src/HOL/Nominal/nominal_datatype.ML

 val Un_assoc = @{thm Un_assoc};
 val Collect_disj_eq = @{thm Collect_disj_eq};
 val Collect_False_empty = @{thm empty_def [THEN sym, THEN eq_reflection]};
 val empty_iff = @{thm empty_iff};
 
-open Datatype_Aux;
 open NominalAtoms;
 
 (** FIXME: Datatype should export this function **)
 
 local
 
-fun dt_recs (DtTFree _) = []
-  | dt_recs (DtType (_, dts)) = maps dt_recs dts
-  | dt_recs (DtRec i) = [i];
-
-fun dt_cases (descr: descr) (_, args, constrs) =
+fun dt_recs (Datatype_Aux.DtTFree _) = []
+  | dt_recs (Datatype_Aux.DtType (_, dts)) = maps dt_recs dts
+  | dt_recs (Datatype_Aux.DtRec i) = [i];
+
+fun dt_cases (descr: Datatype_Aux.descr) (_, args, constrs) =
   let
     fun the_bname i = Long_Name.base_name (#1 (the (AList.lookup (op =) descr i)));
     val bnames = map the_bname (distinct op = (maps dt_recs args));
   in map (fn (c, _) => space_implode "_" (Long_Name.base_name c :: bnames)) constrs end;
 

 
 end;
 
 (* theory data *)
 
-type descr = (int * (string * dtyp list * (string * (dtyp list * dtyp) list) list)) list;
+type descr =
+  (int * (string * Datatype_Aux.dtyp list *
+      (string * (Datatype_Aux.dtyp list * Datatype_Aux.dtyp) list) list)) list;
 
 type nominal_datatype_info =
   {index : int,
    descr : descr,
    rec_names : string list,

     val new_type_names' = map (fn n => n ^ "_Rep") new_type_names;
 
     val (full_new_type_names',thy1) = Datatype.add_datatype config dts'' thy;
 
     val {descr, induct, ...} = Datatype.the_info thy1 (hd full_new_type_names');
-    fun nth_dtyp i = typ_of_dtyp descr (DtRec i);
+    fun nth_dtyp i = Datatype_Aux.typ_of_dtyp descr (Datatype_Aux.DtRec i);
 
     val big_name = space_implode "_" new_type_names;
 
 
     (**** define permutation functions ****)

     val pi = Free ("pi", permT);
     val perm_types = map (fn (i, _) =>
       let val T = nth_dtyp i
       in permT --> T --> T end) descr;
     val perm_names' = Datatype_Prop.indexify_names (map (fn (i, _) =>
-      "perm_" ^ name_of_typ (nth_dtyp i)) descr);
+      "perm_" ^ Datatype_Aux.name_of_typ (nth_dtyp i)) descr);
     val perm_names = replicate (length new_type_names) "Nominal.perm" @
       map (Sign.full_bname thy1) (List.drop (perm_names', length new_type_names));
     val perm_names_types = perm_names ~~ perm_types;
     val perm_names_types' = perm_names' ~~ perm_types;
 
     val perm_eqs = maps (fn (i, (_, _, constrs)) =>
       let val T = nth_dtyp i
       in map (fn (cname, dts) =>
         let
-          val Ts = map (typ_of_dtyp descr) dts;
+          val Ts = map (Datatype_Aux.typ_of_dtyp descr) dts;
           val names = Name.variant_list ["pi"] (Datatype_Prop.make_tnames Ts);
           val args = map Free (names ~~ Ts);
           val c = Const (cname, Ts ---> T);
           fun perm_arg (dt, x) =
             let val T = type_of x
-            in if is_rec_type dt then
+            in if Datatype_Aux.is_rec_type dt then
                 let val Us = binder_types T
                 in list_abs (map (pair "x") Us,
-                  Free (nth perm_names_types' (body_index dt)) $ pi $
+                  Free (nth perm_names_types' (Datatype_Aux.body_index dt)) $ pi $
                     list_comb (x, map (fn (i, U) =>
                       Const ("Nominal.perm", permT --> U --> U) $
                         (Const ("List.rev", permT --> permT) $ pi) $
                         Bound i) ((length Us - 1 downto 0) ~~ Us)))
                 end

     val perm_indnames = Datatype_Prop.make_tnames (map body_type perm_types);
     val perm_fun_def = Simpdata.mk_eq @{thm perm_fun_def};
 
     val unfolded_perm_eq_thms =
       if length descr = length new_type_names then []
-      else map Drule.export_without_context (List.drop (split_conj_thm
+      else map Drule.export_without_context (List.drop (Datatype_Aux.split_conj_thm
         (Goal.prove_global thy2 [] []
           (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
             (map (fn (c as (s, T), x) =>
                let val [T1, T2] = binder_types T
                in HOLogic.mk_eq (Const c $ pi $ Free (x, T2),

 
     val _ = warning "perm_empty_thms";
 
     val perm_empty_thms = maps (fn a =>
       let val permT = mk_permT (Type (a, []))
-      in map Drule.export_without_context (List.take (split_conj_thm
+      in map Drule.export_without_context (List.take (Datatype_Aux.split_conj_thm
         (Goal.prove_global thy2 [] []
           (augment_sort thy2 [pt_class_of thy2 a]
             (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
               (map (fn ((s, T), x) => HOLogic.mk_eq
                   (Const (s, permT --> T --> T) $

         val pi1 = Free ("pi1", permT);
         val pi2 = Free ("pi2", permT);
         val pt_inst = pt_inst_of thy2 a;
         val pt2' = pt_inst RS pt2;
         val pt2_ax = Global_Theory.get_thm thy2 (Long_Name.map_base_name (fn s => "pt_" ^ s ^ "2") a);
-      in List.take (map Drule.export_without_context (split_conj_thm
+      in List.take (map Drule.export_without_context (Datatype_Aux.split_conj_thm
         (Goal.prove_global thy2 [] []
            (augment_sort thy2 [pt_class_of thy2 a]
              (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
                 (map (fn ((s, T), x) =>
                     let val perm = Const (s, permT --> T --> T)

         val at_inst = at_inst_of thy2 a;
         val pt_inst = pt_inst_of thy2 a;
         val pt3' = pt_inst RS pt3;
         val pt3_rev' = at_inst RS (pt_inst RS pt3_rev);
         val pt3_ax = Global_Theory.get_thm thy2 (Long_Name.map_base_name (fn s => "pt_" ^ s ^ "3") a);
-      in List.take (map Drule.export_without_context (split_conj_thm
+      in List.take (map Drule.export_without_context (Datatype_Aux.split_conj_thm
         (Goal.prove_global thy2 [] []
           (augment_sort thy2 [pt_class_of thy2 a] (Logic.mk_implies
              (HOLogic.mk_Trueprop (Const ("Nominal.prm_eq",
                 permT --> permT --> HOLogic.boolT) $ pi1 $ pi2),
               HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj

                [cp_inst RS cp1 RS sym,
                 at_inst RS (pt_inst RS pt_perm_compose) RS sym,
                 at_inst RS (pt_inst RS pt_perm_compose_rev) RS sym]
             end))
         val sort = Sign.minimize_sort thy (Sign.certify_sort thy (cp_class :: pt_class));
-        val thms = split_conj_thm (Goal.prove_global thy [] []
+        val thms = Datatype_Aux.split_conj_thm (Goal.prove_global thy [] []
           (augment_sort thy sort
             (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
               (map (fn ((s, T), x) =>
                   let
                     val pi1 = Free ("pi1", permT1);

 
     (**** Define representing sets ****)
 
     val _ = warning "representing sets";
 
-    val rep_set_names = Datatype_Prop.indexify_names
-      (map (fn (i, _) => name_of_typ (nth_dtyp i) ^ "_set") descr);
+    val rep_set_names =
+      Datatype_Prop.indexify_names
+        (map (fn (i, _) => Datatype_Aux.name_of_typ (nth_dtyp i) ^ "_set") descr);
     val big_rep_name =
       space_implode "_" (Datatype_Prop.indexify_names (map_filter
         (fn (i, ("Nominal.noption", _, _)) => NONE
-          | (i, _) => SOME (name_of_typ (nth_dtyp i))) descr)) ^ "_set";
+          | (i, _) => SOME (Datatype_Aux.name_of_typ (nth_dtyp i))) descr)) ^ "_set";
     val _ = warning ("big_rep_name: " ^ big_rep_name);
 
-    fun strip_option (dtf as DtType ("fun", [dt, DtRec i])) =
+    fun strip_option (dtf as Datatype_Aux.DtType ("fun", [dt, Datatype_Aux.DtRec i])) =
           (case AList.lookup op = descr i of
              SOME ("Nominal.noption", _, [(_, [dt']), _]) =>
                apfst (cons dt) (strip_option dt')
            | _ => ([], dtf))
-      | strip_option (DtType ("fun", [dt, DtType ("Nominal.noption", [dt'])])) =
+      | strip_option (Datatype_Aux.DtType ("fun",
+            [dt, Datatype_Aux.DtType ("Nominal.noption", [dt'])])) =
           apfst (cons dt) (strip_option dt')
       | strip_option dt = ([], dt);
 
-    val dt_atomTs = distinct op = (map (typ_of_dtyp descr)
+    val dt_atomTs = distinct op = (map (Datatype_Aux.typ_of_dtyp descr)
       (maps (fn (_, (_, _, cs)) => maps (maps (fst o strip_option) o snd) cs) descr));
     val dt_atoms = map (fst o dest_Type) dt_atomTs;
 
     fun make_intr s T (cname, cargs) =
       let
         fun mk_prem dt (j, j', prems, ts) =
           let
             val (dts, dt') = strip_option dt;
-            val (dts', dt'') = strip_dtyp dt';
-            val Ts = map (typ_of_dtyp descr) dts;
-            val Us = map (typ_of_dtyp descr) dts';
-            val T = typ_of_dtyp descr dt'';
-            val free = mk_Free "x" (Us ---> T) j;
-            val free' = app_bnds free (length Us);
+            val (dts', dt'') = Datatype_Aux.strip_dtyp dt';
+            val Ts = map (Datatype_Aux.typ_of_dtyp descr) dts;
+            val Us = map (Datatype_Aux.typ_of_dtyp descr) dts';
+            val T = Datatype_Aux.typ_of_dtyp descr dt'';
+            val free = Datatype_Aux.mk_Free "x" (Us ---> T) j;
+            val free' = Datatype_Aux.app_bnds free (length Us);
             fun mk_abs_fun T (i, t) =
               let val U = fastype_of t
               in (i + 1, Const ("Nominal.abs_fun", [T, U, T] --->
-                Type ("Nominal.noption", [U])) $ mk_Free "y" T i $ t)
+                Type ("Nominal.noption", [U])) $ Datatype_Aux.mk_Free "y" T i $ t)
               end
           in (j + 1, j' + length Ts,
             case dt'' of
-                DtRec k => list_all (map (pair "x") Us,
+                Datatype_Aux.DtRec k => list_all (map (pair "x") Us,
                   HOLogic.mk_Trueprop (Free (nth rep_set_names k,
                     T --> HOLogic.boolT) $ free')) :: prems
               | _ => prems,
             snd (fold_rev mk_abs_fun Ts (j', free)) :: ts)
           end;

     val perm_indnames' = map_filter
       (fn (x, (_, ("Nominal.noption", _, _))) => NONE | (x, _) => SOME x)
       (perm_indnames ~~ descr);
 
     fun mk_perm_closed name = map (fn th => Drule.export_without_context (th RS mp))
-      (List.take (split_conj_thm (Goal.prove_global thy4 [] []
+      (List.take (Datatype_Aux.split_conj_thm (Goal.prove_global thy4 [] []
         (augment_sort thy4
           (pt_class_of thy4 name :: map (cp_class_of thy4 name) (remove (op =) name dt_atoms))
           (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj (map
             (fn ((s, T), x) =>
                let

 
     val (ty_idxs, _) = List.foldl
       (fn ((i, ("Nominal.noption", _, _)), p) => p
         | ((i, _), (ty_idxs, j)) => (ty_idxs @ [(i, j)], j + 1)) ([], 0) descr;
 
-    fun reindex (DtType (s, dts)) = DtType (s, map reindex dts)
-      | reindex (DtRec i) = DtRec (the (AList.lookup op = ty_idxs i))
+    fun reindex (Datatype_Aux.DtType (s, dts)) = Datatype_Aux.DtType (s, map reindex dts)
+      | reindex (Datatype_Aux.DtRec i) = Datatype_Aux.DtRec (the (AList.lookup op = ty_idxs i))
       | reindex dt = dt;
 
     fun strip_suffix i s = implode (List.take (raw_explode s, size s - i));  (* FIXME Symbol.explode (?) *)
 
     (** strips the "_Rep" in type names *)

 
     val pdescr = map (fn ((i, (s, dts, constrs)), (_, idxss)) => (i, (s, dts,
       map (fn ((cname, cargs), idxs) => (cname, partition_cargs idxs cargs))
         (constrs ~~ idxss)))) (descr'' ~~ ndescr);
 
-    fun nth_dtyp' i = typ_of_dtyp descr'' (DtRec i);
+    fun nth_dtyp' i = Datatype_Aux.typ_of_dtyp descr'' (Datatype_Aux.DtRec i);
 
     val rep_names = map (fn s =>
       Sign.intern_const thy7 ("Rep_" ^ s)) new_type_names;
     val abs_names = map (fn s =>
       Sign.intern_const thy7 ("Abs_" ^ s)) new_type_names;
 
-    val recTs = get_rec_types descr'';
+    val recTs = Datatype_Aux.get_rec_types descr'';
     val newTs' = take (length new_type_names) recTs';
     val newTs = take (length new_type_names) recTs;
 
     val full_new_type_names = map (Sign.full_bname thy) new_type_names;
 
     fun make_constr_def tname T T' (((cname_rep, _), (cname, cargs)), (cname', mx))
         (thy, defs, eqns) =
       let
         fun constr_arg (dts, dt) (j, l_args, r_args) =
           let
-            val xs = map (fn (dt, i) => mk_Free "x" (typ_of_dtyp descr'' dt) i)
-              (dts ~~ (j upto j + length dts - 1))
-            val x = mk_Free "x" (typ_of_dtyp descr'' dt) (j + length dts)
+            val xs =
+              map (fn (dt, i) => Datatype_Aux.mk_Free "x" (Datatype_Aux.typ_of_dtyp descr'' dt) i)
+                (dts ~~ (j upto j + length dts - 1))
+            val x = Datatype_Aux.mk_Free "x" (Datatype_Aux.typ_of_dtyp descr'' dt) (j + length dts)
           in
             (j + length dts + 1,
              xs @ x :: l_args,
              fold_rev mk_abs_fun xs
                (case dt of
-                  DtRec k => if k < length new_type_names then
-                      Const (nth rep_names k, typ_of_dtyp descr'' dt -->
-                        typ_of_dtyp descr dt) $ x
+                  Datatype_Aux.DtRec k => if k < length new_type_names then
+                      Const (nth rep_names k, Datatype_Aux.typ_of_dtyp descr'' dt -->
+                        Datatype_Aux.typ_of_dtyp descr dt) $ x
                     else error "nested recursion not (yet) supported"
                 | _ => x) :: r_args)
           end
 
         val (_, l_args, r_args) = fold_rev constr_arg cargs (1, [], []);

             let val T = fastype_of t
             in Const ("Nominal.perm", permT --> T --> T) $ pi $ t end;
 
           fun constr_arg (dts, dt) (j, l_args, r_args) =
             let
-              val Ts = map (typ_of_dtyp descr'') dts;
-              val xs = map (fn (T, i) => mk_Free "x" T i)
-                (Ts ~~ (j upto j + length dts - 1))
-              val x = mk_Free "x" (typ_of_dtyp descr'' dt) (j + length dts)
+              val Ts = map (Datatype_Aux.typ_of_dtyp descr'') dts;
+              val xs =
+                map (fn (T, i) => Datatype_Aux.mk_Free "x" T i)
+                  (Ts ~~ (j upto j + length dts - 1));
+              val x =
+                Datatype_Aux.mk_Free "x" (Datatype_Aux.typ_of_dtyp descr'' dt) (j + length dts);
             in
               (j + length dts + 1,
                xs @ x :: l_args,
                map perm (xs @ [x]) @ r_args)
             end

           val cname = Sign.intern_const thy8
             (Long_Name.append tname (Long_Name.base_name cname));
 
           fun make_inj (dts, dt) (j, args1, args2, eqs) =
             let
-              val Ts_idx = map (typ_of_dtyp descr'') dts ~~ (j upto j + length dts - 1);
-              val xs = map (fn (T, i) => mk_Free "x" T i) Ts_idx;
-              val ys = map (fn (T, i) => mk_Free "y" T i) Ts_idx;
-              val x = mk_Free "x" (typ_of_dtyp descr'' dt) (j + length dts);
-              val y = mk_Free "y" (typ_of_dtyp descr'' dt) (j + length dts);
+              val Ts_idx =
+                map (Datatype_Aux.typ_of_dtyp descr'') dts ~~ (j upto j + length dts - 1);
+              val xs = map (fn (T, i) => Datatype_Aux.mk_Free "x" T i) Ts_idx;
+              val ys = map (fn (T, i) => Datatype_Aux.mk_Free "y" T i) Ts_idx;
+              val x =
+                Datatype_Aux.mk_Free "x" (Datatype_Aux.typ_of_dtyp descr'' dt) (j + length dts);
+              val y =
+                Datatype_Aux.mk_Free "y" (Datatype_Aux.typ_of_dtyp descr'' dt) (j + length dts);
             in
               (j + length dts + 1,
                xs @ (x :: args1), ys @ (y :: args2),
                HOLogic.mk_eq
                  (fold_rev mk_abs_fun xs x, fold_rev mk_abs_fun ys y) :: eqs)

             (Long_Name.append tname (Long_Name.base_name cname));
           val atomT = Type (atom, []);
 
           fun process_constr (dts, dt) (j, args1, args2) =
             let
-              val Ts_idx = map (typ_of_dtyp descr'') dts ~~ (j upto j + length dts - 1);
-              val xs = map (fn (T, i) => mk_Free "x" T i) Ts_idx;
-              val x = mk_Free "x" (typ_of_dtyp descr'' dt) (j + length dts);
+              val Ts_idx =
+                map (Datatype_Aux.typ_of_dtyp descr'') dts ~~ (j upto j + length dts - 1);
+              val xs = map (fn (T, i) => Datatype_Aux.mk_Free "x" T i) Ts_idx;
+              val x =
+                Datatype_Aux.mk_Free "x" (Datatype_Aux.typ_of_dtyp descr'' dt) (j + length dts);
             in
               (j + length dts + 1,
                xs @ (x :: args1), fold_rev mk_abs_fun xs x :: args2)
             end;
 

 
     (**** weak induction theorem ****)
 
     fun mk_indrule_lemma (((i, _), T), U) (prems, concls) =
       let
-        val Rep_t = Const (nth rep_names i, T --> U) $ mk_Free "x" T i;
+        val Rep_t = Const (nth rep_names i, T --> U) $ Datatype_Aux.mk_Free "x" T i;
 
         val Abs_t =  Const (nth abs_names i, U --> T);
 
-      in (prems @ [HOLogic.imp $
+      in
+        (prems @ [HOLogic.imp $
             (Const (nth rep_set_names'' i, U --> HOLogic.boolT) $ Rep_t) $
-              (mk_Free "P" (T --> HOLogic.boolT) (i + 1) $ (Abs_t $ Rep_t))],
-          concls @ [mk_Free "P" (T --> HOLogic.boolT) (i + 1) $ mk_Free "x" T i])
+              (Datatype_Aux.mk_Free "P" (T --> HOLogic.boolT) (i + 1) $ (Abs_t $ Rep_t))],
+         concls @
+           [Datatype_Aux.mk_Free "P" (T --> HOLogic.boolT) (i + 1) $ Datatype_Aux.mk_Free "x" T i])
       end;
 
     val (indrule_lemma_prems, indrule_lemma_concls) =
       fold mk_indrule_lemma (descr'' ~~ recTs ~~ recTs') ([], []);
 
     val indrule_lemma = Goal.prove_global thy8 [] []
       (Logic.mk_implies
-        (HOLogic.mk_Trueprop (mk_conj indrule_lemma_prems),
-         HOLogic.mk_Trueprop (mk_conj indrule_lemma_concls))) (fn _ => EVERY
+        (HOLogic.mk_Trueprop (Datatype_Aux.mk_conj indrule_lemma_prems),
+         HOLogic.mk_Trueprop (Datatype_Aux.mk_conj indrule_lemma_concls))) (fn _ => EVERY
            [REPEAT (etac conjE 1),
             REPEAT (EVERY
               [TRY (rtac conjI 1), full_simp_tac (HOL_basic_ss addsimps Rep_inverse_thms) 1,
                etac mp 1, resolve_tac Rep_thms 1])]);
 

     val supp_atm = Global_Theory.get_thms thy8 "supp_atm";
 
     val finite_supp_thms = map (fn atom =>
       let val atomT = Type (atom, [])
       in map Drule.export_without_context (List.take
-        (split_conj_thm (Goal.prove_global thy8 [] []
+        (Datatype_Aux.split_conj_thm (Goal.prove_global thy8 [] []
            (augment_sort thy8 (fs_class_of thy8 atom :: pt_cp_sort)
              (HOLogic.mk_Trueprop
                (foldr1 HOLogic.mk_conj (map (fn (s, T) =>
                  Const ("Finite_Set.finite", HOLogic.mk_setT atomT --> HOLogic.boolT) $
                    (Const ("Nominal.supp", T --> HOLogic.mk_setT atomT) $ Free (s, T)))

               (fresh_const T U $ Free y $ Free x)) (rev xss @ yss)) ys @
           mk_fresh2 (p :: xss) yss;
 
     fun make_ind_prem fsT f k T ((cname, cargs), idxs) =
       let
-        val recs = filter is_rec_type cargs;
-        val Ts = map (typ_of_dtyp descr'') cargs;
-        val recTs' = map (typ_of_dtyp descr'') recs;
+        val recs = filter Datatype_Aux.is_rec_type cargs;
+        val Ts = map (Datatype_Aux.typ_of_dtyp descr'') cargs;
+        val recTs' = map (Datatype_Aux.typ_of_dtyp descr'') recs;
         val tnames = Name.variant_list pnames (Datatype_Prop.make_tnames Ts);
-        val rec_tnames = map fst (filter (is_rec_type o snd) (tnames ~~ cargs));
+        val rec_tnames = map fst (filter (Datatype_Aux.is_rec_type o snd) (tnames ~~ cargs));
         val frees = tnames ~~ Ts;
         val frees' = partition_cargs idxs frees;
         val z = (singleton (Name.variant_list tnames) "z", fsT);
 
         fun mk_prem ((dt, s), T) =
           let
             val (Us, U) = strip_type T;
-            val l = length Us
-          in list_all (z :: map (pair "x") Us, HOLogic.mk_Trueprop
-            (make_pred fsT (body_index dt) U $ Bound l $ app_bnds (Free (s, T)) l))
+            val l = length Us;
+          in
+            list_all (z :: map (pair "x") Us,
+              HOLogic.mk_Trueprop
+                (make_pred fsT (Datatype_Aux.body_index dt) U $ Bound l $
+                  Datatype_Aux.app_bnds (Free (s, T)) l))
           end;
 
         val prems = map mk_prem (recs ~~ rec_tnames ~~ recTs');
         val prems' = map (fn p as (_, T) => HOLogic.mk_Trueprop
             (f T (Free p) (Free z))) (maps fst frees') @

       if length descr'' = 1 then [big_rec_name] else
         map ((curry (op ^) (big_rec_name ^ "_")) o string_of_int)
           (1 upto (length descr''));
     val rec_set_names =  map (Sign.full_bname thy10) rec_set_names';
 
-    val rec_fns = map (uncurry (mk_Free "f"))
+    val rec_fns = map (uncurry (Datatype_Aux.mk_Free "f"))
       (rec_fn_Ts ~~ (1 upto (length rec_fn_Ts)));
     val rec_sets' = map (fn c => list_comb (Free c, rec_fns))
       (rec_set_names' ~~ rec_set_Ts);
     val rec_sets = map (fn c => list_comb (Const c, rec_fns))
       (rec_set_names ~~ rec_set_Ts);

     val rec_ctxt = Free ("z", fsT');
 
     fun make_rec_intr T p rec_set ((cname, cargs), idxs)
         (rec_intr_ts, rec_prems, rec_prems', rec_eq_prems, l) =
       let
-        val Ts = map (typ_of_dtyp descr'') cargs;
+        val Ts = map (Datatype_Aux.typ_of_dtyp descr'') cargs;
         val frees = map (fn i => "x" ^ string_of_int i) (1 upto length Ts) ~~ Ts;
         val frees' = partition_cargs idxs frees;
         val binders = maps fst frees';
         val atomTs = distinct op = (maps (map snd o fst) frees');
         val recs = map_filter
-          (fn ((_, DtRec i), p) => SOME (i, p) | _ => NONE)
+          (fn ((_, Datatype_Aux.DtRec i), p) => SOME (i, p) | _ => NONE)
           (partition_cargs idxs cargs ~~ frees');
         val frees'' = map (fn i => "y" ^ string_of_int i) (1 upto length recs) ~~
           map (fn (i, _) => nth rec_result_Ts i) recs;
         val prems1 = map (fn ((i, (_, x)), y) => HOLogic.mk_Trueprop
           (nth rec_sets' i $ Free x $ Free y)) (recs ~~ frees'');

 
     val (rec_equiv_thms, rec_equiv_thms') = ListPair.unzip (map (fn aT =>
       let
         val permT = mk_permT aT;
         val pi = Free ("pi", permT);
-        val rec_fns_pi = map (mk_perm [] pi o uncurry (mk_Free "f"))
+        val rec_fns_pi = map (mk_perm [] pi o uncurry (Datatype_Aux.mk_Free "f"))
           (rec_fn_Ts ~~ (1 upto (length rec_fn_Ts)));
         val rec_sets_pi = map (fn c => list_comb (Const c, rec_fns_pi))
           (rec_set_names ~~ rec_set_Ts);
         val ps = map (fn ((((T, U), R), R'), i) =>
           let
             val x = Free ("x" ^ string_of_int i, T);
             val y = Free ("y" ^ string_of_int i, U)
           in
             (R $ x $ y, R' $ mk_perm [] pi x $ mk_perm [] pi y)
           end) (recTs ~~ rec_result_Ts ~~ rec_sets ~~ rec_sets_pi ~~ (1 upto length recTs));
-        val ths = map (fn th => Drule.export_without_context (th RS mp)) (split_conj_thm
+        val ths = map (fn th => Drule.export_without_context (th RS mp)) (Datatype_Aux.split_conj_thm
           (Goal.prove_global thy11 [] []
             (augment_sort thy1 pt_cp_sort
               (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj (map HOLogic.mk_imp ps))))
             (fn _ => rtac rec_induct 1 THEN REPEAT
                (simp_tac (Simplifier.global_context thy11 HOL_basic_ss

         val finite = Const ("Finite_Set.finite", aset --> HOLogic.boolT);
         val fins = map (fn (f, T) => HOLogic.mk_Trueprop
           (finite $ (Const ("Nominal.supp", T --> aset) $ f)))
             (rec_fns ~~ rec_fn_Ts)
       in
-        map (fn th => Drule.export_without_context (th RS mp)) (split_conj_thm
+        map (fn th => Drule.export_without_context (th RS mp)) (Datatype_Aux.split_conj_thm
           (Goal.prove_global thy11 []
             (map (augment_sort thy11 fs_cp_sort) fins)
             (augment_sort thy11 fs_cp_sort
               (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
                 (map (fn (((T, U), R), i) =>

     val finite_ctxt_prems = map (fn aT =>
       HOLogic.mk_Trueprop
         (Const ("Finite_Set.finite", HOLogic.mk_setT aT --> HOLogic.boolT) $
            (Const ("Nominal.supp", fsT' --> HOLogic.mk_setT aT) $ rec_ctxt))) dt_atomTs;
 
-    val rec_unique_thms = split_conj_thm (Goal.prove
+    val rec_unique_thms = Datatype_Aux.split_conj_thm (Goal.prove
       (Proof_Context.init_global thy11) (map fst rec_unique_frees)
       (map (augment_sort thy11 fs_cp_sort)
         (flat finite_premss @ finite_ctxt_prems @ rec_prems @ rec_prems'))
       (augment_sort thy11 fs_cp_sort
         (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj rec_unique_concls)))

            val k = length rec_fns;
            val (finite_thss, ths1) = fold_map (fn T => fn xs =>
              apfst (pair T) (chop k xs)) dt_atomTs prems;
            val (finite_ctxt_ths, ths2) = chop (length dt_atomTs) ths1;
            val (P_ind_ths, fcbs) = chop k ths2;
-           val P_ths = map (fn th => th RS mp) (split_conj_thm
+           val P_ths = map (fn th => th RS mp) (Datatype_Aux.split_conj_thm
              (Goal.prove context
                (map fst (rec_unique_frees'' @ rec_unique_frees')) []
                (augment_sort thy11 fs_cp_sort
                  (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
                     (map (fn (((x, y), S), P) => HOLogic.mk_imp