src/HOL/Nominal/nominal_package.ML
author berghofe
Thu Aug 24 15:20:43 2006 +0200 (2006-08-24)
changeset 20411 dd8a1cda2eaf
parent 20397 243293620225
child 20483 04aa552a83bc
permissions -rw-r--r--
Added premises concerning finite support of recursion results to FCBs.
     1 (*  Title:      HOL/Nominal/nominal_package.ML
     2     ID:         $Id$
     3     Author:     Stefan Berghofer and Christian Urban, TU Muenchen
     4 
     5 Nominal datatype package for Isabelle/HOL.
     6 *)
     7 
     8 signature NOMINAL_PACKAGE =
     9 sig
    10   val add_nominal_datatype : bool -> string list -> (string list * bstring * mixfix *
    11     (bstring * string list * mixfix) list) list -> theory -> theory
    12 end
    13 
    14 structure NominalPackage : NOMINAL_PACKAGE =
    15 struct
    16 
    17 open DatatypeAux;
    18 open NominalAtoms;
    19 
    20 (** FIXME: DatatypePackage should export this function **)
    21 
    22 local
    23 
    24 fun dt_recs (DtTFree _) = []
    25   | dt_recs (DtType (_, dts)) = List.concat (map dt_recs dts)
    26   | dt_recs (DtRec i) = [i];
    27 
    28 fun dt_cases (descr: descr) (_, args, constrs) =
    29   let
    30     fun the_bname i = Sign.base_name (#1 (valOf (AList.lookup (op =) descr i)));
    31     val bnames = map the_bname (distinct op = (List.concat (map dt_recs args)));
    32   in map (fn (c, _) => space_implode "_" (Sign.base_name c :: bnames)) constrs end;
    33 
    34 
    35 fun induct_cases descr =
    36   DatatypeProp.indexify_names (List.concat (map (dt_cases descr) (map #2 descr)));
    37 
    38 fun exhaust_cases descr i = dt_cases descr (valOf (AList.lookup (op =) descr i));
    39 
    40 in
    41 
    42 fun mk_case_names_induct descr = RuleCases.case_names (induct_cases descr);
    43 
    44 fun mk_case_names_exhausts descr new =
    45   map (RuleCases.case_names o exhaust_cases descr o #1)
    46     (List.filter (fn ((_, (name, _, _))) => name mem_string new) descr);
    47 
    48 end;
    49 
    50 (*******************************)
    51 
    52 val (_ $ (_ $ (_ $ (distinct_f $ _) $ _))) = hd (prems_of distinct_lemma);
    53 
    54 fun read_typ sign ((Ts, sorts), str) =
    55   let
    56     val T = Type.no_tvars (Sign.read_typ (sign, (AList.lookup op =)
    57       (map (apfst (rpair ~1)) sorts)) str) handle TYPE (msg, _, _) => error msg
    58   in (Ts @ [T], add_typ_tfrees (T, sorts)) end;
    59 
    60 (** taken from HOL/Tools/datatype_aux.ML **)
    61 
    62 fun indtac indrule indnames i st =
    63   let
    64     val ts = HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of indrule));
    65     val ts' = HOLogic.dest_conj (HOLogic.dest_Trueprop
    66       (Logic.strip_imp_concl (List.nth (prems_of st, i - 1))));
    67     val getP = if can HOLogic.dest_imp (hd ts) then
    68       (apfst SOME) o HOLogic.dest_imp else pair NONE;
    69     fun abstr (t1, t2) = (case t1 of
    70         NONE => (case filter (fn Free (s, _) => s mem indnames | _ => false)
    71               (term_frees t2) of
    72             [Free (s, T)] => absfree (s, T, t2)
    73           | _ => sys_error "indtac")
    74       | SOME (_ $ t' $ _) => Abs ("x", fastype_of t', abstract_over (t', t2)))
    75     val cert = cterm_of (Thm.sign_of_thm st);
    76     val Ps = map (cert o head_of o snd o getP) ts;
    77     val indrule' = cterm_instantiate (Ps ~~
    78       (map (cert o abstr o getP) ts')) indrule
    79   in
    80     rtac indrule' i st
    81   end;
    82 
    83 fun mk_subgoal i f st =
    84   let
    85     val cg = List.nth (cprems_of st, i - 1);
    86     val g = term_of cg;
    87     val revcut_rl' = Thm.lift_rule cg revcut_rl;
    88     val v = head_of (Logic.strip_assums_concl (hd (prems_of revcut_rl')));
    89     val ps = Logic.strip_params g;
    90     val cert = cterm_of (sign_of_thm st);
    91   in
    92     compose_tac (false,
    93       Thm.instantiate ([], [(cert v, cert (list_abs (ps,
    94         f (rev ps) (Logic.strip_assums_hyp g) (Logic.strip_assums_concl g))))])
    95       revcut_rl', 2) i st
    96   end;
    97 
    98 (** simplification procedure for sorting permutations **)
    99 
   100 val dj_cp = thm "dj_cp";
   101 
   102 fun dest_permT (Type ("fun", [Type ("List.list", [Type ("*", [T, _])]),
   103       Type ("fun", [_, U])])) = (T, U);
   104 
   105 fun permTs_of (Const ("Nominal.perm", T) $ t $ u) = fst (dest_permT T) :: permTs_of u
   106   | permTs_of _ = [];
   107 
   108 fun perm_simproc' thy ss (Const ("Nominal.perm", T) $ t $ (u as Const ("Nominal.perm", U) $ r $ s)) =
   109       let
   110         val (aT as Type (a, []), S) = dest_permT T;
   111         val (bT as Type (b, []), _) = dest_permT U
   112       in if aT mem permTs_of u andalso aT <> bT then
   113           let
   114             val a' = Sign.base_name a;
   115             val b' = Sign.base_name b;
   116             val cp = PureThy.get_thm thy (Name ("cp_" ^ a' ^ "_" ^ b' ^ "_inst"));
   117             val dj = PureThy.get_thm thy (Name ("dj_" ^ b' ^ "_" ^ a'));
   118             val dj_cp' = [cp, dj] MRS dj_cp;
   119             val cert = SOME o cterm_of thy
   120           in
   121             SOME (mk_meta_eq (Drule.instantiate' [SOME (ctyp_of thy S)]
   122               [cert t, cert r, cert s] dj_cp'))
   123           end
   124         else NONE
   125       end
   126   | perm_simproc' thy ss _ = NONE;
   127 
   128 val perm_simproc =
   129   Simplifier.simproc (the_context ()) "perm_simp" ["pi1 \\<bullet> (pi2 \\<bullet> x)"] perm_simproc';
   130 
   131 val allE_Nil = read_instantiate_sg (the_context()) [("x", "[]")] allE;
   132 
   133 val meta_spec = thm "meta_spec";
   134 
   135 fun projections rule =
   136   ProjectRule.projections (ProofContext.init (Thm.theory_of_thm rule)) rule
   137   |> map (standard #> RuleCases.save rule);
   138 
   139 val supp_prod = thm "supp_prod";
   140 val fresh_prod = thm "fresh_prod";
   141 val supports_fresh = thm "supports_fresh";
   142 val supports_def = thm "Nominal.op supports_def";
   143 val fresh_def = thm "fresh_def";
   144 val supp_def = thm "supp_def";
   145 val rev_simps = thms "rev.simps";
   146 val app_simps = thms "op @.simps";
   147 
   148 fun gen_add_nominal_datatype prep_typ err flat_names new_type_names dts thy =
   149   let
   150     (* this theory is used just for parsing *)
   151   
   152     val tmp_thy = thy |>
   153       Theory.copy |>
   154       Theory.add_types (map (fn (tvs, tname, mx, _) =>
   155         (tname, length tvs, mx)) dts);
   156 
   157     val sign = Theory.sign_of tmp_thy;
   158 
   159     val atoms = atoms_of thy;
   160     val classes = map (NameSpace.map_base (fn s => "pt_" ^ s)) atoms;
   161     val cp_classes = List.concat (map (fn atom1 => map (fn atom2 =>
   162       Sign.intern_class thy ("cp_" ^ Sign.base_name atom1 ^ "_" ^
   163         Sign.base_name atom2)) atoms) atoms);
   164     fun augment_sort S = S union classes;
   165     val augment_sort_typ = map_type_tfree (fn (s, S) => TFree (s, augment_sort S));
   166 
   167     fun prep_constr ((constrs, sorts), (cname, cargs, mx)) =
   168       let val (cargs', sorts') = Library.foldl (prep_typ sign) (([], sorts), cargs)
   169       in (constrs @ [(cname, cargs', mx)], sorts') end
   170 
   171     fun prep_dt_spec ((dts, sorts), (tvs, tname, mx, constrs)) =
   172       let val (constrs', sorts') = Library.foldl prep_constr (([], sorts), constrs)
   173       in (dts @ [(tvs, tname, mx, constrs')], sorts') end
   174 
   175     val (dts', sorts) = Library.foldl prep_dt_spec (([], []), dts);
   176     val sorts' = map (apsnd augment_sort) sorts;
   177     val tyvars = map #1 dts';
   178 
   179     val types_syntax = map (fn (tvs, tname, mx, constrs) => (tname, mx)) dts';
   180     val constr_syntax = map (fn (tvs, tname, mx, constrs) =>
   181       map (fn (cname, cargs, mx) => (cname, mx)) constrs) dts';
   182 
   183     val ps = map (fn (_, n, _, _) =>
   184       (Sign.full_name sign n, Sign.full_name sign (n ^ "_Rep"))) dts;
   185     val rps = map Library.swap ps;
   186 
   187     fun replace_types (Type ("Nominal.ABS", [T, U])) = 
   188           Type ("fun", [T, Type ("Nominal.noption", [replace_types U])])
   189       | replace_types (Type (s, Ts)) =
   190           Type (getOpt (AList.lookup op = ps s, s), map replace_types Ts)
   191       | replace_types T = T;
   192 
   193     val dts'' = map (fn (tvs, tname, mx, constrs) => (tvs, tname ^ "_Rep", NoSyn,
   194       map (fn (cname, cargs, mx) => (cname ^ "_Rep",
   195         map (augment_sort_typ o replace_types) cargs, NoSyn)) constrs)) dts';
   196 
   197     val new_type_names' = map (fn n => n ^ "_Rep") new_type_names;
   198     val full_new_type_names' = map (Sign.full_name (sign_of thy)) new_type_names';
   199 
   200     val ({induction, ...},thy1) =
   201       DatatypePackage.add_datatype_i err flat_names new_type_names' dts'' thy;
   202 
   203     val SOME {descr, ...} = Symtab.lookup
   204       (DatatypePackage.get_datatypes thy1) (hd full_new_type_names');
   205     fun nth_dtyp i = typ_of_dtyp descr sorts' (DtRec i);
   206 
   207     (**** define permutation functions ****)
   208 
   209     val permT = mk_permT (TFree ("'x", HOLogic.typeS));
   210     val pi = Free ("pi", permT);
   211     val perm_types = map (fn (i, _) =>
   212       let val T = nth_dtyp i
   213       in permT --> T --> T end) descr;
   214     val perm_names = replicate (length new_type_names) "Nominal.perm" @
   215       DatatypeProp.indexify_names (map (fn i => Sign.full_name (sign_of thy1)
   216         ("perm_" ^ name_of_typ (nth_dtyp i)))
   217           (length new_type_names upto length descr - 1));
   218     val perm_names_types = perm_names ~~ perm_types;
   219 
   220     val perm_eqs = List.concat (map (fn (i, (_, _, constrs)) =>
   221       let val T = nth_dtyp i
   222       in map (fn (cname, dts) => 
   223         let
   224           val Ts = map (typ_of_dtyp descr sorts') dts;
   225           val names = DatatypeProp.make_tnames Ts;
   226           val args = map Free (names ~~ Ts);
   227           val c = Const (cname, Ts ---> T);
   228           fun perm_arg (dt, x) =
   229             let val T = type_of x
   230             in if is_rec_type dt then
   231                 let val (Us, _) = strip_type T
   232                 in list_abs (map (pair "x") Us,
   233                   Const (List.nth (perm_names_types, body_index dt)) $ pi $
   234                     list_comb (x, map (fn (i, U) =>
   235                       Const ("Nominal.perm", permT --> U --> U) $
   236                         (Const ("List.rev", permT --> permT) $ pi) $
   237                         Bound i) ((length Us - 1 downto 0) ~~ Us)))
   238                 end
   239               else Const ("Nominal.perm", permT --> T --> T) $ pi $ x
   240             end;  
   241         in
   242           (("", HOLogic.mk_Trueprop (HOLogic.mk_eq
   243             (Const (List.nth (perm_names_types, i)) $
   244                Free ("pi", mk_permT (TFree ("'x", HOLogic.typeS))) $
   245                list_comb (c, args),
   246              list_comb (c, map perm_arg (dts ~~ args))))), [])
   247         end) constrs
   248       end) descr);
   249 
   250     val (perm_simps, thy2) = thy1 |>
   251       Theory.add_consts_i (map (fn (s, T) => (Sign.base_name s, T, NoSyn))
   252         (List.drop (perm_names_types, length new_type_names))) |>
   253       PrimrecPackage.add_primrec_unchecked_i "" perm_eqs;
   254 
   255     (**** prove that permutation functions introduced by unfolding are ****)
   256     (**** equivalent to already existing permutation functions         ****)
   257 
   258     val _ = warning ("length descr: " ^ string_of_int (length descr));
   259     val _ = warning ("length new_type_names: " ^ string_of_int (length new_type_names));
   260 
   261     val perm_indnames = DatatypeProp.make_tnames (map body_type perm_types);
   262     val perm_fun_def = PureThy.get_thm thy2 (Name "perm_fun_def");
   263 
   264     val unfolded_perm_eq_thms =
   265       if length descr = length new_type_names then []
   266       else map standard (List.drop (split_conj_thm
   267         (Goal.prove_global thy2 [] []
   268           (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
   269             (map (fn (c as (s, T), x) =>
   270                let val [T1, T2] = binder_types T
   271                in HOLogic.mk_eq (Const c $ pi $ Free (x, T2),
   272                  Const ("Nominal.perm", T) $ pi $ Free (x, T2))
   273                end)
   274              (perm_names_types ~~ perm_indnames))))
   275           (fn _ => EVERY [indtac induction perm_indnames 1,
   276             ALLGOALS (asm_full_simp_tac
   277               (simpset_of thy2 addsimps [perm_fun_def]))])),
   278         length new_type_names));
   279 
   280     (**** prove [] \<bullet> t = t ****)
   281 
   282     val _ = warning "perm_empty_thms";
   283 
   284     val perm_empty_thms = List.concat (map (fn a =>
   285       let val permT = mk_permT (Type (a, []))
   286       in map standard (List.take (split_conj_thm
   287         (Goal.prove_global thy2 [] []
   288           (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
   289             (map (fn ((s, T), x) => HOLogic.mk_eq
   290                 (Const (s, permT --> T --> T) $
   291                    Const ("List.list.Nil", permT) $ Free (x, T),
   292                  Free (x, T)))
   293              (perm_names ~~
   294               map body_type perm_types ~~ perm_indnames))))
   295           (fn _ => EVERY [indtac induction perm_indnames 1,
   296             ALLGOALS (asm_full_simp_tac (simpset_of thy2))])),
   297         length new_type_names))
   298       end)
   299       atoms);
   300 
   301     (**** prove (pi1 @ pi2) \<bullet> t = pi1 \<bullet> (pi2 \<bullet> t) ****)
   302 
   303     val _ = warning "perm_append_thms";
   304 
   305     (*FIXME: these should be looked up statically*)
   306     val at_pt_inst = PureThy.get_thm thy2 (Name "at_pt_inst");
   307     val pt2 = PureThy.get_thm thy2 (Name "pt2");
   308 
   309     val perm_append_thms = List.concat (map (fn a =>
   310       let
   311         val permT = mk_permT (Type (a, []));
   312         val pi1 = Free ("pi1", permT);
   313         val pi2 = Free ("pi2", permT);
   314         val pt_inst = PureThy.get_thm thy2 (Name ("pt_" ^ Sign.base_name a ^ "_inst"));
   315         val pt2' = pt_inst RS pt2;
   316         val pt2_ax = PureThy.get_thm thy2
   317           (Name (NameSpace.map_base (fn s => "pt_" ^ s ^ "2") a));
   318       in List.take (map standard (split_conj_thm
   319         (Goal.prove_global thy2 [] []
   320              (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
   321                 (map (fn ((s, T), x) =>
   322                     let val perm = Const (s, permT --> T --> T)
   323                     in HOLogic.mk_eq
   324                       (perm $ (Const ("List.op @", permT --> permT --> permT) $
   325                          pi1 $ pi2) $ Free (x, T),
   326                        perm $ pi1 $ (perm $ pi2 $ Free (x, T)))
   327                     end)
   328                   (perm_names ~~
   329                    map body_type perm_types ~~ perm_indnames))))
   330            (fn _ => EVERY [indtac induction perm_indnames 1,
   331               ALLGOALS (asm_full_simp_tac (simpset_of thy2 addsimps [pt2', pt2_ax]))]))),
   332          length new_type_names)
   333       end) atoms);
   334 
   335     (**** prove pi1 ~ pi2 ==> pi1 \<bullet> t = pi2 \<bullet> t ****)
   336 
   337     val _ = warning "perm_eq_thms";
   338 
   339     val pt3 = PureThy.get_thm thy2 (Name "pt3");
   340     val pt3_rev = PureThy.get_thm thy2 (Name "pt3_rev");
   341 
   342     val perm_eq_thms = List.concat (map (fn a =>
   343       let
   344         val permT = mk_permT (Type (a, []));
   345         val pi1 = Free ("pi1", permT);
   346         val pi2 = Free ("pi2", permT);
   347         (*FIXME: not robust - better access these theorems using NominalData?*)
   348         val at_inst = PureThy.get_thm thy2 (Name ("at_" ^ Sign.base_name a ^ "_inst"));
   349         val pt_inst = PureThy.get_thm thy2 (Name ("pt_" ^ Sign.base_name a ^ "_inst"));
   350         val pt3' = pt_inst RS pt3;
   351         val pt3_rev' = at_inst RS (pt_inst RS pt3_rev);
   352         val pt3_ax = PureThy.get_thm thy2
   353           (Name (NameSpace.map_base (fn s => "pt_" ^ s ^ "3") a));
   354       in List.take (map standard (split_conj_thm
   355         (Goal.prove_global thy2 [] [] (Logic.mk_implies
   356              (HOLogic.mk_Trueprop (Const ("Nominal.prm_eq",
   357                 permT --> permT --> HOLogic.boolT) $ pi1 $ pi2),
   358               HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
   359                 (map (fn ((s, T), x) =>
   360                     let val perm = Const (s, permT --> T --> T)
   361                     in HOLogic.mk_eq
   362                       (perm $ pi1 $ Free (x, T),
   363                        perm $ pi2 $ Free (x, T))
   364                     end)
   365                   (perm_names ~~
   366                    map body_type perm_types ~~ perm_indnames)))))
   367            (fn _ => EVERY [indtac induction perm_indnames 1,
   368               ALLGOALS (asm_full_simp_tac (simpset_of thy2 addsimps [pt3', pt3_rev', pt3_ax]))]))),
   369          length new_type_names)
   370       end) atoms);
   371 
   372     (**** prove pi1 \<bullet> (pi2 \<bullet> t) = (pi1 \<bullet> pi2) \<bullet> (pi1 \<bullet> t) ****)
   373 
   374     val cp1 = PureThy.get_thm thy2 (Name "cp1");
   375     val dj_cp = PureThy.get_thm thy2 (Name "dj_cp");
   376     val pt_perm_compose = PureThy.get_thm thy2 (Name "pt_perm_compose");
   377     val pt_perm_compose_rev = PureThy.get_thm thy2 (Name "pt_perm_compose_rev");
   378     val dj_perm_perm_forget = PureThy.get_thm thy2 (Name "dj_perm_perm_forget");
   379 
   380     fun composition_instance name1 name2 thy =
   381       let
   382         val name1' = Sign.base_name name1;
   383         val name2' = Sign.base_name name2;
   384         val cp_class = Sign.intern_class thy ("cp_" ^ name1' ^ "_" ^ name2');
   385         val permT1 = mk_permT (Type (name1, []));
   386         val permT2 = mk_permT (Type (name2, []));
   387         val augment = map_type_tfree
   388           (fn (x, S) => TFree (x, cp_class :: S));
   389         val Ts = map (augment o body_type) perm_types;
   390         val cp_inst = PureThy.get_thm thy
   391           (Name ("cp_" ^ name1' ^ "_" ^ name2' ^ "_inst"));
   392         val simps = simpset_of thy addsimps (perm_fun_def ::
   393           (if name1 <> name2 then
   394              let val dj = PureThy.get_thm thy (Name ("dj_" ^ name2' ^ "_" ^ name1'))
   395              in [dj RS (cp_inst RS dj_cp), dj RS dj_perm_perm_forget] end
   396            else
   397              let
   398                val at_inst = PureThy.get_thm thy (Name ("at_" ^ name1' ^ "_inst"));
   399                val pt_inst = PureThy.get_thm thy (Name ("pt_" ^ name1' ^ "_inst"))
   400              in
   401                [cp_inst RS cp1 RS sym,
   402                 at_inst RS (pt_inst RS pt_perm_compose) RS sym,
   403                 at_inst RS (pt_inst RS pt_perm_compose_rev) RS sym]
   404             end))
   405         val thms = split_conj_thm (Goal.prove_global thy [] []
   406             (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
   407               (map (fn ((s, T), x) =>
   408                   let
   409                     val pi1 = Free ("pi1", permT1);
   410                     val pi2 = Free ("pi2", permT2);
   411                     val perm1 = Const (s, permT1 --> T --> T);
   412                     val perm2 = Const (s, permT2 --> T --> T);
   413                     val perm3 = Const ("Nominal.perm", permT1 --> permT2 --> permT2)
   414                   in HOLogic.mk_eq
   415                     (perm1 $ pi1 $ (perm2 $ pi2 $ Free (x, T)),
   416                      perm2 $ (perm3 $ pi1 $ pi2) $ (perm1 $ pi1 $ Free (x, T)))
   417                   end)
   418                 (perm_names ~~ Ts ~~ perm_indnames))))
   419           (fn _ => EVERY [indtac induction perm_indnames 1,
   420              ALLGOALS (asm_full_simp_tac simps)]))
   421       in
   422         foldl (fn ((s, tvs), thy) => AxClass.prove_arity
   423             (s, replicate (length tvs) (cp_class :: classes), [cp_class])
   424             (ClassPackage.intro_classes_tac [] THEN ALLGOALS (resolve_tac thms)) thy)
   425           thy (full_new_type_names' ~~ tyvars)
   426       end;
   427 
   428     val (perm_thmss,thy3) = thy2 |>
   429       fold (fn name1 => fold (composition_instance name1) atoms) atoms |>
   430       curry (Library.foldr (fn ((i, (tyname, args, _)), thy) =>
   431         AxClass.prove_arity (tyname, replicate (length args) classes, classes)
   432         (ClassPackage.intro_classes_tac [] THEN REPEAT (EVERY
   433            [resolve_tac perm_empty_thms 1,
   434             resolve_tac perm_append_thms 1,
   435             resolve_tac perm_eq_thms 1, assume_tac 1])) thy))
   436         (List.take (descr, length new_type_names)) |>
   437       PureThy.add_thmss
   438         [((space_implode "_" new_type_names ^ "_unfolded_perm_eq",
   439           unfolded_perm_eq_thms), [Simplifier.simp_add]),
   440          ((space_implode "_" new_type_names ^ "_perm_empty",
   441           perm_empty_thms), [Simplifier.simp_add]),
   442          ((space_implode "_" new_type_names ^ "_perm_append",
   443           perm_append_thms), [Simplifier.simp_add]),
   444          ((space_implode "_" new_type_names ^ "_perm_eq",
   445           perm_eq_thms), [Simplifier.simp_add])];
   446   
   447     (**** Define representing sets ****)
   448 
   449     val _ = warning "representing sets";
   450 
   451     val rep_set_names = map (Sign.full_name thy3) (DatatypeProp.indexify_names
   452       (map (fn (i, _) => name_of_typ (nth_dtyp i) ^ "_set") descr));
   453     val big_rep_name =
   454       space_implode "_" (DatatypeProp.indexify_names (List.mapPartial
   455         (fn (i, ("Nominal.noption", _, _)) => NONE
   456           | (i, _) => SOME (name_of_typ (nth_dtyp i))) descr)) ^ "_set";
   457     val _ = warning ("big_rep_name: " ^ big_rep_name);
   458 
   459     fun strip_option (dtf as DtType ("fun", [dt, DtRec i])) =
   460           (case AList.lookup op = descr i of
   461              SOME ("Nominal.noption", _, [(_, [dt']), _]) =>
   462                apfst (cons dt) (strip_option dt')
   463            | _ => ([], dtf))
   464       | strip_option (DtType ("fun", [dt, DtType ("Nominal.noption", [dt'])])) =
   465           apfst (cons dt) (strip_option dt')
   466       | strip_option dt = ([], dt);
   467 
   468     val dt_atomTs = distinct op = (map (typ_of_dtyp descr sorts')
   469       (List.concat (map (fn (_, (_, _, cs)) => List.concat
   470         (map (List.concat o map (fst o strip_option) o snd) cs)) descr)));
   471 
   472     fun make_intr s T (cname, cargs) =
   473       let
   474         fun mk_prem (dt, (j, j', prems, ts)) = 
   475           let
   476             val (dts, dt') = strip_option dt;
   477             val (dts', dt'') = strip_dtyp dt';
   478             val Ts = map (typ_of_dtyp descr sorts') dts;
   479             val Us = map (typ_of_dtyp descr sorts') dts';
   480             val T = typ_of_dtyp descr sorts' dt'';
   481             val free = mk_Free "x" (Us ---> T) j;
   482             val free' = app_bnds free (length Us);
   483             fun mk_abs_fun (T, (i, t)) =
   484               let val U = fastype_of t
   485               in (i + 1, Const ("Nominal.abs_fun", [T, U, T] --->
   486                 Type ("Nominal.noption", [U])) $ mk_Free "y" T i $ t)
   487               end
   488           in (j + 1, j' + length Ts,
   489             case dt'' of
   490                 DtRec k => list_all (map (pair "x") Us,
   491                   HOLogic.mk_Trueprop (HOLogic.mk_mem (free',
   492                     Const (List.nth (rep_set_names, k),
   493                       HOLogic.mk_setT T)))) :: prems
   494               | _ => prems,
   495             snd (foldr mk_abs_fun (j', free) Ts) :: ts)
   496           end;
   497 
   498         val (_, _, prems, ts) = foldr mk_prem (1, 1, [], []) cargs;
   499         val concl = HOLogic.mk_Trueprop (HOLogic.mk_mem
   500           (list_comb (Const (cname, map fastype_of ts ---> T), ts),
   501            Const (s, HOLogic.mk_setT T)))
   502       in Logic.list_implies (prems, concl)
   503       end;
   504 
   505     val (intr_ts, ind_consts) =
   506       apfst List.concat (ListPair.unzip (List.mapPartial
   507         (fn ((_, ("Nominal.noption", _, _)), _) => NONE
   508           | ((i, (_, _, constrs)), rep_set_name) =>
   509               let val T = nth_dtyp i
   510               in SOME (map (make_intr rep_set_name T) constrs,
   511                 Const (rep_set_name, HOLogic.mk_setT T))
   512               end)
   513                 (descr ~~ rep_set_names)));
   514 
   515     val (thy4, {raw_induct = rep_induct, intrs = rep_intrs, ...}) =
   516       setmp InductivePackage.quiet_mode false
   517         (InductivePackage.add_inductive_i false true big_rep_name false true false
   518            ind_consts (map (fn x => (("", x), [])) intr_ts) []) thy3;
   519 
   520     (**** Prove that representing set is closed under permutation ****)
   521 
   522     val _ = warning "proving closure under permutation...";
   523 
   524     val perm_indnames' = List.mapPartial
   525       (fn (x, (_, ("Nominal.noption", _, _))) => NONE | (x, _) => SOME x)
   526       (perm_indnames ~~ descr);
   527 
   528     fun mk_perm_closed name = map (fn th => standard (th RS mp))
   529       (List.take (split_conj_thm (Goal.prove_global thy4 [] []
   530         (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj (map
   531            (fn (S, x) =>
   532               let
   533                 val S = map_term_types (map_type_tfree
   534                   (fn (s, cs) => TFree (s, cs union cp_classes))) S;
   535                 val T = HOLogic.dest_setT (fastype_of S);
   536                 val permT = mk_permT (Type (name, []))
   537               in HOLogic.mk_imp (HOLogic.mk_mem (Free (x, T), S),
   538                 HOLogic.mk_mem (Const ("Nominal.perm", permT --> T --> T) $
   539                   Free ("pi", permT) $ Free (x, T), S))
   540               end) (ind_consts ~~ perm_indnames'))))
   541         (fn _ => EVERY (* CU: added perm_fun_def in the final tactic in order to deal with funs *)
   542            [indtac rep_induct [] 1,
   543             ALLGOALS (simp_tac (simpset_of thy4 addsimps
   544               (symmetric perm_fun_def :: PureThy.get_thms thy4 (Name ("abs_perm"))))),
   545             ALLGOALS (resolve_tac rep_intrs 
   546                THEN_ALL_NEW (asm_full_simp_tac (simpset_of thy4 addsimps [perm_fun_def])))])),
   547         length new_type_names));
   548 
   549     (* FIXME: theorems are stored in database for testing only *)
   550     val perm_closed_thmss = map mk_perm_closed atoms;
   551     val (_,thy5) = PureThy.add_thmss [(("perm_closed", List.concat perm_closed_thmss), [])] thy4;
   552 
   553     (**** typedef ****)
   554 
   555     val _ = warning "defining type...";
   556 
   557     val (typedefs, thy6) =
   558       fold_map (fn ((((name, mx), tvs), c), name') => fn thy =>
   559         setmp TypedefPackage.quiet_mode true
   560           (TypedefPackage.add_typedef_i false (SOME name') (name, tvs, mx) c NONE
   561             (rtac exI 1 THEN
   562               QUIET_BREADTH_FIRST (has_fewer_prems 1)
   563               (resolve_tac rep_intrs 1))) thy |> (fn (r, thy) =>
   564         let
   565           val permT = mk_permT (TFree (Name.variant tvs "'a", HOLogic.typeS));
   566           val pi = Free ("pi", permT);
   567           val tvs' = map (fn s => TFree (s, the (AList.lookup op = sorts' s))) tvs;
   568           val T = Type (Sign.intern_type thy name, tvs');
   569           val Const (_, Type (_, [U])) = c
   570         in apfst (pair r o hd)
   571           (PureThy.add_defs_unchecked_i true [(("prm_" ^ name ^ "_def", Logic.mk_equals
   572             (Const ("Nominal.perm", permT --> T --> T) $ pi $ Free ("x", T),
   573              Const (Sign.intern_const thy ("Abs_" ^ name), U --> T) $
   574                (Const ("Nominal.perm", permT --> U --> U) $ pi $
   575                  (Const (Sign.intern_const thy ("Rep_" ^ name), T --> U) $
   576                    Free ("x", T))))), [])] thy)
   577         end))
   578           (types_syntax ~~ tyvars ~~
   579             (List.take (ind_consts, length new_type_names)) ~~ new_type_names) thy5;
   580 
   581     val perm_defs = map snd typedefs;
   582     val Abs_inverse_thms = map (#Abs_inverse o fst) typedefs;
   583     val Rep_inverse_thms = map (#Rep_inverse o fst) typedefs;
   584     val Rep_thms = map (#Rep o fst) typedefs;
   585 
   586     val big_name = space_implode "_" new_type_names;
   587 
   588 
   589     (** prove that new types are in class pt_<name> **)
   590 
   591     val _ = warning "prove that new types are in class pt_<name> ...";
   592 
   593     fun pt_instance ((class, atom), perm_closed_thms) =
   594       fold (fn (((({Abs_inverse, Rep_inverse, Rep, ...},
   595         perm_def), name), tvs), perm_closed) => fn thy =>
   596           AxClass.prove_arity
   597             (Sign.intern_type thy name,
   598               replicate (length tvs) (classes @ cp_classes), [class])
   599             (EVERY [ClassPackage.intro_classes_tac [],
   600               rewrite_goals_tac [perm_def],
   601               asm_full_simp_tac (simpset_of thy addsimps [Rep_inverse]) 1,
   602               asm_full_simp_tac (simpset_of thy addsimps
   603                 [Rep RS perm_closed RS Abs_inverse]) 1,
   604               asm_full_simp_tac (HOL_basic_ss addsimps [PureThy.get_thm thy
   605                 (Name ("pt_" ^ Sign.base_name atom ^ "3"))]) 1]) thy)
   606         (typedefs ~~ new_type_names ~~ tyvars ~~ perm_closed_thms);
   607 
   608 
   609     (** prove that new types are in class cp_<name1>_<name2> **)
   610 
   611     val _ = warning "prove that new types are in class cp_<name1>_<name2> ...";
   612 
   613     fun cp_instance (atom1, perm_closed_thms1) (atom2, perm_closed_thms2) thy =
   614       let
   615         val name = "cp_" ^ Sign.base_name atom1 ^ "_" ^ Sign.base_name atom2;
   616         val class = Sign.intern_class thy name;
   617         val cp1' = PureThy.get_thm thy (Name (name ^ "_inst")) RS cp1
   618       in fold (fn ((((({Abs_inverse, Rep_inverse, Rep, ...},
   619         perm_def), name), tvs), perm_closed1), perm_closed2) => fn thy =>
   620           AxClass.prove_arity
   621             (Sign.intern_type thy name,
   622               replicate (length tvs) (classes @ cp_classes), [class])
   623             (EVERY [ClassPackage.intro_classes_tac [],
   624               rewrite_goals_tac [perm_def],
   625               asm_full_simp_tac (simpset_of thy addsimps
   626                 ((Rep RS perm_closed1 RS Abs_inverse) ::
   627                  (if atom1 = atom2 then []
   628                   else [Rep RS perm_closed2 RS Abs_inverse]))) 1,
   629               cong_tac 1,
   630               rtac refl 1,
   631               rtac cp1' 1]) thy)
   632         (typedefs ~~ new_type_names ~~ tyvars ~~ perm_closed_thms1 ~~
   633           perm_closed_thms2) thy
   634       end;
   635 
   636     val thy7 = fold (fn x => fn thy => thy |>
   637       pt_instance x |>
   638       fold (cp_instance (apfst snd x)) (atoms ~~ perm_closed_thmss))
   639         (classes ~~ atoms ~~ perm_closed_thmss) thy6;
   640 
   641     (**** constructors ****)
   642 
   643     fun mk_abs_fun (x, t) =
   644       let
   645         val T = fastype_of x;
   646         val U = fastype_of t
   647       in
   648         Const ("Nominal.abs_fun", T --> U --> T -->
   649           Type ("Nominal.noption", [U])) $ x $ t
   650       end;
   651 
   652     val (ty_idxs, _) = foldl
   653       (fn ((i, ("Nominal.noption", _, _)), p) => p
   654         | ((i, _), (ty_idxs, j)) => (ty_idxs @ [(i, j)], j + 1)) ([], 0) descr;
   655 
   656     fun reindex (DtType (s, dts)) = DtType (s, map reindex dts)
   657       | reindex (DtRec i) = DtRec (the (AList.lookup op = ty_idxs i))
   658       | reindex dt = dt;
   659 
   660     fun strip_suffix i s = implode (List.take (explode s, size s - i));
   661 
   662     (** strips the "_Rep" in type names *)
   663     fun strip_nth_name i s = 
   664       let val xs = NameSpace.unpack s; 
   665       in NameSpace.pack (Library.nth_map (length xs - i) (strip_suffix 4) xs) end;
   666 
   667     val (descr'', ndescr) = ListPair.unzip (List.mapPartial
   668       (fn (i, ("Nominal.noption", _, _)) => NONE
   669         | (i, (s, dts, constrs)) =>
   670              let
   671                val SOME index = AList.lookup op = ty_idxs i;
   672                val (constrs1, constrs2) = ListPair.unzip
   673                  (map (fn (cname, cargs) => apfst (pair (strip_nth_name 2 (strip_nth_name 1 cname)))
   674                    (foldl_map (fn (dts, dt) =>
   675                      let val (dts', dt') = strip_option dt
   676                      in (dts @ dts' @ [reindex dt'], (length dts, length dts')) end)
   677                        ([], cargs))) constrs)
   678              in SOME ((index, (strip_nth_name 1 s,  map reindex dts, constrs1)),
   679                (index, constrs2))
   680              end) descr);
   681 
   682     val (descr1, descr2) = chop (length new_type_names) descr'';
   683     val descr' = [descr1, descr2];
   684 
   685     fun partition_cargs idxs xs = map (fn (i, j) =>
   686       (List.take (List.drop (xs, i), j), List.nth (xs, i + j))) idxs;
   687 
   688     val pdescr = map (fn ((i, (s, dts, constrs)), (_, idxss)) => (i, (s, dts,
   689       map (fn ((cname, cargs), idxs) => (cname, partition_cargs idxs cargs))
   690         (constrs ~~ idxss)))) (descr'' ~~ ndescr);
   691 
   692     fun nth_dtyp' i = typ_of_dtyp descr'' sorts' (DtRec i);
   693 
   694     val rep_names = map (fn s =>
   695       Sign.intern_const thy7 ("Rep_" ^ s)) new_type_names;
   696     val abs_names = map (fn s =>
   697       Sign.intern_const thy7 ("Abs_" ^ s)) new_type_names;
   698 
   699     val recTs' = List.mapPartial
   700       (fn ((_, ("Nominal.noption", _, _)), T) => NONE
   701         | (_, T) => SOME T) (descr ~~ get_rec_types descr sorts');
   702     val recTs = get_rec_types descr'' sorts';
   703     val newTs' = Library.take (length new_type_names, recTs');
   704     val newTs = Library.take (length new_type_names, recTs);
   705 
   706     val full_new_type_names = map (Sign.full_name (sign_of thy)) new_type_names;
   707 
   708     fun make_constr_def tname T T' ((thy, defs, eqns),
   709         (((cname_rep, _), (cname, cargs)), (cname', mx))) =
   710       let
   711         fun constr_arg ((dts, dt), (j, l_args, r_args)) =
   712           let
   713             val xs = map (fn (dt, i) => mk_Free "x" (typ_of_dtyp descr'' sorts' dt) i)
   714               (dts ~~ (j upto j + length dts - 1))
   715             val x = mk_Free "x" (typ_of_dtyp descr'' sorts' dt) (j + length dts)
   716           in
   717             (j + length dts + 1,
   718              xs @ x :: l_args,
   719              foldr mk_abs_fun
   720                (case dt of
   721                   DtRec k => if k < length new_type_names then
   722                       Const (List.nth (rep_names, k), typ_of_dtyp descr'' sorts' dt -->
   723                         typ_of_dtyp descr sorts' dt) $ x
   724                     else error "nested recursion not (yet) supported"
   725                 | _ => x) xs :: r_args)
   726           end
   727 
   728         val (_, l_args, r_args) = foldr constr_arg (1, [], []) cargs;
   729         val abs_name = Sign.intern_const (Theory.sign_of thy) ("Abs_" ^ tname);
   730         val rep_name = Sign.intern_const (Theory.sign_of thy) ("Rep_" ^ tname);
   731         val constrT = map fastype_of l_args ---> T;
   732         val lhs = list_comb (Const (cname, constrT), l_args);
   733         val rhs = list_comb (Const (cname_rep, map fastype_of r_args ---> T'), r_args);
   734         val def = Logic.mk_equals (lhs, Const (abs_name, T' --> T) $ rhs);
   735         val eqn = HOLogic.mk_Trueprop (HOLogic.mk_eq
   736           (Const (rep_name, T --> T') $ lhs, rhs));
   737         val def_name = (Sign.base_name cname) ^ "_def";
   738         val ([def_thm], thy') = thy |>
   739           Theory.add_consts_i [(cname', constrT, mx)] |>
   740           (PureThy.add_defs_i false o map Thm.no_attributes) [(def_name, def)]
   741       in (thy', defs @ [def_thm], eqns @ [eqn]) end;
   742 
   743     fun dt_constr_defs ((thy, defs, eqns, dist_lemmas), ((((((_, (_, _, constrs)),
   744         (_, (_, _, constrs'))), tname), T), T'), constr_syntax)) =
   745       let
   746         val rep_const = cterm_of thy
   747           (Const (Sign.intern_const thy ("Rep_" ^ tname), T --> T'));
   748         val dist = standard (cterm_instantiate [(cterm_of thy distinct_f, rep_const)] distinct_lemma);
   749         val (thy', defs', eqns') = Library.foldl (make_constr_def tname T T')
   750           ((Theory.add_path tname thy, defs, []), constrs ~~ constrs' ~~ constr_syntax)
   751       in
   752         (parent_path flat_names thy', defs', eqns @ [eqns'], dist_lemmas @ [dist])
   753       end;
   754 
   755     val (thy8, constr_defs, constr_rep_eqns, dist_lemmas) = Library.foldl dt_constr_defs
   756       ((thy7, [], [], []), List.take (descr, length new_type_names) ~~
   757         List.take (pdescr, length new_type_names) ~~
   758         new_type_names ~~ newTs ~~ newTs' ~~ constr_syntax);
   759 
   760     val abs_inject_thms = map (fn tname =>
   761       PureThy.get_thm thy8 (Name ("Abs_" ^ tname ^ "_inject"))) new_type_names;
   762 
   763     val rep_inject_thms = map (fn tname =>
   764       PureThy.get_thm thy8 (Name ("Rep_" ^ tname ^ "_inject"))) new_type_names;
   765 
   766     val rep_thms = map (fn tname =>
   767       PureThy.get_thm thy8 (Name ("Rep_" ^ tname))) new_type_names;
   768 
   769     val rep_inverse_thms = map (fn tname =>
   770       PureThy.get_thm thy8 (Name ("Rep_" ^ tname ^ "_inverse"))) new_type_names;
   771 
   772     (* prove theorem  Rep_i (Constr_j ...) = Constr'_j ...  *)
   773     
   774     fun prove_constr_rep_thm eqn =
   775       let
   776         val inj_thms = map (fn r => r RS iffD1) abs_inject_thms;
   777         val rewrites = constr_defs @ map mk_meta_eq rep_inverse_thms
   778       in Goal.prove_global thy8 [] [] eqn (fn _ => EVERY
   779         [resolve_tac inj_thms 1,
   780          rewrite_goals_tac rewrites,
   781          rtac refl 3,
   782          resolve_tac rep_intrs 2,
   783          REPEAT (resolve_tac rep_thms 1)])
   784       end;
   785 
   786     val constr_rep_thmss = map (map prove_constr_rep_thm) constr_rep_eqns;
   787 
   788     (* prove theorem  pi \<bullet> Rep_i x = Rep_i (pi \<bullet> x) *)
   789 
   790     fun prove_perm_rep_perm (atom, perm_closed_thms) = map (fn th =>
   791       let
   792         val _ $ (_ $ (Rep $ x) $ _) = Logic.unvarify (prop_of th);
   793         val Type ("fun", [T, U]) = fastype_of Rep;
   794         val permT = mk_permT (Type (atom, []));
   795         val pi = Free ("pi", permT);
   796       in
   797         Goal.prove_global thy8 [] [] (HOLogic.mk_Trueprop (HOLogic.mk_eq
   798             (Const ("Nominal.perm", permT --> U --> U) $ pi $ (Rep $ x),
   799              Rep $ (Const ("Nominal.perm", permT --> T --> T) $ pi $ x))))
   800           (fn _ => simp_tac (HOL_basic_ss addsimps (perm_defs @ Abs_inverse_thms @
   801             perm_closed_thms @ Rep_thms)) 1)
   802       end) Rep_thms;
   803 
   804     val perm_rep_perm_thms = List.concat (map prove_perm_rep_perm
   805       (atoms ~~ perm_closed_thmss));
   806 
   807     (* prove distinctness theorems *)
   808 
   809     val distinct_props = setmp DatatypeProp.dtK 1000
   810       (DatatypeProp.make_distincts new_type_names descr' sorts') thy8;
   811 
   812     val dist_rewrites = map (fn (rep_thms, dist_lemma) =>
   813       dist_lemma::(rep_thms @ [In0_eq, In1_eq, In0_not_In1, In1_not_In0]))
   814         (constr_rep_thmss ~~ dist_lemmas);
   815 
   816     fun prove_distinct_thms (_, []) = []
   817       | prove_distinct_thms (p as (rep_thms, dist_lemma), t::ts) =
   818           let
   819             val dist_thm = Goal.prove_global thy8 [] [] t (fn _ =>
   820               simp_tac (simpset_of thy8 addsimps (dist_lemma :: rep_thms)) 1)
   821           in dist_thm::(standard (dist_thm RS not_sym))::
   822             (prove_distinct_thms (p, ts))
   823           end;
   824 
   825     val distinct_thms = map prove_distinct_thms
   826       (constr_rep_thmss ~~ dist_lemmas ~~ distinct_props);
   827 
   828     (** prove equations for permutation functions **)
   829 
   830     val abs_perm = PureThy.get_thms thy8 (Name "abs_perm"); (* FIXME *)
   831 
   832     val perm_simps' = map (fn (((i, (_, _, constrs)), tname), constr_rep_thms) =>
   833       let val T = nth_dtyp' i
   834       in List.concat (map (fn (atom, perm_closed_thms) =>
   835           map (fn ((cname, dts), constr_rep_thm) => 
   836         let
   837           val cname = Sign.intern_const thy8
   838             (NameSpace.append tname (Sign.base_name cname));
   839           val permT = mk_permT (Type (atom, []));
   840           val pi = Free ("pi", permT);
   841 
   842           fun perm t =
   843             let val T = fastype_of t
   844             in Const ("Nominal.perm", permT --> T --> T) $ pi $ t end;
   845 
   846           fun constr_arg ((dts, dt), (j, l_args, r_args)) =
   847             let
   848               val Ts = map (typ_of_dtyp descr'' sorts') dts;
   849               val xs = map (fn (T, i) => mk_Free "x" T i)
   850                 (Ts ~~ (j upto j + length dts - 1))
   851               val x = mk_Free "x" (typ_of_dtyp descr'' sorts' dt) (j + length dts)
   852             in
   853               (j + length dts + 1,
   854                xs @ x :: l_args,
   855                map perm (xs @ [x]) @ r_args)
   856             end
   857 
   858           val (_, l_args, r_args) = foldr constr_arg (1, [], []) dts;
   859           val c = Const (cname, map fastype_of l_args ---> T)
   860         in
   861           Goal.prove_global thy8 [] []
   862             (HOLogic.mk_Trueprop (HOLogic.mk_eq
   863               (perm (list_comb (c, l_args)), list_comb (c, r_args))))
   864             (fn _ => EVERY
   865               [simp_tac (simpset_of thy8 addsimps (constr_rep_thm :: perm_defs)) 1,
   866                simp_tac (HOL_basic_ss addsimps (Rep_thms @ Abs_inverse_thms @
   867                  constr_defs @ perm_closed_thms)) 1,
   868                TRY (simp_tac (HOL_basic_ss addsimps
   869                  (symmetric perm_fun_def :: abs_perm)) 1),
   870                TRY (simp_tac (HOL_basic_ss addsimps
   871                  (perm_fun_def :: perm_defs @ Rep_thms @ Abs_inverse_thms @
   872                     perm_closed_thms)) 1)])
   873         end) (constrs ~~ constr_rep_thms)) (atoms ~~ perm_closed_thmss))
   874       end) (List.take (pdescr, length new_type_names) ~~ new_type_names ~~ constr_rep_thmss);
   875 
   876     (** prove injectivity of constructors **)
   877 
   878     val rep_inject_thms' = map (fn th => th RS sym) rep_inject_thms;
   879     val alpha = PureThy.get_thms thy8 (Name "alpha");
   880     val abs_fresh = PureThy.get_thms thy8 (Name "abs_fresh");
   881 
   882     val inject_thms = map (fn (((i, (_, _, constrs)), tname), constr_rep_thms) =>
   883       let val T = nth_dtyp' i
   884       in List.mapPartial (fn ((cname, dts), constr_rep_thm) =>
   885         if null dts then NONE else SOME
   886         let
   887           val cname = Sign.intern_const thy8
   888             (NameSpace.append tname (Sign.base_name cname));
   889 
   890           fun make_inj ((dts, dt), (j, args1, args2, eqs)) =
   891             let
   892               val Ts_idx = map (typ_of_dtyp descr'' sorts') dts ~~ (j upto j + length dts - 1);
   893               val xs = map (fn (T, i) => mk_Free "x" T i) Ts_idx;
   894               val ys = map (fn (T, i) => mk_Free "y" T i) Ts_idx;
   895               val x = mk_Free "x" (typ_of_dtyp descr'' sorts' dt) (j + length dts);
   896               val y = mk_Free "y" (typ_of_dtyp descr'' sorts' dt) (j + length dts)
   897             in
   898               (j + length dts + 1,
   899                xs @ (x :: args1), ys @ (y :: args2),
   900                HOLogic.mk_eq
   901                  (foldr mk_abs_fun x xs, foldr mk_abs_fun y ys) :: eqs)
   902             end;
   903 
   904           val (_, args1, args2, eqs) = foldr make_inj (1, [], [], []) dts;
   905           val Ts = map fastype_of args1;
   906           val c = Const (cname, Ts ---> T)
   907         in
   908           Goal.prove_global thy8 [] [] (HOLogic.mk_Trueprop (HOLogic.mk_eq
   909               (HOLogic.mk_eq (list_comb (c, args1), list_comb (c, args2)),
   910                foldr1 HOLogic.mk_conj eqs)))
   911             (fn _ => EVERY
   912                [asm_full_simp_tac (simpset_of thy8 addsimps (constr_rep_thm ::
   913                   rep_inject_thms')) 1,
   914                 TRY (asm_full_simp_tac (HOL_basic_ss addsimps (fresh_def :: supp_def ::
   915                   alpha @ abs_perm @ abs_fresh @ rep_inject_thms @
   916                   perm_rep_perm_thms)) 1),
   917                 TRY (asm_full_simp_tac (HOL_basic_ss addsimps (perm_fun_def ::
   918                   expand_fun_eq :: rep_inject_thms @ perm_rep_perm_thms)) 1)])
   919         end) (constrs ~~ constr_rep_thms)
   920       end) (List.take (pdescr, length new_type_names) ~~ new_type_names ~~ constr_rep_thmss);
   921 
   922     (** equations for support and freshness **)
   923 
   924     val (supp_thms, fresh_thms) = ListPair.unzip (map ListPair.unzip
   925       (map (fn ((((i, (_, _, constrs)), tname), inject_thms'), perm_thms') =>
   926       let val T = nth_dtyp' i
   927       in List.concat (map (fn (cname, dts) => map (fn atom =>
   928         let
   929           val cname = Sign.intern_const thy8
   930             (NameSpace.append tname (Sign.base_name cname));
   931           val atomT = Type (atom, []);
   932 
   933           fun process_constr ((dts, dt), (j, args1, args2)) =
   934             let
   935               val Ts_idx = map (typ_of_dtyp descr'' sorts') dts ~~ (j upto j + length dts - 1);
   936               val xs = map (fn (T, i) => mk_Free "x" T i) Ts_idx;
   937               val x = mk_Free "x" (typ_of_dtyp descr'' sorts' dt) (j + length dts)
   938             in
   939               (j + length dts + 1,
   940                xs @ (x :: args1), foldr mk_abs_fun x xs :: args2)
   941             end;
   942 
   943           val (_, args1, args2) = foldr process_constr (1, [], []) dts;
   944           val Ts = map fastype_of args1;
   945           val c = list_comb (Const (cname, Ts ---> T), args1);
   946           fun supp t =
   947             Const ("Nominal.supp", fastype_of t --> HOLogic.mk_setT atomT) $ t;
   948           fun fresh t =
   949             Const ("Nominal.fresh", atomT --> fastype_of t --> HOLogic.boolT) $
   950               Free ("a", atomT) $ t;
   951           val supp_thm = Goal.prove_global thy8 [] []
   952               (HOLogic.mk_Trueprop (HOLogic.mk_eq
   953                 (supp c,
   954                  if null dts then Const ("{}", HOLogic.mk_setT atomT)
   955                  else foldr1 (HOLogic.mk_binop "op Un") (map supp args2))))
   956             (fn _ =>
   957               simp_tac (HOL_basic_ss addsimps (supp_def ::
   958                  Un_assoc :: de_Morgan_conj :: Collect_disj_eq :: finite_Un ::
   959                  symmetric empty_def :: Finites.emptyI :: simp_thms @
   960                  abs_perm @ abs_fresh @ inject_thms' @ perm_thms')) 1)
   961         in
   962           (supp_thm,
   963            Goal.prove_global thy8 [] [] (HOLogic.mk_Trueprop (HOLogic.mk_eq
   964               (fresh c,
   965                if null dts then HOLogic.true_const
   966                else foldr1 HOLogic.mk_conj (map fresh args2))))
   967              (fn _ =>
   968                simp_tac (simpset_of thy8 addsimps [fresh_def, supp_thm]) 1))
   969         end) atoms) constrs)
   970       end) (List.take (pdescr, length new_type_names) ~~ new_type_names ~~ inject_thms ~~ perm_simps')));
   971 
   972     (**** weak induction theorem ****)
   973 
   974     fun mk_indrule_lemma ((prems, concls), (((i, _), T), U)) =
   975       let
   976         val Rep_t = Const (List.nth (rep_names, i), T --> U) $
   977           mk_Free "x" T i;
   978 
   979         val Abs_t =  Const (List.nth (abs_names, i), U --> T)
   980 
   981       in (prems @ [HOLogic.imp $ HOLogic.mk_mem (Rep_t,
   982             Const (List.nth (rep_set_names, i), HOLogic.mk_setT U)) $
   983               (mk_Free "P" (T --> HOLogic.boolT) (i + 1) $ (Abs_t $ Rep_t))],
   984           concls @ [mk_Free "P" (T --> HOLogic.boolT) (i + 1) $ mk_Free "x" T i])
   985       end;
   986 
   987     val (indrule_lemma_prems, indrule_lemma_concls) =
   988       Library.foldl mk_indrule_lemma (([], []), (descr'' ~~ recTs ~~ recTs'));
   989 
   990     val indrule_lemma = Goal.prove_global thy8 [] []
   991       (Logic.mk_implies
   992         (HOLogic.mk_Trueprop (mk_conj indrule_lemma_prems),
   993          HOLogic.mk_Trueprop (mk_conj indrule_lemma_concls))) (fn _ => EVERY
   994            [REPEAT (etac conjE 1),
   995             REPEAT (EVERY
   996               [TRY (rtac conjI 1), full_simp_tac (HOL_basic_ss addsimps Rep_inverse_thms) 1,
   997                etac mp 1, resolve_tac Rep_thms 1])]);
   998 
   999     val Ps = map head_of (HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of indrule_lemma)));
  1000     val frees = if length Ps = 1 then [Free ("P", snd (dest_Var (hd Ps)))] else
  1001       map (Free o apfst fst o dest_Var) Ps;
  1002     val indrule_lemma' = cterm_instantiate
  1003       (map (cterm_of thy8) Ps ~~ map (cterm_of thy8) frees) indrule_lemma;
  1004 
  1005     val Abs_inverse_thms' = map (fn r => r RS subst) Abs_inverse_thms;
  1006 
  1007     val dt_induct_prop = DatatypeProp.make_ind descr' sorts';
  1008     val dt_induct = Goal.prove_global thy8 []
  1009       (Logic.strip_imp_prems dt_induct_prop) (Logic.strip_imp_concl dt_induct_prop)
  1010       (fn prems => EVERY
  1011         [rtac indrule_lemma' 1,
  1012          (DatatypeAux.indtac rep_induct THEN_ALL_NEW ObjectLogic.atomize_tac) 1,
  1013          EVERY (map (fn (prem, r) => (EVERY
  1014            [REPEAT (eresolve_tac Abs_inverse_thms' 1),
  1015             simp_tac (HOL_basic_ss addsimps [symmetric r]) 1,
  1016             DEPTH_SOLVE_1 (ares_tac [prem] 1 ORELSE etac allE 1)]))
  1017                 (prems ~~ constr_defs))]);
  1018 
  1019     val case_names_induct = mk_case_names_induct descr'';
  1020 
  1021     (**** prove that new datatypes have finite support ****)
  1022 
  1023     val _ = warning "proving finite support for the new datatype";
  1024 
  1025     val indnames = DatatypeProp.make_tnames recTs;
  1026 
  1027     val abs_supp = PureThy.get_thms thy8 (Name "abs_supp");
  1028     val supp_atm = PureThy.get_thms thy8 (Name "supp_atm");
  1029 
  1030     val finite_supp_thms = map (fn atom =>
  1031       let val atomT = Type (atom, [])
  1032       in map standard (List.take
  1033         (split_conj_thm (Goal.prove_global thy8 [] [] (HOLogic.mk_Trueprop
  1034            (foldr1 HOLogic.mk_conj (map (fn (s, T) => HOLogic.mk_mem
  1035              (Const ("Nominal.supp", T --> HOLogic.mk_setT atomT) $ Free (s, T),
  1036               Const ("Finite_Set.Finites", HOLogic.mk_setT (HOLogic.mk_setT atomT))))
  1037                (indnames ~~ recTs))))
  1038            (fn _ => indtac dt_induct indnames 1 THEN
  1039             ALLGOALS (asm_full_simp_tac (simpset_of thy8 addsimps
  1040               (abs_supp @ supp_atm @
  1041                PureThy.get_thms thy8 (Name ("fs_" ^ Sign.base_name atom ^ "1")) @
  1042                List.concat supp_thms))))),
  1043          length new_type_names))
  1044       end) atoms;
  1045 
  1046     val simp_atts = replicate (length new_type_names) [Simplifier.simp_add];
  1047 
  1048     val (_, thy9) = thy8 |>
  1049       Theory.add_path big_name |>
  1050       PureThy.add_thms [(("induct_weak", dt_induct), [case_names_induct])] ||>>
  1051       PureThy.add_thmss [(("inducts_weak", projections dt_induct), [case_names_induct])] ||>
  1052       Theory.parent_path ||>>
  1053       DatatypeAux.store_thmss_atts "distinct" new_type_names simp_atts distinct_thms ||>>
  1054       DatatypeAux.store_thmss "constr_rep" new_type_names constr_rep_thmss ||>>
  1055       DatatypeAux.store_thmss_atts "perm" new_type_names simp_atts perm_simps' ||>>
  1056       DatatypeAux.store_thmss "inject" new_type_names inject_thms ||>>
  1057       DatatypeAux.store_thmss "supp" new_type_names supp_thms ||>>
  1058       DatatypeAux.store_thmss_atts "fresh" new_type_names simp_atts fresh_thms ||>
  1059       fold (fn (atom, ths) => fn thy =>
  1060         let val class = Sign.intern_class thy ("fs_" ^ Sign.base_name atom)
  1061         in fold (fn T => AxClass.prove_arity
  1062             (fst (dest_Type T),
  1063               replicate (length sorts) [class], [class])
  1064             (ClassPackage.intro_classes_tac [] THEN resolve_tac ths 1)) newTs thy
  1065         end) (atoms ~~ finite_supp_thms);
  1066 
  1067     (**** strong induction theorem ****)
  1068 
  1069     val pnames = if length descr'' = 1 then ["P"]
  1070       else map (fn i => "P" ^ string_of_int i) (1 upto length descr'');
  1071     val ind_sort = if null dt_atomTs then HOLogic.typeS
  1072       else Sign.certify_sort thy9 (map (fn T => Sign.intern_class thy9 ("fs_" ^
  1073         Sign.base_name (fst (dest_Type T)))) dt_atomTs);
  1074     val fsT = TFree ("'n", ind_sort);
  1075     val fsT' = TFree ("'n", HOLogic.typeS);
  1076 
  1077     val fresh_fs = map (fn (s, T) => (T, Free (s, fsT' --> HOLogic.mk_setT T)))
  1078       (DatatypeProp.indexify_names (replicate (length dt_atomTs) "f") ~~ dt_atomTs);
  1079 
  1080     fun make_pred fsT i T =
  1081       Free (List.nth (pnames, i), fsT --> T --> HOLogic.boolT);
  1082 
  1083     fun mk_fresh1 xs [] = []
  1084       | mk_fresh1 xs ((y as (_, T)) :: ys) = map (fn x => HOLogic.mk_Trueprop
  1085             (HOLogic.mk_not (HOLogic.mk_eq (Free y, Free x))))
  1086               (filter (fn (_, U) => T = U) (rev xs)) @
  1087           mk_fresh1 (y :: xs) ys;
  1088 
  1089     fun mk_fresh2 xss [] = []
  1090       | mk_fresh2 xss ((p as (ys, _)) :: yss) = List.concat (map (fn y as (_, T) =>
  1091             map (fn (_, x as (_, U)) => HOLogic.mk_Trueprop
  1092               (Const ("Nominal.fresh", T --> U --> HOLogic.boolT) $ Free y $ Free x))
  1093                 (rev xss @ yss)) ys) @
  1094           mk_fresh2 (p :: xss) yss;
  1095 
  1096     fun make_ind_prem fsT f k T ((cname, cargs), idxs) =
  1097       let
  1098         val recs = List.filter is_rec_type cargs;
  1099         val Ts = map (typ_of_dtyp descr'' sorts') cargs;
  1100         val recTs' = map (typ_of_dtyp descr'' sorts') recs;
  1101         val tnames = Name.variant_list pnames (DatatypeProp.make_tnames Ts);
  1102         val rec_tnames = map fst (List.filter (is_rec_type o snd) (tnames ~~ cargs));
  1103         val frees = tnames ~~ Ts;
  1104         val frees' = partition_cargs idxs frees;
  1105         val z = (Name.variant tnames "z", fsT);
  1106 
  1107         fun mk_prem ((dt, s), T) =
  1108           let
  1109             val (Us, U) = strip_type T;
  1110             val l = length Us
  1111           in list_all (z :: map (pair "x") Us, HOLogic.mk_Trueprop
  1112             (make_pred fsT (body_index dt) U $ Bound l $ app_bnds (Free (s, T)) l))
  1113           end;
  1114 
  1115         val prems = map mk_prem (recs ~~ rec_tnames ~~ recTs');
  1116         val prems' = map (fn p as (_, T) => HOLogic.mk_Trueprop
  1117             (f T (Free p) (Free z))) (List.concat (map fst frees')) @
  1118           mk_fresh1 [] (List.concat (map fst frees')) @
  1119           mk_fresh2 [] frees'
  1120 
  1121       in list_all_free (frees @ [z], Logic.list_implies (prems' @ prems,
  1122         HOLogic.mk_Trueprop (make_pred fsT k T $ Free z $
  1123           list_comb (Const (cname, Ts ---> T), map Free frees))))
  1124       end;
  1125 
  1126     val ind_prems = List.concat (map (fn (((i, (_, _, constrs)), (_, idxss)), T) =>
  1127       map (make_ind_prem fsT (fn T => fn t => fn u =>
  1128         Const ("Nominal.fresh", T --> fsT --> HOLogic.boolT) $ t $ u) i T)
  1129           (constrs ~~ idxss)) (descr'' ~~ ndescr ~~ recTs));
  1130     val tnames = DatatypeProp.make_tnames recTs;
  1131     val zs = Name.variant_list tnames (replicate (length descr'') "z");
  1132     val ind_concl = HOLogic.mk_Trueprop (foldr1 (HOLogic.mk_binop "op &")
  1133       (map (fn ((((i, _), T), tname), z) =>
  1134         make_pred fsT i T $ Free (z, fsT) $ Free (tname, T))
  1135         (descr'' ~~ recTs ~~ tnames ~~ zs)));
  1136     val induct = Logic.list_implies (ind_prems, ind_concl);
  1137 
  1138     val ind_prems' =
  1139       map (fn (_, f as Free (_, T)) => list_all_free ([("x", fsT')],
  1140         HOLogic.mk_Trueprop (HOLogic.mk_mem (f $ Free ("x", fsT'),
  1141           Const ("Finite_Set.Finites", HOLogic.mk_setT (body_type T)))))) fresh_fs @
  1142       List.concat (map (fn (((i, (_, _, constrs)), (_, idxss)), T) =>
  1143         map (make_ind_prem fsT' (fn T => fn t => fn u => HOLogic.Not $
  1144           HOLogic.mk_mem (t, the (AList.lookup op = fresh_fs T) $ u)) i T)
  1145             (constrs ~~ idxss)) (descr'' ~~ ndescr ~~ recTs));
  1146     val ind_concl' = HOLogic.mk_Trueprop (foldr1 (HOLogic.mk_binop "op &")
  1147       (map (fn ((((i, _), T), tname), z) =>
  1148         make_pred fsT' i T $ Free (z, fsT') $ Free (tname, T))
  1149         (descr'' ~~ recTs ~~ tnames ~~ zs)));
  1150     val induct' = Logic.list_implies (ind_prems', ind_concl');
  1151 
  1152     fun mk_perm Ts (t, u) =
  1153       let
  1154         val T = fastype_of1 (Ts, t);
  1155         val U = fastype_of1 (Ts, u)
  1156       in Const ("Nominal.perm", T --> U --> U) $ t $ u end;
  1157 
  1158     val aux_ind_vars =
  1159       (DatatypeProp.indexify_names (replicate (length dt_atomTs) "pi") ~~
  1160        map mk_permT dt_atomTs) @ [("z", fsT')];
  1161     val aux_ind_Ts = rev (map snd aux_ind_vars);
  1162     val aux_ind_concl = HOLogic.mk_Trueprop (foldr1 (HOLogic.mk_binop "op &")
  1163       (map (fn (((i, _), T), tname) =>
  1164         HOLogic.list_all (aux_ind_vars, make_pred fsT' i T $ Bound 0 $
  1165           foldr (mk_perm aux_ind_Ts) (Free (tname, T))
  1166             (map Bound (length dt_atomTs downto 1))))
  1167         (descr'' ~~ recTs ~~ tnames)));
  1168 
  1169     fun mk_ind_perm i k p l vs j =
  1170       let
  1171         val n = length vs;
  1172         val Ts = map snd vs;
  1173         val T = List.nth (Ts, i - j);
  1174         val pT = NominalAtoms.mk_permT T
  1175       in
  1176         Const ("List.list.Cons", HOLogic.mk_prodT (T, T) --> pT --> pT) $
  1177           (HOLogic.pair_const T T $ Bound (l - j) $ foldr (mk_perm Ts)
  1178             (Bound (i - j))
  1179             (map (mk_ind_perm i k p l vs) (j - 1 downto 0) @
  1180              map Bound (n - k - 1 downto n - k - p))) $
  1181           Const ("List.list.Nil", pT)
  1182       end;
  1183 
  1184     fun mk_fresh i i' j k p l is vs _ _ =
  1185       let
  1186         val n = length vs;
  1187         val Ts = map snd vs;
  1188         val T = List.nth (Ts, n - i - 1 - j);
  1189         val f = the (AList.lookup op = fresh_fs T);
  1190         val U = List.nth (Ts, n - i' - 1);
  1191         val S = HOLogic.mk_setT T;
  1192         val prms' = map Bound (n - k downto n - k - p + 1);
  1193         val prms = map (mk_ind_perm (n - i) k p (n - l) (("a", T) :: vs))
  1194             (j - 1 downto 0) @ prms';
  1195         val bs = rev (List.mapPartial
  1196           (fn ((_, T'), i) => if T = T' then SOME (Bound i) else NONE)
  1197           (List.take (vs, n - k - p - 1) ~~ (1 upto n - k - p - 1)));
  1198         val ts = map (fn i =>
  1199           Const ("Nominal.supp", List.nth (Ts, n - i - 1) --> S) $
  1200             foldr (mk_perm (T :: Ts)) (Bound (n - i)) prms') is
  1201       in
  1202         HOLogic.mk_Trueprop (Const ("Ex", (T --> HOLogic.boolT) --> HOLogic.boolT) $
  1203           Abs ("a", T, HOLogic.Not $
  1204             (Const ("op :", T --> S --> HOLogic.boolT) $ Bound 0 $
  1205               (foldr (fn (t, u) => Const ("insert", T --> S --> S) $ t $ u)
  1206                 (foldl (fn (t, u) => Const ("op Un", S --> S --> S) $ u $ t)
  1207                   (f $ Bound (n - k - p))
  1208                   (Const ("Nominal.supp", U --> S) $
  1209                      foldr (mk_perm (T :: Ts)) (Bound (n - i')) prms :: ts))
  1210                 (foldr (mk_perm (T :: Ts)) (Bound (n - i - j)) prms :: bs)))))
  1211       end;
  1212 
  1213     fun mk_fresh_constr is p vs _ concl =
  1214       let
  1215         val n = length vs;
  1216         val Ts = map snd vs;
  1217         val _ $ (_ $ _ $ t) = concl;
  1218         val c = head_of t;
  1219         val T = body_type (fastype_of c);
  1220         val k = foldr op + 0 (map (fn (_, i) => i + 1) is);
  1221         val ps = map Bound (n - k - 1 downto n - k - p);
  1222         val (_, _, ts, us) = foldl (fn ((_, i), (m, n, ts, us)) =>
  1223           (m - i - 1, n - i,
  1224            ts @ map Bound (n downto n - i + 1) @
  1225              [foldr (mk_perm Ts) (Bound (m - i))
  1226                 (map (mk_ind_perm m k p n vs) (i - 1 downto 0) @ ps)],
  1227            us @ map (fn j => foldr (mk_perm Ts) (Bound j) ps) (m downto m - i)))
  1228           (n - 1, n - k - p - 2, [], []) is
  1229       in
  1230         HOLogic.mk_Trueprop (HOLogic.eq_const T $ list_comb (c, ts) $ list_comb (c, us))
  1231       end;
  1232 
  1233     val abs_fun_finite_supp = PureThy.get_thm thy9 (Name "abs_fun_finite_supp");
  1234 
  1235     val at_finite_select = PureThy.get_thm thy9 (Name "at_finite_select");
  1236 
  1237     val induct_aux_lemmas = List.concat (map (fn Type (s, _) =>
  1238       [PureThy.get_thm thy9 (Name ("pt_" ^ Sign.base_name s ^ "_inst")),
  1239        PureThy.get_thm thy9 (Name ("fs_" ^ Sign.base_name s ^ "1")),
  1240        PureThy.get_thm thy9 (Name ("at_" ^ Sign.base_name s ^ "_inst"))]) dt_atomTs);
  1241 
  1242     val induct_aux_lemmas' = map (fn Type (s, _) =>
  1243       PureThy.get_thm thy9 (Name ("pt_" ^ Sign.base_name s ^ "2")) RS sym) dt_atomTs;
  1244 
  1245     val fresh_aux = PureThy.get_thms thy9 (Name "fresh_aux");
  1246 
  1247     (**********************************************************************
  1248       The subgoals occurring in the proof of induct_aux have the
  1249       following parameters:
  1250 
  1251         x_1 ... x_k p_1 ... p_m z b_1 ... b_n
  1252 
  1253       where
  1254 
  1255         x_i : constructor arguments (introduced by weak induction rule)
  1256         p_i : permutations (one for each atom type in the data type)
  1257         z   : freshness context
  1258         b_i : newly introduced names for binders (sufficiently fresh)
  1259     ***********************************************************************)
  1260 
  1261     val _ = warning "proving strong induction theorem ...";
  1262 
  1263     val induct_aux = Goal.prove_global thy9 [] ind_prems' ind_concl'
  1264       (fn prems => EVERY
  1265         ([mk_subgoal 1 (K (K (K aux_ind_concl))),
  1266           indtac dt_induct tnames 1] @
  1267          List.concat (map (fn ((_, (_, _, constrs)), (_, constrs')) =>
  1268            List.concat (map (fn ((cname, cargs), is) =>
  1269              [simp_tac (HOL_basic_ss addsimps List.concat perm_simps') 1,
  1270               REPEAT (rtac allI 1)] @
  1271              List.concat (map
  1272                (fn ((_, 0), _) => []
  1273                  | ((i, j), k) => List.concat (map (fn j' =>
  1274                      let
  1275                        val DtType (tname, _) = List.nth (cargs, i + j');
  1276                        val atom = Sign.base_name tname
  1277                      in
  1278                        [mk_subgoal 1 (mk_fresh i (i + j) j'
  1279                           (length cargs) (length dt_atomTs)
  1280                           (length cargs + length dt_atomTs + 1 + i - k)
  1281                           (List.mapPartial (fn (i', j) =>
  1282                              if i = i' then NONE else SOME (i' + j)) is)),
  1283                         rtac at_finite_select 1,
  1284                         rtac (PureThy.get_thm thy9 (Name ("at_" ^ atom ^ "_inst"))) 1,
  1285                         asm_full_simp_tac (simpset_of thy9 addsimps
  1286                           [PureThy.get_thm thy9 (Name ("fs_" ^ atom ^ "1"))]) 1,
  1287                         resolve_tac prems 1,
  1288                         etac exE 1,
  1289                         asm_full_simp_tac (simpset_of thy9 addsimps
  1290                           [symmetric fresh_def]) 1]
  1291                      end) (0 upto j - 1))) (is ~~ (0 upto length is - 1))) @
  1292              (if exists (not o equal 0 o snd) is then
  1293                 [mk_subgoal 1 (mk_fresh_constr is (length dt_atomTs)),
  1294                  asm_full_simp_tac (simpset_of thy9 addsimps
  1295                    (List.concat inject_thms @
  1296                     alpha @ abs_perm @ abs_fresh @ [abs_fun_finite_supp] @
  1297                     induct_aux_lemmas)) 1,
  1298                  dtac sym 1,
  1299                  asm_full_simp_tac (simpset_of thy9) 1,
  1300                  REPEAT (etac conjE 1)]
  1301               else
  1302                 []) @
  1303              [(resolve_tac prems THEN_ALL_NEW
  1304                 (atac ORELSE'
  1305                   SUBGOAL (fn (t, i) => case Logic.strip_assums_concl t of
  1306                       _ $ (Const ("Nominal.fresh", _) $ _ $ _) =>
  1307                         asm_simp_tac (simpset_of thy9 addsimps fresh_aux) i
  1308                     | _ =>
  1309                         asm_simp_tac (simpset_of thy9
  1310                         addsimps induct_aux_lemmas'
  1311                         addsimprocs [perm_simproc]) i))) 1])
  1312                (constrs ~~ constrs'))) (descr'' ~~ ndescr)) @
  1313          [REPEAT (eresolve_tac [conjE, allE_Nil] 1),
  1314           REPEAT (etac allE 1),
  1315           REPEAT (TRY (rtac conjI 1) THEN asm_full_simp_tac (simpset_of thy9) 1)]));
  1316 
  1317     val induct_aux' = Thm.instantiate ([],
  1318       map (fn (s, T) =>
  1319         let val pT = TVar (("'n", 0), HOLogic.typeS) --> T --> HOLogic.boolT
  1320         in (cterm_of thy9 (Var ((s, 0), pT)), cterm_of thy9 (Free (s, pT))) end)
  1321           (pnames ~~ recTs) @
  1322       map (fn (_, f) =>
  1323         let val f' = Logic.varify f
  1324         in (cterm_of thy9 f',
  1325           cterm_of thy9 (Const ("Nominal.supp", fastype_of f')))
  1326         end) fresh_fs) induct_aux;
  1327 
  1328     val induct = Goal.prove_global thy9 [] ind_prems ind_concl
  1329       (fn prems => EVERY
  1330          [rtac induct_aux' 1,
  1331           REPEAT (resolve_tac induct_aux_lemmas 1),
  1332           REPEAT ((resolve_tac prems THEN_ALL_NEW
  1333             (etac meta_spec ORELSE' full_simp_tac (HOL_basic_ss addsimps [fresh_def]))) 1)])
  1334 
  1335     val (_, thy10) = thy9 |>
  1336       Theory.add_path big_name |>
  1337       PureThy.add_thms [(("induct'", induct_aux), [])] ||>>
  1338       PureThy.add_thms [(("induct", induct), [case_names_induct])] ||>>
  1339       PureThy.add_thmss [(("inducts", projections induct), [case_names_induct])];
  1340 
  1341     (**** recursion combinator ****)
  1342 
  1343     val _ = warning "defining recursion combinator ...";
  1344 
  1345     val used = foldr add_typ_tfree_names [] recTs;
  1346 
  1347     val (rec_result_Ts', rec_fn_Ts') = DatatypeProp.make_primrec_Ts descr' sorts' used;
  1348 
  1349     val rec_sort = if null dt_atomTs then HOLogic.typeS else 
  1350       let val names = map (Sign.base_name o fst o dest_Type) dt_atomTs
  1351       in Sign.certify_sort thy10 (map (Sign.intern_class thy10)
  1352         (map (fn s => "pt_" ^ s) names @
  1353          List.concat (map (fn s => List.mapPartial (fn s' =>
  1354            if s = s' then NONE
  1355            else SOME ("cp_" ^ s ^ "_" ^ s')) names) names)))
  1356       end;
  1357 
  1358     val rec_result_Ts = map (fn TFree (s, _) => TFree (s, rec_sort)) rec_result_Ts';
  1359     val rec_fn_Ts = map (typ_subst_atomic (rec_result_Ts' ~~ rec_result_Ts)) rec_fn_Ts';
  1360 
  1361     val rec_set_Ts = map (fn (T1, T2) => rec_fn_Ts ---> HOLogic.mk_setT
  1362       (HOLogic.mk_prodT (T1, T2))) (recTs ~~ rec_result_Ts);
  1363 
  1364     val big_rec_name = big_name ^ "_rec_set";
  1365     val rec_set_names = map (Sign.full_name (Theory.sign_of thy10))
  1366       (if length descr'' = 1 then [big_rec_name] else
  1367         (map ((curry (op ^) (big_rec_name ^ "_")) o string_of_int)
  1368           (1 upto (length descr''))));
  1369 
  1370     val rec_fns = map (uncurry (mk_Free "f"))
  1371       (rec_fn_Ts ~~ (1 upto (length rec_fn_Ts)));
  1372     val rec_sets = map (fn c => list_comb (Const c, rec_fns))
  1373       (rec_set_names ~~ rec_set_Ts);
  1374 
  1375     (* introduction rules for graph of recursion function *)
  1376 
  1377     val rec_preds = map (fn (a, T) =>
  1378       Free (a, T --> HOLogic.boolT)) (pnames ~~ rec_result_Ts);
  1379 
  1380     fun mk_fresh3 rs [] = []
  1381       | mk_fresh3 rs ((p as (ys, z)) :: yss) = List.concat (map (fn y as (_, T) =>
  1382             List.mapPartial (fn ((_, (_, x)), r as (_, U)) => if z = x then NONE
  1383               else SOME (HOLogic.mk_Trueprop
  1384                 (Const ("Nominal.fresh", T --> U --> HOLogic.boolT) $ Free y $ Free r)))
  1385                   rs) ys) @
  1386           mk_fresh3 rs yss;
  1387 
  1388     fun make_rec_intr T p rec_set ((rec_intr_ts, rec_prems, rec_prems',
  1389           rec_eq_prems, l), ((cname, cargs), idxs)) =
  1390       let
  1391         val Ts = map (typ_of_dtyp descr'' sorts') cargs;
  1392         val frees = map (fn i => "x" ^ string_of_int i) (1 upto length Ts) ~~ Ts;
  1393         val frees' = partition_cargs idxs frees;
  1394         val atomTs = distinct op = (maps (map snd o fst) frees');
  1395         val recs = List.mapPartial
  1396           (fn ((_, DtRec i), p) => SOME (i, p) | _ => NONE)
  1397           (partition_cargs idxs cargs ~~ frees');
  1398         val frees'' = map (fn i => "y" ^ string_of_int i) (1 upto length recs) ~~
  1399           map (fn (i, _) => List.nth (rec_result_Ts, i)) recs;
  1400         val prems1 = map (fn ((i, (_, x)), y) => HOLogic.mk_Trueprop
  1401           (HOLogic.mk_mem (HOLogic.mk_prod (Free x, Free y),
  1402              List.nth (rec_sets, i)))) (recs ~~ frees'');
  1403         val prems2 =
  1404           map (fn f => map (fn p as (_, T) => HOLogic.mk_Trueprop
  1405             (Const ("Nominal.fresh", T --> fastype_of f --> HOLogic.boolT) $
  1406               Free p $ f)) (List.concat (map fst frees'))) rec_fns;
  1407         val prems3 =
  1408           mk_fresh1 [] (List.concat (map fst frees')) @
  1409           mk_fresh2 [] frees';
  1410         val prems4 = map (fn ((i, _), y) =>
  1411           HOLogic.mk_Trueprop (List.nth (rec_preds, i) $ Free y)) (recs ~~ frees'');
  1412         val prems5 = mk_fresh3 (recs ~~ frees'') frees';
  1413         val prems6 = maps (fn aT => map (fn y as (_, T) => HOLogic.mk_Trueprop
  1414           (HOLogic.mk_mem (Const ("Nominal.supp", T --> HOLogic.mk_setT aT) $ Free y,
  1415              Const ("Finite_Set.Finites", HOLogic.mk_setT (HOLogic.mk_setT aT)))))
  1416                frees'') atomTs;
  1417         val result = list_comb (List.nth (rec_fns, l), map Free (frees @ frees''));
  1418         val result_freshs = map (fn p as (_, T) =>
  1419           Const ("Nominal.fresh", T --> fastype_of result --> HOLogic.boolT) $
  1420             Free p $ result) (List.concat (map fst frees'));
  1421         val P = HOLogic.mk_Trueprop (p $ result)
  1422       in
  1423         (rec_intr_ts @ [Logic.list_implies (List.concat prems2 @ prems3 @ prems1,
  1424            HOLogic.mk_Trueprop (HOLogic.mk_mem
  1425              (HOLogic.mk_prod (list_comb (Const (cname, Ts ---> T), map Free frees),
  1426                result), rec_set)))],
  1427          rec_prems @ [list_all_free (frees @ frees'', Logic.list_implies (prems4, P))],
  1428          if null result_freshs then rec_prems'
  1429          else rec_prems' @ [list_all_free (frees @ frees'',
  1430            Logic.list_implies (List.nth (prems2, l) @ prems3 @ prems5 @ prems6 @ [P],
  1431              HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj result_freshs)))],
  1432          rec_eq_prems @ [List.concat prems2 @ prems3],
  1433          l + 1)
  1434       end;
  1435 
  1436     val (rec_intr_ts, rec_prems, rec_prems', rec_eq_prems, _) =
  1437       Library.foldl (fn (x, ((((d, d'), T), p), rec_set)) =>
  1438         Library.foldl (make_rec_intr T p rec_set) (x, #3 (snd d) ~~ snd d'))
  1439           (([], [], [], [], 0), descr'' ~~ ndescr ~~ recTs ~~ rec_preds ~~ rec_sets);
  1440 
  1441     val (thy11, {intrs = rec_intrs, elims = rec_elims, raw_induct = rec_induct, ...}) =
  1442       setmp InductivePackage.quiet_mode (!quiet_mode)
  1443         (InductivePackage.add_inductive_i false true big_rec_name false false false
  1444            rec_sets (map (fn x => (("", x), [])) rec_intr_ts) []) thy10;
  1445 
  1446     (** equivariance **)
  1447 
  1448     val fresh_bij = PureThy.get_thms thy11 (Name "fresh_bij");
  1449     val perm_bij = PureThy.get_thms thy11 (Name "perm_bij");
  1450 
  1451     val (rec_equiv_thms, rec_equiv_thms') = ListPair.unzip (map (fn aT =>
  1452       let
  1453         val permT = mk_permT aT;
  1454         val pi = Free ("pi", permT);
  1455         val rec_fns_pi = map (curry (mk_perm []) pi o uncurry (mk_Free "f"))
  1456           (rec_fn_Ts ~~ (1 upto (length rec_fn_Ts)));
  1457         val rec_sets_pi = map (fn c => list_comb (Const c, rec_fns_pi))
  1458           (rec_set_names ~~ rec_set_Ts);
  1459         val ps = map (fn ((((T, U), R), R'), i) =>
  1460           let
  1461             val x = Free ("x" ^ string_of_int i, T);
  1462             val y = Free ("y" ^ string_of_int i, U)
  1463           in
  1464             (HOLogic.mk_mem (HOLogic.mk_prod (x, y), R),
  1465              HOLogic.mk_mem (HOLogic.mk_prod (mk_perm [] (pi, x), mk_perm [] (pi, y)), R'))
  1466           end) (recTs ~~ rec_result_Ts ~~ rec_sets ~~ rec_sets_pi ~~ (1 upto length recTs));
  1467         val ths = map (fn th => standard (th RS mp)) (split_conj_thm
  1468           (Goal.prove_global thy11 [] []
  1469             (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj (map HOLogic.mk_imp ps)))
  1470             (fn _ => rtac rec_induct 1 THEN REPEAT
  1471                (NominalPermeq.perm_simp_tac (simpset_of thy11) 1 THEN
  1472                 (resolve_tac rec_intrs THEN_ALL_NEW
  1473                  asm_simp_tac (HOL_ss addsimps (fresh_bij @ perm_bij))) 1))))
  1474         val ths' = map (fn ((P, Q), th) =>
  1475           Goal.prove_global thy11 [] []
  1476             (Logic.mk_implies (HOLogic.mk_Trueprop Q, HOLogic.mk_Trueprop P))
  1477             (fn _ => dtac (Thm.instantiate ([],
  1478                  [(cterm_of thy11 (Var (("pi", 0), permT)),
  1479                    cterm_of thy11 (Const ("List.rev", permT --> permT) $ pi))]) th) 1 THEN
  1480                NominalPermeq.perm_simp_tac HOL_ss 1)) (ps ~~ ths)
  1481       in (ths, ths') end) dt_atomTs);
  1482 
  1483     (** finite support **)
  1484 
  1485     val rec_fin_supp_thms = map (fn aT =>
  1486       let
  1487         val name = Sign.base_name (fst (dest_Type aT));
  1488         val fs_name = PureThy.get_thm thy11 (Name ("fs_" ^ name ^ "1"));
  1489         val aset = HOLogic.mk_setT aT;
  1490         val finites = Const ("Finite_Set.Finites", HOLogic.mk_setT aset);
  1491         val fins = map (fn (f, T) => HOLogic.mk_Trueprop (HOLogic.mk_mem
  1492           (Const ("Nominal.supp", T --> aset) $ f, finites)))
  1493             (rec_fns ~~ rec_fn_Ts)
  1494       in
  1495         map (fn th => standard (th RS mp)) (split_conj_thm
  1496           (Goal.prove_global thy11 [] fins
  1497             (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
  1498               (map (fn (((T, U), R), i) =>
  1499                  let
  1500                    val x = Free ("x" ^ string_of_int i, T);
  1501                    val y = Free ("y" ^ string_of_int i, U)
  1502                  in
  1503                    HOLogic.mk_imp (HOLogic.mk_mem (HOLogic.mk_prod (x, y), R),
  1504                      HOLogic.mk_mem (Const ("Nominal.supp", U --> aset) $ y, finites))
  1505                  end) (recTs ~~ rec_result_Ts ~~ rec_sets ~~ (1 upto length recTs)))))
  1506             (fn fins => (rtac rec_induct THEN_ALL_NEW cut_facts_tac fins) 1 THEN REPEAT
  1507                (NominalPermeq.finite_guess_tac (HOL_ss addsimps [fs_name]) 1))))
  1508       end) dt_atomTs;
  1509 
  1510     (** freshness **)
  1511 
  1512     val perm_fresh_fresh = PureThy.get_thms thy11 (Name "perm_fresh_fresh");
  1513     val perm_swap = PureThy.get_thms thy11 (Name "perm_swap");
  1514 
  1515     fun perm_of_pair (x, y) =
  1516       let
  1517         val T = fastype_of x;
  1518         val pT = mk_permT T
  1519       in Const ("List.list.Cons", HOLogic.mk_prodT (T, T) --> pT --> pT) $
  1520         HOLogic.mk_prod (x, y) $ Const ("List.list.Nil", pT)
  1521       end;
  1522 
  1523     val finite_premss = map (fn aT =>
  1524       map (fn (f, T) => HOLogic.mk_Trueprop (HOLogic.mk_mem
  1525         (Const ("Nominal.supp", T --> HOLogic.mk_setT aT) $ f,
  1526          Const ("Finite_Set.Finites", HOLogic.mk_setT (HOLogic.mk_setT aT)))))
  1527            (rec_fns ~~ rec_fn_Ts)) dt_atomTs;
  1528 
  1529     val rec_fresh_thms = map (fn ((aT, eqvt_ths), finite_prems) =>
  1530       let
  1531         val name = Sign.base_name (fst (dest_Type aT));
  1532         val fs_name = PureThy.get_thm thy11 (Name ("fs_" ^ name ^ "1"));
  1533         val a = Free ("a", aT);
  1534         val freshs = map (fn (f, fT) => HOLogic.mk_Trueprop
  1535             (Const ("Nominal.fresh", aT --> fT --> HOLogic.boolT) $ a $ f))
  1536           (rec_fns ~~ rec_fn_Ts)
  1537       in
  1538         map (fn (((T, U), R), eqvt_th) =>
  1539           let
  1540             val x = Free ("x", T);
  1541             val y = Free ("y", U);
  1542             val y' = Free ("y'", U)
  1543           in
  1544             standard (Goal.prove (Context.init_proof thy11) [] (finite_prems @
  1545                 [HOLogic.mk_Trueprop (HOLogic.mk_mem
  1546                   (HOLogic.mk_prod (x, y), R)),
  1547                  HOLogic.mk_Trueprop (HOLogic.mk_all ("y'", U,
  1548                    HOLogic.mk_imp (HOLogic.mk_mem
  1549                        (HOLogic.mk_prod (x, y'), R),
  1550                      HOLogic.mk_eq (y', y)))),
  1551                  HOLogic.mk_Trueprop (Const ("Nominal.fresh",
  1552                    aT --> T --> HOLogic.boolT) $ a $ x)] @
  1553               freshs)
  1554               (HOLogic.mk_Trueprop (Const ("Nominal.fresh",
  1555                  aT --> U --> HOLogic.boolT) $ a $ y))
  1556               (fn {prems, context} =>
  1557                  let
  1558                    val (finite_prems, rec_prem :: unique_prem ::
  1559                      fresh_prems) = chop (length finite_prems) prems;
  1560                    val unique_prem' = unique_prem RS spec RS mp;
  1561                    val unique = [unique_prem', unique_prem' RS sym] MRS trans;
  1562                    val _ $ (_ $ (_ $ S $ _)) $ _ = prop_of supports_fresh;
  1563                    val tuple = foldr1 HOLogic.mk_prod (x :: rec_fns)
  1564                  in EVERY
  1565                    [rtac (Drule.cterm_instantiate
  1566                       [(cterm_of thy11 S,
  1567                         cterm_of thy11 (Const ("Nominal.supp",
  1568                           fastype_of tuple --> HOLogic.mk_setT aT) $ tuple))]
  1569                       supports_fresh) 1,
  1570                     simp_tac (HOL_basic_ss addsimps
  1571                       [supports_def, symmetric fresh_def, fresh_prod]) 1,
  1572                     REPEAT_DETERM (resolve_tac [allI, impI] 1),
  1573                     REPEAT_DETERM (etac conjE 1),
  1574                     rtac unique 1,
  1575                     SUBPROOF (fn {prems = prems', params = [a, b], ...} => EVERY
  1576                       [cut_facts_tac [rec_prem] 1,
  1577                        rtac (Thm.instantiate ([],
  1578                          [(cterm_of thy11 (Var (("pi", 0), mk_permT aT)),
  1579                            cterm_of thy11 (perm_of_pair (term_of a, term_of b)))]) eqvt_th) 1,
  1580                        asm_simp_tac (HOL_ss addsimps
  1581                          (prems' @ perm_swap @ perm_fresh_fresh)) 1]) context 1,
  1582                     rtac rec_prem 1,
  1583                     simp_tac (HOL_ss addsimps (fs_name ::
  1584                       supp_prod :: finite_Un :: finite_prems)) 1,
  1585                     simp_tac (HOL_ss addsimps (symmetric fresh_def ::
  1586                       fresh_prod :: fresh_prems)) 1]
  1587                  end))
  1588           end) (recTs ~~ rec_result_Ts ~~ rec_sets ~~ eqvt_ths)
  1589       end) (dt_atomTs ~~ rec_equiv_thms' ~~ finite_premss);
  1590 
  1591     (** uniqueness **)
  1592 
  1593     val exists_fresh = PureThy.get_thms thy11 (Name "exists_fresh");
  1594     val fs_atoms = map (fn Type (s, _) => PureThy.get_thm thy11
  1595       (Name ("fs_" ^ Sign.base_name s ^ "1"))) dt_atomTs;
  1596     val fresh_atm = PureThy.get_thms thy11 (Name "fresh_atm");
  1597     val calc_atm = PureThy.get_thms thy11 (Name "calc_atm");
  1598     val fresh_left = PureThy.get_thms thy11 (Name "fresh_left");
  1599 
  1600     val fun_tuple = foldr1 HOLogic.mk_prod rec_fns;
  1601     val fun_tupleT = fastype_of fun_tuple;
  1602     val rec_unique_frees =
  1603       DatatypeProp.indexify_names (replicate (length recTs) "x") ~~ recTs;
  1604     val rec_unique_frees'' = map (fn (s, T) => (s ^ "'", T)) rec_unique_frees;
  1605     val rec_unique_frees' =
  1606       DatatypeProp.indexify_names (replicate (length recTs) "y") ~~ rec_result_Ts;
  1607     val rec_unique_concls = map (fn ((x as (_, T), U), R) =>
  1608         Const ("Ex1", (U --> HOLogic.boolT) --> HOLogic.boolT) $
  1609           Abs ("y", U, HOLogic.mk_mem (HOLogic.pair_const T U $ Free x $ Bound 0, R)))
  1610       (rec_unique_frees ~~ rec_result_Ts ~~ rec_sets);
  1611 
  1612     val induct_aux_rec = Drule.cterm_instantiate
  1613       (map (pairself (cterm_of thy11))
  1614          (map (fn (aT, f) => (Logic.varify f, Abs ("z", HOLogic.unitT,
  1615             Const ("Nominal.supp", fun_tupleT --> HOLogic.mk_setT aT) $ fun_tuple)))
  1616               fresh_fs @
  1617           map (fn (((P, T), (x, U)), Q) =>
  1618            (Var ((P, 0), HOLogic.unitT --> Logic.varifyT T --> HOLogic.boolT),
  1619             Abs ("z", HOLogic.unitT, absfree (x, U, Q))))
  1620               (pnames ~~ recTs ~~ rec_unique_frees ~~ rec_unique_concls) @
  1621           map (fn (s, T) => (Var ((s, 0), Logic.varifyT T), Free (s, T)))
  1622             rec_unique_frees)) induct_aux;
  1623 
  1624     fun obtain_fresh_name vs ths rec_fin_supp T (freshs1, freshs2, ctxt) =
  1625       let
  1626         val p = foldr1 HOLogic.mk_prod (vs @ freshs1);
  1627         val ex = Goal.prove ctxt [] [] (HOLogic.mk_Trueprop
  1628             (HOLogic.exists_const T $ Abs ("x", T,
  1629               Const ("Nominal.fresh", T --> fastype_of p --> HOLogic.boolT) $
  1630                 Bound 0 $ p)))
  1631           (fn _ => EVERY
  1632             [cut_facts_tac ths 1,
  1633              REPEAT_DETERM (dresolve_tac (the (AList.lookup op = rec_fin_supp T)) 1),
  1634              resolve_tac exists_fresh 1,
  1635              asm_simp_tac (HOL_ss addsimps (supp_prod :: finite_Un :: fs_atoms)) 1]);
  1636         val (([cx], ths), ctxt') = Obtain.result
  1637           (fn _ => EVERY
  1638             [etac exE 1,
  1639              full_simp_tac (HOL_ss addsimps (fresh_prod :: fresh_atm)) 1,
  1640              REPEAT (etac conjE 1)])
  1641           [ex] ctxt
  1642       in (freshs1 @ [term_of cx], freshs2 @ ths, ctxt') end;
  1643 
  1644     val rec_unique_thms = split_conj_thm (Goal.prove
  1645       (Context.init_proof thy11) (map fst rec_unique_frees)
  1646       (List.concat finite_premss @ rec_prems @ rec_prems')
  1647       (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj rec_unique_concls))
  1648       (fn {prems, context} =>
  1649          let
  1650            val k = length rec_fns;
  1651            val (finite_thss, ths1) = fold_map (fn T => fn xs =>
  1652              apfst (pair T) (chop k xs)) dt_atomTs prems;
  1653            val (P_ind_ths, ths2) = chop k ths1;
  1654            val P_ths = map (fn th => th RS mp) (split_conj_thm
  1655              (Goal.prove context
  1656                (map fst (rec_unique_frees'' @ rec_unique_frees')) []
  1657                (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
  1658                   (map (fn (((x, y), S), P) => HOLogic.mk_imp (HOLogic.mk_mem
  1659                     (HOLogic.mk_prod (Free x, Free y), S), P $ (Free y)))
  1660                       (rec_unique_frees'' ~~ rec_unique_frees' ~~ rec_sets ~~ rec_preds))))
  1661                (fn _ =>
  1662                   rtac rec_induct 1 THEN
  1663                   REPEAT ((resolve_tac P_ind_ths THEN_ALL_NEW assume_tac) 1))));
  1664            val rec_fin_supp_thms' = map
  1665              (fn (ths, (T, fin_ths)) => (T, map (curry op MRS fin_ths) ths))
  1666              (rec_fin_supp_thms ~~ finite_thss);
  1667            val fcbs = List.concat (map split_conj_thm ths2);
  1668          in EVERY
  1669            ([rtac induct_aux_rec 1] @
  1670             maps (fn (_, finite_ths) =>
  1671               [cut_facts_tac finite_ths 1,
  1672                asm_simp_tac (HOL_ss addsimps [supp_prod, finite_Un]) 1]) finite_thss @
  1673             maps (fn ((_, idxss), elim) => maps (fn idxs =>
  1674               [full_simp_tac (HOL_ss addsimps [symmetric fresh_def, supp_prod, Un_iff]) 1,
  1675                REPEAT_DETERM (eresolve_tac [conjE, ex1E] 1),
  1676                rtac ex1I 1,
  1677                (resolve_tac rec_intrs THEN_ALL_NEW atac) 1,
  1678                rotate_tac ~1 1,
  1679                ((DETERM o etac elim) THEN_ALL_NEW full_simp_tac
  1680                   (HOL_ss addsimps (Pair_eq :: List.concat distinct_thms))) 1] @
  1681                (if null idxs then [] else [etac conjE 1, hyp_subst_tac 1,
  1682                 SUBPROOF (fn {asms, concl, prems = prems', params, context = context', ...} =>
  1683                   let
  1684                     val (_, prem) = split_last prems';
  1685                     val _ $ (_ $ lhs $ rhs) = prop_of prem;
  1686                     val _ $ (_ $ lhs' $ rhs') = term_of concl;
  1687                     val rT = fastype_of lhs';
  1688                     val (c, cargsl) = strip_comb lhs;
  1689                     val cargsl' = partition_cargs idxs cargsl;
  1690                     val boundsl = List.concat (map fst cargsl');
  1691                     val (_, cargsr) = strip_comb rhs;
  1692                     val cargsr' = partition_cargs idxs cargsr;
  1693                     val boundsr = List.concat (map fst cargsr');
  1694                     val (params1, _ :: params2) =
  1695                       chop (length params div 2) (map term_of params);
  1696                     val params' = params1 @ params2;
  1697                     val rec_prems = filter (fn th => case prop_of th of
  1698                       _ $ (Const ("op :", _) $ _ $ _) => true | _ => false) prems';
  1699                     val fresh_prems = filter (fn th => case prop_of th of
  1700                         _ $ (Const ("Nominal.fresh", _) $ _ $ _) => true
  1701                       | _ $ (Const ("Not", _) $ _) => true
  1702                       | _ => false) prems';
  1703                     val Ts = map fastype_of boundsl;
  1704 
  1705                     val _ = warning "step 1: obtaining fresh names";
  1706                     val (freshs1, freshs2, context'') = fold
  1707                       (obtain_fresh_name (rec_fns @ params')
  1708                          (List.concat (map snd finite_thss) @ rec_prems)
  1709                          rec_fin_supp_thms')
  1710                       Ts ([], [], context');
  1711                     val pi1 = map perm_of_pair (boundsl ~~ freshs1);
  1712                     val rpi1 = rev pi1;
  1713                     val pi2 = map perm_of_pair (boundsr ~~ freshs1);
  1714 
  1715                     fun mk_not_sym ths = List.concat (map (fn th =>
  1716                       case prop_of th of
  1717                           _ $ (Const ("Not", _) $ _) => [th, th RS not_sym]
  1718                         | _ => [th]) ths);
  1719                     val fresh_prems' = mk_not_sym fresh_prems;
  1720                     val freshs2' = mk_not_sym freshs2;
  1721 
  1722                     (** as, bs, cs # K as ts, K bs us **)
  1723                     val _ = warning "step 2: as, bs, cs # K as ts, K bs us";
  1724                     val prove_fresh_ss = HOL_ss addsimps
  1725                       (finite_Diff :: List.concat fresh_thms @
  1726                        fs_atoms @ abs_fresh @ abs_supp @ fresh_atm);
  1727                     (* FIXME: avoid asm_full_simp_tac ? *)
  1728                     fun prove_fresh ths y x = Goal.prove context'' [] []
  1729                       (HOLogic.mk_Trueprop (Const ("Nominal.fresh",
  1730                          fastype_of x --> fastype_of y --> HOLogic.boolT) $ x $ y))
  1731                       (fn _ => cut_facts_tac ths 1 THEN asm_full_simp_tac prove_fresh_ss 1);
  1732                     val constr_fresh_thms =
  1733                       map (prove_fresh fresh_prems lhs) boundsl @
  1734                       map (prove_fresh fresh_prems rhs) boundsr @
  1735                       map (prove_fresh freshs2 lhs) freshs1 @
  1736                       map (prove_fresh freshs2 rhs) freshs1;
  1737 
  1738                     (** pi1 o (K as ts) = pi2 o (K bs us) **)
  1739                     val _ = warning "step 3: pi1 o (K as ts) = pi2 o (K bs us)";
  1740                     val pi1_pi2_eq = Goal.prove context'' [] []
  1741                       (HOLogic.mk_Trueprop (HOLogic.mk_eq
  1742                         (foldr (mk_perm []) lhs pi1, foldr (mk_perm []) rhs pi2)))
  1743                       (fn _ => EVERY
  1744                          [cut_facts_tac constr_fresh_thms 1,
  1745                           asm_simp_tac (HOL_basic_ss addsimps perm_fresh_fresh) 1,
  1746                           rtac prem 1]);
  1747 
  1748                     (** pi1 o ts = pi2 o us **)
  1749                     val _ = warning "step 4: pi1 o ts = pi2 o us";
  1750                     val pi1_pi2_eqs = map (fn (t, u) =>
  1751                       Goal.prove context'' [] []
  1752                         (HOLogic.mk_Trueprop (HOLogic.mk_eq
  1753                           (foldr (mk_perm []) t pi1, foldr (mk_perm []) u pi2)))
  1754                         (fn _ => EVERY
  1755                            [cut_facts_tac [pi1_pi2_eq] 1,
  1756                             asm_full_simp_tac (HOL_ss addsimps
  1757                               (calc_atm @ List.concat perm_simps' @
  1758                                fresh_prems' @ freshs2' @ abs_perm @
  1759                                alpha @ List.concat inject_thms)) 1]))
  1760                         (map snd cargsl' ~~ map snd cargsr');
  1761 
  1762                     (** pi1^-1 o pi2 o us = ts **)
  1763                     val _ = warning "step 5: pi1^-1 o pi2 o us = ts";
  1764                     val rpi1_pi2_eqs = map (fn ((t, u), eq) =>
  1765                       Goal.prove context'' [] []
  1766                         (HOLogic.mk_Trueprop (HOLogic.mk_eq
  1767                           (foldr (mk_perm []) u (rpi1 @ pi2), t)))
  1768                         (fn _ => simp_tac (HOL_ss addsimps
  1769                            ((eq RS sym) :: perm_swap)) 1))
  1770                         (map snd cargsl' ~~ map snd cargsr' ~~ pi1_pi2_eqs);
  1771 
  1772                     val (rec_prems1, rec_prems2) =
  1773                       chop (length rec_prems div 2) rec_prems;
  1774 
  1775                     (** (ts, pi1^-1 o pi2 o vs) in rec_set **)
  1776                     val _ = warning "step 6: (ts, pi1^-1 o pi2 o vs) in rec_set";
  1777                     val rec_prems' = map (fn th =>
  1778                       let
  1779                         val _ $ (_ $ (_ $ x $ y) $ S) = prop_of th;
  1780                         val k = find_index (equal S) rec_sets;
  1781                         val pi = rpi1 @ pi2;
  1782                         fun mk_pi z = foldr (mk_perm []) z pi;
  1783                         fun eqvt_tac p =
  1784                           let
  1785                             val U as Type (_, [Type (_, [T, _])]) = fastype_of p;
  1786                             val l = find_index (equal T) dt_atomTs;
  1787                             val th = List.nth (List.nth (rec_equiv_thms', l), k);
  1788                             val th' = Thm.instantiate ([],
  1789                               [(cterm_of thy11 (Var (("pi", 0), U)),
  1790                                 cterm_of thy11 p)]) th;
  1791                           in rtac th' 1 end;
  1792                         val th' = Goal.prove context'' [] []
  1793                           (HOLogic.mk_Trueprop (HOLogic.mk_mem
  1794                             (HOLogic.mk_prod (mk_pi x, mk_pi y), S)))
  1795                           (fn _ => EVERY
  1796                              (map eqvt_tac pi @
  1797                               [simp_tac (HOL_ss addsimps (fresh_prems' @ freshs2' @
  1798                                  perm_swap @ perm_fresh_fresh)) 1,
  1799                                rtac th 1]))
  1800                       in
  1801                         Simplifier.simplify
  1802                           (HOL_basic_ss addsimps rpi1_pi2_eqs) th'
  1803                       end) rec_prems2;
  1804 
  1805                     val ihs = filter (fn th => case prop_of th of
  1806                       _ $ (Const ("All", _) $ _) => true | _ => false) prems';
  1807 
  1808                     (** pi1 o rs = p2 o vs , rs = pi1^-1 o pi2 o vs **)
  1809                     val _ = warning "step 7: pi1 o rs = p2 o vs , rs = pi1^-1 o pi2 o vs";
  1810                     val (rec_eqns1, rec_eqns2) = ListPair.unzip (map (fn (th, ih) =>
  1811                       let
  1812                         val th' = th RS (ih RS spec RS mp) RS sym;
  1813                         val _ $ (_ $ lhs $ rhs) = prop_of th';
  1814                         fun strip_perm (_ $ _ $ t) = strip_perm t
  1815                           | strip_perm t = t;
  1816                       in
  1817                         (Goal.prove context'' [] []
  1818                            (HOLogic.mk_Trueprop (HOLogic.mk_eq
  1819                               (foldr (mk_perm []) lhs pi1,
  1820                                foldr (mk_perm []) (strip_perm rhs) pi2)))
  1821                            (fn _ => simp_tac (HOL_basic_ss addsimps
  1822                               (th' :: perm_swap)) 1),
  1823                          th')
  1824                       end) (rec_prems' ~~ ihs));
  1825 
  1826                     (** as # rs , bs # vs **)
  1827                     val _ = warning "step 8: as # rs , bs # vs";
  1828                     val (rec_freshs1, rec_freshs2) = ListPair.unzip (List.concat
  1829                       (map (fn (((rec_prem, rec_prem'), eqn), ih) =>
  1830                         let
  1831                           val _ $ (_ $ (_ $ x $ (y as Free (_, T))) $ S) =
  1832                             prop_of rec_prem;
  1833                           val _ $ (_ $ (_ $ _ $ y') $ _) = prop_of rec_prem';
  1834                           val k = find_index (equal S) rec_sets;
  1835                           val atoms = List.concat (List.mapPartial
  1836                             (fn ((bs, z), (bs', _)) =>
  1837                               if z = x then NONE else SOME (bs ~~ bs'))
  1838                             (cargsl' ~~ cargsr'))
  1839                         in
  1840                           map (fn (a as Free (_, aT), b) =>
  1841                             let
  1842                               val l = find_index (equal aT) dt_atomTs;
  1843                               val th = Goal.prove context'' [] []
  1844                                 (HOLogic.mk_Trueprop (Const ("Nominal.fresh",
  1845                                   aT --> T --> HOLogic.boolT) $ a $ y))
  1846                                 (fn _ => EVERY
  1847                                    (rtac (List.nth (List.nth (rec_fresh_thms, l), k)) 1 ::
  1848                                     map (fn th => rtac th 1)
  1849                                       (snd (List.nth (finite_thss, l))) @
  1850                                     [rtac rec_prem 1, rtac ih 1,
  1851                                      REPEAT_DETERM (resolve_tac fresh_prems 1)]));
  1852                               val th' = Goal.prove context'' [] []
  1853                                 (HOLogic.mk_Trueprop (Const ("Nominal.fresh",
  1854                                   aT --> T --> HOLogic.boolT) $ b $ y'))
  1855                                 (fn _ => cut_facts_tac [th] 1 THEN
  1856                                     asm_full_simp_tac (HOL_ss addsimps (eqn ::
  1857                                       fresh_left @ fresh_prems' @ freshs2' @
  1858                                       rev_simps @ app_simps @ calc_atm)) 1)
  1859                             in (th, th') end) atoms
  1860                         end) (rec_prems1 ~~ rec_prems2 ~~ rec_eqns2 ~~ ihs)));
  1861 
  1862                     (** as # fK as ts rs , bs # fK bs us vs **)
  1863                     val _ = warning "step 9: as # fK as ts rs , bs # fK bs us vs";
  1864                     fun prove_fresh_result t (a as Free (_, aT)) =
  1865                       Goal.prove context'' [] []
  1866                         (HOLogic.mk_Trueprop (Const ("Nominal.fresh",
  1867                           aT --> rT --> HOLogic.boolT) $ a $ t))
  1868                         (fn _ => EVERY
  1869                            [resolve_tac fcbs 1,
  1870                             REPEAT_DETERM (resolve_tac
  1871                               (fresh_prems @ rec_freshs1 @ rec_freshs2) 1),
  1872                             REPEAT_DETERM (resolve_tac (maps snd rec_fin_supp_thms') 1
  1873                               THEN resolve_tac rec_prems 1),
  1874                             resolve_tac P_ind_ths 1,
  1875                             REPEAT_DETERM (resolve_tac (P_ths @ rec_prems) 1)]);
  1876         
  1877                     val fresh_results =
  1878                       map (prove_fresh_result rhs') (List.concat (map fst cargsl')) @
  1879                       map (prove_fresh_result lhs') (List.concat (map fst cargsr'));
  1880 
  1881                     (** cs # fK as ts rs , cs # fK bs us vs **)
  1882                     val _ = warning "step 10: cs # fK as ts rs , cs # fK bs us vs";
  1883                     fun prove_fresh_result' recs t (a as Free (_, aT)) =
  1884                       Goal.prove context'' [] []
  1885                         (HOLogic.mk_Trueprop (Const ("Nominal.fresh",
  1886                           aT --> rT --> HOLogic.boolT) $ a $ t))
  1887                         (fn _ => EVERY
  1888                           [cut_facts_tac recs 1,
  1889                            REPEAT_DETERM (dresolve_tac
  1890                              (the (AList.lookup op = rec_fin_supp_thms' aT)) 1),
  1891                            NominalPermeq.fresh_guess_tac
  1892                              (HOL_ss addsimps (freshs2 @
  1893                                 fs_atoms @ fresh_atm @
  1894                                 List.concat (map snd finite_thss))) 1]);
  1895 
  1896                     val fresh_results' =
  1897                       map (prove_fresh_result' rec_prems1 rhs') freshs1 @
  1898                       map (prove_fresh_result' rec_prems2 lhs') freshs1;
  1899 
  1900                     (** pi1 o (fK as ts rs) = pi2 o (fK bs us vs) **)
  1901                     val _ = warning "step 11: pi1 o (fK as ts rs) = pi2 o (fK bs us vs)";
  1902                     val pi1_pi2_result = Goal.prove context'' [] []
  1903                       (HOLogic.mk_Trueprop (HOLogic.mk_eq
  1904                         (foldr (mk_perm []) rhs' pi1, foldr (mk_perm []) lhs' pi2)))
  1905                       (fn _ => NominalPermeq.perm_simp_tac (HOL_ss addsimps
  1906                            pi1_pi2_eqs @ rec_eqns1) 1 THEN
  1907                          TRY (simp_tac (HOL_ss addsimps
  1908                            (fresh_prems' @ freshs2' @ calc_atm @ perm_fresh_fresh)) 1));
  1909 
  1910                     val _ = warning "final result";
  1911                     val final = Goal.prove context'' [] [] (term_of concl)
  1912                       (fn _ => cut_facts_tac [pi1_pi2_result RS sym] 1 THEN
  1913                         full_simp_tac (HOL_basic_ss addsimps perm_fresh_fresh @
  1914                           fresh_results @ fresh_results') 1);
  1915                     val final' = ProofContext.export context'' context' [final];
  1916                     val _ = warning "finished!"
  1917                   in
  1918                     resolve_tac final' 1
  1919                   end) context 1])) idxss) (ndescr ~~ rec_elims))
  1920          end));
  1921 
  1922     val rec_total_thms = map (fn r => r RS theI') rec_unique_thms;
  1923 
  1924     (* define primrec combinators *)
  1925 
  1926     val big_reccomb_name = (space_implode "_" new_type_names) ^ "_rec";
  1927     val reccomb_names = map (Sign.full_name thy11)
  1928       (if length descr'' = 1 then [big_reccomb_name] else
  1929         (map ((curry (op ^) (big_reccomb_name ^ "_")) o string_of_int)
  1930           (1 upto (length descr''))));
  1931     val reccombs = map (fn ((name, T), T') => list_comb
  1932       (Const (name, rec_fn_Ts @ [T] ---> T'), rec_fns))
  1933         (reccomb_names ~~ recTs ~~ rec_result_Ts);
  1934 
  1935     val (reccomb_defs, thy12) =
  1936       thy11
  1937       |> Theory.add_consts_i (map (fn ((name, T), T') =>
  1938           (Sign.base_name name, rec_fn_Ts @ [T] ---> T', NoSyn))
  1939           (reccomb_names ~~ recTs ~~ rec_result_Ts))
  1940       |> (PureThy.add_defs_i false o map Thm.no_attributes) (map (fn ((((name, comb), set), T), T') =>
  1941           ((Sign.base_name name) ^ "_def", Logic.mk_equals (comb, absfree ("x", T,
  1942            Const ("The", (T' --> HOLogic.boolT) --> T') $ absfree ("y", T',
  1943              HOLogic.mk_mem (HOLogic.mk_prod (Free ("x", T), Free ("y", T')), set))))))
  1944                (reccomb_names ~~ reccombs ~~ rec_sets ~~ recTs ~~ rec_result_Ts));
  1945 
  1946     (* prove characteristic equations for primrec combinators *)
  1947 
  1948     val rec_thms = map (fn (prems, concl) =>
  1949       let
  1950         val _ $ (_ $ (_ $ x) $ _) = concl;
  1951         val (_, cargs) = strip_comb x;
  1952         val ps = map (fn (x as Free (_, T), i) =>
  1953           (Free ("x" ^ string_of_int i, T), x)) (cargs ~~ (1 upto length cargs));
  1954         val concl' = subst_atomic_types (rec_result_Ts' ~~ rec_result_Ts) concl;
  1955         val prems' = List.concat finite_premss @ rec_prems @ rec_prems' @
  1956           map (subst_atomic ps) prems;
  1957         fun solve rules prems = resolve_tac rules THEN_ALL_NEW
  1958           (resolve_tac prems THEN_ALL_NEW atac)
  1959       in
  1960         Goal.prove_global thy12 [] prems' concl'
  1961           (fn prems => EVERY
  1962             [rewrite_goals_tac reccomb_defs,
  1963              rtac the1_equality 1,
  1964              solve rec_unique_thms prems 1,
  1965              resolve_tac rec_intrs 1,
  1966              REPEAT (solve (prems @ rec_total_thms) prems 1)])
  1967       end) (rec_eq_prems ~~
  1968         DatatypeProp.make_primrecs new_type_names descr' sorts' thy12);
  1969     
  1970     (* FIXME: theorems are stored in database for testing only *)
  1971     val (_, thy13) = thy12 |>
  1972       PureThy.add_thmss
  1973         [(("rec_equiv", List.concat rec_equiv_thms), []),
  1974          (("rec_equiv'", List.concat rec_equiv_thms'), []),
  1975          (("rec_fin_supp", List.concat rec_fin_supp_thms), []),
  1976          (("rec_fresh", List.concat rec_fresh_thms), []),
  1977          (("rec_unique", map standard rec_unique_thms), []),
  1978          (("recs", rec_thms), [])] ||>
  1979       Theory.parent_path;
  1980 
  1981   in
  1982     thy13
  1983   end;
  1984 
  1985 val add_nominal_datatype = gen_add_nominal_datatype read_typ true;
  1986 
  1987 
  1988 (* FIXME: The following stuff should be exported by DatatypePackage *)
  1989 
  1990 local structure P = OuterParse and K = OuterKeyword in
  1991 
  1992 val datatype_decl =
  1993   Scan.option (P.$$$ "(" |-- P.name --| P.$$$ ")") -- P.type_args -- P.name -- P.opt_infix --
  1994     (P.$$$ "=" |-- P.enum1 "|" (P.name -- Scan.repeat P.typ -- P.opt_mixfix));
  1995 
  1996 fun mk_datatype args =
  1997   let
  1998     val names = map (fn ((((NONE, _), t), _), _) => t | ((((SOME t, _), _), _), _) => t) args;
  1999     val specs = map (fn ((((_, vs), t), mx), cons) =>
  2000       (vs, t, mx, map (fn ((x, y), z) => (x, y, z)) cons)) args;
  2001   in add_nominal_datatype false names specs end;
  2002 
  2003 val nominal_datatypeP =
  2004   OuterSyntax.command "nominal_datatype" "define inductive datatypes" K.thy_decl
  2005     (P.and_list1 datatype_decl >> (Toplevel.theory o mk_datatype));
  2006 
  2007 val _ = OuterSyntax.add_parsers [nominal_datatypeP];
  2008 
  2009 end;
  2010 
  2011 end
  2012