src/HOL/BNF/Tools/bnf_fp_def_sugar.ML
author blanchet
Mon Nov 04 10:52:41 2013 +0100 (2013-11-04)
changeset 54235 3aed2ae6eb91
parent 54171 c0b0e1ea839e
child 54236 e00009523727
permissions -rw-r--r--
tuning
     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: 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 * 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     typ list -> typ list list list -> int list list -> int list list -> int list -> thm list list ->
    91     Ctr_Sugar.ctr_sugar list -> term list list -> thm list list -> (thm list -> thm list) ->
    92     local_theory -> gfp_sugar_thms
    93   val co_datatypes: BNF_FP_Util.fp_kind -> (mixfix list -> binding list -> binding list ->
    94       binding list list -> binding list -> (string * sort) list -> typ list * typ list list ->
    95       BNF_Def.bnf list -> local_theory -> BNF_FP_Util.fp_result * local_theory) ->
    96     (bool * bool) * (((((binding * (typ * sort)) list * binding) * (binding * binding)) * mixfix) *
    97       ((((binding * binding) * (binding * typ) list) * (binding * term) list) *
    98         mixfix) list) list ->
    99     local_theory -> local_theory
   100   val parse_co_datatype_cmd: BNF_FP_Util.fp_kind -> (mixfix list -> binding list -> binding list ->
   101       binding list list -> binding list -> (string * sort) list -> typ list * typ list list ->
   102       BNF_Def.bnf list -> local_theory -> BNF_FP_Util.fp_result * local_theory) ->
   103     (local_theory -> local_theory) parser
   104 end;
   105 
   106 structure BNF_FP_Def_Sugar : BNF_FP_DEF_SUGAR =
   107 struct
   108 
   109 open Ctr_Sugar
   110 open BNF_Util
   111 open BNF_Comp
   112 open BNF_Def
   113 open BNF_FP_Util
   114 open BNF_FP_Def_Sugar_Tactics
   115 
   116 val EqN = "Eq_";
   117 
   118 type fp_sugar =
   119   {T: typ,
   120    fp: fp_kind,
   121    index: int,
   122    pre_bnfs: bnf list,
   123    nested_bnfs: bnf list,
   124    nesting_bnfs: bnf list,
   125    fp_res: fp_result,
   126    ctr_defss: thm list list,
   127    ctr_sugars: ctr_sugar list,
   128    co_iterss: term list list,
   129    mapss: thm list list,
   130    co_inducts: thm list,
   131    co_iter_thmsss: thm list list list,
   132    disc_co_itersss: thm list list list,
   133    sel_co_iterssss: thm list list list list};
   134 
   135 fun of_fp_sugar f (fp_sugar as ({index, ...}: fp_sugar)) = nth (f fp_sugar) index;
   136 
   137 fun eq_fp_sugar ({T = T1, fp = fp1, index = index1, fp_res = fp_res1, ...} : fp_sugar,
   138     {T = T2, fp = fp2, index = index2, fp_res = fp_res2, ...} : fp_sugar) =
   139   T1 = T2 andalso fp1 = fp2 andalso index1 = index2 andalso eq_fp_result (fp_res1, fp_res2);
   140 
   141 fun morph_fp_sugar phi ({T, fp, index, pre_bnfs, nested_bnfs, nesting_bnfs, fp_res, ctr_defss,
   142     ctr_sugars, co_iterss, mapss, co_inducts, co_iter_thmsss, disc_co_itersss, sel_co_iterssss}
   143     : fp_sugar) =
   144   {T = Morphism.typ phi T, fp = fp, index = index, pre_bnfs = map (morph_bnf phi) pre_bnfs,
   145     nested_bnfs = map (morph_bnf phi) nested_bnfs, nesting_bnfs = map (morph_bnf phi) nesting_bnfs,
   146    fp_res = morph_fp_result phi fp_res,
   147    ctr_defss = map (map (Morphism.thm phi)) ctr_defss,
   148    ctr_sugars = map (morph_ctr_sugar phi) ctr_sugars,
   149    co_iterss = map (map (Morphism.term phi)) co_iterss,
   150    mapss = map (map (Morphism.thm phi)) mapss,
   151    co_inducts = map (Morphism.thm phi) co_inducts,
   152    co_iter_thmsss = map (map (map (Morphism.thm phi))) co_iter_thmsss,
   153    disc_co_itersss = map (map (map (Morphism.thm phi))) disc_co_itersss,
   154    sel_co_iterssss = map (map (map (map (Morphism.thm phi)))) sel_co_iterssss};
   155 
   156 val transfer_fp_sugar =
   157   morph_fp_sugar o Morphism.thm_morphism o Thm.transfer o Proof_Context.theory_of;
   158 
   159 structure Data = Generic_Data
   160 (
   161   type T = fp_sugar Symtab.table;
   162   val empty = Symtab.empty;
   163   val extend = I;
   164   val merge = Symtab.merge eq_fp_sugar;
   165 );
   166 
   167 fun fp_sugar_of ctxt =
   168   Symtab.lookup (Data.get (Context.Proof ctxt))
   169   #> Option.map (transfer_fp_sugar ctxt);
   170 
   171 fun fp_sugars_of ctxt =
   172   Symtab.fold (cons o transfer_fp_sugar ctxt o snd) (Data.get (Context.Proof ctxt)) [];
   173 
   174 fun co_induct_of (i :: _) = i;
   175 fun strong_co_induct_of [_, s] = s;
   176 
   177 (* TODO: register "sum" and "prod" as datatypes to enable N2M reduction for them *)
   178 
   179 fun register_fp_sugar key fp_sugar =
   180   Local_Theory.declaration {syntax = false, pervasive = true}
   181     (fn phi => Data.map (Symtab.default (key, morph_fp_sugar phi fp_sugar)));
   182 
   183 fun register_fp_sugars fp pre_bnfs nested_bnfs nesting_bnfs (fp_res as {Ts, ...}) ctr_defss
   184     ctr_sugars co_iterss mapss co_inducts co_iter_thmsss disc_co_itersss sel_co_iterssss lthy =
   185   (0, lthy)
   186   |> fold (fn T as Type (s, _) => fn (kk, lthy) => (kk + 1,
   187     register_fp_sugar s {T = T, fp = fp, index = kk, pre_bnfs = pre_bnfs,
   188         nested_bnfs = nested_bnfs, nesting_bnfs = nesting_bnfs, fp_res = fp_res,
   189         ctr_defss = ctr_defss, ctr_sugars = ctr_sugars, co_iterss = co_iterss, mapss = mapss,
   190         co_inducts = co_inducts, co_iter_thmsss = co_iter_thmsss, disc_co_itersss = disc_co_itersss,
   191         sel_co_iterssss = sel_co_iterssss}
   192       lthy)) Ts
   193   |> snd;
   194 
   195 (* This function could produce clashes in contrived examples (e.g., "x.A", "x.x_A", "y.A"). *)
   196 fun quasi_unambiguous_case_names names =
   197   let
   198     val ps = map (`Long_Name.base_name) names;
   199     val dups = Library.duplicates (op =) (map fst ps);
   200     fun underscore s =
   201       let val ss = space_explode Long_Name.separator s in
   202         space_implode "_" (drop (length ss - 2) ss)
   203       end;
   204   in
   205     map (fn (base, full) => if member (op =) dups base then underscore full else base) ps
   206   end;
   207 
   208 val id_def = @{thm id_def};
   209 val mp_conj = @{thm mp_conj};
   210 
   211 val nitpicksimp_attrs = @{attributes [nitpick_simp]};
   212 val code_nitpicksimp_attrs = Code.add_default_eqn_attrib :: nitpicksimp_attrs;
   213 val simp_attrs = @{attributes [simp]};
   214 
   215 fun tvar_subst thy Ts Us =
   216   Vartab.fold (cons o apsnd snd) (fold (Sign.typ_match thy) (Ts ~~ Us) Vartab.empty) [];
   217 
   218 val exists_subtype_in = Term.exists_subtype o member (op =);
   219 
   220 val lists_bmoc = fold (fn xs => fn t => Term.list_comb (t, xs));
   221 
   222 fun flat_rec_arg_args xss =
   223   (* FIXME (once the old datatype package is phased out): The first line below gives the preferred
   224      order. The second line is for compatibility with the old datatype package. *)
   225 (*
   226   flat xss
   227 *)
   228   map hd xss @ maps tl xss;
   229 
   230 fun flat_corec_predss_getterss qss fss = maps (op @) (qss ~~ fss);
   231 
   232 fun flat_corec_preds_predsss_gettersss [] [qss] [fss] = flat_corec_predss_getterss qss fss
   233   | flat_corec_preds_predsss_gettersss (p :: ps) (qss :: qsss) (fss :: fsss) =
   234     p :: flat_corec_predss_getterss qss fss @ flat_corec_preds_predsss_gettersss ps qsss fsss;
   235 
   236 fun mk_tupled_fun x f xs =
   237   if xs = [x] then f else HOLogic.tupled_lambda x (Term.list_comb (f, xs));
   238 
   239 fun mk_uncurried2_fun f xss =
   240   mk_tupled_fun (HOLogic.mk_tuple (map HOLogic.mk_tuple xss)) f (flat_rec_arg_args xss);
   241 
   242 fun mk_flip (x, Type (_, [T1, Type (_, [T2, T3])])) =
   243   Abs ("x", T1, Abs ("y", T2, Var (x, T2 --> T1 --> T3) $ Bound 0 $ Bound 1));
   244 
   245 fun flip_rels lthy n thm =
   246   let
   247     val Rs = Term.add_vars (prop_of thm) [];
   248     val Rs' = rev (drop (length Rs - n) Rs);
   249     val cRs = map (fn f => (certify lthy (Var f), certify lthy (mk_flip f))) Rs';
   250   in
   251     Drule.cterm_instantiate cRs thm
   252   end;
   253 
   254 fun mk_ctor_or_dtor get_T Ts t =
   255   let val Type (_, Ts0) = get_T (fastype_of t) in
   256     Term.subst_atomic_types (Ts0 ~~ Ts) t
   257   end;
   258 
   259 val mk_ctor = mk_ctor_or_dtor range_type;
   260 val mk_dtor = mk_ctor_or_dtor domain_type;
   261 
   262 fun mk_co_iter thy fp fpT Cs t =
   263   let
   264     val (f_Cs, Type (_, [prebody, body])) = strip_fun_type (fastype_of t);
   265     val fpT0 = fp_case fp prebody body;
   266     val Cs0 = distinct (op =) (map (fp_case fp body_type domain_type) f_Cs);
   267     val rho = tvar_subst thy (fpT0 :: Cs0) (fpT :: Cs);
   268   in
   269     Term.subst_TVars rho t
   270   end;
   271 
   272 fun mk_co_iters thy fp fpTs Cs ts0 =
   273   let
   274     val nn = length fpTs;
   275     val (fpTs0, Cs0) =
   276       map ((fp = Greatest_FP ? swap) o dest_funT o snd o strip_typeN nn o fastype_of) ts0
   277       |> split_list;
   278     val rho = tvar_subst thy (fpTs0 @ Cs0) (fpTs @ Cs);
   279   in
   280     map (Term.subst_TVars rho) ts0
   281   end;
   282 
   283 val mk_fp_iter_fun_types = binder_fun_types o fastype_of;
   284 
   285 fun unzip_recT (Type (@{type_name prod}, _)) T = [T]
   286   | unzip_recT _ (Type (@{type_name prod}, Ts)) = Ts
   287   | unzip_recT _ T = [T];
   288 
   289 fun unzip_corecT (Type (@{type_name sum}, _)) T = [T]
   290   | unzip_corecT _ (Type (@{type_name sum}, Ts)) = Ts
   291   | unzip_corecT _ T = [T];
   292 
   293 fun mk_map live Ts Us t =
   294   let val (Type (_, Ts0), Type (_, Us0)) = strip_typeN (live + 1) (fastype_of t) |>> List.last in
   295     Term.subst_atomic_types (Ts0 @ Us0 ~~ Ts @ Us) t
   296   end;
   297 
   298 fun mk_rel live Ts Us t =
   299   let val [Type (_, Ts0), Type (_, Us0)] = binder_types (snd (strip_typeN live (fastype_of t))) in
   300     Term.subst_atomic_types (Ts0 @ Us0 ~~ Ts @ Us) t
   301   end;
   302 
   303 fun build_map_or_rel mk const of_bnf dest lthy build_simple =
   304   let
   305     fun build (TU as (T, U)) =
   306       if T = U then
   307         const T
   308       else
   309         (case TU of
   310           (Type (s, Ts), Type (s', Us)) =>
   311           if s = s' then
   312             let
   313               val bnf = the (bnf_of lthy s);
   314               val live = live_of_bnf bnf;
   315               val mapx = mk live Ts Us (of_bnf bnf);
   316               val TUs' = map dest (fst (strip_typeN live (fastype_of mapx)));
   317             in Term.list_comb (mapx, map build TUs') end
   318           else
   319             build_simple TU
   320         | _ => build_simple TU);
   321   in build end;
   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 val dummy_var_name = "?f"
   327 
   328 fun mk_map_pattern ctxt s =
   329   let
   330     val bnf = the (bnf_of ctxt s);
   331     val mapx = map_of_bnf bnf;
   332     val live = live_of_bnf bnf;
   333     val (f_Ts, _) = strip_typeN live (fastype_of mapx);
   334     val fs = map_index (fn (i, T) => Var ((dummy_var_name, i), T)) f_Ts;
   335   in
   336     (mapx, betapplys (mapx, fs))
   337   end;
   338 
   339 fun dest_map ctxt s call =
   340   let
   341     val (map0, pat) = mk_map_pattern ctxt s;
   342     val (_, tenv) = fo_match ctxt call pat;
   343   in
   344     (map0, Vartab.fold_rev (fn (_, (_, f)) => cons f) tenv [])
   345   end;
   346 
   347 fun liveness_of_fp_bnf n bnf =
   348   (case T_of_bnf bnf of
   349     Type (_, Ts) => map (not o member (op =) (deads_of_bnf bnf)) Ts
   350   | _ => replicate n false);
   351 
   352 fun cannot_merge_types () = error "Mutually recursive types must have the same type parameters";
   353 
   354 fun merge_type_arg T T' = if T = T' then T else cannot_merge_types ();
   355 
   356 fun merge_type_args (As, As') =
   357   if length As = length As' then map2 merge_type_arg As As' else cannot_merge_types ();
   358 
   359 fun reassoc_conjs thm =
   360   reassoc_conjs (thm RS @{thm conj_assoc[THEN iffD1]})
   361   handle THM _ => thm;
   362 
   363 fun type_args_named_constrained_of ((((ncAs, _), _), _), _) = ncAs;
   364 fun type_binding_of ((((_, b), _), _), _) = b;
   365 fun map_binding_of (((_, (b, _)), _), _) = b;
   366 fun rel_binding_of (((_, (_, b)), _), _) = b;
   367 fun mixfix_of ((_, mx), _) = mx;
   368 fun ctr_specs_of (_, ctr_specs) = ctr_specs;
   369 
   370 fun disc_of ((((disc, _), _), _), _) = disc;
   371 fun ctr_of ((((_, ctr), _), _), _) = ctr;
   372 fun args_of (((_, args), _), _) = args;
   373 fun defaults_of ((_, ds), _) = ds;
   374 fun ctr_mixfix_of (_, mx) = mx;
   375 
   376 fun add_nesty_bnf_names Us =
   377   let
   378     fun add (Type (s, Ts)) ss =
   379         let val (needs, ss') = fold_map add Ts ss in
   380           if exists I needs then (true, insert (op =) s ss') else (false, ss')
   381         end
   382       | add T ss = (member (op =) Us T, ss);
   383   in snd oo add end;
   384 
   385 fun nesty_bnfs ctxt ctr_Tsss Us =
   386   map_filter (bnf_of ctxt) (fold (fold (fold (add_nesty_bnf_names Us))) ctr_Tsss []);
   387 
   388 fun indexify proj xs f p = f (find_index (curry op = (proj p)) xs) p;
   389 
   390 type lfp_sugar_thms =
   391   (thm list * thm * Args.src list)
   392   * (thm list list * thm list list * Args.src list)
   393 
   394 type gfp_sugar_thms =
   395   ((thm list * thm) list * Args.src list)
   396   * (thm list list * thm list list * Args.src list)
   397   * (thm list list * thm list list * Args.src list)
   398   * (thm list list * thm list list * Args.src list)
   399   * (thm list list list * thm list list list * Args.src list);
   400 
   401 fun mk_iter_fun_arg_types0 n ms = map2 dest_tupleT ms o dest_sumTN_balanced n o domain_type;
   402 
   403 fun mk_iter_fun_arg_types ctr_Tsss ns mss =
   404   mk_fp_iter_fun_types
   405   #> map3 mk_iter_fun_arg_types0 ns mss
   406   #> map2 (map2 (map2 unzip_recT)) ctr_Tsss;
   407 
   408 fun mk_iters_args_types ctr_Tsss Cs ns mss ctor_iter_fun_Tss lthy =
   409   let
   410     val Css = map2 replicate ns Cs;
   411     val y_Tsss = map3 mk_iter_fun_arg_types0 ns mss (map un_fold_of ctor_iter_fun_Tss);
   412     val g_Tss = map2 (fn C => map (fn y_Ts => y_Ts ---> C)) Cs y_Tsss;
   413 
   414     val ((gss, ysss), lthy) =
   415       lthy
   416       |> mk_Freess "f" g_Tss
   417       ||>> mk_Freesss "x" y_Tsss;
   418 
   419     val y_Tssss = map (map (map single)) y_Tsss;
   420     val yssss = map (map (map single)) ysss;
   421 
   422     val z_Tssss =
   423       map4 (fn n => fn ms => fn ctr_Tss => fn ctor_iter_fun_Ts =>
   424           map3 (fn m => fn ctr_Ts => fn ctor_iter_fun_T =>
   425               map2 unzip_recT ctr_Ts (dest_tupleT m ctor_iter_fun_T))
   426             ms ctr_Tss (dest_sumTN_balanced n (domain_type (co_rec_of ctor_iter_fun_Ts))))
   427         ns mss ctr_Tsss ctor_iter_fun_Tss;
   428 
   429     val z_Tsss' = map (map flat_rec_arg_args) z_Tssss;
   430     val h_Tss = map2 (map2 (curry op --->)) z_Tsss' Css;
   431 
   432     val hss = map2 (map2 retype_free) h_Tss gss;
   433     val zssss_hd = map2 (map2 (map2 (retype_free o hd))) z_Tssss ysss;
   434     val (zssss_tl, lthy) =
   435       lthy
   436       |> mk_Freessss "y" (map (map (map tl)) z_Tssss);
   437     val zssss = map2 (map2 (map2 cons)) zssss_hd zssss_tl;
   438   in
   439     ([(g_Tss, y_Tssss, gss, yssss), (h_Tss, z_Tssss, hss, zssss)], lthy)
   440   end;
   441 
   442 fun mk_coiter_fun_arg_types0 ctr_Tsss Cs ns fun_Ts =
   443   let
   444     (*avoid "'a itself" arguments in coiterators*)
   445     fun repair_arity [[]] = [[@{typ unit}]]
   446       | repair_arity Tss = Tss;
   447 
   448     val ctr_Tsss' = map repair_arity ctr_Tsss;
   449     val f_sum_prod_Ts = map range_type fun_Ts;
   450     val f_prod_Tss = map2 dest_sumTN_balanced ns f_sum_prod_Ts;
   451     val f_Tsss = map2 (map2 (dest_tupleT o length)) ctr_Tsss' f_prod_Tss;
   452     val f_Tssss = map3 (fn C => map2 (map2 (map (curry op --> C) oo unzip_corecT)))
   453       Cs ctr_Tsss' f_Tsss;
   454     val q_Tssss = map (map (map (fn [_] => [] | [_, T] => [mk_pred1T (domain_type T)]))) f_Tssss;
   455   in
   456     (q_Tssss, f_Tsss, f_Tssss, f_sum_prod_Ts)
   457   end;
   458 
   459 fun mk_coiter_p_pred_types Cs ns = map2 (fn n => replicate (Int.max (0, n - 1)) o mk_pred1T) ns Cs;
   460 
   461 fun mk_coiter_fun_arg_types ctr_Tsss Cs ns dtor_coiter =
   462   (mk_coiter_p_pred_types Cs ns,
   463    mk_fp_iter_fun_types dtor_coiter |> mk_coiter_fun_arg_types0 ctr_Tsss Cs ns);
   464 
   465 fun mk_coiters_args_types ctr_Tsss Cs ns dtor_coiter_fun_Tss lthy =
   466   let
   467     val p_Tss = mk_coiter_p_pred_types Cs ns;
   468 
   469     fun mk_types get_Ts =
   470       let
   471         val fun_Ts = map get_Ts dtor_coiter_fun_Tss;
   472         val (q_Tssss, f_Tsss, f_Tssss, f_sum_prod_Ts) = mk_coiter_fun_arg_types0 ctr_Tsss Cs ns fun_Ts;
   473         val pf_Tss = map3 flat_corec_preds_predsss_gettersss p_Tss q_Tssss f_Tssss;
   474       in
   475         (q_Tssss, f_Tsss, f_Tssss, (f_sum_prod_Ts, pf_Tss))
   476       end;
   477 
   478     val (r_Tssss, g_Tsss, g_Tssss, unfold_types) = mk_types un_fold_of;
   479     val (s_Tssss, h_Tsss, h_Tssss, corec_types) = mk_types co_rec_of;
   480 
   481     val ((((Free (z, _), cs), pss), gssss), lthy) =
   482       lthy
   483       |> yield_singleton (mk_Frees "z") dummyT
   484       ||>> mk_Frees "a" Cs
   485       ||>> mk_Freess "p" p_Tss
   486       ||>> mk_Freessss "g" g_Tssss;
   487     val rssss = map (map (map (fn [] => []))) r_Tssss;
   488 
   489     val hssss_hd = map2 (map2 (map2 (fn T :: _ => fn [g] => retype_free T g))) h_Tssss gssss;
   490     val ((sssss, hssss_tl), lthy) =
   491       lthy
   492       |> mk_Freessss "q" s_Tssss
   493       ||>> mk_Freessss "h" (map (map (map tl)) h_Tssss);
   494     val hssss = map2 (map2 (map2 cons)) hssss_hd hssss_tl;
   495 
   496     val cpss = map2 (map o rapp) cs pss;
   497 
   498     fun build_sum_inj mk_inj = build_map lthy (uncurry mk_inj o dest_sumT o snd);
   499 
   500     fun build_dtor_coiter_arg _ [] [cf] = cf
   501       | build_dtor_coiter_arg T [cq] [cf, cf'] =
   502         mk_If cq (build_sum_inj Inl_const (fastype_of cf, T) $ cf)
   503           (build_sum_inj Inr_const (fastype_of cf', T) $ cf');
   504 
   505     fun mk_args qssss fssss f_Tsss =
   506       let
   507         val pfss = map3 flat_corec_preds_predsss_gettersss pss qssss fssss;
   508         val cqssss = map2 (map o map o map o rapp) cs qssss;
   509         val cfssss = map2 (map o map o map o rapp) cs fssss;
   510         val cqfsss = map3 (map3 (map3 build_dtor_coiter_arg)) f_Tsss cqssss cfssss;
   511       in (pfss, cqfsss) end;
   512 
   513     val unfold_args = mk_args rssss gssss g_Tsss;
   514     val corec_args = mk_args sssss hssss h_Tsss;
   515   in
   516     ((z, cs, cpss, [(unfold_args, unfold_types), (corec_args, corec_types)]), lthy)
   517   end;
   518 
   519 fun mk_co_iters_prelims fp ctr_Tsss fpTs Cs ns mss xtor_co_iterss0 lthy =
   520   let
   521     val thy = Proof_Context.theory_of lthy;
   522 
   523     val (xtor_co_iter_fun_Tss, xtor_co_iterss) =
   524       map (mk_co_iters thy fp fpTs Cs #> `(mk_fp_iter_fun_types o hd)) (transpose xtor_co_iterss0)
   525       |> apsnd transpose o apfst transpose o split_list;
   526 
   527     val ((iters_args_types, coiters_args_types), lthy') =
   528       if fp = Least_FP then
   529         mk_iters_args_types ctr_Tsss Cs ns mss xtor_co_iter_fun_Tss lthy |>> (rpair NONE o SOME)
   530       else
   531         mk_coiters_args_types ctr_Tsss Cs ns xtor_co_iter_fun_Tss lthy |>> (pair NONE o SOME)
   532   in
   533     ((xtor_co_iterss, iters_args_types, coiters_args_types), lthy')
   534   end;
   535 
   536 fun mk_iter_body ctor_iter fss xssss =
   537   Term.list_comb (ctor_iter, map2 (mk_sum_caseN_balanced oo map2 mk_uncurried2_fun) fss xssss);
   538 
   539 fun mk_preds_getterss_join c cps sum_prod_T cqfss =
   540   let val n = length cqfss in
   541     Term.lambda c (mk_IfN sum_prod_T cps
   542       (map2 (mk_InN_balanced sum_prod_T n) (map HOLogic.mk_tuple cqfss) (1 upto n)))
   543   end;
   544 
   545 fun mk_coiter_body cs cpss f_sum_prod_Ts cqfsss dtor_coiter =
   546   Term.list_comb (dtor_coiter, map4 mk_preds_getterss_join cs cpss f_sum_prod_Ts cqfsss);
   547 
   548 fun define_co_iters fp fpT Cs binding_specs lthy0 =
   549   let
   550     val thy = Proof_Context.theory_of lthy0;
   551 
   552     val maybe_conceal_def_binding = Thm.def_binding
   553       #> Config.get lthy0 bnf_note_all = false ? Binding.conceal;
   554 
   555     val ((csts, defs), (lthy', lthy)) = lthy0
   556       |> apfst split_list o fold_map (fn (b, spec) =>
   557         Specification.definition (SOME (b, NONE, NoSyn), ((maybe_conceal_def_binding b, []), spec))
   558         #>> apsnd snd) binding_specs
   559       ||> `Local_Theory.restore;
   560 
   561     val phi = Proof_Context.export_morphism lthy lthy';
   562 
   563     val csts' = map (mk_co_iter thy fp fpT Cs o Morphism.term phi) csts;
   564     val defs' = map (Morphism.thm phi) defs;
   565   in
   566     ((csts', defs'), lthy')
   567   end;
   568 
   569 fun define_iters iterNs iter_args_typess' mk_binding fpTs Cs ctor_iters lthy =
   570   let
   571     val nn = length fpTs;
   572 
   573     val fpT_to_C as Type (_, [fpT, _]) = snd (strip_typeN nn (fastype_of (hd ctor_iters)));
   574 
   575     fun generate_iter suf (f_Tss, _, fss, xssss) ctor_iter =
   576       let
   577         val res_T = fold_rev (curry op --->) f_Tss fpT_to_C;
   578         val b = mk_binding suf;
   579         val spec =
   580           mk_Trueprop_eq (lists_bmoc fss (Free (Binding.name_of b, res_T)),
   581             mk_iter_body ctor_iter fss xssss);
   582       in (b, spec) end;
   583   in
   584     define_co_iters Least_FP fpT Cs (map3 generate_iter iterNs iter_args_typess' ctor_iters) lthy
   585   end;
   586 
   587 fun define_coiters coiterNs (_, cs, cpss, coiter_args_typess') mk_binding fpTs Cs dtor_coiters
   588     lthy =
   589   let
   590     val nn = length fpTs;
   591 
   592     val C_to_fpT as Type (_, [_, fpT]) = snd (strip_typeN nn (fastype_of (hd dtor_coiters)));
   593 
   594     fun generate_coiter suf ((pfss, cqfsss), (f_sum_prod_Ts, pf_Tss)) dtor_coiter =
   595       let
   596         val res_T = fold_rev (curry op --->) pf_Tss C_to_fpT;
   597         val b = mk_binding suf;
   598         val spec =
   599           mk_Trueprop_eq (lists_bmoc pfss (Free (Binding.name_of b, res_T)),
   600             mk_coiter_body cs cpss f_sum_prod_Ts cqfsss dtor_coiter);
   601       in (b, spec) end;
   602   in
   603     define_co_iters Greatest_FP fpT Cs
   604       (map3 generate_coiter coiterNs coiter_args_typess' dtor_coiters) lthy
   605   end;
   606 
   607 fun derive_induct_iters_thms_for_types pre_bnfs [fold_args_types, rec_args_types] ctor_induct
   608     ctor_iter_thmss nesting_bnfs nested_bnfs fpTs Cs Xs ctrXs_Tsss ctrss ctr_defss iterss iter_defss
   609     lthy =
   610   let
   611     val iterss' = transpose iterss;
   612     val iter_defss' = transpose iter_defss;
   613 
   614     val [folds, recs] = iterss';
   615     val [fold_defs, rec_defs] = iter_defss';
   616 
   617     val ctr_Tsss = map (map (binder_types o fastype_of)) ctrss;
   618 
   619     val nn = length pre_bnfs;
   620     val ns = map length ctr_Tsss;
   621     val mss = map (map length) ctr_Tsss;
   622 
   623     val pre_map_defs = map map_def_of_bnf pre_bnfs;
   624     val pre_set_defss = map set_defs_of_bnf pre_bnfs;
   625     val nesting_map_idents = map (unfold_thms lthy [id_def] o map_id0_of_bnf) nesting_bnfs;
   626     val nested_map_idents = map (unfold_thms lthy [id_def] o map_id0_of_bnf) nested_bnfs;
   627     val nested_set_maps = maps set_map_of_bnf nested_bnfs;
   628 
   629     val fp_b_names = map base_name_of_typ fpTs;
   630 
   631     val ((((ps, ps'), xsss), us'), names_lthy) =
   632       lthy
   633       |> mk_Frees' "P" (map mk_pred1T fpTs)
   634       ||>> mk_Freesss "x" ctr_Tsss
   635       ||>> Variable.variant_fixes fp_b_names;
   636 
   637     val us = map2 (curry Free) us' fpTs;
   638 
   639     fun mk_sets_nested bnf =
   640       let
   641         val Type (T_name, Us) = T_of_bnf bnf;
   642         val lives = lives_of_bnf bnf;
   643         val sets = sets_of_bnf bnf;
   644         fun mk_set U =
   645           (case find_index (curry op = U) lives of
   646             ~1 => Term.dummy
   647           | i => nth sets i);
   648       in
   649         (T_name, map mk_set Us)
   650       end;
   651 
   652     val setss_nested = map mk_sets_nested nested_bnfs;
   653 
   654     val (induct_thms, induct_thm) =
   655       let
   656         fun mk_set Ts t =
   657           let val Type (_, Ts0) = domain_type (fastype_of t) in
   658             Term.subst_atomic_types (Ts0 ~~ Ts) t
   659           end;
   660 
   661         fun mk_raw_prem_prems _ (x as Free (_, Type _)) (X as TFree _) =
   662             [([], (find_index (curry op = X) Xs + 1, x))]
   663           | mk_raw_prem_prems names_lthy (x as Free (s, Type (T_name, Ts0))) (Type (_, Xs_Ts0)) =
   664             (case AList.lookup (op =) setss_nested T_name of
   665               NONE => []
   666             | SOME raw_sets0 =>
   667               let
   668                 val (Xs_Ts, (Ts, raw_sets)) =
   669                   filter (exists_subtype_in Xs o fst) (Xs_Ts0 ~~ (Ts0 ~~ raw_sets0))
   670                   |> split_list ||> split_list;
   671                 val sets = map (mk_set Ts0) raw_sets;
   672                 val (ys, names_lthy') = names_lthy |> mk_Frees s Ts;
   673                 val xysets = map (pair x) (ys ~~ sets);
   674                 val ppremss = map2 (mk_raw_prem_prems names_lthy') ys Xs_Ts;
   675               in
   676                 flat (map2 (map o apfst o cons) xysets ppremss)
   677               end)
   678           | mk_raw_prem_prems _ _ _ = [];
   679 
   680         fun close_prem_prem xs t =
   681           fold_rev Logic.all (map Free (drop (nn + length xs)
   682             (rev (Term.add_frees t (map dest_Free xs @ ps'))))) t;
   683 
   684         fun mk_prem_prem xs (xysets, (j, x)) =
   685           close_prem_prem xs (Logic.list_implies (map (fn (x', (y, set)) =>
   686               HOLogic.mk_Trueprop (HOLogic.mk_mem (y, set $ x'))) xysets,
   687             HOLogic.mk_Trueprop (nth ps (j - 1) $ x)));
   688 
   689         fun mk_raw_prem phi ctr ctr_Ts ctrXs_Ts =
   690           let
   691             val (xs, names_lthy') = names_lthy |> mk_Frees "x" ctr_Ts;
   692             val pprems = flat (map2 (mk_raw_prem_prems names_lthy') xs ctrXs_Ts);
   693           in (xs, pprems, HOLogic.mk_Trueprop (phi $ Term.list_comb (ctr, xs))) end;
   694 
   695         fun mk_prem (xs, raw_pprems, concl) =
   696           fold_rev Logic.all xs (Logic.list_implies (map (mk_prem_prem xs) raw_pprems, concl));
   697 
   698         val raw_premss = map4 (map3 o mk_raw_prem) ps ctrss ctr_Tsss ctrXs_Tsss;
   699 
   700         val goal =
   701           Library.foldr (Logic.list_implies o apfst (map mk_prem)) (raw_premss,
   702             HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj (map2 (curry op $) ps us)));
   703 
   704         val kksss = map (map (map (fst o snd) o #2)) raw_premss;
   705 
   706         val ctor_induct' = ctor_induct OF (map mk_sumEN_tupled_balanced mss);
   707 
   708         val thm =
   709           Goal.prove_sorry lthy [] [] goal (fn {context = ctxt, ...} =>
   710             mk_induct_tac ctxt nn ns mss kksss (flat ctr_defss) ctor_induct' nested_set_maps
   711               pre_set_defss)
   712           |> singleton (Proof_Context.export names_lthy lthy)
   713           |> Thm.close_derivation;
   714       in
   715         `(conj_dests nn) thm
   716       end;
   717 
   718     val induct_cases = quasi_unambiguous_case_names (maps (map name_of_ctr) ctrss);
   719     val induct_case_names_attr = Attrib.internal (K (Rule_Cases.case_names induct_cases));
   720 
   721     val xctrss = map2 (map2 (curry Term.list_comb)) ctrss xsss;
   722 
   723     fun mk_iter_thmss (_, x_Tssss, fss, _) iters iter_defs ctor_iter_thms =
   724       let
   725         val fiters = map (lists_bmoc fss) iters;
   726 
   727         fun mk_goal fss fiter xctr f xs fxs =
   728           fold_rev (fold_rev Logic.all) (xs :: fss)
   729             (mk_Trueprop_eq (fiter $ xctr, Term.list_comb (f, fxs)));
   730 
   731         fun maybe_tick (T, U) u f =
   732           if try (fst o HOLogic.dest_prodT) U = SOME T then
   733             Term.lambda u (HOLogic.mk_prod (u, f $ u))
   734           else
   735             f;
   736 
   737         fun build_iter (x as Free (_, T)) U =
   738           if T = U then
   739             x
   740           else
   741             build_map lthy (indexify (perhaps (try (snd o HOLogic.dest_prodT)) o snd) Cs
   742               (fn kk => fn TU => maybe_tick TU (nth us kk) (nth fiters kk))) (T, U) $ x;
   743 
   744         val fxsss = map2 (map2 (flat_rec_arg_args oo map2 (map o build_iter))) xsss x_Tssss;
   745 
   746         val goalss = map5 (map4 o mk_goal fss) fiters xctrss fss xsss fxsss;
   747 
   748         val tacss =
   749           map2 (map o mk_iter_tac pre_map_defs (nested_map_idents @ nesting_map_idents) iter_defs)
   750             ctor_iter_thms ctr_defss;
   751 
   752         fun prove goal tac =
   753           Goal.prove_sorry lthy [] [] goal (tac o #context)
   754           |> Thm.close_derivation;
   755       in
   756         map2 (map2 prove) goalss tacss
   757       end;
   758 
   759     val fold_thmss = mk_iter_thmss fold_args_types folds fold_defs (map un_fold_of ctor_iter_thmss);
   760     val rec_thmss = mk_iter_thmss rec_args_types recs rec_defs (map co_rec_of ctor_iter_thmss);
   761   in
   762     ((induct_thms, induct_thm, [induct_case_names_attr]),
   763      (fold_thmss, rec_thmss, code_nitpicksimp_attrs @ simp_attrs))
   764   end;
   765 
   766 fun derive_coinduct_coiters_thms_for_types pre_bnfs (z, cs, cpss,
   767       coiters_args_types as [((pgss, crgsss), _), ((phss, cshsss), _)])
   768     dtor_coinduct dtor_injects dtor_ctors dtor_coiter_thmss nesting_bnfs fpTs Cs Xs ctrXs_Tsss kss
   769     mss ns ctr_defss (ctr_sugars : ctr_sugar list) coiterss coiter_defss export_args lthy =
   770   let
   771     fun mk_ctor_dtor_coiter_thm dtor_inject dtor_ctor coiter =
   772       iffD1 OF [dtor_inject, trans OF [coiter, dtor_ctor RS sym]];
   773 
   774     val ctor_dtor_coiter_thmss =
   775       map3 (map oo mk_ctor_dtor_coiter_thm) dtor_injects dtor_ctors dtor_coiter_thmss;
   776 
   777     val coiterss' = transpose coiterss;
   778     val coiter_defss' = transpose coiter_defss;
   779 
   780     val [unfold_defs, corec_defs] = coiter_defss';
   781 
   782     val nn = length pre_bnfs;
   783 
   784     val pre_map_defs = map map_def_of_bnf pre_bnfs;
   785     val pre_rel_defs = map rel_def_of_bnf pre_bnfs;
   786     val nesting_map_idents = map (unfold_thms lthy [id_def] o map_id0_of_bnf) nesting_bnfs;
   787     val nesting_rel_eqs = map rel_eq_of_bnf nesting_bnfs;
   788 
   789     val fp_b_names = map base_name_of_typ fpTs;
   790 
   791     val ctrss = map #ctrs ctr_sugars;
   792     val discss = map #discs ctr_sugars;
   793     val selsss = map #selss ctr_sugars;
   794     val exhausts = map #exhaust ctr_sugars;
   795     val disc_thmsss = map #disc_thmss ctr_sugars;
   796     val discIss = map #discIs ctr_sugars;
   797     val sel_thmsss = map #sel_thmss ctr_sugars;
   798 
   799     val (((rs, us'), vs'), names_lthy) =
   800       lthy
   801       |> mk_Frees "R" (map (fn T => mk_pred2T T T) fpTs)
   802       ||>> Variable.variant_fixes fp_b_names
   803       ||>> Variable.variant_fixes (map (suffix "'") fp_b_names);
   804 
   805     val us = map2 (curry Free) us' fpTs;
   806     val udiscss = map2 (map o rapp) us discss;
   807     val uselsss = map2 (map o map o rapp) us selsss;
   808 
   809     val vs = map2 (curry Free) vs' fpTs;
   810     val vdiscss = map2 (map o rapp) vs discss;
   811     val vselsss = map2 (map o map o rapp) vs selsss;
   812 
   813     val coinduct_thms_pairs =
   814       let
   815         val uvrs = map3 (fn r => fn u => fn v => r $ u $ v) rs us vs;
   816         val uv_eqs = map2 (curry HOLogic.mk_eq) us vs;
   817         val strong_rs =
   818           map4 (fn u => fn v => fn uvr => fn uv_eq =>
   819             fold_rev Term.lambda [u, v] (HOLogic.mk_disj (uvr, uv_eq))) us vs uvrs uv_eqs;
   820 
   821         fun build_rel rs' T =
   822           (case find_index (curry op = T) fpTs of
   823             ~1 =>
   824             if exists_subtype_in fpTs T then
   825               let
   826                 val Type (s, Ts) = T
   827                 val bnf = the (bnf_of lthy s);
   828                 val live = live_of_bnf bnf;
   829                 val rel = mk_rel live Ts Ts (rel_of_bnf bnf);
   830                 val Ts' = map domain_type (fst (strip_typeN live (fastype_of rel)));
   831               in Term.list_comb (rel, map (build_rel rs') Ts') end
   832             else
   833               HOLogic.eq_const T
   834           | kk => nth rs' kk);
   835 
   836         fun build_rel_app rs' usel vsel = fold rapp [usel, vsel] (build_rel rs' (fastype_of usel));
   837 
   838         fun mk_prem_ctr_concls rs' n k udisc usels vdisc vsels =
   839           (if k = n then [] else [HOLogic.mk_eq (udisc, vdisc)]) @
   840           (if null usels then
   841              []
   842            else
   843              [Library.foldr HOLogic.mk_imp (if n = 1 then [] else [udisc, vdisc],
   844                 Library.foldr1 HOLogic.mk_conj (map2 (build_rel_app rs') usels vsels))]);
   845 
   846         fun mk_prem_concl rs' n udiscs uselss vdiscs vselss =
   847           Library.foldr1 HOLogic.mk_conj
   848             (flat (map5 (mk_prem_ctr_concls rs' n) (1 upto n) udiscs uselss vdiscs vselss))
   849           handle List.Empty => @{term True};
   850 
   851         fun mk_prem rs' uvr u v n udiscs uselss vdiscs vselss =
   852           fold_rev Logic.all [u, v] (Logic.mk_implies (HOLogic.mk_Trueprop uvr,
   853             HOLogic.mk_Trueprop (mk_prem_concl rs' n udiscs uselss vdiscs vselss)));
   854 
   855         val concl =
   856           HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj
   857             (map3 (fn uvr => fn u => fn v => HOLogic.mk_imp (uvr, HOLogic.mk_eq (u, v)))
   858                uvrs us vs));
   859 
   860         fun mk_goal rs' =
   861           Logic.list_implies (map8 (mk_prem rs') uvrs us vs ns udiscss uselsss vdiscss vselsss,
   862             concl);
   863 
   864         val goals = map mk_goal [rs, strong_rs];
   865 
   866         fun prove dtor_coinduct' goal =
   867           Goal.prove_sorry lthy [] [] goal (fn {context = ctxt, ...} =>
   868             mk_coinduct_tac ctxt nesting_rel_eqs nn ns dtor_coinduct' pre_rel_defs dtor_ctors
   869               exhausts ctr_defss disc_thmsss sel_thmsss)
   870           |> singleton (Proof_Context.export names_lthy lthy)
   871           |> Thm.close_derivation;
   872 
   873         fun postproc nn thm =
   874           Thm.permute_prems 0 nn
   875             (if nn = 1 then thm RS mp else funpow nn (fn thm => reassoc_conjs (thm RS mp_conj)) thm)
   876           |> Drule.zero_var_indexes
   877           |> `(conj_dests nn);
   878 
   879         val rel_eqs = map rel_eq_of_bnf pre_bnfs;
   880         val rel_monos = map rel_mono_of_bnf pre_bnfs;
   881         val dtor_coinducts =
   882           [dtor_coinduct, mk_strong_coinduct_thm dtor_coinduct rel_eqs rel_monos lthy];
   883       in
   884         map2 (postproc nn oo prove) dtor_coinducts goals
   885       end;
   886 
   887     fun mk_coinduct_concls ms discs ctrs =
   888       let
   889         fun mk_disc_concl disc = [name_of_disc disc];
   890         fun mk_ctr_concl 0 _ = []
   891           | mk_ctr_concl _ ctor = [name_of_ctr ctor];
   892         val disc_concls = map mk_disc_concl (fst (split_last discs)) @ [[]];
   893         val ctr_concls = map2 mk_ctr_concl ms ctrs;
   894       in
   895         flat (map2 append disc_concls ctr_concls)
   896       end;
   897 
   898     val coinduct_cases = quasi_unambiguous_case_names (map (prefix EqN) fp_b_names);
   899     val coinduct_conclss =
   900       map3 (quasi_unambiguous_case_names ooo mk_coinduct_concls) mss discss ctrss;
   901 
   902     fun mk_maybe_not pos = not pos ? HOLogic.mk_not;
   903 
   904     val fcoiterss' as [gunfolds, hcorecs] =
   905       map2 (fn (pfss, _) => map (lists_bmoc pfss)) (map fst coiters_args_types) coiterss';
   906 
   907     val (unfold_thmss, corec_thmss) =
   908       let
   909         fun mk_goal pfss c cps fcoiter n k ctr m cfs' =
   910           fold_rev (fold_rev Logic.all) ([c] :: pfss)
   911             (Logic.list_implies (seq_conds (HOLogic.mk_Trueprop oo mk_maybe_not) n k cps,
   912                mk_Trueprop_eq (fcoiter $ c, Term.list_comb (ctr, take m cfs'))));
   913 
   914         fun mk_U maybe_mk_sumT =
   915           typ_subst_nonatomic (map2 (fn C => fn fpT => (maybe_mk_sumT fpT C, fpT)) Cs fpTs);
   916 
   917         fun tack z_name (c, u) f =
   918           let val z = Free (z_name, mk_sumT (fastype_of u, fastype_of c)) in
   919             Term.lambda z (mk_sum_case (Term.lambda u u, Term.lambda c (f $ c)) $ z)
   920           end;
   921 
   922         fun build_coiter fcoiters maybe_mk_sumT maybe_tack cqf =
   923           let val T = fastype_of cqf in
   924             if exists_subtype_in Cs T then
   925               let val U = mk_U maybe_mk_sumT T in
   926                 build_map lthy (indexify snd fpTs (fn kk => fn _ =>
   927                   maybe_tack (nth cs kk, nth us kk) (nth fcoiters kk))) (T, U) $ cqf
   928               end
   929             else
   930               cqf
   931           end;
   932 
   933         val crgsss' = map (map (map (build_coiter (un_fold_of fcoiterss') (K I) (K I)))) crgsss;
   934         val cshsss' = map (map (map (build_coiter (co_rec_of fcoiterss') (curry mk_sumT) (tack z))))
   935           cshsss;
   936 
   937         val unfold_goalss = map8 (map4 oooo mk_goal pgss) cs cpss gunfolds ns kss ctrss mss crgsss';
   938         val corec_goalss = map8 (map4 oooo mk_goal phss) cs cpss hcorecs ns kss ctrss mss cshsss';
   939 
   940         val unfold_tacss =
   941           map3 (map oo mk_coiter_tac unfold_defs nesting_map_idents)
   942             (map un_fold_of ctor_dtor_coiter_thmss) pre_map_defs ctr_defss;
   943         val corec_tacss =
   944           map3 (map oo mk_coiter_tac corec_defs nesting_map_idents)
   945             (map co_rec_of ctor_dtor_coiter_thmss) pre_map_defs ctr_defss;
   946 
   947         fun prove goal tac =
   948           Goal.prove_sorry lthy [] [] goal (tac o #context)
   949           |> Thm.close_derivation;
   950 
   951         val unfold_thmss = map2 (map2 prove) unfold_goalss unfold_tacss;
   952         val corec_thmss =
   953           map2 (map2 prove) corec_goalss corec_tacss
   954           |> map (map (unfold_thms lthy @{thms sum_case_if}));
   955       in
   956         (unfold_thmss, corec_thmss)
   957       end;
   958 
   959     val (disc_unfold_iff_thmss, disc_corec_iff_thmss) =
   960       let
   961         fun mk_goal c cps fcoiter n k disc =
   962           mk_Trueprop_eq (disc $ (fcoiter $ c),
   963             if n = 1 then @{const True}
   964             else Library.foldr1 HOLogic.mk_conj (seq_conds mk_maybe_not n k cps));
   965 
   966         val unfold_goalss = map6 (map2 oooo mk_goal) cs cpss gunfolds ns kss discss;
   967         val corec_goalss = map6 (map2 oooo mk_goal) cs cpss hcorecs ns kss discss;
   968 
   969         fun mk_case_split' cp = Drule.instantiate' [] [SOME (certify lthy cp)] @{thm case_split};
   970 
   971         val case_splitss' = map (map mk_case_split') cpss;
   972 
   973         val unfold_tacss =
   974           map3 (map oo mk_disc_coiter_iff_tac) case_splitss' unfold_thmss disc_thmsss;
   975         val corec_tacss =
   976           map3 (map oo mk_disc_coiter_iff_tac) case_splitss' corec_thmss disc_thmsss;
   977 
   978         fun prove goal tac =
   979           Goal.prove_sorry lthy [] [] goal (tac o #context)
   980           |> singleton export_args
   981           |> singleton (Proof_Context.export names_lthy lthy)
   982           |> Thm.close_derivation;
   983 
   984         fun proves [_] [_] = []
   985           | proves goals tacs = map2 prove goals tacs;
   986       in
   987         (map2 proves unfold_goalss unfold_tacss, map2 proves corec_goalss corec_tacss)
   988       end;
   989 
   990     fun mk_disc_coiter_thms coiters discIs = map (op RS) (coiters ~~ discIs);
   991 
   992     val disc_unfold_thmss = map2 mk_disc_coiter_thms unfold_thmss discIss;
   993     val disc_corec_thmss = map2 mk_disc_coiter_thms corec_thmss discIss;
   994 
   995     fun mk_sel_coiter_thm coiter_thm sel sel_thm =
   996       let
   997         val (domT, ranT) = dest_funT (fastype_of sel);
   998         val arg_cong' =
   999           Drule.instantiate' (map (SOME o certifyT lthy) [domT, ranT])
  1000             [NONE, NONE, SOME (certify lthy sel)] arg_cong
  1001           |> Thm.varifyT_global;
  1002         val sel_thm' = sel_thm RSN (2, trans);
  1003       in
  1004         coiter_thm RS arg_cong' RS sel_thm'
  1005       end;
  1006 
  1007     fun mk_sel_coiter_thms coiter_thmss =
  1008       map3 (map3 (map2 o mk_sel_coiter_thm)) coiter_thmss selsss sel_thmsss;
  1009 
  1010     val sel_unfold_thmsss = mk_sel_coiter_thms unfold_thmss;
  1011     val sel_corec_thmsss = mk_sel_coiter_thms corec_thmss;
  1012 
  1013     val coinduct_consumes_attr = Attrib.internal (K (Rule_Cases.consumes nn));
  1014     val coinduct_case_names_attr = Attrib.internal (K (Rule_Cases.case_names coinduct_cases));
  1015     val coinduct_case_concl_attrs =
  1016       map2 (fn casex => fn concls =>
  1017           Attrib.internal (K (Rule_Cases.case_conclusion (casex, concls))))
  1018         coinduct_cases coinduct_conclss;
  1019     val coinduct_case_attrs =
  1020       coinduct_consumes_attr :: coinduct_case_names_attr :: coinduct_case_concl_attrs;
  1021   in
  1022     ((coinduct_thms_pairs, coinduct_case_attrs),
  1023      (unfold_thmss, corec_thmss, code_nitpicksimp_attrs),
  1024      (disc_unfold_thmss, disc_corec_thmss, []),
  1025      (disc_unfold_iff_thmss, disc_corec_iff_thmss, simp_attrs),
  1026      (sel_unfold_thmsss, sel_corec_thmsss, simp_attrs))
  1027   end;
  1028 
  1029 fun define_co_datatypes prepare_constraint prepare_typ prepare_term fp construct_fp
  1030     (wrap_opts as (no_discs_sels, rep_compat), specs) no_defs_lthy0 =
  1031   let
  1032     (* TODO: sanity checks on arguments *)
  1033 
  1034     val _ = if fp = Greatest_FP andalso no_discs_sels then
  1035         error "Cannot define codatatypes without discriminators and selectors"
  1036       else
  1037         ();
  1038 
  1039     fun qualify mandatory fp_b_name =
  1040       Binding.qualify mandatory fp_b_name o (rep_compat ? Binding.qualify false rep_compat_prefix);
  1041 
  1042     val nn = length specs;
  1043     val fp_bs = map type_binding_of specs;
  1044     val fp_b_names = map Binding.name_of fp_bs;
  1045     val fp_common_name = mk_common_name fp_b_names;
  1046     val map_bs = map map_binding_of specs;
  1047     val rel_bs = map rel_binding_of specs;
  1048 
  1049     fun prepare_type_arg (_, (ty, c)) =
  1050       let val TFree (s, _) = prepare_typ no_defs_lthy0 ty in
  1051         TFree (s, prepare_constraint no_defs_lthy0 c)
  1052       end;
  1053 
  1054     val Ass0 = map (map prepare_type_arg o type_args_named_constrained_of) specs;
  1055     val unsorted_Ass0 = map (map (resort_tfree HOLogic.typeS)) Ass0;
  1056     val unsorted_As = Library.foldr1 merge_type_args unsorted_Ass0;
  1057     val num_As = length unsorted_As;
  1058     val set_bss = map (map fst o type_args_named_constrained_of) specs;
  1059 
  1060     val (((Bs0, Cs), Xs), no_defs_lthy) =
  1061       no_defs_lthy0
  1062       |> fold (Variable.declare_typ o resort_tfree dummyS) unsorted_As
  1063       |> mk_TFrees num_As
  1064       ||>> mk_TFrees nn
  1065       ||>> variant_tfrees fp_b_names;
  1066 
  1067     fun add_fake_type spec = Typedecl.basic_typedecl (type_binding_of spec, num_As, mixfix_of spec);
  1068 
  1069     val (fake_T_names, fake_lthy) = fold_map add_fake_type specs no_defs_lthy0;
  1070 
  1071     val qsoty = quote o Syntax.string_of_typ fake_lthy;
  1072 
  1073     val _ = (case duplicates (op =) unsorted_As of [] => ()
  1074       | A :: _ => error ("Duplicate type parameter " ^ qsoty A ^ " in " ^ co_prefix fp ^
  1075           "datatype specification"));
  1076 
  1077     val bad_args =
  1078       map (Logic.type_map (singleton (Variable.polymorphic no_defs_lthy0))) unsorted_As
  1079       |> filter_out Term.is_TVar;
  1080     val _ = null bad_args orelse
  1081       error ("Locally fixed type argument " ^ qsoty (hd bad_args) ^ " in " ^ co_prefix fp ^
  1082         "datatype specification");
  1083 
  1084     val mixfixes = map mixfix_of specs;
  1085 
  1086     val _ = (case duplicates Binding.eq_name fp_bs of [] => ()
  1087       | b :: _ => error ("Duplicate type name declaration " ^ quote (Binding.name_of b)));
  1088 
  1089     val ctr_specss = map ctr_specs_of specs;
  1090 
  1091     val disc_bindingss = map (map disc_of) ctr_specss;
  1092     val ctr_bindingss =
  1093       map2 (fn fp_b_name => map (qualify false fp_b_name o ctr_of)) fp_b_names ctr_specss;
  1094     val ctr_argsss = map (map args_of) ctr_specss;
  1095     val ctr_mixfixess = map (map ctr_mixfix_of) ctr_specss;
  1096 
  1097     val sel_bindingsss = map (map (map fst)) ctr_argsss;
  1098     val fake_ctr_Tsss0 = map (map (map (prepare_typ fake_lthy o snd))) ctr_argsss;
  1099     val raw_sel_defaultsss = map (map defaults_of) ctr_specss;
  1100 
  1101     val (As :: _) :: fake_ctr_Tsss =
  1102       burrow (burrow (Syntax.check_typs fake_lthy)) (Ass0 :: fake_ctr_Tsss0);
  1103     val As' = map dest_TFree As;
  1104 
  1105     val rhs_As' = fold (fold (fold Term.add_tfreesT)) fake_ctr_Tsss [];
  1106     val _ = (case subtract (op =) As' rhs_As' of [] => ()
  1107       | extras => error ("Extra type variables on right-hand side: " ^
  1108           commas (map (qsoty o TFree) extras)));
  1109 
  1110     val fake_Ts = map (fn s => Type (s, As)) fake_T_names;
  1111 
  1112     fun eq_fpT_check (T as Type (s, Ts)) (T' as Type (s', Ts')) =
  1113         s = s' andalso (Ts = Ts' orelse
  1114           error ("Wrong type arguments in " ^ co_prefix fp ^ "recursive type " ^ qsoty T ^
  1115             " (expected " ^ qsoty T' ^ ")"))
  1116       | eq_fpT_check _ _ = false;
  1117 
  1118     fun freeze_fp (T as Type (s, Ts)) =
  1119         (case find_index (eq_fpT_check T) fake_Ts of
  1120           ~1 => Type (s, map freeze_fp Ts)
  1121         | kk => nth Xs kk)
  1122       | freeze_fp T = T;
  1123 
  1124     val unfreeze_fp = Term.typ_subst_atomic (Xs ~~ fake_Ts);
  1125 
  1126     val ctrXs_Tsss = map (map (map freeze_fp)) fake_ctr_Tsss;
  1127     val ctrXs_sum_prod_Ts = map (mk_sumTN_balanced o map HOLogic.mk_tupleT) ctrXs_Tsss;
  1128 
  1129     val fp_eqs =
  1130       map dest_TFree Xs ~~ map (Term.typ_subst_atomic (As ~~ unsorted_As)) ctrXs_sum_prod_Ts;
  1131 
  1132     val rhsXs_As' = fold (fold (fold Term.add_tfreesT)) ctrXs_Tsss [];
  1133     val _ = (case subtract (op =) rhsXs_As' As' of [] => ()
  1134       | extras => List.app (fn extra => warning ("Unused type variable on right-hand side of " ^
  1135           co_prefix fp ^ "datatype definition: " ^ qsoty (TFree extra))) extras);
  1136 
  1137     val (pre_bnfs, (fp_res as {bnfs = fp_bnfs as any_fp_bnf :: _, ctors = ctors0, dtors = dtors0,
  1138            xtor_co_iterss = xtor_co_iterss0, xtor_co_induct, dtor_ctors, ctor_dtors, ctor_injects,
  1139            dtor_injects, xtor_map_thms, xtor_set_thmss, xtor_rel_thms, xtor_co_iter_thmss, ...},
  1140            lthy)) =
  1141       fp_bnf (construct_fp mixfixes map_bs rel_bs set_bss) fp_bs (map dest_TFree unsorted_As) fp_eqs
  1142         no_defs_lthy0
  1143       handle BAD_DEAD (X, X_backdrop) =>
  1144         (case X_backdrop of
  1145           Type (bad_tc, _) =>
  1146           let
  1147             val fake_T = qsoty (unfreeze_fp X);
  1148             val fake_T_backdrop = qsoty (unfreeze_fp X_backdrop);
  1149             fun register_hint () =
  1150               "\nUse the " ^ quote (fst (fst @{command_spec "bnf"})) ^ " command to register " ^
  1151               quote bad_tc ^ " as a bounded natural functor to allow nested (co)recursion through \
  1152               \it";
  1153           in
  1154             if is_some (bnf_of no_defs_lthy bad_tc) orelse
  1155                is_some (fp_sugar_of no_defs_lthy bad_tc) then
  1156               error ("Inadmissible " ^ co_prefix fp ^ "recursive occurrence of type " ^ fake_T ^
  1157                 " in type expression " ^ fake_T_backdrop)
  1158             else if is_some (Datatype_Data.get_info (Proof_Context.theory_of no_defs_lthy)
  1159                 bad_tc) then
  1160               error ("Unsupported " ^ co_prefix fp ^ "recursive occurrence of type " ^ fake_T ^
  1161                 " via the old-style datatype " ^ quote bad_tc ^ " in type expression " ^
  1162                 fake_T_backdrop ^ register_hint ())
  1163             else
  1164               error ("Unsupported " ^ co_prefix fp ^ "recursive occurrence of type " ^ fake_T ^
  1165                 " via type constructor " ^ quote bad_tc ^ " in type expression " ^ fake_T_backdrop ^
  1166                 register_hint ())
  1167           end);
  1168 
  1169     val time = time lthy;
  1170     val timer = time (Timer.startRealTimer ());
  1171 
  1172     val nesting_bnfs = nesty_bnfs lthy ctrXs_Tsss As;
  1173     val nested_bnfs = nesty_bnfs lthy ctrXs_Tsss Xs;
  1174 
  1175     val pre_map_defs = map map_def_of_bnf pre_bnfs;
  1176     val pre_set_defss = map set_defs_of_bnf pre_bnfs;
  1177     val pre_rel_defs = map rel_def_of_bnf pre_bnfs;
  1178     val nesting_set_maps = maps set_map_of_bnf nesting_bnfs;
  1179     val nested_set_maps = maps set_map_of_bnf nested_bnfs;
  1180 
  1181     val live = live_of_bnf any_fp_bnf;
  1182     val _ =
  1183       if live = 0 andalso exists (not o Binding.is_empty) (map_bs @ rel_bs) then
  1184         warning "Map function and relator names ignored"
  1185       else
  1186         ();
  1187 
  1188     val Bs =
  1189       map3 (fn alive => fn A as TFree (_, S) => fn B => if alive then resort_tfree S B else A)
  1190         (liveness_of_fp_bnf num_As any_fp_bnf) As Bs0;
  1191 
  1192     val B_ify = Term.typ_subst_atomic (As ~~ Bs);
  1193 
  1194     val ctors = map (mk_ctor As) ctors0;
  1195     val dtors = map (mk_dtor As) dtors0;
  1196 
  1197     val fpTs = map (domain_type o fastype_of) dtors;
  1198 
  1199     fun massage_simple_notes base =
  1200       filter_out (null o #2)
  1201       #> map (fn (thmN, thms, attrs) =>
  1202         ((qualify true base (Binding.name thmN), attrs), [(thms, [])]));
  1203 
  1204     val massage_multi_notes =
  1205       maps (fn (thmN, thmss, attrs) =>
  1206         map3 (fn fp_b_name => fn Type (T_name, _) => fn thms =>
  1207             ((qualify true fp_b_name (Binding.name thmN), attrs T_name), [(thms, [])]))
  1208           fp_b_names fpTs thmss)
  1209       #> filter_out (null o fst o hd o snd);
  1210 
  1211     val ctr_Tsss = map (map (map (Term.typ_subst_atomic (Xs ~~ fpTs)))) ctrXs_Tsss;
  1212     val ns = map length ctr_Tsss;
  1213     val kss = map (fn n => 1 upto n) ns;
  1214     val mss = map (map length) ctr_Tsss;
  1215 
  1216     val ((xtor_co_iterss, iters_args_types, coiters_args_types), lthy') =
  1217       mk_co_iters_prelims fp ctr_Tsss fpTs Cs ns mss xtor_co_iterss0 lthy;
  1218 
  1219     fun define_ctrs_dtrs_for_type (((((((((((((((((((((((fp_bnf, fp_b), fpT), ctor), dtor),
  1220             xtor_co_iters), ctor_dtor), dtor_ctor), ctor_inject), pre_map_def), pre_set_defs),
  1221           pre_rel_def), fp_map_thm), fp_set_thms), fp_rel_thm), n), ks), ms), ctr_bindings),
  1222         ctr_mixfixes), ctr_Tss), disc_bindings), sel_bindingss), raw_sel_defaultss) no_defs_lthy =
  1223       let
  1224         val fp_b_name = Binding.name_of fp_b;
  1225 
  1226         val dtorT = domain_type (fastype_of ctor);
  1227         val ctr_prod_Ts = map HOLogic.mk_tupleT ctr_Tss;
  1228         val ctr_sum_prod_T = mk_sumTN_balanced ctr_prod_Ts;
  1229 
  1230         val ((((w, xss), yss), u'), names_lthy) =
  1231           no_defs_lthy
  1232           |> yield_singleton (mk_Frees "w") dtorT
  1233           ||>> mk_Freess "x" ctr_Tss
  1234           ||>> mk_Freess "y" (map (map B_ify) ctr_Tss)
  1235           ||>> yield_singleton Variable.variant_fixes fp_b_name;
  1236 
  1237         val u = Free (u', fpT);
  1238 
  1239         val tuple_xs = map HOLogic.mk_tuple xss;
  1240         val tuple_ys = map HOLogic.mk_tuple yss;
  1241 
  1242         val ctr_rhss =
  1243           map3 (fn k => fn xs => fn tuple_x => fold_rev Term.lambda xs (ctor $
  1244             mk_InN_balanced ctr_sum_prod_T n tuple_x k)) ks xss tuple_xs;
  1245 
  1246         val maybe_conceal_def_binding = Thm.def_binding
  1247           #> Config.get no_defs_lthy bnf_note_all = false ? Binding.conceal;
  1248 
  1249         val ((raw_ctrs, raw_ctr_defs), (lthy', lthy)) = no_defs_lthy
  1250           |> apfst split_list o fold_map3 (fn b => fn mx => fn rhs =>
  1251               Local_Theory.define ((b, mx), ((maybe_conceal_def_binding b, []), rhs)) #>> apsnd snd)
  1252             ctr_bindings ctr_mixfixes ctr_rhss
  1253           ||> `Local_Theory.restore;
  1254 
  1255         val phi = Proof_Context.export_morphism lthy lthy';
  1256 
  1257         val ctr_defs = map (Morphism.thm phi) raw_ctr_defs;
  1258         val ctr_defs' =
  1259           map2 (fn m => fn def => mk_unabs_def m (def RS meta_eq_to_obj_eq)) ms ctr_defs;
  1260 
  1261         val ctrs0 = map (Morphism.term phi) raw_ctrs;
  1262         val ctrs = map (mk_ctr As) ctrs0;
  1263 
  1264         fun wrap_ctrs lthy =
  1265           let
  1266             fun exhaust_tac {context = ctxt, prems = _} =
  1267               let
  1268                 val ctor_iff_dtor_thm =
  1269                   let
  1270                     val goal =
  1271                       fold_rev Logic.all [w, u]
  1272                         (mk_Trueprop_eq (HOLogic.mk_eq (u, ctor $ w), HOLogic.mk_eq (dtor $ u, w)));
  1273                   in
  1274                     Goal.prove_sorry lthy [] [] goal (fn {context = ctxt, ...} =>
  1275                       mk_ctor_iff_dtor_tac ctxt (map (SOME o certifyT lthy) [dtorT, fpT])
  1276                         (certify lthy ctor) (certify lthy dtor) ctor_dtor dtor_ctor)
  1277                     |> Thm.close_derivation
  1278                     |> Morphism.thm phi
  1279                   end;
  1280 
  1281                 val sumEN_thm' =
  1282                   unfold_thms lthy @{thms unit_all_eq1}
  1283                     (Drule.instantiate' (map (SOME o certifyT lthy) ctr_prod_Ts) []
  1284                        (mk_sumEN_balanced n))
  1285                   |> Morphism.thm phi;
  1286               in
  1287                 mk_exhaust_tac ctxt n ctr_defs ctor_iff_dtor_thm sumEN_thm'
  1288               end;
  1289 
  1290             val inject_tacss =
  1291               map2 (fn 0 => K [] | _ => fn ctr_def => [fn {context = ctxt, ...} =>
  1292                 mk_inject_tac ctxt ctr_def ctor_inject]) ms ctr_defs;
  1293 
  1294             val half_distinct_tacss =
  1295               map (map (fn (def, def') => fn {context = ctxt, ...} =>
  1296                 mk_half_distinct_tac ctxt ctor_inject [def, def'])) (mk_half_pairss (`I ctr_defs));
  1297 
  1298             val tacss = [exhaust_tac] :: inject_tacss @ half_distinct_tacss;
  1299 
  1300             val sel_defaultss = map (map (apsnd (prepare_term lthy))) raw_sel_defaultss
  1301           in
  1302             wrap_free_constructors tacss (((wrap_opts, ctrs0), standard_binding), (disc_bindings,
  1303               (sel_bindingss, sel_defaultss))) lthy
  1304           end;
  1305 
  1306         fun derive_maps_sets_rels (ctr_sugar, lthy) =
  1307           if live = 0 then
  1308             ((([], [], [], []), ctr_sugar), lthy)
  1309           else
  1310             let
  1311               val rel_flip = rel_flip_of_bnf fp_bnf;
  1312               val nones = replicate live NONE;
  1313 
  1314               val ctor_cong =
  1315                 if fp = Least_FP then
  1316                   Drule.dummy_thm
  1317                 else
  1318                   let val ctor' = mk_ctor Bs ctor in
  1319                     cterm_instantiate_pos [NONE, NONE, SOME (certify lthy ctor')] arg_cong
  1320                   end;
  1321 
  1322               fun mk_cIn ify =
  1323                 certify lthy o (fp = Greatest_FP ? curry (op $) (map_types ify ctor)) oo
  1324                 mk_InN_balanced (ify ctr_sum_prod_T) n;
  1325 
  1326               val cxIns = map2 (mk_cIn I) tuple_xs ks;
  1327               val cyIns = map2 (mk_cIn B_ify) tuple_ys ks;
  1328 
  1329               fun mk_map_thm ctr_def' cxIn =
  1330                 fold_thms lthy [ctr_def']
  1331                   (unfold_thms lthy (pre_map_def ::
  1332                        (if fp = Least_FP then [] else [ctor_dtor, dtor_ctor]) @ sum_prod_thms_map)
  1333                      (cterm_instantiate_pos (nones @ [SOME cxIn])
  1334                         (if fp = Least_FP then fp_map_thm else fp_map_thm RS ctor_cong)))
  1335                 |> singleton (Proof_Context.export names_lthy no_defs_lthy);
  1336 
  1337               fun mk_set_thm fp_set_thm ctr_def' cxIn =
  1338                 fold_thms lthy [ctr_def']
  1339                   (unfold_thms lthy (pre_set_defs @ nested_set_maps @ nesting_set_maps @
  1340                        (if fp = Least_FP then [] else [dtor_ctor]) @ sum_prod_thms_set)
  1341                      (cterm_instantiate_pos [SOME cxIn] fp_set_thm))
  1342                 |> singleton (Proof_Context.export names_lthy no_defs_lthy);
  1343 
  1344               fun mk_set_thms fp_set_thm = map2 (mk_set_thm fp_set_thm) ctr_defs' cxIns;
  1345 
  1346               val map_thms = map2 mk_map_thm ctr_defs' cxIns;
  1347               val set_thmss = map mk_set_thms fp_set_thms;
  1348 
  1349               val rel_infos = (ctr_defs' ~~ cxIns, ctr_defs' ~~ cyIns);
  1350 
  1351               fun mk_rel_thm postproc ctr_defs' cxIn cyIn =
  1352                 fold_thms lthy ctr_defs'
  1353                   (unfold_thms lthy (@{thm Inl_Inr_False} :: pre_rel_def ::
  1354                        (if fp = Least_FP then [] else [dtor_ctor]) @ sum_prod_thms_rel)
  1355                      (cterm_instantiate_pos (nones @ [SOME cxIn, SOME cyIn]) fp_rel_thm))
  1356                 |> postproc
  1357                 |> singleton (Proof_Context.export names_lthy no_defs_lthy);
  1358 
  1359               fun mk_rel_inject_thm ((ctr_def', cxIn), (_, cyIn)) =
  1360                 mk_rel_thm (unfold_thms lthy @{thms eq_sym_Unity_conv}) [ctr_def'] cxIn cyIn;
  1361 
  1362               val rel_inject_thms = map mk_rel_inject_thm (op ~~ rel_infos);
  1363 
  1364               fun mk_half_rel_distinct_thm ((xctr_def', cxIn), (yctr_def', cyIn)) =
  1365                 mk_rel_thm (fn thm => thm RS @{thm eq_False[THEN iffD1]}) [xctr_def', yctr_def']
  1366                   cxIn cyIn;
  1367 
  1368               fun mk_other_half_rel_distinct_thm thm =
  1369                 flip_rels lthy live thm
  1370                 RS (rel_flip RS sym RS @{thm arg_cong[of _ _ Not]} RS iffD2);
  1371 
  1372               val half_rel_distinct_thmss =
  1373                 map (map mk_half_rel_distinct_thm) (mk_half_pairss rel_infos);
  1374               val other_half_rel_distinct_thmss =
  1375                 map (map mk_other_half_rel_distinct_thm) half_rel_distinct_thmss;
  1376               val (rel_distinct_thms, _) =
  1377                 join_halves n half_rel_distinct_thmss other_half_rel_distinct_thmss;
  1378 
  1379               val anonymous_notes =
  1380                 [(map (fn th => th RS @{thm eq_False[THEN iffD2]}) rel_distinct_thms,
  1381                   code_nitpicksimp_attrs),
  1382                  (map2 (fn th => fn 0 => th RS @{thm eq_True[THEN iffD2]} | _ => th)
  1383                     rel_inject_thms ms, code_nitpicksimp_attrs)]
  1384                 |> map (fn (thms, attrs) => ((Binding.empty, attrs), [(thms, [])]));
  1385 
  1386               val notes =
  1387                 [(mapN, map_thms, code_nitpicksimp_attrs @ simp_attrs),
  1388                  (rel_distinctN, rel_distinct_thms, simp_attrs),
  1389                  (rel_injectN, rel_inject_thms, simp_attrs),
  1390                  (setN, flat set_thmss, code_nitpicksimp_attrs @ simp_attrs)]
  1391                 |> massage_simple_notes fp_b_name;
  1392             in
  1393               (((map_thms, rel_inject_thms, rel_distinct_thms, set_thmss), ctr_sugar),
  1394                lthy |> Local_Theory.notes (anonymous_notes @ notes) |> snd)
  1395             end;
  1396 
  1397         fun mk_binding suf = qualify false fp_b_name (Binding.suffix_name ("_" ^ suf) fp_b);
  1398 
  1399         fun massage_res (((maps_sets_rels, ctr_sugar), co_iter_res), lthy) =
  1400           (((maps_sets_rels, (ctrs, xss, ctr_defs, ctr_sugar)), co_iter_res), lthy);
  1401       in
  1402         (wrap_ctrs
  1403          #> derive_maps_sets_rels
  1404          ##>>
  1405            (if fp = Least_FP then define_iters [foldN, recN] (the iters_args_types)
  1406             else define_coiters [unfoldN, corecN] (the coiters_args_types))
  1407              mk_binding fpTs Cs xtor_co_iters
  1408          #> massage_res, lthy')
  1409       end;
  1410 
  1411     fun wrap_types_etc (wrap_types_etcs, lthy) =
  1412       fold_map I wrap_types_etcs lthy
  1413       |>> apsnd split_list o apfst (apsnd split_list4 o apfst split_list4 o split_list)
  1414         o split_list;
  1415 
  1416     fun mk_simp_thms ({injects, distincts, case_thms, ...} : ctr_sugar) un_folds co_recs
  1417         mapsx rel_injects rel_distincts setss =
  1418       injects @ distincts @ case_thms @ co_recs @ un_folds @ mapsx @ rel_injects @ rel_distincts
  1419       @ flat setss;
  1420 
  1421     fun derive_note_induct_iters_thms_for_types
  1422         ((((mapss, rel_injects, rel_distincts, setss), (ctrss, _, ctr_defss, ctr_sugars)),
  1423           (iterss, iter_defss)), lthy) =
  1424       let
  1425         val ((induct_thms, induct_thm, induct_attrs), (fold_thmss, rec_thmss, iter_attrs)) =
  1426           derive_induct_iters_thms_for_types pre_bnfs (the iters_args_types) xtor_co_induct
  1427             xtor_co_iter_thmss nesting_bnfs nested_bnfs fpTs Cs Xs ctrXs_Tsss ctrss ctr_defss iterss
  1428             iter_defss lthy;
  1429 
  1430         val induct_type_attr = Attrib.internal o K o Induct.induct_type;
  1431 
  1432         val simp_thmss =
  1433           map7 mk_simp_thms ctr_sugars fold_thmss rec_thmss mapss rel_injects rel_distincts setss;
  1434 
  1435         val common_notes =
  1436           (if nn > 1 then [(inductN, [induct_thm], induct_attrs)] else [])
  1437           |> massage_simple_notes fp_common_name;
  1438 
  1439         val notes =
  1440           [(foldN, fold_thmss, K iter_attrs),
  1441            (inductN, map single induct_thms, fn T_name => induct_attrs @ [induct_type_attr T_name]),
  1442            (recN, rec_thmss, K iter_attrs),
  1443            (simpsN, simp_thmss, K [])]
  1444           |> massage_multi_notes;
  1445       in
  1446         lthy
  1447         |> Local_Theory.notes (common_notes @ notes) |> snd
  1448         |> register_fp_sugars Least_FP pre_bnfs nested_bnfs nesting_bnfs fp_res ctr_defss ctr_sugars
  1449           iterss mapss [induct_thm] (transpose [fold_thmss, rec_thmss]) [] []
  1450       end;
  1451 
  1452     fun derive_note_coinduct_coiters_thms_for_types
  1453         ((((mapss, rel_injects, rel_distincts, setss), (_, _, ctr_defss, ctr_sugars)),
  1454           (coiterss, coiter_defss)), lthy) =
  1455       let
  1456         val (([(coinduct_thms, coinduct_thm), (strong_coinduct_thms, strong_coinduct_thm)],
  1457               coinduct_attrs),
  1458              (unfold_thmss, corec_thmss, coiter_attrs),
  1459              (disc_unfold_thmss, disc_corec_thmss, disc_coiter_attrs),
  1460              (disc_unfold_iff_thmss, disc_corec_iff_thmss, disc_coiter_iff_attrs),
  1461              (sel_unfold_thmsss, sel_corec_thmsss, sel_coiter_attrs)) =
  1462           derive_coinduct_coiters_thms_for_types pre_bnfs (the coiters_args_types) xtor_co_induct
  1463             dtor_injects dtor_ctors xtor_co_iter_thmss nesting_bnfs fpTs Cs Xs ctrXs_Tsss kss mss ns
  1464             ctr_defss ctr_sugars coiterss coiter_defss (Proof_Context.export lthy' no_defs_lthy)
  1465             lthy;
  1466 
  1467         val sel_unfold_thmss = map flat sel_unfold_thmsss;
  1468         val sel_corec_thmss = map flat sel_corec_thmsss;
  1469 
  1470         val coinduct_type_attr = Attrib.internal o K o Induct.coinduct_type;
  1471 
  1472         val flat_coiter_thms = append oo append;
  1473 
  1474         val simp_thmss =
  1475           map7 mk_simp_thms ctr_sugars
  1476             (map3 flat_coiter_thms disc_unfold_thmss disc_unfold_iff_thmss sel_unfold_thmss)
  1477             (map3 flat_coiter_thms disc_corec_thmss disc_corec_iff_thmss sel_corec_thmss)
  1478             mapss rel_injects rel_distincts setss;
  1479 
  1480         val common_notes =
  1481           (if nn > 1 then
  1482              [(coinductN, [coinduct_thm], coinduct_attrs),
  1483               (strong_coinductN, [strong_coinduct_thm], coinduct_attrs)]
  1484            else
  1485              [])
  1486           |> massage_simple_notes fp_common_name;
  1487 
  1488         val notes =
  1489           [(coinductN, map single coinduct_thms,
  1490             fn T_name => coinduct_attrs @ [coinduct_type_attr T_name]),
  1491            (corecN, corec_thmss, K coiter_attrs),
  1492            (disc_corecN, disc_corec_thmss, K disc_coiter_attrs),
  1493            (disc_corec_iffN, disc_corec_iff_thmss, K disc_coiter_iff_attrs),
  1494            (disc_unfoldN, disc_unfold_thmss, K disc_coiter_attrs),
  1495            (disc_unfold_iffN, disc_unfold_iff_thmss, K disc_coiter_iff_attrs),
  1496            (sel_corecN, sel_corec_thmss, K sel_coiter_attrs),
  1497            (sel_unfoldN, sel_unfold_thmss, K sel_coiter_attrs),
  1498            (simpsN, simp_thmss, K []),
  1499            (strong_coinductN, map single strong_coinduct_thms, K coinduct_attrs),
  1500            (unfoldN, unfold_thmss, K coiter_attrs)]
  1501           |> massage_multi_notes;
  1502 
  1503         fun is_codatatype (Type (s, _)) =
  1504             (case fp_sugar_of lthy s of SOME {fp = Greatest_FP, ...} => true | _ => false)
  1505           | is_codatatype _ = false;
  1506 
  1507         val nitpick_supported = forall (is_codatatype o T_of_bnf) nested_bnfs;
  1508 
  1509         fun register_nitpick fpT ({ctrs, casex, ...} : ctr_sugar) =
  1510           Nitpick_HOL.register_codatatype fpT (fst (dest_Const casex))
  1511             (map (dest_Const o mk_ctr As) ctrs)
  1512           |> Generic_Target.theory_declaration;
  1513       in
  1514         lthy
  1515         |> Local_Theory.notes (common_notes @ notes) |> snd
  1516         |> register_fp_sugars Greatest_FP pre_bnfs nested_bnfs nesting_bnfs fp_res ctr_defss
  1517           ctr_sugars coiterss mapss [coinduct_thm, strong_coinduct_thm]
  1518           (transpose [unfold_thmss, corec_thmss]) (transpose [disc_unfold_thmss, disc_corec_thmss])
  1519           (transpose [sel_unfold_thmsss, sel_corec_thmsss])
  1520         |> nitpick_supported ? fold2 register_nitpick fpTs ctr_sugars
  1521       end;
  1522 
  1523     val lthy'' = lthy'
  1524       |> fold_map define_ctrs_dtrs_for_type (fp_bnfs ~~ fp_bs ~~ fpTs ~~ ctors ~~ dtors ~~
  1525         xtor_co_iterss ~~ ctor_dtors ~~ dtor_ctors ~~ ctor_injects ~~ pre_map_defs ~~
  1526         pre_set_defss ~~ pre_rel_defs ~~ xtor_map_thms ~~ xtor_set_thmss ~~ xtor_rel_thms ~~ ns ~~
  1527         kss ~~ mss ~~ ctr_bindingss ~~ ctr_mixfixess ~~ ctr_Tsss ~~ disc_bindingss ~~
  1528         sel_bindingsss ~~ raw_sel_defaultsss)
  1529       |> wrap_types_etc
  1530       |> fp_case fp derive_note_induct_iters_thms_for_types
  1531            derive_note_coinduct_coiters_thms_for_types;
  1532 
  1533     val timer = time (timer ("Constructors, discriminators, selectors, etc., for the new " ^
  1534       co_prefix fp ^ "datatype"));
  1535   in
  1536     timer; lthy''
  1537   end;
  1538 
  1539 fun co_datatypes x = define_co_datatypes (K I) (K I) (K I) x;
  1540 
  1541 fun co_datatype_cmd x =
  1542   define_co_datatypes Typedecl.read_constraint Syntax.parse_typ Syntax.parse_term x;
  1543 
  1544 val parse_ctr_arg =
  1545   @{keyword "("} |-- parse_binding_colon -- Parse.typ --| @{keyword ")"} ||
  1546   (Parse.typ >> pair Binding.empty);
  1547 
  1548 val parse_defaults =
  1549   @{keyword "("} |-- Parse.reserved "defaults" |-- Scan.repeat parse_bound_term --| @{keyword ")"};
  1550 
  1551 val parse_type_arg_constrained =
  1552   Parse.type_ident -- Scan.option (@{keyword "::"} |-- Parse.!!! Parse.sort);
  1553 
  1554 val parse_type_arg_named_constrained = parse_opt_binding_colon -- parse_type_arg_constrained;
  1555 
  1556 val parse_type_args_named_constrained =
  1557   parse_type_arg_constrained >> (single o pair Binding.empty) ||
  1558   @{keyword "("} |-- Parse.!!! (Parse.list1 parse_type_arg_named_constrained --| @{keyword ")"}) ||
  1559   Scan.succeed [];
  1560 
  1561 val parse_map_rel_binding = Parse.short_ident --| @{keyword ":"} -- parse_binding;
  1562 
  1563 val no_map_rel = (Binding.empty, Binding.empty);
  1564 
  1565 fun extract_map_rel ("map", b) = apfst (K b)
  1566   | extract_map_rel ("rel", b) = apsnd (K b)
  1567   | extract_map_rel (s, _) = error ("Unknown label " ^ quote s ^ " (expected \"map\" or \"rel\")");
  1568 
  1569 val parse_map_rel_bindings =
  1570   @{keyword "("} |-- Scan.repeat parse_map_rel_binding --| @{keyword ")"}
  1571     >> (fn ps => fold extract_map_rel ps no_map_rel) ||
  1572   Scan.succeed no_map_rel;
  1573 
  1574 val parse_ctr_spec =
  1575   parse_opt_binding_colon -- parse_binding -- Scan.repeat parse_ctr_arg --
  1576   Scan.optional parse_defaults [] -- Parse.opt_mixfix;
  1577 
  1578 val parse_spec =
  1579   parse_type_args_named_constrained -- parse_binding -- parse_map_rel_bindings --
  1580   Parse.opt_mixfix -- (@{keyword "="} |-- Parse.enum1 "|" parse_ctr_spec);
  1581 
  1582 val parse_co_datatype = parse_wrap_free_constructors_options -- Parse.and_list1 parse_spec;
  1583 
  1584 fun parse_co_datatype_cmd fp construct_fp = parse_co_datatype >> co_datatype_cmd fp construct_fp;
  1585 
  1586 end;