src/HOL/BNF/Tools/bnf_fp_def_sugar.ML
author blanchet
Thu Apr 25 17:13:24 2013 +0200 (2013-04-25)
changeset 51777 48a0ae342ea0
parent 51769 5c657ca97d99
child 51780 67e4ed510dfb
permissions -rw-r--r--
generate proper attributes for coinduction rules
     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 -> 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 * (typ * sort)) list * binding) * (binding * 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 -> 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 val EqN = "Eq";
    35 
    36 (* This function could produce clashes in contrived examples (e.g., "x.A", "x.x_A", "y.A"). *)
    37 fun quasi_unambiguous_case_names names =
    38   let
    39     val ps = map (`Long_Name.base_name) names;
    40     val dups = Library.duplicates (op =) (map fst ps);
    41     fun underscore s =
    42       let val ss = space_explode Long_Name.separator s in
    43         space_implode "_" (drop (length ss - 2) ss)
    44       end;
    45   in
    46     map (fn (base, full) => if member (op =) dups base then underscore full else base) ps
    47   end;
    48 
    49 val mp_conj = @{thm mp_conj};
    50 
    51 val simp_attrs = @{attributes [simp]};
    52 val code_simp_attrs = Code.add_default_eqn_attrib :: simp_attrs;
    53 
    54 fun split_list4 [] = ([], [], [], [])
    55   | split_list4 ((x1, x2, x3, x4) :: xs) =
    56     let val (xs1, xs2, xs3, xs4) = split_list4 xs;
    57     in (x1 :: xs1, x2 :: xs2, x3 :: xs3, x4 :: xs4) end;
    58 
    59 fun resort_tfree S (TFree (s, _)) = TFree (s, S);
    60 
    61 fun typ_subst inst (T as Type (s, Ts)) =
    62     (case AList.lookup (op =) inst T of
    63       NONE => Type (s, map (typ_subst inst) Ts)
    64     | SOME T' => T')
    65   | typ_subst inst T = the_default T (AList.lookup (op =) inst T);
    66 
    67 fun variant_types ss Ss ctxt =
    68   let
    69     val (tfrees, _) =
    70       fold_map2 (fn s => fn S => Name.variant s #> apfst (rpair S)) ss Ss (Variable.names_of ctxt);
    71     val ctxt' = fold (Variable.declare_constraints o Logic.mk_type o TFree) tfrees ctxt;
    72   in (tfrees, ctxt') end;
    73 
    74 val lists_bmoc = fold (fn xs => fn t => Term.list_comb (t, xs));
    75 
    76 fun mk_tupled_fun x f xs = HOLogic.tupled_lambda x (Term.list_comb (f, xs));
    77 fun mk_uncurried_fun f xs = mk_tupled_fun (HOLogic.mk_tuple xs) f xs;
    78 
    79 fun mk_flip (x, Type (_, [T1, Type (_, [T2, T3])])) =
    80   Abs ("x", T1, Abs ("y", T2, Var (x, T2 --> T1 --> T3) $ Bound 0 $ Bound 1));
    81 
    82 fun flip_rels lthy n thm =
    83   let
    84     val Rs = Term.add_vars (prop_of thm) [];
    85     val Rs' = rev (drop (length Rs - n) Rs);
    86     val cRs = map (fn f => (certify lthy (Var f), certify lthy (mk_flip f))) Rs';
    87   in
    88     Drule.cterm_instantiate cRs thm
    89   end;
    90 
    91 fun mk_ctor_or_dtor get_T Ts t =
    92   let val Type (_, Ts0) = get_T (fastype_of t) in
    93     Term.subst_atomic_types (Ts0 ~~ Ts) t
    94   end;
    95 
    96 val mk_ctor = mk_ctor_or_dtor range_type;
    97 val mk_dtor = mk_ctor_or_dtor domain_type;
    98 
    99 fun mk_rec_like lfp Ts Us t =
   100   let
   101     val (bindings, body) = strip_type (fastype_of t);
   102     val (f_Us, prebody) = split_last bindings;
   103     val Type (_, Ts0) = if lfp then prebody else body;
   104     val Us0 = distinct (op =) (map (if lfp then body_type else domain_type) f_Us);
   105   in
   106     Term.subst_atomic_types (Ts0 @ Us0 ~~ Ts @ Us) t
   107   end;
   108 
   109 fun mk_map live Ts Us t =
   110   let val (Type (_, Ts0), Type (_, Us0)) = strip_typeN (live + 1) (fastype_of t) |>> List.last in
   111     Term.subst_atomic_types (Ts0 @ Us0 ~~ Ts @ Us) t
   112   end;
   113 
   114 fun mk_rel live Ts Us t =
   115   let val [Type (_, Ts0), Type (_, Us0)] = binder_types (snd (strip_typeN live (fastype_of t))) in
   116     Term.subst_atomic_types (Ts0 @ Us0 ~~ Ts @ Us) t
   117   end;
   118 
   119 fun liveness_of_fp_bnf n bnf =
   120   (case T_of_bnf bnf of
   121     Type (_, Ts) => map (not o member (op =) (deads_of_bnf bnf)) Ts
   122   | _ => replicate n false);
   123 
   124 fun cannot_merge_types () = error "Mutually recursive types must have the same type parameters";
   125 
   126 fun merge_type_arg T T' = if T = T' then T else cannot_merge_types ();
   127 
   128 fun merge_type_args (As, As') =
   129   if length As = length As' then map2 merge_type_arg As As' else cannot_merge_types ();
   130 
   131 fun reassoc_conjs thm =
   132   reassoc_conjs (thm RS @{thm conj_assoc[THEN iffD1]})
   133   handle THM _ => thm;
   134 
   135 fun type_args_named_constrained_of ((((ncAs, _), _), _), _) = ncAs;
   136 fun type_binding_of ((((_, b), _), _), _) = b;
   137 fun map_binding_of (((_, (b, _)), _), _) = b;
   138 fun rel_binding_of (((_, (_, b)), _), _) = b;
   139 fun mixfix_of ((_, mx), _) = mx;
   140 fun ctr_specs_of (_, ctr_specs) = ctr_specs;
   141 
   142 fun disc_of ((((disc, _), _), _), _) = disc;
   143 fun ctr_of ((((_, ctr), _), _), _) = ctr;
   144 fun args_of (((_, args), _), _) = args;
   145 fun defaults_of ((_, ds), _) = ds;
   146 fun ctr_mixfix_of (_, mx) = mx;
   147 
   148 fun define_datatypes prepare_constraint prepare_typ prepare_term lfp construct_fp
   149     (wrap_opts as (no_dests, rep_compat), specs) no_defs_lthy0 =
   150   let
   151     (* TODO: sanity checks on arguments *)
   152     (* TODO: integration with function package ("size") *)
   153 
   154     val _ = if not lfp andalso no_dests then error "Cannot define destructor-less codatatypes"
   155       else ();
   156 
   157     fun qualify mandatory fp_b_name =
   158       Binding.qualify mandatory fp_b_name o (rep_compat ? Binding.qualify false rep_compat_prefix);
   159 
   160     val nn = length specs;
   161     val fp_bs = map type_binding_of specs;
   162     val fp_b_names = map Binding.name_of fp_bs;
   163     val fp_common_name = mk_common_name fp_b_names;
   164     val map_bs = map map_binding_of specs;
   165     val rel_bs = map rel_binding_of specs;
   166 
   167     fun prepare_type_arg (_, (ty, c)) =
   168       let val TFree (s, _) = prepare_typ no_defs_lthy0 ty in
   169         TFree (s, prepare_constraint no_defs_lthy0 c)
   170       end;
   171 
   172     val Ass0 = map (map prepare_type_arg o type_args_named_constrained_of) specs;
   173     val unsorted_Ass0 = map (map (resort_tfree HOLogic.typeS)) Ass0;
   174     val unsorted_As = Library.foldr1 merge_type_args unsorted_Ass0;
   175     val set_bss = map (map fst o type_args_named_constrained_of) specs;
   176 
   177     val (((Bs0, Cs), Xs), no_defs_lthy) =
   178       no_defs_lthy0
   179       |> fold (Variable.declare_typ o resort_tfree dummyS) unsorted_As
   180       |> mk_TFrees (length unsorted_As)
   181       ||>> mk_TFrees nn
   182       ||>> apfst (map TFree) o
   183         variant_types (map (prefix "'") fp_b_names) (replicate nn HOLogic.typeS);
   184 
   185     (* TODO: cleaner handling of fake contexts, without "background_theory" *)
   186     (*the "perhaps o try" below helps gracefully handles the case where the new type is defined in a
   187       locale and shadows an existing global type*)
   188 
   189     fun add_fake_type spec =
   190       Sign.add_type no_defs_lthy (type_binding_of spec,
   191         length (type_args_named_constrained_of spec), mixfix_of spec);
   192 
   193     val fake_thy = Theory.copy #> fold add_fake_type specs;
   194     val fake_lthy = Proof_Context.background_theory fake_thy no_defs_lthy;
   195 
   196     fun mk_fake_T b =
   197       Type (fst (Term.dest_Type (Proof_Context.read_type_name fake_lthy true (Binding.name_of b))),
   198         unsorted_As);
   199 
   200     val fake_Ts = map mk_fake_T fp_bs;
   201 
   202     val mixfixes = map mixfix_of specs;
   203 
   204     val _ = (case duplicates Binding.eq_name fp_bs of [] => ()
   205       | b :: _ => error ("Duplicate type name declaration " ^ quote (Binding.name_of b)));
   206 
   207     val ctr_specss = map ctr_specs_of specs;
   208 
   209     val disc_bindingss = map (map disc_of) ctr_specss;
   210     val ctr_bindingss =
   211       map2 (fn fp_b_name => map (qualify false fp_b_name o ctr_of)) fp_b_names ctr_specss;
   212     val ctr_argsss = map (map args_of) ctr_specss;
   213     val ctr_mixfixess = map (map ctr_mixfix_of) ctr_specss;
   214 
   215     val sel_bindingsss = map (map (map fst)) ctr_argsss;
   216     val fake_ctr_Tsss0 = map (map (map (prepare_typ fake_lthy o snd))) ctr_argsss;
   217     val raw_sel_defaultsss = map (map defaults_of) ctr_specss;
   218 
   219     val (As :: _) :: fake_ctr_Tsss =
   220       burrow (burrow (Syntax.check_typs fake_lthy)) (Ass0 :: fake_ctr_Tsss0);
   221 
   222     val _ = (case duplicates (op =) unsorted_As of [] => ()
   223       | A :: _ => error ("Duplicate type parameter " ^
   224           quote (Syntax.string_of_typ no_defs_lthy A)));
   225 
   226     val rhs_As' = fold (fold (fold Term.add_tfreesT)) fake_ctr_Tsss [];
   227     val _ = (case subtract (op =) (map dest_TFree As) rhs_As' of
   228         [] => ()
   229       | A' :: _ => error ("Extra type variable on right-hand side: " ^
   230           quote (Syntax.string_of_typ no_defs_lthy (TFree A'))));
   231 
   232     fun eq_fpT_check (T as Type (s, Us)) (Type (s', Us')) =
   233         s = s' andalso (Us = Us' orelse error ("Illegal occurrence of recursive type " ^
   234           quote (Syntax.string_of_typ fake_lthy T)))
   235       | eq_fpT_check _ _ = false;
   236 
   237     fun freeze_fp (T as Type (s, Us)) =
   238         (case find_index (eq_fpT_check T) fake_Ts of
   239           ~1 => Type (s, map freeze_fp Us)
   240         | kk => nth Xs kk)
   241       | freeze_fp T = T;
   242 
   243     val ctr_TsssXs = map (map (map freeze_fp)) fake_ctr_Tsss;
   244     val ctr_sum_prod_TsXs = map (mk_sumTN_balanced o map HOLogic.mk_tupleT) ctr_TsssXs;
   245 
   246     val fp_eqs =
   247       map dest_TFree Xs ~~ map (Term.typ_subst_atomic (As ~~ unsorted_As)) ctr_sum_prod_TsXs;
   248 
   249     (* TODO: clean up list *)
   250     val (pre_bnfs, ((fp_bnfs as any_fp_bnf :: _, dtors0, ctors0, fp_folds0, fp_recs0, fp_induct,
   251            fp_strong_induct, dtor_ctors, ctor_dtors, ctor_injects, fp_map_thms, fp_set_thmss,
   252            fp_rel_thms, fp_fold_thms, fp_rec_thms), lthy)) =
   253       fp_bnf construct_fp fp_bs mixfixes map_bs rel_bs set_bss (map dest_TFree unsorted_As) fp_eqs
   254         no_defs_lthy0;
   255 
   256     val timer = time (Timer.startRealTimer ());
   257 
   258     fun build_map build_arg (Type (s, Ts)) (Type (_, Us)) =
   259       let
   260         val bnf = the (bnf_of lthy s);
   261         val live = live_of_bnf bnf;
   262         val mapx = mk_map live Ts Us (map_of_bnf bnf);
   263         val TUs' = map dest_funT (fst (strip_typeN live (fastype_of mapx)));
   264       in Term.list_comb (mapx, map build_arg TUs') end;
   265 
   266     fun build_rel_step build_arg (Type (s, Ts)) =
   267       let
   268         val bnf = the (bnf_of lthy s);
   269         val live = live_of_bnf bnf;
   270         val rel = mk_rel live Ts Ts (rel_of_bnf bnf);
   271         val Ts' = map domain_type (fst (strip_typeN live (fastype_of rel)));
   272       in Term.list_comb (rel, map build_arg Ts') end;
   273 
   274     fun add_nesty_bnf_names Us =
   275       let
   276         fun add (Type (s, Ts)) ss =
   277             let val (needs, ss') = fold_map add Ts ss in
   278               if exists I needs then (true, insert (op =) s ss') else (false, ss')
   279             end
   280           | add T ss = (member (op =) Us T, ss);
   281       in snd oo add end;
   282 
   283     fun nesty_bnfs Us =
   284       map_filter (bnf_of lthy) (fold (fold (fold (add_nesty_bnf_names Us))) ctr_TsssXs []);
   285 
   286     val nesting_bnfs = nesty_bnfs As;
   287     val nested_bnfs = nesty_bnfs Xs;
   288 
   289     val pre_map_defs = map map_def_of_bnf pre_bnfs;
   290     val pre_set_defss = map set_defs_of_bnf pre_bnfs;
   291     val pre_rel_defs = map rel_def_of_bnf pre_bnfs;
   292     val nested_map_comps'' = map ((fn thm => thm RS sym) o map_comp_of_bnf) nested_bnfs;
   293     val nested_map_comp's = map map_comp'_of_bnf nested_bnfs;
   294     val nested_map_ids'' = map (unfold_thms lthy @{thms id_def} o map_id_of_bnf) nested_bnfs;
   295     val nesting_map_ids'' = map (unfold_thms lthy @{thms id_def} o map_id_of_bnf) nesting_bnfs;
   296     val nested_set_map's = maps set_map'_of_bnf nested_bnfs;
   297     val nesting_set_map's = maps set_map'_of_bnf nesting_bnfs;
   298 
   299     val live = live_of_bnf any_fp_bnf;
   300 
   301     val Bs =
   302       map3 (fn alive => fn A as TFree (_, S) => fn B => if alive then resort_tfree S B else A)
   303         (liveness_of_fp_bnf (length As) any_fp_bnf) As Bs0;
   304 
   305     val B_ify = Term.typ_subst_atomic (As ~~ Bs);
   306 
   307     val ctors = map (mk_ctor As) ctors0;
   308     val dtors = map (mk_dtor As) dtors0;
   309 
   310     val fpTs = map (domain_type o fastype_of) dtors;
   311 
   312     val exists_fp_subtype = exists_subtype (member (op =) fpTs);
   313     val exists_Cs_subtype = exists_subtype (member (op =) Cs);
   314 
   315     val ctr_Tsss = map (map (map (Term.typ_subst_atomic (Xs ~~ fpTs)))) ctr_TsssXs;
   316     val ns = map length ctr_Tsss;
   317     val kss = map (fn n => 1 upto n) ns;
   318     val mss = map (map length) ctr_Tsss;
   319     val Css = map2 replicate ns Cs;
   320 
   321     val fp_folds as any_fp_fold :: _ = map (mk_rec_like lfp As Cs) fp_folds0;
   322     val fp_recs as any_fp_rec :: _ = map (mk_rec_like lfp As Cs) fp_recs0;
   323 
   324     val fp_fold_fun_Ts = fst (split_last (binder_types (fastype_of any_fp_fold)));
   325     val fp_rec_fun_Ts = fst (split_last (binder_types (fastype_of any_fp_rec)));
   326 
   327     val (((fold_only as (gss, _, _), rec_only as (hss, _, _)),
   328           (cs, cpss, unfold_only as ((pgss, crssss, cgssss), (_, g_Tsss, _)),
   329            corec_only as ((phss, csssss, chssss), (_, h_Tsss, _)))), names_lthy0) =
   330       if lfp then
   331         let
   332           val y_Tsss =
   333             map3 (fn n => fn ms => map2 dest_tupleT ms o dest_sumTN_balanced n o domain_type)
   334               ns mss fp_fold_fun_Ts;
   335           val g_Tss = map2 (map2 (curry (op --->))) y_Tsss Css;
   336 
   337           val ((gss, ysss), lthy) =
   338             lthy
   339             |> mk_Freess "f" g_Tss
   340             ||>> mk_Freesss "x" y_Tsss;
   341 
   342           fun proj_recT proj (Type (s as @{type_name prod}, Ts as [T, U])) =
   343               if member (op =) fpTs T then proj (T, U) else Type (s, map (proj_recT proj) Ts)
   344             | proj_recT proj (Type (s, Ts)) = Type (s, map (proj_recT proj) Ts)
   345             | proj_recT _ T = T;
   346 
   347           fun unzip_recT T =
   348             if exists_fp_subtype T then [proj_recT fst T, proj_recT snd T] else [T];
   349 
   350           val z_Tsss =
   351             map3 (fn n => fn ms => map2 dest_tupleT ms o dest_sumTN_balanced n o domain_type)
   352               ns mss fp_rec_fun_Ts;
   353           val z_Tssss = map (map (map unzip_recT)) z_Tsss;
   354           val h_Tss = map2 (map2 (fold_rev (curry (op --->)))) z_Tssss Css;
   355 
   356           val hss = map2 (map2 retype_free) h_Tss gss;
   357           val zsss = map2 (map2 (map2 retype_free)) z_Tsss ysss;
   358         in
   359           ((((gss, g_Tss, ysss), (hss, h_Tss, zsss)),
   360             ([], [], (([], [], []), ([], [], [])), (([], [], []), ([], [], [])))), lthy)
   361         end
   362       else
   363         let
   364           (*avoid "'a itself" arguments in coiterators and corecursors*)
   365           val mss' =  map (fn [0] => [1] | ms => ms) mss;
   366 
   367           val p_Tss = map2 (fn n => replicate (Int.max (0, n - 1)) o mk_pred1T) ns Cs;
   368 
   369           fun flat_predss_getterss qss fss = maps (op @) (qss ~~ fss);
   370 
   371           fun flat_preds_predsss_gettersss [] [qss] [fss] = flat_predss_getterss qss fss
   372             | flat_preds_predsss_gettersss (p :: ps) (qss :: qsss) (fss :: fsss) =
   373               p :: flat_predss_getterss qss fss @ flat_preds_predsss_gettersss ps qsss fsss;
   374 
   375           fun mk_types maybe_unzipT fun_Ts =
   376             let
   377               val f_sum_prod_Ts = map range_type fun_Ts;
   378               val f_prod_Tss = map2 dest_sumTN_balanced ns f_sum_prod_Ts;
   379               val f_Tsss = map2 (map2 dest_tupleT) mss' f_prod_Tss;
   380               val f_Tssss =
   381                 map2 (fn C => map (map (map (curry (op -->) C) o maybe_unzipT))) Cs f_Tsss;
   382               val q_Tssss =
   383                 map (map (map (fn [_] => [] | [_, C] => [mk_pred1T (domain_type C)]))) f_Tssss;
   384               val pf_Tss = map3 flat_preds_predsss_gettersss p_Tss q_Tssss f_Tssss;
   385             in (q_Tssss, f_sum_prod_Ts, f_Tsss, f_Tssss, pf_Tss) end;
   386 
   387           val (r_Tssss, g_sum_prod_Ts, g_Tsss, g_Tssss, pg_Tss) = mk_types single fp_fold_fun_Ts;
   388 
   389           val (((cs, pss), gssss), lthy) =
   390             lthy
   391             |> mk_Frees "a" Cs
   392             ||>> mk_Freess "p" p_Tss
   393             ||>> mk_Freessss "g" g_Tssss;
   394           val rssss = map (map (map (fn [] => []))) r_Tssss;
   395 
   396           fun proj_corecT proj (Type (s as @{type_name sum}, Ts as [T, U])) =
   397               if member (op =) fpTs T then proj (T, U) else Type (s, map (proj_corecT proj) Ts)
   398             | proj_corecT proj (Type (s, Ts)) = Type (s, map (proj_corecT proj) Ts)
   399             | proj_corecT _ T = T;
   400 
   401           fun unzip_corecT T =
   402             if exists_fp_subtype T then [proj_corecT fst T, proj_corecT snd T] else [T];
   403 
   404           val (s_Tssss, h_sum_prod_Ts, h_Tsss, h_Tssss, ph_Tss) =
   405             mk_types unzip_corecT fp_rec_fun_Ts;
   406 
   407           val hssss_hd = map2 (map2 (map2 (fn T :: _ => fn [g] => retype_free T g))) h_Tssss gssss;
   408           val ((sssss, hssss_tl), lthy) =
   409             lthy
   410             |> mk_Freessss "q" s_Tssss
   411             ||>> mk_Freessss "h" (map (map (map tl)) h_Tssss);
   412           val hssss = map2 (map2 (map2 cons)) hssss_hd hssss_tl;
   413 
   414           val cpss = map2 (map o rapp) cs pss;
   415 
   416           fun mk_terms qssss fssss =
   417             let
   418               val pfss = map3 flat_preds_predsss_gettersss pss qssss fssss;
   419               val cqssss = map2 (map o map o map o rapp) cs qssss;
   420               val cfssss = map2 (map o map o map o rapp) cs fssss;
   421             in (pfss, cqssss, cfssss) end;
   422         in
   423           (((([], [], []), ([], [], [])),
   424             (cs, cpss, (mk_terms rssss gssss, (g_sum_prod_Ts, g_Tsss, pg_Tss)),
   425              (mk_terms sssss hssss, (h_sum_prod_Ts, h_Tsss, ph_Tss)))), lthy)
   426         end;
   427 
   428     fun define_ctrs_case_for_type (((((((((((((((((((((((((fp_bnf, fp_b), fpT), C), ctor), dtor),
   429             fp_fold), fp_rec), ctor_dtor), dtor_ctor), ctor_inject), pre_map_def), pre_set_defs),
   430           pre_rel_def), fp_map_thm), fp_set_thms), fp_rel_thm), n), ks), ms), ctr_bindings),
   431         ctr_mixfixes), ctr_Tss), disc_bindings), sel_bindingss), raw_sel_defaultss) no_defs_lthy =
   432       let
   433         val fp_b_name = Binding.name_of fp_b;
   434 
   435         val dtorT = domain_type (fastype_of ctor);
   436         val ctr_prod_Ts = map HOLogic.mk_tupleT ctr_Tss;
   437         val ctr_sum_prod_T = mk_sumTN_balanced ctr_prod_Ts;
   438         val case_Ts = map (fn Ts => Ts ---> C) ctr_Tss;
   439 
   440         val (((((w, fs), xss), yss), u'), names_lthy) =
   441           no_defs_lthy
   442           |> yield_singleton (mk_Frees "w") dtorT
   443           ||>> mk_Frees "f" case_Ts
   444           ||>> mk_Freess "x" ctr_Tss
   445           ||>> mk_Freess "y" (map (map B_ify) ctr_Tss)
   446           ||>> yield_singleton Variable.variant_fixes fp_b_name;
   447 
   448         val u = Free (u', fpT);
   449 
   450         val tuple_xs = map HOLogic.mk_tuple xss;
   451         val tuple_ys = map HOLogic.mk_tuple yss;
   452 
   453         val ctr_rhss =
   454           map3 (fn k => fn xs => fn tuple_x => fold_rev Term.lambda xs (ctor $
   455             mk_InN_balanced ctr_sum_prod_T n tuple_x k)) ks xss tuple_xs;
   456 
   457         val case_binding = qualify false fp_b_name (Binding.suffix_name ("_" ^ caseN) fp_b);
   458 
   459         val case_rhs =
   460           fold_rev Term.lambda (fs @ [u])
   461             (mk_sum_caseN_balanced (map2 mk_uncurried_fun fs xss) $ (dtor $ u));
   462 
   463         val ((raw_case :: raw_ctrs, raw_case_def :: raw_ctr_defs), (lthy', lthy)) = no_defs_lthy
   464           |> apfst split_list o fold_map3 (fn b => fn mx => fn rhs =>
   465               Local_Theory.define ((b, mx), ((Thm.def_binding b, []), rhs)) #>> apsnd snd)
   466             (case_binding :: ctr_bindings) (NoSyn :: ctr_mixfixes) (case_rhs :: ctr_rhss)
   467           ||> `Local_Theory.restore;
   468 
   469         val phi = Proof_Context.export_morphism lthy lthy';
   470 
   471         val ctr_defs = map (Morphism.thm phi) raw_ctr_defs;
   472         val ctr_defs' =
   473           map2 (fn m => fn def => mk_unabs_def m (def RS meta_eq_to_obj_eq)) ms ctr_defs;
   474         val case_def = Morphism.thm phi raw_case_def;
   475 
   476         val ctrs0 = map (Morphism.term phi) raw_ctrs;
   477         val casex0 = Morphism.term phi raw_case;
   478 
   479         val ctrs = map (mk_ctr As) ctrs0;
   480 
   481         fun wrap lthy =
   482           let
   483             fun exhaust_tac {context = ctxt, prems = _} =
   484               let
   485                 val ctor_iff_dtor_thm =
   486                   let
   487                     val goal =
   488                       fold_rev Logic.all [w, u]
   489                         (mk_Trueprop_eq (HOLogic.mk_eq (u, ctor $ w), HOLogic.mk_eq (dtor $ u, w)));
   490                   in
   491                     Goal.prove_sorry lthy [] [] goal (fn {context = ctxt, ...} =>
   492                       mk_ctor_iff_dtor_tac ctxt (map (SOME o certifyT lthy) [dtorT, fpT])
   493                         (certify lthy ctor) (certify lthy dtor) ctor_dtor dtor_ctor)
   494                     |> Thm.close_derivation
   495                     |> Morphism.thm phi
   496                   end;
   497 
   498                 val sumEN_thm' =
   499                   unfold_thms lthy @{thms all_unit_eq}
   500                     (Drule.instantiate' (map (SOME o certifyT lthy) ctr_prod_Ts) []
   501                        (mk_sumEN_balanced n))
   502                   |> Morphism.thm phi;
   503               in
   504                 mk_exhaust_tac ctxt n ctr_defs ctor_iff_dtor_thm sumEN_thm'
   505               end;
   506 
   507             val inject_tacss =
   508               map2 (fn 0 => K [] | _ => fn ctr_def => [fn {context = ctxt, ...} =>
   509                   mk_inject_tac ctxt ctr_def ctor_inject]) ms ctr_defs;
   510 
   511             val half_distinct_tacss =
   512               map (map (fn (def, def') => fn {context = ctxt, ...} =>
   513                 mk_half_distinct_tac ctxt ctor_inject [def, def'])) (mk_half_pairss (`I ctr_defs));
   514 
   515             val case_tacs =
   516               map3 (fn k => fn m => fn ctr_def => fn {context = ctxt, ...} =>
   517                 mk_case_tac ctxt n k m case_def ctr_def dtor_ctor) ks ms ctr_defs;
   518 
   519             val tacss = [exhaust_tac] :: inject_tacss @ half_distinct_tacss @ [case_tacs];
   520 
   521             val sel_defaultss = map (map (apsnd (prepare_term lthy))) raw_sel_defaultss
   522           in
   523             wrap_datatype tacss (((wrap_opts, ctrs0), casex0), (disc_bindings, (sel_bindingss,
   524               sel_defaultss))) lthy
   525           end;
   526 
   527         fun derive_maps_sets_rels (wrap_res, lthy) =
   528           let
   529             val rel_flip = rel_flip_of_bnf fp_bnf;
   530             val nones = replicate live NONE;
   531 
   532             val ctor_cong =
   533               if lfp then
   534                 Drule.dummy_thm
   535               else
   536                 let val ctor' = mk_ctor Bs ctor in
   537                   cterm_instantiate_pos [NONE, NONE, SOME (certify lthy ctor')] arg_cong
   538                 end;
   539 
   540             fun mk_cIn ify =
   541               certify lthy o (not lfp ? curry (op $) (map_types ify ctor)) oo
   542               mk_InN_balanced (ify ctr_sum_prod_T) n;
   543 
   544             val cxIns = map2 (mk_cIn I) tuple_xs ks;
   545             val cyIns = map2 (mk_cIn B_ify) tuple_ys ks;
   546 
   547             fun mk_map_thm ctr_def' cxIn =
   548               fold_thms lthy [ctr_def']
   549                 (unfold_thms lthy (pre_map_def ::
   550                      (if lfp then [] else [ctor_dtor, dtor_ctor]) @ sum_prod_thms_map)
   551                    (cterm_instantiate_pos (nones @ [SOME cxIn])
   552                       (if lfp then fp_map_thm else fp_map_thm RS ctor_cong)))
   553               |> singleton (Proof_Context.export names_lthy no_defs_lthy);
   554 
   555             fun mk_set_thm fp_set_thm ctr_def' cxIn =
   556               fold_thms lthy [ctr_def']
   557                 (unfold_thms lthy (pre_set_defs @ nested_set_map's @ nesting_set_map's @
   558                      (if lfp then [] else [dtor_ctor]) @ sum_prod_thms_set)
   559                    (cterm_instantiate_pos [SOME cxIn] fp_set_thm))
   560               |> singleton (Proof_Context.export names_lthy no_defs_lthy);
   561 
   562             fun mk_set_thms fp_set_thm = map2 (mk_set_thm fp_set_thm) ctr_defs' cxIns;
   563 
   564             val map_thms = map2 mk_map_thm ctr_defs' cxIns;
   565             val set_thmss = map mk_set_thms fp_set_thms;
   566 
   567             val rel_infos = (ctr_defs' ~~ cxIns, ctr_defs' ~~ cyIns);
   568 
   569             fun mk_rel_thm postproc ctr_defs' cxIn cyIn =
   570               fold_thms lthy ctr_defs'
   571                  (unfold_thms lthy (@{thm Inl_Inr_False} :: pre_rel_def ::
   572                       (if lfp then [] else [dtor_ctor]) @ sum_prod_thms_rel)
   573                     (cterm_instantiate_pos (nones @ [SOME cxIn, SOME cyIn]) fp_rel_thm))
   574               |> postproc
   575               |> singleton (Proof_Context.export names_lthy no_defs_lthy);
   576 
   577             fun mk_rel_inject_thm ((ctr_def', cxIn), (_, cyIn)) =
   578               mk_rel_thm (unfold_thms lthy @{thms eq_sym_Unity_conv}) [ctr_def'] cxIn cyIn;
   579 
   580             val rel_inject_thms = map mk_rel_inject_thm (op ~~ rel_infos);
   581 
   582             fun mk_half_rel_distinct_thm ((xctr_def', cxIn), (yctr_def', cyIn)) =
   583               mk_rel_thm (fn thm => thm RS @{thm eq_False[THEN iffD1]}) [xctr_def', yctr_def']
   584                 cxIn cyIn;
   585 
   586             fun mk_other_half_rel_distinct_thm thm =
   587               flip_rels lthy live thm RS (rel_flip RS sym RS @{thm arg_cong[of _ _ Not]} RS iffD2);
   588 
   589             val half_rel_distinct_thmss =
   590               map (map mk_half_rel_distinct_thm) (mk_half_pairss rel_infos);
   591             val other_half_rel_distinct_thmss =
   592               map (map mk_other_half_rel_distinct_thm) half_rel_distinct_thmss;
   593             val (rel_distinct_thms, _) =
   594               join_halves n half_rel_distinct_thmss other_half_rel_distinct_thmss;
   595 
   596             val notes =
   597               [(mapN, map_thms, code_simp_attrs),
   598                (rel_distinctN, rel_distinct_thms, code_simp_attrs),
   599                (rel_injectN, rel_inject_thms, code_simp_attrs),
   600                (setsN, flat set_thmss, code_simp_attrs)]
   601               |> filter_out (null o #2)
   602               |> map (fn (thmN, thms, attrs) =>
   603                 ((qualify true fp_b_name (Binding.name thmN), attrs), [(thms, [])]));
   604           in
   605             (wrap_res, lthy |> Local_Theory.notes notes |> snd)
   606           end;
   607 
   608         fun define_fold_rec no_defs_lthy =
   609           let
   610             val fpT_to_C = fpT --> C;
   611 
   612             fun build_prod_proj mk_proj (T, U) =
   613               if T = U then
   614                 id_const T
   615               else
   616                 (case (T, U) of
   617                   (Type (s, _), Type (s', _)) =>
   618                   if s = s' then build_map (build_prod_proj mk_proj) T U else mk_proj T
   619                 | _ => mk_proj T);
   620 
   621             (* TODO: Avoid these complications; cf. corec case *)
   622             fun mk_U proj (Type (s as @{type_name prod}, Ts as [T', U])) =
   623                 if member (op =) fpTs T' then proj (T', U) else Type (s, map (mk_U proj) Ts)
   624               | mk_U proj (Type (s, Ts)) = Type (s, map (mk_U proj) Ts)
   625               | mk_U _ T = T;
   626 
   627             fun unzip_rec (x as Free (_, T)) =
   628               if exists_fp_subtype T then
   629                 [build_prod_proj fst_const (T, mk_U fst T) $ x,
   630                  build_prod_proj snd_const (T, mk_U snd T) $ x]
   631               else
   632                 [x];
   633 
   634             fun mk_rec_like_arg f xs = mk_tupled_fun (HOLogic.mk_tuple xs) f (maps unzip_rec xs);
   635 
   636             fun generate_rec_like (suf, fp_rec_like, (fss, f_Tss, xsss)) =
   637               let
   638                 val res_T = fold_rev (curry (op --->)) f_Tss fpT_to_C;
   639                 val binding = qualify false fp_b_name (Binding.suffix_name ("_" ^ suf) fp_b);
   640                 val spec =
   641                   mk_Trueprop_eq (lists_bmoc fss (Free (Binding.name_of binding, res_T)),
   642                     Term.list_comb (fp_rec_like,
   643                       map2 (mk_sum_caseN_balanced oo map2 mk_rec_like_arg) fss xsss));
   644               in (binding, spec) end;
   645 
   646             val rec_like_infos =
   647               [(foldN, fp_fold, fold_only),
   648                (recN, fp_rec, rec_only)];
   649 
   650             val (bindings, specs) = map generate_rec_like rec_like_infos |> split_list;
   651 
   652             val ((csts, defs), (lthy', lthy)) = no_defs_lthy
   653               |> apfst split_list o fold_map2 (fn b => fn spec =>
   654                 Specification.definition (SOME (b, NONE, NoSyn), ((Thm.def_binding b, []), spec))
   655                 #>> apsnd snd) bindings specs
   656               ||> `Local_Theory.restore;
   657 
   658             val phi = Proof_Context.export_morphism lthy lthy';
   659 
   660             val [fold_def, rec_def] = map (Morphism.thm phi) defs;
   661 
   662             val [foldx, recx] = map (mk_rec_like lfp As Cs o Morphism.term phi) csts;
   663           in
   664             ((foldx, recx, fold_def, rec_def), lthy')
   665           end;
   666 
   667         fun define_unfold_corec no_defs_lthy =
   668           let
   669             val B_to_fpT = C --> fpT;
   670 
   671             fun build_sum_inj mk_inj (T, U) =
   672               if T = U then
   673                 id_const T
   674               else
   675                 (case (T, U) of
   676                   (Type (s, _), Type (s', _)) =>
   677                   if s = s' then build_map (build_sum_inj mk_inj) T U
   678                   else uncurry mk_inj (dest_sumT U)
   679                 | _ => uncurry mk_inj (dest_sumT U));
   680 
   681             fun build_dtor_corec_like_arg _ [] [cf] = cf
   682               | build_dtor_corec_like_arg T [cq] [cf, cf'] =
   683                 mk_If cq (build_sum_inj Inl_const (fastype_of cf, T) $ cf)
   684                   (build_sum_inj Inr_const (fastype_of cf', T) $ cf')
   685 
   686             val crgsss = map3 (map3 (map3 build_dtor_corec_like_arg)) g_Tsss crssss cgssss;
   687             val cshsss = map3 (map3 (map3 build_dtor_corec_like_arg)) h_Tsss csssss chssss;
   688 
   689             fun mk_preds_getterss_join c n cps sum_prod_T cqfss =
   690               Term.lambda c (mk_IfN sum_prod_T cps
   691                 (map2 (mk_InN_balanced sum_prod_T n) (map HOLogic.mk_tuple cqfss) (1 upto n)));
   692 
   693             fun generate_corec_like (suf, fp_rec_like, (cqfsss, ((pfss, _, _), (f_sum_prod_Ts, _,
   694                 pf_Tss)))) =
   695               let
   696                 val res_T = fold_rev (curry (op --->)) pf_Tss B_to_fpT;
   697                 val binding = qualify false fp_b_name (Binding.suffix_name ("_" ^ suf) fp_b);
   698                 val spec =
   699                   mk_Trueprop_eq (lists_bmoc pfss (Free (Binding.name_of binding, res_T)),
   700                     Term.list_comb (fp_rec_like,
   701                       map5 mk_preds_getterss_join cs ns cpss f_sum_prod_Ts cqfsss));
   702               in (binding, spec) end;
   703 
   704             val corec_like_infos =
   705               [(unfoldN, fp_fold, (crgsss, unfold_only)),
   706                (corecN, fp_rec, (cshsss, corec_only))];
   707 
   708             val (bindings, specs) = map generate_corec_like corec_like_infos |> split_list;
   709 
   710             val ((csts, defs), (lthy', lthy)) = no_defs_lthy
   711               |> apfst split_list o fold_map2 (fn b => fn spec =>
   712                 Specification.definition (SOME (b, NONE, NoSyn), ((Thm.def_binding b, []), spec))
   713                 #>> apsnd snd) bindings specs
   714               ||> `Local_Theory.restore;
   715 
   716             val phi = Proof_Context.export_morphism lthy lthy';
   717 
   718             val [unfold_def, corec_def] = map (Morphism.thm phi) defs;
   719 
   720             val [unfold, corec] = map (mk_rec_like lfp As Cs o Morphism.term phi) csts;
   721           in
   722             ((unfold, corec, unfold_def, corec_def), lthy')
   723           end;
   724 
   725         val define_rec_likes = if lfp then define_fold_rec else define_unfold_corec;
   726 
   727         fun massage_res ((wrap_res, rec_like_res), lthy) =
   728           (((ctrs, xss, ctr_defs, wrap_res), rec_like_res), lthy);
   729       in
   730         (wrap #> (live > 0 ? derive_maps_sets_rels) ##>> define_rec_likes #> massage_res, lthy')
   731       end;
   732 
   733     fun wrap_types_and_more (wrap_types_and_mores, lthy) =
   734       fold_map I wrap_types_and_mores lthy
   735       |>> apsnd split_list4 o apfst split_list4 o split_list;
   736 
   737     (* TODO: Add map, sets, rel simps *)
   738     val mk_simp_thmss =
   739       map3 (fn (_, _, _, injects, distincts, cases, _, _, _) => fn rec_likes => fn fold_likes =>
   740         injects @ distincts @ cases @ rec_likes @ fold_likes);
   741 
   742     fun derive_induct_fold_rec_thms_for_types (((ctrss, xsss, ctr_defss, wrap_ress), (folds, recs,
   743         fold_defs, rec_defs)), lthy) =
   744       let
   745         val (((ps, ps'), us'), names_lthy) =
   746           lthy
   747           |> mk_Frees' "P" (map mk_pred1T fpTs)
   748           ||>> Variable.variant_fixes fp_b_names;
   749 
   750         val us = map2 (curry Free) us' fpTs;
   751 
   752         fun mk_sets_nested bnf =
   753           let
   754             val Type (T_name, Us) = T_of_bnf bnf;
   755             val lives = lives_of_bnf bnf;
   756             val sets = sets_of_bnf bnf;
   757             fun mk_set U =
   758               (case find_index (curry (op =) U) lives of
   759                 ~1 => Term.dummy
   760               | i => nth sets i);
   761           in
   762             (T_name, map mk_set Us)
   763           end;
   764 
   765         val setss_nested = map mk_sets_nested nested_bnfs;
   766 
   767         val (induct_thms, induct_thm) =
   768           let
   769             fun mk_set Ts t =
   770               let val Type (_, Ts0) = domain_type (fastype_of t) in
   771                 Term.subst_atomic_types (Ts0 ~~ Ts) t
   772               end;
   773 
   774             fun mk_raw_prem_prems names_lthy (x as Free (s, T as Type (T_name, Ts0))) =
   775                 (case find_index (curry (op =) T) fpTs of
   776                   ~1 =>
   777                   (case AList.lookup (op =) setss_nested T_name of
   778                     NONE => []
   779                   | SOME raw_sets0 =>
   780                     let
   781                       val (Ts, raw_sets) =
   782                         split_list (filter (exists_fp_subtype o fst) (Ts0 ~~ raw_sets0));
   783                       val sets = map (mk_set Ts0) raw_sets;
   784                       val (ys, names_lthy') = names_lthy |> mk_Frees s Ts;
   785                       val xysets = map (pair x) (ys ~~ sets);
   786                       val ppremss = map (mk_raw_prem_prems names_lthy') ys;
   787                     in
   788                       flat (map2 (map o apfst o cons) xysets ppremss)
   789                     end)
   790                 | kk => [([], (kk + 1, x))])
   791               | mk_raw_prem_prems _ _ = [];
   792 
   793             fun close_prem_prem xs t =
   794               fold_rev Logic.all (map Free (drop (nn + length xs)
   795                 (rev (Term.add_frees t (map dest_Free xs @ ps'))))) t;
   796 
   797             fun mk_prem_prem xs (xysets, (j, x)) =
   798               close_prem_prem xs (Logic.list_implies (map (fn (x', (y, set)) =>
   799                   HOLogic.mk_Trueprop (HOLogic.mk_mem (y, set $ x'))) xysets,
   800                 HOLogic.mk_Trueprop (nth ps (j - 1) $ x)));
   801 
   802             fun mk_raw_prem phi ctr ctr_Ts =
   803               let
   804                 val (xs, names_lthy') = names_lthy |> mk_Frees "x" ctr_Ts;
   805                 val pprems = maps (mk_raw_prem_prems names_lthy') xs;
   806               in (xs, pprems, HOLogic.mk_Trueprop (phi $ Term.list_comb (ctr, xs))) end;
   807 
   808             fun mk_prem (xs, raw_pprems, concl) =
   809               fold_rev Logic.all xs (Logic.list_implies (map (mk_prem_prem xs) raw_pprems, concl));
   810 
   811             val raw_premss = map3 (map2 o mk_raw_prem) ps ctrss ctr_Tsss;
   812 
   813             val goal =
   814               Library.foldr (Logic.list_implies o apfst (map mk_prem)) (raw_premss,
   815                 HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj (map2 (curry (op $)) ps us)));
   816 
   817             val kksss = map (map (map (fst o snd) o #2)) raw_premss;
   818 
   819             val ctor_induct' = fp_induct OF (map mk_sumEN_tupled_balanced mss);
   820 
   821             val thm =
   822               Goal.prove_sorry lthy [] [] goal (fn {context = ctxt, ...} =>
   823                 mk_induct_tac ctxt nn ns mss kksss (flat ctr_defss) ctor_induct' nested_set_map's
   824                   pre_set_defss)
   825               |> singleton (Proof_Context.export names_lthy lthy)
   826               |> Thm.close_derivation;
   827           in
   828             `(conj_dests nn) thm
   829           end;
   830 
   831         val induct_cases = quasi_unambiguous_case_names (maps (map name_of_ctr) ctrss);
   832 
   833         val (fold_thmss, rec_thmss) =
   834           let
   835             val xctrss = map2 (map2 (curry Term.list_comb)) ctrss xsss;
   836             val gfolds = map (lists_bmoc gss) folds;
   837             val hrecs = map (lists_bmoc hss) recs;
   838 
   839             fun mk_goal fss frec_like xctr f xs fxs =
   840               fold_rev (fold_rev Logic.all) (xs :: fss)
   841                 (mk_Trueprop_eq (frec_like $ xctr, Term.list_comb (f, fxs)));
   842 
   843             fun build_rec_like frec_likes (T, U) =
   844               if T = U then
   845                 id_const T
   846               else
   847                 (case find_index (curry (op =) T) fpTs of
   848                   ~1 => build_map (build_rec_like frec_likes) T U
   849                 | kk => nth frec_likes kk);
   850 
   851             val mk_U = typ_subst (map2 pair fpTs Cs);
   852 
   853             fun intr_rec_likes frec_likes maybe_cons (x as Free (_, T)) =
   854               if exists_fp_subtype T then
   855                 maybe_cons x [build_rec_like frec_likes (T, mk_U T) $ x]
   856               else
   857                 [x];
   858 
   859             val gxsss = map (map (maps (intr_rec_likes gfolds (K I)))) xsss;
   860             val hxsss = map (map (maps (intr_rec_likes hrecs cons))) xsss;
   861 
   862             val fold_goalss = map5 (map4 o mk_goal gss) gfolds xctrss gss xsss gxsss;
   863             val rec_goalss = map5 (map4 o mk_goal hss) hrecs xctrss hss xsss hxsss;
   864 
   865             val fold_tacss =
   866               map2 (map o mk_rec_like_tac pre_map_defs [] nesting_map_ids'' fold_defs) fp_fold_thms
   867                 ctr_defss;
   868             val rec_tacss =
   869               map2 (map o mk_rec_like_tac pre_map_defs nested_map_comp's
   870                 (nested_map_ids'' @ nesting_map_ids'') rec_defs) fp_rec_thms ctr_defss;
   871 
   872             fun prove goal tac =
   873               Goal.prove_sorry lthy [] [] goal (tac o #context)
   874               |> Thm.close_derivation;
   875           in
   876             (map2 (map2 prove) fold_goalss fold_tacss, map2 (map2 prove) rec_goalss rec_tacss)
   877           end;
   878 
   879         val simp_thmss = mk_simp_thmss wrap_ress rec_thmss fold_thmss;
   880 
   881         val induct_case_names_attr = Attrib.internal (K (Rule_Cases.case_names induct_cases));
   882         fun induct_type_attr T_name = Attrib.internal (K (Induct.induct_type T_name));
   883 
   884         val common_notes =
   885           (if nn > 1 then [(inductN, [induct_thm], [induct_case_names_attr])] else [])
   886           |> map (fn (thmN, thms, attrs) =>
   887             ((qualify true fp_common_name (Binding.name thmN), attrs), [(thms, [])]));
   888 
   889         val notes =
   890           [(foldN, fold_thmss, K code_simp_attrs),
   891            (inductN, map single induct_thms,
   892             fn T_name => [induct_case_names_attr, induct_type_attr T_name]),
   893            (recN, rec_thmss, K code_simp_attrs),
   894            (simpsN, simp_thmss, K [])]
   895           |> maps (fn (thmN, thmss, attrs) =>
   896             map3 (fn fp_b_name => fn Type (T_name, _) => fn thms =>
   897               ((qualify true fp_b_name (Binding.name thmN), attrs T_name),
   898                [(thms, [])])) fp_b_names fpTs thmss);
   899       in
   900         lthy |> Local_Theory.notes (common_notes @ notes) |> snd
   901       end;
   902 
   903     fun derive_coinduct_unfold_corec_thms_for_types (((ctrss, _, ctr_defss, wrap_ress), (unfolds,
   904         corecs, unfold_defs, corec_defs)), lthy) =
   905       let
   906         val nesting_rel_eqs = map rel_eq_of_bnf nesting_bnfs;
   907 
   908         val discss = map (map (mk_disc_or_sel As) o #1) wrap_ress;
   909         val selsss = map (map (map (mk_disc_or_sel As)) o #2) wrap_ress;
   910         val exhaust_thms = map #3 wrap_ress;
   911         val disc_thmsss = map #7 wrap_ress;
   912         val discIss = map #8 wrap_ress;
   913         val sel_thmsss = map #9 wrap_ress;
   914 
   915         val (((rs, us'), vs'), names_lthy) =
   916           lthy
   917           |> mk_Frees "R" (map (fn T => mk_pred2T T T) fpTs)
   918           ||>> Variable.variant_fixes fp_b_names
   919           ||>> Variable.variant_fixes (map (suffix "'") fp_b_names);
   920 
   921         val us = map2 (curry Free) us' fpTs;
   922         val udiscss = map2 (map o rapp) us discss;
   923         val uselsss = map2 (map o map o rapp) us selsss;
   924 
   925         val vs = map2 (curry Free) vs' fpTs;
   926         val vdiscss = map2 (map o rapp) vs discss;
   927         val vselsss = map2 (map o map o rapp) vs selsss;
   928 
   929         val ((coinduct_thms, coinduct_thm), (strong_coinduct_thms, strong_coinduct_thm)) =
   930           let
   931             val uvrs = map3 (fn r => fn u => fn v => r $ u $ v) rs us vs;
   932             val uv_eqs = map2 (curry HOLogic.mk_eq) us vs;
   933             val strong_rs =
   934               map4 (fn u => fn v => fn uvr => fn uv_eq =>
   935                 fold_rev Term.lambda [u, v] (HOLogic.mk_disj (uvr, uv_eq))) us vs uvrs uv_eqs;
   936 
   937             fun build_rel rs' T =
   938               (case find_index (curry (op =) T) fpTs of
   939                 ~1 =>
   940                 if exists_fp_subtype T then build_rel_step (build_rel rs') T else HOLogic.eq_const T
   941               | kk => nth rs' kk);
   942 
   943             fun build_rel_app rs' usel vsel =
   944               fold rapp [usel, vsel] (build_rel rs' (fastype_of usel));
   945 
   946             fun mk_prem_ctr_concls rs' n k udisc usels vdisc vsels =
   947               (if k = n then [] else [HOLogic.mk_eq (udisc, vdisc)]) @
   948               (if null usels then
   949                  []
   950                else
   951                  [Library.foldr HOLogic.mk_imp (if n = 1 then [] else [udisc, vdisc],
   952                     Library.foldr1 HOLogic.mk_conj (map2 (build_rel_app rs') usels vsels))]);
   953 
   954             fun mk_prem_concl rs' n udiscs uselss vdiscs vselss =
   955               Library.foldr1 HOLogic.mk_conj
   956                 (flat (map5 (mk_prem_ctr_concls rs' n) (1 upto n) udiscs uselss vdiscs vselss))
   957               handle List.Empty => @{term True};
   958 
   959             fun mk_prem rs' uvr u v n udiscs uselss vdiscs vselss =
   960               fold_rev Logic.all [u, v] (Logic.mk_implies (HOLogic.mk_Trueprop uvr,
   961                 HOLogic.mk_Trueprop (mk_prem_concl rs' n udiscs uselss vdiscs vselss)));
   962 
   963             val concl =
   964               HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj
   965                 (map3 (fn uvr => fn u => fn v => HOLogic.mk_imp (uvr, HOLogic.mk_eq (u, v)))
   966                    uvrs us vs));
   967 
   968             fun mk_goal rs' =
   969               Logic.list_implies (map8 (mk_prem rs') uvrs us vs ns udiscss uselsss vdiscss vselsss,
   970                 concl);
   971 
   972             val goal = mk_goal rs;
   973             val strong_goal = mk_goal strong_rs;
   974 
   975             fun prove dtor_coinduct' goal =
   976               Goal.prove_sorry lthy [] [] goal (fn {context = ctxt, ...} =>
   977                 mk_coinduct_tac ctxt nesting_rel_eqs nn ns dtor_coinduct' pre_rel_defs dtor_ctors
   978                   exhaust_thms ctr_defss disc_thmsss sel_thmsss)
   979               |> singleton (Proof_Context.export names_lthy lthy)
   980               |> Thm.close_derivation;
   981 
   982             fun postproc nn thm =
   983               Thm.permute_prems 0 nn
   984                 (if nn = 1 then thm RS mp
   985                  else funpow nn (fn thm => reassoc_conjs (thm RS mp_conj)) thm)
   986               |> Drule.zero_var_indexes
   987               |> `(conj_dests nn);
   988           in
   989             (postproc nn (prove fp_induct goal), postproc nn (prove fp_strong_induct strong_goal))
   990           end;
   991 
   992         fun mk_coinduct_concls ms discs ctrs =
   993           let
   994             fun mk_disc_concl disc = [name_of_disc disc];
   995             fun mk_ctr_concl 0 _ = []
   996               | mk_ctr_concl _ ctor = [name_of_ctr ctor];
   997             val disc_concls = map mk_disc_concl (fst (split_last discs)) @ [[]];
   998             val ctr_concls = map2 mk_ctr_concl ms ctrs;
   999           in
  1000             flat (map2 append disc_concls ctr_concls)
  1001           end;
  1002 
  1003         val coinduct_cases = quasi_unambiguous_case_names (map (prefix EqN) fp_b_names);
  1004         val coinduct_conclss =
  1005           map3 (quasi_unambiguous_case_names ooo mk_coinduct_concls) mss discss ctrss;
  1006 
  1007         fun mk_maybe_not pos = not pos ? HOLogic.mk_not;
  1008 
  1009         val gunfolds = map (lists_bmoc pgss) unfolds;
  1010         val hcorecs = map (lists_bmoc phss) corecs;
  1011 
  1012         val (unfold_thmss, corec_thmss, safe_unfold_thmss, safe_corec_thmss) =
  1013           let
  1014             fun mk_goal pfss c cps fcorec_like n k ctr m cfs' =
  1015               fold_rev (fold_rev Logic.all) ([c] :: pfss)
  1016                 (Logic.list_implies (seq_conds (HOLogic.mk_Trueprop oo mk_maybe_not) n k cps,
  1017                    mk_Trueprop_eq (fcorec_like $ c, Term.list_comb (ctr, take m cfs'))));
  1018 
  1019             fun build_corec_like fcorec_likes (T, U) =
  1020               if T = U then
  1021                 id_const T
  1022               else
  1023                 (case find_index (curry (op =) U) fpTs of
  1024                   ~1 => build_map (build_corec_like fcorec_likes) T U
  1025                 | kk => nth fcorec_likes kk);
  1026 
  1027             val mk_U = typ_subst (map2 pair Cs fpTs);
  1028 
  1029             fun intr_corec_likes fcorec_likes [] [cf] =
  1030                 let val T = fastype_of cf in
  1031                   if exists_Cs_subtype T then build_corec_like fcorec_likes (T, mk_U T) $ cf else cf
  1032                 end
  1033               | intr_corec_likes fcorec_likes [cq] [cf, cf'] =
  1034                 mk_If cq (intr_corec_likes fcorec_likes [] [cf])
  1035                   (intr_corec_likes fcorec_likes [] [cf']);
  1036 
  1037             val crgsss = map2 (map2 (map2 (intr_corec_likes gunfolds))) crssss cgssss;
  1038             val cshsss = map2 (map2 (map2 (intr_corec_likes hcorecs))) csssss chssss;
  1039 
  1040             val unfold_goalss =
  1041               map8 (map4 oooo mk_goal pgss) cs cpss gunfolds ns kss ctrss mss crgsss;
  1042             val corec_goalss =
  1043               map8 (map4 oooo mk_goal phss) cs cpss hcorecs ns kss ctrss mss cshsss;
  1044 
  1045             fun mk_map_if_distrib bnf =
  1046               let
  1047                 val mapx = map_of_bnf bnf;
  1048                 val live = live_of_bnf bnf;
  1049                 val ((Ts, T), U) = strip_typeN (live + 1) (fastype_of mapx) |>> split_last;
  1050                 val fs = Variable.variant_frees lthy [mapx] (map (pair "f") Ts);
  1051                 val t = Term.list_comb (mapx, map (Var o apfst (rpair 0)) fs);
  1052               in
  1053                 Drule.instantiate' (map (SOME o certifyT lthy) [U, T]) [SOME (certify lthy t)]
  1054                   @{thm if_distrib}
  1055               end;
  1056 
  1057             val nested_map_if_distribs = map mk_map_if_distrib nested_bnfs;
  1058 
  1059             val unfold_tacss =
  1060               map3 (map oo mk_corec_like_tac unfold_defs [] [] nesting_map_ids'' [])
  1061                 fp_fold_thms pre_map_defs ctr_defss;
  1062             val corec_tacss =
  1063               map3 (map oo mk_corec_like_tac corec_defs nested_map_comps'' nested_map_comp's
  1064                   (nested_map_ids'' @ nesting_map_ids'') nested_map_if_distribs)
  1065                 fp_rec_thms pre_map_defs ctr_defss;
  1066 
  1067             fun prove goal tac =
  1068               Goal.prove_sorry lthy [] [] goal (tac o #context) |> Thm.close_derivation;
  1069 
  1070             val unfold_thmss = map2 (map2 prove) unfold_goalss unfold_tacss;
  1071             val corec_thmss = map2 (map2 prove) corec_goalss corec_tacss;
  1072 
  1073             val filter_safesss =
  1074               map2 (map_filter (fn (safes, thm) => if forall I safes then SOME thm else NONE) oo
  1075                 curry (op ~~)) (map2 (map2 (map2 (member (op =)))) cgssss crgsss);
  1076 
  1077             val safe_unfold_thmss = filter_safesss unfold_thmss;
  1078             val safe_corec_thmss = filter_safesss corec_thmss;
  1079           in
  1080             (unfold_thmss, corec_thmss, safe_unfold_thmss, safe_corec_thmss)
  1081           end;
  1082 
  1083         val (disc_unfold_iff_thmss, disc_corec_iff_thmss) =
  1084           let
  1085             fun mk_goal c cps fcorec_like n k disc =
  1086               mk_Trueprop_eq (disc $ (fcorec_like $ c),
  1087                 if n = 1 then @{const True}
  1088                 else Library.foldr1 HOLogic.mk_conj (seq_conds mk_maybe_not n k cps));
  1089 
  1090             val unfold_goalss = map6 (map2 oooo mk_goal) cs cpss gunfolds ns kss discss;
  1091             val corec_goalss = map6 (map2 oooo mk_goal) cs cpss hcorecs ns kss discss;
  1092 
  1093             fun mk_case_split' cp =
  1094               Drule.instantiate' [] [SOME (certify lthy cp)] @{thm case_split};
  1095 
  1096             val case_splitss' = map (map mk_case_split') cpss;
  1097 
  1098             val unfold_tacss =
  1099               map3 (map oo mk_disc_corec_like_iff_tac) case_splitss' unfold_thmss disc_thmsss;
  1100             val corec_tacss =
  1101               map3 (map oo mk_disc_corec_like_iff_tac) case_splitss' corec_thmss disc_thmsss;
  1102 
  1103             fun prove goal tac =
  1104               Goal.prove_sorry lthy [] [] goal (tac o #context)
  1105               |> singleton (Proof_Context.export names_lthy0 no_defs_lthy)
  1106               |> Thm.close_derivation;
  1107 
  1108             fun proves [_] [_] = []
  1109               | proves goals tacs = map2 prove goals tacs;
  1110           in
  1111             (map2 proves unfold_goalss unfold_tacss,
  1112              map2 proves corec_goalss corec_tacss)
  1113           end;
  1114 
  1115         val is_triv_discI = is_triv_implies orf is_concl_refl;
  1116 
  1117         fun mk_disc_corec_like_thms corec_likes discIs =
  1118           map (op RS) (filter_out (is_triv_discI o snd) (corec_likes ~~ discIs));
  1119 
  1120         val disc_unfold_thmss = map2 mk_disc_corec_like_thms unfold_thmss discIss;
  1121         val disc_corec_thmss = map2 mk_disc_corec_like_thms corec_thmss discIss;
  1122 
  1123         fun mk_sel_corec_like_thm corec_like_thm sel sel_thm =
  1124           let
  1125             val (domT, ranT) = dest_funT (fastype_of sel);
  1126             val arg_cong' =
  1127               Drule.instantiate' (map (SOME o certifyT lthy) [domT, ranT])
  1128                 [NONE, NONE, SOME (certify lthy sel)] arg_cong
  1129               |> Thm.varifyT_global;
  1130             val sel_thm' = sel_thm RSN (2, trans);
  1131           in
  1132             corec_like_thm RS arg_cong' RS sel_thm'
  1133           end;
  1134 
  1135         fun mk_sel_corec_like_thms corec_likess =
  1136           map3 (map3 (map2 o mk_sel_corec_like_thm)) corec_likess selsss sel_thmsss |> map flat;
  1137 
  1138         val sel_unfold_thmss = mk_sel_corec_like_thms unfold_thmss;
  1139         val sel_corec_thmss = mk_sel_corec_like_thms corec_thmss;
  1140 
  1141         fun flat_corec_like_thms corec_likes disc_corec_likes sel_corec_likes =
  1142           corec_likes @ disc_corec_likes @ sel_corec_likes;
  1143 
  1144         val simp_thmss =
  1145           mk_simp_thmss wrap_ress
  1146             (map3 flat_corec_like_thms safe_corec_thmss disc_corec_thmss sel_corec_thmss)
  1147             (map3 flat_corec_like_thms safe_unfold_thmss disc_unfold_thmss sel_unfold_thmss);
  1148 
  1149         val anonymous_notes =
  1150           [(flat safe_unfold_thmss @ flat safe_corec_thmss, simp_attrs)]
  1151           |> map (fn (thms, attrs) => ((Binding.empty, attrs), [(thms, [])]));
  1152 
  1153         val coinduct_consumes_attr = Attrib.internal (K (Rule_Cases.consumes nn));
  1154         val coinduct_case_names_attr = Attrib.internal (K (Rule_Cases.case_names coinduct_cases));
  1155         val coinduct_case_concl_attrs =
  1156           map2 (fn casex => fn concls =>
  1157               Attrib.internal (K (Rule_Cases.case_conclusion (casex, concls))))
  1158             coinduct_cases coinduct_conclss;
  1159         val coinduct_case_attrs =
  1160           coinduct_consumes_attr :: coinduct_case_names_attr :: coinduct_case_concl_attrs;
  1161 
  1162         val common_notes =
  1163           (if nn > 1 then
  1164              [(coinductN, [coinduct_thm], coinduct_case_attrs),
  1165               (strong_coinductN, [strong_coinduct_thm], coinduct_case_attrs)]
  1166            else
  1167              [])
  1168           |> map (fn (thmN, thms, attrs) =>
  1169             ((qualify true fp_common_name (Binding.name thmN), attrs), [(thms, [])]));
  1170 
  1171         val notes =
  1172           [(coinductN, map single coinduct_thms, coinduct_case_attrs),
  1173            (corecN, corec_thmss, []),
  1174            (disc_corecN, disc_corec_thmss, simp_attrs),
  1175            (disc_corec_iffN, disc_corec_iff_thmss, simp_attrs),
  1176            (disc_unfoldN, disc_unfold_thmss, simp_attrs),
  1177            (disc_unfold_iffN, disc_unfold_iff_thmss, simp_attrs),
  1178            (sel_corecN, sel_corec_thmss, simp_attrs),
  1179            (sel_unfoldN, sel_unfold_thmss, simp_attrs),
  1180            (simpsN, simp_thmss, []),
  1181            (strong_coinductN, map single strong_coinduct_thms, coinduct_case_attrs),
  1182            (unfoldN, unfold_thmss, [])]
  1183           |> maps (fn (thmN, thmss, attrs) =>
  1184             map_filter (fn (_, []) => NONE | (fp_b_name, thms) =>
  1185               SOME ((qualify true fp_b_name (Binding.name thmN), attrs),
  1186                 [(thms, [])])) (fp_b_names ~~ thmss));
  1187       in
  1188         lthy |> Local_Theory.notes (anonymous_notes @ common_notes @ notes) |> snd
  1189       end;
  1190 
  1191     val lthy' = lthy
  1192       |> fold_map define_ctrs_case_for_type (fp_bnfs ~~ fp_bs ~~ fpTs ~~ Cs ~~ ctors ~~ dtors ~~
  1193         fp_folds ~~ fp_recs ~~ ctor_dtors ~~ dtor_ctors ~~ ctor_injects ~~ pre_map_defs ~~
  1194         pre_set_defss ~~ pre_rel_defs ~~ fp_map_thms ~~ fp_set_thmss ~~ fp_rel_thms ~~ ns ~~ kss ~~
  1195         mss ~~ ctr_bindingss ~~ ctr_mixfixess ~~ ctr_Tsss ~~ disc_bindingss ~~ sel_bindingsss ~~
  1196         raw_sel_defaultsss)
  1197       |> wrap_types_and_more
  1198       |> (if lfp then derive_induct_fold_rec_thms_for_types
  1199           else derive_coinduct_unfold_corec_thms_for_types);
  1200 
  1201     val timer = time (timer ("Constructors, discriminators, selectors, etc., for the new " ^
  1202       (if lfp then "" else "co") ^ "datatype"));
  1203   in
  1204     timer; lthy'
  1205   end;
  1206 
  1207 val datatypes = define_datatypes (K I) (K I) (K I);
  1208 
  1209 val datatype_cmd = define_datatypes Typedecl.read_constraint Syntax.parse_typ Syntax.parse_term;
  1210 
  1211 val parse_ctr_arg =
  1212   @{keyword "("} |-- parse_binding_colon -- Parse.typ --| @{keyword ")"} ||
  1213   (Parse.typ >> pair Binding.empty);
  1214 
  1215 val parse_defaults =
  1216   @{keyword "("} |-- @{keyword "defaults"} |-- Scan.repeat parse_bound_term --| @{keyword ")"};
  1217 
  1218 val parse_type_arg_constrained =
  1219   Parse.type_ident -- Scan.option (@{keyword "::"} |-- Parse.!!! Parse.sort)
  1220 
  1221 val parse_type_arg_named_constrained = parse_opt_binding_colon -- parse_type_arg_constrained
  1222 
  1223 val parse_type_args_named_constrained =
  1224   parse_type_arg_constrained >> (single o pair Binding.empty) ||
  1225   @{keyword "("} |-- Parse.!!! (Parse.list1 parse_type_arg_named_constrained --| @{keyword ")"}) ||
  1226   Scan.succeed [];
  1227 
  1228 val parse_map_rel_binding = Parse.short_ident --| @{keyword ":"} -- Parse.binding;
  1229 
  1230 val no_map_rel = (Binding.empty, Binding.empty);
  1231 
  1232 (* "map" and "rel" are purposedly not registered as keywords, because they are short and nice names
  1233    that we don't want them to be highlighted everywhere because of some obscure feature of the BNF
  1234    package. *)
  1235 fun extract_map_rel ("map", b) = apfst (K b)
  1236   | extract_map_rel ("rel", b) = apsnd (K b)
  1237   | extract_map_rel (s, _) = error ("Expected \"map\" or \"rel\" instead of " ^ quote s);
  1238 
  1239 val parse_map_rel_bindings =
  1240   @{keyword "("} |-- Scan.repeat parse_map_rel_binding --| @{keyword ")"}
  1241     >> (fn ps => fold extract_map_rel ps no_map_rel) ||
  1242   Scan.succeed no_map_rel;
  1243 
  1244 val parse_ctr_spec =
  1245   parse_opt_binding_colon -- Parse.binding -- Scan.repeat parse_ctr_arg --
  1246   Scan.optional parse_defaults [] -- Parse.opt_mixfix;
  1247 
  1248 val parse_spec =
  1249   parse_type_args_named_constrained -- Parse.binding -- parse_map_rel_bindings --
  1250   Parse.opt_mixfix -- (@{keyword "="} |-- Parse.enum1 "|" parse_ctr_spec);
  1251 
  1252 val parse_datatype = parse_wrap_options -- Parse.and_list1 parse_spec;
  1253 
  1254 fun parse_datatype_cmd lfp construct_fp = parse_datatype >> datatype_cmd lfp construct_fp;
  1255 
  1256 end;