src/HOL/Nominal/nominal_package.ML
author urbanc
Thu Nov 10 00:36:26 2005 +0100 (2005-11-10)
changeset 18142 a51ab4152097
parent 18107 ee6b4d3af498
child 18245 65e60434b3c2
permissions -rw-r--r--
called the induction principle "unsafe" instead of "test".
     1 (* $Id$ *)
     2 
     3 signature NOMINAL_PACKAGE =
     4 sig
     5   val add_nominal_datatype : bool -> string list -> (string list * bstring * mixfix *
     6     (bstring * string list * mixfix) list) list -> theory -> theory
     7 end
     8 
     9 structure NominalPackage : NOMINAL_PACKAGE =
    10 struct
    11 
    12 open DatatypeAux;
    13 open NominalAtoms;
    14 
    15 (** FIXME: DatatypePackage should export this function **)
    16 
    17 local
    18 
    19 fun dt_recs (DtTFree _) = []
    20   | dt_recs (DtType (_, dts)) = List.concat (map dt_recs dts)
    21   | dt_recs (DtRec i) = [i];
    22 
    23 fun dt_cases (descr: descr) (_, args, constrs) =
    24   let
    25     fun the_bname i = Sign.base_name (#1 (valOf (AList.lookup (op =) descr i)));
    26     val bnames = map the_bname (distinct (List.concat (map dt_recs args)));
    27   in map (fn (c, _) => space_implode "_" (Sign.base_name c :: bnames)) constrs end;
    28 
    29 
    30 fun induct_cases descr =
    31   DatatypeProp.indexify_names (List.concat (map (dt_cases descr) (map #2 descr)));
    32 
    33 fun exhaust_cases descr i = dt_cases descr (valOf (AList.lookup (op =) descr i));
    34 
    35 in
    36 
    37 fun mk_case_names_induct descr = RuleCases.case_names (induct_cases descr);
    38 
    39 fun mk_case_names_exhausts descr new =
    40   map (RuleCases.case_names o exhaust_cases descr o #1)
    41     (List.filter (fn ((_, (name, _, _))) => name mem_string new) descr);
    42 
    43 end;
    44 
    45 (*******************************)
    46 
    47 val (_ $ (_ $ (_ $ (distinct_f $ _) $ _))) = hd (prems_of distinct_lemma);
    48 
    49 fun read_typ sign ((Ts, sorts), str) =
    50   let
    51     val T = Type.no_tvars (Sign.read_typ (sign, (AList.lookup op =)
    52       (map (apfst (rpair ~1)) sorts)) str) handle TYPE (msg, _, _) => error msg
    53   in (Ts @ [T], add_typ_tfrees (T, sorts)) end;
    54 
    55 (** taken from HOL/Tools/datatype_aux.ML **)
    56 
    57 fun indtac indrule indnames i st =
    58   let
    59     val ts = HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of indrule));
    60     val ts' = HOLogic.dest_conj (HOLogic.dest_Trueprop
    61       (Logic.strip_imp_concl (List.nth (prems_of st, i - 1))));
    62     val getP = if can HOLogic.dest_imp (hd ts) then
    63       (apfst SOME) o HOLogic.dest_imp else pair NONE;
    64     fun abstr (t1, t2) = (case t1 of
    65         NONE => (case filter (fn Free (s, _) => s mem indnames | _ => false)
    66               (term_frees t2) of
    67             [Free (s, T)] => absfree (s, T, t2)
    68           | _ => sys_error "indtac")
    69       | SOME (_ $ t' $ _) => Abs ("x", fastype_of t', abstract_over (t', t2)))
    70     val cert = cterm_of (Thm.sign_of_thm st);
    71     val Ps = map (cert o head_of o snd o getP) ts;
    72     val indrule' = cterm_instantiate (Ps ~~
    73       (map (cert o abstr o getP) ts')) indrule
    74   in
    75     rtac indrule' i st
    76   end;
    77 
    78 fun gen_add_nominal_datatype prep_typ err flat_names new_type_names dts thy =
    79   let
    80     (* this theory is used just for parsing *)
    81   
    82     val tmp_thy = thy |>
    83       Theory.copy |>
    84       Theory.add_types (map (fn (tvs, tname, mx, _) =>
    85         (tname, length tvs, mx)) dts);
    86 
    87     val sign = Theory.sign_of tmp_thy;
    88 
    89     val atoms = atoms_of thy;
    90     val classes = map (NameSpace.map_base (fn s => "pt_" ^ s)) atoms;
    91     val cp_classes = List.concat (map (fn atom1 => map (fn atom2 =>
    92       Sign.intern_class thy ("cp_" ^ Sign.base_name atom1 ^ "_" ^
    93         Sign.base_name atom2)) atoms) atoms);
    94     fun augment_sort S = S union classes;
    95     val augment_sort_typ = map_type_tfree (fn (s, S) => TFree (s, augment_sort S));
    96 
    97     fun prep_constr ((constrs, sorts), (cname, cargs, mx)) =
    98       let val (cargs', sorts') = Library.foldl (prep_typ sign) (([], sorts), cargs)
    99       in (constrs @ [(cname, cargs', mx)], sorts') end
   100 
   101     fun prep_dt_spec ((dts, sorts), (tvs, tname, mx, constrs)) =
   102       let val (constrs', sorts') = Library.foldl prep_constr (([], sorts), constrs)
   103       in (dts @ [(tvs, tname, mx, constrs')], sorts') end
   104 
   105     val (dts', sorts) = Library.foldl prep_dt_spec (([], []), dts);
   106     val sorts' = map (apsnd augment_sort) sorts;
   107     val tyvars = map #1 dts';
   108 
   109     val types_syntax = map (fn (tvs, tname, mx, constrs) => (tname, mx)) dts';
   110     val constr_syntax = map (fn (tvs, tname, mx, constrs) =>
   111       map (fn (cname, cargs, mx) => (cname, mx)) constrs) dts';
   112 
   113     val ps = map (fn (_, n, _, _) =>
   114       (Sign.full_name sign n, Sign.full_name sign (n ^ "_Rep"))) dts;
   115     val rps = map Library.swap ps;
   116 
   117     fun replace_types (Type ("nominal.ABS", [T, U])) = 
   118           Type ("fun", [T, Type ("nominal.nOption", [replace_types U])])
   119       | replace_types (Type (s, Ts)) =
   120           Type (getOpt (AList.lookup op = ps s, s), map replace_types Ts)
   121       | replace_types T = T;
   122 
   123     fun replace_types' (Type (s, Ts)) =
   124           Type (getOpt (AList.lookup op = rps s, s), map replace_types' Ts)
   125       | replace_types' T = T;
   126 
   127     val dts'' = map (fn (tvs, tname, mx, constrs) => (tvs, tname ^ "_Rep", NoSyn,
   128       map (fn (cname, cargs, mx) => (cname,
   129         map (augment_sort_typ o replace_types) cargs, NoSyn)) constrs)) dts';
   130 
   131     val new_type_names' = map (fn n => n ^ "_Rep") new_type_names;
   132     val full_new_type_names' = map (Sign.full_name (sign_of thy)) new_type_names';
   133 
   134     val ({induction, ...},thy1) =
   135       DatatypePackage.add_datatype_i err flat_names new_type_names' dts'' thy;
   136 
   137     val SOME {descr, ...} = Symtab.lookup
   138       (DatatypePackage.get_datatypes thy1) (hd full_new_type_names');
   139     fun nth_dtyp i = typ_of_dtyp descr sorts' (DtRec i);
   140 
   141     val dt_atomTs = filter (fn Type (s, []) => s mem atoms | _ => false)
   142       (get_nonrec_types descr sorts);
   143 
   144     (**** define permutation functions ****)
   145 
   146     val permT = mk_permT (TFree ("'x", HOLogic.typeS));
   147     val pi = Free ("pi", permT);
   148     val perm_types = map (fn (i, _) =>
   149       let val T = nth_dtyp i
   150       in permT --> T --> T end) descr;
   151     val perm_names = replicate (length new_type_names) "nominal.perm" @
   152       DatatypeProp.indexify_names (map (fn i => Sign.full_name (sign_of thy1)
   153         ("perm_" ^ name_of_typ (nth_dtyp i)))
   154           (length new_type_names upto length descr - 1));
   155     val perm_names_types = perm_names ~~ perm_types;
   156 
   157     val perm_eqs = List.concat (map (fn (i, (_, _, constrs)) =>
   158       let val T = nth_dtyp i
   159       in map (fn (cname, dts) => 
   160         let
   161           val Ts = map (typ_of_dtyp descr sorts') dts;
   162           val names = DatatypeProp.make_tnames Ts;
   163           val args = map Free (names ~~ Ts);
   164           val c = Const (cname, Ts ---> T);
   165           fun perm_arg (dt, x) =
   166             let val T = type_of x
   167             in if is_rec_type dt then
   168                 let val (Us, _) = strip_type T
   169                 in list_abs (map (pair "x") Us,
   170                   Const (List.nth (perm_names_types, body_index dt)) $ pi $
   171                     list_comb (x, map (fn (i, U) =>
   172                       Const ("nominal.perm", permT --> U --> U) $
   173                         (Const ("List.rev", permT --> permT) $ pi) $
   174                         Bound i) ((length Us - 1 downto 0) ~~ Us)))
   175                 end
   176               else Const ("nominal.perm", permT --> T --> T) $ pi $ x
   177             end;  
   178         in
   179           (("", HOLogic.mk_Trueprop (HOLogic.mk_eq
   180             (Const (List.nth (perm_names_types, i)) $
   181                Free ("pi", mk_permT (TFree ("'x", HOLogic.typeS))) $
   182                list_comb (c, args),
   183              list_comb (c, map perm_arg (dts ~~ args))))), [])
   184         end) constrs
   185       end) descr);
   186 
   187     val (thy2, perm_simps) = thy1 |>
   188       Theory.add_consts_i (map (fn (s, T) => (Sign.base_name s, T, NoSyn))
   189         (List.drop (perm_names_types, length new_type_names))) |>
   190       PrimrecPackage.add_primrec_i "" perm_eqs;
   191 
   192     (**** prove that permutation functions introduced by unfolding are ****)
   193     (**** equivalent to already existing permutation functions         ****)
   194 
   195     val _ = warning ("length descr: " ^ string_of_int (length descr));
   196     val _ = warning ("length new_type_names: " ^ string_of_int (length new_type_names));
   197 
   198     val perm_indnames = DatatypeProp.make_tnames (map body_type perm_types);
   199     val perm_fun_def = PureThy.get_thm thy2 (Name "perm_fun_def");
   200 
   201     val unfolded_perm_eq_thms =
   202       if length descr = length new_type_names then []
   203       else map standard (List.drop (split_conj_thm
   204         (Goal.prove thy2 [] []
   205           (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
   206             (map (fn (c as (s, T), x) =>
   207                let val [T1, T2] = binder_types T
   208                in HOLogic.mk_eq (Const c $ pi $ Free (x, T2),
   209                  Const ("nominal.perm", T) $ pi $ Free (x, T2))
   210                end)
   211              (perm_names_types ~~ perm_indnames))))
   212           (fn _ => EVERY [indtac induction perm_indnames 1,
   213             ALLGOALS (asm_full_simp_tac
   214               (simpset_of thy2 addsimps [perm_fun_def]))])),
   215         length new_type_names));
   216 
   217     (**** prove [] \<bullet> t = t ****)
   218 
   219     val _ = warning "perm_empty_thms";
   220 
   221     val perm_empty_thms = List.concat (map (fn a =>
   222       let val permT = mk_permT (Type (a, []))
   223       in map standard (List.take (split_conj_thm
   224         (Goal.prove thy2 [] []
   225           (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
   226             (map (fn ((s, T), x) => HOLogic.mk_eq
   227                 (Const (s, permT --> T --> T) $
   228                    Const ("List.list.Nil", permT) $ Free (x, T),
   229                  Free (x, T)))
   230              (perm_names ~~
   231               map body_type perm_types ~~ perm_indnames))))
   232           (fn _ => EVERY [indtac induction perm_indnames 1,
   233             ALLGOALS (asm_full_simp_tac (simpset_of thy2))])),
   234         length new_type_names))
   235       end)
   236       atoms);
   237 
   238     (**** prove (pi1 @ pi2) \<bullet> t = pi1 \<bullet> (pi2 \<bullet> t) ****)
   239 
   240     val _ = warning "perm_append_thms";
   241 
   242     (*FIXME: these should be looked up statically*)
   243     val at_pt_inst = PureThy.get_thm thy2 (Name "at_pt_inst");
   244     val pt2 = PureThy.get_thm thy2 (Name "pt2");
   245 
   246     val perm_append_thms = List.concat (map (fn a =>
   247       let
   248         val permT = mk_permT (Type (a, []));
   249         val pi1 = Free ("pi1", permT);
   250         val pi2 = Free ("pi2", permT);
   251         val pt_inst = PureThy.get_thm thy2 (Name ("pt_" ^ Sign.base_name a ^ "_inst"));
   252         val pt2' = pt_inst RS pt2;
   253         val pt2_ax = PureThy.get_thm thy2
   254           (Name (NameSpace.map_base (fn s => "pt_" ^ s ^ "2") a));
   255       in List.take (map standard (split_conj_thm
   256         (Goal.prove thy2 [] []
   257              (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
   258                 (map (fn ((s, T), x) =>
   259                     let val perm = Const (s, permT --> T --> T)
   260                     in HOLogic.mk_eq
   261                       (perm $ (Const ("List.op @", permT --> permT --> permT) $
   262                          pi1 $ pi2) $ Free (x, T),
   263                        perm $ pi1 $ (perm $ pi2 $ Free (x, T)))
   264                     end)
   265                   (perm_names ~~
   266                    map body_type perm_types ~~ perm_indnames))))
   267            (fn _ => EVERY [indtac induction perm_indnames 1,
   268               ALLGOALS (asm_full_simp_tac (simpset_of thy2 addsimps [pt2', pt2_ax]))]))),
   269          length new_type_names)
   270       end) atoms);
   271 
   272     (**** prove pi1 ~ pi2 ==> pi1 \<bullet> t = pi2 \<bullet> t ****)
   273 
   274     val _ = warning "perm_eq_thms";
   275 
   276     val pt3 = PureThy.get_thm thy2 (Name "pt3");
   277     val pt3_rev = PureThy.get_thm thy2 (Name "pt3_rev");
   278 
   279     val perm_eq_thms = List.concat (map (fn a =>
   280       let
   281         val permT = mk_permT (Type (a, []));
   282         val pi1 = Free ("pi1", permT);
   283         val pi2 = Free ("pi2", permT);
   284         (*FIXME: not robust - better access these theorems using NominalData?*)
   285         val at_inst = PureThy.get_thm thy2 (Name ("at_" ^ Sign.base_name a ^ "_inst"));
   286         val pt_inst = PureThy.get_thm thy2 (Name ("pt_" ^ Sign.base_name a ^ "_inst"));
   287         val pt3' = pt_inst RS pt3;
   288         val pt3_rev' = at_inst RS (pt_inst RS pt3_rev);
   289         val pt3_ax = PureThy.get_thm thy2
   290           (Name (NameSpace.map_base (fn s => "pt_" ^ s ^ "3") a));
   291       in List.take (map standard (split_conj_thm
   292         (Goal.prove thy2 [] [] (Logic.mk_implies
   293              (HOLogic.mk_Trueprop (Const ("nominal.prm_eq",
   294                 permT --> permT --> HOLogic.boolT) $ pi1 $ pi2),
   295               HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
   296                 (map (fn ((s, T), x) =>
   297                     let val perm = Const (s, permT --> T --> T)
   298                     in HOLogic.mk_eq
   299                       (perm $ pi1 $ Free (x, T),
   300                        perm $ pi2 $ Free (x, T))
   301                     end)
   302                   (perm_names ~~
   303                    map body_type perm_types ~~ perm_indnames)))))
   304            (fn _ => EVERY [indtac induction perm_indnames 1,
   305               ALLGOALS (asm_full_simp_tac (simpset_of thy2 addsimps [pt3', pt3_rev', pt3_ax]))]))),
   306          length new_type_names)
   307       end) atoms);
   308 
   309     (**** prove pi1 \<bullet> (pi2 \<bullet> t) = (pi1 \<bullet> pi2) \<bullet> (pi1 \<bullet> t) ****)
   310 
   311     val cp1 = PureThy.get_thm thy2 (Name "cp1");
   312     val dj_cp = PureThy.get_thm thy2 (Name "dj_cp");
   313     val pt_perm_compose = PureThy.get_thm thy2 (Name "pt_perm_compose");
   314     val pt_perm_compose_rev = PureThy.get_thm thy2 (Name "pt_perm_compose_rev");
   315     val dj_perm_perm_forget = PureThy.get_thm thy2 (Name "dj_perm_perm_forget");
   316 
   317     fun composition_instance name1 name2 thy =
   318       let
   319         val name1' = Sign.base_name name1;
   320         val name2' = Sign.base_name name2;
   321         val cp_class = Sign.intern_class thy ("cp_" ^ name1' ^ "_" ^ name2');
   322         val permT1 = mk_permT (Type (name1, []));
   323         val permT2 = mk_permT (Type (name2, []));
   324         val augment = map_type_tfree
   325           (fn (x, S) => TFree (x, cp_class :: S));
   326         val Ts = map (augment o body_type) perm_types;
   327         val cp_inst = PureThy.get_thm thy
   328           (Name ("cp_" ^ name1' ^ "_" ^ name2' ^ "_inst"));
   329         val simps = simpset_of thy addsimps (perm_fun_def ::
   330           (if name1 <> name2 then
   331              let val dj = PureThy.get_thm thy (Name ("dj_" ^ name2' ^ "_" ^ name1'))
   332              in [dj RS (cp_inst RS dj_cp), dj RS dj_perm_perm_forget] end
   333            else
   334              let
   335                val at_inst = PureThy.get_thm thy (Name ("at_" ^ name1' ^ "_inst"));
   336                val pt_inst = PureThy.get_thm thy (Name ("pt_" ^ name1' ^ "_inst"))
   337              in
   338                [cp_inst RS cp1 RS sym,
   339                 at_inst RS (pt_inst RS pt_perm_compose) RS sym,
   340                 at_inst RS (pt_inst RS pt_perm_compose_rev) RS sym]
   341             end))
   342         val thms = split_conj_thm (standard (Goal.prove thy [] []
   343             (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
   344               (map (fn ((s, T), x) =>
   345                   let
   346                     val pi1 = Free ("pi1", permT1);
   347                     val pi2 = Free ("pi2", permT2);
   348                     val perm1 = Const (s, permT1 --> T --> T);
   349                     val perm2 = Const (s, permT2 --> T --> T);
   350                     val perm3 = Const ("nominal.perm", permT1 --> permT2 --> permT2)
   351                   in HOLogic.mk_eq
   352                     (perm1 $ pi1 $ (perm2 $ pi2 $ Free (x, T)),
   353                      perm2 $ (perm3 $ pi1 $ pi2) $ (perm1 $ pi1 $ Free (x, T)))
   354                   end)
   355                 (perm_names ~~ Ts ~~ perm_indnames))))
   356           (fn _ => EVERY [indtac induction perm_indnames 1,
   357              ALLGOALS (asm_full_simp_tac simps)])))
   358       in
   359         foldl (fn ((s, tvs), thy) => AxClass.add_inst_arity_i
   360             (s, replicate (length tvs) (cp_class :: classes), [cp_class])
   361             (AxClass.intro_classes_tac [] THEN ALLGOALS (resolve_tac thms)) thy)
   362           thy (full_new_type_names' ~~ tyvars)
   363       end;
   364 
   365     val (thy3, perm_thmss) = thy2 |>
   366       fold (fn name1 => fold (composition_instance name1) atoms) atoms |>
   367       curry (Library.foldr (fn ((i, (tyname, args, _)), thy) =>
   368         AxClass.add_inst_arity_i (tyname, replicate (length args) classes, classes)
   369         (AxClass.intro_classes_tac [] THEN REPEAT (EVERY
   370            [resolve_tac perm_empty_thms 1,
   371             resolve_tac perm_append_thms 1,
   372             resolve_tac perm_eq_thms 1, assume_tac 1])) thy))
   373         (List.take (descr, length new_type_names)) |>
   374       PureThy.add_thmss
   375         [((space_implode "_" new_type_names ^ "_unfolded_perm_eq",
   376           unfolded_perm_eq_thms), [Simplifier.simp_add_global]),
   377          ((space_implode "_" new_type_names ^ "_perm_empty",
   378           perm_empty_thms), [Simplifier.simp_add_global]),
   379          ((space_implode "_" new_type_names ^ "_perm_append",
   380           perm_append_thms), [Simplifier.simp_add_global]),
   381          ((space_implode "_" new_type_names ^ "_perm_eq",
   382           perm_eq_thms), [Simplifier.simp_add_global])];
   383   
   384     (**** Define representing sets ****)
   385 
   386     val _ = warning "representing sets";
   387 
   388     val rep_set_names = map (Sign.full_name thy3) (DatatypeProp.indexify_names
   389       (map (fn (i, _) => name_of_typ (nth_dtyp i) ^ "_set") descr));
   390     val big_rep_name =
   391       space_implode "_" (DatatypeProp.indexify_names (List.mapPartial
   392         (fn (i, ("nominal.nOption", _, _)) => NONE
   393           | (i, _) => SOME (name_of_typ (nth_dtyp i))) descr)) ^ "_set";
   394     val _ = warning ("big_rep_name: " ^ big_rep_name);
   395 
   396     fun strip_option (dtf as DtType ("fun", [dt, DtRec i])) =
   397           (case AList.lookup op = descr i of
   398              SOME ("nominal.nOption", _, [(_, [dt']), _]) =>
   399                apfst (cons dt) (strip_option dt')
   400            | _ => ([], dtf))
   401       | strip_option dt = ([], dt);
   402 
   403     fun make_intr s T (cname, cargs) =
   404       let
   405         fun mk_prem (dt, (j, j', prems, ts)) = 
   406           let
   407             val (dts, dt') = strip_option dt;
   408             val (dts', dt'') = strip_dtyp dt';
   409             val Ts = map (typ_of_dtyp descr sorts') dts;
   410             val Us = map (typ_of_dtyp descr sorts') dts';
   411             val T = typ_of_dtyp descr sorts' dt'';
   412             val free = mk_Free "x" (Us ---> T) j;
   413             val free' = app_bnds free (length Us);
   414             fun mk_abs_fun (T, (i, t)) =
   415               let val U = fastype_of t
   416               in (i + 1, Const ("nominal.abs_fun", [T, U, T] --->
   417                 Type ("nominal.nOption", [U])) $ mk_Free "y" T i $ t)
   418               end
   419           in (j + 1, j' + length Ts,
   420             case dt'' of
   421                 DtRec k => list_all (map (pair "x") Us,
   422                   HOLogic.mk_Trueprop (HOLogic.mk_mem (free',
   423                     Const (List.nth (rep_set_names, k),
   424                       HOLogic.mk_setT T)))) :: prems
   425               | _ => prems,
   426             snd (foldr mk_abs_fun (j', free) Ts) :: ts)
   427           end;
   428 
   429         val (_, _, prems, ts) = foldr mk_prem (1, 1, [], []) cargs;
   430         val concl = HOLogic.mk_Trueprop (HOLogic.mk_mem
   431           (list_comb (Const (cname, map fastype_of ts ---> T), ts),
   432            Const (s, HOLogic.mk_setT T)))
   433       in Logic.list_implies (prems, concl)
   434       end;
   435 
   436     val (intr_ts, ind_consts) =
   437       apfst List.concat (ListPair.unzip (List.mapPartial
   438         (fn ((_, ("nominal.nOption", _, _)), _) => NONE
   439           | ((i, (_, _, constrs)), rep_set_name) =>
   440               let val T = nth_dtyp i
   441               in SOME (map (make_intr rep_set_name T) constrs,
   442                 Const (rep_set_name, HOLogic.mk_setT T))
   443               end)
   444                 (descr ~~ rep_set_names)));
   445 
   446     val (thy4, {raw_induct = rep_induct, intrs = rep_intrs, ...}) =
   447       setmp InductivePackage.quiet_mode false
   448         (InductivePackage.add_inductive_i false true big_rep_name false true false
   449            ind_consts (map (fn x => (("", x), [])) intr_ts) []) thy3;
   450 
   451     (**** Prove that representing set is closed under permutation ****)
   452 
   453     val _ = warning "proving closure under permutation...";
   454 
   455     val perm_indnames' = List.mapPartial
   456       (fn (x, (_, ("nominal.nOption", _, _))) => NONE | (x, _) => SOME x)
   457       (perm_indnames ~~ descr);
   458 
   459     fun mk_perm_closed name = map (fn th => standard (th RS mp))
   460       (List.take (split_conj_thm (Goal.prove thy4 [] []
   461         (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj (map
   462            (fn (S, x) =>
   463               let
   464                 val S = map_term_types (map_type_tfree
   465                   (fn (s, cs) => TFree (s, cs union cp_classes))) S;
   466                 val T = HOLogic.dest_setT (fastype_of S);
   467                 val permT = mk_permT (Type (name, []))
   468               in HOLogic.mk_imp (HOLogic.mk_mem (Free (x, T), S),
   469                 HOLogic.mk_mem (Const ("nominal.perm", permT --> T --> T) $
   470                   Free ("pi", permT) $ Free (x, T), S))
   471               end) (ind_consts ~~ perm_indnames'))))
   472         (fn _ => EVERY (* CU: added perm_fun_def in the final tactic in order to deal with funs *)
   473            [indtac rep_induct [] 1,
   474             ALLGOALS (simp_tac (simpset_of thy4 addsimps
   475               (symmetric perm_fun_def :: PureThy.get_thms thy4 (Name ("abs_perm"))))),
   476             ALLGOALS (resolve_tac rep_intrs 
   477                THEN_ALL_NEW (asm_full_simp_tac (simpset_of thy4 addsimps [perm_fun_def])))])),
   478         length new_type_names));
   479 
   480     (* FIXME: theorems are stored in database for testing only *)
   481     val perm_closed_thmss = map mk_perm_closed atoms;
   482     val (thy5, _) = PureThy.add_thmss [(("perm_closed",
   483       List.concat perm_closed_thmss), [])] thy4;
   484 
   485     (**** typedef ****)
   486 
   487     val _ = warning "defining type...";
   488 
   489     val (thy6, typedefs) =
   490       foldl_map (fn (thy, ((((name, mx), tvs), c), name')) =>
   491         setmp TypedefPackage.quiet_mode true
   492           (TypedefPackage.add_typedef_i false (SOME name') (name, tvs, mx) c NONE
   493             (rtac exI 1 THEN
   494               QUIET_BREADTH_FIRST (has_fewer_prems 1)
   495               (resolve_tac rep_intrs 1))) thy |> (fn (thy, r) =>
   496         let
   497           val permT = mk_permT (TFree (variant tvs "'a", HOLogic.typeS));
   498           val pi = Free ("pi", permT);
   499           val tvs' = map (fn s => TFree (s, the (AList.lookup op = sorts' s))) tvs;
   500           val T = Type (Sign.intern_type thy name, tvs');
   501           val Const (_, Type (_, [U])) = c
   502         in apsnd (pair r o hd)
   503           (PureThy.add_defs_i true [(("prm_" ^ name ^ "_def", Logic.mk_equals
   504             (Const ("nominal.perm", permT --> T --> T) $ pi $ Free ("x", T),
   505              Const (Sign.intern_const thy ("Abs_" ^ name), U --> T) $
   506                (Const ("nominal.perm", permT --> U --> U) $ pi $
   507                  (Const (Sign.intern_const thy ("Rep_" ^ name), T --> U) $
   508                    Free ("x", T))))), [])] thy)
   509         end))
   510           (thy5, types_syntax ~~ tyvars ~~
   511             (List.take (ind_consts, length new_type_names)) ~~ new_type_names);
   512 
   513     val perm_defs = map snd typedefs;
   514     val Abs_inverse_thms = map (#Abs_inverse o fst) typedefs;
   515     val Rep_inverse_thms = map (#Rep_inverse o fst) typedefs;
   516     val Rep_thms = map (#Rep o fst) typedefs;
   517 
   518     val big_name = space_implode "_" new_type_names;
   519 
   520 
   521     (** prove that new types are in class pt_<name> **)
   522 
   523     val _ = warning "prove that new types are in class pt_<name> ...";
   524 
   525     fun pt_instance ((class, atom), perm_closed_thms) =
   526       fold (fn (((({Abs_inverse, Rep_inverse, Rep, ...},
   527         perm_def), name), tvs), perm_closed) => fn thy =>
   528           AxClass.add_inst_arity_i
   529             (Sign.intern_type thy name,
   530               replicate (length tvs) (classes @ cp_classes), [class])
   531             (EVERY [AxClass.intro_classes_tac [],
   532               rewrite_goals_tac [perm_def],
   533               asm_full_simp_tac (simpset_of thy addsimps [Rep_inverse]) 1,
   534               asm_full_simp_tac (simpset_of thy addsimps
   535                 [Rep RS perm_closed RS Abs_inverse]) 1,
   536               asm_full_simp_tac (HOL_basic_ss addsimps [PureThy.get_thm thy
   537                 (Name ("pt_" ^ Sign.base_name atom ^ "3"))]) 1]) thy)
   538         (typedefs ~~ new_type_names ~~ tyvars ~~ perm_closed_thms);
   539 
   540 
   541     (** prove that new types are in class cp_<name1>_<name2> **)
   542 
   543     val _ = warning "prove that new types are in class cp_<name1>_<name2> ...";
   544 
   545     fun cp_instance (atom1, perm_closed_thms1) (atom2, perm_closed_thms2) thy =
   546       let
   547         val name = "cp_" ^ Sign.base_name atom1 ^ "_" ^ Sign.base_name atom2;
   548         val class = Sign.intern_class thy name;
   549         val cp1' = PureThy.get_thm thy (Name (name ^ "_inst")) RS cp1
   550       in fold (fn ((((({Abs_inverse, Rep_inverse, Rep, ...},
   551         perm_def), name), tvs), perm_closed1), perm_closed2) => fn thy =>
   552           AxClass.add_inst_arity_i
   553             (Sign.intern_type thy name,
   554               replicate (length tvs) (classes @ cp_classes), [class])
   555             (EVERY [AxClass.intro_classes_tac [],
   556               rewrite_goals_tac [perm_def],
   557               asm_full_simp_tac (simpset_of thy addsimps
   558                 ((Rep RS perm_closed1 RS Abs_inverse) ::
   559                  (if atom1 = atom2 then []
   560                   else [Rep RS perm_closed2 RS Abs_inverse]))) 1,
   561               cong_tac 1,
   562               rtac refl 1,
   563               rtac cp1' 1]) thy)
   564         (typedefs ~~ new_type_names ~~ tyvars ~~ perm_closed_thms1 ~~
   565           perm_closed_thms2) thy
   566       end;
   567 
   568     val thy7 = fold (fn x => fn thy => thy |>
   569       pt_instance x |>
   570       fold (cp_instance (apfst snd x)) (atoms ~~ perm_closed_thmss))
   571         (classes ~~ atoms ~~ perm_closed_thmss) thy6;
   572 
   573     (**** constructors ****)
   574 
   575     fun mk_abs_fun (x, t) =
   576       let
   577         val T = fastype_of x;
   578         val U = fastype_of t
   579       in
   580         Const ("nominal.abs_fun", T --> U --> T -->
   581           Type ("nominal.nOption", [U])) $ x $ t
   582       end;
   583 
   584     val (ty_idxs, _) = foldl
   585       (fn ((i, ("nominal.nOption", _, _)), p) => p
   586         | ((i, _), (ty_idxs, j)) => (ty_idxs @ [(i, j)], j + 1)) ([], 0) descr;
   587 
   588     fun reindex (DtType (s, dts)) = DtType (s, map reindex dts)
   589       | reindex (DtRec i) = DtRec (the (AList.lookup op = ty_idxs i))
   590       | reindex dt = dt;
   591 
   592     fun strip_suffix i s = implode (List.take (explode s, size s - i));
   593 
   594     (** strips the "_Rep" in type names *)
   595     fun strip_nth_name i s = 
   596       let val xs = NameSpace.unpack s; 
   597       in NameSpace.pack (Library.nth_map (length xs - i) (strip_suffix 4) xs) end;
   598 
   599     val (descr'', ndescr) = ListPair.unzip (List.mapPartial
   600       (fn (i, ("nominal.nOption", _, _)) => NONE
   601         | (i, (s, dts, constrs)) =>
   602              let
   603                val SOME index = AList.lookup op = ty_idxs i;
   604                val (constrs1, constrs2) = ListPair.unzip
   605                  (map (fn (cname, cargs) => apfst (pair (strip_nth_name 2 cname))
   606                    (foldl_map (fn (dts, dt) =>
   607                      let val (dts', dt') = strip_option dt
   608                      in (dts @ dts' @ [reindex dt'], (length dts, length dts')) end)
   609                        ([], cargs))) constrs)
   610              in SOME ((index, (strip_nth_name 1 s,  map reindex dts, constrs1)),
   611                (index, constrs2))
   612              end) descr);
   613 
   614     val (descr1, descr2) = splitAt (length new_type_names, descr'');
   615     val descr' = [descr1, descr2];
   616 
   617     val typ_of_dtyp' = replace_types' o typ_of_dtyp descr sorts';
   618 
   619     val rep_names = map (fn s =>
   620       Sign.intern_const thy7 ("Rep_" ^ s)) new_type_names;
   621     val abs_names = map (fn s =>
   622       Sign.intern_const thy7 ("Abs_" ^ s)) new_type_names;
   623 
   624     val recTs' = List.mapPartial
   625       (fn ((_, ("nominal.nOption", _, _)), T) => NONE
   626         | (_, T) => SOME T) (descr ~~ get_rec_types descr sorts');
   627     val recTs = get_rec_types descr'' sorts';
   628     val newTs' = Library.take (length new_type_names, recTs');
   629     val newTs = Library.take (length new_type_names, recTs);
   630 
   631     val full_new_type_names = map (Sign.full_name (sign_of thy)) new_type_names;
   632 
   633     fun make_constr_def tname T T' ((thy, defs, eqns), ((cname, cargs), (cname', mx))) =
   634       let
   635         fun constr_arg (dt, (j, l_args, r_args)) =
   636           let
   637             val x' = mk_Free "x" (typ_of_dtyp' dt) j;
   638             val (dts, dt') = strip_option dt;
   639             val xs = map (fn (dt, i) => mk_Free "x" (typ_of_dtyp' dt) i)
   640               (dts ~~ (j upto j + length dts - 1))
   641             val x = mk_Free "x" (typ_of_dtyp' dt') (j + length dts)
   642             val (dts', dt'') = strip_dtyp dt'
   643           in case dt'' of
   644               DtRec k => if k < length new_type_names then
   645                   (j + length dts + 1,
   646                    xs @ x :: l_args,
   647                    foldr mk_abs_fun
   648                      (list_abs (map (pair "z" o typ_of_dtyp') dts',
   649                        Const (List.nth (rep_names, k), typ_of_dtyp' dt'' -->
   650                          typ_of_dtyp descr sorts' dt'') $ app_bnds x (length dts')))
   651                      xs :: r_args)
   652                 else error "nested recursion not (yet) supported"
   653             | _ => (j + 1, x' :: l_args, x' :: r_args)
   654           end
   655 
   656         val (_, l_args, r_args) = foldr constr_arg (1, [], []) cargs;
   657         val abs_name = Sign.intern_const (Theory.sign_of thy) ("Abs_" ^ tname);
   658         val rep_name = Sign.intern_const (Theory.sign_of thy) ("Rep_" ^ tname);
   659         val constrT = map fastype_of l_args ---> T;
   660         val lhs = list_comb (Const (Sign.full_name thy (Sign.base_name cname),
   661           constrT), l_args);
   662         val rhs = list_comb (Const (cname, map fastype_of r_args ---> T'), r_args);
   663         val def = Logic.mk_equals (lhs, Const (abs_name, T' --> T) $ rhs);
   664         val eqn = HOLogic.mk_Trueprop (HOLogic.mk_eq
   665           (Const (rep_name, T --> T') $ lhs, rhs));
   666         val def_name = (Sign.base_name cname) ^ "_def";
   667         val (thy', [def_thm]) = thy |>
   668           Theory.add_consts_i [(cname', constrT, mx)] |>
   669           (PureThy.add_defs_i false o map Thm.no_attributes) [(def_name, def)]
   670       in (thy', defs @ [def_thm], eqns @ [eqn]) end;
   671 
   672     fun dt_constr_defs ((thy, defs, eqns, dist_lemmas),
   673         (((((_, (_, _, constrs)), tname), T), T'), constr_syntax)) =
   674       let
   675         val rep_const = cterm_of thy
   676           (Const (Sign.intern_const thy ("Rep_" ^ tname), T --> T'));
   677         val dist = standard (cterm_instantiate [(cterm_of thy distinct_f, rep_const)] distinct_lemma);
   678         val (thy', defs', eqns') = Library.foldl (make_constr_def tname T T')
   679           ((Theory.add_path tname thy, defs, []), constrs ~~ constr_syntax)
   680       in
   681         (parent_path flat_names thy', defs', eqns @ [eqns'], dist_lemmas @ [dist])
   682       end;
   683 
   684     val (thy8, constr_defs, constr_rep_eqns, dist_lemmas) = Library.foldl dt_constr_defs
   685       ((thy7, [], [], []), List.take (descr, length new_type_names) ~~
   686         new_type_names ~~ newTs ~~ newTs' ~~ constr_syntax);
   687 
   688     val abs_inject_thms = map (fn tname =>
   689       PureThy.get_thm thy8 (Name ("Abs_" ^ tname ^ "_inject"))) new_type_names;
   690 
   691     val rep_inject_thms = map (fn tname =>
   692       PureThy.get_thm thy8 (Name ("Rep_" ^ tname ^ "_inject"))) new_type_names;
   693 
   694     val rep_thms = map (fn tname =>
   695       PureThy.get_thm thy8 (Name ("Rep_" ^ tname))) new_type_names;
   696 
   697     val rep_inverse_thms = map (fn tname =>
   698       PureThy.get_thm thy8 (Name ("Rep_" ^ tname ^ "_inverse"))) new_type_names;
   699 
   700     (* prove theorem  Rep_i (Constr_j ...) = Constr'_j ...  *)
   701     
   702     fun prove_constr_rep_thm eqn =
   703       let
   704         val inj_thms = map (fn r => r RS iffD1) abs_inject_thms;
   705         val rewrites = constr_defs @ map mk_meta_eq rep_inverse_thms
   706       in standard (Goal.prove thy8 [] [] eqn (fn _ => EVERY
   707         [resolve_tac inj_thms 1,
   708          rewrite_goals_tac rewrites,
   709          rtac refl 3,
   710          resolve_tac rep_intrs 2,
   711          REPEAT (resolve_tac rep_thms 1)]))
   712       end;
   713 
   714     val constr_rep_thmss = map (map prove_constr_rep_thm) constr_rep_eqns;
   715 
   716     (* prove theorem  pi \<bullet> Rep_i x = Rep_i (pi \<bullet> x) *)
   717 
   718     fun prove_perm_rep_perm (atom, perm_closed_thms) = map (fn th =>
   719       let
   720         val _ $ (_ $ (Rep $ x) $ _) = Logic.unvarify (prop_of th);
   721         val Type ("fun", [T, U]) = fastype_of Rep;
   722         val permT = mk_permT (Type (atom, []));
   723         val pi = Free ("pi", permT);
   724       in
   725         standard (Goal.prove thy8 [] [] (HOLogic.mk_Trueprop (HOLogic.mk_eq
   726             (Const ("nominal.perm", permT --> U --> U) $ pi $ (Rep $ x),
   727              Rep $ (Const ("nominal.perm", permT --> T --> T) $ pi $ x))))
   728           (fn _ => simp_tac (HOL_basic_ss addsimps (perm_defs @ Abs_inverse_thms @
   729             perm_closed_thms @ Rep_thms)) 1))
   730       end) Rep_thms;
   731 
   732     val perm_rep_perm_thms = List.concat (map prove_perm_rep_perm
   733       (atoms ~~ perm_closed_thmss));
   734 
   735     (* prove distinctness theorems *)
   736 
   737     val distinct_props = setmp DatatypeProp.dtK 1000
   738       (DatatypeProp.make_distincts new_type_names descr' sorts') thy8;
   739 
   740     val dist_rewrites = map (fn (rep_thms, dist_lemma) =>
   741       dist_lemma::(rep_thms @ [In0_eq, In1_eq, In0_not_In1, In1_not_In0]))
   742         (constr_rep_thmss ~~ dist_lemmas);
   743 
   744     fun prove_distinct_thms (_, []) = []
   745       | prove_distinct_thms (p as (rep_thms, dist_lemma), t::ts) =
   746           let
   747             val dist_thm = standard (Goal.prove thy8 [] [] t (fn _ =>
   748               simp_tac (simpset_of thy8 addsimps (dist_lemma :: rep_thms)) 1))
   749           in dist_thm::(standard (dist_thm RS not_sym))::
   750             (prove_distinct_thms (p, ts))
   751           end;
   752 
   753     val distinct_thms = map prove_distinct_thms
   754       (constr_rep_thmss ~~ dist_lemmas ~~ distinct_props);
   755 
   756     (** prove equations for permutation functions **)
   757 
   758     val abs_perm = PureThy.get_thms thy8 (Name "abs_perm"); (* FIXME *)
   759 
   760     val perm_simps' = map (fn (((i, (_, _, constrs)), tname), constr_rep_thms) =>
   761       let val T = replace_types' (nth_dtyp i)
   762       in List.concat (map (fn (atom, perm_closed_thms) =>
   763           map (fn ((cname, dts), constr_rep_thm) => 
   764         let
   765           val cname = Sign.intern_const thy8
   766             (NameSpace.append tname (Sign.base_name cname));
   767           val permT = mk_permT (Type (atom, []));
   768           val pi = Free ("pi", permT);
   769 
   770           fun perm t =
   771             let val T = fastype_of t
   772             in Const ("nominal.perm", permT --> T --> T) $ pi $ t end;
   773 
   774           fun constr_arg (dt, (j, l_args, r_args)) =
   775             let
   776               val x' = mk_Free "x" (typ_of_dtyp' dt) j;
   777               val (dts, dt') = strip_option dt;
   778               val Ts = map typ_of_dtyp' dts;
   779               val xs = map (fn (T, i) => mk_Free "x" T i)
   780                 (Ts ~~ (j upto j + length dts - 1))
   781               val x = mk_Free "x" (typ_of_dtyp' dt') (j + length dts);
   782               val (dts', dt'') = strip_dtyp dt';
   783             in case dt'' of
   784                 DtRec k => if k < length new_type_names then
   785                     (j + length dts + 1,
   786                      xs @ x :: l_args,
   787                      map perm (xs @ [x]) @ r_args)
   788                   else error "nested recursion not (yet) supported"
   789               | _ => (j + 1, x' :: l_args, perm x' :: r_args)
   790             end
   791 
   792           val (_, l_args, r_args) = foldr constr_arg (1, [], []) dts;
   793           val c = Const (cname, map fastype_of l_args ---> T)
   794         in
   795           standard (Goal.prove thy8 [] []
   796             (HOLogic.mk_Trueprop (HOLogic.mk_eq
   797               (perm (list_comb (c, l_args)), list_comb (c, r_args))))
   798             (fn _ => EVERY
   799               [simp_tac (simpset_of thy8 addsimps (constr_rep_thm :: perm_defs)) 1,
   800                simp_tac (HOL_basic_ss addsimps (Rep_thms @ Abs_inverse_thms @
   801                  constr_defs @ perm_closed_thms)) 1,
   802                TRY (simp_tac (HOL_basic_ss addsimps
   803                  (symmetric perm_fun_def :: abs_perm)) 1),
   804                TRY (simp_tac (HOL_basic_ss addsimps
   805                  (perm_fun_def :: perm_defs @ Rep_thms @ Abs_inverse_thms @
   806                     perm_closed_thms)) 1)]))
   807         end) (constrs ~~ constr_rep_thms)) (atoms ~~ perm_closed_thmss))
   808       end) (List.take (descr, length new_type_names) ~~ new_type_names ~~ constr_rep_thmss);
   809 
   810     (** prove injectivity of constructors **)
   811 
   812     val rep_inject_thms' = map (fn th => th RS sym) rep_inject_thms;
   813     val alpha = PureThy.get_thms thy8 (Name "alpha");
   814     val abs_fresh = PureThy.get_thms thy8 (Name "abs_fresh");
   815     val fresh_def = PureThy.get_thm thy8 (Name "fresh_def");
   816     val supp_def = PureThy.get_thm thy8 (Name "supp_def");
   817 
   818     val inject_thms = map (fn (((i, (_, _, constrs)), tname), constr_rep_thms) =>
   819       let val T = replace_types' (nth_dtyp i)
   820       in List.mapPartial (fn ((cname, dts), constr_rep_thm) =>
   821         if null dts then NONE else SOME
   822         let
   823           val cname = Sign.intern_const thy8
   824             (NameSpace.append tname (Sign.base_name cname));
   825 
   826           fun make_inj (dt, (j, args1, args2, eqs)) =
   827             let
   828               val x' = mk_Free "x" (typ_of_dtyp' dt) j;
   829               val y' = mk_Free "y" (typ_of_dtyp' dt) j;
   830               val (dts, dt') = strip_option dt;
   831               val Ts_idx = map typ_of_dtyp' dts ~~ (j upto j + length dts - 1);
   832               val xs = map (fn (T, i) => mk_Free "x" T i) Ts_idx;
   833               val ys = map (fn (T, i) => mk_Free "y" T i) Ts_idx;
   834               val x = mk_Free "x" (typ_of_dtyp' dt') (j + length dts);
   835               val y = mk_Free "y" (typ_of_dtyp' dt') (j + length dts);
   836               val (dts', dt'') = strip_dtyp dt';
   837             in case dt'' of
   838                 DtRec k => if k < length new_type_names then
   839                     (j + length dts + 1,
   840                      xs @ (x :: args1), ys @ (y :: args2),
   841                      HOLogic.mk_eq
   842                        (foldr mk_abs_fun x xs, foldr mk_abs_fun y ys) :: eqs)
   843                   else error "nested recursion not (yet) supported"
   844               | _ => (j + 1, x' :: args1, y' :: args2, HOLogic.mk_eq (x', y') :: eqs)
   845             end;
   846 
   847           val (_, args1, args2, eqs) = foldr make_inj (1, [], [], []) dts;
   848           val Ts = map fastype_of args1;
   849           val c = Const (cname, Ts ---> T)
   850         in
   851           standard (Goal.prove thy8 [] [] (HOLogic.mk_Trueprop (HOLogic.mk_eq
   852               (HOLogic.mk_eq (list_comb (c, args1), list_comb (c, args2)),
   853                foldr1 HOLogic.mk_conj eqs)))
   854             (fn _ => EVERY
   855                [asm_full_simp_tac (simpset_of thy8 addsimps (constr_rep_thm ::
   856                   rep_inject_thms')) 1,
   857                 TRY (asm_full_simp_tac (HOL_basic_ss addsimps (fresh_def :: supp_def ::
   858                   alpha @ abs_perm @ abs_fresh @ rep_inject_thms @
   859                   perm_rep_perm_thms)) 1),
   860                 TRY (asm_full_simp_tac (HOL_basic_ss addsimps (perm_fun_def ::
   861                   expand_fun_eq :: rep_inject_thms @ perm_rep_perm_thms)) 1)]))
   862         end) (constrs ~~ constr_rep_thms)
   863       end) (List.take (descr, length new_type_names) ~~ new_type_names ~~ constr_rep_thmss);
   864 
   865     (** equations for support and freshness **)
   866 
   867     val Un_assoc = PureThy.get_thm thy8 (Name "Un_assoc");
   868     val de_Morgan_conj = PureThy.get_thm thy8 (Name "de_Morgan_conj");
   869     val Collect_disj_eq = PureThy.get_thm thy8 (Name "Collect_disj_eq");
   870     val finite_Un = PureThy.get_thm thy8 (Name "finite_Un");
   871 
   872     val (supp_thms, fresh_thms) = ListPair.unzip (map ListPair.unzip
   873       (map (fn ((((i, (_, _, constrs)), tname), inject_thms'), perm_thms') =>
   874       let val T = replace_types' (nth_dtyp i)
   875       in List.concat (map (fn (cname, dts) => map (fn atom =>
   876         let
   877           val cname = Sign.intern_const thy8
   878             (NameSpace.append tname (Sign.base_name cname));
   879           val atomT = Type (atom, []);
   880 
   881           fun process_constr (dt, (j, args1, args2)) =
   882             let
   883               val x' = mk_Free "x" (typ_of_dtyp' dt) j;
   884               val (dts, dt') = strip_option dt;
   885               val Ts_idx = map typ_of_dtyp' dts ~~ (j upto j + length dts - 1);
   886               val xs = map (fn (T, i) => mk_Free "x" T i) Ts_idx;
   887               val x = mk_Free "x" (typ_of_dtyp' dt') (j + length dts);
   888               val (dts', dt'') = strip_dtyp dt';
   889             in case dt'' of
   890                 DtRec k => if k < length new_type_names then
   891                     (j + length dts + 1,
   892                      xs @ (x :: args1), foldr mk_abs_fun x xs :: args2)
   893                   else error "nested recursion not (yet) supported"
   894               | _ => (j + 1, x' :: args1, x' :: args2)
   895             end;
   896 
   897           val (_, args1, args2) = foldr process_constr (1, [], []) dts;
   898           val Ts = map fastype_of args1;
   899           val c = list_comb (Const (cname, Ts ---> T), args1);
   900           fun supp t =
   901             Const ("nominal.supp", fastype_of t --> HOLogic.mk_setT atomT) $ t;
   902           fun fresh t =
   903             Const ("nominal.fresh", atomT --> fastype_of t --> HOLogic.boolT) $
   904               Free ("a", atomT) $ t;
   905           val supp_thm = standard (Goal.prove thy8 [] []
   906               (HOLogic.mk_Trueprop (HOLogic.mk_eq
   907                 (supp c,
   908                  if null dts then Const ("{}", HOLogic.mk_setT atomT)
   909                  else foldr1 (HOLogic.mk_binop "op Un") (map supp args2))))
   910             (fn _ =>
   911               simp_tac (HOL_basic_ss addsimps (supp_def ::
   912                  Un_assoc :: de_Morgan_conj :: Collect_disj_eq :: finite_Un ::
   913                  symmetric empty_def :: Finites.emptyI :: simp_thms @
   914                  abs_perm @ abs_fresh @ inject_thms' @ perm_thms')) 1))
   915         in
   916           (supp_thm,
   917            standard (Goal.prove thy8 [] [] (HOLogic.mk_Trueprop (HOLogic.mk_eq
   918               (fresh c,
   919                if null dts then HOLogic.true_const
   920                else foldr1 HOLogic.mk_conj (map fresh args2))))
   921              (fn _ =>
   922                simp_tac (simpset_of thy8 addsimps [fresh_def, supp_thm]) 1)))
   923         end) atoms) constrs)
   924       end) (List.take (descr, length new_type_names) ~~ new_type_names ~~ inject_thms ~~ perm_simps')));
   925 
   926     (**** weak induction theorem ****)
   927 
   928     val arities = get_arities descr'';
   929 
   930     fun mk_funs_inv thm =
   931       let
   932         val {sign, prop, ...} = rep_thm thm;
   933         val _ $ (_ $ (Const (_, Type (_, [U, _])) $ _ $ S)) $
   934           (_ $ (_ $ (r $ (a $ _)) $ _)) = Type.freeze prop;
   935         val used = add_term_tfree_names (a, []);
   936 
   937         fun mk_thm i =
   938           let
   939             val Ts = map (TFree o rpair HOLogic.typeS)
   940               (variantlist (replicate i "'t", used));
   941             val f = Free ("f", Ts ---> U)
   942           in standard (Goal.prove sign [] [] (Logic.mk_implies
   943             (HOLogic.mk_Trueprop (HOLogic.list_all
   944                (map (pair "x") Ts, HOLogic.mk_mem (app_bnds f i, S))),
   945              HOLogic.mk_Trueprop (HOLogic.mk_eq (list_abs (map (pair "x") Ts,
   946                r $ (a $ app_bnds f i)), f))))
   947             (fn _ => EVERY [REPEAT (rtac ext 1), REPEAT (etac allE 1), rtac thm 1, atac 1]))
   948           end
   949       in map (fn r => r RS subst) (thm :: map mk_thm arities) end;
   950 
   951     fun mk_indrule_lemma ((prems, concls), (((i, _), T), U)) =
   952       let
   953         val Rep_t = Const (List.nth (rep_names, i), T --> U) $
   954           mk_Free "x" T i;
   955 
   956         val Abs_t =  Const (List.nth (abs_names, i), U --> T)
   957 
   958       in (prems @ [HOLogic.imp $ HOLogic.mk_mem (Rep_t,
   959             Const (List.nth (rep_set_names, i), HOLogic.mk_setT U)) $
   960               (mk_Free "P" (T --> HOLogic.boolT) (i + 1) $ (Abs_t $ Rep_t))],
   961           concls @ [mk_Free "P" (T --> HOLogic.boolT) (i + 1) $ mk_Free "x" T i])
   962       end;
   963 
   964     val (indrule_lemma_prems, indrule_lemma_concls) =
   965       Library.foldl mk_indrule_lemma (([], []), (descr'' ~~ recTs ~~ recTs'));
   966 
   967     val indrule_lemma = standard (Goal.prove thy8 [] []
   968       (Logic.mk_implies
   969         (HOLogic.mk_Trueprop (mk_conj indrule_lemma_prems),
   970          HOLogic.mk_Trueprop (mk_conj indrule_lemma_concls))) (fn _ => EVERY
   971            [REPEAT (etac conjE 1),
   972             REPEAT (EVERY
   973               [TRY (rtac conjI 1), full_simp_tac (HOL_basic_ss addsimps Rep_inverse_thms) 1,
   974                etac mp 1, resolve_tac Rep_thms 1])]));
   975 
   976     val Ps = map head_of (HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of indrule_lemma)));
   977     val frees = if length Ps = 1 then [Free ("P", snd (dest_Var (hd Ps)))] else
   978       map (Free o apfst fst o dest_Var) Ps;
   979     val indrule_lemma' = cterm_instantiate
   980       (map (cterm_of thy8) Ps ~~ map (cterm_of thy8) frees) indrule_lemma;
   981 
   982     val Abs_inverse_thms' = List.concat (map mk_funs_inv Abs_inverse_thms);
   983 
   984     val dt_induct_prop = DatatypeProp.make_ind descr' sorts';
   985     val dt_induct = standard (Goal.prove thy8 []
   986       (Logic.strip_imp_prems dt_induct_prop) (Logic.strip_imp_concl dt_induct_prop)
   987       (fn prems => EVERY
   988         [rtac indrule_lemma' 1,
   989          (DatatypeAux.indtac rep_induct THEN_ALL_NEW ObjectLogic.atomize_tac) 1,
   990          EVERY (map (fn (prem, r) => (EVERY
   991            [REPEAT (eresolve_tac Abs_inverse_thms' 1),
   992             simp_tac (HOL_basic_ss addsimps [symmetric r]) 1,
   993             DEPTH_SOLVE_1 (ares_tac [prem] 1 ORELSE etac allE 1)]))
   994                 (prems ~~ constr_defs))]));
   995 
   996     val case_names_induct = mk_case_names_induct descr'';
   997 
   998     (**** prove that new datatypes have finite support ****)
   999 
  1000     val indnames = DatatypeProp.make_tnames recTs;
  1001 
  1002     val abs_supp = PureThy.get_thms thy8 (Name "abs_supp");
  1003     val supp_atm = PureThy.get_thms thy8 (Name "supp_atm");
  1004 
  1005     val finite_supp_thms = map (fn atom =>
  1006       let val atomT = Type (atom, [])
  1007       in map standard (List.take
  1008         (split_conj_thm (Goal.prove thy8 [] [] (HOLogic.mk_Trueprop
  1009            (foldr1 HOLogic.mk_conj (map (fn (s, T) => HOLogic.mk_mem
  1010              (Const ("nominal.supp", T --> HOLogic.mk_setT atomT) $ Free (s, T),
  1011               Const ("Finite_Set.Finites", HOLogic.mk_setT (HOLogic.mk_setT atomT))))
  1012                (indnames ~~ recTs))))
  1013            (fn _ => indtac dt_induct indnames 1 THEN
  1014             ALLGOALS (asm_full_simp_tac (simpset_of thy8 addsimps
  1015               (abs_supp @ supp_atm @
  1016                PureThy.get_thms thy8 (Name ("fs_" ^ Sign.base_name atom ^ "1")) @
  1017                List.concat supp_thms))))),
  1018          length new_type_names))
  1019       end) atoms;
  1020 
  1021     (**** strong induction theorem ****)
  1022 
  1023     val pnames = if length descr'' = 1 then ["P"]
  1024       else map (fn i => "P" ^ string_of_int i) (1 upto length descr'');
  1025     val ind_sort = map (fn T => Sign.intern_class thy8 ("fs_" ^
  1026       Sign.base_name (fst (dest_Type T)))) dt_atomTs;
  1027     val fsT = TFree ("'n", ind_sort);
  1028 
  1029     fun make_pred i T =
  1030       Free (List.nth (pnames, i), T --> fsT --> HOLogic.boolT);
  1031 
  1032     fun make_ind_prem k T ((cname, cargs), idxs) =
  1033       let
  1034         val recs = List.filter is_rec_type cargs;
  1035         val Ts = map (typ_of_dtyp descr'' sorts') cargs;
  1036         val recTs' = map (typ_of_dtyp descr'' sorts') recs;
  1037         val tnames = variantlist (DatatypeProp.make_tnames Ts, pnames);
  1038         val rec_tnames = map fst (List.filter (is_rec_type o snd) (tnames ~~ cargs));
  1039         val frees = tnames ~~ Ts;
  1040         val z = (variant tnames "z", fsT);
  1041 
  1042         fun mk_prem ((dt, s), T) =
  1043           let
  1044             val (Us, U) = strip_type T;
  1045             val l = length Us
  1046           in list_all (z :: map (pair "x") Us, HOLogic.mk_Trueprop
  1047             (make_pred (body_index dt) U $ app_bnds (Free (s, T)) l $ Bound l))
  1048           end;
  1049 
  1050         val prems = map mk_prem (recs ~~ rec_tnames ~~ recTs');
  1051         val prems' = map (fn p as (_, T) => HOLogic.mk_Trueprop
  1052             (Const ("nominal.fresh", T --> fsT --> HOLogic.boolT) $
  1053               Free p $ Free z))
  1054           (map (curry List.nth frees) (List.concat (map (fn (m, n) =>
  1055              m upto m + n - 1) idxs)))
  1056 
  1057       in list_all_free (z :: frees, Logic.list_implies (prems' @ prems,
  1058         HOLogic.mk_Trueprop (make_pred k T $ 
  1059           list_comb (Const (cname, Ts ---> T), map Free frees) $ Free z)))
  1060       end;
  1061 
  1062     val ind_prems = List.concat (map (fn (((i, (_, _, constrs)), (_, idxss)), T) =>
  1063       map (make_ind_prem i T) (constrs ~~ idxss)) (descr'' ~~ ndescr ~~ recTs));
  1064     val tnames = DatatypeProp.make_tnames recTs;
  1065     val z = (variant tnames "z", fsT);
  1066     val ind_concl = HOLogic.mk_Trueprop (foldr1 (HOLogic.mk_binop "op &")
  1067       (map (fn (((i, _), T), tname) => make_pred i T $ Free (tname, T) $ Free z)
  1068         (descr'' ~~ recTs ~~ tnames)));
  1069     val induct = Logic.list_implies (ind_prems, ind_concl);
  1070 
  1071     val simp_atts = replicate (length new_type_names) [Simplifier.simp_add_global];
  1072 
  1073     val (thy9, _) = thy8 |>
  1074       Theory.add_path big_name |>
  1075       PureThy.add_thms [(("induct_weak", dt_induct), [case_names_induct])] |>>
  1076       Theory.parent_path |>>>
  1077       DatatypeAux.store_thmss_atts "distinct" new_type_names simp_atts distinct_thms |>>>
  1078       DatatypeAux.store_thmss "constr_rep" new_type_names constr_rep_thmss |>>>
  1079       DatatypeAux.store_thmss_atts "perm" new_type_names simp_atts perm_simps' |>>>
  1080       DatatypeAux.store_thmss "inject" new_type_names inject_thms |>>>
  1081       DatatypeAux.store_thmss "supp" new_type_names supp_thms |>>>
  1082       DatatypeAux.store_thmss_atts "fresh" new_type_names simp_atts fresh_thms |>>
  1083       fold (fn (atom, ths) => fn thy =>
  1084         let val class = Sign.intern_class thy ("fs_" ^ Sign.base_name atom)
  1085         in fold (fn T => AxClass.add_inst_arity_i
  1086             (fst (dest_Type T),
  1087               replicate (length sorts) [class], [class])
  1088             (AxClass.intro_classes_tac [] THEN resolve_tac ths 1)) newTs thy
  1089         end) (atoms ~~ finite_supp_thms) |>>
  1090       Theory.add_path big_name |>>>
  1091       PureThy.add_axioms_i [(("induct_unsafe", induct), [case_names_induct])] |>>
  1092       Theory.parent_path;
  1093 
  1094   in
  1095     thy9
  1096   end;
  1097 
  1098 val add_nominal_datatype = gen_add_nominal_datatype read_typ true;
  1099 
  1100 
  1101 (* FIXME: The following stuff should be exported by DatatypePackage *)
  1102 
  1103 local structure P = OuterParse and K = OuterKeyword in
  1104 
  1105 val datatype_decl =
  1106   Scan.option (P.$$$ "(" |-- P.name --| P.$$$ ")") -- P.type_args -- P.name -- P.opt_infix --
  1107     (P.$$$ "=" |-- P.enum1 "|" (P.name -- Scan.repeat P.typ -- P.opt_mixfix));
  1108 
  1109 fun mk_datatype args =
  1110   let
  1111     val names = map (fn ((((NONE, _), t), _), _) => t | ((((SOME t, _), _), _), _) => t) args;
  1112     val specs = map (fn ((((_, vs), t), mx), cons) =>
  1113       (vs, t, mx, map (fn ((x, y), z) => (x, y, z)) cons)) args;
  1114   in add_nominal_datatype false names specs end;
  1115 
  1116 val nominal_datatypeP =
  1117   OuterSyntax.command "nominal_datatype" "define inductive datatypes" K.thy_decl
  1118     (P.and_list1 datatype_decl >> (Toplevel.theory o mk_datatype));
  1119 
  1120 val _ = OuterSyntax.add_parsers [nominal_datatypeP];
  1121 
  1122 end;
  1123 
  1124 end