src/HOL/BNF/Tools/bnf_fp_def_sugar.ML
author blanchet
Tue Oct 02 01:00:18 2012 +0200 (2012-10-02)
changeset 49682 f57af1c46f99
parent 49681 aa66ea552357
child 49683 78a3d5006cf1
permissions -rw-r--r--
removed dead params and dead code
     1 (*  Title:      HOL/BNF/Tools/bnf_fp_def_sugar.ML
     2     Author:     Jasmin Blanchette, TU Muenchen
     3     Copyright   2012
     4 
     5 Sugared datatype and codatatype constructions.
     6 *)
     7 
     8 signature BNF_FP_DEF_SUGAR =
     9 sig
    10   val datatypes: bool ->
    11     (mixfix list -> (string * sort) list option -> binding list -> typ list * typ list list ->
    12       BNF_Def.BNF list -> local_theory -> BNF_FP.fp_result * local_theory) ->
    13     (bool * bool) * ((((typ * sort) list * binding) * mixfix) * ((((binding * binding) *
    14       (binding * typ) list) * (binding * term) list) * mixfix) list) list ->
    15     local_theory -> local_theory
    16   val parse_datatype_cmd: bool ->
    17     (mixfix list -> (string * sort) list option -> binding list -> typ list * typ list list ->
    18       BNF_Def.BNF list -> local_theory -> BNF_FP.fp_result * local_theory) ->
    19     (local_theory -> local_theory) parser
    20 end;
    21 
    22 structure BNF_FP_Def_Sugar : BNF_FP_DEF_SUGAR =
    23 struct
    24 
    25 open BNF_Util
    26 open BNF_Wrap
    27 open BNF_Def
    28 open BNF_FP
    29 open BNF_FP_Def_Sugar_Tactics
    30 
    31 (* This function could produce clashes in contrived examples (e.g., "x.A", "x.x_A", "y.A") *)
    32 fun quasi_unambiguous_case_names names =
    33   let
    34     val ps = map (`Long_Name.base_name) names;
    35     val dups = Library.duplicates (op =) (map fst ps);
    36     fun underscore s =
    37       let val ss = space_explode Long_Name.separator s in
    38         space_implode "_" (drop (length ss - 2) ss)
    39       end;
    40   in
    41     map (fn (base, full) => if member (op =) dups base then underscore full else base) ps
    42   end;
    43 
    44 val mp_conj = @{thm mp_conj};
    45 
    46 val simp_attrs = @{attributes [simp]};
    47 val code_simp_attrs = Code.add_default_eqn_attrib :: simp_attrs;
    48 
    49 fun split_list4 xs = (map #1 xs, map #2 xs, map #3 xs, map #4 xs);
    50 
    51 fun resort_tfree S (TFree (s, _)) = TFree (s, S);
    52 
    53 fun typ_subst inst (T as Type (s, Ts)) =
    54     (case AList.lookup (op =) inst T of
    55       NONE => Type (s, map (typ_subst inst) Ts)
    56     | SOME T' => T')
    57   | typ_subst inst T = the_default T (AList.lookup (op =) inst T);
    58 
    59 fun variant_types ss Ss ctxt =
    60   let
    61     val (tfrees, _) =
    62       fold_map2 (fn s => fn S => Name.variant s #> apfst (rpair S)) ss Ss (Variable.names_of ctxt);
    63     val ctxt' = fold (Variable.declare_constraints o Logic.mk_type o TFree) tfrees ctxt;
    64   in (tfrees, ctxt') end;
    65 
    66 val lists_bmoc = fold (fn xs => fn t => Term.list_comb (t, xs));
    67 
    68 fun mk_tupled_fun x f xs = HOLogic.tupled_lambda x (Term.list_comb (f, xs));
    69 fun mk_uncurried_fun f xs = mk_tupled_fun (HOLogic.mk_tuple xs) f xs;
    70 
    71 fun mk_flip (x, Type (_, [T1, Type (_, [T2, T3])])) =
    72   Abs ("x", T1, Abs ("y", T2, Var (x, T2 --> T1 --> T3) $ Bound 0 $ Bound 1));
    73 
    74 fun flip_rels lthy n thm =
    75   let
    76     val Rs = Term.add_vars (prop_of thm) [];
    77     val Rs' = rev (drop (length Rs - n) Rs);
    78     val cRs = map (fn f => (certify lthy (Var f), certify lthy (mk_flip f))) Rs';
    79   in
    80     Drule.cterm_instantiate cRs thm
    81   end;
    82 
    83 fun mk_ctor_or_dtor get_T Ts t =
    84   let val Type (_, Ts0) = get_T (fastype_of t) in
    85     Term.subst_atomic_types (Ts0 ~~ Ts) t
    86   end;
    87 
    88 val mk_ctor = mk_ctor_or_dtor range_type;
    89 val mk_dtor = mk_ctor_or_dtor domain_type;
    90 
    91 fun mk_rec_like lfp Ts Us t =
    92   let
    93     val (bindings, body) = strip_type (fastype_of t);
    94     val (f_Us, prebody) = split_last bindings;
    95     val Type (_, Ts0) = if lfp then prebody else body;
    96     val Us0 = distinct (op =) (map (if lfp then body_type else domain_type) f_Us);
    97   in
    98     Term.subst_atomic_types (Ts0 @ Us0 ~~ Ts @ Us) t
    99   end;
   100 
   101 fun mk_map live Ts Us t =
   102   let val (Type (_, Ts0), Type (_, Us0)) = strip_typeN (live + 1) (fastype_of t) |>> List.last in
   103     Term.subst_atomic_types (Ts0 @ Us0 ~~ Ts @ Us) t
   104   end;
   105 
   106 fun mk_rel live Ts Us t =
   107   let val [Type (_, Ts0), Type (_, Us0)] = binder_types (snd (strip_typeN live (fastype_of t))) in
   108     Term.subst_atomic_types (Ts0 @ Us0 ~~ Ts @ Us) t
   109   end;
   110 
   111 fun liveness_of_fp_bnf n bnf =
   112   (case T_of_bnf bnf of
   113     Type (_, Ts) => map (not o member (op =) (deads_of_bnf bnf)) Ts
   114   | _ => replicate n false);
   115 
   116 fun tack z_name (c, u) f =
   117   let val z = Free (z_name, mk_sumT (fastype_of u, fastype_of c)) in
   118     Term.lambda z (mk_sum_case (Term.lambda u u, Term.lambda c (f $ c)) $ z)
   119   end;
   120 
   121 fun cannot_merge_types () = error "Mutually recursive types must have the same type parameters";
   122 
   123 fun merge_type_arg T T' = if T = T' then T else cannot_merge_types ();
   124 
   125 fun merge_type_args (As, As') =
   126   if length As = length As' then map2 merge_type_arg As As' else cannot_merge_types ();
   127 
   128 fun reassoc_conjs thm =
   129   reassoc_conjs (thm RS @{thm conj_assoc[THEN iffD1]})
   130   handle THM _ => thm;
   131 
   132 fun type_args_constrained_of (((cAs, _), _), _) = cAs;
   133 fun type_binding_of (((_, b), _), _) = b;
   134 fun mixfix_of ((_, mx), _) = mx;
   135 fun ctr_specs_of (_, ctr_specs) = ctr_specs;
   136 
   137 fun disc_of ((((disc, _), _), _), _) = disc;
   138 fun ctr_of ((((_, ctr), _), _), _) = ctr;
   139 fun args_of (((_, args), _), _) = args;
   140 fun defaults_of ((_, ds), _) = ds;
   141 fun ctr_mixfix_of (_, mx) = mx;
   142 
   143 fun define_datatypes prepare_constraint prepare_typ prepare_term lfp construct_fp
   144     (wrap_opts as (no_dests, rep_compat), specs) no_defs_lthy0 =
   145   let
   146     (* TODO: sanity checks on arguments *)
   147     (* TODO: integration with function package ("size") *)
   148 
   149     val _ = if not lfp andalso no_dests then error "Cannot define destructor-less codatatypes"
   150       else ();
   151 
   152     fun qualify mandatory fp_b_name =
   153       Binding.qualify mandatory fp_b_name o (rep_compat ? Binding.qualify false rep_compat_prefix);
   154 
   155     val nn = length specs;
   156     val fp_bs = map type_binding_of specs;
   157     val fp_b_names = map Binding.name_of fp_bs;
   158     val fp_common_name = mk_common_name fp_b_names;
   159 
   160     fun prepare_type_arg (ty, c) =
   161       let val TFree (s, _) = prepare_typ no_defs_lthy0 ty in
   162         TFree (s, prepare_constraint no_defs_lthy0 c)
   163       end;
   164 
   165     val Ass0 = map (map prepare_type_arg o type_args_constrained_of) specs;
   166     val unsorted_Ass0 = map (map (resort_tfree HOLogic.typeS)) Ass0;
   167     val unsorted_As = Library.foldr1 merge_type_args unsorted_Ass0;
   168 
   169     val (((Bs0, Cs), Xs), no_defs_lthy) =
   170       no_defs_lthy0
   171       |> fold (Variable.declare_typ o resort_tfree dummyS) unsorted_As
   172       |> mk_TFrees (length unsorted_As)
   173       ||>> mk_TFrees nn
   174       ||>> apfst (map TFree) o
   175         variant_types (map (prefix "'") fp_b_names) (replicate nn HOLogic.typeS);
   176 
   177     (* TODO: cleaner handling of fake contexts, without "background_theory" *)
   178     (*the "perhaps o try" below helps gracefully handles the case where the new type is defined in a
   179       locale and shadows an existing global type*)
   180     val fake_thy =
   181       Theory.copy #> fold (fn spec => perhaps (try (Sign.add_type no_defs_lthy
   182         (type_binding_of spec, length (type_args_constrained_of spec), mixfix_of spec)))) specs;
   183     val fake_lthy = Proof_Context.background_theory fake_thy no_defs_lthy;
   184 
   185     fun mk_fake_T b =
   186       Type (fst (Term.dest_Type (Proof_Context.read_type_name fake_lthy true (Binding.name_of b))),
   187         unsorted_As);
   188 
   189     val fake_Ts = map mk_fake_T fp_bs;
   190 
   191     val mixfixes = map mixfix_of specs;
   192 
   193     val _ = (case duplicates Binding.eq_name fp_bs of [] => ()
   194       | b :: _ => error ("Duplicate type name declaration " ^ quote (Binding.name_of b)));
   195 
   196     val ctr_specss = map ctr_specs_of specs;
   197 
   198     val disc_bindingss = map (map disc_of) ctr_specss;
   199     val ctr_bindingss =
   200       map2 (fn fp_b_name => map (qualify false fp_b_name o ctr_of)) fp_b_names ctr_specss;
   201     val ctr_argsss = map (map args_of) ctr_specss;
   202     val ctr_mixfixess = map (map ctr_mixfix_of) ctr_specss;
   203 
   204     val sel_bindingsss = map (map (map fst)) ctr_argsss;
   205     val fake_ctr_Tsss0 = map (map (map (prepare_typ fake_lthy o snd))) ctr_argsss;
   206     val raw_sel_defaultsss = map (map defaults_of) ctr_specss;
   207 
   208     val (As :: _) :: fake_ctr_Tsss =
   209       burrow (burrow (Syntax.check_typs fake_lthy)) (Ass0 :: fake_ctr_Tsss0);
   210 
   211     val _ = (case duplicates (op =) unsorted_As of [] => ()
   212       | A :: _ => error ("Duplicate type parameter " ^
   213           quote (Syntax.string_of_typ no_defs_lthy A)));
   214 
   215     val rhs_As' = fold (fold (fold Term.add_tfreesT)) fake_ctr_Tsss [];
   216     val _ = (case subtract (op =) (map dest_TFree As) rhs_As' of
   217         [] => ()
   218       | A' :: _ => error ("Extra type variable on right-hand side: " ^
   219           quote (Syntax.string_of_typ no_defs_lthy (TFree A'))));
   220 
   221     fun eq_fpT_check (T as Type (s, Us)) (Type (s', Us')) =
   222         s = s' andalso (Us = Us' orelse error ("Illegal occurrence of recursive type " ^
   223           quote (Syntax.string_of_typ fake_lthy T)))
   224       | eq_fpT_check _ _ = false;
   225 
   226     fun freeze_fp (T as Type (s, Us)) =
   227         (case find_index (eq_fpT_check T) fake_Ts of
   228           ~1 => Type (s, map freeze_fp Us)
   229         | kk => nth Xs kk)
   230       | freeze_fp T = T;
   231 
   232     val ctr_TsssXs = map (map (map freeze_fp)) fake_ctr_Tsss;
   233     val ctr_sum_prod_TsXs = map (mk_sumTN_balanced o map HOLogic.mk_tupleT) ctr_TsssXs;
   234 
   235     val fp_eqs =
   236       map dest_TFree Xs ~~ map (Term.typ_subst_atomic (As ~~ unsorted_As)) ctr_sum_prod_TsXs;
   237 
   238     (* TODO: clean up list *)
   239     val (pre_bnfs, ((fp_bnfs as any_fp_bnf :: _, dtors0, ctors0, fp_folds0, fp_recs0, fp_induct,
   240            fp_strong_induct, dtor_ctors, ctor_dtors, ctor_injects, fp_map_thms, fp_set_thmss,
   241            fp_rel_thms, fp_fold_thms, fp_rec_thms), lthy)) =
   242       fp_bnf construct_fp fp_bs mixfixes (map dest_TFree unsorted_As) fp_eqs no_defs_lthy0;
   243 
   244     val timer = time (Timer.startRealTimer ());
   245 
   246     fun build_map build_arg (Type (s, Ts)) (Type (_, Us)) =
   247       let
   248         val bnf = the (bnf_of lthy s);
   249         val live = live_of_bnf bnf;
   250         val mapx = mk_map live Ts Us (map_of_bnf bnf);
   251         val TUs' = map dest_funT (fst (strip_typeN live (fastype_of mapx)));
   252       in Term.list_comb (mapx, map build_arg TUs') end;
   253 
   254     fun build_rel_step build_arg (Type (s, Ts)) =
   255       let
   256         val bnf = the (bnf_of lthy s);
   257         val live = live_of_bnf bnf;
   258         val rel = mk_rel live Ts Ts (rel_of_bnf bnf);
   259         val Ts' = map domain_type (fst (strip_typeN live (fastype_of rel)));
   260       in Term.list_comb (rel, map build_arg Ts') end;
   261 
   262     fun add_nesty_bnf_names Us =
   263       let
   264         fun add (Type (s, Ts)) ss =
   265             let val (needs, ss') = fold_map add Ts ss in
   266               if exists I needs then (true, insert (op =) s ss') else (false, ss')
   267             end
   268           | add T ss = (member (op =) Us T, ss);
   269       in snd oo add end;
   270 
   271     fun nesty_bnfs Us =
   272       map_filter (bnf_of lthy) (fold (fold (fold (add_nesty_bnf_names Us))) ctr_TsssXs []);
   273 
   274     val nesting_bnfs = nesty_bnfs As;
   275     val nested_bnfs = nesty_bnfs Xs;
   276 
   277     val pre_map_defs = map map_def_of_bnf pre_bnfs;
   278     val pre_set_defss = map set_defs_of_bnf pre_bnfs;
   279     val pre_rel_defs = map rel_def_of_bnf pre_bnfs;
   280     val nested_map_comp's = map map_comp'_of_bnf nested_bnfs;
   281     val nested_map_ids'' = map (unfold_thms lthy @{thms id_def} o map_id_of_bnf) nested_bnfs;
   282     val nesting_map_ids'' = map (unfold_thms lthy @{thms id_def} o map_id_of_bnf) nesting_bnfs;
   283     val nested_set_natural's = maps set_natural'_of_bnf nested_bnfs;
   284     val nesting_set_natural's = maps set_natural'_of_bnf nesting_bnfs;
   285 
   286     val live = live_of_bnf any_fp_bnf;
   287 
   288     val Bs =
   289       map3 (fn alive => fn A as TFree (_, S) => fn B => if alive then resort_tfree S B else A)
   290         (liveness_of_fp_bnf (length As) any_fp_bnf) As Bs0;
   291 
   292     val B_ify = Term.typ_subst_atomic (As ~~ Bs);
   293 
   294     val ctors = map (mk_ctor As) ctors0;
   295     val dtors = map (mk_dtor As) dtors0;
   296 
   297     val fpTs = map (domain_type o fastype_of) dtors;
   298 
   299     val exists_fp_subtype = exists_subtype (member (op =) fpTs);
   300     val exists_Cs_subtype = exists_subtype (member (op =) Cs);
   301 
   302     val ctr_Tsss = map (map (map (Term.typ_subst_atomic (Xs ~~ fpTs)))) ctr_TsssXs;
   303     val ns = map length ctr_Tsss;
   304     val kss = map (fn n => 1 upto n) ns;
   305     val mss = map (map length) ctr_Tsss;
   306     val Css = map2 replicate ns Cs;
   307 
   308     val fp_folds as any_fp_fold :: _ = map (mk_rec_like lfp As Cs) fp_folds0;
   309     val fp_recs as any_fp_rec :: _ = map (mk_rec_like lfp As Cs) fp_recs0;
   310 
   311     val fp_fold_fun_Ts = fst (split_last (binder_types (fastype_of any_fp_fold)));
   312     val fp_rec_fun_Ts = fst (split_last (binder_types (fastype_of any_fp_rec)));
   313 
   314     val (((fold_only as (gss, _, _), rec_only as (hss, _, _)),
   315           (zs, cs, cpss, unfold_only as ((pgss, crgsss), _), corec_only as ((phss, cshsss), _))),
   316          names_lthy0) =
   317       if lfp then
   318         let
   319           val y_Tsss =
   320             map3 (fn n => fn ms => map2 dest_tupleT ms o dest_sumTN_balanced n o domain_type)
   321               ns mss fp_fold_fun_Ts;
   322           val g_Tss = map2 (map2 (curry (op --->))) y_Tsss Css;
   323 
   324           val ((gss, ysss), lthy) =
   325             lthy
   326             |> mk_Freess "f" g_Tss
   327             ||>> mk_Freesss "x" y_Tsss;
   328 
   329           fun proj_recT proj (Type (s as @{type_name prod}, Ts as [T, U])) =
   330               if member (op =) fpTs T then proj (T, U) else Type (s, map (proj_recT proj) Ts)
   331             | proj_recT proj (Type (s, Ts)) = Type (s, map (proj_recT proj) Ts)
   332             | proj_recT _ T = T;
   333 
   334           fun unzip_recT T =
   335             if exists_fp_subtype T then [proj_recT fst T, proj_recT snd T] else [T];
   336 
   337           val z_Tsss =
   338             map3 (fn n => fn ms => map2 dest_tupleT ms o dest_sumTN_balanced n o domain_type)
   339               ns mss fp_rec_fun_Ts;
   340           val z_Tssss = map (map (map unzip_recT)) z_Tsss;
   341           val h_Tss = map2 (map2 (fold_rev (curry (op --->)))) z_Tssss Css;
   342 
   343           val hss = map2 (map2 retype_free) h_Tss gss;
   344           val zsss = map2 (map2 (map2 retype_free)) z_Tsss ysss;
   345         in
   346           ((((gss, g_Tss, ysss), (hss, h_Tss, zsss)),
   347             ([], [], [], (([], []), ([], [])), (([], []), ([], [])))), lthy)
   348         end
   349       else
   350         let
   351           (*avoid "'a itself" arguments in coiterators and corecursors*)
   352           val mss' =  map (fn [0] => [1] | ms => ms) mss;
   353 
   354           val p_Tss = map2 (fn n => replicate (Int.max (0, n - 1)) o mk_pred1T) ns Cs;
   355 
   356           fun flat_predss_getterss qss fss = maps (op @) (qss ~~ fss);
   357 
   358           fun flat_preds_predsss_gettersss [] [qss] [fss] = flat_predss_getterss qss fss
   359             | flat_preds_predsss_gettersss (p :: ps) (qss :: qsss) (fss :: fsss) =
   360               p :: flat_predss_getterss qss fss @ flat_preds_predsss_gettersss ps qsss fsss;
   361 
   362           fun mk_types maybe_unzipT fun_Ts =
   363             let
   364               val f_sum_prod_Ts = map range_type fun_Ts;
   365               val f_prod_Tss = map2 dest_sumTN_balanced ns f_sum_prod_Ts;
   366               val f_Tsss = map2 (map2 dest_tupleT) mss' f_prod_Tss;
   367               val f_Tssss =
   368                 map2 (fn C => map (map (map (curry (op -->) C) o maybe_unzipT))) Cs f_Tsss;
   369               val q_Tssss =
   370                 map (map (map (fn [_] => [] | [_, C] => [mk_pred1T (domain_type C)]))) f_Tssss;
   371               val pf_Tss = map3 flat_preds_predsss_gettersss p_Tss q_Tssss f_Tssss;
   372             in (q_Tssss, f_sum_prod_Ts, f_Tsss, f_Tssss, pf_Tss) end;
   373 
   374           val (r_Tssss, g_sum_prod_Ts, g_Tsss, g_Tssss, pg_Tss) = mk_types single fp_fold_fun_Ts;
   375 
   376           val ((((Free (z, _), cs), pss), gssss), lthy) =
   377             lthy
   378             |> yield_singleton (mk_Frees "z") dummyT
   379             ||>> mk_Frees "a" Cs
   380             ||>> mk_Freess "p" p_Tss
   381             ||>> mk_Freessss "g" g_Tssss;
   382           val rssss = map (map (map (fn [] => []))) r_Tssss;
   383 
   384           fun proj_corecT proj (Type (s as @{type_name sum}, Ts as [T, U])) =
   385               if member (op =) fpTs T then proj (T, U) else Type (s, map (proj_corecT proj) Ts)
   386             | proj_corecT proj (Type (s, Ts)) = Type (s, map (proj_corecT proj) Ts)
   387             | proj_corecT _ T = T;
   388 
   389           fun unzip_corecT T =
   390             if exists_fp_subtype T then [proj_corecT fst T, proj_corecT snd T] else [T];
   391 
   392           val (s_Tssss, h_sum_prod_Ts, h_Tsss, h_Tssss, ph_Tss) =
   393             mk_types unzip_corecT fp_rec_fun_Ts;
   394 
   395           val hssss_hd = map2 (map2 (map2 (fn T :: _ => fn [g] => retype_free T g))) h_Tssss gssss;
   396           val ((sssss, hssss_tl), lthy) =
   397             lthy
   398             |> mk_Freessss "q" s_Tssss
   399             ||>> mk_Freessss "h" (map (map (map tl)) h_Tssss);
   400           val hssss = map2 (map2 (map2 cons)) hssss_hd hssss_tl;
   401 
   402           val cpss = map2 (map o rapp) cs pss;
   403 
   404           fun build_sum_inj mk_inj (T, U) =
   405             if T = U then
   406               id_const T
   407             else
   408               (case (T, U) of
   409                 (Type (s, _), Type (s', _)) =>
   410                 if s = s' then build_map (build_sum_inj mk_inj) T U
   411                 else uncurry mk_inj (dest_sumT U)
   412               | _ => uncurry mk_inj (dest_sumT U));
   413 
   414           fun build_dtor_corec_arg _ [] [cf] = cf
   415             | build_dtor_corec_arg T [cq] [cf, cf'] =
   416               mk_If cq (build_sum_inj Inl_const (fastype_of cf, T) $ cf)
   417                 (build_sum_inj Inr_const (fastype_of cf', T) $ cf')
   418 
   419           fun mk_terms f_Tsss qssss fssss =
   420             let
   421               val pfss = map3 flat_preds_predsss_gettersss pss qssss fssss;
   422               val cqssss = map2 (map o map o map o rapp) cs qssss;
   423               val cfssss = map2 (map o map o map o rapp) cs fssss;
   424               val cqfsss = map3 (map3 (map3 build_dtor_corec_arg)) f_Tsss cqssss cfssss;
   425             in (pfss, cqfsss) end;
   426         in
   427           (((([], [], []), ([], [], [])),
   428             ([z], cs, cpss, (mk_terms g_Tsss rssss gssss, (g_sum_prod_Ts, pg_Tss)),
   429              (mk_terms h_Tsss sssss hssss, (h_sum_prod_Ts, ph_Tss)))), lthy)
   430         end;
   431 
   432     fun define_ctrs_case_for_type (((((((((((((((((((((((((fp_bnf, fp_b), fpT), C), ctor), dtor),
   433             fp_fold), fp_rec), ctor_dtor), dtor_ctor), ctor_inject), pre_map_def), pre_set_defs),
   434           pre_rel_def), fp_map_thm), fp_set_thms), fp_rel_thm), n), ks), ms), ctr_bindings),
   435         ctr_mixfixes), ctr_Tss), disc_bindings), sel_bindingss), raw_sel_defaultss) no_defs_lthy =
   436       let
   437         val fp_b_name = Binding.name_of fp_b;
   438 
   439         val dtorT = domain_type (fastype_of ctor);
   440         val ctr_prod_Ts = map HOLogic.mk_tupleT ctr_Tss;
   441         val ctr_sum_prod_T = mk_sumTN_balanced ctr_prod_Ts;
   442         val case_Ts = map (fn Ts => Ts ---> C) ctr_Tss;
   443 
   444         val (((((w, fs), xss), yss), u'), names_lthy) =
   445           no_defs_lthy
   446           |> yield_singleton (mk_Frees "w") dtorT
   447           ||>> mk_Frees "f" case_Ts
   448           ||>> mk_Freess "x" ctr_Tss
   449           ||>> mk_Freess "y" (map (map B_ify) ctr_Tss)
   450           ||>> yield_singleton Variable.variant_fixes fp_b_name;
   451 
   452         val u = Free (u', fpT);
   453 
   454         val tuple_xs = map HOLogic.mk_tuple xss;
   455         val tuple_ys = map HOLogic.mk_tuple yss;
   456 
   457         val ctr_rhss =
   458           map3 (fn k => fn xs => fn tuple_x => fold_rev Term.lambda xs (ctor $
   459             mk_InN_balanced ctr_sum_prod_T n tuple_x k)) ks xss tuple_xs;
   460 
   461         val case_binding = qualify false fp_b_name (Binding.suffix_name ("_" ^ caseN) fp_b);
   462 
   463         val case_rhs =
   464           fold_rev Term.lambda (fs @ [u])
   465             (mk_sum_caseN_balanced (map2 mk_uncurried_fun fs xss) $ (dtor $ u));
   466 
   467         val ((raw_case :: raw_ctrs, raw_case_def :: raw_ctr_defs), (lthy', lthy)) = no_defs_lthy
   468           |> apfst split_list o fold_map3 (fn b => fn mx => fn rhs =>
   469               Local_Theory.define ((b, mx), ((Thm.def_binding b, []), rhs)) #>> apsnd snd)
   470             (case_binding :: ctr_bindings) (NoSyn :: ctr_mixfixes) (case_rhs :: ctr_rhss)
   471           ||> `Local_Theory.restore;
   472 
   473         val phi = Proof_Context.export_morphism lthy lthy';
   474 
   475         val ctr_defs = map (Morphism.thm phi) raw_ctr_defs;
   476         val ctr_defs' =
   477           map2 (fn m => fn def => mk_unabs_def m (def RS meta_eq_to_obj_eq)) ms ctr_defs;
   478         val case_def = Morphism.thm phi raw_case_def;
   479 
   480         val ctrs0 = map (Morphism.term phi) raw_ctrs;
   481         val casex0 = Morphism.term phi raw_case;
   482 
   483         val ctrs = map (mk_ctr As) ctrs0;
   484 
   485         fun wrap lthy =
   486           let
   487             fun exhaust_tac {context = ctxt, ...} =
   488               let
   489                 val ctor_iff_dtor_thm =
   490                   let
   491                     val goal =
   492                       fold_rev Logic.all [w, u]
   493                         (mk_Trueprop_eq (HOLogic.mk_eq (u, ctor $ w), HOLogic.mk_eq (dtor $ u, w)));
   494                   in
   495                     Skip_Proof.prove lthy [] [] goal (fn {context = ctxt, ...} =>
   496                       mk_ctor_iff_dtor_tac ctxt (map (SOME o certifyT lthy) [dtorT, fpT])
   497                         (certify lthy ctor) (certify lthy dtor) ctor_dtor dtor_ctor)
   498                     |> Thm.close_derivation
   499                     |> Morphism.thm phi
   500                   end;
   501 
   502                 val sumEN_thm' =
   503                   unfold_thms lthy @{thms all_unit_eq}
   504                     (Drule.instantiate' (map (SOME o certifyT lthy) ctr_prod_Ts) []
   505                        (mk_sumEN_balanced n))
   506                   |> Morphism.thm phi;
   507               in
   508                 mk_exhaust_tac ctxt n ctr_defs ctor_iff_dtor_thm sumEN_thm'
   509               end;
   510 
   511             val inject_tacss =
   512               map2 (fn 0 => K [] | _ => fn ctr_def => [fn {context = ctxt, ...} =>
   513                   mk_inject_tac ctxt ctr_def ctor_inject]) ms ctr_defs;
   514 
   515             val half_distinct_tacss =
   516               map (map (fn (def, def') => fn {context = ctxt, ...} =>
   517                 mk_half_distinct_tac ctxt ctor_inject [def, def'])) (mk_half_pairss (`I ctr_defs));
   518 
   519             val case_tacs =
   520               map3 (fn k => fn m => fn ctr_def => fn {context = ctxt, ...} =>
   521                 mk_case_tac ctxt n k m case_def ctr_def dtor_ctor) ks ms ctr_defs;
   522 
   523             val tacss = [exhaust_tac] :: inject_tacss @ half_distinct_tacss @ [case_tacs];
   524 
   525             val sel_defaultss = map (map (apsnd (prepare_term lthy))) raw_sel_defaultss
   526           in
   527             wrap_datatype tacss (((wrap_opts, ctrs0), casex0), (disc_bindings, (sel_bindingss,
   528               sel_defaultss))) lthy
   529           end;
   530 
   531         fun derive_maps_sets_rels (wrap_res, lthy) =
   532           let
   533             val rel_flip = rel_flip_of_bnf fp_bnf;
   534             val nones = replicate live NONE;
   535 
   536             val ctor_cong =
   537               if lfp then Drule.dummy_thm
   538               else cterm_instantiate_pos [NONE, NONE, SOME (certify lthy ctor)] arg_cong;
   539 
   540             fun mk_cIn ify =
   541               certify lthy o (not lfp ? curry (op $) (map_types ify ctor)) oo
   542               mk_InN_balanced (ify ctr_sum_prod_T) n;
   543 
   544             val cxIns = map2 (mk_cIn I) tuple_xs ks;
   545             val cyIns = map2 (mk_cIn B_ify) tuple_ys ks;
   546 
   547             fun mk_map_thm ctr_def' cxIn =
   548               fold_thms lthy [ctr_def']
   549                 (unfold_thms lthy (pre_map_def ::
   550                      (if lfp then [] else [ctor_dtor, dtor_ctor]) @ sum_prod_thms_map)
   551                    (cterm_instantiate_pos (nones @ [SOME cxIn])
   552                       (if lfp then fp_map_thm else fp_map_thm RS ctor_cong)))
   553               |> singleton (Proof_Context.export names_lthy no_defs_lthy);
   554 
   555             fun mk_set_thm fp_set_thm ctr_def' cxIn =
   556               fold_thms lthy [ctr_def']
   557                 (unfold_thms lthy (pre_set_defs @ nested_set_natural's @ nesting_set_natural's @
   558                      (if lfp then [] else [dtor_ctor]) @ sum_prod_thms_set)
   559                    (cterm_instantiate_pos [SOME cxIn] fp_set_thm))
   560               |> singleton (Proof_Context.export names_lthy no_defs_lthy);
   561 
   562             fun mk_set_thms fp_set_thm = map2 (mk_set_thm fp_set_thm) ctr_defs' cxIns;
   563 
   564             val map_thms = map2 mk_map_thm ctr_defs' cxIns;
   565             val set_thmss = map mk_set_thms fp_set_thms;
   566 
   567             val rel_infos = (ctr_defs' ~~ cxIns, ctr_defs' ~~ cyIns);
   568 
   569             fun mk_rel_thm postproc ctr_defs' cxIn cyIn =
   570               fold_thms lthy ctr_defs'
   571                  (unfold_thms lthy (pre_rel_def :: (if lfp then [] else [dtor_ctor]) @
   572                       sum_prod_thms_rel)
   573                     (cterm_instantiate_pos (nones @ [SOME cxIn, SOME cyIn]) fp_rel_thm))
   574               |> postproc
   575               |> singleton (Proof_Context.export names_lthy no_defs_lthy);
   576 
   577             fun mk_rel_inject_thm ((ctr_def', cxIn), (_, cyIn)) =
   578               mk_rel_thm (unfold_thms lthy @{thms eq_sym_Unity_conv}) [ctr_def'] cxIn cyIn;
   579 
   580             val rel_inject_thms = map mk_rel_inject_thm (op ~~ rel_infos);
   581 
   582             fun mk_half_rel_distinct_thm ((xctr_def', cxIn), (yctr_def', cyIn)) =
   583               mk_rel_thm (fn thm => thm RS @{thm eq_False[THEN iffD1]}) [xctr_def', yctr_def']
   584                 cxIn cyIn;
   585 
   586             fun mk_other_half_rel_distinct_thm thm =
   587               flip_rels lthy live thm RS (rel_flip RS sym RS @{thm arg_cong[of _ _ Not]} RS iffD2);
   588 
   589             val half_rel_distinct_thmss =
   590               map (map mk_half_rel_distinct_thm) (mk_half_pairss rel_infos);
   591             val other_half_rel_distinct_thmss =
   592               map (map mk_other_half_rel_distinct_thm) half_rel_distinct_thmss;
   593             val (rel_distinct_thms, _) =
   594               join_halves n half_rel_distinct_thmss other_half_rel_distinct_thmss;
   595 
   596             val notes =
   597               [(mapN, map_thms, code_simp_attrs),
   598                (rel_distinctN, rel_distinct_thms, code_simp_attrs),
   599                (rel_injectN, rel_inject_thms, code_simp_attrs),
   600                (setsN, flat set_thmss, code_simp_attrs)]
   601               |> filter_out (null o #2)
   602               |> map (fn (thmN, thms, attrs) =>
   603                 ((qualify true fp_b_name (Binding.name thmN), attrs), [(thms, [])]));
   604           in
   605             (wrap_res, lthy |> Local_Theory.notes notes |> snd)
   606           end;
   607 
   608         fun define_fold_rec no_defs_lthy =
   609           let
   610             val fpT_to_C = fpT --> C;
   611 
   612             fun build_prod_proj mk_proj (T, U) =
   613               if T = U then
   614                 id_const T
   615               else
   616                 (case (T, U) of
   617                   (Type (s, _), Type (s', _)) =>
   618                   if s = s' then build_map (build_prod_proj mk_proj) T U else mk_proj T
   619                 | _ => mk_proj T);
   620 
   621             (* TODO: Avoid these complications; cf. corec case *)
   622             fun mk_U proj (Type (s as @{type_name prod}, Ts as [T', U])) =
   623                 if member (op =) fpTs T' then proj (T', U) else Type (s, map (mk_U proj) Ts)
   624               | mk_U proj (Type (s, Ts)) = Type (s, map (mk_U proj) Ts)
   625               | mk_U _ T = T;
   626 
   627             fun unzip_rec (x as Free (_, T)) =
   628               if exists_fp_subtype T then
   629                 [build_prod_proj fst_const (T, mk_U fst T) $ x,
   630                  build_prod_proj snd_const (T, mk_U snd T) $ x]
   631               else
   632                 [x];
   633 
   634             fun mk_rec_like_arg f xs = mk_tupled_fun (HOLogic.mk_tuple xs) f (maps unzip_rec xs);
   635 
   636             fun generate_rec_like (suf, fp_rec_like, (fss, f_Tss, xsss)) =
   637               let
   638                 val res_T = fold_rev (curry (op --->)) f_Tss fpT_to_C;
   639                 val binding = qualify false fp_b_name (Binding.suffix_name ("_" ^ suf) fp_b);
   640                 val spec =
   641                   mk_Trueprop_eq (lists_bmoc fss (Free (Binding.name_of binding, res_T)),
   642                     Term.list_comb (fp_rec_like,
   643                       map2 (mk_sum_caseN_balanced oo map2 mk_rec_like_arg) fss xsss));
   644               in (binding, spec) end;
   645 
   646             val rec_like_infos =
   647               [(foldN, fp_fold, fold_only),
   648                (recN, fp_rec, rec_only)];
   649 
   650             val (bindings, specs) = map generate_rec_like rec_like_infos |> split_list;
   651 
   652             val ((csts, defs), (lthy', lthy)) = no_defs_lthy
   653               |> apfst split_list o fold_map2 (fn b => fn spec =>
   654                 Specification.definition (SOME (b, NONE, NoSyn), ((Thm.def_binding b, []), spec))
   655                 #>> apsnd snd) bindings specs
   656               ||> `Local_Theory.restore;
   657 
   658             val phi = Proof_Context.export_morphism lthy lthy';
   659 
   660             val [fold_def, rec_def] = map (Morphism.thm phi) defs;
   661 
   662             val [foldx, recx] = map (mk_rec_like lfp As Cs o Morphism.term phi) csts;
   663           in
   664             ((foldx, recx, fold_def, rec_def), lthy)
   665           end;
   666 
   667         fun define_unfold_corec no_defs_lthy =
   668           let
   669             val B_to_fpT = C --> fpT;
   670 
   671             fun mk_preds_getterss_join c n cps sum_prod_T cqfss =
   672               Term.lambda c (mk_IfN sum_prod_T cps
   673                 (map2 (mk_InN_balanced sum_prod_T n) (map HOLogic.mk_tuple cqfss) (1 upto n)));
   674 
   675             fun generate_corec_like (suf, fp_rec_like, ((pfss, cqfsss), (f_sum_prod_Ts,
   676                 pf_Tss))) =
   677               let
   678                 val res_T = fold_rev (curry (op --->)) pf_Tss B_to_fpT;
   679                 val binding = qualify false fp_b_name (Binding.suffix_name ("_" ^ suf) fp_b);
   680                 val spec =
   681                   mk_Trueprop_eq (lists_bmoc pfss (Free (Binding.name_of binding, res_T)),
   682                     Term.list_comb (fp_rec_like,
   683                       map5 mk_preds_getterss_join cs ns cpss f_sum_prod_Ts cqfsss));
   684               in (binding, spec) end;
   685 
   686             val corec_like_infos =
   687               [(unfoldN, fp_fold, unfold_only),
   688                (corecN, fp_rec, corec_only)];
   689 
   690             val (bindings, specs) = map generate_corec_like corec_like_infos |> split_list;
   691 
   692             val ((csts, defs), (lthy', lthy)) = no_defs_lthy
   693               |> apfst split_list o fold_map2 (fn b => fn spec =>
   694                 Specification.definition (SOME (b, NONE, NoSyn), ((Thm.def_binding b, []), spec))
   695                 #>> apsnd snd) bindings specs
   696               ||> `Local_Theory.restore;
   697 
   698             val phi = Proof_Context.export_morphism lthy lthy';
   699 
   700             val [unfold_def, corec_def] = map (Morphism.thm phi) defs;
   701 
   702             val [unfold, corec] = map (mk_rec_like lfp As Cs o Morphism.term phi) csts;
   703           in
   704             ((unfold, corec, unfold_def, corec_def), lthy)
   705           end;
   706 
   707         val define_rec_likes = if lfp then define_fold_rec else define_unfold_corec;
   708 
   709         fun massage_res ((wrap_res, rec_like_res), lthy) =
   710           (((ctrs, xss, ctr_defs, wrap_res), rec_like_res), lthy);
   711       in
   712         (wrap #> (live > 0 ? derive_maps_sets_rels) ##>> define_rec_likes #> massage_res, lthy')
   713       end;
   714 
   715     fun wrap_types_and_more (wrap_types_and_mores, lthy) =
   716       fold_map I wrap_types_and_mores lthy
   717       |>> apsnd split_list4 o apfst split_list4 o split_list;
   718 
   719     (* TODO: Add map, sets, rel simps *)
   720     val mk_simp_thmss =
   721       map3 (fn (_, _, _, injects, distincts, cases, _, _, _) => fn rec_likes => fn fold_likes =>
   722         injects @ distincts @ cases @ rec_likes @ fold_likes);
   723 
   724     fun derive_induct_fold_rec_thms_for_types (((ctrss, xsss, ctr_defss, wrap_ress), (folds, recs,
   725         fold_defs, rec_defs)), lthy) =
   726       let
   727         val (((ps, ps'), us'), names_lthy) =
   728           lthy
   729           |> mk_Frees' "P" (map mk_pred1T fpTs)
   730           ||>> Variable.variant_fixes fp_b_names;
   731 
   732         val us = map2 (curry Free) us' fpTs;
   733 
   734         fun mk_sets_nested bnf =
   735           let
   736             val Type (T_name, Us) = T_of_bnf bnf;
   737             val lives = lives_of_bnf bnf;
   738             val sets = sets_of_bnf bnf;
   739             fun mk_set U =
   740               (case find_index (curry (op =) U) lives of
   741                 ~1 => Term.dummy
   742               | i => nth sets i);
   743           in
   744             (T_name, map mk_set Us)
   745           end;
   746 
   747         val setss_nested = map mk_sets_nested nested_bnfs;
   748 
   749         val (induct_thms, induct_thm) =
   750           let
   751             fun mk_set Ts t =
   752               let val Type (_, Ts0) = domain_type (fastype_of t) in
   753                 Term.subst_atomic_types (Ts0 ~~ Ts) t
   754               end;
   755 
   756             fun mk_raw_prem_prems names_lthy (x as Free (s, T as Type (T_name, Ts0))) =
   757                 (case find_index (curry (op =) T) fpTs of
   758                   ~1 =>
   759                   (case AList.lookup (op =) setss_nested T_name of
   760                     NONE => []
   761                   | SOME raw_sets0 =>
   762                     let
   763                       val (Ts, raw_sets) =
   764                         split_list (filter (exists_fp_subtype o fst) (Ts0 ~~ raw_sets0));
   765                       val sets = map (mk_set Ts0) raw_sets;
   766                       val (ys, names_lthy') = names_lthy |> mk_Frees s Ts;
   767                       val xysets = map (pair x) (ys ~~ sets);
   768                       val ppremss = map (mk_raw_prem_prems names_lthy') ys;
   769                     in
   770                       flat (map2 (map o apfst o cons) xysets ppremss)
   771                     end)
   772                 | kk => [([], (kk + 1, x))])
   773               | mk_raw_prem_prems _ _ = [];
   774 
   775             fun close_prem_prem xs t =
   776               fold_rev Logic.all (map Free (drop (nn + length xs)
   777                 (rev (Term.add_frees t (map dest_Free xs @ ps'))))) t;
   778 
   779             fun mk_prem_prem xs (xysets, (j, x)) =
   780               close_prem_prem xs (Logic.list_implies (map (fn (x', (y, set)) =>
   781                   HOLogic.mk_Trueprop (HOLogic.mk_mem (y, set $ x'))) xysets,
   782                 HOLogic.mk_Trueprop (nth ps (j - 1) $ x)));
   783 
   784             fun mk_raw_prem phi ctr ctr_Ts =
   785               let
   786                 val (xs, names_lthy') = names_lthy |> mk_Frees "x" ctr_Ts;
   787                 val pprems = maps (mk_raw_prem_prems names_lthy') xs;
   788               in (xs, pprems, HOLogic.mk_Trueprop (phi $ Term.list_comb (ctr, xs))) end;
   789 
   790             fun mk_prem (xs, raw_pprems, concl) =
   791               fold_rev Logic.all xs (Logic.list_implies (map (mk_prem_prem xs) raw_pprems, concl));
   792 
   793             val raw_premss = map3 (map2 o mk_raw_prem) ps ctrss ctr_Tsss;
   794 
   795             val goal =
   796               Library.foldr (Logic.list_implies o apfst (map mk_prem)) (raw_premss,
   797                 HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj (map2 (curry (op $)) ps us)));
   798 
   799             val kksss = map (map (map (fst o snd) o #2)) raw_premss;
   800 
   801             val ctor_induct' = fp_induct OF (map mk_sumEN_tupled_balanced mss);
   802 
   803             val thm =
   804               Skip_Proof.prove lthy [] [] goal (fn {context = ctxt, ...} =>
   805                 mk_induct_tac ctxt nn ns mss kksss (flat ctr_defss) ctor_induct'
   806                   nested_set_natural's pre_set_defss)
   807               |> singleton (Proof_Context.export names_lthy lthy)
   808               |> Thm.close_derivation;
   809           in
   810             `(conj_dests nn) thm
   811           end;
   812 
   813         val induct_cases = quasi_unambiguous_case_names (maps (map name_of_ctr) ctrss);
   814 
   815         val (fold_thmss, rec_thmss) =
   816           let
   817             val xctrss = map2 (map2 (curry Term.list_comb)) ctrss xsss;
   818             val gfolds = map (lists_bmoc gss) folds;
   819             val hrecs = map (lists_bmoc hss) recs;
   820 
   821             fun mk_goal fss frec_like xctr f xs fxs =
   822               fold_rev (fold_rev Logic.all) (xs :: fss)
   823                 (mk_Trueprop_eq (frec_like $ xctr, Term.list_comb (f, fxs)));
   824 
   825             fun build_rec_like frec_likes (T, U) =
   826               if T = U then
   827                 id_const T
   828               else
   829                 (case find_index (curry (op =) T) fpTs of
   830                   ~1 => build_map (build_rec_like frec_likes) T U
   831                 | kk => nth frec_likes kk);
   832 
   833             val mk_U = typ_subst (map2 pair fpTs Cs);
   834 
   835             fun intr_rec_likes frec_likes maybe_cons (x as Free (_, T)) =
   836               if exists_fp_subtype T then
   837                 maybe_cons x [build_rec_like frec_likes (T, mk_U T) $ x]
   838               else
   839                 [x];
   840 
   841             val gxsss = map (map (maps (intr_rec_likes gfolds (K I)))) xsss;
   842             val hxsss = map (map (maps (intr_rec_likes hrecs cons))) xsss;
   843 
   844             val fold_goalss = map5 (map4 o mk_goal gss) gfolds xctrss gss xsss gxsss;
   845             val rec_goalss = map5 (map4 o mk_goal hss) hrecs xctrss hss xsss hxsss;
   846 
   847             val fold_tacss =
   848               map2 (map o mk_rec_like_tac pre_map_defs [] nesting_map_ids'' fold_defs) fp_fold_thms
   849                 ctr_defss;
   850             val rec_tacss =
   851               map2 (map o mk_rec_like_tac pre_map_defs nested_map_comp's
   852                 (nested_map_ids'' @ nesting_map_ids'') rec_defs) fp_rec_thms ctr_defss;
   853 
   854             fun prove goal tac =
   855               Skip_Proof.prove lthy [] [] goal (tac o #context)
   856               |> Thm.close_derivation;
   857           in
   858             (map2 (map2 prove) fold_goalss fold_tacss, map2 (map2 prove) rec_goalss rec_tacss)
   859           end;
   860 
   861         val simp_thmss = mk_simp_thmss wrap_ress rec_thmss fold_thmss;
   862 
   863         val induct_case_names_attr = Attrib.internal (K (Rule_Cases.case_names induct_cases));
   864         fun induct_type_attr T_name = Attrib.internal (K (Induct.induct_type T_name));
   865 
   866         val common_notes =
   867           (if nn > 1 then [(inductN, [induct_thm], [induct_case_names_attr])] else [])
   868           |> map (fn (thmN, thms, attrs) =>
   869             ((qualify true fp_common_name (Binding.name thmN), attrs), [(thms, [])]));
   870 
   871         val notes =
   872           [(foldN, fold_thmss, K code_simp_attrs),
   873            (inductN, map single induct_thms,
   874             fn T_name => [induct_case_names_attr, induct_type_attr T_name]),
   875            (recN, rec_thmss, K code_simp_attrs),
   876            (simpsN, simp_thmss, K [])]
   877           |> maps (fn (thmN, thmss, attrs) =>
   878             map3 (fn fp_b_name => fn Type (T_name, _) => fn thms =>
   879               ((qualify true fp_b_name (Binding.name thmN), attrs T_name),
   880                [(thms, [])])) fp_b_names fpTs thmss);
   881       in
   882         lthy |> Local_Theory.notes (common_notes @ notes) |> snd
   883       end;
   884 
   885     fun derive_coinduct_unfold_corec_thms_for_types (((ctrss, _, ctr_defss, wrap_ress), (unfolds,
   886         corecs, unfold_defs, corec_defs)), lthy) =
   887       let
   888         val nesting_rel_eqs = map rel_eq_of_bnf nesting_bnfs;
   889 
   890         val discss = map (map (mk_disc_or_sel As) o #1) wrap_ress;
   891         val selsss = map (map (map (mk_disc_or_sel As)) o #2) wrap_ress;
   892         val exhaust_thms = map #3 wrap_ress;
   893         val disc_thmsss = map #7 wrap_ress;
   894         val discIss = map #8 wrap_ress;
   895         val sel_thmsss = map #9 wrap_ress;
   896 
   897         val (((rs, us'), vs'), names_lthy) =
   898           lthy
   899           |> mk_Frees "R" (map (fn T => mk_pred2T T T) fpTs)
   900           ||>> Variable.variant_fixes fp_b_names
   901           ||>> Variable.variant_fixes (map (suffix "'") fp_b_names);
   902 
   903         val us = map2 (curry Free) us' fpTs;
   904         val udiscss = map2 (map o rapp) us discss;
   905         val uselsss = map2 (map o map o rapp) us selsss;
   906 
   907         val vs = map2 (curry Free) vs' fpTs;
   908         val vdiscss = map2 (map o rapp) vs discss;
   909         val vselsss = map2 (map o map o rapp) vs selsss;
   910 
   911         val ((coinduct_thms, coinduct_thm), (strong_coinduct_thms, strong_coinduct_thm)) =
   912           let
   913             val uvrs = map3 (fn r => fn u => fn v => r $ u $ v) rs us vs;
   914             val uv_eqs = map2 (curry HOLogic.mk_eq) us vs;
   915             val strong_rs =
   916               map4 (fn u => fn v => fn uvr => fn uv_eq =>
   917                 fold_rev Term.lambda [u, v] (HOLogic.mk_disj (uvr, uv_eq))) us vs uvrs uv_eqs;
   918 
   919             fun build_rel rs' T =
   920               (case find_index (curry (op =) T) fpTs of
   921                 ~1 =>
   922                 if exists_fp_subtype T then build_rel_step (build_rel rs') T
   923                 else HOLogic.eq_const T
   924               | kk => nth rs' kk);
   925 
   926             fun build_rel_app rs' usel vsel =
   927               fold rapp [usel, vsel] (build_rel rs' (fastype_of usel));
   928 
   929             fun mk_prem_ctr_concls rs' n k udisc usels vdisc vsels =
   930               (if k = n then [] else [HOLogic.mk_eq (udisc, vdisc)]) @
   931               (if null usels then
   932                  []
   933                else
   934                  [Library.foldr HOLogic.mk_imp (if n = 1 then [] else [udisc, vdisc],
   935                     Library.foldr1 HOLogic.mk_conj (map2 (build_rel_app rs') usels vsels))]);
   936 
   937             fun mk_prem_concl rs' n udiscs uselss vdiscs vselss =
   938               Library.foldr1 HOLogic.mk_conj
   939                 (flat (map5 (mk_prem_ctr_concls rs' n) (1 upto n) udiscs uselss vdiscs vselss))
   940               handle List.Empty => @{term True};
   941 
   942             fun mk_prem rs' uvr u v n udiscs uselss vdiscs vselss =
   943               fold_rev Logic.all [u, v] (Logic.mk_implies (HOLogic.mk_Trueprop uvr,
   944                 HOLogic.mk_Trueprop (mk_prem_concl rs' n udiscs uselss vdiscs vselss)));
   945 
   946             val concl =
   947               HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj
   948                 (map3 (fn uvr => fn u => fn v => HOLogic.mk_imp (uvr, HOLogic.mk_eq (u, v)))
   949                    uvrs us vs));
   950 
   951             fun mk_goal rs' =
   952               Logic.list_implies (map8 (mk_prem rs') uvrs us vs ns udiscss uselsss vdiscss vselsss,
   953                 concl);
   954 
   955             val goal = mk_goal rs;
   956             val strong_goal = mk_goal strong_rs;
   957 
   958             fun prove dtor_coinduct' goal =
   959               Skip_Proof.prove lthy [] [] goal (fn {context = ctxt, ...} =>
   960                 mk_coinduct_tac ctxt nesting_rel_eqs nn ns dtor_coinduct' pre_rel_defs dtor_ctors
   961                   exhaust_thms ctr_defss disc_thmsss sel_thmsss)
   962               |> singleton (Proof_Context.export names_lthy lthy)
   963               |> Thm.close_derivation;
   964 
   965             fun postproc nn thm =
   966               Thm.permute_prems 0 nn
   967                 (if nn = 1 then thm RS mp
   968                  else funpow nn (fn thm => reassoc_conjs (thm RS mp_conj)) thm)
   969               |> Drule.zero_var_indexes
   970               |> `(conj_dests nn);
   971           in
   972             (postproc nn (prove fp_induct goal), postproc nn (prove fp_strong_induct strong_goal))
   973           end;
   974 
   975         fun mk_maybe_not pos = not pos ? HOLogic.mk_not;
   976 
   977         val z = the_single zs;
   978         val gunfolds = map (lists_bmoc pgss) unfolds;
   979         val hcorecs = map (lists_bmoc phss) corecs;
   980 
   981         val (unfold_thmss, corec_thmss, safe_unfold_thmss, safe_corec_thmss) =
   982           let
   983             fun mk_goal pfss c cps fcorec_like n k ctr m cfs' =
   984               fold_rev (fold_rev Logic.all) ([c] :: pfss)
   985                 (Logic.list_implies (seq_conds (HOLogic.mk_Trueprop oo mk_maybe_not) n k cps,
   986                    mk_Trueprop_eq (fcorec_like $ c, Term.list_comb (ctr, take m cfs'))));
   987 
   988             fun build_corec_like fcorec_likes maybe_tack (T, U) =
   989               if T = U then
   990                 id_const T
   991               else
   992                 (case find_index (curry (op =) U) fpTs of
   993                   ~1 => build_map (build_corec_like fcorec_likes maybe_tack) T U
   994                 | kk => maybe_tack (nth cs kk, nth us kk) (nth fcorec_likes kk));
   995 
   996             fun mk_U maybe_mk_sumT =
   997               typ_subst (map2 (fn C => fn fpT => (maybe_mk_sumT fpT C, fpT)) Cs fpTs);
   998 
   999             fun intr_corec_likes fcorec_likes maybe_mk_sumT maybe_tack cqf =
  1000               let val T = fastype_of cqf in
  1001                 if exists_Cs_subtype T then
  1002                   build_corec_like fcorec_likes maybe_tack (T, mk_U maybe_mk_sumT T) $ cqf
  1003                 else
  1004                   cqf
  1005               end;
  1006 
  1007             val crgsss' = map (map (map (intr_corec_likes gunfolds (K I) (K I)))) crgsss;
  1008             val cshsss' =
  1009               map (map (map (intr_corec_likes hcorecs (curry mk_sumT) (tack z)))) cshsss;
  1010 
  1011             val unfold_goalss =
  1012               map8 (map4 oooo mk_goal pgss) cs cpss gunfolds ns kss ctrss mss crgsss';
  1013             val corec_goalss =
  1014               map8 (map4 oooo mk_goal phss) cs cpss hcorecs ns kss ctrss mss cshsss';
  1015 
  1016             val unfold_tacss =
  1017               map3 (map oo mk_corec_like_tac unfold_defs nesting_map_ids'') fp_fold_thms
  1018                 pre_map_defs ctr_defss;
  1019             val corec_tacss =
  1020               map3 (map oo mk_corec_like_tac corec_defs nesting_map_ids'') fp_rec_thms pre_map_defs
  1021                 ctr_defss;
  1022 
  1023             fun prove goal tac =
  1024               Skip_Proof.prove lthy [] [] goal (tac o #context) |> Thm.close_derivation;
  1025 
  1026             val unfold_thmss = map2 (map2 prove) unfold_goalss unfold_tacss;
  1027             val corec_thmss =
  1028               map2 (map2 prove) corec_goalss corec_tacss
  1029               |> map (map (unfold_thms lthy @{thms sum_case_if}));
  1030 
  1031             val unfold_safesss = map2 (map2 (map2 (curry (op =)))) crgsss' crgsss;
  1032             val corec_safesss = map2 (map2 (map2 (curry (op =)))) cshsss' cshsss;
  1033 
  1034             val filter_safesss =
  1035               map2 (map_filter (fn (safes, thm) => if forall I safes then SOME thm else NONE) oo
  1036                 curry (op ~~));
  1037 
  1038             val safe_unfold_thmss = filter_safesss unfold_safesss unfold_thmss;
  1039             val safe_corec_thmss = filter_safesss corec_safesss corec_thmss;
  1040           in
  1041             (unfold_thmss, corec_thmss, safe_unfold_thmss, safe_corec_thmss)
  1042           end;
  1043 
  1044         val (disc_unfold_iff_thmss, disc_corec_iff_thmss) =
  1045           let
  1046             fun mk_goal c cps fcorec_like n k disc =
  1047               mk_Trueprop_eq (disc $ (fcorec_like $ c),
  1048                 if n = 1 then @{const True}
  1049                 else Library.foldr1 HOLogic.mk_conj (seq_conds mk_maybe_not n k cps));
  1050 
  1051             val unfold_goalss = map6 (map2 oooo mk_goal) cs cpss gunfolds ns kss discss;
  1052             val corec_goalss = map6 (map2 oooo mk_goal) cs cpss hcorecs ns kss discss;
  1053 
  1054             fun mk_case_split' cp =
  1055               Drule.instantiate' [] [SOME (certify lthy cp)] @{thm case_split};
  1056 
  1057             val case_splitss' = map (map mk_case_split') cpss;
  1058 
  1059             val unfold_tacss =
  1060               map3 (map oo mk_disc_corec_like_iff_tac) case_splitss' unfold_thmss disc_thmsss;
  1061             val corec_tacss =
  1062               map3 (map oo mk_disc_corec_like_iff_tac) case_splitss' corec_thmss disc_thmsss;
  1063 
  1064             fun prove goal tac =
  1065               Skip_Proof.prove lthy [] [] goal (tac o #context)
  1066               |> singleton (Proof_Context.export names_lthy0 no_defs_lthy)
  1067               |> Thm.close_derivation;
  1068 
  1069             fun proves [_] [_] = []
  1070               | proves goals tacs = map2 prove goals tacs;
  1071           in
  1072             (map2 proves unfold_goalss unfold_tacss,
  1073              map2 proves corec_goalss corec_tacss)
  1074           end;
  1075 
  1076         val is_triv_discI = is_triv_implies orf is_concl_refl;
  1077 
  1078         fun mk_disc_corec_like_thms corec_likes discIs =
  1079           map (op RS) (filter_out (is_triv_discI o snd) (corec_likes ~~ discIs));
  1080 
  1081         val disc_unfold_thmss = map2 mk_disc_corec_like_thms unfold_thmss discIss;
  1082         val disc_corec_thmss = map2 mk_disc_corec_like_thms corec_thmss discIss;
  1083 
  1084         fun mk_sel_corec_like_thm corec_like_thm sel sel_thm =
  1085           let
  1086             val (domT, ranT) = dest_funT (fastype_of sel);
  1087             val arg_cong' =
  1088               Drule.instantiate' (map (SOME o certifyT lthy) [domT, ranT])
  1089                 [NONE, NONE, SOME (certify lthy sel)] arg_cong
  1090               |> Thm.varifyT_global;
  1091             val sel_thm' = sel_thm RSN (2, trans);
  1092           in
  1093             corec_like_thm RS arg_cong' RS sel_thm'
  1094           end;
  1095 
  1096         fun mk_sel_corec_like_thms corec_likess =
  1097           map3 (map3 (map2 o mk_sel_corec_like_thm)) corec_likess selsss sel_thmsss |> map flat;
  1098 
  1099         val sel_unfold_thmss = mk_sel_corec_like_thms unfold_thmss;
  1100         val sel_corec_thmss = mk_sel_corec_like_thms corec_thmss;
  1101 
  1102         fun flat_corec_like_thms corec_likes disc_corec_likes sel_corec_likes =
  1103           corec_likes @ disc_corec_likes @ sel_corec_likes;
  1104 
  1105         val simp_thmss =
  1106           mk_simp_thmss wrap_ress
  1107             (map3 flat_corec_like_thms safe_corec_thmss disc_corec_thmss sel_corec_thmss)
  1108             (map3 flat_corec_like_thms safe_unfold_thmss disc_unfold_thmss sel_unfold_thmss);
  1109 
  1110         val anonymous_notes =
  1111           [(flat safe_unfold_thmss @ flat safe_corec_thmss, simp_attrs)]
  1112           |> map (fn (thms, attrs) => ((Binding.empty, attrs), [(thms, [])]));
  1113 
  1114         val common_notes =
  1115           (if nn > 1 then
  1116              (* FIXME: attribs *)
  1117              [(coinductN, [coinduct_thm], []),
  1118               (strong_coinductN, [strong_coinduct_thm], [])]
  1119            else
  1120              [])
  1121           |> map (fn (thmN, thms, attrs) =>
  1122             ((qualify true fp_common_name (Binding.name thmN), attrs), [(thms, [])]));
  1123 
  1124         val notes =
  1125           [(coinductN, map single coinduct_thms, []), (* FIXME: attribs *)
  1126            (corecN, corec_thmss, []),
  1127            (disc_corecN, disc_corec_thmss, simp_attrs),
  1128            (disc_corec_iffN, disc_corec_iff_thmss, simp_attrs),
  1129            (disc_unfoldN, disc_unfold_thmss, simp_attrs),
  1130            (disc_unfold_iffN, disc_unfold_iff_thmss, simp_attrs),
  1131            (sel_corecN, sel_corec_thmss, simp_attrs),
  1132            (sel_unfoldN, sel_unfold_thmss, simp_attrs),
  1133            (simpsN, simp_thmss, []),
  1134            (strong_coinductN, map single strong_coinduct_thms, []), (* FIXME: attribs *)
  1135            (unfoldN, unfold_thmss, [])]
  1136           |> maps (fn (thmN, thmss, attrs) =>
  1137             map_filter (fn (_, []) => NONE | (fp_b_name, thms) =>
  1138               SOME ((qualify true fp_b_name (Binding.name thmN), attrs),
  1139                 [(thms, [])])) (fp_b_names ~~ thmss));
  1140       in
  1141         lthy |> Local_Theory.notes (anonymous_notes @ common_notes @ notes) |> snd
  1142       end;
  1143 
  1144     val lthy' = lthy
  1145       |> fold_map define_ctrs_case_for_type (fp_bnfs ~~ fp_bs ~~ fpTs ~~ Cs ~~ ctors ~~ dtors ~~
  1146         fp_folds ~~ fp_recs ~~ ctor_dtors ~~ dtor_ctors ~~ ctor_injects ~~ pre_map_defs ~~
  1147         pre_set_defss ~~ pre_rel_defs ~~ fp_map_thms ~~ fp_set_thmss ~~ fp_rel_thms ~~ ns ~~ kss ~~
  1148         mss ~~ ctr_bindingss ~~ ctr_mixfixess ~~ ctr_Tsss ~~ disc_bindingss ~~ sel_bindingsss ~~
  1149         raw_sel_defaultsss)
  1150       |> wrap_types_and_more
  1151       |> (if lfp then derive_induct_fold_rec_thms_for_types
  1152           else derive_coinduct_unfold_corec_thms_for_types);
  1153 
  1154     val timer = time (timer ("Constructors, discriminators, selectors, etc., for the new " ^
  1155       (if lfp then "" else "co") ^ "datatype"));
  1156   in
  1157     timer; lthy'
  1158   end;
  1159 
  1160 val datatypes = define_datatypes (K I) (K I) (K I);
  1161 
  1162 val datatype_cmd = define_datatypes Typedecl.read_constraint Syntax.parse_typ Syntax.read_term;
  1163 
  1164 val parse_ctr_arg =
  1165   @{keyword "("} |-- parse_binding_colon -- Parse.typ --| @{keyword ")"} ||
  1166   (Parse.typ >> pair Binding.empty);
  1167 
  1168 val parse_defaults =
  1169   @{keyword "("} |-- @{keyword "defaults"} |-- Scan.repeat parse_bound_term --| @{keyword ")"};
  1170 
  1171 val parse_single_spec =
  1172   Parse.type_args_constrained -- Parse.binding -- Parse.opt_mixfix --
  1173   (@{keyword "="} |-- Parse.enum1 "|" (parse_opt_binding_colon -- Parse.binding --
  1174     Scan.repeat parse_ctr_arg -- Scan.optional parse_defaults [] -- Parse.opt_mixfix));
  1175 
  1176 val parse_datatype = parse_wrap_options -- Parse.and_list1 parse_single_spec;
  1177 
  1178 fun parse_datatype_cmd lfp construct_fp = parse_datatype >> datatype_cmd lfp construct_fp;
  1179 
  1180 end;