src/HOL/BNF/Tools/bnf_fp_def_sugar.ML
author traytel
Mon Aug 26 13:44:46 2013 +0200 (2013-08-26)
changeset 53203 222ea6acbdd6
parent 53143 ba80154a1118
child 53210 7219c61796c0
permissions -rw-r--r--
moved derivation of ctor_dtor_unfold to sugar (streamlines fp_result interface)
     1 (*  Title:      HOL/BNF/Tools/bnf_fp_def_sugar.ML
     2     Author:     Jasmin Blanchette, TU Muenchen
     3     Copyright   2012, 2013
     4 
     5 Sugared datatype and codatatype constructions.
     6 *)
     7 
     8 signature BNF_FP_DEF_SUGAR =
     9 sig
    10   type fp_sugar =
    11     {T: typ,
    12      fp: BNF_FP_Util.fp_kind,
    13      index: int,
    14      pre_bnfs: BNF_Def.bnf list,
    15      nested_bnfs: BNF_Def.bnf list,
    16      fp_res: BNF_FP_Util.fp_result,
    17      ctr_defss: thm list list,
    18      ctr_sugars: BNF_Ctr_Sugar.ctr_sugar list,
    19      co_iterss: term list list,
    20      co_inducts: thm list,
    21      co_iter_thmsss: thm list list list};
    22 
    23   val of_fp_sugar: (fp_sugar -> 'a list) -> fp_sugar -> 'a
    24   val morph_fp_sugar: morphism -> fp_sugar -> fp_sugar
    25   val fp_sugar_of: Proof.context -> string -> fp_sugar option
    26 
    27   val co_induct_of: 'a list -> 'a
    28   val strong_co_induct_of: 'a list -> 'a
    29 
    30   val tvar_subst: theory -> typ list -> typ list -> ((string * int) * typ) list
    31   val exists_subtype_in: typ list -> typ -> bool
    32   val flat_rec_arg_args: 'a list list -> 'a list
    33   val flat_corec_preds_predsss_gettersss: 'a list -> 'a list list list -> 'a list list list ->
    34     'a list
    35   val mk_co_iter: theory -> BNF_FP_Util.fp_kind -> typ -> typ list -> term -> term
    36   val nesty_bnfs: Proof.context -> typ list list list -> typ list -> BNF_Def.bnf list
    37   val mk_map: int -> typ list -> typ list -> term -> term
    38   val build_map: local_theory -> (typ * typ -> term) -> typ * typ -> term
    39   val dest_map: Proof.context -> string -> term -> term * term list
    40   val dest_ctr: Proof.context -> string -> term -> term * term list
    41   val mk_co_iters_prelims: BNF_FP_Util.fp_kind -> typ list -> typ list -> int list ->
    42     int list list -> term list list -> Proof.context ->
    43     (term list list
    44      * (typ list list * typ list list list list * term list list
    45         * term list list list list) list option
    46      * (string * term list * term list list
    47         * ((term list list * term list list list) * (typ list * typ list list)) list) option)
    48     * Proof.context
    49 
    50   val mk_iter_fun_arg_types: typ list -> int list -> int list list -> term ->
    51     typ list list list list
    52   val mk_coiter_fun_arg_types: typ list -> int list -> int list list -> term ->
    53     typ list list
    54     * (typ list list list list * typ list list list * typ list list list list * typ list)
    55   val define_iters: string list ->
    56     (typ list list * typ list list list list * term list list * term list list list list) list ->
    57     (string -> binding) -> typ list -> typ list -> term list -> Proof.context ->
    58     (term list * thm list) * Proof.context
    59   val define_coiters: string list -> string * term list * term list list
    60     * ((term list list * term list list list) * (typ list * typ list list)) list ->
    61     (string -> binding) -> typ list -> typ list -> term list -> Proof.context ->
    62     (term list * thm list) * Proof.context
    63   val derive_induct_iters_thms_for_types: BNF_Def.bnf list ->
    64     (typ list list * typ list list list list * term list list * term list list list list) list ->
    65     thm -> thm list list -> BNF_Def.bnf list -> BNF_Def.bnf list -> typ list -> typ list ->
    66     typ list -> typ list list list -> term list list -> thm list list -> term list list ->
    67     thm list list -> local_theory ->
    68     (thm list * thm * Args.src list) * (thm list list * Args.src list)
    69     * (thm list list * Args.src list)
    70   val derive_coinduct_coiters_thms_for_types: BNF_Def.bnf list ->
    71     string * term list * term list list * ((term list list * term list list list)
    72       * (typ list * typ list list)) list ->
    73     thm -> thm list -> thm list -> thm list list -> BNF_Def.bnf list -> typ list -> typ list ->
    74     typ list -> int list list -> int list list -> int list -> thm list list ->
    75     BNF_Ctr_Sugar.ctr_sugar list -> term list list -> thm list list -> (thm list -> thm list) ->
    76     local_theory ->
    77     ((thm list * thm) list * Args.src list)
    78     * (thm list list * thm list list * Args.src list)
    79     * (thm list list * thm list list) * (thm list list * thm list list * Args.src list)
    80     * (thm list list * thm list list * Args.src list)
    81     * (thm list list * thm list list * Args.src list)
    82 
    83   val co_datatypes: BNF_FP_Util.fp_kind -> (mixfix list -> binding list -> binding list ->
    84       binding list list -> binding list -> (string * sort) list -> typ list * typ list list ->
    85       BNF_Def.bnf list -> local_theory -> BNF_FP_Util.fp_result * local_theory) ->
    86     (bool * bool) * (((((binding * (typ * sort)) list * binding) * (binding * binding)) * mixfix) *
    87       ((((binding * binding) * (binding * typ) list) * (binding * term) list) *
    88         mixfix) list) list ->
    89     local_theory -> local_theory
    90   val parse_co_datatype_cmd: BNF_FP_Util.fp_kind -> (mixfix list -> binding list -> binding list ->
    91       binding list list -> binding list -> (string * sort) list -> typ list * typ list list ->
    92       BNF_Def.bnf list -> local_theory -> BNF_FP_Util.fp_result * local_theory) ->
    93     (local_theory -> local_theory) parser
    94 end;
    95 
    96 structure BNF_FP_Def_Sugar : BNF_FP_DEF_SUGAR =
    97 struct
    98 
    99 open BNF_Util
   100 open BNF_Ctr_Sugar
   101 open BNF_Def
   102 open BNF_FP_Util
   103 open BNF_FP_Def_Sugar_Tactics
   104 
   105 val EqN = "Eq_";
   106 
   107 type fp_sugar =
   108   {T: typ,
   109    fp: fp_kind,
   110    index: int,
   111    pre_bnfs: bnf list,
   112    nested_bnfs: bnf list,
   113    fp_res: fp_result,
   114    ctr_defss: thm list list,
   115    ctr_sugars: ctr_sugar list,
   116    co_iterss: term list list,
   117    co_inducts: thm list,
   118    co_iter_thmsss: thm list list list};
   119 
   120 fun of_fp_sugar f (fp_sugar as {index, ...}) = nth (f fp_sugar) index;
   121 
   122 fun eq_fp_sugar ({T = T1, fp = fp1, index = index1, fp_res = fp_res1, ...} : fp_sugar,
   123     {T = T2, fp = fp2, index = index2, fp_res = fp_res2, ...} : fp_sugar) =
   124   T1 = T2 andalso fp1 = fp2 andalso index1 = index2 andalso eq_fp_result (fp_res1, fp_res2);
   125 
   126 fun morph_fp_sugar phi {T, fp, index, pre_bnfs, nested_bnfs, fp_res, ctr_defss, ctr_sugars,
   127     co_iterss, co_inducts, co_iter_thmsss} =
   128   {T = Morphism.typ phi T, fp = fp, index = index, pre_bnfs = map (morph_bnf phi) pre_bnfs,
   129     nested_bnfs = map (morph_bnf phi) nested_bnfs, fp_res = morph_fp_result phi fp_res,
   130    ctr_defss = map (map (Morphism.thm phi)) ctr_defss,
   131    ctr_sugars = map (morph_ctr_sugar phi) ctr_sugars,
   132    co_iterss = map (map (Morphism.term phi)) co_iterss,
   133    co_inducts = map (Morphism.thm phi) co_inducts,
   134    co_iter_thmsss = map (map (map (Morphism.thm phi))) co_iter_thmsss};
   135 
   136 structure Data = Generic_Data
   137 (
   138   type T = fp_sugar Symtab.table;
   139   val empty = Symtab.empty;
   140   val extend = I;
   141   val merge = Symtab.merge eq_fp_sugar;
   142 );
   143 
   144 fun fp_sugar_of ctxt =
   145   Symtab.lookup (Data.get (Context.Proof ctxt))
   146   #> Option.map (morph_fp_sugar
   147     (Morphism.thm_morphism (Thm.transfer (Proof_Context.theory_of ctxt))));
   148 
   149 fun co_induct_of (i :: _) = i;
   150 fun strong_co_induct_of [_, s] = s;
   151 
   152 fun register_fp_sugar key fp_sugar =
   153   Local_Theory.declaration {syntax = false, pervasive = true}
   154     (fn phi => Data.map (Symtab.default (key, morph_fp_sugar phi fp_sugar)));
   155 
   156 fun register_fp_sugars fp pre_bnfs nested_bnfs (fp_res as {Ts, ...}) ctr_defss ctr_sugars co_iterss
   157     co_inducts co_iter_thmsss lthy =
   158   (0, lthy)
   159   |> fold (fn T as Type (s, _) => fn (kk, lthy) => (kk + 1,
   160     register_fp_sugar s {T = T, fp = fp, index = kk, pre_bnfs = pre_bnfs,
   161         nested_bnfs = nested_bnfs, fp_res = fp_res, ctr_defss = ctr_defss, ctr_sugars = ctr_sugars,
   162         co_iterss = co_iterss, co_inducts = co_inducts, co_iter_thmsss = co_iter_thmsss}
   163       lthy)) Ts
   164   |> snd;
   165 
   166 (* This function could produce clashes in contrived examples (e.g., "x.A", "x.x_A", "y.A"). *)
   167 fun quasi_unambiguous_case_names names =
   168   let
   169     val ps = map (`Long_Name.base_name) names;
   170     val dups = Library.duplicates (op =) (map fst ps);
   171     fun underscore s =
   172       let val ss = space_explode Long_Name.separator s in
   173         space_implode "_" (drop (length ss - 2) ss)
   174       end;
   175   in
   176     map (fn (base, full) => if member (op =) dups base then underscore full else base) ps
   177   end;
   178 
   179 val id_def = @{thm id_def};
   180 val mp_conj = @{thm mp_conj};
   181 
   182 val simp_attrs = @{attributes [simp]};
   183 val code_simp_attrs = Code.add_default_eqn_attrib :: simp_attrs;
   184 
   185 fun tvar_subst thy Ts Us =
   186   Vartab.fold (cons o apsnd snd) (fold (Sign.typ_match thy) (Ts ~~ Us) Vartab.empty) [];
   187 
   188 val exists_subtype_in = Term.exists_subtype o member (op =);
   189 
   190 val lists_bmoc = fold (fn xs => fn t => Term.list_comb (t, xs));
   191 
   192 fun flat_rec_arg_args xss =
   193   (* FIXME (once the old datatype package is phased out): The first line below gives the preferred
   194      order. The second line is for compatibility with the old datatype package. *)
   195 (*
   196   flat xss
   197 *)
   198   map hd xss @ maps tl xss;
   199 
   200 fun flat_corec_predss_getterss qss fss = maps (op @) (qss ~~ fss);
   201 
   202 fun flat_corec_preds_predsss_gettersss [] [qss] [fss] = flat_corec_predss_getterss qss fss
   203   | flat_corec_preds_predsss_gettersss (p :: ps) (qss :: qsss) (fss :: fsss) =
   204     p :: flat_corec_predss_getterss qss fss @ flat_corec_preds_predsss_gettersss ps qsss fsss;
   205 
   206 fun mk_tupled_fun x f xs = HOLogic.tupled_lambda x (Term.list_comb (f, xs));
   207 fun mk_uncurried2_fun f xss =
   208   mk_tupled_fun (HOLogic.mk_tuple (map HOLogic.mk_tuple xss)) f (flat_rec_arg_args xss);
   209 
   210 fun mk_flip (x, Type (_, [T1, Type (_, [T2, T3])])) =
   211   Abs ("x", T1, Abs ("y", T2, Var (x, T2 --> T1 --> T3) $ Bound 0 $ Bound 1));
   212 
   213 fun flip_rels lthy n thm =
   214   let
   215     val Rs = Term.add_vars (prop_of thm) [];
   216     val Rs' = rev (drop (length Rs - n) Rs);
   217     val cRs = map (fn f => (certify lthy (Var f), certify lthy (mk_flip f))) Rs';
   218   in
   219     Drule.cterm_instantiate cRs thm
   220   end;
   221 
   222 fun mk_ctor_or_dtor get_T Ts t =
   223   let val Type (_, Ts0) = get_T (fastype_of t) in
   224     Term.subst_atomic_types (Ts0 ~~ Ts) t
   225   end;
   226 
   227 val mk_ctor = mk_ctor_or_dtor range_type;
   228 val mk_dtor = mk_ctor_or_dtor domain_type;
   229 
   230 fun mk_co_iter thy fp fpT Cs t =
   231   let
   232     val (f_Cs, Type (_, [prebody, body])) = strip_fun_type (fastype_of t);
   233     val fpT0 = fp_case fp prebody body;
   234     val Cs0 = distinct (op =) (map (fp_case fp body_type domain_type) f_Cs);
   235     val rho = tvar_subst thy (fpT0 :: Cs0) (fpT :: Cs);
   236   in
   237     Term.subst_TVars rho t
   238   end;
   239 
   240 fun mk_co_iters thy fp fpTs Cs ts0 =
   241   let
   242     val nn = length fpTs;
   243     val (fpTs0, Cs0) =
   244       map ((fp = Greatest_FP ? swap) o dest_funT o snd o strip_typeN nn o fastype_of) ts0
   245       |> split_list;
   246     val rho = tvar_subst thy (fpTs0 @ Cs0) (fpTs @ Cs);
   247   in
   248     map (Term.subst_TVars rho) ts0
   249   end;
   250 
   251 val mk_fp_iter_fun_types = binder_fun_types o fastype_of;
   252 
   253 fun unzip_recT Cs (T as Type (@{type_name prod}, Ts as [_, U])) =
   254     if member (op =) Cs U then Ts else [T]
   255   | unzip_recT _ T = [T];
   256 
   257 fun unzip_corecT Cs (T as Type (@{type_name sum}, Ts as [_, U])) =
   258     if member (op =) Cs U then Ts else [T]
   259   | unzip_corecT _ T = [T];
   260 
   261 fun mk_map live Ts Us t =
   262   let val (Type (_, Ts0), Type (_, Us0)) = strip_typeN (live + 1) (fastype_of t) |>> List.last in
   263     Term.subst_atomic_types (Ts0 @ Us0 ~~ Ts @ Us) t
   264   end;
   265 
   266 fun mk_rel live Ts Us t =
   267   let val [Type (_, Ts0), Type (_, Us0)] = binder_types (snd (strip_typeN live (fastype_of t))) in
   268     Term.subst_atomic_types (Ts0 @ Us0 ~~ Ts @ Us) t
   269   end;
   270 
   271 fun build_map lthy build_simple =
   272   let
   273     fun build (TU as (T, U)) =
   274       if T = U then
   275         HOLogic.id_const T
   276       else
   277         (case TU of
   278           (Type (s, Ts), Type (s', Us)) =>
   279           if s = s' then
   280             let
   281               val bnf = the (bnf_of lthy s);
   282               val live = live_of_bnf bnf;
   283               val mapx = mk_map live Ts Us (map_of_bnf bnf);
   284               val TUs' = map dest_funT (fst (strip_typeN live (fastype_of mapx)));
   285             in Term.list_comb (mapx, map build TUs') end
   286           else
   287             build_simple TU
   288         | _ => build_simple TU);
   289   in build end;
   290 
   291 fun fo_match ctxt t pat =
   292   let val thy = Proof_Context.theory_of ctxt in
   293     Pattern.first_order_match thy (pat, t) (Vartab.empty, Vartab.empty)
   294   end;
   295 
   296 val dummy_var_name = "?f"
   297 
   298 fun mk_map_pattern ctxt s =
   299   let
   300     val bnf = the (bnf_of ctxt s);
   301     val mapx = map_of_bnf bnf;
   302     val live = live_of_bnf bnf;
   303     val (f_Ts, _) = strip_typeN live (fastype_of mapx);
   304     val fs = map_index (fn (i, T) => Var ((dummy_var_name, i), T)) f_Ts;
   305   in
   306     (mapx, betapplys (mapx, fs))
   307   end;
   308 
   309 fun dest_map ctxt s call =
   310   let
   311     val (map0, pat) = mk_map_pattern ctxt s;
   312     val (_, tenv) = fo_match ctxt call pat;
   313   in
   314     (map0, Vartab.fold_rev (fn (_, (_, f)) => cons f) tenv [])
   315   end;
   316 
   317 fun dest_ctr ctxt s t =
   318   let
   319     val (f, args) = Term.strip_comb t;
   320   in
   321     (case fp_sugar_of ctxt s of
   322       SOME fp_sugar =>
   323       (case find_first (can (fo_match ctxt f)) (#ctrs (of_fp_sugar #ctr_sugars fp_sugar)) of
   324         SOME f' => (f', args)
   325       | NONE => raise Fail "dest_ctr")
   326     | NONE => raise Fail "dest_ctr")
   327   end;
   328 
   329 fun liveness_of_fp_bnf n bnf =
   330   (case T_of_bnf bnf of
   331     Type (_, Ts) => map (not o member (op =) (deads_of_bnf bnf)) Ts
   332   | _ => replicate n false);
   333 
   334 fun cannot_merge_types () = error "Mutually recursive types must have the same type parameters";
   335 
   336 fun merge_type_arg T T' = if T = T' then T else cannot_merge_types ();
   337 
   338 fun merge_type_args (As, As') =
   339   if length As = length As' then map2 merge_type_arg As As' else cannot_merge_types ();
   340 
   341 fun reassoc_conjs thm =
   342   reassoc_conjs (thm RS @{thm conj_assoc[THEN iffD1]})
   343   handle THM _ => thm;
   344 
   345 fun type_args_named_constrained_of ((((ncAs, _), _), _), _) = ncAs;
   346 fun type_binding_of ((((_, b), _), _), _) = b;
   347 fun map_binding_of (((_, (b, _)), _), _) = b;
   348 fun rel_binding_of (((_, (_, b)), _), _) = b;
   349 fun mixfix_of ((_, mx), _) = mx;
   350 fun ctr_specs_of (_, ctr_specs) = ctr_specs;
   351 
   352 fun disc_of ((((disc, _), _), _), _) = disc;
   353 fun ctr_of ((((_, ctr), _), _), _) = ctr;
   354 fun args_of (((_, args), _), _) = args;
   355 fun defaults_of ((_, ds), _) = ds;
   356 fun ctr_mixfix_of (_, mx) = mx;
   357 
   358 fun add_nesty_bnf_names Us =
   359   let
   360     fun add (Type (s, Ts)) ss =
   361         let val (needs, ss') = fold_map add Ts ss in
   362           if exists I needs then (true, insert (op =) s ss') else (false, ss')
   363         end
   364       | add T ss = (member (op =) Us T, ss);
   365   in snd oo add end;
   366 
   367 fun nesty_bnfs ctxt ctr_Tsss Us =
   368   map_filter (bnf_of ctxt) (fold (fold (fold (add_nesty_bnf_names Us))) ctr_Tsss []);
   369 
   370 fun indexify proj xs f p = f (find_index (curry op = (proj p)) xs) p;
   371 
   372 fun mk_iter_fun_arg_types0 n ms = map2 dest_tupleT ms o dest_sumTN_balanced n o domain_type;
   373 
   374 fun mk_iter_fun_arg_types Cs ns mss =
   375   mk_fp_iter_fun_types
   376   #> map3 mk_iter_fun_arg_types0 ns mss
   377   #> map (map (map (unzip_recT Cs)));
   378 
   379 fun mk_iters_args_types Cs ns mss ctor_iter_fun_Tss lthy =
   380   let
   381     val Css = map2 replicate ns Cs;
   382     val y_Tsss = map3 mk_iter_fun_arg_types0 ns mss (map un_fold_of ctor_iter_fun_Tss);
   383     val g_Tss = map2 (fn C => map (fn y_Ts => y_Ts ---> C)) Cs y_Tsss;
   384 
   385     val ((gss, ysss), lthy) =
   386       lthy
   387       |> mk_Freess "f" g_Tss
   388       ||>> mk_Freesss "x" y_Tsss;
   389 
   390     val y_Tssss = map (map (map single)) y_Tsss;
   391     val yssss = map (map (map single)) ysss;
   392 
   393     val z_Tssss =
   394       map3 (fn n => fn ms => map2 (map (unzip_recT Cs) oo dest_tupleT) ms o
   395         dest_sumTN_balanced n o domain_type o co_rec_of) ns mss ctor_iter_fun_Tss;
   396 
   397     val z_Tsss' = map (map flat_rec_arg_args) z_Tssss;
   398     val h_Tss = map2 (map2 (curry op --->)) z_Tsss' Css;
   399 
   400     val hss = map2 (map2 retype_free) h_Tss gss;
   401     val zssss_hd = map2 (map2 (map2 (retype_free o hd))) z_Tssss ysss;
   402     val (zssss_tl, lthy) =
   403       lthy
   404       |> mk_Freessss "y" (map (map (map tl)) z_Tssss);
   405     val zssss = map2 (map2 (map2 cons)) zssss_hd zssss_tl;
   406   in
   407     ([(g_Tss, y_Tssss, gss, yssss), (h_Tss, z_Tssss, hss, zssss)], lthy)
   408   end;
   409 
   410 fun mk_coiter_fun_arg_types0 Cs ns mss fun_Ts =
   411   let
   412     (*avoid "'a itself" arguments in coiterators and corecursors*)
   413     fun repair_arity [0] = [1]
   414       | repair_arity ms = ms;
   415 
   416     val f_sum_prod_Ts = map range_type fun_Ts;
   417     val f_prod_Tss = map2 dest_sumTN_balanced ns f_sum_prod_Ts;
   418     val f_Tsss = map2 (map2 dest_tupleT o repair_arity) mss f_prod_Tss;
   419     val f_Tssss = map2 (fn C => map (map (map (curry op --> C) o unzip_corecT Cs))) Cs f_Tsss;
   420     val q_Tssss = map (map (map (fn [_] => [] | [_, T] => [mk_pred1T (domain_type T)]))) f_Tssss;
   421   in
   422     (q_Tssss, f_Tsss, f_Tssss, f_sum_prod_Ts)
   423   end;
   424 
   425 fun mk_coiter_p_pred_types Cs ns = map2 (fn n => replicate (Int.max (0, n - 1)) o mk_pred1T) ns Cs;
   426 
   427 fun mk_coiter_fun_arg_types Cs ns mss dtor_coiter =
   428   (mk_coiter_p_pred_types Cs ns,
   429    mk_fp_iter_fun_types dtor_coiter |> mk_coiter_fun_arg_types0 Cs ns mss);
   430 
   431 fun mk_coiters_args_types Cs ns mss dtor_coiter_fun_Tss lthy =
   432   let
   433     val p_Tss = mk_coiter_p_pred_types Cs ns;
   434 
   435     fun mk_types get_Ts =
   436       let
   437         val fun_Ts = map get_Ts dtor_coiter_fun_Tss;
   438         val (q_Tssss, f_Tsss, f_Tssss, f_sum_prod_Ts) = mk_coiter_fun_arg_types0 Cs ns mss fun_Ts;
   439         val pf_Tss = map3 flat_corec_preds_predsss_gettersss p_Tss q_Tssss f_Tssss;
   440       in
   441         (q_Tssss, f_Tsss, f_Tssss, (f_sum_prod_Ts, pf_Tss))
   442       end;
   443 
   444     val (r_Tssss, g_Tsss, g_Tssss, unfold_types) = mk_types un_fold_of;
   445     val (s_Tssss, h_Tsss, h_Tssss, corec_types) = mk_types co_rec_of;
   446 
   447     val ((((Free (z, _), cs), pss), gssss), lthy) =
   448       lthy
   449       |> yield_singleton (mk_Frees "z") dummyT
   450       ||>> mk_Frees "a" Cs
   451       ||>> mk_Freess "p" p_Tss
   452       ||>> mk_Freessss "g" g_Tssss;
   453     val rssss = map (map (map (fn [] => []))) r_Tssss;
   454 
   455     val hssss_hd = map2 (map2 (map2 (fn T :: _ => fn [g] => retype_free T g))) h_Tssss gssss;
   456     val ((sssss, hssss_tl), lthy) =
   457       lthy
   458       |> mk_Freessss "q" s_Tssss
   459       ||>> mk_Freessss "h" (map (map (map tl)) h_Tssss);
   460     val hssss = map2 (map2 (map2 cons)) hssss_hd hssss_tl;
   461 
   462     val cpss = map2 (map o rapp) cs pss;
   463 
   464     fun build_sum_inj mk_inj = build_map lthy (uncurry mk_inj o dest_sumT o snd);
   465 
   466     fun build_dtor_coiter_arg _ [] [cf] = cf
   467       | build_dtor_coiter_arg T [cq] [cf, cf'] =
   468         mk_If cq (build_sum_inj Inl_const (fastype_of cf, T) $ cf)
   469           (build_sum_inj Inr_const (fastype_of cf', T) $ cf');
   470 
   471     fun mk_args qssss fssss f_Tsss =
   472       let
   473         val pfss = map3 flat_corec_preds_predsss_gettersss pss qssss fssss;
   474         val cqssss = map2 (map o map o map o rapp) cs qssss;
   475         val cfssss = map2 (map o map o map o rapp) cs fssss;
   476         val cqfsss = map3 (map3 (map3 build_dtor_coiter_arg)) f_Tsss cqssss cfssss;
   477       in (pfss, cqfsss) end;
   478 
   479     val unfold_args = mk_args rssss gssss g_Tsss;
   480     val corec_args = mk_args sssss hssss h_Tsss;
   481   in
   482     ((z, cs, cpss, [(unfold_args, unfold_types), (corec_args, corec_types)]), lthy)
   483   end;
   484 
   485 fun mk_co_iters_prelims fp fpTs Cs ns mss xtor_co_iterss0 lthy =
   486   let
   487     val thy = Proof_Context.theory_of lthy;
   488 
   489     val (xtor_co_iter_fun_Tss, xtor_co_iterss) =
   490       map (mk_co_iters thy fp fpTs Cs #> `(mk_fp_iter_fun_types o hd)) (transpose xtor_co_iterss0)
   491       |> apsnd transpose o apfst transpose o split_list;
   492 
   493     val ((iters_args_types, coiters_args_types), lthy') =
   494       if fp = Least_FP then
   495         mk_iters_args_types Cs ns mss xtor_co_iter_fun_Tss lthy |>> (rpair NONE o SOME)
   496       else
   497         mk_coiters_args_types Cs ns mss xtor_co_iter_fun_Tss lthy |>> (pair NONE o SOME)
   498   in
   499     ((xtor_co_iterss, iters_args_types, coiters_args_types), lthy')
   500   end;
   501 
   502 fun mk_iter_body ctor_iter fss xssss =
   503   Term.list_comb (ctor_iter, map2 (mk_sum_caseN_balanced oo map2 mk_uncurried2_fun) fss xssss);
   504 
   505 fun mk_preds_getterss_join c cps sum_prod_T cqfss =
   506   let val n = length cqfss in
   507     Term.lambda c (mk_IfN sum_prod_T cps
   508       (map2 (mk_InN_balanced sum_prod_T n) (map HOLogic.mk_tuple cqfss) (1 upto n)))
   509   end;
   510 
   511 fun mk_coiter_body cs cpss f_sum_prod_Ts cqfsss dtor_coiter =
   512   Term.list_comb (dtor_coiter, map4 mk_preds_getterss_join cs cpss f_sum_prod_Ts cqfsss);
   513 
   514 fun define_co_iters fp fpT Cs binding_specs lthy0 =
   515   let
   516     val thy = Proof_Context.theory_of lthy0;
   517 
   518     val ((csts, defs), (lthy', lthy)) = lthy0
   519       |> apfst split_list o fold_map (fn (b, spec) =>
   520         Specification.definition (SOME (b, NONE, NoSyn), ((Thm.def_binding b, []), spec))
   521         #>> apsnd snd) binding_specs
   522       ||> `Local_Theory.restore;
   523 
   524     val phi = Proof_Context.export_morphism lthy lthy';
   525 
   526     val csts' = map (mk_co_iter thy fp fpT Cs o Morphism.term phi) csts;
   527     val defs' = map (Morphism.thm phi) defs;
   528   in
   529     ((csts', defs'), lthy')
   530   end;
   531 
   532 fun define_iters iterNs iter_args_typess' mk_binding fpTs Cs ctor_iters lthy =
   533   let
   534     val nn = length fpTs;
   535 
   536     val fpT_to_C as Type (_, [fpT, _]) = snd (strip_typeN nn (fastype_of (hd ctor_iters)));
   537 
   538     fun generate_iter suf (f_Tss, _, fss, xssss) ctor_iter =
   539       let
   540         val res_T = fold_rev (curry op --->) f_Tss fpT_to_C;
   541         val b = mk_binding suf;
   542         val spec =
   543           mk_Trueprop_eq (lists_bmoc fss (Free (Binding.name_of b, res_T)),
   544             mk_iter_body ctor_iter fss xssss);
   545       in (b, spec) end;
   546   in
   547     define_co_iters Least_FP fpT Cs (map3 generate_iter iterNs iter_args_typess' ctor_iters) lthy
   548   end;
   549 
   550 fun define_coiters coiterNs (_, cs, cpss, coiter_args_typess') mk_binding fpTs Cs dtor_coiters
   551     lthy =
   552   let
   553     val nn = length fpTs;
   554 
   555     val C_to_fpT as Type (_, [_, fpT]) = snd (strip_typeN nn (fastype_of (hd dtor_coiters)));
   556 
   557     fun generate_coiter suf ((pfss, cqfsss), (f_sum_prod_Ts, pf_Tss)) dtor_coiter =
   558       let
   559         val res_T = fold_rev (curry op --->) pf_Tss C_to_fpT;
   560         val b = mk_binding suf;
   561         val spec =
   562           mk_Trueprop_eq (lists_bmoc pfss (Free (Binding.name_of b, res_T)),
   563             mk_coiter_body cs cpss f_sum_prod_Ts cqfsss dtor_coiter);
   564       in (b, spec) end;
   565   in
   566     define_co_iters Greatest_FP fpT Cs
   567       (map3 generate_coiter coiterNs coiter_args_typess' dtor_coiters) lthy
   568   end;
   569 
   570 fun derive_induct_iters_thms_for_types pre_bnfs [fold_args_types, rec_args_types] ctor_induct
   571     ctor_iter_thmss nesting_bnfs nested_bnfs fpTs Cs Xs ctrXs_Tsss ctrss ctr_defss iterss iter_defss
   572     lthy =
   573   let
   574     val iterss' = transpose iterss;
   575     val iter_defss' = transpose iter_defss;
   576 
   577     val [folds, recs] = iterss';
   578     val [fold_defs, rec_defs] = iter_defss';
   579 
   580     val ctr_Tsss = map (map (binder_types o fastype_of)) ctrss;
   581 
   582     val nn = length pre_bnfs;
   583     val ns = map length ctr_Tsss;
   584     val mss = map (map length) ctr_Tsss;
   585 
   586     val pre_map_defs = map map_def_of_bnf pre_bnfs;
   587     val pre_set_defss = map set_defs_of_bnf pre_bnfs;
   588     val nesting_map_ids'' = map (unfold_thms lthy [id_def] o map_id_of_bnf) nesting_bnfs;
   589     val nested_map_ids'' = map (unfold_thms lthy [id_def] o map_id_of_bnf) nested_bnfs;
   590     val nested_set_map's = maps set_map'_of_bnf nested_bnfs;
   591 
   592     val fp_b_names = map base_name_of_typ fpTs;
   593 
   594     val ((((ps, ps'), xsss), us'), names_lthy) =
   595       lthy
   596       |> mk_Frees' "P" (map mk_pred1T fpTs)
   597       ||>> mk_Freesss "x" ctr_Tsss
   598       ||>> Variable.variant_fixes fp_b_names;
   599 
   600     val us = map2 (curry Free) us' fpTs;
   601 
   602     fun mk_sets_nested bnf =
   603       let
   604         val Type (T_name, Us) = T_of_bnf bnf;
   605         val lives = lives_of_bnf bnf;
   606         val sets = sets_of_bnf bnf;
   607         fun mk_set U =
   608           (case find_index (curry op = U) lives of
   609             ~1 => Term.dummy
   610           | i => nth sets i);
   611       in
   612         (T_name, map mk_set Us)
   613       end;
   614 
   615     val setss_nested = map mk_sets_nested nested_bnfs;
   616 
   617     val (induct_thms, induct_thm) =
   618       let
   619         fun mk_set Ts t =
   620           let val Type (_, Ts0) = domain_type (fastype_of t) in
   621             Term.subst_atomic_types (Ts0 ~~ Ts) t
   622           end;
   623 
   624         fun mk_raw_prem_prems _ (x as Free (_, Type _)) (X as TFree _) =
   625             [([], (find_index (curry op = X) Xs + 1, x))]
   626           | mk_raw_prem_prems names_lthy (x as Free (s, Type (T_name, Ts0))) (Type (_, Xs_Ts0)) =
   627             (case AList.lookup (op =) setss_nested T_name of
   628               NONE => []
   629             | SOME raw_sets0 =>
   630               let
   631                 val (Xs_Ts, (Ts, raw_sets)) =
   632                   filter (exists_subtype_in Xs o fst) (Xs_Ts0 ~~ (Ts0 ~~ raw_sets0))
   633                   |> split_list ||> split_list;
   634                 val sets = map (mk_set Ts0) raw_sets;
   635                 val (ys, names_lthy') = names_lthy |> mk_Frees s Ts;
   636                 val xysets = map (pair x) (ys ~~ sets);
   637                 val ppremss = map2 (mk_raw_prem_prems names_lthy') ys Xs_Ts;
   638               in
   639                 flat (map2 (map o apfst o cons) xysets ppremss)
   640               end)
   641           | mk_raw_prem_prems _ _ _ = [];
   642 
   643         fun close_prem_prem xs t =
   644           fold_rev Logic.all (map Free (drop (nn + length xs)
   645             (rev (Term.add_frees t (map dest_Free xs @ ps'))))) t;
   646 
   647         fun mk_prem_prem xs (xysets, (j, x)) =
   648           close_prem_prem xs (Logic.list_implies (map (fn (x', (y, set)) =>
   649               HOLogic.mk_Trueprop (HOLogic.mk_mem (y, set $ x'))) xysets,
   650             HOLogic.mk_Trueprop (nth ps (j - 1) $ x)));
   651 
   652         fun mk_raw_prem phi ctr ctr_Ts ctrXs_Ts =
   653           let
   654             val (xs, names_lthy') = names_lthy |> mk_Frees "x" ctr_Ts;
   655             val pprems = flat (map2 (mk_raw_prem_prems names_lthy') xs ctrXs_Ts);
   656           in (xs, pprems, HOLogic.mk_Trueprop (phi $ Term.list_comb (ctr, xs))) end;
   657 
   658         fun mk_prem (xs, raw_pprems, concl) =
   659           fold_rev Logic.all xs (Logic.list_implies (map (mk_prem_prem xs) raw_pprems, concl));
   660 
   661         val raw_premss = map4 (map3 o mk_raw_prem) ps ctrss ctr_Tsss ctrXs_Tsss;
   662 
   663         val goal =
   664           Library.foldr (Logic.list_implies o apfst (map mk_prem)) (raw_premss,
   665             HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj (map2 (curry op $) ps us)));
   666 
   667         val kksss = map (map (map (fst o snd) o #2)) raw_premss;
   668 
   669         val ctor_induct' = ctor_induct OF (map mk_sumEN_tupled_balanced mss);
   670 
   671         val thm =
   672           Goal.prove_sorry lthy [] [] goal (fn {context = ctxt, ...} =>
   673             mk_induct_tac ctxt nn ns mss kksss (flat ctr_defss) ctor_induct' nested_set_map's
   674               pre_set_defss)
   675           |> singleton (Proof_Context.export names_lthy lthy)
   676           |> Thm.close_derivation;
   677       in
   678         `(conj_dests nn) thm
   679       end;
   680 
   681     val induct_cases = quasi_unambiguous_case_names (maps (map name_of_ctr) ctrss);
   682     val induct_case_names_attr = Attrib.internal (K (Rule_Cases.case_names induct_cases));
   683 
   684     val xctrss = map2 (map2 (curry Term.list_comb)) ctrss xsss;
   685 
   686     fun mk_iter_thmss (_, x_Tssss, fss, _) iters iter_defs ctor_iter_thms =
   687       let
   688         val fiters = map (lists_bmoc fss) iters;
   689 
   690         fun mk_goal fss fiter xctr f xs fxs =
   691           fold_rev (fold_rev Logic.all) (xs :: fss)
   692             (mk_Trueprop_eq (fiter $ xctr, Term.list_comb (f, fxs)));
   693 
   694         fun maybe_tick (T, U) u f =
   695           if try (fst o HOLogic.dest_prodT) U = SOME T then
   696             Term.lambda u (HOLogic.mk_prod (u, f $ u))
   697           else
   698             f;
   699 
   700         fun build_iter (x as Free (_, T)) U =
   701           if T = U then
   702             x
   703           else
   704             build_map lthy (indexify (perhaps (try (snd o HOLogic.dest_prodT)) o snd) Cs
   705               (fn kk => fn TU => maybe_tick TU (nth us kk) (nth fiters kk))) (T, U) $ x;
   706 
   707         val fxsss = map2 (map2 (flat_rec_arg_args oo map2 (map o build_iter))) xsss x_Tssss;
   708 
   709         val goalss = map5 (map4 o mk_goal fss) fiters xctrss fss xsss fxsss;
   710 
   711         val tacss =
   712           map2 (map o mk_iter_tac pre_map_defs (nested_map_ids'' @ nesting_map_ids'') iter_defs)
   713             ctor_iter_thms ctr_defss;
   714 
   715         fun prove goal tac =
   716           Goal.prove_sorry lthy [] [] goal (tac o #context)
   717           |> Thm.close_derivation;
   718       in
   719         map2 (map2 prove) goalss tacss
   720       end;
   721 
   722     val fold_thmss = mk_iter_thmss fold_args_types folds fold_defs (map un_fold_of ctor_iter_thmss);
   723     val rec_thmss = mk_iter_thmss rec_args_types recs rec_defs (map co_rec_of ctor_iter_thmss);
   724   in
   725     ((induct_thms, induct_thm, [induct_case_names_attr]),
   726      (fold_thmss, code_simp_attrs), (rec_thmss, code_simp_attrs))
   727   end;
   728 
   729 fun derive_coinduct_coiters_thms_for_types pre_bnfs (z, cs, cpss,
   730       coiters_args_types as [((pgss, crgsss), _), ((phss, cshsss), _)])
   731     dtor_coinduct dtor_injects dtor_ctors dtor_coiter_thmss nesting_bnfs fpTs Cs As kss mss ns
   732     ctr_defss ctr_sugars coiterss coiter_defss export_args lthy =
   733   let
   734     fun mk_ctor_dtor_coiter_thm dtor_inject dtor_ctor coiter =
   735       iffD1 OF [dtor_inject, trans OF [coiter, dtor_ctor RS sym]];
   736 
   737     val ctor_dtor_coiter_thmss =
   738       map3 (map oo mk_ctor_dtor_coiter_thm) dtor_injects dtor_ctors dtor_coiter_thmss;
   739 
   740     val coiterss' = transpose coiterss;
   741     val coiter_defss' = transpose coiter_defss;
   742 
   743     val [unfold_defs, corec_defs] = coiter_defss';
   744 
   745     val nn = length pre_bnfs;
   746 
   747     val pre_map_defs = map map_def_of_bnf pre_bnfs;
   748     val pre_rel_defs = map rel_def_of_bnf pre_bnfs;
   749     val nesting_map_ids'' = map (unfold_thms lthy [id_def] o map_id_of_bnf) nesting_bnfs;
   750     val nesting_rel_eqs = map rel_eq_of_bnf nesting_bnfs;
   751 
   752     val fp_b_names = map base_name_of_typ fpTs;
   753 
   754     val ctrss = map (map (mk_ctr As) o #ctrs) ctr_sugars;
   755     val discss = map (map (mk_disc_or_sel As) o #discs) ctr_sugars;
   756     val selsss = map (map (map (mk_disc_or_sel As)) o #selss) ctr_sugars;
   757     val exhausts = map #exhaust ctr_sugars;
   758     val disc_thmsss = map #disc_thmss ctr_sugars;
   759     val discIss = map #discIs ctr_sugars;
   760     val sel_thmsss = map #sel_thmss ctr_sugars;
   761 
   762     val (((rs, us'), vs'), names_lthy) =
   763       lthy
   764       |> mk_Frees "R" (map (fn T => mk_pred2T T T) fpTs)
   765       ||>> Variable.variant_fixes fp_b_names
   766       ||>> Variable.variant_fixes (map (suffix "'") fp_b_names);
   767 
   768     val us = map2 (curry Free) us' fpTs;
   769     val udiscss = map2 (map o rapp) us discss;
   770     val uselsss = map2 (map o map o rapp) us selsss;
   771 
   772     val vs = map2 (curry Free) vs' fpTs;
   773     val vdiscss = map2 (map o rapp) vs discss;
   774     val vselsss = map2 (map o map o rapp) vs selsss;
   775 
   776     val coinduct_thms_pairs =
   777       let
   778         val uvrs = map3 (fn r => fn u => fn v => r $ u $ v) rs us vs;
   779         val uv_eqs = map2 (curry HOLogic.mk_eq) us vs;
   780         val strong_rs =
   781           map4 (fn u => fn v => fn uvr => fn uv_eq =>
   782             fold_rev Term.lambda [u, v] (HOLogic.mk_disj (uvr, uv_eq))) us vs uvrs uv_eqs;
   783 
   784         (* TODO: generalize (cf. "build_map") *)
   785         fun build_rel rs' T =
   786           (case find_index (curry op = T) fpTs of
   787             ~1 =>
   788             if exists_subtype_in fpTs T then
   789               let
   790                 val Type (s, Ts) = T
   791                 val bnf = the (bnf_of lthy s);
   792                 val live = live_of_bnf bnf;
   793                 val rel = mk_rel live Ts Ts (rel_of_bnf bnf);
   794                 val Ts' = map domain_type (fst (strip_typeN live (fastype_of rel)));
   795               in Term.list_comb (rel, map (build_rel rs') Ts') end
   796             else
   797               HOLogic.eq_const T
   798           | kk => nth rs' kk);
   799 
   800         fun build_rel_app rs' usel vsel = fold rapp [usel, vsel] (build_rel rs' (fastype_of usel));
   801 
   802         fun mk_prem_ctr_concls rs' n k udisc usels vdisc vsels =
   803           (if k = n then [] else [HOLogic.mk_eq (udisc, vdisc)]) @
   804           (if null usels then
   805              []
   806            else
   807              [Library.foldr HOLogic.mk_imp (if n = 1 then [] else [udisc, vdisc],
   808                 Library.foldr1 HOLogic.mk_conj (map2 (build_rel_app rs') usels vsels))]);
   809 
   810         fun mk_prem_concl rs' n udiscs uselss vdiscs vselss =
   811           Library.foldr1 HOLogic.mk_conj
   812             (flat (map5 (mk_prem_ctr_concls rs' n) (1 upto n) udiscs uselss vdiscs vselss))
   813           handle List.Empty => @{term True};
   814 
   815         fun mk_prem rs' uvr u v n udiscs uselss vdiscs vselss =
   816           fold_rev Logic.all [u, v] (Logic.mk_implies (HOLogic.mk_Trueprop uvr,
   817             HOLogic.mk_Trueprop (mk_prem_concl rs' n udiscs uselss vdiscs vselss)));
   818 
   819         val concl =
   820           HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj
   821             (map3 (fn uvr => fn u => fn v => HOLogic.mk_imp (uvr, HOLogic.mk_eq (u, v)))
   822                uvrs us vs));
   823 
   824         fun mk_goal rs' =
   825           Logic.list_implies (map8 (mk_prem rs') uvrs us vs ns udiscss uselsss vdiscss vselsss,
   826             concl);
   827 
   828         val goals = map mk_goal [rs, strong_rs];
   829 
   830         fun prove dtor_coinduct' goal =
   831           Goal.prove_sorry lthy [] [] goal (fn {context = ctxt, ...} =>
   832             mk_coinduct_tac ctxt nesting_rel_eqs nn ns dtor_coinduct' pre_rel_defs dtor_ctors
   833               exhausts ctr_defss disc_thmsss sel_thmsss)
   834           |> singleton (Proof_Context.export names_lthy lthy)
   835           |> Thm.close_derivation;
   836 
   837         fun postproc nn thm =
   838           Thm.permute_prems 0 nn
   839             (if nn = 1 then thm RS mp else funpow nn (fn thm => reassoc_conjs (thm RS mp_conj)) thm)
   840           |> Drule.zero_var_indexes
   841           |> `(conj_dests nn);
   842 
   843         val rel_eqs = map rel_eq_of_bnf pre_bnfs;
   844         val rel_monos = map rel_mono_of_bnf pre_bnfs;
   845         val dtor_coinducts =
   846           [dtor_coinduct, mk_strong_coinduct_thm dtor_coinduct rel_eqs rel_monos lthy];
   847       in
   848         map2 (postproc nn oo prove) dtor_coinducts goals
   849       end;
   850 
   851     fun mk_coinduct_concls ms discs ctrs =
   852       let
   853         fun mk_disc_concl disc = [name_of_disc disc];
   854         fun mk_ctr_concl 0 _ = []
   855           | mk_ctr_concl _ ctor = [name_of_ctr ctor];
   856         val disc_concls = map mk_disc_concl (fst (split_last discs)) @ [[]];
   857         val ctr_concls = map2 mk_ctr_concl ms ctrs;
   858       in
   859         flat (map2 append disc_concls ctr_concls)
   860       end;
   861 
   862     val coinduct_cases = quasi_unambiguous_case_names (map (prefix EqN) fp_b_names);
   863     val coinduct_conclss =
   864       map3 (quasi_unambiguous_case_names ooo mk_coinduct_concls) mss discss ctrss;
   865 
   866     fun mk_maybe_not pos = not pos ? HOLogic.mk_not;
   867 
   868     val fcoiterss' as [gunfolds, hcorecs] =
   869       map2 (fn (pfss, _) => map (lists_bmoc pfss)) (map fst coiters_args_types) coiterss';
   870 
   871     val (unfold_thmss, corec_thmss, safe_unfold_thmss, safe_corec_thmss) =
   872       let
   873         fun mk_goal pfss c cps fcoiter n k ctr m cfs' =
   874           fold_rev (fold_rev Logic.all) ([c] :: pfss)
   875             (Logic.list_implies (seq_conds (HOLogic.mk_Trueprop oo mk_maybe_not) n k cps,
   876                mk_Trueprop_eq (fcoiter $ c, Term.list_comb (ctr, take m cfs'))));
   877 
   878         fun mk_U maybe_mk_sumT =
   879           typ_subst_nonatomic (map2 (fn C => fn fpT => (maybe_mk_sumT fpT C, fpT)) Cs fpTs);
   880 
   881         fun tack z_name (c, u) f =
   882           let val z = Free (z_name, mk_sumT (fastype_of u, fastype_of c)) in
   883             Term.lambda z (mk_sum_case (Term.lambda u u, Term.lambda c (f $ c)) $ z)
   884           end;
   885 
   886         fun build_coiter fcoiters maybe_mk_sumT maybe_tack cqf =
   887           let val T = fastype_of cqf in
   888             if exists_subtype_in Cs T then
   889               let val U = mk_U maybe_mk_sumT T in
   890                 build_map lthy (indexify snd fpTs (fn kk => fn _ =>
   891                   maybe_tack (nth cs kk, nth us kk) (nth fcoiters kk))) (T, U) $ cqf
   892               end
   893             else
   894               cqf
   895           end;
   896 
   897         val crgsss' = map (map (map (build_coiter (un_fold_of fcoiterss') (K I) (K I)))) crgsss;
   898         val cshsss' = map (map (map (build_coiter (co_rec_of fcoiterss') (curry mk_sumT) (tack z))))
   899           cshsss;
   900 
   901         val unfold_goalss = map8 (map4 oooo mk_goal pgss) cs cpss gunfolds ns kss ctrss mss crgsss';
   902         val corec_goalss = map8 (map4 oooo mk_goal phss) cs cpss hcorecs ns kss ctrss mss cshsss';
   903 
   904         val unfold_tacss =
   905           map3 (map oo mk_coiter_tac unfold_defs nesting_map_ids'')
   906             (map un_fold_of ctor_dtor_coiter_thmss) pre_map_defs ctr_defss;
   907         val corec_tacss =
   908           map3 (map oo mk_coiter_tac corec_defs nesting_map_ids'')
   909             (map co_rec_of ctor_dtor_coiter_thmss) pre_map_defs ctr_defss;
   910 
   911         fun prove goal tac =
   912           Goal.prove_sorry lthy [] [] goal (tac o #context)
   913           |> Thm.close_derivation;
   914 
   915         val unfold_thmss = map2 (map2 prove) unfold_goalss unfold_tacss;
   916         val corec_thmss =
   917           map2 (map2 prove) corec_goalss corec_tacss
   918           |> map (map (unfold_thms lthy @{thms sum_case_if}));
   919 
   920         val unfold_safesss = map2 (map2 (map2 (curry op =))) crgsss' crgsss;
   921         val corec_safesss = map2 (map2 (map2 (curry op =))) cshsss' cshsss;
   922 
   923         val filter_safesss =
   924           map2 (map_filter (fn (safes, thm) => if forall I safes then SOME thm else NONE) oo
   925             curry (op ~~));
   926 
   927         val safe_unfold_thmss = filter_safesss unfold_safesss unfold_thmss;
   928         val safe_corec_thmss = filter_safesss corec_safesss corec_thmss;
   929       in
   930         (unfold_thmss, corec_thmss, safe_unfold_thmss, safe_corec_thmss)
   931       end;
   932 
   933     val (disc_unfold_iff_thmss, disc_corec_iff_thmss) =
   934       let
   935         fun mk_goal c cps fcoiter n k disc =
   936           mk_Trueprop_eq (disc $ (fcoiter $ c),
   937             if n = 1 then @{const True}
   938             else Library.foldr1 HOLogic.mk_conj (seq_conds mk_maybe_not n k cps));
   939 
   940         val unfold_goalss = map6 (map2 oooo mk_goal) cs cpss gunfolds ns kss discss;
   941         val corec_goalss = map6 (map2 oooo mk_goal) cs cpss hcorecs ns kss discss;
   942 
   943         fun mk_case_split' cp = Drule.instantiate' [] [SOME (certify lthy cp)] @{thm case_split};
   944 
   945         val case_splitss' = map (map mk_case_split') cpss;
   946 
   947         val unfold_tacss =
   948           map3 (map oo mk_disc_coiter_iff_tac) case_splitss' unfold_thmss disc_thmsss;
   949         val corec_tacss =
   950           map3 (map oo mk_disc_coiter_iff_tac) case_splitss' corec_thmss disc_thmsss;
   951 
   952         fun prove goal tac =
   953           Goal.prove_sorry lthy [] [] goal (tac o #context)
   954           |> singleton export_args
   955           |> singleton (Proof_Context.export names_lthy lthy)
   956           |> Thm.close_derivation;
   957 
   958         fun proves [_] [_] = []
   959           | proves goals tacs = map2 prove goals tacs;
   960       in
   961         (map2 proves unfold_goalss unfold_tacss, map2 proves corec_goalss corec_tacss)
   962       end;
   963 
   964     val is_triv_discI = is_triv_implies orf is_concl_refl;
   965 
   966     fun mk_disc_coiter_thms coiters discIs =
   967       map (op RS) (filter_out (is_triv_discI o snd) (coiters ~~ discIs));
   968 
   969     val disc_unfold_thmss = map2 mk_disc_coiter_thms unfold_thmss discIss;
   970     val disc_corec_thmss = map2 mk_disc_coiter_thms corec_thmss discIss;
   971 
   972     fun mk_sel_coiter_thm coiter_thm sel sel_thm =
   973       let
   974         val (domT, ranT) = dest_funT (fastype_of sel);
   975         val arg_cong' =
   976           Drule.instantiate' (map (SOME o certifyT lthy) [domT, ranT])
   977             [NONE, NONE, SOME (certify lthy sel)] arg_cong
   978           |> Thm.varifyT_global;
   979         val sel_thm' = sel_thm RSN (2, trans);
   980       in
   981         coiter_thm RS arg_cong' RS sel_thm'
   982       end;
   983 
   984     fun mk_sel_coiter_thms coiter_thmss =
   985       map3 (map3 (map2 o mk_sel_coiter_thm)) coiter_thmss selsss sel_thmsss |> map flat;
   986 
   987     val sel_unfold_thmss = mk_sel_coiter_thms unfold_thmss;
   988     val sel_corec_thmss = mk_sel_coiter_thms corec_thmss;
   989 
   990     val coinduct_consumes_attr = Attrib.internal (K (Rule_Cases.consumes nn));
   991     val coinduct_case_names_attr = Attrib.internal (K (Rule_Cases.case_names coinduct_cases));
   992     val coinduct_case_concl_attrs =
   993       map2 (fn casex => fn concls =>
   994           Attrib.internal (K (Rule_Cases.case_conclusion (casex, concls))))
   995         coinduct_cases coinduct_conclss;
   996     val coinduct_case_attrs =
   997       coinduct_consumes_attr :: coinduct_case_names_attr :: coinduct_case_concl_attrs;
   998   in
   999     ((coinduct_thms_pairs, coinduct_case_attrs),
  1000      (unfold_thmss, corec_thmss, []),
  1001      (safe_unfold_thmss, safe_corec_thmss),
  1002      (disc_unfold_thmss, disc_corec_thmss, simp_attrs),
  1003      (disc_unfold_iff_thmss, disc_corec_iff_thmss, simp_attrs),
  1004      (sel_unfold_thmss, sel_corec_thmss, simp_attrs))
  1005   end;
  1006 
  1007 fun define_co_datatypes prepare_constraint prepare_typ prepare_term fp construct_fp
  1008     (wrap_opts as (no_discs_sels, rep_compat), specs) no_defs_lthy0 =
  1009   let
  1010     (* TODO: sanity checks on arguments *)
  1011 
  1012     val _ = if fp = Greatest_FP andalso no_discs_sels then
  1013         error "Cannot define codatatypes without discriminators and selectors"
  1014       else
  1015         ();
  1016 
  1017     fun qualify mandatory fp_b_name =
  1018       Binding.qualify mandatory fp_b_name o (rep_compat ? Binding.qualify false rep_compat_prefix);
  1019 
  1020     val nn = length specs;
  1021     val fp_bs = map type_binding_of specs;
  1022     val fp_b_names = map Binding.name_of fp_bs;
  1023     val fp_common_name = mk_common_name fp_b_names;
  1024     val map_bs = map2 (fn fp_b_name => qualify false fp_b_name o map_binding_of) fp_b_names specs;
  1025     val rel_bs = map2 (fn fp_b_name => qualify false fp_b_name o rel_binding_of) fp_b_names specs;
  1026 
  1027     fun prepare_type_arg (_, (ty, c)) =
  1028       let val TFree (s, _) = prepare_typ no_defs_lthy0 ty in
  1029         TFree (s, prepare_constraint no_defs_lthy0 c)
  1030       end;
  1031 
  1032     val Ass0 = map (map prepare_type_arg o type_args_named_constrained_of) specs;
  1033     val unsorted_Ass0 = map (map (resort_tfree HOLogic.typeS)) Ass0;
  1034     val unsorted_As = Library.foldr1 merge_type_args unsorted_Ass0;
  1035     val set_bss = map (map fst o type_args_named_constrained_of) specs;
  1036 
  1037     val _ = has_duplicates (op =) unsorted_As andalso
  1038       error ("Duplicate parameters in " ^ co_prefix fp ^ "datatype specification");
  1039 
  1040     val bad_args =
  1041       map (Logic.type_map (singleton (Variable.polymorphic no_defs_lthy0))) unsorted_As
  1042       |> filter_out Term.is_TVar;
  1043     val _ = null bad_args orelse
  1044       error ("Locally fixed type argument " ^
  1045         quote (Syntax.string_of_typ no_defs_lthy0 (hd bad_args)) ^ " in " ^ co_prefix fp ^
  1046         "datatype specification");
  1047 
  1048     val (((Bs0, Cs), Xs), no_defs_lthy) =
  1049       no_defs_lthy0
  1050       |> fold (Variable.declare_typ o resort_tfree dummyS) unsorted_As
  1051       |> mk_TFrees (length unsorted_As)
  1052       ||>> mk_TFrees nn
  1053       ||>> variant_tfrees fp_b_names;
  1054 
  1055     (* TODO: Cleaner handling of fake contexts, without "background_theory". *)
  1056 
  1057     fun add_fake_type spec =
  1058       (*The "qualify" hack below is for the case where the new type shadows an existing global type
  1059         defined in the same theory.*)
  1060       Sign.add_type no_defs_lthy (qualify false "" (type_binding_of spec),
  1061         length (type_args_named_constrained_of spec), mixfix_of spec);
  1062 
  1063     val fake_thy = fold add_fake_type specs;
  1064     val fake_lthy = Proof_Context.background_theory fake_thy no_defs_lthy;
  1065 
  1066     fun mk_fake_T b =
  1067       Type (fst (Term.dest_Type (Proof_Context.read_type_name fake_lthy true (Binding.name_of b))),
  1068         unsorted_As);
  1069 
  1070     val fake_Ts = map mk_fake_T fp_bs;
  1071 
  1072     val mixfixes = map mixfix_of specs;
  1073 
  1074     val _ = (case duplicates Binding.eq_name fp_bs of [] => ()
  1075       | b :: _ => error ("Duplicate type name declaration " ^ quote (Binding.name_of b)));
  1076 
  1077     val ctr_specss = map ctr_specs_of specs;
  1078 
  1079     val disc_bindingss = map (map disc_of) ctr_specss;
  1080     val ctr_bindingss =
  1081       map2 (fn fp_b_name => map (qualify false fp_b_name o ctr_of)) fp_b_names ctr_specss;
  1082     val ctr_argsss = map (map args_of) ctr_specss;
  1083     val ctr_mixfixess = map (map ctr_mixfix_of) ctr_specss;
  1084 
  1085     val sel_bindingsss = map (map (map fst)) ctr_argsss;
  1086     val fake_ctr_Tsss0 = map (map (map (prepare_typ fake_lthy o snd))) ctr_argsss;
  1087     val raw_sel_defaultsss = map (map defaults_of) ctr_specss;
  1088 
  1089     val (As :: _) :: fake_ctr_Tsss =
  1090       burrow (burrow (Syntax.check_typs fake_lthy)) (Ass0 :: fake_ctr_Tsss0);
  1091 
  1092     val _ = (case duplicates (op =) unsorted_As of [] => ()
  1093       | A :: _ => error ("Duplicate type parameter " ^
  1094           quote (Syntax.string_of_typ no_defs_lthy A)));
  1095 
  1096     val rhs_As' = fold (fold (fold Term.add_tfreesT)) fake_ctr_Tsss [];
  1097     val _ = (case subtract (op =) (map dest_TFree As) rhs_As' of
  1098         [] => ()
  1099       | A' :: _ => error ("Extra type variable on right-hand side: " ^
  1100           quote (Syntax.string_of_typ no_defs_lthy (TFree A'))));
  1101 
  1102     fun eq_fpT_check (T as Type (s, Us)) (Type (s', Us')) =
  1103         s = s' andalso (Us = Us' orelse error ("Illegal occurrence of recursive type " ^
  1104           quote (Syntax.string_of_typ fake_lthy T)))
  1105       | eq_fpT_check _ _ = false;
  1106 
  1107     fun freeze_fp (T as Type (s, Us)) =
  1108         (case find_index (eq_fpT_check T) fake_Ts of
  1109           ~1 => Type (s, map freeze_fp Us)
  1110         | kk => nth Xs kk)
  1111       | freeze_fp T = T;
  1112 
  1113     val ctrXs_Tsss = map (map (map freeze_fp)) fake_ctr_Tsss;
  1114     val ctrXs_sum_prod_Ts = map (mk_sumTN_balanced o map HOLogic.mk_tupleT) ctrXs_Tsss;
  1115 
  1116     val fp_eqs =
  1117       map dest_TFree Xs ~~ map (Term.typ_subst_atomic (As ~~ unsorted_As)) ctrXs_sum_prod_Ts;
  1118 
  1119     val (pre_bnfs, (fp_res as {bnfs = fp_bnfs as any_fp_bnf :: _, ctors = ctors0, dtors = dtors0,
  1120            xtor_co_iterss = xtor_co_iterss0, xtor_co_induct, dtor_ctors, ctor_dtors, ctor_injects,
  1121            dtor_injects, xtor_map_thms, xtor_set_thmss, xtor_rel_thms, xtor_co_iter_thmss, ...},
  1122            lthy)) =
  1123       fp_bnf (construct_fp mixfixes map_bs rel_bs set_bss) fp_bs (map dest_TFree unsorted_As) fp_eqs
  1124         no_defs_lthy0;
  1125 
  1126     val time = time lthy;
  1127     val timer = time (Timer.startRealTimer ());
  1128 
  1129     val nesting_bnfs = nesty_bnfs lthy ctrXs_Tsss As;
  1130     val nested_bnfs = nesty_bnfs lthy ctrXs_Tsss Xs;
  1131 
  1132     val pre_map_defs = map map_def_of_bnf pre_bnfs;
  1133     val pre_set_defss = map set_defs_of_bnf pre_bnfs;
  1134     val pre_rel_defs = map rel_def_of_bnf pre_bnfs;
  1135     val nesting_set_map's = maps set_map'_of_bnf nesting_bnfs;
  1136     val nested_set_map's = maps set_map'_of_bnf nested_bnfs;
  1137 
  1138     val live = live_of_bnf any_fp_bnf;
  1139     val _ =
  1140       if live = 0 andalso exists (not o Binding.is_empty) (map_bs @ rel_bs) then
  1141         warning ("Map function and relator names ignored")
  1142       else
  1143         ();
  1144 
  1145     val Bs =
  1146       map3 (fn alive => fn A as TFree (_, S) => fn B => if alive then resort_tfree S B else A)
  1147         (liveness_of_fp_bnf (length As) any_fp_bnf) As Bs0;
  1148 
  1149     val B_ify = Term.typ_subst_atomic (As ~~ Bs);
  1150 
  1151     val ctors = map (mk_ctor As) ctors0;
  1152     val dtors = map (mk_dtor As) dtors0;
  1153 
  1154     val fpTs = map (domain_type o fastype_of) dtors;
  1155 
  1156     fun massage_simple_notes base =
  1157       filter_out (null o #2)
  1158       #> map (fn (thmN, thms, attrs) =>
  1159         ((qualify true base (Binding.name thmN), attrs), [(thms, [])]));
  1160 
  1161     val massage_multi_notes =
  1162       maps (fn (thmN, thmss, attrs) =>
  1163         if forall null thmss then
  1164           []
  1165         else
  1166           map3 (fn fp_b_name => fn Type (T_name, _) => fn thms =>
  1167               ((qualify true fp_b_name (Binding.name thmN), attrs T_name), [(thms, [])]))
  1168             fp_b_names fpTs thmss);
  1169 
  1170     val ctr_Tsss = map (map (map (Term.typ_subst_atomic (Xs ~~ fpTs)))) ctrXs_Tsss;
  1171     val ns = map length ctr_Tsss;
  1172     val kss = map (fn n => 1 upto n) ns;
  1173     val mss = map (map length) ctr_Tsss;
  1174 
  1175     val ((xtor_co_iterss, iters_args_types, coiters_args_types), lthy') =
  1176       mk_co_iters_prelims fp fpTs Cs ns mss xtor_co_iterss0 lthy;
  1177 
  1178     fun define_ctrs_dtrs_for_type (((((((((((((((((((((((fp_bnf, fp_b), fpT), ctor), dtor),
  1179             xtor_co_iters), ctor_dtor), dtor_ctor), ctor_inject), pre_map_def), pre_set_defs),
  1180           pre_rel_def), fp_map_thm), fp_set_thms), fp_rel_thm), n), ks), ms), ctr_bindings),
  1181         ctr_mixfixes), ctr_Tss), disc_bindings), sel_bindingss), raw_sel_defaultss) no_defs_lthy =
  1182       let
  1183         val fp_b_name = Binding.name_of fp_b;
  1184 
  1185         val dtorT = domain_type (fastype_of ctor);
  1186         val ctr_prod_Ts = map HOLogic.mk_tupleT ctr_Tss;
  1187         val ctr_sum_prod_T = mk_sumTN_balanced ctr_prod_Ts;
  1188 
  1189         val ((((w, xss), yss), u'), names_lthy) =
  1190           no_defs_lthy
  1191           |> yield_singleton (mk_Frees "w") dtorT
  1192           ||>> mk_Freess "x" ctr_Tss
  1193           ||>> mk_Freess "y" (map (map B_ify) ctr_Tss)
  1194           ||>> yield_singleton Variable.variant_fixes fp_b_name;
  1195 
  1196         val u = Free (u', fpT);
  1197 
  1198         val tuple_xs = map HOLogic.mk_tuple xss;
  1199         val tuple_ys = map HOLogic.mk_tuple yss;
  1200 
  1201         val ctr_rhss =
  1202           map3 (fn k => fn xs => fn tuple_x => fold_rev Term.lambda xs (ctor $
  1203             mk_InN_balanced ctr_sum_prod_T n tuple_x k)) ks xss tuple_xs;
  1204 
  1205         val ((raw_ctrs, raw_ctr_defs), (lthy', lthy)) = no_defs_lthy
  1206           |> apfst split_list o fold_map3 (fn b => fn mx => fn rhs =>
  1207               Local_Theory.define ((b, mx), ((Thm.def_binding b, []), rhs)) #>> apsnd snd)
  1208             ctr_bindings ctr_mixfixes ctr_rhss
  1209           ||> `Local_Theory.restore;
  1210 
  1211         val phi = Proof_Context.export_morphism lthy lthy';
  1212 
  1213         val ctr_defs = map (Morphism.thm phi) raw_ctr_defs;
  1214         val ctr_defs' =
  1215           map2 (fn m => fn def => mk_unabs_def m (def RS meta_eq_to_obj_eq)) ms ctr_defs;
  1216 
  1217         val ctrs0 = map (Morphism.term phi) raw_ctrs;
  1218         val ctrs = map (mk_ctr As) ctrs0;
  1219 
  1220         fun wrap_ctrs lthy =
  1221           let
  1222             fun exhaust_tac {context = ctxt, prems = _} =
  1223               let
  1224                 val ctor_iff_dtor_thm =
  1225                   let
  1226                     val goal =
  1227                       fold_rev Logic.all [w, u]
  1228                         (mk_Trueprop_eq (HOLogic.mk_eq (u, ctor $ w), HOLogic.mk_eq (dtor $ u, w)));
  1229                   in
  1230                     Goal.prove_sorry lthy [] [] goal (fn {context = ctxt, ...} =>
  1231                       mk_ctor_iff_dtor_tac ctxt (map (SOME o certifyT lthy) [dtorT, fpT])
  1232                         (certify lthy ctor) (certify lthy dtor) ctor_dtor dtor_ctor)
  1233                     |> Thm.close_derivation
  1234                     |> Morphism.thm phi
  1235                   end;
  1236 
  1237                 val sumEN_thm' =
  1238                   unfold_thms lthy @{thms unit_all_eq1}
  1239                     (Drule.instantiate' (map (SOME o certifyT lthy) ctr_prod_Ts) []
  1240                        (mk_sumEN_balanced n))
  1241                   |> Morphism.thm phi;
  1242               in
  1243                 mk_exhaust_tac ctxt n ctr_defs ctor_iff_dtor_thm sumEN_thm'
  1244               end;
  1245 
  1246             val inject_tacss =
  1247               map2 (fn 0 => K [] | _ => fn ctr_def => [fn {context = ctxt, ...} =>
  1248                   mk_inject_tac ctxt ctr_def ctor_inject]) ms ctr_defs;
  1249 
  1250             val half_distinct_tacss =
  1251               map (map (fn (def, def') => fn {context = ctxt, ...} =>
  1252                 mk_half_distinct_tac ctxt ctor_inject [def, def'])) (mk_half_pairss (`I ctr_defs));
  1253 
  1254             val tacss = [exhaust_tac] :: inject_tacss @ half_distinct_tacss;
  1255 
  1256             val sel_defaultss = map (map (apsnd (prepare_term lthy))) raw_sel_defaultss
  1257           in
  1258             wrap_free_constructors tacss (((wrap_opts, ctrs0), standard_binding), (disc_bindings,
  1259               (sel_bindingss, sel_defaultss))) lthy
  1260           end;
  1261 
  1262         fun derive_maps_sets_rels (ctr_sugar, lthy) =
  1263           if live = 0 then
  1264             ((([], [], [], []), ctr_sugar), lthy)
  1265           else
  1266             let
  1267               val rel_flip = rel_flip_of_bnf fp_bnf;
  1268               val nones = replicate live NONE;
  1269 
  1270               val ctor_cong =
  1271                 if fp = Least_FP then
  1272                   Drule.dummy_thm
  1273                 else
  1274                   let val ctor' = mk_ctor Bs ctor in
  1275                     cterm_instantiate_pos [NONE, NONE, SOME (certify lthy ctor')] arg_cong
  1276                   end;
  1277 
  1278               fun mk_cIn ify =
  1279                 certify lthy o (fp = Greatest_FP ? curry (op $) (map_types ify ctor)) oo
  1280                 mk_InN_balanced (ify ctr_sum_prod_T) n;
  1281 
  1282               val cxIns = map2 (mk_cIn I) tuple_xs ks;
  1283               val cyIns = map2 (mk_cIn B_ify) tuple_ys ks;
  1284 
  1285               fun mk_map_thm ctr_def' cxIn =
  1286                 fold_thms lthy [ctr_def']
  1287                   (unfold_thms lthy (pre_map_def ::
  1288                        (if fp = Least_FP then [] else [ctor_dtor, dtor_ctor]) @ sum_prod_thms_map)
  1289                      (cterm_instantiate_pos (nones @ [SOME cxIn])
  1290                         (if fp = Least_FP then fp_map_thm else fp_map_thm RS ctor_cong)))
  1291                 |> singleton (Proof_Context.export names_lthy no_defs_lthy);
  1292 
  1293               fun mk_set_thm fp_set_thm ctr_def' cxIn =
  1294                 fold_thms lthy [ctr_def']
  1295                   (unfold_thms lthy (pre_set_defs @ nested_set_map's @ nesting_set_map's @
  1296                        (if fp = Least_FP then [] else [dtor_ctor]) @ sum_prod_thms_set)
  1297                      (cterm_instantiate_pos [SOME cxIn] fp_set_thm))
  1298                 |> singleton (Proof_Context.export names_lthy no_defs_lthy);
  1299 
  1300               fun mk_set_thms fp_set_thm = map2 (mk_set_thm fp_set_thm) ctr_defs' cxIns;
  1301 
  1302               val map_thms = map2 mk_map_thm ctr_defs' cxIns;
  1303               val set_thmss = map mk_set_thms fp_set_thms;
  1304 
  1305               val rel_infos = (ctr_defs' ~~ cxIns, ctr_defs' ~~ cyIns);
  1306 
  1307               fun mk_rel_thm postproc ctr_defs' cxIn cyIn =
  1308                 fold_thms lthy ctr_defs'
  1309                    (unfold_thms lthy (@{thm Inl_Inr_False} :: pre_rel_def ::
  1310                         (if fp = Least_FP then [] else [dtor_ctor]) @ sum_prod_thms_rel)
  1311                       (cterm_instantiate_pos (nones @ [SOME cxIn, SOME cyIn]) fp_rel_thm))
  1312                 |> postproc
  1313                 |> singleton (Proof_Context.export names_lthy no_defs_lthy);
  1314 
  1315               fun mk_rel_inject_thm ((ctr_def', cxIn), (_, cyIn)) =
  1316                 mk_rel_thm (unfold_thms lthy @{thms eq_sym_Unity_conv}) [ctr_def'] cxIn cyIn;
  1317 
  1318               val rel_inject_thms = map mk_rel_inject_thm (op ~~ rel_infos);
  1319 
  1320               fun mk_half_rel_distinct_thm ((xctr_def', cxIn), (yctr_def', cyIn)) =
  1321                 mk_rel_thm (fn thm => thm RS @{thm eq_False[THEN iffD1]}) [xctr_def', yctr_def']
  1322                   cxIn cyIn;
  1323 
  1324               fun mk_other_half_rel_distinct_thm thm =
  1325                 flip_rels lthy live thm
  1326                 RS (rel_flip RS sym RS @{thm arg_cong[of _ _ Not]} RS iffD2);
  1327 
  1328               val half_rel_distinct_thmss =
  1329                 map (map mk_half_rel_distinct_thm) (mk_half_pairss rel_infos);
  1330               val other_half_rel_distinct_thmss =
  1331                 map (map mk_other_half_rel_distinct_thm) half_rel_distinct_thmss;
  1332               val (rel_distinct_thms, _) =
  1333                 join_halves n half_rel_distinct_thmss other_half_rel_distinct_thmss;
  1334 
  1335               val notes =
  1336                 [(mapN, map_thms, code_simp_attrs),
  1337                  (rel_distinctN, rel_distinct_thms, code_simp_attrs),
  1338                  (rel_injectN, rel_inject_thms, code_simp_attrs),
  1339                  (setsN, flat set_thmss, code_simp_attrs)]
  1340                 |> massage_simple_notes fp_b_name;
  1341             in
  1342               (((map_thms, rel_inject_thms, rel_distinct_thms, set_thmss), ctr_sugar),
  1343                lthy |> Local_Theory.notes notes |> snd)
  1344             end;
  1345 
  1346         fun mk_binding suf = qualify false fp_b_name (Binding.suffix_name ("_" ^ suf) fp_b);
  1347 
  1348         fun massage_res (((maps_sets_rels, ctr_sugar), co_iter_res), lthy) =
  1349           (((maps_sets_rels, (ctrs, xss, ctr_defs, ctr_sugar)), co_iter_res), lthy);
  1350       in
  1351         (wrap_ctrs
  1352          #> derive_maps_sets_rels
  1353          ##>>
  1354            (if fp = Least_FP then define_iters [foldN, recN] (the iters_args_types)
  1355             else define_coiters [unfoldN, corecN] (the coiters_args_types))
  1356              mk_binding fpTs Cs xtor_co_iters
  1357          #> massage_res, lthy')
  1358       end;
  1359 
  1360     fun wrap_types_etc (wrap_types_etcs, lthy) =
  1361       fold_map I wrap_types_etcs lthy
  1362       |>> apsnd split_list o apfst (apsnd split_list4 o apfst split_list4 o split_list)
  1363         o split_list;
  1364 
  1365     val mk_simp_thmss =
  1366       map7 (fn {injects, distincts, case_thms, ...} => fn un_folds => fn co_recs =>
  1367         fn mapsx => fn rel_injects => fn rel_distincts => fn setss =>
  1368           injects @ distincts @ case_thms @ co_recs @ un_folds @ mapsx @ rel_injects
  1369           @ rel_distincts @ flat setss);
  1370 
  1371     fun derive_and_note_induct_iters_thms_for_types
  1372         ((((mapsx, rel_injects, rel_distincts, setss), (ctrss, _, ctr_defss, ctr_sugars)),
  1373           (iterss, iter_defss)), lthy) =
  1374       let
  1375         val ((induct_thms, induct_thm, induct_attrs), (fold_thmss, fold_attrs),
  1376              (rec_thmss, rec_attrs)) =
  1377           derive_induct_iters_thms_for_types pre_bnfs (the iters_args_types) xtor_co_induct
  1378             xtor_co_iter_thmss nesting_bnfs nested_bnfs fpTs Cs Xs ctrXs_Tsss ctrss ctr_defss iterss
  1379             iter_defss lthy;
  1380 
  1381         val induct_type_attr = Attrib.internal o K o Induct.induct_type;
  1382 
  1383         val simp_thmss =
  1384           mk_simp_thmss ctr_sugars fold_thmss rec_thmss mapsx rel_injects rel_distincts setss;
  1385 
  1386         val common_notes =
  1387           (if nn > 1 then [(inductN, [induct_thm], induct_attrs)] else [])
  1388           |> massage_simple_notes fp_common_name;
  1389 
  1390         val notes =
  1391           [(foldN, fold_thmss, K fold_attrs),
  1392            (inductN, map single induct_thms, fn T_name => induct_attrs @ [induct_type_attr T_name]),
  1393            (recN, rec_thmss, K rec_attrs),
  1394            (simpsN, simp_thmss, K [])]
  1395           |> massage_multi_notes;
  1396       in
  1397         lthy
  1398         |> Local_Theory.notes (common_notes @ notes) |> snd
  1399         |> register_fp_sugars Least_FP pre_bnfs nested_bnfs fp_res ctr_defss ctr_sugars iterss
  1400           [induct_thm] (transpose [fold_thmss, rec_thmss])
  1401       end;
  1402 
  1403     fun derive_and_note_coinduct_coiters_thms_for_types
  1404         ((((mapsx, rel_injects, rel_distincts, setss), (_, _, ctr_defss, ctr_sugars)),
  1405           (coiterss, coiter_defss)), lthy) =
  1406       let
  1407         val (([(coinduct_thms, coinduct_thm), (strong_coinduct_thms, strong_coinduct_thm)],
  1408               coinduct_attrs),
  1409              (unfold_thmss, corec_thmss, coiter_attrs),
  1410              (safe_unfold_thmss, safe_corec_thmss),
  1411              (disc_unfold_thmss, disc_corec_thmss, disc_coiter_attrs),
  1412              (disc_unfold_iff_thmss, disc_corec_iff_thmss, disc_coiter_iff_attrs),
  1413              (sel_unfold_thmss, sel_corec_thmss, sel_coiter_attrs)) =
  1414           derive_coinduct_coiters_thms_for_types pre_bnfs (the coiters_args_types) xtor_co_induct
  1415             dtor_injects dtor_ctors xtor_co_iter_thmss nesting_bnfs fpTs Cs As kss mss ns ctr_defss
  1416             ctr_sugars coiterss coiter_defss (Proof_Context.export lthy' no_defs_lthy) lthy;
  1417 
  1418         val coinduct_type_attr = Attrib.internal o K o Induct.coinduct_type;
  1419 
  1420         fun flat_coiter_thms coiters disc_coiters sel_coiters =
  1421           coiters @ disc_coiters @ sel_coiters;
  1422 
  1423         val simp_thmss =
  1424           mk_simp_thmss ctr_sugars
  1425             (map3 flat_coiter_thms safe_unfold_thmss disc_unfold_thmss sel_unfold_thmss)
  1426             (map3 flat_coiter_thms safe_corec_thmss disc_corec_thmss sel_corec_thmss)
  1427             mapsx rel_injects rel_distincts setss;
  1428 
  1429         val anonymous_notes =
  1430           [(flat safe_unfold_thmss @ flat safe_corec_thmss, simp_attrs)]
  1431           |> map (fn (thms, attrs) => ((Binding.empty, attrs), [(thms, [])]));
  1432 
  1433         val common_notes =
  1434           (if nn > 1 then
  1435              [(coinductN, [coinduct_thm], coinduct_attrs),
  1436               (strong_coinductN, [strong_coinduct_thm], coinduct_attrs)]
  1437            else
  1438              [])
  1439           |> massage_simple_notes fp_common_name;
  1440 
  1441         val notes =
  1442           [(coinductN, map single coinduct_thms,
  1443             fn T_name => coinduct_attrs @ [coinduct_type_attr T_name]),
  1444            (corecN, corec_thmss, K coiter_attrs),
  1445            (disc_corecN, disc_corec_thmss, K disc_coiter_attrs),
  1446            (disc_corec_iffN, disc_corec_iff_thmss, K disc_coiter_iff_attrs),
  1447            (disc_unfoldN, disc_unfold_thmss, K disc_coiter_attrs),
  1448            (disc_unfold_iffN, disc_unfold_iff_thmss, K disc_coiter_iff_attrs),
  1449            (sel_corecN, sel_corec_thmss, K sel_coiter_attrs),
  1450            (sel_unfoldN, sel_unfold_thmss, K sel_coiter_attrs),
  1451            (simpsN, simp_thmss, K []),
  1452            (strong_coinductN, map single strong_coinduct_thms, K coinduct_attrs),
  1453            (unfoldN, unfold_thmss, K coiter_attrs)]
  1454           |> massage_multi_notes;
  1455       in
  1456         lthy
  1457         |> Local_Theory.notes (anonymous_notes @ common_notes @ notes) |> snd
  1458         |> register_fp_sugars Greatest_FP pre_bnfs nested_bnfs fp_res ctr_defss ctr_sugars coiterss
  1459           [coinduct_thm, strong_coinduct_thm] (transpose [unfold_thmss, corec_thmss])
  1460       end;
  1461 
  1462     val lthy'' = lthy'
  1463       |> fold_map define_ctrs_dtrs_for_type (fp_bnfs ~~ fp_bs ~~ fpTs ~~ ctors ~~ dtors ~~
  1464         xtor_co_iterss ~~ ctor_dtors ~~ dtor_ctors ~~ ctor_injects ~~ pre_map_defs ~~
  1465         pre_set_defss ~~ pre_rel_defs ~~ xtor_map_thms ~~ xtor_set_thmss ~~ xtor_rel_thms ~~ ns ~~
  1466         kss ~~ mss ~~ ctr_bindingss ~~ ctr_mixfixess ~~ ctr_Tsss ~~ disc_bindingss ~~
  1467         sel_bindingsss ~~ raw_sel_defaultsss)
  1468       |> wrap_types_etc
  1469       |> fp_case fp derive_and_note_induct_iters_thms_for_types
  1470            derive_and_note_coinduct_coiters_thms_for_types;
  1471 
  1472     val timer = time (timer ("Constructors, discriminators, selectors, etc., for the new " ^
  1473       co_prefix fp ^ "datatype"));
  1474   in
  1475     timer; lthy''
  1476   end;
  1477 
  1478 val co_datatypes = define_co_datatypes (K I) (K I) (K I);
  1479 
  1480 val co_datatype_cmd =
  1481   define_co_datatypes Typedecl.read_constraint Syntax.parse_typ Syntax.parse_term;
  1482 
  1483 val parse_ctr_arg =
  1484   @{keyword "("} |-- parse_binding_colon -- Parse.typ --| @{keyword ")"} ||
  1485   (Parse.typ >> pair Binding.empty);
  1486 
  1487 val parse_defaults =
  1488   @{keyword "("} |-- @{keyword "defaults"} |-- Scan.repeat parse_bound_term --| @{keyword ")"};
  1489 
  1490 val parse_type_arg_constrained =
  1491   Parse.type_ident -- Scan.option (@{keyword "::"} |-- Parse.!!! Parse.sort);
  1492 
  1493 val parse_type_arg_named_constrained = parse_opt_binding_colon -- parse_type_arg_constrained;
  1494 
  1495 val parse_type_args_named_constrained =
  1496   parse_type_arg_constrained >> (single o pair Binding.empty) ||
  1497   @{keyword "("} |-- Parse.!!! (Parse.list1 parse_type_arg_named_constrained --| @{keyword ")"}) ||
  1498   Scan.succeed [];
  1499 
  1500 val parse_map_rel_binding = Parse.short_ident --| @{keyword ":"} -- parse_binding;
  1501 
  1502 val no_map_rel = (Binding.empty, Binding.empty);
  1503 
  1504 (* "map" and "rel" are purposedly not registered as keywords, because they are short and nice names
  1505    that we don't want them to be highlighted everywhere. *)
  1506 fun extract_map_rel ("map", b) = apfst (K b)
  1507   | extract_map_rel ("rel", b) = apsnd (K b)
  1508   | extract_map_rel (s, _) = error ("Expected \"map\" or \"rel\" instead of " ^ quote s);
  1509 
  1510 val parse_map_rel_bindings =
  1511   @{keyword "("} |-- Scan.repeat parse_map_rel_binding --| @{keyword ")"}
  1512     >> (fn ps => fold extract_map_rel ps no_map_rel) ||
  1513   Scan.succeed no_map_rel;
  1514 
  1515 val parse_ctr_spec =
  1516   parse_opt_binding_colon -- parse_binding -- Scan.repeat parse_ctr_arg --
  1517   Scan.optional parse_defaults [] -- Parse.opt_mixfix;
  1518 
  1519 val parse_spec =
  1520   parse_type_args_named_constrained -- parse_binding -- parse_map_rel_bindings --
  1521   Parse.opt_mixfix -- (@{keyword "="} |-- Parse.enum1 "|" parse_ctr_spec);
  1522 
  1523 val parse_co_datatype = parse_wrap_free_constructors_options -- Parse.and_list1 parse_spec;
  1524 
  1525 fun parse_co_datatype_cmd fp construct_fp = parse_co_datatype >> co_datatype_cmd fp construct_fp;
  1526 
  1527 end;