src/HOL/Nominal/nominal_package.ML
author wenzelm
Wed Dec 06 01:12:36 2006 +0100 (2006-12-06)
changeset 21669 c68717c16013
parent 21540 f3faed8276e6
child 21858 05f57309170c
permissions -rw-r--r--
removed legacy ML bindings;
     1 (*  Title:      HOL/Nominal/nominal_package.ML
     2     ID:         $Id$
     3     Author:     Stefan Berghofer and Christian Urban, TU Muenchen
     4 
     5 Nominal datatype package for Isabelle/HOL.
     6 *)
     7 
     8 signature NOMINAL_PACKAGE =
     9 sig
    10   val add_nominal_datatype : bool -> string list -> (string list * bstring * mixfix *
    11     (bstring * string list * mixfix) list) list -> theory -> theory
    12   type nominal_datatype_info
    13   val get_nominal_datatypes : theory -> nominal_datatype_info Symtab.table
    14   val get_nominal_datatype : theory -> string -> nominal_datatype_info option
    15   val setup: theory -> theory
    16 end
    17 
    18 structure NominalPackage : NOMINAL_PACKAGE =
    19 struct
    20 
    21 val Finites_emptyI = thm "Finites.emptyI";
    22 val finite_Diff = thm "finite_Diff";
    23 val finite_Un = thm "finite_Un";
    24 val Un_iff = thm "Un_iff";
    25 val In0_eq = thm "In0_eq";
    26 val In1_eq = thm "In1_eq";
    27 val In0_not_In1 = thm "In0_not_In1";
    28 val In1_not_In0 = thm "In1_not_In0";
    29 val Un_assoc = thm "Un_assoc";
    30 val Collect_disj_eq = thm "Collect_disj_eq";
    31 val empty_def = thm "empty_def";
    32 
    33 open DatatypeAux;
    34 open NominalAtoms;
    35 
    36 (** FIXME: DatatypePackage should export this function **)
    37 
    38 local
    39 
    40 fun dt_recs (DtTFree _) = []
    41   | dt_recs (DtType (_, dts)) = List.concat (map dt_recs dts)
    42   | dt_recs (DtRec i) = [i];
    43 
    44 fun dt_cases (descr: descr) (_, args, constrs) =
    45   let
    46     fun the_bname i = Sign.base_name (#1 (valOf (AList.lookup (op =) descr i)));
    47     val bnames = map the_bname (distinct op = (List.concat (map dt_recs args)));
    48   in map (fn (c, _) => space_implode "_" (Sign.base_name c :: bnames)) constrs end;
    49 
    50 
    51 fun induct_cases descr =
    52   DatatypeProp.indexify_names (List.concat (map (dt_cases descr) (map #2 descr)));
    53 
    54 fun exhaust_cases descr i = dt_cases descr (valOf (AList.lookup (op =) descr i));
    55 
    56 in
    57 
    58 fun mk_case_names_induct descr = RuleCases.case_names (induct_cases descr);
    59 
    60 fun mk_case_names_exhausts descr new =
    61   map (RuleCases.case_names o exhaust_cases descr o #1)
    62     (List.filter (fn ((_, (name, _, _))) => name mem_string new) descr);
    63 
    64 end;
    65 
    66 (* data kind 'HOL/nominal_datatypes' *)
    67 
    68 type nominal_datatype_info =
    69   {index : int,
    70    descr : descr,
    71    sorts : (string * sort) list,
    72    rec_names : string list,
    73    rec_rewrites : thm list,
    74    induction : thm,
    75    distinct : thm list,
    76    inject : thm list};
    77 
    78 structure NominalDatatypesData = TheoryDataFun
    79 (struct
    80   val name = "HOL/nominal_datatypes";
    81   type T = nominal_datatype_info Symtab.table;
    82 
    83   val empty = Symtab.empty;
    84   val copy = I;
    85   val extend = I;
    86   fun merge _ tabs : T = Symtab.merge (K true) tabs;
    87 
    88   fun print sg tab =
    89     Pretty.writeln (Pretty.strs ("nominal datatypes:" ::
    90       map #1 (NameSpace.extern_table (Sign.type_space sg, tab))));
    91 end);
    92 
    93 val get_nominal_datatypes = NominalDatatypesData.get;
    94 val put_nominal_datatypes = NominalDatatypesData.put;
    95 val map_nominal_datatypes = NominalDatatypesData.map;
    96 val get_nominal_datatype = Symtab.lookup o get_nominal_datatypes;
    97 val setup = NominalDatatypesData.init;
    98 
    99 (**** make datatype info ****)
   100 
   101 fun make_dt_info descr sorts induct reccomb_names rec_thms
   102     (((i, (_, (tname, _, _))), distinct), inject) =
   103   (tname,
   104    {index = i,
   105     descr = descr,
   106     sorts = sorts,
   107     rec_names = reccomb_names,
   108     rec_rewrites = rec_thms,
   109     induction = induct,
   110     distinct = distinct,
   111     inject = inject});
   112 
   113 (*******************************)
   114 
   115 val (_ $ (_ $ (_ $ (distinct_f $ _) $ _))) = hd (prems_of distinct_lemma);
   116 
   117 fun read_typ sign ((Ts, sorts), str) =
   118   let
   119     val T = Type.no_tvars (Sign.read_typ (sign, (AList.lookup op =)
   120       (map (apfst (rpair ~1)) sorts)) str) handle TYPE (msg, _, _) => error msg
   121   in (Ts @ [T], add_typ_tfrees (T, sorts)) end;
   122 
   123 (** taken from HOL/Tools/datatype_aux.ML **)
   124 
   125 fun indtac indrule indnames i st =
   126   let
   127     val ts = HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of indrule));
   128     val ts' = HOLogic.dest_conj (HOLogic.dest_Trueprop
   129       (Logic.strip_imp_concl (List.nth (prems_of st, i - 1))));
   130     val getP = if can HOLogic.dest_imp (hd ts) then
   131       (apfst SOME) o HOLogic.dest_imp else pair NONE;
   132     fun abstr (t1, t2) = (case t1 of
   133         NONE => (case filter (fn Free (s, _) => s mem indnames | _ => false)
   134               (term_frees t2) of
   135             [Free (s, T)] => absfree (s, T, t2)
   136           | _ => sys_error "indtac")
   137       | SOME (_ $ t') => Abs ("x", fastype_of t', abstract_over (t', t2)))
   138     val cert = cterm_of (Thm.sign_of_thm st);
   139     val Ps = map (cert o head_of o snd o getP) ts;
   140     val indrule' = cterm_instantiate (Ps ~~
   141       (map (cert o abstr o getP) ts')) indrule
   142   in
   143     rtac indrule' i st
   144   end;
   145 
   146 fun mk_subgoal i f st =
   147   let
   148     val cg = List.nth (cprems_of st, i - 1);
   149     val g = term_of cg;
   150     val revcut_rl' = Thm.lift_rule cg revcut_rl;
   151     val v = head_of (Logic.strip_assums_concl (hd (prems_of revcut_rl')));
   152     val ps = Logic.strip_params g;
   153     val cert = cterm_of (sign_of_thm st);
   154   in
   155     compose_tac (false,
   156       Thm.instantiate ([], [(cert v, cert (list_abs (ps,
   157         f (rev ps) (Logic.strip_assums_hyp g) (Logic.strip_assums_concl g))))])
   158       revcut_rl', 2) i st
   159   end;
   160 
   161 (** simplification procedure for sorting permutations **)
   162 
   163 val dj_cp = thm "dj_cp";
   164 
   165 fun dest_permT (Type ("fun", [Type ("List.list", [Type ("*", [T, _])]),
   166       Type ("fun", [_, U])])) = (T, U);
   167 
   168 fun permTs_of (Const ("Nominal.perm", T) $ t $ u) = fst (dest_permT T) :: permTs_of u
   169   | permTs_of _ = [];
   170 
   171 fun perm_simproc' thy ss (Const ("Nominal.perm", T) $ t $ (u as Const ("Nominal.perm", U) $ r $ s)) =
   172       let
   173         val (aT as Type (a, []), S) = dest_permT T;
   174         val (bT as Type (b, []), _) = dest_permT U
   175       in if aT mem permTs_of u andalso aT <> bT then
   176           let
   177             val a' = Sign.base_name a;
   178             val b' = Sign.base_name b;
   179             val cp = PureThy.get_thm thy (Name ("cp_" ^ a' ^ "_" ^ b' ^ "_inst"));
   180             val dj = PureThy.get_thm thy (Name ("dj_" ^ b' ^ "_" ^ a'));
   181             val dj_cp' = [cp, dj] MRS dj_cp;
   182             val cert = SOME o cterm_of thy
   183           in
   184             SOME (mk_meta_eq (Drule.instantiate' [SOME (ctyp_of thy S)]
   185               [cert t, cert r, cert s] dj_cp'))
   186           end
   187         else NONE
   188       end
   189   | perm_simproc' thy ss _ = NONE;
   190 
   191 val perm_simproc =
   192   Simplifier.simproc (the_context ()) "perm_simp" ["pi1 \\<bullet> (pi2 \\<bullet> x)"] perm_simproc';
   193 
   194 val allE_Nil = read_instantiate_sg (the_context()) [("x", "[]")] allE;
   195 
   196 val meta_spec = thm "meta_spec";
   197 
   198 fun projections rule =
   199   ProjectRule.projections (ProofContext.init (Thm.theory_of_thm rule)) rule
   200   |> map (standard #> RuleCases.save rule);
   201 
   202 val supp_prod = thm "supp_prod";
   203 val fresh_prod = thm "fresh_prod";
   204 val supports_fresh = thm "supports_fresh";
   205 val supports_def = thm "Nominal.op supports_def";
   206 val fresh_def = thm "fresh_def";
   207 val supp_def = thm "supp_def";
   208 val rev_simps = thms "rev.simps";
   209 val app_simps = thms "op @.simps";
   210 
   211 val collect_simp = rewrite_rule [mk_meta_eq mem_Collect_eq];
   212 
   213 fun gen_add_nominal_datatype prep_typ err flat_names new_type_names dts thy =
   214   let
   215     (* this theory is used just for parsing *)
   216 
   217     val tmp_thy = thy |>
   218       Theory.copy |>
   219       Theory.add_types (map (fn (tvs, tname, mx, _) =>
   220         (tname, length tvs, mx)) dts);
   221 
   222     val sign = Theory.sign_of tmp_thy;
   223 
   224     val atoms = atoms_of thy;
   225     val classes = map (NameSpace.map_base (fn s => "pt_" ^ s)) atoms;
   226     val cp_classes = List.concat (map (fn atom1 => map (fn atom2 =>
   227       Sign.intern_class thy ("cp_" ^ Sign.base_name atom1 ^ "_" ^
   228         Sign.base_name atom2)) atoms) atoms);
   229     fun augment_sort S = S union classes;
   230     val augment_sort_typ = map_type_tfree (fn (s, S) => TFree (s, augment_sort S));
   231 
   232     fun prep_constr ((constrs, sorts), (cname, cargs, mx)) =
   233       let val (cargs', sorts') = Library.foldl (prep_typ sign) (([], sorts), cargs)
   234       in (constrs @ [(cname, cargs', mx)], sorts') end
   235 
   236     fun prep_dt_spec ((dts, sorts), (tvs, tname, mx, constrs)) =
   237       let val (constrs', sorts') = Library.foldl prep_constr (([], sorts), constrs)
   238       in (dts @ [(tvs, tname, mx, constrs')], sorts') end
   239 
   240     val (dts', sorts) = Library.foldl prep_dt_spec (([], []), dts);
   241     val sorts' = map (apsnd augment_sort) sorts;
   242     val tyvars = map #1 dts';
   243 
   244     val types_syntax = map (fn (tvs, tname, mx, constrs) => (tname, mx)) dts';
   245     val constr_syntax = map (fn (tvs, tname, mx, constrs) =>
   246       map (fn (cname, cargs, mx) => (cname, mx)) constrs) dts';
   247 
   248     val ps = map (fn (_, n, _, _) =>
   249       (Sign.full_name sign n, Sign.full_name sign (n ^ "_Rep"))) dts;
   250     val rps = map Library.swap ps;
   251 
   252     fun replace_types (Type ("Nominal.ABS", [T, U])) =
   253           Type ("fun", [T, Type ("Nominal.noption", [replace_types U])])
   254       | replace_types (Type (s, Ts)) =
   255           Type (getOpt (AList.lookup op = ps s, s), map replace_types Ts)
   256       | replace_types T = T;
   257 
   258     val dts'' = map (fn (tvs, tname, mx, constrs) => (tvs, tname ^ "_Rep", NoSyn,
   259       map (fn (cname, cargs, mx) => (cname ^ "_Rep",
   260         map (augment_sort_typ o replace_types) cargs, NoSyn)) constrs)) dts';
   261 
   262     val new_type_names' = map (fn n => n ^ "_Rep") new_type_names;
   263     val full_new_type_names' = map (Sign.full_name (sign_of thy)) new_type_names';
   264 
   265     val ({induction, ...},thy1) =
   266       DatatypePackage.add_datatype_i err flat_names new_type_names' dts'' thy;
   267 
   268     val SOME {descr, ...} = Symtab.lookup
   269       (DatatypePackage.get_datatypes thy1) (hd full_new_type_names');
   270     fun nth_dtyp i = typ_of_dtyp descr sorts' (DtRec i);
   271 
   272     (**** define permutation functions ****)
   273 
   274     val permT = mk_permT (TFree ("'x", HOLogic.typeS));
   275     val pi = Free ("pi", permT);
   276     val perm_types = map (fn (i, _) =>
   277       let val T = nth_dtyp i
   278       in permT --> T --> T end) descr;
   279     val perm_names = replicate (length new_type_names) "Nominal.perm" @
   280       DatatypeProp.indexify_names (map (fn i => Sign.full_name (sign_of thy1)
   281         ("perm_" ^ name_of_typ (nth_dtyp i)))
   282           (length new_type_names upto length descr - 1));
   283     val perm_names_types = perm_names ~~ perm_types;
   284 
   285     val perm_eqs = List.concat (map (fn (i, (_, _, constrs)) =>
   286       let val T = nth_dtyp i
   287       in map (fn (cname, dts) =>
   288         let
   289           val Ts = map (typ_of_dtyp descr sorts') dts;
   290           val names = DatatypeProp.make_tnames Ts;
   291           val args = map Free (names ~~ Ts);
   292           val c = Const (cname, Ts ---> T);
   293           fun perm_arg (dt, x) =
   294             let val T = type_of x
   295             in if is_rec_type dt then
   296                 let val (Us, _) = strip_type T
   297                 in list_abs (map (pair "x") Us,
   298                   Const (List.nth (perm_names_types, body_index dt)) $ pi $
   299                     list_comb (x, map (fn (i, U) =>
   300                       Const ("Nominal.perm", permT --> U --> U) $
   301                         (Const ("List.rev", permT --> permT) $ pi) $
   302                         Bound i) ((length Us - 1 downto 0) ~~ Us)))
   303                 end
   304               else Const ("Nominal.perm", permT --> T --> T) $ pi $ x
   305             end;
   306         in
   307           (("", HOLogic.mk_Trueprop (HOLogic.mk_eq
   308             (Const (List.nth (perm_names_types, i)) $
   309                Free ("pi", mk_permT (TFree ("'x", HOLogic.typeS))) $
   310                list_comb (c, args),
   311              list_comb (c, map perm_arg (dts ~~ args))))), [])
   312         end) constrs
   313       end) descr);
   314 
   315     val (perm_simps, thy2) = thy1 |>
   316       Theory.add_consts_i (map (fn (s, T) => (Sign.base_name s, T, NoSyn))
   317         (List.drop (perm_names_types, length new_type_names))) |>
   318       PrimrecPackage.add_primrec_unchecked_i "" perm_eqs;
   319 
   320     (**** prove that permutation functions introduced by unfolding are ****)
   321     (**** equivalent to already existing permutation functions         ****)
   322 
   323     val _ = warning ("length descr: " ^ string_of_int (length descr));
   324     val _ = warning ("length new_type_names: " ^ string_of_int (length new_type_names));
   325 
   326     val perm_indnames = DatatypeProp.make_tnames (map body_type perm_types);
   327     val perm_fun_def = PureThy.get_thm thy2 (Name "perm_fun_def");
   328 
   329     val unfolded_perm_eq_thms =
   330       if length descr = length new_type_names then []
   331       else map standard (List.drop (split_conj_thm
   332         (Goal.prove_global thy2 [] []
   333           (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
   334             (map (fn (c as (s, T), x) =>
   335                let val [T1, T2] = binder_types T
   336                in HOLogic.mk_eq (Const c $ pi $ Free (x, T2),
   337                  Const ("Nominal.perm", T) $ pi $ Free (x, T2))
   338                end)
   339              (perm_names_types ~~ perm_indnames))))
   340           (fn _ => EVERY [indtac induction perm_indnames 1,
   341             ALLGOALS (asm_full_simp_tac
   342               (simpset_of thy2 addsimps [perm_fun_def]))])),
   343         length new_type_names));
   344 
   345     (**** prove [] \<bullet> t = t ****)
   346 
   347     val _ = warning "perm_empty_thms";
   348 
   349     val perm_empty_thms = List.concat (map (fn a =>
   350       let val permT = mk_permT (Type (a, []))
   351       in map standard (List.take (split_conj_thm
   352         (Goal.prove_global thy2 [] []
   353           (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
   354             (map (fn ((s, T), x) => HOLogic.mk_eq
   355                 (Const (s, permT --> T --> T) $
   356                    Const ("List.list.Nil", permT) $ Free (x, T),
   357                  Free (x, T)))
   358              (perm_names ~~
   359               map body_type perm_types ~~ perm_indnames))))
   360           (fn _ => EVERY [indtac induction perm_indnames 1,
   361             ALLGOALS (asm_full_simp_tac (simpset_of thy2))])),
   362         length new_type_names))
   363       end)
   364       atoms);
   365 
   366     (**** prove (pi1 @ pi2) \<bullet> t = pi1 \<bullet> (pi2 \<bullet> t) ****)
   367 
   368     val _ = warning "perm_append_thms";
   369 
   370     (*FIXME: these should be looked up statically*)
   371     val at_pt_inst = PureThy.get_thm thy2 (Name "at_pt_inst");
   372     val pt2 = PureThy.get_thm thy2 (Name "pt2");
   373 
   374     val perm_append_thms = List.concat (map (fn a =>
   375       let
   376         val permT = mk_permT (Type (a, []));
   377         val pi1 = Free ("pi1", permT);
   378         val pi2 = Free ("pi2", permT);
   379         val pt_inst = PureThy.get_thm thy2 (Name ("pt_" ^ Sign.base_name a ^ "_inst"));
   380         val pt2' = pt_inst RS pt2;
   381         val pt2_ax = PureThy.get_thm thy2
   382           (Name (NameSpace.map_base (fn s => "pt_" ^ s ^ "2") a));
   383       in List.take (map standard (split_conj_thm
   384         (Goal.prove_global thy2 [] []
   385              (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
   386                 (map (fn ((s, T), x) =>
   387                     let val perm = Const (s, permT --> T --> T)
   388                     in HOLogic.mk_eq
   389                       (perm $ (Const ("List.op @", permT --> permT --> permT) $
   390                          pi1 $ pi2) $ Free (x, T),
   391                        perm $ pi1 $ (perm $ pi2 $ Free (x, T)))
   392                     end)
   393                   (perm_names ~~
   394                    map body_type perm_types ~~ perm_indnames))))
   395            (fn _ => EVERY [indtac induction perm_indnames 1,
   396               ALLGOALS (asm_full_simp_tac (simpset_of thy2 addsimps [pt2', pt2_ax]))]))),
   397          length new_type_names)
   398       end) atoms);
   399 
   400     (**** prove pi1 ~ pi2 ==> pi1 \<bullet> t = pi2 \<bullet> t ****)
   401 
   402     val _ = warning "perm_eq_thms";
   403 
   404     val pt3 = PureThy.get_thm thy2 (Name "pt3");
   405     val pt3_rev = PureThy.get_thm thy2 (Name "pt3_rev");
   406 
   407     val perm_eq_thms = List.concat (map (fn a =>
   408       let
   409         val permT = mk_permT (Type (a, []));
   410         val pi1 = Free ("pi1", permT);
   411         val pi2 = Free ("pi2", permT);
   412         (*FIXME: not robust - better access these theorems using NominalData?*)
   413         val at_inst = PureThy.get_thm thy2 (Name ("at_" ^ Sign.base_name a ^ "_inst"));
   414         val pt_inst = PureThy.get_thm thy2 (Name ("pt_" ^ Sign.base_name a ^ "_inst"));
   415         val pt3' = pt_inst RS pt3;
   416         val pt3_rev' = at_inst RS (pt_inst RS pt3_rev);
   417         val pt3_ax = PureThy.get_thm thy2
   418           (Name (NameSpace.map_base (fn s => "pt_" ^ s ^ "3") a));
   419       in List.take (map standard (split_conj_thm
   420         (Goal.prove_global thy2 [] [] (Logic.mk_implies
   421              (HOLogic.mk_Trueprop (Const ("Nominal.prm_eq",
   422                 permT --> permT --> HOLogic.boolT) $ pi1 $ pi2),
   423               HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
   424                 (map (fn ((s, T), x) =>
   425                     let val perm = Const (s, permT --> T --> T)
   426                     in HOLogic.mk_eq
   427                       (perm $ pi1 $ Free (x, T),
   428                        perm $ pi2 $ Free (x, T))
   429                     end)
   430                   (perm_names ~~
   431                    map body_type perm_types ~~ perm_indnames)))))
   432            (fn _ => EVERY [indtac induction perm_indnames 1,
   433               ALLGOALS (asm_full_simp_tac (simpset_of thy2 addsimps [pt3', pt3_rev', pt3_ax]))]))),
   434          length new_type_names)
   435       end) atoms);
   436 
   437     (**** prove pi1 \<bullet> (pi2 \<bullet> t) = (pi1 \<bullet> pi2) \<bullet> (pi1 \<bullet> t) ****)
   438 
   439     val cp1 = PureThy.get_thm thy2 (Name "cp1");
   440     val dj_cp = PureThy.get_thm thy2 (Name "dj_cp");
   441     val pt_perm_compose = PureThy.get_thm thy2 (Name "pt_perm_compose");
   442     val pt_perm_compose_rev = PureThy.get_thm thy2 (Name "pt_perm_compose_rev");
   443     val dj_perm_perm_forget = PureThy.get_thm thy2 (Name "dj_perm_perm_forget");
   444 
   445     fun composition_instance name1 name2 thy =
   446       let
   447         val name1' = Sign.base_name name1;
   448         val name2' = Sign.base_name name2;
   449         val cp_class = Sign.intern_class thy ("cp_" ^ name1' ^ "_" ^ name2');
   450         val permT1 = mk_permT (Type (name1, []));
   451         val permT2 = mk_permT (Type (name2, []));
   452         val augment = map_type_tfree
   453           (fn (x, S) => TFree (x, cp_class :: S));
   454         val Ts = map (augment o body_type) perm_types;
   455         val cp_inst = PureThy.get_thm thy
   456           (Name ("cp_" ^ name1' ^ "_" ^ name2' ^ "_inst"));
   457         val simps = simpset_of thy addsimps (perm_fun_def ::
   458           (if name1 <> name2 then
   459              let val dj = PureThy.get_thm thy (Name ("dj_" ^ name2' ^ "_" ^ name1'))
   460              in [dj RS (cp_inst RS dj_cp), dj RS dj_perm_perm_forget] end
   461            else
   462              let
   463                val at_inst = PureThy.get_thm thy (Name ("at_" ^ name1' ^ "_inst"));
   464                val pt_inst = PureThy.get_thm thy (Name ("pt_" ^ name1' ^ "_inst"))
   465              in
   466                [cp_inst RS cp1 RS sym,
   467                 at_inst RS (pt_inst RS pt_perm_compose) RS sym,
   468                 at_inst RS (pt_inst RS pt_perm_compose_rev) RS sym]
   469             end))
   470         val thms = split_conj_thm (Goal.prove_global thy [] []
   471             (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
   472               (map (fn ((s, T), x) =>
   473                   let
   474                     val pi1 = Free ("pi1", permT1);
   475                     val pi2 = Free ("pi2", permT2);
   476                     val perm1 = Const (s, permT1 --> T --> T);
   477                     val perm2 = Const (s, permT2 --> T --> T);
   478                     val perm3 = Const ("Nominal.perm", permT1 --> permT2 --> permT2)
   479                   in HOLogic.mk_eq
   480                     (perm1 $ pi1 $ (perm2 $ pi2 $ Free (x, T)),
   481                      perm2 $ (perm3 $ pi1 $ pi2) $ (perm1 $ pi1 $ Free (x, T)))
   482                   end)
   483                 (perm_names ~~ Ts ~~ perm_indnames))))
   484           (fn _ => EVERY [indtac induction perm_indnames 1,
   485              ALLGOALS (asm_full_simp_tac simps)]))
   486       in
   487         foldl (fn ((s, tvs), thy) => AxClass.prove_arity
   488             (s, replicate (length tvs) (cp_class :: classes), [cp_class])
   489             (ClassPackage.intro_classes_tac [] THEN ALLGOALS (resolve_tac thms)) thy)
   490           thy (full_new_type_names' ~~ tyvars)
   491       end;
   492 
   493     val (perm_thmss,thy3) = thy2 |>
   494       fold (fn name1 => fold (composition_instance name1) atoms) atoms |>
   495       curry (Library.foldr (fn ((i, (tyname, args, _)), thy) =>
   496         AxClass.prove_arity (tyname, replicate (length args) classes, classes)
   497         (ClassPackage.intro_classes_tac [] THEN REPEAT (EVERY
   498            [resolve_tac perm_empty_thms 1,
   499             resolve_tac perm_append_thms 1,
   500             resolve_tac perm_eq_thms 1, assume_tac 1])) thy))
   501         (List.take (descr, length new_type_names)) |>
   502       PureThy.add_thmss
   503         [((space_implode "_" new_type_names ^ "_unfolded_perm_eq",
   504           unfolded_perm_eq_thms), [Simplifier.simp_add]),
   505          ((space_implode "_" new_type_names ^ "_perm_empty",
   506           perm_empty_thms), [Simplifier.simp_add]),
   507          ((space_implode "_" new_type_names ^ "_perm_append",
   508           perm_append_thms), [Simplifier.simp_add]),
   509          ((space_implode "_" new_type_names ^ "_perm_eq",
   510           perm_eq_thms), [Simplifier.simp_add])];
   511 
   512     (**** Define representing sets ****)
   513 
   514     val _ = warning "representing sets";
   515 
   516     val rep_set_names = DatatypeProp.indexify_names
   517       (map (fn (i, _) => name_of_typ (nth_dtyp i) ^ "_set") descr);
   518     val big_rep_name =
   519       space_implode "_" (DatatypeProp.indexify_names (List.mapPartial
   520         (fn (i, ("Nominal.noption", _, _)) => NONE
   521           | (i, _) => SOME (name_of_typ (nth_dtyp i))) descr)) ^ "_set";
   522     val _ = warning ("big_rep_name: " ^ big_rep_name);
   523 
   524     fun strip_option (dtf as DtType ("fun", [dt, DtRec i])) =
   525           (case AList.lookup op = descr i of
   526              SOME ("Nominal.noption", _, [(_, [dt']), _]) =>
   527                apfst (cons dt) (strip_option dt')
   528            | _ => ([], dtf))
   529       | strip_option (DtType ("fun", [dt, DtType ("Nominal.noption", [dt'])])) =
   530           apfst (cons dt) (strip_option dt')
   531       | strip_option dt = ([], dt);
   532 
   533     val dt_atomTs = distinct op = (map (typ_of_dtyp descr sorts')
   534       (List.concat (map (fn (_, (_, _, cs)) => List.concat
   535         (map (List.concat o map (fst o strip_option) o snd) cs)) descr)));
   536 
   537     fun make_intr s T (cname, cargs) =
   538       let
   539         fun mk_prem (dt, (j, j', prems, ts)) =
   540           let
   541             val (dts, dt') = strip_option dt;
   542             val (dts', dt'') = strip_dtyp dt';
   543             val Ts = map (typ_of_dtyp descr sorts') dts;
   544             val Us = map (typ_of_dtyp descr sorts') dts';
   545             val T = typ_of_dtyp descr sorts' dt'';
   546             val free = mk_Free "x" (Us ---> T) j;
   547             val free' = app_bnds free (length Us);
   548             fun mk_abs_fun (T, (i, t)) =
   549               let val U = fastype_of t
   550               in (i + 1, Const ("Nominal.abs_fun", [T, U, T] --->
   551                 Type ("Nominal.noption", [U])) $ mk_Free "y" T i $ t)
   552               end
   553           in (j + 1, j' + length Ts,
   554             case dt'' of
   555                 DtRec k => list_all (map (pair "x") Us,
   556                   HOLogic.mk_Trueprop (Free (List.nth (rep_set_names, k),
   557                     T --> HOLogic.boolT) $ free')) :: prems
   558               | _ => prems,
   559             snd (foldr mk_abs_fun (j', free) Ts) :: ts)
   560           end;
   561 
   562         val (_, _, prems, ts) = foldr mk_prem (1, 1, [], []) cargs;
   563         val concl = HOLogic.mk_Trueprop (Free (s, T --> HOLogic.boolT) $
   564           list_comb (Const (cname, map fastype_of ts ---> T), ts))
   565       in Logic.list_implies (prems, concl)
   566       end;
   567 
   568     val (intr_ts, (rep_set_names', recTs')) =
   569       apfst List.concat (apsnd ListPair.unzip (ListPair.unzip (List.mapPartial
   570         (fn ((_, ("Nominal.noption", _, _)), _) => NONE
   571           | ((i, (_, _, constrs)), rep_set_name) =>
   572               let val T = nth_dtyp i
   573               in SOME (map (make_intr rep_set_name T) constrs,
   574                 (rep_set_name, T))
   575               end)
   576                 (descr ~~ rep_set_names))));
   577     val rep_set_names'' = map (Sign.full_name thy3) rep_set_names';
   578 
   579     val ({raw_induct = rep_induct, intrs = rep_intrs, ...}, thy4) =
   580       setmp InductivePackage.quiet_mode false
   581         (TheoryTarget.init NONE #>
   582          InductivePackage.add_inductive_i false big_rep_name false true false
   583            (map (fn (s, T) => (s, SOME (T --> HOLogic.boolT), NoSyn))
   584               (rep_set_names' ~~ recTs'))
   585            [] (map (fn x => (("", []), x)) intr_ts) [] #>
   586          apsnd (ProofContext.theory_of o LocalTheory.exit)) thy3;
   587 
   588     (**** Prove that representing set is closed under permutation ****)
   589 
   590     val _ = warning "proving closure under permutation...";
   591 
   592     val perm_indnames' = List.mapPartial
   593       (fn (x, (_, ("Nominal.noption", _, _))) => NONE | (x, _) => SOME x)
   594       (perm_indnames ~~ descr);
   595 
   596     fun mk_perm_closed name = map (fn th => standard (th RS mp))
   597       (List.take (split_conj_thm (Goal.prove_global thy4 [] []
   598         (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj (map
   599            (fn ((s, T), x) =>
   600               let
   601                 val T = map_type_tfree
   602                   (fn (s, cs) => TFree (s, cs union cp_classes)) T;
   603                 val S = Const (s, T --> HOLogic.boolT);
   604                 val permT = mk_permT (Type (name, []))
   605               in HOLogic.mk_imp (S $ Free (x, T),
   606                 S $ (Const ("Nominal.perm", permT --> T --> T) $
   607                   Free ("pi", permT) $ Free (x, T)))
   608               end) (rep_set_names'' ~~ recTs' ~~ perm_indnames'))))
   609         (fn _ => EVERY (* CU: added perm_fun_def in the final tactic in order to deal with funs *)
   610            [indtac rep_induct [] 1,
   611             ALLGOALS (simp_tac (simpset_of thy4 addsimps
   612               (symmetric perm_fun_def :: PureThy.get_thms thy4 (Name ("abs_perm"))))),
   613             ALLGOALS (resolve_tac rep_intrs
   614                THEN_ALL_NEW (asm_full_simp_tac (simpset_of thy4 addsimps [perm_fun_def])))])),
   615         length new_type_names));
   616 
   617     (* FIXME: theorems are stored in database for testing only *)
   618     val perm_closed_thmss = map mk_perm_closed atoms;
   619     val (_, thy5) = PureThy.add_thmss [(("perm_closed", List.concat perm_closed_thmss), [])] thy4;
   620 
   621     (**** typedef ****)
   622 
   623     val _ = warning "defining type...";
   624 
   625     val (typedefs, thy6) =
   626       thy5
   627       |> fold_map (fn ((((name, mx), tvs), (cname, U)), name') => fn thy =>
   628         setmp TypedefPackage.quiet_mode true
   629           (TypedefPackage.add_typedef_i false (SOME name') (name, tvs, mx)
   630             (Const ("Collect", (U --> HOLogic.boolT) --> HOLogic.mk_setT U) $
   631                Const (cname, U --> HOLogic.boolT)) NONE
   632             (rtac exI 1 THEN rtac CollectI 1 THEN
   633               QUIET_BREADTH_FIRST (has_fewer_prems 1)
   634               (resolve_tac rep_intrs 1))) thy |> (fn ((_, r), thy) =>
   635         let
   636           val permT = mk_permT (TFree (Name.variant tvs "'a", HOLogic.typeS));
   637           val pi = Free ("pi", permT);
   638           val tvs' = map (fn s => TFree (s, the (AList.lookup op = sorts' s))) tvs;
   639           val T = Type (Sign.intern_type thy name, tvs');
   640         in apfst (pair r o hd)
   641           (PureThy.add_defs_unchecked_i true [(("prm_" ^ name ^ "_def", Logic.mk_equals
   642             (Const ("Nominal.perm", permT --> T --> T) $ pi $ Free ("x", T),
   643              Const (Sign.intern_const thy ("Abs_" ^ name), U --> T) $
   644                (Const ("Nominal.perm", permT --> U --> U) $ pi $
   645                  (Const (Sign.intern_const thy ("Rep_" ^ name), T --> U) $
   646                    Free ("x", T))))), [])] thy)
   647         end))
   648           (types_syntax ~~ tyvars ~~
   649             List.take (rep_set_names'' ~~ recTs', length new_type_names) ~~
   650             new_type_names);
   651 
   652     val perm_defs = map snd typedefs;
   653     val Abs_inverse_thms = map (collect_simp o #Abs_inverse o fst) typedefs;
   654     val Rep_inverse_thms = map (#Rep_inverse o fst) typedefs;
   655     val Rep_thms = map (collect_simp o #Rep o fst) typedefs;
   656 
   657     val big_name = space_implode "_" new_type_names;
   658 
   659 
   660     (** prove that new types are in class pt_<name> **)
   661 
   662     val _ = warning "prove that new types are in class pt_<name> ...";
   663 
   664     fun pt_instance ((class, atom), perm_closed_thms) =
   665       fold (fn ((((((Abs_inverse, Rep_inverse), Rep),
   666         perm_def), name), tvs), perm_closed) => fn thy =>
   667           AxClass.prove_arity
   668             (Sign.intern_type thy name,
   669               replicate (length tvs) (classes @ cp_classes), [class])
   670             (EVERY [ClassPackage.intro_classes_tac [],
   671               rewrite_goals_tac [perm_def],
   672               asm_full_simp_tac (simpset_of thy addsimps [Rep_inverse]) 1,
   673               asm_full_simp_tac (simpset_of thy addsimps
   674                 [Rep RS perm_closed RS Abs_inverse]) 1,
   675               asm_full_simp_tac (HOL_basic_ss addsimps [PureThy.get_thm thy
   676                 (Name ("pt_" ^ Sign.base_name atom ^ "3"))]) 1]) thy)
   677         (Abs_inverse_thms ~~ Rep_inverse_thms ~~ Rep_thms ~~ perm_defs ~~
   678            new_type_names ~~ tyvars ~~ perm_closed_thms);
   679 
   680 
   681     (** prove that new types are in class cp_<name1>_<name2> **)
   682 
   683     val _ = warning "prove that new types are in class cp_<name1>_<name2> ...";
   684 
   685     fun cp_instance (atom1, perm_closed_thms1) (atom2, perm_closed_thms2) thy =
   686       let
   687         val name = "cp_" ^ Sign.base_name atom1 ^ "_" ^ Sign.base_name atom2;
   688         val class = Sign.intern_class thy name;
   689         val cp1' = PureThy.get_thm thy (Name (name ^ "_inst")) RS cp1
   690       in fold (fn ((((((Abs_inverse, Rep),
   691         perm_def), name), tvs), perm_closed1), perm_closed2) => fn thy =>
   692           AxClass.prove_arity
   693             (Sign.intern_type thy name,
   694               replicate (length tvs) (classes @ cp_classes), [class])
   695             (EVERY [ClassPackage.intro_classes_tac [],
   696               rewrite_goals_tac [perm_def],
   697               asm_full_simp_tac (simpset_of thy addsimps
   698                 ((Rep RS perm_closed1 RS Abs_inverse) ::
   699                  (if atom1 = atom2 then []
   700                   else [Rep RS perm_closed2 RS Abs_inverse]))) 1,
   701               cong_tac 1,
   702               rtac refl 1,
   703               rtac cp1' 1]) thy)
   704         (Abs_inverse_thms ~~ Rep_thms ~~ perm_defs ~~ new_type_names ~~
   705            tyvars ~~ perm_closed_thms1 ~~ perm_closed_thms2) thy
   706       end;
   707 
   708     val thy7 = fold (fn x => fn thy => thy |>
   709       pt_instance x |>
   710       fold (cp_instance (apfst snd x)) (atoms ~~ perm_closed_thmss))
   711         (classes ~~ atoms ~~ perm_closed_thmss) thy6;
   712 
   713     (**** constructors ****)
   714 
   715     fun mk_abs_fun (x, t) =
   716       let
   717         val T = fastype_of x;
   718         val U = fastype_of t
   719       in
   720         Const ("Nominal.abs_fun", T --> U --> T -->
   721           Type ("Nominal.noption", [U])) $ x $ t
   722       end;
   723 
   724     val (ty_idxs, _) = foldl
   725       (fn ((i, ("Nominal.noption", _, _)), p) => p
   726         | ((i, _), (ty_idxs, j)) => (ty_idxs @ [(i, j)], j + 1)) ([], 0) descr;
   727 
   728     fun reindex (DtType (s, dts)) = DtType (s, map reindex dts)
   729       | reindex (DtRec i) = DtRec (the (AList.lookup op = ty_idxs i))
   730       | reindex dt = dt;
   731 
   732     fun strip_suffix i s = implode (List.take (explode s, size s - i));
   733 
   734     (** strips the "_Rep" in type names *)
   735     fun strip_nth_name i s =
   736       let val xs = NameSpace.unpack s;
   737       in NameSpace.pack (Library.nth_map (length xs - i) (strip_suffix 4) xs) end;
   738 
   739     val (descr'', ndescr) = ListPair.unzip (List.mapPartial
   740       (fn (i, ("Nominal.noption", _, _)) => NONE
   741         | (i, (s, dts, constrs)) =>
   742              let
   743                val SOME index = AList.lookup op = ty_idxs i;
   744                val (constrs1, constrs2) = ListPair.unzip
   745                  (map (fn (cname, cargs) => apfst (pair (strip_nth_name 2 (strip_nth_name 1 cname)))
   746                    (foldl_map (fn (dts, dt) =>
   747                      let val (dts', dt') = strip_option dt
   748                      in (dts @ dts' @ [reindex dt'], (length dts, length dts')) end)
   749                        ([], cargs))) constrs)
   750              in SOME ((index, (strip_nth_name 1 s,  map reindex dts, constrs1)),
   751                (index, constrs2))
   752              end) descr);
   753 
   754     val (descr1, descr2) = chop (length new_type_names) descr'';
   755     val descr' = [descr1, descr2];
   756 
   757     fun partition_cargs idxs xs = map (fn (i, j) =>
   758       (List.take (List.drop (xs, i), j), List.nth (xs, i + j))) idxs;
   759 
   760     val pdescr = map (fn ((i, (s, dts, constrs)), (_, idxss)) => (i, (s, dts,
   761       map (fn ((cname, cargs), idxs) => (cname, partition_cargs idxs cargs))
   762         (constrs ~~ idxss)))) (descr'' ~~ ndescr);
   763 
   764     fun nth_dtyp' i = typ_of_dtyp descr'' sorts' (DtRec i);
   765 
   766     val rep_names = map (fn s =>
   767       Sign.intern_const thy7 ("Rep_" ^ s)) new_type_names;
   768     val abs_names = map (fn s =>
   769       Sign.intern_const thy7 ("Abs_" ^ s)) new_type_names;
   770 
   771     val recTs = get_rec_types descr'' sorts';
   772     val newTs' = Library.take (length new_type_names, recTs');
   773     val newTs = Library.take (length new_type_names, recTs);
   774 
   775     val full_new_type_names = map (Sign.full_name (sign_of thy)) new_type_names;
   776 
   777     fun make_constr_def tname T T' ((thy, defs, eqns),
   778         (((cname_rep, _), (cname, cargs)), (cname', mx))) =
   779       let
   780         fun constr_arg ((dts, dt), (j, l_args, r_args)) =
   781           let
   782             val xs = map (fn (dt, i) => mk_Free "x" (typ_of_dtyp descr'' sorts' dt) i)
   783               (dts ~~ (j upto j + length dts - 1))
   784             val x = mk_Free "x" (typ_of_dtyp descr'' sorts' dt) (j + length dts)
   785           in
   786             (j + length dts + 1,
   787              xs @ x :: l_args,
   788              foldr mk_abs_fun
   789                (case dt of
   790                   DtRec k => if k < length new_type_names then
   791                       Const (List.nth (rep_names, k), typ_of_dtyp descr'' sorts' dt -->
   792                         typ_of_dtyp descr sorts' dt) $ x
   793                     else error "nested recursion not (yet) supported"
   794                 | _ => x) xs :: r_args)
   795           end
   796 
   797         val (_, l_args, r_args) = foldr constr_arg (1, [], []) cargs;
   798         val abs_name = Sign.intern_const (Theory.sign_of thy) ("Abs_" ^ tname);
   799         val rep_name = Sign.intern_const (Theory.sign_of thy) ("Rep_" ^ tname);
   800         val constrT = map fastype_of l_args ---> T;
   801         val lhs = list_comb (Const (cname, constrT), l_args);
   802         val rhs = list_comb (Const (cname_rep, map fastype_of r_args ---> T'), r_args);
   803         val def = Logic.mk_equals (lhs, Const (abs_name, T' --> T) $ rhs);
   804         val eqn = HOLogic.mk_Trueprop (HOLogic.mk_eq
   805           (Const (rep_name, T --> T') $ lhs, rhs));
   806         val def_name = (Sign.base_name cname) ^ "_def";
   807         val ([def_thm], thy') = thy |>
   808           Theory.add_consts_i [(cname', constrT, mx)] |>
   809           (PureThy.add_defs_i false o map Thm.no_attributes) [(def_name, def)]
   810       in (thy', defs @ [def_thm], eqns @ [eqn]) end;
   811 
   812     fun dt_constr_defs ((thy, defs, eqns, dist_lemmas), ((((((_, (_, _, constrs)),
   813         (_, (_, _, constrs'))), tname), T), T'), constr_syntax)) =
   814       let
   815         val rep_const = cterm_of thy
   816           (Const (Sign.intern_const thy ("Rep_" ^ tname), T --> T'));
   817         val dist = standard (cterm_instantiate [(cterm_of thy distinct_f, rep_const)] distinct_lemma);
   818         val (thy', defs', eqns') = Library.foldl (make_constr_def tname T T')
   819           ((Theory.add_path tname thy, defs, []), constrs ~~ constrs' ~~ constr_syntax)
   820       in
   821         (parent_path flat_names thy', defs', eqns @ [eqns'], dist_lemmas @ [dist])
   822       end;
   823 
   824     val (thy8, constr_defs, constr_rep_eqns, dist_lemmas) = Library.foldl dt_constr_defs
   825       ((thy7, [], [], []), List.take (descr, length new_type_names) ~~
   826         List.take (pdescr, length new_type_names) ~~
   827         new_type_names ~~ newTs ~~ newTs' ~~ constr_syntax);
   828 
   829     val abs_inject_thms = map (collect_simp o #Abs_inject o fst) typedefs
   830     val rep_inject_thms = map (#Rep_inject o fst) typedefs
   831 
   832     (* prove theorem  Rep_i (Constr_j ...) = Constr'_j ...  *)
   833 
   834     fun prove_constr_rep_thm eqn =
   835       let
   836         val inj_thms = map (fn r => r RS iffD1) abs_inject_thms;
   837         val rewrites = constr_defs @ map mk_meta_eq Rep_inverse_thms
   838       in Goal.prove_global thy8 [] [] eqn (fn _ => EVERY
   839         [resolve_tac inj_thms 1,
   840          rewrite_goals_tac rewrites,
   841          rtac refl 3,
   842          resolve_tac rep_intrs 2,
   843          REPEAT (resolve_tac Rep_thms 1)])
   844       end;
   845 
   846     val constr_rep_thmss = map (map prove_constr_rep_thm) constr_rep_eqns;
   847 
   848     (* prove theorem  pi \<bullet> Rep_i x = Rep_i (pi \<bullet> x) *)
   849 
   850     fun prove_perm_rep_perm (atom, perm_closed_thms) = map (fn th =>
   851       let
   852         val _ $ (_ $ (Rep $ x)) = Logic.unvarify (prop_of th);
   853         val Type ("fun", [T, U]) = fastype_of Rep;
   854         val permT = mk_permT (Type (atom, []));
   855         val pi = Free ("pi", permT);
   856       in
   857         Goal.prove_global thy8 [] [] (HOLogic.mk_Trueprop (HOLogic.mk_eq
   858             (Const ("Nominal.perm", permT --> U --> U) $ pi $ (Rep $ x),
   859              Rep $ (Const ("Nominal.perm", permT --> T --> T) $ pi $ x))))
   860           (fn _ => simp_tac (HOL_basic_ss addsimps (perm_defs @ Abs_inverse_thms @
   861             perm_closed_thms @ Rep_thms)) 1)
   862       end) Rep_thms;
   863 
   864     val perm_rep_perm_thms = List.concat (map prove_perm_rep_perm
   865       (atoms ~~ perm_closed_thmss));
   866 
   867     (* prove distinctness theorems *)
   868 
   869     val distinct_props = setmp DatatypeProp.dtK 1000
   870       (DatatypeProp.make_distincts new_type_names descr' sorts') thy8;
   871 
   872     val dist_rewrites = map (fn (rep_thms, dist_lemma) =>
   873       dist_lemma::(rep_thms @ [In0_eq, In1_eq, In0_not_In1, In1_not_In0]))
   874         (constr_rep_thmss ~~ dist_lemmas);
   875 
   876     fun prove_distinct_thms (_, []) = []
   877       | prove_distinct_thms (p as (rep_thms, dist_lemma), t::ts) =
   878           let
   879             val dist_thm = Goal.prove_global thy8 [] [] t (fn _ =>
   880               simp_tac (simpset_of thy8 addsimps (dist_lemma :: rep_thms)) 1)
   881           in dist_thm::(standard (dist_thm RS not_sym))::
   882             (prove_distinct_thms (p, ts))
   883           end;
   884 
   885     val distinct_thms = map prove_distinct_thms
   886       (constr_rep_thmss ~~ dist_lemmas ~~ distinct_props);
   887 
   888     (** prove equations for permutation functions **)
   889 
   890     val abs_perm = PureThy.get_thms thy8 (Name "abs_perm"); (* FIXME *)
   891 
   892     val perm_simps' = map (fn (((i, (_, _, constrs)), tname), constr_rep_thms) =>
   893       let val T = nth_dtyp' i
   894       in List.concat (map (fn (atom, perm_closed_thms) =>
   895           map (fn ((cname, dts), constr_rep_thm) =>
   896         let
   897           val cname = Sign.intern_const thy8
   898             (NameSpace.append tname (Sign.base_name cname));
   899           val permT = mk_permT (Type (atom, []));
   900           val pi = Free ("pi", permT);
   901 
   902           fun perm t =
   903             let val T = fastype_of t
   904             in Const ("Nominal.perm", permT --> T --> T) $ pi $ t end;
   905 
   906           fun constr_arg ((dts, dt), (j, l_args, r_args)) =
   907             let
   908               val Ts = map (typ_of_dtyp descr'' sorts') dts;
   909               val xs = map (fn (T, i) => mk_Free "x" T i)
   910                 (Ts ~~ (j upto j + length dts - 1))
   911               val x = mk_Free "x" (typ_of_dtyp descr'' sorts' dt) (j + length dts)
   912             in
   913               (j + length dts + 1,
   914                xs @ x :: l_args,
   915                map perm (xs @ [x]) @ r_args)
   916             end
   917 
   918           val (_, l_args, r_args) = foldr constr_arg (1, [], []) dts;
   919           val c = Const (cname, map fastype_of l_args ---> T)
   920         in
   921           Goal.prove_global thy8 [] []
   922             (HOLogic.mk_Trueprop (HOLogic.mk_eq
   923               (perm (list_comb (c, l_args)), list_comb (c, r_args))))
   924             (fn _ => EVERY
   925               [simp_tac (simpset_of thy8 addsimps (constr_rep_thm :: perm_defs)) 1,
   926                simp_tac (HOL_basic_ss addsimps (Rep_thms @ Abs_inverse_thms @
   927                  constr_defs @ perm_closed_thms)) 1,
   928                TRY (simp_tac (HOL_basic_ss addsimps
   929                  (symmetric perm_fun_def :: abs_perm)) 1),
   930                TRY (simp_tac (HOL_basic_ss addsimps
   931                  (perm_fun_def :: perm_defs @ Rep_thms @ Abs_inverse_thms @
   932                     perm_closed_thms)) 1)])
   933         end) (constrs ~~ constr_rep_thms)) (atoms ~~ perm_closed_thmss))
   934       end) (List.take (pdescr, length new_type_names) ~~ new_type_names ~~ constr_rep_thmss);
   935 
   936     (** prove injectivity of constructors **)
   937 
   938     val rep_inject_thms' = map (fn th => th RS sym) rep_inject_thms;
   939     val alpha = PureThy.get_thms thy8 (Name "alpha");
   940     val abs_fresh = PureThy.get_thms thy8 (Name "abs_fresh");
   941 
   942     val inject_thms = map (fn (((i, (_, _, constrs)), tname), constr_rep_thms) =>
   943       let val T = nth_dtyp' i
   944       in List.mapPartial (fn ((cname, dts), constr_rep_thm) =>
   945         if null dts then NONE else SOME
   946         let
   947           val cname = Sign.intern_const thy8
   948             (NameSpace.append tname (Sign.base_name cname));
   949 
   950           fun make_inj ((dts, dt), (j, args1, args2, eqs)) =
   951             let
   952               val Ts_idx = map (typ_of_dtyp descr'' sorts') dts ~~ (j upto j + length dts - 1);
   953               val xs = map (fn (T, i) => mk_Free "x" T i) Ts_idx;
   954               val ys = map (fn (T, i) => mk_Free "y" T i) Ts_idx;
   955               val x = mk_Free "x" (typ_of_dtyp descr'' sorts' dt) (j + length dts);
   956               val y = mk_Free "y" (typ_of_dtyp descr'' sorts' dt) (j + length dts)
   957             in
   958               (j + length dts + 1,
   959                xs @ (x :: args1), ys @ (y :: args2),
   960                HOLogic.mk_eq
   961                  (foldr mk_abs_fun x xs, foldr mk_abs_fun y ys) :: eqs)
   962             end;
   963 
   964           val (_, args1, args2, eqs) = foldr make_inj (1, [], [], []) dts;
   965           val Ts = map fastype_of args1;
   966           val c = Const (cname, Ts ---> T)
   967         in
   968           Goal.prove_global thy8 [] [] (HOLogic.mk_Trueprop (HOLogic.mk_eq
   969               (HOLogic.mk_eq (list_comb (c, args1), list_comb (c, args2)),
   970                foldr1 HOLogic.mk_conj eqs)))
   971             (fn _ => EVERY
   972                [asm_full_simp_tac (simpset_of thy8 addsimps (constr_rep_thm ::
   973                   rep_inject_thms')) 1,
   974                 TRY (asm_full_simp_tac (HOL_basic_ss addsimps (fresh_def :: supp_def ::
   975                   alpha @ abs_perm @ abs_fresh @ rep_inject_thms @
   976                   perm_rep_perm_thms)) 1),
   977                 TRY (asm_full_simp_tac (HOL_basic_ss addsimps (perm_fun_def ::
   978                   expand_fun_eq :: rep_inject_thms @ perm_rep_perm_thms)) 1)])
   979         end) (constrs ~~ constr_rep_thms)
   980       end) (List.take (pdescr, length new_type_names) ~~ new_type_names ~~ constr_rep_thmss);
   981 
   982     (** equations for support and freshness **)
   983 
   984     val (supp_thms, fresh_thms) = ListPair.unzip (map ListPair.unzip
   985       (map (fn ((((i, (_, _, constrs)), tname), inject_thms'), perm_thms') =>
   986       let val T = nth_dtyp' i
   987       in List.concat (map (fn (cname, dts) => map (fn atom =>
   988         let
   989           val cname = Sign.intern_const thy8
   990             (NameSpace.append tname (Sign.base_name cname));
   991           val atomT = Type (atom, []);
   992 
   993           fun process_constr ((dts, dt), (j, args1, args2)) =
   994             let
   995               val Ts_idx = map (typ_of_dtyp descr'' sorts') dts ~~ (j upto j + length dts - 1);
   996               val xs = map (fn (T, i) => mk_Free "x" T i) Ts_idx;
   997               val x = mk_Free "x" (typ_of_dtyp descr'' sorts' dt) (j + length dts)
   998             in
   999               (j + length dts + 1,
  1000                xs @ (x :: args1), foldr mk_abs_fun x xs :: args2)
  1001             end;
  1002 
  1003           val (_, args1, args2) = foldr process_constr (1, [], []) dts;
  1004           val Ts = map fastype_of args1;
  1005           val c = list_comb (Const (cname, Ts ---> T), args1);
  1006           fun supp t =
  1007             Const ("Nominal.supp", fastype_of t --> HOLogic.mk_setT atomT) $ t;
  1008           fun fresh t =
  1009             Const ("Nominal.fresh", atomT --> fastype_of t --> HOLogic.boolT) $
  1010               Free ("a", atomT) $ t;
  1011           val supp_thm = Goal.prove_global thy8 [] []
  1012               (HOLogic.mk_Trueprop (HOLogic.mk_eq
  1013                 (supp c,
  1014                  if null dts then Const ("{}", HOLogic.mk_setT atomT)
  1015                  else foldr1 (HOLogic.mk_binop "op Un") (map supp args2))))
  1016             (fn _ =>
  1017               simp_tac (HOL_basic_ss addsimps (supp_def ::
  1018                  Un_assoc :: de_Morgan_conj :: Collect_disj_eq :: finite_Un ::
  1019                  symmetric empty_def :: Finites_emptyI :: simp_thms @
  1020                  abs_perm @ abs_fresh @ inject_thms' @ perm_thms')) 1)
  1021         in
  1022           (supp_thm,
  1023            Goal.prove_global thy8 [] [] (HOLogic.mk_Trueprop (HOLogic.mk_eq
  1024               (fresh c,
  1025                if null dts then HOLogic.true_const
  1026                else foldr1 HOLogic.mk_conj (map fresh args2))))
  1027              (fn _ =>
  1028                simp_tac (simpset_of thy8 addsimps [fresh_def, supp_thm]) 1))
  1029         end) atoms) constrs)
  1030       end) (List.take (pdescr, length new_type_names) ~~ new_type_names ~~ inject_thms ~~ perm_simps')));
  1031 
  1032     (**** weak induction theorem ****)
  1033 
  1034     fun mk_indrule_lemma ((prems, concls), (((i, _), T), U)) =
  1035       let
  1036         val Rep_t = Const (List.nth (rep_names, i), T --> U) $
  1037           mk_Free "x" T i;
  1038 
  1039         val Abs_t =  Const (List.nth (abs_names, i), U --> T)
  1040 
  1041       in (prems @ [HOLogic.imp $
  1042             (Const (List.nth (rep_set_names'', i), U --> HOLogic.boolT) $ Rep_t) $
  1043               (mk_Free "P" (T --> HOLogic.boolT) (i + 1) $ (Abs_t $ Rep_t))],
  1044           concls @ [mk_Free "P" (T --> HOLogic.boolT) (i + 1) $ mk_Free "x" T i])
  1045       end;
  1046 
  1047     val (indrule_lemma_prems, indrule_lemma_concls) =
  1048       Library.foldl mk_indrule_lemma (([], []), (descr'' ~~ recTs ~~ recTs'));
  1049 
  1050     val indrule_lemma = Goal.prove_global thy8 [] []
  1051       (Logic.mk_implies
  1052         (HOLogic.mk_Trueprop (mk_conj indrule_lemma_prems),
  1053          HOLogic.mk_Trueprop (mk_conj indrule_lemma_concls))) (fn _ => EVERY
  1054            [REPEAT (etac conjE 1),
  1055             REPEAT (EVERY
  1056               [TRY (rtac conjI 1), full_simp_tac (HOL_basic_ss addsimps Rep_inverse_thms) 1,
  1057                etac mp 1, resolve_tac Rep_thms 1])]);
  1058 
  1059     val Ps = map head_of (HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of indrule_lemma)));
  1060     val frees = if length Ps = 1 then [Free ("P", snd (dest_Var (hd Ps)))] else
  1061       map (Free o apfst fst o dest_Var) Ps;
  1062     val indrule_lemma' = cterm_instantiate
  1063       (map (cterm_of thy8) Ps ~~ map (cterm_of thy8) frees) indrule_lemma;
  1064 
  1065     val Abs_inverse_thms' = map (fn r => r RS subst) Abs_inverse_thms;
  1066 
  1067     val dt_induct_prop = DatatypeProp.make_ind descr' sorts';
  1068     val dt_induct = Goal.prove_global thy8 []
  1069       (Logic.strip_imp_prems dt_induct_prop) (Logic.strip_imp_concl dt_induct_prop)
  1070       (fn prems => EVERY
  1071         [rtac indrule_lemma' 1,
  1072          (DatatypeAux.indtac rep_induct THEN_ALL_NEW ObjectLogic.atomize_tac) 1,
  1073          EVERY (map (fn (prem, r) => (EVERY
  1074            [REPEAT (eresolve_tac Abs_inverse_thms' 1),
  1075             simp_tac (HOL_basic_ss addsimps [symmetric r]) 1,
  1076             DEPTH_SOLVE_1 (ares_tac [prem] 1 ORELSE etac allE 1)]))
  1077                 (prems ~~ constr_defs))]);
  1078 
  1079     val case_names_induct = mk_case_names_induct descr'';
  1080 
  1081     (**** prove that new datatypes have finite support ****)
  1082 
  1083     val _ = warning "proving finite support for the new datatype";
  1084 
  1085     val indnames = DatatypeProp.make_tnames recTs;
  1086 
  1087     val abs_supp = PureThy.get_thms thy8 (Name "abs_supp");
  1088     val supp_atm = PureThy.get_thms thy8 (Name "supp_atm");
  1089 
  1090     val finite_supp_thms = map (fn atom =>
  1091       let val atomT = Type (atom, [])
  1092       in map standard (List.take
  1093         (split_conj_thm (Goal.prove_global thy8 [] [] (HOLogic.mk_Trueprop
  1094            (foldr1 HOLogic.mk_conj (map (fn (s, T) => HOLogic.mk_mem
  1095              (Const ("Nominal.supp", T --> HOLogic.mk_setT atomT) $ Free (s, T),
  1096               Const ("Finite_Set.Finites", HOLogic.mk_setT (HOLogic.mk_setT atomT))))
  1097                (indnames ~~ recTs))))
  1098            (fn _ => indtac dt_induct indnames 1 THEN
  1099             ALLGOALS (asm_full_simp_tac (simpset_of thy8 addsimps
  1100               (abs_supp @ supp_atm @
  1101                PureThy.get_thms thy8 (Name ("fs_" ^ Sign.base_name atom ^ "1")) @
  1102                List.concat supp_thms))))),
  1103          length new_type_names))
  1104       end) atoms;
  1105 
  1106     val simp_atts = replicate (length new_type_names) [Simplifier.simp_add];
  1107 
  1108     val (_, thy9) = thy8 |>
  1109       Theory.add_path big_name |>
  1110       PureThy.add_thms [(("induct_weak", dt_induct), [case_names_induct])] ||>>
  1111       PureThy.add_thmss [(("inducts_weak", projections dt_induct), [case_names_induct])] ||>
  1112       Theory.parent_path ||>>
  1113       DatatypeAux.store_thmss_atts "distinct" new_type_names simp_atts distinct_thms ||>>
  1114       DatatypeAux.store_thmss "constr_rep" new_type_names constr_rep_thmss ||>>
  1115       DatatypeAux.store_thmss_atts "perm" new_type_names simp_atts perm_simps' ||>>
  1116       DatatypeAux.store_thmss "inject" new_type_names inject_thms ||>>
  1117       DatatypeAux.store_thmss "supp" new_type_names supp_thms ||>>
  1118       DatatypeAux.store_thmss_atts "fresh" new_type_names simp_atts fresh_thms ||>
  1119       fold (fn (atom, ths) => fn thy =>
  1120         let val class = Sign.intern_class thy ("fs_" ^ Sign.base_name atom)
  1121         in fold (fn T => AxClass.prove_arity
  1122             (fst (dest_Type T),
  1123               replicate (length sorts) [class], [class])
  1124             (ClassPackage.intro_classes_tac [] THEN resolve_tac ths 1)) newTs thy
  1125         end) (atoms ~~ finite_supp_thms);
  1126 
  1127     (**** strong induction theorem ****)
  1128 
  1129     val pnames = if length descr'' = 1 then ["P"]
  1130       else map (fn i => "P" ^ string_of_int i) (1 upto length descr'');
  1131     val ind_sort = if null dt_atomTs then HOLogic.typeS
  1132       else Sign.certify_sort thy9 (map (fn T => Sign.intern_class thy9 ("fs_" ^
  1133         Sign.base_name (fst (dest_Type T)))) dt_atomTs);
  1134     val fsT = TFree ("'n", ind_sort);
  1135     val fsT' = TFree ("'n", HOLogic.typeS);
  1136 
  1137     val fresh_fs = map (fn (s, T) => (T, Free (s, fsT' --> HOLogic.mk_setT T)))
  1138       (DatatypeProp.indexify_names (replicate (length dt_atomTs) "f") ~~ dt_atomTs);
  1139 
  1140     fun make_pred fsT i T =
  1141       Free (List.nth (pnames, i), fsT --> T --> HOLogic.boolT);
  1142 
  1143     fun mk_fresh1 xs [] = []
  1144       | mk_fresh1 xs ((y as (_, T)) :: ys) = map (fn x => HOLogic.mk_Trueprop
  1145             (HOLogic.mk_not (HOLogic.mk_eq (Free y, Free x))))
  1146               (filter (fn (_, U) => T = U) (rev xs)) @
  1147           mk_fresh1 (y :: xs) ys;
  1148 
  1149     fun mk_fresh2 xss [] = []
  1150       | mk_fresh2 xss ((p as (ys, _)) :: yss) = List.concat (map (fn y as (_, T) =>
  1151             map (fn (_, x as (_, U)) => HOLogic.mk_Trueprop
  1152               (Const ("Nominal.fresh", T --> U --> HOLogic.boolT) $ Free y $ Free x))
  1153                 (rev xss @ yss)) ys) @
  1154           mk_fresh2 (p :: xss) yss;
  1155 
  1156     fun make_ind_prem fsT f k T ((cname, cargs), idxs) =
  1157       let
  1158         val recs = List.filter is_rec_type cargs;
  1159         val Ts = map (typ_of_dtyp descr'' sorts') cargs;
  1160         val recTs' = map (typ_of_dtyp descr'' sorts') recs;
  1161         val tnames = Name.variant_list pnames (DatatypeProp.make_tnames Ts);
  1162         val rec_tnames = map fst (List.filter (is_rec_type o snd) (tnames ~~ cargs));
  1163         val frees = tnames ~~ Ts;
  1164         val frees' = partition_cargs idxs frees;
  1165         val z = (Name.variant tnames "z", fsT);
  1166 
  1167         fun mk_prem ((dt, s), T) =
  1168           let
  1169             val (Us, U) = strip_type T;
  1170             val l = length Us
  1171           in list_all (z :: map (pair "x") Us, HOLogic.mk_Trueprop
  1172             (make_pred fsT (body_index dt) U $ Bound l $ app_bnds (Free (s, T)) l))
  1173           end;
  1174 
  1175         val prems = map mk_prem (recs ~~ rec_tnames ~~ recTs');
  1176         val prems' = map (fn p as (_, T) => HOLogic.mk_Trueprop
  1177             (f T (Free p) (Free z))) (List.concat (map fst frees')) @
  1178           mk_fresh1 [] (List.concat (map fst frees')) @
  1179           mk_fresh2 [] frees'
  1180 
  1181       in list_all_free (frees @ [z], Logic.list_implies (prems' @ prems,
  1182         HOLogic.mk_Trueprop (make_pred fsT k T $ Free z $
  1183           list_comb (Const (cname, Ts ---> T), map Free frees))))
  1184       end;
  1185 
  1186     val ind_prems = List.concat (map (fn (((i, (_, _, constrs)), (_, idxss)), T) =>
  1187       map (make_ind_prem fsT (fn T => fn t => fn u =>
  1188         Const ("Nominal.fresh", T --> fsT --> HOLogic.boolT) $ t $ u) i T)
  1189           (constrs ~~ idxss)) (descr'' ~~ ndescr ~~ recTs));
  1190     val tnames = DatatypeProp.make_tnames recTs;
  1191     val zs = Name.variant_list tnames (replicate (length descr'') "z");
  1192     val ind_concl = HOLogic.mk_Trueprop (foldr1 (HOLogic.mk_binop "op &")
  1193       (map (fn ((((i, _), T), tname), z) =>
  1194         make_pred fsT i T $ Free (z, fsT) $ Free (tname, T))
  1195         (descr'' ~~ recTs ~~ tnames ~~ zs)));
  1196     val induct = Logic.list_implies (ind_prems, ind_concl);
  1197 
  1198     val ind_prems' =
  1199       map (fn (_, f as Free (_, T)) => list_all_free ([("x", fsT')],
  1200         HOLogic.mk_Trueprop (HOLogic.mk_mem (f $ Free ("x", fsT'),
  1201           Const ("Finite_Set.Finites", HOLogic.mk_setT (body_type T)))))) fresh_fs @
  1202       List.concat (map (fn (((i, (_, _, constrs)), (_, idxss)), T) =>
  1203         map (make_ind_prem fsT' (fn T => fn t => fn u => HOLogic.Not $
  1204           HOLogic.mk_mem (t, the (AList.lookup op = fresh_fs T) $ u)) i T)
  1205             (constrs ~~ idxss)) (descr'' ~~ ndescr ~~ recTs));
  1206     val ind_concl' = HOLogic.mk_Trueprop (foldr1 (HOLogic.mk_binop "op &")
  1207       (map (fn ((((i, _), T), tname), z) =>
  1208         make_pred fsT' i T $ Free (z, fsT') $ Free (tname, T))
  1209         (descr'' ~~ recTs ~~ tnames ~~ zs)));
  1210     val induct' = Logic.list_implies (ind_prems', ind_concl');
  1211 
  1212     fun mk_perm Ts (t, u) =
  1213       let
  1214         val T = fastype_of1 (Ts, t);
  1215         val U = fastype_of1 (Ts, u)
  1216       in Const ("Nominal.perm", T --> U --> U) $ t $ u end;
  1217 
  1218     val aux_ind_vars =
  1219       (DatatypeProp.indexify_names (replicate (length dt_atomTs) "pi") ~~
  1220        map mk_permT dt_atomTs) @ [("z", fsT')];
  1221     val aux_ind_Ts = rev (map snd aux_ind_vars);
  1222     val aux_ind_concl = HOLogic.mk_Trueprop (foldr1 (HOLogic.mk_binop "op &")
  1223       (map (fn (((i, _), T), tname) =>
  1224         HOLogic.list_all (aux_ind_vars, make_pred fsT' i T $ Bound 0 $
  1225           foldr (mk_perm aux_ind_Ts) (Free (tname, T))
  1226             (map Bound (length dt_atomTs downto 1))))
  1227         (descr'' ~~ recTs ~~ tnames)));
  1228 
  1229     fun mk_ind_perm i k p l vs j =
  1230       let
  1231         val n = length vs;
  1232         val Ts = map snd vs;
  1233         val T = List.nth (Ts, i - j);
  1234         val pT = NominalAtoms.mk_permT T
  1235       in
  1236         Const ("List.list.Cons", HOLogic.mk_prodT (T, T) --> pT --> pT) $
  1237           (HOLogic.pair_const T T $ Bound (l - j) $ foldr (mk_perm Ts)
  1238             (Bound (i - j))
  1239             (map (mk_ind_perm i k p l vs) (j - 1 downto 0) @
  1240              map Bound (n - k - 1 downto n - k - p))) $
  1241           Const ("List.list.Nil", pT)
  1242       end;
  1243 
  1244     fun mk_fresh i i' j k p l is vs _ _ =
  1245       let
  1246         val n = length vs;
  1247         val Ts = map snd vs;
  1248         val T = List.nth (Ts, n - i - 1 - j);
  1249         val f = the (AList.lookup op = fresh_fs T);
  1250         val U = List.nth (Ts, n - i' - 1);
  1251         val S = HOLogic.mk_setT T;
  1252         val prms' = map Bound (n - k downto n - k - p + 1);
  1253         val prms = map (mk_ind_perm (n - i) k p (n - l) (("a", T) :: vs))
  1254             (j - 1 downto 0) @ prms';
  1255         val bs = rev (List.mapPartial
  1256           (fn ((_, T'), i) => if T = T' then SOME (Bound i) else NONE)
  1257           (List.take (vs, n - k - p - 1) ~~ (1 upto n - k - p - 1)));
  1258         val ts = map (fn i =>
  1259           Const ("Nominal.supp", List.nth (Ts, n - i - 1) --> S) $
  1260             foldr (mk_perm (T :: Ts)) (Bound (n - i)) prms') is
  1261       in
  1262         HOLogic.mk_Trueprop (Const ("Ex", (T --> HOLogic.boolT) --> HOLogic.boolT) $
  1263           Abs ("a", T, HOLogic.Not $
  1264             (Const ("op :", T --> S --> HOLogic.boolT) $ Bound 0 $
  1265               (foldr (fn (t, u) => Const ("insert", T --> S --> S) $ t $ u)
  1266                 (foldl (fn (t, u) => Const ("op Un", S --> S --> S) $ u $ t)
  1267                   (f $ Bound (n - k - p))
  1268                   (Const ("Nominal.supp", U --> S) $
  1269                      foldr (mk_perm (T :: Ts)) (Bound (n - i')) prms :: ts))
  1270                 (foldr (mk_perm (T :: Ts)) (Bound (n - i - j)) prms :: bs)))))
  1271       end;
  1272 
  1273     fun mk_fresh_constr is p vs _ concl =
  1274       let
  1275         val n = length vs;
  1276         val Ts = map snd vs;
  1277         val _ $ (_ $ _ $ t) = concl;
  1278         val c = head_of t;
  1279         val T = body_type (fastype_of c);
  1280         val k = foldr op + 0 (map (fn (_, i) => i + 1) is);
  1281         val ps = map Bound (n - k - 1 downto n - k - p);
  1282         val (_, _, ts, us) = foldl (fn ((_, i), (m, n, ts, us)) =>
  1283           (m - i - 1, n - i,
  1284            ts @ map Bound (n downto n - i + 1) @
  1285              [foldr (mk_perm Ts) (Bound (m - i))
  1286                 (map (mk_ind_perm m k p n vs) (i - 1 downto 0) @ ps)],
  1287            us @ map (fn j => foldr (mk_perm Ts) (Bound j) ps) (m downto m - i)))
  1288           (n - 1, n - k - p - 2, [], []) is
  1289       in
  1290         HOLogic.mk_Trueprop (HOLogic.eq_const T $ list_comb (c, ts) $ list_comb (c, us))
  1291       end;
  1292 
  1293     val abs_fun_finite_supp = PureThy.get_thm thy9 (Name "abs_fun_finite_supp");
  1294 
  1295     val at_finite_select = PureThy.get_thm thy9 (Name "at_finite_select");
  1296 
  1297     val induct_aux_lemmas = List.concat (map (fn Type (s, _) =>
  1298       [PureThy.get_thm thy9 (Name ("pt_" ^ Sign.base_name s ^ "_inst")),
  1299        PureThy.get_thm thy9 (Name ("fs_" ^ Sign.base_name s ^ "1")),
  1300        PureThy.get_thm thy9 (Name ("at_" ^ Sign.base_name s ^ "_inst"))]) dt_atomTs);
  1301 
  1302     val induct_aux_lemmas' = map (fn Type (s, _) =>
  1303       PureThy.get_thm thy9 (Name ("pt_" ^ Sign.base_name s ^ "2")) RS sym) dt_atomTs;
  1304 
  1305     val fresh_aux = PureThy.get_thms thy9 (Name "fresh_aux");
  1306 
  1307     (**********************************************************************
  1308       The subgoals occurring in the proof of induct_aux have the
  1309       following parameters:
  1310 
  1311         x_1 ... x_k p_1 ... p_m z b_1 ... b_n
  1312 
  1313       where
  1314 
  1315         x_i : constructor arguments (introduced by weak induction rule)
  1316         p_i : permutations (one for each atom type in the data type)
  1317         z   : freshness context
  1318         b_i : newly introduced names for binders (sufficiently fresh)
  1319     ***********************************************************************)
  1320 
  1321     val _ = warning "proving strong induction theorem ...";
  1322 
  1323     val induct_aux = Goal.prove_global thy9 [] ind_prems' ind_concl'
  1324       (fn prems => EVERY
  1325         ([mk_subgoal 1 (K (K (K aux_ind_concl))),
  1326           indtac dt_induct tnames 1] @
  1327          List.concat (map (fn ((_, (_, _, constrs)), (_, constrs')) =>
  1328            List.concat (map (fn ((cname, cargs), is) =>
  1329              [simp_tac (HOL_basic_ss addsimps List.concat perm_simps') 1,
  1330               REPEAT (rtac allI 1)] @
  1331              List.concat (map
  1332                (fn ((_, 0), _) => []
  1333                  | ((i, j), k) => List.concat (map (fn j' =>
  1334                      let
  1335                        val DtType (tname, _) = List.nth (cargs, i + j');
  1336                        val atom = Sign.base_name tname
  1337                      in
  1338                        [mk_subgoal 1 (mk_fresh i (i + j) j'
  1339                           (length cargs) (length dt_atomTs)
  1340                           (length cargs + length dt_atomTs + 1 + i - k)
  1341                           (List.mapPartial (fn (i', j) =>
  1342                              if i = i' then NONE else SOME (i' + j)) is)),
  1343                         rtac at_finite_select 1,
  1344                         rtac (PureThy.get_thm thy9 (Name ("at_" ^ atom ^ "_inst"))) 1,
  1345                         asm_full_simp_tac (simpset_of thy9 addsimps
  1346                           [PureThy.get_thm thy9 (Name ("fs_" ^ atom ^ "1"))]) 1,
  1347                         resolve_tac prems 1,
  1348                         etac exE 1,
  1349                         asm_full_simp_tac (simpset_of thy9 addsimps
  1350                           [symmetric fresh_def]) 1]
  1351                      end) (0 upto j - 1))) (is ~~ (0 upto length is - 1))) @
  1352              (if exists (not o equal 0 o snd) is then
  1353                 [mk_subgoal 1 (mk_fresh_constr is (length dt_atomTs)),
  1354                  asm_full_simp_tac (simpset_of thy9 addsimps
  1355                    (List.concat inject_thms @
  1356                     alpha @ abs_perm @ abs_fresh @ [abs_fun_finite_supp] @
  1357                     induct_aux_lemmas)) 1,
  1358                  dtac sym 1,
  1359                  asm_full_simp_tac (simpset_of thy9) 1,
  1360                  REPEAT (etac conjE 1)]
  1361               else
  1362                 []) @
  1363              [(resolve_tac prems THEN_ALL_NEW
  1364                 (atac ORELSE'
  1365                   SUBGOAL (fn (t, i) => case Logic.strip_assums_concl t of
  1366                       _ $ (Const ("Nominal.fresh", _) $ _ $ _) =>
  1367                         asm_simp_tac (simpset_of thy9 addsimps fresh_aux) i
  1368                     | _ =>
  1369                         asm_simp_tac (simpset_of thy9
  1370                         addsimps induct_aux_lemmas'
  1371                         addsimprocs [perm_simproc]) i))) 1])
  1372                (constrs ~~ constrs'))) (descr'' ~~ ndescr)) @
  1373          [REPEAT (eresolve_tac [conjE, allE_Nil] 1),
  1374           REPEAT (etac allE 1),
  1375           REPEAT (TRY (rtac conjI 1) THEN asm_full_simp_tac (simpset_of thy9) 1)]));
  1376 
  1377     val induct_aux' = Thm.instantiate ([],
  1378       map (fn (s, T) =>
  1379         let val pT = TVar (("'n", 0), HOLogic.typeS) --> T --> HOLogic.boolT
  1380         in (cterm_of thy9 (Var ((s, 0), pT)), cterm_of thy9 (Free (s, pT))) end)
  1381           (pnames ~~ recTs) @
  1382       map (fn (_, f) =>
  1383         let val f' = Logic.varify f
  1384         in (cterm_of thy9 f',
  1385           cterm_of thy9 (Const ("Nominal.supp", fastype_of f')))
  1386         end) fresh_fs) induct_aux;
  1387 
  1388     val induct = Goal.prove_global thy9 [] ind_prems ind_concl
  1389       (fn prems => EVERY
  1390          [rtac induct_aux' 1,
  1391           REPEAT (resolve_tac induct_aux_lemmas 1),
  1392           REPEAT ((resolve_tac prems THEN_ALL_NEW
  1393             (etac meta_spec ORELSE' full_simp_tac (HOL_basic_ss addsimps [fresh_def]))) 1)])
  1394 
  1395     val (_, thy10) = thy9 |>
  1396       Theory.add_path big_name |>
  1397       PureThy.add_thms [(("induct'", induct_aux), [])] ||>>
  1398       PureThy.add_thms [(("induct", induct), [case_names_induct])] ||>>
  1399       PureThy.add_thmss [(("inducts", projections induct), [case_names_induct])];
  1400 
  1401     (**** recursion combinator ****)
  1402 
  1403     val _ = warning "defining recursion combinator ...";
  1404 
  1405     val used = foldr add_typ_tfree_names [] recTs;
  1406 
  1407     val (rec_result_Ts', rec_fn_Ts') = DatatypeProp.make_primrec_Ts descr' sorts' used;
  1408 
  1409     val rec_sort = if null dt_atomTs then HOLogic.typeS else
  1410       let val names = map (Sign.base_name o fst o dest_Type) dt_atomTs
  1411       in Sign.certify_sort thy10 (map (Sign.intern_class thy10)
  1412         (map (fn s => "pt_" ^ s) names @
  1413          List.concat (map (fn s => List.mapPartial (fn s' =>
  1414            if s = s' then NONE
  1415            else SOME ("cp_" ^ s ^ "_" ^ s')) names) names)))
  1416       end;
  1417 
  1418     val rec_result_Ts = map (fn TFree (s, _) => TFree (s, rec_sort)) rec_result_Ts';
  1419     val rec_fn_Ts = map (typ_subst_atomic (rec_result_Ts' ~~ rec_result_Ts)) rec_fn_Ts';
  1420 
  1421     val rec_set_Ts = map (fn (T1, T2) =>
  1422       rec_fn_Ts @ [T1, T2] ---> HOLogic.boolT) (recTs ~~ rec_result_Ts);
  1423 
  1424     val big_rec_name = big_name ^ "_rec_set";
  1425     val rec_set_names' =
  1426       if length descr'' = 1 then [big_rec_name] else
  1427         map ((curry (op ^) (big_rec_name ^ "_")) o string_of_int)
  1428           (1 upto (length descr''));
  1429     val rec_set_names =  map (Sign.full_name (Theory.sign_of thy10)) rec_set_names';
  1430 
  1431     val rec_fns = map (uncurry (mk_Free "f"))
  1432       (rec_fn_Ts ~~ (1 upto (length rec_fn_Ts)));
  1433     val rec_sets' = map (fn c => list_comb (Free c, rec_fns))
  1434       (rec_set_names' ~~ rec_set_Ts);
  1435     val rec_sets = map (fn c => list_comb (Const c, rec_fns))
  1436       (rec_set_names ~~ rec_set_Ts);
  1437 
  1438     (* introduction rules for graph of recursion function *)
  1439 
  1440     val rec_preds = map (fn (a, T) =>
  1441       Free (a, T --> HOLogic.boolT)) (pnames ~~ rec_result_Ts);
  1442 
  1443     fun mk_fresh3 rs [] = []
  1444       | mk_fresh3 rs ((p as (ys, z)) :: yss) = List.concat (map (fn y as (_, T) =>
  1445             List.mapPartial (fn ((_, (_, x)), r as (_, U)) => if z = x then NONE
  1446               else SOME (HOLogic.mk_Trueprop
  1447                 (Const ("Nominal.fresh", T --> U --> HOLogic.boolT) $ Free y $ Free r)))
  1448                   rs) ys) @
  1449           mk_fresh3 rs yss;
  1450 
  1451     (* FIXME: avoid collisions with other variable names? *)
  1452     val rec_ctxt = Free ("z", fsT');
  1453 
  1454     fun make_rec_intr T p rec_set ((rec_intr_ts, rec_prems, rec_prems',
  1455           rec_eq_prems, l), ((cname, cargs), idxs)) =
  1456       let
  1457         val Ts = map (typ_of_dtyp descr'' sorts') cargs;
  1458         val frees = map (fn i => "x" ^ string_of_int i) (1 upto length Ts) ~~ Ts;
  1459         val frees' = partition_cargs idxs frees;
  1460         val binders = List.concat (map fst frees');
  1461         val atomTs = distinct op = (maps (map snd o fst) frees');
  1462         val recs = List.mapPartial
  1463           (fn ((_, DtRec i), p) => SOME (i, p) | _ => NONE)
  1464           (partition_cargs idxs cargs ~~ frees');
  1465         val frees'' = map (fn i => "y" ^ string_of_int i) (1 upto length recs) ~~
  1466           map (fn (i, _) => List.nth (rec_result_Ts, i)) recs;
  1467         val prems1 = map (fn ((i, (_, x)), y) => HOLogic.mk_Trueprop
  1468           (List.nth (rec_sets', i) $ Free x $ Free y)) (recs ~~ frees'');
  1469         val prems2 =
  1470           map (fn f => map (fn p as (_, T) => HOLogic.mk_Trueprop
  1471             (Const ("Nominal.fresh", T --> fastype_of f --> HOLogic.boolT) $
  1472               Free p $ f)) binders) rec_fns;
  1473         val prems3 = mk_fresh1 [] binders @ mk_fresh2 [] frees';
  1474         val prems4 = map (fn ((i, _), y) =>
  1475           HOLogic.mk_Trueprop (List.nth (rec_preds, i) $ Free y)) (recs ~~ frees'');
  1476         val prems5 = mk_fresh3 (recs ~~ frees'') frees';
  1477         val prems6 = maps (fn aT => map (fn y as (_, T) => HOLogic.mk_Trueprop
  1478           (HOLogic.mk_mem (Const ("Nominal.supp", T --> HOLogic.mk_setT aT) $ Free y,
  1479              Const ("Finite_Set.Finites", HOLogic.mk_setT (HOLogic.mk_setT aT)))))
  1480                frees'') atomTs;
  1481         val prems7 = map (fn x as (_, T) => HOLogic.mk_Trueprop
  1482           (Const ("Nominal.fresh", T --> fsT' --> HOLogic.boolT) $
  1483              Free x $ rec_ctxt)) binders;
  1484         val result = list_comb (List.nth (rec_fns, l), map Free (frees @ frees''));
  1485         val result_freshs = map (fn p as (_, T) =>
  1486           Const ("Nominal.fresh", T --> fastype_of result --> HOLogic.boolT) $
  1487             Free p $ result) binders;
  1488         val P = HOLogic.mk_Trueprop (p $ result)
  1489       in
  1490         (rec_intr_ts @ [Logic.list_implies (List.concat prems2 @ prems3 @ prems1,
  1491            HOLogic.mk_Trueprop (rec_set $
  1492              list_comb (Const (cname, Ts ---> T), map Free frees) $ result))],
  1493          rec_prems @ [list_all_free (frees @ frees'', Logic.list_implies (prems4, P))],
  1494          rec_prems' @ map (fn fr => list_all_free (frees @ frees'',
  1495            Logic.list_implies (List.nth (prems2, l) @ prems3 @ prems5 @ prems7 @ prems6 @ [P],
  1496              HOLogic.mk_Trueprop fr))) result_freshs,
  1497          rec_eq_prems @ [List.concat prems2 @ prems3],
  1498          l + 1)
  1499       end;
  1500 
  1501     val (rec_intr_ts, rec_prems, rec_prems', rec_eq_prems, _) =
  1502       Library.foldl (fn (x, ((((d, d'), T), p), rec_set)) =>
  1503         Library.foldl (make_rec_intr T p rec_set) (x, #3 (snd d) ~~ snd d'))
  1504           (([], [], [], [], 0), descr'' ~~ ndescr ~~ recTs ~~ rec_preds ~~ rec_sets');
  1505 
  1506     val ({intrs = rec_intrs, elims = rec_elims, raw_induct = rec_induct, ...}, thy11) =
  1507       thy10 |>
  1508       setmp InductivePackage.quiet_mode (!quiet_mode)
  1509         (TheoryTarget.init NONE #>
  1510          InductivePackage.add_inductive_i false big_rec_name false false false
  1511            (map (fn (s, T) => (s, SOME T, NoSyn)) (rec_set_names' ~~ rec_set_Ts))
  1512            (map (apsnd SOME o dest_Free) rec_fns)
  1513            (map (fn x => (("", []), x)) rec_intr_ts) [] #>
  1514          apsnd (ProofContext.theory_of o LocalTheory.exit)) ||>
  1515       PureThy.hide_thms true [NameSpace.append
  1516         (Sign.full_name thy10 big_rec_name) "induct"];
  1517 
  1518     (** equivariance **)
  1519 
  1520     val fresh_bij = PureThy.get_thms thy11 (Name "fresh_bij");
  1521     val perm_bij = PureThy.get_thms thy11 (Name "perm_bij");
  1522 
  1523     val (rec_equiv_thms, rec_equiv_thms') = ListPair.unzip (map (fn aT =>
  1524       let
  1525         val permT = mk_permT aT;
  1526         val pi = Free ("pi", permT);
  1527         val rec_fns_pi = map (curry (mk_perm []) pi o uncurry (mk_Free "f"))
  1528           (rec_fn_Ts ~~ (1 upto (length rec_fn_Ts)));
  1529         val rec_sets_pi = map (fn c => list_comb (Const c, rec_fns_pi))
  1530           (rec_set_names ~~ rec_set_Ts);
  1531         val ps = map (fn ((((T, U), R), R'), i) =>
  1532           let
  1533             val x = Free ("x" ^ string_of_int i, T);
  1534             val y = Free ("y" ^ string_of_int i, U)
  1535           in
  1536             (R $ x $ y, R' $ mk_perm [] (pi, x) $ mk_perm [] (pi, y))
  1537           end) (recTs ~~ rec_result_Ts ~~ rec_sets ~~ rec_sets_pi ~~ (1 upto length recTs));
  1538         val ths = map (fn th => standard (th RS mp)) (split_conj_thm
  1539           (Goal.prove_global thy11 [] []
  1540             (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj (map HOLogic.mk_imp ps)))
  1541             (fn _ => rtac rec_induct 1 THEN REPEAT
  1542                (NominalPermeq.perm_simp_tac (simpset_of thy11) 1 THEN
  1543                 (resolve_tac rec_intrs THEN_ALL_NEW
  1544                  asm_simp_tac (HOL_ss addsimps (fresh_bij @ perm_bij))) 1))))
  1545         val ths' = map (fn ((P, Q), th) =>
  1546           Goal.prove_global thy11 [] []
  1547             (Logic.mk_implies (HOLogic.mk_Trueprop Q, HOLogic.mk_Trueprop P))
  1548             (fn _ => dtac (Thm.instantiate ([],
  1549                  [(cterm_of thy11 (Var (("pi", 0), permT)),
  1550                    cterm_of thy11 (Const ("List.rev", permT --> permT) $ pi))]) th) 1 THEN
  1551                NominalPermeq.perm_simp_tac HOL_ss 1)) (ps ~~ ths)
  1552       in (ths, ths') end) dt_atomTs);
  1553 
  1554     (** finite support **)
  1555 
  1556     val rec_fin_supp_thms = map (fn aT =>
  1557       let
  1558         val name = Sign.base_name (fst (dest_Type aT));
  1559         val fs_name = PureThy.get_thm thy11 (Name ("fs_" ^ name ^ "1"));
  1560         val aset = HOLogic.mk_setT aT;
  1561         val finites = Const ("Finite_Set.Finites", HOLogic.mk_setT aset);
  1562         val fins = map (fn (f, T) => HOLogic.mk_Trueprop (HOLogic.mk_mem
  1563           (Const ("Nominal.supp", T --> aset) $ f, finites)))
  1564             (rec_fns ~~ rec_fn_Ts)
  1565       in
  1566         map (fn th => standard (th RS mp)) (split_conj_thm
  1567           (Goal.prove_global thy11 [] fins
  1568             (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
  1569               (map (fn (((T, U), R), i) =>
  1570                  let
  1571                    val x = Free ("x" ^ string_of_int i, T);
  1572                    val y = Free ("y" ^ string_of_int i, U)
  1573                  in
  1574                    HOLogic.mk_imp (R $ x $ y,
  1575                      HOLogic.mk_mem (Const ("Nominal.supp", U --> aset) $ y, finites))
  1576                  end) (recTs ~~ rec_result_Ts ~~ rec_sets ~~ (1 upto length recTs)))))
  1577             (fn fins => (rtac rec_induct THEN_ALL_NEW cut_facts_tac fins) 1 THEN REPEAT
  1578                (NominalPermeq.finite_guess_tac (HOL_ss addsimps [fs_name]) 1))))
  1579       end) dt_atomTs;
  1580 
  1581     (** freshness **)
  1582 
  1583     val perm_fresh_fresh = PureThy.get_thms thy11 (Name "perm_fresh_fresh");
  1584     val perm_swap = PureThy.get_thms thy11 (Name "perm_swap");
  1585 
  1586     fun perm_of_pair (x, y) =
  1587       let
  1588         val T = fastype_of x;
  1589         val pT = mk_permT T
  1590       in Const ("List.list.Cons", HOLogic.mk_prodT (T, T) --> pT --> pT) $
  1591         HOLogic.mk_prod (x, y) $ Const ("List.list.Nil", pT)
  1592       end;
  1593 
  1594     val finite_premss = map (fn aT =>
  1595       map (fn (f, T) => HOLogic.mk_Trueprop (HOLogic.mk_mem
  1596         (Const ("Nominal.supp", T --> HOLogic.mk_setT aT) $ f,
  1597          Const ("Finite_Set.Finites", HOLogic.mk_setT (HOLogic.mk_setT aT)))))
  1598            (rec_fns ~~ rec_fn_Ts)) dt_atomTs;
  1599 
  1600     val rec_fresh_thms = map (fn ((aT, eqvt_ths), finite_prems) =>
  1601       let
  1602         val name = Sign.base_name (fst (dest_Type aT));
  1603         val fs_name = PureThy.get_thm thy11 (Name ("fs_" ^ name ^ "1"));
  1604         val a = Free ("a", aT);
  1605         val freshs = map (fn (f, fT) => HOLogic.mk_Trueprop
  1606             (Const ("Nominal.fresh", aT --> fT --> HOLogic.boolT) $ a $ f))
  1607           (rec_fns ~~ rec_fn_Ts)
  1608       in
  1609         map (fn (((T, U), R), eqvt_th) =>
  1610           let
  1611             val x = Free ("x", T);
  1612             val y = Free ("y", U);
  1613             val y' = Free ("y'", U)
  1614           in
  1615             standard (Goal.prove (ProofContext.init thy11) [] (finite_prems @
  1616                 [HOLogic.mk_Trueprop (R $ x $ y),
  1617                  HOLogic.mk_Trueprop (HOLogic.mk_all ("y'", U,
  1618                    HOLogic.mk_imp (R $ x $ y', HOLogic.mk_eq (y', y)))),
  1619                  HOLogic.mk_Trueprop (Const ("Nominal.fresh",
  1620                    aT --> T --> HOLogic.boolT) $ a $ x)] @
  1621               freshs)
  1622               (HOLogic.mk_Trueprop (Const ("Nominal.fresh",
  1623                  aT --> U --> HOLogic.boolT) $ a $ y))
  1624               (fn {prems, context} =>
  1625                  let
  1626                    val (finite_prems, rec_prem :: unique_prem ::
  1627                      fresh_prems) = chop (length finite_prems) prems;
  1628                    val unique_prem' = unique_prem RS spec RS mp;
  1629                    val unique = [unique_prem', unique_prem' RS sym] MRS trans;
  1630                    val _ $ (_ $ (_ $ S $ _)) $ _ = prop_of supports_fresh;
  1631                    val tuple = foldr1 HOLogic.mk_prod (x :: rec_fns)
  1632                  in EVERY
  1633                    [rtac (Drule.cterm_instantiate
  1634                       [(cterm_of thy11 S,
  1635                         cterm_of thy11 (Const ("Nominal.supp",
  1636                           fastype_of tuple --> HOLogic.mk_setT aT) $ tuple))]
  1637                       supports_fresh) 1,
  1638                     simp_tac (HOL_basic_ss addsimps
  1639                       [supports_def, symmetric fresh_def, fresh_prod]) 1,
  1640                     REPEAT_DETERM (resolve_tac [allI, impI] 1),
  1641                     REPEAT_DETERM (etac conjE 1),
  1642                     rtac unique 1,
  1643                     SUBPROOF (fn {prems = prems', params = [a, b], ...} => EVERY
  1644                       [cut_facts_tac [rec_prem] 1,
  1645                        rtac (Thm.instantiate ([],
  1646                          [(cterm_of thy11 (Var (("pi", 0), mk_permT aT)),
  1647                            cterm_of thy11 (perm_of_pair (term_of a, term_of b)))]) eqvt_th) 1,
  1648                        asm_simp_tac (HOL_ss addsimps
  1649                          (prems' @ perm_swap @ perm_fresh_fresh)) 1]) context 1,
  1650                     rtac rec_prem 1,
  1651                     simp_tac (HOL_ss addsimps (fs_name ::
  1652                       supp_prod :: finite_Un :: finite_prems)) 1,
  1653                     simp_tac (HOL_ss addsimps (symmetric fresh_def ::
  1654                       fresh_prod :: fresh_prems)) 1]
  1655                  end))
  1656           end) (recTs ~~ rec_result_Ts ~~ rec_sets ~~ eqvt_ths)
  1657       end) (dt_atomTs ~~ rec_equiv_thms' ~~ finite_premss);
  1658 
  1659     (** uniqueness **)
  1660 
  1661     val exists_fresh' = PureThy.get_thms thy11 (Name "exists_fresh'");
  1662     val fs_atoms = map (fn Type (s, _) => PureThy.get_thm thy11
  1663       (Name ("fs_" ^ Sign.base_name s ^ "1"))) dt_atomTs;
  1664     val fresh_atm = PureThy.get_thms thy11 (Name "fresh_atm");
  1665     val calc_atm = PureThy.get_thms thy11 (Name "calc_atm");
  1666     val fresh_left = PureThy.get_thms thy11 (Name "fresh_left");
  1667 
  1668     val fun_tuple = foldr1 HOLogic.mk_prod (rec_ctxt :: rec_fns);
  1669     val fun_tupleT = fastype_of fun_tuple;
  1670     val rec_unique_frees =
  1671       DatatypeProp.indexify_names (replicate (length recTs) "x") ~~ recTs;
  1672     val rec_unique_frees'' = map (fn (s, T) => (s ^ "'", T)) rec_unique_frees;
  1673     val rec_unique_frees' =
  1674       DatatypeProp.indexify_names (replicate (length recTs) "y") ~~ rec_result_Ts;
  1675     val rec_unique_concls = map (fn ((x, U), R) =>
  1676         Const ("Ex1", (U --> HOLogic.boolT) --> HOLogic.boolT) $
  1677           Abs ("y", U, R $ Free x $ Bound 0))
  1678       (rec_unique_frees ~~ rec_result_Ts ~~ rec_sets);
  1679 
  1680     val induct_aux_rec = Drule.cterm_instantiate
  1681       (map (pairself (cterm_of thy11))
  1682          (map (fn (aT, f) => (Logic.varify f, Abs ("z", HOLogic.unitT,
  1683             Const ("Nominal.supp", fun_tupleT --> HOLogic.mk_setT aT) $ fun_tuple)))
  1684               fresh_fs @
  1685           map (fn (((P, T), (x, U)), Q) =>
  1686            (Var ((P, 0), HOLogic.unitT --> Logic.varifyT T --> HOLogic.boolT),
  1687             Abs ("z", HOLogic.unitT, absfree (x, U, Q))))
  1688               (pnames ~~ recTs ~~ rec_unique_frees ~~ rec_unique_concls) @
  1689           map (fn (s, T) => (Var ((s, 0), Logic.varifyT T), Free (s, T)))
  1690             rec_unique_frees)) induct_aux;
  1691 
  1692     fun obtain_fresh_name vs ths rec_fin_supp T (freshs1, freshs2, ctxt) =
  1693       let
  1694         val p = foldr1 HOLogic.mk_prod (vs @ freshs1);
  1695         val ex = Goal.prove ctxt [] [] (HOLogic.mk_Trueprop
  1696             (HOLogic.exists_const T $ Abs ("x", T,
  1697               Const ("Nominal.fresh", T --> fastype_of p --> HOLogic.boolT) $
  1698                 Bound 0 $ p)))
  1699           (fn _ => EVERY
  1700             [cut_facts_tac ths 1,
  1701              REPEAT_DETERM (dresolve_tac (the (AList.lookup op = rec_fin_supp T)) 1),
  1702              resolve_tac exists_fresh' 1,
  1703              asm_simp_tac (HOL_ss addsimps (supp_prod :: finite_Un :: fs_atoms)) 1]);
  1704         val (([cx], ths), ctxt') = Obtain.result
  1705           (fn _ => EVERY
  1706             [etac exE 1,
  1707              full_simp_tac (HOL_ss addsimps (fresh_prod :: fresh_atm)) 1,
  1708              REPEAT (etac conjE 1)])
  1709           [ex] ctxt
  1710       in (freshs1 @ [term_of cx], freshs2 @ ths, ctxt') end;
  1711 
  1712     val finite_ctxt_prems = map (fn aT =>
  1713       HOLogic.mk_Trueprop (HOLogic.mk_mem
  1714         (Const ("Nominal.supp", fsT' --> HOLogic.mk_setT aT) $ rec_ctxt,
  1715          Const ("Finite_Set.Finites", HOLogic.mk_setT (HOLogic.mk_setT aT))))) dt_atomTs;
  1716 
  1717     val rec_unique_thms = split_conj_thm (Goal.prove
  1718       (ProofContext.init thy11) (map fst rec_unique_frees)
  1719       (List.concat finite_premss @ finite_ctxt_prems @ rec_prems @ rec_prems')
  1720       (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj rec_unique_concls))
  1721       (fn {prems, context} =>
  1722          let
  1723            val k = length rec_fns;
  1724            val (finite_thss, ths1) = fold_map (fn T => fn xs =>
  1725              apfst (pair T) (chop k xs)) dt_atomTs prems;
  1726            val (finite_ctxt_ths, ths2) = chop (length dt_atomTs) ths1;
  1727            val (P_ind_ths, fcbs) = chop k ths2;
  1728            val P_ths = map (fn th => th RS mp) (split_conj_thm
  1729              (Goal.prove context
  1730                (map fst (rec_unique_frees'' @ rec_unique_frees')) []
  1731                (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
  1732                   (map (fn (((x, y), S), P) => HOLogic.mk_imp
  1733                     (S $ Free x $ Free y, P $ (Free y)))
  1734                       (rec_unique_frees'' ~~ rec_unique_frees' ~~ rec_sets ~~ rec_preds))))
  1735                (fn _ =>
  1736                   rtac rec_induct 1 THEN
  1737                   REPEAT ((resolve_tac P_ind_ths THEN_ALL_NEW assume_tac) 1))));
  1738            val rec_fin_supp_thms' = map
  1739              (fn (ths, (T, fin_ths)) => (T, map (curry op MRS fin_ths) ths))
  1740              (rec_fin_supp_thms ~~ finite_thss);
  1741          in EVERY
  1742            ([rtac induct_aux_rec 1] @
  1743             maps (fn ((_, finite_ths), finite_th) =>
  1744               [cut_facts_tac (finite_th :: finite_ths) 1,
  1745                asm_simp_tac (HOL_ss addsimps [supp_prod, finite_Un]) 1])
  1746                 (finite_thss ~~ finite_ctxt_ths) @
  1747             maps (fn ((_, idxss), elim) => maps (fn idxs =>
  1748               [full_simp_tac (HOL_ss addsimps [symmetric fresh_def, supp_prod, Un_iff]) 1,
  1749                REPEAT_DETERM (eresolve_tac [conjE, ex1E] 1),
  1750                rtac ex1I 1,
  1751                (resolve_tac rec_intrs THEN_ALL_NEW atac) 1,
  1752                rotate_tac ~1 1,
  1753                ((DETERM o etac elim) THEN_ALL_NEW full_simp_tac
  1754                   (HOL_ss addsimps List.concat distinct_thms)) 1] @
  1755                (if null idxs then [] else [hyp_subst_tac 1,
  1756                 SUBPROOF (fn {asms, concl, prems = prems', params, context = context', ...} =>
  1757                   let
  1758                     val SOME prem = find_first (can (HOLogic.dest_eq o
  1759                       HOLogic.dest_Trueprop o prop_of)) prems';
  1760                     val _ $ (_ $ lhs $ rhs) = prop_of prem;
  1761                     val _ $ (_ $ lhs' $ rhs') = term_of concl;
  1762                     val rT = fastype_of lhs';
  1763                     val (c, cargsl) = strip_comb lhs;
  1764                     val cargsl' = partition_cargs idxs cargsl;
  1765                     val boundsl = List.concat (map fst cargsl');
  1766                     val (_, cargsr) = strip_comb rhs;
  1767                     val cargsr' = partition_cargs idxs cargsr;
  1768                     val boundsr = List.concat (map fst cargsr');
  1769                     val (params1, _ :: params2) =
  1770                       chop (length params div 2) (map term_of params);
  1771                     val params' = params1 @ params2;
  1772                     val rec_prems = filter (fn th => case prop_of th of
  1773                       _ $ (S $ _ $ _) => S mem rec_sets | _ => false) prems';
  1774                     val fresh_prems = filter (fn th => case prop_of th of
  1775                         _ $ (Const ("Nominal.fresh", _) $ _ $ _) => true
  1776                       | _ $ (Const ("Not", _) $ _) => true
  1777                       | _ => false) prems';
  1778                     val Ts = map fastype_of boundsl;
  1779 
  1780                     val _ = warning "step 1: obtaining fresh names";
  1781                     val (freshs1, freshs2, context'') = fold
  1782                       (obtain_fresh_name (rec_ctxt :: rec_fns @ params')
  1783                          (List.concat (map snd finite_thss) @
  1784                             finite_ctxt_ths @ rec_prems)
  1785                          rec_fin_supp_thms')
  1786                       Ts ([], [], context');
  1787                     val pi1 = map perm_of_pair (boundsl ~~ freshs1);
  1788                     val rpi1 = rev pi1;
  1789                     val pi2 = map perm_of_pair (boundsr ~~ freshs1);
  1790                     val rpi2 = rev pi2;
  1791 
  1792                     fun mk_not_sym ths = List.concat (map (fn th =>
  1793                       case prop_of th of
  1794                           _ $ (Const ("Not", _) $ _) => [th, th RS not_sym]
  1795                         | _ => [th]) ths);
  1796                     val fresh_prems' = mk_not_sym fresh_prems;
  1797                     val freshs2' = mk_not_sym freshs2;
  1798 
  1799                     (** as, bs, cs # K as ts, K bs us **)
  1800                     val _ = warning "step 2: as, bs, cs # K as ts, K bs us";
  1801                     val prove_fresh_ss = HOL_ss addsimps
  1802                       (finite_Diff :: List.concat fresh_thms @
  1803                        fs_atoms @ abs_fresh @ abs_supp @ fresh_atm);
  1804                     (* FIXME: avoid asm_full_simp_tac ? *)
  1805                     fun prove_fresh ths y x = Goal.prove context'' [] []
  1806                       (HOLogic.mk_Trueprop (Const ("Nominal.fresh",
  1807                          fastype_of x --> fastype_of y --> HOLogic.boolT) $ x $ y))
  1808                       (fn _ => cut_facts_tac ths 1 THEN asm_full_simp_tac prove_fresh_ss 1);
  1809                     val constr_fresh_thms =
  1810                       map (prove_fresh fresh_prems lhs) boundsl @
  1811                       map (prove_fresh fresh_prems rhs) boundsr @
  1812                       map (prove_fresh freshs2 lhs) freshs1 @
  1813                       map (prove_fresh freshs2 rhs) freshs1;
  1814 
  1815                     (** pi1 o (K as ts) = pi2 o (K bs us) **)
  1816                     val _ = warning "step 3: pi1 o (K as ts) = pi2 o (K bs us)";
  1817                     val pi1_pi2_eq = Goal.prove context'' [] []
  1818                       (HOLogic.mk_Trueprop (HOLogic.mk_eq
  1819                         (foldr (mk_perm []) lhs pi1, foldr (mk_perm []) rhs pi2)))
  1820                       (fn _ => EVERY
  1821                          [cut_facts_tac constr_fresh_thms 1,
  1822                           asm_simp_tac (HOL_basic_ss addsimps perm_fresh_fresh) 1,
  1823                           rtac prem 1]);
  1824 
  1825                     (** pi1 o ts = pi2 o us **)
  1826                     val _ = warning "step 4: pi1 o ts = pi2 o us";
  1827                     val pi1_pi2_eqs = map (fn (t, u) =>
  1828                       Goal.prove context'' [] []
  1829                         (HOLogic.mk_Trueprop (HOLogic.mk_eq
  1830                           (foldr (mk_perm []) t pi1, foldr (mk_perm []) u pi2)))
  1831                         (fn _ => EVERY
  1832                            [cut_facts_tac [pi1_pi2_eq] 1,
  1833                             asm_full_simp_tac (HOL_ss addsimps
  1834                               (calc_atm @ List.concat perm_simps' @
  1835                                fresh_prems' @ freshs2' @ abs_perm @
  1836                                alpha @ List.concat inject_thms)) 1]))
  1837                         (map snd cargsl' ~~ map snd cargsr');
  1838 
  1839                     (** pi1^-1 o pi2 o us = ts **)
  1840                     val _ = warning "step 5: pi1^-1 o pi2 o us = ts";
  1841                     val rpi1_pi2_eqs = map (fn ((t, u), eq) =>
  1842                       Goal.prove context'' [] []
  1843                         (HOLogic.mk_Trueprop (HOLogic.mk_eq
  1844                           (foldr (mk_perm []) u (rpi1 @ pi2), t)))
  1845                         (fn _ => simp_tac (HOL_ss addsimps
  1846                            ((eq RS sym) :: perm_swap)) 1))
  1847                         (map snd cargsl' ~~ map snd cargsr' ~~ pi1_pi2_eqs);
  1848 
  1849                     val (rec_prems1, rec_prems2) =
  1850                       chop (length rec_prems div 2) rec_prems;
  1851 
  1852                     (** (ts, pi1^-1 o pi2 o vs) in rec_set **)
  1853                     val _ = warning "step 6: (ts, pi1^-1 o pi2 o vs) in rec_set";
  1854                     val rec_prems' = map (fn th =>
  1855                       let
  1856                         val _ $ (S $ x $ y) = prop_of th;
  1857                         val k = find_index (equal S) rec_sets;
  1858                         val pi = rpi1 @ pi2;
  1859                         fun mk_pi z = foldr (mk_perm []) z pi;
  1860                         fun eqvt_tac p =
  1861                           let
  1862                             val U as Type (_, [Type (_, [T, _])]) = fastype_of p;
  1863                             val l = find_index (equal T) dt_atomTs;
  1864                             val th = List.nth (List.nth (rec_equiv_thms', l), k);
  1865                             val th' = Thm.instantiate ([],
  1866                               [(cterm_of thy11 (Var (("pi", 0), U)),
  1867                                 cterm_of thy11 p)]) th;
  1868                           in rtac th' 1 end;
  1869                         val th' = Goal.prove context'' [] []
  1870                           (HOLogic.mk_Trueprop (S $ mk_pi x $ mk_pi y))
  1871                           (fn _ => EVERY
  1872                              (map eqvt_tac pi @
  1873                               [simp_tac (HOL_ss addsimps (fresh_prems' @ freshs2' @
  1874                                  perm_swap @ perm_fresh_fresh)) 1,
  1875                                rtac th 1]))
  1876                       in
  1877                         Simplifier.simplify
  1878                           (HOL_basic_ss addsimps rpi1_pi2_eqs) th'
  1879                       end) rec_prems2;
  1880 
  1881                     val ihs = filter (fn th => case prop_of th of
  1882                       _ $ (Const ("All", _) $ _) => true | _ => false) prems';
  1883 
  1884                     (** pi1 o rs = pi2 o vs , rs = pi1^-1 o pi2 o vs **)
  1885                     val _ = warning "step 7: pi1 o rs = pi2 o vs , rs = pi1^-1 o pi2 o vs";
  1886                     val rec_eqns = map (fn (th, ih) =>
  1887                       let
  1888                         val th' = th RS (ih RS spec RS mp) RS sym;
  1889                         val _ $ (_ $ lhs $ rhs) = prop_of th';
  1890                         fun strip_perm (_ $ _ $ t) = strip_perm t
  1891                           | strip_perm t = t;
  1892                       in
  1893                         Goal.prove context'' [] []
  1894                            (HOLogic.mk_Trueprop (HOLogic.mk_eq
  1895                               (foldr (mk_perm []) lhs pi1,
  1896                                foldr (mk_perm []) (strip_perm rhs) pi2)))
  1897                            (fn _ => simp_tac (HOL_basic_ss addsimps
  1898                               (th' :: perm_swap)) 1)
  1899                       end) (rec_prems' ~~ ihs);
  1900 
  1901                     (** as # rs **)
  1902                     val _ = warning "step 8: as # rs";
  1903                     val rec_freshs = List.concat
  1904                       (map (fn (rec_prem, ih) =>
  1905                         let
  1906                           val _ $ (S $ x $ (y as Free (_, T))) =
  1907                             prop_of rec_prem;
  1908                           val k = find_index (equal S) rec_sets;
  1909                           val atoms = List.concat (List.mapPartial (fn (bs, z) =>
  1910                             if z = x then NONE else SOME bs) cargsl')
  1911                         in
  1912                           map (fn a as Free (_, aT) =>
  1913                             let val l = find_index (equal aT) dt_atomTs;
  1914                             in
  1915                               Goal.prove context'' [] []
  1916                                 (HOLogic.mk_Trueprop (Const ("Nominal.fresh",
  1917                                   aT --> T --> HOLogic.boolT) $ a $ y))
  1918                                 (fn _ => EVERY
  1919                                    (rtac (List.nth (List.nth (rec_fresh_thms, l), k)) 1 ::
  1920                                     map (fn th => rtac th 1)
  1921                                       (snd (List.nth (finite_thss, l))) @
  1922                                     [rtac rec_prem 1, rtac ih 1,
  1923                                      REPEAT_DETERM (resolve_tac fresh_prems 1)]))
  1924                             end) atoms
  1925                         end) (rec_prems1 ~~ ihs));
  1926 
  1927                     (** as # fK as ts rs , bs # fK bs us vs **)
  1928                     val _ = warning "step 9: as # fK as ts rs , bs # fK bs us vs";
  1929                     fun prove_fresh_result (a as Free (_, aT)) =
  1930                       Goal.prove context'' [] []
  1931                         (HOLogic.mk_Trueprop (Const ("Nominal.fresh",
  1932                           aT --> rT --> HOLogic.boolT) $ a $ rhs'))
  1933                         (fn _ => EVERY
  1934                            [resolve_tac fcbs 1,
  1935                             REPEAT_DETERM (resolve_tac
  1936                               (fresh_prems @ rec_freshs) 1),
  1937                             REPEAT_DETERM (resolve_tac (maps snd rec_fin_supp_thms') 1
  1938                               THEN resolve_tac rec_prems 1),
  1939                             resolve_tac P_ind_ths 1,
  1940                             REPEAT_DETERM (resolve_tac (P_ths @ rec_prems) 1)]);
  1941 
  1942                     val fresh_results'' = map prove_fresh_result boundsl;
  1943 
  1944                     fun prove_fresh_result'' ((a as Free (_, aT), b), th) =
  1945                       let val th' = Goal.prove context'' [] []
  1946                         (HOLogic.mk_Trueprop (Const ("Nominal.fresh",
  1947                           aT --> rT --> HOLogic.boolT) $
  1948                             foldr (mk_perm []) a (rpi2 @ pi1) $
  1949                             foldr (mk_perm []) rhs' (rpi2 @ pi1)))
  1950                         (fn _ => simp_tac (HOL_ss addsimps fresh_bij) 1 THEN
  1951                            rtac th 1)
  1952                       in
  1953                         Goal.prove context'' [] []
  1954                           (HOLogic.mk_Trueprop (Const ("Nominal.fresh",
  1955                             aT --> rT --> HOLogic.boolT) $ b $ lhs'))
  1956                           (fn _ => EVERY
  1957                              [cut_facts_tac [th'] 1,
  1958                               NominalPermeq.perm_simp_tac (HOL_ss addsimps
  1959                                 (rec_eqns @ pi1_pi2_eqs @ perm_swap)) 1,
  1960                               full_simp_tac (HOL_ss addsimps (calc_atm @
  1961                                 fresh_prems' @ freshs2' @ perm_fresh_fresh)) 1])
  1962                       end;
  1963 
  1964                     val fresh_results = fresh_results'' @ map prove_fresh_result''
  1965                       (boundsl ~~ boundsr ~~ fresh_results'');
  1966 
  1967                     (** cs # fK as ts rs , cs # fK bs us vs **)
  1968                     val _ = warning "step 10: cs # fK as ts rs , cs # fK bs us vs";
  1969                     fun prove_fresh_result' recs t (a as Free (_, aT)) =
  1970                       Goal.prove context'' [] []
  1971                         (HOLogic.mk_Trueprop (Const ("Nominal.fresh",
  1972                           aT --> rT --> HOLogic.boolT) $ a $ t))
  1973                         (fn _ => EVERY
  1974                           [cut_facts_tac recs 1,
  1975                            REPEAT_DETERM (dresolve_tac
  1976                              (the (AList.lookup op = rec_fin_supp_thms' aT)) 1),
  1977                            NominalPermeq.fresh_guess_tac
  1978                              (HOL_ss addsimps (freshs2 @
  1979                                 fs_atoms @ fresh_atm @
  1980                                 List.concat (map snd finite_thss))) 1]);
  1981 
  1982                     val fresh_results' =
  1983                       map (prove_fresh_result' rec_prems1 rhs') freshs1 @
  1984                       map (prove_fresh_result' rec_prems2 lhs') freshs1;
  1985 
  1986                     (** pi1 o (fK as ts rs) = pi2 o (fK bs us vs) **)
  1987                     val _ = warning "step 11: pi1 o (fK as ts rs) = pi2 o (fK bs us vs)";
  1988                     val pi1_pi2_result = Goal.prove context'' [] []
  1989                       (HOLogic.mk_Trueprop (HOLogic.mk_eq
  1990                         (foldr (mk_perm []) rhs' pi1, foldr (mk_perm []) lhs' pi2)))
  1991                       (fn _ => NominalPermeq.perm_simp_tac (HOL_ss addsimps
  1992                            pi1_pi2_eqs @ rec_eqns) 1 THEN
  1993                          TRY (simp_tac (HOL_ss addsimps
  1994                            (fresh_prems' @ freshs2' @ calc_atm @ perm_fresh_fresh)) 1));
  1995 
  1996                     val _ = warning "final result";
  1997                     val final = Goal.prove context'' [] [] (term_of concl)
  1998                       (fn _ => cut_facts_tac [pi1_pi2_result RS sym] 1 THEN
  1999                         full_simp_tac (HOL_basic_ss addsimps perm_fresh_fresh @
  2000                           fresh_results @ fresh_results') 1);
  2001                     val final' = ProofContext.export context'' context' [final];
  2002                     val _ = warning "finished!"
  2003                   in
  2004                     resolve_tac final' 1
  2005                   end) context 1])) idxss) (ndescr ~~ rec_elims))
  2006          end));
  2007 
  2008     val rec_total_thms = map (fn r => r RS theI') rec_unique_thms;
  2009 
  2010     (* define primrec combinators *)
  2011 
  2012     val big_reccomb_name = (space_implode "_" new_type_names) ^ "_rec";
  2013     val reccomb_names = map (Sign.full_name thy11)
  2014       (if length descr'' = 1 then [big_reccomb_name] else
  2015         (map ((curry (op ^) (big_reccomb_name ^ "_")) o string_of_int)
  2016           (1 upto (length descr''))));
  2017     val reccombs = map (fn ((name, T), T') => list_comb
  2018       (Const (name, rec_fn_Ts @ [T] ---> T'), rec_fns))
  2019         (reccomb_names ~~ recTs ~~ rec_result_Ts);
  2020 
  2021     val (reccomb_defs, thy12) =
  2022       thy11
  2023       |> Theory.add_consts_i (map (fn ((name, T), T') =>
  2024           (Sign.base_name name, rec_fn_Ts @ [T] ---> T', NoSyn))
  2025           (reccomb_names ~~ recTs ~~ rec_result_Ts))
  2026       |> (PureThy.add_defs_i false o map Thm.no_attributes) (map (fn ((((name, comb), set), T), T') =>
  2027           ((Sign.base_name name) ^ "_def", Logic.mk_equals (comb, absfree ("x", T,
  2028            Const ("The", (T' --> HOLogic.boolT) --> T') $ absfree ("y", T',
  2029              set $ Free ("x", T) $ Free ("y", T'))))))
  2030                (reccomb_names ~~ reccombs ~~ rec_sets ~~ recTs ~~ rec_result_Ts));
  2031 
  2032     (* prove characteristic equations for primrec combinators *)
  2033 
  2034     val rec_thms = map (fn (prems, concl) =>
  2035       let
  2036         val _ $ (_ $ (_ $ x) $ _) = concl;
  2037         val (_, cargs) = strip_comb x;
  2038         val ps = map (fn (x as Free (_, T), i) =>
  2039           (Free ("x" ^ string_of_int i, T), x)) (cargs ~~ (1 upto length cargs));
  2040         val concl' = subst_atomic_types (rec_result_Ts' ~~ rec_result_Ts) concl;
  2041         val prems' = List.concat finite_premss @ finite_ctxt_prems @
  2042           rec_prems @ rec_prems' @ map (subst_atomic ps) prems;
  2043         fun solve rules prems = resolve_tac rules THEN_ALL_NEW
  2044           (resolve_tac prems THEN_ALL_NEW atac)
  2045       in
  2046         Goal.prove_global thy12 [] prems' concl'
  2047           (fn prems => EVERY
  2048             [rewrite_goals_tac reccomb_defs,
  2049              rtac the1_equality 1,
  2050              solve rec_unique_thms prems 1,
  2051              resolve_tac rec_intrs 1,
  2052              REPEAT (solve (prems @ rec_total_thms) prems 1)])
  2053       end) (rec_eq_prems ~~
  2054         DatatypeProp.make_primrecs new_type_names descr' sorts' thy12);
  2055 
  2056     val dt_infos = map (make_dt_info descr'' sorts induct reccomb_names rec_thms)
  2057       ((0 upto length descr1 - 1) ~~ descr1 ~~ distinct_thms ~~ inject_thms);
  2058 
  2059     (* FIXME: theorems are stored in database for testing only *)
  2060     val (_, thy13) = thy12 |>
  2061       PureThy.add_thmss
  2062         [(("rec_equiv", List.concat rec_equiv_thms), []),
  2063          (("rec_equiv'", List.concat rec_equiv_thms'), []),
  2064          (("rec_fin_supp", List.concat rec_fin_supp_thms), []),
  2065          (("rec_fresh", List.concat rec_fresh_thms), []),
  2066          (("rec_unique", map standard rec_unique_thms), []),
  2067          (("recs", rec_thms), [])] ||>
  2068       Theory.parent_path ||>
  2069       map_nominal_datatypes (fold Symtab.update dt_infos);
  2070 
  2071   in
  2072     thy13
  2073   end;
  2074 
  2075 val add_nominal_datatype = gen_add_nominal_datatype read_typ true;
  2076 
  2077 
  2078 (* FIXME: The following stuff should be exported by DatatypePackage *)
  2079 
  2080 local structure P = OuterParse and K = OuterKeyword in
  2081 
  2082 val datatype_decl =
  2083   Scan.option (P.$$$ "(" |-- P.name --| P.$$$ ")") -- P.type_args -- P.name -- P.opt_infix --
  2084     (P.$$$ "=" |-- P.enum1 "|" (P.name -- Scan.repeat P.typ -- P.opt_mixfix));
  2085 
  2086 fun mk_datatype args =
  2087   let
  2088     val names = map (fn ((((NONE, _), t), _), _) => t | ((((SOME t, _), _), _), _) => t) args;
  2089     val specs = map (fn ((((_, vs), t), mx), cons) =>
  2090       (vs, t, mx, map (fn ((x, y), z) => (x, y, z)) cons)) args;
  2091   in add_nominal_datatype false names specs end;
  2092 
  2093 val nominal_datatypeP =
  2094   OuterSyntax.command "nominal_datatype" "define inductive datatypes" K.thy_decl
  2095     (P.and_list1 datatype_decl >> (Toplevel.theory o mk_datatype));
  2096 
  2097 val _ = OuterSyntax.add_parsers [nominal_datatypeP];
  2098 
  2099 end;
  2100 
  2101 end