disc_corec_thmss, disc_coiter_attrs),
             (disc_unfold_iff_thmss, disc_corec_iff_thmss, disc_coiter_iff_attrs),
             (sel_unfold_thmss, sel_corec_thmss, sel_coiter_attrs)) =
          derive_coinduct_unfold_corec_thms_for_types pre_bnfs fp_folds0 fp_recs0 fp_induct
            fp_strong_induct dtor_ctors fp_fold_thms fp_rec_thms nesting_bnfs nested_bnfs fpTs Cs As
            kss mss ns ctr_defss ctr_sugars unfolds corecs unfold_defs corec_defs lthy;

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

        fun flat_coiter_thms coiters disc_coiters sel_coiters =
          coiters @ disc_coiters @ sel_coiters;

        val simp_thmss =
          mk_simp_thmss ctr_sugars
            (map3 flat_coiter_thms safe_unfold_thmss disc_unfold_thmss sel_unfold_thmss)
            (map3 flat_coiter_thms safe_corec_thmss disc_corec_thmss sel_corec_thmss)
            mapsx rel_injects rel_distincts setss;

        val anonymous_notes =
          [(flat safe_unfold_thmss @ flat safe_corec_thmss, simp_attrs)]
          |> map (fn (thms, attrs) => ((Binding.empty, attrs), [(thms, [])]));

        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;
      in
        lthy
        |> Local_Theory.notes (anonymous_notes @ common_notes @ notes) |> snd
        |> register_fp_sugars false pre_bnfs fp_res ctr_sugars unfolds corecs coinduct_thm
          strong_coinduct_thm unfold_thmss corec_thmss
      end;

    val lthy' = lthy
      |> fold_map define_ctrs_case_for_type (fp_bnfs ~~ fp_bs ~~ fpTs ~~ Cs ~~ ctors ~~ dtors ~~
        fp_folds ~~ fp_recs ~~ ctor_dtors ~~ dtor_ctors ~~ ctor_injects ~~ pre_map_defs ~~
        pre_set_defss ~~ pre_rel_defs ~~ fp_map_thms ~~ fp_set_thmss ~~ fp_rel_thms ~~ ns ~~ kss ~~
        mss ~~ ctr_bindingss ~~ ctr_mixfixess ~~ ctr_Tsss ~~ disc_bindingss ~~ sel_bindingsss ~~
        raw_sel_defaultsss)
      |> wrap_types_etc
      |> (if lfp then derive_and_note_induct_fold_rec_thms_for_types
          else derive_and_note_coinduct_unfold_corec_thms_for_types);

    val timer = time (timer ("Constructors, discriminators, selectors, etc., for the new " ^
      datatype_word lfp));
  in
    timer; lthy'
  end;

val co_datatypes = define_co_datatypes (K I) (K I) (K I);

val co_datatype_cmd =
  define_co_datatypes Typedecl.read_constraint Syntax.parse_typ Syntax.parse_term;

val parse_ctr_arg =
  @{keyword "("} |-- parse_binding_colon -- Parse.typ --| @{keyword ")"} ||
  (Parse.typ >> pair Binding.empty);

val parse_defaults =
  @{keyword "("} |-- @{keyword "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);

(* "map" and "rel" are purposedly not registered as keywords, because they are short and nice names
   that we don't want them to be highlighted everywhere. *)
fun extract_map_rel ("map", b) = apfst (K b)
  | extract_map_rel ("rel", b) = apsnd (K b)
  | extract_map_rel (s, _) = error ("Expected \"map\" or \"rel\" instead of " ^ quote s);

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_options -- Parse.and_list1 parse_spec;

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

end;