src/HOL/BNF/Tools/bnf_fp_def_sugar.ML
author traytel
Wed Aug 21 13:48:25 2013 +0200 (2013-08-21)
changeset 53126 f4d2c64c7aa8
parent 53106 d81211f61a1b
child 53143 ba80154a1118
permissions -rw-r--r--
transfer stored fp_sugar theorems into the "current" theory when retrieving an fp_sugar (avoids non-trivial merges)
     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 list -> BNF_Def.bnf list -> typ list -> typ list -> typ list ->
    74     int list list -> int list list -> int list -> thm list list -> BNF_Ctr_Sugar.ctr_sugar list ->
    75     term list list -> thm list list -> (thm list -> thm list) -> local_theory ->
    76     ((thm list * thm) list * Args.src list)
    77     * (thm list list * thm list list * Args.src list)
    78     * (thm list list * thm list list) * (thm list list * thm list list * Args.src list)
    79     * (thm list list * thm list list * Args.src list)
    80     * (thm list list * thm list list * Args.src list)
    81 
    82   val co_datatypes: BNF_FP_Util.fp_kind -> (mixfix list -> binding list -> binding list ->
    83       binding list list -> binding list -> (string * sort) list -> typ list * typ list list ->
    84       BNF_Def.bnf list -> local_theory -> BNF_FP_Util.fp_result * local_theory) ->
    85     (bool * bool) * (((((binding * (typ * sort)) list * binding) * (binding * binding)) * mixfix) *
    86       ((((binding * binding) * (binding * typ) list) * (binding * term) list) *
    87         mixfix) list) list ->
    88     local_theory -> local_theory
    89   val parse_co_datatype_cmd: BNF_FP_Util.fp_kind -> (mixfix list -> binding list -> binding list ->
    90       binding list list -> binding list -> (string * sort) list -> typ list * typ list list ->
    91       BNF_Def.bnf list -> local_theory -> BNF_FP_Util.fp_result * local_theory) ->
    92     (local_theory -> local_theory) parser
    93 end;
    94 
    95 structure BNF_FP_Def_Sugar : BNF_FP_DEF_SUGAR =
    96 struct
    97 
    98 open BNF_Util
    99 open BNF_Ctr_Sugar
   100 open BNF_Def
   101 open BNF_FP_Util
   102 open BNF_FP_Def_Sugar_Tactics
   103 
   104 val EqN = "Eq_";
   105 
   106 type fp_sugar =
   107   {T: typ,
   108    fp: fp_kind,
   109    index: int,
   110    pre_bnfs: bnf list,
   111    nested_bnfs: bnf list,
   112    fp_res: fp_result,
   113    ctr_defss: thm list list,
   114    ctr_sugars: ctr_sugar list,
   115    co_iterss: term list list,
   116    co_inducts: thm list,
   117    co_iter_thmsss: thm list list list};
   118 
   119 fun of_fp_sugar f (fp_sugar as {index, ...}) = nth (f fp_sugar) index;
   120 
   121 fun eq_fp_sugar ({T = T1, fp = fp1, index = index1, fp_res = fp_res1, ...} : fp_sugar,
   122     {T = T2, fp = fp2, index = index2, fp_res = fp_res2, ...} : fp_sugar) =
   123   T1 = T2 andalso fp1 = fp2 andalso index1 = index2 andalso eq_fp_result (fp_res1, fp_res2);
   124 
   125 fun morph_fp_sugar phi {T, fp, index, pre_bnfs, nested_bnfs, fp_res, ctr_defss, ctr_sugars,
   126     co_iterss, co_inducts, co_iter_thmsss} =
   127   {T = Morphism.typ phi T, fp = fp, index = index, pre_bnfs = map (morph_bnf phi) pre_bnfs,
   128     nested_bnfs = map (morph_bnf phi) nested_bnfs, fp_res = morph_fp_result phi fp_res,
   129    ctr_defss = map (map (Morphism.thm phi)) ctr_defss,
   130    ctr_sugars = map (morph_ctr_sugar phi) ctr_sugars,
   131    co_iterss = map (map (Morphism.term phi)) co_iterss,
   132    co_inducts = map (Morphism.thm phi) co_inducts,
   133    co_iter_thmsss = map (map (map (Morphism.thm phi))) co_iter_thmsss};
   134 
   135 structure Data = Generic_Data
   136 (
   137   type T = fp_sugar Symtab.table;
   138   val empty = Symtab.empty;
   139   val extend = I;
   140   val merge = Symtab.merge eq_fp_sugar;
   141 );
   142 
   143 fun fp_sugar_of ctxt =
   144   Symtab.lookup (Data.get (Context.Proof ctxt))
   145   #> Option.map (morph_fp_sugar
   146     (Morphism.thm_morphism (Thm.transfer (Proof_Context.theory_of ctxt))));
   147 
   148 fun co_induct_of (i :: _) = i;
   149 fun strong_co_induct_of [_, s] = s;
   150 
   151 fun register_fp_sugar key fp_sugar =
   152   Local_Theory.declaration {syntax = false, pervasive = true}
   153     (fn phi => Data.map (Symtab.default (key, morph_fp_sugar phi fp_sugar)));
   154 
   155 fun register_fp_sugars fp pre_bnfs nested_bnfs (fp_res as {Ts, ...}) ctr_defss ctr_sugars co_iterss
   156     co_inducts co_iter_thmsss lthy =
   157   (0, lthy)
   158   |> fold (fn T as Type (s, _) => fn (kk, lthy) => (kk + 1,
   159     register_fp_sugar s {T = T, fp = fp, index = kk, pre_bnfs = pre_bnfs,
   160         nested_bnfs = nested_bnfs, fp_res = fp_res, ctr_defss = ctr_defss, ctr_sugars = ctr_sugars,
   161         co_iterss = co_iterss, co_inducts = co_inducts, co_iter_thmsss = co_iter_thmsss}
   162       lthy)) Ts
   163   |> snd;
   164 
   165 (* This function could produce clashes in contrived examples (e.g., "x.A", "x.x_A", "y.A"). *)
   166 fun quasi_unambiguous_case_names names =
   167   let
   168     val ps = map (`Long_Name.base_name) names;
   169     val dups = Library.duplicates (op =) (map fst ps);
   170     fun underscore s =
   171       let val ss = space_explode Long_Name.separator s in
   172         space_implode "_" (drop (length ss - 2) ss)
   173       end;
   174   in
   175     map (fn (base, full) => if member (op =) dups base then underscore full else base) ps
   176   end;
   177 
   178 val id_def = @{thm id_def};
   179 val mp_conj = @{thm mp_conj};
   180 
   181 val simp_attrs = @{attributes [simp]};
   182 val code_simp_attrs = Code.add_default_eqn_attrib :: simp_attrs;
   183 
   184 fun tvar_subst thy Ts Us =
   185   Vartab.fold (cons o apsnd snd) (fold (Sign.typ_match thy) (Ts ~~ Us) Vartab.empty) [];
   186 
   187 val exists_subtype_in = Term.exists_subtype o member (op =);
   188 
   189 val lists_bmoc = fold (fn xs => fn t => Term.list_comb (t, xs));
   190 
   191 fun flat_rec_arg_args xss =
   192   (* FIXME (once the old datatype package is phased out): The first line below gives the preferred
   193      order. The second line is for compatibility with the old datatype package. *)
   194 (*
   195   flat xss
   196 *)
   197   map hd xss @ maps tl xss;
   198 
   199 fun flat_corec_predss_getterss qss fss = maps (op @) (qss ~~ fss);
   200 
   201 fun flat_corec_preds_predsss_gettersss [] [qss] [fss] = flat_corec_predss_getterss qss fss
   202   | flat_corec_preds_predsss_gettersss (p :: ps) (qss :: qsss) (fss :: fsss) =
   203     p :: flat_corec_predss_getterss qss fss @ flat_corec_preds_predsss_gettersss ps qsss fsss;
   204 
   205 fun mk_tupled_fun x f xs = HOLogic.tupled_lambda x (Term.list_comb (f, xs));
   206 fun mk_uncurried2_fun f xss =
   207   mk_tupled_fun (HOLogic.mk_tuple (map HOLogic.mk_tuple xss)) f (flat_rec_arg_args xss);
   208 
   209 fun mk_flip (x, Type (_, [T1, Type (_, [T2, T3])])) =
   210   Abs ("x", T1, Abs ("y", T2, Var (x, T2 --> T1 --> T3) $ Bound 0 $ Bound 1));
   211 
   212 fun flip_rels lthy n thm =
   213   let
   214     val Rs = Term.add_vars (prop_of thm) [];
   215     val Rs' = rev (drop (length Rs - n) Rs);
   216     val cRs = map (fn f => (certify lthy (Var f), certify lthy (mk_flip f))) Rs';
   217   in
   218     Drule.cterm_instantiate cRs thm
   219   end;
   220 
   221 fun mk_ctor_or_dtor get_T Ts t =
   222   let val Type (_, Ts0) = get_T (fastype_of t) in
   223     Term.subst_atomic_types (Ts0 ~~ Ts) t
   224   end;
   225 
   226 val mk_ctor = mk_ctor_or_dtor range_type;
   227 val mk_dtor = mk_ctor_or_dtor domain_type;
   228 
   229 fun mk_co_iter thy fp fpT Cs t =
   230   let
   231     val (f_Cs, Type (_, [prebody, body])) = strip_fun_type (fastype_of t);
   232     val fpT0 = fp_case fp prebody body;
   233     val Cs0 = distinct (op =) (map (fp_case fp body_type domain_type) f_Cs);
   234     val rho = tvar_subst thy (fpT0 :: Cs0) (fpT :: Cs);
   235   in
   236     Term.subst_TVars rho t
   237   end;
   238 
   239 fun mk_co_iters thy fp fpTs Cs ts0 =
   240   let
   241     val nn = length fpTs;
   242     val (fpTs0, Cs0) =
   243       map ((fp = Greatest_FP ? swap) o dest_funT o snd o strip_typeN nn o fastype_of) ts0
   244       |> split_list;
   245     val rho = tvar_subst thy (fpTs0 @ Cs0) (fpTs @ Cs);
   246   in
   247     map (Term.subst_TVars rho) ts0
   248   end;
   249 
   250 val mk_fp_iter_fun_types = binder_fun_types o fastype_of;
   251 
   252 fun unzip_recT Cs (T as Type (@{type_name prod}, Ts as [_, U])) =
   253     if member (op =) Cs U then Ts else [T]
   254   | unzip_recT _ T = [T];
   255 
   256 fun unzip_corecT Cs (T as Type (@{type_name sum}, Ts as [_, U])) =
   257     if member (op =) Cs U then Ts else [T]
   258   | unzip_corecT _ T = [T];
   259 
   260 fun mk_map live Ts Us t =
   261   let val (Type (_, Ts0), Type (_, Us0)) = strip_typeN (live + 1) (fastype_of t) |>> List.last in
   262     Term.subst_atomic_types (Ts0 @ Us0 ~~ Ts @ Us) t
   263   end;
   264 
   265 fun mk_rel live Ts Us t =
   266   let val [Type (_, Ts0), Type (_, Us0)] = binder_types (snd (strip_typeN live (fastype_of t))) in
   267     Term.subst_atomic_types (Ts0 @ Us0 ~~ Ts @ Us) t
   268   end;
   269 
   270 fun build_map lthy build_simple =
   271   let
   272     fun build (TU as (T, U)) =
   273       if T = U then
   274         HOLogic.id_const T
   275       else
   276         (case TU of
   277           (Type (s, Ts), Type (s', Us)) =>
   278           if s = s' then
   279             let
   280               val bnf = the (bnf_of lthy s);
   281               val live = live_of_bnf bnf;
   282               val mapx = mk_map live Ts Us (map_of_bnf bnf);
   283               val TUs' = map dest_funT (fst (strip_typeN live (fastype_of mapx)));
   284             in Term.list_comb (mapx, map build TUs') end
   285           else
   286             build_simple TU
   287         | _ => build_simple TU);
   288   in build end;
   289 
   290 fun fo_match ctxt t pat =
   291   let val thy = Proof_Context.theory_of ctxt in
   292     Pattern.first_order_match thy (pat, t) (Vartab.empty, Vartab.empty)
   293   end;
   294 
   295 val dummy_var_name = "?f"
   296 
   297 fun mk_map_pattern ctxt s =
   298   let
   299     val bnf = the (bnf_of ctxt s);
   300     val mapx = map_of_bnf bnf;
   301     val live = live_of_bnf bnf;
   302     val (f_Ts, _) = strip_typeN live (fastype_of mapx);
   303     val fs = map_index (fn (i, T) => Var ((dummy_var_name, i), T)) f_Ts;
   304   in
   305     (mapx, betapplys (mapx, fs))
   306   end;
   307 
   308 fun dest_map ctxt s call =
   309   let
   310     val (map0, pat) = mk_map_pattern ctxt s;
   311     val (_, tenv) = fo_match ctxt call pat;
   312   in
   313     (map0, Vartab.fold_rev (fn (_, (_, f)) => cons f) tenv [])
   314   end;
   315 
   316 fun dest_ctr ctxt s t =
   317   let
   318     val (f, args) = Term.strip_comb t;
   319   in
   320     (case fp_sugar_of ctxt s of
   321       SOME fp_sugar =>
   322       (case find_first (can (fo_match ctxt f)) (#ctrs (of_fp_sugar #ctr_sugars fp_sugar)) of
   323         SOME f' => (f', args)
   324       | NONE => raise Fail "dest_ctr")
   325     | NONE => raise Fail "dest_ctr")
   326   end;
   327 
   328 fun liveness_of_fp_bnf n bnf =
   329   (case T_of_bnf bnf of
   330     Type (_, Ts) => map (not o member (op =) (deads_of_bnf bnf)) Ts
   331   | _ => replicate n false);
   332 
   333 fun cannot_merge_types () = error "Mutually recursive types must have the same type parameters";
   334 
   335 fun merge_type_arg T T' = if T = T' then T else cannot_merge_types ();
   336 
   337 fun merge_type_args (As, As') =
   338   if length As = length As' then map2 merge_type_arg As As' else cannot_merge_types ();
   339 
   340 fun reassoc_conjs thm =
   341   reassoc_conjs (thm RS @{thm conj_assoc[THEN iffD1]})
   342   handle THM _ => thm;
   343 
   344 fun type_args_named_constrained_of ((((ncAs, _), _), _), _) = ncAs;
   345 fun type_binding_of ((((_, b), _), _), _) = b;
   346 fun map_binding_of (((_, (b, _)), _), _) = b;
   347 fun rel_binding_of (((_, (_, b)), _), _) = b;
   348 fun mixfix_of ((_, mx), _) = mx;
   349 fun ctr_specs_of (_, ctr_specs) = ctr_specs;
   350 
   351 fun disc_of ((((disc, _), _), _), _) = disc;
   352 fun ctr_of ((((_, ctr), _), _), _) = ctr;
   353 fun args_of (((_, args), _), _) = args;
   354 fun defaults_of ((_, ds), _) = ds;
   355 fun ctr_mixfix_of (_, mx) = mx;
   356 
   357 fun add_nesty_bnf_names Us =
   358   let
   359     fun add (Type (s, Ts)) ss =
   360         let val (needs, ss') = fold_map add Ts ss in
   361           if exists I needs then (true, insert (op =) s ss') else (false, ss')
   362         end
   363       | add T ss = (member (op =) Us T, ss);
   364   in snd oo add end;
   365 
   366 fun nesty_bnfs ctxt ctr_Tsss Us =
   367   map_filter (bnf_of ctxt) (fold (fold (fold (add_nesty_bnf_names Us))) ctr_Tsss []);
   368 
   369 fun indexify proj xs f p = f (find_index (curry op = (proj p)) xs) p;
   370 
   371 fun mk_iter_fun_arg_types0 n ms = map2 dest_tupleT ms o dest_sumTN_balanced n o domain_type;
   372 
   373 fun mk_iter_fun_arg_types Cs ns mss =
   374   mk_fp_iter_fun_types
   375   #> map3 mk_iter_fun_arg_types0 ns mss
   376   #> map (map (map (unzip_recT Cs)));
   377 
   378 fun mk_iters_args_types Cs ns mss ctor_iter_fun_Tss lthy =
   379   let
   380     val Css = map2 replicate ns Cs;
   381     val y_Tsss = map3 mk_iter_fun_arg_types0 ns mss (map un_fold_of ctor_iter_fun_Tss);
   382     val g_Tss = map2 (fn C => map (fn y_Ts => y_Ts ---> C)) Cs y_Tsss;
   383 
   384     val ((gss, ysss), lthy) =
   385       lthy
   386       |> mk_Freess "f" g_Tss
   387       ||>> mk_Freesss "x" y_Tsss;
   388 
   389     val y_Tssss = map (map (map single)) y_Tsss;
   390     val yssss = map (map (map single)) ysss;
   391 
   392     val z_Tssss =
   393       map3 (fn n => fn ms => map2 (map (unzip_recT Cs) oo dest_tupleT) ms o
   394         dest_sumTN_balanced n o domain_type o co_rec_of) ns mss ctor_iter_fun_Tss;
   395 
   396     val z_Tsss' = map (map flat_rec_arg_args) z_Tssss;
   397     val h_Tss = map2 (map2 (curry op --->)) z_Tsss' Css;
   398 
   399     val hss = map2 (map2 retype_free) h_Tss gss;
   400     val zssss_hd = map2 (map2 (map2 (retype_free o hd))) z_Tssss ysss;
   401     val (zssss_tl, lthy) =
   402       lthy
   403       |> mk_Freessss "y" (map (map (map tl)) z_Tssss);
   404     val zssss = map2 (map2 (map2 cons)) zssss_hd zssss_tl;
   405   in
   406     ([(g_Tss, y_Tssss, gss, yssss), (h_Tss, z_Tssss, hss, zssss)], lthy)
   407   end;
   408 
   409 fun mk_coiter_fun_arg_types0 Cs ns mss fun_Ts =
   410   let
   411     (*avoid "'a itself" arguments in coiterators and corecursors*)
   412     fun repair_arity [0] = [1]
   413       | repair_arity ms = ms;
   414 
   415     val f_sum_prod_Ts = map range_type fun_Ts;
   416     val f_prod_Tss = map2 dest_sumTN_balanced ns f_sum_prod_Ts;
   417     val f_Tsss = map2 (map2 dest_tupleT o repair_arity) mss f_prod_Tss;
   418     val f_Tssss = map2 (fn C => map (map (map (curry op --> C) o unzip_corecT Cs))) Cs f_Tsss;
   419     val q_Tssss = map (map (map (fn [_] => [] | [_, T] => [mk_pred1T (domain_type T)]))) f_Tssss;
   420   in
   421     (q_Tssss, f_Tsss, f_Tssss, f_sum_prod_Ts)
   422   end;
   423 
   424 fun mk_coiter_p_pred_types Cs ns = map2 (fn n => replicate (Int.max (0, n - 1)) o mk_pred1T) ns Cs;
   425 
   426 fun mk_coiter_fun_arg_types Cs ns mss dtor_coiter =
   427   (mk_coiter_p_pred_types Cs ns,
   428    mk_fp_iter_fun_types dtor_coiter |> mk_coiter_fun_arg_types0 Cs ns mss);
   429 
   430 fun mk_coiters_args_types Cs ns mss dtor_coiter_fun_Tss lthy =
   431   let
   432     val p_Tss = mk_coiter_p_pred_types Cs ns;
   433 
   434     fun mk_types get_Ts =
   435       let
   436         val fun_Ts = map get_Ts dtor_coiter_fun_Tss;
   437         val (q_Tssss, f_Tsss, f_Tssss, f_sum_prod_Ts) = mk_coiter_fun_arg_types0 Cs ns mss fun_Ts;
   438         val pf_Tss = map3 flat_corec_preds_predsss_gettersss p_Tss q_Tssss f_Tssss;
   439       in
   440         (q_Tssss, f_Tsss, f_Tssss, (f_sum_prod_Ts, pf_Tss))
   441       end;
   442 
   443     val (r_Tssss, g_Tsss, g_Tssss, unfold_types) = mk_types un_fold_of;
   444     val (s_Tssss, h_Tsss, h_Tssss, corec_types) = mk_types co_rec_of;
   445 
   446     val ((((Free (z, _), cs), pss), gssss), lthy) =
   447       lthy
   448       |> yield_singleton (mk_Frees "z") dummyT
   449       ||>> mk_Frees "a" Cs
   450       ||>> mk_Freess "p" p_Tss
   451       ||>> mk_Freessss "g" g_Tssss;
   452     val rssss = map (map (map (fn [] => []))) r_Tssss;
   453 
   454     val hssss_hd = map2 (map2 (map2 (fn T :: _ => fn [g] => retype_free T g))) h_Tssss gssss;
   455     val ((sssss, hssss_tl), lthy) =
   456       lthy
   457       |> mk_Freessss "q" s_Tssss
   458       ||>> mk_Freessss "h" (map (map (map tl)) h_Tssss);
   459     val hssss = map2 (map2 (map2 cons)) hssss_hd hssss_tl;
   460 
   461     val cpss = map2 (map o rapp) cs pss;
   462 
   463     fun build_sum_inj mk_inj = build_map lthy (uncurry mk_inj o dest_sumT o snd);
   464 
   465     fun build_dtor_coiter_arg _ [] [cf] = cf
   466       | build_dtor_coiter_arg T [cq] [cf, cf'] =
   467         mk_If cq (build_sum_inj Inl_const (fastype_of cf, T) $ cf)
   468           (build_sum_inj Inr_const (fastype_of cf', T) $ cf');
   469 
   470     fun mk_args qssss fssss f_Tsss =
   471       let
   472         val pfss = map3 flat_corec_preds_predsss_gettersss pss qssss fssss;
   473         val cqssss = map2 (map o map o map o rapp) cs qssss;
   474         val cfssss = map2 (map o map o map o rapp) cs fssss;
   475         val cqfsss = map3 (map3 (map3 build_dtor_coiter_arg)) f_Tsss cqssss cfssss;
   476       in (pfss, cqfsss) end;
   477 
   478     val unfold_args = mk_args rssss gssss g_Tsss;
   479     val corec_args = mk_args sssss hssss h_Tsss;
   480   in
   481     ((z, cs, cpss, [(unfold_args, unfold_types), (corec_args, corec_types)]), lthy)
   482   end;
   483 
   484 fun mk_co_iters_prelims fp fpTs Cs ns mss xtor_co_iterss0 lthy =
   485   let
   486     val thy = Proof_Context.theory_of lthy;
   487 
   488     val (xtor_co_iter_fun_Tss, xtor_co_iterss) =
   489       map (mk_co_iters thy fp fpTs Cs #> `(mk_fp_iter_fun_types o hd)) (transpose xtor_co_iterss0)
   490       |> apsnd transpose o apfst transpose o split_list;
   491 
   492     val ((iters_args_types, coiters_args_types), lthy') =
   493       if fp = Least_FP then
   494         mk_iters_args_types Cs ns mss xtor_co_iter_fun_Tss lthy |>> (rpair NONE o SOME)
   495       else
   496         mk_coiters_args_types Cs ns mss xtor_co_iter_fun_Tss lthy |>> (pair NONE o SOME)
   497   in
   498     ((xtor_co_iterss, iters_args_types, coiters_args_types), lthy')
   499   end;
   500 
   501 fun mk_iter_body ctor_iter fss xssss =
   502   Term.list_comb (ctor_iter, map2 (mk_sum_caseN_balanced oo map2 mk_uncurried2_fun) fss xssss);
   503 
   504 fun mk_preds_getterss_join c cps sum_prod_T cqfss =
   505   let val n = length cqfss in
   506     Term.lambda c (mk_IfN sum_prod_T cps
   507       (map2 (mk_InN_balanced sum_prod_T n) (map HOLogic.mk_tuple cqfss) (1 upto n)))
   508   end;
   509 
   510 fun mk_coiter_body cs cpss f_sum_prod_Ts cqfsss dtor_coiter =
   511   Term.list_comb (dtor_coiter, map4 mk_preds_getterss_join cs cpss f_sum_prod_Ts cqfsss);
   512 
   513 fun define_co_iters fp fpT Cs binding_specs lthy0 =
   514   let
   515     val thy = Proof_Context.theory_of lthy0;
   516 
   517     val ((csts, defs), (lthy', lthy)) = lthy0
   518       |> apfst split_list o fold_map (fn (b, spec) =>
   519         Specification.definition (SOME (b, NONE, NoSyn), ((Thm.def_binding b, []), spec))
   520         #>> apsnd snd) binding_specs
   521       ||> `Local_Theory.restore;
   522 
   523     val phi = Proof_Context.export_morphism lthy lthy';
   524 
   525     val csts' = map (mk_co_iter thy fp fpT Cs o Morphism.term phi) csts;
   526     val defs' = map (Morphism.thm phi) defs;
   527   in
   528     ((csts', defs'), lthy')
   529   end;
   530 
   531 fun define_iters iterNs iter_args_typess' mk_binding fpTs Cs ctor_iters lthy =
   532   let
   533     val nn = length fpTs;
   534 
   535     val fpT_to_C as Type (_, [fpT, _]) = snd (strip_typeN nn (fastype_of (hd ctor_iters)));
   536 
   537     fun generate_iter suf (f_Tss, _, fss, xssss) ctor_iter =
   538       let
   539         val res_T = fold_rev (curry op --->) f_Tss fpT_to_C;
   540         val b = mk_binding suf;
   541         val spec =
   542           mk_Trueprop_eq (lists_bmoc fss (Free (Binding.name_of b, res_T)),
   543             mk_iter_body ctor_iter fss xssss);
   544       in (b, spec) end;
   545   in
   546     define_co_iters Least_FP fpT Cs (map3 generate_iter iterNs iter_args_typess' ctor_iters) lthy
   547   end;
   548 
   549 fun define_coiters coiterNs (_, cs, cpss, coiter_args_typess') mk_binding fpTs Cs dtor_coiters
   550     lthy =
   551   let
   552     val nn = length fpTs;
   553 
   554     val C_to_fpT as Type (_, [_, fpT]) = snd (strip_typeN nn (fastype_of (hd dtor_coiters)));
   555 
   556     fun generate_coiter suf ((pfss, cqfsss), (f_sum_prod_Ts, pf_Tss)) dtor_coiter =
   557       let
   558         val res_T = fold_rev (curry op --->) pf_Tss C_to_fpT;
   559         val b = mk_binding suf;
   560         val spec =
   561           mk_Trueprop_eq (lists_bmoc pfss (Free (Binding.name_of b, res_T)),
   562             mk_coiter_body cs cpss f_sum_prod_Ts cqfsss dtor_coiter);
   563       in (b, spec) end;
   564   in
   565     define_co_iters Greatest_FP fpT Cs
   566       (map3 generate_coiter coiterNs coiter_args_typess' dtor_coiters) lthy
   567   end;
   568 
   569 fun derive_induct_iters_thms_for_types pre_bnfs [fold_args_types, rec_args_types] ctor_induct
   570     ctor_iter_thmss nesting_bnfs nested_bnfs fpTs Cs Xs ctrXs_Tsss ctrss ctr_defss iterss iter_defss
   571     lthy =
   572   let
   573     val iterss' = transpose iterss;
   574     val iter_defss' = transpose iter_defss;
   575 
   576     val [folds, recs] = iterss';
   577     val [fold_defs, rec_defs] = iter_defss';
   578 
   579     val ctr_Tsss = map (map (binder_types o fastype_of)) ctrss;
   580 
   581     val nn = length pre_bnfs;
   582     val ns = map length ctr_Tsss;
   583     val mss = map (map length) ctr_Tsss;
   584 
   585     val pre_map_defs = map map_def_of_bnf pre_bnfs;
   586     val pre_set_defss = map set_defs_of_bnf pre_bnfs;
   587     val nesting_map_ids'' = map (unfold_thms lthy [id_def] o map_id_of_bnf) nesting_bnfs;
   588     val nested_map_ids'' = map (unfold_thms lthy [id_def] o map_id_of_bnf) nested_bnfs;
   589     val nested_set_map's = maps set_map'_of_bnf nested_bnfs;
   590 
   591     val fp_b_names = map base_name_of_typ fpTs;
   592 
   593     val ((((ps, ps'), xsss), us'), names_lthy) =
   594       lthy
   595       |> mk_Frees' "P" (map mk_pred1T fpTs)
   596       ||>> mk_Freesss "x" ctr_Tsss
   597       ||>> Variable.variant_fixes fp_b_names;
   598 
   599     val us = map2 (curry Free) us' fpTs;
   600 
   601     fun mk_sets_nested bnf =
   602       let
   603         val Type (T_name, Us) = T_of_bnf bnf;
   604         val lives = lives_of_bnf bnf;
   605         val sets = sets_of_bnf bnf;
   606         fun mk_set U =
   607           (case find_index (curry op = U) lives of
   608             ~1 => Term.dummy
   609           | i => nth sets i);
   610       in
   611         (T_name, map mk_set Us)
   612       end;
   613 
   614     val setss_nested = map mk_sets_nested nested_bnfs;
   615 
   616     val (induct_thms, induct_thm) =
   617       let
   618         fun mk_set Ts t =
   619           let val Type (_, Ts0) = domain_type (fastype_of t) in
   620             Term.subst_atomic_types (Ts0 ~~ Ts) t
   621           end;
   622 
   623         fun mk_raw_prem_prems _ (x as Free (_, Type _)) (X as TFree _) =
   624             [([], (find_index (curry op = X) Xs + 1, x))]
   625           | mk_raw_prem_prems names_lthy (x as Free (s, Type (T_name, Ts0))) (Type (_, Xs_Ts0)) =
   626             (case AList.lookup (op =) setss_nested T_name of
   627               NONE => []
   628             | SOME raw_sets0 =>
   629               let
   630                 val (Xs_Ts, (Ts, raw_sets)) =
   631                   filter (exists_subtype_in Xs o fst) (Xs_Ts0 ~~ (Ts0 ~~ raw_sets0))
   632                   |> split_list ||> split_list;
   633                 val sets = map (mk_set Ts0) raw_sets;
   634                 val (ys, names_lthy') = names_lthy |> mk_Frees s Ts;
   635                 val xysets = map (pair x) (ys ~~ sets);
   636                 val ppremss = map2 (mk_raw_prem_prems names_lthy') ys Xs_Ts;
   637               in
   638                 flat (map2 (map o apfst o cons) xysets ppremss)
   639               end)
   640           | mk_raw_prem_prems _ _ _ = [];
   641 
   642         fun close_prem_prem xs t =
   643           fold_rev Logic.all (map Free (drop (nn + length xs)
   644             (rev (Term.add_frees t (map dest_Free xs @ ps'))))) t;
   645 
   646         fun mk_prem_prem xs (xysets, (j, x)) =
   647           close_prem_prem xs (Logic.list_implies (map (fn (x', (y, set)) =>
   648               HOLogic.mk_Trueprop (HOLogic.mk_mem (y, set $ x'))) xysets,
   649             HOLogic.mk_Trueprop (nth ps (j - 1) $ x)));
   650 
   651         fun mk_raw_prem phi ctr ctr_Ts ctrXs_Ts =
   652           let
   653             val (xs, names_lthy') = names_lthy |> mk_Frees "x" ctr_Ts;
   654             val pprems = flat (map2 (mk_raw_prem_prems names_lthy') xs ctrXs_Ts);
   655           in (xs, pprems, HOLogic.mk_Trueprop (phi $ Term.list_comb (ctr, xs))) end;
   656 
   657         fun mk_prem (xs, raw_pprems, concl) =
   658           fold_rev Logic.all xs (Logic.list_implies (map (mk_prem_prem xs) raw_pprems, concl));
   659 
   660         val raw_premss = map4 (map3 o mk_raw_prem) ps ctrss ctr_Tsss ctrXs_Tsss;
   661 
   662         val goal =
   663           Library.foldr (Logic.list_implies o apfst (map mk_prem)) (raw_premss,
   664             HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj (map2 (curry op $) ps us)));
   665 
   666         val kksss = map (map (map (fst o snd) o #2)) raw_premss;
   667 
   668         val ctor_induct' = ctor_induct OF (map mk_sumEN_tupled_balanced mss);
   669 
   670         val thm =
   671           Goal.prove_sorry lthy [] [] goal (fn {context = ctxt, ...} =>
   672             mk_induct_tac ctxt nn ns mss kksss (flat ctr_defss) ctor_induct' nested_set_map's
   673               pre_set_defss)
   674           |> singleton (Proof_Context.export names_lthy lthy)
   675           |> Thm.close_derivation;
   676       in
   677         `(conj_dests nn) thm
   678       end;
   679 
   680     val induct_cases = quasi_unambiguous_case_names (maps (map name_of_ctr) ctrss);
   681     val induct_case_names_attr = Attrib.internal (K (Rule_Cases.case_names induct_cases));
   682 
   683     val xctrss = map2 (map2 (curry Term.list_comb)) ctrss xsss;
   684 
   685     fun mk_iter_thmss (_, x_Tssss, fss, _) iters iter_defs ctor_iter_thms =
   686       let
   687         val fiters = map (lists_bmoc fss) iters;
   688 
   689         fun mk_goal fss fiter xctr f xs fxs =
   690           fold_rev (fold_rev Logic.all) (xs :: fss)
   691             (mk_Trueprop_eq (fiter $ xctr, Term.list_comb (f, fxs)));
   692 
   693         fun maybe_tick (T, U) u f =
   694           if try (fst o HOLogic.dest_prodT) U = SOME T then
   695             Term.lambda u (HOLogic.mk_prod (u, f $ u))
   696           else
   697             f;
   698 
   699         fun build_iter (x as Free (_, T)) U =
   700           if T = U then
   701             x
   702           else
   703             build_map lthy (indexify (perhaps (try (snd o HOLogic.dest_prodT)) o snd) Cs
   704               (fn kk => fn TU => maybe_tick TU (nth us kk) (nth fiters kk))) (T, U) $ x;
   705 
   706         val fxsss = map2 (map2 (flat_rec_arg_args oo map2 (map o build_iter))) xsss x_Tssss;
   707 
   708         val goalss = map5 (map4 o mk_goal fss) fiters xctrss fss xsss fxsss;
   709 
   710         val tacss =
   711           map2 (map o mk_iter_tac pre_map_defs (nested_map_ids'' @ nesting_map_ids'') iter_defs)
   712             ctor_iter_thms ctr_defss;
   713 
   714         fun prove goal tac =
   715           Goal.prove_sorry lthy [] [] goal (tac o #context)
   716           |> Thm.close_derivation;
   717       in
   718         map2 (map2 prove) goalss tacss
   719       end;
   720 
   721     val fold_thmss = mk_iter_thmss fold_args_types folds fold_defs (map un_fold_of ctor_iter_thmss);
   722     val rec_thmss = mk_iter_thmss rec_args_types recs rec_defs (map co_rec_of ctor_iter_thmss);
   723   in
   724     ((induct_thms, induct_thm, [induct_case_names_attr]),
   725      (fold_thmss, code_simp_attrs), (rec_thmss, code_simp_attrs))
   726   end;
   727 
   728 fun derive_coinduct_coiters_thms_for_types pre_bnfs (z, cs, cpss,
   729       coiters_args_types as [((pgss, crgsss), _), ((phss, cshsss), _)])
   730     dtor_coinduct dtor_ctors dtor_coiter_thmss nesting_bnfs fpTs Cs As kss mss ns ctr_defss
   731     ctr_sugars coiterss coiter_defss export_args lthy =
   732   let
   733     val coiterss' = transpose coiterss;
   734     val coiter_defss' = transpose coiter_defss;
   735 
   736     val [unfold_defs, corec_defs] = coiter_defss';
   737 
   738     val nn = length pre_bnfs;
   739 
   740     val pre_map_defs = map map_def_of_bnf pre_bnfs;
   741     val pre_rel_defs = map rel_def_of_bnf pre_bnfs;
   742     val nesting_map_ids'' = map (unfold_thms lthy [id_def] o map_id_of_bnf) nesting_bnfs;
   743     val nesting_rel_eqs = map rel_eq_of_bnf nesting_bnfs;
   744 
   745     val fp_b_names = map base_name_of_typ fpTs;
   746 
   747     val ctrss = map (map (mk_ctr As) o #ctrs) ctr_sugars;
   748     val discss = map (map (mk_disc_or_sel As) o #discs) ctr_sugars;
   749     val selsss = map (map (map (mk_disc_or_sel As)) o #selss) ctr_sugars;
   750     val exhausts = map #exhaust ctr_sugars;
   751     val disc_thmsss = map #disc_thmss ctr_sugars;
   752     val discIss = map #discIs ctr_sugars;
   753     val sel_thmsss = map #sel_thmss ctr_sugars;
   754 
   755     val (((rs, us'), vs'), names_lthy) =
   756       lthy
   757       |> mk_Frees "R" (map (fn T => mk_pred2T T T) fpTs)
   758       ||>> Variable.variant_fixes fp_b_names
   759       ||>> Variable.variant_fixes (map (suffix "'") fp_b_names);
   760 
   761     val us = map2 (curry Free) us' fpTs;
   762     val udiscss = map2 (map o rapp) us discss;
   763     val uselsss = map2 (map o map o rapp) us selsss;
   764 
   765     val vs = map2 (curry Free) vs' fpTs;
   766     val vdiscss = map2 (map o rapp) vs discss;
   767     val vselsss = map2 (map o map o rapp) vs selsss;
   768 
   769     val coinduct_thms_pairs =
   770       let
   771         val uvrs = map3 (fn r => fn u => fn v => r $ u $ v) rs us vs;
   772         val uv_eqs = map2 (curry HOLogic.mk_eq) us vs;
   773         val strong_rs =
   774           map4 (fn u => fn v => fn uvr => fn uv_eq =>
   775             fold_rev Term.lambda [u, v] (HOLogic.mk_disj (uvr, uv_eq))) us vs uvrs uv_eqs;
   776 
   777         (* TODO: generalize (cf. "build_map") *)
   778         fun build_rel rs' T =
   779           (case find_index (curry op = T) fpTs of
   780             ~1 =>
   781             if exists_subtype_in fpTs T then
   782               let
   783                 val Type (s, Ts) = T
   784                 val bnf = the (bnf_of lthy s);
   785                 val live = live_of_bnf bnf;
   786                 val rel = mk_rel live Ts Ts (rel_of_bnf bnf);
   787                 val Ts' = map domain_type (fst (strip_typeN live (fastype_of rel)));
   788               in Term.list_comb (rel, map (build_rel rs') Ts') end
   789             else
   790               HOLogic.eq_const T
   791           | kk => nth rs' kk);
   792 
   793         fun build_rel_app rs' usel vsel = fold rapp [usel, vsel] (build_rel rs' (fastype_of usel));
   794 
   795         fun mk_prem_ctr_concls rs' n k udisc usels vdisc vsels =
   796           (if k = n then [] else [HOLogic.mk_eq (udisc, vdisc)]) @
   797           (if null usels then
   798              []
   799            else
   800              [Library.foldr HOLogic.mk_imp (if n = 1 then [] else [udisc, vdisc],
   801                 Library.foldr1 HOLogic.mk_conj (map2 (build_rel_app rs') usels vsels))]);
   802 
   803         fun mk_prem_concl rs' n udiscs uselss vdiscs vselss =
   804           Library.foldr1 HOLogic.mk_conj
   805             (flat (map5 (mk_prem_ctr_concls rs' n) (1 upto n) udiscs uselss vdiscs vselss))
   806           handle List.Empty => @{term True};
   807 
   808         fun mk_prem rs' uvr u v n udiscs uselss vdiscs vselss =
   809           fold_rev Logic.all [u, v] (Logic.mk_implies (HOLogic.mk_Trueprop uvr,
   810             HOLogic.mk_Trueprop (mk_prem_concl rs' n udiscs uselss vdiscs vselss)));
   811 
   812         val concl =
   813           HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj
   814             (map3 (fn uvr => fn u => fn v => HOLogic.mk_imp (uvr, HOLogic.mk_eq (u, v)))
   815                uvrs us vs));
   816 
   817         fun mk_goal rs' =
   818           Logic.list_implies (map8 (mk_prem rs') uvrs us vs ns udiscss uselsss vdiscss vselsss,
   819             concl);
   820 
   821         val goals = map mk_goal [rs, strong_rs];
   822 
   823         fun prove dtor_coinduct' goal =
   824           Goal.prove_sorry lthy [] [] goal (fn {context = ctxt, ...} =>
   825             mk_coinduct_tac ctxt nesting_rel_eqs nn ns dtor_coinduct' pre_rel_defs dtor_ctors
   826               exhausts ctr_defss disc_thmsss sel_thmsss)
   827           |> singleton (Proof_Context.export names_lthy lthy)
   828           |> Thm.close_derivation;
   829 
   830         fun postproc nn thm =
   831           Thm.permute_prems 0 nn
   832             (if nn = 1 then thm RS mp else funpow nn (fn thm => reassoc_conjs (thm RS mp_conj)) thm)
   833           |> Drule.zero_var_indexes
   834           |> `(conj_dests nn);
   835 
   836         val rel_eqs = map rel_eq_of_bnf pre_bnfs;
   837         val rel_monos = map rel_mono_of_bnf pre_bnfs;
   838         val dtor_coinducts =
   839           [dtor_coinduct, mk_strong_coinduct_thm dtor_coinduct rel_eqs rel_monos lthy];
   840       in
   841         map2 (postproc nn oo prove) dtor_coinducts goals
   842       end;
   843 
   844     fun mk_coinduct_concls ms discs ctrs =
   845       let
   846         fun mk_disc_concl disc = [name_of_disc disc];
   847         fun mk_ctr_concl 0 _ = []
   848           | mk_ctr_concl _ ctor = [name_of_ctr ctor];
   849         val disc_concls = map mk_disc_concl (fst (split_last discs)) @ [[]];
   850         val ctr_concls = map2 mk_ctr_concl ms ctrs;
   851       in
   852         flat (map2 append disc_concls ctr_concls)
   853       end;
   854 
   855     val coinduct_cases = quasi_unambiguous_case_names (map (prefix EqN) fp_b_names);
   856     val coinduct_conclss =
   857       map3 (quasi_unambiguous_case_names ooo mk_coinduct_concls) mss discss ctrss;
   858 
   859     fun mk_maybe_not pos = not pos ? HOLogic.mk_not;
   860 
   861     val fcoiterss' as [gunfolds, hcorecs] =
   862       map2 (fn (pfss, _) => map (lists_bmoc pfss)) (map fst coiters_args_types) coiterss';
   863 
   864     val (unfold_thmss, corec_thmss, safe_unfold_thmss, safe_corec_thmss) =
   865       let
   866         fun mk_goal pfss c cps fcoiter n k ctr m cfs' =
   867           fold_rev (fold_rev Logic.all) ([c] :: pfss)
   868             (Logic.list_implies (seq_conds (HOLogic.mk_Trueprop oo mk_maybe_not) n k cps,
   869                mk_Trueprop_eq (fcoiter $ c, Term.list_comb (ctr, take m cfs'))));
   870 
   871         fun mk_U maybe_mk_sumT =
   872           typ_subst_nonatomic (map2 (fn C => fn fpT => (maybe_mk_sumT fpT C, fpT)) Cs fpTs);
   873 
   874         fun tack z_name (c, u) f =
   875           let val z = Free (z_name, mk_sumT (fastype_of u, fastype_of c)) in
   876             Term.lambda z (mk_sum_case (Term.lambda u u, Term.lambda c (f $ c)) $ z)
   877           end;
   878 
   879         fun build_coiter fcoiters maybe_mk_sumT maybe_tack cqf =
   880           let val T = fastype_of cqf in
   881             if exists_subtype_in Cs T then
   882               let val U = mk_U maybe_mk_sumT T in
   883                 build_map lthy (indexify snd fpTs (fn kk => fn _ =>
   884                   maybe_tack (nth cs kk, nth us kk) (nth fcoiters kk))) (T, U) $ cqf
   885               end
   886             else
   887               cqf
   888           end;
   889 
   890         val crgsss' = map (map (map (build_coiter (un_fold_of fcoiterss') (K I) (K I)))) crgsss;
   891         val cshsss' = map (map (map (build_coiter (co_rec_of fcoiterss') (curry mk_sumT) (tack z))))
   892           cshsss;
   893 
   894         val unfold_goalss = map8 (map4 oooo mk_goal pgss) cs cpss gunfolds ns kss ctrss mss crgsss';
   895         val corec_goalss = map8 (map4 oooo mk_goal phss) cs cpss hcorecs ns kss ctrss mss cshsss';
   896 
   897         val unfold_tacss =
   898           map3 (map oo mk_coiter_tac unfold_defs nesting_map_ids'')
   899             (map un_fold_of dtor_coiter_thmss) pre_map_defs ctr_defss;
   900         val corec_tacss =
   901           map3 (map oo mk_coiter_tac corec_defs nesting_map_ids'')
   902             (map co_rec_of dtor_coiter_thmss) pre_map_defs ctr_defss;
   903 
   904         fun prove goal tac =
   905           Goal.prove_sorry lthy [] [] goal (tac o #context)
   906           |> Thm.close_derivation;
   907 
   908         val unfold_thmss = map2 (map2 prove) unfold_goalss unfold_tacss;
   909         val corec_thmss =
   910           map2 (map2 prove) corec_goalss corec_tacss
   911           |> map (map (unfold_thms lthy @{thms sum_case_if}));
   912 
   913         val unfold_safesss = map2 (map2 (map2 (curry op =))) crgsss' crgsss;
   914         val corec_safesss = map2 (map2 (map2 (curry op =))) cshsss' cshsss;
   915 
   916         val filter_safesss =
   917           map2 (map_filter (fn (safes, thm) => if forall I safes then SOME thm else NONE) oo
   918             curry (op ~~));
   919 
   920         val safe_unfold_thmss = filter_safesss unfold_safesss unfold_thmss;
   921         val safe_corec_thmss = filter_safesss corec_safesss corec_thmss;
   922       in
   923         (unfold_thmss, corec_thmss, safe_unfold_thmss, safe_corec_thmss)
   924       end;
   925 
   926     val (disc_unfold_iff_thmss, disc_corec_iff_thmss) =
   927       let
   928         fun mk_goal c cps fcoiter n k disc =
   929           mk_Trueprop_eq (disc $ (fcoiter $ c),
   930             if n = 1 then @{const True}
   931             else Library.foldr1 HOLogic.mk_conj (seq_conds mk_maybe_not n k cps));
   932 
   933         val unfold_goalss = map6 (map2 oooo mk_goal) cs cpss gunfolds ns kss discss;
   934         val corec_goalss = map6 (map2 oooo mk_goal) cs cpss hcorecs ns kss discss;
   935 
   936         fun mk_case_split' cp = Drule.instantiate' [] [SOME (certify lthy cp)] @{thm case_split};
   937 
   938         val case_splitss' = map (map mk_case_split') cpss;
   939 
   940         val unfold_tacss =
   941           map3 (map oo mk_disc_coiter_iff_tac) case_splitss' unfold_thmss disc_thmsss;
   942         val corec_tacss =
   943           map3 (map oo mk_disc_coiter_iff_tac) case_splitss' corec_thmss disc_thmsss;
   944 
   945         fun prove goal tac =
   946           Goal.prove_sorry lthy [] [] goal (tac o #context)
   947           |> singleton export_args
   948           |> singleton (Proof_Context.export names_lthy lthy)
   949           |> Thm.close_derivation;
   950 
   951         fun proves [_] [_] = []
   952           | proves goals tacs = map2 prove goals tacs;
   953       in
   954         (map2 proves unfold_goalss unfold_tacss, map2 proves corec_goalss corec_tacss)
   955       end;
   956 
   957     val is_triv_discI = is_triv_implies orf is_concl_refl;
   958 
   959     fun mk_disc_coiter_thms coiters discIs =
   960       map (op RS) (filter_out (is_triv_discI o snd) (coiters ~~ discIs));
   961 
   962     val disc_unfold_thmss = map2 mk_disc_coiter_thms unfold_thmss discIss;
   963     val disc_corec_thmss = map2 mk_disc_coiter_thms corec_thmss discIss;
   964 
   965     fun mk_sel_coiter_thm coiter_thm sel sel_thm =
   966       let
   967         val (domT, ranT) = dest_funT (fastype_of sel);
   968         val arg_cong' =
   969           Drule.instantiate' (map (SOME o certifyT lthy) [domT, ranT])
   970             [NONE, NONE, SOME (certify lthy sel)] arg_cong
   971           |> Thm.varifyT_global;
   972         val sel_thm' = sel_thm RSN (2, trans);
   973       in
   974         coiter_thm RS arg_cong' RS sel_thm'
   975       end;
   976 
   977     fun mk_sel_coiter_thms coiter_thmss =
   978       map3 (map3 (map2 o mk_sel_coiter_thm)) coiter_thmss selsss sel_thmsss |> map flat;
   979 
   980     val sel_unfold_thmss = mk_sel_coiter_thms unfold_thmss;
   981     val sel_corec_thmss = mk_sel_coiter_thms corec_thmss;
   982 
   983     val coinduct_consumes_attr = Attrib.internal (K (Rule_Cases.consumes nn));
   984     val coinduct_case_names_attr = Attrib.internal (K (Rule_Cases.case_names coinduct_cases));
   985     val coinduct_case_concl_attrs =
   986       map2 (fn casex => fn concls =>
   987           Attrib.internal (K (Rule_Cases.case_conclusion (casex, concls))))
   988         coinduct_cases coinduct_conclss;
   989     val coinduct_case_attrs =
   990       coinduct_consumes_attr :: coinduct_case_names_attr :: coinduct_case_concl_attrs;
   991   in
   992     ((coinduct_thms_pairs, coinduct_case_attrs),
   993      (unfold_thmss, corec_thmss, []),
   994      (safe_unfold_thmss, safe_corec_thmss),
   995      (disc_unfold_thmss, disc_corec_thmss, simp_attrs),
   996      (disc_unfold_iff_thmss, disc_corec_iff_thmss, simp_attrs),
   997      (sel_unfold_thmss, sel_corec_thmss, simp_attrs))
   998   end;
   999 
  1000 fun define_co_datatypes prepare_constraint prepare_typ prepare_term fp construct_fp
  1001     (wrap_opts as (no_discs_sels, rep_compat), specs) no_defs_lthy0 =
  1002   let
  1003     (* TODO: sanity checks on arguments *)
  1004 
  1005     val _ = if fp = Greatest_FP andalso no_discs_sels then
  1006         error "Cannot define codatatypes without discriminators and selectors"
  1007       else
  1008         ();
  1009 
  1010     fun qualify mandatory fp_b_name =
  1011       Binding.qualify mandatory fp_b_name o (rep_compat ? Binding.qualify false rep_compat_prefix);
  1012 
  1013     val nn = length specs;
  1014     val fp_bs = map type_binding_of specs;
  1015     val fp_b_names = map Binding.name_of fp_bs;
  1016     val fp_common_name = mk_common_name fp_b_names;
  1017     val map_bs = map2 (fn fp_b_name => qualify false fp_b_name o map_binding_of) fp_b_names specs;
  1018     val rel_bs = map2 (fn fp_b_name => qualify false fp_b_name o rel_binding_of) fp_b_names specs;
  1019 
  1020     fun prepare_type_arg (_, (ty, c)) =
  1021       let val TFree (s, _) = prepare_typ no_defs_lthy0 ty in
  1022         TFree (s, prepare_constraint no_defs_lthy0 c)
  1023       end;
  1024 
  1025     val Ass0 = map (map prepare_type_arg o type_args_named_constrained_of) specs;
  1026     val unsorted_Ass0 = map (map (resort_tfree HOLogic.typeS)) Ass0;
  1027     val unsorted_As = Library.foldr1 merge_type_args unsorted_Ass0;
  1028     val set_bss = map (map fst o type_args_named_constrained_of) specs;
  1029 
  1030     val _ = has_duplicates (op =) unsorted_As andalso
  1031       error ("Duplicate parameters in " ^ co_prefix fp ^ "datatype specification");
  1032 
  1033     val bad_args =
  1034       map (Logic.type_map (singleton (Variable.polymorphic no_defs_lthy0))) unsorted_As
  1035       |> filter_out Term.is_TVar;
  1036     val _ = null bad_args orelse
  1037       error ("Locally fixed type argument " ^
  1038         quote (Syntax.string_of_typ no_defs_lthy0 (hd bad_args)) ^ " in " ^ co_prefix fp ^
  1039         "datatype specification");
  1040 
  1041     val (((Bs0, Cs), Xs), no_defs_lthy) =
  1042       no_defs_lthy0
  1043       |> fold (Variable.declare_typ o resort_tfree dummyS) unsorted_As
  1044       |> mk_TFrees (length unsorted_As)
  1045       ||>> mk_TFrees nn
  1046       ||>> variant_tfrees fp_b_names;
  1047 
  1048     (* TODO: Cleaner handling of fake contexts, without "background_theory". *)
  1049 
  1050     fun add_fake_type spec =
  1051       (*The "qualify" hack below is for the case where the new type shadows an existing global type
  1052         defined in the same theory.*)
  1053       Sign.add_type no_defs_lthy (qualify false "" (type_binding_of spec),
  1054         length (type_args_named_constrained_of spec), mixfix_of spec);
  1055 
  1056     val fake_thy = fold add_fake_type specs;
  1057     val fake_lthy = Proof_Context.background_theory fake_thy no_defs_lthy;
  1058 
  1059     fun mk_fake_T b =
  1060       Type (fst (Term.dest_Type (Proof_Context.read_type_name fake_lthy true (Binding.name_of b))),
  1061         unsorted_As);
  1062 
  1063     val fake_Ts = map mk_fake_T fp_bs;
  1064 
  1065     val mixfixes = map mixfix_of specs;
  1066 
  1067     val _ = (case duplicates Binding.eq_name fp_bs of [] => ()
  1068       | b :: _ => error ("Duplicate type name declaration " ^ quote (Binding.name_of b)));
  1069 
  1070     val ctr_specss = map ctr_specs_of specs;
  1071 
  1072     val disc_bindingss = map (map disc_of) ctr_specss;
  1073     val ctr_bindingss =
  1074       map2 (fn fp_b_name => map (qualify false fp_b_name o ctr_of)) fp_b_names ctr_specss;
  1075     val ctr_argsss = map (map args_of) ctr_specss;
  1076     val ctr_mixfixess = map (map ctr_mixfix_of) ctr_specss;
  1077 
  1078     val sel_bindingsss = map (map (map fst)) ctr_argsss;
  1079     val fake_ctr_Tsss0 = map (map (map (prepare_typ fake_lthy o snd))) ctr_argsss;
  1080     val raw_sel_defaultsss = map (map defaults_of) ctr_specss;
  1081 
  1082     val (As :: _) :: fake_ctr_Tsss =
  1083       burrow (burrow (Syntax.check_typs fake_lthy)) (Ass0 :: fake_ctr_Tsss0);
  1084 
  1085     val _ = (case duplicates (op =) unsorted_As of [] => ()
  1086       | A :: _ => error ("Duplicate type parameter " ^
  1087           quote (Syntax.string_of_typ no_defs_lthy A)));
  1088 
  1089     val rhs_As' = fold (fold (fold Term.add_tfreesT)) fake_ctr_Tsss [];
  1090     val _ = (case subtract (op =) (map dest_TFree As) rhs_As' of
  1091         [] => ()
  1092       | A' :: _ => error ("Extra type variable on right-hand side: " ^
  1093           quote (Syntax.string_of_typ no_defs_lthy (TFree A'))));
  1094 
  1095     fun eq_fpT_check (T as Type (s, Us)) (Type (s', Us')) =
  1096         s = s' andalso (Us = Us' orelse error ("Illegal occurrence of recursive type " ^
  1097           quote (Syntax.string_of_typ fake_lthy T)))
  1098       | eq_fpT_check _ _ = false;
  1099 
  1100     fun freeze_fp (T as Type (s, Us)) =
  1101         (case find_index (eq_fpT_check T) fake_Ts of
  1102           ~1 => Type (s, map freeze_fp Us)
  1103         | kk => nth Xs kk)
  1104       | freeze_fp T = T;
  1105 
  1106     val ctrXs_Tsss = map (map (map freeze_fp)) fake_ctr_Tsss;
  1107     val ctrXs_sum_prod_Ts = map (mk_sumTN_balanced o map HOLogic.mk_tupleT) ctrXs_Tsss;
  1108 
  1109     val fp_eqs =
  1110       map dest_TFree Xs ~~ map (Term.typ_subst_atomic (As ~~ unsorted_As)) ctrXs_sum_prod_Ts;
  1111 
  1112     val (pre_bnfs, (fp_res as {bnfs = fp_bnfs as any_fp_bnf :: _, ctors = ctors0, dtors = dtors0,
  1113            xtor_co_iterss = xtor_co_iterss0, xtor_co_induct, dtor_ctors, ctor_dtors, ctor_injects,
  1114            xtor_map_thms, xtor_set_thmss, xtor_rel_thms, xtor_co_iter_thmss, ...}, lthy)) =
  1115       fp_bnf (construct_fp mixfixes map_bs rel_bs set_bss) fp_bs (map dest_TFree unsorted_As) fp_eqs
  1116         no_defs_lthy0;
  1117 
  1118     val timer = time (Timer.startRealTimer ());
  1119 
  1120     val nesting_bnfs = nesty_bnfs lthy ctrXs_Tsss As;
  1121     val nested_bnfs = nesty_bnfs lthy ctrXs_Tsss Xs;
  1122 
  1123     val pre_map_defs = map map_def_of_bnf pre_bnfs;
  1124     val pre_set_defss = map set_defs_of_bnf pre_bnfs;
  1125     val pre_rel_defs = map rel_def_of_bnf pre_bnfs;
  1126     val nesting_set_map's = maps set_map'_of_bnf nesting_bnfs;
  1127     val nested_set_map's = maps set_map'_of_bnf nested_bnfs;
  1128 
  1129     val live = live_of_bnf any_fp_bnf;
  1130     val _ =
  1131       if live = 0 andalso exists (not o Binding.is_empty) (map_bs @ rel_bs) then
  1132         warning ("Map function and relator names ignored")
  1133       else
  1134         ();
  1135 
  1136     val Bs =
  1137       map3 (fn alive => fn A as TFree (_, S) => fn B => if alive then resort_tfree S B else A)
  1138         (liveness_of_fp_bnf (length As) any_fp_bnf) As Bs0;
  1139 
  1140     val B_ify = Term.typ_subst_atomic (As ~~ Bs);
  1141 
  1142     val ctors = map (mk_ctor As) ctors0;
  1143     val dtors = map (mk_dtor As) dtors0;
  1144 
  1145     val fpTs = map (domain_type o fastype_of) dtors;
  1146 
  1147     fun massage_simple_notes base =
  1148       filter_out (null o #2)
  1149       #> map (fn (thmN, thms, attrs) =>
  1150         ((qualify true base (Binding.name thmN), attrs), [(thms, [])]));
  1151 
  1152     val massage_multi_notes =
  1153       maps (fn (thmN, thmss, attrs) =>
  1154         if forall null thmss then
  1155           []
  1156         else
  1157           map3 (fn fp_b_name => fn Type (T_name, _) => fn thms =>
  1158               ((qualify true fp_b_name (Binding.name thmN), attrs T_name), [(thms, [])]))
  1159             fp_b_names fpTs thmss);
  1160 
  1161     val ctr_Tsss = map (map (map (Term.typ_subst_atomic (Xs ~~ fpTs)))) ctrXs_Tsss;
  1162     val ns = map length ctr_Tsss;
  1163     val kss = map (fn n => 1 upto n) ns;
  1164     val mss = map (map length) ctr_Tsss;
  1165 
  1166     val ((xtor_co_iterss, iters_args_types, coiters_args_types), lthy') =
  1167       mk_co_iters_prelims fp fpTs Cs ns mss xtor_co_iterss0 lthy;
  1168 
  1169     fun define_ctrs_dtrs_for_type (((((((((((((((((((((((fp_bnf, fp_b), fpT), ctor), dtor),
  1170             xtor_co_iters), ctor_dtor), dtor_ctor), ctor_inject), pre_map_def), pre_set_defs),
  1171           pre_rel_def), fp_map_thm), fp_set_thms), fp_rel_thm), n), ks), ms), ctr_bindings),
  1172         ctr_mixfixes), ctr_Tss), disc_bindings), sel_bindingss), raw_sel_defaultss) no_defs_lthy =
  1173       let
  1174         val fp_b_name = Binding.name_of fp_b;
  1175 
  1176         val dtorT = domain_type (fastype_of ctor);
  1177         val ctr_prod_Ts = map HOLogic.mk_tupleT ctr_Tss;
  1178         val ctr_sum_prod_T = mk_sumTN_balanced ctr_prod_Ts;
  1179 
  1180         val ((((w, xss), yss), u'), names_lthy) =
  1181           no_defs_lthy
  1182           |> yield_singleton (mk_Frees "w") dtorT
  1183           ||>> mk_Freess "x" ctr_Tss
  1184           ||>> mk_Freess "y" (map (map B_ify) ctr_Tss)
  1185           ||>> yield_singleton Variable.variant_fixes fp_b_name;
  1186 
  1187         val u = Free (u', fpT);
  1188 
  1189         val tuple_xs = map HOLogic.mk_tuple xss;
  1190         val tuple_ys = map HOLogic.mk_tuple yss;
  1191 
  1192         val ctr_rhss =
  1193           map3 (fn k => fn xs => fn tuple_x => fold_rev Term.lambda xs (ctor $
  1194             mk_InN_balanced ctr_sum_prod_T n tuple_x k)) ks xss tuple_xs;
  1195 
  1196         val ((raw_ctrs, raw_ctr_defs), (lthy', lthy)) = no_defs_lthy
  1197           |> apfst split_list o fold_map3 (fn b => fn mx => fn rhs =>
  1198               Local_Theory.define ((b, mx), ((Thm.def_binding b, []), rhs)) #>> apsnd snd)
  1199             ctr_bindings ctr_mixfixes ctr_rhss
  1200           ||> `Local_Theory.restore;
  1201 
  1202         val phi = Proof_Context.export_morphism lthy lthy';
  1203 
  1204         val ctr_defs = map (Morphism.thm phi) raw_ctr_defs;
  1205         val ctr_defs' =
  1206           map2 (fn m => fn def => mk_unabs_def m (def RS meta_eq_to_obj_eq)) ms ctr_defs;
  1207 
  1208         val ctrs0 = map (Morphism.term phi) raw_ctrs;
  1209         val ctrs = map (mk_ctr As) ctrs0;
  1210 
  1211         fun wrap_ctrs lthy =
  1212           let
  1213             fun exhaust_tac {context = ctxt, prems = _} =
  1214               let
  1215                 val ctor_iff_dtor_thm =
  1216                   let
  1217                     val goal =
  1218                       fold_rev Logic.all [w, u]
  1219                         (mk_Trueprop_eq (HOLogic.mk_eq (u, ctor $ w), HOLogic.mk_eq (dtor $ u, w)));
  1220                   in
  1221                     Goal.prove_sorry lthy [] [] goal (fn {context = ctxt, ...} =>
  1222                       mk_ctor_iff_dtor_tac ctxt (map (SOME o certifyT lthy) [dtorT, fpT])
  1223                         (certify lthy ctor) (certify lthy dtor) ctor_dtor dtor_ctor)
  1224                     |> Thm.close_derivation
  1225                     |> Morphism.thm phi
  1226                   end;
  1227 
  1228                 val sumEN_thm' =
  1229                   unfold_thms lthy @{thms unit_all_eq1}
  1230                     (Drule.instantiate' (map (SOME o certifyT lthy) ctr_prod_Ts) []
  1231                        (mk_sumEN_balanced n))
  1232                   |> Morphism.thm phi;
  1233               in
  1234                 mk_exhaust_tac ctxt n ctr_defs ctor_iff_dtor_thm sumEN_thm'
  1235               end;
  1236 
  1237             val inject_tacss =
  1238               map2 (fn 0 => K [] | _ => fn ctr_def => [fn {context = ctxt, ...} =>
  1239                   mk_inject_tac ctxt ctr_def ctor_inject]) ms ctr_defs;
  1240 
  1241             val half_distinct_tacss =
  1242               map (map (fn (def, def') => fn {context = ctxt, ...} =>
  1243                 mk_half_distinct_tac ctxt ctor_inject [def, def'])) (mk_half_pairss (`I ctr_defs));
  1244 
  1245             val tacss = [exhaust_tac] :: inject_tacss @ half_distinct_tacss;
  1246 
  1247             val sel_defaultss = map (map (apsnd (prepare_term lthy))) raw_sel_defaultss
  1248           in
  1249             wrap_free_constructors tacss (((wrap_opts, ctrs0), standard_binding), (disc_bindings,
  1250               (sel_bindingss, sel_defaultss))) lthy
  1251           end;
  1252 
  1253         fun derive_maps_sets_rels (ctr_sugar, lthy) =
  1254           if live = 0 then
  1255             ((([], [], [], []), ctr_sugar), lthy)
  1256           else
  1257             let
  1258               val rel_flip = rel_flip_of_bnf fp_bnf;
  1259               val nones = replicate live NONE;
  1260 
  1261               val ctor_cong =
  1262                 if fp = Least_FP then
  1263                   Drule.dummy_thm
  1264                 else
  1265                   let val ctor' = mk_ctor Bs ctor in
  1266                     cterm_instantiate_pos [NONE, NONE, SOME (certify lthy ctor')] arg_cong
  1267                   end;
  1268 
  1269               fun mk_cIn ify =
  1270                 certify lthy o (fp = Greatest_FP ? curry (op $) (map_types ify ctor)) oo
  1271                 mk_InN_balanced (ify ctr_sum_prod_T) n;
  1272 
  1273               val cxIns = map2 (mk_cIn I) tuple_xs ks;
  1274               val cyIns = map2 (mk_cIn B_ify) tuple_ys ks;
  1275 
  1276               fun mk_map_thm ctr_def' cxIn =
  1277                 fold_thms lthy [ctr_def']
  1278                   (unfold_thms lthy (pre_map_def ::
  1279                        (if fp = Least_FP then [] else [ctor_dtor, dtor_ctor]) @ sum_prod_thms_map)
  1280                      (cterm_instantiate_pos (nones @ [SOME cxIn])
  1281                         (if fp = Least_FP then fp_map_thm else fp_map_thm RS ctor_cong)))
  1282                 |> singleton (Proof_Context.export names_lthy no_defs_lthy);
  1283 
  1284               fun mk_set_thm fp_set_thm ctr_def' cxIn =
  1285                 fold_thms lthy [ctr_def']
  1286                   (unfold_thms lthy (pre_set_defs @ nested_set_map's @ nesting_set_map's @
  1287                        (if fp = Least_FP then [] else [dtor_ctor]) @ sum_prod_thms_set)
  1288                      (cterm_instantiate_pos [SOME cxIn] fp_set_thm))
  1289                 |> singleton (Proof_Context.export names_lthy no_defs_lthy);
  1290 
  1291               fun mk_set_thms fp_set_thm = map2 (mk_set_thm fp_set_thm) ctr_defs' cxIns;
  1292 
  1293               val map_thms = map2 mk_map_thm ctr_defs' cxIns;
  1294               val set_thmss = map mk_set_thms fp_set_thms;
  1295 
  1296               val rel_infos = (ctr_defs' ~~ cxIns, ctr_defs' ~~ cyIns);
  1297 
  1298               fun mk_rel_thm postproc ctr_defs' cxIn cyIn =
  1299                 fold_thms lthy ctr_defs'
  1300                    (unfold_thms lthy (@{thm Inl_Inr_False} :: pre_rel_def ::
  1301                         (if fp = Least_FP then [] else [dtor_ctor]) @ sum_prod_thms_rel)
  1302                       (cterm_instantiate_pos (nones @ [SOME cxIn, SOME cyIn]) fp_rel_thm))
  1303                 |> postproc
  1304                 |> singleton (Proof_Context.export names_lthy no_defs_lthy);
  1305 
  1306               fun mk_rel_inject_thm ((ctr_def', cxIn), (_, cyIn)) =
  1307                 mk_rel_thm (unfold_thms lthy @{thms eq_sym_Unity_conv}) [ctr_def'] cxIn cyIn;
  1308 
  1309               val rel_inject_thms = map mk_rel_inject_thm (op ~~ rel_infos);
  1310 
  1311               fun mk_half_rel_distinct_thm ((xctr_def', cxIn), (yctr_def', cyIn)) =
  1312                 mk_rel_thm (fn thm => thm RS @{thm eq_False[THEN iffD1]}) [xctr_def', yctr_def']
  1313                   cxIn cyIn;
  1314 
  1315               fun mk_other_half_rel_distinct_thm thm =
  1316                 flip_rels lthy live thm
  1317                 RS (rel_flip RS sym RS @{thm arg_cong[of _ _ Not]} RS iffD2);
  1318 
  1319               val half_rel_distinct_thmss =
  1320                 map (map mk_half_rel_distinct_thm) (mk_half_pairss rel_infos);
  1321               val other_half_rel_distinct_thmss =
  1322                 map (map mk_other_half_rel_distinct_thm) half_rel_distinct_thmss;
  1323               val (rel_distinct_thms, _) =
  1324                 join_halves n half_rel_distinct_thmss other_half_rel_distinct_thmss;
  1325 
  1326               val notes =
  1327                 [(mapN, map_thms, code_simp_attrs),
  1328                  (rel_distinctN, rel_distinct_thms, code_simp_attrs),
  1329                  (rel_injectN, rel_inject_thms, code_simp_attrs),
  1330                  (setsN, flat set_thmss, code_simp_attrs)]
  1331                 |> massage_simple_notes fp_b_name;
  1332             in
  1333               (((map_thms, rel_inject_thms, rel_distinct_thms, set_thmss), ctr_sugar),
  1334                lthy |> Local_Theory.notes notes |> snd)
  1335             end;
  1336 
  1337         fun mk_binding suf = qualify false fp_b_name (Binding.suffix_name ("_" ^ suf) fp_b);
  1338 
  1339         fun massage_res (((maps_sets_rels, ctr_sugar), co_iter_res), lthy) =
  1340           (((maps_sets_rels, (ctrs, xss, ctr_defs, ctr_sugar)), co_iter_res), lthy);
  1341       in
  1342         (wrap_ctrs
  1343          #> derive_maps_sets_rels
  1344          ##>>
  1345            (if fp = Least_FP then define_iters [foldN, recN] (the iters_args_types)
  1346             else define_coiters [unfoldN, corecN] (the coiters_args_types))
  1347              mk_binding fpTs Cs xtor_co_iters
  1348          #> massage_res, lthy')
  1349       end;
  1350 
  1351     fun wrap_types_etc (wrap_types_etcs, lthy) =
  1352       fold_map I wrap_types_etcs lthy
  1353       |>> apsnd split_list o apfst (apsnd split_list4 o apfst split_list4 o split_list)
  1354         o split_list;
  1355 
  1356     val mk_simp_thmss =
  1357       map7 (fn {injects, distincts, case_thms, ...} => fn un_folds => fn co_recs =>
  1358         fn mapsx => fn rel_injects => fn rel_distincts => fn setss =>
  1359           injects @ distincts @ case_thms @ co_recs @ un_folds @ mapsx @ rel_injects
  1360           @ rel_distincts @ flat setss);
  1361 
  1362     fun derive_and_note_induct_iters_thms_for_types
  1363         ((((mapsx, rel_injects, rel_distincts, setss), (ctrss, _, ctr_defss, ctr_sugars)),
  1364           (iterss, iter_defss)), lthy) =
  1365       let
  1366         val ((induct_thms, induct_thm, induct_attrs), (fold_thmss, fold_attrs),
  1367              (rec_thmss, rec_attrs)) =
  1368           derive_induct_iters_thms_for_types pre_bnfs (the iters_args_types) xtor_co_induct
  1369             xtor_co_iter_thmss nesting_bnfs nested_bnfs fpTs Cs Xs ctrXs_Tsss ctrss ctr_defss iterss
  1370             iter_defss lthy;
  1371 
  1372         val induct_type_attr = Attrib.internal o K o Induct.induct_type;
  1373 
  1374         val simp_thmss =
  1375           mk_simp_thmss ctr_sugars fold_thmss rec_thmss mapsx rel_injects rel_distincts setss;
  1376 
  1377         val common_notes =
  1378           (if nn > 1 then [(inductN, [induct_thm], induct_attrs)] else [])
  1379           |> massage_simple_notes fp_common_name;
  1380 
  1381         val notes =
  1382           [(foldN, fold_thmss, K fold_attrs),
  1383            (inductN, map single induct_thms, fn T_name => induct_attrs @ [induct_type_attr T_name]),
  1384            (recN, rec_thmss, K rec_attrs),
  1385            (simpsN, simp_thmss, K [])]
  1386           |> massage_multi_notes;
  1387       in
  1388         lthy
  1389         |> Local_Theory.notes (common_notes @ notes) |> snd
  1390         |> register_fp_sugars Least_FP pre_bnfs nested_bnfs fp_res ctr_defss ctr_sugars iterss
  1391           [induct_thm] (transpose [fold_thmss, rec_thmss])
  1392       end;
  1393 
  1394     fun derive_and_note_coinduct_coiters_thms_for_types
  1395         ((((mapsx, rel_injects, rel_distincts, setss), (_, _, ctr_defss, ctr_sugars)),
  1396           (coiterss, coiter_defss)), lthy) =
  1397       let
  1398         val (([(coinduct_thms, coinduct_thm), (strong_coinduct_thms, strong_coinduct_thm)],
  1399               coinduct_attrs),
  1400              (unfold_thmss, corec_thmss, coiter_attrs),
  1401              (safe_unfold_thmss, safe_corec_thmss),
  1402              (disc_unfold_thmss, disc_corec_thmss, disc_coiter_attrs),
  1403              (disc_unfold_iff_thmss, disc_corec_iff_thmss, disc_coiter_iff_attrs),
  1404              (sel_unfold_thmss, sel_corec_thmss, sel_coiter_attrs)) =
  1405           derive_coinduct_coiters_thms_for_types pre_bnfs (the coiters_args_types) xtor_co_induct
  1406             dtor_ctors xtor_co_iter_thmss nesting_bnfs fpTs Cs As kss mss ns ctr_defss ctr_sugars
  1407             coiterss coiter_defss (Proof_Context.export lthy' no_defs_lthy) lthy;
  1408 
  1409         val coinduct_type_attr = Attrib.internal o K o Induct.coinduct_type;
  1410 
  1411         fun flat_coiter_thms coiters disc_coiters sel_coiters =
  1412           coiters @ disc_coiters @ sel_coiters;
  1413 
  1414         val simp_thmss =
  1415           mk_simp_thmss ctr_sugars
  1416             (map3 flat_coiter_thms safe_unfold_thmss disc_unfold_thmss sel_unfold_thmss)
  1417             (map3 flat_coiter_thms safe_corec_thmss disc_corec_thmss sel_corec_thmss)
  1418             mapsx rel_injects rel_distincts setss;
  1419 
  1420         val anonymous_notes =
  1421           [(flat safe_unfold_thmss @ flat safe_corec_thmss, simp_attrs)]
  1422           |> map (fn (thms, attrs) => ((Binding.empty, attrs), [(thms, [])]));
  1423 
  1424         val common_notes =
  1425           (if nn > 1 then
  1426              [(coinductN, [coinduct_thm], coinduct_attrs),
  1427               (strong_coinductN, [strong_coinduct_thm], coinduct_attrs)]
  1428            else
  1429              [])
  1430           |> massage_simple_notes fp_common_name;
  1431 
  1432         val notes =
  1433           [(coinductN, map single coinduct_thms,
  1434             fn T_name => coinduct_attrs @ [coinduct_type_attr T_name]),
  1435            (corecN, corec_thmss, K coiter_attrs),
  1436            (disc_corecN, disc_corec_thmss, K disc_coiter_attrs),
  1437            (disc_corec_iffN, disc_corec_iff_thmss, K disc_coiter_iff_attrs),
  1438            (disc_unfoldN, disc_unfold_thmss, K disc_coiter_attrs),
  1439            (disc_unfold_iffN, disc_unfold_iff_thmss, K disc_coiter_iff_attrs),
  1440            (sel_corecN, sel_corec_thmss, K sel_coiter_attrs),
  1441            (sel_unfoldN, sel_unfold_thmss, K sel_coiter_attrs),
  1442            (simpsN, simp_thmss, K []),
  1443            (strong_coinductN, map single strong_coinduct_thms, K coinduct_attrs),
  1444            (unfoldN, unfold_thmss, K coiter_attrs)]
  1445           |> massage_multi_notes;
  1446       in
  1447         lthy
  1448         |> Local_Theory.notes (anonymous_notes @ common_notes @ notes) |> snd
  1449         |> register_fp_sugars Greatest_FP pre_bnfs nested_bnfs fp_res ctr_defss ctr_sugars coiterss
  1450           [coinduct_thm, strong_coinduct_thm] (transpose [unfold_thmss, corec_thmss])
  1451       end;
  1452 
  1453     val lthy'' = lthy'
  1454       |> fold_map define_ctrs_dtrs_for_type (fp_bnfs ~~ fp_bs ~~ fpTs ~~ ctors ~~ dtors ~~
  1455         xtor_co_iterss ~~ ctor_dtors ~~ dtor_ctors ~~ ctor_injects ~~ pre_map_defs ~~
  1456         pre_set_defss ~~ pre_rel_defs ~~ xtor_map_thms ~~ xtor_set_thmss ~~ xtor_rel_thms ~~ ns ~~
  1457         kss ~~ mss ~~ ctr_bindingss ~~ ctr_mixfixess ~~ ctr_Tsss ~~ disc_bindingss ~~
  1458         sel_bindingsss ~~ raw_sel_defaultsss)
  1459       |> wrap_types_etc
  1460       |> fp_case fp derive_and_note_induct_iters_thms_for_types
  1461            derive_and_note_coinduct_coiters_thms_for_types;
  1462 
  1463     val timer = time (timer ("Constructors, discriminators, selectors, etc., for the new " ^
  1464       co_prefix fp ^ "datatype"));
  1465   in
  1466     timer; lthy''
  1467   end;
  1468 
  1469 val co_datatypes = define_co_datatypes (K I) (K I) (K I);
  1470 
  1471 val co_datatype_cmd =
  1472   define_co_datatypes Typedecl.read_constraint Syntax.parse_typ Syntax.parse_term;
  1473 
  1474 val parse_ctr_arg =
  1475   @{keyword "("} |-- parse_binding_colon -- Parse.typ --| @{keyword ")"} ||
  1476   (Parse.typ >> pair Binding.empty);
  1477 
  1478 val parse_defaults =
  1479   @{keyword "("} |-- @{keyword "defaults"} |-- Scan.repeat parse_bound_term --| @{keyword ")"};
  1480 
  1481 val parse_type_arg_constrained =
  1482   Parse.type_ident -- Scan.option (@{keyword "::"} |-- Parse.!!! Parse.sort);
  1483 
  1484 val parse_type_arg_named_constrained = parse_opt_binding_colon -- parse_type_arg_constrained;
  1485 
  1486 val parse_type_args_named_constrained =
  1487   parse_type_arg_constrained >> (single o pair Binding.empty) ||
  1488   @{keyword "("} |-- Parse.!!! (Parse.list1 parse_type_arg_named_constrained --| @{keyword ")"}) ||
  1489   Scan.succeed [];
  1490 
  1491 val parse_map_rel_binding = Parse.short_ident --| @{keyword ":"} -- parse_binding;
  1492 
  1493 val no_map_rel = (Binding.empty, Binding.empty);
  1494 
  1495 (* "map" and "rel" are purposedly not registered as keywords, because they are short and nice names
  1496    that we don't want them to be highlighted everywhere. *)
  1497 fun extract_map_rel ("map", b) = apfst (K b)
  1498   | extract_map_rel ("rel", b) = apsnd (K b)
  1499   | extract_map_rel (s, _) = error ("Expected \"map\" or \"rel\" instead of " ^ quote s);
  1500 
  1501 val parse_map_rel_bindings =
  1502   @{keyword "("} |-- Scan.repeat parse_map_rel_binding --| @{keyword ")"}
  1503     >> (fn ps => fold extract_map_rel ps no_map_rel) ||
  1504   Scan.succeed no_map_rel;
  1505 
  1506 val parse_ctr_spec =
  1507   parse_opt_binding_colon -- parse_binding -- Scan.repeat parse_ctr_arg --
  1508   Scan.optional parse_defaults [] -- Parse.opt_mixfix;
  1509 
  1510 val parse_spec =
  1511   parse_type_args_named_constrained -- parse_binding -- parse_map_rel_bindings --
  1512   Parse.opt_mixfix -- (@{keyword "="} |-- Parse.enum1 "|" parse_ctr_spec);
  1513 
  1514 val parse_co_datatype = parse_wrap_free_constructors_options -- Parse.and_list1 parse_spec;
  1515 
  1516 fun parse_co_datatype_cmd fp construct_fp = parse_co_datatype >> co_datatype_cmd fp construct_fp;
  1517 
  1518 end;