src/HOL/Nominal/nominal_datatype.ML
author wenzelm
Fri Oct 12 21:22:35 2012 +0200 (2012-10-12)
changeset 49835 31f32ec4d766
parent 49833 1d80798e8d8a
child 51122 386a117925db
permissions -rw-r--r--
discontinued typedef with alternative name;
     1 (*  Title:      HOL/Nominal/nominal_datatype.ML
     2     Author:     Stefan Berghofer and Christian Urban, TU Muenchen
     3 
     4 Nominal datatype package for Isabelle/HOL.
     5 *)
     6 
     7 signature NOMINAL_DATATYPE =
     8 sig
     9   val nominal_datatype : Datatype.config -> Datatype.spec list -> theory -> theory
    10   val nominal_datatype_cmd : Datatype.config -> Datatype.spec_cmd list -> theory -> theory
    11   type descr
    12   type nominal_datatype_info
    13   val get_nominal_datatypes : theory -> nominal_datatype_info Symtab.table
    14   val get_nominal_datatype : theory -> string -> nominal_datatype_info option
    15   val mk_perm: typ list -> term -> term -> term
    16   val perm_of_pair: term * term -> term
    17   val mk_not_sym: thm list -> thm list
    18   val perm_simproc: simproc
    19   val fresh_const: typ -> typ -> term
    20   val fresh_star_const: typ -> typ -> term
    21 end
    22 
    23 structure NominalDatatype : NOMINAL_DATATYPE =
    24 struct
    25 
    26 val finite_emptyI = @{thm finite.emptyI};
    27 val finite_Diff = @{thm finite_Diff};
    28 val finite_Un = @{thm finite_Un};
    29 val Un_iff = @{thm Un_iff};
    30 val In0_eq = @{thm In0_eq};
    31 val In1_eq = @{thm In1_eq};
    32 val In0_not_In1 = @{thm In0_not_In1};
    33 val In1_not_In0 = @{thm In1_not_In0};
    34 val Un_assoc = @{thm Un_assoc};
    35 val Collect_disj_eq = @{thm Collect_disj_eq};
    36 val Collect_False_empty = @{thm empty_def [THEN sym, THEN eq_reflection]};
    37 val empty_iff = @{thm empty_iff};
    38 
    39 open NominalAtoms;
    40 
    41 
    42 (* theory data *)
    43 
    44 type descr =
    45   (int * (string * Datatype.dtyp list *
    46       (string * (Datatype.dtyp list * Datatype.dtyp) list) list)) list;
    47 
    48 type nominal_datatype_info =
    49   {index : int,
    50    descr : descr,
    51    rec_names : string list,
    52    rec_rewrites : thm list,
    53    induction : thm,
    54    distinct : thm list,
    55    inject : thm list};
    56 
    57 structure NominalDatatypesData = Theory_Data
    58 (
    59   type T = nominal_datatype_info Symtab.table;
    60   val empty = Symtab.empty;
    61   val extend = I;
    62   fun merge data = Symtab.merge (K true) data;
    63 );
    64 
    65 val get_nominal_datatypes = NominalDatatypesData.get;
    66 val put_nominal_datatypes = NominalDatatypesData.put;
    67 val map_nominal_datatypes = NominalDatatypesData.map;
    68 val get_nominal_datatype = Symtab.lookup o get_nominal_datatypes;
    69 
    70 
    71 (**** make datatype info ****)
    72 
    73 fun make_dt_info descr induct reccomb_names rec_thms
    74     (i, (((_, (tname, _, _)), distinct), inject)) =
    75   (tname,
    76    {index = i,
    77     descr = descr,
    78     rec_names = reccomb_names,
    79     rec_rewrites = rec_thms,
    80     induction = induct,
    81     distinct = distinct,
    82     inject = inject});
    83 
    84 (*******************************)
    85 
    86 val (_ $ (_ $ (_ $ (distinct_f $ _) $ _))) = hd (prems_of Datatype.distinct_lemma);
    87 
    88 
    89 (** simplification procedure for sorting permutations **)
    90 
    91 val dj_cp = @{thm dj_cp};
    92 
    93 fun dest_permT (Type ("fun", [Type ("List.list", [Type (@{type_name Product_Type.prod}, [T, _])]),
    94       Type ("fun", [_, U])])) = (T, U);
    95 
    96 fun permTs_of (Const ("Nominal.perm", T) $ t $ u) = fst (dest_permT T) :: permTs_of u
    97   | permTs_of _ = [];
    98 
    99 fun perm_simproc' thy ss (Const ("Nominal.perm", T) $ t $ (u as Const ("Nominal.perm", U) $ r $ s)) =
   100       let
   101         val (aT as Type (a, []), S) = dest_permT T;
   102         val (bT as Type (b, []), _) = dest_permT U
   103       in if member (op =) (permTs_of u) aT andalso aT <> bT then
   104           let
   105             val cp = cp_inst_of thy a b;
   106             val dj = dj_thm_of thy b a;
   107             val dj_cp' = [cp, dj] MRS dj_cp;
   108             val cert = SOME o cterm_of thy
   109           in
   110             SOME (mk_meta_eq (Drule.instantiate' [SOME (ctyp_of thy S)]
   111               [cert t, cert r, cert s] dj_cp'))
   112           end
   113         else NONE
   114       end
   115   | perm_simproc' thy ss _ = NONE;
   116 
   117 val perm_simproc =
   118   Simplifier.simproc_global @{theory} "perm_simp" ["pi1 \<bullet> (pi2 \<bullet> x)"] perm_simproc';
   119 
   120 fun projections rule =
   121   Project_Rule.projections (Proof_Context.init_global (Thm.theory_of_thm rule)) rule
   122   |> map (Drule.export_without_context #> Rule_Cases.save rule);
   123 
   124 val supp_prod = @{thm supp_prod};
   125 val fresh_prod = @{thm fresh_prod};
   126 val supports_fresh = @{thm supports_fresh};
   127 val supports_def = Simpdata.mk_eq @{thm Nominal.supports_def};
   128 val fresh_def = Simpdata.mk_eq @{thm fresh_def};
   129 val supp_def = Simpdata.mk_eq @{thm supp_def};
   130 val rev_simps = @{thms rev.simps};
   131 val app_simps = @{thms append.simps};
   132 val at_fin_set_supp = @{thm at_fin_set_supp};
   133 val at_fin_set_fresh = @{thm at_fin_set_fresh};
   134 val abs_fun_eq1 = @{thm abs_fun_eq1};
   135 
   136 val collect_simp = rewrite_rule [mk_meta_eq mem_Collect_eq];
   137 
   138 fun mk_perm Ts t u =
   139   let
   140     val T = fastype_of1 (Ts, t);
   141     val U = fastype_of1 (Ts, u)
   142   in Const ("Nominal.perm", T --> U --> U) $ t $ u end;
   143 
   144 fun perm_of_pair (x, y) =
   145   let
   146     val T = fastype_of x;
   147     val pT = mk_permT T
   148   in Const ("List.list.Cons", HOLogic.mk_prodT (T, T) --> pT --> pT) $
   149     HOLogic.mk_prod (x, y) $ Const ("List.list.Nil", pT)
   150   end;
   151 
   152 fun mk_not_sym ths = maps (fn th => case prop_of th of
   153     _ $ (Const (@{const_name Not}, _) $ (Const (@{const_name HOL.eq}, _) $ _ $ _)) => [th, th RS not_sym]
   154   | _ => [th]) ths;
   155 
   156 fun fresh_const T U = Const ("Nominal.fresh", T --> U --> HOLogic.boolT);
   157 fun fresh_star_const T U =
   158   Const ("Nominal.fresh_star", HOLogic.mk_setT T --> U --> HOLogic.boolT);
   159 
   160 fun gen_nominal_datatype prep_specs config dts thy =
   161   let
   162     val new_type_names = map (fn ((tname, _, _), _) => Binding.name_of tname) dts;
   163 
   164     val (dts', _) = prep_specs dts thy;
   165 
   166     val atoms = atoms_of thy;
   167 
   168     val tyvars = map (fn ((_, tvs, _), _) => tvs) dts';
   169     val sorts = flat tyvars;
   170 
   171     fun inter_sort thy S S' = Sign.inter_sort thy (S, S');
   172     fun augment_sort_typ thy S =
   173       let val S = Sign.minimize_sort thy (Sign.certify_sort thy S)
   174       in map_type_tfree (fn (s, S') => TFree (s,
   175         if member (op = o apsnd fst) sorts s then inter_sort thy S S' else S'))
   176       end;
   177     fun augment_sort thy S = map_types (augment_sort_typ thy S);
   178 
   179     val types_syntax = map (fn ((tname, tvs, mx), constrs) => (tname, mx)) dts';
   180     val constr_syntax = map (fn (_, constrs) =>
   181       map (fn (cname, cargs, mx) => (cname, mx)) constrs) dts';
   182 
   183     val ps = map (fn ((n, _, _), _) =>
   184       (Sign.full_name thy n, Sign.full_name thy (Binding.suffix_name "_Rep" n))) 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 (the_default s (AList.lookup op = ps s), map replace_types Ts)
   191       | replace_types T = T;
   192 
   193     val dts'' = map (fn ((tname, tvs, mx), constrs) =>
   194       ((Binding.suffix_name "_Rep" tname, tvs, NoSyn),
   195         map (fn (cname, cargs, mx) => (Binding.suffix_name "_Rep" cname,
   196           map replace_types cargs, NoSyn)) constrs)) dts';
   197 
   198     val new_type_names' = map (fn n => n ^ "_Rep") new_type_names;
   199 
   200     val (full_new_type_names',thy1) = Datatype.add_datatype config dts'' thy;
   201 
   202     val {descr, induct, ...} = Datatype.the_info thy1 (hd full_new_type_names');
   203     fun nth_dtyp i = Datatype_Aux.typ_of_dtyp descr (Datatype.DtRec i);
   204 
   205     val big_name = space_implode "_" new_type_names;
   206 
   207 
   208     (**** define permutation functions ****)
   209 
   210     val permT = mk_permT (TFree ("'x", HOLogic.typeS));
   211     val pi = Free ("pi", permT);
   212     val perm_types = map (fn (i, _) =>
   213       let val T = nth_dtyp i
   214       in permT --> T --> T end) descr;
   215     val perm_names' = Datatype_Prop.indexify_names (map (fn (i, _) =>
   216       "perm_" ^ Datatype_Aux.name_of_typ (nth_dtyp i)) descr);
   217     val perm_names = replicate (length new_type_names) "Nominal.perm" @
   218       map (Sign.full_bname thy1) (List.drop (perm_names', length new_type_names));
   219     val perm_names_types = perm_names ~~ perm_types;
   220     val perm_names_types' = perm_names' ~~ perm_types;
   221 
   222     val perm_eqs = maps (fn (i, (_, _, constrs)) =>
   223       let val T = nth_dtyp i
   224       in map (fn (cname, dts) =>
   225         let
   226           val Ts = map (Datatype_Aux.typ_of_dtyp descr) dts;
   227           val names = Name.variant_list ["pi"] (Datatype_Prop.make_tnames Ts);
   228           val args = map Free (names ~~ Ts);
   229           val c = Const (cname, Ts ---> T);
   230           fun perm_arg (dt, x) =
   231             let val T = type_of x
   232             in if Datatype_Aux.is_rec_type dt then
   233                 let val Us = binder_types T
   234                 in
   235                   fold_rev (Term.abs o pair "x") Us
   236                     (Free (nth perm_names_types' (Datatype_Aux.body_index dt)) $ pi $
   237                       list_comb (x, map (fn (i, U) =>
   238                         Const ("Nominal.perm", permT --> U --> U) $
   239                           (Const ("List.rev", permT --> permT) $ pi) $
   240                           Bound i) ((length Us - 1 downto 0) ~~ Us)))
   241                 end
   242               else Const ("Nominal.perm", permT --> T --> T) $ pi $ x
   243             end;
   244         in
   245           (Attrib.empty_binding, HOLogic.mk_Trueprop (HOLogic.mk_eq
   246             (Free (nth perm_names_types' i) $
   247                Free ("pi", mk_permT (TFree ("'x", HOLogic.typeS))) $
   248                list_comb (c, args),
   249              list_comb (c, map perm_arg (dts ~~ args)))))
   250         end) constrs
   251       end) descr;
   252 
   253     val (perm_simps, thy2) =
   254       Primrec.add_primrec_overloaded
   255         (map (fn (s, sT) => (s, sT, false))
   256            (List.take (perm_names' ~~ perm_names_types, length new_type_names)))
   257         (map (fn s => (Binding.name s, NONE, NoSyn)) perm_names') perm_eqs thy1;
   258 
   259     (**** prove that permutation functions introduced by unfolding are ****)
   260     (**** equivalent to already existing permutation functions         ****)
   261 
   262     val _ = warning ("length descr: " ^ string_of_int (length descr));
   263     val _ = warning ("length new_type_names: " ^ string_of_int (length new_type_names));
   264 
   265     val perm_indnames = Datatype_Prop.make_tnames (map body_type perm_types);
   266     val perm_fun_def = Simpdata.mk_eq @{thm perm_fun_def};
   267 
   268     val unfolded_perm_eq_thms =
   269       if length descr = length new_type_names then []
   270       else map Drule.export_without_context (List.drop (Datatype_Aux.split_conj_thm
   271         (Goal.prove_global thy2 [] []
   272           (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
   273             (map (fn (c as (s, T), x) =>
   274                let val [T1, T2] = binder_types T
   275                in HOLogic.mk_eq (Const c $ pi $ Free (x, T2),
   276                  Const ("Nominal.perm", T) $ pi $ Free (x, T2))
   277                end)
   278              (perm_names_types ~~ perm_indnames))))
   279           (fn _ => EVERY [Datatype_Aux.ind_tac induct perm_indnames 1,
   280             ALLGOALS (asm_full_simp_tac
   281               (global_simpset_of thy2 addsimps [perm_fun_def]))])),
   282         length new_type_names));
   283 
   284     (**** prove [] \<bullet> t = t ****)
   285 
   286     val _ = warning "perm_empty_thms";
   287 
   288     val perm_empty_thms = maps (fn a =>
   289       let val permT = mk_permT (Type (a, []))
   290       in map Drule.export_without_context (List.take (Datatype_Aux.split_conj_thm
   291         (Goal.prove_global thy2 [] []
   292           (augment_sort thy2 [pt_class_of thy2 a]
   293             (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
   294               (map (fn ((s, T), x) => HOLogic.mk_eq
   295                   (Const (s, permT --> T --> T) $
   296                      Const ("List.list.Nil", permT) $ Free (x, T),
   297                    Free (x, T)))
   298                (perm_names ~~
   299                 map body_type perm_types ~~ perm_indnames)))))
   300           (fn _ => EVERY [Datatype_Aux.ind_tac induct perm_indnames 1,
   301             ALLGOALS (asm_full_simp_tac (global_simpset_of thy2))])),
   302         length new_type_names))
   303       end)
   304       atoms;
   305 
   306     (**** prove (pi1 @ pi2) \<bullet> t = pi1 \<bullet> (pi2 \<bullet> t) ****)
   307 
   308     val _ = warning "perm_append_thms";
   309 
   310     (*FIXME: these should be looked up statically*)
   311     val at_pt_inst = Global_Theory.get_thm thy2 "at_pt_inst";
   312     val pt2 = Global_Theory.get_thm thy2 "pt2";
   313 
   314     val perm_append_thms = maps (fn a =>
   315       let
   316         val permT = mk_permT (Type (a, []));
   317         val pi1 = Free ("pi1", permT);
   318         val pi2 = Free ("pi2", permT);
   319         val pt_inst = pt_inst_of thy2 a;
   320         val pt2' = pt_inst RS pt2;
   321         val pt2_ax = Global_Theory.get_thm thy2 (Long_Name.map_base_name (fn s => "pt_" ^ s ^ "2") a);
   322       in List.take (map Drule.export_without_context (Datatype_Aux.split_conj_thm
   323         (Goal.prove_global thy2 [] []
   324            (augment_sort thy2 [pt_class_of thy2 a]
   325              (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
   326                 (map (fn ((s, T), x) =>
   327                     let val perm = Const (s, permT --> T --> T)
   328                     in HOLogic.mk_eq
   329                       (perm $ (Const ("List.append", permT --> permT --> permT) $
   330                          pi1 $ pi2) $ Free (x, T),
   331                        perm $ pi1 $ (perm $ pi2 $ Free (x, T)))
   332                     end)
   333                   (perm_names ~~
   334                    map body_type perm_types ~~ perm_indnames)))))
   335            (fn _ => EVERY [Datatype_Aux.ind_tac induct perm_indnames 1,
   336               ALLGOALS (asm_full_simp_tac (global_simpset_of thy2 addsimps [pt2', pt2_ax]))]))),
   337          length new_type_names)
   338       end) atoms;
   339 
   340     (**** prove pi1 ~ pi2 ==> pi1 \<bullet> t = pi2 \<bullet> t ****)
   341 
   342     val _ = warning "perm_eq_thms";
   343 
   344     val pt3 = Global_Theory.get_thm thy2 "pt3";
   345     val pt3_rev = Global_Theory.get_thm thy2 "pt3_rev";
   346 
   347     val perm_eq_thms = maps (fn a =>
   348       let
   349         val permT = mk_permT (Type (a, []));
   350         val pi1 = Free ("pi1", permT);
   351         val pi2 = Free ("pi2", permT);
   352         val at_inst = at_inst_of thy2 a;
   353         val pt_inst = pt_inst_of thy2 a;
   354         val pt3' = pt_inst RS pt3;
   355         val pt3_rev' = at_inst RS (pt_inst RS pt3_rev);
   356         val pt3_ax = Global_Theory.get_thm thy2 (Long_Name.map_base_name (fn s => "pt_" ^ s ^ "3") a);
   357       in List.take (map Drule.export_without_context (Datatype_Aux.split_conj_thm
   358         (Goal.prove_global thy2 [] []
   359           (augment_sort thy2 [pt_class_of thy2 a] (Logic.mk_implies
   360              (HOLogic.mk_Trueprop (Const ("Nominal.prm_eq",
   361                 permT --> permT --> HOLogic.boolT) $ pi1 $ pi2),
   362               HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
   363                 (map (fn ((s, T), x) =>
   364                     let val perm = Const (s, permT --> T --> T)
   365                     in HOLogic.mk_eq
   366                       (perm $ pi1 $ Free (x, T),
   367                        perm $ pi2 $ Free (x, T))
   368                     end)
   369                   (perm_names ~~
   370                    map body_type perm_types ~~ perm_indnames))))))
   371            (fn _ => EVERY [Datatype_Aux.ind_tac induct perm_indnames 1,
   372               ALLGOALS (asm_full_simp_tac (global_simpset_of thy2 addsimps [pt3', pt3_rev', pt3_ax]))]))),
   373          length new_type_names)
   374       end) atoms;
   375 
   376     (**** prove pi1 \<bullet> (pi2 \<bullet> t) = (pi1 \<bullet> pi2) \<bullet> (pi1 \<bullet> t) ****)
   377 
   378     val cp1 = Global_Theory.get_thm thy2 "cp1";
   379     val dj_cp = Global_Theory.get_thm thy2 "dj_cp";
   380     val pt_perm_compose = Global_Theory.get_thm thy2 "pt_perm_compose";
   381     val pt_perm_compose_rev = Global_Theory.get_thm thy2 "pt_perm_compose_rev";
   382     val dj_perm_perm_forget = Global_Theory.get_thm thy2 "dj_perm_perm_forget";
   383 
   384     fun composition_instance name1 name2 thy =
   385       let
   386         val cp_class = cp_class_of thy name1 name2;
   387         val pt_class =
   388           if name1 = name2 then [pt_class_of thy name1]
   389           else [];
   390         val permT1 = mk_permT (Type (name1, []));
   391         val permT2 = mk_permT (Type (name2, []));
   392         val Ts = map body_type perm_types;
   393         val cp_inst = cp_inst_of thy name1 name2;
   394         val simps = global_simpset_of thy addsimps (perm_fun_def ::
   395           (if name1 <> name2 then
   396              let val dj = dj_thm_of thy name2 name1
   397              in [dj RS (cp_inst RS dj_cp), dj RS dj_perm_perm_forget] end
   398            else
   399              let
   400                val at_inst = at_inst_of thy name1;
   401                val pt_inst = pt_inst_of thy name1;
   402              in
   403                [cp_inst RS cp1 RS sym,
   404                 at_inst RS (pt_inst RS pt_perm_compose) RS sym,
   405                 at_inst RS (pt_inst RS pt_perm_compose_rev) RS sym]
   406             end))
   407         val sort = Sign.minimize_sort thy (Sign.certify_sort thy (cp_class :: pt_class));
   408         val thms = Datatype_Aux.split_conj_thm (Goal.prove_global thy [] []
   409           (augment_sort thy sort
   410             (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
   411               (map (fn ((s, T), x) =>
   412                   let
   413                     val pi1 = Free ("pi1", permT1);
   414                     val pi2 = Free ("pi2", permT2);
   415                     val perm1 = Const (s, permT1 --> T --> T);
   416                     val perm2 = Const (s, permT2 --> T --> T);
   417                     val perm3 = Const ("Nominal.perm", permT1 --> permT2 --> permT2)
   418                   in HOLogic.mk_eq
   419                     (perm1 $ pi1 $ (perm2 $ pi2 $ Free (x, T)),
   420                      perm2 $ (perm3 $ pi1 $ pi2) $ (perm1 $ pi1 $ Free (x, T)))
   421                   end)
   422                 (perm_names ~~ Ts ~~ perm_indnames)))))
   423           (fn _ => EVERY [Datatype_Aux.ind_tac induct perm_indnames 1,
   424              ALLGOALS (asm_full_simp_tac simps)]))
   425       in
   426         fold (fn (s, tvs) => fn thy => AxClass.prove_arity
   427             (s, map (inter_sort thy sort o snd) tvs, [cp_class])
   428             (Class.intro_classes_tac [] THEN ALLGOALS (resolve_tac thms)) thy)
   429           (full_new_type_names' ~~ tyvars) thy
   430       end;
   431 
   432     val (perm_thmss,thy3) = thy2 |>
   433       fold (fn name1 => fold (composition_instance name1) atoms) atoms |>
   434       fold (fn atom => fn thy =>
   435         let val pt_name = pt_class_of thy atom
   436         in
   437           fold (fn (s, tvs) => fn thy => AxClass.prove_arity
   438               (s, map (inter_sort thy [pt_name] o snd) tvs, [pt_name])
   439               (EVERY
   440                 [Class.intro_classes_tac [],
   441                  resolve_tac perm_empty_thms 1,
   442                  resolve_tac perm_append_thms 1,
   443                  resolve_tac perm_eq_thms 1, assume_tac 1]) thy)
   444             (full_new_type_names' ~~ tyvars) thy
   445         end) atoms |>
   446       Global_Theory.add_thmss
   447         [((Binding.name (space_implode "_" new_type_names ^ "_unfolded_perm_eq"),
   448           unfolded_perm_eq_thms), [Simplifier.simp_add]),
   449          ((Binding.name (space_implode "_" new_type_names ^ "_perm_empty"),
   450           perm_empty_thms), [Simplifier.simp_add]),
   451          ((Binding.name (space_implode "_" new_type_names ^ "_perm_append"),
   452           perm_append_thms), [Simplifier.simp_add]),
   453          ((Binding.name (space_implode "_" new_type_names ^ "_perm_eq"),
   454           perm_eq_thms), [Simplifier.simp_add])];
   455 
   456     (**** Define representing sets ****)
   457 
   458     val _ = warning "representing sets";
   459 
   460     val rep_set_names =
   461       Datatype_Prop.indexify_names
   462         (map (fn (i, _) => Datatype_Aux.name_of_typ (nth_dtyp i) ^ "_set") descr);
   463     val big_rep_name =
   464       space_implode "_" (Datatype_Prop.indexify_names (map_filter
   465         (fn (i, ("Nominal.noption", _, _)) => NONE
   466           | (i, _) => SOME (Datatype_Aux.name_of_typ (nth_dtyp i))) descr)) ^ "_set";
   467     val _ = warning ("big_rep_name: " ^ big_rep_name);
   468 
   469     fun strip_option (dtf as Datatype.DtType ("fun", [dt, Datatype.DtRec i])) =
   470           (case AList.lookup op = descr i of
   471              SOME ("Nominal.noption", _, [(_, [dt']), _]) =>
   472                apfst (cons dt) (strip_option dt')
   473            | _ => ([], dtf))
   474       | strip_option (Datatype.DtType ("fun",
   475             [dt, Datatype.DtType ("Nominal.noption", [dt'])])) =
   476           apfst (cons dt) (strip_option dt')
   477       | strip_option dt = ([], dt);
   478 
   479     val dt_atomTs = distinct op = (map (Datatype_Aux.typ_of_dtyp descr)
   480       (maps (fn (_, (_, _, cs)) => maps (maps (fst o strip_option) o snd) cs) descr));
   481     val dt_atoms = map (fst o dest_Type) dt_atomTs;
   482 
   483     fun make_intr s T (cname, cargs) =
   484       let
   485         fun mk_prem dt (j, j', prems, ts) =
   486           let
   487             val (dts, dt') = strip_option dt;
   488             val (dts', dt'') = Datatype_Aux.strip_dtyp dt';
   489             val Ts = map (Datatype_Aux.typ_of_dtyp descr) dts;
   490             val Us = map (Datatype_Aux.typ_of_dtyp descr) dts';
   491             val T = Datatype_Aux.typ_of_dtyp descr dt'';
   492             val free = Datatype_Aux.mk_Free "x" (Us ---> T) j;
   493             val free' = Datatype_Aux.app_bnds free (length Us);
   494             fun mk_abs_fun T (i, t) =
   495               let val U = fastype_of t
   496               in (i + 1, Const ("Nominal.abs_fun", [T, U, T] --->
   497                 Type ("Nominal.noption", [U])) $ Datatype_Aux.mk_Free "y" T i $ t)
   498               end
   499           in (j + 1, j' + length Ts,
   500             case dt'' of
   501                 Datatype.DtRec k => Logic.list_all (map (pair "x") Us,
   502                   HOLogic.mk_Trueprop (Free (nth rep_set_names k,
   503                     T --> HOLogic.boolT) $ free')) :: prems
   504               | _ => prems,
   505             snd (fold_rev mk_abs_fun Ts (j', free)) :: ts)
   506           end;
   507 
   508         val (_, _, prems, ts) = fold_rev mk_prem cargs (1, 1, [], []);
   509         val concl = HOLogic.mk_Trueprop (Free (s, T --> HOLogic.boolT) $
   510           list_comb (Const (cname, map fastype_of ts ---> T), ts))
   511       in Logic.list_implies (prems, concl)
   512       end;
   513 
   514     val (intr_ts, (rep_set_names', recTs')) =
   515       apfst flat (apsnd ListPair.unzip (ListPair.unzip (map_filter
   516         (fn ((_, ("Nominal.noption", _, _)), _) => NONE
   517           | ((i, (_, _, constrs)), rep_set_name) =>
   518               let val T = nth_dtyp i
   519               in SOME (map (make_intr rep_set_name T) constrs,
   520                 (rep_set_name, T))
   521               end)
   522                 (descr ~~ rep_set_names))));
   523     val rep_set_names'' = map (Sign.full_bname thy3) rep_set_names';
   524 
   525     val ({raw_induct = rep_induct, intrs = rep_intrs, ...}, thy4) =
   526       thy3
   527       |> Sign.map_naming Name_Space.conceal
   528       |> Inductive.add_inductive_global
   529           {quiet_mode = false, verbose = false, alt_name = Binding.name big_rep_name,
   530            coind = false, no_elim = true, no_ind = false, skip_mono = true}
   531           (map (fn (s, T) => ((Binding.name s, T --> HOLogic.boolT), NoSyn))
   532              (rep_set_names' ~~ recTs'))
   533           [] (map (fn x => (Attrib.empty_binding, x)) intr_ts) []
   534       ||> Sign.restore_naming thy3;
   535 
   536     (**** Prove that representing set is closed under permutation ****)
   537 
   538     val _ = warning "proving closure under permutation...";
   539 
   540     val abs_perm = Global_Theory.get_thms thy4 "abs_perm";
   541 
   542     val perm_indnames' = map_filter
   543       (fn (x, (_, ("Nominal.noption", _, _))) => NONE | (x, _) => SOME x)
   544       (perm_indnames ~~ descr);
   545 
   546     fun mk_perm_closed name = map (fn th => Drule.export_without_context (th RS mp))
   547       (List.take (Datatype_Aux.split_conj_thm (Goal.prove_global thy4 [] []
   548         (augment_sort thy4
   549           (pt_class_of thy4 name :: map (cp_class_of thy4 name) (remove (op =) name dt_atoms))
   550           (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj (map
   551             (fn ((s, T), x) =>
   552                let
   553                  val S = Const (s, T --> HOLogic.boolT);
   554                  val permT = mk_permT (Type (name, []))
   555                in HOLogic.mk_imp (S $ Free (x, T),
   556                  S $ (Const ("Nominal.perm", permT --> T --> T) $
   557                    Free ("pi", permT) $ Free (x, T)))
   558                end) (rep_set_names'' ~~ recTs' ~~ perm_indnames')))))
   559         (fn _ => EVERY
   560            [Datatype_Aux.ind_tac rep_induct [] 1,
   561             ALLGOALS (simp_tac (global_simpset_of thy4 addsimps
   562               (Thm.symmetric perm_fun_def :: abs_perm))),
   563             ALLGOALS (resolve_tac rep_intrs THEN_ALL_NEW assume_tac)])),
   564         length new_type_names));
   565 
   566     val perm_closed_thmss = map mk_perm_closed atoms;
   567 
   568     (**** typedef ****)
   569 
   570     val _ = warning "defining type...";
   571 
   572     val (typedefs, thy6) =
   573       thy4
   574       |> fold_map (fn (((name, mx), tvs), (cname, U)) => fn thy =>
   575           Typedef.add_typedef_global
   576             (name, map (fn (v, _) => (v, dummyS)) tvs, mx)  (* FIXME keep constraints!? *)
   577             (Const (@{const_name Collect}, (U --> HOLogic.boolT) --> HOLogic.mk_setT U) $
   578                Const (cname, U --> HOLogic.boolT)) NONE
   579             (rtac exI 1 THEN rtac CollectI 1 THEN
   580               QUIET_BREADTH_FIRST (has_fewer_prems 1)
   581               (resolve_tac rep_intrs 1)) thy |> (fn ((_, r), thy) =>
   582         let
   583           val permT = mk_permT
   584             (TFree (singleton (Name.variant_list (map fst tvs)) "'a", HOLogic.typeS));
   585           val pi = Free ("pi", permT);
   586           val T = Type (Sign.full_name thy name, map TFree tvs);
   587         in apfst (pair r o hd)
   588           (Global_Theory.add_defs_unchecked true
   589             [((Binding.map_name (fn n => "prm_" ^ n ^ "_def") name, Logic.mk_equals
   590               (Const ("Nominal.perm", permT --> T --> T) $ pi $ Free ("x", T),
   591                Const (Sign.intern_const thy ("Abs_" ^ Binding.name_of name), U --> T) $
   592                  (Const ("Nominal.perm", permT --> U --> U) $ pi $
   593                    (Const (Sign.intern_const thy ("Rep_" ^ Binding.name_of name), T --> U) $
   594                      Free ("x", T))))), [])] thy)
   595         end))
   596         (types_syntax ~~ tyvars ~~ List.take (rep_set_names'' ~~ recTs', length new_type_names));
   597 
   598     val perm_defs = map snd typedefs;
   599     val Abs_inverse_thms = map (collect_simp o #Abs_inverse o snd o fst) typedefs;
   600     val Rep_inverse_thms = map (#Rep_inverse o snd o fst) typedefs;
   601     val Rep_thms = map (collect_simp o #Rep o snd o fst) typedefs;
   602 
   603 
   604     (** prove that new types are in class pt_<name> **)
   605 
   606     val _ = warning "prove that new types are in class pt_<name> ...";
   607 
   608     fun pt_instance (atom, perm_closed_thms) =
   609       fold (fn ((((((Abs_inverse, Rep_inverse), Rep),
   610         perm_def), name), tvs), perm_closed) => fn thy =>
   611           let
   612             val pt_class = pt_class_of thy atom;
   613             val sort = Sign.minimize_sort thy (Sign.certify_sort thy
   614               (pt_class :: map (cp_class_of thy atom) (remove (op =) atom dt_atoms)))
   615           in AxClass.prove_arity
   616             (Sign.intern_type thy name,
   617               map (inter_sort thy sort o snd) tvs, [pt_class])
   618             (EVERY [Class.intro_classes_tac [],
   619               rewrite_goals_tac [perm_def],
   620               asm_full_simp_tac (global_simpset_of thy addsimps [Rep_inverse]) 1,
   621               asm_full_simp_tac (global_simpset_of thy addsimps
   622                 [Rep RS perm_closed RS Abs_inverse]) 1,
   623               asm_full_simp_tac (HOL_basic_ss addsimps [Global_Theory.get_thm thy
   624                 ("pt_" ^ Long_Name.base_name atom ^ "3")]) 1]) thy
   625           end)
   626         (Abs_inverse_thms ~~ Rep_inverse_thms ~~ Rep_thms ~~ perm_defs ~~
   627            new_type_names ~~ tyvars ~~ perm_closed_thms);
   628 
   629 
   630     (** prove that new types are in class cp_<name1>_<name2> **)
   631 
   632     val _ = warning "prove that new types are in class cp_<name1>_<name2> ...";
   633 
   634     fun cp_instance (atom1, perm_closed_thms1) (atom2, perm_closed_thms2) thy =
   635       let
   636         val cp_class = cp_class_of thy atom1 atom2;
   637         val sort = Sign.minimize_sort thy (Sign.certify_sort thy
   638           (pt_class_of thy atom1 :: map (cp_class_of thy atom1) (remove (op =) atom1 dt_atoms) @
   639            (if atom1 = atom2 then [cp_class_of thy atom1 atom1] else
   640             pt_class_of thy atom2 :: map (cp_class_of thy atom2) (remove (op =) atom2 dt_atoms))));
   641         val cp1' = cp_inst_of thy atom1 atom2 RS cp1
   642       in fold (fn ((((((Abs_inverse, Rep),
   643         perm_def), name), tvs), perm_closed1), perm_closed2) => fn thy =>
   644           AxClass.prove_arity
   645             (Sign.intern_type thy name,
   646               map (inter_sort thy sort o snd) tvs, [cp_class])
   647             (EVERY [Class.intro_classes_tac [],
   648               rewrite_goals_tac [perm_def],
   649               asm_full_simp_tac (global_simpset_of thy addsimps
   650                 ((Rep RS perm_closed1 RS Abs_inverse) ::
   651                  (if atom1 = atom2 then []
   652                   else [Rep RS perm_closed2 RS Abs_inverse]))) 1,
   653               cong_tac 1,
   654               rtac refl 1,
   655               rtac cp1' 1]) thy)
   656         (Abs_inverse_thms ~~ Rep_thms ~~ perm_defs ~~ new_type_names ~~
   657            tyvars ~~ perm_closed_thms1 ~~ perm_closed_thms2) thy
   658       end;
   659 
   660     val thy7 = fold (fn x => fn thy => thy |>
   661       pt_instance x |>
   662       fold (cp_instance x) (atoms ~~ perm_closed_thmss))
   663         (atoms ~~ perm_closed_thmss) thy6;
   664 
   665     (**** constructors ****)
   666 
   667     fun mk_abs_fun x t =
   668       let
   669         val T = fastype_of x;
   670         val U = fastype_of t
   671       in
   672         Const ("Nominal.abs_fun", T --> U --> T -->
   673           Type ("Nominal.noption", [U])) $ x $ t
   674       end;
   675 
   676     val (ty_idxs, _) = List.foldl
   677       (fn ((i, ("Nominal.noption", _, _)), p) => p
   678         | ((i, _), (ty_idxs, j)) => (ty_idxs @ [(i, j)], j + 1)) ([], 0) descr;
   679 
   680     fun reindex (Datatype.DtType (s, dts)) = Datatype.DtType (s, map reindex dts)
   681       | reindex (Datatype.DtRec i) = Datatype.DtRec (the (AList.lookup op = ty_idxs i))
   682       | reindex dt = dt;
   683 
   684     fun strip_suffix i s = implode (List.take (raw_explode s, size s - i));  (* FIXME Symbol.explode (?) *)
   685 
   686     (** strips the "_Rep" in type names *)
   687     fun strip_nth_name i s =
   688       let val xs = Long_Name.explode s;
   689       in Long_Name.implode (Library.nth_map (length xs - i) (strip_suffix 4) xs) end;
   690 
   691     val (descr'', ndescr) = ListPair.unzip (map_filter
   692       (fn (i, ("Nominal.noption", _, _)) => NONE
   693         | (i, (s, dts, constrs)) =>
   694              let
   695                val SOME index = AList.lookup op = ty_idxs i;
   696                val (constrs2, constrs1) =
   697                  map_split (fn (cname, cargs) =>
   698                    apsnd (pair (strip_nth_name 2 (strip_nth_name 1 cname)))
   699                    (fold_map (fn dt => fn dts =>
   700                      let val (dts', dt') = strip_option dt
   701                      in ((length dts, length dts'), dts @ dts' @ [reindex dt']) end)
   702                        cargs [])) constrs
   703              in SOME ((index, (strip_nth_name 1 s,  map reindex dts, constrs1)),
   704                (index, constrs2))
   705              end) descr);
   706 
   707     val (descr1, descr2) = chop (length new_type_names) descr'';
   708     val descr' = [descr1, descr2];
   709 
   710     fun partition_cargs idxs xs = map (fn (i, j) =>
   711       (List.take (List.drop (xs, i), j), nth xs (i + j))) idxs;
   712 
   713     val pdescr = map (fn ((i, (s, dts, constrs)), (_, idxss)) => (i, (s, dts,
   714       map (fn ((cname, cargs), idxs) => (cname, partition_cargs idxs cargs))
   715         (constrs ~~ idxss)))) (descr'' ~~ ndescr);
   716 
   717     fun nth_dtyp' i = Datatype_Aux.typ_of_dtyp descr'' (Datatype.DtRec i);
   718 
   719     val rep_names = map (fn s =>
   720       Sign.intern_const thy7 ("Rep_" ^ s)) new_type_names;
   721     val abs_names = map (fn s =>
   722       Sign.intern_const thy7 ("Abs_" ^ s)) new_type_names;
   723 
   724     val recTs = Datatype_Aux.get_rec_types descr'';
   725     val newTs' = take (length new_type_names) recTs';
   726     val newTs = take (length new_type_names) recTs;
   727 
   728     val full_new_type_names = map (Sign.full_bname thy) new_type_names;
   729 
   730     fun make_constr_def tname T T' (((cname_rep, _), (cname, cargs)), (cname', mx))
   731         (thy, defs, eqns) =
   732       let
   733         fun constr_arg (dts, dt) (j, l_args, r_args) =
   734           let
   735             val xs =
   736               map (fn (dt, i) => Datatype_Aux.mk_Free "x" (Datatype_Aux.typ_of_dtyp descr'' dt) i)
   737                 (dts ~~ (j upto j + length dts - 1))
   738             val x = Datatype_Aux.mk_Free "x" (Datatype_Aux.typ_of_dtyp descr'' dt) (j + length dts)
   739           in
   740             (j + length dts + 1,
   741              xs @ x :: l_args,
   742              fold_rev mk_abs_fun xs
   743                (case dt of
   744                   Datatype.DtRec k => if k < length new_type_names then
   745                       Const (nth rep_names k, Datatype_Aux.typ_of_dtyp descr'' dt -->
   746                         Datatype_Aux.typ_of_dtyp descr dt) $ x
   747                     else error "nested recursion not (yet) supported"
   748                 | _ => x) :: r_args)
   749           end
   750 
   751         val (_, l_args, r_args) = fold_rev constr_arg cargs (1, [], []);
   752         val abs_name = Sign.intern_const thy ("Abs_" ^ tname);
   753         val rep_name = Sign.intern_const thy ("Rep_" ^ tname);
   754         val constrT = map fastype_of l_args ---> T;
   755         val lhs = list_comb (Const (cname, constrT), l_args);
   756         val rhs = list_comb (Const (cname_rep, map fastype_of r_args ---> T'), r_args);
   757         val def = Logic.mk_equals (lhs, Const (abs_name, T' --> T) $ rhs);
   758         val eqn = HOLogic.mk_Trueprop (HOLogic.mk_eq
   759           (Const (rep_name, T --> T') $ lhs, rhs));
   760         val def_name = (Long_Name.base_name cname) ^ "_def";
   761         val ([def_thm], thy') = thy |>
   762           Sign.add_consts_i [(cname', constrT, mx)] |>
   763           (Global_Theory.add_defs false o map Thm.no_attributes) [(Binding.name def_name, def)]
   764       in (thy', defs @ [def_thm], eqns @ [eqn]) end;
   765 
   766     fun dt_constr_defs ((((((_, (_, _, constrs)),
   767         (_, (_, _, constrs'))), tname), T), T'), constr_syntax) (thy, defs, eqns, dist_lemmas) =
   768       let
   769         val rep_const = cterm_of thy
   770           (Const (Sign.intern_const thy ("Rep_" ^ tname), T --> T'));
   771         val dist =
   772           Drule.export_without_context
   773             (cterm_instantiate [(cterm_of thy distinct_f, rep_const)] Datatype.distinct_lemma);
   774         val (thy', defs', eqns') = fold (make_constr_def tname T T')
   775           (constrs ~~ constrs' ~~ constr_syntax) (Sign.add_path tname thy, defs, [])
   776       in
   777         (Sign.parent_path thy', defs', eqns @ [eqns'], dist_lemmas @ [dist])
   778       end;
   779 
   780     val (thy8, constr_defs, constr_rep_eqns, dist_lemmas) = fold dt_constr_defs
   781       (List.take (descr, length new_type_names) ~~
   782         List.take (pdescr, length new_type_names) ~~
   783         new_type_names ~~ newTs ~~ newTs' ~~ constr_syntax)
   784       (thy7, [], [], []);
   785 
   786     val abs_inject_thms = map (collect_simp o #Abs_inject o snd o fst) typedefs
   787     val rep_inject_thms = map (#Rep_inject o snd o fst) typedefs
   788 
   789     (* prove theorem  Rep_i (Constr_j ...) = Constr'_j ...  *)
   790 
   791     fun prove_constr_rep_thm eqn =
   792       let
   793         val inj_thms = map (fn r => r RS iffD1) abs_inject_thms;
   794         val rewrites = constr_defs @ map mk_meta_eq Rep_inverse_thms
   795       in Goal.prove_global thy8 [] [] eqn (fn _ => EVERY
   796         [resolve_tac inj_thms 1,
   797          rewrite_goals_tac rewrites,
   798          rtac refl 3,
   799          resolve_tac rep_intrs 2,
   800          REPEAT (resolve_tac Rep_thms 1)])
   801       end;
   802 
   803     val constr_rep_thmss = map (map prove_constr_rep_thm) constr_rep_eqns;
   804 
   805     (* prove theorem  pi \<bullet> Rep_i x = Rep_i (pi \<bullet> x) *)
   806 
   807     fun prove_perm_rep_perm (atom, perm_closed_thms) = map (fn th =>
   808       let
   809         val _ $ (_ $ (Rep $ x)) = Logic.unvarify_global (prop_of th);
   810         val Type ("fun", [T, U]) = fastype_of Rep;
   811         val permT = mk_permT (Type (atom, []));
   812         val pi = Free ("pi", permT);
   813       in
   814         Goal.prove_global thy8 [] []
   815           (augment_sort thy8
   816             (pt_class_of thy8 atom :: map (cp_class_of thy8 atom) (remove (op =) atom dt_atoms))
   817             (HOLogic.mk_Trueprop (HOLogic.mk_eq
   818               (Const ("Nominal.perm", permT --> U --> U) $ pi $ (Rep $ x),
   819                Rep $ (Const ("Nominal.perm", permT --> T --> T) $ pi $ x)))))
   820           (fn _ => simp_tac (HOL_basic_ss addsimps (perm_defs @ Abs_inverse_thms @
   821             perm_closed_thms @ Rep_thms)) 1)
   822       end) Rep_thms;
   823 
   824     val perm_rep_perm_thms = maps prove_perm_rep_perm (atoms ~~ perm_closed_thmss);
   825 
   826     (* prove distinctness theorems *)
   827 
   828     val distinct_props = Datatype_Prop.make_distincts descr';
   829     val dist_rewrites = map2 (fn rep_thms => fn dist_lemma =>
   830       dist_lemma :: rep_thms @ [In0_eq, In1_eq, In0_not_In1, In1_not_In0])
   831         constr_rep_thmss dist_lemmas;
   832 
   833     fun prove_distinct_thms _ [] = []
   834       | prove_distinct_thms (p as (rep_thms, dist_lemma)) (t :: ts) =
   835           let
   836             val dist_thm = Goal.prove_global thy8 [] [] t (fn _ =>
   837               simp_tac (global_simpset_of thy8 addsimps (dist_lemma :: rep_thms)) 1)
   838           in
   839             dist_thm :: Drule.export_without_context (dist_thm RS not_sym) ::
   840               prove_distinct_thms p ts
   841           end;
   842 
   843     val distinct_thms = map2 prove_distinct_thms
   844       (constr_rep_thmss ~~ dist_lemmas) distinct_props;
   845 
   846     (** prove equations for permutation functions **)
   847 
   848     val perm_simps' = map (fn (((i, (_, _, constrs)), tname), constr_rep_thms) =>
   849       let val T = nth_dtyp' i
   850       in maps (fn (atom, perm_closed_thms) =>
   851           map (fn ((cname, dts), constr_rep_thm) =>
   852         let
   853           val cname = Sign.intern_const thy8
   854             (Long_Name.append tname (Long_Name.base_name cname));
   855           val permT = mk_permT (Type (atom, []));
   856           val pi = Free ("pi", permT);
   857 
   858           fun perm t =
   859             let val T = fastype_of t
   860             in Const ("Nominal.perm", permT --> T --> T) $ pi $ t end;
   861 
   862           fun constr_arg (dts, dt) (j, l_args, r_args) =
   863             let
   864               val Ts = map (Datatype_Aux.typ_of_dtyp descr'') dts;
   865               val xs =
   866                 map (fn (T, i) => Datatype_Aux.mk_Free "x" T i)
   867                   (Ts ~~ (j upto j + length dts - 1));
   868               val x =
   869                 Datatype_Aux.mk_Free "x" (Datatype_Aux.typ_of_dtyp descr'' dt) (j + length dts);
   870             in
   871               (j + length dts + 1,
   872                xs @ x :: l_args,
   873                map perm (xs @ [x]) @ r_args)
   874             end
   875 
   876           val (_, l_args, r_args) = fold_rev constr_arg dts (1, [], []);
   877           val c = Const (cname, map fastype_of l_args ---> T)
   878         in
   879           Goal.prove_global thy8 [] []
   880             (augment_sort thy8
   881               (pt_class_of thy8 atom :: map (cp_class_of thy8 atom) (remove (op =) atom dt_atoms))
   882               (HOLogic.mk_Trueprop (HOLogic.mk_eq
   883                 (perm (list_comb (c, l_args)), list_comb (c, r_args)))))
   884             (fn _ => EVERY
   885               [simp_tac (global_simpset_of thy8 addsimps (constr_rep_thm :: perm_defs)) 1,
   886                simp_tac (HOL_basic_ss addsimps (Rep_thms @ Abs_inverse_thms @
   887                  constr_defs @ perm_closed_thms)) 1,
   888                TRY (simp_tac (HOL_basic_ss addsimps
   889                  (Thm.symmetric perm_fun_def :: abs_perm)) 1),
   890                TRY (simp_tac (HOL_basic_ss addsimps
   891                  (perm_fun_def :: perm_defs @ Rep_thms @ Abs_inverse_thms @
   892                     perm_closed_thms)) 1)])
   893         end) (constrs ~~ constr_rep_thms)) (atoms ~~ perm_closed_thmss)
   894       end) (List.take (pdescr, length new_type_names) ~~ new_type_names ~~ constr_rep_thmss);
   895 
   896     (** prove injectivity of constructors **)
   897 
   898     val rep_inject_thms' = map (fn th => th RS sym) rep_inject_thms;
   899     val alpha = Global_Theory.get_thms thy8 "alpha";
   900     val abs_fresh = Global_Theory.get_thms thy8 "abs_fresh";
   901 
   902     val pt_cp_sort =
   903       map (pt_class_of thy8) dt_atoms @
   904       maps (fn s => map (cp_class_of thy8 s) (remove (op =) s dt_atoms)) dt_atoms;
   905 
   906     val inject_thms = map (fn (((i, (_, _, constrs)), tname), constr_rep_thms) =>
   907       let val T = nth_dtyp' i
   908       in map_filter (fn ((cname, dts), constr_rep_thm) =>
   909         if null dts then NONE else SOME
   910         let
   911           val cname = Sign.intern_const thy8
   912             (Long_Name.append tname (Long_Name.base_name cname));
   913 
   914           fun make_inj (dts, dt) (j, args1, args2, eqs) =
   915             let
   916               val Ts_idx =
   917                 map (Datatype_Aux.typ_of_dtyp descr'') dts ~~ (j upto j + length dts - 1);
   918               val xs = map (fn (T, i) => Datatype_Aux.mk_Free "x" T i) Ts_idx;
   919               val ys = map (fn (T, i) => Datatype_Aux.mk_Free "y" T i) Ts_idx;
   920               val x =
   921                 Datatype_Aux.mk_Free "x" (Datatype_Aux.typ_of_dtyp descr'' dt) (j + length dts);
   922               val y =
   923                 Datatype_Aux.mk_Free "y" (Datatype_Aux.typ_of_dtyp descr'' dt) (j + length dts);
   924             in
   925               (j + length dts + 1,
   926                xs @ (x :: args1), ys @ (y :: args2),
   927                HOLogic.mk_eq
   928                  (fold_rev mk_abs_fun xs x, fold_rev mk_abs_fun ys y) :: eqs)
   929             end;
   930 
   931           val (_, args1, args2, eqs) = fold_rev make_inj dts (1, [], [], []);
   932           val Ts = map fastype_of args1;
   933           val c = Const (cname, Ts ---> T)
   934         in
   935           Goal.prove_global thy8 [] []
   936             (augment_sort thy8 pt_cp_sort
   937               (HOLogic.mk_Trueprop (HOLogic.mk_eq
   938                 (HOLogic.mk_eq (list_comb (c, args1), list_comb (c, args2)),
   939                  foldr1 HOLogic.mk_conj eqs))))
   940             (fn _ => EVERY
   941                [asm_full_simp_tac (global_simpset_of thy8 addsimps (constr_rep_thm ::
   942                   rep_inject_thms')) 1,
   943                 TRY (asm_full_simp_tac (HOL_basic_ss addsimps (fresh_def :: supp_def ::
   944                   alpha @ abs_perm @ abs_fresh @ rep_inject_thms @
   945                   perm_rep_perm_thms)) 1)])
   946         end) (constrs ~~ constr_rep_thms)
   947       end) (List.take (pdescr, length new_type_names) ~~ new_type_names ~~ constr_rep_thmss);
   948 
   949     (** equations for support and freshness **)
   950 
   951     val (supp_thms, fresh_thms) = ListPair.unzip (map ListPair.unzip
   952       (map (fn ((((i, (_, _, constrs)), tname), inject_thms'), perm_thms') =>
   953       let val T = nth_dtyp' i
   954       in maps (fn (cname, dts) => map (fn atom =>
   955         let
   956           val cname = Sign.intern_const thy8
   957             (Long_Name.append tname (Long_Name.base_name cname));
   958           val atomT = Type (atom, []);
   959 
   960           fun process_constr (dts, dt) (j, args1, args2) =
   961             let
   962               val Ts_idx =
   963                 map (Datatype_Aux.typ_of_dtyp descr'') dts ~~ (j upto j + length dts - 1);
   964               val xs = map (fn (T, i) => Datatype_Aux.mk_Free "x" T i) Ts_idx;
   965               val x =
   966                 Datatype_Aux.mk_Free "x" (Datatype_Aux.typ_of_dtyp descr'' dt) (j + length dts);
   967             in
   968               (j + length dts + 1,
   969                xs @ (x :: args1), fold_rev mk_abs_fun xs x :: args2)
   970             end;
   971 
   972           val (_, args1, args2) = fold_rev process_constr dts (1, [], []);
   973           val Ts = map fastype_of args1;
   974           val c = list_comb (Const (cname, Ts ---> T), args1);
   975           fun supp t =
   976             Const ("Nominal.supp", fastype_of t --> HOLogic.mk_setT atomT) $ t;
   977           fun fresh t = fresh_const atomT (fastype_of t) $ Free ("a", atomT) $ t;
   978           val supp_thm = Goal.prove_global thy8 [] []
   979             (augment_sort thy8 pt_cp_sort
   980               (HOLogic.mk_Trueprop (HOLogic.mk_eq
   981                 (supp c,
   982                  if null dts then HOLogic.mk_set atomT []
   983                  else foldr1 (HOLogic.mk_binop @{const_abbrev union}) (map supp args2)))))
   984             (fn _ =>
   985               simp_tac (HOL_basic_ss addsimps (supp_def ::
   986                  Un_assoc :: @{thm de_Morgan_conj} :: Collect_disj_eq :: finite_Un ::
   987                  Collect_False_empty :: finite_emptyI :: @{thms simp_thms} @
   988                  abs_perm @ abs_fresh @ inject_thms' @ perm_thms')) 1)
   989         in
   990           (supp_thm,
   991            Goal.prove_global thy8 [] [] (augment_sort thy8 pt_cp_sort
   992              (HOLogic.mk_Trueprop (HOLogic.mk_eq
   993                (fresh c,
   994                 if null dts then @{term True}
   995                 else foldr1 HOLogic.mk_conj (map fresh args2)))))
   996              (fn _ =>
   997                simp_tac (HOL_ss addsimps [Un_iff, empty_iff, fresh_def, supp_thm]) 1))
   998         end) atoms) constrs
   999       end) (List.take (pdescr, length new_type_names) ~~ new_type_names ~~ inject_thms ~~ perm_simps')));
  1000 
  1001     (**** weak induction theorem ****)
  1002 
  1003     fun mk_indrule_lemma (((i, _), T), U) (prems, concls) =
  1004       let
  1005         val Rep_t = Const (nth rep_names i, T --> U) $ Datatype_Aux.mk_Free "x" T i;
  1006 
  1007         val Abs_t =  Const (nth abs_names i, U --> T);
  1008 
  1009       in
  1010         (prems @ [HOLogic.imp $
  1011             (Const (nth rep_set_names'' i, U --> HOLogic.boolT) $ Rep_t) $
  1012               (Datatype_Aux.mk_Free "P" (T --> HOLogic.boolT) (i + 1) $ (Abs_t $ Rep_t))],
  1013          concls @
  1014            [Datatype_Aux.mk_Free "P" (T --> HOLogic.boolT) (i + 1) $ Datatype_Aux.mk_Free "x" T i])
  1015       end;
  1016 
  1017     val (indrule_lemma_prems, indrule_lemma_concls) =
  1018       fold mk_indrule_lemma (descr'' ~~ recTs ~~ recTs') ([], []);
  1019 
  1020     val indrule_lemma = Goal.prove_global thy8 [] []
  1021       (Logic.mk_implies
  1022         (HOLogic.mk_Trueprop (Datatype_Aux.mk_conj indrule_lemma_prems),
  1023          HOLogic.mk_Trueprop (Datatype_Aux.mk_conj indrule_lemma_concls))) (fn _ => EVERY
  1024            [REPEAT (etac conjE 1),
  1025             REPEAT (EVERY
  1026               [TRY (rtac conjI 1), full_simp_tac (HOL_basic_ss addsimps Rep_inverse_thms) 1,
  1027                etac mp 1, resolve_tac Rep_thms 1])]);
  1028 
  1029     val Ps = map head_of (HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of indrule_lemma)));
  1030     val frees = if length Ps = 1 then [Free ("P", snd (dest_Var (hd Ps)))] else
  1031       map (Free o apfst fst o dest_Var) Ps;
  1032     val indrule_lemma' = cterm_instantiate
  1033       (map (cterm_of thy8) Ps ~~ map (cterm_of thy8) frees) indrule_lemma;
  1034 
  1035     val Abs_inverse_thms' = map (fn r => r RS subst) Abs_inverse_thms;
  1036 
  1037     val dt_induct_prop = Datatype_Prop.make_ind descr';
  1038     val dt_induct = Goal.prove_global thy8 []
  1039       (Logic.strip_imp_prems dt_induct_prop) (Logic.strip_imp_concl dt_induct_prop)
  1040       (fn {prems, ...} => EVERY
  1041         [rtac indrule_lemma' 1,
  1042          (Datatype_Aux.ind_tac rep_induct [] THEN_ALL_NEW Object_Logic.atomize_prems_tac) 1,
  1043          EVERY (map (fn (prem, r) => (EVERY
  1044            [REPEAT (eresolve_tac Abs_inverse_thms' 1),
  1045             simp_tac (HOL_basic_ss addsimps [Thm.symmetric r]) 1,
  1046             DEPTH_SOLVE_1 (ares_tac [prem] 1 ORELSE etac allE 1)]))
  1047                 (prems ~~ constr_defs))]);
  1048 
  1049     val case_names_induct = Datatype.mk_case_names_induct descr'';
  1050 
  1051     (**** prove that new datatypes have finite support ****)
  1052 
  1053     val _ = warning "proving finite support for the new datatype";
  1054 
  1055     val indnames = Datatype_Prop.make_tnames recTs;
  1056 
  1057     val abs_supp = Global_Theory.get_thms thy8 "abs_supp";
  1058     val supp_atm = Global_Theory.get_thms thy8 "supp_atm";
  1059 
  1060     val finite_supp_thms = map (fn atom =>
  1061       let val atomT = Type (atom, [])
  1062       in map Drule.export_without_context (List.take
  1063         (Datatype_Aux.split_conj_thm (Goal.prove_global thy8 [] []
  1064            (augment_sort thy8 (fs_class_of thy8 atom :: pt_cp_sort)
  1065              (HOLogic.mk_Trueprop
  1066                (foldr1 HOLogic.mk_conj (map (fn (s, T) =>
  1067                  Const ("Finite_Set.finite", HOLogic.mk_setT atomT --> HOLogic.boolT) $
  1068                    (Const ("Nominal.supp", T --> HOLogic.mk_setT atomT) $ Free (s, T)))
  1069                    (indnames ~~ recTs)))))
  1070            (fn _ => Datatype_Aux.ind_tac dt_induct indnames 1 THEN
  1071             ALLGOALS (asm_full_simp_tac (global_simpset_of thy8 addsimps
  1072               (abs_supp @ supp_atm @
  1073                Global_Theory.get_thms thy8 ("fs_" ^ Long_Name.base_name atom ^ "1") @
  1074                flat supp_thms))))),
  1075          length new_type_names))
  1076       end) atoms;
  1077 
  1078     val simp_atts = replicate (length new_type_names) [Simplifier.simp_add];
  1079 
  1080         (* Function to add both the simp and eqvt attributes *)
  1081         (* These two attributes are duplicated on all the types in the mutual nominal datatypes *)
  1082 
  1083     val simp_eqvt_atts = replicate (length new_type_names) [Simplifier.simp_add, NominalThmDecls.eqvt_add];
  1084  
  1085     val (_, thy9) = thy8 |>
  1086       Sign.add_path big_name |>
  1087       Global_Theory.add_thms [((Binding.name "induct", dt_induct), [case_names_induct])] ||>>
  1088       Global_Theory.add_thmss [((Binding.name "inducts", projections dt_induct), [case_names_induct])] ||>
  1089       Sign.parent_path ||>>
  1090       Datatype_Aux.store_thmss_atts "distinct" new_type_names simp_atts distinct_thms ||>>
  1091       Datatype_Aux.store_thmss "constr_rep" new_type_names constr_rep_thmss ||>>
  1092       Datatype_Aux.store_thmss_atts "perm" new_type_names simp_eqvt_atts perm_simps' ||>>
  1093       Datatype_Aux.store_thmss "inject" new_type_names inject_thms ||>>
  1094       Datatype_Aux.store_thmss "supp" new_type_names supp_thms ||>>
  1095       Datatype_Aux.store_thmss_atts "fresh" new_type_names simp_atts fresh_thms ||>
  1096       fold (fn (atom, ths) => fn thy =>
  1097         let
  1098           val class = fs_class_of thy atom;
  1099           val sort = Sign.minimize_sort thy (Sign.certify_sort thy (class :: pt_cp_sort));
  1100         in fold (fn Type (s, Ts) => AxClass.prove_arity
  1101           (s, map (inter_sort thy sort o snd o dest_TFree) Ts, [class])
  1102           (Class.intro_classes_tac [] THEN resolve_tac ths 1)) newTs thy
  1103         end) (atoms ~~ finite_supp_thms);
  1104 
  1105     (**** strong induction theorem ****)
  1106 
  1107     val pnames = if length descr'' = 1 then ["P"]
  1108       else map (fn i => "P" ^ string_of_int i) (1 upto length descr'');
  1109     val ind_sort = if null dt_atomTs then HOLogic.typeS
  1110       else Sign.minimize_sort thy9 (Sign.certify_sort thy9 (map (fs_class_of thy9) dt_atoms));
  1111     val fsT = TFree ("'n", ind_sort);
  1112     val fsT' = TFree ("'n", HOLogic.typeS);
  1113 
  1114     val fresh_fs = map (fn (s, T) => (T, Free (s, fsT' --> HOLogic.mk_setT T)))
  1115       (Datatype_Prop.indexify_names (replicate (length dt_atomTs) "f") ~~ dt_atomTs);
  1116 
  1117     fun make_pred fsT i T = Free (nth pnames i, fsT --> T --> HOLogic.boolT);
  1118 
  1119     fun mk_fresh1 xs [] = []
  1120       | mk_fresh1 xs ((y as (_, T)) :: ys) = map (fn x => HOLogic.mk_Trueprop
  1121             (HOLogic.mk_not (HOLogic.mk_eq (Free y, Free x))))
  1122               (filter (fn (_, U) => T = U) (rev xs)) @
  1123           mk_fresh1 (y :: xs) ys;
  1124 
  1125     fun mk_fresh2 xss [] = []
  1126       | mk_fresh2 xss ((p as (ys, _)) :: yss) = maps (fn y as (_, T) =>
  1127             map (fn (_, x as (_, U)) => HOLogic.mk_Trueprop
  1128               (fresh_const T U $ Free y $ Free x)) (rev xss @ yss)) ys @
  1129           mk_fresh2 (p :: xss) yss;
  1130 
  1131     fun make_ind_prem fsT f k T ((cname, cargs), idxs) =
  1132       let
  1133         val recs = filter Datatype_Aux.is_rec_type cargs;
  1134         val Ts = map (Datatype_Aux.typ_of_dtyp descr'') cargs;
  1135         val recTs' = map (Datatype_Aux.typ_of_dtyp descr'') recs;
  1136         val tnames = Name.variant_list pnames (Datatype_Prop.make_tnames Ts);
  1137         val rec_tnames = map fst (filter (Datatype_Aux.is_rec_type o snd) (tnames ~~ cargs));
  1138         val frees = tnames ~~ Ts;
  1139         val frees' = partition_cargs idxs frees;
  1140         val z = (singleton (Name.variant_list tnames) "z", fsT);
  1141 
  1142         fun mk_prem ((dt, s), T) =
  1143           let
  1144             val (Us, U) = strip_type T;
  1145             val l = length Us;
  1146           in
  1147             Logic.list_all (z :: map (pair "x") Us,
  1148               HOLogic.mk_Trueprop
  1149                 (make_pred fsT (Datatype_Aux.body_index dt) U $ Bound l $
  1150                   Datatype_Aux.app_bnds (Free (s, T)) l))
  1151           end;
  1152 
  1153         val prems = map mk_prem (recs ~~ rec_tnames ~~ recTs');
  1154         val prems' = map (fn p as (_, T) => HOLogic.mk_Trueprop
  1155             (f T (Free p) (Free z))) (maps fst frees') @
  1156           mk_fresh1 [] (maps fst frees') @
  1157           mk_fresh2 [] frees'
  1158 
  1159       in
  1160         fold_rev (Logic.all o Free) (frees @ [z])
  1161           (Logic.list_implies (prems' @ prems,
  1162             HOLogic.mk_Trueprop (make_pred fsT k T $ Free z $
  1163               list_comb (Const (cname, Ts ---> T), map Free frees))))
  1164       end;
  1165 
  1166     val ind_prems = maps (fn (((i, (_, _, constrs)), (_, idxss)), T) =>
  1167       map (make_ind_prem fsT (fn T => fn t => fn u =>
  1168         fresh_const T fsT $ t $ u) i T)
  1169           (constrs ~~ idxss)) (descr'' ~~ ndescr ~~ recTs);
  1170     val tnames = Datatype_Prop.make_tnames recTs;
  1171     val zs = Name.variant_list tnames (replicate (length descr'') "z");
  1172     val ind_concl = HOLogic.mk_Trueprop (foldr1 (HOLogic.mk_binop @{const_name HOL.conj})
  1173       (map (fn ((((i, _), T), tname), z) =>
  1174         make_pred fsT i T $ Free (z, fsT) $ Free (tname, T))
  1175         (descr'' ~~ recTs ~~ tnames ~~ zs)));
  1176     val induct = Logic.list_implies (ind_prems, ind_concl);
  1177 
  1178     val ind_prems' =
  1179       map (fn (_, f as Free (_, T)) => Logic.all (Free ("x", fsT'))
  1180         (HOLogic.mk_Trueprop (Const ("Finite_Set.finite",
  1181           Term.range_type T -->
  1182             HOLogic.boolT) $ (f $ Free ("x", fsT'))))) fresh_fs @
  1183       maps (fn (((i, (_, _, constrs)), (_, idxss)), T) =>
  1184         map (make_ind_prem fsT' (fn T => fn t => fn u => HOLogic.Not $
  1185           HOLogic.mk_mem (t, the (AList.lookup op = fresh_fs T) $ u)) i T)
  1186             (constrs ~~ idxss)) (descr'' ~~ ndescr ~~ recTs);
  1187     val ind_concl' = HOLogic.mk_Trueprop (foldr1 (HOLogic.mk_binop @{const_name HOL.conj})
  1188       (map (fn ((((i, _), T), tname), z) =>
  1189         make_pred fsT' i T $ Free (z, fsT') $ Free (tname, T))
  1190         (descr'' ~~ recTs ~~ tnames ~~ zs)));
  1191     val induct' = Logic.list_implies (ind_prems', ind_concl');
  1192 
  1193     val aux_ind_vars =
  1194       (Datatype_Prop.indexify_names (replicate (length dt_atomTs) "pi") ~~
  1195        map mk_permT dt_atomTs) @ [("z", fsT')];
  1196     val aux_ind_Ts = rev (map snd aux_ind_vars);
  1197     val aux_ind_concl = HOLogic.mk_Trueprop (foldr1 (HOLogic.mk_binop @{const_name HOL.conj})
  1198       (map (fn (((i, _), T), tname) =>
  1199         HOLogic.list_all (aux_ind_vars, make_pred fsT' i T $ Bound 0 $
  1200           fold_rev (mk_perm aux_ind_Ts) (map Bound (length dt_atomTs downto 1))
  1201             (Free (tname, T))))
  1202         (descr'' ~~ recTs ~~ tnames)));
  1203 
  1204     val fin_set_supp = map (fn s =>
  1205       at_inst_of thy9 s RS at_fin_set_supp) dt_atoms;
  1206     val fin_set_fresh = map (fn s =>
  1207       at_inst_of thy9 s RS at_fin_set_fresh) dt_atoms;
  1208     val pt1_atoms = map (fn Type (s, _) =>
  1209       Global_Theory.get_thm thy9 ("pt_" ^ Long_Name.base_name s ^ "1")) dt_atomTs;
  1210     val pt2_atoms = map (fn Type (s, _) =>
  1211       Global_Theory.get_thm thy9 ("pt_" ^ Long_Name.base_name s ^ "2") RS sym) dt_atomTs;
  1212     val exists_fresh' = Global_Theory.get_thms thy9 "exists_fresh'";
  1213     val fs_atoms = Global_Theory.get_thms thy9 "fin_supp";
  1214     val abs_supp = Global_Theory.get_thms thy9 "abs_supp";
  1215     val perm_fresh_fresh = Global_Theory.get_thms thy9 "perm_fresh_fresh";
  1216     val calc_atm = Global_Theory.get_thms thy9 "calc_atm";
  1217     val fresh_atm = Global_Theory.get_thms thy9 "fresh_atm";
  1218     val fresh_left = Global_Theory.get_thms thy9 "fresh_left";
  1219     val perm_swap = Global_Theory.get_thms thy9 "perm_swap";
  1220 
  1221     fun obtain_fresh_name' ths ts T (freshs1, freshs2, ctxt) =
  1222       let
  1223         val p = foldr1 HOLogic.mk_prod (ts @ freshs1);
  1224         val ex = Goal.prove ctxt [] [] (HOLogic.mk_Trueprop
  1225             (HOLogic.exists_const T $ Abs ("x", T,
  1226               fresh_const T (fastype_of p) $
  1227                 Bound 0 $ p)))
  1228           (fn _ => EVERY
  1229             [resolve_tac exists_fresh' 1,
  1230              simp_tac (HOL_ss addsimps (supp_prod :: finite_Un :: fs_atoms @
  1231                fin_set_supp @ ths)) 1]);
  1232         val (([(_, cx)], ths), ctxt') = Obtain.result
  1233           (fn _ => EVERY
  1234             [etac exE 1,
  1235              full_simp_tac (HOL_ss addsimps (fresh_prod :: fresh_atm)) 1,
  1236              REPEAT (etac conjE 1)])
  1237           [ex] ctxt
  1238       in (freshs1 @ [term_of cx], freshs2 @ ths, ctxt') end;
  1239 
  1240     fun fresh_fresh_inst thy a b =
  1241       let
  1242         val T = fastype_of a;
  1243         val SOME th = find_first (fn th => case prop_of th of
  1244             _ $ (_ $ (Const (_, Type (_, [U, _])) $ _ $ _)) $ _ => U = T
  1245           | _ => false) perm_fresh_fresh
  1246       in
  1247         Drule.instantiate' []
  1248           [SOME (cterm_of thy a), NONE, SOME (cterm_of thy b)] th
  1249       end;
  1250 
  1251     val fs_cp_sort =
  1252       map (fs_class_of thy9) dt_atoms @
  1253       maps (fn s => map (cp_class_of thy9 s) (remove (op =) s dt_atoms)) dt_atoms;
  1254 
  1255     (**********************************************************************
  1256       The subgoals occurring in the proof of induct_aux have the
  1257       following parameters:
  1258 
  1259         x_1 ... x_k p_1 ... p_m z
  1260 
  1261       where
  1262 
  1263         x_i : constructor arguments (introduced by weak induction rule)
  1264         p_i : permutations (one for each atom type in the data type)
  1265         z   : freshness context
  1266     ***********************************************************************)
  1267 
  1268     val _ = warning "proving strong induction theorem ...";
  1269 
  1270     val induct_aux = Goal.prove_global thy9 []
  1271         (map (augment_sort thy9 fs_cp_sort) ind_prems')
  1272         (augment_sort thy9 fs_cp_sort ind_concl') (fn {prems, context} =>
  1273       let
  1274         val (prems1, prems2) = chop (length dt_atomTs) prems;
  1275         val ind_ss2 = HOL_ss addsimps
  1276           finite_Diff :: abs_fresh @ abs_supp @ fs_atoms;
  1277         val ind_ss1 = ind_ss2 addsimps fresh_left @ calc_atm @
  1278           fresh_atm @ rev_simps @ app_simps;
  1279         val ind_ss3 = HOL_ss addsimps abs_fun_eq1 ::
  1280           abs_perm @ calc_atm @ perm_swap;
  1281         val ind_ss4 = HOL_basic_ss addsimps fresh_left @ prems1 @
  1282           fin_set_fresh @ calc_atm;
  1283         val ind_ss5 = HOL_basic_ss addsimps pt1_atoms;
  1284         val ind_ss6 = HOL_basic_ss addsimps flat perm_simps';
  1285         val th = Goal.prove context [] []
  1286           (augment_sort thy9 fs_cp_sort aux_ind_concl)
  1287           (fn {context = context1, ...} =>
  1288              EVERY (Datatype_Aux.ind_tac dt_induct tnames 1 ::
  1289                maps (fn ((_, (_, _, constrs)), (_, constrs')) =>
  1290                  map (fn ((cname, cargs), is) =>
  1291                    REPEAT (rtac allI 1) THEN
  1292                    SUBPROOF (fn {prems = iprems, params, concl,
  1293                        context = context2, ...} =>
  1294                      let
  1295                        val concl' = term_of concl;
  1296                        val _ $ (_ $ _ $ u) = concl';
  1297                        val U = fastype_of u;
  1298                        val (xs, params') =
  1299                          chop (length cargs) (map (term_of o #2) params);
  1300                        val Ts = map fastype_of xs;
  1301                        val cnstr = Const (cname, Ts ---> U);
  1302                        val (pis, z) = split_last params';
  1303                        val mk_pi = fold_rev (mk_perm []) pis;
  1304                        val xs' = partition_cargs is xs;
  1305                        val xs'' = map (fn (ts, u) => (map mk_pi ts, mk_pi u)) xs';
  1306                        val ts = maps (fn (ts, u) => ts @ [u]) xs'';
  1307                        val (freshs1, freshs2, context3) = fold (fn t =>
  1308                          let val T = fastype_of t
  1309                          in obtain_fresh_name' prems1
  1310                            (the (AList.lookup op = fresh_fs T) $ z :: ts) T
  1311                          end) (maps fst xs') ([], [], context2);
  1312                        val freshs1' = unflat (map fst xs') freshs1;
  1313                        val freshs2' = map (Simplifier.simplify ind_ss4)
  1314                          (mk_not_sym freshs2);
  1315                        val ind_ss1' = ind_ss1 addsimps freshs2';
  1316                        val ind_ss3' = ind_ss3 addsimps freshs2';
  1317                        val rename_eq =
  1318                          if forall (null o fst) xs' then []
  1319                          else [Goal.prove context3 [] []
  1320                            (HOLogic.mk_Trueprop (HOLogic.mk_eq
  1321                              (list_comb (cnstr, ts),
  1322                               list_comb (cnstr, maps (fn ((bs, t), cs) =>
  1323                                 cs @ [fold_rev (mk_perm []) (map perm_of_pair
  1324                                   (bs ~~ cs)) t]) (xs'' ~~ freshs1')))))
  1325                            (fn _ => EVERY
  1326                               (simp_tac (HOL_ss addsimps flat inject_thms) 1 ::
  1327                                REPEAT (FIRSTGOAL (rtac conjI)) ::
  1328                                maps (fn ((bs, t), cs) =>
  1329                                  if null bs then []
  1330                                  else rtac sym 1 :: maps (fn (b, c) =>
  1331                                    [rtac trans 1, rtac sym 1,
  1332                                     rtac (fresh_fresh_inst thy9 b c) 1,
  1333                                     simp_tac ind_ss1' 1,
  1334                                     simp_tac ind_ss2 1,
  1335                                     simp_tac ind_ss3' 1]) (bs ~~ cs))
  1336                                  (xs'' ~~ freshs1')))];
  1337                        val th = Goal.prove context3 [] [] concl' (fn _ => EVERY
  1338                          [simp_tac (ind_ss6 addsimps rename_eq) 1,
  1339                           cut_facts_tac iprems 1,
  1340                           (resolve_tac prems THEN_ALL_NEW
  1341                             SUBGOAL (fn (t, i) => case Logic.strip_assums_concl t of
  1342                                 _ $ (Const ("Nominal.fresh", _) $ _ $ _) =>
  1343                                   simp_tac ind_ss1' i
  1344                               | _ $ (Const (@{const_name Not}, _) $ _) =>
  1345                                   resolve_tac freshs2' i
  1346                               | _ => asm_simp_tac (HOL_basic_ss addsimps
  1347                                   pt2_atoms addsimprocs [perm_simproc]) i)) 1])
  1348                        val final = Proof_Context.export context3 context2 [th]
  1349                      in
  1350                        resolve_tac final 1
  1351                      end) context1 1) (constrs ~~ constrs')) (descr'' ~~ ndescr)))
  1352       in
  1353         EVERY
  1354           [cut_facts_tac [th] 1,
  1355            REPEAT (eresolve_tac [conjE, @{thm allE_Nil}] 1),
  1356            REPEAT (etac allE 1),
  1357            REPEAT (TRY (rtac conjI 1) THEN asm_full_simp_tac ind_ss5 1)]
  1358       end);
  1359 
  1360     val induct_aux' = Thm.instantiate ([],
  1361       map (fn (s, v as Var (_, T)) =>
  1362         (cterm_of thy9 v, cterm_of thy9 (Free (s, T))))
  1363           (pnames ~~ map head_of (HOLogic.dest_conj
  1364              (HOLogic.dest_Trueprop (concl_of induct_aux)))) @
  1365       map (fn (_, f) =>
  1366         let val f' = Logic.varify_global f
  1367         in (cterm_of thy9 f',
  1368           cterm_of thy9 (Const ("Nominal.supp", fastype_of f')))
  1369         end) fresh_fs) induct_aux;
  1370 
  1371     val induct = Goal.prove_global thy9 []
  1372       (map (augment_sort thy9 fs_cp_sort) ind_prems)
  1373       (augment_sort thy9 fs_cp_sort ind_concl)
  1374       (fn {prems, ...} => EVERY
  1375          [rtac induct_aux' 1,
  1376           REPEAT (resolve_tac fs_atoms 1),
  1377           REPEAT ((resolve_tac prems THEN_ALL_NEW
  1378             (etac @{thm meta_spec} ORELSE' full_simp_tac (HOL_basic_ss addsimps [fresh_def]))) 1)])
  1379 
  1380     val (_, thy10) = thy9 |>
  1381       Sign.add_path big_name |>
  1382       Global_Theory.add_thms [((Binding.name "strong_induct'", induct_aux), [])] ||>>
  1383       Global_Theory.add_thms [((Binding.name "strong_induct", induct), [case_names_induct])] ||>>
  1384       Global_Theory.add_thmss [((Binding.name "strong_inducts", projections induct), [case_names_induct])];
  1385 
  1386     (**** recursion combinator ****)
  1387 
  1388     val _ = warning "defining recursion combinator ...";
  1389 
  1390     val used = fold Term.add_tfree_namesT recTs [];
  1391 
  1392     val (rec_result_Ts', rec_fn_Ts') = Datatype_Prop.make_primrec_Ts descr' used;
  1393 
  1394     val rec_sort = if null dt_atomTs then HOLogic.typeS else
  1395       Sign.minimize_sort thy10 (Sign.certify_sort thy10 pt_cp_sort);
  1396 
  1397     val rec_result_Ts = map (fn TFree (s, _) => TFree (s, rec_sort)) rec_result_Ts';
  1398     val rec_fn_Ts = map (typ_subst_atomic (rec_result_Ts' ~~ rec_result_Ts)) rec_fn_Ts';
  1399 
  1400     val rec_set_Ts = map (fn (T1, T2) =>
  1401       rec_fn_Ts @ [T1, T2] ---> HOLogic.boolT) (recTs ~~ rec_result_Ts);
  1402 
  1403     val big_rec_name = big_name ^ "_rec_set";
  1404     val rec_set_names' =
  1405       if length descr'' = 1 then [big_rec_name] else
  1406         map ((curry (op ^) (big_rec_name ^ "_")) o string_of_int)
  1407           (1 upto (length descr''));
  1408     val rec_set_names =  map (Sign.full_bname thy10) rec_set_names';
  1409 
  1410     val rec_fns = map (uncurry (Datatype_Aux.mk_Free "f"))
  1411       (rec_fn_Ts ~~ (1 upto (length rec_fn_Ts)));
  1412     val rec_sets' = map (fn c => list_comb (Free c, rec_fns))
  1413       (rec_set_names' ~~ rec_set_Ts);
  1414     val rec_sets = map (fn c => list_comb (Const c, rec_fns))
  1415       (rec_set_names ~~ rec_set_Ts);
  1416 
  1417     (* introduction rules for graph of recursion function *)
  1418 
  1419     val rec_preds = map (fn (a, T) =>
  1420       Free (a, T --> HOLogic.boolT)) (pnames ~~ rec_result_Ts);
  1421 
  1422     fun mk_fresh3 rs [] = []
  1423       | mk_fresh3 rs ((p as (ys, z)) :: yss) = maps (fn y as (_, T) =>
  1424             map_filter (fn ((_, (_, x)), r as (_, U)) => if z = x then NONE
  1425               else SOME (HOLogic.mk_Trueprop
  1426                 (fresh_const T U $ Free y $ Free r))) rs) ys @
  1427           mk_fresh3 rs yss;
  1428 
  1429     (* FIXME: avoid collisions with other variable names? *)
  1430     val rec_ctxt = Free ("z", fsT');
  1431 
  1432     fun make_rec_intr T p rec_set ((cname, cargs), idxs)
  1433         (rec_intr_ts, rec_prems, rec_prems', rec_eq_prems, l) =
  1434       let
  1435         val Ts = map (Datatype_Aux.typ_of_dtyp descr'') cargs;
  1436         val frees = map (fn i => "x" ^ string_of_int i) (1 upto length Ts) ~~ Ts;
  1437         val frees' = partition_cargs idxs frees;
  1438         val binders = maps fst frees';
  1439         val atomTs = distinct op = (maps (map snd o fst) frees');
  1440         val recs = map_filter
  1441           (fn ((_, Datatype.DtRec i), p) => SOME (i, p) | _ => NONE)
  1442           (partition_cargs idxs cargs ~~ frees');
  1443         val frees'' = map (fn i => "y" ^ string_of_int i) (1 upto length recs) ~~
  1444           map (fn (i, _) => nth rec_result_Ts i) recs;
  1445         val prems1 = map (fn ((i, (_, x)), y) => HOLogic.mk_Trueprop
  1446           (nth rec_sets' i $ Free x $ Free y)) (recs ~~ frees'');
  1447         val prems2 =
  1448           map (fn f => map (fn p as (_, T) => HOLogic.mk_Trueprop
  1449             (fresh_const T (fastype_of f) $ Free p $ f)) binders) rec_fns;
  1450         val prems3 = mk_fresh1 [] binders @ mk_fresh2 [] frees';
  1451         val prems4 = map (fn ((i, _), y) =>
  1452           HOLogic.mk_Trueprop (nth rec_preds i $ Free y)) (recs ~~ frees'');
  1453         val prems5 = mk_fresh3 (recs ~~ frees'') frees';
  1454         val prems6 = maps (fn aT => map (fn y as (_, T) => HOLogic.mk_Trueprop
  1455           (Const ("Finite_Set.finite", HOLogic.mk_setT aT --> HOLogic.boolT) $
  1456              (Const ("Nominal.supp", T --> HOLogic.mk_setT aT) $ Free y)))
  1457                frees'') atomTs;
  1458         val prems7 = map (fn x as (_, T) => HOLogic.mk_Trueprop
  1459           (fresh_const T fsT' $ Free x $ rec_ctxt)) binders;
  1460         val result = list_comb (nth rec_fns l, map Free (frees @ frees''));
  1461         val result_freshs = map (fn p as (_, T) =>
  1462           fresh_const T (fastype_of result) $ Free p $ result) binders;
  1463         val P = HOLogic.mk_Trueprop (p $ result)
  1464       in
  1465         (rec_intr_ts @ [Logic.list_implies (flat prems2 @ prems3 @ prems1,
  1466            HOLogic.mk_Trueprop (rec_set $
  1467              list_comb (Const (cname, Ts ---> T), map Free frees) $ result))],
  1468          rec_prems @ [fold_rev (Logic.all o Free) (frees @ frees'') (Logic.list_implies (prems4, P))],
  1469          rec_prems' @ map (fn fr => fold_rev (Logic.all o Free) (frees @ frees'')
  1470            (Logic.list_implies (nth prems2 l @ prems3 @ prems5 @ prems7 @ prems6 @ [P],
  1471              HOLogic.mk_Trueprop fr))) result_freshs,
  1472          rec_eq_prems @ [flat prems2 @ prems3],
  1473          l + 1)
  1474       end;
  1475 
  1476     val (rec_intr_ts, rec_prems, rec_prems', rec_eq_prems, _) =
  1477       fold (fn ((((d, d'), T), p), rec_set) =>
  1478         fold (make_rec_intr T p rec_set) (#3 (snd d) ~~ snd d'))
  1479           (descr'' ~~ ndescr ~~ recTs ~~ rec_preds ~~ rec_sets')
  1480           ([], [], [], [], 0);
  1481 
  1482     val ({intrs = rec_intrs, elims = rec_elims, raw_induct = rec_induct, ...}, thy11) =
  1483       thy10
  1484       |> Sign.map_naming Name_Space.conceal
  1485       |> Inductive.add_inductive_global
  1486           {quiet_mode = #quiet config, verbose = false, alt_name = Binding.name big_rec_name,
  1487            coind = false, no_elim = false, no_ind = false, skip_mono = true}
  1488           (map (fn (s, T) => ((Binding.name s, T), NoSyn)) (rec_set_names' ~~ rec_set_Ts))
  1489           (map dest_Free rec_fns)
  1490           (map (fn x => (Attrib.empty_binding, x)) rec_intr_ts) []
  1491       ||> Global_Theory.hide_fact true (Long_Name.append (Sign.full_bname thy10 big_rec_name) "induct")
  1492       ||> Sign.restore_naming thy10;
  1493 
  1494     (** equivariance **)
  1495 
  1496     val fresh_bij = Global_Theory.get_thms thy11 "fresh_bij";
  1497     val perm_bij = Global_Theory.get_thms thy11 "perm_bij";
  1498 
  1499     val (rec_equiv_thms, rec_equiv_thms') = ListPair.unzip (map (fn aT =>
  1500       let
  1501         val permT = mk_permT aT;
  1502         val pi = Free ("pi", permT);
  1503         val rec_fns_pi = map (mk_perm [] pi o uncurry (Datatype_Aux.mk_Free "f"))
  1504           (rec_fn_Ts ~~ (1 upto (length rec_fn_Ts)));
  1505         val rec_sets_pi = map (fn c => list_comb (Const c, rec_fns_pi))
  1506           (rec_set_names ~~ rec_set_Ts);
  1507         val ps = map (fn ((((T, U), R), R'), i) =>
  1508           let
  1509             val x = Free ("x" ^ string_of_int i, T);
  1510             val y = Free ("y" ^ string_of_int i, U)
  1511           in
  1512             (R $ x $ y, R' $ mk_perm [] pi x $ mk_perm [] pi y)
  1513           end) (recTs ~~ rec_result_Ts ~~ rec_sets ~~ rec_sets_pi ~~ (1 upto length recTs));
  1514         val ths = map (fn th => Drule.export_without_context (th RS mp)) (Datatype_Aux.split_conj_thm
  1515           (Goal.prove_global thy11 [] []
  1516             (augment_sort thy1 pt_cp_sort
  1517               (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj (map HOLogic.mk_imp ps))))
  1518             (fn _ => rtac rec_induct 1 THEN REPEAT
  1519                (simp_tac (Simplifier.global_context thy11 HOL_basic_ss
  1520                   addsimps flat perm_simps'
  1521                   addsimprocs [NominalPermeq.perm_simproc_app]) 1 THEN
  1522                 (resolve_tac rec_intrs THEN_ALL_NEW
  1523                  asm_simp_tac (HOL_ss addsimps (fresh_bij @ perm_bij))) 1))))
  1524         val ths' = map (fn ((P, Q), th) =>
  1525           Goal.prove_global thy11 [] []
  1526             (augment_sort thy1 pt_cp_sort
  1527               (Logic.mk_implies (HOLogic.mk_Trueprop Q, HOLogic.mk_Trueprop P)))
  1528             (fn _ => dtac (Thm.instantiate ([],
  1529                  [(cterm_of thy11 (Var (("pi", 0), permT)),
  1530                    cterm_of thy11 (Const ("List.rev", permT --> permT) $ pi))]) th) 1 THEN
  1531                NominalPermeq.perm_simp_tac HOL_ss 1)) (ps ~~ ths)
  1532       in (ths, ths') end) dt_atomTs);
  1533 
  1534     (** finite support **)
  1535 
  1536     val rec_fin_supp_thms = map (fn aT =>
  1537       let
  1538         val name = Long_Name.base_name (fst (dest_Type aT));
  1539         val fs_name = Global_Theory.get_thm thy11 ("fs_" ^ name ^ "1");
  1540         val aset = HOLogic.mk_setT aT;
  1541         val finite = Const ("Finite_Set.finite", aset --> HOLogic.boolT);
  1542         val fins = map (fn (f, T) => HOLogic.mk_Trueprop
  1543           (finite $ (Const ("Nominal.supp", T --> aset) $ f)))
  1544             (rec_fns ~~ rec_fn_Ts)
  1545       in
  1546         map (fn th => Drule.export_without_context (th RS mp)) (Datatype_Aux.split_conj_thm
  1547           (Goal.prove_global thy11 []
  1548             (map (augment_sort thy11 fs_cp_sort) fins)
  1549             (augment_sort thy11 fs_cp_sort
  1550               (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
  1551                 (map (fn (((T, U), R), i) =>
  1552                    let
  1553                      val x = Free ("x" ^ string_of_int i, T);
  1554                      val y = Free ("y" ^ string_of_int i, U)
  1555                    in
  1556                      HOLogic.mk_imp (R $ x $ y,
  1557                        finite $ (Const ("Nominal.supp", U --> aset) $ y))
  1558                    end) (recTs ~~ rec_result_Ts ~~ rec_sets ~~
  1559                      (1 upto length recTs))))))
  1560             (fn {prems = fins, ...} =>
  1561               (rtac rec_induct THEN_ALL_NEW cut_facts_tac fins) 1 THEN REPEAT
  1562                (NominalPermeq.finite_guess_tac (HOL_ss addsimps [fs_name]) 1))))
  1563       end) dt_atomTs;
  1564 
  1565     (** freshness **)
  1566 
  1567     val finite_premss = map (fn aT =>
  1568       map (fn (f, T) => HOLogic.mk_Trueprop
  1569         (Const ("Finite_Set.finite", HOLogic.mk_setT aT --> HOLogic.boolT) $
  1570            (Const ("Nominal.supp", T --> HOLogic.mk_setT aT) $ f)))
  1571            (rec_fns ~~ rec_fn_Ts)) dt_atomTs;
  1572 
  1573     val rec_fns' = map (augment_sort thy11 fs_cp_sort) rec_fns;
  1574 
  1575     val rec_fresh_thms = map (fn ((aT, eqvt_ths), finite_prems) =>
  1576       let
  1577         val name = Long_Name.base_name (fst (dest_Type aT));
  1578         val fs_name = Global_Theory.get_thm thy11 ("fs_" ^ name ^ "1");
  1579         val a = Free ("a", aT);
  1580         val freshs = map (fn (f, fT) => HOLogic.mk_Trueprop
  1581           (fresh_const aT fT $ a $ f)) (rec_fns ~~ rec_fn_Ts)
  1582       in
  1583         map (fn (((T, U), R), eqvt_th) =>
  1584           let
  1585             val x = Free ("x", augment_sort_typ thy11 fs_cp_sort T);
  1586             val y = Free ("y", U);
  1587             val y' = Free ("y'", U)
  1588           in
  1589             Drule.export_without_context (Goal.prove (Proof_Context.init_global thy11) []
  1590               (map (augment_sort thy11 fs_cp_sort)
  1591                 (finite_prems @
  1592                    [HOLogic.mk_Trueprop (R $ x $ y),
  1593                     HOLogic.mk_Trueprop (HOLogic.mk_all ("y'", U,
  1594                       HOLogic.mk_imp (R $ x $ y', HOLogic.mk_eq (y', y)))),
  1595                     HOLogic.mk_Trueprop (fresh_const aT T $ a $ x)] @
  1596                  freshs))
  1597               (HOLogic.mk_Trueprop (fresh_const aT U $ a $ y))
  1598               (fn {prems, context} =>
  1599                  let
  1600                    val (finite_prems, rec_prem :: unique_prem ::
  1601                      fresh_prems) = chop (length finite_prems) prems;
  1602                    val unique_prem' = unique_prem RS spec RS mp;
  1603                    val unique = [unique_prem', unique_prem' RS sym] MRS trans;
  1604                    val _ $ (_ $ (_ $ S $ _)) $ _ = prop_of supports_fresh;
  1605                    val tuple = foldr1 HOLogic.mk_prod (x :: rec_fns')
  1606                  in EVERY
  1607                    [rtac (Drule.cterm_instantiate
  1608                       [(cterm_of thy11 S,
  1609                         cterm_of thy11 (Const ("Nominal.supp",
  1610                           fastype_of tuple --> HOLogic.mk_setT aT) $ tuple))]
  1611                       supports_fresh) 1,
  1612                     simp_tac (HOL_basic_ss addsimps
  1613                       [supports_def, Thm.symmetric fresh_def, fresh_prod]) 1,
  1614                     REPEAT_DETERM (resolve_tac [allI, impI] 1),
  1615                     REPEAT_DETERM (etac conjE 1),
  1616                     rtac unique 1,
  1617                     SUBPROOF (fn {prems = prems', params = [(_, a), (_, b)], ...} => EVERY
  1618                       [cut_facts_tac [rec_prem] 1,
  1619                        rtac (Thm.instantiate ([],
  1620                          [(cterm_of thy11 (Var (("pi", 0), mk_permT aT)),
  1621                            cterm_of thy11 (perm_of_pair (term_of a, term_of b)))]) eqvt_th) 1,
  1622                        asm_simp_tac (HOL_ss addsimps
  1623                          (prems' @ perm_swap @ perm_fresh_fresh)) 1]) context 1,
  1624                     rtac rec_prem 1,
  1625                     simp_tac (HOL_ss addsimps (fs_name ::
  1626                       supp_prod :: finite_Un :: finite_prems)) 1,
  1627                     simp_tac (HOL_ss addsimps (Thm.symmetric fresh_def ::
  1628                       fresh_prod :: fresh_prems)) 1]
  1629                  end))
  1630           end) (recTs ~~ rec_result_Ts ~~ rec_sets ~~ eqvt_ths)
  1631       end) (dt_atomTs ~~ rec_equiv_thms' ~~ finite_premss);
  1632 
  1633     (** uniqueness **)
  1634 
  1635     val fun_tuple = foldr1 HOLogic.mk_prod (rec_ctxt :: rec_fns);
  1636     val fun_tupleT = fastype_of fun_tuple;
  1637     val rec_unique_frees =
  1638       Datatype_Prop.indexify_names (replicate (length recTs) "x") ~~ recTs;
  1639     val rec_unique_frees'' = map (fn (s, T) => (s ^ "'", T)) rec_unique_frees;
  1640     val rec_unique_frees' =
  1641       Datatype_Prop.indexify_names (replicate (length recTs) "y") ~~ rec_result_Ts;
  1642     val rec_unique_concls = map (fn ((x, U), R) =>
  1643         Const (@{const_name Ex1}, (U --> HOLogic.boolT) --> HOLogic.boolT) $
  1644           Abs ("y", U, R $ Free x $ Bound 0))
  1645       (rec_unique_frees ~~ rec_result_Ts ~~ rec_sets);
  1646 
  1647     val induct_aux_rec = Drule.cterm_instantiate
  1648       (map (pairself (cterm_of thy11) o apsnd (augment_sort thy11 fs_cp_sort))
  1649          (map (fn (aT, f) => (Logic.varify_global f, Abs ("z", HOLogic.unitT,
  1650             Const ("Nominal.supp", fun_tupleT --> HOLogic.mk_setT aT) $ fun_tuple)))
  1651               fresh_fs @
  1652           map (fn (((P, T), (x, U)), Q) =>
  1653            (Var ((P, 0), Logic.varifyT_global (fsT' --> T --> HOLogic.boolT)),
  1654             Abs ("z", HOLogic.unitT, absfree (x, U) Q)))
  1655               (pnames ~~ recTs ~~ rec_unique_frees ~~ rec_unique_concls) @
  1656           map (fn (s, T) => (Var ((s, 0), Logic.varifyT_global T), Free (s, T)))
  1657             rec_unique_frees)) induct_aux;
  1658 
  1659     fun obtain_fresh_name vs ths rec_fin_supp T (freshs1, freshs2, ctxt) =
  1660       let
  1661         val p = foldr1 HOLogic.mk_prod (vs @ freshs1);
  1662         val ex = Goal.prove ctxt [] [] (HOLogic.mk_Trueprop
  1663             (HOLogic.exists_const T $ Abs ("x", T,
  1664               fresh_const T (fastype_of p) $ Bound 0 $ p)))
  1665           (fn _ => EVERY
  1666             [cut_facts_tac ths 1,
  1667              REPEAT_DETERM (dresolve_tac (the (AList.lookup op = rec_fin_supp T)) 1),
  1668              resolve_tac exists_fresh' 1,
  1669              asm_simp_tac (HOL_ss addsimps (supp_prod :: finite_Un :: fs_atoms)) 1]);
  1670         val (([(_, cx)], ths), ctxt') = Obtain.result
  1671           (fn _ => EVERY
  1672             [etac exE 1,
  1673              full_simp_tac (HOL_ss addsimps (fresh_prod :: fresh_atm)) 1,
  1674              REPEAT (etac conjE 1)])
  1675           [ex] ctxt
  1676       in (freshs1 @ [term_of cx], freshs2 @ ths, ctxt') end;
  1677 
  1678     val finite_ctxt_prems = map (fn aT =>
  1679       HOLogic.mk_Trueprop
  1680         (Const ("Finite_Set.finite", HOLogic.mk_setT aT --> HOLogic.boolT) $
  1681            (Const ("Nominal.supp", fsT' --> HOLogic.mk_setT aT) $ rec_ctxt))) dt_atomTs;
  1682 
  1683     val rec_unique_thms = Datatype_Aux.split_conj_thm (Goal.prove
  1684       (Proof_Context.init_global thy11) (map fst rec_unique_frees)
  1685       (map (augment_sort thy11 fs_cp_sort)
  1686         (flat finite_premss @ finite_ctxt_prems @ rec_prems @ rec_prems'))
  1687       (augment_sort thy11 fs_cp_sort
  1688         (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj rec_unique_concls)))
  1689       (fn {prems, context} =>
  1690          let
  1691            val k = length rec_fns;
  1692            val (finite_thss, ths1) = fold_map (fn T => fn xs =>
  1693              apfst (pair T) (chop k xs)) dt_atomTs prems;
  1694            val (finite_ctxt_ths, ths2) = chop (length dt_atomTs) ths1;
  1695            val (P_ind_ths, fcbs) = chop k ths2;
  1696            val P_ths = map (fn th => th RS mp) (Datatype_Aux.split_conj_thm
  1697              (Goal.prove context
  1698                (map fst (rec_unique_frees'' @ rec_unique_frees')) []
  1699                (augment_sort thy11 fs_cp_sort
  1700                  (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
  1701                     (map (fn (((x, y), S), P) => HOLogic.mk_imp
  1702                       (S $ Free x $ Free y, P $ (Free y)))
  1703                         (rec_unique_frees'' ~~ rec_unique_frees' ~~
  1704                            rec_sets ~~ rec_preds)))))
  1705                (fn _ =>
  1706                   rtac rec_induct 1 THEN
  1707                   REPEAT ((resolve_tac P_ind_ths THEN_ALL_NEW assume_tac) 1))));
  1708            val rec_fin_supp_thms' = map
  1709              (fn (ths, (T, fin_ths)) => (T, map (curry op MRS fin_ths) ths))
  1710              (rec_fin_supp_thms ~~ finite_thss);
  1711          in EVERY
  1712            ([rtac induct_aux_rec 1] @
  1713             maps (fn ((_, finite_ths), finite_th) =>
  1714               [cut_facts_tac (finite_th :: finite_ths) 1,
  1715                asm_simp_tac (HOL_ss addsimps [supp_prod, finite_Un]) 1])
  1716                 (finite_thss ~~ finite_ctxt_ths) @
  1717             maps (fn ((_, idxss), elim) => maps (fn idxs =>
  1718               [full_simp_tac (HOL_ss addsimps [Thm.symmetric fresh_def, supp_prod, Un_iff]) 1,
  1719                REPEAT_DETERM (eresolve_tac [conjE, ex1E] 1),
  1720                rtac ex1I 1,
  1721                (resolve_tac rec_intrs THEN_ALL_NEW atac) 1,
  1722                rotate_tac ~1 1,
  1723                ((DETERM o etac elim) THEN_ALL_NEW full_simp_tac
  1724                   (HOL_ss addsimps flat distinct_thms)) 1] @
  1725                (if null idxs then [] else [hyp_subst_tac 1,
  1726                 SUBPROOF (fn {asms, concl, prems = prems', params, context = context', ...} =>
  1727                   let
  1728                     val SOME prem = find_first (can (HOLogic.dest_eq o
  1729                       HOLogic.dest_Trueprop o prop_of)) prems';
  1730                     val _ $ (_ $ lhs $ rhs) = prop_of prem;
  1731                     val _ $ (_ $ lhs' $ rhs') = term_of concl;
  1732                     val rT = fastype_of lhs';
  1733                     val (c, cargsl) = strip_comb lhs;
  1734                     val cargsl' = partition_cargs idxs cargsl;
  1735                     val boundsl = maps fst cargsl';
  1736                     val (_, cargsr) = strip_comb rhs;
  1737                     val cargsr' = partition_cargs idxs cargsr;
  1738                     val boundsr = maps fst cargsr';
  1739                     val (params1, _ :: params2) =
  1740                       chop (length params div 2) (map (term_of o #2) params);
  1741                     val params' = params1 @ params2;
  1742                     val rec_prems = filter (fn th => case prop_of th of
  1743                         _ $ p => (case head_of p of
  1744                           Const (s, _) => member (op =) rec_set_names s
  1745                         | _ => false)
  1746                       | _ => false) prems';
  1747                     val fresh_prems = filter (fn th => case prop_of th of
  1748                         _ $ (Const ("Nominal.fresh", _) $ _ $ _) => true
  1749                       | _ $ (Const (@{const_name Not}, _) $ _) => true
  1750                       | _ => false) prems';
  1751                     val Ts = map fastype_of boundsl;
  1752 
  1753                     val _ = warning "step 1: obtaining fresh names";
  1754                     val (freshs1, freshs2, context'') = fold
  1755                       (obtain_fresh_name (rec_ctxt :: rec_fns' @ params')
  1756                          (maps snd finite_thss @ finite_ctxt_ths @ rec_prems)
  1757                          rec_fin_supp_thms')
  1758                       Ts ([], [], context');
  1759                     val pi1 = map perm_of_pair (boundsl ~~ freshs1);
  1760                     val rpi1 = rev pi1;
  1761                     val pi2 = map perm_of_pair (boundsr ~~ freshs1);
  1762                     val rpi2 = rev pi2;
  1763 
  1764                     val fresh_prems' = mk_not_sym fresh_prems;
  1765                     val freshs2' = mk_not_sym freshs2;
  1766 
  1767                     (** as, bs, cs # K as ts, K bs us **)
  1768                     val _ = warning "step 2: as, bs, cs # K as ts, K bs us";
  1769                     val prove_fresh_ss = HOL_ss addsimps
  1770                       (finite_Diff :: flat fresh_thms @
  1771                        fs_atoms @ abs_fresh @ abs_supp @ fresh_atm);
  1772                     (* FIXME: avoid asm_full_simp_tac ? *)
  1773                     fun prove_fresh ths y x = Goal.prove context'' [] []
  1774                       (HOLogic.mk_Trueprop (fresh_const
  1775                          (fastype_of x) (fastype_of y) $ x $ y))
  1776                       (fn _ => cut_facts_tac ths 1 THEN asm_full_simp_tac prove_fresh_ss 1);
  1777                     val constr_fresh_thms =
  1778                       map (prove_fresh fresh_prems lhs) boundsl @
  1779                       map (prove_fresh fresh_prems rhs) boundsr @
  1780                       map (prove_fresh freshs2 lhs) freshs1 @
  1781                       map (prove_fresh freshs2 rhs) freshs1;
  1782 
  1783                     (** pi1 o (K as ts) = pi2 o (K bs us) **)
  1784                     val _ = warning "step 3: pi1 o (K as ts) = pi2 o (K bs us)";
  1785                     val pi1_pi2_eq = Goal.prove context'' [] []
  1786                       (HOLogic.mk_Trueprop (HOLogic.mk_eq
  1787                         (fold_rev (mk_perm []) pi1 lhs, fold_rev (mk_perm []) pi2 rhs)))
  1788                       (fn _ => EVERY
  1789                          [cut_facts_tac constr_fresh_thms 1,
  1790                           asm_simp_tac (HOL_basic_ss addsimps perm_fresh_fresh) 1,
  1791                           rtac prem 1]);
  1792 
  1793                     (** pi1 o ts = pi2 o us **)
  1794                     val _ = warning "step 4: pi1 o ts = pi2 o us";
  1795                     val pi1_pi2_eqs = map (fn (t, u) =>
  1796                       Goal.prove context'' [] []
  1797                         (HOLogic.mk_Trueprop (HOLogic.mk_eq
  1798                           (fold_rev (mk_perm []) pi1 t, fold_rev (mk_perm []) pi2 u)))
  1799                         (fn _ => EVERY
  1800                            [cut_facts_tac [pi1_pi2_eq] 1,
  1801                             asm_full_simp_tac (HOL_ss addsimps
  1802                               (calc_atm @ flat perm_simps' @
  1803                                fresh_prems' @ freshs2' @ abs_perm @
  1804                                alpha @ flat inject_thms)) 1]))
  1805                         (map snd cargsl' ~~ map snd cargsr');
  1806 
  1807                     (** pi1^-1 o pi2 o us = ts **)
  1808                     val _ = warning "step 5: pi1^-1 o pi2 o us = ts";
  1809                     val rpi1_pi2_eqs = map (fn ((t, u), eq) =>
  1810                       Goal.prove context'' [] []
  1811                         (HOLogic.mk_Trueprop (HOLogic.mk_eq
  1812                           (fold_rev (mk_perm []) (rpi1 @ pi2) u, t)))
  1813                         (fn _ => simp_tac (HOL_ss addsimps
  1814                            ((eq RS sym) :: perm_swap)) 1))
  1815                         (map snd cargsl' ~~ map snd cargsr' ~~ pi1_pi2_eqs);
  1816 
  1817                     val (rec_prems1, rec_prems2) =
  1818                       chop (length rec_prems div 2) rec_prems;
  1819 
  1820                     (** (ts, pi1^-1 o pi2 o vs) in rec_set **)
  1821                     val _ = warning "step 6: (ts, pi1^-1 o pi2 o vs) in rec_set";
  1822                     val rec_prems' = map (fn th =>
  1823                       let
  1824                         val _ $ (S $ x $ y) = prop_of th;
  1825                         val Const (s, _) = head_of S;
  1826                         val k = find_index (equal s) rec_set_names;
  1827                         val pi = rpi1 @ pi2;
  1828                         fun mk_pi z = fold_rev (mk_perm []) pi z;
  1829                         fun eqvt_tac p =
  1830                           let
  1831                             val U as Type (_, [Type (_, [T, _])]) = fastype_of p;
  1832                             val l = find_index (equal T) dt_atomTs;
  1833                             val th = nth (nth rec_equiv_thms' l) k;
  1834                             val th' = Thm.instantiate ([],
  1835                               [(cterm_of thy11 (Var (("pi", 0), U)),
  1836                                 cterm_of thy11 p)]) th;
  1837                           in rtac th' 1 end;
  1838                         val th' = Goal.prove context'' [] []
  1839                           (HOLogic.mk_Trueprop (S $ mk_pi x $ mk_pi y))
  1840                           (fn _ => EVERY
  1841                              (map eqvt_tac pi @
  1842                               [simp_tac (HOL_ss addsimps (fresh_prems' @ freshs2' @
  1843                                  perm_swap @ perm_fresh_fresh)) 1,
  1844                                rtac th 1]))
  1845                       in
  1846                         Simplifier.simplify
  1847                           (HOL_basic_ss addsimps rpi1_pi2_eqs) th'
  1848                       end) rec_prems2;
  1849 
  1850                     val ihs = filter (fn th => case prop_of th of
  1851                       _ $ (Const (@{const_name All}, _) $ _) => true | _ => false) prems';
  1852 
  1853                     (** pi1 o rs = pi2 o vs , rs = pi1^-1 o pi2 o vs **)
  1854                     val _ = warning "step 7: pi1 o rs = pi2 o vs , rs = pi1^-1 o pi2 o vs";
  1855                     val rec_eqns = map (fn (th, ih) =>
  1856                       let
  1857                         val th' = th RS (ih RS spec RS mp) RS sym;
  1858                         val _ $ (_ $ lhs $ rhs) = prop_of th';
  1859                         fun strip_perm (_ $ _ $ t) = strip_perm t
  1860                           | strip_perm t = t;
  1861                       in
  1862                         Goal.prove context'' [] []
  1863                            (HOLogic.mk_Trueprop (HOLogic.mk_eq
  1864                               (fold_rev (mk_perm []) pi1 lhs,
  1865                                fold_rev (mk_perm []) pi2 (strip_perm rhs))))
  1866                            (fn _ => simp_tac (HOL_basic_ss addsimps
  1867                               (th' :: perm_swap)) 1)
  1868                       end) (rec_prems' ~~ ihs);
  1869 
  1870                     (** as # rs **)
  1871                     val _ = warning "step 8: as # rs";
  1872                     val rec_freshs =
  1873                       maps (fn (rec_prem, ih) =>
  1874                         let
  1875                           val _ $ (S $ x $ (y as Free (_, T))) =
  1876                             prop_of rec_prem;
  1877                           val k = find_index (equal S) rec_sets;
  1878                           val atoms = flat (map_filter (fn (bs, z) =>
  1879                             if z = x then NONE else SOME bs) cargsl')
  1880                         in
  1881                           map (fn a as Free (_, aT) =>
  1882                             let val l = find_index (equal aT) dt_atomTs;
  1883                             in
  1884                               Goal.prove context'' [] []
  1885                                 (HOLogic.mk_Trueprop (fresh_const aT T $ a $ y))
  1886                                 (fn _ => EVERY
  1887                                    (rtac (nth (nth rec_fresh_thms l) k) 1 ::
  1888                                     map (fn th => rtac th 1)
  1889                                       (snd (nth finite_thss l)) @
  1890                                     [rtac rec_prem 1, rtac ih 1,
  1891                                      REPEAT_DETERM (resolve_tac fresh_prems 1)]))
  1892                             end) atoms
  1893                         end) (rec_prems1 ~~ ihs);
  1894 
  1895                     (** as # fK as ts rs , bs # fK bs us vs **)
  1896                     val _ = warning "step 9: as # fK as ts rs , bs # fK bs us vs";
  1897                     fun prove_fresh_result (a as Free (_, aT)) =
  1898                       Goal.prove context'' [] []
  1899                         (HOLogic.mk_Trueprop (fresh_const aT rT $ a $ rhs'))
  1900                         (fn _ => EVERY
  1901                            [resolve_tac fcbs 1,
  1902                             REPEAT_DETERM (resolve_tac
  1903                               (fresh_prems @ rec_freshs) 1),
  1904                             REPEAT_DETERM (resolve_tac (maps snd rec_fin_supp_thms') 1
  1905                               THEN resolve_tac rec_prems 1),
  1906                             resolve_tac P_ind_ths 1,
  1907                             REPEAT_DETERM (resolve_tac (P_ths @ rec_prems) 1)]);
  1908 
  1909                     val fresh_results'' = map prove_fresh_result boundsl;
  1910 
  1911                     fun prove_fresh_result'' ((a as Free (_, aT), b), th) =
  1912                       let val th' = Goal.prove context'' [] []
  1913                         (HOLogic.mk_Trueprop (fresh_const aT rT $
  1914                             fold_rev (mk_perm []) (rpi2 @ pi1) a $
  1915                             fold_rev (mk_perm []) (rpi2 @ pi1) rhs'))
  1916                         (fn _ => simp_tac (HOL_ss addsimps fresh_bij) 1 THEN
  1917                            rtac th 1)
  1918                       in
  1919                         Goal.prove context'' [] []
  1920                           (HOLogic.mk_Trueprop (fresh_const aT rT $ b $ lhs'))
  1921                           (fn _ => EVERY
  1922                              [cut_facts_tac [th'] 1,
  1923                               full_simp_tac (Simplifier.global_context thy11 HOL_ss
  1924                                 addsimps rec_eqns @ pi1_pi2_eqs @ perm_swap
  1925                                 addsimprocs [NominalPermeq.perm_simproc_app]) 1,
  1926                               full_simp_tac (HOL_ss addsimps (calc_atm @
  1927                                 fresh_prems' @ freshs2' @ perm_fresh_fresh)) 1])
  1928                       end;
  1929 
  1930                     val fresh_results = fresh_results'' @ map prove_fresh_result''
  1931                       (boundsl ~~ boundsr ~~ fresh_results'');
  1932 
  1933                     (** cs # fK as ts rs , cs # fK bs us vs **)
  1934                     val _ = warning "step 10: cs # fK as ts rs , cs # fK bs us vs";
  1935                     fun prove_fresh_result' recs t (a as Free (_, aT)) =
  1936                       Goal.prove context'' [] []
  1937                         (HOLogic.mk_Trueprop (fresh_const aT rT $ a $ t))
  1938                         (fn _ => EVERY
  1939                           [cut_facts_tac recs 1,
  1940                            REPEAT_DETERM (dresolve_tac
  1941                              (the (AList.lookup op = rec_fin_supp_thms' aT)) 1),
  1942                            NominalPermeq.fresh_guess_tac
  1943                              (HOL_ss addsimps (freshs2 @
  1944                                 fs_atoms @ fresh_atm @
  1945                                 maps snd finite_thss)) 1]);
  1946 
  1947                     val fresh_results' =
  1948                       map (prove_fresh_result' rec_prems1 rhs') freshs1 @
  1949                       map (prove_fresh_result' rec_prems2 lhs') freshs1;
  1950 
  1951                     (** pi1 o (fK as ts rs) = pi2 o (fK bs us vs) **)
  1952                     val _ = warning "step 11: pi1 o (fK as ts rs) = pi2 o (fK bs us vs)";
  1953                     val pi1_pi2_result = Goal.prove context'' [] []
  1954                       (HOLogic.mk_Trueprop (HOLogic.mk_eq
  1955                         (fold_rev (mk_perm []) pi1 rhs', fold_rev (mk_perm []) pi2 lhs')))
  1956                       (fn _ => simp_tac (Simplifier.context context'' HOL_ss
  1957                            addsimps pi1_pi2_eqs @ rec_eqns
  1958                            addsimprocs [NominalPermeq.perm_simproc_app]) 1 THEN
  1959                          TRY (simp_tac (HOL_ss addsimps
  1960                            (fresh_prems' @ freshs2' @ calc_atm @ perm_fresh_fresh)) 1));
  1961 
  1962                     val _ = warning "final result";
  1963                     val final = Goal.prove context'' [] [] (term_of concl)
  1964                       (fn _ => cut_facts_tac [pi1_pi2_result RS sym] 1 THEN
  1965                         full_simp_tac (HOL_basic_ss addsimps perm_fresh_fresh @
  1966                           fresh_results @ fresh_results') 1);
  1967                     val final' = Proof_Context.export context'' context' [final];
  1968                     val _ = warning "finished!"
  1969                   in
  1970                     resolve_tac final' 1
  1971                   end) context 1])) idxss) (ndescr ~~ rec_elims))
  1972          end));
  1973 
  1974     val rec_total_thms = map (fn r => r RS @{thm theI'}) rec_unique_thms;
  1975 
  1976     (* define primrec combinators *)
  1977 
  1978     val big_reccomb_name = space_implode "_" new_type_names ^ "_rec";
  1979     val reccomb_names = map (Sign.full_bname thy11)
  1980       (if length descr'' = 1 then [big_reccomb_name] else
  1981         (map ((curry (op ^) (big_reccomb_name ^ "_")) o string_of_int)
  1982           (1 upto (length descr''))));
  1983     val reccombs = map (fn ((name, T), T') => list_comb
  1984       (Const (name, rec_fn_Ts @ [T] ---> T'), rec_fns))
  1985         (reccomb_names ~~ recTs ~~ rec_result_Ts);
  1986 
  1987     val (reccomb_defs, thy12) =
  1988       thy11
  1989       |> Sign.add_consts_i (map (fn ((name, T), T') =>
  1990           (Binding.name (Long_Name.base_name name), rec_fn_Ts @ [T] ---> T', NoSyn))
  1991           (reccomb_names ~~ recTs ~~ rec_result_Ts))
  1992       |> (Global_Theory.add_defs false o map Thm.no_attributes) (map (fn ((((name, comb), set), T), T') =>
  1993           (Binding.name (Long_Name.base_name name ^ "_def"), Logic.mk_equals (comb, absfree ("x", T)
  1994            (Const (@{const_name The}, (T' --> HOLogic.boolT) --> T') $ absfree ("y", T')
  1995              (set $ Free ("x", T) $ Free ("y", T'))))))
  1996                (reccomb_names ~~ reccombs ~~ rec_sets ~~ recTs ~~ rec_result_Ts));
  1997 
  1998     (* prove characteristic equations for primrec combinators *)
  1999 
  2000     val rec_thms = map (fn (prems, concl) =>
  2001       let
  2002         val _ $ (_ $ (_ $ x) $ _) = concl;
  2003         val (_, cargs) = strip_comb x;
  2004         val ps = map (fn (x as Free (_, T), i) =>
  2005           (Free ("x" ^ string_of_int i, T), x)) (cargs ~~ (1 upto length cargs));
  2006         val concl' = subst_atomic_types (rec_result_Ts' ~~ rec_result_Ts) concl;
  2007         val prems' = flat finite_premss @ finite_ctxt_prems @
  2008           rec_prems @ rec_prems' @ map (subst_atomic ps) prems;
  2009         fun solve rules prems = resolve_tac rules THEN_ALL_NEW
  2010           (resolve_tac prems THEN_ALL_NEW atac)
  2011       in
  2012         Goal.prove_global thy12 []
  2013           (map (augment_sort thy12 fs_cp_sort) prems')
  2014           (augment_sort thy12 fs_cp_sort concl')
  2015           (fn {prems, ...} => EVERY
  2016             [rewrite_goals_tac reccomb_defs,
  2017              rtac @{thm the1_equality} 1,
  2018              solve rec_unique_thms prems 1,
  2019              resolve_tac rec_intrs 1,
  2020              REPEAT (solve (prems @ rec_total_thms) prems 1)])
  2021       end) (rec_eq_prems ~~
  2022         Datatype_Prop.make_primrecs reccomb_names descr' thy12);
  2023 
  2024     val dt_infos = map_index (make_dt_info pdescr induct reccomb_names rec_thms)
  2025       (descr1 ~~ distinct_thms ~~ inject_thms);
  2026 
  2027     (* FIXME: theorems are stored in database for testing only *)
  2028     val (_, thy13) = thy12 |>
  2029       Global_Theory.add_thmss
  2030         [((Binding.name "rec_equiv", flat rec_equiv_thms), []),
  2031          ((Binding.name "rec_equiv'", flat rec_equiv_thms'), []),
  2032          ((Binding.name "rec_fin_supp", flat rec_fin_supp_thms), []),
  2033          ((Binding.name "rec_fresh", flat rec_fresh_thms), []),
  2034          ((Binding.name "rec_unique", map Drule.export_without_context rec_unique_thms), []),
  2035          ((Binding.name "recs", rec_thms), [])] ||>
  2036       Sign.parent_path ||>
  2037       map_nominal_datatypes (fold Symtab.update dt_infos);
  2038 
  2039   in
  2040     thy13
  2041   end;
  2042 
  2043 val nominal_datatype = gen_nominal_datatype Datatype.check_specs;
  2044 val nominal_datatype_cmd = gen_nominal_datatype Datatype.read_specs;
  2045 
  2046 val _ =
  2047   Outer_Syntax.command @{command_spec "nominal_datatype"} "define nominal datatypes"
  2048     (Parse.and_list1 Datatype.spec_cmd >>
  2049       (Toplevel.theory o nominal_datatype_cmd Datatype.default_config));
  2050 
  2051 end