src/HOL/Codatatype/Tools/bnf_fp_sugar.ML
author blanchet
Mon Sep 10 17:35:53 2012 +0200 (2012-09-10)
changeset 49255 2ecc533d6697
parent 49254 edc322ac5279
child 49256 df98aeb80a19
permissions -rw-r--r--
use balanced sums for constructors (to gracefully handle 100 constructors or more)
     1 (*  Title:      HOL/Codatatype/Tools/bnf_fp_sugar.ML
     2     Author:     Jasmin Blanchette, TU Muenchen
     3     Copyright   2012
     4 
     5 Sugar for constructing LFPs and GFPs.
     6 *)
     7 
     8 signature BNF_FP_SUGAR =
     9 sig
    10   (* TODO: programmatic interface *)
    11 end;
    12 
    13 structure BNF_FP_Sugar : BNF_FP_SUGAR =
    14 struct
    15 
    16 open BNF_Util
    17 open BNF_Wrap
    18 open BNF_Def
    19 open BNF_FP_Util
    20 open BNF_LFP
    21 open BNF_GFP
    22 open BNF_FP_Sugar_Tactics
    23 
    24 val caseN = "case";
    25 val coitersN = "coiters";
    26 val corecsN = "corecs";
    27 val itersN = "iters";
    28 val recsN = "recs";
    29 
    30 fun split_list8 xs =
    31   (map #1 xs, map #2 xs, map #3 xs, map #4 xs, map #5 xs, map #6 xs, map #7 xs, map #8 xs);
    32 
    33 fun strip_map_type (Type (@{type_name fun}, [T as Type _, T'])) = strip_map_type T' |>> cons T
    34   | strip_map_type T = ([], T);
    35 
    36 fun typ_subst inst (T as Type (s, Ts)) =
    37     (case AList.lookup (op =) inst T of
    38       NONE => Type (s, map (typ_subst inst) Ts)
    39     | SOME T' => T')
    40   | typ_subst inst T = the_default T (AList.lookup (op =) inst T);
    41 
    42 fun retype_free (Free (s, _)) T = Free (s, T);
    43 
    44 val lists_bmoc = fold (fn xs => fn t => Term.list_comb (t, xs))
    45 
    46 fun mk_id T = Const (@{const_name id}, T --> T);
    47 
    48 fun mk_tupled_fun x f xs = HOLogic.tupled_lambda x (Term.list_comb (f, xs));
    49 fun mk_uncurried_fun f xs = mk_tupled_fun (HOLogic.mk_tuple xs) f xs;
    50 fun mk_uncurried2_fun f xss =
    51   mk_tupled_fun (HOLogic.mk_tuple (map HOLogic.mk_tuple xss)) f (flat xss);
    52 
    53 val mk_sumTN_balanced = Balanced_Tree.make mk_sumT;
    54 val dest_sumTN_balanced = Balanced_Tree.dest dest_sumT;
    55 
    56 fun mk_InN_balanced ctxt sum_T Ts t k =
    57   let
    58     val u =
    59       Balanced_Tree.access {left = mk_Inl dummyT, right = mk_Inr dummyT, init = t} (length Ts) k;
    60   in singleton (Type_Infer_Context.infer_types ctxt) (Type.constraint sum_T u) end;
    61 
    62 val mk_sum_caseN_balanced = Balanced_Tree.make mk_sum_case;
    63 
    64 fun mk_sumEN_balanced n =
    65   let
    66     val thm =
    67       Balanced_Tree.make (fn (thm1, thm2) => thm1 RSN (1, thm2 RSN (2, @{thm obj_sumE_f})))
    68         (replicate n asm_rl) OF (replicate n (impI RS allI));
    69     val f as (_, f_T) =
    70       Term.add_vars (prop_of thm) []
    71       |> filter (fn ((s, _), _) => s = "f") |> the_single;
    72     val inst = [pairself (cterm_of @{theory}) (Var f, Abs (Name.uu, domain_type f_T, Bound 0))];
    73   in cterm_instantiate inst thm end;
    74 
    75 fun tick v f = Term.lambda v (HOLogic.mk_prod (v, f $ v));
    76 
    77 fun tack z_name (c, v) f =
    78   let val z = Free (z_name, mk_sumT (fastype_of v, fastype_of c)) in
    79     Term.lambda z (mk_sum_case (Term.lambda v v, Term.lambda c (f $ c)) $ z)
    80   end;
    81 
    82 fun cannot_merge_types () = error "Mutually recursive types must have the same type parameters";
    83 
    84 fun merge_type_arg_constrained ctxt (T, c) (T', c') =
    85   if T = T' then
    86     (case (c, c') of
    87       (_, NONE) => (T, c)
    88     | (NONE, _) => (T, c')
    89     | _ =>
    90       if c = c' then
    91         (T, c)
    92       else
    93         error ("Inconsistent sort constraints for type variable " ^
    94           quote (Syntax.string_of_typ ctxt T)))
    95   else
    96     cannot_merge_types ();
    97 
    98 fun merge_type_args_constrained ctxt (cAs, cAs') =
    99   if length cAs = length cAs' then map2 (merge_type_arg_constrained ctxt) cAs cAs'
   100   else cannot_merge_types ();
   101 
   102 fun type_args_constrained_of (((cAs, _), _), _) = cAs;
   103 val type_args_of = map fst o type_args_constrained_of;
   104 fun type_binder_of (((_, b), _), _) = b;
   105 fun mixfix_of ((_, mx), _) = mx;
   106 fun ctr_specs_of (_, ctr_specs) = ctr_specs;
   107 
   108 fun disc_of (((disc, _), _), _) = disc;
   109 fun ctr_of (((_, ctr), _), _) = ctr;
   110 fun args_of ((_, args), _) = args;
   111 fun ctr_mixfix_of (_, mx) = mx;
   112 
   113 fun prepare_datatype prepare_typ lfp specs fake_lthy no_defs_lthy =
   114   let
   115     val constrained_As =
   116       map (map (apfst (prepare_typ fake_lthy)) o type_args_constrained_of) specs
   117       |> Library.foldr1 (merge_type_args_constrained no_defs_lthy);
   118     val As = map fst constrained_As;
   119     val As' = map dest_TFree As;
   120 
   121     val _ = (case duplicates (op =) As of [] => ()
   122       | A :: _ => error ("Duplicate type parameter " ^
   123           quote (Syntax.string_of_typ no_defs_lthy A)));
   124 
   125     (* TODO: use sort constraints on type args *)
   126 
   127     val N = length specs;
   128 
   129     fun mk_fake_T b =
   130       Type (fst (Term.dest_Type (Proof_Context.read_type_name fake_lthy true (Binding.name_of b))),
   131         As);
   132 
   133     val bs = map type_binder_of specs;
   134     val fakeTs = map mk_fake_T bs;
   135 
   136     val mixfixes = map mixfix_of specs;
   137 
   138     val _ = (case duplicates Binding.eq_name bs of [] => ()
   139       | b :: _ => error ("Duplicate type name declaration " ^ quote (Binding.name_of b)));
   140 
   141     val ctr_specss = map ctr_specs_of specs;
   142 
   143     val disc_binderss = map (map disc_of) ctr_specss;
   144     val ctr_binderss = map (map ctr_of) ctr_specss;
   145     val ctr_argsss = map (map args_of) ctr_specss;
   146     val ctr_mixfixess = map (map ctr_mixfix_of) ctr_specss;
   147 
   148     val sel_bindersss = map (map (map fst)) ctr_argsss;
   149     val fake_ctr_Tsss = map (map (map (prepare_typ fake_lthy o snd))) ctr_argsss;
   150 
   151     val rhs_As' = fold (fold (fold Term.add_tfreesT)) fake_ctr_Tsss [];
   152     val _ = (case subtract (op =) As' rhs_As' of
   153         [] => ()
   154       | A' :: _ => error ("Extra type variables on rhs: " ^
   155           quote (Syntax.string_of_typ no_defs_lthy (TFree A'))));
   156 
   157     val ((Cs, Xs), _) =
   158       no_defs_lthy
   159       |> fold (fold (fn s => Variable.declare_typ (TFree (s, dummyS))) o type_args_of) specs
   160       |> mk_TFrees N
   161       ||>> mk_TFrees N;
   162 
   163     fun eq_fpT (T as Type (s, Us)) (Type (s', Us')) =
   164         s = s' andalso (Us = Us' orelse error ("Illegal occurrence of recursive type " ^
   165           quote (Syntax.string_of_typ fake_lthy T)))
   166       | eq_fpT _ _ = false;
   167 
   168     fun freeze_fp (T as Type (s, Us)) =
   169         (case find_index (eq_fpT T) fakeTs of ~1 => Type (s, map freeze_fp Us) | j => nth Xs j)
   170       | freeze_fp T = T;
   171 
   172     val ctr_TsssXs = map (map (map freeze_fp)) fake_ctr_Tsss;
   173     val ctr_sum_prod_TsXs = map (mk_sumTN_balanced o map HOLogic.mk_tupleT) ctr_TsssXs;
   174 
   175     val eqs = map dest_TFree Xs ~~ ctr_sum_prod_TsXs;
   176 
   177     val (pre_bnfs, ((unfs0, flds0, fp_iters0, fp_recs0, unf_flds, fld_unfs, fld_injects,
   178         fp_iter_thms, fp_rec_thms), lthy)) =
   179       fp_bnf (if lfp then bnf_lfp else bnf_gfp) bs mixfixes As' eqs no_defs_lthy;
   180 
   181     val add_nested_bnf_names =
   182       let
   183         fun add (Type (s, Ts)) ss =
   184             let val (needs, ss') = fold_map add Ts ss in
   185               if exists I needs then (true, insert (op =) s ss') else (false, ss')
   186             end
   187           | add T ss = (member (op =) As T, ss);
   188       in snd oo add end;
   189 
   190     val nested_bnfs =
   191       map_filter (bnf_of lthy) (fold (fold (fold add_nested_bnf_names)) ctr_TsssXs []);
   192 
   193     val timer = time (Timer.startRealTimer ());
   194 
   195     fun mk_unf_or_fld get_T Ts t =
   196       let val Type (_, Ts0) = get_T (fastype_of t) in
   197         Term.subst_atomic_types (Ts0 ~~ Ts) t
   198       end;
   199 
   200     val mk_unf = mk_unf_or_fld domain_type;
   201     val mk_fld = mk_unf_or_fld range_type;
   202 
   203     val unfs = map (mk_unf As) unfs0;
   204     val flds = map (mk_fld As) flds0;
   205 
   206     val fpTs = map (domain_type o fastype_of) unfs;
   207 
   208     val ctr_Tsss = map (map (map (Term.typ_subst_atomic (Xs ~~ fpTs)))) ctr_TsssXs;
   209     val ns = map length ctr_Tsss;
   210     val kss = map (fn n => 1 upto n) ns;
   211     val mss = map (map length) ctr_Tsss;
   212     val Css = map2 replicate ns Cs;
   213 
   214     fun mk_iter_like Ts Us t =
   215       let
   216         val (binders, body) = strip_type (fastype_of t);
   217         val (f_Us, prebody) = split_last binders;
   218         val Type (_, Ts0) = if lfp then prebody else body;
   219         val Us0 = distinct (op =) (map (if lfp then body_type else domain_type) f_Us);
   220       in
   221         Term.subst_atomic_types (Ts0 @ Us0 ~~ Ts @ Us) t
   222       end;
   223 
   224     val fp_iters as fp_iter1 :: _ = map (mk_iter_like As Cs) fp_iters0;
   225     val fp_recs as fp_rec1 :: _ = map (mk_iter_like As Cs) fp_recs0;
   226 
   227     val fp_iter_fun_Ts = fst (split_last (binder_types (fastype_of fp_iter1)));
   228     val fp_rec_fun_Ts = fst (split_last (binder_types (fastype_of fp_rec1)));
   229 
   230     fun dest_rec_pair (T as Type (@{type_name prod}, Us as [_, U])) =
   231         if member (op =) Cs U then Us else [T]
   232       | dest_rec_pair T = [T];
   233 
   234     val ((iter_only as (gss, g_Tss, yssss), rec_only as (hss, h_Tss, zssss)),
   235          (zs, cs, cpss, p_Tss, coiter_only as ((pgss, cgsss), g_sum_prod_Ts, g_prod_Tss, g_Tsss),
   236           corec_only as ((phss, chsss), h_sum_prod_Ts, h_prod_Tss, h_Tsss))) =
   237       if lfp then
   238         let
   239           val y_Tsss =
   240             map3 (fn n => fn ms => map2 dest_tupleT ms o dest_sumTN_balanced n o domain_type)
   241               ns mss fp_iter_fun_Ts;
   242           val g_Tss = map2 (map2 (curry (op --->))) y_Tsss Css;
   243 
   244           val ((gss, ysss), _) =
   245             lthy
   246             |> mk_Freess "f" g_Tss
   247             ||>> mk_Freesss "x" y_Tsss;
   248 
   249           val z_Tssss =
   250             map3 (fn n => fn ms => map2 (map dest_rec_pair oo dest_tupleT) ms o
   251               dest_sumTN_balanced n o domain_type) ns mss fp_rec_fun_Ts;
   252           val h_Tss = map2 (map2 (fold_rev (curry (op --->)))) z_Tssss Css;
   253 
   254           val hss = map2 (map2 retype_free) gss h_Tss;
   255           val (zssss, _) =
   256             lthy
   257             |> mk_Freessss "x" z_Tssss;
   258         in
   259           (((gss, g_Tss, map (map (map single)) ysss), (hss, h_Tss, zssss)),
   260            ([], [], [], [], (([], []), [], [], []), (([], []), [], [], [])))
   261         end
   262       else
   263         let
   264           (*avoid "'a itself" arguments in coiterators and corecursors*)
   265           val mss' =  map (fn [0] => [1] | ms => ms) mss;
   266 
   267           val p_Tss =
   268             map2 (fn C => fn n => replicate (Int.max (0, n - 1)) (C --> HOLogic.boolT)) Cs ns;
   269 
   270           fun zip_preds_getters [] [fs] = fs
   271             | zip_preds_getters (p :: ps) (fs :: fss) = p :: fs @ zip_preds_getters ps fss;
   272 
   273           fun mk_types fun_Ts =
   274             let
   275               val f_sum_prod_Ts = map range_type fun_Ts;
   276               val f_prod_Tss = map2 dest_sumTN_balanced ns f_sum_prod_Ts;
   277               val f_Tsss =
   278                 map3 (fn C => map2 (map (curry (op -->) C) oo dest_tupleT)) Cs mss' f_prod_Tss;
   279               val pf_Tss = map2 zip_preds_getters p_Tss f_Tsss
   280             in (f_sum_prod_Ts, f_prod_Tss, f_Tsss, pf_Tss) end;
   281 
   282           val (g_sum_prod_Ts, g_prod_Tss, g_Tsss, pg_Tss) = mk_types fp_iter_fun_Ts;
   283           val (h_sum_prod_Ts, h_prod_Tss, h_Tsss, ph_Tss) = mk_types fp_rec_fun_Ts;
   284 
   285           val ((((Free (z, _), cs), pss), gsss), _) =
   286             lthy
   287             |> yield_singleton (mk_Frees "z") dummyT
   288             ||>> mk_Frees "a" Cs
   289             ||>> mk_Freess "p" p_Tss
   290             ||>> mk_Freesss "g" g_Tsss;
   291 
   292           val hsss = map2 (map2 (map2 retype_free)) gsss h_Tsss;
   293 
   294           val cpss = map2 (fn c => map (fn p => p $ c)) cs pss;
   295 
   296           fun mk_terms fsss =
   297             let
   298               val pfss = map2 zip_preds_getters pss fsss;
   299               val cfsss = map2 (fn c => map (map (fn f => f $ c))) cs fsss
   300             in (pfss, cfsss) end;
   301         in
   302           ((([], [], []), ([], [], [])),
   303            ([z], cs, cpss, p_Tss, (mk_terms gsss, g_sum_prod_Ts, g_prod_Tss, pg_Tss),
   304             (mk_terms hsss, h_sum_prod_Ts, h_prod_Tss, ph_Tss)))
   305         end;
   306 
   307     fun pour_some_sugar_on_type (((((((((((((((((b, fpT), C), fld), unf), fp_iter), fp_rec),
   308           fld_unf), unf_fld), fld_inject), n), ks), ms), ctr_binders), ctr_mixfixes), ctr_Tss),
   309         disc_binders), sel_binderss) no_defs_lthy =
   310       let
   311         val unfT = domain_type (fastype_of fld);
   312         val ctr_prod_Ts = map HOLogic.mk_tupleT ctr_Tss;
   313         val ctr_sum_prod_T = mk_sumTN_balanced ctr_prod_Ts;
   314         val case_Ts = map (fn Ts => Ts ---> C) ctr_Tss;
   315 
   316         val ((((u, v), fs), xss), _) =
   317           no_defs_lthy
   318           |> yield_singleton (mk_Frees "u") unfT
   319           ||>> yield_singleton (mk_Frees "v") fpT
   320           ||>> mk_Frees "f" case_Ts
   321           ||>> mk_Freess "x" ctr_Tss;
   322 
   323         val ctr_rhss =
   324           map2 (fn k => fn xs =>
   325               fold_rev Term.lambda xs (fld $ mk_InN_balanced no_defs_lthy ctr_sum_prod_T ctr_prod_Ts
   326                 (HOLogic.mk_tuple xs) k))
   327             ks xss;
   328 
   329         val case_binder = Binding.suffix_name ("_" ^ caseN) b;
   330 
   331         val case_rhs =
   332           fold_rev Term.lambda (fs @ [v])
   333             (mk_sum_caseN_balanced (map2 mk_uncurried_fun fs xss) $ (unf $ v));
   334 
   335         val ((raw_case :: raw_ctrs, raw_case_def :: raw_ctr_defs), (lthy', lthy)) = no_defs_lthy
   336           |> apfst split_list o fold_map3 (fn b => fn mx => fn rhs =>
   337                Local_Theory.define ((b, mx), ((Thm.def_binding b, []), rhs)) #>> apsnd snd)
   338              (case_binder :: ctr_binders) (NoSyn :: ctr_mixfixes) (case_rhs :: ctr_rhss)
   339           ||> `Local_Theory.restore;
   340 
   341         (*transforms defined frees into consts (and more)*)
   342         val phi = Proof_Context.export_morphism lthy lthy';
   343 
   344         val ctr_defs = map (Morphism.thm phi) raw_ctr_defs;
   345         val case_def = Morphism.thm phi raw_case_def;
   346 
   347         val ctrs0 = map (Morphism.term phi) raw_ctrs;
   348         val casex0 = Morphism.term phi raw_case;
   349 
   350         val ctrs = map (mk_ctr As) ctrs0;
   351 
   352         fun exhaust_tac {context = ctxt, ...} =
   353           let
   354             val fld_iff_unf_thm =
   355               let
   356                 val goal =
   357                   fold_rev Logic.all [u, v]
   358                     (mk_Trueprop_eq (HOLogic.mk_eq (v, fld $ u), HOLogic.mk_eq (unf $ v, u)));
   359               in
   360                 Skip_Proof.prove lthy [] [] goal (fn {context = ctxt, ...} =>
   361                   mk_fld_iff_unf_tac ctxt (map (SOME o certifyT lthy) [unfT, fpT])
   362                     (certify lthy fld) (certify lthy unf) fld_unf unf_fld)
   363                 |> Thm.close_derivation
   364                 |> Morphism.thm phi
   365               end;
   366 
   367             val sumEN_thm' =
   368               Local_Defs.unfold lthy @{thms all_unit_eq}
   369                 (Drule.instantiate' (map (SOME o certifyT lthy) ctr_prod_Ts) []
   370                    (mk_sumEN_balanced n))
   371               |> Morphism.thm phi;
   372           in
   373             mk_exhaust_tac ctxt n ctr_defs fld_iff_unf_thm sumEN_thm'
   374           end;
   375 
   376         val inject_tacss =
   377           map2 (fn 0 => K [] | _ => fn ctr_def => [fn {context = ctxt, ...} =>
   378               mk_inject_tac ctxt ctr_def fld_inject]) ms ctr_defs;
   379 
   380         val half_distinct_tacss =
   381           map (map (fn (def, def') => fn {context = ctxt, ...} =>
   382             mk_half_distinct_tac ctxt fld_inject [def, def'])) (mk_half_pairss ctr_defs);
   383 
   384         val case_tacs =
   385           map3 (fn k => fn m => fn ctr_def => fn {context = ctxt, ...} =>
   386             mk_case_tac ctxt n k m case_def ctr_def unf_fld) ks ms ctr_defs;
   387 
   388         val tacss = [exhaust_tac] :: inject_tacss @ half_distinct_tacss @ [case_tacs];
   389 
   390         fun some_lfp_sugar no_defs_lthy =
   391           let
   392             val fpT_to_C = fpT --> C;
   393 
   394             fun generate_iter_like (suf, fp_iter_like, (fss, f_Tss, xssss)) =
   395               let
   396                 val res_T = fold_rev (curry (op --->)) f_Tss fpT_to_C;
   397 
   398                 val binder = Binding.suffix_name ("_" ^ suf) b;
   399 
   400                 val spec =
   401                   mk_Trueprop_eq (lists_bmoc fss (Free (Binding.name_of binder, res_T)),
   402                     Term.list_comb (fp_iter_like,
   403                       map2 (mk_sum_caseN_balanced oo map2 mk_uncurried2_fun) fss xssss));
   404               in (binder, spec) end;
   405 
   406             val iter_likes =
   407               [(iterN, fp_iter, iter_only),
   408                (recN, fp_rec, rec_only)];
   409 
   410             val (binders, specs) = map generate_iter_like iter_likes |> split_list;
   411 
   412             val ((csts, defs), (lthy', lthy)) = no_defs_lthy
   413               |> apfst split_list o fold_map2 (fn b => fn spec =>
   414                 Specification.definition (SOME (b, NONE, NoSyn), ((Thm.def_binding b, []), spec))
   415                 #>> apsnd snd) binders specs
   416               ||> `Local_Theory.restore;
   417 
   418             (*transforms defined frees into consts (and more)*)
   419             val phi = Proof_Context.export_morphism lthy lthy';
   420 
   421             val [iter_def, rec_def] = map (Morphism.thm phi) defs;
   422 
   423             val [iter, recx] = map (mk_iter_like As Cs o Morphism.term phi) csts;
   424           in
   425             ((ctrs, iter, recx, v, xss, ctr_defs, iter_def, rec_def), lthy)
   426           end;
   427 
   428         fun some_gfp_sugar no_defs_lthy =
   429           let
   430             val B_to_fpT = C --> fpT;
   431 
   432             fun generate_coiter_like (suf, fp_iter_like, ((pfss, cfsss), f_sum_prod_Ts, f_prod_Tss,
   433                 pf_Tss)) =
   434               let
   435                 val res_T = fold_rev (curry (op --->)) pf_Tss B_to_fpT;
   436 
   437                 val binder = Binding.suffix_name ("_" ^ suf) b;
   438 
   439                 fun mk_preds_getters_join c n cps sum_prod_T prod_Ts cfss =
   440                   Term.lambda c (mk_IfN sum_prod_T cps
   441                     (map2 (mk_InN_balanced no_defs_lthy sum_prod_T prod_Ts)
   442                       (map HOLogic.mk_tuple cfss) (1 upto n)));
   443 
   444                 val spec =
   445                   mk_Trueprop_eq (lists_bmoc pfss (Free (Binding.name_of binder, res_T)),
   446                     Term.list_comb (fp_iter_like,
   447                       map6 mk_preds_getters_join cs ns cpss f_sum_prod_Ts f_prod_Tss cfsss));
   448               in (binder, spec) end;
   449 
   450             val coiter_likes =
   451               [(coiterN, fp_iter, coiter_only),
   452                (corecN, fp_rec, corec_only)];
   453 
   454             val (binders, specs) = map generate_coiter_like coiter_likes |> split_list;
   455 
   456             val ((csts, defs), (lthy', lthy)) = no_defs_lthy
   457               |> apfst split_list o fold_map2 (fn b => fn spec =>
   458                 Specification.definition (SOME (b, NONE, NoSyn), ((Thm.def_binding b, []), spec))
   459                 #>> apsnd snd) binders specs
   460               ||> `Local_Theory.restore;
   461 
   462             (*transforms defined frees into consts (and more)*)
   463             val phi = Proof_Context.export_morphism lthy lthy';
   464 
   465             val [coiter_def, corec_def] = map (Morphism.thm phi) defs;
   466 
   467             val [coiter, corec] = map (mk_iter_like As Cs o Morphism.term phi) csts;
   468           in
   469             ((ctrs, coiter, corec, v, xss, ctr_defs, coiter_def, corec_def), lthy)
   470           end;
   471       in
   472         wrap_datatype tacss ((ctrs0, casex0), (disc_binders, sel_binderss)) lthy'
   473         |> (if lfp then some_lfp_sugar else some_gfp_sugar)
   474       end;
   475 
   476     val pre_map_defs = map map_def_of_bnf pre_bnfs;
   477     val map_ids = map map_id_of_bnf nested_bnfs;
   478 
   479     fun mk_map Ts Us t =
   480       let val (Type (_, Ts0), Type (_, Us0)) = strip_map_type (fastype_of t) |>> List.last in
   481         Term.subst_atomic_types (Ts0 @ Us0 ~~ Ts @ Us) t
   482       end;
   483 
   484     fun build_map build_arg (Type (s, Ts)) (Type (_, Us)) =
   485       let
   486         val map0 = map_of_bnf (the (bnf_of lthy s));
   487         val mapx = mk_map Ts Us map0;
   488         val TUs = map dest_funT (fst (split_last (fst (strip_map_type (fastype_of mapx)))));
   489         val args = map build_arg TUs;
   490       in Term.list_comb (mapx, args) end;
   491 
   492     fun pour_more_sugar_on_lfps ((ctrss, iters, recs, vs, xsss, ctr_defss, iter_defs, rec_defs),
   493         lthy) =
   494       let
   495         val xctrss = map2 (map2 (curry Term.list_comb)) ctrss xsss;
   496         val giters = map (lists_bmoc gss) iters;
   497         val hrecs = map (lists_bmoc hss) recs;
   498 
   499         val (iter_thmss, rec_thmss) =
   500           let
   501             fun mk_goal_iter_like fss fiter_like xctr f xs fxs =
   502               fold_rev (fold_rev Logic.all) (xs :: fss)
   503                 (mk_Trueprop_eq (fiter_like $ xctr, Term.list_comb (f, fxs)));
   504 
   505             fun build_call fiter_likes maybe_tick (T, U) =
   506               if T = U then
   507                 mk_id T
   508               else
   509                 (case find_index (curry (op =) T) fpTs of
   510                   ~1 => build_map (build_call fiter_likes maybe_tick) T U
   511                 | j => maybe_tick (nth vs j) (nth fiter_likes j));
   512 
   513             fun mk_U maybe_prodT =
   514               typ_subst (map2 (fn fpT => fn C => (fpT, maybe_prodT fpT C)) fpTs Cs);
   515 
   516             fun repair_calls fiter_likes maybe_cons maybe_tick maybe_prodT (x as Free (_, T)) =
   517               if member (op =) fpTs T then
   518                 maybe_cons x [build_call fiter_likes (K I) (T, mk_U (K I) T) $ x]
   519               else if exists_subtype (member (op =) fpTs) T then
   520                 [build_call fiter_likes maybe_tick (T, mk_U maybe_prodT T) $ x]
   521               else
   522                 [x];
   523 
   524             val gxsss = map (map (maps (repair_calls giters (K I) (K I) (K I)))) xsss;
   525             val hxsss =
   526               map (map (maps (repair_calls hrecs cons tick (curry HOLogic.mk_prodT)))) xsss;
   527 
   528             val goal_iterss = map5 (map4 o mk_goal_iter_like gss) giters xctrss gss xsss gxsss;
   529             val goal_recss = map5 (map4 o mk_goal_iter_like hss) hrecs xctrss hss xsss hxsss;
   530 
   531             val iter_tacss =
   532               map2 (map o mk_iter_like_tac pre_map_defs map_ids iter_defs) fp_iter_thms ctr_defss;
   533             val rec_tacss =
   534               map2 (map o mk_iter_like_tac pre_map_defs map_ids rec_defs) fp_rec_thms ctr_defss;
   535           in
   536             (map2 (map2 (fn goal => fn tac => Skip_Proof.prove lthy [] [] goal (tac o #context)))
   537                goal_iterss iter_tacss,
   538              map2 (map2 (fn goal => fn tac => Skip_Proof.prove lthy [] [] goal (tac o #context)))
   539                goal_recss rec_tacss)
   540           end;
   541 
   542         val notes =
   543           [(itersN, iter_thmss),
   544            (recsN, rec_thmss)]
   545           |> maps (fn (thmN, thmss) =>
   546             map2 (fn b => fn thms =>
   547                 ((Binding.qualify true (Binding.name_of b) (Binding.name thmN), []), [(thms, [])]))
   548               bs thmss);
   549       in
   550         lthy |> Local_Theory.notes notes |> snd
   551       end;
   552 
   553     fun pour_more_sugar_on_gfps ((ctrss, coiters, corecs, vs, xsss, ctr_defss, coiter_defs,
   554         corec_defs), lthy) =
   555       let
   556         val z = the_single zs;
   557 
   558         val gcoiters = map (lists_bmoc pgss) coiters;
   559         val hcorecs = map (lists_bmoc phss) corecs;
   560 
   561         val (coiter_thmss, corec_thmss) =
   562           let
   563             fun mk_goal_cond pos = HOLogic.mk_Trueprop o (not pos ? HOLogic.mk_not);
   564 
   565             fun mk_goal_coiter_like pfss c cps fcoiter_like n k ctr m cfs' =
   566               fold_rev (fold_rev Logic.all) ([c] :: pfss)
   567                 (Logic.list_implies (seq_conds mk_goal_cond n k cps,
   568                    mk_Trueprop_eq (fcoiter_like $ c, Term.list_comb (ctr, take m cfs'))));
   569 
   570             fun build_call fiter_likes maybe_tack (T, U) =
   571               if T = U then
   572                 mk_id T
   573               else
   574                 (case find_index (curry (op =) U) fpTs of
   575                   ~1 => build_map (build_call fiter_likes maybe_tack) T U
   576                 | j => maybe_tack (nth cs j, nth vs j) (nth fiter_likes j));
   577 
   578             fun mk_U maybe_sumT =
   579               typ_subst (map2 (fn C => fn fpT => (maybe_sumT fpT C, fpT)) Cs fpTs);
   580 
   581             fun repair_calls fiter_likes maybe_sumT maybe_tack
   582                 (cf as Free (_, Type (_, [_, T])) $ _) =
   583               if exists_subtype (member (op =) Cs) T then
   584                 build_call fiter_likes maybe_tack (T, mk_U maybe_sumT T) $ cf
   585               else
   586                 cf;
   587 
   588             val cgsss' = map (map (map (repair_calls gcoiters (K I) (K I)))) cgsss;
   589             val chsss' = map (map (map (repair_calls hcorecs (curry mk_sumT) (tack z)))) chsss;
   590 
   591             val goal_coiterss =
   592               map8 (map4 oooo mk_goal_coiter_like pgss) cs cpss gcoiters ns kss ctrss mss cgsss';
   593             val goal_corecss =
   594               map8 (map4 oooo mk_goal_coiter_like phss) cs cpss hcorecs ns kss ctrss mss chsss';
   595 
   596             val coiter_tacss =
   597               map3 (map oo mk_coiter_like_tac coiter_defs map_ids) fp_iter_thms pre_map_defs
   598                 ctr_defss;
   599             val corec_tacss =
   600               map3 (map oo mk_coiter_like_tac corec_defs map_ids) fp_rec_thms pre_map_defs
   601                 ctr_defss;
   602           in
   603             (map2 (map2 (fn goal => fn tac => Skip_Proof.prove lthy [] [] goal (tac o #context)))
   604                goal_coiterss coiter_tacss,
   605              map2 (map2 (fn goal => fn tac => Skip_Proof.prove lthy [] [] goal (tac o #context)))
   606                goal_corecss corec_tacss)
   607           end;
   608 
   609         val notes =
   610           [(coitersN, coiter_thmss),
   611            (corecsN, corec_thmss)]
   612           |> maps (fn (thmN, thmss) =>
   613             map2 (fn b => fn thms =>
   614                 ((Binding.qualify true (Binding.name_of b) (Binding.name thmN), []), [(thms, [])]))
   615               bs thmss);
   616       in
   617         lthy |> Local_Theory.notes notes |> snd
   618       end;
   619 
   620     val lthy' = lthy
   621       |> fold_map pour_some_sugar_on_type (bs ~~ fpTs ~~ Cs ~~ flds ~~ unfs ~~ fp_iters ~~
   622         fp_recs ~~ fld_unfs ~~ unf_flds ~~ fld_injects ~~ ns ~~ kss ~~ mss ~~ ctr_binderss ~~
   623         ctr_mixfixess ~~ ctr_Tsss ~~ disc_binderss ~~ sel_bindersss)
   624       |>> split_list8
   625       |> (if lfp then pour_more_sugar_on_lfps else pour_more_sugar_on_gfps);
   626 
   627     val timer = time (timer ("Constructors, discriminators, selectors, etc., for the new " ^
   628       (if lfp then "" else "co") ^ "datatype"));
   629   in
   630     (timer; lthy')
   631   end;
   632 
   633 fun datatype_cmd info specs lthy =
   634   let
   635     (* TODO: cleaner handling of fake contexts, without "background_theory" *)
   636     (*the "perhaps o try" below helps gracefully handles the case where the new type is defined in a
   637       locale and shadows an existing global type*)
   638     val fake_thy = Theory.copy
   639       #> fold (fn spec => perhaps (try (Sign.add_type lthy
   640         (type_binder_of spec, length (type_args_constrained_of spec), mixfix_of spec)))) specs;
   641     val fake_lthy = Proof_Context.background_theory fake_thy lthy;
   642   in
   643     prepare_datatype Syntax.read_typ info specs fake_lthy lthy
   644   end;
   645 
   646 val parse_opt_binding_colon = Scan.optional (Parse.binding --| Parse.$$$ ":") no_binder
   647 
   648 val parse_ctr_arg =
   649   Parse.$$$ "(" |-- parse_opt_binding_colon -- Parse.typ --| Parse.$$$ ")" ||
   650   (Parse.typ >> pair no_binder);
   651 
   652 val parse_single_spec =
   653   Parse.type_args_constrained -- Parse.binding -- Parse.opt_mixfix --
   654   (@{keyword "="} |-- Parse.enum1 "|" (parse_opt_binding_colon -- Parse.binding --
   655     Scan.repeat parse_ctr_arg -- Parse.opt_mixfix));
   656 
   657 val _ =
   658   Outer_Syntax.local_theory @{command_spec "data"} "define BNF-based inductive datatypes"
   659     (Parse.and_list1 parse_single_spec >> datatype_cmd true);
   660 
   661 val _ =
   662   Outer_Syntax.local_theory @{command_spec "codata"} "define BNF-based coinductive datatypes"
   663     (Parse.and_list1 parse_single_spec >> datatype_cmd false);
   664 
   665 end;