src/HOL/Nominal/nominal_package.ML
author wenzelm
Sat Oct 06 16:50:04 2007 +0200 (2007-10-06)
changeset 24867 e5b55d7be9bb
parent 24814 0384f48a806e
child 25557 ea6b11021e79
permissions -rw-r--r--
simplified interfaces for outer syntax;
     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 descr
    13   type nominal_datatype_info
    14   val get_nominal_datatypes : theory -> nominal_datatype_info Symtab.table
    15   val get_nominal_datatype : theory -> string -> nominal_datatype_info option
    16   val mk_perm: typ list -> term -> term -> term
    17   val perm_of_pair: term * term -> term
    18   val mk_not_sym: thm list -> thm list
    19   val perm_simproc: simproc
    20 end
    21 
    22 structure NominalPackage : NOMINAL_PACKAGE =
    23 struct
    24 
    25 val finite_emptyI = thm "finite.emptyI";
    26 val finite_Diff = thm "finite_Diff";
    27 val finite_Un = thm "finite_Un";
    28 val Un_iff = thm "Un_iff";
    29 val In0_eq = thm "In0_eq";
    30 val In1_eq = thm "In1_eq";
    31 val In0_not_In1 = thm "In0_not_In1";
    32 val In1_not_In0 = thm "In1_not_In0";
    33 val Un_assoc = thm "Un_assoc";
    34 val Collect_disj_eq = thm "Collect_disj_eq";
    35 val empty_def = thm "empty_def";
    36 val empty_iff = thm "empty_iff";
    37 
    38 open DatatypeAux;
    39 open NominalAtoms;
    40 
    41 (** FIXME: DatatypePackage should export this function **)
    42 
    43 local
    44 
    45 fun dt_recs (DtTFree _) = []
    46   | dt_recs (DtType (_, dts)) = List.concat (map dt_recs dts)
    47   | dt_recs (DtRec i) = [i];
    48 
    49 fun dt_cases (descr: descr) (_, args, constrs) =
    50   let
    51     fun the_bname i = Sign.base_name (#1 (valOf (AList.lookup (op =) descr i)));
    52     val bnames = map the_bname (distinct op = (List.concat (map dt_recs args)));
    53   in map (fn (c, _) => space_implode "_" (Sign.base_name c :: bnames)) constrs end;
    54 
    55 
    56 fun induct_cases descr =
    57   DatatypeProp.indexify_names (List.concat (map (dt_cases descr) (map #2 descr)));
    58 
    59 fun exhaust_cases descr i = dt_cases descr (valOf (AList.lookup (op =) descr i));
    60 
    61 in
    62 
    63 fun mk_case_names_induct descr = RuleCases.case_names (induct_cases descr);
    64 
    65 fun mk_case_names_exhausts descr new =
    66   map (RuleCases.case_names o exhaust_cases descr o #1)
    67     (List.filter (fn ((_, (name, _, _))) => name mem_string new) descr);
    68 
    69 end;
    70 
    71 (* theory data *)
    72 
    73 type descr = (int * (string * dtyp list * (string * (dtyp list * dtyp) list) list)) list;
    74 
    75 type nominal_datatype_info =
    76   {index : int,
    77    descr : descr,
    78    sorts : (string * sort) list,
    79    rec_names : string list,
    80    rec_rewrites : thm list,
    81    induction : thm,
    82    distinct : thm list,
    83    inject : thm list};
    84 
    85 structure NominalDatatypesData = TheoryDataFun
    86 (
    87   type T = nominal_datatype_info Symtab.table;
    88   val empty = Symtab.empty;
    89   val copy = I;
    90   val extend = I;
    91   fun merge _ tabs : T = Symtab.merge (K true) tabs;
    92 );
    93 
    94 val get_nominal_datatypes = NominalDatatypesData.get;
    95 val put_nominal_datatypes = NominalDatatypesData.put;
    96 val map_nominal_datatypes = NominalDatatypesData.map;
    97 val get_nominal_datatype = Symtab.lookup o get_nominal_datatypes;
    98 
    99 
   100 (**** make datatype info ****)
   101 
   102 fun make_dt_info descr sorts induct reccomb_names rec_thms
   103     (((i, (_, (tname, _, _))), distinct), inject) =
   104   (tname,
   105    {index = i,
   106     descr = descr,
   107     sorts = sorts,
   108     rec_names = reccomb_names,
   109     rec_rewrites = rec_thms,
   110     induction = induct,
   111     distinct = distinct,
   112     inject = inject});
   113 
   114 (*******************************)
   115 
   116 val (_ $ (_ $ (_ $ (distinct_f $ _) $ _))) = hd (prems_of distinct_lemma);
   117 
   118 fun read_typ sign ((Ts, sorts), str) =
   119   let
   120     val T = Type.no_tvars (Sign.read_def_typ (sign, (AList.lookup op =)
   121       (map (apfst (rpair ~1)) sorts)) str) handle TYPE (msg, _, _) => error msg
   122   in (Ts @ [T], add_typ_tfrees (T, sorts)) end;
   123 
   124 (** taken from HOL/Tools/datatype_aux.ML **)
   125 
   126 fun indtac indrule indnames i st =
   127   let
   128     val ts = HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of indrule));
   129     val ts' = HOLogic.dest_conj (HOLogic.dest_Trueprop
   130       (Logic.strip_imp_concl (List.nth (prems_of st, i - 1))));
   131     val getP = if can HOLogic.dest_imp (hd ts) then
   132       (apfst SOME) o HOLogic.dest_imp else pair NONE;
   133     fun abstr (t1, t2) = (case t1 of
   134         NONE => (case filter (fn Free (s, _) => s mem indnames | _ => false)
   135               (term_frees t2) of
   136             [Free (s, T)] => absfree (s, T, t2)
   137           | _ => sys_error "indtac")
   138       | SOME (_ $ t') => Abs ("x", fastype_of t', abstract_over (t', t2)))
   139     val cert = cterm_of (Thm.theory_of_thm st);
   140     val Ps = map (cert o head_of o snd o getP) ts;
   141     val indrule' = cterm_instantiate (Ps ~~
   142       (map (cert o abstr o getP) ts')) indrule
   143   in
   144     rtac indrule' i st
   145   end;
   146 
   147 fun mk_subgoal i f st =
   148   let
   149     val cg = List.nth (cprems_of st, i - 1);
   150     val g = term_of cg;
   151     val revcut_rl' = Thm.lift_rule cg revcut_rl;
   152     val v = head_of (Logic.strip_assums_concl (hd (prems_of revcut_rl')));
   153     val ps = Logic.strip_params g;
   154     val cert = cterm_of (Thm.theory_of_thm st);
   155   in
   156     compose_tac (false,
   157       Thm.instantiate ([], [(cert v, cert (list_abs (ps,
   158         f (rev ps) (Logic.strip_assums_hyp g) (Logic.strip_assums_concl g))))])
   159       revcut_rl', 2) i st
   160   end;
   161 
   162 (** simplification procedure for sorting permutations **)
   163 
   164 val dj_cp = thm "dj_cp";
   165 
   166 fun dest_permT (Type ("fun", [Type ("List.list", [Type ("*", [T, _])]),
   167       Type ("fun", [_, U])])) = (T, U);
   168 
   169 fun permTs_of (Const ("Nominal.perm", T) $ t $ u) = fst (dest_permT T) :: permTs_of u
   170   | permTs_of _ = [];
   171 
   172 fun perm_simproc' thy ss (Const ("Nominal.perm", T) $ t $ (u as Const ("Nominal.perm", U) $ r $ s)) =
   173       let
   174         val (aT as Type (a, []), S) = dest_permT T;
   175         val (bT as Type (b, []), _) = dest_permT U
   176       in if aT mem permTs_of u andalso aT <> bT then
   177           let
   178             val a' = Sign.base_name a;
   179             val b' = Sign.base_name b;
   180             val cp = PureThy.get_thm thy (Name ("cp_" ^ a' ^ "_" ^ b' ^ "_inst"));
   181             val dj = PureThy.get_thm thy (Name ("dj_" ^ b' ^ "_" ^ a'));
   182             val dj_cp' = [cp, dj] MRS dj_cp;
   183             val cert = SOME o cterm_of thy
   184           in
   185             SOME (mk_meta_eq (Drule.instantiate' [SOME (ctyp_of thy S)]
   186               [cert t, cert r, cert s] dj_cp'))
   187           end
   188         else NONE
   189       end
   190   | perm_simproc' thy ss _ = NONE;
   191 
   192 val perm_simproc =
   193   Simplifier.simproc (the_context ()) "perm_simp" ["pi1 \\<bullet> (pi2 \\<bullet> x)"] perm_simproc';
   194 
   195 val allE_Nil = read_instantiate_sg (the_context()) [("x", "[]")] allE;
   196 
   197 val meta_spec = thm "meta_spec";
   198 
   199 fun projections rule =
   200   ProjectRule.projections (ProofContext.init (Thm.theory_of_thm rule)) rule
   201   |> map (standard #> RuleCases.save rule);
   202 
   203 val supp_prod = thm "supp_prod";
   204 val fresh_prod = thm "fresh_prod";
   205 val supports_fresh = thm "supports_fresh";
   206 val supports_def = thm "Nominal.supports_def";
   207 val fresh_def = thm "fresh_def";
   208 val supp_def = thm "supp_def";
   209 val rev_simps = thms "rev.simps";
   210 val app_simps = thms "append.simps";
   211 
   212 val collect_simp = rewrite_rule [mk_meta_eq mem_Collect_eq];
   213 
   214 fun mk_perm Ts t u =
   215   let
   216     val T = fastype_of1 (Ts, t);
   217     val U = fastype_of1 (Ts, u)
   218   in Const ("Nominal.perm", T --> U --> U) $ t $ u end;
   219 
   220 fun perm_of_pair (x, y) =
   221   let
   222     val T = fastype_of x;
   223     val pT = mk_permT T
   224   in Const ("List.list.Cons", HOLogic.mk_prodT (T, T) --> pT --> pT) $
   225     HOLogic.mk_prod (x, y) $ Const ("List.list.Nil", pT)
   226   end;
   227 
   228 fun mk_not_sym ths = maps (fn th => case prop_of th of
   229     _ $ (Const ("Not", _) $ _) => [th, th RS not_sym]
   230   | _ => [th]) ths;
   231 
   232 fun gen_add_nominal_datatype prep_typ err flat_names new_type_names dts thy =
   233   let
   234     (* this theory is used just for parsing *)
   235 
   236     val tmp_thy = thy |>
   237       Theory.copy |>
   238       Sign.add_types (map (fn (tvs, tname, mx, _) =>
   239         (tname, length tvs, mx)) dts);
   240 
   241     val atoms = atoms_of thy;
   242     val classes = map (NameSpace.map_base (fn s => "pt_" ^ s)) atoms;
   243     val cp_classes = List.concat (map (fn atom1 => map (fn atom2 =>
   244       Sign.intern_class thy ("cp_" ^ Sign.base_name atom1 ^ "_" ^
   245         Sign.base_name atom2)) atoms) atoms);
   246     fun augment_sort S = S union classes;
   247     val augment_sort_typ = map_type_tfree (fn (s, S) => TFree (s, augment_sort S));
   248 
   249     fun prep_constr ((constrs, sorts), (cname, cargs, mx)) =
   250       let val (cargs', sorts') = Library.foldl (prep_typ tmp_thy) (([], sorts), cargs)
   251       in (constrs @ [(cname, cargs', mx)], sorts') end
   252 
   253     fun prep_dt_spec ((dts, sorts), (tvs, tname, mx, constrs)) =
   254       let val (constrs', sorts') = Library.foldl prep_constr (([], sorts), constrs)
   255       in (dts @ [(tvs, tname, mx, constrs')], sorts') end
   256 
   257     val (dts', sorts) = Library.foldl prep_dt_spec (([], []), dts);
   258     val sorts' = map (apsnd augment_sort) sorts;
   259     val tyvars = map #1 dts';
   260 
   261     val types_syntax = map (fn (tvs, tname, mx, constrs) => (tname, mx)) dts';
   262     val constr_syntax = map (fn (tvs, tname, mx, constrs) =>
   263       map (fn (cname, cargs, mx) => (cname, mx)) constrs) dts';
   264 
   265     val ps = map (fn (_, n, _, _) =>
   266       (Sign.full_name tmp_thy n, Sign.full_name tmp_thy (n ^ "_Rep"))) dts;
   267     val rps = map Library.swap ps;
   268 
   269     fun replace_types (Type ("Nominal.ABS", [T, U])) =
   270           Type ("fun", [T, Type ("Nominal.noption", [replace_types U])])
   271       | replace_types (Type (s, Ts)) =
   272           Type (getOpt (AList.lookup op = ps s, s), map replace_types Ts)
   273       | replace_types T = T;
   274 
   275     val dts'' = map (fn (tvs, tname, mx, constrs) => (tvs, tname ^ "_Rep", NoSyn,
   276       map (fn (cname, cargs, mx) => (cname ^ "_Rep",
   277         map (augment_sort_typ o replace_types) cargs, NoSyn)) constrs)) dts';
   278 
   279     val new_type_names' = map (fn n => n ^ "_Rep") new_type_names;
   280     val full_new_type_names' = map (Sign.full_name thy) new_type_names';
   281 
   282     val ({induction, ...},thy1) =
   283       DatatypePackage.add_datatype_i err flat_names new_type_names' dts'' thy;
   284 
   285     val SOME {descr, ...} = Symtab.lookup
   286       (DatatypePackage.get_datatypes thy1) (hd full_new_type_names');
   287     fun nth_dtyp i = typ_of_dtyp descr sorts' (DtRec i);
   288 
   289     (**** define permutation functions ****)
   290 
   291     val permT = mk_permT (TFree ("'x", HOLogic.typeS));
   292     val pi = Free ("pi", permT);
   293     val perm_types = map (fn (i, _) =>
   294       let val T = nth_dtyp i
   295       in permT --> T --> T end) descr;
   296     val perm_names = replicate (length new_type_names) "Nominal.perm" @
   297       DatatypeProp.indexify_names (map (fn i => Sign.full_name thy1
   298         ("perm_" ^ name_of_typ (nth_dtyp i)))
   299           (length new_type_names upto length descr - 1));
   300     val perm_names_types = perm_names ~~ perm_types;
   301 
   302     val perm_eqs = List.concat (map (fn (i, (_, _, constrs)) =>
   303       let val T = nth_dtyp i
   304       in map (fn (cname, dts) =>
   305         let
   306           val Ts = map (typ_of_dtyp descr sorts') dts;
   307           val names = DatatypeProp.make_tnames Ts;
   308           val args = map Free (names ~~ Ts);
   309           val c = Const (cname, Ts ---> T);
   310           fun perm_arg (dt, x) =
   311             let val T = type_of x
   312             in if is_rec_type dt then
   313                 let val (Us, _) = strip_type T
   314                 in list_abs (map (pair "x") Us,
   315                   Const (List.nth (perm_names_types, body_index dt)) $ pi $
   316                     list_comb (x, map (fn (i, U) =>
   317                       Const ("Nominal.perm", permT --> U --> U) $
   318                         (Const ("List.rev", permT --> permT) $ pi) $
   319                         Bound i) ((length Us - 1 downto 0) ~~ Us)))
   320                 end
   321               else Const ("Nominal.perm", permT --> T --> T) $ pi $ x
   322             end;
   323         in
   324           (("", HOLogic.mk_Trueprop (HOLogic.mk_eq
   325             (Const (List.nth (perm_names_types, i)) $
   326                Free ("pi", mk_permT (TFree ("'x", HOLogic.typeS))) $
   327                list_comb (c, args),
   328              list_comb (c, map perm_arg (dts ~~ args))))), [])
   329         end) constrs
   330       end) descr);
   331 
   332     val (perm_simps, thy2) = thy1 |>
   333       Sign.add_consts_i (map (fn (s, T) => (Sign.base_name s, T, NoSyn))
   334         (List.drop (perm_names_types, length new_type_names))) |>
   335       PrimrecPackage.add_primrec_unchecked_i "" perm_eqs;
   336 
   337     (**** prove that permutation functions introduced by unfolding are ****)
   338     (**** equivalent to already existing permutation functions         ****)
   339 
   340     val _ = warning ("length descr: " ^ string_of_int (length descr));
   341     val _ = warning ("length new_type_names: " ^ string_of_int (length new_type_names));
   342 
   343     val perm_indnames = DatatypeProp.make_tnames (map body_type perm_types);
   344     val perm_fun_def = PureThy.get_thm thy2 (Name "perm_fun_def");
   345 
   346     val unfolded_perm_eq_thms =
   347       if length descr = length new_type_names then []
   348       else map standard (List.drop (split_conj_thm
   349         (Goal.prove_global thy2 [] []
   350           (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
   351             (map (fn (c as (s, T), x) =>
   352                let val [T1, T2] = binder_types T
   353                in HOLogic.mk_eq (Const c $ pi $ Free (x, T2),
   354                  Const ("Nominal.perm", T) $ pi $ Free (x, T2))
   355                end)
   356              (perm_names_types ~~ perm_indnames))))
   357           (fn _ => EVERY [indtac induction perm_indnames 1,
   358             ALLGOALS (asm_full_simp_tac
   359               (simpset_of thy2 addsimps [perm_fun_def]))])),
   360         length new_type_names));
   361 
   362     (**** prove [] \<bullet> t = t ****)
   363 
   364     val _ = warning "perm_empty_thms";
   365 
   366     val perm_empty_thms = List.concat (map (fn a =>
   367       let val permT = mk_permT (Type (a, []))
   368       in map standard (List.take (split_conj_thm
   369         (Goal.prove_global thy2 [] []
   370           (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
   371             (map (fn ((s, T), x) => HOLogic.mk_eq
   372                 (Const (s, permT --> T --> T) $
   373                    Const ("List.list.Nil", permT) $ Free (x, T),
   374                  Free (x, T)))
   375              (perm_names ~~
   376               map body_type perm_types ~~ perm_indnames))))
   377           (fn _ => EVERY [indtac induction perm_indnames 1,
   378             ALLGOALS (asm_full_simp_tac (simpset_of thy2))])),
   379         length new_type_names))
   380       end)
   381       atoms);
   382 
   383     (**** prove (pi1 @ pi2) \<bullet> t = pi1 \<bullet> (pi2 \<bullet> t) ****)
   384 
   385     val _ = warning "perm_append_thms";
   386 
   387     (*FIXME: these should be looked up statically*)
   388     val at_pt_inst = PureThy.get_thm thy2 (Name "at_pt_inst");
   389     val pt2 = PureThy.get_thm thy2 (Name "pt2");
   390 
   391     val perm_append_thms = List.concat (map (fn a =>
   392       let
   393         val permT = mk_permT (Type (a, []));
   394         val pi1 = Free ("pi1", permT);
   395         val pi2 = Free ("pi2", permT);
   396         val pt_inst = PureThy.get_thm thy2 (Name ("pt_" ^ Sign.base_name a ^ "_inst"));
   397         val pt2' = pt_inst RS pt2;
   398         val pt2_ax = PureThy.get_thm thy2
   399           (Name (NameSpace.map_base (fn s => "pt_" ^ s ^ "2") a));
   400       in List.take (map standard (split_conj_thm
   401         (Goal.prove_global thy2 [] []
   402              (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
   403                 (map (fn ((s, T), x) =>
   404                     let val perm = Const (s, permT --> T --> T)
   405                     in HOLogic.mk_eq
   406                       (perm $ (Const ("List.append", permT --> permT --> permT) $
   407                          pi1 $ pi2) $ Free (x, T),
   408                        perm $ pi1 $ (perm $ pi2 $ Free (x, T)))
   409                     end)
   410                   (perm_names ~~
   411                    map body_type perm_types ~~ perm_indnames))))
   412            (fn _ => EVERY [indtac induction perm_indnames 1,
   413               ALLGOALS (asm_full_simp_tac (simpset_of thy2 addsimps [pt2', pt2_ax]))]))),
   414          length new_type_names)
   415       end) atoms);
   416 
   417     (**** prove pi1 ~ pi2 ==> pi1 \<bullet> t = pi2 \<bullet> t ****)
   418 
   419     val _ = warning "perm_eq_thms";
   420 
   421     val pt3 = PureThy.get_thm thy2 (Name "pt3");
   422     val pt3_rev = PureThy.get_thm thy2 (Name "pt3_rev");
   423 
   424     val perm_eq_thms = List.concat (map (fn a =>
   425       let
   426         val permT = mk_permT (Type (a, []));
   427         val pi1 = Free ("pi1", permT);
   428         val pi2 = Free ("pi2", permT);
   429         (*FIXME: not robust - better access these theorems using NominalData?*)
   430         val at_inst = PureThy.get_thm thy2 (Name ("at_" ^ Sign.base_name a ^ "_inst"));
   431         val pt_inst = PureThy.get_thm thy2 (Name ("pt_" ^ Sign.base_name a ^ "_inst"));
   432         val pt3' = pt_inst RS pt3;
   433         val pt3_rev' = at_inst RS (pt_inst RS pt3_rev);
   434         val pt3_ax = PureThy.get_thm thy2
   435           (Name (NameSpace.map_base (fn s => "pt_" ^ s ^ "3") a));
   436       in List.take (map standard (split_conj_thm
   437         (Goal.prove_global thy2 [] [] (Logic.mk_implies
   438              (HOLogic.mk_Trueprop (Const ("Nominal.prm_eq",
   439                 permT --> permT --> HOLogic.boolT) $ pi1 $ pi2),
   440               HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
   441                 (map (fn ((s, T), x) =>
   442                     let val perm = Const (s, permT --> T --> T)
   443                     in HOLogic.mk_eq
   444                       (perm $ pi1 $ Free (x, T),
   445                        perm $ pi2 $ Free (x, T))
   446                     end)
   447                   (perm_names ~~
   448                    map body_type perm_types ~~ perm_indnames)))))
   449            (fn _ => EVERY [indtac induction perm_indnames 1,
   450               ALLGOALS (asm_full_simp_tac (simpset_of thy2 addsimps [pt3', pt3_rev', pt3_ax]))]))),
   451          length new_type_names)
   452       end) atoms);
   453 
   454     (**** prove pi1 \<bullet> (pi2 \<bullet> t) = (pi1 \<bullet> pi2) \<bullet> (pi1 \<bullet> t) ****)
   455 
   456     val cp1 = PureThy.get_thm thy2 (Name "cp1");
   457     val dj_cp = PureThy.get_thm thy2 (Name "dj_cp");
   458     val pt_perm_compose = PureThy.get_thm thy2 (Name "pt_perm_compose");
   459     val pt_perm_compose_rev = PureThy.get_thm thy2 (Name "pt_perm_compose_rev");
   460     val dj_perm_perm_forget = PureThy.get_thm thy2 (Name "dj_perm_perm_forget");
   461 
   462     fun composition_instance name1 name2 thy =
   463       let
   464         val name1' = Sign.base_name name1;
   465         val name2' = Sign.base_name name2;
   466         val cp_class = Sign.intern_class thy ("cp_" ^ name1' ^ "_" ^ name2');
   467         val permT1 = mk_permT (Type (name1, []));
   468         val permT2 = mk_permT (Type (name2, []));
   469         val augment = map_type_tfree
   470           (fn (x, S) => TFree (x, cp_class :: S));
   471         val Ts = map (augment o body_type) perm_types;
   472         val cp_inst = PureThy.get_thm thy
   473           (Name ("cp_" ^ name1' ^ "_" ^ name2' ^ "_inst"));
   474         val simps = simpset_of thy addsimps (perm_fun_def ::
   475           (if name1 <> name2 then
   476              let val dj = PureThy.get_thm thy (Name ("dj_" ^ name2' ^ "_" ^ name1'))
   477              in [dj RS (cp_inst RS dj_cp), dj RS dj_perm_perm_forget] end
   478            else
   479              let
   480                val at_inst = PureThy.get_thm thy (Name ("at_" ^ name1' ^ "_inst"));
   481                val pt_inst = PureThy.get_thm thy (Name ("pt_" ^ name1' ^ "_inst"))
   482              in
   483                [cp_inst RS cp1 RS sym,
   484                 at_inst RS (pt_inst RS pt_perm_compose) RS sym,
   485                 at_inst RS (pt_inst RS pt_perm_compose_rev) RS sym]
   486             end))
   487         val thms = split_conj_thm (Goal.prove_global thy [] []
   488             (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
   489               (map (fn ((s, T), x) =>
   490                   let
   491                     val pi1 = Free ("pi1", permT1);
   492                     val pi2 = Free ("pi2", permT2);
   493                     val perm1 = Const (s, permT1 --> T --> T);
   494                     val perm2 = Const (s, permT2 --> T --> T);
   495                     val perm3 = Const ("Nominal.perm", permT1 --> permT2 --> permT2)
   496                   in HOLogic.mk_eq
   497                     (perm1 $ pi1 $ (perm2 $ pi2 $ Free (x, T)),
   498                      perm2 $ (perm3 $ pi1 $ pi2) $ (perm1 $ pi1 $ Free (x, T)))
   499                   end)
   500                 (perm_names ~~ Ts ~~ perm_indnames))))
   501           (fn _ => EVERY [indtac induction perm_indnames 1,
   502              ALLGOALS (asm_full_simp_tac simps)]))
   503       in
   504         foldl (fn ((s, tvs), thy) => AxClass.prove_arity
   505             (s, replicate (length tvs) (cp_class :: classes), [cp_class])
   506             (Class.intro_classes_tac [] THEN ALLGOALS (resolve_tac thms)) thy)
   507           thy (full_new_type_names' ~~ tyvars)
   508       end;
   509 
   510     val (perm_thmss,thy3) = thy2 |>
   511       fold (fn name1 => fold (composition_instance name1) atoms) atoms |>
   512       curry (Library.foldr (fn ((i, (tyname, args, _)), thy) =>
   513         AxClass.prove_arity (tyname, replicate (length args) classes, classes)
   514         (Class.intro_classes_tac [] THEN REPEAT (EVERY
   515            [resolve_tac perm_empty_thms 1,
   516             resolve_tac perm_append_thms 1,
   517             resolve_tac perm_eq_thms 1, assume_tac 1])) thy))
   518         (List.take (descr, length new_type_names)) |>
   519       PureThy.add_thmss
   520         [((space_implode "_" new_type_names ^ "_unfolded_perm_eq",
   521           unfolded_perm_eq_thms), [Simplifier.simp_add]),
   522          ((space_implode "_" new_type_names ^ "_perm_empty",
   523           perm_empty_thms), [Simplifier.simp_add]),
   524          ((space_implode "_" new_type_names ^ "_perm_append",
   525           perm_append_thms), [Simplifier.simp_add]),
   526          ((space_implode "_" new_type_names ^ "_perm_eq",
   527           perm_eq_thms), [Simplifier.simp_add])];
   528 
   529     (**** Define representing sets ****)
   530 
   531     val _ = warning "representing sets";
   532 
   533     val rep_set_names = DatatypeProp.indexify_names
   534       (map (fn (i, _) => name_of_typ (nth_dtyp i) ^ "_set") descr);
   535     val big_rep_name =
   536       space_implode "_" (DatatypeProp.indexify_names (List.mapPartial
   537         (fn (i, ("Nominal.noption", _, _)) => NONE
   538           | (i, _) => SOME (name_of_typ (nth_dtyp i))) descr)) ^ "_set";
   539     val _ = warning ("big_rep_name: " ^ big_rep_name);
   540 
   541     fun strip_option (dtf as DtType ("fun", [dt, DtRec i])) =
   542           (case AList.lookup op = descr i of
   543              SOME ("Nominal.noption", _, [(_, [dt']), _]) =>
   544                apfst (cons dt) (strip_option dt')
   545            | _ => ([], dtf))
   546       | strip_option (DtType ("fun", [dt, DtType ("Nominal.noption", [dt'])])) =
   547           apfst (cons dt) (strip_option dt')
   548       | strip_option dt = ([], dt);
   549 
   550     val dt_atomTs = distinct op = (map (typ_of_dtyp descr sorts')
   551       (List.concat (map (fn (_, (_, _, cs)) => List.concat
   552         (map (List.concat o map (fst o strip_option) o snd) cs)) descr)));
   553 
   554     fun make_intr s T (cname, cargs) =
   555       let
   556         fun mk_prem (dt, (j, j', prems, ts)) =
   557           let
   558             val (dts, dt') = strip_option dt;
   559             val (dts', dt'') = strip_dtyp dt';
   560             val Ts = map (typ_of_dtyp descr sorts') dts;
   561             val Us = map (typ_of_dtyp descr sorts') dts';
   562             val T = typ_of_dtyp descr sorts' dt'';
   563             val free = mk_Free "x" (Us ---> T) j;
   564             val free' = app_bnds free (length Us);
   565             fun mk_abs_fun (T, (i, t)) =
   566               let val U = fastype_of t
   567               in (i + 1, Const ("Nominal.abs_fun", [T, U, T] --->
   568                 Type ("Nominal.noption", [U])) $ mk_Free "y" T i $ t)
   569               end
   570           in (j + 1, j' + length Ts,
   571             case dt'' of
   572                 DtRec k => list_all (map (pair "x") Us,
   573                   HOLogic.mk_Trueprop (Free (List.nth (rep_set_names, k),
   574                     T --> HOLogic.boolT) $ free')) :: prems
   575               | _ => prems,
   576             snd (foldr mk_abs_fun (j', free) Ts) :: ts)
   577           end;
   578 
   579         val (_, _, prems, ts) = foldr mk_prem (1, 1, [], []) cargs;
   580         val concl = HOLogic.mk_Trueprop (Free (s, T --> HOLogic.boolT) $
   581           list_comb (Const (cname, map fastype_of ts ---> T), ts))
   582       in Logic.list_implies (prems, concl)
   583       end;
   584 
   585     val (intr_ts, (rep_set_names', recTs')) =
   586       apfst List.concat (apsnd ListPair.unzip (ListPair.unzip (List.mapPartial
   587         (fn ((_, ("Nominal.noption", _, _)), _) => NONE
   588           | ((i, (_, _, constrs)), rep_set_name) =>
   589               let val T = nth_dtyp i
   590               in SOME (map (make_intr rep_set_name T) constrs,
   591                 (rep_set_name, T))
   592               end)
   593                 (descr ~~ rep_set_names))));
   594     val rep_set_names'' = map (Sign.full_name thy3) rep_set_names';
   595 
   596     val ({raw_induct = rep_induct, intrs = rep_intrs, ...}, thy4) =
   597       setmp InductivePackage.quiet_mode false
   598         (InductivePackage.add_inductive_global {verbose = false, kind = Thm.internalK,
   599             alt_name = big_rep_name, coind = false, no_elim = true, no_ind = false}
   600            (map (fn (s, T) => ((s, T --> HOLogic.boolT), NoSyn))
   601               (rep_set_names' ~~ recTs'))
   602            [] (map (fn x => (("", []), x)) intr_ts) []) thy3;
   603 
   604     (**** Prove that representing set is closed under permutation ****)
   605 
   606     val _ = warning "proving closure under permutation...";
   607 
   608     val perm_indnames' = List.mapPartial
   609       (fn (x, (_, ("Nominal.noption", _, _))) => NONE | (x, _) => SOME x)
   610       (perm_indnames ~~ descr);
   611 
   612     fun mk_perm_closed name = map (fn th => standard (th RS mp))
   613       (List.take (split_conj_thm (Goal.prove_global thy4 [] []
   614         (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj (map
   615            (fn ((s, T), x) =>
   616               let
   617                 val T = map_type_tfree
   618                   (fn (s, cs) => TFree (s, cs union cp_classes)) T;
   619                 val S = Const (s, T --> HOLogic.boolT);
   620                 val permT = mk_permT (Type (name, []))
   621               in HOLogic.mk_imp (S $ Free (x, T),
   622                 S $ (Const ("Nominal.perm", permT --> T --> T) $
   623                   Free ("pi", permT) $ Free (x, T)))
   624               end) (rep_set_names'' ~~ recTs' ~~ perm_indnames'))))
   625         (fn _ => EVERY (* CU: added perm_fun_def in the final tactic in order to deal with funs *)
   626            [indtac rep_induct [] 1,
   627             ALLGOALS (simp_tac (simpset_of thy4 addsimps
   628               (symmetric perm_fun_def :: PureThy.get_thms thy4 (Name ("abs_perm"))))),
   629             ALLGOALS (resolve_tac rep_intrs
   630                THEN_ALL_NEW (asm_full_simp_tac (simpset_of thy4 addsimps [perm_fun_def])))])),
   631         length new_type_names));
   632 
   633     (* FIXME: theorems are stored in database for testing only *)
   634     val perm_closed_thmss = map mk_perm_closed atoms;
   635     val (_, thy5) = PureThy.add_thmss [(("perm_closed", List.concat perm_closed_thmss), [])] thy4;
   636 
   637     (**** typedef ****)
   638 
   639     val _ = warning "defining type...";
   640 
   641     val (typedefs, thy6) =
   642       thy5
   643       |> fold_map (fn ((((name, mx), tvs), (cname, U)), name') => fn thy =>
   644         setmp TypedefPackage.quiet_mode true
   645           (TypedefPackage.add_typedef_i false (SOME name') (name, tvs, mx)
   646             (Const ("Collect", (U --> HOLogic.boolT) --> HOLogic.mk_setT U) $
   647                Const (cname, U --> HOLogic.boolT)) NONE
   648             (rtac exI 1 THEN rtac CollectI 1 THEN
   649               QUIET_BREADTH_FIRST (has_fewer_prems 1)
   650               (resolve_tac rep_intrs 1))) thy |> (fn ((_, r), thy) =>
   651         let
   652           val permT = mk_permT (TFree (Name.variant tvs "'a", HOLogic.typeS));
   653           val pi = Free ("pi", permT);
   654           val tvs' = map (fn s => TFree (s, the (AList.lookup op = sorts' s))) tvs;
   655           val T = Type (Sign.intern_type thy name, tvs');
   656         in apfst (pair r o hd)
   657           (PureThy.add_defs_unchecked_i true [(("prm_" ^ name ^ "_def", Logic.mk_equals
   658             (Const ("Nominal.perm", permT --> T --> T) $ pi $ Free ("x", T),
   659              Const (Sign.intern_const thy ("Abs_" ^ name), U --> T) $
   660                (Const ("Nominal.perm", permT --> U --> U) $ pi $
   661                  (Const (Sign.intern_const thy ("Rep_" ^ name), T --> U) $
   662                    Free ("x", T))))), [])] thy)
   663         end))
   664           (types_syntax ~~ tyvars ~~
   665             List.take (rep_set_names'' ~~ recTs', length new_type_names) ~~
   666             new_type_names);
   667 
   668     val perm_defs = map snd typedefs;
   669     val Abs_inverse_thms = map (collect_simp o #Abs_inverse o fst) typedefs;
   670     val Rep_inverse_thms = map (#Rep_inverse o fst) typedefs;
   671     val Rep_thms = map (collect_simp o #Rep o fst) typedefs;
   672 
   673     val big_name = space_implode "_" new_type_names;
   674 
   675 
   676     (** prove that new types are in class pt_<name> **)
   677 
   678     val _ = warning "prove that new types are in class pt_<name> ...";
   679 
   680     fun pt_instance ((class, atom), perm_closed_thms) =
   681       fold (fn ((((((Abs_inverse, Rep_inverse), Rep),
   682         perm_def), name), tvs), perm_closed) => fn thy =>
   683           AxClass.prove_arity
   684             (Sign.intern_type thy name,
   685               replicate (length tvs) (classes @ cp_classes), [class])
   686             (EVERY [Class.intro_classes_tac [],
   687               rewrite_goals_tac [perm_def],
   688               asm_full_simp_tac (simpset_of thy addsimps [Rep_inverse]) 1,
   689               asm_full_simp_tac (simpset_of thy addsimps
   690                 [Rep RS perm_closed RS Abs_inverse]) 1,
   691               asm_full_simp_tac (HOL_basic_ss addsimps [PureThy.get_thm thy
   692                 (Name ("pt_" ^ Sign.base_name atom ^ "3"))]) 1]) thy)
   693         (Abs_inverse_thms ~~ Rep_inverse_thms ~~ Rep_thms ~~ perm_defs ~~
   694            new_type_names ~~ tyvars ~~ perm_closed_thms);
   695 
   696 
   697     (** prove that new types are in class cp_<name1>_<name2> **)
   698 
   699     val _ = warning "prove that new types are in class cp_<name1>_<name2> ...";
   700 
   701     fun cp_instance (atom1, perm_closed_thms1) (atom2, perm_closed_thms2) thy =
   702       let
   703         val name = "cp_" ^ Sign.base_name atom1 ^ "_" ^ Sign.base_name atom2;
   704         val class = Sign.intern_class thy name;
   705         val cp1' = PureThy.get_thm thy (Name (name ^ "_inst")) RS cp1
   706       in fold (fn ((((((Abs_inverse, Rep),
   707         perm_def), name), tvs), perm_closed1), perm_closed2) => fn thy =>
   708           AxClass.prove_arity
   709             (Sign.intern_type thy name,
   710               replicate (length tvs) (classes @ cp_classes), [class])
   711             (EVERY [Class.intro_classes_tac [],
   712               rewrite_goals_tac [perm_def],
   713               asm_full_simp_tac (simpset_of thy addsimps
   714                 ((Rep RS perm_closed1 RS Abs_inverse) ::
   715                  (if atom1 = atom2 then []
   716                   else [Rep RS perm_closed2 RS Abs_inverse]))) 1,
   717               cong_tac 1,
   718               rtac refl 1,
   719               rtac cp1' 1]) thy)
   720         (Abs_inverse_thms ~~ Rep_thms ~~ perm_defs ~~ new_type_names ~~
   721            tyvars ~~ perm_closed_thms1 ~~ perm_closed_thms2) thy
   722       end;
   723 
   724     val thy7 = fold (fn x => fn thy => thy |>
   725       pt_instance x |>
   726       fold (cp_instance (apfst snd x)) (atoms ~~ perm_closed_thmss))
   727         (classes ~~ atoms ~~ perm_closed_thmss) thy6;
   728 
   729     (**** constructors ****)
   730 
   731     fun mk_abs_fun (x, t) =
   732       let
   733         val T = fastype_of x;
   734         val U = fastype_of t
   735       in
   736         Const ("Nominal.abs_fun", T --> U --> T -->
   737           Type ("Nominal.noption", [U])) $ x $ t
   738       end;
   739 
   740     val (ty_idxs, _) = foldl
   741       (fn ((i, ("Nominal.noption", _, _)), p) => p
   742         | ((i, _), (ty_idxs, j)) => (ty_idxs @ [(i, j)], j + 1)) ([], 0) descr;
   743 
   744     fun reindex (DtType (s, dts)) = DtType (s, map reindex dts)
   745       | reindex (DtRec i) = DtRec (the (AList.lookup op = ty_idxs i))
   746       | reindex dt = dt;
   747 
   748     fun strip_suffix i s = implode (List.take (explode s, size s - i));
   749 
   750     (** strips the "_Rep" in type names *)
   751     fun strip_nth_name i s =
   752       let val xs = NameSpace.explode s;
   753       in NameSpace.implode (Library.nth_map (length xs - i) (strip_suffix 4) xs) end;
   754 
   755     val (descr'', ndescr) = ListPair.unzip (List.mapPartial
   756       (fn (i, ("Nominal.noption", _, _)) => NONE
   757         | (i, (s, dts, constrs)) =>
   758              let
   759                val SOME index = AList.lookup op = ty_idxs i;
   760                val (constrs1, constrs2) = ListPair.unzip
   761                  (map (fn (cname, cargs) => apfst (pair (strip_nth_name 2 (strip_nth_name 1 cname)))
   762                    (foldl_map (fn (dts, dt) =>
   763                      let val (dts', dt') = strip_option dt
   764                      in (dts @ dts' @ [reindex dt'], (length dts, length dts')) end)
   765                        ([], cargs))) constrs)
   766              in SOME ((index, (strip_nth_name 1 s,  map reindex dts, constrs1)),
   767                (index, constrs2))
   768              end) descr);
   769 
   770     val (descr1, descr2) = chop (length new_type_names) descr'';
   771     val descr' = [descr1, descr2];
   772 
   773     fun partition_cargs idxs xs = map (fn (i, j) =>
   774       (List.take (List.drop (xs, i), j), List.nth (xs, i + j))) idxs;
   775 
   776     val pdescr = map (fn ((i, (s, dts, constrs)), (_, idxss)) => (i, (s, dts,
   777       map (fn ((cname, cargs), idxs) => (cname, partition_cargs idxs cargs))
   778         (constrs ~~ idxss)))) (descr'' ~~ ndescr);
   779 
   780     fun nth_dtyp' i = typ_of_dtyp descr'' sorts' (DtRec i);
   781 
   782     val rep_names = map (fn s =>
   783       Sign.intern_const thy7 ("Rep_" ^ s)) new_type_names;
   784     val abs_names = map (fn s =>
   785       Sign.intern_const thy7 ("Abs_" ^ s)) new_type_names;
   786 
   787     val recTs = get_rec_types descr'' sorts';
   788     val newTs' = Library.take (length new_type_names, recTs');
   789     val newTs = Library.take (length new_type_names, recTs);
   790 
   791     val full_new_type_names = map (Sign.full_name thy) new_type_names;
   792 
   793     fun make_constr_def tname T T' ((thy, defs, eqns),
   794         (((cname_rep, _), (cname, cargs)), (cname', mx))) =
   795       let
   796         fun constr_arg ((dts, dt), (j, l_args, r_args)) =
   797           let
   798             val xs = map (fn (dt, i) => mk_Free "x" (typ_of_dtyp descr'' sorts' dt) i)
   799               (dts ~~ (j upto j + length dts - 1))
   800             val x = mk_Free "x" (typ_of_dtyp descr'' sorts' dt) (j + length dts)
   801           in
   802             (j + length dts + 1,
   803              xs @ x :: l_args,
   804              foldr mk_abs_fun
   805                (case dt of
   806                   DtRec k => if k < length new_type_names then
   807                       Const (List.nth (rep_names, k), typ_of_dtyp descr'' sorts' dt -->
   808                         typ_of_dtyp descr sorts' dt) $ x
   809                     else error "nested recursion not (yet) supported"
   810                 | _ => x) xs :: r_args)
   811           end
   812 
   813         val (_, l_args, r_args) = foldr constr_arg (1, [], []) cargs;
   814         val abs_name = Sign.intern_const thy ("Abs_" ^ tname);
   815         val rep_name = Sign.intern_const thy ("Rep_" ^ tname);
   816         val constrT = map fastype_of l_args ---> T;
   817         val lhs = list_comb (Const (cname, constrT), l_args);
   818         val rhs = list_comb (Const (cname_rep, map fastype_of r_args ---> T'), r_args);
   819         val def = Logic.mk_equals (lhs, Const (abs_name, T' --> T) $ rhs);
   820         val eqn = HOLogic.mk_Trueprop (HOLogic.mk_eq
   821           (Const (rep_name, T --> T') $ lhs, rhs));
   822         val def_name = (Sign.base_name cname) ^ "_def";
   823         val ([def_thm], thy') = thy |>
   824           Sign.add_consts_i [(cname', constrT, mx)] |>
   825           (PureThy.add_defs_i false o map Thm.no_attributes) [(def_name, def)]
   826       in (thy', defs @ [def_thm], eqns @ [eqn]) end;
   827 
   828     fun dt_constr_defs ((thy, defs, eqns, dist_lemmas), ((((((_, (_, _, constrs)),
   829         (_, (_, _, constrs'))), tname), T), T'), constr_syntax)) =
   830       let
   831         val rep_const = cterm_of thy
   832           (Const (Sign.intern_const thy ("Rep_" ^ tname), T --> T'));
   833         val dist = standard (cterm_instantiate [(cterm_of thy distinct_f, rep_const)] distinct_lemma);
   834         val (thy', defs', eqns') = Library.foldl (make_constr_def tname T T')
   835           ((Sign.add_path tname thy, defs, []), constrs ~~ constrs' ~~ constr_syntax)
   836       in
   837         (parent_path flat_names thy', defs', eqns @ [eqns'], dist_lemmas @ [dist])
   838       end;
   839 
   840     val (thy8, constr_defs, constr_rep_eqns, dist_lemmas) = Library.foldl dt_constr_defs
   841       ((thy7, [], [], []), List.take (descr, length new_type_names) ~~
   842         List.take (pdescr, length new_type_names) ~~
   843         new_type_names ~~ newTs ~~ newTs' ~~ constr_syntax);
   844 
   845     val abs_inject_thms = map (collect_simp o #Abs_inject o fst) typedefs
   846     val rep_inject_thms = map (#Rep_inject o fst) typedefs
   847 
   848     (* prove theorem  Rep_i (Constr_j ...) = Constr'_j ...  *)
   849 
   850     fun prove_constr_rep_thm eqn =
   851       let
   852         val inj_thms = map (fn r => r RS iffD1) abs_inject_thms;
   853         val rewrites = constr_defs @ map mk_meta_eq Rep_inverse_thms
   854       in Goal.prove_global thy8 [] [] eqn (fn _ => EVERY
   855         [resolve_tac inj_thms 1,
   856          rewrite_goals_tac rewrites,
   857          rtac refl 3,
   858          resolve_tac rep_intrs 2,
   859          REPEAT (resolve_tac Rep_thms 1)])
   860       end;
   861 
   862     val constr_rep_thmss = map (map prove_constr_rep_thm) constr_rep_eqns;
   863 
   864     (* prove theorem  pi \<bullet> Rep_i x = Rep_i (pi \<bullet> x) *)
   865 
   866     fun prove_perm_rep_perm (atom, perm_closed_thms) = map (fn th =>
   867       let
   868         val _ $ (_ $ (Rep $ x)) = Logic.unvarify (prop_of th);
   869         val Type ("fun", [T, U]) = fastype_of Rep;
   870         val permT = mk_permT (Type (atom, []));
   871         val pi = Free ("pi", permT);
   872       in
   873         Goal.prove_global thy8 [] [] (HOLogic.mk_Trueprop (HOLogic.mk_eq
   874             (Const ("Nominal.perm", permT --> U --> U) $ pi $ (Rep $ x),
   875              Rep $ (Const ("Nominal.perm", permT --> T --> T) $ pi $ x))))
   876           (fn _ => simp_tac (HOL_basic_ss addsimps (perm_defs @ Abs_inverse_thms @
   877             perm_closed_thms @ Rep_thms)) 1)
   878       end) Rep_thms;
   879 
   880     val perm_rep_perm_thms = List.concat (map prove_perm_rep_perm
   881       (atoms ~~ perm_closed_thmss));
   882 
   883     (* prove distinctness theorems *)
   884 
   885     val distinctness_limit = Config.get_thy thy8 DatatypeProp.distinctness_limit;
   886     val thy8' = Config.put_thy DatatypeProp.distinctness_limit 1000 thy8;
   887     val distinct_props = DatatypeProp.make_distincts new_type_names descr' sorts' thy8';
   888     val thy8 = Config.put_thy DatatypeProp.distinctness_limit distinctness_limit thy8';
   889 
   890     val dist_rewrites = map (fn (rep_thms, dist_lemma) =>
   891       dist_lemma::(rep_thms @ [In0_eq, In1_eq, In0_not_In1, In1_not_In0]))
   892         (constr_rep_thmss ~~ dist_lemmas);
   893 
   894     fun prove_distinct_thms (_, []) = []
   895       | prove_distinct_thms (p as (rep_thms, dist_lemma), t::ts) =
   896           let
   897             val dist_thm = Goal.prove_global thy8 [] [] t (fn _ =>
   898               simp_tac (simpset_of thy8 addsimps (dist_lemma :: rep_thms)) 1)
   899           in dist_thm::(standard (dist_thm RS not_sym))::
   900             (prove_distinct_thms (p, ts))
   901           end;
   902 
   903     val distinct_thms = map prove_distinct_thms
   904       (constr_rep_thmss ~~ dist_lemmas ~~ distinct_props);
   905 
   906     (** prove equations for permutation functions **)
   907 
   908     val abs_perm = PureThy.get_thms thy8 (Name "abs_perm"); (* FIXME *)
   909 
   910     val perm_simps' = map (fn (((i, (_, _, constrs)), tname), constr_rep_thms) =>
   911       let val T = nth_dtyp' i
   912       in List.concat (map (fn (atom, perm_closed_thms) =>
   913           map (fn ((cname, dts), constr_rep_thm) =>
   914         let
   915           val cname = Sign.intern_const thy8
   916             (NameSpace.append tname (Sign.base_name cname));
   917           val permT = mk_permT (Type (atom, []));
   918           val pi = Free ("pi", permT);
   919 
   920           fun perm t =
   921             let val T = fastype_of t
   922             in Const ("Nominal.perm", permT --> T --> T) $ pi $ t end;
   923 
   924           fun constr_arg ((dts, dt), (j, l_args, r_args)) =
   925             let
   926               val Ts = map (typ_of_dtyp descr'' sorts') dts;
   927               val xs = map (fn (T, i) => mk_Free "x" T i)
   928                 (Ts ~~ (j upto j + length dts - 1))
   929               val x = mk_Free "x" (typ_of_dtyp descr'' sorts' dt) (j + length dts)
   930             in
   931               (j + length dts + 1,
   932                xs @ x :: l_args,
   933                map perm (xs @ [x]) @ r_args)
   934             end
   935 
   936           val (_, l_args, r_args) = foldr constr_arg (1, [], []) dts;
   937           val c = Const (cname, map fastype_of l_args ---> T)
   938         in
   939           Goal.prove_global thy8 [] []
   940             (HOLogic.mk_Trueprop (HOLogic.mk_eq
   941               (perm (list_comb (c, l_args)), list_comb (c, r_args))))
   942             (fn _ => EVERY
   943               [simp_tac (simpset_of thy8 addsimps (constr_rep_thm :: perm_defs)) 1,
   944                simp_tac (HOL_basic_ss addsimps (Rep_thms @ Abs_inverse_thms @
   945                  constr_defs @ perm_closed_thms)) 1,
   946                TRY (simp_tac (HOL_basic_ss addsimps
   947                  (symmetric perm_fun_def :: abs_perm)) 1),
   948                TRY (simp_tac (HOL_basic_ss addsimps
   949                  (perm_fun_def :: perm_defs @ Rep_thms @ Abs_inverse_thms @
   950                     perm_closed_thms)) 1)])
   951         end) (constrs ~~ constr_rep_thms)) (atoms ~~ perm_closed_thmss))
   952       end) (List.take (pdescr, length new_type_names) ~~ new_type_names ~~ constr_rep_thmss);
   953 
   954     (** prove injectivity of constructors **)
   955 
   956     val rep_inject_thms' = map (fn th => th RS sym) rep_inject_thms;
   957     val alpha = PureThy.get_thms thy8 (Name "alpha");
   958     val abs_fresh = PureThy.get_thms thy8 (Name "abs_fresh");
   959 
   960     val inject_thms = map (fn (((i, (_, _, constrs)), tname), constr_rep_thms) =>
   961       let val T = nth_dtyp' i
   962       in List.mapPartial (fn ((cname, dts), constr_rep_thm) =>
   963         if null dts then NONE else SOME
   964         let
   965           val cname = Sign.intern_const thy8
   966             (NameSpace.append tname (Sign.base_name cname));
   967 
   968           fun make_inj ((dts, dt), (j, args1, args2, eqs)) =
   969             let
   970               val Ts_idx = map (typ_of_dtyp descr'' sorts') dts ~~ (j upto j + length dts - 1);
   971               val xs = map (fn (T, i) => mk_Free "x" T i) Ts_idx;
   972               val ys = map (fn (T, i) => mk_Free "y" T i) Ts_idx;
   973               val x = mk_Free "x" (typ_of_dtyp descr'' sorts' dt) (j + length dts);
   974               val y = mk_Free "y" (typ_of_dtyp descr'' sorts' dt) (j + length dts)
   975             in
   976               (j + length dts + 1,
   977                xs @ (x :: args1), ys @ (y :: args2),
   978                HOLogic.mk_eq
   979                  (foldr mk_abs_fun x xs, foldr mk_abs_fun y ys) :: eqs)
   980             end;
   981 
   982           val (_, args1, args2, eqs) = foldr make_inj (1, [], [], []) dts;
   983           val Ts = map fastype_of args1;
   984           val c = Const (cname, Ts ---> T)
   985         in
   986           Goal.prove_global thy8 [] [] (HOLogic.mk_Trueprop (HOLogic.mk_eq
   987               (HOLogic.mk_eq (list_comb (c, args1), list_comb (c, args2)),
   988                foldr1 HOLogic.mk_conj eqs)))
   989             (fn _ => EVERY
   990                [asm_full_simp_tac (simpset_of thy8 addsimps (constr_rep_thm ::
   991                   rep_inject_thms')) 1,
   992                 TRY (asm_full_simp_tac (HOL_basic_ss addsimps (fresh_def :: supp_def ::
   993                   alpha @ abs_perm @ abs_fresh @ rep_inject_thms @
   994                   perm_rep_perm_thms)) 1),
   995                 TRY (asm_full_simp_tac (HOL_basic_ss addsimps (perm_fun_def ::
   996                   expand_fun_eq :: rep_inject_thms @ perm_rep_perm_thms)) 1)])
   997         end) (constrs ~~ constr_rep_thms)
   998       end) (List.take (pdescr, length new_type_names) ~~ new_type_names ~~ constr_rep_thmss);
   999 
  1000     (** equations for support and freshness **)
  1001 
  1002     val (supp_thms, fresh_thms) = ListPair.unzip (map ListPair.unzip
  1003       (map (fn ((((i, (_, _, constrs)), tname), inject_thms'), perm_thms') =>
  1004       let val T = nth_dtyp' i
  1005       in List.concat (map (fn (cname, dts) => map (fn atom =>
  1006         let
  1007           val cname = Sign.intern_const thy8
  1008             (NameSpace.append tname (Sign.base_name cname));
  1009           val atomT = Type (atom, []);
  1010 
  1011           fun process_constr ((dts, dt), (j, args1, args2)) =
  1012             let
  1013               val Ts_idx = map (typ_of_dtyp descr'' sorts') dts ~~ (j upto j + length dts - 1);
  1014               val xs = map (fn (T, i) => mk_Free "x" T i) Ts_idx;
  1015               val x = mk_Free "x" (typ_of_dtyp descr'' sorts' dt) (j + length dts)
  1016             in
  1017               (j + length dts + 1,
  1018                xs @ (x :: args1), foldr mk_abs_fun x xs :: args2)
  1019             end;
  1020 
  1021           val (_, args1, args2) = foldr process_constr (1, [], []) dts;
  1022           val Ts = map fastype_of args1;
  1023           val c = list_comb (Const (cname, Ts ---> T), args1);
  1024           fun supp t =
  1025             Const ("Nominal.supp", fastype_of t --> HOLogic.mk_setT atomT) $ t;
  1026           fun fresh t =
  1027             Const ("Nominal.fresh", atomT --> fastype_of t --> HOLogic.boolT) $
  1028               Free ("a", atomT) $ t;
  1029           val supp_thm = Goal.prove_global thy8 [] []
  1030               (HOLogic.mk_Trueprop (HOLogic.mk_eq
  1031                 (supp c,
  1032                  if null dts then Const ("{}", HOLogic.mk_setT atomT)
  1033                  else foldr1 (HOLogic.mk_binop "op Un") (map supp args2))))
  1034             (fn _ =>
  1035               simp_tac (HOL_basic_ss addsimps (supp_def ::
  1036                  Un_assoc :: de_Morgan_conj :: Collect_disj_eq :: finite_Un ::
  1037                  symmetric empty_def :: finite_emptyI :: simp_thms @
  1038                  abs_perm @ abs_fresh @ inject_thms' @ perm_thms')) 1)
  1039         in
  1040           (supp_thm,
  1041            Goal.prove_global thy8 [] [] (HOLogic.mk_Trueprop (HOLogic.mk_eq
  1042               (fresh c,
  1043                if null dts then HOLogic.true_const
  1044                else foldr1 HOLogic.mk_conj (map fresh args2))))
  1045              (fn _ =>
  1046                simp_tac (HOL_ss addsimps [Un_iff, empty_iff, fresh_def, supp_thm]) 1))
  1047         end) atoms) constrs)
  1048       end) (List.take (pdescr, length new_type_names) ~~ new_type_names ~~ inject_thms ~~ perm_simps')));
  1049 
  1050     (**** weak induction theorem ****)
  1051 
  1052     fun mk_indrule_lemma ((prems, concls), (((i, _), T), U)) =
  1053       let
  1054         val Rep_t = Const (List.nth (rep_names, i), T --> U) $
  1055           mk_Free "x" T i;
  1056 
  1057         val Abs_t =  Const (List.nth (abs_names, i), U --> T)
  1058 
  1059       in (prems @ [HOLogic.imp $
  1060             (Const (List.nth (rep_set_names'', i), U --> HOLogic.boolT) $ Rep_t) $
  1061               (mk_Free "P" (T --> HOLogic.boolT) (i + 1) $ (Abs_t $ Rep_t))],
  1062           concls @ [mk_Free "P" (T --> HOLogic.boolT) (i + 1) $ mk_Free "x" T i])
  1063       end;
  1064 
  1065     val (indrule_lemma_prems, indrule_lemma_concls) =
  1066       Library.foldl mk_indrule_lemma (([], []), (descr'' ~~ recTs ~~ recTs'));
  1067 
  1068     val indrule_lemma = Goal.prove_global thy8 [] []
  1069       (Logic.mk_implies
  1070         (HOLogic.mk_Trueprop (mk_conj indrule_lemma_prems),
  1071          HOLogic.mk_Trueprop (mk_conj indrule_lemma_concls))) (fn _ => EVERY
  1072            [REPEAT (etac conjE 1),
  1073             REPEAT (EVERY
  1074               [TRY (rtac conjI 1), full_simp_tac (HOL_basic_ss addsimps Rep_inverse_thms) 1,
  1075                etac mp 1, resolve_tac Rep_thms 1])]);
  1076 
  1077     val Ps = map head_of (HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of indrule_lemma)));
  1078     val frees = if length Ps = 1 then [Free ("P", snd (dest_Var (hd Ps)))] else
  1079       map (Free o apfst fst o dest_Var) Ps;
  1080     val indrule_lemma' = cterm_instantiate
  1081       (map (cterm_of thy8) Ps ~~ map (cterm_of thy8) frees) indrule_lemma;
  1082 
  1083     val Abs_inverse_thms' = map (fn r => r RS subst) Abs_inverse_thms;
  1084 
  1085     val dt_induct_prop = DatatypeProp.make_ind descr' sorts';
  1086     val dt_induct = Goal.prove_global thy8 []
  1087       (Logic.strip_imp_prems dt_induct_prop) (Logic.strip_imp_concl dt_induct_prop)
  1088       (fn prems => EVERY
  1089         [rtac indrule_lemma' 1,
  1090          (DatatypeAux.indtac rep_induct THEN_ALL_NEW ObjectLogic.atomize_prems_tac) 1,
  1091          EVERY (map (fn (prem, r) => (EVERY
  1092            [REPEAT (eresolve_tac Abs_inverse_thms' 1),
  1093             simp_tac (HOL_basic_ss addsimps [symmetric r]) 1,
  1094             DEPTH_SOLVE_1 (ares_tac [prem] 1 ORELSE etac allE 1)]))
  1095                 (prems ~~ constr_defs))]);
  1096 
  1097     val case_names_induct = mk_case_names_induct descr'';
  1098 
  1099     (**** prove that new datatypes have finite support ****)
  1100 
  1101     val _ = warning "proving finite support for the new datatype";
  1102 
  1103     val indnames = DatatypeProp.make_tnames recTs;
  1104 
  1105     val abs_supp = PureThy.get_thms thy8 (Name "abs_supp");
  1106     val supp_atm = PureThy.get_thms thy8 (Name "supp_atm");
  1107 
  1108     val finite_supp_thms = map (fn atom =>
  1109       let val atomT = Type (atom, [])
  1110       in map standard (List.take
  1111         (split_conj_thm (Goal.prove_global thy8 [] [] (HOLogic.mk_Trueprop
  1112            (foldr1 HOLogic.mk_conj (map (fn (s, T) =>
  1113              Const ("Finite_Set.finite", HOLogic.mk_setT atomT --> HOLogic.boolT) $
  1114                (Const ("Nominal.supp", T --> HOLogic.mk_setT atomT) $ Free (s, T)))
  1115                (indnames ~~ recTs))))
  1116            (fn _ => indtac dt_induct indnames 1 THEN
  1117             ALLGOALS (asm_full_simp_tac (simpset_of thy8 addsimps
  1118               (abs_supp @ supp_atm @
  1119                PureThy.get_thms thy8 (Name ("fs_" ^ Sign.base_name atom ^ "1")) @
  1120                List.concat supp_thms))))),
  1121          length new_type_names))
  1122       end) atoms;
  1123 
  1124     val simp_atts = replicate (length new_type_names) [Simplifier.simp_add];
  1125 
  1126 	(* Function to add both the simp and eqvt attributes *)
  1127         (* These two attributes are duplicated on all the types in the mutual nominal datatypes *)
  1128 
  1129     val simp_eqvt_atts = replicate (length new_type_names) [Simplifier.simp_add, NominalThmDecls.eqvt_add];
  1130  
  1131     val (_, thy9) = thy8 |>
  1132       Sign.add_path big_name |>
  1133       PureThy.add_thms [(("weak_induct", dt_induct), [case_names_induct])] ||>>
  1134       PureThy.add_thmss [(("weak_inducts", projections dt_induct), [case_names_induct])] ||>
  1135       Sign.parent_path ||>>
  1136       DatatypeAux.store_thmss_atts "distinct" new_type_names simp_atts distinct_thms ||>>
  1137       DatatypeAux.store_thmss "constr_rep" new_type_names constr_rep_thmss ||>>
  1138       DatatypeAux.store_thmss_atts "perm" new_type_names simp_eqvt_atts perm_simps' ||>>
  1139       DatatypeAux.store_thmss "inject" new_type_names inject_thms ||>>
  1140       DatatypeAux.store_thmss "supp" new_type_names supp_thms ||>>
  1141       DatatypeAux.store_thmss_atts "fresh" new_type_names simp_atts fresh_thms ||>
  1142       fold (fn (atom, ths) => fn thy =>
  1143         let val class = Sign.intern_class thy ("fs_" ^ Sign.base_name atom)
  1144         in fold (fn T => AxClass.prove_arity
  1145             (fst (dest_Type T),
  1146               replicate (length sorts) [class], [class])
  1147             (Class.intro_classes_tac [] THEN resolve_tac ths 1)) newTs thy
  1148         end) (atoms ~~ finite_supp_thms);
  1149 
  1150     (**** strong induction theorem ****)
  1151 
  1152     val pnames = if length descr'' = 1 then ["P"]
  1153       else map (fn i => "P" ^ string_of_int i) (1 upto length descr'');
  1154     val ind_sort = if null dt_atomTs then HOLogic.typeS
  1155       else Sign.certify_sort thy9 (map (fn T => Sign.intern_class thy9 ("fs_" ^
  1156         Sign.base_name (fst (dest_Type T)))) dt_atomTs);
  1157     val fsT = TFree ("'n", ind_sort);
  1158     val fsT' = TFree ("'n", HOLogic.typeS);
  1159 
  1160     val fresh_fs = map (fn (s, T) => (T, Free (s, fsT' --> HOLogic.mk_setT T)))
  1161       (DatatypeProp.indexify_names (replicate (length dt_atomTs) "f") ~~ dt_atomTs);
  1162 
  1163     fun make_pred fsT i T =
  1164       Free (List.nth (pnames, i), fsT --> T --> HOLogic.boolT);
  1165 
  1166     fun mk_fresh1 xs [] = []
  1167       | mk_fresh1 xs ((y as (_, T)) :: ys) = map (fn x => HOLogic.mk_Trueprop
  1168             (HOLogic.mk_not (HOLogic.mk_eq (Free y, Free x))))
  1169               (filter (fn (_, U) => T = U) (rev xs)) @
  1170           mk_fresh1 (y :: xs) ys;
  1171 
  1172     fun mk_fresh2 xss [] = []
  1173       | mk_fresh2 xss ((p as (ys, _)) :: yss) = List.concat (map (fn y as (_, T) =>
  1174             map (fn (_, x as (_, U)) => HOLogic.mk_Trueprop
  1175               (Const ("Nominal.fresh", T --> U --> HOLogic.boolT) $ Free y $ Free x))
  1176                 (rev xss @ yss)) ys) @
  1177           mk_fresh2 (p :: xss) yss;
  1178 
  1179     fun make_ind_prem fsT f k T ((cname, cargs), idxs) =
  1180       let
  1181         val recs = List.filter is_rec_type cargs;
  1182         val Ts = map (typ_of_dtyp descr'' sorts') cargs;
  1183         val recTs' = map (typ_of_dtyp descr'' sorts') recs;
  1184         val tnames = Name.variant_list pnames (DatatypeProp.make_tnames Ts);
  1185         val rec_tnames = map fst (List.filter (is_rec_type o snd) (tnames ~~ cargs));
  1186         val frees = tnames ~~ Ts;
  1187         val frees' = partition_cargs idxs frees;
  1188         val z = (Name.variant tnames "z", fsT);
  1189 
  1190         fun mk_prem ((dt, s), T) =
  1191           let
  1192             val (Us, U) = strip_type T;
  1193             val l = length Us
  1194           in list_all (z :: map (pair "x") Us, HOLogic.mk_Trueprop
  1195             (make_pred fsT (body_index dt) U $ Bound l $ app_bnds (Free (s, T)) l))
  1196           end;
  1197 
  1198         val prems = map mk_prem (recs ~~ rec_tnames ~~ recTs');
  1199         val prems' = map (fn p as (_, T) => HOLogic.mk_Trueprop
  1200             (f T (Free p) (Free z))) (List.concat (map fst frees')) @
  1201           mk_fresh1 [] (List.concat (map fst frees')) @
  1202           mk_fresh2 [] frees'
  1203 
  1204       in list_all_free (frees @ [z], Logic.list_implies (prems' @ prems,
  1205         HOLogic.mk_Trueprop (make_pred fsT k T $ Free z $
  1206           list_comb (Const (cname, Ts ---> T), map Free frees))))
  1207       end;
  1208 
  1209     val ind_prems = List.concat (map (fn (((i, (_, _, constrs)), (_, idxss)), T) =>
  1210       map (make_ind_prem fsT (fn T => fn t => fn u =>
  1211         Const ("Nominal.fresh", T --> fsT --> HOLogic.boolT) $ t $ u) i T)
  1212           (constrs ~~ idxss)) (descr'' ~~ ndescr ~~ recTs));
  1213     val tnames = DatatypeProp.make_tnames recTs;
  1214     val zs = Name.variant_list tnames (replicate (length descr'') "z");
  1215     val ind_concl = HOLogic.mk_Trueprop (foldr1 (HOLogic.mk_binop "op &")
  1216       (map (fn ((((i, _), T), tname), z) =>
  1217         make_pred fsT i T $ Free (z, fsT) $ Free (tname, T))
  1218         (descr'' ~~ recTs ~~ tnames ~~ zs)));
  1219     val induct = Logic.list_implies (ind_prems, ind_concl);
  1220 
  1221     val ind_prems' =
  1222       map (fn (_, f as Free (_, T)) => list_all_free ([("x", fsT')],
  1223         HOLogic.mk_Trueprop (Const ("Finite_Set.finite", body_type T -->
  1224           HOLogic.boolT) $ (f $ Free ("x", fsT'))))) fresh_fs @
  1225       List.concat (map (fn (((i, (_, _, constrs)), (_, idxss)), T) =>
  1226         map (make_ind_prem fsT' (fn T => fn t => fn u => HOLogic.Not $
  1227           HOLogic.mk_mem (t, the (AList.lookup op = fresh_fs T) $ u)) i T)
  1228             (constrs ~~ idxss)) (descr'' ~~ ndescr ~~ recTs));
  1229     val ind_concl' = HOLogic.mk_Trueprop (foldr1 (HOLogic.mk_binop "op &")
  1230       (map (fn ((((i, _), T), tname), z) =>
  1231         make_pred fsT' i T $ Free (z, fsT') $ Free (tname, T))
  1232         (descr'' ~~ recTs ~~ tnames ~~ zs)));
  1233     val induct' = Logic.list_implies (ind_prems', ind_concl');
  1234 
  1235     val aux_ind_vars =
  1236       (DatatypeProp.indexify_names (replicate (length dt_atomTs) "pi") ~~
  1237        map mk_permT dt_atomTs) @ [("z", fsT')];
  1238     val aux_ind_Ts = rev (map snd aux_ind_vars);
  1239     val aux_ind_concl = HOLogic.mk_Trueprop (foldr1 (HOLogic.mk_binop "op &")
  1240       (map (fn (((i, _), T), tname) =>
  1241         HOLogic.list_all (aux_ind_vars, make_pred fsT' i T $ Bound 0 $
  1242           fold_rev (mk_perm aux_ind_Ts) (map Bound (length dt_atomTs downto 1))
  1243             (Free (tname, T))))
  1244         (descr'' ~~ recTs ~~ tnames)));
  1245 
  1246     fun mk_ind_perm i k p l vs j =
  1247       let
  1248         val n = length vs;
  1249         val Ts = map snd vs;
  1250         val T = List.nth (Ts, i - j);
  1251         val pT = NominalAtoms.mk_permT T
  1252       in
  1253         Const ("List.list.Cons", HOLogic.mk_prodT (T, T) --> pT --> pT) $
  1254           (HOLogic.pair_const T T $ Bound (l - j) $ fold_rev (mk_perm Ts)
  1255             (map (mk_ind_perm i k p l vs) (j - 1 downto 0) @
  1256              map Bound (n - k - 1 downto n - k - p))
  1257             (Bound (i - j))) $
  1258           Const ("List.list.Nil", pT)
  1259       end;
  1260 
  1261     fun mk_fresh i i' j k p l is vs _ _ =
  1262       let
  1263         val n = length vs;
  1264         val Ts = map snd vs;
  1265         val T = List.nth (Ts, n - i - 1 - j);
  1266         val f = the (AList.lookup op = fresh_fs T);
  1267         val U = List.nth (Ts, n - i' - 1);
  1268         val S = HOLogic.mk_setT T;
  1269         val prms' = map Bound (n - k downto n - k - p + 1);
  1270         val prms = map (mk_ind_perm (n - i) k p (n - l) (("a", T) :: vs))
  1271             (j - 1 downto 0) @ prms';
  1272         val bs = rev (List.mapPartial
  1273           (fn ((_, T'), i) => if T = T' then SOME (Bound i) else NONE)
  1274           (List.take (vs, n - k - p - 1) ~~ (1 upto n - k - p - 1)));
  1275         val ts = map (fn i =>
  1276           Const ("Nominal.supp", List.nth (Ts, n - i - 1) --> S) $
  1277             fold_rev (mk_perm (T :: Ts)) prms' (Bound (n - i))) is
  1278       in
  1279         HOLogic.mk_Trueprop (Const ("Ex", (T --> HOLogic.boolT) --> HOLogic.boolT) $
  1280           Abs ("a", T, HOLogic.Not $
  1281             (Const ("op :", T --> S --> HOLogic.boolT) $ Bound 0 $
  1282               (foldr (fn (t, u) => Const ("insert", T --> S --> S) $ t $ u)
  1283                 (foldl (fn (t, u) => Const ("op Un", S --> S --> S) $ u $ t)
  1284                   (f $ Bound (n - k - p))
  1285                   (Const ("Nominal.supp", U --> S) $
  1286                      fold_rev (mk_perm (T :: Ts)) prms (Bound (n - i')) :: ts))
  1287                 (fold_rev (mk_perm (T :: Ts)) prms (Bound (n - i - j)) :: bs)))))
  1288       end;
  1289 
  1290     fun mk_fresh_constr is p vs _ concl =
  1291       let
  1292         val n = length vs;
  1293         val Ts = map snd vs;
  1294         val _ $ (_ $ _ $ t) = concl;
  1295         val c = head_of t;
  1296         val T = body_type (fastype_of c);
  1297         val k = foldr op + 0 (map (fn (_, i) => i + 1) is);
  1298         val ps = map Bound (n - k - 1 downto n - k - p);
  1299         val (_, _, ts, us) = foldl (fn ((_, i), (m, n, ts, us)) =>
  1300           (m - i - 1, n - i,
  1301            ts @ map Bound (n downto n - i + 1) @ [fold_rev (mk_perm Ts)
  1302              (map (mk_ind_perm m k p n vs) (i - 1 downto 0) @ ps)
  1303              (Bound (m - i))],
  1304            us @ map (fn j => fold_rev (mk_perm Ts) ps (Bound j)) (m downto m - i)))
  1305           (n - 1, n - k - p - 2, [], []) is
  1306       in
  1307         HOLogic.mk_Trueprop (HOLogic.eq_const T $ list_comb (c, ts) $ list_comb (c, us))
  1308       end;
  1309 
  1310     val abs_fun_finite_supp = PureThy.get_thm thy9 (Name "abs_fun_finite_supp");
  1311 
  1312     val at_finite_select = PureThy.get_thm thy9 (Name "at_finite_select");
  1313 
  1314     val induct_aux_lemmas = List.concat (map (fn Type (s, _) =>
  1315       [PureThy.get_thm thy9 (Name ("pt_" ^ Sign.base_name s ^ "_inst")),
  1316        PureThy.get_thm thy9 (Name ("fs_" ^ Sign.base_name s ^ "1")),
  1317        PureThy.get_thm thy9 (Name ("at_" ^ Sign.base_name s ^ "_inst"))]) dt_atomTs);
  1318 
  1319     val induct_aux_lemmas' = map (fn Type (s, _) =>
  1320       PureThy.get_thm thy9 (Name ("pt_" ^ Sign.base_name s ^ "2")) RS sym) dt_atomTs;
  1321 
  1322     val fresh_aux = PureThy.get_thms thy9 (Name "fresh_aux");
  1323 
  1324     (**********************************************************************
  1325       The subgoals occurring in the proof of induct_aux have the
  1326       following parameters:
  1327 
  1328         x_1 ... x_k p_1 ... p_m z b_1 ... b_n
  1329 
  1330       where
  1331 
  1332         x_i : constructor arguments (introduced by weak induction rule)
  1333         p_i : permutations (one for each atom type in the data type)
  1334         z   : freshness context
  1335         b_i : newly introduced names for binders (sufficiently fresh)
  1336     ***********************************************************************)
  1337 
  1338     val _ = warning "proving strong induction theorem ...";
  1339 
  1340     val induct_aux = Goal.prove_global thy9 [] ind_prems' ind_concl'
  1341       (fn prems => EVERY
  1342         ([mk_subgoal 1 (K (K (K aux_ind_concl))),
  1343           indtac dt_induct tnames 1] @
  1344          List.concat (map (fn ((_, (_, _, constrs)), (_, constrs')) =>
  1345            List.concat (map (fn ((cname, cargs), is) =>
  1346              [simp_tac (HOL_basic_ss addsimps List.concat perm_simps') 1,
  1347               REPEAT (rtac allI 1)] @
  1348              List.concat (map
  1349                (fn ((_, 0), _) => []
  1350                  | ((i, j), k) => List.concat (map (fn j' =>
  1351                      let
  1352                        val DtType (tname, _) = List.nth (cargs, i + j');
  1353                        val atom = Sign.base_name tname
  1354                      in
  1355                        [mk_subgoal 1 (mk_fresh i (i + j) j'
  1356                           (length cargs) (length dt_atomTs)
  1357                           (length cargs + length dt_atomTs + 1 + i - k)
  1358                           (List.mapPartial (fn (i', j) =>
  1359                              if i = i' then NONE else SOME (i' + j)) is)),
  1360                         rtac at_finite_select 1,
  1361                         rtac (PureThy.get_thm thy9 (Name ("at_" ^ atom ^ "_inst"))) 1,
  1362                         asm_full_simp_tac (simpset_of thy9 addsimps
  1363                           [PureThy.get_thm thy9 (Name ("fs_" ^ atom ^ "1"))]) 1,
  1364                         resolve_tac prems 1,
  1365                         etac exE 1,
  1366                         asm_full_simp_tac (simpset_of thy9 addsimps
  1367                           [symmetric fresh_def]) 1]
  1368                      end) (0 upto j - 1))) (is ~~ (0 upto length is - 1))) @
  1369              (if exists (not o equal 0 o snd) is then
  1370                 [mk_subgoal 1 (mk_fresh_constr is (length dt_atomTs)),
  1371                  asm_full_simp_tac (simpset_of thy9 addsimps
  1372                    (List.concat inject_thms @
  1373                     alpha @ abs_perm @ abs_fresh @ [abs_fun_finite_supp] @
  1374                     induct_aux_lemmas)) 1,
  1375                  dtac sym 1,
  1376                  asm_full_simp_tac (simpset_of thy9) 1,
  1377                  REPEAT (etac conjE 1)]
  1378               else
  1379                 []) @
  1380              [(resolve_tac prems THEN_ALL_NEW
  1381                 (atac ORELSE'
  1382                   SUBGOAL (fn (t, i) => case Logic.strip_assums_concl t of
  1383                       _ $ (Const ("Nominal.fresh", _) $ _ $ _) =>
  1384                         asm_simp_tac (simpset_of thy9 addsimps fresh_aux) i
  1385                     | _ =>
  1386                         asm_simp_tac (simpset_of thy9
  1387                         addsimps induct_aux_lemmas'
  1388                         addsimprocs [perm_simproc]) i))) 1])
  1389                (constrs ~~ constrs'))) (descr'' ~~ ndescr)) @
  1390          [REPEAT (eresolve_tac [conjE, allE_Nil] 1),
  1391           REPEAT (etac allE 1),
  1392           REPEAT (TRY (rtac conjI 1) THEN asm_full_simp_tac (simpset_of thy9) 1)]));
  1393 
  1394     val induct_aux' = Thm.instantiate ([],
  1395       map (fn (s, T) =>
  1396         let val pT = TVar (("'n", 0), HOLogic.typeS) --> T --> HOLogic.boolT
  1397         in (cterm_of thy9 (Var ((s, 0), pT)), cterm_of thy9 (Free (s, pT))) end)
  1398           (pnames ~~ recTs) @
  1399       map (fn (_, f) =>
  1400         let val f' = Logic.varify f
  1401         in (cterm_of thy9 f',
  1402           cterm_of thy9 (Const ("Nominal.supp", fastype_of f')))
  1403         end) fresh_fs) induct_aux;
  1404 
  1405     val induct = Goal.prove_global thy9 [] ind_prems ind_concl
  1406       (fn prems => EVERY
  1407          [rtac induct_aux' 1,
  1408           REPEAT (resolve_tac induct_aux_lemmas 1),
  1409           REPEAT ((resolve_tac prems THEN_ALL_NEW
  1410             (etac meta_spec ORELSE' full_simp_tac (HOL_basic_ss addsimps [fresh_def]))) 1)])
  1411 
  1412     val (_, thy10) = thy9 |>
  1413       Sign.add_path big_name |>
  1414       PureThy.add_thms [(("induct'", induct_aux), [])] ||>>
  1415       PureThy.add_thms [(("induct", induct), [case_names_induct])] ||>>
  1416       PureThy.add_thmss [(("inducts", projections induct), [case_names_induct])];
  1417 
  1418     (**** recursion combinator ****)
  1419 
  1420     val _ = warning "defining recursion combinator ...";
  1421 
  1422     val used = foldr add_typ_tfree_names [] recTs;
  1423 
  1424     val (rec_result_Ts', rec_fn_Ts') = DatatypeProp.make_primrec_Ts descr' sorts' used;
  1425 
  1426     val rec_sort = if null dt_atomTs then HOLogic.typeS else
  1427       let val names = map (Sign.base_name o fst o dest_Type) dt_atomTs
  1428       in Sign.certify_sort thy10 (map (Sign.intern_class thy10)
  1429         (map (fn s => "pt_" ^ s) names @
  1430          List.concat (map (fn s => List.mapPartial (fn s' =>
  1431            if s = s' then NONE
  1432            else SOME ("cp_" ^ s ^ "_" ^ s')) names) names)))
  1433       end;
  1434 
  1435     val rec_result_Ts = map (fn TFree (s, _) => TFree (s, rec_sort)) rec_result_Ts';
  1436     val rec_fn_Ts = map (typ_subst_atomic (rec_result_Ts' ~~ rec_result_Ts)) rec_fn_Ts';
  1437 
  1438     val rec_set_Ts = map (fn (T1, T2) =>
  1439       rec_fn_Ts @ [T1, T2] ---> HOLogic.boolT) (recTs ~~ rec_result_Ts);
  1440 
  1441     val big_rec_name = big_name ^ "_rec_set";
  1442     val rec_set_names' =
  1443       if length descr'' = 1 then [big_rec_name] else
  1444         map ((curry (op ^) (big_rec_name ^ "_")) o string_of_int)
  1445           (1 upto (length descr''));
  1446     val rec_set_names =  map (Sign.full_name thy10) rec_set_names';
  1447 
  1448     val rec_fns = map (uncurry (mk_Free "f"))
  1449       (rec_fn_Ts ~~ (1 upto (length rec_fn_Ts)));
  1450     val rec_sets' = map (fn c => list_comb (Free c, rec_fns))
  1451       (rec_set_names' ~~ rec_set_Ts);
  1452     val rec_sets = map (fn c => list_comb (Const c, rec_fns))
  1453       (rec_set_names ~~ rec_set_Ts);
  1454 
  1455     (* introduction rules for graph of recursion function *)
  1456 
  1457     val rec_preds = map (fn (a, T) =>
  1458       Free (a, T --> HOLogic.boolT)) (pnames ~~ rec_result_Ts);
  1459 
  1460     fun mk_fresh3 rs [] = []
  1461       | mk_fresh3 rs ((p as (ys, z)) :: yss) = List.concat (map (fn y as (_, T) =>
  1462             List.mapPartial (fn ((_, (_, x)), r as (_, U)) => if z = x then NONE
  1463               else SOME (HOLogic.mk_Trueprop
  1464                 (Const ("Nominal.fresh", T --> U --> HOLogic.boolT) $ Free y $ Free r)))
  1465                   rs) ys) @
  1466           mk_fresh3 rs yss;
  1467 
  1468     (* FIXME: avoid collisions with other variable names? *)
  1469     val rec_ctxt = Free ("z", fsT');
  1470 
  1471     fun make_rec_intr T p rec_set ((rec_intr_ts, rec_prems, rec_prems',
  1472           rec_eq_prems, l), ((cname, cargs), idxs)) =
  1473       let
  1474         val Ts = map (typ_of_dtyp descr'' sorts') cargs;
  1475         val frees = map (fn i => "x" ^ string_of_int i) (1 upto length Ts) ~~ Ts;
  1476         val frees' = partition_cargs idxs frees;
  1477         val binders = List.concat (map fst frees');
  1478         val atomTs = distinct op = (maps (map snd o fst) frees');
  1479         val recs = List.mapPartial
  1480           (fn ((_, DtRec i), p) => SOME (i, p) | _ => NONE)
  1481           (partition_cargs idxs cargs ~~ frees');
  1482         val frees'' = map (fn i => "y" ^ string_of_int i) (1 upto length recs) ~~
  1483           map (fn (i, _) => List.nth (rec_result_Ts, i)) recs;
  1484         val prems1 = map (fn ((i, (_, x)), y) => HOLogic.mk_Trueprop
  1485           (List.nth (rec_sets', i) $ Free x $ Free y)) (recs ~~ frees'');
  1486         val prems2 =
  1487           map (fn f => map (fn p as (_, T) => HOLogic.mk_Trueprop
  1488             (Const ("Nominal.fresh", T --> fastype_of f --> HOLogic.boolT) $
  1489               Free p $ f)) binders) rec_fns;
  1490         val prems3 = mk_fresh1 [] binders @ mk_fresh2 [] frees';
  1491         val prems4 = map (fn ((i, _), y) =>
  1492           HOLogic.mk_Trueprop (List.nth (rec_preds, i) $ Free y)) (recs ~~ frees'');
  1493         val prems5 = mk_fresh3 (recs ~~ frees'') frees';
  1494         val prems6 = maps (fn aT => map (fn y as (_, T) => HOLogic.mk_Trueprop
  1495           (Const ("Finite_Set.finite", HOLogic.mk_setT aT --> HOLogic.boolT) $
  1496              (Const ("Nominal.supp", T --> HOLogic.mk_setT aT) $ Free y)))
  1497                frees'') atomTs;
  1498         val prems7 = map (fn x as (_, T) => HOLogic.mk_Trueprop
  1499           (Const ("Nominal.fresh", T --> fsT' --> HOLogic.boolT) $
  1500              Free x $ rec_ctxt)) binders;
  1501         val result = list_comb (List.nth (rec_fns, l), map Free (frees @ frees''));
  1502         val result_freshs = map (fn p as (_, T) =>
  1503           Const ("Nominal.fresh", T --> fastype_of result --> HOLogic.boolT) $
  1504             Free p $ result) binders;
  1505         val P = HOLogic.mk_Trueprop (p $ result)
  1506       in
  1507         (rec_intr_ts @ [Logic.list_implies (List.concat prems2 @ prems3 @ prems1,
  1508            HOLogic.mk_Trueprop (rec_set $
  1509              list_comb (Const (cname, Ts ---> T), map Free frees) $ result))],
  1510          rec_prems @ [list_all_free (frees @ frees'', Logic.list_implies (prems4, P))],
  1511          rec_prems' @ map (fn fr => list_all_free (frees @ frees'',
  1512            Logic.list_implies (List.nth (prems2, l) @ prems3 @ prems5 @ prems7 @ prems6 @ [P],
  1513              HOLogic.mk_Trueprop fr))) result_freshs,
  1514          rec_eq_prems @ [List.concat prems2 @ prems3],
  1515          l + 1)
  1516       end;
  1517 
  1518     val (rec_intr_ts, rec_prems, rec_prems', rec_eq_prems, _) =
  1519       Library.foldl (fn (x, ((((d, d'), T), p), rec_set)) =>
  1520         Library.foldl (make_rec_intr T p rec_set) (x, #3 (snd d) ~~ snd d'))
  1521           (([], [], [], [], 0), descr'' ~~ ndescr ~~ recTs ~~ rec_preds ~~ rec_sets');
  1522 
  1523     val ({intrs = rec_intrs, elims = rec_elims, raw_induct = rec_induct, ...}, thy11) =
  1524       thy10 |>
  1525       setmp InductivePackage.quiet_mode (!quiet_mode)
  1526         (InductivePackage.add_inductive_global {verbose = false, kind = Thm.internalK,
  1527             alt_name = big_rec_name, coind = false, no_elim = false, no_ind = false}
  1528            (map (fn (s, T) => ((s, T), NoSyn)) (rec_set_names' ~~ rec_set_Ts))
  1529            (map dest_Free rec_fns)
  1530            (map (fn x => (("", []), x)) rec_intr_ts) []) ||>
  1531       PureThy.hide_thms true [NameSpace.append
  1532         (Sign.full_name thy10 big_rec_name) "induct"];
  1533 
  1534     (** equivariance **)
  1535 
  1536     val fresh_bij = PureThy.get_thms thy11 (Name "fresh_bij");
  1537     val perm_bij = PureThy.get_thms thy11 (Name "perm_bij");
  1538 
  1539     val (rec_equiv_thms, rec_equiv_thms') = ListPair.unzip (map (fn aT =>
  1540       let
  1541         val permT = mk_permT aT;
  1542         val pi = Free ("pi", permT);
  1543         val rec_fns_pi = map (mk_perm [] pi o uncurry (mk_Free "f"))
  1544           (rec_fn_Ts ~~ (1 upto (length rec_fn_Ts)));
  1545         val rec_sets_pi = map (fn c => list_comb (Const c, rec_fns_pi))
  1546           (rec_set_names ~~ rec_set_Ts);
  1547         val ps = map (fn ((((T, U), R), R'), i) =>
  1548           let
  1549             val x = Free ("x" ^ string_of_int i, T);
  1550             val y = Free ("y" ^ string_of_int i, U)
  1551           in
  1552             (R $ x $ y, R' $ mk_perm [] pi x $ mk_perm [] pi y)
  1553           end) (recTs ~~ rec_result_Ts ~~ rec_sets ~~ rec_sets_pi ~~ (1 upto length recTs));
  1554         val ths = map (fn th => standard (th RS mp)) (split_conj_thm
  1555           (Goal.prove_global thy11 [] []
  1556             (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj (map HOLogic.mk_imp ps)))
  1557             (fn _ => rtac rec_induct 1 THEN REPEAT
  1558                (NominalPermeq.perm_simp_tac (HOL_basic_ss addsimps flat perm_simps') 1 THEN
  1559                 (resolve_tac rec_intrs THEN_ALL_NEW
  1560                  asm_simp_tac (HOL_ss addsimps (fresh_bij @ perm_bij))) 1))))
  1561         val ths' = map (fn ((P, Q), th) =>
  1562           Goal.prove_global thy11 [] []
  1563             (Logic.mk_implies (HOLogic.mk_Trueprop Q, HOLogic.mk_Trueprop P))
  1564             (fn _ => dtac (Thm.instantiate ([],
  1565                  [(cterm_of thy11 (Var (("pi", 0), permT)),
  1566                    cterm_of thy11 (Const ("List.rev", permT --> permT) $ pi))]) th) 1 THEN
  1567                NominalPermeq.perm_simp_tac HOL_ss 1)) (ps ~~ ths)
  1568       in (ths, ths') end) dt_atomTs);
  1569 
  1570     (** finite support **)
  1571 
  1572     val rec_fin_supp_thms = map (fn aT =>
  1573       let
  1574         val name = Sign.base_name (fst (dest_Type aT));
  1575         val fs_name = PureThy.get_thm thy11 (Name ("fs_" ^ name ^ "1"));
  1576         val aset = HOLogic.mk_setT aT;
  1577         val finite = Const ("Finite_Set.finite", aset --> HOLogic.boolT);
  1578         val fins = map (fn (f, T) => HOLogic.mk_Trueprop
  1579           (finite $ (Const ("Nominal.supp", T --> aset) $ f)))
  1580             (rec_fns ~~ rec_fn_Ts)
  1581       in
  1582         map (fn th => standard (th RS mp)) (split_conj_thm
  1583           (Goal.prove_global thy11 [] fins
  1584             (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
  1585               (map (fn (((T, U), R), i) =>
  1586                  let
  1587                    val x = Free ("x" ^ string_of_int i, T);
  1588                    val y = Free ("y" ^ string_of_int i, U)
  1589                  in
  1590                    HOLogic.mk_imp (R $ x $ y,
  1591                      finite $ (Const ("Nominal.supp", U --> aset) $ y))
  1592                  end) (recTs ~~ rec_result_Ts ~~ rec_sets ~~ (1 upto length recTs)))))
  1593             (fn fins => (rtac rec_induct THEN_ALL_NEW cut_facts_tac fins) 1 THEN REPEAT
  1594                (NominalPermeq.finite_guess_tac (HOL_ss addsimps [fs_name]) 1))))
  1595       end) dt_atomTs;
  1596 
  1597     (** freshness **)
  1598 
  1599     val perm_fresh_fresh = PureThy.get_thms thy11 (Name "perm_fresh_fresh");
  1600     val perm_swap = PureThy.get_thms thy11 (Name "perm_swap");
  1601 
  1602     val finite_premss = map (fn aT =>
  1603       map (fn (f, T) => HOLogic.mk_Trueprop
  1604         (Const ("Finite_Set.finite", HOLogic.mk_setT aT --> HOLogic.boolT) $
  1605            (Const ("Nominal.supp", T --> HOLogic.mk_setT aT) $ f)))
  1606            (rec_fns ~~ rec_fn_Ts)) dt_atomTs;
  1607 
  1608     val rec_fresh_thms = map (fn ((aT, eqvt_ths), finite_prems) =>
  1609       let
  1610         val name = Sign.base_name (fst (dest_Type aT));
  1611         val fs_name = PureThy.get_thm thy11 (Name ("fs_" ^ name ^ "1"));
  1612         val a = Free ("a", aT);
  1613         val freshs = map (fn (f, fT) => HOLogic.mk_Trueprop
  1614             (Const ("Nominal.fresh", aT --> fT --> HOLogic.boolT) $ a $ f))
  1615           (rec_fns ~~ rec_fn_Ts)
  1616       in
  1617         map (fn (((T, U), R), eqvt_th) =>
  1618           let
  1619             val x = Free ("x", T);
  1620             val y = Free ("y", U);
  1621             val y' = Free ("y'", U)
  1622           in
  1623             standard (Goal.prove (ProofContext.init thy11) [] (finite_prems @
  1624                 [HOLogic.mk_Trueprop (R $ x $ y),
  1625                  HOLogic.mk_Trueprop (HOLogic.mk_all ("y'", U,
  1626                    HOLogic.mk_imp (R $ x $ y', HOLogic.mk_eq (y', y)))),
  1627                  HOLogic.mk_Trueprop (Const ("Nominal.fresh",
  1628                    aT --> T --> HOLogic.boolT) $ a $ x)] @
  1629               freshs)
  1630               (HOLogic.mk_Trueprop (Const ("Nominal.fresh",
  1631                  aT --> U --> HOLogic.boolT) $ a $ y))
  1632               (fn {prems, context} =>
  1633                  let
  1634                    val (finite_prems, rec_prem :: unique_prem ::
  1635                      fresh_prems) = chop (length finite_prems) prems;
  1636                    val unique_prem' = unique_prem RS spec RS mp;
  1637                    val unique = [unique_prem', unique_prem' RS sym] MRS trans;
  1638                    val _ $ (_ $ (_ $ S $ _)) $ _ = prop_of supports_fresh;
  1639                    val tuple = foldr1 HOLogic.mk_prod (x :: rec_fns)
  1640                  in EVERY
  1641                    [rtac (Drule.cterm_instantiate
  1642                       [(cterm_of thy11 S,
  1643                         cterm_of thy11 (Const ("Nominal.supp",
  1644                           fastype_of tuple --> HOLogic.mk_setT aT) $ tuple))]
  1645                       supports_fresh) 1,
  1646                     simp_tac (HOL_basic_ss addsimps
  1647                       [supports_def, symmetric fresh_def, fresh_prod]) 1,
  1648                     REPEAT_DETERM (resolve_tac [allI, impI] 1),
  1649                     REPEAT_DETERM (etac conjE 1),
  1650                     rtac unique 1,
  1651                     SUBPROOF (fn {prems = prems', params = [a, b], ...} => EVERY
  1652                       [cut_facts_tac [rec_prem] 1,
  1653                        rtac (Thm.instantiate ([],
  1654                          [(cterm_of thy11 (Var (("pi", 0), mk_permT aT)),
  1655                            cterm_of thy11 (perm_of_pair (term_of a, term_of b)))]) eqvt_th) 1,
  1656                        asm_simp_tac (HOL_ss addsimps
  1657                          (prems' @ perm_swap @ perm_fresh_fresh)) 1]) context 1,
  1658                     rtac rec_prem 1,
  1659                     simp_tac (HOL_ss addsimps (fs_name ::
  1660                       supp_prod :: finite_Un :: finite_prems)) 1,
  1661                     simp_tac (HOL_ss addsimps (symmetric fresh_def ::
  1662                       fresh_prod :: fresh_prems)) 1]
  1663                  end))
  1664           end) (recTs ~~ rec_result_Ts ~~ rec_sets ~~ eqvt_ths)
  1665       end) (dt_atomTs ~~ rec_equiv_thms' ~~ finite_premss);
  1666 
  1667     (** uniqueness **)
  1668 
  1669     val exists_fresh' = PureThy.get_thms thy11 (Name "exists_fresh'");
  1670     val fs_atoms = map (fn Type (s, _) => PureThy.get_thm thy11
  1671       (Name ("fs_" ^ Sign.base_name s ^ "1"))) dt_atomTs;
  1672     val fresh_atm = PureThy.get_thms thy11 (Name "fresh_atm");
  1673     val calc_atm = PureThy.get_thms thy11 (Name "calc_atm");
  1674     val fresh_left = PureThy.get_thms thy11 (Name "fresh_left");
  1675 
  1676     val fun_tuple = foldr1 HOLogic.mk_prod (rec_ctxt :: rec_fns);
  1677     val fun_tupleT = fastype_of fun_tuple;
  1678     val rec_unique_frees =
  1679       DatatypeProp.indexify_names (replicate (length recTs) "x") ~~ recTs;
  1680     val rec_unique_frees'' = map (fn (s, T) => (s ^ "'", T)) rec_unique_frees;
  1681     val rec_unique_frees' =
  1682       DatatypeProp.indexify_names (replicate (length recTs) "y") ~~ rec_result_Ts;
  1683     val rec_unique_concls = map (fn ((x, U), R) =>
  1684         Const ("Ex1", (U --> HOLogic.boolT) --> HOLogic.boolT) $
  1685           Abs ("y", U, R $ Free x $ Bound 0))
  1686       (rec_unique_frees ~~ rec_result_Ts ~~ rec_sets);
  1687 
  1688     val induct_aux_rec = Drule.cterm_instantiate
  1689       (map (pairself (cterm_of thy11))
  1690          (map (fn (aT, f) => (Logic.varify f, Abs ("z", HOLogic.unitT,
  1691             Const ("Nominal.supp", fun_tupleT --> HOLogic.mk_setT aT) $ fun_tuple)))
  1692               fresh_fs @
  1693           map (fn (((P, T), (x, U)), Q) =>
  1694            (Var ((P, 0), HOLogic.unitT --> Logic.varifyT T --> HOLogic.boolT),
  1695             Abs ("z", HOLogic.unitT, absfree (x, U, Q))))
  1696               (pnames ~~ recTs ~~ rec_unique_frees ~~ rec_unique_concls) @
  1697           map (fn (s, T) => (Var ((s, 0), Logic.varifyT T), Free (s, T)))
  1698             rec_unique_frees)) induct_aux;
  1699 
  1700     fun obtain_fresh_name vs ths rec_fin_supp T (freshs1, freshs2, ctxt) =
  1701       let
  1702         val p = foldr1 HOLogic.mk_prod (vs @ freshs1);
  1703         val ex = Goal.prove ctxt [] [] (HOLogic.mk_Trueprop
  1704             (HOLogic.exists_const T $ Abs ("x", T,
  1705               Const ("Nominal.fresh", T --> fastype_of p --> HOLogic.boolT) $
  1706                 Bound 0 $ p)))
  1707           (fn _ => EVERY
  1708             [cut_facts_tac ths 1,
  1709              REPEAT_DETERM (dresolve_tac (the (AList.lookup op = rec_fin_supp T)) 1),
  1710              resolve_tac exists_fresh' 1,
  1711              asm_simp_tac (HOL_ss addsimps (supp_prod :: finite_Un :: fs_atoms)) 1]);
  1712         val (([cx], ths), ctxt') = Obtain.result
  1713           (fn _ => EVERY
  1714             [etac exE 1,
  1715              full_simp_tac (HOL_ss addsimps (fresh_prod :: fresh_atm)) 1,
  1716              REPEAT (etac conjE 1)])
  1717           [ex] ctxt
  1718       in (freshs1 @ [term_of cx], freshs2 @ ths, ctxt') end;
  1719 
  1720     val finite_ctxt_prems = map (fn aT =>
  1721       HOLogic.mk_Trueprop
  1722         (Const ("Finite_Set.finite", HOLogic.mk_setT aT --> HOLogic.boolT) $
  1723            (Const ("Nominal.supp", fsT' --> HOLogic.mk_setT aT) $ rec_ctxt))) dt_atomTs;
  1724 
  1725     val rec_unique_thms = split_conj_thm (Goal.prove
  1726       (ProofContext.init thy11) (map fst rec_unique_frees)
  1727       (List.concat finite_premss @ finite_ctxt_prems @ rec_prems @ rec_prems')
  1728       (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj rec_unique_concls))
  1729       (fn {prems, context} =>
  1730          let
  1731            val k = length rec_fns;
  1732            val (finite_thss, ths1) = fold_map (fn T => fn xs =>
  1733              apfst (pair T) (chop k xs)) dt_atomTs prems;
  1734            val (finite_ctxt_ths, ths2) = chop (length dt_atomTs) ths1;
  1735            val (P_ind_ths, fcbs) = chop k ths2;
  1736            val P_ths = map (fn th => th RS mp) (split_conj_thm
  1737              (Goal.prove context
  1738                (map fst (rec_unique_frees'' @ rec_unique_frees')) []
  1739                (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
  1740                   (map (fn (((x, y), S), P) => HOLogic.mk_imp
  1741                     (S $ Free x $ Free y, P $ (Free y)))
  1742                       (rec_unique_frees'' ~~ rec_unique_frees' ~~ rec_sets ~~ rec_preds))))
  1743                (fn _ =>
  1744                   rtac rec_induct 1 THEN
  1745                   REPEAT ((resolve_tac P_ind_ths THEN_ALL_NEW assume_tac) 1))));
  1746            val rec_fin_supp_thms' = map
  1747              (fn (ths, (T, fin_ths)) => (T, map (curry op MRS fin_ths) ths))
  1748              (rec_fin_supp_thms ~~ finite_thss);
  1749          in EVERY
  1750            ([rtac induct_aux_rec 1] @
  1751             maps (fn ((_, finite_ths), finite_th) =>
  1752               [cut_facts_tac (finite_th :: finite_ths) 1,
  1753                asm_simp_tac (HOL_ss addsimps [supp_prod, finite_Un]) 1])
  1754                 (finite_thss ~~ finite_ctxt_ths) @
  1755             maps (fn ((_, idxss), elim) => maps (fn idxs =>
  1756               [full_simp_tac (HOL_ss addsimps [symmetric fresh_def, supp_prod, Un_iff]) 1,
  1757                REPEAT_DETERM (eresolve_tac [conjE, ex1E] 1),
  1758                rtac ex1I 1,
  1759                (resolve_tac rec_intrs THEN_ALL_NEW atac) 1,
  1760                rotate_tac ~1 1,
  1761                ((DETERM o etac elim) THEN_ALL_NEW full_simp_tac
  1762                   (HOL_ss addsimps List.concat distinct_thms)) 1] @
  1763                (if null idxs then [] else [hyp_subst_tac 1,
  1764                 SUBPROOF (fn {asms, concl, prems = prems', params, context = context', ...} =>
  1765                   let
  1766                     val SOME prem = find_first (can (HOLogic.dest_eq o
  1767                       HOLogic.dest_Trueprop o prop_of)) prems';
  1768                     val _ $ (_ $ lhs $ rhs) = prop_of prem;
  1769                     val _ $ (_ $ lhs' $ rhs') = term_of concl;
  1770                     val rT = fastype_of lhs';
  1771                     val (c, cargsl) = strip_comb lhs;
  1772                     val cargsl' = partition_cargs idxs cargsl;
  1773                     val boundsl = List.concat (map fst cargsl');
  1774                     val (_, cargsr) = strip_comb rhs;
  1775                     val cargsr' = partition_cargs idxs cargsr;
  1776                     val boundsr = List.concat (map fst cargsr');
  1777                     val (params1, _ :: params2) =
  1778                       chop (length params div 2) (map term_of params);
  1779                     val params' = params1 @ params2;
  1780                     val rec_prems = filter (fn th => case prop_of th of
  1781                       _ $ (S $ _ $ _) => S mem rec_sets | _ => false) prems';
  1782                     val fresh_prems = filter (fn th => case prop_of th of
  1783                         _ $ (Const ("Nominal.fresh", _) $ _ $ _) => true
  1784                       | _ $ (Const ("Not", _) $ _) => true
  1785                       | _ => false) prems';
  1786                     val Ts = map fastype_of boundsl;
  1787 
  1788                     val _ = warning "step 1: obtaining fresh names";
  1789                     val (freshs1, freshs2, context'') = fold
  1790                       (obtain_fresh_name (rec_ctxt :: rec_fns @ params')
  1791                          (List.concat (map snd finite_thss) @
  1792                             finite_ctxt_ths @ rec_prems)
  1793                          rec_fin_supp_thms')
  1794                       Ts ([], [], context');
  1795                     val pi1 = map perm_of_pair (boundsl ~~ freshs1);
  1796                     val rpi1 = rev pi1;
  1797                     val pi2 = map perm_of_pair (boundsr ~~ freshs1);
  1798                     val rpi2 = rev pi2;
  1799 
  1800                     val fresh_prems' = mk_not_sym fresh_prems;
  1801                     val freshs2' = mk_not_sym freshs2;
  1802 
  1803                     (** as, bs, cs # K as ts, K bs us **)
  1804                     val _ = warning "step 2: as, bs, cs # K as ts, K bs us";
  1805                     val prove_fresh_ss = HOL_ss addsimps
  1806                       (finite_Diff :: List.concat fresh_thms @
  1807                        fs_atoms @ abs_fresh @ abs_supp @ fresh_atm);
  1808                     (* FIXME: avoid asm_full_simp_tac ? *)
  1809                     fun prove_fresh ths y x = Goal.prove context'' [] []
  1810                       (HOLogic.mk_Trueprop (Const ("Nominal.fresh",
  1811                          fastype_of x --> fastype_of y --> HOLogic.boolT) $ x $ y))
  1812                       (fn _ => cut_facts_tac ths 1 THEN asm_full_simp_tac prove_fresh_ss 1);
  1813                     val constr_fresh_thms =
  1814                       map (prove_fresh fresh_prems lhs) boundsl @
  1815                       map (prove_fresh fresh_prems rhs) boundsr @
  1816                       map (prove_fresh freshs2 lhs) freshs1 @
  1817                       map (prove_fresh freshs2 rhs) freshs1;
  1818 
  1819                     (** pi1 o (K as ts) = pi2 o (K bs us) **)
  1820                     val _ = warning "step 3: pi1 o (K as ts) = pi2 o (K bs us)";
  1821                     val pi1_pi2_eq = Goal.prove context'' [] []
  1822                       (HOLogic.mk_Trueprop (HOLogic.mk_eq
  1823                         (fold_rev (mk_perm []) pi1 lhs, fold_rev (mk_perm []) pi2 rhs)))
  1824                       (fn _ => EVERY
  1825                          [cut_facts_tac constr_fresh_thms 1,
  1826                           asm_simp_tac (HOL_basic_ss addsimps perm_fresh_fresh) 1,
  1827                           rtac prem 1]);
  1828 
  1829                     (** pi1 o ts = pi2 o us **)
  1830                     val _ = warning "step 4: pi1 o ts = pi2 o us";
  1831                     val pi1_pi2_eqs = map (fn (t, u) =>
  1832                       Goal.prove context'' [] []
  1833                         (HOLogic.mk_Trueprop (HOLogic.mk_eq
  1834                           (fold_rev (mk_perm []) pi1 t, fold_rev (mk_perm []) pi2 u)))
  1835                         (fn _ => EVERY
  1836                            [cut_facts_tac [pi1_pi2_eq] 1,
  1837                             asm_full_simp_tac (HOL_ss addsimps
  1838                               (calc_atm @ List.concat perm_simps' @
  1839                                fresh_prems' @ freshs2' @ abs_perm @
  1840                                alpha @ List.concat inject_thms)) 1]))
  1841                         (map snd cargsl' ~~ map snd cargsr');
  1842 
  1843                     (** pi1^-1 o pi2 o us = ts **)
  1844                     val _ = warning "step 5: pi1^-1 o pi2 o us = ts";
  1845                     val rpi1_pi2_eqs = map (fn ((t, u), eq) =>
  1846                       Goal.prove context'' [] []
  1847                         (HOLogic.mk_Trueprop (HOLogic.mk_eq
  1848                           (fold_rev (mk_perm []) (rpi1 @ pi2) u, t)))
  1849                         (fn _ => simp_tac (HOL_ss addsimps
  1850                            ((eq RS sym) :: perm_swap)) 1))
  1851                         (map snd cargsl' ~~ map snd cargsr' ~~ pi1_pi2_eqs);
  1852 
  1853                     val (rec_prems1, rec_prems2) =
  1854                       chop (length rec_prems div 2) rec_prems;
  1855 
  1856                     (** (ts, pi1^-1 o pi2 o vs) in rec_set **)
  1857                     val _ = warning "step 6: (ts, pi1^-1 o pi2 o vs) in rec_set";
  1858                     val rec_prems' = map (fn th =>
  1859                       let
  1860                         val _ $ (S $ x $ y) = prop_of th;
  1861                         val k = find_index (equal S) rec_sets;
  1862                         val pi = rpi1 @ pi2;
  1863                         fun mk_pi z = fold_rev (mk_perm []) pi z;
  1864                         fun eqvt_tac p =
  1865                           let
  1866                             val U as Type (_, [Type (_, [T, _])]) = fastype_of p;
  1867                             val l = find_index (equal T) dt_atomTs;
  1868                             val th = List.nth (List.nth (rec_equiv_thms', l), k);
  1869                             val th' = Thm.instantiate ([],
  1870                               [(cterm_of thy11 (Var (("pi", 0), U)),
  1871                                 cterm_of thy11 p)]) th;
  1872                           in rtac th' 1 end;
  1873                         val th' = Goal.prove context'' [] []
  1874                           (HOLogic.mk_Trueprop (S $ mk_pi x $ mk_pi y))
  1875                           (fn _ => EVERY
  1876                              (map eqvt_tac pi @
  1877                               [simp_tac (HOL_ss addsimps (fresh_prems' @ freshs2' @
  1878                                  perm_swap @ perm_fresh_fresh)) 1,
  1879                                rtac th 1]))
  1880                       in
  1881                         Simplifier.simplify
  1882                           (HOL_basic_ss addsimps rpi1_pi2_eqs) th'
  1883                       end) rec_prems2;
  1884 
  1885                     val ihs = filter (fn th => case prop_of th of
  1886                       _ $ (Const ("All", _) $ _) => true | _ => false) prems';
  1887 
  1888                     (** pi1 o rs = pi2 o vs , rs = pi1^-1 o pi2 o vs **)
  1889                     val _ = warning "step 7: pi1 o rs = pi2 o vs , rs = pi1^-1 o pi2 o vs";
  1890                     val rec_eqns = map (fn (th, ih) =>
  1891                       let
  1892                         val th' = th RS (ih RS spec RS mp) RS sym;
  1893                         val _ $ (_ $ lhs $ rhs) = prop_of th';
  1894                         fun strip_perm (_ $ _ $ t) = strip_perm t
  1895                           | strip_perm t = t;
  1896                       in
  1897                         Goal.prove context'' [] []
  1898                            (HOLogic.mk_Trueprop (HOLogic.mk_eq
  1899                               (fold_rev (mk_perm []) pi1 lhs,
  1900                                fold_rev (mk_perm []) pi2 (strip_perm rhs))))
  1901                            (fn _ => simp_tac (HOL_basic_ss addsimps
  1902                               (th' :: perm_swap)) 1)
  1903                       end) (rec_prems' ~~ ihs);
  1904 
  1905                     (** as # rs **)
  1906                     val _ = warning "step 8: as # rs";
  1907                     val rec_freshs = List.concat
  1908                       (map (fn (rec_prem, ih) =>
  1909                         let
  1910                           val _ $ (S $ x $ (y as Free (_, T))) =
  1911                             prop_of rec_prem;
  1912                           val k = find_index (equal S) rec_sets;
  1913                           val atoms = List.concat (List.mapPartial (fn (bs, z) =>
  1914                             if z = x then NONE else SOME bs) cargsl')
  1915                         in
  1916                           map (fn a as Free (_, aT) =>
  1917                             let val l = find_index (equal aT) dt_atomTs;
  1918                             in
  1919                               Goal.prove context'' [] []
  1920                                 (HOLogic.mk_Trueprop (Const ("Nominal.fresh",
  1921                                   aT --> T --> HOLogic.boolT) $ a $ y))
  1922                                 (fn _ => EVERY
  1923                                    (rtac (List.nth (List.nth (rec_fresh_thms, l), k)) 1 ::
  1924                                     map (fn th => rtac th 1)
  1925                                       (snd (List.nth (finite_thss, l))) @
  1926                                     [rtac rec_prem 1, rtac ih 1,
  1927                                      REPEAT_DETERM (resolve_tac fresh_prems 1)]))
  1928                             end) atoms
  1929                         end) (rec_prems1 ~~ ihs));
  1930 
  1931                     (** as # fK as ts rs , bs # fK bs us vs **)
  1932                     val _ = warning "step 9: as # fK as ts rs , bs # fK bs us vs";
  1933                     fun prove_fresh_result (a as Free (_, aT)) =
  1934                       Goal.prove context'' [] []
  1935                         (HOLogic.mk_Trueprop (Const ("Nominal.fresh",
  1936                           aT --> rT --> HOLogic.boolT) $ a $ rhs'))
  1937                         (fn _ => EVERY
  1938                            [resolve_tac fcbs 1,
  1939                             REPEAT_DETERM (resolve_tac
  1940                               (fresh_prems @ rec_freshs) 1),
  1941                             REPEAT_DETERM (resolve_tac (maps snd rec_fin_supp_thms') 1
  1942                               THEN resolve_tac rec_prems 1),
  1943                             resolve_tac P_ind_ths 1,
  1944                             REPEAT_DETERM (resolve_tac (P_ths @ rec_prems) 1)]);
  1945 
  1946                     val fresh_results'' = map prove_fresh_result boundsl;
  1947 
  1948                     fun prove_fresh_result'' ((a as Free (_, aT), b), th) =
  1949                       let val th' = Goal.prove context'' [] []
  1950                         (HOLogic.mk_Trueprop (Const ("Nominal.fresh",
  1951                           aT --> rT --> HOLogic.boolT) $
  1952                             fold_rev (mk_perm []) (rpi2 @ pi1) a $
  1953                             fold_rev (mk_perm []) (rpi2 @ pi1) rhs'))
  1954                         (fn _ => simp_tac (HOL_ss addsimps fresh_bij) 1 THEN
  1955                            rtac th 1)
  1956                       in
  1957                         Goal.prove context'' [] []
  1958                           (HOLogic.mk_Trueprop (Const ("Nominal.fresh",
  1959                             aT --> rT --> HOLogic.boolT) $ b $ lhs'))
  1960                           (fn _ => EVERY
  1961                              [cut_facts_tac [th'] 1,
  1962                               NominalPermeq.perm_simp_tac (HOL_ss addsimps
  1963                                 (rec_eqns @ pi1_pi2_eqs @ perm_swap)) 1,
  1964                               full_simp_tac (HOL_ss addsimps (calc_atm @
  1965                                 fresh_prems' @ freshs2' @ perm_fresh_fresh)) 1])
  1966                       end;
  1967 
  1968                     val fresh_results = fresh_results'' @ map prove_fresh_result''
  1969                       (boundsl ~~ boundsr ~~ fresh_results'');
  1970 
  1971                     (** cs # fK as ts rs , cs # fK bs us vs **)
  1972                     val _ = warning "step 10: cs # fK as ts rs , cs # fK bs us vs";
  1973                     fun prove_fresh_result' recs t (a as Free (_, aT)) =
  1974                       Goal.prove context'' [] []
  1975                         (HOLogic.mk_Trueprop (Const ("Nominal.fresh",
  1976                           aT --> rT --> HOLogic.boolT) $ a $ t))
  1977                         (fn _ => EVERY
  1978                           [cut_facts_tac recs 1,
  1979                            REPEAT_DETERM (dresolve_tac
  1980                              (the (AList.lookup op = rec_fin_supp_thms' aT)) 1),
  1981                            NominalPermeq.fresh_guess_tac
  1982                              (HOL_ss addsimps (freshs2 @
  1983                                 fs_atoms @ fresh_atm @
  1984                                 List.concat (map snd finite_thss))) 1]);
  1985 
  1986                     val fresh_results' =
  1987                       map (prove_fresh_result' rec_prems1 rhs') freshs1 @
  1988                       map (prove_fresh_result' rec_prems2 lhs') freshs1;
  1989 
  1990                     (** pi1 o (fK as ts rs) = pi2 o (fK bs us vs) **)
  1991                     val _ = warning "step 11: pi1 o (fK as ts rs) = pi2 o (fK bs us vs)";
  1992                     val pi1_pi2_result = Goal.prove context'' [] []
  1993                       (HOLogic.mk_Trueprop (HOLogic.mk_eq
  1994                         (fold_rev (mk_perm []) pi1 rhs', fold_rev (mk_perm []) pi2 lhs')))
  1995                       (fn _ => NominalPermeq.perm_simp_tac (HOL_ss addsimps
  1996                            pi1_pi2_eqs @ rec_eqns) 1 THEN
  1997                          TRY (simp_tac (HOL_ss addsimps
  1998                            (fresh_prems' @ freshs2' @ calc_atm @ perm_fresh_fresh)) 1));
  1999 
  2000                     val _ = warning "final result";
  2001                     val final = Goal.prove context'' [] [] (term_of concl)
  2002                       (fn _ => cut_facts_tac [pi1_pi2_result RS sym] 1 THEN
  2003                         full_simp_tac (HOL_basic_ss addsimps perm_fresh_fresh @
  2004                           fresh_results @ fresh_results') 1);
  2005                     val final' = ProofContext.export context'' context' [final];
  2006                     val _ = warning "finished!"
  2007                   in
  2008                     resolve_tac final' 1
  2009                   end) context 1])) idxss) (ndescr ~~ rec_elims))
  2010          end));
  2011 
  2012     val rec_total_thms = map (fn r => r RS theI') rec_unique_thms;
  2013 
  2014     (* define primrec combinators *)
  2015 
  2016     val big_reccomb_name = (space_implode "_" new_type_names) ^ "_rec";
  2017     val reccomb_names = map (Sign.full_name thy11)
  2018       (if length descr'' = 1 then [big_reccomb_name] else
  2019         (map ((curry (op ^) (big_reccomb_name ^ "_")) o string_of_int)
  2020           (1 upto (length descr''))));
  2021     val reccombs = map (fn ((name, T), T') => list_comb
  2022       (Const (name, rec_fn_Ts @ [T] ---> T'), rec_fns))
  2023         (reccomb_names ~~ recTs ~~ rec_result_Ts);
  2024 
  2025     val (reccomb_defs, thy12) =
  2026       thy11
  2027       |> Sign.add_consts_i (map (fn ((name, T), T') =>
  2028           (Sign.base_name name, rec_fn_Ts @ [T] ---> T', NoSyn))
  2029           (reccomb_names ~~ recTs ~~ rec_result_Ts))
  2030       |> (PureThy.add_defs_i false o map Thm.no_attributes) (map (fn ((((name, comb), set), T), T') =>
  2031           ((Sign.base_name name) ^ "_def", Logic.mk_equals (comb, absfree ("x", T,
  2032            Const ("The", (T' --> HOLogic.boolT) --> T') $ absfree ("y", T',
  2033              set $ Free ("x", T) $ Free ("y", T'))))))
  2034                (reccomb_names ~~ reccombs ~~ rec_sets ~~ recTs ~~ rec_result_Ts));
  2035 
  2036     (* prove characteristic equations for primrec combinators *)
  2037 
  2038     val rec_thms = map (fn (prems, concl) =>
  2039       let
  2040         val _ $ (_ $ (_ $ x) $ _) = concl;
  2041         val (_, cargs) = strip_comb x;
  2042         val ps = map (fn (x as Free (_, T), i) =>
  2043           (Free ("x" ^ string_of_int i, T), x)) (cargs ~~ (1 upto length cargs));
  2044         val concl' = subst_atomic_types (rec_result_Ts' ~~ rec_result_Ts) concl;
  2045         val prems' = List.concat finite_premss @ finite_ctxt_prems @
  2046           rec_prems @ rec_prems' @ map (subst_atomic ps) prems;
  2047         fun solve rules prems = resolve_tac rules THEN_ALL_NEW
  2048           (resolve_tac prems THEN_ALL_NEW atac)
  2049       in
  2050         Goal.prove_global thy12 [] prems' concl'
  2051           (fn prems => EVERY
  2052             [rewrite_goals_tac reccomb_defs,
  2053              rtac the1_equality 1,
  2054              solve rec_unique_thms prems 1,
  2055              resolve_tac rec_intrs 1,
  2056              REPEAT (solve (prems @ rec_total_thms) prems 1)])
  2057       end) (rec_eq_prems ~~
  2058         DatatypeProp.make_primrecs new_type_names descr' sorts' thy12);
  2059 
  2060     val dt_infos = map (make_dt_info pdescr sorts induct reccomb_names rec_thms)
  2061       ((0 upto length descr1 - 1) ~~ descr1 ~~ distinct_thms ~~ inject_thms);
  2062 
  2063     (* FIXME: theorems are stored in database for testing only *)
  2064     val (_, thy13) = thy12 |>
  2065       PureThy.add_thmss
  2066         [(("rec_equiv", List.concat rec_equiv_thms), []),
  2067          (("rec_equiv'", List.concat rec_equiv_thms'), []),
  2068          (("rec_fin_supp", List.concat rec_fin_supp_thms), []),
  2069          (("rec_fresh", List.concat rec_fresh_thms), []),
  2070          (("rec_unique", map standard rec_unique_thms), []),
  2071          (("recs", rec_thms), [])] ||>
  2072       Sign.parent_path ||>
  2073       map_nominal_datatypes (fold Symtab.update dt_infos);
  2074 
  2075   in
  2076     thy13
  2077   end;
  2078 
  2079 val add_nominal_datatype = gen_add_nominal_datatype read_typ true;
  2080 
  2081 
  2082 (* FIXME: The following stuff should be exported by DatatypePackage *)
  2083 
  2084 local structure P = OuterParse and K = OuterKeyword in
  2085 
  2086 val datatype_decl =
  2087   Scan.option (P.$$$ "(" |-- P.name --| P.$$$ ")") -- P.type_args -- P.name -- P.opt_infix --
  2088     (P.$$$ "=" |-- P.enum1 "|" (P.name -- Scan.repeat P.typ -- P.opt_mixfix));
  2089 
  2090 fun mk_datatype args =
  2091   let
  2092     val names = map (fn ((((NONE, _), t), _), _) => t | ((((SOME t, _), _), _), _) => t) args;
  2093     val specs = map (fn ((((_, vs), t), mx), cons) =>
  2094       (vs, t, mx, map (fn ((x, y), z) => (x, y, z)) cons)) args;
  2095   in add_nominal_datatype false names specs end;
  2096 
  2097 val _ =
  2098   OuterSyntax.command "nominal_datatype" "define inductive datatypes" K.thy_decl
  2099     (P.and_list1 datatype_decl >> (Toplevel.theory o mk_datatype));
  2100 
  2101 end;
  2102 
  2103 end