(*  Title:      HOL/Nominal/nominal.ML

     Author:     Stefan Berghofer and Christian Urban, TU Muenchen

 Nominal datatype package for Isabelle/HOL.

 *)

 signature NOMINAL =

 sig

   val add_nominal_datatype : Datatype.config -> string list ->

     (string list * bstring * mixfix *

       (bstring * string list * mixfix) list) list -> theory -> theory

   type descr

   type nominal_datatype_info

   val get_nominal_datatypes : theory -> nominal_datatype_info Symtab.table

   val get_nominal_datatype : theory -> string -> nominal_datatype_info option

   val mk_perm: typ list -> term -> term -> term

   val perm_of_pair: term * term -> term

   val mk_not_sym: thm list -> thm list

   val perm_simproc: simproc

   val fresh_const: typ -> typ -> term

   val fresh_star_const: typ -> typ -> term

 end

 structure Nominal : NOMINAL =

 struct

 val finite_emptyI = thm "finite.emptyI";

 val finite_Diff = thm "finite_Diff";

 val finite_Un = thm "finite_Un";

 val Un_iff = thm "Un_iff";

 val In0_eq = thm "In0_eq";

 val In1_eq = thm "In1_eq";

 val In0_not_In1 = thm "In0_not_In1";

 val In1_not_In0 = thm "In1_not_In0";

 val Un_assoc = thm "Un_assoc";

 val Collect_disj_eq = thm "Collect_disj_eq";

 val empty_def = thm "empty_def";

 val empty_iff = thm "empty_iff";

 open DatatypeAux;

 open NominalAtoms;

 (** FIXME: Datatype should export this function **)

 local

 fun dt_recs (DtTFree _) = []

   | dt_recs (DtType (_, dts)) = List.concat (map dt_recs dts)

   | dt_recs (DtRec i) = [i];

 fun dt_cases (descr: descr) (_, args, constrs) =

   let

     fun the_bname i = Long_Name.base_name (#1 (valOf (AList.lookup (op =) descr i)));

     val bnames = map the_bname (distinct op = (List.concat (map dt_recs args)));

   in map (fn (c, _) => space_implode "_" (Long_Name.base_name c :: bnames)) constrs end;

 fun induct_cases descr =

   DatatypeProp.indexify_names (List.concat (map (dt_cases descr) (map #2 descr)));

 fun exhaust_cases descr i = dt_cases descr (valOf (AList.lookup (op =) descr i));

 in

 fun mk_case_names_induct descr = RuleCases.case_names (induct_cases descr);

 fun mk_case_names_exhausts descr new =

   map (RuleCases.case_names o exhaust_cases descr o #1)

     (List.filter (fn ((_, (name, _, _))) => name mem_string new) descr);

 end;

 (* theory data *)

 type descr = (int * (string * dtyp list * (string * (dtyp list * dtyp) list) list)) list;

 type nominal_datatype_info =

   {index : int,

    descr : descr,

    sorts : (string * sort) list,

    rec_names : string list,

    rec_rewrites : thm list,

    induction : thm,

    distinct : thm list,

    inject : thm list};

 structure NominalDatatypesData = TheoryDataFun

 (

   type T = nominal_datatype_info Symtab.table;

   val empty = Symtab.empty;

   val copy = I;

   val extend = I;

   fun merge _ tabs : T = Symtab.merge (K true) tabs;

 );

 val get_nominal_datatypes = NominalDatatypesData.get;

 val put_nominal_datatypes = NominalDatatypesData.put;

 val map_nominal_datatypes = NominalDatatypesData.map;

 val get_nominal_datatype = Symtab.lookup o get_nominal_datatypes;

 (**** make datatype info ****)

 fun make_dt_info descr sorts induct reccomb_names rec_thms

     (((i, (_, (tname, _, _))), distinct), inject) =

   (tname,

    {index = i,

     descr = descr,

     sorts = sorts,

     rec_names = reccomb_names,

     rec_rewrites = rec_thms,

     induction = induct,

     distinct = distinct,

     inject = inject});

 (*******************************)

 val (_ $ (_ $ (_ $ (distinct_f $ _) $ _))) = hd (prems_of distinct_lemma);

 (** simplification procedure for sorting permutations **)

 val dj_cp = thm "dj_cp";

 fun dest_permT (Type ("fun", [Type ("List.list", [Type ("*", [T, _])]),

       Type ("fun", [_, U])])) = (T, U);

 fun permTs_of (Const ("Nominal.perm", T) $ t $ u) = fst (dest_permT T) :: permTs_of u

   | permTs_of _ = [];

 fun perm_simproc' thy ss (Const ("Nominal.perm", T) $ t $ (u as Const ("Nominal.perm", U) $ r $ s)) =

       let

         val (aT as Type (a, []), S) = dest_permT T;

         val (bT as Type (b, []), _) = dest_permT U

       in if aT mem permTs_of u andalso aT <> bT then

           let

             val cp = cp_inst_of thy a b;

             val dj = dj_thm_of thy b a;

             val dj_cp' = [cp, dj] MRS dj_cp;

             val cert = SOME o cterm_of thy

           in

             SOME (mk_meta_eq (Drule.instantiate' [SOME (ctyp_of thy S)]

               [cert t, cert r, cert s] dj_cp'))

           end

         else NONE

       end

   | perm_simproc' thy ss _ = NONE;

 val perm_simproc =

   Simplifier.simproc (the_context ()) "perm_simp" ["pi1 \<bullet> (pi2 \<bullet> x)"] perm_simproc';

 val meta_spec = thm "meta_spec";

 fun projections rule =

   ProjectRule.projections (ProofContext.init (Thm.theory_of_thm rule)) rule

   |> map (standard #> RuleCases.save rule);

 val supp_prod = thm "supp_prod";

 val fresh_prod = thm "fresh_prod";

 val supports_fresh = thm "supports_fresh";

 val supports_def = thm "Nominal.supports_def";

 val fresh_def = thm "fresh_def";

 val supp_def = thm "supp_def";

 val rev_simps = thms "rev.simps";

 val app_simps = thms "append.simps";

 val at_fin_set_supp = thm "at_fin_set_supp";

 val at_fin_set_fresh = thm "at_fin_set_fresh";

 val abs_fun_eq1 = thm "abs_fun_eq1";

 val collect_simp = rewrite_rule [mk_meta_eq mem_Collect_eq];

 fun mk_perm Ts t u =

   let

     val T = fastype_of1 (Ts, t);

     val U = fastype_of1 (Ts, u)

   in Const ("Nominal.perm", T --> U --> U) $ t $ u end;

 fun perm_of_pair (x, y) =

   let

     val T = fastype_of x;

     val pT = mk_permT T

   in Const ("List.list.Cons", HOLogic.mk_prodT (T, T) --> pT --> pT) $

     HOLogic.mk_prod (x, y) $ Const ("List.list.Nil", pT)

   end;

 fun mk_not_sym ths = maps (fn th => case prop_of th of

     _ $ (Const ("Not", _) $ (Const ("op =", _) $ _ $ _)) => [th, th RS not_sym]

   | _ => [th]) ths;

 fun fresh_const T U = Const ("Nominal.fresh", T --> U --> HOLogic.boolT);

 fun fresh_star_const T U =

   Const ("Nominal.fresh_star", HOLogic.mk_setT T --> U --> HOLogic.boolT);

 fun gen_add_nominal_datatype prep_typ config new_type_names dts thy =

   let

     (* this theory is used just for parsing *)

     val tmp_thy = thy |>

       Theory.copy |>

       Sign.add_types (map (fn (tvs, tname, mx, _) =>

         (Binding.name tname, length tvs, mx)) dts);

     val atoms = atoms_of thy;

     fun prep_constr ((constrs, sorts), (cname, cargs, mx)) =

       let val (cargs', sorts') = Library.foldl (prep_typ tmp_thy) (([], sorts), cargs)

       in (constrs @ [(cname, cargs', mx)], sorts') end

     fun prep_dt_spec ((dts, sorts), (tvs, tname, mx, constrs)) =

       let val (constrs', sorts') = Library.foldl prep_constr (([], sorts), constrs)

       in (dts @ [(tvs, tname, mx, constrs')], sorts') end

     val (dts', sorts) = Library.foldl prep_dt_spec (([], []), dts);

     val tyvars = map (map (fn s =>

       (s, the (AList.lookup (op =) sorts s))) o #1) dts';

     fun inter_sort thy S S' = Type.inter_sort (Sign.tsig_of thy) (S, S');

     fun augment_sort_typ thy S =

       let val S = Sign.certify_sort thy S

       in map_type_tfree (fn (s, S') => TFree (s,

         if member (op = o apsnd fst) sorts s then inter_sort thy S S' else S'))

       end;

     fun augment_sort thy S = map_types (augment_sort_typ thy S);

     val types_syntax = map (fn (tvs, tname, mx, constrs) => (tname, mx)) dts';

     val constr_syntax = map (fn (tvs, tname, mx, constrs) =>

       map (fn (cname, cargs, mx) => (cname, mx)) constrs) dts';

     val ps = map (fn (_, n, _, _) =>

       (Sign.full_bname tmp_thy n, Sign.full_bname tmp_thy (n ^ "_Rep"))) dts;

     val rps = map Library.swap ps;

     fun replace_types (Type ("Nominal.ABS", [T, U])) =

           Type ("fun", [T, Type ("Nominal.noption", [replace_types U])])

       | replace_types (Type (s, Ts)) =

           Type (getOpt (AList.lookup op = ps s, s), map replace_types Ts)

       | replace_types T = T;

     val dts'' = map (fn (tvs, tname, mx, constrs) => (tvs, Binding.name (tname ^ "_Rep"), NoSyn,

       map (fn (cname, cargs, mx) => (Binding.name (cname ^ "_Rep"),

         map replace_types cargs, NoSyn)) constrs)) dts';

     val new_type_names' = map (fn n => n ^ "_Rep") new_type_names;

     val (full_new_type_names',thy1) =

       Datatype.add_datatype config new_type_names' dts'' thy;

     val {descr, induction, ...} =

       Datatype.the_info thy1 (hd full_new_type_names');

     fun nth_dtyp i = typ_of_dtyp descr sorts (DtRec i);

     val big_name = space_implode "_" new_type_names;

     (**** define permutation functions ****)

     val permT = mk_permT (TFree ("'x", HOLogic.typeS));

     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' = DatatypeProp.indexify_names (map (fn (i, _) =>

       "perm_" ^ 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 sorts) dts;

           val names = Name.variant_list ["pi"] (DatatypeProp.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

                 let val (Us, _) = strip_type T

                 in list_abs (map (pair "x") Us,

                   Free (nth perm_names_types' (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

               else Const ("Nominal.perm", permT --> T --> T) $ pi $ x

             end;

         in

           (Attrib.empty_binding, HOLogic.mk_Trueprop (HOLogic.mk_eq

             (Free (nth perm_names_types' i) $

                Free ("pi", mk_permT (TFree ("'x", HOLogic.typeS))) $

                list_comb (c, args),

              list_comb (c, map perm_arg (dts ~~ args)))))

         end) constrs

       end) descr;

     val (perm_simps, thy2) =

       Primrec.add_primrec_overloaded

         (map (fn (s, sT) => (s, sT, false))

            (List.take (perm_names' ~~ perm_names_types, length new_type_names)))

         (map (fn s => (Binding.name s, NONE, NoSyn)) perm_names') perm_eqs thy1;

     (**** prove that permutation functions introduced by unfolding are ****)

     (**** equivalent to already existing permutation functions         ****)

     val _ = warning ("length descr: " ^ string_of_int (length descr));

     val _ = warning ("length new_type_names: " ^ string_of_int (length new_type_names));

     val perm_indnames = DatatypeProp.make_tnames (map body_type perm_types);

     val perm_fun_def = PureThy.get_thm thy2 "perm_fun_def";

     val unfolded_perm_eq_thms =

       if length descr = length new_type_names then []

       else map standard (List.drop (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),

                  Const ("Nominal.perm", T) $ pi $ Free (x, T2))

                end)

              (perm_names_types ~~ perm_indnames))))

           (fn _ => EVERY [indtac induction perm_indnames 1,

             ALLGOALS (asm_full_simp_tac

               (simpset_of thy2 addsimps [perm_fun_def]))])),

         length new_type_names));

     (**** prove [] \<bullet> t = t ****)

     val _ = warning "perm_empty_thms";

     val perm_empty_thms = List.concat (map (fn a =>

       let val permT = mk_permT (Type (a, []))

       in map standard (List.take (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) $

                      Const ("List.list.Nil", permT) $ Free (x, T),

                    Free (x, T)))

                (perm_names ~~

                 map body_type perm_types ~~ perm_indnames)))))

           (fn _ => EVERY [indtac induction perm_indnames 1,

             ALLGOALS (asm_full_simp_tac (simpset_of thy2))])),

         length new_type_names))

       end)

       atoms);

     (**** prove (pi1 @ pi2) \<bullet> t = pi1 \<bullet> (pi2 \<bullet> t) ****)

     val _ = warning "perm_append_thms";

     (*FIXME: these should be looked up statically*)

     val at_pt_inst = PureThy.get_thm thy2 "at_pt_inst";

     val pt2 = PureThy.get_thm thy2 "pt2";

     val perm_append_thms = List.concat (map (fn a =>

       let

         val permT = mk_permT (Type (a, []));

         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 = PureThy.get_thm thy2 (Long_Name.map_base_name (fn s => "pt_" ^ s ^ "2") a);

       in List.take (map standard (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)

                     in HOLogic.mk_eq

                       (perm $ (Const ("List.append", permT --> permT --> permT) $

                          pi1 $ pi2) $ Free (x, T),

                        perm $ pi1 $ (perm $ pi2 $ Free (x, T)))

                     end)

                   (perm_names ~~

                    map body_type perm_types ~~ perm_indnames)))))

            (fn _ => EVERY [indtac induction perm_indnames 1,

               ALLGOALS (asm_full_simp_tac (simpset_of thy2 addsimps [pt2', pt2_ax]))]))),

          length new_type_names)

       end) atoms);

     (**** prove pi1 ~ pi2 ==> pi1 \<bullet> t = pi2 \<bullet> t ****)

     val _ = warning "perm_eq_thms";

     val pt3 = PureThy.get_thm thy2 "pt3";

     val pt3_rev = PureThy.get_thm thy2 "pt3_rev";

     val perm_eq_thms = List.concat (map (fn a =>

       let

         val permT = mk_permT (Type (a, []));

         val pi1 = Free ("pi1", permT);

         val pi2 = Free ("pi2", permT);

         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 = PureThy.get_thm thy2 (Long_Name.map_base_name (fn s => "pt_" ^ s ^ "3") a);

       in List.take (map standard (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

                 (map (fn ((s, T), x) =>

                     let val perm = Const (s, permT --> T --> T)

                     in HOLogic.mk_eq

                       (perm $ pi1 $ Free (x, T),

                        perm $ pi2 $ Free (x, T))

                     end)

                   (perm_names ~~

                    map body_type perm_types ~~ perm_indnames))))))

            (fn _ => EVERY [indtac induction perm_indnames 1,

               ALLGOALS (asm_full_simp_tac (simpset_of thy2 addsimps [pt3', pt3_rev', pt3_ax]))]))),

          length new_type_names)

       end) atoms);

     (**** prove pi1 \<bullet> (pi2 \<bullet> t) = (pi1 \<bullet> pi2) \<bullet> (pi1 \<bullet> t) ****)

     val cp1 = PureThy.get_thm thy2 "cp1";

     val dj_cp = PureThy.get_thm thy2 "dj_cp";

     val pt_perm_compose = PureThy.get_thm thy2 "pt_perm_compose";

     val pt_perm_compose_rev = PureThy.get_thm thy2 "pt_perm_compose_rev";

     val dj_perm_perm_forget = PureThy.get_thm thy2 "dj_perm_perm_forget";

     fun composition_instance name1 name2 thy =

       let

         val cp_class = cp_class_of thy name1 name2;

         val pt_class =

           if name1 = name2 then [pt_class_of thy name1]

           else [];

         val permT1 = mk_permT (Type (name1, []));

         val permT2 = mk_permT (Type (name2, []));

         val Ts = map body_type perm_types;

         val cp_inst = cp_inst_of thy name1 name2;

         val simps = simpset_of thy addsimps (perm_fun_def ::

           (if name1 <> name2 then

              let val dj = dj_thm_of thy name2 name1

              in [dj RS (cp_inst RS dj_cp), dj RS dj_perm_perm_forget] end

            else

              let

                val at_inst = at_inst_of thy name1;

                val pt_inst = pt_inst_of thy name1;

              in

                [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.certify_sort thy (cp_class :: pt_class);

         val thms = 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);

                     val pi2 = Free ("pi2", permT2);

                     val perm1 = Const (s, permT1 --> T --> T);

                     val perm2 = Const (s, permT2 --> T --> T);

                     val perm3 = Const ("Nominal.perm", permT1 --> permT2 --> permT2)

                   in HOLogic.mk_eq

                     (perm1 $ pi1 $ (perm2 $ pi2 $ Free (x, T)),

                      perm2 $ (perm3 $ pi1 $ pi2) $ (perm1 $ pi1 $ Free (x, T)))

                   end)

                 (perm_names ~~ Ts ~~ perm_indnames)))))

           (fn _ => EVERY [indtac induction perm_indnames 1,

              ALLGOALS (asm_full_simp_tac simps)]))

       in

         fold (fn (s, tvs) => fn thy => AxClass.prove_arity

             (s, map (inter_sort thy sort o snd) tvs, [cp_class])

             (Class.intro_classes_tac [] THEN ALLGOALS (resolve_tac thms)) thy)

           (full_new_type_names' ~~ tyvars) thy

       end;

     val (perm_thmss,thy3) = thy2 |>

       fold (fn name1 => fold (composition_instance name1) atoms) atoms |>

       fold (fn atom => fn thy =>

         let val pt_name = pt_class_of thy atom

         in

           fold (fn (s, tvs) => fn thy => AxClass.prove_arity

               (s, map (inter_sort thy [pt_name] o snd) tvs, [pt_name])

               (EVERY

                 [Class.intro_classes_tac [],

                  resolve_tac perm_empty_thms 1,

                  resolve_tac perm_append_thms 1,

                  resolve_tac perm_eq_thms 1, assume_tac 1]) thy)

             (full_new_type_names' ~~ tyvars) thy

         end) atoms |>

       PureThy.add_thmss

         [((Binding.name (space_implode "_" new_type_names ^ "_unfolded_perm_eq"),

           unfolded_perm_eq_thms), [Simplifier.simp_add]),

          ((Binding.name (space_implode "_" new_type_names ^ "_perm_empty"),

           perm_empty_thms), [Simplifier.simp_add]),

          ((Binding.name (space_implode "_" new_type_names ^ "_perm_append"),

           perm_append_thms), [Simplifier.simp_add]),

          ((Binding.name (space_implode "_" new_type_names ^ "_perm_eq"),

           perm_eq_thms), [Simplifier.simp_add])];

     (**** Define representing sets ****)

     val _ = warning "representing sets";

     val rep_set_names = DatatypeProp.indexify_names

       (map (fn (i, _) => name_of_typ (nth_dtyp i) ^ "_set") descr);

     val big_rep_name =

       space_implode "_" (DatatypeProp.indexify_names (List.mapPartial

         (fn (i, ("Nominal.noption", _, _)) => NONE

           | (i, _) => SOME (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])) =

           (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'])])) =

           apfst (cons dt) (strip_option dt')

       | strip_option dt = ([], dt);

     val dt_atomTs = distinct op = (map (typ_of_dtyp descr sorts)

       (List.concat (map (fn (_, (_, _, cs)) => List.concat

         (map (List.concat o map (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 sorts) dts;

             val Us = map (typ_of_dtyp descr sorts) dts';

             val T = typ_of_dtyp descr sorts dt'';

             val free = mk_Free "x" (Us ---> T) j;

             val free' = 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)

               end

           in (j + 1, j' + length Ts,

             case dt'' of

                 DtRec k => list_all (map (pair "x") Us,

                   HOLogic.mk_Trueprop (Free (List.nth (rep_set_names, k),

                     T --> HOLogic.boolT) $ free')) :: prems

               | _ => prems,

             snd (List.foldr mk_abs_fun (j', free) Ts) :: ts)

           end;

         val (_, _, prems, ts) = List.foldr mk_prem (1, 1, [], []) cargs;

         val concl = HOLogic.mk_Trueprop (Free (s, T --> HOLogic.boolT) $

           list_comb (Const (cname, map fastype_of ts ---> T), ts))

       in Logic.list_implies (prems, concl)

       end;

     val (intr_ts, (rep_set_names', recTs')) =

       apfst List.concat (apsnd ListPair.unzip (ListPair.unzip (List.mapPartial

         (fn ((_, ("Nominal.noption", _, _)), _) => NONE

           | ((i, (_, _, constrs)), rep_set_name) =>

               let val T = nth_dtyp i

               in SOME (map (make_intr rep_set_name T) constrs,

                 (rep_set_name, T))

               end)

                 (descr ~~ rep_set_names))));

     val rep_set_names'' = map (Sign.full_bname thy3) rep_set_names';

     val ({raw_induct = rep_induct, intrs = rep_intrs, ...}, thy4) =

         Inductive.add_inductive_global (serial_string ())

           {quiet_mode = false, verbose = false, kind = Thm.internalK,

            alt_name = Binding.name big_rep_name, coind = false, no_elim = true, no_ind = false,

            skip_mono = true, fork_mono = false}

           (map (fn (s, T) => ((Binding.name s, T --> HOLogic.boolT), NoSyn))

              (rep_set_names' ~~ recTs'))

           [] (map (fn x => (Attrib.empty_binding, x)) intr_ts) [] thy3;

     (**** Prove that representing set is closed under permutation ****)

     val _ = warning "proving closure under permutation...";

     val abs_perm = PureThy.get_thms thy4 "abs_perm";

     val perm_indnames' = List.mapPartial

       (fn (x, (_, ("Nominal.noption", _, _))) => NONE | (x, _) => SOME x)

       (perm_indnames ~~ descr);

     fun mk_perm_closed name = map (fn th => standard (th RS mp))

       (List.take (split_conj_thm (Goal.prove_global thy4 [] []

         (augment_sort thy4

           (pt_class_of thy4 name :: map (cp_class_of thy4 name) (dt_atoms \ name))

           (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj (map

             (fn ((s, T), x) =>

                let

                  val S = Const (s, T --> HOLogic.boolT);

                  val permT = mk_permT (Type (name, []))

                in HOLogic.mk_imp (S $ Free (x, T),

                  S $ (Const ("Nominal.perm", permT --> T --> T) $

                    Free ("pi", permT) $ Free (x, T)))

                end) (rep_set_names'' ~~ recTs' ~~ perm_indnames')))))

         (fn _ => EVERY

            [indtac rep_induct [] 1,

             ALLGOALS (simp_tac (simpset_of thy4 addsimps

               (symmetric perm_fun_def :: abs_perm))),

             ALLGOALS (resolve_tac rep_intrs THEN_ALL_NEW assume_tac)])),

         length new_type_names));

     val perm_closed_thmss = map mk_perm_closed atoms;

     (**** typedef ****)

     val _ = warning "defining type...";

     val (typedefs, thy6) =

       thy4

       |> fold_map (fn ((((name, mx), tvs), (cname, U)), name') => fn thy =>

           Typedef.add_typedef false (SOME (Binding.name name'))

             (Binding.name name, map fst tvs, mx)

             (Const ("Collect", (U --> HOLogic.boolT) --> HOLogic.mk_setT U) $

                Const (cname, U --> HOLogic.boolT)) NONE

             (rtac exI 1 THEN rtac CollectI 1 THEN

               QUIET_BREADTH_FIRST (has_fewer_prems 1)

               (resolve_tac rep_intrs 1)) thy |> (fn ((_, r), thy) =>

         let

           val permT = mk_permT

             (TFree (Name.variant (map fst tvs) "'a", HOLogic.typeS));

           val pi = Free ("pi", permT);

           val T = Type (Sign.intern_type thy name, map TFree tvs);

         in apfst (pair r o hd)

           (PureThy.add_defs_unchecked true [((Binding.name ("prm_" ^ name ^ "_def"), Logic.mk_equals

             (Const ("Nominal.perm", permT --> T --> T) $ pi $ Free ("x", T),

              Const (Sign.intern_const thy ("Abs_" ^ name), U --> T) $

                (Const ("Nominal.perm", permT --> U --> U) $ pi $

                  (Const (Sign.intern_const thy ("Rep_" ^ name), T --> U) $

                    Free ("x", T))))), [])] thy)

         end))

           (types_syntax ~~ tyvars ~~

             List.take (rep_set_names'' ~~ recTs', length new_type_names) ~~

             new_type_names);

     val perm_defs = map snd typedefs;

     val Abs_inverse_thms = map (collect_simp o #Abs_inverse o fst) typedefs;

     val Rep_inverse_thms = map (#Rep_inverse o fst) typedefs;

     val Rep_thms = map (collect_simp o #Rep o fst) typedefs;

     (** prove that new types are in class pt_<name> **)

     val _ = warning "prove that new types are in class pt_<name> ...";

     fun pt_instance (atom, perm_closed_thms) =

       fold (fn ((((((Abs_inverse, Rep_inverse), Rep),

         perm_def), name), tvs), perm_closed) => fn thy =>

           let

             val pt_class = pt_class_of thy atom;

             val sort = Sign.certify_sort thy

               (pt_class :: map (cp_class_of thy atom) (dt_atoms \ atom))

           in AxClass.prove_arity

             (Sign.intern_type thy name,

               map (inter_sort thy sort o snd) tvs, [pt_class])

             (EVERY [Class.intro_classes_tac [],

               rewrite_goals_tac [perm_def],

               asm_full_simp_tac (simpset_of thy addsimps [Rep_inverse]) 1,

               asm_full_simp_tac (simpset_of thy addsimps

                 [Rep RS perm_closed RS Abs_inverse]) 1,

               asm_full_simp_tac (HOL_basic_ss addsimps [PureThy.get_thm thy

                 ("pt_" ^ Long_Name.base_name atom ^ "3")]) 1]) thy

           end)

         (Abs_inverse_thms ~~ Rep_inverse_thms ~~ Rep_thms ~~ perm_defs ~~

            new_type_names ~~ tyvars ~~ perm_closed_thms);

     (** prove that new types are in class cp_<name1>_<name2> **)

     val _ = warning "prove that new types are in class cp_<name1>_<name2> ...";

     fun cp_instance (atom1, perm_closed_thms1) (atom2, perm_closed_thms2) thy =

       let

         val cp_class = cp_class_of thy atom1 atom2;

         val sort = Sign.certify_sort thy

           (pt_class_of thy atom1 :: map (cp_class_of thy atom1) (dt_atoms \ atom1) @

            (if atom1 = atom2 then [cp_class_of thy atom1 atom1] else

             pt_class_of thy atom2 :: map (cp_class_of thy atom2) (dt_atoms \ atom2)));

         val cp1' = cp_inst_of thy atom1 atom2 RS cp1

       in fold (fn ((((((Abs_inverse, Rep),

         perm_def), name), tvs), perm_closed1), perm_closed2) => fn thy =>

           AxClass.prove_arity

             (Sign.intern_type thy name,

               map (inter_sort thy sort o snd) tvs, [cp_class])

             (EVERY [Class.intro_classes_tac [],

               rewrite_goals_tac [perm_def],

               asm_full_simp_tac (simpset_of thy addsimps

                 ((Rep RS perm_closed1 RS Abs_inverse) ::

                  (if atom1 = atom2 then []

                   else [Rep RS perm_closed2 RS Abs_inverse]))) 1,

               cong_tac 1,

               rtac refl 1,

               rtac cp1' 1]) thy)

         (Abs_inverse_thms ~~ Rep_thms ~~ perm_defs ~~ new_type_names ~~

            tyvars ~~ perm_closed_thms1 ~~ perm_closed_thms2) thy

       end;

     val thy7 = fold (fn x => fn thy => thy |>

       pt_instance x |>

       fold (cp_instance x) (atoms ~~ perm_closed_thmss))

         (atoms ~~ perm_closed_thmss) thy6;

     (**** constructors ****)

     fun mk_abs_fun (x, t) =

       let

         val T = fastype_of x;

         val U = fastype_of t

       in

         Const ("Nominal.abs_fun", T --> U --> T -->

           Type ("Nominal.noption", [U])) $ x $ t

       end;

     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))

       | reindex dt = dt;

     fun strip_suffix i s = implode (List.take (explode s, size s - i));

     (** strips the "_Rep" in type names *)

     fun strip_nth_name i s =

       let val xs = Long_Name.explode s;

       in Long_Name.implode (Library.nth_map (length xs - i) (strip_suffix 4) xs) end;

     val (descr'', ndescr) = ListPair.unzip (map_filter

       (fn (i, ("Nominal.noption", _, _)) => NONE

         | (i, (s, dts, constrs)) =>

              let

                val SOME index = AList.lookup op = ty_idxs i;

                val (constrs2, constrs1) =

                  map_split (fn (cname, cargs) =>

                    apsnd (pair (strip_nth_name 2 (strip_nth_name 1 cname)))

                    (fold_map (fn dt => fn dts =>

                      let val (dts', dt') = strip_option dt

                      in ((length dts, length dts'), dts @ dts' @ [reindex dt']) end)

                        cargs [])) constrs

              in SOME ((index, (strip_nth_name 1 s,  map reindex dts, constrs1)),

                (index, constrs2))

              end) descr);

     val (descr1, descr2) = chop (length new_type_names) descr'';

     val descr' = [descr1, descr2];

     fun partition_cargs idxs xs = map (fn (i, j) =>

       (List.take (List.drop (xs, i), j), List.nth (xs, i + j))) idxs;

     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'' sorts (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'' sorts;

     val newTs' = Library.take (length new_type_names, recTs');

     val newTs = Library.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' ((thy, defs, eqns),

         (((cname_rep, _), (cname, cargs)), (cname', mx))) =

       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'' sorts dt) i)

               (dts ~~ (j upto j + length dts - 1))

             val x = mk_Free "x" (typ_of_dtyp descr'' sorts dt) (j + length dts)

           in

             (j + length dts + 1,

              xs @ x :: l_args,

              List.foldr mk_abs_fun

                (case dt of

                   DtRec k => if k < length new_type_names then

                       Const (List.nth (rep_names, k), typ_of_dtyp descr'' sorts dt -->

                         typ_of_dtyp descr sorts dt) $ x

                     else error "nested recursion not (yet) supported"

                 | _ => x) xs :: r_args)

           end

         val (_, l_args, r_args) = List.foldr constr_arg (1, [], []) cargs;

         val abs_name = Sign.intern_const thy ("Abs_" ^ tname);

         val rep_name = Sign.intern_const thy ("Rep_" ^ tname);

         val constrT = map fastype_of l_args ---> T;

         val lhs = list_comb (Const (cname, constrT), l_args);

         val rhs = list_comb (Const (cname_rep, map fastype_of r_args ---> T'), r_args);

         val def = Logic.mk_equals (lhs, Const (abs_name, T' --> T) $ rhs);

         val eqn = HOLogic.mk_Trueprop (HOLogic.mk_eq

           (Const (rep_name, T --> T') $ lhs, rhs));

         val def_name = (Long_Name.base_name cname) ^ "_def";

         val ([def_thm], thy') = thy |>

           Sign.add_consts_i [(Binding.name cname', constrT, mx)] |>

           (PureThy.add_defs false o map Thm.no_attributes) [(Binding.name def_name, def)]

       in (thy', defs @ [def_thm], eqns @ [eqn]) end;

     fun dt_constr_defs ((thy, defs, eqns, dist_lemmas), ((((((_, (_, _, constrs)),

         (_, (_, _, constrs'))), tname), T), T'), constr_syntax)) =

       let

         val rep_const = cterm_of thy

           (Const (Sign.intern_const thy ("Rep_" ^ tname), T --> T'));

         val dist = standard (cterm_instantiate [(cterm_of thy distinct_f, rep_const)] distinct_lemma);

         val (thy', defs', eqns') = Library.foldl (make_constr_def tname T T')

           ((Sign.add_path tname thy, defs, []), constrs ~~ constrs' ~~ constr_syntax)

       in

         (parent_path (#flat_names config) thy', defs', eqns @ [eqns'], dist_lemmas @ [dist])

       end;

     val (thy8, constr_defs, constr_rep_eqns, dist_lemmas) = Library.foldl dt_constr_defs

       ((thy7, [], [], []), List.take (descr, length new_type_names) ~~

         List.take (pdescr, length new_type_names) ~~

         new_type_names ~~ newTs ~~ newTs' ~~ constr_syntax);

     val abs_inject_thms = map (collect_simp o #Abs_inject o fst) typedefs

     val rep_inject_thms = map (#Rep_inject o fst) typedefs

     (* prove theorem  Rep_i (Constr_j ...) = Constr'_j ...  *)

     fun prove_constr_rep_thm eqn =

       let

         val inj_thms = map (fn r => r RS iffD1) abs_inject_thms;

         val rewrites = constr_defs @ map mk_meta_eq Rep_inverse_thms

       in Goal.prove_global thy8 [] [] eqn (fn _ => EVERY

         [resolve_tac inj_thms 1,

          rewrite_goals_tac rewrites,

          rtac refl 3,

          resolve_tac rep_intrs 2,

          REPEAT (resolve_tac Rep_thms 1)])

       end;

     val constr_rep_thmss = map (map prove_constr_rep_thm) constr_rep_eqns;

     (* prove theorem  pi \<bullet> Rep_i x = Rep_i (pi \<bullet> x) *)

     fun prove_perm_rep_perm (atom, perm_closed_thms) = map (fn th =>

       let

         val _ $ (_ $ (Rep $ x)) = Logic.unvarify (prop_of th);

         val Type ("fun", [T, U]) = fastype_of Rep;

         val permT = mk_permT (Type (atom, []));

         val pi = Free ("pi", permT);

       in

         Goal.prove_global thy8 [] []

           (augment_sort thy8

             (pt_class_of thy8 atom :: map (cp_class_of thy8 atom) (dt_atoms \ atom))

             (HOLogic.mk_Trueprop (HOLogic.mk_eq

               (Const ("Nominal.perm", permT --> U --> U) $ pi $ (Rep $ x),

                Rep $ (Const ("Nominal.perm", permT --> T --> T) $ pi $ x)))))

           (fn _ => simp_tac (HOL_basic_ss addsimps (perm_defs @ Abs_inverse_thms @

             perm_closed_thms @ Rep_thms)) 1)

       end) Rep_thms;

     val perm_rep_perm_thms = List.concat (map prove_perm_rep_perm

       (atoms ~~ perm_closed_thmss));

     (* prove distinctness theorems *)

     val distinct_props = DatatypeProp.make_distincts descr' sorts;

     val dist_rewrites = map2 (fn rep_thms => fn dist_lemma =>

       dist_lemma :: rep_thms @ [In0_eq, In1_eq, In0_not_In1, In1_not_In0])

         constr_rep_thmss dist_lemmas;

     fun prove_distinct_thms _ (_, []) = []

       | prove_distinct_thms (p as (rep_thms, dist_lemma)) (k, t :: ts) =

           let

             val dist_thm = Goal.prove_global thy8 [] [] t (fn _ =>

               simp_tac (simpset_of thy8 addsimps (dist_lemma :: rep_thms)) 1)

           in dist_thm :: standard (dist_thm RS not_sym) ::

             prove_distinct_thms p (k, ts)

           end;

     val distinct_thms = map2 prove_distinct_thms

       (constr_rep_thmss ~~ dist_lemmas) distinct_props;

     (** prove equations for permutation functions **)

     val perm_simps' = map (fn (((i, (_, _, constrs)), tname), constr_rep_thms) =>

       let val T = nth_dtyp' i

       in List.concat (map (fn (atom, perm_closed_thms) =>

           map (fn ((cname, dts), constr_rep_thm) =>

         let

           val cname = Sign.intern_const thy8

             (Long_Name.append tname (Long_Name.base_name cname));

           val permT = mk_permT (Type (atom, []));

           val pi = Free ("pi", permT);

           fun perm t =

             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'' sorts) 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'' sorts dt) (j + length dts)

             in

               (j + length dts + 1,

                xs @ x :: l_args,

                map perm (xs @ [x]) @ r_args)

             end

           val (_, l_args, r_args) = List.foldr constr_arg (1, [], []) dts;

           val c = Const (cname, map fastype_of l_args ---> T)

         in

           Goal.prove_global thy8 [] []

             (augment_sort thy8

               (pt_class_of thy8 atom :: map (cp_class_of thy8 atom) (dt_atoms \ atom))

               (HOLogic.mk_Trueprop (HOLogic.mk_eq

                 (perm (list_comb (c, l_args)), list_comb (c, r_args)))))

             (fn _ => EVERY

               [simp_tac (simpset_of thy8 addsimps (constr_rep_thm :: perm_defs)) 1,

                simp_tac (HOL_basic_ss addsimps (Rep_thms @ Abs_inverse_thms @

                  constr_defs @ perm_closed_thms)) 1,

                TRY (simp_tac (HOL_basic_ss addsimps

                  (symmetric perm_fun_def :: abs_perm)) 1),

                TRY (simp_tac (HOL_basic_ss addsimps

                  (perm_fun_def :: perm_defs @ Rep_thms @ Abs_inverse_thms @

                     perm_closed_thms)) 1)])

         end) (constrs ~~ constr_rep_thms)) (atoms ~~ perm_closed_thmss))

       end) (List.take (pdescr, length new_type_names) ~~ new_type_names ~~ constr_rep_thmss);

     (** prove injectivity of constructors **)

     val rep_inject_thms' = map (fn th => th RS sym) rep_inject_thms;

     val alpha = PureThy.get_thms thy8 "alpha";

     val abs_fresh = PureThy.get_thms thy8 "abs_fresh";

     val pt_cp_sort =

       map (pt_class_of thy8) dt_atoms @

       maps (fn s => map (cp_class_of thy8 s) (dt_atoms \ s)) dt_atoms;

     val inject_thms = map (fn (((i, (_, _, constrs)), tname), constr_rep_thms) =>

       let val T = nth_dtyp' i

       in List.mapPartial (fn ((cname, dts), constr_rep_thm) =>

         if null dts then NONE else SOME

         let

           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'' sorts) 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'' sorts dt) (j + length dts);

               val y = mk_Free "y" (typ_of_dtyp descr'' sorts dt) (j + length dts)

             in

               (j + length dts + 1,

                xs @ (x :: args1), ys @ (y :: args2),

                HOLogic.mk_eq

                  (List.foldr mk_abs_fun x xs, List.foldr mk_abs_fun y ys) :: eqs)

             end;

           val (_, args1, args2, eqs) = List.foldr make_inj (1, [], [], []) dts;

           val Ts = map fastype_of args1;

           val c = Const (cname, Ts ---> T)

         in

           Goal.prove_global thy8 [] []

             (augment_sort thy8 pt_cp_sort

               (HOLogic.mk_Trueprop (HOLogic.mk_eq

                 (HOLogic.mk_eq (list_comb (c, args1), list_comb (c, args2)),

                  foldr1 HOLogic.mk_conj eqs))))

             (fn _ => EVERY

                [asm_full_simp_tac (simpset_of thy8 addsimps (constr_rep_thm ::

                   rep_inject_thms')) 1,

                 TRY (asm_full_simp_tac (HOL_basic_ss addsimps (fresh_def :: supp_def ::

                   alpha @ abs_perm @ abs_fresh @ rep_inject_thms @

                   perm_rep_perm_thms)) 1)])

         end) (constrs ~~ constr_rep_thms)

       end) (List.take (pdescr, length new_type_names) ~~ new_type_names ~~ constr_rep_thmss);

     (** equations for support and freshness **)

     val (supp_thms, fresh_thms) = ListPair.unzip (map ListPair.unzip

       (map (fn ((((i, (_, _, constrs)), tname), inject_thms'), perm_thms') =>

       let val T = nth_dtyp' i

       in List.concat (map (fn (cname, dts) => map (fn atom =>

         let

           val cname = Sign.intern_const thy8

             (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'' sorts) 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'' sorts dt) (j + length dts)

             in

               (j + length dts + 1,

                xs @ (x :: args1), List.foldr mk_abs_fun x xs :: args2)

             end;

           val (_, args1, args2) = List.foldr process_constr (1, [], []) dts;

           val Ts = map fastype_of args1;

           val c = list_comb (Const (cname, Ts ---> T), args1);

           fun supp t =

             Const ("Nominal.supp", fastype_of t --> HOLogic.mk_setT atomT) $ t;

           fun fresh t = fresh_const atomT (fastype_of t) $ Free ("a", atomT) $ t;

           val supp_thm = Goal.prove_global thy8 [] []

             (augment_sort thy8 pt_cp_sort

               (HOLogic.mk_Trueprop (HOLogic.mk_eq

                 (supp c,

                  if null dts then HOLogic.mk_set atomT []

                  else foldr1 (HOLogic.mk_binop @{const_name Un}) (map supp args2)))))

             (fn _ =>

               simp_tac (HOL_basic_ss addsimps (supp_def ::

                  Un_assoc :: de_Morgan_conj :: Collect_disj_eq :: finite_Un ::

                  symmetric empty_def :: finite_emptyI :: simp_thms @

                  abs_perm @ abs_fresh @ inject_thms' @ perm_thms')) 1)

         in

           (supp_thm,

            Goal.prove_global thy8 [] [] (augment_sort thy8 pt_cp_sort

              (HOLogic.mk_Trueprop (HOLogic.mk_eq

                (fresh c,

                 if null dts then HOLogic.true_const

                 else foldr1 HOLogic.mk_conj (map fresh args2)))))

              (fn _ =>

                simp_tac (HOL_ss addsimps [Un_iff, empty_iff, fresh_def, supp_thm]) 1))

         end) atoms) constrs)

       end) (List.take (pdescr, length new_type_names) ~~ new_type_names ~~ inject_thms ~~ perm_simps')));

     (**** weak induction theorem ****)

     fun mk_indrule_lemma ((prems, concls), (((i, _), T), U)) =

       let

         val Rep_t = Const (List.nth (rep_names, i), T --> U) $

           mk_Free "x" T i;

         val Abs_t =  Const (List.nth (abs_names, i), U --> T)

       in (prems @ [HOLogic.imp $

             (Const (List.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])

       end;

     val (indrule_lemma_prems, indrule_lemma_concls) =

       Library.foldl 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

            [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 Ps = map head_of (HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of indrule_lemma)));

     val frees = if length Ps = 1 then [Free ("P", snd (dest_Var (hd Ps)))] else

       map (Free o apfst fst o dest_Var) Ps;

     val indrule_lemma' = cterm_instantiate

       (map (cterm_of thy8) Ps ~~ map (cterm_of thy8) frees) indrule_lemma;

     val Abs_inverse_thms' = map (fn r => r RS subst) Abs_inverse_thms;

     val dt_induct_prop = DatatypeProp.make_ind descr' sorts;

     val dt_induct = Goal.prove_global thy8 []

       (Logic.strip_imp_prems dt_induct_prop) (Logic.strip_imp_concl dt_induct_prop)

       (fn {prems, ...} => EVERY

         [rtac indrule_lemma' 1,

          (indtac rep_induct [] THEN_ALL_NEW ObjectLogic.atomize_prems_tac) 1,

          EVERY (map (fn (prem, r) => (EVERY

            [REPEAT (eresolve_tac Abs_inverse_thms' 1),

             simp_tac (HOL_basic_ss addsimps [symmetric r]) 1,

             DEPTH_SOLVE_1 (ares_tac [prem] 1 ORELSE etac allE 1)]))

                 (prems ~~ constr_defs))]);

     val case_names_induct = mk_case_names_induct descr'';

     (**** prove that new datatypes have finite support ****)

     val _ = warning "proving finite support for the new datatype";

     val indnames = DatatypeProp.make_tnames recTs;

     val abs_supp = PureThy.get_thms thy8 "abs_supp";

     val supp_atm = PureThy.get_thms thy8 "supp_atm";

     val finite_supp_thms = map (fn atom =>

       let val atomT = Type (atom, [])

       in map standard (List.take

         (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)))

                    (indnames ~~ recTs)))))

            (fn _ => indtac dt_induct indnames 1 THEN

             ALLGOALS (asm_full_simp_tac (simpset_of thy8 addsimps

               (abs_supp @ supp_atm @

                PureThy.get_thms thy8 ("fs_" ^ Long_Name.base_name atom ^ "1") @

                List.concat supp_thms))))),

          length new_type_names))

       end) atoms;

     val simp_atts = replicate (length new_type_names) [Simplifier.simp_add];

 	(* Function to add both the simp and eqvt attributes *)

         (* These two attributes are duplicated on all the types in the mutual nominal datatypes *)

     val simp_eqvt_atts = replicate (length new_type_names) [Simplifier.simp_add, NominalThmDecls.eqvt_add];

     val (_, thy9) = thy8 |>

       Sign.add_path big_name |>

       PureThy.add_thms [((Binding.name "induct", dt_induct), [case_names_induct])] ||>>

       PureThy.add_thmss [((Binding.name "inducts", projections dt_induct), [case_names_induct])] ||>

       Sign.parent_path ||>>

       DatatypeAux.store_thmss_atts "distinct" new_type_names simp_atts distinct_thms ||>>

       DatatypeAux.store_thmss "constr_rep" new_type_names constr_rep_thmss ||>>

       DatatypeAux.store_thmss_atts "perm" new_type_names simp_eqvt_atts perm_simps' ||>>

       DatatypeAux.store_thmss "inject" new_type_names inject_thms ||>>

       DatatypeAux.store_thmss "supp" new_type_names supp_thms ||>>

       DatatypeAux.store_thmss_atts "fresh" new_type_names simp_atts fresh_thms ||>

       fold (fn (atom, ths) => fn thy =>

         let

           val class = fs_class_of thy atom;

           val sort = Sign.certify_sort thy (class :: pt_cp_sort)

         in fold (fn Type (s, Ts) => AxClass.prove_arity

           (s, map (inter_sort thy sort o snd o dest_TFree) Ts, [class])

           (Class.intro_classes_tac [] THEN resolve_tac ths 1)) newTs thy

         end) (atoms ~~ finite_supp_thms);