src/HOL/Codatatype/Tools/bnf_fp_sugar.ML
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generate iter/rec goals
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(*  Title:      HOL/Codatatype/Tools/bnf_fp_sugar.ML
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    Author:     Jasmin Blanchette, TU Muenchen
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    Copyright   2012
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Sugar for constructing LFPs and GFPs.
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*)
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signature BNF_FP_SUGAR =
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sig
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end;
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structure BNF_FP_Sugar : BNF_FP_SUGAR =
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struct
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open BNF_Util
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open BNF_Wrap
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open BNF_FP_Util
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open BNF_LFP
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open BNF_GFP
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open BNF_FP_Sugar_Tactics
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val caseN = "case";
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val itersN = "iters";
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val recsN = "recs";
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fun retype_free (Free (s, _)) T = Free (s, T);
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fun flat_list_comb (f, xss) = fold (fn xs => fn t => Term.list_comb (t, xs)) xss f
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fun mk_tupled_fun x f xs = HOLogic.tupled_lambda x (Term.list_comb (f, xs));
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fun mk_uncurried_fun f xs = mk_tupled_fun (HOLogic.mk_tuple xs) f xs;
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fun mk_uncurried2_fun f xss =
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  mk_tupled_fun (HOLogic.mk_tuple (map HOLogic.mk_tuple xss)) f (flat xss);
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fun cannot_merge_types () = error "Mutually recursive types must have the same type parameters";
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fun merge_type_arg_constrained ctxt (T, c) (T', c') =
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  if T = T' then
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    (case (c, c') of
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      (_, NONE) => (T, c)
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    | (NONE, _) => (T, c')
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    | _ =>
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      if c = c' then
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        (T, c)
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      else
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        error ("Inconsistent sort constraints for type variable " ^
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          quote (Syntax.string_of_typ ctxt T)))
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  else
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    cannot_merge_types ();
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fun merge_type_args_constrained ctxt (cAs, cAs') =
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  if length cAs = length cAs' then map2 (merge_type_arg_constrained ctxt) cAs cAs'
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  else cannot_merge_types ();
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fun type_args_constrained_of (((cAs, _), _), _) = cAs;
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val type_args_of = map fst o type_args_constrained_of;
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fun type_binder_of (((_, b), _), _) = b;
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fun mixfix_of ((_, mx), _) = mx;
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fun ctr_specs_of (_, ctr_specs) = ctr_specs;
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fun disc_of (((disc, _), _), _) = disc;
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fun ctr_of (((_, ctr), _), _) = ctr;
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fun args_of ((_, args), _) = args;
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fun ctr_mixfix_of (_, mx) = mx;
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fun prepare_datatype prepare_typ gfp specs fake_lthy no_defs_lthy =
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  let
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    val constrained_As =
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      map (map (apfst (prepare_typ fake_lthy)) o type_args_constrained_of) specs
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      |> Library.foldr1 (merge_type_args_constrained no_defs_lthy);
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    val As = map fst constrained_As;
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    val As' = map dest_TFree As;
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    val _ = (case duplicates (op =) As of [] => ()
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      | A :: _ => error ("Duplicate type parameter " ^
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          quote (Syntax.string_of_typ no_defs_lthy A)));
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    (* TODO: use sort constraints on type args *)
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    val N = length specs;
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    fun mk_fake_T b =
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      Type (fst (Term.dest_Type (Proof_Context.read_type_name fake_lthy true (Binding.name_of b))),
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        As);
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    val bs = map type_binder_of specs;
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    val fakeTs = map mk_fake_T bs;
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    val mixfixes = map mixfix_of specs;
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    val _ = (case duplicates Binding.eq_name bs of [] => ()
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      | b :: _ => error ("Duplicate type name declaration " ^ quote (Binding.name_of b)));
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    val ctr_specss = map ctr_specs_of specs;
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    val disc_binderss = map (map disc_of) ctr_specss;
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    val ctr_binderss = map (map ctr_of) ctr_specss;
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    val ctr_argsss = map (map args_of) ctr_specss;
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    val ctr_mixfixess = map (map ctr_mixfix_of) ctr_specss;
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    val sel_bindersss = map (map (map fst)) ctr_argsss;
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    val fake_ctr_Tsss = map (map (map (prepare_typ fake_lthy o snd))) ctr_argsss;
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    val rhs_As' = fold (fold (fold Term.add_tfreesT)) fake_ctr_Tsss [];
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    val _ = (case subtract (op =) As' rhs_As' of
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        [] => ()
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      | A' :: _ => error ("Extra type variables on rhs: " ^
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          quote (Syntax.string_of_typ no_defs_lthy (TFree A'))));
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    val ((Cs, Xs), _) =
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      no_defs_lthy
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      |> fold (fold (fn s => Variable.declare_typ (TFree (s, dummyS))) o type_args_of) specs
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      |> mk_TFrees N
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      ||>> mk_TFrees N;
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    fun eq_fpT (T as Type (s, Us)) (Type (s', Us')) =
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        s = s' andalso (Us = Us' orelse error ("Illegal occurrence of recursive type " ^
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          quote (Syntax.string_of_typ fake_lthy T)))
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      | eq_fpT _ _ = false;
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    fun freeze_fp (T as Type (s, Us)) =
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        (case find_index (eq_fpT T) fakeTs of ~1 => Type (s, map freeze_fp Us) | j => nth Xs j)
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      | freeze_fp T = T;
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    val ctr_TsssXs = map (map (map freeze_fp)) fake_ctr_Tsss;
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    val sum_prod_TsXs = map (mk_sumTN o map HOLogic.mk_tupleT) ctr_TsssXs;
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    val eqs = map dest_TFree Xs ~~ sum_prod_TsXs;
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    val ((unfs0, flds0, fp_iters0, fp_recs0, unf_flds, fld_unfs, fld_injects), lthy) =
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      fp_bnf (if gfp then bnf_gfp else bnf_lfp) bs mixfixes As' eqs no_defs_lthy;
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    val timer = time (Timer.startRealTimer ());
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    fun mk_unf_or_fld get_T Ts t =
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      let val Type (_, Ts0) = get_T (fastype_of t) in
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        Term.subst_atomic_types (Ts0 ~~ Ts) t
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      end;
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    val mk_unf = mk_unf_or_fld domain_type;
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    val mk_fld = mk_unf_or_fld range_type;
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    val unfs = map (mk_unf As) unfs0;
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    val flds = map (mk_fld As) flds0;
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    val fpTs = map (domain_type o fastype_of) unfs;
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    val is_fpT = member (op =) fpTs;
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    val ctr_Tsss = map (map (map (Term.typ_subst_atomic (Xs ~~ fpTs)))) ctr_TsssXs;
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    val ns = map length ctr_Tsss;
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    val mss = map (map length) ctr_Tsss;
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    val Css = map2 replicate ns Cs;
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    val Cs' = flat Css;
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    fun mk_iter_or_rec Ts Us c =
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      let
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        val (binders, body) = strip_type (fastype_of c);
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        val (fst_binders, last_binder) = split_last binders;
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        val Type (_, Ts0) = if gfp then body else last_binder;
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        val Us0 = map (if gfp then domain_type else body_type) fst_binders;
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      in
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        Term.subst_atomic_types (Ts0 @ Us0 ~~ Ts @ Us) c
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      end;
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    val fp_iters as fp_iter1 :: _ = map (mk_iter_or_rec As Cs) fp_iters0;
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    val fp_recs as fp_rec1 :: _ = map (mk_iter_or_rec As Cs) fp_recs0;
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    val fp_y_Ts = map domain_type (fst (split_last (binder_types (fastype_of fp_iter1))));
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    val y_Tsss = map3 (fn ms => map2 dest_tupleT ms oo dest_sumTN) mss ns fp_y_Ts;
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    val g_Tss = map2 (map2 (curry (op --->))) y_Tsss Css;
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    fun dest_rec_pair (T as Type (@{type_name prod}, Us as [_, U])) =
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        if member (op =) Cs U then Us else [T]
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      | dest_rec_pair T = [T];
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    val fp_z_Ts = map domain_type (fst (split_last (binder_types (fastype_of fp_rec1))));
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    val z_Tssss =
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      map3 (fn ms => map2 (map dest_rec_pair oo dest_tupleT) ms oo dest_sumTN) mss ns fp_z_Ts;
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    val h_Tss = map2 (map2 (fold_rev (curry (op --->)))) z_Tssss Css;
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    val ((gss, ysss), _) =
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      lthy
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      |> mk_Freess "f" g_Tss
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      ||>> mk_Freesss "x" y_Tsss;
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    val hss = map2 (map2 retype_free) gss h_Tss;
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    val (zssss, _) =
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      lthy
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      |> mk_Freessss "x" z_Tssss;
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    fun pour_sugar_on_type ((((((((((((((b, fpT), C), fld), unf), fp_iter), fp_rec), fld_unf),
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          unf_fld), fld_inject), ctr_binders), ctr_mixfixes), ctr_Tss), disc_binders), sel_binderss)
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        no_defs_lthy =
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      let
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        val n = length ctr_Tss;
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        val ks = 1 upto n;
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        val ms = map length ctr_Tss;
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        val unfT = domain_type (fastype_of fld);
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        val prod_Ts = map HOLogic.mk_tupleT ctr_Tss;
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        val case_Ts = map (fn Ts => Ts ---> C) ctr_Tss;
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        val ((((u, v), fs), xss), _) =
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          no_defs_lthy
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          |> yield_singleton (mk_Frees "u") unfT
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          ||>> yield_singleton (mk_Frees "v") fpT
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          ||>> mk_Frees "f" case_Ts
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          ||>> mk_Freess "x" ctr_Tss;
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        val ctr_rhss =
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          map2 (fn k => fn xs =>
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            fold_rev Term.lambda xs (fld $ mk_InN prod_Ts (HOLogic.mk_tuple xs) k)) ks xss;
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        val case_binder = Binding.suffix_name ("_" ^ caseN) b;
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        val case_rhs =
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          fold_rev Term.lambda (fs @ [v]) (mk_sum_caseN (map2 mk_uncurried_fun fs xss) $ (unf $ v));
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        val ((raw_case :: raw_ctrs, raw_case_def :: raw_ctr_defs), (lthy', lthy)) = no_defs_lthy
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          |> apfst split_list o fold_map3 (fn b => fn mx => fn rhs =>
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               Local_Theory.define ((b, mx), ((Thm.def_binding b, []), rhs)) #>> apsnd snd)
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             (case_binder :: ctr_binders) (NoSyn :: ctr_mixfixes) (case_rhs :: ctr_rhss)
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          ||> `Local_Theory.restore;
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        (*transforms defined frees into consts (and more)*)
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        val phi = Proof_Context.export_morphism lthy lthy';
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        val ctr_defs = map (Morphism.thm phi) raw_ctr_defs;
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        val case_def = Morphism.thm phi raw_case_def;
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        val ctrs0 = map (Morphism.term phi) raw_ctrs;
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        val casex0 = Morphism.term phi raw_case;
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        val ctrs = map (mk_ctr As) ctrs0;
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        fun exhaust_tac {context = ctxt, ...} =
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          let
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            val fld_iff_unf_thm =
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              let
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                val goal =
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                  fold_rev Logic.all [u, v]
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                    (mk_Trueprop_eq (HOLogic.mk_eq (v, fld $ u), HOLogic.mk_eq (unf $ v, u)));
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              in
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                Skip_Proof.prove lthy [] [] goal (fn {context = ctxt, ...} =>
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                  mk_fld_iff_unf_tac ctxt (map (SOME o certifyT lthy) [unfT, fpT])
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                    (certify lthy fld) (certify lthy unf) fld_unf unf_fld)
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                |> Thm.close_derivation
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                |> Morphism.thm phi
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              end;
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            val sumEN_thm' =
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              Local_Defs.unfold lthy @{thms all_unit_eq}
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                (Drule.instantiate' (map (SOME o certifyT lthy) prod_Ts) [] (mk_sumEN n))
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              |> Morphism.thm phi;
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          in
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            mk_exhaust_tac ctxt n ctr_defs fld_iff_unf_thm sumEN_thm'
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          end;
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        val inject_tacss =
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          map2 (fn 0 => K []
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                 | _ => fn ctr_def => [fn {context = ctxt, ...} =>
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                     mk_inject_tac ctxt ctr_def fld_inject])
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            ms ctr_defs;
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        val half_distinct_tacss =
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          map (map (fn (def, def') => fn {context = ctxt, ...} =>
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            mk_half_distinct_tac ctxt fld_inject [def, def'])) (mk_half_pairss ctr_defs);
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        val case_tacs =
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          map3 (fn k => fn m => fn ctr_def => fn {context = ctxt, ...} =>
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            mk_case_tac ctxt n k m case_def ctr_def unf_fld) ks ms ctr_defs;
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        val tacss = [exhaust_tac] :: inject_tacss @ half_distinct_tacss @ [case_tacs];
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        fun sugar_datatype no_defs_lthy =
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          let
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            val iter_T = flat g_Tss ---> fpT --> C;
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            val rec_T = flat h_Tss ---> fpT --> C;
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            val iter_binder = Binding.suffix_name ("_" ^ iterN) b;
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            val rec_binder = Binding.suffix_name ("_" ^ recN) b;
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            val iter_free = Free (Binding.name_of iter_binder, iter_T);
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            val rec_free = Free (Binding.name_of rec_binder, rec_T);
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            val iter_spec =
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              mk_Trueprop_eq (flat_list_comb (iter_free, gss),
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                Term.list_comb (fp_iter, map2 (mk_sum_caseN oo map2 mk_uncurried_fun) gss ysss));
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            val rec_spec =
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              mk_Trueprop_eq (flat_list_comb (rec_free, hss),
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                Term.list_comb (fp_rec, map2 (mk_sum_caseN oo map2 mk_uncurried2_fun) hss zssss));
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            val (([raw_iter, raw_rec], [raw_iter_def, raw_rec_def]), (lthy', lthy)) = no_defs_lthy
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              |> apfst split_list o fold_map2 (fn b => fn spec =>
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                Specification.definition (SOME (b, NONE, NoSyn), ((Thm.def_binding b, []), spec))
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                #>> apsnd snd) [iter_binder, rec_binder] [iter_spec, rec_spec]
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              ||> `Local_Theory.restore;
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            (*transforms defined frees into consts (and more)*)
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            val phi = Proof_Context.export_morphism lthy lthy';
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            val iter_def = Morphism.thm phi raw_iter_def;
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            val rec_def = Morphism.thm phi raw_rec_def;
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            val iter0 = Morphism.term phi raw_iter;
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            val rec0 = Morphism.term phi raw_rec;
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            val iter = mk_iter_or_rec As Cs' iter0;
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            val recx = mk_iter_or_rec As Cs' rec0;
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          in
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            ([[ctrs], [[iter]], [[recx]], xss], lthy)
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          end;
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        fun sugar_codatatype no_defs_lthy = ([], no_defs_lthy);
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      in
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        wrap_datatype tacss ((ctrs0, casex0), (disc_binders, sel_binderss)) lthy'
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        |> (if gfp then sugar_codatatype else sugar_datatype)
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      end;
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    fun pour_more_sugar_on_datatypes ([[ctrss], [[iters]], [[recs]], xsss], lthy) =
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      let
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        val xctrss = map2 (map2 (curry Term.list_comb)) ctrss xsss;
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        val giters = map (fn iter => flat_list_comb (iter, gss)) iters;
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        val hrecs = map (fn recx => flat_list_comb (recx, hss)) recs;
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        val (iter_thmss, rec_thmss) =
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          let
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            fun mk_goal_iter_or_rec fss fc xctr f xs xs' =
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              mk_Trueprop_eq (fc $ xctr, Term.list_comb (f, xs'));
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            fun fix_iter_free (x as Free (_, T)) =
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              (case find_index (eq_fpT T) fpTs of ~1 => x | j => nth giters j $ x);
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            fun fix_rec_free (x as Free (_, T)) =
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              (case find_index (eq_fpT T) fpTs of ~1 => [x] | j => [x, nth hrecs j $ x]);
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            val iter_xsss = map (map (map fix_iter_free)) xsss;
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            val rec_xsss = map (map (maps fix_rec_free)) xsss;
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            val goal_iterss =
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              map5 (map4 o mk_goal_iter_or_rec gss) giters xctrss gss xsss iter_xsss;
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            val goal_recss =
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              map5 (map4 o mk_goal_iter_or_rec hss) hrecs xctrss hss xsss rec_xsss;
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            val iter_tacss =
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              map (map (K (fn _ => Skip_Proof.cheat_tac (Proof_Context.theory_of lthy)))) goal_iterss;
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              (* ### map (map mk_iter_or_rec_tac); (* needs ctr_def, iter_def, fld_iter *) *)
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            val rec_tacss =
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              map (map (K (fn _ => Skip_Proof.cheat_tac (Proof_Context.theory_of lthy)))) goal_recss;
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          in
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            (map2 (map2 (Skip_Proof.prove lthy [] [])) goal_iterss iter_tacss,
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             map2 (map2 (Skip_Proof.prove lthy [] [])) goal_recss rec_tacss)
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          end;
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        val notes =
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          [(itersN, iter_thmss),
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           (recsN, rec_thmss)]
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          |> maps (fn (thmN, thmss) =>
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            map2 (fn b => fn thms =>
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                ((Binding.qualify true (Binding.name_of b) (Binding.name thmN), []), [(thms, [])]))
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              bs thmss);
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      in
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        lthy |> Local_Theory.notes notes |> snd
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      end;
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    val lthy' = lthy
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      |> fold_map pour_sugar_on_type (bs ~~ fpTs ~~ Cs ~~ flds ~~ unfs ~~ fp_iters ~~ fp_recs ~~
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        fld_unfs ~~ unf_flds ~~ fld_injects ~~ ctr_binderss ~~ ctr_mixfixess ~~ ctr_Tsss ~~
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        disc_binderss ~~ sel_bindersss)
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      |> (if gfp then snd else pour_more_sugar_on_datatypes o apfst transpose);
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    val timer = time (timer ("Constructors, discriminators, selectors, etc., for the new " ^
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      (if gfp then "co" else "") ^ "datatype"));
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  in
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    (timer; lthy')
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  end;
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fun datatype_cmd info specs lthy =
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  let
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    (*the "perhaps o try" below helps gracefully handles the case where the new type is defined in a
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      locale and shadows an existing global type*)
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    val fake_thy = Theory.copy
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      #> fold (fn spec => perhaps (try (Sign.add_type lthy
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        (type_binder_of spec, length (type_args_constrained_of spec), mixfix_of spec)))) specs;
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    val fake_lthy = Proof_Context.background_theory fake_thy lthy;
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  in
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    prepare_datatype Syntax.read_typ info specs fake_lthy lthy
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  end;
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val parse_opt_binding_colon = Scan.optional (Parse.binding --| Parse.$$$ ":") no_binder
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val parse_ctr_arg =
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  Parse.$$$ "(" |-- parse_opt_binding_colon -- Parse.typ --| Parse.$$$ ")" ||
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  (Parse.typ >> pair no_binder);
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val parse_single_spec =
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  Parse.type_args_constrained -- Parse.binding -- Parse.opt_mixfix --
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  (@{keyword "="} |-- Parse.enum1 "|" (parse_opt_binding_colon -- Parse.binding --
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    Scan.repeat parse_ctr_arg -- Parse.opt_mixfix));
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val _ =
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  Outer_Syntax.local_theory @{command_spec "data"} "define BNF-based inductive datatypes"
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    (Parse.and_list1 parse_single_spec >> datatype_cmd false);
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val _ =
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  Outer_Syntax.local_theory @{command_spec "codata"} "define BNF-based coinductive datatypes"
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    (Parse.and_list1 parse_single_spec >> datatype_cmd true);
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end;