src/HOL/Nominal/nominal_package.ML
author berghofe
Thu Mar 23 18:14:06 2006 +0100 (2006-03-23)
changeset 19322 bf84bdf05f14
parent 19275 3d10de7a8ca7
child 19403 5c15cd397a82
permissions -rw-r--r--
Replaced iteration combinator by recursion combinator.
     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 op = (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 mk_subgoal i f st =
    79   let
    80     val cg = List.nth (cprems_of st, i - 1);
    81     val g = term_of cg;
    82     val revcut_rl' = Thm.lift_rule cg revcut_rl;
    83     val v = head_of (Logic.strip_assums_concl (hd (prems_of revcut_rl')));
    84     val ps = Logic.strip_params g;
    85     val cert = cterm_of (sign_of_thm st);
    86   in
    87     compose_tac (false,
    88       Thm.instantiate ([], [(cert v, cert (list_abs (ps,
    89         f (rev ps) (Logic.strip_assums_hyp g) (Logic.strip_assums_concl g))))])
    90       revcut_rl', 2) i st
    91   end;
    92 
    93 (** simplification procedure for sorting permutations **)
    94 
    95 val dj_cp = thm "dj_cp";
    96 
    97 fun dest_permT (Type ("fun", [Type ("List.list", [Type ("*", [T, _])]),
    98       Type ("fun", [_, U])])) = (T, U);
    99 
   100 fun permTs_of (Const ("nominal.perm", T) $ t $ u) = fst (dest_permT T) :: permTs_of u
   101   | permTs_of _ = [];
   102 
   103 fun perm_simproc' thy ss (Const ("nominal.perm", T) $ t $ (u as Const ("nominal.perm", U) $ r $ s)) =
   104       let
   105         val (aT as Type (a, []), S) = dest_permT T;
   106         val (bT as Type (b, []), _) = dest_permT U
   107       in if aT mem permTs_of u andalso aT <> bT then
   108           let
   109             val a' = Sign.base_name a;
   110             val b' = Sign.base_name b;
   111             val cp = PureThy.get_thm thy (Name ("cp_" ^ a' ^ "_" ^ b' ^ "_inst"));
   112             val dj = PureThy.get_thm thy (Name ("dj_" ^ b' ^ "_" ^ a'));
   113             val dj_cp' = [cp, dj] MRS dj_cp;
   114             val cert = SOME o cterm_of thy
   115           in
   116             SOME (mk_meta_eq (Drule.instantiate' [SOME (ctyp_of thy S)]
   117               [cert t, cert r, cert s] dj_cp'))
   118           end
   119         else NONE
   120       end
   121   | perm_simproc' thy ss _ = NONE;
   122 
   123 val perm_simproc =
   124   Simplifier.simproc (the_context ()) "perm_simp" ["pi1 \\<bullet> (pi2 \\<bullet> x)"] perm_simproc';
   125 
   126 val allE_Nil = read_instantiate_sg (the_context()) [("x", "[]")] allE;
   127 
   128 val meta_spec = thm "meta_spec";
   129 
   130 fun projections rule =
   131   ProjectRule.projections rule
   132   |> map (standard #> RuleCases.save rule);
   133 
   134 fun norm_sort thy = Sorts.norm_sort (snd (#classes (Type.rep_tsig (Sign.tsig_of thy))));
   135 
   136 fun gen_add_nominal_datatype prep_typ err flat_names new_type_names dts thy =
   137   let
   138     (* this theory is used just for parsing *)
   139   
   140     val tmp_thy = thy |>
   141       Theory.copy |>
   142       Theory.add_types (map (fn (tvs, tname, mx, _) =>
   143         (tname, length tvs, mx)) dts);
   144 
   145     val sign = Theory.sign_of tmp_thy;
   146 
   147     val atoms = atoms_of thy;
   148     val classes = map (NameSpace.map_base (fn s => "pt_" ^ s)) atoms;
   149     val cp_classes = List.concat (map (fn atom1 => map (fn atom2 =>
   150       Sign.intern_class thy ("cp_" ^ Sign.base_name atom1 ^ "_" ^
   151         Sign.base_name atom2)) atoms) atoms);
   152     fun augment_sort S = S union classes;
   153     val augment_sort_typ = map_type_tfree (fn (s, S) => TFree (s, augment_sort S));
   154 
   155     fun prep_constr ((constrs, sorts), (cname, cargs, mx)) =
   156       let val (cargs', sorts') = Library.foldl (prep_typ sign) (([], sorts), cargs)
   157       in (constrs @ [(cname, cargs', mx)], sorts') end
   158 
   159     fun prep_dt_spec ((dts, sorts), (tvs, tname, mx, constrs)) =
   160       let val (constrs', sorts') = Library.foldl prep_constr (([], sorts), constrs)
   161       in (dts @ [(tvs, tname, mx, constrs')], sorts') end
   162 
   163     val (dts', sorts) = Library.foldl prep_dt_spec (([], []), dts);
   164     val sorts' = map (apsnd augment_sort) sorts;
   165     val tyvars = map #1 dts';
   166 
   167     val types_syntax = map (fn (tvs, tname, mx, constrs) => (tname, mx)) dts';
   168     val constr_syntax = map (fn (tvs, tname, mx, constrs) =>
   169       map (fn (cname, cargs, mx) => (cname, mx)) constrs) dts';
   170 
   171     val ps = map (fn (_, n, _, _) =>
   172       (Sign.full_name sign n, Sign.full_name sign (n ^ "_Rep"))) dts;
   173     val rps = map Library.swap ps;
   174 
   175     fun replace_types (Type ("nominal.ABS", [T, U])) = 
   176           Type ("fun", [T, Type ("nominal.noption", [replace_types U])])
   177       | replace_types (Type (s, Ts)) =
   178           Type (getOpt (AList.lookup op = ps s, s), map replace_types Ts)
   179       | replace_types T = T;
   180 
   181     fun replace_types' (Type (s, Ts)) =
   182           Type (getOpt (AList.lookup op = rps s, s), map replace_types' Ts)
   183       | replace_types' T = T;
   184 
   185     val dts'' = map (fn (tvs, tname, mx, constrs) => (tvs, tname ^ "_Rep", NoSyn,
   186       map (fn (cname, cargs, mx) => (cname,
   187         map (augment_sort_typ o replace_types) cargs, NoSyn)) constrs)) dts';
   188 
   189     val new_type_names' = map (fn n => n ^ "_Rep") new_type_names;
   190     val full_new_type_names' = map (Sign.full_name (sign_of thy)) new_type_names';
   191 
   192     val ({induction, ...},thy1) =
   193       DatatypePackage.add_datatype_i err flat_names new_type_names' dts'' thy;
   194 
   195     val SOME {descr, ...} = Symtab.lookup
   196       (DatatypePackage.get_datatypes thy1) (hd full_new_type_names');
   197     fun nth_dtyp i = typ_of_dtyp descr sorts' (DtRec i);
   198 
   199     (**** define permutation functions ****)
   200 
   201     val permT = mk_permT (TFree ("'x", HOLogic.typeS));
   202     val pi = Free ("pi", permT);
   203     val perm_types = map (fn (i, _) =>
   204       let val T = nth_dtyp i
   205       in permT --> T --> T end) descr;
   206     val perm_names = replicate (length new_type_names) "nominal.perm" @
   207       DatatypeProp.indexify_names (map (fn i => Sign.full_name (sign_of thy1)
   208         ("perm_" ^ name_of_typ (nth_dtyp i)))
   209           (length new_type_names upto length descr - 1));
   210     val perm_names_types = perm_names ~~ perm_types;
   211 
   212     val perm_eqs = List.concat (map (fn (i, (_, _, constrs)) =>
   213       let val T = nth_dtyp i
   214       in map (fn (cname, dts) => 
   215         let
   216           val Ts = map (typ_of_dtyp descr sorts') dts;
   217           val names = DatatypeProp.make_tnames Ts;
   218           val args = map Free (names ~~ Ts);
   219           val c = Const (cname, Ts ---> T);
   220           fun perm_arg (dt, x) =
   221             let val T = type_of x
   222             in if is_rec_type dt then
   223                 let val (Us, _) = strip_type T
   224                 in list_abs (map (pair "x") Us,
   225                   Const (List.nth (perm_names_types, body_index dt)) $ pi $
   226                     list_comb (x, map (fn (i, U) =>
   227                       Const ("nominal.perm", permT --> U --> U) $
   228                         (Const ("List.rev", permT --> permT) $ pi) $
   229                         Bound i) ((length Us - 1 downto 0) ~~ Us)))
   230                 end
   231               else Const ("nominal.perm", permT --> T --> T) $ pi $ x
   232             end;  
   233         in
   234           (("", HOLogic.mk_Trueprop (HOLogic.mk_eq
   235             (Const (List.nth (perm_names_types, i)) $
   236                Free ("pi", mk_permT (TFree ("'x", HOLogic.typeS))) $
   237                list_comb (c, args),
   238              list_comb (c, map perm_arg (dts ~~ args))))), [])
   239         end) constrs
   240       end) descr);
   241 
   242     val (thy2, perm_simps) = thy1 |>
   243       Theory.add_consts_i (map (fn (s, T) => (Sign.base_name s, T, NoSyn))
   244         (List.drop (perm_names_types, length new_type_names))) |>
   245       PrimrecPackage.add_primrec_i "" perm_eqs;
   246 
   247     (**** prove that permutation functions introduced by unfolding are ****)
   248     (**** equivalent to already existing permutation functions         ****)
   249 
   250     val _ = warning ("length descr: " ^ string_of_int (length descr));
   251     val _ = warning ("length new_type_names: " ^ string_of_int (length new_type_names));
   252 
   253     val perm_indnames = DatatypeProp.make_tnames (map body_type perm_types);
   254     val perm_fun_def = PureThy.get_thm thy2 (Name "perm_fun_def");
   255 
   256     val unfolded_perm_eq_thms =
   257       if length descr = length new_type_names then []
   258       else map standard (List.drop (split_conj_thm
   259         (Goal.prove thy2 [] []
   260           (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
   261             (map (fn (c as (s, T), x) =>
   262                let val [T1, T2] = binder_types T
   263                in HOLogic.mk_eq (Const c $ pi $ Free (x, T2),
   264                  Const ("nominal.perm", T) $ pi $ Free (x, T2))
   265                end)
   266              (perm_names_types ~~ perm_indnames))))
   267           (fn _ => EVERY [indtac induction perm_indnames 1,
   268             ALLGOALS (asm_full_simp_tac
   269               (simpset_of thy2 addsimps [perm_fun_def]))])),
   270         length new_type_names));
   271 
   272     (**** prove [] \<bullet> t = t ****)
   273 
   274     val _ = warning "perm_empty_thms";
   275 
   276     val perm_empty_thms = List.concat (map (fn a =>
   277       let val permT = mk_permT (Type (a, []))
   278       in map standard (List.take (split_conj_thm
   279         (Goal.prove thy2 [] []
   280           (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
   281             (map (fn ((s, T), x) => HOLogic.mk_eq
   282                 (Const (s, permT --> T --> T) $
   283                    Const ("List.list.Nil", permT) $ Free (x, T),
   284                  Free (x, T)))
   285              (perm_names ~~
   286               map body_type perm_types ~~ perm_indnames))))
   287           (fn _ => EVERY [indtac induction perm_indnames 1,
   288             ALLGOALS (asm_full_simp_tac (simpset_of thy2))])),
   289         length new_type_names))
   290       end)
   291       atoms);
   292 
   293     (**** prove (pi1 @ pi2) \<bullet> t = pi1 \<bullet> (pi2 \<bullet> t) ****)
   294 
   295     val _ = warning "perm_append_thms";
   296 
   297     (*FIXME: these should be looked up statically*)
   298     val at_pt_inst = PureThy.get_thm thy2 (Name "at_pt_inst");
   299     val pt2 = PureThy.get_thm thy2 (Name "pt2");
   300 
   301     val perm_append_thms = List.concat (map (fn a =>
   302       let
   303         val permT = mk_permT (Type (a, []));
   304         val pi1 = Free ("pi1", permT);
   305         val pi2 = Free ("pi2", permT);
   306         val pt_inst = PureThy.get_thm thy2 (Name ("pt_" ^ Sign.base_name a ^ "_inst"));
   307         val pt2' = pt_inst RS pt2;
   308         val pt2_ax = PureThy.get_thm thy2
   309           (Name (NameSpace.map_base (fn s => "pt_" ^ s ^ "2") a));
   310       in List.take (map standard (split_conj_thm
   311         (Goal.prove thy2 [] []
   312              (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
   313                 (map (fn ((s, T), x) =>
   314                     let val perm = Const (s, permT --> T --> T)
   315                     in HOLogic.mk_eq
   316                       (perm $ (Const ("List.op @", permT --> permT --> permT) $
   317                          pi1 $ pi2) $ Free (x, T),
   318                        perm $ pi1 $ (perm $ pi2 $ Free (x, T)))
   319                     end)
   320                   (perm_names ~~
   321                    map body_type perm_types ~~ perm_indnames))))
   322            (fn _ => EVERY [indtac induction perm_indnames 1,
   323               ALLGOALS (asm_full_simp_tac (simpset_of thy2 addsimps [pt2', pt2_ax]))]))),
   324          length new_type_names)
   325       end) atoms);
   326 
   327     (**** prove pi1 ~ pi2 ==> pi1 \<bullet> t = pi2 \<bullet> t ****)
   328 
   329     val _ = warning "perm_eq_thms";
   330 
   331     val pt3 = PureThy.get_thm thy2 (Name "pt3");
   332     val pt3_rev = PureThy.get_thm thy2 (Name "pt3_rev");
   333 
   334     val perm_eq_thms = List.concat (map (fn a =>
   335       let
   336         val permT = mk_permT (Type (a, []));
   337         val pi1 = Free ("pi1", permT);
   338         val pi2 = Free ("pi2", permT);
   339         (*FIXME: not robust - better access these theorems using NominalData?*)
   340         val at_inst = PureThy.get_thm thy2 (Name ("at_" ^ Sign.base_name a ^ "_inst"));
   341         val pt_inst = PureThy.get_thm thy2 (Name ("pt_" ^ Sign.base_name a ^ "_inst"));
   342         val pt3' = pt_inst RS pt3;
   343         val pt3_rev' = at_inst RS (pt_inst RS pt3_rev);
   344         val pt3_ax = PureThy.get_thm thy2
   345           (Name (NameSpace.map_base (fn s => "pt_" ^ s ^ "3") a));
   346       in List.take (map standard (split_conj_thm
   347         (Goal.prove thy2 [] [] (Logic.mk_implies
   348              (HOLogic.mk_Trueprop (Const ("nominal.prm_eq",
   349                 permT --> permT --> HOLogic.boolT) $ pi1 $ pi2),
   350               HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
   351                 (map (fn ((s, T), x) =>
   352                     let val perm = Const (s, permT --> T --> T)
   353                     in HOLogic.mk_eq
   354                       (perm $ pi1 $ Free (x, T),
   355                        perm $ pi2 $ Free (x, T))
   356                     end)
   357                   (perm_names ~~
   358                    map body_type perm_types ~~ perm_indnames)))))
   359            (fn _ => EVERY [indtac induction perm_indnames 1,
   360               ALLGOALS (asm_full_simp_tac (simpset_of thy2 addsimps [pt3', pt3_rev', pt3_ax]))]))),
   361          length new_type_names)
   362       end) atoms);
   363 
   364     (**** prove pi1 \<bullet> (pi2 \<bullet> t) = (pi1 \<bullet> pi2) \<bullet> (pi1 \<bullet> t) ****)
   365 
   366     val cp1 = PureThy.get_thm thy2 (Name "cp1");
   367     val dj_cp = PureThy.get_thm thy2 (Name "dj_cp");
   368     val pt_perm_compose = PureThy.get_thm thy2 (Name "pt_perm_compose");
   369     val pt_perm_compose_rev = PureThy.get_thm thy2 (Name "pt_perm_compose_rev");
   370     val dj_perm_perm_forget = PureThy.get_thm thy2 (Name "dj_perm_perm_forget");
   371 
   372     fun composition_instance name1 name2 thy =
   373       let
   374         val name1' = Sign.base_name name1;
   375         val name2' = Sign.base_name name2;
   376         val cp_class = Sign.intern_class thy ("cp_" ^ name1' ^ "_" ^ name2');
   377         val permT1 = mk_permT (Type (name1, []));
   378         val permT2 = mk_permT (Type (name2, []));
   379         val augment = map_type_tfree
   380           (fn (x, S) => TFree (x, cp_class :: S));
   381         val Ts = map (augment o body_type) perm_types;
   382         val cp_inst = PureThy.get_thm thy
   383           (Name ("cp_" ^ name1' ^ "_" ^ name2' ^ "_inst"));
   384         val simps = simpset_of thy addsimps (perm_fun_def ::
   385           (if name1 <> name2 then
   386              let val dj = PureThy.get_thm thy (Name ("dj_" ^ name2' ^ "_" ^ name1'))
   387              in [dj RS (cp_inst RS dj_cp), dj RS dj_perm_perm_forget] end
   388            else
   389              let
   390                val at_inst = PureThy.get_thm thy (Name ("at_" ^ name1' ^ "_inst"));
   391                val pt_inst = PureThy.get_thm thy (Name ("pt_" ^ name1' ^ "_inst"))
   392              in
   393                [cp_inst RS cp1 RS sym,
   394                 at_inst RS (pt_inst RS pt_perm_compose) RS sym,
   395                 at_inst RS (pt_inst RS pt_perm_compose_rev) RS sym]
   396             end))
   397         val thms = split_conj_thm (standard (Goal.prove thy [] []
   398             (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
   399               (map (fn ((s, T), x) =>
   400                   let
   401                     val pi1 = Free ("pi1", permT1);
   402                     val pi2 = Free ("pi2", permT2);
   403                     val perm1 = Const (s, permT1 --> T --> T);
   404                     val perm2 = Const (s, permT2 --> T --> T);
   405                     val perm3 = Const ("nominal.perm", permT1 --> permT2 --> permT2)
   406                   in HOLogic.mk_eq
   407                     (perm1 $ pi1 $ (perm2 $ pi2 $ Free (x, T)),
   408                      perm2 $ (perm3 $ pi1 $ pi2) $ (perm1 $ pi1 $ Free (x, T)))
   409                   end)
   410                 (perm_names ~~ Ts ~~ perm_indnames))))
   411           (fn _ => EVERY [indtac induction perm_indnames 1,
   412              ALLGOALS (asm_full_simp_tac simps)])))
   413       in
   414         foldl (fn ((s, tvs), thy) => AxClass.prove_arity
   415             (s, replicate (length tvs) (cp_class :: classes), [cp_class])
   416             (ClassPackage.intro_classes_tac [] THEN ALLGOALS (resolve_tac thms)) thy)
   417           thy (full_new_type_names' ~~ tyvars)
   418       end;
   419 
   420     val (perm_thmss,thy3) = thy2 |>
   421       fold (fn name1 => fold (composition_instance name1) atoms) atoms |>
   422       curry (Library.foldr (fn ((i, (tyname, args, _)), thy) =>
   423         AxClass.prove_arity (tyname, replicate (length args) classes, classes)
   424         (ClassPackage.intro_classes_tac [] THEN REPEAT (EVERY
   425            [resolve_tac perm_empty_thms 1,
   426             resolve_tac perm_append_thms 1,
   427             resolve_tac perm_eq_thms 1, assume_tac 1])) thy))
   428         (List.take (descr, length new_type_names)) |>
   429       PureThy.add_thmss
   430         [((space_implode "_" new_type_names ^ "_unfolded_perm_eq",
   431           unfolded_perm_eq_thms), [Simplifier.simp_add]),
   432          ((space_implode "_" new_type_names ^ "_perm_empty",
   433           perm_empty_thms), [Simplifier.simp_add]),
   434          ((space_implode "_" new_type_names ^ "_perm_append",
   435           perm_append_thms), [Simplifier.simp_add]),
   436          ((space_implode "_" new_type_names ^ "_perm_eq",
   437           perm_eq_thms), [Simplifier.simp_add])];
   438   
   439     (**** Define representing sets ****)
   440 
   441     val _ = warning "representing sets";
   442 
   443     val rep_set_names = map (Sign.full_name thy3) (DatatypeProp.indexify_names
   444       (map (fn (i, _) => name_of_typ (nth_dtyp i) ^ "_set") descr));
   445     val big_rep_name =
   446       space_implode "_" (DatatypeProp.indexify_names (List.mapPartial
   447         (fn (i, ("nominal.noption", _, _)) => NONE
   448           | (i, _) => SOME (name_of_typ (nth_dtyp i))) descr)) ^ "_set";
   449     val _ = warning ("big_rep_name: " ^ big_rep_name);
   450 
   451     fun strip_option (dtf as DtType ("fun", [dt, DtRec i])) =
   452           (case AList.lookup op = descr i of
   453              SOME ("nominal.noption", _, [(_, [dt']), _]) =>
   454                apfst (cons dt) (strip_option dt')
   455            | _ => ([], dtf))
   456       | strip_option (DtType ("fun", [dt, DtType ("nominal.noption", [dt'])])) =
   457           apfst (cons dt) (strip_option dt')
   458       | strip_option dt = ([], dt);
   459 
   460     val dt_atomTs = distinct op = (map (typ_of_dtyp descr sorts')
   461       (List.concat (map (fn (_, (_, _, cs)) => List.concat
   462         (map (List.concat o map (fst o strip_option) o snd) cs)) descr)));
   463 
   464     fun make_intr s T (cname, cargs) =
   465       let
   466         fun mk_prem (dt, (j, j', prems, ts)) = 
   467           let
   468             val (dts, dt') = strip_option dt;
   469             val (dts', dt'') = strip_dtyp dt';
   470             val Ts = map (typ_of_dtyp descr sorts') dts;
   471             val Us = map (typ_of_dtyp descr sorts') dts';
   472             val T = typ_of_dtyp descr sorts' dt'';
   473             val free = mk_Free "x" (Us ---> T) j;
   474             val free' = app_bnds free (length Us);
   475             fun mk_abs_fun (T, (i, t)) =
   476               let val U = fastype_of t
   477               in (i + 1, Const ("nominal.abs_fun", [T, U, T] --->
   478                 Type ("nominal.noption", [U])) $ mk_Free "y" T i $ t)
   479               end
   480           in (j + 1, j' + length Ts,
   481             case dt'' of
   482                 DtRec k => list_all (map (pair "x") Us,
   483                   HOLogic.mk_Trueprop (HOLogic.mk_mem (free',
   484                     Const (List.nth (rep_set_names, k),
   485                       HOLogic.mk_setT T)))) :: prems
   486               | _ => prems,
   487             snd (foldr mk_abs_fun (j', free) Ts) :: ts)
   488           end;
   489 
   490         val (_, _, prems, ts) = foldr mk_prem (1, 1, [], []) cargs;
   491         val concl = HOLogic.mk_Trueprop (HOLogic.mk_mem
   492           (list_comb (Const (cname, map fastype_of ts ---> T), ts),
   493            Const (s, HOLogic.mk_setT T)))
   494       in Logic.list_implies (prems, concl)
   495       end;
   496 
   497     val (intr_ts, ind_consts) =
   498       apfst List.concat (ListPair.unzip (List.mapPartial
   499         (fn ((_, ("nominal.noption", _, _)), _) => NONE
   500           | ((i, (_, _, constrs)), rep_set_name) =>
   501               let val T = nth_dtyp i
   502               in SOME (map (make_intr rep_set_name T) constrs,
   503                 Const (rep_set_name, HOLogic.mk_setT T))
   504               end)
   505                 (descr ~~ rep_set_names)));
   506 
   507     val (thy4, {raw_induct = rep_induct, intrs = rep_intrs, ...}) =
   508       setmp InductivePackage.quiet_mode false
   509         (InductivePackage.add_inductive_i false true big_rep_name false true false
   510            ind_consts (map (fn x => (("", x), [])) intr_ts) []) thy3;
   511 
   512     (**** Prove that representing set is closed under permutation ****)
   513 
   514     val _ = warning "proving closure under permutation...";
   515 
   516     val perm_indnames' = List.mapPartial
   517       (fn (x, (_, ("nominal.noption", _, _))) => NONE | (x, _) => SOME x)
   518       (perm_indnames ~~ descr);
   519 
   520     fun mk_perm_closed name = map (fn th => standard (th RS mp))
   521       (List.take (split_conj_thm (Goal.prove thy4 [] []
   522         (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj (map
   523            (fn (S, x) =>
   524               let
   525                 val S = map_term_types (map_type_tfree
   526                   (fn (s, cs) => TFree (s, cs union cp_classes))) S;
   527                 val T = HOLogic.dest_setT (fastype_of S);
   528                 val permT = mk_permT (Type (name, []))
   529               in HOLogic.mk_imp (HOLogic.mk_mem (Free (x, T), S),
   530                 HOLogic.mk_mem (Const ("nominal.perm", permT --> T --> T) $
   531                   Free ("pi", permT) $ Free (x, T), S))
   532               end) (ind_consts ~~ perm_indnames'))))
   533         (fn _ => EVERY (* CU: added perm_fun_def in the final tactic in order to deal with funs *)
   534            [indtac rep_induct [] 1,
   535             ALLGOALS (simp_tac (simpset_of thy4 addsimps
   536               (symmetric perm_fun_def :: PureThy.get_thms thy4 (Name ("abs_perm"))))),
   537             ALLGOALS (resolve_tac rep_intrs 
   538                THEN_ALL_NEW (asm_full_simp_tac (simpset_of thy4 addsimps [perm_fun_def])))])),
   539         length new_type_names));
   540 
   541     (* FIXME: theorems are stored in database for testing only *)
   542     val perm_closed_thmss = map mk_perm_closed atoms;
   543     val (_,thy5) = PureThy.add_thmss [(("perm_closed", List.concat perm_closed_thmss), [])] thy4;
   544 
   545     (**** typedef ****)
   546 
   547     val _ = warning "defining type...";
   548 
   549     val (typedefs, thy6) =
   550       fold_map (fn ((((name, mx), tvs), c), name') => fn thy =>
   551         setmp TypedefPackage.quiet_mode true
   552           (TypedefPackage.add_typedef_i false (SOME name') (name, tvs, mx) c NONE
   553             (rtac exI 1 THEN
   554               QUIET_BREADTH_FIRST (has_fewer_prems 1)
   555               (resolve_tac rep_intrs 1))) thy |> (fn (thy, r) =>
   556         let
   557           val permT = mk_permT (TFree (variant tvs "'a", HOLogic.typeS));
   558           val pi = Free ("pi", permT);
   559           val tvs' = map (fn s => TFree (s, the (AList.lookup op = sorts' s))) tvs;
   560           val T = Type (Sign.intern_type thy name, tvs');
   561           val Const (_, Type (_, [U])) = c
   562         in apfst (pair r o hd)
   563           (PureThy.add_defs_i true [(("prm_" ^ name ^ "_def", Logic.mk_equals
   564             (Const ("nominal.perm", permT --> T --> T) $ pi $ Free ("x", T),
   565              Const (Sign.intern_const thy ("Abs_" ^ name), U --> T) $
   566                (Const ("nominal.perm", permT --> U --> U) $ pi $
   567                  (Const (Sign.intern_const thy ("Rep_" ^ name), T --> U) $
   568                    Free ("x", T))))), [])] thy)
   569         end))
   570           (types_syntax ~~ tyvars ~~
   571             (List.take (ind_consts, length new_type_names)) ~~ new_type_names) thy5;
   572 
   573     val perm_defs = map snd typedefs;
   574     val Abs_inverse_thms = map (#Abs_inverse o fst) typedefs;
   575     val Rep_inverse_thms = map (#Rep_inverse o fst) typedefs;
   576     val Rep_thms = map (#Rep o fst) typedefs;
   577 
   578     val big_name = space_implode "_" new_type_names;
   579 
   580 
   581     (** prove that new types are in class pt_<name> **)
   582 
   583     val _ = warning "prove that new types are in class pt_<name> ...";
   584 
   585     fun pt_instance ((class, atom), perm_closed_thms) =
   586       fold (fn (((({Abs_inverse, Rep_inverse, Rep, ...},
   587         perm_def), name), tvs), perm_closed) => fn thy =>
   588           AxClass.prove_arity
   589             (Sign.intern_type thy name,
   590               replicate (length tvs) (classes @ cp_classes), [class])
   591             (EVERY [ClassPackage.intro_classes_tac [],
   592               rewrite_goals_tac [perm_def],
   593               asm_full_simp_tac (simpset_of thy addsimps [Rep_inverse]) 1,
   594               asm_full_simp_tac (simpset_of thy addsimps
   595                 [Rep RS perm_closed RS Abs_inverse]) 1,
   596               asm_full_simp_tac (HOL_basic_ss addsimps [PureThy.get_thm thy
   597                 (Name ("pt_" ^ Sign.base_name atom ^ "3"))]) 1]) thy)
   598         (typedefs ~~ new_type_names ~~ tyvars ~~ perm_closed_thms);
   599 
   600 
   601     (** prove that new types are in class cp_<name1>_<name2> **)
   602 
   603     val _ = warning "prove that new types are in class cp_<name1>_<name2> ...";
   604 
   605     fun cp_instance (atom1, perm_closed_thms1) (atom2, perm_closed_thms2) thy =
   606       let
   607         val name = "cp_" ^ Sign.base_name atom1 ^ "_" ^ Sign.base_name atom2;
   608         val class = Sign.intern_class thy name;
   609         val cp1' = PureThy.get_thm thy (Name (name ^ "_inst")) RS cp1
   610       in fold (fn ((((({Abs_inverse, Rep_inverse, Rep, ...},
   611         perm_def), name), tvs), perm_closed1), perm_closed2) => fn thy =>
   612           AxClass.prove_arity
   613             (Sign.intern_type thy name,
   614               replicate (length tvs) (classes @ cp_classes), [class])
   615             (EVERY [ClassPackage.intro_classes_tac [],
   616               rewrite_goals_tac [perm_def],
   617               asm_full_simp_tac (simpset_of thy addsimps
   618                 ((Rep RS perm_closed1 RS Abs_inverse) ::
   619                  (if atom1 = atom2 then []
   620                   else [Rep RS perm_closed2 RS Abs_inverse]))) 1,
   621               cong_tac 1,
   622               rtac refl 1,
   623               rtac cp1' 1]) thy)
   624         (typedefs ~~ new_type_names ~~ tyvars ~~ perm_closed_thms1 ~~
   625           perm_closed_thms2) thy
   626       end;
   627 
   628     val thy7 = fold (fn x => fn thy => thy |>
   629       pt_instance x |>
   630       fold (cp_instance (apfst snd x)) (atoms ~~ perm_closed_thmss))
   631         (classes ~~ atoms ~~ perm_closed_thmss) thy6;
   632 
   633     (**** constructors ****)
   634 
   635     fun mk_abs_fun (x, t) =
   636       let
   637         val T = fastype_of x;
   638         val U = fastype_of t
   639       in
   640         Const ("nominal.abs_fun", T --> U --> T -->
   641           Type ("nominal.noption", [U])) $ x $ t
   642       end;
   643 
   644     val (ty_idxs, _) = foldl
   645       (fn ((i, ("nominal.noption", _, _)), p) => p
   646         | ((i, _), (ty_idxs, j)) => (ty_idxs @ [(i, j)], j + 1)) ([], 0) descr;
   647 
   648     fun reindex (DtType (s, dts)) = DtType (s, map reindex dts)
   649       | reindex (DtRec i) = DtRec (the (AList.lookup op = ty_idxs i))
   650       | reindex dt = dt;
   651 
   652     fun strip_suffix i s = implode (List.take (explode s, size s - i));
   653 
   654     (** strips the "_Rep" in type names *)
   655     fun strip_nth_name i s = 
   656       let val xs = NameSpace.unpack s; 
   657       in NameSpace.pack (Library.nth_map (length xs - i) (strip_suffix 4) xs) end;
   658 
   659     val (descr'', ndescr) = ListPair.unzip (List.mapPartial
   660       (fn (i, ("nominal.noption", _, _)) => NONE
   661         | (i, (s, dts, constrs)) =>
   662              let
   663                val SOME index = AList.lookup op = ty_idxs i;
   664                val (constrs1, constrs2) = ListPair.unzip
   665                  (map (fn (cname, cargs) => apfst (pair (strip_nth_name 2 cname))
   666                    (foldl_map (fn (dts, dt) =>
   667                      let val (dts', dt') = strip_option dt
   668                      in (dts @ dts' @ [reindex dt'], (length dts, length dts')) end)
   669                        ([], cargs))) constrs)
   670              in SOME ((index, (strip_nth_name 1 s,  map reindex dts, constrs1)),
   671                (index, constrs2))
   672              end) descr);
   673 
   674     val (descr1, descr2) = splitAt (length new_type_names, descr'');
   675     val descr' = [descr1, descr2];
   676 
   677     val typ_of_dtyp' = replace_types' o typ_of_dtyp descr sorts';
   678 
   679     val rep_names = map (fn s =>
   680       Sign.intern_const thy7 ("Rep_" ^ s)) new_type_names;
   681     val abs_names = map (fn s =>
   682       Sign.intern_const thy7 ("Abs_" ^ s)) new_type_names;
   683 
   684     val recTs' = List.mapPartial
   685       (fn ((_, ("nominal.noption", _, _)), T) => NONE
   686         | (_, T) => SOME T) (descr ~~ get_rec_types descr sorts');
   687     val recTs = get_rec_types descr'' sorts';
   688     val newTs' = Library.take (length new_type_names, recTs');
   689     val newTs = Library.take (length new_type_names, recTs);
   690 
   691     val full_new_type_names = map (Sign.full_name (sign_of thy)) new_type_names;
   692 
   693     fun make_constr_def tname T T' ((thy, defs, eqns), ((cname, cargs), (cname', mx))) =
   694       let
   695         fun constr_arg (dt, (j, l_args, r_args)) =
   696           let
   697             val x' = mk_Free "x" (typ_of_dtyp' dt) j;
   698             val (dts, dt') = strip_option dt;
   699             val xs = map (fn (dt, i) => mk_Free "x" (typ_of_dtyp' dt) i)
   700               (dts ~~ (j upto j + length dts - 1))
   701             val x = mk_Free "x" (typ_of_dtyp' dt') (j + length dts)
   702             val (dts', dt'') = strip_dtyp dt'
   703           in
   704             (j + length dts + 1,
   705              xs @ x :: l_args,
   706              foldr mk_abs_fun
   707                (case dt'' of
   708                   DtRec k => if k < length new_type_names then
   709                       list_abs (map (pair "z" o typ_of_dtyp') dts',
   710                         Const (List.nth (rep_names, k), typ_of_dtyp' dt'' -->
   711                           typ_of_dtyp descr sorts' dt'') $ app_bnds x (length dts'))
   712                     else error "nested recursion not (yet) supported"
   713                 | _ => x) xs :: r_args)
   714           end
   715 
   716         val (_, l_args, r_args) = foldr constr_arg (1, [], []) cargs;
   717         val abs_name = Sign.intern_const (Theory.sign_of thy) ("Abs_" ^ tname);
   718         val rep_name = Sign.intern_const (Theory.sign_of thy) ("Rep_" ^ tname);
   719         val constrT = map fastype_of l_args ---> T;
   720         val lhs = list_comb (Const (Sign.full_name thy (Sign.base_name cname),
   721           constrT), l_args);
   722         val rhs = list_comb (Const (cname, map fastype_of r_args ---> T'), r_args);
   723         val def = Logic.mk_equals (lhs, Const (abs_name, T' --> T) $ rhs);
   724         val eqn = HOLogic.mk_Trueprop (HOLogic.mk_eq
   725           (Const (rep_name, T --> T') $ lhs, rhs));
   726         val def_name = (Sign.base_name cname) ^ "_def";
   727         val ([def_thm], thy') = thy |>
   728           Theory.add_consts_i [(cname', constrT, mx)] |>
   729           (PureThy.add_defs_i false o map Thm.no_attributes) [(def_name, def)]
   730       in (thy', defs @ [def_thm], eqns @ [eqn]) end;
   731 
   732     fun dt_constr_defs ((thy, defs, eqns, dist_lemmas),
   733         (((((_, (_, _, constrs)), tname), T), T'), constr_syntax)) =
   734       let
   735         val rep_const = cterm_of thy
   736           (Const (Sign.intern_const thy ("Rep_" ^ tname), T --> T'));
   737         val dist = standard (cterm_instantiate [(cterm_of thy distinct_f, rep_const)] distinct_lemma);
   738         val (thy', defs', eqns') = Library.foldl (make_constr_def tname T T')
   739           ((Theory.add_path tname thy, defs, []), constrs ~~ constr_syntax)
   740       in
   741         (parent_path flat_names thy', defs', eqns @ [eqns'], dist_lemmas @ [dist])
   742       end;
   743 
   744     val (thy8, constr_defs, constr_rep_eqns, dist_lemmas) = Library.foldl dt_constr_defs
   745       ((thy7, [], [], []), List.take (descr, length new_type_names) ~~
   746         new_type_names ~~ newTs ~~ newTs' ~~ constr_syntax);
   747 
   748     val abs_inject_thms = map (fn tname =>
   749       PureThy.get_thm thy8 (Name ("Abs_" ^ tname ^ "_inject"))) new_type_names;
   750 
   751     val rep_inject_thms = map (fn tname =>
   752       PureThy.get_thm thy8 (Name ("Rep_" ^ tname ^ "_inject"))) new_type_names;
   753 
   754     val rep_thms = map (fn tname =>
   755       PureThy.get_thm thy8 (Name ("Rep_" ^ tname))) new_type_names;
   756 
   757     val rep_inverse_thms = map (fn tname =>
   758       PureThy.get_thm thy8 (Name ("Rep_" ^ tname ^ "_inverse"))) new_type_names;
   759 
   760     (* prove theorem  Rep_i (Constr_j ...) = Constr'_j ...  *)
   761     
   762     fun prove_constr_rep_thm eqn =
   763       let
   764         val inj_thms = map (fn r => r RS iffD1) abs_inject_thms;
   765         val rewrites = constr_defs @ map mk_meta_eq rep_inverse_thms
   766       in standard (Goal.prove thy8 [] [] eqn (fn _ => EVERY
   767         [resolve_tac inj_thms 1,
   768          rewrite_goals_tac rewrites,
   769          rtac refl 3,
   770          resolve_tac rep_intrs 2,
   771          REPEAT (resolve_tac rep_thms 1)]))
   772       end;
   773 
   774     val constr_rep_thmss = map (map prove_constr_rep_thm) constr_rep_eqns;
   775 
   776     (* prove theorem  pi \<bullet> Rep_i x = Rep_i (pi \<bullet> x) *)
   777 
   778     fun prove_perm_rep_perm (atom, perm_closed_thms) = map (fn th =>
   779       let
   780         val _ $ (_ $ (Rep $ x) $ _) = Logic.unvarify (prop_of th);
   781         val Type ("fun", [T, U]) = fastype_of Rep;
   782         val permT = mk_permT (Type (atom, []));
   783         val pi = Free ("pi", permT);
   784       in
   785         standard (Goal.prove thy8 [] [] (HOLogic.mk_Trueprop (HOLogic.mk_eq
   786             (Const ("nominal.perm", permT --> U --> U) $ pi $ (Rep $ x),
   787              Rep $ (Const ("nominal.perm", permT --> T --> T) $ pi $ x))))
   788           (fn _ => simp_tac (HOL_basic_ss addsimps (perm_defs @ Abs_inverse_thms @
   789             perm_closed_thms @ Rep_thms)) 1))
   790       end) Rep_thms;
   791 
   792     val perm_rep_perm_thms = List.concat (map prove_perm_rep_perm
   793       (atoms ~~ perm_closed_thmss));
   794 
   795     (* prove distinctness theorems *)
   796 
   797     val distinct_props = setmp DatatypeProp.dtK 1000
   798       (DatatypeProp.make_distincts new_type_names descr' sorts') thy8;
   799 
   800     val dist_rewrites = map (fn (rep_thms, dist_lemma) =>
   801       dist_lemma::(rep_thms @ [In0_eq, In1_eq, In0_not_In1, In1_not_In0]))
   802         (constr_rep_thmss ~~ dist_lemmas);
   803 
   804     fun prove_distinct_thms (_, []) = []
   805       | prove_distinct_thms (p as (rep_thms, dist_lemma), t::ts) =
   806           let
   807             val dist_thm = standard (Goal.prove thy8 [] [] t (fn _ =>
   808               simp_tac (simpset_of thy8 addsimps (dist_lemma :: rep_thms)) 1))
   809           in dist_thm::(standard (dist_thm RS not_sym))::
   810             (prove_distinct_thms (p, ts))
   811           end;
   812 
   813     val distinct_thms = map prove_distinct_thms
   814       (constr_rep_thmss ~~ dist_lemmas ~~ distinct_props);
   815 
   816     (** prove equations for permutation functions **)
   817 
   818     val abs_perm = PureThy.get_thms thy8 (Name "abs_perm"); (* FIXME *)
   819 
   820     val perm_simps' = map (fn (((i, (_, _, constrs)), tname), constr_rep_thms) =>
   821       let val T = replace_types' (nth_dtyp i)
   822       in List.concat (map (fn (atom, perm_closed_thms) =>
   823           map (fn ((cname, dts), constr_rep_thm) => 
   824         let
   825           val cname = Sign.intern_const thy8
   826             (NameSpace.append tname (Sign.base_name cname));
   827           val permT = mk_permT (Type (atom, []));
   828           val pi = Free ("pi", permT);
   829 
   830           fun perm t =
   831             let val T = fastype_of t
   832             in Const ("nominal.perm", permT --> T --> T) $ pi $ t end;
   833 
   834           fun constr_arg (dt, (j, l_args, r_args)) =
   835             let
   836               val x' = mk_Free "x" (typ_of_dtyp' dt) j;
   837               val (dts, dt') = strip_option dt;
   838               val Ts = map typ_of_dtyp' dts;
   839               val xs = map (fn (T, i) => mk_Free "x" T i)
   840                 (Ts ~~ (j upto j + length dts - 1))
   841               val x = mk_Free "x" (typ_of_dtyp' dt') (j + length dts);
   842               val (dts', dt'') = strip_dtyp dt';
   843             in
   844               (j + length dts + 1,
   845                xs @ x :: l_args,
   846                map perm (xs @ [x]) @ r_args)
   847             end
   848 
   849           val (_, l_args, r_args) = foldr constr_arg (1, [], []) dts;
   850           val c = Const (cname, map fastype_of l_args ---> T)
   851         in
   852           standard (Goal.prove thy8 [] []
   853             (HOLogic.mk_Trueprop (HOLogic.mk_eq
   854               (perm (list_comb (c, l_args)), list_comb (c, r_args))))
   855             (fn _ => EVERY
   856               [simp_tac (simpset_of thy8 addsimps (constr_rep_thm :: perm_defs)) 1,
   857                simp_tac (HOL_basic_ss addsimps (Rep_thms @ Abs_inverse_thms @
   858                  constr_defs @ perm_closed_thms)) 1,
   859                TRY (simp_tac (HOL_basic_ss addsimps
   860                  (symmetric perm_fun_def :: abs_perm)) 1),
   861                TRY (simp_tac (HOL_basic_ss addsimps
   862                  (perm_fun_def :: perm_defs @ Rep_thms @ Abs_inverse_thms @
   863                     perm_closed_thms)) 1)]))
   864         end) (constrs ~~ constr_rep_thms)) (atoms ~~ perm_closed_thmss))
   865       end) (List.take (descr, length new_type_names) ~~ new_type_names ~~ constr_rep_thmss);
   866 
   867     (** prove injectivity of constructors **)
   868 
   869     val rep_inject_thms' = map (fn th => th RS sym) rep_inject_thms;
   870     val alpha = PureThy.get_thms thy8 (Name "alpha");
   871     val abs_fresh = PureThy.get_thms thy8 (Name "abs_fresh");
   872     val fresh_def = PureThy.get_thm thy8 (Name "fresh_def");
   873     val supp_def = PureThy.get_thm thy8 (Name "supp_def");
   874 
   875     val inject_thms = map (fn (((i, (_, _, constrs)), tname), constr_rep_thms) =>
   876       let val T = replace_types' (nth_dtyp i)
   877       in List.mapPartial (fn ((cname, dts), constr_rep_thm) =>
   878         if null dts then NONE else SOME
   879         let
   880           val cname = Sign.intern_const thy8
   881             (NameSpace.append tname (Sign.base_name cname));
   882 
   883           fun make_inj (dt, (j, args1, args2, eqs)) =
   884             let
   885               val x' = mk_Free "x" (typ_of_dtyp' dt) j;
   886               val y' = mk_Free "y" (typ_of_dtyp' dt) j;
   887               val (dts, dt') = strip_option dt;
   888               val Ts_idx = map typ_of_dtyp' dts ~~ (j upto j + length dts - 1);
   889               val xs = map (fn (T, i) => mk_Free "x" T i) Ts_idx;
   890               val ys = map (fn (T, i) => mk_Free "y" T i) Ts_idx;
   891               val x = mk_Free "x" (typ_of_dtyp' dt') (j + length dts);
   892               val y = mk_Free "y" (typ_of_dtyp' dt') (j + length dts);
   893               val (dts', dt'') = strip_dtyp dt';
   894             in
   895               (j + length dts + 1,
   896                xs @ (x :: args1), ys @ (y :: args2),
   897                HOLogic.mk_eq
   898                  (foldr mk_abs_fun x xs, foldr mk_abs_fun y ys) :: eqs)
   899             end;
   900 
   901           val (_, args1, args2, eqs) = foldr make_inj (1, [], [], []) dts;
   902           val Ts = map fastype_of args1;
   903           val c = Const (cname, Ts ---> T)
   904         in
   905           standard (Goal.prove thy8 [] [] (HOLogic.mk_Trueprop (HOLogic.mk_eq
   906               (HOLogic.mk_eq (list_comb (c, args1), list_comb (c, args2)),
   907                foldr1 HOLogic.mk_conj eqs)))
   908             (fn _ => EVERY
   909                [asm_full_simp_tac (simpset_of thy8 addsimps (constr_rep_thm ::
   910                   rep_inject_thms')) 1,
   911                 TRY (asm_full_simp_tac (HOL_basic_ss addsimps (fresh_def :: supp_def ::
   912                   alpha @ abs_perm @ abs_fresh @ rep_inject_thms @
   913                   perm_rep_perm_thms)) 1),
   914                 TRY (asm_full_simp_tac (HOL_basic_ss addsimps (perm_fun_def ::
   915                   expand_fun_eq :: rep_inject_thms @ perm_rep_perm_thms)) 1)]))
   916         end) (constrs ~~ constr_rep_thms)
   917       end) (List.take (descr, length new_type_names) ~~ new_type_names ~~ constr_rep_thmss);
   918 
   919     (** equations for support and freshness **)
   920 
   921     val Un_assoc = PureThy.get_thm thy8 (Name "Un_assoc");
   922     val de_Morgan_conj = PureThy.get_thm thy8 (Name "de_Morgan_conj");
   923     val Collect_disj_eq = PureThy.get_thm thy8 (Name "Collect_disj_eq");
   924     val finite_Un = PureThy.get_thm thy8 (Name "finite_Un");
   925 
   926     val (supp_thms, fresh_thms) = ListPair.unzip (map ListPair.unzip
   927       (map (fn ((((i, (_, _, constrs)), tname), inject_thms'), perm_thms') =>
   928       let val T = replace_types' (nth_dtyp i)
   929       in List.concat (map (fn (cname, dts) => map (fn atom =>
   930         let
   931           val cname = Sign.intern_const thy8
   932             (NameSpace.append tname (Sign.base_name cname));
   933           val atomT = Type (atom, []);
   934 
   935           fun process_constr (dt, (j, args1, args2)) =
   936             let
   937               val x' = mk_Free "x" (typ_of_dtyp' dt) j;
   938               val (dts, dt') = strip_option dt;
   939               val Ts_idx = map typ_of_dtyp' dts ~~ (j upto j + length dts - 1);
   940               val xs = map (fn (T, i) => mk_Free "x" T i) Ts_idx;
   941               val x = mk_Free "x" (typ_of_dtyp' dt') (j + length dts);
   942               val (dts', dt'') = strip_dtyp dt';
   943             in
   944               (j + length dts + 1,
   945                xs @ (x :: args1), foldr mk_abs_fun x xs :: args2)
   946             end;
   947 
   948           val (_, args1, args2) = foldr process_constr (1, [], []) dts;
   949           val Ts = map fastype_of args1;
   950           val c = list_comb (Const (cname, Ts ---> T), args1);
   951           fun supp t =
   952             Const ("nominal.supp", fastype_of t --> HOLogic.mk_setT atomT) $ t;
   953           fun fresh t =
   954             Const ("nominal.fresh", atomT --> fastype_of t --> HOLogic.boolT) $
   955               Free ("a", atomT) $ t;
   956           val supp_thm = standard (Goal.prove thy8 [] []
   957               (HOLogic.mk_Trueprop (HOLogic.mk_eq
   958                 (supp c,
   959                  if null dts then Const ("{}", HOLogic.mk_setT atomT)
   960                  else foldr1 (HOLogic.mk_binop "op Un") (map supp args2))))
   961             (fn _ =>
   962               simp_tac (HOL_basic_ss addsimps (supp_def ::
   963                  Un_assoc :: de_Morgan_conj :: Collect_disj_eq :: finite_Un ::
   964                  symmetric empty_def :: Finites.emptyI :: simp_thms @
   965                  abs_perm @ abs_fresh @ inject_thms' @ perm_thms')) 1))
   966         in
   967           (supp_thm,
   968            standard (Goal.prove thy8 [] [] (HOLogic.mk_Trueprop (HOLogic.mk_eq
   969               (fresh c,
   970                if null dts then HOLogic.true_const
   971                else foldr1 HOLogic.mk_conj (map fresh args2))))
   972              (fn _ =>
   973                simp_tac (simpset_of thy8 addsimps [fresh_def, supp_thm]) 1)))
   974         end) atoms) constrs)
   975       end) (List.take (descr, length new_type_names) ~~ new_type_names ~~ inject_thms ~~ perm_simps')));
   976 
   977     (**** weak induction theorem ****)
   978 
   979     val arities = get_arities descr'';
   980 
   981     fun mk_funs_inv thm =
   982       let
   983         val {sign, prop, ...} = rep_thm thm;
   984         val _ $ (_ $ (Const (_, Type (_, [U, _])) $ _ $ S)) $
   985           (_ $ (_ $ (r $ (a $ _)) $ _)) = Type.freeze prop;
   986         val used = add_term_tfree_names (a, []);
   987 
   988         fun mk_thm i =
   989           let
   990             val Ts = map (TFree o rpair HOLogic.typeS)
   991               (variantlist (replicate i "'t", used));
   992             val f = Free ("f", Ts ---> U)
   993           in standard (Goal.prove sign [] [] (Logic.mk_implies
   994             (HOLogic.mk_Trueprop (HOLogic.list_all
   995                (map (pair "x") Ts, HOLogic.mk_mem (app_bnds f i, S))),
   996              HOLogic.mk_Trueprop (HOLogic.mk_eq (list_abs (map (pair "x") Ts,
   997                r $ (a $ app_bnds f i)), f))))
   998             (fn _ => EVERY [REPEAT (rtac ext 1), REPEAT (etac allE 1), rtac thm 1, atac 1]))
   999           end
  1000       in map (fn r => r RS subst) (thm :: map mk_thm arities) end;
  1001 
  1002     fun mk_indrule_lemma ((prems, concls), (((i, _), T), U)) =
  1003       let
  1004         val Rep_t = Const (List.nth (rep_names, i), T --> U) $
  1005           mk_Free "x" T i;
  1006 
  1007         val Abs_t =  Const (List.nth (abs_names, i), U --> T)
  1008 
  1009       in (prems @ [HOLogic.imp $ HOLogic.mk_mem (Rep_t,
  1010             Const (List.nth (rep_set_names, i), HOLogic.mk_setT U)) $
  1011               (mk_Free "P" (T --> HOLogic.boolT) (i + 1) $ (Abs_t $ Rep_t))],
  1012           concls @ [mk_Free "P" (T --> HOLogic.boolT) (i + 1) $ mk_Free "x" T i])
  1013       end;
  1014 
  1015     val (indrule_lemma_prems, indrule_lemma_concls) =
  1016       Library.foldl mk_indrule_lemma (([], []), (descr'' ~~ recTs ~~ recTs'));
  1017 
  1018     val indrule_lemma = standard (Goal.prove thy8 [] []
  1019       (Logic.mk_implies
  1020         (HOLogic.mk_Trueprop (mk_conj indrule_lemma_prems),
  1021          HOLogic.mk_Trueprop (mk_conj indrule_lemma_concls))) (fn _ => EVERY
  1022            [REPEAT (etac conjE 1),
  1023             REPEAT (EVERY
  1024               [TRY (rtac conjI 1), full_simp_tac (HOL_basic_ss addsimps Rep_inverse_thms) 1,
  1025                etac mp 1, resolve_tac Rep_thms 1])]));
  1026 
  1027     val Ps = map head_of (HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of indrule_lemma)));
  1028     val frees = if length Ps = 1 then [Free ("P", snd (dest_Var (hd Ps)))] else
  1029       map (Free o apfst fst o dest_Var) Ps;
  1030     val indrule_lemma' = cterm_instantiate
  1031       (map (cterm_of thy8) Ps ~~ map (cterm_of thy8) frees) indrule_lemma;
  1032 
  1033     val Abs_inverse_thms' = List.concat (map mk_funs_inv Abs_inverse_thms);
  1034 
  1035     val dt_induct_prop = DatatypeProp.make_ind descr' sorts';
  1036     val dt_induct = standard (Goal.prove thy8 []
  1037       (Logic.strip_imp_prems dt_induct_prop) (Logic.strip_imp_concl dt_induct_prop)
  1038       (fn prems => EVERY
  1039         [rtac indrule_lemma' 1,
  1040          (DatatypeAux.indtac rep_induct THEN_ALL_NEW ObjectLogic.atomize_tac) 1,
  1041          EVERY (map (fn (prem, r) => (EVERY
  1042            [REPEAT (eresolve_tac Abs_inverse_thms' 1),
  1043             simp_tac (HOL_basic_ss addsimps [symmetric r]) 1,
  1044             DEPTH_SOLVE_1 (ares_tac [prem] 1 ORELSE etac allE 1)]))
  1045                 (prems ~~ constr_defs))]));
  1046 
  1047     val case_names_induct = mk_case_names_induct descr'';
  1048 
  1049     (**** prove that new datatypes have finite support ****)
  1050 
  1051     val _ = warning "proving finite support for the new datatype";
  1052 
  1053     val indnames = DatatypeProp.make_tnames recTs;
  1054 
  1055     val abs_supp = PureThy.get_thms thy8 (Name "abs_supp");
  1056     val supp_atm = PureThy.get_thms thy8 (Name "supp_atm");
  1057 
  1058     val finite_supp_thms = map (fn atom =>
  1059       let val atomT = Type (atom, [])
  1060       in map standard (List.take
  1061         (split_conj_thm (Goal.prove thy8 [] [] (HOLogic.mk_Trueprop
  1062            (foldr1 HOLogic.mk_conj (map (fn (s, T) => HOLogic.mk_mem
  1063              (Const ("nominal.supp", T --> HOLogic.mk_setT atomT) $ Free (s, T),
  1064               Const ("Finite_Set.Finites", HOLogic.mk_setT (HOLogic.mk_setT atomT))))
  1065                (indnames ~~ recTs))))
  1066            (fn _ => indtac dt_induct indnames 1 THEN
  1067             ALLGOALS (asm_full_simp_tac (simpset_of thy8 addsimps
  1068               (abs_supp @ supp_atm @
  1069                PureThy.get_thms thy8 (Name ("fs_" ^ Sign.base_name atom ^ "1")) @
  1070                List.concat supp_thms))))),
  1071          length new_type_names))
  1072       end) atoms;
  1073 
  1074     val simp_atts = replicate (length new_type_names) [Simplifier.simp_add];
  1075 
  1076     val (_, thy9) = thy8 |>
  1077       Theory.add_path big_name |>
  1078       PureThy.add_thms [(("induct_weak", dt_induct), [case_names_induct])] ||>>
  1079       PureThy.add_thmss [(("inducts_weak", projections dt_induct), [case_names_induct])] ||>
  1080       Theory.parent_path ||>>
  1081       DatatypeAux.store_thmss_atts "distinct" new_type_names simp_atts distinct_thms ||>>
  1082       DatatypeAux.store_thmss "constr_rep" new_type_names constr_rep_thmss ||>>
  1083       DatatypeAux.store_thmss_atts "perm" new_type_names simp_atts perm_simps' ||>>
  1084       DatatypeAux.store_thmss "inject" new_type_names inject_thms ||>>
  1085       DatatypeAux.store_thmss "supp" new_type_names supp_thms ||>>
  1086       DatatypeAux.store_thmss_atts "fresh" new_type_names simp_atts fresh_thms ||>
  1087       fold (fn (atom, ths) => fn thy =>
  1088         let val class = Sign.intern_class thy ("fs_" ^ Sign.base_name atom)
  1089         in fold (fn T => AxClass.prove_arity
  1090             (fst (dest_Type T),
  1091               replicate (length sorts) [class], [class])
  1092             (ClassPackage.intro_classes_tac [] THEN resolve_tac ths 1)) newTs thy
  1093         end) (atoms ~~ finite_supp_thms);
  1094 
  1095     (**** strong induction theorem ****)
  1096 
  1097     val pnames = if length descr'' = 1 then ["P"]
  1098       else map (fn i => "P" ^ string_of_int i) (1 upto length descr'');
  1099     val ind_sort = if null dt_atomTs then HOLogic.typeS
  1100       else norm_sort thy9 (map (fn T => Sign.intern_class thy9 ("fs_" ^
  1101         Sign.base_name (fst (dest_Type T)))) dt_atomTs);
  1102     val fsT = TFree ("'n", ind_sort);
  1103     val fsT' = TFree ("'n", HOLogic.typeS);
  1104 
  1105     val fresh_fs = map (fn (s, T) => (T, Free (s, fsT' --> HOLogic.mk_setT T)))
  1106       (DatatypeProp.indexify_names (replicate (length dt_atomTs) "f") ~~ dt_atomTs);
  1107 
  1108     fun make_pred fsT i T =
  1109       Free (List.nth (pnames, i), fsT --> T --> HOLogic.boolT);
  1110 
  1111     fun make_ind_prem fsT f k T ((cname, cargs), idxs) =
  1112       let
  1113         val recs = List.filter is_rec_type cargs;
  1114         val Ts = map (typ_of_dtyp descr'' sorts') cargs;
  1115         val recTs' = map (typ_of_dtyp descr'' sorts') recs;
  1116         val tnames = variantlist (DatatypeProp.make_tnames Ts, pnames);
  1117         val rec_tnames = map fst (List.filter (is_rec_type o snd) (tnames ~~ cargs));
  1118         val frees = tnames ~~ Ts;
  1119         val z = (variant tnames "z", fsT);
  1120 
  1121         fun mk_prem ((dt, s), T) =
  1122           let
  1123             val (Us, U) = strip_type T;
  1124             val l = length Us
  1125           in list_all (z :: map (pair "x") Us, HOLogic.mk_Trueprop
  1126             (make_pred fsT (body_index dt) U $ Bound l $ app_bnds (Free (s, T)) l))
  1127           end;
  1128 
  1129         val prems = map mk_prem (recs ~~ rec_tnames ~~ recTs');
  1130         val prems' = map (fn p as (_, T) => HOLogic.mk_Trueprop
  1131             (f T (Free p) (Free z)))
  1132           (map (curry List.nth frees) (List.concat (map (fn (m, n) =>
  1133              m upto m + n - 1) idxs)))
  1134 
  1135       in list_all_free (frees @ [z], Logic.list_implies (prems' @ prems,
  1136         HOLogic.mk_Trueprop (make_pred fsT k T $ Free z $
  1137           list_comb (Const (cname, Ts ---> T), map Free frees))))
  1138       end;
  1139 
  1140     val ind_prems = List.concat (map (fn (((i, (_, _, constrs)), (_, idxss)), T) =>
  1141       map (make_ind_prem fsT (fn T => fn t => fn u =>
  1142         Const ("nominal.fresh", T --> fsT --> HOLogic.boolT) $ t $ u) i T)
  1143           (constrs ~~ idxss)) (descr'' ~~ ndescr ~~ recTs));
  1144     val tnames = DatatypeProp.make_tnames recTs;
  1145     val zs = variantlist (replicate (length descr'') "z", tnames);
  1146     val ind_concl = HOLogic.mk_Trueprop (foldr1 (HOLogic.mk_binop "op &")
  1147       (map (fn ((((i, _), T), tname), z) =>
  1148         make_pred fsT i T $ Free (z, fsT) $ Free (tname, T))
  1149         (descr'' ~~ recTs ~~ tnames ~~ zs)));
  1150     val induct = Logic.list_implies (ind_prems, ind_concl);
  1151 
  1152     val ind_prems' =
  1153       map (fn (_, f as Free (_, T)) => list_all_free ([("x", fsT')],
  1154         HOLogic.mk_Trueprop (HOLogic.mk_mem (f $ Free ("x", fsT'),
  1155           Const ("Finite_Set.Finites", HOLogic.mk_setT (body_type T)))))) fresh_fs @
  1156       List.concat (map (fn (((i, (_, _, constrs)), (_, idxss)), T) =>
  1157         map (make_ind_prem fsT' (fn T => fn t => fn u => HOLogic.Not $
  1158           HOLogic.mk_mem (t, the (AList.lookup op = fresh_fs T) $ u)) i T)
  1159             (constrs ~~ idxss)) (descr'' ~~ ndescr ~~ recTs));
  1160     val ind_concl' = HOLogic.mk_Trueprop (foldr1 (HOLogic.mk_binop "op &")
  1161       (map (fn ((((i, _), T), tname), z) =>
  1162         make_pred fsT' i T $ Free (z, fsT') $ Free (tname, T))
  1163         (descr'' ~~ recTs ~~ tnames ~~ zs)));
  1164     val induct' = Logic.list_implies (ind_prems', ind_concl');
  1165 
  1166     fun mk_perm Ts (t, u) =
  1167       let
  1168         val T = fastype_of1 (Ts, t);
  1169         val U = fastype_of1 (Ts, u)
  1170       in Const ("nominal.perm", T --> U --> U) $ t $ u end;
  1171 
  1172     val aux_ind_vars =
  1173       (DatatypeProp.indexify_names (replicate (length dt_atomTs) "pi") ~~
  1174        map mk_permT dt_atomTs) @ [("z", fsT')];
  1175     val aux_ind_Ts = rev (map snd aux_ind_vars);
  1176     val aux_ind_concl = HOLogic.mk_Trueprop (foldr1 (HOLogic.mk_binop "op &")
  1177       (map (fn (((i, _), T), tname) =>
  1178         HOLogic.list_all (aux_ind_vars, make_pred fsT' i T $ Bound 0 $
  1179           foldr (mk_perm aux_ind_Ts) (Free (tname, T))
  1180             (map Bound (length dt_atomTs downto 1))))
  1181         (descr'' ~~ recTs ~~ tnames)));
  1182 
  1183     fun mk_ind_perm i k p l vs j =
  1184       let
  1185         val n = length vs;
  1186         val Ts = map snd vs;
  1187         val T = List.nth (Ts, i - j);
  1188         val pT = NominalAtoms.mk_permT T
  1189       in
  1190         Const ("List.list.Cons", HOLogic.mk_prodT (T, T) --> pT --> pT) $
  1191           (HOLogic.pair_const T T $ Bound (l - j) $ foldr (mk_perm Ts)
  1192             (Bound (i - j))
  1193             (map (mk_ind_perm i k p l vs) (j - 1 downto 0) @
  1194              map Bound (n - k - 1 downto n - k - p))) $
  1195           Const ("List.list.Nil", pT)
  1196       end;
  1197 
  1198     fun mk_fresh i i' j k p l vs _ _ =
  1199       let
  1200         val n = length vs;
  1201         val Ts = map snd vs;
  1202         val T = List.nth (Ts, n - i - 1 - j);
  1203         val f = the (AList.lookup op = fresh_fs T);
  1204         val U = List.nth (Ts, n - i' - 1);
  1205         val S = HOLogic.mk_setT T;
  1206         val prms = map (mk_ind_perm (n - i) k p (n - l) (("a", T) :: vs))
  1207             (j - 1 downto 0) @
  1208           map Bound (n - k downto n - k - p + 1)
  1209       in
  1210         HOLogic.mk_Trueprop (Const ("Ex", (T --> HOLogic.boolT) --> HOLogic.boolT) $
  1211           Abs ("a", T, HOLogic.Not $
  1212             (Const ("op :", T --> S --> HOLogic.boolT) $ Bound 0 $
  1213               (Const ("insert", T --> S --> S) $
  1214                 (foldr (mk_perm (T :: Ts)) (Bound (n - i - j)) prms) $
  1215                 (Const ("op Un", S --> S --> S) $ (f $ Bound (n - k - p)) $
  1216                    (Const ("nominal.supp", U --> S) $
  1217                      foldr (mk_perm (T :: Ts)) (Bound (n - i')) prms))))))
  1218       end;
  1219 
  1220     fun mk_fresh_constr is p vs _ concl =
  1221       let
  1222         val n = length vs;
  1223         val Ts = map snd vs;
  1224         val _ $ (_ $ _ $ t) = concl;
  1225         val c = head_of t;
  1226         val T = body_type (fastype_of c);
  1227         val k = foldr op + 0 (map (fn (_, i) => i + 1) is);
  1228         val ps = map Bound (n - k - 1 downto n - k - p);
  1229         val (_, _, ts, us) = foldl (fn ((_, i), (m, n, ts, us)) =>
  1230           (m - i - 1, n - i,
  1231            ts @ map Bound (n downto n - i + 1) @
  1232              [foldr (mk_perm Ts) (Bound (m - i))
  1233                 (map (mk_ind_perm m k p n vs) (i - 1 downto 0) @ ps)],
  1234            us @ map (fn j => foldr (mk_perm Ts) (Bound j) ps) (m downto m - i)))
  1235           (n - 1, n - k - p - 2, [], []) is
  1236       in
  1237         HOLogic.mk_Trueprop (HOLogic.eq_const T $ list_comb (c, ts) $ list_comb (c, us))
  1238       end;
  1239 
  1240     val abs_fun_finite_supp = PureThy.get_thm thy9 (Name "abs_fun_finite_supp");
  1241 
  1242     val at_finite_select = PureThy.get_thm thy9 (Name "at_finite_select");
  1243 
  1244     val induct_aux_lemmas = List.concat (map (fn Type (s, _) =>
  1245       [PureThy.get_thm thy9 (Name ("pt_" ^ Sign.base_name s ^ "_inst")),
  1246        PureThy.get_thm thy9 (Name ("fs_" ^ Sign.base_name s ^ "1")),
  1247        PureThy.get_thm thy9 (Name ("at_" ^ Sign.base_name s ^ "_inst"))]) dt_atomTs);
  1248 
  1249     val induct_aux_lemmas' = map (fn Type (s, _) =>
  1250       PureThy.get_thm thy9 (Name ("pt_" ^ Sign.base_name s ^ "2")) RS sym) dt_atomTs;
  1251 
  1252     val induct_aux = standard (Goal.prove thy9 [] ind_prems' ind_concl'
  1253       (fn prems => EVERY
  1254         ([mk_subgoal 1 (K (K (K aux_ind_concl))),
  1255           indtac dt_induct tnames 1] @
  1256          List.concat (map (fn ((_, (_, _, constrs)), (_, constrs')) =>
  1257            List.concat (map (fn ((cname, cargs), is) =>
  1258              [simp_tac (HOL_basic_ss addsimps List.concat perm_simps') 1,
  1259               REPEAT (rtac allI 1)] @
  1260              List.concat (map
  1261                (fn ((_, 0), _) => []
  1262                  | ((i, j), k) => List.concat (map (fn j' =>
  1263                      let
  1264                        val DtType (tname, _) = List.nth (cargs, i + j');
  1265                        val atom = Sign.base_name tname
  1266                      in
  1267                        [mk_subgoal 1 (mk_fresh i (i + j) j'
  1268                           (length cargs) (length dt_atomTs)
  1269                           (length cargs + length dt_atomTs + 1 + i - k)),
  1270                         rtac at_finite_select 1,
  1271                         rtac (PureThy.get_thm thy9 (Name ("at_" ^ atom ^ "_inst"))) 1,
  1272                         asm_full_simp_tac (simpset_of thy9 addsimps
  1273                           [PureThy.get_thm thy9 (Name ("fs_" ^ atom ^ "1"))]) 1,
  1274                         resolve_tac prems 1,
  1275                         etac exE 1,
  1276                         asm_full_simp_tac (simpset_of thy9 addsimps
  1277                           [symmetric fresh_def]) 1]
  1278                      end) (0 upto j - 1))) (is ~~ (0 upto length is - 1))) @
  1279              (if exists (not o equal 0 o snd) is then
  1280                 [mk_subgoal 1 (mk_fresh_constr is (length dt_atomTs)),
  1281                  asm_full_simp_tac (simpset_of thy9 addsimps
  1282                    (List.concat inject_thms @
  1283                     alpha @ abs_perm @ abs_fresh @ [abs_fun_finite_supp] @
  1284                     induct_aux_lemmas)) 1,
  1285                  dtac sym 1,
  1286                  asm_full_simp_tac (simpset_of thy9
  1287                    addsimps induct_aux_lemmas'
  1288                    addsimprocs [perm_simproc]) 1,
  1289                  REPEAT (etac conjE 1)]
  1290               else
  1291                 []) @
  1292              [(resolve_tac prems THEN_ALL_NEW
  1293                 (atac ORELSE' ((REPEAT o etac allE) THEN' atac))) 1])
  1294                (constrs ~~ constrs'))) (descr'' ~~ ndescr)) @
  1295          [REPEAT (eresolve_tac [conjE, allE_Nil] 1),
  1296           REPEAT (etac allE 1),
  1297           REPEAT (TRY (rtac conjI 1) THEN asm_full_simp_tac (simpset_of thy9) 1)])));
  1298 
  1299     val induct_aux' = Thm.instantiate ([],
  1300       map (fn (s, T) =>
  1301         let val pT = TVar (("'n", 0), HOLogic.typeS) --> T --> HOLogic.boolT
  1302         in (cterm_of thy9 (Var ((s, 0), pT)), cterm_of thy9 (Free (s, pT))) end)
  1303           (pnames ~~ recTs) @
  1304       map (fn (_, f) =>
  1305         let val f' = Logic.varify f
  1306         in (cterm_of thy9 f',
  1307           cterm_of thy9 (Const ("nominal.supp", fastype_of f')))
  1308         end) fresh_fs) induct_aux;
  1309 
  1310     val induct = standard (Goal.prove thy9 [] ind_prems ind_concl
  1311       (fn prems => EVERY
  1312          [rtac induct_aux' 1,
  1313           REPEAT (resolve_tac induct_aux_lemmas 1),
  1314           REPEAT ((resolve_tac prems THEN_ALL_NEW
  1315             (etac meta_spec ORELSE' full_simp_tac (HOL_basic_ss addsimps [fresh_def]))) 1)]))
  1316 
  1317     val (_, thy10) = thy9 |>
  1318       Theory.add_path big_name |>
  1319       PureThy.add_thms [(("induct'", induct_aux), [])] ||>>
  1320       PureThy.add_thms [(("induct", induct), [case_names_induct])] ||>>
  1321       PureThy.add_thmss [(("inducts", projections induct), [case_names_induct])] ||>
  1322       Theory.parent_path;
  1323 
  1324     (**** recursion combinator ****)
  1325 
  1326     val _ = warning "defining recursion combinator ...";
  1327 
  1328     val used = foldr add_typ_tfree_names [] recTs;
  1329 
  1330     val (rec_result_Ts, rec_fn_Ts) = DatatypeProp.make_primrec_Ts descr' sorts' used;
  1331 
  1332     val permTs = map mk_permT dt_atomTs;
  1333     val perms = map Free
  1334       (DatatypeProp.indexify_names (replicate (length permTs) "pi") ~~ permTs);
  1335 
  1336     val rec_set_Ts = map (fn (T1, T2) => rec_fn_Ts ---> HOLogic.mk_setT
  1337       (HOLogic.mk_prodT (T1, permTs ---> T2))) (recTs ~~ rec_result_Ts);
  1338 
  1339     val big_rec_name = big_name ^ "_rec_set";
  1340     val rec_set_names = map (Sign.full_name (Theory.sign_of thy10))
  1341       (if length descr'' = 1 then [big_rec_name] else
  1342         (map ((curry (op ^) (big_rec_name ^ "_")) o string_of_int)
  1343           (1 upto (length descr''))));
  1344 
  1345     val rec_fns = map (uncurry (mk_Free "f"))
  1346       (rec_fn_Ts ~~ (1 upto (length rec_fn_Ts)));
  1347     val rec_sets = map (fn c => list_comb (Const c, rec_fns))
  1348       (rec_set_names ~~ rec_set_Ts);
  1349 
  1350     (* introduction rules for graph of recursion function *)
  1351 
  1352     fun partition_cargs idxs xs = map (fn (i, j) =>
  1353       (List.take (List.drop (xs, i), j), List.nth (xs, i + j))) idxs;
  1354 
  1355     fun mk_fresh_fun (a, t) = Const ("nominal.fresh_fun",
  1356       (fastype_of a --> fastype_of t) --> fastype_of t) $ lambda a t;
  1357 
  1358     fun make_rec_intr T rec_set ((rec_intr_ts, l), ((cname, cargs), idxs)) =
  1359       let
  1360         fun mk_prem ((dts, (dt, U)), (j, k, prems, t1s, t2s, t3s, atoms)) =
  1361           let
  1362             val free1 = mk_Free "x" U (j + length dts);
  1363             val Us = map snd dts;
  1364             val xs = Us ~~ (j upto j + length dts - 1);
  1365             val frees = map (uncurry (mk_Free "x")) xs;
  1366             val frees' = map (uncurry (mk_Free "x'")) xs;
  1367             val frees'' = Us ~~ (frees ~~ frees');
  1368             val pis = map (fn (T, p) =>
  1369               case filter (equal T o fst) frees'' of
  1370                 [] => p
  1371               | xs => HOLogic.mk_binop "List.op @" (p,
  1372                 HOLogic.mk_list (HOLogic.mk_prod o snd)
  1373                   (HOLogic.mk_prodT (T, T)) xs))
  1374                   (dt_atomTs ~~ perms)
  1375           in (case dt of
  1376              DtRec m =>
  1377                let val free2 = mk_Free "y"
  1378                  (permTs ---> List.nth (rec_result_Ts, m)) k
  1379                in (j + length dts + 1, k + 1,
  1380                    HOLogic.mk_Trueprop
  1381                      (HOLogic.mk_mem (HOLogic.mk_prod
  1382                        (free1, free2),
  1383                          List.nth (rec_sets, m))) :: prems,
  1384                    frees @ free1 :: t1s,
  1385                    frees' @ foldr (mk_perm []) free1 pis :: t2s,
  1386                    list_comb (free2, pis) :: t3s,
  1387                    frees' @ atoms)
  1388                end
  1389            | _ => (j + length dts + 1, k, prems,
  1390                frees @ free1 :: t1s,
  1391                frees' @ foldr (mk_perm []) free1 pis :: t2s,
  1392                t3s,
  1393                frees' @ atoms))
  1394           end;
  1395 
  1396         val Ts = map (typ_of_dtyp descr'' sorts') cargs;
  1397         val (_, _, prems, t1s, t2s, t3s, atoms) = foldr mk_prem (1, 1, [], [], [], [], [])
  1398           (partition_cargs idxs (cargs ~~ Ts))
  1399 
  1400       in (rec_intr_ts @ [Logic.list_implies (prems, HOLogic.mk_Trueprop (HOLogic.mk_mem
  1401         (HOLogic.mk_prod (list_comb (Const (cname, Ts ---> T), t1s),
  1402           foldr (uncurry lambda)
  1403             (foldr mk_fresh_fun
  1404               (list_comb (List.nth (rec_fns, l), t2s @ t3s)) atoms)
  1405             perms), rec_set)))], l + 1)
  1406       end;
  1407 
  1408     val (rec_intr_ts, _) = Library.foldl (fn (x, (((d, d'), T), rec_set)) =>
  1409       Library.foldl (make_rec_intr T rec_set) (x, #3 (snd d) ~~ snd d'))
  1410         (([], 0), descr'' ~~ ndescr ~~ recTs ~~ rec_sets);
  1411 
  1412     val (thy11, {intrs = rec_intrs, elims = rec_elims, ...}) =
  1413       setmp InductivePackage.quiet_mode (!quiet_mode)
  1414         (InductivePackage.add_inductive_i false true big_rec_name false false false
  1415            rec_sets (map (fn x => (("", x), [])) rec_intr_ts) []) thy10;
  1416 
  1417   in
  1418     thy11
  1419   end;
  1420 
  1421 val add_nominal_datatype = gen_add_nominal_datatype read_typ true;
  1422 
  1423 
  1424 (* FIXME: The following stuff should be exported by DatatypePackage *)
  1425 
  1426 local structure P = OuterParse and K = OuterKeyword in
  1427 
  1428 val datatype_decl =
  1429   Scan.option (P.$$$ "(" |-- P.name --| P.$$$ ")") -- P.type_args -- P.name -- P.opt_infix --
  1430     (P.$$$ "=" |-- P.enum1 "|" (P.name -- Scan.repeat P.typ -- P.opt_mixfix));
  1431 
  1432 fun mk_datatype args =
  1433   let
  1434     val names = map (fn ((((NONE, _), t), _), _) => t | ((((SOME t, _), _), _), _) => t) args;
  1435     val specs = map (fn ((((_, vs), t), mx), cons) =>
  1436       (vs, t, mx, map (fn ((x, y), z) => (x, y, z)) cons)) args;
  1437   in add_nominal_datatype false names specs end;
  1438 
  1439 val nominal_datatypeP =
  1440   OuterSyntax.command "nominal_datatype" "define inductive datatypes" K.thy_decl
  1441     (P.and_list1 datatype_decl >> (Toplevel.theory o mk_datatype));
  1442 
  1443 val _ = OuterSyntax.add_parsers [nominal_datatypeP];
  1444 
  1445 end;
  1446 
  1447 end
  1448