src/HOL/Nominal/nominal_package.ML
author haftmann
Fri Jul 21 14:48:17 2006 +0200 (2006-07-21)
changeset 20179 a2f3f523c84b
parent 20145 922b4e7b8efd
child 20267 1154363129be
permissions -rw-r--r--
adaption to argument change in primrec_package
     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 (ProofContext.init (Thm.theory_of_thm rule)) rule
   137   |> map (standard #> RuleCases.save rule);
   138 
   139 fun gen_add_nominal_datatype prep_typ err flat_names new_type_names dts thy =
   140   let
   141     (* this theory is used just for parsing *)
   142   
   143     val tmp_thy = thy |>
   144       Theory.copy |>
   145       Theory.add_types (map (fn (tvs, tname, mx, _) =>
   146         (tname, length tvs, mx)) dts);
   147 
   148     val sign = Theory.sign_of tmp_thy;
   149 
   150     val atoms = atoms_of thy;
   151     val classes = map (NameSpace.map_base (fn s => "pt_" ^ s)) atoms;
   152     val cp_classes = List.concat (map (fn atom1 => map (fn atom2 =>
   153       Sign.intern_class thy ("cp_" ^ Sign.base_name atom1 ^ "_" ^
   154         Sign.base_name atom2)) atoms) atoms);
   155     fun augment_sort S = S union classes;
   156     val augment_sort_typ = map_type_tfree (fn (s, S) => TFree (s, augment_sort S));
   157 
   158     fun prep_constr ((constrs, sorts), (cname, cargs, mx)) =
   159       let val (cargs', sorts') = Library.foldl (prep_typ sign) (([], sorts), cargs)
   160       in (constrs @ [(cname, cargs', mx)], sorts') end
   161 
   162     fun prep_dt_spec ((dts, sorts), (tvs, tname, mx, constrs)) =
   163       let val (constrs', sorts') = Library.foldl prep_constr (([], sorts), constrs)
   164       in (dts @ [(tvs, tname, mx, constrs')], sorts') end
   165 
   166     val (dts', sorts) = Library.foldl prep_dt_spec (([], []), dts);
   167     val sorts' = map (apsnd augment_sort) sorts;
   168     val tyvars = map #1 dts';
   169 
   170     val types_syntax = map (fn (tvs, tname, mx, constrs) => (tname, mx)) dts';
   171     val constr_syntax = map (fn (tvs, tname, mx, constrs) =>
   172       map (fn (cname, cargs, mx) => (cname, mx)) constrs) dts';
   173 
   174     val ps = map (fn (_, n, _, _) =>
   175       (Sign.full_name sign n, Sign.full_name sign (n ^ "_Rep"))) dts;
   176     val rps = map Library.swap ps;
   177 
   178     fun replace_types (Type ("Nominal.ABS", [T, U])) = 
   179           Type ("fun", [T, Type ("Nominal.noption", [replace_types U])])
   180       | replace_types (Type (s, Ts)) =
   181           Type (getOpt (AList.lookup op = ps s, s), map replace_types Ts)
   182       | replace_types T = T;
   183 
   184     val dts'' = map (fn (tvs, tname, mx, constrs) => (tvs, tname ^ "_Rep", NoSyn,
   185       map (fn (cname, cargs, mx) => (cname ^ "_Rep",
   186         map (augment_sort_typ o replace_types) cargs, NoSyn)) constrs)) dts';
   187 
   188     val new_type_names' = map (fn n => n ^ "_Rep") new_type_names;
   189     val full_new_type_names' = map (Sign.full_name (sign_of thy)) new_type_names';
   190 
   191     val ({induction, ...},thy1) =
   192       DatatypePackage.add_datatype_i err flat_names new_type_names' dts'' thy;
   193 
   194     val SOME {descr, ...} = Symtab.lookup
   195       (DatatypePackage.get_datatypes thy1) (hd full_new_type_names');
   196     fun nth_dtyp i = typ_of_dtyp descr sorts' (DtRec i);
   197 
   198     (**** define permutation functions ****)
   199 
   200     val permT = mk_permT (TFree ("'x", HOLogic.typeS));
   201     val pi = Free ("pi", permT);
   202     val perm_types = map (fn (i, _) =>
   203       let val T = nth_dtyp i
   204       in permT --> T --> T end) descr;
   205     val perm_names = replicate (length new_type_names) "Nominal.perm" @
   206       DatatypeProp.indexify_names (map (fn i => Sign.full_name (sign_of thy1)
   207         ("perm_" ^ name_of_typ (nth_dtyp i)))
   208           (length new_type_names upto length descr - 1));
   209     val perm_names_types = perm_names ~~ perm_types;
   210 
   211     val perm_eqs = List.concat (map (fn (i, (_, _, constrs)) =>
   212       let val T = nth_dtyp i
   213       in map (fn (cname, dts) => 
   214         let
   215           val Ts = map (typ_of_dtyp descr sorts') dts;
   216           val names = DatatypeProp.make_tnames Ts;
   217           val args = map Free (names ~~ Ts);
   218           val c = Const (cname, Ts ---> T);
   219           fun perm_arg (dt, x) =
   220             let val T = type_of x
   221             in if is_rec_type dt then
   222                 let val (Us, _) = strip_type T
   223                 in list_abs (map (pair "x") Us,
   224                   Const (List.nth (perm_names_types, body_index dt)) $ pi $
   225                     list_comb (x, map (fn (i, U) =>
   226                       Const ("Nominal.perm", permT --> U --> U) $
   227                         (Const ("List.rev", permT --> permT) $ pi) $
   228                         Bound i) ((length Us - 1 downto 0) ~~ Us)))
   229                 end
   230               else Const ("Nominal.perm", permT --> T --> T) $ pi $ x
   231             end;  
   232         in
   233           (("", HOLogic.mk_Trueprop (HOLogic.mk_eq
   234             (Const (List.nth (perm_names_types, i)) $
   235                Free ("pi", mk_permT (TFree ("'x", HOLogic.typeS))) $
   236                list_comb (c, args),
   237              list_comb (c, map perm_arg (dts ~~ args))))), [])
   238         end) constrs
   239       end) descr);
   240 
   241     val (perm_simps, thy2) = thy1 |>
   242       Theory.add_consts_i (map (fn (s, T) => (Sign.base_name s, T, NoSyn))
   243         (List.drop (perm_names_types, length new_type_names))) |>
   244       PrimrecPackage.add_primrec_unchecked_i "" perm_eqs;
   245 
   246     (**** prove that permutation functions introduced by unfolding are ****)
   247     (**** equivalent to already existing permutation functions         ****)
   248 
   249     val _ = warning ("length descr: " ^ string_of_int (length descr));
   250     val _ = warning ("length new_type_names: " ^ string_of_int (length new_type_names));
   251 
   252     val perm_indnames = DatatypeProp.make_tnames (map body_type perm_types);
   253     val perm_fun_def = PureThy.get_thm thy2 (Name "perm_fun_def");
   254 
   255     val unfolded_perm_eq_thms =
   256       if length descr = length new_type_names then []
   257       else map standard (List.drop (split_conj_thm
   258         (Goal.prove_global thy2 [] []
   259           (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
   260             (map (fn (c as (s, T), x) =>
   261                let val [T1, T2] = binder_types T
   262                in HOLogic.mk_eq (Const c $ pi $ Free (x, T2),
   263                  Const ("Nominal.perm", T) $ pi $ Free (x, T2))
   264                end)
   265              (perm_names_types ~~ perm_indnames))))
   266           (fn _ => EVERY [indtac induction perm_indnames 1,
   267             ALLGOALS (asm_full_simp_tac
   268               (simpset_of thy2 addsimps [perm_fun_def]))])),
   269         length new_type_names));
   270 
   271     (**** prove [] \<bullet> t = t ****)
   272 
   273     val _ = warning "perm_empty_thms";
   274 
   275     val perm_empty_thms = List.concat (map (fn a =>
   276       let val permT = mk_permT (Type (a, []))
   277       in map standard (List.take (split_conj_thm
   278         (Goal.prove_global thy2 [] []
   279           (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
   280             (map (fn ((s, T), x) => HOLogic.mk_eq
   281                 (Const (s, permT --> T --> T) $
   282                    Const ("List.list.Nil", permT) $ Free (x, T),
   283                  Free (x, T)))
   284              (perm_names ~~
   285               map body_type perm_types ~~ perm_indnames))))
   286           (fn _ => EVERY [indtac induction perm_indnames 1,
   287             ALLGOALS (asm_full_simp_tac (simpset_of thy2))])),
   288         length new_type_names))
   289       end)
   290       atoms);
   291 
   292     (**** prove (pi1 @ pi2) \<bullet> t = pi1 \<bullet> (pi2 \<bullet> t) ****)
   293 
   294     val _ = warning "perm_append_thms";
   295 
   296     (*FIXME: these should be looked up statically*)
   297     val at_pt_inst = PureThy.get_thm thy2 (Name "at_pt_inst");
   298     val pt2 = PureThy.get_thm thy2 (Name "pt2");
   299 
   300     val perm_append_thms = List.concat (map (fn a =>
   301       let
   302         val permT = mk_permT (Type (a, []));
   303         val pi1 = Free ("pi1", permT);
   304         val pi2 = Free ("pi2", permT);
   305         val pt_inst = PureThy.get_thm thy2 (Name ("pt_" ^ Sign.base_name a ^ "_inst"));
   306         val pt2' = pt_inst RS pt2;
   307         val pt2_ax = PureThy.get_thm thy2
   308           (Name (NameSpace.map_base (fn s => "pt_" ^ s ^ "2") a));
   309       in List.take (map standard (split_conj_thm
   310         (Goal.prove_global thy2 [] []
   311              (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
   312                 (map (fn ((s, T), x) =>
   313                     let val perm = Const (s, permT --> T --> T)
   314                     in HOLogic.mk_eq
   315                       (perm $ (Const ("List.op @", permT --> permT --> permT) $
   316                          pi1 $ pi2) $ Free (x, T),
   317                        perm $ pi1 $ (perm $ pi2 $ Free (x, T)))
   318                     end)
   319                   (perm_names ~~
   320                    map body_type perm_types ~~ perm_indnames))))
   321            (fn _ => EVERY [indtac induction perm_indnames 1,
   322               ALLGOALS (asm_full_simp_tac (simpset_of thy2 addsimps [pt2', pt2_ax]))]))),
   323          length new_type_names)
   324       end) atoms);
   325 
   326     (**** prove pi1 ~ pi2 ==> pi1 \<bullet> t = pi2 \<bullet> t ****)
   327 
   328     val _ = warning "perm_eq_thms";
   329 
   330     val pt3 = PureThy.get_thm thy2 (Name "pt3");
   331     val pt3_rev = PureThy.get_thm thy2 (Name "pt3_rev");
   332 
   333     val perm_eq_thms = List.concat (map (fn a =>
   334       let
   335         val permT = mk_permT (Type (a, []));
   336         val pi1 = Free ("pi1", permT);
   337         val pi2 = Free ("pi2", permT);
   338         (*FIXME: not robust - better access these theorems using NominalData?*)
   339         val at_inst = PureThy.get_thm thy2 (Name ("at_" ^ Sign.base_name a ^ "_inst"));
   340         val pt_inst = PureThy.get_thm thy2 (Name ("pt_" ^ Sign.base_name a ^ "_inst"));
   341         val pt3' = pt_inst RS pt3;
   342         val pt3_rev' = at_inst RS (pt_inst RS pt3_rev);
   343         val pt3_ax = PureThy.get_thm thy2
   344           (Name (NameSpace.map_base (fn s => "pt_" ^ s ^ "3") a));
   345       in List.take (map standard (split_conj_thm
   346         (Goal.prove_global thy2 [] [] (Logic.mk_implies
   347              (HOLogic.mk_Trueprop (Const ("Nominal.prm_eq",
   348                 permT --> permT --> HOLogic.boolT) $ pi1 $ pi2),
   349               HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
   350                 (map (fn ((s, T), x) =>
   351                     let val perm = Const (s, permT --> T --> T)
   352                     in HOLogic.mk_eq
   353                       (perm $ pi1 $ Free (x, T),
   354                        perm $ pi2 $ Free (x, T))
   355                     end)
   356                   (perm_names ~~
   357                    map body_type perm_types ~~ perm_indnames)))))
   358            (fn _ => EVERY [indtac induction perm_indnames 1,
   359               ALLGOALS (asm_full_simp_tac (simpset_of thy2 addsimps [pt3', pt3_rev', pt3_ax]))]))),
   360          length new_type_names)
   361       end) atoms);
   362 
   363     (**** prove pi1 \<bullet> (pi2 \<bullet> t) = (pi1 \<bullet> pi2) \<bullet> (pi1 \<bullet> t) ****)
   364 
   365     val cp1 = PureThy.get_thm thy2 (Name "cp1");
   366     val dj_cp = PureThy.get_thm thy2 (Name "dj_cp");
   367     val pt_perm_compose = PureThy.get_thm thy2 (Name "pt_perm_compose");
   368     val pt_perm_compose_rev = PureThy.get_thm thy2 (Name "pt_perm_compose_rev");
   369     val dj_perm_perm_forget = PureThy.get_thm thy2 (Name "dj_perm_perm_forget");
   370 
   371     fun composition_instance name1 name2 thy =
   372       let
   373         val name1' = Sign.base_name name1;
   374         val name2' = Sign.base_name name2;
   375         val cp_class = Sign.intern_class thy ("cp_" ^ name1' ^ "_" ^ name2');
   376         val permT1 = mk_permT (Type (name1, []));
   377         val permT2 = mk_permT (Type (name2, []));
   378         val augment = map_type_tfree
   379           (fn (x, S) => TFree (x, cp_class :: S));
   380         val Ts = map (augment o body_type) perm_types;
   381         val cp_inst = PureThy.get_thm thy
   382           (Name ("cp_" ^ name1' ^ "_" ^ name2' ^ "_inst"));
   383         val simps = simpset_of thy addsimps (perm_fun_def ::
   384           (if name1 <> name2 then
   385              let val dj = PureThy.get_thm thy (Name ("dj_" ^ name2' ^ "_" ^ name1'))
   386              in [dj RS (cp_inst RS dj_cp), dj RS dj_perm_perm_forget] end
   387            else
   388              let
   389                val at_inst = PureThy.get_thm thy (Name ("at_" ^ name1' ^ "_inst"));
   390                val pt_inst = PureThy.get_thm thy (Name ("pt_" ^ name1' ^ "_inst"))
   391              in
   392                [cp_inst RS cp1 RS sym,
   393                 at_inst RS (pt_inst RS pt_perm_compose) RS sym,
   394                 at_inst RS (pt_inst RS pt_perm_compose_rev) RS sym]
   395             end))
   396         val thms = split_conj_thm (Goal.prove_global thy [] []
   397             (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
   398               (map (fn ((s, T), x) =>
   399                   let
   400                     val pi1 = Free ("pi1", permT1);
   401                     val pi2 = Free ("pi2", permT2);
   402                     val perm1 = Const (s, permT1 --> T --> T);
   403                     val perm2 = Const (s, permT2 --> T --> T);
   404                     val perm3 = Const ("Nominal.perm", permT1 --> permT2 --> permT2)
   405                   in HOLogic.mk_eq
   406                     (perm1 $ pi1 $ (perm2 $ pi2 $ Free (x, T)),
   407                      perm2 $ (perm3 $ pi1 $ pi2) $ (perm1 $ pi1 $ Free (x, T)))
   408                   end)
   409                 (perm_names ~~ Ts ~~ perm_indnames))))
   410           (fn _ => EVERY [indtac induction perm_indnames 1,
   411              ALLGOALS (asm_full_simp_tac simps)]))
   412       in
   413         foldl (fn ((s, tvs), thy) => AxClass.prove_arity
   414             (s, replicate (length tvs) (cp_class :: classes), [cp_class])
   415             (ClassPackage.intro_classes_tac [] THEN ALLGOALS (resolve_tac thms)) thy)
   416           thy (full_new_type_names' ~~ tyvars)
   417       end;
   418 
   419     val (perm_thmss,thy3) = thy2 |>
   420       fold (fn name1 => fold (composition_instance name1) atoms) atoms |>
   421       curry (Library.foldr (fn ((i, (tyname, args, _)), thy) =>
   422         AxClass.prove_arity (tyname, replicate (length args) classes, classes)
   423         (ClassPackage.intro_classes_tac [] THEN REPEAT (EVERY
   424            [resolve_tac perm_empty_thms 1,
   425             resolve_tac perm_append_thms 1,
   426             resolve_tac perm_eq_thms 1, assume_tac 1])) thy))
   427         (List.take (descr, length new_type_names)) |>
   428       PureThy.add_thmss
   429         [((space_implode "_" new_type_names ^ "_unfolded_perm_eq",
   430           unfolded_perm_eq_thms), [Simplifier.simp_add]),
   431          ((space_implode "_" new_type_names ^ "_perm_empty",
   432           perm_empty_thms), [Simplifier.simp_add]),
   433          ((space_implode "_" new_type_names ^ "_perm_append",
   434           perm_append_thms), [Simplifier.simp_add]),
   435          ((space_implode "_" new_type_names ^ "_perm_eq",
   436           perm_eq_thms), [Simplifier.simp_add])];
   437   
   438     (**** Define representing sets ****)
   439 
   440     val _ = warning "representing sets";
   441 
   442     val rep_set_names = map (Sign.full_name thy3) (DatatypeProp.indexify_names
   443       (map (fn (i, _) => name_of_typ (nth_dtyp i) ^ "_set") descr));
   444     val big_rep_name =
   445       space_implode "_" (DatatypeProp.indexify_names (List.mapPartial
   446         (fn (i, ("Nominal.noption", _, _)) => NONE
   447           | (i, _) => SOME (name_of_typ (nth_dtyp i))) descr)) ^ "_set";
   448     val _ = warning ("big_rep_name: " ^ big_rep_name);
   449 
   450     fun strip_option (dtf as DtType ("fun", [dt, DtRec i])) =
   451           (case AList.lookup op = descr i of
   452              SOME ("Nominal.noption", _, [(_, [dt']), _]) =>
   453                apfst (cons dt) (strip_option dt')
   454            | _ => ([], dtf))
   455       | strip_option (DtType ("fun", [dt, DtType ("Nominal.noption", [dt'])])) =
   456           apfst (cons dt) (strip_option dt')
   457       | strip_option dt = ([], dt);
   458 
   459     val dt_atomTs = distinct op = (map (typ_of_dtyp descr sorts')
   460       (List.concat (map (fn (_, (_, _, cs)) => List.concat
   461         (map (List.concat o map (fst o strip_option) o snd) cs)) descr)));
   462 
   463     fun make_intr s T (cname, cargs) =
   464       let
   465         fun mk_prem (dt, (j, j', prems, ts)) = 
   466           let
   467             val (dts, dt') = strip_option dt;
   468             val (dts', dt'') = strip_dtyp dt';
   469             val Ts = map (typ_of_dtyp descr sorts') dts;
   470             val Us = map (typ_of_dtyp descr sorts') dts';
   471             val T = typ_of_dtyp descr sorts' dt'';
   472             val free = mk_Free "x" (Us ---> T) j;
   473             val free' = app_bnds free (length Us);
   474             fun mk_abs_fun (T, (i, t)) =
   475               let val U = fastype_of t
   476               in (i + 1, Const ("Nominal.abs_fun", [T, U, T] --->
   477                 Type ("Nominal.noption", [U])) $ mk_Free "y" T i $ t)
   478               end
   479           in (j + 1, j' + length Ts,
   480             case dt'' of
   481                 DtRec k => list_all (map (pair "x") Us,
   482                   HOLogic.mk_Trueprop (HOLogic.mk_mem (free',
   483                     Const (List.nth (rep_set_names, k),
   484                       HOLogic.mk_setT T)))) :: prems
   485               | _ => prems,
   486             snd (foldr mk_abs_fun (j', free) Ts) :: ts)
   487           end;
   488 
   489         val (_, _, prems, ts) = foldr mk_prem (1, 1, [], []) cargs;
   490         val concl = HOLogic.mk_Trueprop (HOLogic.mk_mem
   491           (list_comb (Const (cname, map fastype_of ts ---> T), ts),
   492            Const (s, HOLogic.mk_setT T)))
   493       in Logic.list_implies (prems, concl)
   494       end;
   495 
   496     val (intr_ts, ind_consts) =
   497       apfst List.concat (ListPair.unzip (List.mapPartial
   498         (fn ((_, ("Nominal.noption", _, _)), _) => NONE
   499           | ((i, (_, _, constrs)), rep_set_name) =>
   500               let val T = nth_dtyp i
   501               in SOME (map (make_intr rep_set_name T) constrs,
   502                 Const (rep_set_name, HOLogic.mk_setT T))
   503               end)
   504                 (descr ~~ rep_set_names)));
   505 
   506     val (thy4, {raw_induct = rep_induct, intrs = rep_intrs, ...}) =
   507       setmp InductivePackage.quiet_mode false
   508         (InductivePackage.add_inductive_i false true big_rep_name false true false
   509            ind_consts (map (fn x => (("", x), [])) intr_ts) []) thy3;
   510 
   511     (**** Prove that representing set is closed under permutation ****)
   512 
   513     val _ = warning "proving closure under permutation...";
   514 
   515     val perm_indnames' = List.mapPartial
   516       (fn (x, (_, ("Nominal.noption", _, _))) => NONE | (x, _) => SOME x)
   517       (perm_indnames ~~ descr);
   518 
   519     fun mk_perm_closed name = map (fn th => standard (th RS mp))
   520       (List.take (split_conj_thm (Goal.prove_global thy4 [] []
   521         (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj (map
   522            (fn (S, x) =>
   523               let
   524                 val S = map_term_types (map_type_tfree
   525                   (fn (s, cs) => TFree (s, cs union cp_classes))) S;
   526                 val T = HOLogic.dest_setT (fastype_of S);
   527                 val permT = mk_permT (Type (name, []))
   528               in HOLogic.mk_imp (HOLogic.mk_mem (Free (x, T), S),
   529                 HOLogic.mk_mem (Const ("Nominal.perm", permT --> T --> T) $
   530                   Free ("pi", permT) $ Free (x, T), S))
   531               end) (ind_consts ~~ perm_indnames'))))
   532         (fn _ => EVERY (* CU: added perm_fun_def in the final tactic in order to deal with funs *)
   533            [indtac rep_induct [] 1,
   534             ALLGOALS (simp_tac (simpset_of thy4 addsimps
   535               (symmetric perm_fun_def :: PureThy.get_thms thy4 (Name ("abs_perm"))))),
   536             ALLGOALS (resolve_tac rep_intrs 
   537                THEN_ALL_NEW (asm_full_simp_tac (simpset_of thy4 addsimps [perm_fun_def])))])),
   538         length new_type_names));
   539 
   540     (* FIXME: theorems are stored in database for testing only *)
   541     val perm_closed_thmss = map mk_perm_closed atoms;
   542     val (_,thy5) = PureThy.add_thmss [(("perm_closed", List.concat perm_closed_thmss), [])] thy4;
   543 
   544     (**** typedef ****)
   545 
   546     val _ = warning "defining type...";
   547 
   548     val (typedefs, thy6) =
   549       fold_map (fn ((((name, mx), tvs), c), name') => fn thy =>
   550         setmp TypedefPackage.quiet_mode true
   551           (TypedefPackage.add_typedef_i false (SOME name') (name, tvs, mx) c NONE
   552             (rtac exI 1 THEN
   553               QUIET_BREADTH_FIRST (has_fewer_prems 1)
   554               (resolve_tac rep_intrs 1))) thy |> (fn (r, thy) =>
   555         let
   556           val permT = mk_permT (TFree (Name.variant tvs "'a", HOLogic.typeS));
   557           val pi = Free ("pi", permT);
   558           val tvs' = map (fn s => TFree (s, the (AList.lookup op = sorts' s))) tvs;
   559           val T = Type (Sign.intern_type thy name, tvs');
   560           val Const (_, Type (_, [U])) = c
   561         in apfst (pair r o hd)
   562           (PureThy.add_defs_unchecked_i true [(("prm_" ^ name ^ "_def", Logic.mk_equals
   563             (Const ("Nominal.perm", permT --> T --> T) $ pi $ Free ("x", T),
   564              Const (Sign.intern_const thy ("Abs_" ^ name), U --> T) $
   565                (Const ("Nominal.perm", permT --> U --> U) $ pi $
   566                  (Const (Sign.intern_const thy ("Rep_" ^ name), T --> U) $
   567                    Free ("x", T))))), [])] thy)
   568         end))
   569           (types_syntax ~~ tyvars ~~
   570             (List.take (ind_consts, length new_type_names)) ~~ new_type_names) thy5;
   571 
   572     val perm_defs = map snd typedefs;
   573     val Abs_inverse_thms = map (#Abs_inverse o fst) typedefs;
   574     val Rep_inverse_thms = map (#Rep_inverse o fst) typedefs;
   575     val Rep_thms = map (#Rep o fst) typedefs;
   576 
   577     val big_name = space_implode "_" new_type_names;
   578 
   579 
   580     (** prove that new types are in class pt_<name> **)
   581 
   582     val _ = warning "prove that new types are in class pt_<name> ...";
   583 
   584     fun pt_instance ((class, atom), perm_closed_thms) =
   585       fold (fn (((({Abs_inverse, Rep_inverse, Rep, ...},
   586         perm_def), name), tvs), perm_closed) => fn thy =>
   587           AxClass.prove_arity
   588             (Sign.intern_type thy name,
   589               replicate (length tvs) (classes @ cp_classes), [class])
   590             (EVERY [ClassPackage.intro_classes_tac [],
   591               rewrite_goals_tac [perm_def],
   592               asm_full_simp_tac (simpset_of thy addsimps [Rep_inverse]) 1,
   593               asm_full_simp_tac (simpset_of thy addsimps
   594                 [Rep RS perm_closed RS Abs_inverse]) 1,
   595               asm_full_simp_tac (HOL_basic_ss addsimps [PureThy.get_thm thy
   596                 (Name ("pt_" ^ Sign.base_name atom ^ "3"))]) 1]) thy)
   597         (typedefs ~~ new_type_names ~~ tyvars ~~ perm_closed_thms);
   598 
   599 
   600     (** prove that new types are in class cp_<name1>_<name2> **)
   601 
   602     val _ = warning "prove that new types are in class cp_<name1>_<name2> ...";
   603 
   604     fun cp_instance (atom1, perm_closed_thms1) (atom2, perm_closed_thms2) thy =
   605       let
   606         val name = "cp_" ^ Sign.base_name atom1 ^ "_" ^ Sign.base_name atom2;
   607         val class = Sign.intern_class thy name;
   608         val cp1' = PureThy.get_thm thy (Name (name ^ "_inst")) RS cp1
   609       in fold (fn ((((({Abs_inverse, Rep_inverse, Rep, ...},
   610         perm_def), name), tvs), perm_closed1), perm_closed2) => fn thy =>
   611           AxClass.prove_arity
   612             (Sign.intern_type thy name,
   613               replicate (length tvs) (classes @ cp_classes), [class])
   614             (EVERY [ClassPackage.intro_classes_tac [],
   615               rewrite_goals_tac [perm_def],
   616               asm_full_simp_tac (simpset_of thy addsimps
   617                 ((Rep RS perm_closed1 RS Abs_inverse) ::
   618                  (if atom1 = atom2 then []
   619                   else [Rep RS perm_closed2 RS Abs_inverse]))) 1,
   620               cong_tac 1,
   621               rtac refl 1,
   622               rtac cp1' 1]) thy)
   623         (typedefs ~~ new_type_names ~~ tyvars ~~ perm_closed_thms1 ~~
   624           perm_closed_thms2) thy
   625       end;
   626 
   627     val thy7 = fold (fn x => fn thy => thy |>
   628       pt_instance x |>
   629       fold (cp_instance (apfst snd x)) (atoms ~~ perm_closed_thmss))
   630         (classes ~~ atoms ~~ perm_closed_thmss) thy6;
   631 
   632     (**** constructors ****)
   633 
   634     fun mk_abs_fun (x, t) =
   635       let
   636         val T = fastype_of x;
   637         val U = fastype_of t
   638       in
   639         Const ("Nominal.abs_fun", T --> U --> T -->
   640           Type ("Nominal.noption", [U])) $ x $ t
   641       end;
   642 
   643     val (ty_idxs, _) = foldl
   644       (fn ((i, ("Nominal.noption", _, _)), p) => p
   645         | ((i, _), (ty_idxs, j)) => (ty_idxs @ [(i, j)], j + 1)) ([], 0) descr;
   646 
   647     fun reindex (DtType (s, dts)) = DtType (s, map reindex dts)
   648       | reindex (DtRec i) = DtRec (the (AList.lookup op = ty_idxs i))
   649       | reindex dt = dt;
   650 
   651     fun strip_suffix i s = implode (List.take (explode s, size s - i));
   652 
   653     (** strips the "_Rep" in type names *)
   654     fun strip_nth_name i s = 
   655       let val xs = NameSpace.unpack s; 
   656       in NameSpace.pack (Library.nth_map (length xs - i) (strip_suffix 4) xs) end;
   657 
   658     val (descr'', ndescr) = ListPair.unzip (List.mapPartial
   659       (fn (i, ("Nominal.noption", _, _)) => NONE
   660         | (i, (s, dts, constrs)) =>
   661              let
   662                val SOME index = AList.lookup op = ty_idxs i;
   663                val (constrs1, constrs2) = ListPair.unzip
   664                  (map (fn (cname, cargs) => apfst (pair (strip_nth_name 2 (strip_nth_name 1 cname)))
   665                    (foldl_map (fn (dts, dt) =>
   666                      let val (dts', dt') = strip_option dt
   667                      in (dts @ dts' @ [reindex dt'], (length dts, length dts')) end)
   668                        ([], cargs))) constrs)
   669              in SOME ((index, (strip_nth_name 1 s,  map reindex dts, constrs1)),
   670                (index, constrs2))
   671              end) descr);
   672 
   673     val (descr1, descr2) = chop (length new_type_names) descr'';
   674     val descr' = [descr1, descr2];
   675 
   676     fun partition_cargs idxs xs = map (fn (i, j) =>
   677       (List.take (List.drop (xs, i), j), List.nth (xs, i + j))) idxs;
   678 
   679     val pdescr = map (fn ((i, (s, dts, constrs)), (_, idxss)) => (i, (s, dts,
   680       map (fn ((cname, cargs), idxs) => (cname, partition_cargs idxs cargs))
   681         (constrs ~~ idxss)))) (descr'' ~~ ndescr);
   682 
   683     fun nth_dtyp' i = typ_of_dtyp descr'' sorts' (DtRec i);
   684 
   685     val rep_names = map (fn s =>
   686       Sign.intern_const thy7 ("Rep_" ^ s)) new_type_names;
   687     val abs_names = map (fn s =>
   688       Sign.intern_const thy7 ("Abs_" ^ s)) new_type_names;
   689 
   690     val recTs' = List.mapPartial
   691       (fn ((_, ("Nominal.noption", _, _)), T) => NONE
   692         | (_, T) => SOME T) (descr ~~ get_rec_types descr sorts');
   693     val recTs = get_rec_types descr'' sorts';
   694     val newTs' = Library.take (length new_type_names, recTs');
   695     val newTs = Library.take (length new_type_names, recTs);
   696 
   697     val full_new_type_names = map (Sign.full_name (sign_of thy)) new_type_names;
   698 
   699     fun make_constr_def tname T T' ((thy, defs, eqns),
   700         (((cname_rep, _), (cname, cargs)), (cname', mx))) =
   701       let
   702         fun constr_arg ((dts, dt), (j, l_args, r_args)) =
   703           let
   704             val xs = map (fn (dt, i) => mk_Free "x" (typ_of_dtyp descr'' sorts' dt) i)
   705               (dts ~~ (j upto j + length dts - 1))
   706             val x = mk_Free "x" (typ_of_dtyp descr'' sorts' dt) (j + length dts)
   707           in
   708             (j + length dts + 1,
   709              xs @ x :: l_args,
   710              foldr mk_abs_fun
   711                (case dt of
   712                   DtRec k => if k < length new_type_names then
   713                       Const (List.nth (rep_names, k), typ_of_dtyp descr'' sorts' dt -->
   714                         typ_of_dtyp descr sorts' dt) $ x
   715                     else error "nested recursion not (yet) supported"
   716                 | _ => x) xs :: r_args)
   717           end
   718 
   719         val (_, l_args, r_args) = foldr constr_arg (1, [], []) cargs;
   720         val abs_name = Sign.intern_const (Theory.sign_of thy) ("Abs_" ^ tname);
   721         val rep_name = Sign.intern_const (Theory.sign_of thy) ("Rep_" ^ tname);
   722         val constrT = map fastype_of l_args ---> T;
   723         val lhs = list_comb (Const (cname, constrT), l_args);
   724         val rhs = list_comb (Const (cname_rep, map fastype_of r_args ---> T'), r_args);
   725         val def = Logic.mk_equals (lhs, Const (abs_name, T' --> T) $ rhs);
   726         val eqn = HOLogic.mk_Trueprop (HOLogic.mk_eq
   727           (Const (rep_name, T --> T') $ lhs, rhs));
   728         val def_name = (Sign.base_name cname) ^ "_def";
   729         val ([def_thm], thy') = thy |>
   730           Theory.add_consts_i [(cname', constrT, mx)] |>
   731           (PureThy.add_defs_i false o map Thm.no_attributes) [(def_name, def)]
   732       in (thy', defs @ [def_thm], eqns @ [eqn]) end;
   733 
   734     fun dt_constr_defs ((thy, defs, eqns, dist_lemmas), ((((((_, (_, _, constrs)),
   735         (_, (_, _, constrs'))), tname), T), T'), constr_syntax)) =
   736       let
   737         val rep_const = cterm_of thy
   738           (Const (Sign.intern_const thy ("Rep_" ^ tname), T --> T'));
   739         val dist = standard (cterm_instantiate [(cterm_of thy distinct_f, rep_const)] distinct_lemma);
   740         val (thy', defs', eqns') = Library.foldl (make_constr_def tname T T')
   741           ((Theory.add_path tname thy, defs, []), constrs ~~ constrs' ~~ constr_syntax)
   742       in
   743         (parent_path flat_names thy', defs', eqns @ [eqns'], dist_lemmas @ [dist])
   744       end;
   745 
   746     val (thy8, constr_defs, constr_rep_eqns, dist_lemmas) = Library.foldl dt_constr_defs
   747       ((thy7, [], [], []), List.take (descr, length new_type_names) ~~
   748         List.take (pdescr, length new_type_names) ~~
   749         new_type_names ~~ newTs ~~ newTs' ~~ constr_syntax);
   750 
   751     val abs_inject_thms = map (fn tname =>
   752       PureThy.get_thm thy8 (Name ("Abs_" ^ tname ^ "_inject"))) new_type_names;
   753 
   754     val rep_inject_thms = map (fn tname =>
   755       PureThy.get_thm thy8 (Name ("Rep_" ^ tname ^ "_inject"))) new_type_names;
   756 
   757     val rep_thms = map (fn tname =>
   758       PureThy.get_thm thy8 (Name ("Rep_" ^ tname))) new_type_names;
   759 
   760     val rep_inverse_thms = map (fn tname =>
   761       PureThy.get_thm thy8 (Name ("Rep_" ^ tname ^ "_inverse"))) new_type_names;
   762 
   763     (* prove theorem  Rep_i (Constr_j ...) = Constr'_j ...  *)
   764     
   765     fun prove_constr_rep_thm eqn =
   766       let
   767         val inj_thms = map (fn r => r RS iffD1) abs_inject_thms;
   768         val rewrites = constr_defs @ map mk_meta_eq rep_inverse_thms
   769       in Goal.prove_global thy8 [] [] eqn (fn _ => EVERY
   770         [resolve_tac inj_thms 1,
   771          rewrite_goals_tac rewrites,
   772          rtac refl 3,
   773          resolve_tac rep_intrs 2,
   774          REPEAT (resolve_tac rep_thms 1)])
   775       end;
   776 
   777     val constr_rep_thmss = map (map prove_constr_rep_thm) constr_rep_eqns;
   778 
   779     (* prove theorem  pi \<bullet> Rep_i x = Rep_i (pi \<bullet> x) *)
   780 
   781     fun prove_perm_rep_perm (atom, perm_closed_thms) = map (fn th =>
   782       let
   783         val _ $ (_ $ (Rep $ x) $ _) = Logic.unvarify (prop_of th);
   784         val Type ("fun", [T, U]) = fastype_of Rep;
   785         val permT = mk_permT (Type (atom, []));
   786         val pi = Free ("pi", permT);
   787       in
   788         Goal.prove_global thy8 [] [] (HOLogic.mk_Trueprop (HOLogic.mk_eq
   789             (Const ("Nominal.perm", permT --> U --> U) $ pi $ (Rep $ x),
   790              Rep $ (Const ("Nominal.perm", permT --> T --> T) $ pi $ x))))
   791           (fn _ => simp_tac (HOL_basic_ss addsimps (perm_defs @ Abs_inverse_thms @
   792             perm_closed_thms @ Rep_thms)) 1)
   793       end) Rep_thms;
   794 
   795     val perm_rep_perm_thms = List.concat (map prove_perm_rep_perm
   796       (atoms ~~ perm_closed_thmss));
   797 
   798     (* prove distinctness theorems *)
   799 
   800     val distinct_props = setmp DatatypeProp.dtK 1000
   801       (DatatypeProp.make_distincts new_type_names descr' sorts') thy8;
   802 
   803     val dist_rewrites = map (fn (rep_thms, dist_lemma) =>
   804       dist_lemma::(rep_thms @ [In0_eq, In1_eq, In0_not_In1, In1_not_In0]))
   805         (constr_rep_thmss ~~ dist_lemmas);
   806 
   807     fun prove_distinct_thms (_, []) = []
   808       | prove_distinct_thms (p as (rep_thms, dist_lemma), t::ts) =
   809           let
   810             val dist_thm = Goal.prove_global thy8 [] [] t (fn _ =>
   811               simp_tac (simpset_of thy8 addsimps (dist_lemma :: rep_thms)) 1)
   812           in dist_thm::(standard (dist_thm RS not_sym))::
   813             (prove_distinct_thms (p, ts))
   814           end;
   815 
   816     val distinct_thms = map prove_distinct_thms
   817       (constr_rep_thmss ~~ dist_lemmas ~~ distinct_props);
   818 
   819     (** prove equations for permutation functions **)
   820 
   821     val abs_perm = PureThy.get_thms thy8 (Name "abs_perm"); (* FIXME *)
   822 
   823     val perm_simps' = map (fn (((i, (_, _, constrs)), tname), constr_rep_thms) =>
   824       let val T = nth_dtyp' i
   825       in List.concat (map (fn (atom, perm_closed_thms) =>
   826           map (fn ((cname, dts), constr_rep_thm) => 
   827         let
   828           val cname = Sign.intern_const thy8
   829             (NameSpace.append tname (Sign.base_name cname));
   830           val permT = mk_permT (Type (atom, []));
   831           val pi = Free ("pi", permT);
   832 
   833           fun perm t =
   834             let val T = fastype_of t
   835             in Const ("Nominal.perm", permT --> T --> T) $ pi $ t end;
   836 
   837           fun constr_arg ((dts, dt), (j, l_args, r_args)) =
   838             let
   839               val Ts = map (typ_of_dtyp descr'' sorts') dts;
   840               val xs = map (fn (T, i) => mk_Free "x" T i)
   841                 (Ts ~~ (j upto j + length dts - 1))
   842               val x = mk_Free "x" (typ_of_dtyp descr'' sorts' dt) (j + length dts)
   843             in
   844               (j + length dts + 1,
   845                xs @ x :: l_args,
   846                map perm (xs @ [x]) @ r_args)
   847             end
   848 
   849           val (_, l_args, r_args) = foldr constr_arg (1, [], []) dts;
   850           val c = Const (cname, map fastype_of l_args ---> T)
   851         in
   852           Goal.prove_global thy8 [] []
   853             (HOLogic.mk_Trueprop (HOLogic.mk_eq
   854               (perm (list_comb (c, l_args)), list_comb (c, r_args))))
   855             (fn _ => EVERY
   856               [simp_tac (simpset_of thy8 addsimps (constr_rep_thm :: perm_defs)) 1,
   857                simp_tac (HOL_basic_ss addsimps (Rep_thms @ Abs_inverse_thms @
   858                  constr_defs @ perm_closed_thms)) 1,
   859                TRY (simp_tac (HOL_basic_ss addsimps
   860                  (symmetric perm_fun_def :: abs_perm)) 1),
   861                TRY (simp_tac (HOL_basic_ss addsimps
   862                  (perm_fun_def :: perm_defs @ Rep_thms @ Abs_inverse_thms @
   863                     perm_closed_thms)) 1)])
   864         end) (constrs ~~ constr_rep_thms)) (atoms ~~ perm_closed_thmss))
   865       end) (List.take (pdescr, length new_type_names) ~~ new_type_names ~~ constr_rep_thmss);
   866 
   867     (** prove injectivity of constructors **)
   868 
   869     val rep_inject_thms' = map (fn th => th RS sym) rep_inject_thms;
   870     val alpha = PureThy.get_thms thy8 (Name "alpha");
   871     val abs_fresh = PureThy.get_thms thy8 (Name "abs_fresh");
   872     val fresh_def = PureThy.get_thm thy8 (Name "fresh_def");
   873     val supp_def = PureThy.get_thm thy8 (Name "supp_def");
   874 
   875     val inject_thms = map (fn (((i, (_, _, constrs)), tname), constr_rep_thms) =>
   876       let val T = nth_dtyp' i
   877       in List.mapPartial (fn ((cname, dts), constr_rep_thm) =>
   878         if null dts then NONE else SOME
   879         let
   880           val cname = Sign.intern_const thy8
   881             (NameSpace.append tname (Sign.base_name cname));
   882 
   883           fun make_inj ((dts, dt), (j, args1, args2, eqs)) =
   884             let
   885               val Ts_idx = map (typ_of_dtyp descr'' sorts') dts ~~ (j upto j + length dts - 1);
   886               val xs = map (fn (T, i) => mk_Free "x" T i) Ts_idx;
   887               val ys = map (fn (T, i) => mk_Free "y" T i) Ts_idx;
   888               val x = mk_Free "x" (typ_of_dtyp descr'' sorts' dt) (j + length dts);
   889               val y = mk_Free "y" (typ_of_dtyp descr'' sorts' dt) (j + length dts)
   890             in
   891               (j + length dts + 1,
   892                xs @ (x :: args1), ys @ (y :: args2),
   893                HOLogic.mk_eq
   894                  (foldr mk_abs_fun x xs, foldr mk_abs_fun y ys) :: eqs)
   895             end;
   896 
   897           val (_, args1, args2, eqs) = foldr make_inj (1, [], [], []) dts;
   898           val Ts = map fastype_of args1;
   899           val c = Const (cname, Ts ---> T)
   900         in
   901           Goal.prove_global thy8 [] [] (HOLogic.mk_Trueprop (HOLogic.mk_eq
   902               (HOLogic.mk_eq (list_comb (c, args1), list_comb (c, args2)),
   903                foldr1 HOLogic.mk_conj eqs)))
   904             (fn _ => EVERY
   905                [asm_full_simp_tac (simpset_of thy8 addsimps (constr_rep_thm ::
   906                   rep_inject_thms')) 1,
   907                 TRY (asm_full_simp_tac (HOL_basic_ss addsimps (fresh_def :: supp_def ::
   908                   alpha @ abs_perm @ abs_fresh @ rep_inject_thms @
   909                   perm_rep_perm_thms)) 1),
   910                 TRY (asm_full_simp_tac (HOL_basic_ss addsimps (perm_fun_def ::
   911                   expand_fun_eq :: rep_inject_thms @ perm_rep_perm_thms)) 1)])
   912         end) (constrs ~~ constr_rep_thms)
   913       end) (List.take (pdescr, length new_type_names) ~~ new_type_names ~~ constr_rep_thmss);
   914 
   915     (** equations for support and freshness **)
   916 
   917     val Un_assoc = PureThy.get_thm thy8 (Name "Un_assoc");
   918     val de_Morgan_conj = PureThy.get_thm thy8 (Name "de_Morgan_conj");
   919     val Collect_disj_eq = PureThy.get_thm thy8 (Name "Collect_disj_eq");
   920     val finite_Un = PureThy.get_thm thy8 (Name "finite_Un");
   921 
   922     val (supp_thms, fresh_thms) = ListPair.unzip (map ListPair.unzip
   923       (map (fn ((((i, (_, _, constrs)), tname), inject_thms'), perm_thms') =>
   924       let val T = nth_dtyp' i
   925       in List.concat (map (fn (cname, dts) => map (fn atom =>
   926         let
   927           val cname = Sign.intern_const thy8
   928             (NameSpace.append tname (Sign.base_name cname));
   929           val atomT = Type (atom, []);
   930 
   931           fun process_constr ((dts, dt), (j, args1, args2)) =
   932             let
   933               val Ts_idx = map (typ_of_dtyp descr'' sorts') dts ~~ (j upto j + length dts - 1);
   934               val xs = map (fn (T, i) => mk_Free "x" T i) Ts_idx;
   935               val x = mk_Free "x" (typ_of_dtyp descr'' sorts' dt) (j + length dts)
   936             in
   937               (j + length dts + 1,
   938                xs @ (x :: args1), foldr mk_abs_fun x xs :: args2)
   939             end;
   940 
   941           val (_, args1, args2) = foldr process_constr (1, [], []) dts;
   942           val Ts = map fastype_of args1;
   943           val c = list_comb (Const (cname, Ts ---> T), args1);
   944           fun supp t =
   945             Const ("Nominal.supp", fastype_of t --> HOLogic.mk_setT atomT) $ t;
   946           fun fresh t =
   947             Const ("Nominal.fresh", atomT --> fastype_of t --> HOLogic.boolT) $
   948               Free ("a", atomT) $ t;
   949           val supp_thm = Goal.prove_global thy8 [] []
   950               (HOLogic.mk_Trueprop (HOLogic.mk_eq
   951                 (supp c,
   952                  if null dts then Const ("{}", HOLogic.mk_setT atomT)
   953                  else foldr1 (HOLogic.mk_binop "op Un") (map supp args2))))
   954             (fn _ =>
   955               simp_tac (HOL_basic_ss addsimps (supp_def ::
   956                  Un_assoc :: de_Morgan_conj :: Collect_disj_eq :: finite_Un ::
   957                  symmetric empty_def :: Finites.emptyI :: simp_thms @
   958                  abs_perm @ abs_fresh @ inject_thms' @ perm_thms')) 1)
   959         in
   960           (supp_thm,
   961            Goal.prove_global thy8 [] [] (HOLogic.mk_Trueprop (HOLogic.mk_eq
   962               (fresh c,
   963                if null dts then HOLogic.true_const
   964                else foldr1 HOLogic.mk_conj (map fresh args2))))
   965              (fn _ =>
   966                simp_tac (simpset_of thy8 addsimps [fresh_def, supp_thm]) 1))
   967         end) atoms) constrs)
   968       end) (List.take (pdescr, length new_type_names) ~~ new_type_names ~~ inject_thms ~~ perm_simps')));
   969 
   970     (**** weak induction theorem ****)
   971 
   972     fun mk_indrule_lemma ((prems, concls), (((i, _), T), U)) =
   973       let
   974         val Rep_t = Const (List.nth (rep_names, i), T --> U) $
   975           mk_Free "x" T i;
   976 
   977         val Abs_t =  Const (List.nth (abs_names, i), U --> T)
   978 
   979       in (prems @ [HOLogic.imp $ HOLogic.mk_mem (Rep_t,
   980             Const (List.nth (rep_set_names, i), HOLogic.mk_setT U)) $
   981               (mk_Free "P" (T --> HOLogic.boolT) (i + 1) $ (Abs_t $ Rep_t))],
   982           concls @ [mk_Free "P" (T --> HOLogic.boolT) (i + 1) $ mk_Free "x" T i])
   983       end;
   984 
   985     val (indrule_lemma_prems, indrule_lemma_concls) =
   986       Library.foldl mk_indrule_lemma (([], []), (descr'' ~~ recTs ~~ recTs'));
   987 
   988     val indrule_lemma = Goal.prove_global thy8 [] []
   989       (Logic.mk_implies
   990         (HOLogic.mk_Trueprop (mk_conj indrule_lemma_prems),
   991          HOLogic.mk_Trueprop (mk_conj indrule_lemma_concls))) (fn _ => EVERY
   992            [REPEAT (etac conjE 1),
   993             REPEAT (EVERY
   994               [TRY (rtac conjI 1), full_simp_tac (HOL_basic_ss addsimps Rep_inverse_thms) 1,
   995                etac mp 1, resolve_tac Rep_thms 1])]);
   996 
   997     val Ps = map head_of (HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of indrule_lemma)));
   998     val frees = if length Ps = 1 then [Free ("P", snd (dest_Var (hd Ps)))] else
   999       map (Free o apfst fst o dest_Var) Ps;
  1000     val indrule_lemma' = cterm_instantiate
  1001       (map (cterm_of thy8) Ps ~~ map (cterm_of thy8) frees) indrule_lemma;
  1002 
  1003     val Abs_inverse_thms' = map (fn r => r RS subst) Abs_inverse_thms;
  1004 
  1005     val dt_induct_prop = DatatypeProp.make_ind descr' sorts';
  1006     val dt_induct = Goal.prove_global thy8 []
  1007       (Logic.strip_imp_prems dt_induct_prop) (Logic.strip_imp_concl dt_induct_prop)
  1008       (fn prems => EVERY
  1009         [rtac indrule_lemma' 1,
  1010          (DatatypeAux.indtac rep_induct THEN_ALL_NEW ObjectLogic.atomize_tac) 1,
  1011          EVERY (map (fn (prem, r) => (EVERY
  1012            [REPEAT (eresolve_tac Abs_inverse_thms' 1),
  1013             simp_tac (HOL_basic_ss addsimps [symmetric r]) 1,
  1014             DEPTH_SOLVE_1 (ares_tac [prem] 1 ORELSE etac allE 1)]))
  1015                 (prems ~~ constr_defs))]);
  1016 
  1017     val case_names_induct = mk_case_names_induct descr'';
  1018 
  1019     (**** prove that new datatypes have finite support ****)
  1020 
  1021     val _ = warning "proving finite support for the new datatype";
  1022 
  1023     val indnames = DatatypeProp.make_tnames recTs;
  1024 
  1025     val abs_supp = PureThy.get_thms thy8 (Name "abs_supp");
  1026     val supp_atm = PureThy.get_thms thy8 (Name "supp_atm");
  1027 
  1028     val finite_supp_thms = map (fn atom =>
  1029       let val atomT = Type (atom, [])
  1030       in map standard (List.take
  1031         (split_conj_thm (Goal.prove_global thy8 [] [] (HOLogic.mk_Trueprop
  1032            (foldr1 HOLogic.mk_conj (map (fn (s, T) => HOLogic.mk_mem
  1033              (Const ("Nominal.supp", T --> HOLogic.mk_setT atomT) $ Free (s, T),
  1034               Const ("Finite_Set.Finites", HOLogic.mk_setT (HOLogic.mk_setT atomT))))
  1035                (indnames ~~ recTs))))
  1036            (fn _ => indtac dt_induct indnames 1 THEN
  1037             ALLGOALS (asm_full_simp_tac (simpset_of thy8 addsimps
  1038               (abs_supp @ supp_atm @
  1039                PureThy.get_thms thy8 (Name ("fs_" ^ Sign.base_name atom ^ "1")) @
  1040                List.concat supp_thms))))),
  1041          length new_type_names))
  1042       end) atoms;
  1043 
  1044     val simp_atts = replicate (length new_type_names) [Simplifier.simp_add];
  1045 
  1046     val (_, thy9) = thy8 |>
  1047       Theory.add_path big_name |>
  1048       PureThy.add_thms [(("induct_weak", dt_induct), [case_names_induct])] ||>>
  1049       PureThy.add_thmss [(("inducts_weak", projections dt_induct), [case_names_induct])] ||>
  1050       Theory.parent_path ||>>
  1051       DatatypeAux.store_thmss_atts "distinct" new_type_names simp_atts distinct_thms ||>>
  1052       DatatypeAux.store_thmss "constr_rep" new_type_names constr_rep_thmss ||>>
  1053       DatatypeAux.store_thmss_atts "perm" new_type_names simp_atts perm_simps' ||>>
  1054       DatatypeAux.store_thmss "inject" new_type_names inject_thms ||>>
  1055       DatatypeAux.store_thmss "supp" new_type_names supp_thms ||>>
  1056       DatatypeAux.store_thmss_atts "fresh" new_type_names simp_atts fresh_thms ||>
  1057       fold (fn (atom, ths) => fn thy =>
  1058         let val class = Sign.intern_class thy ("fs_" ^ Sign.base_name atom)
  1059         in fold (fn T => AxClass.prove_arity
  1060             (fst (dest_Type T),
  1061               replicate (length sorts) [class], [class])
  1062             (ClassPackage.intro_classes_tac [] THEN resolve_tac ths 1)) newTs thy
  1063         end) (atoms ~~ finite_supp_thms);
  1064 
  1065     (**** strong induction theorem ****)
  1066 
  1067     val pnames = if length descr'' = 1 then ["P"]
  1068       else map (fn i => "P" ^ string_of_int i) (1 upto length descr'');
  1069     val ind_sort = if null dt_atomTs then HOLogic.typeS
  1070       else Sign.certify_sort thy9 (map (fn T => Sign.intern_class thy9 ("fs_" ^
  1071         Sign.base_name (fst (dest_Type T)))) dt_atomTs);
  1072     val fsT = TFree ("'n", ind_sort);
  1073     val fsT' = TFree ("'n", HOLogic.typeS);
  1074 
  1075     val fresh_fs = map (fn (s, T) => (T, Free (s, fsT' --> HOLogic.mk_setT T)))
  1076       (DatatypeProp.indexify_names (replicate (length dt_atomTs) "f") ~~ dt_atomTs);
  1077 
  1078     fun make_pred fsT i T =
  1079       Free (List.nth (pnames, i), fsT --> T --> HOLogic.boolT);
  1080 
  1081     fun mk_fresh1 xs [] = []
  1082       | mk_fresh1 xs ((y as (_, T)) :: ys) = map (fn x => HOLogic.mk_Trueprop
  1083             (HOLogic.mk_not (HOLogic.mk_eq (Free y, Free x))))
  1084               (filter (fn (_, U) => T = U) (rev xs)) @
  1085           mk_fresh1 (y :: xs) ys;
  1086 
  1087     fun mk_fresh2 xss [] = []
  1088       | mk_fresh2 xss ((p as (ys, _)) :: yss) = List.concat (map (fn y as (_, T) =>
  1089             map (fn (_, x as (_, U)) => HOLogic.mk_Trueprop
  1090               (Const ("Nominal.fresh", T --> U --> HOLogic.boolT) $ Free y $ Free x))
  1091                 (rev xss @ yss)) ys) @
  1092           mk_fresh2 (p :: xss) yss;
  1093 
  1094     fun make_ind_prem fsT f k T ((cname, cargs), idxs) =
  1095       let
  1096         val recs = List.filter is_rec_type cargs;
  1097         val Ts = map (typ_of_dtyp descr'' sorts') cargs;
  1098         val recTs' = map (typ_of_dtyp descr'' sorts') recs;
  1099         val tnames = Name.variant_list pnames (DatatypeProp.make_tnames Ts);
  1100         val rec_tnames = map fst (List.filter (is_rec_type o snd) (tnames ~~ cargs));
  1101         val frees = tnames ~~ Ts;
  1102         val frees' = partition_cargs idxs frees;
  1103         val z = (Name.variant tnames "z", fsT);
  1104 
  1105         fun mk_prem ((dt, s), T) =
  1106           let
  1107             val (Us, U) = strip_type T;
  1108             val l = length Us
  1109           in list_all (z :: map (pair "x") Us, HOLogic.mk_Trueprop
  1110             (make_pred fsT (body_index dt) U $ Bound l $ app_bnds (Free (s, T)) l))
  1111           end;
  1112 
  1113         val prems = map mk_prem (recs ~~ rec_tnames ~~ recTs');
  1114         val prems' = map (fn p as (_, T) => HOLogic.mk_Trueprop
  1115             (f T (Free p) (Free z))) (List.concat (map fst frees')) @
  1116           mk_fresh1 [] (List.concat (map fst frees')) @
  1117           mk_fresh2 [] frees'
  1118 
  1119       in list_all_free (frees @ [z], Logic.list_implies (prems' @ prems,
  1120         HOLogic.mk_Trueprop (make_pred fsT k T $ Free z $
  1121           list_comb (Const (cname, Ts ---> T), map Free frees))))
  1122       end;
  1123 
  1124     val ind_prems = List.concat (map (fn (((i, (_, _, constrs)), (_, idxss)), T) =>
  1125       map (make_ind_prem fsT (fn T => fn t => fn u =>
  1126         Const ("Nominal.fresh", T --> fsT --> HOLogic.boolT) $ t $ u) i T)
  1127           (constrs ~~ idxss)) (descr'' ~~ ndescr ~~ recTs));
  1128     val tnames = DatatypeProp.make_tnames recTs;
  1129     val zs = Name.variant_list tnames (replicate (length descr'') "z");
  1130     val ind_concl = HOLogic.mk_Trueprop (foldr1 (HOLogic.mk_binop "op &")
  1131       (map (fn ((((i, _), T), tname), z) =>
  1132         make_pred fsT i T $ Free (z, fsT) $ Free (tname, T))
  1133         (descr'' ~~ recTs ~~ tnames ~~ zs)));
  1134     val induct = Logic.list_implies (ind_prems, ind_concl);
  1135 
  1136     val ind_prems' =
  1137       map (fn (_, f as Free (_, T)) => list_all_free ([("x", fsT')],
  1138         HOLogic.mk_Trueprop (HOLogic.mk_mem (f $ Free ("x", fsT'),
  1139           Const ("Finite_Set.Finites", HOLogic.mk_setT (body_type T)))))) fresh_fs @
  1140       List.concat (map (fn (((i, (_, _, constrs)), (_, idxss)), T) =>
  1141         map (make_ind_prem fsT' (fn T => fn t => fn u => HOLogic.Not $
  1142           HOLogic.mk_mem (t, the (AList.lookup op = fresh_fs T) $ u)) i T)
  1143             (constrs ~~ idxss)) (descr'' ~~ ndescr ~~ recTs));
  1144     val ind_concl' = HOLogic.mk_Trueprop (foldr1 (HOLogic.mk_binop "op &")
  1145       (map (fn ((((i, _), T), tname), z) =>
  1146         make_pred fsT' i T $ Free (z, fsT') $ Free (tname, T))
  1147         (descr'' ~~ recTs ~~ tnames ~~ zs)));
  1148     val induct' = Logic.list_implies (ind_prems', ind_concl');
  1149 
  1150     fun mk_perm Ts (t, u) =
  1151       let
  1152         val T = fastype_of1 (Ts, t);
  1153         val U = fastype_of1 (Ts, u)
  1154       in Const ("Nominal.perm", T --> U --> U) $ t $ u end;
  1155 
  1156     val aux_ind_vars =
  1157       (DatatypeProp.indexify_names (replicate (length dt_atomTs) "pi") ~~
  1158        map mk_permT dt_atomTs) @ [("z", fsT')];
  1159     val aux_ind_Ts = rev (map snd aux_ind_vars);
  1160     val aux_ind_concl = HOLogic.mk_Trueprop (foldr1 (HOLogic.mk_binop "op &")
  1161       (map (fn (((i, _), T), tname) =>
  1162         HOLogic.list_all (aux_ind_vars, make_pred fsT' i T $ Bound 0 $
  1163           foldr (mk_perm aux_ind_Ts) (Free (tname, T))
  1164             (map Bound (length dt_atomTs downto 1))))
  1165         (descr'' ~~ recTs ~~ tnames)));
  1166 
  1167     fun mk_ind_perm i k p l vs j =
  1168       let
  1169         val n = length vs;
  1170         val Ts = map snd vs;
  1171         val T = List.nth (Ts, i - j);
  1172         val pT = NominalAtoms.mk_permT T
  1173       in
  1174         Const ("List.list.Cons", HOLogic.mk_prodT (T, T) --> pT --> pT) $
  1175           (HOLogic.pair_const T T $ Bound (l - j) $ foldr (mk_perm Ts)
  1176             (Bound (i - j))
  1177             (map (mk_ind_perm i k p l vs) (j - 1 downto 0) @
  1178              map Bound (n - k - 1 downto n - k - p))) $
  1179           Const ("List.list.Nil", pT)
  1180       end;
  1181 
  1182     fun mk_fresh i i' j k p l is vs _ _ =
  1183       let
  1184         val n = length vs;
  1185         val Ts = map snd vs;
  1186         val T = List.nth (Ts, n - i - 1 - j);
  1187         val f = the (AList.lookup op = fresh_fs T);
  1188         val U = List.nth (Ts, n - i' - 1);
  1189         val S = HOLogic.mk_setT T;
  1190         val prms' = map Bound (n - k downto n - k - p + 1);
  1191         val prms = map (mk_ind_perm (n - i) k p (n - l) (("a", T) :: vs))
  1192             (j - 1 downto 0) @ prms';
  1193         val bs = rev (List.mapPartial
  1194           (fn ((_, T'), i) => if T = T' then SOME (Bound i) else NONE)
  1195           (List.take (vs, n - k - p - 1) ~~ (1 upto n - k - p - 1)));
  1196         val ts = map (fn i =>
  1197           Const ("Nominal.supp", List.nth (Ts, n - i - 1) --> S) $
  1198             foldr (mk_perm (T :: Ts)) (Bound (n - i)) prms') is
  1199       in
  1200         HOLogic.mk_Trueprop (Const ("Ex", (T --> HOLogic.boolT) --> HOLogic.boolT) $
  1201           Abs ("a", T, HOLogic.Not $
  1202             (Const ("op :", T --> S --> HOLogic.boolT) $ Bound 0 $
  1203               (foldr (fn (t, u) => Const ("insert", T --> S --> S) $ t $ u)
  1204                 (foldl (fn (t, u) => Const ("op Un", S --> S --> S) $ u $ t)
  1205                   (f $ Bound (n - k - p))
  1206                   (Const ("Nominal.supp", U --> S) $
  1207                      foldr (mk_perm (T :: Ts)) (Bound (n - i')) prms :: ts))
  1208                 (foldr (mk_perm (T :: Ts)) (Bound (n - i - j)) prms :: bs)))))
  1209       end;
  1210 
  1211     fun mk_fresh_constr is p vs _ concl =
  1212       let
  1213         val n = length vs;
  1214         val Ts = map snd vs;
  1215         val _ $ (_ $ _ $ t) = concl;
  1216         val c = head_of t;
  1217         val T = body_type (fastype_of c);
  1218         val k = foldr op + 0 (map (fn (_, i) => i + 1) is);
  1219         val ps = map Bound (n - k - 1 downto n - k - p);
  1220         val (_, _, ts, us) = foldl (fn ((_, i), (m, n, ts, us)) =>
  1221           (m - i - 1, n - i,
  1222            ts @ map Bound (n downto n - i + 1) @
  1223              [foldr (mk_perm Ts) (Bound (m - i))
  1224                 (map (mk_ind_perm m k p n vs) (i - 1 downto 0) @ ps)],
  1225            us @ map (fn j => foldr (mk_perm Ts) (Bound j) ps) (m downto m - i)))
  1226           (n - 1, n - k - p - 2, [], []) is
  1227       in
  1228         HOLogic.mk_Trueprop (HOLogic.eq_const T $ list_comb (c, ts) $ list_comb (c, us))
  1229       end;
  1230 
  1231     val abs_fun_finite_supp = PureThy.get_thm thy9 (Name "abs_fun_finite_supp");
  1232 
  1233     val at_finite_select = PureThy.get_thm thy9 (Name "at_finite_select");
  1234 
  1235     val induct_aux_lemmas = List.concat (map (fn Type (s, _) =>
  1236       [PureThy.get_thm thy9 (Name ("pt_" ^ Sign.base_name s ^ "_inst")),
  1237        PureThy.get_thm thy9 (Name ("fs_" ^ Sign.base_name s ^ "1")),
  1238        PureThy.get_thm thy9 (Name ("at_" ^ Sign.base_name s ^ "_inst"))]) dt_atomTs);
  1239 
  1240     val induct_aux_lemmas' = map (fn Type (s, _) =>
  1241       PureThy.get_thm thy9 (Name ("pt_" ^ Sign.base_name s ^ "2")) RS sym) dt_atomTs;
  1242 
  1243     val fresh_aux = PureThy.get_thms thy9 (Name "fresh_aux");
  1244 
  1245     (**********************************************************************
  1246       The subgoals occurring in the proof of induct_aux have the
  1247       following parameters:
  1248 
  1249         x_1 ... x_k p_1 ... p_m z b_1 ... b_n
  1250 
  1251       where
  1252 
  1253         x_i : constructor arguments (introduced by weak induction rule)
  1254         p_i : permutations (one for each atom type in the data type)
  1255         z   : freshness context
  1256         b_i : newly introduced names for binders (sufficiently fresh)
  1257     ***********************************************************************)
  1258 
  1259     val _ = warning "proving strong induction theorem ...";
  1260 
  1261     val induct_aux = Goal.prove_global thy9 [] ind_prems' ind_concl'
  1262       (fn prems => EVERY
  1263         ([mk_subgoal 1 (K (K (K aux_ind_concl))),
  1264           indtac dt_induct tnames 1] @
  1265          List.concat (map (fn ((_, (_, _, constrs)), (_, constrs')) =>
  1266            List.concat (map (fn ((cname, cargs), is) =>
  1267              [simp_tac (HOL_basic_ss addsimps List.concat perm_simps') 1,
  1268               REPEAT (rtac allI 1)] @
  1269              List.concat (map
  1270                (fn ((_, 0), _) => []
  1271                  | ((i, j), k) => List.concat (map (fn j' =>
  1272                      let
  1273                        val DtType (tname, _) = List.nth (cargs, i + j');
  1274                        val atom = Sign.base_name tname
  1275                      in
  1276                        [mk_subgoal 1 (mk_fresh i (i + j) j'
  1277                           (length cargs) (length dt_atomTs)
  1278                           (length cargs + length dt_atomTs + 1 + i - k)
  1279                           (List.mapPartial (fn (i', j) =>
  1280                              if i = i' then NONE else SOME (i' + j)) is)),
  1281                         rtac at_finite_select 1,
  1282                         rtac (PureThy.get_thm thy9 (Name ("at_" ^ atom ^ "_inst"))) 1,
  1283                         asm_full_simp_tac (simpset_of thy9 addsimps
  1284                           [PureThy.get_thm thy9 (Name ("fs_" ^ atom ^ "1"))]) 1,
  1285                         resolve_tac prems 1,
  1286                         etac exE 1,
  1287                         asm_full_simp_tac (simpset_of thy9 addsimps
  1288                           [symmetric fresh_def]) 1]
  1289                      end) (0 upto j - 1))) (is ~~ (0 upto length is - 1))) @
  1290              (if exists (not o equal 0 o snd) is then
  1291                 [mk_subgoal 1 (mk_fresh_constr is (length dt_atomTs)),
  1292                  asm_full_simp_tac (simpset_of thy9 addsimps
  1293                    (List.concat inject_thms @
  1294                     alpha @ abs_perm @ abs_fresh @ [abs_fun_finite_supp] @
  1295                     induct_aux_lemmas)) 1,
  1296                  dtac sym 1,
  1297                  asm_full_simp_tac (simpset_of thy9) 1,
  1298                  REPEAT (etac conjE 1)]
  1299               else
  1300                 []) @
  1301              [(resolve_tac prems THEN_ALL_NEW
  1302                 (atac ORELSE'
  1303                   SUBGOAL (fn (t, i) => case Logic.strip_assums_concl t of
  1304                       _ $ (Const ("Nominal.fresh", _) $ _ $ _) =>
  1305                         asm_simp_tac (simpset_of thy9 addsimps fresh_aux) i
  1306                     | _ =>
  1307                         asm_simp_tac (simpset_of thy9
  1308                         addsimps induct_aux_lemmas'
  1309                         addsimprocs [perm_simproc]) i))) 1])
  1310                (constrs ~~ constrs'))) (descr'' ~~ ndescr)) @
  1311          [REPEAT (eresolve_tac [conjE, allE_Nil] 1),
  1312           REPEAT (etac allE 1),
  1313           REPEAT (TRY (rtac conjI 1) THEN asm_full_simp_tac (simpset_of thy9) 1)]));
  1314 
  1315     val induct_aux' = Thm.instantiate ([],
  1316       map (fn (s, T) =>
  1317         let val pT = TVar (("'n", 0), HOLogic.typeS) --> T --> HOLogic.boolT
  1318         in (cterm_of thy9 (Var ((s, 0), pT)), cterm_of thy9 (Free (s, pT))) end)
  1319           (pnames ~~ recTs) @
  1320       map (fn (_, f) =>
  1321         let val f' = Logic.varify f
  1322         in (cterm_of thy9 f',
  1323           cterm_of thy9 (Const ("Nominal.supp", fastype_of f')))
  1324         end) fresh_fs) induct_aux;
  1325 
  1326     val induct = Goal.prove_global thy9 [] ind_prems ind_concl
  1327       (fn prems => EVERY
  1328          [rtac induct_aux' 1,
  1329           REPEAT (resolve_tac induct_aux_lemmas 1),
  1330           REPEAT ((resolve_tac prems THEN_ALL_NEW
  1331             (etac meta_spec ORELSE' full_simp_tac (HOL_basic_ss addsimps [fresh_def]))) 1)])
  1332 
  1333     val (_, thy10) = thy9 |>
  1334       Theory.add_path big_name |>
  1335       PureThy.add_thms [(("induct'", induct_aux), [])] ||>>
  1336       PureThy.add_thms [(("induct", induct), [case_names_induct])] ||>>
  1337       PureThy.add_thmss [(("inducts", projections induct), [case_names_induct])] ||>
  1338       Theory.parent_path;
  1339 
  1340     (**** recursion combinator ****)
  1341 
  1342     val _ = warning "defining recursion combinator ...";
  1343 
  1344     val used = foldr add_typ_tfree_names [] recTs;
  1345 
  1346     val (rec_result_Ts', rec_fn_Ts') = DatatypeProp.make_primrec_Ts descr' sorts' used;
  1347 
  1348     val rec_sort = if null dt_atomTs then HOLogic.typeS else 
  1349       let val names = map (Sign.base_name o fst o dest_Type) dt_atomTs
  1350       in Sign.certify_sort thy10 (map (Sign.intern_class thy10)
  1351         (map (fn s => "pt_" ^ s) names @
  1352          List.concat (map (fn s => List.mapPartial (fn s' =>
  1353            if s = s' then NONE
  1354            else SOME ("cp_" ^ s ^ "_" ^ s')) names) names)))
  1355       end;
  1356 
  1357     val rec_result_Ts = map (fn TFree (s, _) => TFree (s, rec_sort)) rec_result_Ts';
  1358     val rec_fn_Ts = map (typ_subst_atomic (rec_result_Ts' ~~ rec_result_Ts)) rec_fn_Ts';
  1359 
  1360     val rec_set_Ts = map (fn (T1, T2) => rec_fn_Ts ---> HOLogic.mk_setT
  1361       (HOLogic.mk_prodT (T1, T2))) (recTs ~~ rec_result_Ts);
  1362 
  1363     val big_rec_name = big_name ^ "_rec_set";
  1364     val rec_set_names = map (Sign.full_name (Theory.sign_of thy10))
  1365       (if length descr'' = 1 then [big_rec_name] else
  1366         (map ((curry (op ^) (big_rec_name ^ "_")) o string_of_int)
  1367           (1 upto (length descr''))));
  1368 
  1369     val rec_fns = map (uncurry (mk_Free "f"))
  1370       (rec_fn_Ts ~~ (1 upto (length rec_fn_Ts)));
  1371     val rec_sets = map (fn c => list_comb (Const c, rec_fns))
  1372       (rec_set_names ~~ rec_set_Ts);
  1373 
  1374     (* introduction rules for graph of recursion function *)
  1375 
  1376     val rec_preds = map (fn (a, T) =>
  1377       Free (a, T --> HOLogic.boolT)) (pnames ~~ rec_result_Ts);
  1378 
  1379     fun make_rec_intr T p rec_set
  1380           ((rec_intr_ts, rec_prems, rec_prems', l), ((cname, cargs), idxs)) =
  1381       let
  1382         val Ts = map (typ_of_dtyp descr'' sorts') cargs;
  1383         val frees = map (fn i => "x" ^ string_of_int i) (1 upto length Ts) ~~ Ts;
  1384         val frees' = partition_cargs idxs frees;
  1385         val recs = List.mapPartial
  1386           (fn ((_, DtRec i), p) => SOME (i, p) | _ => NONE)
  1387           (partition_cargs idxs cargs ~~ frees');
  1388         val frees'' = map (fn i => "y" ^ string_of_int i) (1 upto length recs) ~~
  1389           map (fn (i, _) => List.nth (rec_result_Ts, i)) recs;
  1390         val prems1 = map (fn ((i, (_, x)), y) => HOLogic.mk_Trueprop
  1391           (HOLogic.mk_mem (HOLogic.mk_prod (Free x, Free y),
  1392              List.nth (rec_sets, i)))) (recs ~~ frees'');
  1393         val prems2 =
  1394           map (fn f => map (fn p as (_, T) => HOLogic.mk_Trueprop
  1395             (Const ("Nominal.fresh", T --> fastype_of f --> HOLogic.boolT) $
  1396               Free p $ f)) (List.concat (map fst frees'))) rec_fns;
  1397         val prems3 =
  1398           mk_fresh1 [] (List.concat (map fst frees')) @
  1399           mk_fresh2 [] frees';
  1400         val prems4 = map (fn ((i, _), y) =>
  1401           HOLogic.mk_Trueprop (List.nth (rec_preds, i) $ Free y)) (recs ~~ frees'');
  1402         val result = list_comb (List.nth (rec_fns, l), map Free (frees @ frees''));
  1403         val rec_freshs = map (fn p as (_, T) =>
  1404           Const ("Nominal.fresh", T --> fastype_of result --> HOLogic.boolT) $
  1405             Free p $ result) (List.concat (map (fst o snd) recs));
  1406         val P = HOLogic.mk_Trueprop (p $ result)
  1407       in
  1408         (rec_intr_ts @ [Logic.list_implies (List.concat prems2 @ prems3 @ prems1,
  1409            HOLogic.mk_Trueprop (HOLogic.mk_mem
  1410              (HOLogic.mk_prod (list_comb (Const (cname, Ts ---> T), map Free frees),
  1411                result), rec_set)))],
  1412          rec_prems @ [list_all_free (frees @ frees'', Logic.list_implies (prems4, P))],
  1413          if null rec_freshs then rec_prems'
  1414          else rec_prems' @ [list_all_free (frees @ frees'',
  1415            Logic.list_implies (List.nth (prems2, l) @ prems3 @ [P],
  1416              HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj rec_freshs)))],
  1417          l + 1)
  1418       end;
  1419 
  1420     val (rec_intr_ts, rec_prems, rec_prems', _) =
  1421       Library.foldl (fn (x, ((((d, d'), T), p), rec_set)) =>
  1422         Library.foldl (make_rec_intr T p rec_set) (x, #3 (snd d) ~~ snd d'))
  1423           (([], [], [], 0), descr'' ~~ ndescr ~~ recTs ~~ rec_preds ~~ rec_sets);
  1424 
  1425     val (thy11, {intrs = rec_intrs, elims = rec_elims, raw_induct = rec_induct, ...}) =
  1426       setmp InductivePackage.quiet_mode (!quiet_mode)
  1427         (InductivePackage.add_inductive_i false true big_rec_name false false false
  1428            rec_sets (map (fn x => (("", x), [])) rec_intr_ts) []) thy10;
  1429 
  1430     (** equivariance **)
  1431 
  1432     val fresh_bij = PureThy.get_thms thy11 (Name "fresh_bij");
  1433     val perm_bij = PureThy.get_thms thy11 (Name "perm_bij");
  1434 
  1435     val (rec_equiv_thms, rec_equiv_thms') = ListPair.unzip (map (fn aT =>
  1436       let
  1437         val permT = mk_permT aT;
  1438         val pi = Free ("pi", permT);
  1439         val rec_fns_pi = map (curry (mk_perm []) pi o uncurry (mk_Free "f"))
  1440           (rec_fn_Ts ~~ (1 upto (length rec_fn_Ts)));
  1441         val rec_sets_pi = map (fn c => list_comb (Const c, rec_fns_pi))
  1442           (rec_set_names ~~ rec_set_Ts);
  1443         val ps = map (fn ((((T, U), R), R'), i) =>
  1444           let
  1445             val x = Free ("x" ^ string_of_int i, T);
  1446             val y = Free ("y" ^ string_of_int i, U)
  1447           in
  1448             (HOLogic.mk_mem (HOLogic.mk_prod (x, y), R),
  1449              HOLogic.mk_mem (HOLogic.mk_prod (mk_perm [] (pi, x), mk_perm [] (pi, y)), R'))
  1450           end) (recTs ~~ rec_result_Ts ~~ rec_sets ~~ rec_sets_pi ~~ (1 upto length recTs));
  1451         val ths = map (fn th => standard (th RS mp)) (split_conj_thm
  1452           (Goal.prove_global thy11 [] []
  1453             (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj (map HOLogic.mk_imp ps)))
  1454             (fn _ => rtac rec_induct 1 THEN REPEAT
  1455                (NominalPermeq.perm_simp_tac (simpset_of thy11) 1 THEN
  1456                 (resolve_tac rec_intrs THEN_ALL_NEW
  1457                  asm_simp_tac (HOL_ss addsimps (fresh_bij @ perm_bij))) 1))))
  1458         val ths' = map (fn ((P, Q), th) =>
  1459           Goal.prove_global thy11 [] []
  1460             (Logic.mk_implies (HOLogic.mk_Trueprop Q, HOLogic.mk_Trueprop P))
  1461             (fn _ => dtac (Thm.instantiate ([],
  1462                  [(cterm_of thy11 (Var (("pi", 0), permT)),
  1463                    cterm_of thy11 (Const ("List.rev", permT --> permT) $ pi))]) th) 1 THEN
  1464                NominalPermeq.perm_simp_tac HOL_ss 1)) (ps ~~ ths)
  1465       in (ths, ths') end) dt_atomTs);
  1466 
  1467     (** finite support **)
  1468 
  1469     val rec_fin_supp_thms = map (fn aT =>
  1470       let
  1471         val name = Sign.base_name (fst (dest_Type aT));
  1472         val fs_name = PureThy.get_thm thy11 (Name ("fs_" ^ name ^ "1"));
  1473         val aset = HOLogic.mk_setT aT;
  1474         val finites = Const ("Finite_Set.Finites", HOLogic.mk_setT aset);
  1475         val fins = map (fn (f, T) => HOLogic.mk_Trueprop (HOLogic.mk_mem
  1476           (Const ("Nominal.supp", T --> aset) $ f, finites)))
  1477             (rec_fns ~~ rec_fn_Ts)
  1478       in
  1479         map (fn th => standard (th RS mp)) (split_conj_thm
  1480           (Goal.prove_global thy11 [] fins
  1481             (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
  1482               (map (fn (((T, U), R), i) =>
  1483                  let
  1484                    val x = Free ("x" ^ string_of_int i, T);
  1485                    val y = Free ("y" ^ string_of_int i, U)
  1486                  in
  1487                    HOLogic.mk_imp (HOLogic.mk_mem (HOLogic.mk_prod (x, y), R),
  1488                      HOLogic.mk_mem (Const ("Nominal.supp", U --> aset) $ y, finites))
  1489                  end) (recTs ~~ rec_result_Ts ~~ rec_sets ~~ (1 upto length recTs)))))
  1490             (fn fins => (rtac rec_induct THEN_ALL_NEW cut_facts_tac fins) 1 THEN REPEAT
  1491                (NominalPermeq.finite_guess_tac (HOL_ss addsimps [fs_name]) 1))))
  1492       end) dt_atomTs;
  1493 
  1494     (** uniqueness **)
  1495 
  1496     val fresh_prems = List.concat (map (fn aT =>
  1497       map (fn (f, T) => HOLogic.mk_Trueprop (HOLogic.mk_mem
  1498         (Const ("Nominal.supp", T --> HOLogic.mk_setT aT) $ f,
  1499          Const ("Finite_Set.Finites", HOLogic.mk_setT (HOLogic.mk_setT aT)))))
  1500            (rec_fns ~~ rec_fn_Ts)) dt_atomTs);
  1501 
  1502     val fun_tuple = foldr1 HOLogic.mk_prod rec_fns;
  1503     val fun_tupleT = fastype_of fun_tuple;
  1504     val rec_unique_frees =
  1505       DatatypeProp.indexify_names (replicate (length recTs) "x") ~~ recTs;
  1506     val rec_unique_concls = map (fn ((x as (_, T), U), R) =>
  1507         Const ("Ex1", (U --> HOLogic.boolT) --> HOLogic.boolT) $
  1508           Abs ("y", U, HOLogic.mk_mem (HOLogic.pair_const T U $ Free x $ Bound 0, R)))
  1509       (rec_unique_frees ~~ rec_result_Ts ~~ rec_sets);
  1510     val induct_aux_rec = Drule.cterm_instantiate
  1511       (map (pairself (cterm_of thy11))
  1512          (map (fn (aT, f) => (Logic.varify f, Abs ("z", HOLogic.unitT,
  1513             Const ("Nominal.supp", fun_tupleT --> HOLogic.mk_setT aT) $ fun_tuple)))
  1514               fresh_fs @
  1515           map (fn (((P, T), (x, U)), Q) =>
  1516            (Var ((P, 0), HOLogic.unitT --> Logic.varifyT T --> HOLogic.boolT),
  1517             Abs ("z", HOLogic.unitT, absfree (x, U, Q))))
  1518               (pnames ~~ recTs ~~ rec_unique_frees ~~ rec_unique_concls) @
  1519           map (fn (s, T) => (Var ((s, 0), Logic.varifyT T), Free (s, T)))
  1520             rec_unique_frees)) induct_aux;
  1521 
  1522     val rec_unique = map standard (split_conj_thm (Goal.prove_global thy11 []
  1523       (fresh_prems @ rec_prems @ rec_prems')
  1524       (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj rec_unique_concls))
  1525       (fn ths => EVERY
  1526          [rtac induct_aux_rec 1,
  1527           print_tac "after application of induction theorem",
  1528           setmp quick_and_dirty true (SkipProof.cheat_tac thy11) (** FIXME !! **)])));
  1529     
  1530     (* FIXME: theorems are stored in database for testing only *)
  1531     val (_, thy12) = thy11 |>
  1532       Theory.add_path big_name |>
  1533       PureThy.add_thmss
  1534         [(("rec_equiv", List.concat rec_equiv_thms), []),
  1535          (("rec_equiv'", List.concat rec_equiv_thms'), []),
  1536          (("rec_fin_supp", List.concat rec_fin_supp_thms), []),
  1537          (("rec_unique", rec_unique), [])] ||>
  1538       Theory.parent_path;
  1539 
  1540   in
  1541     thy12
  1542   end;
  1543 
  1544 val add_nominal_datatype = gen_add_nominal_datatype read_typ true;
  1545 
  1546 
  1547 (* FIXME: The following stuff should be exported by DatatypePackage *)
  1548 
  1549 local structure P = OuterParse and K = OuterKeyword in
  1550 
  1551 val datatype_decl =
  1552   Scan.option (P.$$$ "(" |-- P.name --| P.$$$ ")") -- P.type_args -- P.name -- P.opt_infix --
  1553     (P.$$$ "=" |-- P.enum1 "|" (P.name -- Scan.repeat P.typ -- P.opt_mixfix));
  1554 
  1555 fun mk_datatype args =
  1556   let
  1557     val names = map (fn ((((NONE, _), t), _), _) => t | ((((SOME t, _), _), _), _) => t) args;
  1558     val specs = map (fn ((((_, vs), t), mx), cons) =>
  1559       (vs, t, mx, map (fn ((x, y), z) => (x, y, z)) cons)) args;
  1560   in add_nominal_datatype false names specs end;
  1561 
  1562 val nominal_datatypeP =
  1563   OuterSyntax.command "nominal_datatype" "define inductive datatypes" K.thy_decl
  1564     (P.and_list1 datatype_decl >> (Toplevel.theory o mk_datatype));
  1565 
  1566 val _ = OuterSyntax.add_parsers [nominal_datatypeP];
  1567 
  1568 end;
  1569 
  1570 end
  1571