src/HOL/Nominal/nominal_package.ML
author wenzelm
Thu Jan 05 17:16:38 2006 +0100 (2006-01-05)
changeset 18582 4f4cc426b440
parent 18579 002d371401f5
child 18658 317a6f0ef8b9
permissions -rw-r--r--
provide projections of induct_weak, induct_unsafe;
     1 (* $Id$ *)
     2 
     3 signature NOMINAL_PACKAGE =
     4 sig
     5   val add_nominal_datatype : bool -> string list -> (string list * bstring * mixfix *
     6     (bstring * string list * mixfix) list) list -> theory -> theory
     7 end
     8 
     9 structure NominalPackage : NOMINAL_PACKAGE =
    10 struct
    11 
    12 open DatatypeAux;
    13 open NominalAtoms;
    14 
    15 (** FIXME: DatatypePackage should export this function **)
    16 
    17 local
    18 
    19 fun dt_recs (DtTFree _) = []
    20   | dt_recs (DtType (_, dts)) = List.concat (map dt_recs dts)
    21   | dt_recs (DtRec i) = [i];
    22 
    23 fun dt_cases (descr: descr) (_, args, constrs) =
    24   let
    25     fun the_bname i = Sign.base_name (#1 (valOf (AList.lookup (op =) descr i)));
    26     val bnames = map the_bname (distinct (List.concat (map dt_recs args)));
    27   in map (fn (c, _) => space_implode "_" (Sign.base_name c :: bnames)) constrs end;
    28 
    29 
    30 fun induct_cases descr =
    31   DatatypeProp.indexify_names (List.concat (map (dt_cases descr) (map #2 descr)));
    32 
    33 fun exhaust_cases descr i = dt_cases descr (valOf (AList.lookup (op =) descr i));
    34 
    35 in
    36 
    37 fun mk_case_names_induct descr = RuleCases.case_names (induct_cases descr);
    38 
    39 fun mk_case_names_exhausts descr new =
    40   map (RuleCases.case_names o exhaust_cases descr o #1)
    41     (List.filter (fn ((_, (name, _, _))) => name mem_string new) descr);
    42 
    43 end;
    44 
    45 (*******************************)
    46 
    47 val (_ $ (_ $ (_ $ (distinct_f $ _) $ _))) = hd (prems_of distinct_lemma);
    48 
    49 fun read_typ sign ((Ts, sorts), str) =
    50   let
    51     val T = Type.no_tvars (Sign.read_typ (sign, (AList.lookup op =)
    52       (map (apfst (rpair ~1)) sorts)) str) handle TYPE (msg, _, _) => error msg
    53   in (Ts @ [T], add_typ_tfrees (T, sorts)) end;
    54 
    55 (** taken from HOL/Tools/datatype_aux.ML **)
    56 
    57 fun indtac indrule indnames i st =
    58   let
    59     val ts = HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of indrule));
    60     val ts' = HOLogic.dest_conj (HOLogic.dest_Trueprop
    61       (Logic.strip_imp_concl (List.nth (prems_of st, i - 1))));
    62     val getP = if can HOLogic.dest_imp (hd ts) then
    63       (apfst SOME) o HOLogic.dest_imp else pair NONE;
    64     fun abstr (t1, t2) = (case t1 of
    65         NONE => (case filter (fn Free (s, _) => s mem indnames | _ => false)
    66               (term_frees t2) of
    67             [Free (s, T)] => absfree (s, T, t2)
    68           | _ => sys_error "indtac")
    69       | SOME (_ $ t' $ _) => Abs ("x", fastype_of t', abstract_over (t', t2)))
    70     val cert = cterm_of (Thm.sign_of_thm st);
    71     val Ps = map (cert o head_of o snd o getP) ts;
    72     val indrule' = cterm_instantiate (Ps ~~
    73       (map (cert o abstr o getP) ts')) indrule
    74   in
    75     rtac indrule' i st
    76   end;
    77 
    78 fun projections rule =
    79   ProjectRule.projections rule
    80   |> map (standard #> RuleCases.save rule);
    81 
    82 fun gen_add_nominal_datatype prep_typ err flat_names new_type_names dts thy =
    83   let
    84     (* this theory is used just for parsing *)
    85   
    86     val tmp_thy = thy |>
    87       Theory.copy |>
    88       Theory.add_types (map (fn (tvs, tname, mx, _) =>
    89         (tname, length tvs, mx)) dts);
    90 
    91     val sign = Theory.sign_of tmp_thy;
    92 
    93     val atoms = atoms_of thy;
    94     val classes = map (NameSpace.map_base (fn s => "pt_" ^ s)) atoms;
    95     val cp_classes = List.concat (map (fn atom1 => map (fn atom2 =>
    96       Sign.intern_class thy ("cp_" ^ Sign.base_name atom1 ^ "_" ^
    97         Sign.base_name atom2)) atoms) atoms);
    98     fun augment_sort S = S union classes;
    99     val augment_sort_typ = map_type_tfree (fn (s, S) => TFree (s, augment_sort S));
   100 
   101     fun prep_constr ((constrs, sorts), (cname, cargs, mx)) =
   102       let val (cargs', sorts') = Library.foldl (prep_typ sign) (([], sorts), cargs)
   103       in (constrs @ [(cname, cargs', mx)], sorts') end
   104 
   105     fun prep_dt_spec ((dts, sorts), (tvs, tname, mx, constrs)) =
   106       let val (constrs', sorts') = Library.foldl prep_constr (([], sorts), constrs)
   107       in (dts @ [(tvs, tname, mx, constrs')], sorts') end
   108 
   109     val (dts', sorts) = Library.foldl prep_dt_spec (([], []), dts);
   110     val sorts' = map (apsnd augment_sort) sorts;
   111     val tyvars = map #1 dts';
   112 
   113     val types_syntax = map (fn (tvs, tname, mx, constrs) => (tname, mx)) dts';
   114     val constr_syntax = map (fn (tvs, tname, mx, constrs) =>
   115       map (fn (cname, cargs, mx) => (cname, mx)) constrs) dts';
   116 
   117     val ps = map (fn (_, n, _, _) =>
   118       (Sign.full_name sign n, Sign.full_name sign (n ^ "_Rep"))) dts;
   119     val rps = map Library.swap ps;
   120 
   121     fun replace_types (Type ("nominal.ABS", [T, U])) = 
   122           Type ("fun", [T, Type ("nominal.noption", [replace_types U])])
   123       | replace_types (Type (s, Ts)) =
   124           Type (getOpt (AList.lookup op = ps s, s), map replace_types Ts)
   125       | replace_types T = T;
   126 
   127     fun replace_types' (Type (s, Ts)) =
   128           Type (getOpt (AList.lookup op = rps s, s), map replace_types' Ts)
   129       | replace_types' T = T;
   130 
   131     val dts'' = map (fn (tvs, tname, mx, constrs) => (tvs, tname ^ "_Rep", NoSyn,
   132       map (fn (cname, cargs, mx) => (cname,
   133         map (augment_sort_typ o replace_types) cargs, NoSyn)) constrs)) dts';
   134 
   135     val new_type_names' = map (fn n => n ^ "_Rep") new_type_names;
   136     val full_new_type_names' = map (Sign.full_name (sign_of thy)) new_type_names';
   137 
   138     val ({induction, ...},thy1) =
   139       DatatypePackage.add_datatype_i err flat_names new_type_names' dts'' thy;
   140 
   141     val SOME {descr, ...} = Symtab.lookup
   142       (DatatypePackage.get_datatypes thy1) (hd full_new_type_names');
   143     fun nth_dtyp i = typ_of_dtyp descr sorts' (DtRec i);
   144 
   145     (**** define permutation functions ****)
   146 
   147     val permT = mk_permT (TFree ("'x", HOLogic.typeS));
   148     val pi = Free ("pi", permT);
   149     val perm_types = map (fn (i, _) =>
   150       let val T = nth_dtyp i
   151       in permT --> T --> T end) descr;
   152     val perm_names = replicate (length new_type_names) "nominal.perm" @
   153       DatatypeProp.indexify_names (map (fn i => Sign.full_name (sign_of thy1)
   154         ("perm_" ^ name_of_typ (nth_dtyp i)))
   155           (length new_type_names upto length descr - 1));
   156     val perm_names_types = perm_names ~~ perm_types;
   157 
   158     val perm_eqs = List.concat (map (fn (i, (_, _, constrs)) =>
   159       let val T = nth_dtyp i
   160       in map (fn (cname, dts) => 
   161         let
   162           val Ts = map (typ_of_dtyp descr sorts') dts;
   163           val names = DatatypeProp.make_tnames Ts;
   164           val args = map Free (names ~~ Ts);
   165           val c = Const (cname, Ts ---> T);
   166           fun perm_arg (dt, x) =
   167             let val T = type_of x
   168             in if is_rec_type dt then
   169                 let val (Us, _) = strip_type T
   170                 in list_abs (map (pair "x") Us,
   171                   Const (List.nth (perm_names_types, body_index dt)) $ pi $
   172                     list_comb (x, map (fn (i, U) =>
   173                       Const ("nominal.perm", permT --> U --> U) $
   174                         (Const ("List.rev", permT --> permT) $ pi) $
   175                         Bound i) ((length Us - 1 downto 0) ~~ Us)))
   176                 end
   177               else Const ("nominal.perm", permT --> T --> T) $ pi $ x
   178             end;  
   179         in
   180           (("", HOLogic.mk_Trueprop (HOLogic.mk_eq
   181             (Const (List.nth (perm_names_types, i)) $
   182                Free ("pi", mk_permT (TFree ("'x", HOLogic.typeS))) $
   183                list_comb (c, args),
   184              list_comb (c, map perm_arg (dts ~~ args))))), [])
   185         end) constrs
   186       end) descr);
   187 
   188     val (thy2, perm_simps) = thy1 |>
   189       Theory.add_consts_i (map (fn (s, T) => (Sign.base_name s, T, NoSyn))
   190         (List.drop (perm_names_types, length new_type_names))) |>
   191       PrimrecPackage.add_primrec_i "" perm_eqs;
   192 
   193     (**** prove that permutation functions introduced by unfolding are ****)
   194     (**** equivalent to already existing permutation functions         ****)
   195 
   196     val _ = warning ("length descr: " ^ string_of_int (length descr));
   197     val _ = warning ("length new_type_names: " ^ string_of_int (length new_type_names));
   198 
   199     val perm_indnames = DatatypeProp.make_tnames (map body_type perm_types);
   200     val perm_fun_def = PureThy.get_thm thy2 (Name "perm_fun_def");
   201 
   202     val unfolded_perm_eq_thms =
   203       if length descr = length new_type_names then []
   204       else map standard (List.drop (split_conj_thm
   205         (Goal.prove thy2 [] []
   206           (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
   207             (map (fn (c as (s, T), x) =>
   208                let val [T1, T2] = binder_types T
   209                in HOLogic.mk_eq (Const c $ pi $ Free (x, T2),
   210                  Const ("nominal.perm", T) $ pi $ Free (x, T2))
   211                end)
   212              (perm_names_types ~~ perm_indnames))))
   213           (fn _ => EVERY [indtac induction perm_indnames 1,
   214             ALLGOALS (asm_full_simp_tac
   215               (simpset_of thy2 addsimps [perm_fun_def]))])),
   216         length new_type_names));
   217 
   218     (**** prove [] \<bullet> t = t ****)
   219 
   220     val _ = warning "perm_empty_thms";
   221 
   222     val perm_empty_thms = List.concat (map (fn a =>
   223       let val permT = mk_permT (Type (a, []))
   224       in map standard (List.take (split_conj_thm
   225         (Goal.prove thy2 [] []
   226           (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
   227             (map (fn ((s, T), x) => HOLogic.mk_eq
   228                 (Const (s, permT --> T --> T) $
   229                    Const ("List.list.Nil", permT) $ Free (x, T),
   230                  Free (x, T)))
   231              (perm_names ~~
   232               map body_type perm_types ~~ perm_indnames))))
   233           (fn _ => EVERY [indtac induction perm_indnames 1,
   234             ALLGOALS (asm_full_simp_tac (simpset_of thy2))])),
   235         length new_type_names))
   236       end)
   237       atoms);
   238 
   239     (**** prove (pi1 @ pi2) \<bullet> t = pi1 \<bullet> (pi2 \<bullet> t) ****)
   240 
   241     val _ = warning "perm_append_thms";
   242 
   243     (*FIXME: these should be looked up statically*)
   244     val at_pt_inst = PureThy.get_thm thy2 (Name "at_pt_inst");
   245     val pt2 = PureThy.get_thm thy2 (Name "pt2");
   246 
   247     val perm_append_thms = List.concat (map (fn a =>
   248       let
   249         val permT = mk_permT (Type (a, []));
   250         val pi1 = Free ("pi1", permT);
   251         val pi2 = Free ("pi2", permT);
   252         val pt_inst = PureThy.get_thm thy2 (Name ("pt_" ^ Sign.base_name a ^ "_inst"));
   253         val pt2' = pt_inst RS pt2;
   254         val pt2_ax = PureThy.get_thm thy2
   255           (Name (NameSpace.map_base (fn s => "pt_" ^ s ^ "2") a));
   256       in List.take (map standard (split_conj_thm
   257         (Goal.prove thy2 [] []
   258              (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
   259                 (map (fn ((s, T), x) =>
   260                     let val perm = Const (s, permT --> T --> T)
   261                     in HOLogic.mk_eq
   262                       (perm $ (Const ("List.op @", permT --> permT --> permT) $
   263                          pi1 $ pi2) $ Free (x, T),
   264                        perm $ pi1 $ (perm $ pi2 $ Free (x, T)))
   265                     end)
   266                   (perm_names ~~
   267                    map body_type perm_types ~~ perm_indnames))))
   268            (fn _ => EVERY [indtac induction perm_indnames 1,
   269               ALLGOALS (asm_full_simp_tac (simpset_of thy2 addsimps [pt2', pt2_ax]))]))),
   270          length new_type_names)
   271       end) atoms);
   272 
   273     (**** prove pi1 ~ pi2 ==> pi1 \<bullet> t = pi2 \<bullet> t ****)
   274 
   275     val _ = warning "perm_eq_thms";
   276 
   277     val pt3 = PureThy.get_thm thy2 (Name "pt3");
   278     val pt3_rev = PureThy.get_thm thy2 (Name "pt3_rev");
   279 
   280     val perm_eq_thms = List.concat (map (fn a =>
   281       let
   282         val permT = mk_permT (Type (a, []));
   283         val pi1 = Free ("pi1", permT);
   284         val pi2 = Free ("pi2", permT);
   285         (*FIXME: not robust - better access these theorems using NominalData?*)
   286         val at_inst = PureThy.get_thm thy2 (Name ("at_" ^ Sign.base_name a ^ "_inst"));
   287         val pt_inst = PureThy.get_thm thy2 (Name ("pt_" ^ Sign.base_name a ^ "_inst"));
   288         val pt3' = pt_inst RS pt3;
   289         val pt3_rev' = at_inst RS (pt_inst RS pt3_rev);
   290         val pt3_ax = PureThy.get_thm thy2
   291           (Name (NameSpace.map_base (fn s => "pt_" ^ s ^ "3") a));
   292       in List.take (map standard (split_conj_thm
   293         (Goal.prove thy2 [] [] (Logic.mk_implies
   294              (HOLogic.mk_Trueprop (Const ("nominal.prm_eq",
   295                 permT --> permT --> HOLogic.boolT) $ pi1 $ pi2),
   296               HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
   297                 (map (fn ((s, T), x) =>
   298                     let val perm = Const (s, permT --> T --> T)
   299                     in HOLogic.mk_eq
   300                       (perm $ pi1 $ Free (x, T),
   301                        perm $ pi2 $ Free (x, T))
   302                     end)
   303                   (perm_names ~~
   304                    map body_type perm_types ~~ perm_indnames)))))
   305            (fn _ => EVERY [indtac induction perm_indnames 1,
   306               ALLGOALS (asm_full_simp_tac (simpset_of thy2 addsimps [pt3', pt3_rev', pt3_ax]))]))),
   307          length new_type_names)
   308       end) atoms);
   309 
   310     (**** prove pi1 \<bullet> (pi2 \<bullet> t) = (pi1 \<bullet> pi2) \<bullet> (pi1 \<bullet> t) ****)
   311 
   312     val cp1 = PureThy.get_thm thy2 (Name "cp1");
   313     val dj_cp = PureThy.get_thm thy2 (Name "dj_cp");
   314     val pt_perm_compose = PureThy.get_thm thy2 (Name "pt_perm_compose");
   315     val pt_perm_compose_rev = PureThy.get_thm thy2 (Name "pt_perm_compose_rev");
   316     val dj_perm_perm_forget = PureThy.get_thm thy2 (Name "dj_perm_perm_forget");
   317 
   318     fun composition_instance name1 name2 thy =
   319       let
   320         val name1' = Sign.base_name name1;
   321         val name2' = Sign.base_name name2;
   322         val cp_class = Sign.intern_class thy ("cp_" ^ name1' ^ "_" ^ name2');
   323         val permT1 = mk_permT (Type (name1, []));
   324         val permT2 = mk_permT (Type (name2, []));
   325         val augment = map_type_tfree
   326           (fn (x, S) => TFree (x, cp_class :: S));
   327         val Ts = map (augment o body_type) perm_types;
   328         val cp_inst = PureThy.get_thm thy
   329           (Name ("cp_" ^ name1' ^ "_" ^ name2' ^ "_inst"));
   330         val simps = simpset_of thy addsimps (perm_fun_def ::
   331           (if name1 <> name2 then
   332              let val dj = PureThy.get_thm thy (Name ("dj_" ^ name2' ^ "_" ^ name1'))
   333              in [dj RS (cp_inst RS dj_cp), dj RS dj_perm_perm_forget] end
   334            else
   335              let
   336                val at_inst = PureThy.get_thm thy (Name ("at_" ^ name1' ^ "_inst"));
   337                val pt_inst = PureThy.get_thm thy (Name ("pt_" ^ name1' ^ "_inst"))
   338              in
   339                [cp_inst RS cp1 RS sym,
   340                 at_inst RS (pt_inst RS pt_perm_compose) RS sym,
   341                 at_inst RS (pt_inst RS pt_perm_compose_rev) RS sym]
   342             end))
   343         val thms = split_conj_thm (standard (Goal.prove thy [] []
   344             (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj
   345               (map (fn ((s, T), x) =>
   346                   let
   347                     val pi1 = Free ("pi1", permT1);
   348                     val pi2 = Free ("pi2", permT2);
   349                     val perm1 = Const (s, permT1 --> T --> T);
   350                     val perm2 = Const (s, permT2 --> T --> T);
   351                     val perm3 = Const ("nominal.perm", permT1 --> permT2 --> permT2)
   352                   in HOLogic.mk_eq
   353                     (perm1 $ pi1 $ (perm2 $ pi2 $ Free (x, T)),
   354                      perm2 $ (perm3 $ pi1 $ pi2) $ (perm1 $ pi1 $ Free (x, T)))
   355                   end)
   356                 (perm_names ~~ Ts ~~ perm_indnames))))
   357           (fn _ => EVERY [indtac induction perm_indnames 1,
   358              ALLGOALS (asm_full_simp_tac simps)])))
   359       in
   360         foldl (fn ((s, tvs), thy) => AxClass.add_inst_arity_i
   361             (s, replicate (length tvs) (cp_class :: classes), [cp_class])
   362             (AxClass.intro_classes_tac [] THEN ALLGOALS (resolve_tac thms)) thy)
   363           thy (full_new_type_names' ~~ tyvars)
   364       end;
   365 
   366     val (perm_thmss,thy3) = thy2 |>
   367       fold (fn name1 => fold (composition_instance name1) atoms) atoms |>
   368       curry (Library.foldr (fn ((i, (tyname, args, _)), thy) =>
   369         AxClass.add_inst_arity_i (tyname, replicate (length args) classes, classes)
   370         (AxClass.intro_classes_tac [] THEN REPEAT (EVERY
   371            [resolve_tac perm_empty_thms 1,
   372             resolve_tac perm_append_thms 1,
   373             resolve_tac perm_eq_thms 1, assume_tac 1])) thy))
   374         (List.take (descr, length new_type_names)) |>
   375       PureThy.add_thmss
   376         [((space_implode "_" new_type_names ^ "_unfolded_perm_eq",
   377           unfolded_perm_eq_thms), [Simplifier.simp_add_global]),
   378          ((space_implode "_" new_type_names ^ "_perm_empty",
   379           perm_empty_thms), [Simplifier.simp_add_global]),
   380          ((space_implode "_" new_type_names ^ "_perm_append",
   381           perm_append_thms), [Simplifier.simp_add_global]),
   382          ((space_implode "_" new_type_names ^ "_perm_eq",
   383           perm_eq_thms), [Simplifier.simp_add_global])];
   384   
   385     (**** Define representing sets ****)
   386 
   387     val _ = warning "representing sets";
   388 
   389     val rep_set_names = map (Sign.full_name thy3) (DatatypeProp.indexify_names
   390       (map (fn (i, _) => name_of_typ (nth_dtyp i) ^ "_set") descr));
   391     val big_rep_name =
   392       space_implode "_" (DatatypeProp.indexify_names (List.mapPartial
   393         (fn (i, ("nominal.noption", _, _)) => NONE
   394           | (i, _) => SOME (name_of_typ (nth_dtyp i))) descr)) ^ "_set";
   395     val _ = warning ("big_rep_name: " ^ big_rep_name);
   396 
   397     fun strip_option (dtf as DtType ("fun", [dt, DtRec i])) =
   398           (case AList.lookup op = descr i of
   399              SOME ("nominal.noption", _, [(_, [dt']), _]) =>
   400                apfst (cons dt) (strip_option dt')
   401            | _ => ([], dtf))
   402       | strip_option (DtType ("fun", [dt, DtType ("nominal.noption", [dt'])])) =
   403           apfst (cons dt) (strip_option dt')
   404       | strip_option dt = ([], dt);
   405 
   406     val dt_atomTs = distinct (map (typ_of_dtyp descr sorts')
   407       (List.concat (map (fn (_, (_, _, cs)) => List.concat
   408         (map (List.concat o map (fst o strip_option) o snd) cs)) descr)));
   409 
   410     fun make_intr s T (cname, cargs) =
   411       let
   412         fun mk_prem (dt, (j, j', prems, ts)) = 
   413           let
   414             val (dts, dt') = strip_option dt;
   415             val (dts', dt'') = strip_dtyp dt';
   416             val Ts = map (typ_of_dtyp descr sorts') dts;
   417             val Us = map (typ_of_dtyp descr sorts') dts';
   418             val T = typ_of_dtyp descr sorts' dt'';
   419             val free = mk_Free "x" (Us ---> T) j;
   420             val free' = app_bnds free (length Us);
   421             fun mk_abs_fun (T, (i, t)) =
   422               let val U = fastype_of t
   423               in (i + 1, Const ("nominal.abs_fun", [T, U, T] --->
   424                 Type ("nominal.noption", [U])) $ mk_Free "y" T i $ t)
   425               end
   426           in (j + 1, j' + length Ts,
   427             case dt'' of
   428                 DtRec k => list_all (map (pair "x") Us,
   429                   HOLogic.mk_Trueprop (HOLogic.mk_mem (free',
   430                     Const (List.nth (rep_set_names, k),
   431                       HOLogic.mk_setT T)))) :: prems
   432               | _ => prems,
   433             snd (foldr mk_abs_fun (j', free) Ts) :: ts)
   434           end;
   435 
   436         val (_, _, prems, ts) = foldr mk_prem (1, 1, [], []) cargs;
   437         val concl = HOLogic.mk_Trueprop (HOLogic.mk_mem
   438           (list_comb (Const (cname, map fastype_of ts ---> T), ts),
   439            Const (s, HOLogic.mk_setT T)))
   440       in Logic.list_implies (prems, concl)
   441       end;
   442 
   443     val (intr_ts, ind_consts) =
   444       apfst List.concat (ListPair.unzip (List.mapPartial
   445         (fn ((_, ("nominal.noption", _, _)), _) => NONE
   446           | ((i, (_, _, constrs)), rep_set_name) =>
   447               let val T = nth_dtyp i
   448               in SOME (map (make_intr rep_set_name T) constrs,
   449                 Const (rep_set_name, HOLogic.mk_setT T))
   450               end)
   451                 (descr ~~ rep_set_names)));
   452 
   453     val (thy4, {raw_induct = rep_induct, intrs = rep_intrs, ...}) =
   454       setmp InductivePackage.quiet_mode false
   455         (InductivePackage.add_inductive_i false true big_rep_name false true false
   456            ind_consts (map (fn x => (("", x), [])) intr_ts) []) thy3;
   457 
   458     (**** Prove that representing set is closed under permutation ****)
   459 
   460     val _ = warning "proving closure under permutation...";
   461 
   462     val perm_indnames' = List.mapPartial
   463       (fn (x, (_, ("nominal.noption", _, _))) => NONE | (x, _) => SOME x)
   464       (perm_indnames ~~ descr);
   465 
   466     fun mk_perm_closed name = map (fn th => standard (th RS mp))
   467       (List.take (split_conj_thm (Goal.prove thy4 [] []
   468         (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj (map
   469            (fn (S, x) =>
   470               let
   471                 val S = map_term_types (map_type_tfree
   472                   (fn (s, cs) => TFree (s, cs union cp_classes))) S;
   473                 val T = HOLogic.dest_setT (fastype_of S);
   474                 val permT = mk_permT (Type (name, []))
   475               in HOLogic.mk_imp (HOLogic.mk_mem (Free (x, T), S),
   476                 HOLogic.mk_mem (Const ("nominal.perm", permT --> T --> T) $
   477                   Free ("pi", permT) $ Free (x, T), S))
   478               end) (ind_consts ~~ perm_indnames'))))
   479         (fn _ => EVERY (* CU: added perm_fun_def in the final tactic in order to deal with funs *)
   480            [indtac rep_induct [] 1,
   481             ALLGOALS (simp_tac (simpset_of thy4 addsimps
   482               (symmetric perm_fun_def :: PureThy.get_thms thy4 (Name ("abs_perm"))))),
   483             ALLGOALS (resolve_tac rep_intrs 
   484                THEN_ALL_NEW (asm_full_simp_tac (simpset_of thy4 addsimps [perm_fun_def])))])),
   485         length new_type_names));
   486 
   487     (* FIXME: theorems are stored in database for testing only *)
   488     val perm_closed_thmss = map mk_perm_closed atoms;
   489     val (_,thy5) = PureThy.add_thmss [(("perm_closed", List.concat perm_closed_thmss), [])] thy4;
   490 
   491     (**** typedef ****)
   492 
   493     val _ = warning "defining type...";
   494 
   495     val (typedefs, thy6) =
   496       fold_map (fn ((((name, mx), tvs), c), name') => fn thy =>
   497         setmp TypedefPackage.quiet_mode true
   498           (TypedefPackage.add_typedef_i false (SOME name') (name, tvs, mx) c NONE
   499             (rtac exI 1 THEN
   500               QUIET_BREADTH_FIRST (has_fewer_prems 1)
   501               (resolve_tac rep_intrs 1))) thy |> (fn (thy, r) =>
   502         let
   503           val permT = mk_permT (TFree (variant tvs "'a", HOLogic.typeS));
   504           val pi = Free ("pi", permT);
   505           val tvs' = map (fn s => TFree (s, the (AList.lookup op = sorts' s))) tvs;
   506           val T = Type (Sign.intern_type thy name, tvs');
   507           val Const (_, Type (_, [U])) = c
   508         in apfst (pair r o hd)
   509           (PureThy.add_defs_i true [(("prm_" ^ name ^ "_def", Logic.mk_equals
   510             (Const ("nominal.perm", permT --> T --> T) $ pi $ Free ("x", T),
   511              Const (Sign.intern_const thy ("Abs_" ^ name), U --> T) $
   512                (Const ("nominal.perm", permT --> U --> U) $ pi $
   513                  (Const (Sign.intern_const thy ("Rep_" ^ name), T --> U) $
   514                    Free ("x", T))))), [])] thy)
   515         end))
   516           (types_syntax ~~ tyvars ~~
   517             (List.take (ind_consts, length new_type_names)) ~~ new_type_names) thy5;
   518 
   519     val perm_defs = map snd typedefs;
   520     val Abs_inverse_thms = map (#Abs_inverse o fst) typedefs;
   521     val Rep_inverse_thms = map (#Rep_inverse o fst) typedefs;
   522     val Rep_thms = map (#Rep o fst) typedefs;
   523 
   524     val big_name = space_implode "_" new_type_names;
   525 
   526 
   527     (** prove that new types are in class pt_<name> **)
   528 
   529     val _ = warning "prove that new types are in class pt_<name> ...";
   530 
   531     fun pt_instance ((class, atom), perm_closed_thms) =
   532       fold (fn (((({Abs_inverse, Rep_inverse, Rep, ...},
   533         perm_def), name), tvs), perm_closed) => fn thy =>
   534           AxClass.add_inst_arity_i
   535             (Sign.intern_type thy name,
   536               replicate (length tvs) (classes @ cp_classes), [class])
   537             (EVERY [AxClass.intro_classes_tac [],
   538               rewrite_goals_tac [perm_def],
   539               asm_full_simp_tac (simpset_of thy addsimps [Rep_inverse]) 1,
   540               asm_full_simp_tac (simpset_of thy addsimps
   541                 [Rep RS perm_closed RS Abs_inverse]) 1,
   542               asm_full_simp_tac (HOL_basic_ss addsimps [PureThy.get_thm thy
   543                 (Name ("pt_" ^ Sign.base_name atom ^ "3"))]) 1]) thy)
   544         (typedefs ~~ new_type_names ~~ tyvars ~~ perm_closed_thms);
   545 
   546 
   547     (** prove that new types are in class cp_<name1>_<name2> **)
   548 
   549     val _ = warning "prove that new types are in class cp_<name1>_<name2> ...";
   550 
   551     fun cp_instance (atom1, perm_closed_thms1) (atom2, perm_closed_thms2) thy =
   552       let
   553         val name = "cp_" ^ Sign.base_name atom1 ^ "_" ^ Sign.base_name atom2;
   554         val class = Sign.intern_class thy name;
   555         val cp1' = PureThy.get_thm thy (Name (name ^ "_inst")) RS cp1
   556       in fold (fn ((((({Abs_inverse, Rep_inverse, Rep, ...},
   557         perm_def), name), tvs), perm_closed1), perm_closed2) => fn thy =>
   558           AxClass.add_inst_arity_i
   559             (Sign.intern_type thy name,
   560               replicate (length tvs) (classes @ cp_classes), [class])
   561             (EVERY [AxClass.intro_classes_tac [],
   562               rewrite_goals_tac [perm_def],
   563               asm_full_simp_tac (simpset_of thy addsimps
   564                 ((Rep RS perm_closed1 RS Abs_inverse) ::
   565                  (if atom1 = atom2 then []
   566                   else [Rep RS perm_closed2 RS Abs_inverse]))) 1,
   567               cong_tac 1,
   568               rtac refl 1,
   569               rtac cp1' 1]) thy)
   570         (typedefs ~~ new_type_names ~~ tyvars ~~ perm_closed_thms1 ~~
   571           perm_closed_thms2) thy
   572       end;
   573 
   574     val thy7 = fold (fn x => fn thy => thy |>
   575       pt_instance x |>
   576       fold (cp_instance (apfst snd x)) (atoms ~~ perm_closed_thmss))
   577         (classes ~~ atoms ~~ perm_closed_thmss) thy6;
   578 
   579     (**** constructors ****)
   580 
   581     fun mk_abs_fun (x, t) =
   582       let
   583         val T = fastype_of x;
   584         val U = fastype_of t
   585       in
   586         Const ("nominal.abs_fun", T --> U --> T -->
   587           Type ("nominal.noption", [U])) $ x $ t
   588       end;
   589 
   590     val (ty_idxs, _) = foldl
   591       (fn ((i, ("nominal.noption", _, _)), p) => p
   592         | ((i, _), (ty_idxs, j)) => (ty_idxs @ [(i, j)], j + 1)) ([], 0) descr;
   593 
   594     fun reindex (DtType (s, dts)) = DtType (s, map reindex dts)
   595       | reindex (DtRec i) = DtRec (the (AList.lookup op = ty_idxs i))
   596       | reindex dt = dt;
   597 
   598     fun strip_suffix i s = implode (List.take (explode s, size s - i));
   599 
   600     (** strips the "_Rep" in type names *)
   601     fun strip_nth_name i s = 
   602       let val xs = NameSpace.unpack s; 
   603       in NameSpace.pack (Library.nth_map (length xs - i) (strip_suffix 4) xs) end;
   604 
   605     val (descr'', ndescr) = ListPair.unzip (List.mapPartial
   606       (fn (i, ("nominal.noption", _, _)) => NONE
   607         | (i, (s, dts, constrs)) =>
   608              let
   609                val SOME index = AList.lookup op = ty_idxs i;
   610                val (constrs1, constrs2) = ListPair.unzip
   611                  (map (fn (cname, cargs) => apfst (pair (strip_nth_name 2 cname))
   612                    (foldl_map (fn (dts, dt) =>
   613                      let val (dts', dt') = strip_option dt
   614                      in (dts @ dts' @ [reindex dt'], (length dts, length dts')) end)
   615                        ([], cargs))) constrs)
   616              in SOME ((index, (strip_nth_name 1 s,  map reindex dts, constrs1)),
   617                (index, constrs2))
   618              end) descr);
   619 
   620     val (descr1, descr2) = splitAt (length new_type_names, descr'');
   621     val descr' = [descr1, descr2];
   622 
   623     val typ_of_dtyp' = replace_types' o typ_of_dtyp descr sorts';
   624 
   625     val rep_names = map (fn s =>
   626       Sign.intern_const thy7 ("Rep_" ^ s)) new_type_names;
   627     val abs_names = map (fn s =>
   628       Sign.intern_const thy7 ("Abs_" ^ s)) new_type_names;
   629 
   630     val recTs' = List.mapPartial
   631       (fn ((_, ("nominal.noption", _, _)), T) => NONE
   632         | (_, T) => SOME T) (descr ~~ get_rec_types descr sorts');
   633     val recTs = get_rec_types descr'' sorts';
   634     val newTs' = Library.take (length new_type_names, recTs');
   635     val newTs = Library.take (length new_type_names, recTs);
   636 
   637     val full_new_type_names = map (Sign.full_name (sign_of thy)) new_type_names;
   638 
   639     fun make_constr_def tname T T' ((thy, defs, eqns), ((cname, cargs), (cname', mx))) =
   640       let
   641         fun constr_arg (dt, (j, l_args, r_args)) =
   642           let
   643             val x' = mk_Free "x" (typ_of_dtyp' dt) j;
   644             val (dts, dt') = strip_option dt;
   645             val xs = map (fn (dt, i) => mk_Free "x" (typ_of_dtyp' dt) i)
   646               (dts ~~ (j upto j + length dts - 1))
   647             val x = mk_Free "x" (typ_of_dtyp' dt') (j + length dts)
   648             val (dts', dt'') = strip_dtyp dt'
   649           in
   650             (j + length dts + 1,
   651              xs @ x :: l_args,
   652              foldr mk_abs_fun
   653                (case dt'' of
   654                   DtRec k => if k < length new_type_names then
   655                       list_abs (map (pair "z" o typ_of_dtyp') dts',
   656                         Const (List.nth (rep_names, k), typ_of_dtyp' dt'' -->
   657                           typ_of_dtyp descr sorts' dt'') $ app_bnds x (length dts'))
   658                     else error "nested recursion not (yet) supported"
   659                 | _ => x) xs :: r_args)
   660           end
   661 
   662         val (_, l_args, r_args) = foldr constr_arg (1, [], []) cargs;
   663         val abs_name = Sign.intern_const (Theory.sign_of thy) ("Abs_" ^ tname);
   664         val rep_name = Sign.intern_const (Theory.sign_of thy) ("Rep_" ^ tname);
   665         val constrT = map fastype_of l_args ---> T;
   666         val lhs = list_comb (Const (Sign.full_name thy (Sign.base_name cname),
   667           constrT), l_args);
   668         val rhs = list_comb (Const (cname, map fastype_of r_args ---> T'), r_args);
   669         val def = Logic.mk_equals (lhs, Const (abs_name, T' --> T) $ rhs);
   670         val eqn = HOLogic.mk_Trueprop (HOLogic.mk_eq
   671           (Const (rep_name, T --> T') $ lhs, rhs));
   672         val def_name = (Sign.base_name cname) ^ "_def";
   673         val ([def_thm], thy') = thy |>
   674           Theory.add_consts_i [(cname', constrT, mx)] |>
   675           (PureThy.add_defs_i false o map Thm.no_attributes) [(def_name, def)]
   676       in (thy', defs @ [def_thm], eqns @ [eqn]) end;
   677 
   678     fun dt_constr_defs ((thy, defs, eqns, dist_lemmas),
   679         (((((_, (_, _, constrs)), tname), T), T'), constr_syntax)) =
   680       let
   681         val rep_const = cterm_of thy
   682           (Const (Sign.intern_const thy ("Rep_" ^ tname), T --> T'));
   683         val dist = standard (cterm_instantiate [(cterm_of thy distinct_f, rep_const)] distinct_lemma);
   684         val (thy', defs', eqns') = Library.foldl (make_constr_def tname T T')
   685           ((Theory.add_path tname thy, defs, []), constrs ~~ constr_syntax)
   686       in
   687         (parent_path flat_names thy', defs', eqns @ [eqns'], dist_lemmas @ [dist])
   688       end;
   689 
   690     val (thy8, constr_defs, constr_rep_eqns, dist_lemmas) = Library.foldl dt_constr_defs
   691       ((thy7, [], [], []), List.take (descr, length new_type_names) ~~
   692         new_type_names ~~ newTs ~~ newTs' ~~ constr_syntax);
   693 
   694     val abs_inject_thms = map (fn tname =>
   695       PureThy.get_thm thy8 (Name ("Abs_" ^ tname ^ "_inject"))) new_type_names;
   696 
   697     val rep_inject_thms = map (fn tname =>
   698       PureThy.get_thm thy8 (Name ("Rep_" ^ tname ^ "_inject"))) new_type_names;
   699 
   700     val rep_thms = map (fn tname =>
   701       PureThy.get_thm thy8 (Name ("Rep_" ^ tname))) new_type_names;
   702 
   703     val rep_inverse_thms = map (fn tname =>
   704       PureThy.get_thm thy8 (Name ("Rep_" ^ tname ^ "_inverse"))) new_type_names;
   705 
   706     (* prove theorem  Rep_i (Constr_j ...) = Constr'_j ...  *)
   707     
   708     fun prove_constr_rep_thm eqn =
   709       let
   710         val inj_thms = map (fn r => r RS iffD1) abs_inject_thms;
   711         val rewrites = constr_defs @ map mk_meta_eq rep_inverse_thms
   712       in standard (Goal.prove thy8 [] [] eqn (fn _ => EVERY
   713         [resolve_tac inj_thms 1,
   714          rewrite_goals_tac rewrites,
   715          rtac refl 3,
   716          resolve_tac rep_intrs 2,
   717          REPEAT (resolve_tac rep_thms 1)]))
   718       end;
   719 
   720     val constr_rep_thmss = map (map prove_constr_rep_thm) constr_rep_eqns;
   721 
   722     (* prove theorem  pi \<bullet> Rep_i x = Rep_i (pi \<bullet> x) *)
   723 
   724     fun prove_perm_rep_perm (atom, perm_closed_thms) = map (fn th =>
   725       let
   726         val _ $ (_ $ (Rep $ x) $ _) = Logic.unvarify (prop_of th);
   727         val Type ("fun", [T, U]) = fastype_of Rep;
   728         val permT = mk_permT (Type (atom, []));
   729         val pi = Free ("pi", permT);
   730       in
   731         standard (Goal.prove thy8 [] [] (HOLogic.mk_Trueprop (HOLogic.mk_eq
   732             (Const ("nominal.perm", permT --> U --> U) $ pi $ (Rep $ x),
   733              Rep $ (Const ("nominal.perm", permT --> T --> T) $ pi $ x))))
   734           (fn _ => simp_tac (HOL_basic_ss addsimps (perm_defs @ Abs_inverse_thms @
   735             perm_closed_thms @ Rep_thms)) 1))
   736       end) Rep_thms;
   737 
   738     val perm_rep_perm_thms = List.concat (map prove_perm_rep_perm
   739       (atoms ~~ perm_closed_thmss));
   740 
   741     (* prove distinctness theorems *)
   742 
   743     val distinct_props = setmp DatatypeProp.dtK 1000
   744       (DatatypeProp.make_distincts new_type_names descr' sorts') thy8;
   745 
   746     val dist_rewrites = map (fn (rep_thms, dist_lemma) =>
   747       dist_lemma::(rep_thms @ [In0_eq, In1_eq, In0_not_In1, In1_not_In0]))
   748         (constr_rep_thmss ~~ dist_lemmas);
   749 
   750     fun prove_distinct_thms (_, []) = []
   751       | prove_distinct_thms (p as (rep_thms, dist_lemma), t::ts) =
   752           let
   753             val dist_thm = standard (Goal.prove thy8 [] [] t (fn _ =>
   754               simp_tac (simpset_of thy8 addsimps (dist_lemma :: rep_thms)) 1))
   755           in dist_thm::(standard (dist_thm RS not_sym))::
   756             (prove_distinct_thms (p, ts))
   757           end;
   758 
   759     val distinct_thms = map prove_distinct_thms
   760       (constr_rep_thmss ~~ dist_lemmas ~~ distinct_props);
   761 
   762     (** prove equations for permutation functions **)
   763 
   764     val abs_perm = PureThy.get_thms thy8 (Name "abs_perm"); (* FIXME *)
   765 
   766     val perm_simps' = map (fn (((i, (_, _, constrs)), tname), constr_rep_thms) =>
   767       let val T = replace_types' (nth_dtyp i)
   768       in List.concat (map (fn (atom, perm_closed_thms) =>
   769           map (fn ((cname, dts), constr_rep_thm) => 
   770         let
   771           val cname = Sign.intern_const thy8
   772             (NameSpace.append tname (Sign.base_name cname));
   773           val permT = mk_permT (Type (atom, []));
   774           val pi = Free ("pi", permT);
   775 
   776           fun perm t =
   777             let val T = fastype_of t
   778             in Const ("nominal.perm", permT --> T --> T) $ pi $ t end;
   779 
   780           fun constr_arg (dt, (j, l_args, r_args)) =
   781             let
   782               val x' = mk_Free "x" (typ_of_dtyp' dt) j;
   783               val (dts, dt') = strip_option dt;
   784               val Ts = map typ_of_dtyp' dts;
   785               val xs = map (fn (T, i) => mk_Free "x" T i)
   786                 (Ts ~~ (j upto j + length dts - 1))
   787               val x = mk_Free "x" (typ_of_dtyp' dt') (j + length dts);
   788               val (dts', dt'') = strip_dtyp dt';
   789             in
   790               (j + length dts + 1,
   791                xs @ x :: l_args,
   792                map perm (xs @ [x]) @ r_args)
   793             end
   794 
   795           val (_, l_args, r_args) = foldr constr_arg (1, [], []) dts;
   796           val c = Const (cname, map fastype_of l_args ---> T)
   797         in
   798           standard (Goal.prove thy8 [] []
   799             (HOLogic.mk_Trueprop (HOLogic.mk_eq
   800               (perm (list_comb (c, l_args)), list_comb (c, r_args))))
   801             (fn _ => EVERY
   802               [simp_tac (simpset_of thy8 addsimps (constr_rep_thm :: perm_defs)) 1,
   803                simp_tac (HOL_basic_ss addsimps (Rep_thms @ Abs_inverse_thms @
   804                  constr_defs @ perm_closed_thms)) 1,
   805                TRY (simp_tac (HOL_basic_ss addsimps
   806                  (symmetric perm_fun_def :: abs_perm)) 1),
   807                TRY (simp_tac (HOL_basic_ss addsimps
   808                  (perm_fun_def :: perm_defs @ Rep_thms @ Abs_inverse_thms @
   809                     perm_closed_thms)) 1)]))
   810         end) (constrs ~~ constr_rep_thms)) (atoms ~~ perm_closed_thmss))
   811       end) (List.take (descr, length new_type_names) ~~ new_type_names ~~ constr_rep_thmss);
   812 
   813     (** prove injectivity of constructors **)
   814 
   815     val rep_inject_thms' = map (fn th => th RS sym) rep_inject_thms;
   816     val alpha = PureThy.get_thms thy8 (Name "alpha");
   817     val abs_fresh = PureThy.get_thms thy8 (Name "abs_fresh");
   818     val fresh_def = PureThy.get_thm thy8 (Name "fresh_def");
   819     val supp_def = PureThy.get_thm thy8 (Name "supp_def");
   820 
   821     val inject_thms = map (fn (((i, (_, _, constrs)), tname), constr_rep_thms) =>
   822       let val T = replace_types' (nth_dtyp i)
   823       in List.mapPartial (fn ((cname, dts), constr_rep_thm) =>
   824         if null dts then NONE else SOME
   825         let
   826           val cname = Sign.intern_const thy8
   827             (NameSpace.append tname (Sign.base_name cname));
   828 
   829           fun make_inj (dt, (j, args1, args2, eqs)) =
   830             let
   831               val x' = mk_Free "x" (typ_of_dtyp' dt) j;
   832               val y' = mk_Free "y" (typ_of_dtyp' dt) j;
   833               val (dts, dt') = strip_option dt;
   834               val Ts_idx = map typ_of_dtyp' dts ~~ (j upto j + length dts - 1);
   835               val xs = map (fn (T, i) => mk_Free "x" T i) Ts_idx;
   836               val ys = map (fn (T, i) => mk_Free "y" T i) Ts_idx;
   837               val x = mk_Free "x" (typ_of_dtyp' dt') (j + length dts);
   838               val y = mk_Free "y" (typ_of_dtyp' dt') (j + length dts);
   839               val (dts', dt'') = strip_dtyp dt';
   840             in
   841               (j + length dts + 1,
   842                xs @ (x :: args1), ys @ (y :: args2),
   843                HOLogic.mk_eq
   844                  (foldr mk_abs_fun x xs, foldr mk_abs_fun y ys) :: eqs)
   845             end;
   846 
   847           val (_, args1, args2, eqs) = foldr make_inj (1, [], [], []) dts;
   848           val Ts = map fastype_of args1;
   849           val c = Const (cname, Ts ---> T)
   850         in
   851           standard (Goal.prove thy8 [] [] (HOLogic.mk_Trueprop (HOLogic.mk_eq
   852               (HOLogic.mk_eq (list_comb (c, args1), list_comb (c, args2)),
   853                foldr1 HOLogic.mk_conj eqs)))
   854             (fn _ => EVERY
   855                [asm_full_simp_tac (simpset_of thy8 addsimps (constr_rep_thm ::
   856                   rep_inject_thms')) 1,
   857                 TRY (asm_full_simp_tac (HOL_basic_ss addsimps (fresh_def :: supp_def ::
   858                   alpha @ abs_perm @ abs_fresh @ rep_inject_thms @
   859                   perm_rep_perm_thms)) 1),
   860                 TRY (asm_full_simp_tac (HOL_basic_ss addsimps (perm_fun_def ::
   861                   expand_fun_eq :: rep_inject_thms @ perm_rep_perm_thms)) 1)]))
   862         end) (constrs ~~ constr_rep_thms)
   863       end) (List.take (descr, length new_type_names) ~~ new_type_names ~~ constr_rep_thmss);
   864 
   865     (** equations for support and freshness **)
   866 
   867     val Un_assoc = PureThy.get_thm thy8 (Name "Un_assoc");
   868     val de_Morgan_conj = PureThy.get_thm thy8 (Name "de_Morgan_conj");
   869     val Collect_disj_eq = PureThy.get_thm thy8 (Name "Collect_disj_eq");
   870     val finite_Un = PureThy.get_thm thy8 (Name "finite_Un");
   871 
   872     val (supp_thms, fresh_thms) = ListPair.unzip (map ListPair.unzip
   873       (map (fn ((((i, (_, _, constrs)), tname), inject_thms'), perm_thms') =>
   874       let val T = replace_types' (nth_dtyp i)
   875       in List.concat (map (fn (cname, dts) => map (fn atom =>
   876         let
   877           val cname = Sign.intern_const thy8
   878             (NameSpace.append tname (Sign.base_name cname));
   879           val atomT = Type (atom, []);
   880 
   881           fun process_constr (dt, (j, args1, args2)) =
   882             let
   883               val x' = mk_Free "x" (typ_of_dtyp' dt) j;
   884               val (dts, dt') = strip_option dt;
   885               val Ts_idx = map typ_of_dtyp' dts ~~ (j upto j + length dts - 1);
   886               val xs = map (fn (T, i) => mk_Free "x" T i) Ts_idx;
   887               val x = mk_Free "x" (typ_of_dtyp' dt') (j + length dts);
   888               val (dts', dt'') = strip_dtyp dt';
   889             in
   890               (j + length dts + 1,
   891                xs @ (x :: args1), foldr mk_abs_fun x xs :: args2)
   892             end;
   893 
   894           val (_, args1, args2) = foldr process_constr (1, [], []) dts;
   895           val Ts = map fastype_of args1;
   896           val c = list_comb (Const (cname, Ts ---> T), args1);
   897           fun supp t =
   898             Const ("nominal.supp", fastype_of t --> HOLogic.mk_setT atomT) $ t;
   899           fun fresh t =
   900             Const ("nominal.fresh", atomT --> fastype_of t --> HOLogic.boolT) $
   901               Free ("a", atomT) $ t;
   902           val supp_thm = standard (Goal.prove thy8 [] []
   903               (HOLogic.mk_Trueprop (HOLogic.mk_eq
   904                 (supp c,
   905                  if null dts then Const ("{}", HOLogic.mk_setT atomT)
   906                  else foldr1 (HOLogic.mk_binop "op Un") (map supp args2))))
   907             (fn _ =>
   908               simp_tac (HOL_basic_ss addsimps (supp_def ::
   909                  Un_assoc :: de_Morgan_conj :: Collect_disj_eq :: finite_Un ::
   910                  symmetric empty_def :: Finites.emptyI :: simp_thms @
   911                  abs_perm @ abs_fresh @ inject_thms' @ perm_thms')) 1))
   912         in
   913           (supp_thm,
   914            standard (Goal.prove thy8 [] [] (HOLogic.mk_Trueprop (HOLogic.mk_eq
   915               (fresh c,
   916                if null dts then HOLogic.true_const
   917                else foldr1 HOLogic.mk_conj (map fresh args2))))
   918              (fn _ =>
   919                simp_tac (simpset_of thy8 addsimps [fresh_def, supp_thm]) 1)))
   920         end) atoms) constrs)
   921       end) (List.take (descr, length new_type_names) ~~ new_type_names ~~ inject_thms ~~ perm_simps')));
   922 
   923     (**** weak induction theorem ****)
   924 
   925     val arities = get_arities descr'';
   926 
   927     fun mk_funs_inv thm =
   928       let
   929         val {sign, prop, ...} = rep_thm thm;
   930         val _ $ (_ $ (Const (_, Type (_, [U, _])) $ _ $ S)) $
   931           (_ $ (_ $ (r $ (a $ _)) $ _)) = Type.freeze prop;
   932         val used = add_term_tfree_names (a, []);
   933 
   934         fun mk_thm i =
   935           let
   936             val Ts = map (TFree o rpair HOLogic.typeS)
   937               (variantlist (replicate i "'t", used));
   938             val f = Free ("f", Ts ---> U)
   939           in standard (Goal.prove sign [] [] (Logic.mk_implies
   940             (HOLogic.mk_Trueprop (HOLogic.list_all
   941                (map (pair "x") Ts, HOLogic.mk_mem (app_bnds f i, S))),
   942              HOLogic.mk_Trueprop (HOLogic.mk_eq (list_abs (map (pair "x") Ts,
   943                r $ (a $ app_bnds f i)), f))))
   944             (fn _ => EVERY [REPEAT (rtac ext 1), REPEAT (etac allE 1), rtac thm 1, atac 1]))
   945           end
   946       in map (fn r => r RS subst) (thm :: map mk_thm arities) end;
   947 
   948     fun mk_indrule_lemma ((prems, concls), (((i, _), T), U)) =
   949       let
   950         val Rep_t = Const (List.nth (rep_names, i), T --> U) $
   951           mk_Free "x" T i;
   952 
   953         val Abs_t =  Const (List.nth (abs_names, i), U --> T)
   954 
   955       in (prems @ [HOLogic.imp $ HOLogic.mk_mem (Rep_t,
   956             Const (List.nth (rep_set_names, i), HOLogic.mk_setT U)) $
   957               (mk_Free "P" (T --> HOLogic.boolT) (i + 1) $ (Abs_t $ Rep_t))],
   958           concls @ [mk_Free "P" (T --> HOLogic.boolT) (i + 1) $ mk_Free "x" T i])
   959       end;
   960 
   961     val (indrule_lemma_prems, indrule_lemma_concls) =
   962       Library.foldl mk_indrule_lemma (([], []), (descr'' ~~ recTs ~~ recTs'));
   963 
   964     val indrule_lemma = standard (Goal.prove thy8 [] []
   965       (Logic.mk_implies
   966         (HOLogic.mk_Trueprop (mk_conj indrule_lemma_prems),
   967          HOLogic.mk_Trueprop (mk_conj indrule_lemma_concls))) (fn _ => EVERY
   968            [REPEAT (etac conjE 1),
   969             REPEAT (EVERY
   970               [TRY (rtac conjI 1), full_simp_tac (HOL_basic_ss addsimps Rep_inverse_thms) 1,
   971                etac mp 1, resolve_tac Rep_thms 1])]));
   972 
   973     val Ps = map head_of (HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of indrule_lemma)));
   974     val frees = if length Ps = 1 then [Free ("P", snd (dest_Var (hd Ps)))] else
   975       map (Free o apfst fst o dest_Var) Ps;
   976     val indrule_lemma' = cterm_instantiate
   977       (map (cterm_of thy8) Ps ~~ map (cterm_of thy8) frees) indrule_lemma;
   978 
   979     val Abs_inverse_thms' = List.concat (map mk_funs_inv Abs_inverse_thms);
   980 
   981     val dt_induct_prop = DatatypeProp.make_ind descr' sorts';
   982     val dt_induct = standard (Goal.prove thy8 []
   983       (Logic.strip_imp_prems dt_induct_prop) (Logic.strip_imp_concl dt_induct_prop)
   984       (fn prems => EVERY
   985         [rtac indrule_lemma' 1,
   986          (DatatypeAux.indtac rep_induct THEN_ALL_NEW ObjectLogic.atomize_tac) 1,
   987          EVERY (map (fn (prem, r) => (EVERY
   988            [REPEAT (eresolve_tac Abs_inverse_thms' 1),
   989             simp_tac (HOL_basic_ss addsimps [symmetric r]) 1,
   990             DEPTH_SOLVE_1 (ares_tac [prem] 1 ORELSE etac allE 1)]))
   991                 (prems ~~ constr_defs))]));
   992 
   993     val case_names_induct = mk_case_names_induct descr'';
   994 
   995     (**** prove that new datatypes have finite support ****)
   996 
   997     val _ = warning "proving finite support for the new datatype";
   998 
   999     val indnames = DatatypeProp.make_tnames recTs;
  1000 
  1001     val abs_supp = PureThy.get_thms thy8 (Name "abs_supp");
  1002     val supp_atm = PureThy.get_thms thy8 (Name "supp_atm");
  1003 
  1004     val finite_supp_thms = map (fn atom =>
  1005       let val atomT = Type (atom, [])
  1006       in map standard (List.take
  1007         (split_conj_thm (Goal.prove thy8 [] [] (HOLogic.mk_Trueprop
  1008            (foldr1 HOLogic.mk_conj (map (fn (s, T) => HOLogic.mk_mem
  1009              (Const ("nominal.supp", T --> HOLogic.mk_setT atomT) $ Free (s, T),
  1010               Const ("Finite_Set.Finites", HOLogic.mk_setT (HOLogic.mk_setT atomT))))
  1011                (indnames ~~ recTs))))
  1012            (fn _ => indtac dt_induct indnames 1 THEN
  1013             ALLGOALS (asm_full_simp_tac (simpset_of thy8 addsimps
  1014               (abs_supp @ supp_atm @
  1015                PureThy.get_thms thy8 (Name ("fs_" ^ Sign.base_name atom ^ "1")) @
  1016                List.concat supp_thms))))),
  1017          length new_type_names))
  1018       end) atoms;
  1019 
  1020     (**** strong induction theorem ****)
  1021 
  1022     val pnames = if length descr'' = 1 then ["P"]
  1023       else map (fn i => "P" ^ string_of_int i) (1 upto length descr'');
  1024     val ind_sort = if null dt_atomTs then HOLogic.typeS
  1025       else map (fn T => Sign.intern_class thy8 ("fs_" ^
  1026         Sign.base_name (fst (dest_Type T)))) dt_atomTs;
  1027     val fsT = TFree ("'n", ind_sort);
  1028 
  1029     fun make_pred i T =
  1030       Free (List.nth (pnames, i), fsT --> T --> HOLogic.boolT);
  1031 
  1032     fun make_ind_prem k T ((cname, cargs), idxs) =
  1033       let
  1034         val recs = List.filter is_rec_type cargs;
  1035         val Ts = map (typ_of_dtyp descr'' sorts') cargs;
  1036         val recTs' = map (typ_of_dtyp descr'' sorts') recs;
  1037         val tnames = variantlist (DatatypeProp.make_tnames Ts, pnames);
  1038         val rec_tnames = map fst (List.filter (is_rec_type o snd) (tnames ~~ cargs));
  1039         val frees = tnames ~~ Ts;
  1040         val z = (variant tnames "z", fsT);
  1041 
  1042         fun mk_prem ((dt, s), T) =
  1043           let
  1044             val (Us, U) = strip_type T;
  1045             val l = length Us
  1046           in list_all (z :: map (pair "x") Us, HOLogic.mk_Trueprop
  1047             (make_pred (body_index dt) U $ Bound l $ app_bnds (Free (s, T)) l))
  1048           end;
  1049 
  1050         val prems = map mk_prem (recs ~~ rec_tnames ~~ recTs');
  1051         val prems' = map (fn p as (_, T) => HOLogic.mk_Trueprop
  1052             (Const ("nominal.fresh", T --> fsT --> HOLogic.boolT) $
  1053               Free p $ Free z))
  1054           (map (curry List.nth frees) (List.concat (map (fn (m, n) =>
  1055              m upto m + n - 1) idxs)))
  1056 
  1057       in list_all_free (frees @ [z], Logic.list_implies (prems' @ prems,
  1058         HOLogic.mk_Trueprop (make_pred k T $ Free z $
  1059           list_comb (Const (cname, Ts ---> T), map Free frees))))
  1060       end;
  1061 
  1062     val ind_prems = List.concat (map (fn (((i, (_, _, constrs)), (_, idxss)), T) =>
  1063       map (make_ind_prem i T) (constrs ~~ idxss)) (descr'' ~~ ndescr ~~ recTs));
  1064     val tnames = DatatypeProp.make_tnames recTs;
  1065     val z = (variant tnames "z", fsT);
  1066     val ind_concl = HOLogic.mk_Trueprop (foldr1 (HOLogic.mk_binop "op &")
  1067       (map (fn (((i, _), T), tname) => make_pred i T $ Free z $ Free (tname, T))
  1068         (descr'' ~~ recTs ~~ tnames)));
  1069     val induct = Logic.list_implies (ind_prems, ind_concl);
  1070 
  1071     val simp_atts = replicate (length new_type_names) [Simplifier.simp_add_global];
  1072 
  1073     val ((_, [induct_rule]), thy9) = thy8 |>
  1074       Theory.add_path big_name |>
  1075       PureThy.add_thms [(("induct_weak", dt_induct), [case_names_induct])] ||>
  1076       (PureThy.add_thmss [(("inducts_weak", projections dt_induct), [case_names_induct])] #> snd)
  1077       ||> Theory.parent_path ||>>
  1078       DatatypeAux.store_thmss_atts "distinct" new_type_names simp_atts distinct_thms ||>>
  1079       DatatypeAux.store_thmss "constr_rep" new_type_names constr_rep_thmss ||>>
  1080       DatatypeAux.store_thmss_atts "perm" new_type_names simp_atts perm_simps' ||>>
  1081       DatatypeAux.store_thmss "inject" new_type_names inject_thms ||>>
  1082       DatatypeAux.store_thmss "supp" new_type_names supp_thms ||>>
  1083       DatatypeAux.store_thmss_atts "fresh" new_type_names simp_atts fresh_thms ||>
  1084       fold (fn (atom, ths) => fn thy =>
  1085         let val class = Sign.intern_class thy ("fs_" ^ Sign.base_name atom)
  1086         in fold (fn T => AxClass.add_inst_arity_i
  1087             (fst (dest_Type T),
  1088               replicate (length sorts) [class], [class])
  1089             (AxClass.intro_classes_tac [] THEN resolve_tac ths 1)) newTs thy
  1090         end) (atoms ~~ finite_supp_thms) ||>
  1091       Theory.add_path big_name ||>>
  1092       PureThy.add_axioms_i [(("induct_unsafe", induct), [case_names_induct])]
  1093     val thy10 = thy9
  1094       |> PureThy.add_thmss [(("inducts_unsafe", projections induct_rule), [case_names_induct])]
  1095       |> snd
  1096       |> Theory.parent_path;
  1097   in
  1098     thy10
  1099   end;
  1100 
  1101 val add_nominal_datatype = gen_add_nominal_datatype read_typ true;
  1102 
  1103 
  1104 (* FIXME: The following stuff should be exported by DatatypePackage *)
  1105 
  1106 local structure P = OuterParse and K = OuterKeyword in
  1107 
  1108 val datatype_decl =
  1109   Scan.option (P.$$$ "(" |-- P.name --| P.$$$ ")") -- P.type_args -- P.name -- P.opt_infix --
  1110     (P.$$$ "=" |-- P.enum1 "|" (P.name -- Scan.repeat P.typ -- P.opt_mixfix));
  1111 
  1112 fun mk_datatype args =
  1113   let
  1114     val names = map (fn ((((NONE, _), t), _), _) => t | ((((SOME t, _), _), _), _) => t) args;
  1115     val specs = map (fn ((((_, vs), t), mx), cons) =>
  1116       (vs, t, mx, map (fn ((x, y), z) => (x, y, z)) cons)) args;
  1117   in add_nominal_datatype false names specs end;
  1118 
  1119 val nominal_datatypeP =
  1120   OuterSyntax.command "nominal_datatype" "define inductive datatypes" K.thy_decl
  1121     (P.and_list1 datatype_decl >> (Toplevel.theory o mk_datatype));
  1122 
  1123 val _ = OuterSyntax.add_parsers [nominal_datatypeP];
  1124 
  1125 end;
  1126 
  1127 end
  1128