(*  Title:      HOL/Nominal/nominal_datatype.ML

     Author:     Stefan Berghofer and Christian Urban, TU Muenchen

 Nominal datatype package for Isabelle/HOL.

 *)

 signature NOMINAL_DATATYPE =

 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 NominalDatatype : NOMINAL_DATATYPE =

 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 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) =

   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;

 fun induct_cases descr =

   Datatype_Prop.indexify_names (maps (dt_cases descr) (map #2 descr));

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

 in

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

 fun mk_case_names_exhausts descr new =

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

     (filter (fn ((_, (name, _, _))) => member (op =) new name) 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 = Theory_Data

 (

   type T = nominal_datatype_info Symtab.table;

   val empty = Symtab.empty;

   val extend = I;

   fun merge data = Symtab.merge (K true) data;

 );

 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 (@{type_name Product_Type.prod}, [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 member (op =) (permTs_of u) aT 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_global @{theory} "perm_simp" ["pi1 \<bullet> (pi2 \<bullet> x)"] perm_simproc';

 fun projections rule =

   Project_Rule.projections (Proof_Context.init_global (Thm.theory_of_thm rule)) rule

   |> map (Drule.export_without_context #> Rule_Cases.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 (@{const_name Not}, _) $ (Const (@{const_name HOL.eq}, _) $ _ $ _)) => [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_global (map (fn (tvs, tname, mx, _) =>

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

     val atoms = atoms_of thy;

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

       let val (cargs', sorts') = fold_map (prep_typ tmp_thy) cargs sorts

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

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

       let val (constrs', sorts') = fold prep_constr constrs ([], sorts)

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

     val (dts', sorts) = fold 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' = Sign.inter_sort thy (S, S');

     fun augment_sort_typ thy S =

       let val S = Sign.minimize_sort thy (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 (the_default s (AList.lookup op = ps 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, induct, ...} =

       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' = Datatype_Prop.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"] (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

                 let val Us = binder_types 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 = Datatype_Prop.make_tnames (map body_type perm_types);

     val perm_fun_def = Global_Theory.get_thm thy2 "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

         (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 induct perm_indnames 1,

             ALLGOALS (asm_full_simp_tac

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

         length new_type_names));

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

     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

         (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 induct perm_indnames 1,

             ALLGOALS (asm_full_simp_tac (global_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 = Global_Theory.get_thm thy2 "at_pt_inst";

     val pt2 = Global_Theory.get_thm thy2 "pt2";

     val perm_append_thms = maps (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 = 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

         (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 induct perm_indnames 1,

               ALLGOALS (asm_full_simp_tac (global_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 = Global_Theory.get_thm thy2 "pt3";

     val pt3_rev = Global_Theory.get_thm thy2 "pt3_rev";

     val perm_eq_thms = maps (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 = 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

         (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 induct perm_indnames 1,

               ALLGOALS (asm_full_simp_tac (global_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 = Global_Theory.get_thm thy2 "cp1";

     val dj_cp = Global_Theory.get_thm thy2 "dj_cp";

     val pt_perm_compose = Global_Theory.get_thm thy2 "pt_perm_compose";

     val pt_perm_compose_rev = Global_Theory.get_thm thy2 "pt_perm_compose_rev";

     val dj_perm_perm_forget = Global_Theory.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 = global_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.minimize_sort thy (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 induct 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 |>

       Global_Theory.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 = Datatype_Prop.indexify_names

       (map (fn (i, _) => 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";

     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)

       (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 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 (nth rep_set_names k,

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

               | _ => prems,

             snd (fold_rev mk_abs_fun Ts (j', free)) :: ts)

           end;

         val (_, _, prems, ts) = fold_rev mk_prem cargs (1, 1, [], []);

         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 flat (apsnd ListPair.unzip (ListPair.unzip (map_filter

         (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) =

       thy3

       |> Sign.map_naming Name_Space.conceal

       |> Inductive.add_inductive_global

           {quiet_mode = false, verbose = false, 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) []

       ||> Sign.restore_naming thy3;

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

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

     val abs_perm = Global_Theory.get_thms thy4 "abs_perm";

     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 [] []

         (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 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 (global_simpset_of thy4 addsimps

               (Thm.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_global false (SOME (Binding.name name'))

             (Binding.name name, map (fn (v, _) => (v, dummyS)) tvs, mx)  (* FIXME keep constraints!? *)

             (Const (@{const_name 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 (singleton (Name.variant_list (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)

           (Global_Theory.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 snd o fst) typedefs;

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

     val Rep_thms = map (collect_simp o #Rep o snd 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.minimize_sort thy (Sign.certify_sort thy

               (pt_class :: map (cp_class_of thy atom) (remove (op =) atom dt_atoms)))

           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 (global_simpset_of thy addsimps [Rep_inverse]) 1,

               asm_full_simp_tac (global_simpset_of thy addsimps

                 [Rep RS perm_closed RS Abs_inverse]) 1,

               asm_full_simp_tac (HOL_basic_ss addsimps [Global_Theory.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.minimize_sort thy (Sign.certify_sort thy

           (pt_class_of thy atom1 :: map (cp_class_of thy atom1) (remove (op =) atom1 dt_atoms) @

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

             pt_class_of thy atom2 :: map (cp_class_of thy atom2) (remove (op =) atom2 dt_atoms))));

         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 (global_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 (raw_explode s, size s - i));  (* FIXME Symbol.explode (?) *)

     (** 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), 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' = 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'' 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,

              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'' sorts dt -->

                         typ_of_dtyp descr sorts dt) $ x

                     else error "nested recursion not (yet) supported"

                 | _ => x) :: r_args)

           end

         val (_, l_args, r_args) = fold_rev constr_arg cargs (1, [], []);

         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)] |>

           (Global_Theory.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 ((((((_, (_, _, constrs)),

         (_, (_, _, constrs'))), tname), T), T'), constr_syntax) (thy, defs, eqns, dist_lemmas) =

       let

         val rep_const = cterm_of thy

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

         val dist =

           Drule.export_without_context

             (cterm_instantiate [(cterm_of thy distinct_f, rep_const)] distinct_lemma);

         val (thy', defs', eqns') = fold (make_constr_def tname T T')

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

       in

         (Sign.parent_path thy', defs', eqns @ [eqns'], dist_lemmas @ [dist])

       end;

     val (thy8, constr_defs, constr_rep_eqns, dist_lemmas) = fold dt_constr_defs

       (List.take (descr, length new_type_names) ~~

         List.take (pdescr, length new_type_names) ~~

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

       (thy7, [], [], []);

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

     val rep_inject_thms = map (#Rep_inject o snd 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_global (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) (remove (op =) atom dt_atoms))

             (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 = maps prove_perm_rep_perm (atoms ~~ perm_closed_thmss);

     (* prove distinctness theorems *)

     val distinct_props = Datatype_Prop.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 (global_simpset_of thy8 addsimps (dist_lemma :: rep_thms)) 1)

           in

             dist_thm :: Drule.export_without_context (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 maps (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) = fold_rev constr_arg dts (1, [], []);

           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) (remove (op =) atom dt_atoms))

               (HOLogic.mk_Trueprop (HOLogic.mk_eq

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

             (fn _ => EVERY

               [simp_tac (global_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

                  (Thm.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 = Global_Theory.get_thms thy8 "alpha";

     val abs_fresh = Global_Theory.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) (remove (op =) s dt_atoms)) dt_atoms;

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

       let val T = nth_dtyp' i

       in map_filter (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

                  (fold_rev mk_abs_fun xs x, fold_rev mk_abs_fun ys y) :: eqs)

             end;

           val (_, args1, args2, eqs) = fold_rev make_inj dts (1, [], [], []);

           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 (global_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 maps (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), fold_rev mk_abs_fun xs x :: args2)

             end;

           val (_, args1, args2) = fold_rev process_constr dts (1, [], []);

           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_abbrev union}) (map supp args2)))))

             (fn _ =>

               simp_tac (HOL_basic_ss addsimps (supp_def ::

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

                  Collect_False_empty :: 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 (((i, _), T), U) (prems, concls) =

       let

         val Rep_t = Const (nth rep_names i, T --> U) $ mk_Free "x" T i;

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

       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])

       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

            [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 = Datatype_Prop.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 Object_Logic.atomize_prems_tac) 1,

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

            [REPEAT (eresolve_tac Abs_inverse_thms' 1),

             simp_tac (HOL_basic_ss addsimps [Thm.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 = Datatype_Prop.make_tnames recTs;

     val abs_supp = Global_Theory.get_thms thy8 "abs_supp";

     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 [] []

            (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 (global_simpset_of thy8 addsimps

               (abs_supp @ supp_atm @

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

                flat 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 |>

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

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

       Sign.parent_path ||>>

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

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

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

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

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

       Datatype_Aux.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.minimize_sort thy (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);

     (**** strong induction theorem ****)

     val pnames = if length descr'' = 1 then ["P"]

       else map (fn i => "P" ^ string_of_int i) (1 upto length descr'');

     val ind_sort = if null dt_atomTs then HOLogic.typeS

       else Sign.minimize_sort thy9 (Sign.certify_sort thy9 (map (fs_class_of thy9) dt_atoms));

     val fsT = TFree ("'n", ind_sort);

     val fsT' = TFree ("'n", HOLogic.typeS);

     val fresh_fs = map (fn (s, T) => (T, Free (s, fsT' --> HOLogic.mk_setT T)))

       (Datatype_Prop.indexify_names (replicate (length dt_atomTs) "f") ~~ dt_atomTs);

     fun make_pred fsT i T = Free (nth pnames i, fsT --> T --> HOLogic.boolT);

     fun mk_fresh1 xs [] = []

       | mk_fresh1 xs ((y as (_, T)) :: ys) = map (fn x => HOLogic.mk_Trueprop

             (HOLogic.mk_not (HOLogic.mk_eq (Free y, Free x))))

               (filter (fn (_, U) => T = U) (rev xs)) @

           mk_fresh1 (y :: xs) ys;

     fun mk_fresh2 xss [] = []

       | mk_fresh2 xss ((p as (ys, _)) :: yss) = maps (fn y as (_, T) =>

             map (fn (_, x as (_, U)) => HOLogic.mk_Trueprop

               (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'' sorts) cargs;

         val recTs' = map (typ_of_dtyp descr'' sorts) 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 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))

           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') @

           mk_fresh1 [] (maps fst frees') @

           mk_fresh2 [] frees'

       in list_all_free (frees @ [z], Logic.list_implies (prems' @ prems,

         HOLogic.mk_Trueprop (make_pred fsT k T $ Free z $

           list_comb (Const (cname, Ts ---> T), map Free frees))))

       end;

     val ind_prems = maps (fn (((i, (_, _, constrs)), (_, idxss)), T) =>

       map (make_ind_prem fsT (fn T => fn t => fn u =>

         fresh_const T fsT $ t $ u) i T)

           (constrs ~~ idxss)) (descr'' ~~ ndescr ~~ recTs);

     val tnames = Datatype_Prop.make_tnames recTs;

     val zs = Name.variant_list tnames (replicate (length descr'') "z");

     val ind_concl = HOLogic.mk_Trueprop (foldr1 (HOLogic.mk_binop @{const_name HOL.conj})

       (map (fn ((((i, _), T), tname), z) =>

         make_pred fsT i T $ Free (z, fsT) $ Free (tname, T))

         (descr'' ~~ recTs ~~ tnames ~~ zs)));

     val induct = Logic.list_implies (ind_prems, ind_concl);

     val ind_prems' =

       map (fn (_, f as Free (_, T)) => list_all_free ([("x", fsT')],

         HOLogic.mk_Trueprop (Const ("Finite_Set.finite",

           (List.last (binder_types T) --> HOLogic.boolT) -->

             HOLogic.boolT) $ (f $ Free ("x", fsT'))))) fresh_fs @

       maps (fn (((i, (_, _, constrs)), (_, idxss)), T) =>

         map (make_ind_prem fsT' (fn T => fn t => fn u => HOLogic.Not $

           HOLogic.mk_mem (t, the (AList.lookup op = fresh_fs T) $ u)) i T)

             (constrs ~~ idxss)) (descr'' ~~ ndescr ~~ recTs);

     val ind_concl' = HOLogic.mk_Trueprop (foldr1 (HOLogic.mk_binop @{const_name HOL.conj})

       (map (fn ((((i, _), T), tname), z) =>

         make_pred fsT' i T $ Free (z, fsT') $ Free (tname, T))

         (descr'' ~~ recTs ~~ tnames ~~ zs)));

     val induct' = Logic.list_implies (ind_prems', ind_concl');

     val aux_ind_vars =

       (Datatype_Prop.indexify_names (replicate (length dt_atomTs) "pi") ~~

        map mk_permT dt_atomTs) @ [("z", fsT')];

     val aux_ind_Ts = rev (map snd aux_ind_vars);

     val aux_ind_concl = HOLogic.mk_Trueprop (foldr1 (HOLogic.mk_binop @{const_name HOL.conj})

       (map (fn (((i, _), T), tname) =>

         HOLogic.list_all (aux_ind_vars, make_pred fsT' i T $ Bound 0 $

           fold_rev (mk_perm aux_ind_Ts) (map Bound (length dt_atomTs downto 1))

             (Free (tname, T))))

         (descr'' ~~ recTs ~~ tnames)));

     val fin_set_supp = map (fn s =>

       at_inst_of thy9 s RS at_fin_set_supp) dt_atoms;

     val fin_set_fresh = map (fn s =>

       at_inst_of thy9 s RS at_fin_set_fresh) dt_atoms;

     val pt1_atoms = map (fn Type (s, _) =>

       Global_Theory.get_thm thy9 ("pt_" ^ Long_Name.base_name s ^ "1")) dt_atomTs;

     val pt2_atoms = map (fn Type (s, _) =>

       Global_Theory.get_thm thy9 ("pt_" ^ Long_Name.base_name s ^ "2") RS sym) dt_atomTs;

     val exists_fresh' = Global_Theory.get_thms thy9 "exists_fresh'";

     val fs_atoms = Global_Theory.get_thms thy9 "fin_supp";

     val abs_supp = Global_Theory.get_thms thy9 "abs_supp";

     val perm_fresh_fresh = Global_Theory.get_thms thy9 "perm_fresh_fresh";

     val calc_atm = Global_Theory.get_thms thy9 "calc_atm";

     val fresh_atm = Global_Theory.get_thms thy9 "fresh_atm";

     val fresh_left = Global_Theory.get_thms thy9 "fresh_left";

     val perm_swap = Global_Theory.get_thms thy9 "perm_swap";

     fun obtain_fresh_name' ths ts T (freshs1, freshs2, ctxt) =

       let

         val p = foldr1 HOLogic.mk_prod (ts @ freshs1);

         val ex = Goal.prove ctxt [] [] (HOLogic.mk_Trueprop

             (HOLogic.exists_const T $ Abs ("x", T,

               fresh_const T (fastype_of p) $

                 Bound 0 $ p)))

           (fn _ => EVERY

             [resolve_tac exists_fresh' 1,

              simp_tac (HOL_ss addsimps (supp_prod :: finite_Un :: fs_atoms @

                fin_set_supp @ ths)) 1]);

         val (([(_, cx)], ths), ctxt') = Obtain.result

           (fn _ => EVERY

             [etac exE 1,

              full_simp_tac (HOL_ss addsimps (fresh_prod :: fresh_atm)) 1,

              REPEAT (etac conjE 1)])

           [ex] ctxt

       in (freshs1 @ [term_of cx], freshs2 @ ths, ctxt') end;

     fun fresh_fresh_inst thy a b =

       let

         val T = fastype_of a;

         val SOME th = find_first (fn th => case prop_of th of

             _ $ (_ $ (Const (_, Type (_, [U, _])) $ _ $ _)) $ _ => U = T

           | _ => false) perm_fresh_fresh

       in

         Drule.instantiate' []

           [SOME (cterm_of thy a), NONE, SOME (cterm_of thy b)] th

       end;

     val fs_cp_sort =

       map (fs_class_of thy9) dt_atoms @

       maps (fn s => map (cp_class_of thy9 s) (remove (op =) s dt_atoms)) dt_atoms;

     (**********************************************************************

       The subgoals occurring in the proof of induct_aux have the

       following parameters:

         x_1 ... x_k p_1 ... p_m z

       where

         x_i : constructor arguments (introduced by weak induction rule)

         p_i : permutations (one for each atom type in the data type)

         z   : freshness context

     ***********************************************************************)

     val _ = warning "proving strong induction theorem ...";

     val induct_aux = Goal.prove_global thy9 []

         (map (augment_sort thy9 fs_cp_sort) ind_prems')

         (augment_sort thy9 fs_cp_sort ind_concl') (fn {prems, context} =>

       let

         val (prems1, prems2) = chop (length dt_atomTs) prems;

         val ind_ss2 = HOL_ss addsimps

           finite_Diff :: abs_fresh @ abs_supp @ fs_atoms;

         val ind_ss1 = ind_ss2 addsimps fresh_left @ calc_atm @

           fresh_atm @ rev_simps @ app_simps;

         val ind_ss3 = HOL_ss addsimps abs_fun_eq1 ::

           abs_perm @ calc_atm @ perm_swap;

         val ind_ss4 = HOL_basic_ss addsimps fresh_left @ prems1 @

           fin_set_fresh @ calc_atm;

         val ind_ss5 = HOL_basic_ss addsimps pt1_atoms;

         val ind_ss6 = HOL_basic_ss addsimps flat perm_simps';

         val th = Goal.prove context [] []

           (augment_sort thy9 fs_cp_sort aux_ind_concl)

           (fn {context = context1, ...} =>

              EVERY (indtac dt_induct tnames 1 ::

                maps (fn ((_, (_, _, constrs)), (_, constrs')) =>

                  map (fn ((cname, cargs), is) =>

                    REPEAT (rtac allI 1) THEN

                    SUBPROOF (fn {prems = iprems, params, concl,

                        context = context2, ...} =>

                      let

                        val concl' = term_of concl;

                        val _ $ (_ $ _ $ u) = concl';

                        val U = fastype_of u;

                        val (xs, params') =

                          chop (length cargs) (map (term_of o #2) params);

                        val Ts = map fastype_of xs;

                        val cnstr = Const (cname, Ts ---> U);

                        val (pis, z) = split_last params';

                        val mk_pi = fold_rev (mk_perm []) pis;

                        val xs' = partition_cargs is xs;

                        val xs'' = map (fn (ts, u) => (map mk_pi ts, mk_pi u)) xs';

                        val ts = maps (fn (ts, u) => ts @ [u]) xs'';

                        val (freshs1, freshs2, context3) = fold (fn t =>

                          let val T = fastype_of t

                          in obtain_fresh_name' prems1

                            (the (AList.lookup op = fresh_fs T) $ z :: ts) T

                          end) (maps fst xs') ([], [], context2);

                        val freshs1' = unflat (map fst xs') freshs1;

                        val freshs2' = map (Simplifier.simplify ind_ss4)

                          (mk_not_sym freshs2);

                        val ind_ss1' = ind_ss1 addsimps freshs2';

                        val ind_ss3' = ind_ss3 addsimps freshs2';

                        val rename_eq =

                          if forall (null o fst) xs' then []

                          else [Goal.prove context3 [] []

                            (HOLogic.mk_Trueprop (HOLogic.mk_eq

                              (list_comb (cnstr, ts),

                               list_comb (cnstr, maps (fn ((bs, t), cs) =>

                                 cs @ [fold_rev (mk_perm []) (map perm_of_pair

                                   (bs ~~ cs)) t]) (xs'' ~~ freshs1')))))

                            (fn _ => EVERY

                               (simp_tac (HOL_ss addsimps flat inject_thms) 1 ::

                                REPEAT (FIRSTGOAL (rtac conjI)) ::

                                maps (fn ((bs, t), cs) =>

                                  if null bs then []

                                  else rtac sym 1 :: maps (fn (b, c) =>

                                    [rtac trans 1, rtac sym 1,

                                     rtac (fresh_fresh_inst thy9 b c) 1,

                                     simp_tac ind_ss1' 1,

                                     simp_tac ind_ss2 1,

                                     simp_tac ind_ss3' 1]) (bs ~~ cs))

                                  (xs'' ~~ freshs1')))];

                        val th = Goal.prove context3 [] [] concl' (fn _ => EVERY

                          [simp_tac (ind_ss6 addsimps rename_eq) 1,

                           cut_facts_tac iprems 1,

                           (resolve_tac prems THEN_ALL_NEW

                             SUBGOAL (fn (t, i) => case Logic.strip_assums_concl t of

                                 _ $ (Const ("Nominal.fresh", _) $ _ $ _) =>

                                   simp_tac ind_ss1' i

                               | _ $ (Const (@{const_name Not}, _) $ _) =>

                                   resolve_tac freshs2' i

                               | _ => asm_simp_tac (HOL_basic_ss addsimps

                                   pt2_atoms addsimprocs [perm_simproc]) i)) 1])

                        val final = Proof_Context.export context3 context2 [th]

                      in

                        resolve_tac final 1

                      end) context1 1) (constrs ~~ constrs')) (descr'' ~~ ndescr)))

       in

         EVERY

           [cut_facts_tac [th] 1,

            REPEAT (eresolve_tac [conjE, @{thm allE_Nil}] 1),

            REPEAT (etac allE 1),

            REPEAT (TRY (rtac conjI 1) THEN asm_full_simp_tac ind_ss5 1)]

       end);

     val induct_aux' = Thm.instantiate ([],

       map (fn (s, v as Var (_, T)) =>

         (cterm_of thy9 v, cterm_of thy9 (Free (s, T))))

           (pnames ~~ map head_of (HOLogic.dest_conj

              (HOLogic.dest_Trueprop (concl_of induct_aux)))) @

       map (fn (_, f) =>

         let val f' = Logic.varify_global f

         in (cterm_of thy9 f',

           cterm_of thy9 (Const ("Nominal.supp", fastype_of f')))

         end) fresh_fs) induct_aux;

     val induct = Goal.prove_global thy9 []

       (map (augment_sort thy9 fs_cp_sort) ind_prems)

       (augment_sort thy9 fs_cp_sort ind_concl)

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

          [rtac induct_aux' 1,

           REPEAT (resolve_tac fs_atoms 1),

           REPEAT ((resolve_tac prems THEN_ALL_NEW

             (etac @{thm meta_spec} ORELSE' full_simp_tac (HOL_basic_ss addsimps [fresh_def]))) 1)])

     val (_, thy10) = thy9 |>

       Sign.add_path big_name |>

       Global_Theory.add_thms [((Binding.name "strong_induct'", induct_aux), [])] ||>>

       Global_Theory.add_thms [((Binding.name "strong_induct", induct), [case_names_induct])] ||>>

       Global_Theory.add_thmss [((Binding.name "strong_inducts", projections induct), [case_names_induct])];

     (**** recursion combinator ****)

     val _ = warning "defining recursion combinator ...";

     val used = List.foldr OldTerm.add_typ_tfree_names [] recTs;

     val (rec_result_Ts', rec_fn_Ts') = Datatype_Prop.make_primrec_Ts descr' sorts used;

     val rec_sort = if null dt_atomTs then HOLogic.typeS else

       Sign.minimize_sort thy10 (Sign.certify_sort thy10 pt_cp_sort);

     val rec_result_Ts = map (fn TFree (s, _) => TFree (s, rec_sort)) rec_result_Ts';

     val rec_fn_Ts = map (typ_subst_atomic (rec_result_Ts' ~~ rec_result_Ts)) rec_fn_Ts';

     val rec_set_Ts = map (fn (T1, T2) =>

       rec_fn_Ts @ [T1, T2] ---> HOLogic.boolT) (recTs ~~ rec_result_Ts);

     val big_rec_name = big_name ^ "_rec_set";

     val rec_set_names' =

       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"))

       (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);

     (* introduction rules for graph of recursion function *)

     val rec_preds = map (fn (a, T) =>

       Free (a, T --> HOLogic.boolT)) (pnames ~~ rec_result_Ts);

     fun mk_fresh3 rs [] = []

       | mk_fresh3 rs ((p as (ys, z)) :: yss) = maps (fn y as (_, T) =>

             map_filter (fn ((_, (_, x)), r as (_, U)) => if z = x then NONE

               else SOME (HOLogic.mk_Trueprop

                 (fresh_const T U $ Free y $ Free r))) rs) ys @

           mk_fresh3 rs yss;

     (* FIXME: avoid collisions with other variable names? *)

     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'' sorts) 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)

           (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 prems2 =

           map (fn f => map (fn p as (_, T) => HOLogic.mk_Trueprop

             (fresh_const T (fastype_of f) $ Free p $ f)) binders) rec_fns;

         val prems3 = mk_fresh1 [] binders @ mk_fresh2 [] frees';

         val prems4 = map (fn ((i, _), y) =>

           HOLogic.mk_Trueprop (nth rec_preds i $ Free y)) (recs ~~ frees'');

         val prems5 = mk_fresh3 (recs ~~ frees'') frees';

         val prems6 = maps (fn aT => map (fn y as (_, T) => HOLogic.mk_Trueprop

           (Const ("Finite_Set.finite", HOLogic.mk_setT aT --> HOLogic.boolT) $

              (Const ("Nominal.supp", T --> HOLogic.mk_setT aT) $ Free y)))

                frees'') atomTs;

         val prems7 = map (fn x as (_, T) => HOLogic.mk_Trueprop

           (fresh_const T fsT' $ Free x $ rec_ctxt)) binders;

         val result = list_comb (nth rec_fns l, map Free (frees @ frees''));

         val result_freshs = map (fn p as (_, T) =>

           fresh_const T (fastype_of result) $ Free p $ result) binders;

         val P = HOLogic.mk_Trueprop (p $ result)

       in

         (rec_intr_ts @ [Logic.list_implies (flat prems2 @ prems3 @ prems1,

            HOLogic.mk_Trueprop (rec_set $

              list_comb (Const (cname, Ts ---> T), map Free frees) $ result))],

          rec_prems @ [list_all_free (frees @ frees'', Logic.list_implies (prems4, P))],

          rec_prems' @ map (fn fr => list_all_free (frees @ frees'',

            Logic.list_implies (nth prems2 l @ prems3 @ prems5 @ prems7 @ prems6 @ [P],

              HOLogic.mk_Trueprop fr))) result_freshs,

          rec_eq_prems @ [flat prems2 @ prems3],

          l + 1)

       end;

     val (rec_intr_ts, rec_prems, rec_prems', rec_eq_prems, _) =

       fold (fn ((((d, d'), T), p), rec_set) =>

         fold (make_rec_intr T p rec_set) (#3 (snd d) ~~ snd d'))

           (descr'' ~~ ndescr ~~ recTs ~~ rec_preds ~~ rec_sets')

           ([], [], [], [], 0);

     val ({intrs = rec_intrs, elims = rec_elims, raw_induct = rec_induct, ...}, thy11) =

       thy10

       |> Sign.map_naming Name_Space.conceal

       |> Inductive.add_inductive_global

           {quiet_mode = #quiet config, verbose = false, alt_name = Binding.name big_rec_name,

            coind = false, no_elim = false, no_ind = false, skip_mono = true, fork_mono = false}

           (map (fn (s, T) => ((Binding.name s, T), NoSyn)) (rec_set_names' ~~ rec_set_Ts))

           (map dest_Free rec_fns)

           (map (fn x => (Attrib.empty_binding, x)) rec_intr_ts) []

       ||> Global_Theory.hide_fact true (Long_Name.append (Sign.full_bname thy10 big_rec_name) "induct")

       ||> Sign.restore_naming thy10;

     (** equivariance **)

     val fresh_bij = Global_Theory.get_thms thy11 "fresh_bij";

     val perm_bij = Global_Theory.get_thms thy11 "perm_bij";

     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"))

           (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

           (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

                   addsimps flat perm_simps'

                   addsimprocs [NominalPermeq.perm_simproc_app]) 1 THEN

                 (resolve_tac rec_intrs THEN_ALL_NEW

                  asm_simp_tac (HOL_ss addsimps (fresh_bij @ perm_bij))) 1))))

         val ths' = map (fn ((P, Q), th) =>

           Goal.prove_global thy11 [] []

             (augment_sort thy1 pt_cp_sort

               (Logic.mk_implies (HOLogic.mk_Trueprop Q, HOLogic.mk_Trueprop P)))

             (fn _ => dtac (Thm.instantiate ([],

                  [(cterm_of thy11 (Var (("pi", 0), permT)),

                    cterm_of thy11 (Const ("List.rev", permT --> permT) $ pi))]) th) 1 THEN

                NominalPermeq.perm_simp_tac HOL_ss 1)) (ps ~~ ths)

       in (ths, ths') end) dt_atomTs);

     (** finite support **)

     val rec_fin_supp_thms = map (fn aT =>

       let

         val name = Long_Name.base_name (fst (dest_Type aT));

         val fs_name = Global_Theory.get_thm thy11 ("fs_" ^ name ^ "1");

         val aset = HOLogic.mk_setT aT;

         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

           (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) =>

                    let

                      val x = Free ("x" ^ string_of_int i, T);

                      val y = Free ("y" ^ string_of_int i, U)

                    in

                      HOLogic.mk_imp (R $ x $ y,

                        finite $ (Const ("Nominal.supp", U --> aset) $ y))

                    end) (recTs ~~ rec_result_Ts ~~ rec_sets ~~

                      (1 upto length recTs))))))

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

               (rtac rec_induct THEN_ALL_NEW cut_facts_tac fins) 1 THEN REPEAT

                (NominalPermeq.finite_guess_tac (HOL_ss addsimps [fs_name]) 1))))

       end) dt_atomTs;

     (** freshness **)

     val finite_premss = map (fn aT =>

       map (fn (f, T) => HOLogic.mk_Trueprop

         (Const ("Finite_Set.finite", HOLogic.mk_setT aT --> HOLogic.boolT) $

            (Const ("Nominal.supp", T --> HOLogic.mk_setT aT) $ f)))

            (rec_fns ~~ rec_fn_Ts)) dt_atomTs;

     val rec_fns' = map (augment_sort thy11 fs_cp_sort) rec_fns;

     val rec_fresh_thms = map (fn ((aT, eqvt_ths), finite_prems) =>

       let

         val name = Long_Name.base_name (fst (dest_Type aT));

         val fs_name = Global_Theory.get_thm thy11 ("fs_" ^ name ^ "1");

         val a = Free ("a", aT);

         val freshs = map (fn (f, fT) => HOLogic.mk_Trueprop

           (fresh_const aT fT $ a $ f)) (rec_fns ~~ rec_fn_Ts)

       in

         map (fn (((T, U), R), eqvt_th) =>

           let

             val x = Free ("x", augment_sort_typ thy11 fs_cp_sort T);

             val y = Free ("y", U);

             val y' = Free ("y'", U)

           in

             Drule.export_without_context (Goal.prove (Proof_Context.init_global thy11) []

               (map (augment_sort thy11 fs_cp_sort)

                 (finite_prems @

                    [HOLogic.mk_Trueprop (R $ x $ y),

                     HOLogic.mk_Trueprop (HOLogic.mk_all ("y'", U,

                       HOLogic.mk_imp (R $ x $ y', HOLogic.mk_eq (y', y)))),

                     HOLogic.mk_Trueprop (fresh_const aT T $ a $ x)] @

                  freshs))

               (HOLogic.mk_Trueprop (fresh_const aT U $ a $ y))

               (fn {prems, context} =>

                  let

                    val (finite_prems, rec_prem :: unique_prem ::

                      fresh_prems) = chop (length finite_prems) prems;

                    val unique_prem' = unique_prem RS spec RS mp;

                    val unique = [unique_prem', unique_prem' RS sym] MRS trans;

                    val _ $ (_ $ (_ $ S $ _)) $ _ = prop_of supports_fresh;

                    val tuple = foldr1 HOLogic.mk_prod (x :: rec_fns')

                  in EVERY

                    [rtac (Drule.cterm_instantiate

                       [(cterm_of thy11 S,

                         cterm_of thy11 (Const ("Nominal.supp",

                           fastype_of tuple --> HOLogic.mk_setT aT) $ tuple))]

                       supports_fresh) 1,

                     simp_tac (HOL_basic_ss addsimps

                       [supports_def, Thm.symmetric fresh_def, fresh_prod]) 1,

                     REPEAT_DETERM (resolve_tac [allI, impI] 1),

                     REPEAT_DETERM (etac conjE 1),

                     rtac unique 1,

                     SUBPROOF (fn {prems = prems', params = [(_, a), (_, b)], ...} => EVERY

                       [cut_facts_tac [rec_prem] 1,

                        rtac (Thm.instantiate ([],

                          [(cterm_of thy11 (Var (("pi", 0), mk_permT aT)),

                            cterm_of thy11 (perm_of_pair (term_of a, term_of b)))]) eqvt_th) 1,

                        asm_simp_tac (HOL_ss addsimps

                          (prems' @ perm_swap @ perm_fresh_fresh)) 1]) context 1,

                     rtac rec_prem 1,

                     simp_tac (HOL_ss addsimps (fs_name ::

                       supp_prod :: finite_Un :: finite_prems)) 1,

                     simp_tac (HOL_ss addsimps (Thm.symmetric fresh_def ::

                       fresh_prod :: fresh_prems)) 1]

                  end))

           end) (recTs ~~ rec_result_Ts ~~ rec_sets ~~ eqvt_ths)

       end) (dt_atomTs ~~ rec_equiv_thms' ~~ finite_premss);

     (** uniqueness **)

     val fun_tuple = foldr1 HOLogic.mk_prod (rec_ctxt :: rec_fns);

     val fun_tupleT = fastype_of fun_tuple;

     val rec_unique_frees =

       Datatype_Prop.indexify_names (replicate (length recTs) "x") ~~ recTs;

     val rec_unique_frees'' = map (fn (s, T) => (s ^ "'", T)) rec_unique_frees;

     val rec_unique_frees' =

       Datatype_Prop.indexify_names (replicate (length recTs) "y") ~~ rec_result_Ts;

     val rec_unique_concls = map (fn ((x, U), R) =>

         Const (@{const_name Ex1}, (U --> HOLogic.boolT) --> HOLogic.boolT) $

           Abs ("y", U, R $ Free x $ Bound 0))

       (rec_unique_frees ~~ rec_result_Ts ~~ rec_sets);

     val induct_aux_rec = Drule.cterm_instantiate

       (map (pairself (cterm_of thy11) o apsnd (augment_sort thy11 fs_cp_sort))

          (map (fn (aT, f) => (Logic.varify_global f, Abs ("z", HOLogic.unitT,

             Const ("Nominal.supp", fun_tupleT --> HOLogic.mk_setT aT) $ fun_tuple)))

               fresh_fs @

           map (fn (((P, T), (x, U)), Q) =>

            (Var ((P, 0), Logic.varifyT_global (fsT' --> T --> HOLogic.boolT)),

             Abs ("z", HOLogic.unitT, absfree (x, U, Q))))

               (pnames ~~ recTs ~~ rec_unique_frees ~~ rec_unique_concls) @

           map (fn (s, T) => (Var ((s, 0), Logic.varifyT_global T), Free (s, T)))

             rec_unique_frees)) induct_aux;

     fun obtain_fresh_name vs ths rec_fin_supp T (freshs1, freshs2, ctxt) =

       let

         val p = foldr1 HOLogic.mk_prod (vs @ freshs1);

         val ex = Goal.prove ctxt [] [] (HOLogic.mk_Trueprop

             (HOLogic.exists_const T $ Abs ("x", T,

               fresh_const T (fastype_of p) $ Bound 0 $ p)))

           (fn _ => EVERY

             [cut_facts_tac ths 1,

              REPEAT_DETERM (dresolve_tac (the (AList.lookup op = rec_fin_supp T)) 1),

              resolve_tac exists_fresh' 1,

              asm_simp_tac (HOL_ss addsimps (supp_prod :: finite_Un :: fs_atoms)) 1]);

         val (([(_, cx)], ths), ctxt') = Obtain.result

           (fn _ => EVERY

             [etac exE 1,

              full_simp_tac (HOL_ss addsimps (fresh_prod :: fresh_atm)) 1,

              REPEAT (etac conjE 1)])

           [ex] ctxt

       in (freshs1 @ [term_of cx], freshs2 @ ths, ctxt') end;

     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

       (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)))

       (fn {prems, context} =>

          let

            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

              (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

                       (S $ Free x $ Free y, P $ (Free y)))

                         (rec_unique_frees'' ~~ rec_unique_frees' ~~

                            rec_sets ~~ rec_preds)))))

                (fn _ =>

                   rtac rec_induct 1 THEN

                   REPEAT ((resolve_tac P_ind_ths THEN_ALL_NEW assume_tac) 1))));

            val rec_fin_supp_thms' = map

              (fn (ths, (T, fin_ths)) => (T, map (curry op MRS fin_ths) ths))

              (rec_fin_supp_thms ~~ finite_thss);

          in EVERY

            ([rtac induct_aux_rec 1] @

             maps (fn ((_, finite_ths), finite_th) =>

               [cut_facts_tac (finite_th :: finite_ths) 1,

                asm_simp_tac (HOL_ss addsimps [supp_prod, finite_Un]) 1])

                 (finite_thss ~~ finite_ctxt_ths) @

             maps (fn ((_, idxss), elim) => maps (fn idxs =>

               [full_simp_tac (HOL_ss addsimps [Thm.symmetric fresh_def, supp_prod, Un_iff]) 1,

                REPEAT_DETERM (eresolve_tac [conjE, ex1E] 1),

                rtac ex1I 1,

                (resolve_tac rec_intrs THEN_ALL_NEW atac) 1,

                rotate_tac ~1 1,

                ((DETERM o etac elim) THEN_ALL_NEW full_simp_tac

                   (HOL_ss addsimps flat distinct_thms)) 1] @

                (if null idxs then [] else [hyp_subst_tac 1,

                 SUBPROOF (fn {asms, concl, prems = prems', params, context = context', ...} =>

                   let

                     val SOME prem = find_first (can (HOLogic.dest_eq o

                       HOLogic.dest_Trueprop o prop_of)) prems';

                     val _ $ (_ $ lhs $ rhs) = prop_of prem;

                     val _ $ (_ $ lhs' $ rhs') = term_of concl;

                     val rT = fastype_of lhs';

                     val (c, cargsl) = strip_comb lhs;

                     val cargsl' = partition_cargs idxs cargsl;

                     val boundsl = maps fst cargsl';

                     val (_, cargsr) = strip_comb rhs;

                     val cargsr' = partition_cargs idxs cargsr;

                     val boundsr = maps fst cargsr';

                     val (params1, _ :: params2) =

                       chop (length params div 2) (map (term_of o #2) params);

                     val params' = params1 @ params2;

                     val rec_prems = filter (fn th => case prop_of th of

                         _ $ p => (case head_of p of

                           Const (s, _) => member (op =) rec_set_names s

                         | _ => false)

                       | _ => false) prems';

                     val fresh_prems = filter (fn th => case prop_of th of

                         _ $ (Const ("Nominal.fresh", _) $ _ $ _) => true

                       | _ $ (Const (@{const_name Not}, _) $ _) => true

                       | _ => false) prems';

                     val Ts = map fastype_of boundsl;

                     val _ = warning "step 1: obtaining fresh names";

                     val (freshs1, freshs2, context'') = fold

                       (obtain_fresh_name (rec_ctxt :: rec_fns' @ params')

                          (maps snd finite_thss @ finite_ctxt_ths @ rec_prems)

                          rec_fin_supp_thms')

                       Ts ([], [], context');

                     val pi1 = map perm_of_pair (boundsl ~~ freshs1);

                     val rpi1 = rev pi1;

                     val pi2 = map perm_of_pair (boundsr ~~ freshs1);

                     val rpi2 = rev pi2;

                     val fresh_prems' = mk_not_sym fresh_prems;

                     val freshs2' = mk_not_sym freshs2;

                     (** as, bs, cs # K as ts, K bs us **)

                     val _ = warning "step 2: as, bs, cs # K as ts, K bs us";

                     val prove_fresh_ss = HOL_ss addsimps

                       (finite_Diff :: flat fresh_thms @

                        fs_atoms @ abs_fresh @ abs_supp @ fresh_atm);

                     (* FIXME: avoid asm_full_simp_tac ? *)

                     fun prove_fresh ths y x = Goal.prove context'' [] []

                       (HOLogic.mk_Trueprop (fresh_const

                          (fastype_of x) (fastype_of y) $ x $ y))

                       (fn _ => cut_facts_tac ths 1 THEN asm_full_simp_tac prove_fresh_ss 1);

                     val constr_fresh_thms =

                       map (prove_fresh fresh_prems lhs) boundsl @

                       map (prove_fresh fresh_prems rhs) boundsr @

                       map (prove_fresh freshs2 lhs) freshs1 @

                       map (prove_fresh freshs2 rhs) freshs1;

                     (** pi1 o (K as ts) = pi2 o (K bs us) **)

                     val _ = warning "step 3: pi1 o (K as ts) = pi2 o (K bs us)";

                     val pi1_pi2_eq = Goal.prove context'' [] []

                       (HOLogic.mk_Trueprop (HOLogic.mk_eq

                         (fold_rev (mk_perm []) pi1 lhs, fold_rev (mk_perm []) pi2 rhs)))

                       (fn _ => EVERY

                          [cut_facts_tac constr_fresh_thms 1,

                           asm_simp_tac (HOL_basic_ss addsimps perm_fresh_fresh) 1,

                           rtac prem 1]);

                     (** pi1 o ts = pi2 o us **)

                     val _ = warning "step 4: pi1 o ts = pi2 o us";

                     val pi1_pi2_eqs = map (fn (t, u) =>

                       Goal.prove context'' [] []

                         (HOLogic.mk_Trueprop (HOLogic.mk_eq

                           (fold_rev (mk_perm []) pi1 t, fold_rev (mk_perm []) pi2 u)))

                         (fn _ => EVERY

                            [cut_facts_tac [pi1_pi2_eq] 1,

                             asm_full_simp_tac (HOL_ss addsimps

                               (calc_atm @ flat perm_simps' @

                                fresh_prems' @ freshs2' @ abs_perm @

                                alpha @ flat inject_thms)) 1]))

                         (map snd cargsl' ~~ map snd cargsr');

                     (** pi1^-1 o pi2 o us = ts **)

                     val _ = warning "step 5: pi1^-1 o pi2 o us = ts";

                     val rpi1_pi2_eqs = map (fn ((t, u), eq) =>

                       Goal.prove context'' [] []

                         (HOLogic.mk_Trueprop (HOLogic.mk_eq

                           (fold_rev (mk_perm []) (rpi1 @ pi2) u, t)))

                         (fn _ => simp_tac (HOL_ss addsimps

                            ((eq RS sym) :: perm_swap)) 1))

                         (map snd cargsl' ~~ map snd cargsr' ~~ pi1_pi2_eqs);

                     val (rec_prems1, rec_prems2) =

                       chop (length rec_prems div 2) rec_prems;

                     (** (ts, pi1^-1 o pi2 o vs) in rec_set **)

                     val _ = warning "step 6: (ts, pi1^-1 o pi2 o vs) in rec_set";

                     val rec_prems' = map (fn th =>

                       let

                         val _ $ (S $ x $ y) = prop_of th;

                         val Const (s, _) = head_of S;

                         val k = find_index (equal s) rec_set_names;

                         val pi = rpi1 @ pi2;

                         fun mk_pi z = fold_rev (mk_perm []) pi z;

                         fun eqvt_tac p =

                           let

                             val U as Type (_, [Type (_, [T, _])]) = fastype_of p;

                             val l = find_index (equal T) dt_atomTs;

                             val th = nth (nth rec_equiv_thms' l) k;

                             val th' = Thm.instantiate ([],

                               [(cterm_of thy11 (Var (("pi", 0), U)),

                                 cterm_of thy11 p)]) th;

                           in rtac th' 1 end;

                         val th' = Goal.prove context'' [] []

                           (HOLogic.mk_Trueprop (S $ mk_pi x $ mk_pi y))

                           (fn _ => EVERY

                              (map eqvt_tac pi @

                               [simp_tac (HOL_ss addsimps (fresh_prems' @ freshs2' @

                                  perm_swap @ perm_fresh_fresh)) 1,

                                rtac th 1]))

                       in

                         Simplifier.simplify

                           (HOL_basic_ss addsimps rpi1_pi2_eqs) th'

                       end) rec_prems2;

                     val ihs = filter (fn th => case prop_of th of

                       _ $ (Const (@{const_name All}, _) $ _) => true | _ => false) prems';

                     (** pi1 o rs = pi2 o vs , rs = pi1^-1 o pi2 o vs **)

                     val _ = warning "step 7: pi1 o rs = pi2 o vs , rs = pi1^-1 o pi2 o vs";

                     val rec_eqns = map (fn (th, ih) =>

                       let

                         val th' = th RS (ih RS spec RS mp) RS sym;

                         val _ $ (_ $ lhs $ rhs) = prop_of th';

                         fun strip_perm (_ $ _ $ t) = strip_perm t

                           | strip_perm t = t;

                       in

                         Goal.prove context'' [] []

                            (HOLogic.mk_Trueprop (HOLogic.mk_eq

                               (fold_rev (mk_perm []) pi1 lhs,

                                fold_rev (mk_perm []) pi2 (strip_perm rhs))))

                            (fn _ => simp_tac (HOL_basic_ss addsimps

                               (th' :: perm_swap)) 1)

                       end) (rec_prems' ~~ ihs);

                     (** as # rs **)

                     val _ = warning "step 8: as # rs";

                     val rec_freshs =

                       maps (fn (rec_prem, ih) =>

                         let

                           val _ $ (S $ x $ (y as Free (_, T))) =

                             prop_of rec_prem;

                           val k = find_index (equal S) rec_sets;

                           val atoms = flat (map_filter (fn (bs, z) =>

                             if z = x then NONE else SOME bs) cargsl')

                         in

                           map (fn a as Free (_, aT) =>

                             let val l = find_index (equal aT) dt_atomTs;

                             in

                               Goal.prove context'' [] []

                                 (HOLogic.mk_Trueprop (fresh_const aT T $ a $ y))

                                 (fn _ => EVERY

                                    (rtac (nth (nth rec_fresh_thms l) k) 1 ::

                                     map (fn th => rtac th 1)

                                       (snd (nth finite_thss l)) @

                                     [rtac rec_prem 1, rtac ih 1,

                                      REPEAT_DETERM (resolve_tac fresh_prems 1)]))

                             end) atoms

                         end) (rec_prems1 ~~ ihs);

                     (** as # fK as ts rs , bs # fK bs us vs **)

                     val _ = warning "step 9: as # fK as ts rs , bs # fK bs us vs";

                     fun prove_fresh_result (a as Free (_, aT)) =

                       Goal.prove context'' [] []

                         (HOLogic.mk_Trueprop (fresh_const aT rT $ a $ rhs'))

                         (fn _ => EVERY

                            [resolve_tac fcbs 1,

                             REPEAT_DETERM (resolve_tac

                               (fresh_prems @ rec_freshs) 1),

                             REPEAT_DETERM (resolve_tac (maps snd rec_fin_supp_thms') 1

                               THEN resolve_tac rec_prems 1),

                             resolve_tac P_ind_ths 1,

                             REPEAT_DETERM (resolve_tac (P_ths @ rec_prems) 1)]);

                     val fresh_results'' = map prove_fresh_result boundsl;

                     fun prove_fresh_result'' ((a as Free (_, aT), b), th) =

                       let val th' = Goal.prove context'' [] []

                         (HOLogic.mk_Trueprop (fresh_const aT rT $

                             fold_rev (mk_perm []) (rpi2 @ pi1) a $

                             fold_rev (mk_perm []) (rpi2 @ pi1) rhs'))

                         (fn _ => simp_tac (HOL_ss addsimps fresh_bij) 1 THEN

                            rtac th 1)

                       in

                         Goal.prove context'' [] []

                           (HOLogic.mk_Trueprop (fresh_const aT rT $ b $ lhs'))

                           (fn _ => EVERY

                              [cut_facts_tac [th'] 1,

                               full_simp_tac (Simplifier.global_context thy11 HOL_ss

                                 addsimps rec_eqns @ pi1_pi2_eqs @ perm_swap

                                 addsimprocs [NominalPermeq.perm_simproc_app]) 1,

                               full_simp_tac (HOL_ss addsimps (calc_atm @

                                 fresh_prems' @ freshs2' @ perm_fresh_fresh)) 1])

                       end;

                     val fresh_results = fresh_results'' @ map prove_fresh_result''

                       (boundsl ~~ boundsr ~~ fresh_results'');

                     (** cs # fK as ts rs , cs # fK bs us vs **)

                     val _ = warning "step 10: cs # fK as ts rs , cs # fK bs us vs";

                     fun prove_fresh_result' recs t (a as Free (_, aT)) =

                       Goal.prove context'' [] []

                         (HOLogic.mk_Trueprop (fresh_const aT rT $ a $ t))

                         (fn _ => EVERY

                           [cut_facts_tac recs 1,

                            REPEAT_DETERM (dresolve_tac

                              (the (AList.lookup op = rec_fin_supp_thms' aT)) 1),

                            NominalPermeq.fresh_guess_tac

                              (HOL_ss addsimps (freshs2 @

                                 fs_atoms @ fresh_atm @

                                 maps snd finite_thss)) 1]);

                     val fresh_results' =

                       map (prove_fresh_result' rec_prems1 rhs') freshs1 @

                       map (prove_fresh_result' rec_prems2 lhs') freshs1;

                     (** pi1 o (fK as ts rs) = pi2 o (fK bs us vs) **)

                     val _ = warning "step 11: pi1 o (fK as ts rs) = pi2 o (fK bs us vs)";

                     val pi1_pi2_result = Goal.prove context'' [] []

                       (HOLogic.mk_Trueprop (HOLogic.mk_eq

                         (fold_rev (mk_perm []) pi1 rhs', fold_rev (mk_perm []) pi2 lhs')))

                       (fn _ => simp_tac (Simplifier.context context'' HOL_ss

                            addsimps pi1_pi2_eqs @ rec_eqns

                            addsimprocs [NominalPermeq.perm_simproc_app]) 1 THEN

                          TRY (simp_tac (HOL_ss addsimps

                            (fresh_prems' @ freshs2' @ calc_atm @ perm_fresh_fresh)) 1));

                     val _ = warning "final result";

                     val final = Goal.prove context'' [] [] (term_of concl)

                       (fn _ => cut_facts_tac [pi1_pi2_result RS sym] 1 THEN

                         full_simp_tac (HOL_basic_ss addsimps perm_fresh_fresh @

                           fresh_results @ fresh_results') 1);

                     val final' = Proof_Context.export context'' context' [final];

                     val _ = warning "finished!"

                   in

                     resolve_tac final' 1

                   end) context 1])) idxss) (ndescr ~~ rec_elims))

          end));

     val rec_total_thms = map (fn r => r RS theI') rec_unique_thms;

     (* define primrec combinators *)

     val big_reccomb_name = (space_implode "_" new_type_names) ^ "_rec";

     val reccomb_names = map (Sign.full_bname thy11)

       (if length descr'' = 1 then [big_reccomb_name] else

         (map ((curry (op ^) (big_reccomb_name ^ "_")) o string_of_int)

           (1 upto (length descr''))));

     val reccombs = map (fn ((name, T), T') => list_comb

       (Const (name, rec_fn_Ts @ [T] ---> T'), rec_fns))

         (reccomb_names ~~ recTs ~~ rec_result_Ts);

     val (reccomb_defs, thy12) =

       thy11

       |> Sign.add_consts_i (map (fn ((name, T), T') =>

           (Binding.name (Long_Name.base_name name), rec_fn_Ts @ [T] ---> T', NoSyn))

           (reccomb_names ~~ recTs ~~ rec_result_Ts))

       |> (Global_Theory.add_defs false o map Thm.no_attributes) (map (fn ((((name, comb), set), T), T') =>

           (Binding.name (Long_Name.base_name name ^ "_def"), Logic.mk_equals (comb, absfree ("x", T,

            Const (@{const_name The}, (T' --> HOLogic.boolT) --> T') $ absfree ("y", T',

              set $ Free ("x", T) $ Free ("y", T'))))))

                (reccomb_names ~~ reccombs ~~ rec_sets ~~ recTs ~~ rec_result_Ts));

     (* prove characteristic equations for primrec combinators *)

     val rec_thms = map (fn (prems, concl) =>

       let

         val _ $ (_ $ (_ $ x) $ _) = concl;

         val (_, cargs) = strip_comb x;

         val ps = map (fn (x as Free (_, T), i) =>

           (Free ("x" ^ string_of_int i, T), x)) (cargs ~~ (1 upto length cargs));

         val concl' = subst_atomic_types (rec_result_Ts' ~~ rec_result_Ts) concl;

         val prems' = flat finite_premss @ finite_ctxt_prems @

           rec_prems @ rec_prems' @ map (subst_atomic ps) prems;

         fun solve rules prems = resolve_tac rules THEN_ALL_NEW

           (resolve_tac prems THEN_ALL_NEW atac)

       in

         Goal.prove_global thy12 []

           (map (augment_sort thy12 fs_cp_sort) prems')

           (augment_sort thy12 fs_cp_sort concl')

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

             [rewrite_goals_tac reccomb_defs,

              rtac the1_equality 1,

              solve rec_unique_thms prems 1,

              resolve_tac rec_intrs 1,

              REPEAT (solve (prems @ rec_total_thms) prems 1)])

       end) (rec_eq_prems ~~

         Datatype_Prop.make_primrecs new_type_names descr' sorts thy12);

     val dt_infos = map_index (make_dt_info pdescr sorts induct reccomb_names rec_thms)

       (descr1 ~~ distinct_thms ~~ inject_thms);

     (* FIXME: theorems are stored in database for testing only *)

     val (_, thy13) = thy12 |>

       Global_Theory.add_thmss

         [((Binding.name "rec_equiv", flat rec_equiv_thms), []),

          ((Binding.name "rec_equiv'", flat rec_equiv_thms'), []),

          ((Binding.name "rec_fin_supp", flat rec_fin_supp_thms), []),

          ((Binding.name "rec_fresh", flat rec_fresh_thms), []),

          ((Binding.name "rec_unique", map Drule.export_without_context rec_unique_thms), []),

          ((Binding.name "recs", rec_thms), [])] ||>

       Sign.parent_path ||>

       map_nominal_datatypes (fold Symtab.update dt_infos);

   in

     thy13

   end;

 val add_nominal_datatype = gen_add_nominal_datatype Datatype.read_typ;

 (* FIXME: The following stuff should be exported by Datatype *)

 val datatype_decl =

   Scan.option (Parse.$$$ "(" |-- Parse.name --| Parse.$$$ ")") --

     Parse.type_args -- Parse.name -- Parse.opt_mixfix --

     (Parse.$$$ "=" |-- Parse.enum1 "|" (Parse.name -- Scan.repeat Parse.typ -- Parse.opt_mixfix));

 fun mk_datatype args =

   let

     val names = map (fn ((((NONE, _), t), _), _) => t | ((((SOME t, _), _), _), _) => t) args;

     val specs = map (fn ((((_, vs), t), mx), cons) =>

       (vs, t, mx, map (fn ((x, y), z) => (x, y, z)) cons)) args;

   in add_nominal_datatype Datatype.default_config names specs end;

 val _ =

   Outer_Syntax.command "nominal_datatype" "define inductive datatypes" Keyword.thy_decl

     (Parse.and_list1 datatype_decl >> (Toplevel.theory o mk_datatype));

 end