(*  Title:      HOL/Nominal/nominal_atoms.ML

     ID:         $Id$

     Author:     Christian Urban and Stefan Berghofer, TU Muenchen

 Declaration of atom types to be used in nominal datatypes.

 *)

 signature NOMINAL_ATOMS =

 sig

   val create_nom_typedecls : string list -> theory -> theory

   type atom_info

   val get_atom_infos : theory -> atom_info Symtab.table

   val get_atom_info : theory -> string -> atom_info option

   val atoms_of : theory -> string list

   val mk_permT : typ -> typ

 end

 structure NominalAtoms : NOMINAL_ATOMS =

 struct

 val finite_emptyI = @{thm "finite.emptyI"};

 val Collect_const = @{thm "Collect_const"};

 val inductive_forall_def = @{thm "induct_forall_def"};

 (* theory data *)

 type atom_info =

   {pt_class : string,

    fs_class : string,

    cp_classes : (string * string) list};

 structure NominalData = TheoryDataFun

 (

   type T = atom_info Symtab.table;

   val empty = Symtab.empty;

   val copy = I;

   val extend = I;

   fun merge _ x = Symtab.merge (K true) x;

 );

 fun make_atom_info ((pt_class, fs_class), cp_classes) =

   {pt_class = pt_class,

    fs_class = fs_class,

    cp_classes = cp_classes};

 val get_atom_infos = NominalData.get;

 val get_atom_info = Symtab.lookup o NominalData.get;

 fun atoms_of thy = map fst (Symtab.dest (NominalData.get thy));

 fun mk_permT T = HOLogic.listT (HOLogic.mk_prodT (T, T));

 fun mk_Cons x xs =

   let val T = fastype_of x

   in Const ("List.list.Cons", T --> HOLogic.listT T --> HOLogic.listT T) $ x $ xs end;

 (* this function sets up all matters related to atom-  *)

 (* kinds; the user specifies a list of atom-kind names *)

 (* atom_decl <ak1> ... <akn>                           *)

 fun create_nom_typedecls ak_names thy =

   let

     val (_,thy1) = 

     fold_map (fn ak => fn thy => 

           let val dt = ([],ak,NoSyn,[(ak,[@{typ nat}],NoSyn)]) 

               val ({inject,case_thms,...},thy1) = DatatypePackage.add_datatype_i true false [ak] [dt] thy

               val inject_flat = Library.flat inject

               val ak_type = Type (Sign.intern_type thy1 ak,[])

               val ak_sign = Sign.intern_const thy1 ak 

               val inj_type = @{typ nat}-->ak_type

               val inj_on_type = inj_type-->(@{typ "nat set"})-->@{typ bool}  

               (* first statement *)

               val stmnt1 = HOLogic.mk_Trueprop 

                   (Const (@{const_name "inj_on"},inj_on_type) $ 

                          Const (ak_sign,inj_type) $ HOLogic.mk_UNIV @{typ nat})

               val simp1 = @{thm inj_on_def}::inject_flat

               val proof1 = fn _ => EVERY [simp_tac (HOL_basic_ss addsimps simp1) 1,

                                           rtac @{thm ballI} 1,

                                           rtac @{thm ballI} 1,

                                           rtac @{thm impI} 1,

                                           atac 1]

               val (inj_thm,thy2) = 

                    PureThy.add_thms [((ak^"_inj",Goal.prove_global thy1 [] [] stmnt1 proof1), [])] thy1

               (* second statement *)

               val y = Free ("y",ak_type)  

               val stmnt2 = HOLogic.mk_Trueprop

                   (HOLogic.mk_exists ("x",@{typ nat},HOLogic.mk_eq (y,Const (ak_sign,inj_type) $ Bound 0)))

               val proof2 = fn _ => EVERY [case_tac "y" 1,

                                           asm_simp_tac (HOL_basic_ss addsimps simp1) 1,

                                           rtac @{thm exI} 1,

                                           rtac @{thm refl} 1]

               (* third statement *)

               val (inject_thm,thy3) =

                   PureThy.add_thms [((ak^"_injection",Goal.prove_global thy2 [] [] stmnt2 proof2), [])] thy2

               val stmnt3 = HOLogic.mk_Trueprop

                            (HOLogic.mk_not

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

                                   HOLogic.mk_UNIV ak_type))

               val simp2 = [@{thm image_def},@{thm bex_UNIV}]@inject_thm

               val simp3 = [@{thm UNIV_def}]

               val proof3 = fn _ => EVERY [cut_facts_tac inj_thm 1,

                                           dtac @{thm range_inj_infinite} 1,

                                           asm_full_simp_tac (HOL_basic_ss addsimps simp2) 1,

                                           simp_tac (HOL_basic_ss addsimps simp3) 1]  

               val (inf_thm,thy4) =  

                     PureThy.add_thms [((ak^"_infinite",Goal.prove_global thy1 [] [] stmnt3 proof3), [])] thy3

           in 

             ((inj_thm,inject_thm,inf_thm),thy4)

           end) ak_names thy

     (* produces a list consisting of pairs:         *)

     (*  fst component is the atom-kind name         *)

     (*  snd component is its type                   *)

     val full_ak_names = map (Sign.intern_type thy1) ak_names;

     val ak_names_types = ak_names ~~ map (Type o rpair []) full_ak_names;

     (* adds for every atom-kind an axiom             *)

     (* <ak>_infinite: infinite (UNIV::<ak_type> set) *)

     val (inf_axs,thy2) = PureThy.add_axioms_i (map (fn (ak_name, T) =>

       let 

     val name = ak_name ^ "_infinite"

         val axiom = HOLogic.mk_Trueprop (HOLogic.mk_not

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

                        HOLogic.mk_UNIV T))

       in

         ((name, axiom), []) 

       end) ak_names_types) thy1;

     (* declares a swapping function for every atom-kind, it is         *)

     (* const swap_<ak> :: <akT> * <akT> => <akT> => <akT>              *)

     (* swap_<ak> (a,b) c = (if a=c then b (else if b=c then a else c)) *)

     (* overloades then the general swap-function                       *) 

     val (swap_eqs, thy3) = fold_map (fn (ak_name, T) => fn thy =>

       let

         val swapT = HOLogic.mk_prodT (T, T) --> T --> T;

         val swap_name = Sign.full_name thy ("swap_" ^ ak_name);

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

         val b = Free ("b", T);

         val c = Free ("c", T);

         val ab = Free ("ab", HOLogic.mk_prodT (T, T))

         val cif = Const ("HOL.If", HOLogic.boolT --> T --> T --> T);

         val cswap_akname = Const (swap_name, swapT);

         val cswap = Const ("Nominal.swap", swapT)

         val name = "swap_"^ak_name^"_def";

         val def1 = HOLogic.mk_Trueprop (HOLogic.mk_eq

                 (cswap_akname $ HOLogic.mk_prod (a,b) $ c,

                     cif $ HOLogic.mk_eq (a,c) $ b $ (cif $ HOLogic.mk_eq (b,c) $ a $ c)))

         val def2 = Logic.mk_equals (cswap $ ab $ c, cswap_akname $ ab $ c)

       in

         thy |> Sign.add_consts_i [("swap_" ^ ak_name, swapT, NoSyn)] 

             |> PureThy.add_defs_unchecked_i true [((name, def2),[])]

             |> snd

             |> PrimrecPackage.add_primrec_unchecked_i "" [(("", def1),[])]

       end) ak_names_types thy2;

     (* declares a permutation function for every atom-kind acting  *)

     (* on such atoms                                               *)

     (* const <ak>_prm_<ak> :: (<akT> * <akT>)list => akT => akT    *)

     (* <ak>_prm_<ak> []     a = a                                  *)

     (* <ak>_prm_<ak> (x#xs) a = swap_<ak> x (perm xs a)            *)

     val (prm_eqs, thy4) = fold_map (fn (ak_name, T) => fn thy =>

       let

         val swapT = HOLogic.mk_prodT (T, T) --> T --> T;

         val swap_name = Sign.full_name thy ("swap_" ^ ak_name)

         val prmT = mk_permT T --> T --> T;

         val prm_name = ak_name ^ "_prm_" ^ ak_name;

         val qu_prm_name = Sign.full_name thy prm_name;

         val x  = Free ("x", HOLogic.mk_prodT (T, T));

         val xs = Free ("xs", mk_permT T);

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

         val cnil  = Const ("List.list.Nil", mk_permT T);

         val def1 = HOLogic.mk_Trueprop (HOLogic.mk_eq (Const (qu_prm_name, prmT) $ cnil $ a, a));

         val def2 = HOLogic.mk_Trueprop (HOLogic.mk_eq

                    (Const (qu_prm_name, prmT) $ mk_Cons x xs $ a,

                     Const (swap_name, swapT) $ x $ (Const (qu_prm_name, prmT) $ xs $ a)));

       in

         thy |> Sign.add_consts_i [(prm_name, mk_permT T --> T --> T, NoSyn)] 

             |> PrimrecPackage.add_primrec_unchecked_i "" [(("", def1), []),(("", def2), [])]

       end) ak_names_types thy3;

     (* defines permutation functions for all combinations of atom-kinds; *)

     (* there are a trivial cases and non-trivial cases                   *)

     (* non-trivial case:                                                 *)

     (* <ak>_prm_<ak>_def:  perm pi a == <ak>_prm_<ak> pi a               *)

     (* trivial case with <ak> != <ak'>                                   *)

     (* <ak>_prm<ak'>_def[simp]:  perm pi a == a                          *)

     (*                                                                   *)

     (* the trivial cases are added to the simplifier, while the non-     *)

     (* have their own rules proved below                                 *)  

     val (perm_defs, thy5) = fold_map (fn (ak_name, T) => fn thy =>

       fold_map (fn (ak_name', T') => fn thy' =>

         let

           val perm_def_name = ak_name ^ "_prm_" ^ ak_name';

           val pi = Free ("pi", mk_permT T);

           val a  = Free ("a", T');

           val cperm = Const ("Nominal.perm", mk_permT T --> T' --> T');

           val cperm_def = Const (Sign.full_name thy' perm_def_name, mk_permT T --> T' --> T');

           val name = ak_name ^ "_prm_" ^ ak_name' ^ "_def";

           val def = Logic.mk_equals

                     (cperm $ pi $ a, if ak_name = ak_name' then cperm_def $ pi $ a else a)

         in

           PureThy.add_defs_unchecked_i true [((name, def),[])] thy'

         end) ak_names_types thy) ak_names_types thy4;

     (* proves that every atom-kind is an instance of at *)

     (* lemma at_<ak>_inst:                              *)

     (* at TYPE(<ak>)                                    *)

     val (prm_cons_thms,thy6) = 

       thy5 |> PureThy.add_thms (map (fn (ak_name, T) =>

       let

         val ak_name_qu = Sign.full_name thy5 (ak_name);

         val i_type = Type(ak_name_qu,[]);

 	val cat = Const ("Nominal.at",(Term.itselfT i_type)  --> HOLogic.boolT);

         val at_type = Logic.mk_type i_type;

         val simp_s = HOL_ss addsimps PureThy.get_thmss thy5

                                   [Name "at_def",

                                    Name (ak_name ^ "_prm_" ^ ak_name ^ "_def"),

                                    Name (ak_name ^ "_prm_" ^ ak_name ^ ".simps"),

                                    Name ("swap_" ^ ak_name ^ "_def"),

                                    Name ("swap_" ^ ak_name ^ ".simps"),

                                    Name (ak_name ^ "_infinite")]

 	val name = "at_"^ak_name^ "_inst";

         val statement = HOLogic.mk_Trueprop (cat $ at_type);

         val proof = fn _ => simp_tac simp_s 1

       in 

         ((name, Goal.prove_global thy5 [] [] statement proof), []) 

       end) ak_names_types);

     (* declares a perm-axclass for every atom-kind               *)

     (* axclass pt_<ak>                                           *)

     (* pt_<ak>1[simp]: perm [] x = x                             *)

     (* pt_<ak>2:       perm (pi1@pi2) x = perm pi1 (perm pi2 x)  *)

     (* pt_<ak>3:       pi1 ~ pi2 ==> perm pi1 x = perm pi2 x     *)

      val (pt_ax_classes,thy7) =  fold_map (fn (ak_name, T) => fn thy =>

       let 

 	  val cl_name = "pt_"^ak_name;

           val ty = TFree("'a",["HOL.type"]);

           val x   = Free ("x", ty);

           val pi1 = Free ("pi1", mk_permT T);

           val pi2 = Free ("pi2", mk_permT T);

           val cperm = Const ("Nominal.perm", mk_permT T --> ty --> ty);

           val cnil  = Const ("List.list.Nil", mk_permT T);

           val cappend = Const ("List.append",mk_permT T --> mk_permT T --> mk_permT T);

           val cprm_eq = Const ("Nominal.prm_eq",mk_permT T --> mk_permT T --> HOLogic.boolT);

           (* nil axiom *)

           val axiom1 = HOLogic.mk_Trueprop (HOLogic.mk_eq 

                        (cperm $ cnil $ x, x));

           (* append axiom *)

           val axiom2 = HOLogic.mk_Trueprop (HOLogic.mk_eq

                        (cperm $ (cappend $ pi1 $ pi2) $ x, cperm $ pi1 $ (cperm $ pi2 $ x)));

           (* perm-eq axiom *)

           val axiom3 = Logic.mk_implies

                        (HOLogic.mk_Trueprop (cprm_eq $ pi1 $ pi2),

                         HOLogic.mk_Trueprop (HOLogic.mk_eq (cperm $ pi1 $ x, cperm $ pi2 $ x)));

       in

           AxClass.define_class (cl_name, ["HOL.type"]) []

                 [((cl_name ^ "1", [Simplifier.simp_add]), [axiom1]),

                  ((cl_name ^ "2", []), [axiom2]),                           

                  ((cl_name ^ "3", []), [axiom3])] thy                          

       end) ak_names_types thy6;

     (* proves that every pt_<ak>-type together with <ak>-type *)

     (* instance of pt                                         *)

     (* lemma pt_<ak>_inst:                                    *)

     (* pt TYPE('x::pt_<ak>) TYPE(<ak>)                        *)

     val (prm_inst_thms,thy8) = 

       thy7 |> PureThy.add_thms (map (fn (ak_name, T) =>

       let

         val ak_name_qu = Sign.full_name thy7 ak_name;

         val pt_name_qu = Sign.full_name thy7 ("pt_"^ak_name);

         val i_type1 = TFree("'x",[pt_name_qu]);

         val i_type2 = Type(ak_name_qu,[]);

 	val cpt = Const ("Nominal.pt",(Term.itselfT i_type1)-->(Term.itselfT i_type2)-->HOLogic.boolT);

         val pt_type = Logic.mk_type i_type1;

         val at_type = Logic.mk_type i_type2;

         val simp_s = HOL_ss addsimps PureThy.get_thmss thy7

                                   [Name "pt_def",

                                    Name ("pt_" ^ ak_name ^ "1"),

                                    Name ("pt_" ^ ak_name ^ "2"),

                                    Name ("pt_" ^ ak_name ^ "3")];

 	val name = "pt_"^ak_name^ "_inst";

         val statement = HOLogic.mk_Trueprop (cpt $ pt_type $ at_type);

         val proof = fn _ => simp_tac simp_s 1;

       in 

         ((name, Goal.prove_global thy7 [] [] statement proof), []) 

       end) ak_names_types);

      (* declares an fs-axclass for every atom-kind       *)

      (* axclass fs_<ak>                                  *)

      (* fs_<ak>1: finite ((supp x)::<ak> set)            *)

      val (fs_ax_classes,thy11) =  fold_map (fn (ak_name, T) => fn thy =>

        let 

 	  val cl_name = "fs_"^ak_name;

 	  val pt_name = Sign.full_name thy ("pt_"^ak_name);

           val ty = TFree("'a",["HOL.type"]);

           val x   = Free ("x", ty);

           val csupp    = Const ("Nominal.supp", ty --> HOLogic.mk_setT T);

           val cfinite  = Const ("Finite_Set.finite", HOLogic.mk_setT T --> HOLogic.boolT)

           val axiom1   = HOLogic.mk_Trueprop (cfinite $ (csupp $ x));

        in  

         AxClass.define_class (cl_name, [pt_name]) [] [((cl_name ^ "1", []), [axiom1])] thy            

        end) ak_names_types thy8; 

      (* proves that every fs_<ak>-type together with <ak>-type   *)

      (* instance of fs-type                                      *)

      (* lemma abst_<ak>_inst:                                    *)

      (* fs TYPE('x::pt_<ak>) TYPE (<ak>)                         *)

      val (fs_inst_thms,thy12) = 

        thy11 |> PureThy.add_thms (map (fn (ak_name, T) =>

        let

          val ak_name_qu = Sign.full_name thy11 ak_name;

          val fs_name_qu = Sign.full_name thy11 ("fs_"^ak_name);

          val i_type1 = TFree("'x",[fs_name_qu]);

          val i_type2 = Type(ak_name_qu,[]);

  	 val cfs = Const ("Nominal.fs", 

                                  (Term.itselfT i_type1)-->(Term.itselfT i_type2)-->HOLogic.boolT);

          val fs_type = Logic.mk_type i_type1;

          val at_type = Logic.mk_type i_type2;

 	 val simp_s = HOL_ss addsimps PureThy.get_thmss thy11

                                    [Name "fs_def",

                                     Name ("fs_" ^ ak_name ^ "1")];

 	 val name = "fs_"^ak_name^ "_inst";

          val statement = HOLogic.mk_Trueprop (cfs $ fs_type $ at_type);

          val proof = fn _ => simp_tac simp_s 1;

        in 

          ((name, Goal.prove_global thy11 [] [] statement proof), []) 

        end) ak_names_types);

        (* declares for every atom-kind combination an axclass            *)

        (* cp_<ak1>_<ak2> giving a composition property                   *)

        (* cp_<ak1>_<ak2>1: pi1 o pi2 o x = (pi1 o pi2) o (pi1 o x)       *)

         val (cp_ax_classes,thy12b) = fold_map (fn (ak_name, T) => fn thy =>

 	 fold_map (fn (ak_name', T') => fn thy' =>

 	     let

 	       val cl_name = "cp_"^ak_name^"_"^ak_name';

 	       val ty = TFree("'a",["HOL.type"]);

                val x   = Free ("x", ty);

                val pi1 = Free ("pi1", mk_permT T);

 	       val pi2 = Free ("pi2", mk_permT T');                  

 	       val cperm1 = Const ("Nominal.perm", mk_permT T  --> ty --> ty);

                val cperm2 = Const ("Nominal.perm", mk_permT T' --> ty --> ty);

                val cperm3 = Const ("Nominal.perm", mk_permT T  --> mk_permT T' --> mk_permT T');

                val ax1   = HOLogic.mk_Trueprop 

 			   (HOLogic.mk_eq (cperm1 $ pi1 $ (cperm2 $ pi2 $ x), 

                                            cperm2 $ (cperm3 $ pi1 $ pi2) $ (cperm1 $ pi1 $ x)));

 	       in  

 		 AxClass.define_class (cl_name, ["HOL.type"]) [] [((cl_name ^ "1", []), [ax1])] thy'  

 	       end) ak_names_types thy) ak_names_types thy12;

         (* proves for every <ak>-combination a cp_<ak1>_<ak2>_inst theorem;     *)

         (* lemma cp_<ak1>_<ak2>_inst:                                           *)

         (* cp TYPE('a::cp_<ak1>_<ak2>) TYPE(<ak1>) TYPE(<ak2>)                  *)

         val (cp_thms,thy12c) = fold_map (fn (ak_name, T) => fn thy =>

 	 fold_map (fn (ak_name', T') => fn thy' =>

            let

              val ak_name_qu  = Sign.full_name thy' (ak_name);

 	     val ak_name_qu' = Sign.full_name thy' (ak_name');

              val cp_name_qu  = Sign.full_name thy' ("cp_"^ak_name^"_"^ak_name');

              val i_type0 = TFree("'a",[cp_name_qu]);

              val i_type1 = Type(ak_name_qu,[]);

              val i_type2 = Type(ak_name_qu',[]);

 	     val ccp = Const ("Nominal.cp",

                              (Term.itselfT i_type0)-->(Term.itselfT i_type1)-->

                                                       (Term.itselfT i_type2)-->HOLogic.boolT);

              val at_type  = Logic.mk_type i_type1;

              val at_type' = Logic.mk_type i_type2;

 	     val cp_type  = Logic.mk_type i_type0;

              val simp_s   = HOL_basic_ss addsimps PureThy.get_thmss thy' [(Name "cp_def")];

 	     val cp1      = PureThy.get_thm thy' (Name ("cp_"^ak_name^"_"^ak_name'^"1"));

 	     val name = "cp_"^ak_name^ "_"^ak_name'^"_inst";

              val statement = HOLogic.mk_Trueprop (ccp $ cp_type $ at_type $ at_type');

              val proof = fn _ => EVERY [simp_tac simp_s 1, 

                                         rtac allI 1, rtac allI 1, rtac allI 1,

                                         rtac cp1 1];

 	   in

 	     PureThy.add_thms [((name, Goal.prove_global thy' [] [] statement proof), [])] thy'

 	   end) 

            ak_names_types thy) ak_names_types thy12b;

         (* proves for every non-trivial <ak>-combination a disjointness   *)

         (* theorem; i.e. <ak1> != <ak2>                                   *)

         (* lemma ds_<ak1>_<ak2>:                                          *)

         (* dj TYPE(<ak1>) TYPE(<ak2>)                                     *)

         val (dj_thms, thy12d) = fold_map (fn (ak_name,T) => fn thy =>

 	  fold_map (fn (ak_name',T') => fn thy' =>

           (if not (ak_name = ak_name') 

            then 

 	       let

 		 val ak_name_qu  = Sign.full_name thy' ak_name;

 	         val ak_name_qu' = Sign.full_name thy' ak_name';

                  val i_type1 = Type(ak_name_qu,[]);

                  val i_type2 = Type(ak_name_qu',[]);

 	         val cdj = Const ("Nominal.disjoint",

                            (Term.itselfT i_type1)-->(Term.itselfT i_type2)-->HOLogic.boolT);

                  val at_type  = Logic.mk_type i_type1;

                  val at_type' = Logic.mk_type i_type2;

                  val simp_s = HOL_ss addsimps PureThy.get_thmss thy' 

 					   [Name "disjoint_def",

                                             Name (ak_name^"_prm_"^ak_name'^"_def"),

                                             Name (ak_name'^"_prm_"^ak_name^"_def")];

 	         val name = "dj_"^ak_name^"_"^ak_name';

                  val statement = HOLogic.mk_Trueprop (cdj $ at_type $ at_type');

                  val proof = fn _ => simp_tac simp_s 1;

 	       in

 		PureThy.add_thms [((name, Goal.prove_global thy' [] [] statement proof), [])] thy'

 	       end

            else 

             ([],thy')))  (* do nothing branch, if ak_name = ak_name' *) 

 	    ak_names_types thy) ak_names_types thy12c;

      (********  pt_<ak> class instances  ********)

      (*=========================================*)

      (* some abbreviations for theorems *)

       val pt1           = @{thm "pt1"};

       val pt2           = @{thm "pt2"};

       val pt3           = @{thm "pt3"};

       val at_pt_inst    = @{thm "at_pt_inst"};

       val pt_set_inst   = @{thm "pt_set_inst"}; 

       val pt_unit_inst  = @{thm "pt_unit_inst"};

       val pt_prod_inst  = @{thm "pt_prod_inst"}; 

       val pt_nprod_inst = @{thm "pt_nprod_inst"}; 

       val pt_list_inst  = @{thm "pt_list_inst"};

       val pt_optn_inst  = @{thm "pt_option_inst"};

       val pt_noptn_inst = @{thm "pt_noption_inst"};

       val pt_fun_inst   = @{thm "pt_fun_inst"};

      (* for all atom-kind combinations <ak>/<ak'> show that        *)

      (* every <ak> is an instance of pt_<ak'>; the proof for       *)

      (* ak!=ak' is by definition; the case ak=ak' uses at_pt_inst. *)

      val thy13 = fold (fn ak_name => fn thy =>

 	fold (fn ak_name' => fn thy' =>

          let

            val qu_name =  Sign.full_name thy' ak_name';

            val cls_name = Sign.full_name thy' ("pt_"^ak_name);

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

            val proof1 = EVERY [Class.intro_classes_tac [],

                                  rtac ((at_inst RS at_pt_inst) RS pt1) 1,

                                  rtac ((at_inst RS at_pt_inst) RS pt2) 1,

                                  rtac ((at_inst RS at_pt_inst) RS pt3) 1,

                                  atac 1];

            val simp_s = HOL_basic_ss addsimps 

                         PureThy.get_thmss thy' [Name (ak_name^"_prm_"^ak_name'^"_def")];  

            val proof2 = EVERY [Class.intro_classes_tac [], REPEAT (asm_simp_tac simp_s 1)];

          in

            thy'

            |> AxClass.prove_arity (qu_name,[],[cls_name])

               (if ak_name = ak_name' then proof1 else proof2)

          end) ak_names thy) ak_names thy12c;

      (* show that                       *)

      (*      fun(pt_<ak>,pt_<ak>)       *)

      (*      noption(pt_<ak>)           *)

      (*      option(pt_<ak>)            *)

      (*      list(pt_<ak>)              *)

      (*      *(pt_<ak>,pt_<ak>)         *)

      (*      nprod(pt_<ak>,pt_<ak>)     *)

      (*      unit                       *)

      (*      set(pt_<ak>)               *)

      (* are instances of pt_<ak>        *)

      val thy18 = fold (fn ak_name => fn thy =>

        let

           val cls_name = Sign.full_name thy ("pt_"^ak_name);

           val at_thm   = PureThy.get_thm thy (Name ("at_"^ak_name^"_inst"));

           val pt_inst  = PureThy.get_thm thy (Name ("pt_"^ak_name^"_inst"));

           fun pt_proof thm = 

               EVERY [Class.intro_classes_tac [],

                      rtac (thm RS pt1) 1, rtac (thm RS pt2) 1, rtac (thm RS pt3) 1, atac 1];

           val pt_thm_fun   = at_thm RS (pt_inst RS (pt_inst RS pt_fun_inst));

           val pt_thm_noptn = pt_inst RS pt_noptn_inst; 

           val pt_thm_optn  = pt_inst RS pt_optn_inst; 

           val pt_thm_list  = pt_inst RS pt_list_inst;

           val pt_thm_prod  = pt_inst RS (pt_inst RS pt_prod_inst);

           val pt_thm_nprod = pt_inst RS (pt_inst RS pt_nprod_inst);

           val pt_thm_unit  = pt_unit_inst;

           val pt_thm_set   = pt_inst RS pt_set_inst

        in

         thy

         |> AxClass.prove_arity ("fun",[[cls_name],[cls_name]],[cls_name]) (pt_proof pt_thm_fun)

         |> AxClass.prove_arity ("Nominal.noption",[[cls_name]],[cls_name]) (pt_proof pt_thm_noptn) 

         |> AxClass.prove_arity ("Datatype.option",[[cls_name]],[cls_name]) (pt_proof pt_thm_optn)

         |> AxClass.prove_arity ("List.list",[[cls_name]],[cls_name]) (pt_proof pt_thm_list)

         |> AxClass.prove_arity ("*",[[cls_name],[cls_name]],[cls_name]) (pt_proof pt_thm_prod)

         |> AxClass.prove_arity ("Nominal.nprod",[[cls_name],[cls_name]],[cls_name]) 

                                     (pt_proof pt_thm_nprod)

         |> AxClass.prove_arity ("Product_Type.unit",[],[cls_name]) (pt_proof pt_thm_unit)

         |> AxClass.prove_arity ("set",[[cls_name]],[cls_name]) (pt_proof pt_thm_set)

      end) ak_names thy13; 

        (********  fs_<ak> class instances  ********)

        (*=========================================*)

        (* abbreviations for some lemmas *)

        val fs1            = @{thm "fs1"};

        val fs_at_inst     = @{thm "fs_at_inst"};

        val fs_unit_inst   = @{thm "fs_unit_inst"};

        val fs_prod_inst   = @{thm "fs_prod_inst"};

        val fs_nprod_inst  = @{thm "fs_nprod_inst"};

        val fs_list_inst   = @{thm "fs_list_inst"};

        val fs_option_inst = @{thm "fs_option_inst"};

        val dj_supp        = @{thm "dj_supp"};

        (* shows that <ak> is an instance of fs_<ak>     *)

        (* uses the theorem at_<ak>_inst                 *)

        val thy20 = fold (fn ak_name => fn thy =>

         fold (fn ak_name' => fn thy' =>

         let

            val qu_name =  Sign.full_name thy' ak_name';

            val qu_class = Sign.full_name thy' ("fs_"^ak_name);

            val proof =

                (if ak_name = ak_name'

                 then

                   let val at_thm = PureThy.get_thm thy' (Name ("at_"^ak_name^"_inst"));

                   in  EVERY [Class.intro_classes_tac [],

                              rtac ((at_thm RS fs_at_inst) RS fs1) 1] end

                 else

                   let val dj_inst = PureThy.get_thm thy' (Name ("dj_"^ak_name'^"_"^ak_name));

                       val simp_s = HOL_basic_ss addsimps [dj_inst RS dj_supp, finite_emptyI];

                   in EVERY [Class.intro_classes_tac [], asm_simp_tac simp_s 1] end)

         in

          AxClass.prove_arity (qu_name,[],[qu_class]) proof thy'

         end) ak_names thy) ak_names thy18;

        (* shows that                  *)

        (*    unit                     *)

        (*    *(fs_<ak>,fs_<ak>)       *)

        (*    nprod(fs_<ak>,fs_<ak>)   *)

        (*    list(fs_<ak>)            *)

        (*    option(fs_<ak>)          *) 

        (* are instances of fs_<ak>    *)

        val thy24 = fold (fn ak_name => fn thy => 

         let

           val cls_name = Sign.full_name thy ("fs_"^ak_name);

           val fs_inst  = PureThy.get_thm thy (Name ("fs_"^ak_name^"_inst"));

           fun fs_proof thm = EVERY [Class.intro_classes_tac [], rtac (thm RS fs1) 1];

           val fs_thm_unit  = fs_unit_inst;

           val fs_thm_prod  = fs_inst RS (fs_inst RS fs_prod_inst);

           val fs_thm_nprod = fs_inst RS (fs_inst RS fs_nprod_inst);

           val fs_thm_list  = fs_inst RS fs_list_inst;

           val fs_thm_optn  = fs_inst RS fs_option_inst;

         in 

          thy

          |> AxClass.prove_arity ("Product_Type.unit",[],[cls_name]) (fs_proof fs_thm_unit) 

          |> AxClass.prove_arity ("*",[[cls_name],[cls_name]],[cls_name]) (fs_proof fs_thm_prod) 

          |> AxClass.prove_arity ("Nominal.nprod",[[cls_name],[cls_name]],[cls_name]) 

                                      (fs_proof fs_thm_nprod) 

          |> AxClass.prove_arity ("List.list",[[cls_name]],[cls_name]) (fs_proof fs_thm_list)

          |> AxClass.prove_arity ("Datatype.option",[[cls_name]],[cls_name]) (fs_proof fs_thm_optn)

         end) ak_names thy20;

        (********  cp_<ak>_<ai> class instances  ********)

        (*==============================================*)

        (* abbreviations for some lemmas *)

        val cp1             = @{thm "cp1"};

        val cp_unit_inst    = @{thm "cp_unit_inst"};

        val cp_bool_inst    = @{thm "cp_bool_inst"};

        val cp_prod_inst    = @{thm "cp_prod_inst"};

        val cp_list_inst    = @{thm "cp_list_inst"};

        val cp_fun_inst     = @{thm "cp_fun_inst"};

        val cp_option_inst  = @{thm "cp_option_inst"};

        val cp_noption_inst = @{thm "cp_noption_inst"};

        val cp_set_inst     = @{thm "cp_set_inst"};

        val pt_perm_compose = @{thm "pt_perm_compose"};

        val dj_pp_forget    = @{thm "dj_perm_perm_forget"};

        (* shows that <aj> is an instance of cp_<ak>_<ai>  *)

        (* for every  <ak>/<ai>-combination                *)

        val thy25 = fold (fn ak_name => fn thy =>

          fold (fn ak_name' => fn thy' =>

           fold (fn ak_name'' => fn thy'' =>

             let

               val name =  Sign.full_name thy'' ak_name;

               val cls_name = Sign.full_name thy'' ("cp_"^ak_name'^"_"^ak_name'');

               val proof =

                 (if (ak_name'=ak_name'') then 

                   (let

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

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

                   in

 		   EVERY [Class.intro_classes_tac [],

                           rtac (at_inst RS (pt_inst RS pt_perm_compose)) 1]

                   end)

 		else

 		  (let

                      val dj_inst  = PureThy.get_thm thy'' (Name ("dj_"^ak_name''^"_"^ak_name'));

 		     val simp_s = HOL_basic_ss addsimps

                                         ((dj_inst RS dj_pp_forget)::

                                          (PureThy.get_thmss thy''

                                            [Name (ak_name' ^"_prm_"^ak_name^"_def"),

                                             Name (ak_name''^"_prm_"^ak_name^"_def")]));

                   in

                     EVERY [Class.intro_classes_tac [], simp_tac simp_s 1]

                   end))

               in

                 AxClass.prove_arity (name,[],[cls_name]) proof thy''

               end) ak_names thy') ak_names thy) ak_names thy24;

        (* shows that                                                    *) 

        (*      units                                                    *) 

        (*      products                                                 *)

        (*      lists                                                    *)

        (*      functions                                                *)

        (*      options                                                  *)

        (*      noptions                                                 *)

        (*      sets                                                     *)

        (* are instances of cp_<ak>_<ai> for every <ak>/<ai>-combination *)

        val thy26 = fold (fn ak_name => fn thy =>

 	fold (fn ak_name' => fn thy' =>

         let

             val cls_name = Sign.full_name thy' ("cp_"^ak_name^"_"^ak_name');

             val cp_inst  = PureThy.get_thm thy' (Name ("cp_"^ak_name^"_"^ak_name'^"_inst"));

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

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

             fun cp_proof thm  = EVERY [Class.intro_classes_tac [],rtac (thm RS cp1) 1];

             val cp_thm_unit = cp_unit_inst;

             val cp_thm_prod = cp_inst RS (cp_inst RS cp_prod_inst);

             val cp_thm_list = cp_inst RS cp_list_inst;

             val cp_thm_fun  = at_inst RS (pt_inst RS (cp_inst RS (cp_inst RS cp_fun_inst)));

             val cp_thm_optn = cp_inst RS cp_option_inst;

             val cp_thm_noptn = cp_inst RS cp_noption_inst;

             val cp_thm_set = cp_inst RS cp_set_inst;

         in

          thy'

          |> AxClass.prove_arity ("Product_Type.unit",[],[cls_name]) (cp_proof cp_thm_unit)

 	 |> AxClass.prove_arity ("*",[[cls_name],[cls_name]],[cls_name]) (cp_proof cp_thm_prod)

          |> AxClass.prove_arity ("List.list",[[cls_name]],[cls_name]) (cp_proof cp_thm_list)

          |> AxClass.prove_arity ("fun",[[cls_name],[cls_name]],[cls_name]) (cp_proof cp_thm_fun)

          |> AxClass.prove_arity ("Datatype.option",[[cls_name]],[cls_name]) (cp_proof cp_thm_optn)

          |> AxClass.prove_arity ("Nominal.noption",[[cls_name]],[cls_name]) (cp_proof cp_thm_noptn)

          |> AxClass.prove_arity ("set",[[cls_name]],[cls_name]) (cp_proof cp_thm_set)

         end) ak_names thy) ak_names thy25;

      (* show that discrete nominal types are permutation types, finitely     *)

      (* supported and have the commutation property                          *)

      (* discrete types have a permutation operation defined as pi o x = x;   *)

      (* which renders the proofs to be simple "simp_all"-proofs.             *)

      val thy32 =

         let

 	  fun discrete_pt_inst discrete_ty defn =

 	     fold (fn ak_name => fn thy =>

 	     let

 	       val qu_class = Sign.full_name thy ("pt_"^ak_name);

 	       val simp_s = HOL_basic_ss addsimps [defn];

                val proof = EVERY [Class.intro_classes_tac [], REPEAT (asm_simp_tac simp_s 1)];

              in 

 	       AxClass.prove_arity (discrete_ty,[],[qu_class]) proof thy

              end) ak_names;

           fun discrete_fs_inst discrete_ty defn = 

 	     fold (fn ak_name => fn thy =>

 	     let

 	       val qu_class = Sign.full_name thy ("fs_"^ak_name);

 	       val supp_def = @{thm "Nominal.supp_def"};

                val simp_s = HOL_ss addsimps [supp_def,Collect_const,finite_emptyI,defn];

                val proof = EVERY [Class.intro_classes_tac [], asm_simp_tac simp_s 1];

              in 

 	       AxClass.prove_arity (discrete_ty,[],[qu_class]) proof thy

              end) ak_names;

           fun discrete_cp_inst discrete_ty defn = 

 	     fold (fn ak_name' => (fold (fn ak_name => fn thy =>

 	     let

 	       val qu_class = Sign.full_name thy ("cp_"^ak_name^"_"^ak_name');

 	       val supp_def = @{thm "Nominal.supp_def"};

                val simp_s = HOL_ss addsimps [defn];

                val proof = EVERY [Class.intro_classes_tac [], asm_simp_tac simp_s 1];

              in

 	       AxClass.prove_arity (discrete_ty,[],[qu_class]) proof thy

              end) ak_names)) ak_names;

         in

          thy26

          |> discrete_pt_inst "nat"  @{thm "perm_nat_def"}

          |> discrete_fs_inst "nat"  @{thm "perm_nat_def"}

          |> discrete_cp_inst "nat"  @{thm "perm_nat_def"}

          |> discrete_pt_inst "bool" @{thm "perm_bool"}

          |> discrete_fs_inst "bool" @{thm "perm_bool"}

          |> discrete_cp_inst "bool" @{thm "perm_bool"}

          |> discrete_pt_inst "IntDef.int" @{thm "perm_int_def"}

          |> discrete_fs_inst "IntDef.int" @{thm "perm_int_def"}

          |> discrete_cp_inst "IntDef.int" @{thm "perm_int_def"}

          |> discrete_pt_inst "List.char" @{thm "perm_char_def"}

          |> discrete_fs_inst "List.char" @{thm "perm_char_def"}

          |> discrete_cp_inst "List.char" @{thm "perm_char_def"}

         end;

        (* abbreviations for some lemmas *)

        (*===============================*)

        val abs_fun_pi          = @{thm "Nominal.abs_fun_pi"};

        val abs_fun_pi_ineq     = @{thm "Nominal.abs_fun_pi_ineq"};

        val abs_fun_eq          = @{thm "Nominal.abs_fun_eq"};

        val abs_fun_eq'         = @{thm "Nominal.abs_fun_eq'"};

        val abs_fun_fresh       = @{thm "Nominal.abs_fun_fresh"};

        val abs_fun_fresh'      = @{thm "Nominal.abs_fun_fresh'"};

        val dj_perm_forget      = @{thm "Nominal.dj_perm_forget"};

        val dj_pp_forget        = @{thm "Nominal.dj_perm_perm_forget"};

        val fresh_iff           = @{thm "Nominal.fresh_abs_fun_iff"};

        val fresh_iff_ineq      = @{thm "Nominal.fresh_abs_fun_iff_ineq"};

        val abs_fun_supp        = @{thm "Nominal.abs_fun_supp"};

        val abs_fun_supp_ineq   = @{thm "Nominal.abs_fun_supp_ineq"};

        val pt_swap_bij         = @{thm "Nominal.pt_swap_bij"};

        val pt_swap_bij'        = @{thm "Nominal.pt_swap_bij'"};

        val pt_fresh_fresh      = @{thm "Nominal.pt_fresh_fresh"};

        val pt_bij              = @{thm "Nominal.pt_bij"};

        val pt_perm_compose     = @{thm "Nominal.pt_perm_compose"};

        val pt_perm_compose'    = @{thm "Nominal.pt_perm_compose'"};

        val perm_app            = @{thm "Nominal.pt_fun_app_eq"};

        val at_fresh            = @{thm "Nominal.at_fresh"};

        val at_fresh_ineq       = @{thm "Nominal.at_fresh_ineq"};

        val at_calc             = @{thms "Nominal.at_calc"};

        val at_swap_simps       = @{thms "Nominal.at_swap_simps"};

        val at_supp             = @{thm "Nominal.at_supp"};

        val dj_supp             = @{thm "Nominal.dj_supp"};

        val fresh_left_ineq     = @{thm "Nominal.pt_fresh_left_ineq"};

        val fresh_left          = @{thm "Nominal.pt_fresh_left"};

        val fresh_right_ineq    = @{thm "Nominal.pt_fresh_right_ineq"};

        val fresh_right         = @{thm "Nominal.pt_fresh_right"};

        val fresh_bij_ineq      = @{thm "Nominal.pt_fresh_bij_ineq"};

        val fresh_bij           = @{thm "Nominal.pt_fresh_bij"};

        val fresh_eqvt          = @{thm "Nominal.pt_fresh_eqvt"};

        val fresh_eqvt_ineq     = @{thm "Nominal.pt_fresh_eqvt_ineq"};

        val set_diff_eqvt       = @{thm "Nominal.pt_set_diff_eqvt"};

        val in_eqvt             = @{thm "Nominal.pt_in_eqvt"};

        val eq_eqvt             = @{thm "Nominal.pt_eq_eqvt"};

        val all_eqvt            = @{thm "Nominal.pt_all_eqvt"};

        val ex_eqvt             = @{thm "Nominal.pt_ex_eqvt"};

        val pt_pi_rev           = @{thm "Nominal.pt_pi_rev"};

        val pt_rev_pi           = @{thm "Nominal.pt_rev_pi"};

        val at_exists_fresh     = @{thm "Nominal.at_exists_fresh"};

        val at_exists_fresh'    = @{thm "Nominal.at_exists_fresh'"};

        val fresh_perm_app_ineq = @{thm "Nominal.pt_fresh_perm_app_ineq"};

        val fresh_perm_app      = @{thm "Nominal.pt_fresh_perm_app"};	

        val fresh_aux           = @{thm "Nominal.pt_fresh_aux"};  

        val pt_perm_supp_ineq   = @{thm "Nominal.pt_perm_supp_ineq"};

        val pt_perm_supp        = @{thm "Nominal.pt_perm_supp"};

        (* Now we collect and instantiate some lemmas w.r.t. all atom      *)

        (* types; this allows for example to use abs_perm (which is a      *)

        (* collection of theorems) instead of thm abs_fun_pi with explicit *)

        (* instantiations.                                                 *)

        val (_, thy33) =

          let

              (* takes a theorem thm and a list of theorems [t1,..,tn]            *)

              (* produces a list of theorems of the form [t1 RS thm,..,tn RS thm] *) 

              fun instR thm thms = map (fn ti => ti RS thm) thms;

              (* takes two theorem lists (hopefully of the same length ;o)                *)

              (* produces a list of theorems of the form                                  *)

              (* [t1 RS m1,..,tn RS mn] where [t1,..,tn] is thms1 and [m1,..,mn] is thms2 *) 

              fun inst_zip thms1 thms2 = map (fn (t1,t2) => t1 RS t2) (thms1 ~~ thms2);

              (* takes a theorem list of the form [l1,...,ln]              *)

              (* and a list of theorem lists of the form                   *)

              (* [[h11,...,h1m],....,[hk1,....,hkm]                        *)

              (* produces the list of theorem lists                        *)

              (* [[l1 RS h11,...,l1 RS h1m],...,[ln RS hk1,...,ln RS hkm]] *)

              fun inst_mult thms thmss = map (fn (t,ts) => instR t ts) (thms ~~ thmss);

              (* FIXME: these lists do not need to be created dynamically again *)

              (* list of all at_inst-theorems *)

              val ats = map (fn ak => PureThy.get_thm thy32 (Name ("at_"^ak^"_inst"))) ak_names

              (* list of all pt_inst-theorems *)

              val pts = map (fn ak => PureThy.get_thm thy32 (Name ("pt_"^ak^"_inst"))) ak_names

              (* list of all cp_inst-theorems as a collection of lists*)

              val cps = 

 		 let fun cps_fun ak1 ak2 = PureThy.get_thm thy32 (Name ("cp_"^ak1^"_"^ak2^"_inst"))

 		 in map (fn aki => (map (cps_fun aki) ak_names)) ak_names end; 

              (* list of all cp_inst-theorems that have different atom types *)

              val cps' = 

 		let fun cps'_fun ak1 ak2 = 

 		if ak1=ak2 then NONE else SOME(PureThy.get_thm thy32 (Name ("cp_"^ak1^"_"^ak2^"_inst")))

 		in map (fn aki => (List.mapPartial I (map (cps'_fun aki) ak_names))) ak_names end;

              (* list of all dj_inst-theorems *)

              val djs = 

 	       let fun djs_fun (ak1,ak2) = 

 		     if ak1=ak2 then NONE else SOME(PureThy.get_thm thy32 (Name ("dj_"^ak2^"_"^ak1)))

 	       in List.mapPartial I (map djs_fun (Library.product ak_names ak_names)) end;

              (* list of all fs_inst-theorems *)

              val fss = map (fn ak => PureThy.get_thm thy32 (Name ("fs_"^ak^"_inst"))) ak_names

              (* list of all at_inst-theorems *)

              val fs_axs = map (fn ak => PureThy.get_thm thy32 (Name ("fs_"^ak^"1"))) ak_names

              fun inst_pt thms = Library.flat (map (fn ti => instR ti pts) thms);

              fun inst_at thms = Library.flat (map (fn ti => instR ti ats) thms);

              fun inst_fs thms = Library.flat (map (fn ti => instR ti fss) thms);

              fun inst_cp thms cps = Library.flat (inst_mult thms cps);

 	     fun inst_pt_at thms = inst_zip ats (inst_pt thms);

              fun inst_dj thms = Library.flat (map (fn ti => instR ti djs) thms);

 	     fun inst_pt_pt_at_cp thms = inst_cp (inst_zip ats (inst_zip pts (inst_pt thms))) cps;

              fun inst_pt_at_fs thms = inst_zip (inst_fs [fs1]) (inst_zip ats (inst_pt thms));

 	     fun inst_pt_pt_at_cp thms =

 		 let val i_pt_pt_at = inst_zip ats (inst_zip pts (inst_pt thms));

                      val i_pt_pt_at_cp = inst_cp i_pt_pt_at cps';

 		 in i_pt_pt_at_cp end;

              fun inst_pt_pt_at_cp_dj thms = inst_zip djs (inst_pt_pt_at_cp thms);

            in

             thy32 

 	    |>   PureThy.add_thmss [(("alpha", inst_pt_at [abs_fun_eq]),[])]

             ||>> PureThy.add_thmss [(("alpha'", inst_pt_at [abs_fun_eq']),[])]

             ||>> PureThy.add_thmss [(("alpha_fresh", inst_pt_at [abs_fun_fresh]),[])]

             ||>> PureThy.add_thmss [(("alpha_fresh'", inst_pt_at [abs_fun_fresh']),[])]

             ||>> PureThy.add_thmss [(("perm_swap", inst_pt_at [pt_swap_bij] @ inst_pt_at [pt_swap_bij']),[])]

             ||>> PureThy.add_thmss [(("swap_simps", inst_at at_swap_simps),[])]	 

             ||>> PureThy.add_thmss 

 	      let val thms1 = inst_pt_at [pt_pi_rev];

 		  val thms2 = inst_pt_at [pt_rev_pi];

               in [(("perm_pi_simp",thms1 @ thms2),[])] end

             ||>> PureThy.add_thmss [(("perm_fresh_fresh", inst_pt_at [pt_fresh_fresh]),[])]

             ||>> PureThy.add_thmss [(("perm_bij", inst_pt_at [pt_bij]),[])]

             ||>> PureThy.add_thmss 

 	      let val thms1 = inst_pt_at [pt_perm_compose];

 		  val thms2 = instR cp1 (Library.flat cps');

               in [(("perm_compose",thms1 @ thms2),[])] end

             ||>> PureThy.add_thmss [(("perm_compose'",inst_pt_at [pt_perm_compose']),[])] 

             ||>> PureThy.add_thmss [(("perm_app", inst_pt_at [perm_app]),[])]

             ||>> PureThy.add_thmss [(("supp_atm", (inst_at [at_supp]) @ (inst_dj [dj_supp])),[])]

             ||>> PureThy.add_thmss [(("exists_fresh", inst_at [at_exists_fresh]),[])]

             ||>> PureThy.add_thmss [(("exists_fresh'", inst_at [at_exists_fresh']),[])]

             ||>> PureThy.add_thmss

               let

                 val thms1 = inst_pt_at [all_eqvt];

                 val thms2 = map (fold_rule [inductive_forall_def]) thms1

               in

                 [(("all_eqvt", thms1 @ thms2), [NominalThmDecls.eqvt_force_add])]

               end

             ||>> PureThy.add_thmss [(("ex_eqvt", inst_pt_at [ex_eqvt]),[NominalThmDecls.eqvt_force_add])]

             ||>> PureThy.add_thmss 

 	      let val thms1 = inst_at [at_fresh]

 		  val thms2 = inst_dj [at_fresh_ineq]

 	      in [(("fresh_atm", thms1 @ thms2),[])] end

             ||>> PureThy.add_thmss

 	      let val thms1 = filter

                 (fn th => case prop_of th of _ $ _ $ Var _ => true | _ => false)

                 (List.concat (List.concat perm_defs))

               in [(("calc_atm", (inst_at at_calc) @ thms1),[])] end

             ||>> PureThy.add_thmss

 	      let val thms1 = inst_pt_at [abs_fun_pi]

 		  and thms2 = inst_pt_pt_at_cp [abs_fun_pi_ineq]

 	      in [(("abs_perm", thms1 @ thms2),[NominalThmDecls.eqvt_add])] end

             ||>> PureThy.add_thmss

 	      let val thms1 = inst_dj [dj_perm_forget]

 		  and thms2 = inst_dj [dj_pp_forget]

               in [(("perm_dj", thms1 @ thms2),[])] end

             ||>> PureThy.add_thmss

 	      let val thms1 = inst_pt_at_fs [fresh_iff]

                   and thms2 = inst_pt_at [fresh_iff]

 		  and thms3 = inst_pt_pt_at_cp_dj [fresh_iff_ineq]

 	    in [(("abs_fresh", thms1 @ thms2 @ thms3),[])] end

 	    ||>> PureThy.add_thmss

 	      let val thms1 = inst_pt_at [abs_fun_supp]

 		  and thms2 = inst_pt_at_fs [abs_fun_supp]

 		  and thms3 = inst_pt_pt_at_cp_dj [abs_fun_supp_ineq]

 	      in [(("abs_supp", thms1 @ thms2 @ thms3),[])] end

             ||>> PureThy.add_thmss

 	      let val thms1 = inst_pt_at [fresh_left]

 		  and thms2 = inst_pt_pt_at_cp [fresh_left_ineq]

 	      in [(("fresh_left", thms1 @ thms2),[])] end

             ||>> PureThy.add_thmss

 	      let val thms1 = inst_pt_at [fresh_right]

 		  and thms2 = inst_pt_pt_at_cp [fresh_right_ineq]

 	      in [(("fresh_right", thms1 @ thms2),[])] end

             ||>> PureThy.add_thmss

 	      let val thms1 = inst_pt_at [fresh_bij]

  		  and thms2 = inst_pt_pt_at_cp [fresh_bij_ineq]

 	      in [(("fresh_bij", thms1 @ thms2),[])] end

             ||>> PureThy.add_thmss

 	      let val thms1 = inst_pt_at [fresh_eqvt]

                   and thms2 = inst_pt_pt_at_cp_dj [fresh_eqvt_ineq]

 	      in [(("fresh_eqvt", thms1 @ thms2),[NominalThmDecls.eqvt_add])] end

             ||>> PureThy.add_thmss

 	      let val thms1 = inst_pt_at [in_eqvt]

 	      in [(("in_eqvt", thms1),[NominalThmDecls.eqvt_add])] end

   	    ||>> PureThy.add_thmss

 	      let val thms1 = inst_pt_at [eq_eqvt]

 	      in [(("eq_eqvt", thms1),[NominalThmDecls.eqvt_add])] end

 	    ||>> PureThy.add_thmss

 	      let val thms1 = inst_pt_at [set_diff_eqvt]

 	      in [(("set_diff_eqvt", thms1),[NominalThmDecls.eqvt_add])] end

             ||>> PureThy.add_thmss

 	      let val thms1 = inst_pt_at [fresh_aux]

 		  and thms2 = inst_pt_pt_at_cp_dj [fresh_perm_app_ineq] 

 	      in  [(("fresh_aux", thms1 @ thms2),[])] end  

             ||>> PureThy.add_thmss

 	      let val thms1 = inst_pt_at [fresh_perm_app]

 		  and thms2 = inst_pt_pt_at_cp_dj [fresh_perm_app_ineq] 

 	      in  [(("fresh_perm_app", thms1 @ thms2),[])] end 

             ||>> PureThy.add_thmss

 	      let val thms1 = inst_pt_at [pt_perm_supp]

 		  and thms2 = inst_pt_pt_at_cp [pt_perm_supp_ineq] 

 	      in  [(("supp_eqvt", thms1 @ thms2),[NominalThmDecls.eqvt_add])] end  

             ||>> PureThy.add_thmss [(("fin_supp",fs_axs),[])]

 	   end;

     in 

       NominalData.map (fold Symtab.update (full_ak_names ~~ map make_atom_info

         (pt_ax_classes ~~

          fs_ax_classes ~~

          map (fn cls => full_ak_names ~~ cls) cp_ax_classes))) thy33

     end;

 (* syntax und parsing *)

 structure P = OuterParse and K = OuterKeyword;

 val _ =

   OuterSyntax.command "atom_decl" "Declare new kinds of atoms" K.thy_decl

     (Scan.repeat1 P.name >> (Toplevel.theory o create_nom_typedecls));

 end;