src/HOL/BNF/Tools/bnf_fp_def_sugar.ML
author blanchet
Mon Sep 09 13:47:58 2013 +0200 (2013-09-09)
changeset 53475 185ad6cf6576
parent 53329 c31c0c311cf0
child 53476 eb3865c3ee58
permissions -rw-r--r--
enriched data structure with necessary theorems
     1 (*  Title:      HOL/BNF/Tools/bnf_fp_def_sugar.ML
     2     Author:     Jasmin Blanchette, TU Muenchen
     3     Copyright   2012, 2013
     4 
     5 Sugared datatype and codatatype constructions.
     6 *)
     7 
     8 signature BNF_FP_DEF_SUGAR =
     9 sig
    10   type fp_sugar =
    11     {T: typ,
    12      fp: BNF_FP_Util.fp_kind,
    13      index: int,
    14      pre_bnfs: BNF_Def.bnf list,
    15      nested_bnfs: BNF_Def.bnf list,
    16      nesting_bnfs: BNF_Def.bnf list,
    17      fp_res: BNF_FP_Util.fp_result,
    18      ctr_defss: thm list list,
    19      ctr_sugars: BNF_Ctr_Sugar.ctr_sugar list,
    20      co_iterss: term list list,
    21      co_inducts: thm list,
    22      co_iter_thmsss: thm list list list,
    23      disc_co_itersss: thm list list list,
    24      sel_co_iterssss: thm list list list list};
    25 
    26   val of_fp_sugar: (fp_sugar -> 'a list) -> fp_sugar -> 'a
    27   val morph_fp_sugar: morphism -> fp_sugar -> fp_sugar
    28   val fp_sugar_of: Proof.context -> string -> fp_sugar option
    29 
    30   val co_induct_of: 'a list -> 'a
    31   val strong_co_induct_of: 'a list -> 'a
    32 
    33   val tvar_subst: theory -> typ list -> typ list -> ((string * int) * typ) list
    34   val exists_subtype_in: typ list -> typ -> bool
    35   val flat_rec_arg_args: 'a list list -> 'a list
    36   val flat_corec_preds_predsss_gettersss: 'a list -> 'a list list list -> 'a list list list ->
    37     'a list
    38   val mk_co_iter: theory -> BNF_FP_Util.fp_kind -> typ -> typ list -> term -> term
    39   val nesty_bnfs: Proof.context -> typ list list list -> typ list -> BNF_Def.bnf list
    40   val mk_map: int -> typ list -> typ list -> term -> term
    41   val mk_rel: int -> typ list -> typ list -> term -> term
    42   val build_map: local_theory -> (typ * typ -> term) -> typ * typ -> term
    43   val build_rel: local_theory -> (typ * typ -> term) -> typ * typ -> term
    44   val dest_map: Proof.context -> string -> term -> term * term list
    45   val dest_ctr: Proof.context -> string -> term -> term * term list
    46   val mk_co_iters_prelims: BNF_FP_Util.fp_kind -> typ list -> typ list -> int list ->
    47     int list list -> term list list -> Proof.context ->
    48     (term list list
    49      * (typ list list * typ list list list list * term list list
    50         * term list list list list) list option
    51      * (string * term list * term list list
    52         * ((term list list * term list list list) * (typ list * typ list list)) list) option)
    53     * Proof.context
    54 
    55   val mk_iter_fun_arg_types: typ list -> int list -> int list list -> term ->
    56     typ list list list list
    57   val mk_coiter_fun_arg_types: typ list -> int list -> int list list -> term ->
    58     typ list list
    59     * (typ list list list list * typ list list list * typ list list list list * typ list)
    60   val define_iters: string list ->
    61     (typ list list * typ list list list list * term list list * term list list list list) list ->
    62     (string -> binding) -> typ list -> typ list -> term list -> Proof.context ->
    63     (term list * thm list) * Proof.context
    64   val define_coiters: string list -> string * term list * term list list
    65     * ((term list list * term list list list) * (typ list * typ list list)) list ->
    66     (string -> binding) -> typ list -> typ list -> term list -> Proof.context ->
    67     (term list * thm list) * Proof.context
    68   val derive_induct_iters_thms_for_types: BNF_Def.bnf list ->
    69     (typ list list * typ list list list list * term list list * term list list list list) list ->
    70     thm -> thm list list -> BNF_Def.bnf list -> BNF_Def.bnf list -> typ list -> typ list ->
    71     typ list -> typ list list list -> term list list -> thm list list -> term list list ->
    72     thm list list -> local_theory ->
    73     (thm list * thm * Args.src list) * (thm list list * Args.src list)
    74     * (thm list list * Args.src list)
    75   val derive_coinduct_coiters_thms_for_types: BNF_Def.bnf list ->
    76     string * term list * term list list * ((term list list * term list list list)
    77       * (typ list * typ list list)) list ->
    78     thm -> thm list -> thm list -> thm list list -> BNF_Def.bnf list -> typ list -> typ list ->
    79     int list list -> int list list -> int list -> thm list list -> BNF_Ctr_Sugar.ctr_sugar list ->
    80     term list list -> thm list list -> (thm list -> thm list) -> local_theory ->
    81     ((thm list * thm) list * Args.src list)
    82     * (thm list list * thm list list * Args.src list)
    83     * (thm list list * thm list list) * (thm list list * thm list list * Args.src list)
    84     * (thm list list * thm list list * Args.src list)
    85     * (thm list list list * thm list list list * Args.src list)
    86 
    87   val co_datatypes: BNF_FP_Util.fp_kind -> (mixfix list -> binding list -> binding list ->
    88       binding list list -> binding list -> (string * sort) list -> typ list * typ list list ->
    89       BNF_Def.bnf list -> local_theory -> BNF_FP_Util.fp_result * local_theory) ->
    90     (bool * bool) * (((((binding * (typ * sort)) list * binding) * (binding * binding)) * mixfix) *
    91       ((((binding * binding) * (binding * typ) list) * (binding * term) list) *
    92         mixfix) list) list ->
    93     local_theory -> local_theory
    94   val parse_co_datatype_cmd: BNF_FP_Util.fp_kind -> (mixfix list -> binding list -> binding list ->
    95       binding list list -> binding list -> (string * sort) list -> typ list * typ list list ->
    96       BNF_Def.bnf list -> local_theory -> BNF_FP_Util.fp_result * local_theory) ->
    97     (local_theory -> local_theory) parser
    98 end;
    99 
   100 structure BNF_FP_Def_Sugar : BNF_FP_DEF_SUGAR =
   101 struct
   102 
   103 open BNF_Util
   104 open BNF_Ctr_Sugar
   105 open BNF_Comp
   106 open BNF_Def
   107 open BNF_FP_Util
   108 open BNF_FP_Def_Sugar_Tactics
   109 
   110 val EqN = "Eq_";
   111 
   112 type fp_sugar =
   113   {T: typ,
   114    fp: fp_kind,
   115    index: int,
   116    pre_bnfs: bnf list,
   117    nested_bnfs: bnf list,
   118    nesting_bnfs: bnf list,
   119    fp_res: fp_result,
   120    ctr_defss: thm list list,
   121    ctr_sugars: ctr_sugar list,
   122    co_iterss: term list list,
   123    co_inducts: thm list,
   124    co_iter_thmsss: thm list list list,
   125    disc_co_itersss: thm list list list,
   126    sel_co_iterssss: thm list list list list};
   127 
   128 fun of_fp_sugar f (fp_sugar as {index, ...}) = nth (f fp_sugar) index;
   129 
   130 fun eq_fp_sugar ({T = T1, fp = fp1, index = index1, fp_res = fp_res1, ...} : fp_sugar,
   131     {T = T2, fp = fp2, index = index2, fp_res = fp_res2, ...} : fp_sugar) =
   132   T1 = T2 andalso fp1 = fp2 andalso index1 = index2 andalso eq_fp_result (fp_res1, fp_res2);
   133 
   134 fun morph_fp_sugar phi {T, fp, index, pre_bnfs, nested_bnfs, nesting_bnfs, fp_res, ctr_defss,
   135     ctr_sugars, co_iterss, co_inducts, co_iter_thmsss, disc_co_itersss, sel_co_iterssss} =
   136   {T = Morphism.typ phi T, fp = fp, index = index, pre_bnfs = map (morph_bnf phi) pre_bnfs,
   137     nested_bnfs = map (morph_bnf phi) nested_bnfs, nesting_bnfs = map (morph_bnf phi) nesting_bnfs,
   138    fp_res = morph_fp_result phi fp_res,
   139    ctr_defss = map (map (Morphism.thm phi)) ctr_defss,
   140    ctr_sugars = map (morph_ctr_sugar phi) ctr_sugars,
   141    co_iterss = map (map (Morphism.term phi)) co_iterss,
   142    co_inducts = map (Morphism.thm phi) co_inducts,
   143    co_iter_thmsss = map (map (map (Morphism.thm phi))) co_iter_thmsss,
   144    disc_co_itersss = map (map (map (Morphism.thm phi))) disc_co_itersss,
   145    sel_co_iterssss = map (map (map (map (Morphism.thm phi)))) sel_co_iterssss};
   146 
   147 structure Data = Generic_Data
   148 (
   149   type T = fp_sugar Symtab.table;
   150   val empty = Symtab.empty;
   151   val extend = I;
   152   val merge = Symtab.merge eq_fp_sugar;
   153 );
   154 
   155 fun fp_sugar_of ctxt =
   156   Symtab.lookup (Data.get (Context.Proof ctxt))
   157   #> Option.map (morph_fp_sugar
   158     (Morphism.thm_morphism (Thm.transfer (Proof_Context.theory_of ctxt))));
   159 
   160 fun co_induct_of (i :: _) = i;
   161 fun strong_co_induct_of [_, s] = s;
   162 
   163 (* TODO: register "sum" and "prod" as datatypes to enable N2M reduction for them *)
   164 
   165 fun register_fp_sugar key fp_sugar =
   166   Local_Theory.declaration {syntax = false, pervasive = true}
   167     (fn phi => Data.map (Symtab.default (key, morph_fp_sugar phi fp_sugar)));
   168 
   169 fun register_fp_sugars fp pre_bnfs nested_bnfs nesting_bnfs (fp_res as {Ts, ...}) ctr_defss
   170     ctr_sugars co_iterss co_inducts co_iter_thmsss disc_co_itersss sel_co_iterssss lthy =
   171   (0, lthy)
   172   |> fold (fn T as Type (s, _) => fn (kk, lthy) => (kk + 1,
   173     register_fp_sugar s {T = T, fp = fp, index = kk, pre_bnfs = pre_bnfs,
   174         nested_bnfs = nested_bnfs, nesting_bnfs = nesting_bnfs, fp_res = fp_res,
   175         ctr_defss = ctr_defss, ctr_sugars = ctr_sugars, co_iterss = co_iterss,
   176         co_inducts = co_inducts, co_iter_thmsss = co_iter_thmsss, disc_co_itersss = disc_co_itersss,
   177         sel_co_iterssss = sel_co_iterssss}
   178       lthy)) Ts
   179   |> snd;
   180 
   181 (* This function could produce clashes in contrived examples (e.g., "x.A", "x.x_A", "y.A"). *)
   182 fun quasi_unambiguous_case_names names =
   183   let
   184     val ps = map (`Long_Name.base_name) names;
   185     val dups = Library.duplicates (op =) (map fst ps);
   186     fun underscore s =
   187       let val ss = space_explode Long_Name.separator s in
   188         space_implode "_" (drop (length ss - 2) ss)
   189       end;
   190   in
   191     map (fn (base, full) => if member (op =) dups base then underscore full else base) ps
   192   end;
   193 
   194 val id_def = @{thm id_def};
   195 val mp_conj = @{thm mp_conj};
   196 
   197 val simp_attrs = @{attributes [simp]};
   198 val code_simp_attrs = Code.add_default_eqn_attrib :: simp_attrs;
   199 
   200 fun tvar_subst thy Ts Us =
   201   Vartab.fold (cons o apsnd snd) (fold (Sign.typ_match thy) (Ts ~~ Us) Vartab.empty) [];
   202 
   203 val exists_subtype_in = Term.exists_subtype o member (op =);
   204 
   205 val lists_bmoc = fold (fn xs => fn t => Term.list_comb (t, xs));
   206 
   207 fun flat_rec_arg_args xss =
   208   (* FIXME (once the old datatype package is phased out): The first line below gives the preferred
   209      order. The second line is for compatibility with the old datatype package. *)
   210 (*
   211   flat xss
   212 *)
   213   map hd xss @ maps tl xss;
   214 
   215 fun flat_corec_predss_getterss qss fss = maps (op @) (qss ~~ fss);
   216 
   217 fun flat_corec_preds_predsss_gettersss [] [qss] [fss] = flat_corec_predss_getterss qss fss
   218   | flat_corec_preds_predsss_gettersss (p :: ps) (qss :: qsss) (fss :: fsss) =
   219     p :: flat_corec_predss_getterss qss fss @ flat_corec_preds_predsss_gettersss ps qsss fsss;
   220 
   221 fun mk_tupled_fun x f xs = HOLogic.tupled_lambda x (Term.list_comb (f, xs));
   222 fun mk_uncurried2_fun f xss =
   223   mk_tupled_fun (HOLogic.mk_tuple (map HOLogic.mk_tuple xss)) f (flat_rec_arg_args xss);
   224 
   225 fun mk_flip (x, Type (_, [T1, Type (_, [T2, T3])])) =
   226   Abs ("x", T1, Abs ("y", T2, Var (x, T2 --> T1 --> T3) $ Bound 0 $ Bound 1));
   227 
   228 fun flip_rels lthy n thm =
   229   let
   230     val Rs = Term.add_vars (prop_of thm) [];
   231     val Rs' = rev (drop (length Rs - n) Rs);
   232     val cRs = map (fn f => (certify lthy (Var f), certify lthy (mk_flip f))) Rs';
   233   in
   234     Drule.cterm_instantiate cRs thm
   235   end;
   236 
   237 fun mk_ctor_or_dtor get_T Ts t =
   238   let val Type (_, Ts0) = get_T (fastype_of t) in
   239     Term.subst_atomic_types (Ts0 ~~ Ts) t
   240   end;
   241 
   242 val mk_ctor = mk_ctor_or_dtor range_type;
   243 val mk_dtor = mk_ctor_or_dtor domain_type;
   244 
   245 fun mk_co_iter thy fp fpT Cs t =
   246   let
   247     val (f_Cs, Type (_, [prebody, body])) = strip_fun_type (fastype_of t);
   248     val fpT0 = fp_case fp prebody body;
   249     val Cs0 = distinct (op =) (map (fp_case fp body_type domain_type) f_Cs);
   250     val rho = tvar_subst thy (fpT0 :: Cs0) (fpT :: Cs);
   251   in
   252     Term.subst_TVars rho t
   253   end;
   254 
   255 fun mk_co_iters thy fp fpTs Cs ts0 =
   256   let
   257     val nn = length fpTs;
   258     val (fpTs0, Cs0) =
   259       map ((fp = Greatest_FP ? swap) o dest_funT o snd o strip_typeN nn o fastype_of) ts0
   260       |> split_list;
   261     val rho = tvar_subst thy (fpTs0 @ Cs0) (fpTs @ Cs);
   262   in
   263     map (Term.subst_TVars rho) ts0
   264   end;
   265 
   266 val mk_fp_iter_fun_types = binder_fun_types o fastype_of;
   267 
   268 fun unzip_recT Cs (T as Type (@{type_name prod}, Ts as [_, U])) =
   269     if member (op =) Cs U then Ts else [T]
   270   | unzip_recT _ T = [T];
   271 
   272 fun unzip_corecT Cs (T as Type (@{type_name sum}, Ts as [_, U])) =
   273     if member (op =) Cs U then Ts else [T]
   274   | unzip_corecT _ T = [T];
   275 
   276 fun mk_map live Ts Us t =
   277   let val (Type (_, Ts0), Type (_, Us0)) = strip_typeN (live + 1) (fastype_of t) |>> List.last in
   278     Term.subst_atomic_types (Ts0 @ Us0 ~~ Ts @ Us) t
   279   end;
   280 
   281 fun mk_rel live Ts Us t =
   282   let val [Type (_, Ts0), Type (_, Us0)] = binder_types (snd (strip_typeN live (fastype_of t))) in
   283     Term.subst_atomic_types (Ts0 @ Us0 ~~ Ts @ Us) t
   284   end;
   285 
   286 local
   287 
   288 fun build_map_or_rel mk const of_bnf dest lthy build_simple =
   289   let
   290     fun build (TU as (T, U)) =
   291       if T = U then
   292         const T
   293       else
   294         (case TU of
   295           (Type (s, Ts), Type (s', Us)) =>
   296           if s = s' then
   297             let
   298               val bnf = the (bnf_of lthy s);
   299               val live = live_of_bnf bnf;
   300               val mapx = mk live Ts Us (of_bnf bnf);
   301               val TUs' = map dest (fst (strip_typeN live (fastype_of mapx)));
   302             in Term.list_comb (mapx, map build TUs') end
   303           else
   304             build_simple TU
   305         | _ => build_simple TU);
   306   in build end;
   307 
   308 in
   309 
   310 val build_map = build_map_or_rel mk_map HOLogic.id_const map_of_bnf dest_funT;
   311 val build_rel = build_map_or_rel mk_rel HOLogic.eq_const rel_of_bnf dest_pred2T;
   312 
   313 end;
   314 
   315 fun fo_match ctxt t pat =
   316   let val thy = Proof_Context.theory_of ctxt in
   317     Pattern.first_order_match thy (pat, t) (Vartab.empty, Vartab.empty)
   318   end;
   319 
   320 val dummy_var_name = "?f"
   321 
   322 fun mk_map_pattern ctxt s =
   323   let
   324     val bnf = the (bnf_of ctxt s);
   325     val mapx = map_of_bnf bnf;
   326     val live = live_of_bnf bnf;
   327     val (f_Ts, _) = strip_typeN live (fastype_of mapx);
   328     val fs = map_index (fn (i, T) => Var ((dummy_var_name, i), T)) f_Ts;
   329   in
   330     (mapx, betapplys (mapx, fs))
   331   end;
   332 
   333 fun dest_map ctxt s call =
   334   let
   335     val (map0, pat) = mk_map_pattern ctxt s;
   336     val (_, tenv) = fo_match ctxt call pat;
   337   in
   338     (map0, Vartab.fold_rev (fn (_, (_, f)) => cons f) tenv [])
   339   end;
   340 
   341 fun dest_ctr ctxt s t =
   342   let
   343     val (f, args) = Term.strip_comb t;
   344   in
   345     (case fp_sugar_of ctxt s of
   346       SOME fp_sugar =>
   347       (case find_first (can (fo_match ctxt f)) (#ctrs (of_fp_sugar #ctr_sugars fp_sugar)) of
   348         SOME f' => (f', args)
   349       | NONE => raise Fail "dest_ctr")
   350     | NONE => raise Fail "dest_ctr")
   351   end;
   352 
   353 fun liveness_of_fp_bnf n bnf =
   354   (case T_of_bnf bnf of
   355     Type (_, Ts) => map (not o member (op =) (deads_of_bnf bnf)) Ts
   356   | _ => replicate n false);
   357 
   358 fun cannot_merge_types () = error "Mutually recursive types must have the same type parameters";
   359 
   360 fun merge_type_arg T T' = if T = T' then T else cannot_merge_types ();
   361 
   362 fun merge_type_args (As, As') =
   363   if length As = length As' then map2 merge_type_arg As As' else cannot_merge_types ();
   364 
   365 fun reassoc_conjs thm =
   366   reassoc_conjs (thm RS @{thm conj_assoc[THEN iffD1]})
   367   handle THM _ => thm;
   368 
   369 fun type_args_named_constrained_of ((((ncAs, _), _), _), _) = ncAs;
   370 fun type_binding_of ((((_, b), _), _), _) = b;
   371 fun map_binding_of (((_, (b, _)), _), _) = b;
   372 fun rel_binding_of (((_, (_, b)), _), _) = b;
   373 fun mixfix_of ((_, mx), _) = mx;
   374 fun ctr_specs_of (_, ctr_specs) = ctr_specs;
   375 
   376 fun disc_of ((((disc, _), _), _), _) = disc;
   377 fun ctr_of ((((_, ctr), _), _), _) = ctr;
   378 fun args_of (((_, args), _), _) = args;
   379 fun defaults_of ((_, ds), _) = ds;
   380 fun ctr_mixfix_of (_, mx) = mx;
   381 
   382 fun add_nesty_bnf_names Us =
   383   let
   384     fun add (Type (s, Ts)) ss =
   385         let val (needs, ss') = fold_map add Ts ss in
   386           if exists I needs then (true, insert (op =) s ss') else (false, ss')
   387         end
   388       | add T ss = (member (op =) Us T, ss);
   389   in snd oo add end;
   390 
   391 fun nesty_bnfs ctxt ctr_Tsss Us =
   392   map_filter (bnf_of ctxt) (fold (fold (fold (add_nesty_bnf_names Us))) ctr_Tsss []);
   393 
   394 fun indexify proj xs f p = f (find_index (curry op = (proj p)) xs) p;
   395 
   396 fun mk_iter_fun_arg_types0 n ms = map2 dest_tupleT ms o dest_sumTN_balanced n o domain_type;
   397 
   398 fun mk_iter_fun_arg_types Cs ns mss =
   399   mk_fp_iter_fun_types
   400   #> map3 mk_iter_fun_arg_types0 ns mss
   401   #> map (map (map (unzip_recT Cs)));
   402 
   403 fun mk_iters_args_types Cs ns mss ctor_iter_fun_Tss lthy =
   404   let
   405     val Css = map2 replicate ns Cs;
   406     val y_Tsss = map3 mk_iter_fun_arg_types0 ns mss (map un_fold_of ctor_iter_fun_Tss);
   407     val g_Tss = map2 (fn C => map (fn y_Ts => y_Ts ---> C)) Cs y_Tsss;
   408 
   409     val ((gss, ysss), lthy) =
   410       lthy
   411       |> mk_Freess "f" g_Tss
   412       ||>> mk_Freesss "x" y_Tsss;
   413 
   414     val y_Tssss = map (map (map single)) y_Tsss;
   415     val yssss = map (map (map single)) ysss;
   416 
   417     val z_Tssss =
   418       map3 (fn n => fn ms => map2 (map (unzip_recT Cs) oo dest_tupleT) ms o
   419         dest_sumTN_balanced n o domain_type o co_rec_of) ns mss ctor_iter_fun_Tss;
   420 
   421     val z_Tsss' = map (map flat_rec_arg_args) z_Tssss;
   422     val h_Tss = map2 (map2 (curry op --->)) z_Tsss' Css;
   423 
   424     val hss = map2 (map2 retype_free) h_Tss gss;
   425     val zssss_hd = map2 (map2 (map2 (retype_free o hd))) z_Tssss ysss;
   426     val (zssss_tl, lthy) =
   427       lthy
   428       |> mk_Freessss "y" (map (map (map tl)) z_Tssss);
   429     val zssss = map2 (map2 (map2 cons)) zssss_hd zssss_tl;
   430   in
   431     ([(g_Tss, y_Tssss, gss, yssss), (h_Tss, z_Tssss, hss, zssss)], lthy)
   432   end;
   433 
   434 fun mk_coiter_fun_arg_types0 Cs ns mss fun_Ts =
   435   let
   436     (*avoid "'a itself" arguments in coiterators and corecursors*)
   437     fun repair_arity [0] = [1]
   438       | repair_arity ms = ms;
   439 
   440     val f_sum_prod_Ts = map range_type fun_Ts;
   441     val f_prod_Tss = map2 dest_sumTN_balanced ns f_sum_prod_Ts;
   442     val f_Tsss = map2 (map2 dest_tupleT o repair_arity) mss f_prod_Tss;
   443     val f_Tssss = map2 (fn C => map (map (map (curry op --> C) o unzip_corecT Cs))) Cs f_Tsss;
   444     val q_Tssss = map (map (map (fn [_] => [] | [_, T] => [mk_pred1T (domain_type T)]))) f_Tssss;
   445   in
   446     (q_Tssss, f_Tsss, f_Tssss, f_sum_prod_Ts)
   447   end;
   448 
   449 fun mk_coiter_p_pred_types Cs ns = map2 (fn n => replicate (Int.max (0, n - 1)) o mk_pred1T) ns Cs;
   450 
   451 fun mk_coiter_fun_arg_types Cs ns mss dtor_coiter =
   452   (mk_coiter_p_pred_types Cs ns,
   453    mk_fp_iter_fun_types dtor_coiter |> mk_coiter_fun_arg_types0 Cs ns mss);
   454 
   455 fun mk_coiters_args_types Cs ns mss dtor_coiter_fun_Tss lthy =
   456   let
   457     val p_Tss = mk_coiter_p_pred_types Cs ns;
   458 
   459     fun mk_types get_Ts =
   460       let
   461         val fun_Ts = map get_Ts dtor_coiter_fun_Tss;
   462         val (q_Tssss, f_Tsss, f_Tssss, f_sum_prod_Ts) = mk_coiter_fun_arg_types0 Cs ns mss fun_Ts;
   463         val pf_Tss = map3 flat_corec_preds_predsss_gettersss p_Tss q_Tssss f_Tssss;
   464       in
   465         (q_Tssss, f_Tsss, f_Tssss, (f_sum_prod_Ts, pf_Tss))
   466       end;
   467 
   468     val (r_Tssss, g_Tsss, g_Tssss, unfold_types) = mk_types un_fold_of;
   469     val (s_Tssss, h_Tsss, h_Tssss, corec_types) = mk_types co_rec_of;
   470 
   471     val ((((Free (z, _), cs), pss), gssss), lthy) =
   472       lthy
   473       |> yield_singleton (mk_Frees "z") dummyT
   474       ||>> mk_Frees "a" Cs
   475       ||>> mk_Freess "p" p_Tss
   476       ||>> mk_Freessss "g" g_Tssss;
   477     val rssss = map (map (map (fn [] => []))) r_Tssss;
   478 
   479     val hssss_hd = map2 (map2 (map2 (fn T :: _ => fn [g] => retype_free T g))) h_Tssss gssss;
   480     val ((sssss, hssss_tl), lthy) =
   481       lthy
   482       |> mk_Freessss "q" s_Tssss
   483       ||>> mk_Freessss "h" (map (map (map tl)) h_Tssss);
   484     val hssss = map2 (map2 (map2 cons)) hssss_hd hssss_tl;
   485 
   486     val cpss = map2 (map o rapp) cs pss;
   487 
   488     fun build_sum_inj mk_inj = build_map lthy (uncurry mk_inj o dest_sumT o snd);
   489 
   490     fun build_dtor_coiter_arg _ [] [cf] = cf
   491       | build_dtor_coiter_arg T [cq] [cf, cf'] =
   492         mk_If cq (build_sum_inj Inl_const (fastype_of cf, T) $ cf)
   493           (build_sum_inj Inr_const (fastype_of cf', T) $ cf');
   494 
   495     fun mk_args qssss fssss f_Tsss =
   496       let
   497         val pfss = map3 flat_corec_preds_predsss_gettersss pss qssss fssss;
   498         val cqssss = map2 (map o map o map o rapp) cs qssss;
   499         val cfssss = map2 (map o map o map o rapp) cs fssss;
   500         val cqfsss = map3 (map3 (map3 build_dtor_coiter_arg)) f_Tsss cqssss cfssss;
   501       in (pfss, cqfsss) end;
   502 
   503     val unfold_args = mk_args rssss gssss g_Tsss;
   504     val corec_args = mk_args sssss hssss h_Tsss;
   505   in
   506     ((z, cs, cpss, [(unfold_args, unfold_types), (corec_args, corec_types)]), lthy)
   507   end;
   508 
   509 fun mk_co_iters_prelims fp fpTs Cs ns mss xtor_co_iterss0 lthy =
   510   let
   511     val thy = Proof_Context.theory_of lthy;
   512 
   513     val (xtor_co_iter_fun_Tss, xtor_co_iterss) =
   514       map (mk_co_iters thy fp fpTs Cs #> `(mk_fp_iter_fun_types o hd)) (transpose xtor_co_iterss0)
   515       |> apsnd transpose o apfst transpose o split_list;
   516 
   517     val ((iters_args_types, coiters_args_types), lthy') =
   518       if fp = Least_FP then
   519         mk_iters_args_types Cs ns mss xtor_co_iter_fun_Tss lthy |>> (rpair NONE o SOME)
   520       else
   521         mk_coiters_args_types Cs ns mss xtor_co_iter_fun_Tss lthy |>> (pair NONE o SOME)
   522   in
   523     ((xtor_co_iterss, iters_args_types, coiters_args_types), lthy')
   524   end;
   525 
   526 fun mk_iter_body ctor_iter fss xssss =
   527   Term.list_comb (ctor_iter, map2 (mk_sum_caseN_balanced oo map2 mk_uncurried2_fun) fss xssss);
   528 
   529 fun mk_preds_getterss_join c cps sum_prod_T cqfss =
   530   let val n = length cqfss in
   531     Term.lambda c (mk_IfN sum_prod_T cps
   532       (map2 (mk_InN_balanced sum_prod_T n) (map HOLogic.mk_tuple cqfss) (1 upto n)))
   533   end;
   534 
   535 fun mk_coiter_body cs cpss f_sum_prod_Ts cqfsss dtor_coiter =
   536   Term.list_comb (dtor_coiter, map4 mk_preds_getterss_join cs cpss f_sum_prod_Ts cqfsss);
   537 
   538 fun define_co_iters fp fpT Cs binding_specs lthy0 =
   539   let
   540     val thy = Proof_Context.theory_of lthy0;
   541 
   542     val ((csts, defs), (lthy', lthy)) = lthy0
   543       |> apfst split_list o fold_map (fn (b, spec) =>
   544         Specification.definition (SOME (b, NONE, NoSyn), ((Thm.def_binding b, []), spec))
   545         #>> apsnd snd) binding_specs
   546       ||> `Local_Theory.restore;
   547 
   548     val phi = Proof_Context.export_morphism lthy lthy';
   549 
   550     val csts' = map (mk_co_iter thy fp fpT Cs o Morphism.term phi) csts;
   551     val defs' = map (Morphism.thm phi) defs;
   552   in
   553     ((csts', defs'), lthy')
   554   end;
   555 
   556 fun define_iters iterNs iter_args_typess' mk_binding fpTs Cs ctor_iters lthy =
   557   let
   558     val nn = length fpTs;
   559 
   560     val fpT_to_C as Type (_, [fpT, _]) = snd (strip_typeN nn (fastype_of (hd ctor_iters)));
   561 
   562     fun generate_iter suf (f_Tss, _, fss, xssss) ctor_iter =
   563       let
   564         val res_T = fold_rev (curry op --->) f_Tss fpT_to_C;
   565         val b = mk_binding suf;
   566         val spec =
   567           mk_Trueprop_eq (lists_bmoc fss (Free (Binding.name_of b, res_T)),
   568             mk_iter_body ctor_iter fss xssss);
   569       in (b, spec) end;
   570   in
   571     define_co_iters Least_FP fpT Cs (map3 generate_iter iterNs iter_args_typess' ctor_iters) lthy
   572   end;
   573 
   574 fun define_coiters coiterNs (_, cs, cpss, coiter_args_typess') mk_binding fpTs Cs dtor_coiters
   575     lthy =
   576   let
   577     val nn = length fpTs;
   578 
   579     val C_to_fpT as Type (_, [_, fpT]) = snd (strip_typeN nn (fastype_of (hd dtor_coiters)));
   580 
   581     fun generate_coiter suf ((pfss, cqfsss), (f_sum_prod_Ts, pf_Tss)) dtor_coiter =
   582       let
   583         val res_T = fold_rev (curry op --->) pf_Tss C_to_fpT;
   584         val b = mk_binding suf;
   585         val spec =
   586           mk_Trueprop_eq (lists_bmoc pfss (Free (Binding.name_of b, res_T)),
   587             mk_coiter_body cs cpss f_sum_prod_Ts cqfsss dtor_coiter);
   588       in (b, spec) end;
   589   in
   590     define_co_iters Greatest_FP fpT Cs
   591       (map3 generate_coiter coiterNs coiter_args_typess' dtor_coiters) lthy
   592   end;
   593 
   594 fun derive_induct_iters_thms_for_types pre_bnfs [fold_args_types, rec_args_types] ctor_induct
   595     ctor_iter_thmss nesting_bnfs nested_bnfs fpTs Cs Xs ctrXs_Tsss ctrss ctr_defss iterss iter_defss
   596     lthy =
   597   let
   598     val iterss' = transpose iterss;
   599     val iter_defss' = transpose iter_defss;
   600 
   601     val [folds, recs] = iterss';
   602     val [fold_defs, rec_defs] = iter_defss';
   603 
   604     val ctr_Tsss = map (map (binder_types o fastype_of)) ctrss;
   605 
   606     val nn = length pre_bnfs;
   607     val ns = map length ctr_Tsss;
   608     val mss = map (map length) ctr_Tsss;
   609 
   610     val pre_map_defs = map map_def_of_bnf pre_bnfs;
   611     val pre_set_defss = map set_defs_of_bnf pre_bnfs;
   612     val nesting_map_idents = map (unfold_thms lthy [id_def] o map_id0_of_bnf) nesting_bnfs;
   613     val nested_map_idents = map (unfold_thms lthy [id_def] o map_id0_of_bnf) nested_bnfs;
   614     val nested_set_maps = maps set_map_of_bnf nested_bnfs;
   615 
   616     val fp_b_names = map base_name_of_typ fpTs;
   617 
   618     val ((((ps, ps'), xsss), us'), names_lthy) =
   619       lthy
   620       |> mk_Frees' "P" (map mk_pred1T fpTs)
   621       ||>> mk_Freesss "x" ctr_Tsss
   622       ||>> Variable.variant_fixes fp_b_names;
   623 
   624     val us = map2 (curry Free) us' fpTs;
   625 
   626     fun mk_sets_nested bnf =
   627       let
   628         val Type (T_name, Us) = T_of_bnf bnf;
   629         val lives = lives_of_bnf bnf;
   630         val sets = sets_of_bnf bnf;
   631         fun mk_set U =
   632           (case find_index (curry op = U) lives of
   633             ~1 => Term.dummy
   634           | i => nth sets i);
   635       in
   636         (T_name, map mk_set Us)
   637       end;
   638 
   639     val setss_nested = map mk_sets_nested nested_bnfs;
   640 
   641     val (induct_thms, induct_thm) =
   642       let
   643         fun mk_set Ts t =
   644           let val Type (_, Ts0) = domain_type (fastype_of t) in
   645             Term.subst_atomic_types (Ts0 ~~ Ts) t
   646           end;
   647 
   648         fun mk_raw_prem_prems _ (x as Free (_, Type _)) (X as TFree _) =
   649             [([], (find_index (curry op = X) Xs + 1, x))]
   650           | mk_raw_prem_prems names_lthy (x as Free (s, Type (T_name, Ts0))) (Type (_, Xs_Ts0)) =
   651             (case AList.lookup (op =) setss_nested T_name of
   652               NONE => []
   653             | SOME raw_sets0 =>
   654               let
   655                 val (Xs_Ts, (Ts, raw_sets)) =
   656                   filter (exists_subtype_in Xs o fst) (Xs_Ts0 ~~ (Ts0 ~~ raw_sets0))
   657                   |> split_list ||> split_list;
   658                 val sets = map (mk_set Ts0) raw_sets;
   659                 val (ys, names_lthy') = names_lthy |> mk_Frees s Ts;
   660                 val xysets = map (pair x) (ys ~~ sets);
   661                 val ppremss = map2 (mk_raw_prem_prems names_lthy') ys Xs_Ts;
   662               in
   663                 flat (map2 (map o apfst o cons) xysets ppremss)
   664               end)
   665           | mk_raw_prem_prems _ _ _ = [];
   666 
   667         fun close_prem_prem xs t =
   668           fold_rev Logic.all (map Free (drop (nn + length xs)
   669             (rev (Term.add_frees t (map dest_Free xs @ ps'))))) t;
   670 
   671         fun mk_prem_prem xs (xysets, (j, x)) =
   672           close_prem_prem xs (Logic.list_implies (map (fn (x', (y, set)) =>
   673               HOLogic.mk_Trueprop (HOLogic.mk_mem (y, set $ x'))) xysets,
   674             HOLogic.mk_Trueprop (nth ps (j - 1) $ x)));
   675 
   676         fun mk_raw_prem phi ctr ctr_Ts ctrXs_Ts =
   677           let
   678             val (xs, names_lthy') = names_lthy |> mk_Frees "x" ctr_Ts;
   679             val pprems = flat (map2 (mk_raw_prem_prems names_lthy') xs ctrXs_Ts);
   680           in (xs, pprems, HOLogic.mk_Trueprop (phi $ Term.list_comb (ctr, xs))) end;
   681 
   682         fun mk_prem (xs, raw_pprems, concl) =
   683           fold_rev Logic.all xs (Logic.list_implies (map (mk_prem_prem xs) raw_pprems, concl));
   684 
   685         val raw_premss = map4 (map3 o mk_raw_prem) ps ctrss ctr_Tsss ctrXs_Tsss;
   686 
   687         val goal =
   688           Library.foldr (Logic.list_implies o apfst (map mk_prem)) (raw_premss,
   689             HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj (map2 (curry op $) ps us)));
   690 
   691         val kksss = map (map (map (fst o snd) o #2)) raw_premss;
   692 
   693         val ctor_induct' = ctor_induct OF (map mk_sumEN_tupled_balanced mss);
   694 
   695         val thm =
   696           Goal.prove_sorry lthy [] [] goal (fn {context = ctxt, ...} =>
   697             mk_induct_tac ctxt nn ns mss kksss (flat ctr_defss) ctor_induct' nested_set_maps
   698               pre_set_defss)
   699           |> singleton (Proof_Context.export names_lthy lthy)
   700           |> Thm.close_derivation;
   701       in
   702         `(conj_dests nn) thm
   703       end;
   704 
   705     val induct_cases = quasi_unambiguous_case_names (maps (map name_of_ctr) ctrss);
   706     val induct_case_names_attr = Attrib.internal (K (Rule_Cases.case_names induct_cases));
   707 
   708     val xctrss = map2 (map2 (curry Term.list_comb)) ctrss xsss;
   709 
   710     fun mk_iter_thmss (_, x_Tssss, fss, _) iters iter_defs ctor_iter_thms =
   711       let
   712         val fiters = map (lists_bmoc fss) iters;
   713 
   714         fun mk_goal fss fiter xctr f xs fxs =
   715           fold_rev (fold_rev Logic.all) (xs :: fss)
   716             (mk_Trueprop_eq (fiter $ xctr, Term.list_comb (f, fxs)));
   717 
   718         fun maybe_tick (T, U) u f =
   719           if try (fst o HOLogic.dest_prodT) U = SOME T then
   720             Term.lambda u (HOLogic.mk_prod (u, f $ u))
   721           else
   722             f;
   723 
   724         fun build_iter (x as Free (_, T)) U =
   725           if T = U then
   726             x
   727           else
   728             build_map lthy (indexify (perhaps (try (snd o HOLogic.dest_prodT)) o snd) Cs
   729               (fn kk => fn TU => maybe_tick TU (nth us kk) (nth fiters kk))) (T, U) $ x;
   730 
   731         val fxsss = map2 (map2 (flat_rec_arg_args oo map2 (map o build_iter))) xsss x_Tssss;
   732 
   733         val goalss = map5 (map4 o mk_goal fss) fiters xctrss fss xsss fxsss;
   734 
   735         val tacss =
   736           map2 (map o mk_iter_tac pre_map_defs (nested_map_idents @ nesting_map_idents) iter_defs)
   737             ctor_iter_thms ctr_defss;
   738 
   739         fun prove goal tac =
   740           Goal.prove_sorry lthy [] [] goal (tac o #context)
   741           |> Thm.close_derivation;
   742       in
   743         map2 (map2 prove) goalss tacss
   744       end;
   745 
   746     val fold_thmss = mk_iter_thmss fold_args_types folds fold_defs (map un_fold_of ctor_iter_thmss);
   747     val rec_thmss = mk_iter_thmss rec_args_types recs rec_defs (map co_rec_of ctor_iter_thmss);
   748   in
   749     ((induct_thms, induct_thm, [induct_case_names_attr]),
   750      (fold_thmss, code_simp_attrs), (rec_thmss, code_simp_attrs))
   751   end;
   752 
   753 fun derive_coinduct_coiters_thms_for_types pre_bnfs (z, cs, cpss,
   754       coiters_args_types as [((pgss, crgsss), _), ((phss, cshsss), _)])
   755     dtor_coinduct dtor_injects dtor_ctors dtor_coiter_thmss nesting_bnfs fpTs Cs kss mss ns
   756     ctr_defss ctr_sugars coiterss coiter_defss export_args lthy =
   757   let
   758     fun mk_ctor_dtor_coiter_thm dtor_inject dtor_ctor coiter =
   759       iffD1 OF [dtor_inject, trans OF [coiter, dtor_ctor RS sym]];
   760 
   761     val ctor_dtor_coiter_thmss =
   762       map3 (map oo mk_ctor_dtor_coiter_thm) dtor_injects dtor_ctors dtor_coiter_thmss;
   763 
   764     val coiterss' = transpose coiterss;
   765     val coiter_defss' = transpose coiter_defss;
   766 
   767     val [unfold_defs, corec_defs] = coiter_defss';
   768 
   769     val nn = length pre_bnfs;
   770 
   771     val pre_map_defs = map map_def_of_bnf pre_bnfs;
   772     val pre_rel_defs = map rel_def_of_bnf pre_bnfs;
   773     val nesting_map_idents = map (unfold_thms lthy [id_def] o map_id0_of_bnf) nesting_bnfs;
   774     val nesting_rel_eqs = map rel_eq_of_bnf nesting_bnfs;
   775 
   776     val fp_b_names = map base_name_of_typ fpTs;
   777 
   778     val ctrss = map #ctrs ctr_sugars;
   779     val discss = map #discs ctr_sugars;
   780     val selsss = map #selss ctr_sugars;
   781     val exhausts = map #exhaust ctr_sugars;
   782     val disc_thmsss = map #disc_thmss ctr_sugars;
   783     val discIss = map #discIs ctr_sugars;
   784     val sel_thmsss = map #sel_thmss ctr_sugars;
   785 
   786     val (((rs, us'), vs'), names_lthy) =
   787       lthy
   788       |> mk_Frees "R" (map (fn T => mk_pred2T T T) fpTs)
   789       ||>> Variable.variant_fixes fp_b_names
   790       ||>> Variable.variant_fixes (map (suffix "'") fp_b_names);
   791 
   792     val us = map2 (curry Free) us' fpTs;
   793     val udiscss = map2 (map o rapp) us discss;
   794     val uselsss = map2 (map o map o rapp) us selsss;
   795 
   796     val vs = map2 (curry Free) vs' fpTs;
   797     val vdiscss = map2 (map o rapp) vs discss;
   798     val vselsss = map2 (map o map o rapp) vs selsss;
   799 
   800     val coinduct_thms_pairs =
   801       let
   802         val uvrs = map3 (fn r => fn u => fn v => r $ u $ v) rs us vs;
   803         val uv_eqs = map2 (curry HOLogic.mk_eq) us vs;
   804         val strong_rs =
   805           map4 (fn u => fn v => fn uvr => fn uv_eq =>
   806             fold_rev Term.lambda [u, v] (HOLogic.mk_disj (uvr, uv_eq))) us vs uvrs uv_eqs;
   807 
   808         (* TODO: generalize (cf. "build_map") *)
   809         fun build_rel rs' T =
   810           (case find_index (curry op = T) fpTs of
   811             ~1 =>
   812             if exists_subtype_in fpTs T then
   813               let
   814                 val Type (s, Ts) = T
   815                 val bnf = the (bnf_of lthy s);
   816                 val live = live_of_bnf bnf;
   817                 val rel = mk_rel live Ts Ts (rel_of_bnf bnf);
   818                 val Ts' = map domain_type (fst (strip_typeN live (fastype_of rel)));
   819               in Term.list_comb (rel, map (build_rel rs') Ts') end
   820             else
   821               HOLogic.eq_const T
   822           | kk => nth rs' kk);
   823 
   824         fun build_rel_app rs' usel vsel = fold rapp [usel, vsel] (build_rel rs' (fastype_of usel));
   825 
   826         fun mk_prem_ctr_concls rs' n k udisc usels vdisc vsels =
   827           (if k = n then [] else [HOLogic.mk_eq (udisc, vdisc)]) @
   828           (if null usels then
   829              []
   830            else
   831              [Library.foldr HOLogic.mk_imp (if n = 1 then [] else [udisc, vdisc],
   832                 Library.foldr1 HOLogic.mk_conj (map2 (build_rel_app rs') usels vsels))]);
   833 
   834         fun mk_prem_concl rs' n udiscs uselss vdiscs vselss =
   835           Library.foldr1 HOLogic.mk_conj
   836             (flat (map5 (mk_prem_ctr_concls rs' n) (1 upto n) udiscs uselss vdiscs vselss))
   837           handle List.Empty => @{term True};
   838 
   839         fun mk_prem rs' uvr u v n udiscs uselss vdiscs vselss =
   840           fold_rev Logic.all [u, v] (Logic.mk_implies (HOLogic.mk_Trueprop uvr,
   841             HOLogic.mk_Trueprop (mk_prem_concl rs' n udiscs uselss vdiscs vselss)));
   842 
   843         val concl =
   844           HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj
   845             (map3 (fn uvr => fn u => fn v => HOLogic.mk_imp (uvr, HOLogic.mk_eq (u, v)))
   846                uvrs us vs));
   847 
   848         fun mk_goal rs' =
   849           Logic.list_implies (map8 (mk_prem rs') uvrs us vs ns udiscss uselsss vdiscss vselsss,
   850             concl);
   851 
   852         val goals = map mk_goal [rs, strong_rs];
   853 
   854         fun prove dtor_coinduct' goal =
   855           Goal.prove_sorry lthy [] [] goal (fn {context = ctxt, ...} =>
   856             mk_coinduct_tac ctxt nesting_rel_eqs nn ns dtor_coinduct' pre_rel_defs dtor_ctors
   857               exhausts ctr_defss disc_thmsss sel_thmsss)
   858           |> singleton (Proof_Context.export names_lthy lthy)
   859           |> Thm.close_derivation;
   860 
   861         fun postproc nn thm =
   862           Thm.permute_prems 0 nn
   863             (if nn = 1 then thm RS mp else funpow nn (fn thm => reassoc_conjs (thm RS mp_conj)) thm)
   864           |> Drule.zero_var_indexes
   865           |> `(conj_dests nn);
   866 
   867         val rel_eqs = map rel_eq_of_bnf pre_bnfs;
   868         val rel_monos = map rel_mono_of_bnf pre_bnfs;
   869         val dtor_coinducts =
   870           [dtor_coinduct, mk_strong_coinduct_thm dtor_coinduct rel_eqs rel_monos lthy];
   871       in
   872         map2 (postproc nn oo prove) dtor_coinducts goals
   873       end;
   874 
   875     fun mk_coinduct_concls ms discs ctrs =
   876       let
   877         fun mk_disc_concl disc = [name_of_disc disc];
   878         fun mk_ctr_concl 0 _ = []
   879           | mk_ctr_concl _ ctor = [name_of_ctr ctor];
   880         val disc_concls = map mk_disc_concl (fst (split_last discs)) @ [[]];
   881         val ctr_concls = map2 mk_ctr_concl ms ctrs;
   882       in
   883         flat (map2 append disc_concls ctr_concls)
   884       end;
   885 
   886     val coinduct_cases = quasi_unambiguous_case_names (map (prefix EqN) fp_b_names);
   887     val coinduct_conclss =
   888       map3 (quasi_unambiguous_case_names ooo mk_coinduct_concls) mss discss ctrss;
   889 
   890     fun mk_maybe_not pos = not pos ? HOLogic.mk_not;
   891 
   892     val fcoiterss' as [gunfolds, hcorecs] =
   893       map2 (fn (pfss, _) => map (lists_bmoc pfss)) (map fst coiters_args_types) coiterss';
   894 
   895     val (unfold_thmss, corec_thmss, safe_unfold_thmss, safe_corec_thmss) =
   896       let
   897         fun mk_goal pfss c cps fcoiter n k ctr m cfs' =
   898           fold_rev (fold_rev Logic.all) ([c] :: pfss)
   899             (Logic.list_implies (seq_conds (HOLogic.mk_Trueprop oo mk_maybe_not) n k cps,
   900                mk_Trueprop_eq (fcoiter $ c, Term.list_comb (ctr, take m cfs'))));
   901 
   902         fun mk_U maybe_mk_sumT =
   903           typ_subst_nonatomic (map2 (fn C => fn fpT => (maybe_mk_sumT fpT C, fpT)) Cs fpTs);
   904 
   905         fun tack z_name (c, u) f =
   906           let val z = Free (z_name, mk_sumT (fastype_of u, fastype_of c)) in
   907             Term.lambda z (mk_sum_case (Term.lambda u u, Term.lambda c (f $ c)) $ z)
   908           end;
   909 
   910         fun build_coiter fcoiters maybe_mk_sumT maybe_tack cqf =
   911           let val T = fastype_of cqf in
   912             if exists_subtype_in Cs T then
   913               let val U = mk_U maybe_mk_sumT T in
   914                 build_map lthy (indexify snd fpTs (fn kk => fn _ =>
   915                   maybe_tack (nth cs kk, nth us kk) (nth fcoiters kk))) (T, U) $ cqf
   916               end
   917             else
   918               cqf
   919           end;
   920 
   921         val crgsss' = map (map (map (build_coiter (un_fold_of fcoiterss') (K I) (K I)))) crgsss;
   922         val cshsss' = map (map (map (build_coiter (co_rec_of fcoiterss') (curry mk_sumT) (tack z))))
   923           cshsss;
   924 
   925         val unfold_goalss = map8 (map4 oooo mk_goal pgss) cs cpss gunfolds ns kss ctrss mss crgsss';
   926         val corec_goalss = map8 (map4 oooo mk_goal phss) cs cpss hcorecs ns kss ctrss mss cshsss';
   927 
   928         val unfold_tacss =
   929           map3 (map oo mk_coiter_tac unfold_defs nesting_map_idents)
   930             (map un_fold_of ctor_dtor_coiter_thmss) pre_map_defs ctr_defss;
   931         val corec_tacss =
   932           map3 (map oo mk_coiter_tac corec_defs nesting_map_idents)
   933             (map co_rec_of ctor_dtor_coiter_thmss) pre_map_defs ctr_defss;
   934 
   935         fun prove goal tac =
   936           Goal.prove_sorry lthy [] [] goal (tac o #context)
   937           |> Thm.close_derivation;
   938 
   939         val unfold_thmss = map2 (map2 prove) unfold_goalss unfold_tacss;
   940         val corec_thmss =
   941           map2 (map2 prove) corec_goalss corec_tacss
   942           |> map (map (unfold_thms lthy @{thms sum_case_if}));
   943 
   944         val unfold_safesss = map2 (map2 (map2 (curry op =))) crgsss' crgsss;
   945         val corec_safesss = map2 (map2 (map2 (curry op =))) cshsss' cshsss;
   946 
   947         val filter_safesss =
   948           map2 (map_filter (fn (safes, thm) => if forall I safes then SOME thm else NONE) oo
   949             curry (op ~~));
   950 
   951         val safe_unfold_thmss = filter_safesss unfold_safesss unfold_thmss;
   952         val safe_corec_thmss = filter_safesss corec_safesss corec_thmss;
   953       in
   954         (unfold_thmss, corec_thmss, safe_unfold_thmss, safe_corec_thmss)
   955       end;
   956 
   957     val (disc_unfold_iff_thmss, disc_corec_iff_thmss) =
   958       let
   959         fun mk_goal c cps fcoiter n k disc =
   960           mk_Trueprop_eq (disc $ (fcoiter $ c),
   961             if n = 1 then @{const True}
   962             else Library.foldr1 HOLogic.mk_conj (seq_conds mk_maybe_not n k cps));
   963 
   964         val unfold_goalss = map6 (map2 oooo mk_goal) cs cpss gunfolds ns kss discss;
   965         val corec_goalss = map6 (map2 oooo mk_goal) cs cpss hcorecs ns kss discss;
   966 
   967         fun mk_case_split' cp = Drule.instantiate' [] [SOME (certify lthy cp)] @{thm case_split};
   968 
   969         val case_splitss' = map (map mk_case_split') cpss;
   970 
   971         val unfold_tacss =
   972           map3 (map oo mk_disc_coiter_iff_tac) case_splitss' unfold_thmss disc_thmsss;
   973         val corec_tacss =
   974           map3 (map oo mk_disc_coiter_iff_tac) case_splitss' corec_thmss disc_thmsss;
   975 
   976         fun prove goal tac =
   977           Goal.prove_sorry lthy [] [] goal (tac o #context)
   978           |> singleton export_args
   979           |> singleton (Proof_Context.export names_lthy lthy)
   980           |> Thm.close_derivation;
   981 
   982         fun proves [_] [_] = []
   983           | proves goals tacs = map2 prove goals tacs;
   984       in
   985         (map2 proves unfold_goalss unfold_tacss, map2 proves corec_goalss corec_tacss)
   986       end;
   987 
   988     val is_triv_discI = is_triv_implies orf is_concl_refl;
   989 
   990     fun mk_disc_coiter_thms coiters discIs =
   991       map (op RS) (filter_out (is_triv_discI o snd) (coiters ~~ discIs));
   992 
   993     val disc_unfold_thmss = map2 mk_disc_coiter_thms unfold_thmss discIss;
   994     val disc_corec_thmss = map2 mk_disc_coiter_thms corec_thmss discIss;
   995 
   996     fun mk_sel_coiter_thm coiter_thm sel sel_thm =
   997       let
   998         val (domT, ranT) = dest_funT (fastype_of sel);
   999         val arg_cong' =
  1000           Drule.instantiate' (map (SOME o certifyT lthy) [domT, ranT])
  1001             [NONE, NONE, SOME (certify lthy sel)] arg_cong
  1002           |> Thm.varifyT_global;
  1003         val sel_thm' = sel_thm RSN (2, trans);
  1004       in
  1005         coiter_thm RS arg_cong' RS sel_thm'
  1006       end;
  1007 
  1008     fun mk_sel_coiter_thms coiter_thmss =
  1009       map3 (map3 (map2 o mk_sel_coiter_thm)) coiter_thmss selsss sel_thmsss;
  1010 
  1011     val sel_unfold_thmsss = mk_sel_coiter_thms unfold_thmss;
  1012     val sel_corec_thmsss = mk_sel_coiter_thms corec_thmss;
  1013 
  1014     val coinduct_consumes_attr = Attrib.internal (K (Rule_Cases.consumes nn));
  1015     val coinduct_case_names_attr = Attrib.internal (K (Rule_Cases.case_names coinduct_cases));
  1016     val coinduct_case_concl_attrs =
  1017       map2 (fn casex => fn concls =>
  1018           Attrib.internal (K (Rule_Cases.case_conclusion (casex, concls))))
  1019         coinduct_cases coinduct_conclss;
  1020     val coinduct_case_attrs =
  1021       coinduct_consumes_attr :: coinduct_case_names_attr :: coinduct_case_concl_attrs;
  1022   in
  1023     ((coinduct_thms_pairs, coinduct_case_attrs),
  1024      (unfold_thmss, corec_thmss, []),
  1025      (safe_unfold_thmss, safe_corec_thmss),
  1026      (disc_unfold_thmss, disc_corec_thmss, simp_attrs),
  1027      (disc_unfold_iff_thmss, disc_corec_iff_thmss, simp_attrs),
  1028      (sel_unfold_thmsss, sel_corec_thmsss, simp_attrs))
  1029   end;
  1030 
  1031 fun define_co_datatypes prepare_constraint prepare_typ prepare_term fp construct_fp
  1032     (wrap_opts as (no_discs_sels, rep_compat), specs) no_defs_lthy0 =
  1033   let
  1034     (* TODO: sanity checks on arguments *)
  1035 
  1036     val _ = if fp = Greatest_FP andalso no_discs_sels then
  1037         error "Cannot define codatatypes without discriminators and selectors"
  1038       else
  1039         ();
  1040 
  1041     fun qualify mandatory fp_b_name =
  1042       Binding.qualify mandatory fp_b_name o (rep_compat ? Binding.qualify false rep_compat_prefix);
  1043 
  1044     val nn = length specs;
  1045     val fp_bs = map type_binding_of specs;
  1046     val fp_b_names = map Binding.name_of fp_bs;
  1047     val fp_common_name = mk_common_name fp_b_names;
  1048     val map_bs = map map_binding_of specs;
  1049     val rel_bs = map rel_binding_of specs;
  1050 
  1051     fun prepare_type_arg (_, (ty, c)) =
  1052       let val TFree (s, _) = prepare_typ no_defs_lthy0 ty in
  1053         TFree (s, prepare_constraint no_defs_lthy0 c)
  1054       end;
  1055 
  1056     val Ass0 = map (map prepare_type_arg o type_args_named_constrained_of) specs;
  1057     val unsorted_Ass0 = map (map (resort_tfree HOLogic.typeS)) Ass0;
  1058     val unsorted_As = Library.foldr1 merge_type_args unsorted_Ass0;
  1059     val num_As = length unsorted_As;
  1060     val set_bss = map (map fst o type_args_named_constrained_of) specs;
  1061 
  1062     val (((Bs0, Cs), Xs), no_defs_lthy) =
  1063       no_defs_lthy0
  1064       |> fold (Variable.declare_typ o resort_tfree dummyS) unsorted_As
  1065       |> mk_TFrees num_As
  1066       ||>> mk_TFrees nn
  1067       ||>> variant_tfrees fp_b_names;
  1068 
  1069     fun add_fake_type spec = Typedecl.basic_typedecl (type_binding_of spec, num_As, mixfix_of spec);
  1070 
  1071     val (fake_T_names, fake_lthy) = fold_map add_fake_type specs no_defs_lthy0;
  1072 
  1073     val qsoty = quote o Syntax.string_of_typ fake_lthy;
  1074 
  1075     val _ = (case duplicates (op =) unsorted_As of [] => ()
  1076       | A :: _ => error ("Duplicate type parameter " ^ qsoty A ^ " in " ^ co_prefix fp ^
  1077           "datatype specification"));
  1078 
  1079     val bad_args =
  1080       map (Logic.type_map (singleton (Variable.polymorphic no_defs_lthy0))) unsorted_As
  1081       |> filter_out Term.is_TVar;
  1082     val _ = null bad_args orelse
  1083       error ("Locally fixed type argument " ^ qsoty (hd bad_args) ^ " in " ^ co_prefix fp ^
  1084         "datatype specification");
  1085 
  1086     val mixfixes = map mixfix_of specs;
  1087 
  1088     val _ = (case duplicates Binding.eq_name fp_bs of [] => ()
  1089       | b :: _ => error ("Duplicate type name declaration " ^ quote (Binding.name_of b)));
  1090 
  1091     val ctr_specss = map ctr_specs_of specs;
  1092 
  1093     val disc_bindingss = map (map disc_of) ctr_specss;
  1094     val ctr_bindingss =
  1095       map2 (fn fp_b_name => map (qualify false fp_b_name o ctr_of)) fp_b_names ctr_specss;
  1096     val ctr_argsss = map (map args_of) ctr_specss;
  1097     val ctr_mixfixess = map (map ctr_mixfix_of) ctr_specss;
  1098 
  1099     val sel_bindingsss = map (map (map fst)) ctr_argsss;
  1100     val fake_ctr_Tsss0 = map (map (map (prepare_typ fake_lthy o snd))) ctr_argsss;
  1101     val raw_sel_defaultsss = map (map defaults_of) ctr_specss;
  1102 
  1103     val (As :: _) :: fake_ctr_Tsss =
  1104       burrow (burrow (Syntax.check_typs fake_lthy)) (Ass0 :: fake_ctr_Tsss0);
  1105     val As' = map dest_TFree As;
  1106 
  1107     val rhs_As' = fold (fold (fold Term.add_tfreesT)) fake_ctr_Tsss [];
  1108     val _ = (case subtract (op =) As' rhs_As' of [] => ()
  1109       | extras => error ("Extra type variables on right-hand side: " ^
  1110           commas (map (qsoty o TFree) extras)));
  1111 
  1112     val fake_Ts = map (fn s => Type (s, As)) fake_T_names;
  1113 
  1114     fun eq_fpT_check (T as Type (s, Ts)) (T' as Type (s', Ts')) =
  1115         s = s' andalso (Ts = Ts' orelse
  1116           error ("Wrong type arguments in " ^ co_prefix fp ^ "recursive type " ^ qsoty T ^
  1117             " (expected " ^ qsoty T' ^ ")"))
  1118       | eq_fpT_check _ _ = false;
  1119 
  1120     fun freeze_fp (T as Type (s, Ts)) =
  1121         (case find_index (eq_fpT_check T) fake_Ts of
  1122           ~1 => Type (s, map freeze_fp Ts)
  1123         | kk => nth Xs kk)
  1124       | freeze_fp T = T;
  1125 
  1126     val unfreeze_fp = Term.typ_subst_atomic (Xs ~~ fake_Ts);
  1127 
  1128     val ctrXs_Tsss = map (map (map freeze_fp)) fake_ctr_Tsss;
  1129     val ctrXs_sum_prod_Ts = map (mk_sumTN_balanced o map HOLogic.mk_tupleT) ctrXs_Tsss;
  1130 
  1131     val fp_eqs =
  1132       map dest_TFree Xs ~~ map (Term.typ_subst_atomic (As ~~ unsorted_As)) ctrXs_sum_prod_Ts;
  1133 
  1134     val rhsXs_As' = fold (fold (fold Term.add_tfreesT)) ctrXs_Tsss [];
  1135     val _ = (case subtract (op =) rhsXs_As' As' of [] => ()
  1136       | extras => List.app (fn extra => warning ("Unused type variable on right-hand side of " ^
  1137           co_prefix fp ^ "datatype definition: " ^ qsoty (TFree extra))) extras);
  1138 
  1139     val (pre_bnfs, (fp_res as {bnfs = fp_bnfs as any_fp_bnf :: _, ctors = ctors0, dtors = dtors0,
  1140            xtor_co_iterss = xtor_co_iterss0, xtor_co_induct, dtor_ctors, ctor_dtors, ctor_injects,
  1141            dtor_injects, xtor_map_thms, xtor_set_thmss, xtor_rel_thms, xtor_co_iter_thmss, ...},
  1142            lthy)) =
  1143       fp_bnf (construct_fp mixfixes map_bs rel_bs set_bss) fp_bs (map dest_TFree unsorted_As) fp_eqs
  1144         no_defs_lthy0
  1145       handle BAD_DEAD (X, X_backdrop) =>
  1146         (case X_backdrop of
  1147           Type (bad_tc, _) =>
  1148           let
  1149             val fake_T = qsoty (unfreeze_fp X);
  1150             val fake_T_backdrop = qsoty (unfreeze_fp X_backdrop);
  1151             fun register_hint () =
  1152               "\nUse the " ^ quote (fst (fst @{command_spec "bnf"})) ^ " command to register " ^
  1153               quote bad_tc ^ " as a bounded natural functor to allow nested (co)recursion through \
  1154               \it";
  1155           in
  1156             if is_some (bnf_of no_defs_lthy bad_tc) orelse
  1157                is_some (fp_sugar_of no_defs_lthy bad_tc) then
  1158               error ("Inadmissible " ^ co_prefix fp ^ "recursive occurrence of type " ^ fake_T ^
  1159                 " in type expression " ^ fake_T_backdrop)
  1160             else if is_some (Datatype_Data.get_info (Proof_Context.theory_of no_defs_lthy)
  1161                 bad_tc) then
  1162               error ("Unsupported " ^ co_prefix fp ^ "recursive occurrence of type " ^ fake_T ^
  1163                 " via the old-style datatype " ^ quote bad_tc ^ " in type expression " ^
  1164                 fake_T_backdrop ^ register_hint ())
  1165             else
  1166               error ("Unsupported " ^ co_prefix fp ^ "recursive occurrence of type " ^ fake_T ^
  1167                 " via type constructor " ^ quote bad_tc ^ " in type expression " ^ fake_T_backdrop ^
  1168                 register_hint ())
  1169           end);
  1170 
  1171     val time = time lthy;
  1172     val timer = time (Timer.startRealTimer ());
  1173 
  1174     val nesting_bnfs = nesty_bnfs lthy ctrXs_Tsss As;
  1175     val nested_bnfs = nesty_bnfs lthy ctrXs_Tsss Xs;
  1176 
  1177     val pre_map_defs = map map_def_of_bnf pre_bnfs;
  1178     val pre_set_defss = map set_defs_of_bnf pre_bnfs;
  1179     val pre_rel_defs = map rel_def_of_bnf pre_bnfs;
  1180     val nesting_set_maps = maps set_map_of_bnf nesting_bnfs;
  1181     val nested_set_maps = maps set_map_of_bnf nested_bnfs;
  1182 
  1183     val live = live_of_bnf any_fp_bnf;
  1184     val _ =
  1185       if live = 0 andalso exists (not o Binding.is_empty) (map_bs @ rel_bs) then
  1186         warning "Map function and relator names ignored"
  1187       else
  1188         ();
  1189 
  1190     val Bs =
  1191       map3 (fn alive => fn A as TFree (_, S) => fn B => if alive then resort_tfree S B else A)
  1192         (liveness_of_fp_bnf num_As any_fp_bnf) As Bs0;
  1193 
  1194     val B_ify = Term.typ_subst_atomic (As ~~ Bs);
  1195 
  1196     val ctors = map (mk_ctor As) ctors0;
  1197     val dtors = map (mk_dtor As) dtors0;
  1198 
  1199     val fpTs = map (domain_type o fastype_of) dtors;
  1200 
  1201     fun massage_simple_notes base =
  1202       filter_out (null o #2)
  1203       #> map (fn (thmN, thms, attrs) =>
  1204         ((qualify true base (Binding.name thmN), attrs), [(thms, [])]));
  1205 
  1206     val massage_multi_notes =
  1207       maps (fn (thmN, thmss, attrs) =>
  1208         if forall null thmss then
  1209           []
  1210         else
  1211           map3 (fn fp_b_name => fn Type (T_name, _) => fn thms =>
  1212               ((qualify true fp_b_name (Binding.name thmN), attrs T_name), [(thms, [])]))
  1213             fp_b_names fpTs thmss);
  1214 
  1215     val ctr_Tsss = map (map (map (Term.typ_subst_atomic (Xs ~~ fpTs)))) ctrXs_Tsss;
  1216     val ns = map length ctr_Tsss;
  1217     val kss = map (fn n => 1 upto n) ns;
  1218     val mss = map (map length) ctr_Tsss;
  1219 
  1220     val ((xtor_co_iterss, iters_args_types, coiters_args_types), lthy') =
  1221       mk_co_iters_prelims fp fpTs Cs ns mss xtor_co_iterss0 lthy;
  1222 
  1223     fun define_ctrs_dtrs_for_type (((((((((((((((((((((((fp_bnf, fp_b), fpT), ctor), dtor),
  1224             xtor_co_iters), ctor_dtor), dtor_ctor), ctor_inject), pre_map_def), pre_set_defs),
  1225           pre_rel_def), fp_map_thm), fp_set_thms), fp_rel_thm), n), ks), ms), ctr_bindings),
  1226         ctr_mixfixes), ctr_Tss), disc_bindings), sel_bindingss), raw_sel_defaultss) no_defs_lthy =
  1227       let
  1228         val fp_b_name = Binding.name_of fp_b;
  1229 
  1230         val dtorT = domain_type (fastype_of ctor);
  1231         val ctr_prod_Ts = map HOLogic.mk_tupleT ctr_Tss;
  1232         val ctr_sum_prod_T = mk_sumTN_balanced ctr_prod_Ts;
  1233 
  1234         val ((((w, xss), yss), u'), names_lthy) =
  1235           no_defs_lthy
  1236           |> yield_singleton (mk_Frees "w") dtorT
  1237           ||>> mk_Freess "x" ctr_Tss
  1238           ||>> mk_Freess "y" (map (map B_ify) ctr_Tss)
  1239           ||>> yield_singleton Variable.variant_fixes fp_b_name;
  1240 
  1241         val u = Free (u', fpT);
  1242 
  1243         val tuple_xs = map HOLogic.mk_tuple xss;
  1244         val tuple_ys = map HOLogic.mk_tuple yss;
  1245 
  1246         val ctr_rhss =
  1247           map3 (fn k => fn xs => fn tuple_x => fold_rev Term.lambda xs (ctor $
  1248             mk_InN_balanced ctr_sum_prod_T n tuple_x k)) ks xss tuple_xs;
  1249 
  1250         val ((raw_ctrs, raw_ctr_defs), (lthy', lthy)) = no_defs_lthy
  1251           |> apfst split_list o fold_map3 (fn b => fn mx => fn rhs =>
  1252               Local_Theory.define ((b, mx), ((Thm.def_binding b, []), rhs)) #>> apsnd snd)
  1253             ctr_bindings ctr_mixfixes ctr_rhss
  1254           ||> `Local_Theory.restore;
  1255 
  1256         val phi = Proof_Context.export_morphism lthy lthy';
  1257 
  1258         val ctr_defs = map (Morphism.thm phi) raw_ctr_defs;
  1259         val ctr_defs' =
  1260           map2 (fn m => fn def => mk_unabs_def m (def RS meta_eq_to_obj_eq)) ms ctr_defs;
  1261 
  1262         val ctrs0 = map (Morphism.term phi) raw_ctrs;
  1263         val ctrs = map (mk_ctr As) ctrs0;
  1264 
  1265         fun wrap_ctrs lthy =
  1266           let
  1267             fun exhaust_tac {context = ctxt, prems = _} =
  1268               let
  1269                 val ctor_iff_dtor_thm =
  1270                   let
  1271                     val goal =
  1272                       fold_rev Logic.all [w, u]
  1273                         (mk_Trueprop_eq (HOLogic.mk_eq (u, ctor $ w), HOLogic.mk_eq (dtor $ u, w)));
  1274                   in
  1275                     Goal.prove_sorry lthy [] [] goal (fn {context = ctxt, ...} =>
  1276                       mk_ctor_iff_dtor_tac ctxt (map (SOME o certifyT lthy) [dtorT, fpT])
  1277                         (certify lthy ctor) (certify lthy dtor) ctor_dtor dtor_ctor)
  1278                     |> Thm.close_derivation
  1279                     |> Morphism.thm phi
  1280                   end;
  1281 
  1282                 val sumEN_thm' =
  1283                   unfold_thms lthy @{thms unit_all_eq1}
  1284                     (Drule.instantiate' (map (SOME o certifyT lthy) ctr_prod_Ts) []
  1285                        (mk_sumEN_balanced n))
  1286                   |> Morphism.thm phi;
  1287               in
  1288                 mk_exhaust_tac ctxt n ctr_defs ctor_iff_dtor_thm sumEN_thm'
  1289               end;
  1290 
  1291             val inject_tacss =
  1292               map2 (fn 0 => K [] | _ => fn ctr_def => [fn {context = ctxt, ...} =>
  1293                 mk_inject_tac ctxt ctr_def ctor_inject]) ms ctr_defs;
  1294 
  1295             val half_distinct_tacss =
  1296               map (map (fn (def, def') => fn {context = ctxt, ...} =>
  1297                 mk_half_distinct_tac ctxt ctor_inject [def, def'])) (mk_half_pairss (`I ctr_defs));
  1298 
  1299             val tacss = [exhaust_tac] :: inject_tacss @ half_distinct_tacss;
  1300 
  1301             val sel_defaultss = map (map (apsnd (prepare_term lthy))) raw_sel_defaultss
  1302           in
  1303             wrap_free_constructors tacss (((wrap_opts, ctrs0), standard_binding), (disc_bindings,
  1304               (sel_bindingss, sel_defaultss))) lthy
  1305           end;
  1306 
  1307         fun derive_maps_sets_rels (ctr_sugar, lthy) =
  1308           if live = 0 then
  1309             ((([], [], [], []), ctr_sugar), lthy)
  1310           else
  1311             let
  1312               val rel_flip = rel_flip_of_bnf fp_bnf;
  1313               val nones = replicate live NONE;
  1314 
  1315               val ctor_cong =
  1316                 if fp = Least_FP then
  1317                   Drule.dummy_thm
  1318                 else
  1319                   let val ctor' = mk_ctor Bs ctor in
  1320                     cterm_instantiate_pos [NONE, NONE, SOME (certify lthy ctor')] arg_cong
  1321                   end;
  1322 
  1323               fun mk_cIn ify =
  1324                 certify lthy o (fp = Greatest_FP ? curry (op $) (map_types ify ctor)) oo
  1325                 mk_InN_balanced (ify ctr_sum_prod_T) n;
  1326 
  1327               val cxIns = map2 (mk_cIn I) tuple_xs ks;
  1328               val cyIns = map2 (mk_cIn B_ify) tuple_ys ks;
  1329 
  1330               fun mk_map_thm ctr_def' cxIn =
  1331                 fold_thms lthy [ctr_def']
  1332                   (unfold_thms lthy (pre_map_def ::
  1333                        (if fp = Least_FP then [] else [ctor_dtor, dtor_ctor]) @ sum_prod_thms_map)
  1334                      (cterm_instantiate_pos (nones @ [SOME cxIn])
  1335                         (if fp = Least_FP then fp_map_thm else fp_map_thm RS ctor_cong)))
  1336                 |> singleton (Proof_Context.export names_lthy no_defs_lthy);
  1337 
  1338               fun mk_set_thm fp_set_thm ctr_def' cxIn =
  1339                 fold_thms lthy [ctr_def']
  1340                   (unfold_thms lthy (pre_set_defs @ nested_set_maps @ nesting_set_maps @
  1341                        (if fp = Least_FP then [] else [dtor_ctor]) @ sum_prod_thms_set)
  1342                      (cterm_instantiate_pos [SOME cxIn] fp_set_thm))
  1343                 |> singleton (Proof_Context.export names_lthy no_defs_lthy);
  1344 
  1345               fun mk_set_thms fp_set_thm = map2 (mk_set_thm fp_set_thm) ctr_defs' cxIns;
  1346 
  1347               val map_thms = map2 mk_map_thm ctr_defs' cxIns;
  1348               val set_thmss = map mk_set_thms fp_set_thms;
  1349 
  1350               val rel_infos = (ctr_defs' ~~ cxIns, ctr_defs' ~~ cyIns);
  1351 
  1352               fun mk_rel_thm postproc ctr_defs' cxIn cyIn =
  1353                 fold_thms lthy ctr_defs'
  1354                   (unfold_thms lthy (@{thm Inl_Inr_False} :: pre_rel_def ::
  1355                        (if fp = Least_FP then [] else [dtor_ctor]) @ sum_prod_thms_rel)
  1356                      (cterm_instantiate_pos (nones @ [SOME cxIn, SOME cyIn]) fp_rel_thm))
  1357                 |> postproc
  1358                 |> singleton (Proof_Context.export names_lthy no_defs_lthy);
  1359 
  1360               fun mk_rel_inject_thm ((ctr_def', cxIn), (_, cyIn)) =
  1361                 mk_rel_thm (unfold_thms lthy @{thms eq_sym_Unity_conv}) [ctr_def'] cxIn cyIn;
  1362 
  1363               val rel_inject_thms = map mk_rel_inject_thm (op ~~ rel_infos);
  1364 
  1365               fun mk_half_rel_distinct_thm ((xctr_def', cxIn), (yctr_def', cyIn)) =
  1366                 mk_rel_thm (fn thm => thm RS @{thm eq_False[THEN iffD1]}) [xctr_def', yctr_def']
  1367                   cxIn cyIn;
  1368 
  1369               fun mk_other_half_rel_distinct_thm thm =
  1370                 flip_rels lthy live thm
  1371                 RS (rel_flip RS sym RS @{thm arg_cong[of _ _ Not]} RS iffD2);
  1372 
  1373               val half_rel_distinct_thmss =
  1374                 map (map mk_half_rel_distinct_thm) (mk_half_pairss rel_infos);
  1375               val other_half_rel_distinct_thmss =
  1376                 map (map mk_other_half_rel_distinct_thm) half_rel_distinct_thmss;
  1377               val (rel_distinct_thms, _) =
  1378                 join_halves n half_rel_distinct_thmss other_half_rel_distinct_thmss;
  1379 
  1380               val notes =
  1381                 [(mapN, map_thms, code_simp_attrs),
  1382                  (rel_distinctN, rel_distinct_thms, code_simp_attrs),
  1383                  (rel_injectN, rel_inject_thms, code_simp_attrs),
  1384                  (setsN, flat set_thmss, code_simp_attrs)]
  1385                 |> massage_simple_notes fp_b_name;
  1386             in
  1387               (((map_thms, rel_inject_thms, rel_distinct_thms, set_thmss), ctr_sugar),
  1388                lthy |> Local_Theory.notes notes |> snd)
  1389             end;
  1390 
  1391         fun mk_binding suf = qualify false fp_b_name (Binding.suffix_name ("_" ^ suf) fp_b);
  1392 
  1393         fun massage_res (((maps_sets_rels, ctr_sugar), co_iter_res), lthy) =
  1394           (((maps_sets_rels, (ctrs, xss, ctr_defs, ctr_sugar)), co_iter_res), lthy);
  1395       in
  1396         (wrap_ctrs
  1397          #> derive_maps_sets_rels
  1398          ##>>
  1399            (if fp = Least_FP then define_iters [foldN, recN] (the iters_args_types)
  1400             else define_coiters [unfoldN, corecN] (the coiters_args_types))
  1401              mk_binding fpTs Cs xtor_co_iters
  1402          #> massage_res, lthy')
  1403       end;
  1404 
  1405     fun wrap_types_etc (wrap_types_etcs, lthy) =
  1406       fold_map I wrap_types_etcs lthy
  1407       |>> apsnd split_list o apfst (apsnd split_list4 o apfst split_list4 o split_list)
  1408         o split_list;
  1409 
  1410     val mk_simp_thmss =
  1411       map7 (fn {injects, distincts, case_thms, ...} => fn un_folds => fn co_recs =>
  1412         fn mapsx => fn rel_injects => fn rel_distincts => fn setss =>
  1413           injects @ distincts @ case_thms @ co_recs @ un_folds @ mapsx @ rel_injects
  1414           @ rel_distincts @ flat setss);
  1415 
  1416     fun derive_and_note_induct_iters_thms_for_types
  1417         ((((mapsx, rel_injects, rel_distincts, setss), (ctrss, _, ctr_defss, ctr_sugars)),
  1418           (iterss, iter_defss)), lthy) =
  1419       let
  1420         val ((induct_thms, induct_thm, induct_attrs), (fold_thmss, fold_attrs),
  1421              (rec_thmss, rec_attrs)) =
  1422           derive_induct_iters_thms_for_types pre_bnfs (the iters_args_types) xtor_co_induct
  1423             xtor_co_iter_thmss nesting_bnfs nested_bnfs fpTs Cs Xs ctrXs_Tsss ctrss ctr_defss iterss
  1424             iter_defss lthy;
  1425 
  1426         val induct_type_attr = Attrib.internal o K o Induct.induct_type;
  1427 
  1428         val simp_thmss =
  1429           mk_simp_thmss ctr_sugars fold_thmss rec_thmss mapsx rel_injects rel_distincts setss;
  1430 
  1431         val common_notes =
  1432           (if nn > 1 then [(inductN, [induct_thm], induct_attrs)] else [])
  1433           |> massage_simple_notes fp_common_name;
  1434 
  1435         val notes =
  1436           [(foldN, fold_thmss, K fold_attrs),
  1437            (inductN, map single induct_thms, fn T_name => induct_attrs @ [induct_type_attr T_name]),
  1438            (recN, rec_thmss, K rec_attrs),
  1439            (simpsN, simp_thmss, K [])]
  1440           |> massage_multi_notes;
  1441       in
  1442         lthy
  1443         |> Local_Theory.notes (common_notes @ notes) |> snd
  1444         |> register_fp_sugars Least_FP pre_bnfs nested_bnfs nesting_bnfs fp_res ctr_defss ctr_sugars
  1445           iterss [induct_thm] (transpose [fold_thmss, rec_thmss]) [] []
  1446       end;
  1447 
  1448     fun derive_and_note_coinduct_coiters_thms_for_types
  1449         ((((mapsx, rel_injects, rel_distincts, setss), (_, _, ctr_defss, ctr_sugars)),
  1450           (coiterss, coiter_defss)), lthy) =
  1451       let
  1452         val (([(coinduct_thms, coinduct_thm), (strong_coinduct_thms, strong_coinduct_thm)],
  1453               coinduct_attrs),
  1454              (unfold_thmss, corec_thmss, coiter_attrs),
  1455              (safe_unfold_thmss, safe_corec_thmss),
  1456              (disc_unfold_thmss, disc_corec_thmss, disc_coiter_attrs),
  1457              (disc_unfold_iff_thmss, disc_corec_iff_thmss, disc_coiter_iff_attrs),
  1458              (sel_unfold_thmsss, sel_corec_thmsss, sel_coiter_attrs)) =
  1459           derive_coinduct_coiters_thms_for_types pre_bnfs (the coiters_args_types) xtor_co_induct
  1460             dtor_injects dtor_ctors xtor_co_iter_thmss nesting_bnfs fpTs Cs kss mss ns ctr_defss
  1461             ctr_sugars coiterss coiter_defss (Proof_Context.export lthy' no_defs_lthy) lthy;
  1462 
  1463         val sel_unfold_thmss = map flat sel_unfold_thmsss;
  1464         val sel_corec_thmss = map flat sel_corec_thmsss;
  1465 
  1466         val coinduct_type_attr = Attrib.internal o K o Induct.coinduct_type;
  1467 
  1468         fun flat_coiter_thms coiters disc_coiters sel_coiters =
  1469           coiters @ disc_coiters @ sel_coiters;
  1470 
  1471         val simp_thmss =
  1472           mk_simp_thmss ctr_sugars
  1473             (map3 flat_coiter_thms safe_unfold_thmss disc_unfold_thmss sel_unfold_thmss)
  1474             (map3 flat_coiter_thms safe_corec_thmss disc_corec_thmss sel_corec_thmss)
  1475             mapsx rel_injects rel_distincts setss;
  1476 
  1477         val anonymous_notes =
  1478           [(flat safe_unfold_thmss @ flat safe_corec_thmss, simp_attrs)]
  1479           |> map (fn (thms, attrs) => ((Binding.empty, attrs), [(thms, [])]));
  1480 
  1481         val common_notes =
  1482           (if nn > 1 then
  1483              [(coinductN, [coinduct_thm], coinduct_attrs),
  1484               (strong_coinductN, [strong_coinduct_thm], coinduct_attrs)]
  1485            else
  1486              [])
  1487           |> massage_simple_notes fp_common_name;
  1488 
  1489         val notes =
  1490           [(coinductN, map single coinduct_thms,
  1491             fn T_name => coinduct_attrs @ [coinduct_type_attr T_name]),
  1492            (corecN, corec_thmss, K coiter_attrs),
  1493            (disc_corecN, disc_corec_thmss, K disc_coiter_attrs),
  1494            (disc_corec_iffN, disc_corec_iff_thmss, K disc_coiter_iff_attrs),
  1495            (disc_unfoldN, disc_unfold_thmss, K disc_coiter_attrs),
  1496            (disc_unfold_iffN, disc_unfold_iff_thmss, K disc_coiter_iff_attrs),
  1497            (sel_corecN, sel_corec_thmss, K sel_coiter_attrs),
  1498            (sel_unfoldN, sel_unfold_thmss, K sel_coiter_attrs),
  1499            (simpsN, simp_thmss, K []),
  1500            (strong_coinductN, map single strong_coinduct_thms, K coinduct_attrs),
  1501            (unfoldN, unfold_thmss, K coiter_attrs)]
  1502           |> massage_multi_notes;
  1503       in
  1504         lthy
  1505         |> Local_Theory.notes (anonymous_notes @ common_notes @ notes) |> snd
  1506         |> register_fp_sugars Greatest_FP pre_bnfs nested_bnfs nesting_bnfs fp_res ctr_defss
  1507           ctr_sugars coiterss [coinduct_thm, strong_coinduct_thm]
  1508           (transpose [unfold_thmss, corec_thmss]) (transpose [disc_unfold_thmss, disc_corec_thmss])
  1509           (transpose [sel_unfold_thmsss, sel_corec_thmsss])
  1510       end;
  1511 
  1512     val lthy'' = lthy'
  1513       |> fold_map define_ctrs_dtrs_for_type (fp_bnfs ~~ fp_bs ~~ fpTs ~~ ctors ~~ dtors ~~
  1514         xtor_co_iterss ~~ ctor_dtors ~~ dtor_ctors ~~ ctor_injects ~~ pre_map_defs ~~
  1515         pre_set_defss ~~ pre_rel_defs ~~ xtor_map_thms ~~ xtor_set_thmss ~~ xtor_rel_thms ~~ ns ~~
  1516         kss ~~ mss ~~ ctr_bindingss ~~ ctr_mixfixess ~~ ctr_Tsss ~~ disc_bindingss ~~
  1517         sel_bindingsss ~~ raw_sel_defaultsss)
  1518       |> wrap_types_etc
  1519       |> fp_case fp derive_and_note_induct_iters_thms_for_types
  1520            derive_and_note_coinduct_coiters_thms_for_types;
  1521 
  1522     val timer = time (timer ("Constructors, discriminators, selectors, etc., for the new " ^
  1523       co_prefix fp ^ "datatype"));
  1524   in
  1525     timer; lthy''
  1526   end;
  1527 
  1528 val co_datatypes = define_co_datatypes (K I) (K I) (K I);
  1529 
  1530 val co_datatype_cmd =
  1531   define_co_datatypes Typedecl.read_constraint Syntax.parse_typ Syntax.parse_term;
  1532 
  1533 val parse_ctr_arg =
  1534   @{keyword "("} |-- parse_binding_colon -- Parse.typ --| @{keyword ")"} ||
  1535   (Parse.typ >> pair Binding.empty);
  1536 
  1537 val parse_defaults =
  1538   @{keyword "("} |-- @{keyword "defaults"} |-- Scan.repeat parse_bound_term --| @{keyword ")"};
  1539 
  1540 val parse_type_arg_constrained =
  1541   Parse.type_ident -- Scan.option (@{keyword "::"} |-- Parse.!!! Parse.sort);
  1542 
  1543 val parse_type_arg_named_constrained = parse_opt_binding_colon -- parse_type_arg_constrained;
  1544 
  1545 val parse_type_args_named_constrained =
  1546   parse_type_arg_constrained >> (single o pair Binding.empty) ||
  1547   @{keyword "("} |-- Parse.!!! (Parse.list1 parse_type_arg_named_constrained --| @{keyword ")"}) ||
  1548   Scan.succeed [];
  1549 
  1550 val parse_map_rel_binding = Parse.short_ident --| @{keyword ":"} -- parse_binding;
  1551 
  1552 val no_map_rel = (Binding.empty, Binding.empty);
  1553 
  1554 (* "map" and "rel" are purposedly not registered as keywords, because they are short and nice names
  1555    that we don't want them to be highlighted everywhere. *)
  1556 fun extract_map_rel ("map", b) = apfst (K b)
  1557   | extract_map_rel ("rel", b) = apsnd (K b)
  1558   | extract_map_rel (s, _) = error ("Unknown label " ^ quote s ^ " (expected \"map\" or \"rel\")");
  1559 
  1560 val parse_map_rel_bindings =
  1561   @{keyword "("} |-- Scan.repeat parse_map_rel_binding --| @{keyword ")"}
  1562     >> (fn ps => fold extract_map_rel ps no_map_rel) ||
  1563   Scan.succeed no_map_rel;
  1564 
  1565 val parse_ctr_spec =
  1566   parse_opt_binding_colon -- parse_binding -- Scan.repeat parse_ctr_arg --
  1567   Scan.optional parse_defaults [] -- Parse.opt_mixfix;
  1568 
  1569 val parse_spec =
  1570   parse_type_args_named_constrained -- parse_binding -- parse_map_rel_bindings --
  1571   Parse.opt_mixfix -- (@{keyword "="} |-- Parse.enum1 "|" parse_ctr_spec);
  1572 
  1573 val parse_co_datatype = parse_wrap_free_constructors_options -- Parse.and_list1 parse_spec;
  1574 
  1575 fun parse_co_datatype_cmd fp construct_fp = parse_co_datatype >> co_datatype_cmd fp construct_fp;
  1576 
  1577 end;