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

fun ctr_mixfix_of (_, mx) = mx;

fun prepare_datatype prepare_typ prepare_term lfp (no_dests, specs) fake_lthy no_defs_lthy =

  let

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

      else ();

    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 raw_sel_defaultsss = map (map defaults_of) ctr_specss;

    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, 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) 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_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 [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_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 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), 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, 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_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 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,

                      map5 mk_preds_getterss_join cs ns cpss f_sum_prod_Ts cqfsss));

              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;

        val sel_defaultss = map (map (apsnd (prepare_term lthy'))) raw_sel_defaultss;

      in

        wrap_datatype tacss (((no_dests, ctrs0), casex0), (disc_binders, (sel_binderss,

          sel_defaultss))) 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)))

               goal_iterss iter_tacss,

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

               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 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

                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 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 (repair_calls gcoiters (K I) (K I)))) crgsss;

            val cshsss' = map (map (map (repair_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 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) |> 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 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 ~~ raw_sel_defaultsss)

      |>> 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 lfp (bundle as (_, 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 Syntax.read_term lfp bundle fake_lthy lthy

  end;

val parse_opt_binding_colon = Scan.optional (Parse.binding --| @{keyword ":"}) no_binder

val parse_ctr_arg =

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

  (Parse.typ >> pair no_binder);

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;

val _ =

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

    (parse_datatype >> datatype_cmd true);

val _ =

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

    (parse_datatype >> datatype_cmd false);

end;