src/HOL/Nominal/nominal_package.ML
author berghofe
Fri Apr 28 15:54:34 2006 +0200 (2006-04-28)
changeset 19494 2e909d5309f4
parent 19489 4441b637871b
child 19635 f7aa7d174343
permissions -rw-r--r--
Renamed "nominal" theory to "Nominal".
     1 (*  Title:      HOL/Nominal/nominal_package.ML
     2     ID:         $Id$
     3     Author:     Stefan Berghofer and Christian Urban, TU Muenchen
     4 
     5 Nominal datatype package for Isabelle/HOL.
     6 *)
     7 
     8 signature NOMINAL_PACKAGE =
     9 sig
    10   val add_nominal_datatype : bool -> string list -> (string list * bstring * mixfix *
    11     (bstring * string list * mixfix) list) list -> theory -> theory
    12 end
    13 
    14 structure NominalPackage : NOMINAL_PACKAGE =
    15 struct
    16 
    17 open DatatypeAux;
    18 open NominalAtoms;
    19 
    20 (** FIXME: DatatypePackage should export this function **)
    21 
    22 local
    23 
    24 fun dt_recs (DtTFree _) = []
    25   | dt_recs (DtType (_, dts)) = List.concat (map dt_recs dts)
    26   | dt_recs (DtRec i) = [i];
    27 
    28 fun dt_cases (descr: descr) (_, args, constrs) =
    29   let
    30     fun the_bname i = Sign.base_name (#1 (valOf (AList.lookup (op =) descr i)));
    31     val bnames = map the_bname (distinct op = (List.concat (map dt_recs args)));
    32   in map (fn (c, _) => space_implode "_" (Sign.base_name c :: bnames)) constrs end;
    33 
    34 
    35 fun induct_cases descr =
    36   DatatypeProp.indexify_names (List.concat (map (dt_cases descr) (map #2 descr)));
    37 
    38 fun exhaust_cases descr i = dt_cases descr (valOf (AList.lookup (op =) descr i));
    39 
    40 in
    41 
    42 fun mk_case_names_induct descr = RuleCases.case_names (induct_cases descr);
    43 
    44 fun mk_case_names_exhausts descr new =
    45   map (RuleCases.case_names o exhaust_cases descr o #1)
    46     (List.filter (fn ((_, (name, _, _))) => name mem_string new) descr);
    47 
    48 end;
    49 
    50 (*******************************)
    51 
    52 val (_ $ (_ $ (_ $ (distinct_f $ _) $ _))) = hd (prems_of distinct_lemma);
    53 
    54 fun read_typ sign ((Ts, sorts), str) =
    55   let
    56     val T = Type.no_tvars (Sign.read_typ (sign, (AList.lookup op =)
    57       (map (apfst (rpair ~1)) sorts)) str) handle TYPE (msg, _, _) => error msg
    58   in (Ts @ [T], add_typ_tfrees (T, sorts)) end;
    59 
    60 (** taken from HOL/Tools/datatype_aux.ML **)
    61 
    62 fun indtac indrule indnames i st =
    63   let
    64     val ts = HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of indrule));
    65     val ts' = HOLogic.dest_conj (HOLogic.dest_Trueprop
    66       (Logic.strip_imp_concl (List.nth (prems_of st, i - 1))));
    67     val getP = if can HOLogic.dest_imp (hd ts) then
    68       (apfst SOME) o HOLogic.dest_imp else pair NONE;
    69     fun abstr (t1, t2) = (case t1 of
    70         NONE => (case filter (fn Free (s, _) => s mem indnames | _ => false)
    71               (term_frees t2) of
    72             [Free (s, T)] => absfree (s, T, t2)
    73           | _ => sys_error "indtac")
    74       | SOME (_ $ t' $ _) => Abs ("x", fastype_of t', abstract_over (t', t2)))
    75     val cert = cterm_of (Thm.sign_of_thm st);
    76     val Ps = map (cert o head_of o snd o getP) ts;
    77     val indrule' = cterm_instantiate (Ps ~~
    78       (map (cert o abstr o getP) ts')) indrule
    79   in
    80     rtac indrule' i st
    81   end;
    82 
    83 fun mk_subgoal i f st =
    84   let
    85     val cg = List.nth (cprems_of st, i - 1);
    86     val g = term_of cg;
    87     val revcut_rl' = Thm.lift_rule cg revcut_rl;
    88     val v = head_of (Logic.strip_assums_concl (hd (prems_of revcut_rl')));
    89     val ps = Logic.strip_params g;
    90     val cert = cterm_of (sign_of_thm st);
    91   in
    92     compose_tac (false,
    93       Thm.instantiate ([], [(cert v, cert (list_abs (ps,
    94         f (rev ps) (Logic.strip_assums_hyp g) (Logic.strip_assums_concl g))))])
    95       revcut_rl', 2) i st
    96   end;
    97 
    98 (** simplification procedure for sorting permutations **)
    99 
   100 val dj_cp = thm "dj_cp";
   101 
   102 fun dest_permT (Type ("fun", [Type ("List.list", [Type ("*", [T, _])]),
   103       Type ("fun", [_, U])])) = (T, U);
   104 
   105 fun permTs_of (Const ("Nominal.perm", T) $ t $ u) = fst (dest_permT T) :: permTs_of u
   106   | permTs_of _ = [];
   107 
   108 fun perm_simproc' thy ss (Const ("Nominal.perm", T) $ t $ (u as Const ("Nominal.perm", U) $ r $ s)) =
   109       let
   110         val (aT as Type (a, []), S) = dest_permT T;
   111         val (bT as Type (b, []), _) = dest_permT U
   112       in if aT mem permTs_of u andalso aT <> bT then
   113           let
   114             val a' = Sign.base_name a;
   115             val b' = Sign.base_name b;
   116             val cp = PureThy.get_thm thy (Name ("cp_" ^ a' ^ "_" ^ b' ^ "_inst"));
   117             val dj = PureThy.get_thm thy (Name ("dj_" ^ b' ^ "_" ^ a'));
   118             val dj_cp' = [cp, dj] MRS dj_cp;
   119             val cert = SOME o cterm_of thy
   120           in
   121             SOME (mk_meta_eq (Drule.instantiate' [SOME (ctyp_of thy S)]
   122               [cert t, cert r, cert s] dj_cp'))
   123           end
   124         else NONE
   125       end
   126   | perm_simproc' thy ss _ = NONE;
   127 
   128 val perm_simproc =
   129   Simplifier.simproc (the_context ()) "perm_simp" ["pi1 \\<bullet> (pi2 \\<bullet> x)"] perm_simproc';
   130 
   131 val allE_Nil = read_instantiate_sg (the_context()) [("x", "[]")] allE;
   132 
   133 val meta_spec = thm "meta_spec";
   134 
   135 fun projections rule =
   136   ProjectRule.projections rule
   137   |> map (standard #> RuleCases.save rule);
   138 
   139 fun norm_sort thy = Sorts.norm_sort (snd (#classes (Type.rep_tsig (Sign.tsig_of thy))));
   140 
   141 fun gen_add_nominal_datatype prep_typ err flat_names new_type_names dts thy =
   142   let
   143     (* this theory is used just for parsing *)
   144   
   145     val tmp_thy = thy |>
   146       Theory.copy |>
   147       Theory.add_types (map (fn (tvs, tname, mx, _) =>
   148         (tname, length tvs, mx)) dts);
   149 
   150     val sign = Theory.sign_of tmp_thy;
   151 
   152     val atoms = atoms_of thy;
   153     val classes = map (NameSpace.map_base (fn s => "pt_" ^ s)) atoms;
   154     val cp_classes = List.concat (map (fn atom1 => map (fn atom2 =>
   155       Sign.intern_class thy ("cp_" ^ Sign.base_name atom1 ^ "_" ^
   156         Sign.base_name atom2)) atoms) atoms);
   157     fun augment_sort S = S union classes;
   158     val augment_sort_typ = map_type_tfree (fn (s, S) => TFree (s, augment_sort S));
   159 
   160     fun prep_constr ((constrs, sorts), (cname, cargs, mx)) =
   161       let val (cargs', sorts') = Library.foldl (prep_typ sign) (([], sorts), cargs)
   162       in (constrs @ [(cname, cargs', mx)], sorts') end
   163 
   164     fun prep_dt_spec ((dts, sorts), (tvs, tname, mx, constrs)) =
   165       let val (constrs', sorts') = Library.foldl prep_constr (([], sorts), constrs)
   166       in (dts @ [(tvs, tname, mx, constrs')], sorts') end
   167 
   168     val (dts', sorts) = Library.foldl prep_dt_spec (([], []), dts);
   169     val sorts' = map (apsnd augment_sort) sorts;
   170     val tyvars = map #1 dts';
   171 
   172     val types_syntax = map (fn (tvs, tname, mx, constrs) => (tname, mx)) dts';
   173     val constr_syntax = map (fn (tvs, tname, mx, constrs) =>
   174       map (fn (cname, cargs, mx) => (cname, mx)) constrs) dts';
   175 
   176     val ps = map (fn (_, n, _, _) =>
   177       (Sign.full_name sign n, Sign.full_name sign (n ^ "_Rep"))) dts;
   178     val rps = map Library.swap ps;
   179 
   180     fun replace_types (Type ("Nominal.ABS", [T, U])) = 
   181           Type ("fun", [T, Type ("Nominal.noption", [replace_types U])])
   182       | replace_types (Type (s, Ts)) =
   183           Type (getOpt (AList.lookup op = ps s, s), map replace_types Ts)
   184       | replace_types T = T;
   185 
   186     fun replace_types' (Type (s, Ts)) =
   187           Type (getOpt (AList.lookup op = rps s, s), map replace_types' Ts)
   188       | replace_types' T = T;
   189 
   190     val dts'' = map (fn (tvs, tname, mx, constrs) => (tvs, tname ^ "_Rep", NoSyn,
   191       map (fn (cname, cargs, mx) => (cname,
   192         map (augment_sort_typ o replace_types) cargs, NoSyn)) constrs)) dts';
   193 
   194     val new_type_names' = map (fn n => n ^ "_Rep") new_type_names;
   195     val full_new_type_names' = map (Sign.full_name (sign_of thy)) new_type_names';
   196 
   197     val ({induction, ...},thy1) =
   198       DatatypePackage.add_datatype_i err flat_names new_type_names' dts'' thy;
   199 
   200     val SOME {descr, ...} = Symtab.lookup
   201       (DatatypePackage.get_datatypes thy1) (hd full_new_type_names');
   202     fun nth_dtyp i = typ_of_dtyp descr sorts' (DtRec i);
   203 
   204     (**** define permutation functions ****)
   205 
   206     val permT = mk_permT (TFree ("'x", HOLogic.typeS));
   207     val pi = Free ("pi", permT);
   208     val perm_types = map (fn (i, _) =>
   209       let val T = nth_dtyp i
   210       in permT --> T --> T end) descr;
   211     val perm_names = replicate (length new_type_names) "Nominal.perm" @
   212       DatatypeProp.indexify_names (map (fn i => Sign.full_name (sign_of thy1)
   213         ("perm_" ^ name_of_typ (nth_dtyp i)))
   214           (length new_type_names upto length descr - 1));
   215     val perm_names_types = perm_names ~~ perm_types;
   216 
   217     val perm_eqs = List.concat (map (fn (i, (_, _, constrs)) =>
   218       let val T = nth_dtyp i
   219       in map (fn (cname, dts) => 
   220         let
   221           val Ts = map (typ_of_dtyp descr sorts') dts;
   222           val names = DatatypeProp.make_tnames Ts;
   223           val args = map Free (names ~~ Ts);
   224           val c = Const (cname, Ts ---> T);
   225           fun perm_arg (dt, x) =
   226             let val T = type_of x
   227             in if is_rec_type dt then
   228                 let val (Us, _) = strip_type T
   229                 in list_abs (map (pair "x") Us,
   230                   Const (List.nth (perm_names_types, body_index dt)) $ pi $
   231                     list_comb (x, map (fn (i, U) =>
   232                       Const ("Nominal.perm", permT --> U --> U) $
   233                         (Const ("List.rev", permT --> permT) $ pi) $
   234                         Bound i) ((length Us - 1 downto 0) ~~ Us)))
   235                 end
   236               else Const ("Nominal.perm", permT --> T --> T) $ pi $ x
   237             end;  
   238         in
   239           (("", HOLogic.mk_Trueprop (HOLogic.mk_eq
   240             (Const (List.nth (perm_names_types, i)) $
   241                Free ("pi", mk_permT (TFree ("'x", HOLogic.typeS))) $
   242                list_comb (c, args),
   243              list_comb (c, map perm_arg (dts ~~ args))))), [])
   244         end) constrs
   245       end) descr);
   246 
   247     val (thy2, perm_simps) = thy1 |>
   248       Theory.add_consts_i (map (fn (s, T) => (Sign.base_name s, T, NoSyn))
   249         (List.drop (perm_names_types, length new_type_names))) |>
   250       PrimrecPackage.add_primrec_i "" perm_eqs;
   251 
   252     (**** prove that permutation functions introduced by unfolding are ****)
   253     (**** equivalent to already existing permutation functions         ****)
   254 
   255     val _ = warning ("length descr: " ^ string_of_int (length descr));
   256     val _ = warning ("length new_type_names: " ^ string_of_int (length new_type_names));
   257 
   258     val perm_indnames = DatatypeProp.make_tnames (map body_type perm_types);
   259     val perm_fun_def = PureThy.get_thm thy2 (Name "perm_fun_def");
   260 
   261     val unfolded_perm_eq_thms =
   262       if length descr = length new_type_names then []
   263       else map standard (List.drop (split_conj_thm
   264         (Goal.prove thy2 [] []
   265           (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
   266             (map (fn (c as (s, T), x) =>
   267                let val [T1, T2] = binder_types T
   268                in HOLogic.mk_eq (Const c $ pi $ Free (x, T2),
   269                  Const ("Nominal.perm", T) $ pi $ Free (x, T2))
   270                end)
   271              (perm_names_types ~~ perm_indnames))))
   272           (fn _ => EVERY [indtac induction perm_indnames 1,
   273             ALLGOALS (asm_full_simp_tac
   274               (simpset_of thy2 addsimps [perm_fun_def]))])),
   275         length new_type_names));
   276 
   277     (**** prove [] \<bullet> t = t ****)
   278 
   279     val _ = warning "perm_empty_thms";
   280 
   281     val perm_empty_thms = List.concat (map (fn a =>
   282       let val permT = mk_permT (Type (a, []))
   283       in map standard (List.take (split_conj_thm
   284         (Goal.prove thy2 [] []
   285           (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
   286             (map (fn ((s, T), x) => HOLogic.mk_eq
   287                 (Const (s, permT --> T --> T) $
   288                    Const ("List.list.Nil", permT) $ Free (x, T),
   289                  Free (x, T)))
   290              (perm_names ~~
   291               map body_type perm_types ~~ perm_indnames))))
   292           (fn _ => EVERY [indtac induction perm_indnames 1,
   293             ALLGOALS (asm_full_simp_tac (simpset_of thy2))])),
   294         length new_type_names))
   295       end)
   296       atoms);
   297 
   298     (**** prove (pi1 @ pi2) \<bullet> t = pi1 \<bullet> (pi2 \<bullet> t) ****)
   299 
   300     val _ = warning "perm_append_thms";
   301 
   302     (*FIXME: these should be looked up statically*)
   303     val at_pt_inst = PureThy.get_thm thy2 (Name "at_pt_inst");
   304     val pt2 = PureThy.get_thm thy2 (Name "pt2");
   305 
   306     val perm_append_thms = List.concat (map (fn a =>
   307       let
   308         val permT = mk_permT (Type (a, []));
   309         val pi1 = Free ("pi1", permT);
   310         val pi2 = Free ("pi2", permT);
   311         val pt_inst = PureThy.get_thm thy2 (Name ("pt_" ^ Sign.base_name a ^ "_inst"));
   312         val pt2' = pt_inst RS pt2;
   313         val pt2_ax = PureThy.get_thm thy2
   314           (Name (NameSpace.map_base (fn s => "pt_" ^ s ^ "2") a));
   315       in List.take (map standard (split_conj_thm
   316         (Goal.prove thy2 [] []
   317              (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
   318                 (map (fn ((s, T), x) =>
   319                     let val perm = Const (s, permT --> T --> T)
   320                     in HOLogic.mk_eq
   321                       (perm $ (Const ("List.op @", permT --> permT --> permT) $
   322                          pi1 $ pi2) $ Free (x, T),
   323                        perm $ pi1 $ (perm $ pi2 $ Free (x, T)))
   324                     end)
   325                   (perm_names ~~
   326                    map body_type perm_types ~~ perm_indnames))))
   327            (fn _ => EVERY [indtac induction perm_indnames 1,
   328               ALLGOALS (asm_full_simp_tac (simpset_of thy2 addsimps [pt2', pt2_ax]))]))),
   329          length new_type_names)
   330       end) atoms);
   331 
   332     (**** prove pi1 ~ pi2 ==> pi1 \<bullet> t = pi2 \<bullet> t ****)
   333 
   334     val _ = warning "perm_eq_thms";
   335 
   336     val pt3 = PureThy.get_thm thy2 (Name "pt3");
   337     val pt3_rev = PureThy.get_thm thy2 (Name "pt3_rev");
   338 
   339     val perm_eq_thms = List.concat (map (fn a =>
   340       let
   341         val permT = mk_permT (Type (a, []));
   342         val pi1 = Free ("pi1", permT);
   343         val pi2 = Free ("pi2", permT);
   344         (*FIXME: not robust - better access these theorems using NominalData?*)
   345         val at_inst = PureThy.get_thm thy2 (Name ("at_" ^ Sign.base_name a ^ "_inst"));
   346         val pt_inst = PureThy.get_thm thy2 (Name ("pt_" ^ Sign.base_name a ^ "_inst"));
   347         val pt3' = pt_inst RS pt3;
   348         val pt3_rev' = at_inst RS (pt_inst RS pt3_rev);
   349         val pt3_ax = PureThy.get_thm thy2
   350           (Name (NameSpace.map_base (fn s => "pt_" ^ s ^ "3") a));
   351       in List.take (map standard (split_conj_thm
   352         (Goal.prove thy2 [] [] (Logic.mk_implies
   353              (HOLogic.mk_Trueprop (Const ("Nominal.prm_eq",
   354                 permT --> permT --> HOLogic.boolT) $ pi1 $ pi2),
   355               HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
   356                 (map (fn ((s, T), x) =>
   357                     let val perm = Const (s, permT --> T --> T)
   358                     in HOLogic.mk_eq
   359                       (perm $ pi1 $ Free (x, T),
   360                        perm $ pi2 $ Free (x, T))
   361                     end)
   362                   (perm_names ~~
   363                    map body_type perm_types ~~ perm_indnames)))))
   364            (fn _ => EVERY [indtac induction perm_indnames 1,
   365               ALLGOALS (asm_full_simp_tac (simpset_of thy2 addsimps [pt3', pt3_rev', pt3_ax]))]))),
   366          length new_type_names)
   367       end) atoms);
   368 
   369     (**** prove pi1 \<bullet> (pi2 \<bullet> t) = (pi1 \<bullet> pi2) \<bullet> (pi1 \<bullet> t) ****)
   370 
   371     val cp1 = PureThy.get_thm thy2 (Name "cp1");
   372     val dj_cp = PureThy.get_thm thy2 (Name "dj_cp");
   373     val pt_perm_compose = PureThy.get_thm thy2 (Name "pt_perm_compose");
   374     val pt_perm_compose_rev = PureThy.get_thm thy2 (Name "pt_perm_compose_rev");
   375     val dj_perm_perm_forget = PureThy.get_thm thy2 (Name "dj_perm_perm_forget");
   376 
   377     fun composition_instance name1 name2 thy =
   378       let
   379         val name1' = Sign.base_name name1;
   380         val name2' = Sign.base_name name2;
   381         val cp_class = Sign.intern_class thy ("cp_" ^ name1' ^ "_" ^ name2');
   382         val permT1 = mk_permT (Type (name1, []));
   383         val permT2 = mk_permT (Type (name2, []));
   384         val augment = map_type_tfree
   385           (fn (x, S) => TFree (x, cp_class :: S));
   386         val Ts = map (augment o body_type) perm_types;
   387         val cp_inst = PureThy.get_thm thy
   388           (Name ("cp_" ^ name1' ^ "_" ^ name2' ^ "_inst"));
   389         val simps = simpset_of thy addsimps (perm_fun_def ::
   390           (if name1 <> name2 then
   391              let val dj = PureThy.get_thm thy (Name ("dj_" ^ name2' ^ "_" ^ name1'))
   392              in [dj RS (cp_inst RS dj_cp), dj RS dj_perm_perm_forget] end
   393            else
   394              let
   395                val at_inst = PureThy.get_thm thy (Name ("at_" ^ name1' ^ "_inst"));
   396                val pt_inst = PureThy.get_thm thy (Name ("pt_" ^ name1' ^ "_inst"))
   397              in
   398                [cp_inst RS cp1 RS sym,
   399                 at_inst RS (pt_inst RS pt_perm_compose) RS sym,
   400                 at_inst RS (pt_inst RS pt_perm_compose_rev) RS sym]
   401             end))
   402         val thms = split_conj_thm (standard (Goal.prove thy [] []
   403             (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
   404               (map (fn ((s, T), x) =>
   405                   let
   406                     val pi1 = Free ("pi1", permT1);
   407                     val pi2 = Free ("pi2", permT2);
   408                     val perm1 = Const (s, permT1 --> T --> T);
   409                     val perm2 = Const (s, permT2 --> T --> T);
   410                     val perm3 = Const ("Nominal.perm", permT1 --> permT2 --> permT2)
   411                   in HOLogic.mk_eq
   412                     (perm1 $ pi1 $ (perm2 $ pi2 $ Free (x, T)),
   413                      perm2 $ (perm3 $ pi1 $ pi2) $ (perm1 $ pi1 $ Free (x, T)))
   414                   end)
   415                 (perm_names ~~ Ts ~~ perm_indnames))))
   416           (fn _ => EVERY [indtac induction perm_indnames 1,
   417              ALLGOALS (asm_full_simp_tac simps)])))
   418       in
   419         foldl (fn ((s, tvs), thy) => AxClass.prove_arity
   420             (s, replicate (length tvs) (cp_class :: classes), [cp_class])
   421             (ClassPackage.intro_classes_tac [] THEN ALLGOALS (resolve_tac thms)) thy)
   422           thy (full_new_type_names' ~~ tyvars)
   423       end;
   424 
   425     val (perm_thmss,thy3) = thy2 |>
   426       fold (fn name1 => fold (composition_instance name1) atoms) atoms |>
   427       curry (Library.foldr (fn ((i, (tyname, args, _)), thy) =>
   428         AxClass.prove_arity (tyname, replicate (length args) classes, classes)
   429         (ClassPackage.intro_classes_tac [] THEN REPEAT (EVERY
   430            [resolve_tac perm_empty_thms 1,
   431             resolve_tac perm_append_thms 1,
   432             resolve_tac perm_eq_thms 1, assume_tac 1])) thy))
   433         (List.take (descr, length new_type_names)) |>
   434       PureThy.add_thmss
   435         [((space_implode "_" new_type_names ^ "_unfolded_perm_eq",
   436           unfolded_perm_eq_thms), [Simplifier.simp_add]),
   437          ((space_implode "_" new_type_names ^ "_perm_empty",
   438           perm_empty_thms), [Simplifier.simp_add]),
   439          ((space_implode "_" new_type_names ^ "_perm_append",
   440           perm_append_thms), [Simplifier.simp_add]),
   441          ((space_implode "_" new_type_names ^ "_perm_eq",
   442           perm_eq_thms), [Simplifier.simp_add])];
   443   
   444     (**** Define representing sets ****)
   445 
   446     val _ = warning "representing sets";
   447 
   448     val rep_set_names = map (Sign.full_name thy3) (DatatypeProp.indexify_names
   449       (map (fn (i, _) => name_of_typ (nth_dtyp i) ^ "_set") descr));
   450     val big_rep_name =
   451       space_implode "_" (DatatypeProp.indexify_names (List.mapPartial
   452         (fn (i, ("Nominal.noption", _, _)) => NONE
   453           | (i, _) => SOME (name_of_typ (nth_dtyp i))) descr)) ^ "_set";
   454     val _ = warning ("big_rep_name: " ^ big_rep_name);
   455 
   456     fun strip_option (dtf as DtType ("fun", [dt, DtRec i])) =
   457           (case AList.lookup op = descr i of
   458              SOME ("Nominal.noption", _, [(_, [dt']), _]) =>
   459                apfst (cons dt) (strip_option dt')
   460            | _ => ([], dtf))
   461       | strip_option (DtType ("fun", [dt, DtType ("Nominal.noption", [dt'])])) =
   462           apfst (cons dt) (strip_option dt')
   463       | strip_option dt = ([], dt);
   464 
   465     val dt_atomTs = distinct op = (map (typ_of_dtyp descr sorts')
   466       (List.concat (map (fn (_, (_, _, cs)) => List.concat
   467         (map (List.concat o map (fst o strip_option) o snd) cs)) descr)));
   468 
   469     fun make_intr s T (cname, cargs) =
   470       let
   471         fun mk_prem (dt, (j, j', prems, ts)) = 
   472           let
   473             val (dts, dt') = strip_option dt;
   474             val (dts', dt'') = strip_dtyp dt';
   475             val Ts = map (typ_of_dtyp descr sorts') dts;
   476             val Us = map (typ_of_dtyp descr sorts') dts';
   477             val T = typ_of_dtyp descr sorts' dt'';
   478             val free = mk_Free "x" (Us ---> T) j;
   479             val free' = app_bnds free (length Us);
   480             fun mk_abs_fun (T, (i, t)) =
   481               let val U = fastype_of t
   482               in (i + 1, Const ("Nominal.abs_fun", [T, U, T] --->
   483                 Type ("Nominal.noption", [U])) $ mk_Free "y" T i $ t)
   484               end
   485           in (j + 1, j' + length Ts,
   486             case dt'' of
   487                 DtRec k => list_all (map (pair "x") Us,
   488                   HOLogic.mk_Trueprop (HOLogic.mk_mem (free',
   489                     Const (List.nth (rep_set_names, k),
   490                       HOLogic.mk_setT T)))) :: prems
   491               | _ => prems,
   492             snd (foldr mk_abs_fun (j', free) Ts) :: ts)
   493           end;
   494 
   495         val (_, _, prems, ts) = foldr mk_prem (1, 1, [], []) cargs;
   496         val concl = HOLogic.mk_Trueprop (HOLogic.mk_mem
   497           (list_comb (Const (cname, map fastype_of ts ---> T), ts),
   498            Const (s, HOLogic.mk_setT T)))
   499       in Logic.list_implies (prems, concl)
   500       end;
   501 
   502     val (intr_ts, ind_consts) =
   503       apfst List.concat (ListPair.unzip (List.mapPartial
   504         (fn ((_, ("Nominal.noption", _, _)), _) => NONE
   505           | ((i, (_, _, constrs)), rep_set_name) =>
   506               let val T = nth_dtyp i
   507               in SOME (map (make_intr rep_set_name T) constrs,
   508                 Const (rep_set_name, HOLogic.mk_setT T))
   509               end)
   510                 (descr ~~ rep_set_names)));
   511 
   512     val (thy4, {raw_induct = rep_induct, intrs = rep_intrs, ...}) =
   513       setmp InductivePackage.quiet_mode false
   514         (InductivePackage.add_inductive_i false true big_rep_name false true false
   515            ind_consts (map (fn x => (("", x), [])) intr_ts) []) thy3;
   516 
   517     (**** Prove that representing set is closed under permutation ****)
   518 
   519     val _ = warning "proving closure under permutation...";
   520 
   521     val perm_indnames' = List.mapPartial
   522       (fn (x, (_, ("Nominal.noption", _, _))) => NONE | (x, _) => SOME x)
   523       (perm_indnames ~~ descr);
   524 
   525     fun mk_perm_closed name = map (fn th => standard (th RS mp))
   526       (List.take (split_conj_thm (Goal.prove thy4 [] []
   527         (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj (map
   528            (fn (S, x) =>
   529               let
   530                 val S = map_term_types (map_type_tfree
   531                   (fn (s, cs) => TFree (s, cs union cp_classes))) S;
   532                 val T = HOLogic.dest_setT (fastype_of S);
   533                 val permT = mk_permT (Type (name, []))
   534               in HOLogic.mk_imp (HOLogic.mk_mem (Free (x, T), S),
   535                 HOLogic.mk_mem (Const ("Nominal.perm", permT --> T --> T) $
   536                   Free ("pi", permT) $ Free (x, T), S))
   537               end) (ind_consts ~~ perm_indnames'))))
   538         (fn _ => EVERY (* CU: added perm_fun_def in the final tactic in order to deal with funs *)
   539            [indtac rep_induct [] 1,
   540             ALLGOALS (simp_tac (simpset_of thy4 addsimps
   541               (symmetric perm_fun_def :: PureThy.get_thms thy4 (Name ("abs_perm"))))),
   542             ALLGOALS (resolve_tac rep_intrs 
   543                THEN_ALL_NEW (asm_full_simp_tac (simpset_of thy4 addsimps [perm_fun_def])))])),
   544         length new_type_names));
   545 
   546     (* FIXME: theorems are stored in database for testing only *)
   547     val perm_closed_thmss = map mk_perm_closed atoms;
   548     val (_,thy5) = PureThy.add_thmss [(("perm_closed", List.concat perm_closed_thmss), [])] thy4;
   549 
   550     (**** typedef ****)
   551 
   552     val _ = warning "defining type...";
   553 
   554     val (typedefs, thy6) =
   555       fold_map (fn ((((name, mx), tvs), c), name') => fn thy =>
   556         setmp TypedefPackage.quiet_mode true
   557           (TypedefPackage.add_typedef_i false (SOME name') (name, tvs, mx) c NONE
   558             (rtac exI 1 THEN
   559               QUIET_BREADTH_FIRST (has_fewer_prems 1)
   560               (resolve_tac rep_intrs 1))) thy |> (fn (r, thy) =>
   561         let
   562           val permT = mk_permT (TFree (variant tvs "'a", HOLogic.typeS));
   563           val pi = Free ("pi", permT);
   564           val tvs' = map (fn s => TFree (s, the (AList.lookup op = sorts' s))) tvs;
   565           val T = Type (Sign.intern_type thy name, tvs');
   566           val Const (_, Type (_, [U])) = c
   567         in apfst (pair r o hd)
   568           (PureThy.add_defs_i true [(("prm_" ^ name ^ "_def", Logic.mk_equals
   569             (Const ("Nominal.perm", permT --> T --> T) $ pi $ Free ("x", T),
   570              Const (Sign.intern_const thy ("Abs_" ^ name), U --> T) $
   571                (Const ("Nominal.perm", permT --> U --> U) $ pi $
   572                  (Const (Sign.intern_const thy ("Rep_" ^ name), T --> U) $
   573                    Free ("x", T))))), [])] thy)
   574         end))
   575           (types_syntax ~~ tyvars ~~
   576             (List.take (ind_consts, length new_type_names)) ~~ new_type_names) thy5;
   577 
   578     val perm_defs = map snd typedefs;
   579     val Abs_inverse_thms = map (#Abs_inverse o fst) typedefs;
   580     val Rep_inverse_thms = map (#Rep_inverse o fst) typedefs;
   581     val Rep_thms = map (#Rep o fst) typedefs;
   582 
   583     val big_name = space_implode "_" new_type_names;
   584 
   585 
   586     (** prove that new types are in class pt_<name> **)
   587 
   588     val _ = warning "prove that new types are in class pt_<name> ...";
   589 
   590     fun pt_instance ((class, atom), perm_closed_thms) =
   591       fold (fn (((({Abs_inverse, Rep_inverse, Rep, ...},
   592         perm_def), name), tvs), perm_closed) => fn thy =>
   593           AxClass.prove_arity
   594             (Sign.intern_type thy name,
   595               replicate (length tvs) (classes @ cp_classes), [class])
   596             (EVERY [ClassPackage.intro_classes_tac [],
   597               rewrite_goals_tac [perm_def],
   598               asm_full_simp_tac (simpset_of thy addsimps [Rep_inverse]) 1,
   599               asm_full_simp_tac (simpset_of thy addsimps
   600                 [Rep RS perm_closed RS Abs_inverse]) 1,
   601               asm_full_simp_tac (HOL_basic_ss addsimps [PureThy.get_thm thy
   602                 (Name ("pt_" ^ Sign.base_name atom ^ "3"))]) 1]) thy)
   603         (typedefs ~~ new_type_names ~~ tyvars ~~ perm_closed_thms);
   604 
   605 
   606     (** prove that new types are in class cp_<name1>_<name2> **)
   607 
   608     val _ = warning "prove that new types are in class cp_<name1>_<name2> ...";
   609 
   610     fun cp_instance (atom1, perm_closed_thms1) (atom2, perm_closed_thms2) thy =
   611       let
   612         val name = "cp_" ^ Sign.base_name atom1 ^ "_" ^ Sign.base_name atom2;
   613         val class = Sign.intern_class thy name;
   614         val cp1' = PureThy.get_thm thy (Name (name ^ "_inst")) RS cp1
   615       in fold (fn ((((({Abs_inverse, Rep_inverse, Rep, ...},
   616         perm_def), name), tvs), perm_closed1), perm_closed2) => fn thy =>
   617           AxClass.prove_arity
   618             (Sign.intern_type thy name,
   619               replicate (length tvs) (classes @ cp_classes), [class])
   620             (EVERY [ClassPackage.intro_classes_tac [],
   621               rewrite_goals_tac [perm_def],
   622               asm_full_simp_tac (simpset_of thy addsimps
   623                 ((Rep RS perm_closed1 RS Abs_inverse) ::
   624                  (if atom1 = atom2 then []
   625                   else [Rep RS perm_closed2 RS Abs_inverse]))) 1,
   626               cong_tac 1,
   627               rtac refl 1,
   628               rtac cp1' 1]) thy)
   629         (typedefs ~~ new_type_names ~~ tyvars ~~ perm_closed_thms1 ~~
   630           perm_closed_thms2) thy
   631       end;
   632 
   633     val thy7 = fold (fn x => fn thy => thy |>
   634       pt_instance x |>
   635       fold (cp_instance (apfst snd x)) (atoms ~~ perm_closed_thmss))
   636         (classes ~~ atoms ~~ perm_closed_thmss) thy6;
   637 
   638     (**** constructors ****)
   639 
   640     fun mk_abs_fun (x, t) =
   641       let
   642         val T = fastype_of x;
   643         val U = fastype_of t
   644       in
   645         Const ("Nominal.abs_fun", T --> U --> T -->
   646           Type ("Nominal.noption", [U])) $ x $ t
   647       end;
   648 
   649     val (ty_idxs, _) = foldl
   650       (fn ((i, ("Nominal.noption", _, _)), p) => p
   651         | ((i, _), (ty_idxs, j)) => (ty_idxs @ [(i, j)], j + 1)) ([], 0) descr;
   652 
   653     fun reindex (DtType (s, dts)) = DtType (s, map reindex dts)
   654       | reindex (DtRec i) = DtRec (the (AList.lookup op = ty_idxs i))
   655       | reindex dt = dt;
   656 
   657     fun strip_suffix i s = implode (List.take (explode s, size s - i));
   658 
   659     (** strips the "_Rep" in type names *)
   660     fun strip_nth_name i s = 
   661       let val xs = NameSpace.unpack s; 
   662       in NameSpace.pack (Library.nth_map (length xs - i) (strip_suffix 4) xs) end;
   663 
   664     val (descr'', ndescr) = ListPair.unzip (List.mapPartial
   665       (fn (i, ("Nominal.noption", _, _)) => NONE
   666         | (i, (s, dts, constrs)) =>
   667              let
   668                val SOME index = AList.lookup op = ty_idxs i;
   669                val (constrs1, constrs2) = ListPair.unzip
   670                  (map (fn (cname, cargs) => apfst (pair (strip_nth_name 2 cname))
   671                    (foldl_map (fn (dts, dt) =>
   672                      let val (dts', dt') = strip_option dt
   673                      in (dts @ dts' @ [reindex dt'], (length dts, length dts')) end)
   674                        ([], cargs))) constrs)
   675              in SOME ((index, (strip_nth_name 1 s,  map reindex dts, constrs1)),
   676                (index, constrs2))
   677              end) descr);
   678 
   679     val (descr1, descr2) = chop (length new_type_names) descr'';
   680     val descr' = [descr1, descr2];
   681 
   682     val typ_of_dtyp' = replace_types' o typ_of_dtyp descr sorts';
   683 
   684     val rep_names = map (fn s =>
   685       Sign.intern_const thy7 ("Rep_" ^ s)) new_type_names;
   686     val abs_names = map (fn s =>
   687       Sign.intern_const thy7 ("Abs_" ^ s)) new_type_names;
   688 
   689     val recTs' = List.mapPartial
   690       (fn ((_, ("Nominal.noption", _, _)), T) => NONE
   691         | (_, T) => SOME T) (descr ~~ get_rec_types descr sorts');
   692     val recTs = get_rec_types descr'' sorts';
   693     val newTs' = Library.take (length new_type_names, recTs');
   694     val newTs = Library.take (length new_type_names, recTs);
   695 
   696     val full_new_type_names = map (Sign.full_name (sign_of thy)) new_type_names;
   697 
   698     fun make_constr_def tname T T' ((thy, defs, eqns), ((cname, cargs), (cname', mx))) =
   699       let
   700         fun constr_arg (dt, (j, l_args, r_args)) =
   701           let
   702             val x' = mk_Free "x" (typ_of_dtyp' dt) j;
   703             val (dts, dt') = strip_option dt;
   704             val xs = map (fn (dt, i) => mk_Free "x" (typ_of_dtyp' dt) i)
   705               (dts ~~ (j upto j + length dts - 1))
   706             val x = mk_Free "x" (typ_of_dtyp' dt') (j + length dts)
   707             val (dts', dt'') = strip_dtyp dt'
   708           in
   709             (j + length dts + 1,
   710              xs @ x :: l_args,
   711              foldr mk_abs_fun
   712                (case dt'' of
   713                   DtRec k => if k < length new_type_names then
   714                       list_abs (map (pair "z" o typ_of_dtyp') dts',
   715                         Const (List.nth (rep_names, k), typ_of_dtyp' dt'' -->
   716                           typ_of_dtyp descr sorts' dt'') $ app_bnds x (length dts'))
   717                     else error "nested recursion not (yet) supported"
   718                 | _ => x) xs :: r_args)
   719           end
   720 
   721         val (_, l_args, r_args) = foldr constr_arg (1, [], []) cargs;
   722         val abs_name = Sign.intern_const (Theory.sign_of thy) ("Abs_" ^ tname);
   723         val rep_name = Sign.intern_const (Theory.sign_of thy) ("Rep_" ^ tname);
   724         val constrT = map fastype_of l_args ---> T;
   725         val lhs = list_comb (Const (Sign.full_name thy (Sign.base_name cname),
   726           constrT), l_args);
   727         val rhs = list_comb (Const (cname, map fastype_of r_args ---> T'), r_args);
   728         val def = Logic.mk_equals (lhs, Const (abs_name, T' --> T) $ rhs);
   729         val eqn = HOLogic.mk_Trueprop (HOLogic.mk_eq
   730           (Const (rep_name, T --> T') $ lhs, rhs));
   731         val def_name = (Sign.base_name cname) ^ "_def";
   732         val ([def_thm], thy') = thy |>
   733           Theory.add_consts_i [(cname', constrT, mx)] |>
   734           (PureThy.add_defs_i false o map Thm.no_attributes) [(def_name, def)]
   735       in (thy', defs @ [def_thm], eqns @ [eqn]) end;
   736 
   737     fun dt_constr_defs ((thy, defs, eqns, dist_lemmas),
   738         (((((_, (_, _, constrs)), tname), T), T'), constr_syntax)) =
   739       let
   740         val rep_const = cterm_of thy
   741           (Const (Sign.intern_const thy ("Rep_" ^ tname), T --> T'));
   742         val dist = standard (cterm_instantiate [(cterm_of thy distinct_f, rep_const)] distinct_lemma);
   743         val (thy', defs', eqns') = Library.foldl (make_constr_def tname T T')
   744           ((Theory.add_path tname thy, defs, []), constrs ~~ constr_syntax)
   745       in
   746         (parent_path flat_names thy', defs', eqns @ [eqns'], dist_lemmas @ [dist])
   747       end;
   748 
   749     val (thy8, constr_defs, constr_rep_eqns, dist_lemmas) = Library.foldl dt_constr_defs
   750       ((thy7, [], [], []), List.take (descr, length new_type_names) ~~
   751         new_type_names ~~ newTs ~~ newTs' ~~ constr_syntax);
   752 
   753     val abs_inject_thms = map (fn tname =>
   754       PureThy.get_thm thy8 (Name ("Abs_" ^ tname ^ "_inject"))) new_type_names;
   755 
   756     val rep_inject_thms = map (fn tname =>
   757       PureThy.get_thm thy8 (Name ("Rep_" ^ tname ^ "_inject"))) new_type_names;
   758 
   759     val rep_thms = map (fn tname =>
   760       PureThy.get_thm thy8 (Name ("Rep_" ^ tname))) new_type_names;
   761 
   762     val rep_inverse_thms = map (fn tname =>
   763       PureThy.get_thm thy8 (Name ("Rep_" ^ tname ^ "_inverse"))) new_type_names;
   764 
   765     (* prove theorem  Rep_i (Constr_j ...) = Constr'_j ...  *)
   766     
   767     fun prove_constr_rep_thm eqn =
   768       let
   769         val inj_thms = map (fn r => r RS iffD1) abs_inject_thms;
   770         val rewrites = constr_defs @ map mk_meta_eq rep_inverse_thms
   771       in standard (Goal.prove thy8 [] [] eqn (fn _ => EVERY
   772         [resolve_tac inj_thms 1,
   773          rewrite_goals_tac rewrites,
   774          rtac refl 3,
   775          resolve_tac rep_intrs 2,
   776          REPEAT (resolve_tac rep_thms 1)]))
   777       end;
   778 
   779     val constr_rep_thmss = map (map prove_constr_rep_thm) constr_rep_eqns;
   780 
   781     (* prove theorem  pi \<bullet> Rep_i x = Rep_i (pi \<bullet> x) *)
   782 
   783     fun prove_perm_rep_perm (atom, perm_closed_thms) = map (fn th =>
   784       let
   785         val _ $ (_ $ (Rep $ x) $ _) = Logic.unvarify (prop_of th);
   786         val Type ("fun", [T, U]) = fastype_of Rep;
   787         val permT = mk_permT (Type (atom, []));
   788         val pi = Free ("pi", permT);
   789       in
   790         standard (Goal.prove thy8 [] [] (HOLogic.mk_Trueprop (HOLogic.mk_eq
   791             (Const ("Nominal.perm", permT --> U --> U) $ pi $ (Rep $ x),
   792              Rep $ (Const ("Nominal.perm", permT --> T --> T) $ pi $ x))))
   793           (fn _ => simp_tac (HOL_basic_ss addsimps (perm_defs @ Abs_inverse_thms @
   794             perm_closed_thms @ Rep_thms)) 1))
   795       end) Rep_thms;
   796 
   797     val perm_rep_perm_thms = List.concat (map prove_perm_rep_perm
   798       (atoms ~~ perm_closed_thmss));
   799 
   800     (* prove distinctness theorems *)
   801 
   802     val distinct_props = setmp DatatypeProp.dtK 1000
   803       (DatatypeProp.make_distincts new_type_names descr' sorts') thy8;
   804 
   805     val dist_rewrites = map (fn (rep_thms, dist_lemma) =>
   806       dist_lemma::(rep_thms @ [In0_eq, In1_eq, In0_not_In1, In1_not_In0]))
   807         (constr_rep_thmss ~~ dist_lemmas);
   808 
   809     fun prove_distinct_thms (_, []) = []
   810       | prove_distinct_thms (p as (rep_thms, dist_lemma), t::ts) =
   811           let
   812             val dist_thm = standard (Goal.prove thy8 [] [] t (fn _ =>
   813               simp_tac (simpset_of thy8 addsimps (dist_lemma :: rep_thms)) 1))
   814           in dist_thm::(standard (dist_thm RS not_sym))::
   815             (prove_distinct_thms (p, ts))
   816           end;
   817 
   818     val distinct_thms = map prove_distinct_thms
   819       (constr_rep_thmss ~~ dist_lemmas ~~ distinct_props);
   820 
   821     (** prove equations for permutation functions **)
   822 
   823     val abs_perm = PureThy.get_thms thy8 (Name "abs_perm"); (* FIXME *)
   824 
   825     val perm_simps' = map (fn (((i, (_, _, constrs)), tname), constr_rep_thms) =>
   826       let val T = replace_types' (nth_dtyp i)
   827       in List.concat (map (fn (atom, perm_closed_thms) =>
   828           map (fn ((cname, dts), constr_rep_thm) => 
   829         let
   830           val cname = Sign.intern_const thy8
   831             (NameSpace.append tname (Sign.base_name cname));
   832           val permT = mk_permT (Type (atom, []));
   833           val pi = Free ("pi", permT);
   834 
   835           fun perm t =
   836             let val T = fastype_of t
   837             in Const ("Nominal.perm", permT --> T --> T) $ pi $ t end;
   838 
   839           fun constr_arg (dt, (j, l_args, r_args)) =
   840             let
   841               val x' = mk_Free "x" (typ_of_dtyp' dt) j;
   842               val (dts, dt') = strip_option dt;
   843               val Ts = map typ_of_dtyp' dts;
   844               val xs = map (fn (T, i) => mk_Free "x" T i)
   845                 (Ts ~~ (j upto j + length dts - 1))
   846               val x = mk_Free "x" (typ_of_dtyp' dt') (j + length dts);
   847               val (dts', dt'') = strip_dtyp dt';
   848             in
   849               (j + length dts + 1,
   850                xs @ x :: l_args,
   851                map perm (xs @ [x]) @ r_args)
   852             end
   853 
   854           val (_, l_args, r_args) = foldr constr_arg (1, [], []) dts;
   855           val c = Const (cname, map fastype_of l_args ---> T)
   856         in
   857           standard (Goal.prove thy8 [] []
   858             (HOLogic.mk_Trueprop (HOLogic.mk_eq
   859               (perm (list_comb (c, l_args)), list_comb (c, r_args))))
   860             (fn _ => EVERY
   861               [simp_tac (simpset_of thy8 addsimps (constr_rep_thm :: perm_defs)) 1,
   862                simp_tac (HOL_basic_ss addsimps (Rep_thms @ Abs_inverse_thms @
   863                  constr_defs @ perm_closed_thms)) 1,
   864                TRY (simp_tac (HOL_basic_ss addsimps
   865                  (symmetric perm_fun_def :: abs_perm)) 1),
   866                TRY (simp_tac (HOL_basic_ss addsimps
   867                  (perm_fun_def :: perm_defs @ Rep_thms @ Abs_inverse_thms @
   868                     perm_closed_thms)) 1)]))
   869         end) (constrs ~~ constr_rep_thms)) (atoms ~~ perm_closed_thmss))
   870       end) (List.take (descr, length new_type_names) ~~ new_type_names ~~ constr_rep_thmss);
   871 
   872     (** prove injectivity of constructors **)
   873 
   874     val rep_inject_thms' = map (fn th => th RS sym) rep_inject_thms;
   875     val alpha = PureThy.get_thms thy8 (Name "alpha");
   876     val abs_fresh = PureThy.get_thms thy8 (Name "abs_fresh");
   877     val fresh_def = PureThy.get_thm thy8 (Name "fresh_def");
   878     val supp_def = PureThy.get_thm thy8 (Name "supp_def");
   879 
   880     val inject_thms = map (fn (((i, (_, _, constrs)), tname), constr_rep_thms) =>
   881       let val T = replace_types' (nth_dtyp i)
   882       in List.mapPartial (fn ((cname, dts), constr_rep_thm) =>
   883         if null dts then NONE else SOME
   884         let
   885           val cname = Sign.intern_const thy8
   886             (NameSpace.append tname (Sign.base_name cname));
   887 
   888           fun make_inj (dt, (j, args1, args2, eqs)) =
   889             let
   890               val x' = mk_Free "x" (typ_of_dtyp' dt) j;
   891               val y' = mk_Free "y" (typ_of_dtyp' dt) j;
   892               val (dts, dt') = strip_option dt;
   893               val Ts_idx = map typ_of_dtyp' dts ~~ (j upto j + length dts - 1);
   894               val xs = map (fn (T, i) => mk_Free "x" T i) Ts_idx;
   895               val ys = map (fn (T, i) => mk_Free "y" T i) Ts_idx;
   896               val x = mk_Free "x" (typ_of_dtyp' dt') (j + length dts);
   897               val y = mk_Free "y" (typ_of_dtyp' dt') (j + length dts);
   898               val (dts', dt'') = strip_dtyp dt';
   899             in
   900               (j + length dts + 1,
   901                xs @ (x :: args1), ys @ (y :: args2),
   902                HOLogic.mk_eq
   903                  (foldr mk_abs_fun x xs, foldr mk_abs_fun y ys) :: eqs)
   904             end;
   905 
   906           val (_, args1, args2, eqs) = foldr make_inj (1, [], [], []) dts;
   907           val Ts = map fastype_of args1;
   908           val c = Const (cname, Ts ---> T)
   909         in
   910           standard (Goal.prove thy8 [] [] (HOLogic.mk_Trueprop (HOLogic.mk_eq
   911               (HOLogic.mk_eq (list_comb (c, args1), list_comb (c, args2)),
   912                foldr1 HOLogic.mk_conj eqs)))
   913             (fn _ => EVERY
   914                [asm_full_simp_tac (simpset_of thy8 addsimps (constr_rep_thm ::
   915                   rep_inject_thms')) 1,
   916                 TRY (asm_full_simp_tac (HOL_basic_ss addsimps (fresh_def :: supp_def ::
   917                   alpha @ abs_perm @ abs_fresh @ rep_inject_thms @
   918                   perm_rep_perm_thms)) 1),
   919                 TRY (asm_full_simp_tac (HOL_basic_ss addsimps (perm_fun_def ::
   920                   expand_fun_eq :: rep_inject_thms @ perm_rep_perm_thms)) 1)]))
   921         end) (constrs ~~ constr_rep_thms)
   922       end) (List.take (descr, length new_type_names) ~~ new_type_names ~~ constr_rep_thmss);
   923 
   924     (** equations for support and freshness **)
   925 
   926     val Un_assoc = PureThy.get_thm thy8 (Name "Un_assoc");
   927     val de_Morgan_conj = PureThy.get_thm thy8 (Name "de_Morgan_conj");
   928     val Collect_disj_eq = PureThy.get_thm thy8 (Name "Collect_disj_eq");
   929     val finite_Un = PureThy.get_thm thy8 (Name "finite_Un");
   930 
   931     val (supp_thms, fresh_thms) = ListPair.unzip (map ListPair.unzip
   932       (map (fn ((((i, (_, _, constrs)), tname), inject_thms'), perm_thms') =>
   933       let val T = replace_types' (nth_dtyp i)
   934       in List.concat (map (fn (cname, dts) => map (fn atom =>
   935         let
   936           val cname = Sign.intern_const thy8
   937             (NameSpace.append tname (Sign.base_name cname));
   938           val atomT = Type (atom, []);
   939 
   940           fun process_constr (dt, (j, args1, args2)) =
   941             let
   942               val x' = mk_Free "x" (typ_of_dtyp' dt) j;
   943               val (dts, dt') = strip_option dt;
   944               val Ts_idx = map typ_of_dtyp' dts ~~ (j upto j + length dts - 1);
   945               val xs = map (fn (T, i) => mk_Free "x" T i) Ts_idx;
   946               val x = mk_Free "x" (typ_of_dtyp' dt') (j + length dts);
   947               val (dts', dt'') = strip_dtyp dt';
   948             in
   949               (j + length dts + 1,
   950                xs @ (x :: args1), foldr mk_abs_fun x xs :: args2)
   951             end;
   952 
   953           val (_, args1, args2) = foldr process_constr (1, [], []) dts;
   954           val Ts = map fastype_of args1;
   955           val c = list_comb (Const (cname, Ts ---> T), args1);
   956           fun supp t =
   957             Const ("Nominal.supp", fastype_of t --> HOLogic.mk_setT atomT) $ t;
   958           fun fresh t =
   959             Const ("Nominal.fresh", atomT --> fastype_of t --> HOLogic.boolT) $
   960               Free ("a", atomT) $ t;
   961           val supp_thm = standard (Goal.prove thy8 [] []
   962               (HOLogic.mk_Trueprop (HOLogic.mk_eq
   963                 (supp c,
   964                  if null dts then Const ("{}", HOLogic.mk_setT atomT)
   965                  else foldr1 (HOLogic.mk_binop "op Un") (map supp args2))))
   966             (fn _ =>
   967               simp_tac (HOL_basic_ss addsimps (supp_def ::
   968                  Un_assoc :: de_Morgan_conj :: Collect_disj_eq :: finite_Un ::
   969                  symmetric empty_def :: Finites.emptyI :: simp_thms @
   970                  abs_perm @ abs_fresh @ inject_thms' @ perm_thms')) 1))
   971         in
   972           (supp_thm,
   973            standard (Goal.prove thy8 [] [] (HOLogic.mk_Trueprop (HOLogic.mk_eq
   974               (fresh c,
   975                if null dts then HOLogic.true_const
   976                else foldr1 HOLogic.mk_conj (map fresh args2))))
   977              (fn _ =>
   978                simp_tac (simpset_of thy8 addsimps [fresh_def, supp_thm]) 1)))
   979         end) atoms) constrs)
   980       end) (List.take (descr, length new_type_names) ~~ new_type_names ~~ inject_thms ~~ perm_simps')));
   981 
   982     (**** weak induction theorem ****)
   983 
   984     val arities = get_arities descr'';
   985 
   986     fun mk_funs_inv thm =
   987       let
   988         val {sign, prop, ...} = rep_thm thm;
   989         val _ $ (_ $ (Const (_, Type (_, [U, _])) $ _ $ S)) $
   990           (_ $ (_ $ (r $ (a $ _)) $ _)) = Type.freeze prop;
   991         val used = add_term_tfree_names (a, []);
   992 
   993         fun mk_thm i =
   994           let
   995             val Ts = map (TFree o rpair HOLogic.typeS)
   996               (variantlist (replicate i "'t", used));
   997             val f = Free ("f", Ts ---> U)
   998           in standard (Goal.prove sign [] [] (Logic.mk_implies
   999             (HOLogic.mk_Trueprop (HOLogic.list_all
  1000                (map (pair "x") Ts, HOLogic.mk_mem (app_bnds f i, S))),
  1001              HOLogic.mk_Trueprop (HOLogic.mk_eq (list_abs (map (pair "x") Ts,
  1002                r $ (a $ app_bnds f i)), f))))
  1003             (fn _ => EVERY [REPEAT (rtac ext 1), REPEAT (etac allE 1), rtac thm 1, atac 1]))
  1004           end
  1005       in map (fn r => r RS subst) (thm :: map mk_thm arities) end;
  1006 
  1007     fun mk_indrule_lemma ((prems, concls), (((i, _), T), U)) =
  1008       let
  1009         val Rep_t = Const (List.nth (rep_names, i), T --> U) $
  1010           mk_Free "x" T i;
  1011 
  1012         val Abs_t =  Const (List.nth (abs_names, i), U --> T)
  1013 
  1014       in (prems @ [HOLogic.imp $ HOLogic.mk_mem (Rep_t,
  1015             Const (List.nth (rep_set_names, i), HOLogic.mk_setT U)) $
  1016               (mk_Free "P" (T --> HOLogic.boolT) (i + 1) $ (Abs_t $ Rep_t))],
  1017           concls @ [mk_Free "P" (T --> HOLogic.boolT) (i + 1) $ mk_Free "x" T i])
  1018       end;
  1019 
  1020     val (indrule_lemma_prems, indrule_lemma_concls) =
  1021       Library.foldl mk_indrule_lemma (([], []), (descr'' ~~ recTs ~~ recTs'));
  1022 
  1023     val indrule_lemma = standard (Goal.prove thy8 [] []
  1024       (Logic.mk_implies
  1025         (HOLogic.mk_Trueprop (mk_conj indrule_lemma_prems),
  1026          HOLogic.mk_Trueprop (mk_conj indrule_lemma_concls))) (fn _ => EVERY
  1027            [REPEAT (etac conjE 1),
  1028             REPEAT (EVERY
  1029               [TRY (rtac conjI 1), full_simp_tac (HOL_basic_ss addsimps Rep_inverse_thms) 1,
  1030                etac mp 1, resolve_tac Rep_thms 1])]));
  1031 
  1032     val Ps = map head_of (HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of indrule_lemma)));
  1033     val frees = if length Ps = 1 then [Free ("P", snd (dest_Var (hd Ps)))] else
  1034       map (Free o apfst fst o dest_Var) Ps;
  1035     val indrule_lemma' = cterm_instantiate
  1036       (map (cterm_of thy8) Ps ~~ map (cterm_of thy8) frees) indrule_lemma;
  1037 
  1038     val Abs_inverse_thms' = List.concat (map mk_funs_inv Abs_inverse_thms);
  1039 
  1040     val dt_induct_prop = DatatypeProp.make_ind descr' sorts';
  1041     val dt_induct = standard (Goal.prove thy8 []
  1042       (Logic.strip_imp_prems dt_induct_prop) (Logic.strip_imp_concl dt_induct_prop)
  1043       (fn prems => EVERY
  1044         [rtac indrule_lemma' 1,
  1045          (DatatypeAux.indtac rep_induct THEN_ALL_NEW ObjectLogic.atomize_tac) 1,
  1046          EVERY (map (fn (prem, r) => (EVERY
  1047            [REPEAT (eresolve_tac Abs_inverse_thms' 1),
  1048             simp_tac (HOL_basic_ss addsimps [symmetric r]) 1,
  1049             DEPTH_SOLVE_1 (ares_tac [prem] 1 ORELSE etac allE 1)]))
  1050                 (prems ~~ constr_defs))]));
  1051 
  1052     val case_names_induct = mk_case_names_induct descr'';
  1053 
  1054     (**** prove that new datatypes have finite support ****)
  1055 
  1056     val _ = warning "proving finite support for the new datatype";
  1057 
  1058     val indnames = DatatypeProp.make_tnames recTs;
  1059 
  1060     val abs_supp = PureThy.get_thms thy8 (Name "abs_supp");
  1061     val supp_atm = PureThy.get_thms thy8 (Name "supp_atm");
  1062 
  1063     val finite_supp_thms = map (fn atom =>
  1064       let val atomT = Type (atom, [])
  1065       in map standard (List.take
  1066         (split_conj_thm (Goal.prove thy8 [] [] (HOLogic.mk_Trueprop
  1067            (foldr1 HOLogic.mk_conj (map (fn (s, T) => HOLogic.mk_mem
  1068              (Const ("Nominal.supp", T --> HOLogic.mk_setT atomT) $ Free (s, T),
  1069               Const ("Finite_Set.Finites", HOLogic.mk_setT (HOLogic.mk_setT atomT))))
  1070                (indnames ~~ recTs))))
  1071            (fn _ => indtac dt_induct indnames 1 THEN
  1072             ALLGOALS (asm_full_simp_tac (simpset_of thy8 addsimps
  1073               (abs_supp @ supp_atm @
  1074                PureThy.get_thms thy8 (Name ("fs_" ^ Sign.base_name atom ^ "1")) @
  1075                List.concat supp_thms))))),
  1076          length new_type_names))
  1077       end) atoms;
  1078 
  1079     val simp_atts = replicate (length new_type_names) [Simplifier.simp_add];
  1080 
  1081     val (_, thy9) = thy8 |>
  1082       Theory.add_path big_name |>
  1083       PureThy.add_thms [(("induct_weak", dt_induct), [case_names_induct])] ||>>
  1084       PureThy.add_thmss [(("inducts_weak", projections dt_induct), [case_names_induct])] ||>
  1085       Theory.parent_path ||>>
  1086       DatatypeAux.store_thmss_atts "distinct" new_type_names simp_atts distinct_thms ||>>
  1087       DatatypeAux.store_thmss "constr_rep" new_type_names constr_rep_thmss ||>>
  1088       DatatypeAux.store_thmss_atts "perm" new_type_names simp_atts perm_simps' ||>>
  1089       DatatypeAux.store_thmss "inject" new_type_names inject_thms ||>>
  1090       DatatypeAux.store_thmss "supp" new_type_names supp_thms ||>>
  1091       DatatypeAux.store_thmss_atts "fresh" new_type_names simp_atts fresh_thms ||>
  1092       fold (fn (atom, ths) => fn thy =>
  1093         let val class = Sign.intern_class thy ("fs_" ^ Sign.base_name atom)
  1094         in fold (fn T => AxClass.prove_arity
  1095             (fst (dest_Type T),
  1096               replicate (length sorts) [class], [class])
  1097             (ClassPackage.intro_classes_tac [] THEN resolve_tac ths 1)) newTs thy
  1098         end) (atoms ~~ finite_supp_thms);
  1099 
  1100     (**** strong induction theorem ****)
  1101 
  1102     val pnames = if length descr'' = 1 then ["P"]
  1103       else map (fn i => "P" ^ string_of_int i) (1 upto length descr'');
  1104     val ind_sort = if null dt_atomTs then HOLogic.typeS
  1105       else norm_sort thy9 (map (fn T => Sign.intern_class thy9 ("fs_" ^
  1106         Sign.base_name (fst (dest_Type T)))) dt_atomTs);
  1107     val fsT = TFree ("'n", ind_sort);
  1108     val fsT' = TFree ("'n", HOLogic.typeS);
  1109 
  1110     val fresh_fs = map (fn (s, T) => (T, Free (s, fsT' --> HOLogic.mk_setT T)))
  1111       (DatatypeProp.indexify_names (replicate (length dt_atomTs) "f") ~~ dt_atomTs);
  1112 
  1113     fun make_pred fsT i T =
  1114       Free (List.nth (pnames, i), fsT --> T --> HOLogic.boolT);
  1115 
  1116     fun make_ind_prem fsT f k T ((cname, cargs), idxs) =
  1117       let
  1118         val recs = List.filter is_rec_type cargs;
  1119         val Ts = map (typ_of_dtyp descr'' sorts') cargs;
  1120         val recTs' = map (typ_of_dtyp descr'' sorts') recs;
  1121         val tnames = variantlist (DatatypeProp.make_tnames Ts, pnames);
  1122         val rec_tnames = map fst (List.filter (is_rec_type o snd) (tnames ~~ cargs));
  1123         val frees = tnames ~~ Ts;
  1124         val z = (variant tnames "z", fsT);
  1125 
  1126         fun mk_prem ((dt, s), T) =
  1127           let
  1128             val (Us, U) = strip_type T;
  1129             val l = length Us
  1130           in list_all (z :: map (pair "x") Us, HOLogic.mk_Trueprop
  1131             (make_pred fsT (body_index dt) U $ Bound l $ app_bnds (Free (s, T)) l))
  1132           end;
  1133 
  1134         val prems = map mk_prem (recs ~~ rec_tnames ~~ recTs');
  1135         val prems' = map (fn p as (_, T) => HOLogic.mk_Trueprop
  1136             (f T (Free p) (Free z)))
  1137           (map (curry List.nth frees) (List.concat (map (fn (m, n) =>
  1138              m upto m + n - 1) idxs)))
  1139 
  1140       in list_all_free (frees @ [z], Logic.list_implies (prems' @ prems,
  1141         HOLogic.mk_Trueprop (make_pred fsT k T $ Free z $
  1142           list_comb (Const (cname, Ts ---> T), map Free frees))))
  1143       end;
  1144 
  1145     val ind_prems = List.concat (map (fn (((i, (_, _, constrs)), (_, idxss)), T) =>
  1146       map (make_ind_prem fsT (fn T => fn t => fn u =>
  1147         Const ("Nominal.fresh", T --> fsT --> HOLogic.boolT) $ t $ u) i T)
  1148           (constrs ~~ idxss)) (descr'' ~~ ndescr ~~ recTs));
  1149     val tnames = DatatypeProp.make_tnames recTs;
  1150     val zs = variantlist (replicate (length descr'') "z", tnames);
  1151     val ind_concl = HOLogic.mk_Trueprop (foldr1 (HOLogic.mk_binop "op &")
  1152       (map (fn ((((i, _), T), tname), z) =>
  1153         make_pred fsT i T $ Free (z, fsT) $ Free (tname, T))
  1154         (descr'' ~~ recTs ~~ tnames ~~ zs)));
  1155     val induct = Logic.list_implies (ind_prems, ind_concl);
  1156 
  1157     val ind_prems' =
  1158       map (fn (_, f as Free (_, T)) => list_all_free ([("x", fsT')],
  1159         HOLogic.mk_Trueprop (HOLogic.mk_mem (f $ Free ("x", fsT'),
  1160           Const ("Finite_Set.Finites", HOLogic.mk_setT (body_type T)))))) fresh_fs @
  1161       List.concat (map (fn (((i, (_, _, constrs)), (_, idxss)), T) =>
  1162         map (make_ind_prem fsT' (fn T => fn t => fn u => HOLogic.Not $
  1163           HOLogic.mk_mem (t, the (AList.lookup op = fresh_fs T) $ u)) i T)
  1164             (constrs ~~ idxss)) (descr'' ~~ ndescr ~~ recTs));
  1165     val ind_concl' = HOLogic.mk_Trueprop (foldr1 (HOLogic.mk_binop "op &")
  1166       (map (fn ((((i, _), T), tname), z) =>
  1167         make_pred fsT' i T $ Free (z, fsT') $ Free (tname, T))
  1168         (descr'' ~~ recTs ~~ tnames ~~ zs)));
  1169     val induct' = Logic.list_implies (ind_prems', ind_concl');
  1170 
  1171     fun mk_perm Ts (t, u) =
  1172       let
  1173         val T = fastype_of1 (Ts, t);
  1174         val U = fastype_of1 (Ts, u)
  1175       in Const ("Nominal.perm", T --> U --> U) $ t $ u end;
  1176 
  1177     val aux_ind_vars =
  1178       (DatatypeProp.indexify_names (replicate (length dt_atomTs) "pi") ~~
  1179        map mk_permT dt_atomTs) @ [("z", fsT')];
  1180     val aux_ind_Ts = rev (map snd aux_ind_vars);
  1181     val aux_ind_concl = HOLogic.mk_Trueprop (foldr1 (HOLogic.mk_binop "op &")
  1182       (map (fn (((i, _), T), tname) =>
  1183         HOLogic.list_all (aux_ind_vars, make_pred fsT' i T $ Bound 0 $
  1184           foldr (mk_perm aux_ind_Ts) (Free (tname, T))
  1185             (map Bound (length dt_atomTs downto 1))))
  1186         (descr'' ~~ recTs ~~ tnames)));
  1187 
  1188     fun mk_ind_perm i k p l vs j =
  1189       let
  1190         val n = length vs;
  1191         val Ts = map snd vs;
  1192         val T = List.nth (Ts, i - j);
  1193         val pT = NominalAtoms.mk_permT T
  1194       in
  1195         Const ("List.list.Cons", HOLogic.mk_prodT (T, T) --> pT --> pT) $
  1196           (HOLogic.pair_const T T $ Bound (l - j) $ foldr (mk_perm Ts)
  1197             (Bound (i - j))
  1198             (map (mk_ind_perm i k p l vs) (j - 1 downto 0) @
  1199              map Bound (n - k - 1 downto n - k - p))) $
  1200           Const ("List.list.Nil", pT)
  1201       end;
  1202 
  1203     fun mk_fresh i i' j k p l vs _ _ =
  1204       let
  1205         val n = length vs;
  1206         val Ts = map snd vs;
  1207         val T = List.nth (Ts, n - i - 1 - j);
  1208         val f = the (AList.lookup op = fresh_fs T);
  1209         val U = List.nth (Ts, n - i' - 1);
  1210         val S = HOLogic.mk_setT T;
  1211         val prms = map (mk_ind_perm (n - i) k p (n - l) (("a", T) :: vs))
  1212             (j - 1 downto 0) @
  1213           map Bound (n - k downto n - k - p + 1)
  1214       in
  1215         HOLogic.mk_Trueprop (Const ("Ex", (T --> HOLogic.boolT) --> HOLogic.boolT) $
  1216           Abs ("a", T, HOLogic.Not $
  1217             (Const ("op :", T --> S --> HOLogic.boolT) $ Bound 0 $
  1218               (Const ("insert", T --> S --> S) $
  1219                 (foldr (mk_perm (T :: Ts)) (Bound (n - i - j)) prms) $
  1220                 (Const ("op Un", S --> S --> S) $ (f $ Bound (n - k - p)) $
  1221                    (Const ("Nominal.supp", U --> S) $
  1222                      foldr (mk_perm (T :: Ts)) (Bound (n - i')) prms))))))
  1223       end;
  1224 
  1225     fun mk_fresh_constr is p vs _ concl =
  1226       let
  1227         val n = length vs;
  1228         val Ts = map snd vs;
  1229         val _ $ (_ $ _ $ t) = concl;
  1230         val c = head_of t;
  1231         val T = body_type (fastype_of c);
  1232         val k = foldr op + 0 (map (fn (_, i) => i + 1) is);
  1233         val ps = map Bound (n - k - 1 downto n - k - p);
  1234         val (_, _, ts, us) = foldl (fn ((_, i), (m, n, ts, us)) =>
  1235           (m - i - 1, n - i,
  1236            ts @ map Bound (n downto n - i + 1) @
  1237              [foldr (mk_perm Ts) (Bound (m - i))
  1238                 (map (mk_ind_perm m k p n vs) (i - 1 downto 0) @ ps)],
  1239            us @ map (fn j => foldr (mk_perm Ts) (Bound j) ps) (m downto m - i)))
  1240           (n - 1, n - k - p - 2, [], []) is
  1241       in
  1242         HOLogic.mk_Trueprop (HOLogic.eq_const T $ list_comb (c, ts) $ list_comb (c, us))
  1243       end;
  1244 
  1245     val abs_fun_finite_supp = PureThy.get_thm thy9 (Name "abs_fun_finite_supp");
  1246 
  1247     val at_finite_select = PureThy.get_thm thy9 (Name "at_finite_select");
  1248 
  1249     val induct_aux_lemmas = List.concat (map (fn Type (s, _) =>
  1250       [PureThy.get_thm thy9 (Name ("pt_" ^ Sign.base_name s ^ "_inst")),
  1251        PureThy.get_thm thy9 (Name ("fs_" ^ Sign.base_name s ^ "1")),
  1252        PureThy.get_thm thy9 (Name ("at_" ^ Sign.base_name s ^ "_inst"))]) dt_atomTs);
  1253 
  1254     val induct_aux_lemmas' = map (fn Type (s, _) =>
  1255       PureThy.get_thm thy9 (Name ("pt_" ^ Sign.base_name s ^ "2")) RS sym) dt_atomTs;
  1256 
  1257     val induct_aux = standard (Goal.prove thy9 [] ind_prems' ind_concl'
  1258       (fn prems => EVERY
  1259         ([mk_subgoal 1 (K (K (K aux_ind_concl))),
  1260           indtac dt_induct tnames 1] @
  1261          List.concat (map (fn ((_, (_, _, constrs)), (_, constrs')) =>
  1262            List.concat (map (fn ((cname, cargs), is) =>
  1263              [simp_tac (HOL_basic_ss addsimps List.concat perm_simps') 1,
  1264               REPEAT (rtac allI 1)] @
  1265              List.concat (map
  1266                (fn ((_, 0), _) => []
  1267                  | ((i, j), k) => List.concat (map (fn j' =>
  1268                      let
  1269                        val DtType (tname, _) = List.nth (cargs, i + j');
  1270                        val atom = Sign.base_name tname
  1271                      in
  1272                        [mk_subgoal 1 (mk_fresh i (i + j) j'
  1273                           (length cargs) (length dt_atomTs)
  1274                           (length cargs + length dt_atomTs + 1 + i - k)),
  1275                         rtac at_finite_select 1,
  1276                         rtac (PureThy.get_thm thy9 (Name ("at_" ^ atom ^ "_inst"))) 1,
  1277                         asm_full_simp_tac (simpset_of thy9 addsimps
  1278                           [PureThy.get_thm thy9 (Name ("fs_" ^ atom ^ "1"))]) 1,
  1279                         resolve_tac prems 1,
  1280                         etac exE 1,
  1281                         asm_full_simp_tac (simpset_of thy9 addsimps
  1282                           [symmetric fresh_def]) 1]
  1283                      end) (0 upto j - 1))) (is ~~ (0 upto length is - 1))) @
  1284              (if exists (not o equal 0 o snd) is then
  1285                 [mk_subgoal 1 (mk_fresh_constr is (length dt_atomTs)),
  1286                  asm_full_simp_tac (simpset_of thy9 addsimps
  1287                    (List.concat inject_thms @
  1288                     alpha @ abs_perm @ abs_fresh @ [abs_fun_finite_supp] @
  1289                     induct_aux_lemmas)) 1,
  1290                  dtac sym 1,
  1291                  asm_full_simp_tac (simpset_of thy9
  1292                    addsimps induct_aux_lemmas'
  1293                    addsimprocs [perm_simproc]) 1,
  1294                  REPEAT (etac conjE 1)]
  1295               else
  1296                 []) @
  1297              [(resolve_tac prems THEN_ALL_NEW
  1298                 (atac ORELSE' ((REPEAT o etac allE) THEN' atac))) 1])
  1299                (constrs ~~ constrs'))) (descr'' ~~ ndescr)) @
  1300          [REPEAT (eresolve_tac [conjE, allE_Nil] 1),
  1301           REPEAT (etac allE 1),
  1302           REPEAT (TRY (rtac conjI 1) THEN asm_full_simp_tac (simpset_of thy9) 1)])));
  1303 
  1304     val induct_aux' = Thm.instantiate ([],
  1305       map (fn (s, T) =>
  1306         let val pT = TVar (("'n", 0), HOLogic.typeS) --> T --> HOLogic.boolT
  1307         in (cterm_of thy9 (Var ((s, 0), pT)), cterm_of thy9 (Free (s, pT))) end)
  1308           (pnames ~~ recTs) @
  1309       map (fn (_, f) =>
  1310         let val f' = Logic.varify f
  1311         in (cterm_of thy9 f',
  1312           cterm_of thy9 (Const ("Nominal.supp", fastype_of f')))
  1313         end) fresh_fs) induct_aux;
  1314 
  1315     val induct = standard (Goal.prove thy9 [] ind_prems ind_concl
  1316       (fn prems => EVERY
  1317          [rtac induct_aux' 1,
  1318           REPEAT (resolve_tac induct_aux_lemmas 1),
  1319           REPEAT ((resolve_tac prems THEN_ALL_NEW
  1320             (etac meta_spec ORELSE' full_simp_tac (HOL_basic_ss addsimps [fresh_def]))) 1)]))
  1321 
  1322     val (_, thy10) = thy9 |>
  1323       Theory.add_path big_name |>
  1324       PureThy.add_thms [(("induct'", induct_aux), [])] ||>>
  1325       PureThy.add_thms [(("induct", induct), [case_names_induct])] ||>>
  1326       PureThy.add_thmss [(("inducts", projections induct), [case_names_induct])] ||>
  1327       Theory.parent_path;
  1328 
  1329     (**** recursion combinator ****)
  1330 
  1331     val _ = warning "defining recursion combinator ...";
  1332 
  1333     val used = foldr add_typ_tfree_names [] recTs;
  1334 
  1335     val (rec_result_Ts, rec_fn_Ts) = DatatypeProp.make_primrec_Ts descr' sorts' used;
  1336 
  1337     val permTs = map mk_permT dt_atomTs;
  1338     val perms = map Free
  1339       (DatatypeProp.indexify_names (replicate (length permTs) "pi") ~~ permTs);
  1340 
  1341     val rec_set_Ts = map (fn (T1, T2) => rec_fn_Ts ---> HOLogic.mk_setT
  1342       (HOLogic.mk_prodT (T1, permTs ---> T2))) (recTs ~~ rec_result_Ts);
  1343 
  1344     val big_rec_name = big_name ^ "_rec_set";
  1345     val rec_set_names = map (Sign.full_name (Theory.sign_of thy10))
  1346       (if length descr'' = 1 then [big_rec_name] else
  1347         (map ((curry (op ^) (big_rec_name ^ "_")) o string_of_int)
  1348           (1 upto (length descr''))));
  1349 
  1350     val rec_fns = map (uncurry (mk_Free "f"))
  1351       (rec_fn_Ts ~~ (1 upto (length rec_fn_Ts)));
  1352     val rec_sets = map (fn c => list_comb (Const c, rec_fns))
  1353       (rec_set_names ~~ rec_set_Ts);
  1354 
  1355     (* introduction rules for graph of recursion function *)
  1356 
  1357     fun partition_cargs idxs xs = map (fn (i, j) =>
  1358       (List.take (List.drop (xs, i), j), List.nth (xs, i + j))) idxs;
  1359 
  1360     fun mk_fresh_fun (a, t) = Const ("Nominal.fresh_fun",
  1361       (fastype_of a --> fastype_of t) --> fastype_of t) $ lambda a t;
  1362 
  1363     fun make_rec_intr T rec_set ((rec_intr_ts, l), ((cname, cargs), idxs)) =
  1364       let
  1365         fun mk_prem ((dts, (dt, U)), (j, k, prems, t1s, t2s, t3s, atoms)) =
  1366           let
  1367             val free1 = mk_Free "x" U (j + length dts);
  1368             val Us = map snd dts;
  1369             val xs = Us ~~ (j upto j + length dts - 1);
  1370             val frees = map (uncurry (mk_Free "x")) xs;
  1371             val frees' = map (uncurry (mk_Free "x'")) xs;
  1372             val frees'' = Us ~~ (frees ~~ frees');
  1373             val pis = map (fn (T, p) =>
  1374               case filter (equal T o fst) frees'' of
  1375                 [] => p
  1376               | xs => HOLogic.mk_binop "List.op @" (p,
  1377                 HOLogic.mk_list (HOLogic.mk_prod o snd)
  1378                   (HOLogic.mk_prodT (T, T)) xs))
  1379                   (dt_atomTs ~~ perms)
  1380           in (case dt of
  1381              DtRec m =>
  1382                let val free2 = mk_Free "y"
  1383                  (permTs ---> List.nth (rec_result_Ts, m)) k
  1384                in (j + length dts + 1, k + 1,
  1385                    HOLogic.mk_Trueprop
  1386                      (HOLogic.mk_mem (HOLogic.mk_prod
  1387                        (free1, free2),
  1388                          List.nth (rec_sets, m))) :: prems,
  1389                    frees @ free1 :: t1s,
  1390                    frees' @ foldr (mk_perm []) free1 pis :: t2s,
  1391                    list_comb (free2, pis) :: t3s,
  1392                    frees' @ atoms)
  1393                end
  1394            | _ => (j + length dts + 1, k, prems,
  1395                frees @ free1 :: t1s,
  1396                frees' @ foldr (mk_perm []) free1 pis :: t2s,
  1397                t3s,
  1398                frees' @ atoms))
  1399           end;
  1400 
  1401         val Ts = map (typ_of_dtyp descr'' sorts') cargs;
  1402         val (_, _, prems, t1s, t2s, t3s, atoms) = foldr mk_prem (1, 1, [], [], [], [], [])
  1403           (partition_cargs idxs (cargs ~~ Ts))
  1404 
  1405       in (rec_intr_ts @ [Logic.list_implies (prems, HOLogic.mk_Trueprop (HOLogic.mk_mem
  1406         (HOLogic.mk_prod (list_comb (Const (cname, Ts ---> T), t1s),
  1407           foldr (uncurry lambda)
  1408             (foldr mk_fresh_fun
  1409               (list_comb (List.nth (rec_fns, l), t2s @ t3s)) atoms)
  1410             perms), rec_set)))], l + 1)
  1411       end;
  1412 
  1413     val (rec_intr_ts, _) = Library.foldl (fn (x, (((d, d'), T), rec_set)) =>
  1414       Library.foldl (make_rec_intr T rec_set) (x, #3 (snd d) ~~ snd d'))
  1415         (([], 0), descr'' ~~ ndescr ~~ recTs ~~ rec_sets);
  1416 
  1417     val (thy11, {intrs = rec_intrs, elims = rec_elims, ...}) =
  1418       setmp InductivePackage.quiet_mode (!quiet_mode)
  1419         (InductivePackage.add_inductive_i false true big_rec_name false false false
  1420            rec_sets (map (fn x => (("", x), [])) rec_intr_ts) []) thy10;
  1421 
  1422   in
  1423     thy11
  1424   end;
  1425 
  1426 val add_nominal_datatype = gen_add_nominal_datatype read_typ true;
  1427 
  1428 
  1429 (* FIXME: The following stuff should be exported by DatatypePackage *)
  1430 
  1431 local structure P = OuterParse and K = OuterKeyword in
  1432 
  1433 val datatype_decl =
  1434   Scan.option (P.$$$ "(" |-- P.name --| P.$$$ ")") -- P.type_args -- P.name -- P.opt_infix --
  1435     (P.$$$ "=" |-- P.enum1 "|" (P.name -- Scan.repeat P.typ -- P.opt_mixfix));
  1436 
  1437 fun mk_datatype args =
  1438   let
  1439     val names = map (fn ((((NONE, _), t), _), _) => t | ((((SOME t, _), _), _), _) => t) args;
  1440     val specs = map (fn ((((_, vs), t), mx), cons) =>
  1441       (vs, t, mx, map (fn ((x, y), z) => (x, y, z)) cons)) args;
  1442   in add_nominal_datatype false names specs end;
  1443 
  1444 val nominal_datatypeP =
  1445   OuterSyntax.command "nominal_datatype" "define inductive datatypes" K.thy_decl
  1446     (P.and_list1 datatype_decl >> (Toplevel.theory o mk_datatype));
  1447 
  1448 val _ = OuterSyntax.add_parsers [nominal_datatypeP];
  1449 
  1450 end;
  1451 
  1452 end
  1453