src/HOL/BNF/Tools/bnf_fp_def_sugar.ML
author blanchet
Tue Aug 06 15:50:23 2013 +0200 (2013-08-06)
changeset 52872 fd14b0ead643
parent 52871 94ab1f8151e2
child 52898 2af1caada147
permissions -rw-r--r--
tuning
     1 (*  Title:      HOL/BNF/Tools/bnf_fp_def_sugar.ML
     2     Author:     Jasmin Blanchette, TU Muenchen
     3     Copyright   2012, 2013
     4 
     5 Sugared datatype and codatatype constructions.
     6 *)
     7 
     8 signature BNF_FP_DEF_SUGAR =
     9 sig
    10   type fp_sugar =
    11     {T: typ,
    12      fp: BNF_FP_Util.fp_kind,
    13      index: int,
    14      pre_bnfs: BNF_Def.bnf list,
    15      nested_bnfs: BNF_Def.bnf list,
    16      fp_res: BNF_FP_Util.fp_result,
    17      ctr_defss: thm list list,
    18      ctr_sugars: BNF_Ctr_Sugar.ctr_sugar list,
    19      co_iterss: term list list,
    20      co_inducts: thm list,
    21      co_iter_thmsss: thm list list list};
    22 
    23   val of_fp_sugar: (fp_sugar -> 'a list) -> fp_sugar -> 'a
    24   val morph_fp_sugar: morphism -> fp_sugar -> fp_sugar
    25   val fp_sugar_of: Proof.context -> string -> fp_sugar option
    26 
    27   val tvar_subst: theory -> typ list -> typ list -> ((string * int) * typ) list
    28   val exists_subtype_in: typ list -> typ -> bool
    29   val flat_rec_arg_args: 'a list list -> 'a list
    30   val flat_corec_preds_predsss_gettersss: 'a list -> 'a list list list -> 'a list list list ->
    31     'a list
    32   val mk_co_iter: theory -> BNF_FP_Util.fp_kind -> typ -> typ list -> term -> term
    33   val nesty_bnfs: Proof.context -> typ list list list -> typ list -> BNF_Def.bnf list
    34   val mk_map: int -> typ list -> typ list -> term -> term
    35   val build_map: local_theory -> (typ * typ -> term) -> typ * typ -> term
    36   val mk_co_iters_prelims: BNF_FP_Util.fp_kind -> typ list -> typ list -> int list ->
    37     int list list -> term list list -> Proof.context ->
    38     (term list list
    39      * (typ list list * typ list list list list * term list list
    40         * term list list list list) list option
    41      * (string * term list * term list list
    42         * ((term list list * term list list list) * (typ list * typ list list)) list) option)
    43     * Proof.context
    44 
    45   val mk_iter_fun_arg_types: typ list -> int list -> int list list -> term ->
    46     typ list list list list
    47   val mk_coiter_fun_arg_types: typ list -> int list -> int list list -> term ->
    48     typ list list list list * typ list list list * typ list list list list * typ list
    49   val define_iters: string list ->
    50     (typ list list * typ list list list list * term list list * term list list list list) list ->
    51     (string -> binding) -> typ list -> typ list -> term list -> Proof.context ->
    52     (term list * thm list) * Proof.context
    53   val define_coiters: string list -> string * term list * term list list
    54     * ((term list list * term list list list) * (typ list * typ list list)) list ->
    55     (string -> binding) -> typ list -> typ list -> term list -> Proof.context ->
    56     (term list * thm list) * Proof.context
    57   val derive_induct_iters_thms_for_types: BNF_Def.bnf list ->
    58     (typ list list * typ list list list list * term list list * term list list list list) list ->
    59     thm list -> thm list list -> BNF_Def.bnf list -> BNF_Def.bnf list -> typ list -> typ list ->
    60     typ list -> typ list list list -> term list list -> thm list list -> term list list ->
    61     thm list list -> local_theory ->
    62     (thm list * thm * Args.src list) * (thm list list * Args.src list)
    63     * (thm list list * Args.src list)
    64   val derive_coinduct_coiters_thms_for_types: BNF_Def.bnf list ->
    65     string * term list * term list list * ((term list list * term list list list)
    66       * (typ list * typ list list)) list ->
    67     thm list -> thm list -> thm list list -> BNF_Def.bnf list -> BNF_Def.bnf list -> typ list ->
    68     typ list -> typ list -> int list list -> int list list -> int list -> thm list list ->
    69     BNF_Ctr_Sugar.ctr_sugar list -> term list list -> thm list list -> (thm list -> thm list) ->
    70     local_theory ->
    71     ((thm list * thm) list * Args.src list)
    72     * (thm list list * thm list list * Args.src list)
    73     * (thm list list * thm list list) * (thm list list * thm list list * Args.src list)
    74     * (thm list list * thm list list * Args.src list)
    75     * (thm list list * thm list list * Args.src list)
    76 
    77   val co_datatypes: BNF_FP_Util.fp_kind -> (mixfix list -> binding list -> binding list ->
    78       binding list list -> binding list -> (string * sort) list -> typ list * typ list list ->
    79       BNF_Def.bnf list -> local_theory -> BNF_FP_Util.fp_result * local_theory) ->
    80     (bool * bool) * (((((binding * (typ * sort)) list * binding) * (binding * binding)) * mixfix) *
    81       ((((binding * binding) * (binding * typ) list) * (binding * term) list) *
    82         mixfix) list) list ->
    83     local_theory -> local_theory
    84   val parse_co_datatype_cmd: BNF_FP_Util.fp_kind -> (mixfix list -> binding list -> binding list ->
    85       binding list list -> binding list -> (string * sort) list -> typ list * typ list list ->
    86       BNF_Def.bnf list -> local_theory -> BNF_FP_Util.fp_result * local_theory) ->
    87     (local_theory -> local_theory) parser
    88 end;
    89 
    90 structure BNF_FP_Def_Sugar : BNF_FP_DEF_SUGAR =
    91 struct
    92 
    93 open BNF_Util
    94 open BNF_Ctr_Sugar
    95 open BNF_Def
    96 open BNF_FP_Util
    97 open BNF_FP_Def_Sugar_Tactics
    98 
    99 val EqN = "Eq_";
   100 
   101 type fp_sugar =
   102   {T: typ,
   103    fp: fp_kind,
   104    index: int,
   105    pre_bnfs: bnf list,
   106    nested_bnfs: bnf list,
   107    fp_res: fp_result,
   108    ctr_defss: thm list list,
   109    ctr_sugars: ctr_sugar list,
   110    co_iterss: term list list,
   111    co_inducts: thm list,
   112    co_iter_thmsss: thm list list list};
   113 
   114 fun of_fp_sugar f (fp_sugar as {index, ...}) = nth (f fp_sugar) index;
   115 
   116 fun eq_fp_sugar ({T = T1, fp = fp1, index = index1, fp_res = fp_res1, ...} : fp_sugar,
   117     {T = T2, fp = fp2, index = index2, fp_res = fp_res2, ...} : fp_sugar) =
   118   T1 = T2 andalso fp1 = fp2 andalso index1 = index2 andalso eq_fp_result (fp_res1, fp_res2);
   119 
   120 fun morph_fp_sugar phi {T, fp, index, pre_bnfs, nested_bnfs, fp_res, ctr_defss, ctr_sugars,
   121     co_iterss, co_inducts, co_iter_thmsss} =
   122   {T = Morphism.typ phi T, fp = fp, index = index, pre_bnfs = map (morph_bnf phi) pre_bnfs,
   123     nested_bnfs = map (morph_bnf phi) nested_bnfs, fp_res = morph_fp_result phi fp_res,
   124    ctr_defss = map (map (Morphism.thm phi)) ctr_defss,
   125    ctr_sugars = map (morph_ctr_sugar phi) ctr_sugars,
   126    co_iterss = map (map (Morphism.term phi)) co_iterss,
   127    co_inducts = map (Morphism.thm phi) co_inducts,
   128    co_iter_thmsss = map (map (map (Morphism.thm phi))) co_iter_thmsss};
   129 
   130 structure Data = Generic_Data
   131 (
   132   type T = fp_sugar Symtab.table;
   133   val empty = Symtab.empty;
   134   val extend = I;
   135   val merge = Symtab.merge eq_fp_sugar;
   136 );
   137 
   138 val fp_sugar_of = Symtab.lookup o Data.get o Context.Proof;
   139 
   140 fun register_fp_sugar key fp_sugar =
   141   Local_Theory.declaration {syntax = false, pervasive = true}
   142     (fn phi => Data.map (Symtab.default (key, morph_fp_sugar phi fp_sugar)));
   143 
   144 fun register_fp_sugars fp pre_bnfs nested_bnfs (fp_res as {Ts, ...}) ctr_defss ctr_sugars co_iterss
   145     co_inducts co_iter_thmsss lthy =
   146   (0, lthy)
   147   |> fold (fn T as Type (s, _) => fn (kk, lthy) => (kk + 1,
   148     register_fp_sugar s {T = T, fp = fp, index = kk, pre_bnfs = pre_bnfs,
   149         nested_bnfs = nested_bnfs, fp_res = fp_res, ctr_defss = ctr_defss, ctr_sugars = ctr_sugars,
   150         co_iterss = co_iterss, co_inducts = co_inducts, co_iter_thmsss = co_iter_thmsss}
   151       lthy)) Ts
   152   |> snd;
   153 
   154 (* This function could produce clashes in contrived examples (e.g., "x.A", "x.x_A", "y.A"). *)
   155 fun quasi_unambiguous_case_names names =
   156   let
   157     val ps = map (`Long_Name.base_name) names;
   158     val dups = Library.duplicates (op =) (map fst ps);
   159     fun underscore s =
   160       let val ss = space_explode Long_Name.separator s in
   161         space_implode "_" (drop (length ss - 2) ss)
   162       end;
   163   in
   164     map (fn (base, full) => if member (op =) dups base then underscore full else base) ps
   165   end;
   166 
   167 val id_def = @{thm id_def};
   168 val mp_conj = @{thm mp_conj};
   169 
   170 val simp_attrs = @{attributes [simp]};
   171 val code_simp_attrs = Code.add_default_eqn_attrib :: simp_attrs;
   172 
   173 fun tvar_subst thy Ts Us =
   174   Vartab.fold (cons o apsnd snd) (fold (Sign.typ_match thy) (Ts ~~ Us) Vartab.empty) [];
   175 
   176 val exists_subtype_in = Term.exists_subtype o member (op =);
   177 
   178 fun resort_tfree S (TFree (s, _)) = TFree (s, S);
   179 
   180 fun typ_subst_nonatomic inst (T as Type (s, Ts)) =
   181     (case AList.lookup (op =) inst T of
   182       NONE => Type (s, map (typ_subst_nonatomic inst) Ts)
   183     | SOME T' => T')
   184   | typ_subst_nonatomic inst T = the_default T (AList.lookup (op =) inst T);
   185 
   186 val lists_bmoc = fold (fn xs => fn t => Term.list_comb (t, xs));
   187 
   188 fun flat_rec_arg_args xss =
   189   (* FIXME (once the old datatype package is phased out): The first line below gives the preferred
   190      order. The second line is for compatibility with the old datatype package. *)
   191 (*
   192   flat xss
   193 *)
   194   map hd xss @ maps tl xss;
   195 
   196 fun flat_corec_predss_getterss qss fss = maps (op @) (qss ~~ fss);
   197 
   198 fun flat_corec_preds_predsss_gettersss [] [qss] [fss] = flat_corec_predss_getterss qss fss
   199   | flat_corec_preds_predsss_gettersss (p :: ps) (qss :: qsss) (fss :: fsss) =
   200     p :: flat_corec_predss_getterss qss fss @ flat_corec_preds_predsss_gettersss ps qsss fsss;
   201 
   202 fun mk_tupled_fun x f xs = HOLogic.tupled_lambda x (Term.list_comb (f, xs));
   203 fun mk_uncurried_fun f xs = mk_tupled_fun (HOLogic.mk_tuple xs) f xs;
   204 fun mk_uncurried2_fun f xss =
   205   mk_tupled_fun (HOLogic.mk_tuple (map HOLogic.mk_tuple xss)) f (flat_rec_arg_args xss);
   206 
   207 fun mk_flip (x, Type (_, [T1, Type (_, [T2, T3])])) =
   208   Abs ("x", T1, Abs ("y", T2, Var (x, T2 --> T1 --> T3) $ Bound 0 $ Bound 1));
   209 
   210 fun flip_rels lthy n thm =
   211   let
   212     val Rs = Term.add_vars (prop_of thm) [];
   213     val Rs' = rev (drop (length Rs - n) Rs);
   214     val cRs = map (fn f => (certify lthy (Var f), certify lthy (mk_flip f))) Rs';
   215   in
   216     Drule.cterm_instantiate cRs thm
   217   end;
   218 
   219 fun mk_ctor_or_dtor get_T Ts t =
   220   let val Type (_, Ts0) = get_T (fastype_of t) in
   221     Term.subst_atomic_types (Ts0 ~~ Ts) t
   222   end;
   223 
   224 val mk_ctor = mk_ctor_or_dtor range_type;
   225 val mk_dtor = mk_ctor_or_dtor domain_type;
   226 
   227 fun mk_co_iter thy fp fpT Cs t =
   228   let
   229     val (binders, body) = strip_type (fastype_of t);
   230     val (f_Cs, prebody) = split_last binders;
   231     val fpT0 = if fp = Least_FP then prebody else body;
   232     val Cs0 = distinct (op =) (map (if fp = Least_FP then body_type else domain_type) f_Cs);
   233     val rho = tvar_subst thy (fpT0 :: Cs0) (fpT :: Cs);
   234   in
   235     Term.subst_TVars rho t
   236   end;
   237 
   238 fun mk_co_iters thy fp fpTs Cs ts0 =
   239   let
   240     val nn = length fpTs;
   241     val (fpTs0, Cs0) =
   242       map ((fp = Greatest_FP ? swap) o dest_funT o snd o strip_typeN nn o fastype_of) ts0
   243       |> split_list;
   244     val rho = tvar_subst thy (fpTs0 @ Cs0) (fpTs @ Cs);
   245   in
   246     map (Term.subst_TVars rho) ts0
   247   end;
   248 
   249 val mk_fp_iter_fun_types = fst o split_last o binder_types o fastype_of;
   250 
   251 fun unzip_recT Cs (T as Type (@{type_name prod}, Ts as [_, U])) =
   252     if member (op =) Cs U then Ts else [T]
   253   | unzip_recT _ T = [T];
   254 
   255 fun unzip_corecT Cs (T as Type (@{type_name sum}, Ts as [_, U])) =
   256     if member (op =) Cs U then Ts else [T]
   257   | unzip_corecT _ T = [T];
   258 
   259 fun mk_map live Ts Us t =
   260   let val (Type (_, Ts0), Type (_, Us0)) = strip_typeN (live + 1) (fastype_of t) |>> List.last in
   261     Term.subst_atomic_types (Ts0 @ Us0 ~~ Ts @ Us) t
   262   end;
   263 
   264 fun mk_rel live Ts Us t =
   265   let val [Type (_, Ts0), Type (_, Us0)] = binder_types (snd (strip_typeN live (fastype_of t))) in
   266     Term.subst_atomic_types (Ts0 @ Us0 ~~ Ts @ Us) t
   267   end;
   268 
   269 fun build_map lthy build_simple =
   270   let
   271     fun build (TU as (T, U)) =
   272       if T = U then
   273         id_const T
   274       else
   275         (case TU of
   276           (Type (s, Ts), Type (s', Us)) =>
   277           if s = s' then
   278             let
   279               val bnf = the (bnf_of lthy s);
   280               val live = live_of_bnf bnf;
   281               val mapx = mk_map live Ts Us (map_of_bnf bnf);
   282               val TUs' = map dest_funT (fst (strip_typeN live (fastype_of mapx)));
   283             in Term.list_comb (mapx, map build TUs') end
   284           else
   285             build_simple TU
   286         | _ => build_simple TU);
   287   in build end;
   288 
   289 fun liveness_of_fp_bnf n bnf =
   290   (case T_of_bnf bnf of
   291     Type (_, Ts) => map (not o member (op =) (deads_of_bnf bnf)) Ts
   292   | _ => replicate n false);
   293 
   294 fun cannot_merge_types () = error "Mutually recursive types must have the same type parameters";
   295 
   296 fun merge_type_arg T T' = if T = T' then T else cannot_merge_types ();
   297 
   298 fun merge_type_args (As, As') =
   299   if length As = length As' then map2 merge_type_arg As As' else cannot_merge_types ();
   300 
   301 fun reassoc_conjs thm =
   302   reassoc_conjs (thm RS @{thm conj_assoc[THEN iffD1]})
   303   handle THM _ => thm;
   304 
   305 fun type_args_named_constrained_of ((((ncAs, _), _), _), _) = ncAs;
   306 fun type_binding_of ((((_, b), _), _), _) = b;
   307 fun map_binding_of (((_, (b, _)), _), _) = b;
   308 fun rel_binding_of (((_, (_, b)), _), _) = b;
   309 fun mixfix_of ((_, mx), _) = mx;
   310 fun ctr_specs_of (_, ctr_specs) = ctr_specs;
   311 
   312 fun disc_of ((((disc, _), _), _), _) = disc;
   313 fun ctr_of ((((_, ctr), _), _), _) = ctr;
   314 fun args_of (((_, args), _), _) = args;
   315 fun defaults_of ((_, ds), _) = ds;
   316 fun ctr_mixfix_of (_, mx) = mx;
   317 
   318 fun add_nesty_bnf_names Us =
   319   let
   320     fun add (Type (s, Ts)) ss =
   321         let val (needs, ss') = fold_map add Ts ss in
   322           if exists I needs then (true, insert (op =) s ss') else (false, ss')
   323         end
   324       | add T ss = (member (op =) Us T, ss);
   325   in snd oo add end;
   326 
   327 fun nesty_bnfs ctxt ctr_Tsss Us =
   328   map_filter (bnf_of ctxt) (fold (fold (fold (add_nesty_bnf_names Us))) ctr_Tsss []);
   329 
   330 fun indexify proj xs f p = f (find_index (curry (op =) (proj p)) xs) p;
   331 
   332 fun mk_iter_fun_arg_types0 n ms = map2 dest_tupleT ms o dest_sumTN_balanced n o domain_type;
   333 
   334 fun mk_iter_fun_arg_types Cs ns mss =
   335   mk_fp_iter_fun_types
   336   #> map3 mk_iter_fun_arg_types0 ns mss
   337   #> map (map (map (unzip_recT Cs)));
   338 
   339 fun mk_iters_args_types Cs ns mss ctor_iter_fun_Tss lthy =
   340   let
   341     val Css = map2 replicate ns Cs;
   342     val y_Tsss = map3 mk_iter_fun_arg_types0 ns mss (map un_fold_of ctor_iter_fun_Tss);
   343     val g_Tss = map2 (fn C => map (fn y_Ts => y_Ts ---> C)) Cs y_Tsss;
   344 
   345     val ((gss, ysss), lthy) =
   346       lthy
   347       |> mk_Freess "f" g_Tss
   348       ||>> mk_Freesss "x" y_Tsss;
   349 
   350     val y_Tssss = map (map (map single)) y_Tsss;
   351     val yssss = map (map (map single)) ysss;
   352 
   353     val z_Tssss =
   354       map3 (fn n => fn ms => map2 (map (unzip_recT Cs) oo dest_tupleT) ms o
   355         dest_sumTN_balanced n o domain_type o co_rec_of) ns mss ctor_iter_fun_Tss;
   356 
   357     val z_Tsss' = map (map flat_rec_arg_args) z_Tssss;
   358     val h_Tss = map2 (map2 (curry (op --->))) z_Tsss' Css;
   359 
   360     val hss = map2 (map2 retype_free) h_Tss gss;
   361     val zssss_hd = map2 (map2 (map2 (retype_free o hd))) z_Tssss ysss;
   362     val (zssss_tl, lthy) =
   363       lthy
   364       |> mk_Freessss "y" (map (map (map tl)) z_Tssss);
   365     val zssss = map2 (map2 (map2 cons)) zssss_hd zssss_tl;
   366   in
   367     ([(g_Tss, y_Tssss, gss, yssss), (h_Tss, z_Tssss, hss, zssss)], lthy)
   368   end;
   369 
   370 fun mk_coiter_fun_arg_types0 Cs ns mss fun_Ts =
   371   let
   372     (*avoid "'a itself" arguments in coiterators and corecursors*)
   373     fun repair_arity [0] = [1]
   374       | repair_arity ms = ms;
   375 
   376     val f_sum_prod_Ts = map range_type fun_Ts;
   377     val f_prod_Tss = map2 dest_sumTN_balanced ns f_sum_prod_Ts;
   378     val f_Tsss = map2 (map2 dest_tupleT o repair_arity) mss f_prod_Tss;
   379     val f_Tssss = map2 (fn C => map (map (map (curry (op -->) C) o unzip_corecT Cs))) Cs f_Tsss;
   380     val q_Tssss = map (map (map (fn [_] => [] | [_, T] => [mk_pred1T (domain_type T)]))) f_Tssss;
   381   in
   382     (q_Tssss, f_Tsss, f_Tssss, f_sum_prod_Ts)
   383   end;
   384 
   385 fun mk_coiter_fun_arg_types Cs ns mss =
   386   mk_fp_iter_fun_types
   387   #> mk_coiter_fun_arg_types0 Cs ns mss;
   388 
   389 fun mk_coiters_args_types Cs ns mss dtor_coiter_fun_Tss lthy =
   390   let
   391     val p_Tss = map2 (fn n => replicate (Int.max (0, n - 1)) o mk_pred1T) ns Cs;
   392 
   393     fun mk_types get_Ts =
   394       let
   395         val fun_Ts = map get_Ts dtor_coiter_fun_Tss;
   396         val (q_Tssss, f_Tsss, f_Tssss, f_sum_prod_Ts) = mk_coiter_fun_arg_types0 Cs ns mss fun_Ts;
   397         val pf_Tss = map3 flat_corec_preds_predsss_gettersss p_Tss q_Tssss f_Tssss;
   398       in
   399         (q_Tssss, f_Tsss, f_Tssss, (f_sum_prod_Ts, pf_Tss))
   400       end;
   401 
   402     val (r_Tssss, g_Tsss, g_Tssss, unfold_types) = mk_types un_fold_of;
   403     val (s_Tssss, h_Tsss, h_Tssss, corec_types) = mk_types co_rec_of;
   404 
   405     val ((((Free (z, _), cs), pss), gssss), lthy) =
   406       lthy
   407       |> yield_singleton (mk_Frees "z") dummyT
   408       ||>> mk_Frees "a" Cs
   409       ||>> mk_Freess "p" p_Tss
   410       ||>> mk_Freessss "g" g_Tssss;
   411     val rssss = map (map (map (fn [] => []))) r_Tssss;
   412 
   413     val hssss_hd = map2 (map2 (map2 (fn T :: _ => fn [g] => retype_free T g))) h_Tssss gssss;
   414     val ((sssss, hssss_tl), lthy) =
   415       lthy
   416       |> mk_Freessss "q" s_Tssss
   417       ||>> mk_Freessss "h" (map (map (map tl)) h_Tssss);
   418     val hssss = map2 (map2 (map2 cons)) hssss_hd hssss_tl;
   419 
   420     val cpss = map2 (map o rapp) cs pss;
   421 
   422     fun build_sum_inj mk_inj = build_map lthy (uncurry mk_inj o dest_sumT o snd);
   423 
   424     fun build_dtor_coiter_arg _ [] [cf] = cf
   425       | build_dtor_coiter_arg T [cq] [cf, cf'] =
   426         mk_If cq (build_sum_inj Inl_const (fastype_of cf, T) $ cf)
   427           (build_sum_inj Inr_const (fastype_of cf', T) $ cf');
   428 
   429     fun mk_args qssss fssss f_Tsss =
   430       let
   431         val pfss = map3 flat_corec_preds_predsss_gettersss pss qssss fssss;
   432         val cqssss = map2 (map o map o map o rapp) cs qssss;
   433         val cfssss = map2 (map o map o map o rapp) cs fssss;
   434         val cqfsss = map3 (map3 (map3 build_dtor_coiter_arg)) f_Tsss cqssss cfssss;
   435       in (pfss, cqfsss) end;
   436 
   437     val unfold_args = mk_args rssss gssss g_Tsss;
   438     val corec_args = mk_args sssss hssss h_Tsss;
   439   in
   440     ((z, cs, cpss, [(unfold_args, unfold_types), (corec_args, corec_types)]), lthy)
   441   end;
   442 
   443 fun mk_co_iters_prelims fp fpTs Cs ns mss xtor_co_iterss0 lthy =
   444   let
   445     val thy = Proof_Context.theory_of lthy;
   446 
   447     val (xtor_co_iter_fun_Tss, xtor_co_iterss) =
   448       map (mk_co_iters thy fp fpTs Cs #> `(mk_fp_iter_fun_types o hd)) (transpose xtor_co_iterss0)
   449       |> apsnd transpose o apfst transpose o split_list;
   450 
   451     val ((iters_args_types, coiters_args_types), lthy') =
   452       if fp = Least_FP then
   453         mk_iters_args_types Cs ns mss xtor_co_iter_fun_Tss lthy |>> (rpair NONE o SOME)
   454       else
   455         mk_coiters_args_types Cs ns mss xtor_co_iter_fun_Tss lthy |>> (pair NONE o SOME)
   456   in
   457     ((xtor_co_iterss, iters_args_types, coiters_args_types), lthy')
   458   end;
   459 
   460 fun mk_iter_body ctor_iter fss xssss =
   461   Term.list_comb (ctor_iter, map2 (mk_sum_caseN_balanced oo map2 mk_uncurried2_fun) fss xssss);
   462 
   463 fun mk_preds_getterss_join c cps sum_prod_T cqfss =
   464   let val n = length cqfss in
   465     Term.lambda c (mk_IfN sum_prod_T cps
   466       (map2 (mk_InN_balanced sum_prod_T n) (map HOLogic.mk_tuple cqfss) (1 upto n)))
   467   end;
   468 
   469 fun mk_coiter_body cs cpss f_sum_prod_Ts cqfsss dtor_coiter =
   470   Term.list_comb (dtor_coiter, map4 mk_preds_getterss_join cs cpss f_sum_prod_Ts cqfsss);
   471 
   472 fun define_co_iters fp fpT Cs binding_specs lthy0 =
   473   let
   474     val thy = Proof_Context.theory_of lthy0;
   475 
   476     val ((csts, defs), (lthy', lthy)) = lthy0
   477       |> apfst split_list o fold_map (fn (b, spec) =>
   478         Specification.definition (SOME (b, NONE, NoSyn), ((Thm.def_binding b, []), spec))
   479         #>> apsnd snd) binding_specs
   480       ||> `Local_Theory.restore;
   481 
   482     val phi = Proof_Context.export_morphism lthy lthy';
   483 
   484     val csts' = map (mk_co_iter thy fp fpT Cs o Morphism.term phi) csts;
   485     val defs' = map (Morphism.thm phi) defs;
   486   in
   487     ((csts', defs'), lthy')
   488   end;
   489 
   490 fun define_iters iterNs iter_args_typess' mk_binding fpTs Cs ctor_iters lthy =
   491   let
   492     val nn = length fpTs;
   493 
   494     val fpT_to_C as Type (_, [fpT, _]) = snd (strip_typeN nn (fastype_of (hd ctor_iters)));
   495 
   496     fun generate_iter suf (f_Tss, _, fss, xssss) ctor_iter =
   497       let
   498         val res_T = fold_rev (curry (op --->)) f_Tss fpT_to_C;
   499         val b = mk_binding suf;
   500         val spec =
   501           mk_Trueprop_eq (lists_bmoc fss (Free (Binding.name_of b, res_T)),
   502             mk_iter_body ctor_iter fss xssss);
   503       in (b, spec) end;
   504   in
   505     define_co_iters Least_FP fpT Cs (map3 generate_iter iterNs iter_args_typess' ctor_iters) lthy
   506   end;
   507 
   508 fun define_coiters coiterNs (_, cs, cpss, coiter_args_typess') mk_binding fpTs Cs dtor_coiters
   509     lthy =
   510   let
   511     val nn = length fpTs;
   512 
   513     val C_to_fpT as Type (_, [_, fpT]) = snd (strip_typeN nn (fastype_of (hd dtor_coiters)));
   514 
   515     fun generate_coiter suf ((pfss, cqfsss), (f_sum_prod_Ts, pf_Tss)) dtor_coiter =
   516       let
   517         val res_T = fold_rev (curry (op --->)) pf_Tss C_to_fpT;
   518         val b = mk_binding suf;
   519         val spec =
   520           mk_Trueprop_eq (lists_bmoc pfss (Free (Binding.name_of b, res_T)),
   521             mk_coiter_body cs cpss f_sum_prod_Ts cqfsss dtor_coiter);
   522       in (b, spec) end;
   523   in
   524     define_co_iters Greatest_FP fpT Cs
   525       (map3 generate_coiter coiterNs coiter_args_typess' dtor_coiters) lthy
   526   end;
   527 
   528 fun derive_induct_iters_thms_for_types pre_bnfs [fold_args_types, rec_args_types] [ctor_induct]
   529     ctor_iter_thmss nesting_bnfs nested_bnfs fpTs Cs Xs ctrXs_Tsss ctrss ctr_defss iterss iter_defss
   530     lthy =
   531   let
   532     val iterss' = transpose iterss;
   533     val iter_defss' = transpose iter_defss;
   534 
   535     val [folds, recs] = iterss';
   536     val [fold_defs, rec_defs] = iter_defss';
   537 
   538     val ctr_Tsss = map (map (binder_types o fastype_of)) ctrss;
   539 
   540     val nn = length pre_bnfs;
   541     val ns = map length ctr_Tsss;
   542     val mss = map (map length) ctr_Tsss;
   543 
   544     val pre_map_defs = map map_def_of_bnf pre_bnfs;
   545     val pre_set_defss = map set_defs_of_bnf pre_bnfs;
   546     val nesting_map_ids'' = map (unfold_thms lthy [id_def] o map_id_of_bnf) nesting_bnfs;
   547     val nested_map_ids'' = map (unfold_thms lthy [id_def] o map_id_of_bnf) nested_bnfs;
   548     val nested_set_map's = maps set_map'_of_bnf nested_bnfs;
   549 
   550     val fp_b_names = map base_name_of_typ fpTs;
   551 
   552     val ((((ps, ps'), xsss), us'), names_lthy) =
   553       lthy
   554       |> mk_Frees' "P" (map mk_pred1T fpTs)
   555       ||>> mk_Freesss "x" ctr_Tsss
   556       ||>> Variable.variant_fixes fp_b_names;
   557 
   558     val us = map2 (curry Free) us' fpTs;
   559 
   560     fun mk_sets_nested bnf =
   561       let
   562         val Type (T_name, Us) = T_of_bnf bnf;
   563         val lives = lives_of_bnf bnf;
   564         val sets = sets_of_bnf bnf;
   565         fun mk_set U =
   566           (case find_index (curry (op =) U) lives of
   567             ~1 => Term.dummy
   568           | i => nth sets i);
   569       in
   570         (T_name, map mk_set Us)
   571       end;
   572 
   573     val setss_nested = map mk_sets_nested nested_bnfs;
   574 
   575     val (induct_thms, induct_thm) =
   576       let
   577         fun mk_set Ts t =
   578           let val Type (_, Ts0) = domain_type (fastype_of t) in
   579             Term.subst_atomic_types (Ts0 ~~ Ts) t
   580           end;
   581 
   582         fun mk_raw_prem_prems _ (x as Free (_, Type _)) (X as TFree _) =
   583             [([], (find_index (curry (op =) X) Xs + 1, x))]
   584           | mk_raw_prem_prems names_lthy (x as Free (s, Type (T_name, Ts0))) (Type (_, Xs_Ts0)) =
   585             (case AList.lookup (op =) setss_nested T_name of
   586               NONE => []
   587             | SOME raw_sets0 =>
   588               let
   589                 val (Xs_Ts, (Ts, raw_sets)) =
   590                   filter (exists_subtype_in Xs o fst) (Xs_Ts0 ~~ (Ts0 ~~ raw_sets0))
   591                   |> split_list ||> split_list;
   592                 val sets = map (mk_set Ts0) raw_sets;
   593                 val (ys, names_lthy') = names_lthy |> mk_Frees s Ts;
   594                 val xysets = map (pair x) (ys ~~ sets);
   595                 val ppremss = map2 (mk_raw_prem_prems names_lthy') ys Xs_Ts;
   596               in
   597                 flat (map2 (map o apfst o cons) xysets ppremss)
   598               end)
   599           | mk_raw_prem_prems _ _ _ = [];
   600 
   601         fun close_prem_prem xs t =
   602           fold_rev Logic.all (map Free (drop (nn + length xs)
   603             (rev (Term.add_frees t (map dest_Free xs @ ps'))))) t;
   604 
   605         fun mk_prem_prem xs (xysets, (j, x)) =
   606           close_prem_prem xs (Logic.list_implies (map (fn (x', (y, set)) =>
   607               HOLogic.mk_Trueprop (HOLogic.mk_mem (y, set $ x'))) xysets,
   608             HOLogic.mk_Trueprop (nth ps (j - 1) $ x)));
   609 
   610         fun mk_raw_prem phi ctr ctr_Ts ctrXs_Ts =
   611           let
   612             val (xs, names_lthy') = names_lthy |> mk_Frees "x" ctr_Ts;
   613             val pprems = flat (map2 (mk_raw_prem_prems names_lthy') xs ctrXs_Ts);
   614           in (xs, pprems, HOLogic.mk_Trueprop (phi $ Term.list_comb (ctr, xs))) end;
   615 
   616         fun mk_prem (xs, raw_pprems, concl) =
   617           fold_rev Logic.all xs (Logic.list_implies (map (mk_prem_prem xs) raw_pprems, concl));
   618 
   619         val raw_premss = map4 (map3 o mk_raw_prem) ps ctrss ctr_Tsss ctrXs_Tsss;
   620 
   621         val goal =
   622           Library.foldr (Logic.list_implies o apfst (map mk_prem)) (raw_premss,
   623             HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj (map2 (curry (op $)) ps us)));
   624 
   625         val kksss = map (map (map (fst o snd) o #2)) raw_premss;
   626 
   627         val ctor_induct' = ctor_induct OF (map mk_sumEN_tupled_balanced mss);
   628 
   629         val thm =
   630           Goal.prove_sorry lthy [] [] goal (fn {context = ctxt, ...} =>
   631             mk_induct_tac ctxt nn ns mss kksss (flat ctr_defss) ctor_induct' nested_set_map's
   632               pre_set_defss)
   633           |> singleton (Proof_Context.export names_lthy lthy)
   634           |> Thm.close_derivation;
   635       in
   636         `(conj_dests nn) thm
   637       end;
   638 
   639     val induct_cases = quasi_unambiguous_case_names (maps (map name_of_ctr) ctrss);
   640     val induct_case_names_attr = Attrib.internal (K (Rule_Cases.case_names induct_cases));
   641 
   642     val xctrss = map2 (map2 (curry Term.list_comb)) ctrss xsss;
   643 
   644     fun mk_iter_thmss (_, x_Tssss, fss, _) iters iter_defs ctor_iter_thms =
   645       let
   646         val fiters = map (lists_bmoc fss) iters;
   647 
   648         fun mk_goal fss fiter xctr f xs fxs =
   649           fold_rev (fold_rev Logic.all) (xs :: fss)
   650             (mk_Trueprop_eq (fiter $ xctr, Term.list_comb (f, fxs)));
   651 
   652         fun maybe_tick (T, U) u f =
   653           if try (fst o HOLogic.dest_prodT) U = SOME T then
   654             Term.lambda u (HOLogic.mk_prod (u, f $ u))
   655           else
   656             f;
   657 
   658         fun build_iter (x as Free (_, T)) U =
   659           if T = U then
   660             x
   661           else
   662             build_map lthy (indexify (perhaps (try (snd o HOLogic.dest_prodT)) o snd) Cs
   663               (fn kk => fn TU => maybe_tick TU (nth us kk) (nth fiters kk))) (T, U) $ x;
   664 
   665         val fxsss = map2 (map2 (flat_rec_arg_args oo map2 (map o build_iter))) xsss x_Tssss;
   666 
   667         val goalss = map5 (map4 o mk_goal fss) fiters xctrss fss xsss fxsss;
   668 
   669         val tacss =
   670           map2 (map o mk_iter_tac pre_map_defs (nested_map_ids'' @ nesting_map_ids'') iter_defs)
   671             ctor_iter_thms ctr_defss;
   672 
   673         fun prove goal tac =
   674           Goal.prove_sorry lthy [] [] goal (tac o #context)
   675           |> Thm.close_derivation;
   676       in
   677         map2 (map2 prove) goalss tacss
   678       end;
   679 
   680     val fold_thmss = mk_iter_thmss fold_args_types folds fold_defs (map un_fold_of ctor_iter_thmss);
   681     val rec_thmss = mk_iter_thmss rec_args_types recs rec_defs (map co_rec_of ctor_iter_thmss);
   682   in
   683     ((induct_thms, induct_thm, [induct_case_names_attr]),
   684      (fold_thmss, code_simp_attrs), (rec_thmss, code_simp_attrs))
   685   end;
   686 
   687 fun derive_coinduct_coiters_thms_for_types pre_bnfs (z, cs, cpss,
   688       coiters_args_types as [((pgss, crgsss), _), ((phss, cshsss), _)])
   689     dtor_coinducts dtor_ctors dtor_coiter_thmss nesting_bnfs nested_bnfs fpTs Cs As kss mss ns
   690     ctr_defss ctr_sugars coiterss coiter_defss export_args lthy =
   691   let
   692     val coiterss' = transpose coiterss;
   693     val coiter_defss' = transpose coiter_defss;
   694 
   695     val [unfold_defs, corec_defs] = coiter_defss';
   696 
   697     val nn = length pre_bnfs;
   698 
   699     val pre_map_defs = map map_def_of_bnf pre_bnfs;
   700     val pre_rel_defs = map rel_def_of_bnf pre_bnfs;
   701     val nesting_map_ids'' = map (unfold_thms lthy [id_def] o map_id_of_bnf) nesting_bnfs;
   702     val nesting_rel_eqs = map rel_eq_of_bnf nesting_bnfs;
   703     val nested_map_comp's = map map_comp'_of_bnf nested_bnfs;
   704     val nested_map_ids'' = map (unfold_thms lthy [id_def] o map_id_of_bnf) nested_bnfs;
   705 
   706     val fp_b_names = map base_name_of_typ fpTs;
   707 
   708     val ctrss = map (map (mk_ctr As) o #ctrs) ctr_sugars;
   709     val discss = map (map (mk_disc_or_sel As) o #discs) ctr_sugars;
   710     val selsss = map (map (map (mk_disc_or_sel As)) o #selss) ctr_sugars;
   711     val exhausts = map #exhaust ctr_sugars;
   712     val disc_thmsss = map #disc_thmss ctr_sugars;
   713     val discIss = map #discIs ctr_sugars;
   714     val sel_thmsss = map #sel_thmss ctr_sugars;
   715 
   716     val (((rs, us'), vs'), names_lthy) =
   717       lthy
   718       |> mk_Frees "R" (map (fn T => mk_pred2T T T) fpTs)
   719       ||>> Variable.variant_fixes fp_b_names
   720       ||>> Variable.variant_fixes (map (suffix "'") fp_b_names);
   721 
   722     val us = map2 (curry Free) us' fpTs;
   723     val udiscss = map2 (map o rapp) us discss;
   724     val uselsss = map2 (map o map o rapp) us selsss;
   725 
   726     val vs = map2 (curry Free) vs' fpTs;
   727     val vdiscss = map2 (map o rapp) vs discss;
   728     val vselsss = map2 (map o map o rapp) vs selsss;
   729 
   730     val coinduct_thms_pairs =
   731       let
   732         val uvrs = map3 (fn r => fn u => fn v => r $ u $ v) rs us vs;
   733         val uv_eqs = map2 (curry HOLogic.mk_eq) us vs;
   734         val strong_rs =
   735           map4 (fn u => fn v => fn uvr => fn uv_eq =>
   736             fold_rev Term.lambda [u, v] (HOLogic.mk_disj (uvr, uv_eq))) us vs uvrs uv_eqs;
   737 
   738         (* TODO: generalize (cf. "build_map") *)
   739         fun build_rel rs' T =
   740           (case find_index (curry (op =) T) fpTs of
   741             ~1 =>
   742             if exists_subtype_in fpTs T then
   743               let
   744                 val Type (s, Ts) = T
   745                 val bnf = the (bnf_of lthy s);
   746                 val live = live_of_bnf bnf;
   747                 val rel = mk_rel live Ts Ts (rel_of_bnf bnf);
   748                 val Ts' = map domain_type (fst (strip_typeN live (fastype_of rel)));
   749               in Term.list_comb (rel, map (build_rel rs') Ts') end
   750             else
   751               HOLogic.eq_const T
   752           | kk => nth rs' kk);
   753 
   754         fun build_rel_app rs' usel vsel = fold rapp [usel, vsel] (build_rel rs' (fastype_of usel));
   755 
   756         fun mk_prem_ctr_concls rs' n k udisc usels vdisc vsels =
   757           (if k = n then [] else [HOLogic.mk_eq (udisc, vdisc)]) @
   758           (if null usels then
   759              []
   760            else
   761              [Library.foldr HOLogic.mk_imp (if n = 1 then [] else [udisc, vdisc],
   762                 Library.foldr1 HOLogic.mk_conj (map2 (build_rel_app rs') usels vsels))]);
   763 
   764         fun mk_prem_concl rs' n udiscs uselss vdiscs vselss =
   765           Library.foldr1 HOLogic.mk_conj
   766             (flat (map5 (mk_prem_ctr_concls rs' n) (1 upto n) udiscs uselss vdiscs vselss))
   767           handle List.Empty => @{term True};
   768 
   769         fun mk_prem rs' uvr u v n udiscs uselss vdiscs vselss =
   770           fold_rev Logic.all [u, v] (Logic.mk_implies (HOLogic.mk_Trueprop uvr,
   771             HOLogic.mk_Trueprop (mk_prem_concl rs' n udiscs uselss vdiscs vselss)));
   772 
   773         val concl =
   774           HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj
   775             (map3 (fn uvr => fn u => fn v => HOLogic.mk_imp (uvr, HOLogic.mk_eq (u, v)))
   776                uvrs us vs));
   777 
   778         fun mk_goal rs' =
   779           Logic.list_implies (map8 (mk_prem rs') uvrs us vs ns udiscss uselsss vdiscss vselsss,
   780             concl);
   781 
   782         val goals = map mk_goal [rs, strong_rs];
   783 
   784         fun prove dtor_coinduct' goal =
   785           Goal.prove_sorry lthy [] [] goal (fn {context = ctxt, ...} =>
   786             mk_coinduct_tac ctxt nesting_rel_eqs nn ns dtor_coinduct' pre_rel_defs dtor_ctors
   787               exhausts ctr_defss disc_thmsss sel_thmsss)
   788           |> singleton (Proof_Context.export names_lthy lthy)
   789           |> Thm.close_derivation;
   790 
   791         fun postproc nn thm =
   792           Thm.permute_prems 0 nn
   793             (if nn = 1 then thm RS mp else funpow nn (fn thm => reassoc_conjs (thm RS mp_conj)) thm)
   794           |> Drule.zero_var_indexes
   795           |> `(conj_dests nn);
   796       in
   797         map2 (postproc nn oo prove) dtor_coinducts goals
   798       end;
   799 
   800     fun mk_coinduct_concls ms discs ctrs =
   801       let
   802         fun mk_disc_concl disc = [name_of_disc disc];
   803         fun mk_ctr_concl 0 _ = []
   804           | mk_ctr_concl _ ctor = [name_of_ctr ctor];
   805         val disc_concls = map mk_disc_concl (fst (split_last discs)) @ [[]];
   806         val ctr_concls = map2 mk_ctr_concl ms ctrs;
   807       in
   808         flat (map2 append disc_concls ctr_concls)
   809       end;
   810 
   811     val coinduct_cases = quasi_unambiguous_case_names (map (prefix EqN) fp_b_names);
   812     val coinduct_conclss =
   813       map3 (quasi_unambiguous_case_names ooo mk_coinduct_concls) mss discss ctrss;
   814 
   815     fun mk_maybe_not pos = not pos ? HOLogic.mk_not;
   816 
   817     val fcoiterss' as [gunfolds, hcorecs] =
   818       map2 (fn (pfss, _) => map (lists_bmoc pfss)) (map fst coiters_args_types) coiterss';
   819 
   820     val (unfold_thmss, corec_thmss, safe_unfold_thmss, safe_corec_thmss) =
   821       let
   822         fun mk_goal pfss c cps fcoiter n k ctr m cfs' =
   823           fold_rev (fold_rev Logic.all) ([c] :: pfss)
   824             (Logic.list_implies (seq_conds (HOLogic.mk_Trueprop oo mk_maybe_not) n k cps,
   825                mk_Trueprop_eq (fcoiter $ c, Term.list_comb (ctr, take m cfs'))));
   826 
   827         fun mk_U maybe_mk_sumT =
   828           typ_subst_nonatomic (map2 (fn C => fn fpT => (maybe_mk_sumT fpT C, fpT)) Cs fpTs);
   829 
   830         fun tack z_name (c, u) f =
   831           let val z = Free (z_name, mk_sumT (fastype_of u, fastype_of c)) in
   832             Term.lambda z (mk_sum_case (Term.lambda u u, Term.lambda c (f $ c)) $ z)
   833           end;
   834 
   835         fun build_coiter fcoiters maybe_mk_sumT maybe_tack cqf =
   836           let val T = fastype_of cqf in
   837             if exists_subtype_in Cs T then
   838               let val U = mk_U maybe_mk_sumT T in
   839                 build_map lthy (indexify snd fpTs (fn kk => fn TU =>
   840                   maybe_tack (nth cs kk, nth us kk) (nth fcoiters kk))) (T, U) $ cqf
   841               end
   842             else
   843               cqf
   844           end;
   845 
   846         val crgsss' = map (map (map (build_coiter (un_fold_of fcoiterss') (K I) (K I)))) crgsss;
   847         val cshsss' = map (map (map (build_coiter (co_rec_of fcoiterss') (curry mk_sumT) (tack z))))
   848           cshsss;
   849 
   850         val unfold_goalss = map8 (map4 oooo mk_goal pgss) cs cpss gunfolds ns kss ctrss mss crgsss';
   851         val corec_goalss = map8 (map4 oooo mk_goal phss) cs cpss hcorecs ns kss ctrss mss cshsss';
   852 
   853         fun mk_map_if_distrib bnf =
   854           let
   855             val mapx = map_of_bnf bnf;
   856             val live = live_of_bnf bnf;
   857             val ((Ts, T), U) = strip_typeN (live + 1) (fastype_of mapx) |>> split_last;
   858             val fs = Variable.variant_frees lthy [mapx] (map (pair "f") Ts);
   859             val t = Term.list_comb (mapx, map (Var o apfst (rpair 0)) fs);
   860           in
   861             Drule.instantiate' (map (SOME o certifyT lthy) [U, T]) [SOME (certify lthy t)]
   862               @{thm if_distrib}
   863           end;
   864 
   865         val nested_map_if_distribs = map mk_map_if_distrib nested_bnfs;
   866 
   867         val unfold_tacss =
   868           map3 (map oo mk_coiter_tac unfold_defs nesting_map_ids'')
   869             (map un_fold_of dtor_coiter_thmss) pre_map_defs ctr_defss;
   870         val corec_tacss =
   871           map3 (map oo mk_coiter_tac corec_defs nesting_map_ids'')
   872             (map co_rec_of dtor_coiter_thmss) pre_map_defs ctr_defss;
   873 
   874         fun prove goal tac =
   875           Goal.prove_sorry lthy [] [] goal (tac o #context)
   876           |> Thm.close_derivation;
   877 
   878         val unfold_thmss = map2 (map2 prove) unfold_goalss unfold_tacss;
   879         val corec_thmss =
   880           map2 (map2 prove) corec_goalss corec_tacss
   881           |> map (map (unfold_thms lthy @{thms sum_case_if}));
   882 
   883         val unfold_safesss = map2 (map2 (map2 (curry (op =)))) crgsss' crgsss;
   884         val corec_safesss = map2 (map2 (map2 (curry (op =)))) cshsss' cshsss;
   885 
   886         val filter_safesss =
   887           map2 (map_filter (fn (safes, thm) => if forall I safes then SOME thm else NONE) oo
   888             curry (op ~~));
   889 
   890         val safe_unfold_thmss = filter_safesss unfold_safesss unfold_thmss;
   891         val safe_corec_thmss = filter_safesss corec_safesss corec_thmss;
   892       in
   893         (unfold_thmss, corec_thmss, safe_unfold_thmss, safe_corec_thmss)
   894       end;
   895 
   896     val (disc_unfold_iff_thmss, disc_corec_iff_thmss) =
   897       let
   898         fun mk_goal c cps fcoiter n k disc =
   899           mk_Trueprop_eq (disc $ (fcoiter $ c),
   900             if n = 1 then @{const True}
   901             else Library.foldr1 HOLogic.mk_conj (seq_conds mk_maybe_not n k cps));
   902 
   903         val unfold_goalss = map6 (map2 oooo mk_goal) cs cpss gunfolds ns kss discss;
   904         val corec_goalss = map6 (map2 oooo mk_goal) cs cpss hcorecs ns kss discss;
   905 
   906         fun mk_case_split' cp = Drule.instantiate' [] [SOME (certify lthy cp)] @{thm case_split};
   907 
   908         val case_splitss' = map (map mk_case_split') cpss;
   909 
   910         val unfold_tacss =
   911           map3 (map oo mk_disc_coiter_iff_tac) case_splitss' unfold_thmss disc_thmsss;
   912         val corec_tacss =
   913           map3 (map oo mk_disc_coiter_iff_tac) case_splitss' corec_thmss disc_thmsss;
   914 
   915         fun prove goal tac =
   916           Goal.prove_sorry lthy [] [] goal (tac o #context)
   917           |> singleton export_args
   918           |> singleton (Proof_Context.export names_lthy lthy)
   919           |> Thm.close_derivation;
   920 
   921         fun proves [_] [_] = []
   922           | proves goals tacs = map2 prove goals tacs;
   923       in
   924         (map2 proves unfold_goalss unfold_tacss, map2 proves corec_goalss corec_tacss)
   925       end;
   926 
   927     val is_triv_discI = is_triv_implies orf is_concl_refl;
   928 
   929     fun mk_disc_coiter_thms coiters discIs =
   930       map (op RS) (filter_out (is_triv_discI o snd) (coiters ~~ discIs));
   931 
   932     val disc_unfold_thmss = map2 mk_disc_coiter_thms unfold_thmss discIss;
   933     val disc_corec_thmss = map2 mk_disc_coiter_thms corec_thmss discIss;
   934 
   935     fun mk_sel_coiter_thm coiter_thm sel sel_thm =
   936       let
   937         val (domT, ranT) = dest_funT (fastype_of sel);
   938         val arg_cong' =
   939           Drule.instantiate' (map (SOME o certifyT lthy) [domT, ranT])
   940             [NONE, NONE, SOME (certify lthy sel)] arg_cong
   941           |> Thm.varifyT_global;
   942         val sel_thm' = sel_thm RSN (2, trans);
   943       in
   944         coiter_thm RS arg_cong' RS sel_thm'
   945       end;
   946 
   947     fun mk_sel_coiter_thms coiter_thmss =
   948       map3 (map3 (map2 o mk_sel_coiter_thm)) coiter_thmss selsss sel_thmsss |> map flat;
   949 
   950     val sel_unfold_thmss = mk_sel_coiter_thms unfold_thmss;
   951     val sel_corec_thmss = mk_sel_coiter_thms corec_thmss;
   952 
   953     val coinduct_consumes_attr = Attrib.internal (K (Rule_Cases.consumes nn));
   954     val coinduct_case_names_attr = Attrib.internal (K (Rule_Cases.case_names coinduct_cases));
   955     val coinduct_case_concl_attrs =
   956       map2 (fn casex => fn concls =>
   957           Attrib.internal (K (Rule_Cases.case_conclusion (casex, concls))))
   958         coinduct_cases coinduct_conclss;
   959     val coinduct_case_attrs =
   960       coinduct_consumes_attr :: coinduct_case_names_attr :: coinduct_case_concl_attrs;
   961   in
   962     ((coinduct_thms_pairs, coinduct_case_attrs),
   963      (unfold_thmss, corec_thmss, []),
   964      (safe_unfold_thmss, safe_corec_thmss),
   965      (disc_unfold_thmss, disc_corec_thmss, simp_attrs),
   966      (disc_unfold_iff_thmss, disc_corec_iff_thmss, simp_attrs),
   967      (sel_unfold_thmss, sel_corec_thmss, simp_attrs))
   968   end;
   969 
   970 fun define_co_datatypes prepare_constraint prepare_typ prepare_term fp construct_fp
   971     (wrap_opts as (no_dests, rep_compat), specs) no_defs_lthy0 =
   972   let
   973     (* TODO: sanity checks on arguments *)
   974 
   975     val _ = if fp = Greatest_FP andalso no_dests then
   976         error "Cannot define destructor-less codatatypes"
   977       else
   978         ();
   979 
   980     fun qualify mandatory fp_b_name =
   981       Binding.qualify mandatory fp_b_name o (rep_compat ? Binding.qualify false rep_compat_prefix);
   982 
   983     val nn = length specs;
   984     val fp_bs = map type_binding_of specs;
   985     val fp_b_names = map Binding.name_of fp_bs;
   986     val fp_common_name = mk_common_name fp_b_names;
   987     val map_bs = map map_binding_of specs;
   988     val rel_bs = map rel_binding_of specs;
   989 
   990     fun prepare_type_arg (_, (ty, c)) =
   991       let val TFree (s, _) = prepare_typ no_defs_lthy0 ty in
   992         TFree (s, prepare_constraint no_defs_lthy0 c)
   993       end;
   994 
   995     val Ass0 = map (map prepare_type_arg o type_args_named_constrained_of) specs;
   996     val unsorted_Ass0 = map (map (resort_tfree HOLogic.typeS)) Ass0;
   997     val unsorted_As = Library.foldr1 merge_type_args unsorted_Ass0;
   998     val set_bss = map (map fst o type_args_named_constrained_of) specs;
   999 
  1000     val (((Bs0, Cs), Xs), no_defs_lthy) =
  1001       no_defs_lthy0
  1002       |> fold (Variable.declare_typ o resort_tfree dummyS) unsorted_As
  1003       |> mk_TFrees (length unsorted_As)
  1004       ||>> mk_TFrees nn
  1005       ||>> variant_tfrees fp_b_names;
  1006 
  1007     (* TODO: Cleaner handling of fake contexts, without "background_theory". The case where the new
  1008        type is defined in a locale and shadows an existing global type is currently not handled. *)
  1009 
  1010     fun add_fake_type spec =
  1011       Sign.add_type no_defs_lthy (type_binding_of spec,
  1012         length (type_args_named_constrained_of spec), mixfix_of spec);
  1013 
  1014     val fake_thy = fold add_fake_type specs;
  1015     val fake_lthy = Proof_Context.background_theory fake_thy no_defs_lthy;
  1016 
  1017     fun mk_fake_T b =
  1018       Type (fst (Term.dest_Type (Proof_Context.read_type_name fake_lthy true (Binding.name_of b))),
  1019         unsorted_As);
  1020 
  1021     val fake_Ts = map mk_fake_T fp_bs;
  1022 
  1023     val mixfixes = map mixfix_of specs;
  1024 
  1025     val _ = (case duplicates Binding.eq_name fp_bs of [] => ()
  1026       | b :: _ => error ("Duplicate type name declaration " ^ quote (Binding.name_of b)));
  1027 
  1028     val ctr_specss = map ctr_specs_of specs;
  1029 
  1030     val disc_bindingss = map (map disc_of) ctr_specss;
  1031     val ctr_bindingss =
  1032       map2 (fn fp_b_name => map (qualify false fp_b_name o ctr_of)) fp_b_names ctr_specss;
  1033     val ctr_argsss = map (map args_of) ctr_specss;
  1034     val ctr_mixfixess = map (map ctr_mixfix_of) ctr_specss;
  1035 
  1036     val sel_bindingsss = map (map (map fst)) ctr_argsss;
  1037     val fake_ctr_Tsss0 = map (map (map (prepare_typ fake_lthy o snd))) ctr_argsss;
  1038     val raw_sel_defaultsss = map (map defaults_of) ctr_specss;
  1039 
  1040     val (As :: _) :: fake_ctr_Tsss =
  1041       burrow (burrow (Syntax.check_typs fake_lthy)) (Ass0 :: fake_ctr_Tsss0);
  1042 
  1043     val _ = (case duplicates (op =) unsorted_As of [] => ()
  1044       | A :: _ => error ("Duplicate type parameter " ^
  1045           quote (Syntax.string_of_typ no_defs_lthy A)));
  1046 
  1047     val rhs_As' = fold (fold (fold Term.add_tfreesT)) fake_ctr_Tsss [];
  1048     val _ = (case subtract (op =) (map dest_TFree As) rhs_As' of
  1049         [] => ()
  1050       | A' :: _ => error ("Extra type variable on right-hand side: " ^
  1051           quote (Syntax.string_of_typ no_defs_lthy (TFree A'))));
  1052 
  1053     fun eq_fpT_check (T as Type (s, Us)) (Type (s', Us')) =
  1054         s = s' andalso (Us = Us' orelse error ("Illegal occurrence of recursive type " ^
  1055           quote (Syntax.string_of_typ fake_lthy T)))
  1056       | eq_fpT_check _ _ = false;
  1057 
  1058     fun freeze_fp (T as Type (s, Us)) =
  1059         (case find_index (eq_fpT_check T) fake_Ts of
  1060           ~1 => Type (s, map freeze_fp Us)
  1061         | kk => nth Xs kk)
  1062       | freeze_fp T = T;
  1063 
  1064     val ctrXs_Tsss = map (map (map freeze_fp)) fake_ctr_Tsss;
  1065     val ctrXs_sum_prod_Ts = map (mk_sumTN_balanced o map HOLogic.mk_tupleT) ctrXs_Tsss;
  1066 
  1067     val fp_eqs =
  1068       map dest_TFree Xs ~~ map (Term.typ_subst_atomic (As ~~ unsorted_As)) ctrXs_sum_prod_Ts;
  1069 
  1070     val (pre_bnfs, (fp_res as {bnfs = fp_bnfs as any_fp_bnf :: _, ctors = ctors0, dtors = dtors0,
  1071            xtor_co_iterss = xtor_co_iterss0, xtor_co_inducts, dtor_ctors, ctor_dtors, ctor_injects,
  1072            xtor_map_thms, xtor_set_thmss, xtor_rel_thms, xtor_co_iter_thmss, ...}, lthy)) =
  1073       fp_bnf (construct_fp mixfixes map_bs rel_bs set_bss) fp_bs (map dest_TFree unsorted_As) fp_eqs
  1074         no_defs_lthy0;
  1075 
  1076     val timer = time (Timer.startRealTimer ());
  1077 
  1078     val nesting_bnfs = nesty_bnfs lthy ctrXs_Tsss As;
  1079     val nested_bnfs = nesty_bnfs lthy ctrXs_Tsss Xs;
  1080 
  1081     val pre_map_defs = map map_def_of_bnf pre_bnfs;
  1082     val pre_set_defss = map set_defs_of_bnf pre_bnfs;
  1083     val pre_rel_defs = map rel_def_of_bnf pre_bnfs;
  1084     val nesting_set_map's = maps set_map'_of_bnf nesting_bnfs;
  1085     val nested_set_map's = maps set_map'_of_bnf nested_bnfs;
  1086 
  1087     val live = live_of_bnf any_fp_bnf;
  1088 
  1089     val Bs =
  1090       map3 (fn alive => fn A as TFree (_, S) => fn B => if alive then resort_tfree S B else A)
  1091         (liveness_of_fp_bnf (length As) any_fp_bnf) As Bs0;
  1092 
  1093     val B_ify = Term.typ_subst_atomic (As ~~ Bs);
  1094 
  1095     val ctors = map (mk_ctor As) ctors0;
  1096     val dtors = map (mk_dtor As) dtors0;
  1097 
  1098     val fpTs = map (domain_type o fastype_of) dtors;
  1099 
  1100     fun massage_simple_notes base =
  1101       filter_out (null o #2)
  1102       #> map (fn (thmN, thms, attrs) =>
  1103         ((qualify true base (Binding.name thmN), attrs), [(thms, [])]));
  1104 
  1105     val massage_multi_notes =
  1106       maps (fn (thmN, thmss, attrs) =>
  1107         if forall null thmss then
  1108           []
  1109         else
  1110           map3 (fn fp_b_name => fn Type (T_name, _) => fn thms =>
  1111               ((qualify true fp_b_name (Binding.name thmN), attrs T_name), [(thms, [])]))
  1112             fp_b_names fpTs thmss);
  1113 
  1114     val ctr_Tsss = map (map (map (Term.typ_subst_atomic (Xs ~~ fpTs)))) ctrXs_Tsss;
  1115     val ns = map length ctr_Tsss;
  1116     val kss = map (fn n => 1 upto n) ns;
  1117     val mss = map (map length) ctr_Tsss;
  1118 
  1119     val ((xtor_co_iterss, iters_args_types, coiters_args_types), lthy') =
  1120       mk_co_iters_prelims fp fpTs Cs ns mss xtor_co_iterss0 lthy;
  1121 
  1122     fun define_ctrs_case_for_type ((((((((((((((((((((((((fp_bnf, fp_b), fpT), C), ctor), dtor),
  1123             xtor_co_iters), ctor_dtor), dtor_ctor), ctor_inject), pre_map_def), pre_set_defs),
  1124           pre_rel_def), fp_map_thm), fp_set_thms), fp_rel_thm), n), ks), ms), ctr_bindings),
  1125         ctr_mixfixes), ctr_Tss), disc_bindings), sel_bindingss), raw_sel_defaultss) no_defs_lthy =
  1126       let
  1127         val fp_b_name = Binding.name_of fp_b;
  1128 
  1129         val dtorT = domain_type (fastype_of ctor);
  1130         val ctr_prod_Ts = map HOLogic.mk_tupleT ctr_Tss;
  1131         val ctr_sum_prod_T = mk_sumTN_balanced ctr_prod_Ts;
  1132         val case_Ts = map (fn Ts => Ts ---> C) ctr_Tss;
  1133 
  1134         val (((((w, fs), xss), yss), u'), names_lthy) =
  1135           no_defs_lthy
  1136           |> yield_singleton (mk_Frees "w") dtorT
  1137           ||>> mk_Frees "f" case_Ts
  1138           ||>> mk_Freess "x" ctr_Tss
  1139           ||>> mk_Freess "y" (map (map B_ify) ctr_Tss)
  1140           ||>> yield_singleton Variable.variant_fixes fp_b_name;
  1141 
  1142         val u = Free (u', fpT);
  1143 
  1144         val tuple_xs = map HOLogic.mk_tuple xss;
  1145         val tuple_ys = map HOLogic.mk_tuple yss;
  1146 
  1147         val ctr_rhss =
  1148           map3 (fn k => fn xs => fn tuple_x => fold_rev Term.lambda xs (ctor $
  1149             mk_InN_balanced ctr_sum_prod_T n tuple_x k)) ks xss tuple_xs;
  1150 
  1151         val case_binding = qualify false fp_b_name (Binding.suffix_name ("_" ^ caseN) fp_b);
  1152 
  1153         val case_rhs =
  1154           fold_rev Term.lambda (fs @ [u])
  1155             (mk_sum_caseN_balanced (map2 mk_uncurried_fun fs xss) $ (dtor $ u));
  1156 
  1157         val ((raw_case :: raw_ctrs, raw_case_def :: raw_ctr_defs), (lthy', lthy)) = no_defs_lthy
  1158           |> apfst split_list o fold_map3 (fn b => fn mx => fn rhs =>
  1159               Local_Theory.define ((b, mx), ((Thm.def_binding b, []), rhs)) #>> apsnd snd)
  1160             (case_binding :: ctr_bindings) (NoSyn :: ctr_mixfixes) (case_rhs :: ctr_rhss)
  1161           ||> `Local_Theory.restore;
  1162 
  1163         val phi = Proof_Context.export_morphism lthy lthy';
  1164 
  1165         val ctr_defs = map (Morphism.thm phi) raw_ctr_defs;
  1166         val ctr_defs' =
  1167           map2 (fn m => fn def => mk_unabs_def m (def RS meta_eq_to_obj_eq)) ms ctr_defs;
  1168         val case_def = Morphism.thm phi raw_case_def;
  1169 
  1170         val ctrs0 = map (Morphism.term phi) raw_ctrs;
  1171         val casex0 = Morphism.term phi raw_case;
  1172 
  1173         val ctrs = map (mk_ctr As) ctrs0;
  1174 
  1175         fun wrap_ctrs lthy =
  1176           let
  1177             fun exhaust_tac {context = ctxt, prems = _} =
  1178               let
  1179                 val ctor_iff_dtor_thm =
  1180                   let
  1181                     val goal =
  1182                       fold_rev Logic.all [w, u]
  1183                         (mk_Trueprop_eq (HOLogic.mk_eq (u, ctor $ w), HOLogic.mk_eq (dtor $ u, w)));
  1184                   in
  1185                     Goal.prove_sorry lthy [] [] goal (fn {context = ctxt, ...} =>
  1186                       mk_ctor_iff_dtor_tac ctxt (map (SOME o certifyT lthy) [dtorT, fpT])
  1187                         (certify lthy ctor) (certify lthy dtor) ctor_dtor dtor_ctor)
  1188                     |> Thm.close_derivation
  1189                     |> Morphism.thm phi
  1190                   end;
  1191 
  1192                 val sumEN_thm' =
  1193                   unfold_thms lthy @{thms unit_all_eq1}
  1194                     (Drule.instantiate' (map (SOME o certifyT lthy) ctr_prod_Ts) []
  1195                        (mk_sumEN_balanced n))
  1196                   |> Morphism.thm phi;
  1197               in
  1198                 mk_exhaust_tac ctxt n ctr_defs ctor_iff_dtor_thm sumEN_thm'
  1199               end;
  1200 
  1201             val inject_tacss =
  1202               map2 (fn 0 => K [] | _ => fn ctr_def => [fn {context = ctxt, ...} =>
  1203                   mk_inject_tac ctxt ctr_def ctor_inject]) ms ctr_defs;
  1204 
  1205             val half_distinct_tacss =
  1206               map (map (fn (def, def') => fn {context = ctxt, ...} =>
  1207                 mk_half_distinct_tac ctxt ctor_inject [def, def'])) (mk_half_pairss (`I ctr_defs));
  1208 
  1209             val case_tacs =
  1210               map3 (fn k => fn m => fn ctr_def => fn {context = ctxt, ...} =>
  1211                 mk_case_tac ctxt n k m case_def ctr_def dtor_ctor) ks ms ctr_defs;
  1212 
  1213             val tacss = [exhaust_tac] :: inject_tacss @ half_distinct_tacss @ [case_tacs];
  1214 
  1215             val sel_defaultss = map (map (apsnd (prepare_term lthy))) raw_sel_defaultss
  1216           in
  1217             wrap_free_constructors tacss (((wrap_opts, ctrs0), casex0), (disc_bindings,
  1218               (sel_bindingss, sel_defaultss))) lthy
  1219           end;
  1220 
  1221         fun derive_maps_sets_rels (ctr_sugar, lthy) =
  1222           if live = 0 then
  1223             ((([], [], [], []), ctr_sugar), lthy)
  1224           else
  1225             let
  1226               val rel_flip = rel_flip_of_bnf fp_bnf;
  1227               val nones = replicate live NONE;
  1228 
  1229               val ctor_cong =
  1230                 if fp = Least_FP then
  1231                   Drule.dummy_thm
  1232                 else
  1233                   let val ctor' = mk_ctor Bs ctor in
  1234                     cterm_instantiate_pos [NONE, NONE, SOME (certify lthy ctor')] arg_cong
  1235                   end;
  1236 
  1237               fun mk_cIn ify =
  1238                 certify lthy o (fp = Greatest_FP ? curry (op $) (map_types ify ctor)) oo
  1239                 mk_InN_balanced (ify ctr_sum_prod_T) n;
  1240 
  1241               val cxIns = map2 (mk_cIn I) tuple_xs ks;
  1242               val cyIns = map2 (mk_cIn B_ify) tuple_ys ks;
  1243 
  1244               fun mk_map_thm ctr_def' cxIn =
  1245                 fold_thms lthy [ctr_def']
  1246                   (unfold_thms lthy (pre_map_def ::
  1247                        (if fp = Least_FP then [] else [ctor_dtor, dtor_ctor]) @ sum_prod_thms_map)
  1248                      (cterm_instantiate_pos (nones @ [SOME cxIn])
  1249                         (if fp = Least_FP then fp_map_thm else fp_map_thm RS ctor_cong)))
  1250                 |> singleton (Proof_Context.export names_lthy no_defs_lthy);
  1251 
  1252               fun mk_set_thm fp_set_thm ctr_def' cxIn =
  1253                 fold_thms lthy [ctr_def']
  1254                   (unfold_thms lthy (pre_set_defs @ nested_set_map's @ nesting_set_map's @
  1255                        (if fp = Least_FP then [] else [dtor_ctor]) @ sum_prod_thms_set)
  1256                      (cterm_instantiate_pos [SOME cxIn] fp_set_thm))
  1257                 |> singleton (Proof_Context.export names_lthy no_defs_lthy);
  1258 
  1259               fun mk_set_thms fp_set_thm = map2 (mk_set_thm fp_set_thm) ctr_defs' cxIns;
  1260 
  1261               val map_thms = map2 mk_map_thm ctr_defs' cxIns;
  1262               val set_thmss = map mk_set_thms fp_set_thms;
  1263 
  1264               val rel_infos = (ctr_defs' ~~ cxIns, ctr_defs' ~~ cyIns);
  1265 
  1266               fun mk_rel_thm postproc ctr_defs' cxIn cyIn =
  1267                 fold_thms lthy ctr_defs'
  1268                    (unfold_thms lthy (@{thm Inl_Inr_False} :: pre_rel_def ::
  1269                         (if fp = Least_FP then [] else [dtor_ctor]) @ sum_prod_thms_rel)
  1270                       (cterm_instantiate_pos (nones @ [SOME cxIn, SOME cyIn]) fp_rel_thm))
  1271                 |> postproc
  1272                 |> singleton (Proof_Context.export names_lthy no_defs_lthy);
  1273 
  1274               fun mk_rel_inject_thm ((ctr_def', cxIn), (_, cyIn)) =
  1275                 mk_rel_thm (unfold_thms lthy @{thms eq_sym_Unity_conv}) [ctr_def'] cxIn cyIn;
  1276 
  1277               val rel_inject_thms = map mk_rel_inject_thm (op ~~ rel_infos);
  1278 
  1279               fun mk_half_rel_distinct_thm ((xctr_def', cxIn), (yctr_def', cyIn)) =
  1280                 mk_rel_thm (fn thm => thm RS @{thm eq_False[THEN iffD1]}) [xctr_def', yctr_def']
  1281                   cxIn cyIn;
  1282 
  1283               fun mk_other_half_rel_distinct_thm thm =
  1284                 flip_rels lthy live thm
  1285                 RS (rel_flip RS sym RS @{thm arg_cong[of _ _ Not]} RS iffD2);
  1286 
  1287               val half_rel_distinct_thmss =
  1288                 map (map mk_half_rel_distinct_thm) (mk_half_pairss rel_infos);
  1289               val other_half_rel_distinct_thmss =
  1290                 map (map mk_other_half_rel_distinct_thm) half_rel_distinct_thmss;
  1291               val (rel_distinct_thms, _) =
  1292                 join_halves n half_rel_distinct_thmss other_half_rel_distinct_thmss;
  1293 
  1294               val notes =
  1295                 [(mapN, map_thms, code_simp_attrs),
  1296                  (rel_distinctN, rel_distinct_thms, code_simp_attrs),
  1297                  (rel_injectN, rel_inject_thms, code_simp_attrs),
  1298                  (setsN, flat set_thmss, code_simp_attrs)]
  1299                 |> massage_simple_notes fp_b_name;
  1300             in
  1301               (((map_thms, rel_inject_thms, rel_distinct_thms, set_thmss), ctr_sugar),
  1302                lthy |> Local_Theory.notes notes |> snd)
  1303             end;
  1304 
  1305         fun mk_binding suf = qualify false fp_b_name (Binding.suffix_name ("_" ^ suf) fp_b);
  1306 
  1307         fun massage_res (((maps_sets_rels, ctr_sugar), co_iter_res), lthy) =
  1308           (((maps_sets_rels, (ctrs, xss, ctr_defs, ctr_sugar)), co_iter_res), lthy);
  1309       in
  1310         (wrap_ctrs
  1311          #> derive_maps_sets_rels
  1312          ##>>
  1313            (if fp = Least_FP then define_iters [foldN, recN] (the iters_args_types)
  1314             else define_coiters [unfoldN, corecN] (the coiters_args_types))
  1315              mk_binding fpTs Cs xtor_co_iters
  1316          #> massage_res, lthy')
  1317       end;
  1318 
  1319     fun wrap_types_etc (wrap_types_etcs, lthy) =
  1320       fold_map I wrap_types_etcs lthy
  1321       |>> apsnd split_list o apfst (apsnd split_list4 o apfst split_list4 o split_list)
  1322         o split_list;
  1323 
  1324     val mk_simp_thmss =
  1325       map7 (fn {injects, distincts, case_thms, ...} => fn un_folds => fn co_recs =>
  1326         fn mapsx => fn rel_injects => fn rel_distincts => fn setss =>
  1327           injects @ distincts @ case_thms @ co_recs @ un_folds @ mapsx @ rel_injects
  1328           @ rel_distincts @ flat setss);
  1329 
  1330     fun derive_and_note_induct_iters_thms_for_types
  1331         ((((mapsx, rel_injects, rel_distincts, setss), (ctrss, _, ctr_defss, ctr_sugars)),
  1332           (iterss, iter_defss)), lthy) =
  1333       let
  1334         val ((induct_thms, induct_thm, induct_attrs), (fold_thmss, fold_attrs),
  1335              (rec_thmss, rec_attrs)) =
  1336           derive_induct_iters_thms_for_types pre_bnfs (the iters_args_types) xtor_co_inducts
  1337             xtor_co_iter_thmss nesting_bnfs nested_bnfs fpTs Cs Xs ctrXs_Tsss ctrss ctr_defss iterss
  1338             iter_defss lthy;
  1339 
  1340         val induct_type_attr = Attrib.internal o K o Induct.induct_type;
  1341 
  1342         val simp_thmss =
  1343           mk_simp_thmss ctr_sugars fold_thmss rec_thmss mapsx rel_injects rel_distincts setss;
  1344 
  1345         val common_notes =
  1346           (if nn > 1 then [(inductN, [induct_thm], induct_attrs)] else [])
  1347           |> massage_simple_notes fp_common_name;
  1348 
  1349         val notes =
  1350           [(foldN, fold_thmss, K fold_attrs),
  1351            (inductN, map single induct_thms, fn T_name => induct_attrs @ [induct_type_attr T_name]),
  1352            (recN, rec_thmss, K rec_attrs),
  1353            (simpsN, simp_thmss, K [])]
  1354           |> massage_multi_notes;
  1355       in
  1356         lthy
  1357         |> Local_Theory.notes (common_notes @ notes) |> snd
  1358         |> register_fp_sugars Least_FP pre_bnfs nested_bnfs fp_res ctr_defss ctr_sugars iterss
  1359           [induct_thm] (transpose [fold_thmss, rec_thmss])
  1360       end;
  1361 
  1362     fun derive_and_note_coinduct_coiters_thms_for_types
  1363         ((((mapsx, rel_injects, rel_distincts, setss), (_, _, ctr_defss, ctr_sugars)),
  1364           (coiterss, coiter_defss)), lthy) =
  1365       let
  1366         val (([(coinduct_thms, coinduct_thm), (strong_coinduct_thms, strong_coinduct_thm)],
  1367               coinduct_attrs),
  1368              (unfold_thmss, corec_thmss, coiter_attrs),
  1369              (safe_unfold_thmss, safe_corec_thmss),
  1370              (disc_unfold_thmss, disc_corec_thmss, disc_coiter_attrs),
  1371              (disc_unfold_iff_thmss, disc_corec_iff_thmss, disc_coiter_iff_attrs),
  1372              (sel_unfold_thmss, sel_corec_thmss, sel_coiter_attrs)) =
  1373           derive_coinduct_coiters_thms_for_types pre_bnfs (the coiters_args_types) xtor_co_inducts
  1374             dtor_ctors xtor_co_iter_thmss nesting_bnfs nested_bnfs fpTs Cs As kss mss ns ctr_defss
  1375             ctr_sugars coiterss coiter_defss (Proof_Context.export lthy' no_defs_lthy)
  1376             lthy;
  1377 
  1378         val coinduct_type_attr = Attrib.internal o K o Induct.coinduct_type;
  1379 
  1380         fun flat_coiter_thms coiters disc_coiters sel_coiters =
  1381           coiters @ disc_coiters @ sel_coiters;
  1382 
  1383         val simp_thmss =
  1384           mk_simp_thmss ctr_sugars
  1385             (map3 flat_coiter_thms safe_unfold_thmss disc_unfold_thmss sel_unfold_thmss)
  1386             (map3 flat_coiter_thms safe_corec_thmss disc_corec_thmss sel_corec_thmss)
  1387             mapsx rel_injects rel_distincts setss;
  1388 
  1389         val anonymous_notes =
  1390           [(flat safe_unfold_thmss @ flat safe_corec_thmss, simp_attrs)]
  1391           |> map (fn (thms, attrs) => ((Binding.empty, attrs), [(thms, [])]));
  1392 
  1393         val common_notes =
  1394           (if nn > 1 then
  1395              [(coinductN, [coinduct_thm], coinduct_attrs),
  1396               (strong_coinductN, [strong_coinduct_thm], coinduct_attrs)]
  1397            else
  1398              [])
  1399           |> massage_simple_notes fp_common_name;
  1400 
  1401         val notes =
  1402           [(coinductN, map single coinduct_thms,
  1403             fn T_name => coinduct_attrs @ [coinduct_type_attr T_name]),
  1404            (corecN, corec_thmss, K coiter_attrs),
  1405            (disc_corecN, disc_corec_thmss, K disc_coiter_attrs),
  1406            (disc_corec_iffN, disc_corec_iff_thmss, K disc_coiter_iff_attrs),
  1407            (disc_unfoldN, disc_unfold_thmss, K disc_coiter_attrs),
  1408            (disc_unfold_iffN, disc_unfold_iff_thmss, K disc_coiter_iff_attrs),
  1409            (sel_corecN, sel_corec_thmss, K sel_coiter_attrs),
  1410            (sel_unfoldN, sel_unfold_thmss, K sel_coiter_attrs),
  1411            (simpsN, simp_thmss, K []),
  1412            (strong_coinductN, map single strong_coinduct_thms, K coinduct_attrs),
  1413            (unfoldN, unfold_thmss, K coiter_attrs)]
  1414           |> massage_multi_notes;
  1415       in
  1416         lthy
  1417         |> Local_Theory.notes (anonymous_notes @ common_notes @ notes) |> snd
  1418         |> register_fp_sugars Greatest_FP pre_bnfs nested_bnfs fp_res ctr_defss ctr_sugars coiterss
  1419           [coinduct_thm, strong_coinduct_thm] (transpose [unfold_thmss, corec_thmss])
  1420       end;
  1421 
  1422     val lthy'' = lthy'
  1423       |> fold_map define_ctrs_case_for_type (fp_bnfs ~~ fp_bs ~~ fpTs ~~ Cs ~~ ctors ~~ dtors ~~
  1424         xtor_co_iterss ~~ ctor_dtors ~~ dtor_ctors ~~ ctor_injects ~~ pre_map_defs ~~
  1425         pre_set_defss ~~ pre_rel_defs ~~ xtor_map_thms ~~ xtor_set_thmss ~~ xtor_rel_thms ~~ ns ~~
  1426         kss ~~ mss ~~ ctr_bindingss ~~ ctr_mixfixess ~~ ctr_Tsss ~~ disc_bindingss ~~
  1427         sel_bindingsss ~~ raw_sel_defaultsss)
  1428       |> wrap_types_etc
  1429       |> (if fp = Least_FP then derive_and_note_induct_iters_thms_for_types
  1430           else derive_and_note_coinduct_coiters_thms_for_types);
  1431 
  1432     val timer = time (timer ("Constructors, discriminators, selectors, etc., for the new " ^
  1433       datatype_word fp));
  1434   in
  1435     timer; lthy''
  1436   end;
  1437 
  1438 val co_datatypes = define_co_datatypes (K I) (K I) (K I);
  1439 
  1440 val co_datatype_cmd =
  1441   define_co_datatypes Typedecl.read_constraint Syntax.parse_typ Syntax.parse_term;
  1442 
  1443 val parse_ctr_arg =
  1444   @{keyword "("} |-- parse_binding_colon -- Parse.typ --| @{keyword ")"} ||
  1445   (Parse.typ >> pair Binding.empty);
  1446 
  1447 val parse_defaults =
  1448   @{keyword "("} |-- @{keyword "defaults"} |-- Scan.repeat parse_bound_term --| @{keyword ")"};
  1449 
  1450 val parse_type_arg_constrained =
  1451   Parse.type_ident -- Scan.option (@{keyword "::"} |-- Parse.!!! Parse.sort);
  1452 
  1453 val parse_type_arg_named_constrained = parse_opt_binding_colon -- parse_type_arg_constrained;
  1454 
  1455 val parse_type_args_named_constrained =
  1456   parse_type_arg_constrained >> (single o pair Binding.empty) ||
  1457   @{keyword "("} |-- Parse.!!! (Parse.list1 parse_type_arg_named_constrained --| @{keyword ")"}) ||
  1458   Scan.succeed [];
  1459 
  1460 val parse_map_rel_binding = Parse.short_ident --| @{keyword ":"} -- parse_binding;
  1461 
  1462 val no_map_rel = (Binding.empty, Binding.empty);
  1463 
  1464 (* "map" and "rel" are purposedly not registered as keywords, because they are short and nice names
  1465    that we don't want them to be highlighted everywhere. *)
  1466 fun extract_map_rel ("map", b) = apfst (K b)
  1467   | extract_map_rel ("rel", b) = apsnd (K b)
  1468   | extract_map_rel (s, _) = error ("Expected \"map\" or \"rel\" instead of " ^ quote s);
  1469 
  1470 val parse_map_rel_bindings =
  1471   @{keyword "("} |-- Scan.repeat parse_map_rel_binding --| @{keyword ")"}
  1472     >> (fn ps => fold extract_map_rel ps no_map_rel) ||
  1473   Scan.succeed no_map_rel;
  1474 
  1475 val parse_ctr_spec =
  1476   parse_opt_binding_colon -- parse_binding -- Scan.repeat parse_ctr_arg --
  1477   Scan.optional parse_defaults [] -- Parse.opt_mixfix;
  1478 
  1479 val parse_spec =
  1480   parse_type_args_named_constrained -- parse_binding -- parse_map_rel_bindings --
  1481   Parse.opt_mixfix -- (@{keyword "="} |-- Parse.enum1 "|" parse_ctr_spec);
  1482 
  1483 val parse_co_datatype = parse_wrap_free_constructors_options -- Parse.and_list1 parse_spec;
  1484 
  1485 fun parse_co_datatype_cmd fp construct_fp = parse_co_datatype >> co_datatype_cmd fp construct_fp;
  1486 
  1487 end;