src/HOL/BNF/Tools/bnf_fp_def_sugar.ML
author blanchet
Fri Mar 08 14:15:39 2013 +0100 (2013-03-08)
changeset 51380 cac8c9a636b6
parent 50170 8155e280f239
child 51551 88d1d19fb74f
permissions -rw-r--r--
proper type inference for default values
     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 [] = ([], [], [], [])
    50   | split_list4 ((x1, x2, x3, x4) :: xs) =
    51     let val (xs1, xs2, xs3, xs4) = split_list4 xs;
    52     in (x1 :: xs1, x2 :: xs2, x3 :: xs3, x4 :: xs4) end;
    53 
    54 fun resort_tfree S (TFree (s, _)) = TFree (s, S);
    55 
    56 fun typ_subst inst (T as Type (s, Ts)) =
    57     (case AList.lookup (op =) inst T of
    58       NONE => Type (s, map (typ_subst inst) Ts)
    59     | SOME T' => T')
    60   | typ_subst inst T = the_default T (AList.lookup (op =) inst T);
    61 
    62 fun variant_types ss Ss ctxt =
    63   let
    64     val (tfrees, _) =
    65       fold_map2 (fn s => fn S => Name.variant s #> apfst (rpair S)) ss Ss (Variable.names_of ctxt);
    66     val ctxt' = fold (Variable.declare_constraints o Logic.mk_type o TFree) tfrees ctxt;
    67   in (tfrees, ctxt') end;
    68 
    69 val lists_bmoc = fold (fn xs => fn t => Term.list_comb (t, xs));
    70 
    71 fun mk_tupled_fun x f xs = HOLogic.tupled_lambda x (Term.list_comb (f, xs));
    72 fun mk_uncurried_fun f xs = mk_tupled_fun (HOLogic.mk_tuple xs) f xs;
    73 
    74 fun mk_flip (x, Type (_, [T1, Type (_, [T2, T3])])) =
    75   Abs ("x", T1, Abs ("y", T2, Var (x, T2 --> T1 --> T3) $ Bound 0 $ Bound 1));
    76 
    77 fun flip_rels lthy n thm =
    78   let
    79     val Rs = Term.add_vars (prop_of thm) [];
    80     val Rs' = rev (drop (length Rs - n) Rs);
    81     val cRs = map (fn f => (certify lthy (Var f), certify lthy (mk_flip f))) Rs';
    82   in
    83     Drule.cterm_instantiate cRs thm
    84   end;
    85 
    86 fun mk_ctor_or_dtor get_T Ts t =
    87   let val Type (_, Ts0) = get_T (fastype_of t) in
    88     Term.subst_atomic_types (Ts0 ~~ Ts) t
    89   end;
    90 
    91 val mk_ctor = mk_ctor_or_dtor range_type;
    92 val mk_dtor = mk_ctor_or_dtor domain_type;
    93 
    94 fun mk_rec_like lfp Ts Us t =
    95   let
    96     val (bindings, body) = strip_type (fastype_of t);
    97     val (f_Us, prebody) = split_last bindings;
    98     val Type (_, Ts0) = if lfp then prebody else body;
    99     val Us0 = distinct (op =) (map (if lfp then body_type else domain_type) f_Us);
   100   in
   101     Term.subst_atomic_types (Ts0 @ Us0 ~~ Ts @ Us) t
   102   end;
   103 
   104 fun mk_map live Ts Us t =
   105   let val (Type (_, Ts0), Type (_, Us0)) = strip_typeN (live + 1) (fastype_of t) |>> List.last in
   106     Term.subst_atomic_types (Ts0 @ Us0 ~~ Ts @ Us) t
   107   end;
   108 
   109 fun mk_rel live Ts Us t =
   110   let val [Type (_, Ts0), Type (_, Us0)] = binder_types (snd (strip_typeN live (fastype_of t))) in
   111     Term.subst_atomic_types (Ts0 @ Us0 ~~ Ts @ Us) t
   112   end;
   113 
   114 fun liveness_of_fp_bnf n bnf =
   115   (case T_of_bnf bnf of
   116     Type (_, Ts) => map (not o member (op =) (deads_of_bnf bnf)) Ts
   117   | _ => replicate n false);
   118 
   119 fun cannot_merge_types () = error "Mutually recursive types must have the same type parameters";
   120 
   121 fun merge_type_arg T T' = if T = T' then T else cannot_merge_types ();
   122 
   123 fun merge_type_args (As, As') =
   124   if length As = length As' then map2 merge_type_arg As As' else cannot_merge_types ();
   125 
   126 fun reassoc_conjs thm =
   127   reassoc_conjs (thm RS @{thm conj_assoc[THEN iffD1]})
   128   handle THM _ => thm;
   129 
   130 fun type_args_constrained_of (((cAs, _), _), _) = cAs;
   131 fun type_binding_of (((_, b), _), _) = b;
   132 fun mixfix_of ((_, mx), _) = mx;
   133 fun ctr_specs_of (_, ctr_specs) = ctr_specs;
   134 
   135 fun disc_of ((((disc, _), _), _), _) = disc;
   136 fun ctr_of ((((_, ctr), _), _), _) = ctr;
   137 fun args_of (((_, args), _), _) = args;
   138 fun defaults_of ((_, ds), _) = ds;
   139 fun ctr_mixfix_of (_, mx) = mx;
   140 
   141 fun define_datatypes prepare_constraint prepare_typ prepare_term lfp construct_fp
   142     (wrap_opts as (no_dests, rep_compat), specs) no_defs_lthy0 =
   143   let
   144     (* TODO: sanity checks on arguments *)
   145     (* TODO: integration with function package ("size") *)
   146 
   147     val _ = if not lfp andalso no_dests then error "Cannot define destructor-less codatatypes"
   148       else ();
   149 
   150     fun qualify mandatory fp_b_name =
   151       Binding.qualify mandatory fp_b_name o (rep_compat ? Binding.qualify false rep_compat_prefix);
   152 
   153     val nn = length specs;
   154     val fp_bs = map type_binding_of specs;
   155     val fp_b_names = map Binding.name_of fp_bs;
   156     val fp_common_name = mk_common_name fp_b_names;
   157 
   158     fun prepare_type_arg (ty, c) =
   159       let val TFree (s, _) = prepare_typ no_defs_lthy0 ty in
   160         TFree (s, prepare_constraint no_defs_lthy0 c)
   161       end;
   162 
   163     val Ass0 = map (map prepare_type_arg o type_args_constrained_of) specs;
   164     val unsorted_Ass0 = map (map (resort_tfree HOLogic.typeS)) Ass0;
   165     val unsorted_As = Library.foldr1 merge_type_args unsorted_Ass0;
   166 
   167     val (((Bs0, Cs), Xs), no_defs_lthy) =
   168       no_defs_lthy0
   169       |> fold (Variable.declare_typ o resort_tfree dummyS) unsorted_As
   170       |> mk_TFrees (length unsorted_As)
   171       ||>> mk_TFrees nn
   172       ||>> apfst (map TFree) o
   173         variant_types (map (prefix "'") fp_b_names) (replicate nn HOLogic.typeS);
   174 
   175     (* TODO: cleaner handling of fake contexts, without "background_theory" *)
   176     (*the "perhaps o try" below helps gracefully handles the case where the new type is defined in a
   177       locale and shadows an existing global type*)
   178     val fake_thy =
   179       Theory.copy #> fold (fn spec => perhaps (try (Sign.add_type no_defs_lthy
   180         (type_binding_of spec, length (type_args_constrained_of spec), mixfix_of spec)))) specs;
   181     val fake_lthy = Proof_Context.background_theory fake_thy no_defs_lthy;
   182 
   183     fun mk_fake_T b =
   184       Type (fst (Term.dest_Type (Proof_Context.read_type_name fake_lthy true (Binding.name_of b))),
   185         unsorted_As);
   186 
   187     val fake_Ts = map mk_fake_T fp_bs;
   188 
   189     val mixfixes = map mixfix_of specs;
   190 
   191     val _ = (case duplicates Binding.eq_name fp_bs of [] => ()
   192       | b :: _ => error ("Duplicate type name declaration " ^ quote (Binding.name_of b)));
   193 
   194     val ctr_specss = map ctr_specs_of specs;
   195 
   196     val disc_bindingss = map (map disc_of) ctr_specss;
   197     val ctr_bindingss =
   198       map2 (fn fp_b_name => map (qualify false fp_b_name o ctr_of)) fp_b_names ctr_specss;
   199     val ctr_argsss = map (map args_of) ctr_specss;
   200     val ctr_mixfixess = map (map ctr_mixfix_of) ctr_specss;
   201 
   202     val sel_bindingsss = map (map (map fst)) ctr_argsss;
   203     val fake_ctr_Tsss0 = map (map (map (prepare_typ fake_lthy o snd))) ctr_argsss;
   204     val raw_sel_defaultsss = map (map defaults_of) ctr_specss;
   205 
   206     val (As :: _) :: fake_ctr_Tsss =
   207       burrow (burrow (Syntax.check_typs fake_lthy)) (Ass0 :: fake_ctr_Tsss0);
   208 
   209     val _ = (case duplicates (op =) unsorted_As of [] => ()
   210       | A :: _ => error ("Duplicate type parameter " ^
   211           quote (Syntax.string_of_typ no_defs_lthy A)));
   212 
   213     val rhs_As' = fold (fold (fold Term.add_tfreesT)) fake_ctr_Tsss [];
   214     val _ = (case subtract (op =) (map dest_TFree As) rhs_As' of
   215         [] => ()
   216       | A' :: _ => error ("Extra type variable on right-hand side: " ^
   217           quote (Syntax.string_of_typ no_defs_lthy (TFree A'))));
   218 
   219     fun eq_fpT_check (T as Type (s, Us)) (Type (s', Us')) =
   220         s = s' andalso (Us = Us' orelse error ("Illegal occurrence of recursive type " ^
   221           quote (Syntax.string_of_typ fake_lthy T)))
   222       | eq_fpT_check _ _ = false;
   223 
   224     fun freeze_fp (T as Type (s, Us)) =
   225         (case find_index (eq_fpT_check T) fake_Ts of
   226           ~1 => Type (s, map freeze_fp Us)
   227         | kk => nth Xs kk)
   228       | freeze_fp T = T;
   229 
   230     val ctr_TsssXs = map (map (map freeze_fp)) fake_ctr_Tsss;
   231     val ctr_sum_prod_TsXs = map (mk_sumTN_balanced o map HOLogic.mk_tupleT) ctr_TsssXs;
   232 
   233     val fp_eqs =
   234       map dest_TFree Xs ~~ map (Term.typ_subst_atomic (As ~~ unsorted_As)) ctr_sum_prod_TsXs;
   235 
   236     (* TODO: clean up list *)
   237     val (pre_bnfs, ((fp_bnfs as any_fp_bnf :: _, dtors0, ctors0, fp_folds0, fp_recs0, fp_induct,
   238            fp_strong_induct, dtor_ctors, ctor_dtors, ctor_injects, fp_map_thms, fp_set_thmss,
   239            fp_rel_thms, fp_fold_thms, fp_rec_thms), lthy)) =
   240       fp_bnf construct_fp fp_bs mixfixes (map dest_TFree unsorted_As) fp_eqs no_defs_lthy0;
   241 
   242     val timer = time (Timer.startRealTimer ());
   243 
   244     fun build_map build_arg (Type (s, Ts)) (Type (_, Us)) =
   245       let
   246         val bnf = the (bnf_of lthy s);
   247         val live = live_of_bnf bnf;
   248         val mapx = mk_map live Ts Us (map_of_bnf bnf);
   249         val TUs' = map dest_funT (fst (strip_typeN live (fastype_of mapx)));
   250       in Term.list_comb (mapx, map build_arg TUs') end;
   251 
   252     fun build_rel_step build_arg (Type (s, Ts)) =
   253       let
   254         val bnf = the (bnf_of lthy s);
   255         val live = live_of_bnf bnf;
   256         val rel = mk_rel live Ts Ts (rel_of_bnf bnf);
   257         val Ts' = map domain_type (fst (strip_typeN live (fastype_of rel)));
   258       in Term.list_comb (rel, map build_arg Ts') end;
   259 
   260     fun add_nesty_bnf_names Us =
   261       let
   262         fun add (Type (s, Ts)) ss =
   263             let val (needs, ss') = fold_map add Ts ss in
   264               if exists I needs then (true, insert (op =) s ss') else (false, ss')
   265             end
   266           | add T ss = (member (op =) Us T, ss);
   267       in snd oo add end;
   268 
   269     fun nesty_bnfs Us =
   270       map_filter (bnf_of lthy) (fold (fold (fold (add_nesty_bnf_names Us))) ctr_TsssXs []);
   271 
   272     val nesting_bnfs = nesty_bnfs As;
   273     val nested_bnfs = nesty_bnfs Xs;
   274 
   275     val pre_map_defs = map map_def_of_bnf pre_bnfs;
   276     val pre_set_defss = map set_defs_of_bnf pre_bnfs;
   277     val pre_rel_defs = map rel_def_of_bnf pre_bnfs;
   278     val nested_map_comps'' = map ((fn thm => thm RS sym) o map_comp_of_bnf) nested_bnfs;
   279     val nested_map_comp's = map map_comp'_of_bnf nested_bnfs;
   280     val nested_map_ids'' = map (unfold_thms lthy @{thms id_def} o map_id_of_bnf) nested_bnfs;
   281     val nesting_map_ids'' = map (unfold_thms lthy @{thms id_def} o map_id_of_bnf) nesting_bnfs;
   282     val nested_set_natural's = maps set_natural'_of_bnf nested_bnfs;
   283     val nesting_set_natural's = maps set_natural'_of_bnf nesting_bnfs;
   284 
   285     val live = live_of_bnf any_fp_bnf;
   286 
   287     val Bs =
   288       map3 (fn alive => fn A as TFree (_, S) => fn B => if alive then resort_tfree S B else A)
   289         (liveness_of_fp_bnf (length As) any_fp_bnf) As Bs0;
   290 
   291     val B_ify = Term.typ_subst_atomic (As ~~ Bs);
   292 
   293     val ctors = map (mk_ctor As) ctors0;
   294     val dtors = map (mk_dtor As) dtors0;
   295 
   296     val fpTs = map (domain_type o fastype_of) dtors;
   297 
   298     val exists_fp_subtype = exists_subtype (member (op =) fpTs);
   299     val exists_Cs_subtype = exists_subtype (member (op =) Cs);
   300 
   301     val ctr_Tsss = map (map (map (Term.typ_subst_atomic (Xs ~~ fpTs)))) ctr_TsssXs;
   302     val ns = map length ctr_Tsss;
   303     val kss = map (fn n => 1 upto n) ns;
   304     val mss = map (map length) ctr_Tsss;
   305     val Css = map2 replicate ns Cs;
   306 
   307     val fp_folds as any_fp_fold :: _ = map (mk_rec_like lfp As Cs) fp_folds0;
   308     val fp_recs as any_fp_rec :: _ = map (mk_rec_like lfp As Cs) fp_recs0;
   309 
   310     val fp_fold_fun_Ts = fst (split_last (binder_types (fastype_of any_fp_fold)));
   311     val fp_rec_fun_Ts = fst (split_last (binder_types (fastype_of any_fp_rec)));
   312 
   313     val (((fold_only as (gss, _, _), rec_only as (hss, _, _)),
   314           (cs, cpss, unfold_only as ((pgss, crssss, cgssss), (_, g_Tsss, _)),
   315            corec_only as ((phss, csssss, chssss), (_, h_Tsss, _)))), names_lthy0) =
   316       if lfp then
   317         let
   318           val y_Tsss =
   319             map3 (fn n => fn ms => map2 dest_tupleT ms o dest_sumTN_balanced n o domain_type)
   320               ns mss fp_fold_fun_Ts;
   321           val g_Tss = map2 (map2 (curry (op --->))) y_Tsss Css;
   322 
   323           val ((gss, ysss), lthy) =
   324             lthy
   325             |> mk_Freess "f" g_Tss
   326             ||>> mk_Freesss "x" y_Tsss;
   327 
   328           fun proj_recT proj (Type (s as @{type_name prod}, Ts as [T, U])) =
   329               if member (op =) fpTs T then proj (T, U) else Type (s, map (proj_recT proj) Ts)
   330             | proj_recT proj (Type (s, Ts)) = Type (s, map (proj_recT proj) Ts)
   331             | proj_recT _ T = T;
   332 
   333           fun unzip_recT T =
   334             if exists_fp_subtype T then [proj_recT fst T, proj_recT snd T] else [T];
   335 
   336           val z_Tsss =
   337             map3 (fn n => fn ms => map2 dest_tupleT ms o dest_sumTN_balanced n o domain_type)
   338               ns mss fp_rec_fun_Ts;
   339           val z_Tssss = map (map (map unzip_recT)) z_Tsss;
   340           val h_Tss = map2 (map2 (fold_rev (curry (op --->)))) z_Tssss Css;
   341 
   342           val hss = map2 (map2 retype_free) h_Tss gss;
   343           val zsss = map2 (map2 (map2 retype_free)) z_Tsss ysss;
   344         in
   345           ((((gss, g_Tss, ysss), (hss, h_Tss, zsss)),
   346             ([], [], (([], [], []), ([], [], [])), (([], [], []), ([], [], [])))), lthy)
   347         end
   348       else
   349         let
   350           (*avoid "'a itself" arguments in coiterators and corecursors*)
   351           val mss' =  map (fn [0] => [1] | ms => ms) mss;
   352 
   353           val p_Tss = map2 (fn n => replicate (Int.max (0, n - 1)) o mk_pred1T) ns Cs;
   354 
   355           fun flat_predss_getterss qss fss = maps (op @) (qss ~~ fss);
   356 
   357           fun flat_preds_predsss_gettersss [] [qss] [fss] = flat_predss_getterss qss fss
   358             | flat_preds_predsss_gettersss (p :: ps) (qss :: qsss) (fss :: fsss) =
   359               p :: flat_predss_getterss qss fss @ flat_preds_predsss_gettersss ps qsss fsss;
   360 
   361           fun mk_types maybe_unzipT fun_Ts =
   362             let
   363               val f_sum_prod_Ts = map range_type fun_Ts;
   364               val f_prod_Tss = map2 dest_sumTN_balanced ns f_sum_prod_Ts;
   365               val f_Tsss = map2 (map2 dest_tupleT) mss' f_prod_Tss;
   366               val f_Tssss =
   367                 map2 (fn C => map (map (map (curry (op -->) C) o maybe_unzipT))) Cs f_Tsss;
   368               val q_Tssss =
   369                 map (map (map (fn [_] => [] | [_, C] => [mk_pred1T (domain_type C)]))) f_Tssss;
   370               val pf_Tss = map3 flat_preds_predsss_gettersss p_Tss q_Tssss f_Tssss;
   371             in (q_Tssss, f_sum_prod_Ts, f_Tsss, f_Tssss, pf_Tss) end;
   372 
   373           val (r_Tssss, g_sum_prod_Ts, g_Tsss, g_Tssss, pg_Tss) = mk_types single fp_fold_fun_Ts;
   374 
   375           val (((cs, pss), gssss), lthy) =
   376             lthy
   377             |> mk_Frees "a" Cs
   378             ||>> mk_Freess "p" p_Tss
   379             ||>> mk_Freessss "g" g_Tssss;
   380           val rssss = map (map (map (fn [] => []))) r_Tssss;
   381 
   382           fun proj_corecT proj (Type (s as @{type_name sum}, Ts as [T, U])) =
   383               if member (op =) fpTs T then proj (T, U) else Type (s, map (proj_corecT proj) Ts)
   384             | proj_corecT proj (Type (s, Ts)) = Type (s, map (proj_corecT proj) Ts)
   385             | proj_corecT _ T = T;
   386 
   387           fun unzip_corecT T =
   388             if exists_fp_subtype T then [proj_corecT fst T, proj_corecT snd T] else [T];
   389 
   390           val (s_Tssss, h_sum_prod_Ts, h_Tsss, h_Tssss, ph_Tss) =
   391             mk_types unzip_corecT fp_rec_fun_Ts;
   392 
   393           val hssss_hd = map2 (map2 (map2 (fn T :: _ => fn [g] => retype_free T g))) h_Tssss gssss;
   394           val ((sssss, hssss_tl), lthy) =
   395             lthy
   396             |> mk_Freessss "q" s_Tssss
   397             ||>> mk_Freessss "h" (map (map (map tl)) h_Tssss);
   398           val hssss = map2 (map2 (map2 cons)) hssss_hd hssss_tl;
   399 
   400           val cpss = map2 (map o rapp) cs pss;
   401 
   402           fun mk_terms qssss fssss =
   403             let
   404               val pfss = map3 flat_preds_predsss_gettersss pss qssss fssss;
   405               val cqssss = map2 (map o map o map o rapp) cs qssss;
   406               val cfssss = map2 (map o map o map o rapp) cs fssss;
   407             in (pfss, cqssss, cfssss) end;
   408         in
   409           (((([], [], []), ([], [], [])),
   410             (cs, cpss, (mk_terms rssss gssss, (g_sum_prod_Ts, g_Tsss, pg_Tss)),
   411              (mk_terms sssss hssss, (h_sum_prod_Ts, h_Tsss, ph_Tss)))), lthy)
   412         end;
   413 
   414     fun define_ctrs_case_for_type (((((((((((((((((((((((((fp_bnf, fp_b), fpT), C), ctor), dtor),
   415             fp_fold), fp_rec), ctor_dtor), dtor_ctor), ctor_inject), pre_map_def), pre_set_defs),
   416           pre_rel_def), fp_map_thm), fp_set_thms), fp_rel_thm), n), ks), ms), ctr_bindings),
   417         ctr_mixfixes), ctr_Tss), disc_bindings), sel_bindingss), raw_sel_defaultss) no_defs_lthy =
   418       let
   419         val fp_b_name = Binding.name_of fp_b;
   420 
   421         val dtorT = domain_type (fastype_of ctor);
   422         val ctr_prod_Ts = map HOLogic.mk_tupleT ctr_Tss;
   423         val ctr_sum_prod_T = mk_sumTN_balanced ctr_prod_Ts;
   424         val case_Ts = map (fn Ts => Ts ---> C) ctr_Tss;
   425 
   426         val (((((w, fs), xss), yss), u'), names_lthy) =
   427           no_defs_lthy
   428           |> yield_singleton (mk_Frees "w") dtorT
   429           ||>> mk_Frees "f" case_Ts
   430           ||>> mk_Freess "x" ctr_Tss
   431           ||>> mk_Freess "y" (map (map B_ify) ctr_Tss)
   432           ||>> yield_singleton Variable.variant_fixes fp_b_name;
   433 
   434         val u = Free (u', fpT);
   435 
   436         val tuple_xs = map HOLogic.mk_tuple xss;
   437         val tuple_ys = map HOLogic.mk_tuple yss;
   438 
   439         val ctr_rhss =
   440           map3 (fn k => fn xs => fn tuple_x => fold_rev Term.lambda xs (ctor $
   441             mk_InN_balanced ctr_sum_prod_T n tuple_x k)) ks xss tuple_xs;
   442 
   443         val case_binding = qualify false fp_b_name (Binding.suffix_name ("_" ^ caseN) fp_b);
   444 
   445         val case_rhs =
   446           fold_rev Term.lambda (fs @ [u])
   447             (mk_sum_caseN_balanced (map2 mk_uncurried_fun fs xss) $ (dtor $ u));
   448 
   449         val ((raw_case :: raw_ctrs, raw_case_def :: raw_ctr_defs), (lthy', lthy)) = no_defs_lthy
   450           |> apfst split_list o fold_map3 (fn b => fn mx => fn rhs =>
   451               Local_Theory.define ((b, mx), ((Thm.def_binding b, []), rhs)) #>> apsnd snd)
   452             (case_binding :: ctr_bindings) (NoSyn :: ctr_mixfixes) (case_rhs :: ctr_rhss)
   453           ||> `Local_Theory.restore;
   454 
   455         val phi = Proof_Context.export_morphism lthy lthy';
   456 
   457         val ctr_defs = map (Morphism.thm phi) raw_ctr_defs;
   458         val ctr_defs' =
   459           map2 (fn m => fn def => mk_unabs_def m (def RS meta_eq_to_obj_eq)) ms ctr_defs;
   460         val case_def = Morphism.thm phi raw_case_def;
   461 
   462         val ctrs0 = map (Morphism.term phi) raw_ctrs;
   463         val casex0 = Morphism.term phi raw_case;
   464 
   465         val ctrs = map (mk_ctr As) ctrs0;
   466 
   467         fun wrap lthy =
   468           let
   469             fun exhaust_tac {context = ctxt, prems = _} =
   470               let
   471                 val ctor_iff_dtor_thm =
   472                   let
   473                     val goal =
   474                       fold_rev Logic.all [w, u]
   475                         (mk_Trueprop_eq (HOLogic.mk_eq (u, ctor $ w), HOLogic.mk_eq (dtor $ u, w)));
   476                   in
   477                     Skip_Proof.prove lthy [] [] goal (fn {context = ctxt, ...} =>
   478                       mk_ctor_iff_dtor_tac ctxt (map (SOME o certifyT lthy) [dtorT, fpT])
   479                         (certify lthy ctor) (certify lthy dtor) ctor_dtor dtor_ctor)
   480                     |> Thm.close_derivation
   481                     |> Morphism.thm phi
   482                   end;
   483 
   484                 val sumEN_thm' =
   485                   unfold_thms lthy @{thms all_unit_eq}
   486                     (Drule.instantiate' (map (SOME o certifyT lthy) ctr_prod_Ts) []
   487                        (mk_sumEN_balanced n))
   488                   |> Morphism.thm phi;
   489               in
   490                 mk_exhaust_tac ctxt n ctr_defs ctor_iff_dtor_thm sumEN_thm'
   491               end;
   492 
   493             val inject_tacss =
   494               map2 (fn 0 => K [] | _ => fn ctr_def => [fn {context = ctxt, ...} =>
   495                   mk_inject_tac ctxt ctr_def ctor_inject]) ms ctr_defs;
   496 
   497             val half_distinct_tacss =
   498               map (map (fn (def, def') => fn {context = ctxt, ...} =>
   499                 mk_half_distinct_tac ctxt ctor_inject [def, def'])) (mk_half_pairss (`I ctr_defs));
   500 
   501             val case_tacs =
   502               map3 (fn k => fn m => fn ctr_def => fn {context = ctxt, ...} =>
   503                 mk_case_tac ctxt n k m case_def ctr_def dtor_ctor) ks ms ctr_defs;
   504 
   505             val tacss = [exhaust_tac] :: inject_tacss @ half_distinct_tacss @ [case_tacs];
   506 
   507             val sel_defaultss = map (map (apsnd (prepare_term lthy))) raw_sel_defaultss
   508           in
   509             wrap_datatype tacss (((wrap_opts, ctrs0), casex0), (disc_bindings, (sel_bindingss,
   510               sel_defaultss))) lthy
   511           end;
   512 
   513         fun derive_maps_sets_rels (wrap_res, lthy) =
   514           let
   515             val rel_flip = rel_flip_of_bnf fp_bnf;
   516             val nones = replicate live NONE;
   517 
   518             val ctor_cong =
   519               if lfp then Drule.dummy_thm
   520               else cterm_instantiate_pos [NONE, NONE, SOME (certify lthy ctor)] arg_cong;
   521 
   522             fun mk_cIn ify =
   523               certify lthy o (not lfp ? curry (op $) (map_types ify ctor)) oo
   524               mk_InN_balanced (ify ctr_sum_prod_T) n;
   525 
   526             val cxIns = map2 (mk_cIn I) tuple_xs ks;
   527             val cyIns = map2 (mk_cIn B_ify) tuple_ys ks;
   528 
   529             fun mk_map_thm ctr_def' cxIn =
   530               fold_thms lthy [ctr_def']
   531                 (unfold_thms lthy (pre_map_def ::
   532                      (if lfp then [] else [ctor_dtor, dtor_ctor]) @ sum_prod_thms_map)
   533                    (cterm_instantiate_pos (nones @ [SOME cxIn])
   534                       (if lfp then fp_map_thm else fp_map_thm RS ctor_cong)))
   535               |> singleton (Proof_Context.export names_lthy no_defs_lthy);
   536 
   537             fun mk_set_thm fp_set_thm ctr_def' cxIn =
   538               fold_thms lthy [ctr_def']
   539                 (unfold_thms lthy (pre_set_defs @ nested_set_natural's @ nesting_set_natural's @
   540                      (if lfp then [] else [dtor_ctor]) @ sum_prod_thms_set)
   541                    (cterm_instantiate_pos [SOME cxIn] fp_set_thm))
   542               |> singleton (Proof_Context.export names_lthy no_defs_lthy);
   543 
   544             fun mk_set_thms fp_set_thm = map2 (mk_set_thm fp_set_thm) ctr_defs' cxIns;
   545 
   546             val map_thms = map2 mk_map_thm ctr_defs' cxIns;
   547             val set_thmss = map mk_set_thms fp_set_thms;
   548 
   549             val rel_infos = (ctr_defs' ~~ cxIns, ctr_defs' ~~ cyIns);
   550 
   551             fun mk_rel_thm postproc ctr_defs' cxIn cyIn =
   552               fold_thms lthy ctr_defs'
   553                  (unfold_thms lthy (pre_rel_def :: (if lfp then [] else [dtor_ctor]) @
   554                       sum_prod_thms_rel)
   555                     (cterm_instantiate_pos (nones @ [SOME cxIn, SOME cyIn]) fp_rel_thm))
   556               |> postproc
   557               |> singleton (Proof_Context.export names_lthy no_defs_lthy);
   558 
   559             fun mk_rel_inject_thm ((ctr_def', cxIn), (_, cyIn)) =
   560               mk_rel_thm (unfold_thms lthy @{thms eq_sym_Unity_conv}) [ctr_def'] cxIn cyIn;
   561 
   562             val rel_inject_thms = map mk_rel_inject_thm (op ~~ rel_infos);
   563 
   564             fun mk_half_rel_distinct_thm ((xctr_def', cxIn), (yctr_def', cyIn)) =
   565               mk_rel_thm (fn thm => thm RS @{thm eq_False[THEN iffD1]}) [xctr_def', yctr_def']
   566                 cxIn cyIn;
   567 
   568             fun mk_other_half_rel_distinct_thm thm =
   569               flip_rels lthy live thm RS (rel_flip RS sym RS @{thm arg_cong[of _ _ Not]} RS iffD2);
   570 
   571             val half_rel_distinct_thmss =
   572               map (map mk_half_rel_distinct_thm) (mk_half_pairss rel_infos);
   573             val other_half_rel_distinct_thmss =
   574               map (map mk_other_half_rel_distinct_thm) half_rel_distinct_thmss;
   575             val (rel_distinct_thms, _) =
   576               join_halves n half_rel_distinct_thmss other_half_rel_distinct_thmss;
   577 
   578             val notes =
   579               [(mapN, map_thms, code_simp_attrs),
   580                (rel_distinctN, rel_distinct_thms, code_simp_attrs),
   581                (rel_injectN, rel_inject_thms, code_simp_attrs),
   582                (setsN, flat set_thmss, code_simp_attrs)]
   583               |> filter_out (null o #2)
   584               |> map (fn (thmN, thms, attrs) =>
   585                 ((qualify true fp_b_name (Binding.name thmN), attrs), [(thms, [])]));
   586           in
   587             (wrap_res, lthy |> Local_Theory.notes notes |> snd)
   588           end;
   589 
   590         fun define_fold_rec no_defs_lthy =
   591           let
   592             val fpT_to_C = fpT --> C;
   593 
   594             fun build_prod_proj mk_proj (T, U) =
   595               if T = U then
   596                 id_const T
   597               else
   598                 (case (T, U) of
   599                   (Type (s, _), Type (s', _)) =>
   600                   if s = s' then build_map (build_prod_proj mk_proj) T U else mk_proj T
   601                 | _ => mk_proj T);
   602 
   603             (* TODO: Avoid these complications; cf. corec case *)
   604             fun mk_U proj (Type (s as @{type_name prod}, Ts as [T', U])) =
   605                 if member (op =) fpTs T' then proj (T', U) else Type (s, map (mk_U proj) Ts)
   606               | mk_U proj (Type (s, Ts)) = Type (s, map (mk_U proj) Ts)
   607               | mk_U _ T = T;
   608 
   609             fun unzip_rec (x as Free (_, T)) =
   610               if exists_fp_subtype T then
   611                 [build_prod_proj fst_const (T, mk_U fst T) $ x,
   612                  build_prod_proj snd_const (T, mk_U snd T) $ x]
   613               else
   614                 [x];
   615 
   616             fun mk_rec_like_arg f xs = mk_tupled_fun (HOLogic.mk_tuple xs) f (maps unzip_rec xs);
   617 
   618             fun generate_rec_like (suf, fp_rec_like, (fss, f_Tss, xsss)) =
   619               let
   620                 val res_T = fold_rev (curry (op --->)) f_Tss fpT_to_C;
   621                 val binding = qualify false fp_b_name (Binding.suffix_name ("_" ^ suf) fp_b);
   622                 val spec =
   623                   mk_Trueprop_eq (lists_bmoc fss (Free (Binding.name_of binding, res_T)),
   624                     Term.list_comb (fp_rec_like,
   625                       map2 (mk_sum_caseN_balanced oo map2 mk_rec_like_arg) fss xsss));
   626               in (binding, spec) end;
   627 
   628             val rec_like_infos =
   629               [(foldN, fp_fold, fold_only),
   630                (recN, fp_rec, rec_only)];
   631 
   632             val (bindings, specs) = map generate_rec_like rec_like_infos |> split_list;
   633 
   634             val ((csts, defs), (lthy', lthy)) = no_defs_lthy
   635               |> apfst split_list o fold_map2 (fn b => fn spec =>
   636                 Specification.definition (SOME (b, NONE, NoSyn), ((Thm.def_binding b, []), spec))
   637                 #>> apsnd snd) bindings specs
   638               ||> `Local_Theory.restore;
   639 
   640             val phi = Proof_Context.export_morphism lthy lthy';
   641 
   642             val [fold_def, rec_def] = map (Morphism.thm phi) defs;
   643 
   644             val [foldx, recx] = map (mk_rec_like lfp As Cs o Morphism.term phi) csts;
   645           in
   646             ((foldx, recx, fold_def, rec_def), lthy')
   647           end;
   648 
   649         fun define_unfold_corec no_defs_lthy =
   650           let
   651             val B_to_fpT = C --> fpT;
   652 
   653             fun build_sum_inj mk_inj (T, U) =
   654               if T = U then
   655                 id_const T
   656               else
   657                 (case (T, U) of
   658                   (Type (s, _), Type (s', _)) =>
   659                   if s = s' then build_map (build_sum_inj mk_inj) T U
   660                   else uncurry mk_inj (dest_sumT U)
   661                 | _ => uncurry mk_inj (dest_sumT U));
   662 
   663             fun build_dtor_corec_like_arg _ [] [cf] = cf
   664               | build_dtor_corec_like_arg T [cq] [cf, cf'] =
   665                 mk_If cq (build_sum_inj Inl_const (fastype_of cf, T) $ cf)
   666                   (build_sum_inj Inr_const (fastype_of cf', T) $ cf')
   667 
   668             val crgsss = map3 (map3 (map3 build_dtor_corec_like_arg)) g_Tsss crssss cgssss;
   669             val cshsss = map3 (map3 (map3 build_dtor_corec_like_arg)) h_Tsss csssss chssss;
   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, (cqfsss, ((pfss, _, _), (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, (crgsss, unfold_only)),
   688                (corecN, fp_rec, (cshsss, 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 else HOLogic.eq_const T
   923               | kk => nth rs' kk);
   924 
   925             fun build_rel_app rs' usel vsel =
   926               fold rapp [usel, vsel] (build_rel rs' (fastype_of usel));
   927 
   928             fun mk_prem_ctr_concls rs' n k udisc usels vdisc vsels =
   929               (if k = n then [] else [HOLogic.mk_eq (udisc, vdisc)]) @
   930               (if null usels then
   931                  []
   932                else
   933                  [Library.foldr HOLogic.mk_imp (if n = 1 then [] else [udisc, vdisc],
   934                     Library.foldr1 HOLogic.mk_conj (map2 (build_rel_app rs') usels vsels))]);
   935 
   936             fun mk_prem_concl rs' n udiscs uselss vdiscs vselss =
   937               Library.foldr1 HOLogic.mk_conj
   938                 (flat (map5 (mk_prem_ctr_concls rs' n) (1 upto n) udiscs uselss vdiscs vselss))
   939               handle List.Empty => @{term True};
   940 
   941             fun mk_prem rs' uvr u v n udiscs uselss vdiscs vselss =
   942               fold_rev Logic.all [u, v] (Logic.mk_implies (HOLogic.mk_Trueprop uvr,
   943                 HOLogic.mk_Trueprop (mk_prem_concl rs' n udiscs uselss vdiscs vselss)));
   944 
   945             val concl =
   946               HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj
   947                 (map3 (fn uvr => fn u => fn v => HOLogic.mk_imp (uvr, HOLogic.mk_eq (u, v)))
   948                    uvrs us vs));
   949 
   950             fun mk_goal rs' =
   951               Logic.list_implies (map8 (mk_prem rs') uvrs us vs ns udiscss uselsss vdiscss vselsss,
   952                 concl);
   953 
   954             val goal = mk_goal rs;
   955             val strong_goal = mk_goal strong_rs;
   956 
   957             fun prove dtor_coinduct' goal =
   958               Skip_Proof.prove lthy [] [] goal (fn {context = ctxt, ...} =>
   959                 mk_coinduct_tac ctxt nesting_rel_eqs nn ns dtor_coinduct' pre_rel_defs dtor_ctors
   960                   exhaust_thms ctr_defss disc_thmsss sel_thmsss)
   961               |> singleton (Proof_Context.export names_lthy lthy)
   962               |> Thm.close_derivation;
   963 
   964             fun postproc nn thm =
   965               Thm.permute_prems 0 nn
   966                 (if nn = 1 then thm RS mp
   967                  else funpow nn (fn thm => reassoc_conjs (thm RS mp_conj)) thm)
   968               |> Drule.zero_var_indexes
   969               |> `(conj_dests nn);
   970           in
   971             (postproc nn (prove fp_induct goal), postproc nn (prove fp_strong_induct strong_goal))
   972           end;
   973 
   974         fun mk_maybe_not pos = not pos ? HOLogic.mk_not;
   975 
   976         val gunfolds = map (lists_bmoc pgss) unfolds;
   977         val hcorecs = map (lists_bmoc phss) corecs;
   978 
   979         val (unfold_thmss, corec_thmss, safe_unfold_thmss, safe_corec_thmss) =
   980           let
   981             fun mk_goal pfss c cps fcorec_like n k ctr m cfs' =
   982               fold_rev (fold_rev Logic.all) ([c] :: pfss)
   983                 (Logic.list_implies (seq_conds (HOLogic.mk_Trueprop oo mk_maybe_not) n k cps,
   984                    mk_Trueprop_eq (fcorec_like $ c, Term.list_comb (ctr, take m cfs'))));
   985 
   986             fun build_corec_like fcorec_likes (T, U) =
   987               if T = U then
   988                 id_const T
   989               else
   990                 (case find_index (curry (op =) U) fpTs of
   991                   ~1 => build_map (build_corec_like fcorec_likes) T U
   992                 | kk => nth fcorec_likes kk);
   993 
   994             val mk_U = typ_subst (map2 pair Cs fpTs);
   995 
   996             fun intr_corec_likes fcorec_likes [] [cf] =
   997                 let val T = fastype_of cf in
   998                   if exists_Cs_subtype T then build_corec_like fcorec_likes (T, mk_U T) $ cf else cf
   999                 end
  1000               | intr_corec_likes fcorec_likes [cq] [cf, cf'] =
  1001                 mk_If cq (intr_corec_likes fcorec_likes [] [cf])
  1002                   (intr_corec_likes fcorec_likes [] [cf']);
  1003 
  1004             val crgsss = map2 (map2 (map2 (intr_corec_likes gunfolds))) crssss cgssss;
  1005             val cshsss = map2 (map2 (map2 (intr_corec_likes hcorecs))) csssss chssss;
  1006 
  1007             val unfold_goalss =
  1008               map8 (map4 oooo mk_goal pgss) cs cpss gunfolds ns kss ctrss mss crgsss;
  1009             val corec_goalss =
  1010               map8 (map4 oooo mk_goal phss) cs cpss hcorecs ns kss ctrss mss cshsss;
  1011 
  1012             fun mk_map_if_distrib bnf =
  1013               let
  1014                 val mapx = map_of_bnf bnf;
  1015                 val live = live_of_bnf bnf;
  1016                 val ((Ts, T), U) = strip_typeN (live + 1) (fastype_of mapx) |>> split_last;
  1017                 val fs = Variable.variant_frees lthy [mapx] (map (pair "f") Ts);
  1018                 val t = Term.list_comb (mapx, map (Var o apfst (rpair 0)) fs);
  1019               in
  1020                 Drule.instantiate' (map (SOME o certifyT lthy) [U, T]) [SOME (certify lthy t)]
  1021                   @{thm if_distrib}
  1022               end;
  1023 
  1024             val nested_map_if_distribs = map mk_map_if_distrib nested_bnfs;
  1025 
  1026             val unfold_tacss =
  1027               map3 (map oo mk_corec_like_tac unfold_defs [] [] nesting_map_ids'' [])
  1028                 fp_fold_thms pre_map_defs ctr_defss;
  1029             val corec_tacss =
  1030               map3 (map oo mk_corec_like_tac corec_defs nested_map_comps'' nested_map_comp's
  1031                   (nested_map_ids'' @ nesting_map_ids'') nested_map_if_distribs)
  1032                 fp_rec_thms pre_map_defs ctr_defss;
  1033 
  1034             fun prove goal tac =
  1035               Skip_Proof.prove lthy [] [] goal (tac o #context) |> Thm.close_derivation;
  1036 
  1037             val unfold_thmss = map2 (map2 prove) unfold_goalss unfold_tacss;
  1038             val corec_thmss = map2 (map2 prove) corec_goalss corec_tacss;
  1039 
  1040             val filter_safesss =
  1041               map2 (map_filter (fn (safes, thm) => if forall I safes then SOME thm else NONE) oo
  1042                 curry (op ~~)) (map2 (map2 (map2 (member (op =)))) cgssss crgsss);
  1043 
  1044             val safe_unfold_thmss = filter_safesss unfold_thmss;
  1045             val safe_corec_thmss = filter_safesss corec_thmss;
  1046           in
  1047             (unfold_thmss, corec_thmss, safe_unfold_thmss, safe_corec_thmss)
  1048           end;
  1049 
  1050         val (disc_unfold_iff_thmss, disc_corec_iff_thmss) =
  1051           let
  1052             fun mk_goal c cps fcorec_like n k disc =
  1053               mk_Trueprop_eq (disc $ (fcorec_like $ c),
  1054                 if n = 1 then @{const True}
  1055                 else Library.foldr1 HOLogic.mk_conj (seq_conds mk_maybe_not n k cps));
  1056 
  1057             val unfold_goalss = map6 (map2 oooo mk_goal) cs cpss gunfolds ns kss discss;
  1058             val corec_goalss = map6 (map2 oooo mk_goal) cs cpss hcorecs ns kss discss;
  1059 
  1060             fun mk_case_split' cp =
  1061               Drule.instantiate' [] [SOME (certify lthy cp)] @{thm case_split};
  1062 
  1063             val case_splitss' = map (map mk_case_split') cpss;
  1064 
  1065             val unfold_tacss =
  1066               map3 (map oo mk_disc_corec_like_iff_tac) case_splitss' unfold_thmss disc_thmsss;
  1067             val corec_tacss =
  1068               map3 (map oo mk_disc_corec_like_iff_tac) case_splitss' corec_thmss disc_thmsss;
  1069 
  1070             fun prove goal tac =
  1071               Skip_Proof.prove lthy [] [] goal (tac o #context)
  1072               |> singleton (Proof_Context.export names_lthy0 no_defs_lthy)
  1073               |> Thm.close_derivation;
  1074 
  1075             fun proves [_] [_] = []
  1076               | proves goals tacs = map2 prove goals tacs;
  1077           in
  1078             (map2 proves unfold_goalss unfold_tacss,
  1079              map2 proves corec_goalss corec_tacss)
  1080           end;
  1081 
  1082         val is_triv_discI = is_triv_implies orf is_concl_refl;
  1083 
  1084         fun mk_disc_corec_like_thms corec_likes discIs =
  1085           map (op RS) (filter_out (is_triv_discI o snd) (corec_likes ~~ discIs));
  1086 
  1087         val disc_unfold_thmss = map2 mk_disc_corec_like_thms unfold_thmss discIss;
  1088         val disc_corec_thmss = map2 mk_disc_corec_like_thms corec_thmss discIss;
  1089 
  1090         fun mk_sel_corec_like_thm corec_like_thm sel sel_thm =
  1091           let
  1092             val (domT, ranT) = dest_funT (fastype_of sel);
  1093             val arg_cong' =
  1094               Drule.instantiate' (map (SOME o certifyT lthy) [domT, ranT])
  1095                 [NONE, NONE, SOME (certify lthy sel)] arg_cong
  1096               |> Thm.varifyT_global;
  1097             val sel_thm' = sel_thm RSN (2, trans);
  1098           in
  1099             corec_like_thm RS arg_cong' RS sel_thm'
  1100           end;
  1101 
  1102         fun mk_sel_corec_like_thms corec_likess =
  1103           map3 (map3 (map2 o mk_sel_corec_like_thm)) corec_likess selsss sel_thmsss |> map flat;
  1104 
  1105         val sel_unfold_thmss = mk_sel_corec_like_thms unfold_thmss;
  1106         val sel_corec_thmss = mk_sel_corec_like_thms corec_thmss;
  1107 
  1108         fun flat_corec_like_thms corec_likes disc_corec_likes sel_corec_likes =
  1109           corec_likes @ disc_corec_likes @ sel_corec_likes;
  1110 
  1111         val simp_thmss =
  1112           mk_simp_thmss wrap_ress
  1113             (map3 flat_corec_like_thms safe_corec_thmss disc_corec_thmss sel_corec_thmss)
  1114             (map3 flat_corec_like_thms safe_unfold_thmss disc_unfold_thmss sel_unfold_thmss);
  1115 
  1116         val anonymous_notes =
  1117           [(flat safe_unfold_thmss @ flat safe_corec_thmss, simp_attrs)]
  1118           |> map (fn (thms, attrs) => ((Binding.empty, attrs), [(thms, [])]));
  1119 
  1120         val common_notes =
  1121           (if nn > 1 then
  1122              (* FIXME: attribs *)
  1123              [(coinductN, [coinduct_thm], []),
  1124               (strong_coinductN, [strong_coinduct_thm], [])]
  1125            else
  1126              [])
  1127           |> map (fn (thmN, thms, attrs) =>
  1128             ((qualify true fp_common_name (Binding.name thmN), attrs), [(thms, [])]));
  1129 
  1130         val notes =
  1131           [(coinductN, map single coinduct_thms, []), (* FIXME: attribs *)
  1132            (corecN, corec_thmss, []),
  1133            (disc_corecN, disc_corec_thmss, simp_attrs),
  1134            (disc_corec_iffN, disc_corec_iff_thmss, simp_attrs),
  1135            (disc_unfoldN, disc_unfold_thmss, simp_attrs),
  1136            (disc_unfold_iffN, disc_unfold_iff_thmss, simp_attrs),
  1137            (sel_corecN, sel_corec_thmss, simp_attrs),
  1138            (sel_unfoldN, sel_unfold_thmss, simp_attrs),
  1139            (simpsN, simp_thmss, []),
  1140            (strong_coinductN, map single strong_coinduct_thms, []), (* FIXME: attribs *)
  1141            (unfoldN, unfold_thmss, [])]
  1142           |> maps (fn (thmN, thmss, attrs) =>
  1143             map_filter (fn (_, []) => NONE | (fp_b_name, thms) =>
  1144               SOME ((qualify true fp_b_name (Binding.name thmN), attrs),
  1145                 [(thms, [])])) (fp_b_names ~~ thmss));
  1146       in
  1147         lthy |> Local_Theory.notes (anonymous_notes @ common_notes @ notes) |> snd
  1148       end;
  1149 
  1150     val lthy' = lthy
  1151       |> fold_map define_ctrs_case_for_type (fp_bnfs ~~ fp_bs ~~ fpTs ~~ Cs ~~ ctors ~~ dtors ~~
  1152         fp_folds ~~ fp_recs ~~ ctor_dtors ~~ dtor_ctors ~~ ctor_injects ~~ pre_map_defs ~~
  1153         pre_set_defss ~~ pre_rel_defs ~~ fp_map_thms ~~ fp_set_thmss ~~ fp_rel_thms ~~ ns ~~ kss ~~
  1154         mss ~~ ctr_bindingss ~~ ctr_mixfixess ~~ ctr_Tsss ~~ disc_bindingss ~~ sel_bindingsss ~~
  1155         raw_sel_defaultsss)
  1156       |> wrap_types_and_more
  1157       |> (if lfp then derive_induct_fold_rec_thms_for_types
  1158           else derive_coinduct_unfold_corec_thms_for_types);
  1159 
  1160     val timer = time (timer ("Constructors, discriminators, selectors, etc., for the new " ^
  1161       (if lfp then "" else "co") ^ "datatype"));
  1162   in
  1163     timer; lthy'
  1164   end;
  1165 
  1166 val datatypes = define_datatypes (K I) (K I) (K I);
  1167 
  1168 val datatype_cmd = define_datatypes Typedecl.read_constraint Syntax.parse_typ Syntax.parse_term;
  1169 
  1170 val parse_ctr_arg =
  1171   @{keyword "("} |-- parse_binding_colon -- Parse.typ --| @{keyword ")"} ||
  1172   (Parse.typ >> pair Binding.empty);
  1173 
  1174 val parse_defaults =
  1175   @{keyword "("} |-- @{keyword "defaults"} |-- Scan.repeat parse_bound_term --| @{keyword ")"};
  1176 
  1177 val parse_single_spec =
  1178   Parse.type_args_constrained -- Parse.binding -- Parse.opt_mixfix --
  1179   (@{keyword "="} |-- Parse.enum1 "|" (parse_opt_binding_colon -- Parse.binding --
  1180     Scan.repeat parse_ctr_arg -- Scan.optional parse_defaults [] -- Parse.opt_mixfix));
  1181 
  1182 val parse_datatype = parse_wrap_options -- Parse.and_list1 parse_single_spec;
  1183 
  1184 fun parse_datatype_cmd lfp construct_fp = parse_datatype >> datatype_cmd lfp construct_fp;
  1185 
  1186 end;