src/HOL/BNF/Tools/bnf_fp_def_sugar.ML
author blanchet
Thu Sep 26 10:00:07 2013 +0200 (2013-09-26)
changeset 53907 d177eb989c65
parent 53901 3d93e8b4ae02
child 53974 612505263257
permissions -rw-r--r--
added data query function
     1 (*  Title:      HOL/BNF/Tools/bnf_fp_def_sugar.ML
     2     Author:     Jasmin Blanchette, TU Muenchen
     3     Copyright   2012, 2013
     4 
     5 Sugared datatype and codatatype constructions.
     6 *)
     7 
     8 signature BNF_FP_DEF_SUGAR =
     9 sig
    10   type fp_sugar =
    11     {T: typ,
    12      fp: BNF_FP_Util.fp_kind,
    13      index: int,
    14      pre_bnfs: BNF_Def.bnf list,
    15      nested_bnfs: BNF_Def.bnf list,
    16      nesting_bnfs: BNF_Def.bnf list,
    17      fp_res: BNF_FP_Util.fp_result,
    18      ctr_defss: thm list list,
    19      ctr_sugars: BNF_Ctr_Sugar.ctr_sugar list,
    20      co_iterss: term list list,
    21      mapss: thm list list,
    22      co_inducts: thm list,
    23      co_iter_thmsss: thm list list list,
    24      disc_co_itersss: thm list list list,
    25      sel_co_iterssss: thm list list list list};
    26 
    27   val of_fp_sugar: (fp_sugar -> 'a list) -> fp_sugar -> 'a
    28   val morph_fp_sugar: morphism -> fp_sugar -> fp_sugar
    29   val fp_sugar_of: Proof.context -> string -> fp_sugar option
    30   val fp_sugars_of: Proof.context -> fp_sugar list
    31 
    32   val co_induct_of: 'a list -> 'a
    33   val strong_co_induct_of: 'a list -> 'a
    34 
    35   val tvar_subst: theory -> typ list -> typ list -> ((string * int) * typ) list
    36   val exists_subtype_in: typ list -> typ -> bool
    37   val flat_rec_arg_args: 'a list list -> 'a list
    38   val flat_corec_preds_predsss_gettersss: 'a list -> 'a list list list -> 'a list list list ->
    39     'a list
    40   val mk_co_iter: theory -> BNF_FP_Util.fp_kind -> typ -> typ list -> term -> term
    41   val nesty_bnfs: Proof.context -> typ list list list -> typ list -> BNF_Def.bnf list
    42   val mk_map: int -> typ list -> typ list -> term -> term
    43   val mk_rel: int -> typ list -> typ list -> term -> term
    44   val build_map: local_theory -> (typ * typ -> term) -> typ * typ -> term
    45   val build_rel: local_theory -> (typ * typ -> term) -> typ * typ -> term
    46   val dest_map: Proof.context -> string -> term -> term * term list
    47   val dest_ctr: Proof.context -> string -> term -> term * term list
    48 
    49   type lfp_sugar_thms =
    50     (thm list * thm * Args.src list)
    51     * (thm list list * thm list list * Args.src list)
    52 
    53   type gfp_sugar_thms =
    54     ((thm list * thm) list * Args.src list)
    55     * (thm list list * thm list list * Args.src list)
    56     * (thm list list * thm list list) * (thm list list * thm list list * Args.src list)
    57     * (thm list list * thm list list * Args.src list)
    58     * (thm list list list * thm list list list * Args.src list)
    59 
    60   val mk_co_iters_prelims: BNF_FP_Util.fp_kind -> typ list list list -> typ list -> typ list ->
    61     int list -> int list list -> term list list -> Proof.context ->
    62     (term list list
    63      * (typ list list * typ list list list list * term list list
    64         * term list list list list) list option
    65      * (string * term list * term list list
    66         * ((term list list * term list list list) * (typ list * typ list list)) list) option)
    67     * Proof.context
    68   val mk_iter_fun_arg_types: typ list list list -> int list -> int list list -> term ->
    69     typ list list list list
    70   val mk_coiter_fun_arg_types: typ list list list -> typ list -> int list -> term ->
    71     typ list list
    72     * (typ list list list list * typ list list list * typ list list list list * typ list)
    73   val define_iters: string list ->
    74     (typ list list * typ list list list list * term list list * term list list list list) list ->
    75     (string -> binding) -> typ list -> typ list -> term list -> Proof.context ->
    76     (term list * thm list) * Proof.context
    77   val define_coiters: string list -> string * term list * term list list
    78     * ((term list list * term list list list) * (typ list * typ list list)) list ->
    79     (string -> binding) -> typ list -> typ list -> term list -> Proof.context ->
    80     (term list * thm list) * Proof.context
    81   val derive_induct_iters_thms_for_types: BNF_Def.bnf list ->
    82     (typ list list * typ list list list list * term list list * term list list list list) list ->
    83     thm -> thm list list -> BNF_Def.bnf list -> BNF_Def.bnf list -> typ list -> typ list ->
    84     typ list -> typ list list list -> term list list -> thm list list -> term list list ->
    85     thm list list -> local_theory -> lfp_sugar_thms
    86   val derive_coinduct_coiters_thms_for_types: BNF_Def.bnf list ->
    87     string * term list * term list list * ((term list list * term list list list)
    88       * (typ list * typ list list)) list ->
    89     thm -> thm list -> thm list -> thm list list -> BNF_Def.bnf list -> typ list -> typ list ->
    90     int list list -> int list list -> int list -> thm list list -> BNF_Ctr_Sugar.ctr_sugar list ->
    91     term list list -> thm list list -> (thm list -> thm list) -> local_theory -> gfp_sugar_thms
    92   val co_datatypes: BNF_FP_Util.fp_kind -> (mixfix list -> binding list -> binding list ->
    93       binding list list -> binding list -> (string * sort) list -> typ list * typ list list ->
    94       BNF_Def.bnf list -> local_theory -> BNF_FP_Util.fp_result * local_theory) ->
    95     (bool * bool) * (((((binding * (typ * sort)) list * binding) * (binding * binding)) * mixfix) *
    96       ((((binding * binding) * (binding * typ) list) * (binding * term) list) *
    97         mixfix) list) list ->
    98     local_theory -> local_theory
    99   val parse_co_datatype_cmd: BNF_FP_Util.fp_kind -> (mixfix list -> binding list -> binding list ->
   100       binding list list -> binding list -> (string * sort) list -> typ list * typ list list ->
   101       BNF_Def.bnf list -> local_theory -> BNF_FP_Util.fp_result * local_theory) ->
   102     (local_theory -> local_theory) parser
   103 end;
   104 
   105 structure BNF_FP_Def_Sugar : BNF_FP_DEF_SUGAR =
   106 struct
   107 
   108 open BNF_Util
   109 open BNF_Ctr_Sugar
   110 open BNF_Comp
   111 open BNF_Def
   112 open BNF_FP_Util
   113 open BNF_FP_Def_Sugar_Tactics
   114 
   115 val EqN = "Eq_";
   116 
   117 type fp_sugar =
   118   {T: typ,
   119    fp: fp_kind,
   120    index: int,
   121    pre_bnfs: bnf list,
   122    nested_bnfs: bnf list,
   123    nesting_bnfs: bnf list,
   124    fp_res: fp_result,
   125    ctr_defss: thm list list,
   126    ctr_sugars: ctr_sugar list,
   127    co_iterss: term list list,
   128    mapss: thm list list,
   129    co_inducts: thm list,
   130    co_iter_thmsss: thm list list list,
   131    disc_co_itersss: thm list list list,
   132    sel_co_iterssss: thm list list list list};
   133 
   134 fun of_fp_sugar f (fp_sugar as {index, ...}) = nth (f fp_sugar) index;
   135 
   136 fun eq_fp_sugar ({T = T1, fp = fp1, index = index1, fp_res = fp_res1, ...} : fp_sugar,
   137     {T = T2, fp = fp2, index = index2, fp_res = fp_res2, ...} : fp_sugar) =
   138   T1 = T2 andalso fp1 = fp2 andalso index1 = index2 andalso eq_fp_result (fp_res1, fp_res2);
   139 
   140 fun morph_fp_sugar phi {T, fp, index, pre_bnfs, nested_bnfs, nesting_bnfs, fp_res, ctr_defss,
   141     ctr_sugars, co_iterss, mapss, co_inducts, co_iter_thmsss, disc_co_itersss, sel_co_iterssss} =
   142   {T = Morphism.typ phi T, fp = fp, index = index, pre_bnfs = map (morph_bnf phi) pre_bnfs,
   143     nested_bnfs = map (morph_bnf phi) nested_bnfs, nesting_bnfs = map (morph_bnf phi) nesting_bnfs,
   144    fp_res = morph_fp_result phi fp_res,
   145    ctr_defss = map (map (Morphism.thm phi)) ctr_defss,
   146    ctr_sugars = map (morph_ctr_sugar phi) ctr_sugars,
   147    co_iterss = map (map (Morphism.term phi)) co_iterss,
   148    mapss = map (map (Morphism.thm phi)) mapss,
   149    co_inducts = map (Morphism.thm phi) co_inducts,
   150    co_iter_thmsss = map (map (map (Morphism.thm phi))) co_iter_thmsss,
   151    disc_co_itersss = map (map (map (Morphism.thm phi))) disc_co_itersss,
   152    sel_co_iterssss = map (map (map (map (Morphism.thm phi)))) sel_co_iterssss};
   153 
   154 val transfer_fp_sugar =
   155   morph_fp_sugar o Morphism.thm_morphism o Thm.transfer o Proof_Context.theory_of;
   156 
   157 structure Data = Generic_Data
   158 (
   159   type T = fp_sugar Symtab.table;
   160   val empty = Symtab.empty;
   161   val extend = I;
   162   val merge = Symtab.merge eq_fp_sugar;
   163 );
   164 
   165 fun fp_sugar_of ctxt =
   166   Symtab.lookup (Data.get (Context.Proof ctxt))
   167   #> Option.map (transfer_fp_sugar ctxt);
   168 
   169 fun fp_sugars_of ctxt =
   170   Symtab.fold (cons o transfer_fp_sugar ctxt o snd) (Data.get (Context.Proof ctxt)) [];
   171 
   172 fun co_induct_of (i :: _) = i;
   173 fun strong_co_induct_of [_, s] = s;
   174 
   175 (* TODO: register "sum" and "prod" as datatypes to enable N2M reduction for them *)
   176 
   177 fun register_fp_sugar key fp_sugar =
   178   Local_Theory.declaration {syntax = false, pervasive = true}
   179     (fn phi => Data.map (Symtab.default (key, morph_fp_sugar phi fp_sugar)));
   180 
   181 fun register_fp_sugars fp pre_bnfs nested_bnfs nesting_bnfs (fp_res as {Ts, ...}) ctr_defss
   182     ctr_sugars co_iterss mapss co_inducts co_iter_thmsss disc_co_itersss sel_co_iterssss lthy =
   183   (0, lthy)
   184   |> fold (fn T as Type (s, _) => fn (kk, lthy) => (kk + 1,
   185     register_fp_sugar s {T = T, fp = fp, index = kk, pre_bnfs = pre_bnfs,
   186         nested_bnfs = nested_bnfs, nesting_bnfs = nesting_bnfs, fp_res = fp_res,
   187         ctr_defss = ctr_defss, ctr_sugars = ctr_sugars, co_iterss = co_iterss, mapss = mapss,
   188         co_inducts = co_inducts, co_iter_thmsss = co_iter_thmsss, disc_co_itersss = disc_co_itersss,
   189         sel_co_iterssss = sel_co_iterssss}
   190       lthy)) Ts
   191   |> snd;
   192 
   193 (* This function could produce clashes in contrived examples (e.g., "x.A", "x.x_A", "y.A"). *)
   194 fun quasi_unambiguous_case_names names =
   195   let
   196     val ps = map (`Long_Name.base_name) names;
   197     val dups = Library.duplicates (op =) (map fst ps);
   198     fun underscore s =
   199       let val ss = space_explode Long_Name.separator s in
   200         space_implode "_" (drop (length ss - 2) ss)
   201       end;
   202   in
   203     map (fn (base, full) => if member (op =) dups base then underscore full else base) ps
   204   end;
   205 
   206 val id_def = @{thm id_def};
   207 val mp_conj = @{thm mp_conj};
   208 
   209 val nitpick_attrs = @{attributes [nitpick_simp]};
   210 val code_nitpick_simp_attrs = Code.add_default_eqn_attrib :: nitpick_attrs;
   211 val simp_attrs = @{attributes [simp]};
   212 
   213 fun tvar_subst thy Ts Us =
   214   Vartab.fold (cons o apsnd snd) (fold (Sign.typ_match thy) (Ts ~~ Us) Vartab.empty) [];
   215 
   216 val exists_subtype_in = Term.exists_subtype o member (op =);
   217 
   218 val lists_bmoc = fold (fn xs => fn t => Term.list_comb (t, xs));
   219 
   220 fun flat_rec_arg_args xss =
   221   (* FIXME (once the old datatype package is phased out): The first line below gives the preferred
   222      order. The second line is for compatibility with the old datatype package. *)
   223 (*
   224   flat xss
   225 *)
   226   map hd xss @ maps tl xss;
   227 
   228 fun flat_corec_predss_getterss qss fss = maps (op @) (qss ~~ fss);
   229 
   230 fun flat_corec_preds_predsss_gettersss [] [qss] [fss] = flat_corec_predss_getterss qss fss
   231   | flat_corec_preds_predsss_gettersss (p :: ps) (qss :: qsss) (fss :: fsss) =
   232     p :: flat_corec_predss_getterss qss fss @ flat_corec_preds_predsss_gettersss ps qsss fsss;
   233 
   234 fun mk_tupled_fun x f xs = HOLogic.tupled_lambda x (Term.list_comb (f, xs));
   235 fun mk_uncurried2_fun f xss =
   236   mk_tupled_fun (HOLogic.mk_tuple (map HOLogic.mk_tuple xss)) f (flat_rec_arg_args xss);
   237 
   238 fun mk_flip (x, Type (_, [T1, Type (_, [T2, T3])])) =
   239   Abs ("x", T1, Abs ("y", T2, Var (x, T2 --> T1 --> T3) $ Bound 0 $ Bound 1));
   240 
   241 fun flip_rels lthy n thm =
   242   let
   243     val Rs = Term.add_vars (prop_of thm) [];
   244     val Rs' = rev (drop (length Rs - n) Rs);
   245     val cRs = map (fn f => (certify lthy (Var f), certify lthy (mk_flip f))) Rs';
   246   in
   247     Drule.cterm_instantiate cRs thm
   248   end;
   249 
   250 fun mk_ctor_or_dtor get_T Ts t =
   251   let val Type (_, Ts0) = get_T (fastype_of t) in
   252     Term.subst_atomic_types (Ts0 ~~ Ts) t
   253   end;
   254 
   255 val mk_ctor = mk_ctor_or_dtor range_type;
   256 val mk_dtor = mk_ctor_or_dtor domain_type;
   257 
   258 fun mk_co_iter thy fp fpT Cs t =
   259   let
   260     val (f_Cs, Type (_, [prebody, body])) = strip_fun_type (fastype_of t);
   261     val fpT0 = fp_case fp prebody body;
   262     val Cs0 = distinct (op =) (map (fp_case fp body_type domain_type) f_Cs);
   263     val rho = tvar_subst thy (fpT0 :: Cs0) (fpT :: Cs);
   264   in
   265     Term.subst_TVars rho t
   266   end;
   267 
   268 fun mk_co_iters thy fp fpTs Cs ts0 =
   269   let
   270     val nn = length fpTs;
   271     val (fpTs0, Cs0) =
   272       map ((fp = Greatest_FP ? swap) o dest_funT o snd o strip_typeN nn o fastype_of) ts0
   273       |> split_list;
   274     val rho = tvar_subst thy (fpTs0 @ Cs0) (fpTs @ Cs);
   275   in
   276     map (Term.subst_TVars rho) ts0
   277   end;
   278 
   279 val mk_fp_iter_fun_types = binder_fun_types o fastype_of;
   280 
   281 fun unzip_recT (Type (@{type_name prod}, _)) T = [T]
   282   | unzip_recT _ (Type (@{type_name prod}, Ts)) = Ts
   283   | unzip_recT _ T = [T];
   284 
   285 fun unzip_corecT (Type (@{type_name sum}, _)) T = [T]
   286   | unzip_corecT _ (Type (@{type_name sum}, Ts)) = Ts
   287   | unzip_corecT _ T = [T];
   288 
   289 fun mk_map live Ts Us t =
   290   let val (Type (_, Ts0), Type (_, Us0)) = strip_typeN (live + 1) (fastype_of t) |>> List.last in
   291     Term.subst_atomic_types (Ts0 @ Us0 ~~ Ts @ Us) t
   292   end;
   293 
   294 fun mk_rel live Ts Us t =
   295   let val [Type (_, Ts0), Type (_, Us0)] = binder_types (snd (strip_typeN live (fastype_of t))) in
   296     Term.subst_atomic_types (Ts0 @ Us0 ~~ Ts @ Us) t
   297   end;
   298 
   299 local
   300 
   301 fun build_map_or_rel mk const of_bnf dest lthy build_simple =
   302   let
   303     fun build (TU as (T, U)) =
   304       if T = U then
   305         const T
   306       else
   307         (case TU of
   308           (Type (s, Ts), Type (s', Us)) =>
   309           if s = s' then
   310             let
   311               val bnf = the (bnf_of lthy s);
   312               val live = live_of_bnf bnf;
   313               val mapx = mk live Ts Us (of_bnf bnf);
   314               val TUs' = map dest (fst (strip_typeN live (fastype_of mapx)));
   315             in Term.list_comb (mapx, map build TUs') end
   316           else
   317             build_simple TU
   318         | _ => build_simple TU);
   319   in build end;
   320 
   321 in
   322 
   323 val build_map = build_map_or_rel mk_map HOLogic.id_const map_of_bnf dest_funT;
   324 val build_rel = build_map_or_rel mk_rel HOLogic.eq_const rel_of_bnf dest_pred2T;
   325 
   326 end;
   327 
   328 val dummy_var_name = "?f"
   329 
   330 fun mk_map_pattern ctxt s =
   331   let
   332     val bnf = the (bnf_of ctxt s);
   333     val mapx = map_of_bnf bnf;
   334     val live = live_of_bnf bnf;
   335     val (f_Ts, _) = strip_typeN live (fastype_of mapx);
   336     val fs = map_index (fn (i, T) => Var ((dummy_var_name, i), T)) f_Ts;
   337   in
   338     (mapx, betapplys (mapx, fs))
   339   end;
   340 
   341 fun dest_map ctxt s call =
   342   let
   343     val (map0, pat) = mk_map_pattern ctxt s;
   344     val (_, tenv) = fo_match ctxt call pat;
   345   in
   346     (map0, Vartab.fold_rev (fn (_, (_, f)) => cons f) tenv [])
   347   end;
   348 
   349 fun liveness_of_fp_bnf n bnf =
   350   (case T_of_bnf bnf of
   351     Type (_, Ts) => map (not o member (op =) (deads_of_bnf bnf)) Ts
   352   | _ => replicate n false);
   353 
   354 fun cannot_merge_types () = error "Mutually recursive types must have the same type parameters";
   355 
   356 fun merge_type_arg T T' = if T = T' then T else cannot_merge_types ();
   357 
   358 fun merge_type_args (As, As') =
   359   if length As = length As' then map2 merge_type_arg As As' else cannot_merge_types ();
   360 
   361 fun reassoc_conjs thm =
   362   reassoc_conjs (thm RS @{thm conj_assoc[THEN iffD1]})
   363   handle THM _ => thm;
   364 
   365 fun type_args_named_constrained_of ((((ncAs, _), _), _), _) = ncAs;
   366 fun type_binding_of ((((_, b), _), _), _) = b;
   367 fun map_binding_of (((_, (b, _)), _), _) = b;
   368 fun rel_binding_of (((_, (_, b)), _), _) = b;
   369 fun mixfix_of ((_, mx), _) = mx;
   370 fun ctr_specs_of (_, ctr_specs) = ctr_specs;
   371 
   372 fun disc_of ((((disc, _), _), _), _) = disc;
   373 fun ctr_of ((((_, ctr), _), _), _) = ctr;
   374 fun args_of (((_, args), _), _) = args;
   375 fun defaults_of ((_, ds), _) = ds;
   376 fun ctr_mixfix_of (_, mx) = mx;
   377 
   378 fun add_nesty_bnf_names Us =
   379   let
   380     fun add (Type (s, Ts)) ss =
   381         let val (needs, ss') = fold_map add Ts ss in
   382           if exists I needs then (true, insert (op =) s ss') else (false, ss')
   383         end
   384       | add T ss = (member (op =) Us T, ss);
   385   in snd oo add end;
   386 
   387 fun nesty_bnfs ctxt ctr_Tsss Us =
   388   map_filter (bnf_of ctxt) (fold (fold (fold (add_nesty_bnf_names Us))) ctr_Tsss []);
   389 
   390 fun indexify proj xs f p = f (find_index (curry op = (proj p)) xs) p;
   391 
   392 type lfp_sugar_thms =
   393   (thm list * thm * Args.src list)
   394   * (thm list list * thm list list * Args.src list)
   395 
   396 type gfp_sugar_thms =
   397   ((thm list * thm) list * Args.src list)
   398   * (thm list list * thm list list * Args.src list)
   399   * (thm list list * thm list list) * (thm list list * thm list list * Args.src list)
   400   * (thm list list * thm list list * Args.src list)
   401   * (thm list list list * thm list list list * Args.src list);
   402 
   403 fun mk_iter_fun_arg_types0 n ms = map2 dest_tupleT ms o dest_sumTN_balanced n o domain_type;
   404 
   405 fun mk_iter_fun_arg_types ctr_Tsss ns mss =
   406   mk_fp_iter_fun_types
   407   #> map3 mk_iter_fun_arg_types0 ns mss
   408   #> map2 (map2 (map2 unzip_recT)) ctr_Tsss;
   409 
   410 fun mk_iters_args_types ctr_Tsss Cs ns mss ctor_iter_fun_Tss lthy =
   411   let
   412     val Css = map2 replicate ns Cs;
   413     val y_Tsss = map3 mk_iter_fun_arg_types0 ns mss (map un_fold_of ctor_iter_fun_Tss);
   414     val g_Tss = map2 (fn C => map (fn y_Ts => y_Ts ---> C)) Cs y_Tsss;
   415 
   416     val ((gss, ysss), lthy) =
   417       lthy
   418       |> mk_Freess "f" g_Tss
   419       ||>> mk_Freesss "x" y_Tsss;
   420 
   421     val y_Tssss = map (map (map single)) y_Tsss;
   422     val yssss = map (map (map single)) ysss;
   423 
   424     val z_Tssss =
   425       map4 (fn n => fn ms => fn ctr_Tss => fn ctor_iter_fun_Ts =>
   426           map3 (fn m => fn ctr_Ts => fn ctor_iter_fun_T =>
   427               map2 unzip_recT ctr_Ts (dest_tupleT m ctor_iter_fun_T))
   428             ms ctr_Tss (dest_sumTN_balanced n (domain_type (co_rec_of ctor_iter_fun_Ts))))
   429         ns mss ctr_Tsss ctor_iter_fun_Tss;
   430 
   431     val z_Tsss' = map (map flat_rec_arg_args) z_Tssss;
   432     val h_Tss = map2 (map2 (curry op --->)) z_Tsss' Css;
   433 
   434     val hss = map2 (map2 retype_free) h_Tss gss;
   435     val zssss_hd = map2 (map2 (map2 (retype_free o hd))) z_Tssss ysss;
   436     val (zssss_tl, lthy) =
   437       lthy
   438       |> mk_Freessss "y" (map (map (map tl)) z_Tssss);
   439     val zssss = map2 (map2 (map2 cons)) zssss_hd zssss_tl;
   440   in
   441     ([(g_Tss, y_Tssss, gss, yssss), (h_Tss, z_Tssss, hss, zssss)], lthy)
   442   end;
   443 
   444 fun mk_coiter_fun_arg_types0 ctr_Tsss Cs ns fun_Ts =
   445   let
   446     (*avoid "'a itself" arguments in coiterators*)
   447     fun repair_arity [[]] = [[@{typ unit}]]
   448       | repair_arity Tss = Tss;
   449 
   450     val ctr_Tsss' = map repair_arity ctr_Tsss;
   451     val f_sum_prod_Ts = map range_type fun_Ts;
   452     val f_prod_Tss = map2 dest_sumTN_balanced ns f_sum_prod_Ts;
   453     val f_Tsss = map2 (map2 (dest_tupleT o length)) ctr_Tsss' f_prod_Tss;
   454     val f_Tssss = map3 (fn C => map2 (map2 (map (curry op --> C) oo unzip_corecT)))
   455       Cs ctr_Tsss' f_Tsss;
   456     val q_Tssss = map (map (map (fn [_] => [] | [_, T] => [mk_pred1T (domain_type T)]))) f_Tssss;
   457   in
   458     (q_Tssss, f_Tsss, f_Tssss, f_sum_prod_Ts)
   459   end;
   460 
   461 fun mk_coiter_p_pred_types Cs ns = map2 (fn n => replicate (Int.max (0, n - 1)) o mk_pred1T) ns Cs;
   462 
   463 fun mk_coiter_fun_arg_types ctr_Tsss Cs ns dtor_coiter =
   464   (mk_coiter_p_pred_types Cs ns,
   465    mk_fp_iter_fun_types dtor_coiter |> mk_coiter_fun_arg_types0 ctr_Tsss Cs ns);
   466 
   467 fun mk_coiters_args_types ctr_Tsss Cs ns dtor_coiter_fun_Tss lthy =
   468   let
   469     val p_Tss = mk_coiter_p_pred_types Cs ns;
   470 
   471     fun mk_types get_Ts =
   472       let
   473         val fun_Ts = map get_Ts dtor_coiter_fun_Tss;
   474         val (q_Tssss, f_Tsss, f_Tssss, f_sum_prod_Ts) = mk_coiter_fun_arg_types0 ctr_Tsss Cs ns fun_Ts;
   475         val pf_Tss = map3 flat_corec_preds_predsss_gettersss p_Tss q_Tssss f_Tssss;
   476       in
   477         (q_Tssss, f_Tsss, f_Tssss, (f_sum_prod_Ts, pf_Tss))
   478       end;
   479 
   480     val (r_Tssss, g_Tsss, g_Tssss, unfold_types) = mk_types un_fold_of;
   481     val (s_Tssss, h_Tsss, h_Tssss, corec_types) = mk_types co_rec_of;
   482 
   483     val ((((Free (z, _), cs), pss), gssss), lthy) =
   484       lthy
   485       |> yield_singleton (mk_Frees "z") dummyT
   486       ||>> mk_Frees "a" Cs
   487       ||>> mk_Freess "p" p_Tss
   488       ||>> mk_Freessss "g" g_Tssss;
   489     val rssss = map (map (map (fn [] => []))) r_Tssss;
   490 
   491     val hssss_hd = map2 (map2 (map2 (fn T :: _ => fn [g] => retype_free T g))) h_Tssss gssss;
   492     val ((sssss, hssss_tl), lthy) =
   493       lthy
   494       |> mk_Freessss "q" s_Tssss
   495       ||>> mk_Freessss "h" (map (map (map tl)) h_Tssss);
   496     val hssss = map2 (map2 (map2 cons)) hssss_hd hssss_tl;
   497 
   498     val cpss = map2 (map o rapp) cs pss;
   499 
   500     fun build_sum_inj mk_inj = build_map lthy (uncurry mk_inj o dest_sumT o snd);
   501 
   502     fun build_dtor_coiter_arg _ [] [cf] = cf
   503       | build_dtor_coiter_arg T [cq] [cf, cf'] =
   504         mk_If cq (build_sum_inj Inl_const (fastype_of cf, T) $ cf)
   505           (build_sum_inj Inr_const (fastype_of cf', T) $ cf');
   506 
   507     fun mk_args qssss fssss f_Tsss =
   508       let
   509         val pfss = map3 flat_corec_preds_predsss_gettersss pss qssss fssss;
   510         val cqssss = map2 (map o map o map o rapp) cs qssss;
   511         val cfssss = map2 (map o map o map o rapp) cs fssss;
   512         val cqfsss = map3 (map3 (map3 build_dtor_coiter_arg)) f_Tsss cqssss cfssss;
   513       in (pfss, cqfsss) end;
   514 
   515     val unfold_args = mk_args rssss gssss g_Tsss;
   516     val corec_args = mk_args sssss hssss h_Tsss;
   517   in
   518     ((z, cs, cpss, [(unfold_args, unfold_types), (corec_args, corec_types)]), lthy)
   519   end;
   520 
   521 fun mk_co_iters_prelims fp ctr_Tsss fpTs Cs ns mss xtor_co_iterss0 lthy =
   522   let
   523     val thy = Proof_Context.theory_of lthy;
   524 
   525     val (xtor_co_iter_fun_Tss, xtor_co_iterss) =
   526       map (mk_co_iters thy fp fpTs Cs #> `(mk_fp_iter_fun_types o hd)) (transpose xtor_co_iterss0)
   527       |> apsnd transpose o apfst transpose o split_list;
   528 
   529     val ((iters_args_types, coiters_args_types), lthy') =
   530       if fp = Least_FP then
   531         mk_iters_args_types ctr_Tsss Cs ns mss xtor_co_iter_fun_Tss lthy |>> (rpair NONE o SOME)
   532       else
   533         mk_coiters_args_types ctr_Tsss Cs ns xtor_co_iter_fun_Tss lthy |>> (pair NONE o SOME)
   534   in
   535     ((xtor_co_iterss, iters_args_types, coiters_args_types), lthy')
   536   end;
   537 
   538 fun mk_iter_body ctor_iter fss xssss =
   539   Term.list_comb (ctor_iter, map2 (mk_sum_caseN_balanced oo map2 mk_uncurried2_fun) fss xssss);
   540 
   541 fun mk_preds_getterss_join c cps sum_prod_T cqfss =
   542   let val n = length cqfss in
   543     Term.lambda c (mk_IfN sum_prod_T cps
   544       (map2 (mk_InN_balanced sum_prod_T n) (map HOLogic.mk_tuple cqfss) (1 upto n)))
   545   end;
   546 
   547 fun mk_coiter_body cs cpss f_sum_prod_Ts cqfsss dtor_coiter =
   548   Term.list_comb (dtor_coiter, map4 mk_preds_getterss_join cs cpss f_sum_prod_Ts cqfsss);
   549 
   550 fun define_co_iters fp fpT Cs binding_specs lthy0 =
   551   let
   552     val thy = Proof_Context.theory_of lthy0;
   553 
   554     val maybe_conceal_def_binding = Thm.def_binding
   555       #> Config.get lthy0 bnf_note_all = false ? Binding.conceal;
   556 
   557     val ((csts, defs), (lthy', lthy)) = lthy0
   558       |> apfst split_list o fold_map (fn (b, spec) =>
   559         Specification.definition (SOME (b, NONE, NoSyn), ((maybe_conceal_def_binding b, []), spec))
   560         #>> apsnd snd) binding_specs
   561       ||> `Local_Theory.restore;
   562 
   563     val phi = Proof_Context.export_morphism lthy lthy';
   564 
   565     val csts' = map (mk_co_iter thy fp fpT Cs o Morphism.term phi) csts;
   566     val defs' = map (Morphism.thm phi) defs;
   567   in
   568     ((csts', defs'), lthy')
   569   end;
   570 
   571 fun define_iters iterNs iter_args_typess' mk_binding fpTs Cs ctor_iters lthy =
   572   let
   573     val nn = length fpTs;
   574 
   575     val fpT_to_C as Type (_, [fpT, _]) = snd (strip_typeN nn (fastype_of (hd ctor_iters)));
   576 
   577     fun generate_iter suf (f_Tss, _, fss, xssss) ctor_iter =
   578       let
   579         val res_T = fold_rev (curry op --->) f_Tss fpT_to_C;
   580         val b = mk_binding suf;
   581         val spec =
   582           mk_Trueprop_eq (lists_bmoc fss (Free (Binding.name_of b, res_T)),
   583             mk_iter_body ctor_iter fss xssss);
   584       in (b, spec) end;
   585   in
   586     define_co_iters Least_FP fpT Cs (map3 generate_iter iterNs iter_args_typess' ctor_iters) lthy
   587   end;
   588 
   589 fun define_coiters coiterNs (_, cs, cpss, coiter_args_typess') mk_binding fpTs Cs dtor_coiters
   590     lthy =
   591   let
   592     val nn = length fpTs;
   593 
   594     val C_to_fpT as Type (_, [_, fpT]) = snd (strip_typeN nn (fastype_of (hd dtor_coiters)));
   595 
   596     fun generate_coiter suf ((pfss, cqfsss), (f_sum_prod_Ts, pf_Tss)) dtor_coiter =
   597       let
   598         val res_T = fold_rev (curry op --->) pf_Tss C_to_fpT;
   599         val b = mk_binding suf;
   600         val spec =
   601           mk_Trueprop_eq (lists_bmoc pfss (Free (Binding.name_of b, res_T)),
   602             mk_coiter_body cs cpss f_sum_prod_Ts cqfsss dtor_coiter);
   603       in (b, spec) end;
   604   in
   605     define_co_iters Greatest_FP fpT Cs
   606       (map3 generate_coiter coiterNs coiter_args_typess' dtor_coiters) lthy
   607   end;
   608 
   609 fun derive_induct_iters_thms_for_types pre_bnfs [fold_args_types, rec_args_types] ctor_induct
   610     ctor_iter_thmss nesting_bnfs nested_bnfs fpTs Cs Xs ctrXs_Tsss ctrss ctr_defss iterss iter_defss
   611     lthy =
   612   let
   613     val iterss' = transpose iterss;
   614     val iter_defss' = transpose iter_defss;
   615 
   616     val [folds, recs] = iterss';
   617     val [fold_defs, rec_defs] = iter_defss';
   618 
   619     val ctr_Tsss = map (map (binder_types o fastype_of)) ctrss;
   620 
   621     val nn = length pre_bnfs;
   622     val ns = map length ctr_Tsss;
   623     val mss = map (map length) ctr_Tsss;
   624 
   625     val pre_map_defs = map map_def_of_bnf pre_bnfs;
   626     val pre_set_defss = map set_defs_of_bnf pre_bnfs;
   627     val nesting_map_idents = map (unfold_thms lthy [id_def] o map_id0_of_bnf) nesting_bnfs;
   628     val nested_map_idents = map (unfold_thms lthy [id_def] o map_id0_of_bnf) nested_bnfs;
   629     val nested_set_maps = maps set_map_of_bnf nested_bnfs;
   630 
   631     val fp_b_names = map base_name_of_typ fpTs;
   632 
   633     val ((((ps, ps'), xsss), us'), names_lthy) =
   634       lthy
   635       |> mk_Frees' "P" (map mk_pred1T fpTs)
   636       ||>> mk_Freesss "x" ctr_Tsss
   637       ||>> Variable.variant_fixes fp_b_names;
   638 
   639     val us = map2 (curry Free) us' fpTs;
   640 
   641     fun mk_sets_nested bnf =
   642       let
   643         val Type (T_name, Us) = T_of_bnf bnf;
   644         val lives = lives_of_bnf bnf;
   645         val sets = sets_of_bnf bnf;
   646         fun mk_set U =
   647           (case find_index (curry op = U) lives of
   648             ~1 => Term.dummy
   649           | i => nth sets i);
   650       in
   651         (T_name, map mk_set Us)
   652       end;
   653 
   654     val setss_nested = map mk_sets_nested nested_bnfs;
   655 
   656     val (induct_thms, induct_thm) =
   657       let
   658         fun mk_set Ts t =
   659           let val Type (_, Ts0) = domain_type (fastype_of t) in
   660             Term.subst_atomic_types (Ts0 ~~ Ts) t
   661           end;
   662 
   663         fun mk_raw_prem_prems _ (x as Free (_, Type _)) (X as TFree _) =
   664             [([], (find_index (curry op = X) Xs + 1, x))]
   665           | mk_raw_prem_prems names_lthy (x as Free (s, Type (T_name, Ts0))) (Type (_, Xs_Ts0)) =
   666             (case AList.lookup (op =) setss_nested T_name of
   667               NONE => []
   668             | SOME raw_sets0 =>
   669               let
   670                 val (Xs_Ts, (Ts, raw_sets)) =
   671                   filter (exists_subtype_in Xs o fst) (Xs_Ts0 ~~ (Ts0 ~~ raw_sets0))
   672                   |> split_list ||> split_list;
   673                 val sets = map (mk_set Ts0) raw_sets;
   674                 val (ys, names_lthy') = names_lthy |> mk_Frees s Ts;
   675                 val xysets = map (pair x) (ys ~~ sets);
   676                 val ppremss = map2 (mk_raw_prem_prems names_lthy') ys Xs_Ts;
   677               in
   678                 flat (map2 (map o apfst o cons) xysets ppremss)
   679               end)
   680           | mk_raw_prem_prems _ _ _ = [];
   681 
   682         fun close_prem_prem xs t =
   683           fold_rev Logic.all (map Free (drop (nn + length xs)
   684             (rev (Term.add_frees t (map dest_Free xs @ ps'))))) t;
   685 
   686         fun mk_prem_prem xs (xysets, (j, x)) =
   687           close_prem_prem xs (Logic.list_implies (map (fn (x', (y, set)) =>
   688               HOLogic.mk_Trueprop (HOLogic.mk_mem (y, set $ x'))) xysets,
   689             HOLogic.mk_Trueprop (nth ps (j - 1) $ x)));
   690 
   691         fun mk_raw_prem phi ctr ctr_Ts ctrXs_Ts =
   692           let
   693             val (xs, names_lthy') = names_lthy |> mk_Frees "x" ctr_Ts;
   694             val pprems = flat (map2 (mk_raw_prem_prems names_lthy') xs ctrXs_Ts);
   695           in (xs, pprems, HOLogic.mk_Trueprop (phi $ Term.list_comb (ctr, xs))) end;
   696 
   697         fun mk_prem (xs, raw_pprems, concl) =
   698           fold_rev Logic.all xs (Logic.list_implies (map (mk_prem_prem xs) raw_pprems, concl));
   699 
   700         val raw_premss = map4 (map3 o mk_raw_prem) ps ctrss ctr_Tsss ctrXs_Tsss;
   701 
   702         val goal =
   703           Library.foldr (Logic.list_implies o apfst (map mk_prem)) (raw_premss,
   704             HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj (map2 (curry op $) ps us)));
   705 
   706         val kksss = map (map (map (fst o snd) o #2)) raw_premss;
   707 
   708         val ctor_induct' = ctor_induct OF (map mk_sumEN_tupled_balanced mss);
   709 
   710         val thm =
   711           Goal.prove_sorry lthy [] [] goal (fn {context = ctxt, ...} =>
   712             mk_induct_tac ctxt nn ns mss kksss (flat ctr_defss) ctor_induct' nested_set_maps
   713               pre_set_defss)
   714           |> singleton (Proof_Context.export names_lthy lthy)
   715           |> Thm.close_derivation;
   716       in
   717         `(conj_dests nn) thm
   718       end;
   719 
   720     val induct_cases = quasi_unambiguous_case_names (maps (map name_of_ctr) ctrss);
   721     val induct_case_names_attr = Attrib.internal (K (Rule_Cases.case_names induct_cases));
   722 
   723     val xctrss = map2 (map2 (curry Term.list_comb)) ctrss xsss;
   724 
   725     fun mk_iter_thmss (_, x_Tssss, fss, _) iters iter_defs ctor_iter_thms =
   726       let
   727         val fiters = map (lists_bmoc fss) iters;
   728 
   729         fun mk_goal fss fiter xctr f xs fxs =
   730           fold_rev (fold_rev Logic.all) (xs :: fss)
   731             (mk_Trueprop_eq (fiter $ xctr, Term.list_comb (f, fxs)));
   732 
   733         fun maybe_tick (T, U) u f =
   734           if try (fst o HOLogic.dest_prodT) U = SOME T then
   735             Term.lambda u (HOLogic.mk_prod (u, f $ u))
   736           else
   737             f;
   738 
   739         fun build_iter (x as Free (_, T)) U =
   740           if T = U then
   741             x
   742           else
   743             build_map lthy (indexify (perhaps (try (snd o HOLogic.dest_prodT)) o snd) Cs
   744               (fn kk => fn TU => maybe_tick TU (nth us kk) (nth fiters kk))) (T, U) $ x;
   745 
   746         val fxsss = map2 (map2 (flat_rec_arg_args oo map2 (map o build_iter))) xsss x_Tssss;
   747 
   748         val goalss = map5 (map4 o mk_goal fss) fiters xctrss fss xsss fxsss;
   749 
   750         val tacss =
   751           map2 (map o mk_iter_tac pre_map_defs (nested_map_idents @ nesting_map_idents) iter_defs)
   752             ctor_iter_thms ctr_defss;
   753 
   754         fun prove goal tac =
   755           Goal.prove_sorry lthy [] [] goal (tac o #context)
   756           |> Thm.close_derivation;
   757       in
   758         map2 (map2 prove) goalss tacss
   759       end;
   760 
   761     val fold_thmss = mk_iter_thmss fold_args_types folds fold_defs (map un_fold_of ctor_iter_thmss);
   762     val rec_thmss = mk_iter_thmss rec_args_types recs rec_defs (map co_rec_of ctor_iter_thmss);
   763   in
   764     ((induct_thms, induct_thm, [induct_case_names_attr]),
   765      (fold_thmss, rec_thmss, code_nitpick_simp_attrs @ simp_attrs))
   766   end;
   767 
   768 fun derive_coinduct_coiters_thms_for_types pre_bnfs (z, cs, cpss,
   769       coiters_args_types as [((pgss, crgsss), _), ((phss, cshsss), _)])
   770     dtor_coinduct dtor_injects dtor_ctors dtor_coiter_thmss nesting_bnfs fpTs Cs kss mss ns
   771     ctr_defss ctr_sugars coiterss coiter_defss export_args lthy =
   772   let
   773     fun mk_ctor_dtor_coiter_thm dtor_inject dtor_ctor coiter =
   774       iffD1 OF [dtor_inject, trans OF [coiter, dtor_ctor RS sym]];
   775 
   776     val ctor_dtor_coiter_thmss =
   777       map3 (map oo mk_ctor_dtor_coiter_thm) dtor_injects dtor_ctors dtor_coiter_thmss;
   778 
   779     val coiterss' = transpose coiterss;
   780     val coiter_defss' = transpose coiter_defss;
   781 
   782     val [unfold_defs, corec_defs] = coiter_defss';
   783 
   784     val nn = length pre_bnfs;
   785 
   786     val pre_map_defs = map map_def_of_bnf pre_bnfs;
   787     val pre_rel_defs = map rel_def_of_bnf pre_bnfs;
   788     val nesting_map_idents = map (unfold_thms lthy [id_def] o map_id0_of_bnf) nesting_bnfs;
   789     val nesting_rel_eqs = map rel_eq_of_bnf nesting_bnfs;
   790 
   791     val fp_b_names = map base_name_of_typ fpTs;
   792 
   793     val ctrss = map #ctrs ctr_sugars;
   794     val discss = map #discs ctr_sugars;
   795     val selsss = map #selss ctr_sugars;
   796     val exhausts = map #exhaust ctr_sugars;
   797     val disc_thmsss = map #disc_thmss ctr_sugars;
   798     val discIss = map #discIs ctr_sugars;
   799     val sel_thmsss = map #sel_thmss ctr_sugars;
   800 
   801     val (((rs, us'), vs'), names_lthy) =
   802       lthy
   803       |> mk_Frees "R" (map (fn T => mk_pred2T T T) fpTs)
   804       ||>> Variable.variant_fixes fp_b_names
   805       ||>> Variable.variant_fixes (map (suffix "'") fp_b_names);
   806 
   807     val us = map2 (curry Free) us' fpTs;
   808     val udiscss = map2 (map o rapp) us discss;
   809     val uselsss = map2 (map o map o rapp) us selsss;
   810 
   811     val vs = map2 (curry Free) vs' fpTs;
   812     val vdiscss = map2 (map o rapp) vs discss;
   813     val vselsss = map2 (map o map o rapp) vs selsss;
   814 
   815     val coinduct_thms_pairs =
   816       let
   817         val uvrs = map3 (fn r => fn u => fn v => r $ u $ v) rs us vs;
   818         val uv_eqs = map2 (curry HOLogic.mk_eq) us vs;
   819         val strong_rs =
   820           map4 (fn u => fn v => fn uvr => fn uv_eq =>
   821             fold_rev Term.lambda [u, v] (HOLogic.mk_disj (uvr, uv_eq))) us vs uvrs uv_eqs;
   822 
   823         (* TODO: generalize (cf. "build_map") *)
   824         fun build_rel rs' T =
   825           (case find_index (curry op = T) fpTs of
   826             ~1 =>
   827             if exists_subtype_in fpTs T then
   828               let
   829                 val Type (s, Ts) = T
   830                 val bnf = the (bnf_of lthy s);
   831                 val live = live_of_bnf bnf;
   832                 val rel = mk_rel live Ts Ts (rel_of_bnf bnf);
   833                 val Ts' = map domain_type (fst (strip_typeN live (fastype_of rel)));
   834               in Term.list_comb (rel, map (build_rel rs') Ts') end
   835             else
   836               HOLogic.eq_const T
   837           | kk => nth rs' kk);
   838 
   839         fun build_rel_app rs' usel vsel = fold rapp [usel, vsel] (build_rel rs' (fastype_of usel));
   840 
   841         fun mk_prem_ctr_concls rs' n k udisc usels vdisc vsels =
   842           (if k = n then [] else [HOLogic.mk_eq (udisc, vdisc)]) @
   843           (if null usels then
   844              []
   845            else
   846              [Library.foldr HOLogic.mk_imp (if n = 1 then [] else [udisc, vdisc],
   847                 Library.foldr1 HOLogic.mk_conj (map2 (build_rel_app rs') usels vsels))]);
   848 
   849         fun mk_prem_concl rs' n udiscs uselss vdiscs vselss =
   850           Library.foldr1 HOLogic.mk_conj
   851             (flat (map5 (mk_prem_ctr_concls rs' n) (1 upto n) udiscs uselss vdiscs vselss))
   852           handle List.Empty => @{term True};
   853 
   854         fun mk_prem rs' uvr u v n udiscs uselss vdiscs vselss =
   855           fold_rev Logic.all [u, v] (Logic.mk_implies (HOLogic.mk_Trueprop uvr,
   856             HOLogic.mk_Trueprop (mk_prem_concl rs' n udiscs uselss vdiscs vselss)));
   857 
   858         val concl =
   859           HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj
   860             (map3 (fn uvr => fn u => fn v => HOLogic.mk_imp (uvr, HOLogic.mk_eq (u, v)))
   861                uvrs us vs));
   862 
   863         fun mk_goal rs' =
   864           Logic.list_implies (map8 (mk_prem rs') uvrs us vs ns udiscss uselsss vdiscss vselsss,
   865             concl);
   866 
   867         val goals = map mk_goal [rs, strong_rs];
   868 
   869         fun prove dtor_coinduct' goal =
   870           Goal.prove_sorry lthy [] [] goal (fn {context = ctxt, ...} =>
   871             mk_coinduct_tac ctxt nesting_rel_eqs nn ns dtor_coinduct' pre_rel_defs dtor_ctors
   872               exhausts ctr_defss disc_thmsss sel_thmsss)
   873           |> singleton (Proof_Context.export names_lthy lthy)
   874           |> Thm.close_derivation;
   875 
   876         fun postproc nn thm =
   877           Thm.permute_prems 0 nn
   878             (if nn = 1 then thm RS mp else funpow nn (fn thm => reassoc_conjs (thm RS mp_conj)) thm)
   879           |> Drule.zero_var_indexes
   880           |> `(conj_dests nn);
   881 
   882         val rel_eqs = map rel_eq_of_bnf pre_bnfs;
   883         val rel_monos = map rel_mono_of_bnf pre_bnfs;
   884         val dtor_coinducts =
   885           [dtor_coinduct, mk_strong_coinduct_thm dtor_coinduct rel_eqs rel_monos lthy];
   886       in
   887         map2 (postproc nn oo prove) dtor_coinducts goals
   888       end;
   889 
   890     fun mk_coinduct_concls ms discs ctrs =
   891       let
   892         fun mk_disc_concl disc = [name_of_disc disc];
   893         fun mk_ctr_concl 0 _ = []
   894           | mk_ctr_concl _ ctor = [name_of_ctr ctor];
   895         val disc_concls = map mk_disc_concl (fst (split_last discs)) @ [[]];
   896         val ctr_concls = map2 mk_ctr_concl ms ctrs;
   897       in
   898         flat (map2 append disc_concls ctr_concls)
   899       end;
   900 
   901     val coinduct_cases = quasi_unambiguous_case_names (map (prefix EqN) fp_b_names);
   902     val coinduct_conclss =
   903       map3 (quasi_unambiguous_case_names ooo mk_coinduct_concls) mss discss ctrss;
   904 
   905     fun mk_maybe_not pos = not pos ? HOLogic.mk_not;
   906 
   907     val fcoiterss' as [gunfolds, hcorecs] =
   908       map2 (fn (pfss, _) => map (lists_bmoc pfss)) (map fst coiters_args_types) coiterss';
   909 
   910     val (unfold_thmss, corec_thmss, safe_unfold_thmss, safe_corec_thmss) =
   911       let
   912         fun mk_goal pfss c cps fcoiter n k ctr m cfs' =
   913           fold_rev (fold_rev Logic.all) ([c] :: pfss)
   914             (Logic.list_implies (seq_conds (HOLogic.mk_Trueprop oo mk_maybe_not) n k cps,
   915                mk_Trueprop_eq (fcoiter $ c, Term.list_comb (ctr, take m cfs'))));
   916 
   917         fun mk_U maybe_mk_sumT =
   918           typ_subst_nonatomic (map2 (fn C => fn fpT => (maybe_mk_sumT fpT C, fpT)) Cs fpTs);
   919 
   920         fun tack z_name (c, u) f =
   921           let val z = Free (z_name, mk_sumT (fastype_of u, fastype_of c)) in
   922             Term.lambda z (mk_sum_case (Term.lambda u u, Term.lambda c (f $ c)) $ z)
   923           end;
   924 
   925         fun build_coiter fcoiters maybe_mk_sumT maybe_tack cqf =
   926           let val T = fastype_of cqf in
   927             if exists_subtype_in Cs T then
   928               let val U = mk_U maybe_mk_sumT T in
   929                 build_map lthy (indexify snd fpTs (fn kk => fn _ =>
   930                   maybe_tack (nth cs kk, nth us kk) (nth fcoiters kk))) (T, U) $ cqf
   931               end
   932             else
   933               cqf
   934           end;
   935 
   936         val crgsss' = map (map (map (build_coiter (un_fold_of fcoiterss') (K I) (K I)))) crgsss;
   937         val cshsss' = map (map (map (build_coiter (co_rec_of fcoiterss') (curry mk_sumT) (tack z))))
   938           cshsss;
   939 
   940         val unfold_goalss = map8 (map4 oooo mk_goal pgss) cs cpss gunfolds ns kss ctrss mss crgsss';
   941         val corec_goalss = map8 (map4 oooo mk_goal phss) cs cpss hcorecs ns kss ctrss mss cshsss';
   942 
   943         val unfold_tacss =
   944           map3 (map oo mk_coiter_tac unfold_defs nesting_map_idents)
   945             (map un_fold_of ctor_dtor_coiter_thmss) pre_map_defs ctr_defss;
   946         val corec_tacss =
   947           map3 (map oo mk_coiter_tac corec_defs nesting_map_idents)
   948             (map co_rec_of ctor_dtor_coiter_thmss) pre_map_defs ctr_defss;
   949 
   950         fun prove goal tac =
   951           Goal.prove_sorry lthy [] [] goal (tac o #context)
   952           |> Thm.close_derivation;
   953 
   954         val unfold_thmss = map2 (map2 prove) unfold_goalss unfold_tacss;
   955         val corec_thmss =
   956           map2 (map2 prove) corec_goalss corec_tacss
   957           |> map (map (unfold_thms lthy @{thms sum_case_if}));
   958 
   959         val unfold_safesss = map2 (map2 (map2 (curry op =))) crgsss' crgsss;
   960         val corec_safesss = map2 (map2 (map2 (curry op =))) cshsss' cshsss;
   961 
   962         val filter_safesss =
   963           map2 (map_filter (fn (safes, thm) => if forall I safes then SOME thm else NONE) oo
   964             curry (op ~~));
   965 
   966         val safe_unfold_thmss = filter_safesss unfold_safesss unfold_thmss;
   967         val safe_corec_thmss = filter_safesss corec_safesss corec_thmss;
   968       in
   969         (unfold_thmss, corec_thmss, safe_unfold_thmss, safe_corec_thmss)
   970       end;
   971 
   972     val (disc_unfold_iff_thmss, disc_corec_iff_thmss) =
   973       let
   974         fun mk_goal c cps fcoiter n k disc =
   975           mk_Trueprop_eq (disc $ (fcoiter $ c),
   976             if n = 1 then @{const True}
   977             else Library.foldr1 HOLogic.mk_conj (seq_conds mk_maybe_not n k cps));
   978 
   979         val unfold_goalss = map6 (map2 oooo mk_goal) cs cpss gunfolds ns kss discss;
   980         val corec_goalss = map6 (map2 oooo mk_goal) cs cpss hcorecs ns kss discss;
   981 
   982         fun mk_case_split' cp = Drule.instantiate' [] [SOME (certify lthy cp)] @{thm case_split};
   983 
   984         val case_splitss' = map (map mk_case_split') cpss;
   985 
   986         val unfold_tacss =
   987           map3 (map oo mk_disc_coiter_iff_tac) case_splitss' unfold_thmss disc_thmsss;
   988         val corec_tacss =
   989           map3 (map oo mk_disc_coiter_iff_tac) case_splitss' corec_thmss disc_thmsss;
   990 
   991         fun prove goal tac =
   992           Goal.prove_sorry lthy [] [] goal (tac o #context)
   993           |> singleton export_args
   994           |> singleton (Proof_Context.export names_lthy lthy)
   995           |> Thm.close_derivation;
   996 
   997         fun proves [_] [_] = []
   998           | proves goals tacs = map2 prove goals tacs;
   999       in
  1000         (map2 proves unfold_goalss unfold_tacss, map2 proves corec_goalss corec_tacss)
  1001       end;
  1002 
  1003     fun mk_disc_coiter_thms coiters discIs = map (op RS) (coiters ~~ discIs);
  1004 
  1005     val disc_unfold_thmss = map2 mk_disc_coiter_thms unfold_thmss discIss;
  1006     val disc_corec_thmss = map2 mk_disc_coiter_thms corec_thmss discIss;
  1007 
  1008     fun mk_sel_coiter_thm coiter_thm sel sel_thm =
  1009       let
  1010         val (domT, ranT) = dest_funT (fastype_of sel);
  1011         val arg_cong' =
  1012           Drule.instantiate' (map (SOME o certifyT lthy) [domT, ranT])
  1013             [NONE, NONE, SOME (certify lthy sel)] arg_cong
  1014           |> Thm.varifyT_global;
  1015         val sel_thm' = sel_thm RSN (2, trans);
  1016       in
  1017         coiter_thm RS arg_cong' RS sel_thm'
  1018       end;
  1019 
  1020     fun mk_sel_coiter_thms coiter_thmss =
  1021       map3 (map3 (map2 o mk_sel_coiter_thm)) coiter_thmss selsss sel_thmsss;
  1022 
  1023     val sel_unfold_thmsss = mk_sel_coiter_thms unfold_thmss;
  1024     val sel_corec_thmsss = mk_sel_coiter_thms corec_thmss;
  1025 
  1026     val coinduct_consumes_attr = Attrib.internal (K (Rule_Cases.consumes nn));
  1027     val coinduct_case_names_attr = Attrib.internal (K (Rule_Cases.case_names coinduct_cases));
  1028     val coinduct_case_concl_attrs =
  1029       map2 (fn casex => fn concls =>
  1030           Attrib.internal (K (Rule_Cases.case_conclusion (casex, concls))))
  1031         coinduct_cases coinduct_conclss;
  1032     val coinduct_case_attrs =
  1033       coinduct_consumes_attr :: coinduct_case_names_attr :: coinduct_case_concl_attrs;
  1034   in
  1035     ((coinduct_thms_pairs, coinduct_case_attrs),
  1036      (unfold_thmss, corec_thmss, code_nitpick_simp_attrs),
  1037      (safe_unfold_thmss, safe_corec_thmss),
  1038      (disc_unfold_thmss, disc_corec_thmss, []),
  1039      (disc_unfold_iff_thmss, disc_corec_iff_thmss, simp_attrs),
  1040      (sel_unfold_thmsss, sel_corec_thmsss, simp_attrs))
  1041   end;
  1042 
  1043 fun define_co_datatypes prepare_constraint prepare_typ prepare_term fp construct_fp
  1044     (wrap_opts as (no_discs_sels, rep_compat), specs) no_defs_lthy0 =
  1045   let
  1046     (* TODO: sanity checks on arguments *)
  1047 
  1048     val _ = if fp = Greatest_FP andalso no_discs_sels then
  1049         error "Cannot define codatatypes without discriminators and selectors"
  1050       else
  1051         ();
  1052 
  1053     fun qualify mandatory fp_b_name =
  1054       Binding.qualify mandatory fp_b_name o (rep_compat ? Binding.qualify false rep_compat_prefix);
  1055 
  1056     val nn = length specs;
  1057     val fp_bs = map type_binding_of specs;
  1058     val fp_b_names = map Binding.name_of fp_bs;
  1059     val fp_common_name = mk_common_name fp_b_names;
  1060     val map_bs = map map_binding_of specs;
  1061     val rel_bs = map rel_binding_of specs;
  1062 
  1063     fun prepare_type_arg (_, (ty, c)) =
  1064       let val TFree (s, _) = prepare_typ no_defs_lthy0 ty in
  1065         TFree (s, prepare_constraint no_defs_lthy0 c)
  1066       end;
  1067 
  1068     val Ass0 = map (map prepare_type_arg o type_args_named_constrained_of) specs;
  1069     val unsorted_Ass0 = map (map (resort_tfree HOLogic.typeS)) Ass0;
  1070     val unsorted_As = Library.foldr1 merge_type_args unsorted_Ass0;
  1071     val num_As = length unsorted_As;
  1072     val set_bss = map (map fst o type_args_named_constrained_of) specs;
  1073 
  1074     val (((Bs0, Cs), Xs), no_defs_lthy) =
  1075       no_defs_lthy0
  1076       |> fold (Variable.declare_typ o resort_tfree dummyS) unsorted_As
  1077       |> mk_TFrees num_As
  1078       ||>> mk_TFrees nn
  1079       ||>> variant_tfrees fp_b_names;
  1080 
  1081     fun add_fake_type spec = Typedecl.basic_typedecl (type_binding_of spec, num_As, mixfix_of spec);
  1082 
  1083     val (fake_T_names, fake_lthy) = fold_map add_fake_type specs no_defs_lthy0;
  1084 
  1085     val qsoty = quote o Syntax.string_of_typ fake_lthy;
  1086 
  1087     val _ = (case duplicates (op =) unsorted_As of [] => ()
  1088       | A :: _ => error ("Duplicate type parameter " ^ qsoty A ^ " in " ^ co_prefix fp ^
  1089           "datatype specification"));
  1090 
  1091     val bad_args =
  1092       map (Logic.type_map (singleton (Variable.polymorphic no_defs_lthy0))) unsorted_As
  1093       |> filter_out Term.is_TVar;
  1094     val _ = null bad_args orelse
  1095       error ("Locally fixed type argument " ^ qsoty (hd bad_args) ^ " in " ^ co_prefix fp ^
  1096         "datatype specification");
  1097 
  1098     val mixfixes = map mixfix_of specs;
  1099 
  1100     val _ = (case duplicates Binding.eq_name fp_bs of [] => ()
  1101       | b :: _ => error ("Duplicate type name declaration " ^ quote (Binding.name_of b)));
  1102 
  1103     val ctr_specss = map ctr_specs_of specs;
  1104 
  1105     val disc_bindingss = map (map disc_of) ctr_specss;
  1106     val ctr_bindingss =
  1107       map2 (fn fp_b_name => map (qualify false fp_b_name o ctr_of)) fp_b_names ctr_specss;
  1108     val ctr_argsss = map (map args_of) ctr_specss;
  1109     val ctr_mixfixess = map (map ctr_mixfix_of) ctr_specss;
  1110 
  1111     val sel_bindingsss = map (map (map fst)) ctr_argsss;
  1112     val fake_ctr_Tsss0 = map (map (map (prepare_typ fake_lthy o snd))) ctr_argsss;
  1113     val raw_sel_defaultsss = map (map defaults_of) ctr_specss;
  1114 
  1115     val (As :: _) :: fake_ctr_Tsss =
  1116       burrow (burrow (Syntax.check_typs fake_lthy)) (Ass0 :: fake_ctr_Tsss0);
  1117     val As' = map dest_TFree As;
  1118 
  1119     val rhs_As' = fold (fold (fold Term.add_tfreesT)) fake_ctr_Tsss [];
  1120     val _ = (case subtract (op =) As' rhs_As' of [] => ()
  1121       | extras => error ("Extra type variables on right-hand side: " ^
  1122           commas (map (qsoty o TFree) extras)));
  1123 
  1124     val fake_Ts = map (fn s => Type (s, As)) fake_T_names;
  1125 
  1126     fun eq_fpT_check (T as Type (s, Ts)) (T' as Type (s', Ts')) =
  1127         s = s' andalso (Ts = Ts' orelse
  1128           error ("Wrong type arguments in " ^ co_prefix fp ^ "recursive type " ^ qsoty T ^
  1129             " (expected " ^ qsoty T' ^ ")"))
  1130       | eq_fpT_check _ _ = false;
  1131 
  1132     fun freeze_fp (T as Type (s, Ts)) =
  1133         (case find_index (eq_fpT_check T) fake_Ts of
  1134           ~1 => Type (s, map freeze_fp Ts)
  1135         | kk => nth Xs kk)
  1136       | freeze_fp T = T;
  1137 
  1138     val unfreeze_fp = Term.typ_subst_atomic (Xs ~~ fake_Ts);
  1139 
  1140     val ctrXs_Tsss = map (map (map freeze_fp)) fake_ctr_Tsss;
  1141     val ctrXs_sum_prod_Ts = map (mk_sumTN_balanced o map HOLogic.mk_tupleT) ctrXs_Tsss;
  1142 
  1143     val fp_eqs =
  1144       map dest_TFree Xs ~~ map (Term.typ_subst_atomic (As ~~ unsorted_As)) ctrXs_sum_prod_Ts;
  1145 
  1146     val rhsXs_As' = fold (fold (fold Term.add_tfreesT)) ctrXs_Tsss [];
  1147     val _ = (case subtract (op =) rhsXs_As' As' of [] => ()
  1148       | extras => List.app (fn extra => warning ("Unused type variable on right-hand side of " ^
  1149           co_prefix fp ^ "datatype definition: " ^ qsoty (TFree extra))) extras);
  1150 
  1151     val (pre_bnfs, (fp_res as {bnfs = fp_bnfs as any_fp_bnf :: _, ctors = ctors0, dtors = dtors0,
  1152            xtor_co_iterss = xtor_co_iterss0, xtor_co_induct, dtor_ctors, ctor_dtors, ctor_injects,
  1153            dtor_injects, xtor_map_thms, xtor_set_thmss, xtor_rel_thms, xtor_co_iter_thmss, ...},
  1154            lthy)) =
  1155       fp_bnf (construct_fp mixfixes map_bs rel_bs set_bss) fp_bs (map dest_TFree unsorted_As) fp_eqs
  1156         no_defs_lthy0
  1157       handle BAD_DEAD (X, X_backdrop) =>
  1158         (case X_backdrop of
  1159           Type (bad_tc, _) =>
  1160           let
  1161             val fake_T = qsoty (unfreeze_fp X);
  1162             val fake_T_backdrop = qsoty (unfreeze_fp X_backdrop);
  1163             fun register_hint () =
  1164               "\nUse the " ^ quote (fst (fst @{command_spec "bnf"})) ^ " command to register " ^
  1165               quote bad_tc ^ " as a bounded natural functor to allow nested (co)recursion through \
  1166               \it";
  1167           in
  1168             if is_some (bnf_of no_defs_lthy bad_tc) orelse
  1169                is_some (fp_sugar_of no_defs_lthy bad_tc) then
  1170               error ("Inadmissible " ^ co_prefix fp ^ "recursive occurrence of type " ^ fake_T ^
  1171                 " in type expression " ^ fake_T_backdrop)
  1172             else if is_some (Datatype_Data.get_info (Proof_Context.theory_of no_defs_lthy)
  1173                 bad_tc) then
  1174               error ("Unsupported " ^ co_prefix fp ^ "recursive occurrence of type " ^ fake_T ^
  1175                 " via the old-style datatype " ^ quote bad_tc ^ " in type expression " ^
  1176                 fake_T_backdrop ^ register_hint ())
  1177             else
  1178               error ("Unsupported " ^ co_prefix fp ^ "recursive occurrence of type " ^ fake_T ^
  1179                 " via type constructor " ^ quote bad_tc ^ " in type expression " ^ fake_T_backdrop ^
  1180                 register_hint ())
  1181           end);
  1182 
  1183     val time = time lthy;
  1184     val timer = time (Timer.startRealTimer ());
  1185 
  1186     val nesting_bnfs = nesty_bnfs lthy ctrXs_Tsss As;
  1187     val nested_bnfs = nesty_bnfs lthy ctrXs_Tsss Xs;
  1188 
  1189     val pre_map_defs = map map_def_of_bnf pre_bnfs;
  1190     val pre_set_defss = map set_defs_of_bnf pre_bnfs;
  1191     val pre_rel_defs = map rel_def_of_bnf pre_bnfs;
  1192     val nesting_set_maps = maps set_map_of_bnf nesting_bnfs;
  1193     val nested_set_maps = maps set_map_of_bnf nested_bnfs;
  1194 
  1195     val live = live_of_bnf any_fp_bnf;
  1196     val _ =
  1197       if live = 0 andalso exists (not o Binding.is_empty) (map_bs @ rel_bs) then
  1198         warning "Map function and relator names ignored"
  1199       else
  1200         ();
  1201 
  1202     val Bs =
  1203       map3 (fn alive => fn A as TFree (_, S) => fn B => if alive then resort_tfree S B else A)
  1204         (liveness_of_fp_bnf num_As any_fp_bnf) As Bs0;
  1205 
  1206     val B_ify = Term.typ_subst_atomic (As ~~ Bs);
  1207 
  1208     val ctors = map (mk_ctor As) ctors0;
  1209     val dtors = map (mk_dtor As) dtors0;
  1210 
  1211     val fpTs = map (domain_type o fastype_of) dtors;
  1212 
  1213     fun massage_simple_notes base =
  1214       filter_out (null o #2)
  1215       #> map (fn (thmN, thms, attrs) =>
  1216         ((qualify true base (Binding.name thmN), attrs), [(thms, [])]));
  1217 
  1218     val massage_multi_notes =
  1219       maps (fn (thmN, thmss, attrs) =>
  1220         map3 (fn fp_b_name => fn Type (T_name, _) => fn thms =>
  1221             ((qualify true fp_b_name (Binding.name thmN), attrs T_name), [(thms, [])]))
  1222           fp_b_names fpTs thmss)
  1223       #> filter_out (null o fst o hd o snd);
  1224 
  1225     val ctr_Tsss = map (map (map (Term.typ_subst_atomic (Xs ~~ fpTs)))) ctrXs_Tsss;
  1226     val ns = map length ctr_Tsss;
  1227     val kss = map (fn n => 1 upto n) ns;
  1228     val mss = map (map length) ctr_Tsss;
  1229 
  1230     val ((xtor_co_iterss, iters_args_types, coiters_args_types), lthy') =
  1231       mk_co_iters_prelims fp ctr_Tsss fpTs Cs ns mss xtor_co_iterss0 lthy;
  1232 
  1233     fun define_ctrs_dtrs_for_type (((((((((((((((((((((((fp_bnf, fp_b), fpT), ctor), dtor),
  1234             xtor_co_iters), ctor_dtor), dtor_ctor), ctor_inject), pre_map_def), pre_set_defs),
  1235           pre_rel_def), fp_map_thm), fp_set_thms), fp_rel_thm), n), ks), ms), ctr_bindings),
  1236         ctr_mixfixes), ctr_Tss), disc_bindings), sel_bindingss), raw_sel_defaultss) no_defs_lthy =
  1237       let
  1238         val fp_b_name = Binding.name_of fp_b;
  1239 
  1240         val dtorT = domain_type (fastype_of ctor);
  1241         val ctr_prod_Ts = map HOLogic.mk_tupleT ctr_Tss;
  1242         val ctr_sum_prod_T = mk_sumTN_balanced ctr_prod_Ts;
  1243 
  1244         val ((((w, xss), yss), u'), names_lthy) =
  1245           no_defs_lthy
  1246           |> yield_singleton (mk_Frees "w") dtorT
  1247           ||>> mk_Freess "x" ctr_Tss
  1248           ||>> mk_Freess "y" (map (map B_ify) ctr_Tss)
  1249           ||>> yield_singleton Variable.variant_fixes fp_b_name;
  1250 
  1251         val u = Free (u', fpT);
  1252 
  1253         val tuple_xs = map HOLogic.mk_tuple xss;
  1254         val tuple_ys = map HOLogic.mk_tuple yss;
  1255 
  1256         val ctr_rhss =
  1257           map3 (fn k => fn xs => fn tuple_x => fold_rev Term.lambda xs (ctor $
  1258             mk_InN_balanced ctr_sum_prod_T n tuple_x k)) ks xss tuple_xs;
  1259 
  1260         val maybe_conceal_def_binding = Thm.def_binding
  1261           #> Config.get no_defs_lthy bnf_note_all = false ? Binding.conceal;
  1262 
  1263         val ((raw_ctrs, raw_ctr_defs), (lthy', lthy)) = no_defs_lthy
  1264           |> apfst split_list o fold_map3 (fn b => fn mx => fn rhs =>
  1265               Local_Theory.define ((b, mx), ((maybe_conceal_def_binding b, []), rhs)) #>> apsnd snd)
  1266             ctr_bindings ctr_mixfixes ctr_rhss
  1267           ||> `Local_Theory.restore;
  1268 
  1269         val phi = Proof_Context.export_morphism lthy lthy';
  1270 
  1271         val ctr_defs = map (Morphism.thm phi) raw_ctr_defs;
  1272         val ctr_defs' =
  1273           map2 (fn m => fn def => mk_unabs_def m (def RS meta_eq_to_obj_eq)) ms ctr_defs;
  1274 
  1275         val ctrs0 = map (Morphism.term phi) raw_ctrs;
  1276         val ctrs = map (mk_ctr As) ctrs0;
  1277 
  1278         fun wrap_ctrs lthy =
  1279           let
  1280             fun exhaust_tac {context = ctxt, prems = _} =
  1281               let
  1282                 val ctor_iff_dtor_thm =
  1283                   let
  1284                     val goal =
  1285                       fold_rev Logic.all [w, u]
  1286                         (mk_Trueprop_eq (HOLogic.mk_eq (u, ctor $ w), HOLogic.mk_eq (dtor $ u, w)));
  1287                   in
  1288                     Goal.prove_sorry lthy [] [] goal (fn {context = ctxt, ...} =>
  1289                       mk_ctor_iff_dtor_tac ctxt (map (SOME o certifyT lthy) [dtorT, fpT])
  1290                         (certify lthy ctor) (certify lthy dtor) ctor_dtor dtor_ctor)
  1291                     |> Thm.close_derivation
  1292                     |> Morphism.thm phi
  1293                   end;
  1294 
  1295                 val sumEN_thm' =
  1296                   unfold_thms lthy @{thms unit_all_eq1}
  1297                     (Drule.instantiate' (map (SOME o certifyT lthy) ctr_prod_Ts) []
  1298                        (mk_sumEN_balanced n))
  1299                   |> Morphism.thm phi;
  1300               in
  1301                 mk_exhaust_tac ctxt n ctr_defs ctor_iff_dtor_thm sumEN_thm'
  1302               end;
  1303 
  1304             val inject_tacss =
  1305               map2 (fn 0 => K [] | _ => fn ctr_def => [fn {context = ctxt, ...} =>
  1306                 mk_inject_tac ctxt ctr_def ctor_inject]) ms ctr_defs;
  1307 
  1308             val half_distinct_tacss =
  1309               map (map (fn (def, def') => fn {context = ctxt, ...} =>
  1310                 mk_half_distinct_tac ctxt ctor_inject [def, def'])) (mk_half_pairss (`I ctr_defs));
  1311 
  1312             val tacss = [exhaust_tac] :: inject_tacss @ half_distinct_tacss;
  1313 
  1314             val sel_defaultss = map (map (apsnd (prepare_term lthy))) raw_sel_defaultss
  1315           in
  1316             wrap_free_constructors tacss (((wrap_opts, ctrs0), standard_binding), (disc_bindings,
  1317               (sel_bindingss, sel_defaultss))) lthy
  1318           end;
  1319 
  1320         fun derive_maps_sets_rels (ctr_sugar, lthy) =
  1321           if live = 0 then
  1322             ((([], [], [], []), ctr_sugar), lthy)
  1323           else
  1324             let
  1325               val rel_flip = rel_flip_of_bnf fp_bnf;
  1326               val nones = replicate live NONE;
  1327 
  1328               val ctor_cong =
  1329                 if fp = Least_FP then
  1330                   Drule.dummy_thm
  1331                 else
  1332                   let val ctor' = mk_ctor Bs ctor in
  1333                     cterm_instantiate_pos [NONE, NONE, SOME (certify lthy ctor')] arg_cong
  1334                   end;
  1335 
  1336               fun mk_cIn ify =
  1337                 certify lthy o (fp = Greatest_FP ? curry (op $) (map_types ify ctor)) oo
  1338                 mk_InN_balanced (ify ctr_sum_prod_T) n;
  1339 
  1340               val cxIns = map2 (mk_cIn I) tuple_xs ks;
  1341               val cyIns = map2 (mk_cIn B_ify) tuple_ys ks;
  1342 
  1343               fun mk_map_thm ctr_def' cxIn =
  1344                 fold_thms lthy [ctr_def']
  1345                   (unfold_thms lthy (pre_map_def ::
  1346                        (if fp = Least_FP then [] else [ctor_dtor, dtor_ctor]) @ sum_prod_thms_map)
  1347                      (cterm_instantiate_pos (nones @ [SOME cxIn])
  1348                         (if fp = Least_FP then fp_map_thm else fp_map_thm RS ctor_cong)))
  1349                 |> singleton (Proof_Context.export names_lthy no_defs_lthy);
  1350 
  1351               fun mk_set_thm fp_set_thm ctr_def' cxIn =
  1352                 fold_thms lthy [ctr_def']
  1353                   (unfold_thms lthy (pre_set_defs @ nested_set_maps @ nesting_set_maps @
  1354                        (if fp = Least_FP then [] else [dtor_ctor]) @ sum_prod_thms_set)
  1355                      (cterm_instantiate_pos [SOME cxIn] fp_set_thm))
  1356                 |> singleton (Proof_Context.export names_lthy no_defs_lthy);
  1357 
  1358               fun mk_set_thms fp_set_thm = map2 (mk_set_thm fp_set_thm) ctr_defs' cxIns;
  1359 
  1360               val map_thms = map2 mk_map_thm ctr_defs' cxIns;
  1361               val set_thmss = map mk_set_thms fp_set_thms;
  1362 
  1363               val rel_infos = (ctr_defs' ~~ cxIns, ctr_defs' ~~ cyIns);
  1364 
  1365               fun mk_rel_thm postproc ctr_defs' cxIn cyIn =
  1366                 fold_thms lthy ctr_defs'
  1367                   (unfold_thms lthy (@{thm Inl_Inr_False} :: pre_rel_def ::
  1368                        (if fp = Least_FP then [] else [dtor_ctor]) @ sum_prod_thms_rel)
  1369                      (cterm_instantiate_pos (nones @ [SOME cxIn, SOME cyIn]) fp_rel_thm))
  1370                 |> postproc
  1371                 |> singleton (Proof_Context.export names_lthy no_defs_lthy);
  1372 
  1373               fun mk_rel_inject_thm ((ctr_def', cxIn), (_, cyIn)) =
  1374                 mk_rel_thm (unfold_thms lthy @{thms eq_sym_Unity_conv}) [ctr_def'] cxIn cyIn;
  1375 
  1376               val rel_inject_thms = map mk_rel_inject_thm (op ~~ rel_infos);
  1377 
  1378               fun mk_half_rel_distinct_thm ((xctr_def', cxIn), (yctr_def', cyIn)) =
  1379                 mk_rel_thm (fn thm => thm RS @{thm eq_False[THEN iffD1]}) [xctr_def', yctr_def']
  1380                   cxIn cyIn;
  1381 
  1382               fun mk_other_half_rel_distinct_thm thm =
  1383                 flip_rels lthy live thm
  1384                 RS (rel_flip RS sym RS @{thm arg_cong[of _ _ Not]} RS iffD2);
  1385 
  1386               val half_rel_distinct_thmss =
  1387                 map (map mk_half_rel_distinct_thm) (mk_half_pairss rel_infos);
  1388               val other_half_rel_distinct_thmss =
  1389                 map (map mk_other_half_rel_distinct_thm) half_rel_distinct_thmss;
  1390               val (rel_distinct_thms, _) =
  1391                 join_halves n half_rel_distinct_thmss other_half_rel_distinct_thmss;
  1392 
  1393               val notes =
  1394                 [(mapN, map_thms, code_nitpick_simp_attrs @ simp_attrs),
  1395                  (rel_distinctN, rel_distinct_thms, code_nitpick_simp_attrs @ simp_attrs),
  1396                  (rel_injectN, rel_inject_thms, code_nitpick_simp_attrs @ simp_attrs),
  1397                  (setN, flat set_thmss, code_nitpick_simp_attrs @ simp_attrs)]
  1398                 |> massage_simple_notes fp_b_name;
  1399             in
  1400               (((map_thms, rel_inject_thms, rel_distinct_thms, set_thmss), ctr_sugar),
  1401                lthy |> Local_Theory.notes notes |> snd)
  1402             end;
  1403 
  1404         fun mk_binding suf = qualify false fp_b_name (Binding.suffix_name ("_" ^ suf) fp_b);
  1405 
  1406         fun massage_res (((maps_sets_rels, ctr_sugar), co_iter_res), lthy) =
  1407           (((maps_sets_rels, (ctrs, xss, ctr_defs, ctr_sugar)), co_iter_res), lthy);
  1408       in
  1409         (wrap_ctrs
  1410          #> derive_maps_sets_rels
  1411          ##>>
  1412            (if fp = Least_FP then define_iters [foldN, recN] (the iters_args_types)
  1413             else define_coiters [unfoldN, corecN] (the coiters_args_types))
  1414              mk_binding fpTs Cs xtor_co_iters
  1415          #> massage_res, lthy')
  1416       end;
  1417 
  1418     fun wrap_types_etc (wrap_types_etcs, lthy) =
  1419       fold_map I wrap_types_etcs lthy
  1420       |>> apsnd split_list o apfst (apsnd split_list4 o apfst split_list4 o split_list)
  1421         o split_list;
  1422 
  1423     fun mk_simp_thms {injects, distincts, case_thms, ...} un_folds co_recs mapsx rel_injects
  1424         rel_distincts setss =
  1425       injects @ distincts @ case_thms @ co_recs @ un_folds @ mapsx @ rel_injects @ rel_distincts
  1426       @ flat setss;
  1427 
  1428     fun derive_note_induct_iters_thms_for_types
  1429         ((((mapss, rel_injects, rel_distincts, setss), (ctrss, _, ctr_defss, ctr_sugars)),
  1430           (iterss, iter_defss)), lthy) =
  1431       let
  1432         val ((induct_thms, induct_thm, induct_attrs), (fold_thmss, rec_thmss, iter_attrs)) =
  1433           derive_induct_iters_thms_for_types pre_bnfs (the iters_args_types) xtor_co_induct
  1434             xtor_co_iter_thmss nesting_bnfs nested_bnfs fpTs Cs Xs ctrXs_Tsss ctrss ctr_defss iterss
  1435             iter_defss lthy;
  1436 
  1437         val induct_type_attr = Attrib.internal o K o Induct.induct_type;
  1438 
  1439         val simp_thmss =
  1440           map7 mk_simp_thms ctr_sugars fold_thmss rec_thmss mapss rel_injects rel_distincts setss;
  1441 
  1442         val common_notes =
  1443           (if nn > 1 then [(inductN, [induct_thm], induct_attrs)] else [])
  1444           |> massage_simple_notes fp_common_name;
  1445 
  1446         val notes =
  1447           [(foldN, fold_thmss, K iter_attrs),
  1448            (inductN, map single induct_thms, fn T_name => induct_attrs @ [induct_type_attr T_name]),
  1449            (recN, rec_thmss, K iter_attrs),
  1450            (simpsN, simp_thmss, K [])]
  1451           |> massage_multi_notes;
  1452       in
  1453         lthy
  1454         |> Local_Theory.notes (common_notes @ notes) |> snd
  1455         |> register_fp_sugars Least_FP pre_bnfs nested_bnfs nesting_bnfs fp_res ctr_defss ctr_sugars
  1456           iterss mapss [induct_thm] (transpose [fold_thmss, rec_thmss]) [] []
  1457       end;
  1458 
  1459     fun derive_note_coinduct_coiters_thms_for_types
  1460         ((((mapss, rel_injects, rel_distincts, setss), (_, _, ctr_defss, ctr_sugars)),
  1461           (coiterss, coiter_defss)), lthy) =
  1462       let
  1463         val (([(coinduct_thms, coinduct_thm), (strong_coinduct_thms, strong_coinduct_thm)],
  1464               coinduct_attrs),
  1465              (unfold_thmss, corec_thmss, coiter_attrs),
  1466              (safe_unfold_thmss, safe_corec_thmss),
  1467              (disc_unfold_thmss, disc_corec_thmss, disc_coiter_attrs),
  1468              (disc_unfold_iff_thmss, disc_corec_iff_thmss, disc_coiter_iff_attrs),
  1469              (sel_unfold_thmsss, sel_corec_thmsss, sel_coiter_attrs)) =
  1470           derive_coinduct_coiters_thms_for_types pre_bnfs (the coiters_args_types) xtor_co_induct
  1471             dtor_injects dtor_ctors xtor_co_iter_thmss nesting_bnfs fpTs Cs kss mss ns ctr_defss
  1472             ctr_sugars coiterss coiter_defss (Proof_Context.export lthy' no_defs_lthy) lthy;
  1473 
  1474         val sel_unfold_thmss = map flat sel_unfold_thmsss;
  1475         val sel_corec_thmss = map flat sel_corec_thmsss;
  1476 
  1477         val coinduct_type_attr = Attrib.internal o K o Induct.coinduct_type;
  1478 
  1479         fun flat_coiter_thms coiters disc_coiters disc_coiter_iffs sel_coiters =
  1480           coiters @ disc_coiters @ disc_coiter_iffs @ sel_coiters;
  1481 
  1482         val simp_thmss =
  1483           map7 mk_simp_thms ctr_sugars
  1484             (map4 flat_coiter_thms safe_unfold_thmss disc_unfold_thmss disc_unfold_iff_thmss
  1485                sel_unfold_thmss)
  1486             (map4 flat_coiter_thms safe_corec_thmss disc_corec_thmss disc_corec_iff_thmss
  1487                sel_corec_thmss)
  1488             mapss rel_injects rel_distincts setss;
  1489 
  1490         val anonymous_notes =
  1491           [(flat safe_unfold_thmss @ flat safe_corec_thmss, simp_attrs)]
  1492           |> map (fn (thms, attrs) => ((Binding.empty, attrs), [(thms, [])]));
  1493 
  1494         val common_notes =
  1495           (if nn > 1 then
  1496              [(coinductN, [coinduct_thm], coinduct_attrs),
  1497               (strong_coinductN, [strong_coinduct_thm], coinduct_attrs)]
  1498            else
  1499              [])
  1500           |> massage_simple_notes fp_common_name;
  1501 
  1502         val notes =
  1503           [(coinductN, map single coinduct_thms,
  1504             fn T_name => coinduct_attrs @ [coinduct_type_attr T_name]),
  1505            (corecN, corec_thmss, K coiter_attrs),
  1506            (disc_corecN, disc_corec_thmss, K disc_coiter_attrs),
  1507            (disc_corec_iffN, disc_corec_iff_thmss, K disc_coiter_iff_attrs),
  1508            (disc_unfoldN, disc_unfold_thmss, K disc_coiter_attrs),
  1509            (disc_unfold_iffN, disc_unfold_iff_thmss, K disc_coiter_iff_attrs),
  1510            (sel_corecN, sel_corec_thmss, K sel_coiter_attrs),
  1511            (sel_unfoldN, sel_unfold_thmss, K sel_coiter_attrs),
  1512            (simpsN, simp_thmss, K []),
  1513            (strong_coinductN, map single strong_coinduct_thms, K coinduct_attrs),
  1514            (unfoldN, unfold_thmss, K coiter_attrs)]
  1515           |> massage_multi_notes;
  1516 
  1517         fun register_nitpick fpT {ctrs, casex, ...} =
  1518           Nitpick_HOL.register_codatatype fpT (fst (dest_Const casex))
  1519             (map (dest_Const o mk_ctr As) ctrs)
  1520           |> Generic_Target.theory_declaration;
  1521       in
  1522         lthy
  1523         |> Local_Theory.notes (anonymous_notes @ common_notes @ notes) |> snd
  1524         |> register_fp_sugars Greatest_FP pre_bnfs nested_bnfs nesting_bnfs fp_res ctr_defss
  1525           ctr_sugars coiterss mapss [coinduct_thm, strong_coinduct_thm]
  1526           (transpose [unfold_thmss, corec_thmss]) (transpose [disc_unfold_thmss, disc_corec_thmss])
  1527           (transpose [sel_unfold_thmsss, sel_corec_thmsss])
  1528         |> fold2 register_nitpick fpTs ctr_sugars
  1529       end;
  1530 
  1531     val lthy'' = lthy'
  1532       |> fold_map define_ctrs_dtrs_for_type (fp_bnfs ~~ fp_bs ~~ fpTs ~~ ctors ~~ dtors ~~
  1533         xtor_co_iterss ~~ ctor_dtors ~~ dtor_ctors ~~ ctor_injects ~~ pre_map_defs ~~
  1534         pre_set_defss ~~ pre_rel_defs ~~ xtor_map_thms ~~ xtor_set_thmss ~~ xtor_rel_thms ~~ ns ~~
  1535         kss ~~ mss ~~ ctr_bindingss ~~ ctr_mixfixess ~~ ctr_Tsss ~~ disc_bindingss ~~
  1536         sel_bindingsss ~~ raw_sel_defaultsss)
  1537       |> wrap_types_etc
  1538       |> fp_case fp derive_note_induct_iters_thms_for_types
  1539            derive_note_coinduct_coiters_thms_for_types;
  1540 
  1541     val timer = time (timer ("Constructors, discriminators, selectors, etc., for the new " ^
  1542       co_prefix fp ^ "datatype"));
  1543   in
  1544     timer; lthy''
  1545   end;
  1546 
  1547 val co_datatypes = define_co_datatypes (K I) (K I) (K I);
  1548 
  1549 val co_datatype_cmd =
  1550   define_co_datatypes Typedecl.read_constraint Syntax.parse_typ Syntax.parse_term;
  1551 
  1552 val parse_ctr_arg =
  1553   @{keyword "("} |-- parse_binding_colon -- Parse.typ --| @{keyword ")"} ||
  1554   (Parse.typ >> pair Binding.empty);
  1555 
  1556 val parse_defaults =
  1557   @{keyword "("} |-- Parse.reserved "defaults" |-- Scan.repeat parse_bound_term --| @{keyword ")"};
  1558 
  1559 val parse_type_arg_constrained =
  1560   Parse.type_ident -- Scan.option (@{keyword "::"} |-- Parse.!!! Parse.sort);
  1561 
  1562 val parse_type_arg_named_constrained = parse_opt_binding_colon -- parse_type_arg_constrained;
  1563 
  1564 val parse_type_args_named_constrained =
  1565   parse_type_arg_constrained >> (single o pair Binding.empty) ||
  1566   @{keyword "("} |-- Parse.!!! (Parse.list1 parse_type_arg_named_constrained --| @{keyword ")"}) ||
  1567   Scan.succeed [];
  1568 
  1569 val parse_map_rel_binding = Parse.short_ident --| @{keyword ":"} -- parse_binding;
  1570 
  1571 val no_map_rel = (Binding.empty, Binding.empty);
  1572 
  1573 fun extract_map_rel ("map", b) = apfst (K b)
  1574   | extract_map_rel ("rel", b) = apsnd (K b)
  1575   | extract_map_rel (s, _) = error ("Unknown label " ^ quote s ^ " (expected \"map\" or \"rel\")");
  1576 
  1577 val parse_map_rel_bindings =
  1578   @{keyword "("} |-- Scan.repeat parse_map_rel_binding --| @{keyword ")"}
  1579     >> (fn ps => fold extract_map_rel ps no_map_rel) ||
  1580   Scan.succeed no_map_rel;
  1581 
  1582 val parse_ctr_spec =
  1583   parse_opt_binding_colon -- parse_binding -- Scan.repeat parse_ctr_arg --
  1584   Scan.optional parse_defaults [] -- Parse.opt_mixfix;
  1585 
  1586 val parse_spec =
  1587   parse_type_args_named_constrained -- parse_binding -- parse_map_rel_bindings --
  1588   Parse.opt_mixfix -- (@{keyword "="} |-- Parse.enum1 "|" parse_ctr_spec);
  1589 
  1590 val parse_co_datatype = parse_wrap_free_constructors_options -- Parse.and_list1 parse_spec;
  1591 
  1592 fun parse_co_datatype_cmd fp construct_fp = parse_co_datatype >> co_datatype_cmd fp construct_fp;
  1593 
  1594 end;