src/HOL/BNF/Tools/bnf_fp_def_sugar.ML
author blanchet
Mon Oct 01 11:04:30 2012 +0200 (2012-10-01)
changeset 49672 902b24e0ffb4
parent 49671 61729b149397
child 49681 aa66ea552357
permissions -rw-r--r--
fixed recursor definition for datatypes with inner products (e.g. "'a trm" from the lambda-term example)
     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 tick u f = Term.lambda u (HOLogic.mk_prod (u, f $ u));
   117 
   118 fun tack z_name (c, u) f =
   119   let val z = Free (z_name, mk_sumT (fastype_of u, fastype_of c)) in
   120     Term.lambda z (mk_sum_case (Term.lambda u u, Term.lambda c (f $ c)) $ z)
   121   end;
   122 
   123 fun cannot_merge_types () = error "Mutually recursive types must have the same type parameters";
   124 
   125 fun merge_type_arg T T' = if T = T' then T else cannot_merge_types ();
   126 
   127 fun merge_type_args (As, As') =
   128   if length As = length As' then map2 merge_type_arg As As' else cannot_merge_types ();
   129 
   130 fun reassoc_conjs thm =
   131   reassoc_conjs (thm RS @{thm conj_assoc[THEN iffD1]})
   132   handle THM _ => thm;
   133 
   134 fun type_args_constrained_of (((cAs, _), _), _) = cAs;
   135 fun type_binding_of (((_, b), _), _) = b;
   136 fun mixfix_of ((_, mx), _) = mx;
   137 fun ctr_specs_of (_, ctr_specs) = ctr_specs;
   138 
   139 fun disc_of ((((disc, _), _), _), _) = disc;
   140 fun ctr_of ((((_, ctr), _), _), _) = ctr;
   141 fun args_of (((_, args), _), _) = args;
   142 fun defaults_of ((_, ds), _) = ds;
   143 fun ctr_mixfix_of (_, mx) = mx;
   144 
   145 fun define_datatypes prepare_constraint prepare_typ prepare_term lfp construct_fp
   146     (wrap_opts as (no_dests, rep_compat), specs) no_defs_lthy0 =
   147   let
   148     (* TODO: sanity checks on arguments *)
   149     (* TODO: integration with function package ("size") *)
   150 
   151     val _ = if not lfp andalso no_dests then error "Cannot define destructor-less codatatypes"
   152       else ();
   153 
   154     fun qualify mandatory fp_b_name =
   155       Binding.qualify mandatory fp_b_name o (rep_compat ? Binding.qualify false rep_compat_prefix);
   156 
   157     val nn = length specs;
   158     val fp_bs = map type_binding_of specs;
   159     val fp_b_names = map Binding.name_of fp_bs;
   160     val fp_common_name = mk_common_name fp_b_names;
   161 
   162     fun prepare_type_arg (ty, c) =
   163       let val TFree (s, _) = prepare_typ no_defs_lthy0 ty in
   164         TFree (s, prepare_constraint no_defs_lthy0 c)
   165       end;
   166 
   167     val Ass0 = map (map prepare_type_arg o type_args_constrained_of) specs;
   168     val unsorted_Ass0 = map (map (resort_tfree HOLogic.typeS)) Ass0;
   169     val unsorted_As = Library.foldr1 merge_type_args unsorted_Ass0;
   170 
   171     val (((Bs0, Cs), Xs), no_defs_lthy) =
   172       no_defs_lthy0
   173       |> fold (Variable.declare_typ o resort_tfree dummyS) unsorted_As
   174       |> mk_TFrees (length unsorted_As)
   175       ||>> mk_TFrees nn
   176       ||>> apfst (map TFree) o
   177         variant_types (map (prefix "'") fp_b_names) (replicate nn HOLogic.typeS);
   178 
   179     (* TODO: cleaner handling of fake contexts, without "background_theory" *)
   180     (*the "perhaps o try" below helps gracefully handles the case where the new type is defined in a
   181       locale and shadows an existing global type*)
   182     val fake_thy =
   183       Theory.copy #> fold (fn spec => perhaps (try (Sign.add_type no_defs_lthy
   184         (type_binding_of spec, length (type_args_constrained_of spec), mixfix_of spec)))) specs;
   185     val fake_lthy = Proof_Context.background_theory fake_thy no_defs_lthy;
   186 
   187     fun mk_fake_T b =
   188       Type (fst (Term.dest_Type (Proof_Context.read_type_name fake_lthy true (Binding.name_of b))),
   189         unsorted_As);
   190 
   191     val fake_Ts = map mk_fake_T fp_bs;
   192 
   193     val mixfixes = map mixfix_of specs;
   194 
   195     val _ = (case duplicates Binding.eq_name fp_bs of [] => ()
   196       | b :: _ => error ("Duplicate type name declaration " ^ quote (Binding.name_of b)));
   197 
   198     val ctr_specss = map ctr_specs_of specs;
   199 
   200     val disc_bindingss = map (map disc_of) ctr_specss;
   201     val ctr_bindingss =
   202       map2 (fn fp_b_name => map (qualify false fp_b_name o ctr_of)) fp_b_names ctr_specss;
   203     val ctr_argsss = map (map args_of) ctr_specss;
   204     val ctr_mixfixess = map (map ctr_mixfix_of) ctr_specss;
   205 
   206     val sel_bindingsss = map (map (map fst)) ctr_argsss;
   207     val fake_ctr_Tsss0 = map (map (map (prepare_typ fake_lthy o snd))) ctr_argsss;
   208     val raw_sel_defaultsss = map (map defaults_of) ctr_specss;
   209 
   210     val (As :: _) :: fake_ctr_Tsss =
   211       burrow (burrow (Syntax.check_typs fake_lthy)) (Ass0 :: fake_ctr_Tsss0);
   212 
   213     val _ = (case duplicates (op =) unsorted_As of [] => ()
   214       | A :: _ => error ("Duplicate type parameter " ^
   215           quote (Syntax.string_of_typ no_defs_lthy A)));
   216 
   217     val rhs_As' = fold (fold (fold Term.add_tfreesT)) fake_ctr_Tsss [];
   218     val _ = (case subtract (op =) (map dest_TFree As) rhs_As' of
   219         [] => ()
   220       | A' :: _ => error ("Extra type variable on right-hand side: " ^
   221           quote (Syntax.string_of_typ no_defs_lthy (TFree A'))));
   222 
   223     fun eq_fpT_check (T as Type (s, Us)) (Type (s', Us')) =
   224         s = s' andalso (Us = Us' orelse error ("Illegal occurrence of recursive type " ^
   225           quote (Syntax.string_of_typ fake_lthy T)))
   226       | eq_fpT_check _ _ = false;
   227 
   228     fun freeze_fp (T as Type (s, Us)) =
   229         (case find_index (eq_fpT_check T) fake_Ts of
   230           ~1 => Type (s, map freeze_fp Us)
   231         | kk => nth Xs kk)
   232       | freeze_fp T = T;
   233 
   234     val ctr_TsssXs = map (map (map freeze_fp)) fake_ctr_Tsss;
   235     val ctr_sum_prod_TsXs = map (mk_sumTN_balanced o map HOLogic.mk_tupleT) ctr_TsssXs;
   236 
   237     val fp_eqs =
   238       map dest_TFree Xs ~~ map (Term.typ_subst_atomic (As ~~ unsorted_As)) ctr_sum_prod_TsXs;
   239 
   240     (* TODO: clean up list *)
   241     val (pre_bnfs, ((fp_bnfs as any_fp_bnf :: _, dtors0, ctors0, fp_folds0, fp_recs0, fp_induct,
   242            fp_strong_induct, dtor_ctors, ctor_dtors, ctor_injects, fp_map_thms, fp_set_thmss,
   243            fp_rel_thms, fp_fold_thms, fp_rec_thms), lthy)) =
   244       fp_bnf construct_fp fp_bs mixfixes (map dest_TFree unsorted_As) fp_eqs no_defs_lthy0;
   245 
   246     val timer = time (Timer.startRealTimer ());
   247 
   248     fun build_map build_arg (Type (s, Ts)) (Type (_, Us)) =
   249       let
   250         val bnf = the (bnf_of lthy s);
   251         val live = live_of_bnf bnf;
   252         val mapx = mk_map live Ts Us (map_of_bnf bnf);
   253         val TUs' = map dest_funT (fst (strip_typeN live (fastype_of mapx)));
   254       in Term.list_comb (mapx, map build_arg TUs') end;
   255 
   256     fun build_rel_step build_arg (Type (s, Ts)) =
   257       let
   258         val bnf = the (bnf_of lthy s);
   259         val live = live_of_bnf bnf;
   260         val rel = mk_rel live Ts Ts (rel_of_bnf bnf);
   261         val Ts' = map domain_type (fst (strip_typeN live (fastype_of rel)));
   262       in Term.list_comb (rel, map build_arg Ts') end;
   263 
   264     fun add_nesty_bnf_names Us =
   265       let
   266         fun add (Type (s, Ts)) ss =
   267             let val (needs, ss') = fold_map add Ts ss in
   268               if exists I needs then (true, insert (op =) s ss') else (false, ss')
   269             end
   270           | add T ss = (member (op =) Us T, ss);
   271       in snd oo add end;
   272 
   273     fun nesty_bnfs Us =
   274       map_filter (bnf_of lthy) (fold (fold (fold (add_nesty_bnf_names Us))) ctr_TsssXs []);
   275 
   276     val nesting_bnfs = nesty_bnfs As;
   277     val nested_bnfs = nesty_bnfs Xs;
   278 
   279     val pre_map_defs = map map_def_of_bnf pre_bnfs;
   280     val pre_set_defss = map set_defs_of_bnf pre_bnfs;
   281     val pre_rel_defs = map rel_def_of_bnf pre_bnfs;
   282     val nested_map_comp's = map map_comp'_of_bnf nested_bnfs;
   283     val nested_map_ids'' = map (unfold_thms lthy @{thms id_def} o map_id_of_bnf) nested_bnfs;
   284     val nesting_map_ids'' = map (unfold_thms lthy @{thms id_def} o map_id_of_bnf) nesting_bnfs;
   285     val nested_set_natural's = maps set_natural'_of_bnf nested_bnfs;
   286     val nesting_set_natural's = maps set_natural'_of_bnf nesting_bnfs;
   287 
   288     val live = live_of_bnf any_fp_bnf;
   289 
   290     val Bs =
   291       map3 (fn alive => fn A as TFree (_, S) => fn B => if alive then resort_tfree S B else A)
   292         (liveness_of_fp_bnf (length As) any_fp_bnf) As Bs0;
   293 
   294     val B_ify = Term.typ_subst_atomic (As ~~ Bs);
   295 
   296     val ctors = map (mk_ctor As) ctors0;
   297     val dtors = map (mk_dtor As) dtors0;
   298 
   299     val fpTs = map (domain_type o fastype_of) dtors;
   300 
   301     val exists_fp_subtype = exists_subtype (member (op =) fpTs);
   302     val exists_Cs_subtype = exists_subtype (member (op =) Cs);
   303 
   304     val ctr_Tsss = map (map (map (Term.typ_subst_atomic (Xs ~~ fpTs)))) ctr_TsssXs;
   305     val ns = map length ctr_Tsss;
   306     val kss = map (fn n => 1 upto n) ns;
   307     val mss = map (map length) ctr_Tsss;
   308     val Css = map2 replicate ns Cs;
   309 
   310     val fp_folds as any_fp_fold :: _ = map (mk_rec_like lfp As Cs) fp_folds0;
   311     val fp_recs as any_fp_rec :: _ = map (mk_rec_like lfp As Cs) fp_recs0;
   312 
   313     val fp_fold_fun_Ts = fst (split_last (binder_types (fastype_of any_fp_fold)));
   314     val fp_rec_fun_Ts = fst (split_last (binder_types (fastype_of any_fp_rec)));
   315 
   316     val (((fold_only as (gss, _, _), rec_only as (hss, _, _)),
   317           (zs, cs, cpss, unfold_only as ((pgss, crgsss), _), corec_only as ((phss, cshsss), _))),
   318          names_lthy0) =
   319       if lfp then
   320         let
   321           val y_Tsss =
   322             map3 (fn n => fn ms => map2 dest_tupleT ms o dest_sumTN_balanced n o domain_type)
   323               ns mss fp_fold_fun_Ts;
   324           val g_Tss = map2 (map2 (curry (op --->))) y_Tsss Css;
   325 
   326           val ((gss, ysss), lthy) =
   327             lthy
   328             |> mk_Freess "f" g_Tss
   329             ||>> mk_Freesss "x" y_Tsss;
   330 
   331           fun proj_recT proj (Type (s as @{type_name prod}, Ts as [T, U])) =
   332               if member (op =) fpTs T then proj (T, U) else Type (s, map (proj_recT proj) Ts)
   333             | proj_recT proj (Type (s, Ts)) = Type (s, map (proj_recT proj) Ts)
   334             | proj_recT _ T = T;
   335 
   336           fun unzip_recT T =
   337             if exists_fp_subtype T then [proj_recT fst T, proj_recT snd T] else [T];
   338 
   339           val z_Tsss =
   340             map3 (fn n => fn ms => map2 dest_tupleT ms o dest_sumTN_balanced n o domain_type)
   341               ns mss fp_rec_fun_Ts;
   342           val z_Tssss = map (map (map unzip_recT)) z_Tsss;
   343           val h_Tss = map2 (map2 (fold_rev (curry (op --->)))) z_Tssss Css;
   344 
   345           val hss = map2 (map2 retype_free) h_Tss gss;
   346           val zsss = map2 (map2 (map2 retype_free)) z_Tsss ysss;
   347         in
   348           ((((gss, g_Tss, ysss), (hss, h_Tss, zsss)),
   349             ([], [], [], (([], []), ([], [])), (([], []), ([], [])))), lthy)
   350         end
   351       else
   352         let
   353           (*avoid "'a itself" arguments in coiterators and corecursors*)
   354           val mss' =  map (fn [0] => [1] | ms => ms) mss;
   355 
   356           val p_Tss = map2 (fn n => replicate (Int.max (0, n - 1)) o mk_pred1T) ns Cs;
   357 
   358           fun flat_predss_getterss qss fss = maps (op @) (qss ~~ fss);
   359 
   360           fun flat_preds_predsss_gettersss [] [qss] [fss] = flat_predss_getterss qss fss
   361             | flat_preds_predsss_gettersss (p :: ps) (qss :: qsss) (fss :: fsss) =
   362               p :: flat_predss_getterss qss fss @ flat_preds_predsss_gettersss ps qsss fsss;
   363 
   364           fun mk_types maybe_dest_sumT fun_Ts =
   365             let
   366               val f_sum_prod_Ts = map range_type fun_Ts;
   367               val f_prod_Tss = map2 dest_sumTN_balanced ns f_sum_prod_Ts;
   368               val f_Tssss =
   369                 map3 (fn C => map2 (map (map (curry (op -->) C) o maybe_dest_sumT) oo dest_tupleT))
   370                   Cs mss' f_prod_Tss;
   371               val q_Tssss =
   372                 map (map (map (fn [_] => [] | [_, C] => [mk_pred1T (domain_type C)]))) f_Tssss;
   373               val pf_Tss = map3 flat_preds_predsss_gettersss p_Tss q_Tssss f_Tssss;
   374             in (q_Tssss, f_sum_prod_Ts, f_Tssss, pf_Tss) end;
   375 
   376           val (r_Tssss, g_sum_prod_Ts, g_Tssss, pg_Tss) = mk_types single fp_fold_fun_Ts;
   377 
   378           val ((((Free (z, _), cs), pss), gssss), lthy) =
   379             lthy
   380             |> yield_singleton (mk_Frees "z") dummyT
   381             ||>> mk_Frees "a" Cs
   382             ||>> mk_Freess "p" p_Tss
   383             ||>> mk_Freessss "g" g_Tssss;
   384           val rssss = map (map (map (fn [] => []))) r_Tssss;
   385 
   386           fun dest_corec_sumT (T as Type (@{type_name sum}, Us as [_, U])) =
   387               if member (op =) Cs U then Us else [T]
   388             | dest_corec_sumT T = [T];
   389 
   390           val (s_Tssss, h_sum_prod_Ts, h_Tssss, ph_Tss) = mk_types dest_corec_sumT fp_rec_fun_Ts;
   391 
   392           val hssss_hd = map2 (map2 (map2 (fn T :: _ => fn [g] => retype_free T g))) h_Tssss gssss;
   393           val ((sssss, hssss_tl), lthy) =
   394             lthy
   395             |> mk_Freessss "q" s_Tssss
   396             ||>> mk_Freessss "h" (map (map (map tl)) h_Tssss);
   397           val hssss = map2 (map2 (map2 cons)) hssss_hd hssss_tl;
   398 
   399           val cpss = map2 (fn c => map (fn p => p $ c)) cs pss;
   400 
   401           fun mk_preds_getters_join [] [cf] = cf
   402             | mk_preds_getters_join [cq] [cf, cf'] =
   403               mk_If cq (mk_Inl (fastype_of cf') cf) (mk_Inr (fastype_of cf) cf');
   404 
   405           fun mk_terms qssss fssss =
   406             let
   407               val pfss = map3 flat_preds_predsss_gettersss pss qssss fssss;
   408               val cqssss = map2 (fn c => map (map (map (fn f => f $ c)))) cs qssss;
   409               val cfssss = map2 (fn c => map (map (map (fn f => f $ c)))) cs fssss;
   410               val cqfsss = map2 (map2 (map2 mk_preds_getters_join)) cqssss cfssss;
   411             in (pfss, cqfsss) end;
   412         in
   413           (((([], [], []), ([], [], [])),
   414             ([z], cs, cpss, (mk_terms rssss gssss, (g_sum_prod_Ts, pg_Tss)),
   415              (mk_terms sssss hssss, (h_sum_prod_Ts, ph_Tss)))), lthy)
   416         end;
   417 
   418     fun define_ctrs_case_for_type (((((((((((((((((((((((((fp_bnf, fp_b), fpT), C), ctor), dtor),
   419             fp_fold), fp_rec), ctor_dtor), dtor_ctor), ctor_inject), pre_map_def), pre_set_defs),
   420           pre_rel_def), fp_map_thm), fp_set_thms), fp_rel_thm), n), ks), ms), ctr_bindings),
   421         ctr_mixfixes), ctr_Tss), disc_bindings), sel_bindingss), raw_sel_defaultss) no_defs_lthy =
   422       let
   423         val fp_b_name = Binding.name_of fp_b;
   424 
   425         val dtorT = domain_type (fastype_of ctor);
   426         val ctr_prod_Ts = map HOLogic.mk_tupleT ctr_Tss;
   427         val ctr_sum_prod_T = mk_sumTN_balanced ctr_prod_Ts;
   428         val case_Ts = map (fn Ts => Ts ---> C) ctr_Tss;
   429 
   430         val (((((w, fs), xss), yss), u'), names_lthy) =
   431           no_defs_lthy
   432           |> yield_singleton (mk_Frees "w") dtorT
   433           ||>> mk_Frees "f" case_Ts
   434           ||>> mk_Freess "x" ctr_Tss
   435           ||>> mk_Freess "y" (map (map B_ify) ctr_Tss)
   436           ||>> yield_singleton Variable.variant_fixes fp_b_name;
   437 
   438         val u = Free (u', fpT);
   439 
   440         val tuple_xs = map HOLogic.mk_tuple xss;
   441         val tuple_ys = map HOLogic.mk_tuple yss;
   442 
   443         val ctr_rhss =
   444           map3 (fn k => fn xs => fn tuple_x => fold_rev Term.lambda xs (ctor $
   445             mk_InN_balanced ctr_sum_prod_T n tuple_x k)) ks xss tuple_xs;
   446 
   447         val case_binding = qualify false fp_b_name (Binding.suffix_name ("_" ^ caseN) fp_b);
   448 
   449         val case_rhs =
   450           fold_rev Term.lambda (fs @ [u])
   451             (mk_sum_caseN_balanced (map2 mk_uncurried_fun fs xss) $ (dtor $ u));
   452 
   453         val ((raw_case :: raw_ctrs, raw_case_def :: raw_ctr_defs), (lthy', lthy)) = no_defs_lthy
   454           |> apfst split_list o fold_map3 (fn b => fn mx => fn rhs =>
   455               Local_Theory.define ((b, mx), ((Thm.def_binding b, []), rhs)) #>> apsnd snd)
   456             (case_binding :: ctr_bindings) (NoSyn :: ctr_mixfixes) (case_rhs :: ctr_rhss)
   457           ||> `Local_Theory.restore;
   458 
   459         val phi = Proof_Context.export_morphism lthy lthy';
   460 
   461         val ctr_defs = map (Morphism.thm phi) raw_ctr_defs;
   462         val ctr_defs' =
   463           map2 (fn m => fn def => mk_unabs_def m (def RS meta_eq_to_obj_eq)) ms ctr_defs;
   464         val case_def = Morphism.thm phi raw_case_def;
   465 
   466         val ctrs0 = map (Morphism.term phi) raw_ctrs;
   467         val casex0 = Morphism.term phi raw_case;
   468 
   469         val ctrs = map (mk_ctr As) ctrs0;
   470 
   471         fun wrap lthy =
   472           let
   473             fun exhaust_tac {context = ctxt, ...} =
   474               let
   475                 val ctor_iff_dtor_thm =
   476                   let
   477                     val goal =
   478                       fold_rev Logic.all [w, u]
   479                         (mk_Trueprop_eq (HOLogic.mk_eq (u, ctor $ w), HOLogic.mk_eq (dtor $ u, w)));
   480                   in
   481                     Skip_Proof.prove lthy [] [] goal (fn {context = ctxt, ...} =>
   482                       mk_ctor_iff_dtor_tac ctxt (map (SOME o certifyT lthy) [dtorT, fpT])
   483                         (certify lthy ctor) (certify lthy dtor) ctor_dtor dtor_ctor)
   484                     |> Thm.close_derivation
   485                     |> Morphism.thm phi
   486                   end;
   487 
   488                 val sumEN_thm' =
   489                   unfold_thms lthy @{thms all_unit_eq}
   490                     (Drule.instantiate' (map (SOME o certifyT lthy) ctr_prod_Ts) []
   491                        (mk_sumEN_balanced n))
   492                   |> Morphism.thm phi;
   493               in
   494                 mk_exhaust_tac ctxt n ctr_defs ctor_iff_dtor_thm sumEN_thm'
   495               end;
   496 
   497             val inject_tacss =
   498               map2 (fn 0 => K [] | _ => fn ctr_def => [fn {context = ctxt, ...} =>
   499                   mk_inject_tac ctxt ctr_def ctor_inject]) ms ctr_defs;
   500 
   501             val half_distinct_tacss =
   502               map (map (fn (def, def') => fn {context = ctxt, ...} =>
   503                 mk_half_distinct_tac ctxt ctor_inject [def, def'])) (mk_half_pairss (`I ctr_defs));
   504 
   505             val case_tacs =
   506               map3 (fn k => fn m => fn ctr_def => fn {context = ctxt, ...} =>
   507                 mk_case_tac ctxt n k m case_def ctr_def dtor_ctor) ks ms ctr_defs;
   508 
   509             val tacss = [exhaust_tac] :: inject_tacss @ half_distinct_tacss @ [case_tacs];
   510 
   511             val sel_defaultss = map (map (apsnd (prepare_term lthy))) raw_sel_defaultss
   512           in
   513             wrap_datatype tacss (((wrap_opts, ctrs0), casex0), (disc_bindings, (sel_bindingss,
   514               sel_defaultss))) lthy
   515           end;
   516 
   517         fun derive_maps_sets_rels (wrap_res, lthy) =
   518           let
   519             val rel_flip = rel_flip_of_bnf fp_bnf;
   520             val nones = replicate live NONE;
   521 
   522             val ctor_cong =
   523               if lfp then Drule.dummy_thm
   524               else cterm_instantiate_pos [NONE, NONE, SOME (certify lthy ctor)] arg_cong;
   525 
   526             fun mk_cIn ify =
   527               certify lthy o (not lfp ? curry (op $) (map_types ify ctor)) oo
   528               mk_InN_balanced (ify ctr_sum_prod_T) n;
   529 
   530             val cxIns = map2 (mk_cIn I) tuple_xs ks;
   531             val cyIns = map2 (mk_cIn B_ify) tuple_ys ks;
   532 
   533             fun mk_map_thm ctr_def' xs cxIn =
   534               fold_thms lthy [ctr_def']
   535                 (unfold_thms lthy (pre_map_def ::
   536                      (if lfp then [] else [ctor_dtor, dtor_ctor]) @ sum_prod_thms_map)
   537                    (cterm_instantiate_pos (nones @ [SOME cxIn])
   538                       (if lfp then fp_map_thm else fp_map_thm RS ctor_cong)))
   539               |> singleton (Proof_Context.export names_lthy no_defs_lthy);
   540 
   541             fun mk_set_thm fp_set_thm ctr_def' xs cxIn =
   542               fold_thms lthy [ctr_def']
   543                 (unfold_thms lthy (pre_set_defs @ nested_set_natural's @ nesting_set_natural's @
   544                      (if lfp then [] else [dtor_ctor]) @ sum_prod_thms_set)
   545                    (cterm_instantiate_pos [SOME cxIn] fp_set_thm))
   546               |> singleton (Proof_Context.export names_lthy no_defs_lthy);
   547 
   548             fun mk_set_thms fp_set_thm = map3 (mk_set_thm fp_set_thm) ctr_defs' xss cxIns;
   549 
   550             val map_thms = map3 mk_map_thm ctr_defs' xss cxIns;
   551             val set_thmss = map mk_set_thms fp_set_thms;
   552 
   553             val rel_infos = (ctr_defs' ~~ xss ~~ cxIns, ctr_defs' ~~ yss ~~ cyIns);
   554 
   555             fun mk_rel_thm postproc ctr_defs' xs cxIn ys cyIn =
   556               fold_thms lthy ctr_defs'
   557                  (unfold_thms lthy (pre_rel_def :: (if lfp then [] else [dtor_ctor]) @
   558                       sum_prod_thms_rel)
   559                     (cterm_instantiate_pos (nones @ [SOME cxIn, SOME cyIn]) fp_rel_thm))
   560               |> postproc
   561               |> singleton (Proof_Context.export names_lthy no_defs_lthy);
   562 
   563             fun mk_rel_inject_thm (((ctr_def', xs), cxIn), ((_, ys), cyIn)) =
   564               mk_rel_thm (unfold_thms lthy @{thms eq_sym_Unity_conv}) [ctr_def'] xs cxIn ys cyIn;
   565 
   566             val rel_inject_thms = map mk_rel_inject_thm (op ~~ rel_infos);
   567 
   568             fun mk_half_rel_distinct_thm (((xctr_def', xs), cxIn), ((yctr_def', ys), cyIn)) =
   569               mk_rel_thm (fn thm => thm RS @{thm eq_False[THEN iffD1]}) [xctr_def', yctr_def']
   570                 xs cxIn ys cyIn;
   571 
   572             fun mk_other_half_rel_distinct_thm thm =
   573               flip_rels lthy live thm RS (rel_flip RS sym RS @{thm arg_cong[of _ _ Not]} RS iffD2);
   574 
   575             val half_rel_distinct_thmss =
   576               map (map mk_half_rel_distinct_thm) (mk_half_pairss rel_infos);
   577             val other_half_rel_distinct_thmss =
   578               map (map mk_other_half_rel_distinct_thm) half_rel_distinct_thmss;
   579             val (rel_distinct_thms, _) =
   580               join_halves n half_rel_distinct_thmss other_half_rel_distinct_thmss;
   581 
   582             val notes =
   583               [(mapN, map_thms, code_simp_attrs),
   584                (rel_distinctN, rel_distinct_thms, code_simp_attrs),
   585                (rel_injectN, rel_inject_thms, code_simp_attrs),
   586                (setsN, flat set_thmss, code_simp_attrs)]
   587               |> filter_out (null o #2)
   588               |> map (fn (thmN, thms, attrs) =>
   589                 ((qualify true fp_b_name (Binding.name thmN), attrs), [(thms, [])]));
   590           in
   591             (wrap_res, lthy |> Local_Theory.notes notes |> snd)
   592           end;
   593 
   594         fun define_fold_rec no_defs_lthy =
   595           let
   596             val fpT_to_C = fpT --> C;
   597 
   598             fun build_ctor_rec_arg mk_proj (T, U) =
   599               if T = U then
   600                 id_const T
   601               else
   602                 (case (T, U) of
   603                   (Type (s, _), Type (s', _)) =>
   604                   if s = s' then build_map (build_ctor_rec_arg mk_proj) T U else mk_proj T
   605                 | _ => mk_proj T);
   606 
   607             fun mk_U proj (Type (s as @{type_name prod}, Ts as [T', U])) =
   608                 if member (op =) fpTs T' then proj (T', U) else Type (s, map (mk_U proj) Ts)
   609               | mk_U proj (Type (s, Ts)) = Type (s, map (mk_U proj) Ts)
   610               | mk_U _ T = T;
   611 
   612             fun unzip_rec (x as Free (_, T)) =
   613               if exists_fp_subtype T then
   614                 [build_ctor_rec_arg fst_const (T, mk_U fst T) $ x,
   615                  build_ctor_rec_arg snd_const (T, mk_U snd T) $ x]
   616               else
   617                 [x];
   618 
   619             fun mk_rec_like_arg f xs = mk_tupled_fun (HOLogic.mk_tuple xs) f (maps unzip_rec xs);
   620 
   621             fun generate_rec_like (suf, fp_rec_like, (fss, f_Tss, xsss)) =
   622               let
   623                 val res_T = fold_rev (curry (op --->)) f_Tss fpT_to_C;
   624                 val binding = qualify false fp_b_name (Binding.suffix_name ("_" ^ suf) fp_b);
   625                 val spec =
   626                   mk_Trueprop_eq (lists_bmoc fss (Free (Binding.name_of binding, res_T)),
   627                     Term.list_comb (fp_rec_like,
   628                       map2 (mk_sum_caseN_balanced oo map2 mk_rec_like_arg) fss xsss));
   629               in (binding, spec) end;
   630 
   631             val rec_like_infos =
   632               [(foldN, fp_fold, fold_only),
   633                (recN, fp_rec, rec_only)];
   634 
   635             val (bindings, specs) = map generate_rec_like rec_like_infos |> split_list;
   636 
   637             val ((csts, defs), (lthy', lthy)) = no_defs_lthy
   638               |> apfst split_list o fold_map2 (fn b => fn spec =>
   639                 Specification.definition (SOME (b, NONE, NoSyn), ((Thm.def_binding b, []), spec))
   640                 #>> apsnd snd) bindings specs
   641               ||> `Local_Theory.restore;
   642 
   643             val phi = Proof_Context.export_morphism lthy lthy';
   644 
   645             val [fold_def, rec_def] = map (Morphism.thm phi) defs;
   646 
   647             val [foldx, recx] = map (mk_rec_like lfp As Cs o Morphism.term phi) csts;
   648           in
   649             ((foldx, recx, fold_def, rec_def), lthy)
   650           end;
   651 
   652         fun define_unfold_corec no_defs_lthy =
   653           let
   654             val B_to_fpT = C --> fpT;
   655 
   656             fun mk_preds_getterss_join c n cps sum_prod_T cqfss =
   657               Term.lambda c (mk_IfN sum_prod_T cps
   658                 (map2 (mk_InN_balanced sum_prod_T n) (map HOLogic.mk_tuple cqfss) (1 upto n)));
   659 
   660             fun generate_corec_like (suf, fp_rec_like, ((pfss, cqfsss), (f_sum_prod_Ts,
   661                 pf_Tss))) =
   662               let
   663                 val res_T = fold_rev (curry (op --->)) pf_Tss B_to_fpT;
   664                 val binding = qualify false fp_b_name (Binding.suffix_name ("_" ^ suf) fp_b);
   665                 val spec =
   666                   mk_Trueprop_eq (lists_bmoc pfss (Free (Binding.name_of binding, res_T)),
   667                     Term.list_comb (fp_rec_like,
   668                       map5 mk_preds_getterss_join cs ns cpss f_sum_prod_Ts cqfsss));
   669               in (binding, spec) end;
   670 
   671             val corec_like_infos =
   672               [(unfoldN, fp_fold, unfold_only),
   673                (corecN, fp_rec, corec_only)];
   674 
   675             val (bindings, specs) = map generate_corec_like corec_like_infos |> split_list;
   676 
   677             val ((csts, defs), (lthy', lthy)) = no_defs_lthy
   678               |> apfst split_list o fold_map2 (fn b => fn spec =>
   679                 Specification.definition (SOME (b, NONE, NoSyn), ((Thm.def_binding b, []), spec))
   680                 #>> apsnd snd) bindings specs
   681               ||> `Local_Theory.restore;
   682 
   683             val phi = Proof_Context.export_morphism lthy lthy';
   684 
   685             val [unfold_def, corec_def] = map (Morphism.thm phi) defs;
   686 
   687             val [unfold, corec] = map (mk_rec_like lfp As Cs o Morphism.term phi) csts;
   688           in
   689             ((unfold, corec, unfold_def, corec_def), lthy)
   690           end;
   691 
   692         val define_rec_likes = if lfp then define_fold_rec else define_unfold_corec;
   693 
   694         fun massage_res ((wrap_res, rec_like_res), lthy) =
   695           (((ctrs, xss, ctr_defs, wrap_res), rec_like_res), lthy);
   696       in
   697         (wrap #> (live > 0 ? derive_maps_sets_rels) ##>> define_rec_likes #> massage_res, lthy')
   698       end;
   699 
   700     fun wrap_types_and_more (wrap_types_and_mores, lthy) =
   701       fold_map I wrap_types_and_mores lthy
   702       |>> apsnd split_list4 o apfst split_list4 o split_list;
   703 
   704     (* TODO: Add map, sets, rel simps *)
   705     val mk_simp_thmss =
   706       map3 (fn (_, _, _, injects, distincts, cases, _, _, _) => fn rec_likes => fn fold_likes =>
   707         injects @ distincts @ cases @ rec_likes @ fold_likes);
   708 
   709     fun derive_induct_fold_rec_thms_for_types (((ctrss, xsss, ctr_defss, wrap_ress), (folds, recs,
   710         fold_defs, rec_defs)), lthy) =
   711       let
   712         val (((ps, ps'), us'), names_lthy) =
   713           lthy
   714           |> mk_Frees' "P" (map mk_pred1T fpTs)
   715           ||>> Variable.variant_fixes fp_b_names;
   716 
   717         val us = map2 (curry Free) us' fpTs;
   718 
   719         fun mk_sets_nested bnf =
   720           let
   721             val Type (T_name, Us) = T_of_bnf bnf;
   722             val lives = lives_of_bnf bnf;
   723             val sets = sets_of_bnf bnf;
   724             fun mk_set U =
   725               (case find_index (curry (op =) U) lives of
   726                 ~1 => Term.dummy
   727               | i => nth sets i);
   728           in
   729             (T_name, map mk_set Us)
   730           end;
   731 
   732         val setss_nested = map mk_sets_nested nested_bnfs;
   733 
   734         val (induct_thms, induct_thm) =
   735           let
   736             fun mk_set Ts t =
   737               let val Type (_, Ts0) = domain_type (fastype_of t) in
   738                 Term.subst_atomic_types (Ts0 ~~ Ts) t
   739               end;
   740 
   741             fun mk_raw_prem_prems names_lthy (x as Free (s, T as Type (T_name, Ts0))) =
   742                 (case find_index (curry (op =) T) fpTs of
   743                   ~1 =>
   744                   (case AList.lookup (op =) setss_nested T_name of
   745                     NONE => []
   746                   | SOME raw_sets0 =>
   747                     let
   748                       val (Ts, raw_sets) =
   749                         split_list (filter (exists_fp_subtype o fst) (Ts0 ~~ raw_sets0));
   750                       val sets = map (mk_set Ts0) raw_sets;
   751                       val (ys, names_lthy') = names_lthy |> mk_Frees s Ts;
   752                       val xysets = map (pair x) (ys ~~ sets);
   753                       val ppremss = map (mk_raw_prem_prems names_lthy') ys;
   754                     in
   755                       flat (map2 (map o apfst o cons) xysets ppremss)
   756                     end)
   757                 | kk => [([], (kk + 1, x))])
   758               | mk_raw_prem_prems _ _ = [];
   759 
   760             fun close_prem_prem xs t =
   761               fold_rev Logic.all (map Free (drop (nn + length xs)
   762                 (rev (Term.add_frees t (map dest_Free xs @ ps'))))) t;
   763 
   764             fun mk_prem_prem xs (xysets, (j, x)) =
   765               close_prem_prem xs (Logic.list_implies (map (fn (x', (y, set)) =>
   766                   HOLogic.mk_Trueprop (HOLogic.mk_mem (y, set $ x'))) xysets,
   767                 HOLogic.mk_Trueprop (nth ps (j - 1) $ x)));
   768 
   769             fun mk_raw_prem phi ctr ctr_Ts =
   770               let
   771                 val (xs, names_lthy') = names_lthy |> mk_Frees "x" ctr_Ts;
   772                 val pprems = maps (mk_raw_prem_prems names_lthy') xs;
   773               in (xs, pprems, HOLogic.mk_Trueprop (phi $ Term.list_comb (ctr, xs))) end;
   774 
   775             fun mk_prem (xs, raw_pprems, concl) =
   776               fold_rev Logic.all xs (Logic.list_implies (map (mk_prem_prem xs) raw_pprems, concl));
   777 
   778             val raw_premss = map3 (map2 o mk_raw_prem) ps ctrss ctr_Tsss;
   779 
   780             val goal =
   781               Library.foldr (Logic.list_implies o apfst (map mk_prem)) (raw_premss,
   782                 HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj (map2 (curry (op $)) ps us)));
   783 
   784             val kksss = map (map (map (fst o snd) o #2)) raw_premss;
   785 
   786             val ctor_induct' = fp_induct OF (map mk_sumEN_tupled_balanced mss);
   787 
   788             val thm =
   789               Skip_Proof.prove lthy [] [] goal (fn {context = ctxt, ...} =>
   790                 mk_induct_tac ctxt nn ns mss kksss (flat ctr_defss) ctor_induct'
   791                   nested_set_natural's pre_set_defss)
   792               |> singleton (Proof_Context.export names_lthy lthy)
   793               |> Thm.close_derivation;
   794           in
   795             `(conj_dests nn) thm
   796           end;
   797 
   798         val induct_cases = quasi_unambiguous_case_names (maps (map name_of_ctr) ctrss);
   799 
   800         val (fold_thmss, rec_thmss) =
   801           let
   802             val xctrss = map2 (map2 (curry Term.list_comb)) ctrss xsss;
   803             val gfolds = map (lists_bmoc gss) folds;
   804             val hrecs = map (lists_bmoc hss) recs;
   805 
   806             fun mk_goal fss frec_like xctr f xs fxs =
   807               fold_rev (fold_rev Logic.all) (xs :: fss)
   808                 (mk_Trueprop_eq (frec_like $ xctr, Term.list_comb (f, fxs)));
   809 
   810             fun build_rec_like frec_likes maybe_tick (T, U) =
   811               if T = U then
   812                 id_const T
   813               else
   814                 (case find_index (curry (op =) T) fpTs of
   815                   ~1 => build_map (build_rec_like frec_likes maybe_tick) T U
   816                 | kk => maybe_tick (nth us kk) (nth frec_likes kk));
   817 
   818             fun mk_U maybe_mk_prodT =
   819               typ_subst (map2 (fn fpT => fn C => (fpT, maybe_mk_prodT fpT C)) fpTs Cs);
   820 
   821             fun intr_rec_likes frec_likes maybe_cons maybe_tick maybe_mk_prodT (x as Free (_, T)) =
   822               if exists_fp_subtype T then
   823                 maybe_cons x [build_rec_like frec_likes (K I) (T, mk_U (K I) T) $ x]
   824               else
   825                 [x];
   826 
   827             val gxsss = map (map (maps (intr_rec_likes gfolds (K I) (K I) (K I)))) xsss;
   828             val hxsss =
   829               map (map (maps (intr_rec_likes hrecs cons tick (curry HOLogic.mk_prodT)))) xsss;
   830 
   831             val fold_goalss = map5 (map4 o mk_goal gss) gfolds xctrss gss xsss gxsss;
   832             val rec_goalss = map5 (map4 o mk_goal hss) hrecs xctrss hss xsss hxsss;
   833 
   834             val fold_tacss =
   835               map2 (map o mk_rec_like_tac pre_map_defs [] nesting_map_ids'' fold_defs) fp_fold_thms
   836                 ctr_defss;
   837             val rec_tacss =
   838               map2 (map o mk_rec_like_tac pre_map_defs nested_map_comp's
   839                 (nested_map_ids'' @ nesting_map_ids'') rec_defs) fp_rec_thms ctr_defss;
   840 
   841             fun prove goal tac =
   842               Skip_Proof.prove lthy [] [] goal (tac o #context)
   843               |> Thm.close_derivation;
   844           in
   845             (map2 (map2 prove) fold_goalss fold_tacss, map2 (map2 prove) rec_goalss rec_tacss)
   846           end;
   847 
   848         val simp_thmss = mk_simp_thmss wrap_ress rec_thmss fold_thmss;
   849 
   850         val induct_case_names_attr = Attrib.internal (K (Rule_Cases.case_names induct_cases));
   851         fun induct_type_attr T_name = Attrib.internal (K (Induct.induct_type T_name));
   852 
   853         val common_notes =
   854           (if nn > 1 then [(inductN, [induct_thm], [induct_case_names_attr])] else [])
   855           |> map (fn (thmN, thms, attrs) =>
   856             ((qualify true fp_common_name (Binding.name thmN), attrs), [(thms, [])]));
   857 
   858         val notes =
   859           [(foldN, fold_thmss, K code_simp_attrs),
   860            (inductN, map single induct_thms,
   861             fn T_name => [induct_case_names_attr, induct_type_attr T_name]),
   862            (recN, rec_thmss, K code_simp_attrs),
   863            (simpsN, simp_thmss, K [])]
   864           |> maps (fn (thmN, thmss, attrs) =>
   865             map3 (fn fp_b_name => fn Type (T_name, _) => fn thms =>
   866               ((qualify true fp_b_name (Binding.name thmN), attrs T_name),
   867                [(thms, [])])) fp_b_names fpTs thmss);
   868       in
   869         lthy |> Local_Theory.notes (common_notes @ notes) |> snd
   870       end;
   871 
   872     fun derive_coinduct_unfold_corec_thms_for_types (((ctrss, _, ctr_defss, wrap_ress), (unfolds,
   873         corecs, unfold_defs, corec_defs)), lthy) =
   874       let
   875         val nesting_rel_eqs = map rel_eq_of_bnf nesting_bnfs;
   876 
   877         val discss = map (map (mk_disc_or_sel As) o #1) wrap_ress;
   878         val selsss = map (map (map (mk_disc_or_sel As)) o #2) wrap_ress;
   879         val exhaust_thms = map #3 wrap_ress;
   880         val disc_thmsss = map #7 wrap_ress;
   881         val discIss = map #8 wrap_ress;
   882         val sel_thmsss = map #9 wrap_ress;
   883 
   884         val (((rs, us'), vs'), names_lthy) =
   885           lthy
   886           |> mk_Frees "R" (map (fn T => mk_pred2T T T) fpTs)
   887           ||>> Variable.variant_fixes fp_b_names
   888           ||>> Variable.variant_fixes (map (suffix "'") fp_b_names);
   889 
   890         val us = map2 (curry Free) us' fpTs;
   891         val udiscss = map2 (map o rapp) us discss;
   892         val uselsss = map2 (map o map o rapp) us selsss;
   893 
   894         val vs = map2 (curry Free) vs' fpTs;
   895         val vdiscss = map2 (map o rapp) vs discss;
   896         val vselsss = map2 (map o map o rapp) vs selsss;
   897 
   898         val ((coinduct_thms, coinduct_thm), (strong_coinduct_thms, strong_coinduct_thm)) =
   899           let
   900             val uvrs = map3 (fn r => fn u => fn v => r $ u $ v) rs us vs;
   901             val uv_eqs = map2 (curry HOLogic.mk_eq) us vs;
   902             val strong_rs =
   903               map4 (fn u => fn v => fn uvr => fn uv_eq =>
   904                 fold_rev Term.lambda [u, v] (HOLogic.mk_disj (uvr, uv_eq))) us vs uvrs uv_eqs;
   905 
   906             fun build_rel rs' T =
   907               (case find_index (curry (op =) T) fpTs of
   908                 ~1 =>
   909                 if exists_fp_subtype T then build_rel_step (build_rel rs') T
   910                 else HOLogic.eq_const T
   911               | kk => nth rs' kk);
   912 
   913             fun build_rel_app rs' usel vsel =
   914               fold rapp [usel, vsel] (build_rel rs' (fastype_of usel));
   915 
   916             fun mk_prem_ctr_concls rs' n k udisc usels vdisc vsels =
   917               (if k = n then [] else [HOLogic.mk_eq (udisc, vdisc)]) @
   918               (if null usels then
   919                  []
   920                else
   921                  [Library.foldr HOLogic.mk_imp (if n = 1 then [] else [udisc, vdisc],
   922                     Library.foldr1 HOLogic.mk_conj (map2 (build_rel_app rs') usels vsels))]);
   923 
   924             fun mk_prem_concl rs' n udiscs uselss vdiscs vselss =
   925               Library.foldr1 HOLogic.mk_conj
   926                 (flat (map5 (mk_prem_ctr_concls rs' n) (1 upto n) udiscs uselss vdiscs vselss))
   927               handle List.Empty => @{term True};
   928 
   929             fun mk_prem rs' uvr u v n udiscs uselss vdiscs vselss =
   930               fold_rev Logic.all [u, v] (Logic.mk_implies (HOLogic.mk_Trueprop uvr,
   931                 HOLogic.mk_Trueprop (mk_prem_concl rs' n udiscs uselss vdiscs vselss)));
   932 
   933             val concl =
   934               HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj
   935                 (map3 (fn uvr => fn u => fn v => HOLogic.mk_imp (uvr, HOLogic.mk_eq (u, v)))
   936                    uvrs us vs));
   937 
   938             fun mk_goal rs' =
   939               Logic.list_implies (map8 (mk_prem rs') uvrs us vs ns udiscss uselsss vdiscss vselsss,
   940                 concl);
   941 
   942             val goal = mk_goal rs;
   943             val strong_goal = mk_goal strong_rs;
   944 
   945             fun prove dtor_coinduct' goal =
   946               Skip_Proof.prove lthy [] [] goal (fn {context = ctxt, ...} =>
   947                 mk_coinduct_tac ctxt nesting_rel_eqs nn ns dtor_coinduct' pre_rel_defs dtor_ctors
   948                   exhaust_thms ctr_defss disc_thmsss sel_thmsss)
   949               |> singleton (Proof_Context.export names_lthy lthy)
   950               |> Thm.close_derivation;
   951 
   952             fun postproc nn thm =
   953               Thm.permute_prems 0 nn
   954                 (if nn = 1 then thm RS mp
   955                  else funpow nn (fn thm => reassoc_conjs (thm RS mp_conj)) thm)
   956               |> Drule.zero_var_indexes
   957               |> `(conj_dests nn);
   958           in
   959             (postproc nn (prove fp_induct goal), postproc nn (prove fp_strong_induct strong_goal))
   960           end;
   961 
   962         fun mk_maybe_not pos = not pos ? HOLogic.mk_not;
   963 
   964         val z = the_single zs;
   965         val gunfolds = map (lists_bmoc pgss) unfolds;
   966         val hcorecs = map (lists_bmoc phss) corecs;
   967 
   968         val (unfold_thmss, corec_thmss, safe_unfold_thmss, safe_corec_thmss) =
   969           let
   970             fun mk_goal pfss c cps fcorec_like n k ctr m cfs' =
   971               fold_rev (fold_rev Logic.all) ([c] :: pfss)
   972                 (Logic.list_implies (seq_conds (HOLogic.mk_Trueprop oo mk_maybe_not) n k cps,
   973                    mk_Trueprop_eq (fcorec_like $ c, Term.list_comb (ctr, take m cfs'))));
   974 
   975             fun build_corec_like fcorec_likes maybe_tack (T, U) =
   976               if T = U then
   977                 id_const T
   978               else
   979                 (case find_index (curry (op =) U) fpTs of
   980                   ~1 => build_map (build_corec_like fcorec_likes maybe_tack) T U
   981                 | kk => maybe_tack (nth cs kk, nth us kk) (nth fcorec_likes kk));
   982 
   983             fun mk_U maybe_mk_sumT =
   984               typ_subst (map2 (fn C => fn fpT => (maybe_mk_sumT fpT C, fpT)) Cs fpTs);
   985 
   986             fun intr_corec_likes fcorec_likes maybe_mk_sumT maybe_tack cqf =
   987               let val T = fastype_of cqf in
   988                 if exists_Cs_subtype T then
   989                   build_corec_like fcorec_likes maybe_tack (T, mk_U maybe_mk_sumT T) $ cqf
   990                 else
   991                   cqf
   992               end;
   993 
   994             val crgsss' = map (map (map (intr_corec_likes gunfolds (K I) (K I)))) crgsss;
   995             val cshsss' =
   996               map (map (map (intr_corec_likes hcorecs (curry mk_sumT) (tack z)))) cshsss;
   997 
   998             val unfold_goalss =
   999               map8 (map4 oooo mk_goal pgss) cs cpss gunfolds ns kss ctrss mss crgsss';
  1000             val corec_goalss =
  1001               map8 (map4 oooo mk_goal phss) cs cpss hcorecs ns kss ctrss mss cshsss';
  1002 
  1003             val unfold_tacss =
  1004               map3 (map oo mk_corec_like_tac unfold_defs nesting_map_ids'') fp_fold_thms
  1005                 pre_map_defs ctr_defss;
  1006             val corec_tacss =
  1007               map3 (map oo mk_corec_like_tac corec_defs nesting_map_ids'') fp_rec_thms pre_map_defs
  1008                 ctr_defss;
  1009 
  1010             fun prove goal tac =
  1011               Skip_Proof.prove lthy [] [] goal (tac o #context) |> Thm.close_derivation;
  1012 
  1013             val unfold_thmss = map2 (map2 prove) unfold_goalss unfold_tacss;
  1014             val corec_thmss =
  1015               map2 (map2 prove) corec_goalss corec_tacss
  1016               |> map (map (unfold_thms lthy @{thms sum_case_if}));
  1017 
  1018             val unfold_safesss = map2 (map2 (map2 (curry (op =)))) crgsss' crgsss;
  1019             val corec_safesss = map2 (map2 (map2 (curry (op =)))) cshsss' cshsss;
  1020 
  1021             val filter_safesss =
  1022               map2 (map_filter (fn (safes, thm) => if forall I safes then SOME thm else NONE) oo
  1023                 curry (op ~~));
  1024 
  1025             val safe_unfold_thmss = filter_safesss unfold_safesss unfold_thmss;
  1026             val safe_corec_thmss = filter_safesss corec_safesss corec_thmss;
  1027           in
  1028             (unfold_thmss, corec_thmss, safe_unfold_thmss, safe_corec_thmss)
  1029           end;
  1030 
  1031         val (disc_unfold_iff_thmss, disc_corec_iff_thmss) =
  1032           let
  1033             fun mk_goal c cps fcorec_like n k disc =
  1034               mk_Trueprop_eq (disc $ (fcorec_like $ c),
  1035                 if n = 1 then @{const True}
  1036                 else Library.foldr1 HOLogic.mk_conj (seq_conds mk_maybe_not n k cps));
  1037 
  1038             val unfold_goalss = map6 (map2 oooo mk_goal) cs cpss gunfolds ns kss discss;
  1039             val corec_goalss = map6 (map2 oooo mk_goal) cs cpss hcorecs ns kss discss;
  1040 
  1041             fun mk_case_split' cp =
  1042               Drule.instantiate' [] [SOME (certify lthy cp)] @{thm case_split};
  1043 
  1044             val case_splitss' = map (map mk_case_split') cpss;
  1045 
  1046             val unfold_tacss =
  1047               map3 (map oo mk_disc_corec_like_iff_tac) case_splitss' unfold_thmss disc_thmsss;
  1048             val corec_tacss =
  1049               map3 (map oo mk_disc_corec_like_iff_tac) case_splitss' corec_thmss disc_thmsss;
  1050 
  1051             fun prove goal tac =
  1052               Skip_Proof.prove lthy [] [] goal (tac o #context)
  1053               |> singleton (Proof_Context.export names_lthy0 no_defs_lthy)
  1054               |> Thm.close_derivation;
  1055 
  1056             fun proves [_] [_] = []
  1057               | proves goals tacs = map2 prove goals tacs;
  1058           in
  1059             (map2 proves unfold_goalss unfold_tacss,
  1060              map2 proves corec_goalss corec_tacss)
  1061           end;
  1062 
  1063         val is_triv_discI = is_triv_implies orf is_concl_refl;
  1064 
  1065         fun mk_disc_corec_like_thms corec_likes discIs =
  1066           map (op RS) (filter_out (is_triv_discI o snd) (corec_likes ~~ discIs));
  1067 
  1068         val disc_unfold_thmss = map2 mk_disc_corec_like_thms unfold_thmss discIss;
  1069         val disc_corec_thmss = map2 mk_disc_corec_like_thms corec_thmss discIss;
  1070 
  1071         fun mk_sel_corec_like_thm corec_like_thm sel sel_thm =
  1072           let
  1073             val (domT, ranT) = dest_funT (fastype_of sel);
  1074             val arg_cong' =
  1075               Drule.instantiate' (map (SOME o certifyT lthy) [domT, ranT])
  1076                 [NONE, NONE, SOME (certify lthy sel)] arg_cong
  1077               |> Thm.varifyT_global;
  1078             val sel_thm' = sel_thm RSN (2, trans);
  1079           in
  1080             corec_like_thm RS arg_cong' RS sel_thm'
  1081           end;
  1082 
  1083         fun mk_sel_corec_like_thms corec_likess =
  1084           map3 (map3 (map2 o mk_sel_corec_like_thm)) corec_likess selsss sel_thmsss |> map flat;
  1085 
  1086         val sel_unfold_thmss = mk_sel_corec_like_thms unfold_thmss;
  1087         val sel_corec_thmss = mk_sel_corec_like_thms corec_thmss;
  1088 
  1089         fun flat_corec_like_thms corec_likes disc_corec_likes sel_corec_likes =
  1090           corec_likes @ disc_corec_likes @ sel_corec_likes;
  1091 
  1092         val simp_thmss =
  1093           mk_simp_thmss wrap_ress
  1094             (map3 flat_corec_like_thms safe_corec_thmss disc_corec_thmss sel_corec_thmss)
  1095             (map3 flat_corec_like_thms safe_unfold_thmss disc_unfold_thmss sel_unfold_thmss);
  1096 
  1097         val anonymous_notes =
  1098           [(flat safe_unfold_thmss @ flat safe_corec_thmss, simp_attrs)]
  1099           |> map (fn (thms, attrs) => ((Binding.empty, attrs), [(thms, [])]));
  1100 
  1101         val common_notes =
  1102           (if nn > 1 then
  1103              (* FIXME: attribs *)
  1104              [(coinductN, [coinduct_thm], []),
  1105               (strong_coinductN, [strong_coinduct_thm], [])]
  1106            else
  1107              [])
  1108           |> map (fn (thmN, thms, attrs) =>
  1109             ((qualify true fp_common_name (Binding.name thmN), attrs), [(thms, [])]));
  1110 
  1111         val notes =
  1112           [(coinductN, map single coinduct_thms, []), (* FIXME: attribs *)
  1113            (corecN, corec_thmss, []),
  1114            (disc_corecN, disc_corec_thmss, simp_attrs),
  1115            (disc_corec_iffN, disc_corec_iff_thmss, simp_attrs),
  1116            (disc_unfoldN, disc_unfold_thmss, simp_attrs),
  1117            (disc_unfold_iffN, disc_unfold_iff_thmss, simp_attrs),
  1118            (sel_corecN, sel_corec_thmss, simp_attrs),
  1119            (sel_unfoldN, sel_unfold_thmss, simp_attrs),
  1120            (simpsN, simp_thmss, []),
  1121            (strong_coinductN, map single strong_coinduct_thms, []), (* FIXME: attribs *)
  1122            (unfoldN, unfold_thmss, [])]
  1123           |> maps (fn (thmN, thmss, attrs) =>
  1124             map_filter (fn (_, []) => NONE | (fp_b_name, thms) =>
  1125               SOME ((qualify true fp_b_name (Binding.name thmN), attrs),
  1126                 [(thms, [])])) (fp_b_names ~~ thmss));
  1127       in
  1128         lthy |> Local_Theory.notes (anonymous_notes @ common_notes @ notes) |> snd
  1129       end;
  1130 
  1131     val lthy' = lthy
  1132       |> fold_map define_ctrs_case_for_type (fp_bnfs ~~ fp_bs ~~ fpTs ~~ Cs ~~ ctors ~~ dtors ~~
  1133         fp_folds ~~ fp_recs ~~ ctor_dtors ~~ dtor_ctors ~~ ctor_injects ~~ pre_map_defs ~~
  1134         pre_set_defss ~~ pre_rel_defs ~~ fp_map_thms ~~ fp_set_thmss ~~ fp_rel_thms ~~ ns ~~ kss ~~
  1135         mss ~~ ctr_bindingss ~~ ctr_mixfixess ~~ ctr_Tsss ~~ disc_bindingss ~~ sel_bindingsss ~~
  1136         raw_sel_defaultsss)
  1137       |> wrap_types_and_more
  1138       |> (if lfp then derive_induct_fold_rec_thms_for_types
  1139           else derive_coinduct_unfold_corec_thms_for_types);
  1140 
  1141     val timer = time (timer ("Constructors, discriminators, selectors, etc., for the new " ^
  1142       (if lfp then "" else "co") ^ "datatype"));
  1143   in
  1144     timer; lthy'
  1145   end;
  1146 
  1147 val datatypes = define_datatypes (K I) (K I) (K I);
  1148 
  1149 val datatype_cmd = define_datatypes Typedecl.read_constraint Syntax.parse_typ Syntax.read_term;
  1150 
  1151 val parse_ctr_arg =
  1152   @{keyword "("} |-- parse_binding_colon -- Parse.typ --| @{keyword ")"} ||
  1153   (Parse.typ >> pair Binding.empty);
  1154 
  1155 val parse_defaults =
  1156   @{keyword "("} |-- @{keyword "defaults"} |-- Scan.repeat parse_bound_term --| @{keyword ")"};
  1157 
  1158 val parse_single_spec =
  1159   Parse.type_args_constrained -- Parse.binding -- Parse.opt_mixfix --
  1160   (@{keyword "="} |-- Parse.enum1 "|" (parse_opt_binding_colon -- Parse.binding --
  1161     Scan.repeat parse_ctr_arg -- Scan.optional parse_defaults [] -- Parse.opt_mixfix));
  1162 
  1163 val parse_datatype = parse_wrap_options -- Parse.and_list1 parse_single_spec;
  1164 
  1165 fun parse_datatype_cmd lfp construct_fp = parse_datatype >> datatype_cmd lfp construct_fp;
  1166 
  1167 end;