src/HOL/BNF/Tools/bnf_fp_def_sugar.ML
author blanchet
Wed Apr 24 13:16:21 2013 +0200 (2013-04-24)
changeset 51758 55963309557b
parent 51757 7babcb61aa5c
child 51766 f19a4d0ab1bf
permissions -rw-r--r--
honor user-specified name for map function
     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 ->
    12       binding list list -> typ list * typ list list -> BNF_Def.BNF list -> local_theory ->
    13       BNF_FP.fp_result * local_theory) ->
    14     (bool * bool) * ((((binding * (binding * (typ * sort)) list) * binding) * mixfix) *
    15       ((((binding * binding) * (binding * typ) list) * (binding * term) list) *
    16         mixfix) list) list ->
    17     local_theory -> local_theory
    18   val parse_datatype_cmd: bool ->
    19     (mixfix list -> (string * sort) list option -> binding list -> binding list ->
    20       binding list list -> typ list * typ list list -> BNF_Def.BNF list -> local_theory ->
    21       BNF_FP.fp_result * local_theory) ->
    22     (local_theory -> local_theory) parser
    23 end;
    24 
    25 structure BNF_FP_Def_Sugar : BNF_FP_DEF_SUGAR =
    26 struct
    27 
    28 open BNF_Util
    29 open BNF_Wrap
    30 open BNF_Def
    31 open BNF_FP
    32 open BNF_FP_Def_Sugar_Tactics
    33 
    34 (* This function could produce clashes in contrived examples (e.g., "x.A", "x.x_A", "y.A") *)
    35 fun quasi_unambiguous_case_names names =
    36   let
    37     val ps = map (`Long_Name.base_name) names;
    38     val dups = Library.duplicates (op =) (map fst ps);
    39     fun underscore s =
    40       let val ss = space_explode Long_Name.separator s in
    41         space_implode "_" (drop (length ss - 2) ss)
    42       end;
    43   in
    44     map (fn (base, full) => if member (op =) dups base then underscore full else base) ps
    45   end;
    46 
    47 val mp_conj = @{thm mp_conj};
    48 
    49 val simp_attrs = @{attributes [simp]};
    50 val code_simp_attrs = Code.add_default_eqn_attrib :: simp_attrs;
    51 
    52 fun split_list4 [] = ([], [], [], [])
    53   | split_list4 ((x1, x2, x3, x4) :: xs) =
    54     let val (xs1, xs2, xs3, xs4) = split_list4 xs;
    55     in (x1 :: xs1, x2 :: xs2, x3 :: xs3, x4 :: xs4) end;
    56 
    57 fun resort_tfree S (TFree (s, _)) = TFree (s, S);
    58 
    59 fun typ_subst inst (T as Type (s, Ts)) =
    60     (case AList.lookup (op =) inst T of
    61       NONE => Type (s, map (typ_subst inst) Ts)
    62     | SOME T' => T')
    63   | typ_subst inst T = the_default T (AList.lookup (op =) inst T);
    64 
    65 fun variant_types ss Ss ctxt =
    66   let
    67     val (tfrees, _) =
    68       fold_map2 (fn s => fn S => Name.variant s #> apfst (rpair S)) ss Ss (Variable.names_of ctxt);
    69     val ctxt' = fold (Variable.declare_constraints o Logic.mk_type o TFree) tfrees ctxt;
    70   in (tfrees, ctxt') end;
    71 
    72 val lists_bmoc = fold (fn xs => fn t => Term.list_comb (t, xs));
    73 
    74 fun mk_tupled_fun x f xs = HOLogic.tupled_lambda x (Term.list_comb (f, xs));
    75 fun mk_uncurried_fun f xs = mk_tupled_fun (HOLogic.mk_tuple xs) f xs;
    76 
    77 fun mk_flip (x, Type (_, [T1, Type (_, [T2, T3])])) =
    78   Abs ("x", T1, Abs ("y", T2, Var (x, T2 --> T1 --> T3) $ Bound 0 $ Bound 1));
    79 
    80 fun flip_rels lthy n thm =
    81   let
    82     val Rs = Term.add_vars (prop_of thm) [];
    83     val Rs' = rev (drop (length Rs - n) Rs);
    84     val cRs = map (fn f => (certify lthy (Var f), certify lthy (mk_flip f))) Rs';
    85   in
    86     Drule.cterm_instantiate cRs thm
    87   end;
    88 
    89 fun mk_ctor_or_dtor get_T Ts t =
    90   let val Type (_, Ts0) = get_T (fastype_of t) in
    91     Term.subst_atomic_types (Ts0 ~~ Ts) t
    92   end;
    93 
    94 val mk_ctor = mk_ctor_or_dtor range_type;
    95 val mk_dtor = mk_ctor_or_dtor domain_type;
    96 
    97 fun mk_rec_like lfp Ts Us t =
    98   let
    99     val (bindings, body) = strip_type (fastype_of t);
   100     val (f_Us, prebody) = split_last bindings;
   101     val Type (_, Ts0) = if lfp then prebody else body;
   102     val Us0 = distinct (op =) (map (if lfp then body_type else domain_type) f_Us);
   103   in
   104     Term.subst_atomic_types (Ts0 @ Us0 ~~ Ts @ Us) t
   105   end;
   106 
   107 fun mk_map live Ts Us t =
   108   let val (Type (_, Ts0), Type (_, Us0)) = strip_typeN (live + 1) (fastype_of t) |>> List.last in
   109     Term.subst_atomic_types (Ts0 @ Us0 ~~ Ts @ Us) t
   110   end;
   111 
   112 fun mk_rel live Ts Us t =
   113   let val [Type (_, Ts0), Type (_, Us0)] = binder_types (snd (strip_typeN live (fastype_of t))) in
   114     Term.subst_atomic_types (Ts0 @ Us0 ~~ Ts @ Us) t
   115   end;
   116 
   117 fun liveness_of_fp_bnf n bnf =
   118   (case T_of_bnf bnf of
   119     Type (_, Ts) => map (not o member (op =) (deads_of_bnf bnf)) Ts
   120   | _ => replicate n false);
   121 
   122 fun cannot_merge_types () = error "Mutually recursive types must have the same type parameters";
   123 
   124 fun merge_type_arg T T' = if T = T' then T else cannot_merge_types ();
   125 
   126 fun merge_type_args (As, As') =
   127   if length As = length As' then map2 merge_type_arg As As' else cannot_merge_types ();
   128 
   129 fun reassoc_conjs thm =
   130   reassoc_conjs (thm RS @{thm conj_assoc[THEN iffD1]})
   131   handle THM _ => thm;
   132 
   133 fun map_binding_of ((((b, _), _), _), _) = b;
   134 fun type_args_named_constrained_of ((((_, ncAs), _), _), _) = ncAs;
   135 fun type_binding_of (((_, b), _), _) = b;
   136 fun mixfix_of ((_, mx), _) = mx;
   137 fun ctr_specs_of (_, ctr_specs) = ctr_specs;
   138 
   139 fun disc_of ((((disc, _), _), _), _) = disc;
   140 fun ctr_of ((((_, ctr), _), _), _) = ctr;
   141 fun args_of (((_, args), _), _) = args;
   142 fun defaults_of ((_, ds), _) = ds;
   143 fun ctr_mixfix_of (_, mx) = mx;
   144 
   145 fun define_datatypes prepare_constraint prepare_typ prepare_term lfp construct_fp
   146     (wrap_opts as (no_dests, rep_compat), specs) no_defs_lthy0 =
   147   let
   148     (* TODO: sanity checks on arguments *)
   149     (* TODO: integration with function package ("size") *)
   150 
   151     val _ = if not lfp andalso no_dests then error "Cannot define destructor-less codatatypes"
   152       else ();
   153 
   154     fun qualify mandatory fp_b_name =
   155       Binding.qualify mandatory fp_b_name o (rep_compat ? Binding.qualify false rep_compat_prefix);
   156 
   157     val nn = length specs;
   158     val fp_bs = map type_binding_of specs;
   159     val fp_b_names = map Binding.name_of fp_bs;
   160     val fp_common_name = mk_common_name fp_b_names;
   161     val map_bs = map map_binding_of specs;
   162 
   163     fun prepare_type_arg (_, (ty, c)) =
   164       let val TFree (s, _) = prepare_typ no_defs_lthy0 ty in
   165         TFree (s, prepare_constraint no_defs_lthy0 c)
   166       end;
   167 
   168     val Ass0 = map (map prepare_type_arg o type_args_named_constrained_of) specs;
   169     val unsorted_Ass0 = map (map (resort_tfree HOLogic.typeS)) Ass0;
   170     val unsorted_As = Library.foldr1 merge_type_args unsorted_Ass0;
   171     val set_bss = map (map fst o type_args_named_constrained_of) specs;
   172 
   173     val (((Bs0, Cs), Xs), no_defs_lthy) =
   174       no_defs_lthy0
   175       |> fold (Variable.declare_typ o resort_tfree dummyS) unsorted_As
   176       |> mk_TFrees (length unsorted_As)
   177       ||>> mk_TFrees nn
   178       ||>> apfst (map TFree) o
   179         variant_types (map (prefix "'") fp_b_names) (replicate nn HOLogic.typeS);
   180 
   181     (* TODO: cleaner handling of fake contexts, without "background_theory" *)
   182     (*the "perhaps o try" below helps gracefully handles the case where the new type is defined in a
   183       locale and shadows an existing global type*)
   184     val fake_thy =
   185       Theory.copy #> fold (fn spec => perhaps (try (Sign.add_type no_defs_lthy
   186           (type_binding_of spec, length (type_args_named_constrained_of spec), mixfix_of spec))))
   187         specs;
   188     val fake_lthy = Proof_Context.background_theory fake_thy no_defs_lthy;
   189 
   190     fun mk_fake_T b =
   191       Type (fst (Term.dest_Type (Proof_Context.read_type_name fake_lthy true (Binding.name_of b))),
   192         unsorted_As);
   193 
   194     val fake_Ts = map mk_fake_T fp_bs;
   195 
   196     val mixfixes = map mixfix_of specs;
   197 
   198     val _ = (case duplicates Binding.eq_name fp_bs of [] => ()
   199       | b :: _ => error ("Duplicate type name declaration " ^ quote (Binding.name_of b)));
   200 
   201     val ctr_specss = map ctr_specs_of specs;
   202 
   203     val disc_bindingss = map (map disc_of) ctr_specss;
   204     val ctr_bindingss =
   205       map2 (fn fp_b_name => map (qualify false fp_b_name o ctr_of)) fp_b_names ctr_specss;
   206     val ctr_argsss = map (map args_of) ctr_specss;
   207     val ctr_mixfixess = map (map ctr_mixfix_of) ctr_specss;
   208 
   209     val sel_bindingsss = map (map (map fst)) ctr_argsss;
   210     val fake_ctr_Tsss0 = map (map (map (prepare_typ fake_lthy o snd))) ctr_argsss;
   211     val raw_sel_defaultsss = map (map defaults_of) ctr_specss;
   212 
   213     val (As :: _) :: fake_ctr_Tsss =
   214       burrow (burrow (Syntax.check_typs fake_lthy)) (Ass0 :: fake_ctr_Tsss0);
   215 
   216     val _ = (case duplicates (op =) unsorted_As of [] => ()
   217       | A :: _ => error ("Duplicate type parameter " ^
   218           quote (Syntax.string_of_typ no_defs_lthy A)));
   219 
   220     val rhs_As' = fold (fold (fold Term.add_tfreesT)) fake_ctr_Tsss [];
   221     val _ = (case subtract (op =) (map dest_TFree As) rhs_As' of
   222         [] => ()
   223       | A' :: _ => error ("Extra type variable on right-hand side: " ^
   224           quote (Syntax.string_of_typ no_defs_lthy (TFree A'))));
   225 
   226     fun eq_fpT_check (T as Type (s, Us)) (Type (s', Us')) =
   227         s = s' andalso (Us = Us' orelse error ("Illegal occurrence of recursive type " ^
   228           quote (Syntax.string_of_typ fake_lthy T)))
   229       | eq_fpT_check _ _ = false;
   230 
   231     fun freeze_fp (T as Type (s, Us)) =
   232         (case find_index (eq_fpT_check T) fake_Ts of
   233           ~1 => Type (s, map freeze_fp Us)
   234         | kk => nth Xs kk)
   235       | freeze_fp T = T;
   236 
   237     val ctr_TsssXs = map (map (map freeze_fp)) fake_ctr_Tsss;
   238     val ctr_sum_prod_TsXs = map (mk_sumTN_balanced o map HOLogic.mk_tupleT) ctr_TsssXs;
   239 
   240     val fp_eqs =
   241       map dest_TFree Xs ~~ map (Term.typ_subst_atomic (As ~~ unsorted_As)) ctr_sum_prod_TsXs;
   242 
   243     (* TODO: clean up list *)
   244     val (pre_bnfs, ((fp_bnfs as any_fp_bnf :: _, dtors0, ctors0, fp_folds0, fp_recs0, fp_induct,
   245            fp_strong_induct, dtor_ctors, ctor_dtors, ctor_injects, fp_map_thms, fp_set_thmss,
   246            fp_rel_thms, fp_fold_thms, fp_rec_thms), lthy)) =
   247       fp_bnf construct_fp fp_bs mixfixes map_bs set_bss (map dest_TFree unsorted_As) fp_eqs
   248         no_defs_lthy0;
   249 
   250     val timer = time (Timer.startRealTimer ());
   251 
   252     fun build_map build_arg (Type (s, Ts)) (Type (_, Us)) =
   253       let
   254         val bnf = the (bnf_of lthy s);
   255         val live = live_of_bnf bnf;
   256         val mapx = mk_map live Ts Us (map_of_bnf bnf);
   257         val TUs' = map dest_funT (fst (strip_typeN live (fastype_of mapx)));
   258       in Term.list_comb (mapx, map build_arg TUs') end;
   259 
   260     fun build_rel_step build_arg (Type (s, Ts)) =
   261       let
   262         val bnf = the (bnf_of lthy s);
   263         val live = live_of_bnf bnf;
   264         val rel = mk_rel live Ts Ts (rel_of_bnf bnf);
   265         val Ts' = map domain_type (fst (strip_typeN live (fastype_of rel)));
   266       in Term.list_comb (rel, map build_arg Ts') end;
   267 
   268     fun add_nesty_bnf_names Us =
   269       let
   270         fun add (Type (s, Ts)) ss =
   271             let val (needs, ss') = fold_map add Ts ss in
   272               if exists I needs then (true, insert (op =) s ss') else (false, ss')
   273             end
   274           | add T ss = (member (op =) Us T, ss);
   275       in snd oo add end;
   276 
   277     fun nesty_bnfs Us =
   278       map_filter (bnf_of lthy) (fold (fold (fold (add_nesty_bnf_names Us))) ctr_TsssXs []);
   279 
   280     val nesting_bnfs = nesty_bnfs As;
   281     val nested_bnfs = nesty_bnfs Xs;
   282 
   283     val pre_map_defs = map map_def_of_bnf pre_bnfs;
   284     val pre_set_defss = map set_defs_of_bnf pre_bnfs;
   285     val pre_rel_defs = map rel_def_of_bnf pre_bnfs;
   286     val nested_map_comps'' = map ((fn thm => thm RS sym) o map_comp_of_bnf) nested_bnfs;
   287     val nested_map_comp's = map map_comp'_of_bnf nested_bnfs;
   288     val nested_map_ids'' = map (unfold_thms lthy @{thms id_def} o map_id_of_bnf) nested_bnfs;
   289     val nesting_map_ids'' = map (unfold_thms lthy @{thms id_def} o map_id_of_bnf) nesting_bnfs;
   290     val nested_set_natural's = maps set_natural'_of_bnf nested_bnfs;
   291     val nesting_set_natural's = maps set_natural'_of_bnf nesting_bnfs;
   292 
   293     val live = live_of_bnf any_fp_bnf;
   294 
   295     val Bs =
   296       map3 (fn alive => fn A as TFree (_, S) => fn B => if alive then resort_tfree S B else A)
   297         (liveness_of_fp_bnf (length As) any_fp_bnf) As Bs0;
   298 
   299     val B_ify = Term.typ_subst_atomic (As ~~ Bs);
   300 
   301     val ctors = map (mk_ctor As) ctors0;
   302     val dtors = map (mk_dtor As) dtors0;
   303 
   304     val fpTs = map (domain_type o fastype_of) dtors;
   305 
   306     val exists_fp_subtype = exists_subtype (member (op =) fpTs);
   307     val exists_Cs_subtype = exists_subtype (member (op =) Cs);
   308 
   309     val ctr_Tsss = map (map (map (Term.typ_subst_atomic (Xs ~~ fpTs)))) ctr_TsssXs;
   310     val ns = map length ctr_Tsss;
   311     val kss = map (fn n => 1 upto n) ns;
   312     val mss = map (map length) ctr_Tsss;
   313     val Css = map2 replicate ns Cs;
   314 
   315     val fp_folds as any_fp_fold :: _ = map (mk_rec_like lfp As Cs) fp_folds0;
   316     val fp_recs as any_fp_rec :: _ = map (mk_rec_like lfp As Cs) fp_recs0;
   317 
   318     val fp_fold_fun_Ts = fst (split_last (binder_types (fastype_of any_fp_fold)));
   319     val fp_rec_fun_Ts = fst (split_last (binder_types (fastype_of any_fp_rec)));
   320 
   321     val (((fold_only as (gss, _, _), rec_only as (hss, _, _)),
   322           (cs, cpss, unfold_only as ((pgss, crssss, cgssss), (_, g_Tsss, _)),
   323            corec_only as ((phss, csssss, chssss), (_, h_Tsss, _)))), names_lthy0) =
   324       if lfp then
   325         let
   326           val y_Tsss =
   327             map3 (fn n => fn ms => map2 dest_tupleT ms o dest_sumTN_balanced n o domain_type)
   328               ns mss fp_fold_fun_Ts;
   329           val g_Tss = map2 (map2 (curry (op --->))) y_Tsss Css;
   330 
   331           val ((gss, ysss), lthy) =
   332             lthy
   333             |> mk_Freess "f" g_Tss
   334             ||>> mk_Freesss "x" y_Tsss;
   335 
   336           fun proj_recT proj (Type (s as @{type_name prod}, Ts as [T, U])) =
   337               if member (op =) fpTs T then proj (T, U) else Type (s, map (proj_recT proj) Ts)
   338             | proj_recT proj (Type (s, Ts)) = Type (s, map (proj_recT proj) Ts)
   339             | proj_recT _ T = T;
   340 
   341           fun unzip_recT T =
   342             if exists_fp_subtype T then [proj_recT fst T, proj_recT snd T] else [T];
   343 
   344           val z_Tsss =
   345             map3 (fn n => fn ms => map2 dest_tupleT ms o dest_sumTN_balanced n o domain_type)
   346               ns mss fp_rec_fun_Ts;
   347           val z_Tssss = map (map (map unzip_recT)) z_Tsss;
   348           val h_Tss = map2 (map2 (fold_rev (curry (op --->)))) z_Tssss Css;
   349 
   350           val hss = map2 (map2 retype_free) h_Tss gss;
   351           val zsss = map2 (map2 (map2 retype_free)) z_Tsss ysss;
   352         in
   353           ((((gss, g_Tss, ysss), (hss, h_Tss, zsss)),
   354             ([], [], (([], [], []), ([], [], [])), (([], [], []), ([], [], [])))), lthy)
   355         end
   356       else
   357         let
   358           (*avoid "'a itself" arguments in coiterators and corecursors*)
   359           val mss' =  map (fn [0] => [1] | ms => ms) mss;
   360 
   361           val p_Tss = map2 (fn n => replicate (Int.max (0, n - 1)) o mk_pred1T) ns Cs;
   362 
   363           fun flat_predss_getterss qss fss = maps (op @) (qss ~~ fss);
   364 
   365           fun flat_preds_predsss_gettersss [] [qss] [fss] = flat_predss_getterss qss fss
   366             | flat_preds_predsss_gettersss (p :: ps) (qss :: qsss) (fss :: fsss) =
   367               p :: flat_predss_getterss qss fss @ flat_preds_predsss_gettersss ps qsss fsss;
   368 
   369           fun mk_types maybe_unzipT fun_Ts =
   370             let
   371               val f_sum_prod_Ts = map range_type fun_Ts;
   372               val f_prod_Tss = map2 dest_sumTN_balanced ns f_sum_prod_Ts;
   373               val f_Tsss = map2 (map2 dest_tupleT) mss' f_prod_Tss;
   374               val f_Tssss =
   375                 map2 (fn C => map (map (map (curry (op -->) C) o maybe_unzipT))) Cs f_Tsss;
   376               val q_Tssss =
   377                 map (map (map (fn [_] => [] | [_, C] => [mk_pred1T (domain_type C)]))) f_Tssss;
   378               val pf_Tss = map3 flat_preds_predsss_gettersss p_Tss q_Tssss f_Tssss;
   379             in (q_Tssss, f_sum_prod_Ts, f_Tsss, f_Tssss, pf_Tss) end;
   380 
   381           val (r_Tssss, g_sum_prod_Ts, g_Tsss, g_Tssss, pg_Tss) = mk_types single fp_fold_fun_Ts;
   382 
   383           val (((cs, pss), gssss), lthy) =
   384             lthy
   385             |> mk_Frees "a" Cs
   386             ||>> mk_Freess "p" p_Tss
   387             ||>> mk_Freessss "g" g_Tssss;
   388           val rssss = map (map (map (fn [] => []))) r_Tssss;
   389 
   390           fun proj_corecT proj (Type (s as @{type_name sum}, Ts as [T, U])) =
   391               if member (op =) fpTs T then proj (T, U) else Type (s, map (proj_corecT proj) Ts)
   392             | proj_corecT proj (Type (s, Ts)) = Type (s, map (proj_corecT proj) Ts)
   393             | proj_corecT _ T = T;
   394 
   395           fun unzip_corecT T =
   396             if exists_fp_subtype T then [proj_corecT fst T, proj_corecT snd T] else [T];
   397 
   398           val (s_Tssss, h_sum_prod_Ts, h_Tsss, h_Tssss, ph_Tss) =
   399             mk_types unzip_corecT fp_rec_fun_Ts;
   400 
   401           val hssss_hd = map2 (map2 (map2 (fn T :: _ => fn [g] => retype_free T g))) h_Tssss gssss;
   402           val ((sssss, hssss_tl), lthy) =
   403             lthy
   404             |> mk_Freessss "q" s_Tssss
   405             ||>> mk_Freessss "h" (map (map (map tl)) h_Tssss);
   406           val hssss = map2 (map2 (map2 cons)) hssss_hd hssss_tl;
   407 
   408           val cpss = map2 (map o rapp) cs pss;
   409 
   410           fun mk_terms qssss fssss =
   411             let
   412               val pfss = map3 flat_preds_predsss_gettersss pss qssss fssss;
   413               val cqssss = map2 (map o map o map o rapp) cs qssss;
   414               val cfssss = map2 (map o map o map o rapp) cs fssss;
   415             in (pfss, cqssss, cfssss) end;
   416         in
   417           (((([], [], []), ([], [], [])),
   418             (cs, cpss, (mk_terms rssss gssss, (g_sum_prod_Ts, g_Tsss, pg_Tss)),
   419              (mk_terms sssss hssss, (h_sum_prod_Ts, h_Tsss, ph_Tss)))), lthy)
   420         end;
   421 
   422     fun define_ctrs_case_for_type (((((((((((((((((((((((((fp_bnf, fp_b), fpT), C), ctor), dtor),
   423             fp_fold), fp_rec), ctor_dtor), dtor_ctor), ctor_inject), pre_map_def), pre_set_defs),
   424           pre_rel_def), fp_map_thm), fp_set_thms), fp_rel_thm), n), ks), ms), ctr_bindings),
   425         ctr_mixfixes), ctr_Tss), disc_bindings), sel_bindingss), raw_sel_defaultss) no_defs_lthy =
   426       let
   427         val fp_b_name = Binding.name_of fp_b;
   428 
   429         val dtorT = domain_type (fastype_of ctor);
   430         val ctr_prod_Ts = map HOLogic.mk_tupleT ctr_Tss;
   431         val ctr_sum_prod_T = mk_sumTN_balanced ctr_prod_Ts;
   432         val case_Ts = map (fn Ts => Ts ---> C) ctr_Tss;
   433 
   434         val (((((w, fs), xss), yss), u'), names_lthy) =
   435           no_defs_lthy
   436           |> yield_singleton (mk_Frees "w") dtorT
   437           ||>> mk_Frees "f" case_Ts
   438           ||>> mk_Freess "x" ctr_Tss
   439           ||>> mk_Freess "y" (map (map B_ify) ctr_Tss)
   440           ||>> yield_singleton Variable.variant_fixes fp_b_name;
   441 
   442         val u = Free (u', fpT);
   443 
   444         val tuple_xs = map HOLogic.mk_tuple xss;
   445         val tuple_ys = map HOLogic.mk_tuple yss;
   446 
   447         val ctr_rhss =
   448           map3 (fn k => fn xs => fn tuple_x => fold_rev Term.lambda xs (ctor $
   449             mk_InN_balanced ctr_sum_prod_T n tuple_x k)) ks xss tuple_xs;
   450 
   451         val case_binding = qualify false fp_b_name (Binding.suffix_name ("_" ^ caseN) fp_b);
   452 
   453         val case_rhs =
   454           fold_rev Term.lambda (fs @ [u])
   455             (mk_sum_caseN_balanced (map2 mk_uncurried_fun fs xss) $ (dtor $ u));
   456 
   457         val ((raw_case :: raw_ctrs, raw_case_def :: raw_ctr_defs), (lthy', lthy)) = no_defs_lthy
   458           |> apfst split_list o fold_map3 (fn b => fn mx => fn rhs =>
   459               Local_Theory.define ((b, mx), ((Thm.def_binding b, []), rhs)) #>> apsnd snd)
   460             (case_binding :: ctr_bindings) (NoSyn :: ctr_mixfixes) (case_rhs :: ctr_rhss)
   461           ||> `Local_Theory.restore;
   462 
   463         val phi = Proof_Context.export_morphism lthy lthy';
   464 
   465         val ctr_defs = map (Morphism.thm phi) raw_ctr_defs;
   466         val ctr_defs' =
   467           map2 (fn m => fn def => mk_unabs_def m (def RS meta_eq_to_obj_eq)) ms ctr_defs;
   468         val case_def = Morphism.thm phi raw_case_def;
   469 
   470         val ctrs0 = map (Morphism.term phi) raw_ctrs;
   471         val casex0 = Morphism.term phi raw_case;
   472 
   473         val ctrs = map (mk_ctr As) ctrs0;
   474 
   475         fun wrap lthy =
   476           let
   477             fun exhaust_tac {context = ctxt, prems = _} =
   478               let
   479                 val ctor_iff_dtor_thm =
   480                   let
   481                     val goal =
   482                       fold_rev Logic.all [w, u]
   483                         (mk_Trueprop_eq (HOLogic.mk_eq (u, ctor $ w), HOLogic.mk_eq (dtor $ u, w)));
   484                   in
   485                     Goal.prove_sorry lthy [] [] goal (fn {context = ctxt, ...} =>
   486                       mk_ctor_iff_dtor_tac ctxt (map (SOME o certifyT lthy) [dtorT, fpT])
   487                         (certify lthy ctor) (certify lthy dtor) ctor_dtor dtor_ctor)
   488                     |> Thm.close_derivation
   489                     |> Morphism.thm phi
   490                   end;
   491 
   492                 val sumEN_thm' =
   493                   unfold_thms lthy @{thms all_unit_eq}
   494                     (Drule.instantiate' (map (SOME o certifyT lthy) ctr_prod_Ts) []
   495                        (mk_sumEN_balanced n))
   496                   |> Morphism.thm phi;
   497               in
   498                 mk_exhaust_tac ctxt n ctr_defs ctor_iff_dtor_thm sumEN_thm'
   499               end;
   500 
   501             val inject_tacss =
   502               map2 (fn 0 => K [] | _ => fn ctr_def => [fn {context = ctxt, ...} =>
   503                   mk_inject_tac ctxt ctr_def ctor_inject]) ms ctr_defs;
   504 
   505             val half_distinct_tacss =
   506               map (map (fn (def, def') => fn {context = ctxt, ...} =>
   507                 mk_half_distinct_tac ctxt ctor_inject [def, def'])) (mk_half_pairss (`I ctr_defs));
   508 
   509             val case_tacs =
   510               map3 (fn k => fn m => fn ctr_def => fn {context = ctxt, ...} =>
   511                 mk_case_tac ctxt n k m case_def ctr_def dtor_ctor) ks ms ctr_defs;
   512 
   513             val tacss = [exhaust_tac] :: inject_tacss @ half_distinct_tacss @ [case_tacs];
   514 
   515             val sel_defaultss = map (map (apsnd (prepare_term lthy))) raw_sel_defaultss
   516           in
   517             wrap_datatype tacss (((wrap_opts, ctrs0), casex0), (disc_bindings, (sel_bindingss,
   518               sel_defaultss))) lthy
   519           end;
   520 
   521         fun derive_maps_sets_rels (wrap_res, lthy) =
   522           let
   523             val rel_flip = rel_flip_of_bnf fp_bnf;
   524             val nones = replicate live NONE;
   525 
   526             val ctor_cong =
   527               if lfp then
   528                 Drule.dummy_thm
   529               else
   530                 let val ctor' = mk_ctor Bs ctor in
   531                   cterm_instantiate_pos [NONE, NONE, SOME (certify lthy ctor')] arg_cong
   532                 end;
   533 
   534             fun mk_cIn ify =
   535               certify lthy o (not lfp ? curry (op $) (map_types ify ctor)) oo
   536               mk_InN_balanced (ify ctr_sum_prod_T) n;
   537 
   538             val cxIns = map2 (mk_cIn I) tuple_xs ks;
   539             val cyIns = map2 (mk_cIn B_ify) tuple_ys ks;
   540 
   541             fun mk_map_thm ctr_def' cxIn =
   542               fold_thms lthy [ctr_def']
   543                 (unfold_thms lthy (pre_map_def ::
   544                      (if lfp then [] else [ctor_dtor, dtor_ctor]) @ sum_prod_thms_map)
   545                    (cterm_instantiate_pos (nones @ [SOME cxIn])
   546                       (if lfp then fp_map_thm else fp_map_thm RS ctor_cong)))
   547               |> singleton (Proof_Context.export names_lthy no_defs_lthy);
   548 
   549             fun mk_set_thm fp_set_thm ctr_def' cxIn =
   550               fold_thms lthy [ctr_def']
   551                 (unfold_thms lthy (pre_set_defs @ nested_set_natural's @ nesting_set_natural's @
   552                      (if lfp then [] else [dtor_ctor]) @ sum_prod_thms_set)
   553                    (cterm_instantiate_pos [SOME cxIn] fp_set_thm))
   554               |> singleton (Proof_Context.export names_lthy no_defs_lthy);
   555 
   556             fun mk_set_thms fp_set_thm = map2 (mk_set_thm fp_set_thm) ctr_defs' cxIns;
   557 
   558             val map_thms = map2 mk_map_thm ctr_defs' cxIns;
   559             val set_thmss = map mk_set_thms fp_set_thms;
   560 
   561             val rel_infos = (ctr_defs' ~~ cxIns, ctr_defs' ~~ cyIns);
   562 
   563             fun mk_rel_thm postproc ctr_defs' cxIn cyIn =
   564               fold_thms lthy ctr_defs'
   565                  (unfold_thms lthy (@{thm Inl_Inr_False} :: pre_rel_def ::
   566                       (if lfp then [] else [dtor_ctor]) @ sum_prod_thms_rel)
   567                     (cterm_instantiate_pos (nones @ [SOME cxIn, SOME cyIn]) fp_rel_thm))
   568               |> postproc
   569               |> singleton (Proof_Context.export names_lthy no_defs_lthy);
   570 
   571             fun mk_rel_inject_thm ((ctr_def', cxIn), (_, cyIn)) =
   572               mk_rel_thm (unfold_thms lthy @{thms eq_sym_Unity_conv}) [ctr_def'] cxIn cyIn;
   573 
   574             val rel_inject_thms = map mk_rel_inject_thm (op ~~ rel_infos);
   575 
   576             fun mk_half_rel_distinct_thm ((xctr_def', cxIn), (yctr_def', cyIn)) =
   577               mk_rel_thm (fn thm => thm RS @{thm eq_False[THEN iffD1]}) [xctr_def', yctr_def']
   578                 cxIn cyIn;
   579 
   580             fun mk_other_half_rel_distinct_thm thm =
   581               flip_rels lthy live thm RS (rel_flip RS sym RS @{thm arg_cong[of _ _ Not]} RS iffD2);
   582 
   583             val half_rel_distinct_thmss =
   584               map (map mk_half_rel_distinct_thm) (mk_half_pairss rel_infos);
   585             val other_half_rel_distinct_thmss =
   586               map (map mk_other_half_rel_distinct_thm) half_rel_distinct_thmss;
   587             val (rel_distinct_thms, _) =
   588               join_halves n half_rel_distinct_thmss other_half_rel_distinct_thmss;
   589 
   590             val notes =
   591               [(mapN, map_thms, code_simp_attrs),
   592                (rel_distinctN, rel_distinct_thms, code_simp_attrs),
   593                (rel_injectN, rel_inject_thms, code_simp_attrs),
   594                (setsN, flat set_thmss, code_simp_attrs)]
   595               |> filter_out (null o #2)
   596               |> map (fn (thmN, thms, attrs) =>
   597                 ((qualify true fp_b_name (Binding.name thmN), attrs), [(thms, [])]));
   598           in
   599             (wrap_res, lthy |> Local_Theory.notes notes |> snd)
   600           end;
   601 
   602         fun define_fold_rec no_defs_lthy =
   603           let
   604             val fpT_to_C = fpT --> C;
   605 
   606             fun build_prod_proj mk_proj (T, U) =
   607               if T = U then
   608                 id_const T
   609               else
   610                 (case (T, U) of
   611                   (Type (s, _), Type (s', _)) =>
   612                   if s = s' then build_map (build_prod_proj mk_proj) T U else mk_proj T
   613                 | _ => mk_proj T);
   614 
   615             (* TODO: Avoid these complications; cf. corec case *)
   616             fun mk_U proj (Type (s as @{type_name prod}, Ts as [T', U])) =
   617                 if member (op =) fpTs T' then proj (T', U) else Type (s, map (mk_U proj) Ts)
   618               | mk_U proj (Type (s, Ts)) = Type (s, map (mk_U proj) Ts)
   619               | mk_U _ T = T;
   620 
   621             fun unzip_rec (x as Free (_, T)) =
   622               if exists_fp_subtype T then
   623                 [build_prod_proj fst_const (T, mk_U fst T) $ x,
   624                  build_prod_proj snd_const (T, mk_U snd T) $ x]
   625               else
   626                 [x];
   627 
   628             fun mk_rec_like_arg f xs = mk_tupled_fun (HOLogic.mk_tuple xs) f (maps unzip_rec xs);
   629 
   630             fun generate_rec_like (suf, fp_rec_like, (fss, f_Tss, xsss)) =
   631               let
   632                 val res_T = fold_rev (curry (op --->)) f_Tss fpT_to_C;
   633                 val binding = qualify false fp_b_name (Binding.suffix_name ("_" ^ suf) fp_b);
   634                 val spec =
   635                   mk_Trueprop_eq (lists_bmoc fss (Free (Binding.name_of binding, res_T)),
   636                     Term.list_comb (fp_rec_like,
   637                       map2 (mk_sum_caseN_balanced oo map2 mk_rec_like_arg) fss xsss));
   638               in (binding, spec) end;
   639 
   640             val rec_like_infos =
   641               [(foldN, fp_fold, fold_only),
   642                (recN, fp_rec, rec_only)];
   643 
   644             val (bindings, specs) = map generate_rec_like rec_like_infos |> split_list;
   645 
   646             val ((csts, defs), (lthy', lthy)) = no_defs_lthy
   647               |> apfst split_list o fold_map2 (fn b => fn spec =>
   648                 Specification.definition (SOME (b, NONE, NoSyn), ((Thm.def_binding b, []), spec))
   649                 #>> apsnd snd) bindings specs
   650               ||> `Local_Theory.restore;
   651 
   652             val phi = Proof_Context.export_morphism lthy lthy';
   653 
   654             val [fold_def, rec_def] = map (Morphism.thm phi) defs;
   655 
   656             val [foldx, recx] = map (mk_rec_like lfp As Cs o Morphism.term phi) csts;
   657           in
   658             ((foldx, recx, fold_def, rec_def), lthy')
   659           end;
   660 
   661         fun define_unfold_corec no_defs_lthy =
   662           let
   663             val B_to_fpT = C --> fpT;
   664 
   665             fun build_sum_inj mk_inj (T, U) =
   666               if T = U then
   667                 id_const T
   668               else
   669                 (case (T, U) of
   670                   (Type (s, _), Type (s', _)) =>
   671                   if s = s' then build_map (build_sum_inj mk_inj) T U
   672                   else uncurry mk_inj (dest_sumT U)
   673                 | _ => uncurry mk_inj (dest_sumT U));
   674 
   675             fun build_dtor_corec_like_arg _ [] [cf] = cf
   676               | build_dtor_corec_like_arg T [cq] [cf, cf'] =
   677                 mk_If cq (build_sum_inj Inl_const (fastype_of cf, T) $ cf)
   678                   (build_sum_inj Inr_const (fastype_of cf', T) $ cf')
   679 
   680             val crgsss = map3 (map3 (map3 build_dtor_corec_like_arg)) g_Tsss crssss cgssss;
   681             val cshsss = map3 (map3 (map3 build_dtor_corec_like_arg)) h_Tsss csssss chssss;
   682 
   683             fun mk_preds_getterss_join c n cps sum_prod_T cqfss =
   684               Term.lambda c (mk_IfN sum_prod_T cps
   685                 (map2 (mk_InN_balanced sum_prod_T n) (map HOLogic.mk_tuple cqfss) (1 upto n)));
   686 
   687             fun generate_corec_like (suf, fp_rec_like, (cqfsss, ((pfss, _, _), (f_sum_prod_Ts, _,
   688                 pf_Tss)))) =
   689               let
   690                 val res_T = fold_rev (curry (op --->)) pf_Tss B_to_fpT;
   691                 val binding = qualify false fp_b_name (Binding.suffix_name ("_" ^ suf) fp_b);
   692                 val spec =
   693                   mk_Trueprop_eq (lists_bmoc pfss (Free (Binding.name_of binding, res_T)),
   694                     Term.list_comb (fp_rec_like,
   695                       map5 mk_preds_getterss_join cs ns cpss f_sum_prod_Ts cqfsss));
   696               in (binding, spec) end;
   697 
   698             val corec_like_infos =
   699               [(unfoldN, fp_fold, (crgsss, unfold_only)),
   700                (corecN, fp_rec, (cshsss, corec_only))];
   701 
   702             val (bindings, specs) = map generate_corec_like corec_like_infos |> split_list;
   703 
   704             val ((csts, defs), (lthy', lthy)) = no_defs_lthy
   705               |> apfst split_list o fold_map2 (fn b => fn spec =>
   706                 Specification.definition (SOME (b, NONE, NoSyn), ((Thm.def_binding b, []), spec))
   707                 #>> apsnd snd) bindings specs
   708               ||> `Local_Theory.restore;
   709 
   710             val phi = Proof_Context.export_morphism lthy lthy';
   711 
   712             val [unfold_def, corec_def] = map (Morphism.thm phi) defs;
   713 
   714             val [unfold, corec] = map (mk_rec_like lfp As Cs o Morphism.term phi) csts;
   715           in
   716             ((unfold, corec, unfold_def, corec_def), lthy')
   717           end;
   718 
   719         val define_rec_likes = if lfp then define_fold_rec else define_unfold_corec;
   720 
   721         fun massage_res ((wrap_res, rec_like_res), lthy) =
   722           (((ctrs, xss, ctr_defs, wrap_res), rec_like_res), lthy);
   723       in
   724         (wrap #> (live > 0 ? derive_maps_sets_rels) ##>> define_rec_likes #> massage_res, lthy')
   725       end;
   726 
   727     fun wrap_types_and_more (wrap_types_and_mores, lthy) =
   728       fold_map I wrap_types_and_mores lthy
   729       |>> apsnd split_list4 o apfst split_list4 o split_list;
   730 
   731     (* TODO: Add map, sets, rel simps *)
   732     val mk_simp_thmss =
   733       map3 (fn (_, _, _, injects, distincts, cases, _, _, _) => fn rec_likes => fn fold_likes =>
   734         injects @ distincts @ cases @ rec_likes @ fold_likes);
   735 
   736     fun derive_induct_fold_rec_thms_for_types (((ctrss, xsss, ctr_defss, wrap_ress), (folds, recs,
   737         fold_defs, rec_defs)), lthy) =
   738       let
   739         val (((ps, ps'), us'), names_lthy) =
   740           lthy
   741           |> mk_Frees' "P" (map mk_pred1T fpTs)
   742           ||>> Variable.variant_fixes fp_b_names;
   743 
   744         val us = map2 (curry Free) us' fpTs;
   745 
   746         fun mk_sets_nested bnf =
   747           let
   748             val Type (T_name, Us) = T_of_bnf bnf;
   749             val lives = lives_of_bnf bnf;
   750             val sets = sets_of_bnf bnf;
   751             fun mk_set U =
   752               (case find_index (curry (op =) U) lives of
   753                 ~1 => Term.dummy
   754               | i => nth sets i);
   755           in
   756             (T_name, map mk_set Us)
   757           end;
   758 
   759         val setss_nested = map mk_sets_nested nested_bnfs;
   760 
   761         val (induct_thms, induct_thm) =
   762           let
   763             fun mk_set Ts t =
   764               let val Type (_, Ts0) = domain_type (fastype_of t) in
   765                 Term.subst_atomic_types (Ts0 ~~ Ts) t
   766               end;
   767 
   768             fun mk_raw_prem_prems names_lthy (x as Free (s, T as Type (T_name, Ts0))) =
   769                 (case find_index (curry (op =) T) fpTs of
   770                   ~1 =>
   771                   (case AList.lookup (op =) setss_nested T_name of
   772                     NONE => []
   773                   | SOME raw_sets0 =>
   774                     let
   775                       val (Ts, raw_sets) =
   776                         split_list (filter (exists_fp_subtype o fst) (Ts0 ~~ raw_sets0));
   777                       val sets = map (mk_set Ts0) raw_sets;
   778                       val (ys, names_lthy') = names_lthy |> mk_Frees s Ts;
   779                       val xysets = map (pair x) (ys ~~ sets);
   780                       val ppremss = map (mk_raw_prem_prems names_lthy') ys;
   781                     in
   782                       flat (map2 (map o apfst o cons) xysets ppremss)
   783                     end)
   784                 | kk => [([], (kk + 1, x))])
   785               | mk_raw_prem_prems _ _ = [];
   786 
   787             fun close_prem_prem xs t =
   788               fold_rev Logic.all (map Free (drop (nn + length xs)
   789                 (rev (Term.add_frees t (map dest_Free xs @ ps'))))) t;
   790 
   791             fun mk_prem_prem xs (xysets, (j, x)) =
   792               close_prem_prem xs (Logic.list_implies (map (fn (x', (y, set)) =>
   793                   HOLogic.mk_Trueprop (HOLogic.mk_mem (y, set $ x'))) xysets,
   794                 HOLogic.mk_Trueprop (nth ps (j - 1) $ x)));
   795 
   796             fun mk_raw_prem phi ctr ctr_Ts =
   797               let
   798                 val (xs, names_lthy') = names_lthy |> mk_Frees "x" ctr_Ts;
   799                 val pprems = maps (mk_raw_prem_prems names_lthy') xs;
   800               in (xs, pprems, HOLogic.mk_Trueprop (phi $ Term.list_comb (ctr, xs))) end;
   801 
   802             fun mk_prem (xs, raw_pprems, concl) =
   803               fold_rev Logic.all xs (Logic.list_implies (map (mk_prem_prem xs) raw_pprems, concl));
   804 
   805             val raw_premss = map3 (map2 o mk_raw_prem) ps ctrss ctr_Tsss;
   806 
   807             val goal =
   808               Library.foldr (Logic.list_implies o apfst (map mk_prem)) (raw_premss,
   809                 HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj (map2 (curry (op $)) ps us)));
   810 
   811             val kksss = map (map (map (fst o snd) o #2)) raw_premss;
   812 
   813             val ctor_induct' = fp_induct OF (map mk_sumEN_tupled_balanced mss);
   814 
   815             val thm =
   816               Goal.prove_sorry lthy [] [] goal (fn {context = ctxt, ...} =>
   817                 mk_induct_tac ctxt nn ns mss kksss (flat ctr_defss) ctor_induct'
   818                   nested_set_natural's pre_set_defss)
   819               |> singleton (Proof_Context.export names_lthy lthy)
   820               |> Thm.close_derivation;
   821           in
   822             `(conj_dests nn) thm
   823           end;
   824 
   825         val induct_cases = quasi_unambiguous_case_names (maps (map name_of_ctr) ctrss);
   826 
   827         val (fold_thmss, rec_thmss) =
   828           let
   829             val xctrss = map2 (map2 (curry Term.list_comb)) ctrss xsss;
   830             val gfolds = map (lists_bmoc gss) folds;
   831             val hrecs = map (lists_bmoc hss) recs;
   832 
   833             fun mk_goal fss frec_like xctr f xs fxs =
   834               fold_rev (fold_rev Logic.all) (xs :: fss)
   835                 (mk_Trueprop_eq (frec_like $ xctr, Term.list_comb (f, fxs)));
   836 
   837             fun build_rec_like frec_likes (T, U) =
   838               if T = U then
   839                 id_const T
   840               else
   841                 (case find_index (curry (op =) T) fpTs of
   842                   ~1 => build_map (build_rec_like frec_likes) T U
   843                 | kk => nth frec_likes kk);
   844 
   845             val mk_U = typ_subst (map2 pair fpTs Cs);
   846 
   847             fun intr_rec_likes frec_likes maybe_cons (x as Free (_, T)) =
   848               if exists_fp_subtype T then
   849                 maybe_cons x [build_rec_like frec_likes (T, mk_U T) $ x]
   850               else
   851                 [x];
   852 
   853             val gxsss = map (map (maps (intr_rec_likes gfolds (K I)))) xsss;
   854             val hxsss = map (map (maps (intr_rec_likes hrecs cons))) xsss;
   855 
   856             val fold_goalss = map5 (map4 o mk_goal gss) gfolds xctrss gss xsss gxsss;
   857             val rec_goalss = map5 (map4 o mk_goal hss) hrecs xctrss hss xsss hxsss;
   858 
   859             val fold_tacss =
   860               map2 (map o mk_rec_like_tac pre_map_defs [] nesting_map_ids'' fold_defs) fp_fold_thms
   861                 ctr_defss;
   862             val rec_tacss =
   863               map2 (map o mk_rec_like_tac pre_map_defs nested_map_comp's
   864                 (nested_map_ids'' @ nesting_map_ids'') rec_defs) fp_rec_thms ctr_defss;
   865 
   866             fun prove goal tac =
   867               Goal.prove_sorry lthy [] [] goal (tac o #context)
   868               |> Thm.close_derivation;
   869           in
   870             (map2 (map2 prove) fold_goalss fold_tacss, map2 (map2 prove) rec_goalss rec_tacss)
   871           end;
   872 
   873         val simp_thmss = mk_simp_thmss wrap_ress rec_thmss fold_thmss;
   874 
   875         val induct_case_names_attr = Attrib.internal (K (Rule_Cases.case_names induct_cases));
   876         fun induct_type_attr T_name = Attrib.internal (K (Induct.induct_type T_name));
   877 
   878         val common_notes =
   879           (if nn > 1 then [(inductN, [induct_thm], [induct_case_names_attr])] else [])
   880           |> map (fn (thmN, thms, attrs) =>
   881             ((qualify true fp_common_name (Binding.name thmN), attrs), [(thms, [])]));
   882 
   883         val notes =
   884           [(foldN, fold_thmss, K code_simp_attrs),
   885            (inductN, map single induct_thms,
   886             fn T_name => [induct_case_names_attr, induct_type_attr T_name]),
   887            (recN, rec_thmss, K code_simp_attrs),
   888            (simpsN, simp_thmss, K [])]
   889           |> maps (fn (thmN, thmss, attrs) =>
   890             map3 (fn fp_b_name => fn Type (T_name, _) => fn thms =>
   891               ((qualify true fp_b_name (Binding.name thmN), attrs T_name),
   892                [(thms, [])])) fp_b_names fpTs thmss);
   893       in
   894         lthy |> Local_Theory.notes (common_notes @ notes) |> snd
   895       end;
   896 
   897     fun derive_coinduct_unfold_corec_thms_for_types (((ctrss, _, ctr_defss, wrap_ress), (unfolds,
   898         corecs, unfold_defs, corec_defs)), lthy) =
   899       let
   900         val nesting_rel_eqs = map rel_eq_of_bnf nesting_bnfs;
   901 
   902         val discss = map (map (mk_disc_or_sel As) o #1) wrap_ress;
   903         val selsss = map (map (map (mk_disc_or_sel As)) o #2) wrap_ress;
   904         val exhaust_thms = map #3 wrap_ress;
   905         val disc_thmsss = map #7 wrap_ress;
   906         val discIss = map #8 wrap_ress;
   907         val sel_thmsss = map #9 wrap_ress;
   908 
   909         val (((rs, us'), vs'), names_lthy) =
   910           lthy
   911           |> mk_Frees "R" (map (fn T => mk_pred2T T T) fpTs)
   912           ||>> Variable.variant_fixes fp_b_names
   913           ||>> Variable.variant_fixes (map (suffix "'") fp_b_names);
   914 
   915         val us = map2 (curry Free) us' fpTs;
   916         val udiscss = map2 (map o rapp) us discss;
   917         val uselsss = map2 (map o map o rapp) us selsss;
   918 
   919         val vs = map2 (curry Free) vs' fpTs;
   920         val vdiscss = map2 (map o rapp) vs discss;
   921         val vselsss = map2 (map o map o rapp) vs selsss;
   922 
   923         val ((coinduct_thms, coinduct_thm), (strong_coinduct_thms, strong_coinduct_thm)) =
   924           let
   925             val uvrs = map3 (fn r => fn u => fn v => r $ u $ v) rs us vs;
   926             val uv_eqs = map2 (curry HOLogic.mk_eq) us vs;
   927             val strong_rs =
   928               map4 (fn u => fn v => fn uvr => fn uv_eq =>
   929                 fold_rev Term.lambda [u, v] (HOLogic.mk_disj (uvr, uv_eq))) us vs uvrs uv_eqs;
   930 
   931             fun build_rel rs' T =
   932               (case find_index (curry (op =) T) fpTs of
   933                 ~1 =>
   934                 if exists_fp_subtype T then build_rel_step (build_rel rs') T else HOLogic.eq_const T
   935               | kk => nth rs' kk);
   936 
   937             fun build_rel_app rs' usel vsel =
   938               fold rapp [usel, vsel] (build_rel rs' (fastype_of usel));
   939 
   940             fun mk_prem_ctr_concls rs' n k udisc usels vdisc vsels =
   941               (if k = n then [] else [HOLogic.mk_eq (udisc, vdisc)]) @
   942               (if null usels then
   943                  []
   944                else
   945                  [Library.foldr HOLogic.mk_imp (if n = 1 then [] else [udisc, vdisc],
   946                     Library.foldr1 HOLogic.mk_conj (map2 (build_rel_app rs') usels vsels))]);
   947 
   948             fun mk_prem_concl rs' n udiscs uselss vdiscs vselss =
   949               Library.foldr1 HOLogic.mk_conj
   950                 (flat (map5 (mk_prem_ctr_concls rs' n) (1 upto n) udiscs uselss vdiscs vselss))
   951               handle List.Empty => @{term True};
   952 
   953             fun mk_prem rs' uvr u v n udiscs uselss vdiscs vselss =
   954               fold_rev Logic.all [u, v] (Logic.mk_implies (HOLogic.mk_Trueprop uvr,
   955                 HOLogic.mk_Trueprop (mk_prem_concl rs' n udiscs uselss vdiscs vselss)));
   956 
   957             val concl =
   958               HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj
   959                 (map3 (fn uvr => fn u => fn v => HOLogic.mk_imp (uvr, HOLogic.mk_eq (u, v)))
   960                    uvrs us vs));
   961 
   962             fun mk_goal rs' =
   963               Logic.list_implies (map8 (mk_prem rs') uvrs us vs ns udiscss uselsss vdiscss vselsss,
   964                 concl);
   965 
   966             val goal = mk_goal rs;
   967             val strong_goal = mk_goal strong_rs;
   968 
   969             fun prove dtor_coinduct' goal =
   970               Goal.prove_sorry lthy [] [] goal (fn {context = ctxt, ...} =>
   971                 mk_coinduct_tac ctxt nesting_rel_eqs nn ns dtor_coinduct' pre_rel_defs dtor_ctors
   972                   exhaust_thms ctr_defss disc_thmsss sel_thmsss)
   973               |> singleton (Proof_Context.export names_lthy lthy)
   974               |> Thm.close_derivation;
   975 
   976             fun postproc nn thm =
   977               Thm.permute_prems 0 nn
   978                 (if nn = 1 then thm RS mp
   979                  else funpow nn (fn thm => reassoc_conjs (thm RS mp_conj)) thm)
   980               |> Drule.zero_var_indexes
   981               |> `(conj_dests nn);
   982           in
   983             (postproc nn (prove fp_induct goal), postproc nn (prove fp_strong_induct strong_goal))
   984           end;
   985 
   986         fun mk_maybe_not pos = not pos ? HOLogic.mk_not;
   987 
   988         val gunfolds = map (lists_bmoc pgss) unfolds;
   989         val hcorecs = map (lists_bmoc phss) corecs;
   990 
   991         val (unfold_thmss, corec_thmss, safe_unfold_thmss, safe_corec_thmss) =
   992           let
   993             fun mk_goal pfss c cps fcorec_like n k ctr m cfs' =
   994               fold_rev (fold_rev Logic.all) ([c] :: pfss)
   995                 (Logic.list_implies (seq_conds (HOLogic.mk_Trueprop oo mk_maybe_not) n k cps,
   996                    mk_Trueprop_eq (fcorec_like $ c, Term.list_comb (ctr, take m cfs'))));
   997 
   998             fun build_corec_like fcorec_likes (T, U) =
   999               if T = U then
  1000                 id_const T
  1001               else
  1002                 (case find_index (curry (op =) U) fpTs of
  1003                   ~1 => build_map (build_corec_like fcorec_likes) T U
  1004                 | kk => nth fcorec_likes kk);
  1005 
  1006             val mk_U = typ_subst (map2 pair Cs fpTs);
  1007 
  1008             fun intr_corec_likes fcorec_likes [] [cf] =
  1009                 let val T = fastype_of cf in
  1010                   if exists_Cs_subtype T then build_corec_like fcorec_likes (T, mk_U T) $ cf else cf
  1011                 end
  1012               | intr_corec_likes fcorec_likes [cq] [cf, cf'] =
  1013                 mk_If cq (intr_corec_likes fcorec_likes [] [cf])
  1014                   (intr_corec_likes fcorec_likes [] [cf']);
  1015 
  1016             val crgsss = map2 (map2 (map2 (intr_corec_likes gunfolds))) crssss cgssss;
  1017             val cshsss = map2 (map2 (map2 (intr_corec_likes hcorecs))) csssss chssss;
  1018 
  1019             val unfold_goalss =
  1020               map8 (map4 oooo mk_goal pgss) cs cpss gunfolds ns kss ctrss mss crgsss;
  1021             val corec_goalss =
  1022               map8 (map4 oooo mk_goal phss) cs cpss hcorecs ns kss ctrss mss cshsss;
  1023 
  1024             fun mk_map_if_distrib bnf =
  1025               let
  1026                 val mapx = map_of_bnf bnf;
  1027                 val live = live_of_bnf bnf;
  1028                 val ((Ts, T), U) = strip_typeN (live + 1) (fastype_of mapx) |>> split_last;
  1029                 val fs = Variable.variant_frees lthy [mapx] (map (pair "f") Ts);
  1030                 val t = Term.list_comb (mapx, map (Var o apfst (rpair 0)) fs);
  1031               in
  1032                 Drule.instantiate' (map (SOME o certifyT lthy) [U, T]) [SOME (certify lthy t)]
  1033                   @{thm if_distrib}
  1034               end;
  1035 
  1036             val nested_map_if_distribs = map mk_map_if_distrib nested_bnfs;
  1037 
  1038             val unfold_tacss =
  1039               map3 (map oo mk_corec_like_tac unfold_defs [] [] nesting_map_ids'' [])
  1040                 fp_fold_thms pre_map_defs ctr_defss;
  1041             val corec_tacss =
  1042               map3 (map oo mk_corec_like_tac corec_defs nested_map_comps'' nested_map_comp's
  1043                   (nested_map_ids'' @ nesting_map_ids'') nested_map_if_distribs)
  1044                 fp_rec_thms pre_map_defs ctr_defss;
  1045 
  1046             fun prove goal tac =
  1047               Goal.prove_sorry lthy [] [] goal (tac o #context) |> Thm.close_derivation;
  1048 
  1049             val unfold_thmss = map2 (map2 prove) unfold_goalss unfold_tacss;
  1050             val corec_thmss = map2 (map2 prove) corec_goalss corec_tacss;
  1051 
  1052             val filter_safesss =
  1053               map2 (map_filter (fn (safes, thm) => if forall I safes then SOME thm else NONE) oo
  1054                 curry (op ~~)) (map2 (map2 (map2 (member (op =)))) cgssss crgsss);
  1055 
  1056             val safe_unfold_thmss = filter_safesss unfold_thmss;
  1057             val safe_corec_thmss = filter_safesss corec_thmss;
  1058           in
  1059             (unfold_thmss, corec_thmss, safe_unfold_thmss, safe_corec_thmss)
  1060           end;
  1061 
  1062         val (disc_unfold_iff_thmss, disc_corec_iff_thmss) =
  1063           let
  1064             fun mk_goal c cps fcorec_like n k disc =
  1065               mk_Trueprop_eq (disc $ (fcorec_like $ c),
  1066                 if n = 1 then @{const True}
  1067                 else Library.foldr1 HOLogic.mk_conj (seq_conds mk_maybe_not n k cps));
  1068 
  1069             val unfold_goalss = map6 (map2 oooo mk_goal) cs cpss gunfolds ns kss discss;
  1070             val corec_goalss = map6 (map2 oooo mk_goal) cs cpss hcorecs ns kss discss;
  1071 
  1072             fun mk_case_split' cp =
  1073               Drule.instantiate' [] [SOME (certify lthy cp)] @{thm case_split};
  1074 
  1075             val case_splitss' = map (map mk_case_split') cpss;
  1076 
  1077             val unfold_tacss =
  1078               map3 (map oo mk_disc_corec_like_iff_tac) case_splitss' unfold_thmss disc_thmsss;
  1079             val corec_tacss =
  1080               map3 (map oo mk_disc_corec_like_iff_tac) case_splitss' corec_thmss disc_thmsss;
  1081 
  1082             fun prove goal tac =
  1083               Goal.prove_sorry lthy [] [] goal (tac o #context)
  1084               |> singleton (Proof_Context.export names_lthy0 no_defs_lthy)
  1085               |> Thm.close_derivation;
  1086 
  1087             fun proves [_] [_] = []
  1088               | proves goals tacs = map2 prove goals tacs;
  1089           in
  1090             (map2 proves unfold_goalss unfold_tacss,
  1091              map2 proves corec_goalss corec_tacss)
  1092           end;
  1093 
  1094         val is_triv_discI = is_triv_implies orf is_concl_refl;
  1095 
  1096         fun mk_disc_corec_like_thms corec_likes discIs =
  1097           map (op RS) (filter_out (is_triv_discI o snd) (corec_likes ~~ discIs));
  1098 
  1099         val disc_unfold_thmss = map2 mk_disc_corec_like_thms unfold_thmss discIss;
  1100         val disc_corec_thmss = map2 mk_disc_corec_like_thms corec_thmss discIss;
  1101 
  1102         fun mk_sel_corec_like_thm corec_like_thm sel sel_thm =
  1103           let
  1104             val (domT, ranT) = dest_funT (fastype_of sel);
  1105             val arg_cong' =
  1106               Drule.instantiate' (map (SOME o certifyT lthy) [domT, ranT])
  1107                 [NONE, NONE, SOME (certify lthy sel)] arg_cong
  1108               |> Thm.varifyT_global;
  1109             val sel_thm' = sel_thm RSN (2, trans);
  1110           in
  1111             corec_like_thm RS arg_cong' RS sel_thm'
  1112           end;
  1113 
  1114         fun mk_sel_corec_like_thms corec_likess =
  1115           map3 (map3 (map2 o mk_sel_corec_like_thm)) corec_likess selsss sel_thmsss |> map flat;
  1116 
  1117         val sel_unfold_thmss = mk_sel_corec_like_thms unfold_thmss;
  1118         val sel_corec_thmss = mk_sel_corec_like_thms corec_thmss;
  1119 
  1120         fun flat_corec_like_thms corec_likes disc_corec_likes sel_corec_likes =
  1121           corec_likes @ disc_corec_likes @ sel_corec_likes;
  1122 
  1123         val simp_thmss =
  1124           mk_simp_thmss wrap_ress
  1125             (map3 flat_corec_like_thms safe_corec_thmss disc_corec_thmss sel_corec_thmss)
  1126             (map3 flat_corec_like_thms safe_unfold_thmss disc_unfold_thmss sel_unfold_thmss);
  1127 
  1128         val anonymous_notes =
  1129           [(flat safe_unfold_thmss @ flat safe_corec_thmss, simp_attrs)]
  1130           |> map (fn (thms, attrs) => ((Binding.empty, attrs), [(thms, [])]));
  1131 
  1132         val common_notes =
  1133           (if nn > 1 then
  1134              (* FIXME: attribs *)
  1135              [(coinductN, [coinduct_thm], []),
  1136               (strong_coinductN, [strong_coinduct_thm], [])]
  1137            else
  1138              [])
  1139           |> map (fn (thmN, thms, attrs) =>
  1140             ((qualify true fp_common_name (Binding.name thmN), attrs), [(thms, [])]));
  1141 
  1142         val notes =
  1143           [(coinductN, map single coinduct_thms, []), (* FIXME: attribs *)
  1144            (corecN, corec_thmss, []),
  1145            (disc_corecN, disc_corec_thmss, simp_attrs),
  1146            (disc_corec_iffN, disc_corec_iff_thmss, simp_attrs),
  1147            (disc_unfoldN, disc_unfold_thmss, simp_attrs),
  1148            (disc_unfold_iffN, disc_unfold_iff_thmss, simp_attrs),
  1149            (sel_corecN, sel_corec_thmss, simp_attrs),
  1150            (sel_unfoldN, sel_unfold_thmss, simp_attrs),
  1151            (simpsN, simp_thmss, []),
  1152            (strong_coinductN, map single strong_coinduct_thms, []), (* FIXME: attribs *)
  1153            (unfoldN, unfold_thmss, [])]
  1154           |> maps (fn (thmN, thmss, attrs) =>
  1155             map_filter (fn (_, []) => NONE | (fp_b_name, thms) =>
  1156               SOME ((qualify true fp_b_name (Binding.name thmN), attrs),
  1157                 [(thms, [])])) (fp_b_names ~~ thmss));
  1158       in
  1159         lthy |> Local_Theory.notes (anonymous_notes @ common_notes @ notes) |> snd
  1160       end;
  1161 
  1162     val lthy' = lthy
  1163       |> fold_map define_ctrs_case_for_type (fp_bnfs ~~ fp_bs ~~ fpTs ~~ Cs ~~ ctors ~~ dtors ~~
  1164         fp_folds ~~ fp_recs ~~ ctor_dtors ~~ dtor_ctors ~~ ctor_injects ~~ pre_map_defs ~~
  1165         pre_set_defss ~~ pre_rel_defs ~~ fp_map_thms ~~ fp_set_thmss ~~ fp_rel_thms ~~ ns ~~ kss ~~
  1166         mss ~~ ctr_bindingss ~~ ctr_mixfixess ~~ ctr_Tsss ~~ disc_bindingss ~~ sel_bindingsss ~~
  1167         raw_sel_defaultsss)
  1168       |> wrap_types_and_more
  1169       |> (if lfp then derive_induct_fold_rec_thms_for_types
  1170           else derive_coinduct_unfold_corec_thms_for_types);
  1171 
  1172     val timer = time (timer ("Constructors, discriminators, selectors, etc., for the new " ^
  1173       (if lfp then "" else "co") ^ "datatype"));
  1174   in
  1175     timer; lthy'
  1176   end;
  1177 
  1178 val datatypes = define_datatypes (K I) (K I) (K I);
  1179 
  1180 val datatype_cmd = define_datatypes Typedecl.read_constraint Syntax.parse_typ Syntax.parse_term;
  1181 
  1182 val parse_ctr_arg =
  1183   @{keyword "("} |-- parse_binding_colon -- Parse.typ --| @{keyword ")"} ||
  1184   (Parse.typ >> pair Binding.empty);
  1185 
  1186 val parse_defaults =
  1187   @{keyword "("} |-- @{keyword "defaults"} |-- Scan.repeat parse_bound_term --| @{keyword ")"};
  1188 
  1189 val parse_type_arg_constrained =
  1190   Parse.type_ident -- Scan.option (@{keyword "::"} |-- Parse.!!! Parse.sort)
  1191 
  1192 val parse_type_arg_named_constrained = parse_opt_binding_colon -- parse_type_arg_constrained
  1193 
  1194 val parse_type_args_named_constrained =
  1195   parse_type_arg_constrained >> (single o pair Binding.empty) ||
  1196   @{keyword "("} |-- Parse.!!! (Parse.list1 parse_type_arg_named_constrained --| @{keyword ")"}) ||
  1197   Scan.succeed [];
  1198 
  1199 val parse_single_spec =
  1200   parse_opt_binding_colon -- parse_type_args_named_constrained -- Parse.binding --
  1201   Parse.opt_mixfix -- (@{keyword "="} |-- Parse.enum1 "|" (parse_opt_binding_colon --
  1202       Parse.binding -- Scan.repeat parse_ctr_arg -- Scan.optional parse_defaults [] --
  1203     Parse.opt_mixfix));
  1204 
  1205 val parse_datatype = parse_wrap_options -- Parse.and_list1 parse_single_spec;
  1206 
  1207 fun parse_datatype_cmd lfp construct_fp = parse_datatype >> datatype_cmd lfp construct_fp;
  1208 
  1209 end;