src/HOL/Nominal/nominal_atoms.ML
author berghofe
Wed May 07 10:59:35 2008 +0200 (2008-05-07 ago)
changeset 26820 2909150bd614
parent 26773 ba8b1a8a12a7
child 27095 c1c27955d7dd
permissions -rw-r--r--
- Deleted arity proofs for set
- Produce specific instances of theorems insert_eqvt, set_eqvt and perm_set_eq
     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_i 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 _ => EVERY [case_tac "y" 1,
    98                                           asm_simp_tac (HOL_basic_ss addsimps simp1) 1,
    99                                           rtac @{thm exI} 1,
   100                                           rtac @{thm refl} 1]
   101 
   102               (* third statement *)
   103               val (inject_thm,thy3) =
   104                   PureThy.add_thms [((ak^"_injection",Goal.prove_global thy2 [] [] stmnt2 proof2), [])] thy2
   105   
   106               val stmnt3 = HOLogic.mk_Trueprop
   107                            (HOLogic.mk_not
   108                               (Const ("Finite_Set.finite", HOLogic.mk_setT ak_type --> HOLogic.boolT) $
   109                                   HOLogic.mk_UNIV ak_type))
   110              
   111               val simp2 = [@{thm image_def},@{thm bex_UNIV}]@inject_thm
   112               val simp3 = [@{thm UNIV_def}]
   113 
   114               val proof3 = fn _ => EVERY [cut_facts_tac inj_thm 1,
   115                                           dtac @{thm range_inj_infinite} 1,
   116                                           asm_full_simp_tac (HOL_basic_ss addsimps simp2) 1,
   117                                           simp_tac (HOL_basic_ss addsimps simp3) 1]  
   118            
   119               val (inf_thm,thy4) =  
   120                     PureThy.add_thms [((ak^"_infinite",Goal.prove_global thy3 [] [] stmnt3 proof3), [])] thy3
   121           in 
   122             ((inj_thm,inject_thm,inf_thm),thy4)
   123           end) ak_names thy
   124 
   125     (* produces a list consisting of pairs:         *)
   126     (*  fst component is the atom-kind name         *)
   127     (*  snd component is its type                   *)
   128     val full_ak_names = map (Sign.intern_type thy1) ak_names;
   129     val ak_names_types = ak_names ~~ map (Type o rpair []) full_ak_names;
   130      
   131     (* declares a swapping function for every atom-kind, it is         *)
   132     (* const swap_<ak> :: <akT> * <akT> => <akT> => <akT>              *)
   133     (* swap_<ak> (a,b) c = (if a=c then b (else if b=c then a else c)) *)
   134     (* overloades then the general swap-function                       *) 
   135     val (swap_eqs, thy3) = fold_map (fn (ak_name, T) => fn thy =>
   136       let
   137         val swapT = HOLogic.mk_prodT (T, T) --> T --> T;
   138         val swap_name = Sign.full_name thy ("swap_" ^ ak_name);
   139         val a = Free ("a", T);
   140         val b = Free ("b", T);
   141         val c = Free ("c", T);
   142         val ab = Free ("ab", HOLogic.mk_prodT (T, T))
   143         val cif = Const ("HOL.If", HOLogic.boolT --> T --> T --> T);
   144         val cswap_akname = Const (swap_name, swapT);
   145         val cswap = Const ("Nominal.swap", swapT)
   146 
   147         val name = "swap_"^ak_name^"_def";
   148         val def1 = HOLogic.mk_Trueprop (HOLogic.mk_eq
   149                 (cswap_akname $ HOLogic.mk_prod (a,b) $ c,
   150                     cif $ HOLogic.mk_eq (a,c) $ b $ (cif $ HOLogic.mk_eq (b,c) $ a $ c)))
   151         val def2 = Logic.mk_equals (cswap $ ab $ c, cswap_akname $ ab $ c)
   152       in
   153         thy |> Sign.add_consts_i [("swap_" ^ ak_name, swapT, NoSyn)] 
   154             |> PureThy.add_defs_unchecked_i true [((name, def2),[])]
   155             |> snd
   156             |> OldPrimrecPackage.add_primrec_unchecked_i "" [(("", def1),[])]
   157       end) ak_names_types thy1;
   158     
   159     (* declares a permutation function for every atom-kind acting  *)
   160     (* on such atoms                                               *)
   161     (* const <ak>_prm_<ak> :: (<akT> * <akT>)list => akT => akT    *)
   162     (* <ak>_prm_<ak> []     a = a                                  *)
   163     (* <ak>_prm_<ak> (x#xs) a = swap_<ak> x (perm xs a)            *)
   164     val (prm_eqs, thy4) = fold_map (fn (ak_name, T) => fn thy =>
   165       let
   166         val swapT = HOLogic.mk_prodT (T, T) --> T --> T;
   167         val swap_name = Sign.full_name thy ("swap_" ^ ak_name)
   168         val prmT = mk_permT T --> T --> T;
   169         val prm_name = ak_name ^ "_prm_" ^ ak_name;
   170         val qu_prm_name = Sign.full_name thy prm_name;
   171         val x  = Free ("x", HOLogic.mk_prodT (T, T));
   172         val xs = Free ("xs", mk_permT T);
   173         val a  = Free ("a", T) ;
   174 
   175         val cnil  = Const ("List.list.Nil", mk_permT T);
   176         
   177         val def1 = HOLogic.mk_Trueprop (HOLogic.mk_eq (Const (qu_prm_name, prmT) $ cnil $ a, a));
   178 
   179         val def2 = HOLogic.mk_Trueprop (HOLogic.mk_eq
   180                    (Const (qu_prm_name, prmT) $ mk_Cons x xs $ a,
   181                     Const (swap_name, swapT) $ x $ (Const (qu_prm_name, prmT) $ xs $ a)));
   182       in
   183         thy |> Sign.add_consts_i [(prm_name, mk_permT T --> T --> T, NoSyn)] 
   184             |> OldPrimrecPackage.add_primrec_unchecked_i "" [(("", def1), []),(("", def2), [])]
   185       end) ak_names_types thy3;
   186     
   187     (* defines permutation functions for all combinations of atom-kinds; *)
   188     (* there are a trivial cases and non-trivial cases                   *)
   189     (* non-trivial case:                                                 *)
   190     (* <ak>_prm_<ak>_def:  perm pi a == <ak>_prm_<ak> pi a               *)
   191     (* trivial case with <ak> != <ak'>                                   *)
   192     (* <ak>_prm<ak'>_def[simp]:  perm pi a == a                          *)
   193     (*                                                                   *)
   194     (* the trivial cases are added to the simplifier, while the non-     *)
   195     (* have their own rules proved below                                 *)  
   196     val (perm_defs, thy5) = fold_map (fn (ak_name, T) => fn thy =>
   197       fold_map (fn (ak_name', T') => fn thy' =>
   198         let
   199           val perm_def_name = ak_name ^ "_prm_" ^ ak_name';
   200           val pi = Free ("pi", mk_permT T);
   201           val a  = Free ("a", T');
   202           val cperm = Const ("Nominal.perm", mk_permT T --> T' --> T');
   203           val cperm_def = Const (Sign.full_name thy' perm_def_name, mk_permT T --> T' --> T');
   204 
   205           val name = ak_name ^ "_prm_" ^ ak_name' ^ "_def";
   206           val def = Logic.mk_equals
   207                     (cperm $ pi $ a, if ak_name = ak_name' then cperm_def $ pi $ a else a)
   208         in
   209           PureThy.add_defs_unchecked_i true [((name, def),[])] thy'
   210         end) ak_names_types thy) ak_names_types thy4;
   211     
   212     (* proves that every atom-kind is an instance of at *)
   213     (* lemma at_<ak>_inst:                              *)
   214     (* at TYPE(<ak>)                                    *)
   215     val (prm_cons_thms,thy6) = 
   216       thy5 |> PureThy.add_thms (map (fn (ak_name, T) =>
   217       let
   218         val ak_name_qu = Sign.full_name thy5 (ak_name);
   219         val i_type = Type(ak_name_qu,[]);
   220         val cat = Const ("Nominal.at",(Term.itselfT i_type)  --> HOLogic.boolT);
   221         val at_type = Logic.mk_type i_type;
   222         val simp_s = HOL_ss addsimps maps (PureThy.get_thms thy5)
   223                                   ["at_def",
   224                                    ak_name ^ "_prm_" ^ ak_name ^ "_def",
   225                                    ak_name ^ "_prm_" ^ ak_name ^ ".simps",
   226                                    "swap_" ^ ak_name ^ "_def",
   227                                    "swap_" ^ ak_name ^ ".simps",
   228                                    ak_name ^ "_infinite"]
   229             
   230         val name = "at_"^ak_name^ "_inst";
   231         val statement = HOLogic.mk_Trueprop (cat $ at_type);
   232 
   233         val proof = fn _ => simp_tac simp_s 1
   234 
   235       in 
   236         ((name, Goal.prove_global thy5 [] [] statement proof), []) 
   237       end) ak_names_types);
   238 
   239     (* declares a perm-axclass for every atom-kind               *)
   240     (* axclass pt_<ak>                                           *)
   241     (* pt_<ak>1[simp]: perm [] x = x                             *)
   242     (* pt_<ak>2:       perm (pi1@pi2) x = perm pi1 (perm pi2 x)  *)
   243     (* pt_<ak>3:       pi1 ~ pi2 ==> perm pi1 x = perm pi2 x     *)
   244      val (pt_ax_classes,thy7) =  fold_map (fn (ak_name, T) => fn thy =>
   245       let 
   246           val cl_name = "pt_"^ak_name;
   247           val ty = TFree("'a",["HOL.type"]);
   248           val x   = Free ("x", ty);
   249           val pi1 = Free ("pi1", mk_permT T);
   250           val pi2 = Free ("pi2", mk_permT T);
   251           val cperm = Const ("Nominal.perm", mk_permT T --> ty --> ty);
   252           val cnil  = Const ("List.list.Nil", mk_permT T);
   253           val cappend = Const ("List.append",mk_permT T --> mk_permT T --> mk_permT T);
   254           val cprm_eq = Const ("Nominal.prm_eq",mk_permT T --> mk_permT T --> HOLogic.boolT);
   255           (* nil axiom *)
   256           val axiom1 = HOLogic.mk_Trueprop (HOLogic.mk_eq 
   257                        (cperm $ cnil $ x, x));
   258           (* append axiom *)
   259           val axiom2 = HOLogic.mk_Trueprop (HOLogic.mk_eq
   260                        (cperm $ (cappend $ pi1 $ pi2) $ x, cperm $ pi1 $ (cperm $ pi2 $ x)));
   261           (* perm-eq axiom *)
   262           val axiom3 = Logic.mk_implies
   263                        (HOLogic.mk_Trueprop (cprm_eq $ pi1 $ pi2),
   264                         HOLogic.mk_Trueprop (HOLogic.mk_eq (cperm $ pi1 $ x, cperm $ pi2 $ x)));
   265       in
   266           AxClass.define_class (cl_name, ["HOL.type"]) []
   267                 [((cl_name ^ "1", [Simplifier.simp_add]), [axiom1]),
   268                  ((cl_name ^ "2", []), [axiom2]),                           
   269                  ((cl_name ^ "3", []), [axiom3])] thy                          
   270       end) ak_names_types thy6;
   271 
   272     (* proves that every pt_<ak>-type together with <ak>-type *)
   273     (* instance of pt                                         *)
   274     (* lemma pt_<ak>_inst:                                    *)
   275     (* pt TYPE('x::pt_<ak>) TYPE(<ak>)                        *)
   276     val (prm_inst_thms,thy8) = 
   277       thy7 |> PureThy.add_thms (map (fn (ak_name, T) =>
   278       let
   279         val ak_name_qu = Sign.full_name thy7 ak_name;
   280         val pt_name_qu = Sign.full_name thy7 ("pt_"^ak_name);
   281         val i_type1 = TFree("'x",[pt_name_qu]);
   282         val i_type2 = Type(ak_name_qu,[]);
   283         val cpt = Const ("Nominal.pt",(Term.itselfT i_type1)-->(Term.itselfT i_type2)-->HOLogic.boolT);
   284         val pt_type = Logic.mk_type i_type1;
   285         val at_type = Logic.mk_type i_type2;
   286         val simp_s = HOL_ss addsimps maps (PureThy.get_thms thy7)
   287                                   ["pt_def",
   288                                    "pt_" ^ ak_name ^ "1",
   289                                    "pt_" ^ ak_name ^ "2",
   290                                    "pt_" ^ ak_name ^ "3"];
   291 
   292         val name = "pt_"^ak_name^ "_inst";
   293         val statement = HOLogic.mk_Trueprop (cpt $ pt_type $ at_type);
   294 
   295         val proof = fn _ => simp_tac simp_s 1;
   296       in 
   297         ((name, Goal.prove_global thy7 [] [] statement proof), []) 
   298       end) ak_names_types);
   299 
   300      (* declares an fs-axclass for every atom-kind       *)
   301      (* axclass fs_<ak>                                  *)
   302      (* fs_<ak>1: finite ((supp x)::<ak> set)            *)
   303      val (fs_ax_classes,thy11) =  fold_map (fn (ak_name, T) => fn thy =>
   304        let 
   305           val cl_name = "fs_"^ak_name;
   306           val pt_name = Sign.full_name thy ("pt_"^ak_name);
   307           val ty = TFree("'a",["HOL.type"]);
   308           val x   = Free ("x", ty);
   309           val csupp    = Const ("Nominal.supp", ty --> HOLogic.mk_setT T);
   310           val cfinite  = Const ("Finite_Set.finite", HOLogic.mk_setT T --> HOLogic.boolT)
   311           
   312           val axiom1   = HOLogic.mk_Trueprop (cfinite $ (csupp $ x));
   313 
   314        in  
   315         AxClass.define_class (cl_name, [pt_name]) [] [((cl_name ^ "1", []), [axiom1])] thy            
   316        end) ak_names_types thy8; 
   317          
   318      (* proves that every fs_<ak>-type together with <ak>-type   *)
   319      (* instance of fs-type                                      *)
   320      (* lemma abst_<ak>_inst:                                    *)
   321      (* fs TYPE('x::pt_<ak>) TYPE (<ak>)                         *)
   322      val (fs_inst_thms,thy12) = 
   323        thy11 |> PureThy.add_thms (map (fn (ak_name, T) =>
   324        let
   325          val ak_name_qu = Sign.full_name thy11 ak_name;
   326          val fs_name_qu = Sign.full_name thy11 ("fs_"^ak_name);
   327          val i_type1 = TFree("'x",[fs_name_qu]);
   328          val i_type2 = Type(ak_name_qu,[]);
   329          val cfs = Const ("Nominal.fs", 
   330                                  (Term.itselfT i_type1)-->(Term.itselfT i_type2)-->HOLogic.boolT);
   331          val fs_type = Logic.mk_type i_type1;
   332          val at_type = Logic.mk_type i_type2;
   333          val simp_s = HOL_ss addsimps maps (PureThy.get_thms thy11)
   334                                    ["fs_def",
   335                                     "fs_" ^ ak_name ^ "1"];
   336     
   337          val name = "fs_"^ak_name^ "_inst";
   338          val statement = HOLogic.mk_Trueprop (cfs $ fs_type $ at_type);
   339 
   340          val proof = fn _ => simp_tac simp_s 1;
   341        in 
   342          ((name, Goal.prove_global thy11 [] [] statement proof), []) 
   343        end) ak_names_types);
   344 
   345        (* declares for every atom-kind combination an axclass            *)
   346        (* cp_<ak1>_<ak2> giving a composition property                   *)
   347        (* cp_<ak1>_<ak2>1: pi1 o pi2 o x = (pi1 o pi2) o (pi1 o x)       *)
   348         val (cp_ax_classes,thy12b) = fold_map (fn (ak_name, T) => fn thy =>
   349          fold_map (fn (ak_name', T') => fn thy' =>
   350              let
   351                val cl_name = "cp_"^ak_name^"_"^ak_name';
   352                val ty = TFree("'a",["HOL.type"]);
   353                val x   = Free ("x", ty);
   354                val pi1 = Free ("pi1", mk_permT T);
   355                val pi2 = Free ("pi2", mk_permT T');                  
   356                val cperm1 = Const ("Nominal.perm", mk_permT T  --> ty --> ty);
   357                val cperm2 = Const ("Nominal.perm", mk_permT T' --> ty --> ty);
   358                val cperm3 = Const ("Nominal.perm", mk_permT T  --> mk_permT T' --> mk_permT T');
   359 
   360                val ax1   = HOLogic.mk_Trueprop 
   361                            (HOLogic.mk_eq (cperm1 $ pi1 $ (cperm2 $ pi2 $ x), 
   362                                            cperm2 $ (cperm3 $ pi1 $ pi2) $ (cperm1 $ pi1 $ x)));
   363                in  
   364                  AxClass.define_class (cl_name, ["HOL.type"]) [] [((cl_name ^ "1", []), [ax1])] thy'  
   365                end) ak_names_types thy) ak_names_types thy12;
   366 
   367         (* proves for every <ak>-combination a cp_<ak1>_<ak2>_inst theorem;     *)
   368         (* lemma cp_<ak1>_<ak2>_inst:                                           *)
   369         (* cp TYPE('a::cp_<ak1>_<ak2>) TYPE(<ak1>) TYPE(<ak2>)                  *)
   370         val (cp_thms,thy12c) = fold_map (fn (ak_name, T) => fn thy =>
   371          fold_map (fn (ak_name', T') => fn thy' =>
   372            let
   373              val ak_name_qu  = Sign.full_name thy' (ak_name);
   374              val ak_name_qu' = Sign.full_name thy' (ak_name');
   375              val cp_name_qu  = Sign.full_name thy' ("cp_"^ak_name^"_"^ak_name');
   376              val i_type0 = TFree("'a",[cp_name_qu]);
   377              val i_type1 = Type(ak_name_qu,[]);
   378              val i_type2 = Type(ak_name_qu',[]);
   379              val ccp = Const ("Nominal.cp",
   380                              (Term.itselfT i_type0)-->(Term.itselfT i_type1)-->
   381                                                       (Term.itselfT i_type2)-->HOLogic.boolT);
   382              val at_type  = Logic.mk_type i_type1;
   383              val at_type' = Logic.mk_type i_type2;
   384              val cp_type  = Logic.mk_type i_type0;
   385              val simp_s   = HOL_basic_ss addsimps maps (PureThy.get_thms thy') ["cp_def"];
   386              val cp1      = PureThy.get_thm thy' ("cp_" ^ ak_name ^ "_" ^ ak_name' ^ "1");
   387 
   388              val name = "cp_"^ak_name^ "_"^ak_name'^"_inst";
   389              val statement = HOLogic.mk_Trueprop (ccp $ cp_type $ at_type $ at_type');
   390 
   391              val proof = fn _ => EVERY [simp_tac simp_s 1, 
   392                                         rtac allI 1, rtac allI 1, rtac allI 1,
   393                                         rtac cp1 1];
   394            in
   395              PureThy.add_thms [((name, Goal.prove_global thy' [] [] statement proof), [])] thy'
   396            end) 
   397            ak_names_types thy) ak_names_types thy12b;
   398        
   399         (* proves for every non-trivial <ak>-combination a disjointness   *)
   400         (* theorem; i.e. <ak1> != <ak2>                                   *)
   401         (* lemma ds_<ak1>_<ak2>:                                          *)
   402         (* dj TYPE(<ak1>) TYPE(<ak2>)                                     *)
   403         val (dj_thms, thy12d) = fold_map (fn (ak_name,T) => fn thy =>
   404           fold_map (fn (ak_name',T') => fn thy' =>
   405           (if not (ak_name = ak_name') 
   406            then 
   407                let
   408                  val ak_name_qu  = Sign.full_name thy' ak_name;
   409                  val ak_name_qu' = Sign.full_name thy' ak_name';
   410                  val i_type1 = Type(ak_name_qu,[]);
   411                  val i_type2 = Type(ak_name_qu',[]);
   412                  val cdj = Const ("Nominal.disjoint",
   413                            (Term.itselfT i_type1)-->(Term.itselfT i_type2)-->HOLogic.boolT);
   414                  val at_type  = Logic.mk_type i_type1;
   415                  val at_type' = Logic.mk_type i_type2;
   416                  val simp_s = HOL_ss addsimps maps (PureThy.get_thms thy')
   417                                            ["disjoint_def",
   418                                             ak_name ^ "_prm_" ^ ak_name' ^ "_def",
   419                                             ak_name' ^ "_prm_" ^ ak_name ^ "_def"];
   420 
   421                  val name = "dj_"^ak_name^"_"^ak_name';
   422                  val statement = HOLogic.mk_Trueprop (cdj $ at_type $ at_type');
   423 
   424                  val proof = fn _ => simp_tac simp_s 1;
   425                in
   426                 PureThy.add_thms [((name, Goal.prove_global thy' [] [] statement proof), [])] thy'
   427                end
   428            else 
   429             ([],thy')))  (* do nothing branch, if ak_name = ak_name' *) 
   430             ak_names_types thy) ak_names_types thy12c;
   431 
   432      (********  pt_<ak> class instances  ********)
   433      (*=========================================*)
   434      (* some abbreviations for theorems *)
   435       val pt1           = @{thm "pt1"};
   436       val pt2           = @{thm "pt2"};
   437       val pt3           = @{thm "pt3"};
   438       val at_pt_inst    = @{thm "at_pt_inst"};
   439       val pt_unit_inst  = @{thm "pt_unit_inst"};
   440       val pt_prod_inst  = @{thm "pt_prod_inst"}; 
   441       val pt_nprod_inst = @{thm "pt_nprod_inst"}; 
   442       val pt_list_inst  = @{thm "pt_list_inst"};
   443       val pt_optn_inst  = @{thm "pt_option_inst"};
   444       val pt_noptn_inst = @{thm "pt_noption_inst"};
   445       val pt_fun_inst   = @{thm "pt_fun_inst"};
   446 
   447      (* for all atom-kind combinations <ak>/<ak'> show that        *)
   448      (* every <ak> is an instance of pt_<ak'>; the proof for       *)
   449      (* ak!=ak' is by definition; the case ak=ak' uses at_pt_inst. *)
   450      val thy13 = fold (fn ak_name => fn thy =>
   451         fold (fn ak_name' => fn thy' =>
   452          let
   453            val qu_name =  Sign.full_name thy' ak_name';
   454            val cls_name = Sign.full_name thy' ("pt_"^ak_name);
   455            val at_inst  = PureThy.get_thm thy' ("at_" ^ ak_name' ^ "_inst");
   456 
   457            val proof1 = EVERY [Class.intro_classes_tac [],
   458                                  rtac ((at_inst RS at_pt_inst) RS pt1) 1,
   459                                  rtac ((at_inst RS at_pt_inst) RS pt2) 1,
   460                                  rtac ((at_inst RS at_pt_inst) RS pt3) 1,
   461                                  atac 1];
   462            val simp_s = HOL_basic_ss addsimps 
   463                         maps (PureThy.get_thms thy') [ak_name ^ "_prm_" ^ ak_name' ^ "_def"];  
   464            val proof2 = EVERY [Class.intro_classes_tac [], REPEAT (asm_simp_tac simp_s 1)];
   465 
   466          in
   467            thy'
   468            |> AxClass.prove_arity (qu_name,[],[cls_name])
   469               (if ak_name = ak_name' then proof1 else proof2)
   470          end) ak_names thy) ak_names thy12d;
   471 
   472      (* show that                       *)
   473      (*      fun(pt_<ak>,pt_<ak>)       *)
   474      (*      noption(pt_<ak>)           *)
   475      (*      option(pt_<ak>)            *)
   476      (*      list(pt_<ak>)              *)
   477      (*      *(pt_<ak>,pt_<ak>)         *)
   478      (*      nprod(pt_<ak>,pt_<ak>)     *)
   479      (*      unit                       *)
   480      (*      set(pt_<ak>)               *)
   481      (* are instances of pt_<ak>        *)
   482      val thy18 = fold (fn ak_name => fn thy =>
   483        let
   484           val cls_name = Sign.full_name thy ("pt_"^ak_name);
   485           val at_thm   = PureThy.get_thm thy ("at_"^ak_name^"_inst");
   486           val pt_inst  = PureThy.get_thm thy ("pt_"^ak_name^"_inst");
   487 
   488           fun pt_proof thm = 
   489               EVERY [Class.intro_classes_tac [],
   490                      rtac (thm RS pt1) 1, rtac (thm RS pt2) 1, rtac (thm RS pt3) 1, atac 1];
   491 
   492           val pt_thm_fun   = at_thm RS (pt_inst RS (pt_inst RS pt_fun_inst));
   493           val pt_thm_noptn = pt_inst RS pt_noptn_inst; 
   494           val pt_thm_optn  = pt_inst RS pt_optn_inst; 
   495           val pt_thm_list  = pt_inst RS pt_list_inst;
   496           val pt_thm_prod  = pt_inst RS (pt_inst RS pt_prod_inst);
   497           val pt_thm_nprod = pt_inst RS (pt_inst RS pt_nprod_inst);
   498           val pt_thm_unit  = pt_unit_inst;
   499        in
   500         thy
   501         |> AxClass.prove_arity ("fun",[[cls_name],[cls_name]],[cls_name]) (pt_proof pt_thm_fun)
   502         |> AxClass.prove_arity ("Nominal.noption",[[cls_name]],[cls_name]) (pt_proof pt_thm_noptn) 
   503         |> AxClass.prove_arity ("Datatype.option",[[cls_name]],[cls_name]) (pt_proof pt_thm_optn)
   504         |> AxClass.prove_arity ("List.list",[[cls_name]],[cls_name]) (pt_proof pt_thm_list)
   505         |> AxClass.prove_arity ("*",[[cls_name],[cls_name]],[cls_name]) (pt_proof pt_thm_prod)
   506         |> AxClass.prove_arity ("Nominal.nprod",[[cls_name],[cls_name]],[cls_name]) 
   507                                     (pt_proof pt_thm_nprod)
   508         |> AxClass.prove_arity ("Product_Type.unit",[],[cls_name]) (pt_proof pt_thm_unit)
   509      end) ak_names thy13; 
   510 
   511        (********  fs_<ak> class instances  ********)
   512        (*=========================================*)
   513        (* abbreviations for some lemmas *)
   514        val fs1            = @{thm "fs1"};
   515        val fs_at_inst     = @{thm "fs_at_inst"};
   516        val fs_unit_inst   = @{thm "fs_unit_inst"};
   517        val fs_prod_inst   = @{thm "fs_prod_inst"};
   518        val fs_nprod_inst  = @{thm "fs_nprod_inst"};
   519        val fs_list_inst   = @{thm "fs_list_inst"};
   520        val fs_option_inst = @{thm "fs_option_inst"};
   521        val dj_supp        = @{thm "dj_supp"};
   522 
   523        (* shows that <ak> is an instance of fs_<ak>     *)
   524        (* uses the theorem at_<ak>_inst                 *)
   525        val thy20 = fold (fn ak_name => fn thy =>
   526         fold (fn ak_name' => fn thy' =>
   527         let
   528            val qu_name =  Sign.full_name thy' ak_name';
   529            val qu_class = Sign.full_name thy' ("fs_"^ak_name);
   530            val proof =
   531                (if ak_name = ak_name'
   532                 then
   533                   let val at_thm = PureThy.get_thm thy' ("at_"^ak_name^"_inst");
   534                   in  EVERY [Class.intro_classes_tac [],
   535                              rtac ((at_thm RS fs_at_inst) RS fs1) 1] end
   536                 else
   537                   let val dj_inst = PureThy.get_thm thy' ("dj_"^ak_name'^"_"^ak_name);
   538                       val simp_s = HOL_basic_ss addsimps [dj_inst RS dj_supp, finite_emptyI];
   539                   in EVERY [Class.intro_classes_tac [], asm_simp_tac simp_s 1] end)
   540         in
   541          AxClass.prove_arity (qu_name,[],[qu_class]) proof thy'
   542         end) ak_names thy) ak_names thy18;
   543 
   544        (* shows that                  *)
   545        (*    unit                     *)
   546        (*    *(fs_<ak>,fs_<ak>)       *)
   547        (*    nprod(fs_<ak>,fs_<ak>)   *)
   548        (*    list(fs_<ak>)            *)
   549        (*    option(fs_<ak>)          *) 
   550        (* are instances of fs_<ak>    *)
   551 
   552        val thy24 = fold (fn ak_name => fn thy => 
   553         let
   554           val cls_name = Sign.full_name thy ("fs_"^ak_name);
   555           val fs_inst  = PureThy.get_thm thy ("fs_"^ak_name^"_inst");
   556           fun fs_proof thm = EVERY [Class.intro_classes_tac [], rtac (thm RS fs1) 1];
   557 
   558           val fs_thm_unit  = fs_unit_inst;
   559           val fs_thm_prod  = fs_inst RS (fs_inst RS fs_prod_inst);
   560           val fs_thm_nprod = fs_inst RS (fs_inst RS fs_nprod_inst);
   561           val fs_thm_list  = fs_inst RS fs_list_inst;
   562           val fs_thm_optn  = fs_inst RS fs_option_inst;
   563         in 
   564          thy
   565          |> AxClass.prove_arity ("Product_Type.unit",[],[cls_name]) (fs_proof fs_thm_unit) 
   566          |> AxClass.prove_arity ("*",[[cls_name],[cls_name]],[cls_name]) (fs_proof fs_thm_prod) 
   567          |> AxClass.prove_arity ("Nominal.nprod",[[cls_name],[cls_name]],[cls_name]) 
   568                                      (fs_proof fs_thm_nprod) 
   569          |> AxClass.prove_arity ("List.list",[[cls_name]],[cls_name]) (fs_proof fs_thm_list)
   570          |> AxClass.prove_arity ("Datatype.option",[[cls_name]],[cls_name]) (fs_proof fs_thm_optn)
   571         end) ak_names thy20;
   572 
   573        (********  cp_<ak>_<ai> class instances  ********)
   574        (*==============================================*)
   575        (* abbreviations for some lemmas *)
   576        val cp1             = @{thm "cp1"};
   577        val cp_unit_inst    = @{thm "cp_unit_inst"};
   578        val cp_bool_inst    = @{thm "cp_bool_inst"};
   579        val cp_prod_inst    = @{thm "cp_prod_inst"};
   580        val cp_list_inst    = @{thm "cp_list_inst"};
   581        val cp_fun_inst     = @{thm "cp_fun_inst"};
   582        val cp_option_inst  = @{thm "cp_option_inst"};
   583        val cp_noption_inst = @{thm "cp_noption_inst"};
   584        val pt_perm_compose = @{thm "pt_perm_compose"};
   585 
   586        val dj_pp_forget    = @{thm "dj_perm_perm_forget"};
   587 
   588        (* shows that <aj> is an instance of cp_<ak>_<ai>  *)
   589        (* for every  <ak>/<ai>-combination                *)
   590        val thy25 = fold (fn ak_name => fn thy =>
   591          fold (fn ak_name' => fn thy' =>
   592           fold (fn ak_name'' => fn thy'' =>
   593             let
   594               val name =  Sign.full_name thy'' ak_name;
   595               val cls_name = Sign.full_name thy'' ("cp_"^ak_name'^"_"^ak_name'');
   596               val proof =
   597                 (if (ak_name'=ak_name'') then 
   598                   (let
   599                     val pt_inst  = PureThy.get_thm thy'' ("pt_"^ak_name''^"_inst");
   600                     val at_inst  = PureThy.get_thm thy'' ("at_"^ak_name''^"_inst");
   601                   in
   602                    EVERY [Class.intro_classes_tac [],
   603                           rtac (at_inst RS (pt_inst RS pt_perm_compose)) 1]
   604                   end)
   605                 else
   606                   (let
   607                      val dj_inst  = PureThy.get_thm thy'' ("dj_"^ak_name''^"_"^ak_name');
   608                      val simp_s = HOL_basic_ss addsimps
   609                                         ((dj_inst RS dj_pp_forget)::
   610                                          (maps (PureThy.get_thms thy'')
   611                                            [ak_name' ^"_prm_"^ak_name^"_def",
   612                                             ak_name''^"_prm_"^ak_name^"_def"]));
   613                   in
   614                     EVERY [Class.intro_classes_tac [], simp_tac simp_s 1]
   615                   end))
   616               in
   617                 AxClass.prove_arity (name,[],[cls_name]) proof thy''
   618               end) ak_names thy') ak_names thy) ak_names thy24;
   619 
   620        (* shows that                                                    *) 
   621        (*      units                                                    *) 
   622        (*      products                                                 *)
   623        (*      lists                                                    *)
   624        (*      functions                                                *)
   625        (*      options                                                  *)
   626        (*      noptions                                                 *)
   627        (*      sets                                                     *)
   628        (* are instances of cp_<ak>_<ai> for every <ak>/<ai>-combination *)
   629        val thy26 = fold (fn ak_name => fn thy =>
   630         fold (fn ak_name' => fn thy' =>
   631         let
   632             val cls_name = Sign.full_name thy' ("cp_"^ak_name^"_"^ak_name');
   633             val cp_inst  = PureThy.get_thm thy' ("cp_"^ak_name^"_"^ak_name'^"_inst");
   634             val pt_inst  = PureThy.get_thm thy' ("pt_"^ak_name^"_inst");
   635             val at_inst  = PureThy.get_thm thy' ("at_"^ak_name^"_inst");
   636 
   637             fun cp_proof thm  = EVERY [Class.intro_classes_tac [],rtac (thm RS cp1) 1];
   638           
   639             val cp_thm_unit = cp_unit_inst;
   640             val cp_thm_prod = cp_inst RS (cp_inst RS cp_prod_inst);
   641             val cp_thm_list = cp_inst RS cp_list_inst;
   642             val cp_thm_fun  = at_inst RS (pt_inst RS (cp_inst RS (cp_inst RS cp_fun_inst)));
   643             val cp_thm_optn = cp_inst RS cp_option_inst;
   644             val cp_thm_noptn = cp_inst RS cp_noption_inst;
   645         in
   646          thy'
   647          |> AxClass.prove_arity ("Product_Type.unit",[],[cls_name]) (cp_proof cp_thm_unit)
   648          |> AxClass.prove_arity ("*",[[cls_name],[cls_name]],[cls_name]) (cp_proof cp_thm_prod)
   649          |> AxClass.prove_arity ("List.list",[[cls_name]],[cls_name]) (cp_proof cp_thm_list)
   650          |> AxClass.prove_arity ("fun",[[cls_name],[cls_name]],[cls_name]) (cp_proof cp_thm_fun)
   651          |> AxClass.prove_arity ("Datatype.option",[[cls_name]],[cls_name]) (cp_proof cp_thm_optn)
   652          |> AxClass.prove_arity ("Nominal.noption",[[cls_name]],[cls_name]) (cp_proof cp_thm_noptn)
   653         end) ak_names thy) ak_names thy25;
   654 
   655      (* show that discrete nominal types are permutation types, finitely     *)
   656      (* supported and have the commutation property                          *)
   657      (* discrete types have a permutation operation defined as pi o x = x;   *)
   658      (* which renders the proofs to be simple "simp_all"-proofs.             *)
   659      val thy32 =
   660         let
   661           fun discrete_pt_inst discrete_ty defn =
   662              fold (fn ak_name => fn thy =>
   663              let
   664                val qu_class = Sign.full_name thy ("pt_"^ak_name);
   665                val simp_s = HOL_basic_ss addsimps [defn];
   666                val proof = EVERY [Class.intro_classes_tac [], REPEAT (asm_simp_tac simp_s 1)];
   667              in 
   668                AxClass.prove_arity (discrete_ty,[],[qu_class]) proof thy
   669              end) ak_names;
   670 
   671           fun discrete_fs_inst discrete_ty defn = 
   672              fold (fn ak_name => fn thy =>
   673              let
   674                val qu_class = Sign.full_name thy ("fs_"^ak_name);
   675                val supp_def = @{thm "Nominal.supp_def"};
   676                val simp_s = HOL_ss addsimps [supp_def,Collect_const,finite_emptyI,defn];
   677                val proof = EVERY [Class.intro_classes_tac [], asm_simp_tac simp_s 1];
   678              in 
   679                AxClass.prove_arity (discrete_ty,[],[qu_class]) proof thy
   680              end) ak_names;
   681 
   682           fun discrete_cp_inst discrete_ty defn = 
   683              fold (fn ak_name' => (fold (fn ak_name => fn thy =>
   684              let
   685                val qu_class = Sign.full_name thy ("cp_"^ak_name^"_"^ak_name');
   686                val supp_def = @{thm "Nominal.supp_def"};
   687                val simp_s = HOL_ss addsimps [defn];
   688                val proof = EVERY [Class.intro_classes_tac [], asm_simp_tac simp_s 1];
   689              in
   690                AxClass.prove_arity (discrete_ty,[],[qu_class]) proof thy
   691              end) ak_names)) ak_names;
   692 
   693         in
   694          thy26
   695          |> discrete_pt_inst "nat"  @{thm "perm_nat_def"}
   696          |> discrete_fs_inst "nat"  @{thm "perm_nat_def"}
   697          |> discrete_cp_inst "nat"  @{thm "perm_nat_def"}
   698          |> discrete_pt_inst "bool" @{thm "perm_bool"}
   699          |> discrete_fs_inst "bool" @{thm "perm_bool"}
   700          |> discrete_cp_inst "bool" @{thm "perm_bool"}
   701          |> discrete_pt_inst @{type_name "Int.int"} @{thm "perm_int_def"}
   702          |> discrete_fs_inst @{type_name "Int.int"} @{thm "perm_int_def"}
   703          |> discrete_cp_inst @{type_name "Int.int"} @{thm "perm_int_def"}
   704          |> discrete_pt_inst "List.char" @{thm "perm_char_def"}
   705          |> discrete_fs_inst "List.char" @{thm "perm_char_def"}
   706          |> discrete_cp_inst "List.char" @{thm "perm_char_def"}
   707         end;
   708 
   709 
   710        (* abbreviations for some lemmas *)
   711        (*===============================*)
   712        val abs_fun_pi          = @{thm "Nominal.abs_fun_pi"};
   713        val abs_fun_pi_ineq     = @{thm "Nominal.abs_fun_pi_ineq"};
   714        val abs_fun_eq          = @{thm "Nominal.abs_fun_eq"};
   715        val abs_fun_eq'         = @{thm "Nominal.abs_fun_eq'"};
   716        val abs_fun_fresh       = @{thm "Nominal.abs_fun_fresh"};
   717        val abs_fun_fresh'      = @{thm "Nominal.abs_fun_fresh'"};
   718        val dj_perm_forget      = @{thm "Nominal.dj_perm_forget"};
   719        val dj_pp_forget        = @{thm "Nominal.dj_perm_perm_forget"};
   720        val fresh_iff           = @{thm "Nominal.fresh_abs_fun_iff"};
   721        val fresh_iff_ineq      = @{thm "Nominal.fresh_abs_fun_iff_ineq"};
   722        val abs_fun_supp        = @{thm "Nominal.abs_fun_supp"};
   723        val abs_fun_supp_ineq   = @{thm "Nominal.abs_fun_supp_ineq"};
   724        val pt_swap_bij         = @{thm "Nominal.pt_swap_bij"};
   725        val pt_swap_bij'        = @{thm "Nominal.pt_swap_bij'"};
   726        val pt_fresh_fresh      = @{thm "Nominal.pt_fresh_fresh"};
   727        val pt_bij              = @{thm "Nominal.pt_bij"};
   728        val pt_perm_compose     = @{thm "Nominal.pt_perm_compose"};
   729        val pt_perm_compose'    = @{thm "Nominal.pt_perm_compose'"};
   730        val perm_app            = @{thm "Nominal.pt_fun_app_eq"};
   731        val at_fresh            = @{thm "Nominal.at_fresh"};
   732        val at_fresh_ineq       = @{thm "Nominal.at_fresh_ineq"};
   733        val at_calc             = @{thms "Nominal.at_calc"};
   734        val at_swap_simps       = @{thms "Nominal.at_swap_simps"};
   735        val at_supp             = @{thm "Nominal.at_supp"};
   736        val dj_supp             = @{thm "Nominal.dj_supp"};
   737        val fresh_left_ineq     = @{thm "Nominal.pt_fresh_left_ineq"};
   738        val fresh_left          = @{thm "Nominal.pt_fresh_left"};
   739        val fresh_right_ineq    = @{thm "Nominal.pt_fresh_right_ineq"};
   740        val fresh_right         = @{thm "Nominal.pt_fresh_right"};
   741        val fresh_bij_ineq      = @{thm "Nominal.pt_fresh_bij_ineq"};
   742        val fresh_bij           = @{thm "Nominal.pt_fresh_bij"};
   743        val fresh_star_bij_ineq = @{thms "Nominal.pt_fresh_star_bij_ineq"};
   744        val fresh_star_bij      = @{thms "Nominal.pt_fresh_star_bij"};
   745        val fresh_eqvt          = @{thm "Nominal.pt_fresh_eqvt"};
   746        val fresh_eqvt_ineq     = @{thm "Nominal.pt_fresh_eqvt_ineq"};
   747        val set_diff_eqvt       = @{thm "Nominal.pt_set_diff_eqvt"};
   748        val in_eqvt             = @{thm "Nominal.pt_in_eqvt"};
   749        val eq_eqvt             = @{thm "Nominal.pt_eq_eqvt"};
   750        val all_eqvt            = @{thm "Nominal.pt_all_eqvt"};
   751        val ex_eqvt             = @{thm "Nominal.pt_ex_eqvt"};
   752        val pt_pi_rev           = @{thm "Nominal.pt_pi_rev"};
   753        val pt_rev_pi           = @{thm "Nominal.pt_rev_pi"};
   754        val at_exists_fresh     = @{thm "Nominal.at_exists_fresh"};
   755        val at_exists_fresh'    = @{thm "Nominal.at_exists_fresh'"};
   756        val fresh_perm_app_ineq = @{thm "Nominal.pt_fresh_perm_app_ineq"};
   757        val fresh_perm_app      = @{thm "Nominal.pt_fresh_perm_app"};    
   758        val fresh_aux           = @{thm "Nominal.pt_fresh_aux"};  
   759        val pt_perm_supp_ineq   = @{thm "Nominal.pt_perm_supp_ineq"};
   760        val pt_perm_supp        = @{thm "Nominal.pt_perm_supp"};
   761        val subseteq_eqvt       = @{thm "Nominal.pt_subseteq_eqvt"};
   762        val insert_eqvt         = @{thm "Nominal.insert_eqvt"};
   763        val set_eqvt            = @{thm "Nominal.set_eqvt"};
   764        val perm_set_eq         = @{thm "Nominal.perm_set_eq"};
   765 
   766        (* Now we collect and instantiate some lemmas w.r.t. all atom      *)
   767        (* types; this allows for example to use abs_perm (which is a      *)
   768        (* collection of theorems) instead of thm abs_fun_pi with explicit *)
   769        (* instantiations.                                                 *)
   770        val (_, thy33) =
   771          let
   772 
   773              (* takes a theorem thm and a list of theorems [t1,..,tn]            *)
   774              (* produces a list of theorems of the form [t1 RS thm,..,tn RS thm] *) 
   775              fun instR thm thms = map (fn ti => ti RS thm) thms;
   776 
   777 	     (* takes a theorem thm and a list of theorems [(t1,m1),..,(tn,mn)]  *)
   778              (* produces a list of theorems of the form [[t1,m1] MRS thm,..,[tn,mn] MRS thm] *) 
   779              fun instRR thm thms = map (fn (ti,mi) => [ti,mi] MRS thm) thms;
   780 
   781              (* takes two theorem lists (hopefully of the same length ;o)                *)
   782              (* produces a list of theorems of the form                                  *)
   783              (* [t1 RS m1,..,tn RS mn] where [t1,..,tn] is thms1 and [m1,..,mn] is thms2 *) 
   784              fun inst_zip thms1 thms2 = map (fn (t1,t2) => t1 RS t2) (thms1 ~~ thms2);
   785 
   786              (* takes a theorem list of the form [l1,...,ln]              *)
   787              (* and a list of theorem lists of the form                   *)
   788              (* [[h11,...,h1m],....,[hk1,....,hkm]                        *)
   789              (* produces the list of theorem lists                        *)
   790              (* [[l1 RS h11,...,l1 RS h1m],...,[ln RS hk1,...,ln RS hkm]] *)
   791              fun inst_mult thms thmss = map (fn (t,ts) => instR t ts) (thms ~~ thmss);
   792 
   793              (* FIXME: these lists do not need to be created dynamically again *)
   794 
   795              
   796              (* list of all at_inst-theorems *)
   797              val ats = map (fn ak => PureThy.get_thm thy32 ("at_"^ak^"_inst")) ak_names
   798              (* list of all pt_inst-theorems *)
   799              val pts = map (fn ak => PureThy.get_thm thy32 ("pt_"^ak^"_inst")) ak_names
   800              (* list of all cp_inst-theorems as a collection of lists*)
   801              val cps = 
   802                  let fun cps_fun ak1 ak2 =  PureThy.get_thm thy32 ("cp_"^ak1^"_"^ak2^"_inst")
   803                  in map (fn aki => (map (cps_fun aki) ak_names)) ak_names end; 
   804              (* list of all cp_inst-theorems that have different atom types *)
   805              val cps' = 
   806                 let fun cps'_fun ak1 ak2 = 
   807                 if ak1=ak2 then NONE else SOME (PureThy.get_thm thy32 ("cp_"^ak1^"_"^ak2^"_inst"))
   808                 in map (fn aki => (List.mapPartial I (map (cps'_fun aki) ak_names))) ak_names end;
   809              (* list of all dj_inst-theorems *)
   810              val djs = 
   811                let fun djs_fun ak1 ak2 = 
   812                      if ak1=ak2 then NONE else SOME(PureThy.get_thm thy32 ("dj_"^ak2^"_"^ak1))
   813                in map_filter I (map_product djs_fun ak_names ak_names) end;
   814              (* list of all fs_inst-theorems *)
   815              val fss = map (fn ak => PureThy.get_thm thy32 ("fs_"^ak^"_inst")) ak_names
   816              (* list of all at_inst-theorems *)
   817              val fs_axs = map (fn ak => PureThy.get_thm thy32 ("fs_"^ak^"1")) ak_names
   818 
   819              fun inst_pt thms = maps (fn ti => instR ti pts) thms;
   820              fun inst_at thms = maps (fn ti => instR ti ats) thms;
   821              fun inst_fs thms = maps (fn ti => instR ti fss) thms;
   822              fun inst_cp thms cps = flat (inst_mult thms cps);
   823              fun inst_pt_at thms = maps (fn ti => instRR ti (pts ~~ ats)) thms;
   824              fun inst_dj thms = maps (fn ti => instR ti djs) thms;
   825              fun inst_pt_pt_at_cp thms = inst_cp (inst_zip ats (inst_zip pts (inst_pt thms))) cps;
   826              fun inst_pt_at_fs thms = inst_zip (inst_fs [fs1]) (inst_zip ats (inst_pt thms));
   827              fun inst_pt_pt_at_cp thms =
   828                  let val i_pt_pt_at = inst_zip ats (inst_zip pts (inst_pt thms));
   829                      val i_pt_pt_at_cp = inst_cp i_pt_pt_at cps';
   830                  in i_pt_pt_at_cp end;
   831              fun inst_pt_pt_at_cp_dj thms = inst_zip djs (inst_pt_pt_at_cp thms);
   832            in
   833             thy32 
   834             |>   PureThy.add_thmss [(("alpha", inst_pt_at [abs_fun_eq]),[])]
   835             ||>> PureThy.add_thmss [(("alpha'", inst_pt_at [abs_fun_eq']),[])]
   836             ||>> PureThy.add_thmss [(("alpha_fresh", inst_pt_at [abs_fun_fresh]),[])]
   837             ||>> PureThy.add_thmss [(("alpha_fresh'", inst_pt_at [abs_fun_fresh']),[])]
   838             ||>> PureThy.add_thmss [(("perm_swap", inst_pt_at [pt_swap_bij] @ inst_pt_at [pt_swap_bij']),[])]
   839             ||>> PureThy.add_thmss [(("swap_simps", inst_at at_swap_simps),[])]  
   840             ||>> PureThy.add_thmss 
   841               let val thms1 = inst_pt_at [pt_pi_rev];
   842                   val thms2 = inst_pt_at [pt_rev_pi];
   843               in [(("perm_pi_simp",thms1 @ thms2),[])] end
   844             ||>> PureThy.add_thmss [(("perm_fresh_fresh", inst_pt_at [pt_fresh_fresh]),[])]
   845             ||>> PureThy.add_thmss [(("perm_bij", inst_pt_at [pt_bij]),[])]
   846             ||>> PureThy.add_thmss 
   847               let val thms1 = inst_pt_at [pt_perm_compose];
   848                   val thms2 = instR cp1 (Library.flat cps');
   849               in [(("perm_compose",thms1 @ thms2),[])] end
   850             ||>> PureThy.add_thmss [(("perm_compose'",inst_pt_at [pt_perm_compose']),[])] 
   851             ||>> PureThy.add_thmss [(("perm_app", inst_pt_at [perm_app]),[])]
   852             ||>> PureThy.add_thmss [(("supp_atm", (inst_at [at_supp]) @ (inst_dj [dj_supp])),[])]
   853             ||>> PureThy.add_thmss [(("exists_fresh", inst_at [at_exists_fresh]),[])]
   854             ||>> PureThy.add_thmss [(("exists_fresh'", inst_at [at_exists_fresh']),[])]
   855             ||>> PureThy.add_thmss
   856               let
   857                 val thms1 = inst_pt_at [all_eqvt];
   858                 val thms2 = map (fold_rule [inductive_forall_def]) thms1
   859               in
   860                 [(("all_eqvt", thms1 @ thms2), [NominalThmDecls.eqvt_force_add])]
   861               end
   862             ||>> PureThy.add_thmss [(("ex_eqvt", inst_pt_at [ex_eqvt]),[NominalThmDecls.eqvt_force_add])]
   863             ||>> PureThy.add_thmss 
   864               let val thms1 = inst_at [at_fresh]
   865                   val thms2 = inst_dj [at_fresh_ineq]
   866               in [(("fresh_atm", thms1 @ thms2),[])] end
   867             ||>> PureThy.add_thmss
   868               let val thms1 = filter
   869                 (fn th => case prop_of th of _ $ _ $ Var _ => true | _ => false)
   870                 (List.concat (List.concat perm_defs))
   871               in [(("calc_atm", (inst_at at_calc) @ thms1),[])] end
   872             ||>> PureThy.add_thmss
   873               let val thms1 = inst_pt_at [abs_fun_pi]
   874                   and thms2 = inst_pt_pt_at_cp [abs_fun_pi_ineq]
   875               in [(("abs_perm", thms1 @ thms2),[NominalThmDecls.eqvt_add])] end
   876             ||>> PureThy.add_thmss
   877               let val thms1 = inst_dj [dj_perm_forget]
   878                   and thms2 = inst_dj [dj_pp_forget]
   879               in [(("perm_dj", thms1 @ thms2),[])] end
   880             ||>> PureThy.add_thmss
   881               let val thms1 = inst_pt_at_fs [fresh_iff]
   882                   and thms2 = inst_pt_at [fresh_iff]
   883                   and thms3 = inst_pt_pt_at_cp_dj [fresh_iff_ineq]
   884             in [(("abs_fresh", thms1 @ thms2 @ thms3),[])] end
   885             ||>> PureThy.add_thmss
   886               let val thms1 = inst_pt_at [abs_fun_supp]
   887                   and thms2 = inst_pt_at_fs [abs_fun_supp]
   888                   and thms3 = inst_pt_pt_at_cp_dj [abs_fun_supp_ineq]
   889               in [(("abs_supp", thms1 @ thms2 @ thms3),[])] end
   890             ||>> PureThy.add_thmss
   891               let val thms1 = inst_pt_at [fresh_left]
   892                   and thms2 = inst_pt_pt_at_cp [fresh_left_ineq]
   893               in [(("fresh_left", thms1 @ thms2),[])] end
   894             ||>> PureThy.add_thmss
   895               let val thms1 = inst_pt_at [fresh_right]
   896                   and thms2 = inst_pt_pt_at_cp [fresh_right_ineq]
   897               in [(("fresh_right", thms1 @ thms2),[])] end
   898             ||>> PureThy.add_thmss
   899               let val thms1 = inst_pt_at [fresh_bij]
   900                   and thms2 = inst_pt_pt_at_cp [fresh_bij_ineq]
   901               in [(("fresh_bij", thms1 @ thms2),[])] end
   902             ||>> PureThy.add_thmss
   903               let val thms1 = inst_pt_at fresh_star_bij
   904                   and thms2 = flat (map (fn ti => inst_pt_pt_at_cp [ti]) fresh_star_bij_ineq);
   905               in [(("fresh_star_bij", thms1 @ thms2),[])] end
   906             ||>> PureThy.add_thmss
   907               let val thms1 = inst_pt_at [fresh_eqvt]
   908                   and thms2 = inst_pt_pt_at_cp_dj [fresh_eqvt_ineq]
   909               in [(("fresh_eqvt", thms1 @ thms2),[NominalThmDecls.eqvt_add])] end
   910             ||>> PureThy.add_thmss
   911               let val thms1 = inst_pt_at [in_eqvt]
   912               in [(("in_eqvt", thms1),[NominalThmDecls.eqvt_add])] end
   913             ||>> PureThy.add_thmss
   914               let val thms1 = inst_pt_at [eq_eqvt]
   915               in [(("eq_eqvt", thms1),[NominalThmDecls.eqvt_add])] end
   916             ||>> PureThy.add_thmss
   917               let val thms1 = inst_pt_at [set_diff_eqvt]
   918               in [(("set_diff_eqvt", thms1),[NominalThmDecls.eqvt_add])] end
   919             ||>> PureThy.add_thmss
   920               let val thms1 = inst_pt_at [subseteq_eqvt]
   921               in [(("subseteq_eqvt", thms1),[NominalThmDecls.eqvt_add])] end
   922             ||>> PureThy.add_thmss [(("insert_eqvt", inst_pt_at [insert_eqvt]), [NominalThmDecls.eqvt_add])]
   923             ||>> PureThy.add_thmss [(("set_eqvt", inst_pt_at [set_eqvt]), [NominalThmDecls.eqvt_add])]
   924             ||>> PureThy.add_thmss [(("perm_set_eq", inst_pt_at [perm_set_eq]), [])]
   925             ||>> PureThy.add_thmss
   926               let val thms1 = inst_pt_at [fresh_aux]
   927                   and thms2 = inst_pt_pt_at_cp_dj [fresh_perm_app_ineq] 
   928               in  [(("fresh_aux", thms1 @ thms2),[])] end  
   929             ||>> PureThy.add_thmss
   930               let val thms1 = inst_pt_at [fresh_perm_app]
   931                   and thms2 = inst_pt_pt_at_cp_dj [fresh_perm_app_ineq] 
   932               in  [(("fresh_perm_app", thms1 @ thms2),[])] end 
   933             ||>> PureThy.add_thmss
   934               let val thms1 = inst_pt_at [pt_perm_supp]
   935                   and thms2 = inst_pt_pt_at_cp [pt_perm_supp_ineq] 
   936               in  [(("supp_eqvt", thms1 @ thms2),[NominalThmDecls.eqvt_add])] end  
   937             ||>> PureThy.add_thmss [(("fin_supp",fs_axs),[])]
   938            end;
   939 
   940     in 
   941       NominalData.map (fold Symtab.update (full_ak_names ~~ map make_atom_info
   942         (pt_ax_classes ~~
   943          fs_ax_classes ~~
   944          map (fn cls => full_ak_names ~~ cls) cp_ax_classes))) thy33
   945     end;
   946 
   947 
   948 (* syntax und parsing *)
   949 structure P = OuterParse and K = OuterKeyword;
   950 
   951 val _ =
   952   OuterSyntax.command "atom_decl" "Declare new kinds of atoms" K.thy_decl
   953     (Scan.repeat1 P.name >> (Toplevel.theory o create_nom_typedecls));
   954 
   955 end;