(*  Title:      HOL/Codatatype/Tools/bnf_fp_sugar.ML

     Author:     Jasmin Blanchette, TU Muenchen

     Copyright   2012

 Sugar for constructing LFPs and GFPs.

 *)

 signature BNF_FP_SUGAR =

 sig

   val datatyp: bool ->

     (mixfix list -> (string * sort) list option -> binding list -> typ list * typ list list ->

       BNF_Def.BNF list -> local_theory ->

       (term list * term list * term list * term list * thm * thm list * thm list * thm list *

          thm list * thm list) * local_theory) ->

     bool * ((((typ * sort) list * binding) * mixfix) * ((((binding * binding) *

       (binding * typ) list) * (binding * term) list) * mixfix) list) list ->

     local_theory -> local_theory

   val parse_datatype_cmd: bool ->

     (mixfix list -> (string * sort) list option -> binding list -> typ list * typ list list ->

       BNF_Def.BNF list -> local_theory ->

       (term list * term list * term list * term list * thm * thm list * thm list * thm list *

          thm list * thm list) * local_theory) ->

     (local_theory -> local_theory) parser

 end;

 structure BNF_FP_Sugar : BNF_FP_SUGAR =

 struct

 open BNF_Util

 open BNF_Wrap

 open BNF_Def

 open BNF_FP_Util

 open BNF_FP_Sugar_Tactics

 val simp_attrs = @{attributes [simp]};

 fun split_list10 xs =

   (map #1 xs, map #2 xs, map #3 xs, map #4 xs, map #5 xs, map #6 xs, map #7 xs, map #8 xs,

    map #9 xs, map #10 xs);

 fun strip_map_type (Type (@{type_name fun}, [T as Type _, T'])) = strip_map_type T' |>> cons T

   | strip_map_type T = ([], T);

 fun resort_tfree S (TFree (s, _)) = TFree (s, S);

 fun typ_subst inst (T as Type (s, Ts)) =

     (case AList.lookup (op =) inst T of

       NONE => Type (s, map (typ_subst inst) Ts)

     | SOME T' => T')

   | typ_subst inst T = the_default T (AList.lookup (op =) inst T);

 val lists_bmoc = fold (fn xs => fn t => Term.list_comb (t, xs));

 fun mk_tupled_fun x f xs = HOLogic.tupled_lambda x (Term.list_comb (f, xs));

 fun mk_uncurried_fun f xs = mk_tupled_fun (HOLogic.mk_tuple xs) f xs;

 fun mk_uncurried2_fun f xss =

   mk_tupled_fun (HOLogic.mk_tuple (map HOLogic.mk_tuple xss)) f (flat xss);

 fun tick v f = Term.lambda v (HOLogic.mk_prod (v, f $ v));

 fun tack z_name (c, v) f =

   let val z = Free (z_name, mk_sumT (fastype_of v, fastype_of c)) in

     Term.lambda z (mk_sum_case (Term.lambda v v, Term.lambda c (f $ c)) $ z)

   end;

 fun cannot_merge_types () = error "Mutually recursive types must have the same type parameters";

 fun merge_type_arg T T' = if T = T' then T else cannot_merge_types ();

 fun merge_type_args (As, As') =

   if length As = length As' then map2 merge_type_arg As As' else cannot_merge_types ();

 fun type_args_constrained_of (((cAs, _), _), _) = cAs;

 fun type_binding_of (((_, b), _), _) = b;

 fun mixfix_of ((_, mx), _) = mx;

 fun ctr_specs_of (_, ctr_specs) = ctr_specs;

 fun disc_of ((((disc, _), _), _), _) = disc;

 fun ctr_of ((((_, ctr), _), _), _) = ctr;

 fun args_of (((_, args), _), _) = args;

 fun defaults_of ((_, ds), _) = ds;

 fun ctr_mixfix_of (_, mx) = mx;

 fun define_datatype prepare_constraint prepare_typ prepare_term lfp construct (no_dests, specs)

     no_defs_lthy0 =

   let

     (* TODO: sanity checks on arguments *)

     val _ = if not lfp andalso no_dests then error "Cannot define destructor-less codatatypes"

       else ();

     val nn = length specs;

     val fp_bs = map type_binding_of specs;

     val fp_common_name = mk_common_name fp_bs;

     fun prepare_type_arg (ty, c) =

       let val TFree (s, _) = prepare_typ no_defs_lthy0 ty in

         TFree (s, prepare_constraint no_defs_lthy0 c)

       end;

     val Ass0 = map (map prepare_type_arg o type_args_constrained_of) specs;

     val unsorted_Ass0 = map (map (resort_tfree HOLogic.typeS)) Ass0;

     val unsorted_As = Library.foldr1 merge_type_args unsorted_Ass0;

     val ((Bs, Cs), no_defs_lthy) =

       no_defs_lthy0

       |> fold (Variable.declare_typ o resort_tfree dummyS) unsorted_As

       |> mk_TFrees nn

       ||>> mk_TFrees nn;

     (* TODO: cleaner handling of fake contexts, without "background_theory" *)

     (*the "perhaps o try" below helps gracefully handles the case where the new type is defined in a

       locale and shadows an existing global type*)

     val fake_thy =

       Theory.copy #> fold (fn spec => perhaps (try (Sign.add_type no_defs_lthy

         (type_binding_of spec, length (type_args_constrained_of spec), mixfix_of spec)))) specs;

     val fake_lthy = Proof_Context.background_theory fake_thy no_defs_lthy;

     fun mk_fake_T b =

       Type (fst (Term.dest_Type (Proof_Context.read_type_name fake_lthy true (Binding.name_of b))),

         unsorted_As);

     val fake_Ts = map mk_fake_T fp_bs;

     val mixfixes = map mixfix_of specs;

     val _ = (case duplicates Binding.eq_name fp_bs of [] => ()

       | b :: _ => error ("Duplicate type name declaration " ^ quote (Binding.name_of b)));

     val ctr_specss = map ctr_specs_of specs;

     val disc_bindingss = map (map disc_of) ctr_specss;

     val ctr_bindingss =

       map2 (fn fp_b => map (Binding.qualify false (Binding.name_of fp_b) o ctr_of))

         fp_bs ctr_specss;

     val ctr_argsss = map (map args_of) ctr_specss;

     val ctr_mixfixess = map (map ctr_mixfix_of) ctr_specss;

     val sel_bindingsss = map (map (map fst)) ctr_argsss;

     val fake_ctr_Tsss0 = map (map (map (prepare_typ fake_lthy o snd))) ctr_argsss;

     val raw_sel_defaultsss = map (map defaults_of) ctr_specss;

     val (As :: _) :: fake_ctr_Tsss =

       burrow (burrow (Syntax.check_typs fake_lthy)) (Ass0 :: fake_ctr_Tsss0);

     val _ = (case duplicates (op =) unsorted_As of [] => ()

       | A :: _ => error ("Duplicate type parameter " ^

           quote (Syntax.string_of_typ no_defs_lthy A)));

     val rhs_As' = fold (fold (fold Term.add_tfreesT)) fake_ctr_Tsss [];

     val _ = (case subtract (op =) (map dest_TFree As) rhs_As' of

         [] => ()

       | A' :: _ => error ("Extra type variable on right-hand side: " ^

           quote (Syntax.string_of_typ no_defs_lthy (TFree A'))));

     fun eq_fpT (T as Type (s, Us)) (Type (s', Us')) =

         s = s' andalso (Us = Us' orelse error ("Illegal occurrence of recursive type " ^

           quote (Syntax.string_of_typ fake_lthy T)))

       | eq_fpT _ _ = false;

     fun freeze_fp (T as Type (s, Us)) =

         (case find_index (eq_fpT T) fake_Ts of ~1 => Type (s, map freeze_fp Us) | j => nth Bs j)

       | freeze_fp T = T;

     val ctr_TsssBs = map (map (map freeze_fp)) fake_ctr_Tsss;

     val ctr_sum_prod_TsBs = map (mk_sumTN_balanced o map HOLogic.mk_tupleT) ctr_TsssBs;

     val fp_eqs =

       map dest_TFree Bs ~~ map (Term.typ_subst_atomic (As ~~ unsorted_As)) ctr_sum_prod_TsBs;

     val (pre_bnfs, ((unfs0, flds0, fp_iters0, fp_recs0, fp_induct, unf_flds, fld_unfs, fld_injects,

         fp_iter_thms, fp_rec_thms), lthy)) =

       fp_bnf construct fp_bs mixfixes (map dest_TFree unsorted_As) fp_eqs no_defs_lthy0;

     fun add_nesty_bnf_names Us =

       let

         fun add (Type (s, Ts)) ss =

             let val (needs, ss') = fold_map add Ts ss in

               if exists I needs then (true, insert (op =) s ss') else (false, ss')

             end

           | add T ss = (member (op =) Us T, ss);

       in snd oo add end;

     fun nesty_bnfs Us =

       map_filter (bnf_of lthy) (fold (fold (fold (add_nesty_bnf_names Us))) ctr_TsssBs []);

     val nesting_bnfs = nesty_bnfs As;

     val nested_bnfs = nesty_bnfs Bs;

     val timer = time (Timer.startRealTimer ());

     fun mk_unf_or_fld get_T Ts t =

       let val Type (_, Ts0) = get_T (fastype_of t) in

         Term.subst_atomic_types (Ts0 ~~ Ts) t

       end;

     val mk_unf = mk_unf_or_fld domain_type;

     val mk_fld = mk_unf_or_fld range_type;

     val unfs = map (mk_unf As) unfs0;

     val flds = map (mk_fld As) flds0;

     val fpTs = map (domain_type o fastype_of) unfs;

     val exists_fp_subtype = exists_subtype (member (op =) fpTs);

     val ctr_Tsss = map (map (map (Term.typ_subst_atomic (Bs ~~ fpTs)))) ctr_TsssBs;

     val ns = map length ctr_Tsss;

     val kss = map (fn n => 1 upto n) ns;

     val mss = map (map length) ctr_Tsss;

     val Css = map2 replicate ns Cs;

     fun mk_iter_like Ts Us t =

       let

         val (bindings, body) = strip_type (fastype_of t);

         val (f_Us, prebody) = split_last bindings;

         val Type (_, Ts0) = if lfp then prebody else body;

         val Us0 = distinct (op =) (map (if lfp then body_type else domain_type) f_Us);

       in

         Term.subst_atomic_types (Ts0 @ Us0 ~~ Ts @ Us) t

       end;

     val fp_iters as fp_iter1 :: _ = map (mk_iter_like As Cs) fp_iters0;

     val fp_recs as fp_rec1 :: _ = map (mk_iter_like As Cs) fp_recs0;

     val fp_iter_fun_Ts = fst (split_last (binder_types (fastype_of fp_iter1)));

     val fp_rec_fun_Ts = fst (split_last (binder_types (fastype_of fp_rec1)));

     val ((iter_only as (gss, _, _), rec_only as (hss, _, _)),

          (zs, cs, cpss, coiter_only as ((pgss, crgsss), _), corec_only as ((phss, cshsss), _))) =

       if lfp then

         let

           val y_Tsss =

             map3 (fn n => fn ms => map2 dest_tupleT ms o dest_sumTN_balanced n o domain_type)

               ns mss fp_iter_fun_Ts;

           val g_Tss = map2 (map2 (curry (op --->))) y_Tsss Css;

           val ((gss, ysss), _) =

             lthy

             |> mk_Freess "f" g_Tss

             ||>> mk_Freesss "x" y_Tsss;

           val yssss = map (map (map single)) ysss;

           fun dest_rec_prodT (T as Type (@{type_name prod}, Us as [_, U])) =

               if member (op =) Cs U then Us else [T]

             | dest_rec_prodT T = [T];

           val z_Tssss =

             map3 (fn n => fn ms => map2 (map dest_rec_prodT oo dest_tupleT) ms o

               dest_sumTN_balanced n o domain_type) ns mss fp_rec_fun_Ts;

           val h_Tss = map2 (map2 (fold_rev (curry (op --->)))) z_Tssss Css;

           val hss = map2 (map2 retype_free) h_Tss gss;

           val zssss_hd = map2 (map2 (map2 (retype_free o hd))) z_Tssss ysss;

           val (zssss_tl, _) =

             lthy

             |> mk_Freessss "y" (map (map (map tl)) z_Tssss);

           val zssss = map2 (map2 (map2 cons)) zssss_hd zssss_tl;

         in

           (((gss, g_Tss, yssss), (hss, h_Tss, zssss)),

            ([], [], [], (([], []), ([], [])), (([], []), ([], []))))

         end

       else

         let

           (*avoid "'a itself" arguments in coiterators and corecursors*)

           val mss' =  map (fn [0] => [1] | ms => ms) mss;

           val p_Tss = map2 (fn n => replicate (Int.max (0, n - 1)) o mk_predT) ns Cs;

           fun zip_predss_getterss qss fss = maps (op @) (qss ~~ fss);

           fun zip_preds_predsss_gettersss [] [qss] [fss] = zip_predss_getterss qss fss

             | zip_preds_predsss_gettersss (p :: ps) (qss :: qsss) (fss :: fsss) =

               p :: zip_predss_getterss qss fss @ zip_preds_predsss_gettersss ps qsss fsss;

           fun mk_types maybe_dest_sumT fun_Ts =

             let

               val f_sum_prod_Ts = map range_type fun_Ts;

               val f_prod_Tss = map2 dest_sumTN_balanced ns f_sum_prod_Ts;

               val f_Tssss =

                 map3 (fn C => map2 (map (map (curry (op -->) C) o maybe_dest_sumT) oo dest_tupleT))

                   Cs mss' f_prod_Tss;

               val q_Tssss =

                 map (map (map (fn [_] => [] | [_, C] => [mk_predT (domain_type C)]))) f_Tssss;

               val pf_Tss = map3 zip_preds_predsss_gettersss p_Tss q_Tssss f_Tssss;

             in (q_Tssss, f_sum_prod_Ts, f_Tssss, pf_Tss) end;

           val (r_Tssss, g_sum_prod_Ts, g_Tssss, pg_Tss) = mk_types single fp_iter_fun_Ts;

           val ((((Free (z, _), cs), pss), gssss), _) =

             lthy

             |> yield_singleton (mk_Frees "z") dummyT

             ||>> mk_Frees "a" Cs

             ||>> mk_Freess "p" p_Tss

             ||>> mk_Freessss "g" g_Tssss;

           val rssss = map (map (map (fn [] => []))) r_Tssss;

           fun dest_corec_sumT (T as Type (@{type_name sum}, Us as [_, U])) =

               if member (op =) Cs U then Us else [T]

             | dest_corec_sumT T = [T];

           val (s_Tssss, h_sum_prod_Ts, h_Tssss, ph_Tss) = mk_types dest_corec_sumT fp_rec_fun_Ts;

           val hssss_hd = map2 (map2 (map2 (fn T :: _ => fn [g] => retype_free T g))) h_Tssss gssss;

           val ((sssss, hssss_tl), _) =

             lthy

             |> mk_Freessss "q" s_Tssss

             ||>> mk_Freessss "h" (map (map (map tl)) h_Tssss);

           val hssss = map2 (map2 (map2 cons)) hssss_hd hssss_tl;

           val cpss = map2 (fn c => map (fn p => p $ c)) cs pss;

           fun mk_preds_getters_join [] [cf] = cf

             | mk_preds_getters_join [cq] [cf, cf'] =

               mk_If cq (mk_Inl (fastype_of cf') cf) (mk_Inr (fastype_of cf) cf');

           fun mk_terms qssss fssss =

             let

               val pfss = map3 zip_preds_predsss_gettersss pss qssss fssss;

               val cqssss = map2 (fn c => map (map (map (fn f => f $ c)))) cs qssss;

               val cfssss = map2 (fn c => map (map (map (fn f => f $ c)))) cs fssss;

               val cqfsss = map2 (map2 (map2 mk_preds_getters_join)) cqssss cfssss;

             in (pfss, cqfsss) end;

         in

           ((([], [], []), ([], [], [])),

            ([z], cs, cpss, (mk_terms rssss gssss, (g_sum_prod_Ts, pg_Tss)),

             (mk_terms sssss hssss, (h_sum_prod_Ts, ph_Tss))))

         end;

     fun define_ctrs_case_for_type ((((((((((((((((((fp_b, fpT), C), fld), unf), fp_iter), fp_rec),

           fld_unf), unf_fld), fld_inject), n), ks), ms), ctr_bindings), ctr_mixfixes), ctr_Tss),

         disc_bindings), sel_bindingss), raw_sel_defaultss) no_defs_lthy =

       let

         val unfT = domain_type (fastype_of fld);

         val ctr_prod_Ts = map HOLogic.mk_tupleT ctr_Tss;

         val ctr_sum_prod_T = mk_sumTN_balanced ctr_prod_Ts;

         val case_Ts = map (fn Ts => Ts ---> C) ctr_Tss;

         val ((((u, fs), xss), v'), _) =

           no_defs_lthy

           |> yield_singleton (mk_Frees "u") unfT

           ||>> mk_Frees "f" case_Ts

           ||>> mk_Freess "x" ctr_Tss

           ||>> yield_singleton (Variable.variant_fixes) (Binding.name_of fp_b);

         val v = Free (v', fpT);

         val ctr_rhss =

           map2 (fn k => fn xs => fold_rev Term.lambda xs (fld $

             mk_InN_balanced ctr_sum_prod_T n (HOLogic.mk_tuple xs) k)) ks xss;

         val case_binding = Binding.suffix_name ("_" ^ caseN) fp_b;

         val case_rhs =

           fold_rev Term.lambda (fs @ [v])

             (mk_sum_caseN_balanced (map2 mk_uncurried_fun fs xss) $ (unf $ v));

         val ((raw_case :: raw_ctrs, raw_case_def :: raw_ctr_defs), (lthy', lthy)) = no_defs_lthy

           |> apfst split_list o fold_map3 (fn b => fn mx => fn rhs =>

               Local_Theory.define ((b, mx), ((Thm.def_binding b, []), rhs)) #>> apsnd snd)

             (case_binding :: ctr_bindings) (NoSyn :: ctr_mixfixes) (case_rhs :: ctr_rhss)

           ||> `Local_Theory.restore;

         val phi = Proof_Context.export_morphism lthy lthy';

         val ctr_defs = map (Morphism.thm phi) raw_ctr_defs;

         val case_def = Morphism.thm phi raw_case_def;

         val ctrs0 = map (Morphism.term phi) raw_ctrs;

         val casex0 = Morphism.term phi raw_case;

         val ctrs = map (mk_ctr As) ctrs0;

         fun exhaust_tac {context = ctxt, ...} =

           let

             val fld_iff_unf_thm =

               let

                 val goal =

                   fold_rev Logic.all [u, v]

                     (mk_Trueprop_eq (HOLogic.mk_eq (v, fld $ u), HOLogic.mk_eq (unf $ v, u)));

               in

                 Skip_Proof.prove lthy [] [] goal (fn {context = ctxt, ...} =>

                   mk_fld_iff_unf_tac ctxt (map (SOME o certifyT lthy) [unfT, fpT])

                     (certify lthy fld) (certify lthy unf) fld_unf unf_fld)

                 |> Thm.close_derivation

                 |> Morphism.thm phi

               end;

             val sumEN_thm' =

               Local_Defs.unfold lthy @{thms all_unit_eq}

                 (Drule.instantiate' (map (SOME o certifyT lthy) ctr_prod_Ts) []

                    (mk_sumEN_balanced n))

               |> Morphism.thm phi;

           in

             mk_exhaust_tac ctxt n ctr_defs fld_iff_unf_thm sumEN_thm'

           end;

         val inject_tacss =

           map2 (fn 0 => K [] | _ => fn ctr_def => [fn {context = ctxt, ...} =>

               mk_inject_tac ctxt ctr_def fld_inject]) ms ctr_defs;

         val half_distinct_tacss =

           map (map (fn (def, def') => fn {context = ctxt, ...} =>

             mk_half_distinct_tac ctxt fld_inject [def, def'])) (mk_half_pairss ctr_defs);

         val case_tacs =

           map3 (fn k => fn m => fn ctr_def => fn {context = ctxt, ...} =>

             mk_case_tac ctxt n k m case_def ctr_def unf_fld) ks ms ctr_defs;

         val tacss = [exhaust_tac] :: inject_tacss @ half_distinct_tacss @ [case_tacs];

         fun define_iter_rec ((selss0, discIs, sel_thmss), no_defs_lthy) =

           let

             val fpT_to_C = fpT --> C;

             fun generate_iter_like (suf, fp_iter_like, (fss, f_Tss, xssss)) =

               let

                 val res_T = fold_rev (curry (op --->)) f_Tss fpT_to_C;

                 val binding = Binding.suffix_name ("_" ^ suf) fp_b;

                 val spec =

                   mk_Trueprop_eq (lists_bmoc fss (Free (Binding.name_of binding, res_T)),

                     Term.list_comb (fp_iter_like,

                       map2 (mk_sum_caseN_balanced oo map2 mk_uncurried2_fun) fss xssss));

               in (binding, spec) end;

             val iter_like_infos =

               [(iterN, fp_iter, iter_only),

                (recN, fp_rec, rec_only)];

             val (bindings, specs) = map generate_iter_like iter_like_infos |> split_list;

             val ((csts, defs), (lthy', lthy)) = no_defs_lthy

               |> apfst split_list o fold_map2 (fn b => fn spec =>

                 Specification.definition (SOME (b, NONE, NoSyn), ((Thm.def_binding b, []), spec))

                 #>> apsnd snd) bindings specs

               ||> `Local_Theory.restore;

             val phi = Proof_Context.export_morphism lthy lthy';

             val [iter_def, rec_def] = map (Morphism.thm phi) defs;

             val [iter, recx] = map (mk_iter_like As Cs o Morphism.term phi) csts;

           in

             ((ctrs, selss0, iter, recx, xss, ctr_defs, discIs, sel_thmss, iter_def, rec_def), lthy)

           end;

         fun define_coiter_corec ((selss0, discIs, sel_thmss), no_defs_lthy) =

           let

             val B_to_fpT = C --> fpT;

             fun mk_preds_getterss_join c n cps sum_prod_T cqfss =

               Term.lambda c (mk_IfN sum_prod_T cps

                 (map2 (mk_InN_balanced sum_prod_T n) (map HOLogic.mk_tuple cqfss) (1 upto n)));

             fun generate_coiter_like (suf, fp_iter_like, ((pfss, cqfsss), (f_sum_prod_Ts,

                 pf_Tss))) =

               let

                 val res_T = fold_rev (curry (op --->)) pf_Tss B_to_fpT;

                 val binding = Binding.suffix_name ("_" ^ suf) fp_b;

                 val spec =

                   mk_Trueprop_eq (lists_bmoc pfss (Free (Binding.name_of binding, res_T)),

                     Term.list_comb (fp_iter_like,

                       map5 mk_preds_getterss_join cs ns cpss f_sum_prod_Ts cqfsss));

               in (binding, spec) end;

             val coiter_like_infos =

               [(coiterN, fp_iter, coiter_only),

                (corecN, fp_rec, corec_only)];

             val (bindings, specs) = map generate_coiter_like coiter_like_infos |> split_list;

             val ((csts, defs), (lthy', lthy)) = no_defs_lthy

               |> apfst split_list o fold_map2 (fn b => fn spec =>

                 Specification.definition (SOME (b, NONE, NoSyn), ((Thm.def_binding b, []), spec))

                 #>> apsnd snd) bindings specs

               ||> `Local_Theory.restore;

             val phi = Proof_Context.export_morphism lthy lthy';

             val [coiter_def, corec_def] = map (Morphism.thm phi) defs;

             val [coiter, corec] = map (mk_iter_like As Cs o Morphism.term phi) csts;

           in

             ((ctrs, selss0, coiter, corec, xss, ctr_defs, discIs, sel_thmss, coiter_def, corec_def),

              lthy)

           end;

         fun wrap lthy =

           let val sel_defaultss = map (map (apsnd (prepare_term lthy))) raw_sel_defaultss in

             wrap_datatype tacss (((no_dests, ctrs0), casex0), (disc_bindings, (sel_bindingss,

               sel_defaultss))) lthy

           end;

         val define_iter_likes = if lfp then define_iter_rec else define_coiter_corec;

       in

         ((wrap, define_iter_likes), lthy')

       end;

     val pre_map_defs = map map_def_of_bnf pre_bnfs;

     val pre_set_defss = map set_defs_of_bnf pre_bnfs;

     val nested_set_natural's = maps set_natural'_of_bnf nested_bnfs;

     val nesting_map_ids = map map_id_of_bnf nesting_bnfs;

     fun mk_map Ts Us t =

       let val (Type (_, Ts0), Type (_, Us0)) = strip_map_type (fastype_of t) |>> List.last in

         Term.subst_atomic_types (Ts0 @ Us0 ~~ Ts @ Us) t

       end;

     fun build_map build_arg (Type (s, Ts)) (Type (_, Us)) =

       let

         val map0 = map_of_bnf (the (bnf_of lthy s));

         val mapx = mk_map Ts Us map0;

         val TUs = map dest_funT (fst (split_last (fst (strip_map_type (fastype_of mapx)))));

         val args = map build_arg TUs;

       in Term.list_comb (mapx, args) end;

     fun derive_induct_iter_rec_thms_for_types ((ctrss, _, iters, recs, xsss, ctr_defss, _, _,

         iter_defs, rec_defs), lthy) =

       let

         val (((phis, phis'), vs'), names_lthy) =

           lthy

           |> mk_Frees' "P" (map mk_predT fpTs)

           ||>> Variable.variant_fixes (map Binding.name_of fp_bs);

         val vs = map2 (curry Free) vs' fpTs;

         fun mk_sets_nested bnf =

           let

             val Type (T_name, Us) = T_of_bnf bnf;

             val lives = lives_of_bnf bnf;

             val sets = sets_of_bnf bnf;

             fun mk_set U =

               (case find_index (curry (op =) U) lives of

                 ~1 => Term.dummy

               | i => nth sets i);

           in

             (T_name, map mk_set Us)

           end;

         val setss_nested = map mk_sets_nested nested_bnfs;

         val (induct_thms, induct_thm) =

           let

             fun mk_set Ts t =

               let val Type (_, Ts0) = domain_type (fastype_of t) in

                 Term.subst_atomic_types (Ts0 ~~ Ts) t

               end;

             fun mk_raw_prem_prems names_lthy (x as Free (s, T as Type (T_name, Ts0))) =

                 (case find_index (curry (op =) T) fpTs of

                   ~1 =>

                   (case AList.lookup (op =) setss_nested T_name of

                     NONE => []

                   | SOME raw_sets0 =>

                     let

                       val (Ts, raw_sets) =

                         split_list (filter (exists_fp_subtype o fst) (Ts0 ~~ raw_sets0));

                       val sets = map (mk_set Ts0) raw_sets;

                       val (ys, names_lthy') = names_lthy |> mk_Frees s Ts;

                       val xysets = map (pair x) (ys ~~ sets);

                       val ppremss = map (mk_raw_prem_prems names_lthy') ys;

                     in

                       flat (map2 (map o apfst o cons) xysets ppremss)

                     end)

                 | i => [([], (i, x))])

               | mk_raw_prem_prems _ _ = [];

             fun close_prem_prem x' t =

               fold_rev Logic.all

                 (map Free (drop (nn + 1) (rev (Term.add_frees t (x' :: phis'))))) t;

             fun mk_prem_prem x (xysets, (i, x')) =

               close_prem_prem (dest_Free x)

                 (Logic.list_implies (map (fn (x'', (y, set)) =>

                      HOLogic.mk_Trueprop (HOLogic.mk_mem (y, set $ x''))) xysets,

                    HOLogic.mk_Trueprop (nth phis i $ x')));

             fun mk_raw_prem phi ctr ctr_Ts =

               let

                 val (xs, names_lthy') = names_lthy |> mk_Frees "x" ctr_Ts;

                 val pprems =

                   maps (fn x => map (mk_prem_prem x) (mk_raw_prem_prems names_lthy' x)) xs;

               in

                 (xs, pprems, HOLogic.mk_Trueprop (phi $ Term.list_comb (ctr, xs)))

               end;

             val raw_premss = map3 (map2 o mk_raw_prem) phis ctrss ctr_Tsss;

             fun mk_prem (xs, pprems, concl) =

               fold_rev Logic.all xs (Logic.list_implies (pprems, concl));

             val goal =

               Library.foldr (Logic.list_implies o apfst (map mk_prem)) (raw_premss,

                 HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj

                   (map2 (curry (op $)) phis vs)));

             (* ### WRONG *)

             val rss = map (map (length o #2)) raw_premss;

             val ppisss = map (map (fn r => map (pair r) (1 upto r))) rss;

             val fld_induct' = fp_induct OF (map mk_sumEN_tupled_balanced mss);

             val induct_thm =

               Skip_Proof.prove lthy [] [] goal (fn {context = ctxt, ...} =>

                 mk_induct_tac ctxt ns mss ppisss (flat ctr_defss) fld_induct' nested_set_natural's

                   pre_set_defss)

               |> singleton (Proof_Context.export names_lthy lthy)

           in

             `(conj_dests nn) induct_thm

           end;

         val (iter_thmss, rec_thmss) =

           let

             val xctrss = map2 (map2 (curry Term.list_comb)) ctrss xsss;

             val giters = map (lists_bmoc gss) iters;

             val hrecs = map (lists_bmoc hss) recs;

             fun mk_goal_iter_like fss fiter_like xctr f xs fxs =

               fold_rev (fold_rev Logic.all) (xs :: fss)

                 (mk_Trueprop_eq (fiter_like $ xctr, Term.list_comb (f, fxs)));

             fun build_call fiter_likes maybe_tick (T, U) =

               if T = U then

                 id_const T

               else

                 (case find_index (curry (op =) T) fpTs of

                   ~1 => build_map (build_call fiter_likes maybe_tick) T U

                 | j => maybe_tick (nth vs j) (nth fiter_likes j));

             fun mk_U maybe_mk_prodT =

               typ_subst (map2 (fn fpT => fn C => (fpT, maybe_mk_prodT fpT C)) fpTs Cs);

             fun intr_calls fiter_likes maybe_cons maybe_tick maybe_mk_prodT (x as Free (_, T)) =

               if member (op =) fpTs T then

                 maybe_cons x [build_call fiter_likes (K I) (T, mk_U (K I) T) $ x]

               else if exists_fp_subtype T then

                 [build_call fiter_likes maybe_tick (T, mk_U maybe_mk_prodT T) $ x]

               else

                 [x];

             val gxsss = map (map (maps (intr_calls giters (K I) (K I) (K I)))) xsss;

             val hxsss = map (map (maps (intr_calls hrecs cons tick (curry HOLogic.mk_prodT)))) xsss;

             val goal_iterss = map5 (map4 o mk_goal_iter_like gss) giters xctrss gss xsss gxsss;

             val goal_recss = map5 (map4 o mk_goal_iter_like hss) hrecs xctrss hss xsss hxsss;

             val iter_tacss =

               map2 (map o mk_iter_like_tac pre_map_defs nesting_map_ids iter_defs) fp_iter_thms

                 ctr_defss;

             val rec_tacss =

               map2 (map o mk_iter_like_tac pre_map_defs nesting_map_ids rec_defs) fp_rec_thms

                 ctr_defss;

           in

             (map2 (map2 (fn goal => fn tac => Skip_Proof.prove lthy [] [] goal (tac o #context)))

                goal_iterss iter_tacss,

              map2 (map2 (fn goal => fn tac => Skip_Proof.prove lthy [] [] goal (tac o #context)))

                goal_recss rec_tacss)

           end;

         val common_notes =

           (if nn > 1 then [(inductN, [induct_thm], [])] (* FIXME: attribs *) else [])

           |> map (fn (thmN, thms, attrs) =>

               ((Binding.qualify true fp_common_name (Binding.name thmN), attrs), [(thms, [])]));

         val notes =

           [(inductN, map single induct_thms, []), (* FIXME: attribs *)

            (itersN, iter_thmss, simp_attrs),

            (recsN, rec_thmss, Code.add_default_eqn_attrib :: simp_attrs)]

           |> maps (fn (thmN, thmss, attrs) =>

             map2 (fn b => fn thms =>

               ((Binding.qualify true (Binding.name_of b) (Binding.name thmN), attrs),

                 [(thms, [])])) fp_bs thmss);

       in

         lthy |> Local_Theory.notes (common_notes @ notes) |> snd

       end;

     fun derive_coinduct_coiter_corec_thms_for_types ((ctrss, selsss, coiters, corecs, _, ctr_defss,

         discIss, sel_thmsss, coiter_defs, corec_defs), lthy) =

       let

         val (vs', _) =

           lthy

           |> Variable.variant_fixes (map Binding.name_of fp_bs);

         val vs = map2 (curry Free) vs' fpTs;

         val (coinduct_thms, coinduct_thm) =

           let

             val coinduct_thm = fp_induct;

           in

             `(conj_dests nn) coinduct_thm

           end;

         val (coiter_thmss, corec_thmss) =

           let

             val z = the_single zs;

             val gcoiters = map (lists_bmoc pgss) coiters;

             val hcorecs = map (lists_bmoc phss) corecs;

             fun mk_goal_cond pos = HOLogic.mk_Trueprop o (not pos ? HOLogic.mk_not);

             fun mk_goal_coiter_like pfss c cps fcoiter_like n k ctr m cfs' =

               fold_rev (fold_rev Logic.all) ([c] :: pfss)

                 (Logic.list_implies (seq_conds mk_goal_cond n k cps,

                    mk_Trueprop_eq (fcoiter_like $ c, Term.list_comb (ctr, take m cfs'))));

             fun build_call fiter_likes maybe_tack (T, U) =

               if T = U then

                 id_const T

               else

                 (case find_index (curry (op =) U) fpTs of

                   ~1 => build_map (build_call fiter_likes maybe_tack) T U

                 | j => maybe_tack (nth cs j, nth vs j) (nth fiter_likes j));

             fun mk_U maybe_mk_sumT =

               typ_subst (map2 (fn C => fn fpT => (maybe_mk_sumT fpT C, fpT)) Cs fpTs);

             fun intr_calls fiter_likes maybe_mk_sumT maybe_tack cqf =

               let val T = fastype_of cqf in

                 if exists_subtype (member (op =) Cs) T then

                   build_call fiter_likes maybe_tack (T, mk_U maybe_mk_sumT T) $ cqf

                 else

                   cqf

               end;

             val crgsss' = map (map (map (intr_calls gcoiters (K I) (K I)))) crgsss;

             val cshsss' = map (map (map (intr_calls hcorecs (curry mk_sumT) (tack z)))) cshsss;

             val goal_coiterss =

               map8 (map4 oooo mk_goal_coiter_like pgss) cs cpss gcoiters ns kss ctrss mss crgsss';

             val goal_corecss =

               map8 (map4 oooo mk_goal_coiter_like phss) cs cpss hcorecs ns kss ctrss mss cshsss';

             val coiter_tacss =

               map3 (map oo mk_coiter_like_tac coiter_defs nesting_map_ids) fp_iter_thms pre_map_defs

                 ctr_defss;

             val corec_tacss =

               map3 (map oo mk_coiter_like_tac corec_defs nesting_map_ids) fp_rec_thms pre_map_defs

                 ctr_defss;

           in

             (map2 (map2 (fn goal => fn tac =>

                  Skip_Proof.prove lthy [] [] goal (tac o #context) |> Thm.close_derivation))

                goal_coiterss coiter_tacss,

              map2 (map2 (fn goal => fn tac =>

                  Skip_Proof.prove lthy [] [] goal (tac o #context)

                  |> Local_Defs.unfold lthy @{thms sum_case_if} |> Thm.close_derivation))

                goal_corecss corec_tacss)

           end;

         fun mk_disc_coiter_like_thms [_] = K []

           | mk_disc_coiter_like_thms thms = map2 (curry (op RS)) thms;

         val disc_coiter_thmss = map2 mk_disc_coiter_like_thms coiter_thmss discIss;

         val disc_corec_thmss = map2 mk_disc_coiter_like_thms corec_thmss discIss;

         fun mk_sel_coiter_like_thm coiter_like_thm sel0 sel_thm =

           let

             val (domT, ranT) = dest_funT (fastype_of sel0);

             val arg_cong' =

               Drule.instantiate' (map (SOME o certifyT lthy) [domT, ranT])

                 [NONE, NONE, SOME (certify lthy sel0)] arg_cong

               |> Thm.varifyT_global;

             val sel_thm' = sel_thm RSN (2, trans);

           in

             coiter_like_thm RS arg_cong' RS sel_thm'

           end;

         val sel_coiter_thmsss =

           map3 (map3 (map2 o mk_sel_coiter_like_thm)) coiter_thmss selsss sel_thmsss;

         val sel_corec_thmsss =

           map3 (map3 (map2 o mk_sel_coiter_like_thm)) corec_thmss selsss sel_thmsss;

         val common_notes =

           (if nn > 1 then [(coinductN, [coinduct_thm], [])] (* FIXME: attribs *) else [])

           |> map (fn (thmN, thms, attrs) =>

               ((Binding.qualify true fp_common_name (Binding.name thmN), attrs), [(thms, [])]));

         val notes =

           [(coinductN, map single coinduct_thms, []), (* FIXME: attribs *)

            (coitersN, coiter_thmss, []),

            (disc_coitersN, disc_coiter_thmss, []),

            (sel_coitersN, map flat sel_coiter_thmsss, []),

            (corecsN, corec_thmss, []),

            (disc_corecsN, disc_corec_thmss, []),

            (sel_corecsN, map flat sel_corec_thmsss, [])]

           |> maps (fn (thmN, thmss, attrs) =>

             map_filter (fn (_, []) => NONE | (b, thms) =>

               SOME ((Binding.qualify true (Binding.name_of b) (Binding.name thmN), attrs),

                 [(thms, [])])) (fp_bs ~~ thmss));

       in

         lthy |> Local_Theory.notes (common_notes @ notes) |> snd

       end;

     fun wrap_types_and_define_iter_likes ((wraps, define_iter_likess), lthy) =

       fold_map2 (curry (op o)) define_iter_likess wraps lthy |>> split_list10

     val lthy' = lthy

       |> fold_map define_ctrs_case_for_type (fp_bs ~~ fpTs ~~ Cs ~~ flds ~~ unfs ~~ fp_iters ~~

         fp_recs ~~ fld_unfs ~~ unf_flds ~~ fld_injects ~~ ns ~~ kss ~~ mss ~~ ctr_bindingss ~~

         ctr_mixfixess ~~ ctr_Tsss ~~ disc_bindingss ~~ sel_bindingsss ~~ raw_sel_defaultsss)

       |>> split_list |> wrap_types_and_define_iter_likes

       |> (if lfp then derive_induct_iter_rec_thms_for_types

           else derive_coinduct_coiter_corec_thms_for_types);

     val timer = time (timer ("Constructors, discriminators, selectors, etc., for the new " ^

       (if lfp then "" else "co") ^ "datatype"));

   in

     timer; lthy'

   end;

 val datatyp = define_datatype (K I) (K I) (K I);

 val datatype_cmd = define_datatype Typedecl.read_constraint Syntax.parse_typ Syntax.read_term;

 val parse_binding_colon = Parse.binding --| @{keyword ":"};

 val parse_opt_binding_colon = Scan.optional parse_binding_colon no_binding;

 val parse_ctr_arg =

   @{keyword "("} |-- parse_binding_colon -- Parse.typ --| @{keyword ")"} ||

   (Parse.typ >> pair no_binding);

 val parse_defaults =

   @{keyword "("} |-- @{keyword "defaults"} |-- Scan.repeat parse_bound_term --| @{keyword ")"};

 val parse_single_spec =

   Parse.type_args_constrained -- Parse.binding -- Parse.opt_mixfix --

   (@{keyword "="} |-- Parse.enum1 "|" (parse_opt_binding_colon -- Parse.binding --

     Scan.repeat parse_ctr_arg -- Scan.optional parse_defaults [] -- Parse.opt_mixfix));

 val parse_datatype = parse_wrap_options -- Parse.and_list1 parse_single_spec;

 fun parse_datatype_cmd lfp construct = parse_datatype >> datatype_cmd lfp construct;

 end;