src/HOL/BNF/Tools/bnf_fp_def_sugar.ML
author blanchet
Fri Jun 07 09:30:13 2013 +0200 (2013-06-07)
changeset 52334 705bc4f5fc70
parent 52330 8ce7fba90bb3
child 52335 0e3f949792ca
permissions -rw-r--r--
tuning
     1 (*  Title:      HOL/BNF/Tools/bnf_fp_def_sugar.ML
     2     Author:     Jasmin Blanchette, TU Muenchen
     3     Copyright   2012
     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      un_folds: term list,
    19      co_recs: term list,
    20      co_induct: thm,
    21      strong_co_induct: thm,
    22      un_fold_thmss: thm list list,
    23      co_rec_thmss: thm list list};
    24 
    25   val of_fp_sugar: (fp_sugar -> 'a list) -> fp_sugar -> 'a
    26   val morph_fp_sugar: morphism -> fp_sugar -> fp_sugar
    27   val fp_sugar_of: Proof.context -> string -> fp_sugar option
    28 
    29   val tvar_subst: theory -> typ list -> typ list -> ((string * int) * typ) list
    30   val exists_subtype_in: typ list -> typ -> bool
    31   val flat_rec: 'a list list -> '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_co_iters_prelims: BNF_FP_Util.fp_kind -> typ list -> typ list -> int list ->
    35     int list list -> term list list -> Proof.context ->
    36     (term list list
    37      * (typ list list * typ list list list list * term list list
    38         * term list list list list) list option
    39      * (term list * term list list
    40         * ((term list list * term list list list list * term list list list list)
    41            * (typ list * typ list list list * typ list list)) list) option)
    42     * Proof.context
    43   val mk_map: int -> typ list -> typ list -> term -> term
    44   val build_map: local_theory -> (typ * typ -> term) -> typ * typ -> term
    45 
    46   val mk_iter_fun_arg_typessss: typ list -> int list -> int list list -> term ->
    47     typ list list list list
    48   val define_iters: string list ->
    49     (typ list list * typ list list list list * term list list * term list list list list) list ->
    50     (string -> binding) -> typ list -> typ list -> term list -> Proof.context ->
    51     (term list * thm list) * Proof.context
    52   val define_coiters: string list -> term list * term list list
    53     * ((term list list * term list list list list * term list list list list)
    54        * (typ list * typ list list 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 -> term list list ->
    58     (typ list list * typ list list list list * term list list * term list list list list) list ->
    59     thm -> 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 * thm list * 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 -> term list list -> thm ->
    65     thm -> thm list -> thm list list -> BNF_Def.bnf list -> BNF_Def.bnf list -> typ list ->
    66     typ list -> typ list -> int list list -> int list list -> int list -> thm list list ->
    67     BNF_Ctr_Sugar.ctr_sugar list -> term list list -> thm list list -> local_theory ->
    68     (thm * thm list * thm * thm list * Args.src list)
    69     * (thm list list * thm list list * Args.src list)
    70     * (thm list list * thm list list) * (thm list list * thm list list * Args.src list)
    71     * (thm list list * thm list list * Args.src list)
    72     * (thm list list * thm list list * Args.src list)
    73 
    74   val co_datatypes: BNF_FP_Util.fp_kind -> (mixfix list -> binding list -> binding list ->
    75       binding list list -> binding list -> (string * sort) list -> typ list * typ list list ->
    76       BNF_Def.bnf list -> local_theory -> BNF_FP_Util.fp_result * local_theory) ->
    77     (bool * bool) * (((((binding * (typ * sort)) list * binding) * (binding * binding)) * mixfix) *
    78       ((((binding * binding) * (binding * typ) list) * (binding * term) list) *
    79         mixfix) list) list ->
    80     local_theory -> local_theory
    81   val parse_co_datatype_cmd: BNF_FP_Util.fp_kind -> (mixfix list -> binding list -> binding list ->
    82       binding list list -> binding list -> (string * sort) list -> typ list * typ list list ->
    83       BNF_Def.bnf list -> local_theory -> BNF_FP_Util.fp_result * local_theory) ->
    84     (local_theory -> local_theory) parser
    85 end;
    86 
    87 structure BNF_FP_Def_Sugar : BNF_FP_DEF_SUGAR =
    88 struct
    89 
    90 open BNF_Util
    91 open BNF_Ctr_Sugar
    92 open BNF_Def
    93 open BNF_FP_Util
    94 open BNF_FP_Def_Sugar_Tactics
    95 
    96 val EqN = "Eq_";
    97 
    98 type fp_sugar =
    99   {T: typ,
   100    fp: fp_kind,
   101    index: int,
   102    pre_bnfs: bnf list,
   103    fp_res: fp_result,
   104    ctr_defss: thm list list,
   105    ctr_sugars: ctr_sugar list,
   106    un_folds: term list,
   107    co_recs: term list,
   108    co_induct: thm,
   109    strong_co_induct: thm,
   110    un_fold_thmss: thm list list,
   111    co_rec_thmss: thm list list};
   112 
   113 fun of_fp_sugar f (fp_sugar as {index, ...}) = nth (f fp_sugar) index;
   114 
   115 fun eq_fp_sugar ({T = T1, fp = fp1, index = index1, fp_res = fp_res1, ...} : fp_sugar,
   116     {T = T2, fp = fp2, index = index2, fp_res = fp_res2, ...} : fp_sugar) =
   117   T1 = T2 andalso fp1 = fp2 andalso index1 = index2 andalso eq_fp_result (fp_res1, fp_res2);
   118 
   119 fun morph_fp_sugar phi {T, fp, index, pre_bnfs, fp_res, ctr_defss, ctr_sugars, un_folds,
   120     co_recs, co_induct, strong_co_induct, un_fold_thmss, co_rec_thmss} =
   121   {T = Morphism.typ phi T, fp = fp, index = index, pre_bnfs = map (morph_bnf phi)
   122    pre_bnfs, fp_res = morph_fp_result phi fp_res,
   123    ctr_defss = map (map (Morphism.thm phi)) ctr_defss,
   124    ctr_sugars = map (morph_ctr_sugar phi) ctr_sugars, un_folds = map (Morphism.term phi) un_folds,
   125    co_recs = map (Morphism.term phi) co_recs, co_induct = Morphism.thm phi co_induct,
   126    strong_co_induct = Morphism.thm phi strong_co_induct,
   127    un_fold_thmss = map (map (Morphism.thm phi)) un_fold_thmss,
   128    co_rec_thmss = map (map (Morphism.thm phi)) co_rec_thmss};
   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.update_new (key, morph_fp_sugar phi fp_sugar)));
   143 
   144 fun register_fp_sugars fp pre_bnfs (fp_res as {Ts, ...}) ctr_defss ctr_sugars [un_folds, co_recs]
   145     co_induct strong_co_induct [un_fold_thmss, co_rec_thmss] 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, fp_res = fp_res,
   149       ctr_defss = ctr_defss, ctr_sugars = ctr_sugars, un_folds = un_folds, co_recs = co_recs,
   150       co_induct = co_induct, strong_co_induct = strong_co_induct, un_fold_thmss = un_fold_thmss,
   151       co_rec_thmss = co_rec_thmss} 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 mp_conj = @{thm mp_conj};
   168 
   169 val simp_attrs = @{attributes [simp]};
   170 val code_simp_attrs = Code.add_default_eqn_attrib :: simp_attrs;
   171 
   172 fun tvar_subst thy Ts Us =
   173   Vartab.fold (cons o apsnd snd) (fold (Sign.typ_match thy) (Ts ~~ Us) Vartab.empty) [];
   174 
   175 val exists_subtype_in = Term.exists_subtype o member (op =);
   176 
   177 fun resort_tfree S (TFree (s, _)) = TFree (s, S);
   178 
   179 fun typ_subst_nonatomic inst (T as Type (s, Ts)) =
   180     (case AList.lookup (op =) inst T of
   181       NONE => Type (s, map (typ_subst_nonatomic inst) Ts)
   182     | SOME T' => T')
   183   | typ_subst_nonatomic inst T = the_default T (AList.lookup (op =) inst T);
   184 
   185 val lists_bmoc = fold (fn xs => fn t => Term.list_comb (t, xs));
   186 
   187 fun flat_rec xss =
   188   (* The first line below gives the preferred order. The second line is for compatibility with the
   189      old datatype package: *)
   190 (*
   191   flat xss
   192 *)
   193   map hd xss @ maps tl xss;
   194 
   195 fun flat_predss_getterss qss fss = maps (op @) (qss ~~ fss);
   196 
   197 fun flat_preds_predsss_gettersss [] [qss] [fss] = flat_predss_getterss qss fss
   198   | flat_preds_predsss_gettersss (p :: ps) (qss :: qsss) (fss :: fsss) =
   199     p :: flat_predss_getterss qss fss @ flat_preds_predsss_gettersss ps qsss fsss;
   200 
   201 fun mk_tupled_fun x f xs = HOLogic.tupled_lambda x (Term.list_comb (f, xs));
   202 fun mk_uncurried_fun f xs = mk_tupled_fun (HOLogic.mk_tuple xs) f xs;
   203 fun mk_uncurried2_fun f xss =
   204   mk_tupled_fun (HOLogic.mk_tuple (map HOLogic.mk_tuple xss)) f (flat_rec xss);
   205 
   206 fun mk_flip (x, Type (_, [T1, Type (_, [T2, T3])])) =
   207   Abs ("x", T1, Abs ("y", T2, Var (x, T2 --> T1 --> T3) $ Bound 0 $ Bound 1));
   208 
   209 fun flip_rels lthy n thm =
   210   let
   211     val Rs = Term.add_vars (prop_of thm) [];
   212     val Rs' = rev (drop (length Rs - n) Rs);
   213     val cRs = map (fn f => (certify lthy (Var f), certify lthy (mk_flip f))) Rs';
   214   in
   215     Drule.cterm_instantiate cRs thm
   216   end;
   217 
   218 fun mk_ctor_or_dtor get_T Ts t =
   219   let val Type (_, Ts0) = get_T (fastype_of t) in
   220     Term.subst_atomic_types (Ts0 ~~ Ts) t
   221   end;
   222 
   223 val mk_ctor = mk_ctor_or_dtor range_type;
   224 val mk_dtor = mk_ctor_or_dtor domain_type;
   225 
   226 fun mk_co_iter thy fp fpT Cs t =
   227   let
   228     val (binders, body) = strip_type (fastype_of t);
   229     val (f_Cs, prebody) = split_last binders;
   230     val fpT0 = if fp = Least_FP then prebody else body;
   231     val Cs0 = distinct (op =) (map (if fp = Least_FP then body_type else domain_type) f_Cs);
   232     val rho = tvar_subst thy (fpT0 :: Cs0) (fpT :: Cs);
   233   in
   234     Term.subst_TVars rho t
   235   end;
   236 
   237 fun mk_co_iters thy fp fpTs Cs ts0 =
   238   let
   239     val nn = length fpTs;
   240     val (fpTs0, Cs0) =
   241       map ((fp = Greatest_FP ? swap) o dest_funT o snd o strip_typeN nn o fastype_of) ts0
   242       |> split_list;
   243     val rho = tvar_subst thy (fpTs0 @ Cs0) (fpTs @ Cs);
   244   in
   245     map (Term.subst_TVars rho) ts0
   246   end;
   247 
   248 val mk_fp_iter_fun_types = fst o split_last o binder_types o fastype_of;
   249 
   250 fun project_co_recT special_Tname Cs proj =
   251   let
   252     fun project (Type (s, Ts as [T, U])) =
   253         if s = special_Tname andalso member (op =) Cs U then proj (T, U)
   254         else Type (s, map project Ts)
   255       | project (Type (s, Ts)) = Type (s, map project Ts)
   256       | project T = T;
   257   in project end;
   258 
   259 val project_corecT = project_co_recT @{type_name sum};
   260 
   261 fun unzip_recT Cs (T as Type (@{type_name prod}, Ts as [_, U])) =
   262     if member (op =) Cs U then Ts else [T]
   263   | unzip_recT _ T = [T];
   264 
   265 fun mk_fun_arg_typess n ms = map2 dest_tupleT ms o dest_sumTN_balanced n o domain_type;
   266 
   267 fun mk_iter_fun_arg_typessss Cs ns mss =
   268   mk_fp_iter_fun_types
   269   #> map3 mk_fun_arg_typess ns mss
   270   #> map (map (map (unzip_recT Cs)));
   271 
   272 fun mk_iters_args_types Cs ns mss [ctor_fold_fun_Ts, ctor_rec_fun_Ts] lthy =
   273   let
   274     val Css = map2 replicate ns Cs;
   275     val y_Tsss = map3 mk_fun_arg_typess ns mss ctor_fold_fun_Ts;
   276     val g_Tss = map2 (fn C => map (fn y_Ts => y_Ts ---> C)) Cs y_Tsss;
   277 
   278     val ((gss, ysss), lthy) =
   279       lthy
   280       |> mk_Freess "f" g_Tss
   281       ||>> mk_Freesss "x" y_Tsss;
   282 
   283     val y_Tssss = map (map (map single)) y_Tsss;
   284     val yssss = map (map (map single)) ysss;
   285 
   286     val z_Tssss =
   287       map3 (fn n => fn ms => map2 (map (unzip_recT Cs) oo dest_tupleT) ms o
   288         dest_sumTN_balanced n o domain_type) ns mss ctor_rec_fun_Ts;
   289 
   290     val z_Tsss' = map (map flat_rec) z_Tssss;
   291     val h_Tss = map2 (map2 (curry (op --->))) z_Tsss' Css;
   292 
   293     val hss = map2 (map2 retype_free) h_Tss gss;
   294     val zssss_hd = map2 (map2 (map2 (retype_free o hd))) z_Tssss ysss;
   295     val (zssss_tl, lthy) =
   296       lthy
   297       |> mk_Freessss "y" (map (map (map tl)) z_Tssss);
   298     val zssss = map2 (map2 (map2 cons)) zssss_hd zssss_tl;
   299   in
   300     ([(g_Tss, y_Tssss, gss, yssss), (h_Tss, z_Tssss, hss, zssss)], lthy)
   301   end;
   302 
   303 fun mk_coiters_args_types Cs ns mss [dtor_unfold_fun_Ts, dtor_corec_fun_Ts] lthy =
   304   let
   305     (*avoid "'a itself" arguments in coiterators and corecursors*)
   306     fun repair_arity [0] = [1]
   307       | repair_arity ms = ms;
   308 
   309     fun unzip_corecT T =
   310       if exists_subtype_in Cs T then [project_corecT Cs fst T, project_corecT Cs snd T]
   311       else [T];
   312 
   313     val p_Tss = map2 (fn n => replicate (Int.max (0, n - 1)) o mk_pred1T) ns Cs;
   314 
   315     fun mk_types maybe_unzipT fun_Ts =
   316       let
   317         val f_sum_prod_Ts = map range_type fun_Ts;
   318         val f_prod_Tss = map2 dest_sumTN_balanced ns f_sum_prod_Ts;
   319         val f_Tsss = map2 (map2 dest_tupleT o repair_arity) mss f_prod_Tss;
   320         val f_Tssss = map2 (fn C => map (map (map (curry (op -->) C) o maybe_unzipT))) Cs f_Tsss;
   321         val q_Tssss =
   322           map (map (map (fn [_] => [] | [_, T] => [mk_pred1T (domain_type T)]))) f_Tssss;
   323         val pf_Tss = map3 flat_preds_predsss_gettersss p_Tss q_Tssss f_Tssss;
   324       in (q_Tssss, f_Tssss, (f_sum_prod_Ts, f_Tsss, pf_Tss)) end;
   325 
   326     val (r_Tssss, g_Tssss, unfold_types) = mk_types single dtor_unfold_fun_Ts;
   327     val (s_Tssss, h_Tssss, corec_types) = mk_types unzip_corecT dtor_corec_fun_Ts;
   328 
   329     val (((cs, pss), gssss), lthy) =
   330       lthy
   331       |> mk_Frees "a" Cs
   332       ||>> mk_Freess "p" p_Tss
   333       ||>> mk_Freessss "g" g_Tssss;
   334     val rssss = map (map (map (fn [] => []))) r_Tssss;
   335 
   336     val hssss_hd = map2 (map2 (map2 (fn T :: _ => fn [g] => retype_free T g))) h_Tssss gssss;
   337     val ((sssss, hssss_tl), lthy) =
   338       lthy
   339       |> mk_Freessss "q" s_Tssss
   340       ||>> mk_Freessss "h" (map (map (map tl)) h_Tssss);
   341     val hssss = map2 (map2 (map2 cons)) hssss_hd hssss_tl;
   342 
   343     val cpss = map2 (map o rapp) cs pss;
   344 
   345     fun mk_args qssss fssss =
   346       let
   347         val pfss = map3 flat_preds_predsss_gettersss pss qssss fssss;
   348         val cqssss = map2 (map o map o map o rapp) cs qssss;
   349         val cfssss = map2 (map o map o map o rapp) cs fssss;
   350       in (pfss, cqssss, cfssss) end;
   351 
   352     val unfold_args = mk_args rssss gssss;
   353     val corec_args = mk_args sssss hssss;
   354   in
   355     ((cs, cpss, [(unfold_args, unfold_types), (corec_args, corec_types)]), lthy)
   356   end;
   357 
   358 fun mk_co_iters_prelims fp fpTs Cs ns mss xtor_co_iterss0 lthy =
   359   let
   360     val thy = Proof_Context.theory_of lthy;
   361 
   362     val (xtor_co_iter_fun_Tss', xtor_co_iterss') =
   363       map (mk_co_iters thy fp fpTs Cs #> `(mk_fp_iter_fun_types o hd)) (transpose xtor_co_iterss0)
   364       |> split_list;
   365 
   366     val ((iters_args_types, coiters_args_types), lthy') =
   367       if fp = Least_FP then
   368         mk_iters_args_types Cs ns mss xtor_co_iter_fun_Tss' lthy |>> (rpair NONE o SOME)
   369       else
   370         mk_coiters_args_types Cs ns mss xtor_co_iter_fun_Tss' lthy |>> (pair NONE o SOME)
   371   in
   372     ((transpose xtor_co_iterss', iters_args_types, coiters_args_types), lthy')
   373   end;
   374 
   375 fun mk_map live Ts Us t =
   376   let val (Type (_, Ts0), Type (_, Us0)) = strip_typeN (live + 1) (fastype_of t) |>> List.last in
   377     Term.subst_atomic_types (Ts0 @ Us0 ~~ Ts @ Us) t
   378   end;
   379 
   380 fun mk_rel live Ts Us t =
   381   let val [Type (_, Ts0), Type (_, Us0)] = binder_types (snd (strip_typeN live (fastype_of t))) in
   382     Term.subst_atomic_types (Ts0 @ Us0 ~~ Ts @ Us) t
   383   end;
   384 
   385 fun liveness_of_fp_bnf n bnf =
   386   (case T_of_bnf bnf of
   387     Type (_, Ts) => map (not o member (op =) (deads_of_bnf bnf)) Ts
   388   | _ => replicate n false);
   389 
   390 fun cannot_merge_types () = error "Mutually recursive types must have the same type parameters";
   391 
   392 fun merge_type_arg T T' = if T = T' then T else cannot_merge_types ();
   393 
   394 fun merge_type_args (As, As') =
   395   if length As = length As' then map2 merge_type_arg As As' else cannot_merge_types ();
   396 
   397 fun reassoc_conjs thm =
   398   reassoc_conjs (thm RS @{thm conj_assoc[THEN iffD1]})
   399   handle THM _ => thm;
   400 
   401 fun type_args_named_constrained_of ((((ncAs, _), _), _), _) = ncAs;
   402 fun type_binding_of ((((_, b), _), _), _) = b;
   403 fun map_binding_of (((_, (b, _)), _), _) = b;
   404 fun rel_binding_of (((_, (_, b)), _), _) = b;
   405 fun mixfix_of ((_, mx), _) = mx;
   406 fun ctr_specs_of (_, ctr_specs) = ctr_specs;
   407 
   408 fun disc_of ((((disc, _), _), _), _) = disc;
   409 fun ctr_of ((((_, ctr), _), _), _) = ctr;
   410 fun args_of (((_, args), _), _) = args;
   411 fun defaults_of ((_, ds), _) = ds;
   412 fun ctr_mixfix_of (_, mx) = mx;
   413 
   414 fun add_nesty_bnf_names Us =
   415   let
   416     fun add (Type (s, Ts)) ss =
   417         let val (needs, ss') = fold_map add Ts ss in
   418           if exists I needs then (true, insert (op =) s ss') else (false, ss')
   419         end
   420       | add T ss = (member (op =) Us T, ss);
   421   in snd oo add end;
   422 
   423 fun nesty_bnfs ctxt ctr_Tsss Us =
   424   map_filter (bnf_of ctxt) (fold (fold (fold (add_nesty_bnf_names Us))) ctr_Tsss []);
   425 
   426 fun indexify proj xs f p = f (find_index (curry (op =) (proj p)) xs) p;
   427 
   428 fun build_map lthy build_simple =
   429   let
   430     fun build (TU as (T, U)) =
   431       if T = U then
   432         id_const T
   433       else
   434         (case TU of
   435           (Type (s, Ts), Type (s', Us)) =>
   436           if s = s' then
   437             let
   438               val bnf = the (bnf_of lthy s);
   439               val live = live_of_bnf bnf;
   440               val mapx = mk_map live Ts Us (map_of_bnf bnf);
   441               val TUs' = map dest_funT (fst (strip_typeN live (fastype_of mapx)));
   442             in Term.list_comb (mapx, map build TUs') end
   443           else
   444             build_simple TU
   445         | _ => build_simple TU);
   446   in build end;
   447 
   448 fun mk_iter_body ctor_iter fss xssss =
   449   Term.list_comb (ctor_iter, map2 (mk_sum_caseN_balanced oo map2 mk_uncurried2_fun) fss xssss);
   450 
   451 fun mk_preds_getterss_join c cps sum_prod_T cqfss =
   452   let val n = length cqfss in
   453     Term.lambda c (mk_IfN sum_prod_T cps
   454       (map2 (mk_InN_balanced sum_prod_T n) (map HOLogic.mk_tuple cqfss) (1 upto n)))
   455   end;
   456 
   457 fun mk_coiter_body lthy cs cpss f_sum_prod_Ts f_Tsss cqssss cfssss dtor_coiter =
   458   let
   459     fun build_sum_inj mk_inj = build_map lthy (uncurry mk_inj o dest_sumT o snd);
   460 
   461     fun build_dtor_coiter_arg _ [] [cf] = cf
   462       | build_dtor_coiter_arg T [cq] [cf, cf'] =
   463         mk_If cq (build_sum_inj Inl_const (fastype_of cf, T) $ cf)
   464           (build_sum_inj Inr_const (fastype_of cf', T) $ cf')
   465 
   466     val cqfsss = map3 (map3 (map3 build_dtor_coiter_arg)) f_Tsss cqssss cfssss;
   467   in
   468     Term.list_comb (dtor_coiter, map4 mk_preds_getterss_join cs cpss f_sum_prod_Ts cqfsss)
   469   end;
   470 
   471 fun define_co_iters fp fpT Cs binding_specs lthy0 =
   472   let
   473     val thy = Proof_Context.theory_of lthy0;
   474 
   475     val ((csts, defs), (lthy', lthy)) = lthy0
   476       |> apfst split_list o fold_map (fn (b, spec) =>
   477         Specification.definition (SOME (b, NONE, NoSyn), ((Thm.def_binding b, []), spec))
   478         #>> apsnd snd) binding_specs
   479       ||> `Local_Theory.restore;
   480 
   481     val phi = Proof_Context.export_morphism lthy lthy';
   482 
   483     val csts' = map (mk_co_iter thy fp fpT Cs o Morphism.term phi) csts;
   484     val defs' = map (Morphism.thm phi) defs;
   485   in
   486     ((csts', defs'), lthy')
   487   end;
   488 
   489 fun define_iters iterNs iter_args_typess' mk_binding fpTs Cs ctor_iters lthy =
   490   let
   491     val nn = length fpTs;
   492 
   493     val fpT_to_C as Type (_, [fpT, _]) = snd (strip_typeN nn (fastype_of (hd ctor_iters)));
   494 
   495     fun generate_iter suf (f_Tss, _, fss, xssss) ctor_iter =
   496       let
   497         val res_T = fold_rev (curry (op --->)) f_Tss fpT_to_C;
   498         val b = mk_binding suf;
   499         val spec =
   500           mk_Trueprop_eq (lists_bmoc fss (Free (Binding.name_of b, res_T)),
   501             mk_iter_body ctor_iter fss xssss);
   502       in (b, spec) end;
   503   in
   504     define_co_iters Least_FP fpT Cs (map3 generate_iter iterNs iter_args_typess' ctor_iters) lthy
   505   end;
   506 
   507 fun define_coiters coiterNs (cs, cpss, coiter_args_typess') mk_binding fpTs Cs dtor_coiters lthy =
   508   let
   509     val nn = length fpTs;
   510 
   511     val C_to_fpT as Type (_, [_, fpT]) = snd (strip_typeN nn (fastype_of (hd dtor_coiters)));
   512 
   513     fun generate_coiter suf ((pfss, cqssss, cfssss), (f_sum_prod_Ts, f_Tsss, pf_Tss)) dtor_coiter =
   514       let
   515         val res_T = fold_rev (curry (op --->)) pf_Tss C_to_fpT;
   516         val b = mk_binding suf;
   517         val spec =
   518           mk_Trueprop_eq (lists_bmoc pfss (Free (Binding.name_of b, res_T)),
   519             mk_coiter_body lthy cs cpss f_sum_prod_Ts f_Tsss cqssss cfssss dtor_coiter);
   520       in (b, spec) end;
   521   in
   522     define_co_iters Greatest_FP fpT Cs
   523       (map3 generate_coiter coiterNs coiter_args_typess' dtor_coiters) lthy
   524   end;
   525 
   526 fun derive_induct_iters_thms_for_types pre_bnfs (ctor_iters1 :: _) [fold_args_types, rec_args_types]
   527     ctor_induct [ctor_fold_thms, ctor_rec_thms] nesting_bnfs nested_bnfs fpTs Cs Xs ctrXs_Tsss ctrss
   528     ctr_defss [folds, recs] [fold_defs, rec_defs] lthy =
   529   let
   530     val ctr_Tsss = map (map (binder_types o fastype_of)) ctrss;
   531 
   532     val nn = length pre_bnfs;
   533     val ns = map length ctr_Tsss;
   534     val mss = map (map length) ctr_Tsss;
   535 
   536     val pre_map_defs = map map_def_of_bnf pre_bnfs;
   537     val pre_set_defss = map set_defs_of_bnf pre_bnfs;
   538     val nesting_map_ids'' = map (unfold_thms lthy @{thms id_def} o map_id_of_bnf) nesting_bnfs;
   539     val nested_map_ids'' = map (unfold_thms lthy @{thms id_def} o map_id_of_bnf) nested_bnfs;
   540     val nested_set_map's = maps set_map'_of_bnf nested_bnfs;
   541 
   542     val fp_b_names = map base_name_of_typ fpTs;
   543 
   544     val ctor_fold_fun_Ts = mk_fp_iter_fun_types (un_fold_of ctor_iters1);
   545     val ctor_rec_fun_Ts = mk_fp_iter_fun_types (co_rec_of ctor_iters1);
   546 
   547     val ((((ps, ps'), xsss), us'), names_lthy) =
   548       lthy
   549       |> mk_Frees' "P" (map mk_pred1T fpTs)
   550       ||>> mk_Freesss "x" ctr_Tsss
   551       ||>> Variable.variant_fixes fp_b_names;
   552 
   553     val us = map2 (curry Free) us' fpTs;
   554 
   555     fun mk_sets_nested bnf =
   556       let
   557         val Type (T_name, Us) = T_of_bnf bnf;
   558         val lives = lives_of_bnf bnf;
   559         val sets = sets_of_bnf bnf;
   560         fun mk_set U =
   561           (case find_index (curry (op =) U) lives of
   562             ~1 => Term.dummy
   563           | i => nth sets i);
   564       in
   565         (T_name, map mk_set Us)
   566       end;
   567 
   568     val setss_nested = map mk_sets_nested nested_bnfs;
   569 
   570     val (induct_thms, induct_thm) =
   571       let
   572         fun mk_set Ts t =
   573           let val Type (_, Ts0) = domain_type (fastype_of t) in
   574             Term.subst_atomic_types (Ts0 ~~ Ts) t
   575           end;
   576 
   577         fun mk_raw_prem_prems _ (x as Free (_, Type _)) (X as TFree _) =
   578             [([], (find_index (curry (op =) X) Xs + 1, x))]
   579           | mk_raw_prem_prems names_lthy (x as Free (s, Type (T_name, Ts0))) (Type (_, Xs_Ts0)) =
   580             (case AList.lookup (op =) setss_nested T_name of
   581               NONE => []
   582             | SOME raw_sets0 =>
   583               let
   584                 val (Xs_Ts, (Ts, raw_sets)) =
   585                   filter (exists_subtype_in Xs o fst) (Xs_Ts0 ~~ (Ts0 ~~ raw_sets0))
   586                   |> split_list ||> split_list;
   587                 val sets = map (mk_set Ts0) raw_sets;
   588                 val (ys, names_lthy') = names_lthy |> mk_Frees s Ts;
   589                 val xysets = map (pair x) (ys ~~ sets);
   590                 val ppremss = map2 (mk_raw_prem_prems names_lthy') ys Xs_Ts;
   591               in
   592                 flat (map2 (map o apfst o cons) xysets ppremss)
   593               end)
   594           | mk_raw_prem_prems _ _ _ = [];
   595 
   596         fun close_prem_prem xs t =
   597           fold_rev Logic.all (map Free (drop (nn + length xs)
   598             (rev (Term.add_frees t (map dest_Free xs @ ps'))))) t;
   599 
   600         fun mk_prem_prem xs (xysets, (j, x)) =
   601           close_prem_prem xs (Logic.list_implies (map (fn (x', (y, set)) =>
   602               HOLogic.mk_Trueprop (HOLogic.mk_mem (y, set $ x'))) xysets,
   603             HOLogic.mk_Trueprop (nth ps (j - 1) $ x)));
   604 
   605         fun mk_raw_prem phi ctr ctr_Ts ctrXs_Ts =
   606           let
   607             val (xs, names_lthy') = names_lthy |> mk_Frees "x" ctr_Ts;
   608             val pprems = flat (map2 (mk_raw_prem_prems names_lthy') xs ctrXs_Ts);
   609           in (xs, pprems, HOLogic.mk_Trueprop (phi $ Term.list_comb (ctr, xs))) end;
   610 
   611         fun mk_prem (xs, raw_pprems, concl) =
   612           fold_rev Logic.all xs (Logic.list_implies (map (mk_prem_prem xs) raw_pprems, concl));
   613 
   614         val raw_premss = map4 (map3 o mk_raw_prem) ps ctrss ctr_Tsss ctrXs_Tsss;
   615 
   616         val goal =
   617           Library.foldr (Logic.list_implies o apfst (map mk_prem)) (raw_premss,
   618             HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj (map2 (curry (op $)) ps us)));
   619 
   620         val kksss = map (map (map (fst o snd) o #2)) raw_premss;
   621 
   622         val ctor_induct' = ctor_induct OF (map mk_sumEN_tupled_balanced mss);
   623 
   624         val thm =
   625           Goal.prove_sorry lthy [] [] goal (fn {context = ctxt, ...} =>
   626             mk_induct_tac ctxt nn ns mss kksss (flat ctr_defss) ctor_induct' nested_set_map's
   627               pre_set_defss)
   628           |> singleton (Proof_Context.export names_lthy lthy)
   629           |> Thm.close_derivation;
   630       in
   631         `(conj_dests nn) thm
   632       end;
   633 
   634     val induct_cases = quasi_unambiguous_case_names (maps (map name_of_ctr) ctrss);
   635     val induct_case_names_attr = Attrib.internal (K (Rule_Cases.case_names induct_cases));
   636 
   637     val xctrss = map2 (map2 (curry Term.list_comb)) ctrss xsss;
   638 
   639     fun mk_iter_thmss (_, x_Tssss, fss, _) iters iter_defs ctor_iter_thms =
   640       let
   641         val fiters = map (lists_bmoc fss) iters;
   642 
   643         fun mk_goal fss fiter xctr f xs fxs =
   644           fold_rev (fold_rev Logic.all) (xs :: fss)
   645             (mk_Trueprop_eq (fiter $ xctr, Term.list_comb (f, fxs)));
   646 
   647         fun maybe_tick (T, U) u f =
   648           if try (fst o HOLogic.dest_prodT) U = SOME T then
   649             Term.lambda u (HOLogic.mk_prod (u, f $ u))
   650           else
   651             f;
   652 
   653         fun unzip_iters (x as Free (_, T)) =
   654           map (fn U => if U = T then x else
   655             build_map lthy (indexify (perhaps (try (snd o HOLogic.dest_prodT)) o snd) Cs
   656               (fn kk => fn TU => maybe_tick TU (nth us kk) (nth fiters kk))) (T, U) $ x);
   657 
   658         val fxsss = map2 (map2 (flat_rec oo map2 unzip_iters)) xsss x_Tssss;
   659 
   660         val goalss = map5 (map4 o mk_goal fss) fiters xctrss fss xsss fxsss;
   661 
   662         val tacss =
   663           map2 (map o mk_iter_tac pre_map_defs (nested_map_ids'' @ nesting_map_ids'') iter_defs)
   664             ctor_iter_thms ctr_defss;
   665 
   666         fun prove goal tac =
   667           Goal.prove_sorry lthy [] [] goal (tac o #context)
   668           |> Thm.close_derivation;
   669       in
   670         map2 (map2 prove) goalss tacss
   671       end;
   672 
   673     val fold_thmss = mk_iter_thmss fold_args_types folds fold_defs ctor_fold_thms;
   674     val rec_thmss = mk_iter_thmss rec_args_types recs rec_defs ctor_rec_thms;
   675   in
   676     ((induct_thm, induct_thms, [induct_case_names_attr]),
   677      (fold_thmss, code_simp_attrs), (rec_thmss, code_simp_attrs))
   678   end;
   679 
   680 fun derive_coinduct_coiters_thms_for_types pre_bnfs (dtor_coiters1 :: _) dtor_coinduct
   681     dtor_strong_induct dtor_ctors [dtor_unfold_thms, dtor_corec_thms] nesting_bnfs nested_bnfs fpTs
   682     Cs As kss mss ns ctr_defss ctr_sugars [unfolds, corecs] [unfold_defs, corec_defs] lthy =
   683   let
   684     val nn = length pre_bnfs;
   685 
   686     val pre_map_defs = map map_def_of_bnf pre_bnfs;
   687     val pre_rel_defs = map rel_def_of_bnf pre_bnfs;
   688     val nesting_map_ids'' = map (unfold_thms lthy @{thms id_def} o map_id_of_bnf) nesting_bnfs;
   689     val nesting_rel_eqs = map rel_eq_of_bnf nesting_bnfs;
   690     val nested_map_comp's = map map_comp'_of_bnf nested_bnfs;
   691     val nested_map_comps'' = map ((fn thm => thm RS sym) o map_comp_of_bnf) nested_bnfs;
   692     val nested_map_ids'' = map (unfold_thms lthy @{thms id_def} o map_id_of_bnf) nested_bnfs;
   693 
   694     val fp_b_names = map base_name_of_typ fpTs;
   695 
   696     val dtor_unfold_fun_Ts = mk_fp_iter_fun_types (un_fold_of dtor_coiters1);
   697     val dtor_corec_fun_Ts = mk_fp_iter_fun_types (co_rec_of dtor_coiters1);
   698 
   699     val ctrss = map (map (mk_ctr As) o #ctrs) ctr_sugars;
   700     val discss = map (map (mk_disc_or_sel As) o #discs) ctr_sugars;
   701     val selsss = map (map (map (mk_disc_or_sel As)) o #selss) ctr_sugars;
   702     val exhausts = map #exhaust ctr_sugars;
   703     val disc_thmsss = map #disc_thmss ctr_sugars;
   704     val discIss = map #discIs ctr_sugars;
   705     val sel_thmsss = map #sel_thmss ctr_sugars;
   706 
   707     val ((cs, cpss, [((pgss, crssss, cgssss), _), ((phss, csssss, chssss), _)]), names_lthy0) =
   708       mk_coiters_args_types Cs ns mss [dtor_unfold_fun_Ts, dtor_corec_fun_Ts] lthy;
   709 
   710     val (((rs, us'), vs'), names_lthy) =
   711       names_lthy0
   712       |> mk_Frees "R" (map (fn T => mk_pred2T T T) fpTs)
   713       ||>> Variable.variant_fixes fp_b_names
   714       ||>> Variable.variant_fixes (map (suffix "'") fp_b_names);
   715 
   716     val us = map2 (curry Free) us' fpTs;
   717     val udiscss = map2 (map o rapp) us discss;
   718     val uselsss = map2 (map o map o rapp) us selsss;
   719 
   720     val vs = map2 (curry Free) vs' fpTs;
   721     val vdiscss = map2 (map o rapp) vs discss;
   722     val vselsss = map2 (map o map o rapp) vs selsss;
   723 
   724     val ((coinduct_thms, coinduct_thm), (strong_coinduct_thms, strong_coinduct_thm)) =
   725       let
   726         val uvrs = map3 (fn r => fn u => fn v => r $ u $ v) rs us vs;
   727         val uv_eqs = map2 (curry HOLogic.mk_eq) us vs;
   728         val strong_rs =
   729           map4 (fn u => fn v => fn uvr => fn uv_eq =>
   730             fold_rev Term.lambda [u, v] (HOLogic.mk_disj (uvr, uv_eq))) us vs uvrs uv_eqs;
   731 
   732         fun build_rel rs' T =
   733           (case find_index (curry (op =) T) fpTs of
   734             ~1 =>
   735             if exists_subtype_in fpTs T then
   736               let
   737                 val Type (s, Ts) = T
   738                 val bnf = the (bnf_of lthy s);
   739                 val live = live_of_bnf bnf;
   740                 val rel = mk_rel live Ts Ts (rel_of_bnf bnf);
   741                 val Ts' = map domain_type (fst (strip_typeN live (fastype_of rel)));
   742               in Term.list_comb (rel, map (build_rel rs') Ts') end
   743             else
   744               HOLogic.eq_const T
   745           | kk => nth rs' kk);
   746 
   747         fun build_rel_app rs' usel vsel = fold rapp [usel, vsel] (build_rel rs' (fastype_of usel));
   748 
   749         fun mk_prem_ctr_concls rs' n k udisc usels vdisc vsels =
   750           (if k = n then [] else [HOLogic.mk_eq (udisc, vdisc)]) @
   751           (if null usels then
   752              []
   753            else
   754              [Library.foldr HOLogic.mk_imp (if n = 1 then [] else [udisc, vdisc],
   755                 Library.foldr1 HOLogic.mk_conj (map2 (build_rel_app rs') usels vsels))]);
   756 
   757         fun mk_prem_concl rs' n udiscs uselss vdiscs vselss =
   758           Library.foldr1 HOLogic.mk_conj
   759             (flat (map5 (mk_prem_ctr_concls rs' n) (1 upto n) udiscs uselss vdiscs vselss))
   760           handle List.Empty => @{term True};
   761 
   762         fun mk_prem rs' uvr u v n udiscs uselss vdiscs vselss =
   763           fold_rev Logic.all [u, v] (Logic.mk_implies (HOLogic.mk_Trueprop uvr,
   764             HOLogic.mk_Trueprop (mk_prem_concl rs' n udiscs uselss vdiscs vselss)));
   765 
   766         val concl =
   767           HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj
   768             (map3 (fn uvr => fn u => fn v => HOLogic.mk_imp (uvr, HOLogic.mk_eq (u, v)))
   769                uvrs us vs));
   770 
   771         fun mk_goal rs' =
   772           Logic.list_implies (map8 (mk_prem rs') uvrs us vs ns udiscss uselsss vdiscss vselsss,
   773             concl);
   774 
   775         val goal = mk_goal rs;
   776         val strong_goal = mk_goal strong_rs;
   777 
   778         fun prove dtor_coinduct' goal =
   779           Goal.prove_sorry lthy [] [] goal (fn {context = ctxt, ...} =>
   780             mk_coinduct_tac ctxt nesting_rel_eqs nn ns dtor_coinduct' pre_rel_defs dtor_ctors
   781               exhausts ctr_defss disc_thmsss sel_thmsss)
   782           |> singleton (Proof_Context.export names_lthy lthy)
   783           |> Thm.close_derivation;
   784 
   785         fun postproc nn thm =
   786           Thm.permute_prems 0 nn
   787             (if nn = 1 then thm RS mp else funpow nn (fn thm => reassoc_conjs (thm RS mp_conj)) thm)
   788           |> Drule.zero_var_indexes
   789           |> `(conj_dests nn);
   790       in
   791         (postproc nn (prove dtor_coinduct goal), postproc nn (prove dtor_strong_induct strong_goal))
   792       end;
   793 
   794     fun mk_coinduct_concls ms discs ctrs =
   795       let
   796         fun mk_disc_concl disc = [name_of_disc disc];
   797         fun mk_ctr_concl 0 _ = []
   798           | mk_ctr_concl _ ctor = [name_of_ctr ctor];
   799         val disc_concls = map mk_disc_concl (fst (split_last discs)) @ [[]];
   800         val ctr_concls = map2 mk_ctr_concl ms ctrs;
   801       in
   802         flat (map2 append disc_concls ctr_concls)
   803       end;
   804 
   805     val coinduct_cases = quasi_unambiguous_case_names (map (prefix EqN) fp_b_names);
   806     val coinduct_conclss =
   807       map3 (quasi_unambiguous_case_names ooo mk_coinduct_concls) mss discss ctrss;
   808 
   809     fun mk_maybe_not pos = not pos ? HOLogic.mk_not;
   810 
   811     val gunfolds = map (lists_bmoc pgss) unfolds;
   812     val hcorecs = map (lists_bmoc phss) corecs;
   813 
   814     val (unfold_thmss, corec_thmss, safe_unfold_thmss, safe_corec_thmss) =
   815       let
   816         fun mk_goal pfss c cps fcoiter n k ctr m cfs' =
   817           fold_rev (fold_rev Logic.all) ([c] :: pfss)
   818             (Logic.list_implies (seq_conds (HOLogic.mk_Trueprop oo mk_maybe_not) n k cps,
   819                mk_Trueprop_eq (fcoiter $ c, Term.list_comb (ctr, take m cfs'))));
   820 
   821         (* TODO: get rid of "mk_U" *)
   822         val mk_U = typ_subst_nonatomic (map2 pair Cs fpTs);
   823 
   824         fun intr_coiters fcoiters [] [cf] =
   825             let val T = fastype_of cf in
   826               if exists_subtype_in Cs T then
   827                 build_map lthy (indexify fst Cs (K o nth fcoiters)) (T, mk_U T) $ cf
   828               else
   829                 cf
   830             end
   831           | intr_coiters fcoiters [cq] [cf, cf'] =
   832             mk_If cq (intr_coiters fcoiters [] [cf]) (intr_coiters fcoiters [] [cf']);
   833 
   834         val crgsss = map2 (map2 (map2 (intr_coiters gunfolds))) crssss cgssss;
   835         val cshsss = map2 (map2 (map2 (intr_coiters hcorecs))) csssss chssss;
   836 
   837         val unfold_goalss = map8 (map4 oooo mk_goal pgss) cs cpss gunfolds ns kss ctrss mss crgsss;
   838         val corec_goalss = map8 (map4 oooo mk_goal phss) cs cpss hcorecs ns kss ctrss mss cshsss;
   839 
   840         fun mk_map_if_distrib bnf =
   841           let
   842             val mapx = map_of_bnf bnf;
   843             val live = live_of_bnf bnf;
   844             val ((Ts, T), U) = strip_typeN (live + 1) (fastype_of mapx) |>> split_last;
   845             val fs = Variable.variant_frees lthy [mapx] (map (pair "f") Ts);
   846             val t = Term.list_comb (mapx, map (Var o apfst (rpair 0)) fs);
   847           in
   848             Drule.instantiate' (map (SOME o certifyT lthy) [U, T]) [SOME (certify lthy t)]
   849               @{thm if_distrib}
   850           end;
   851 
   852         val nested_map_if_distribs = map mk_map_if_distrib nested_bnfs;
   853 
   854         val unfold_tacss =
   855           map3 (map oo mk_coiter_tac unfold_defs [] [] nesting_map_ids'' [])
   856             dtor_unfold_thms pre_map_defs ctr_defss;
   857         val corec_tacss =
   858           map3 (map oo mk_coiter_tac corec_defs nested_map_comps'' nested_map_comp's
   859               (nested_map_ids'' @ nesting_map_ids'') nested_map_if_distribs)
   860             dtor_corec_thms pre_map_defs ctr_defss;
   861 
   862         fun prove goal tac =
   863           Goal.prove_sorry lthy [] [] goal (tac o #context)
   864           |> Thm.close_derivation;
   865 
   866         val unfold_thmss = map2 (map2 prove) unfold_goalss unfold_tacss;
   867         val corec_thmss = map2 (map2 prove) corec_goalss corec_tacss;
   868 
   869         val filter_safesss =
   870           map2 (map_filter (fn (safes, thm) => if forall I safes then SOME thm else NONE) oo
   871             curry (op ~~)) (map2 (map2 (map2 (member (op =)))) cgssss crgsss);
   872 
   873         val safe_unfold_thmss = filter_safesss unfold_thmss;
   874         val safe_corec_thmss = filter_safesss corec_thmss;
   875       in
   876         (unfold_thmss, corec_thmss, safe_unfold_thmss, safe_corec_thmss)
   877       end;
   878 
   879     val (disc_unfold_iff_thmss, disc_corec_iff_thmss) =
   880       let
   881         fun mk_goal c cps fcoiter n k disc =
   882           mk_Trueprop_eq (disc $ (fcoiter $ c),
   883             if n = 1 then @{const True}
   884             else Library.foldr1 HOLogic.mk_conj (seq_conds mk_maybe_not n k cps));
   885 
   886         val unfold_goalss = map6 (map2 oooo mk_goal) cs cpss gunfolds ns kss discss;
   887         val corec_goalss = map6 (map2 oooo mk_goal) cs cpss hcorecs ns kss discss;
   888 
   889         fun mk_case_split' cp = Drule.instantiate' [] [SOME (certify lthy cp)] @{thm case_split};
   890 
   891         val case_splitss' = map (map mk_case_split') cpss;
   892 
   893         val unfold_tacss =
   894           map3 (map oo mk_disc_coiter_iff_tac) case_splitss' unfold_thmss disc_thmsss;
   895         val corec_tacss =
   896           map3 (map oo mk_disc_coiter_iff_tac) case_splitss' corec_thmss disc_thmsss;
   897 
   898         fun prove goal tac =
   899           Goal.prove_sorry lthy [] [] goal (tac o #context)
   900           |> singleton (Proof_Context.export names_lthy lthy)
   901           |> Thm.close_derivation;
   902 
   903         fun proves [_] [_] = []
   904           | proves goals tacs = map2 prove goals tacs;
   905       in
   906         (map2 proves unfold_goalss unfold_tacss, map2 proves corec_goalss corec_tacss)
   907       end;
   908 
   909     val is_triv_discI = is_triv_implies orf is_concl_refl;
   910 
   911     fun mk_disc_coiter_thms coiters discIs =
   912       map (op RS) (filter_out (is_triv_discI o snd) (coiters ~~ discIs));
   913 
   914     val disc_unfold_thmss = map2 mk_disc_coiter_thms unfold_thmss discIss;
   915     val disc_corec_thmss = map2 mk_disc_coiter_thms corec_thmss discIss;
   916 
   917     fun mk_sel_coiter_thm coiter_thm sel sel_thm =
   918       let
   919         val (domT, ranT) = dest_funT (fastype_of sel);
   920         val arg_cong' =
   921           Drule.instantiate' (map (SOME o certifyT lthy) [domT, ranT])
   922             [NONE, NONE, SOME (certify lthy sel)] arg_cong
   923           |> Thm.varifyT_global;
   924         val sel_thm' = sel_thm RSN (2, trans);
   925       in
   926         coiter_thm RS arg_cong' RS sel_thm'
   927       end;
   928 
   929     fun mk_sel_coiter_thms coiter_thmss =
   930       map3 (map3 (map2 o mk_sel_coiter_thm)) coiter_thmss selsss sel_thmsss |> map flat;
   931 
   932     val sel_unfold_thmss = mk_sel_coiter_thms unfold_thmss;
   933     val sel_corec_thmss = mk_sel_coiter_thms corec_thmss;
   934 
   935     val coinduct_consumes_attr = Attrib.internal (K (Rule_Cases.consumes nn));
   936     val coinduct_case_names_attr = Attrib.internal (K (Rule_Cases.case_names coinduct_cases));
   937     val coinduct_case_concl_attrs =
   938       map2 (fn casex => fn concls =>
   939           Attrib.internal (K (Rule_Cases.case_conclusion (casex, concls))))
   940         coinduct_cases coinduct_conclss;
   941     val coinduct_case_attrs =
   942       coinduct_consumes_attr :: coinduct_case_names_attr :: coinduct_case_concl_attrs;
   943   in
   944     ((coinduct_thm, coinduct_thms, strong_coinduct_thm, strong_coinduct_thms, coinduct_case_attrs),
   945      (unfold_thmss, corec_thmss, []),
   946      (safe_unfold_thmss, safe_corec_thmss),
   947      (disc_unfold_thmss, disc_corec_thmss, simp_attrs),
   948      (disc_unfold_iff_thmss, disc_corec_iff_thmss, simp_attrs),
   949      (sel_unfold_thmss, sel_corec_thmss, simp_attrs))
   950   end;
   951 
   952 fun define_co_datatypes prepare_constraint prepare_typ prepare_term fp construct_fp
   953     (wrap_opts as (no_dests, rep_compat), specs) no_defs_lthy0 =
   954   let
   955     (* TODO: sanity checks on arguments *)
   956 
   957     val _ = if fp = Greatest_FP andalso no_dests then
   958         error "Cannot define destructor-less codatatypes"
   959       else
   960         ();
   961 
   962     fun qualify mandatory fp_b_name =
   963       Binding.qualify mandatory fp_b_name o (rep_compat ? Binding.qualify false rep_compat_prefix);
   964 
   965     val nn = length specs;
   966     val fp_bs = map type_binding_of specs;
   967     val fp_b_names = map Binding.name_of fp_bs;
   968     val fp_common_name = mk_common_name fp_b_names;
   969     val map_bs = map map_binding_of specs;
   970     val rel_bs = map rel_binding_of specs;
   971 
   972     fun prepare_type_arg (_, (ty, c)) =
   973       let val TFree (s, _) = prepare_typ no_defs_lthy0 ty in
   974         TFree (s, prepare_constraint no_defs_lthy0 c)
   975       end;
   976 
   977     val Ass0 = map (map prepare_type_arg o type_args_named_constrained_of) specs;
   978     val unsorted_Ass0 = map (map (resort_tfree HOLogic.typeS)) Ass0;
   979     val unsorted_As = Library.foldr1 merge_type_args unsorted_Ass0;
   980     val set_bss = map (map fst o type_args_named_constrained_of) specs;
   981 
   982     val (((Bs0, Cs), Xs), no_defs_lthy) =
   983       no_defs_lthy0
   984       |> fold (Variable.declare_typ o resort_tfree dummyS) unsorted_As
   985       |> mk_TFrees (length unsorted_As)
   986       ||>> mk_TFrees nn
   987       ||>> variant_tfrees fp_b_names;
   988 
   989     (* TODO: Cleaner handling of fake contexts, without "background_theory". The case where the new
   990        type is defined in a locale and shadows an existing global type is currently not handled. *)
   991 
   992     fun add_fake_type spec =
   993       Sign.add_type no_defs_lthy (type_binding_of spec,
   994         length (type_args_named_constrained_of spec), mixfix_of spec);
   995 
   996     val fake_thy = Theory.copy #> fold add_fake_type specs;
   997     val fake_lthy = Proof_Context.background_theory fake_thy no_defs_lthy;
   998 
   999     fun mk_fake_T b =
  1000       Type (fst (Term.dest_Type (Proof_Context.read_type_name fake_lthy true (Binding.name_of b))),
  1001         unsorted_As);
  1002 
  1003     val fake_Ts = map mk_fake_T fp_bs;
  1004 
  1005     val mixfixes = map mixfix_of specs;
  1006 
  1007     val _ = (case duplicates Binding.eq_name fp_bs of [] => ()
  1008       | b :: _ => error ("Duplicate type name declaration " ^ quote (Binding.name_of b)));
  1009 
  1010     val ctr_specss = map ctr_specs_of specs;
  1011 
  1012     val disc_bindingss = map (map disc_of) ctr_specss;
  1013     val ctr_bindingss =
  1014       map2 (fn fp_b_name => map (qualify false fp_b_name o ctr_of)) fp_b_names ctr_specss;
  1015     val ctr_argsss = map (map args_of) ctr_specss;
  1016     val ctr_mixfixess = map (map ctr_mixfix_of) ctr_specss;
  1017 
  1018     val sel_bindingsss = map (map (map fst)) ctr_argsss;
  1019     val fake_ctr_Tsss0 = map (map (map (prepare_typ fake_lthy o snd))) ctr_argsss;
  1020     val raw_sel_defaultsss = map (map defaults_of) ctr_specss;
  1021 
  1022     val (As :: _) :: fake_ctr_Tsss =
  1023       burrow (burrow (Syntax.check_typs fake_lthy)) (Ass0 :: fake_ctr_Tsss0);
  1024 
  1025     val _ = (case duplicates (op =) unsorted_As of [] => ()
  1026       | A :: _ => error ("Duplicate type parameter " ^
  1027           quote (Syntax.string_of_typ no_defs_lthy A)));
  1028 
  1029     val rhs_As' = fold (fold (fold Term.add_tfreesT)) fake_ctr_Tsss [];
  1030     val _ = (case subtract (op =) (map dest_TFree As) rhs_As' of
  1031         [] => ()
  1032       | A' :: _ => error ("Extra type variable on right-hand side: " ^
  1033           quote (Syntax.string_of_typ no_defs_lthy (TFree A'))));
  1034 
  1035     fun eq_fpT_check (T as Type (s, Us)) (Type (s', Us')) =
  1036         s = s' andalso (Us = Us' orelse error ("Illegal occurrence of recursive type " ^
  1037           quote (Syntax.string_of_typ fake_lthy T)))
  1038       | eq_fpT_check _ _ = false;
  1039 
  1040     fun freeze_fp (T as Type (s, Us)) =
  1041         (case find_index (eq_fpT_check T) fake_Ts of
  1042           ~1 => Type (s, map freeze_fp Us)
  1043         | kk => nth Xs kk)
  1044       | freeze_fp T = T;
  1045 
  1046     val ctrXs_Tsss = map (map (map freeze_fp)) fake_ctr_Tsss;
  1047     val ctrXs_sum_prod_Ts = map (mk_sumTN_balanced o map HOLogic.mk_tupleT) ctrXs_Tsss;
  1048 
  1049     val fp_eqs =
  1050       map dest_TFree Xs ~~ map (Term.typ_subst_atomic (As ~~ unsorted_As)) ctrXs_sum_prod_Ts;
  1051 
  1052     val (pre_bnfs, (fp_res as {bnfs = fp_bnfs as any_fp_bnf :: _, ctors = ctors0, dtors = dtors0,
  1053            xtor_co_iterss = xtor_co_iterss0, xtor_co_induct, xtor_strong_co_induct, dtor_ctors,
  1054            ctor_dtors, ctor_injects, xtor_map_thms, xtor_set_thmss, xtor_rel_thms,
  1055            xtor_co_iter_thmss, ...}, lthy)) =
  1056       fp_bnf (construct_fp mixfixes map_bs rel_bs set_bss) fp_bs (map dest_TFree unsorted_As) fp_eqs
  1057         no_defs_lthy0;
  1058 
  1059     val timer = time (Timer.startRealTimer ());
  1060 
  1061     val nesting_bnfs = nesty_bnfs lthy ctrXs_Tsss As;
  1062     val nested_bnfs = nesty_bnfs lthy ctrXs_Tsss Xs;
  1063 
  1064     val pre_map_defs = map map_def_of_bnf pre_bnfs;
  1065     val pre_set_defss = map set_defs_of_bnf pre_bnfs;
  1066     val pre_rel_defs = map rel_def_of_bnf pre_bnfs;
  1067     val nesting_set_map's = maps set_map'_of_bnf nesting_bnfs;
  1068     val nested_set_map's = maps set_map'_of_bnf nested_bnfs;
  1069 
  1070     val live = live_of_bnf any_fp_bnf;
  1071 
  1072     val Bs =
  1073       map3 (fn alive => fn A as TFree (_, S) => fn B => if alive then resort_tfree S B else A)
  1074         (liveness_of_fp_bnf (length As) any_fp_bnf) As Bs0;
  1075 
  1076     val B_ify = Term.typ_subst_atomic (As ~~ Bs);
  1077 
  1078     val ctors = map (mk_ctor As) ctors0;
  1079     val dtors = map (mk_dtor As) dtors0;
  1080 
  1081     val fpTs = map (domain_type o fastype_of) dtors;
  1082 
  1083     fun massage_simple_notes base =
  1084       filter_out (null o #2)
  1085       #> map (fn (thmN, thms, attrs) =>
  1086         ((qualify true base (Binding.name thmN), attrs), [(thms, [])]));
  1087 
  1088     val massage_multi_notes =
  1089       maps (fn (thmN, thmss, attrs) =>
  1090         if forall null thmss then
  1091           []
  1092         else
  1093           map3 (fn fp_b_name => fn Type (T_name, _) => fn thms =>
  1094               ((qualify true fp_b_name (Binding.name thmN), attrs T_name), [(thms, [])]))
  1095             fp_b_names fpTs thmss);
  1096 
  1097     val ctr_Tsss = map (map (map (Term.typ_subst_atomic (Xs ~~ fpTs)))) ctrXs_Tsss;
  1098     val ns = map length ctr_Tsss;
  1099     val kss = map (fn n => 1 upto n) ns;
  1100     val mss = map (map length) ctr_Tsss;
  1101 
  1102     val ((xtor_co_iterss, iters_args_types, coiters_args_types), lthy) =
  1103       mk_co_iters_prelims fp fpTs Cs ns mss xtor_co_iterss0 lthy;
  1104 
  1105     fun define_ctrs_case_for_type ((((((((((((((((((((((((fp_bnf, fp_b), fpT), C), ctor), dtor),
  1106               xtor_co_iters), ctor_dtor), dtor_ctor), ctor_inject), pre_map_def), pre_set_defs),
  1107             pre_rel_def), fp_map_thm), fp_set_thms), fp_rel_thm), n), ks), ms), ctr_bindings),
  1108           ctr_mixfixes), ctr_Tss), disc_bindings), sel_bindingss), raw_sel_defaultss) no_defs_lthy =
  1109       let
  1110         val fp_b_name = Binding.name_of fp_b;
  1111 
  1112         val dtorT = domain_type (fastype_of ctor);
  1113         val ctr_prod_Ts = map HOLogic.mk_tupleT ctr_Tss;
  1114         val ctr_sum_prod_T = mk_sumTN_balanced ctr_prod_Ts;
  1115         val case_Ts = map (fn Ts => Ts ---> C) ctr_Tss;
  1116 
  1117         val (((((w, fs), xss), yss), u'), names_lthy) =
  1118           no_defs_lthy
  1119           |> yield_singleton (mk_Frees "w") dtorT
  1120           ||>> mk_Frees "f" case_Ts
  1121           ||>> mk_Freess "x" ctr_Tss
  1122           ||>> mk_Freess "y" (map (map B_ify) ctr_Tss)
  1123           ||>> yield_singleton Variable.variant_fixes fp_b_name;
  1124 
  1125         val u = Free (u', fpT);
  1126 
  1127         val tuple_xs = map HOLogic.mk_tuple xss;
  1128         val tuple_ys = map HOLogic.mk_tuple yss;
  1129 
  1130         val ctr_rhss =
  1131           map3 (fn k => fn xs => fn tuple_x => fold_rev Term.lambda xs (ctor $
  1132             mk_InN_balanced ctr_sum_prod_T n tuple_x k)) ks xss tuple_xs;
  1133 
  1134         val case_binding = qualify false fp_b_name (Binding.suffix_name ("_" ^ caseN) fp_b);
  1135 
  1136         val case_rhs =
  1137           fold_rev Term.lambda (fs @ [u])
  1138             (mk_sum_caseN_balanced (map2 mk_uncurried_fun fs xss) $ (dtor $ u));
  1139 
  1140         val ((raw_case :: raw_ctrs, raw_case_def :: raw_ctr_defs), (lthy', lthy)) = no_defs_lthy
  1141           |> apfst split_list o fold_map3 (fn b => fn mx => fn rhs =>
  1142               Local_Theory.define ((b, mx), ((Thm.def_binding b, []), rhs)) #>> apsnd snd)
  1143             (case_binding :: ctr_bindings) (NoSyn :: ctr_mixfixes) (case_rhs :: ctr_rhss)
  1144           ||> `Local_Theory.restore;
  1145 
  1146         val phi = Proof_Context.export_morphism lthy lthy';
  1147 
  1148         val ctr_defs = map (Morphism.thm phi) raw_ctr_defs;
  1149         val ctr_defs' =
  1150           map2 (fn m => fn def => mk_unabs_def m (def RS meta_eq_to_obj_eq)) ms ctr_defs;
  1151         val case_def = Morphism.thm phi raw_case_def;
  1152 
  1153         val ctrs0 = map (Morphism.term phi) raw_ctrs;
  1154         val casex0 = Morphism.term phi raw_case;
  1155 
  1156         val ctrs = map (mk_ctr As) ctrs0;
  1157 
  1158         fun wrap_ctrs lthy =
  1159           let
  1160             fun exhaust_tac {context = ctxt, prems = _} =
  1161               let
  1162                 val ctor_iff_dtor_thm =
  1163                   let
  1164                     val goal =
  1165                       fold_rev Logic.all [w, u]
  1166                         (mk_Trueprop_eq (HOLogic.mk_eq (u, ctor $ w), HOLogic.mk_eq (dtor $ u, w)));
  1167                   in
  1168                     Goal.prove_sorry lthy [] [] goal (fn {context = ctxt, ...} =>
  1169                       mk_ctor_iff_dtor_tac ctxt (map (SOME o certifyT lthy) [dtorT, fpT])
  1170                         (certify lthy ctor) (certify lthy dtor) ctor_dtor dtor_ctor)
  1171                     |> Thm.close_derivation
  1172                     |> Morphism.thm phi
  1173                   end;
  1174 
  1175                 val sumEN_thm' =
  1176                   unfold_thms lthy @{thms all_unit_eq}
  1177                     (Drule.instantiate' (map (SOME o certifyT lthy) ctr_prod_Ts) []
  1178                        (mk_sumEN_balanced n))
  1179                   |> Morphism.thm phi;
  1180               in
  1181                 mk_exhaust_tac ctxt n ctr_defs ctor_iff_dtor_thm sumEN_thm'
  1182               end;
  1183 
  1184             val inject_tacss =
  1185               map2 (fn 0 => K [] | _ => fn ctr_def => [fn {context = ctxt, ...} =>
  1186                   mk_inject_tac ctxt ctr_def ctor_inject]) ms ctr_defs;
  1187 
  1188             val half_distinct_tacss =
  1189               map (map (fn (def, def') => fn {context = ctxt, ...} =>
  1190                 mk_half_distinct_tac ctxt ctor_inject [def, def'])) (mk_half_pairss (`I ctr_defs));
  1191 
  1192             val case_tacs =
  1193               map3 (fn k => fn m => fn ctr_def => fn {context = ctxt, ...} =>
  1194                 mk_case_tac ctxt n k m case_def ctr_def dtor_ctor) ks ms ctr_defs;
  1195 
  1196             val tacss = [exhaust_tac] :: inject_tacss @ half_distinct_tacss @ [case_tacs];
  1197 
  1198             val sel_defaultss = map (map (apsnd (prepare_term lthy))) raw_sel_defaultss
  1199           in
  1200             wrap_free_constructors tacss (((wrap_opts, ctrs0), casex0), (disc_bindings,
  1201               (sel_bindingss, sel_defaultss))) lthy
  1202           end;
  1203 
  1204         fun derive_maps_sets_rels (ctr_sugar, lthy) =
  1205           if live = 0 then
  1206             ((([], [], [], []), ctr_sugar), lthy)
  1207           else
  1208             let
  1209               val rel_flip = rel_flip_of_bnf fp_bnf;
  1210               val nones = replicate live NONE;
  1211 
  1212               val ctor_cong =
  1213                 if fp = Least_FP then
  1214                   Drule.dummy_thm
  1215                 else
  1216                   let val ctor' = mk_ctor Bs ctor in
  1217                     cterm_instantiate_pos [NONE, NONE, SOME (certify lthy ctor')] arg_cong
  1218                   end;
  1219 
  1220               fun mk_cIn ify =
  1221                 certify lthy o (fp = Greatest_FP ? curry (op $) (map_types ify ctor)) oo
  1222                 mk_InN_balanced (ify ctr_sum_prod_T) n;
  1223 
  1224               val cxIns = map2 (mk_cIn I) tuple_xs ks;
  1225               val cyIns = map2 (mk_cIn B_ify) tuple_ys ks;
  1226 
  1227               fun mk_map_thm ctr_def' cxIn =
  1228                 fold_thms lthy [ctr_def']
  1229                   (unfold_thms lthy (pre_map_def ::
  1230                        (if fp = Least_FP then [] else [ctor_dtor, dtor_ctor]) @ sum_prod_thms_map)
  1231                      (cterm_instantiate_pos (nones @ [SOME cxIn])
  1232                         (if fp = Least_FP then fp_map_thm else fp_map_thm RS ctor_cong)))
  1233                 |> singleton (Proof_Context.export names_lthy no_defs_lthy);
  1234 
  1235               fun mk_set_thm fp_set_thm ctr_def' cxIn =
  1236                 fold_thms lthy [ctr_def']
  1237                   (unfold_thms lthy (pre_set_defs @ nested_set_map's @ nesting_set_map's @
  1238                        (if fp = Least_FP then [] else [dtor_ctor]) @ sum_prod_thms_set)
  1239                      (cterm_instantiate_pos [SOME cxIn] fp_set_thm))
  1240                 |> singleton (Proof_Context.export names_lthy no_defs_lthy);
  1241 
  1242               fun mk_set_thms fp_set_thm = map2 (mk_set_thm fp_set_thm) ctr_defs' cxIns;
  1243 
  1244               val map_thms = map2 mk_map_thm ctr_defs' cxIns;
  1245               val set_thmss = map mk_set_thms fp_set_thms;
  1246 
  1247               val rel_infos = (ctr_defs' ~~ cxIns, ctr_defs' ~~ cyIns);
  1248 
  1249               fun mk_rel_thm postproc ctr_defs' cxIn cyIn =
  1250                 fold_thms lthy ctr_defs'
  1251                    (unfold_thms lthy (@{thm Inl_Inr_False} :: pre_rel_def ::
  1252                         (if fp = Least_FP then [] else [dtor_ctor]) @ sum_prod_thms_rel)
  1253                       (cterm_instantiate_pos (nones @ [SOME cxIn, SOME cyIn]) fp_rel_thm))
  1254                 |> postproc
  1255                 |> singleton (Proof_Context.export names_lthy no_defs_lthy);
  1256 
  1257               fun mk_rel_inject_thm ((ctr_def', cxIn), (_, cyIn)) =
  1258                 mk_rel_thm (unfold_thms lthy @{thms eq_sym_Unity_conv}) [ctr_def'] cxIn cyIn;
  1259 
  1260               val rel_inject_thms = map mk_rel_inject_thm (op ~~ rel_infos);
  1261 
  1262               fun mk_half_rel_distinct_thm ((xctr_def', cxIn), (yctr_def', cyIn)) =
  1263                 mk_rel_thm (fn thm => thm RS @{thm eq_False[THEN iffD1]}) [xctr_def', yctr_def']
  1264                   cxIn cyIn;
  1265 
  1266               fun mk_other_half_rel_distinct_thm thm =
  1267                 flip_rels lthy live thm
  1268                 RS (rel_flip RS sym RS @{thm arg_cong[of _ _ Not]} RS iffD2);
  1269 
  1270               val half_rel_distinct_thmss =
  1271                 map (map mk_half_rel_distinct_thm) (mk_half_pairss rel_infos);
  1272               val other_half_rel_distinct_thmss =
  1273                 map (map mk_other_half_rel_distinct_thm) half_rel_distinct_thmss;
  1274               val (rel_distinct_thms, _) =
  1275                 join_halves n half_rel_distinct_thmss other_half_rel_distinct_thmss;
  1276 
  1277               val notes =
  1278                 [(mapN, map_thms, code_simp_attrs),
  1279                  (rel_distinctN, rel_distinct_thms, code_simp_attrs),
  1280                  (rel_injectN, rel_inject_thms, code_simp_attrs),
  1281                  (setsN, flat set_thmss, code_simp_attrs)]
  1282                 |> massage_simple_notes fp_b_name;
  1283             in
  1284               (((map_thms, rel_inject_thms, rel_distinct_thms, set_thmss), ctr_sugar),
  1285                lthy |> Local_Theory.notes notes |> snd)
  1286             end;
  1287 
  1288         fun mk_binding suf = qualify false fp_b_name (Binding.suffix_name ("_" ^ suf) fp_b);
  1289 
  1290         fun massage_res (((maps_sets_rels, ctr_sugar), co_iter_res), lthy) =
  1291           (((maps_sets_rels, (ctrs, xss, ctr_defs, ctr_sugar)), co_iter_res), lthy);
  1292       in
  1293         (wrap_ctrs
  1294          #> derive_maps_sets_rels
  1295          ##>>
  1296            (if fp = Least_FP then define_iters [foldN, recN] (the iters_args_types)
  1297             else define_coiters [unfoldN, corecN] (the coiters_args_types))
  1298              mk_binding fpTs Cs xtor_co_iters
  1299          #> massage_res, lthy')
  1300       end;
  1301 
  1302     fun wrap_types_etc (wrap_types_etcs, lthy) =
  1303       fold_map I wrap_types_etcs lthy
  1304       |>> apsnd (apsnd transpose o apfst transpose o split_list)
  1305         o apfst (apsnd split_list4 o apfst split_list4 o split_list) o split_list;
  1306 
  1307     val mk_simp_thmss =
  1308       map7 (fn {injects, distincts, case_thms, ...} => fn un_folds => fn co_recs =>
  1309         fn mapsx => fn rel_injects => fn rel_distincts => fn setss =>
  1310           injects @ distincts @ case_thms @ co_recs @ un_folds @ mapsx @ rel_injects
  1311           @ rel_distincts @ flat setss);
  1312 
  1313     fun derive_and_note_induct_iters_thms_for_types
  1314         ((((mapsx, rel_injects, rel_distincts, setss), (ctrss, _, ctr_defss, ctr_sugars)),
  1315           (iterss', iter_defss')), lthy) =
  1316       let
  1317         val ((induct_thm, induct_thms, induct_attrs), (fold_thmss, fold_attrs),
  1318              (rec_thmss, rec_attrs)) =
  1319           derive_induct_iters_thms_for_types pre_bnfs xtor_co_iterss (the iters_args_types)
  1320             xtor_co_induct (transpose xtor_co_iter_thmss) nesting_bnfs nested_bnfs fpTs Cs Xs
  1321             ctrXs_Tsss ctrss ctr_defss iterss' iter_defss' lthy;
  1322 
  1323         val induct_type_attr = Attrib.internal o K o Induct.induct_type;
  1324 
  1325         val simp_thmss =
  1326           mk_simp_thmss ctr_sugars fold_thmss rec_thmss mapsx rel_injects rel_distincts setss;
  1327 
  1328         val common_notes =
  1329           (if nn > 1 then [(inductN, [induct_thm], induct_attrs)] else [])
  1330           |> massage_simple_notes fp_common_name;
  1331 
  1332         val notes =
  1333           [(foldN, fold_thmss, K fold_attrs),
  1334            (inductN, map single induct_thms, fn T_name => induct_attrs @ [induct_type_attr T_name]),
  1335            (recN, rec_thmss, K rec_attrs),
  1336            (simpsN, simp_thmss, K [])]
  1337           |> massage_multi_notes;
  1338       in
  1339         lthy
  1340         |> Local_Theory.notes (common_notes @ notes) |> snd
  1341         |> register_fp_sugars Least_FP pre_bnfs fp_res ctr_defss ctr_sugars iterss' induct_thm
  1342           induct_thm [fold_thmss, rec_thmss]
  1343       end;
  1344 
  1345     fun derive_and_note_coinduct_coiters_thms_for_types
  1346         ((((mapsx, rel_injects, rel_distincts, setss), (_, _, ctr_defss, ctr_sugars)),
  1347           (coiterss', coiter_defss')), lthy) =
  1348       let
  1349         val ((coinduct_thm, coinduct_thms, strong_coinduct_thm, strong_coinduct_thms,
  1350               coinduct_attrs),
  1351              (unfold_thmss, corec_thmss, coiter_attrs),
  1352              (safe_unfold_thmss, safe_corec_thmss),
  1353              (disc_unfold_thmss, disc_corec_thmss, disc_coiter_attrs),
  1354              (disc_unfold_iff_thmss, disc_corec_iff_thmss, disc_coiter_iff_attrs),
  1355              (sel_unfold_thmss, sel_corec_thmss, sel_coiter_attrs)) =
  1356           derive_coinduct_coiters_thms_for_types pre_bnfs xtor_co_iterss xtor_co_induct
  1357             xtor_strong_co_induct dtor_ctors (transpose xtor_co_iter_thmss) nesting_bnfs nested_bnfs
  1358             fpTs Cs As kss mss ns ctr_defss ctr_sugars coiterss' coiter_defss' lthy;
  1359 
  1360         val coinduct_type_attr = Attrib.internal o K o Induct.coinduct_type;
  1361 
  1362         fun flat_coiter_thms coiters disc_coiters sel_coiters =
  1363           coiters @ disc_coiters @ sel_coiters;
  1364 
  1365         val simp_thmss =
  1366           mk_simp_thmss ctr_sugars
  1367             (map3 flat_coiter_thms safe_unfold_thmss disc_unfold_thmss sel_unfold_thmss)
  1368             (map3 flat_coiter_thms safe_corec_thmss disc_corec_thmss sel_corec_thmss)
  1369             mapsx rel_injects rel_distincts setss;
  1370 
  1371         val anonymous_notes =
  1372           [(flat safe_unfold_thmss @ flat safe_corec_thmss, simp_attrs)]
  1373           |> map (fn (thms, attrs) => ((Binding.empty, attrs), [(thms, [])]));
  1374 
  1375         val common_notes =
  1376           (if nn > 1 then
  1377              [(coinductN, [coinduct_thm], coinduct_attrs),
  1378               (strong_coinductN, [strong_coinduct_thm], coinduct_attrs)]
  1379            else
  1380              [])
  1381           |> massage_simple_notes fp_common_name;
  1382 
  1383         val notes =
  1384           [(coinductN, map single coinduct_thms,
  1385             fn T_name => coinduct_attrs @ [coinduct_type_attr T_name]),
  1386            (corecN, corec_thmss, K coiter_attrs),
  1387            (disc_corecN, disc_corec_thmss, K disc_coiter_attrs),
  1388            (disc_corec_iffN, disc_corec_iff_thmss, K disc_coiter_iff_attrs),
  1389            (disc_unfoldN, disc_unfold_thmss, K disc_coiter_attrs),
  1390            (disc_unfold_iffN, disc_unfold_iff_thmss, K disc_coiter_iff_attrs),
  1391            (sel_corecN, sel_corec_thmss, K sel_coiter_attrs),
  1392            (sel_unfoldN, sel_unfold_thmss, K sel_coiter_attrs),
  1393            (simpsN, simp_thmss, K []),
  1394            (strong_coinductN, map single strong_coinduct_thms, K coinduct_attrs),
  1395            (unfoldN, unfold_thmss, K coiter_attrs)]
  1396           |> massage_multi_notes;
  1397       in
  1398         lthy
  1399         |> Local_Theory.notes (anonymous_notes @ common_notes @ notes) |> snd
  1400         |> register_fp_sugars Greatest_FP pre_bnfs fp_res ctr_defss ctr_sugars coiterss'
  1401           coinduct_thm strong_coinduct_thm [unfold_thmss, corec_thmss]
  1402       end;
  1403 
  1404     val lthy' = lthy
  1405       |> fold_map define_ctrs_case_for_type (fp_bnfs ~~ fp_bs ~~ fpTs ~~ Cs ~~ ctors ~~ dtors ~~
  1406         xtor_co_iterss ~~ ctor_dtors ~~ dtor_ctors ~~ ctor_injects ~~ pre_map_defs ~~
  1407         pre_set_defss ~~ pre_rel_defs ~~ xtor_map_thms ~~ xtor_set_thmss ~~ xtor_rel_thms ~~ ns ~~
  1408         kss ~~ mss ~~ ctr_bindingss ~~ ctr_mixfixess ~~ ctr_Tsss ~~ disc_bindingss ~~
  1409         sel_bindingsss ~~ raw_sel_defaultsss)
  1410       |> wrap_types_etc
  1411       |> (if fp = Least_FP then derive_and_note_induct_iters_thms_for_types
  1412           else derive_and_note_coinduct_coiters_thms_for_types);
  1413 
  1414     val timer = time (timer ("Constructors, discriminators, selectors, etc., for the new " ^
  1415       datatype_word fp));
  1416   in
  1417     timer; lthy'
  1418   end;
  1419 
  1420 val co_datatypes = define_co_datatypes (K I) (K I) (K I);
  1421 
  1422 val co_datatype_cmd =
  1423   define_co_datatypes Typedecl.read_constraint Syntax.parse_typ Syntax.parse_term;
  1424 
  1425 val parse_ctr_arg =
  1426   @{keyword "("} |-- parse_binding_colon -- Parse.typ --| @{keyword ")"} ||
  1427   (Parse.typ >> pair Binding.empty);
  1428 
  1429 val parse_defaults =
  1430   @{keyword "("} |-- @{keyword "defaults"} |-- Scan.repeat parse_bound_term --| @{keyword ")"};
  1431 
  1432 val parse_type_arg_constrained =
  1433   Parse.type_ident -- Scan.option (@{keyword "::"} |-- Parse.!!! Parse.sort);
  1434 
  1435 val parse_type_arg_named_constrained = parse_opt_binding_colon -- parse_type_arg_constrained;
  1436 
  1437 val parse_type_args_named_constrained =
  1438   parse_type_arg_constrained >> (single o pair Binding.empty) ||
  1439   @{keyword "("} |-- Parse.!!! (Parse.list1 parse_type_arg_named_constrained --| @{keyword ")"}) ||
  1440   Scan.succeed [];
  1441 
  1442 val parse_map_rel_binding = Parse.short_ident --| @{keyword ":"} -- parse_binding;
  1443 
  1444 val no_map_rel = (Binding.empty, Binding.empty);
  1445 
  1446 (* "map" and "rel" are purposedly not registered as keywords, because they are short and nice names
  1447    that we don't want them to be highlighted everywhere. *)
  1448 fun extract_map_rel ("map", b) = apfst (K b)
  1449   | extract_map_rel ("rel", b) = apsnd (K b)
  1450   | extract_map_rel (s, _) = error ("Expected \"map\" or \"rel\" instead of " ^ quote s);
  1451 
  1452 val parse_map_rel_bindings =
  1453   @{keyword "("} |-- Scan.repeat parse_map_rel_binding --| @{keyword ")"}
  1454     >> (fn ps => fold extract_map_rel ps no_map_rel) ||
  1455   Scan.succeed no_map_rel;
  1456 
  1457 val parse_ctr_spec =
  1458   parse_opt_binding_colon -- parse_binding -- Scan.repeat parse_ctr_arg --
  1459   Scan.optional parse_defaults [] -- Parse.opt_mixfix;
  1460 
  1461 val parse_spec =
  1462   parse_type_args_named_constrained -- parse_binding -- parse_map_rel_bindings --
  1463   Parse.opt_mixfix -- (@{keyword "="} |-- Parse.enum1 "|" parse_ctr_spec);
  1464 
  1465 val parse_co_datatype = parse_wrap_options -- Parse.and_list1 parse_spec;
  1466 
  1467 fun parse_co_datatype_cmd fp construct_fp = parse_co_datatype >> co_datatype_cmd fp construct_fp;
  1468 
  1469 end;