src/HOL/Nominal/nominal_package.ML
author haftmann
Wed Sep 06 10:01:04 2006 +0200 (2006-09-06)
changeset 20483 04aa552a83bc
parent 20411 dd8a1cda2eaf
child 20548 8ef25fe585a8
permissions -rw-r--r--
TypedefPackage.add_typedef_* now yields name of introduced type constructor
     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 val supp_prod = thm "supp_prod";
   140 val fresh_prod = thm "fresh_prod";
   141 val supports_fresh = thm "supports_fresh";
   142 val supports_def = thm "Nominal.op supports_def";
   143 val fresh_def = thm "fresh_def";
   144 val supp_def = thm "supp_def";
   145 val rev_simps = thms "rev.simps";
   146 val app_simps = thms "op @.simps";
   147 
   148 fun gen_add_nominal_datatype prep_typ err flat_names new_type_names dts thy =
   149   let
   150     (* this theory is used just for parsing *)
   151   
   152     val tmp_thy = thy |>
   153       Theory.copy |>
   154       Theory.add_types (map (fn (tvs, tname, mx, _) =>
   155         (tname, length tvs, mx)) dts);
   156 
   157     val sign = Theory.sign_of tmp_thy;
   158 
   159     val atoms = atoms_of thy;
   160     val classes = map (NameSpace.map_base (fn s => "pt_" ^ s)) atoms;
   161     val cp_classes = List.concat (map (fn atom1 => map (fn atom2 =>
   162       Sign.intern_class thy ("cp_" ^ Sign.base_name atom1 ^ "_" ^
   163         Sign.base_name atom2)) atoms) atoms);
   164     fun augment_sort S = S union classes;
   165     val augment_sort_typ = map_type_tfree (fn (s, S) => TFree (s, augment_sort S));
   166 
   167     fun prep_constr ((constrs, sorts), (cname, cargs, mx)) =
   168       let val (cargs', sorts') = Library.foldl (prep_typ sign) (([], sorts), cargs)
   169       in (constrs @ [(cname, cargs', mx)], sorts') end
   170 
   171     fun prep_dt_spec ((dts, sorts), (tvs, tname, mx, constrs)) =
   172       let val (constrs', sorts') = Library.foldl prep_constr (([], sorts), constrs)
   173       in (dts @ [(tvs, tname, mx, constrs')], sorts') end
   174 
   175     val (dts', sorts) = Library.foldl prep_dt_spec (([], []), dts);
   176     val sorts' = map (apsnd augment_sort) sorts;
   177     val tyvars = map #1 dts';
   178 
   179     val types_syntax = map (fn (tvs, tname, mx, constrs) => (tname, mx)) dts';
   180     val constr_syntax = map (fn (tvs, tname, mx, constrs) =>
   181       map (fn (cname, cargs, mx) => (cname, mx)) constrs) dts';
   182 
   183     val ps = map (fn (_, n, _, _) =>
   184       (Sign.full_name sign n, Sign.full_name sign (n ^ "_Rep"))) dts;
   185     val rps = map Library.swap ps;
   186 
   187     fun replace_types (Type ("Nominal.ABS", [T, U])) = 
   188           Type ("fun", [T, Type ("Nominal.noption", [replace_types U])])
   189       | replace_types (Type (s, Ts)) =
   190           Type (getOpt (AList.lookup op = ps s, s), map replace_types Ts)
   191       | replace_types T = T;
   192 
   193     val dts'' = map (fn (tvs, tname, mx, constrs) => (tvs, tname ^ "_Rep", NoSyn,
   194       map (fn (cname, cargs, mx) => (cname ^ "_Rep",
   195         map (augment_sort_typ o replace_types) cargs, NoSyn)) constrs)) dts';
   196 
   197     val new_type_names' = map (fn n => n ^ "_Rep") new_type_names;
   198     val full_new_type_names' = map (Sign.full_name (sign_of thy)) new_type_names';
   199 
   200     val ({induction, ...},thy1) =
   201       DatatypePackage.add_datatype_i err flat_names new_type_names' dts'' thy;
   202 
   203     val SOME {descr, ...} = Symtab.lookup
   204       (DatatypePackage.get_datatypes thy1) (hd full_new_type_names');
   205     fun nth_dtyp i = typ_of_dtyp descr sorts' (DtRec i);
   206 
   207     (**** define permutation functions ****)
   208 
   209     val permT = mk_permT (TFree ("'x", HOLogic.typeS));
   210     val pi = Free ("pi", permT);
   211     val perm_types = map (fn (i, _) =>
   212       let val T = nth_dtyp i
   213       in permT --> T --> T end) descr;
   214     val perm_names = replicate (length new_type_names) "Nominal.perm" @
   215       DatatypeProp.indexify_names (map (fn i => Sign.full_name (sign_of thy1)
   216         ("perm_" ^ name_of_typ (nth_dtyp i)))
   217           (length new_type_names upto length descr - 1));
   218     val perm_names_types = perm_names ~~ perm_types;
   219 
   220     val perm_eqs = List.concat (map (fn (i, (_, _, constrs)) =>
   221       let val T = nth_dtyp i
   222       in map (fn (cname, dts) => 
   223         let
   224           val Ts = map (typ_of_dtyp descr sorts') dts;
   225           val names = DatatypeProp.make_tnames Ts;
   226           val args = map Free (names ~~ Ts);
   227           val c = Const (cname, Ts ---> T);
   228           fun perm_arg (dt, x) =
   229             let val T = type_of x
   230             in if is_rec_type dt then
   231                 let val (Us, _) = strip_type T
   232                 in list_abs (map (pair "x") Us,
   233                   Const (List.nth (perm_names_types, body_index dt)) $ pi $
   234                     list_comb (x, map (fn (i, U) =>
   235                       Const ("Nominal.perm", permT --> U --> U) $
   236                         (Const ("List.rev", permT --> permT) $ pi) $
   237                         Bound i) ((length Us - 1 downto 0) ~~ Us)))
   238                 end
   239               else Const ("Nominal.perm", permT --> T --> T) $ pi $ x
   240             end;  
   241         in
   242           (("", HOLogic.mk_Trueprop (HOLogic.mk_eq
   243             (Const (List.nth (perm_names_types, i)) $
   244                Free ("pi", mk_permT (TFree ("'x", HOLogic.typeS))) $
   245                list_comb (c, args),
   246              list_comb (c, map perm_arg (dts ~~ args))))), [])
   247         end) constrs
   248       end) descr);
   249 
   250     val (perm_simps, thy2) = thy1 |>
   251       Theory.add_consts_i (map (fn (s, T) => (Sign.base_name s, T, NoSyn))
   252         (List.drop (perm_names_types, length new_type_names))) |>
   253       PrimrecPackage.add_primrec_unchecked_i "" perm_eqs;
   254 
   255     (**** prove that permutation functions introduced by unfolding are ****)
   256     (**** equivalent to already existing permutation functions         ****)
   257 
   258     val _ = warning ("length descr: " ^ string_of_int (length descr));
   259     val _ = warning ("length new_type_names: " ^ string_of_int (length new_type_names));
   260 
   261     val perm_indnames = DatatypeProp.make_tnames (map body_type perm_types);
   262     val perm_fun_def = PureThy.get_thm thy2 (Name "perm_fun_def");
   263 
   264     val unfolded_perm_eq_thms =
   265       if length descr = length new_type_names then []
   266       else map standard (List.drop (split_conj_thm
   267         (Goal.prove_global thy2 [] []
   268           (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
   269             (map (fn (c as (s, T), x) =>
   270                let val [T1, T2] = binder_types T
   271                in HOLogic.mk_eq (Const c $ pi $ Free (x, T2),
   272                  Const ("Nominal.perm", T) $ pi $ Free (x, T2))
   273                end)
   274              (perm_names_types ~~ perm_indnames))))
   275           (fn _ => EVERY [indtac induction perm_indnames 1,
   276             ALLGOALS (asm_full_simp_tac
   277               (simpset_of thy2 addsimps [perm_fun_def]))])),
   278         length new_type_names));
   279 
   280     (**** prove [] \<bullet> t = t ****)
   281 
   282     val _ = warning "perm_empty_thms";
   283 
   284     val perm_empty_thms = List.concat (map (fn a =>
   285       let val permT = mk_permT (Type (a, []))
   286       in map standard (List.take (split_conj_thm
   287         (Goal.prove_global thy2 [] []
   288           (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
   289             (map (fn ((s, T), x) => HOLogic.mk_eq
   290                 (Const (s, permT --> T --> T) $
   291                    Const ("List.list.Nil", permT) $ Free (x, T),
   292                  Free (x, T)))
   293              (perm_names ~~
   294               map body_type perm_types ~~ perm_indnames))))
   295           (fn _ => EVERY [indtac induction perm_indnames 1,
   296             ALLGOALS (asm_full_simp_tac (simpset_of thy2))])),
   297         length new_type_names))
   298       end)
   299       atoms);
   300 
   301     (**** prove (pi1 @ pi2) \<bullet> t = pi1 \<bullet> (pi2 \<bullet> t) ****)
   302 
   303     val _ = warning "perm_append_thms";
   304 
   305     (*FIXME: these should be looked up statically*)
   306     val at_pt_inst = PureThy.get_thm thy2 (Name "at_pt_inst");
   307     val pt2 = PureThy.get_thm thy2 (Name "pt2");
   308 
   309     val perm_append_thms = List.concat (map (fn a =>
   310       let
   311         val permT = mk_permT (Type (a, []));
   312         val pi1 = Free ("pi1", permT);
   313         val pi2 = Free ("pi2", permT);
   314         val pt_inst = PureThy.get_thm thy2 (Name ("pt_" ^ Sign.base_name a ^ "_inst"));
   315         val pt2' = pt_inst RS pt2;
   316         val pt2_ax = PureThy.get_thm thy2
   317           (Name (NameSpace.map_base (fn s => "pt_" ^ s ^ "2") a));
   318       in List.take (map standard (split_conj_thm
   319         (Goal.prove_global thy2 [] []
   320              (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
   321                 (map (fn ((s, T), x) =>
   322                     let val perm = Const (s, permT --> T --> T)
   323                     in HOLogic.mk_eq
   324                       (perm $ (Const ("List.op @", permT --> permT --> permT) $
   325                          pi1 $ pi2) $ Free (x, T),
   326                        perm $ pi1 $ (perm $ pi2 $ Free (x, T)))
   327                     end)
   328                   (perm_names ~~
   329                    map body_type perm_types ~~ perm_indnames))))
   330            (fn _ => EVERY [indtac induction perm_indnames 1,
   331               ALLGOALS (asm_full_simp_tac (simpset_of thy2 addsimps [pt2', pt2_ax]))]))),
   332          length new_type_names)
   333       end) atoms);
   334 
   335     (**** prove pi1 ~ pi2 ==> pi1 \<bullet> t = pi2 \<bullet> t ****)
   336 
   337     val _ = warning "perm_eq_thms";
   338 
   339     val pt3 = PureThy.get_thm thy2 (Name "pt3");
   340     val pt3_rev = PureThy.get_thm thy2 (Name "pt3_rev");
   341 
   342     val perm_eq_thms = List.concat (map (fn a =>
   343       let
   344         val permT = mk_permT (Type (a, []));
   345         val pi1 = Free ("pi1", permT);
   346         val pi2 = Free ("pi2", permT);
   347         (*FIXME: not robust - better access these theorems using NominalData?*)
   348         val at_inst = PureThy.get_thm thy2 (Name ("at_" ^ Sign.base_name a ^ "_inst"));
   349         val pt_inst = PureThy.get_thm thy2 (Name ("pt_" ^ Sign.base_name a ^ "_inst"));
   350         val pt3' = pt_inst RS pt3;
   351         val pt3_rev' = at_inst RS (pt_inst RS pt3_rev);
   352         val pt3_ax = PureThy.get_thm thy2
   353           (Name (NameSpace.map_base (fn s => "pt_" ^ s ^ "3") a));
   354       in List.take (map standard (split_conj_thm
   355         (Goal.prove_global thy2 [] [] (Logic.mk_implies
   356              (HOLogic.mk_Trueprop (Const ("Nominal.prm_eq",
   357                 permT --> permT --> HOLogic.boolT) $ pi1 $ pi2),
   358               HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
   359                 (map (fn ((s, T), x) =>
   360                     let val perm = Const (s, permT --> T --> T)
   361                     in HOLogic.mk_eq
   362                       (perm $ pi1 $ Free (x, T),
   363                        perm $ pi2 $ Free (x, T))
   364                     end)
   365                   (perm_names ~~
   366                    map body_type perm_types ~~ perm_indnames)))))
   367            (fn _ => EVERY [indtac induction perm_indnames 1,
   368               ALLGOALS (asm_full_simp_tac (simpset_of thy2 addsimps [pt3', pt3_rev', pt3_ax]))]))),
   369          length new_type_names)
   370       end) atoms);
   371 
   372     (**** prove pi1 \<bullet> (pi2 \<bullet> t) = (pi1 \<bullet> pi2) \<bullet> (pi1 \<bullet> t) ****)
   373 
   374     val cp1 = PureThy.get_thm thy2 (Name "cp1");
   375     val dj_cp = PureThy.get_thm thy2 (Name "dj_cp");
   376     val pt_perm_compose = PureThy.get_thm thy2 (Name "pt_perm_compose");
   377     val pt_perm_compose_rev = PureThy.get_thm thy2 (Name "pt_perm_compose_rev");
   378     val dj_perm_perm_forget = PureThy.get_thm thy2 (Name "dj_perm_perm_forget");
   379 
   380     fun composition_instance name1 name2 thy =
   381       let
   382         val name1' = Sign.base_name name1;
   383         val name2' = Sign.base_name name2;
   384         val cp_class = Sign.intern_class thy ("cp_" ^ name1' ^ "_" ^ name2');
   385         val permT1 = mk_permT (Type (name1, []));
   386         val permT2 = mk_permT (Type (name2, []));
   387         val augment = map_type_tfree
   388           (fn (x, S) => TFree (x, cp_class :: S));
   389         val Ts = map (augment o body_type) perm_types;
   390         val cp_inst = PureThy.get_thm thy
   391           (Name ("cp_" ^ name1' ^ "_" ^ name2' ^ "_inst"));
   392         val simps = simpset_of thy addsimps (perm_fun_def ::
   393           (if name1 <> name2 then
   394              let val dj = PureThy.get_thm thy (Name ("dj_" ^ name2' ^ "_" ^ name1'))
   395              in [dj RS (cp_inst RS dj_cp), dj RS dj_perm_perm_forget] end
   396            else
   397              let
   398                val at_inst = PureThy.get_thm thy (Name ("at_" ^ name1' ^ "_inst"));
   399                val pt_inst = PureThy.get_thm thy (Name ("pt_" ^ name1' ^ "_inst"))
   400              in
   401                [cp_inst RS cp1 RS sym,
   402                 at_inst RS (pt_inst RS pt_perm_compose) RS sym,
   403                 at_inst RS (pt_inst RS pt_perm_compose_rev) RS sym]
   404             end))
   405         val thms = split_conj_thm (Goal.prove_global thy [] []
   406             (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
   407               (map (fn ((s, T), x) =>
   408                   let
   409                     val pi1 = Free ("pi1", permT1);
   410                     val pi2 = Free ("pi2", permT2);
   411                     val perm1 = Const (s, permT1 --> T --> T);
   412                     val perm2 = Const (s, permT2 --> T --> T);
   413                     val perm3 = Const ("Nominal.perm", permT1 --> permT2 --> permT2)
   414                   in HOLogic.mk_eq
   415                     (perm1 $ pi1 $ (perm2 $ pi2 $ Free (x, T)),
   416                      perm2 $ (perm3 $ pi1 $ pi2) $ (perm1 $ pi1 $ Free (x, T)))
   417                   end)
   418                 (perm_names ~~ Ts ~~ perm_indnames))))
   419           (fn _ => EVERY [indtac induction perm_indnames 1,
   420              ALLGOALS (asm_full_simp_tac simps)]))
   421       in
   422         foldl (fn ((s, tvs), thy) => AxClass.prove_arity
   423             (s, replicate (length tvs) (cp_class :: classes), [cp_class])
   424             (ClassPackage.intro_classes_tac [] THEN ALLGOALS (resolve_tac thms)) thy)
   425           thy (full_new_type_names' ~~ tyvars)
   426       end;
   427 
   428     val (perm_thmss,thy3) = thy2 |>
   429       fold (fn name1 => fold (composition_instance name1) atoms) atoms |>
   430       curry (Library.foldr (fn ((i, (tyname, args, _)), thy) =>
   431         AxClass.prove_arity (tyname, replicate (length args) classes, classes)
   432         (ClassPackage.intro_classes_tac [] THEN REPEAT (EVERY
   433            [resolve_tac perm_empty_thms 1,
   434             resolve_tac perm_append_thms 1,
   435             resolve_tac perm_eq_thms 1, assume_tac 1])) thy))
   436         (List.take (descr, length new_type_names)) |>
   437       PureThy.add_thmss
   438         [((space_implode "_" new_type_names ^ "_unfolded_perm_eq",
   439           unfolded_perm_eq_thms), [Simplifier.simp_add]),
   440          ((space_implode "_" new_type_names ^ "_perm_empty",
   441           perm_empty_thms), [Simplifier.simp_add]),
   442          ((space_implode "_" new_type_names ^ "_perm_append",
   443           perm_append_thms), [Simplifier.simp_add]),
   444          ((space_implode "_" new_type_names ^ "_perm_eq",
   445           perm_eq_thms), [Simplifier.simp_add])];
   446   
   447     (**** Define representing sets ****)
   448 
   449     val _ = warning "representing sets";
   450 
   451     val rep_set_names = map (Sign.full_name thy3) (DatatypeProp.indexify_names
   452       (map (fn (i, _) => name_of_typ (nth_dtyp i) ^ "_set") descr));
   453     val big_rep_name =
   454       space_implode "_" (DatatypeProp.indexify_names (List.mapPartial
   455         (fn (i, ("Nominal.noption", _, _)) => NONE
   456           | (i, _) => SOME (name_of_typ (nth_dtyp i))) descr)) ^ "_set";
   457     val _ = warning ("big_rep_name: " ^ big_rep_name);
   458 
   459     fun strip_option (dtf as DtType ("fun", [dt, DtRec i])) =
   460           (case AList.lookup op = descr i of
   461              SOME ("Nominal.noption", _, [(_, [dt']), _]) =>
   462                apfst (cons dt) (strip_option dt')
   463            | _ => ([], dtf))
   464       | strip_option (DtType ("fun", [dt, DtType ("Nominal.noption", [dt'])])) =
   465           apfst (cons dt) (strip_option dt')
   466       | strip_option dt = ([], dt);
   467 
   468     val dt_atomTs = distinct op = (map (typ_of_dtyp descr sorts')
   469       (List.concat (map (fn (_, (_, _, cs)) => List.concat
   470         (map (List.concat o map (fst o strip_option) o snd) cs)) descr)));
   471 
   472     fun make_intr s T (cname, cargs) =
   473       let
   474         fun mk_prem (dt, (j, j', prems, ts)) = 
   475           let
   476             val (dts, dt') = strip_option dt;
   477             val (dts', dt'') = strip_dtyp dt';
   478             val Ts = map (typ_of_dtyp descr sorts') dts;
   479             val Us = map (typ_of_dtyp descr sorts') dts';
   480             val T = typ_of_dtyp descr sorts' dt'';
   481             val free = mk_Free "x" (Us ---> T) j;
   482             val free' = app_bnds free (length Us);
   483             fun mk_abs_fun (T, (i, t)) =
   484               let val U = fastype_of t
   485               in (i + 1, Const ("Nominal.abs_fun", [T, U, T] --->
   486                 Type ("Nominal.noption", [U])) $ mk_Free "y" T i $ t)
   487               end
   488           in (j + 1, j' + length Ts,
   489             case dt'' of
   490                 DtRec k => list_all (map (pair "x") Us,
   491                   HOLogic.mk_Trueprop (HOLogic.mk_mem (free',
   492                     Const (List.nth (rep_set_names, k),
   493                       HOLogic.mk_setT T)))) :: prems
   494               | _ => prems,
   495             snd (foldr mk_abs_fun (j', free) Ts) :: ts)
   496           end;
   497 
   498         val (_, _, prems, ts) = foldr mk_prem (1, 1, [], []) cargs;
   499         val concl = HOLogic.mk_Trueprop (HOLogic.mk_mem
   500           (list_comb (Const (cname, map fastype_of ts ---> T), ts),
   501            Const (s, HOLogic.mk_setT T)))
   502       in Logic.list_implies (prems, concl)
   503       end;
   504 
   505     val (intr_ts, ind_consts) =
   506       apfst List.concat (ListPair.unzip (List.mapPartial
   507         (fn ((_, ("Nominal.noption", _, _)), _) => NONE
   508           | ((i, (_, _, constrs)), rep_set_name) =>
   509               let val T = nth_dtyp i
   510               in SOME (map (make_intr rep_set_name T) constrs,
   511                 Const (rep_set_name, HOLogic.mk_setT T))
   512               end)
   513                 (descr ~~ rep_set_names)));
   514 
   515     val (thy4, {raw_induct = rep_induct, intrs = rep_intrs, ...}) =
   516       setmp InductivePackage.quiet_mode false
   517         (InductivePackage.add_inductive_i false true big_rep_name false true false
   518            ind_consts (map (fn x => (("", x), [])) intr_ts) []) thy3;
   519 
   520     (**** Prove that representing set is closed under permutation ****)
   521 
   522     val _ = warning "proving closure under permutation...";
   523 
   524     val perm_indnames' = List.mapPartial
   525       (fn (x, (_, ("Nominal.noption", _, _))) => NONE | (x, _) => SOME x)
   526       (perm_indnames ~~ descr);
   527 
   528     fun mk_perm_closed name = map (fn th => standard (th RS mp))
   529       (List.take (split_conj_thm (Goal.prove_global thy4 [] []
   530         (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj (map
   531            (fn (S, x) =>
   532               let
   533                 val S = map_term_types (map_type_tfree
   534                   (fn (s, cs) => TFree (s, cs union cp_classes))) S;
   535                 val T = HOLogic.dest_setT (fastype_of S);
   536                 val permT = mk_permT (Type (name, []))
   537               in HOLogic.mk_imp (HOLogic.mk_mem (Free (x, T), S),
   538                 HOLogic.mk_mem (Const ("Nominal.perm", permT --> T --> T) $
   539                   Free ("pi", permT) $ Free (x, T), S))
   540               end) (ind_consts ~~ perm_indnames'))))
   541         (fn _ => EVERY (* CU: added perm_fun_def in the final tactic in order to deal with funs *)
   542            [indtac rep_induct [] 1,
   543             ALLGOALS (simp_tac (simpset_of thy4 addsimps
   544               (symmetric perm_fun_def :: PureThy.get_thms thy4 (Name ("abs_perm"))))),
   545             ALLGOALS (resolve_tac rep_intrs 
   546                THEN_ALL_NEW (asm_full_simp_tac (simpset_of thy4 addsimps [perm_fun_def])))])),
   547         length new_type_names));
   548 
   549     (* FIXME: theorems are stored in database for testing only *)
   550     val perm_closed_thmss = map mk_perm_closed atoms;
   551     val (_, thy5) = PureThy.add_thmss [(("perm_closed", List.concat perm_closed_thmss), [])] thy4;
   552 
   553     (**** typedef ****)
   554 
   555     val _ = warning "defining type...";
   556 
   557     val (typedefs, thy6) =
   558       thy5
   559       |> fold_map (fn ((((name, mx), tvs), c), name') => fn thy =>
   560         setmp TypedefPackage.quiet_mode true
   561           (TypedefPackage.add_typedef_i false (SOME name') (name, tvs, mx) c NONE
   562             (rtac exI 1 THEN
   563               QUIET_BREADTH_FIRST (has_fewer_prems 1)
   564               (resolve_tac rep_intrs 1))) thy |> (fn ((_, r), thy) =>
   565         let
   566           val permT = mk_permT (TFree (Name.variant tvs "'a", HOLogic.typeS));
   567           val pi = Free ("pi", permT);
   568           val tvs' = map (fn s => TFree (s, the (AList.lookup op = sorts' s))) tvs;
   569           val T = Type (Sign.intern_type thy name, tvs');
   570           val Const (_, Type (_, [U])) = c
   571         in apfst (pair r o hd)
   572           (PureThy.add_defs_unchecked_i true [(("prm_" ^ name ^ "_def", Logic.mk_equals
   573             (Const ("Nominal.perm", permT --> T --> T) $ pi $ Free ("x", T),
   574              Const (Sign.intern_const thy ("Abs_" ^ name), U --> T) $
   575                (Const ("Nominal.perm", permT --> U --> U) $ pi $
   576                  (Const (Sign.intern_const thy ("Rep_" ^ name), T --> U) $
   577                    Free ("x", T))))), [])] thy)
   578         end))
   579           (types_syntax ~~ tyvars ~~
   580             (List.take (ind_consts, length new_type_names)) ~~ new_type_names);
   581 
   582     val perm_defs = map snd typedefs;
   583     val Abs_inverse_thms = map (#Abs_inverse o fst) typedefs;
   584     val Rep_inverse_thms = map (#Rep_inverse o fst) typedefs;
   585     val Rep_thms = map (#Rep o fst) typedefs;
   586 
   587     val big_name = space_implode "_" new_type_names;
   588 
   589 
   590     (** prove that new types are in class pt_<name> **)
   591 
   592     val _ = warning "prove that new types are in class pt_<name> ...";
   593 
   594     fun pt_instance ((class, atom), perm_closed_thms) =
   595       fold (fn (((({Abs_inverse, Rep_inverse, Rep, ...},
   596         perm_def), name), tvs), perm_closed) => fn thy =>
   597           AxClass.prove_arity
   598             (Sign.intern_type thy name,
   599               replicate (length tvs) (classes @ cp_classes), [class])
   600             (EVERY [ClassPackage.intro_classes_tac [],
   601               rewrite_goals_tac [perm_def],
   602               asm_full_simp_tac (simpset_of thy addsimps [Rep_inverse]) 1,
   603               asm_full_simp_tac (simpset_of thy addsimps
   604                 [Rep RS perm_closed RS Abs_inverse]) 1,
   605               asm_full_simp_tac (HOL_basic_ss addsimps [PureThy.get_thm thy
   606                 (Name ("pt_" ^ Sign.base_name atom ^ "3"))]) 1]) thy)
   607         (typedefs ~~ new_type_names ~~ tyvars ~~ perm_closed_thms);
   608 
   609 
   610     (** prove that new types are in class cp_<name1>_<name2> **)
   611 
   612     val _ = warning "prove that new types are in class cp_<name1>_<name2> ...";
   613 
   614     fun cp_instance (atom1, perm_closed_thms1) (atom2, perm_closed_thms2) thy =
   615       let
   616         val name = "cp_" ^ Sign.base_name atom1 ^ "_" ^ Sign.base_name atom2;
   617         val class = Sign.intern_class thy name;
   618         val cp1' = PureThy.get_thm thy (Name (name ^ "_inst")) RS cp1
   619       in fold (fn ((((({Abs_inverse, Rep_inverse, Rep, ...},
   620         perm_def), name), tvs), perm_closed1), perm_closed2) => fn thy =>
   621           AxClass.prove_arity
   622             (Sign.intern_type thy name,
   623               replicate (length tvs) (classes @ cp_classes), [class])
   624             (EVERY [ClassPackage.intro_classes_tac [],
   625               rewrite_goals_tac [perm_def],
   626               asm_full_simp_tac (simpset_of thy addsimps
   627                 ((Rep RS perm_closed1 RS Abs_inverse) ::
   628                  (if atom1 = atom2 then []
   629                   else [Rep RS perm_closed2 RS Abs_inverse]))) 1,
   630               cong_tac 1,
   631               rtac refl 1,
   632               rtac cp1' 1]) thy)
   633         (typedefs ~~ new_type_names ~~ tyvars ~~ perm_closed_thms1 ~~
   634           perm_closed_thms2) thy
   635       end;
   636 
   637     val thy7 = fold (fn x => fn thy => thy |>
   638       pt_instance x |>
   639       fold (cp_instance (apfst snd x)) (atoms ~~ perm_closed_thmss))
   640         (classes ~~ atoms ~~ perm_closed_thmss) thy6;
   641 
   642     (**** constructors ****)
   643 
   644     fun mk_abs_fun (x, t) =
   645       let
   646         val T = fastype_of x;
   647         val U = fastype_of t
   648       in
   649         Const ("Nominal.abs_fun", T --> U --> T -->
   650           Type ("Nominal.noption", [U])) $ x $ t
   651       end;
   652 
   653     val (ty_idxs, _) = foldl
   654       (fn ((i, ("Nominal.noption", _, _)), p) => p
   655         | ((i, _), (ty_idxs, j)) => (ty_idxs @ [(i, j)], j + 1)) ([], 0) descr;
   656 
   657     fun reindex (DtType (s, dts)) = DtType (s, map reindex dts)
   658       | reindex (DtRec i) = DtRec (the (AList.lookup op = ty_idxs i))
   659       | reindex dt = dt;
   660 
   661     fun strip_suffix i s = implode (List.take (explode s, size s - i));
   662 
   663     (** strips the "_Rep" in type names *)
   664     fun strip_nth_name i s = 
   665       let val xs = NameSpace.unpack s; 
   666       in NameSpace.pack (Library.nth_map (length xs - i) (strip_suffix 4) xs) end;
   667 
   668     val (descr'', ndescr) = ListPair.unzip (List.mapPartial
   669       (fn (i, ("Nominal.noption", _, _)) => NONE
   670         | (i, (s, dts, constrs)) =>
   671              let
   672                val SOME index = AList.lookup op = ty_idxs i;
   673                val (constrs1, constrs2) = ListPair.unzip
   674                  (map (fn (cname, cargs) => apfst (pair (strip_nth_name 2 (strip_nth_name 1 cname)))
   675                    (foldl_map (fn (dts, dt) =>
   676                      let val (dts', dt') = strip_option dt
   677                      in (dts @ dts' @ [reindex dt'], (length dts, length dts')) end)
   678                        ([], cargs))) constrs)
   679              in SOME ((index, (strip_nth_name 1 s,  map reindex dts, constrs1)),
   680                (index, constrs2))
   681              end) descr);
   682 
   683     val (descr1, descr2) = chop (length new_type_names) descr'';
   684     val descr' = [descr1, descr2];
   685 
   686     fun partition_cargs idxs xs = map (fn (i, j) =>
   687       (List.take (List.drop (xs, i), j), List.nth (xs, i + j))) idxs;
   688 
   689     val pdescr = map (fn ((i, (s, dts, constrs)), (_, idxss)) => (i, (s, dts,
   690       map (fn ((cname, cargs), idxs) => (cname, partition_cargs idxs cargs))
   691         (constrs ~~ idxss)))) (descr'' ~~ ndescr);
   692 
   693     fun nth_dtyp' i = typ_of_dtyp descr'' sorts' (DtRec i);
   694 
   695     val rep_names = map (fn s =>
   696       Sign.intern_const thy7 ("Rep_" ^ s)) new_type_names;
   697     val abs_names = map (fn s =>
   698       Sign.intern_const thy7 ("Abs_" ^ s)) new_type_names;
   699 
   700     val recTs' = List.mapPartial
   701       (fn ((_, ("Nominal.noption", _, _)), T) => NONE
   702         | (_, T) => SOME T) (descr ~~ get_rec_types descr sorts');
   703     val recTs = get_rec_types descr'' sorts';
   704     val newTs' = Library.take (length new_type_names, recTs');
   705     val newTs = Library.take (length new_type_names, recTs);
   706 
   707     val full_new_type_names = map (Sign.full_name (sign_of thy)) new_type_names;
   708 
   709     fun make_constr_def tname T T' ((thy, defs, eqns),
   710         (((cname_rep, _), (cname, cargs)), (cname', mx))) =
   711       let
   712         fun constr_arg ((dts, dt), (j, l_args, r_args)) =
   713           let
   714             val xs = map (fn (dt, i) => mk_Free "x" (typ_of_dtyp descr'' sorts' dt) i)
   715               (dts ~~ (j upto j + length dts - 1))
   716             val x = mk_Free "x" (typ_of_dtyp descr'' sorts' dt) (j + length dts)
   717           in
   718             (j + length dts + 1,
   719              xs @ x :: l_args,
   720              foldr mk_abs_fun
   721                (case dt of
   722                   DtRec k => if k < length new_type_names then
   723                       Const (List.nth (rep_names, k), typ_of_dtyp descr'' sorts' dt -->
   724                         typ_of_dtyp descr sorts' dt) $ x
   725                     else error "nested recursion not (yet) supported"
   726                 | _ => x) xs :: r_args)
   727           end
   728 
   729         val (_, l_args, r_args) = foldr constr_arg (1, [], []) cargs;
   730         val abs_name = Sign.intern_const (Theory.sign_of thy) ("Abs_" ^ tname);
   731         val rep_name = Sign.intern_const (Theory.sign_of thy) ("Rep_" ^ tname);
   732         val constrT = map fastype_of l_args ---> T;
   733         val lhs = list_comb (Const (cname, constrT), l_args);
   734         val rhs = list_comb (Const (cname_rep, map fastype_of r_args ---> T'), r_args);
   735         val def = Logic.mk_equals (lhs, Const (abs_name, T' --> T) $ rhs);
   736         val eqn = HOLogic.mk_Trueprop (HOLogic.mk_eq
   737           (Const (rep_name, T --> T') $ lhs, rhs));
   738         val def_name = (Sign.base_name cname) ^ "_def";
   739         val ([def_thm], thy') = thy |>
   740           Theory.add_consts_i [(cname', constrT, mx)] |>
   741           (PureThy.add_defs_i false o map Thm.no_attributes) [(def_name, def)]
   742       in (thy', defs @ [def_thm], eqns @ [eqn]) end;
   743 
   744     fun dt_constr_defs ((thy, defs, eqns, dist_lemmas), ((((((_, (_, _, constrs)),
   745         (_, (_, _, constrs'))), tname), T), T'), constr_syntax)) =
   746       let
   747         val rep_const = cterm_of thy
   748           (Const (Sign.intern_const thy ("Rep_" ^ tname), T --> T'));
   749         val dist = standard (cterm_instantiate [(cterm_of thy distinct_f, rep_const)] distinct_lemma);
   750         val (thy', defs', eqns') = Library.foldl (make_constr_def tname T T')
   751           ((Theory.add_path tname thy, defs, []), constrs ~~ constrs' ~~ constr_syntax)
   752       in
   753         (parent_path flat_names thy', defs', eqns @ [eqns'], dist_lemmas @ [dist])
   754       end;
   755 
   756     val (thy8, constr_defs, constr_rep_eqns, dist_lemmas) = Library.foldl dt_constr_defs
   757       ((thy7, [], [], []), List.take (descr, length new_type_names) ~~
   758         List.take (pdescr, length new_type_names) ~~
   759         new_type_names ~~ newTs ~~ newTs' ~~ constr_syntax);
   760 
   761     val abs_inject_thms = map (fn tname =>
   762       PureThy.get_thm thy8 (Name ("Abs_" ^ tname ^ "_inject"))) new_type_names;
   763 
   764     val rep_inject_thms = map (fn tname =>
   765       PureThy.get_thm thy8 (Name ("Rep_" ^ tname ^ "_inject"))) new_type_names;
   766 
   767     val rep_thms = map (fn tname =>
   768       PureThy.get_thm thy8 (Name ("Rep_" ^ tname))) new_type_names;
   769 
   770     val rep_inverse_thms = map (fn tname =>
   771       PureThy.get_thm thy8 (Name ("Rep_" ^ tname ^ "_inverse"))) new_type_names;
   772 
   773     (* prove theorem  Rep_i (Constr_j ...) = Constr'_j ...  *)
   774     
   775     fun prove_constr_rep_thm eqn =
   776       let
   777         val inj_thms = map (fn r => r RS iffD1) abs_inject_thms;
   778         val rewrites = constr_defs @ map mk_meta_eq rep_inverse_thms
   779       in Goal.prove_global thy8 [] [] eqn (fn _ => EVERY
   780         [resolve_tac inj_thms 1,
   781          rewrite_goals_tac rewrites,
   782          rtac refl 3,
   783          resolve_tac rep_intrs 2,
   784          REPEAT (resolve_tac rep_thms 1)])
   785       end;
   786 
   787     val constr_rep_thmss = map (map prove_constr_rep_thm) constr_rep_eqns;
   788 
   789     (* prove theorem  pi \<bullet> Rep_i x = Rep_i (pi \<bullet> x) *)
   790 
   791     fun prove_perm_rep_perm (atom, perm_closed_thms) = map (fn th =>
   792       let
   793         val _ $ (_ $ (Rep $ x) $ _) = Logic.unvarify (prop_of th);
   794         val Type ("fun", [T, U]) = fastype_of Rep;
   795         val permT = mk_permT (Type (atom, []));
   796         val pi = Free ("pi", permT);
   797       in
   798         Goal.prove_global thy8 [] [] (HOLogic.mk_Trueprop (HOLogic.mk_eq
   799             (Const ("Nominal.perm", permT --> U --> U) $ pi $ (Rep $ x),
   800              Rep $ (Const ("Nominal.perm", permT --> T --> T) $ pi $ x))))
   801           (fn _ => simp_tac (HOL_basic_ss addsimps (perm_defs @ Abs_inverse_thms @
   802             perm_closed_thms @ Rep_thms)) 1)
   803       end) Rep_thms;
   804 
   805     val perm_rep_perm_thms = List.concat (map prove_perm_rep_perm
   806       (atoms ~~ perm_closed_thmss));
   807 
   808     (* prove distinctness theorems *)
   809 
   810     val distinct_props = setmp DatatypeProp.dtK 1000
   811       (DatatypeProp.make_distincts new_type_names descr' sorts') thy8;
   812 
   813     val dist_rewrites = map (fn (rep_thms, dist_lemma) =>
   814       dist_lemma::(rep_thms @ [In0_eq, In1_eq, In0_not_In1, In1_not_In0]))
   815         (constr_rep_thmss ~~ dist_lemmas);
   816 
   817     fun prove_distinct_thms (_, []) = []
   818       | prove_distinct_thms (p as (rep_thms, dist_lemma), t::ts) =
   819           let
   820             val dist_thm = Goal.prove_global thy8 [] [] t (fn _ =>
   821               simp_tac (simpset_of thy8 addsimps (dist_lemma :: rep_thms)) 1)
   822           in dist_thm::(standard (dist_thm RS not_sym))::
   823             (prove_distinct_thms (p, ts))
   824           end;
   825 
   826     val distinct_thms = map prove_distinct_thms
   827       (constr_rep_thmss ~~ dist_lemmas ~~ distinct_props);
   828 
   829     (** prove equations for permutation functions **)
   830 
   831     val abs_perm = PureThy.get_thms thy8 (Name "abs_perm"); (* FIXME *)
   832 
   833     val perm_simps' = map (fn (((i, (_, _, constrs)), tname), constr_rep_thms) =>
   834       let val T = nth_dtyp' i
   835       in List.concat (map (fn (atom, perm_closed_thms) =>
   836           map (fn ((cname, dts), constr_rep_thm) => 
   837         let
   838           val cname = Sign.intern_const thy8
   839             (NameSpace.append tname (Sign.base_name cname));
   840           val permT = mk_permT (Type (atom, []));
   841           val pi = Free ("pi", permT);
   842 
   843           fun perm t =
   844             let val T = fastype_of t
   845             in Const ("Nominal.perm", permT --> T --> T) $ pi $ t end;
   846 
   847           fun constr_arg ((dts, dt), (j, l_args, r_args)) =
   848             let
   849               val Ts = map (typ_of_dtyp descr'' sorts') dts;
   850               val xs = map (fn (T, i) => mk_Free "x" T i)
   851                 (Ts ~~ (j upto j + length dts - 1))
   852               val x = mk_Free "x" (typ_of_dtyp descr'' sorts' dt) (j + length dts)
   853             in
   854               (j + length dts + 1,
   855                xs @ x :: l_args,
   856                map perm (xs @ [x]) @ r_args)
   857             end
   858 
   859           val (_, l_args, r_args) = foldr constr_arg (1, [], []) dts;
   860           val c = Const (cname, map fastype_of l_args ---> T)
   861         in
   862           Goal.prove_global thy8 [] []
   863             (HOLogic.mk_Trueprop (HOLogic.mk_eq
   864               (perm (list_comb (c, l_args)), list_comb (c, r_args))))
   865             (fn _ => EVERY
   866               [simp_tac (simpset_of thy8 addsimps (constr_rep_thm :: perm_defs)) 1,
   867                simp_tac (HOL_basic_ss addsimps (Rep_thms @ Abs_inverse_thms @
   868                  constr_defs @ perm_closed_thms)) 1,
   869                TRY (simp_tac (HOL_basic_ss addsimps
   870                  (symmetric perm_fun_def :: abs_perm)) 1),
   871                TRY (simp_tac (HOL_basic_ss addsimps
   872                  (perm_fun_def :: perm_defs @ Rep_thms @ Abs_inverse_thms @
   873                     perm_closed_thms)) 1)])
   874         end) (constrs ~~ constr_rep_thms)) (atoms ~~ perm_closed_thmss))
   875       end) (List.take (pdescr, length new_type_names) ~~ new_type_names ~~ constr_rep_thmss);
   876 
   877     (** prove injectivity of constructors **)
   878 
   879     val rep_inject_thms' = map (fn th => th RS sym) rep_inject_thms;
   880     val alpha = PureThy.get_thms thy8 (Name "alpha");
   881     val abs_fresh = PureThy.get_thms thy8 (Name "abs_fresh");
   882 
   883     val inject_thms = map (fn (((i, (_, _, constrs)), tname), constr_rep_thms) =>
   884       let val T = nth_dtyp' i
   885       in List.mapPartial (fn ((cname, dts), constr_rep_thm) =>
   886         if null dts then NONE else SOME
   887         let
   888           val cname = Sign.intern_const thy8
   889             (NameSpace.append tname (Sign.base_name cname));
   890 
   891           fun make_inj ((dts, dt), (j, args1, args2, eqs)) =
   892             let
   893               val Ts_idx = map (typ_of_dtyp descr'' sorts') dts ~~ (j upto j + length dts - 1);
   894               val xs = map (fn (T, i) => mk_Free "x" T i) Ts_idx;
   895               val ys = map (fn (T, i) => mk_Free "y" T i) Ts_idx;
   896               val x = mk_Free "x" (typ_of_dtyp descr'' sorts' dt) (j + length dts);
   897               val y = mk_Free "y" (typ_of_dtyp descr'' sorts' dt) (j + length dts)
   898             in
   899               (j + length dts + 1,
   900                xs @ (x :: args1), ys @ (y :: args2),
   901                HOLogic.mk_eq
   902                  (foldr mk_abs_fun x xs, foldr mk_abs_fun y ys) :: eqs)
   903             end;
   904 
   905           val (_, args1, args2, eqs) = foldr make_inj (1, [], [], []) dts;
   906           val Ts = map fastype_of args1;
   907           val c = Const (cname, Ts ---> T)
   908         in
   909           Goal.prove_global thy8 [] [] (HOLogic.mk_Trueprop (HOLogic.mk_eq
   910               (HOLogic.mk_eq (list_comb (c, args1), list_comb (c, args2)),
   911                foldr1 HOLogic.mk_conj eqs)))
   912             (fn _ => EVERY
   913                [asm_full_simp_tac (simpset_of thy8 addsimps (constr_rep_thm ::
   914                   rep_inject_thms')) 1,
   915                 TRY (asm_full_simp_tac (HOL_basic_ss addsimps (fresh_def :: supp_def ::
   916                   alpha @ abs_perm @ abs_fresh @ rep_inject_thms @
   917                   perm_rep_perm_thms)) 1),
   918                 TRY (asm_full_simp_tac (HOL_basic_ss addsimps (perm_fun_def ::
   919                   expand_fun_eq :: rep_inject_thms @ perm_rep_perm_thms)) 1)])
   920         end) (constrs ~~ constr_rep_thms)
   921       end) (List.take (pdescr, length new_type_names) ~~ new_type_names ~~ constr_rep_thmss);
   922 
   923     (** equations for support and freshness **)
   924 
   925     val (supp_thms, fresh_thms) = ListPair.unzip (map ListPair.unzip
   926       (map (fn ((((i, (_, _, constrs)), tname), inject_thms'), perm_thms') =>
   927       let val T = nth_dtyp' i
   928       in List.concat (map (fn (cname, dts) => map (fn atom =>
   929         let
   930           val cname = Sign.intern_const thy8
   931             (NameSpace.append tname (Sign.base_name cname));
   932           val atomT = Type (atom, []);
   933 
   934           fun process_constr ((dts, dt), (j, args1, args2)) =
   935             let
   936               val Ts_idx = map (typ_of_dtyp descr'' sorts') dts ~~ (j upto j + length dts - 1);
   937               val xs = map (fn (T, i) => mk_Free "x" T i) Ts_idx;
   938               val x = mk_Free "x" (typ_of_dtyp descr'' sorts' dt) (j + length dts)
   939             in
   940               (j + length dts + 1,
   941                xs @ (x :: args1), foldr mk_abs_fun x xs :: args2)
   942             end;
   943 
   944           val (_, args1, args2) = foldr process_constr (1, [], []) dts;
   945           val Ts = map fastype_of args1;
   946           val c = list_comb (Const (cname, Ts ---> T), args1);
   947           fun supp t =
   948             Const ("Nominal.supp", fastype_of t --> HOLogic.mk_setT atomT) $ t;
   949           fun fresh t =
   950             Const ("Nominal.fresh", atomT --> fastype_of t --> HOLogic.boolT) $
   951               Free ("a", atomT) $ t;
   952           val supp_thm = Goal.prove_global thy8 [] []
   953               (HOLogic.mk_Trueprop (HOLogic.mk_eq
   954                 (supp c,
   955                  if null dts then Const ("{}", HOLogic.mk_setT atomT)
   956                  else foldr1 (HOLogic.mk_binop "op Un") (map supp args2))))
   957             (fn _ =>
   958               simp_tac (HOL_basic_ss addsimps (supp_def ::
   959                  Un_assoc :: de_Morgan_conj :: Collect_disj_eq :: finite_Un ::
   960                  symmetric empty_def :: Finites.emptyI :: simp_thms @
   961                  abs_perm @ abs_fresh @ inject_thms' @ perm_thms')) 1)
   962         in
   963           (supp_thm,
   964            Goal.prove_global thy8 [] [] (HOLogic.mk_Trueprop (HOLogic.mk_eq
   965               (fresh c,
   966                if null dts then HOLogic.true_const
   967                else foldr1 HOLogic.mk_conj (map fresh args2))))
   968              (fn _ =>
   969                simp_tac (simpset_of thy8 addsimps [fresh_def, supp_thm]) 1))
   970         end) atoms) constrs)
   971       end) (List.take (pdescr, length new_type_names) ~~ new_type_names ~~ inject_thms ~~ perm_simps')));
   972 
   973     (**** weak induction theorem ****)
   974 
   975     fun mk_indrule_lemma ((prems, concls), (((i, _), T), U)) =
   976       let
   977         val Rep_t = Const (List.nth (rep_names, i), T --> U) $
   978           mk_Free "x" T i;
   979 
   980         val Abs_t =  Const (List.nth (abs_names, i), U --> T)
   981 
   982       in (prems @ [HOLogic.imp $ HOLogic.mk_mem (Rep_t,
   983             Const (List.nth (rep_set_names, i), HOLogic.mk_setT U)) $
   984               (mk_Free "P" (T --> HOLogic.boolT) (i + 1) $ (Abs_t $ Rep_t))],
   985           concls @ [mk_Free "P" (T --> HOLogic.boolT) (i + 1) $ mk_Free "x" T i])
   986       end;
   987 
   988     val (indrule_lemma_prems, indrule_lemma_concls) =
   989       Library.foldl mk_indrule_lemma (([], []), (descr'' ~~ recTs ~~ recTs'));
   990 
   991     val indrule_lemma = Goal.prove_global thy8 [] []
   992       (Logic.mk_implies
   993         (HOLogic.mk_Trueprop (mk_conj indrule_lemma_prems),
   994          HOLogic.mk_Trueprop (mk_conj indrule_lemma_concls))) (fn _ => EVERY
   995            [REPEAT (etac conjE 1),
   996             REPEAT (EVERY
   997               [TRY (rtac conjI 1), full_simp_tac (HOL_basic_ss addsimps Rep_inverse_thms) 1,
   998                etac mp 1, resolve_tac Rep_thms 1])]);
   999 
  1000     val Ps = map head_of (HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of indrule_lemma)));
  1001     val frees = if length Ps = 1 then [Free ("P", snd (dest_Var (hd Ps)))] else
  1002       map (Free o apfst fst o dest_Var) Ps;
  1003     val indrule_lemma' = cterm_instantiate
  1004       (map (cterm_of thy8) Ps ~~ map (cterm_of thy8) frees) indrule_lemma;
  1005 
  1006     val Abs_inverse_thms' = map (fn r => r RS subst) Abs_inverse_thms;
  1007 
  1008     val dt_induct_prop = DatatypeProp.make_ind descr' sorts';
  1009     val dt_induct = Goal.prove_global thy8 []
  1010       (Logic.strip_imp_prems dt_induct_prop) (Logic.strip_imp_concl dt_induct_prop)
  1011       (fn prems => EVERY
  1012         [rtac indrule_lemma' 1,
  1013          (DatatypeAux.indtac rep_induct THEN_ALL_NEW ObjectLogic.atomize_tac) 1,
  1014          EVERY (map (fn (prem, r) => (EVERY
  1015            [REPEAT (eresolve_tac Abs_inverse_thms' 1),
  1016             simp_tac (HOL_basic_ss addsimps [symmetric r]) 1,
  1017             DEPTH_SOLVE_1 (ares_tac [prem] 1 ORELSE etac allE 1)]))
  1018                 (prems ~~ constr_defs))]);
  1019 
  1020     val case_names_induct = mk_case_names_induct descr'';
  1021 
  1022     (**** prove that new datatypes have finite support ****)
  1023 
  1024     val _ = warning "proving finite support for the new datatype";
  1025 
  1026     val indnames = DatatypeProp.make_tnames recTs;
  1027 
  1028     val abs_supp = PureThy.get_thms thy8 (Name "abs_supp");
  1029     val supp_atm = PureThy.get_thms thy8 (Name "supp_atm");
  1030 
  1031     val finite_supp_thms = map (fn atom =>
  1032       let val atomT = Type (atom, [])
  1033       in map standard (List.take
  1034         (split_conj_thm (Goal.prove_global thy8 [] [] (HOLogic.mk_Trueprop
  1035            (foldr1 HOLogic.mk_conj (map (fn (s, T) => HOLogic.mk_mem
  1036              (Const ("Nominal.supp", T --> HOLogic.mk_setT atomT) $ Free (s, T),
  1037               Const ("Finite_Set.Finites", HOLogic.mk_setT (HOLogic.mk_setT atomT))))
  1038                (indnames ~~ recTs))))
  1039            (fn _ => indtac dt_induct indnames 1 THEN
  1040             ALLGOALS (asm_full_simp_tac (simpset_of thy8 addsimps
  1041               (abs_supp @ supp_atm @
  1042                PureThy.get_thms thy8 (Name ("fs_" ^ Sign.base_name atom ^ "1")) @
  1043                List.concat supp_thms))))),
  1044          length new_type_names))
  1045       end) atoms;
  1046 
  1047     val simp_atts = replicate (length new_type_names) [Simplifier.simp_add];
  1048 
  1049     val (_, thy9) = thy8 |>
  1050       Theory.add_path big_name |>
  1051       PureThy.add_thms [(("induct_weak", dt_induct), [case_names_induct])] ||>>
  1052       PureThy.add_thmss [(("inducts_weak", projections dt_induct), [case_names_induct])] ||>
  1053       Theory.parent_path ||>>
  1054       DatatypeAux.store_thmss_atts "distinct" new_type_names simp_atts distinct_thms ||>>
  1055       DatatypeAux.store_thmss "constr_rep" new_type_names constr_rep_thmss ||>>
  1056       DatatypeAux.store_thmss_atts "perm" new_type_names simp_atts perm_simps' ||>>
  1057       DatatypeAux.store_thmss "inject" new_type_names inject_thms ||>>
  1058       DatatypeAux.store_thmss "supp" new_type_names supp_thms ||>>
  1059       DatatypeAux.store_thmss_atts "fresh" new_type_names simp_atts fresh_thms ||>
  1060       fold (fn (atom, ths) => fn thy =>
  1061         let val class = Sign.intern_class thy ("fs_" ^ Sign.base_name atom)
  1062         in fold (fn T => AxClass.prove_arity
  1063             (fst (dest_Type T),
  1064               replicate (length sorts) [class], [class])
  1065             (ClassPackage.intro_classes_tac [] THEN resolve_tac ths 1)) newTs thy
  1066         end) (atoms ~~ finite_supp_thms);
  1067 
  1068     (**** strong induction theorem ****)
  1069 
  1070     val pnames = if length descr'' = 1 then ["P"]
  1071       else map (fn i => "P" ^ string_of_int i) (1 upto length descr'');
  1072     val ind_sort = if null dt_atomTs then HOLogic.typeS
  1073       else Sign.certify_sort thy9 (map (fn T => Sign.intern_class thy9 ("fs_" ^
  1074         Sign.base_name (fst (dest_Type T)))) dt_atomTs);
  1075     val fsT = TFree ("'n", ind_sort);
  1076     val fsT' = TFree ("'n", HOLogic.typeS);
  1077 
  1078     val fresh_fs = map (fn (s, T) => (T, Free (s, fsT' --> HOLogic.mk_setT T)))
  1079       (DatatypeProp.indexify_names (replicate (length dt_atomTs) "f") ~~ dt_atomTs);
  1080 
  1081     fun make_pred fsT i T =
  1082       Free (List.nth (pnames, i), fsT --> T --> HOLogic.boolT);
  1083 
  1084     fun mk_fresh1 xs [] = []
  1085       | mk_fresh1 xs ((y as (_, T)) :: ys) = map (fn x => HOLogic.mk_Trueprop
  1086             (HOLogic.mk_not (HOLogic.mk_eq (Free y, Free x))))
  1087               (filter (fn (_, U) => T = U) (rev xs)) @
  1088           mk_fresh1 (y :: xs) ys;
  1089 
  1090     fun mk_fresh2 xss [] = []
  1091       | mk_fresh2 xss ((p as (ys, _)) :: yss) = List.concat (map (fn y as (_, T) =>
  1092             map (fn (_, x as (_, U)) => HOLogic.mk_Trueprop
  1093               (Const ("Nominal.fresh", T --> U --> HOLogic.boolT) $ Free y $ Free x))
  1094                 (rev xss @ yss)) ys) @
  1095           mk_fresh2 (p :: xss) yss;
  1096 
  1097     fun make_ind_prem fsT f k T ((cname, cargs), idxs) =
  1098       let
  1099         val recs = List.filter is_rec_type cargs;
  1100         val Ts = map (typ_of_dtyp descr'' sorts') cargs;
  1101         val recTs' = map (typ_of_dtyp descr'' sorts') recs;
  1102         val tnames = Name.variant_list pnames (DatatypeProp.make_tnames Ts);
  1103         val rec_tnames = map fst (List.filter (is_rec_type o snd) (tnames ~~ cargs));
  1104         val frees = tnames ~~ Ts;
  1105         val frees' = partition_cargs idxs frees;
  1106         val z = (Name.variant tnames "z", fsT);
  1107 
  1108         fun mk_prem ((dt, s), T) =
  1109           let
  1110             val (Us, U) = strip_type T;
  1111             val l = length Us
  1112           in list_all (z :: map (pair "x") Us, HOLogic.mk_Trueprop
  1113             (make_pred fsT (body_index dt) U $ Bound l $ app_bnds (Free (s, T)) l))
  1114           end;
  1115 
  1116         val prems = map mk_prem (recs ~~ rec_tnames ~~ recTs');
  1117         val prems' = map (fn p as (_, T) => HOLogic.mk_Trueprop
  1118             (f T (Free p) (Free z))) (List.concat (map fst frees')) @
  1119           mk_fresh1 [] (List.concat (map fst frees')) @
  1120           mk_fresh2 [] frees'
  1121 
  1122       in list_all_free (frees @ [z], Logic.list_implies (prems' @ prems,
  1123         HOLogic.mk_Trueprop (make_pred fsT k T $ Free z $
  1124           list_comb (Const (cname, Ts ---> T), map Free frees))))
  1125       end;
  1126 
  1127     val ind_prems = List.concat (map (fn (((i, (_, _, constrs)), (_, idxss)), T) =>
  1128       map (make_ind_prem fsT (fn T => fn t => fn u =>
  1129         Const ("Nominal.fresh", T --> fsT --> HOLogic.boolT) $ t $ u) i T)
  1130           (constrs ~~ idxss)) (descr'' ~~ ndescr ~~ recTs));
  1131     val tnames = DatatypeProp.make_tnames recTs;
  1132     val zs = Name.variant_list tnames (replicate (length descr'') "z");
  1133     val ind_concl = HOLogic.mk_Trueprop (foldr1 (HOLogic.mk_binop "op &")
  1134       (map (fn ((((i, _), T), tname), z) =>
  1135         make_pred fsT i T $ Free (z, fsT) $ Free (tname, T))
  1136         (descr'' ~~ recTs ~~ tnames ~~ zs)));
  1137     val induct = Logic.list_implies (ind_prems, ind_concl);
  1138 
  1139     val ind_prems' =
  1140       map (fn (_, f as Free (_, T)) => list_all_free ([("x", fsT')],
  1141         HOLogic.mk_Trueprop (HOLogic.mk_mem (f $ Free ("x", fsT'),
  1142           Const ("Finite_Set.Finites", HOLogic.mk_setT (body_type T)))))) fresh_fs @
  1143       List.concat (map (fn (((i, (_, _, constrs)), (_, idxss)), T) =>
  1144         map (make_ind_prem fsT' (fn T => fn t => fn u => HOLogic.Not $
  1145           HOLogic.mk_mem (t, the (AList.lookup op = fresh_fs T) $ u)) i T)
  1146             (constrs ~~ idxss)) (descr'' ~~ ndescr ~~ recTs));
  1147     val ind_concl' = HOLogic.mk_Trueprop (foldr1 (HOLogic.mk_binop "op &")
  1148       (map (fn ((((i, _), T), tname), z) =>
  1149         make_pred fsT' i T $ Free (z, fsT') $ Free (tname, T))
  1150         (descr'' ~~ recTs ~~ tnames ~~ zs)));
  1151     val induct' = Logic.list_implies (ind_prems', ind_concl');
  1152 
  1153     fun mk_perm Ts (t, u) =
  1154       let
  1155         val T = fastype_of1 (Ts, t);
  1156         val U = fastype_of1 (Ts, u)
  1157       in Const ("Nominal.perm", T --> U --> U) $ t $ u end;
  1158 
  1159     val aux_ind_vars =
  1160       (DatatypeProp.indexify_names (replicate (length dt_atomTs) "pi") ~~
  1161        map mk_permT dt_atomTs) @ [("z", fsT')];
  1162     val aux_ind_Ts = rev (map snd aux_ind_vars);
  1163     val aux_ind_concl = HOLogic.mk_Trueprop (foldr1 (HOLogic.mk_binop "op &")
  1164       (map (fn (((i, _), T), tname) =>
  1165         HOLogic.list_all (aux_ind_vars, make_pred fsT' i T $ Bound 0 $
  1166           foldr (mk_perm aux_ind_Ts) (Free (tname, T))
  1167             (map Bound (length dt_atomTs downto 1))))
  1168         (descr'' ~~ recTs ~~ tnames)));
  1169 
  1170     fun mk_ind_perm i k p l vs j =
  1171       let
  1172         val n = length vs;
  1173         val Ts = map snd vs;
  1174         val T = List.nth (Ts, i - j);
  1175         val pT = NominalAtoms.mk_permT T
  1176       in
  1177         Const ("List.list.Cons", HOLogic.mk_prodT (T, T) --> pT --> pT) $
  1178           (HOLogic.pair_const T T $ Bound (l - j) $ foldr (mk_perm Ts)
  1179             (Bound (i - j))
  1180             (map (mk_ind_perm i k p l vs) (j - 1 downto 0) @
  1181              map Bound (n - k - 1 downto n - k - p))) $
  1182           Const ("List.list.Nil", pT)
  1183       end;
  1184 
  1185     fun mk_fresh i i' j k p l is vs _ _ =
  1186       let
  1187         val n = length vs;
  1188         val Ts = map snd vs;
  1189         val T = List.nth (Ts, n - i - 1 - j);
  1190         val f = the (AList.lookup op = fresh_fs T);
  1191         val U = List.nth (Ts, n - i' - 1);
  1192         val S = HOLogic.mk_setT T;
  1193         val prms' = map Bound (n - k downto n - k - p + 1);
  1194         val prms = map (mk_ind_perm (n - i) k p (n - l) (("a", T) :: vs))
  1195             (j - 1 downto 0) @ prms';
  1196         val bs = rev (List.mapPartial
  1197           (fn ((_, T'), i) => if T = T' then SOME (Bound i) else NONE)
  1198           (List.take (vs, n - k - p - 1) ~~ (1 upto n - k - p - 1)));
  1199         val ts = map (fn i =>
  1200           Const ("Nominal.supp", List.nth (Ts, n - i - 1) --> S) $
  1201             foldr (mk_perm (T :: Ts)) (Bound (n - i)) prms') is
  1202       in
  1203         HOLogic.mk_Trueprop (Const ("Ex", (T --> HOLogic.boolT) --> HOLogic.boolT) $
  1204           Abs ("a", T, HOLogic.Not $
  1205             (Const ("op :", T --> S --> HOLogic.boolT) $ Bound 0 $
  1206               (foldr (fn (t, u) => Const ("insert", T --> S --> S) $ t $ u)
  1207                 (foldl (fn (t, u) => Const ("op Un", S --> S --> S) $ u $ t)
  1208                   (f $ Bound (n - k - p))
  1209                   (Const ("Nominal.supp", U --> S) $
  1210                      foldr (mk_perm (T :: Ts)) (Bound (n - i')) prms :: ts))
  1211                 (foldr (mk_perm (T :: Ts)) (Bound (n - i - j)) prms :: bs)))))
  1212       end;
  1213 
  1214     fun mk_fresh_constr is p vs _ concl =
  1215       let
  1216         val n = length vs;
  1217         val Ts = map snd vs;
  1218         val _ $ (_ $ _ $ t) = concl;
  1219         val c = head_of t;
  1220         val T = body_type (fastype_of c);
  1221         val k = foldr op + 0 (map (fn (_, i) => i + 1) is);
  1222         val ps = map Bound (n - k - 1 downto n - k - p);
  1223         val (_, _, ts, us) = foldl (fn ((_, i), (m, n, ts, us)) =>
  1224           (m - i - 1, n - i,
  1225            ts @ map Bound (n downto n - i + 1) @
  1226              [foldr (mk_perm Ts) (Bound (m - i))
  1227                 (map (mk_ind_perm m k p n vs) (i - 1 downto 0) @ ps)],
  1228            us @ map (fn j => foldr (mk_perm Ts) (Bound j) ps) (m downto m - i)))
  1229           (n - 1, n - k - p - 2, [], []) is
  1230       in
  1231         HOLogic.mk_Trueprop (HOLogic.eq_const T $ list_comb (c, ts) $ list_comb (c, us))
  1232       end;
  1233 
  1234     val abs_fun_finite_supp = PureThy.get_thm thy9 (Name "abs_fun_finite_supp");
  1235 
  1236     val at_finite_select = PureThy.get_thm thy9 (Name "at_finite_select");
  1237 
  1238     val induct_aux_lemmas = List.concat (map (fn Type (s, _) =>
  1239       [PureThy.get_thm thy9 (Name ("pt_" ^ Sign.base_name s ^ "_inst")),
  1240        PureThy.get_thm thy9 (Name ("fs_" ^ Sign.base_name s ^ "1")),
  1241        PureThy.get_thm thy9 (Name ("at_" ^ Sign.base_name s ^ "_inst"))]) dt_atomTs);
  1242 
  1243     val induct_aux_lemmas' = map (fn Type (s, _) =>
  1244       PureThy.get_thm thy9 (Name ("pt_" ^ Sign.base_name s ^ "2")) RS sym) dt_atomTs;
  1245 
  1246     val fresh_aux = PureThy.get_thms thy9 (Name "fresh_aux");
  1247 
  1248     (**********************************************************************
  1249       The subgoals occurring in the proof of induct_aux have the
  1250       following parameters:
  1251 
  1252         x_1 ... x_k p_1 ... p_m z b_1 ... b_n
  1253 
  1254       where
  1255 
  1256         x_i : constructor arguments (introduced by weak induction rule)
  1257         p_i : permutations (one for each atom type in the data type)
  1258         z   : freshness context
  1259         b_i : newly introduced names for binders (sufficiently fresh)
  1260     ***********************************************************************)
  1261 
  1262     val _ = warning "proving strong induction theorem ...";
  1263 
  1264     val induct_aux = Goal.prove_global thy9 [] ind_prems' ind_concl'
  1265       (fn prems => EVERY
  1266         ([mk_subgoal 1 (K (K (K aux_ind_concl))),
  1267           indtac dt_induct tnames 1] @
  1268          List.concat (map (fn ((_, (_, _, constrs)), (_, constrs')) =>
  1269            List.concat (map (fn ((cname, cargs), is) =>
  1270              [simp_tac (HOL_basic_ss addsimps List.concat perm_simps') 1,
  1271               REPEAT (rtac allI 1)] @
  1272              List.concat (map
  1273                (fn ((_, 0), _) => []
  1274                  | ((i, j), k) => List.concat (map (fn j' =>
  1275                      let
  1276                        val DtType (tname, _) = List.nth (cargs, i + j');
  1277                        val atom = Sign.base_name tname
  1278                      in
  1279                        [mk_subgoal 1 (mk_fresh i (i + j) j'
  1280                           (length cargs) (length dt_atomTs)
  1281                           (length cargs + length dt_atomTs + 1 + i - k)
  1282                           (List.mapPartial (fn (i', j) =>
  1283                              if i = i' then NONE else SOME (i' + j)) is)),
  1284                         rtac at_finite_select 1,
  1285                         rtac (PureThy.get_thm thy9 (Name ("at_" ^ atom ^ "_inst"))) 1,
  1286                         asm_full_simp_tac (simpset_of thy9 addsimps
  1287                           [PureThy.get_thm thy9 (Name ("fs_" ^ atom ^ "1"))]) 1,
  1288                         resolve_tac prems 1,
  1289                         etac exE 1,
  1290                         asm_full_simp_tac (simpset_of thy9 addsimps
  1291                           [symmetric fresh_def]) 1]
  1292                      end) (0 upto j - 1))) (is ~~ (0 upto length is - 1))) @
  1293              (if exists (not o equal 0 o snd) is then
  1294                 [mk_subgoal 1 (mk_fresh_constr is (length dt_atomTs)),
  1295                  asm_full_simp_tac (simpset_of thy9 addsimps
  1296                    (List.concat inject_thms @
  1297                     alpha @ abs_perm @ abs_fresh @ [abs_fun_finite_supp] @
  1298                     induct_aux_lemmas)) 1,
  1299                  dtac sym 1,
  1300                  asm_full_simp_tac (simpset_of thy9) 1,
  1301                  REPEAT (etac conjE 1)]
  1302               else
  1303                 []) @
  1304              [(resolve_tac prems THEN_ALL_NEW
  1305                 (atac ORELSE'
  1306                   SUBGOAL (fn (t, i) => case Logic.strip_assums_concl t of
  1307                       _ $ (Const ("Nominal.fresh", _) $ _ $ _) =>
  1308                         asm_simp_tac (simpset_of thy9 addsimps fresh_aux) i
  1309                     | _ =>
  1310                         asm_simp_tac (simpset_of thy9
  1311                         addsimps induct_aux_lemmas'
  1312                         addsimprocs [perm_simproc]) i))) 1])
  1313                (constrs ~~ constrs'))) (descr'' ~~ ndescr)) @
  1314          [REPEAT (eresolve_tac [conjE, allE_Nil] 1),
  1315           REPEAT (etac allE 1),
  1316           REPEAT (TRY (rtac conjI 1) THEN asm_full_simp_tac (simpset_of thy9) 1)]));
  1317 
  1318     val induct_aux' = Thm.instantiate ([],
  1319       map (fn (s, T) =>
  1320         let val pT = TVar (("'n", 0), HOLogic.typeS) --> T --> HOLogic.boolT
  1321         in (cterm_of thy9 (Var ((s, 0), pT)), cterm_of thy9 (Free (s, pT))) end)
  1322           (pnames ~~ recTs) @
  1323       map (fn (_, f) =>
  1324         let val f' = Logic.varify f
  1325         in (cterm_of thy9 f',
  1326           cterm_of thy9 (Const ("Nominal.supp", fastype_of f')))
  1327         end) fresh_fs) induct_aux;
  1328 
  1329     val induct = Goal.prove_global thy9 [] ind_prems ind_concl
  1330       (fn prems => EVERY
  1331          [rtac induct_aux' 1,
  1332           REPEAT (resolve_tac induct_aux_lemmas 1),
  1333           REPEAT ((resolve_tac prems THEN_ALL_NEW
  1334             (etac meta_spec ORELSE' full_simp_tac (HOL_basic_ss addsimps [fresh_def]))) 1)])
  1335 
  1336     val (_, thy10) = thy9 |>
  1337       Theory.add_path big_name |>
  1338       PureThy.add_thms [(("induct'", induct_aux), [])] ||>>
  1339       PureThy.add_thms [(("induct", induct), [case_names_induct])] ||>>
  1340       PureThy.add_thmss [(("inducts", projections induct), [case_names_induct])];
  1341 
  1342     (**** recursion combinator ****)
  1343 
  1344     val _ = warning "defining recursion combinator ...";
  1345 
  1346     val used = foldr add_typ_tfree_names [] recTs;
  1347 
  1348     val (rec_result_Ts', rec_fn_Ts') = DatatypeProp.make_primrec_Ts descr' sorts' used;
  1349 
  1350     val rec_sort = if null dt_atomTs then HOLogic.typeS else 
  1351       let val names = map (Sign.base_name o fst o dest_Type) dt_atomTs
  1352       in Sign.certify_sort thy10 (map (Sign.intern_class thy10)
  1353         (map (fn s => "pt_" ^ s) names @
  1354          List.concat (map (fn s => List.mapPartial (fn s' =>
  1355            if s = s' then NONE
  1356            else SOME ("cp_" ^ s ^ "_" ^ s')) names) names)))
  1357       end;
  1358 
  1359     val rec_result_Ts = map (fn TFree (s, _) => TFree (s, rec_sort)) rec_result_Ts';
  1360     val rec_fn_Ts = map (typ_subst_atomic (rec_result_Ts' ~~ rec_result_Ts)) rec_fn_Ts';
  1361 
  1362     val rec_set_Ts = map (fn (T1, T2) => rec_fn_Ts ---> HOLogic.mk_setT
  1363       (HOLogic.mk_prodT (T1, T2))) (recTs ~~ rec_result_Ts);
  1364 
  1365     val big_rec_name = big_name ^ "_rec_set";
  1366     val rec_set_names = map (Sign.full_name (Theory.sign_of thy10))
  1367       (if length descr'' = 1 then [big_rec_name] else
  1368         (map ((curry (op ^) (big_rec_name ^ "_")) o string_of_int)
  1369           (1 upto (length descr''))));
  1370 
  1371     val rec_fns = map (uncurry (mk_Free "f"))
  1372       (rec_fn_Ts ~~ (1 upto (length rec_fn_Ts)));
  1373     val rec_sets = map (fn c => list_comb (Const c, rec_fns))
  1374       (rec_set_names ~~ rec_set_Ts);
  1375 
  1376     (* introduction rules for graph of recursion function *)
  1377 
  1378     val rec_preds = map (fn (a, T) =>
  1379       Free (a, T --> HOLogic.boolT)) (pnames ~~ rec_result_Ts);
  1380 
  1381     fun mk_fresh3 rs [] = []
  1382       | mk_fresh3 rs ((p as (ys, z)) :: yss) = List.concat (map (fn y as (_, T) =>
  1383             List.mapPartial (fn ((_, (_, x)), r as (_, U)) => if z = x then NONE
  1384               else SOME (HOLogic.mk_Trueprop
  1385                 (Const ("Nominal.fresh", T --> U --> HOLogic.boolT) $ Free y $ Free r)))
  1386                   rs) ys) @
  1387           mk_fresh3 rs yss;
  1388 
  1389     fun make_rec_intr T p rec_set ((rec_intr_ts, rec_prems, rec_prems',
  1390           rec_eq_prems, l), ((cname, cargs), idxs)) =
  1391       let
  1392         val Ts = map (typ_of_dtyp descr'' sorts') cargs;
  1393         val frees = map (fn i => "x" ^ string_of_int i) (1 upto length Ts) ~~ Ts;
  1394         val frees' = partition_cargs idxs frees;
  1395         val atomTs = distinct op = (maps (map snd o fst) frees');
  1396         val recs = List.mapPartial
  1397           (fn ((_, DtRec i), p) => SOME (i, p) | _ => NONE)
  1398           (partition_cargs idxs cargs ~~ frees');
  1399         val frees'' = map (fn i => "y" ^ string_of_int i) (1 upto length recs) ~~
  1400           map (fn (i, _) => List.nth (rec_result_Ts, i)) recs;
  1401         val prems1 = map (fn ((i, (_, x)), y) => HOLogic.mk_Trueprop
  1402           (HOLogic.mk_mem (HOLogic.mk_prod (Free x, Free y),
  1403              List.nth (rec_sets, i)))) (recs ~~ frees'');
  1404         val prems2 =
  1405           map (fn f => map (fn p as (_, T) => HOLogic.mk_Trueprop
  1406             (Const ("Nominal.fresh", T --> fastype_of f --> HOLogic.boolT) $
  1407               Free p $ f)) (List.concat (map fst frees'))) rec_fns;
  1408         val prems3 =
  1409           mk_fresh1 [] (List.concat (map fst frees')) @
  1410           mk_fresh2 [] frees';
  1411         val prems4 = map (fn ((i, _), y) =>
  1412           HOLogic.mk_Trueprop (List.nth (rec_preds, i) $ Free y)) (recs ~~ frees'');
  1413         val prems5 = mk_fresh3 (recs ~~ frees'') frees';
  1414         val prems6 = maps (fn aT => map (fn y as (_, T) => HOLogic.mk_Trueprop
  1415           (HOLogic.mk_mem (Const ("Nominal.supp", T --> HOLogic.mk_setT aT) $ Free y,
  1416              Const ("Finite_Set.Finites", HOLogic.mk_setT (HOLogic.mk_setT aT)))))
  1417                frees'') atomTs;
  1418         val result = list_comb (List.nth (rec_fns, l), map Free (frees @ frees''));
  1419         val result_freshs = map (fn p as (_, T) =>
  1420           Const ("Nominal.fresh", T --> fastype_of result --> HOLogic.boolT) $
  1421             Free p $ result) (List.concat (map fst frees'));
  1422         val P = HOLogic.mk_Trueprop (p $ result)
  1423       in
  1424         (rec_intr_ts @ [Logic.list_implies (List.concat prems2 @ prems3 @ prems1,
  1425            HOLogic.mk_Trueprop (HOLogic.mk_mem
  1426              (HOLogic.mk_prod (list_comb (Const (cname, Ts ---> T), map Free frees),
  1427                result), rec_set)))],
  1428          rec_prems @ [list_all_free (frees @ frees'', Logic.list_implies (prems4, P))],
  1429          if null result_freshs then rec_prems'
  1430          else rec_prems' @ [list_all_free (frees @ frees'',
  1431            Logic.list_implies (List.nth (prems2, l) @ prems3 @ prems5 @ prems6 @ [P],
  1432              HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj result_freshs)))],
  1433          rec_eq_prems @ [List.concat prems2 @ prems3],
  1434          l + 1)
  1435       end;
  1436 
  1437     val (rec_intr_ts, rec_prems, rec_prems', rec_eq_prems, _) =
  1438       Library.foldl (fn (x, ((((d, d'), T), p), rec_set)) =>
  1439         Library.foldl (make_rec_intr T p rec_set) (x, #3 (snd d) ~~ snd d'))
  1440           (([], [], [], [], 0), descr'' ~~ ndescr ~~ recTs ~~ rec_preds ~~ rec_sets);
  1441 
  1442     val (thy11, {intrs = rec_intrs, elims = rec_elims, raw_induct = rec_induct, ...}) =
  1443       setmp InductivePackage.quiet_mode (!quiet_mode)
  1444         (InductivePackage.add_inductive_i false true big_rec_name false false false
  1445            rec_sets (map (fn x => (("", x), [])) rec_intr_ts) []) thy10;
  1446 
  1447     (** equivariance **)
  1448 
  1449     val fresh_bij = PureThy.get_thms thy11 (Name "fresh_bij");
  1450     val perm_bij = PureThy.get_thms thy11 (Name "perm_bij");
  1451 
  1452     val (rec_equiv_thms, rec_equiv_thms') = ListPair.unzip (map (fn aT =>
  1453       let
  1454         val permT = mk_permT aT;
  1455         val pi = Free ("pi", permT);
  1456         val rec_fns_pi = map (curry (mk_perm []) pi o uncurry (mk_Free "f"))
  1457           (rec_fn_Ts ~~ (1 upto (length rec_fn_Ts)));
  1458         val rec_sets_pi = map (fn c => list_comb (Const c, rec_fns_pi))
  1459           (rec_set_names ~~ rec_set_Ts);
  1460         val ps = map (fn ((((T, U), R), R'), i) =>
  1461           let
  1462             val x = Free ("x" ^ string_of_int i, T);
  1463             val y = Free ("y" ^ string_of_int i, U)
  1464           in
  1465             (HOLogic.mk_mem (HOLogic.mk_prod (x, y), R),
  1466              HOLogic.mk_mem (HOLogic.mk_prod (mk_perm [] (pi, x), mk_perm [] (pi, y)), R'))
  1467           end) (recTs ~~ rec_result_Ts ~~ rec_sets ~~ rec_sets_pi ~~ (1 upto length recTs));
  1468         val ths = map (fn th => standard (th RS mp)) (split_conj_thm
  1469           (Goal.prove_global thy11 [] []
  1470             (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj (map HOLogic.mk_imp ps)))
  1471             (fn _ => rtac rec_induct 1 THEN REPEAT
  1472                (NominalPermeq.perm_simp_tac (simpset_of thy11) 1 THEN
  1473                 (resolve_tac rec_intrs THEN_ALL_NEW
  1474                  asm_simp_tac (HOL_ss addsimps (fresh_bij @ perm_bij))) 1))))
  1475         val ths' = map (fn ((P, Q), th) =>
  1476           Goal.prove_global thy11 [] []
  1477             (Logic.mk_implies (HOLogic.mk_Trueprop Q, HOLogic.mk_Trueprop P))
  1478             (fn _ => dtac (Thm.instantiate ([],
  1479                  [(cterm_of thy11 (Var (("pi", 0), permT)),
  1480                    cterm_of thy11 (Const ("List.rev", permT --> permT) $ pi))]) th) 1 THEN
  1481                NominalPermeq.perm_simp_tac HOL_ss 1)) (ps ~~ ths)
  1482       in (ths, ths') end) dt_atomTs);
  1483 
  1484     (** finite support **)
  1485 
  1486     val rec_fin_supp_thms = map (fn aT =>
  1487       let
  1488         val name = Sign.base_name (fst (dest_Type aT));
  1489         val fs_name = PureThy.get_thm thy11 (Name ("fs_" ^ name ^ "1"));
  1490         val aset = HOLogic.mk_setT aT;
  1491         val finites = Const ("Finite_Set.Finites", HOLogic.mk_setT aset);
  1492         val fins = map (fn (f, T) => HOLogic.mk_Trueprop (HOLogic.mk_mem
  1493           (Const ("Nominal.supp", T --> aset) $ f, finites)))
  1494             (rec_fns ~~ rec_fn_Ts)
  1495       in
  1496         map (fn th => standard (th RS mp)) (split_conj_thm
  1497           (Goal.prove_global thy11 [] fins
  1498             (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
  1499               (map (fn (((T, U), R), i) =>
  1500                  let
  1501                    val x = Free ("x" ^ string_of_int i, T);
  1502                    val y = Free ("y" ^ string_of_int i, U)
  1503                  in
  1504                    HOLogic.mk_imp (HOLogic.mk_mem (HOLogic.mk_prod (x, y), R),
  1505                      HOLogic.mk_mem (Const ("Nominal.supp", U --> aset) $ y, finites))
  1506                  end) (recTs ~~ rec_result_Ts ~~ rec_sets ~~ (1 upto length recTs)))))
  1507             (fn fins => (rtac rec_induct THEN_ALL_NEW cut_facts_tac fins) 1 THEN REPEAT
  1508                (NominalPermeq.finite_guess_tac (HOL_ss addsimps [fs_name]) 1))))
  1509       end) dt_atomTs;
  1510 
  1511     (** freshness **)
  1512 
  1513     val perm_fresh_fresh = PureThy.get_thms thy11 (Name "perm_fresh_fresh");
  1514     val perm_swap = PureThy.get_thms thy11 (Name "perm_swap");
  1515 
  1516     fun perm_of_pair (x, y) =
  1517       let
  1518         val T = fastype_of x;
  1519         val pT = mk_permT T
  1520       in Const ("List.list.Cons", HOLogic.mk_prodT (T, T) --> pT --> pT) $
  1521         HOLogic.mk_prod (x, y) $ Const ("List.list.Nil", pT)
  1522       end;
  1523 
  1524     val finite_premss = map (fn aT =>
  1525       map (fn (f, T) => HOLogic.mk_Trueprop (HOLogic.mk_mem
  1526         (Const ("Nominal.supp", T --> HOLogic.mk_setT aT) $ f,
  1527          Const ("Finite_Set.Finites", HOLogic.mk_setT (HOLogic.mk_setT aT)))))
  1528            (rec_fns ~~ rec_fn_Ts)) dt_atomTs;
  1529 
  1530     val rec_fresh_thms = map (fn ((aT, eqvt_ths), finite_prems) =>
  1531       let
  1532         val name = Sign.base_name (fst (dest_Type aT));
  1533         val fs_name = PureThy.get_thm thy11 (Name ("fs_" ^ name ^ "1"));
  1534         val a = Free ("a", aT);
  1535         val freshs = map (fn (f, fT) => HOLogic.mk_Trueprop
  1536             (Const ("Nominal.fresh", aT --> fT --> HOLogic.boolT) $ a $ f))
  1537           (rec_fns ~~ rec_fn_Ts)
  1538       in
  1539         map (fn (((T, U), R), eqvt_th) =>
  1540           let
  1541             val x = Free ("x", T);
  1542             val y = Free ("y", U);
  1543             val y' = Free ("y'", U)
  1544           in
  1545             standard (Goal.prove (Context.init_proof thy11) [] (finite_prems @
  1546                 [HOLogic.mk_Trueprop (HOLogic.mk_mem
  1547                   (HOLogic.mk_prod (x, y), R)),
  1548                  HOLogic.mk_Trueprop (HOLogic.mk_all ("y'", U,
  1549                    HOLogic.mk_imp (HOLogic.mk_mem
  1550                        (HOLogic.mk_prod (x, y'), R),
  1551                      HOLogic.mk_eq (y', y)))),
  1552                  HOLogic.mk_Trueprop (Const ("Nominal.fresh",
  1553                    aT --> T --> HOLogic.boolT) $ a $ x)] @
  1554               freshs)
  1555               (HOLogic.mk_Trueprop (Const ("Nominal.fresh",
  1556                  aT --> U --> HOLogic.boolT) $ a $ y))
  1557               (fn {prems, context} =>
  1558                  let
  1559                    val (finite_prems, rec_prem :: unique_prem ::
  1560                      fresh_prems) = chop (length finite_prems) prems;
  1561                    val unique_prem' = unique_prem RS spec RS mp;
  1562                    val unique = [unique_prem', unique_prem' RS sym] MRS trans;
  1563                    val _ $ (_ $ (_ $ S $ _)) $ _ = prop_of supports_fresh;
  1564                    val tuple = foldr1 HOLogic.mk_prod (x :: rec_fns)
  1565                  in EVERY
  1566                    [rtac (Drule.cterm_instantiate
  1567                       [(cterm_of thy11 S,
  1568                         cterm_of thy11 (Const ("Nominal.supp",
  1569                           fastype_of tuple --> HOLogic.mk_setT aT) $ tuple))]
  1570                       supports_fresh) 1,
  1571                     simp_tac (HOL_basic_ss addsimps
  1572                       [supports_def, symmetric fresh_def, fresh_prod]) 1,
  1573                     REPEAT_DETERM (resolve_tac [allI, impI] 1),
  1574                     REPEAT_DETERM (etac conjE 1),
  1575                     rtac unique 1,
  1576                     SUBPROOF (fn {prems = prems', params = [a, b], ...} => EVERY
  1577                       [cut_facts_tac [rec_prem] 1,
  1578                        rtac (Thm.instantiate ([],
  1579                          [(cterm_of thy11 (Var (("pi", 0), mk_permT aT)),
  1580                            cterm_of thy11 (perm_of_pair (term_of a, term_of b)))]) eqvt_th) 1,
  1581                        asm_simp_tac (HOL_ss addsimps
  1582                          (prems' @ perm_swap @ perm_fresh_fresh)) 1]) context 1,
  1583                     rtac rec_prem 1,
  1584                     simp_tac (HOL_ss addsimps (fs_name ::
  1585                       supp_prod :: finite_Un :: finite_prems)) 1,
  1586                     simp_tac (HOL_ss addsimps (symmetric fresh_def ::
  1587                       fresh_prod :: fresh_prems)) 1]
  1588                  end))
  1589           end) (recTs ~~ rec_result_Ts ~~ rec_sets ~~ eqvt_ths)
  1590       end) (dt_atomTs ~~ rec_equiv_thms' ~~ finite_premss);
  1591 
  1592     (** uniqueness **)
  1593 
  1594     val exists_fresh = PureThy.get_thms thy11 (Name "exists_fresh");
  1595     val fs_atoms = map (fn Type (s, _) => PureThy.get_thm thy11
  1596       (Name ("fs_" ^ Sign.base_name s ^ "1"))) dt_atomTs;
  1597     val fresh_atm = PureThy.get_thms thy11 (Name "fresh_atm");
  1598     val calc_atm = PureThy.get_thms thy11 (Name "calc_atm");
  1599     val fresh_left = PureThy.get_thms thy11 (Name "fresh_left");
  1600 
  1601     val fun_tuple = foldr1 HOLogic.mk_prod rec_fns;
  1602     val fun_tupleT = fastype_of fun_tuple;
  1603     val rec_unique_frees =
  1604       DatatypeProp.indexify_names (replicate (length recTs) "x") ~~ recTs;
  1605     val rec_unique_frees'' = map (fn (s, T) => (s ^ "'", T)) rec_unique_frees;
  1606     val rec_unique_frees' =
  1607       DatatypeProp.indexify_names (replicate (length recTs) "y") ~~ rec_result_Ts;
  1608     val rec_unique_concls = map (fn ((x as (_, T), U), R) =>
  1609         Const ("Ex1", (U --> HOLogic.boolT) --> HOLogic.boolT) $
  1610           Abs ("y", U, HOLogic.mk_mem (HOLogic.pair_const T U $ Free x $ Bound 0, R)))
  1611       (rec_unique_frees ~~ rec_result_Ts ~~ rec_sets);
  1612 
  1613     val induct_aux_rec = Drule.cterm_instantiate
  1614       (map (pairself (cterm_of thy11))
  1615          (map (fn (aT, f) => (Logic.varify f, Abs ("z", HOLogic.unitT,
  1616             Const ("Nominal.supp", fun_tupleT --> HOLogic.mk_setT aT) $ fun_tuple)))
  1617               fresh_fs @
  1618           map (fn (((P, T), (x, U)), Q) =>
  1619            (Var ((P, 0), HOLogic.unitT --> Logic.varifyT T --> HOLogic.boolT),
  1620             Abs ("z", HOLogic.unitT, absfree (x, U, Q))))
  1621               (pnames ~~ recTs ~~ rec_unique_frees ~~ rec_unique_concls) @
  1622           map (fn (s, T) => (Var ((s, 0), Logic.varifyT T), Free (s, T)))
  1623             rec_unique_frees)) induct_aux;
  1624 
  1625     fun obtain_fresh_name vs ths rec_fin_supp T (freshs1, freshs2, ctxt) =
  1626       let
  1627         val p = foldr1 HOLogic.mk_prod (vs @ freshs1);
  1628         val ex = Goal.prove ctxt [] [] (HOLogic.mk_Trueprop
  1629             (HOLogic.exists_const T $ Abs ("x", T,
  1630               Const ("Nominal.fresh", T --> fastype_of p --> HOLogic.boolT) $
  1631                 Bound 0 $ p)))
  1632           (fn _ => EVERY
  1633             [cut_facts_tac ths 1,
  1634              REPEAT_DETERM (dresolve_tac (the (AList.lookup op = rec_fin_supp T)) 1),
  1635              resolve_tac exists_fresh 1,
  1636              asm_simp_tac (HOL_ss addsimps (supp_prod :: finite_Un :: fs_atoms)) 1]);
  1637         val (([cx], ths), ctxt') = Obtain.result
  1638           (fn _ => EVERY
  1639             [etac exE 1,
  1640              full_simp_tac (HOL_ss addsimps (fresh_prod :: fresh_atm)) 1,
  1641              REPEAT (etac conjE 1)])
  1642           [ex] ctxt
  1643       in (freshs1 @ [term_of cx], freshs2 @ ths, ctxt') end;
  1644 
  1645     val rec_unique_thms = split_conj_thm (Goal.prove
  1646       (Context.init_proof thy11) (map fst rec_unique_frees)
  1647       (List.concat finite_premss @ rec_prems @ rec_prems')
  1648       (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj rec_unique_concls))
  1649       (fn {prems, context} =>
  1650          let
  1651            val k = length rec_fns;
  1652            val (finite_thss, ths1) = fold_map (fn T => fn xs =>
  1653              apfst (pair T) (chop k xs)) dt_atomTs prems;
  1654            val (P_ind_ths, ths2) = chop k ths1;
  1655            val P_ths = map (fn th => th RS mp) (split_conj_thm
  1656              (Goal.prove context
  1657                (map fst (rec_unique_frees'' @ rec_unique_frees')) []
  1658                (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
  1659                   (map (fn (((x, y), S), P) => HOLogic.mk_imp (HOLogic.mk_mem
  1660                     (HOLogic.mk_prod (Free x, Free y), S), P $ (Free y)))
  1661                       (rec_unique_frees'' ~~ rec_unique_frees' ~~ rec_sets ~~ rec_preds))))
  1662                (fn _ =>
  1663                   rtac rec_induct 1 THEN
  1664                   REPEAT ((resolve_tac P_ind_ths THEN_ALL_NEW assume_tac) 1))));
  1665            val rec_fin_supp_thms' = map
  1666              (fn (ths, (T, fin_ths)) => (T, map (curry op MRS fin_ths) ths))
  1667              (rec_fin_supp_thms ~~ finite_thss);
  1668            val fcbs = List.concat (map split_conj_thm ths2);
  1669          in EVERY
  1670            ([rtac induct_aux_rec 1] @
  1671             maps (fn (_, finite_ths) =>
  1672               [cut_facts_tac finite_ths 1,
  1673                asm_simp_tac (HOL_ss addsimps [supp_prod, finite_Un]) 1]) finite_thss @
  1674             maps (fn ((_, idxss), elim) => maps (fn idxs =>
  1675               [full_simp_tac (HOL_ss addsimps [symmetric fresh_def, supp_prod, Un_iff]) 1,
  1676                REPEAT_DETERM (eresolve_tac [conjE, ex1E] 1),
  1677                rtac ex1I 1,
  1678                (resolve_tac rec_intrs THEN_ALL_NEW atac) 1,
  1679                rotate_tac ~1 1,
  1680                ((DETERM o etac elim) THEN_ALL_NEW full_simp_tac
  1681                   (HOL_ss addsimps (Pair_eq :: List.concat distinct_thms))) 1] @
  1682                (if null idxs then [] else [etac conjE 1, hyp_subst_tac 1,
  1683                 SUBPROOF (fn {asms, concl, prems = prems', params, context = context', ...} =>
  1684                   let
  1685                     val (_, prem) = split_last prems';
  1686                     val _ $ (_ $ lhs $ rhs) = prop_of prem;
  1687                     val _ $ (_ $ lhs' $ rhs') = term_of concl;
  1688                     val rT = fastype_of lhs';
  1689                     val (c, cargsl) = strip_comb lhs;
  1690                     val cargsl' = partition_cargs idxs cargsl;
  1691                     val boundsl = List.concat (map fst cargsl');
  1692                     val (_, cargsr) = strip_comb rhs;
  1693                     val cargsr' = partition_cargs idxs cargsr;
  1694                     val boundsr = List.concat (map fst cargsr');
  1695                     val (params1, _ :: params2) =
  1696                       chop (length params div 2) (map term_of params);
  1697                     val params' = params1 @ params2;
  1698                     val rec_prems = filter (fn th => case prop_of th of
  1699                       _ $ (Const ("op :", _) $ _ $ _) => true | _ => false) prems';
  1700                     val fresh_prems = filter (fn th => case prop_of th of
  1701                         _ $ (Const ("Nominal.fresh", _) $ _ $ _) => true
  1702                       | _ $ (Const ("Not", _) $ _) => true
  1703                       | _ => false) prems';
  1704                     val Ts = map fastype_of boundsl;
  1705 
  1706                     val _ = warning "step 1: obtaining fresh names";
  1707                     val (freshs1, freshs2, context'') = fold
  1708                       (obtain_fresh_name (rec_fns @ params')
  1709                          (List.concat (map snd finite_thss) @ rec_prems)
  1710                          rec_fin_supp_thms')
  1711                       Ts ([], [], context');
  1712                     val pi1 = map perm_of_pair (boundsl ~~ freshs1);
  1713                     val rpi1 = rev pi1;
  1714                     val pi2 = map perm_of_pair (boundsr ~~ freshs1);
  1715 
  1716                     fun mk_not_sym ths = List.concat (map (fn th =>
  1717                       case prop_of th of
  1718                           _ $ (Const ("Not", _) $ _) => [th, th RS not_sym]
  1719                         | _ => [th]) ths);
  1720                     val fresh_prems' = mk_not_sym fresh_prems;
  1721                     val freshs2' = mk_not_sym freshs2;
  1722 
  1723                     (** as, bs, cs # K as ts, K bs us **)
  1724                     val _ = warning "step 2: as, bs, cs # K as ts, K bs us";
  1725                     val prove_fresh_ss = HOL_ss addsimps
  1726                       (finite_Diff :: List.concat fresh_thms @
  1727                        fs_atoms @ abs_fresh @ abs_supp @ fresh_atm);
  1728                     (* FIXME: avoid asm_full_simp_tac ? *)
  1729                     fun prove_fresh ths y x = Goal.prove context'' [] []
  1730                       (HOLogic.mk_Trueprop (Const ("Nominal.fresh",
  1731                          fastype_of x --> fastype_of y --> HOLogic.boolT) $ x $ y))
  1732                       (fn _ => cut_facts_tac ths 1 THEN asm_full_simp_tac prove_fresh_ss 1);
  1733                     val constr_fresh_thms =
  1734                       map (prove_fresh fresh_prems lhs) boundsl @
  1735                       map (prove_fresh fresh_prems rhs) boundsr @
  1736                       map (prove_fresh freshs2 lhs) freshs1 @
  1737                       map (prove_fresh freshs2 rhs) freshs1;
  1738 
  1739                     (** pi1 o (K as ts) = pi2 o (K bs us) **)
  1740                     val _ = warning "step 3: pi1 o (K as ts) = pi2 o (K bs us)";
  1741                     val pi1_pi2_eq = Goal.prove context'' [] []
  1742                       (HOLogic.mk_Trueprop (HOLogic.mk_eq
  1743                         (foldr (mk_perm []) lhs pi1, foldr (mk_perm []) rhs pi2)))
  1744                       (fn _ => EVERY
  1745                          [cut_facts_tac constr_fresh_thms 1,
  1746                           asm_simp_tac (HOL_basic_ss addsimps perm_fresh_fresh) 1,
  1747                           rtac prem 1]);
  1748 
  1749                     (** pi1 o ts = pi2 o us **)
  1750                     val _ = warning "step 4: pi1 o ts = pi2 o us";
  1751                     val pi1_pi2_eqs = map (fn (t, u) =>
  1752                       Goal.prove context'' [] []
  1753                         (HOLogic.mk_Trueprop (HOLogic.mk_eq
  1754                           (foldr (mk_perm []) t pi1, foldr (mk_perm []) u pi2)))
  1755                         (fn _ => EVERY
  1756                            [cut_facts_tac [pi1_pi2_eq] 1,
  1757                             asm_full_simp_tac (HOL_ss addsimps
  1758                               (calc_atm @ List.concat perm_simps' @
  1759                                fresh_prems' @ freshs2' @ abs_perm @
  1760                                alpha @ List.concat inject_thms)) 1]))
  1761                         (map snd cargsl' ~~ map snd cargsr');
  1762 
  1763                     (** pi1^-1 o pi2 o us = ts **)
  1764                     val _ = warning "step 5: pi1^-1 o pi2 o us = ts";
  1765                     val rpi1_pi2_eqs = map (fn ((t, u), eq) =>
  1766                       Goal.prove context'' [] []
  1767                         (HOLogic.mk_Trueprop (HOLogic.mk_eq
  1768                           (foldr (mk_perm []) u (rpi1 @ pi2), t)))
  1769                         (fn _ => simp_tac (HOL_ss addsimps
  1770                            ((eq RS sym) :: perm_swap)) 1))
  1771                         (map snd cargsl' ~~ map snd cargsr' ~~ pi1_pi2_eqs);
  1772 
  1773                     val (rec_prems1, rec_prems2) =
  1774                       chop (length rec_prems div 2) rec_prems;
  1775 
  1776                     (** (ts, pi1^-1 o pi2 o vs) in rec_set **)
  1777                     val _ = warning "step 6: (ts, pi1^-1 o pi2 o vs) in rec_set";
  1778                     val rec_prems' = map (fn th =>
  1779                       let
  1780                         val _ $ (_ $ (_ $ x $ y) $ S) = prop_of th;
  1781                         val k = find_index (equal S) rec_sets;
  1782                         val pi = rpi1 @ pi2;
  1783                         fun mk_pi z = foldr (mk_perm []) z pi;
  1784                         fun eqvt_tac p =
  1785                           let
  1786                             val U as Type (_, [Type (_, [T, _])]) = fastype_of p;
  1787                             val l = find_index (equal T) dt_atomTs;
  1788                             val th = List.nth (List.nth (rec_equiv_thms', l), k);
  1789                             val th' = Thm.instantiate ([],
  1790                               [(cterm_of thy11 (Var (("pi", 0), U)),
  1791                                 cterm_of thy11 p)]) th;
  1792                           in rtac th' 1 end;
  1793                         val th' = Goal.prove context'' [] []
  1794                           (HOLogic.mk_Trueprop (HOLogic.mk_mem
  1795                             (HOLogic.mk_prod (mk_pi x, mk_pi y), S)))
  1796                           (fn _ => EVERY
  1797                              (map eqvt_tac pi @
  1798                               [simp_tac (HOL_ss addsimps (fresh_prems' @ freshs2' @
  1799                                  perm_swap @ perm_fresh_fresh)) 1,
  1800                                rtac th 1]))
  1801                       in
  1802                         Simplifier.simplify
  1803                           (HOL_basic_ss addsimps rpi1_pi2_eqs) th'
  1804                       end) rec_prems2;
  1805 
  1806                     val ihs = filter (fn th => case prop_of th of
  1807                       _ $ (Const ("All", _) $ _) => true | _ => false) prems';
  1808 
  1809                     (** pi1 o rs = p2 o vs , rs = pi1^-1 o pi2 o vs **)
  1810                     val _ = warning "step 7: pi1 o rs = p2 o vs , rs = pi1^-1 o pi2 o vs";
  1811                     val (rec_eqns1, rec_eqns2) = ListPair.unzip (map (fn (th, ih) =>
  1812                       let
  1813                         val th' = th RS (ih RS spec RS mp) RS sym;
  1814                         val _ $ (_ $ lhs $ rhs) = prop_of th';
  1815                         fun strip_perm (_ $ _ $ t) = strip_perm t
  1816                           | strip_perm t = t;
  1817                       in
  1818                         (Goal.prove context'' [] []
  1819                            (HOLogic.mk_Trueprop (HOLogic.mk_eq
  1820                               (foldr (mk_perm []) lhs pi1,
  1821                                foldr (mk_perm []) (strip_perm rhs) pi2)))
  1822                            (fn _ => simp_tac (HOL_basic_ss addsimps
  1823                               (th' :: perm_swap)) 1),
  1824                          th')
  1825                       end) (rec_prems' ~~ ihs));
  1826 
  1827                     (** as # rs , bs # vs **)
  1828                     val _ = warning "step 8: as # rs , bs # vs";
  1829                     val (rec_freshs1, rec_freshs2) = ListPair.unzip (List.concat
  1830                       (map (fn (((rec_prem, rec_prem'), eqn), ih) =>
  1831                         let
  1832                           val _ $ (_ $ (_ $ x $ (y as Free (_, T))) $ S) =
  1833                             prop_of rec_prem;
  1834                           val _ $ (_ $ (_ $ _ $ y') $ _) = prop_of rec_prem';
  1835                           val k = find_index (equal S) rec_sets;
  1836                           val atoms = List.concat (List.mapPartial
  1837                             (fn ((bs, z), (bs', _)) =>
  1838                               if z = x then NONE else SOME (bs ~~ bs'))
  1839                             (cargsl' ~~ cargsr'))
  1840                         in
  1841                           map (fn (a as Free (_, aT), b) =>
  1842                             let
  1843                               val l = find_index (equal aT) dt_atomTs;
  1844                               val th = Goal.prove context'' [] []
  1845                                 (HOLogic.mk_Trueprop (Const ("Nominal.fresh",
  1846                                   aT --> T --> HOLogic.boolT) $ a $ y))
  1847                                 (fn _ => EVERY
  1848                                    (rtac (List.nth (List.nth (rec_fresh_thms, l), k)) 1 ::
  1849                                     map (fn th => rtac th 1)
  1850                                       (snd (List.nth (finite_thss, l))) @
  1851                                     [rtac rec_prem 1, rtac ih 1,
  1852                                      REPEAT_DETERM (resolve_tac fresh_prems 1)]));
  1853                               val th' = Goal.prove context'' [] []
  1854                                 (HOLogic.mk_Trueprop (Const ("Nominal.fresh",
  1855                                   aT --> T --> HOLogic.boolT) $ b $ y'))
  1856                                 (fn _ => cut_facts_tac [th] 1 THEN
  1857                                     asm_full_simp_tac (HOL_ss addsimps (eqn ::
  1858                                       fresh_left @ fresh_prems' @ freshs2' @
  1859                                       rev_simps @ app_simps @ calc_atm)) 1)
  1860                             in (th, th') end) atoms
  1861                         end) (rec_prems1 ~~ rec_prems2 ~~ rec_eqns2 ~~ ihs)));
  1862 
  1863                     (** as # fK as ts rs , bs # fK bs us vs **)
  1864                     val _ = warning "step 9: as # fK as ts rs , bs # fK bs us vs";
  1865                     fun prove_fresh_result t (a as Free (_, aT)) =
  1866                       Goal.prove context'' [] []
  1867                         (HOLogic.mk_Trueprop (Const ("Nominal.fresh",
  1868                           aT --> rT --> HOLogic.boolT) $ a $ t))
  1869                         (fn _ => EVERY
  1870                            [resolve_tac fcbs 1,
  1871                             REPEAT_DETERM (resolve_tac
  1872                               (fresh_prems @ rec_freshs1 @ rec_freshs2) 1),
  1873                             REPEAT_DETERM (resolve_tac (maps snd rec_fin_supp_thms') 1
  1874                               THEN resolve_tac rec_prems 1),
  1875                             resolve_tac P_ind_ths 1,
  1876                             REPEAT_DETERM (resolve_tac (P_ths @ rec_prems) 1)]);
  1877         
  1878                     val fresh_results =
  1879                       map (prove_fresh_result rhs') (List.concat (map fst cargsl')) @
  1880                       map (prove_fresh_result lhs') (List.concat (map fst cargsr'));
  1881 
  1882                     (** cs # fK as ts rs , cs # fK bs us vs **)
  1883                     val _ = warning "step 10: cs # fK as ts rs , cs # fK bs us vs";
  1884                     fun prove_fresh_result' recs t (a as Free (_, aT)) =
  1885                       Goal.prove context'' [] []
  1886                         (HOLogic.mk_Trueprop (Const ("Nominal.fresh",
  1887                           aT --> rT --> HOLogic.boolT) $ a $ t))
  1888                         (fn _ => EVERY
  1889                           [cut_facts_tac recs 1,
  1890                            REPEAT_DETERM (dresolve_tac
  1891                              (the (AList.lookup op = rec_fin_supp_thms' aT)) 1),
  1892                            NominalPermeq.fresh_guess_tac
  1893                              (HOL_ss addsimps (freshs2 @
  1894                                 fs_atoms @ fresh_atm @
  1895                                 List.concat (map snd finite_thss))) 1]);
  1896 
  1897                     val fresh_results' =
  1898                       map (prove_fresh_result' rec_prems1 rhs') freshs1 @
  1899                       map (prove_fresh_result' rec_prems2 lhs') freshs1;
  1900 
  1901                     (** pi1 o (fK as ts rs) = pi2 o (fK bs us vs) **)
  1902                     val _ = warning "step 11: pi1 o (fK as ts rs) = pi2 o (fK bs us vs)";
  1903                     val pi1_pi2_result = Goal.prove context'' [] []
  1904                       (HOLogic.mk_Trueprop (HOLogic.mk_eq
  1905                         (foldr (mk_perm []) rhs' pi1, foldr (mk_perm []) lhs' pi2)))
  1906                       (fn _ => NominalPermeq.perm_simp_tac (HOL_ss addsimps
  1907                            pi1_pi2_eqs @ rec_eqns1) 1 THEN
  1908                          TRY (simp_tac (HOL_ss addsimps
  1909                            (fresh_prems' @ freshs2' @ calc_atm @ perm_fresh_fresh)) 1));
  1910 
  1911                     val _ = warning "final result";
  1912                     val final = Goal.prove context'' [] [] (term_of concl)
  1913                       (fn _ => cut_facts_tac [pi1_pi2_result RS sym] 1 THEN
  1914                         full_simp_tac (HOL_basic_ss addsimps perm_fresh_fresh @
  1915                           fresh_results @ fresh_results') 1);
  1916                     val final' = ProofContext.export context'' context' [final];
  1917                     val _ = warning "finished!"
  1918                   in
  1919                     resolve_tac final' 1
  1920                   end) context 1])) idxss) (ndescr ~~ rec_elims))
  1921          end));
  1922 
  1923     val rec_total_thms = map (fn r => r RS theI') rec_unique_thms;
  1924 
  1925     (* define primrec combinators *)
  1926 
  1927     val big_reccomb_name = (space_implode "_" new_type_names) ^ "_rec";
  1928     val reccomb_names = map (Sign.full_name thy11)
  1929       (if length descr'' = 1 then [big_reccomb_name] else
  1930         (map ((curry (op ^) (big_reccomb_name ^ "_")) o string_of_int)
  1931           (1 upto (length descr''))));
  1932     val reccombs = map (fn ((name, T), T') => list_comb
  1933       (Const (name, rec_fn_Ts @ [T] ---> T'), rec_fns))
  1934         (reccomb_names ~~ recTs ~~ rec_result_Ts);
  1935 
  1936     val (reccomb_defs, thy12) =
  1937       thy11
  1938       |> Theory.add_consts_i (map (fn ((name, T), T') =>
  1939           (Sign.base_name name, rec_fn_Ts @ [T] ---> T', NoSyn))
  1940           (reccomb_names ~~ recTs ~~ rec_result_Ts))
  1941       |> (PureThy.add_defs_i false o map Thm.no_attributes) (map (fn ((((name, comb), set), T), T') =>
  1942           ((Sign.base_name name) ^ "_def", Logic.mk_equals (comb, absfree ("x", T,
  1943            Const ("The", (T' --> HOLogic.boolT) --> T') $ absfree ("y", T',
  1944              HOLogic.mk_mem (HOLogic.mk_prod (Free ("x", T), Free ("y", T')), set))))))
  1945                (reccomb_names ~~ reccombs ~~ rec_sets ~~ recTs ~~ rec_result_Ts));
  1946 
  1947     (* prove characteristic equations for primrec combinators *)
  1948 
  1949     val rec_thms = map (fn (prems, concl) =>
  1950       let
  1951         val _ $ (_ $ (_ $ x) $ _) = concl;
  1952         val (_, cargs) = strip_comb x;
  1953         val ps = map (fn (x as Free (_, T), i) =>
  1954           (Free ("x" ^ string_of_int i, T), x)) (cargs ~~ (1 upto length cargs));
  1955         val concl' = subst_atomic_types (rec_result_Ts' ~~ rec_result_Ts) concl;
  1956         val prems' = List.concat finite_premss @ rec_prems @ rec_prems' @
  1957           map (subst_atomic ps) prems;
  1958         fun solve rules prems = resolve_tac rules THEN_ALL_NEW
  1959           (resolve_tac prems THEN_ALL_NEW atac)
  1960       in
  1961         Goal.prove_global thy12 [] prems' concl'
  1962           (fn prems => EVERY
  1963             [rewrite_goals_tac reccomb_defs,
  1964              rtac the1_equality 1,
  1965              solve rec_unique_thms prems 1,
  1966              resolve_tac rec_intrs 1,
  1967              REPEAT (solve (prems @ rec_total_thms) prems 1)])
  1968       end) (rec_eq_prems ~~
  1969         DatatypeProp.make_primrecs new_type_names descr' sorts' thy12);
  1970     
  1971     (* FIXME: theorems are stored in database for testing only *)
  1972     val (_, thy13) = thy12 |>
  1973       PureThy.add_thmss
  1974         [(("rec_equiv", List.concat rec_equiv_thms), []),
  1975          (("rec_equiv'", List.concat rec_equiv_thms'), []),
  1976          (("rec_fin_supp", List.concat rec_fin_supp_thms), []),
  1977          (("rec_fresh", List.concat rec_fresh_thms), []),
  1978          (("rec_unique", map standard rec_unique_thms), []),
  1979          (("recs", rec_thms), [])] ||>
  1980       Theory.parent_path;
  1981 
  1982   in
  1983     thy13
  1984   end;
  1985 
  1986 val add_nominal_datatype = gen_add_nominal_datatype read_typ true;
  1987 
  1988 
  1989 (* FIXME: The following stuff should be exported by DatatypePackage *)
  1990 
  1991 local structure P = OuterParse and K = OuterKeyword in
  1992 
  1993 val datatype_decl =
  1994   Scan.option (P.$$$ "(" |-- P.name --| P.$$$ ")") -- P.type_args -- P.name -- P.opt_infix --
  1995     (P.$$$ "=" |-- P.enum1 "|" (P.name -- Scan.repeat P.typ -- P.opt_mixfix));
  1996 
  1997 fun mk_datatype args =
  1998   let
  1999     val names = map (fn ((((NONE, _), t), _), _) => t | ((((SOME t, _), _), _), _) => t) args;
  2000     val specs = map (fn ((((_, vs), t), mx), cons) =>
  2001       (vs, t, mx, map (fn ((x, y), z) => (x, y, z)) cons)) args;
  2002   in add_nominal_datatype false names specs end;
  2003 
  2004 val nominal_datatypeP =
  2005   OuterSyntax.command "nominal_datatype" "define inductive datatypes" K.thy_decl
  2006     (P.and_list1 datatype_decl >> (Toplevel.theory o mk_datatype));
  2007 
  2008 val _ = OuterSyntax.add_parsers [nominal_datatypeP];
  2009 
  2010 end;
  2011 
  2012 end
  2013