src/HOL/BNF/Tools/bnf_fp_def_sugar.ML
author blanchet
Fri Jun 07 17:04:55 2013 +0100 (2013-06-07)
changeset 52349 07fd21c01e93
parent 52348 740923a6e530
child 52351 40136fcb5e7a
permissions -rw-r--r--
code simplifications (cf. 78a3d5006cf1)
     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      fp_res: BNF_FP_Util.fp_result,
    16      ctr_defss: thm list list,
    17      ctr_sugars: BNF_Ctr_Sugar.ctr_sugar list,
    18      co_iterss: term list list,
    19      co_inducts: thm list,
    20      co_iter_thmsss: thm list list list};
    21 
    22   val of_fp_sugar: (fp_sugar -> 'a list) -> fp_sugar -> 'a
    23   val morph_fp_sugar: morphism -> fp_sugar -> fp_sugar
    24   val fp_sugar_of: Proof.context -> string -> fp_sugar option
    25 
    26   val tvar_subst: theory -> typ list -> typ list -> ((string * int) * typ) list
    27   val exists_subtype_in: typ list -> typ -> bool
    28   val flat_rec: 'a list list -> 'a list
    29   val mk_co_iter: theory -> BNF_FP_Util.fp_kind -> typ -> typ list -> term -> term
    30   val nesty_bnfs: Proof.context -> typ list list list -> typ list -> BNF_Def.bnf list
    31   val mk_map: int -> typ list -> typ list -> term -> term
    32   val build_map: local_theory -> (typ * typ -> term) -> typ * typ -> term
    33   val mk_co_iters_prelims: BNF_FP_Util.fp_kind -> typ list -> typ list -> int list ->
    34     int list list -> term list list -> Proof.context ->
    35     (term list list
    36      * (typ list list * typ list list list list * term list list
    37         * term list list list list) list option
    38      * (string * term list * term list list
    39         * ((term list list * term list list list) * (typ list * typ list list)) list) option)
    40     * Proof.context
    41 
    42   val mk_iter_fun_arg_typessss: typ list -> int list -> int list list -> term ->
    43     typ list list list list
    44   val define_iters: string list ->
    45     (typ list list * typ list list list list * term list list * term list list list list) list ->
    46     (string -> binding) -> typ list -> typ list -> term list -> Proof.context ->
    47     (term list * thm list) * Proof.context
    48   val define_coiters: string list -> string * term list * term list list
    49     * ((term list list * term list list list) * (typ list * typ list list)) list ->
    50     (string -> binding) -> typ list -> typ list -> term list -> Proof.context ->
    51     (term list * thm list) * Proof.context
    52   val derive_induct_iters_thms_for_types: BNF_Def.bnf list ->
    53     (typ list list * typ list list list list * term list list * term list list list list) list ->
    54     thm list -> thm list list -> BNF_Def.bnf list -> BNF_Def.bnf list -> typ list -> typ list ->
    55     typ list -> typ list list list -> term list list -> thm list list -> term list list ->
    56     thm list list -> local_theory ->
    57     (thm list * thm * Args.src list) * (thm list list * Args.src list)
    58     * (thm list list * Args.src list)
    59   val derive_coinduct_coiters_thms_for_types: BNF_Def.bnf list ->
    60     string * term list * term list list * ((term list list * term list list list)
    61       * (typ list * typ list list list * typ list list)) list ->
    62     thm list -> thm list -> thm list list -> BNF_Def.bnf list -> BNF_Def.bnf list -> typ list ->
    63     typ list -> typ list -> int list list -> int list list -> int list -> thm list list ->
    64     BNF_Ctr_Sugar.ctr_sugar list -> term list list -> thm list list -> local_theory ->
    65     ((thm list * thm) list * Args.src list)
    66     * (thm list list * thm list list * Args.src list)
    67     * (thm list list * thm list list) * (thm list list * thm list list * Args.src list)
    68     * (thm list list * thm list list * Args.src list)
    69     * (thm list list * thm list list * Args.src list)
    70 
    71   val co_datatypes: BNF_FP_Util.fp_kind -> (mixfix list -> binding list -> binding list ->
    72       binding list list -> binding list -> (string * sort) list -> typ list * typ list list ->
    73       BNF_Def.bnf list -> local_theory -> BNF_FP_Util.fp_result * local_theory) ->
    74     (bool * bool) * (((((binding * (typ * sort)) list * binding) * (binding * binding)) * mixfix) *
    75       ((((binding * binding) * (binding * typ) list) * (binding * term) list) *
    76         mixfix) list) list ->
    77     local_theory -> local_theory
    78   val parse_co_datatype_cmd: BNF_FP_Util.fp_kind -> (mixfix list -> binding list -> binding list ->
    79       binding list list -> binding list -> (string * sort) list -> typ list * typ list list ->
    80       BNF_Def.bnf list -> local_theory -> BNF_FP_Util.fp_result * local_theory) ->
    81     (local_theory -> local_theory) parser
    82 end;
    83 
    84 structure BNF_FP_Def_Sugar : BNF_FP_DEF_SUGAR =
    85 struct
    86 
    87 open BNF_Util
    88 open BNF_Ctr_Sugar
    89 open BNF_Def
    90 open BNF_FP_Util
    91 open BNF_FP_Def_Sugar_Tactics
    92 
    93 val EqN = "Eq_";
    94 
    95 type fp_sugar =
    96   {T: typ,
    97    fp: fp_kind,
    98    index: int,
    99    pre_bnfs: bnf list,
   100    fp_res: fp_result,
   101    ctr_defss: thm list list,
   102    ctr_sugars: ctr_sugar list,
   103    co_iterss: term list list,
   104    co_inducts: thm list,
   105    co_iter_thmsss: thm list list list};
   106 
   107 fun of_fp_sugar f (fp_sugar as {index, ...}) = nth (f fp_sugar) index;
   108 
   109 fun eq_fp_sugar ({T = T1, fp = fp1, index = index1, fp_res = fp_res1, ...} : fp_sugar,
   110     {T = T2, fp = fp2, index = index2, fp_res = fp_res2, ...} : fp_sugar) =
   111   T1 = T2 andalso fp1 = fp2 andalso index1 = index2 andalso eq_fp_result (fp_res1, fp_res2);
   112 
   113 fun morph_fp_sugar phi {T, fp, index, pre_bnfs, fp_res, ctr_defss, ctr_sugars, co_iterss,
   114     co_inducts, co_iter_thmsss} =
   115   {T = Morphism.typ phi T, fp = fp, index = index, pre_bnfs = map (morph_bnf phi)
   116    pre_bnfs, fp_res = morph_fp_result phi fp_res,
   117    ctr_defss = map (map (Morphism.thm phi)) ctr_defss,
   118    ctr_sugars = map (morph_ctr_sugar phi) ctr_sugars,
   119    co_iterss = map (map (Morphism.term phi)) co_iterss,
   120    co_inducts = map (Morphism.thm phi) co_inducts,
   121    co_iter_thmsss = map (map (map (Morphism.thm phi))) co_iter_thmsss};
   122 
   123 structure Data = Generic_Data
   124 (
   125   type T = fp_sugar Symtab.table;
   126   val empty = Symtab.empty;
   127   val extend = I;
   128   val merge = Symtab.merge eq_fp_sugar;
   129 );
   130 
   131 val fp_sugar_of = Symtab.lookup o Data.get o Context.Proof;
   132 
   133 fun register_fp_sugar key fp_sugar =
   134   Local_Theory.declaration {syntax = false, pervasive = true}
   135     (fn phi => Data.map (Symtab.update_new (key, morph_fp_sugar phi fp_sugar)));
   136 
   137 fun register_fp_sugars fp pre_bnfs (fp_res as {Ts, ...}) ctr_defss ctr_sugars co_iterss co_inducts
   138     co_iter_thmsss lthy =
   139   (0, lthy)
   140   |> fold (fn T as Type (s, _) => fn (kk, lthy) => (kk + 1,
   141     register_fp_sugar s {T = T, fp = fp, index = kk, pre_bnfs = pre_bnfs, fp_res = fp_res,
   142         ctr_defss = ctr_defss, ctr_sugars = ctr_sugars, co_iterss = co_iterss,
   143         co_inducts = co_inducts, co_iter_thmsss = co_iter_thmsss}
   144       lthy)) Ts
   145   |> snd;
   146 
   147 (* This function could produce clashes in contrived examples (e.g., "x.A", "x.x_A", "y.A"). *)
   148 fun quasi_unambiguous_case_names names =
   149   let
   150     val ps = map (`Long_Name.base_name) names;
   151     val dups = Library.duplicates (op =) (map fst ps);
   152     fun underscore s =
   153       let val ss = space_explode Long_Name.separator s in
   154         space_implode "_" (drop (length ss - 2) ss)
   155       end;
   156   in
   157     map (fn (base, full) => if member (op =) dups base then underscore full else base) ps
   158   end;
   159 
   160 val mp_conj = @{thm mp_conj};
   161 
   162 val simp_attrs = @{attributes [simp]};
   163 val code_simp_attrs = Code.add_default_eqn_attrib :: simp_attrs;
   164 
   165 fun tvar_subst thy Ts Us =
   166   Vartab.fold (cons o apsnd snd) (fold (Sign.typ_match thy) (Ts ~~ Us) Vartab.empty) [];
   167 
   168 val exists_subtype_in = Term.exists_subtype o member (op =);
   169 
   170 fun resort_tfree S (TFree (s, _)) = TFree (s, S);
   171 
   172 fun typ_subst_nonatomic inst (T as Type (s, Ts)) =
   173     (case AList.lookup (op =) inst T of
   174       NONE => Type (s, map (typ_subst_nonatomic inst) Ts)
   175     | SOME T' => T')
   176   | typ_subst_nonatomic inst T = the_default T (AList.lookup (op =) inst T);
   177 
   178 val lists_bmoc = fold (fn xs => fn t => Term.list_comb (t, xs));
   179 
   180 fun flat_rec xss =
   181   (* The first line below gives the preferred order. The second line is for compatibility with the
   182      old datatype package: *)
   183 (*
   184   flat xss
   185 *)
   186   map hd xss @ maps tl xss;
   187 
   188 fun flat_predss_getterss qss fss = maps (op @) (qss ~~ fss);
   189 
   190 fun flat_preds_predsss_gettersss [] [qss] [fss] = flat_predss_getterss qss fss
   191   | flat_preds_predsss_gettersss (p :: ps) (qss :: qsss) (fss :: fsss) =
   192     p :: flat_predss_getterss qss fss @ flat_preds_predsss_gettersss ps qsss fsss;
   193 
   194 fun mk_tupled_fun x f xs = HOLogic.tupled_lambda x (Term.list_comb (f, xs));
   195 fun mk_uncurried_fun f xs = mk_tupled_fun (HOLogic.mk_tuple xs) f xs;
   196 fun mk_uncurried2_fun f xss =
   197   mk_tupled_fun (HOLogic.mk_tuple (map HOLogic.mk_tuple xss)) f (flat_rec xss);
   198 
   199 fun mk_flip (x, Type (_, [T1, Type (_, [T2, T3])])) =
   200   Abs ("x", T1, Abs ("y", T2, Var (x, T2 --> T1 --> T3) $ Bound 0 $ Bound 1));
   201 
   202 fun flip_rels lthy n thm =
   203   let
   204     val Rs = Term.add_vars (prop_of thm) [];
   205     val Rs' = rev (drop (length Rs - n) Rs);
   206     val cRs = map (fn f => (certify lthy (Var f), certify lthy (mk_flip f))) Rs';
   207   in
   208     Drule.cterm_instantiate cRs thm
   209   end;
   210 
   211 fun mk_ctor_or_dtor get_T Ts t =
   212   let val Type (_, Ts0) = get_T (fastype_of t) in
   213     Term.subst_atomic_types (Ts0 ~~ Ts) t
   214   end;
   215 
   216 val mk_ctor = mk_ctor_or_dtor range_type;
   217 val mk_dtor = mk_ctor_or_dtor domain_type;
   218 
   219 fun mk_co_iter thy fp fpT Cs t =
   220   let
   221     val (binders, body) = strip_type (fastype_of t);
   222     val (f_Cs, prebody) = split_last binders;
   223     val fpT0 = if fp = Least_FP then prebody else body;
   224     val Cs0 = distinct (op =) (map (if fp = Least_FP then body_type else domain_type) f_Cs);
   225     val rho = tvar_subst thy (fpT0 :: Cs0) (fpT :: Cs);
   226   in
   227     Term.subst_TVars rho t
   228   end;
   229 
   230 fun mk_co_iters thy fp fpTs Cs ts0 =
   231   let
   232     val nn = length fpTs;
   233     val (fpTs0, Cs0) =
   234       map ((fp = Greatest_FP ? swap) o dest_funT o snd o strip_typeN nn o fastype_of) ts0
   235       |> split_list;
   236     val rho = tvar_subst thy (fpTs0 @ Cs0) (fpTs @ Cs);
   237   in
   238     map (Term.subst_TVars rho) ts0
   239   end;
   240 
   241 val mk_fp_iter_fun_types = fst o split_last o binder_types o fastype_of;
   242 
   243 fun unzip_recT Cs (T as Type (@{type_name prod}, Ts as [_, U])) =
   244     if member (op =) Cs U then Ts else [T]
   245   | unzip_recT _ T = [T];
   246 
   247 fun mk_map live Ts Us t =
   248   let val (Type (_, Ts0), Type (_, Us0)) = strip_typeN (live + 1) (fastype_of t) |>> List.last in
   249     Term.subst_atomic_types (Ts0 @ Us0 ~~ Ts @ Us) t
   250   end;
   251 
   252 fun mk_rel live Ts Us t =
   253   let val [Type (_, Ts0), Type (_, Us0)] = binder_types (snd (strip_typeN live (fastype_of t))) in
   254     Term.subst_atomic_types (Ts0 @ Us0 ~~ Ts @ Us) t
   255   end;
   256 
   257 fun build_map lthy build_simple =
   258   let
   259     fun build (TU as (T, U)) =
   260       if T = U then
   261         id_const T
   262       else
   263         (case TU of
   264           (Type (s, Ts), Type (s', Us)) =>
   265           if s = s' then
   266             let
   267               val bnf = the (bnf_of lthy s);
   268               val live = live_of_bnf bnf;
   269               val mapx = mk_map live Ts Us (map_of_bnf bnf);
   270               val TUs' = map dest_funT (fst (strip_typeN live (fastype_of mapx)));
   271             in Term.list_comb (mapx, map build TUs') end
   272           else
   273             build_simple TU
   274         | _ => build_simple TU);
   275   in build end;
   276 
   277 fun liveness_of_fp_bnf n bnf =
   278   (case T_of_bnf bnf of
   279     Type (_, Ts) => map (not o member (op =) (deads_of_bnf bnf)) Ts
   280   | _ => replicate n false);
   281 
   282 fun cannot_merge_types () = error "Mutually recursive types must have the same type parameters";
   283 
   284 fun merge_type_arg T T' = if T = T' then T else cannot_merge_types ();
   285 
   286 fun merge_type_args (As, As') =
   287   if length As = length As' then map2 merge_type_arg As As' else cannot_merge_types ();
   288 
   289 fun reassoc_conjs thm =
   290   reassoc_conjs (thm RS @{thm conj_assoc[THEN iffD1]})
   291   handle THM _ => thm;
   292 
   293 fun type_args_named_constrained_of ((((ncAs, _), _), _), _) = ncAs;
   294 fun type_binding_of ((((_, b), _), _), _) = b;
   295 fun map_binding_of (((_, (b, _)), _), _) = b;
   296 fun rel_binding_of (((_, (_, b)), _), _) = b;
   297 fun mixfix_of ((_, mx), _) = mx;
   298 fun ctr_specs_of (_, ctr_specs) = ctr_specs;
   299 
   300 fun disc_of ((((disc, _), _), _), _) = disc;
   301 fun ctr_of ((((_, ctr), _), _), _) = ctr;
   302 fun args_of (((_, args), _), _) = args;
   303 fun defaults_of ((_, ds), _) = ds;
   304 fun ctr_mixfix_of (_, mx) = mx;
   305 
   306 fun add_nesty_bnf_names Us =
   307   let
   308     fun add (Type (s, Ts)) ss =
   309         let val (needs, ss') = fold_map add Ts ss in
   310           if exists I needs then (true, insert (op =) s ss') else (false, ss')
   311         end
   312       | add T ss = (member (op =) Us T, ss);
   313   in snd oo add end;
   314 
   315 fun nesty_bnfs ctxt ctr_Tsss Us =
   316   map_filter (bnf_of ctxt) (fold (fold (fold (add_nesty_bnf_names Us))) ctr_Tsss []);
   317 
   318 fun indexify proj xs f p = f (find_index (curry (op =) (proj p)) xs) p;
   319 
   320 fun mk_fun_arg_typess n ms = map2 dest_tupleT ms o dest_sumTN_balanced n o domain_type;
   321 
   322 fun mk_iter_fun_arg_typessss Cs ns mss =
   323   mk_fp_iter_fun_types
   324   #> map3 mk_fun_arg_typess ns mss
   325   #> map (map (map (unzip_recT Cs)));
   326 
   327 fun mk_iters_args_types Cs ns mss ctor_iter_fun_Tss lthy =
   328   let
   329     val Css = map2 replicate ns Cs;
   330     val y_Tsss = map3 mk_fun_arg_typess ns mss (map un_fold_of ctor_iter_fun_Tss);
   331     val g_Tss = map2 (fn C => map (fn y_Ts => y_Ts ---> C)) Cs y_Tsss;
   332 
   333     val ((gss, ysss), lthy) =
   334       lthy
   335       |> mk_Freess "f" g_Tss
   336       ||>> mk_Freesss "x" y_Tsss;
   337 
   338     val y_Tssss = map (map (map single)) y_Tsss;
   339     val yssss = map (map (map single)) ysss;
   340 
   341     val z_Tssss =
   342       map3 (fn n => fn ms => map2 (map (unzip_recT Cs) oo dest_tupleT) ms o
   343         dest_sumTN_balanced n o domain_type o co_rec_of) ns mss ctor_iter_fun_Tss;
   344 
   345     val z_Tsss' = map (map flat_rec) z_Tssss;
   346     val h_Tss = map2 (map2 (curry (op --->))) z_Tsss' Css;
   347 
   348     val hss = map2 (map2 retype_free) h_Tss gss;
   349     val zssss_hd = map2 (map2 (map2 (retype_free o hd))) z_Tssss ysss;
   350     val (zssss_tl, lthy) =
   351       lthy
   352       |> mk_Freessss "y" (map (map (map tl)) z_Tssss);
   353     val zssss = map2 (map2 (map2 cons)) zssss_hd zssss_tl;
   354   in
   355     ([(g_Tss, y_Tssss, gss, yssss), (h_Tss, z_Tssss, hss, zssss)], lthy)
   356   end;
   357 
   358 fun mk_coiters_args_types Cs ns mss dtor_coiter_fun_Tss lthy =
   359   let
   360     (*avoid "'a itself" arguments in coiterators and corecursors*)
   361     fun repair_arity [0] = [1]
   362       | repair_arity ms = ms;
   363 
   364     fun unzip_corecT (T as Type (@{type_name sum}, Ts as [_, U])) =
   365         if member (op =) Cs U then Ts else [T]
   366       | unzip_corecT T = [T];
   367 
   368     val p_Tss = map2 (fn n => replicate (Int.max (0, n - 1)) o mk_pred1T) ns Cs;
   369 
   370     fun mk_types maybe_unzipT get_Ts =
   371       let
   372         val fun_Ts = map get_Ts dtor_coiter_fun_Tss;
   373         val f_sum_prod_Ts = map range_type fun_Ts;
   374         val f_prod_Tss = map2 dest_sumTN_balanced ns f_sum_prod_Ts;
   375         val f_Tsss = map2 (map2 dest_tupleT o repair_arity) mss f_prod_Tss;
   376         val f_Tssss = map2 (fn C => map (map (map (curry (op -->) C) o maybe_unzipT))) Cs f_Tsss;
   377         val q_Tssss =
   378           map (map (map (fn [_] => [] | [_, T] => [mk_pred1T (domain_type T)]))) f_Tssss;
   379         val pf_Tss = map3 flat_preds_predsss_gettersss p_Tss q_Tssss f_Tssss;
   380       in (q_Tssss, f_Tsss, f_Tssss, (f_sum_prod_Ts, pf_Tss)) end;
   381 
   382     val (r_Tssss, g_Tsss, g_Tssss, unfold_types) = mk_types single un_fold_of;
   383     val (s_Tssss, h_Tsss, h_Tssss, corec_types) = mk_types unzip_corecT co_rec_of;
   384 
   385     val ((((Free (z, _), cs), pss), gssss), lthy) =
   386       lthy
   387       |> yield_singleton (mk_Frees "z") dummyT
   388       ||>> mk_Frees "a" Cs
   389       ||>> mk_Freess "p" p_Tss
   390       ||>> mk_Freessss "g" g_Tssss;
   391     val rssss = map (map (map (fn [] => []))) r_Tssss;
   392 
   393     val hssss_hd = map2 (map2 (map2 (fn T :: _ => fn [g] => retype_free T g))) h_Tssss gssss;
   394     val ((sssss, hssss_tl), lthy) =
   395       lthy
   396       |> mk_Freessss "q" s_Tssss
   397       ||>> mk_Freessss "h" (map (map (map tl)) h_Tssss);
   398     val hssss = map2 (map2 (map2 cons)) hssss_hd hssss_tl;
   399 
   400     val cpss = map2 (map o rapp) cs pss;
   401 
   402     fun build_sum_inj mk_inj = build_map lthy (uncurry mk_inj o dest_sumT o snd);
   403 
   404     fun build_dtor_coiter_arg _ [] [cf] = cf
   405       | build_dtor_coiter_arg T [cq] [cf, cf'] =
   406         mk_If cq (build_sum_inj Inl_const (fastype_of cf, T) $ cf)
   407           (build_sum_inj Inr_const (fastype_of cf', T) $ cf');
   408 
   409     fun mk_args qssss fssss f_Tsss =
   410       let
   411         val pfss = map3 flat_preds_predsss_gettersss pss qssss fssss;
   412         val cqssss = map2 (map o map o map o rapp) cs qssss;
   413         val cfssss = map2 (map o map o map o rapp) cs fssss;
   414         val cqfsss = map3 (map3 (map3 build_dtor_coiter_arg)) f_Tsss cqssss cfssss;
   415       in (pfss, cqfsss) end;
   416 
   417     val unfold_args = mk_args rssss gssss g_Tsss;
   418     val corec_args = mk_args sssss hssss h_Tsss;
   419   in
   420     ((z, cs, cpss, [(unfold_args, unfold_types), (corec_args, corec_types)]), lthy)
   421   end;
   422 
   423 fun mk_co_iters_prelims fp fpTs Cs ns mss xtor_co_iterss0 lthy =
   424   let
   425     val thy = Proof_Context.theory_of lthy;
   426 
   427     val (xtor_co_iter_fun_Tss, xtor_co_iterss) =
   428       map (mk_co_iters thy fp fpTs Cs #> `(mk_fp_iter_fun_types o hd)) (transpose xtor_co_iterss0)
   429       |> apsnd transpose o apfst transpose o split_list;
   430 
   431     val ((iters_args_types, coiters_args_types), lthy') =
   432       if fp = Least_FP then
   433         mk_iters_args_types Cs ns mss xtor_co_iter_fun_Tss lthy |>> (rpair NONE o SOME)
   434       else
   435         mk_coiters_args_types Cs ns mss xtor_co_iter_fun_Tss lthy |>> (pair NONE o SOME)
   436   in
   437     ((xtor_co_iterss, iters_args_types, coiters_args_types), lthy')
   438   end;
   439 
   440 fun mk_iter_body ctor_iter fss xssss =
   441   Term.list_comb (ctor_iter, map2 (mk_sum_caseN_balanced oo map2 mk_uncurried2_fun) fss xssss);
   442 
   443 fun mk_preds_getterss_join c cps sum_prod_T cqfss =
   444   let val n = length cqfss in
   445     Term.lambda c (mk_IfN sum_prod_T cps
   446       (map2 (mk_InN_balanced sum_prod_T n) (map HOLogic.mk_tuple cqfss) (1 upto n)))
   447   end;
   448 
   449 fun mk_coiter_body cs cpss f_sum_prod_Ts cqfsss dtor_coiter =
   450   Term.list_comb (dtor_coiter, map4 mk_preds_getterss_join cs cpss f_sum_prod_Ts cqfsss);
   451 
   452 fun define_co_iters fp fpT Cs binding_specs lthy0 =
   453   let
   454     val thy = Proof_Context.theory_of lthy0;
   455 
   456     val ((csts, defs), (lthy', lthy)) = lthy0
   457       |> apfst split_list o fold_map (fn (b, spec) =>
   458         Specification.definition (SOME (b, NONE, NoSyn), ((Thm.def_binding b, []), spec))
   459         #>> apsnd snd) binding_specs
   460       ||> `Local_Theory.restore;
   461 
   462     val phi = Proof_Context.export_morphism lthy lthy';
   463 
   464     val csts' = map (mk_co_iter thy fp fpT Cs o Morphism.term phi) csts;
   465     val defs' = map (Morphism.thm phi) defs;
   466   in
   467     ((csts', defs'), lthy')
   468   end;
   469 
   470 fun define_iters iterNs iter_args_typess' mk_binding fpTs Cs ctor_iters lthy =
   471   let
   472     val nn = length fpTs;
   473 
   474     val fpT_to_C as Type (_, [fpT, _]) = snd (strip_typeN nn (fastype_of (hd ctor_iters)));
   475 
   476     fun generate_iter suf (f_Tss, _, fss, xssss) ctor_iter =
   477       let
   478         val res_T = fold_rev (curry (op --->)) f_Tss fpT_to_C;
   479         val b = mk_binding suf;
   480         val spec =
   481           mk_Trueprop_eq (lists_bmoc fss (Free (Binding.name_of b, res_T)),
   482             mk_iter_body ctor_iter fss xssss);
   483       in (b, spec) end;
   484   in
   485     define_co_iters Least_FP fpT Cs (map3 generate_iter iterNs iter_args_typess' ctor_iters) lthy
   486   end;
   487 
   488 fun define_coiters coiterNs (_, cs, cpss, coiter_args_typess') mk_binding fpTs Cs dtor_coiters
   489     lthy =
   490   let
   491     val nn = length fpTs;
   492 
   493     val C_to_fpT as Type (_, [_, fpT]) = snd (strip_typeN nn (fastype_of (hd dtor_coiters)));
   494 
   495     fun generate_coiter suf ((pfss, cqfsss), (f_sum_prod_Ts, pf_Tss)) dtor_coiter =
   496       let
   497         val res_T = fold_rev (curry (op --->)) pf_Tss C_to_fpT;
   498         val b = mk_binding suf;
   499         val spec =
   500           mk_Trueprop_eq (lists_bmoc pfss (Free (Binding.name_of b, res_T)),
   501             mk_coiter_body cs cpss f_sum_prod_Ts cqfsss dtor_coiter);
   502       in (b, spec) end;
   503   in
   504     define_co_iters Greatest_FP fpT Cs
   505       (map3 generate_coiter coiterNs coiter_args_typess' dtor_coiters) lthy
   506   end;
   507 
   508 fun derive_induct_iters_thms_for_types pre_bnfs [fold_args_types, rec_args_types] [ctor_induct]
   509     ctor_iter_thmss nesting_bnfs nested_bnfs fpTs Cs Xs ctrXs_Tsss ctrss ctr_defss iterss iter_defss
   510     lthy =
   511   let
   512     val iterss' = transpose iterss;
   513     val iter_defss' = transpose iter_defss;
   514 
   515     val [folds, recs] = iterss';
   516     val [fold_defs, rec_defs] = iter_defss';
   517 
   518     val ctr_Tsss = map (map (binder_types o fastype_of)) ctrss;
   519 
   520     val nn = length pre_bnfs;
   521     val ns = map length ctr_Tsss;
   522     val mss = map (map length) ctr_Tsss;
   523 
   524     val pre_map_defs = map map_def_of_bnf pre_bnfs;
   525     val pre_set_defss = map set_defs_of_bnf pre_bnfs;
   526     val nesting_map_ids'' = map (unfold_thms lthy @{thms id_def} o map_id_of_bnf) nesting_bnfs;
   527     val nested_map_ids'' = map (unfold_thms lthy @{thms id_def} o map_id_of_bnf) nested_bnfs;
   528     val nested_set_map's = maps set_map'_of_bnf nested_bnfs;
   529 
   530     val fp_b_names = map base_name_of_typ fpTs;
   531 
   532     val ((((ps, ps'), xsss), us'), names_lthy) =
   533       lthy
   534       |> mk_Frees' "P" (map mk_pred1T fpTs)
   535       ||>> mk_Freesss "x" ctr_Tsss
   536       ||>> Variable.variant_fixes fp_b_names;
   537 
   538     val us = map2 (curry Free) us' fpTs;
   539 
   540     fun mk_sets_nested bnf =
   541       let
   542         val Type (T_name, Us) = T_of_bnf bnf;
   543         val lives = lives_of_bnf bnf;
   544         val sets = sets_of_bnf bnf;
   545         fun mk_set U =
   546           (case find_index (curry (op =) U) lives of
   547             ~1 => Term.dummy
   548           | i => nth sets i);
   549       in
   550         (T_name, map mk_set Us)
   551       end;
   552 
   553     val setss_nested = map mk_sets_nested nested_bnfs;
   554 
   555     val (induct_thms, induct_thm) =
   556       let
   557         fun mk_set Ts t =
   558           let val Type (_, Ts0) = domain_type (fastype_of t) in
   559             Term.subst_atomic_types (Ts0 ~~ Ts) t
   560           end;
   561 
   562         fun mk_raw_prem_prems _ (x as Free (_, Type _)) (X as TFree _) =
   563             [([], (find_index (curry (op =) X) Xs + 1, x))]
   564           | mk_raw_prem_prems names_lthy (x as Free (s, Type (T_name, Ts0))) (Type (_, Xs_Ts0)) =
   565             (case AList.lookup (op =) setss_nested T_name of
   566               NONE => []
   567             | SOME raw_sets0 =>
   568               let
   569                 val (Xs_Ts, (Ts, raw_sets)) =
   570                   filter (exists_subtype_in Xs o fst) (Xs_Ts0 ~~ (Ts0 ~~ raw_sets0))
   571                   |> split_list ||> split_list;
   572                 val sets = map (mk_set Ts0) raw_sets;
   573                 val (ys, names_lthy') = names_lthy |> mk_Frees s Ts;
   574                 val xysets = map (pair x) (ys ~~ sets);
   575                 val ppremss = map2 (mk_raw_prem_prems names_lthy') ys Xs_Ts;
   576               in
   577                 flat (map2 (map o apfst o cons) xysets ppremss)
   578               end)
   579           | mk_raw_prem_prems _ _ _ = [];
   580 
   581         fun close_prem_prem xs t =
   582           fold_rev Logic.all (map Free (drop (nn + length xs)
   583             (rev (Term.add_frees t (map dest_Free xs @ ps'))))) t;
   584 
   585         fun mk_prem_prem xs (xysets, (j, x)) =
   586           close_prem_prem xs (Logic.list_implies (map (fn (x', (y, set)) =>
   587               HOLogic.mk_Trueprop (HOLogic.mk_mem (y, set $ x'))) xysets,
   588             HOLogic.mk_Trueprop (nth ps (j - 1) $ x)));
   589 
   590         fun mk_raw_prem phi ctr ctr_Ts ctrXs_Ts =
   591           let
   592             val (xs, names_lthy') = names_lthy |> mk_Frees "x" ctr_Ts;
   593             val pprems = flat (map2 (mk_raw_prem_prems names_lthy') xs ctrXs_Ts);
   594           in (xs, pprems, HOLogic.mk_Trueprop (phi $ Term.list_comb (ctr, xs))) end;
   595 
   596         fun mk_prem (xs, raw_pprems, concl) =
   597           fold_rev Logic.all xs (Logic.list_implies (map (mk_prem_prem xs) raw_pprems, concl));
   598 
   599         val raw_premss = map4 (map3 o mk_raw_prem) ps ctrss ctr_Tsss ctrXs_Tsss;
   600 
   601         val goal =
   602           Library.foldr (Logic.list_implies o apfst (map mk_prem)) (raw_premss,
   603             HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj (map2 (curry (op $)) ps us)));
   604 
   605         val kksss = map (map (map (fst o snd) o #2)) raw_premss;
   606 
   607         val ctor_induct' = ctor_induct OF (map mk_sumEN_tupled_balanced mss);
   608 
   609         val thm =
   610           Goal.prove_sorry lthy [] [] goal (fn {context = ctxt, ...} =>
   611             mk_induct_tac ctxt nn ns mss kksss (flat ctr_defss) ctor_induct' nested_set_map's
   612               pre_set_defss)
   613           |> singleton (Proof_Context.export names_lthy lthy)
   614           |> Thm.close_derivation;
   615       in
   616         `(conj_dests nn) thm
   617       end;
   618 
   619     val induct_cases = quasi_unambiguous_case_names (maps (map name_of_ctr) ctrss);
   620     val induct_case_names_attr = Attrib.internal (K (Rule_Cases.case_names induct_cases));
   621 
   622     val xctrss = map2 (map2 (curry Term.list_comb)) ctrss xsss;
   623 
   624     fun mk_iter_thmss (_, x_Tssss, fss, _) iters iter_defs ctor_iter_thms =
   625       let
   626         val fiters = map (lists_bmoc fss) iters;
   627 
   628         fun mk_goal fss fiter xctr f xs fxs =
   629           fold_rev (fold_rev Logic.all) (xs :: fss)
   630             (mk_Trueprop_eq (fiter $ xctr, Term.list_comb (f, fxs)));
   631 
   632         fun maybe_tick (T, U) u f =
   633           if try (fst o HOLogic.dest_prodT) U = SOME T then
   634             Term.lambda u (HOLogic.mk_prod (u, f $ u))
   635           else
   636             f;
   637 
   638         fun unzip_iters (x as Free (_, T)) =
   639           map (fn U => if U = T then x else
   640             build_map lthy (indexify (perhaps (try (snd o HOLogic.dest_prodT)) o snd) Cs
   641               (fn kk => fn TU => maybe_tick TU (nth us kk) (nth fiters kk))) (T, U) $ x);
   642 
   643         val fxsss = map2 (map2 (flat_rec oo map2 unzip_iters)) xsss x_Tssss;
   644 
   645         val goalss = map5 (map4 o mk_goal fss) fiters xctrss fss xsss fxsss;
   646 
   647         val tacss =
   648           map2 (map o mk_iter_tac pre_map_defs (nested_map_ids'' @ nesting_map_ids'') iter_defs)
   649             ctor_iter_thms ctr_defss;
   650 
   651         fun prove goal tac =
   652           Goal.prove_sorry lthy [] [] goal (tac o #context)
   653           |> Thm.close_derivation;
   654       in
   655         map2 (map2 prove) goalss tacss
   656       end;
   657 
   658     val fold_thmss = mk_iter_thmss fold_args_types folds fold_defs (map un_fold_of ctor_iter_thmss);
   659     val rec_thmss = mk_iter_thmss rec_args_types recs rec_defs (map co_rec_of ctor_iter_thmss);
   660   in
   661     ((induct_thms, induct_thm, [induct_case_names_attr]),
   662      (fold_thmss, code_simp_attrs), (rec_thmss, code_simp_attrs))
   663   end;
   664 
   665 fun derive_coinduct_coiters_thms_for_types pre_bnfs (z, cs, cpss,
   666       [(unfold_args as (pgss, crgsss), _), (corec_args as (phss, cshsss), _)])
   667     dtor_coinducts dtor_ctors dtor_coiter_thmss nesting_bnfs nested_bnfs fpTs Cs As kss mss ns
   668     ctr_defss ctr_sugars coiterss coiter_defss lthy =
   669   let
   670     val coiterss' = transpose coiterss;
   671     val coiter_defss' = transpose coiter_defss;
   672 
   673     val [unfold_defs, corec_defs] = coiter_defss';
   674 
   675     val nn = length pre_bnfs;
   676 
   677     val pre_map_defs = map map_def_of_bnf pre_bnfs;
   678     val pre_rel_defs = map rel_def_of_bnf pre_bnfs;
   679     val nesting_map_ids'' = map (unfold_thms lthy @{thms id_def} o map_id_of_bnf) nesting_bnfs;
   680     val nesting_rel_eqs = map rel_eq_of_bnf nesting_bnfs;
   681     val nested_map_comp's = map map_comp'_of_bnf nested_bnfs;
   682     val nested_map_ids'' = map (unfold_thms lthy @{thms id_def} o map_id_of_bnf) nested_bnfs;
   683 
   684     val fp_b_names = map base_name_of_typ fpTs;
   685 
   686     val ctrss = map (map (mk_ctr As) o #ctrs) ctr_sugars;
   687     val discss = map (map (mk_disc_or_sel As) o #discs) ctr_sugars;
   688     val selsss = map (map (map (mk_disc_or_sel As)) o #selss) ctr_sugars;
   689     val exhausts = map #exhaust ctr_sugars;
   690     val disc_thmsss = map #disc_thmss ctr_sugars;
   691     val discIss = map #discIs ctr_sugars;
   692     val sel_thmsss = map #sel_thmss ctr_sugars;
   693 
   694     val (((rs, us'), vs'), names_lthy) =
   695       lthy
   696       |> mk_Frees "R" (map (fn T => mk_pred2T T T) fpTs)
   697       ||>> Variable.variant_fixes fp_b_names
   698       ||>> Variable.variant_fixes (map (suffix "'") fp_b_names);
   699 
   700     val us = map2 (curry Free) us' fpTs;
   701     val udiscss = map2 (map o rapp) us discss;
   702     val uselsss = map2 (map o map o rapp) us selsss;
   703 
   704     val vs = map2 (curry Free) vs' fpTs;
   705     val vdiscss = map2 (map o rapp) vs discss;
   706     val vselsss = map2 (map o map o rapp) vs selsss;
   707 
   708     val coinduct_thms_pairs =
   709       let
   710         val uvrs = map3 (fn r => fn u => fn v => r $ u $ v) rs us vs;
   711         val uv_eqs = map2 (curry HOLogic.mk_eq) us vs;
   712         val strong_rs =
   713           map4 (fn u => fn v => fn uvr => fn uv_eq =>
   714             fold_rev Term.lambda [u, v] (HOLogic.mk_disj (uvr, uv_eq))) us vs uvrs uv_eqs;
   715 
   716         (* TODO: generalize (cf. "build_map") *)
   717         fun build_rel rs' T =
   718           (case find_index (curry (op =) T) fpTs of
   719             ~1 =>
   720             if exists_subtype_in fpTs T then
   721               let
   722                 val Type (s, Ts) = T
   723                 val bnf = the (bnf_of lthy s);
   724                 val live = live_of_bnf bnf;
   725                 val rel = mk_rel live Ts Ts (rel_of_bnf bnf);
   726                 val Ts' = map domain_type (fst (strip_typeN live (fastype_of rel)));
   727               in Term.list_comb (rel, map (build_rel rs') Ts') end
   728             else
   729               HOLogic.eq_const T
   730           | kk => nth rs' kk);
   731 
   732         fun build_rel_app rs' usel vsel = fold rapp [usel, vsel] (build_rel rs' (fastype_of usel));
   733 
   734         fun mk_prem_ctr_concls rs' n k udisc usels vdisc vsels =
   735           (if k = n then [] else [HOLogic.mk_eq (udisc, vdisc)]) @
   736           (if null usels then
   737              []
   738            else
   739              [Library.foldr HOLogic.mk_imp (if n = 1 then [] else [udisc, vdisc],
   740                 Library.foldr1 HOLogic.mk_conj (map2 (build_rel_app rs') usels vsels))]);
   741 
   742         fun mk_prem_concl rs' n udiscs uselss vdiscs vselss =
   743           Library.foldr1 HOLogic.mk_conj
   744             (flat (map5 (mk_prem_ctr_concls rs' n) (1 upto n) udiscs uselss vdiscs vselss))
   745           handle List.Empty => @{term True};
   746 
   747         fun mk_prem rs' uvr u v n udiscs uselss vdiscs vselss =
   748           fold_rev Logic.all [u, v] (Logic.mk_implies (HOLogic.mk_Trueprop uvr,
   749             HOLogic.mk_Trueprop (mk_prem_concl rs' n udiscs uselss vdiscs vselss)));
   750 
   751         val concl =
   752           HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj
   753             (map3 (fn uvr => fn u => fn v => HOLogic.mk_imp (uvr, HOLogic.mk_eq (u, v)))
   754                uvrs us vs));
   755 
   756         fun mk_goal rs' =
   757           Logic.list_implies (map8 (mk_prem rs') uvrs us vs ns udiscss uselsss vdiscss vselsss,
   758             concl);
   759 
   760         val goals = map mk_goal [rs, strong_rs];
   761 
   762         fun prove dtor_coinduct' goal =
   763           Goal.prove_sorry lthy [] [] goal (fn {context = ctxt, ...} =>
   764             mk_coinduct_tac ctxt nesting_rel_eqs nn ns dtor_coinduct' pre_rel_defs dtor_ctors
   765               exhausts ctr_defss disc_thmsss sel_thmsss)
   766           |> singleton (Proof_Context.export names_lthy lthy)
   767           |> Thm.close_derivation;
   768 
   769         fun postproc nn thm =
   770           Thm.permute_prems 0 nn
   771             (if nn = 1 then thm RS mp else funpow nn (fn thm => reassoc_conjs (thm RS mp_conj)) thm)
   772           |> Drule.zero_var_indexes
   773           |> `(conj_dests nn);
   774       in
   775         map2 (postproc nn oo prove) dtor_coinducts goals
   776       end;
   777 
   778     fun mk_coinduct_concls ms discs ctrs =
   779       let
   780         fun mk_disc_concl disc = [name_of_disc disc];
   781         fun mk_ctr_concl 0 _ = []
   782           | mk_ctr_concl _ ctor = [name_of_ctr ctor];
   783         val disc_concls = map mk_disc_concl (fst (split_last discs)) @ [[]];
   784         val ctr_concls = map2 mk_ctr_concl ms ctrs;
   785       in
   786         flat (map2 append disc_concls ctr_concls)
   787       end;
   788 
   789     val coinduct_cases = quasi_unambiguous_case_names (map (prefix EqN) fp_b_names);
   790     val coinduct_conclss =
   791       map3 (quasi_unambiguous_case_names ooo mk_coinduct_concls) mss discss ctrss;
   792 
   793     fun mk_maybe_not pos = not pos ? HOLogic.mk_not;
   794 
   795     val fcoiterss' as [gunfolds, hcorecs] =
   796       map2 (fn (pfss, _) => map (lists_bmoc pfss)) [unfold_args, corec_args] coiterss';
   797 
   798     val (unfold_thmss, corec_thmss, safe_unfold_thmss, safe_corec_thmss) =
   799       let
   800         fun mk_goal pfss c cps fcoiter n k ctr m cfs' =
   801           fold_rev (fold_rev Logic.all) ([c] :: pfss)
   802             (Logic.list_implies (seq_conds (HOLogic.mk_Trueprop oo mk_maybe_not) n k cps,
   803                mk_Trueprop_eq (fcoiter $ c, Term.list_comb (ctr, take m cfs'))));
   804 
   805         fun build_coiter fcoiters maybe_tack (T, U) =
   806           if T = U then
   807              id_const T
   808           else
   809             (case find_index (curry (op =) U) fpTs of
   810               ~1 => build_map lthy (build_coiter fcoiters maybe_tack) (T, U)
   811             | kk => maybe_tack (nth cs kk, nth us kk) (nth fcoiters kk));
   812  
   813         fun mk_U maybe_mk_sumT =
   814           typ_subst_nonatomic (map2 (fn C => fn fpT => (maybe_mk_sumT fpT C, fpT)) Cs fpTs);
   815 
   816         fun tack z_name (c, u) f =
   817           let val z = Free (z_name, mk_sumT (fastype_of u, fastype_of c)) in
   818             Term.lambda z (mk_sum_case (Term.lambda u u, Term.lambda c (f $ c)) $ z)
   819           end;
   820 
   821         fun intr_coiters fcoiters maybe_mk_sumT maybe_tack cqf =
   822           let val T = fastype_of cqf in
   823             if exists_subtype_in Cs T then
   824               build_coiter fcoiters maybe_tack (T, mk_U maybe_mk_sumT T) $ cqf
   825             else
   826               cqf
   827           end;
   828 
   829         val crgsss' =
   830           (fn fcoiters => fn (_, cqfsss) =>
   831               map (map (map (intr_coiters fcoiters (K I) (K I)))) cqfsss)
   832             (un_fold_of fcoiterss') unfold_args;
   833         val cshsss' =
   834           (fn fcoiters => fn (_, cqfsss) =>
   835               map (map (map (intr_coiters fcoiters (curry mk_sumT) (tack z)))) cqfsss)
   836             (co_rec_of fcoiterss') corec_args;
   837 
   838 (*###
   839         val [crgsss', cshsss'] =
   840           map2 (fn fcoiters => fn (_, cqssss, cfssss) =>
   841               map2 (map2 (map2 (intr_coiters fcoiters))) cqssss cfssss)
   842             fcoiterss' [unfold_args, corec_args];
   843 *)
   844 
   845         val unfold_goalss = map8 (map4 oooo mk_goal pgss) cs cpss gunfolds ns kss ctrss mss crgsss';
   846         val corec_goalss = map8 (map4 oooo mk_goal phss) cs cpss hcorecs ns kss ctrss mss cshsss';
   847 
   848         fun mk_map_if_distrib bnf =
   849           let
   850             val mapx = map_of_bnf bnf;
   851             val live = live_of_bnf bnf;
   852             val ((Ts, T), U) = strip_typeN (live + 1) (fastype_of mapx) |>> split_last;
   853             val fs = Variable.variant_frees lthy [mapx] (map (pair "f") Ts);
   854             val t = Term.list_comb (mapx, map (Var o apfst (rpair 0)) fs);
   855           in
   856             Drule.instantiate' (map (SOME o certifyT lthy) [U, T]) [SOME (certify lthy t)]
   857               @{thm if_distrib}
   858           end;
   859 
   860         val nested_map_if_distribs = map mk_map_if_distrib nested_bnfs;
   861 
   862         val unfold_tacss =
   863           map3 (map oo mk_coiter_tac unfold_defs nesting_map_ids'')
   864             (map un_fold_of dtor_coiter_thmss) pre_map_defs ctr_defss;
   865         val corec_tacss =
   866           map3 (map oo mk_coiter_tac corec_defs nesting_map_ids'')
   867             (map co_rec_of dtor_coiter_thmss) pre_map_defs ctr_defss;
   868 
   869         fun prove goal tac =
   870           Goal.prove_sorry lthy [] [] goal (tac o #context)
   871           |> Thm.close_derivation;
   872 
   873         val unfold_thmss = map2 (map2 prove) unfold_goalss unfold_tacss;
   874         val corec_thmss =
   875           map2 (map2 prove) corec_goalss corec_tacss
   876           |> map (map (unfold_thms lthy @{thms sum_case_if}));
   877 
   878         val unfold_safesss = map2 (map2 (map2 (curry (op =)))) crgsss' crgsss;
   879         val corec_safesss = map2 (map2 (map2 (curry (op =)))) cshsss' cshsss;
   880 
   881         val filter_safesss =
   882           map2 (map_filter (fn (safes, thm) => if forall I safes then SOME thm else NONE) oo
   883             curry (op ~~));
   884 
   885         val safe_unfold_thmss = filter_safesss unfold_safesss unfold_thmss;
   886         val safe_corec_thmss = filter_safesss corec_safesss corec_thmss;
   887       in
   888         (unfold_thmss, corec_thmss, safe_unfold_thmss, safe_corec_thmss)
   889       end;
   890 
   891     val (disc_unfold_iff_thmss, disc_corec_iff_thmss) =
   892       let
   893         fun mk_goal c cps fcoiter n k disc =
   894           mk_Trueprop_eq (disc $ (fcoiter $ c),
   895             if n = 1 then @{const True}
   896             else Library.foldr1 HOLogic.mk_conj (seq_conds mk_maybe_not n k cps));
   897 
   898         val unfold_goalss = map6 (map2 oooo mk_goal) cs cpss gunfolds ns kss discss;
   899         val corec_goalss = map6 (map2 oooo mk_goal) cs cpss hcorecs ns kss discss;
   900 
   901         fun mk_case_split' cp = Drule.instantiate' [] [SOME (certify lthy cp)] @{thm case_split};
   902 
   903         val case_splitss' = map (map mk_case_split') cpss;
   904 
   905         val unfold_tacss =
   906           map3 (map oo mk_disc_coiter_iff_tac) case_splitss' unfold_thmss disc_thmsss;
   907         val corec_tacss =
   908           map3 (map oo mk_disc_coiter_iff_tac) case_splitss' corec_thmss disc_thmsss;
   909 
   910         fun prove goal tac =
   911           Goal.prove_sorry lthy [] [] goal (tac o #context)
   912           |> singleton (Proof_Context.export names_lthy lthy)
   913           |> Thm.close_derivation;
   914 
   915         fun proves [_] [_] = []
   916           | proves goals tacs = map2 prove goals tacs;
   917       in
   918         (map2 proves unfold_goalss unfold_tacss, map2 proves corec_goalss corec_tacss)
   919       end;
   920 
   921     val is_triv_discI = is_triv_implies orf is_concl_refl;
   922 
   923     fun mk_disc_coiter_thms coiters discIs =
   924       map (op RS) (filter_out (is_triv_discI o snd) (coiters ~~ discIs));
   925 
   926     val disc_unfold_thmss = map2 mk_disc_coiter_thms unfold_thmss discIss;
   927     val disc_corec_thmss = map2 mk_disc_coiter_thms corec_thmss discIss;
   928 
   929     fun mk_sel_coiter_thm coiter_thm sel sel_thm =
   930       let
   931         val (domT, ranT) = dest_funT (fastype_of sel);
   932         val arg_cong' =
   933           Drule.instantiate' (map (SOME o certifyT lthy) [domT, ranT])
   934             [NONE, NONE, SOME (certify lthy sel)] arg_cong
   935           |> Thm.varifyT_global;
   936         val sel_thm' = sel_thm RSN (2, trans);
   937       in
   938         coiter_thm RS arg_cong' RS sel_thm'
   939       end;
   940 
   941     fun mk_sel_coiter_thms coiter_thmss =
   942       map3 (map3 (map2 o mk_sel_coiter_thm)) coiter_thmss selsss sel_thmsss |> map flat;
   943 
   944     val sel_unfold_thmss = mk_sel_coiter_thms unfold_thmss;
   945     val sel_corec_thmss = mk_sel_coiter_thms corec_thmss;
   946 
   947     val coinduct_consumes_attr = Attrib.internal (K (Rule_Cases.consumes nn));
   948     val coinduct_case_names_attr = Attrib.internal (K (Rule_Cases.case_names coinduct_cases));
   949     val coinduct_case_concl_attrs =
   950       map2 (fn casex => fn concls =>
   951           Attrib.internal (K (Rule_Cases.case_conclusion (casex, concls))))
   952         coinduct_cases coinduct_conclss;
   953     val coinduct_case_attrs =
   954       coinduct_consumes_attr :: coinduct_case_names_attr :: coinduct_case_concl_attrs;
   955   in
   956     ((coinduct_thms_pairs, coinduct_case_attrs),
   957      (unfold_thmss, corec_thmss, []),
   958      (safe_unfold_thmss, safe_corec_thmss),
   959      (disc_unfold_thmss, disc_corec_thmss, simp_attrs),
   960      (disc_unfold_iff_thmss, disc_corec_iff_thmss, simp_attrs),
   961      (sel_unfold_thmss, sel_corec_thmss, simp_attrs))
   962   end;
   963 
   964 fun define_co_datatypes prepare_constraint prepare_typ prepare_term fp construct_fp
   965     (wrap_opts as (no_dests, rep_compat), specs) no_defs_lthy0 =
   966   let
   967     (* TODO: sanity checks on arguments *)
   968 
   969     val _ = if fp = Greatest_FP andalso no_dests then
   970         error "Cannot define destructor-less codatatypes"
   971       else
   972         ();
   973 
   974     fun qualify mandatory fp_b_name =
   975       Binding.qualify mandatory fp_b_name o (rep_compat ? Binding.qualify false rep_compat_prefix);
   976 
   977     val nn = length specs;
   978     val fp_bs = map type_binding_of specs;
   979     val fp_b_names = map Binding.name_of fp_bs;
   980     val fp_common_name = mk_common_name fp_b_names;
   981     val map_bs = map map_binding_of specs;
   982     val rel_bs = map rel_binding_of specs;
   983 
   984     fun prepare_type_arg (_, (ty, c)) =
   985       let val TFree (s, _) = prepare_typ no_defs_lthy0 ty in
   986         TFree (s, prepare_constraint no_defs_lthy0 c)
   987       end;
   988 
   989     val Ass0 = map (map prepare_type_arg o type_args_named_constrained_of) specs;
   990     val unsorted_Ass0 = map (map (resort_tfree HOLogic.typeS)) Ass0;
   991     val unsorted_As = Library.foldr1 merge_type_args unsorted_Ass0;
   992     val set_bss = map (map fst o type_args_named_constrained_of) specs;
   993 
   994     val (((Bs0, Cs), Xs), no_defs_lthy) =
   995       no_defs_lthy0
   996       |> fold (Variable.declare_typ o resort_tfree dummyS) unsorted_As
   997       |> mk_TFrees (length unsorted_As)
   998       ||>> mk_TFrees nn
   999       ||>> variant_tfrees fp_b_names;
  1000 
  1001     (* TODO: Cleaner handling of fake contexts, without "background_theory". The case where the new
  1002        type is defined in a locale and shadows an existing global type is currently not handled. *)
  1003 
  1004     fun add_fake_type spec =
  1005       Sign.add_type no_defs_lthy (type_binding_of spec,
  1006         length (type_args_named_constrained_of spec), mixfix_of spec);
  1007 
  1008     val fake_thy = Theory.copy #> fold add_fake_type specs;
  1009     val fake_lthy = Proof_Context.background_theory fake_thy no_defs_lthy;
  1010 
  1011     fun mk_fake_T b =
  1012       Type (fst (Term.dest_Type (Proof_Context.read_type_name fake_lthy true (Binding.name_of b))),
  1013         unsorted_As);
  1014 
  1015     val fake_Ts = map mk_fake_T fp_bs;
  1016 
  1017     val mixfixes = map mixfix_of specs;
  1018 
  1019     val _ = (case duplicates Binding.eq_name fp_bs of [] => ()
  1020       | b :: _ => error ("Duplicate type name declaration " ^ quote (Binding.name_of b)));
  1021 
  1022     val ctr_specss = map ctr_specs_of specs;
  1023 
  1024     val disc_bindingss = map (map disc_of) ctr_specss;
  1025     val ctr_bindingss =
  1026       map2 (fn fp_b_name => map (qualify false fp_b_name o ctr_of)) fp_b_names ctr_specss;
  1027     val ctr_argsss = map (map args_of) ctr_specss;
  1028     val ctr_mixfixess = map (map ctr_mixfix_of) ctr_specss;
  1029 
  1030     val sel_bindingsss = map (map (map fst)) ctr_argsss;
  1031     val fake_ctr_Tsss0 = map (map (map (prepare_typ fake_lthy o snd))) ctr_argsss;
  1032     val raw_sel_defaultsss = map (map defaults_of) ctr_specss;
  1033 
  1034     val (As :: _) :: fake_ctr_Tsss =
  1035       burrow (burrow (Syntax.check_typs fake_lthy)) (Ass0 :: fake_ctr_Tsss0);
  1036 
  1037     val _ = (case duplicates (op =) unsorted_As of [] => ()
  1038       | A :: _ => error ("Duplicate type parameter " ^
  1039           quote (Syntax.string_of_typ no_defs_lthy A)));
  1040 
  1041     val rhs_As' = fold (fold (fold Term.add_tfreesT)) fake_ctr_Tsss [];
  1042     val _ = (case subtract (op =) (map dest_TFree As) rhs_As' of
  1043         [] => ()
  1044       | A' :: _ => error ("Extra type variable on right-hand side: " ^
  1045           quote (Syntax.string_of_typ no_defs_lthy (TFree A'))));
  1046 
  1047     fun eq_fpT_check (T as Type (s, Us)) (Type (s', Us')) =
  1048         s = s' andalso (Us = Us' orelse error ("Illegal occurrence of recursive type " ^
  1049           quote (Syntax.string_of_typ fake_lthy T)))
  1050       | eq_fpT_check _ _ = false;
  1051 
  1052     fun freeze_fp (T as Type (s, Us)) =
  1053         (case find_index (eq_fpT_check T) fake_Ts of
  1054           ~1 => Type (s, map freeze_fp Us)
  1055         | kk => nth Xs kk)
  1056       | freeze_fp T = T;
  1057 
  1058     val ctrXs_Tsss = map (map (map freeze_fp)) fake_ctr_Tsss;
  1059     val ctrXs_sum_prod_Ts = map (mk_sumTN_balanced o map HOLogic.mk_tupleT) ctrXs_Tsss;
  1060 
  1061     val fp_eqs =
  1062       map dest_TFree Xs ~~ map (Term.typ_subst_atomic (As ~~ unsorted_As)) ctrXs_sum_prod_Ts;
  1063 
  1064     val (pre_bnfs, (fp_res as {bnfs = fp_bnfs as any_fp_bnf :: _, ctors = ctors0, dtors = dtors0,
  1065            xtor_co_iterss = xtor_co_iterss0, xtor_co_inducts, dtor_ctors, ctor_dtors, ctor_injects,
  1066            xtor_map_thms, xtor_set_thmss, xtor_rel_thms, xtor_co_iter_thmss, ...}, lthy)) =
  1067       fp_bnf (construct_fp mixfixes map_bs rel_bs set_bss) fp_bs (map dest_TFree unsorted_As) fp_eqs
  1068         no_defs_lthy0;
  1069 
  1070     val timer = time (Timer.startRealTimer ());
  1071 
  1072     val nesting_bnfs = nesty_bnfs lthy ctrXs_Tsss As;
  1073     val nested_bnfs = nesty_bnfs lthy ctrXs_Tsss Xs;
  1074 
  1075     val pre_map_defs = map map_def_of_bnf pre_bnfs;
  1076     val pre_set_defss = map set_defs_of_bnf pre_bnfs;
  1077     val pre_rel_defs = map rel_def_of_bnf pre_bnfs;
  1078     val nesting_set_map's = maps set_map'_of_bnf nesting_bnfs;
  1079     val nested_set_map's = maps set_map'_of_bnf nested_bnfs;
  1080 
  1081     val live = live_of_bnf any_fp_bnf;
  1082 
  1083     val Bs =
  1084       map3 (fn alive => fn A as TFree (_, S) => fn B => if alive then resort_tfree S B else A)
  1085         (liveness_of_fp_bnf (length As) any_fp_bnf) As Bs0;
  1086 
  1087     val B_ify = Term.typ_subst_atomic (As ~~ Bs);
  1088 
  1089     val ctors = map (mk_ctor As) ctors0;
  1090     val dtors = map (mk_dtor As) dtors0;
  1091 
  1092     val fpTs = map (domain_type o fastype_of) dtors;
  1093 
  1094     fun massage_simple_notes base =
  1095       filter_out (null o #2)
  1096       #> map (fn (thmN, thms, attrs) =>
  1097         ((qualify true base (Binding.name thmN), attrs), [(thms, [])]));
  1098 
  1099     val massage_multi_notes =
  1100       maps (fn (thmN, thmss, attrs) =>
  1101         if forall null thmss then
  1102           []
  1103         else
  1104           map3 (fn fp_b_name => fn Type (T_name, _) => fn thms =>
  1105               ((qualify true fp_b_name (Binding.name thmN), attrs T_name), [(thms, [])]))
  1106             fp_b_names fpTs thmss);
  1107 
  1108     val ctr_Tsss = map (map (map (Term.typ_subst_atomic (Xs ~~ fpTs)))) ctrXs_Tsss;
  1109     val ns = map length ctr_Tsss;
  1110     val kss = map (fn n => 1 upto n) ns;
  1111     val mss = map (map length) ctr_Tsss;
  1112 
  1113     val ((xtor_co_iterss, iters_args_types, coiters_args_types), lthy) =
  1114       mk_co_iters_prelims fp fpTs Cs ns mss xtor_co_iterss0 lthy;
  1115 
  1116     fun define_ctrs_case_for_type ((((((((((((((((((((((((fp_bnf, fp_b), fpT), C), ctor), dtor),
  1117             xtor_co_iters), ctor_dtor), dtor_ctor), ctor_inject), pre_map_def), pre_set_defs),
  1118           pre_rel_def), fp_map_thm), fp_set_thms), fp_rel_thm), n), ks), ms), ctr_bindings),
  1119         ctr_mixfixes), ctr_Tss), disc_bindings), sel_bindingss), raw_sel_defaultss) no_defs_lthy =
  1120       let
  1121         val fp_b_name = Binding.name_of fp_b;
  1122 
  1123         val dtorT = domain_type (fastype_of ctor);
  1124         val ctr_prod_Ts = map HOLogic.mk_tupleT ctr_Tss;
  1125         val ctr_sum_prod_T = mk_sumTN_balanced ctr_prod_Ts;
  1126         val case_Ts = map (fn Ts => Ts ---> C) ctr_Tss;
  1127 
  1128         val (((((w, fs), xss), yss), u'), names_lthy) =
  1129           no_defs_lthy
  1130           |> yield_singleton (mk_Frees "w") dtorT
  1131           ||>> mk_Frees "f" case_Ts
  1132           ||>> mk_Freess "x" ctr_Tss
  1133           ||>> mk_Freess "y" (map (map B_ify) ctr_Tss)
  1134           ||>> yield_singleton Variable.variant_fixes fp_b_name;
  1135 
  1136         val u = Free (u', fpT);
  1137 
  1138         val tuple_xs = map HOLogic.mk_tuple xss;
  1139         val tuple_ys = map HOLogic.mk_tuple yss;
  1140 
  1141         val ctr_rhss =
  1142           map3 (fn k => fn xs => fn tuple_x => fold_rev Term.lambda xs (ctor $
  1143             mk_InN_balanced ctr_sum_prod_T n tuple_x k)) ks xss tuple_xs;
  1144 
  1145         val case_binding = qualify false fp_b_name (Binding.suffix_name ("_" ^ caseN) fp_b);
  1146 
  1147         val case_rhs =
  1148           fold_rev Term.lambda (fs @ [u])
  1149             (mk_sum_caseN_balanced (map2 mk_uncurried_fun fs xss) $ (dtor $ u));
  1150 
  1151         val ((raw_case :: raw_ctrs, raw_case_def :: raw_ctr_defs), (lthy', lthy)) = no_defs_lthy
  1152           |> apfst split_list o fold_map3 (fn b => fn mx => fn rhs =>
  1153               Local_Theory.define ((b, mx), ((Thm.def_binding b, []), rhs)) #>> apsnd snd)
  1154             (case_binding :: ctr_bindings) (NoSyn :: ctr_mixfixes) (case_rhs :: ctr_rhss)
  1155           ||> `Local_Theory.restore;
  1156 
  1157         val phi = Proof_Context.export_morphism lthy lthy';
  1158 
  1159         val ctr_defs = map (Morphism.thm phi) raw_ctr_defs;
  1160         val ctr_defs' =
  1161           map2 (fn m => fn def => mk_unabs_def m (def RS meta_eq_to_obj_eq)) ms ctr_defs;
  1162         val case_def = Morphism.thm phi raw_case_def;
  1163 
  1164         val ctrs0 = map (Morphism.term phi) raw_ctrs;
  1165         val casex0 = Morphism.term phi raw_case;
  1166 
  1167         val ctrs = map (mk_ctr As) ctrs0;
  1168 
  1169         fun wrap_ctrs lthy =
  1170           let
  1171             fun exhaust_tac {context = ctxt, prems = _} =
  1172               let
  1173                 val ctor_iff_dtor_thm =
  1174                   let
  1175                     val goal =
  1176                       fold_rev Logic.all [w, u]
  1177                         (mk_Trueprop_eq (HOLogic.mk_eq (u, ctor $ w), HOLogic.mk_eq (dtor $ u, w)));
  1178                   in
  1179                     Goal.prove_sorry lthy [] [] goal (fn {context = ctxt, ...} =>
  1180                       mk_ctor_iff_dtor_tac ctxt (map (SOME o certifyT lthy) [dtorT, fpT])
  1181                         (certify lthy ctor) (certify lthy dtor) ctor_dtor dtor_ctor)
  1182                     |> Thm.close_derivation
  1183                     |> Morphism.thm phi
  1184                   end;
  1185 
  1186                 val sumEN_thm' =
  1187                   unfold_thms lthy @{thms all_unit_eq}
  1188                     (Drule.instantiate' (map (SOME o certifyT lthy) ctr_prod_Ts) []
  1189                        (mk_sumEN_balanced n))
  1190                   |> Morphism.thm phi;
  1191               in
  1192                 mk_exhaust_tac ctxt n ctr_defs ctor_iff_dtor_thm sumEN_thm'
  1193               end;
  1194 
  1195             val inject_tacss =
  1196               map2 (fn 0 => K [] | _ => fn ctr_def => [fn {context = ctxt, ...} =>
  1197                   mk_inject_tac ctxt ctr_def ctor_inject]) ms ctr_defs;
  1198 
  1199             val half_distinct_tacss =
  1200               map (map (fn (def, def') => fn {context = ctxt, ...} =>
  1201                 mk_half_distinct_tac ctxt ctor_inject [def, def'])) (mk_half_pairss (`I ctr_defs));
  1202 
  1203             val case_tacs =
  1204               map3 (fn k => fn m => fn ctr_def => fn {context = ctxt, ...} =>
  1205                 mk_case_tac ctxt n k m case_def ctr_def dtor_ctor) ks ms ctr_defs;
  1206 
  1207             val tacss = [exhaust_tac] :: inject_tacss @ half_distinct_tacss @ [case_tacs];
  1208 
  1209             val sel_defaultss = map (map (apsnd (prepare_term lthy))) raw_sel_defaultss
  1210           in
  1211             wrap_free_constructors tacss (((wrap_opts, ctrs0), casex0), (disc_bindings,
  1212               (sel_bindingss, sel_defaultss))) lthy
  1213           end;
  1214 
  1215         fun derive_maps_sets_rels (ctr_sugar, lthy) =
  1216           if live = 0 then
  1217             ((([], [], [], []), ctr_sugar), lthy)
  1218           else
  1219             let
  1220               val rel_flip = rel_flip_of_bnf fp_bnf;
  1221               val nones = replicate live NONE;
  1222 
  1223               val ctor_cong =
  1224                 if fp = Least_FP then
  1225                   Drule.dummy_thm
  1226                 else
  1227                   let val ctor' = mk_ctor Bs ctor in
  1228                     cterm_instantiate_pos [NONE, NONE, SOME (certify lthy ctor')] arg_cong
  1229                   end;
  1230 
  1231               fun mk_cIn ify =
  1232                 certify lthy o (fp = Greatest_FP ? curry (op $) (map_types ify ctor)) oo
  1233                 mk_InN_balanced (ify ctr_sum_prod_T) n;
  1234 
  1235               val cxIns = map2 (mk_cIn I) tuple_xs ks;
  1236               val cyIns = map2 (mk_cIn B_ify) tuple_ys ks;
  1237 
  1238               fun mk_map_thm ctr_def' cxIn =
  1239                 fold_thms lthy [ctr_def']
  1240                   (unfold_thms lthy (pre_map_def ::
  1241                        (if fp = Least_FP then [] else [ctor_dtor, dtor_ctor]) @ sum_prod_thms_map)
  1242                      (cterm_instantiate_pos (nones @ [SOME cxIn])
  1243                         (if fp = Least_FP then fp_map_thm else fp_map_thm RS ctor_cong)))
  1244                 |> singleton (Proof_Context.export names_lthy no_defs_lthy);
  1245 
  1246               fun mk_set_thm fp_set_thm ctr_def' cxIn =
  1247                 fold_thms lthy [ctr_def']
  1248                   (unfold_thms lthy (pre_set_defs @ nested_set_map's @ nesting_set_map's @
  1249                        (if fp = Least_FP then [] else [dtor_ctor]) @ sum_prod_thms_set)
  1250                      (cterm_instantiate_pos [SOME cxIn] fp_set_thm))
  1251                 |> singleton (Proof_Context.export names_lthy no_defs_lthy);
  1252 
  1253               fun mk_set_thms fp_set_thm = map2 (mk_set_thm fp_set_thm) ctr_defs' cxIns;
  1254 
  1255               val map_thms = map2 mk_map_thm ctr_defs' cxIns;
  1256               val set_thmss = map mk_set_thms fp_set_thms;
  1257 
  1258               val rel_infos = (ctr_defs' ~~ cxIns, ctr_defs' ~~ cyIns);
  1259 
  1260               fun mk_rel_thm postproc ctr_defs' cxIn cyIn =
  1261                 fold_thms lthy ctr_defs'
  1262                    (unfold_thms lthy (@{thm Inl_Inr_False} :: pre_rel_def ::
  1263                         (if fp = Least_FP then [] else [dtor_ctor]) @ sum_prod_thms_rel)
  1264                       (cterm_instantiate_pos (nones @ [SOME cxIn, SOME cyIn]) fp_rel_thm))
  1265                 |> postproc
  1266                 |> singleton (Proof_Context.export names_lthy no_defs_lthy);
  1267 
  1268               fun mk_rel_inject_thm ((ctr_def', cxIn), (_, cyIn)) =
  1269                 mk_rel_thm (unfold_thms lthy @{thms eq_sym_Unity_conv}) [ctr_def'] cxIn cyIn;
  1270 
  1271               val rel_inject_thms = map mk_rel_inject_thm (op ~~ rel_infos);
  1272 
  1273               fun mk_half_rel_distinct_thm ((xctr_def', cxIn), (yctr_def', cyIn)) =
  1274                 mk_rel_thm (fn thm => thm RS @{thm eq_False[THEN iffD1]}) [xctr_def', yctr_def']
  1275                   cxIn cyIn;
  1276 
  1277               fun mk_other_half_rel_distinct_thm thm =
  1278                 flip_rels lthy live thm
  1279                 RS (rel_flip RS sym RS @{thm arg_cong[of _ _ Not]} RS iffD2);
  1280 
  1281               val half_rel_distinct_thmss =
  1282                 map (map mk_half_rel_distinct_thm) (mk_half_pairss rel_infos);
  1283               val other_half_rel_distinct_thmss =
  1284                 map (map mk_other_half_rel_distinct_thm) half_rel_distinct_thmss;
  1285               val (rel_distinct_thms, _) =
  1286                 join_halves n half_rel_distinct_thmss other_half_rel_distinct_thmss;
  1287 
  1288               val notes =
  1289                 [(mapN, map_thms, code_simp_attrs),
  1290                  (rel_distinctN, rel_distinct_thms, code_simp_attrs),
  1291                  (rel_injectN, rel_inject_thms, code_simp_attrs),
  1292                  (setsN, flat set_thmss, code_simp_attrs)]
  1293                 |> massage_simple_notes fp_b_name;
  1294             in
  1295               (((map_thms, rel_inject_thms, rel_distinct_thms, set_thmss), ctr_sugar),
  1296                lthy |> Local_Theory.notes notes |> snd)
  1297             end;
  1298 
  1299         fun mk_binding suf = qualify false fp_b_name (Binding.suffix_name ("_" ^ suf) fp_b);
  1300 
  1301         fun massage_res (((maps_sets_rels, ctr_sugar), co_iter_res), lthy) =
  1302           (((maps_sets_rels, (ctrs, xss, ctr_defs, ctr_sugar)), co_iter_res), lthy);
  1303       in
  1304         (wrap_ctrs
  1305          #> derive_maps_sets_rels
  1306          ##>>
  1307            (if fp = Least_FP then define_iters [foldN, recN] (the iters_args_types)
  1308             else define_coiters [unfoldN, corecN] (the coiters_args_types))
  1309              mk_binding fpTs Cs xtor_co_iters
  1310          #> massage_res, lthy')
  1311       end;
  1312 
  1313     fun wrap_types_etc (wrap_types_etcs, lthy) =
  1314       fold_map I wrap_types_etcs lthy
  1315       |>> apsnd split_list o apfst (apsnd split_list4 o apfst split_list4 o split_list)
  1316         o split_list;
  1317 
  1318     val mk_simp_thmss =
  1319       map7 (fn {injects, distincts, case_thms, ...} => fn un_folds => fn co_recs =>
  1320         fn mapsx => fn rel_injects => fn rel_distincts => fn setss =>
  1321           injects @ distincts @ case_thms @ co_recs @ un_folds @ mapsx @ rel_injects
  1322           @ rel_distincts @ flat setss);
  1323 
  1324     fun derive_and_note_induct_iters_thms_for_types
  1325         ((((mapsx, rel_injects, rel_distincts, setss), (ctrss, _, ctr_defss, ctr_sugars)),
  1326           (iterss, iter_defss)), lthy) =
  1327       let
  1328         val ((induct_thms, induct_thm, induct_attrs), (fold_thmss, fold_attrs),
  1329              (rec_thmss, rec_attrs)) =
  1330           derive_induct_iters_thms_for_types pre_bnfs (the iters_args_types) xtor_co_inducts
  1331             xtor_co_iter_thmss nesting_bnfs nested_bnfs fpTs Cs Xs ctrXs_Tsss ctrss ctr_defss iterss
  1332             iter_defss lthy;
  1333 
  1334         val induct_type_attr = Attrib.internal o K o Induct.induct_type;
  1335 
  1336         val simp_thmss =
  1337           mk_simp_thmss ctr_sugars fold_thmss rec_thmss mapsx rel_injects rel_distincts setss;
  1338 
  1339         val common_notes =
  1340           (if nn > 1 then [(inductN, [induct_thm], induct_attrs)] else [])
  1341           |> massage_simple_notes fp_common_name;
  1342 
  1343         val notes =
  1344           [(foldN, fold_thmss, K fold_attrs),
  1345            (inductN, map single induct_thms, fn T_name => induct_attrs @ [induct_type_attr T_name]),
  1346            (recN, rec_thmss, K rec_attrs),
  1347            (simpsN, simp_thmss, K [])]
  1348           |> massage_multi_notes;
  1349       in
  1350         lthy
  1351         |> Local_Theory.notes (common_notes @ notes) |> snd
  1352         |> register_fp_sugars Least_FP pre_bnfs fp_res ctr_defss ctr_sugars iterss [induct_thm]
  1353           (transpose [fold_thmss, rec_thmss])
  1354       end;
  1355 
  1356     fun derive_and_note_coinduct_coiters_thms_for_types
  1357         ((((mapsx, rel_injects, rel_distincts, setss), (_, _, ctr_defss, ctr_sugars)),
  1358           (coiterss, coiter_defss)), lthy) =
  1359       let
  1360         val (([(coinduct_thms, coinduct_thm), (strong_coinduct_thms, strong_coinduct_thm)],
  1361               coinduct_attrs),
  1362              (unfold_thmss, corec_thmss, coiter_attrs),
  1363              (safe_unfold_thmss, safe_corec_thmss),
  1364              (disc_unfold_thmss, disc_corec_thmss, disc_coiter_attrs),
  1365              (disc_unfold_iff_thmss, disc_corec_iff_thmss, disc_coiter_iff_attrs),
  1366              (sel_unfold_thmss, sel_corec_thmss, sel_coiter_attrs)) =
  1367           derive_coinduct_coiters_thms_for_types pre_bnfs (the coiters_args_types) xtor_co_inducts
  1368             dtor_ctors xtor_co_iter_thmss nesting_bnfs nested_bnfs fpTs Cs As kss mss ns ctr_defss
  1369             ctr_sugars coiterss coiter_defss lthy;
  1370 
  1371         val coinduct_type_attr = Attrib.internal o K o Induct.coinduct_type;
  1372 
  1373         fun flat_coiter_thms coiters disc_coiters sel_coiters =
  1374           coiters @ disc_coiters @ sel_coiters;
  1375 
  1376         val simp_thmss =
  1377           mk_simp_thmss ctr_sugars
  1378             (map3 flat_coiter_thms safe_unfold_thmss disc_unfold_thmss sel_unfold_thmss)
  1379             (map3 flat_coiter_thms safe_corec_thmss disc_corec_thmss sel_corec_thmss)
  1380             mapsx rel_injects rel_distincts setss;
  1381 
  1382         val anonymous_notes =
  1383           [(flat safe_unfold_thmss @ flat safe_corec_thmss, simp_attrs)]
  1384           |> map (fn (thms, attrs) => ((Binding.empty, attrs), [(thms, [])]));
  1385 
  1386         val common_notes =
  1387           (if nn > 1 then
  1388              [(coinductN, [coinduct_thm], coinduct_attrs),
  1389               (strong_coinductN, [strong_coinduct_thm], coinduct_attrs)]
  1390            else
  1391              [])
  1392           |> massage_simple_notes fp_common_name;
  1393 
  1394         val notes =
  1395           [(coinductN, map single coinduct_thms,
  1396             fn T_name => coinduct_attrs @ [coinduct_type_attr T_name]),
  1397            (corecN, corec_thmss, K coiter_attrs),
  1398            (disc_corecN, disc_corec_thmss, K disc_coiter_attrs),
  1399            (disc_corec_iffN, disc_corec_iff_thmss, K disc_coiter_iff_attrs),
  1400            (disc_unfoldN, disc_unfold_thmss, K disc_coiter_attrs),
  1401            (disc_unfold_iffN, disc_unfold_iff_thmss, K disc_coiter_iff_attrs),
  1402            (sel_corecN, sel_corec_thmss, K sel_coiter_attrs),
  1403            (sel_unfoldN, sel_unfold_thmss, K sel_coiter_attrs),
  1404            (simpsN, simp_thmss, K []),
  1405            (strong_coinductN, map single strong_coinduct_thms, K coinduct_attrs),
  1406            (unfoldN, unfold_thmss, K coiter_attrs)]
  1407           |> massage_multi_notes;
  1408       in
  1409         lthy
  1410         |> Local_Theory.notes (anonymous_notes @ common_notes @ notes) |> snd
  1411         |> register_fp_sugars Greatest_FP pre_bnfs fp_res ctr_defss ctr_sugars coiterss
  1412           [coinduct_thm, strong_coinduct_thm] (transpose [unfold_thmss, corec_thmss])
  1413       end;
  1414 
  1415     val lthy' = lthy
  1416       |> fold_map define_ctrs_case_for_type (fp_bnfs ~~ fp_bs ~~ fpTs ~~ Cs ~~ ctors ~~ dtors ~~
  1417         xtor_co_iterss ~~ ctor_dtors ~~ dtor_ctors ~~ ctor_injects ~~ pre_map_defs ~~
  1418         pre_set_defss ~~ pre_rel_defs ~~ xtor_map_thms ~~ xtor_set_thmss ~~ xtor_rel_thms ~~ ns ~~
  1419         kss ~~ mss ~~ ctr_bindingss ~~ ctr_mixfixess ~~ ctr_Tsss ~~ disc_bindingss ~~
  1420         sel_bindingsss ~~ raw_sel_defaultsss)
  1421       |> wrap_types_etc
  1422       |> (if fp = Least_FP then derive_and_note_induct_iters_thms_for_types
  1423           else derive_and_note_coinduct_coiters_thms_for_types);
  1424 
  1425     val timer = time (timer ("Constructors, discriminators, selectors, etc., for the new " ^
  1426       datatype_word fp));
  1427   in
  1428     timer; lthy'
  1429   end;
  1430 
  1431 val co_datatypes = define_co_datatypes (K I) (K I) (K I);
  1432 
  1433 val co_datatype_cmd =
  1434   define_co_datatypes Typedecl.read_constraint Syntax.parse_typ Syntax.parse_term;
  1435 
  1436 val parse_ctr_arg =
  1437   @{keyword "("} |-- parse_binding_colon -- Parse.typ --| @{keyword ")"} ||
  1438   (Parse.typ >> pair Binding.empty);
  1439 
  1440 val parse_defaults =
  1441   @{keyword "("} |-- @{keyword "defaults"} |-- Scan.repeat parse_bound_term --| @{keyword ")"};
  1442 
  1443 val parse_type_arg_constrained =
  1444   Parse.type_ident -- Scan.option (@{keyword "::"} |-- Parse.!!! Parse.sort);
  1445 
  1446 val parse_type_arg_named_constrained = parse_opt_binding_colon -- parse_type_arg_constrained;
  1447 
  1448 val parse_type_args_named_constrained =
  1449   parse_type_arg_constrained >> (single o pair Binding.empty) ||
  1450   @{keyword "("} |-- Parse.!!! (Parse.list1 parse_type_arg_named_constrained --| @{keyword ")"}) ||
  1451   Scan.succeed [];
  1452 
  1453 val parse_map_rel_binding = Parse.short_ident --| @{keyword ":"} -- parse_binding;
  1454 
  1455 val no_map_rel = (Binding.empty, Binding.empty);
  1456 
  1457 (* "map" and "rel" are purposedly not registered as keywords, because they are short and nice names
  1458    that we don't want them to be highlighted everywhere. *)
  1459 fun extract_map_rel ("map", b) = apfst (K b)
  1460   | extract_map_rel ("rel", b) = apsnd (K b)
  1461   | extract_map_rel (s, _) = error ("Expected \"map\" or \"rel\" instead of " ^ quote s);
  1462 
  1463 val parse_map_rel_bindings =
  1464   @{keyword "("} |-- Scan.repeat parse_map_rel_binding --| @{keyword ")"}
  1465     >> (fn ps => fold extract_map_rel ps no_map_rel) ||
  1466   Scan.succeed no_map_rel;
  1467 
  1468 val parse_ctr_spec =
  1469   parse_opt_binding_colon -- parse_binding -- Scan.repeat parse_ctr_arg --
  1470   Scan.optional parse_defaults [] -- Parse.opt_mixfix;
  1471 
  1472 val parse_spec =
  1473   parse_type_args_named_constrained -- parse_binding -- parse_map_rel_bindings --
  1474   Parse.opt_mixfix -- (@{keyword "="} |-- Parse.enum1 "|" parse_ctr_spec);
  1475 
  1476 val parse_co_datatype = parse_wrap_options -- Parse.and_list1 parse_spec;
  1477 
  1478 fun parse_co_datatype_cmd fp construct_fp = parse_co_datatype >> co_datatype_cmd fp construct_fp;
  1479 
  1480 end;