src/HOL/Nominal/nominal_datatype.ML
author wenzelm
Fri Dec 02 15:23:27 2011 +0100 (2011-12-02)
changeset 45741 088256c289e7
parent 45740 132a3e1c0fe5
child 45822 843dc212f69e
permissions -rw-r--r--
eliminated some legacy operations;
     1 (*  Title:      HOL/Nominal/nominal_datatype.ML
     2     Author:     Stefan Berghofer and Christian Urban, TU Muenchen
     3 
     4 Nominal datatype package for Isabelle/HOL.
     5 *)
     6 
     7 signature NOMINAL_DATATYPE =
     8 sig
     9   val add_nominal_datatype : Datatype.config ->
    10     (string list * binding * mixfix * (binding * string list * mixfix) list) list ->
    11     theory -> theory
    12   type descr
    13   type nominal_datatype_info
    14   val get_nominal_datatypes : theory -> nominal_datatype_info Symtab.table
    15   val get_nominal_datatype : theory -> string -> nominal_datatype_info option
    16   val mk_perm: typ list -> term -> term -> term
    17   val perm_of_pair: term * term -> term
    18   val mk_not_sym: thm list -> thm list
    19   val perm_simproc: simproc
    20   val fresh_const: typ -> typ -> term
    21   val fresh_star_const: typ -> typ -> term
    22 end
    23 
    24 structure NominalDatatype : NOMINAL_DATATYPE =
    25 struct
    26 
    27 val finite_emptyI = @{thm finite.emptyI};
    28 val finite_Diff = @{thm finite_Diff};
    29 val finite_Un = @{thm finite_Un};
    30 val Un_iff = @{thm Un_iff};
    31 val In0_eq = @{thm In0_eq};
    32 val In1_eq = @{thm In1_eq};
    33 val In0_not_In1 = @{thm In0_not_In1};
    34 val In1_not_In0 = @{thm In1_not_In0};
    35 val Un_assoc = @{thm Un_assoc};
    36 val Collect_disj_eq = @{thm Collect_disj_eq};
    37 val Collect_False_empty = @{thm empty_def [THEN sym, THEN eq_reflection]};
    38 val empty_iff = @{thm empty_iff};
    39 
    40 open Datatype_Aux;
    41 open NominalAtoms;
    42 
    43 (** FIXME: Datatype should export this function **)
    44 
    45 local
    46 
    47 fun dt_recs (DtTFree _) = []
    48   | dt_recs (DtType (_, dts)) = maps dt_recs dts
    49   | dt_recs (DtRec i) = [i];
    50 
    51 fun dt_cases (descr: descr) (_, args, constrs) =
    52   let
    53     fun the_bname i = Long_Name.base_name (#1 (the (AList.lookup (op =) descr i)));
    54     val bnames = map the_bname (distinct op = (maps dt_recs args));
    55   in map (fn (c, _) => space_implode "_" (Long_Name.base_name c :: bnames)) constrs end;
    56 
    57 
    58 fun induct_cases descr =
    59   Datatype_Prop.indexify_names (maps (dt_cases descr) (map #2 descr));
    60 
    61 fun exhaust_cases descr i = dt_cases descr (the (AList.lookup (op =) descr i));
    62 
    63 in
    64 
    65 fun mk_case_names_induct descr = Rule_Cases.case_names (induct_cases descr);
    66 
    67 fun mk_case_names_exhausts descr new =
    68   map (Rule_Cases.case_names o exhaust_cases descr o #1)
    69     (filter (fn ((_, (name, _, _))) => member (op =) new name) descr);
    70 
    71 end;
    72 
    73 (* theory data *)
    74 
    75 type descr = (int * (string * dtyp list * (string * (dtyp list * dtyp) list) list)) list;
    76 
    77 type nominal_datatype_info =
    78   {index : int,
    79    descr : descr,
    80    sorts : (string * sort) list,
    81    rec_names : string list,
    82    rec_rewrites : thm list,
    83    induction : thm,
    84    distinct : thm list,
    85    inject : thm list};
    86 
    87 structure NominalDatatypesData = Theory_Data
    88 (
    89   type T = nominal_datatype_info Symtab.table;
    90   val empty = Symtab.empty;
    91   val extend = I;
    92   fun merge data = Symtab.merge (K true) data;
    93 );
    94 
    95 val get_nominal_datatypes = NominalDatatypesData.get;
    96 val put_nominal_datatypes = NominalDatatypesData.put;
    97 val map_nominal_datatypes = NominalDatatypesData.map;
    98 val get_nominal_datatype = Symtab.lookup o get_nominal_datatypes;
    99 
   100 
   101 (**** make datatype info ****)
   102 
   103 fun make_dt_info descr sorts induct reccomb_names rec_thms
   104     (i, (((_, (tname, _, _)), distinct), inject)) =
   105   (tname,
   106    {index = i,
   107     descr = descr,
   108     sorts = sorts,
   109     rec_names = reccomb_names,
   110     rec_rewrites = rec_thms,
   111     induction = induct,
   112     distinct = distinct,
   113     inject = inject});
   114 
   115 (*******************************)
   116 
   117 val (_ $ (_ $ (_ $ (distinct_f $ _) $ _))) = hd (prems_of distinct_lemma);
   118 
   119 
   120 (** simplification procedure for sorting permutations **)
   121 
   122 val dj_cp = @{thm dj_cp};
   123 
   124 fun dest_permT (Type ("fun", [Type ("List.list", [Type (@{type_name Product_Type.prod}, [T, _])]),
   125       Type ("fun", [_, U])])) = (T, U);
   126 
   127 fun permTs_of (Const ("Nominal.perm", T) $ t $ u) = fst (dest_permT T) :: permTs_of u
   128   | permTs_of _ = [];
   129 
   130 fun perm_simproc' thy ss (Const ("Nominal.perm", T) $ t $ (u as Const ("Nominal.perm", U) $ r $ s)) =
   131       let
   132         val (aT as Type (a, []), S) = dest_permT T;
   133         val (bT as Type (b, []), _) = dest_permT U
   134       in if member (op =) (permTs_of u) aT andalso aT <> bT then
   135           let
   136             val cp = cp_inst_of thy a b;
   137             val dj = dj_thm_of thy b a;
   138             val dj_cp' = [cp, dj] MRS dj_cp;
   139             val cert = SOME o cterm_of thy
   140           in
   141             SOME (mk_meta_eq (Drule.instantiate' [SOME (ctyp_of thy S)]
   142               [cert t, cert r, cert s] dj_cp'))
   143           end
   144         else NONE
   145       end
   146   | perm_simproc' thy ss _ = NONE;
   147 
   148 val perm_simproc =
   149   Simplifier.simproc_global @{theory} "perm_simp" ["pi1 \<bullet> (pi2 \<bullet> x)"] perm_simproc';
   150 
   151 fun projections rule =
   152   Project_Rule.projections (Proof_Context.init_global (Thm.theory_of_thm rule)) rule
   153   |> map (Drule.export_without_context #> Rule_Cases.save rule);
   154 
   155 val supp_prod = @{thm supp_prod};
   156 val fresh_prod = @{thm fresh_prod};
   157 val supports_fresh = @{thm supports_fresh};
   158 val supports_def = Simpdata.mk_eq @{thm Nominal.supports_def};
   159 val fresh_def = Simpdata.mk_eq @{thm fresh_def};
   160 val supp_def = Simpdata.mk_eq @{thm supp_def};
   161 val rev_simps = @{thms rev.simps};
   162 val app_simps = @{thms append.simps};
   163 val at_fin_set_supp = @{thm at_fin_set_supp};
   164 val at_fin_set_fresh = @{thm at_fin_set_fresh};
   165 val abs_fun_eq1 = @{thm abs_fun_eq1};
   166 
   167 val collect_simp = rewrite_rule [mk_meta_eq mem_Collect_eq];
   168 
   169 fun mk_perm Ts t u =
   170   let
   171     val T = fastype_of1 (Ts, t);
   172     val U = fastype_of1 (Ts, u)
   173   in Const ("Nominal.perm", T --> U --> U) $ t $ u end;
   174 
   175 fun perm_of_pair (x, y) =
   176   let
   177     val T = fastype_of x;
   178     val pT = mk_permT T
   179   in Const ("List.list.Cons", HOLogic.mk_prodT (T, T) --> pT --> pT) $
   180     HOLogic.mk_prod (x, y) $ Const ("List.list.Nil", pT)
   181   end;
   182 
   183 fun mk_not_sym ths = maps (fn th => case prop_of th of
   184     _ $ (Const (@{const_name Not}, _) $ (Const (@{const_name HOL.eq}, _) $ _ $ _)) => [th, th RS not_sym]
   185   | _ => [th]) ths;
   186 
   187 fun fresh_const T U = Const ("Nominal.fresh", T --> U --> HOLogic.boolT);
   188 fun fresh_star_const T U =
   189   Const ("Nominal.fresh_star", HOLogic.mk_setT T --> U --> HOLogic.boolT);
   190 
   191 fun gen_add_nominal_datatype prep_typ config dts thy =
   192   let
   193     val new_type_names = map (Binding.name_of o #2) dts;
   194 
   195 
   196     (* this theory is used just for parsing *)
   197 
   198     val tmp_thy = thy |>
   199       Theory.copy |>
   200       Sign.add_types_global (map (fn (tvs, tname, mx, _) => (tname, length tvs, mx)) dts);
   201 
   202     val atoms = atoms_of thy;
   203 
   204     fun prep_constr (cname, cargs, mx) (constrs, sorts) =
   205       let val (cargs', sorts') = fold_map (prep_typ tmp_thy) cargs sorts
   206       in (constrs @ [(cname, cargs', mx)], sorts') end
   207 
   208     fun prep_dt_spec (tvs, tname, mx, constrs) (dts, sorts) =
   209       let val (constrs', sorts') = fold prep_constr constrs ([], sorts)
   210       in (dts @ [(tvs, tname, mx, constrs')], sorts') end
   211 
   212     val (dts', sorts) = fold prep_dt_spec dts ([], []);
   213     val tyvars = map (map (fn s =>
   214       (s, the (AList.lookup (op =) sorts s))) o #1) dts';
   215 
   216     fun inter_sort thy S S' = Sign.inter_sort thy (S, S');
   217     fun augment_sort_typ thy S =
   218       let val S = Sign.minimize_sort thy (Sign.certify_sort thy S)
   219       in map_type_tfree (fn (s, S') => TFree (s,
   220         if member (op = o apsnd fst) sorts s then inter_sort thy S S' else S'))
   221       end;
   222     fun augment_sort thy S = map_types (augment_sort_typ thy S);
   223 
   224     val types_syntax = map (fn (tvs, tname, mx, constrs) => (tname, mx)) dts';
   225     val constr_syntax = map (fn (tvs, tname, mx, constrs) =>
   226       map (fn (cname, cargs, mx) => (cname, mx)) constrs) dts';
   227 
   228     val ps = map (fn (_, n, _, _) =>
   229       (Sign.full_name tmp_thy n, Sign.full_name tmp_thy (Binding.suffix_name "_Rep" n))) dts;
   230     val rps = map Library.swap ps;
   231 
   232     fun replace_types (Type ("Nominal.ABS", [T, U])) =
   233           Type ("fun", [T, Type ("Nominal.noption", [replace_types U])])
   234       | replace_types (Type (s, Ts)) =
   235           Type (the_default s (AList.lookup op = ps s), map replace_types Ts)
   236       | replace_types T = T;
   237 
   238     val dts'' = map (fn (tvs, tname, mx, constrs) =>
   239       (tvs, Binding.suffix_name "_Rep" tname, NoSyn,
   240         map (fn (cname, cargs, mx) => (Binding.suffix_name "_Rep" cname,
   241           map replace_types cargs, NoSyn)) constrs)) dts';
   242 
   243     val new_type_names' = map (fn n => n ^ "_Rep") new_type_names;
   244 
   245     val (full_new_type_names',thy1) = Datatype.add_datatype config dts'' thy;
   246 
   247     val {descr, induct, ...} = Datatype.the_info thy1 (hd full_new_type_names');
   248     fun nth_dtyp i = typ_of_dtyp descr sorts (DtRec i);
   249 
   250     val big_name = space_implode "_" new_type_names;
   251 
   252 
   253     (**** define permutation functions ****)
   254 
   255     val permT = mk_permT (TFree ("'x", HOLogic.typeS));
   256     val pi = Free ("pi", permT);
   257     val perm_types = map (fn (i, _) =>
   258       let val T = nth_dtyp i
   259       in permT --> T --> T end) descr;
   260     val perm_names' = Datatype_Prop.indexify_names (map (fn (i, _) =>
   261       "perm_" ^ name_of_typ (nth_dtyp i)) descr);
   262     val perm_names = replicate (length new_type_names) "Nominal.perm" @
   263       map (Sign.full_bname thy1) (List.drop (perm_names', length new_type_names));
   264     val perm_names_types = perm_names ~~ perm_types;
   265     val perm_names_types' = perm_names' ~~ perm_types;
   266 
   267     val perm_eqs = maps (fn (i, (_, _, constrs)) =>
   268       let val T = nth_dtyp i
   269       in map (fn (cname, dts) =>
   270         let
   271           val Ts = map (typ_of_dtyp descr sorts) dts;
   272           val names = Name.variant_list ["pi"] (Datatype_Prop.make_tnames Ts);
   273           val args = map Free (names ~~ Ts);
   274           val c = Const (cname, Ts ---> T);
   275           fun perm_arg (dt, x) =
   276             let val T = type_of x
   277             in if is_rec_type dt then
   278                 let val Us = binder_types T
   279                 in list_abs (map (pair "x") Us,
   280                   Free (nth perm_names_types' (body_index dt)) $ pi $
   281                     list_comb (x, map (fn (i, U) =>
   282                       Const ("Nominal.perm", permT --> U --> U) $
   283                         (Const ("List.rev", permT --> permT) $ pi) $
   284                         Bound i) ((length Us - 1 downto 0) ~~ Us)))
   285                 end
   286               else Const ("Nominal.perm", permT --> T --> T) $ pi $ x
   287             end;
   288         in
   289           (Attrib.empty_binding, HOLogic.mk_Trueprop (HOLogic.mk_eq
   290             (Free (nth perm_names_types' i) $
   291                Free ("pi", mk_permT (TFree ("'x", HOLogic.typeS))) $
   292                list_comb (c, args),
   293              list_comb (c, map perm_arg (dts ~~ args)))))
   294         end) constrs
   295       end) descr;
   296 
   297     val (perm_simps, thy2) =
   298       Primrec.add_primrec_overloaded
   299         (map (fn (s, sT) => (s, sT, false))
   300            (List.take (perm_names' ~~ perm_names_types, length new_type_names)))
   301         (map (fn s => (Binding.name s, NONE, NoSyn)) perm_names') perm_eqs thy1;
   302 
   303     (**** prove that permutation functions introduced by unfolding are ****)
   304     (**** equivalent to already existing permutation functions         ****)
   305 
   306     val _ = warning ("length descr: " ^ string_of_int (length descr));
   307     val _ = warning ("length new_type_names: " ^ string_of_int (length new_type_names));
   308 
   309     val perm_indnames = Datatype_Prop.make_tnames (map body_type perm_types);
   310     val perm_fun_def = Simpdata.mk_eq @{thm perm_fun_def};
   311 
   312     val unfolded_perm_eq_thms =
   313       if length descr = length new_type_names then []
   314       else map Drule.export_without_context (List.drop (split_conj_thm
   315         (Goal.prove_global thy2 [] []
   316           (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
   317             (map (fn (c as (s, T), x) =>
   318                let val [T1, T2] = binder_types T
   319                in HOLogic.mk_eq (Const c $ pi $ Free (x, T2),
   320                  Const ("Nominal.perm", T) $ pi $ Free (x, T2))
   321                end)
   322              (perm_names_types ~~ perm_indnames))))
   323           (fn _ => EVERY [Datatype_Aux.ind_tac induct perm_indnames 1,
   324             ALLGOALS (asm_full_simp_tac
   325               (global_simpset_of thy2 addsimps [perm_fun_def]))])),
   326         length new_type_names));
   327 
   328     (**** prove [] \<bullet> t = t ****)
   329 
   330     val _ = warning "perm_empty_thms";
   331 
   332     val perm_empty_thms = maps (fn a =>
   333       let val permT = mk_permT (Type (a, []))
   334       in map Drule.export_without_context (List.take (split_conj_thm
   335         (Goal.prove_global thy2 [] []
   336           (augment_sort thy2 [pt_class_of thy2 a]
   337             (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
   338               (map (fn ((s, T), x) => HOLogic.mk_eq
   339                   (Const (s, permT --> T --> T) $
   340                      Const ("List.list.Nil", permT) $ Free (x, T),
   341                    Free (x, T)))
   342                (perm_names ~~
   343                 map body_type perm_types ~~ perm_indnames)))))
   344           (fn _ => EVERY [Datatype_Aux.ind_tac induct perm_indnames 1,
   345             ALLGOALS (asm_full_simp_tac (global_simpset_of thy2))])),
   346         length new_type_names))
   347       end)
   348       atoms;
   349 
   350     (**** prove (pi1 @ pi2) \<bullet> t = pi1 \<bullet> (pi2 \<bullet> t) ****)
   351 
   352     val _ = warning "perm_append_thms";
   353 
   354     (*FIXME: these should be looked up statically*)
   355     val at_pt_inst = Global_Theory.get_thm thy2 "at_pt_inst";
   356     val pt2 = Global_Theory.get_thm thy2 "pt2";
   357 
   358     val perm_append_thms = maps (fn a =>
   359       let
   360         val permT = mk_permT (Type (a, []));
   361         val pi1 = Free ("pi1", permT);
   362         val pi2 = Free ("pi2", permT);
   363         val pt_inst = pt_inst_of thy2 a;
   364         val pt2' = pt_inst RS pt2;
   365         val pt2_ax = Global_Theory.get_thm thy2 (Long_Name.map_base_name (fn s => "pt_" ^ s ^ "2") a);
   366       in List.take (map Drule.export_without_context (split_conj_thm
   367         (Goal.prove_global thy2 [] []
   368            (augment_sort thy2 [pt_class_of thy2 a]
   369              (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
   370                 (map (fn ((s, T), x) =>
   371                     let val perm = Const (s, permT --> T --> T)
   372                     in HOLogic.mk_eq
   373                       (perm $ (Const ("List.append", permT --> permT --> permT) $
   374                          pi1 $ pi2) $ Free (x, T),
   375                        perm $ pi1 $ (perm $ pi2 $ Free (x, T)))
   376                     end)
   377                   (perm_names ~~
   378                    map body_type perm_types ~~ perm_indnames)))))
   379            (fn _ => EVERY [Datatype_Aux.ind_tac induct perm_indnames 1,
   380               ALLGOALS (asm_full_simp_tac (global_simpset_of thy2 addsimps [pt2', pt2_ax]))]))),
   381          length new_type_names)
   382       end) atoms;
   383 
   384     (**** prove pi1 ~ pi2 ==> pi1 \<bullet> t = pi2 \<bullet> t ****)
   385 
   386     val _ = warning "perm_eq_thms";
   387 
   388     val pt3 = Global_Theory.get_thm thy2 "pt3";
   389     val pt3_rev = Global_Theory.get_thm thy2 "pt3_rev";
   390 
   391     val perm_eq_thms = maps (fn a =>
   392       let
   393         val permT = mk_permT (Type (a, []));
   394         val pi1 = Free ("pi1", permT);
   395         val pi2 = Free ("pi2", permT);
   396         val at_inst = at_inst_of thy2 a;
   397         val pt_inst = pt_inst_of thy2 a;
   398         val pt3' = pt_inst RS pt3;
   399         val pt3_rev' = at_inst RS (pt_inst RS pt3_rev);
   400         val pt3_ax = Global_Theory.get_thm thy2 (Long_Name.map_base_name (fn s => "pt_" ^ s ^ "3") a);
   401       in List.take (map Drule.export_without_context (split_conj_thm
   402         (Goal.prove_global thy2 [] []
   403           (augment_sort thy2 [pt_class_of thy2 a] (Logic.mk_implies
   404              (HOLogic.mk_Trueprop (Const ("Nominal.prm_eq",
   405                 permT --> permT --> HOLogic.boolT) $ pi1 $ pi2),
   406               HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
   407                 (map (fn ((s, T), x) =>
   408                     let val perm = Const (s, permT --> T --> T)
   409                     in HOLogic.mk_eq
   410                       (perm $ pi1 $ Free (x, T),
   411                        perm $ pi2 $ Free (x, T))
   412                     end)
   413                   (perm_names ~~
   414                    map body_type perm_types ~~ perm_indnames))))))
   415            (fn _ => EVERY [Datatype_Aux.ind_tac induct perm_indnames 1,
   416               ALLGOALS (asm_full_simp_tac (global_simpset_of thy2 addsimps [pt3', pt3_rev', pt3_ax]))]))),
   417          length new_type_names)
   418       end) atoms;
   419 
   420     (**** prove pi1 \<bullet> (pi2 \<bullet> t) = (pi1 \<bullet> pi2) \<bullet> (pi1 \<bullet> t) ****)
   421 
   422     val cp1 = Global_Theory.get_thm thy2 "cp1";
   423     val dj_cp = Global_Theory.get_thm thy2 "dj_cp";
   424     val pt_perm_compose = Global_Theory.get_thm thy2 "pt_perm_compose";
   425     val pt_perm_compose_rev = Global_Theory.get_thm thy2 "pt_perm_compose_rev";
   426     val dj_perm_perm_forget = Global_Theory.get_thm thy2 "dj_perm_perm_forget";
   427 
   428     fun composition_instance name1 name2 thy =
   429       let
   430         val cp_class = cp_class_of thy name1 name2;
   431         val pt_class =
   432           if name1 = name2 then [pt_class_of thy name1]
   433           else [];
   434         val permT1 = mk_permT (Type (name1, []));
   435         val permT2 = mk_permT (Type (name2, []));
   436         val Ts = map body_type perm_types;
   437         val cp_inst = cp_inst_of thy name1 name2;
   438         val simps = global_simpset_of thy addsimps (perm_fun_def ::
   439           (if name1 <> name2 then
   440              let val dj = dj_thm_of thy name2 name1
   441              in [dj RS (cp_inst RS dj_cp), dj RS dj_perm_perm_forget] end
   442            else
   443              let
   444                val at_inst = at_inst_of thy name1;
   445                val pt_inst = pt_inst_of thy name1;
   446              in
   447                [cp_inst RS cp1 RS sym,
   448                 at_inst RS (pt_inst RS pt_perm_compose) RS sym,
   449                 at_inst RS (pt_inst RS pt_perm_compose_rev) RS sym]
   450             end))
   451         val sort = Sign.minimize_sort thy (Sign.certify_sort thy (cp_class :: pt_class));
   452         val thms = split_conj_thm (Goal.prove_global thy [] []
   453           (augment_sort thy sort
   454             (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
   455               (map (fn ((s, T), x) =>
   456                   let
   457                     val pi1 = Free ("pi1", permT1);
   458                     val pi2 = Free ("pi2", permT2);
   459                     val perm1 = Const (s, permT1 --> T --> T);
   460                     val perm2 = Const (s, permT2 --> T --> T);
   461                     val perm3 = Const ("Nominal.perm", permT1 --> permT2 --> permT2)
   462                   in HOLogic.mk_eq
   463                     (perm1 $ pi1 $ (perm2 $ pi2 $ Free (x, T)),
   464                      perm2 $ (perm3 $ pi1 $ pi2) $ (perm1 $ pi1 $ Free (x, T)))
   465                   end)
   466                 (perm_names ~~ Ts ~~ perm_indnames)))))
   467           (fn _ => EVERY [Datatype_Aux.ind_tac induct perm_indnames 1,
   468              ALLGOALS (asm_full_simp_tac simps)]))
   469       in
   470         fold (fn (s, tvs) => fn thy => AxClass.prove_arity
   471             (s, map (inter_sort thy sort o snd) tvs, [cp_class])
   472             (Class.intro_classes_tac [] THEN ALLGOALS (resolve_tac thms)) thy)
   473           (full_new_type_names' ~~ tyvars) thy
   474       end;
   475 
   476     val (perm_thmss,thy3) = thy2 |>
   477       fold (fn name1 => fold (composition_instance name1) atoms) atoms |>
   478       fold (fn atom => fn thy =>
   479         let val pt_name = pt_class_of thy atom
   480         in
   481           fold (fn (s, tvs) => fn thy => AxClass.prove_arity
   482               (s, map (inter_sort thy [pt_name] o snd) tvs, [pt_name])
   483               (EVERY
   484                 [Class.intro_classes_tac [],
   485                  resolve_tac perm_empty_thms 1,
   486                  resolve_tac perm_append_thms 1,
   487                  resolve_tac perm_eq_thms 1, assume_tac 1]) thy)
   488             (full_new_type_names' ~~ tyvars) thy
   489         end) atoms |>
   490       Global_Theory.add_thmss
   491         [((Binding.name (space_implode "_" new_type_names ^ "_unfolded_perm_eq"),
   492           unfolded_perm_eq_thms), [Simplifier.simp_add]),
   493          ((Binding.name (space_implode "_" new_type_names ^ "_perm_empty"),
   494           perm_empty_thms), [Simplifier.simp_add]),
   495          ((Binding.name (space_implode "_" new_type_names ^ "_perm_append"),
   496           perm_append_thms), [Simplifier.simp_add]),
   497          ((Binding.name (space_implode "_" new_type_names ^ "_perm_eq"),
   498           perm_eq_thms), [Simplifier.simp_add])];
   499 
   500     (**** Define representing sets ****)
   501 
   502     val _ = warning "representing sets";
   503 
   504     val rep_set_names = Datatype_Prop.indexify_names
   505       (map (fn (i, _) => name_of_typ (nth_dtyp i) ^ "_set") descr);
   506     val big_rep_name =
   507       space_implode "_" (Datatype_Prop.indexify_names (map_filter
   508         (fn (i, ("Nominal.noption", _, _)) => NONE
   509           | (i, _) => SOME (name_of_typ (nth_dtyp i))) descr)) ^ "_set";
   510     val _ = warning ("big_rep_name: " ^ big_rep_name);
   511 
   512     fun strip_option (dtf as DtType ("fun", [dt, DtRec i])) =
   513           (case AList.lookup op = descr i of
   514              SOME ("Nominal.noption", _, [(_, [dt']), _]) =>
   515                apfst (cons dt) (strip_option dt')
   516            | _ => ([], dtf))
   517       | strip_option (DtType ("fun", [dt, DtType ("Nominal.noption", [dt'])])) =
   518           apfst (cons dt) (strip_option dt')
   519       | strip_option dt = ([], dt);
   520 
   521     val dt_atomTs = distinct op = (map (typ_of_dtyp descr sorts)
   522       (maps (fn (_, (_, _, cs)) => maps (maps (fst o strip_option) o snd) cs) descr));
   523     val dt_atoms = map (fst o dest_Type) dt_atomTs;
   524 
   525     fun make_intr s T (cname, cargs) =
   526       let
   527         fun mk_prem dt (j, j', prems, ts) =
   528           let
   529             val (dts, dt') = strip_option dt;
   530             val (dts', dt'') = strip_dtyp dt';
   531             val Ts = map (typ_of_dtyp descr sorts) dts;
   532             val Us = map (typ_of_dtyp descr sorts) dts';
   533             val T = typ_of_dtyp descr sorts dt'';
   534             val free = mk_Free "x" (Us ---> T) j;
   535             val free' = app_bnds free (length Us);
   536             fun mk_abs_fun T (i, t) =
   537               let val U = fastype_of t
   538               in (i + 1, Const ("Nominal.abs_fun", [T, U, T] --->
   539                 Type ("Nominal.noption", [U])) $ mk_Free "y" T i $ t)
   540               end
   541           in (j + 1, j' + length Ts,
   542             case dt'' of
   543                 DtRec k => list_all (map (pair "x") Us,
   544                   HOLogic.mk_Trueprop (Free (nth rep_set_names k,
   545                     T --> HOLogic.boolT) $ free')) :: prems
   546               | _ => prems,
   547             snd (fold_rev mk_abs_fun Ts (j', free)) :: ts)
   548           end;
   549 
   550         val (_, _, prems, ts) = fold_rev mk_prem cargs (1, 1, [], []);
   551         val concl = HOLogic.mk_Trueprop (Free (s, T --> HOLogic.boolT) $
   552           list_comb (Const (cname, map fastype_of ts ---> T), ts))
   553       in Logic.list_implies (prems, concl)
   554       end;
   555 
   556     val (intr_ts, (rep_set_names', recTs')) =
   557       apfst flat (apsnd ListPair.unzip (ListPair.unzip (map_filter
   558         (fn ((_, ("Nominal.noption", _, _)), _) => NONE
   559           | ((i, (_, _, constrs)), rep_set_name) =>
   560               let val T = nth_dtyp i
   561               in SOME (map (make_intr rep_set_name T) constrs,
   562                 (rep_set_name, T))
   563               end)
   564                 (descr ~~ rep_set_names))));
   565     val rep_set_names'' = map (Sign.full_bname thy3) rep_set_names';
   566 
   567     val ({raw_induct = rep_induct, intrs = rep_intrs, ...}, thy4) =
   568       thy3
   569       |> Sign.map_naming Name_Space.conceal
   570       |> Inductive.add_inductive_global
   571           {quiet_mode = false, verbose = false, alt_name = Binding.name big_rep_name,
   572            coind = false, no_elim = true, no_ind = false, skip_mono = true, fork_mono = false}
   573           (map (fn (s, T) => ((Binding.name s, T --> HOLogic.boolT), NoSyn))
   574              (rep_set_names' ~~ recTs'))
   575           [] (map (fn x => (Attrib.empty_binding, x)) intr_ts) []
   576       ||> Sign.restore_naming thy3;
   577 
   578     (**** Prove that representing set is closed under permutation ****)
   579 
   580     val _ = warning "proving closure under permutation...";
   581 
   582     val abs_perm = Global_Theory.get_thms thy4 "abs_perm";
   583 
   584     val perm_indnames' = map_filter
   585       (fn (x, (_, ("Nominal.noption", _, _))) => NONE | (x, _) => SOME x)
   586       (perm_indnames ~~ descr);
   587 
   588     fun mk_perm_closed name = map (fn th => Drule.export_without_context (th RS mp))
   589       (List.take (split_conj_thm (Goal.prove_global thy4 [] []
   590         (augment_sort thy4
   591           (pt_class_of thy4 name :: map (cp_class_of thy4 name) (remove (op =) name dt_atoms))
   592           (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj (map
   593             (fn ((s, T), x) =>
   594                let
   595                  val S = Const (s, T --> HOLogic.boolT);
   596                  val permT = mk_permT (Type (name, []))
   597                in HOLogic.mk_imp (S $ Free (x, T),
   598                  S $ (Const ("Nominal.perm", permT --> T --> T) $
   599                    Free ("pi", permT) $ Free (x, T)))
   600                end) (rep_set_names'' ~~ recTs' ~~ perm_indnames')))))
   601         (fn _ => EVERY
   602            [Datatype_Aux.ind_tac rep_induct [] 1,
   603             ALLGOALS (simp_tac (global_simpset_of thy4 addsimps
   604               (Thm.symmetric perm_fun_def :: abs_perm))),
   605             ALLGOALS (resolve_tac rep_intrs THEN_ALL_NEW assume_tac)])),
   606         length new_type_names));
   607 
   608     val perm_closed_thmss = map mk_perm_closed atoms;
   609 
   610     (**** typedef ****)
   611 
   612     val _ = warning "defining type...";
   613 
   614     val (typedefs, thy6) =
   615       thy4
   616       |> fold_map (fn (((name, mx), tvs), (cname, U)) => fn thy =>
   617           Typedef.add_typedef_global false NONE
   618             (name, map (fn (v, _) => (v, dummyS)) tvs, mx)  (* FIXME keep constraints!? *)
   619             (Const (@{const_name Collect}, (U --> HOLogic.boolT) --> HOLogic.mk_setT U) $
   620                Const (cname, U --> HOLogic.boolT)) NONE
   621             (rtac exI 1 THEN rtac CollectI 1 THEN
   622               QUIET_BREADTH_FIRST (has_fewer_prems 1)
   623               (resolve_tac rep_intrs 1)) thy |> (fn ((_, r), thy) =>
   624         let
   625           val permT = mk_permT
   626             (TFree (singleton (Name.variant_list (map fst tvs)) "'a", HOLogic.typeS));
   627           val pi = Free ("pi", permT);
   628           val T = Type (Sign.full_name thy name, map TFree tvs);
   629         in apfst (pair r o hd)
   630           (Global_Theory.add_defs_unchecked true
   631             [((Binding.map_name (fn n => "prm_" ^ n ^ "_def") name, Logic.mk_equals
   632               (Const ("Nominal.perm", permT --> T --> T) $ pi $ Free ("x", T),
   633                Const (Sign.intern_const thy ("Abs_" ^ Binding.name_of name), U --> T) $
   634                  (Const ("Nominal.perm", permT --> U --> U) $ pi $
   635                    (Const (Sign.intern_const thy ("Rep_" ^ Binding.name_of name), T --> U) $
   636                      Free ("x", T))))), [])] thy)
   637         end))
   638         (types_syntax ~~ tyvars ~~ List.take (rep_set_names'' ~~ recTs', length new_type_names));
   639 
   640     val perm_defs = map snd typedefs;
   641     val Abs_inverse_thms = map (collect_simp o #Abs_inverse o snd o fst) typedefs;
   642     val Rep_inverse_thms = map (#Rep_inverse o snd o fst) typedefs;
   643     val Rep_thms = map (collect_simp o #Rep o snd o fst) typedefs;
   644 
   645 
   646     (** prove that new types are in class pt_<name> **)
   647 
   648     val _ = warning "prove that new types are in class pt_<name> ...";
   649 
   650     fun pt_instance (atom, perm_closed_thms) =
   651       fold (fn ((((((Abs_inverse, Rep_inverse), Rep),
   652         perm_def), name), tvs), perm_closed) => fn thy =>
   653           let
   654             val pt_class = pt_class_of thy atom;
   655             val sort = Sign.minimize_sort thy (Sign.certify_sort thy
   656               (pt_class :: map (cp_class_of thy atom) (remove (op =) atom dt_atoms)))
   657           in AxClass.prove_arity
   658             (Sign.intern_type thy name,
   659               map (inter_sort thy sort o snd) tvs, [pt_class])
   660             (EVERY [Class.intro_classes_tac [],
   661               rewrite_goals_tac [perm_def],
   662               asm_full_simp_tac (global_simpset_of thy addsimps [Rep_inverse]) 1,
   663               asm_full_simp_tac (global_simpset_of thy addsimps
   664                 [Rep RS perm_closed RS Abs_inverse]) 1,
   665               asm_full_simp_tac (HOL_basic_ss addsimps [Global_Theory.get_thm thy
   666                 ("pt_" ^ Long_Name.base_name atom ^ "3")]) 1]) thy
   667           end)
   668         (Abs_inverse_thms ~~ Rep_inverse_thms ~~ Rep_thms ~~ perm_defs ~~
   669            new_type_names ~~ tyvars ~~ perm_closed_thms);
   670 
   671 
   672     (** prove that new types are in class cp_<name1>_<name2> **)
   673 
   674     val _ = warning "prove that new types are in class cp_<name1>_<name2> ...";
   675 
   676     fun cp_instance (atom1, perm_closed_thms1) (atom2, perm_closed_thms2) thy =
   677       let
   678         val cp_class = cp_class_of thy atom1 atom2;
   679         val sort = Sign.minimize_sort thy (Sign.certify_sort thy
   680           (pt_class_of thy atom1 :: map (cp_class_of thy atom1) (remove (op =) atom1 dt_atoms) @
   681            (if atom1 = atom2 then [cp_class_of thy atom1 atom1] else
   682             pt_class_of thy atom2 :: map (cp_class_of thy atom2) (remove (op =) atom2 dt_atoms))));
   683         val cp1' = cp_inst_of thy atom1 atom2 RS cp1
   684       in fold (fn ((((((Abs_inverse, Rep),
   685         perm_def), name), tvs), perm_closed1), perm_closed2) => fn thy =>
   686           AxClass.prove_arity
   687             (Sign.intern_type thy name,
   688               map (inter_sort thy sort o snd) tvs, [cp_class])
   689             (EVERY [Class.intro_classes_tac [],
   690               rewrite_goals_tac [perm_def],
   691               asm_full_simp_tac (global_simpset_of thy addsimps
   692                 ((Rep RS perm_closed1 RS Abs_inverse) ::
   693                  (if atom1 = atom2 then []
   694                   else [Rep RS perm_closed2 RS Abs_inverse]))) 1,
   695               cong_tac 1,
   696               rtac refl 1,
   697               rtac cp1' 1]) thy)
   698         (Abs_inverse_thms ~~ Rep_thms ~~ perm_defs ~~ new_type_names ~~
   699            tyvars ~~ perm_closed_thms1 ~~ perm_closed_thms2) thy
   700       end;
   701 
   702     val thy7 = fold (fn x => fn thy => thy |>
   703       pt_instance x |>
   704       fold (cp_instance x) (atoms ~~ perm_closed_thmss))
   705         (atoms ~~ perm_closed_thmss) thy6;
   706 
   707     (**** constructors ****)
   708 
   709     fun mk_abs_fun x t =
   710       let
   711         val T = fastype_of x;
   712         val U = fastype_of t
   713       in
   714         Const ("Nominal.abs_fun", T --> U --> T -->
   715           Type ("Nominal.noption", [U])) $ x $ t
   716       end;
   717 
   718     val (ty_idxs, _) = List.foldl
   719       (fn ((i, ("Nominal.noption", _, _)), p) => p
   720         | ((i, _), (ty_idxs, j)) => (ty_idxs @ [(i, j)], j + 1)) ([], 0) descr;
   721 
   722     fun reindex (DtType (s, dts)) = DtType (s, map reindex dts)
   723       | reindex (DtRec i) = DtRec (the (AList.lookup op = ty_idxs i))
   724       | reindex dt = dt;
   725 
   726     fun strip_suffix i s = implode (List.take (raw_explode s, size s - i));  (* FIXME Symbol.explode (?) *)
   727 
   728     (** strips the "_Rep" in type names *)
   729     fun strip_nth_name i s =
   730       let val xs = Long_Name.explode s;
   731       in Long_Name.implode (Library.nth_map (length xs - i) (strip_suffix 4) xs) end;
   732 
   733     val (descr'', ndescr) = ListPair.unzip (map_filter
   734       (fn (i, ("Nominal.noption", _, _)) => NONE
   735         | (i, (s, dts, constrs)) =>
   736              let
   737                val SOME index = AList.lookup op = ty_idxs i;
   738                val (constrs2, constrs1) =
   739                  map_split (fn (cname, cargs) =>
   740                    apsnd (pair (strip_nth_name 2 (strip_nth_name 1 cname)))
   741                    (fold_map (fn dt => fn dts =>
   742                      let val (dts', dt') = strip_option dt
   743                      in ((length dts, length dts'), dts @ dts' @ [reindex dt']) end)
   744                        cargs [])) constrs
   745              in SOME ((index, (strip_nth_name 1 s,  map reindex dts, constrs1)),
   746                (index, constrs2))
   747              end) descr);
   748 
   749     val (descr1, descr2) = chop (length new_type_names) descr'';
   750     val descr' = [descr1, descr2];
   751 
   752     fun partition_cargs idxs xs = map (fn (i, j) =>
   753       (List.take (List.drop (xs, i), j), nth xs (i + j))) idxs;
   754 
   755     val pdescr = map (fn ((i, (s, dts, constrs)), (_, idxss)) => (i, (s, dts,
   756       map (fn ((cname, cargs), idxs) => (cname, partition_cargs idxs cargs))
   757         (constrs ~~ idxss)))) (descr'' ~~ ndescr);
   758 
   759     fun nth_dtyp' i = typ_of_dtyp descr'' sorts (DtRec i);
   760 
   761     val rep_names = map (fn s =>
   762       Sign.intern_const thy7 ("Rep_" ^ s)) new_type_names;
   763     val abs_names = map (fn s =>
   764       Sign.intern_const thy7 ("Abs_" ^ s)) new_type_names;
   765 
   766     val recTs = get_rec_types descr'' sorts;
   767     val newTs' = take (length new_type_names) recTs';
   768     val newTs = take (length new_type_names) recTs;
   769 
   770     val full_new_type_names = map (Sign.full_bname thy) new_type_names;
   771 
   772     fun make_constr_def tname T T' (((cname_rep, _), (cname, cargs)), (cname', mx))
   773         (thy, defs, eqns) =
   774       let
   775         fun constr_arg (dts, dt) (j, l_args, r_args) =
   776           let
   777             val xs = map (fn (dt, i) => mk_Free "x" (typ_of_dtyp descr'' sorts dt) i)
   778               (dts ~~ (j upto j + length dts - 1))
   779             val x = mk_Free "x" (typ_of_dtyp descr'' sorts dt) (j + length dts)
   780           in
   781             (j + length dts + 1,
   782              xs @ x :: l_args,
   783              fold_rev mk_abs_fun xs
   784                (case dt of
   785                   DtRec k => if k < length new_type_names then
   786                       Const (nth rep_names k, typ_of_dtyp descr'' sorts dt -->
   787                         typ_of_dtyp descr sorts dt) $ x
   788                     else error "nested recursion not (yet) supported"
   789                 | _ => x) :: r_args)
   790           end
   791 
   792         val (_, l_args, r_args) = fold_rev constr_arg cargs (1, [], []);
   793         val abs_name = Sign.intern_const thy ("Abs_" ^ tname);
   794         val rep_name = Sign.intern_const thy ("Rep_" ^ tname);
   795         val constrT = map fastype_of l_args ---> T;
   796         val lhs = list_comb (Const (cname, constrT), l_args);
   797         val rhs = list_comb (Const (cname_rep, map fastype_of r_args ---> T'), r_args);
   798         val def = Logic.mk_equals (lhs, Const (abs_name, T' --> T) $ rhs);
   799         val eqn = HOLogic.mk_Trueprop (HOLogic.mk_eq
   800           (Const (rep_name, T --> T') $ lhs, rhs));
   801         val def_name = (Long_Name.base_name cname) ^ "_def";
   802         val ([def_thm], thy') = thy |>
   803           Sign.add_consts_i [(cname', constrT, mx)] |>
   804           (Global_Theory.add_defs false o map Thm.no_attributes) [(Binding.name def_name, def)]
   805       in (thy', defs @ [def_thm], eqns @ [eqn]) end;
   806 
   807     fun dt_constr_defs ((((((_, (_, _, constrs)),
   808         (_, (_, _, constrs'))), tname), T), T'), constr_syntax) (thy, defs, eqns, dist_lemmas) =
   809       let
   810         val rep_const = cterm_of thy
   811           (Const (Sign.intern_const thy ("Rep_" ^ tname), T --> T'));
   812         val dist =
   813           Drule.export_without_context
   814             (cterm_instantiate [(cterm_of thy distinct_f, rep_const)] distinct_lemma);
   815         val (thy', defs', eqns') = fold (make_constr_def tname T T')
   816           (constrs ~~ constrs' ~~ constr_syntax) (Sign.add_path tname thy, defs, [])
   817       in
   818         (Sign.parent_path thy', defs', eqns @ [eqns'], dist_lemmas @ [dist])
   819       end;
   820 
   821     val (thy8, constr_defs, constr_rep_eqns, dist_lemmas) = fold dt_constr_defs
   822       (List.take (descr, length new_type_names) ~~
   823         List.take (pdescr, length new_type_names) ~~
   824         new_type_names ~~ newTs ~~ newTs' ~~ constr_syntax)
   825       (thy7, [], [], []);
   826 
   827     val abs_inject_thms = map (collect_simp o #Abs_inject o snd o fst) typedefs
   828     val rep_inject_thms = map (#Rep_inject o snd o fst) typedefs
   829 
   830     (* prove theorem  Rep_i (Constr_j ...) = Constr'_j ...  *)
   831 
   832     fun prove_constr_rep_thm eqn =
   833       let
   834         val inj_thms = map (fn r => r RS iffD1) abs_inject_thms;
   835         val rewrites = constr_defs @ map mk_meta_eq Rep_inverse_thms
   836       in Goal.prove_global thy8 [] [] eqn (fn _ => EVERY
   837         [resolve_tac inj_thms 1,
   838          rewrite_goals_tac rewrites,
   839          rtac refl 3,
   840          resolve_tac rep_intrs 2,
   841          REPEAT (resolve_tac Rep_thms 1)])
   842       end;
   843 
   844     val constr_rep_thmss = map (map prove_constr_rep_thm) constr_rep_eqns;
   845 
   846     (* prove theorem  pi \<bullet> Rep_i x = Rep_i (pi \<bullet> x) *)
   847 
   848     fun prove_perm_rep_perm (atom, perm_closed_thms) = map (fn th =>
   849       let
   850         val _ $ (_ $ (Rep $ x)) = Logic.unvarify_global (prop_of th);
   851         val Type ("fun", [T, U]) = fastype_of Rep;
   852         val permT = mk_permT (Type (atom, []));
   853         val pi = Free ("pi", permT);
   854       in
   855         Goal.prove_global thy8 [] []
   856           (augment_sort thy8
   857             (pt_class_of thy8 atom :: map (cp_class_of thy8 atom) (remove (op =) atom dt_atoms))
   858             (HOLogic.mk_Trueprop (HOLogic.mk_eq
   859               (Const ("Nominal.perm", permT --> U --> U) $ pi $ (Rep $ x),
   860                Rep $ (Const ("Nominal.perm", permT --> T --> T) $ pi $ x)))))
   861           (fn _ => simp_tac (HOL_basic_ss addsimps (perm_defs @ Abs_inverse_thms @
   862             perm_closed_thms @ Rep_thms)) 1)
   863       end) Rep_thms;
   864 
   865     val perm_rep_perm_thms = maps prove_perm_rep_perm (atoms ~~ perm_closed_thmss);
   866 
   867     (* prove distinctness theorems *)
   868 
   869     val distinct_props = Datatype_Prop.make_distincts descr' sorts;
   870     val dist_rewrites = map2 (fn rep_thms => fn dist_lemma =>
   871       dist_lemma :: rep_thms @ [In0_eq, In1_eq, In0_not_In1, In1_not_In0])
   872         constr_rep_thmss dist_lemmas;
   873 
   874     fun prove_distinct_thms _ (_, []) = []
   875       | prove_distinct_thms (p as (rep_thms, dist_lemma)) (k, t :: ts) =
   876           let
   877             val dist_thm = Goal.prove_global thy8 [] [] t (fn _ =>
   878               simp_tac (global_simpset_of thy8 addsimps (dist_lemma :: rep_thms)) 1)
   879           in
   880             dist_thm :: Drule.export_without_context (dist_thm RS not_sym) ::
   881               prove_distinct_thms p (k, ts)
   882           end;
   883 
   884     val distinct_thms = map2 prove_distinct_thms
   885       (constr_rep_thmss ~~ dist_lemmas) distinct_props;
   886 
   887     (** prove equations for permutation functions **)
   888 
   889     val perm_simps' = map (fn (((i, (_, _, constrs)), tname), constr_rep_thms) =>
   890       let val T = nth_dtyp' i
   891       in maps (fn (atom, perm_closed_thms) =>
   892           map (fn ((cname, dts), constr_rep_thm) =>
   893         let
   894           val cname = Sign.intern_const thy8
   895             (Long_Name.append tname (Long_Name.base_name cname));
   896           val permT = mk_permT (Type (atom, []));
   897           val pi = Free ("pi", permT);
   898 
   899           fun perm t =
   900             let val T = fastype_of t
   901             in Const ("Nominal.perm", permT --> T --> T) $ pi $ t end;
   902 
   903           fun constr_arg (dts, dt) (j, l_args, r_args) =
   904             let
   905               val Ts = map (typ_of_dtyp descr'' sorts) dts;
   906               val xs = map (fn (T, i) => mk_Free "x" T i)
   907                 (Ts ~~ (j upto j + length dts - 1))
   908               val x = mk_Free "x" (typ_of_dtyp descr'' sorts dt) (j + length dts)
   909             in
   910               (j + length dts + 1,
   911                xs @ x :: l_args,
   912                map perm (xs @ [x]) @ r_args)
   913             end
   914 
   915           val (_, l_args, r_args) = fold_rev constr_arg dts (1, [], []);
   916           val c = Const (cname, map fastype_of l_args ---> T)
   917         in
   918           Goal.prove_global thy8 [] []
   919             (augment_sort thy8
   920               (pt_class_of thy8 atom :: map (cp_class_of thy8 atom) (remove (op =) atom dt_atoms))
   921               (HOLogic.mk_Trueprop (HOLogic.mk_eq
   922                 (perm (list_comb (c, l_args)), list_comb (c, r_args)))))
   923             (fn _ => EVERY
   924               [simp_tac (global_simpset_of thy8 addsimps (constr_rep_thm :: perm_defs)) 1,
   925                simp_tac (HOL_basic_ss addsimps (Rep_thms @ Abs_inverse_thms @
   926                  constr_defs @ perm_closed_thms)) 1,
   927                TRY (simp_tac (HOL_basic_ss addsimps
   928                  (Thm.symmetric perm_fun_def :: abs_perm)) 1),
   929                TRY (simp_tac (HOL_basic_ss addsimps
   930                  (perm_fun_def :: perm_defs @ Rep_thms @ Abs_inverse_thms @
   931                     perm_closed_thms)) 1)])
   932         end) (constrs ~~ constr_rep_thms)) (atoms ~~ perm_closed_thmss)
   933       end) (List.take (pdescr, length new_type_names) ~~ new_type_names ~~ constr_rep_thmss);
   934 
   935     (** prove injectivity of constructors **)
   936 
   937     val rep_inject_thms' = map (fn th => th RS sym) rep_inject_thms;
   938     val alpha = Global_Theory.get_thms thy8 "alpha";
   939     val abs_fresh = Global_Theory.get_thms thy8 "abs_fresh";
   940 
   941     val pt_cp_sort =
   942       map (pt_class_of thy8) dt_atoms @
   943       maps (fn s => map (cp_class_of thy8 s) (remove (op =) s dt_atoms)) dt_atoms;
   944 
   945     val inject_thms = map (fn (((i, (_, _, constrs)), tname), constr_rep_thms) =>
   946       let val T = nth_dtyp' i
   947       in map_filter (fn ((cname, dts), constr_rep_thm) =>
   948         if null dts then NONE else SOME
   949         let
   950           val cname = Sign.intern_const thy8
   951             (Long_Name.append tname (Long_Name.base_name cname));
   952 
   953           fun make_inj (dts, dt) (j, args1, args2, eqs) =
   954             let
   955               val Ts_idx = map (typ_of_dtyp descr'' sorts) dts ~~ (j upto j + length dts - 1);
   956               val xs = map (fn (T, i) => mk_Free "x" T i) Ts_idx;
   957               val ys = map (fn (T, i) => mk_Free "y" T i) Ts_idx;
   958               val x = mk_Free "x" (typ_of_dtyp descr'' sorts dt) (j + length dts);
   959               val y = mk_Free "y" (typ_of_dtyp descr'' sorts dt) (j + length dts)
   960             in
   961               (j + length dts + 1,
   962                xs @ (x :: args1), ys @ (y :: args2),
   963                HOLogic.mk_eq
   964                  (fold_rev mk_abs_fun xs x, fold_rev mk_abs_fun ys y) :: eqs)
   965             end;
   966 
   967           val (_, args1, args2, eqs) = fold_rev make_inj dts (1, [], [], []);
   968           val Ts = map fastype_of args1;
   969           val c = Const (cname, Ts ---> T)
   970         in
   971           Goal.prove_global thy8 [] []
   972             (augment_sort thy8 pt_cp_sort
   973               (HOLogic.mk_Trueprop (HOLogic.mk_eq
   974                 (HOLogic.mk_eq (list_comb (c, args1), list_comb (c, args2)),
   975                  foldr1 HOLogic.mk_conj eqs))))
   976             (fn _ => EVERY
   977                [asm_full_simp_tac (global_simpset_of thy8 addsimps (constr_rep_thm ::
   978                   rep_inject_thms')) 1,
   979                 TRY (asm_full_simp_tac (HOL_basic_ss addsimps (fresh_def :: supp_def ::
   980                   alpha @ abs_perm @ abs_fresh @ rep_inject_thms @
   981                   perm_rep_perm_thms)) 1)])
   982         end) (constrs ~~ constr_rep_thms)
   983       end) (List.take (pdescr, length new_type_names) ~~ new_type_names ~~ constr_rep_thmss);
   984 
   985     (** equations for support and freshness **)
   986 
   987     val (supp_thms, fresh_thms) = ListPair.unzip (map ListPair.unzip
   988       (map (fn ((((i, (_, _, constrs)), tname), inject_thms'), perm_thms') =>
   989       let val T = nth_dtyp' i
   990       in maps (fn (cname, dts) => map (fn atom =>
   991         let
   992           val cname = Sign.intern_const thy8
   993             (Long_Name.append tname (Long_Name.base_name cname));
   994           val atomT = Type (atom, []);
   995 
   996           fun process_constr (dts, dt) (j, args1, args2) =
   997             let
   998               val Ts_idx = map (typ_of_dtyp descr'' sorts) dts ~~ (j upto j + length dts - 1);
   999               val xs = map (fn (T, i) => mk_Free "x" T i) Ts_idx;
  1000               val x = mk_Free "x" (typ_of_dtyp descr'' sorts dt) (j + length dts)
  1001             in
  1002               (j + length dts + 1,
  1003                xs @ (x :: args1), fold_rev mk_abs_fun xs x :: args2)
  1004             end;
  1005 
  1006           val (_, args1, args2) = fold_rev process_constr dts (1, [], []);
  1007           val Ts = map fastype_of args1;
  1008           val c = list_comb (Const (cname, Ts ---> T), args1);
  1009           fun supp t =
  1010             Const ("Nominal.supp", fastype_of t --> HOLogic.mk_setT atomT) $ t;
  1011           fun fresh t = fresh_const atomT (fastype_of t) $ Free ("a", atomT) $ t;
  1012           val supp_thm = Goal.prove_global thy8 [] []
  1013             (augment_sort thy8 pt_cp_sort
  1014               (HOLogic.mk_Trueprop (HOLogic.mk_eq
  1015                 (supp c,
  1016                  if null dts then HOLogic.mk_set atomT []
  1017                  else foldr1 (HOLogic.mk_binop @{const_abbrev union}) (map supp args2)))))
  1018             (fn _ =>
  1019               simp_tac (HOL_basic_ss addsimps (supp_def ::
  1020                  Un_assoc :: de_Morgan_conj :: Collect_disj_eq :: finite_Un ::
  1021                  Collect_False_empty :: finite_emptyI :: @{thms simp_thms} @
  1022                  abs_perm @ abs_fresh @ inject_thms' @ perm_thms')) 1)
  1023         in
  1024           (supp_thm,
  1025            Goal.prove_global thy8 [] [] (augment_sort thy8 pt_cp_sort
  1026              (HOLogic.mk_Trueprop (HOLogic.mk_eq
  1027                (fresh c,
  1028                 if null dts then @{term True}
  1029                 else foldr1 HOLogic.mk_conj (map fresh args2)))))
  1030              (fn _ =>
  1031                simp_tac (HOL_ss addsimps [Un_iff, empty_iff, fresh_def, supp_thm]) 1))
  1032         end) atoms) constrs
  1033       end) (List.take (pdescr, length new_type_names) ~~ new_type_names ~~ inject_thms ~~ perm_simps')));
  1034 
  1035     (**** weak induction theorem ****)
  1036 
  1037     fun mk_indrule_lemma (((i, _), T), U) (prems, concls) =
  1038       let
  1039         val Rep_t = Const (nth rep_names i, T --> U) $ mk_Free "x" T i;
  1040 
  1041         val Abs_t =  Const (nth abs_names i, U --> T);
  1042 
  1043       in (prems @ [HOLogic.imp $
  1044             (Const (nth rep_set_names'' i, U --> HOLogic.boolT) $ Rep_t) $
  1045               (mk_Free "P" (T --> HOLogic.boolT) (i + 1) $ (Abs_t $ Rep_t))],
  1046           concls @ [mk_Free "P" (T --> HOLogic.boolT) (i + 1) $ mk_Free "x" T i])
  1047       end;
  1048 
  1049     val (indrule_lemma_prems, indrule_lemma_concls) =
  1050       fold mk_indrule_lemma (descr'' ~~ recTs ~~ recTs') ([], []);
  1051 
  1052     val indrule_lemma = Goal.prove_global thy8 [] []
  1053       (Logic.mk_implies
  1054         (HOLogic.mk_Trueprop (mk_conj indrule_lemma_prems),
  1055          HOLogic.mk_Trueprop (mk_conj indrule_lemma_concls))) (fn _ => EVERY
  1056            [REPEAT (etac conjE 1),
  1057             REPEAT (EVERY
  1058               [TRY (rtac conjI 1), full_simp_tac (HOL_basic_ss addsimps Rep_inverse_thms) 1,
  1059                etac mp 1, resolve_tac Rep_thms 1])]);
  1060 
  1061     val Ps = map head_of (HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of indrule_lemma)));
  1062     val frees = if length Ps = 1 then [Free ("P", snd (dest_Var (hd Ps)))] else
  1063       map (Free o apfst fst o dest_Var) Ps;
  1064     val indrule_lemma' = cterm_instantiate
  1065       (map (cterm_of thy8) Ps ~~ map (cterm_of thy8) frees) indrule_lemma;
  1066 
  1067     val Abs_inverse_thms' = map (fn r => r RS subst) Abs_inverse_thms;
  1068 
  1069     val dt_induct_prop = Datatype_Prop.make_ind descr' sorts;
  1070     val dt_induct = Goal.prove_global thy8 []
  1071       (Logic.strip_imp_prems dt_induct_prop) (Logic.strip_imp_concl dt_induct_prop)
  1072       (fn {prems, ...} => EVERY
  1073         [rtac indrule_lemma' 1,
  1074          (Datatype_Aux.ind_tac rep_induct [] THEN_ALL_NEW Object_Logic.atomize_prems_tac) 1,
  1075          EVERY (map (fn (prem, r) => (EVERY
  1076            [REPEAT (eresolve_tac Abs_inverse_thms' 1),
  1077             simp_tac (HOL_basic_ss addsimps [Thm.symmetric r]) 1,
  1078             DEPTH_SOLVE_1 (ares_tac [prem] 1 ORELSE etac allE 1)]))
  1079                 (prems ~~ constr_defs))]);
  1080 
  1081     val case_names_induct = mk_case_names_induct descr'';
  1082 
  1083     (**** prove that new datatypes have finite support ****)
  1084 
  1085     val _ = warning "proving finite support for the new datatype";
  1086 
  1087     val indnames = Datatype_Prop.make_tnames recTs;
  1088 
  1089     val abs_supp = Global_Theory.get_thms thy8 "abs_supp";
  1090     val supp_atm = Global_Theory.get_thms thy8 "supp_atm";
  1091 
  1092     val finite_supp_thms = map (fn atom =>
  1093       let val atomT = Type (atom, [])
  1094       in map Drule.export_without_context (List.take
  1095         (split_conj_thm (Goal.prove_global thy8 [] []
  1096            (augment_sort thy8 (fs_class_of thy8 atom :: pt_cp_sort)
  1097              (HOLogic.mk_Trueprop
  1098                (foldr1 HOLogic.mk_conj (map (fn (s, T) =>
  1099                  Const ("Finite_Set.finite", HOLogic.mk_setT atomT --> HOLogic.boolT) $
  1100                    (Const ("Nominal.supp", T --> HOLogic.mk_setT atomT) $ Free (s, T)))
  1101                    (indnames ~~ recTs)))))
  1102            (fn _ => Datatype_Aux.ind_tac dt_induct indnames 1 THEN
  1103             ALLGOALS (asm_full_simp_tac (global_simpset_of thy8 addsimps
  1104               (abs_supp @ supp_atm @
  1105                Global_Theory.get_thms thy8 ("fs_" ^ Long_Name.base_name atom ^ "1") @
  1106                flat supp_thms))))),
  1107          length new_type_names))
  1108       end) atoms;
  1109 
  1110     val simp_atts = replicate (length new_type_names) [Simplifier.simp_add];
  1111 
  1112         (* Function to add both the simp and eqvt attributes *)
  1113         (* These two attributes are duplicated on all the types in the mutual nominal datatypes *)
  1114 
  1115     val simp_eqvt_atts = replicate (length new_type_names) [Simplifier.simp_add, NominalThmDecls.eqvt_add];
  1116  
  1117     val (_, thy9) = thy8 |>
  1118       Sign.add_path big_name |>
  1119       Global_Theory.add_thms [((Binding.name "induct", dt_induct), [case_names_induct])] ||>>
  1120       Global_Theory.add_thmss [((Binding.name "inducts", projections dt_induct), [case_names_induct])] ||>
  1121       Sign.parent_path ||>>
  1122       Datatype_Aux.store_thmss_atts "distinct" new_type_names simp_atts distinct_thms ||>>
  1123       Datatype_Aux.store_thmss "constr_rep" new_type_names constr_rep_thmss ||>>
  1124       Datatype_Aux.store_thmss_atts "perm" new_type_names simp_eqvt_atts perm_simps' ||>>
  1125       Datatype_Aux.store_thmss "inject" new_type_names inject_thms ||>>
  1126       Datatype_Aux.store_thmss "supp" new_type_names supp_thms ||>>
  1127       Datatype_Aux.store_thmss_atts "fresh" new_type_names simp_atts fresh_thms ||>
  1128       fold (fn (atom, ths) => fn thy =>
  1129         let
  1130           val class = fs_class_of thy atom;
  1131           val sort = Sign.minimize_sort thy (Sign.certify_sort thy (class :: pt_cp_sort));
  1132         in fold (fn Type (s, Ts) => AxClass.prove_arity
  1133           (s, map (inter_sort thy sort o snd o dest_TFree) Ts, [class])
  1134           (Class.intro_classes_tac [] THEN resolve_tac ths 1)) newTs thy
  1135         end) (atoms ~~ finite_supp_thms);
  1136 
  1137     (**** strong induction theorem ****)
  1138 
  1139     val pnames = if length descr'' = 1 then ["P"]
  1140       else map (fn i => "P" ^ string_of_int i) (1 upto length descr'');
  1141     val ind_sort = if null dt_atomTs then HOLogic.typeS
  1142       else Sign.minimize_sort thy9 (Sign.certify_sort thy9 (map (fs_class_of thy9) dt_atoms));
  1143     val fsT = TFree ("'n", ind_sort);
  1144     val fsT' = TFree ("'n", HOLogic.typeS);
  1145 
  1146     val fresh_fs = map (fn (s, T) => (T, Free (s, fsT' --> HOLogic.mk_setT T)))
  1147       (Datatype_Prop.indexify_names (replicate (length dt_atomTs) "f") ~~ dt_atomTs);
  1148 
  1149     fun make_pred fsT i T = Free (nth pnames i, fsT --> T --> HOLogic.boolT);
  1150 
  1151     fun mk_fresh1 xs [] = []
  1152       | mk_fresh1 xs ((y as (_, T)) :: ys) = map (fn x => HOLogic.mk_Trueprop
  1153             (HOLogic.mk_not (HOLogic.mk_eq (Free y, Free x))))
  1154               (filter (fn (_, U) => T = U) (rev xs)) @
  1155           mk_fresh1 (y :: xs) ys;
  1156 
  1157     fun mk_fresh2 xss [] = []
  1158       | mk_fresh2 xss ((p as (ys, _)) :: yss) = maps (fn y as (_, T) =>
  1159             map (fn (_, x as (_, U)) => HOLogic.mk_Trueprop
  1160               (fresh_const T U $ Free y $ Free x)) (rev xss @ yss)) ys @
  1161           mk_fresh2 (p :: xss) yss;
  1162 
  1163     fun make_ind_prem fsT f k T ((cname, cargs), idxs) =
  1164       let
  1165         val recs = filter is_rec_type cargs;
  1166         val Ts = map (typ_of_dtyp descr'' sorts) cargs;
  1167         val recTs' = map (typ_of_dtyp descr'' sorts) recs;
  1168         val tnames = Name.variant_list pnames (Datatype_Prop.make_tnames Ts);
  1169         val rec_tnames = map fst (filter (is_rec_type o snd) (tnames ~~ cargs));
  1170         val frees = tnames ~~ Ts;
  1171         val frees' = partition_cargs idxs frees;
  1172         val z = (singleton (Name.variant_list tnames) "z", fsT);
  1173 
  1174         fun mk_prem ((dt, s), T) =
  1175           let
  1176             val (Us, U) = strip_type T;
  1177             val l = length Us
  1178           in list_all (z :: map (pair "x") Us, HOLogic.mk_Trueprop
  1179             (make_pred fsT (body_index dt) U $ Bound l $ app_bnds (Free (s, T)) l))
  1180           end;
  1181 
  1182         val prems = map mk_prem (recs ~~ rec_tnames ~~ recTs');
  1183         val prems' = map (fn p as (_, T) => HOLogic.mk_Trueprop
  1184             (f T (Free p) (Free z))) (maps fst frees') @
  1185           mk_fresh1 [] (maps fst frees') @
  1186           mk_fresh2 [] frees'
  1187 
  1188       in list_all_free (frees @ [z], Logic.list_implies (prems' @ prems,
  1189         HOLogic.mk_Trueprop (make_pred fsT k T $ Free z $
  1190           list_comb (Const (cname, Ts ---> T), map Free frees))))
  1191       end;
  1192 
  1193     val ind_prems = maps (fn (((i, (_, _, constrs)), (_, idxss)), T) =>
  1194       map (make_ind_prem fsT (fn T => fn t => fn u =>
  1195         fresh_const T fsT $ t $ u) i T)
  1196           (constrs ~~ idxss)) (descr'' ~~ ndescr ~~ recTs);
  1197     val tnames = Datatype_Prop.make_tnames recTs;
  1198     val zs = Name.variant_list tnames (replicate (length descr'') "z");
  1199     val ind_concl = HOLogic.mk_Trueprop (foldr1 (HOLogic.mk_binop @{const_name HOL.conj})
  1200       (map (fn ((((i, _), T), tname), z) =>
  1201         make_pred fsT i T $ Free (z, fsT) $ Free (tname, T))
  1202         (descr'' ~~ recTs ~~ tnames ~~ zs)));
  1203     val induct = Logic.list_implies (ind_prems, ind_concl);
  1204 
  1205     val ind_prems' =
  1206       map (fn (_, f as Free (_, T)) => list_all_free ([("x", fsT')],
  1207         HOLogic.mk_Trueprop (Const ("Finite_Set.finite",
  1208           Term.range_type T -->
  1209             HOLogic.boolT) $ (f $ Free ("x", fsT'))))) fresh_fs @
  1210       maps (fn (((i, (_, _, constrs)), (_, idxss)), T) =>
  1211         map (make_ind_prem fsT' (fn T => fn t => fn u => HOLogic.Not $
  1212           HOLogic.mk_mem (t, the (AList.lookup op = fresh_fs T) $ u)) i T)
  1213             (constrs ~~ idxss)) (descr'' ~~ ndescr ~~ recTs);
  1214     val ind_concl' = HOLogic.mk_Trueprop (foldr1 (HOLogic.mk_binop @{const_name HOL.conj})
  1215       (map (fn ((((i, _), T), tname), z) =>
  1216         make_pred fsT' i T $ Free (z, fsT') $ Free (tname, T))
  1217         (descr'' ~~ recTs ~~ tnames ~~ zs)));
  1218     val induct' = Logic.list_implies (ind_prems', ind_concl');
  1219 
  1220     val aux_ind_vars =
  1221       (Datatype_Prop.indexify_names (replicate (length dt_atomTs) "pi") ~~
  1222        map mk_permT dt_atomTs) @ [("z", fsT')];
  1223     val aux_ind_Ts = rev (map snd aux_ind_vars);
  1224     val aux_ind_concl = HOLogic.mk_Trueprop (foldr1 (HOLogic.mk_binop @{const_name HOL.conj})
  1225       (map (fn (((i, _), T), tname) =>
  1226         HOLogic.list_all (aux_ind_vars, make_pred fsT' i T $ Bound 0 $
  1227           fold_rev (mk_perm aux_ind_Ts) (map Bound (length dt_atomTs downto 1))
  1228             (Free (tname, T))))
  1229         (descr'' ~~ recTs ~~ tnames)));
  1230 
  1231     val fin_set_supp = map (fn s =>
  1232       at_inst_of thy9 s RS at_fin_set_supp) dt_atoms;
  1233     val fin_set_fresh = map (fn s =>
  1234       at_inst_of thy9 s RS at_fin_set_fresh) dt_atoms;
  1235     val pt1_atoms = map (fn Type (s, _) =>
  1236       Global_Theory.get_thm thy9 ("pt_" ^ Long_Name.base_name s ^ "1")) dt_atomTs;
  1237     val pt2_atoms = map (fn Type (s, _) =>
  1238       Global_Theory.get_thm thy9 ("pt_" ^ Long_Name.base_name s ^ "2") RS sym) dt_atomTs;
  1239     val exists_fresh' = Global_Theory.get_thms thy9 "exists_fresh'";
  1240     val fs_atoms = Global_Theory.get_thms thy9 "fin_supp";
  1241     val abs_supp = Global_Theory.get_thms thy9 "abs_supp";
  1242     val perm_fresh_fresh = Global_Theory.get_thms thy9 "perm_fresh_fresh";
  1243     val calc_atm = Global_Theory.get_thms thy9 "calc_atm";
  1244     val fresh_atm = Global_Theory.get_thms thy9 "fresh_atm";
  1245     val fresh_left = Global_Theory.get_thms thy9 "fresh_left";
  1246     val perm_swap = Global_Theory.get_thms thy9 "perm_swap";
  1247 
  1248     fun obtain_fresh_name' ths ts T (freshs1, freshs2, ctxt) =
  1249       let
  1250         val p = foldr1 HOLogic.mk_prod (ts @ freshs1);
  1251         val ex = Goal.prove ctxt [] [] (HOLogic.mk_Trueprop
  1252             (HOLogic.exists_const T $ Abs ("x", T,
  1253               fresh_const T (fastype_of p) $
  1254                 Bound 0 $ p)))
  1255           (fn _ => EVERY
  1256             [resolve_tac exists_fresh' 1,
  1257              simp_tac (HOL_ss addsimps (supp_prod :: finite_Un :: fs_atoms @
  1258                fin_set_supp @ ths)) 1]);
  1259         val (([(_, cx)], ths), ctxt') = Obtain.result
  1260           (fn _ => EVERY
  1261             [etac exE 1,
  1262              full_simp_tac (HOL_ss addsimps (fresh_prod :: fresh_atm)) 1,
  1263              REPEAT (etac conjE 1)])
  1264           [ex] ctxt
  1265       in (freshs1 @ [term_of cx], freshs2 @ ths, ctxt') end;
  1266 
  1267     fun fresh_fresh_inst thy a b =
  1268       let
  1269         val T = fastype_of a;
  1270         val SOME th = find_first (fn th => case prop_of th of
  1271             _ $ (_ $ (Const (_, Type (_, [U, _])) $ _ $ _)) $ _ => U = T
  1272           | _ => false) perm_fresh_fresh
  1273       in
  1274         Drule.instantiate' []
  1275           [SOME (cterm_of thy a), NONE, SOME (cterm_of thy b)] th
  1276       end;
  1277 
  1278     val fs_cp_sort =
  1279       map (fs_class_of thy9) dt_atoms @
  1280       maps (fn s => map (cp_class_of thy9 s) (remove (op =) s dt_atoms)) dt_atoms;
  1281 
  1282     (**********************************************************************
  1283       The subgoals occurring in the proof of induct_aux have the
  1284       following parameters:
  1285 
  1286         x_1 ... x_k p_1 ... p_m z
  1287 
  1288       where
  1289 
  1290         x_i : constructor arguments (introduced by weak induction rule)
  1291         p_i : permutations (one for each atom type in the data type)
  1292         z   : freshness context
  1293     ***********************************************************************)
  1294 
  1295     val _ = warning "proving strong induction theorem ...";
  1296 
  1297     val induct_aux = Goal.prove_global thy9 []
  1298         (map (augment_sort thy9 fs_cp_sort) ind_prems')
  1299         (augment_sort thy9 fs_cp_sort ind_concl') (fn {prems, context} =>
  1300       let
  1301         val (prems1, prems2) = chop (length dt_atomTs) prems;
  1302         val ind_ss2 = HOL_ss addsimps
  1303           finite_Diff :: abs_fresh @ abs_supp @ fs_atoms;
  1304         val ind_ss1 = ind_ss2 addsimps fresh_left @ calc_atm @
  1305           fresh_atm @ rev_simps @ app_simps;
  1306         val ind_ss3 = HOL_ss addsimps abs_fun_eq1 ::
  1307           abs_perm @ calc_atm @ perm_swap;
  1308         val ind_ss4 = HOL_basic_ss addsimps fresh_left @ prems1 @
  1309           fin_set_fresh @ calc_atm;
  1310         val ind_ss5 = HOL_basic_ss addsimps pt1_atoms;
  1311         val ind_ss6 = HOL_basic_ss addsimps flat perm_simps';
  1312         val th = Goal.prove context [] []
  1313           (augment_sort thy9 fs_cp_sort aux_ind_concl)
  1314           (fn {context = context1, ...} =>
  1315              EVERY (Datatype_Aux.ind_tac dt_induct tnames 1 ::
  1316                maps (fn ((_, (_, _, constrs)), (_, constrs')) =>
  1317                  map (fn ((cname, cargs), is) =>
  1318                    REPEAT (rtac allI 1) THEN
  1319                    SUBPROOF (fn {prems = iprems, params, concl,
  1320                        context = context2, ...} =>
  1321                      let
  1322                        val concl' = term_of concl;
  1323                        val _ $ (_ $ _ $ u) = concl';
  1324                        val U = fastype_of u;
  1325                        val (xs, params') =
  1326                          chop (length cargs) (map (term_of o #2) params);
  1327                        val Ts = map fastype_of xs;
  1328                        val cnstr = Const (cname, Ts ---> U);
  1329                        val (pis, z) = split_last params';
  1330                        val mk_pi = fold_rev (mk_perm []) pis;
  1331                        val xs' = partition_cargs is xs;
  1332                        val xs'' = map (fn (ts, u) => (map mk_pi ts, mk_pi u)) xs';
  1333                        val ts = maps (fn (ts, u) => ts @ [u]) xs'';
  1334                        val (freshs1, freshs2, context3) = fold (fn t =>
  1335                          let val T = fastype_of t
  1336                          in obtain_fresh_name' prems1
  1337                            (the (AList.lookup op = fresh_fs T) $ z :: ts) T
  1338                          end) (maps fst xs') ([], [], context2);
  1339                        val freshs1' = unflat (map fst xs') freshs1;
  1340                        val freshs2' = map (Simplifier.simplify ind_ss4)
  1341                          (mk_not_sym freshs2);
  1342                        val ind_ss1' = ind_ss1 addsimps freshs2';
  1343                        val ind_ss3' = ind_ss3 addsimps freshs2';
  1344                        val rename_eq =
  1345                          if forall (null o fst) xs' then []
  1346                          else [Goal.prove context3 [] []
  1347                            (HOLogic.mk_Trueprop (HOLogic.mk_eq
  1348                              (list_comb (cnstr, ts),
  1349                               list_comb (cnstr, maps (fn ((bs, t), cs) =>
  1350                                 cs @ [fold_rev (mk_perm []) (map perm_of_pair
  1351                                   (bs ~~ cs)) t]) (xs'' ~~ freshs1')))))
  1352                            (fn _ => EVERY
  1353                               (simp_tac (HOL_ss addsimps flat inject_thms) 1 ::
  1354                                REPEAT (FIRSTGOAL (rtac conjI)) ::
  1355                                maps (fn ((bs, t), cs) =>
  1356                                  if null bs then []
  1357                                  else rtac sym 1 :: maps (fn (b, c) =>
  1358                                    [rtac trans 1, rtac sym 1,
  1359                                     rtac (fresh_fresh_inst thy9 b c) 1,
  1360                                     simp_tac ind_ss1' 1,
  1361                                     simp_tac ind_ss2 1,
  1362                                     simp_tac ind_ss3' 1]) (bs ~~ cs))
  1363                                  (xs'' ~~ freshs1')))];
  1364                        val th = Goal.prove context3 [] [] concl' (fn _ => EVERY
  1365                          [simp_tac (ind_ss6 addsimps rename_eq) 1,
  1366                           cut_facts_tac iprems 1,
  1367                           (resolve_tac prems THEN_ALL_NEW
  1368                             SUBGOAL (fn (t, i) => case Logic.strip_assums_concl t of
  1369                                 _ $ (Const ("Nominal.fresh", _) $ _ $ _) =>
  1370                                   simp_tac ind_ss1' i
  1371                               | _ $ (Const (@{const_name Not}, _) $ _) =>
  1372                                   resolve_tac freshs2' i
  1373                               | _ => asm_simp_tac (HOL_basic_ss addsimps
  1374                                   pt2_atoms addsimprocs [perm_simproc]) i)) 1])
  1375                        val final = Proof_Context.export context3 context2 [th]
  1376                      in
  1377                        resolve_tac final 1
  1378                      end) context1 1) (constrs ~~ constrs')) (descr'' ~~ ndescr)))
  1379       in
  1380         EVERY
  1381           [cut_facts_tac [th] 1,
  1382            REPEAT (eresolve_tac [conjE, @{thm allE_Nil}] 1),
  1383            REPEAT (etac allE 1),
  1384            REPEAT (TRY (rtac conjI 1) THEN asm_full_simp_tac ind_ss5 1)]
  1385       end);
  1386 
  1387     val induct_aux' = Thm.instantiate ([],
  1388       map (fn (s, v as Var (_, T)) =>
  1389         (cterm_of thy9 v, cterm_of thy9 (Free (s, T))))
  1390           (pnames ~~ map head_of (HOLogic.dest_conj
  1391              (HOLogic.dest_Trueprop (concl_of induct_aux)))) @
  1392       map (fn (_, f) =>
  1393         let val f' = Logic.varify_global f
  1394         in (cterm_of thy9 f',
  1395           cterm_of thy9 (Const ("Nominal.supp", fastype_of f')))
  1396         end) fresh_fs) induct_aux;
  1397 
  1398     val induct = Goal.prove_global thy9 []
  1399       (map (augment_sort thy9 fs_cp_sort) ind_prems)
  1400       (augment_sort thy9 fs_cp_sort ind_concl)
  1401       (fn {prems, ...} => EVERY
  1402          [rtac induct_aux' 1,
  1403           REPEAT (resolve_tac fs_atoms 1),
  1404           REPEAT ((resolve_tac prems THEN_ALL_NEW
  1405             (etac @{thm meta_spec} ORELSE' full_simp_tac (HOL_basic_ss addsimps [fresh_def]))) 1)])
  1406 
  1407     val (_, thy10) = thy9 |>
  1408       Sign.add_path big_name |>
  1409       Global_Theory.add_thms [((Binding.name "strong_induct'", induct_aux), [])] ||>>
  1410       Global_Theory.add_thms [((Binding.name "strong_induct", induct), [case_names_induct])] ||>>
  1411       Global_Theory.add_thmss [((Binding.name "strong_inducts", projections induct), [case_names_induct])];
  1412 
  1413     (**** recursion combinator ****)
  1414 
  1415     val _ = warning "defining recursion combinator ...";
  1416 
  1417     val used = fold Term.add_tfree_namesT recTs [];
  1418 
  1419     val (rec_result_Ts', rec_fn_Ts') = Datatype_Prop.make_primrec_Ts descr' sorts used;
  1420 
  1421     val rec_sort = if null dt_atomTs then HOLogic.typeS else
  1422       Sign.minimize_sort thy10 (Sign.certify_sort thy10 pt_cp_sort);
  1423 
  1424     val rec_result_Ts = map (fn TFree (s, _) => TFree (s, rec_sort)) rec_result_Ts';
  1425     val rec_fn_Ts = map (typ_subst_atomic (rec_result_Ts' ~~ rec_result_Ts)) rec_fn_Ts';
  1426 
  1427     val rec_set_Ts = map (fn (T1, T2) =>
  1428       rec_fn_Ts @ [T1, T2] ---> HOLogic.boolT) (recTs ~~ rec_result_Ts);
  1429 
  1430     val big_rec_name = big_name ^ "_rec_set";
  1431     val rec_set_names' =
  1432       if length descr'' = 1 then [big_rec_name] else
  1433         map ((curry (op ^) (big_rec_name ^ "_")) o string_of_int)
  1434           (1 upto (length descr''));
  1435     val rec_set_names =  map (Sign.full_bname thy10) rec_set_names';
  1436 
  1437     val rec_fns = map (uncurry (mk_Free "f"))
  1438       (rec_fn_Ts ~~ (1 upto (length rec_fn_Ts)));
  1439     val rec_sets' = map (fn c => list_comb (Free c, rec_fns))
  1440       (rec_set_names' ~~ rec_set_Ts);
  1441     val rec_sets = map (fn c => list_comb (Const c, rec_fns))
  1442       (rec_set_names ~~ rec_set_Ts);
  1443 
  1444     (* introduction rules for graph of recursion function *)
  1445 
  1446     val rec_preds = map (fn (a, T) =>
  1447       Free (a, T --> HOLogic.boolT)) (pnames ~~ rec_result_Ts);
  1448 
  1449     fun mk_fresh3 rs [] = []
  1450       | mk_fresh3 rs ((p as (ys, z)) :: yss) = maps (fn y as (_, T) =>
  1451             map_filter (fn ((_, (_, x)), r as (_, U)) => if z = x then NONE
  1452               else SOME (HOLogic.mk_Trueprop
  1453                 (fresh_const T U $ Free y $ Free r))) rs) ys @
  1454           mk_fresh3 rs yss;
  1455 
  1456     (* FIXME: avoid collisions with other variable names? *)
  1457     val rec_ctxt = Free ("z", fsT');
  1458 
  1459     fun make_rec_intr T p rec_set ((cname, cargs), idxs)
  1460         (rec_intr_ts, rec_prems, rec_prems', rec_eq_prems, l) =
  1461       let
  1462         val Ts = map (typ_of_dtyp descr'' sorts) cargs;
  1463         val frees = map (fn i => "x" ^ string_of_int i) (1 upto length Ts) ~~ Ts;
  1464         val frees' = partition_cargs idxs frees;
  1465         val binders = maps fst frees';
  1466         val atomTs = distinct op = (maps (map snd o fst) frees');
  1467         val recs = map_filter
  1468           (fn ((_, DtRec i), p) => SOME (i, p) | _ => NONE)
  1469           (partition_cargs idxs cargs ~~ frees');
  1470         val frees'' = map (fn i => "y" ^ string_of_int i) (1 upto length recs) ~~
  1471           map (fn (i, _) => nth rec_result_Ts i) recs;
  1472         val prems1 = map (fn ((i, (_, x)), y) => HOLogic.mk_Trueprop
  1473           (nth rec_sets' i $ Free x $ Free y)) (recs ~~ frees'');
  1474         val prems2 =
  1475           map (fn f => map (fn p as (_, T) => HOLogic.mk_Trueprop
  1476             (fresh_const T (fastype_of f) $ Free p $ f)) binders) rec_fns;
  1477         val prems3 = mk_fresh1 [] binders @ mk_fresh2 [] frees';
  1478         val prems4 = map (fn ((i, _), y) =>
  1479           HOLogic.mk_Trueprop (nth rec_preds i $ Free y)) (recs ~~ frees'');
  1480         val prems5 = mk_fresh3 (recs ~~ frees'') frees';
  1481         val prems6 = maps (fn aT => map (fn y as (_, T) => HOLogic.mk_Trueprop
  1482           (Const ("Finite_Set.finite", HOLogic.mk_setT aT --> HOLogic.boolT) $
  1483              (Const ("Nominal.supp", T --> HOLogic.mk_setT aT) $ Free y)))
  1484                frees'') atomTs;
  1485         val prems7 = map (fn x as (_, T) => HOLogic.mk_Trueprop
  1486           (fresh_const T fsT' $ Free x $ rec_ctxt)) binders;
  1487         val result = list_comb (nth rec_fns l, map Free (frees @ frees''));
  1488         val result_freshs = map (fn p as (_, T) =>
  1489           fresh_const T (fastype_of result) $ Free p $ result) binders;
  1490         val P = HOLogic.mk_Trueprop (p $ result)
  1491       in
  1492         (rec_intr_ts @ [Logic.list_implies (flat prems2 @ prems3 @ prems1,
  1493            HOLogic.mk_Trueprop (rec_set $
  1494              list_comb (Const (cname, Ts ---> T), map Free frees) $ result))],
  1495          rec_prems @ [list_all_free (frees @ frees'', Logic.list_implies (prems4, P))],
  1496          rec_prems' @ map (fn fr => list_all_free (frees @ frees'',
  1497            Logic.list_implies (nth prems2 l @ prems3 @ prems5 @ prems7 @ prems6 @ [P],
  1498              HOLogic.mk_Trueprop fr))) result_freshs,
  1499          rec_eq_prems @ [flat prems2 @ prems3],
  1500          l + 1)
  1501       end;
  1502 
  1503     val (rec_intr_ts, rec_prems, rec_prems', rec_eq_prems, _) =
  1504       fold (fn ((((d, d'), T), p), rec_set) =>
  1505         fold (make_rec_intr T p rec_set) (#3 (snd d) ~~ snd d'))
  1506           (descr'' ~~ ndescr ~~ recTs ~~ rec_preds ~~ rec_sets')
  1507           ([], [], [], [], 0);
  1508 
  1509     val ({intrs = rec_intrs, elims = rec_elims, raw_induct = rec_induct, ...}, thy11) =
  1510       thy10
  1511       |> Sign.map_naming Name_Space.conceal
  1512       |> Inductive.add_inductive_global
  1513           {quiet_mode = #quiet config, verbose = false, alt_name = Binding.name big_rec_name,
  1514            coind = false, no_elim = false, no_ind = false, skip_mono = true, fork_mono = false}
  1515           (map (fn (s, T) => ((Binding.name s, T), NoSyn)) (rec_set_names' ~~ rec_set_Ts))
  1516           (map dest_Free rec_fns)
  1517           (map (fn x => (Attrib.empty_binding, x)) rec_intr_ts) []
  1518       ||> Global_Theory.hide_fact true (Long_Name.append (Sign.full_bname thy10 big_rec_name) "induct")
  1519       ||> Sign.restore_naming thy10;
  1520 
  1521     (** equivariance **)
  1522 
  1523     val fresh_bij = Global_Theory.get_thms thy11 "fresh_bij";
  1524     val perm_bij = Global_Theory.get_thms thy11 "perm_bij";
  1525 
  1526     val (rec_equiv_thms, rec_equiv_thms') = ListPair.unzip (map (fn aT =>
  1527       let
  1528         val permT = mk_permT aT;
  1529         val pi = Free ("pi", permT);
  1530         val rec_fns_pi = map (mk_perm [] pi o uncurry (mk_Free "f"))
  1531           (rec_fn_Ts ~~ (1 upto (length rec_fn_Ts)));
  1532         val rec_sets_pi = map (fn c => list_comb (Const c, rec_fns_pi))
  1533           (rec_set_names ~~ rec_set_Ts);
  1534         val ps = map (fn ((((T, U), R), R'), i) =>
  1535           let
  1536             val x = Free ("x" ^ string_of_int i, T);
  1537             val y = Free ("y" ^ string_of_int i, U)
  1538           in
  1539             (R $ x $ y, R' $ mk_perm [] pi x $ mk_perm [] pi y)
  1540           end) (recTs ~~ rec_result_Ts ~~ rec_sets ~~ rec_sets_pi ~~ (1 upto length recTs));
  1541         val ths = map (fn th => Drule.export_without_context (th RS mp)) (split_conj_thm
  1542           (Goal.prove_global thy11 [] []
  1543             (augment_sort thy1 pt_cp_sort
  1544               (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj (map HOLogic.mk_imp ps))))
  1545             (fn _ => rtac rec_induct 1 THEN REPEAT
  1546                (simp_tac (Simplifier.global_context thy11 HOL_basic_ss
  1547                   addsimps flat perm_simps'
  1548                   addsimprocs [NominalPermeq.perm_simproc_app]) 1 THEN
  1549                 (resolve_tac rec_intrs THEN_ALL_NEW
  1550                  asm_simp_tac (HOL_ss addsimps (fresh_bij @ perm_bij))) 1))))
  1551         val ths' = map (fn ((P, Q), th) =>
  1552           Goal.prove_global thy11 [] []
  1553             (augment_sort thy1 pt_cp_sort
  1554               (Logic.mk_implies (HOLogic.mk_Trueprop Q, HOLogic.mk_Trueprop P)))
  1555             (fn _ => dtac (Thm.instantiate ([],
  1556                  [(cterm_of thy11 (Var (("pi", 0), permT)),
  1557                    cterm_of thy11 (Const ("List.rev", permT --> permT) $ pi))]) th) 1 THEN
  1558                NominalPermeq.perm_simp_tac HOL_ss 1)) (ps ~~ ths)
  1559       in (ths, ths') end) dt_atomTs);
  1560 
  1561     (** finite support **)
  1562 
  1563     val rec_fin_supp_thms = map (fn aT =>
  1564       let
  1565         val name = Long_Name.base_name (fst (dest_Type aT));
  1566         val fs_name = Global_Theory.get_thm thy11 ("fs_" ^ name ^ "1");
  1567         val aset = HOLogic.mk_setT aT;
  1568         val finite = Const ("Finite_Set.finite", aset --> HOLogic.boolT);
  1569         val fins = map (fn (f, T) => HOLogic.mk_Trueprop
  1570           (finite $ (Const ("Nominal.supp", T --> aset) $ f)))
  1571             (rec_fns ~~ rec_fn_Ts)
  1572       in
  1573         map (fn th => Drule.export_without_context (th RS mp)) (split_conj_thm
  1574           (Goal.prove_global thy11 []
  1575             (map (augment_sort thy11 fs_cp_sort) fins)
  1576             (augment_sort thy11 fs_cp_sort
  1577               (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
  1578                 (map (fn (((T, U), R), i) =>
  1579                    let
  1580                      val x = Free ("x" ^ string_of_int i, T);
  1581                      val y = Free ("y" ^ string_of_int i, U)
  1582                    in
  1583                      HOLogic.mk_imp (R $ x $ y,
  1584                        finite $ (Const ("Nominal.supp", U --> aset) $ y))
  1585                    end) (recTs ~~ rec_result_Ts ~~ rec_sets ~~
  1586                      (1 upto length recTs))))))
  1587             (fn {prems = fins, ...} =>
  1588               (rtac rec_induct THEN_ALL_NEW cut_facts_tac fins) 1 THEN REPEAT
  1589                (NominalPermeq.finite_guess_tac (HOL_ss addsimps [fs_name]) 1))))
  1590       end) dt_atomTs;
  1591 
  1592     (** freshness **)
  1593 
  1594     val finite_premss = map (fn aT =>
  1595       map (fn (f, T) => HOLogic.mk_Trueprop
  1596         (Const ("Finite_Set.finite", HOLogic.mk_setT aT --> HOLogic.boolT) $
  1597            (Const ("Nominal.supp", T --> HOLogic.mk_setT aT) $ f)))
  1598            (rec_fns ~~ rec_fn_Ts)) dt_atomTs;
  1599 
  1600     val rec_fns' = map (augment_sort thy11 fs_cp_sort) rec_fns;
  1601 
  1602     val rec_fresh_thms = map (fn ((aT, eqvt_ths), finite_prems) =>
  1603       let
  1604         val name = Long_Name.base_name (fst (dest_Type aT));
  1605         val fs_name = Global_Theory.get_thm thy11 ("fs_" ^ name ^ "1");
  1606         val a = Free ("a", aT);
  1607         val freshs = map (fn (f, fT) => HOLogic.mk_Trueprop
  1608           (fresh_const aT fT $ a $ f)) (rec_fns ~~ rec_fn_Ts)
  1609       in
  1610         map (fn (((T, U), R), eqvt_th) =>
  1611           let
  1612             val x = Free ("x", augment_sort_typ thy11 fs_cp_sort T);
  1613             val y = Free ("y", U);
  1614             val y' = Free ("y'", U)
  1615           in
  1616             Drule.export_without_context (Goal.prove (Proof_Context.init_global thy11) []
  1617               (map (augment_sort thy11 fs_cp_sort)
  1618                 (finite_prems @
  1619                    [HOLogic.mk_Trueprop (R $ x $ y),
  1620                     HOLogic.mk_Trueprop (HOLogic.mk_all ("y'", U,
  1621                       HOLogic.mk_imp (R $ x $ y', HOLogic.mk_eq (y', y)))),
  1622                     HOLogic.mk_Trueprop (fresh_const aT T $ a $ x)] @
  1623                  freshs))
  1624               (HOLogic.mk_Trueprop (fresh_const aT U $ a $ y))
  1625               (fn {prems, context} =>
  1626                  let
  1627                    val (finite_prems, rec_prem :: unique_prem ::
  1628                      fresh_prems) = chop (length finite_prems) prems;
  1629                    val unique_prem' = unique_prem RS spec RS mp;
  1630                    val unique = [unique_prem', unique_prem' RS sym] MRS trans;
  1631                    val _ $ (_ $ (_ $ S $ _)) $ _ = prop_of supports_fresh;
  1632                    val tuple = foldr1 HOLogic.mk_prod (x :: rec_fns')
  1633                  in EVERY
  1634                    [rtac (Drule.cterm_instantiate
  1635                       [(cterm_of thy11 S,
  1636                         cterm_of thy11 (Const ("Nominal.supp",
  1637                           fastype_of tuple --> HOLogic.mk_setT aT) $ tuple))]
  1638                       supports_fresh) 1,
  1639                     simp_tac (HOL_basic_ss addsimps
  1640                       [supports_def, Thm.symmetric fresh_def, fresh_prod]) 1,
  1641                     REPEAT_DETERM (resolve_tac [allI, impI] 1),
  1642                     REPEAT_DETERM (etac conjE 1),
  1643                     rtac unique 1,
  1644                     SUBPROOF (fn {prems = prems', params = [(_, a), (_, b)], ...} => EVERY
  1645                       [cut_facts_tac [rec_prem] 1,
  1646                        rtac (Thm.instantiate ([],
  1647                          [(cterm_of thy11 (Var (("pi", 0), mk_permT aT)),
  1648                            cterm_of thy11 (perm_of_pair (term_of a, term_of b)))]) eqvt_th) 1,
  1649                        asm_simp_tac (HOL_ss addsimps
  1650                          (prems' @ perm_swap @ perm_fresh_fresh)) 1]) context 1,
  1651                     rtac rec_prem 1,
  1652                     simp_tac (HOL_ss addsimps (fs_name ::
  1653                       supp_prod :: finite_Un :: finite_prems)) 1,
  1654                     simp_tac (HOL_ss addsimps (Thm.symmetric fresh_def ::
  1655                       fresh_prod :: fresh_prems)) 1]
  1656                  end))
  1657           end) (recTs ~~ rec_result_Ts ~~ rec_sets ~~ eqvt_ths)
  1658       end) (dt_atomTs ~~ rec_equiv_thms' ~~ finite_premss);
  1659 
  1660     (** uniqueness **)
  1661 
  1662     val fun_tuple = foldr1 HOLogic.mk_prod (rec_ctxt :: rec_fns);
  1663     val fun_tupleT = fastype_of fun_tuple;
  1664     val rec_unique_frees =
  1665       Datatype_Prop.indexify_names (replicate (length recTs) "x") ~~ recTs;
  1666     val rec_unique_frees'' = map (fn (s, T) => (s ^ "'", T)) rec_unique_frees;
  1667     val rec_unique_frees' =
  1668       Datatype_Prop.indexify_names (replicate (length recTs) "y") ~~ rec_result_Ts;
  1669     val rec_unique_concls = map (fn ((x, U), R) =>
  1670         Const (@{const_name Ex1}, (U --> HOLogic.boolT) --> HOLogic.boolT) $
  1671           Abs ("y", U, R $ Free x $ Bound 0))
  1672       (rec_unique_frees ~~ rec_result_Ts ~~ rec_sets);
  1673 
  1674     val induct_aux_rec = Drule.cterm_instantiate
  1675       (map (pairself (cterm_of thy11) o apsnd (augment_sort thy11 fs_cp_sort))
  1676          (map (fn (aT, f) => (Logic.varify_global f, Abs ("z", HOLogic.unitT,
  1677             Const ("Nominal.supp", fun_tupleT --> HOLogic.mk_setT aT) $ fun_tuple)))
  1678               fresh_fs @
  1679           map (fn (((P, T), (x, U)), Q) =>
  1680            (Var ((P, 0), Logic.varifyT_global (fsT' --> T --> HOLogic.boolT)),
  1681             Abs ("z", HOLogic.unitT, absfree (x, U) Q)))
  1682               (pnames ~~ recTs ~~ rec_unique_frees ~~ rec_unique_concls) @
  1683           map (fn (s, T) => (Var ((s, 0), Logic.varifyT_global T), Free (s, T)))
  1684             rec_unique_frees)) induct_aux;
  1685 
  1686     fun obtain_fresh_name vs ths rec_fin_supp T (freshs1, freshs2, ctxt) =
  1687       let
  1688         val p = foldr1 HOLogic.mk_prod (vs @ freshs1);
  1689         val ex = Goal.prove ctxt [] [] (HOLogic.mk_Trueprop
  1690             (HOLogic.exists_const T $ Abs ("x", T,
  1691               fresh_const T (fastype_of p) $ Bound 0 $ p)))
  1692           (fn _ => EVERY
  1693             [cut_facts_tac ths 1,
  1694              REPEAT_DETERM (dresolve_tac (the (AList.lookup op = rec_fin_supp T)) 1),
  1695              resolve_tac exists_fresh' 1,
  1696              asm_simp_tac (HOL_ss addsimps (supp_prod :: finite_Un :: fs_atoms)) 1]);
  1697         val (([(_, cx)], ths), ctxt') = Obtain.result
  1698           (fn _ => EVERY
  1699             [etac exE 1,
  1700              full_simp_tac (HOL_ss addsimps (fresh_prod :: fresh_atm)) 1,
  1701              REPEAT (etac conjE 1)])
  1702           [ex] ctxt
  1703       in (freshs1 @ [term_of cx], freshs2 @ ths, ctxt') end;
  1704 
  1705     val finite_ctxt_prems = map (fn aT =>
  1706       HOLogic.mk_Trueprop
  1707         (Const ("Finite_Set.finite", HOLogic.mk_setT aT --> HOLogic.boolT) $
  1708            (Const ("Nominal.supp", fsT' --> HOLogic.mk_setT aT) $ rec_ctxt))) dt_atomTs;
  1709 
  1710     val rec_unique_thms = split_conj_thm (Goal.prove
  1711       (Proof_Context.init_global thy11) (map fst rec_unique_frees)
  1712       (map (augment_sort thy11 fs_cp_sort)
  1713         (flat finite_premss @ finite_ctxt_prems @ rec_prems @ rec_prems'))
  1714       (augment_sort thy11 fs_cp_sort
  1715         (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj rec_unique_concls)))
  1716       (fn {prems, context} =>
  1717          let
  1718            val k = length rec_fns;
  1719            val (finite_thss, ths1) = fold_map (fn T => fn xs =>
  1720              apfst (pair T) (chop k xs)) dt_atomTs prems;
  1721            val (finite_ctxt_ths, ths2) = chop (length dt_atomTs) ths1;
  1722            val (P_ind_ths, fcbs) = chop k ths2;
  1723            val P_ths = map (fn th => th RS mp) (split_conj_thm
  1724              (Goal.prove context
  1725                (map fst (rec_unique_frees'' @ rec_unique_frees')) []
  1726                (augment_sort thy11 fs_cp_sort
  1727                  (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
  1728                     (map (fn (((x, y), S), P) => HOLogic.mk_imp
  1729                       (S $ Free x $ Free y, P $ (Free y)))
  1730                         (rec_unique_frees'' ~~ rec_unique_frees' ~~
  1731                            rec_sets ~~ rec_preds)))))
  1732                (fn _ =>
  1733                   rtac rec_induct 1 THEN
  1734                   REPEAT ((resolve_tac P_ind_ths THEN_ALL_NEW assume_tac) 1))));
  1735            val rec_fin_supp_thms' = map
  1736              (fn (ths, (T, fin_ths)) => (T, map (curry op MRS fin_ths) ths))
  1737              (rec_fin_supp_thms ~~ finite_thss);
  1738          in EVERY
  1739            ([rtac induct_aux_rec 1] @
  1740             maps (fn ((_, finite_ths), finite_th) =>
  1741               [cut_facts_tac (finite_th :: finite_ths) 1,
  1742                asm_simp_tac (HOL_ss addsimps [supp_prod, finite_Un]) 1])
  1743                 (finite_thss ~~ finite_ctxt_ths) @
  1744             maps (fn ((_, idxss), elim) => maps (fn idxs =>
  1745               [full_simp_tac (HOL_ss addsimps [Thm.symmetric fresh_def, supp_prod, Un_iff]) 1,
  1746                REPEAT_DETERM (eresolve_tac [conjE, ex1E] 1),
  1747                rtac ex1I 1,
  1748                (resolve_tac rec_intrs THEN_ALL_NEW atac) 1,
  1749                rotate_tac ~1 1,
  1750                ((DETERM o etac elim) THEN_ALL_NEW full_simp_tac
  1751                   (HOL_ss addsimps flat distinct_thms)) 1] @
  1752                (if null idxs then [] else [hyp_subst_tac 1,
  1753                 SUBPROOF (fn {asms, concl, prems = prems', params, context = context', ...} =>
  1754                   let
  1755                     val SOME prem = find_first (can (HOLogic.dest_eq o
  1756                       HOLogic.dest_Trueprop o prop_of)) prems';
  1757                     val _ $ (_ $ lhs $ rhs) = prop_of prem;
  1758                     val _ $ (_ $ lhs' $ rhs') = term_of concl;
  1759                     val rT = fastype_of lhs';
  1760                     val (c, cargsl) = strip_comb lhs;
  1761                     val cargsl' = partition_cargs idxs cargsl;
  1762                     val boundsl = maps fst cargsl';
  1763                     val (_, cargsr) = strip_comb rhs;
  1764                     val cargsr' = partition_cargs idxs cargsr;
  1765                     val boundsr = maps fst cargsr';
  1766                     val (params1, _ :: params2) =
  1767                       chop (length params div 2) (map (term_of o #2) params);
  1768                     val params' = params1 @ params2;
  1769                     val rec_prems = filter (fn th => case prop_of th of
  1770                         _ $ p => (case head_of p of
  1771                           Const (s, _) => member (op =) rec_set_names s
  1772                         | _ => false)
  1773                       | _ => false) prems';
  1774                     val fresh_prems = filter (fn th => case prop_of th of
  1775                         _ $ (Const ("Nominal.fresh", _) $ _ $ _) => true
  1776                       | _ $ (Const (@{const_name Not}, _) $ _) => true
  1777                       | _ => false) prems';
  1778                     val Ts = map fastype_of boundsl;
  1779 
  1780                     val _ = warning "step 1: obtaining fresh names";
  1781                     val (freshs1, freshs2, context'') = fold
  1782                       (obtain_fresh_name (rec_ctxt :: rec_fns' @ params')
  1783                          (maps snd finite_thss @ finite_ctxt_ths @ rec_prems)
  1784                          rec_fin_supp_thms')
  1785                       Ts ([], [], context');
  1786                     val pi1 = map perm_of_pair (boundsl ~~ freshs1);
  1787                     val rpi1 = rev pi1;
  1788                     val pi2 = map perm_of_pair (boundsr ~~ freshs1);
  1789                     val rpi2 = rev pi2;
  1790 
  1791                     val fresh_prems' = mk_not_sym fresh_prems;
  1792                     val freshs2' = mk_not_sym freshs2;
  1793 
  1794                     (** as, bs, cs # K as ts, K bs us **)
  1795                     val _ = warning "step 2: as, bs, cs # K as ts, K bs us";
  1796                     val prove_fresh_ss = HOL_ss addsimps
  1797                       (finite_Diff :: flat fresh_thms @
  1798                        fs_atoms @ abs_fresh @ abs_supp @ fresh_atm);
  1799                     (* FIXME: avoid asm_full_simp_tac ? *)
  1800                     fun prove_fresh ths y x = Goal.prove context'' [] []
  1801                       (HOLogic.mk_Trueprop (fresh_const
  1802                          (fastype_of x) (fastype_of y) $ x $ y))
  1803                       (fn _ => cut_facts_tac ths 1 THEN asm_full_simp_tac prove_fresh_ss 1);
  1804                     val constr_fresh_thms =
  1805                       map (prove_fresh fresh_prems lhs) boundsl @
  1806                       map (prove_fresh fresh_prems rhs) boundsr @
  1807                       map (prove_fresh freshs2 lhs) freshs1 @
  1808                       map (prove_fresh freshs2 rhs) freshs1;
  1809 
  1810                     (** pi1 o (K as ts) = pi2 o (K bs us) **)
  1811                     val _ = warning "step 3: pi1 o (K as ts) = pi2 o (K bs us)";
  1812                     val pi1_pi2_eq = Goal.prove context'' [] []
  1813                       (HOLogic.mk_Trueprop (HOLogic.mk_eq
  1814                         (fold_rev (mk_perm []) pi1 lhs, fold_rev (mk_perm []) pi2 rhs)))
  1815                       (fn _ => EVERY
  1816                          [cut_facts_tac constr_fresh_thms 1,
  1817                           asm_simp_tac (HOL_basic_ss addsimps perm_fresh_fresh) 1,
  1818                           rtac prem 1]);
  1819 
  1820                     (** pi1 o ts = pi2 o us **)
  1821                     val _ = warning "step 4: pi1 o ts = pi2 o us";
  1822                     val pi1_pi2_eqs = map (fn (t, u) =>
  1823                       Goal.prove context'' [] []
  1824                         (HOLogic.mk_Trueprop (HOLogic.mk_eq
  1825                           (fold_rev (mk_perm []) pi1 t, fold_rev (mk_perm []) pi2 u)))
  1826                         (fn _ => EVERY
  1827                            [cut_facts_tac [pi1_pi2_eq] 1,
  1828                             asm_full_simp_tac (HOL_ss addsimps
  1829                               (calc_atm @ flat perm_simps' @
  1830                                fresh_prems' @ freshs2' @ abs_perm @
  1831                                alpha @ flat inject_thms)) 1]))
  1832                         (map snd cargsl' ~~ map snd cargsr');
  1833 
  1834                     (** pi1^-1 o pi2 o us = ts **)
  1835                     val _ = warning "step 5: pi1^-1 o pi2 o us = ts";
  1836                     val rpi1_pi2_eqs = map (fn ((t, u), eq) =>
  1837                       Goal.prove context'' [] []
  1838                         (HOLogic.mk_Trueprop (HOLogic.mk_eq
  1839                           (fold_rev (mk_perm []) (rpi1 @ pi2) u, t)))
  1840                         (fn _ => simp_tac (HOL_ss addsimps
  1841                            ((eq RS sym) :: perm_swap)) 1))
  1842                         (map snd cargsl' ~~ map snd cargsr' ~~ pi1_pi2_eqs);
  1843 
  1844                     val (rec_prems1, rec_prems2) =
  1845                       chop (length rec_prems div 2) rec_prems;
  1846 
  1847                     (** (ts, pi1^-1 o pi2 o vs) in rec_set **)
  1848                     val _ = warning "step 6: (ts, pi1^-1 o pi2 o vs) in rec_set";
  1849                     val rec_prems' = map (fn th =>
  1850                       let
  1851                         val _ $ (S $ x $ y) = prop_of th;
  1852                         val Const (s, _) = head_of S;
  1853                         val k = find_index (equal s) rec_set_names;
  1854                         val pi = rpi1 @ pi2;
  1855                         fun mk_pi z = fold_rev (mk_perm []) pi z;
  1856                         fun eqvt_tac p =
  1857                           let
  1858                             val U as Type (_, [Type (_, [T, _])]) = fastype_of p;
  1859                             val l = find_index (equal T) dt_atomTs;
  1860                             val th = nth (nth rec_equiv_thms' l) k;
  1861                             val th' = Thm.instantiate ([],
  1862                               [(cterm_of thy11 (Var (("pi", 0), U)),
  1863                                 cterm_of thy11 p)]) th;
  1864                           in rtac th' 1 end;
  1865                         val th' = Goal.prove context'' [] []
  1866                           (HOLogic.mk_Trueprop (S $ mk_pi x $ mk_pi y))
  1867                           (fn _ => EVERY
  1868                              (map eqvt_tac pi @
  1869                               [simp_tac (HOL_ss addsimps (fresh_prems' @ freshs2' @
  1870                                  perm_swap @ perm_fresh_fresh)) 1,
  1871                                rtac th 1]))
  1872                       in
  1873                         Simplifier.simplify
  1874                           (HOL_basic_ss addsimps rpi1_pi2_eqs) th'
  1875                       end) rec_prems2;
  1876 
  1877                     val ihs = filter (fn th => case prop_of th of
  1878                       _ $ (Const (@{const_name All}, _) $ _) => true | _ => false) prems';
  1879 
  1880                     (** pi1 o rs = pi2 o vs , rs = pi1^-1 o pi2 o vs **)
  1881                     val _ = warning "step 7: pi1 o rs = pi2 o vs , rs = pi1^-1 o pi2 o vs";
  1882                     val rec_eqns = map (fn (th, ih) =>
  1883                       let
  1884                         val th' = th RS (ih RS spec RS mp) RS sym;
  1885                         val _ $ (_ $ lhs $ rhs) = prop_of th';
  1886                         fun strip_perm (_ $ _ $ t) = strip_perm t
  1887                           | strip_perm t = t;
  1888                       in
  1889                         Goal.prove context'' [] []
  1890                            (HOLogic.mk_Trueprop (HOLogic.mk_eq
  1891                               (fold_rev (mk_perm []) pi1 lhs,
  1892                                fold_rev (mk_perm []) pi2 (strip_perm rhs))))
  1893                            (fn _ => simp_tac (HOL_basic_ss addsimps
  1894                               (th' :: perm_swap)) 1)
  1895                       end) (rec_prems' ~~ ihs);
  1896 
  1897                     (** as # rs **)
  1898                     val _ = warning "step 8: as # rs";
  1899                     val rec_freshs =
  1900                       maps (fn (rec_prem, ih) =>
  1901                         let
  1902                           val _ $ (S $ x $ (y as Free (_, T))) =
  1903                             prop_of rec_prem;
  1904                           val k = find_index (equal S) rec_sets;
  1905                           val atoms = flat (map_filter (fn (bs, z) =>
  1906                             if z = x then NONE else SOME bs) cargsl')
  1907                         in
  1908                           map (fn a as Free (_, aT) =>
  1909                             let val l = find_index (equal aT) dt_atomTs;
  1910                             in
  1911                               Goal.prove context'' [] []
  1912                                 (HOLogic.mk_Trueprop (fresh_const aT T $ a $ y))
  1913                                 (fn _ => EVERY
  1914                                    (rtac (nth (nth rec_fresh_thms l) k) 1 ::
  1915                                     map (fn th => rtac th 1)
  1916                                       (snd (nth finite_thss l)) @
  1917                                     [rtac rec_prem 1, rtac ih 1,
  1918                                      REPEAT_DETERM (resolve_tac fresh_prems 1)]))
  1919                             end) atoms
  1920                         end) (rec_prems1 ~~ ihs);
  1921 
  1922                     (** as # fK as ts rs , bs # fK bs us vs **)
  1923                     val _ = warning "step 9: as # fK as ts rs , bs # fK bs us vs";
  1924                     fun prove_fresh_result (a as Free (_, aT)) =
  1925                       Goal.prove context'' [] []
  1926                         (HOLogic.mk_Trueprop (fresh_const aT rT $ a $ rhs'))
  1927                         (fn _ => EVERY
  1928                            [resolve_tac fcbs 1,
  1929                             REPEAT_DETERM (resolve_tac
  1930                               (fresh_prems @ rec_freshs) 1),
  1931                             REPEAT_DETERM (resolve_tac (maps snd rec_fin_supp_thms') 1
  1932                               THEN resolve_tac rec_prems 1),
  1933                             resolve_tac P_ind_ths 1,
  1934                             REPEAT_DETERM (resolve_tac (P_ths @ rec_prems) 1)]);
  1935 
  1936                     val fresh_results'' = map prove_fresh_result boundsl;
  1937 
  1938                     fun prove_fresh_result'' ((a as Free (_, aT), b), th) =
  1939                       let val th' = Goal.prove context'' [] []
  1940                         (HOLogic.mk_Trueprop (fresh_const aT rT $
  1941                             fold_rev (mk_perm []) (rpi2 @ pi1) a $
  1942                             fold_rev (mk_perm []) (rpi2 @ pi1) rhs'))
  1943                         (fn _ => simp_tac (HOL_ss addsimps fresh_bij) 1 THEN
  1944                            rtac th 1)
  1945                       in
  1946                         Goal.prove context'' [] []
  1947                           (HOLogic.mk_Trueprop (fresh_const aT rT $ b $ lhs'))
  1948                           (fn _ => EVERY
  1949                              [cut_facts_tac [th'] 1,
  1950                               full_simp_tac (Simplifier.global_context thy11 HOL_ss
  1951                                 addsimps rec_eqns @ pi1_pi2_eqs @ perm_swap
  1952                                 addsimprocs [NominalPermeq.perm_simproc_app]) 1,
  1953                               full_simp_tac (HOL_ss addsimps (calc_atm @
  1954                                 fresh_prems' @ freshs2' @ perm_fresh_fresh)) 1])
  1955                       end;
  1956 
  1957                     val fresh_results = fresh_results'' @ map prove_fresh_result''
  1958                       (boundsl ~~ boundsr ~~ fresh_results'');
  1959 
  1960                     (** cs # fK as ts rs , cs # fK bs us vs **)
  1961                     val _ = warning "step 10: cs # fK as ts rs , cs # fK bs us vs";
  1962                     fun prove_fresh_result' recs t (a as Free (_, aT)) =
  1963                       Goal.prove context'' [] []
  1964                         (HOLogic.mk_Trueprop (fresh_const aT rT $ a $ t))
  1965                         (fn _ => EVERY
  1966                           [cut_facts_tac recs 1,
  1967                            REPEAT_DETERM (dresolve_tac
  1968                              (the (AList.lookup op = rec_fin_supp_thms' aT)) 1),
  1969                            NominalPermeq.fresh_guess_tac
  1970                              (HOL_ss addsimps (freshs2 @
  1971                                 fs_atoms @ fresh_atm @
  1972                                 maps snd finite_thss)) 1]);
  1973 
  1974                     val fresh_results' =
  1975                       map (prove_fresh_result' rec_prems1 rhs') freshs1 @
  1976                       map (prove_fresh_result' rec_prems2 lhs') freshs1;
  1977 
  1978                     (** pi1 o (fK as ts rs) = pi2 o (fK bs us vs) **)
  1979                     val _ = warning "step 11: pi1 o (fK as ts rs) = pi2 o (fK bs us vs)";
  1980                     val pi1_pi2_result = Goal.prove context'' [] []
  1981                       (HOLogic.mk_Trueprop (HOLogic.mk_eq
  1982                         (fold_rev (mk_perm []) pi1 rhs', fold_rev (mk_perm []) pi2 lhs')))
  1983                       (fn _ => simp_tac (Simplifier.context context'' HOL_ss
  1984                            addsimps pi1_pi2_eqs @ rec_eqns
  1985                            addsimprocs [NominalPermeq.perm_simproc_app]) 1 THEN
  1986                          TRY (simp_tac (HOL_ss addsimps
  1987                            (fresh_prems' @ freshs2' @ calc_atm @ perm_fresh_fresh)) 1));
  1988 
  1989                     val _ = warning "final result";
  1990                     val final = Goal.prove context'' [] [] (term_of concl)
  1991                       (fn _ => cut_facts_tac [pi1_pi2_result RS sym] 1 THEN
  1992                         full_simp_tac (HOL_basic_ss addsimps perm_fresh_fresh @
  1993                           fresh_results @ fresh_results') 1);
  1994                     val final' = Proof_Context.export context'' context' [final];
  1995                     val _ = warning "finished!"
  1996                   in
  1997                     resolve_tac final' 1
  1998                   end) context 1])) idxss) (ndescr ~~ rec_elims))
  1999          end));
  2000 
  2001     val rec_total_thms = map (fn r => r RS theI') rec_unique_thms;
  2002 
  2003     (* define primrec combinators *)
  2004 
  2005     val big_reccomb_name = space_implode "_" new_type_names ^ "_rec";
  2006     val reccomb_names = map (Sign.full_bname thy11)
  2007       (if length descr'' = 1 then [big_reccomb_name] else
  2008         (map ((curry (op ^) (big_reccomb_name ^ "_")) o string_of_int)
  2009           (1 upto (length descr''))));
  2010     val reccombs = map (fn ((name, T), T') => list_comb
  2011       (Const (name, rec_fn_Ts @ [T] ---> T'), rec_fns))
  2012         (reccomb_names ~~ recTs ~~ rec_result_Ts);
  2013 
  2014     val (reccomb_defs, thy12) =
  2015       thy11
  2016       |> Sign.add_consts_i (map (fn ((name, T), T') =>
  2017           (Binding.name (Long_Name.base_name name), rec_fn_Ts @ [T] ---> T', NoSyn))
  2018           (reccomb_names ~~ recTs ~~ rec_result_Ts))
  2019       |> (Global_Theory.add_defs false o map Thm.no_attributes) (map (fn ((((name, comb), set), T), T') =>
  2020           (Binding.name (Long_Name.base_name name ^ "_def"), Logic.mk_equals (comb, absfree ("x", T)
  2021            (Const (@{const_name The}, (T' --> HOLogic.boolT) --> T') $ absfree ("y", T')
  2022              (set $ Free ("x", T) $ Free ("y", T'))))))
  2023                (reccomb_names ~~ reccombs ~~ rec_sets ~~ recTs ~~ rec_result_Ts));
  2024 
  2025     (* prove characteristic equations for primrec combinators *)
  2026 
  2027     val rec_thms = map (fn (prems, concl) =>
  2028       let
  2029         val _ $ (_ $ (_ $ x) $ _) = concl;
  2030         val (_, cargs) = strip_comb x;
  2031         val ps = map (fn (x as Free (_, T), i) =>
  2032           (Free ("x" ^ string_of_int i, T), x)) (cargs ~~ (1 upto length cargs));
  2033         val concl' = subst_atomic_types (rec_result_Ts' ~~ rec_result_Ts) concl;
  2034         val prems' = flat finite_premss @ finite_ctxt_prems @
  2035           rec_prems @ rec_prems' @ map (subst_atomic ps) prems;
  2036         fun solve rules prems = resolve_tac rules THEN_ALL_NEW
  2037           (resolve_tac prems THEN_ALL_NEW atac)
  2038       in
  2039         Goal.prove_global thy12 []
  2040           (map (augment_sort thy12 fs_cp_sort) prems')
  2041           (augment_sort thy12 fs_cp_sort concl')
  2042           (fn {prems, ...} => EVERY
  2043             [rewrite_goals_tac reccomb_defs,
  2044              rtac the1_equality 1,
  2045              solve rec_unique_thms prems 1,
  2046              resolve_tac rec_intrs 1,
  2047              REPEAT (solve (prems @ rec_total_thms) prems 1)])
  2048       end) (rec_eq_prems ~~
  2049         Datatype_Prop.make_primrecs new_type_names descr' sorts thy12);
  2050 
  2051     val dt_infos = map_index (make_dt_info pdescr sorts induct reccomb_names rec_thms)
  2052       (descr1 ~~ distinct_thms ~~ inject_thms);
  2053 
  2054     (* FIXME: theorems are stored in database for testing only *)
  2055     val (_, thy13) = thy12 |>
  2056       Global_Theory.add_thmss
  2057         [((Binding.name "rec_equiv", flat rec_equiv_thms), []),
  2058          ((Binding.name "rec_equiv'", flat rec_equiv_thms'), []),
  2059          ((Binding.name "rec_fin_supp", flat rec_fin_supp_thms), []),
  2060          ((Binding.name "rec_fresh", flat rec_fresh_thms), []),
  2061          ((Binding.name "rec_unique", map Drule.export_without_context rec_unique_thms), []),
  2062          ((Binding.name "recs", rec_thms), [])] ||>
  2063       Sign.parent_path ||>
  2064       map_nominal_datatypes (fold Symtab.update dt_infos);
  2065 
  2066   in
  2067     thy13
  2068   end;
  2069 
  2070 val add_nominal_datatype = gen_add_nominal_datatype Datatype.read_typ;
  2071 
  2072 val _ =
  2073   Outer_Syntax.command "nominal_datatype" "define inductive datatypes" Keyword.thy_decl
  2074     (Parse.and_list1 Datatype.parse_decl
  2075       >> (Toplevel.theory o add_nominal_datatype Datatype.default_config));
  2076 
  2077 end