src/HOL/Nominal/nominal_atoms.ML
author wenzelm
Thu Jun 26 15:06:25 2008 +0200 (2008-06-26)
changeset 27370 8e8f96dfaf63
parent 27216 dc1455f96f56
child 27399 1fb3d1219c12
permissions -rw-r--r--
Args.context;
     1 (*  title:      HOL/Nominal/nominal_atoms.ML
     2     ID:         $Id$
     3     Author:     Christian Urban and Stefan Berghofer, TU Muenchen
     4 
     5 Declaration of atom types to be used in nominal datatypes.
     6 *)
     7 
     8 signature NOMINAL_ATOMS =
     9 sig
    10   val create_nom_typedecls : string list -> theory -> theory
    11   type atom_info
    12   val get_atom_infos : theory -> atom_info Symtab.table
    13   val get_atom_info : theory -> string -> atom_info option
    14   val atoms_of : theory -> string list
    15   val mk_permT : typ -> typ
    16 end
    17 
    18 structure NominalAtoms : NOMINAL_ATOMS =
    19 struct
    20 
    21 val finite_emptyI = @{thm "finite.emptyI"};
    22 val Collect_const = @{thm "Collect_const"};
    23 
    24 val inductive_forall_def = @{thm "induct_forall_def"};
    25 
    26 
    27 (* theory data *)
    28 
    29 type atom_info =
    30   {pt_class : string,
    31    fs_class : string,
    32    cp_classes : (string * string) list};
    33 
    34 structure NominalData = TheoryDataFun
    35 (
    36   type T = atom_info Symtab.table;
    37   val empty = Symtab.empty;
    38   val copy = I;
    39   val extend = I;
    40   fun merge _ x = Symtab.merge (K true) x;
    41 );
    42 
    43 fun make_atom_info ((pt_class, fs_class), cp_classes) =
    44   {pt_class = pt_class,
    45    fs_class = fs_class,
    46    cp_classes = cp_classes};
    47 
    48 val get_atom_infos = NominalData.get;
    49 val get_atom_info = Symtab.lookup o NominalData.get;
    50 
    51 fun atoms_of thy = map fst (Symtab.dest (NominalData.get thy));
    52 
    53 fun mk_permT T = HOLogic.listT (HOLogic.mk_prodT (T, T));
    54 
    55 fun mk_Cons x xs =
    56   let val T = fastype_of x
    57   in Const ("List.list.Cons", T --> HOLogic.listT T --> HOLogic.listT T) $ x $ xs end;
    58 
    59 (* this function sets up all matters related to atom-  *)
    60 (* kinds; the user specifies a list of atom-kind names *)
    61 (* atom_decl <ak1> ... <akn>                           *)
    62 fun create_nom_typedecls ak_names thy =
    63   let
    64     
    65     val (_,thy1) = 
    66     fold_map (fn ak => fn thy => 
    67           let val dt = ([],ak,NoSyn,[(ak,[@{typ nat}],NoSyn)]) 
    68               val ({inject,case_thms,...},thy1) = DatatypePackage.add_datatype true false [ak] [dt] thy
    69               val inject_flat = Library.flat inject
    70               val ak_type = Type (Sign.intern_type thy1 ak,[])
    71               val ak_sign = Sign.intern_const thy1 ak 
    72               
    73               val inj_type = @{typ nat}-->ak_type
    74               val inj_on_type = inj_type-->(@{typ "nat set"})-->@{typ bool}  
    75 
    76               (* first statement *)
    77               val stmnt1 = HOLogic.mk_Trueprop 
    78                   (Const (@{const_name "inj_on"},inj_on_type) $ 
    79                          Const (ak_sign,inj_type) $ HOLogic.mk_UNIV @{typ nat})
    80 
    81               val simp1 = @{thm inj_on_def}::inject_flat
    82               
    83               val proof1 = fn _ => EVERY [simp_tac (HOL_basic_ss addsimps simp1) 1,
    84                                           rtac @{thm ballI} 1,
    85                                           rtac @{thm ballI} 1,
    86                                           rtac @{thm impI} 1,
    87                                           atac 1]
    88              
    89               val (inj_thm,thy2) = 
    90                    PureThy.add_thms [((ak^"_inj",Goal.prove_global thy1 [] [] stmnt1 proof1), [])] thy1
    91               
    92               (* second statement *)
    93               val y = Free ("y",ak_type)  
    94               val stmnt2 = HOLogic.mk_Trueprop
    95                   (HOLogic.mk_exists ("x",@{typ nat},HOLogic.mk_eq (y,Const (ak_sign,inj_type) $ Bound 0)))
    96 
    97               val proof2 = fn {prems, context} =>
    98                 InductTacs.case_tac context "y" 1 THEN
    99                 asm_simp_tac (HOL_basic_ss addsimps simp1) 1 THEN
   100                 rtac @{thm exI} 1 THEN
   101                 rtac @{thm refl} 1
   102 
   103               (* third statement *)
   104               val (inject_thm,thy3) =
   105                   PureThy.add_thms [((ak^"_injection",Goal.prove_global thy2 [] [] stmnt2 proof2), [])] thy2
   106   
   107               val stmnt3 = HOLogic.mk_Trueprop
   108                            (HOLogic.mk_not
   109                               (Const ("Finite_Set.finite", HOLogic.mk_setT ak_type --> HOLogic.boolT) $
   110                                   HOLogic.mk_UNIV ak_type))
   111              
   112               val simp2 = [@{thm image_def},@{thm bex_UNIV}]@inject_thm
   113               val simp3 = [@{thm UNIV_def}]
   114 
   115               val proof3 = fn _ => EVERY [cut_facts_tac inj_thm 1,
   116                                           dtac @{thm range_inj_infinite} 1,
   117                                           asm_full_simp_tac (HOL_basic_ss addsimps simp2) 1,
   118                                           simp_tac (HOL_basic_ss addsimps simp3) 1]  
   119            
   120               val (inf_thm,thy4) =  
   121                     PureThy.add_thms [((ak^"_infinite",Goal.prove_global thy3 [] [] stmnt3 proof3), [])] thy3
   122           in 
   123             ((inj_thm,inject_thm,inf_thm),thy4)
   124           end) ak_names thy
   125 
   126     (* produces a list consisting of pairs:         *)
   127     (*  fst component is the atom-kind name         *)
   128     (*  snd component is its type                   *)
   129     val full_ak_names = map (Sign.intern_type thy1) ak_names;
   130     val ak_names_types = ak_names ~~ map (Type o rpair []) full_ak_names;
   131      
   132     (* declares a swapping function for every atom-kind, it is         *)
   133     (* const swap_<ak> :: <akT> * <akT> => <akT> => <akT>              *)
   134     (* swap_<ak> (a,b) c = (if a=c then b (else if b=c then a else c)) *)
   135     (* overloades then the general swap-function                       *) 
   136     val (swap_eqs, thy3) = fold_map (fn (ak_name, T) => fn thy =>
   137       let
   138         val swapT = HOLogic.mk_prodT (T, T) --> T --> T;
   139         val swap_name = Sign.full_name thy ("swap_" ^ ak_name);
   140         val a = Free ("a", T);
   141         val b = Free ("b", T);
   142         val c = Free ("c", T);
   143         val ab = Free ("ab", HOLogic.mk_prodT (T, T))
   144         val cif = Const ("HOL.If", HOLogic.boolT --> T --> T --> T);
   145         val cswap_akname = Const (swap_name, swapT);
   146         val cswap = Const ("Nominal.swap", swapT)
   147 
   148         val name = "swap_"^ak_name^"_def";
   149         val def1 = HOLogic.mk_Trueprop (HOLogic.mk_eq
   150                 (cswap_akname $ HOLogic.mk_prod (a,b) $ c,
   151                     cif $ HOLogic.mk_eq (a,c) $ b $ (cif $ HOLogic.mk_eq (b,c) $ a $ c)))
   152         val def2 = Logic.mk_equals (cswap $ ab $ c, cswap_akname $ ab $ c)
   153       in
   154         thy |> Sign.add_consts_i [("swap_" ^ ak_name, swapT, NoSyn)] 
   155             |> PureThy.add_defs_unchecked_i true [((name, def2),[])]
   156             |> snd
   157             |> OldPrimrecPackage.add_primrec_unchecked_i "" [(("", def1),[])]
   158       end) ak_names_types thy1;
   159     
   160     (* declares a permutation function for every atom-kind acting  *)
   161     (* on such atoms                                               *)
   162     (* const <ak>_prm_<ak> :: (<akT> * <akT>)list => akT => akT    *)
   163     (* <ak>_prm_<ak> []     a = a                                  *)
   164     (* <ak>_prm_<ak> (x#xs) a = swap_<ak> x (perm xs a)            *)
   165     val (prm_eqs, thy4) = fold_map (fn (ak_name, T) => fn thy =>
   166       let
   167         val swapT = HOLogic.mk_prodT (T, T) --> T --> T;
   168         val swap_name = Sign.full_name thy ("swap_" ^ ak_name)
   169         val prmT = mk_permT T --> T --> T;
   170         val prm_name = ak_name ^ "_prm_" ^ ak_name;
   171         val qu_prm_name = Sign.full_name thy prm_name;
   172         val x  = Free ("x", HOLogic.mk_prodT (T, T));
   173         val xs = Free ("xs", mk_permT T);
   174         val a  = Free ("a", T) ;
   175 
   176         val cnil  = Const ("List.list.Nil", mk_permT T);
   177         
   178         val def1 = HOLogic.mk_Trueprop (HOLogic.mk_eq (Const (qu_prm_name, prmT) $ cnil $ a, a));
   179 
   180         val def2 = HOLogic.mk_Trueprop (HOLogic.mk_eq
   181                    (Const (qu_prm_name, prmT) $ mk_Cons x xs $ a,
   182                     Const (swap_name, swapT) $ x $ (Const (qu_prm_name, prmT) $ xs $ a)));
   183       in
   184         thy |> Sign.add_consts_i [(prm_name, mk_permT T --> T --> T, NoSyn)] 
   185             |> OldPrimrecPackage.add_primrec_unchecked_i "" [(("", def1), []),(("", def2), [])]
   186       end) ak_names_types thy3;
   187     
   188     (* defines permutation functions for all combinations of atom-kinds; *)
   189     (* there are a trivial cases and non-trivial cases                   *)
   190     (* non-trivial case:                                                 *)
   191     (* <ak>_prm_<ak>_def:  perm pi a == <ak>_prm_<ak> pi a               *)
   192     (* trivial case with <ak> != <ak'>                                   *)
   193     (* <ak>_prm<ak'>_def[simp]:  perm pi a == a                          *)
   194     (*                                                                   *)
   195     (* the trivial cases are added to the simplifier, while the non-     *)
   196     (* have their own rules proved below                                 *)  
   197     val (perm_defs, thy5) = fold_map (fn (ak_name, T) => fn thy =>
   198       fold_map (fn (ak_name', T') => fn thy' =>
   199         let
   200           val perm_def_name = ak_name ^ "_prm_" ^ ak_name';
   201           val pi = Free ("pi", mk_permT T);
   202           val a  = Free ("a", T');
   203           val cperm = Const ("Nominal.perm", mk_permT T --> T' --> T');
   204           val cperm_def = Const (Sign.full_name thy' perm_def_name, mk_permT T --> T' --> T');
   205 
   206           val name = ak_name ^ "_prm_" ^ ak_name' ^ "_def";
   207           val def = Logic.mk_equals
   208                     (cperm $ pi $ a, if ak_name = ak_name' then cperm_def $ pi $ a else a)
   209         in
   210           PureThy.add_defs_unchecked_i true [((name, def),[])] thy'
   211         end) ak_names_types thy) ak_names_types thy4;
   212     
   213     (* proves that every atom-kind is an instance of at *)
   214     (* lemma at_<ak>_inst:                              *)
   215     (* at TYPE(<ak>)                                    *)
   216     val (prm_cons_thms,thy6) = 
   217       thy5 |> PureThy.add_thms (map (fn (ak_name, T) =>
   218       let
   219         val ak_name_qu = Sign.full_name thy5 (ak_name);
   220         val i_type = Type(ak_name_qu,[]);
   221         val cat = Const ("Nominal.at",(Term.itselfT i_type)  --> HOLogic.boolT);
   222         val at_type = Logic.mk_type i_type;
   223         val simp_s = HOL_ss addsimps maps (PureThy.get_thms thy5)
   224                                   ["at_def",
   225                                    ak_name ^ "_prm_" ^ ak_name ^ "_def",
   226                                    ak_name ^ "_prm_" ^ ak_name ^ ".simps",
   227                                    "swap_" ^ ak_name ^ "_def",
   228                                    "swap_" ^ ak_name ^ ".simps",
   229                                    ak_name ^ "_infinite"]
   230             
   231         val name = "at_"^ak_name^ "_inst";
   232         val statement = HOLogic.mk_Trueprop (cat $ at_type);
   233 
   234         val proof = fn _ => simp_tac simp_s 1
   235 
   236       in 
   237         ((name, Goal.prove_global thy5 [] [] statement proof), []) 
   238       end) ak_names_types);
   239 
   240     (* declares a perm-axclass for every atom-kind               *)
   241     (* axclass pt_<ak>                                           *)
   242     (* pt_<ak>1[simp]: perm [] x = x                             *)
   243     (* pt_<ak>2:       perm (pi1@pi2) x = perm pi1 (perm pi2 x)  *)
   244     (* pt_<ak>3:       pi1 ~ pi2 ==> perm pi1 x = perm pi2 x     *)
   245      val (pt_ax_classes,thy7) =  fold_map (fn (ak_name, T) => fn thy =>
   246       let 
   247           val cl_name = "pt_"^ak_name;
   248           val ty = TFree("'a",["HOL.type"]);
   249           val x   = Free ("x", ty);
   250           val pi1 = Free ("pi1", mk_permT T);
   251           val pi2 = Free ("pi2", mk_permT T);
   252           val cperm = Const ("Nominal.perm", mk_permT T --> ty --> ty);
   253           val cnil  = Const ("List.list.Nil", mk_permT T);
   254           val cappend = Const ("List.append",mk_permT T --> mk_permT T --> mk_permT T);
   255           val cprm_eq = Const ("Nominal.prm_eq",mk_permT T --> mk_permT T --> HOLogic.boolT);
   256           (* nil axiom *)
   257           val axiom1 = HOLogic.mk_Trueprop (HOLogic.mk_eq 
   258                        (cperm $ cnil $ x, x));
   259           (* append axiom *)
   260           val axiom2 = HOLogic.mk_Trueprop (HOLogic.mk_eq
   261                        (cperm $ (cappend $ pi1 $ pi2) $ x, cperm $ pi1 $ (cperm $ pi2 $ x)));
   262           (* perm-eq axiom *)
   263           val axiom3 = Logic.mk_implies
   264                        (HOLogic.mk_Trueprop (cprm_eq $ pi1 $ pi2),
   265                         HOLogic.mk_Trueprop (HOLogic.mk_eq (cperm $ pi1 $ x, cperm $ pi2 $ x)));
   266       in
   267           AxClass.define_class (cl_name, ["HOL.type"]) []
   268                 [((cl_name ^ "1", [Simplifier.simp_add]), [axiom1]),
   269                  ((cl_name ^ "2", []), [axiom2]),                           
   270                  ((cl_name ^ "3", []), [axiom3])] thy                          
   271       end) ak_names_types thy6;
   272 
   273     (* proves that every pt_<ak>-type together with <ak>-type *)
   274     (* instance of pt                                         *)
   275     (* lemma pt_<ak>_inst:                                    *)
   276     (* pt TYPE('x::pt_<ak>) TYPE(<ak>)                        *)
   277     val (prm_inst_thms,thy8) = 
   278       thy7 |> PureThy.add_thms (map (fn (ak_name, T) =>
   279       let
   280         val ak_name_qu = Sign.full_name thy7 ak_name;
   281         val pt_name_qu = Sign.full_name thy7 ("pt_"^ak_name);
   282         val i_type1 = TFree("'x",[pt_name_qu]);
   283         val i_type2 = Type(ak_name_qu,[]);
   284         val cpt = Const ("Nominal.pt",(Term.itselfT i_type1)-->(Term.itselfT i_type2)-->HOLogic.boolT);
   285         val pt_type = Logic.mk_type i_type1;
   286         val at_type = Logic.mk_type i_type2;
   287         val simp_s = HOL_ss addsimps maps (PureThy.get_thms thy7)
   288                                   ["pt_def",
   289                                    "pt_" ^ ak_name ^ "1",
   290                                    "pt_" ^ ak_name ^ "2",
   291                                    "pt_" ^ ak_name ^ "3"];
   292 
   293         val name = "pt_"^ak_name^ "_inst";
   294         val statement = HOLogic.mk_Trueprop (cpt $ pt_type $ at_type);
   295 
   296         val proof = fn _ => simp_tac simp_s 1;
   297       in 
   298         ((name, Goal.prove_global thy7 [] [] statement proof), []) 
   299       end) ak_names_types);
   300 
   301      (* declares an fs-axclass for every atom-kind       *)
   302      (* axclass fs_<ak>                                  *)
   303      (* fs_<ak>1: finite ((supp x)::<ak> set)            *)
   304      val (fs_ax_classes,thy11) =  fold_map (fn (ak_name, T) => fn thy =>
   305        let 
   306           val cl_name = "fs_"^ak_name;
   307           val pt_name = Sign.full_name thy ("pt_"^ak_name);
   308           val ty = TFree("'a",["HOL.type"]);
   309           val x   = Free ("x", ty);
   310           val csupp    = Const ("Nominal.supp", ty --> HOLogic.mk_setT T);
   311           val cfinite  = Const ("Finite_Set.finite", HOLogic.mk_setT T --> HOLogic.boolT)
   312           
   313           val axiom1   = HOLogic.mk_Trueprop (cfinite $ (csupp $ x));
   314 
   315        in  
   316         AxClass.define_class (cl_name, [pt_name]) [] [((cl_name ^ "1", []), [axiom1])] thy            
   317        end) ak_names_types thy8; 
   318          
   319      (* proves that every fs_<ak>-type together with <ak>-type   *)
   320      (* instance of fs-type                                      *)
   321      (* lemma abst_<ak>_inst:                                    *)
   322      (* fs TYPE('x::pt_<ak>) TYPE (<ak>)                         *)
   323      val (fs_inst_thms,thy12) = 
   324        thy11 |> PureThy.add_thms (map (fn (ak_name, T) =>
   325        let
   326          val ak_name_qu = Sign.full_name thy11 ak_name;
   327          val fs_name_qu = Sign.full_name thy11 ("fs_"^ak_name);
   328          val i_type1 = TFree("'x",[fs_name_qu]);
   329          val i_type2 = Type(ak_name_qu,[]);
   330          val cfs = Const ("Nominal.fs", 
   331                                  (Term.itselfT i_type1)-->(Term.itselfT i_type2)-->HOLogic.boolT);
   332          val fs_type = Logic.mk_type i_type1;
   333          val at_type = Logic.mk_type i_type2;
   334          val simp_s = HOL_ss addsimps maps (PureThy.get_thms thy11)
   335                                    ["fs_def",
   336                                     "fs_" ^ ak_name ^ "1"];
   337     
   338          val name = "fs_"^ak_name^ "_inst";
   339          val statement = HOLogic.mk_Trueprop (cfs $ fs_type $ at_type);
   340 
   341          val proof = fn _ => simp_tac simp_s 1;
   342        in 
   343          ((name, Goal.prove_global thy11 [] [] statement proof), []) 
   344        end) ak_names_types);
   345 
   346        (* declares for every atom-kind combination an axclass            *)
   347        (* cp_<ak1>_<ak2> giving a composition property                   *)
   348        (* cp_<ak1>_<ak2>1: pi1 o pi2 o x = (pi1 o pi2) o (pi1 o x)       *)
   349         val (cp_ax_classes,thy12b) = fold_map (fn (ak_name, T) => fn thy =>
   350          fold_map (fn (ak_name', T') => fn thy' =>
   351              let
   352                val cl_name = "cp_"^ak_name^"_"^ak_name';
   353                val ty = TFree("'a",["HOL.type"]);
   354                val x   = Free ("x", ty);
   355                val pi1 = Free ("pi1", mk_permT T);
   356                val pi2 = Free ("pi2", mk_permT T');                  
   357                val cperm1 = Const ("Nominal.perm", mk_permT T  --> ty --> ty);
   358                val cperm2 = Const ("Nominal.perm", mk_permT T' --> ty --> ty);
   359                val cperm3 = Const ("Nominal.perm", mk_permT T  --> mk_permT T' --> mk_permT T');
   360 
   361                val ax1   = HOLogic.mk_Trueprop 
   362                            (HOLogic.mk_eq (cperm1 $ pi1 $ (cperm2 $ pi2 $ x), 
   363                                            cperm2 $ (cperm3 $ pi1 $ pi2) $ (cperm1 $ pi1 $ x)));
   364                in  
   365                  AxClass.define_class (cl_name, ["HOL.type"]) [] [((cl_name ^ "1", []), [ax1])] thy'  
   366                end) ak_names_types thy) ak_names_types thy12;
   367 
   368         (* proves for every <ak>-combination a cp_<ak1>_<ak2>_inst theorem;     *)
   369         (* lemma cp_<ak1>_<ak2>_inst:                                           *)
   370         (* cp TYPE('a::cp_<ak1>_<ak2>) TYPE(<ak1>) TYPE(<ak2>)                  *)
   371         val (cp_thms,thy12c) = fold_map (fn (ak_name, T) => fn thy =>
   372          fold_map (fn (ak_name', T') => fn thy' =>
   373            let
   374              val ak_name_qu  = Sign.full_name thy' (ak_name);
   375              val ak_name_qu' = Sign.full_name thy' (ak_name');
   376              val cp_name_qu  = Sign.full_name thy' ("cp_"^ak_name^"_"^ak_name');
   377              val i_type0 = TFree("'a",[cp_name_qu]);
   378              val i_type1 = Type(ak_name_qu,[]);
   379              val i_type2 = Type(ak_name_qu',[]);
   380              val ccp = Const ("Nominal.cp",
   381                              (Term.itselfT i_type0)-->(Term.itselfT i_type1)-->
   382                                                       (Term.itselfT i_type2)-->HOLogic.boolT);
   383              val at_type  = Logic.mk_type i_type1;
   384              val at_type' = Logic.mk_type i_type2;
   385              val cp_type  = Logic.mk_type i_type0;
   386              val simp_s   = HOL_basic_ss addsimps maps (PureThy.get_thms thy') ["cp_def"];
   387              val cp1      = PureThy.get_thm thy' ("cp_" ^ ak_name ^ "_" ^ ak_name' ^ "1");
   388 
   389              val name = "cp_"^ak_name^ "_"^ak_name'^"_inst";
   390              val statement = HOLogic.mk_Trueprop (ccp $ cp_type $ at_type $ at_type');
   391 
   392              val proof = fn _ => EVERY [simp_tac simp_s 1, 
   393                                         rtac allI 1, rtac allI 1, rtac allI 1,
   394                                         rtac cp1 1];
   395            in
   396              PureThy.add_thms [((name, Goal.prove_global thy' [] [] statement proof), [])] thy'
   397            end) 
   398            ak_names_types thy) ak_names_types thy12b;
   399        
   400         (* proves for every non-trivial <ak>-combination a disjointness   *)
   401         (* theorem; i.e. <ak1> != <ak2>                                   *)
   402         (* lemma ds_<ak1>_<ak2>:                                          *)
   403         (* dj TYPE(<ak1>) TYPE(<ak2>)                                     *)
   404         val (dj_thms, thy12d) = fold_map (fn (ak_name,T) => fn thy =>
   405           fold_map (fn (ak_name',T') => fn thy' =>
   406           (if not (ak_name = ak_name') 
   407            then 
   408                let
   409                  val ak_name_qu  = Sign.full_name thy' ak_name;
   410                  val ak_name_qu' = Sign.full_name thy' ak_name';
   411                  val i_type1 = Type(ak_name_qu,[]);
   412                  val i_type2 = Type(ak_name_qu',[]);
   413                  val cdj = Const ("Nominal.disjoint",
   414                            (Term.itselfT i_type1)-->(Term.itselfT i_type2)-->HOLogic.boolT);
   415                  val at_type  = Logic.mk_type i_type1;
   416                  val at_type' = Logic.mk_type i_type2;
   417                  val simp_s = HOL_ss addsimps maps (PureThy.get_thms thy')
   418                                            ["disjoint_def",
   419                                             ak_name ^ "_prm_" ^ ak_name' ^ "_def",
   420                                             ak_name' ^ "_prm_" ^ ak_name ^ "_def"];
   421 
   422                  val name = "dj_"^ak_name^"_"^ak_name';
   423                  val statement = HOLogic.mk_Trueprop (cdj $ at_type $ at_type');
   424 
   425                  val proof = fn _ => simp_tac simp_s 1;
   426                in
   427                 PureThy.add_thms [((name, Goal.prove_global thy' [] [] statement proof), [])] thy'
   428                end
   429            else 
   430             ([],thy')))  (* do nothing branch, if ak_name = ak_name' *) 
   431             ak_names_types thy) ak_names_types thy12c;
   432 
   433      (********  pt_<ak> class instances  ********)
   434      (*=========================================*)
   435      (* some abbreviations for theorems *)
   436       val pt1           = @{thm "pt1"};
   437       val pt2           = @{thm "pt2"};
   438       val pt3           = @{thm "pt3"};
   439       val at_pt_inst    = @{thm "at_pt_inst"};
   440       val pt_unit_inst  = @{thm "pt_unit_inst"};
   441       val pt_prod_inst  = @{thm "pt_prod_inst"}; 
   442       val pt_nprod_inst = @{thm "pt_nprod_inst"}; 
   443       val pt_list_inst  = @{thm "pt_list_inst"};
   444       val pt_optn_inst  = @{thm "pt_option_inst"};
   445       val pt_noptn_inst = @{thm "pt_noption_inst"};
   446       val pt_fun_inst   = @{thm "pt_fun_inst"};
   447 
   448      (* for all atom-kind combinations <ak>/<ak'> show that        *)
   449      (* every <ak> is an instance of pt_<ak'>; the proof for       *)
   450      (* ak!=ak' is by definition; the case ak=ak' uses at_pt_inst. *)
   451      val thy13 = fold (fn ak_name => fn thy =>
   452         fold (fn ak_name' => fn thy' =>
   453          let
   454            val qu_name =  Sign.full_name thy' ak_name';
   455            val cls_name = Sign.full_name thy' ("pt_"^ak_name);
   456            val at_inst  = PureThy.get_thm thy' ("at_" ^ ak_name' ^ "_inst");
   457 
   458            val proof1 = EVERY [Class.intro_classes_tac [],
   459                                  rtac ((at_inst RS at_pt_inst) RS pt1) 1,
   460                                  rtac ((at_inst RS at_pt_inst) RS pt2) 1,
   461                                  rtac ((at_inst RS at_pt_inst) RS pt3) 1,
   462                                  atac 1];
   463            val simp_s = HOL_basic_ss addsimps 
   464                         maps (PureThy.get_thms thy') [ak_name ^ "_prm_" ^ ak_name' ^ "_def"];  
   465            val proof2 = EVERY [Class.intro_classes_tac [], REPEAT (asm_simp_tac simp_s 1)];
   466 
   467          in
   468            thy'
   469            |> AxClass.prove_arity (qu_name,[],[cls_name])
   470               (if ak_name = ak_name' then proof1 else proof2)
   471          end) ak_names thy) ak_names thy12d;
   472 
   473      (* show that                       *)
   474      (*      fun(pt_<ak>,pt_<ak>)       *)
   475      (*      noption(pt_<ak>)           *)
   476      (*      option(pt_<ak>)            *)
   477      (*      list(pt_<ak>)              *)
   478      (*      *(pt_<ak>,pt_<ak>)         *)
   479      (*      nprod(pt_<ak>,pt_<ak>)     *)
   480      (*      unit                       *)
   481      (*      set(pt_<ak>)               *)
   482      (* are instances of pt_<ak>        *)
   483      val thy18 = fold (fn ak_name => fn thy =>
   484        let
   485           val cls_name = Sign.full_name thy ("pt_"^ak_name);
   486           val at_thm   = PureThy.get_thm thy ("at_"^ak_name^"_inst");
   487           val pt_inst  = PureThy.get_thm thy ("pt_"^ak_name^"_inst");
   488 
   489           fun pt_proof thm = 
   490               EVERY [Class.intro_classes_tac [],
   491                      rtac (thm RS pt1) 1, rtac (thm RS pt2) 1, rtac (thm RS pt3) 1, atac 1];
   492 
   493           val pt_thm_fun   = at_thm RS (pt_inst RS (pt_inst RS pt_fun_inst));
   494           val pt_thm_noptn = pt_inst RS pt_noptn_inst; 
   495           val pt_thm_optn  = pt_inst RS pt_optn_inst; 
   496           val pt_thm_list  = pt_inst RS pt_list_inst;
   497           val pt_thm_prod  = pt_inst RS (pt_inst RS pt_prod_inst);
   498           val pt_thm_nprod = pt_inst RS (pt_inst RS pt_nprod_inst);
   499           val pt_thm_unit  = pt_unit_inst;
   500        in
   501         thy
   502         |> AxClass.prove_arity ("fun",[[cls_name],[cls_name]],[cls_name]) (pt_proof pt_thm_fun)
   503         |> AxClass.prove_arity ("Nominal.noption",[[cls_name]],[cls_name]) (pt_proof pt_thm_noptn) 
   504         |> AxClass.prove_arity ("Datatype.option",[[cls_name]],[cls_name]) (pt_proof pt_thm_optn)
   505         |> AxClass.prove_arity ("List.list",[[cls_name]],[cls_name]) (pt_proof pt_thm_list)
   506         |> AxClass.prove_arity ("*",[[cls_name],[cls_name]],[cls_name]) (pt_proof pt_thm_prod)
   507         |> AxClass.prove_arity ("Nominal.nprod",[[cls_name],[cls_name]],[cls_name]) 
   508                                     (pt_proof pt_thm_nprod)
   509         |> AxClass.prove_arity ("Product_Type.unit",[],[cls_name]) (pt_proof pt_thm_unit)
   510      end) ak_names thy13; 
   511 
   512        (********  fs_<ak> class instances  ********)
   513        (*=========================================*)
   514        (* abbreviations for some lemmas *)
   515        val fs1            = @{thm "fs1"};
   516        val fs_at_inst     = @{thm "fs_at_inst"};
   517        val fs_unit_inst   = @{thm "fs_unit_inst"};
   518        val fs_prod_inst   = @{thm "fs_prod_inst"};
   519        val fs_nprod_inst  = @{thm "fs_nprod_inst"};
   520        val fs_list_inst   = @{thm "fs_list_inst"};
   521        val fs_option_inst = @{thm "fs_option_inst"};
   522        val dj_supp        = @{thm "dj_supp"};
   523 
   524        (* shows that <ak> is an instance of fs_<ak>     *)
   525        (* uses the theorem at_<ak>_inst                 *)
   526        val thy20 = fold (fn ak_name => fn thy =>
   527         fold (fn ak_name' => fn thy' =>
   528         let
   529            val qu_name =  Sign.full_name thy' ak_name';
   530            val qu_class = Sign.full_name thy' ("fs_"^ak_name);
   531            val proof =
   532                (if ak_name = ak_name'
   533                 then
   534                   let val at_thm = PureThy.get_thm thy' ("at_"^ak_name^"_inst");
   535                   in  EVERY [Class.intro_classes_tac [],
   536                              rtac ((at_thm RS fs_at_inst) RS fs1) 1] end
   537                 else
   538                   let val dj_inst = PureThy.get_thm thy' ("dj_"^ak_name'^"_"^ak_name);
   539                       val simp_s = HOL_basic_ss addsimps [dj_inst RS dj_supp, finite_emptyI];
   540                   in EVERY [Class.intro_classes_tac [], asm_simp_tac simp_s 1] end)
   541         in
   542          AxClass.prove_arity (qu_name,[],[qu_class]) proof thy'
   543         end) ak_names thy) ak_names thy18;
   544 
   545        (* shows that                  *)
   546        (*    unit                     *)
   547        (*    *(fs_<ak>,fs_<ak>)       *)
   548        (*    nprod(fs_<ak>,fs_<ak>)   *)
   549        (*    list(fs_<ak>)            *)
   550        (*    option(fs_<ak>)          *) 
   551        (* are instances of fs_<ak>    *)
   552 
   553        val thy24 = fold (fn ak_name => fn thy => 
   554         let
   555           val cls_name = Sign.full_name thy ("fs_"^ak_name);
   556           val fs_inst  = PureThy.get_thm thy ("fs_"^ak_name^"_inst");
   557           fun fs_proof thm = EVERY [Class.intro_classes_tac [], rtac (thm RS fs1) 1];
   558 
   559           val fs_thm_unit  = fs_unit_inst;
   560           val fs_thm_prod  = fs_inst RS (fs_inst RS fs_prod_inst);
   561           val fs_thm_nprod = fs_inst RS (fs_inst RS fs_nprod_inst);
   562           val fs_thm_list  = fs_inst RS fs_list_inst;
   563           val fs_thm_optn  = fs_inst RS fs_option_inst;
   564         in 
   565          thy
   566          |> AxClass.prove_arity ("Product_Type.unit",[],[cls_name]) (fs_proof fs_thm_unit) 
   567          |> AxClass.prove_arity ("*",[[cls_name],[cls_name]],[cls_name]) (fs_proof fs_thm_prod) 
   568          |> AxClass.prove_arity ("Nominal.nprod",[[cls_name],[cls_name]],[cls_name]) 
   569                                      (fs_proof fs_thm_nprod) 
   570          |> AxClass.prove_arity ("List.list",[[cls_name]],[cls_name]) (fs_proof fs_thm_list)
   571          |> AxClass.prove_arity ("Datatype.option",[[cls_name]],[cls_name]) (fs_proof fs_thm_optn)
   572         end) ak_names thy20;
   573 
   574        (********  cp_<ak>_<ai> class instances  ********)
   575        (*==============================================*)
   576        (* abbreviations for some lemmas *)
   577        val cp1             = @{thm "cp1"};
   578        val cp_unit_inst    = @{thm "cp_unit_inst"};
   579        val cp_bool_inst    = @{thm "cp_bool_inst"};
   580        val cp_prod_inst    = @{thm "cp_prod_inst"};
   581        val cp_list_inst    = @{thm "cp_list_inst"};
   582        val cp_fun_inst     = @{thm "cp_fun_inst"};
   583        val cp_option_inst  = @{thm "cp_option_inst"};
   584        val cp_noption_inst = @{thm "cp_noption_inst"};
   585        val pt_perm_compose = @{thm "pt_perm_compose"};
   586 
   587        val dj_pp_forget    = @{thm "dj_perm_perm_forget"};
   588 
   589        (* shows that <aj> is an instance of cp_<ak>_<ai>  *)
   590        (* for every  <ak>/<ai>-combination                *)
   591        val thy25 = fold (fn ak_name => fn thy =>
   592          fold (fn ak_name' => fn thy' =>
   593           fold (fn ak_name'' => fn thy'' =>
   594             let
   595               val name =  Sign.full_name thy'' ak_name;
   596               val cls_name = Sign.full_name thy'' ("cp_"^ak_name'^"_"^ak_name'');
   597               val proof =
   598                 (if (ak_name'=ak_name'') then 
   599                   (let
   600                     val pt_inst  = PureThy.get_thm thy'' ("pt_"^ak_name''^"_inst");
   601                     val at_inst  = PureThy.get_thm thy'' ("at_"^ak_name''^"_inst");
   602                   in
   603                    EVERY [Class.intro_classes_tac [],
   604                           rtac (at_inst RS (pt_inst RS pt_perm_compose)) 1]
   605                   end)
   606                 else
   607                   (let
   608                      val dj_inst  = PureThy.get_thm thy'' ("dj_"^ak_name''^"_"^ak_name');
   609                      val simp_s = HOL_basic_ss addsimps
   610                                         ((dj_inst RS dj_pp_forget)::
   611                                          (maps (PureThy.get_thms thy'')
   612                                            [ak_name' ^"_prm_"^ak_name^"_def",
   613                                             ak_name''^"_prm_"^ak_name^"_def"]));
   614                   in
   615                     EVERY [Class.intro_classes_tac [], simp_tac simp_s 1]
   616                   end))
   617               in
   618                 AxClass.prove_arity (name,[],[cls_name]) proof thy''
   619               end) ak_names thy') ak_names thy) ak_names thy24;
   620 
   621        (* shows that                                                    *) 
   622        (*      units                                                    *) 
   623        (*      products                                                 *)
   624        (*      lists                                                    *)
   625        (*      functions                                                *)
   626        (*      options                                                  *)
   627        (*      noptions                                                 *)
   628        (*      sets                                                     *)
   629        (* are instances of cp_<ak>_<ai> for every <ak>/<ai>-combination *)
   630        val thy26 = fold (fn ak_name => fn thy =>
   631         fold (fn ak_name' => fn thy' =>
   632         let
   633             val cls_name = Sign.full_name thy' ("cp_"^ak_name^"_"^ak_name');
   634             val cp_inst  = PureThy.get_thm thy' ("cp_"^ak_name^"_"^ak_name'^"_inst");
   635             val pt_inst  = PureThy.get_thm thy' ("pt_"^ak_name^"_inst");
   636             val at_inst  = PureThy.get_thm thy' ("at_"^ak_name^"_inst");
   637 
   638             fun cp_proof thm  = EVERY [Class.intro_classes_tac [],rtac (thm RS cp1) 1];
   639           
   640             val cp_thm_unit = cp_unit_inst;
   641             val cp_thm_prod = cp_inst RS (cp_inst RS cp_prod_inst);
   642             val cp_thm_list = cp_inst RS cp_list_inst;
   643             val cp_thm_fun  = at_inst RS (pt_inst RS (cp_inst RS (cp_inst RS cp_fun_inst)));
   644             val cp_thm_optn = cp_inst RS cp_option_inst;
   645             val cp_thm_noptn = cp_inst RS cp_noption_inst;
   646         in
   647          thy'
   648          |> AxClass.prove_arity ("Product_Type.unit",[],[cls_name]) (cp_proof cp_thm_unit)
   649          |> AxClass.prove_arity ("*",[[cls_name],[cls_name]],[cls_name]) (cp_proof cp_thm_prod)
   650          |> AxClass.prove_arity ("List.list",[[cls_name]],[cls_name]) (cp_proof cp_thm_list)
   651          |> AxClass.prove_arity ("fun",[[cls_name],[cls_name]],[cls_name]) (cp_proof cp_thm_fun)
   652          |> AxClass.prove_arity ("Datatype.option",[[cls_name]],[cls_name]) (cp_proof cp_thm_optn)
   653          |> AxClass.prove_arity ("Nominal.noption",[[cls_name]],[cls_name]) (cp_proof cp_thm_noptn)
   654         end) ak_names thy) ak_names thy25;
   655 
   656      (* show that discrete nominal types are permutation types, finitely     *)
   657      (* supported and have the commutation property                          *)
   658      (* discrete types have a permutation operation defined as pi o x = x;   *)
   659      (* which renders the proofs to be simple "simp_all"-proofs.             *)
   660      val thy32 =
   661         let
   662           fun discrete_pt_inst discrete_ty defn =
   663              fold (fn ak_name => fn thy =>
   664              let
   665                val qu_class = Sign.full_name thy ("pt_"^ak_name);
   666                val simp_s = HOL_basic_ss addsimps [defn];
   667                val proof = EVERY [Class.intro_classes_tac [], REPEAT (asm_simp_tac simp_s 1)];
   668              in 
   669                AxClass.prove_arity (discrete_ty,[],[qu_class]) proof thy
   670              end) ak_names;
   671 
   672           fun discrete_fs_inst discrete_ty defn = 
   673              fold (fn ak_name => fn thy =>
   674              let
   675                val qu_class = Sign.full_name thy ("fs_"^ak_name);
   676                val supp_def = @{thm "Nominal.supp_def"};
   677                val simp_s = HOL_ss addsimps [supp_def,Collect_const,finite_emptyI,defn];
   678                val proof = EVERY [Class.intro_classes_tac [], asm_simp_tac simp_s 1];
   679              in 
   680                AxClass.prove_arity (discrete_ty,[],[qu_class]) proof thy
   681              end) ak_names;
   682 
   683           fun discrete_cp_inst discrete_ty defn = 
   684              fold (fn ak_name' => (fold (fn ak_name => fn thy =>
   685              let
   686                val qu_class = Sign.full_name thy ("cp_"^ak_name^"_"^ak_name');
   687                val supp_def = @{thm "Nominal.supp_def"};
   688                val simp_s = HOL_ss addsimps [defn];
   689                val proof = EVERY [Class.intro_classes_tac [], asm_simp_tac simp_s 1];
   690              in
   691                AxClass.prove_arity (discrete_ty,[],[qu_class]) proof thy
   692              end) ak_names)) ak_names;
   693 
   694         in
   695          thy26
   696          |> discrete_pt_inst "nat"  @{thm "perm_nat_def"}
   697          |> discrete_fs_inst "nat"  @{thm "perm_nat_def"}
   698          |> discrete_cp_inst "nat"  @{thm "perm_nat_def"}
   699          |> discrete_pt_inst "bool" @{thm "perm_bool"}
   700          |> discrete_fs_inst "bool" @{thm "perm_bool"}
   701          |> discrete_cp_inst "bool" @{thm "perm_bool"}
   702          |> discrete_pt_inst @{type_name "Int.int"} @{thm "perm_int_def"}
   703          |> discrete_fs_inst @{type_name "Int.int"} @{thm "perm_int_def"}
   704          |> discrete_cp_inst @{type_name "Int.int"} @{thm "perm_int_def"}
   705          |> discrete_pt_inst "List.char" @{thm "perm_char_def"}
   706          |> discrete_fs_inst "List.char" @{thm "perm_char_def"}
   707          |> discrete_cp_inst "List.char" @{thm "perm_char_def"}
   708         end;
   709 
   710 
   711        (* abbreviations for some lemmas *)
   712        (*===============================*)
   713        val abs_fun_pi          = @{thm "Nominal.abs_fun_pi"};
   714        val abs_fun_pi_ineq     = @{thm "Nominal.abs_fun_pi_ineq"};
   715        val abs_fun_eq          = @{thm "Nominal.abs_fun_eq"};
   716        val abs_fun_eq'         = @{thm "Nominal.abs_fun_eq'"};
   717        val abs_fun_fresh       = @{thm "Nominal.abs_fun_fresh"};
   718        val abs_fun_fresh'      = @{thm "Nominal.abs_fun_fresh'"};
   719        val dj_perm_forget      = @{thm "Nominal.dj_perm_forget"};
   720        val dj_pp_forget        = @{thm "Nominal.dj_perm_perm_forget"};
   721        val fresh_iff           = @{thm "Nominal.fresh_abs_fun_iff"};
   722        val fresh_iff_ineq      = @{thm "Nominal.fresh_abs_fun_iff_ineq"};
   723        val abs_fun_supp        = @{thm "Nominal.abs_fun_supp"};
   724        val abs_fun_supp_ineq   = @{thm "Nominal.abs_fun_supp_ineq"};
   725        val pt_swap_bij         = @{thm "Nominal.pt_swap_bij"};
   726        val pt_swap_bij'        = @{thm "Nominal.pt_swap_bij'"};
   727        val pt_fresh_fresh      = @{thm "Nominal.pt_fresh_fresh"};
   728        val pt_bij              = @{thm "Nominal.pt_bij"};
   729        val pt_perm_compose     = @{thm "Nominal.pt_perm_compose"};
   730        val pt_perm_compose'    = @{thm "Nominal.pt_perm_compose'"};
   731        val perm_app            = @{thm "Nominal.pt_fun_app_eq"};
   732        val at_fresh            = @{thm "Nominal.at_fresh"};
   733        val at_fresh_ineq       = @{thm "Nominal.at_fresh_ineq"};
   734        val at_calc             = @{thms "Nominal.at_calc"};
   735        val at_swap_simps       = @{thms "Nominal.at_swap_simps"};
   736        val at_supp             = @{thm "Nominal.at_supp"};
   737        val dj_supp             = @{thm "Nominal.dj_supp"};
   738        val fresh_left_ineq     = @{thm "Nominal.pt_fresh_left_ineq"};
   739        val fresh_left          = @{thm "Nominal.pt_fresh_left"};
   740        val fresh_right_ineq    = @{thm "Nominal.pt_fresh_right_ineq"};
   741        val fresh_right         = @{thm "Nominal.pt_fresh_right"};
   742        val fresh_bij_ineq      = @{thm "Nominal.pt_fresh_bij_ineq"};
   743        val fresh_bij           = @{thm "Nominal.pt_fresh_bij"};
   744        val fresh_star_bij_ineq = @{thms "Nominal.pt_fresh_star_bij_ineq"};
   745        val fresh_star_bij      = @{thms "Nominal.pt_fresh_star_bij"};
   746        val fresh_eqvt          = @{thm "Nominal.pt_fresh_eqvt"};
   747        val fresh_eqvt_ineq     = @{thm "Nominal.pt_fresh_eqvt_ineq"};
   748        val set_diff_eqvt       = @{thm "Nominal.pt_set_diff_eqvt"};
   749        val in_eqvt             = @{thm "Nominal.pt_in_eqvt"};
   750        val eq_eqvt             = @{thm "Nominal.pt_eq_eqvt"};
   751        val all_eqvt            = @{thm "Nominal.pt_all_eqvt"};
   752        val ex_eqvt             = @{thm "Nominal.pt_ex_eqvt"};
   753        val pt_pi_rev           = @{thm "Nominal.pt_pi_rev"};
   754        val pt_rev_pi           = @{thm "Nominal.pt_rev_pi"};
   755        val at_exists_fresh     = @{thm "Nominal.at_exists_fresh"};
   756        val at_exists_fresh'    = @{thm "Nominal.at_exists_fresh'"};
   757        val fresh_perm_app_ineq = @{thm "Nominal.pt_fresh_perm_app_ineq"};
   758        val fresh_perm_app      = @{thm "Nominal.pt_fresh_perm_app"};    
   759        val fresh_aux           = @{thm "Nominal.pt_fresh_aux"};  
   760        val pt_perm_supp_ineq   = @{thm "Nominal.pt_perm_supp_ineq"};
   761        val pt_perm_supp        = @{thm "Nominal.pt_perm_supp"};
   762        val subseteq_eqvt       = @{thm "Nominal.pt_subseteq_eqvt"};
   763        val insert_eqvt         = @{thm "Nominal.insert_eqvt"};
   764        val set_eqvt            = @{thm "Nominal.set_eqvt"};
   765        val perm_set_eq         = @{thm "Nominal.perm_set_eq"};
   766 
   767        (* Now we collect and instantiate some lemmas w.r.t. all atom      *)
   768        (* types; this allows for example to use abs_perm (which is a      *)
   769        (* collection of theorems) instead of thm abs_fun_pi with explicit *)
   770        (* instantiations.                                                 *)
   771        val (_, thy33) =
   772          let
   773 
   774              (* takes a theorem thm and a list of theorems [t1,..,tn]            *)
   775              (* produces a list of theorems of the form [t1 RS thm,..,tn RS thm] *) 
   776              fun instR thm thms = map (fn ti => ti RS thm) thms;
   777 
   778 	     (* takes a theorem thm and a list of theorems [(t1,m1),..,(tn,mn)]  *)
   779              (* produces a list of theorems of the form [[t1,m1] MRS thm,..,[tn,mn] MRS thm] *) 
   780              fun instRR thm thms = map (fn (ti,mi) => [ti,mi] MRS thm) thms;
   781 
   782              (* takes two theorem lists (hopefully of the same length ;o)                *)
   783              (* produces a list of theorems of the form                                  *)
   784              (* [t1 RS m1,..,tn RS mn] where [t1,..,tn] is thms1 and [m1,..,mn] is thms2 *) 
   785              fun inst_zip thms1 thms2 = map (fn (t1,t2) => t1 RS t2) (thms1 ~~ thms2);
   786 
   787              (* takes a theorem list of the form [l1,...,ln]              *)
   788              (* and a list of theorem lists of the form                   *)
   789              (* [[h11,...,h1m],....,[hk1,....,hkm]                        *)
   790              (* produces the list of theorem lists                        *)
   791              (* [[l1 RS h11,...,l1 RS h1m],...,[ln RS hk1,...,ln RS hkm]] *)
   792              fun inst_mult thms thmss = map (fn (t,ts) => instR t ts) (thms ~~ thmss);
   793 
   794              (* FIXME: these lists do not need to be created dynamically again *)
   795 
   796              
   797              (* list of all at_inst-theorems *)
   798              val ats = map (fn ak => PureThy.get_thm thy32 ("at_"^ak^"_inst")) ak_names
   799              (* list of all pt_inst-theorems *)
   800              val pts = map (fn ak => PureThy.get_thm thy32 ("pt_"^ak^"_inst")) ak_names
   801              (* list of all cp_inst-theorems as a collection of lists*)
   802              val cps = 
   803                  let fun cps_fun ak1 ak2 =  PureThy.get_thm thy32 ("cp_"^ak1^"_"^ak2^"_inst")
   804                  in map (fn aki => (map (cps_fun aki) ak_names)) ak_names end; 
   805              (* list of all cp_inst-theorems that have different atom types *)
   806              val cps' = 
   807                 let fun cps'_fun ak1 ak2 = 
   808                 if ak1=ak2 then NONE else SOME (PureThy.get_thm thy32 ("cp_"^ak1^"_"^ak2^"_inst"))
   809                 in map (fn aki => (List.mapPartial I (map (cps'_fun aki) ak_names))) ak_names end;
   810              (* list of all dj_inst-theorems *)
   811              val djs = 
   812                let fun djs_fun ak1 ak2 = 
   813                      if ak1=ak2 then NONE else SOME(PureThy.get_thm thy32 ("dj_"^ak2^"_"^ak1))
   814                in map_filter I (map_product djs_fun ak_names ak_names) end;
   815              (* list of all fs_inst-theorems *)
   816              val fss = map (fn ak => PureThy.get_thm thy32 ("fs_"^ak^"_inst")) ak_names
   817              (* list of all at_inst-theorems *)
   818              val fs_axs = map (fn ak => PureThy.get_thm thy32 ("fs_"^ak^"1")) ak_names
   819 
   820              fun inst_pt thms = maps (fn ti => instR ti pts) thms;
   821              fun inst_at thms = maps (fn ti => instR ti ats) thms;
   822              fun inst_fs thms = maps (fn ti => instR ti fss) thms;
   823              fun inst_cp thms cps = flat (inst_mult thms cps);
   824              fun inst_pt_at thms = maps (fn ti => instRR ti (pts ~~ ats)) thms;
   825              fun inst_dj thms = maps (fn ti => instR ti djs) thms;
   826              fun inst_pt_pt_at_cp thms = inst_cp (inst_zip ats (inst_zip pts (inst_pt thms))) cps;
   827              fun inst_pt_at_fs thms = inst_zip (inst_fs [fs1]) (inst_zip ats (inst_pt thms));
   828              fun inst_pt_pt_at_cp thms =
   829                  let val i_pt_pt_at = inst_zip ats (inst_zip pts (inst_pt thms));
   830                      val i_pt_pt_at_cp = inst_cp i_pt_pt_at cps';
   831                  in i_pt_pt_at_cp end;
   832              fun inst_pt_pt_at_cp_dj thms = inst_zip djs (inst_pt_pt_at_cp thms);
   833            in
   834             thy32 
   835             |>   PureThy.add_thmss [(("alpha", inst_pt_at [abs_fun_eq]),[])]
   836             ||>> PureThy.add_thmss [(("alpha'", inst_pt_at [abs_fun_eq']),[])]
   837             ||>> PureThy.add_thmss [(("alpha_fresh", inst_pt_at [abs_fun_fresh]),[])]
   838             ||>> PureThy.add_thmss [(("alpha_fresh'", inst_pt_at [abs_fun_fresh']),[])]
   839             ||>> PureThy.add_thmss [(("perm_swap", inst_pt_at [pt_swap_bij] @ inst_pt_at [pt_swap_bij']),[])]
   840             ||>> PureThy.add_thmss [(("swap_simps", inst_at at_swap_simps),[])]  
   841             ||>> PureThy.add_thmss 
   842               let val thms1 = inst_pt_at [pt_pi_rev];
   843                   val thms2 = inst_pt_at [pt_rev_pi];
   844               in [(("perm_pi_simp",thms1 @ thms2),[])] end
   845             ||>> PureThy.add_thmss [(("perm_fresh_fresh", inst_pt_at [pt_fresh_fresh]),[])]
   846             ||>> PureThy.add_thmss [(("perm_bij", inst_pt_at [pt_bij]),[])]
   847             ||>> PureThy.add_thmss 
   848               let val thms1 = inst_pt_at [pt_perm_compose];
   849                   val thms2 = instR cp1 (Library.flat cps');
   850               in [(("perm_compose",thms1 @ thms2),[])] end
   851             ||>> PureThy.add_thmss [(("perm_compose'",inst_pt_at [pt_perm_compose']),[])] 
   852             ||>> PureThy.add_thmss [(("perm_app", inst_pt_at [perm_app]),[])]
   853             ||>> PureThy.add_thmss [(("supp_atm", (inst_at [at_supp]) @ (inst_dj [dj_supp])),[])]
   854             ||>> PureThy.add_thmss [(("exists_fresh", inst_at [at_exists_fresh]),[])]
   855             ||>> PureThy.add_thmss [(("exists_fresh'", inst_at [at_exists_fresh']),[])]
   856             ||>> PureThy.add_thmss
   857               let
   858                 val thms1 = inst_pt_at [all_eqvt];
   859                 val thms2 = map (fold_rule [inductive_forall_def]) thms1
   860               in
   861                 [(("all_eqvt", thms1 @ thms2), [NominalThmDecls.eqvt_force_add])]
   862               end
   863             ||>> PureThy.add_thmss [(("ex_eqvt", inst_pt_at [ex_eqvt]),[NominalThmDecls.eqvt_force_add])]
   864             ||>> PureThy.add_thmss 
   865               let val thms1 = inst_at [at_fresh]
   866                   val thms2 = inst_dj [at_fresh_ineq]
   867               in [(("fresh_atm", thms1 @ thms2),[])] end
   868             ||>> PureThy.add_thmss
   869               let val thms1 = filter
   870                 (fn th => case prop_of th of _ $ _ $ Var _ => true | _ => false)
   871                 (List.concat (List.concat perm_defs))
   872               in [(("calc_atm", (inst_at at_calc) @ thms1),[])] end
   873             ||>> PureThy.add_thmss
   874               let val thms1 = inst_pt_at [abs_fun_pi]
   875                   and thms2 = inst_pt_pt_at_cp [abs_fun_pi_ineq]
   876               in [(("abs_perm", thms1 @ thms2),[NominalThmDecls.eqvt_add])] end
   877             ||>> PureThy.add_thmss
   878               let val thms1 = inst_dj [dj_perm_forget]
   879                   and thms2 = inst_dj [dj_pp_forget]
   880               in [(("perm_dj", thms1 @ thms2),[])] end
   881             ||>> PureThy.add_thmss
   882               let val thms1 = inst_pt_at_fs [fresh_iff]
   883                   and thms2 = inst_pt_at [fresh_iff]
   884                   and thms3 = inst_pt_pt_at_cp_dj [fresh_iff_ineq]
   885             in [(("abs_fresh", thms1 @ thms2 @ thms3),[])] end
   886             ||>> PureThy.add_thmss
   887               let val thms1 = inst_pt_at [abs_fun_supp]
   888                   and thms2 = inst_pt_at_fs [abs_fun_supp]
   889                   and thms3 = inst_pt_pt_at_cp_dj [abs_fun_supp_ineq]
   890               in [(("abs_supp", thms1 @ thms2 @ thms3),[])] end
   891             ||>> PureThy.add_thmss
   892               let val thms1 = inst_pt_at [fresh_left]
   893                   and thms2 = inst_pt_pt_at_cp [fresh_left_ineq]
   894               in [(("fresh_left", thms1 @ thms2),[])] end
   895             ||>> PureThy.add_thmss
   896               let val thms1 = inst_pt_at [fresh_right]
   897                   and thms2 = inst_pt_pt_at_cp [fresh_right_ineq]
   898               in [(("fresh_right", thms1 @ thms2),[])] end
   899             ||>> PureThy.add_thmss
   900               let val thms1 = inst_pt_at [fresh_bij]
   901                   and thms2 = inst_pt_pt_at_cp [fresh_bij_ineq]
   902               in [(("fresh_bij", thms1 @ thms2),[])] end
   903             ||>> PureThy.add_thmss
   904               let val thms1 = inst_pt_at fresh_star_bij
   905                   and thms2 = flat (map (fn ti => inst_pt_pt_at_cp [ti]) fresh_star_bij_ineq);
   906               in [(("fresh_star_bij", thms1 @ thms2),[])] end
   907             ||>> PureThy.add_thmss
   908               let val thms1 = inst_pt_at [fresh_eqvt]
   909                   and thms2 = inst_pt_pt_at_cp_dj [fresh_eqvt_ineq]
   910               in [(("fresh_eqvt", thms1 @ thms2),[NominalThmDecls.eqvt_add])] end
   911             ||>> PureThy.add_thmss
   912               let val thms1 = inst_pt_at [in_eqvt]
   913               in [(("in_eqvt", thms1),[NominalThmDecls.eqvt_add])] end
   914             ||>> PureThy.add_thmss
   915               let val thms1 = inst_pt_at [eq_eqvt]
   916               in [(("eq_eqvt", thms1),[NominalThmDecls.eqvt_add])] end
   917             ||>> PureThy.add_thmss
   918               let val thms1 = inst_pt_at [set_diff_eqvt]
   919               in [(("set_diff_eqvt", thms1),[NominalThmDecls.eqvt_add])] end
   920             ||>> PureThy.add_thmss
   921               let val thms1 = inst_pt_at [subseteq_eqvt]
   922               in [(("subseteq_eqvt", thms1),[NominalThmDecls.eqvt_add])] end
   923             ||>> PureThy.add_thmss [(("insert_eqvt", inst_pt_at [insert_eqvt]), [NominalThmDecls.eqvt_add])]
   924             ||>> PureThy.add_thmss [(("set_eqvt", inst_pt_at [set_eqvt]), [NominalThmDecls.eqvt_add])]
   925             ||>> PureThy.add_thmss [(("perm_set_eq", inst_pt_at [perm_set_eq]), [])]
   926             ||>> PureThy.add_thmss
   927               let val thms1 = inst_pt_at [fresh_aux]
   928                   and thms2 = inst_pt_pt_at_cp_dj [fresh_perm_app_ineq] 
   929               in  [(("fresh_aux", thms1 @ thms2),[])] end  
   930             ||>> PureThy.add_thmss
   931               let val thms1 = inst_pt_at [fresh_perm_app]
   932                   and thms2 = inst_pt_pt_at_cp_dj [fresh_perm_app_ineq] 
   933               in  [(("fresh_perm_app", thms1 @ thms2),[])] end 
   934             ||>> PureThy.add_thmss
   935               let val thms1 = inst_pt_at [pt_perm_supp]
   936                   and thms2 = inst_pt_pt_at_cp [pt_perm_supp_ineq] 
   937               in  [(("supp_eqvt", thms1 @ thms2),[NominalThmDecls.eqvt_add])] end  
   938             ||>> PureThy.add_thmss [(("fin_supp",fs_axs),[])]
   939            end;
   940 
   941     in 
   942       NominalData.map (fold Symtab.update (full_ak_names ~~ map make_atom_info
   943         (pt_ax_classes ~~
   944          fs_ax_classes ~~
   945          map (fn cls => full_ak_names ~~ cls) cp_ax_classes))) thy33
   946     end;
   947 
   948 
   949 (* syntax und parsing *)
   950 structure P = OuterParse and K = OuterKeyword;
   951 
   952 val _ =
   953   OuterSyntax.command "atom_decl" "Declare new kinds of atoms" K.thy_decl
   954     (Scan.repeat1 P.name >> (Toplevel.theory o create_nom_typedecls));
   955 
   956 end;