src/HOL/Codatatype/Tools/bnf_fp_sugar.ML
author blanchet
Sat Sep 15 21:10:26 2012 +0200 (2012-09-15)
changeset 49389 da621dc65146
parent 49381 be09db8426cb
child 49392 e1f325ab9503
permissions -rw-r--r--
tuning
     1 (*  Title:      HOL/Codatatype/Tools/bnf_fp_sugar.ML
     2     Author:     Jasmin Blanchette, TU Muenchen
     3     Copyright   2012
     4 
     5 Sugared datatype and codatatype constructions.
     6 *)
     7 
     8 signature BNF_FP_SUGAR =
     9 sig
    10   val datatyp: bool ->
    11     (mixfix list -> (string * sort) list option -> binding list -> typ list * typ list list ->
    12       BNF_Def.BNF list -> local_theory ->
    13       (term list * term list * term list * term list * thm * thm list * thm list * thm list *
    14          thm list * thm list) * local_theory) ->
    15     bool * ((((typ * sort) list * binding) * mixfix) * ((((binding * binding) *
    16       (binding * typ) list) * (binding * term) list) * mixfix) list) list ->
    17     local_theory -> local_theory
    18   val parse_datatype_cmd: bool ->
    19     (mixfix list -> (string * sort) list option -> binding list -> typ list * typ list list ->
    20       BNF_Def.BNF list -> local_theory ->
    21       (term list * term list * term list * term list * thm * thm list * thm list * thm list *
    22          thm list * thm list) * local_theory) ->
    23     (local_theory -> local_theory) parser
    24 end;
    25 
    26 structure BNF_FP_Sugar : BNF_FP_SUGAR =
    27 struct
    28 
    29 open BNF_Util
    30 open BNF_Wrap
    31 open BNF_Def
    32 open BNF_FP_Util
    33 open BNF_FP_Sugar_Tactics
    34 
    35 val simp_attrs = @{attributes [simp]};
    36 
    37 fun split_list10 xs =
    38   (map #1 xs, map #2 xs, map #3 xs, map #4 xs, map #5 xs, map #6 xs, map #7 xs, map #8 xs,
    39    map #9 xs, map #10 xs);
    40 
    41 fun strip_map_type @{type_name fun} (Type (_, [T, Type (_, [T', T''])])) = ([T, T'], T'')
    42   | strip_map_type _ T = strip_type T;
    43 
    44 fun resort_tfree S (TFree (s, _)) = TFree (s, S);
    45 
    46 fun typ_subst inst (T as Type (s, Ts)) =
    47     (case AList.lookup (op =) inst T of
    48       NONE => Type (s, map (typ_subst inst) Ts)
    49     | SOME T' => T')
    50   | typ_subst inst T = the_default T (AList.lookup (op =) inst T);
    51 
    52 val lists_bmoc = fold (fn xs => fn t => Term.list_comb (t, xs));
    53 
    54 fun mk_tupled_fun x f xs = HOLogic.tupled_lambda x (Term.list_comb (f, xs));
    55 fun mk_uncurried_fun f xs = mk_tupled_fun (HOLogic.mk_tuple xs) f xs;
    56 fun mk_uncurried2_fun f xss =
    57   mk_tupled_fun (HOLogic.mk_tuple (map HOLogic.mk_tuple xss)) f (flat xss);
    58 
    59 fun tick v f = Term.lambda v (HOLogic.mk_prod (v, f $ v));
    60 
    61 fun tack z_name (c, v) f =
    62   let val z = Free (z_name, mk_sumT (fastype_of v, fastype_of c)) in
    63     Term.lambda z (mk_sum_case (Term.lambda v v, Term.lambda c (f $ c)) $ z)
    64   end;
    65 
    66 fun cannot_merge_types () = error "Mutually recursive types must have the same type parameters";
    67 
    68 fun merge_type_arg T T' = if T = T' then T else cannot_merge_types ();
    69 
    70 fun merge_type_args (As, As') =
    71   if length As = length As' then map2 merge_type_arg As As' else cannot_merge_types ();
    72 
    73 fun type_args_constrained_of (((cAs, _), _), _) = cAs;
    74 fun type_binding_of (((_, b), _), _) = b;
    75 fun mixfix_of ((_, mx), _) = mx;
    76 fun ctr_specs_of (_, ctr_specs) = ctr_specs;
    77 
    78 fun disc_of ((((disc, _), _), _), _) = disc;
    79 fun ctr_of ((((_, ctr), _), _), _) = ctr;
    80 fun args_of (((_, args), _), _) = args;
    81 fun defaults_of ((_, ds), _) = ds;
    82 fun ctr_mixfix_of (_, mx) = mx;
    83 
    84 fun define_datatype prepare_constraint prepare_typ prepare_term lfp construct (no_dests, specs)
    85     no_defs_lthy0 =
    86   let
    87     (* TODO: sanity checks on arguments *)
    88 
    89     val _ = if not lfp andalso no_dests then error "Cannot define destructor-less codatatypes"
    90       else ();
    91 
    92     val nn = length specs;
    93     val fp_bs = map type_binding_of specs;
    94     val fp_common_name = mk_common_name fp_bs;
    95 
    96     fun prepare_type_arg (ty, c) =
    97       let val TFree (s, _) = prepare_typ no_defs_lthy0 ty in
    98         TFree (s, prepare_constraint no_defs_lthy0 c)
    99       end;
   100 
   101     val Ass0 = map (map prepare_type_arg o type_args_constrained_of) specs;
   102     val unsorted_Ass0 = map (map (resort_tfree HOLogic.typeS)) Ass0;
   103     val unsorted_As = Library.foldr1 merge_type_args unsorted_Ass0;
   104 
   105     val ((Bs, Cs), no_defs_lthy) =
   106       no_defs_lthy0
   107       |> fold (Variable.declare_typ o resort_tfree dummyS) unsorted_As
   108       |> mk_TFrees nn
   109       ||>> mk_TFrees nn;
   110 
   111     (* TODO: cleaner handling of fake contexts, without "background_theory" *)
   112     (*the "perhaps o try" below helps gracefully handles the case where the new type is defined in a
   113       locale and shadows an existing global type*)
   114     val fake_thy =
   115       Theory.copy #> fold (fn spec => perhaps (try (Sign.add_type no_defs_lthy
   116         (type_binding_of spec, length (type_args_constrained_of spec), mixfix_of spec)))) specs;
   117     val fake_lthy = Proof_Context.background_theory fake_thy no_defs_lthy;
   118 
   119     fun mk_fake_T b =
   120       Type (fst (Term.dest_Type (Proof_Context.read_type_name fake_lthy true (Binding.name_of b))),
   121         unsorted_As);
   122 
   123     val fake_Ts = map mk_fake_T fp_bs;
   124 
   125     val mixfixes = map mixfix_of specs;
   126 
   127     val _ = (case duplicates Binding.eq_name fp_bs of [] => ()
   128       | b :: _ => error ("Duplicate type name declaration " ^ quote (Binding.name_of b)));
   129 
   130     val ctr_specss = map ctr_specs_of specs;
   131 
   132     val disc_bindingss = map (map disc_of) ctr_specss;
   133     val ctr_bindingss =
   134       map2 (fn fp_b => map (Binding.qualify false (Binding.name_of fp_b) o ctr_of))
   135         fp_bs ctr_specss;
   136     val ctr_argsss = map (map args_of) ctr_specss;
   137     val ctr_mixfixess = map (map ctr_mixfix_of) ctr_specss;
   138 
   139     val sel_bindingsss = map (map (map fst)) ctr_argsss;
   140     val fake_ctr_Tsss0 = map (map (map (prepare_typ fake_lthy o snd))) ctr_argsss;
   141     val raw_sel_defaultsss = map (map defaults_of) ctr_specss;
   142 
   143     val (As :: _) :: fake_ctr_Tsss =
   144       burrow (burrow (Syntax.check_typs fake_lthy)) (Ass0 :: fake_ctr_Tsss0);
   145 
   146     val _ = (case duplicates (op =) unsorted_As of [] => ()
   147       | A :: _ => error ("Duplicate type parameter " ^
   148           quote (Syntax.string_of_typ no_defs_lthy A)));
   149 
   150     val rhs_As' = fold (fold (fold Term.add_tfreesT)) fake_ctr_Tsss [];
   151     val _ = (case subtract (op =) (map dest_TFree As) rhs_As' of
   152         [] => ()
   153       | A' :: _ => error ("Extra type variable on right-hand side: " ^
   154           quote (Syntax.string_of_typ no_defs_lthy (TFree A'))));
   155 
   156     fun eq_fpT (T as Type (s, Us)) (Type (s', Us')) =
   157         s = s' andalso (Us = Us' orelse error ("Illegal occurrence of recursive type " ^
   158           quote (Syntax.string_of_typ fake_lthy T)))
   159       | eq_fpT _ _ = false;
   160 
   161     fun freeze_fp (T as Type (s, Us)) =
   162         (case find_index (eq_fpT T) fake_Ts of ~1 => Type (s, map freeze_fp Us) | j => nth Bs j)
   163       | freeze_fp T = T;
   164 
   165     val ctr_TsssBs = map (map (map freeze_fp)) fake_ctr_Tsss;
   166     val ctr_sum_prod_TsBs = map (mk_sumTN_balanced o map HOLogic.mk_tupleT) ctr_TsssBs;
   167 
   168     val fp_eqs =
   169       map dest_TFree Bs ~~ map (Term.typ_subst_atomic (As ~~ unsorted_As)) ctr_sum_prod_TsBs;
   170 
   171     val (pre_bnfs, ((unfs0, flds0, fp_iters0, fp_recs0, fp_induct, unf_flds, fld_unfs, fld_injects,
   172         fp_iter_thms, fp_rec_thms), lthy)) =
   173       fp_bnf construct fp_bs mixfixes (map dest_TFree unsorted_As) fp_eqs no_defs_lthy0;
   174 
   175     fun add_nesty_bnf_names Us =
   176       let
   177         fun add (Type (s, Ts)) ss =
   178             let val (needs, ss') = fold_map add Ts ss in
   179               if exists I needs then (true, insert (op =) s ss') else (false, ss')
   180             end
   181           | add T ss = (member (op =) Us T, ss);
   182       in snd oo add end;
   183 
   184     fun nesty_bnfs Us =
   185       map_filter (bnf_of lthy) (fold (fold (fold (add_nesty_bnf_names Us))) ctr_TsssBs []);
   186 
   187     val nesting_bnfs = nesty_bnfs As;
   188     val nested_bnfs = nesty_bnfs Bs;
   189 
   190     val timer = time (Timer.startRealTimer ());
   191 
   192     fun mk_unf_or_fld get_T Ts t =
   193       let val Type (_, Ts0) = get_T (fastype_of t) in
   194         Term.subst_atomic_types (Ts0 ~~ Ts) t
   195       end;
   196 
   197     val mk_unf = mk_unf_or_fld domain_type;
   198     val mk_fld = mk_unf_or_fld range_type;
   199 
   200     val unfs = map (mk_unf As) unfs0;
   201     val flds = map (mk_fld As) flds0;
   202 
   203     val fpTs = map (domain_type o fastype_of) unfs;
   204 
   205     val exists_fp_subtype = exists_subtype (member (op =) fpTs);
   206 
   207     val ctr_Tsss = map (map (map (Term.typ_subst_atomic (Bs ~~ fpTs)))) ctr_TsssBs;
   208     val ns = map length ctr_Tsss;
   209     val kss = map (fn n => 1 upto n) ns;
   210     val mss = map (map length) ctr_Tsss;
   211     val Css = map2 replicate ns Cs;
   212 
   213     fun mk_iter_like Ts Us t =
   214       let
   215         val (bindings, body) = strip_type (fastype_of t);
   216         val (f_Us, prebody) = split_last bindings;
   217         val Type (_, Ts0) = if lfp then prebody else body;
   218         val Us0 = distinct (op =) (map (if lfp then body_type else domain_type) f_Us);
   219       in
   220         Term.subst_atomic_types (Ts0 @ Us0 ~~ Ts @ Us) t
   221       end;
   222 
   223     val fp_iters as fp_iter1 :: _ = map (mk_iter_like As Cs) fp_iters0;
   224     val fp_recs as fp_rec1 :: _ = map (mk_iter_like As Cs) fp_recs0;
   225 
   226     val fp_iter_fun_Ts = fst (split_last (binder_types (fastype_of fp_iter1)));
   227     val fp_rec_fun_Ts = fst (split_last (binder_types (fastype_of fp_rec1)));
   228 
   229     val ((iter_only as (gss, _, _), rec_only as (hss, _, _)),
   230          (zs, cs, cpss, coiter_only as ((pgss, crgsss), _), corec_only as ((phss, cshsss), _))) =
   231       if lfp then
   232         let
   233           val y_Tsss =
   234             map3 (fn n => fn ms => map2 dest_tupleT ms o dest_sumTN_balanced n o domain_type)
   235               ns mss fp_iter_fun_Ts;
   236           val g_Tss = map2 (map2 (curry (op --->))) y_Tsss Css;
   237 
   238           val ((gss, ysss), _) =
   239             lthy
   240             |> mk_Freess "f" g_Tss
   241             ||>> mk_Freesss "x" y_Tsss;
   242           val yssss = map (map (map single)) ysss;
   243 
   244           fun dest_rec_prodT (T as Type (@{type_name prod}, Us as [_, U])) =
   245               if member (op =) Cs U then Us else [T]
   246             | dest_rec_prodT T = [T];
   247 
   248           val z_Tssss =
   249             map3 (fn n => fn ms => map2 (map dest_rec_prodT oo dest_tupleT) ms o
   250               dest_sumTN_balanced n o domain_type) ns mss fp_rec_fun_Ts;
   251           val h_Tss = map2 (map2 (fold_rev (curry (op --->)))) z_Tssss Css;
   252 
   253           val hss = map2 (map2 retype_free) h_Tss gss;
   254           val zssss_hd = map2 (map2 (map2 (retype_free o hd))) z_Tssss ysss;
   255           val (zssss_tl, _) =
   256             lthy
   257             |> mk_Freessss "y" (map (map (map tl)) z_Tssss);
   258           val zssss = map2 (map2 (map2 cons)) zssss_hd zssss_tl;
   259         in
   260           (((gss, g_Tss, yssss), (hss, h_Tss, zssss)),
   261            ([], [], [], (([], []), ([], [])), (([], []), ([], []))))
   262         end
   263       else
   264         let
   265           (*avoid "'a itself" arguments in coiterators and corecursors*)
   266           val mss' =  map (fn [0] => [1] | ms => ms) mss;
   267 
   268           val p_Tss = map2 (fn n => replicate (Int.max (0, n - 1)) o mk_predT) ns Cs;
   269 
   270           fun zip_predss_getterss qss fss = maps (op @) (qss ~~ fss);
   271 
   272           fun zip_preds_predsss_gettersss [] [qss] [fss] = zip_predss_getterss qss fss
   273             | zip_preds_predsss_gettersss (p :: ps) (qss :: qsss) (fss :: fsss) =
   274               p :: zip_predss_getterss qss fss @ zip_preds_predsss_gettersss ps qsss fsss;
   275 
   276           fun mk_types maybe_dest_sumT fun_Ts =
   277             let
   278               val f_sum_prod_Ts = map range_type fun_Ts;
   279               val f_prod_Tss = map2 dest_sumTN_balanced ns f_sum_prod_Ts;
   280               val f_Tssss =
   281                 map3 (fn C => map2 (map (map (curry (op -->) C) o maybe_dest_sumT) oo dest_tupleT))
   282                   Cs mss' f_prod_Tss;
   283               val q_Tssss =
   284                 map (map (map (fn [_] => [] | [_, C] => [mk_predT (domain_type C)]))) f_Tssss;
   285               val pf_Tss = map3 zip_preds_predsss_gettersss p_Tss q_Tssss f_Tssss;
   286             in (q_Tssss, f_sum_prod_Ts, f_Tssss, pf_Tss) end;
   287 
   288           val (r_Tssss, g_sum_prod_Ts, g_Tssss, pg_Tss) = mk_types single fp_iter_fun_Ts;
   289 
   290           val ((((Free (z, _), cs), pss), gssss), _) =
   291             lthy
   292             |> yield_singleton (mk_Frees "z") dummyT
   293             ||>> mk_Frees "a" Cs
   294             ||>> mk_Freess "p" p_Tss
   295             ||>> mk_Freessss "g" g_Tssss;
   296           val rssss = map (map (map (fn [] => []))) r_Tssss;
   297 
   298           fun dest_corec_sumT (T as Type (@{type_name sum}, Us as [_, U])) =
   299               if member (op =) Cs U then Us else [T]
   300             | dest_corec_sumT T = [T];
   301 
   302           val (s_Tssss, h_sum_prod_Ts, h_Tssss, ph_Tss) = mk_types dest_corec_sumT fp_rec_fun_Ts;
   303 
   304           val hssss_hd = map2 (map2 (map2 (fn T :: _ => fn [g] => retype_free T g))) h_Tssss gssss;
   305           val ((sssss, hssss_tl), _) =
   306             lthy
   307             |> mk_Freessss "q" s_Tssss
   308             ||>> mk_Freessss "h" (map (map (map tl)) h_Tssss);
   309           val hssss = map2 (map2 (map2 cons)) hssss_hd hssss_tl;
   310 
   311           val cpss = map2 (fn c => map (fn p => p $ c)) cs pss;
   312 
   313           fun mk_preds_getters_join [] [cf] = cf
   314             | mk_preds_getters_join [cq] [cf, cf'] =
   315               mk_If cq (mk_Inl (fastype_of cf') cf) (mk_Inr (fastype_of cf) cf');
   316 
   317           fun mk_terms qssss fssss =
   318             let
   319               val pfss = map3 zip_preds_predsss_gettersss pss qssss fssss;
   320               val cqssss = map2 (fn c => map (map (map (fn f => f $ c)))) cs qssss;
   321               val cfssss = map2 (fn c => map (map (map (fn f => f $ c)))) cs fssss;
   322               val cqfsss = map2 (map2 (map2 mk_preds_getters_join)) cqssss cfssss;
   323             in (pfss, cqfsss) end;
   324         in
   325           ((([], [], []), ([], [], [])),
   326            ([z], cs, cpss, (mk_terms rssss gssss, (g_sum_prod_Ts, pg_Tss)),
   327             (mk_terms sssss hssss, (h_sum_prod_Ts, ph_Tss))))
   328         end;
   329 
   330     fun define_ctrs_case_for_type ((((((((((((((((((fp_b, fpT), C), fld), unf), fp_iter), fp_rec),
   331           fld_unf), unf_fld), fld_inject), n), ks), ms), ctr_bindings), ctr_mixfixes), ctr_Tss),
   332         disc_bindings), sel_bindingss), raw_sel_defaultss) no_defs_lthy =
   333       let
   334         val unfT = domain_type (fastype_of fld);
   335         val ctr_prod_Ts = map HOLogic.mk_tupleT ctr_Tss;
   336         val ctr_sum_prod_T = mk_sumTN_balanced ctr_prod_Ts;
   337         val case_Ts = map (fn Ts => Ts ---> C) ctr_Tss;
   338 
   339         val ((((u, fs), xss), v'), _) =
   340           no_defs_lthy
   341           |> yield_singleton (mk_Frees "u") unfT
   342           ||>> mk_Frees "f" case_Ts
   343           ||>> mk_Freess "x" ctr_Tss
   344           ||>> yield_singleton (Variable.variant_fixes) (Binding.name_of fp_b);
   345 
   346         val v = Free (v', fpT);
   347 
   348         val ctr_rhss =
   349           map2 (fn k => fn xs => fold_rev Term.lambda xs (fld $
   350             mk_InN_balanced ctr_sum_prod_T n (HOLogic.mk_tuple xs) k)) ks xss;
   351 
   352         val case_binding = Binding.suffix_name ("_" ^ caseN) fp_b;
   353 
   354         val case_rhs =
   355           fold_rev Term.lambda (fs @ [v])
   356             (mk_sum_caseN_balanced (map2 mk_uncurried_fun fs xss) $ (unf $ v));
   357 
   358         val ((raw_case :: raw_ctrs, raw_case_def :: raw_ctr_defs), (lthy', lthy)) = no_defs_lthy
   359           |> apfst split_list o fold_map3 (fn b => fn mx => fn rhs =>
   360               Local_Theory.define ((b, mx), ((Thm.def_binding b, []), rhs)) #>> apsnd snd)
   361             (case_binding :: ctr_bindings) (NoSyn :: ctr_mixfixes) (case_rhs :: ctr_rhss)
   362           ||> `Local_Theory.restore;
   363 
   364         val phi = Proof_Context.export_morphism lthy lthy';
   365 
   366         val ctr_defs = map (Morphism.thm phi) raw_ctr_defs;
   367         val case_def = Morphism.thm phi raw_case_def;
   368 
   369         val ctrs0 = map (Morphism.term phi) raw_ctrs;
   370         val casex0 = Morphism.term phi raw_case;
   371 
   372         val ctrs = map (mk_ctr As) ctrs0;
   373 
   374         fun exhaust_tac {context = ctxt, ...} =
   375           let
   376             val fld_iff_unf_thm =
   377               let
   378                 val goal =
   379                   fold_rev Logic.all [u, v]
   380                     (mk_Trueprop_eq (HOLogic.mk_eq (v, fld $ u), HOLogic.mk_eq (unf $ v, u)));
   381               in
   382                 Skip_Proof.prove lthy [] [] goal (fn {context = ctxt, ...} =>
   383                   mk_fld_iff_unf_tac ctxt (map (SOME o certifyT lthy) [unfT, fpT])
   384                     (certify lthy fld) (certify lthy unf) fld_unf unf_fld)
   385                 |> Thm.close_derivation
   386                 |> Morphism.thm phi
   387               end;
   388 
   389             val sumEN_thm' =
   390               Local_Defs.unfold lthy @{thms all_unit_eq}
   391                 (Drule.instantiate' (map (SOME o certifyT lthy) ctr_prod_Ts) []
   392                    (mk_sumEN_balanced n))
   393               |> Morphism.thm phi;
   394           in
   395             mk_exhaust_tac ctxt n ctr_defs fld_iff_unf_thm sumEN_thm'
   396           end;
   397 
   398         val inject_tacss =
   399           map2 (fn 0 => K [] | _ => fn ctr_def => [fn {context = ctxt, ...} =>
   400               mk_inject_tac ctxt ctr_def fld_inject]) ms ctr_defs;
   401 
   402         val half_distinct_tacss =
   403           map (map (fn (def, def') => fn {context = ctxt, ...} =>
   404             mk_half_distinct_tac ctxt fld_inject [def, def'])) (mk_half_pairss ctr_defs);
   405 
   406         val case_tacs =
   407           map3 (fn k => fn m => fn ctr_def => fn {context = ctxt, ...} =>
   408             mk_case_tac ctxt n k m case_def ctr_def unf_fld) ks ms ctr_defs;
   409 
   410         val tacss = [exhaust_tac] :: inject_tacss @ half_distinct_tacss @ [case_tacs];
   411 
   412         fun define_iter_rec ((selss0, discIs, sel_thmss), no_defs_lthy) =
   413           let
   414             val fpT_to_C = fpT --> C;
   415 
   416             fun generate_iter_like (suf, fp_iter_like, (fss, f_Tss, xssss)) =
   417               let
   418                 val res_T = fold_rev (curry (op --->)) f_Tss fpT_to_C;
   419                 val binding = Binding.suffix_name ("_" ^ suf) fp_b;
   420                 val spec =
   421                   mk_Trueprop_eq (lists_bmoc fss (Free (Binding.name_of binding, res_T)),
   422                     Term.list_comb (fp_iter_like,
   423                       map2 (mk_sum_caseN_balanced oo map2 mk_uncurried2_fun) fss xssss));
   424               in (binding, spec) end;
   425 
   426             val iter_like_infos =
   427               [(iterN, fp_iter, iter_only),
   428                (recN, fp_rec, rec_only)];
   429 
   430             val (bindings, specs) = map generate_iter_like iter_like_infos |> split_list;
   431 
   432             val ((csts, defs), (lthy', lthy)) = no_defs_lthy
   433               |> apfst split_list o fold_map2 (fn b => fn spec =>
   434                 Specification.definition (SOME (b, NONE, NoSyn), ((Thm.def_binding b, []), spec))
   435                 #>> apsnd snd) bindings specs
   436               ||> `Local_Theory.restore;
   437 
   438             val phi = Proof_Context.export_morphism lthy lthy';
   439 
   440             val [iter_def, rec_def] = map (Morphism.thm phi) defs;
   441 
   442             val [iter, recx] = map (mk_iter_like As Cs o Morphism.term phi) csts;
   443           in
   444             ((ctrs, selss0, iter, recx, xss, ctr_defs, discIs, sel_thmss, iter_def, rec_def), lthy)
   445           end;
   446 
   447         fun define_coiter_corec ((selss0, discIs, sel_thmss), no_defs_lthy) =
   448           let
   449             val B_to_fpT = C --> fpT;
   450 
   451             fun mk_preds_getterss_join c n cps sum_prod_T cqfss =
   452               Term.lambda c (mk_IfN sum_prod_T cps
   453                 (map2 (mk_InN_balanced sum_prod_T n) (map HOLogic.mk_tuple cqfss) (1 upto n)));
   454 
   455             fun generate_coiter_like (suf, fp_iter_like, ((pfss, cqfsss), (f_sum_prod_Ts,
   456                 pf_Tss))) =
   457               let
   458                 val res_T = fold_rev (curry (op --->)) pf_Tss B_to_fpT;
   459                 val binding = Binding.suffix_name ("_" ^ suf) fp_b;
   460                 val spec =
   461                   mk_Trueprop_eq (lists_bmoc pfss (Free (Binding.name_of binding, res_T)),
   462                     Term.list_comb (fp_iter_like,
   463                       map5 mk_preds_getterss_join cs ns cpss f_sum_prod_Ts cqfsss));
   464               in (binding, spec) end;
   465 
   466             val coiter_like_infos =
   467               [(coiterN, fp_iter, coiter_only),
   468                (corecN, fp_rec, corec_only)];
   469 
   470             val (bindings, specs) = map generate_coiter_like coiter_like_infos |> split_list;
   471 
   472             val ((csts, defs), (lthy', lthy)) = no_defs_lthy
   473               |> apfst split_list o fold_map2 (fn b => fn spec =>
   474                 Specification.definition (SOME (b, NONE, NoSyn), ((Thm.def_binding b, []), spec))
   475                 #>> apsnd snd) bindings specs
   476               ||> `Local_Theory.restore;
   477 
   478             val phi = Proof_Context.export_morphism lthy lthy';
   479 
   480             val [coiter_def, corec_def] = map (Morphism.thm phi) defs;
   481 
   482             val [coiter, corec] = map (mk_iter_like As Cs o Morphism.term phi) csts;
   483           in
   484             ((ctrs, selss0, coiter, corec, xss, ctr_defs, discIs, sel_thmss, coiter_def, corec_def),
   485              lthy)
   486           end;
   487 
   488         fun wrap lthy =
   489           let val sel_defaultss = map (map (apsnd (prepare_term lthy))) raw_sel_defaultss in
   490             wrap_datatype tacss (((no_dests, ctrs0), casex0), (disc_bindings, (sel_bindingss,
   491               sel_defaultss))) lthy
   492           end;
   493 
   494         val define_iter_likes = if lfp then define_iter_rec else define_coiter_corec;
   495       in
   496         ((wrap, define_iter_likes), lthy')
   497       end;
   498 
   499     val pre_map_defs = map map_def_of_bnf pre_bnfs;
   500     val pre_set_defss = map set_defs_of_bnf pre_bnfs;
   501     val nested_set_natural's = maps set_natural'_of_bnf nested_bnfs;
   502     val nesting_map_ids = map map_id_of_bnf nesting_bnfs;
   503 
   504     fun mk_map s Ts Us t =
   505       let val (Type (_, Ts0), Type (_, Us0)) = strip_map_type s (fastype_of t) |>> List.last in
   506         Term.subst_atomic_types (Ts0 @ Us0 ~~ Ts @ Us) t
   507       end;
   508 
   509     fun build_map build_arg (T as Type (s, Ts)) (U as Type (_, Us)) =
   510       let
   511         val map0 = map_of_bnf (the (bnf_of lthy s));
   512         val mapx = mk_map s Ts Us map0;
   513         val TUs = map dest_funT (fst (split_last (fst (strip_map_type s (fastype_of mapx)))));
   514         val args = map build_arg TUs;
   515       in Term.list_comb (mapx, args) end;
   516 
   517     fun derive_induct_iter_rec_thms_for_types ((ctrss, _, iters, recs, xsss, ctr_defss, _, _,
   518         iter_defs, rec_defs), lthy) =
   519       let
   520         val (((phis, phis'), vs'), names_lthy) =
   521           lthy
   522           |> mk_Frees' "P" (map mk_predT fpTs)
   523           ||>> Variable.variant_fixes (map Binding.name_of fp_bs);
   524 
   525         val vs = map2 (curry Free) vs' fpTs;
   526 
   527         fun mk_sets_nested bnf =
   528           let
   529             val Type (T_name, Us) = T_of_bnf bnf;
   530             val lives = lives_of_bnf bnf;
   531             val sets = sets_of_bnf bnf;
   532             fun mk_set U =
   533               (case find_index (curry (op =) U) lives of
   534                 ~1 => Term.dummy
   535               | i => nth sets i);
   536           in
   537             (T_name, map mk_set Us)
   538           end;
   539 
   540         val setss_nested = map mk_sets_nested nested_bnfs;
   541 
   542         val (induct_thms, induct_thm) =
   543           let
   544             fun mk_set Ts t =
   545               let val Type (_, Ts0) = domain_type (fastype_of t) in
   546                 Term.subst_atomic_types (Ts0 ~~ Ts) t
   547               end;
   548 
   549             fun mk_raw_prem_prems names_lthy (x as Free (s, T as Type (T_name, Ts0))) =
   550                 (case find_index (curry (op =) T) fpTs of
   551                   ~1 =>
   552                   (case AList.lookup (op =) setss_nested T_name of
   553                     NONE => []
   554                   | SOME raw_sets0 =>
   555                     let
   556                       val (Ts, raw_sets) =
   557                         split_list (filter (exists_fp_subtype o fst) (Ts0 ~~ raw_sets0));
   558                       val sets = map (mk_set Ts0) raw_sets;
   559                       val (ys, names_lthy') = names_lthy |> mk_Frees s Ts;
   560                       val xysets = map (pair x) (ys ~~ sets);
   561                       val ppremss = map (mk_raw_prem_prems names_lthy') ys;
   562                     in
   563                       flat (map2 (map o apfst o cons) xysets ppremss)
   564                     end)
   565                 | i => [([], (i + 1, x))])
   566               | mk_raw_prem_prems _ _ = [];
   567 
   568             fun close_prem_prem xs t =
   569               fold_rev Logic.all (map Free (drop (nn + length xs)
   570                 (rev (Term.add_frees t (map dest_Free xs @ phis'))))) t;
   571 
   572             fun mk_prem_prem xs (xysets, (j, x)) =
   573               close_prem_prem xs (Logic.list_implies (map (fn (x', (y, set)) =>
   574                   HOLogic.mk_Trueprop (HOLogic.mk_mem (y, set $ x'))) xysets,
   575                 HOLogic.mk_Trueprop (nth phis (j - 1) $ x)));
   576 
   577             fun mk_raw_prem phi ctr ctr_Ts =
   578               let
   579                 val (xs, names_lthy') = names_lthy |> mk_Frees "x" ctr_Ts;
   580                 val pprems = maps (mk_raw_prem_prems names_lthy') xs;
   581               in (xs, pprems, HOLogic.mk_Trueprop (phi $ Term.list_comb (ctr, xs))) end;
   582 
   583             fun mk_prem (xs, raw_pprems, concl) =
   584               fold_rev Logic.all xs (Logic.list_implies (map (mk_prem_prem xs) raw_pprems, concl));
   585 
   586             val raw_premss = map3 (map2 o mk_raw_prem) phis ctrss ctr_Tsss;
   587 
   588             val goal =
   589               Library.foldr (Logic.list_implies o apfst (map mk_prem)) (raw_premss,
   590                 HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj
   591                   (map2 (curry (op $)) phis vs)));
   592 
   593             fun mk_raw_prem_prems_indices pprems =
   594               let
   595                 fun has_index kk (_, (kk', _)) = kk' = kk;
   596                 val buckets = Library.partition_list has_index 1 nn pprems;
   597                 val pps = map length buckets;
   598               in
   599                 map (fn pprem as (xysets, (kk, _)) =>
   600                   ((nth pps (kk - 1), find_index (curry (op =) pprem) (nth buckets (kk - 1)) + 1),
   601                    (length xysets, kk))) pprems
   602               end;
   603 
   604             val ppjjqqkksss = map (map (mk_raw_prem_prems_indices o #2)) raw_premss;
   605 
   606             val fld_induct' = fp_induct OF (map mk_sumEN_tupled_balanced mss);
   607 
   608             val induct_thm =
   609               Skip_Proof.prove lthy [] [] goal (fn {context = ctxt, ...} =>
   610                 mk_induct_tac ctxt ns mss ppjjqqkksss (flat ctr_defss) fld_induct'
   611                   nested_set_natural's pre_set_defss)
   612               |> singleton (Proof_Context.export names_lthy lthy)
   613           in
   614             `(conj_dests nn) induct_thm
   615           end;
   616 
   617         val (iter_thmss, rec_thmss) =
   618           let
   619             val xctrss = map2 (map2 (curry Term.list_comb)) ctrss xsss;
   620             val giters = map (lists_bmoc gss) iters;
   621             val hrecs = map (lists_bmoc hss) recs;
   622 
   623             fun mk_goal_iter_like fss fiter_like xctr f xs fxs =
   624               fold_rev (fold_rev Logic.all) (xs :: fss)
   625                 (mk_Trueprop_eq (fiter_like $ xctr, Term.list_comb (f, fxs)));
   626 
   627             fun build_call fiter_likes maybe_tick (T, U) =
   628               if T = U then
   629                 id_const T
   630               else
   631                 (case find_index (curry (op =) T) fpTs of
   632                   ~1 => build_map (build_call fiter_likes maybe_tick) T U
   633                 | j => maybe_tick (nth vs j) (nth fiter_likes j));
   634 
   635             fun mk_U maybe_mk_prodT =
   636               typ_subst (map2 (fn fpT => fn C => (fpT, maybe_mk_prodT fpT C)) fpTs Cs);
   637 
   638             fun intr_calls fiter_likes maybe_cons maybe_tick maybe_mk_prodT (x as Free (_, T)) =
   639               if member (op =) fpTs T then
   640                 maybe_cons x [build_call fiter_likes (K I) (T, mk_U (K I) T) $ x]
   641               else if exists_fp_subtype T then
   642                 [build_call fiter_likes maybe_tick (T, mk_U maybe_mk_prodT T) $ x]
   643               else
   644                 [x];
   645 
   646             val gxsss = map (map (maps (intr_calls giters (K I) (K I) (K I)))) xsss;
   647             val hxsss = map (map (maps (intr_calls hrecs cons tick (curry HOLogic.mk_prodT)))) xsss;
   648 
   649             val goal_iterss = map5 (map4 o mk_goal_iter_like gss) giters xctrss gss xsss gxsss;
   650             val goal_recss = map5 (map4 o mk_goal_iter_like hss) hrecs xctrss hss xsss hxsss;
   651 
   652             val iter_tacss =
   653               map2 (map o mk_iter_like_tac pre_map_defs nesting_map_ids iter_defs) fp_iter_thms
   654                 ctr_defss;
   655             val rec_tacss =
   656               map2 (map o mk_iter_like_tac pre_map_defs nesting_map_ids rec_defs) fp_rec_thms
   657                 ctr_defss;
   658           in
   659             (map2 (map2 (fn goal => fn tac => Skip_Proof.prove lthy [] [] goal (tac o #context)))
   660                goal_iterss iter_tacss,
   661              map2 (map2 (fn goal => fn tac => Skip_Proof.prove lthy [] [] goal (tac o #context)))
   662                goal_recss rec_tacss)
   663           end;
   664 
   665         val common_notes =
   666           (if nn > 1 then [(inductN, [induct_thm], [])] (* FIXME: attribs *) else [])
   667           |> map (fn (thmN, thms, attrs) =>
   668               ((Binding.qualify true fp_common_name (Binding.name thmN), attrs), [(thms, [])]));
   669 
   670         val notes =
   671           [(inductN, map single induct_thms, []), (* FIXME: attribs *)
   672            (itersN, iter_thmss, simp_attrs),
   673            (recsN, rec_thmss, Code.add_default_eqn_attrib :: simp_attrs)]
   674           |> maps (fn (thmN, thmss, attrs) =>
   675             map2 (fn b => fn thms =>
   676               ((Binding.qualify true (Binding.name_of b) (Binding.name thmN), attrs),
   677                 [(thms, [])])) fp_bs thmss);
   678       in
   679         lthy |> Local_Theory.notes (common_notes @ notes) |> snd
   680       end;
   681 
   682     fun derive_coinduct_coiter_corec_thms_for_types ((ctrss, selsss, coiters, corecs, _, ctr_defss,
   683         discIss, sel_thmsss, coiter_defs, corec_defs), lthy) =
   684       let
   685         val (vs', _) =
   686           lthy
   687           |> Variable.variant_fixes (map Binding.name_of fp_bs);
   688 
   689         val vs = map2 (curry Free) vs' fpTs;
   690 
   691         val (coinduct_thms, coinduct_thm) =
   692           let
   693             val coinduct_thm = fp_induct;
   694           in
   695             `(conj_dests nn) coinduct_thm
   696           end;
   697 
   698         val (coiter_thmss, corec_thmss) =
   699           let
   700             val z = the_single zs;
   701             val gcoiters = map (lists_bmoc pgss) coiters;
   702             val hcorecs = map (lists_bmoc phss) corecs;
   703 
   704             fun mk_goal_cond pos = HOLogic.mk_Trueprop o (not pos ? HOLogic.mk_not);
   705 
   706             fun mk_goal_coiter_like pfss c cps fcoiter_like n k ctr m cfs' =
   707               fold_rev (fold_rev Logic.all) ([c] :: pfss)
   708                 (Logic.list_implies (seq_conds mk_goal_cond n k cps,
   709                    mk_Trueprop_eq (fcoiter_like $ c, Term.list_comb (ctr, take m cfs'))));
   710 
   711             fun build_call fiter_likes maybe_tack (T, U) =
   712               if T = U then
   713                 id_const T
   714               else
   715                 (case find_index (curry (op =) U) fpTs of
   716                   ~1 => build_map (build_call fiter_likes maybe_tack) T U
   717                 | j => maybe_tack (nth cs j, nth vs j) (nth fiter_likes j));
   718 
   719             fun mk_U maybe_mk_sumT =
   720               typ_subst (map2 (fn C => fn fpT => (maybe_mk_sumT fpT C, fpT)) Cs fpTs);
   721 
   722             fun intr_calls fiter_likes maybe_mk_sumT maybe_tack cqf =
   723               let val T = fastype_of cqf in
   724                 if exists_subtype (member (op =) Cs) T then
   725                   build_call fiter_likes maybe_tack (T, mk_U maybe_mk_sumT T) $ cqf
   726                 else
   727                   cqf
   728               end;
   729 
   730             val crgsss' = map (map (map (intr_calls gcoiters (K I) (K I)))) crgsss;
   731             val cshsss' = map (map (map (intr_calls hcorecs (curry mk_sumT) (tack z)))) cshsss;
   732 
   733             val goal_coiterss =
   734               map8 (map4 oooo mk_goal_coiter_like pgss) cs cpss gcoiters ns kss ctrss mss crgsss';
   735             val goal_corecss =
   736               map8 (map4 oooo mk_goal_coiter_like phss) cs cpss hcorecs ns kss ctrss mss cshsss';
   737 
   738             val coiter_tacss =
   739               map3 (map oo mk_coiter_like_tac coiter_defs nesting_map_ids) fp_iter_thms pre_map_defs
   740                 ctr_defss;
   741             val corec_tacss =
   742               map3 (map oo mk_coiter_like_tac corec_defs nesting_map_ids) fp_rec_thms pre_map_defs
   743                 ctr_defss;
   744           in
   745             (map2 (map2 (fn goal => fn tac =>
   746                  Skip_Proof.prove lthy [] [] goal (tac o #context) |> Thm.close_derivation))
   747                goal_coiterss coiter_tacss,
   748              map2 (map2 (fn goal => fn tac =>
   749                  Skip_Proof.prove lthy [] [] goal (tac o #context)
   750                  |> Local_Defs.unfold lthy @{thms sum_case_if} |> Thm.close_derivation))
   751                goal_corecss corec_tacss)
   752           end;
   753 
   754         fun mk_disc_coiter_like_thms [_] = K []
   755           | mk_disc_coiter_like_thms thms = map2 (curry (op RS)) thms;
   756 
   757         val disc_coiter_thmss = map2 mk_disc_coiter_like_thms coiter_thmss discIss;
   758         val disc_corec_thmss = map2 mk_disc_coiter_like_thms corec_thmss discIss;
   759 
   760         fun mk_sel_coiter_like_thm coiter_like_thm sel0 sel_thm =
   761           let
   762             val (domT, ranT) = dest_funT (fastype_of sel0);
   763             val arg_cong' =
   764               Drule.instantiate' (map (SOME o certifyT lthy) [domT, ranT])
   765                 [NONE, NONE, SOME (certify lthy sel0)] arg_cong
   766               |> Thm.varifyT_global;
   767             val sel_thm' = sel_thm RSN (2, trans);
   768           in
   769             coiter_like_thm RS arg_cong' RS sel_thm'
   770           end;
   771 
   772         val sel_coiter_thmsss =
   773           map3 (map3 (map2 o mk_sel_coiter_like_thm)) coiter_thmss selsss sel_thmsss;
   774         val sel_corec_thmsss =
   775           map3 (map3 (map2 o mk_sel_coiter_like_thm)) corec_thmss selsss sel_thmsss;
   776 
   777         val common_notes =
   778           (if nn > 1 then [(coinductN, [coinduct_thm], [])] (* FIXME: attribs *) else [])
   779           |> map (fn (thmN, thms, attrs) =>
   780               ((Binding.qualify true fp_common_name (Binding.name thmN), attrs), [(thms, [])]));
   781 
   782         val notes =
   783           [(coinductN, map single coinduct_thms, []), (* FIXME: attribs *)
   784            (coitersN, coiter_thmss, []),
   785            (disc_coitersN, disc_coiter_thmss, []),
   786            (sel_coitersN, map flat sel_coiter_thmsss, []),
   787            (corecsN, corec_thmss, []),
   788            (disc_corecsN, disc_corec_thmss, []),
   789            (sel_corecsN, map flat sel_corec_thmsss, [])]
   790           |> maps (fn (thmN, thmss, attrs) =>
   791             map_filter (fn (_, []) => NONE | (b, thms) =>
   792               SOME ((Binding.qualify true (Binding.name_of b) (Binding.name thmN), attrs),
   793                 [(thms, [])])) (fp_bs ~~ thmss));
   794       in
   795         lthy |> Local_Theory.notes (common_notes @ notes) |> snd
   796       end;
   797 
   798     fun wrap_types_and_define_iter_likes ((wraps, define_iter_likess), lthy) =
   799       fold_map2 (curry (op o)) define_iter_likess wraps lthy |>> split_list10
   800 
   801     val lthy' = lthy
   802       |> fold_map define_ctrs_case_for_type (fp_bs ~~ fpTs ~~ Cs ~~ flds ~~ unfs ~~ fp_iters ~~
   803         fp_recs ~~ fld_unfs ~~ unf_flds ~~ fld_injects ~~ ns ~~ kss ~~ mss ~~ ctr_bindingss ~~
   804         ctr_mixfixess ~~ ctr_Tsss ~~ disc_bindingss ~~ sel_bindingsss ~~ raw_sel_defaultsss)
   805       |>> split_list |> wrap_types_and_define_iter_likes
   806       |> (if lfp then derive_induct_iter_rec_thms_for_types
   807           else derive_coinduct_coiter_corec_thms_for_types);
   808 
   809     val timer = time (timer ("Constructors, discriminators, selectors, etc., for the new " ^
   810       (if lfp then "" else "co") ^ "datatype"));
   811   in
   812     timer; lthy'
   813   end;
   814 
   815 val datatyp = define_datatype (K I) (K I) (K I);
   816 
   817 val datatype_cmd = define_datatype Typedecl.read_constraint Syntax.parse_typ Syntax.read_term;
   818 
   819 val parse_binding_colon = Parse.binding --| @{keyword ":"};
   820 val parse_opt_binding_colon = Scan.optional parse_binding_colon no_binding;
   821 
   822 val parse_ctr_arg =
   823   @{keyword "("} |-- parse_binding_colon -- Parse.typ --| @{keyword ")"} ||
   824   (Parse.typ >> pair no_binding);
   825 
   826 val parse_defaults =
   827   @{keyword "("} |-- @{keyword "defaults"} |-- Scan.repeat parse_bound_term --| @{keyword ")"};
   828 
   829 val parse_single_spec =
   830   Parse.type_args_constrained -- Parse.binding -- Parse.opt_mixfix --
   831   (@{keyword "="} |-- Parse.enum1 "|" (parse_opt_binding_colon -- Parse.binding --
   832     Scan.repeat parse_ctr_arg -- Scan.optional parse_defaults [] -- Parse.opt_mixfix));
   833 
   834 val parse_datatype = parse_wrap_options -- Parse.and_list1 parse_single_spec;
   835 
   836 fun parse_datatype_cmd lfp construct = parse_datatype >> datatype_cmd lfp construct;
   837 
   838 end;