src/HOL/Nominal/nominal_package.ML
author berghofe
Wed Nov 02 16:37:39 2005 +0100 (2005-11-02)
changeset 18068 e8c3d371594e
parent 18067 8b9848d150ba
child 18104 dbe58b104cb9
permissions -rw-r--r--
Moved atom stuff to new file nominal_atoms.ML
     1 (* $Id$ *)
     2 
     3 signature NOMINAL_PACKAGE =
     4 sig
     5   val add_nominal_datatype : bool -> string list -> (string list * bstring * mixfix *
     6     (bstring * string list * mixfix) list) list -> theory -> theory
     7 end
     8 
     9 structure NominalPackage : NOMINAL_PACKAGE =
    10 struct
    11 
    12 open DatatypeAux;
    13 open NominalAtoms;
    14 
    15 (** FIXME: DatatypePackage should export this function **)
    16 
    17 local
    18 
    19 fun dt_recs (DtTFree _) = []
    20   | dt_recs (DtType (_, dts)) = List.concat (map dt_recs dts)
    21   | dt_recs (DtRec i) = [i];
    22 
    23 fun dt_cases (descr: descr) (_, args, constrs) =
    24   let
    25     fun the_bname i = Sign.base_name (#1 (valOf (AList.lookup (op =) descr i)));
    26     val bnames = map the_bname (distinct (List.concat (map dt_recs args)));
    27   in map (fn (c, _) => space_implode "_" (Sign.base_name c :: bnames)) constrs end;
    28 
    29 
    30 fun induct_cases descr =
    31   DatatypeProp.indexify_names (List.concat (map (dt_cases descr) (map #2 descr)));
    32 
    33 fun exhaust_cases descr i = dt_cases descr (valOf (AList.lookup (op =) descr i));
    34 
    35 in
    36 
    37 fun mk_case_names_induct descr = RuleCases.case_names (induct_cases descr);
    38 
    39 fun mk_case_names_exhausts descr new =
    40   map (RuleCases.case_names o exhaust_cases descr o #1)
    41     (List.filter (fn ((_, (name, _, _))) => name mem_string new) descr);
    42 
    43 end;
    44 
    45 (*******************************)
    46 
    47 val (_ $ (_ $ (_ $ (distinct_f $ _) $ _))) = hd (prems_of distinct_lemma);
    48 
    49 fun read_typ sign ((Ts, sorts), str) =
    50   let
    51     val T = Type.no_tvars (Sign.read_typ (sign, (AList.lookup op =)
    52       (map (apfst (rpair ~1)) sorts)) str) handle TYPE (msg, _, _) => error msg
    53   in (Ts @ [T], add_typ_tfrees (T, sorts)) end;
    54 
    55 (** taken from HOL/Tools/datatype_aux.ML **)
    56 
    57 fun indtac indrule indnames i st =
    58   let
    59     val ts = HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of indrule));
    60     val ts' = HOLogic.dest_conj (HOLogic.dest_Trueprop
    61       (Logic.strip_imp_concl (List.nth (prems_of st, i - 1))));
    62     val getP = if can HOLogic.dest_imp (hd ts) then
    63       (apfst SOME) o HOLogic.dest_imp else pair NONE;
    64     fun abstr (t1, t2) = (case t1 of
    65         NONE => (case filter (fn Free (s, _) => s mem indnames | _ => false)
    66               (term_frees t2) of
    67             [Free (s, T)] => absfree (s, T, t2)
    68           | _ => sys_error "indtac")
    69       | SOME (_ $ t' $ _) => Abs ("x", fastype_of t', abstract_over (t', t2)))
    70     val cert = cterm_of (Thm.sign_of_thm st);
    71     val Ps = map (cert o head_of o snd o getP) ts;
    72     val indrule' = cterm_instantiate (Ps ~~
    73       (map (cert o abstr o getP) ts')) indrule
    74   in
    75     rtac indrule' i st
    76   end;
    77 
    78 fun gen_add_nominal_datatype prep_typ err flat_names new_type_names dts thy =
    79   let
    80     (* this theory is used just for parsing *)
    81   
    82     val tmp_thy = thy |>
    83       Theory.copy |>
    84       Theory.add_types (map (fn (tvs, tname, mx, _) =>
    85         (tname, length tvs, mx)) dts);
    86 
    87     val sign = Theory.sign_of tmp_thy;
    88 
    89     val atoms = atoms_of thy;
    90     val classes = map (NameSpace.map_base (fn s => "pt_" ^ s)) atoms;
    91     val cp_classes = List.concat (map (fn atom1 => map (fn atom2 =>
    92       Sign.intern_class thy ("cp_" ^ Sign.base_name atom1 ^ "_" ^
    93         Sign.base_name atom2)) atoms) atoms);
    94     fun augment_sort S = S union classes;
    95     val augment_sort_typ = map_type_tfree (fn (s, S) => TFree (s, augment_sort S));
    96 
    97     fun prep_constr ((constrs, sorts), (cname, cargs, mx)) =
    98       let val (cargs', sorts') = Library.foldl (prep_typ sign) (([], sorts), cargs)
    99       in (constrs @ [(cname, cargs', mx)], sorts') end
   100 
   101     fun prep_dt_spec ((dts, sorts), (tvs, tname, mx, constrs)) =
   102       let val (constrs', sorts') = Library.foldl prep_constr (([], sorts), constrs)
   103       in (dts @ [(tvs, tname, mx, constrs')], sorts') end
   104 
   105     val (dts', sorts) = Library.foldl prep_dt_spec (([], []), dts);
   106     val sorts' = map (apsnd augment_sort) sorts;
   107     val tyvars = map #1 dts';
   108 
   109     val types_syntax = map (fn (tvs, tname, mx, constrs) => (tname, mx)) dts';
   110     val constr_syntax = map (fn (tvs, tname, mx, constrs) =>
   111       map (fn (cname, cargs, mx) => (cname, mx)) constrs) dts';
   112 
   113     val ps = map (fn (_, n, _, _) =>
   114       (Sign.full_name sign n, Sign.full_name sign (n ^ "_Rep"))) dts;
   115     val rps = map Library.swap ps;
   116 
   117     fun replace_types (Type ("nominal.ABS", [T, U])) = 
   118           Type ("fun", [T, Type ("nominal.nOption", [replace_types U])])
   119       | replace_types (Type (s, Ts)) =
   120           Type (getOpt (AList.lookup op = ps s, s), map replace_types Ts)
   121       | replace_types T = T;
   122 
   123     fun replace_types' (Type (s, Ts)) =
   124           Type (getOpt (AList.lookup op = rps s, s), map replace_types' Ts)
   125       | replace_types' T = T;
   126 
   127     val dts'' = map (fn (tvs, tname, mx, constrs) => (tvs, tname ^ "_Rep", NoSyn,
   128       map (fn (cname, cargs, mx) => (cname,
   129         map (augment_sort_typ o replace_types) cargs, NoSyn)) constrs)) dts';
   130 
   131     val new_type_names' = map (fn n => n ^ "_Rep") new_type_names;
   132     val full_new_type_names' = map (Sign.full_name (sign_of thy)) new_type_names';
   133 
   134     val ({induction, ...},thy1) =
   135       DatatypePackage.add_datatype_i err flat_names new_type_names' dts'' thy;
   136 
   137     val SOME {descr, ...} = Symtab.lookup
   138       (DatatypePackage.get_datatypes thy1) (hd full_new_type_names');
   139     val typ_of_dtyp = typ_of_dtyp descr sorts';
   140     fun nth_dtyp i = typ_of_dtyp (DtRec i);
   141 
   142     (**** define permutation functions ****)
   143 
   144     val permT = mk_permT (TFree ("'x", HOLogic.typeS));
   145     val pi = Free ("pi", permT);
   146     val perm_types = map (fn (i, _) =>
   147       let val T = nth_dtyp i
   148       in permT --> T --> T end) descr;
   149     val perm_names = replicate (length new_type_names) "nominal.perm" @
   150       DatatypeProp.indexify_names (map (fn i => Sign.full_name (sign_of thy1)
   151         ("perm_" ^ name_of_typ (nth_dtyp i)))
   152           (length new_type_names upto length descr - 1));
   153     val perm_names_types = perm_names ~~ perm_types;
   154 
   155     val perm_eqs = List.concat (map (fn (i, (_, _, constrs)) =>
   156       let val T = nth_dtyp i
   157       in map (fn (cname, dts) => 
   158         let
   159           val Ts = map typ_of_dtyp dts;
   160           val names = DatatypeProp.make_tnames Ts;
   161           val args = map Free (names ~~ Ts);
   162           val c = Const (cname, Ts ---> T);
   163           fun perm_arg (dt, x) =
   164             let val T = type_of x
   165             in if is_rec_type dt then
   166                 let val (Us, _) = strip_type T
   167                 in list_abs (map (pair "x") Us,
   168                   Const (List.nth (perm_names_types, body_index dt)) $ pi $
   169                     list_comb (x, map (fn (i, U) =>
   170                       Const ("nominal.perm", permT --> U --> U) $
   171                         (Const ("List.rev", permT --> permT) $ pi) $
   172                         Bound i) ((length Us - 1 downto 0) ~~ Us)))
   173                 end
   174               else Const ("nominal.perm", permT --> T --> T) $ pi $ x
   175             end;  
   176         in
   177           (("", HOLogic.mk_Trueprop (HOLogic.mk_eq
   178             (Const (List.nth (perm_names_types, i)) $
   179                Free ("pi", mk_permT (TFree ("'x", HOLogic.typeS))) $
   180                list_comb (c, args),
   181              list_comb (c, map perm_arg (dts ~~ args))))), [])
   182         end) constrs
   183       end) descr);
   184 
   185     val (thy2, perm_simps) = thy1 |>
   186       Theory.add_consts_i (map (fn (s, T) => (Sign.base_name s, T, NoSyn))
   187         (List.drop (perm_names_types, length new_type_names))) |>
   188       PrimrecPackage.add_primrec_i "" perm_eqs;
   189 
   190     (**** prove that permutation functions introduced by unfolding are ****)
   191     (**** equivalent to already existing permutation functions         ****)
   192 
   193     val _ = warning ("length descr: " ^ string_of_int (length descr));
   194     val _ = warning ("length new_type_names: " ^ string_of_int (length new_type_names));
   195 
   196     val perm_indnames = DatatypeProp.make_tnames (map body_type perm_types);
   197     val perm_fun_def = PureThy.get_thm thy2 (Name "perm_fun_def");
   198 
   199     val unfolded_perm_eq_thms =
   200       if length descr = length new_type_names then []
   201       else map standard (List.drop (split_conj_thm
   202         (Goal.prove thy2 [] []
   203           (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
   204             (map (fn (c as (s, T), x) =>
   205                let val [T1, T2] = binder_types T
   206                in HOLogic.mk_eq (Const c $ pi $ Free (x, T2),
   207                  Const ("nominal.perm", T) $ pi $ Free (x, T2))
   208                end)
   209              (perm_names_types ~~ perm_indnames))))
   210           (fn _ => EVERY [indtac induction perm_indnames 1,
   211             ALLGOALS (asm_full_simp_tac
   212               (simpset_of thy2 addsimps [perm_fun_def]))])),
   213         length new_type_names));
   214 
   215     (**** prove [] \<bullet> t = t ****)
   216 
   217     val _ = warning "perm_empty_thms";
   218 
   219     val perm_empty_thms = List.concat (map (fn a =>
   220       let val permT = mk_permT (Type (a, []))
   221       in map standard (List.take (split_conj_thm
   222         (Goal.prove thy2 [] []
   223           (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
   224             (map (fn ((s, T), x) => HOLogic.mk_eq
   225                 (Const (s, permT --> T --> T) $
   226                    Const ("List.list.Nil", permT) $ Free (x, T),
   227                  Free (x, T)))
   228              (perm_names ~~
   229               map body_type perm_types ~~ perm_indnames))))
   230           (fn _ => EVERY [indtac induction perm_indnames 1,
   231             ALLGOALS (asm_full_simp_tac (simpset_of thy2))])),
   232         length new_type_names))
   233       end)
   234       atoms);
   235 
   236     (**** prove (pi1 @ pi2) \<bullet> t = pi1 \<bullet> (pi2 \<bullet> t) ****)
   237 
   238     val _ = warning "perm_append_thms";
   239 
   240     (*FIXME: these should be looked up statically*)
   241     val at_pt_inst = PureThy.get_thm thy2 (Name "at_pt_inst");
   242     val pt2 = PureThy.get_thm thy2 (Name "pt2");
   243 
   244     val perm_append_thms = List.concat (map (fn a =>
   245       let
   246         val permT = mk_permT (Type (a, []));
   247         val pi1 = Free ("pi1", permT);
   248         val pi2 = Free ("pi2", permT);
   249         val pt_inst = PureThy.get_thm thy2 (Name ("pt_" ^ Sign.base_name a ^ "_inst"));
   250         val pt2' = pt_inst RS pt2;
   251         val pt2_ax = PureThy.get_thm thy2
   252           (Name (NameSpace.map_base (fn s => "pt_" ^ s ^ "2") a));
   253       in List.take (map standard (split_conj_thm
   254         (Goal.prove thy2 [] []
   255              (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
   256                 (map (fn ((s, T), x) =>
   257                     let val perm = Const (s, permT --> T --> T)
   258                     in HOLogic.mk_eq
   259                       (perm $ (Const ("List.op @", permT --> permT --> permT) $
   260                          pi1 $ pi2) $ Free (x, T),
   261                        perm $ pi1 $ (perm $ pi2 $ Free (x, T)))
   262                     end)
   263                   (perm_names ~~
   264                    map body_type perm_types ~~ perm_indnames))))
   265            (fn _ => EVERY [indtac induction perm_indnames 1,
   266               ALLGOALS (asm_full_simp_tac (simpset_of thy2 addsimps [pt2', pt2_ax]))]))),
   267          length new_type_names)
   268       end) atoms);
   269 
   270     (**** prove pi1 ~ pi2 ==> pi1 \<bullet> t = pi2 \<bullet> t ****)
   271 
   272     val _ = warning "perm_eq_thms";
   273 
   274     val pt3 = PureThy.get_thm thy2 (Name "pt3");
   275     val pt3_rev = PureThy.get_thm thy2 (Name "pt3_rev");
   276 
   277     val perm_eq_thms = List.concat (map (fn a =>
   278       let
   279         val permT = mk_permT (Type (a, []));
   280         val pi1 = Free ("pi1", permT);
   281         val pi2 = Free ("pi2", permT);
   282         (*FIXME: not robust - better access these theorems using NominalData?*)
   283         val at_inst = PureThy.get_thm thy2 (Name ("at_" ^ Sign.base_name a ^ "_inst"));
   284         val pt_inst = PureThy.get_thm thy2 (Name ("pt_" ^ Sign.base_name a ^ "_inst"));
   285         val pt3' = pt_inst RS pt3;
   286         val pt3_rev' = at_inst RS (pt_inst RS pt3_rev);
   287         val pt3_ax = PureThy.get_thm thy2
   288           (Name (NameSpace.map_base (fn s => "pt_" ^ s ^ "3") a));
   289       in List.take (map standard (split_conj_thm
   290         (Goal.prove thy2 [] [] (Logic.mk_implies
   291              (HOLogic.mk_Trueprop (Const ("nominal.prm_eq",
   292                 permT --> permT --> HOLogic.boolT) $ pi1 $ pi2),
   293               HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
   294                 (map (fn ((s, T), x) =>
   295                     let val perm = Const (s, permT --> T --> T)
   296                     in HOLogic.mk_eq
   297                       (perm $ pi1 $ Free (x, T),
   298                        perm $ pi2 $ Free (x, T))
   299                     end)
   300                   (perm_names ~~
   301                    map body_type perm_types ~~ perm_indnames)))))
   302            (fn _ => EVERY [indtac induction perm_indnames 1,
   303               ALLGOALS (asm_full_simp_tac (simpset_of thy2 addsimps [pt3', pt3_rev', pt3_ax]))]))),
   304          length new_type_names)
   305       end) atoms);
   306 
   307     (**** prove pi1 \<bullet> (pi2 \<bullet> t) = (pi1 \<bullet> pi2) \<bullet> (pi1 \<bullet> t) ****)
   308 
   309     val cp1 = PureThy.get_thm thy2 (Name "cp1");
   310     val dj_cp = PureThy.get_thm thy2 (Name "dj_cp");
   311     val pt_perm_compose = PureThy.get_thm thy2 (Name "pt_perm_compose");
   312     val pt_perm_compose_rev = PureThy.get_thm thy2 (Name "pt_perm_compose_rev");
   313     val dj_perm_perm_forget = PureThy.get_thm thy2 (Name "dj_perm_perm_forget");
   314 
   315     fun composition_instance name1 name2 thy =
   316       let
   317         val name1' = Sign.base_name name1;
   318         val name2' = Sign.base_name name2;
   319         val cp_class = Sign.intern_class thy ("cp_" ^ name1' ^ "_" ^ name2');
   320         val permT1 = mk_permT (Type (name1, []));
   321         val permT2 = mk_permT (Type (name2, []));
   322         val augment = map_type_tfree
   323           (fn (x, S) => TFree (x, cp_class :: S));
   324         val Ts = map (augment o body_type) perm_types;
   325         val cp_inst = PureThy.get_thm thy
   326           (Name ("cp_" ^ name1' ^ "_" ^ name2' ^ "_inst"));
   327         val simps = simpset_of thy addsimps (perm_fun_def ::
   328           (if name1 <> name2 then
   329              let val dj = PureThy.get_thm thy (Name ("dj_" ^ name2' ^ "_" ^ name1'))
   330              in [dj RS (cp_inst RS dj_cp), dj RS dj_perm_perm_forget] end
   331            else
   332              let
   333                val at_inst = PureThy.get_thm thy (Name ("at_" ^ name1' ^ "_inst"));
   334                val pt_inst = PureThy.get_thm thy (Name ("pt_" ^ name1' ^ "_inst"))
   335              in
   336                [cp_inst RS cp1 RS sym,
   337                 at_inst RS (pt_inst RS pt_perm_compose) RS sym,
   338                 at_inst RS (pt_inst RS pt_perm_compose_rev) RS sym]
   339             end))
   340         val thms = split_conj_thm (standard (Goal.prove thy [] []
   341             (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
   342               (map (fn ((s, T), x) =>
   343                   let
   344                     val pi1 = Free ("pi1", permT1);
   345                     val pi2 = Free ("pi2", permT2);
   346                     val perm1 = Const (s, permT1 --> T --> T);
   347                     val perm2 = Const (s, permT2 --> T --> T);
   348                     val perm3 = Const ("nominal.perm", permT1 --> permT2 --> permT2)
   349                   in HOLogic.mk_eq
   350                     (perm1 $ pi1 $ (perm2 $ pi2 $ Free (x, T)),
   351                      perm2 $ (perm3 $ pi1 $ pi2) $ (perm1 $ pi1 $ Free (x, T)))
   352                   end)
   353                 (perm_names ~~ Ts ~~ perm_indnames))))
   354           (fn _ => EVERY [indtac induction perm_indnames 1,
   355              ALLGOALS (asm_full_simp_tac simps)])))
   356       in
   357         foldl (fn ((s, tvs), thy) => AxClass.add_inst_arity_i
   358             (s, replicate (length tvs) (cp_class :: classes), [cp_class])
   359             (AxClass.intro_classes_tac [] THEN ALLGOALS (resolve_tac thms)) thy)
   360           thy (full_new_type_names' ~~ tyvars)
   361       end;
   362 
   363     val (thy3, perm_thmss) = thy2 |>
   364       fold (fn name1 => fold (composition_instance name1) atoms) atoms |>
   365       curry (Library.foldr (fn ((i, (tyname, args, _)), thy) =>
   366         AxClass.add_inst_arity_i (tyname, replicate (length args) classes, classes)
   367         (AxClass.intro_classes_tac [] THEN REPEAT (EVERY
   368            [resolve_tac perm_empty_thms 1,
   369             resolve_tac perm_append_thms 1,
   370             resolve_tac perm_eq_thms 1, assume_tac 1])) thy))
   371         (List.take (descr, length new_type_names)) |>
   372       PureThy.add_thmss
   373         [((space_implode "_" new_type_names ^ "_unfolded_perm_eq",
   374           unfolded_perm_eq_thms), [Simplifier.simp_add_global]),
   375          ((space_implode "_" new_type_names ^ "_perm_empty",
   376           perm_empty_thms), [Simplifier.simp_add_global]),
   377          ((space_implode "_" new_type_names ^ "_perm_append",
   378           perm_append_thms), [Simplifier.simp_add_global]),
   379          ((space_implode "_" new_type_names ^ "_perm_eq",
   380           perm_eq_thms), [Simplifier.simp_add_global])];
   381   
   382     (**** Define representing sets ****)
   383 
   384     val _ = warning "representing sets";
   385 
   386     val rep_set_names = map (Sign.full_name thy3) (DatatypeProp.indexify_names
   387       (map (fn (i, _) => name_of_typ (nth_dtyp i) ^ "_set") descr));
   388     val big_rep_name =
   389       space_implode "_" (DatatypeProp.indexify_names (List.mapPartial
   390         (fn (i, ("nominal.nOption", _, _)) => NONE
   391           | (i, _) => SOME (name_of_typ (nth_dtyp i))) descr)) ^ "_set";
   392     val _ = warning ("big_rep_name: " ^ big_rep_name);
   393 
   394     fun strip_option (dtf as DtType ("fun", [dt, DtRec i])) =
   395           (case AList.lookup op = descr i of
   396              SOME ("nominal.nOption", _, [(_, [dt']), _]) =>
   397                apfst (cons dt) (strip_option dt')
   398            | _ => ([], dtf))
   399       | strip_option dt = ([], dt);
   400 
   401     fun make_intr s T (cname, cargs) =
   402       let
   403         fun mk_prem (dt, (j, j', prems, ts)) = 
   404           let
   405             val (dts, dt') = strip_option dt;
   406             val (dts', dt'') = strip_dtyp dt';
   407             val Ts = map typ_of_dtyp dts;
   408             val Us = map typ_of_dtyp dts';
   409             val T = typ_of_dtyp dt'';
   410             val free = mk_Free "x" (Us ---> T) j;
   411             val free' = app_bnds free (length Us);
   412             fun mk_abs_fun (T, (i, t)) =
   413               let val U = fastype_of t
   414               in (i + 1, Const ("nominal.abs_fun", [T, U, T] --->
   415                 Type ("nominal.nOption", [U])) $ mk_Free "y" T i $ t)
   416               end
   417           in (j + 1, j' + length Ts,
   418             case dt'' of
   419                 DtRec k => list_all (map (pair "x") Us,
   420                   HOLogic.mk_Trueprop (HOLogic.mk_mem (free',
   421                     Const (List.nth (rep_set_names, k),
   422                       HOLogic.mk_setT T)))) :: prems
   423               | _ => prems,
   424             snd (foldr mk_abs_fun (j', free) Ts) :: ts)
   425           end;
   426 
   427         val (_, _, prems, ts) = foldr mk_prem (1, 1, [], []) cargs;
   428         val concl = HOLogic.mk_Trueprop (HOLogic.mk_mem
   429           (list_comb (Const (cname, map fastype_of ts ---> T), ts),
   430            Const (s, HOLogic.mk_setT T)))
   431       in Logic.list_implies (prems, concl)
   432       end;
   433 
   434     val (intr_ts, ind_consts) =
   435       apfst List.concat (ListPair.unzip (List.mapPartial
   436         (fn ((_, ("nominal.nOption", _, _)), _) => NONE
   437           | ((i, (_, _, constrs)), rep_set_name) =>
   438               let val T = nth_dtyp i
   439               in SOME (map (make_intr rep_set_name T) constrs,
   440                 Const (rep_set_name, HOLogic.mk_setT T))
   441               end)
   442                 (descr ~~ rep_set_names)));
   443 
   444     val (thy4, {raw_induct = rep_induct, intrs = rep_intrs, ...}) =
   445       setmp InductivePackage.quiet_mode false
   446         (InductivePackage.add_inductive_i false true big_rep_name false true false
   447            ind_consts (map (fn x => (("", x), [])) intr_ts) []) thy3;
   448 
   449     (**** Prove that representing set is closed under permutation ****)
   450 
   451     val _ = warning "proving closure under permutation...";
   452 
   453     val perm_indnames' = List.mapPartial
   454       (fn (x, (_, ("nominal.nOption", _, _))) => NONE | (x, _) => SOME x)
   455       (perm_indnames ~~ descr);
   456 
   457     fun mk_perm_closed name = map (fn th => standard (th RS mp))
   458       (List.take (split_conj_thm (Goal.prove thy4 [] []
   459         (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj (map
   460            (fn (S, x) =>
   461               let
   462                 val S = map_term_types (map_type_tfree
   463                   (fn (s, cs) => TFree (s, cs union cp_classes))) S;
   464                 val T = HOLogic.dest_setT (fastype_of S);
   465                 val permT = mk_permT (Type (name, []))
   466               in HOLogic.mk_imp (HOLogic.mk_mem (Free (x, T), S),
   467                 HOLogic.mk_mem (Const ("nominal.perm", permT --> T --> T) $
   468                   Free ("pi", permT) $ Free (x, T), S))
   469               end) (ind_consts ~~ perm_indnames'))))
   470         (fn _ => EVERY (* CU: added perm_fun_def in the final tactic in order to deal with funs *)
   471            [indtac rep_induct [] 1,
   472             ALLGOALS (simp_tac (simpset_of thy4 addsimps
   473               (symmetric perm_fun_def :: PureThy.get_thms thy4 (Name ("abs_perm"))))),
   474             ALLGOALS (resolve_tac rep_intrs 
   475                THEN_ALL_NEW (asm_full_simp_tac (simpset_of thy4 addsimps [perm_fun_def])))])),
   476         length new_type_names));
   477 
   478     (* FIXME: theorems are stored in database for testing only *)
   479     val perm_closed_thmss = map mk_perm_closed atoms;
   480     val (thy5, _) = PureThy.add_thmss [(("perm_closed",
   481       List.concat perm_closed_thmss), [])] thy4;
   482 
   483     (**** typedef ****)
   484 
   485     val _ = warning "defining type...";
   486 
   487     val (thy6, typedefs) =
   488       foldl_map (fn (thy, ((((name, mx), tvs), c), name')) =>
   489         setmp TypedefPackage.quiet_mode true
   490           (TypedefPackage.add_typedef_i false (SOME name') (name, tvs, mx) c NONE
   491             (rtac exI 1 THEN
   492               QUIET_BREADTH_FIRST (has_fewer_prems 1)
   493               (resolve_tac rep_intrs 1))) thy |> (fn (thy, r) =>
   494         let
   495           val permT = mk_permT (TFree (variant tvs "'a", HOLogic.typeS));
   496           val pi = Free ("pi", permT);
   497           val tvs' = map (fn s => TFree (s, the (AList.lookup op = sorts' s))) tvs;
   498           val T = Type (Sign.intern_type thy name, tvs');
   499           val Const (_, Type (_, [U])) = c
   500         in apsnd (pair r o hd)
   501           (PureThy.add_defs_i true [(("prm_" ^ name ^ "_def", Logic.mk_equals
   502             (Const ("nominal.perm", permT --> T --> T) $ pi $ Free ("x", T),
   503              Const (Sign.intern_const thy ("Abs_" ^ name), U --> T) $
   504                (Const ("nominal.perm", permT --> U --> U) $ pi $
   505                  (Const (Sign.intern_const thy ("Rep_" ^ name), T --> U) $
   506                    Free ("x", T))))), [])] thy)
   507         end))
   508           (thy5, types_syntax ~~ tyvars ~~
   509             (List.take (ind_consts, length new_type_names)) ~~ new_type_names);
   510 
   511     val perm_defs = map snd typedefs;
   512     val Abs_inverse_thms = map (#Abs_inverse o fst) typedefs;
   513     val Rep_inverse_thms = map (#Rep_inverse o fst) typedefs;
   514     val Rep_thms = map (#Rep o fst) typedefs;
   515 
   516     val big_name = space_implode "_" new_type_names;
   517 
   518 
   519     (** prove that new types are in class pt_<name> **)
   520 
   521     val _ = warning "prove that new types are in class pt_<name> ...";
   522 
   523     fun pt_instance ((class, atom), perm_closed_thms) =
   524       fold (fn (((({Abs_inverse, Rep_inverse, Rep, ...},
   525         perm_def), name), tvs), perm_closed) => fn thy =>
   526           AxClass.add_inst_arity_i
   527             (Sign.intern_type thy name,
   528               replicate (length tvs) (classes @ cp_classes), [class])
   529             (EVERY [AxClass.intro_classes_tac [],
   530               rewrite_goals_tac [perm_def],
   531               asm_full_simp_tac (simpset_of thy addsimps [Rep_inverse]) 1,
   532               asm_full_simp_tac (simpset_of thy addsimps
   533                 [Rep RS perm_closed RS Abs_inverse]) 1,
   534               asm_full_simp_tac (HOL_basic_ss addsimps [PureThy.get_thm thy
   535                 (Name ("pt_" ^ Sign.base_name atom ^ "3"))]) 1]) thy)
   536         (typedefs ~~ new_type_names ~~ tyvars ~~ perm_closed_thms);
   537 
   538 
   539     (** prove that new types are in class cp_<name1>_<name2> **)
   540 
   541     val _ = warning "prove that new types are in class cp_<name1>_<name2> ...";
   542 
   543     fun cp_instance (atom1, perm_closed_thms1) (atom2, perm_closed_thms2) thy =
   544       let
   545         val name = "cp_" ^ Sign.base_name atom1 ^ "_" ^ Sign.base_name atom2;
   546         val class = Sign.intern_class thy name;
   547         val cp1' = PureThy.get_thm thy (Name (name ^ "_inst")) RS cp1
   548       in fold (fn ((((({Abs_inverse, Rep_inverse, Rep, ...},
   549         perm_def), name), tvs), perm_closed1), perm_closed2) => fn thy =>
   550           AxClass.add_inst_arity_i
   551             (Sign.intern_type thy name,
   552               replicate (length tvs) (classes @ cp_classes), [class])
   553             (EVERY [AxClass.intro_classes_tac [],
   554               rewrite_goals_tac [perm_def],
   555               asm_full_simp_tac (simpset_of thy addsimps
   556                 ((Rep RS perm_closed1 RS Abs_inverse) ::
   557                  (if atom1 = atom2 then []
   558                   else [Rep RS perm_closed2 RS Abs_inverse]))) 1,
   559               cong_tac 1,
   560               rtac refl 1,
   561               rtac cp1' 1]) thy)
   562         (typedefs ~~ new_type_names ~~ tyvars ~~ perm_closed_thms1 ~~
   563           perm_closed_thms2) thy
   564       end;
   565 
   566     val thy7 = fold (fn x => fn thy => thy |>
   567       pt_instance x |>
   568       fold (cp_instance (apfst snd x)) (atoms ~~ perm_closed_thmss))
   569         (classes ~~ atoms ~~ perm_closed_thmss) thy6;
   570 
   571     (**** constructors ****)
   572 
   573     fun mk_abs_fun (x, t) =
   574       let
   575         val T = fastype_of x;
   576         val U = fastype_of t
   577       in
   578         Const ("nominal.abs_fun", T --> U --> T -->
   579           Type ("nominal.nOption", [U])) $ x $ t
   580       end;
   581 
   582     val (ty_idxs, _) = foldl
   583       (fn ((i, ("nominal.nOption", _, _)), p) => p
   584         | ((i, _), (ty_idxs, j)) => (ty_idxs @ [(i, j)], j + 1)) ([], 0) descr;
   585 
   586     fun reindex (DtType (s, dts)) = DtType (s, map reindex dts)
   587       | reindex (DtRec i) = DtRec (the (AList.lookup op = ty_idxs i))
   588       | reindex dt = dt;
   589 
   590     fun strip_suffix i s = implode (List.take (explode s, size s - i));
   591 
   592     (** strips the "_Rep" in type names *)
   593     fun strip_nth_name i s = 
   594       let val xs = NameSpace.unpack s; 
   595       in NameSpace.pack (Library.nth_map (length xs - i) (strip_suffix 4) xs) end;
   596 
   597     val descr'' = List.mapPartial
   598       (fn (i, ("nominal.nOption", _, _)) => NONE
   599         | (i, (s, dts, constrs)) => SOME (the (AList.lookup op = ty_idxs i),
   600             (strip_nth_name 1 s,  map reindex dts,
   601              map (fn (cname, cargs) =>
   602                (strip_nth_name 2 cname,
   603                 foldl (fn (dt, dts) =>
   604                   let val (dts', dt') = strip_option dt
   605                   in (dts @ dts' @ [reindex dt']) end) [] cargs)) constrs))) descr;
   606 
   607     val (descr1, descr2) = splitAt (length new_type_names, descr'');
   608     val descr' = [descr1, descr2];
   609 
   610     val typ_of_dtyp' = replace_types' o typ_of_dtyp;
   611 
   612     val rep_names = map (fn s =>
   613       Sign.intern_const thy7 ("Rep_" ^ s)) new_type_names;
   614     val abs_names = map (fn s =>
   615       Sign.intern_const thy7 ("Abs_" ^ s)) new_type_names;
   616 
   617     val recTs' = List.mapPartial
   618       (fn ((_, ("nominal.nOption", _, _)), T) => NONE
   619         | (_, T) => SOME T) (descr ~~ get_rec_types descr sorts');
   620     val recTs = get_rec_types (List.concat descr') sorts';
   621     val newTs' = Library.take (length new_type_names, recTs');
   622     val newTs = Library.take (length new_type_names, recTs);
   623 
   624     val full_new_type_names = map (Sign.full_name (sign_of thy)) new_type_names;
   625 
   626     fun make_constr_def tname T T' ((thy, defs, eqns), ((cname, cargs), (cname', mx))) =
   627       let
   628         fun constr_arg (dt, (j, l_args, r_args)) =
   629           let
   630             val x' = mk_Free "x" (typ_of_dtyp' dt) j;
   631             val (dts, dt') = strip_option dt;
   632             val xs = map (fn (dt, i) => mk_Free "x" (typ_of_dtyp' dt) i)
   633               (dts ~~ (j upto j + length dts - 1))
   634             val x = mk_Free "x" (typ_of_dtyp' dt') (j + length dts)
   635             val (dts', dt'') = strip_dtyp dt'
   636           in case dt'' of
   637               DtRec k => if k < length new_type_names then
   638                   (j + length dts + 1,
   639                    xs @ x :: l_args,
   640                    foldr mk_abs_fun
   641                      (list_abs (map (pair "z" o typ_of_dtyp') dts',
   642                        Const (List.nth (rep_names, k), typ_of_dtyp' dt'' -->
   643                          typ_of_dtyp dt'') $ app_bnds x (length dts')))
   644                      xs :: r_args)
   645                 else error "nested recursion not (yet) supported"
   646             | _ => (j + 1, x' :: l_args, x' :: r_args)
   647           end
   648 
   649         val (_, l_args, r_args) = foldr constr_arg (1, [], []) cargs;
   650         val abs_name = Sign.intern_const (Theory.sign_of thy) ("Abs_" ^ tname);
   651         val rep_name = Sign.intern_const (Theory.sign_of thy) ("Rep_" ^ tname);
   652         val constrT = map fastype_of l_args ---> T;
   653         val lhs = list_comb (Const (Sign.full_name thy (Sign.base_name cname),
   654           constrT), l_args);
   655         val rhs = list_comb (Const (cname, map fastype_of r_args ---> T'), r_args);
   656         val def = Logic.mk_equals (lhs, Const (abs_name, T' --> T) $ rhs);
   657         val eqn = HOLogic.mk_Trueprop (HOLogic.mk_eq
   658           (Const (rep_name, T --> T') $ lhs, rhs));
   659         val def_name = (Sign.base_name cname) ^ "_def";
   660         val (thy', [def_thm]) = thy |>
   661           Theory.add_consts_i [(cname', constrT, mx)] |>
   662           (PureThy.add_defs_i false o map Thm.no_attributes) [(def_name, def)]
   663       in (thy', defs @ [def_thm], eqns @ [eqn]) end;
   664 
   665     fun dt_constr_defs ((thy, defs, eqns, dist_lemmas),
   666         (((((_, (_, _, constrs)), tname), T), T'), constr_syntax)) =
   667       let
   668         val rep_const = cterm_of thy
   669           (Const (Sign.intern_const thy ("Rep_" ^ tname), T --> T'));
   670         val dist = standard (cterm_instantiate [(cterm_of thy distinct_f, rep_const)] distinct_lemma);
   671         val (thy', defs', eqns') = Library.foldl (make_constr_def tname T T')
   672           ((Theory.add_path tname thy, defs, []), constrs ~~ constr_syntax)
   673       in
   674         (parent_path flat_names thy', defs', eqns @ [eqns'], dist_lemmas @ [dist])
   675       end;
   676 
   677     val (thy8, constr_defs, constr_rep_eqns, dist_lemmas) = Library.foldl dt_constr_defs
   678       ((thy7, [], [], []), List.take (descr, length new_type_names) ~~
   679         new_type_names ~~ newTs ~~ newTs' ~~ constr_syntax);
   680 
   681     val abs_inject_thms = map (fn tname =>
   682       PureThy.get_thm thy8 (Name ("Abs_" ^ tname ^ "_inject"))) new_type_names;
   683 
   684     val rep_inject_thms = map (fn tname =>
   685       PureThy.get_thm thy8 (Name ("Rep_" ^ tname ^ "_inject"))) new_type_names;
   686 
   687     val rep_thms = map (fn tname =>
   688       PureThy.get_thm thy8 (Name ("Rep_" ^ tname))) new_type_names;
   689 
   690     val rep_inverse_thms = map (fn tname =>
   691       PureThy.get_thm thy8 (Name ("Rep_" ^ tname ^ "_inverse"))) new_type_names;
   692 
   693     (* prove theorem  Rep_i (Constr_j ...) = Constr'_j ...  *)
   694     
   695     fun prove_constr_rep_thm eqn =
   696       let
   697         val inj_thms = map (fn r => r RS iffD1) abs_inject_thms;
   698         val rewrites = constr_defs @ map mk_meta_eq rep_inverse_thms
   699       in standard (Goal.prove thy8 [] [] eqn (fn _ => EVERY
   700         [resolve_tac inj_thms 1,
   701          rewrite_goals_tac rewrites,
   702          rtac refl 3,
   703          resolve_tac rep_intrs 2,
   704          REPEAT (resolve_tac rep_thms 1)]))
   705       end;
   706 
   707     val constr_rep_thmss = map (map prove_constr_rep_thm) constr_rep_eqns;
   708 
   709     (* prove theorem  pi \<bullet> Rep_i x = Rep_i (pi \<bullet> x) *)
   710 
   711     fun prove_perm_rep_perm (atom, perm_closed_thms) = map (fn th =>
   712       let
   713         val _ $ (_ $ (Rep $ x) $ _) = Logic.unvarify (prop_of th);
   714         val Type ("fun", [T, U]) = fastype_of Rep;
   715         val permT = mk_permT (Type (atom, []));
   716         val pi = Free ("pi", permT);
   717       in
   718         standard (Goal.prove thy8 [] [] (HOLogic.mk_Trueprop (HOLogic.mk_eq
   719             (Const ("nominal.perm", permT --> U --> U) $ pi $ (Rep $ x),
   720              Rep $ (Const ("nominal.perm", permT --> T --> T) $ pi $ x))))
   721           (fn _ => simp_tac (HOL_basic_ss addsimps (perm_defs @ Abs_inverse_thms @
   722             perm_closed_thms @ Rep_thms)) 1))
   723       end) Rep_thms;
   724 
   725     val perm_rep_perm_thms = List.concat (map prove_perm_rep_perm
   726       (atoms ~~ perm_closed_thmss));
   727 
   728     (* prove distinctness theorems *)
   729 
   730     val distinct_props = setmp DatatypeProp.dtK 1000
   731       (DatatypeProp.make_distincts new_type_names descr' sorts') thy8;
   732 
   733     val dist_rewrites = map (fn (rep_thms, dist_lemma) =>
   734       dist_lemma::(rep_thms @ [In0_eq, In1_eq, In0_not_In1, In1_not_In0]))
   735         (constr_rep_thmss ~~ dist_lemmas);
   736 
   737     fun prove_distinct_thms (_, []) = []
   738       | prove_distinct_thms (p as (rep_thms, dist_lemma), t::ts) =
   739           let
   740             val dist_thm = standard (Goal.prove thy8 [] [] t (fn _ =>
   741               simp_tac (simpset_of thy8 addsimps (dist_lemma :: rep_thms)) 1))
   742           in dist_thm::(standard (dist_thm RS not_sym))::
   743             (prove_distinct_thms (p, ts))
   744           end;
   745 
   746     val distinct_thms = map prove_distinct_thms
   747       (constr_rep_thmss ~~ dist_lemmas ~~ distinct_props);
   748 
   749     (** prove equations for permutation functions **)
   750 
   751     val abs_perm = PureThy.get_thms thy8 (Name "abs_perm"); (* FIXME *)
   752 
   753     val perm_simps' = map (fn (((i, (_, _, constrs)), tname), constr_rep_thms) =>
   754       let val T = replace_types' (nth_dtyp i)
   755       in List.concat (map (fn (atom, perm_closed_thms) =>
   756           map (fn ((cname, dts), constr_rep_thm) => 
   757         let
   758           val cname = Sign.intern_const thy8
   759             (NameSpace.append tname (Sign.base_name cname));
   760           val permT = mk_permT (Type (atom, []));
   761           val pi = Free ("pi", permT);
   762 
   763           fun perm t =
   764             let val T = fastype_of t
   765             in Const ("nominal.perm", permT --> T --> T) $ pi $ t end;
   766 
   767           fun constr_arg (dt, (j, l_args, r_args)) =
   768             let
   769               val x' = mk_Free "x" (typ_of_dtyp' dt) j;
   770               val (dts, dt') = strip_option dt;
   771               val Ts = map typ_of_dtyp' dts;
   772               val xs = map (fn (T, i) => mk_Free "x" T i)
   773                 (Ts ~~ (j upto j + length dts - 1))
   774               val x = mk_Free "x" (typ_of_dtyp' dt') (j + length dts);
   775               val (dts', dt'') = strip_dtyp dt';
   776             in case dt'' of
   777                 DtRec k => if k < length new_type_names then
   778                     (j + length dts + 1,
   779                      xs @ x :: l_args,
   780                      map perm (xs @ [x]) @ r_args)
   781                   else error "nested recursion not (yet) supported"
   782               | _ => (j + 1, x' :: l_args, perm x' :: r_args)
   783             end
   784 
   785           val (_, l_args, r_args) = foldr constr_arg (1, [], []) dts;
   786           val c = Const (cname, map fastype_of l_args ---> T)
   787         in
   788           standard (Goal.prove thy8 [] []
   789             (HOLogic.mk_Trueprop (HOLogic.mk_eq
   790               (perm (list_comb (c, l_args)), list_comb (c, r_args))))
   791             (fn _ => EVERY
   792               [simp_tac (simpset_of thy8 addsimps (constr_rep_thm :: perm_defs)) 1,
   793                simp_tac (HOL_basic_ss addsimps (Rep_thms @ Abs_inverse_thms @
   794                  constr_defs @ perm_closed_thms)) 1,
   795                TRY (simp_tac (HOL_basic_ss addsimps
   796                  (symmetric perm_fun_def :: abs_perm)) 1),
   797                TRY (simp_tac (HOL_basic_ss addsimps
   798                  (perm_fun_def :: perm_defs @ Rep_thms @ Abs_inverse_thms @
   799                     perm_closed_thms)) 1)]))
   800         end) (constrs ~~ constr_rep_thms)) (atoms ~~ perm_closed_thmss))
   801       end) (List.take (descr, length new_type_names) ~~ new_type_names ~~ constr_rep_thmss);
   802 
   803     (** prove injectivity of constructors **)
   804 
   805     val rep_inject_thms' = map (fn th => th RS sym) rep_inject_thms;
   806     val alpha = PureThy.get_thms thy8 (Name "alpha");
   807     val abs_fresh = PureThy.get_thms thy8 (Name "abs_fresh");
   808     val fresh_def = PureThy.get_thm thy8 (Name "fresh_def");
   809     val supp_def = PureThy.get_thm thy8 (Name "supp_def");
   810 
   811     val inject_thms = map (fn (((i, (_, _, constrs)), tname), constr_rep_thms) =>
   812       let val T = replace_types' (nth_dtyp i)
   813       in List.mapPartial (fn ((cname, dts), constr_rep_thm) =>
   814         if null dts then NONE else SOME
   815         let
   816           val cname = Sign.intern_const thy8
   817             (NameSpace.append tname (Sign.base_name cname));
   818 
   819           fun make_inj (dt, (j, args1, args2, eqs)) =
   820             let
   821               val x' = mk_Free "x" (typ_of_dtyp' dt) j;
   822               val y' = mk_Free "y" (typ_of_dtyp' dt) j;
   823               val (dts, dt') = strip_option dt;
   824               val Ts_idx = map typ_of_dtyp' dts ~~ (j upto j + length dts - 1);
   825               val xs = map (fn (T, i) => mk_Free "x" T i) Ts_idx;
   826               val ys = map (fn (T, i) => mk_Free "y" T i) Ts_idx;
   827               val x = mk_Free "x" (typ_of_dtyp' dt') (j + length dts);
   828               val y = mk_Free "y" (typ_of_dtyp' dt') (j + length dts);
   829               val (dts', dt'') = strip_dtyp dt';
   830             in case dt'' of
   831                 DtRec k => if k < length new_type_names then
   832                     (j + length dts + 1,
   833                      xs @ (x :: args1), ys @ (y :: args2),
   834                      HOLogic.mk_eq
   835                        (foldr mk_abs_fun x xs, foldr mk_abs_fun y ys) :: eqs)
   836                   else error "nested recursion not (yet) supported"
   837               | _ => (j + 1, x' :: args1, y' :: args2, HOLogic.mk_eq (x', y') :: eqs)
   838             end;
   839 
   840           val (_, args1, args2, eqs) = foldr make_inj (1, [], [], []) dts;
   841           val Ts = map fastype_of args1;
   842           val c = Const (cname, Ts ---> T)
   843         in
   844           standard (Goal.prove thy8 [] [] (HOLogic.mk_Trueprop (HOLogic.mk_eq
   845               (HOLogic.mk_eq (list_comb (c, args1), list_comb (c, args2)),
   846                foldr1 HOLogic.mk_conj eqs)))
   847             (fn _ => EVERY
   848                [asm_full_simp_tac (simpset_of thy8 addsimps (constr_rep_thm ::
   849                   rep_inject_thms')) 1,
   850                 TRY (asm_full_simp_tac (HOL_basic_ss addsimps (fresh_def :: supp_def ::
   851                   alpha @ abs_perm @ abs_fresh @ rep_inject_thms @
   852                   perm_rep_perm_thms)) 1),
   853                 TRY (asm_full_simp_tac (HOL_basic_ss addsimps (perm_fun_def ::
   854                   expand_fun_eq :: rep_inject_thms @ perm_rep_perm_thms)) 1)]))
   855         end) (constrs ~~ constr_rep_thms)
   856       end) (List.take (descr, length new_type_names) ~~ new_type_names ~~ constr_rep_thmss);
   857 
   858     (** equations for support and freshness **)
   859 
   860     val Un_assoc = PureThy.get_thm thy8 (Name "Un_assoc");
   861     val de_Morgan_conj = PureThy.get_thm thy8 (Name "de_Morgan_conj");
   862     val Collect_disj_eq = PureThy.get_thm thy8 (Name "Collect_disj_eq");
   863     val finite_Un = PureThy.get_thm thy8 (Name "finite_Un");
   864 
   865     val (supp_thms, fresh_thms) = ListPair.unzip (map ListPair.unzip
   866       (map (fn ((((i, (_, _, constrs)), tname), inject_thms'), perm_thms') =>
   867       let val T = replace_types' (nth_dtyp i)
   868       in List.concat (map (fn (cname, dts) => map (fn atom =>
   869         let
   870           val cname = Sign.intern_const thy8
   871             (NameSpace.append tname (Sign.base_name cname));
   872           val atomT = Type (atom, []);
   873 
   874           fun process_constr (dt, (j, args1, args2)) =
   875             let
   876               val x' = mk_Free "x" (typ_of_dtyp' dt) j;
   877               val (dts, dt') = strip_option dt;
   878               val Ts_idx = map typ_of_dtyp' dts ~~ (j upto j + length dts - 1);
   879               val xs = map (fn (T, i) => mk_Free "x" T i) Ts_idx;
   880               val x = mk_Free "x" (typ_of_dtyp' dt') (j + length dts);
   881               val (dts', dt'') = strip_dtyp dt';
   882             in case dt'' of
   883                 DtRec k => if k < length new_type_names then
   884                     (j + length dts + 1,
   885                      xs @ (x :: args1), foldr mk_abs_fun x xs :: args2)
   886                   else error "nested recursion not (yet) supported"
   887               | _ => (j + 1, x' :: args1, x' :: args2)
   888             end;
   889 
   890           val (_, args1, args2) = foldr process_constr (1, [], []) dts;
   891           val Ts = map fastype_of args1;
   892           val c = list_comb (Const (cname, Ts ---> T), args1);
   893           fun supp t =
   894             Const ("nominal.supp", fastype_of t --> HOLogic.mk_setT atomT) $ t;
   895           fun fresh t =
   896             Const ("nominal.fresh", atomT --> fastype_of t --> HOLogic.boolT) $
   897               Free ("a", atomT) $ t;
   898           val supp_thm = standard (Goal.prove thy8 [] []
   899               (HOLogic.mk_Trueprop (HOLogic.mk_eq
   900                 (supp c,
   901                  if null dts then Const ("{}", HOLogic.mk_setT atomT)
   902                  else foldr1 (HOLogic.mk_binop "op Un") (map supp args2))))
   903             (fn _ =>
   904               simp_tac (HOL_basic_ss addsimps (supp_def ::
   905                  Un_assoc :: de_Morgan_conj :: Collect_disj_eq :: finite_Un ::
   906                  symmetric empty_def :: Finites.emptyI :: simp_thms @
   907                  abs_perm @ abs_fresh @ inject_thms' @ perm_thms')) 1))
   908         in
   909           (supp_thm,
   910            standard (Goal.prove thy8 [] [] (HOLogic.mk_Trueprop (HOLogic.mk_eq
   911               (fresh c,
   912                if null dts then HOLogic.true_const
   913                else foldr1 HOLogic.mk_conj (map fresh args2))))
   914              (fn _ =>
   915                simp_tac (simpset_of thy8 addsimps [fresh_def, supp_thm]) 1)))
   916         end) atoms) constrs)
   917       end) (List.take (descr, length new_type_names) ~~ new_type_names ~~ inject_thms ~~ perm_simps')));
   918 
   919     (**** Induction theorem ****)
   920 
   921     val arities = get_arities (List.concat descr');
   922 
   923     fun mk_funs_inv thm =
   924       let
   925         val {sign, prop, ...} = rep_thm thm;
   926         val _ $ (_ $ (Const (_, Type (_, [U, _])) $ _ $ S)) $
   927           (_ $ (_ $ (r $ (a $ _)) $ _)) = Type.freeze prop;
   928         val used = add_term_tfree_names (a, []);
   929 
   930         fun mk_thm i =
   931           let
   932             val Ts = map (TFree o rpair HOLogic.typeS)
   933               (variantlist (replicate i "'t", used));
   934             val f = Free ("f", Ts ---> U)
   935           in standard (Goal.prove sign [] [] (Logic.mk_implies
   936             (HOLogic.mk_Trueprop (HOLogic.list_all
   937                (map (pair "x") Ts, HOLogic.mk_mem (app_bnds f i, S))),
   938              HOLogic.mk_Trueprop (HOLogic.mk_eq (list_abs (map (pair "x") Ts,
   939                r $ (a $ app_bnds f i)), f))))
   940             (fn _ => EVERY [REPEAT (rtac ext 1), REPEAT (etac allE 1), rtac thm 1, atac 1]))
   941           end
   942       in map (fn r => r RS subst) (thm :: map mk_thm arities) end;
   943 
   944     fun mk_indrule_lemma ((prems, concls), (((i, _), T), U)) =
   945       let
   946         val Rep_t = Const (List.nth (rep_names, i), T --> U) $
   947           mk_Free "x" T i;
   948 
   949         val Abs_t =  Const (List.nth (abs_names, i), U --> T)
   950 
   951       in (prems @ [HOLogic.imp $ HOLogic.mk_mem (Rep_t,
   952             Const (List.nth (rep_set_names, i), HOLogic.mk_setT U)) $
   953               (mk_Free "P" (T --> HOLogic.boolT) (i + 1) $ (Abs_t $ Rep_t))],
   954           concls @ [mk_Free "P" (T --> HOLogic.boolT) (i + 1) $ mk_Free "x" T i])
   955       end;
   956 
   957     val (indrule_lemma_prems, indrule_lemma_concls) =
   958       Library.foldl mk_indrule_lemma (([], []), (List.concat descr' ~~ recTs ~~ recTs'));
   959 
   960     val indrule_lemma = standard (Goal.prove thy8 [] []
   961       (Logic.mk_implies
   962         (HOLogic.mk_Trueprop (mk_conj indrule_lemma_prems),
   963          HOLogic.mk_Trueprop (mk_conj indrule_lemma_concls))) (fn _ => EVERY
   964            [REPEAT (etac conjE 1),
   965             REPEAT (EVERY
   966               [TRY (rtac conjI 1), full_simp_tac (HOL_basic_ss addsimps Rep_inverse_thms) 1,
   967                etac mp 1, resolve_tac Rep_thms 1])]));
   968 
   969     val Ps = map head_of (HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of indrule_lemma)));
   970     val frees = if length Ps = 1 then [Free ("P", snd (dest_Var (hd Ps)))] else
   971       map (Free o apfst fst o dest_Var) Ps;
   972     val indrule_lemma' = cterm_instantiate
   973       (map (cterm_of thy8) Ps ~~ map (cterm_of thy8) frees) indrule_lemma;
   974 
   975     val Abs_inverse_thms' = List.concat (map mk_funs_inv Abs_inverse_thms);
   976 
   977     val dt_induct_prop = DatatypeProp.make_ind descr' sorts';
   978     val dt_induct = standard (Goal.prove thy8 []
   979       (Logic.strip_imp_prems dt_induct_prop) (Logic.strip_imp_concl dt_induct_prop)
   980       (fn prems => EVERY
   981         [rtac indrule_lemma' 1,
   982          (DatatypeAux.indtac rep_induct THEN_ALL_NEW ObjectLogic.atomize_tac) 1,
   983          EVERY (map (fn (prem, r) => (EVERY
   984            [REPEAT (eresolve_tac Abs_inverse_thms' 1),
   985             simp_tac (HOL_basic_ss addsimps [symmetric r]) 1,
   986             DEPTH_SOLVE_1 (ares_tac [prem] 1 ORELSE etac allE 1)]))
   987                 (prems ~~ constr_defs))]));
   988 
   989     val case_names_induct = mk_case_names_induct (List.concat descr');
   990 
   991     (**** prove that new datatypes have finite support ****)
   992 
   993     val indnames = DatatypeProp.make_tnames recTs;
   994 
   995     val abs_supp = PureThy.get_thms thy8 (Name "abs_supp");
   996     val supp_atm = PureThy.get_thms thy8 (Name "supp_atm");
   997 
   998     val finite_supp_thms = map (fn atom =>
   999       let val atomT = Type (atom, [])
  1000       in map standard (List.take
  1001         (split_conj_thm (Goal.prove thy8 [] [] (HOLogic.mk_Trueprop
  1002            (foldr1 HOLogic.mk_conj (map (fn (s, T) => HOLogic.mk_mem
  1003              (Const ("nominal.supp", T --> HOLogic.mk_setT atomT) $ Free (s, T),
  1004               Const ("Finite_Set.Finites", HOLogic.mk_setT (HOLogic.mk_setT atomT))))
  1005                (indnames ~~ recTs))))
  1006            (fn _ => indtac dt_induct indnames 1 THEN
  1007             ALLGOALS (asm_full_simp_tac (simpset_of thy8 addsimps
  1008               (abs_supp @ supp_atm @
  1009                PureThy.get_thms thy8 (Name ("fs_" ^ Sign.base_name atom ^ "1")) @
  1010                List.concat supp_thms))))),
  1011          length new_type_names))
  1012       end) atoms;
  1013 
  1014     val (thy9, _) = thy8 |>
  1015       Theory.add_path big_name |>
  1016       PureThy.add_thms [(("induct_weak", dt_induct), [case_names_induct])] |>>
  1017       Theory.parent_path |>>>
  1018       DatatypeAux.store_thmss "distinct" new_type_names distinct_thms |>>>
  1019       DatatypeAux.store_thmss "constr_rep" new_type_names constr_rep_thmss |>>>
  1020       DatatypeAux.store_thmss "perm" new_type_names perm_simps' |>>>
  1021       DatatypeAux.store_thmss "inject" new_type_names inject_thms |>>>
  1022       DatatypeAux.store_thmss "supp" new_type_names supp_thms |>>>
  1023       DatatypeAux.store_thmss "fresh" new_type_names fresh_thms |>>
  1024       fold (fn (atom, ths) => fn thy =>
  1025         let val class = Sign.intern_class thy ("fs_" ^ Sign.base_name atom)
  1026         in fold (fn T => AxClass.add_inst_arity_i
  1027             (fst (dest_Type T),
  1028               replicate (length sorts) [class], [class])
  1029             (AxClass.intro_classes_tac [] THEN resolve_tac ths 1)) newTs thy
  1030         end) (atoms ~~ finite_supp_thms);
  1031 
  1032   in
  1033     thy9
  1034   end;
  1035 
  1036 val add_nominal_datatype = gen_add_nominal_datatype read_typ true;
  1037 
  1038 
  1039 (* FIXME: The following stuff should be exported by DatatypePackage *)
  1040 
  1041 local structure P = OuterParse and K = OuterKeyword in
  1042 
  1043 val datatype_decl =
  1044   Scan.option (P.$$$ "(" |-- P.name --| P.$$$ ")") -- P.type_args -- P.name -- P.opt_infix --
  1045     (P.$$$ "=" |-- P.enum1 "|" (P.name -- Scan.repeat P.typ -- P.opt_mixfix));
  1046 
  1047 fun mk_datatype args =
  1048   let
  1049     val names = map (fn ((((NONE, _), t), _), _) => t | ((((SOME t, _), _), _), _) => t) args;
  1050     val specs = map (fn ((((_, vs), t), mx), cons) =>
  1051       (vs, t, mx, map (fn ((x, y), z) => (x, y, z)) cons)) args;
  1052   in add_nominal_datatype false names specs end;
  1053 
  1054 val nominal_datatypeP =
  1055   OuterSyntax.command "nominal_datatype" "define inductive datatypes" K.thy_decl
  1056     (P.and_list1 datatype_decl >> (Toplevel.theory o mk_datatype));
  1057 
  1058 val _ = OuterSyntax.add_parsers [nominal_datatypeP];
  1059 
  1060 end;
  1061 
  1062 end