src/HOL/BNF/Tools/bnf_fp_def_sugar.ML
author blanchet
Mon Oct 01 10:34:58 2012 +0200 (2012-10-01)
changeset 49670 c7a034d01936
parent 49636 b7256a88a84b
child 49671 61729b149397
permissions -rw-r--r--
changed the type of the recursor for nested recursion
     1 (*  Title:      HOL/BNF/Tools/bnf_fp_def_sugar.ML
     2     Author:     Jasmin Blanchette, TU Muenchen
     3     Copyright   2012
     4 
     5 Sugared datatype and codatatype constructions.
     6 *)
     7 
     8 signature BNF_FP_DEF_SUGAR =
     9 sig
    10   val datatypes: bool ->
    11     (mixfix list -> (string * sort) list option -> binding list -> typ list * typ list list ->
    12       BNF_Def.BNF list -> local_theory -> BNF_FP.fp_result * local_theory) ->
    13     (bool * bool) * ((((typ * sort) list * binding) * mixfix) * ((((binding * binding) *
    14       (binding * typ) list) * (binding * term) list) * mixfix) list) list ->
    15     local_theory -> local_theory
    16   val parse_datatype_cmd: bool ->
    17     (mixfix list -> (string * sort) list option -> binding list -> typ list * typ list list ->
    18       BNF_Def.BNF list -> local_theory -> BNF_FP.fp_result * local_theory) ->
    19     (local_theory -> local_theory) parser
    20 end;
    21 
    22 structure BNF_FP_Def_Sugar : BNF_FP_DEF_SUGAR =
    23 struct
    24 
    25 open BNF_Util
    26 open BNF_Wrap
    27 open BNF_Def
    28 open BNF_FP
    29 open BNF_FP_Def_Sugar_Tactics
    30 
    31 (* This function could produce clashes in contrived examples (e.g., "x.A", "x.x_A", "y.A") *)
    32 fun quasi_unambiguous_case_names names =
    33   let
    34     val ps = map (`Long_Name.base_name) names;
    35     val dups = Library.duplicates (op =) (map fst ps);
    36     fun underscore s =
    37       let val ss = space_explode Long_Name.separator s in
    38         space_implode "_" (drop (length ss - 2) ss)
    39       end;
    40   in
    41     map (fn (base, full) => if member (op =) dups base then underscore full else base) ps
    42   end;
    43 
    44 val mp_conj = @{thm mp_conj};
    45 
    46 val simp_attrs = @{attributes [simp]};
    47 val code_simp_attrs = Code.add_default_eqn_attrib :: simp_attrs;
    48 
    49 fun split_list4 xs = (map #1 xs, map #2 xs, map #3 xs, map #4 xs);
    50 
    51 fun resort_tfree S (TFree (s, _)) = TFree (s, S);
    52 
    53 fun typ_subst inst (T as Type (s, Ts)) =
    54     (case AList.lookup (op =) inst T of
    55       NONE => Type (s, map (typ_subst inst) Ts)
    56     | SOME T' => T')
    57   | typ_subst inst T = the_default T (AList.lookup (op =) inst T);
    58 
    59 fun variant_types ss Ss ctxt =
    60   let
    61     val (tfrees, _) =
    62       fold_map2 (fn s => fn S => Name.variant s #> apfst (rpair S)) ss Ss (Variable.names_of ctxt);
    63     val ctxt' = fold (Variable.declare_constraints o Logic.mk_type o TFree) tfrees ctxt;
    64   in (tfrees, ctxt') end;
    65 
    66 val lists_bmoc = fold (fn xs => fn t => Term.list_comb (t, xs));
    67 
    68 fun mk_tupled_fun x f xs = HOLogic.tupled_lambda x (Term.list_comb (f, xs));
    69 fun mk_uncurried_fun f xs = mk_tupled_fun (HOLogic.mk_tuple xs) f xs;
    70 
    71 fun mk_flip (x, Type (_, [T1, Type (_, [T2, T3])])) =
    72   Abs ("x", T1, Abs ("y", T2, Var (x, T2 --> T1 --> T3) $ Bound 0 $ Bound 1));
    73 
    74 fun flip_rels lthy n thm =
    75   let
    76     val Rs = Term.add_vars (prop_of thm) [];
    77     val Rs' = rev (drop (length Rs - n) Rs);
    78     val cRs = map (fn f => (certify lthy (Var f), certify lthy (mk_flip f))) Rs';
    79   in
    80     Drule.cterm_instantiate cRs thm
    81   end;
    82 
    83 fun mk_ctor_or_dtor get_T Ts t =
    84   let val Type (_, Ts0) = get_T (fastype_of t) in
    85     Term.subst_atomic_types (Ts0 ~~ Ts) t
    86   end;
    87 
    88 val mk_ctor = mk_ctor_or_dtor range_type;
    89 val mk_dtor = mk_ctor_or_dtor domain_type;
    90 
    91 fun mk_rec_like lfp Ts Us t =
    92   let
    93     val (bindings, body) = strip_type (fastype_of t);
    94     val (f_Us, prebody) = split_last bindings;
    95     val Type (_, Ts0) = if lfp then prebody else body;
    96     val Us0 = distinct (op =) (map (if lfp then body_type else domain_type) f_Us);
    97   in
    98     Term.subst_atomic_types (Ts0 @ Us0 ~~ Ts @ Us) t
    99   end;
   100 
   101 fun mk_map live Ts Us t =
   102   let val (Type (_, Ts0), Type (_, Us0)) = strip_typeN (live + 1) (fastype_of t) |>> List.last in
   103     Term.subst_atomic_types (Ts0 @ Us0 ~~ Ts @ Us) t
   104   end;
   105 
   106 fun mk_rel live Ts Us t =
   107   let val [Type (_, Ts0), Type (_, Us0)] = binder_types (snd (strip_typeN live (fastype_of t))) in
   108     Term.subst_atomic_types (Ts0 @ Us0 ~~ Ts @ Us) t
   109   end;
   110 
   111 fun liveness_of_fp_bnf n bnf =
   112   (case T_of_bnf bnf of
   113     Type (_, Ts) => map (not o member (op =) (deads_of_bnf bnf)) Ts
   114   | _ => replicate n false);
   115 
   116 fun tick u f = Term.lambda u (HOLogic.mk_prod (u, f $ u));
   117 
   118 fun tack z_name (c, u) f =
   119   let val z = Free (z_name, mk_sumT (fastype_of u, fastype_of c)) in
   120     Term.lambda z (mk_sum_case (Term.lambda u u, Term.lambda c (f $ c)) $ z)
   121   end;
   122 
   123 fun cannot_merge_types () = error "Mutually recursive types must have the same type parameters";
   124 
   125 fun merge_type_arg T T' = if T = T' then T else cannot_merge_types ();
   126 
   127 fun merge_type_args (As, As') =
   128   if length As = length As' then map2 merge_type_arg As As' else cannot_merge_types ();
   129 
   130 fun reassoc_conjs thm =
   131   reassoc_conjs (thm RS @{thm conj_assoc[THEN iffD1]})
   132   handle THM _ => thm;
   133 
   134 fun type_args_constrained_of (((cAs, _), _), _) = cAs;
   135 fun type_binding_of (((_, b), _), _) = b;
   136 fun mixfix_of ((_, mx), _) = mx;
   137 fun ctr_specs_of (_, ctr_specs) = ctr_specs;
   138 
   139 fun disc_of ((((disc, _), _), _), _) = disc;
   140 fun ctr_of ((((_, ctr), _), _), _) = ctr;
   141 fun args_of (((_, args), _), _) = args;
   142 fun defaults_of ((_, ds), _) = ds;
   143 fun ctr_mixfix_of (_, mx) = mx;
   144 
   145 fun define_datatypes prepare_constraint prepare_typ prepare_term lfp construct_fp
   146     (wrap_opts as (no_dests, rep_compat), specs) no_defs_lthy0 =
   147   let
   148     (* TODO: sanity checks on arguments *)
   149     (* TODO: integration with function package ("size") *)
   150 
   151     val _ = if not lfp andalso no_dests then error "Cannot define destructor-less codatatypes"
   152       else ();
   153 
   154     fun qualify mandatory fp_b_name =
   155       Binding.qualify mandatory fp_b_name o (rep_compat ? Binding.qualify false rep_compat_prefix);
   156 
   157     val nn = length specs;
   158     val fp_bs = map type_binding_of specs;
   159     val fp_b_names = map Binding.name_of fp_bs;
   160     val fp_common_name = mk_common_name fp_b_names;
   161 
   162     fun prepare_type_arg (ty, c) =
   163       let val TFree (s, _) = prepare_typ no_defs_lthy0 ty in
   164         TFree (s, prepare_constraint no_defs_lthy0 c)
   165       end;
   166 
   167     val Ass0 = map (map prepare_type_arg o type_args_constrained_of) specs;
   168     val unsorted_Ass0 = map (map (resort_tfree HOLogic.typeS)) Ass0;
   169     val unsorted_As = Library.foldr1 merge_type_args unsorted_Ass0;
   170 
   171     val (((Bs0, Cs), Xs), no_defs_lthy) =
   172       no_defs_lthy0
   173       |> fold (Variable.declare_typ o resort_tfree dummyS) unsorted_As
   174       |> mk_TFrees (length unsorted_As)
   175       ||>> mk_TFrees nn
   176       ||>> apfst (map TFree) o
   177         variant_types (map (prefix "'") fp_b_names) (replicate nn HOLogic.typeS);
   178 
   179     (* TODO: cleaner handling of fake contexts, without "background_theory" *)
   180     (*the "perhaps o try" below helps gracefully handles the case where the new type is defined in a
   181       locale and shadows an existing global type*)
   182     val fake_thy =
   183       Theory.copy #> fold (fn spec => perhaps (try (Sign.add_type no_defs_lthy
   184         (type_binding_of spec, length (type_args_constrained_of spec), mixfix_of spec)))) specs;
   185     val fake_lthy = Proof_Context.background_theory fake_thy no_defs_lthy;
   186 
   187     fun mk_fake_T b =
   188       Type (fst (Term.dest_Type (Proof_Context.read_type_name fake_lthy true (Binding.name_of b))),
   189         unsorted_As);
   190 
   191     val fake_Ts = map mk_fake_T fp_bs;
   192 
   193     val mixfixes = map mixfix_of specs;
   194 
   195     val _ = (case duplicates Binding.eq_name fp_bs of [] => ()
   196       | b :: _ => error ("Duplicate type name declaration " ^ quote (Binding.name_of b)));
   197 
   198     val ctr_specss = map ctr_specs_of specs;
   199 
   200     val disc_bindingss = map (map disc_of) ctr_specss;
   201     val ctr_bindingss =
   202       map2 (fn fp_b_name => map (qualify false fp_b_name o ctr_of)) fp_b_names ctr_specss;
   203     val ctr_argsss = map (map args_of) ctr_specss;
   204     val ctr_mixfixess = map (map ctr_mixfix_of) ctr_specss;
   205 
   206     val sel_bindingsss = map (map (map fst)) ctr_argsss;
   207     val fake_ctr_Tsss0 = map (map (map (prepare_typ fake_lthy o snd))) ctr_argsss;
   208     val raw_sel_defaultsss = map (map defaults_of) ctr_specss;
   209 
   210     val (As :: _) :: fake_ctr_Tsss =
   211       burrow (burrow (Syntax.check_typs fake_lthy)) (Ass0 :: fake_ctr_Tsss0);
   212 
   213     val _ = (case duplicates (op =) unsorted_As of [] => ()
   214       | A :: _ => error ("Duplicate type parameter " ^
   215           quote (Syntax.string_of_typ no_defs_lthy A)));
   216 
   217     val rhs_As' = fold (fold (fold Term.add_tfreesT)) fake_ctr_Tsss [];
   218     val _ = (case subtract (op =) (map dest_TFree As) rhs_As' of
   219         [] => ()
   220       | A' :: _ => error ("Extra type variable on right-hand side: " ^
   221           quote (Syntax.string_of_typ no_defs_lthy (TFree A'))));
   222 
   223     fun eq_fpT_check (T as Type (s, Us)) (Type (s', Us')) =
   224         s = s' andalso (Us = Us' orelse error ("Illegal occurrence of recursive type " ^
   225           quote (Syntax.string_of_typ fake_lthy T)))
   226       | eq_fpT_check _ _ = false;
   227 
   228     fun freeze_fp (T as Type (s, Us)) =
   229         (case find_index (eq_fpT_check T) fake_Ts of
   230           ~1 => Type (s, map freeze_fp Us)
   231         | kk => nth Xs kk)
   232       | freeze_fp T = T;
   233 
   234     val ctr_TsssXs = map (map (map freeze_fp)) fake_ctr_Tsss;
   235     val ctr_sum_prod_TsXs = map (mk_sumTN_balanced o map HOLogic.mk_tupleT) ctr_TsssXs;
   236 
   237     val fp_eqs =
   238       map dest_TFree Xs ~~ map (Term.typ_subst_atomic (As ~~ unsorted_As)) ctr_sum_prod_TsXs;
   239 
   240     (* TODO: clean up list *)
   241     val (pre_bnfs, ((fp_bnfs as any_fp_bnf :: _, dtors0, ctors0, fp_folds0, fp_recs0, fp_induct,
   242            fp_strong_induct, dtor_ctors, ctor_dtors, ctor_injects, fp_map_thms, fp_set_thmss,
   243            fp_rel_thms, fp_fold_thms, fp_rec_thms), lthy)) =
   244       fp_bnf construct_fp fp_bs mixfixes (map dest_TFree unsorted_As) fp_eqs no_defs_lthy0;
   245 
   246     val timer = time (Timer.startRealTimer ());
   247 
   248     fun build_map build_arg (Type (s, Ts)) (Type (_, Us)) =
   249       let
   250         val bnf = the (bnf_of lthy s);
   251         val live = live_of_bnf bnf;
   252         val mapx = mk_map live Ts Us (map_of_bnf bnf);
   253         val TUs' = map dest_funT (fst (strip_typeN live (fastype_of mapx)));
   254       in Term.list_comb (mapx, map build_arg TUs') end;
   255 
   256     fun build_rel_step build_arg (Type (s, Ts)) =
   257       let
   258         val bnf = the (bnf_of lthy s);
   259         val live = live_of_bnf bnf;
   260         val rel = mk_rel live Ts Ts (rel_of_bnf bnf);
   261         val Ts' = map domain_type (fst (strip_typeN live (fastype_of rel)));
   262       in Term.list_comb (rel, map build_arg Ts') end;
   263 
   264     fun add_nesty_bnf_names Us =
   265       let
   266         fun add (Type (s, Ts)) ss =
   267             let val (needs, ss') = fold_map add Ts ss in
   268               if exists I needs then (true, insert (op =) s ss') else (false, ss')
   269             end
   270           | add T ss = (member (op =) Us T, ss);
   271       in snd oo add end;
   272 
   273     fun nesty_bnfs Us =
   274       map_filter (bnf_of lthy) (fold (fold (fold (add_nesty_bnf_names Us))) ctr_TsssXs []);
   275 
   276     val nesting_bnfs = nesty_bnfs As;
   277     val nested_bnfs = nesty_bnfs Xs;
   278 
   279     val pre_map_defs = map map_def_of_bnf pre_bnfs;
   280     val pre_set_defss = map set_defs_of_bnf pre_bnfs;
   281     val pre_rel_defs = map rel_def_of_bnf pre_bnfs;
   282     val nested_map_comp's = map map_comp'_of_bnf nested_bnfs;
   283     val nested_map_ids'' = map (unfold_thms lthy @{thms id_def} o map_id_of_bnf) nested_bnfs;
   284     val nesting_map_ids = map map_id_of_bnf nesting_bnfs;
   285     val nesting_map_ids'' = map (unfold_thms lthy @{thms id_def}) nesting_map_ids;
   286     val nested_set_natural's = maps set_natural'_of_bnf nested_bnfs;
   287     val nesting_set_natural's = maps set_natural'_of_bnf nesting_bnfs;
   288 
   289     val live = live_of_bnf any_fp_bnf;
   290 
   291     val Bs =
   292       map3 (fn alive => fn A as TFree (_, S) => fn B => if alive then resort_tfree S B else A)
   293         (liveness_of_fp_bnf (length As) any_fp_bnf) As Bs0;
   294 
   295     val B_ify = Term.typ_subst_atomic (As ~~ Bs);
   296 
   297     val ctors = map (mk_ctor As) ctors0;
   298     val dtors = map (mk_dtor As) dtors0;
   299 
   300     val fpTs = map (domain_type o fastype_of) dtors;
   301 
   302     val exists_fp_subtype = exists_subtype (member (op =) fpTs);
   303     val exists_Cs_subtype = exists_subtype (member (op =) Cs);
   304 
   305     val ctr_Tsss = map (map (map (Term.typ_subst_atomic (Xs ~~ fpTs)))) ctr_TsssXs;
   306     val ns = map length ctr_Tsss;
   307     val kss = map (fn n => 1 upto n) ns;
   308     val mss = map (map length) ctr_Tsss;
   309     val Css = map2 replicate ns Cs;
   310 
   311     val fp_folds as any_fp_fold :: _ = map (mk_rec_like lfp As Cs) fp_folds0;
   312     val fp_recs as any_fp_rec :: _ = map (mk_rec_like lfp As Cs) fp_recs0;
   313 
   314     val fp_fold_fun_Ts = fst (split_last (binder_types (fastype_of any_fp_fold)));
   315     val fp_rec_fun_Ts = fst (split_last (binder_types (fastype_of any_fp_rec)));
   316 
   317     val (((fold_only as (gss, _, _), rec_only as (hss, _, _)),
   318           (zs, cs, cpss, unfold_only as ((pgss, crgsss), _), corec_only as ((phss, cshsss), _))),
   319          names_lthy0) =
   320       if lfp then
   321         let
   322           val y_Tsss =
   323             map3 (fn n => fn ms => map2 dest_tupleT ms o dest_sumTN_balanced n o domain_type)
   324               ns mss fp_fold_fun_Ts;
   325           val g_Tss = map2 (map2 (curry (op --->))) y_Tsss Css;
   326 
   327           val ((gss, ysss), lthy) =
   328             lthy
   329             |> mk_Freess "f" g_Tss
   330             ||>> mk_Freesss "x" y_Tsss;
   331 
   332           fun proj_recT proj (Type (s as @{type_name prod}, Ts as [T, U])) =
   333               if member (op =) fpTs T then proj (T, U) else Type (s, map (proj_recT proj) Ts)
   334             | proj_recT proj (Type (s, Ts)) = Type (s, map (proj_recT proj) Ts)
   335             | proj_recT _ T = T;
   336 
   337           fun unzip_recT T =
   338             if exists_fp_subtype T then [proj_recT fst T, proj_recT snd T] else [T];
   339 
   340           val z_Tsss =
   341             map3 (fn n => fn ms => map2 dest_tupleT ms o dest_sumTN_balanced n o domain_type)
   342               ns mss fp_rec_fun_Ts;
   343           val z_Tssss = map (map (map unzip_recT)) z_Tsss;
   344           val h_Tss = map2 (map2 (fold_rev (curry (op --->)))) z_Tssss Css;
   345 
   346           val hss = map2 (map2 retype_free) h_Tss gss;
   347           val zsss = map2 (map2 (map2 retype_free)) z_Tsss ysss;
   348         in
   349           ((((gss, g_Tss, ysss), (hss, h_Tss, zsss)),
   350             ([], [], [], (([], []), ([], [])), (([], []), ([], [])))), lthy)
   351         end
   352       else
   353         let
   354           (*avoid "'a itself" arguments in coiterators and corecursors*)
   355           val mss' =  map (fn [0] => [1] | ms => ms) mss;
   356 
   357           val p_Tss = map2 (fn n => replicate (Int.max (0, n - 1)) o mk_pred1T) ns Cs;
   358 
   359           fun flat_predss_getterss qss fss = maps (op @) (qss ~~ fss);
   360 
   361           fun flat_preds_predsss_gettersss [] [qss] [fss] = flat_predss_getterss qss fss
   362             | flat_preds_predsss_gettersss (p :: ps) (qss :: qsss) (fss :: fsss) =
   363               p :: flat_predss_getterss qss fss @ flat_preds_predsss_gettersss ps qsss fsss;
   364 
   365           fun mk_types maybe_dest_sumT fun_Ts =
   366             let
   367               val f_sum_prod_Ts = map range_type fun_Ts;
   368               val f_prod_Tss = map2 dest_sumTN_balanced ns f_sum_prod_Ts;
   369               val f_Tssss =
   370                 map3 (fn C => map2 (map (map (curry (op -->) C) o maybe_dest_sumT) oo dest_tupleT))
   371                   Cs mss' f_prod_Tss;
   372               val q_Tssss =
   373                 map (map (map (fn [_] => [] | [_, C] => [mk_pred1T (domain_type C)]))) f_Tssss;
   374               val pf_Tss = map3 flat_preds_predsss_gettersss p_Tss q_Tssss f_Tssss;
   375             in (q_Tssss, f_sum_prod_Ts, f_Tssss, pf_Tss) end;
   376 
   377           val (r_Tssss, g_sum_prod_Ts, g_Tssss, pg_Tss) = mk_types single fp_fold_fun_Ts;
   378 
   379           val ((((Free (z, _), cs), pss), gssss), lthy) =
   380             lthy
   381             |> yield_singleton (mk_Frees "z") dummyT
   382             ||>> mk_Frees "a" Cs
   383             ||>> mk_Freess "p" p_Tss
   384             ||>> mk_Freessss "g" g_Tssss;
   385           val rssss = map (map (map (fn [] => []))) r_Tssss;
   386 
   387           fun dest_corec_sumT (T as Type (@{type_name sum}, Us as [_, U])) =
   388               if member (op =) Cs U then Us else [T]
   389             | dest_corec_sumT T = [T];
   390 
   391           val (s_Tssss, h_sum_prod_Ts, h_Tssss, ph_Tss) = mk_types dest_corec_sumT fp_rec_fun_Ts;
   392 
   393           val hssss_hd = map2 (map2 (map2 (fn T :: _ => fn [g] => retype_free T g))) h_Tssss gssss;
   394           val ((sssss, hssss_tl), lthy) =
   395             lthy
   396             |> mk_Freessss "q" s_Tssss
   397             ||>> mk_Freessss "h" (map (map (map tl)) h_Tssss);
   398           val hssss = map2 (map2 (map2 cons)) hssss_hd hssss_tl;
   399 
   400           val cpss = map2 (fn c => map (fn p => p $ c)) cs pss;
   401 
   402           fun mk_preds_getters_join [] [cf] = cf
   403             | mk_preds_getters_join [cq] [cf, cf'] =
   404               mk_If cq (mk_Inl (fastype_of cf') cf) (mk_Inr (fastype_of cf) cf');
   405 
   406           fun mk_terms qssss fssss =
   407             let
   408               val pfss = map3 flat_preds_predsss_gettersss pss qssss fssss;
   409               val cqssss = map2 (fn c => map (map (map (fn f => f $ c)))) cs qssss;
   410               val cfssss = map2 (fn c => map (map (map (fn f => f $ c)))) cs fssss;
   411               val cqfsss = map2 (map2 (map2 mk_preds_getters_join)) cqssss cfssss;
   412             in (pfss, cqfsss) end;
   413         in
   414           (((([], [], []), ([], [], [])),
   415             ([z], cs, cpss, (mk_terms rssss gssss, (g_sum_prod_Ts, pg_Tss)),
   416              (mk_terms sssss hssss, (h_sum_prod_Ts, ph_Tss)))), lthy)
   417         end;
   418 
   419     fun define_ctrs_case_for_type (((((((((((((((((((((((((fp_bnf, fp_b), fpT), C), ctor), dtor),
   420             fp_fold), fp_rec), ctor_dtor), dtor_ctor), ctor_inject), pre_map_def), pre_set_defs),
   421           pre_rel_def), fp_map_thm), fp_set_thms), fp_rel_thm), n), ks), ms), ctr_bindings),
   422         ctr_mixfixes), ctr_Tss), disc_bindings), sel_bindingss), raw_sel_defaultss) no_defs_lthy =
   423       let
   424         val fp_b_name = Binding.name_of fp_b;
   425 
   426         val dtorT = domain_type (fastype_of ctor);
   427         val ctr_prod_Ts = map HOLogic.mk_tupleT ctr_Tss;
   428         val ctr_sum_prod_T = mk_sumTN_balanced ctr_prod_Ts;
   429         val case_Ts = map (fn Ts => Ts ---> C) ctr_Tss;
   430 
   431         val (((((w, fs), xss), yss), u'), names_lthy) =
   432           no_defs_lthy
   433           |> yield_singleton (mk_Frees "w") dtorT
   434           ||>> mk_Frees "f" case_Ts
   435           ||>> mk_Freess "x" ctr_Tss
   436           ||>> mk_Freess "y" (map (map B_ify) ctr_Tss)
   437           ||>> yield_singleton Variable.variant_fixes fp_b_name;
   438 
   439         val u = Free (u', fpT);
   440 
   441         val tuple_xs = map HOLogic.mk_tuple xss;
   442         val tuple_ys = map HOLogic.mk_tuple yss;
   443 
   444         val ctr_rhss =
   445           map3 (fn k => fn xs => fn tuple_x => fold_rev Term.lambda xs (ctor $
   446             mk_InN_balanced ctr_sum_prod_T n tuple_x k)) ks xss tuple_xs;
   447 
   448         val case_binding = qualify false fp_b_name (Binding.suffix_name ("_" ^ caseN) fp_b);
   449 
   450         val case_rhs =
   451           fold_rev Term.lambda (fs @ [u])
   452             (mk_sum_caseN_balanced (map2 mk_uncurried_fun fs xss) $ (dtor $ u));
   453 
   454         val ((raw_case :: raw_ctrs, raw_case_def :: raw_ctr_defs), (lthy', lthy)) = no_defs_lthy
   455           |> apfst split_list o fold_map3 (fn b => fn mx => fn rhs =>
   456               Local_Theory.define ((b, mx), ((Thm.def_binding b, []), rhs)) #>> apsnd snd)
   457             (case_binding :: ctr_bindings) (NoSyn :: ctr_mixfixes) (case_rhs :: ctr_rhss)
   458           ||> `Local_Theory.restore;
   459 
   460         val phi = Proof_Context.export_morphism lthy lthy';
   461 
   462         val ctr_defs = map (Morphism.thm phi) raw_ctr_defs;
   463         val ctr_defs' =
   464           map2 (fn m => fn def => mk_unabs_def m (def RS meta_eq_to_obj_eq)) ms ctr_defs;
   465         val case_def = Morphism.thm phi raw_case_def;
   466 
   467         val ctrs0 = map (Morphism.term phi) raw_ctrs;
   468         val casex0 = Morphism.term phi raw_case;
   469 
   470         val ctrs = map (mk_ctr As) ctrs0;
   471 
   472         fun wrap lthy =
   473           let
   474             fun exhaust_tac {context = ctxt, ...} =
   475               let
   476                 val ctor_iff_dtor_thm =
   477                   let
   478                     val goal =
   479                       fold_rev Logic.all [w, u]
   480                         (mk_Trueprop_eq (HOLogic.mk_eq (u, ctor $ w), HOLogic.mk_eq (dtor $ u, w)));
   481                   in
   482                     Skip_Proof.prove lthy [] [] goal (fn {context = ctxt, ...} =>
   483                       mk_ctor_iff_dtor_tac ctxt (map (SOME o certifyT lthy) [dtorT, fpT])
   484                         (certify lthy ctor) (certify lthy dtor) ctor_dtor dtor_ctor)
   485                     |> Thm.close_derivation
   486                     |> Morphism.thm phi
   487                   end;
   488 
   489                 val sumEN_thm' =
   490                   unfold_thms lthy @{thms all_unit_eq}
   491                     (Drule.instantiate' (map (SOME o certifyT lthy) ctr_prod_Ts) []
   492                        (mk_sumEN_balanced n))
   493                   |> Morphism.thm phi;
   494               in
   495                 mk_exhaust_tac ctxt n ctr_defs ctor_iff_dtor_thm sumEN_thm'
   496               end;
   497 
   498             val inject_tacss =
   499               map2 (fn 0 => K [] | _ => fn ctr_def => [fn {context = ctxt, ...} =>
   500                   mk_inject_tac ctxt ctr_def ctor_inject]) ms ctr_defs;
   501 
   502             val half_distinct_tacss =
   503               map (map (fn (def, def') => fn {context = ctxt, ...} =>
   504                 mk_half_distinct_tac ctxt ctor_inject [def, def'])) (mk_half_pairss (`I ctr_defs));
   505 
   506             val case_tacs =
   507               map3 (fn k => fn m => fn ctr_def => fn {context = ctxt, ...} =>
   508                 mk_case_tac ctxt n k m case_def ctr_def dtor_ctor) ks ms ctr_defs;
   509 
   510             val tacss = [exhaust_tac] :: inject_tacss @ half_distinct_tacss @ [case_tacs];
   511 
   512             val sel_defaultss = map (map (apsnd (prepare_term lthy))) raw_sel_defaultss
   513           in
   514             wrap_datatype tacss (((wrap_opts, ctrs0), casex0), (disc_bindings, (sel_bindingss,
   515               sel_defaultss))) lthy
   516           end;
   517 
   518         fun derive_maps_sets_rels (wrap_res, lthy) =
   519           let
   520             val rel_flip = rel_flip_of_bnf fp_bnf;
   521             val nones = replicate live NONE;
   522 
   523             val ctor_cong =
   524               if lfp then Drule.dummy_thm
   525               else cterm_instantiate_pos [NONE, NONE, SOME (certify lthy ctor)] arg_cong;
   526 
   527             fun mk_cIn ify =
   528               certify lthy o (not lfp ? curry (op $) (map_types ify ctor)) oo
   529               mk_InN_balanced (ify ctr_sum_prod_T) n;
   530 
   531             val cxIns = map2 (mk_cIn I) tuple_xs ks;
   532             val cyIns = map2 (mk_cIn B_ify) tuple_ys ks;
   533 
   534             fun mk_map_thm ctr_def' xs cxIn =
   535               fold_thms lthy [ctr_def']
   536                 (unfold_thms lthy (pre_map_def ::
   537                      (if lfp then [] else [ctor_dtor, dtor_ctor]) @ sum_prod_thms_map)
   538                    (cterm_instantiate_pos (nones @ [SOME cxIn])
   539                       (if lfp then fp_map_thm else fp_map_thm RS ctor_cong)))
   540               |> singleton (Proof_Context.export names_lthy no_defs_lthy);
   541 
   542             fun mk_set_thm fp_set_thm ctr_def' xs cxIn =
   543               fold_thms lthy [ctr_def']
   544                 (unfold_thms lthy (pre_set_defs @ nested_set_natural's @ nesting_set_natural's @
   545                      (if lfp then [] else [dtor_ctor]) @ sum_prod_thms_set)
   546                    (cterm_instantiate_pos [SOME cxIn] fp_set_thm))
   547               |> singleton (Proof_Context.export names_lthy no_defs_lthy);
   548 
   549             fun mk_set_thms fp_set_thm = map3 (mk_set_thm fp_set_thm) ctr_defs' xss cxIns;
   550 
   551             val map_thms = map3 mk_map_thm ctr_defs' xss cxIns;
   552             val set_thmss = map mk_set_thms fp_set_thms;
   553 
   554             val rel_infos = (ctr_defs' ~~ xss ~~ cxIns, ctr_defs' ~~ yss ~~ cyIns);
   555 
   556             fun mk_rel_thm postproc ctr_defs' xs cxIn ys cyIn =
   557               fold_thms lthy ctr_defs'
   558                  (unfold_thms lthy (pre_rel_def :: (if lfp then [] else [dtor_ctor]) @
   559                       sum_prod_thms_rel)
   560                     (cterm_instantiate_pos (nones @ [SOME cxIn, SOME cyIn]) fp_rel_thm))
   561               |> postproc
   562               |> singleton (Proof_Context.export names_lthy no_defs_lthy);
   563 
   564             fun mk_rel_inject_thm (((ctr_def', xs), cxIn), ((_, ys), cyIn)) =
   565               mk_rel_thm (unfold_thms lthy @{thms eq_sym_Unity_conv}) [ctr_def'] xs cxIn ys cyIn;
   566 
   567             val rel_inject_thms = map mk_rel_inject_thm (op ~~ rel_infos);
   568 
   569             fun mk_half_rel_distinct_thm (((xctr_def', xs), cxIn), ((yctr_def', ys), cyIn)) =
   570               mk_rel_thm (fn thm => thm RS @{thm eq_False[THEN iffD1]}) [xctr_def', yctr_def']
   571                 xs cxIn ys cyIn;
   572 
   573             fun mk_other_half_rel_distinct_thm thm =
   574               flip_rels lthy live thm RS (rel_flip RS sym RS @{thm arg_cong[of _ _ Not]} RS iffD2);
   575 
   576             val half_rel_distinct_thmss =
   577               map (map mk_half_rel_distinct_thm) (mk_half_pairss rel_infos);
   578             val other_half_rel_distinct_thmss =
   579               map (map mk_other_half_rel_distinct_thm) half_rel_distinct_thmss;
   580             val (rel_distinct_thms, _) =
   581               join_halves n half_rel_distinct_thmss other_half_rel_distinct_thmss;
   582 
   583             val notes =
   584               [(mapN, map_thms, code_simp_attrs),
   585                (rel_distinctN, rel_distinct_thms, code_simp_attrs),
   586                (rel_injectN, rel_inject_thms, code_simp_attrs),
   587                (setsN, flat set_thmss, code_simp_attrs)]
   588               |> filter_out (null o #2)
   589               |> map (fn (thmN, thms, attrs) =>
   590                 ((qualify true fp_b_name (Binding.name thmN), attrs), [(thms, [])]));
   591           in
   592             (wrap_res, lthy |> Local_Theory.notes notes |> snd)
   593           end;
   594 
   595         fun define_fold_rec no_defs_lthy =
   596           let
   597             val fpT_to_C = fpT --> C;
   598 
   599             fun build_ctor_rec_arg mk_proj (T, U) =
   600               if T = U then
   601                 id_const T
   602               else
   603                 (case (T, U) of
   604                   (Type (s, _), Type (s', _)) =>
   605                   if s = s' then build_map (build_ctor_rec_arg mk_proj) T U else mk_proj T
   606                 | _ => mk_proj T);
   607 
   608             fun mk_U proj (T as Type (@{type_name prod}, [T', U])) =
   609                 if member (op =) fpTs T' then proj (T', U) else T
   610               | mk_U proj (Type (s, Ts)) = Type (s, map (mk_U proj) Ts)
   611               | mk_U _ T = T;
   612 
   613             fun unzip_rec (x as Free (_, T)) =
   614               if exists_fp_subtype T then
   615                 [build_ctor_rec_arg fst_const (T, mk_U fst T) $ x,
   616                  build_ctor_rec_arg snd_const (T, mk_U snd T) $ x]
   617               else
   618                 [x];
   619 
   620             fun mk_rec_like_arg f xs = mk_tupled_fun (HOLogic.mk_tuple xs) f (maps unzip_rec xs);
   621 
   622             fun generate_rec_like (suf, fp_rec_like, (fss, f_Tss, xsss)) =
   623               let
   624                 val res_T = fold_rev (curry (op --->)) f_Tss fpT_to_C;
   625                 val binding = qualify false fp_b_name (Binding.suffix_name ("_" ^ suf) fp_b);
   626                 val spec =
   627                   mk_Trueprop_eq (lists_bmoc fss (Free (Binding.name_of binding, res_T)),
   628                     Term.list_comb (fp_rec_like,
   629                       map2 (mk_sum_caseN_balanced oo map2 mk_rec_like_arg) fss xsss));
   630               in (binding, spec) end;
   631 
   632             val rec_like_infos =
   633               [(foldN, fp_fold, fold_only),
   634                (recN, fp_rec, rec_only)];
   635 
   636             val (bindings, specs) = map generate_rec_like rec_like_infos |> split_list;
   637 
   638             val ((csts, defs), (lthy', lthy)) = no_defs_lthy
   639               |> apfst split_list o fold_map2 (fn b => fn spec =>
   640                 Specification.definition (SOME (b, NONE, NoSyn), ((Thm.def_binding b, []), spec))
   641                 #>> apsnd snd) bindings specs
   642               ||> `Local_Theory.restore;
   643 
   644             val phi = Proof_Context.export_morphism lthy lthy';
   645 
   646             val [fold_def, rec_def] = map (Morphism.thm phi) defs;
   647 
   648             val [foldx, recx] = map (mk_rec_like lfp As Cs o Morphism.term phi) csts;
   649           in
   650             ((foldx, recx, fold_def, rec_def), lthy)
   651           end;
   652 
   653         fun define_unfold_corec no_defs_lthy =
   654           let
   655             val B_to_fpT = C --> fpT;
   656 
   657             fun mk_preds_getterss_join c n cps sum_prod_T cqfss =
   658               Term.lambda c (mk_IfN sum_prod_T cps
   659                 (map2 (mk_InN_balanced sum_prod_T n) (map HOLogic.mk_tuple cqfss) (1 upto n)));
   660 
   661             fun generate_corec_like (suf, fp_rec_like, ((pfss, cqfsss), (f_sum_prod_Ts,
   662                 pf_Tss))) =
   663               let
   664                 val res_T = fold_rev (curry (op --->)) pf_Tss B_to_fpT;
   665                 val binding = qualify false fp_b_name (Binding.suffix_name ("_" ^ suf) fp_b);
   666                 val spec =
   667                   mk_Trueprop_eq (lists_bmoc pfss (Free (Binding.name_of binding, res_T)),
   668                     Term.list_comb (fp_rec_like,
   669                       map5 mk_preds_getterss_join cs ns cpss f_sum_prod_Ts cqfsss));
   670               in (binding, spec) end;
   671 
   672             val corec_like_infos =
   673               [(unfoldN, fp_fold, unfold_only),
   674                (corecN, fp_rec, corec_only)];
   675 
   676             val (bindings, specs) = map generate_corec_like corec_like_infos |> split_list;
   677 
   678             val ((csts, defs), (lthy', lthy)) = no_defs_lthy
   679               |> apfst split_list o fold_map2 (fn b => fn spec =>
   680                 Specification.definition (SOME (b, NONE, NoSyn), ((Thm.def_binding b, []), spec))
   681                 #>> apsnd snd) bindings specs
   682               ||> `Local_Theory.restore;
   683 
   684             val phi = Proof_Context.export_morphism lthy lthy';
   685 
   686             val [unfold_def, corec_def] = map (Morphism.thm phi) defs;
   687 
   688             val [unfold, corec] = map (mk_rec_like lfp As Cs o Morphism.term phi) csts;
   689           in
   690             ((unfold, corec, unfold_def, corec_def), lthy)
   691           end;
   692 
   693         val define_rec_likes = if lfp then define_fold_rec else define_unfold_corec;
   694 
   695         fun massage_res ((wrap_res, rec_like_res), lthy) =
   696           (((ctrs, xss, ctr_defs, wrap_res), rec_like_res), lthy);
   697       in
   698         (wrap #> (live > 0 ? derive_maps_sets_rels) ##>> define_rec_likes #> massage_res, lthy')
   699       end;
   700 
   701     fun wrap_types_and_more (wrap_types_and_mores, lthy) =
   702       fold_map I wrap_types_and_mores lthy
   703       |>> apsnd split_list4 o apfst split_list4 o split_list;
   704 
   705     (* TODO: Add map, sets, rel simps *)
   706     val mk_simp_thmss =
   707       map3 (fn (_, _, _, injects, distincts, cases, _, _, _) => fn rec_likes => fn fold_likes =>
   708         injects @ distincts @ cases @ rec_likes @ fold_likes);
   709 
   710     fun derive_induct_fold_rec_thms_for_types (((ctrss, xsss, ctr_defss, wrap_ress), (folds, recs,
   711         fold_defs, rec_defs)), lthy) =
   712       let
   713         val (((ps, ps'), us'), names_lthy) =
   714           lthy
   715           |> mk_Frees' "P" (map mk_pred1T fpTs)
   716           ||>> Variable.variant_fixes fp_b_names;
   717 
   718         val us = map2 (curry Free) us' fpTs;
   719 
   720         fun mk_sets_nested bnf =
   721           let
   722             val Type (T_name, Us) = T_of_bnf bnf;
   723             val lives = lives_of_bnf bnf;
   724             val sets = sets_of_bnf bnf;
   725             fun mk_set U =
   726               (case find_index (curry (op =) U) lives of
   727                 ~1 => Term.dummy
   728               | i => nth sets i);
   729           in
   730             (T_name, map mk_set Us)
   731           end;
   732 
   733         val setss_nested = map mk_sets_nested nested_bnfs;
   734 
   735         val (induct_thms, induct_thm) =
   736           let
   737             fun mk_set Ts t =
   738               let val Type (_, Ts0) = domain_type (fastype_of t) in
   739                 Term.subst_atomic_types (Ts0 ~~ Ts) t
   740               end;
   741 
   742             fun mk_raw_prem_prems names_lthy (x as Free (s, T as Type (T_name, Ts0))) =
   743                 (case find_index (curry (op =) T) fpTs of
   744                   ~1 =>
   745                   (case AList.lookup (op =) setss_nested T_name of
   746                     NONE => []
   747                   | SOME raw_sets0 =>
   748                     let
   749                       val (Ts, raw_sets) =
   750                         split_list (filter (exists_fp_subtype o fst) (Ts0 ~~ raw_sets0));
   751                       val sets = map (mk_set Ts0) raw_sets;
   752                       val (ys, names_lthy') = names_lthy |> mk_Frees s Ts;
   753                       val xysets = map (pair x) (ys ~~ sets);
   754                       val ppremss = map (mk_raw_prem_prems names_lthy') ys;
   755                     in
   756                       flat (map2 (map o apfst o cons) xysets ppremss)
   757                     end)
   758                 | kk => [([], (kk + 1, x))])
   759               | mk_raw_prem_prems _ _ = [];
   760 
   761             fun close_prem_prem xs t =
   762               fold_rev Logic.all (map Free (drop (nn + length xs)
   763                 (rev (Term.add_frees t (map dest_Free xs @ ps'))))) t;
   764 
   765             fun mk_prem_prem xs (xysets, (j, x)) =
   766               close_prem_prem xs (Logic.list_implies (map (fn (x', (y, set)) =>
   767                   HOLogic.mk_Trueprop (HOLogic.mk_mem (y, set $ x'))) xysets,
   768                 HOLogic.mk_Trueprop (nth ps (j - 1) $ x)));
   769 
   770             fun mk_raw_prem phi ctr ctr_Ts =
   771               let
   772                 val (xs, names_lthy') = names_lthy |> mk_Frees "x" ctr_Ts;
   773                 val pprems = maps (mk_raw_prem_prems names_lthy') xs;
   774               in (xs, pprems, HOLogic.mk_Trueprop (phi $ Term.list_comb (ctr, xs))) end;
   775 
   776             fun mk_prem (xs, raw_pprems, concl) =
   777               fold_rev Logic.all xs (Logic.list_implies (map (mk_prem_prem xs) raw_pprems, concl));
   778 
   779             val raw_premss = map3 (map2 o mk_raw_prem) ps ctrss ctr_Tsss;
   780 
   781             val goal =
   782               Library.foldr (Logic.list_implies o apfst (map mk_prem)) (raw_premss,
   783                 HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj (map2 (curry (op $)) ps us)));
   784 
   785             val kksss = map (map (map (fst o snd) o #2)) raw_premss;
   786 
   787             val ctor_induct' = fp_induct OF (map mk_sumEN_tupled_balanced mss);
   788 
   789             val thm =
   790               Skip_Proof.prove lthy [] [] goal (fn {context = ctxt, ...} =>
   791                 mk_induct_tac ctxt nn ns mss kksss (flat ctr_defss) ctor_induct'
   792                   nested_set_natural's pre_set_defss)
   793               |> singleton (Proof_Context.export names_lthy lthy)
   794               |> Thm.close_derivation;
   795           in
   796             `(conj_dests nn) thm
   797           end;
   798 
   799         val induct_cases = quasi_unambiguous_case_names (maps (map name_of_ctr) ctrss);
   800 
   801         val (fold_thmss, rec_thmss) =
   802           let
   803             val xctrss = map2 (map2 (curry Term.list_comb)) ctrss xsss;
   804             val gfolds = map (lists_bmoc gss) folds;
   805             val hrecs = map (lists_bmoc hss) recs;
   806 
   807             fun mk_goal fss frec_like xctr f xs fxs =
   808               fold_rev (fold_rev Logic.all) (xs :: fss)
   809                 (mk_Trueprop_eq (frec_like $ xctr, Term.list_comb (f, fxs)));
   810 
   811             fun build_rec_like frec_likes maybe_tick (T, U) =
   812               if T = U then
   813                 id_const T
   814               else
   815                 (case find_index (curry (op =) T) fpTs of
   816                   ~1 => build_map (build_rec_like frec_likes maybe_tick) T U
   817                 | kk => maybe_tick (nth us kk) (nth frec_likes kk));
   818 
   819             fun mk_U maybe_mk_prodT =
   820               typ_subst (map2 (fn fpT => fn C => (fpT, maybe_mk_prodT fpT C)) fpTs Cs);
   821 
   822             fun intr_rec_likes frec_likes maybe_cons maybe_tick maybe_mk_prodT (x as Free (_, T)) =
   823               if exists_fp_subtype T then
   824                 maybe_cons x [build_rec_like frec_likes (K I) (T, mk_U (K I) T) $ x]
   825               else
   826                 [x];
   827 
   828             val gxsss = map (map (maps (intr_rec_likes gfolds (K I) (K I) (K I)))) xsss;
   829             val hxsss =
   830               map (map (maps (intr_rec_likes hrecs cons tick (curry HOLogic.mk_prodT)))) xsss;
   831 
   832             val fold_goalss = map5 (map4 o mk_goal gss) gfolds xctrss gss xsss gxsss;
   833             val rec_goalss = map5 (map4 o mk_goal hss) hrecs xctrss hss xsss hxsss;
   834 
   835             val fold_tacss =
   836               map2 (map o mk_rec_like_tac pre_map_defs [] nesting_map_ids'' fold_defs) fp_fold_thms
   837                 ctr_defss;
   838             val rec_tacss =
   839               map2 (map o mk_rec_like_tac pre_map_defs nested_map_comp's
   840                 (nested_map_ids'' @ nesting_map_ids'') rec_defs) fp_rec_thms ctr_defss;
   841 
   842             fun prove goal tac =
   843               Skip_Proof.prove lthy [] [] goal (tac o #context)
   844               |> Thm.close_derivation;
   845           in
   846             (map2 (map2 prove) fold_goalss fold_tacss, map2 (map2 prove) rec_goalss rec_tacss)
   847           end;
   848 
   849         val simp_thmss = mk_simp_thmss wrap_ress rec_thmss fold_thmss;
   850 
   851         val induct_case_names_attr = Attrib.internal (K (Rule_Cases.case_names induct_cases));
   852         fun induct_type_attr T_name = Attrib.internal (K (Induct.induct_type T_name));
   853 
   854         val common_notes =
   855           (if nn > 1 then [(inductN, [induct_thm], [induct_case_names_attr])] else [])
   856           |> map (fn (thmN, thms, attrs) =>
   857             ((qualify true fp_common_name (Binding.name thmN), attrs), [(thms, [])]));
   858 
   859         val notes =
   860           [(foldN, fold_thmss, K code_simp_attrs),
   861            (inductN, map single induct_thms,
   862             fn T_name => [induct_case_names_attr, induct_type_attr T_name]),
   863            (recN, rec_thmss, K code_simp_attrs),
   864            (simpsN, simp_thmss, K [])]
   865           |> maps (fn (thmN, thmss, attrs) =>
   866             map3 (fn fp_b_name => fn Type (T_name, _) => fn thms =>
   867               ((qualify true fp_b_name (Binding.name thmN), attrs T_name),
   868                [(thms, [])])) fp_b_names fpTs thmss);
   869       in
   870         lthy |> Local_Theory.notes (common_notes @ notes) |> snd
   871       end;
   872 
   873     fun derive_coinduct_unfold_corec_thms_for_types (((ctrss, _, ctr_defss, wrap_ress), (unfolds,
   874         corecs, unfold_defs, corec_defs)), lthy) =
   875       let
   876         val nesting_rel_eqs = map rel_eq_of_bnf nesting_bnfs;
   877 
   878         val discss = map (map (mk_disc_or_sel As) o #1) wrap_ress;
   879         val selsss = map (map (map (mk_disc_or_sel As)) o #2) wrap_ress;
   880         val exhaust_thms = map #3 wrap_ress;
   881         val disc_thmsss = map #7 wrap_ress;
   882         val discIss = map #8 wrap_ress;
   883         val sel_thmsss = map #9 wrap_ress;
   884 
   885         val (((rs, us'), vs'), names_lthy) =
   886           lthy
   887           |> mk_Frees "R" (map (fn T => mk_pred2T T T) fpTs)
   888           ||>> Variable.variant_fixes fp_b_names
   889           ||>> Variable.variant_fixes (map (suffix "'") fp_b_names);
   890 
   891         val us = map2 (curry Free) us' fpTs;
   892         val udiscss = map2 (map o rapp) us discss;
   893         val uselsss = map2 (map o map o rapp) us selsss;
   894 
   895         val vs = map2 (curry Free) vs' fpTs;
   896         val vdiscss = map2 (map o rapp) vs discss;
   897         val vselsss = map2 (map o map o rapp) vs selsss;
   898 
   899         val ((coinduct_thms, coinduct_thm), (strong_coinduct_thms, strong_coinduct_thm)) =
   900           let
   901             val uvrs = map3 (fn r => fn u => fn v => r $ u $ v) rs us vs;
   902             val uv_eqs = map2 (curry HOLogic.mk_eq) us vs;
   903             val strong_rs =
   904               map4 (fn u => fn v => fn uvr => fn uv_eq =>
   905                 fold_rev Term.lambda [u, v] (HOLogic.mk_disj (uvr, uv_eq))) us vs uvrs uv_eqs;
   906 
   907             fun build_rel rs' T =
   908               (case find_index (curry (op =) T) fpTs of
   909                 ~1 =>
   910                 if exists_fp_subtype T then build_rel_step (build_rel rs') T
   911                 else HOLogic.eq_const T
   912               | kk => nth rs' kk);
   913 
   914             fun build_rel_app rs' usel vsel =
   915               fold rapp [usel, vsel] (build_rel rs' (fastype_of usel));
   916 
   917             fun mk_prem_ctr_concls rs' n k udisc usels vdisc vsels =
   918               (if k = n then [] else [HOLogic.mk_eq (udisc, vdisc)]) @
   919               (if null usels then
   920                  []
   921                else
   922                  [Library.foldr HOLogic.mk_imp (if n = 1 then [] else [udisc, vdisc],
   923                     Library.foldr1 HOLogic.mk_conj (map2 (build_rel_app rs') usels vsels))]);
   924 
   925             fun mk_prem_concl rs' n udiscs uselss vdiscs vselss =
   926               Library.foldr1 HOLogic.mk_conj
   927                 (flat (map5 (mk_prem_ctr_concls rs' n) (1 upto n) udiscs uselss vdiscs vselss))
   928               handle List.Empty => @{term True};
   929 
   930             fun mk_prem rs' uvr u v n udiscs uselss vdiscs vselss =
   931               fold_rev Logic.all [u, v] (Logic.mk_implies (HOLogic.mk_Trueprop uvr,
   932                 HOLogic.mk_Trueprop (mk_prem_concl rs' n udiscs uselss vdiscs vselss)));
   933 
   934             val concl =
   935               HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj
   936                 (map3 (fn uvr => fn u => fn v => HOLogic.mk_imp (uvr, HOLogic.mk_eq (u, v)))
   937                    uvrs us vs));
   938 
   939             fun mk_goal rs' =
   940               Logic.list_implies (map8 (mk_prem rs') uvrs us vs ns udiscss uselsss vdiscss vselsss,
   941                 concl);
   942 
   943             val goal = mk_goal rs;
   944             val strong_goal = mk_goal strong_rs;
   945 
   946             fun prove dtor_coinduct' goal =
   947               Skip_Proof.prove lthy [] [] goal (fn {context = ctxt, ...} =>
   948                 mk_coinduct_tac ctxt nesting_rel_eqs nn ns dtor_coinduct' pre_rel_defs dtor_ctors
   949                   exhaust_thms ctr_defss disc_thmsss sel_thmsss)
   950               |> singleton (Proof_Context.export names_lthy lthy)
   951               |> Thm.close_derivation;
   952 
   953             fun postproc nn thm =
   954               Thm.permute_prems 0 nn
   955                 (if nn = 1 then thm RS mp
   956                  else funpow nn (fn thm => reassoc_conjs (thm RS mp_conj)) thm)
   957               |> Drule.zero_var_indexes
   958               |> `(conj_dests nn);
   959           in
   960             (postproc nn (prove fp_induct goal), postproc nn (prove fp_strong_induct strong_goal))
   961           end;
   962 
   963         fun mk_maybe_not pos = not pos ? HOLogic.mk_not;
   964 
   965         val z = the_single zs;
   966         val gunfolds = map (lists_bmoc pgss) unfolds;
   967         val hcorecs = map (lists_bmoc phss) corecs;
   968 
   969         val (unfold_thmss, corec_thmss, safe_unfold_thmss, safe_corec_thmss) =
   970           let
   971             fun mk_goal pfss c cps fcorec_like n k ctr m cfs' =
   972               fold_rev (fold_rev Logic.all) ([c] :: pfss)
   973                 (Logic.list_implies (seq_conds (HOLogic.mk_Trueprop oo mk_maybe_not) n k cps,
   974                    mk_Trueprop_eq (fcorec_like $ c, Term.list_comb (ctr, take m cfs'))));
   975 
   976             fun build_corec_like fcorec_likes maybe_tack (T, U) =
   977               if T = U then
   978                 id_const T
   979               else
   980                 (case find_index (curry (op =) U) fpTs of
   981                   ~1 => build_map (build_corec_like fcorec_likes maybe_tack) T U
   982                 | kk => maybe_tack (nth cs kk, nth us kk) (nth fcorec_likes kk));
   983 
   984             fun mk_U maybe_mk_sumT =
   985               typ_subst (map2 (fn C => fn fpT => (maybe_mk_sumT fpT C, fpT)) Cs fpTs);
   986 
   987             fun intr_corec_likes fcorec_likes maybe_mk_sumT maybe_tack cqf =
   988               let val T = fastype_of cqf in
   989                 if exists_Cs_subtype T then
   990                   build_corec_like fcorec_likes maybe_tack (T, mk_U maybe_mk_sumT T) $ cqf
   991                 else
   992                   cqf
   993               end;
   994 
   995             val crgsss' = map (map (map (intr_corec_likes gunfolds (K I) (K I)))) crgsss;
   996             val cshsss' =
   997               map (map (map (intr_corec_likes hcorecs (curry mk_sumT) (tack z)))) cshsss;
   998 
   999             val unfold_goalss =
  1000               map8 (map4 oooo mk_goal pgss) cs cpss gunfolds ns kss ctrss mss crgsss';
  1001             val corec_goalss =
  1002               map8 (map4 oooo mk_goal phss) cs cpss hcorecs ns kss ctrss mss cshsss';
  1003 
  1004             val unfold_tacss =
  1005               map3 (map oo mk_corec_like_tac unfold_defs nesting_map_ids) fp_fold_thms pre_map_defs
  1006                 ctr_defss;
  1007             val corec_tacss =
  1008               map3 (map oo mk_corec_like_tac corec_defs nesting_map_ids) fp_rec_thms pre_map_defs
  1009                 ctr_defss;
  1010 
  1011             fun prove goal tac =
  1012               Skip_Proof.prove lthy [] [] goal (tac o #context) |> Thm.close_derivation;
  1013 
  1014             val unfold_thmss = map2 (map2 prove) unfold_goalss unfold_tacss;
  1015             val corec_thmss =
  1016               map2 (map2 prove) corec_goalss corec_tacss
  1017               |> map (map (unfold_thms lthy @{thms sum_case_if}));
  1018 
  1019             val unfold_safesss = map2 (map2 (map2 (curry (op =)))) crgsss' crgsss;
  1020             val corec_safesss = map2 (map2 (map2 (curry (op =)))) cshsss' cshsss;
  1021 
  1022             val filter_safesss =
  1023               map2 (map_filter (fn (safes, thm) => if forall I safes then SOME thm else NONE) oo
  1024                 curry (op ~~));
  1025 
  1026             val safe_unfold_thmss = filter_safesss unfold_safesss unfold_thmss;
  1027             val safe_corec_thmss = filter_safesss corec_safesss corec_thmss;
  1028           in
  1029             (unfold_thmss, corec_thmss, safe_unfold_thmss, safe_corec_thmss)
  1030           end;
  1031 
  1032         val (disc_unfold_iff_thmss, disc_corec_iff_thmss) =
  1033           let
  1034             fun mk_goal c cps fcorec_like n k disc =
  1035               mk_Trueprop_eq (disc $ (fcorec_like $ c),
  1036                 if n = 1 then @{const True}
  1037                 else Library.foldr1 HOLogic.mk_conj (seq_conds mk_maybe_not n k cps));
  1038 
  1039             val unfold_goalss = map6 (map2 oooo mk_goal) cs cpss gunfolds ns kss discss;
  1040             val corec_goalss = map6 (map2 oooo mk_goal) cs cpss hcorecs ns kss discss;
  1041 
  1042             fun mk_case_split' cp =
  1043               Drule.instantiate' [] [SOME (certify lthy cp)] @{thm case_split};
  1044 
  1045             val case_splitss' = map (map mk_case_split') cpss;
  1046 
  1047             val unfold_tacss =
  1048               map3 (map oo mk_disc_corec_like_iff_tac) case_splitss' unfold_thmss disc_thmsss;
  1049             val corec_tacss =
  1050               map3 (map oo mk_disc_corec_like_iff_tac) case_splitss' corec_thmss disc_thmsss;
  1051 
  1052             fun prove goal tac =
  1053               Skip_Proof.prove lthy [] [] goal (tac o #context)
  1054               |> singleton (Proof_Context.export names_lthy0 no_defs_lthy)
  1055               |> Thm.close_derivation;
  1056 
  1057             fun proves [_] [_] = []
  1058               | proves goals tacs = map2 prove goals tacs;
  1059           in
  1060             (map2 proves unfold_goalss unfold_tacss,
  1061              map2 proves corec_goalss corec_tacss)
  1062           end;
  1063 
  1064         val is_triv_discI = is_triv_implies orf is_concl_refl;
  1065 
  1066         fun mk_disc_corec_like_thms corec_likes discIs =
  1067           map (op RS) (filter_out (is_triv_discI o snd) (corec_likes ~~ discIs));
  1068 
  1069         val disc_unfold_thmss = map2 mk_disc_corec_like_thms unfold_thmss discIss;
  1070         val disc_corec_thmss = map2 mk_disc_corec_like_thms corec_thmss discIss;
  1071 
  1072         fun mk_sel_corec_like_thm corec_like_thm sel sel_thm =
  1073           let
  1074             val (domT, ranT) = dest_funT (fastype_of sel);
  1075             val arg_cong' =
  1076               Drule.instantiate' (map (SOME o certifyT lthy) [domT, ranT])
  1077                 [NONE, NONE, SOME (certify lthy sel)] arg_cong
  1078               |> Thm.varifyT_global;
  1079             val sel_thm' = sel_thm RSN (2, trans);
  1080           in
  1081             corec_like_thm RS arg_cong' RS sel_thm'
  1082           end;
  1083 
  1084         fun mk_sel_corec_like_thms corec_likess =
  1085           map3 (map3 (map2 o mk_sel_corec_like_thm)) corec_likess selsss sel_thmsss |> map flat;
  1086 
  1087         val sel_unfold_thmss = mk_sel_corec_like_thms unfold_thmss;
  1088         val sel_corec_thmss = mk_sel_corec_like_thms corec_thmss;
  1089 
  1090         fun flat_corec_like_thms corec_likes disc_corec_likes sel_corec_likes =
  1091           corec_likes @ disc_corec_likes @ sel_corec_likes;
  1092 
  1093         val simp_thmss =
  1094           mk_simp_thmss wrap_ress
  1095             (map3 flat_corec_like_thms safe_corec_thmss disc_corec_thmss sel_corec_thmss)
  1096             (map3 flat_corec_like_thms safe_unfold_thmss disc_unfold_thmss sel_unfold_thmss);
  1097 
  1098         val anonymous_notes =
  1099           [(flat safe_unfold_thmss @ flat safe_corec_thmss, simp_attrs)]
  1100           |> map (fn (thms, attrs) => ((Binding.empty, attrs), [(thms, [])]));
  1101 
  1102         val common_notes =
  1103           (if nn > 1 then
  1104              (* FIXME: attribs *)
  1105              [(coinductN, [coinduct_thm], []),
  1106               (strong_coinductN, [strong_coinduct_thm], [])]
  1107            else
  1108              [])
  1109           |> map (fn (thmN, thms, attrs) =>
  1110             ((qualify true fp_common_name (Binding.name thmN), attrs), [(thms, [])]));
  1111 
  1112         val notes =
  1113           [(coinductN, map single coinduct_thms, []), (* FIXME: attribs *)
  1114            (corecN, corec_thmss, []),
  1115            (disc_corecN, disc_corec_thmss, simp_attrs),
  1116            (disc_corec_iffN, disc_corec_iff_thmss, simp_attrs),
  1117            (disc_unfoldN, disc_unfold_thmss, simp_attrs),
  1118            (disc_unfold_iffN, disc_unfold_iff_thmss, simp_attrs),
  1119            (sel_corecN, sel_corec_thmss, simp_attrs),
  1120            (sel_unfoldN, sel_unfold_thmss, simp_attrs),
  1121            (simpsN, simp_thmss, []),
  1122            (strong_coinductN, map single strong_coinduct_thms, []), (* FIXME: attribs *)
  1123            (unfoldN, unfold_thmss, [])]
  1124           |> maps (fn (thmN, thmss, attrs) =>
  1125             map_filter (fn (_, []) => NONE | (fp_b_name, thms) =>
  1126               SOME ((qualify true fp_b_name (Binding.name thmN), attrs),
  1127                 [(thms, [])])) (fp_b_names ~~ thmss));
  1128       in
  1129         lthy |> Local_Theory.notes (anonymous_notes @ common_notes @ notes) |> snd
  1130       end;
  1131 
  1132     val lthy' = lthy
  1133       |> fold_map define_ctrs_case_for_type (fp_bnfs ~~ fp_bs ~~ fpTs ~~ Cs ~~ ctors ~~ dtors ~~
  1134         fp_folds ~~ fp_recs ~~ ctor_dtors ~~ dtor_ctors ~~ ctor_injects ~~ pre_map_defs ~~
  1135         pre_set_defss ~~ pre_rel_defs ~~ fp_map_thms ~~ fp_set_thmss ~~ fp_rel_thms ~~ ns ~~ kss ~~
  1136         mss ~~ ctr_bindingss ~~ ctr_mixfixess ~~ ctr_Tsss ~~ disc_bindingss ~~ sel_bindingsss ~~
  1137         raw_sel_defaultsss)
  1138       |> wrap_types_and_more
  1139       |> (if lfp then derive_induct_fold_rec_thms_for_types
  1140           else derive_coinduct_unfold_corec_thms_for_types);
  1141 
  1142     val timer = time (timer ("Constructors, discriminators, selectors, etc., for the new " ^
  1143       (if lfp then "" else "co") ^ "datatype"));
  1144   in
  1145     timer; lthy'
  1146   end;
  1147 
  1148 val datatypes = define_datatypes (K I) (K I) (K I);
  1149 
  1150 val datatype_cmd = define_datatypes Typedecl.read_constraint Syntax.parse_typ Syntax.read_term;
  1151 
  1152 val parse_ctr_arg =
  1153   @{keyword "("} |-- parse_binding_colon -- Parse.typ --| @{keyword ")"} ||
  1154   (Parse.typ >> pair Binding.empty);
  1155 
  1156 val parse_defaults =
  1157   @{keyword "("} |-- @{keyword "defaults"} |-- Scan.repeat parse_bound_term --| @{keyword ")"};
  1158 
  1159 val parse_single_spec =
  1160   Parse.type_args_constrained -- Parse.binding -- Parse.opt_mixfix --
  1161   (@{keyword "="} |-- Parse.enum1 "|" (parse_opt_binding_colon -- Parse.binding --
  1162     Scan.repeat parse_ctr_arg -- Scan.optional parse_defaults [] -- Parse.opt_mixfix));
  1163 
  1164 val parse_datatype = parse_wrap_options -- Parse.and_list1 parse_single_spec;
  1165 
  1166 fun parse_datatype_cmd lfp construct_fp = parse_datatype >> datatype_cmd lfp construct_fp;
  1167 
  1168 end;