src/HOL/BNF/Tools/bnf_fp_def_sugar.ML
author blanchet
Fri Sep 13 02:26:59 2013 +0200 (2013-09-13)
changeset 53591 b6e2993fd0d3
parent 53569 b4db0ade27bd
child 53592 5a7bf8c859f6
permissions -rw-r--r--
don't wrongly destroy sum types in coiterators
     1 (*  Title:      HOL/BNF/Tools/bnf_fp_def_sugar.ML
     2     Author:     Jasmin Blanchette, TU Muenchen
     3     Copyright   2012, 2013
     4 
     5 Sugared datatype and codatatype constructions.
     6 *)
     7 
     8 signature BNF_FP_DEF_SUGAR =
     9 sig
    10   type fp_sugar =
    11     {T: typ,
    12      fp: BNF_FP_Util.fp_kind,
    13      index: int,
    14      pre_bnfs: BNF_Def.bnf list,
    15      nested_bnfs: BNF_Def.bnf list,
    16      nesting_bnfs: BNF_Def.bnf list,
    17      fp_res: BNF_FP_Util.fp_result,
    18      ctr_defss: thm list list,
    19      ctr_sugars: BNF_Ctr_Sugar.ctr_sugar list,
    20      co_iterss: term list list,
    21      mapss: thm list list,
    22      co_inducts: thm list,
    23      co_iter_thmsss: thm list list list,
    24      disc_co_itersss: thm list list list,
    25      sel_co_iterssss: thm list list list list};
    26 
    27   val of_fp_sugar: (fp_sugar -> 'a list) -> fp_sugar -> 'a
    28   val morph_fp_sugar: morphism -> fp_sugar -> fp_sugar
    29   val fp_sugar_of: Proof.context -> string -> fp_sugar option
    30 
    31   val co_induct_of: 'a list -> 'a
    32   val strong_co_induct_of: 'a list -> 'a
    33 
    34   val tvar_subst: theory -> typ list -> typ list -> ((string * int) * typ) list
    35   val exists_subtype_in: typ list -> typ -> bool
    36   val flat_rec_arg_args: 'a list list -> 'a list
    37   val flat_corec_preds_predsss_gettersss: 'a list -> 'a list list list -> 'a list list list ->
    38     'a list
    39   val mk_co_iter: theory -> BNF_FP_Util.fp_kind -> typ -> typ list -> term -> term
    40   val nesty_bnfs: Proof.context -> typ list list list -> typ list -> BNF_Def.bnf list
    41   val mk_map: int -> typ list -> typ list -> term -> term
    42   val mk_rel: int -> typ list -> typ list -> term -> term
    43   val build_map: local_theory -> (typ * typ -> term) -> typ * typ -> term
    44   val build_rel: local_theory -> (typ * typ -> term) -> typ * typ -> term
    45   val dest_map: Proof.context -> string -> term -> term * term list
    46   val dest_ctr: Proof.context -> string -> term -> term * term list
    47   val mk_co_iters_prelims: BNF_FP_Util.fp_kind -> typ list list list -> typ list -> typ list ->
    48     int list -> int list list -> term list list -> Proof.context ->
    49     (term list list
    50      * (typ list list * typ list list list list * term list list
    51         * term list list list list) list option
    52      * (string * term list * term list list
    53         * ((term list list * term list list list) * (typ list * typ list list)) list) option)
    54     * Proof.context
    55 
    56   val mk_iter_fun_arg_types: typ list -> int list -> int list list -> term ->
    57     typ list list list list
    58   val mk_coiter_fun_arg_types: typ list list list -> typ list -> int list -> term ->
    59     typ list list
    60     * (typ list list list list * typ list list list * typ list list list list * typ list)
    61   val define_iters: string list ->
    62     (typ list list * typ list list list list * term list list * term list list list list) list ->
    63     (string -> binding) -> typ list -> typ list -> term list -> Proof.context ->
    64     (term list * thm list) * Proof.context
    65   val define_coiters: string list -> string * term list * term list list
    66     * ((term list list * term list list list) * (typ list * typ list list)) list ->
    67     (string -> binding) -> typ list -> typ list -> term list -> Proof.context ->
    68     (term list * thm list) * Proof.context
    69   val derive_induct_iters_thms_for_types: BNF_Def.bnf list ->
    70     (typ list list * typ list list list list * term list list * term list list list list) list ->
    71     thm -> thm list list -> BNF_Def.bnf list -> BNF_Def.bnf list -> typ list -> typ list ->
    72     typ list -> typ list list list -> term list list -> thm list list -> term list list ->
    73     thm list list -> local_theory ->
    74     (thm list * thm * Args.src list) * (thm list list * Args.src list)
    75     * (thm list list * Args.src list)
    76   val derive_coinduct_coiters_thms_for_types: BNF_Def.bnf list ->
    77     string * term list * term list list * ((term list list * term list list list)
    78       * (typ list * typ list list)) list ->
    79     thm -> thm list -> thm list -> thm list list -> BNF_Def.bnf list -> typ list -> typ list ->
    80     int list list -> int list list -> int list -> thm list list -> BNF_Ctr_Sugar.ctr_sugar list ->
    81     term list list -> thm list list -> (thm list -> thm list) -> local_theory ->
    82     ((thm list * thm) list * Args.src list)
    83     * (thm list list * thm list list * Args.src list)
    84     * (thm list list * thm list list) * (thm list list * thm list list * Args.src list)
    85     * (thm list list * thm list list * Args.src list)
    86     * (thm list list list * thm list list list * Args.src list)
    87 
    88   val co_datatypes: BNF_FP_Util.fp_kind -> (mixfix list -> binding list -> binding list ->
    89       binding list list -> binding list -> (string * sort) list -> typ list * typ list list ->
    90       BNF_Def.bnf list -> local_theory -> BNF_FP_Util.fp_result * local_theory) ->
    91     (bool * bool) * (((((binding * (typ * sort)) list * binding) * (binding * binding)) * mixfix) *
    92       ((((binding * binding) * (binding * typ) list) * (binding * term) list) *
    93         mixfix) list) list ->
    94     local_theory -> local_theory
    95   val parse_co_datatype_cmd: BNF_FP_Util.fp_kind -> (mixfix list -> binding list -> binding list ->
    96       binding list list -> binding list -> (string * sort) list -> typ list * typ list list ->
    97       BNF_Def.bnf list -> local_theory -> BNF_FP_Util.fp_result * local_theory) ->
    98     (local_theory -> local_theory) parser
    99 end;
   100 
   101 structure BNF_FP_Def_Sugar : BNF_FP_DEF_SUGAR =
   102 struct
   103 
   104 open BNF_Util
   105 open BNF_Ctr_Sugar
   106 open BNF_Comp
   107 open BNF_Def
   108 open BNF_FP_Util
   109 open BNF_FP_Def_Sugar_Tactics
   110 
   111 val EqN = "Eq_";
   112 
   113 type fp_sugar =
   114   {T: typ,
   115    fp: fp_kind,
   116    index: int,
   117    pre_bnfs: bnf list,
   118    nested_bnfs: bnf list,
   119    nesting_bnfs: bnf list,
   120    fp_res: fp_result,
   121    ctr_defss: thm list list,
   122    ctr_sugars: ctr_sugar list,
   123    co_iterss: term list list,
   124    mapss: thm list list,
   125    co_inducts: thm list,
   126    co_iter_thmsss: thm list list list,
   127    disc_co_itersss: thm list list list,
   128    sel_co_iterssss: thm list list list list};
   129 
   130 fun of_fp_sugar f (fp_sugar as {index, ...}) = nth (f fp_sugar) index;
   131 
   132 fun eq_fp_sugar ({T = T1, fp = fp1, index = index1, fp_res = fp_res1, ...} : fp_sugar,
   133     {T = T2, fp = fp2, index = index2, fp_res = fp_res2, ...} : fp_sugar) =
   134   T1 = T2 andalso fp1 = fp2 andalso index1 = index2 andalso eq_fp_result (fp_res1, fp_res2);
   135 
   136 fun morph_fp_sugar phi {T, fp, index, pre_bnfs, nested_bnfs, nesting_bnfs, fp_res, ctr_defss,
   137     ctr_sugars, co_iterss, mapss, co_inducts, co_iter_thmsss, disc_co_itersss, sel_co_iterssss} =
   138   {T = Morphism.typ phi T, fp = fp, index = index, pre_bnfs = map (morph_bnf phi) pre_bnfs,
   139     nested_bnfs = map (morph_bnf phi) nested_bnfs, nesting_bnfs = map (morph_bnf phi) nesting_bnfs,
   140    fp_res = morph_fp_result phi fp_res,
   141    ctr_defss = map (map (Morphism.thm phi)) ctr_defss,
   142    ctr_sugars = map (morph_ctr_sugar phi) ctr_sugars,
   143    co_iterss = map (map (Morphism.term phi)) co_iterss,
   144    mapss = map (map (Morphism.thm phi)) mapss,
   145    co_inducts = map (Morphism.thm phi) co_inducts,
   146    co_iter_thmsss = map (map (map (Morphism.thm phi))) co_iter_thmsss,
   147    disc_co_itersss = map (map (map (Morphism.thm phi))) disc_co_itersss,
   148    sel_co_iterssss = map (map (map (map (Morphism.thm phi)))) sel_co_iterssss};
   149 
   150 structure Data = Generic_Data
   151 (
   152   type T = fp_sugar Symtab.table;
   153   val empty = Symtab.empty;
   154   val extend = I;
   155   val merge = Symtab.merge eq_fp_sugar;
   156 );
   157 
   158 fun fp_sugar_of ctxt =
   159   Symtab.lookup (Data.get (Context.Proof ctxt))
   160   #> Option.map (morph_fp_sugar
   161     (Morphism.thm_morphism (Thm.transfer (Proof_Context.theory_of ctxt))));
   162 
   163 fun co_induct_of (i :: _) = i;
   164 fun strong_co_induct_of [_, s] = s;
   165 
   166 (* TODO: register "sum" and "prod" as datatypes to enable N2M reduction for them *)
   167 
   168 fun register_fp_sugar key fp_sugar =
   169   Local_Theory.declaration {syntax = false, pervasive = true}
   170     (fn phi => Data.map (Symtab.default (key, morph_fp_sugar phi fp_sugar)));
   171 
   172 fun register_fp_sugars fp pre_bnfs nested_bnfs nesting_bnfs (fp_res as {Ts, ...}) ctr_defss
   173     ctr_sugars co_iterss mapss co_inducts co_iter_thmsss disc_co_itersss sel_co_iterssss lthy =
   174   (0, lthy)
   175   |> fold (fn T as Type (s, _) => fn (kk, lthy) => (kk + 1,
   176     register_fp_sugar s {T = T, fp = fp, index = kk, pre_bnfs = pre_bnfs,
   177         nested_bnfs = nested_bnfs, nesting_bnfs = nesting_bnfs, fp_res = fp_res,
   178         ctr_defss = ctr_defss, ctr_sugars = ctr_sugars, co_iterss = co_iterss, mapss = mapss,
   179         co_inducts = co_inducts, co_iter_thmsss = co_iter_thmsss, disc_co_itersss = disc_co_itersss,
   180         sel_co_iterssss = sel_co_iterssss}
   181       lthy)) Ts
   182   |> snd;
   183 
   184 (* This function could produce clashes in contrived examples (e.g., "x.A", "x.x_A", "y.A"). *)
   185 fun quasi_unambiguous_case_names names =
   186   let
   187     val ps = map (`Long_Name.base_name) names;
   188     val dups = Library.duplicates (op =) (map fst ps);
   189     fun underscore s =
   190       let val ss = space_explode Long_Name.separator s in
   191         space_implode "_" (drop (length ss - 2) ss)
   192       end;
   193   in
   194     map (fn (base, full) => if member (op =) dups base then underscore full else base) ps
   195   end;
   196 
   197 val id_def = @{thm id_def};
   198 val mp_conj = @{thm mp_conj};
   199 
   200 val simp_attrs = @{attributes [simp]};
   201 val code_simp_attrs = Code.add_default_eqn_attrib :: simp_attrs;
   202 
   203 fun tvar_subst thy Ts Us =
   204   Vartab.fold (cons o apsnd snd) (fold (Sign.typ_match thy) (Ts ~~ Us) Vartab.empty) [];
   205 
   206 val exists_subtype_in = Term.exists_subtype o member (op =);
   207 
   208 val lists_bmoc = fold (fn xs => fn t => Term.list_comb (t, xs));
   209 
   210 fun flat_rec_arg_args xss =
   211   (* FIXME (once the old datatype package is phased out): The first line below gives the preferred
   212      order. The second line is for compatibility with the old datatype package. *)
   213 (*
   214   flat xss
   215 *)
   216   map hd xss @ maps tl xss;
   217 
   218 fun flat_corec_predss_getterss qss fss = maps (op @) (qss ~~ fss);
   219 
   220 fun flat_corec_preds_predsss_gettersss [] [qss] [fss] = flat_corec_predss_getterss qss fss
   221   | flat_corec_preds_predsss_gettersss (p :: ps) (qss :: qsss) (fss :: fsss) =
   222     p :: flat_corec_predss_getterss qss fss @ flat_corec_preds_predsss_gettersss ps qsss fsss;
   223 
   224 fun mk_tupled_fun x f xs = HOLogic.tupled_lambda x (Term.list_comb (f, xs));
   225 fun mk_uncurried2_fun f xss =
   226   mk_tupled_fun (HOLogic.mk_tuple (map HOLogic.mk_tuple xss)) f (flat_rec_arg_args xss);
   227 
   228 fun mk_flip (x, Type (_, [T1, Type (_, [T2, T3])])) =
   229   Abs ("x", T1, Abs ("y", T2, Var (x, T2 --> T1 --> T3) $ Bound 0 $ Bound 1));
   230 
   231 fun flip_rels lthy n thm =
   232   let
   233     val Rs = Term.add_vars (prop_of thm) [];
   234     val Rs' = rev (drop (length Rs - n) Rs);
   235     val cRs = map (fn f => (certify lthy (Var f), certify lthy (mk_flip f))) Rs';
   236   in
   237     Drule.cterm_instantiate cRs thm
   238   end;
   239 
   240 fun mk_ctor_or_dtor get_T Ts t =
   241   let val Type (_, Ts0) = get_T (fastype_of t) in
   242     Term.subst_atomic_types (Ts0 ~~ Ts) t
   243   end;
   244 
   245 val mk_ctor = mk_ctor_or_dtor range_type;
   246 val mk_dtor = mk_ctor_or_dtor domain_type;
   247 
   248 fun mk_co_iter thy fp fpT Cs t =
   249   let
   250     val (f_Cs, Type (_, [prebody, body])) = strip_fun_type (fastype_of t);
   251     val fpT0 = fp_case fp prebody body;
   252     val Cs0 = distinct (op =) (map (fp_case fp body_type domain_type) f_Cs);
   253     val rho = tvar_subst thy (fpT0 :: Cs0) (fpT :: Cs);
   254   in
   255     Term.subst_TVars rho t
   256   end;
   257 
   258 fun mk_co_iters thy fp fpTs Cs ts0 =
   259   let
   260     val nn = length fpTs;
   261     val (fpTs0, Cs0) =
   262       map ((fp = Greatest_FP ? swap) o dest_funT o snd o strip_typeN nn o fastype_of) ts0
   263       |> split_list;
   264     val rho = tvar_subst thy (fpTs0 @ Cs0) (fpTs @ Cs);
   265   in
   266     map (Term.subst_TVars rho) ts0
   267   end;
   268 
   269 val mk_fp_iter_fun_types = binder_fun_types o fastype_of;
   270 
   271 (* ### FIXME? *)
   272 fun unzip_recT Cs (T as Type (@{type_name prod}, Ts as [_, U])) =
   273     if member (op =) Cs U then Ts else [T]
   274   | unzip_recT _ T = [T];
   275 
   276 fun unzip_corecT (Type (@{type_name sum}, _)) T = [T]
   277   | unzip_corecT _ (T as Type (@{type_name sum}, Ts)) = Ts
   278   | unzip_corecT _ T = [T];
   279 
   280 fun mk_map live Ts Us t =
   281   let val (Type (_, Ts0), Type (_, Us0)) = strip_typeN (live + 1) (fastype_of t) |>> List.last in
   282     Term.subst_atomic_types (Ts0 @ Us0 ~~ Ts @ Us) t
   283   end;
   284 
   285 fun mk_rel live Ts Us t =
   286   let val [Type (_, Ts0), Type (_, Us0)] = binder_types (snd (strip_typeN live (fastype_of t))) in
   287     Term.subst_atomic_types (Ts0 @ Us0 ~~ Ts @ Us) t
   288   end;
   289 
   290 local
   291 
   292 fun build_map_or_rel mk const of_bnf dest lthy build_simple =
   293   let
   294     fun build (TU as (T, U)) =
   295       if T = U then
   296         const T
   297       else
   298         (case TU of
   299           (Type (s, Ts), Type (s', Us)) =>
   300           if s = s' then
   301             let
   302               val bnf = the (bnf_of lthy s);
   303               val live = live_of_bnf bnf;
   304               val mapx = mk live Ts Us (of_bnf bnf);
   305               val TUs' = map dest (fst (strip_typeN live (fastype_of mapx)));
   306             in Term.list_comb (mapx, map build TUs') end
   307           else
   308             build_simple TU
   309         | _ => build_simple TU);
   310   in build end;
   311 
   312 in
   313 
   314 val build_map = build_map_or_rel mk_map HOLogic.id_const map_of_bnf dest_funT;
   315 val build_rel = build_map_or_rel mk_rel HOLogic.eq_const rel_of_bnf dest_pred2T;
   316 
   317 end;
   318 
   319 fun fo_match ctxt t pat =
   320   let val thy = Proof_Context.theory_of ctxt in
   321     Pattern.first_order_match thy (pat, t) (Vartab.empty, Vartab.empty)
   322   end;
   323 
   324 val dummy_var_name = "?f"
   325 
   326 fun mk_map_pattern ctxt s =
   327   let
   328     val bnf = the (bnf_of ctxt s);
   329     val mapx = map_of_bnf bnf;
   330     val live = live_of_bnf bnf;
   331     val (f_Ts, _) = strip_typeN live (fastype_of mapx);
   332     val fs = map_index (fn (i, T) => Var ((dummy_var_name, i), T)) f_Ts;
   333   in
   334     (mapx, betapplys (mapx, fs))
   335   end;
   336 
   337 fun dest_map ctxt s call =
   338   let
   339     val (map0, pat) = mk_map_pattern ctxt s;
   340     val (_, tenv) = fo_match ctxt call pat;
   341   in
   342     (map0, Vartab.fold_rev (fn (_, (_, f)) => cons f) tenv [])
   343   end;
   344 
   345 fun dest_ctr ctxt s t =
   346   let
   347     val (f, args) = Term.strip_comb t;
   348   in
   349     (case fp_sugar_of ctxt s of
   350       SOME fp_sugar =>
   351       (case find_first (can (fo_match ctxt f)) (#ctrs (of_fp_sugar #ctr_sugars fp_sugar)) of
   352         SOME f' => (f', args)
   353       | NONE => raise Fail "dest_ctr")
   354     | NONE => raise Fail "dest_ctr")
   355   end;
   356 
   357 fun liveness_of_fp_bnf n bnf =
   358   (case T_of_bnf bnf of
   359     Type (_, Ts) => map (not o member (op =) (deads_of_bnf bnf)) Ts
   360   | _ => replicate n false);
   361 
   362 fun cannot_merge_types () = error "Mutually recursive types must have the same type parameters";
   363 
   364 fun merge_type_arg T T' = if T = T' then T else cannot_merge_types ();
   365 
   366 fun merge_type_args (As, As') =
   367   if length As = length As' then map2 merge_type_arg As As' else cannot_merge_types ();
   368 
   369 fun reassoc_conjs thm =
   370   reassoc_conjs (thm RS @{thm conj_assoc[THEN iffD1]})
   371   handle THM _ => thm;
   372 
   373 fun type_args_named_constrained_of ((((ncAs, _), _), _), _) = ncAs;
   374 fun type_binding_of ((((_, b), _), _), _) = b;
   375 fun map_binding_of (((_, (b, _)), _), _) = b;
   376 fun rel_binding_of (((_, (_, b)), _), _) = b;
   377 fun mixfix_of ((_, mx), _) = mx;
   378 fun ctr_specs_of (_, ctr_specs) = ctr_specs;
   379 
   380 fun disc_of ((((disc, _), _), _), _) = disc;
   381 fun ctr_of ((((_, ctr), _), _), _) = ctr;
   382 fun args_of (((_, args), _), _) = args;
   383 fun defaults_of ((_, ds), _) = ds;
   384 fun ctr_mixfix_of (_, mx) = mx;
   385 
   386 fun add_nesty_bnf_names Us =
   387   let
   388     fun add (Type (s, Ts)) ss =
   389         let val (needs, ss') = fold_map add Ts ss in
   390           if exists I needs then (true, insert (op =) s ss') else (false, ss')
   391         end
   392       | add T ss = (member (op =) Us T, ss);
   393   in snd oo add end;
   394 
   395 fun nesty_bnfs ctxt ctr_Tsss Us =
   396   map_filter (bnf_of ctxt) (fold (fold (fold (add_nesty_bnf_names Us))) ctr_Tsss []);
   397 
   398 fun indexify proj xs f p = f (find_index (curry op = (proj p)) xs) p;
   399 
   400 fun mk_iter_fun_arg_types0 n ms = map2 dest_tupleT ms o dest_sumTN_balanced n o domain_type;
   401 
   402 fun mk_iter_fun_arg_types Cs ns mss =
   403   mk_fp_iter_fun_types
   404   #> map3 mk_iter_fun_arg_types0 ns mss
   405   #> map (map (map (unzip_recT Cs)));
   406 
   407 fun mk_iters_args_types Cs ns mss ctor_iter_fun_Tss lthy =
   408   let
   409     val Css = map2 replicate ns Cs;
   410     val y_Tsss = map3 mk_iter_fun_arg_types0 ns mss (map un_fold_of ctor_iter_fun_Tss);
   411     val g_Tss = map2 (fn C => map (fn y_Ts => y_Ts ---> C)) Cs y_Tsss;
   412 
   413     val ((gss, ysss), lthy) =
   414       lthy
   415       |> mk_Freess "f" g_Tss
   416       ||>> mk_Freesss "x" y_Tsss;
   417 
   418     val y_Tssss = map (map (map single)) y_Tsss;
   419     val yssss = map (map (map single)) ysss;
   420 
   421     val z_Tssss =
   422       map3 (fn n => fn ms => map2 (map (unzip_recT Cs) oo dest_tupleT) ms o
   423         dest_sumTN_balanced n o domain_type o co_rec_of) ns mss ctor_iter_fun_Tss;
   424 
   425     val z_Tsss' = map (map flat_rec_arg_args) z_Tssss;
   426     val h_Tss = map2 (map2 (curry op --->)) z_Tsss' Css;
   427 
   428     val hss = map2 (map2 retype_free) h_Tss gss;
   429     val zssss_hd = map2 (map2 (map2 (retype_free o hd))) z_Tssss ysss;
   430     val (zssss_tl, lthy) =
   431       lthy
   432       |> mk_Freessss "y" (map (map (map tl)) z_Tssss);
   433     val zssss = map2 (map2 (map2 cons)) zssss_hd zssss_tl;
   434   in
   435     ([(g_Tss, y_Tssss, gss, yssss), (h_Tss, z_Tssss, hss, zssss)], lthy)
   436   end;
   437 
   438 fun mk_coiter_fun_arg_types0 ctr_Tsss Cs ns fun_Ts =
   439   let
   440     (*avoid "'a itself" arguments in coiterators*)
   441     fun repair_arity [[]] = [[@{typ unit}]]
   442       | repair_arity Tss = Tss;
   443 
   444     val ctr_Tsss' = map repair_arity ctr_Tsss;
   445     val f_sum_prod_Ts = map range_type fun_Ts;
   446     val f_prod_Tss = map2 dest_sumTN_balanced ns f_sum_prod_Ts;
   447     val f_Tsss = map2 (map2 (dest_tupleT o length)) ctr_Tsss' f_prod_Tss;
   448     val f_Tssss = map3 (fn C => map2 (map2 (map (curry op --> C) oo unzip_corecT)))
   449       Cs ctr_Tsss' f_Tsss;
   450     val q_Tssss = map (map (map (fn [_] => [] | [_, T] => [mk_pred1T (domain_type T)]))) f_Tssss;
   451   in
   452     (q_Tssss, f_Tsss, f_Tssss, f_sum_prod_Ts)
   453   end;
   454 
   455 fun mk_coiter_p_pred_types Cs ns = map2 (fn n => replicate (Int.max (0, n - 1)) o mk_pred1T) ns Cs;
   456 
   457 fun mk_coiter_fun_arg_types ctr_Tsss Cs ns dtor_coiter =
   458   (mk_coiter_p_pred_types Cs ns,
   459    mk_fp_iter_fun_types dtor_coiter |> mk_coiter_fun_arg_types0 ctr_Tsss Cs ns);
   460 
   461 fun mk_coiters_args_types ctr_Tsss Cs ns mss dtor_coiter_fun_Tss lthy =
   462   let
   463     val p_Tss = mk_coiter_p_pred_types Cs ns;
   464 
   465     fun mk_types get_Ts =
   466       let
   467         val fun_Ts = map get_Ts dtor_coiter_fun_Tss;
   468         val (q_Tssss, f_Tsss, f_Tssss, f_sum_prod_Ts) = mk_coiter_fun_arg_types0 ctr_Tsss Cs ns fun_Ts;
   469         val pf_Tss = map3 flat_corec_preds_predsss_gettersss p_Tss q_Tssss f_Tssss;
   470       in
   471         (q_Tssss, f_Tsss, f_Tssss, (f_sum_prod_Ts, pf_Tss))
   472       end;
   473 
   474     val (r_Tssss, g_Tsss, g_Tssss, unfold_types) = mk_types un_fold_of;
   475     val (s_Tssss, h_Tsss, h_Tssss, corec_types) = mk_types co_rec_of;
   476 
   477     val ((((Free (z, _), cs), pss), gssss), lthy) =
   478       lthy
   479       |> yield_singleton (mk_Frees "z") dummyT
   480       ||>> mk_Frees "a" Cs
   481       ||>> mk_Freess "p" p_Tss
   482       ||>> mk_Freessss "g" g_Tssss;
   483     val rssss = map (map (map (fn [] => []))) r_Tssss;
   484 
   485     val hssss_hd = map2 (map2 (map2 (fn T :: _ => fn [g] => retype_free T g))) h_Tssss gssss;
   486     val ((sssss, hssss_tl), lthy) =
   487       lthy
   488       |> mk_Freessss "q" s_Tssss
   489       ||>> mk_Freessss "h" (map (map (map tl)) h_Tssss);
   490     val hssss = map2 (map2 (map2 cons)) hssss_hd hssss_tl;
   491 
   492     val cpss = map2 (map o rapp) cs pss;
   493 
   494     fun build_sum_inj mk_inj = build_map lthy (uncurry mk_inj o dest_sumT o snd);
   495 
   496     fun build_dtor_coiter_arg _ [] [cf] = cf
   497       | build_dtor_coiter_arg T [cq] [cf, cf'] =
   498         mk_If cq (build_sum_inj Inl_const (fastype_of cf, T) $ cf)
   499           (build_sum_inj Inr_const (fastype_of cf', T) $ cf');
   500 
   501     fun mk_args qssss fssss f_Tsss =
   502       let
   503         val pfss = map3 flat_corec_preds_predsss_gettersss pss qssss fssss;
   504         val cqssss = map2 (map o map o map o rapp) cs qssss;
   505         val cfssss = map2 (map o map o map o rapp) cs fssss;
   506         val cqfsss = map3 (map3 (map3 build_dtor_coiter_arg)) f_Tsss cqssss cfssss;
   507       in (pfss, cqfsss) end;
   508 
   509     val unfold_args = mk_args rssss gssss g_Tsss;
   510     val corec_args = mk_args sssss hssss h_Tsss;
   511   in
   512     ((z, cs, cpss, [(unfold_args, unfold_types), (corec_args, corec_types)]), lthy)
   513   end;
   514 
   515 fun mk_co_iters_prelims fp ctr_Tsss fpTs Cs ns mss xtor_co_iterss0 lthy =
   516   let
   517     val thy = Proof_Context.theory_of lthy;
   518 
   519     val (xtor_co_iter_fun_Tss, xtor_co_iterss) =
   520       map (mk_co_iters thy fp fpTs Cs #> `(mk_fp_iter_fun_types o hd)) (transpose xtor_co_iterss0)
   521       |> apsnd transpose o apfst transpose o split_list;
   522 
   523     val ((iters_args_types, coiters_args_types), lthy') =
   524       if fp = Least_FP then
   525         mk_iters_args_types Cs ns mss xtor_co_iter_fun_Tss lthy |>> (rpair NONE o SOME)
   526       else
   527         mk_coiters_args_types ctr_Tsss Cs ns mss xtor_co_iter_fun_Tss lthy |>> (pair NONE o SOME)
   528   in
   529     ((xtor_co_iterss, iters_args_types, coiters_args_types), lthy')
   530   end;
   531 
   532 fun mk_iter_body ctor_iter fss xssss =
   533   Term.list_comb (ctor_iter, map2 (mk_sum_caseN_balanced oo map2 mk_uncurried2_fun) fss xssss);
   534 
   535 fun mk_preds_getterss_join c cps sum_prod_T cqfss =
   536   let val n = length cqfss in
   537     Term.lambda c (mk_IfN sum_prod_T cps
   538       (map2 (mk_InN_balanced sum_prod_T n) (map HOLogic.mk_tuple cqfss) (1 upto n)))
   539   end;
   540 
   541 fun mk_coiter_body cs cpss f_sum_prod_Ts cqfsss dtor_coiter =
   542   Term.list_comb (dtor_coiter, map4 mk_preds_getterss_join cs cpss f_sum_prod_Ts cqfsss);
   543 
   544 fun define_co_iters fp fpT Cs binding_specs lthy0 =
   545   let
   546     val thy = Proof_Context.theory_of lthy0;
   547 
   548     val maybe_conceal_def_binding = Thm.def_binding
   549       #> Config.get lthy0 bnf_note_all = false ? Binding.conceal;
   550 
   551     val ((csts, defs), (lthy', lthy)) = lthy0
   552       |> apfst split_list o fold_map (fn (b, spec) =>
   553         Specification.definition (SOME (b, NONE, NoSyn), ((maybe_conceal_def_binding b, []), spec))
   554         #>> apsnd snd) binding_specs
   555       ||> `Local_Theory.restore;
   556 
   557     val phi = Proof_Context.export_morphism lthy lthy';
   558 
   559     val csts' = map (mk_co_iter thy fp fpT Cs o Morphism.term phi) csts;
   560     val defs' = map (Morphism.thm phi) defs;
   561   in
   562     ((csts', defs'), lthy')
   563   end;
   564 
   565 fun define_iters iterNs iter_args_typess' mk_binding fpTs Cs ctor_iters lthy =
   566   let
   567     val nn = length fpTs;
   568 
   569     val fpT_to_C as Type (_, [fpT, _]) = snd (strip_typeN nn (fastype_of (hd ctor_iters)));
   570 
   571     fun generate_iter suf (f_Tss, _, fss, xssss) ctor_iter =
   572       let
   573         val res_T = fold_rev (curry op --->) f_Tss fpT_to_C;
   574         val b = mk_binding suf;
   575         val spec =
   576           mk_Trueprop_eq (lists_bmoc fss (Free (Binding.name_of b, res_T)),
   577             mk_iter_body ctor_iter fss xssss);
   578       in (b, spec) end;
   579   in
   580     define_co_iters Least_FP fpT Cs (map3 generate_iter iterNs iter_args_typess' ctor_iters) lthy
   581   end;
   582 
   583 fun define_coiters coiterNs (_, cs, cpss, coiter_args_typess') mk_binding fpTs Cs dtor_coiters
   584     lthy =
   585   let
   586     val nn = length fpTs;
   587 
   588     val C_to_fpT as Type (_, [_, fpT]) = snd (strip_typeN nn (fastype_of (hd dtor_coiters)));
   589 
   590     fun generate_coiter suf ((pfss, cqfsss), (f_sum_prod_Ts, pf_Tss)) dtor_coiter =
   591       let
   592         val res_T = fold_rev (curry op --->) pf_Tss C_to_fpT;
   593         val b = mk_binding suf;
   594         val spec =
   595           mk_Trueprop_eq (lists_bmoc pfss (Free (Binding.name_of b, res_T)),
   596             mk_coiter_body cs cpss f_sum_prod_Ts cqfsss dtor_coiter);
   597       in (b, spec) end;
   598   in
   599     define_co_iters Greatest_FP fpT Cs
   600       (map3 generate_coiter coiterNs coiter_args_typess' dtor_coiters) lthy
   601   end;
   602 
   603 fun derive_induct_iters_thms_for_types pre_bnfs [fold_args_types, rec_args_types] ctor_induct
   604     ctor_iter_thmss nesting_bnfs nested_bnfs fpTs Cs Xs ctrXs_Tsss ctrss ctr_defss iterss iter_defss
   605     lthy =
   606   let
   607     val iterss' = transpose iterss;
   608     val iter_defss' = transpose iter_defss;
   609 
   610     val [folds, recs] = iterss';
   611     val [fold_defs, rec_defs] = iter_defss';
   612 
   613     val ctr_Tsss = map (map (binder_types o fastype_of)) ctrss;
   614 
   615     val nn = length pre_bnfs;
   616     val ns = map length ctr_Tsss;
   617     val mss = map (map length) ctr_Tsss;
   618 
   619     val pre_map_defs = map map_def_of_bnf pre_bnfs;
   620     val pre_set_defss = map set_defs_of_bnf pre_bnfs;
   621     val nesting_map_idents = map (unfold_thms lthy [id_def] o map_id0_of_bnf) nesting_bnfs;
   622     val nested_map_idents = map (unfold_thms lthy [id_def] o map_id0_of_bnf) nested_bnfs;
   623     val nested_set_maps = maps set_map_of_bnf nested_bnfs;
   624 
   625     val fp_b_names = map base_name_of_typ fpTs;
   626 
   627     val ((((ps, ps'), xsss), us'), names_lthy) =
   628       lthy
   629       |> mk_Frees' "P" (map mk_pred1T fpTs)
   630       ||>> mk_Freesss "x" ctr_Tsss
   631       ||>> Variable.variant_fixes fp_b_names;
   632 
   633     val us = map2 (curry Free) us' fpTs;
   634 
   635     fun mk_sets_nested bnf =
   636       let
   637         val Type (T_name, Us) = T_of_bnf bnf;
   638         val lives = lives_of_bnf bnf;
   639         val sets = sets_of_bnf bnf;
   640         fun mk_set U =
   641           (case find_index (curry op = U) lives of
   642             ~1 => Term.dummy
   643           | i => nth sets i);
   644       in
   645         (T_name, map mk_set Us)
   646       end;
   647 
   648     val setss_nested = map mk_sets_nested nested_bnfs;
   649 
   650     val (induct_thms, induct_thm) =
   651       let
   652         fun mk_set Ts t =
   653           let val Type (_, Ts0) = domain_type (fastype_of t) in
   654             Term.subst_atomic_types (Ts0 ~~ Ts) t
   655           end;
   656 
   657         fun mk_raw_prem_prems _ (x as Free (_, Type _)) (X as TFree _) =
   658             [([], (find_index (curry op = X) Xs + 1, x))]
   659           | mk_raw_prem_prems names_lthy (x as Free (s, Type (T_name, Ts0))) (Type (_, Xs_Ts0)) =
   660             (case AList.lookup (op =) setss_nested T_name of
   661               NONE => []
   662             | SOME raw_sets0 =>
   663               let
   664                 val (Xs_Ts, (Ts, raw_sets)) =
   665                   filter (exists_subtype_in Xs o fst) (Xs_Ts0 ~~ (Ts0 ~~ raw_sets0))
   666                   |> split_list ||> split_list;
   667                 val sets = map (mk_set Ts0) raw_sets;
   668                 val (ys, names_lthy') = names_lthy |> mk_Frees s Ts;
   669                 val xysets = map (pair x) (ys ~~ sets);
   670                 val ppremss = map2 (mk_raw_prem_prems names_lthy') ys Xs_Ts;
   671               in
   672                 flat (map2 (map o apfst o cons) xysets ppremss)
   673               end)
   674           | mk_raw_prem_prems _ _ _ = [];
   675 
   676         fun close_prem_prem xs t =
   677           fold_rev Logic.all (map Free (drop (nn + length xs)
   678             (rev (Term.add_frees t (map dest_Free xs @ ps'))))) t;
   679 
   680         fun mk_prem_prem xs (xysets, (j, x)) =
   681           close_prem_prem xs (Logic.list_implies (map (fn (x', (y, set)) =>
   682               HOLogic.mk_Trueprop (HOLogic.mk_mem (y, set $ x'))) xysets,
   683             HOLogic.mk_Trueprop (nth ps (j - 1) $ x)));
   684 
   685         fun mk_raw_prem phi ctr ctr_Ts ctrXs_Ts =
   686           let
   687             val (xs, names_lthy') = names_lthy |> mk_Frees "x" ctr_Ts;
   688             val pprems = flat (map2 (mk_raw_prem_prems names_lthy') xs ctrXs_Ts);
   689           in (xs, pprems, HOLogic.mk_Trueprop (phi $ Term.list_comb (ctr, xs))) end;
   690 
   691         fun mk_prem (xs, raw_pprems, concl) =
   692           fold_rev Logic.all xs (Logic.list_implies (map (mk_prem_prem xs) raw_pprems, concl));
   693 
   694         val raw_premss = map4 (map3 o mk_raw_prem) ps ctrss ctr_Tsss ctrXs_Tsss;
   695 
   696         val goal =
   697           Library.foldr (Logic.list_implies o apfst (map mk_prem)) (raw_premss,
   698             HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj (map2 (curry op $) ps us)));
   699 
   700         val kksss = map (map (map (fst o snd) o #2)) raw_premss;
   701 
   702         val ctor_induct' = ctor_induct OF (map mk_sumEN_tupled_balanced mss);
   703 
   704         val thm =
   705           Goal.prove_sorry lthy [] [] goal (fn {context = ctxt, ...} =>
   706             mk_induct_tac ctxt nn ns mss kksss (flat ctr_defss) ctor_induct' nested_set_maps
   707               pre_set_defss)
   708           |> singleton (Proof_Context.export names_lthy lthy)
   709           |> Thm.close_derivation;
   710       in
   711         `(conj_dests nn) thm
   712       end;
   713 
   714     val induct_cases = quasi_unambiguous_case_names (maps (map name_of_ctr) ctrss);
   715     val induct_case_names_attr = Attrib.internal (K (Rule_Cases.case_names induct_cases));
   716 
   717     val xctrss = map2 (map2 (curry Term.list_comb)) ctrss xsss;
   718 
   719     fun mk_iter_thmss (_, x_Tssss, fss, _) iters iter_defs ctor_iter_thms =
   720       let
   721         val fiters = map (lists_bmoc fss) iters;
   722 
   723         fun mk_goal fss fiter xctr f xs fxs =
   724           fold_rev (fold_rev Logic.all) (xs :: fss)
   725             (mk_Trueprop_eq (fiter $ xctr, Term.list_comb (f, fxs)));
   726 
   727         fun maybe_tick (T, U) u f =
   728           if try (fst o HOLogic.dest_prodT) U = SOME T then
   729             Term.lambda u (HOLogic.mk_prod (u, f $ u))
   730           else
   731             f;
   732 
   733         fun build_iter (x as Free (_, T)) U =
   734           if T = U then
   735             x
   736           else
   737             build_map lthy (indexify (perhaps (try (snd o HOLogic.dest_prodT)) o snd) Cs
   738               (fn kk => fn TU => maybe_tick TU (nth us kk) (nth fiters kk))) (T, U) $ x;
   739 
   740         val fxsss = map2 (map2 (flat_rec_arg_args oo map2 (map o build_iter))) xsss x_Tssss;
   741 
   742         val goalss = map5 (map4 o mk_goal fss) fiters xctrss fss xsss fxsss;
   743 
   744         val tacss =
   745           map2 (map o mk_iter_tac pre_map_defs (nested_map_idents @ nesting_map_idents) iter_defs)
   746             ctor_iter_thms ctr_defss;
   747 
   748         fun prove goal tac =
   749           Goal.prove_sorry lthy [] [] goal (tac o #context)
   750           |> Thm.close_derivation;
   751       in
   752         map2 (map2 prove) goalss tacss
   753       end;
   754 
   755     val fold_thmss = mk_iter_thmss fold_args_types folds fold_defs (map un_fold_of ctor_iter_thmss);
   756     val rec_thmss = mk_iter_thmss rec_args_types recs rec_defs (map co_rec_of ctor_iter_thmss);
   757   in
   758     ((induct_thms, induct_thm, [induct_case_names_attr]),
   759      (fold_thmss, code_simp_attrs), (rec_thmss, code_simp_attrs))
   760   end;
   761 
   762 fun derive_coinduct_coiters_thms_for_types pre_bnfs (z, cs, cpss,
   763       coiters_args_types as [((pgss, crgsss), _), ((phss, cshsss), _)])
   764     dtor_coinduct dtor_injects dtor_ctors dtor_coiter_thmss nesting_bnfs fpTs Cs kss mss ns
   765     ctr_defss ctr_sugars coiterss coiter_defss export_args lthy =
   766   let
   767     fun mk_ctor_dtor_coiter_thm dtor_inject dtor_ctor coiter =
   768       iffD1 OF [dtor_inject, trans OF [coiter, dtor_ctor RS sym]];
   769 
   770     val ctor_dtor_coiter_thmss =
   771       map3 (map oo mk_ctor_dtor_coiter_thm) dtor_injects dtor_ctors dtor_coiter_thmss;
   772 
   773     val coiterss' = transpose coiterss;
   774     val coiter_defss' = transpose coiter_defss;
   775 
   776     val [unfold_defs, corec_defs] = coiter_defss';
   777 
   778     val nn = length pre_bnfs;
   779 
   780     val pre_map_defs = map map_def_of_bnf pre_bnfs;
   781     val pre_rel_defs = map rel_def_of_bnf pre_bnfs;
   782     val nesting_map_idents = map (unfold_thms lthy [id_def] o map_id0_of_bnf) nesting_bnfs;
   783     val nesting_rel_eqs = map rel_eq_of_bnf nesting_bnfs;
   784 
   785     val fp_b_names = map base_name_of_typ fpTs;
   786 
   787     val ctrss = map #ctrs ctr_sugars;
   788     val discss = map #discs ctr_sugars;
   789     val selsss = map #selss ctr_sugars;
   790     val exhausts = map #exhaust ctr_sugars;
   791     val disc_thmsss = map #disc_thmss ctr_sugars;
   792     val discIss = map #discIs ctr_sugars;
   793     val sel_thmsss = map #sel_thmss ctr_sugars;
   794 
   795     val (((rs, us'), vs'), names_lthy) =
   796       lthy
   797       |> mk_Frees "R" (map (fn T => mk_pred2T T T) fpTs)
   798       ||>> Variable.variant_fixes fp_b_names
   799       ||>> Variable.variant_fixes (map (suffix "'") fp_b_names);
   800 
   801     val us = map2 (curry Free) us' fpTs;
   802     val udiscss = map2 (map o rapp) us discss;
   803     val uselsss = map2 (map o map o rapp) us selsss;
   804 
   805     val vs = map2 (curry Free) vs' fpTs;
   806     val vdiscss = map2 (map o rapp) vs discss;
   807     val vselsss = map2 (map o map o rapp) vs selsss;
   808 
   809     val coinduct_thms_pairs =
   810       let
   811         val uvrs = map3 (fn r => fn u => fn v => r $ u $ v) rs us vs;
   812         val uv_eqs = map2 (curry HOLogic.mk_eq) us vs;
   813         val strong_rs =
   814           map4 (fn u => fn v => fn uvr => fn uv_eq =>
   815             fold_rev Term.lambda [u, v] (HOLogic.mk_disj (uvr, uv_eq))) us vs uvrs uv_eqs;
   816 
   817         (* TODO: generalize (cf. "build_map") *)
   818         fun build_rel rs' T =
   819           (case find_index (curry op = T) fpTs of
   820             ~1 =>
   821             if exists_subtype_in fpTs T then
   822               let
   823                 val Type (s, Ts) = T
   824                 val bnf = the (bnf_of lthy s);
   825                 val live = live_of_bnf bnf;
   826                 val rel = mk_rel live Ts Ts (rel_of_bnf bnf);
   827                 val Ts' = map domain_type (fst (strip_typeN live (fastype_of rel)));
   828               in Term.list_comb (rel, map (build_rel rs') Ts') end
   829             else
   830               HOLogic.eq_const T
   831           | kk => nth rs' kk);
   832 
   833         fun build_rel_app rs' usel vsel = fold rapp [usel, vsel] (build_rel rs' (fastype_of usel));
   834 
   835         fun mk_prem_ctr_concls rs' n k udisc usels vdisc vsels =
   836           (if k = n then [] else [HOLogic.mk_eq (udisc, vdisc)]) @
   837           (if null usels then
   838              []
   839            else
   840              [Library.foldr HOLogic.mk_imp (if n = 1 then [] else [udisc, vdisc],
   841                 Library.foldr1 HOLogic.mk_conj (map2 (build_rel_app rs') usels vsels))]);
   842 
   843         fun mk_prem_concl rs' n udiscs uselss vdiscs vselss =
   844           Library.foldr1 HOLogic.mk_conj
   845             (flat (map5 (mk_prem_ctr_concls rs' n) (1 upto n) udiscs uselss vdiscs vselss))
   846           handle List.Empty => @{term True};
   847 
   848         fun mk_prem rs' uvr u v n udiscs uselss vdiscs vselss =
   849           fold_rev Logic.all [u, v] (Logic.mk_implies (HOLogic.mk_Trueprop uvr,
   850             HOLogic.mk_Trueprop (mk_prem_concl rs' n udiscs uselss vdiscs vselss)));
   851 
   852         val concl =
   853           HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj
   854             (map3 (fn uvr => fn u => fn v => HOLogic.mk_imp (uvr, HOLogic.mk_eq (u, v)))
   855                uvrs us vs));
   856 
   857         fun mk_goal rs' =
   858           Logic.list_implies (map8 (mk_prem rs') uvrs us vs ns udiscss uselsss vdiscss vselsss,
   859             concl);
   860 
   861         val goals = map mk_goal [rs, strong_rs];
   862 
   863         fun prove dtor_coinduct' goal =
   864           Goal.prove_sorry lthy [] [] goal (fn {context = ctxt, ...} =>
   865             mk_coinduct_tac ctxt nesting_rel_eqs nn ns dtor_coinduct' pre_rel_defs dtor_ctors
   866               exhausts ctr_defss disc_thmsss sel_thmsss)
   867           |> singleton (Proof_Context.export names_lthy lthy)
   868           |> Thm.close_derivation;
   869 
   870         fun postproc nn thm =
   871           Thm.permute_prems 0 nn
   872             (if nn = 1 then thm RS mp else funpow nn (fn thm => reassoc_conjs (thm RS mp_conj)) thm)
   873           |> Drule.zero_var_indexes
   874           |> `(conj_dests nn);
   875 
   876         val rel_eqs = map rel_eq_of_bnf pre_bnfs;
   877         val rel_monos = map rel_mono_of_bnf pre_bnfs;
   878         val dtor_coinducts =
   879           [dtor_coinduct, mk_strong_coinduct_thm dtor_coinduct rel_eqs rel_monos lthy];
   880       in
   881         map2 (postproc nn oo prove) dtor_coinducts goals
   882       end;
   883 
   884     fun mk_coinduct_concls ms discs ctrs =
   885       let
   886         fun mk_disc_concl disc = [name_of_disc disc];
   887         fun mk_ctr_concl 0 _ = []
   888           | mk_ctr_concl _ ctor = [name_of_ctr ctor];
   889         val disc_concls = map mk_disc_concl (fst (split_last discs)) @ [[]];
   890         val ctr_concls = map2 mk_ctr_concl ms ctrs;
   891       in
   892         flat (map2 append disc_concls ctr_concls)
   893       end;
   894 
   895     val coinduct_cases = quasi_unambiguous_case_names (map (prefix EqN) fp_b_names);
   896     val coinduct_conclss =
   897       map3 (quasi_unambiguous_case_names ooo mk_coinduct_concls) mss discss ctrss;
   898 
   899     fun mk_maybe_not pos = not pos ? HOLogic.mk_not;
   900 
   901     val fcoiterss' as [gunfolds, hcorecs] =
   902       map2 (fn (pfss, _) => map (lists_bmoc pfss)) (map fst coiters_args_types) coiterss';
   903 
   904     val (unfold_thmss, corec_thmss, safe_unfold_thmss, safe_corec_thmss) =
   905       let
   906         fun mk_goal pfss c cps fcoiter n k ctr m cfs' =
   907           fold_rev (fold_rev Logic.all) ([c] :: pfss)
   908             (Logic.list_implies (seq_conds (HOLogic.mk_Trueprop oo mk_maybe_not) n k cps,
   909                mk_Trueprop_eq (fcoiter $ c, Term.list_comb (ctr, take m cfs'))));
   910 
   911         fun mk_U maybe_mk_sumT =
   912           typ_subst_nonatomic (map2 (fn C => fn fpT => (maybe_mk_sumT fpT C, fpT)) Cs fpTs);
   913 
   914         fun tack z_name (c, u) f =
   915           let val z = Free (z_name, mk_sumT (fastype_of u, fastype_of c)) in
   916             Term.lambda z (mk_sum_case (Term.lambda u u, Term.lambda c (f $ c)) $ z)
   917           end;
   918 
   919         fun build_coiter fcoiters maybe_mk_sumT maybe_tack cqf =
   920           let val T = fastype_of cqf in
   921             if exists_subtype_in Cs T then
   922               let val U = mk_U maybe_mk_sumT T in
   923                 build_map lthy (indexify snd fpTs (fn kk => fn _ =>
   924                   maybe_tack (nth cs kk, nth us kk) (nth fcoiters kk))) (T, U) $ cqf
   925               end
   926             else
   927               cqf
   928           end;
   929 
   930         val crgsss' = map (map (map (build_coiter (un_fold_of fcoiterss') (K I) (K I)))) crgsss;
   931         val cshsss' = map (map (map (build_coiter (co_rec_of fcoiterss') (curry mk_sumT) (tack z))))
   932           cshsss;
   933 
   934         val unfold_goalss = map8 (map4 oooo mk_goal pgss) cs cpss gunfolds ns kss ctrss mss crgsss';
   935         val corec_goalss = map8 (map4 oooo mk_goal phss) cs cpss hcorecs ns kss ctrss mss cshsss';
   936 
   937         val unfold_tacss =
   938           map3 (map oo mk_coiter_tac unfold_defs nesting_map_idents)
   939             (map un_fold_of ctor_dtor_coiter_thmss) pre_map_defs ctr_defss;
   940         val corec_tacss =
   941           map3 (map oo mk_coiter_tac corec_defs nesting_map_idents)
   942             (map co_rec_of ctor_dtor_coiter_thmss) pre_map_defs ctr_defss;
   943 
   944         fun prove goal tac =
   945           Goal.prove_sorry lthy [] [] goal (tac o #context)
   946           |> Thm.close_derivation;
   947 
   948         val unfold_thmss = map2 (map2 prove) unfold_goalss unfold_tacss;
   949         val corec_thmss =
   950           map2 (map2 prove) corec_goalss corec_tacss
   951           |> map (map (unfold_thms lthy @{thms sum_case_if}));
   952 
   953         val unfold_safesss = map2 (map2 (map2 (curry op =))) crgsss' crgsss;
   954         val corec_safesss = map2 (map2 (map2 (curry op =))) cshsss' cshsss;
   955 
   956         val filter_safesss =
   957           map2 (map_filter (fn (safes, thm) => if forall I safes then SOME thm else NONE) oo
   958             curry (op ~~));
   959 
   960         val safe_unfold_thmss = filter_safesss unfold_safesss unfold_thmss;
   961         val safe_corec_thmss = filter_safesss corec_safesss corec_thmss;
   962       in
   963         (unfold_thmss, corec_thmss, safe_unfold_thmss, safe_corec_thmss)
   964       end;
   965 
   966     val (disc_unfold_iff_thmss, disc_corec_iff_thmss) =
   967       let
   968         fun mk_goal c cps fcoiter n k disc =
   969           mk_Trueprop_eq (disc $ (fcoiter $ c),
   970             if n = 1 then @{const True}
   971             else Library.foldr1 HOLogic.mk_conj (seq_conds mk_maybe_not n k cps));
   972 
   973         val unfold_goalss = map6 (map2 oooo mk_goal) cs cpss gunfolds ns kss discss;
   974         val corec_goalss = map6 (map2 oooo mk_goal) cs cpss hcorecs ns kss discss;
   975 
   976         fun mk_case_split' cp = Drule.instantiate' [] [SOME (certify lthy cp)] @{thm case_split};
   977 
   978         val case_splitss' = map (map mk_case_split') cpss;
   979 
   980         val unfold_tacss =
   981           map3 (map oo mk_disc_coiter_iff_tac) case_splitss' unfold_thmss disc_thmsss;
   982         val corec_tacss =
   983           map3 (map oo mk_disc_coiter_iff_tac) case_splitss' corec_thmss disc_thmsss;
   984 
   985         fun prove goal tac =
   986           Goal.prove_sorry lthy [] [] goal (tac o #context)
   987           |> singleton export_args
   988           |> singleton (Proof_Context.export names_lthy lthy)
   989           |> Thm.close_derivation;
   990 
   991         fun proves [_] [_] = []
   992           | proves goals tacs = map2 prove goals tacs;
   993       in
   994         (map2 proves unfold_goalss unfold_tacss, map2 proves corec_goalss corec_tacss)
   995       end;
   996 
   997     val is_triv_discI = is_triv_implies orf is_concl_refl;
   998 
   999     fun mk_disc_coiter_thms coiters discIs =
  1000       map (op RS) (filter_out (is_triv_discI o snd) (coiters ~~ discIs));
  1001 
  1002     val disc_unfold_thmss = map2 mk_disc_coiter_thms unfold_thmss discIss;
  1003     val disc_corec_thmss = map2 mk_disc_coiter_thms corec_thmss discIss;
  1004 
  1005     fun mk_sel_coiter_thm coiter_thm sel sel_thm =
  1006       let
  1007         val (domT, ranT) = dest_funT (fastype_of sel);
  1008         val arg_cong' =
  1009           Drule.instantiate' (map (SOME o certifyT lthy) [domT, ranT])
  1010             [NONE, NONE, SOME (certify lthy sel)] arg_cong
  1011           |> Thm.varifyT_global;
  1012         val sel_thm' = sel_thm RSN (2, trans);
  1013       in
  1014         coiter_thm RS arg_cong' RS sel_thm'
  1015       end;
  1016 
  1017     fun mk_sel_coiter_thms coiter_thmss =
  1018       map3 (map3 (map2 o mk_sel_coiter_thm)) coiter_thmss selsss sel_thmsss;
  1019 
  1020     val sel_unfold_thmsss = mk_sel_coiter_thms unfold_thmss;
  1021     val sel_corec_thmsss = mk_sel_coiter_thms corec_thmss;
  1022 
  1023     val coinduct_consumes_attr = Attrib.internal (K (Rule_Cases.consumes nn));
  1024     val coinduct_case_names_attr = Attrib.internal (K (Rule_Cases.case_names coinduct_cases));
  1025     val coinduct_case_concl_attrs =
  1026       map2 (fn casex => fn concls =>
  1027           Attrib.internal (K (Rule_Cases.case_conclusion (casex, concls))))
  1028         coinduct_cases coinduct_conclss;
  1029     val coinduct_case_attrs =
  1030       coinduct_consumes_attr :: coinduct_case_names_attr :: coinduct_case_concl_attrs;
  1031   in
  1032     ((coinduct_thms_pairs, coinduct_case_attrs),
  1033      (unfold_thmss, corec_thmss, []),
  1034      (safe_unfold_thmss, safe_corec_thmss),
  1035      (disc_unfold_thmss, disc_corec_thmss, simp_attrs),
  1036      (disc_unfold_iff_thmss, disc_corec_iff_thmss, simp_attrs),
  1037      (sel_unfold_thmsss, sel_corec_thmsss, simp_attrs))
  1038   end;
  1039 
  1040 fun define_co_datatypes prepare_constraint prepare_typ prepare_term fp construct_fp
  1041     (wrap_opts as (no_discs_sels, rep_compat), specs) no_defs_lthy0 =
  1042   let
  1043     (* TODO: sanity checks on arguments *)
  1044 
  1045     val _ = if fp = Greatest_FP andalso no_discs_sels then
  1046         error "Cannot define codatatypes without discriminators and selectors"
  1047       else
  1048         ();
  1049 
  1050     fun qualify mandatory fp_b_name =
  1051       Binding.qualify mandatory fp_b_name o (rep_compat ? Binding.qualify false rep_compat_prefix);
  1052 
  1053     val nn = length specs;
  1054     val fp_bs = map type_binding_of specs;
  1055     val fp_b_names = map Binding.name_of fp_bs;
  1056     val fp_common_name = mk_common_name fp_b_names;
  1057     val map_bs = map map_binding_of specs;
  1058     val rel_bs = map rel_binding_of specs;
  1059 
  1060     fun prepare_type_arg (_, (ty, c)) =
  1061       let val TFree (s, _) = prepare_typ no_defs_lthy0 ty in
  1062         TFree (s, prepare_constraint no_defs_lthy0 c)
  1063       end;
  1064 
  1065     val Ass0 = map (map prepare_type_arg o type_args_named_constrained_of) specs;
  1066     val unsorted_Ass0 = map (map (resort_tfree HOLogic.typeS)) Ass0;
  1067     val unsorted_As = Library.foldr1 merge_type_args unsorted_Ass0;
  1068     val num_As = length unsorted_As;
  1069     val set_bss = map (map fst o type_args_named_constrained_of) specs;
  1070 
  1071     val (((Bs0, Cs), Xs), no_defs_lthy) =
  1072       no_defs_lthy0
  1073       |> fold (Variable.declare_typ o resort_tfree dummyS) unsorted_As
  1074       |> mk_TFrees num_As
  1075       ||>> mk_TFrees nn
  1076       ||>> variant_tfrees fp_b_names;
  1077 
  1078     fun add_fake_type spec = Typedecl.basic_typedecl (type_binding_of spec, num_As, mixfix_of spec);
  1079 
  1080     val (fake_T_names, fake_lthy) = fold_map add_fake_type specs no_defs_lthy0;
  1081 
  1082     val qsoty = quote o Syntax.string_of_typ fake_lthy;
  1083 
  1084     val _ = (case duplicates (op =) unsorted_As of [] => ()
  1085       | A :: _ => error ("Duplicate type parameter " ^ qsoty A ^ " in " ^ co_prefix fp ^
  1086           "datatype specification"));
  1087 
  1088     val bad_args =
  1089       map (Logic.type_map (singleton (Variable.polymorphic no_defs_lthy0))) unsorted_As
  1090       |> filter_out Term.is_TVar;
  1091     val _ = null bad_args orelse
  1092       error ("Locally fixed type argument " ^ qsoty (hd bad_args) ^ " in " ^ co_prefix fp ^
  1093         "datatype specification");
  1094 
  1095     val mixfixes = map mixfix_of specs;
  1096 
  1097     val _ = (case duplicates Binding.eq_name fp_bs of [] => ()
  1098       | b :: _ => error ("Duplicate type name declaration " ^ quote (Binding.name_of b)));
  1099 
  1100     val ctr_specss = map ctr_specs_of specs;
  1101 
  1102     val disc_bindingss = map (map disc_of) ctr_specss;
  1103     val ctr_bindingss =
  1104       map2 (fn fp_b_name => map (qualify false fp_b_name o ctr_of)) fp_b_names ctr_specss;
  1105     val ctr_argsss = map (map args_of) ctr_specss;
  1106     val ctr_mixfixess = map (map ctr_mixfix_of) ctr_specss;
  1107 
  1108     val sel_bindingsss = map (map (map fst)) ctr_argsss;
  1109     val fake_ctr_Tsss0 = map (map (map (prepare_typ fake_lthy o snd))) ctr_argsss;
  1110     val raw_sel_defaultsss = map (map defaults_of) ctr_specss;
  1111 
  1112     val (As :: _) :: fake_ctr_Tsss =
  1113       burrow (burrow (Syntax.check_typs fake_lthy)) (Ass0 :: fake_ctr_Tsss0);
  1114     val As' = map dest_TFree As;
  1115 
  1116     val rhs_As' = fold (fold (fold Term.add_tfreesT)) fake_ctr_Tsss [];
  1117     val _ = (case subtract (op =) As' rhs_As' of [] => ()
  1118       | extras => error ("Extra type variables on right-hand side: " ^
  1119           commas (map (qsoty o TFree) extras)));
  1120 
  1121     val fake_Ts = map (fn s => Type (s, As)) fake_T_names;
  1122 
  1123     fun eq_fpT_check (T as Type (s, Ts)) (T' as Type (s', Ts')) =
  1124         s = s' andalso (Ts = Ts' orelse
  1125           error ("Wrong type arguments in " ^ co_prefix fp ^ "recursive type " ^ qsoty T ^
  1126             " (expected " ^ qsoty T' ^ ")"))
  1127       | eq_fpT_check _ _ = false;
  1128 
  1129     fun freeze_fp (T as Type (s, Ts)) =
  1130         (case find_index (eq_fpT_check T) fake_Ts of
  1131           ~1 => Type (s, map freeze_fp Ts)
  1132         | kk => nth Xs kk)
  1133       | freeze_fp T = T;
  1134 
  1135     val unfreeze_fp = Term.typ_subst_atomic (Xs ~~ fake_Ts);
  1136 
  1137     val ctrXs_Tsss = map (map (map freeze_fp)) fake_ctr_Tsss;
  1138     val ctrXs_sum_prod_Ts = map (mk_sumTN_balanced o map HOLogic.mk_tupleT) ctrXs_Tsss;
  1139 
  1140     val fp_eqs =
  1141       map dest_TFree Xs ~~ map (Term.typ_subst_atomic (As ~~ unsorted_As)) ctrXs_sum_prod_Ts;
  1142 
  1143     val rhsXs_As' = fold (fold (fold Term.add_tfreesT)) ctrXs_Tsss [];
  1144     val _ = (case subtract (op =) rhsXs_As' As' of [] => ()
  1145       | extras => List.app (fn extra => warning ("Unused type variable on right-hand side of " ^
  1146           co_prefix fp ^ "datatype definition: " ^ qsoty (TFree extra))) extras);
  1147 
  1148     val (pre_bnfs, (fp_res as {bnfs = fp_bnfs as any_fp_bnf :: _, ctors = ctors0, dtors = dtors0,
  1149            xtor_co_iterss = xtor_co_iterss0, xtor_co_induct, dtor_ctors, ctor_dtors, ctor_injects,
  1150            dtor_injects, xtor_map_thms, xtor_set_thmss, xtor_rel_thms, xtor_co_iter_thmss, ...},
  1151            lthy)) =
  1152       fp_bnf (construct_fp mixfixes map_bs rel_bs set_bss) fp_bs (map dest_TFree unsorted_As) fp_eqs
  1153         no_defs_lthy0
  1154       handle BAD_DEAD (X, X_backdrop) =>
  1155         (case X_backdrop of
  1156           Type (bad_tc, _) =>
  1157           let
  1158             val fake_T = qsoty (unfreeze_fp X);
  1159             val fake_T_backdrop = qsoty (unfreeze_fp X_backdrop);
  1160             fun register_hint () =
  1161               "\nUse the " ^ quote (fst (fst @{command_spec "bnf"})) ^ " command to register " ^
  1162               quote bad_tc ^ " as a bounded natural functor to allow nested (co)recursion through \
  1163               \it";
  1164           in
  1165             if is_some (bnf_of no_defs_lthy bad_tc) orelse
  1166                is_some (fp_sugar_of no_defs_lthy bad_tc) then
  1167               error ("Inadmissible " ^ co_prefix fp ^ "recursive occurrence of type " ^ fake_T ^
  1168                 " in type expression " ^ fake_T_backdrop)
  1169             else if is_some (Datatype_Data.get_info (Proof_Context.theory_of no_defs_lthy)
  1170                 bad_tc) then
  1171               error ("Unsupported " ^ co_prefix fp ^ "recursive occurrence of type " ^ fake_T ^
  1172                 " via the old-style datatype " ^ quote bad_tc ^ " in type expression " ^
  1173                 fake_T_backdrop ^ register_hint ())
  1174             else
  1175               error ("Unsupported " ^ co_prefix fp ^ "recursive occurrence of type " ^ fake_T ^
  1176                 " via type constructor " ^ quote bad_tc ^ " in type expression " ^ fake_T_backdrop ^
  1177                 register_hint ())
  1178           end);
  1179 
  1180     val time = time lthy;
  1181     val timer = time (Timer.startRealTimer ());
  1182 
  1183     val nesting_bnfs = nesty_bnfs lthy ctrXs_Tsss As;
  1184     val nested_bnfs = nesty_bnfs lthy ctrXs_Tsss Xs;
  1185 
  1186     val pre_map_defs = map map_def_of_bnf pre_bnfs;
  1187     val pre_set_defss = map set_defs_of_bnf pre_bnfs;
  1188     val pre_rel_defs = map rel_def_of_bnf pre_bnfs;
  1189     val nesting_set_maps = maps set_map_of_bnf nesting_bnfs;
  1190     val nested_set_maps = maps set_map_of_bnf nested_bnfs;
  1191 
  1192     val live = live_of_bnf any_fp_bnf;
  1193     val _ =
  1194       if live = 0 andalso exists (not o Binding.is_empty) (map_bs @ rel_bs) then
  1195         warning "Map function and relator names ignored"
  1196       else
  1197         ();
  1198 
  1199     val Bs =
  1200       map3 (fn alive => fn A as TFree (_, S) => fn B => if alive then resort_tfree S B else A)
  1201         (liveness_of_fp_bnf num_As any_fp_bnf) As Bs0;
  1202 
  1203     val B_ify = Term.typ_subst_atomic (As ~~ Bs);
  1204 
  1205     val ctors = map (mk_ctor As) ctors0;
  1206     val dtors = map (mk_dtor As) dtors0;
  1207 
  1208     val fpTs = map (domain_type o fastype_of) dtors;
  1209 
  1210     fun massage_simple_notes base =
  1211       filter_out (null o #2)
  1212       #> map (fn (thmN, thms, attrs) =>
  1213         ((qualify true base (Binding.name thmN), attrs), [(thms, [])]));
  1214 
  1215     val massage_multi_notes =
  1216       maps (fn (thmN, thmss, attrs) =>
  1217         if forall null thmss then
  1218           []
  1219         else
  1220           map3 (fn fp_b_name => fn Type (T_name, _) => fn thms =>
  1221               ((qualify true fp_b_name (Binding.name thmN), attrs T_name), [(thms, [])]))
  1222             fp_b_names fpTs thmss);
  1223 
  1224     val ctr_Tsss = map (map (map (Term.typ_subst_atomic (Xs ~~ fpTs)))) ctrXs_Tsss;
  1225     val ns = map length ctr_Tsss;
  1226     val kss = map (fn n => 1 upto n) ns;
  1227     val mss = map (map length) ctr_Tsss;
  1228 
  1229     val ((xtor_co_iterss, iters_args_types, coiters_args_types), lthy') =
  1230       mk_co_iters_prelims fp ctr_Tsss fpTs Cs ns mss xtor_co_iterss0 lthy;
  1231 
  1232     fun define_ctrs_dtrs_for_type (((((((((((((((((((((((fp_bnf, fp_b), fpT), ctor), dtor),
  1233             xtor_co_iters), ctor_dtor), dtor_ctor), ctor_inject), pre_map_def), pre_set_defs),
  1234           pre_rel_def), fp_map_thm), fp_set_thms), fp_rel_thm), n), ks), ms), ctr_bindings),
  1235         ctr_mixfixes), ctr_Tss), disc_bindings), sel_bindingss), raw_sel_defaultss) no_defs_lthy =
  1236       let
  1237         val fp_b_name = Binding.name_of fp_b;
  1238 
  1239         val dtorT = domain_type (fastype_of ctor);
  1240         val ctr_prod_Ts = map HOLogic.mk_tupleT ctr_Tss;
  1241         val ctr_sum_prod_T = mk_sumTN_balanced ctr_prod_Ts;
  1242 
  1243         val ((((w, xss), yss), u'), names_lthy) =
  1244           no_defs_lthy
  1245           |> yield_singleton (mk_Frees "w") dtorT
  1246           ||>> mk_Freess "x" ctr_Tss
  1247           ||>> mk_Freess "y" (map (map B_ify) ctr_Tss)
  1248           ||>> yield_singleton Variable.variant_fixes fp_b_name;
  1249 
  1250         val u = Free (u', fpT);
  1251 
  1252         val tuple_xs = map HOLogic.mk_tuple xss;
  1253         val tuple_ys = map HOLogic.mk_tuple yss;
  1254 
  1255         val ctr_rhss =
  1256           map3 (fn k => fn xs => fn tuple_x => fold_rev Term.lambda xs (ctor $
  1257             mk_InN_balanced ctr_sum_prod_T n tuple_x k)) ks xss tuple_xs;
  1258 
  1259         val maybe_conceal_def_binding = Thm.def_binding
  1260           #> Config.get no_defs_lthy bnf_note_all = false ? Binding.conceal;
  1261 
  1262         val ((raw_ctrs, raw_ctr_defs), (lthy', lthy)) = no_defs_lthy
  1263           |> apfst split_list o fold_map3 (fn b => fn mx => fn rhs =>
  1264               Local_Theory.define ((b, mx), ((maybe_conceal_def_binding b, []), rhs)) #>> apsnd snd)
  1265             ctr_bindings ctr_mixfixes ctr_rhss
  1266           ||> `Local_Theory.restore;
  1267 
  1268         val phi = Proof_Context.export_morphism lthy lthy';
  1269 
  1270         val ctr_defs = map (Morphism.thm phi) raw_ctr_defs;
  1271         val ctr_defs' =
  1272           map2 (fn m => fn def => mk_unabs_def m (def RS meta_eq_to_obj_eq)) ms ctr_defs;
  1273 
  1274         val ctrs0 = map (Morphism.term phi) raw_ctrs;
  1275         val ctrs = map (mk_ctr As) ctrs0;
  1276 
  1277         fun wrap_ctrs lthy =
  1278           let
  1279             fun exhaust_tac {context = ctxt, prems = _} =
  1280               let
  1281                 val ctor_iff_dtor_thm =
  1282                   let
  1283                     val goal =
  1284                       fold_rev Logic.all [w, u]
  1285                         (mk_Trueprop_eq (HOLogic.mk_eq (u, ctor $ w), HOLogic.mk_eq (dtor $ u, w)));
  1286                   in
  1287                     Goal.prove_sorry lthy [] [] goal (fn {context = ctxt, ...} =>
  1288                       mk_ctor_iff_dtor_tac ctxt (map (SOME o certifyT lthy) [dtorT, fpT])
  1289                         (certify lthy ctor) (certify lthy dtor) ctor_dtor dtor_ctor)
  1290                     |> Thm.close_derivation
  1291                     |> Morphism.thm phi
  1292                   end;
  1293 
  1294                 val sumEN_thm' =
  1295                   unfold_thms lthy @{thms unit_all_eq1}
  1296                     (Drule.instantiate' (map (SOME o certifyT lthy) ctr_prod_Ts) []
  1297                        (mk_sumEN_balanced n))
  1298                   |> Morphism.thm phi;
  1299               in
  1300                 mk_exhaust_tac ctxt n ctr_defs ctor_iff_dtor_thm sumEN_thm'
  1301               end;
  1302 
  1303             val inject_tacss =
  1304               map2 (fn 0 => K [] | _ => fn ctr_def => [fn {context = ctxt, ...} =>
  1305                 mk_inject_tac ctxt ctr_def ctor_inject]) ms ctr_defs;
  1306 
  1307             val half_distinct_tacss =
  1308               map (map (fn (def, def') => fn {context = ctxt, ...} =>
  1309                 mk_half_distinct_tac ctxt ctor_inject [def, def'])) (mk_half_pairss (`I ctr_defs));
  1310 
  1311             val tacss = [exhaust_tac] :: inject_tacss @ half_distinct_tacss;
  1312 
  1313             val sel_defaultss = map (map (apsnd (prepare_term lthy))) raw_sel_defaultss
  1314           in
  1315             wrap_free_constructors tacss (((wrap_opts, ctrs0), standard_binding), (disc_bindings,
  1316               (sel_bindingss, sel_defaultss))) lthy
  1317           end;
  1318 
  1319         fun derive_maps_sets_rels (ctr_sugar, lthy) =
  1320           if live = 0 then
  1321             ((([], [], [], []), ctr_sugar), lthy)
  1322           else
  1323             let
  1324               val rel_flip = rel_flip_of_bnf fp_bnf;
  1325               val nones = replicate live NONE;
  1326 
  1327               val ctor_cong =
  1328                 if fp = Least_FP then
  1329                   Drule.dummy_thm
  1330                 else
  1331                   let val ctor' = mk_ctor Bs ctor in
  1332                     cterm_instantiate_pos [NONE, NONE, SOME (certify lthy ctor')] arg_cong
  1333                   end;
  1334 
  1335               fun mk_cIn ify =
  1336                 certify lthy o (fp = Greatest_FP ? curry (op $) (map_types ify ctor)) oo
  1337                 mk_InN_balanced (ify ctr_sum_prod_T) n;
  1338 
  1339               val cxIns = map2 (mk_cIn I) tuple_xs ks;
  1340               val cyIns = map2 (mk_cIn B_ify) tuple_ys ks;
  1341 
  1342               fun mk_map_thm ctr_def' cxIn =
  1343                 fold_thms lthy [ctr_def']
  1344                   (unfold_thms lthy (pre_map_def ::
  1345                        (if fp = Least_FP then [] else [ctor_dtor, dtor_ctor]) @ sum_prod_thms_map)
  1346                      (cterm_instantiate_pos (nones @ [SOME cxIn])
  1347                         (if fp = Least_FP then fp_map_thm else fp_map_thm RS ctor_cong)))
  1348                 |> singleton (Proof_Context.export names_lthy no_defs_lthy);
  1349 
  1350               fun mk_set_thm fp_set_thm ctr_def' cxIn =
  1351                 fold_thms lthy [ctr_def']
  1352                   (unfold_thms lthy (pre_set_defs @ nested_set_maps @ nesting_set_maps @
  1353                        (if fp = Least_FP then [] else [dtor_ctor]) @ sum_prod_thms_set)
  1354                      (cterm_instantiate_pos [SOME cxIn] fp_set_thm))
  1355                 |> singleton (Proof_Context.export names_lthy no_defs_lthy);
  1356 
  1357               fun mk_set_thms fp_set_thm = map2 (mk_set_thm fp_set_thm) ctr_defs' cxIns;
  1358 
  1359               val map_thms = map2 mk_map_thm ctr_defs' cxIns;
  1360               val set_thmss = map mk_set_thms fp_set_thms;
  1361 
  1362               val rel_infos = (ctr_defs' ~~ cxIns, ctr_defs' ~~ cyIns);
  1363 
  1364               fun mk_rel_thm postproc ctr_defs' cxIn cyIn =
  1365                 fold_thms lthy ctr_defs'
  1366                   (unfold_thms lthy (@{thm Inl_Inr_False} :: pre_rel_def ::
  1367                        (if fp = Least_FP then [] else [dtor_ctor]) @ sum_prod_thms_rel)
  1368                      (cterm_instantiate_pos (nones @ [SOME cxIn, SOME cyIn]) fp_rel_thm))
  1369                 |> postproc
  1370                 |> singleton (Proof_Context.export names_lthy no_defs_lthy);
  1371 
  1372               fun mk_rel_inject_thm ((ctr_def', cxIn), (_, cyIn)) =
  1373                 mk_rel_thm (unfold_thms lthy @{thms eq_sym_Unity_conv}) [ctr_def'] cxIn cyIn;
  1374 
  1375               val rel_inject_thms = map mk_rel_inject_thm (op ~~ rel_infos);
  1376 
  1377               fun mk_half_rel_distinct_thm ((xctr_def', cxIn), (yctr_def', cyIn)) =
  1378                 mk_rel_thm (fn thm => thm RS @{thm eq_False[THEN iffD1]}) [xctr_def', yctr_def']
  1379                   cxIn cyIn;
  1380 
  1381               fun mk_other_half_rel_distinct_thm thm =
  1382                 flip_rels lthy live thm
  1383                 RS (rel_flip RS sym RS @{thm arg_cong[of _ _ Not]} RS iffD2);
  1384 
  1385               val half_rel_distinct_thmss =
  1386                 map (map mk_half_rel_distinct_thm) (mk_half_pairss rel_infos);
  1387               val other_half_rel_distinct_thmss =
  1388                 map (map mk_other_half_rel_distinct_thm) half_rel_distinct_thmss;
  1389               val (rel_distinct_thms, _) =
  1390                 join_halves n half_rel_distinct_thmss other_half_rel_distinct_thmss;
  1391 
  1392               val notes =
  1393                 [(mapN, map_thms, code_simp_attrs),
  1394                  (rel_distinctN, rel_distinct_thms, code_simp_attrs),
  1395                  (rel_injectN, rel_inject_thms, code_simp_attrs),
  1396                  (setsN, flat set_thmss, code_simp_attrs)]
  1397                 |> massage_simple_notes fp_b_name;
  1398             in
  1399               (((map_thms, rel_inject_thms, rel_distinct_thms, set_thmss), ctr_sugar),
  1400                lthy |> Local_Theory.notes notes |> snd)
  1401             end;
  1402 
  1403         fun mk_binding suf = qualify false fp_b_name (Binding.suffix_name ("_" ^ suf) fp_b);
  1404 
  1405         fun massage_res (((maps_sets_rels, ctr_sugar), co_iter_res), lthy) =
  1406           (((maps_sets_rels, (ctrs, xss, ctr_defs, ctr_sugar)), co_iter_res), lthy);
  1407       in
  1408         (wrap_ctrs
  1409          #> derive_maps_sets_rels
  1410          ##>>
  1411            (if fp = Least_FP then define_iters [foldN, recN] (the iters_args_types)
  1412             else define_coiters [unfoldN, corecN] (the coiters_args_types))
  1413              mk_binding fpTs Cs xtor_co_iters
  1414          #> massage_res, lthy')
  1415       end;
  1416 
  1417     fun wrap_types_etc (wrap_types_etcs, lthy) =
  1418       fold_map I wrap_types_etcs lthy
  1419       |>> apsnd split_list o apfst (apsnd split_list4 o apfst split_list4 o split_list)
  1420         o split_list;
  1421 
  1422     val mk_simp_thmss =
  1423       map7 (fn {injects, distincts, case_thms, ...} => fn un_folds => fn co_recs =>
  1424         fn mapsx => fn rel_injects => fn rel_distincts => fn setss =>
  1425           injects @ distincts @ case_thms @ co_recs @ un_folds @ mapsx @ rel_injects
  1426           @ rel_distincts @ flat setss);
  1427 
  1428     fun derive_and_note_induct_iters_thms_for_types
  1429         ((((mapss, rel_injects, rel_distincts, setss), (ctrss, _, ctr_defss, ctr_sugars)),
  1430           (iterss, iter_defss)), lthy) =
  1431       let
  1432         val ((induct_thms, induct_thm, induct_attrs), (fold_thmss, fold_attrs),
  1433              (rec_thmss, rec_attrs)) =
  1434           derive_induct_iters_thms_for_types pre_bnfs (the iters_args_types) xtor_co_induct
  1435             xtor_co_iter_thmss nesting_bnfs nested_bnfs fpTs Cs Xs ctrXs_Tsss ctrss ctr_defss iterss
  1436             iter_defss lthy;
  1437 
  1438         val induct_type_attr = Attrib.internal o K o Induct.induct_type;
  1439 
  1440         val simp_thmss =
  1441           mk_simp_thmss ctr_sugars fold_thmss rec_thmss mapss rel_injects rel_distincts setss;
  1442 
  1443         val common_notes =
  1444           (if nn > 1 then [(inductN, [induct_thm], induct_attrs)] else [])
  1445           |> massage_simple_notes fp_common_name;
  1446 
  1447         val notes =
  1448           [(foldN, fold_thmss, K fold_attrs),
  1449            (inductN, map single induct_thms, fn T_name => induct_attrs @ [induct_type_attr T_name]),
  1450            (recN, rec_thmss, K rec_attrs),
  1451            (simpsN, simp_thmss, K [])]
  1452           |> massage_multi_notes;
  1453       in
  1454         lthy
  1455         |> Local_Theory.notes (common_notes @ notes) |> snd
  1456         |> register_fp_sugars Least_FP pre_bnfs nested_bnfs nesting_bnfs fp_res ctr_defss ctr_sugars
  1457           iterss mapss [induct_thm] (transpose [fold_thmss, rec_thmss]) [] []
  1458       end;
  1459 
  1460     fun derive_and_note_coinduct_coiters_thms_for_types
  1461         ((((mapss, rel_injects, rel_distincts, setss), (_, _, ctr_defss, ctr_sugars)),
  1462           (coiterss, coiter_defss)), lthy) =
  1463       let
  1464         val (([(coinduct_thms, coinduct_thm), (strong_coinduct_thms, strong_coinduct_thm)],
  1465               coinduct_attrs),
  1466              (unfold_thmss, corec_thmss, coiter_attrs),
  1467              (safe_unfold_thmss, safe_corec_thmss),
  1468              (disc_unfold_thmss, disc_corec_thmss, disc_coiter_attrs),
  1469              (disc_unfold_iff_thmss, disc_corec_iff_thmss, disc_coiter_iff_attrs),
  1470              (sel_unfold_thmsss, sel_corec_thmsss, sel_coiter_attrs)) =
  1471           derive_coinduct_coiters_thms_for_types pre_bnfs (the coiters_args_types) xtor_co_induct
  1472             dtor_injects dtor_ctors xtor_co_iter_thmss nesting_bnfs fpTs Cs kss mss ns ctr_defss
  1473             ctr_sugars coiterss coiter_defss (Proof_Context.export lthy' no_defs_lthy) lthy;
  1474 
  1475         val sel_unfold_thmss = map flat sel_unfold_thmsss;
  1476         val sel_corec_thmss = map flat sel_corec_thmsss;
  1477 
  1478         val coinduct_type_attr = Attrib.internal o K o Induct.coinduct_type;
  1479 
  1480         fun flat_coiter_thms coiters disc_coiters sel_coiters =
  1481           coiters @ disc_coiters @ sel_coiters;
  1482 
  1483         val simp_thmss =
  1484           mk_simp_thmss ctr_sugars
  1485             (map3 flat_coiter_thms safe_unfold_thmss disc_unfold_thmss sel_unfold_thmss)
  1486             (map3 flat_coiter_thms safe_corec_thmss disc_corec_thmss sel_corec_thmss)
  1487             mapss rel_injects rel_distincts setss;
  1488 
  1489         val anonymous_notes =
  1490           [(flat safe_unfold_thmss @ flat safe_corec_thmss, simp_attrs)]
  1491           |> map (fn (thms, attrs) => ((Binding.empty, attrs), [(thms, [])]));
  1492 
  1493         val common_notes =
  1494           (if nn > 1 then
  1495              [(coinductN, [coinduct_thm], coinduct_attrs),
  1496               (strong_coinductN, [strong_coinduct_thm], coinduct_attrs)]
  1497            else
  1498              [])
  1499           |> massage_simple_notes fp_common_name;
  1500 
  1501         val notes =
  1502           [(coinductN, map single coinduct_thms,
  1503             fn T_name => coinduct_attrs @ [coinduct_type_attr T_name]),
  1504            (corecN, corec_thmss, K coiter_attrs),
  1505            (disc_corecN, disc_corec_thmss, K disc_coiter_attrs),
  1506            (disc_corec_iffN, disc_corec_iff_thmss, K disc_coiter_iff_attrs),
  1507            (disc_unfoldN, disc_unfold_thmss, K disc_coiter_attrs),
  1508            (disc_unfold_iffN, disc_unfold_iff_thmss, K disc_coiter_iff_attrs),
  1509            (sel_corecN, sel_corec_thmss, K sel_coiter_attrs),
  1510            (sel_unfoldN, sel_unfold_thmss, K sel_coiter_attrs),
  1511            (simpsN, simp_thmss, K []),
  1512            (strong_coinductN, map single strong_coinduct_thms, K coinduct_attrs),
  1513            (unfoldN, unfold_thmss, K coiter_attrs)]
  1514           |> massage_multi_notes;
  1515       in
  1516         lthy
  1517         |> Local_Theory.notes (anonymous_notes @ common_notes @ notes) |> snd
  1518         |> register_fp_sugars Greatest_FP pre_bnfs nested_bnfs nesting_bnfs fp_res ctr_defss
  1519           ctr_sugars coiterss mapss [coinduct_thm, strong_coinduct_thm]
  1520           (transpose [unfold_thmss, corec_thmss]) (transpose [disc_unfold_thmss, disc_corec_thmss])
  1521           (transpose [sel_unfold_thmsss, sel_corec_thmsss])
  1522       end;
  1523 
  1524     val lthy'' = lthy'
  1525       |> fold_map define_ctrs_dtrs_for_type (fp_bnfs ~~ fp_bs ~~ fpTs ~~ ctors ~~ dtors ~~
  1526         xtor_co_iterss ~~ ctor_dtors ~~ dtor_ctors ~~ ctor_injects ~~ pre_map_defs ~~
  1527         pre_set_defss ~~ pre_rel_defs ~~ xtor_map_thms ~~ xtor_set_thmss ~~ xtor_rel_thms ~~ ns ~~
  1528         kss ~~ mss ~~ ctr_bindingss ~~ ctr_mixfixess ~~ ctr_Tsss ~~ disc_bindingss ~~
  1529         sel_bindingsss ~~ raw_sel_defaultsss)
  1530       |> wrap_types_etc
  1531       |> fp_case fp derive_and_note_induct_iters_thms_for_types
  1532            derive_and_note_coinduct_coiters_thms_for_types;
  1533 
  1534     val timer = time (timer ("Constructors, discriminators, selectors, etc., for the new " ^
  1535       co_prefix fp ^ "datatype"));
  1536   in
  1537     timer; lthy''
  1538   end;
  1539 
  1540 val co_datatypes = define_co_datatypes (K I) (K I) (K I);
  1541 
  1542 val co_datatype_cmd =
  1543   define_co_datatypes Typedecl.read_constraint Syntax.parse_typ Syntax.parse_term;
  1544 
  1545 val parse_ctr_arg =
  1546   @{keyword "("} |-- parse_binding_colon -- Parse.typ --| @{keyword ")"} ||
  1547   (Parse.typ >> pair Binding.empty);
  1548 
  1549 val parse_defaults =
  1550   @{keyword "("} |-- Parse.reserved "defaults" |-- Scan.repeat parse_bound_term --| @{keyword ")"};
  1551 
  1552 val parse_type_arg_constrained =
  1553   Parse.type_ident -- Scan.option (@{keyword "::"} |-- Parse.!!! Parse.sort);
  1554 
  1555 val parse_type_arg_named_constrained = parse_opt_binding_colon -- parse_type_arg_constrained;
  1556 
  1557 val parse_type_args_named_constrained =
  1558   parse_type_arg_constrained >> (single o pair Binding.empty) ||
  1559   @{keyword "("} |-- Parse.!!! (Parse.list1 parse_type_arg_named_constrained --| @{keyword ")"}) ||
  1560   Scan.succeed [];
  1561 
  1562 val parse_map_rel_binding = Parse.short_ident --| @{keyword ":"} -- parse_binding;
  1563 
  1564 val no_map_rel = (Binding.empty, Binding.empty);
  1565 
  1566 fun extract_map_rel ("map", b) = apfst (K b)
  1567   | extract_map_rel ("rel", b) = apsnd (K b)
  1568   | extract_map_rel (s, _) = error ("Unknown label " ^ quote s ^ " (expected \"map\" or \"rel\")");
  1569 
  1570 val parse_map_rel_bindings =
  1571   @{keyword "("} |-- Scan.repeat parse_map_rel_binding --| @{keyword ")"}
  1572     >> (fn ps => fold extract_map_rel ps no_map_rel) ||
  1573   Scan.succeed no_map_rel;
  1574 
  1575 val parse_ctr_spec =
  1576   parse_opt_binding_colon -- parse_binding -- Scan.repeat parse_ctr_arg --
  1577   Scan.optional parse_defaults [] -- Parse.opt_mixfix;
  1578 
  1579 val parse_spec =
  1580   parse_type_args_named_constrained -- parse_binding -- parse_map_rel_bindings --
  1581   Parse.opt_mixfix -- (@{keyword "="} |-- Parse.enum1 "|" parse_ctr_spec);
  1582 
  1583 val parse_co_datatype = parse_wrap_free_constructors_options -- Parse.and_list1 parse_spec;
  1584 
  1585 fun parse_co_datatype_cmd fp construct_fp = parse_co_datatype >> co_datatype_cmd fp construct_fp;
  1586 
  1587 end;