(*  Title:      HOL/Nominal/nominal_package.ML

    ID:         $Id$

    Author:     Stefan Berghofer and Christian Urban, TU Muenchen

Nominal datatype package for Isabelle/HOL.

*)

signature NOMINAL_PACKAGE =

sig

  val add_nominal_datatype : bool -> string list -> (string list * bstring * mixfix *

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

end

structure NominalPackage : NOMINAL_PACKAGE =

struct

open DatatypeAux;

open NominalAtoms;

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

local

fun dt_recs (DtTFree _) = []

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

  | dt_recs (DtRec i) = [i];

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

  let

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

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

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

fun induct_cases descr =

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

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

in

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

fun mk_case_names_exhausts descr new =

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

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

end;

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

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

fun read_typ sign ((Ts, sorts), str) =

  let

    val T = Type.no_tvars (Sign.read_typ (sign, (AList.lookup op =)

      (map (apfst (rpair ~1)) sorts)) str) handle TYPE (msg, _, _) => error msg

  in (Ts @ [T], add_typ_tfrees (T, sorts)) end;

(** taken from HOL/Tools/datatype_aux.ML **)

fun indtac indrule indnames i st =

  let

    val ts = HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of indrule));

    val ts' = HOLogic.dest_conj (HOLogic.dest_Trueprop

      (Logic.strip_imp_concl (List.nth (prems_of st, i - 1))));

    val getP = if can HOLogic.dest_imp (hd ts) then

      (apfst SOME) o HOLogic.dest_imp else pair NONE;

    fun abstr (t1, t2) = (case t1 of

        NONE => (case filter (fn Free (s, _) => s mem indnames | _ => false)

              (term_frees t2) of

            [Free (s, T)] => absfree (s, T, t2)

          | _ => sys_error "indtac")

      | SOME (_ $ t' $ _) => Abs ("x", fastype_of t', abstract_over (t', t2)))

    val cert = cterm_of (Thm.sign_of_thm st);

    val Ps = map (cert o head_of o snd o getP) ts;

    val indrule' = cterm_instantiate (Ps ~~

      (map (cert o abstr o getP) ts')) indrule

  in

    rtac indrule' i st

  end;

fun mk_subgoal i f st =

  let

    val cg = List.nth (cprems_of st, i - 1);

    val g = term_of cg;

    val revcut_rl' = Thm.lift_rule cg revcut_rl;

    val v = head_of (Logic.strip_assums_concl (hd (prems_of revcut_rl')));

    val ps = Logic.strip_params g;

    val cert = cterm_of (sign_of_thm st);

  in

    compose_tac (false,

      Thm.instantiate ([], [(cert v, cert (list_abs (ps,

        f (rev ps) (Logic.strip_assums_hyp g) (Logic.strip_assums_concl g))))])

      revcut_rl', 2) i st

  end;

(** simplification procedure for sorting permutations **)

val dj_cp = thm "dj_cp";

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

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

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

  | permTs_of _ = [];

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

      let

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

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

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

          let

            val a' = Sign.base_name a;

            val b' = Sign.base_name b;

            val cp = PureThy.get_thm thy (Name ("cp_" ^ a' ^ "_" ^ b' ^ "_inst"));

            val dj = PureThy.get_thm thy (Name ("dj_" ^ b' ^ "_" ^ a'));

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

            val cert = SOME o cterm_of thy

          in

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

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

          end

        else NONE

      end

  | perm_simproc' thy ss _ = NONE;

val perm_simproc =

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

val allE_Nil = read_instantiate_sg (the_context()) [("x", "[]")] allE;

val meta_spec = thm "meta_spec";

fun projections rule =

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

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

fun gen_add_nominal_datatype prep_typ err flat_names new_type_names dts thy =

  let

    (* this theory is used just for parsing *)

    val tmp_thy = thy |>

      Theory.copy |>

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

        (tname, length tvs, mx)) dts);

    val sign = Theory.sign_of tmp_thy;

    val atoms = atoms_of thy;

    val classes = map (NameSpace.map_base (fn s => "pt_" ^ s)) atoms;

    val cp_classes = List.concat (map (fn atom1 => map (fn atom2 =>

      Sign.intern_class thy ("cp_" ^ Sign.base_name atom1 ^ "_" ^

        Sign.base_name atom2)) atoms) atoms);

    fun augment_sort S = S union classes;

    val augment_sort_typ = map_type_tfree (fn (s, S) => TFree (s, augment_sort S));

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

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

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

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

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

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

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

    val sorts' = map (apsnd augment_sort) sorts;

    val tyvars = map #1 dts';

    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_name sign n, Sign.full_name sign (n ^ "_Rep"))) dts;

    val rps = map Library.swap ps;

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

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

      | replace_types (Type (s, Ts)) =

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

      | replace_types T = T;

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

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

        map (augment_sort_typ o replace_types) cargs, NoSyn)) constrs)) dts';

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

    val full_new_type_names' = map (Sign.full_name (sign_of thy)) new_type_names';

    val ({induction, ...},thy1) =

      DatatypePackage.add_datatype_i err flat_names new_type_names' dts'' thy;

    val SOME {descr, ...} = Symtab.lookup

      (DatatypePackage.get_datatypes thy1) (hd full_new_type_names');

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

    (**** 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 = replicate (length new_type_names) "Nominal.perm" @

      DatatypeProp.indexify_names (map (fn i => Sign.full_name (sign_of thy1)

        ("perm_" ^ name_of_typ (nth_dtyp i)))

          (length new_type_names upto length descr - 1));

    val perm_names_types = perm_names ~~ perm_types;

    val perm_eqs = List.concat (map (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 = DatatypeProp.make_tnames Ts;

          val args = map Free (names ~~ Ts);

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

          fun perm_arg (dt, x) =

            let val T = type_of x

            in if is_rec_type dt then

                let val (Us, _) = strip_type T

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

                  Const (List.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

          (("", HOLogic.mk_Trueprop (HOLogic.mk_eq

            (Const (List.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) = thy1 |>

      Theory.add_consts_i (map (fn (s, T) => (Sign.base_name s, T, NoSyn))

        (List.drop (perm_names_types, length new_type_names))) |>

      PrimrecPackage.add_primrec_unchecked_i "" perm_eqs;

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

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

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

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

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

    val perm_fun_def = PureThy.get_thm thy2 (Name "perm_fun_def");

    val unfolded_perm_eq_thms =

      if length descr = length new_type_names then []

      else map standard (List.drop (split_conj_thm

        (Goal.prove_global thy2 [] []

          (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj

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

               let val [T1, T2] = binder_types T

               in HOLogic.mk_eq (Const c $ pi $ Free (x, T2),

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

               end)

             (perm_names_types ~~ perm_indnames))))

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

            ALLGOALS (asm_full_simp_tac

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

        length new_type_names));

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

    val _ = warning "perm_empty_thms";

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

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

      in map standard (List.take (split_conj_thm

        (Goal.prove_global thy2 [] []

          (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj

            (map (fn ((s, T), x) => HOLogic.mk_eq

                (Const (s, permT --> T --> T) $

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

                 Free (x, T)))

             (perm_names ~~

              map body_type perm_types ~~ perm_indnames))))

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

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

        length new_type_names))

      end)

      atoms);

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

    val _ = warning "perm_append_thms";

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

    val at_pt_inst = PureThy.get_thm thy2 (Name "at_pt_inst");

    val pt2 = PureThy.get_thm thy2 (Name "pt2");

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

      let

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

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

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

        val pt_inst = PureThy.get_thm thy2 (Name ("pt_" ^ Sign.base_name a ^ "_inst"));

        val pt2' = pt_inst RS pt2;

        val pt2_ax = PureThy.get_thm thy2

          (Name (NameSpace.map_base (fn s => "pt_" ^ s ^ "2") a));

      in List.take (map standard (split_conj_thm

        (Goal.prove_global thy2 [] []

             (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.op @", permT --> permT --> permT) $

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

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

                    end)

                  (perm_names ~~

                   map body_type perm_types ~~ perm_indnames))))

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

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

         length new_type_names)

      end) atoms);

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

    val _ = warning "perm_eq_thms";

    val pt3 = PureThy.get_thm thy2 (Name "pt3");

    val pt3_rev = PureThy.get_thm thy2 (Name "pt3_rev");

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

      let

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

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

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

        (*FIXME: not robust - better access these theorems using NominalData?*)

        val at_inst = PureThy.get_thm thy2 (Name ("at_" ^ Sign.base_name a ^ "_inst"));

        val pt_inst = PureThy.get_thm thy2 (Name ("pt_" ^ Sign.base_name a ^ "_inst"));

        val pt3' = pt_inst RS pt3;

        val pt3_rev' = at_inst RS (pt_inst RS pt3_rev);

        val pt3_ax = PureThy.get_thm thy2

          (Name (NameSpace.map_base (fn s => "pt_" ^ s ^ "3") a));

      in List.take (map standard (split_conj_thm

        (Goal.prove_global thy2 [] [] (Logic.mk_implies

             (HOLogic.mk_Trueprop (Const ("Nominal.prm_eq",

                permT --> permT --> HOLogic.boolT) $ pi1 $ pi2),

              HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj

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

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

                    in HOLogic.mk_eq

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

                       perm $ pi2 $ Free (x, T))

                    end)

                  (perm_names ~~

                   map body_type perm_types ~~ perm_indnames)))))

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

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

         length new_type_names)

      end) atoms);

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

    val cp1 = PureThy.get_thm thy2 (Name "cp1");

    val dj_cp = PureThy.get_thm thy2 (Name "dj_cp");

    val pt_perm_compose = PureThy.get_thm thy2 (Name "pt_perm_compose");

    val pt_perm_compose_rev = PureThy.get_thm thy2 (Name "pt_perm_compose_rev");

    val dj_perm_perm_forget = PureThy.get_thm thy2 (Name "dj_perm_perm_forget");

    fun composition_instance name1 name2 thy =

      let

        val name1' = Sign.base_name name1;

        val name2' = Sign.base_name name2;

        val cp_class = Sign.intern_class thy ("cp_" ^ name1' ^ "_" ^ name2');

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

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

        val augment = map_type_tfree

          (fn (x, S) => TFree (x, cp_class :: S));

        val Ts = map (augment o body_type) perm_types;

        val cp_inst = PureThy.get_thm thy

          (Name ("cp_" ^ name1' ^ "_" ^ name2' ^ "_inst"));

        val simps = simpset_of thy addsimps (perm_fun_def ::

          (if name1 <> name2 then

             let val dj = PureThy.get_thm thy (Name ("dj_" ^ name2' ^ "_" ^ name1'))

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

           else

             let

               val at_inst = PureThy.get_thm thy (Name ("at_" ^ name1' ^ "_inst"));

               val pt_inst = PureThy.get_thm thy (Name ("pt_" ^ name1' ^ "_inst"))

             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 thms = split_conj_thm (Goal.prove_global thy [] []

            (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj

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

                  let

                    val pi1 = Free ("pi1", permT1);

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

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

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

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

                  in HOLogic.mk_eq

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

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

                  end)

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

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

             ALLGOALS (asm_full_simp_tac simps)]))

      in

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

            (s, replicate (length tvs) (cp_class :: classes), [cp_class])

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

          thy (full_new_type_names' ~~ tyvars)

      end;

    val (perm_thmss,thy3) = thy2 |>

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

      curry (Library.foldr (fn ((i, (tyname, args, _)), thy) =>

        AxClass.prove_arity (tyname, replicate (length args) classes, classes)

        (ClassPackage.intro_classes_tac [] THEN REPEAT (EVERY

           [resolve_tac perm_empty_thms 1,

            resolve_tac perm_append_thms 1,

            resolve_tac perm_eq_thms 1, assume_tac 1])) thy))

        (List.take (descr, length new_type_names)) |>

      PureThy.add_thmss

        [((space_implode "_" new_type_names ^ "_unfolded_perm_eq",

          unfolded_perm_eq_thms), [Simplifier.simp_add]),

         ((space_implode "_" new_type_names ^ "_perm_empty",

          perm_empty_thms), [Simplifier.simp_add]),

         ((space_implode "_" new_type_names ^ "_perm_append",

          perm_append_thms), [Simplifier.simp_add]),

         ((space_implode "_" new_type_names ^ "_perm_eq",

          perm_eq_thms), [Simplifier.simp_add])];

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

    val _ = warning "representing sets";

    val rep_set_names = map (Sign.full_name thy3) (DatatypeProp.indexify_names

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

    val big_rep_name =

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

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

          | (i, _) => SOME (name_of_typ (nth_dtyp i))) descr)) ^ "_set";

    val _ = warning ("big_rep_name: " ^ big_rep_name);

    fun strip_option (dtf as DtType ("fun", [dt, DtRec i])) =

          (case AList.lookup op = descr i of

             SOME ("Nominal.noption", _, [(_, [dt']), _]) =>

               apfst (cons dt) (strip_option dt')

           | _ => ([], dtf))

      | strip_option (DtType ("fun", [dt, DtType ("Nominal.noption", [dt'])])) =

          apfst (cons dt) (strip_option dt')

      | strip_option dt = ([], dt);

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

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

        (map (List.concat o map (fst o strip_option) o snd) cs)) descr)));

    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 (HOLogic.mk_mem (free',

                    Const (List.nth (rep_set_names, k),

                      HOLogic.mk_setT T)))) :: prems

              | _ => prems,

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

          end;

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

        val concl = HOLogic.mk_Trueprop (HOLogic.mk_mem

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

           Const (s, HOLogic.mk_setT T)))

      in Logic.list_implies (prems, concl)

      end;

    val (intr_ts, ind_consts) =

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

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

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

              let val T = nth_dtyp i

              in SOME (map (make_intr rep_set_name T) constrs,

                Const (rep_set_name, HOLogic.mk_setT T))

              end)

                (descr ~~ rep_set_names)));

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

      setmp InductivePackage.quiet_mode false

        (InductivePackage.add_inductive_i false true big_rep_name false true false

           ind_consts (map (fn x => (("", x), [])) intr_ts) []) thy3;

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

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

    val perm_indnames' = List.mapPartial

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

      (perm_indnames ~~ descr);

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

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

        (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj (map

           (fn (S, x) =>

              let

                val S = map_term_types (map_type_tfree

                  (fn (s, cs) => TFree (s, cs union cp_classes))) S;

                val T = HOLogic.dest_setT (fastype_of S);

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

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

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

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

              end) (ind_consts ~~ perm_indnames'))))

        (fn _ => EVERY (* CU: added perm_fun_def in the final tactic in order to deal with funs *)

           [indtac rep_induct [] 1,

            ALLGOALS (simp_tac (simpset_of thy4 addsimps

              (symmetric perm_fun_def :: PureThy.get_thms thy4 (Name ("abs_perm"))))),

            ALLGOALS (resolve_tac rep_intrs 

               THEN_ALL_NEW (asm_full_simp_tac (simpset_of thy4 addsimps [perm_fun_def])))])),

        length new_type_names));

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

    val perm_closed_thmss = map mk_perm_closed atoms;

    val (_,thy5) = PureThy.add_thmss [(("perm_closed", List.concat perm_closed_thmss), [])] thy4;

    (**** typedef ****)

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

    val (typedefs, thy6) =

      fold_map (fn ((((name, mx), tvs), c), name') => fn thy =>

        setmp TypedefPackage.quiet_mode true

          (TypedefPackage.add_typedef_i false (SOME name') (name, tvs, mx) c NONE

            (rtac exI 1 THEN

              QUIET_BREADTH_FIRST (has_fewer_prems 1)

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

        let

          val permT = mk_permT (TFree (Name.variant tvs "'a", HOLogic.typeS));

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

          val tvs' = map (fn s => TFree (s, the (AList.lookup op = sorts' s))) tvs;

          val T = Type (Sign.intern_type thy name, tvs');

          val Const (_, Type (_, [U])) = c

        in apfst (pair r o hd)

          (PureThy.add_defs_unchecked_i true [(("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 (ind_consts, length new_type_names)) ~~ new_type_names) thy5;

    val perm_defs = map snd typedefs;

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

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

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

    val big_name = space_implode "_" new_type_names;

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

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

    fun pt_instance ((class, atom), perm_closed_thms) =

      fold (fn (((({Abs_inverse, Rep_inverse, Rep, ...},

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

          AxClass.prove_arity

            (Sign.intern_type thy name,

              replicate (length tvs) (classes @ cp_classes), [class])

            (EVERY [ClassPackage.intro_classes_tac [],

              rewrite_goals_tac [perm_def],

              asm_full_simp_tac (simpset_of thy addsimps [Rep_inverse]) 1,

              asm_full_simp_tac (simpset_of thy addsimps

                [Rep RS perm_closed RS Abs_inverse]) 1,

              asm_full_simp_tac (HOL_basic_ss addsimps [PureThy.get_thm thy

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

        (typedefs ~~ 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 name = "cp_" ^ Sign.base_name atom1 ^ "_" ^ Sign.base_name atom2;

        val class = Sign.intern_class thy name;

        val cp1' = PureThy.get_thm thy (Name (name ^ "_inst")) RS cp1

      in fold (fn ((((({Abs_inverse, Rep_inverse, Rep, ...},

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

          AxClass.prove_arity

            (Sign.intern_type thy name,

              replicate (length tvs) (classes @ cp_classes), [class])

            (EVERY [ClassPackage.intro_classes_tac [],

              rewrite_goals_tac [perm_def],

              asm_full_simp_tac (simpset_of thy addsimps

                ((Rep RS perm_closed1 RS Abs_inverse) ::

                 (if atom1 = atom2 then []

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

              cong_tac 1,

              rtac refl 1,

              rtac cp1' 1]) thy)

        (typedefs ~~ 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 (apfst snd x)) (atoms ~~ perm_closed_thmss))

        (classes ~~ 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, _) = foldl

      (fn ((i, ("Nominal.noption", _, _)), p) => p

        | ((i, _), (ty_idxs, j)) => (ty_idxs @ [(i, j)], j + 1)) ([], 0) descr;

    fun reindex (DtType (s, dts)) = DtType (s, map reindex dts)

      | reindex (DtRec i) = DtRec (the (AList.lookup op = ty_idxs i))

      | reindex dt = dt;

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

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

    fun strip_nth_name i s = 

      let val xs = NameSpace.unpack s; 

      in NameSpace.pack (Library.nth_map (length xs - i) (strip_suffix 4) xs) end;

    val (descr'', ndescr) = ListPair.unzip (List.mapPartial

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

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

             let

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

               val (constrs1, constrs2) = ListPair.unzip

                 (map (fn (cname, cargs) => apfst (pair (strip_nth_name 2 (strip_nth_name 1 cname)))

                   (foldl_map (fn (dts, dt) =>

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

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

                       ([], cargs))) constrs)

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

               (index, constrs2))

             end) descr);

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

    val descr' = [descr1, descr2];

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

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

    val pdescr = map (fn ((i, (s, dts, constrs)), (_, idxss)) => (i, (s, dts,

      map (fn ((cname, cargs), idxs) => (cname, partition_cargs idxs cargs))

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

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

    val rep_names = map (fn s =>

      Sign.intern_const thy7 ("Rep_" ^ s)) new_type_names;

    val abs_names = map (fn s =>

      Sign.intern_const thy7 ("Abs_" ^ s)) new_type_names;

    val recTs' = List.mapPartial

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

        | (_, T) => SOME T) (descr ~~ get_rec_types descr sorts');

    val recTs = get_rec_types descr'' sorts';

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

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

    val full_new_type_names = map (Sign.full_name (sign_of thy)) new_type_names;

    fun make_constr_def tname T T' ((thy, defs, eqns),

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

      let

        fun constr_arg ((dts, dt), (j, l_args, r_args)) =

          let

            val xs = map (fn (dt, i) => mk_Free "x" (typ_of_dtyp descr'' sorts' dt) i)

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

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

          in

            (j + length dts + 1,

             xs @ x :: l_args,

             foldr mk_abs_fun

               (case dt of

                  DtRec k => if k < length new_type_names then

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

                        typ_of_dtyp descr sorts' dt) $ x

                    else error "nested recursion not (yet) supported"

                | _ => x) xs :: r_args)

          end

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

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

        val rep_name = Sign.intern_const (Theory.sign_of 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 = (Sign.base_name cname) ^ "_def";

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

          Theory.add_consts_i [(cname', constrT, mx)] |>

          (PureThy.add_defs_i false o map Thm.no_attributes) [(def_name, def)]

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

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

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

      let

        val rep_const = cterm_of thy

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

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

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

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

      in

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

      end;

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

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

        List.take (pdescr, length new_type_names) ~~

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

    val abs_inject_thms = map (fn tname =>

      PureThy.get_thm thy8 (Name ("Abs_" ^ tname ^ "_inject"))) new_type_names;

    val rep_inject_thms = map (fn tname =>

      PureThy.get_thm thy8 (Name ("Rep_" ^ tname ^ "_inject"))) new_type_names;

    val rep_thms = map (fn tname =>

      PureThy.get_thm thy8 (Name ("Rep_" ^ tname))) new_type_names;

    val rep_inverse_thms = map (fn tname =>

      PureThy.get_thm thy8 (Name ("Rep_" ^ tname ^ "_inverse"))) new_type_names;

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

    fun prove_constr_rep_thm eqn =

      let

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

        val rewrites = constr_defs @ map mk_meta_eq rep_inverse_thms

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

        [resolve_tac inj_thms 1,

         rewrite_goals_tac rewrites,

         rtac refl 3,

         resolve_tac rep_intrs 2,

         REPEAT (resolve_tac rep_thms 1)])

      end;

    val constr_rep_thmss = map (map prove_constr_rep_thm) constr_rep_eqns;

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

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

      let

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

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

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

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

      in

        Goal.prove_global thy8 [] [] (HOLogic.mk_Trueprop (HOLogic.mk_eq

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

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

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

            perm_closed_thms @ Rep_thms)) 1)

      end) Rep_thms;

    val perm_rep_perm_thms = List.concat (map prove_perm_rep_perm

      (atoms ~~ perm_closed_thmss));

    (* prove distinctness theorems *)

    val distinct_props = setmp DatatypeProp.dtK 1000

      (DatatypeProp.make_distincts new_type_names descr' sorts') thy8;

    val dist_rewrites = map (fn (rep_thms, 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), t::ts) =

          let

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

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

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

            (prove_distinct_thms (p, ts))

          end;

    val distinct_thms = map prove_distinct_thms

      (constr_rep_thmss ~~ dist_lemmas ~~ distinct_props);

    (** prove equations for permutation functions **)

    val abs_perm = PureThy.get_thms thy8 (Name "abs_perm"); (* FIXME *)

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

      let val T = nth_dtyp' i

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

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

        let

          val cname = Sign.intern_const thy8

            (NameSpace.append tname (Sign.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) = foldr constr_arg (1, [], []) dts;

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

        in

          Goal.prove_global thy8 [] []

            (HOLogic.mk_Trueprop (HOLogic.mk_eq

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

            (fn _ => EVERY

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

               simp_tac (HOL_basic_ss addsimps (Rep_thms @ Abs_inverse_thms @

                 constr_defs @ perm_closed_thms)) 1,

               TRY (simp_tac (HOL_basic_ss addsimps

                 (symmetric perm_fun_def :: abs_perm)) 1),

               TRY (simp_tac (HOL_basic_ss addsimps

                 (perm_fun_def :: perm_defs @ Rep_thms @ Abs_inverse_thms @

                    perm_closed_thms)) 1)])

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

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

    (** prove injectivity of constructors **)

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

    val alpha = PureThy.get_thms thy8 (Name "alpha");

    val abs_fresh = PureThy.get_thms thy8 (Name "abs_fresh");

    val fresh_def = PureThy.get_thm thy8 (Name "fresh_def");

    val supp_def = PureThy.get_thm thy8 (Name "supp_def");

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

      let val T = nth_dtyp' i

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

        if null dts then NONE else SOME

        let

          val cname = Sign.intern_const thy8

            (NameSpace.append tname (Sign.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

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

            end;

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

          val Ts = map fastype_of args1;

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

        in

          Goal.prove_global thy8 [] [] (HOLogic.mk_Trueprop (HOLogic.mk_eq

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

               foldr1 HOLogic.mk_conj eqs)))

            (fn _ => EVERY

               [asm_full_simp_tac (simpset_of thy8 addsimps (constr_rep_thm ::

                  rep_inject_thms')) 1,

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

                  alpha @ abs_perm @ abs_fresh @ rep_inject_thms @

                  perm_rep_perm_thms)) 1),

                TRY (asm_full_simp_tac (HOL_basic_ss addsimps (perm_fun_def ::

                  expand_fun_eq :: 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 Un_assoc = PureThy.get_thm thy8 (Name "Un_assoc");

    val de_Morgan_conj = PureThy.get_thm thy8 (Name "de_Morgan_conj");

    val Collect_disj_eq = PureThy.get_thm thy8 (Name "Collect_disj_eq");

    val finite_Un = PureThy.get_thm thy8 (Name "finite_Un");

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

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

      let val T = nth_dtyp' i

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

        let

          val cname = Sign.intern_const thy8

            (NameSpace.append tname (Sign.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), foldr mk_abs_fun x xs :: args2)

            end;

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

          val Ts = map fastype_of args1;

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

          fun supp t =

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

          fun fresh t =

            Const ("Nominal.fresh", atomT --> fastype_of t --> HOLogic.boolT) $

              Free ("a", atomT) $ t;

          val supp_thm = Goal.prove_global thy8 [] []

              (HOLogic.mk_Trueprop (HOLogic.mk_eq

                (supp c,

                 if null dts then Const ("{}", HOLogic.mk_setT atomT)

                 else foldr1 (HOLogic.mk_binop "op Un") (map supp args2))))

            (fn _ =>

              simp_tac (HOL_basic_ss addsimps (supp_def ::

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

                 symmetric empty_def :: Finites.emptyI :: simp_thms @

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

        in

          (supp_thm,

           Goal.prove_global thy8 [] [] (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 (simpset_of thy8 addsimps [fresh_def, supp_thm]) 1))

        end) atoms) constrs)

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

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

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

      let

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

          mk_Free "x" T i;

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

      in (prems @ [HOLogic.imp $ HOLogic.mk_mem (Rep_t,

            Const (List.nth (rep_set_names, i), HOLogic.mk_setT U)) $

              (mk_Free "P" (T --> HOLogic.boolT) (i + 1) $ (Abs_t $ Rep_t))],

          concls @ [mk_Free "P" (T --> HOLogic.boolT) (i + 1) $ mk_Free "x" T i])

      end;

    val (indrule_lemma_prems, indrule_lemma_concls) =

      Library.foldl mk_indrule_lemma (([], []), (descr'' ~~ recTs ~~ recTs'));

    val indrule_lemma = Goal.prove_global thy8 [] []

      (Logic.mk_implies

        (HOLogic.mk_Trueprop (mk_conj indrule_lemma_prems),

         HOLogic.mk_Trueprop (mk_conj indrule_lemma_concls))) (fn _ => EVERY

           [REPEAT (etac conjE 1),

            REPEAT (EVERY

              [TRY (rtac conjI 1), full_simp_tac (HOL_basic_ss addsimps Rep_inverse_thms) 1,