src/HOL/Tools/BNF/bnf_fp_def_sugar.ML
author blanchet
Mon Feb 17 13:31:42 2014 +0100 (2014-02-17)
changeset 55535 10194808430d
parent 55486 8609527278f2
child 55539 0819931d652d
permissions -rw-r--r--
name derivations in 'primrec' for code extraction from proof terms
     1 (*  Title:      HOL/Tools/BNF/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_inductss: thm list list,
    24      co_iter_thmsss: thm list list list,
    25      disc_co_itersss: thm list list list,
    26      sel_co_iterssss: thm list list list list};
    27 
    28   val of_fp_sugar: (fp_sugar -> 'a list) -> fp_sugar -> 'a
    29   val morph_fp_sugar: morphism -> fp_sugar -> fp_sugar
    30   val transfer_fp_sugar: Proof.context -> fp_sugar -> fp_sugar
    31   val fp_sugar_of: Proof.context -> string -> fp_sugar option
    32   val fp_sugars_of: Proof.context -> fp_sugar list
    33 
    34   val co_induct_of: 'a list -> 'a
    35   val strong_co_induct_of: 'a list -> 'a
    36 
    37   val tvar_subst: theory -> typ list -> typ list -> ((string * int) * typ) list
    38   val flat_rec_arg_args: 'a list list -> 'a list
    39   val flat_corec_preds_predsss_gettersss: 'a list -> 'a list list list -> 'a list list list ->
    40     'a list
    41   val mk_co_iter: theory -> BNF_FP_Util.fp_kind -> typ -> typ list -> term -> term
    42   val nesty_bnfs: Proof.context -> typ list list list -> typ list -> BNF_Def.bnf list
    43 
    44   type lfp_sugar_thms =
    45     (thm list * thm * Args.src list)
    46     * (thm list list * thm list list * Args.src list)
    47 
    48   val morph_lfp_sugar_thms: morphism -> lfp_sugar_thms -> lfp_sugar_thms
    49   val transfer_lfp_sugar_thms: Proof.context -> lfp_sugar_thms -> lfp_sugar_thms
    50 
    51   type gfp_sugar_thms =
    52     ((thm list * thm) list * Args.src list)
    53     * (thm list list * thm list list * Args.src list)
    54     * (thm list list * thm list list * Args.src list)
    55     * (thm list list * thm list list * Args.src list)
    56     * (thm list list list * thm list list list * Args.src list)
    57 
    58   val morph_gfp_sugar_thms: morphism -> gfp_sugar_thms -> gfp_sugar_thms
    59   val transfer_gfp_sugar_thms: Proof.context -> gfp_sugar_thms -> gfp_sugar_thms
    60 
    61   val mk_co_iters_prelims: BNF_FP_Util.fp_kind -> typ list list list -> typ list -> typ list ->
    62     int list -> int list list -> term list list -> Proof.context ->
    63     (term list list
    64      * (typ list list * typ list list list list * term list list
    65         * term list list list list) list option
    66      * (string * term list * term list list
    67         * ((term list list * term list list list) * typ list) list) option)
    68     * Proof.context
    69   val mk_iter_fun_arg_types: typ list list list -> int list -> int list list -> term ->
    70     typ list list list list
    71   val mk_coiter_fun_arg_types: typ list list list -> typ list -> int list -> term ->
    72     typ list list
    73     * (typ list list list list * typ list list list * typ list list list list * typ list)
    74   val define_iters: string list ->
    75     (typ list list * typ list list list list * term list list * term list list list list) list ->
    76     (string -> binding) -> typ list -> typ list -> term list -> Proof.context ->
    77     (term list * thm list) * Proof.context
    78   val define_coiters: string list -> string * term list * term list list
    79     * ((term list list * term list list list) * typ list) list ->
    80     (string -> binding) -> typ list -> typ list -> term list -> Proof.context ->
    81     (term list * thm list) * Proof.context
    82   val derive_induct_iters_thms_for_types: BNF_Def.bnf list ->
    83     (typ list list * typ list list list list * term list list * term list list list list) list ->
    84     thm -> thm list list -> BNF_Def.bnf list -> BNF_Def.bnf list -> typ list -> typ list ->
    85     typ list -> typ list list list -> term list list -> thm list list -> term list list ->
    86     thm list list -> local_theory -> lfp_sugar_thms
    87   val derive_coinduct_coiters_thms_for_types: BNF_Def.bnf list ->
    88     string * term list * term list list * ((term list list * term list list list)
    89       * typ list) list ->
    90     thm -> thm list -> thm list -> thm list list -> BNF_Def.bnf list -> typ list -> typ list ->
    91     typ list -> typ list list list -> int list list -> int list list -> int list -> thm list list ->
    92     Ctr_Sugar.ctr_sugar list -> term list list -> thm list list -> (thm list -> thm list) ->
    93     local_theory -> gfp_sugar_thms
    94 
    95   type co_datatype_spec =
    96     ((((binding * (typ * sort)) list * binding) * (binding * binding)) * mixfix)
    97     * ((binding, binding * typ, term) Ctr_Sugar.ctr_spec * mixfix) list
    98 
    99   val co_datatypes: 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     (bool * bool) * co_datatype_spec list ->
   103     local_theory -> local_theory
   104   val parse_co_datatype_cmd: BNF_FP_Util.fp_kind -> (mixfix list -> binding list -> binding list ->
   105       binding list list -> binding list -> (string * sort) list -> typ list * typ list list ->
   106       BNF_Def.bnf list -> local_theory -> BNF_FP_Util.fp_result * local_theory) ->
   107     (local_theory -> local_theory) parser
   108 end;
   109 
   110 structure BNF_FP_Def_Sugar : BNF_FP_DEF_SUGAR =
   111 struct
   112 
   113 open Ctr_Sugar
   114 open BNF_Util
   115 open BNF_Comp
   116 open BNF_Def
   117 open BNF_FP_Util
   118 open BNF_FP_Def_Sugar_Tactics
   119 
   120 val EqN = "Eq_";
   121 
   122 type fp_sugar =
   123   {T: typ,
   124    fp: fp_kind,
   125    index: int,
   126    pre_bnfs: bnf list,
   127    nested_bnfs: bnf list,
   128    nesting_bnfs: bnf list,
   129    fp_res: fp_result,
   130    ctr_defss: thm list list,
   131    ctr_sugars: ctr_sugar list,
   132    co_iterss: term list list,
   133    mapss: thm list list,
   134    co_inducts: thm list,
   135    co_inductss: thm list list,
   136    co_iter_thmsss: thm list list list,
   137    disc_co_itersss: thm list list list,
   138    sel_co_iterssss: thm list list list list};
   139 
   140 fun of_fp_sugar f (fp_sugar as ({index, ...}: fp_sugar)) = nth (f fp_sugar) index;
   141 
   142 fun morph_fp_sugar phi ({T, fp, index, pre_bnfs, nested_bnfs, nesting_bnfs, fp_res, ctr_defss,
   143     ctr_sugars, co_iterss, mapss, co_inducts, co_inductss, co_iter_thmsss, disc_co_itersss,
   144     sel_co_iterssss} : fp_sugar) =
   145   {T = Morphism.typ phi T, fp = fp, index = index, pre_bnfs = map (morph_bnf phi) pre_bnfs,
   146     nested_bnfs = map (morph_bnf phi) nested_bnfs, nesting_bnfs = map (morph_bnf phi) nesting_bnfs,
   147    fp_res = morph_fp_result phi fp_res,
   148    ctr_defss = map (map (Morphism.thm phi)) ctr_defss,
   149    ctr_sugars = map (morph_ctr_sugar phi) ctr_sugars,
   150    co_iterss = map (map (Morphism.term phi)) co_iterss,
   151    mapss = map (map (Morphism.thm phi)) mapss,
   152    co_inducts = map (Morphism.thm phi) co_inducts,
   153    co_inductss = map (map (Morphism.thm phi)) co_inductss,
   154    co_iter_thmsss = map (map (map (Morphism.thm phi))) co_iter_thmsss,
   155    disc_co_itersss = map (map (map (Morphism.thm phi))) disc_co_itersss,
   156    sel_co_iterssss = map (map (map (map (Morphism.thm phi)))) sel_co_iterssss};
   157 
   158 val transfer_fp_sugar =
   159   morph_fp_sugar o Morphism.transfer_morphism o Proof_Context.theory_of;
   160 
   161 structure Data = Generic_Data
   162 (
   163   type T = fp_sugar Symtab.table;
   164   val empty = Symtab.empty;
   165   val extend = I;
   166   fun merge data : T = Symtab.merge (K true) data;
   167 );
   168 
   169 fun fp_sugar_of ctxt =
   170   Symtab.lookup (Data.get (Context.Proof ctxt))
   171   #> Option.map (transfer_fp_sugar ctxt);
   172 
   173 fun fp_sugars_of ctxt =
   174   Symtab.fold (cons o transfer_fp_sugar ctxt o snd) (Data.get (Context.Proof ctxt)) [];
   175 
   176 fun co_induct_of (i :: _) = i;
   177 fun strong_co_induct_of [_, s] = s;
   178 
   179 (* TODO: register "sum" and "prod" as datatypes to enable N2M reduction for them *)
   180 
   181 fun register_fp_sugar key fp_sugar =
   182   Local_Theory.declaration {syntax = false, pervasive = true}
   183     (fn phi => Data.map (Symtab.update (key, morph_fp_sugar phi fp_sugar)));
   184 
   185 fun register_fp_sugars fp pre_bnfs nested_bnfs nesting_bnfs (fp_res as {Ts, ...}) ctr_defss
   186     ctr_sugars co_iterss mapss co_inducts co_inductss co_iter_thmsss disc_co_itersss
   187     sel_co_iterssss lthy =
   188   (0, lthy)
   189   |> fold (fn T as Type (s, _) => fn (kk, lthy) => (kk + 1,
   190     register_fp_sugar s {T = T, fp = fp, index = kk, pre_bnfs = pre_bnfs, nested_bnfs = nested_bnfs,
   191         nesting_bnfs = nesting_bnfs, fp_res = fp_res, ctr_defss = ctr_defss,
   192         ctr_sugars = ctr_sugars, co_iterss = co_iterss, mapss = mapss, co_inducts = co_inducts,
   193         co_inductss = co_inductss, co_iter_thmsss = co_iter_thmsss,
   194         disc_co_itersss = disc_co_itersss, sel_co_iterssss = sel_co_iterssss}
   195       lthy)) Ts
   196   |> snd;
   197 
   198 (* This function could produce (fairly harmless) clashes in contrived examples (e.g., "x.A",
   199    "x.x_A", "y.A"). *)
   200 fun quasi_unambiguous_case_names names =
   201   let
   202     val ps = map (`Long_Name.base_name) names;
   203     val dups = Library.duplicates (op =) (map fst ps);
   204     fun underscore s =
   205       let val ss = space_explode Long_Name.separator s in
   206         space_implode "_" (drop (length ss - 2) ss)
   207       end;
   208   in
   209     map (fn (base, full) => if member (op =) dups base then underscore full else base) ps
   210   end;
   211 
   212 val id_def = @{thm id_def};
   213 val mp_conj = @{thm mp_conj};
   214 
   215 val fundefcong_attrs = @{attributes [fundef_cong]};
   216 val nitpicksimp_attrs = @{attributes [nitpick_simp]};
   217 val code_nitpicksimp_attrs = Code.add_default_eqn_attrib :: nitpicksimp_attrs;
   218 val simp_attrs = @{attributes [simp]};
   219 
   220 fun tvar_subst thy Ts Us =
   221   Vartab.fold (cons o apsnd snd) (fold (Sign.typ_match thy) (Ts ~~ Us) Vartab.empty) [];
   222 
   223 val lists_bmoc = fold (fn xs => fn t => Term.list_comb (t, xs));
   224 
   225 fun flat_rec_arg_args xss =
   226   (* FIXME (once the old datatype package is phased out): The first line below gives the preferred
   227      order. The second line is for compatibility with the old datatype package. *)
   228   (* flat xss *)
   229   map hd xss @ maps tl xss;
   230 
   231 fun flat_corec_predss_getterss qss fss = maps (op @) (qss ~~ fss);
   232 
   233 fun flat_corec_preds_predsss_gettersss [] [qss] [fss] = flat_corec_predss_getterss qss fss
   234   | flat_corec_preds_predsss_gettersss (p :: ps) (qss :: qsss) (fss :: fsss) =
   235     p :: flat_corec_predss_getterss qss fss @ flat_corec_preds_predsss_gettersss ps qsss fsss;
   236 
   237 fun mk_tupled_fun x f xs =
   238   if xs = [x] then f else HOLogic.tupled_lambda x (Term.list_comb (f, xs));
   239 
   240 fun mk_uncurried2_fun f xss =
   241   mk_tupled_fun (HOLogic.mk_tuple (map HOLogic.mk_tuple xss)) f (flat_rec_arg_args xss);
   242 
   243 fun mk_flip (x, Type (_, [T1, Type (_, [T2, T3])])) =
   244   Abs ("x", T1, Abs ("y", T2, Var (x, T2 --> T1 --> T3) $ Bound 0 $ Bound 1));
   245 
   246 fun flip_rels lthy n thm =
   247   let
   248     val Rs = Term.add_vars (prop_of thm) [];
   249     val Rs' = rev (drop (length Rs - n) Rs);
   250     val cRs = map (fn f => (certify lthy (Var f), certify lthy (mk_flip f))) Rs';
   251   in
   252     Drule.cterm_instantiate cRs thm
   253   end;
   254 
   255 fun mk_ctor_or_dtor get_T Ts t =
   256   let val Type (_, Ts0) = get_T (fastype_of t) in
   257     Term.subst_atomic_types (Ts0 ~~ Ts) t
   258   end;
   259 
   260 val mk_ctor = mk_ctor_or_dtor range_type;
   261 val mk_dtor = mk_ctor_or_dtor domain_type;
   262 
   263 fun mk_co_iter thy fp fpT Cs t =
   264   let
   265     val (f_Cs, Type (_, [prebody, body])) = strip_fun_type (fastype_of t);
   266     val fpT0 = fp_case fp prebody body;
   267     val Cs0 = distinct (op =) (map (fp_case fp body_type domain_type) f_Cs);
   268     val rho = tvar_subst thy (fpT0 :: Cs0) (fpT :: Cs);
   269   in
   270     Term.subst_TVars rho t
   271   end;
   272 
   273 fun mk_co_iters thy fp fpTs Cs ts0 =
   274   let
   275     val nn = length fpTs;
   276     val (fpTs0, Cs0) =
   277       map ((fp = Greatest_FP ? swap) o dest_funT o snd o strip_typeN nn o fastype_of) ts0
   278       |> split_list;
   279     val rho = tvar_subst thy (fpTs0 @ Cs0) (fpTs @ Cs);
   280   in
   281     map (Term.subst_TVars rho) ts0
   282   end;
   283 
   284 val mk_fp_iter_fun_types = binder_fun_types o fastype_of;
   285 
   286 fun unzip_recT (Type (@{type_name prod}, _)) T = [T]
   287   | unzip_recT _ (Type (@{type_name prod}, Ts)) = Ts
   288   | unzip_recT _ T = [T];
   289 
   290 fun unzip_corecT (Type (@{type_name sum}, _)) T = [T]
   291   | unzip_corecT _ (Type (@{type_name sum}, Ts)) = Ts
   292   | unzip_corecT _ T = [T];
   293 
   294 fun liveness_of_fp_bnf n bnf =
   295   (case T_of_bnf bnf of
   296     Type (_, Ts) => map (not o member (op =) (deads_of_bnf bnf)) Ts
   297   | _ => replicate n false);
   298 
   299 fun cannot_merge_types () = error "Mutually recursive types must have the same type parameters";
   300 
   301 fun merge_type_arg T T' = if T = T' then T else cannot_merge_types ();
   302 
   303 fun merge_type_args (As, As') =
   304   if length As = length As' then map2 merge_type_arg As As' else cannot_merge_types ();
   305 
   306 fun reassoc_conjs thm =
   307   reassoc_conjs (thm RS @{thm conj_assoc[THEN iffD1]})
   308   handle THM _ => thm;
   309 
   310 type co_datatype_spec =
   311   ((((binding * (typ * sort)) list * binding) * (binding * binding)) * mixfix)
   312   * ((binding, binding * typ, term) ctr_spec * mixfix) list;
   313 
   314 fun type_args_named_constrained_of_spec ((((ncAs, _), _), _), _) = ncAs;
   315 fun type_binding_of_spec ((((_, b), _), _), _) = b;
   316 fun map_binding_of_spec (((_, (b, _)), _), _) = b;
   317 fun rel_binding_of_spec (((_, (_, b)), _), _) = b;
   318 fun mixfix_of_spec ((_, mx), _) = mx;
   319 fun mixfixed_ctr_specs_of_spec (_, mx_ctr_specs) = mx_ctr_specs;
   320 
   321 fun add_nesty_bnf_names Us =
   322   let
   323     fun add (Type (s, Ts)) ss =
   324         let val (needs, ss') = fold_map add Ts ss in
   325           if exists I needs then (true, insert (op =) s ss') else (false, ss')
   326         end
   327       | add T ss = (member (op =) Us T, ss);
   328   in snd oo add end;
   329 
   330 fun nesty_bnfs ctxt ctr_Tsss Us =
   331   map_filter (bnf_of ctxt) (fold (fold (fold (add_nesty_bnf_names Us))) ctr_Tsss []);
   332 
   333 fun indexify proj xs f p = f (find_index (curry (op =) (proj p)) xs) p;
   334 
   335 type lfp_sugar_thms =
   336   (thm list * thm * Args.src list)
   337   * (thm list list * thm list list * Args.src list)
   338 
   339 fun morph_lfp_sugar_thms phi ((inducts, induct, induct_attrs), (foldss, recss, iter_attrs)) =
   340   ((map (Morphism.thm phi) inducts, Morphism.thm phi induct, induct_attrs),
   341    (map (map (Morphism.thm phi)) foldss, map (map (Morphism.thm phi)) recss, iter_attrs))
   342   : lfp_sugar_thms;
   343 
   344 val transfer_lfp_sugar_thms =
   345   morph_lfp_sugar_thms o Morphism.transfer_morphism o Proof_Context.theory_of;
   346 
   347 type gfp_sugar_thms =
   348   ((thm list * thm) list * Args.src list)
   349   * (thm list list * thm list list * Args.src list)
   350   * (thm list list * thm list list * Args.src list)
   351   * (thm list list * thm list list * Args.src list)
   352   * (thm list list list * thm list list list * Args.src list);
   353 
   354 fun morph_gfp_sugar_thms phi ((coinducts_pairs, coinduct_attrs),
   355     (unfoldss, corecss, coiter_attrs), (disc_unfoldss, disc_corecss, disc_iter_attrs),
   356     (disc_unfold_iffss, disc_corec_iffss, disc_iter_iff_attrs),
   357     (sel_unfoldsss, sel_corecsss, sel_iter_attrs)) =
   358   ((map (apfst (map (Morphism.thm phi)) o apsnd (Morphism.thm phi)) coinducts_pairs,
   359     coinduct_attrs),
   360    (map (map (Morphism.thm phi)) unfoldss, map (map (Morphism.thm phi)) corecss, coiter_attrs),
   361    (map (map (Morphism.thm phi)) disc_unfoldss, map (map (Morphism.thm phi)) disc_corecss,
   362     disc_iter_attrs),
   363    (map (map (Morphism.thm phi)) disc_unfold_iffss, map (map (Morphism.thm phi)) disc_corec_iffss,
   364     disc_iter_iff_attrs),
   365    (map (map (map (Morphism.thm phi))) sel_unfoldsss,
   366     map (map (map (Morphism.thm phi))) sel_corecsss, sel_iter_attrs)) : gfp_sugar_thms;
   367 
   368 val transfer_gfp_sugar_thms =
   369   morph_gfp_sugar_thms o Morphism.transfer_morphism o Proof_Context.theory_of;
   370 
   371 fun mk_iter_fun_arg_types0 n ms = map2 dest_tupleT ms o dest_sumTN_balanced n o domain_type;
   372 
   373 fun mk_iter_fun_arg_types ctr_Tsss ns mss =
   374   mk_fp_iter_fun_types
   375   #> map3 mk_iter_fun_arg_types0 ns mss
   376   #> map2 (map2 (map2 unzip_recT)) ctr_Tsss;
   377 
   378 fun mk_iters_args_types ctr_Tsss Cs ns mss ctor_iter_fun_Tss lthy =
   379   let
   380     val Css = map2 replicate ns Cs;
   381     val y_Tsss = map3 mk_iter_fun_arg_types0 ns mss (map un_fold_of ctor_iter_fun_Tss);
   382     val g_Tss = map2 (fn C => map (fn y_Ts => y_Ts ---> C)) Cs y_Tsss;
   383 
   384     val ((gss, ysss), lthy) =
   385       lthy
   386       |> mk_Freess "f" g_Tss
   387       ||>> mk_Freesss "x" y_Tsss;
   388 
   389     val y_Tssss = map (map (map single)) y_Tsss;
   390     val yssss = map (map (map single)) ysss;
   391 
   392     val z_Tssss =
   393       map4 (fn n => fn ms => fn ctr_Tss => fn ctor_iter_fun_Ts =>
   394           map3 (fn m => fn ctr_Ts => fn ctor_iter_fun_T =>
   395               map2 unzip_recT ctr_Ts (dest_tupleT m ctor_iter_fun_T))
   396             ms ctr_Tss (dest_sumTN_balanced n (domain_type (co_rec_of ctor_iter_fun_Ts))))
   397         ns mss ctr_Tsss ctor_iter_fun_Tss;
   398 
   399     val z_Tsss' = map (map flat_rec_arg_args) z_Tssss;
   400     val h_Tss = map2 (map2 (curry (op --->))) z_Tsss' Css;
   401 
   402     val hss = map2 (map2 retype_free) h_Tss gss;
   403     val zssss_hd = map2 (map2 (map2 (retype_free o hd))) z_Tssss ysss;
   404     val (zssss_tl, lthy) =
   405       lthy
   406       |> mk_Freessss "y" (map (map (map tl)) z_Tssss);
   407     val zssss = map2 (map2 (map2 cons)) zssss_hd zssss_tl;
   408   in
   409     ([(g_Tss, y_Tssss, gss, yssss), (h_Tss, z_Tssss, hss, zssss)], lthy)
   410   end;
   411 
   412 fun mk_coiter_fun_arg_types0 ctr_Tsss Cs ns fun_Ts =
   413   let
   414     (*avoid "'a itself" arguments in coiterators*)
   415     fun repair_arity [[]] = [[@{typ unit}]]
   416       | repair_arity Tss = Tss;
   417 
   418     val ctr_Tsss' = map repair_arity ctr_Tsss;
   419     val f_sum_prod_Ts = map range_type fun_Ts;
   420     val f_prod_Tss = map2 dest_sumTN_balanced ns f_sum_prod_Ts;
   421     val f_Tsss = map2 (map2 (dest_tupleT o length)) ctr_Tsss' f_prod_Tss;
   422     val f_Tssss = map3 (fn C => map2 (map2 (map (curry (op -->) C) oo unzip_corecT)))
   423       Cs ctr_Tsss' f_Tsss;
   424     val q_Tssss = map (map (map (fn [_] => [] | [_, T] => [mk_pred1T (domain_type T)]))) f_Tssss;
   425   in
   426     (q_Tssss, f_Tsss, f_Tssss, f_sum_prod_Ts)
   427   end;
   428 
   429 fun mk_coiter_p_pred_types Cs ns = map2 (fn n => replicate (Int.max (0, n - 1)) o mk_pred1T) ns Cs;
   430 
   431 fun mk_coiter_fun_arg_types ctr_Tsss Cs ns dtor_coiter =
   432   (mk_coiter_p_pred_types Cs ns,
   433    mk_fp_iter_fun_types dtor_coiter |> mk_coiter_fun_arg_types0 ctr_Tsss Cs ns);
   434 
   435 fun mk_coiters_args_types ctr_Tsss Cs ns dtor_coiter_fun_Tss lthy =
   436   let
   437     val p_Tss = mk_coiter_p_pred_types Cs ns;
   438 
   439     fun mk_types get_Ts =
   440       let
   441         val fun_Ts = map get_Ts dtor_coiter_fun_Tss;
   442         val (q_Tssss, f_Tsss, f_Tssss, f_sum_prod_Ts) = mk_coiter_fun_arg_types0 ctr_Tsss Cs ns fun_Ts;
   443       in
   444         (q_Tssss, f_Tsss, f_Tssss, f_sum_prod_Ts)
   445       end;
   446 
   447     val (r_Tssss, g_Tsss, g_Tssss, unfold_types) = mk_types un_fold_of;
   448     val (s_Tssss, h_Tsss, h_Tssss, corec_types) = mk_types co_rec_of;
   449 
   450     val ((((Free (z, _), cs), pss), gssss), lthy) =
   451       lthy
   452       |> yield_singleton (mk_Frees "z") dummyT
   453       ||>> mk_Frees "a" Cs
   454       ||>> mk_Freess "p" p_Tss
   455       ||>> mk_Freessss "g" g_Tssss;
   456     val rssss = map (map (map (fn [] => []))) r_Tssss;
   457 
   458     val hssss_hd = map2 (map2 (map2 (fn T :: _ => fn [g] => retype_free T g))) h_Tssss gssss;
   459     val ((sssss, hssss_tl), lthy) =
   460       lthy
   461       |> mk_Freessss "q" s_Tssss
   462       ||>> mk_Freessss "h" (map (map (map tl)) h_Tssss);
   463     val hssss = map2 (map2 (map2 cons)) hssss_hd hssss_tl;
   464 
   465     val cpss = map2 (map o rapp) cs pss;
   466 
   467     fun build_sum_inj mk_inj = build_map lthy (uncurry mk_inj o dest_sumT o snd);
   468 
   469     fun build_dtor_coiter_arg _ [] [cf] = cf
   470       | build_dtor_coiter_arg T [cq] [cf, cf'] =
   471         mk_If cq (build_sum_inj Inl_const (fastype_of cf, T) $ cf)
   472           (build_sum_inj Inr_const (fastype_of cf', T) $ cf');
   473 
   474     fun mk_args qssss fssss f_Tsss =
   475       let
   476         val pfss = map3 flat_corec_preds_predsss_gettersss pss qssss fssss;
   477         val cqssss = map2 (map o map o map o rapp) cs qssss;
   478         val cfssss = map2 (map o map o map o rapp) cs fssss;
   479         val cqfsss = map3 (map3 (map3 build_dtor_coiter_arg)) f_Tsss cqssss cfssss;
   480       in (pfss, cqfsss) end;
   481 
   482     val unfold_args = mk_args rssss gssss g_Tsss;
   483     val corec_args = mk_args sssss hssss h_Tsss;
   484   in
   485     ((z, cs, cpss, [(unfold_args, unfold_types), (corec_args, corec_types)]), lthy)
   486   end;
   487 
   488 fun mk_co_iters_prelims fp ctr_Tsss fpTs Cs ns mss xtor_co_iterss0 lthy =
   489   let
   490     val thy = Proof_Context.theory_of lthy;
   491 
   492     val (xtor_co_iter_fun_Tss, xtor_co_iterss) =
   493       map (mk_co_iters thy fp fpTs Cs #> `(mk_fp_iter_fun_types o hd)) (transpose xtor_co_iterss0)
   494       |> apsnd transpose o apfst transpose o split_list;
   495 
   496     val ((iters_args_types, coiters_args_types), lthy') =
   497       if fp = Least_FP then
   498         mk_iters_args_types ctr_Tsss Cs ns mss xtor_co_iter_fun_Tss lthy |>> (rpair NONE o SOME)
   499       else
   500         mk_coiters_args_types ctr_Tsss Cs ns xtor_co_iter_fun_Tss lthy |>> (pair NONE o SOME);
   501   in
   502     ((xtor_co_iterss, iters_args_types, coiters_args_types), lthy')
   503   end;
   504 
   505 fun mk_preds_getterss_join c cps sum_prod_T cqfss =
   506   let val n = length cqfss in
   507     Term.lambda c (mk_IfN sum_prod_T cps
   508       (map2 (mk_InN_balanced sum_prod_T n) (map HOLogic.mk_tuple cqfss) (1 upto n)))
   509   end;
   510 
   511 fun define_co_iters fp fpT Cs binding_specs lthy0 =
   512   let
   513     val thy = Proof_Context.theory_of lthy0;
   514 
   515     val maybe_conceal_def_binding = Thm.def_binding
   516       #> Config.get lthy0 bnf_note_all = false ? Binding.conceal;
   517 
   518     val ((csts, defs), (lthy', lthy)) = lthy0
   519       |> apfst split_list o fold_map (fn (b, rhs) =>
   520         Local_Theory.define ((b, NoSyn), ((maybe_conceal_def_binding b, []), rhs))
   521         #>> apsnd snd) binding_specs
   522       ||> `Local_Theory.restore;
   523 
   524     val phi = Proof_Context.export_morphism lthy lthy';
   525 
   526     val csts' = map (mk_co_iter thy fp fpT Cs o Morphism.term phi) csts;
   527     val defs' = map (Morphism.thm phi) defs;
   528   in
   529     ((csts', defs'), lthy')
   530   end;
   531 
   532 fun define_iters iterNs iter_args_typess' mk_binding fpTs Cs ctor_iters lthy =
   533   let
   534     val nn = length fpTs;
   535 
   536     val Type (_, [fpT, _]) = snd (strip_typeN nn (fastype_of (hd ctor_iters)));
   537 
   538     fun generate_iter pre (_, _, fss, xssss) ctor_iter =
   539       (mk_binding pre,
   540        fold_rev (fold_rev Term.lambda) fss (Term.list_comb (ctor_iter,
   541          map2 (mk_case_sumN_balanced oo map2 mk_uncurried2_fun) fss xssss)));
   542   in
   543     define_co_iters Least_FP fpT Cs (map3 generate_iter iterNs iter_args_typess' ctor_iters) lthy
   544   end;
   545 
   546 fun define_coiters coiterNs (_, cs, cpss, coiter_args_typess') mk_binding fpTs Cs dtor_coiters
   547     lthy =
   548   let
   549     val nn = length fpTs;
   550 
   551     val Type (_, [_, fpT]) = snd (strip_typeN nn (fastype_of (hd dtor_coiters)));
   552 
   553     fun generate_coiter pre ((pfss, cqfsss), f_sum_prod_Ts) dtor_coiter =
   554       (mk_binding pre,
   555        fold_rev (fold_rev Term.lambda) pfss (Term.list_comb (dtor_coiter,
   556          map4 mk_preds_getterss_join cs cpss f_sum_prod_Ts cqfsss)));
   557   in
   558     define_co_iters Greatest_FP fpT Cs
   559       (map3 generate_coiter coiterNs coiter_args_typess' dtor_coiters) lthy
   560   end;
   561 
   562 fun derive_induct_iters_thms_for_types pre_bnfs [fold_args_types, rec_args_types] ctor_induct
   563     ctor_iter_thmss nesting_bnfs nested_bnfs fpTs Cs Xs ctrXs_Tsss ctrss ctr_defss iterss iter_defss
   564     lthy =
   565   let
   566     val iterss' = transpose iterss;
   567     val iter_defss' = transpose iter_defss;
   568 
   569     val [folds, recs] = iterss';
   570     val [fold_defs, rec_defs] = iter_defss';
   571 
   572     val ctr_Tsss = map (map (binder_types o fastype_of)) ctrss;
   573 
   574     val nn = length pre_bnfs;
   575     val ns = map length ctr_Tsss;
   576     val mss = map (map length) ctr_Tsss;
   577 
   578     val pre_map_defs = map map_def_of_bnf pre_bnfs;
   579     val pre_set_defss = map set_defs_of_bnf pre_bnfs;
   580     val nesting_map_idents = map (unfold_thms lthy [id_def] o map_id0_of_bnf) nesting_bnfs;
   581     val nested_map_idents = map (unfold_thms lthy [id_def] o map_id0_of_bnf) nested_bnfs;
   582     val nested_set_maps = maps set_map_of_bnf nested_bnfs;
   583 
   584     val fp_b_names = map base_name_of_typ fpTs;
   585 
   586     val ((((ps, ps'), xsss), us'), names_lthy) =
   587       lthy
   588       |> mk_Frees' "P" (map mk_pred1T fpTs)
   589       ||>> mk_Freesss "x" ctr_Tsss
   590       ||>> Variable.variant_fixes fp_b_names;
   591 
   592     val us = map2 (curry Free) us' fpTs;
   593 
   594     fun mk_sets_nested bnf =
   595       let
   596         val Type (T_name, Us) = T_of_bnf bnf;
   597         val lives = lives_of_bnf bnf;
   598         val sets = sets_of_bnf bnf;
   599         fun mk_set U =
   600           (case find_index (curry (op =) U) lives of
   601             ~1 => Term.dummy
   602           | i => nth sets i);
   603       in
   604         (T_name, map mk_set Us)
   605       end;
   606 
   607     val setss_nested = map mk_sets_nested nested_bnfs;
   608 
   609     val (induct_thms, induct_thm) =
   610       let
   611         fun mk_set Ts t =
   612           let val Type (_, Ts0) = domain_type (fastype_of t) in
   613             Term.subst_atomic_types (Ts0 ~~ Ts) t
   614           end;
   615 
   616         fun mk_raw_prem_prems _ (x as Free (_, Type _)) (X as TFree _) =
   617             [([], (find_index (curry (op =) X) Xs + 1, x))]
   618           | mk_raw_prem_prems names_lthy (x as Free (s, Type (T_name, Ts0))) (Type (_, Xs_Ts0)) =
   619             (case AList.lookup (op =) setss_nested T_name of
   620               NONE => []
   621             | SOME raw_sets0 =>
   622               let
   623                 val (Xs_Ts, (Ts, raw_sets)) =
   624                   filter (exists_subtype_in Xs o fst) (Xs_Ts0 ~~ (Ts0 ~~ raw_sets0))
   625                   |> split_list ||> split_list;
   626                 val sets = map (mk_set Ts0) raw_sets;
   627                 val (ys, names_lthy') = names_lthy |> mk_Frees s Ts;
   628                 val xysets = map (pair x) (ys ~~ sets);
   629                 val ppremss = map2 (mk_raw_prem_prems names_lthy') ys Xs_Ts;
   630               in
   631                 flat (map2 (map o apfst o cons) xysets ppremss)
   632               end)
   633           | mk_raw_prem_prems _ _ _ = [];
   634 
   635         fun close_prem_prem xs t =
   636           fold_rev Logic.all (map Free (drop (nn + length xs)
   637             (rev (Term.add_frees t (map dest_Free xs @ ps'))))) t;
   638 
   639         fun mk_prem_prem xs (xysets, (j, x)) =
   640           close_prem_prem xs (Logic.list_implies (map (fn (x', (y, set)) =>
   641               HOLogic.mk_Trueprop (HOLogic.mk_mem (y, set $ x'))) xysets,
   642             HOLogic.mk_Trueprop (nth ps (j - 1) $ x)));
   643 
   644         fun mk_raw_prem phi ctr ctr_Ts ctrXs_Ts =
   645           let
   646             val (xs, names_lthy') = names_lthy |> mk_Frees "x" ctr_Ts;
   647             val pprems = flat (map2 (mk_raw_prem_prems names_lthy') xs ctrXs_Ts);
   648           in (xs, pprems, HOLogic.mk_Trueprop (phi $ Term.list_comb (ctr, xs))) end;
   649 
   650         fun mk_prem (xs, raw_pprems, concl) =
   651           fold_rev Logic.all xs (Logic.list_implies (map (mk_prem_prem xs) raw_pprems, concl));
   652 
   653         val raw_premss = map4 (map3 o mk_raw_prem) ps ctrss ctr_Tsss ctrXs_Tsss;
   654 
   655         val goal =
   656           Library.foldr (Logic.list_implies o apfst (map mk_prem)) (raw_premss,
   657             HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj (map2 (curry (op $)) ps us)));
   658 
   659         val kksss = map (map (map (fst o snd) o #2)) raw_premss;
   660 
   661         val ctor_induct' = ctor_induct OF (map mk_sumEN_tupled_balanced mss);
   662 
   663         val thm =
   664           Goal.prove_sorry lthy [] [] goal (fn {context = ctxt, ...} =>
   665             mk_induct_tac ctxt nn ns mss kksss (flat ctr_defss) ctor_induct' nested_set_maps
   666               pre_set_defss)
   667           |> singleton (Proof_Context.export names_lthy lthy)
   668           (* for "datatype_realizer.ML": *)
   669           |> Thm.name_derivation (fst (dest_Type (hd fpTs)) ^ Long_Name.separator ^
   670             (if nn > 1 then space_implode "_" (tl fp_b_names) ^ Long_Name.separator else "") ^
   671             inductN);
   672       in
   673         `(conj_dests nn) thm
   674       end;
   675 
   676     val induct_cases = quasi_unambiguous_case_names (maps (map name_of_ctr) ctrss);
   677     val induct_case_names_attr = Attrib.internal (K (Rule_Cases.case_names induct_cases));
   678 
   679     val xctrss = map2 (map2 (curry Term.list_comb)) ctrss xsss;
   680 
   681     fun mk_iter_thmss (_, x_Tssss, fss, _) iters iter_defs ctor_iter_thms =
   682       let
   683         val fiters = map (lists_bmoc fss) iters;
   684 
   685         fun mk_goal fss fiter xctr f xs fxs =
   686           fold_rev (fold_rev Logic.all) (xs :: fss)
   687             (mk_Trueprop_eq (fiter $ xctr, Term.list_comb (f, fxs)));
   688 
   689         fun maybe_tick (T, U) u f =
   690           if try (fst o HOLogic.dest_prodT) U = SOME T then
   691             Term.lambda u (HOLogic.mk_prod (u, f $ u))
   692           else
   693             f;
   694 
   695         fun build_iter (x as Free (_, T)) U =
   696           if T = U then
   697             x
   698           else
   699             build_map lthy (indexify (perhaps (try (snd o HOLogic.dest_prodT)) o snd) Cs
   700               (fn kk => fn TU => maybe_tick TU (nth us kk) (nth fiters kk))) (T, U) $ x;
   701 
   702         val fxsss = map2 (map2 (flat_rec_arg_args oo map2 (map o build_iter))) xsss x_Tssss;
   703 
   704         val goalss = map5 (map4 o mk_goal fss) fiters xctrss fss xsss fxsss;
   705 
   706         val tacss =
   707           map2 (map o mk_iter_tac pre_map_defs (nested_map_idents @ nesting_map_idents) iter_defs)
   708             ctor_iter_thms ctr_defss;
   709 
   710         fun prove goal tac =
   711           Goal.prove_sorry lthy [] [] goal (tac o #context)
   712           |> Thm.close_derivation;
   713       in
   714         map2 (map2 prove) goalss tacss
   715       end;
   716 
   717     val fold_thmss = mk_iter_thmss fold_args_types folds fold_defs (map un_fold_of ctor_iter_thmss);
   718     val rec_thmss = mk_iter_thmss rec_args_types recs rec_defs (map co_rec_of ctor_iter_thmss);
   719   in
   720     ((induct_thms, induct_thm, [induct_case_names_attr]),
   721      (fold_thmss, rec_thmss, code_nitpicksimp_attrs @ simp_attrs))
   722   end;
   723 
   724 fun derive_coinduct_coiters_thms_for_types pre_bnfs (z, cs, cpss,
   725       coiters_args_types as [((pgss, crgsss), _), ((phss, cshsss), _)])
   726     dtor_coinduct dtor_injects dtor_ctors dtor_coiter_thmss nesting_bnfs fpTs Cs Xs ctrXs_Tsss kss
   727     mss ns ctr_defss (ctr_sugars : ctr_sugar list) coiterss coiter_defss export_args lthy =
   728   let
   729     fun mk_ctor_dtor_coiter_thm dtor_inject dtor_ctor coiter =
   730       iffD1 OF [dtor_inject, trans OF [coiter, dtor_ctor RS sym]];
   731 
   732     val ctor_dtor_coiter_thmss =
   733       map3 (map oo mk_ctor_dtor_coiter_thm) dtor_injects dtor_ctors dtor_coiter_thmss;
   734 
   735     val coiterss' = transpose coiterss;
   736     val coiter_defss' = transpose coiter_defss;
   737 
   738     val [unfold_defs, corec_defs] = coiter_defss';
   739 
   740     val nn = length pre_bnfs;
   741 
   742     val pre_map_defs = map map_def_of_bnf pre_bnfs;
   743     val pre_rel_defs = map rel_def_of_bnf pre_bnfs;
   744     val nesting_map_idents = map (unfold_thms lthy [id_def] o map_id0_of_bnf) nesting_bnfs;
   745     val nesting_rel_eqs = map rel_eq_of_bnf nesting_bnfs;
   746 
   747     val fp_b_names = map base_name_of_typ fpTs;
   748 
   749     val ctrss = map #ctrs ctr_sugars;
   750     val discss = map #discs ctr_sugars;
   751     val selsss = map #selss ctr_sugars;
   752     val exhausts = map #exhaust ctr_sugars;
   753     val disc_thmsss = map #disc_thmss ctr_sugars;
   754     val discIss = map #discIs ctr_sugars;
   755     val sel_thmsss = map #sel_thmss ctr_sugars;
   756 
   757     val (((rs, us'), vs'), names_lthy) =
   758       lthy
   759       |> mk_Frees "R" (map (fn T => mk_pred2T T T) fpTs)
   760       ||>> Variable.variant_fixes fp_b_names
   761       ||>> Variable.variant_fixes (map (suffix "'") fp_b_names);
   762 
   763     val us = map2 (curry Free) us' fpTs;
   764     val udiscss = map2 (map o rapp) us discss;
   765     val uselsss = map2 (map o map o rapp) us selsss;
   766 
   767     val vs = map2 (curry Free) vs' fpTs;
   768     val vdiscss = map2 (map o rapp) vs discss;
   769     val vselsss = map2 (map o map o rapp) vs selsss;
   770 
   771     val coinduct_thms_pairs =
   772       let
   773         val uvrs = map3 (fn r => fn u => fn v => r $ u $ v) rs us vs;
   774         val uv_eqs = map2 (curry HOLogic.mk_eq) us vs;
   775         val strong_rs =
   776           map4 (fn u => fn v => fn uvr => fn uv_eq =>
   777             fold_rev Term.lambda [u, v] (HOLogic.mk_disj (uvr, uv_eq))) us vs uvrs uv_eqs;
   778 
   779         fun build_the_rel rs' T Xs_T =
   780           build_rel lthy (fn (_, X) => nth rs' (find_index (curry (op =) X) Xs)) (T, Xs_T)
   781           |> Term.subst_atomic_types (Xs ~~ fpTs);
   782 
   783         fun build_rel_app rs' usel vsel Xs_T =
   784           fold rapp [usel, vsel] (build_the_rel rs' (fastype_of usel) Xs_T);
   785 
   786         fun mk_prem_ctr_concls rs' n k udisc usels vdisc vsels ctrXs_Ts =
   787           (if k = n then [] else [HOLogic.mk_eq (udisc, vdisc)]) @
   788           (if null usels then
   789              []
   790            else
   791              [Library.foldr HOLogic.mk_imp (if n = 1 then [] else [udisc, vdisc],
   792                 Library.foldr1 HOLogic.mk_conj (map3 (build_rel_app rs') usels vsels ctrXs_Ts))]);
   793 
   794         fun mk_prem_concl rs' n udiscs uselss vdiscs vselss ctrXs_Tss =
   795           Library.foldr1 HOLogic.mk_conj (flat (map6 (mk_prem_ctr_concls rs' n)
   796             (1 upto n) udiscs uselss vdiscs vselss ctrXs_Tss))
   797           handle List.Empty => @{term True};
   798 
   799         fun mk_prem rs' uvr u v n udiscs uselss vdiscs vselss ctrXs_Tss =
   800           fold_rev Logic.all [u, v] (Logic.mk_implies (HOLogic.mk_Trueprop uvr,
   801             HOLogic.mk_Trueprop (mk_prem_concl rs' n udiscs uselss vdiscs vselss ctrXs_Tss)));
   802 
   803         val concl =
   804           HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj
   805             (map3 (fn uvr => fn u => fn v => HOLogic.mk_imp (uvr, HOLogic.mk_eq (u, v)))
   806                uvrs us vs));
   807 
   808         fun mk_goal rs' =
   809           Logic.list_implies (map9 (mk_prem rs') uvrs us vs ns udiscss uselsss vdiscss vselsss
   810             ctrXs_Tsss, concl);
   811 
   812         val goals = map mk_goal [rs, strong_rs];
   813 
   814         fun prove dtor_coinduct' goal =
   815           Goal.prove_sorry lthy [] [] goal (fn {context = ctxt, ...} =>
   816             mk_coinduct_tac ctxt nesting_rel_eqs nn ns dtor_coinduct' pre_rel_defs dtor_ctors
   817               exhausts ctr_defss disc_thmsss sel_thmsss)
   818           |> singleton (Proof_Context.export names_lthy lthy)
   819           |> Thm.close_derivation;
   820 
   821         fun postproc nn thm =
   822           Thm.permute_prems 0 nn
   823             (if nn = 1 then thm RS mp else funpow nn (fn thm => reassoc_conjs (thm RS mp_conj)) thm)
   824           |> Drule.zero_var_indexes
   825           |> `(conj_dests nn);
   826 
   827         val rel_eqs = map rel_eq_of_bnf pre_bnfs;
   828         val rel_monos = map rel_mono_of_bnf pre_bnfs;
   829         val dtor_coinducts =
   830           [dtor_coinduct, mk_strong_coinduct_thm dtor_coinduct rel_eqs rel_monos lthy];
   831       in
   832         map2 (postproc nn oo prove) dtor_coinducts goals
   833       end;
   834 
   835     fun mk_coinduct_concls ms discs ctrs =
   836       let
   837         fun mk_disc_concl disc = [name_of_disc disc];
   838         fun mk_ctr_concl 0 _ = []
   839           | mk_ctr_concl _ ctor = [name_of_ctr ctor];
   840         val disc_concls = map mk_disc_concl (fst (split_last discs)) @ [[]];
   841         val ctr_concls = map2 mk_ctr_concl ms ctrs;
   842       in
   843         flat (map2 append disc_concls ctr_concls)
   844       end;
   845 
   846     val coinduct_cases = quasi_unambiguous_case_names (map (prefix EqN) fp_b_names);
   847     val coinduct_conclss =
   848       map3 (quasi_unambiguous_case_names ooo mk_coinduct_concls) mss discss ctrss;
   849 
   850     fun mk_maybe_not pos = not pos ? HOLogic.mk_not;
   851 
   852     val fcoiterss' as [gunfolds, hcorecs] =
   853       map2 (fn (pfss, _) => map (lists_bmoc pfss)) (map fst coiters_args_types) coiterss';
   854 
   855     val (unfold_thmss, corec_thmss) =
   856       let
   857         fun mk_goal pfss c cps fcoiter n k ctr m cfs' =
   858           fold_rev (fold_rev Logic.all) ([c] :: pfss)
   859             (Logic.list_implies (seq_conds (HOLogic.mk_Trueprop oo mk_maybe_not) n k cps,
   860                mk_Trueprop_eq (fcoiter $ c, Term.list_comb (ctr, take m cfs'))));
   861 
   862         fun mk_U maybe_mk_sumT =
   863           typ_subst_nonatomic (map2 (fn C => fn fpT => (maybe_mk_sumT fpT C, fpT)) Cs fpTs);
   864 
   865         fun tack z_name (c, u) f =
   866           let val z = Free (z_name, mk_sumT (fastype_of u, fastype_of c)) in
   867             Term.lambda z (mk_case_sum (Term.lambda u u, Term.lambda c (f $ c)) $ z)
   868           end;
   869 
   870         fun build_coiter fcoiters maybe_mk_sumT maybe_tack cqf =
   871           let val T = fastype_of cqf in
   872             if exists_subtype_in Cs T then
   873               let val U = mk_U maybe_mk_sumT T in
   874                 build_map lthy (indexify fst (map2 maybe_mk_sumT fpTs Cs) (fn kk => fn _ =>
   875                   maybe_tack (nth cs kk, nth us kk) (nth fcoiters kk))) (T, U) $ cqf
   876               end
   877             else
   878               cqf
   879           end;
   880 
   881         val crgsss' = map (map (map (build_coiter (un_fold_of fcoiterss') (K I) (K I)))) crgsss;
   882         val cshsss' = map (map (map (build_coiter (co_rec_of fcoiterss') (curry mk_sumT) (tack z))))
   883           cshsss;
   884 
   885         val unfold_goalss = map8 (map4 oooo mk_goal pgss) cs cpss gunfolds ns kss ctrss mss crgsss';
   886         val corec_goalss = map8 (map4 oooo mk_goal phss) cs cpss hcorecs ns kss ctrss mss cshsss';
   887 
   888         val unfold_tacss =
   889           map3 (map oo mk_coiter_tac unfold_defs nesting_map_idents)
   890             (map un_fold_of ctor_dtor_coiter_thmss) pre_map_defs ctr_defss;
   891         val corec_tacss =
   892           map3 (map oo mk_coiter_tac corec_defs nesting_map_idents)
   893             (map co_rec_of ctor_dtor_coiter_thmss) pre_map_defs ctr_defss;
   894 
   895         fun prove goal tac =
   896           Goal.prove_sorry lthy [] [] goal (tac o #context)
   897           |> Thm.close_derivation;
   898 
   899         val unfold_thmss = map2 (map2 prove) unfold_goalss unfold_tacss;
   900         val corec_thmss =
   901           map2 (map2 prove) corec_goalss corec_tacss
   902           |> map (map (unfold_thms lthy @{thms case_sum_if}));
   903       in
   904         (unfold_thmss, corec_thmss)
   905       end;
   906 
   907     val (disc_unfold_iff_thmss, disc_corec_iff_thmss) =
   908       let
   909         fun mk_goal c cps fcoiter n k disc =
   910           mk_Trueprop_eq (disc $ (fcoiter $ c),
   911             if n = 1 then @{const True}
   912             else Library.foldr1 HOLogic.mk_conj (seq_conds mk_maybe_not n k cps));
   913 
   914         val unfold_goalss = map6 (map2 oooo mk_goal) cs cpss gunfolds ns kss discss;
   915         val corec_goalss = map6 (map2 oooo mk_goal) cs cpss hcorecs ns kss discss;
   916 
   917         fun mk_case_split' cp = Drule.instantiate' [] [SOME (certify lthy cp)] @{thm case_split};
   918 
   919         val case_splitss' = map (map mk_case_split') cpss;
   920 
   921         val unfold_tacss =
   922           map3 (map oo mk_disc_coiter_iff_tac) case_splitss' unfold_thmss disc_thmsss;
   923         val corec_tacss =
   924           map3 (map oo mk_disc_coiter_iff_tac) case_splitss' corec_thmss disc_thmsss;
   925 
   926         fun prove goal tac =
   927           Goal.prove_sorry lthy [] [] goal (tac o #context)
   928           |> singleton export_args
   929           |> singleton (Proof_Context.export names_lthy lthy)
   930           |> Thm.close_derivation;
   931 
   932         fun proves [_] [_] = []
   933           | proves goals tacs = map2 prove goals tacs;
   934       in
   935         (map2 proves unfold_goalss unfold_tacss, map2 proves corec_goalss corec_tacss)
   936       end;
   937 
   938     fun mk_disc_coiter_thms coiters discIs = map (op RS) (coiters ~~ discIs);
   939 
   940     val disc_unfold_thmss = map2 mk_disc_coiter_thms unfold_thmss discIss;
   941     val disc_corec_thmss = map2 mk_disc_coiter_thms corec_thmss discIss;
   942 
   943     fun mk_sel_coiter_thm coiter_thm sel sel_thm =
   944       let
   945         val (domT, ranT) = dest_funT (fastype_of sel);
   946         val arg_cong' =
   947           Drule.instantiate' (map (SOME o certifyT lthy) [domT, ranT])
   948             [NONE, NONE, SOME (certify lthy sel)] arg_cong
   949           |> Thm.varifyT_global;
   950         val sel_thm' = sel_thm RSN (2, trans);
   951       in
   952         coiter_thm RS arg_cong' RS sel_thm'
   953       end;
   954 
   955     fun mk_sel_coiter_thms coiter_thmss =
   956       map3 (map3 (map2 o mk_sel_coiter_thm)) coiter_thmss selsss sel_thmsss;
   957 
   958     val sel_unfold_thmsss = mk_sel_coiter_thms unfold_thmss;
   959     val sel_corec_thmsss = mk_sel_coiter_thms corec_thmss;
   960 
   961     val coinduct_consumes_attr = Attrib.internal (K (Rule_Cases.consumes nn));
   962     val coinduct_case_names_attr = Attrib.internal (K (Rule_Cases.case_names coinduct_cases));
   963     val coinduct_case_concl_attrs =
   964       map2 (fn casex => fn concls =>
   965           Attrib.internal (K (Rule_Cases.case_conclusion (casex, concls))))
   966         coinduct_cases coinduct_conclss;
   967     val coinduct_case_attrs =
   968       coinduct_consumes_attr :: coinduct_case_names_attr :: coinduct_case_concl_attrs;
   969   in
   970     ((coinduct_thms_pairs, coinduct_case_attrs),
   971      (unfold_thmss, corec_thmss, code_nitpicksimp_attrs),
   972      (disc_unfold_thmss, disc_corec_thmss, []),
   973      (disc_unfold_iff_thmss, disc_corec_iff_thmss, simp_attrs),
   974      (sel_unfold_thmsss, sel_corec_thmsss, simp_attrs))
   975   end;
   976 
   977 fun define_co_datatypes prepare_constraint prepare_typ prepare_term fp construct_fp
   978     (wrap_opts as (no_discs_sels, _), specs) no_defs_lthy0 =
   979   let
   980     (* TODO: sanity checks on arguments *)
   981 
   982     val _ = if fp = Greatest_FP andalso no_discs_sels then
   983         error "Cannot define codatatypes without discriminators and selectors"
   984       else
   985         ();
   986 
   987     val nn = length specs;
   988     val fp_bs = map type_binding_of_spec specs;
   989     val fp_b_names = map Binding.name_of fp_bs;
   990     val fp_common_name = mk_common_name fp_b_names;
   991     val map_bs = map map_binding_of_spec specs;
   992     val rel_bs = map rel_binding_of_spec specs;
   993 
   994     fun prepare_type_arg (_, (ty, c)) =
   995       let val TFree (s, _) = prepare_typ no_defs_lthy0 ty in
   996         TFree (s, prepare_constraint no_defs_lthy0 c)
   997       end;
   998 
   999     val Ass0 = map (map prepare_type_arg o type_args_named_constrained_of_spec) specs;
  1000     val unsorted_Ass0 = map (map (resort_tfree HOLogic.typeS)) Ass0;
  1001     val unsorted_As = Library.foldr1 merge_type_args unsorted_Ass0;
  1002     val num_As = length unsorted_As;
  1003     val set_bss = map (map fst o type_args_named_constrained_of_spec) specs;
  1004 
  1005     val (((Bs0, Cs), Xs), no_defs_lthy) =
  1006       no_defs_lthy0
  1007       |> fold (Variable.declare_typ o resort_tfree dummyS) unsorted_As
  1008       |> mk_TFrees num_As
  1009       ||>> mk_TFrees nn
  1010       ||>> variant_tfrees fp_b_names;
  1011 
  1012     fun add_fake_type spec =
  1013       Typedecl.basic_typedecl (type_binding_of_spec spec, num_As, mixfix_of_spec spec);
  1014 
  1015     val (fake_T_names, fake_lthy) = fold_map add_fake_type specs no_defs_lthy0;
  1016 
  1017     val qsoty = quote o Syntax.string_of_typ fake_lthy;
  1018 
  1019     val _ = (case Library.duplicates (op =) unsorted_As of [] => ()
  1020       | A :: _ => error ("Duplicate type parameter " ^ qsoty A ^ " in " ^ co_prefix fp ^
  1021           "datatype specification"));
  1022 
  1023     val bad_args =
  1024       map (Logic.type_map (singleton (Variable.polymorphic no_defs_lthy0))) unsorted_As
  1025       |> filter_out Term.is_TVar;
  1026     val _ = null bad_args orelse
  1027       error ("Locally fixed type argument " ^ qsoty (hd bad_args) ^ " in " ^ co_prefix fp ^
  1028         "datatype specification");
  1029 
  1030     val mixfixes = map mixfix_of_spec specs;
  1031 
  1032     val _ = (case Library.duplicates Binding.eq_name fp_bs of [] => ()
  1033       | b :: _ => error ("Duplicate type name declaration " ^ quote (Binding.name_of b)));
  1034 
  1035     val mx_ctr_specss = map mixfixed_ctr_specs_of_spec specs;
  1036     val ctr_specss = map (map fst) mx_ctr_specss;
  1037     val ctr_mixfixess = map (map snd) mx_ctr_specss;
  1038 
  1039     val disc_bindingss = map (map disc_of_ctr_spec) ctr_specss;
  1040     val ctr_bindingss =
  1041       map2 (fn fp_b_name => map (Binding.qualify false fp_b_name o ctr_of_ctr_spec)) fp_b_names
  1042         ctr_specss;
  1043     val ctr_argsss = map (map args_of_ctr_spec) ctr_specss;
  1044 
  1045     val sel_bindingsss = map (map (map fst)) ctr_argsss;
  1046     val fake_ctr_Tsss0 = map (map (map (prepare_typ fake_lthy o snd))) ctr_argsss;
  1047     val raw_sel_defaultsss = map (map sel_defaults_of_ctr_spec) ctr_specss;
  1048 
  1049     val (As :: _) :: fake_ctr_Tsss =
  1050       burrow (burrow (Syntax.check_typs fake_lthy)) (Ass0 :: fake_ctr_Tsss0);
  1051     val As' = map dest_TFree As;
  1052 
  1053     val rhs_As' = fold (fold (fold Term.add_tfreesT)) fake_ctr_Tsss [];
  1054     val _ = (case subtract (op =) As' rhs_As' of [] => ()
  1055       | extras => error ("Extra type variables on right-hand side: " ^
  1056           commas (map (qsoty o TFree) extras)));
  1057 
  1058     val fake_Ts = map (fn s => Type (s, As)) fake_T_names;
  1059 
  1060     fun eq_fpT_check (T as Type (s, Ts)) (T' as Type (s', Ts')) =
  1061         s = s' andalso (Ts = Ts' orelse
  1062           error ("Wrong type arguments in " ^ co_prefix fp ^ "recursive type " ^ qsoty T ^
  1063             " (expected " ^ qsoty T' ^ ")"))
  1064       | eq_fpT_check _ _ = false;
  1065 
  1066     fun freeze_fp (T as Type (s, Ts)) =
  1067         (case find_index (eq_fpT_check T) fake_Ts of
  1068           ~1 => Type (s, map freeze_fp Ts)
  1069         | kk => nth Xs kk)
  1070       | freeze_fp T = T;
  1071 
  1072     val unfreeze_fp = Term.typ_subst_atomic (Xs ~~ fake_Ts);
  1073 
  1074     val ctrXs_Tsss = map (map (map freeze_fp)) fake_ctr_Tsss;
  1075     val ctrXs_sum_prod_Ts = map (mk_sumTN_balanced o map HOLogic.mk_tupleT) ctrXs_Tsss;
  1076 
  1077     val fp_eqs =
  1078       map dest_TFree Xs ~~ map (Term.typ_subst_atomic (As ~~ unsorted_As)) ctrXs_sum_prod_Ts;
  1079 
  1080     val rhsXs_As' = fold (fold (fold Term.add_tfreesT)) ctrXs_Tsss [];
  1081     val _ = (case subtract (op =) rhsXs_As' As' of [] => ()
  1082       | extras => List.app (fn extra => warning ("Unused type variable on right-hand side of " ^
  1083           co_prefix fp ^ "datatype definition: " ^ qsoty (TFree extra))) extras);
  1084 
  1085     val (pre_bnfs, (fp_res as {bnfs = fp_bnfs as any_fp_bnf :: _, ctors = ctors0, dtors = dtors0,
  1086            xtor_co_iterss = xtor_co_iterss0, xtor_co_induct, dtor_ctors, ctor_dtors, ctor_injects,
  1087            dtor_injects, xtor_map_thms, xtor_set_thmss, xtor_rel_thms, xtor_co_iter_thmss, ...},
  1088            lthy)) =
  1089       fp_bnf (construct_fp mixfixes map_bs rel_bs set_bss) fp_bs (map dest_TFree unsorted_As) fp_eqs
  1090         no_defs_lthy0
  1091       handle BAD_DEAD (X, X_backdrop) =>
  1092         (case X_backdrop of
  1093           Type (bad_tc, _) =>
  1094           let
  1095             val fake_T = qsoty (unfreeze_fp X);
  1096             val fake_T_backdrop = qsoty (unfreeze_fp X_backdrop);
  1097             fun register_hint () =
  1098               "\nUse the " ^ quote (fst (fst @{command_spec "bnf"})) ^ " command to register " ^
  1099               quote bad_tc ^ " as a bounded natural functor to allow nested (co)recursion through \
  1100               \it";
  1101           in
  1102             if is_some (bnf_of no_defs_lthy bad_tc) orelse
  1103                is_some (fp_sugar_of no_defs_lthy bad_tc) then
  1104               error ("Inadmissible " ^ co_prefix fp ^ "recursive occurrence of type " ^ fake_T ^
  1105                 " in type expression " ^ fake_T_backdrop)
  1106             else if is_some (Datatype_Data.get_info (Proof_Context.theory_of no_defs_lthy)
  1107                 bad_tc) then
  1108               error ("Unsupported " ^ co_prefix fp ^ "recursive occurrence of type " ^ fake_T ^
  1109                 " via the old-style datatype " ^ quote bad_tc ^ " in type expression " ^
  1110                 fake_T_backdrop ^ register_hint ())
  1111             else
  1112               error ("Unsupported " ^ co_prefix fp ^ "recursive occurrence of type " ^ fake_T ^
  1113                 " via type constructor " ^ quote bad_tc ^ " in type expression " ^ fake_T_backdrop ^
  1114                 register_hint ())
  1115           end);
  1116 
  1117     val time = time lthy;
  1118     val timer = time (Timer.startRealTimer ());
  1119 
  1120     val nesting_bnfs = nesty_bnfs lthy ctrXs_Tsss As;
  1121     val nested_bnfs = nesty_bnfs lthy ctrXs_Tsss Xs;
  1122 
  1123     val pre_map_defs = map map_def_of_bnf pre_bnfs;
  1124     val pre_set_defss = map set_defs_of_bnf pre_bnfs;
  1125     val pre_rel_defs = map rel_def_of_bnf pre_bnfs;
  1126     val nesting_set_maps = maps set_map_of_bnf nesting_bnfs;
  1127     val nested_set_maps = maps set_map_of_bnf nested_bnfs;
  1128 
  1129     val live = live_of_bnf any_fp_bnf;
  1130     val _ =
  1131       if live = 0 andalso exists (not o Binding.is_empty) (map_bs @ rel_bs) then
  1132         warning "Map function and relator names ignored"
  1133       else
  1134         ();
  1135 
  1136     val Bs =
  1137       map3 (fn alive => fn A as TFree (_, S) => fn B => if alive then resort_tfree S B else A)
  1138         (liveness_of_fp_bnf num_As any_fp_bnf) As Bs0;
  1139 
  1140     val B_ify = Term.typ_subst_atomic (As ~~ Bs);
  1141 
  1142     val ctors = map (mk_ctor As) ctors0;
  1143     val dtors = map (mk_dtor As) dtors0;
  1144 
  1145     val fpTs = map (domain_type o fastype_of) dtors;
  1146 
  1147     fun massage_simple_notes base =
  1148       filter_out (null o #2)
  1149       #> map (fn (thmN, thms, attrs) =>
  1150         ((Binding.qualify true base (Binding.name thmN), attrs), [(thms, [])]));
  1151 
  1152     val massage_multi_notes =
  1153       maps (fn (thmN, thmss, attrs) =>
  1154         map3 (fn fp_b_name => fn Type (T_name, _) => fn thms =>
  1155             ((Binding.qualify true fp_b_name (Binding.name thmN), attrs T_name), [(thms, [])]))
  1156           fp_b_names fpTs thmss)
  1157       #> filter_out (null o fst o hd o snd);
  1158 
  1159     val ctr_Tsss = map (map (map (Term.typ_subst_atomic (Xs ~~ fpTs)))) ctrXs_Tsss;
  1160     val ns = map length ctr_Tsss;
  1161     val kss = map (fn n => 1 upto n) ns;
  1162     val mss = map (map length) ctr_Tsss;
  1163 
  1164     val ((xtor_co_iterss, iters_args_types, coiters_args_types), lthy') =
  1165       mk_co_iters_prelims fp ctr_Tsss fpTs Cs ns mss xtor_co_iterss0 lthy;
  1166 
  1167     fun define_ctrs_dtrs_for_type (((((((((((((((((((((((fp_bnf, fp_b), fpT), ctor), dtor),
  1168             xtor_co_iters), ctor_dtor), dtor_ctor), ctor_inject), pre_map_def), pre_set_defs),
  1169           pre_rel_def), fp_map_thm), fp_set_thms), fp_rel_thm), n), ks), ms), ctr_bindings),
  1170         ctr_mixfixes), ctr_Tss), disc_bindings), sel_bindingss), raw_sel_defaultss) no_defs_lthy =
  1171       let
  1172         val fp_b_name = Binding.name_of fp_b;
  1173 
  1174         val dtorT = domain_type (fastype_of ctor);
  1175         val ctr_prod_Ts = map HOLogic.mk_tupleT ctr_Tss;
  1176         val ctr_sum_prod_T = mk_sumTN_balanced ctr_prod_Ts;
  1177 
  1178         val ((((w, xss), yss), u'), names_lthy) =
  1179           no_defs_lthy
  1180           |> yield_singleton (mk_Frees "w") dtorT
  1181           ||>> mk_Freess "x" ctr_Tss
  1182           ||>> mk_Freess "y" (map (map B_ify) ctr_Tss)
  1183           ||>> yield_singleton Variable.variant_fixes fp_b_name;
  1184 
  1185         val u = Free (u', fpT);
  1186 
  1187         val tuple_xs = map HOLogic.mk_tuple xss;
  1188         val tuple_ys = map HOLogic.mk_tuple yss;
  1189 
  1190         val ctr_rhss =
  1191           map3 (fn k => fn xs => fn tuple_x => fold_rev Term.lambda xs (ctor $
  1192             mk_InN_balanced ctr_sum_prod_T n tuple_x k)) ks xss tuple_xs;
  1193 
  1194         val maybe_conceal_def_binding = Thm.def_binding
  1195           #> Config.get no_defs_lthy bnf_note_all = false ? Binding.conceal;
  1196 
  1197         val ((raw_ctrs, raw_ctr_defs), (lthy', lthy)) = no_defs_lthy
  1198           |> apfst split_list o fold_map3 (fn b => fn mx => fn rhs =>
  1199               Local_Theory.define ((b, mx), ((maybe_conceal_def_binding b, []), rhs)) #>> apsnd snd)
  1200             ctr_bindings ctr_mixfixes ctr_rhss
  1201           ||> `Local_Theory.restore;
  1202 
  1203         val phi = Proof_Context.export_morphism lthy lthy';
  1204 
  1205         val ctr_defs = map (Morphism.thm phi) raw_ctr_defs;
  1206         val ctr_defs' =
  1207           map2 (fn m => fn def => mk_unabs_def m (def RS meta_eq_to_obj_eq)) ms ctr_defs;
  1208 
  1209         val ctrs0 = map (Morphism.term phi) raw_ctrs;
  1210         val ctrs = map (mk_ctr As) ctrs0;
  1211 
  1212         fun wrap_ctrs lthy =
  1213           let
  1214             fun exhaust_tac {context = ctxt, prems = _} =
  1215               let
  1216                 val ctor_iff_dtor_thm =
  1217                   let
  1218                     val goal =
  1219                       fold_rev Logic.all [w, u]
  1220                         (mk_Trueprop_eq (HOLogic.mk_eq (u, ctor $ w), HOLogic.mk_eq (dtor $ u, w)));
  1221                   in
  1222                     Goal.prove_sorry lthy [] [] goal (fn {context = ctxt, ...} =>
  1223                       mk_ctor_iff_dtor_tac ctxt (map (SOME o certifyT lthy) [dtorT, fpT])
  1224                         (certify lthy ctor) (certify lthy dtor) ctor_dtor dtor_ctor)
  1225                     |> Morphism.thm phi
  1226                     |> Thm.close_derivation
  1227                   end;
  1228 
  1229                 val sumEN_thm' =
  1230                   unfold_thms lthy @{thms unit_all_eq1}
  1231                     (Drule.instantiate' (map (SOME o certifyT lthy) ctr_prod_Ts) []
  1232                        (mk_sumEN_balanced n))
  1233                   |> Morphism.thm phi;
  1234               in
  1235                 mk_exhaust_tac ctxt n ctr_defs ctor_iff_dtor_thm sumEN_thm'
  1236               end;
  1237 
  1238             val inject_tacss =
  1239               map2 (fn 0 => K [] | _ => fn ctr_def => [fn {context = ctxt, ...} =>
  1240                 mk_inject_tac ctxt ctr_def ctor_inject]) ms ctr_defs;
  1241 
  1242             val half_distinct_tacss =
  1243               map (map (fn (def, def') => fn {context = ctxt, ...} =>
  1244                 mk_half_distinct_tac ctxt ctor_inject [def, def'])) (mk_half_pairss (`I ctr_defs));
  1245 
  1246             val tacss = [exhaust_tac] :: inject_tacss @ half_distinct_tacss;
  1247 
  1248             val sel_defaultss = map (map (apsnd (prepare_term lthy))) raw_sel_defaultss
  1249 
  1250             fun ctr_spec_of disc_b ctr0 sel_bs sel_defs = (((disc_b, ctr0), sel_bs), sel_defs);
  1251             val ctr_specs = map4 ctr_spec_of disc_bindings ctrs0 sel_bindingss sel_defaultss;
  1252           in
  1253             free_constructors tacss ((wrap_opts, standard_binding), ctr_specs) lthy
  1254           end;
  1255 
  1256         fun derive_maps_sets_rels (ctr_sugar as {case_cong, ...} : ctr_sugar, lthy) =
  1257           if live = 0 then
  1258             ((([], [], [], []), ctr_sugar), lthy)
  1259           else
  1260             let
  1261               val rel_flip = rel_flip_of_bnf fp_bnf;
  1262               val nones = replicate live NONE;
  1263 
  1264               val ctor_cong =
  1265                 if fp = Least_FP then
  1266                   Drule.dummy_thm
  1267                 else
  1268                   let val ctor' = mk_ctor Bs ctor in
  1269                     cterm_instantiate_pos [NONE, NONE, SOME (certify lthy ctor')] arg_cong
  1270                   end;
  1271 
  1272               fun mk_cIn ify =
  1273                 certify lthy o (fp = Greatest_FP ? curry (op $) (map_types ify ctor)) oo
  1274                 mk_InN_balanced (ify ctr_sum_prod_T) n;
  1275 
  1276               val cxIns = map2 (mk_cIn I) tuple_xs ks;
  1277               val cyIns = map2 (mk_cIn B_ify) tuple_ys ks;
  1278 
  1279               fun mk_map_thm ctr_def' cxIn =
  1280                 fold_thms lthy [ctr_def']
  1281                   (unfold_thms lthy (pre_map_def ::
  1282                        (if fp = Least_FP then [] else [ctor_dtor, dtor_ctor]) @ sum_prod_thms_map)
  1283                      (cterm_instantiate_pos (nones @ [SOME cxIn])
  1284                         (if fp = Least_FP then fp_map_thm else fp_map_thm RS ctor_cong)))
  1285                 |> singleton (Proof_Context.export names_lthy no_defs_lthy);
  1286 
  1287               fun mk_set_thm fp_set_thm ctr_def' cxIn =
  1288                 fold_thms lthy [ctr_def']
  1289                   (unfold_thms lthy (pre_set_defs @ nested_set_maps @ nesting_set_maps @
  1290                        (if fp = Least_FP then [] else [dtor_ctor]) @ sum_prod_thms_set)
  1291                      (cterm_instantiate_pos [SOME cxIn] fp_set_thm))
  1292                 |> singleton (Proof_Context.export names_lthy no_defs_lthy);
  1293 
  1294               fun mk_set_thms fp_set_thm = map2 (mk_set_thm fp_set_thm) ctr_defs' cxIns;
  1295 
  1296               val map_thms = map2 mk_map_thm ctr_defs' cxIns;
  1297               val set_thmss = map mk_set_thms fp_set_thms;
  1298               val set_thms = flat set_thmss;
  1299 
  1300               val rel_infos = (ctr_defs' ~~ cxIns, ctr_defs' ~~ cyIns);
  1301 
  1302               fun mk_rel_thm postproc ctr_defs' cxIn cyIn =
  1303                 fold_thms lthy ctr_defs'
  1304                   (unfold_thms lthy (@{thm Inl_Inr_False} :: pre_rel_def ::
  1305                        (if fp = Least_FP then [] else [dtor_ctor]) @ sum_prod_thms_rel)
  1306                      (cterm_instantiate_pos (nones @ [SOME cxIn, SOME cyIn]) fp_rel_thm))
  1307                 |> postproc
  1308                 |> singleton (Proof_Context.export names_lthy no_defs_lthy);
  1309 
  1310               fun mk_rel_inject_thm ((ctr_def', cxIn), (_, cyIn)) =
  1311                 mk_rel_thm (unfold_thms lthy @{thms eq_sym_Unity_conv}) [ctr_def'] cxIn cyIn;
  1312 
  1313               val rel_inject_thms = map mk_rel_inject_thm (op ~~ rel_infos);
  1314 
  1315               fun mk_half_rel_distinct_thm ((xctr_def', cxIn), (yctr_def', cyIn)) =
  1316                 mk_rel_thm (fn thm => thm RS @{thm eq_False[THEN iffD1]}) [xctr_def', yctr_def']
  1317                   cxIn cyIn;
  1318 
  1319               fun mk_other_half_rel_distinct_thm thm =
  1320                 flip_rels lthy live thm
  1321                 RS (rel_flip RS sym RS @{thm arg_cong[of _ _ Not]} RS iffD2);
  1322 
  1323               val half_rel_distinct_thmss =
  1324                 map (map mk_half_rel_distinct_thm) (mk_half_pairss rel_infos);
  1325               val other_half_rel_distinct_thmss =
  1326                 map (map mk_other_half_rel_distinct_thm) half_rel_distinct_thmss;
  1327               val (rel_distinct_thms, _) =
  1328                 join_halves n half_rel_distinct_thmss other_half_rel_distinct_thmss;
  1329 
  1330               val rel_eq_thms =
  1331                 map (fn th => th RS @{thm eq_False[THEN iffD2]}) rel_distinct_thms @
  1332                 map2 (fn th => fn 0 => th RS @{thm eq_True[THEN iffD2]} | _ => th)
  1333                   rel_inject_thms ms;
  1334 
  1335               val anonymous_notes =
  1336                 [([case_cong], fundefcong_attrs),
  1337                  (rel_eq_thms, code_nitpicksimp_attrs)]
  1338                 |> map (fn (thms, attrs) => ((Binding.empty, attrs), [(thms, [])]));
  1339 
  1340               val notes =
  1341                 [(mapN, map_thms, code_nitpicksimp_attrs @ simp_attrs),
  1342                  (rel_distinctN, rel_distinct_thms, simp_attrs),
  1343                  (rel_injectN, rel_inject_thms, simp_attrs),
  1344                  (setN, set_thms, code_nitpicksimp_attrs @ simp_attrs)]
  1345                 |> massage_simple_notes fp_b_name;
  1346             in
  1347               (((map_thms, rel_inject_thms, rel_distinct_thms, set_thmss), ctr_sugar),
  1348                lthy
  1349                |> Spec_Rules.add Spec_Rules.Equational (`(single o lhs_head_of o hd) map_thms)
  1350                |> Spec_Rules.add Spec_Rules.Equational (`(single o lhs_head_of o hd) rel_eq_thms)
  1351                |> Spec_Rules.add Spec_Rules.Equational (`(single o lhs_head_of o hd) set_thms)
  1352                |> Local_Theory.notes (anonymous_notes @ notes)
  1353                |> snd)
  1354             end;
  1355 
  1356         fun mk_binding pre = Binding.qualify false fp_b_name (Binding.prefix_name (pre ^ "_") fp_b);
  1357 
  1358         fun massage_res (((maps_sets_rels, ctr_sugar), co_iter_res), lthy) =
  1359           (((maps_sets_rels, (ctrs, xss, ctr_defs, ctr_sugar)), co_iter_res), lthy);
  1360       in
  1361         (wrap_ctrs
  1362          #> derive_maps_sets_rels
  1363          ##>>
  1364            (if fp = Least_FP then define_iters [foldN, recN] (the iters_args_types)
  1365             else define_coiters [unfoldN, corecN] (the coiters_args_types))
  1366              mk_binding fpTs Cs xtor_co_iters
  1367          #> massage_res, lthy')
  1368       end;
  1369 
  1370     fun wrap_types_etc (wrap_types_etcs, lthy) =
  1371       fold_map I wrap_types_etcs lthy
  1372       |>> apsnd split_list o apfst (apsnd split_list4 o apfst split_list4 o split_list)
  1373         o split_list;
  1374 
  1375     fun mk_simp_thms ({injects, distincts, case_thms, ...} : ctr_sugar) un_folds co_recs
  1376         mapsx rel_injects rel_distincts setss =
  1377       injects @ distincts @ case_thms @ co_recs @ un_folds @ mapsx @ rel_injects @ rel_distincts
  1378       @ flat setss;
  1379 
  1380     fun derive_note_induct_iters_thms_for_types
  1381         ((((mapss, rel_injects, rel_distincts, setss), (ctrss, _, ctr_defss, ctr_sugars)),
  1382           (iterss, iter_defss)), lthy) =
  1383       let
  1384         val ((induct_thms, induct_thm, induct_attrs), (fold_thmss, rec_thmss, iter_attrs)) =
  1385           derive_induct_iters_thms_for_types pre_bnfs (the iters_args_types) xtor_co_induct
  1386             xtor_co_iter_thmss nesting_bnfs nested_bnfs fpTs Cs Xs ctrXs_Tsss ctrss ctr_defss iterss
  1387             iter_defss lthy;
  1388 
  1389         val induct_type_attr = Attrib.internal o K o Induct.induct_type;
  1390 
  1391         val simp_thmss =
  1392           map7 mk_simp_thms ctr_sugars fold_thmss rec_thmss mapss rel_injects rel_distincts setss;
  1393 
  1394         val common_notes =
  1395           (if nn > 1 then [(inductN, [induct_thm], induct_attrs)] else [])
  1396           |> massage_simple_notes fp_common_name;
  1397 
  1398         val notes =
  1399           [(foldN, fold_thmss, K iter_attrs),
  1400            (inductN, map single induct_thms, fn T_name => induct_attrs @ [induct_type_attr T_name]),
  1401            (recN, rec_thmss, K iter_attrs),
  1402            (simpsN, simp_thmss, K [])]
  1403           |> massage_multi_notes;
  1404       in
  1405         lthy
  1406         |> Spec_Rules.add Spec_Rules.Equational (map un_fold_of iterss, flat fold_thmss)
  1407         |> Spec_Rules.add Spec_Rules.Equational (map co_rec_of iterss, flat rec_thmss)
  1408         |> Local_Theory.notes (common_notes @ notes) |> snd
  1409         |> register_fp_sugars Least_FP pre_bnfs nested_bnfs nesting_bnfs fp_res ctr_defss ctr_sugars
  1410           iterss mapss [induct_thm] (transpose [induct_thms]) (transpose [fold_thmss, rec_thmss])
  1411           [] []
  1412       end;
  1413 
  1414     fun derive_note_coinduct_coiters_thms_for_types
  1415         ((((mapss, rel_injects, rel_distincts, setss), (_, _, ctr_defss, ctr_sugars)),
  1416           (coiterss, coiter_defss)), lthy) =
  1417       let
  1418         val (([(coinduct_thms, coinduct_thm), (strong_coinduct_thms, strong_coinduct_thm)],
  1419               coinduct_attrs),
  1420              (unfold_thmss, corec_thmss, coiter_attrs),
  1421              (disc_unfold_thmss, disc_corec_thmss, disc_coiter_attrs),
  1422              (disc_unfold_iff_thmss, disc_corec_iff_thmss, disc_coiter_iff_attrs),
  1423              (sel_unfold_thmsss, sel_corec_thmsss, sel_coiter_attrs)) =
  1424           derive_coinduct_coiters_thms_for_types pre_bnfs (the coiters_args_types) xtor_co_induct
  1425             dtor_injects dtor_ctors xtor_co_iter_thmss nesting_bnfs fpTs Cs Xs ctrXs_Tsss kss mss ns
  1426             ctr_defss ctr_sugars coiterss coiter_defss (Proof_Context.export lthy' no_defs_lthy)
  1427             lthy;
  1428 
  1429         val sel_unfold_thmss = map flat sel_unfold_thmsss;
  1430         val sel_corec_thmss = map flat sel_corec_thmsss;
  1431 
  1432         val coinduct_type_attr = Attrib.internal o K o Induct.coinduct_type;
  1433 
  1434         val flat_coiter_thms = append oo append;
  1435 
  1436         val simp_thmss =
  1437           map7 mk_simp_thms ctr_sugars
  1438             (map3 flat_coiter_thms disc_unfold_thmss disc_unfold_iff_thmss sel_unfold_thmss)
  1439             (map3 flat_coiter_thms disc_corec_thmss disc_corec_iff_thmss sel_corec_thmss)
  1440             mapss rel_injects rel_distincts setss;
  1441 
  1442         val common_notes =
  1443           (if nn > 1 then
  1444              [(coinductN, [coinduct_thm], coinduct_attrs),
  1445               (strong_coinductN, [strong_coinduct_thm], coinduct_attrs)]
  1446            else
  1447              [])
  1448           |> massage_simple_notes fp_common_name;
  1449 
  1450         val notes =
  1451           [(coinductN, map single coinduct_thms,
  1452             fn T_name => coinduct_attrs @ [coinduct_type_attr T_name]),
  1453            (corecN, corec_thmss, K coiter_attrs),
  1454            (disc_corecN, disc_corec_thmss, K disc_coiter_attrs),
  1455            (disc_corec_iffN, disc_corec_iff_thmss, K disc_coiter_iff_attrs),
  1456            (disc_unfoldN, disc_unfold_thmss, K disc_coiter_attrs),
  1457            (disc_unfold_iffN, disc_unfold_iff_thmss, K disc_coiter_iff_attrs),
  1458            (sel_corecN, sel_corec_thmss, K sel_coiter_attrs),
  1459            (sel_unfoldN, sel_unfold_thmss, K sel_coiter_attrs),
  1460            (simpsN, simp_thmss, K []),
  1461            (strong_coinductN, map single strong_coinduct_thms, K coinduct_attrs),
  1462            (unfoldN, unfold_thmss, K coiter_attrs)]
  1463           |> massage_multi_notes;
  1464 
  1465         (* TODO: code theorems *)
  1466         fun add_spec_rules coiter_of sel_thmss ctr_thmss =
  1467           fold (curry (Spec_Rules.add Spec_Rules.Equational) (map coiter_of coiterss))
  1468             [flat sel_thmss, flat ctr_thmss]
  1469       in
  1470         lthy
  1471         |> add_spec_rules un_fold_of sel_unfold_thmss unfold_thmss
  1472         |> add_spec_rules co_rec_of sel_corec_thmss corec_thmss
  1473         |> Local_Theory.notes (common_notes @ notes) |> snd
  1474         |> register_fp_sugars Greatest_FP pre_bnfs nested_bnfs nesting_bnfs fp_res ctr_defss
  1475           ctr_sugars coiterss mapss [coinduct_thm, strong_coinduct_thm]
  1476           (transpose [coinduct_thms, strong_coinduct_thms])
  1477           (transpose [unfold_thmss, corec_thmss]) (transpose [disc_unfold_thmss, disc_corec_thmss])
  1478           (transpose [sel_unfold_thmsss, sel_corec_thmsss])
  1479       end;
  1480 
  1481     val lthy'' = lthy'
  1482       |> fold_map define_ctrs_dtrs_for_type (fp_bnfs ~~ fp_bs ~~ fpTs ~~ ctors ~~ dtors ~~
  1483         xtor_co_iterss ~~ ctor_dtors ~~ dtor_ctors ~~ ctor_injects ~~ pre_map_defs ~~
  1484         pre_set_defss ~~ pre_rel_defs ~~ xtor_map_thms ~~ xtor_set_thmss ~~ xtor_rel_thms ~~ ns ~~
  1485         kss ~~ mss ~~ ctr_bindingss ~~ ctr_mixfixess ~~ ctr_Tsss ~~ disc_bindingss ~~
  1486         sel_bindingsss ~~ raw_sel_defaultsss)
  1487       |> wrap_types_etc
  1488       |> fp_case fp derive_note_induct_iters_thms_for_types
  1489            derive_note_coinduct_coiters_thms_for_types;
  1490 
  1491     val timer = time (timer ("Constructors, discriminators, selectors, etc., for the new " ^
  1492       co_prefix fp ^ "datatype"));
  1493   in
  1494     timer; lthy''
  1495   end;
  1496 
  1497 fun co_datatypes x = define_co_datatypes (K I) (K I) (K I) x;
  1498 
  1499 fun co_datatype_cmd x =
  1500   define_co_datatypes Typedecl.read_constraint Syntax.parse_typ Syntax.parse_term x;
  1501 
  1502 val parse_type_arg_constrained =
  1503   Parse.type_ident -- Scan.option (@{keyword "::"} |-- Parse.!!! Parse.sort);
  1504 
  1505 val parse_type_arg_named_constrained = parse_opt_binding_colon -- parse_type_arg_constrained;
  1506 
  1507 (*FIXME: use parse_type_args_named_constrained from BNF_Util and thus 
  1508   allow users to kill certain arguments of a (co)datatype*)
  1509 val parse_type_args_named_constrained =
  1510   parse_type_arg_constrained >> (single o pair Binding.empty) ||
  1511   @{keyword "("} |-- Parse.!!! (Parse.list1 parse_type_arg_named_constrained --| @{keyword ")"}) ||
  1512   Scan.succeed [];
  1513 
  1514 val parse_ctr_arg =
  1515   @{keyword "("} |-- parse_binding_colon -- Parse.typ --| @{keyword ")"} ||
  1516   (Parse.typ >> pair Binding.empty);
  1517 
  1518 val parse_ctr_specs =
  1519   Parse.enum1 "|" (parse_ctr_spec parse_binding parse_ctr_arg -- Parse.opt_mixfix);
  1520 
  1521 val parse_spec =
  1522   parse_type_args_named_constrained -- parse_binding -- parse_map_rel_bindings --
  1523   Parse.opt_mixfix -- (@{keyword "="} |-- parse_ctr_specs);
  1524 
  1525 val parse_co_datatype = parse_ctr_options -- Parse.and_list1 parse_spec;
  1526 
  1527 fun parse_co_datatype_cmd fp construct_fp = parse_co_datatype >> co_datatype_cmd fp construct_fp;
  1528 
  1529 end;