(*  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

   (* TODO: programmatic interface *)

 end;

 structure BNF_FP_Sugar : BNF_FP_SUGAR =

 struct

 open BNF_Util

 open BNF_Wrap

 open BNF_Def

 open BNF_FP_Util

 open BNF_LFP

 open BNF_GFP

 open BNF_FP_Sugar_Tactics

 val caseN = "case";

 val coitersN = "coiters";

 val corecsN = "corecs";

 val disc_coitersN = "disc_coiters";

 val disc_corecsN = "disc_corecs";

 val itersN = "iters";

 val recsN = "recs";

 val sel_coitersN = "sel_coiters";

 val sel_corecsN = "sel_corecs";

 fun split_list11 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, map #11 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 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);

 fun retype_free (Free (s, _)) T = Free (s, T);

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

 fun mk_predT T = T --> HOLogic.boolT;

 fun mk_id T = Const (@{const_name id}, T --> T);

 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_constrained ctxt (T, c) (T', c') =

   if T = T' then

     (case (c, c') of

       (_, NONE) => (T, c)

     | (NONE, _) => (T, c')

     | _ =>

       if c = c' then

         (T, c)

       else

         error ("Inconsistent sort constraints for type variable " ^

           quote (Syntax.string_of_typ ctxt T)))

   else

     cannot_merge_types ();

 fun merge_type_args_constrained ctxt (cAs, cAs') =

   if length cAs = length cAs' then map2 (merge_type_arg_constrained ctxt) cAs cAs'

   else cannot_merge_types ();

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

 val type_args_of = map fst o type_args_constrained_of;

 fun type_binder_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 ctr_mixfix_of (_, mx) = mx;

 fun prepare_datatype prepare_typ lfp specs fake_lthy no_defs_lthy =

   let

     val constrained_As =

       map (map (apfst (prepare_typ fake_lthy)) o type_args_constrained_of) specs

       |> Library.foldr1 (merge_type_args_constrained no_defs_lthy);

     val As = map fst constrained_As;

     val As' = map dest_TFree As;

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

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

           quote (Syntax.string_of_typ no_defs_lthy A)));

     (* TODO: use sort constraints on type args *)

     val N = length specs;

     fun mk_fake_T b =

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

         As);

     val bs = map type_binder_of specs;

     val fakeTs = map mk_fake_T bs;

     val mixfixes = map mixfix_of specs;

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

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

     val ctr_specss = map ctr_specs_of specs;

     val disc_binderss = map (map disc_of) ctr_specss;

     val ctr_binderss = map (map ctr_of) ctr_specss;

     val ctr_argsss = map (map args_of) ctr_specss;

     val ctr_mixfixess = map (map ctr_mixfix_of) ctr_specss;

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

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

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

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

         [] => ()

       | A' :: _ => error ("Extra type variables on rhs: " ^

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

     val ((Cs, Xs), _) =

       no_defs_lthy

       |> fold (fold (fn s => Variable.declare_typ (TFree (s, dummyS))) o type_args_of) specs

       |> mk_TFrees N

       ||>> mk_TFrees N;

     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) fakeTs of ~1 => Type (s, map freeze_fp Us) | j => nth Xs j)

       | freeze_fp T = T;

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

     val ctr_sum_prod_TsXs = map (mk_sumTN_balanced o map HOLogic.mk_tupleT) ctr_TsssXs;

     val eqs = map dest_TFree Xs ~~ ctr_sum_prod_TsXs;

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

         fp_iter_thms, fp_rec_thms), lthy)) =

       fp_bnf (if lfp then bnf_lfp else bnf_gfp) bs mixfixes As' eqs no_defs_lthy;

     val add_nested_bnf_names =

       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 =) As T, ss);

       in snd oo add end;

     val nested_bnfs =

       map_filter (bnf_of lthy) (fold (fold (fold add_nested_bnf_names)) ctr_TsssXs []);

     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 ctr_Tsss = map (map (map (Term.typ_subst_atomic (Xs ~~ fpTs)))) ctr_TsssXs;

     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 (binders, body) = strip_type (fastype_of t);

         val (f_Us, prebody) = split_last binders;

         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, _, cgssss), _),

           corec_only as ((phss, _, chssss), _))) =

       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) gss h_Tss;

           val zssss_hd = map2 (map2 (map2 (fn y => fn T :: _ => retype_free y T))) ysss z_Tssss;

           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_getterss qss fss = maps (op @) (qss ~~ fss);

           fun zip_preds_gettersss [] [qss] [fss] = zip_getterss qss fss

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

               p :: zip_getterss qss fss @ zip_preds_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_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 [g] => fn T :: _ => retype_free g T))) gssss h_Tssss;

           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_terms qssss fssss =

             let

               val pfss = map3 zip_preds_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;

             in (pfss, cqssss, cfssss) 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 pour_some_sugar_on_type (((((((((((((((((b, fpT), C), fld), unf), fp_iter), fp_rec),

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

         disc_binders), sel_binderss) 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, v), fs), xss), _) =

           no_defs_lthy

           |> yield_singleton (mk_Frees "u") unfT

           ||>> yield_singleton (mk_Frees "v") fpT

           ||>> mk_Frees "f" case_Ts

           ||>> mk_Freess "x" ctr_Tss;

         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_binder = Binding.suffix_name ("_" ^ caseN) 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_binder :: ctr_binders) (NoSyn :: ctr_mixfixes) (case_rhs :: ctr_rhss)

           ||> `Local_Theory.restore;

         (*transforms defined frees into consts (and more)*)

         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 some_lfp_sugar ((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 binder = Binding.suffix_name ("_" ^ suf) b;

                 val spec =

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

                     Term.list_comb (fp_iter_like,

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

               in (binder, spec) end;

             val iter_like_bundles =

               [(iterN, fp_iter, iter_only),

                (recN, fp_rec, rec_only)];

             val (binders, specs) = map generate_iter_like iter_like_bundles |> 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) binders specs

               ||> `Local_Theory.restore;

             (*transforms defined frees into consts (and more)*)

             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, v, xss, ctr_defs, discIs, sel_thmss, iter_def, rec_def),

              lthy)

           end;

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

           let

             val B_to_fpT = C --> fpT;

             fun mk_getters_join [] [cf] = cf

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

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

             fun mk_preds_gettersss_join c n cps sum_prod_T cqsss cfsss =

               Term.lambda c (mk_IfN sum_prod_T cps

                 (map2 (mk_InN_balanced sum_prod_T n)

                    (map2 (HOLogic.mk_tuple oo map2 mk_getters_join) cqsss cfsss) (1 upto n)));

             fun generate_coiter_like (suf, fp_iter_like, ((pfss, cqssss, cfssss), (f_sum_prod_Ts,

                 pf_Tss))) =

               let

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

                 val binder = Binding.suffix_name ("_" ^ suf) b;

                 val spec =

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

                     Term.list_comb (fp_iter_like,

                       map6 mk_preds_gettersss_join cs ns cpss f_sum_prod_Ts cqssss cfssss));

               in (binder, spec) end;

             val coiter_like_bundles =

               [(coiterN, fp_iter, coiter_only),

                (corecN, fp_rec, corec_only)];

             val (binders, specs) = map generate_coiter_like coiter_like_bundles |> 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) binders specs

               ||> `Local_Theory.restore;

             (*transforms defined frees into consts (and more)*)

             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, v, xss, ctr_defs, discIs, sel_thmss, coiter_def,

               corec_def), lthy)

           end;

       in

         wrap_datatype tacss ((ctrs0, casex0), (disc_binders, sel_binderss)) lthy'

         |> (if lfp then some_lfp_sugar else some_gfp_sugar)

       end;

     val pre_map_defs = map map_def_of_bnf pre_bnfs;

     val map_ids = map map_id_of_bnf nested_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 pour_more_sugar_on_lfps ((ctrss, _, iters, recs, vs, xsss, ctr_defss, _, _, iter_defs,

         rec_defs), lthy) =

       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;

         val (iter_thmss, rec_thmss) =

           let

             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

                 mk_id 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 repair_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_subtype (member (op =) fpTs) T then

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

               else

                 [x];

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

             val hxsss =

               map (map (maps (repair_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 map_ids iter_defs) fp_iter_thms ctr_defss;

             val rec_tacss =

               map2 (map o mk_iter_like_tac pre_map_defs map_ids rec_defs) fp_rec_thms ctr_defss;

           in

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

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

                goal_iterss iter_tacss,

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

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

                goal_recss rec_tacss)

           end;

         val notes =

           [(itersN, iter_thmss),

            (recsN, rec_thmss)]

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

             map2 (fn b => fn thms =>

                 ((Binding.qualify true (Binding.name_of b) (Binding.name thmN), []), [(thms, [])]))

               bs thmss);

       in

         lthy |> Local_Theory.notes notes |> snd

       end;

     fun pour_more_sugar_on_gfps ((ctrss, selsss, coiters, corecs, vs, _, ctr_defss, discIss,

         sel_thmsss, coiter_defs, corec_defs), lthy) =

       let

         val z = the_single zs;

         val gcoiters = map (lists_bmoc pgss) coiters;

         val hcorecs = map (lists_bmoc phss) corecs;

         val (coiter_thmss, corec_thmss) =

           let

             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 cfss' =

               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, lists_bmoc (take m cfss') ctr)));

             fun build_call fiter_likes maybe_tack (T, U) =

               if T = U then

                 mk_id 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 repair_calls fiter_likes maybe_mk_sumT maybe_tack

                 (cf as Free (_, Type (_, [_, T])) $ _) =

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

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

               else

                 cf;

             val cgssss' =

               map (map (map (map (repair_calls gcoiters (K I) (K I))))) cgssss;

             val chssss' =

               map (map (map (map (repair_calls hcorecs (curry mk_sumT) (tack z))))) chssss;

             val goal_coiterss =

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

             val goal_corecss =

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

             val coiter_tacss =

               map3 (map oo mk_coiter_like_tac coiter_defs map_ids) fp_iter_thms pre_map_defs

                 ctr_defss;

             val corec_tacss =

               map3 (map oo mk_coiter_like_tac corec_defs 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)))

                goal_coiterss coiter_tacss,

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

                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 notes =

           [(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) =>

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

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

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

       in

         lthy |> Local_Theory.notes notes |> snd

       end;

     val lthy' = lthy

       |> fold_map pour_some_sugar_on_type (bs ~~ fpTs ~~ Cs ~~ flds ~~ unfs ~~ fp_iters ~~

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

         ctr_mixfixess ~~ ctr_Tsss ~~ disc_binderss ~~ sel_bindersss)

       |>> split_list11

       |> (if lfp then pour_more_sugar_on_lfps else pour_more_sugar_on_gfps);

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

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

   in

     (timer; lthy')

   end;

 fun datatype_cmd info specs lthy =

   let

     (* 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 lthy

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

     val fake_lthy = Proof_Context.background_theory fake_thy lthy;

   in

     prepare_datatype Syntax.read_typ info specs fake_lthy lthy

   end;

 val parse_opt_binding_colon = Scan.optional (Parse.binding --| Parse.$$$ ":") no_binder

 val parse_ctr_arg =

   Parse.$$$ "(" |-- parse_opt_binding_colon -- Parse.typ --| Parse.$$$ ")" ||

   (Parse.typ >> pair no_binder);

 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 -- Parse.opt_mixfix));

 val _ =

   Outer_Syntax.local_theory @{command_spec "data"} "define BNF-based inductive datatypes"

     (Parse.and_list1 parse_single_spec >> datatype_cmd true);

 val _ =

   Outer_Syntax.local_theory @{command_spec "codata"} "define BNF-based coinductive datatypes"

     (Parse.and_list1 parse_single_spec >> datatype_cmd false);

 end;