src/HOL/BNF/Tools/bnf_fp_def_sugar.ML
author blanchet
Wed Apr 24 13:16:20 2013 +0200 (2013-04-24)
changeset 51757 7babcb61aa5c
parent 51756 b0bae7bd236c
child 51758 55963309557b
permissions -rw-r--r--
honor user-specified set function names
     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 -> binding list list ->
    12       typ list * typ list list -> BNF_Def.BNF list -> local_theory ->
    13       BNF_FP.fp_result * local_theory) ->
    14     (bool * bool) * (((((binding * typ) * sort) list * binding) * mixfix) * ((((binding * binding) *
    15       (binding * typ) list) * (binding * term) list) * mixfix) list) list ->
    16     local_theory -> local_theory
    17   val parse_datatype_cmd: bool ->
    18     (mixfix list -> (string * sort) list option -> binding list -> binding list list ->
    19       typ list * typ list list -> BNF_Def.BNF list -> local_theory ->
    20       BNF_FP.fp_result * local_theory) ->
    21     (local_theory -> local_theory) parser
    22 end;
    23 
    24 structure BNF_FP_Def_Sugar : BNF_FP_DEF_SUGAR =
    25 struct
    26 
    27 open BNF_Util
    28 open BNF_Wrap
    29 open BNF_Def
    30 open BNF_FP
    31 open BNF_FP_Def_Sugar_Tactics
    32 
    33 (* This function could produce clashes in contrived examples (e.g., "x.A", "x.x_A", "y.A") *)
    34 fun quasi_unambiguous_case_names names =
    35   let
    36     val ps = map (`Long_Name.base_name) names;
    37     val dups = Library.duplicates (op =) (map fst ps);
    38     fun underscore s =
    39       let val ss = space_explode Long_Name.separator s in
    40         space_implode "_" (drop (length ss - 2) ss)
    41       end;
    42   in
    43     map (fn (base, full) => if member (op =) dups base then underscore full else base) ps
    44   end;
    45 
    46 val mp_conj = @{thm mp_conj};
    47 
    48 val simp_attrs = @{attributes [simp]};
    49 val code_simp_attrs = Code.add_default_eqn_attrib :: simp_attrs;
    50 
    51 fun split_list4 [] = ([], [], [], [])
    52   | split_list4 ((x1, x2, x3, x4) :: xs) =
    53     let val (xs1, xs2, xs3, xs4) = split_list4 xs;
    54     in (x1 :: xs1, x2 :: xs2, x3 :: xs3, x4 :: xs4) end;
    55 
    56 fun resort_tfree S (TFree (s, _)) = TFree (s, S);
    57 
    58 fun typ_subst inst (T as Type (s, Ts)) =
    59     (case AList.lookup (op =) inst T of
    60       NONE => Type (s, map (typ_subst inst) Ts)
    61     | SOME T' => T')
    62   | typ_subst inst T = the_default T (AList.lookup (op =) inst T);
    63 
    64 fun variant_types ss Ss ctxt =
    65   let
    66     val (tfrees, _) =
    67       fold_map2 (fn s => fn S => Name.variant s #> apfst (rpair S)) ss Ss (Variable.names_of ctxt);
    68     val ctxt' = fold (Variable.declare_constraints o Logic.mk_type o TFree) tfrees ctxt;
    69   in (tfrees, ctxt') end;
    70 
    71 val lists_bmoc = fold (fn xs => fn t => Term.list_comb (t, xs));
    72 
    73 fun mk_tupled_fun x f xs = HOLogic.tupled_lambda x (Term.list_comb (f, xs));
    74 fun mk_uncurried_fun f xs = mk_tupled_fun (HOLogic.mk_tuple xs) f xs;
    75 
    76 fun mk_flip (x, Type (_, [T1, Type (_, [T2, T3])])) =
    77   Abs ("x", T1, Abs ("y", T2, Var (x, T2 --> T1 --> T3) $ Bound 0 $ Bound 1));
    78 
    79 fun flip_rels lthy n thm =
    80   let
    81     val Rs = Term.add_vars (prop_of thm) [];
    82     val Rs' = rev (drop (length Rs - n) Rs);
    83     val cRs = map (fn f => (certify lthy (Var f), certify lthy (mk_flip f))) Rs';
    84   in
    85     Drule.cterm_instantiate cRs thm
    86   end;
    87 
    88 fun mk_ctor_or_dtor get_T Ts t =
    89   let val Type (_, Ts0) = get_T (fastype_of t) in
    90     Term.subst_atomic_types (Ts0 ~~ Ts) t
    91   end;
    92 
    93 val mk_ctor = mk_ctor_or_dtor range_type;
    94 val mk_dtor = mk_ctor_or_dtor domain_type;
    95 
    96 fun mk_rec_like lfp Ts Us t =
    97   let
    98     val (bindings, body) = strip_type (fastype_of t);
    99     val (f_Us, prebody) = split_last bindings;
   100     val Type (_, Ts0) = if lfp then prebody else body;
   101     val Us0 = distinct (op =) (map (if lfp then body_type else domain_type) f_Us);
   102   in
   103     Term.subst_atomic_types (Ts0 @ Us0 ~~ Ts @ Us) t
   104   end;
   105 
   106 fun mk_map live Ts Us t =
   107   let val (Type (_, Ts0), Type (_, Us0)) = strip_typeN (live + 1) (fastype_of t) |>> List.last in
   108     Term.subst_atomic_types (Ts0 @ Us0 ~~ Ts @ Us) t
   109   end;
   110 
   111 fun mk_rel live Ts Us t =
   112   let val [Type (_, Ts0), Type (_, Us0)] = binder_types (snd (strip_typeN live (fastype_of t))) in
   113     Term.subst_atomic_types (Ts0 @ Us0 ~~ Ts @ Us) t
   114   end;
   115 
   116 fun liveness_of_fp_bnf n bnf =
   117   (case T_of_bnf bnf of
   118     Type (_, Ts) => map (not o member (op =) (deads_of_bnf bnf)) Ts
   119   | _ => replicate n false);
   120 
   121 fun cannot_merge_types () = error "Mutually recursive types must have the same type parameters";
   122 
   123 fun merge_type_arg T T' = if T = T' then T else cannot_merge_types ();
   124 
   125 fun merge_type_args (As, As') =
   126   if length As = length As' then map2 merge_type_arg As As' else cannot_merge_types ();
   127 
   128 fun reassoc_conjs thm =
   129   reassoc_conjs (thm RS @{thm conj_assoc[THEN iffD1]})
   130   handle THM _ => thm;
   131 
   132 fun type_args_constrained_of (((cAs, _), _), _) = cAs;
   133 fun type_binding_of (((_, b), _), _) = b;
   134 fun mixfix_of ((_, mx), _) = mx;
   135 fun ctr_specs_of (_, ctr_specs) = ctr_specs;
   136 
   137 fun disc_of ((((disc, _), _), _), _) = disc;
   138 fun ctr_of ((((_, ctr), _), _), _) = ctr;
   139 fun args_of (((_, args), _), _) = args;
   140 fun defaults_of ((_, ds), _) = ds;
   141 fun ctr_mixfix_of (_, mx) = mx;
   142 
   143 fun define_datatypes prepare_constraint prepare_typ prepare_term lfp construct_fp
   144     (wrap_opts as (no_dests, rep_compat), specs) no_defs_lthy0 =
   145   let
   146     (* TODO: sanity checks on arguments *)
   147     (* TODO: integration with function package ("size") *)
   148 
   149     val _ = if not lfp andalso no_dests then error "Cannot define destructor-less codatatypes"
   150       else ();
   151 
   152     fun qualify mandatory fp_b_name =
   153       Binding.qualify mandatory fp_b_name o (rep_compat ? Binding.qualify false rep_compat_prefix);
   154 
   155     val nn = length specs;
   156     val fp_bs = map type_binding_of specs;
   157     val fp_b_names = map Binding.name_of fp_bs;
   158     val fp_common_name = mk_common_name fp_b_names;
   159 
   160     fun prepare_type_arg ((_, ty), c) =
   161       let val TFree (s, _) = prepare_typ no_defs_lthy0 ty in
   162         TFree (s, prepare_constraint no_defs_lthy0 c)
   163       end;
   164 
   165     val Ass0 = map (map prepare_type_arg o type_args_constrained_of) specs;
   166     val unsorted_Ass0 = map (map (resort_tfree HOLogic.typeS)) Ass0;
   167     val unsorted_As = Library.foldr1 merge_type_args unsorted_Ass0;
   168     val set_bss = map (map (fst o fst) o type_args_constrained_of) specs;
   169 
   170     val (((Bs0, Cs), Xs), no_defs_lthy) =
   171       no_defs_lthy0
   172       |> fold (Variable.declare_typ o resort_tfree dummyS) unsorted_As
   173       |> mk_TFrees (length unsorted_As)
   174       ||>> mk_TFrees nn
   175       ||>> apfst (map TFree) o
   176         variant_types (map (prefix "'") fp_b_names) (replicate nn HOLogic.typeS);
   177 
   178     (* TODO: cleaner handling of fake contexts, without "background_theory" *)
   179     (*the "perhaps o try" below helps gracefully handles the case where the new type is defined in a
   180       locale and shadows an existing global type*)
   181     val fake_thy =
   182       Theory.copy #> fold (fn spec => perhaps (try (Sign.add_type no_defs_lthy
   183         (type_binding_of spec, length (type_args_constrained_of spec), mixfix_of spec)))) specs;
   184     val fake_lthy = Proof_Context.background_theory fake_thy no_defs_lthy;
   185 
   186     fun mk_fake_T b =
   187       Type (fst (Term.dest_Type (Proof_Context.read_type_name fake_lthy true (Binding.name_of b))),
   188         unsorted_As);
   189 
   190     val fake_Ts = map mk_fake_T fp_bs;
   191 
   192     val mixfixes = map mixfix_of specs;
   193 
   194     val _ = (case duplicates Binding.eq_name fp_bs of [] => ()
   195       | b :: _ => error ("Duplicate type name declaration " ^ quote (Binding.name_of b)));
   196 
   197     val ctr_specss = map ctr_specs_of specs;
   198 
   199     val disc_bindingss = map (map disc_of) ctr_specss;
   200     val ctr_bindingss =
   201       map2 (fn fp_b_name => map (qualify false fp_b_name o ctr_of)) fp_b_names ctr_specss;
   202     val ctr_argsss = map (map args_of) ctr_specss;
   203     val ctr_mixfixess = map (map ctr_mixfix_of) ctr_specss;
   204 
   205     val sel_bindingsss = map (map (map fst)) ctr_argsss;
   206     val fake_ctr_Tsss0 = map (map (map (prepare_typ fake_lthy o snd))) ctr_argsss;
   207     val raw_sel_defaultsss = map (map defaults_of) ctr_specss;
   208 
   209     val (As :: _) :: fake_ctr_Tsss =
   210       burrow (burrow (Syntax.check_typs fake_lthy)) (Ass0 :: fake_ctr_Tsss0);
   211 
   212     val _ = (case duplicates (op =) unsorted_As of [] => ()
   213       | A :: _ => error ("Duplicate type parameter " ^
   214           quote (Syntax.string_of_typ no_defs_lthy A)));
   215 
   216     val rhs_As' = fold (fold (fold Term.add_tfreesT)) fake_ctr_Tsss [];
   217     val _ = (case subtract (op =) (map dest_TFree As) rhs_As' of
   218         [] => ()
   219       | A' :: _ => error ("Extra type variable on right-hand side: " ^
   220           quote (Syntax.string_of_typ no_defs_lthy (TFree A'))));
   221 
   222     fun eq_fpT_check (T as Type (s, Us)) (Type (s', Us')) =
   223         s = s' andalso (Us = Us' orelse error ("Illegal occurrence of recursive type " ^
   224           quote (Syntax.string_of_typ fake_lthy T)))
   225       | eq_fpT_check _ _ = false;
   226 
   227     fun freeze_fp (T as Type (s, Us)) =
   228         (case find_index (eq_fpT_check T) fake_Ts of
   229           ~1 => Type (s, map freeze_fp Us)
   230         | kk => nth Xs kk)
   231       | freeze_fp T = T;
   232 
   233     val ctr_TsssXs = map (map (map freeze_fp)) fake_ctr_Tsss;
   234     val ctr_sum_prod_TsXs = map (mk_sumTN_balanced o map HOLogic.mk_tupleT) ctr_TsssXs;
   235 
   236     val fp_eqs =
   237       map dest_TFree Xs ~~ map (Term.typ_subst_atomic (As ~~ unsorted_As)) ctr_sum_prod_TsXs;
   238 
   239     (* TODO: clean up list *)
   240     val (pre_bnfs, ((fp_bnfs as any_fp_bnf :: _, dtors0, ctors0, fp_folds0, fp_recs0, fp_induct,
   241            fp_strong_induct, dtor_ctors, ctor_dtors, ctor_injects, fp_map_thms, fp_set_thmss,
   242            fp_rel_thms, fp_fold_thms, fp_rec_thms), lthy)) =
   243       fp_bnf construct_fp fp_bs mixfixes set_bss (map dest_TFree unsorted_As) fp_eqs no_defs_lthy0;
   244 
   245     val timer = time (Timer.startRealTimer ());
   246 
   247     fun build_map build_arg (Type (s, Ts)) (Type (_, Us)) =
   248       let
   249         val bnf = the (bnf_of lthy s);
   250         val live = live_of_bnf bnf;
   251         val mapx = mk_map live Ts Us (map_of_bnf bnf);
   252         val TUs' = map dest_funT (fst (strip_typeN live (fastype_of mapx)));
   253       in Term.list_comb (mapx, map build_arg TUs') end;
   254 
   255     fun build_rel_step build_arg (Type (s, Ts)) =
   256       let
   257         val bnf = the (bnf_of lthy s);
   258         val live = live_of_bnf bnf;
   259         val rel = mk_rel live Ts Ts (rel_of_bnf bnf);
   260         val Ts' = map domain_type (fst (strip_typeN live (fastype_of rel)));
   261       in Term.list_comb (rel, map build_arg Ts') end;
   262 
   263     fun add_nesty_bnf_names Us =
   264       let
   265         fun add (Type (s, Ts)) ss =
   266             let val (needs, ss') = fold_map add Ts ss in
   267               if exists I needs then (true, insert (op =) s ss') else (false, ss')
   268             end
   269           | add T ss = (member (op =) Us T, ss);
   270       in snd oo add end;
   271 
   272     fun nesty_bnfs Us =
   273       map_filter (bnf_of lthy) (fold (fold (fold (add_nesty_bnf_names Us))) ctr_TsssXs []);
   274 
   275     val nesting_bnfs = nesty_bnfs As;
   276     val nested_bnfs = nesty_bnfs Xs;
   277 
   278     val pre_map_defs = map map_def_of_bnf pre_bnfs;
   279     val pre_set_defss = map set_defs_of_bnf pre_bnfs;
   280     val pre_rel_defs = map rel_def_of_bnf pre_bnfs;
   281     val nested_map_comps'' = map ((fn thm => thm RS sym) o map_comp_of_bnf) nested_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           (cs, cpss, unfold_only as ((pgss, crssss, cgssss), (_, g_Tsss, _)),
   318            corec_only as ((phss, csssss, chssss), (_, h_Tsss, _)))), 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_unzipT 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_Tsss = map2 (map2 dest_tupleT) mss' f_prod_Tss;
   369               val f_Tssss =
   370                 map2 (fn C => map (map (map (curry (op -->) C) o maybe_unzipT))) Cs f_Tsss;
   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_Tsss, f_Tssss, pf_Tss) end;
   375 
   376           val (r_Tssss, g_sum_prod_Ts, g_Tsss, g_Tssss, pg_Tss) = mk_types single fp_fold_fun_Ts;
   377 
   378           val (((cs, pss), gssss), lthy) =
   379             lthy
   380             |> mk_Frees "a" Cs
   381             ||>> mk_Freess "p" p_Tss
   382             ||>> mk_Freessss "g" g_Tssss;
   383           val rssss = map (map (map (fn [] => []))) r_Tssss;
   384 
   385           fun proj_corecT proj (Type (s as @{type_name sum}, Ts as [T, U])) =
   386               if member (op =) fpTs T then proj (T, U) else Type (s, map (proj_corecT proj) Ts)
   387             | proj_corecT proj (Type (s, Ts)) = Type (s, map (proj_corecT proj) Ts)
   388             | proj_corecT _ T = T;
   389 
   390           fun unzip_corecT T =
   391             if exists_fp_subtype T then [proj_corecT fst T, proj_corecT snd T] else [T];
   392 
   393           val (s_Tssss, h_sum_prod_Ts, h_Tsss, h_Tssss, ph_Tss) =
   394             mk_types unzip_corecT fp_rec_fun_Ts;
   395 
   396           val hssss_hd = map2 (map2 (map2 (fn T :: _ => fn [g] => retype_free T g))) h_Tssss gssss;
   397           val ((sssss, hssss_tl), lthy) =
   398             lthy
   399             |> mk_Freessss "q" s_Tssss
   400             ||>> mk_Freessss "h" (map (map (map tl)) h_Tssss);
   401           val hssss = map2 (map2 (map2 cons)) hssss_hd hssss_tl;
   402 
   403           val cpss = map2 (map o rapp) cs pss;
   404 
   405           fun mk_terms qssss fssss =
   406             let
   407               val pfss = map3 flat_preds_predsss_gettersss pss qssss fssss;
   408               val cqssss = map2 (map o map o map o rapp) cs qssss;
   409               val cfssss = map2 (map o map o map o rapp) cs fssss;
   410             in (pfss, cqssss, cfssss) end;
   411         in
   412           (((([], [], []), ([], [], [])),
   413             (cs, cpss, (mk_terms rssss gssss, (g_sum_prod_Ts, g_Tsss, pg_Tss)),
   414              (mk_terms sssss hssss, (h_sum_prod_Ts, h_Tsss, ph_Tss)))), lthy)
   415         end;
   416 
   417     fun define_ctrs_case_for_type (((((((((((((((((((((((((fp_bnf, fp_b), fpT), C), ctor), dtor),
   418             fp_fold), fp_rec), ctor_dtor), dtor_ctor), ctor_inject), pre_map_def), pre_set_defs),
   419           pre_rel_def), fp_map_thm), fp_set_thms), fp_rel_thm), n), ks), ms), ctr_bindings),
   420         ctr_mixfixes), ctr_Tss), disc_bindings), sel_bindingss), raw_sel_defaultss) no_defs_lthy =
   421       let
   422         val fp_b_name = Binding.name_of fp_b;
   423 
   424         val dtorT = domain_type (fastype_of ctor);
   425         val ctr_prod_Ts = map HOLogic.mk_tupleT ctr_Tss;
   426         val ctr_sum_prod_T = mk_sumTN_balanced ctr_prod_Ts;
   427         val case_Ts = map (fn Ts => Ts ---> C) ctr_Tss;
   428 
   429         val (((((w, fs), xss), yss), u'), names_lthy) =
   430           no_defs_lthy
   431           |> yield_singleton (mk_Frees "w") dtorT
   432           ||>> mk_Frees "f" case_Ts
   433           ||>> mk_Freess "x" ctr_Tss
   434           ||>> mk_Freess "y" (map (map B_ify) ctr_Tss)
   435           ||>> yield_singleton Variable.variant_fixes fp_b_name;
   436 
   437         val u = Free (u', fpT);
   438 
   439         val tuple_xs = map HOLogic.mk_tuple xss;
   440         val tuple_ys = map HOLogic.mk_tuple yss;
   441 
   442         val ctr_rhss =
   443           map3 (fn k => fn xs => fn tuple_x => fold_rev Term.lambda xs (ctor $
   444             mk_InN_balanced ctr_sum_prod_T n tuple_x k)) ks xss tuple_xs;
   445 
   446         val case_binding = qualify false fp_b_name (Binding.suffix_name ("_" ^ caseN) fp_b);
   447 
   448         val case_rhs =
   449           fold_rev Term.lambda (fs @ [u])
   450             (mk_sum_caseN_balanced (map2 mk_uncurried_fun fs xss) $ (dtor $ u));
   451 
   452         val ((raw_case :: raw_ctrs, raw_case_def :: raw_ctr_defs), (lthy', lthy)) = no_defs_lthy
   453           |> apfst split_list o fold_map3 (fn b => fn mx => fn rhs =>
   454               Local_Theory.define ((b, mx), ((Thm.def_binding b, []), rhs)) #>> apsnd snd)
   455             (case_binding :: ctr_bindings) (NoSyn :: ctr_mixfixes) (case_rhs :: ctr_rhss)
   456           ||> `Local_Theory.restore;
   457 
   458         val phi = Proof_Context.export_morphism lthy lthy';
   459 
   460         val ctr_defs = map (Morphism.thm phi) raw_ctr_defs;
   461         val ctr_defs' =
   462           map2 (fn m => fn def => mk_unabs_def m (def RS meta_eq_to_obj_eq)) ms ctr_defs;
   463         val case_def = Morphism.thm phi raw_case_def;
   464 
   465         val ctrs0 = map (Morphism.term phi) raw_ctrs;
   466         val casex0 = Morphism.term phi raw_case;
   467 
   468         val ctrs = map (mk_ctr As) ctrs0;
   469 
   470         fun wrap lthy =
   471           let
   472             fun exhaust_tac {context = ctxt, prems = _} =
   473               let
   474                 val ctor_iff_dtor_thm =
   475                   let
   476                     val goal =
   477                       fold_rev Logic.all [w, u]
   478                         (mk_Trueprop_eq (HOLogic.mk_eq (u, ctor $ w), HOLogic.mk_eq (dtor $ u, w)));
   479                   in
   480                     Goal.prove_sorry lthy [] [] goal (fn {context = ctxt, ...} =>
   481                       mk_ctor_iff_dtor_tac ctxt (map (SOME o certifyT lthy) [dtorT, fpT])
   482                         (certify lthy ctor) (certify lthy dtor) ctor_dtor dtor_ctor)
   483                     |> Thm.close_derivation
   484                     |> Morphism.thm phi
   485                   end;
   486 
   487                 val sumEN_thm' =
   488                   unfold_thms lthy @{thms all_unit_eq}
   489                     (Drule.instantiate' (map (SOME o certifyT lthy) ctr_prod_Ts) []
   490                        (mk_sumEN_balanced n))
   491                   |> Morphism.thm phi;
   492               in
   493                 mk_exhaust_tac ctxt n ctr_defs ctor_iff_dtor_thm sumEN_thm'
   494               end;
   495 
   496             val inject_tacss =
   497               map2 (fn 0 => K [] | _ => fn ctr_def => [fn {context = ctxt, ...} =>
   498                   mk_inject_tac ctxt ctr_def ctor_inject]) ms ctr_defs;
   499 
   500             val half_distinct_tacss =
   501               map (map (fn (def, def') => fn {context = ctxt, ...} =>
   502                 mk_half_distinct_tac ctxt ctor_inject [def, def'])) (mk_half_pairss (`I ctr_defs));
   503 
   504             val case_tacs =
   505               map3 (fn k => fn m => fn ctr_def => fn {context = ctxt, ...} =>
   506                 mk_case_tac ctxt n k m case_def ctr_def dtor_ctor) ks ms ctr_defs;
   507 
   508             val tacss = [exhaust_tac] :: inject_tacss @ half_distinct_tacss @ [case_tacs];
   509 
   510             val sel_defaultss = map (map (apsnd (prepare_term lthy))) raw_sel_defaultss
   511           in
   512             wrap_datatype tacss (((wrap_opts, ctrs0), casex0), (disc_bindings, (sel_bindingss,
   513               sel_defaultss))) lthy
   514           end;
   515 
   516         fun derive_maps_sets_rels (wrap_res, lthy) =
   517           let
   518             val rel_flip = rel_flip_of_bnf fp_bnf;
   519             val nones = replicate live NONE;
   520 
   521             val ctor_cong =
   522               if lfp then
   523                 Drule.dummy_thm
   524               else
   525                 let val ctor' = mk_ctor Bs ctor in
   526                   cterm_instantiate_pos [NONE, NONE, SOME (certify lthy ctor')] arg_cong
   527                 end;
   528 
   529             fun mk_cIn ify =
   530               certify lthy o (not lfp ? curry (op $) (map_types ify ctor)) oo
   531               mk_InN_balanced (ify ctr_sum_prod_T) n;
   532 
   533             val cxIns = map2 (mk_cIn I) tuple_xs ks;
   534             val cyIns = map2 (mk_cIn B_ify) tuple_ys ks;
   535 
   536             fun mk_map_thm ctr_def' cxIn =
   537               fold_thms lthy [ctr_def']
   538                 (unfold_thms lthy (pre_map_def ::
   539                      (if lfp then [] else [ctor_dtor, dtor_ctor]) @ sum_prod_thms_map)
   540                    (cterm_instantiate_pos (nones @ [SOME cxIn])
   541                       (if lfp then fp_map_thm else fp_map_thm RS ctor_cong)))
   542               |> singleton (Proof_Context.export names_lthy no_defs_lthy);
   543 
   544             fun mk_set_thm fp_set_thm ctr_def' cxIn =
   545               fold_thms lthy [ctr_def']
   546                 (unfold_thms lthy (pre_set_defs @ nested_set_natural's @ nesting_set_natural's @
   547                      (if lfp then [] else [dtor_ctor]) @ sum_prod_thms_set)
   548                    (cterm_instantiate_pos [SOME cxIn] fp_set_thm))
   549               |> singleton (Proof_Context.export names_lthy no_defs_lthy);
   550 
   551             fun mk_set_thms fp_set_thm = map2 (mk_set_thm fp_set_thm) ctr_defs' cxIns;
   552 
   553             val map_thms = map2 mk_map_thm ctr_defs' cxIns;
   554             val set_thmss = map mk_set_thms fp_set_thms;
   555 
   556             val rel_infos = (ctr_defs' ~~ cxIns, ctr_defs' ~~ cyIns);
   557 
   558             fun mk_rel_thm postproc ctr_defs' cxIn cyIn =
   559               fold_thms lthy ctr_defs'
   560                  (unfold_thms lthy (@{thm Inl_Inr_False} :: pre_rel_def ::
   561                       (if lfp then [] else [dtor_ctor]) @ sum_prod_thms_rel)
   562                     (cterm_instantiate_pos (nones @ [SOME cxIn, SOME cyIn]) fp_rel_thm))
   563               |> postproc
   564               |> singleton (Proof_Context.export names_lthy no_defs_lthy);
   565 
   566             fun mk_rel_inject_thm ((ctr_def', cxIn), (_, cyIn)) =
   567               mk_rel_thm (unfold_thms lthy @{thms eq_sym_Unity_conv}) [ctr_def'] cxIn cyIn;
   568 
   569             val rel_inject_thms = map mk_rel_inject_thm (op ~~ rel_infos);
   570 
   571             fun mk_half_rel_distinct_thm ((xctr_def', cxIn), (yctr_def', cyIn)) =
   572               mk_rel_thm (fn thm => thm RS @{thm eq_False[THEN iffD1]}) [xctr_def', yctr_def']
   573                 cxIn cyIn;
   574 
   575             fun mk_other_half_rel_distinct_thm thm =
   576               flip_rels lthy live thm RS (rel_flip RS sym RS @{thm arg_cong[of _ _ Not]} RS iffD2);
   577 
   578             val half_rel_distinct_thmss =
   579               map (map mk_half_rel_distinct_thm) (mk_half_pairss rel_infos);
   580             val other_half_rel_distinct_thmss =
   581               map (map mk_other_half_rel_distinct_thm) half_rel_distinct_thmss;
   582             val (rel_distinct_thms, _) =
   583               join_halves n half_rel_distinct_thmss other_half_rel_distinct_thmss;
   584 
   585             val notes =
   586               [(mapN, map_thms, code_simp_attrs),
   587                (rel_distinctN, rel_distinct_thms, code_simp_attrs),
   588                (rel_injectN, rel_inject_thms, code_simp_attrs),
   589                (setsN, flat set_thmss, code_simp_attrs)]
   590               |> filter_out (null o #2)
   591               |> map (fn (thmN, thms, attrs) =>
   592                 ((qualify true fp_b_name (Binding.name thmN), attrs), [(thms, [])]));
   593           in
   594             (wrap_res, lthy |> Local_Theory.notes notes |> snd)
   595           end;
   596 
   597         fun define_fold_rec no_defs_lthy =
   598           let
   599             val fpT_to_C = fpT --> C;
   600 
   601             fun build_prod_proj mk_proj (T, U) =
   602               if T = U then
   603                 id_const T
   604               else
   605                 (case (T, U) of
   606                   (Type (s, _), Type (s', _)) =>
   607                   if s = s' then build_map (build_prod_proj mk_proj) T U else mk_proj T
   608                 | _ => mk_proj T);
   609 
   610             (* TODO: Avoid these complications; cf. corec case *)
   611             fun mk_U proj (Type (s as @{type_name prod}, Ts as [T', U])) =
   612                 if member (op =) fpTs T' then proj (T', U) else Type (s, map (mk_U proj) Ts)
   613               | mk_U proj (Type (s, Ts)) = Type (s, map (mk_U proj) Ts)
   614               | mk_U _ T = T;
   615 
   616             fun unzip_rec (x as Free (_, T)) =
   617               if exists_fp_subtype T then
   618                 [build_prod_proj fst_const (T, mk_U fst T) $ x,
   619                  build_prod_proj snd_const (T, mk_U snd T) $ x]
   620               else
   621                 [x];
   622 
   623             fun mk_rec_like_arg f xs = mk_tupled_fun (HOLogic.mk_tuple xs) f (maps unzip_rec xs);
   624 
   625             fun generate_rec_like (suf, fp_rec_like, (fss, f_Tss, xsss)) =
   626               let
   627                 val res_T = fold_rev (curry (op --->)) f_Tss fpT_to_C;
   628                 val binding = qualify false fp_b_name (Binding.suffix_name ("_" ^ suf) fp_b);
   629                 val spec =
   630                   mk_Trueprop_eq (lists_bmoc fss (Free (Binding.name_of binding, res_T)),
   631                     Term.list_comb (fp_rec_like,
   632                       map2 (mk_sum_caseN_balanced oo map2 mk_rec_like_arg) fss xsss));
   633               in (binding, spec) end;
   634 
   635             val rec_like_infos =
   636               [(foldN, fp_fold, fold_only),
   637                (recN, fp_rec, rec_only)];
   638 
   639             val (bindings, specs) = map generate_rec_like rec_like_infos |> split_list;
   640 
   641             val ((csts, defs), (lthy', lthy)) = no_defs_lthy
   642               |> apfst split_list o fold_map2 (fn b => fn spec =>
   643                 Specification.definition (SOME (b, NONE, NoSyn), ((Thm.def_binding b, []), spec))
   644                 #>> apsnd snd) bindings specs
   645               ||> `Local_Theory.restore;
   646 
   647             val phi = Proof_Context.export_morphism lthy lthy';
   648 
   649             val [fold_def, rec_def] = map (Morphism.thm phi) defs;
   650 
   651             val [foldx, recx] = map (mk_rec_like lfp As Cs o Morphism.term phi) csts;
   652           in
   653             ((foldx, recx, fold_def, rec_def), lthy')
   654           end;
   655 
   656         fun define_unfold_corec no_defs_lthy =
   657           let
   658             val B_to_fpT = C --> fpT;
   659 
   660             fun build_sum_inj mk_inj (T, U) =
   661               if T = U then
   662                 id_const T
   663               else
   664                 (case (T, U) of
   665                   (Type (s, _), Type (s', _)) =>
   666                   if s = s' then build_map (build_sum_inj mk_inj) T U
   667                   else uncurry mk_inj (dest_sumT U)
   668                 | _ => uncurry mk_inj (dest_sumT U));
   669 
   670             fun build_dtor_corec_like_arg _ [] [cf] = cf
   671               | build_dtor_corec_like_arg T [cq] [cf, cf'] =
   672                 mk_If cq (build_sum_inj Inl_const (fastype_of cf, T) $ cf)
   673                   (build_sum_inj Inr_const (fastype_of cf', T) $ cf')
   674 
   675             val crgsss = map3 (map3 (map3 build_dtor_corec_like_arg)) g_Tsss crssss cgssss;
   676             val cshsss = map3 (map3 (map3 build_dtor_corec_like_arg)) h_Tsss csssss chssss;
   677 
   678             fun mk_preds_getterss_join c n cps sum_prod_T cqfss =
   679               Term.lambda c (mk_IfN sum_prod_T cps
   680                 (map2 (mk_InN_balanced sum_prod_T n) (map HOLogic.mk_tuple cqfss) (1 upto n)));
   681 
   682             fun generate_corec_like (suf, fp_rec_like, (cqfsss, ((pfss, _, _), (f_sum_prod_Ts, _,
   683                 pf_Tss)))) =
   684               let
   685                 val res_T = fold_rev (curry (op --->)) pf_Tss B_to_fpT;
   686                 val binding = qualify false fp_b_name (Binding.suffix_name ("_" ^ suf) fp_b);
   687                 val spec =
   688                   mk_Trueprop_eq (lists_bmoc pfss (Free (Binding.name_of binding, res_T)),
   689                     Term.list_comb (fp_rec_like,
   690                       map5 mk_preds_getterss_join cs ns cpss f_sum_prod_Ts cqfsss));
   691               in (binding, spec) end;
   692 
   693             val corec_like_infos =
   694               [(unfoldN, fp_fold, (crgsss, unfold_only)),
   695                (corecN, fp_rec, (cshsss, corec_only))];
   696 
   697             val (bindings, specs) = map generate_corec_like corec_like_infos |> split_list;
   698 
   699             val ((csts, defs), (lthy', lthy)) = no_defs_lthy
   700               |> apfst split_list o fold_map2 (fn b => fn spec =>
   701                 Specification.definition (SOME (b, NONE, NoSyn), ((Thm.def_binding b, []), spec))
   702                 #>> apsnd snd) bindings specs
   703               ||> `Local_Theory.restore;
   704 
   705             val phi = Proof_Context.export_morphism lthy lthy';
   706 
   707             val [unfold_def, corec_def] = map (Morphism.thm phi) defs;
   708 
   709             val [unfold, corec] = map (mk_rec_like lfp As Cs o Morphism.term phi) csts;
   710           in
   711             ((unfold, corec, unfold_def, corec_def), lthy')
   712           end;
   713 
   714         val define_rec_likes = if lfp then define_fold_rec else define_unfold_corec;
   715 
   716         fun massage_res ((wrap_res, rec_like_res), lthy) =
   717           (((ctrs, xss, ctr_defs, wrap_res), rec_like_res), lthy);
   718       in
   719         (wrap #> (live > 0 ? derive_maps_sets_rels) ##>> define_rec_likes #> massage_res, lthy')
   720       end;
   721 
   722     fun wrap_types_and_more (wrap_types_and_mores, lthy) =
   723       fold_map I wrap_types_and_mores lthy
   724       |>> apsnd split_list4 o apfst split_list4 o split_list;
   725 
   726     (* TODO: Add map, sets, rel simps *)
   727     val mk_simp_thmss =
   728       map3 (fn (_, _, _, injects, distincts, cases, _, _, _) => fn rec_likes => fn fold_likes =>
   729         injects @ distincts @ cases @ rec_likes @ fold_likes);
   730 
   731     fun derive_induct_fold_rec_thms_for_types (((ctrss, xsss, ctr_defss, wrap_ress), (folds, recs,
   732         fold_defs, rec_defs)), lthy) =
   733       let
   734         val (((ps, ps'), us'), names_lthy) =
   735           lthy
   736           |> mk_Frees' "P" (map mk_pred1T fpTs)
   737           ||>> Variable.variant_fixes fp_b_names;
   738 
   739         val us = map2 (curry Free) us' fpTs;
   740 
   741         fun mk_sets_nested bnf =
   742           let
   743             val Type (T_name, Us) = T_of_bnf bnf;
   744             val lives = lives_of_bnf bnf;
   745             val sets = sets_of_bnf bnf;
   746             fun mk_set U =
   747               (case find_index (curry (op =) U) lives of
   748                 ~1 => Term.dummy
   749               | i => nth sets i);
   750           in
   751             (T_name, map mk_set Us)
   752           end;
   753 
   754         val setss_nested = map mk_sets_nested nested_bnfs;
   755 
   756         val (induct_thms, induct_thm) =
   757           let
   758             fun mk_set Ts t =
   759               let val Type (_, Ts0) = domain_type (fastype_of t) in
   760                 Term.subst_atomic_types (Ts0 ~~ Ts) t
   761               end;
   762 
   763             fun mk_raw_prem_prems names_lthy (x as Free (s, T as Type (T_name, Ts0))) =
   764                 (case find_index (curry (op =) T) fpTs of
   765                   ~1 =>
   766                   (case AList.lookup (op =) setss_nested T_name of
   767                     NONE => []
   768                   | SOME raw_sets0 =>
   769                     let
   770                       val (Ts, raw_sets) =
   771                         split_list (filter (exists_fp_subtype o fst) (Ts0 ~~ raw_sets0));
   772                       val sets = map (mk_set Ts0) raw_sets;
   773                       val (ys, names_lthy') = names_lthy |> mk_Frees s Ts;
   774                       val xysets = map (pair x) (ys ~~ sets);
   775                       val ppremss = map (mk_raw_prem_prems names_lthy') ys;
   776                     in
   777                       flat (map2 (map o apfst o cons) xysets ppremss)
   778                     end)
   779                 | kk => [([], (kk + 1, x))])
   780               | mk_raw_prem_prems _ _ = [];
   781 
   782             fun close_prem_prem xs t =
   783               fold_rev Logic.all (map Free (drop (nn + length xs)
   784                 (rev (Term.add_frees t (map dest_Free xs @ ps'))))) t;
   785 
   786             fun mk_prem_prem xs (xysets, (j, x)) =
   787               close_prem_prem xs (Logic.list_implies (map (fn (x', (y, set)) =>
   788                   HOLogic.mk_Trueprop (HOLogic.mk_mem (y, set $ x'))) xysets,
   789                 HOLogic.mk_Trueprop (nth ps (j - 1) $ x)));
   790 
   791             fun mk_raw_prem phi ctr ctr_Ts =
   792               let
   793                 val (xs, names_lthy') = names_lthy |> mk_Frees "x" ctr_Ts;
   794                 val pprems = maps (mk_raw_prem_prems names_lthy') xs;
   795               in (xs, pprems, HOLogic.mk_Trueprop (phi $ Term.list_comb (ctr, xs))) end;
   796 
   797             fun mk_prem (xs, raw_pprems, concl) =
   798               fold_rev Logic.all xs (Logic.list_implies (map (mk_prem_prem xs) raw_pprems, concl));
   799 
   800             val raw_premss = map3 (map2 o mk_raw_prem) ps ctrss ctr_Tsss;
   801 
   802             val goal =
   803               Library.foldr (Logic.list_implies o apfst (map mk_prem)) (raw_premss,
   804                 HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj (map2 (curry (op $)) ps us)));
   805 
   806             val kksss = map (map (map (fst o snd) o #2)) raw_premss;
   807 
   808             val ctor_induct' = fp_induct OF (map mk_sumEN_tupled_balanced mss);
   809 
   810             val thm =
   811               Goal.prove_sorry lthy [] [] goal (fn {context = ctxt, ...} =>
   812                 mk_induct_tac ctxt nn ns mss kksss (flat ctr_defss) ctor_induct'
   813                   nested_set_natural's pre_set_defss)
   814               |> singleton (Proof_Context.export names_lthy lthy)
   815               |> Thm.close_derivation;
   816           in
   817             `(conj_dests nn) thm
   818           end;
   819 
   820         val induct_cases = quasi_unambiguous_case_names (maps (map name_of_ctr) ctrss);
   821 
   822         val (fold_thmss, rec_thmss) =
   823           let
   824             val xctrss = map2 (map2 (curry Term.list_comb)) ctrss xsss;
   825             val gfolds = map (lists_bmoc gss) folds;
   826             val hrecs = map (lists_bmoc hss) recs;
   827 
   828             fun mk_goal fss frec_like xctr f xs fxs =
   829               fold_rev (fold_rev Logic.all) (xs :: fss)
   830                 (mk_Trueprop_eq (frec_like $ xctr, Term.list_comb (f, fxs)));
   831 
   832             fun build_rec_like frec_likes (T, U) =
   833               if T = U then
   834                 id_const T
   835               else
   836                 (case find_index (curry (op =) T) fpTs of
   837                   ~1 => build_map (build_rec_like frec_likes) T U
   838                 | kk => nth frec_likes kk);
   839 
   840             val mk_U = typ_subst (map2 pair fpTs Cs);
   841 
   842             fun intr_rec_likes frec_likes maybe_cons (x as Free (_, T)) =
   843               if exists_fp_subtype T then
   844                 maybe_cons x [build_rec_like frec_likes (T, mk_U T) $ x]
   845               else
   846                 [x];
   847 
   848             val gxsss = map (map (maps (intr_rec_likes gfolds (K I)))) xsss;
   849             val hxsss = map (map (maps (intr_rec_likes hrecs cons))) xsss;
   850 
   851             val fold_goalss = map5 (map4 o mk_goal gss) gfolds xctrss gss xsss gxsss;
   852             val rec_goalss = map5 (map4 o mk_goal hss) hrecs xctrss hss xsss hxsss;
   853 
   854             val fold_tacss =
   855               map2 (map o mk_rec_like_tac pre_map_defs [] nesting_map_ids'' fold_defs) fp_fold_thms
   856                 ctr_defss;
   857             val rec_tacss =
   858               map2 (map o mk_rec_like_tac pre_map_defs nested_map_comp's
   859                 (nested_map_ids'' @ nesting_map_ids'') rec_defs) fp_rec_thms ctr_defss;
   860 
   861             fun prove goal tac =
   862               Goal.prove_sorry lthy [] [] goal (tac o #context)
   863               |> Thm.close_derivation;
   864           in
   865             (map2 (map2 prove) fold_goalss fold_tacss, map2 (map2 prove) rec_goalss rec_tacss)
   866           end;
   867 
   868         val simp_thmss = mk_simp_thmss wrap_ress rec_thmss fold_thmss;
   869 
   870         val induct_case_names_attr = Attrib.internal (K (Rule_Cases.case_names induct_cases));
   871         fun induct_type_attr T_name = Attrib.internal (K (Induct.induct_type T_name));
   872 
   873         val common_notes =
   874           (if nn > 1 then [(inductN, [induct_thm], [induct_case_names_attr])] else [])
   875           |> map (fn (thmN, thms, attrs) =>
   876             ((qualify true fp_common_name (Binding.name thmN), attrs), [(thms, [])]));
   877 
   878         val notes =
   879           [(foldN, fold_thmss, K code_simp_attrs),
   880            (inductN, map single induct_thms,
   881             fn T_name => [induct_case_names_attr, induct_type_attr T_name]),
   882            (recN, rec_thmss, K code_simp_attrs),
   883            (simpsN, simp_thmss, K [])]
   884           |> maps (fn (thmN, thmss, attrs) =>
   885             map3 (fn fp_b_name => fn Type (T_name, _) => fn thms =>
   886               ((qualify true fp_b_name (Binding.name thmN), attrs T_name),
   887                [(thms, [])])) fp_b_names fpTs thmss);
   888       in
   889         lthy |> Local_Theory.notes (common_notes @ notes) |> snd
   890       end;
   891 
   892     fun derive_coinduct_unfold_corec_thms_for_types (((ctrss, _, ctr_defss, wrap_ress), (unfolds,
   893         corecs, unfold_defs, corec_defs)), lthy) =
   894       let
   895         val nesting_rel_eqs = map rel_eq_of_bnf nesting_bnfs;
   896 
   897         val discss = map (map (mk_disc_or_sel As) o #1) wrap_ress;
   898         val selsss = map (map (map (mk_disc_or_sel As)) o #2) wrap_ress;
   899         val exhaust_thms = map #3 wrap_ress;
   900         val disc_thmsss = map #7 wrap_ress;
   901         val discIss = map #8 wrap_ress;
   902         val sel_thmsss = map #9 wrap_ress;
   903 
   904         val (((rs, us'), vs'), names_lthy) =
   905           lthy
   906           |> mk_Frees "R" (map (fn T => mk_pred2T T T) fpTs)
   907           ||>> Variable.variant_fixes fp_b_names
   908           ||>> Variable.variant_fixes (map (suffix "'") fp_b_names);
   909 
   910         val us = map2 (curry Free) us' fpTs;
   911         val udiscss = map2 (map o rapp) us discss;
   912         val uselsss = map2 (map o map o rapp) us selsss;
   913 
   914         val vs = map2 (curry Free) vs' fpTs;
   915         val vdiscss = map2 (map o rapp) vs discss;
   916         val vselsss = map2 (map o map o rapp) vs selsss;
   917 
   918         val ((coinduct_thms, coinduct_thm), (strong_coinduct_thms, strong_coinduct_thm)) =
   919           let
   920             val uvrs = map3 (fn r => fn u => fn v => r $ u $ v) rs us vs;
   921             val uv_eqs = map2 (curry HOLogic.mk_eq) us vs;
   922             val strong_rs =
   923               map4 (fn u => fn v => fn uvr => fn uv_eq =>
   924                 fold_rev Term.lambda [u, v] (HOLogic.mk_disj (uvr, uv_eq))) us vs uvrs uv_eqs;
   925 
   926             fun build_rel rs' T =
   927               (case find_index (curry (op =) T) fpTs of
   928                 ~1 =>
   929                 if exists_fp_subtype T then build_rel_step (build_rel rs') T else HOLogic.eq_const T
   930               | kk => nth rs' kk);
   931 
   932             fun build_rel_app rs' usel vsel =
   933               fold rapp [usel, vsel] (build_rel rs' (fastype_of usel));
   934 
   935             fun mk_prem_ctr_concls rs' n k udisc usels vdisc vsels =
   936               (if k = n then [] else [HOLogic.mk_eq (udisc, vdisc)]) @
   937               (if null usels then
   938                  []
   939                else
   940                  [Library.foldr HOLogic.mk_imp (if n = 1 then [] else [udisc, vdisc],
   941                     Library.foldr1 HOLogic.mk_conj (map2 (build_rel_app rs') usels vsels))]);
   942 
   943             fun mk_prem_concl rs' n udiscs uselss vdiscs vselss =
   944               Library.foldr1 HOLogic.mk_conj
   945                 (flat (map5 (mk_prem_ctr_concls rs' n) (1 upto n) udiscs uselss vdiscs vselss))
   946               handle List.Empty => @{term True};
   947 
   948             fun mk_prem rs' uvr u v n udiscs uselss vdiscs vselss =
   949               fold_rev Logic.all [u, v] (Logic.mk_implies (HOLogic.mk_Trueprop uvr,
   950                 HOLogic.mk_Trueprop (mk_prem_concl rs' n udiscs uselss vdiscs vselss)));
   951 
   952             val concl =
   953               HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj
   954                 (map3 (fn uvr => fn u => fn v => HOLogic.mk_imp (uvr, HOLogic.mk_eq (u, v)))
   955                    uvrs us vs));
   956 
   957             fun mk_goal rs' =
   958               Logic.list_implies (map8 (mk_prem rs') uvrs us vs ns udiscss uselsss vdiscss vselsss,
   959                 concl);
   960 
   961             val goal = mk_goal rs;
   962             val strong_goal = mk_goal strong_rs;
   963 
   964             fun prove dtor_coinduct' goal =
   965               Goal.prove_sorry lthy [] [] goal (fn {context = ctxt, ...} =>
   966                 mk_coinduct_tac ctxt nesting_rel_eqs nn ns dtor_coinduct' pre_rel_defs dtor_ctors
   967                   exhaust_thms ctr_defss disc_thmsss sel_thmsss)
   968               |> singleton (Proof_Context.export names_lthy lthy)
   969               |> Thm.close_derivation;
   970 
   971             fun postproc nn thm =
   972               Thm.permute_prems 0 nn
   973                 (if nn = 1 then thm RS mp
   974                  else funpow nn (fn thm => reassoc_conjs (thm RS mp_conj)) thm)
   975               |> Drule.zero_var_indexes
   976               |> `(conj_dests nn);
   977           in
   978             (postproc nn (prove fp_induct goal), postproc nn (prove fp_strong_induct strong_goal))
   979           end;
   980 
   981         fun mk_maybe_not pos = not pos ? HOLogic.mk_not;
   982 
   983         val gunfolds = map (lists_bmoc pgss) unfolds;
   984         val hcorecs = map (lists_bmoc phss) corecs;
   985 
   986         val (unfold_thmss, corec_thmss, safe_unfold_thmss, safe_corec_thmss) =
   987           let
   988             fun mk_goal pfss c cps fcorec_like n k ctr m cfs' =
   989               fold_rev (fold_rev Logic.all) ([c] :: pfss)
   990                 (Logic.list_implies (seq_conds (HOLogic.mk_Trueprop oo mk_maybe_not) n k cps,
   991                    mk_Trueprop_eq (fcorec_like $ c, Term.list_comb (ctr, take m cfs'))));
   992 
   993             fun build_corec_like fcorec_likes (T, U) =
   994               if T = U then
   995                 id_const T
   996               else
   997                 (case find_index (curry (op =) U) fpTs of
   998                   ~1 => build_map (build_corec_like fcorec_likes) T U
   999                 | kk => nth fcorec_likes kk);
  1000 
  1001             val mk_U = typ_subst (map2 pair Cs fpTs);
  1002 
  1003             fun intr_corec_likes fcorec_likes [] [cf] =
  1004                 let val T = fastype_of cf in
  1005                   if exists_Cs_subtype T then build_corec_like fcorec_likes (T, mk_U T) $ cf else cf
  1006                 end
  1007               | intr_corec_likes fcorec_likes [cq] [cf, cf'] =
  1008                 mk_If cq (intr_corec_likes fcorec_likes [] [cf])
  1009                   (intr_corec_likes fcorec_likes [] [cf']);
  1010 
  1011             val crgsss = map2 (map2 (map2 (intr_corec_likes gunfolds))) crssss cgssss;
  1012             val cshsss = map2 (map2 (map2 (intr_corec_likes hcorecs))) csssss chssss;
  1013 
  1014             val unfold_goalss =
  1015               map8 (map4 oooo mk_goal pgss) cs cpss gunfolds ns kss ctrss mss crgsss;
  1016             val corec_goalss =
  1017               map8 (map4 oooo mk_goal phss) cs cpss hcorecs ns kss ctrss mss cshsss;
  1018 
  1019             fun mk_map_if_distrib bnf =
  1020               let
  1021                 val mapx = map_of_bnf bnf;
  1022                 val live = live_of_bnf bnf;
  1023                 val ((Ts, T), U) = strip_typeN (live + 1) (fastype_of mapx) |>> split_last;
  1024                 val fs = Variable.variant_frees lthy [mapx] (map (pair "f") Ts);
  1025                 val t = Term.list_comb (mapx, map (Var o apfst (rpair 0)) fs);
  1026               in
  1027                 Drule.instantiate' (map (SOME o certifyT lthy) [U, T]) [SOME (certify lthy t)]
  1028                   @{thm if_distrib}
  1029               end;
  1030 
  1031             val nested_map_if_distribs = map mk_map_if_distrib nested_bnfs;
  1032 
  1033             val unfold_tacss =
  1034               map3 (map oo mk_corec_like_tac unfold_defs [] [] nesting_map_ids'' [])
  1035                 fp_fold_thms pre_map_defs ctr_defss;
  1036             val corec_tacss =
  1037               map3 (map oo mk_corec_like_tac corec_defs nested_map_comps'' nested_map_comp's
  1038                   (nested_map_ids'' @ nesting_map_ids'') nested_map_if_distribs)
  1039                 fp_rec_thms pre_map_defs ctr_defss;
  1040 
  1041             fun prove goal tac =
  1042               Goal.prove_sorry lthy [] [] goal (tac o #context) |> Thm.close_derivation;
  1043 
  1044             val unfold_thmss = map2 (map2 prove) unfold_goalss unfold_tacss;
  1045             val corec_thmss = map2 (map2 prove) corec_goalss corec_tacss;
  1046 
  1047             val filter_safesss =
  1048               map2 (map_filter (fn (safes, thm) => if forall I safes then SOME thm else NONE) oo
  1049                 curry (op ~~)) (map2 (map2 (map2 (member (op =)))) cgssss crgsss);
  1050 
  1051             val safe_unfold_thmss = filter_safesss unfold_thmss;
  1052             val safe_corec_thmss = filter_safesss corec_thmss;
  1053           in
  1054             (unfold_thmss, corec_thmss, safe_unfold_thmss, safe_corec_thmss)
  1055           end;
  1056 
  1057         val (disc_unfold_iff_thmss, disc_corec_iff_thmss) =
  1058           let
  1059             fun mk_goal c cps fcorec_like n k disc =
  1060               mk_Trueprop_eq (disc $ (fcorec_like $ c),
  1061                 if n = 1 then @{const True}
  1062                 else Library.foldr1 HOLogic.mk_conj (seq_conds mk_maybe_not n k cps));
  1063 
  1064             val unfold_goalss = map6 (map2 oooo mk_goal) cs cpss gunfolds ns kss discss;
  1065             val corec_goalss = map6 (map2 oooo mk_goal) cs cpss hcorecs ns kss discss;
  1066 
  1067             fun mk_case_split' cp =
  1068               Drule.instantiate' [] [SOME (certify lthy cp)] @{thm case_split};
  1069 
  1070             val case_splitss' = map (map mk_case_split') cpss;
  1071 
  1072             val unfold_tacss =
  1073               map3 (map oo mk_disc_corec_like_iff_tac) case_splitss' unfold_thmss disc_thmsss;
  1074             val corec_tacss =
  1075               map3 (map oo mk_disc_corec_like_iff_tac) case_splitss' corec_thmss disc_thmsss;
  1076 
  1077             fun prove goal tac =
  1078               Goal.prove_sorry lthy [] [] goal (tac o #context)
  1079               |> singleton (Proof_Context.export names_lthy0 no_defs_lthy)
  1080               |> Thm.close_derivation;
  1081 
  1082             fun proves [_] [_] = []
  1083               | proves goals tacs = map2 prove goals tacs;
  1084           in
  1085             (map2 proves unfold_goalss unfold_tacss,
  1086              map2 proves corec_goalss corec_tacss)
  1087           end;
  1088 
  1089         val is_triv_discI = is_triv_implies orf is_concl_refl;
  1090 
  1091         fun mk_disc_corec_like_thms corec_likes discIs =
  1092           map (op RS) (filter_out (is_triv_discI o snd) (corec_likes ~~ discIs));
  1093 
  1094         val disc_unfold_thmss = map2 mk_disc_corec_like_thms unfold_thmss discIss;
  1095         val disc_corec_thmss = map2 mk_disc_corec_like_thms corec_thmss discIss;
  1096 
  1097         fun mk_sel_corec_like_thm corec_like_thm sel sel_thm =
  1098           let
  1099             val (domT, ranT) = dest_funT (fastype_of sel);
  1100             val arg_cong' =
  1101               Drule.instantiate' (map (SOME o certifyT lthy) [domT, ranT])
  1102                 [NONE, NONE, SOME (certify lthy sel)] arg_cong
  1103               |> Thm.varifyT_global;
  1104             val sel_thm' = sel_thm RSN (2, trans);
  1105           in
  1106             corec_like_thm RS arg_cong' RS sel_thm'
  1107           end;
  1108 
  1109         fun mk_sel_corec_like_thms corec_likess =
  1110           map3 (map3 (map2 o mk_sel_corec_like_thm)) corec_likess selsss sel_thmsss |> map flat;
  1111 
  1112         val sel_unfold_thmss = mk_sel_corec_like_thms unfold_thmss;
  1113         val sel_corec_thmss = mk_sel_corec_like_thms corec_thmss;
  1114 
  1115         fun flat_corec_like_thms corec_likes disc_corec_likes sel_corec_likes =
  1116           corec_likes @ disc_corec_likes @ sel_corec_likes;
  1117 
  1118         val simp_thmss =
  1119           mk_simp_thmss wrap_ress
  1120             (map3 flat_corec_like_thms safe_corec_thmss disc_corec_thmss sel_corec_thmss)
  1121             (map3 flat_corec_like_thms safe_unfold_thmss disc_unfold_thmss sel_unfold_thmss);
  1122 
  1123         val anonymous_notes =
  1124           [(flat safe_unfold_thmss @ flat safe_corec_thmss, simp_attrs)]
  1125           |> map (fn (thms, attrs) => ((Binding.empty, attrs), [(thms, [])]));
  1126 
  1127         val common_notes =
  1128           (if nn > 1 then
  1129              (* FIXME: attribs *)
  1130              [(coinductN, [coinduct_thm], []),
  1131               (strong_coinductN, [strong_coinduct_thm], [])]
  1132            else
  1133              [])
  1134           |> map (fn (thmN, thms, attrs) =>
  1135             ((qualify true fp_common_name (Binding.name thmN), attrs), [(thms, [])]));
  1136 
  1137         val notes =
  1138           [(coinductN, map single coinduct_thms, []), (* FIXME: attribs *)
  1139            (corecN, corec_thmss, []),
  1140            (disc_corecN, disc_corec_thmss, simp_attrs),
  1141            (disc_corec_iffN, disc_corec_iff_thmss, simp_attrs),
  1142            (disc_unfoldN, disc_unfold_thmss, simp_attrs),
  1143            (disc_unfold_iffN, disc_unfold_iff_thmss, simp_attrs),
  1144            (sel_corecN, sel_corec_thmss, simp_attrs),
  1145            (sel_unfoldN, sel_unfold_thmss, simp_attrs),
  1146            (simpsN, simp_thmss, []),
  1147            (strong_coinductN, map single strong_coinduct_thms, []), (* FIXME: attribs *)
  1148            (unfoldN, unfold_thmss, [])]
  1149           |> maps (fn (thmN, thmss, attrs) =>
  1150             map_filter (fn (_, []) => NONE | (fp_b_name, thms) =>
  1151               SOME ((qualify true fp_b_name (Binding.name thmN), attrs),
  1152                 [(thms, [])])) (fp_b_names ~~ thmss));
  1153       in
  1154         lthy |> Local_Theory.notes (anonymous_notes @ common_notes @ notes) |> snd
  1155       end;
  1156 
  1157     val lthy' = lthy
  1158       |> fold_map define_ctrs_case_for_type (fp_bnfs ~~ fp_bs ~~ fpTs ~~ Cs ~~ ctors ~~ dtors ~~
  1159         fp_folds ~~ fp_recs ~~ ctor_dtors ~~ dtor_ctors ~~ ctor_injects ~~ pre_map_defs ~~
  1160         pre_set_defss ~~ pre_rel_defs ~~ fp_map_thms ~~ fp_set_thmss ~~ fp_rel_thms ~~ ns ~~ kss ~~
  1161         mss ~~ ctr_bindingss ~~ ctr_mixfixess ~~ ctr_Tsss ~~ disc_bindingss ~~ sel_bindingsss ~~
  1162         raw_sel_defaultsss)
  1163       |> wrap_types_and_more
  1164       |> (if lfp then derive_induct_fold_rec_thms_for_types
  1165           else derive_coinduct_unfold_corec_thms_for_types);
  1166 
  1167     val timer = time (timer ("Constructors, discriminators, selectors, etc., for the new " ^
  1168       (if lfp then "" else "co") ^ "datatype"));
  1169   in
  1170     timer; lthy'
  1171   end;
  1172 
  1173 val datatypes = define_datatypes (K I) (K I) (K I);
  1174 
  1175 val datatype_cmd = define_datatypes Typedecl.read_constraint Syntax.parse_typ Syntax.parse_term;
  1176 
  1177 val parse_ctr_arg =
  1178   @{keyword "("} |-- parse_binding_colon -- Parse.typ --| @{keyword ")"} ||
  1179   (Parse.typ >> pair Binding.empty);
  1180 
  1181 val parse_defaults =
  1182   @{keyword "("} |-- @{keyword "defaults"} |-- Scan.repeat parse_bound_term --| @{keyword ")"};
  1183 
  1184 (* Identical to unexported function of the same name in "Pure/Isar/parse.ML" *)
  1185 fun parse_type_arguments arg =
  1186   arg >> single || @{keyword "("} |-- Parse.!!! (Parse.list1 arg --| @{keyword ")"}) ||
  1187   Scan.succeed [];
  1188 
  1189 val parse_type_arg_constrained =
  1190   parse_opt_binding_colon -- Parse.type_ident --
  1191   Scan.option (@{keyword "::"} |-- Parse.!!! Parse.sort)
  1192 
  1193 val parse_type_args_constrained = parse_type_arguments parse_type_arg_constrained;
  1194 
  1195 val parse_single_spec =
  1196   parse_type_args_constrained -- Parse.binding -- Parse.opt_mixfix --
  1197   (@{keyword "="} |-- Parse.enum1 "|" (parse_opt_binding_colon -- Parse.binding --
  1198     Scan.repeat parse_ctr_arg -- Scan.optional parse_defaults [] -- Parse.opt_mixfix));
  1199 
  1200 val parse_datatype = parse_wrap_options -- Parse.and_list1 parse_single_spec;
  1201 
  1202 fun parse_datatype_cmd lfp construct_fp = parse_datatype >> datatype_cmd lfp construct_fp;
  1203 
  1204 end;