(*  Title:      HOL/Tools/BNF/bnf_fp_def_sugar.ML

     Author:     Jasmin Blanchette, TU Muenchen

     Copyright   2012, 2013

 Sugared datatype and codatatype constructions.

 *)

 signature BNF_FP_DEF_SUGAR =

 sig

   val fp_sugar_of: Proof.context -> string -> BNF_FP_Util.fp_sugar option

   val fp_sugars_of: Proof.context -> BNF_FP_Util.fp_sugar list

   val fp_sugar_interpretation: (BNF_FP_Util.fp_sugar list -> theory -> theory) -> theory -> theory

   val register_fp_sugars: BNF_FP_Util.fp_sugar list -> local_theory -> local_theory

   val co_induct_of: 'a list -> 'a

   val strong_co_induct_of: 'a list -> 'a

   val flat_corec_preds_predsss_gettersss: 'a list -> 'a list list list -> 'a list list list ->

     'a list

   val nesty_bnfs: Proof.context -> typ list list list -> typ list -> BNF_Def.bnf list

   type lfp_sugar_thms =

     (thm list * thm * Args.src list) * (thm list list * Args.src list)

   val morph_lfp_sugar_thms: morphism -> lfp_sugar_thms -> lfp_sugar_thms

   val transfer_lfp_sugar_thms: Proof.context -> lfp_sugar_thms -> lfp_sugar_thms

   type gfp_sugar_thms =

     ((thm list * thm) list * Args.src list)

     * (thm list list * Args.src list)

     * (thm list list * Args.src list)

     * (thm list list * Args.src list)

     * (thm list list list * Args.src list)

   val morph_gfp_sugar_thms: morphism -> gfp_sugar_thms -> gfp_sugar_thms

   val transfer_gfp_sugar_thms: Proof.context -> gfp_sugar_thms -> gfp_sugar_thms

   val mk_co_recs_prelims: BNF_Util.fp_kind -> typ list list list -> typ list -> typ list ->

      typ list -> typ list -> int list -> int list list -> term list -> Proof.context ->

      (term list

       * (typ list list * typ list list list list * term list list * term list list list list) option

       * (string * term list * term list list

          * ((term list list * term list list list) * typ list)) option)

      * Proof.context

   val repair_nullary_single_ctr: typ list list -> typ list list

   val mk_corec_p_pred_types: typ list -> int list -> typ list list

   val mk_corec_fun_arg_types: typ list list list -> typ list -> typ list -> typ list -> int list ->

     int list list -> term ->

     typ list list

     * (typ list list list list * typ list list list * typ list list list list * typ list)

   val define_rec:

     typ list list * typ list list list list * term list list * term list list list list ->

     (string -> binding) -> typ list -> typ list -> term list -> term -> Proof.context ->

     (term * thm) * Proof.context

   val define_corec: 'a * term list * term list list

       * ((term list list * term list list list) * typ list) -> (string -> binding) -> 'b list ->

     typ list -> term list -> term -> local_theory -> (term * thm) * local_theory

   val derive_induct_recs_thms_for_types: BNF_Def.bnf list ->

      ('a * typ list list list list * term list list * 'b) option -> thm -> thm list ->

      BNF_Def.bnf list -> BNF_Def.bnf list -> typ list -> typ list -> typ list ->

      typ list list list -> thm list -> thm list -> thm list -> term list list -> thm list list ->

      term list -> thm list -> Proof.context -> lfp_sugar_thms

   val derive_coinduct_corecs_thms_for_types: BNF_Def.bnf list ->

     string * term list * term list list * ((term list list * term list list list) * typ list) ->

     thm -> thm list -> thm list -> thm list -> BNF_Def.bnf list -> typ list -> typ list ->

     typ list -> typ list list list -> int list list -> int list list -> int list -> thm list ->

     thm list -> (thm -> thm) -> thm list list -> Ctr_Sugar.ctr_sugar list -> term list ->

     thm list -> (thm list -> thm list) -> Proof.context -> gfp_sugar_thms

   type co_datatype_spec =

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

     * ((binding, binding * typ, term) Ctr_Sugar.ctr_spec * mixfix) list

   val co_datatypes: BNF_Util.fp_kind -> (mixfix list -> binding list -> binding list ->

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

       BNF_Def.bnf list -> BNF_Comp.absT_info list -> local_theory ->

       BNF_FP_Util.fp_result * local_theory) ->

     (bool * bool) * co_datatype_spec list ->

     local_theory -> local_theory

   val parse_co_datatype_cmd: BNF_Util.fp_kind -> (mixfix list -> binding list -> binding list ->

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

       BNF_Def.bnf list -> BNF_Comp.absT_info list -> local_theory ->

       BNF_FP_Util.fp_result * local_theory) ->

     (local_theory -> local_theory) parser

 end;

 structure BNF_FP_Def_Sugar : BNF_FP_DEF_SUGAR =

 struct

 open Ctr_Sugar

 open BNF_FP_Rec_Sugar_Util

 open BNF_Util

 open BNF_Comp

 open BNF_Def

 open BNF_FP_Util

 open BNF_FP_Def_Sugar_Tactics

 open BNF_LFP_Size

 val EqN = "Eq_";

 structure Data = Generic_Data

 (

   type T = fp_sugar Symtab.table;

   val empty = Symtab.empty;

   val extend = I;

   fun merge data : T = Symtab.merge (K true) data;

 );

 fun fp_sugar_of ctxt =

   Symtab.lookup (Data.get (Context.Proof ctxt))

   #> Option.map (transfer_fp_sugar ctxt);

 fun fp_sugars_of ctxt =

   Symtab.fold (cons o transfer_fp_sugar ctxt o snd) (Data.get (Context.Proof ctxt)) [];

 fun co_induct_of (i :: _) = i;

 fun strong_co_induct_of [_, s] = s;

 structure FP_Sugar_Interpretation = Interpretation

 (

   type T = fp_sugar list;

   val eq: T * T -> bool = op = o pairself (map #T);

 );

 (* FIXME naming *)

 fun with_repaired_path f (fp_sugars as ({T = Type (s, _), ...} : fp_sugar) :: _) thy =

   thy

   (*|> Sign.root_path*)

   (*|> Sign.add_path (Long_Name.qualifier s)*)

   |> f fp_sugars

   (*|> Sign.restore_naming thy*);

 fun fp_sugar_interpretation f = FP_Sugar_Interpretation.interpretation (with_repaired_path f);

 fun register_fp_sugars (fp_sugars as {fp, ...} :: _) =

   fold (fn fp_sugar as {T = Type (s, _), ...} =>

       Local_Theory.declaration {syntax = false, pervasive = true}

         (fn phi => Data.map (Symtab.update (s, morph_fp_sugar phi fp_sugar))))

     fp_sugars

   #> fp = Least_FP ? generate_lfp_size fp_sugars

   #> Local_Theory.background_theory (FP_Sugar_Interpretation.data fp_sugars);

 fun register_as_fp_sugars Ts BTs Xs fp pre_bnfs absT_infos nested_bnfs nesting_bnfs fp_res

     ctrXs_Tsss ctr_defss ctr_sugars co_recs co_rec_defs mapss common_co_inducts co_inductss

     co_rec_thmss disc_co_recss sel_co_recsss rel_injectss rel_distinctss =

   let

     val fp_sugars =

       map_index (fn (kk, T) =>

         {T = T, BT = nth BTs kk, X = nth Xs kk, fp = fp, fp_res = fp_res, fp_res_index = kk,

          pre_bnf = nth pre_bnfs kk, absT_info = nth absT_infos kk, nested_bnfs = nested_bnfs,

          nesting_bnfs = nesting_bnfs, ctrXs_Tss = nth ctrXs_Tsss kk, ctr_defs = nth ctr_defss kk,

          ctr_sugar = nth ctr_sugars kk, co_rec = nth co_recs kk, co_rec_def = nth co_rec_defs kk,

          maps = nth mapss kk, common_co_inducts = common_co_inducts,

          co_inducts = nth co_inductss kk, co_rec_thms = nth co_rec_thmss kk,

          disc_co_recs = nth disc_co_recss kk, sel_co_recss = nth sel_co_recsss kk,

          rel_injects = nth rel_injectss kk, rel_distincts = nth rel_distinctss kk}) Ts

   in

     register_fp_sugars fp_sugars

   end;

 (* This function could produce (fairly harmless) clashes in contrived examples (e.g., "x.A",

    "x.x_A", "y.A"). *)

 fun quasi_unambiguous_case_names names =

   let

     val ps = map (`Long_Name.base_name) names;

     val dups = Library.duplicates (op =) (map fst ps);

     fun underscore s =

       let val ss = space_explode Long_Name.separator s in

         space_implode "_" (drop (length ss - 2) ss)

       end;

   in

     map (fn (base, full) => if member (op =) dups base then underscore full else base) ps

   end;

 val id_def = @{thm id_def};

 val mp_conj = @{thm mp_conj};

 val fundefcong_attrs = @{attributes [fundef_cong]};

 val nitpicksimp_attrs = @{attributes [nitpick_simp]};

 val code_nitpicksimp_attrs = Code.add_default_eqn_attrib :: nitpicksimp_attrs;

 val simp_attrs = @{attributes [simp]};

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

 fun flat_corec_predss_getterss qss gss = maps (op @) (qss ~~ gss);

 fun flat_corec_preds_predsss_gettersss [] [qss] [gss] = flat_corec_predss_getterss qss gss

   | flat_corec_preds_predsss_gettersss (p :: ps) (qss :: qsss) (gss :: gsss) =

     p :: flat_corec_predss_getterss qss gss @ flat_corec_preds_predsss_gettersss ps qsss gsss;

 fun mk_flip (x, Type (_, [T1, Type (_, [T2, T3])])) =

   Abs ("x", T1, Abs ("y", T2, Var (x, T2 --> T1 --> T3) $ Bound 0 $ Bound 1));

 fun flip_rels lthy n thm =

   let

     val Rs = Term.add_vars (prop_of thm) [];

     val Rs' = rev (drop (length Rs - n) Rs);

     val cRs = map (fn f => (certify lthy (Var f), certify lthy (mk_flip f))) Rs';

   in

     Drule.cterm_instantiate cRs thm

   end;

 fun mk_ctor_or_dtor get_T Ts t =

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

     Term.subst_atomic_types (Ts0 ~~ Ts) t

   end;

 val mk_ctor = mk_ctor_or_dtor range_type;

 val mk_dtor = mk_ctor_or_dtor domain_type;

 fun mk_xtor_co_recs thy fp fpTs Cs ts0 =

   let

     val nn = length fpTs;

     val (fpTs0, Cs0) =

       map ((fp = Greatest_FP ? swap) o dest_funT o snd o strip_typeN nn o fastype_of) ts0

       |> split_list;

     val rho = tvar_subst thy (fpTs0 @ Cs0) (fpTs @ Cs);

   in

     map (Term.subst_TVars rho) ts0

   end;

 fun unzip_recT (Type (@{type_name prod}, _)) T = [T]

   | unzip_recT _ (Type (@{type_name prod}, Ts)) = Ts

   | unzip_recT _ T = [T];

 fun unzip_corecT (Type (@{type_name sum}, _)) T = [T]

   | unzip_corecT _ (Type (@{type_name sum}, Ts)) = Ts

   | unzip_corecT _ T = [T];

 fun liveness_of_fp_bnf n bnf =

   (case T_of_bnf bnf of

     Type (_, Ts) => map (not o member (op =) (deads_of_bnf bnf)) Ts

   | _ => replicate n false);

 fun cannot_merge_types fp =

   error ("Mutually " ^ co_prefix fp ^ "recursive types must have the same type parameters");

 fun merge_type_arg fp T T' = if T = T' then T else cannot_merge_types fp;

 fun merge_type_args fp (As, As') =

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

 fun reassoc_conjs thm =

   reassoc_conjs (thm RS @{thm conj_assoc[THEN iffD1]})

   handle THM _ => thm;

 type co_datatype_spec =

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

   * ((binding, binding * typ, term) ctr_spec * mixfix) list;

 fun type_args_named_constrained_of_spec ((((ncAs, _), _), _), _) = ncAs;

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

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

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

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

 fun mixfixed_ctr_specs_of_spec (_, mx_ctr_specs) = mx_ctr_specs;

 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 ctxt ctr_Tsss Us =

   map_filter (bnf_of ctxt) (fold (fold (fold (add_nesty_bnf_names Us))) ctr_Tsss []);

 fun indexify proj xs f p = f (find_index (curry (op =) (proj p)) xs) p;

 type lfp_sugar_thms =

   (thm list * thm * Args.src list) * (thm list list * Args.src list);

 fun morph_lfp_sugar_thms phi ((inducts, induct, induct_attrs), (recss, rec_attrs)) =

   ((map (Morphism.thm phi) inducts, Morphism.thm phi induct, induct_attrs),

    (map (map (Morphism.thm phi)) recss, rec_attrs))

   : lfp_sugar_thms;

 val transfer_lfp_sugar_thms =

   morph_lfp_sugar_thms o Morphism.transfer_morphism o Proof_Context.theory_of;

 type gfp_sugar_thms =

   ((thm list * thm) list * Args.src list)

   * (thm list list * Args.src list)

   * (thm list list * Args.src list)

   * (thm list list * Args.src list)

   * (thm list list list * Args.src list);

 fun morph_gfp_sugar_thms phi ((coinducts_pairs, coinduct_attrs),

     (corecss, corec_attrs), (disc_corecss, disc_corec_attrs),

     (disc_corec_iffss, disc_corec_iff_attrs), (sel_corecsss, sel_corec_attrs)) =

   ((map (apfst (map (Morphism.thm phi)) o apsnd (Morphism.thm phi)) coinducts_pairs,

     coinduct_attrs),

    (map (map (Morphism.thm phi)) corecss, corec_attrs),

    (map (map (Morphism.thm phi)) disc_corecss, disc_corec_attrs),

    (map (map (Morphism.thm phi)) disc_corec_iffss, disc_corec_iff_attrs),

    (map (map (map (Morphism.thm phi))) sel_corecsss, sel_corec_attrs)) : gfp_sugar_thms;

 val transfer_gfp_sugar_thms =

   morph_gfp_sugar_thms o Morphism.transfer_morphism o Proof_Context.theory_of;

 fun mk_recs_args_types ctr_Tsss Cs absTs repTs ns mss ctor_rec_fun_Ts lthy =

   let

     val Css = map2 replicate ns Cs;

     val x_Tssss =

       map6 (fn absT => fn repT => fn n => fn ms => fn ctr_Tss => fn ctor_rec_fun_T =>

           map2 (map2 unzip_recT)

             ctr_Tss (dest_absumprodT absT repT n ms (domain_type ctor_rec_fun_T)))

         absTs repTs ns mss ctr_Tsss ctor_rec_fun_Ts;

     val x_Tsss' = map (map flat_rec_arg_args) x_Tssss;

     val f_Tss = map2 (map2 (curry (op --->))) x_Tsss' Css;

     val ((fss, xssss), lthy) =

       lthy

       |> mk_Freess "f" f_Tss

       ||>> mk_Freessss "x" x_Tssss;

   in

     ((f_Tss, x_Tssss, fss, xssss), lthy)

   end;

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

 fun repair_nullary_single_ctr [[]] = [[HOLogic.unitT]]

   | repair_nullary_single_ctr Tss = Tss;

 fun mk_corec_fun_arg_types0 ctr_Tsss Cs absTs repTs ns mss fun_Ts =

   let

     val ctr_Tsss' = map repair_nullary_single_ctr ctr_Tsss;

     val g_absTs = map range_type fun_Ts;

     val g_Tsss = map repair_nullary_single_ctr (map5 dest_absumprodT absTs repTs ns mss g_absTs);

     val g_Tssss = map3 (fn C => map2 (map2 (map (curry (op -->) C) oo unzip_corecT)))

       Cs ctr_Tsss' g_Tsss;

     val q_Tssss = map (map (map (fn [_] => [] | [_, T] => [mk_pred1T (domain_type T)]))) g_Tssss;

   in

     (q_Tssss, g_Tsss, g_Tssss, g_absTs)

   end;

 fun mk_corec_p_pred_types Cs ns = map2 (fn n => replicate (Int.max (0, n - 1)) o mk_pred1T) ns Cs;

 fun mk_corec_fun_arg_types ctr_Tsss Cs absTs repTs ns mss dtor_corec =

   (mk_corec_p_pred_types Cs ns,

    mk_corec_fun_arg_types0 ctr_Tsss Cs absTs repTs ns mss

      (binder_fun_types (fastype_of dtor_corec)));

 fun mk_corecs_args_types ctr_Tsss Cs absTs repTs ns mss dtor_corec_fun_Ts lthy =

   let

     val p_Tss = mk_corec_p_pred_types Cs ns;

     val (q_Tssss, g_Tsss, g_Tssss, corec_types) =

       mk_corec_fun_arg_types0 ctr_Tsss Cs absTs repTs ns mss dtor_corec_fun_Ts;

     val (((((Free (x, _), cs), pss), qssss), gssss), lthy) =

       lthy

       |> yield_singleton (mk_Frees "x") dummyT

       ||>> mk_Frees "a" Cs

       ||>> mk_Freess "p" p_Tss

       ||>> mk_Freessss "q" q_Tssss

       ||>> mk_Freessss "g" g_Tssss;

     val cpss = map2 (map o rapp) cs pss;

     fun build_sum_inj mk_inj = build_map lthy (uncurry mk_inj o dest_sumT o snd);

     fun build_dtor_corec_arg _ [] [cg] = cg

       | build_dtor_corec_arg T [cq] [cg, cg'] =

         mk_If cq (build_sum_inj Inl_const (fastype_of cg, T) $ cg)

           (build_sum_inj Inr_const (fastype_of cg', T) $ cg');

     val pgss = map3 flat_corec_preds_predsss_gettersss pss qssss gssss;

     val cqssss = map2 (map o map o map o rapp) cs qssss;

     val cgssss = map2 (map o map o map o rapp) cs gssss;

     val cqgsss = map3 (map3 (map3 build_dtor_corec_arg)) g_Tsss cqssss cgssss;

   in

     ((x, cs, cpss, ((pgss, cqgsss), corec_types)), lthy)

   end;

 fun mk_co_recs_prelims fp ctr_Tsss fpTs Cs absTs repTs ns mss xtor_co_recs0 lthy =

   let

     val thy = Proof_Context.theory_of lthy;

     val (xtor_co_rec_fun_Ts, xtor_co_recs) =

       mk_xtor_co_recs thy fp fpTs Cs xtor_co_recs0 |> `(binder_fun_types o fastype_of o hd);

     val ((recs_args_types, corecs_args_types), lthy') =

       if fp = Least_FP then

         mk_recs_args_types ctr_Tsss Cs absTs repTs ns mss xtor_co_rec_fun_Ts lthy

         |>> (rpair NONE o SOME)

       else

         mk_corecs_args_types ctr_Tsss Cs absTs repTs ns mss xtor_co_rec_fun_Ts lthy

         |>> (pair NONE o SOME);

   in

     ((xtor_co_recs, recs_args_types, corecs_args_types), lthy')

   end;

 fun mk_preds_getterss_join c cps absT abs cqgss =

   let

     val n = length cqgss;

     val ts = map2 (mk_absumprod absT abs n) (1 upto n) cqgss;

   in

     Term.lambda c (mk_IfN absT cps ts)

   end;

 fun define_co_rec fp fpT Cs b rhs lthy0 =

   let

     val thy = Proof_Context.theory_of lthy0;

     val maybe_conceal_def_binding = Thm.def_binding

       #> Config.get lthy0 bnf_note_all = false ? Binding.conceal;

     val ((cst, (_, def)), (lthy', lthy)) = lthy0

       |> Local_Theory.define ((b, NoSyn), ((maybe_conceal_def_binding b, []), rhs))

       ||> `Local_Theory.restore;

     val phi = Proof_Context.export_morphism lthy lthy';

     val cst' = mk_co_rec thy fp fpT Cs (Morphism.term phi cst);

     val def' = Morphism.thm phi def;

   in

     ((cst', def'), lthy')

   end;

 fun define_rec (_, _, fss, xssss) mk_binding fpTs Cs reps ctor_rec =

   let

     val nn = length fpTs;

     val (ctor_rec_absTs, fpT) = strip_typeN nn (fastype_of ctor_rec)

       |>> map domain_type ||> domain_type;

   in

     define_co_rec Least_FP fpT Cs (mk_binding recN)

       (fold_rev (fold_rev Term.lambda) fss (Term.list_comb (ctor_rec,

          map4 (fn ctor_rec_absT => fn rep => fn fs => fn xsss =>

              mk_case_absumprod ctor_rec_absT rep fs (map (map HOLogic.mk_tuple) xsss)

                (map flat_rec_arg_args xsss))

            ctor_rec_absTs reps fss xssss)))

   end;

 fun define_corec (_, cs, cpss, ((pgss, cqgsss), f_absTs)) mk_binding fpTs Cs abss dtor_corec =

   let

     val nn = length fpTs;

     val fpT = range_type (snd (strip_typeN nn (fastype_of dtor_corec)));

   in

     define_co_rec Greatest_FP fpT Cs (mk_binding corecN)

       (fold_rev (fold_rev Term.lambda) pgss (Term.list_comb (dtor_corec,

          map5 mk_preds_getterss_join cs cpss f_absTs abss cqgsss)))

   end;

 fun derive_induct_recs_thms_for_types pre_bnfs rec_args_typess ctor_induct ctor_rec_thms

     nesting_bnfs nested_bnfs fpTs Cs Xs ctrXs_Tsss fp_abs_inverses fp_type_definitions abs_inverses

     ctrss ctr_defss recs rec_defs lthy =

   let

     val ctr_Tsss = map (map (binder_types o fastype_of)) ctrss;

     val nn = length pre_bnfs;

     val ns = map length ctr_Tsss;

     val mss = map (map length) ctr_Tsss;

     val pre_map_defs = map map_def_of_bnf pre_bnfs;

     val pre_set_defss = map set_defs_of_bnf pre_bnfs;

     val nesting_map_idents = map (unfold_thms lthy [id_def] o map_id0_of_bnf) nesting_bnfs;

     val nested_map_idents = map (unfold_thms lthy [id_def] o map_id0_of_bnf) nested_bnfs;

     val nested_set_maps = maps set_map_of_bnf nested_bnfs;

     val fp_b_names = map base_name_of_typ fpTs;

     val ((((ps, ps'), xsss), us'), names_lthy) =

       lthy

       |> mk_Frees' "P" (map mk_pred1T fpTs)

       ||>> mk_Freesss "x" ctr_Tsss

       ||>> Variable.variant_fixes fp_b_names;

     val us = map2 (curry Free) us' 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 _ (x as Free (_, Type _)) (X as TFree _) =

             [([], (find_index (curry (op =) X) Xs + 1, x))]

           | mk_raw_prem_prems names_lthy (x as Free (s, Type (T_name, Ts0))) (Type (_, Xs_Ts0)) =

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

               NONE => []

             | SOME raw_sets0 =>

               let

                 val (Xs_Ts, (Ts, raw_sets)) =

                   filter (exists_subtype_in Xs o fst) (Xs_Ts0 ~~ (Ts0 ~~ raw_sets0))

                   |> split_list ||> split_list;

                 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 = map2 (mk_raw_prem_prems names_lthy') ys Xs_Ts;

               in

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

               end)

           | mk_raw_prem_prems _ _ _ = [];

         fun close_prem_prem xs t =

           fold_rev Logic.all (map Free (drop (nn + length xs)

             (rev (Term.add_frees t (map dest_Free xs @ ps'))))) t;

         fun mk_prem_prem xs (xysets, (j, x)) =

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

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

             HOLogic.mk_Trueprop (nth ps (j - 1) $ x)));

         fun mk_raw_prem phi ctr ctr_Ts ctrXs_Ts =

           let

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

             val pprems = flat (map2 (mk_raw_prem_prems names_lthy') xs ctrXs_Ts);

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

         fun mk_prem (xs, raw_pprems, concl) =

           fold_rev Logic.all xs (Logic.list_implies (map (mk_prem_prem xs) raw_pprems, concl));

         val raw_premss = map4 (map3 o mk_raw_prem) ps ctrss ctr_Tsss ctrXs_Tsss;

         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 $)) ps us)));

         val kksss = map (map (map (fst o snd) o #2)) raw_premss;

         val ctor_induct' = ctor_induct OF (map2 mk_absumprodE fp_type_definitions mss);

         val thm =

           Goal.prove_sorry lthy [] [] goal (fn {context = ctxt, ...} =>

             mk_induct_tac ctxt nn ns mss kksss (flat ctr_defss) ctor_induct' fp_abs_inverses

               abs_inverses nested_set_maps pre_set_defss)

           |> singleton (Proof_Context.export names_lthy lthy)

           (* for "datatype_realizer.ML": *)

           |> Thm.name_derivation (fst (dest_Type (hd fpTs)) ^ Long_Name.separator ^

             (if nn > 1 then space_implode "_" (tl fp_b_names) ^ Long_Name.separator else "") ^

             inductN);

       in

         `(conj_dests nn) thm

       end;

     val induct_cases = quasi_unambiguous_case_names (maps (map name_of_ctr) ctrss);

     val induct_case_names_attr = Attrib.internal (K (Rule_Cases.case_names induct_cases));

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

     fun mk_rec_thmss (_, x_Tssss, fss, _) recs rec_defs ctor_rec_thms =

       let

         val frecs = map (lists_bmoc fss) recs;

         fun mk_goal frec xctr f xs fxs =

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

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

         fun maybe_tick (T, U) u f =

           if try (fst o HOLogic.dest_prodT) U = SOME T then

             Term.lambda u (HOLogic.mk_prod (u, f $ u))

           else

             f;

         fun build_rec (x as Free (_, T)) U =

           if T = U then

             x

           else

             build_map lthy (indexify (perhaps (try (snd o HOLogic.dest_prodT)) o snd) Cs

               (fn kk => fn TU => maybe_tick TU (nth us kk) (nth frecs kk))) (T, U) $ x;

         val fxsss = map2 (map2 (flat_rec_arg_args oo map2 (map o build_rec))) xsss x_Tssss;

         val goalss = map5 (map4 o mk_goal) frecs xctrss fss xsss fxsss;

         val tacss =

           map4 (map ooo mk_rec_tac pre_map_defs (nested_map_idents @ nesting_map_idents) rec_defs)

             ctor_rec_thms fp_abs_inverses abs_inverses ctr_defss;

         fun prove goal tac =

           Goal.prove_sorry lthy [] [] goal (tac o #context)

           |> Thm.close_derivation;

       in

         map2 (map2 prove) goalss tacss

       end;

     val rec_thmss = mk_rec_thmss (the rec_args_typess) recs rec_defs ctor_rec_thms;

   in

     ((induct_thms, induct_thm, [induct_case_names_attr]),

      (rec_thmss, code_nitpicksimp_attrs @ simp_attrs))

   end;

 fun derive_coinduct_corecs_thms_for_types pre_bnfs (x, cs, cpss, ((pgss, cqgsss), _))

     dtor_coinduct dtor_injects dtor_ctors dtor_corec_thms nesting_bnfs fpTs Cs Xs ctrXs_Tsss kss

     mss ns fp_abs_inverses abs_inverses mk_vimage2p ctr_defss (ctr_sugars : ctr_sugar list)

     corecs corec_defs export_args lthy =

   let

     fun mk_ctor_dtor_corec_thm dtor_inject dtor_ctor corec =

       iffD1 OF [dtor_inject, trans OF [corec, dtor_ctor RS sym]];

     val ctor_dtor_corec_thms = map3 mk_ctor_dtor_corec_thm dtor_injects dtor_ctors dtor_corec_thms;

     val nn = length pre_bnfs;

     val pre_map_defs = map map_def_of_bnf pre_bnfs;

     val pre_rel_defs = map rel_def_of_bnf pre_bnfs;

     val nesting_map_idents = map (unfold_thms lthy [id_def] o map_id0_of_bnf) nesting_bnfs;

     val nesting_rel_eqs = map rel_eq_of_bnf nesting_bnfs;

     val fp_b_names = map base_name_of_typ fpTs;

     val ctrss = map #ctrs ctr_sugars;

     val discss = map #discs ctr_sugars;

     val selsss = map #selss ctr_sugars;

     val exhausts = map #exhaust ctr_sugars;

     val disc_thmsss = map #disc_thmss ctr_sugars;

     val discIss = map #discIs ctr_sugars;

     val sel_thmsss = map #sel_thmss ctr_sugars;

     val (((rs, us'), vs'), names_lthy) =

       lthy

       |> mk_Frees "R" (map (fn T => mk_pred2T T T) fpTs)

       ||>> Variable.variant_fixes fp_b_names

       ||>> Variable.variant_fixes (map (suffix "'") fp_b_names);

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

     val udiscss = map2 (map o rapp) us discss;

     val uselsss = map2 (map o map o rapp) us selsss;

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

     val vdiscss = map2 (map o rapp) vs discss;

     val vselsss = map2 (map o map o rapp) vs selsss;

     val coinduct_thms_pairs =

       let

         val uvrs = map3 (fn r => fn u => fn v => r $ u $ v) rs us vs;

         val uv_eqs = map2 (curry HOLogic.mk_eq) us vs;

         val strong_rs =

           map4 (fn u => fn v => fn uvr => fn uv_eq =>

             fold_rev Term.lambda [u, v] (HOLogic.mk_disj (uvr, uv_eq))) us vs uvrs uv_eqs;

         fun build_the_rel rs' T Xs_T =

           build_rel lthy (fn (_, X) => nth rs' (find_index (curry (op =) X) Xs)) (T, Xs_T)

           |> Term.subst_atomic_types (Xs ~~ fpTs);

         fun build_rel_app rs' usel vsel Xs_T =

           fold rapp [usel, vsel] (build_the_rel rs' (fastype_of usel) Xs_T);

         fun mk_prem_ctr_concls rs' n k udisc usels vdisc vsels ctrXs_Ts =

           (if k = n then [] else [HOLogic.mk_eq (udisc, vdisc)]) @

           (if null usels then

              []

            else

              [Library.foldr HOLogic.mk_imp (if n = 1 then [] else [udisc, vdisc],

                 Library.foldr1 HOLogic.mk_conj (map3 (build_rel_app rs') usels vsels ctrXs_Ts))]);

         fun mk_prem_concl rs' n udiscs uselss vdiscs vselss ctrXs_Tss =

           Library.foldr1 HOLogic.mk_conj (flat (map6 (mk_prem_ctr_concls rs' n)

             (1 upto n) udiscs uselss vdiscs vselss ctrXs_Tss))

           handle List.Empty => @{term True};

         fun mk_prem rs' uvr u v n udiscs uselss vdiscs vselss ctrXs_Tss =

           fold_rev Logic.all [u, v] (Logic.mk_implies (HOLogic.mk_Trueprop uvr,

             HOLogic.mk_Trueprop (mk_prem_concl rs' n udiscs uselss vdiscs vselss ctrXs_Tss)));

         val concl =

           HOLogic.mk_Trueprop (Library.foldr1 HOLogic.mk_conj

             (map3 (fn uvr => fn u => fn v => HOLogic.mk_imp (uvr, HOLogic.mk_eq (u, v)))

                uvrs us vs));

         fun mk_goal rs' =

           Logic.list_implies (map9 (mk_prem rs') uvrs us vs ns udiscss uselsss vdiscss vselsss

             ctrXs_Tsss, concl);

         val goals = map mk_goal [rs, strong_rs];

         fun prove dtor_coinduct' goal =

           Goal.prove_sorry lthy [] [] goal (fn {context = ctxt, ...} =>

             mk_coinduct_tac ctxt nesting_rel_eqs nn ns dtor_coinduct' pre_rel_defs fp_abs_inverses

               abs_inverses dtor_ctors exhausts ctr_defss disc_thmsss sel_thmsss)

           |> singleton (Proof_Context.export names_lthy lthy)

           |> Thm.close_derivation;

         fun postproc nn thm =

           Thm.permute_prems 0 nn

             (if nn = 1 then thm RS mp else funpow nn (fn thm => reassoc_conjs (thm RS mp_conj)) thm)

           |> Drule.zero_var_indexes

           |> `(conj_dests nn);

         val rel_eqs = map rel_eq_of_bnf pre_bnfs;

         val rel_monos = map rel_mono_of_bnf pre_bnfs;

         val dtor_coinducts =

           [dtor_coinduct, mk_strong_coinduct_thm dtor_coinduct rel_eqs rel_monos mk_vimage2p lthy]

       in

         map2 (postproc nn oo prove) dtor_coinducts goals

       end;

     fun mk_coinduct_concls ms discs ctrs =

       let

         fun mk_disc_concl disc = [name_of_disc disc];

         fun mk_ctr_concl 0 _ = []

           | mk_ctr_concl _ ctor = [name_of_ctr ctor];

         val disc_concls = map mk_disc_concl (fst (split_last discs)) @ [[]];

         val ctr_concls = map2 mk_ctr_concl ms ctrs;

       in

         flat (map2 append disc_concls ctr_concls)

       end;

     val coinduct_cases = quasi_unambiguous_case_names (map (prefix EqN) fp_b_names);

     val coinduct_conclss =

       map3 (quasi_unambiguous_case_names ooo mk_coinduct_concls) mss discss ctrss;

     fun mk_maybe_not pos = not pos ? HOLogic.mk_not;

     val gcorecs = map (lists_bmoc pgss) corecs;

     val corec_thmss =

       let

         fun mk_goal c cps gcorec n k ctr m cfs' =

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

             (Logic.list_implies (seq_conds (HOLogic.mk_Trueprop oo mk_maybe_not) n k cps,

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

         val mk_U = typ_subst_nonatomic (map2 (fn C => fn fpT => (mk_sumT (fpT, C), fpT)) Cs fpTs);

         fun tack (c, u) f =

           let val x' = Free (x, mk_sumT (fastype_of u, fastype_of c)) in

             Term.lambda x' (mk_case_sum (Term.lambda u u, Term.lambda c (f $ c)) $ x')

           end;

         fun build_corec cqg =

           let val T = fastype_of cqg in

             if exists_subtype_in Cs T then

               let val U = mk_U T in

                 build_map lthy (indexify fst (map2 (curry mk_sumT) fpTs Cs) (fn kk => fn _ =>

                   tack (nth cs kk, nth us kk) (nth gcorecs kk))) (T, U) $ cqg

               end

             else

               cqg

           end;

         val cqgsss' = map (map (map build_corec)) cqgsss;

         val goalss = map8 (map4 oooo mk_goal) cs cpss gcorecs ns kss ctrss mss cqgsss';

         val tacss =

           map4 (map ooo mk_corec_tac corec_defs nesting_map_idents)

             ctor_dtor_corec_thms pre_map_defs abs_inverses ctr_defss;

         fun prove goal tac =

           Goal.prove_sorry lthy [] [] goal (tac o #context)

           |> Thm.close_derivation;

       in

         map2 (map2 prove) goalss tacss

         |> map (map (unfold_thms lthy @{thms case_sum_if}))

       end;

     val disc_corec_iff_thmss =

       let

         fun mk_goal c cps gcorec n k disc =

           mk_Trueprop_eq (disc $ (gcorec $ c),

             if n = 1 then @{const True}

             else Library.foldr1 HOLogic.mk_conj (seq_conds mk_maybe_not n k cps));

         val goalss = map6 (map2 oooo mk_goal) cs cpss gcorecs ns kss discss;

         fun mk_case_split' cp = Drule.instantiate' [] [SOME (certify lthy cp)] @{thm case_split};

         val case_splitss' = map (map mk_case_split') cpss;

         val tacss = map3 (map oo mk_disc_corec_iff_tac) case_splitss' corec_thmss disc_thmsss;

         fun prove goal tac =

           Goal.prove_sorry lthy [] [] goal (tac o #context)

           |> singleton export_args

           |> singleton (Proof_Context.export names_lthy lthy)

           |> Thm.close_derivation;

         fun proves [_] [_] = []

           | proves goals tacs = map2 prove goals tacs;

       in

         map2 proves goalss tacss

       end;

     fun mk_disc_corec_thms corecs discIs = map (op RS) (corecs ~~ discIs);

     val disc_corec_thmss = map2 mk_disc_corec_thms corec_thmss discIss;

     fun mk_sel_corec_thm corec_thm sel sel_thm =

       let

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

         val arg_cong' =

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

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

           |> Thm.varifyT_global;

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

       in

         corec_thm RS arg_cong' RS sel_thm'

       end;

     fun mk_sel_corec_thms corec_thmss =

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

     val sel_corec_thmsss = mk_sel_corec_thms corec_thmss;

     val coinduct_consumes_attr = Attrib.internal (K (Rule_Cases.consumes nn));

     val coinduct_case_names_attr = Attrib.internal (K (Rule_Cases.case_names coinduct_cases));

     val coinduct_case_concl_attrs =

       map2 (fn casex => fn concls =>

           Attrib.internal (K (Rule_Cases.case_conclusion (casex, concls))))

         coinduct_cases coinduct_conclss;

     val coinduct_case_attrs =

       coinduct_consumes_attr :: coinduct_case_names_attr :: coinduct_case_concl_attrs;

   in

     ((coinduct_thms_pairs, coinduct_case_attrs),

      (corec_thmss, code_nitpicksimp_attrs),

      (disc_corec_thmss, []),

      (disc_corec_iff_thmss, simp_attrs),

      (sel_corec_thmsss, simp_attrs))

   end;

 fun define_co_datatypes prepare_constraint prepare_typ prepare_term fp construct_fp

     (wrap_opts as (no_discs_sels, _), specs) no_defs_lthy0 =

   let

     (* TODO: sanity checks on arguments *)

     val _ = if fp = Greatest_FP andalso no_discs_sels then

         error "Cannot define codatatypes without discriminators and selectors"

       else

         ();

     val nn = length specs;

     val fp_bs = map type_binding_of_spec specs;

     val fp_b_names = map Binding.name_of fp_bs;

     val fp_common_name = mk_common_name fp_b_names;

     val map_bs = map map_binding_of_spec specs;

     val rel_bs = map rel_binding_of_spec specs;

     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_named_constrained_of_spec) specs;

     val unsorted_Ass0 = map (map (resort_tfree @{sort type})) Ass0;

     val unsorted_As = Library.foldr1 (merge_type_args fp) unsorted_Ass0;

     val num_As = length unsorted_As;

     val set_boss = map (map fst o type_args_named_constrained_of_spec) specs;

     val set_bss = map (map (the_default Binding.empty)) set_boss;

     val (((Bs0, Cs), Xs), no_defs_lthy) =

       no_defs_lthy0

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

       |> mk_TFrees num_As

       ||>> mk_TFrees nn

       ||>> variant_tfrees fp_b_names;

     fun add_fake_type spec =

       Typedecl.basic_typedecl (type_binding_of_spec spec, num_As, mixfix_of_spec spec);

     val (fake_T_names, fake_lthy) = fold_map add_fake_type specs no_defs_lthy0;

     val qsoty = quote o Syntax.string_of_typ fake_lthy;

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

       | A :: _ => error ("Duplicate type parameter " ^ qsoty A ^ " in " ^ co_prefix fp ^

           "datatype specification"));

     val bad_args =

       map (Logic.type_map (singleton (Variable.polymorphic no_defs_lthy0))) unsorted_As

       |> filter_out Term.is_TVar;

     val _ = null bad_args orelse

       error ("Locally fixed type argument " ^ qsoty (hd bad_args) ^ " in " ^ co_prefix fp ^

         "datatype specification");

     val mixfixes = map mixfix_of_spec specs;

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

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

     val mx_ctr_specss = map mixfixed_ctr_specs_of_spec specs;

     val ctr_specss = map (map fst) mx_ctr_specss;

     val ctr_mixfixess = map (map snd) mx_ctr_specss;

     val disc_bindingss = map (map disc_of_ctr_spec) ctr_specss;

     val ctr_bindingss =

       map2 (fn fp_b_name => map (Binding.qualify false fp_b_name o ctr_of_ctr_spec)) fp_b_names

         ctr_specss;

     val ctr_argsss = map (map args_of_ctr_spec) 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 sel_defaults_of_ctr_spec) ctr_specss;

     val (As :: _) :: fake_ctr_Tsss =

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

     val As' = map dest_TFree As;

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

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

       | extras => error ("Extra type variables on right-hand side: " ^

           commas (map (qsoty o TFree) extras)));

     val fake_Ts = map (fn s => Type (s, As)) fake_T_names;

     fun eq_fpT_check (T as Type (s, Ts)) (T' as Type (s', Ts')) =

         s = s' andalso (Ts = Ts' orelse

           error ("Wrong type arguments in " ^ co_prefix fp ^ "recursive type " ^ qsoty T ^

             " (expected " ^ qsoty T' ^ ")"))

       | eq_fpT_check _ _ = false;

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

         (case find_index (eq_fpT_check T) fake_Ts of

           ~1 => Type (s, map freeze_fp Ts)

         | kk => nth Xs kk)

       | freeze_fp T = T;

     val unfreeze_fp = Term.typ_subst_atomic (Xs ~~ fake_Ts);

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

     val ctrXs_repTs = map mk_sumprodT_balanced ctrXs_Tsss;

     val fp_eqs =

       map dest_TFree Xs ~~ map (Term.typ_subst_atomic (As ~~ unsorted_As)) ctrXs_repTs;

     val rhsXs_As' = fold (fold (fold Term.add_tfreesT)) ctrXs_Tsss [];

     val _ = (case subtract (op =) rhsXs_As' As' of [] => ()

       | extras => List.app (fn extra => warning ("Unused type variable on right-hand side of " ^

           co_prefix fp ^ "datatype definition: " ^ qsoty (TFree extra))) extras);

     val killed_As =

       map_filter (fn (A, set_bos) => if exists is_none set_bos then SOME A else NONE)

         (unsorted_As ~~ transpose set_boss);

     val ((pre_bnfs, absT_infos), (fp_res as {bnfs = fp_bnfs as any_fp_bnf :: _, ctors = ctors0,

              dtors = dtors0, xtor_co_recs = xtor_co_recs0, xtor_co_induct, dtor_ctors,

              ctor_dtors, ctor_injects, dtor_injects, xtor_map_thms, xtor_set_thmss, xtor_rel_thms,

              xtor_co_rec_thms, ...},

            lthy)) =

       fp_bnf (construct_fp mixfixes map_bs rel_bs set_bss) fp_bs (map dest_TFree unsorted_As)

         (map dest_TFree killed_As) fp_eqs no_defs_lthy0

       handle BAD_DEAD (X, X_backdrop) =>

         (case X_backdrop of

           Type (bad_tc, _) =>

           let

             val fake_T = qsoty (unfreeze_fp X);

             val fake_T_backdrop = qsoty (unfreeze_fp X_backdrop);

             fun register_hint () =

               "\nUse the " ^ quote (fst (fst @{command_spec "bnf"})) ^ " command to register " ^

               quote bad_tc ^ " as a bounded natural functor to allow nested (co)recursion through \

               \it";

           in

             if is_some (bnf_of no_defs_lthy bad_tc) orelse

                is_some (fp_sugar_of no_defs_lthy bad_tc) then

               error ("Inadmissible " ^ co_prefix fp ^ "recursive occurrence of type " ^ fake_T ^

                 " in type expression " ^ fake_T_backdrop)

             else if is_some (Datatype_Data.get_info (Proof_Context.theory_of no_defs_lthy)

                 bad_tc) then

               error ("Unsupported " ^ co_prefix fp ^ "recursive occurrence of type " ^ fake_T ^

                 " via the old-style datatype " ^ quote bad_tc ^ " in type expression " ^

                 fake_T_backdrop ^ register_hint ())

             else

               error ("Unsupported " ^ co_prefix fp ^ "recursive occurrence of type " ^ fake_T ^

                 " via type constructor " ^ quote bad_tc ^ " in type expression " ^ fake_T_backdrop ^

                 register_hint ())

           end);

     val abss = map #abs absT_infos;

     val reps = map #rep absT_infos;

     val absTs = map #absT absT_infos;

     val repTs = map #repT absT_infos;

     val abs_injects = map #abs_inject absT_infos;

     val abs_inverses = map #abs_inverse absT_infos;

     val type_definitions = map #type_definition absT_infos;

     val time = time lthy;

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

     val nesting_bnfs = nesty_bnfs lthy ctrXs_Tsss As;

     val nested_bnfs = nesty_bnfs lthy ctrXs_Tsss Xs;

     val pre_map_defs = map map_def_of_bnf pre_bnfs;

     val pre_set_defss = map set_defs_of_bnf pre_bnfs;

     val pre_rel_defs = map rel_def_of_bnf pre_bnfs;

     val nesting_set_maps = maps set_map_of_bnf nesting_bnfs;

     val nested_set_maps = maps set_map_of_bnf nested_bnfs;

     val live = live_of_bnf any_fp_bnf;

     val _ =

       if live = 0 andalso exists (not o Binding.is_empty) (map_bs @ rel_bs) then

         warning "Map function and relator names ignored"

       else

         ();

     val Bs =

       map3 (fn alive => fn A as TFree (_, S) => fn B => if alive then resort_tfree S B else A)

         (liveness_of_fp_bnf num_As any_fp_bnf) As Bs0;

     val B_ify = Term.typ_subst_atomic (As ~~ Bs);

     val ctors = map (mk_ctor As) ctors0;

     val dtors = map (mk_dtor As) dtors0;

     val fpTs = map (domain_type o fastype_of) dtors;

     val fpBTs = map B_ify fpTs;

     fun massage_simple_notes base =

       filter_out (null o #2)

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

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

     val massage_multi_notes =

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

         map3 (fn fp_b_name => fn Type (T_name, _) => fn thms =>

             ((Binding.qualify true fp_b_name (Binding.name thmN), attrs T_name), [(thms, [])]))

           fp_b_names fpTs thmss)

       #> filter_out (null o fst o hd o snd);

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

     val ns = map length ctr_Tsss;

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

     val mss = map (map length) ctr_Tsss;

     val ((xtor_co_recs, recs_args_types, corecs_args_types), lthy') =

       mk_co_recs_prelims fp ctr_Tsss fpTs Cs absTs repTs ns mss xtor_co_recs0 lthy;

     fun define_ctrs_dtrs_for_type (((((((((((((((((((((((((((fp_bnf, fp_b), fpT), ctor), dtor),

               xtor_co_rec), ctor_dtor), dtor_ctor), ctor_inject), pre_map_def), pre_set_defs),

             pre_rel_def), fp_map_thm), fp_set_thms), fp_rel_thm), n), ks), ms), abs),

           abs_inject), abs_inverse), type_definition), ctr_bindings), ctr_mixfixes), ctr_Tss),

         disc_bindings), sel_bindingss), raw_sel_defaultss) no_defs_lthy =

       let

         val fp_b_name = Binding.name_of fp_b;

         val ctr_absT = domain_type (fastype_of ctor);

         val ((((w, xss), yss), u'), names_lthy) =

           no_defs_lthy

           |> yield_singleton (mk_Frees "w") ctr_absT

           ||>> mk_Freess "x" ctr_Tss

           ||>> mk_Freess "y" (map (map B_ify) ctr_Tss)

           ||>> yield_singleton Variable.variant_fixes fp_b_name;

         val u = Free (u', fpT);

         val ctr_rhss =

           map2 (fn k => fn xs => fold_rev Term.lambda xs (ctor $ mk_absumprod ctr_absT abs n k xs))

             ks xss;

         val maybe_conceal_def_binding = Thm.def_binding

           #> Config.get no_defs_lthy bnf_note_all = false ? Binding.conceal;

         val ((raw_ctrs, 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), ((maybe_conceal_def_binding b, []), rhs)) #>> apsnd snd)

             ctr_bindings ctr_mixfixes ctr_rhss

           ||> `Local_Theory.restore;

         val phi = Proof_Context.export_morphism lthy lthy';

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

         val ctr_defs' =

           map2 (fn m => fn def => mk_unabs_def m (def RS meta_eq_to_obj_eq)) ms ctr_defs;

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

         val ctrs = map (mk_ctr As) ctrs0;

         fun wrap_ctrs lthy =

           let

             fun exhaust_tac {context = ctxt, prems = _} =

               let

                 val ctor_iff_dtor_thm =

                   let

                     val goal =

                       fold_rev Logic.all [w, u]

                         (mk_Trueprop_eq (HOLogic.mk_eq (u, ctor $ w), HOLogic.mk_eq (dtor $ u, w)));

                   in

                     Goal.prove_sorry lthy [] [] goal (fn {context = ctxt, ...} =>

                       mk_ctor_iff_dtor_tac ctxt (map (SOME o certifyT lthy) [ctr_absT, fpT])

                         (certify lthy ctor) (certify lthy dtor) ctor_dtor dtor_ctor)

                     |> Morphism.thm phi

                     |> Thm.close_derivation

                   end;

                 val sumEN_thm' =

                   unfold_thms lthy @{thms unit_all_eq1} (mk_absumprodE type_definition ms)

                   |> Morphism.thm phi;

               in

                 mk_exhaust_tac ctxt n ctr_defs ctor_iff_dtor_thm sumEN_thm'

               end;

             val inject_tacss =

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

                 mk_inject_tac ctxt ctr_def ctor_inject abs_inject]) ctr_defs ms;

             val half_distinct_tacss =

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

                   mk_half_distinct_tac ctxt ctor_inject abs_inject [def, def']))

                 (mk_half_pairss (`I ctr_defs));

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

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

             fun ctr_spec_of disc_b ctr0 sel_bs sel_defs = (((disc_b, ctr0), sel_bs), sel_defs);

             val ctr_specs = map4 ctr_spec_of disc_bindings ctrs0 sel_bindingss sel_defaultss;

           in

             free_constructors tacss ((wrap_opts, standard_binding), ctr_specs) lthy

           end;

         fun derive_maps_sets_rels (ctr_sugar as {case_cong, ...} : ctr_sugar, lthy) =

           if live = 0 then

             ((([], [], [], []), ctr_sugar), lthy)

           else

             let

               val rel_flip = rel_flip_of_bnf fp_bnf;

               val nones = replicate live NONE;

               val ctor_cong =

                 if fp = Least_FP then

                   Drule.dummy_thm

                 else

                   let val ctor' = mk_ctor Bs ctor in

                     cterm_instantiate_pos [NONE, NONE, SOME (certify lthy ctor')] arg_cong

                   end;

               fun mk_cIn ctor k xs =

                 let val absT = domain_type (fastype_of ctor) in

                   mk_absumprod absT abs n k xs

                   |> fp = Greatest_FP ? curry (op $) ctor

                   |> certify lthy

                 end;

               val cxIns = map2 (mk_cIn ctor) ks xss;

               val cyIns = map2 (mk_cIn (Term.map_types B_ify ctor)) ks yss;

               fun mk_map_thm ctr_def' cxIn =

                 fold_thms lthy [ctr_def']

                   (unfold_thms lthy (o_apply :: pre_map_def ::

                        (if fp = Least_FP then [] else [ctor_dtor, dtor_ctor]) @ sumprod_thms_map @

                        abs_inverses)

                      (cterm_instantiate_pos (nones @ [SOME cxIn])

                         (if fp = Least_FP then fp_map_thm else fp_map_thm RS ctor_cong)))

                 |> singleton (Proof_Context.export names_lthy no_defs_lthy);

               fun mk_set_thm fp_set_thm ctr_def' cxIn =

                 fold_thms lthy [ctr_def']

                   (unfold_thms lthy (pre_set_defs @ nested_set_maps @ nesting_set_maps @

                        (if fp = Least_FP then [] else [dtor_ctor]) @ sumprod_thms_set @

                        abs_inverses)

                      (cterm_instantiate_pos [SOME cxIn] fp_set_thm))

                 |> singleton (Proof_Context.export names_lthy no_defs_lthy);

               fun mk_set_thms fp_set_thm = map2 (mk_set_thm fp_set_thm) ctr_defs' cxIns;

               val map_thms = map2 mk_map_thm ctr_defs' cxIns;

               val set_thmss = map mk_set_thms fp_set_thms;

               val set_thms = flat set_thmss;

               val rel_infos = (ctr_defs' ~~ cxIns, ctr_defs' ~~ cyIns);

               fun mk_rel_thm postproc ctr_defs' cxIn cyIn =

                 fold_thms lthy ctr_defs'

                   (unfold_thms lthy (pre_rel_def :: abs_inverse ::

                        (if fp = Least_FP then [] else [dtor_ctor]) @ sumprod_thms_rel @

                        @{thms vimage2p_def sum.inject Inl_Inr_False})

                      (cterm_instantiate_pos (nones @ [SOME cxIn, SOME cyIn]) fp_rel_thm))

                 |> postproc

                 |> singleton (Proof_Context.export names_lthy no_defs_lthy);

               fun mk_rel_inject_thm ((ctr_def', cxIn), (_, cyIn)) =

                 mk_rel_thm (unfold_thms lthy @{thms eq_sym_Unity_conv}) [ctr_def'] cxIn cyIn;

               val rel_inject_thms = map mk_rel_inject_thm (op ~~ rel_infos);

               fun mk_half_rel_distinct_thm ((xctr_def', cxIn), (yctr_def', cyIn)) =

                 mk_rel_thm (fn thm => thm RS @{thm eq_False[THEN iffD1]}) [xctr_def', yctr_def']

                   cxIn cyIn;

               fun mk_other_half_rel_distinct_thm thm =

                 flip_rels lthy live thm

                 RS (rel_flip RS sym RS @{thm arg_cong[of _ _ Not]} RS iffD2);

               val half_rel_distinct_thmss =

                 map (map mk_half_rel_distinct_thm) (mk_half_pairss rel_infos);

               val other_half_rel_distinct_thmss =

                 map (map mk_other_half_rel_distinct_thm) half_rel_distinct_thmss;

               val (rel_distinct_thms, _) =

                 join_halves n half_rel_distinct_thmss other_half_rel_distinct_thmss;

               val rel_eq_thms =

                 map (fn th => th RS @{thm eq_False[THEN iffD2]}) rel_distinct_thms @

                 map2 (fn th => fn 0 => th RS @{thm eq_True[THEN iffD2]} | _ => th)

                   rel_inject_thms ms;

               val anonymous_notes =

                 [([case_cong], fundefcong_attrs),

                  (rel_eq_thms, code_nitpicksimp_attrs)]

                 |> map (fn (thms, attrs) => ((Binding.empty, attrs), [(thms, [])]));

               val notes =

                 [(mapN, map_thms, code_nitpicksimp_attrs @ simp_attrs),

                  (rel_distinctN, rel_distinct_thms, simp_attrs),

                  (rel_injectN, rel_inject_thms, simp_attrs),

                  (setN, set_thms, code_nitpicksimp_attrs @ simp_attrs)]

                 |> massage_simple_notes fp_b_name;

             in

               (((map_thms, rel_inject_thms, rel_distinct_thms, set_thmss), ctr_sugar),

                lthy

                |> Spec_Rules.add Spec_Rules.Equational (`(single o lhs_head_of o hd) map_thms)

                |> fp = Least_FP

                  ? Spec_Rules.add Spec_Rules.Equational (`(single o lhs_head_of o hd) rel_eq_thms)

                |> Spec_Rules.add Spec_Rules.Equational (`(single o lhs_head_of o hd) set_thms)

                |> Local_Theory.notes (anonymous_notes @ notes)

                |> snd)

             end;

         fun mk_binding pre = Binding.qualify false fp_b_name (Binding.prefix_name (pre ^ "_") fp_b);

         fun massage_res (((maps_sets_rels, ctr_sugar), co_rec_res), lthy) =

           (((maps_sets_rels, (ctrs, xss, ctr_defs, ctr_sugar)), co_rec_res), lthy);

       in

         (wrap_ctrs

          #> derive_maps_sets_rels

          ##>>

            (if fp = Least_FP then define_rec (the recs_args_types) mk_binding fpTs Cs reps

            else define_corec (the corecs_args_types) mk_binding fpTs Cs abss) xtor_co_rec

          #> massage_res, lthy')

       end;

     fun wrap_types_etc (wrap_types_etcs, lthy) =

       fold_map I wrap_types_etcs lthy

       |>> apsnd split_list o apfst (apsnd split_list4 o apfst split_list4 o split_list)

         o split_list;

     fun mk_simp_thms ({injects, distincts, case_thms, ...} : ctr_sugar) co_recs mapsx rel_injects

         rel_distincts setss =

       injects @ distincts @ case_thms @ co_recs @ mapsx @ rel_injects @ rel_distincts @ flat setss;

     fun derive_note_induct_recs_thms_for_types

         ((((mapss, rel_injectss, rel_distinctss, setss), (ctrss, _, ctr_defss, ctr_sugars)),

           (recs, rec_defs)), lthy) =

       let

         val ((induct_thms, induct_thm, induct_attrs), (rec_thmss, rec_attrs)) =

           derive_induct_recs_thms_for_types pre_bnfs recs_args_types xtor_co_induct xtor_co_rec_thms

             nesting_bnfs nested_bnfs fpTs Cs Xs ctrXs_Tsss abs_inverses type_definitions

             abs_inverses ctrss ctr_defss recs rec_defs lthy;

         val induct_type_attr = Attrib.internal o K o Induct.induct_type;

         val simp_thmss =

           map6 mk_simp_thms ctr_sugars rec_thmss mapss rel_injectss rel_distinctss setss;

         val common_notes =

           (if nn > 1 then [(inductN, [induct_thm], induct_attrs)] else [])

           |> massage_simple_notes fp_common_name;

         val notes =

           [(inductN, map single induct_thms, fn T_name => induct_attrs @ [induct_type_attr T_name]),

            (recN, rec_thmss, K rec_attrs),

            (simpsN, simp_thmss, K [])]

           |> massage_multi_notes;

       in

         lthy

         |> Spec_Rules.add Spec_Rules.Equational (recs, flat rec_thmss)

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

         |> register_as_fp_sugars fpTs fpBTs Xs Least_FP pre_bnfs absT_infos nested_bnfs nesting_bnfs

           fp_res ctrXs_Tsss ctr_defss ctr_sugars recs rec_defs mapss [induct_thm]

           (map single induct_thms) rec_thmss (replicate nn []) (replicate nn []) rel_injectss

           rel_distinctss

       end;

     fun derive_note_coinduct_corecs_thms_for_types

         ((((mapss, rel_injectss, rel_distinctss, setss), (_, _, ctr_defss, ctr_sugars)),

           (corecs, corec_defs)), lthy) =

       let

         val (([(coinduct_thms, coinduct_thm), (strong_coinduct_thms, strong_coinduct_thm)],

               coinduct_attrs),

              (corec_thmss, corec_attrs), (disc_corec_thmss, disc_corec_attrs),

              (disc_corec_iff_thmss, disc_corec_iff_attrs), (sel_corec_thmsss, sel_corec_attrs)) =

           derive_coinduct_corecs_thms_for_types pre_bnfs (the corecs_args_types) xtor_co_induct

             dtor_injects dtor_ctors xtor_co_rec_thms nesting_bnfs fpTs Cs Xs ctrXs_Tsss kss mss ns

             abs_inverses abs_inverses I ctr_defss ctr_sugars corecs corec_defs

             (Proof_Context.export lthy' no_defs_lthy) lthy;

         val sel_corec_thmss = map flat sel_corec_thmsss;

         val coinduct_type_attr = Attrib.internal o K o Induct.coinduct_type;

         val flat_corec_thms = append oo append;

         val simp_thmss =

           map6 mk_simp_thms ctr_sugars

             (map3 flat_corec_thms disc_corec_thmss disc_corec_iff_thmss sel_corec_thmss)

             mapss rel_injectss rel_distinctss setss;

         val common_notes =

           (if nn > 1 then

              [(coinductN, [coinduct_thm], coinduct_attrs),

               (strong_coinductN, [strong_coinduct_thm], coinduct_attrs)]

            else

              [])

           |> massage_simple_notes fp_common_name;

         val notes =

           [(coinductN, map single coinduct_thms,

             fn T_name => coinduct_attrs @ [coinduct_type_attr T_name]),

            (corecN, corec_thmss, K corec_attrs),

            (disc_corecN, disc_corec_thmss, K disc_corec_attrs),

            (disc_corec_iffN, disc_corec_iff_thmss, K disc_corec_iff_attrs),

            (sel_corecN, sel_corec_thmss, K sel_corec_attrs),

            (simpsN, simp_thmss, K []),

            (strong_coinductN, map single strong_coinduct_thms, K coinduct_attrs)]

           |> massage_multi_notes;

       in

         lthy

         (* TODO: code theorems *)

         |> fold (curry (Spec_Rules.add Spec_Rules.Equational) corecs)

           [flat sel_corec_thmss, flat corec_thmss]

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

         |> register_as_fp_sugars fpTs fpBTs Xs Greatest_FP pre_bnfs absT_infos nested_bnfs

           nesting_bnfs fp_res ctrXs_Tsss ctr_defss ctr_sugars corecs corec_defs mapss

           [coinduct_thm, strong_coinduct_thm] (transpose [coinduct_thms, strong_coinduct_thms])

           corec_thmss disc_corec_thmss sel_corec_thmsss rel_injectss rel_distinctss

       end;

     val lthy'' = lthy'

       |> fold_map define_ctrs_dtrs_for_type (fp_bnfs ~~ fp_bs ~~ fpTs ~~ ctors ~~ dtors ~~

         xtor_co_recs ~~ ctor_dtors ~~ dtor_ctors ~~ ctor_injects ~~ pre_map_defs ~~ pre_set_defss ~~

         pre_rel_defs ~~ xtor_map_thms ~~ xtor_set_thmss ~~ xtor_rel_thms ~~ ns ~~ kss ~~ mss ~~

         abss ~~ abs_injects ~~ abs_inverses ~~ type_definitions ~~ ctr_bindingss ~~ ctr_mixfixess ~~

         ctr_Tsss ~~ disc_bindingss ~~ sel_bindingsss ~~ raw_sel_defaultsss)

       |> wrap_types_etc

       |> case_fp fp derive_note_induct_recs_thms_for_types

            derive_note_coinduct_corecs_thms_for_types;

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

       co_prefix fp ^ "datatype"));

   in

     timer; lthy''

   end;

 fun co_datatypes x = define_co_datatypes (K I) (K I) (K I) x;

 fun co_datatype_cmd x =

   define_co_datatypes Typedecl.read_constraint Syntax.parse_typ Syntax.parse_term x;

 val parse_ctr_arg =

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

   (Parse.typ >> pair Binding.empty);

 val parse_ctr_specs =

   Parse.enum1 "|" (parse_ctr_spec parse_binding parse_ctr_arg -- Parse.opt_mixfix);

 val parse_spec =

   parse_type_args_named_constrained -- parse_binding -- parse_map_rel_bindings --

   Parse.opt_mixfix -- (@{keyword "="} |-- parse_ctr_specs);

 val parse_co_datatype = parse_ctr_options -- Parse.and_list1 parse_spec;

 fun parse_co_datatype_cmd fp construct_fp = parse_co_datatype >> co_datatype_cmd fp construct_fp;

 val _ = Context.>> (Context.map_theory FP_Sugar_Interpretation.init);

 end;