src/HOL/BNF/Tools/bnf_fp_def_sugar.ML
author blanchet
Tue Oct 02 01:00:18 2012 +0200 (2012-10-02)
changeset 49683 78a3d5006cf1
parent 49682 f57af1c46f99
child 49693 393d7242adaf
permissions -rw-r--r--
continued changing type of corec type
     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 cannot_merge_types () = error "Mutually recursive types must have the same type parameters";
   117 
   118 fun merge_type_arg T T' = if T = T' then T else cannot_merge_types ();
   119 
   120 fun merge_type_args (As, As') =
   121   if length As = length As' then map2 merge_type_arg As As' else cannot_merge_types ();
   122 
   123 fun reassoc_conjs thm =
   124   reassoc_conjs (thm RS @{thm conj_assoc[THEN iffD1]})
   125   handle THM _ => thm;
   126 
   127 fun type_args_constrained_of (((cAs, _), _), _) = cAs;
   128 fun type_binding_of (((_, b), _), _) = b;
   129 fun mixfix_of ((_, mx), _) = mx;
   130 fun ctr_specs_of (_, ctr_specs) = ctr_specs;
   131 
   132 fun disc_of ((((disc, _), _), _), _) = disc;
   133 fun ctr_of ((((_, ctr), _), _), _) = ctr;
   134 fun args_of (((_, args), _), _) = args;
   135 fun defaults_of ((_, ds), _) = ds;
   136 fun ctr_mixfix_of (_, mx) = mx;
   137 
   138 fun define_datatypes prepare_constraint prepare_typ prepare_term lfp construct_fp
   139     (wrap_opts as (no_dests, rep_compat), specs) no_defs_lthy0 =
   140   let
   141     (* TODO: sanity checks on arguments *)
   142     (* TODO: integration with function package ("size") *)
   143 
   144     val _ = if not lfp andalso no_dests then error "Cannot define destructor-less codatatypes"
   145       else ();
   146 
   147     fun qualify mandatory fp_b_name =
   148       Binding.qualify mandatory fp_b_name o (rep_compat ? Binding.qualify false rep_compat_prefix);
   149 
   150     val nn = length specs;
   151     val fp_bs = map type_binding_of specs;
   152     val fp_b_names = map Binding.name_of fp_bs;
   153     val fp_common_name = mk_common_name fp_b_names;
   154 
   155     fun prepare_type_arg (ty, c) =
   156       let val TFree (s, _) = prepare_typ no_defs_lthy0 ty in
   157         TFree (s, prepare_constraint no_defs_lthy0 c)
   158       end;
   159 
   160     val Ass0 = map (map prepare_type_arg o type_args_constrained_of) specs;
   161     val unsorted_Ass0 = map (map (resort_tfree HOLogic.typeS)) Ass0;
   162     val unsorted_As = Library.foldr1 merge_type_args unsorted_Ass0;
   163 
   164     val (((Bs0, Cs), Xs), no_defs_lthy) =
   165       no_defs_lthy0
   166       |> fold (Variable.declare_typ o resort_tfree dummyS) unsorted_As
   167       |> mk_TFrees (length unsorted_As)
   168       ||>> mk_TFrees nn
   169       ||>> apfst (map TFree) o
   170         variant_types (map (prefix "'") fp_b_names) (replicate nn HOLogic.typeS);
   171 
   172     (* TODO: cleaner handling of fake contexts, without "background_theory" *)
   173     (*the "perhaps o try" below helps gracefully handles the case where the new type is defined in a
   174       locale and shadows an existing global type*)
   175     val fake_thy =
   176       Theory.copy #> fold (fn spec => perhaps (try (Sign.add_type no_defs_lthy
   177         (type_binding_of spec, length (type_args_constrained_of spec), mixfix_of spec)))) specs;
   178     val fake_lthy = Proof_Context.background_theory fake_thy no_defs_lthy;
   179 
   180     fun mk_fake_T b =
   181       Type (fst (Term.dest_Type (Proof_Context.read_type_name fake_lthy true (Binding.name_of b))),
   182         unsorted_As);
   183 
   184     val fake_Ts = map mk_fake_T fp_bs;
   185 
   186     val mixfixes = map mixfix_of specs;
   187 
   188     val _ = (case duplicates Binding.eq_name fp_bs of [] => ()
   189       | b :: _ => error ("Duplicate type name declaration " ^ quote (Binding.name_of b)));
   190 
   191     val ctr_specss = map ctr_specs_of specs;
   192 
   193     val disc_bindingss = map (map disc_of) ctr_specss;
   194     val ctr_bindingss =
   195       map2 (fn fp_b_name => map (qualify false fp_b_name o ctr_of)) fp_b_names ctr_specss;
   196     val ctr_argsss = map (map args_of) ctr_specss;
   197     val ctr_mixfixess = map (map ctr_mixfix_of) ctr_specss;
   198 
   199     val sel_bindingsss = map (map (map fst)) ctr_argsss;
   200     val fake_ctr_Tsss0 = map (map (map (prepare_typ fake_lthy o snd))) ctr_argsss;
   201     val raw_sel_defaultsss = map (map defaults_of) ctr_specss;
   202 
   203     val (As :: _) :: fake_ctr_Tsss =
   204       burrow (burrow (Syntax.check_typs fake_lthy)) (Ass0 :: fake_ctr_Tsss0);
   205 
   206     val _ = (case duplicates (op =) unsorted_As of [] => ()
   207       | A :: _ => error ("Duplicate type parameter " ^
   208           quote (Syntax.string_of_typ no_defs_lthy A)));
   209 
   210     val rhs_As' = fold (fold (fold Term.add_tfreesT)) fake_ctr_Tsss [];
   211     val _ = (case subtract (op =) (map dest_TFree As) rhs_As' of
   212         [] => ()
   213       | A' :: _ => error ("Extra type variable on right-hand side: " ^
   214           quote (Syntax.string_of_typ no_defs_lthy (TFree A'))));
   215 
   216     fun eq_fpT_check (T as Type (s, Us)) (Type (s', Us')) =
   217         s = s' andalso (Us = Us' orelse error ("Illegal occurrence of recursive type " ^
   218           quote (Syntax.string_of_typ fake_lthy T)))
   219       | eq_fpT_check _ _ = false;
   220 
   221     fun freeze_fp (T as Type (s, Us)) =
   222         (case find_index (eq_fpT_check T) fake_Ts of
   223           ~1 => Type (s, map freeze_fp Us)
   224         | kk => nth Xs kk)
   225       | freeze_fp T = T;
   226 
   227     val ctr_TsssXs = map (map (map freeze_fp)) fake_ctr_Tsss;
   228     val ctr_sum_prod_TsXs = map (mk_sumTN_balanced o map HOLogic.mk_tupleT) ctr_TsssXs;
   229 
   230     val fp_eqs =
   231       map dest_TFree Xs ~~ map (Term.typ_subst_atomic (As ~~ unsorted_As)) ctr_sum_prod_TsXs;
   232 
   233     (* TODO: clean up list *)
   234     val (pre_bnfs, ((fp_bnfs as any_fp_bnf :: _, dtors0, ctors0, fp_folds0, fp_recs0, fp_induct,
   235            fp_strong_induct, dtor_ctors, ctor_dtors, ctor_injects, fp_map_thms, fp_set_thmss,
   236            fp_rel_thms, fp_fold_thms, fp_rec_thms), lthy)) =
   237       fp_bnf construct_fp fp_bs mixfixes (map dest_TFree unsorted_As) fp_eqs no_defs_lthy0;
   238 
   239     val timer = time (Timer.startRealTimer ());
   240 
   241     fun build_map build_arg (Type (s, Ts)) (Type (_, Us)) =
   242       let
   243         val bnf = the (bnf_of lthy s);
   244         val live = live_of_bnf bnf;
   245         val mapx = mk_map live Ts Us (map_of_bnf bnf);
   246         val TUs' = map dest_funT (fst (strip_typeN live (fastype_of mapx)));
   247       in Term.list_comb (mapx, map build_arg TUs') end;
   248 
   249     fun build_rel_step build_arg (Type (s, Ts)) =
   250       let
   251         val bnf = the (bnf_of lthy s);
   252         val live = live_of_bnf bnf;
   253         val rel = mk_rel live Ts Ts (rel_of_bnf bnf);
   254         val Ts' = map domain_type (fst (strip_typeN live (fastype_of rel)));
   255       in Term.list_comb (rel, map build_arg Ts') end;
   256 
   257     fun add_nesty_bnf_names Us =
   258       let
   259         fun add (Type (s, Ts)) ss =
   260             let val (needs, ss') = fold_map add Ts ss in
   261               if exists I needs then (true, insert (op =) s ss') else (false, ss')
   262             end
   263           | add T ss = (member (op =) Us T, ss);
   264       in snd oo add end;
   265 
   266     fun nesty_bnfs Us =
   267       map_filter (bnf_of lthy) (fold (fold (fold (add_nesty_bnf_names Us))) ctr_TsssXs []);
   268 
   269     val nesting_bnfs = nesty_bnfs As;
   270     val nested_bnfs = nesty_bnfs Xs;
   271 
   272     val pre_map_defs = map map_def_of_bnf pre_bnfs;
   273     val pre_set_defss = map set_defs_of_bnf pre_bnfs;
   274     val pre_rel_defs = map rel_def_of_bnf pre_bnfs;
   275     val nested_map_comps'' = map ((fn thm => thm RS sym) o map_comp_of_bnf) nested_bnfs;
   276     val nested_map_comp's = map map_comp'_of_bnf nested_bnfs;
   277     val nested_map_ids'' = map (unfold_thms lthy @{thms id_def} o map_id_of_bnf) nested_bnfs;
   278     val nesting_map_ids'' = map (unfold_thms lthy @{thms id_def} o map_id_of_bnf) nesting_bnfs;
   279     val nested_set_natural's = maps set_natural'_of_bnf nested_bnfs;
   280     val nesting_set_natural's = maps set_natural'_of_bnf nesting_bnfs;
   281 
   282     val live = live_of_bnf any_fp_bnf;
   283 
   284     val Bs =
   285       map3 (fn alive => fn A as TFree (_, S) => fn B => if alive then resort_tfree S B else A)
   286         (liveness_of_fp_bnf (length As) any_fp_bnf) As Bs0;
   287 
   288     val B_ify = Term.typ_subst_atomic (As ~~ Bs);
   289 
   290     val ctors = map (mk_ctor As) ctors0;
   291     val dtors = map (mk_dtor As) dtors0;
   292 
   293     val fpTs = map (domain_type o fastype_of) dtors;
   294 
   295     val exists_fp_subtype = exists_subtype (member (op =) fpTs);
   296     val exists_Cs_subtype = exists_subtype (member (op =) Cs);
   297 
   298     val ctr_Tsss = map (map (map (Term.typ_subst_atomic (Xs ~~ fpTs)))) ctr_TsssXs;
   299     val ns = map length ctr_Tsss;
   300     val kss = map (fn n => 1 upto n) ns;
   301     val mss = map (map length) ctr_Tsss;
   302     val Css = map2 replicate ns Cs;
   303 
   304     val fp_folds as any_fp_fold :: _ = map (mk_rec_like lfp As Cs) fp_folds0;
   305     val fp_recs as any_fp_rec :: _ = map (mk_rec_like lfp As Cs) fp_recs0;
   306 
   307     val fp_fold_fun_Ts = fst (split_last (binder_types (fastype_of any_fp_fold)));
   308     val fp_rec_fun_Ts = fst (split_last (binder_types (fastype_of any_fp_rec)));
   309 
   310     val (((fold_only as (gss, _, _), rec_only as (hss, _, _)),
   311           (cs, cpss, unfold_only as ((pgss, crssss, cgssss), (_, g_Tsss, _)),
   312            corec_only as ((phss, csssss, chssss), (_, h_Tsss, _)))), names_lthy0) =
   313       if lfp then
   314         let
   315           val y_Tsss =
   316             map3 (fn n => fn ms => map2 dest_tupleT ms o dest_sumTN_balanced n o domain_type)
   317               ns mss fp_fold_fun_Ts;
   318           val g_Tss = map2 (map2 (curry (op --->))) y_Tsss Css;
   319 
   320           val ((gss, ysss), lthy) =
   321             lthy
   322             |> mk_Freess "f" g_Tss
   323             ||>> mk_Freesss "x" y_Tsss;
   324 
   325           fun proj_recT proj (Type (s as @{type_name prod}, Ts as [T, U])) =
   326               if member (op =) fpTs T then proj (T, U) else Type (s, map (proj_recT proj) Ts)
   327             | proj_recT proj (Type (s, Ts)) = Type (s, map (proj_recT proj) Ts)
   328             | proj_recT _ T = T;
   329 
   330           fun unzip_recT T =
   331             if exists_fp_subtype T then [proj_recT fst T, proj_recT snd T] else [T];
   332 
   333           val z_Tsss =
   334             map3 (fn n => fn ms => map2 dest_tupleT ms o dest_sumTN_balanced n o domain_type)
   335               ns mss fp_rec_fun_Ts;
   336           val z_Tssss = map (map (map unzip_recT)) z_Tsss;
   337           val h_Tss = map2 (map2 (fold_rev (curry (op --->)))) z_Tssss Css;
   338 
   339           val hss = map2 (map2 retype_free) h_Tss gss;
   340           val zsss = map2 (map2 (map2 retype_free)) z_Tsss ysss;
   341         in
   342           ((((gss, g_Tss, ysss), (hss, h_Tss, zsss)),
   343             ([], [], (([], [], []), ([], [], [])), (([], [], []), ([], [], [])))), lthy)
   344         end
   345       else
   346         let
   347           (*avoid "'a itself" arguments in coiterators and corecursors*)
   348           val mss' =  map (fn [0] => [1] | ms => ms) mss;
   349 
   350           val p_Tss = map2 (fn n => replicate (Int.max (0, n - 1)) o mk_pred1T) ns Cs;
   351 
   352           fun flat_predss_getterss qss fss = maps (op @) (qss ~~ fss);
   353 
   354           fun flat_preds_predsss_gettersss [] [qss] [fss] = flat_predss_getterss qss fss
   355             | flat_preds_predsss_gettersss (p :: ps) (qss :: qsss) (fss :: fsss) =
   356               p :: flat_predss_getterss qss fss @ flat_preds_predsss_gettersss ps qsss fsss;
   357 
   358           fun mk_types maybe_unzipT fun_Ts =
   359             let
   360               val f_sum_prod_Ts = map range_type fun_Ts;
   361               val f_prod_Tss = map2 dest_sumTN_balanced ns f_sum_prod_Ts;
   362               val f_Tsss = map2 (map2 dest_tupleT) mss' f_prod_Tss;
   363               val f_Tssss =
   364                 map2 (fn C => map (map (map (curry (op -->) C) o maybe_unzipT))) Cs f_Tsss;
   365               val q_Tssss =
   366                 map (map (map (fn [_] => [] | [_, C] => [mk_pred1T (domain_type C)]))) f_Tssss;
   367               val pf_Tss = map3 flat_preds_predsss_gettersss p_Tss q_Tssss f_Tssss;
   368             in (q_Tssss, f_sum_prod_Ts, f_Tsss, f_Tssss, pf_Tss) end;
   369 
   370           val (r_Tssss, g_sum_prod_Ts, g_Tsss, g_Tssss, pg_Tss) = mk_types single fp_fold_fun_Ts;
   371 
   372           val (((cs, pss), gssss), lthy) =
   373             lthy
   374             |> mk_Frees "a" Cs
   375             ||>> mk_Freess "p" p_Tss
   376             ||>> mk_Freessss "g" g_Tssss;
   377           val rssss = map (map (map (fn [] => []))) r_Tssss;
   378 
   379           fun proj_corecT proj (Type (s as @{type_name sum}, Ts as [T, U])) =
   380               if member (op =) fpTs T then proj (T, U) else Type (s, map (proj_corecT proj) Ts)
   381             | proj_corecT proj (Type (s, Ts)) = Type (s, map (proj_corecT proj) Ts)
   382             | proj_corecT _ T = T;
   383 
   384           fun unzip_corecT T =
   385             if exists_fp_subtype T then [proj_corecT fst T, proj_corecT snd T] else [T];
   386 
   387           val (s_Tssss, h_sum_prod_Ts, h_Tsss, h_Tssss, ph_Tss) =
   388             mk_types unzip_corecT fp_rec_fun_Ts;
   389 
   390           val hssss_hd = map2 (map2 (map2 (fn T :: _ => fn [g] => retype_free T g))) h_Tssss gssss;
   391           val ((sssss, hssss_tl), lthy) =
   392             lthy
   393             |> mk_Freessss "q" s_Tssss
   394             ||>> mk_Freessss "h" (map (map (map tl)) h_Tssss);
   395           val hssss = map2 (map2 (map2 cons)) hssss_hd hssss_tl;
   396 
   397           val cpss = map2 (map o rapp) cs pss;
   398 
   399           fun mk_terms qssss fssss =
   400             let
   401               val pfss = map3 flat_preds_predsss_gettersss pss qssss fssss;
   402               val cqssss = map2 (map o map o map o rapp) cs qssss;
   403               val cfssss = map2 (map o map o map o rapp) cs fssss;
   404             in (pfss, cqssss, cfssss) end;
   405         in
   406           (((([], [], []), ([], [], [])),
   407             (cs, cpss, (mk_terms rssss gssss, (g_sum_prod_Ts, g_Tsss, pg_Tss)),
   408              (mk_terms sssss hssss, (h_sum_prod_Ts, h_Tsss, ph_Tss)))), lthy)
   409         end;
   410 
   411     fun define_ctrs_case_for_type (((((((((((((((((((((((((fp_bnf, fp_b), fpT), C), ctor), dtor),
   412             fp_fold), fp_rec), ctor_dtor), dtor_ctor), ctor_inject), pre_map_def), pre_set_defs),
   413           pre_rel_def), fp_map_thm), fp_set_thms), fp_rel_thm), n), ks), ms), ctr_bindings),
   414         ctr_mixfixes), ctr_Tss), disc_bindings), sel_bindingss), raw_sel_defaultss) no_defs_lthy =
   415       let
   416         val fp_b_name = Binding.name_of fp_b;
   417 
   418         val dtorT = domain_type (fastype_of ctor);
   419         val ctr_prod_Ts = map HOLogic.mk_tupleT ctr_Tss;
   420         val ctr_sum_prod_T = mk_sumTN_balanced ctr_prod_Ts;
   421         val case_Ts = map (fn Ts => Ts ---> C) ctr_Tss;
   422 
   423         val (((((w, fs), xss), yss), u'), names_lthy) =
   424           no_defs_lthy
   425           |> yield_singleton (mk_Frees "w") dtorT
   426           ||>> mk_Frees "f" case_Ts
   427           ||>> mk_Freess "x" ctr_Tss
   428           ||>> mk_Freess "y" (map (map B_ify) ctr_Tss)
   429           ||>> yield_singleton Variable.variant_fixes fp_b_name;
   430 
   431         val u = Free (u', fpT);
   432 
   433         val tuple_xs = map HOLogic.mk_tuple xss;
   434         val tuple_ys = map HOLogic.mk_tuple yss;
   435 
   436         val ctr_rhss =
   437           map3 (fn k => fn xs => fn tuple_x => fold_rev Term.lambda xs (ctor $
   438             mk_InN_balanced ctr_sum_prod_T n tuple_x k)) ks xss tuple_xs;
   439 
   440         val case_binding = qualify false fp_b_name (Binding.suffix_name ("_" ^ caseN) fp_b);
   441 
   442         val case_rhs =
   443           fold_rev Term.lambda (fs @ [u])
   444             (mk_sum_caseN_balanced (map2 mk_uncurried_fun fs xss) $ (dtor $ u));
   445 
   446         val ((raw_case :: raw_ctrs, raw_case_def :: raw_ctr_defs), (lthy', lthy)) = no_defs_lthy
   447           |> apfst split_list o fold_map3 (fn b => fn mx => fn rhs =>
   448               Local_Theory.define ((b, mx), ((Thm.def_binding b, []), rhs)) #>> apsnd snd)
   449             (case_binding :: ctr_bindings) (NoSyn :: ctr_mixfixes) (case_rhs :: ctr_rhss)
   450           ||> `Local_Theory.restore;
   451 
   452         val phi = Proof_Context.export_morphism lthy lthy';
   453 
   454         val ctr_defs = map (Morphism.thm phi) raw_ctr_defs;
   455         val ctr_defs' =
   456           map2 (fn m => fn def => mk_unabs_def m (def RS meta_eq_to_obj_eq)) ms ctr_defs;
   457         val case_def = Morphism.thm phi raw_case_def;
   458 
   459         val ctrs0 = map (Morphism.term phi) raw_ctrs;
   460         val casex0 = Morphism.term phi raw_case;
   461 
   462         val ctrs = map (mk_ctr As) ctrs0;
   463 
   464         fun wrap lthy =
   465           let
   466             fun exhaust_tac {context = ctxt, ...} =
   467               let
   468                 val ctor_iff_dtor_thm =
   469                   let
   470                     val goal =
   471                       fold_rev Logic.all [w, u]
   472                         (mk_Trueprop_eq (HOLogic.mk_eq (u, ctor $ w), HOLogic.mk_eq (dtor $ u, w)));
   473                   in
   474                     Skip_Proof.prove lthy [] [] goal (fn {context = ctxt, ...} =>
   475                       mk_ctor_iff_dtor_tac ctxt (map (SOME o certifyT lthy) [dtorT, fpT])
   476                         (certify lthy ctor) (certify lthy dtor) ctor_dtor dtor_ctor)
   477                     |> Thm.close_derivation
   478                     |> Morphism.thm phi
   479                   end;
   480 
   481                 val sumEN_thm' =
   482                   unfold_thms lthy @{thms all_unit_eq}
   483                     (Drule.instantiate' (map (SOME o certifyT lthy) ctr_prod_Ts) []
   484                        (mk_sumEN_balanced n))
   485                   |> Morphism.thm phi;
   486               in
   487                 mk_exhaust_tac ctxt n ctr_defs ctor_iff_dtor_thm sumEN_thm'
   488               end;
   489 
   490             val inject_tacss =
   491               map2 (fn 0 => K [] | _ => fn ctr_def => [fn {context = ctxt, ...} =>
   492                   mk_inject_tac ctxt ctr_def ctor_inject]) ms ctr_defs;
   493 
   494             val half_distinct_tacss =
   495               map (map (fn (def, def') => fn {context = ctxt, ...} =>
   496                 mk_half_distinct_tac ctxt ctor_inject [def, def'])) (mk_half_pairss (`I ctr_defs));
   497 
   498             val case_tacs =
   499               map3 (fn k => fn m => fn ctr_def => fn {context = ctxt, ...} =>
   500                 mk_case_tac ctxt n k m case_def ctr_def dtor_ctor) ks ms ctr_defs;
   501 
   502             val tacss = [exhaust_tac] :: inject_tacss @ half_distinct_tacss @ [case_tacs];
   503 
   504             val sel_defaultss = map (map (apsnd (prepare_term lthy))) raw_sel_defaultss
   505           in
   506             wrap_datatype tacss (((wrap_opts, ctrs0), casex0), (disc_bindings, (sel_bindingss,
   507               sel_defaultss))) lthy
   508           end;
   509 
   510         fun derive_maps_sets_rels (wrap_res, lthy) =
   511           let
   512             val rel_flip = rel_flip_of_bnf fp_bnf;
   513             val nones = replicate live NONE;
   514 
   515             val ctor_cong =
   516               if lfp then Drule.dummy_thm
   517               else cterm_instantiate_pos [NONE, NONE, SOME (certify lthy ctor)] arg_cong;
   518 
   519             fun mk_cIn ify =
   520               certify lthy o (not lfp ? curry (op $) (map_types ify ctor)) oo
   521               mk_InN_balanced (ify ctr_sum_prod_T) n;
   522 
   523             val cxIns = map2 (mk_cIn I) tuple_xs ks;
   524             val cyIns = map2 (mk_cIn B_ify) tuple_ys ks;
   525 
   526             fun mk_map_thm ctr_def' cxIn =
   527               fold_thms lthy [ctr_def']
   528                 (unfold_thms lthy (pre_map_def ::
   529                      (if lfp then [] else [ctor_dtor, dtor_ctor]) @ sum_prod_thms_map)
   530                    (cterm_instantiate_pos (nones @ [SOME cxIn])
   531                       (if lfp then fp_map_thm else fp_map_thm RS ctor_cong)))
   532               |> singleton (Proof_Context.export names_lthy no_defs_lthy);
   533 
   534             fun mk_set_thm fp_set_thm ctr_def' cxIn =
   535               fold_thms lthy [ctr_def']
   536                 (unfold_thms lthy (pre_set_defs @ nested_set_natural's @ nesting_set_natural's @
   537                      (if lfp then [] else [dtor_ctor]) @ sum_prod_thms_set)
   538                    (cterm_instantiate_pos [SOME cxIn] fp_set_thm))
   539               |> singleton (Proof_Context.export names_lthy no_defs_lthy);
   540 
   541             fun mk_set_thms fp_set_thm = map2 (mk_set_thm fp_set_thm) ctr_defs' cxIns;
   542 
   543             val map_thms = map2 mk_map_thm ctr_defs' cxIns;
   544             val set_thmss = map mk_set_thms fp_set_thms;
   545 
   546             val rel_infos = (ctr_defs' ~~ cxIns, ctr_defs' ~~ cyIns);
   547 
   548             fun mk_rel_thm postproc ctr_defs' cxIn cyIn =
   549               fold_thms lthy ctr_defs'
   550                  (unfold_thms lthy (pre_rel_def :: (if lfp then [] else [dtor_ctor]) @
   551                       sum_prod_thms_rel)
   552                     (cterm_instantiate_pos (nones @ [SOME cxIn, SOME cyIn]) fp_rel_thm))
   553               |> postproc
   554               |> singleton (Proof_Context.export names_lthy no_defs_lthy);
   555 
   556             fun mk_rel_inject_thm ((ctr_def', cxIn), (_, cyIn)) =
   557               mk_rel_thm (unfold_thms lthy @{thms eq_sym_Unity_conv}) [ctr_def'] cxIn cyIn;
   558 
   559             val rel_inject_thms = map mk_rel_inject_thm (op ~~ rel_infos);
   560 
   561             fun mk_half_rel_distinct_thm ((xctr_def', cxIn), (yctr_def', cyIn)) =
   562               mk_rel_thm (fn thm => thm RS @{thm eq_False[THEN iffD1]}) [xctr_def', yctr_def']
   563                 cxIn cyIn;
   564 
   565             fun mk_other_half_rel_distinct_thm thm =
   566               flip_rels lthy live thm RS (rel_flip RS sym RS @{thm arg_cong[of _ _ Not]} RS iffD2);
   567 
   568             val half_rel_distinct_thmss =
   569               map (map mk_half_rel_distinct_thm) (mk_half_pairss rel_infos);
   570             val other_half_rel_distinct_thmss =
   571               map (map mk_other_half_rel_distinct_thm) half_rel_distinct_thmss;
   572             val (rel_distinct_thms, _) =
   573               join_halves n half_rel_distinct_thmss other_half_rel_distinct_thmss;
   574 
   575             val notes =
   576               [(mapN, map_thms, code_simp_attrs),
   577                (rel_distinctN, rel_distinct_thms, code_simp_attrs),
   578                (rel_injectN, rel_inject_thms, code_simp_attrs),
   579                (setsN, flat set_thmss, code_simp_attrs)]
   580               |> filter_out (null o #2)
   581               |> map (fn (thmN, thms, attrs) =>
   582                 ((qualify true fp_b_name (Binding.name thmN), attrs), [(thms, [])]));
   583           in
   584             (wrap_res, lthy |> Local_Theory.notes notes |> snd)
   585           end;
   586 
   587         fun define_fold_rec no_defs_lthy =
   588           let
   589             val fpT_to_C = fpT --> C;
   590 
   591             fun build_prod_proj mk_proj (T, U) =
   592               if T = U then
   593                 id_const T
   594               else
   595                 (case (T, U) of
   596                   (Type (s, _), Type (s', _)) =>
   597                   if s = s' then build_map (build_prod_proj mk_proj) T U else mk_proj T
   598                 | _ => mk_proj T);
   599 
   600             (* TODO: Avoid these complications; cf. corec case *)
   601             fun mk_U proj (Type (s as @{type_name prod}, Ts as [T', U])) =
   602                 if member (op =) fpTs T' then proj (T', U) else Type (s, map (mk_U proj) Ts)
   603               | mk_U proj (Type (s, Ts)) = Type (s, map (mk_U proj) Ts)
   604               | mk_U _ T = T;
   605 
   606             fun unzip_rec (x as Free (_, T)) =
   607               if exists_fp_subtype T then
   608                 [build_prod_proj fst_const (T, mk_U fst T) $ x,
   609                  build_prod_proj snd_const (T, mk_U snd T) $ x]
   610               else
   611                 [x];
   612 
   613             fun mk_rec_like_arg f xs = mk_tupled_fun (HOLogic.mk_tuple xs) f (maps unzip_rec xs);
   614 
   615             fun generate_rec_like (suf, fp_rec_like, (fss, f_Tss, xsss)) =
   616               let
   617                 val res_T = fold_rev (curry (op --->)) f_Tss fpT_to_C;
   618                 val binding = qualify false fp_b_name (Binding.suffix_name ("_" ^ suf) fp_b);
   619                 val spec =
   620                   mk_Trueprop_eq (lists_bmoc fss (Free (Binding.name_of binding, res_T)),
   621                     Term.list_comb (fp_rec_like,
   622                       map2 (mk_sum_caseN_balanced oo map2 mk_rec_like_arg) fss xsss));
   623               in (binding, spec) end;
   624 
   625             val rec_like_infos =
   626               [(foldN, fp_fold, fold_only),
   627                (recN, fp_rec, rec_only)];
   628 
   629             val (bindings, specs) = map generate_rec_like rec_like_infos |> split_list;
   630 
   631             val ((csts, defs), (lthy', lthy)) = no_defs_lthy
   632               |> apfst split_list o fold_map2 (fn b => fn spec =>
   633                 Specification.definition (SOME (b, NONE, NoSyn), ((Thm.def_binding b, []), spec))
   634                 #>> apsnd snd) bindings specs
   635               ||> `Local_Theory.restore;
   636 
   637             val phi = Proof_Context.export_morphism lthy lthy';
   638 
   639             val [fold_def, rec_def] = map (Morphism.thm phi) defs;
   640 
   641             val [foldx, recx] = map (mk_rec_like lfp As Cs o Morphism.term phi) csts;
   642           in
   643             ((foldx, recx, fold_def, rec_def), lthy)
   644           end;
   645 
   646         fun define_unfold_corec no_defs_lthy =
   647           let
   648             val B_to_fpT = C --> fpT;
   649 
   650             fun build_sum_inj mk_inj (T, U) =
   651               if T = U then
   652                 id_const T
   653               else
   654                 (case (T, U) of
   655                   (Type (s, _), Type (s', _)) =>
   656                   if s = s' then build_map (build_sum_inj mk_inj) T U
   657                   else uncurry mk_inj (dest_sumT U)
   658                 | _ => uncurry mk_inj (dest_sumT U));
   659 
   660             fun build_dtor_corec_like_arg _ [] [cf] = cf
   661               | build_dtor_corec_like_arg T [cq] [cf, cf'] =
   662                 mk_If cq (build_sum_inj Inl_const (fastype_of cf, T) $ cf)
   663                   (build_sum_inj Inr_const (fastype_of cf', T) $ cf')
   664 
   665             val crgsss = map3 (map3 (map3 build_dtor_corec_like_arg)) g_Tsss crssss cgssss;
   666             val cshsss = map3 (map3 (map3 build_dtor_corec_like_arg)) h_Tsss csssss chssss;
   667 
   668             fun mk_preds_getterss_join c n cps sum_prod_T cqfss =
   669               Term.lambda c (mk_IfN sum_prod_T cps
   670                 (map2 (mk_InN_balanced sum_prod_T n) (map HOLogic.mk_tuple cqfss) (1 upto n)));
   671 
   672             fun generate_corec_like (suf, fp_rec_like, (cqfsss, ((pfss, _, _), (f_sum_prod_Ts, _,
   673                 pf_Tss)))) =
   674               let
   675                 val res_T = fold_rev (curry (op --->)) pf_Tss B_to_fpT;
   676                 val binding = qualify false fp_b_name (Binding.suffix_name ("_" ^ suf) fp_b);
   677                 val spec =
   678                   mk_Trueprop_eq (lists_bmoc pfss (Free (Binding.name_of binding, res_T)),
   679                     Term.list_comb (fp_rec_like,
   680                       map5 mk_preds_getterss_join cs ns cpss f_sum_prod_Ts cqfsss));
   681               in (binding, spec) end;
   682 
   683             val corec_like_infos =
   684               [(unfoldN, fp_fold, (crgsss, unfold_only)),
   685                (corecN, fp_rec, (cshsss, corec_only))];
   686 
   687             val (bindings, specs) = map generate_corec_like corec_like_infos |> split_list;
   688 
   689             val ((csts, defs), (lthy', lthy)) = no_defs_lthy
   690               |> apfst split_list o fold_map2 (fn b => fn spec =>
   691                 Specification.definition (SOME (b, NONE, NoSyn), ((Thm.def_binding b, []), spec))
   692                 #>> apsnd snd) bindings specs
   693               ||> `Local_Theory.restore;
   694 
   695             val phi = Proof_Context.export_morphism lthy lthy';
   696 
   697             val [unfold_def, corec_def] = map (Morphism.thm phi) defs;
   698 
   699             val [unfold, corec] = map (mk_rec_like lfp As Cs o Morphism.term phi) csts;
   700           in
   701             ((unfold, corec, unfold_def, corec_def), lthy)
   702           end;
   703 
   704         val define_rec_likes = if lfp then define_fold_rec else define_unfold_corec;
   705 
   706         fun massage_res ((wrap_res, rec_like_res), lthy) =
   707           (((ctrs, xss, ctr_defs, wrap_res), rec_like_res), lthy);
   708       in
   709         (wrap #> (live > 0 ? derive_maps_sets_rels) ##>> define_rec_likes #> massage_res, lthy')
   710       end;
   711 
   712     fun wrap_types_and_more (wrap_types_and_mores, lthy) =
   713       fold_map I wrap_types_and_mores lthy
   714       |>> apsnd split_list4 o apfst split_list4 o split_list;
   715 
   716     (* TODO: Add map, sets, rel simps *)
   717     val mk_simp_thmss =
   718       map3 (fn (_, _, _, injects, distincts, cases, _, _, _) => fn rec_likes => fn fold_likes =>
   719         injects @ distincts @ cases @ rec_likes @ fold_likes);
   720 
   721     fun derive_induct_fold_rec_thms_for_types (((ctrss, xsss, ctr_defss, wrap_ress), (folds, recs,
   722         fold_defs, rec_defs)), lthy) =
   723       let
   724         val (((ps, ps'), us'), names_lthy) =
   725           lthy
   726           |> mk_Frees' "P" (map mk_pred1T fpTs)
   727           ||>> Variable.variant_fixes fp_b_names;
   728 
   729         val us = map2 (curry Free) us' fpTs;
   730 
   731         fun mk_sets_nested bnf =
   732           let
   733             val Type (T_name, Us) = T_of_bnf bnf;
   734             val lives = lives_of_bnf bnf;
   735             val sets = sets_of_bnf bnf;
   736             fun mk_set U =
   737               (case find_index (curry (op =) U) lives of
   738                 ~1 => Term.dummy
   739               | i => nth sets i);
   740           in
   741             (T_name, map mk_set Us)
   742           end;
   743 
   744         val setss_nested = map mk_sets_nested nested_bnfs;
   745 
   746         val (induct_thms, induct_thm) =
   747           let
   748             fun mk_set Ts t =
   749               let val Type (_, Ts0) = domain_type (fastype_of t) in
   750                 Term.subst_atomic_types (Ts0 ~~ Ts) t
   751               end;
   752 
   753             fun mk_raw_prem_prems names_lthy (x as Free (s, T as Type (T_name, Ts0))) =
   754                 (case find_index (curry (op =) T) fpTs of
   755                   ~1 =>
   756                   (case AList.lookup (op =) setss_nested T_name of
   757                     NONE => []
   758                   | SOME raw_sets0 =>
   759                     let
   760                       val (Ts, raw_sets) =
   761                         split_list (filter (exists_fp_subtype o fst) (Ts0 ~~ raw_sets0));
   762                       val sets = map (mk_set Ts0) raw_sets;
   763                       val (ys, names_lthy') = names_lthy |> mk_Frees s Ts;
   764                       val xysets = map (pair x) (ys ~~ sets);
   765                       val ppremss = map (mk_raw_prem_prems names_lthy') ys;
   766                     in
   767                       flat (map2 (map o apfst o cons) xysets ppremss)
   768                     end)
   769                 | kk => [([], (kk + 1, x))])
   770               | mk_raw_prem_prems _ _ = [];
   771 
   772             fun close_prem_prem xs t =
   773               fold_rev Logic.all (map Free (drop (nn + length xs)
   774                 (rev (Term.add_frees t (map dest_Free xs @ ps'))))) t;
   775 
   776             fun mk_prem_prem xs (xysets, (j, x)) =
   777               close_prem_prem xs (Logic.list_implies (map (fn (x', (y, set)) =>
   778                   HOLogic.mk_Trueprop (HOLogic.mk_mem (y, set $ x'))) xysets,
   779                 HOLogic.mk_Trueprop (nth ps (j - 1) $ x)));
   780 
   781             fun mk_raw_prem phi ctr ctr_Ts =
   782               let
   783                 val (xs, names_lthy') = names_lthy |> mk_Frees "x" ctr_Ts;
   784                 val pprems = maps (mk_raw_prem_prems names_lthy') xs;
   785               in (xs, pprems, HOLogic.mk_Trueprop (phi $ Term.list_comb (ctr, xs))) end;
   786 
   787             fun mk_prem (xs, raw_pprems, concl) =
   788               fold_rev Logic.all xs (Logic.list_implies (map (mk_prem_prem xs) raw_pprems, concl));
   789 
   790             val raw_premss = map3 (map2 o mk_raw_prem) ps ctrss ctr_Tsss;
   791 
   792             val goal =
   793               Library.foldr (Logic.list_implies o apfst (map mk_prem)) (raw_premss,
   794                 HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj (map2 (curry (op $)) ps us)));
   795 
   796             val kksss = map (map (map (fst o snd) o #2)) raw_premss;
   797 
   798             val ctor_induct' = fp_induct OF (map mk_sumEN_tupled_balanced mss);
   799 
   800             val thm =
   801               Skip_Proof.prove lthy [] [] goal (fn {context = ctxt, ...} =>
   802                 mk_induct_tac ctxt nn ns mss kksss (flat ctr_defss) ctor_induct'
   803                   nested_set_natural's pre_set_defss)
   804               |> singleton (Proof_Context.export names_lthy lthy)
   805               |> Thm.close_derivation;
   806           in
   807             `(conj_dests nn) thm
   808           end;
   809 
   810         val induct_cases = quasi_unambiguous_case_names (maps (map name_of_ctr) ctrss);
   811 
   812         val (fold_thmss, rec_thmss) =
   813           let
   814             val xctrss = map2 (map2 (curry Term.list_comb)) ctrss xsss;
   815             val gfolds = map (lists_bmoc gss) folds;
   816             val hrecs = map (lists_bmoc hss) recs;
   817 
   818             fun mk_goal fss frec_like xctr f xs fxs =
   819               fold_rev (fold_rev Logic.all) (xs :: fss)
   820                 (mk_Trueprop_eq (frec_like $ xctr, Term.list_comb (f, fxs)));
   821 
   822             fun build_rec_like frec_likes (T, U) =
   823               if T = U then
   824                 id_const T
   825               else
   826                 (case find_index (curry (op =) T) fpTs of
   827                   ~1 => build_map (build_rec_like frec_likes) T U
   828                 | kk => nth frec_likes kk);
   829 
   830             val mk_U = typ_subst (map2 pair fpTs Cs);
   831 
   832             fun intr_rec_likes frec_likes maybe_cons (x as Free (_, T)) =
   833               if exists_fp_subtype T then
   834                 maybe_cons x [build_rec_like frec_likes (T, mk_U T) $ x]
   835               else
   836                 [x];
   837 
   838             val gxsss = map (map (maps (intr_rec_likes gfolds (K I)))) xsss;
   839             val hxsss = map (map (maps (intr_rec_likes hrecs cons))) xsss;
   840 
   841             val fold_goalss = map5 (map4 o mk_goal gss) gfolds xctrss gss xsss gxsss;
   842             val rec_goalss = map5 (map4 o mk_goal hss) hrecs xctrss hss xsss hxsss;
   843 
   844             val fold_tacss =
   845               map2 (map o mk_rec_like_tac pre_map_defs [] nesting_map_ids'' fold_defs) fp_fold_thms
   846                 ctr_defss;
   847             val rec_tacss =
   848               map2 (map o mk_rec_like_tac pre_map_defs nested_map_comp's
   849                 (nested_map_ids'' @ nesting_map_ids'') rec_defs) fp_rec_thms ctr_defss;
   850 
   851             fun prove goal tac =
   852               Skip_Proof.prove lthy [] [] goal (tac o #context)
   853               |> Thm.close_derivation;
   854           in
   855             (map2 (map2 prove) fold_goalss fold_tacss, map2 (map2 prove) rec_goalss rec_tacss)
   856           end;
   857 
   858         val simp_thmss = mk_simp_thmss wrap_ress rec_thmss fold_thmss;
   859 
   860         val induct_case_names_attr = Attrib.internal (K (Rule_Cases.case_names induct_cases));
   861         fun induct_type_attr T_name = Attrib.internal (K (Induct.induct_type T_name));
   862 
   863         val common_notes =
   864           (if nn > 1 then [(inductN, [induct_thm], [induct_case_names_attr])] else [])
   865           |> map (fn (thmN, thms, attrs) =>
   866             ((qualify true fp_common_name (Binding.name thmN), attrs), [(thms, [])]));
   867 
   868         val notes =
   869           [(foldN, fold_thmss, K code_simp_attrs),
   870            (inductN, map single induct_thms,
   871             fn T_name => [induct_case_names_attr, induct_type_attr T_name]),
   872            (recN, rec_thmss, K code_simp_attrs),
   873            (simpsN, simp_thmss, K [])]
   874           |> maps (fn (thmN, thmss, attrs) =>
   875             map3 (fn fp_b_name => fn Type (T_name, _) => fn thms =>
   876               ((qualify true fp_b_name (Binding.name thmN), attrs T_name),
   877                [(thms, [])])) fp_b_names fpTs thmss);
   878       in
   879         lthy |> Local_Theory.notes (common_notes @ notes) |> snd
   880       end;
   881 
   882     fun derive_coinduct_unfold_corec_thms_for_types (((ctrss, _, ctr_defss, wrap_ress), (unfolds,
   883         corecs, unfold_defs, corec_defs)), lthy) =
   884       let
   885         val nesting_rel_eqs = map rel_eq_of_bnf nesting_bnfs;
   886 
   887         val discss = map (map (mk_disc_or_sel As) o #1) wrap_ress;
   888         val selsss = map (map (map (mk_disc_or_sel As)) o #2) wrap_ress;
   889         val exhaust_thms = map #3 wrap_ress;
   890         val disc_thmsss = map #7 wrap_ress;
   891         val discIss = map #8 wrap_ress;
   892         val sel_thmsss = map #9 wrap_ress;
   893 
   894         val (((rs, us'), vs'), names_lthy) =
   895           lthy
   896           |> mk_Frees "R" (map (fn T => mk_pred2T T T) fpTs)
   897           ||>> Variable.variant_fixes fp_b_names
   898           ||>> Variable.variant_fixes (map (suffix "'") fp_b_names);
   899 
   900         val us = map2 (curry Free) us' fpTs;
   901         val udiscss = map2 (map o rapp) us discss;
   902         val uselsss = map2 (map o map o rapp) us selsss;
   903 
   904         val vs = map2 (curry Free) vs' fpTs;
   905         val vdiscss = map2 (map o rapp) vs discss;
   906         val vselsss = map2 (map o map o rapp) vs selsss;
   907 
   908         val ((coinduct_thms, coinduct_thm), (strong_coinduct_thms, strong_coinduct_thm)) =
   909           let
   910             val uvrs = map3 (fn r => fn u => fn v => r $ u $ v) rs us vs;
   911             val uv_eqs = map2 (curry HOLogic.mk_eq) us vs;
   912             val strong_rs =
   913               map4 (fn u => fn v => fn uvr => fn uv_eq =>
   914                 fold_rev Term.lambda [u, v] (HOLogic.mk_disj (uvr, uv_eq))) us vs uvrs uv_eqs;
   915 
   916             fun build_rel rs' T =
   917               (case find_index (curry (op =) T) fpTs of
   918                 ~1 =>
   919                 if exists_fp_subtype T then build_rel_step (build_rel rs') T else HOLogic.eq_const T
   920               | kk => nth rs' kk);
   921 
   922             fun build_rel_app rs' usel vsel =
   923               fold rapp [usel, vsel] (build_rel rs' (fastype_of usel));
   924 
   925             fun mk_prem_ctr_concls rs' n k udisc usels vdisc vsels =
   926               (if k = n then [] else [HOLogic.mk_eq (udisc, vdisc)]) @
   927               (if null usels then
   928                  []
   929                else
   930                  [Library.foldr HOLogic.mk_imp (if n = 1 then [] else [udisc, vdisc],
   931                     Library.foldr1 HOLogic.mk_conj (map2 (build_rel_app rs') usels vsels))]);
   932 
   933             fun mk_prem_concl rs' n udiscs uselss vdiscs vselss =
   934               Library.foldr1 HOLogic.mk_conj
   935                 (flat (map5 (mk_prem_ctr_concls rs' n) (1 upto n) udiscs uselss vdiscs vselss))
   936               handle List.Empty => @{term True};
   937 
   938             fun mk_prem rs' uvr u v n udiscs uselss vdiscs vselss =
   939               fold_rev Logic.all [u, v] (Logic.mk_implies (HOLogic.mk_Trueprop uvr,
   940                 HOLogic.mk_Trueprop (mk_prem_concl rs' n udiscs uselss vdiscs vselss)));
   941 
   942             val concl =
   943               HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj
   944                 (map3 (fn uvr => fn u => fn v => HOLogic.mk_imp (uvr, HOLogic.mk_eq (u, v)))
   945                    uvrs us vs));
   946 
   947             fun mk_goal rs' =
   948               Logic.list_implies (map8 (mk_prem rs') uvrs us vs ns udiscss uselsss vdiscss vselsss,
   949                 concl);
   950 
   951             val goal = mk_goal rs;
   952             val strong_goal = mk_goal strong_rs;
   953 
   954             fun prove dtor_coinduct' goal =
   955               Skip_Proof.prove lthy [] [] goal (fn {context = ctxt, ...} =>
   956                 mk_coinduct_tac ctxt nesting_rel_eqs nn ns dtor_coinduct' pre_rel_defs dtor_ctors
   957                   exhaust_thms ctr_defss disc_thmsss sel_thmsss)
   958               |> singleton (Proof_Context.export names_lthy lthy)
   959               |> Thm.close_derivation;
   960 
   961             fun postproc nn thm =
   962               Thm.permute_prems 0 nn
   963                 (if nn = 1 then thm RS mp
   964                  else funpow nn (fn thm => reassoc_conjs (thm RS mp_conj)) thm)
   965               |> Drule.zero_var_indexes
   966               |> `(conj_dests nn);
   967           in
   968             (postproc nn (prove fp_induct goal), postproc nn (prove fp_strong_induct strong_goal))
   969           end;
   970 
   971         fun mk_maybe_not pos = not pos ? HOLogic.mk_not;
   972 
   973         val gunfolds = map (lists_bmoc pgss) unfolds;
   974         val hcorecs = map (lists_bmoc phss) corecs;
   975 
   976         val (unfold_thmss, corec_thmss, safe_unfold_thmss, safe_corec_thmss) =
   977           let
   978             fun mk_goal pfss c cps fcorec_like n k ctr m cfs' =
   979               fold_rev (fold_rev Logic.all) ([c] :: pfss)
   980                 (Logic.list_implies (seq_conds (HOLogic.mk_Trueprop oo mk_maybe_not) n k cps,
   981                    mk_Trueprop_eq (fcorec_like $ c, Term.list_comb (ctr, take m cfs'))));
   982 
   983             fun build_corec_like fcorec_likes (T, U) =
   984               if T = U then
   985                 id_const T
   986               else
   987                 (case find_index (curry (op =) U) fpTs of
   988                   ~1 => build_map (build_corec_like fcorec_likes) T U
   989                 | kk => nth fcorec_likes kk);
   990 
   991             val mk_U = typ_subst (map2 pair Cs fpTs);
   992 
   993             fun intr_corec_likes fcorec_likes [] [cf] =
   994                 let val T = fastype_of cf in
   995                   if exists_Cs_subtype T then build_corec_like fcorec_likes (T, mk_U T) $ cf else cf
   996                 end
   997               | intr_corec_likes fcorec_likes [cq] [cf, cf'] =
   998                 mk_If cq (intr_corec_likes fcorec_likes [] [cf])
   999                   (intr_corec_likes fcorec_likes [] [cf']);
  1000 
  1001             val crgsss = map2 (map2 (map2 (intr_corec_likes gunfolds))) crssss cgssss;
  1002             val cshsss = map2 (map2 (map2 (intr_corec_likes hcorecs))) csssss chssss;
  1003 
  1004             val unfold_goalss =
  1005               map8 (map4 oooo mk_goal pgss) cs cpss gunfolds ns kss ctrss mss crgsss;
  1006             val corec_goalss =
  1007               map8 (map4 oooo mk_goal phss) cs cpss hcorecs ns kss ctrss mss cshsss;
  1008 
  1009             fun mk_map_if_distrib bnf =
  1010               let
  1011                 val mapx = map_of_bnf bnf;
  1012                 val live = live_of_bnf bnf;
  1013                 val ((Ts, T), U) = strip_typeN (live + 1) (fastype_of mapx) |>> split_last;
  1014                 val fs = Variable.variant_frees lthy [mapx] (map (pair "f") Ts);
  1015                 val t = Term.list_comb (mapx, map (Var o apfst (rpair 0)) fs);
  1016               in
  1017                 Drule.instantiate' (map (SOME o certifyT lthy) [U, T]) [SOME (certify lthy t)]
  1018                   @{thm if_distrib}
  1019               end;
  1020 
  1021             val nested_map_if_distribs = map mk_map_if_distrib nested_bnfs;
  1022 
  1023             val unfold_tacss =
  1024               map3 (map oo mk_corec_like_tac unfold_defs [] [] nesting_map_ids'' [])
  1025                 fp_fold_thms pre_map_defs ctr_defss;
  1026             val corec_tacss =
  1027               map3 (map oo mk_corec_like_tac corec_defs nested_map_comps'' nested_map_comp's
  1028                   (nested_map_ids'' @ nesting_map_ids'') nested_map_if_distribs)
  1029                 fp_rec_thms pre_map_defs ctr_defss;
  1030 
  1031             fun prove goal tac =
  1032               Skip_Proof.prove lthy [] [] goal (tac o #context) |> Thm.close_derivation;
  1033 
  1034             val unfold_thmss = map2 (map2 prove) unfold_goalss unfold_tacss;
  1035             val corec_thmss = map2 (map2 prove) corec_goalss corec_tacss;
  1036 
  1037             val filter_safesss =
  1038               map2 (map_filter (fn (safes, thm) => if forall I safes then SOME thm else NONE) oo
  1039                 curry (op ~~)) (map2 (map2 (map2 (member (op =)))) cgssss crgsss);
  1040 
  1041             val safe_unfold_thmss = filter_safesss unfold_thmss;
  1042             val safe_corec_thmss = filter_safesss corec_thmss;
  1043           in
  1044             (unfold_thmss, corec_thmss, safe_unfold_thmss, safe_corec_thmss)
  1045           end;
  1046 
  1047         val (disc_unfold_iff_thmss, disc_corec_iff_thmss) =
  1048           let
  1049             fun mk_goal c cps fcorec_like n k disc =
  1050               mk_Trueprop_eq (disc $ (fcorec_like $ c),
  1051                 if n = 1 then @{const True}
  1052                 else Library.foldr1 HOLogic.mk_conj (seq_conds mk_maybe_not n k cps));
  1053 
  1054             val unfold_goalss = map6 (map2 oooo mk_goal) cs cpss gunfolds ns kss discss;
  1055             val corec_goalss = map6 (map2 oooo mk_goal) cs cpss hcorecs ns kss discss;
  1056 
  1057             fun mk_case_split' cp =
  1058               Drule.instantiate' [] [SOME (certify lthy cp)] @{thm case_split};
  1059 
  1060             val case_splitss' = map (map mk_case_split') cpss;
  1061 
  1062             val unfold_tacss =
  1063               map3 (map oo mk_disc_corec_like_iff_tac) case_splitss' unfold_thmss disc_thmsss;
  1064             val corec_tacss =
  1065               map3 (map oo mk_disc_corec_like_iff_tac) case_splitss' corec_thmss disc_thmsss;
  1066 
  1067             fun prove goal tac =
  1068               Skip_Proof.prove lthy [] [] goal (tac o #context)
  1069               |> singleton (Proof_Context.export names_lthy0 no_defs_lthy)
  1070               |> Thm.close_derivation;
  1071 
  1072             fun proves [_] [_] = []
  1073               | proves goals tacs = map2 prove goals tacs;
  1074           in
  1075             (map2 proves unfold_goalss unfold_tacss,
  1076              map2 proves corec_goalss corec_tacss)
  1077           end;
  1078 
  1079         val is_triv_discI = is_triv_implies orf is_concl_refl;
  1080 
  1081         fun mk_disc_corec_like_thms corec_likes discIs =
  1082           map (op RS) (filter_out (is_triv_discI o snd) (corec_likes ~~ discIs));
  1083 
  1084         val disc_unfold_thmss = map2 mk_disc_corec_like_thms unfold_thmss discIss;
  1085         val disc_corec_thmss = map2 mk_disc_corec_like_thms corec_thmss discIss;
  1086 
  1087         fun mk_sel_corec_like_thm corec_like_thm sel sel_thm =
  1088           let
  1089             val (domT, ranT) = dest_funT (fastype_of sel);
  1090             val arg_cong' =
  1091               Drule.instantiate' (map (SOME o certifyT lthy) [domT, ranT])
  1092                 [NONE, NONE, SOME (certify lthy sel)] arg_cong
  1093               |> Thm.varifyT_global;
  1094             val sel_thm' = sel_thm RSN (2, trans);
  1095           in
  1096             corec_like_thm RS arg_cong' RS sel_thm'
  1097           end;
  1098 
  1099         fun mk_sel_corec_like_thms corec_likess =
  1100           map3 (map3 (map2 o mk_sel_corec_like_thm)) corec_likess selsss sel_thmsss |> map flat;
  1101 
  1102         val sel_unfold_thmss = mk_sel_corec_like_thms unfold_thmss;
  1103         val sel_corec_thmss = mk_sel_corec_like_thms corec_thmss;
  1104 
  1105         fun flat_corec_like_thms corec_likes disc_corec_likes sel_corec_likes =
  1106           corec_likes @ disc_corec_likes @ sel_corec_likes;
  1107 
  1108         val simp_thmss =
  1109           mk_simp_thmss wrap_ress
  1110             (map3 flat_corec_like_thms safe_corec_thmss disc_corec_thmss sel_corec_thmss)
  1111             (map3 flat_corec_like_thms safe_unfold_thmss disc_unfold_thmss sel_unfold_thmss);
  1112 
  1113         val anonymous_notes =
  1114           [(flat safe_unfold_thmss @ flat safe_corec_thmss, simp_attrs)]
  1115           |> map (fn (thms, attrs) => ((Binding.empty, attrs), [(thms, [])]));
  1116 
  1117         val common_notes =
  1118           (if nn > 1 then
  1119              (* FIXME: attribs *)
  1120              [(coinductN, [coinduct_thm], []),
  1121               (strong_coinductN, [strong_coinduct_thm], [])]
  1122            else
  1123              [])
  1124           |> map (fn (thmN, thms, attrs) =>
  1125             ((qualify true fp_common_name (Binding.name thmN), attrs), [(thms, [])]));
  1126 
  1127         val notes =
  1128           [(coinductN, map single coinduct_thms, []), (* FIXME: attribs *)
  1129            (corecN, corec_thmss, []),
  1130            (disc_corecN, disc_corec_thmss, simp_attrs),
  1131            (disc_corec_iffN, disc_corec_iff_thmss, simp_attrs),
  1132            (disc_unfoldN, disc_unfold_thmss, simp_attrs),
  1133            (disc_unfold_iffN, disc_unfold_iff_thmss, simp_attrs),
  1134            (sel_corecN, sel_corec_thmss, simp_attrs),
  1135            (sel_unfoldN, sel_unfold_thmss, simp_attrs),
  1136            (simpsN, simp_thmss, []),
  1137            (strong_coinductN, map single strong_coinduct_thms, []), (* FIXME: attribs *)
  1138            (unfoldN, unfold_thmss, [])]
  1139           |> maps (fn (thmN, thmss, attrs) =>
  1140             map_filter (fn (_, []) => NONE | (fp_b_name, thms) =>
  1141               SOME ((qualify true fp_b_name (Binding.name thmN), attrs),
  1142                 [(thms, [])])) (fp_b_names ~~ thmss));
  1143       in
  1144         lthy |> Local_Theory.notes (anonymous_notes @ common_notes @ notes) |> snd
  1145       end;
  1146 
  1147     val lthy' = lthy
  1148       |> fold_map define_ctrs_case_for_type (fp_bnfs ~~ fp_bs ~~ fpTs ~~ Cs ~~ ctors ~~ dtors ~~
  1149         fp_folds ~~ fp_recs ~~ ctor_dtors ~~ dtor_ctors ~~ ctor_injects ~~ pre_map_defs ~~
  1150         pre_set_defss ~~ pre_rel_defs ~~ fp_map_thms ~~ fp_set_thmss ~~ fp_rel_thms ~~ ns ~~ kss ~~
  1151         mss ~~ ctr_bindingss ~~ ctr_mixfixess ~~ ctr_Tsss ~~ disc_bindingss ~~ sel_bindingsss ~~
  1152         raw_sel_defaultsss)
  1153       |> wrap_types_and_more
  1154       |> (if lfp then derive_induct_fold_rec_thms_for_types
  1155           else derive_coinduct_unfold_corec_thms_for_types);
  1156 
  1157     val timer = time (timer ("Constructors, discriminators, selectors, etc., for the new " ^
  1158       (if lfp then "" else "co") ^ "datatype"));
  1159   in
  1160     timer; lthy'
  1161   end;
  1162 
  1163 val datatypes = define_datatypes (K I) (K I) (K I);
  1164 
  1165 val datatype_cmd = define_datatypes Typedecl.read_constraint Syntax.parse_typ Syntax.read_term;
  1166 
  1167 val parse_ctr_arg =
  1168   @{keyword "("} |-- parse_binding_colon -- Parse.typ --| @{keyword ")"} ||
  1169   (Parse.typ >> pair Binding.empty);
  1170 
  1171 val parse_defaults =
  1172   @{keyword "("} |-- @{keyword "defaults"} |-- Scan.repeat parse_bound_term --| @{keyword ")"};
  1173 
  1174 val parse_single_spec =
  1175   Parse.type_args_constrained -- Parse.binding -- Parse.opt_mixfix --
  1176   (@{keyword "="} |-- Parse.enum1 "|" (parse_opt_binding_colon -- Parse.binding --
  1177     Scan.repeat parse_ctr_arg -- Scan.optional parse_defaults [] -- Parse.opt_mixfix));
  1178 
  1179 val parse_datatype = parse_wrap_options -- Parse.and_list1 parse_single_spec;
  1180 
  1181 fun parse_datatype_cmd lfp construct_fp = parse_datatype >> datatype_cmd lfp construct_fp;
  1182 
  1183 end;