isabelle: src/HOL/Nominal/nominal_package.ML@3d10de7a8ca7

     1 (* $Id$ *)

     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

     9 structure NominalPackage : NOMINAL_PACKAGE =

    10 struct

    12 open DatatypeAux;

    13 open NominalAtoms;

    15 (** FIXME: DatatypePackage should export this function **)

    17 local

    19 fun dt_recs (DtTFree _) = []

    20   | dt_recs (DtType (_, dts)) = List.concat (map dt_recs dts)

    21   | dt_recs (DtRec i) = [i];

    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 op = (List.concat (map dt_recs args)));

    27   in map (fn (c, _) => space_implode "_" (Sign.base_name c :: bnames)) constrs end;

    30 fun induct_cases descr =

    31   DatatypeProp.indexify_names (List.concat (map (dt_cases descr) (map #2 descr)));

    33 fun exhaust_cases descr i = dt_cases descr (valOf (AList.lookup (op =) descr i));

    35 in

    37 fun mk_case_names_induct descr = RuleCases.case_names (induct_cases descr);

    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);

    43 end;

    45 (*******************************)

    47 val (_ $ (_ $ (_ $ (distinct_f $ _) $ _))) = hd (prems_of distinct_lemma);

    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;

    55 (** taken from HOL/Tools/datatype_aux.ML **)

    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;

    78 fun mk_subgoal i f st =

    79   let

    80     val cg = List.nth (cprems_of st, i - 1);

    81     val g = term_of cg;

    82     val revcut_rl' = Thm.lift_rule cg revcut_rl;

    83     val v = head_of (Logic.strip_assums_concl (hd (prems_of revcut_rl')));

    84     val ps = Logic.strip_params g;

    85     val cert = cterm_of (sign_of_thm st);

    86   in

    87     compose_tac (false,

    88       Thm.instantiate ([], [(cert v, cert (list_abs (ps,

    89         f (rev ps) (Logic.strip_assums_hyp g) (Logic.strip_assums_concl g))))])

    90       revcut_rl', 2) i st

    91   end;

    93 (** simplification procedure for sorting permutations **)

    95 val dj_cp = thm "dj_cp";

    97 fun dest_permT (Type ("fun", [Type ("List.list", [Type ("*", [T, _])]),

    98       Type ("fun", [_, U])])) = (T, U);

   100 fun permTs_of (Const ("nominal.perm", T) $ t $ u) = fst (dest_permT T) :: permTs_of u

   101   | permTs_of _ = [];

   103 fun perm_simproc' thy ss (Const ("nominal.perm", T) $ t $ (u as Const ("nominal.perm", U) $ r $ s)) =

   104       let

   105         val (aT as Type (a, []), S) = dest_permT T;

   106         val (bT as Type (b, []), _) = dest_permT U

   107       in if aT mem permTs_of u andalso aT <> bT then

   108           let

   109             val a' = Sign.base_name a;

   110             val b' = Sign.base_name b;

   111             val cp = PureThy.get_thm thy (Name ("cp_" ^ a' ^ "_" ^ b' ^ "_inst"));

   112             val dj = PureThy.get_thm thy (Name ("dj_" ^ b' ^ "_" ^ a'));

   113             val dj_cp' = [cp, dj] MRS dj_cp;

   114             val cert = SOME o cterm_of thy

   115           in

   116             SOME (mk_meta_eq (Drule.instantiate' [SOME (ctyp_of thy S)]

   117               [cert t, cert r, cert s] dj_cp'))

   118           end

   119         else NONE

   120       end

   121   | perm_simproc' thy ss _ = NONE;

   123 val perm_simproc =

   124   Simplifier.simproc (the_context ()) "perm_simp" ["pi1 \\<bullet> (pi2 \\<bullet> x)"] perm_simproc';

   126 val allE_Nil = read_instantiate_sg (the_context()) [("x", "[]")] allE;

   128 val meta_spec = thm "meta_spec";

   130 fun projections rule =

   131   ProjectRule.projections rule

   132   |> map (standard #> RuleCases.save rule);

   134 fun norm_sort thy = Sorts.norm_sort (snd (#classes (Type.rep_tsig (Sign.tsig_of thy))));

   136 fun gen_add_nominal_datatype prep_typ err flat_names new_type_names dts thy =

   137   let

   138     (* this theory is used just for parsing *)

   140     val tmp_thy = thy |>

   141       Theory.copy |>

   142       Theory.add_types (map (fn (tvs, tname, mx, _) =>

   143         (tname, length tvs, mx)) dts);

   145     val sign = Theory.sign_of tmp_thy;

   147     val atoms = atoms_of thy;

   148     val classes = map (NameSpace.map_base (fn s => "pt_" ^ s)) atoms;

   149     val cp_classes = List.concat (map (fn atom1 => map (fn atom2 =>

   150       Sign.intern_class thy ("cp_" ^ Sign.base_name atom1 ^ "_" ^

   151         Sign.base_name atom2)) atoms) atoms);

   152     fun augment_sort S = S union classes;

   153     val augment_sort_typ = map_type_tfree (fn (s, S) => TFree (s, augment_sort S));

   155     fun prep_constr ((constrs, sorts), (cname, cargs, mx)) =

   156       let val (cargs', sorts') = Library.foldl (prep_typ sign) (([], sorts), cargs)

   157       in (constrs @ [(cname, cargs', mx)], sorts') end

   159     fun prep_dt_spec ((dts, sorts), (tvs, tname, mx, constrs)) =

   160       let val (constrs', sorts') = Library.foldl prep_constr (([], sorts), constrs)

   161       in (dts @ [(tvs, tname, mx, constrs')], sorts') end

   163     val (dts', sorts) = Library.foldl prep_dt_spec (([], []), dts);

   164     val sorts' = map (apsnd augment_sort) sorts;

   165     val tyvars = map #1 dts';

   167     val types_syntax = map (fn (tvs, tname, mx, constrs) => (tname, mx)) dts';

   168     val constr_syntax = map (fn (tvs, tname, mx, constrs) =>

   169       map (fn (cname, cargs, mx) => (cname, mx)) constrs) dts';

   171     val ps = map (fn (_, n, _, _) =>

   172       (Sign.full_name sign n, Sign.full_name sign (n ^ "_Rep"))) dts;

   173     val rps = map Library.swap ps;

   175     fun replace_types (Type ("nominal.ABS", [T, U])) =

   176           Type ("fun", [T, Type ("nominal.noption", [replace_types U])])

   177       | replace_types (Type (s, Ts)) =

   178           Type (getOpt (AList.lookup op = ps s, s), map replace_types Ts)

   179       | replace_types T = T;

   181     fun replace_types' (Type (s, Ts)) =

   182           Type (getOpt (AList.lookup op = rps s, s), map replace_types' Ts)

   183       | replace_types' T = T;

   185     val dts'' = map (fn (tvs, tname, mx, constrs) => (tvs, tname ^ "_Rep", NoSyn,

   186       map (fn (cname, cargs, mx) => (cname,

   187         map (augment_sort_typ o replace_types) cargs, NoSyn)) constrs)) dts';

   189     val new_type_names' = map (fn n => n ^ "_Rep") new_type_names;

   190     val full_new_type_names' = map (Sign.full_name (sign_of thy)) new_type_names';

   192     val ({induction, ...},thy1) =

   193       DatatypePackage.add_datatype_i err flat_names new_type_names' dts'' thy;

   195     val SOME {descr, ...} = Symtab.lookup

   196       (DatatypePackage.get_datatypes thy1) (hd full_new_type_names');

   197     fun nth_dtyp i = typ_of_dtyp descr sorts' (DtRec i);

   199     (**** define permutation functions ****)

   201     val permT = mk_permT (TFree ("'x", HOLogic.typeS));

   202     val pi = Free ("pi", permT);

   203     val perm_types = map (fn (i, _) =>

   204       let val T = nth_dtyp i

   205       in permT --> T --> T end) descr;

   206     val perm_names = replicate (length new_type_names) "nominal.perm" @

   207       DatatypeProp.indexify_names (map (fn i => Sign.full_name (sign_of thy1)

   208         ("perm_" ^ name_of_typ (nth_dtyp i)))

   209           (length new_type_names upto length descr - 1));

   210     val perm_names_types = perm_names ~~ perm_types;

   212     val perm_eqs = List.concat (map (fn (i, (_, _, constrs)) =>

   213       let val T = nth_dtyp i

   214       in map (fn (cname, dts) =>

   215         let

   216           val Ts = map (typ_of_dtyp descr sorts') dts;

   217           val names = DatatypeProp.make_tnames Ts;

   218           val args = map Free (names ~~ Ts);

   219           val c = Const (cname, Ts ---> T);

   220           fun perm_arg (dt, x) =

   221             let val T = type_of x

   222             in if is_rec_type dt then

   223                 let val (Us, _) = strip_type T

   224                 in list_abs (map (pair "x") Us,

   225                   Const (List.nth (perm_names_types, body_index dt)) $ pi $

   226                     list_comb (x, map (fn (i, U) =>

   227                       Const ("nominal.perm", permT --> U --> U) $

   228                         (Const ("List.rev", permT --> permT) $ pi) $

   229                         Bound i) ((length Us - 1 downto 0) ~~ Us)))

   230                 end

   231               else Const ("nominal.perm", permT --> T --> T) $ pi $ x

   232             end;

   233         in

   234           (("", HOLogic.mk_Trueprop (HOLogic.mk_eq

   235             (Const (List.nth (perm_names_types, i)) $

   236                Free ("pi", mk_permT (TFree ("'x", HOLogic.typeS))) $

   237                list_comb (c, args),

   238              list_comb (c, map perm_arg (dts ~~ args))))), [])

   239         end) constrs

   240       end) descr);

   242     val (thy2, perm_simps) = thy1 |>

   243       Theory.add_consts_i (map (fn (s, T) => (Sign.base_name s, T, NoSyn))

   244         (List.drop (perm_names_types, length new_type_names))) |>

   245       PrimrecPackage.add_primrec_i "" perm_eqs;

   247     (**** prove that permutation functions introduced by unfolding are ****)

   248     (**** equivalent to already existing permutation functions         ****)

   250     val _ = warning ("length descr: " ^ string_of_int (length descr));

   251     val _ = warning ("length new_type_names: " ^ string_of_int (length new_type_names));

   253     val perm_indnames = DatatypeProp.make_tnames (map body_type perm_types);

   254     val perm_fun_def = PureThy.get_thm thy2 (Name "perm_fun_def");

   256     val unfolded_perm_eq_thms =

   257       if length descr = length new_type_names then []

   258       else map standard (List.drop (split_conj_thm

   259         (Goal.prove thy2 [] []

   260           (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj

   261             (map (fn (c as (s, T), x) =>

   262                let val [T1, T2] = binder_types T

   263                in HOLogic.mk_eq (Const c $ pi $ Free (x, T2),

   264                  Const ("nominal.perm", T) $ pi $ Free (x, T2))

   265                end)

   266              (perm_names_types ~~ perm_indnames))))

   267           (fn _ => EVERY [indtac induction perm_indnames 1,

   268             ALLGOALS (asm_full_simp_tac

   269               (simpset_of thy2 addsimps [perm_fun_def]))])),

   270         length new_type_names));

   272     (**** prove [] \<bullet> t = t ****)

   274     val _ = warning "perm_empty_thms";

   276     val perm_empty_thms = List.concat (map (fn a =>

   277       let val permT = mk_permT (Type (a, []))

   278       in map standard (List.take (split_conj_thm

   279         (Goal.prove thy2 [] []

   280           (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj

   281             (map (fn ((s, T), x) => HOLogic.mk_eq

   282                 (Const (s, permT --> T --> T) $

   283                    Const ("List.list.Nil", permT) $ Free (x, T),

   284                  Free (x, T)))

   285              (perm_names ~~

   286               map body_type perm_types ~~ perm_indnames))))

   287           (fn _ => EVERY [indtac induction perm_indnames 1,

   288             ALLGOALS (asm_full_simp_tac (simpset_of thy2))])),

   289         length new_type_names))

   290       end)

   291       atoms);

   293     (**** prove (pi1 @ pi2) \<bullet> t = pi1 \<bullet> (pi2 \<bullet> t) ****)

   295     val _ = warning "perm_append_thms";

   297     (*FIXME: these should be looked up statically*)

   298     val at_pt_inst = PureThy.get_thm thy2 (Name "at_pt_inst");

   299     val pt2 = PureThy.get_thm thy2 (Name "pt2");

   301     val perm_append_thms = List.concat (map (fn a =>

   302       let

   303         val permT = mk_permT (Type (a, []));

   304         val pi1 = Free ("pi1", permT);

   305         val pi2 = Free ("pi2", permT);

   306         val pt_inst = PureThy.get_thm thy2 (Name ("pt_" ^ Sign.base_name a ^ "_inst"));

   307         val pt2' = pt_inst RS pt2;

   308         val pt2_ax = PureThy.get_thm thy2

   309           (Name (NameSpace.map_base (fn s => "pt_" ^ s ^ "2") a));

   310       in List.take (map standard (split_conj_thm

   311         (Goal.prove thy2 [] []

   312              (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj

   313                 (map (fn ((s, T), x) =>

   314                     let val perm = Const (s, permT --> T --> T)

   315                     in HOLogic.mk_eq

   316                       (perm $ (Const ("List.op @", permT --> permT --> permT) $

   317                          pi1 $ pi2) $ Free (x, T),

   318                        perm $ pi1 $ (perm $ pi2 $ Free (x, T)))

   319                     end)

   320                   (perm_names ~~

   321                    map body_type perm_types ~~ perm_indnames))))

   322            (fn _ => EVERY [indtac induction perm_indnames 1,

   323               ALLGOALS (asm_full_simp_tac (simpset_of thy2 addsimps [pt2', pt2_ax]))]))),

   324          length new_type_names)

   325       end) atoms);

   327     (**** prove pi1 ~ pi2 ==> pi1 \<bullet> t = pi2 \<bullet> t ****)

   329     val _ = warning "perm_eq_thms";

   331     val pt3 = PureThy.get_thm thy2 (Name "pt3");

   332     val pt3_rev = PureThy.get_thm thy2 (Name "pt3_rev");

   334     val perm_eq_thms = List.concat (map (fn a =>

   335       let

   336         val permT = mk_permT (Type (a, []));

   337         val pi1 = Free ("pi1", permT);

   338         val pi2 = Free ("pi2", permT);

   339         (*FIXME: not robust - better access these theorems using NominalData?*)

   340         val at_inst = PureThy.get_thm thy2 (Name ("at_" ^ Sign.base_name a ^ "_inst"));

   341         val pt_inst = PureThy.get_thm thy2 (Name ("pt_" ^ Sign.base_name a ^ "_inst"));

   342         val pt3' = pt_inst RS pt3;

   343         val pt3_rev' = at_inst RS (pt_inst RS pt3_rev);

   344         val pt3_ax = PureThy.get_thm thy2

   345           (Name (NameSpace.map_base (fn s => "pt_" ^ s ^ "3") a));

   346       in List.take (map standard (split_conj_thm

   347         (Goal.prove thy2 [] [] (Logic.mk_implies

   348              (HOLogic.mk_Trueprop (Const ("nominal.prm_eq",

   349                 permT --> permT --> HOLogic.boolT) $ pi1 $ pi2),

   350               HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj

   351                 (map (fn ((s, T), x) =>

   352                     let val perm = Const (s, permT --> T --> T)

   353                     in HOLogic.mk_eq

   354                       (perm $ pi1 $ Free (x, T),

   355                        perm $ pi2 $ Free (x, T))

   356                     end)

   357                   (perm_names ~~

   358                    map body_type perm_types ~~ perm_indnames)))))

   359            (fn _ => EVERY [indtac induction perm_indnames 1,

   360               ALLGOALS (asm_full_simp_tac (simpset_of thy2 addsimps [pt3', pt3_rev', pt3_ax]))]))),

   361          length new_type_names)

   362       end) atoms);

   364     (**** prove pi1 \<bullet> (pi2 \<bullet> t) = (pi1 \<bullet> pi2) \<bullet> (pi1 \<bullet> t) ****)

   366     val cp1 = PureThy.get_thm thy2 (Name "cp1");

   367     val dj_cp = PureThy.get_thm thy2 (Name "dj_cp");

   368     val pt_perm_compose = PureThy.get_thm thy2 (Name "pt_perm_compose");

   369     val pt_perm_compose_rev = PureThy.get_thm thy2 (Name "pt_perm_compose_rev");

   370     val dj_perm_perm_forget = PureThy.get_thm thy2 (Name "dj_perm_perm_forget");

   372     fun composition_instance name1 name2 thy =

   373       let

   374         val name1' = Sign.base_name name1;

   375         val name2' = Sign.base_name name2;

   376         val cp_class = Sign.intern_class thy ("cp_" ^ name1' ^ "_" ^ name2');

   377         val permT1 = mk_permT (Type (name1, []));

   378         val permT2 = mk_permT (Type (name2, []));

   379         val augment = map_type_tfree

   380           (fn (x, S) => TFree (x, cp_class :: S));

   381         val Ts = map (augment o body_type) perm_types;

   382         val cp_inst = PureThy.get_thm thy

   383           (Name ("cp_" ^ name1' ^ "_" ^ name2' ^ "_inst"));

   384         val simps = simpset_of thy addsimps (perm_fun_def ::

   385           (if name1 <> name2 then

   386              let val dj = PureThy.get_thm thy (Name ("dj_" ^ name2' ^ "_" ^ name1'))

   387              in [dj RS (cp_inst RS dj_cp), dj RS dj_perm_perm_forget] end

   388            else

   389              let

   390                val at_inst = PureThy.get_thm thy (Name ("at_" ^ name1' ^ "_inst"));

   391                val pt_inst = PureThy.get_thm thy (Name ("pt_" ^ name1' ^ "_inst"))

   392              in

   393                [cp_inst RS cp1 RS sym,

   394                 at_inst RS (pt_inst RS pt_perm_compose) RS sym,

   395                 at_inst RS (pt_inst RS pt_perm_compose_rev) RS sym]

   396             end))

   397         val thms = split_conj_thm (standard (Goal.prove thy [] []

   398             (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj

   399               (map (fn ((s, T), x) =>

   400                   let

   401                     val pi1 = Free ("pi1", permT1);

   402                     val pi2 = Free ("pi2", permT2);

   403                     val perm1 = Const (s, permT1 --> T --> T);

   404                     val perm2 = Const (s, permT2 --> T --> T);

   405                     val perm3 = Const ("nominal.perm", permT1 --> permT2 --> permT2)

   406                   in HOLogic.mk_eq

   407                     (perm1 $ pi1 $ (perm2 $ pi2 $ Free (x, T)),

   408                      perm2 $ (perm3 $ pi1 $ pi2) $ (perm1 $ pi1 $ Free (x, T)))

   409                   end)

   410                 (perm_names ~~ Ts ~~ perm_indnames))))

   411           (fn _ => EVERY [indtac induction perm_indnames 1,

   412              ALLGOALS (asm_full_simp_tac simps)])))

   413       in

   414         foldl (fn ((s, tvs), thy) => AxClass.prove_arity

   415             (s, replicate (length tvs) (cp_class :: classes), [cp_class])

   416             (ClassPackage.intro_classes_tac [] THEN ALLGOALS (resolve_tac thms)) thy)

   417           thy (full_new_type_names' ~~ tyvars)

   418       end;

   420     val (perm_thmss,thy3) = thy2 |>

   421       fold (fn name1 => fold (composition_instance name1) atoms) atoms |>

   422       curry (Library.foldr (fn ((i, (tyname, args, _)), thy) =>

   423         AxClass.prove_arity (tyname, replicate (length args) classes, classes)

   424         (ClassPackage.intro_classes_tac [] THEN REPEAT (EVERY

   425            [resolve_tac perm_empty_thms 1,

   426             resolve_tac perm_append_thms 1,

   427             resolve_tac perm_eq_thms 1, assume_tac 1])) thy))

   428         (List.take (descr, length new_type_names)) |>

   429       PureThy.add_thmss

   430         [((space_implode "_" new_type_names ^ "_unfolded_perm_eq",

   431           unfolded_perm_eq_thms), [Simplifier.simp_add]),

   432          ((space_implode "_" new_type_names ^ "_perm_empty",

   433           perm_empty_thms), [Simplifier.simp_add]),

   434          ((space_implode "_" new_type_names ^ "_perm_append",

   435           perm_append_thms), [Simplifier.simp_add]),

   436          ((space_implode "_" new_type_names ^ "_perm_eq",

   437           perm_eq_thms), [Simplifier.simp_add])];

   439     (**** Define representing sets ****)

   441     val _ = warning "representing sets";

   443     val rep_set_names = map (Sign.full_name thy3) (DatatypeProp.indexify_names

   444       (map (fn (i, _) => name_of_typ (nth_dtyp i) ^ "_set") descr));

   445     val big_rep_name =

   446       space_implode "_" (DatatypeProp.indexify_names (List.mapPartial

   447         (fn (i, ("nominal.noption", _, _)) => NONE

   448           | (i, _) => SOME (name_of_typ (nth_dtyp i))) descr)) ^ "_set";

   449     val _ = warning ("big_rep_name: " ^ big_rep_name);

   451     fun strip_option (dtf as DtType ("fun", [dt, DtRec i])) =

   452           (case AList.lookup op = descr i of

   453              SOME ("nominal.noption", _, [(_, [dt']), _]) =>

   454                apfst (cons dt) (strip_option dt')

   455            | _ => ([], dtf))

   456       | strip_option (DtType ("fun", [dt, DtType ("nominal.noption", [dt'])])) =

   457           apfst (cons dt) (strip_option dt')

   458       | strip_option dt = ([], dt);

   460     val dt_atomTs = distinct op = (map (typ_of_dtyp descr sorts')

   461       (List.concat (map (fn (_, (_, _, cs)) => List.concat

   462         (map (List.concat o map (fst o strip_option) o snd) cs)) descr)));

   464     fun make_intr s T (cname, cargs) =

   465       let

   466         fun mk_prem (dt, (j, j', prems, ts)) =

   467           let

   468             val (dts, dt') = strip_option dt;

   469             val (dts', dt'') = strip_dtyp dt';

   470             val Ts = map (typ_of_dtyp descr sorts') dts;

   471             val Us = map (typ_of_dtyp descr sorts') dts';

   472             val T = typ_of_dtyp descr sorts' dt'';

   473             val free = mk_Free "x" (Us ---> T) j;

   474             val free' = app_bnds free (length Us);

   475             fun mk_abs_fun (T, (i, t)) =

   476               let val U = fastype_of t

   477               in (i + 1, Const ("nominal.abs_fun", [T, U, T] --->

   478                 Type ("nominal.noption", [U])) $ mk_Free "y" T i $ t)

   479               end

   480           in (j + 1, j' + length Ts,

   481             case dt'' of

   482                 DtRec k => list_all (map (pair "x") Us,

   483                   HOLogic.mk_Trueprop (HOLogic.mk_mem (free',

   484                     Const (List.nth (rep_set_names, k),

   485                       HOLogic.mk_setT T)))) :: prems

   486               | _ => prems,

   487             snd (foldr mk_abs_fun (j', free) Ts) :: ts)

   488           end;

   490         val (_, _, prems, ts) = foldr mk_prem (1, 1, [], []) cargs;

   491         val concl = HOLogic.mk_Trueprop (HOLogic.mk_mem

   492           (list_comb (Const (cname, map fastype_of ts ---> T), ts),

   493            Const (s, HOLogic.mk_setT T)))

   494       in Logic.list_implies (prems, concl)

   495       end;

   497     val (intr_ts, ind_consts) =

   498       apfst List.concat (ListPair.unzip (List.mapPartial

   499         (fn ((_, ("nominal.noption", _, _)), _) => NONE

   500           | ((i, (_, _, constrs)), rep_set_name) =>

   501               let val T = nth_dtyp i

   502               in SOME (map (make_intr rep_set_name T) constrs,

   503                 Const (rep_set_name, HOLogic.mk_setT T))

   504               end)

   505                 (descr ~~ rep_set_names)));

   507     val (thy4, {raw_induct = rep_induct, intrs = rep_intrs, ...}) =

   508       setmp InductivePackage.quiet_mode false

   509         (InductivePackage.add_inductive_i false true big_rep_name false true false

   510            ind_consts (map (fn x => (("", x), [])) intr_ts) []) thy3;

   512     (**** Prove that representing set is closed under permutation ****)

   514     val _ = warning "proving closure under permutation...";

   516     val perm_indnames' = List.mapPartial

   517       (fn (x, (_, ("nominal.noption", _, _))) => NONE | (x, _) => SOME x)

   518       (perm_indnames ~~ descr);

   520     fun mk_perm_closed name = map (fn th => standard (th RS mp))

   521       (List.take (split_conj_thm (Goal.prove thy4 [] []

   522         (HOLogic.mk_Trueprop (foldr1 HOLogic.mk_conj (map

   523            (fn (S, x) =>

   524               let

   525                 val S = map_term_types (map_type_tfree

   526                   (fn (s, cs) => TFree (s, cs union cp_classes))) S;

   527                 val T = HOLogic.dest_setT (fastype_of S);

   528                 val permT = mk_permT (Type (name, []))

   529               in HOLogic.mk_imp (HOLogic.mk_mem (Free (x, T), S),

   530                 HOLogic.mk_mem (Const ("nominal.perm", permT --> T --> T) $

   531                   Free ("pi", permT) $ Free (x, T), S))

   532               end) (ind_consts ~~ perm_indnames'))))

   533         (fn _ => EVERY (* CU: added perm_fun_def in the final tactic in order to deal with funs *)

   534            [indtac rep_induct [] 1,

   535             ALLGOALS (simp_tac (simpset_of thy4 addsimps

   536               (symmetric perm_fun_def :: PureThy.get_thms thy4 (Name ("abs_perm"))))),

   537             ALLGOALS (resolve_tac rep_intrs

   538                THEN_ALL_NEW (asm_full_simp_tac (simpset_of thy4 addsimps [perm_fun_def])))])),

   539         length new_type_names));

   541     (* FIXME: theorems are stored in database for testing only *)

   542     val perm_closed_thmss = map mk_perm_closed atoms;

   543     val (_,thy5) = PureThy.add_thmss [(("perm_closed", List.concat perm_closed_thmss), [])] thy4;

   545     (**** typedef ****)

   547     val _ = warning "defining type...";

   549     val (typedefs, thy6) =

   550       fold_map (fn ((((name, mx), tvs), c), name') => fn thy =>

   551         setmp TypedefPackage.quiet_mode true

   552           (TypedefPackage.add_typedef_i false (SOME name') (name, tvs, mx) c NONE

   553             (rtac exI 1 THEN

   554               QUIET_BREADTH_FIRST (has_fewer_prems 1)

   555               (resolve_tac rep_intrs 1))) thy |> (fn (thy, r) =>

   556         let

   557           val permT = mk_permT (TFree (variant tvs "'a", HOLogic.typeS));

   558           val pi = Free ("pi", permT);

   559           val tvs' = map (fn s => TFree (s, the (AList.lookup op = sorts' s))) tvs;

   560           val T = Type (Sign.intern_type thy name, tvs');

   561           val Const (_, Type (_, [U])) = c

   562         in apfst (pair r o hd)

   563           (PureThy.add_defs_i true [(("prm_" ^ name ^ "_def", Logic.mk_equals

   564             (Const ("nominal.perm", permT --> T --> T) $ pi $ Free ("x", T),

   565              Const (Sign.intern_const thy ("Abs_" ^ name), U --> T) $

   566                (Const ("nominal.perm", permT --> U --> U) $ pi $

   567                  (Const (Sign.intern_const thy ("Rep_" ^ name), T --> U) $

   568                    Free ("x", T))))), [])] thy)

   569         end))

   570           (types_syntax ~~ tyvars ~~

   571             (List.take (ind_consts, length new_type_names)) ~~ new_type_names) thy5;

   573     val perm_defs = map snd typedefs;

   574     val Abs_inverse_thms = map (#Abs_inverse o fst) typedefs;

   575     val Rep_inverse_thms = map (#Rep_inverse o fst) typedefs;

   576     val Rep_thms = map (#Rep o fst) typedefs;

   578     val big_name = space_implode "_" new_type_names;

   581     (** prove that new types are in class pt_<name> **)

   583     val _ = warning "prove that new types are in class pt_<name> ...";

   585     fun pt_instance ((class, atom), perm_closed_thms) =

   586       fold (fn (((({Abs_inverse, Rep_inverse, Rep, ...},

   587         perm_def), name), tvs), perm_closed) => fn thy =>

   588           AxClass.prove_arity

   589             (Sign.intern_type thy name,

   590               replicate (length tvs) (classes @ cp_classes), [class])

   591             (EVERY [ClassPackage.intro_classes_tac [],

   592               rewrite_goals_tac [perm_def],

   593               asm_full_simp_tac (simpset_of thy addsimps [Rep_inverse]) 1,

   594               asm_full_simp_tac (simpset_of thy addsimps

   595                 [Rep RS perm_closed RS Abs_inverse]) 1,

   596               asm_full_simp_tac (HOL_basic_ss addsimps [PureThy.get_thm thy

   597                 (Name ("pt_" ^ Sign.base_name atom ^ "3"))]) 1]) thy)

   598         (typedefs ~~ new_type_names ~~ tyvars ~~ perm_closed_thms);

   601     (** prove that new types are in class cp_<name1>_<name2> **)

   603     val _ = warning "prove that new types are in class cp_<name1>_<name2> ...";

   605     fun cp_instance (atom1, perm_closed_thms1) (atom2, perm_closed_thms2) thy =

   606       let

   607         val name = "cp_" ^ Sign.base_name atom1 ^ "_" ^ Sign.base_name atom2;

   608         val class = Sign.intern_class thy name;

   609         val cp1' = PureThy.get_thm thy (Name (name ^ "_inst")) RS cp1

   610       in fold (fn ((((({Abs_inverse, Rep_inverse, Rep, ...},

   611         perm_def), name), tvs), perm_closed1), perm_closed2) => fn thy =>

   612           AxClass.prove_arity

   613             (Sign.intern_type thy name,

   614               replicate (length tvs) (classes @ cp_classes), [class])

   615             (EVERY [ClassPackage.intro_classes_tac [],

   616               rewrite_goals_tac [perm_def],

   617               asm_full_simp_tac (simpset_of thy addsimps

   618                 ((Rep RS perm_closed1 RS Abs_inverse) ::

   619                  (if atom1 = atom2 then []

   620                   else [Rep RS perm_closed2 RS Abs_inverse]))) 1,

   621               cong_tac 1,

   622               rtac refl 1,

   623               rtac cp1' 1]) thy)

   624         (typedefs ~~ new_type_names ~~ tyvars ~~ perm_closed_thms1 ~~

   625           perm_closed_thms2) thy

   626       end;

   628     val thy7 = fold (fn x => fn thy => thy |>

   629       pt_instance x |>

   630       fold (cp_instance (apfst snd x)) (atoms ~~ perm_closed_thmss))

   631         (classes ~~ atoms ~~ perm_closed_thmss) thy6;

   633     (**** constructors ****)

   635     fun mk_abs_fun (x, t) =

   636       let

   637         val T = fastype_of x;

   638         val U = fastype_of t

   639       in

   640         Const ("nominal.abs_fun", T --> U --> T -->

   641           Type ("nominal.noption", [U])) $ x $ t

   642       end;

   644     val (ty_idxs, _) = foldl

   645       (fn ((i, ("nominal.noption", _, _)), p) => p

   646         | ((i, _), (ty_idxs, j)) => (ty_idxs @ [(i, j)], j + 1)) ([], 0) descr;

   648     fun reindex (DtType (s, dts)) = DtType (s, map reindex dts)

   649       | reindex (DtRec i) = DtRec (the (AList.lookup op = ty_idxs i))

   650       | reindex dt = dt;

   652     fun strip_suffix i s = implode (List.take (explode s, size s - i));

   654     (** strips the "_Rep" in type names *)

   655     fun strip_nth_name i s =

   656       let val xs = NameSpace.unpack s;

   657       in NameSpace.pack (Library.nth_map (length xs - i) (strip_suffix 4) xs) end;

   659     val (descr'', ndescr) = ListPair.unzip (List.mapPartial

   660       (fn (i, ("nominal.noption", _, _)) => NONE

   661         | (i, (s, dts, constrs)) =>

   662              let

   663                val SOME index = AList.lookup op = ty_idxs i;

   664                val (constrs1, constrs2) = ListPair.unzip

   665                  (map (fn (cname, cargs) => apfst (pair (strip_nth_name 2 cname))

   666                    (foldl_map (fn (dts, dt) =>

   667                      let val (dts', dt') = strip_option dt

   668                      in (dts @ dts' @ [reindex dt'], (length dts, length dts')) end)

   669                        ([], cargs))) constrs)

   670              in SOME ((index, (strip_nth_name 1 s,  map reindex dts, constrs1)),

   671                (index, constrs2))

   672              end) descr);

   674     val (descr1, descr2) = splitAt (length new_type_names, descr'');

   675     val descr' = [descr1, descr2];

   677     val typ_of_dtyp' = replace_types' o typ_of_dtyp descr sorts';

   679     val rep_names = map (fn s =>

   680       Sign.intern_const thy7 ("Rep_" ^ s)) new_type_names;

   681     val abs_names = map (fn s =>

   682       Sign.intern_const thy7 ("Abs_" ^ s)) new_type_names;

   684     val recTs' = List.mapPartial

   685       (fn ((_, ("nominal.noption", _, _)), T) => NONE

   686         | (_, T) => SOME T) (descr ~~ get_rec_types descr sorts');

   687     val recTs = get_rec_types descr'' sorts';

   688     val newTs' = Library.take (length new_type_names, recTs');

   689     val newTs = Library.take (length new_type_names, recTs);

   691     val full_new_type_names = map (Sign.full_name (sign_of thy)) new_type_names;

   693     fun make_constr_def tname T T' ((thy, defs, eqns), ((cname, cargs), (cname', mx))) =

   694       let

   695         fun constr_arg (dt, (j, l_args, r_args)) =

   696           let

   697             val x' = mk_Free "x" (typ_of_dtyp' dt) j;

   698             val (dts, dt') = strip_option dt;

   699             val xs = map (fn (dt, i) => mk_Free "x" (typ_of_dtyp' dt) i)

   700               (dts ~~ (j upto j + length dts - 1))

   701             val x = mk_Free "x" (typ_of_dtyp' dt') (j + length dts)

   702             val (dts', dt'') = strip_dtyp dt'

   703           in

   704             (j + length dts + 1,

   705              xs @ x :: l_args,

   706              foldr mk_abs_fun

   707                (case dt'' of

   708                   DtRec k => if k < length new_type_names then

   709                       list_abs (map (pair "z" o typ_of_dtyp') dts',

   710                         Const (List.nth (rep_names, k), typ_of_dtyp' dt'' -->

   711                           typ_of_dtyp descr sorts' dt'') $ app_bnds x (length dts'))

   712                     else error "nested recursion not (yet) supported"

   713                 | _ => x) xs :: r_args)

   714           end

   716         val (_, l_args, r_args) = foldr constr_arg (1, [], []) cargs;

   717         val abs_name = Sign.intern_const (Theory.sign_of thy) ("Abs_" ^ tname);

   718         val rep_name = Sign.intern_const (Theory.sign_of thy) ("Rep_" ^ tname);

   719         val constrT = map fastype_of l_args ---> T;

   720         val lhs = list_comb (Const (Sign.full_name thy (Sign.base_name cname),

   721           constrT), l_args);

   722         val rhs = list_comb (Const (cname, map fastype_of r_args ---> T'), r_args);

   723         val def = Logic.mk_equals (lhs, Const (abs_name, T' --> T) $ rhs);

   724         val eqn = HOLogic.mk_Trueprop (HOLogic.mk_eq

   725           (Const (rep_name, T --> T') $ lhs, rhs));

   726         val def_name = (Sign.base_name cname) ^ "_def";

   727         val ([def_thm], thy') = thy |>

   728           Theory.add_consts_i [(cname', constrT, mx)] |>

   729           (PureThy.add_defs_i false o map Thm.no_attributes) [(def_name, def)]

   730       in (thy', defs @ [def_thm], eqns @ [eqn]) end;

   732     fun dt_constr_defs ((thy, defs, eqns, dist_lemmas),

   733         (((((_, (_, _, constrs)), tname), T), T'), constr_syntax)) =

   734       let

   735         val rep_const = cterm_of thy

   736           (Const (Sign.intern_const thy ("Rep_" ^ tname), T --> T'));

   737         val dist = standard (cterm_instantiate [(cterm_of thy distinct_f, rep_const)] distinct_lemma);

   738         val (thy', defs', eqns') = Library.foldl (make_constr_def tname T T')

   739           ((Theory.add_path tname thy, defs, []), constrs ~~ constr_syntax)

   740       in

   741         (parent_path flat_names thy', defs', eqns @ [eqns'], dist_lemmas @ [dist])

   742       end;

   744     val (thy8, constr_defs, constr_rep_eqns, dist_lemmas) = Library.foldl dt_constr_defs

   745       ((thy7, [], [], []), List.take (descr, length new_type_names) ~~

   746         new_type_names ~~ newTs ~~ newTs' ~~ constr_syntax);

   748     val abs_inject_thms = map (fn tname =>

   749       PureThy.get_thm thy8 (Name ("Abs_" ^ tname ^ "_inject"))) new_type_names;

   751     val rep_inject_thms = map (fn tname =>

   752       PureThy.get_thm thy8 (Name ("Rep_" ^ tname ^ "_inject"))) new_type_names;

   754     val rep_thms = map (fn tname =>

   755       PureThy.get_thm thy8 (Name ("Rep_" ^ tname))) new_type_names;

   757     val rep_inverse_thms = map (fn tname =>

   758       PureThy.get_thm thy8 (Name ("Rep_" ^ tname ^ "_inverse"))) new_type_names;

   760     (* prove theorem  Rep_i (Constr_j ...) = Constr'_j ...  *)

   762     fun prove_constr_rep_thm eqn =

   763       let

   764         val inj_thms = map (fn r => r RS iffD1) abs_inject_thms;

   765         val rewrites = constr_defs @ map mk_meta_eq rep_inverse_thms

   766       in standard (Goal.prove thy8 [] [] eqn (fn _ => EVERY

   767         [resolve_tac inj_thms 1,

   768          rewrite_goals_tac rewrites,

   769          rtac refl 3,

   770          resolve_tac rep_intrs 2,

   771          REPEAT (resolve_tac rep_thms 1)]))

   772       end;

   774     val constr_rep_thmss = map (map prove_constr_rep_thm) constr_rep_eqns;

   776     (* prove theorem  pi \<bullet> Rep_i x = Rep_i (pi \<bullet> x) *)

   778     fun prove_perm_rep_perm (atom, perm_closed_thms) = map (fn th =>

   779       let

   780         val _ $ (_ $ (Rep $ x) $ _) = Logic.unvarify (prop_of th);

   781         val Type ("fun", [T, U]) = fastype_of Rep;

   782         val permT = mk_permT (Type (atom, []));

   783         val pi = Free ("pi", permT);

   784       in

   785         standard (Goal.prove thy8 [] [] (HOLogic.mk_Trueprop (HOLogic.mk_eq

   786             (Const ("nominal.perm", permT --> U --> U) $ pi $ (Rep $ x),

   787              Rep $ (Const ("nominal.perm", permT --> T --> T) $ pi $ x))))

   788           (fn _ => simp_tac (HOL_basic_ss addsimps (perm_defs @ Abs_inverse_thms @

   789             perm_closed_thms @ Rep_thms)) 1))

   790       end) Rep_thms;

   792     val perm_rep_perm_thms = List.concat (map prove_perm_rep_perm

   793       (atoms ~~ perm_closed_thmss));

   795     (* prove distinctness theorems *)

   797     val distinct_props = setmp DatatypeProp.dtK 1000

   798       (DatatypeProp.make_distincts new_type_names descr' sorts') thy8;

   800     val dist_rewrites = map (fn (rep_thms, dist_lemma) =>

   801       dist_lemma::(rep_thms @ [In0_eq, In1_eq, In0_not_In1, In1_not_In0]))

   802         (constr_rep_thmss ~~ dist_lemmas);

   804     fun prove_distinct_thms (_, []) = []

   805       | prove_distinct_thms (p as (rep_thms, dist_lemma), t::ts) =

   806           let

   807             val dist_thm = standard (Goal.prove thy8 [] [] t (fn _ =>

   808               simp_tac (simpset_of thy8 addsimps (dist_lemma :: rep_thms)) 1))

   809           in dist_thm::(standard (dist_thm RS not_sym))::

   810             (prove_distinct_thms (p, ts))

   811           end;

   813     val distinct_thms = map prove_distinct_thms

   814       (constr_rep_thmss ~~ dist_lemmas ~~ distinct_props);

   816     (** prove equations for permutation functions **)

   818     val abs_perm = PureThy.get_thms thy8 (Name "abs_perm"); (* FIXME *)

   820     val perm_simps' = map (fn (((i, (_, _, constrs)), tname), constr_rep_thms) =>

   821       let val T = replace_types' (nth_dtyp i)

   822       in List.concat (map (fn (atom, perm_closed_thms) =>

   823           map (fn ((cname, dts), constr_rep_thm) =>

   824         let

   825           val cname = Sign.intern_const thy8

   826             (NameSpace.append tname (Sign.base_name cname));

   827           val permT = mk_permT (Type (atom, []));

   828           val pi = Free ("pi", permT);

   830           fun perm t =

   831             let val T = fastype_of t

   832             in Const ("nominal.perm", permT --> T --> T) $ pi $ t end;

   834           fun constr_arg (dt, (j, l_args, r_args)) =

   835             let

   836               val x' = mk_Free "x" (typ_of_dtyp' dt) j;

   837               val (dts, dt') = strip_option dt;

   838               val Ts = map typ_of_dtyp' dts;

   839               val xs = map (fn (T, i) => mk_Free "x" T i)

   840                 (Ts ~~ (j upto j + length dts - 1))

   841               val x = mk_Free "x" (typ_of_dtyp' dt') (j + length dts);

   842               val (dts', dt'') = strip_dtyp dt';

   843             in

   844               (j + length dts + 1,

   845                xs @ x :: l_args,

   846                map perm (xs @ [x]) @ r_args)

   847             end

   849           val (_, l_args, r_args) = foldr constr_arg (1, [], []) dts;

   850           val c = Const (cname, map fastype_of l_args ---> T)

   851         in

   852           standard (Goal.prove thy8 [] []

   853             (HOLogic.mk_Trueprop (HOLogic.mk_eq

   854               (perm (list_comb (c, l_args)), list_comb (c, r_args))))

   855             (fn _ => EVERY

   856               [simp_tac (simpset_of thy8 addsimps (constr_rep_thm :: perm_defs)) 1,

   857                simp_tac (HOL_basic_ss addsimps (Rep_thms @ Abs_inverse_thms @

   858                  constr_defs @ perm_closed_thms)) 1,

   859                TRY (simp_tac (HOL_basic_ss addsimps

   860                  (symmetric perm_fun_def :: abs_perm)) 1),

   861                TRY (simp_tac (HOL_basic_ss addsimps

   862                  (perm_fun_def :: perm_defs @ Rep_thms @ Abs_inverse_thms @

   863                     perm_closed_thms)) 1)]))

   864         end) (constrs ~~ constr_rep_thms)) (atoms ~~ perm_closed_thmss))

   865       end) (List.take (descr, length new_type_names) ~~ new_type_names ~~ constr_rep_thmss);

   867     (** prove injectivity of constructors **)

   869     val rep_inject_thms' = map (fn th => th RS sym) rep_inject_thms;

   870     val alpha = PureThy.get_thms thy8 (Name "alpha");

   871     val abs_fresh = PureThy.get_thms thy8 (Name "abs_fresh");

   872     val fresh_def = PureThy.get_thm thy8 (Name "fresh_def");

   873     val supp_def = PureThy.get_thm thy8 (Name "supp_def");

   875     val inject_thms = map (fn (((i, (_, _, constrs)), tname), constr_rep_thms) =>

   876       let val T = replace_types' (nth_dtyp i)

   877       in List.mapPartial (fn ((cname, dts), constr_rep_thm) =>

   878         if null dts then NONE else SOME

   879         let

   880           val cname = Sign.intern_const thy8

   881             (NameSpace.append tname (Sign.base_name cname));

   883           fun make_inj (dt, (j, args1, args2, eqs)) =

   884             let

   885               val x' = mk_Free "x" (typ_of_dtyp' dt) j;

   886               val y' = mk_Free "y" (typ_of_dtyp' dt) j;

   887               val (dts, dt') = strip_option dt;

   888               val Ts_idx = map typ_of_dtyp' dts ~~ (j upto j + length dts - 1);

   889               val xs = map (fn (T, i) => mk_Free "x" T i) Ts_idx;

   890               val ys = map (fn (T, i) => mk_Free "y" T i) Ts_idx;

   891               val x = mk_Free "x" (typ_of_dtyp' dt') (j + length dts);

   892               val y = mk_Free "y" (typ_of_dtyp' dt') (j + length dts);

   893               val (dts', dt'') = strip_dtyp dt';

   894             in

   895               (j + length dts + 1,

   896                xs @ (x :: args1), ys @ (y :: args2),

   897                HOLogic.mk_eq

   898                  (foldr mk_abs_fun x xs, foldr mk_abs_fun y ys) :: eqs)

   899             end;

   901           val (_, args1, args2, eqs) = foldr make_inj (1, [], [], []) dts;

   902           val Ts = map fastype_of args1;

   903           val c = Const (cname, Ts ---> T)

   904         in

   905           standard (Goal.prove thy8 [] [] (HOLogic.mk_Trueprop (HOLogic.mk_eq

   906               (HOLogic.mk_eq (list_comb (c, args1), list_comb (c, args2)),

   907                foldr1 HOLogic.mk_conj eqs)))

   908             (fn _ => EVERY

   909                [asm_full_simp_tac (simpset_of thy8 addsimps (constr_rep_thm ::

   910                   rep_inject_thms')) 1,

   911                 TRY (asm_full_simp_tac (HOL_basic_ss addsimps (fresh_def :: supp_def ::

   912                   alpha @ abs_perm @ abs_fresh @ rep_inject_thms @

   913                   perm_rep_perm_thms)) 1),

   914                 TRY (asm_full_simp_tac (HOL_basic_ss addsimps (perm_fun_def ::

   915                   expand_fun_eq :: rep_inject_thms @ perm_rep_perm_thms)) 1)]))

   916         end) (constrs ~~ constr_rep_thms)

   917       end) (List.take (descr, length new_type_names) ~~ new_type_names ~~ constr_rep_thmss);

   919     (** equations for support and freshness **)

   921     val Un_assoc = PureThy.get_thm thy8 (Name "Un_assoc");

   922     val de_Morgan_conj = PureThy.get_thm thy8 (Name "de_Morgan_conj");

   923     val Collect_disj_eq = PureThy.get_thm thy8 (Name "Collect_disj_eq");

   924     val finite_Un = PureThy.get_thm thy8 (Name "finite_Un");

   926     val (supp_thms, fresh_thms) = ListPair.unzip (map ListPair.unzip

   927       (map (fn ((((i, (_, _, constrs)), tname), inject_thms'), perm_thms') =>

   928       let val T = replace_types' (nth_dtyp i)

   929       in List.concat (map (fn (cname, dts) => map (fn atom =>

   930         let

   931           val cname = Sign.intern_const thy8

   932             (NameSpace.append tname (Sign.base_name cname));

   933           val atomT = Type (atom, []);

   935           fun process_constr (dt, (j, args1, args2)) =

   936             let

   937               val x' = mk_Free "x" (typ_of_dtyp' dt) j;

   938               val (dts, dt') = strip_option dt;

   939               val Ts_idx = map typ_of_dtyp' dts ~~ (j upto j + length dts - 1);

   940               val xs = map (fn (T, i) => mk_Free "x" T i) Ts_idx;

   941               val x = mk_Free "x" (typ_of_dtyp' dt') (j + length dts);

   942               val (dts', dt'') = strip_dtyp dt';

   943             in

   944               (j + length dts + 1,

   945                xs @ (x :: args1), foldr mk_abs_fun x xs :: args2)

   946             end;

   948           val (_, args1, args2) = foldr process_constr (1, [], []) dts;

   949           val Ts = map fastype_of args1;

   950           val c = list_comb (Const (cname, Ts ---> T), args1);

   951           fun supp t =

   952             Const ("nominal.supp", fastype_of t --> HOLogic.mk_setT atomT) $ t;

   953           fun fresh t =

   954             Const ("nominal.fresh", atomT --> fastype_of t --> HOLogic.boolT) $

   955               Free ("a", atomT) $ t;

   956           val supp_thm = standard (Goal.prove thy8 [] []

   957               (HOLogic.mk_Trueprop (HOLogic.mk_eq

   958                 (supp c,

   959                  if null dts then Const ("{}", HOLogic.mk_setT atomT)

   960                  else foldr1 (HOLogic.mk_binop "op Un") (map supp args2))))

   961             (fn _ =>

   962               simp_tac (HOL_basic_ss addsimps (supp_def ::

   963                  Un_assoc :: de_Morgan_conj :: Collect_disj_eq :: finite_Un ::

   964                  symmetric empty_def :: Finites.emptyI :: simp_thms @

   965                  abs_perm @ abs_fresh @ inject_thms' @ perm_thms')) 1))

   966         in

   967           (supp_thm,

   968            standard (Goal.prove thy8 [] [] (HOLogic.mk_Trueprop (HOLogic.mk_eq

   969               (fresh c,

   970                if null dts then HOLogic.true_const

   971                else foldr1 HOLogic.mk_conj (map fresh args2))))

   972              (fn _ =>

   973                simp_tac (simpset_of thy8 addsimps [fresh_def, supp_thm]) 1)))

   974         end) atoms) constrs)

   975       end) (List.take (descr, length new_type_names) ~~ new_type_names ~~ inject_thms ~~ perm_simps')));

   977     (**** weak induction theorem ****)

   979     val arities = get_arities descr'';

   981     fun mk_funs_inv thm =

   982       let

   983         val {sign, prop, ...} = rep_thm thm;

   984         val _ $ (_ $ (Const (_, Type (_, [U, _])) $ _ $ S)) $

   985           (_ $ (_ $ (r $ (a $ _)) $ _)) = Type.freeze prop;

   986         val used = add_term_tfree_names (a, []);

   988         fun mk_thm i =

   989           let

   990             val Ts = map (TFree o rpair HOLogic.typeS)

   991               (variantlist (replicate i "'t", used));

   992             val f = Free ("f", Ts ---> U)

   993           in standard (Goal.prove sign [] [] (Logic.mk_implies

   994             (HOLogic.mk_Trueprop (HOLogic.list_all

   995                (map (pair "x") Ts, HOLogic.mk_mem (app_bnds f i, S))),

   996              HOLogic.mk_Trueprop (HOLogic.mk_eq (list_abs (map (pair "x") Ts,

   997                r $ (a $ app_bnds f i)), f))))

   998             (fn _ => EVERY [REPEAT (rtac ext 1), REPEAT (etac allE 1), rtac thm 1, atac 1]))

   999           end

  1000       in map (fn r => r RS subst) (thm :: map mk_thm arities) end;

  1002     fun mk_indrule_lemma ((prems, concls), (((i, _), T), U)) =

  1003       let

  1004         val Rep_t = Const (List.nth (rep_names, i), T --> U) $

  1005           mk_Free "x" T i;

  1007         val Abs_t =  Const (List.nth (abs_names, i), U --> T)

  1009       in (prems @ [HOLogic.imp $ HOLogic.mk_mem (Rep_t,

  1010             Const (List.nth (rep_set_names, i), HOLogic.mk_setT U)) $

  1011               (mk_Free "P" (T --> HOLogic.boolT) (i + 1) $ (Abs_t $ Rep_t))],

  1012           concls @ [mk_Free "P" (T --> HOLogic.boolT) (i + 1) $ mk_Free "x" T i])

  1013       end;

  1015     val (indrule_lemma_prems, indrule_lemma_concls) =

  1016       Library.foldl mk_indrule_lemma (([], []), (descr'' ~~ recTs ~~ recTs'));

  1018     val indrule_lemma = standard (Goal.prove thy8 [] []

  1019       (Logic.mk_implies

  1020         (HOLogic.mk_Trueprop (mk_conj indrule_lemma_prems),

  1021          HOLogic.mk_Trueprop (mk_conj indrule_lemma_concls))) (fn _ => EVERY

  1022            [REPEAT (etac conjE 1),

  1023             REPEAT (EVERY

  1024               [TRY (rtac conjI 1), full_simp_tac (HOL_basic_ss addsimps Rep_inverse_thms) 1,

  1025                etac mp 1, resolve_tac Rep_thms 1])]));

  1027     val Ps = map head_of (HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of indrule_lemma)));

  1028     val frees = if length Ps = 1 then [Free ("P", snd (dest_Var (hd Ps)))] else

  1029       map (Free o apfst fst o dest_Var) Ps;

  1030     val indrule_lemma' = cterm_instantiate

  1031       (map (cterm_of thy8) Ps ~~ map (cterm_of thy8) frees) indrule_lemma;

  1033     val Abs_inverse_thms' = List.concat (map mk_funs_inv Abs_inverse_thms);

  1035     val dt_induct_prop = DatatypeProp.make_ind descr' sorts';

  1036     val dt_induct = standard (Goal.prove thy8 []

  1037       (Logic.strip_imp_prems dt_induct_prop) (Logic.strip_imp_concl dt_induct_prop)

  1038       (fn prems => EVERY

  1039         [rtac indrule_lemma' 1,

  1040          (DatatypeAux.indtac rep_induct THEN_ALL_NEW ObjectLogic.atomize_tac) 1,

  1041          EVERY (map (fn (prem, r) => (EVERY

  1042            [REPEAT (eresolve_tac Abs_inverse_thms' 1),

  1043             simp_tac (HOL_basic_ss addsimps [symmetric r]) 1,

  1044             DEPTH_SOLVE_1 (ares_tac [prem] 1 ORELSE etac allE 1)]))

  1045                 (prems ~~ constr_defs))]));

  1047     val case_names_induct = mk_case_names_induct descr'';

  1049     (**** prove that new datatypes have finite support ****)

  1051     val _ = warning "proving finite support for the new datatype";

  1053     val indnames = DatatypeProp.make_tnames recTs;

  1055     val abs_supp = PureThy.get_thms thy8 (Name "abs_supp");

  1056     val supp_atm = PureThy.get_thms thy8 (Name "supp_atm");

  1058     val finite_supp_thms = map (fn atom =>

  1059       let val atomT = Type (atom, [])

  1060       in map standard (List.take

  1061         (split_conj_thm (Goal.prove thy8 [] [] (HOLogic.mk_Trueprop

  1062            (foldr1 HOLogic.mk_conj (map (fn (s, T) => HOLogic.mk_mem

  1063              (Const ("nominal.supp", T --> HOLogic.mk_setT atomT) $ Free (s, T),

  1064               Const ("Finite_Set.Finites", HOLogic.mk_setT (HOLogic.mk_setT atomT))))

  1065                (indnames ~~ recTs))))

  1066            (fn _ => indtac dt_induct indnames 1 THEN

  1067             ALLGOALS (asm_full_simp_tac (simpset_of thy8 addsimps

  1068               (abs_supp @ supp_atm @

  1069                PureThy.get_thms thy8 (Name ("fs_" ^ Sign.base_name atom ^ "1")) @

  1070                List.concat supp_thms))))),

  1071          length new_type_names))

  1072       end) atoms;

  1074     val simp_atts = replicate (length new_type_names) [Simplifier.simp_add];

  1076     val (_, thy9) = thy8 |>

  1077       Theory.add_path big_name |>

  1078       PureThy.add_thms [(("induct_weak", dt_induct), [case_names_induct])] ||>>

  1079       PureThy.add_thmss [(("inducts_weak", projections dt_induct), [case_names_induct])] ||>

  1080       Theory.parent_path ||>>

  1081       DatatypeAux.store_thmss_atts "distinct" new_type_names simp_atts distinct_thms ||>>

  1082       DatatypeAux.store_thmss "constr_rep" new_type_names constr_rep_thmss ||>>

  1083       DatatypeAux.store_thmss_atts "perm" new_type_names simp_atts perm_simps' ||>>

  1084       DatatypeAux.store_thmss "inject" new_type_names inject_thms ||>>

  1085       DatatypeAux.store_thmss "supp" new_type_names supp_thms ||>>

  1086       DatatypeAux.store_thmss_atts "fresh" new_type_names simp_atts fresh_thms ||>

  1087       fold (fn (atom, ths) => fn thy =>

  1088         let val class = Sign.intern_class thy ("fs_" ^ Sign.base_name atom)

  1089         in fold (fn T => AxClass.prove_arity

  1090             (fst (dest_Type T),

  1091               replicate (length sorts) [class], [class])

  1092             (ClassPackage.intro_classes_tac [] THEN resolve_tac ths 1)) newTs thy

  1093         end) (atoms ~~ finite_supp_thms);

  1095     (**** strong induction theorem ****)

  1097     val pnames = if length descr'' = 1 then ["P"]

  1098       else map (fn i => "P" ^ string_of_int i) (1 upto length descr'');

  1099     val ind_sort = if null dt_atomTs then HOLogic.typeS

  1100       else norm_sort thy9 (map (fn T => Sign.intern_class thy9 ("fs_" ^

  1101         Sign.base_name (fst (dest_Type T)))) dt_atomTs);

  1102     val fsT = TFree ("'n", ind_sort);

  1103     val fsT' = TFree ("'n", HOLogic.typeS);

  1105     val fresh_fs = map (fn (s, T) => (T, Free (s, fsT' --> HOLogic.mk_setT T)))

  1106       (DatatypeProp.indexify_names (replicate (length dt_atomTs) "f") ~~ dt_atomTs);

  1108     fun make_pred fsT i T =

  1109       Free (List.nth (pnames, i), fsT --> T --> HOLogic.boolT);

  1111     fun make_ind_prem fsT f k T ((cname, cargs), idxs) =

  1112       let

  1113         val recs = List.filter is_rec_type cargs;

  1114         val Ts = map (typ_of_dtyp descr'' sorts') cargs;

  1115         val recTs' = map (typ_of_dtyp descr'' sorts') recs;

  1116         val tnames = variantlist (DatatypeProp.make_tnames Ts, pnames);

  1117         val rec_tnames = map fst (List.filter (is_rec_type o snd) (tnames ~~ cargs));

  1118         val frees = tnames ~~ Ts;

  1119         val z = (variant tnames "z", fsT);

  1121         fun mk_prem ((dt, s), T) =

  1122           let

  1123             val (Us, U) = strip_type T;

  1124             val l = length Us

  1125           in list_all (z :: map (pair "x") Us, HOLogic.mk_Trueprop

  1126             (make_pred fsT (body_index dt) U $ Bound l $ app_bnds (Free (s, T)) l))

  1127           end;

  1129         val prems = map mk_prem (recs ~~ rec_tnames ~~ recTs');

  1130         val prems' = map (fn p as (_, T) => HOLogic.mk_Trueprop

  1131             (f T (Free p) (Free z)))

  1132           (map (curry List.nth frees) (List.concat (map (fn (m, n) =>

  1133              m upto m + n - 1) idxs)))

  1135       in list_all_free (frees @ [z], Logic.list_implies (prems' @ prems,

  1136         HOLogic.mk_Trueprop (make_pred fsT k T $ Free z $

  1137           list_comb (Const (cname, Ts ---> T), map Free frees))))

  1138       end;

  1140     val ind_prems = List.concat (map (fn (((i, (_, _, constrs)), (_, idxss)), T) =>

  1141       map (make_ind_prem fsT (fn T => fn t => fn u =>

  1142         Const ("nominal.fresh", T --> fsT --> HOLogic.boolT) $ t $ u) i T)

  1143           (constrs ~~ idxss)) (descr'' ~~ ndescr ~~ recTs));

  1144     val tnames = DatatypeProp.make_tnames recTs;

  1145     val zs = variantlist (replicate (length descr'') "z", tnames);

  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);

  1152     val ind_prems' =

  1153       map (fn (_, f as Free (_, T)) => list_all_free ([("x", fsT')],

  1154         HOLogic.mk_Trueprop (HOLogic.mk_mem (f $ Free ("x", fsT'),

  1155           Const ("Finite_Set.Finites", HOLogic.mk_setT (body_type T)))))) fresh_fs @

  1156       List.concat (map (fn (((i, (_, _, constrs)), (_, idxss)), T) =>

  1157         map (make_ind_prem fsT' (fn T => fn t => fn u => HOLogic.Not $

  1158           HOLogic.mk_mem (t, the (AList.lookup op = fresh_fs T) $ u)) i T)

  1159             (constrs ~~ idxss)) (descr'' ~~ ndescr ~~ recTs));

  1160     val ind_concl' = HOLogic.mk_Trueprop (foldr1 (HOLogic.mk_binop "op &")

  1161       (map (fn ((((i, _), T), tname), z) =>

  1162         make_pred fsT' i T $ Free (z, fsT') $ Free (tname, T))

  1163         (descr'' ~~ recTs ~~ tnames ~~ zs)));

  1164     val induct' = Logic.list_implies (ind_prems', ind_concl');

  1166     fun mk_perm Ts (t, u) =

  1167       let

  1168         val T = fastype_of1 (Ts, t);

  1169         val U = fastype_of1 (Ts, u)

  1170       in Const ("nominal.perm", T --> U --> U) $ t $ u end;

  1172     val aux_ind_vars =

  1173       (DatatypeProp.indexify_names (replicate (length dt_atomTs) "pi") ~~

  1174        map mk_permT dt_atomTs) @ [("z", fsT')];

  1175     val aux_ind_Ts = rev (map snd aux_ind_vars);

  1176     val aux_ind_concl = HOLogic.mk_Trueprop (foldr1 (HOLogic.mk_binop "op &")

  1177       (map (fn (((i, _), T), tname) =>

  1178         HOLogic.list_all (aux_ind_vars, make_pred fsT' i T $ Bound 0 $

  1179           foldr (mk_perm aux_ind_Ts) (Free (tname, T))

  1180             (map Bound (length dt_atomTs downto 1))))

  1181         (descr'' ~~ recTs ~~ tnames)));

  1183     fun mk_ind_perm i k p l vs j =

  1184       let

  1185         val n = length vs;

  1186         val Ts = map snd vs;

  1187         val T = List.nth (Ts, i - j);

  1188         val pT = NominalAtoms.mk_permT T

  1189       in

  1190         Const ("List.list.Cons", HOLogic.mk_prodT (T, T) --> pT --> pT) $

  1191           (HOLogic.pair_const T T $ Bound (l - j) $ foldr (mk_perm Ts)

  1192             (Bound (i - j))

  1193             (map (mk_ind_perm i k p l vs) (j - 1 downto 0) @

  1194              map Bound (n - k - 1 downto n - k - p))) $

  1195           Const ("List.list.Nil", pT)

  1196       end;

  1198     fun mk_fresh i i' j k p l vs _ _ =

  1199       let

  1200         val n = length vs;

  1201         val Ts = map snd vs;

  1202         val T = List.nth (Ts, n - i - 1 - j);

  1203         val f = the (AList.lookup op = fresh_fs T);

  1204         val U = List.nth (Ts, n - i' - 1);

  1205         val S = HOLogic.mk_setT T;

  1206         val prms = map (mk_ind_perm (n - i) k p (n - l) (("a", T) :: vs))

  1207             (j - 1 downto 0) @

  1208           map Bound (n - k downto n - k - p + 1)

  1209       in

  1210         HOLogic.mk_Trueprop (Const ("Ex", (T --> HOLogic.boolT) --> HOLogic.boolT) $

  1211           Abs ("a", T, HOLogic.Not $

  1212             (Const ("op :", T --> S --> HOLogic.boolT) $ Bound 0 $

  1213               (Const ("insert", T --> S --> S) $

  1214                 (foldr (mk_perm (T :: Ts)) (Bound (n - i - j)) prms) $

  1215                 (Const ("op Un", S --> S --> S) $ (f $ Bound (n - k - p)) $

  1216                    (Const ("nominal.supp", U --> S) $

  1217                      foldr (mk_perm (T :: Ts)) (Bound (n - i')) prms))))))

  1218       end;

  1220     fun mk_fresh_constr is p vs _ concl =

  1221       let

  1222         val n = length vs;

  1223         val Ts = map snd vs;

  1224         val _ $ (_ $ _ $ t) = concl;

  1225         val c = head_of t;

  1226         val T = body_type (fastype_of c);

  1227         val k = foldr op + 0 (map (fn (_, i) => i + 1) is);

  1228         val ps = map Bound (n - k - 1 downto n - k - p);

  1229         val (_, _, ts, us) = foldl (fn ((_, i), (m, n, ts, us)) =>

  1230           (m - i - 1, n - i,

  1231            ts @ map Bound (n downto n - i + 1) @

  1232              [foldr (mk_perm Ts) (Bound (m - i))

  1233                 (map (mk_ind_perm m k p n vs) (i - 1 downto 0) @ ps)],

  1234            us @ map (fn j => foldr (mk_perm Ts) (Bound j) ps) (m downto m - i)))

  1235           (n - 1, n - k - p - 2, [], []) is

  1236       in

  1237         HOLogic.mk_Trueprop (HOLogic.eq_const T $ list_comb (c, ts) $ list_comb (c, us))

  1238       end;

  1240     val abs_fun_finite_supp = PureThy.get_thm thy9 (Name "abs_fun_finite_supp");

  1242     val at_finite_select = PureThy.get_thm thy9 (Name "at_finite_select");

  1244     val induct_aux_lemmas = List.concat (map (fn Type (s, _) =>

  1245       [PureThy.get_thm thy9 (Name ("pt_" ^ Sign.base_name s ^ "_inst")),

  1246        PureThy.get_thm thy9 (Name ("fs_" ^ Sign.base_name s ^ "1")),

  1247        PureThy.get_thm thy9 (Name ("at_" ^ Sign.base_name s ^ "_inst"))]) dt_atomTs);

  1249     val induct_aux_lemmas' = map (fn Type (s, _) =>

  1250       PureThy.get_thm thy9 (Name ("pt_" ^ Sign.base_name s ^ "2")) RS sym) dt_atomTs;

  1252     val induct_aux = standard (Goal.prove thy9 [] ind_prems' ind_concl'

  1253       (fn prems => EVERY

  1254         ([mk_subgoal 1 (K (K (K aux_ind_concl))),

  1255           indtac dt_induct tnames 1] @

  1256          List.concat (map (fn ((_, (_, _, constrs)), (_, constrs')) =>

  1257            List.concat (map (fn ((cname, cargs), is) =>

  1258              [simp_tac (HOL_basic_ss addsimps List.concat perm_simps') 1,

  1259               REPEAT (rtac allI 1)] @

  1260              List.concat (map

  1261                (fn ((_, 0), _) => []

  1262                  | ((i, j), k) => List.concat (map (fn j' =>

  1263                      let

  1264                        val DtType (tname, _) = List.nth (cargs, i + j');

  1265                        val atom = Sign.base_name tname

  1266                      in

  1267                        [mk_subgoal 1 (mk_fresh i (i + j) j'

  1268                           (length cargs) (length dt_atomTs)

  1269                           (length cargs + length dt_atomTs + 1 + i - k)),

  1270                         rtac at_finite_select 1,

  1271                         rtac (PureThy.get_thm thy9 (Name ("at_" ^ atom ^ "_inst"))) 1,

  1272                         asm_full_simp_tac (simpset_of thy9 addsimps

  1273                           [PureThy.get_thm thy9 (Name ("fs_" ^ atom ^ "1"))]) 1,

  1274                         resolve_tac prems 1,

  1275                         etac exE 1,

  1276                         asm_full_simp_tac (simpset_of thy9 addsimps

  1277                           [symmetric fresh_def]) 1]

  1278                      end) (0 upto j - 1))) (is ~~ (0 upto length is - 1))) @

  1279              (if exists (not o equal 0 o snd) is then

  1280                 [mk_subgoal 1 (mk_fresh_constr is (length dt_atomTs)),

  1281                  asm_full_simp_tac (simpset_of thy9 addsimps

  1282                    (List.concat inject_thms @

  1283                     alpha @ abs_perm @ abs_fresh @ [abs_fun_finite_supp] @

  1284                     induct_aux_lemmas)) 1,

  1285                  dtac sym 1,

  1286                  asm_full_simp_tac (simpset_of thy9

  1287                    addsimps induct_aux_lemmas'

  1288                    addsimprocs [perm_simproc]) 1,

  1289                  REPEAT (etac conjE 1)]

  1290               else

  1291                 []) @

  1292              [(resolve_tac prems THEN_ALL_NEW

  1293                 (atac ORELSE' ((REPEAT o etac allE) THEN' atac))) 1])

  1294                (constrs ~~ constrs'))) (descr'' ~~ ndescr)) @

  1295          [REPEAT (eresolve_tac [conjE, allE_Nil] 1),

  1296           REPEAT (etac allE 1),

  1297           REPEAT (TRY (rtac conjI 1) THEN asm_full_simp_tac (simpset_of thy9) 1)])));

  1299     val induct_aux' = Thm.instantiate ([],

  1300       map (fn (s, T) =>

  1301         let val pT = TVar (("'n", 0), HOLogic.typeS) --> T --> HOLogic.boolT

  1302         in (cterm_of thy9 (Var ((s, 0), pT)), cterm_of thy9 (Free (s, pT))) end)

  1303           (pnames ~~ recTs) @

  1304       map (fn (_, f) =>

  1305         let val f' = Logic.varify f

  1306         in (cterm_of thy9 f',

  1307           cterm_of thy9 (Const ("nominal.supp", fastype_of f')))

  1308         end) fresh_fs) induct_aux;

  1310     val induct = standard (Goal.prove thy9 [] ind_prems ind_concl

  1311       (fn prems => EVERY

  1312          [rtac induct_aux' 1,

  1313           REPEAT (resolve_tac induct_aux_lemmas 1),

  1314           REPEAT ((resolve_tac prems THEN_ALL_NEW

  1315             (etac meta_spec ORELSE' full_simp_tac (HOL_basic_ss addsimps [fresh_def]))) 1)]))

  1317     val (_, thy10) = thy9 |>

  1318       Theory.add_path big_name |>

  1319       PureThy.add_thms [(("induct'", induct_aux), [])] ||>>

  1320       PureThy.add_thms [(("induct", induct), [case_names_induct])] ||>>

  1321       PureThy.add_thmss [(("inducts", projections induct), [case_names_induct])] ||>

  1322       Theory.parent_path;

  1324     (**** iteration combinator ****)

  1326     val _ = warning "defining iteration combinator ...";

  1328     val used = foldr add_typ_tfree_names [] recTs;

  1329     val rec_result_Ts = map TFree (variantlist (replicate (length descr'') "'t", used) ~~

  1330       replicate (length descr'') HOLogic.typeS);

  1332     val iter_fn_Ts = List.concat (map (fn (i, (_, _, constrs)) =>

  1333       map (fn (_, cargs) =>

  1334         let

  1335           val Ts = map (typ_of_dtyp descr'' sorts') cargs;

  1336           fun mk_argT (dt, T) =

  1337             if is_rec_type dt then List.nth (rec_result_Ts, body_index dt)

  1338             else T;

  1339           val argTs = map mk_argT (cargs ~~ Ts)

  1340         in argTs ---> List.nth (rec_result_Ts, i)

  1341         end) constrs) descr'');

  1343     val permTs = map mk_permT dt_atomTs;

  1344     val perms = map Free

  1345       (DatatypeProp.indexify_names (replicate (length permTs) "pi") ~~ permTs);

  1347     val iter_set_Ts = map (fn (T1, T2) => iter_fn_Ts ---> HOLogic.mk_setT

  1348       (HOLogic.mk_prodT (T1, permTs ---> T2))) (recTs ~~ rec_result_Ts);

  1350     val big_iter_name = big_name ^ "_iter_set";

  1351     val iter_set_names = map (Sign.full_name (Theory.sign_of thy10))

  1352       (if length descr'' = 1 then [big_iter_name] else

  1353         (map ((curry (op ^) (big_iter_name ^ "_")) o string_of_int)

  1354           (1 upto (length descr''))));

  1356     val iter_fns = map (uncurry (mk_Free "f"))

  1357       (iter_fn_Ts ~~ (1 upto (length iter_fn_Ts)));

  1358     val iter_sets = map (fn c => list_comb (Const c, iter_fns))

  1359       (iter_set_names ~~ iter_set_Ts);

  1361     (* introduction rules for graph of iteration function *)

  1363     fun partition_cargs idxs xs = map (fn (i, j) =>

  1364       (List.take (List.drop (xs, i), j), List.nth (xs, i + j))) idxs;

  1366     fun mk_fresh_fun (a, t) = Const ("nominal.fresh_fun",

  1367       (fastype_of a --> fastype_of t) --> fastype_of t) $ lambda a t;

  1369     fun make_iter_intr T iter_set ((iter_intr_ts, l), ((cname, cargs), idxs)) =

  1370       let

  1371         fun mk_prem ((dts, (dt, U)), (j, k, prems, t1s, t2s, atoms)) =

  1372           let

  1373             val free1 = mk_Free "x" U (j + length dts);

  1374             val Us = map snd dts;

  1375             val xs = Us ~~ (j upto j + length dts - 1);

  1376             val frees = map (uncurry (mk_Free "x")) xs;

  1377             val frees' = map (uncurry (mk_Free "x'")) xs;

  1378             val frees'' = Us ~~ (frees ~~ frees');

  1379             val pis = map (fn (T, p) =>

  1380               case filter (equal T o fst) frees'' of

  1381                 [] => p

  1382               | xs => HOLogic.mk_binop "List.op @" (p,

  1383                 HOLogic.mk_list (HOLogic.mk_prod o snd)

  1384                   (HOLogic.mk_prodT (T, T)) xs))

  1385                   (dt_atomTs ~~ perms)

  1386           in (case dt of

  1387              DtRec m =>

  1388                let val free2 = mk_Free "y"

  1389                  (permTs ---> List.nth (rec_result_Ts, m)) k

  1390                in (j + length dts + 1, k + 1,

  1391                    HOLogic.mk_Trueprop

  1392                      (HOLogic.mk_mem (HOLogic.mk_prod

  1393                        (free1, free2),

  1394                          List.nth (iter_sets, m))) :: prems,

  1395                    frees @ free1 :: t1s,

  1396                    frees' @ list_comb (free2, pis) :: t2s,

  1397                    frees' @ atoms)

  1398                end

  1399            | _ => (j + length dts + 1, k, prems,

  1400                frees @ free1 :: t1s,

  1401                frees' @ foldr (mk_perm []) free1 pis :: t2s,

  1402                frees' @ atoms))

  1403           end;

  1405         val Ts = map (typ_of_dtyp descr'' sorts') cargs;

  1406         val (_, _, prems, t1s, t2s, atoms) = foldr mk_prem (1, 1, [], [], [], [])

  1407           (partition_cargs idxs (cargs ~~ Ts))

  1409       in (iter_intr_ts @ [Logic.list_implies (prems, HOLogic.mk_Trueprop (HOLogic.mk_mem

  1410         (HOLogic.mk_prod (list_comb (Const (cname, Ts ---> T), t1s),

  1411           foldr (uncurry lambda)

  1412             (foldr mk_fresh_fun

  1413               (list_comb (List.nth (iter_fns, l), t2s)) atoms)

  1414             perms), iter_set)))], l + 1)

  1415       end;

  1417     val (iter_intr_ts, _) = Library.foldl (fn (x, (((d, d'), T), iter_set)) =>

  1418       Library.foldl (make_iter_intr T iter_set) (x, #3 (snd d) ~~ snd d'))

  1419         (([], 0), descr'' ~~ ndescr ~~ recTs ~~ iter_sets);

  1421     val (thy11, {intrs = iter_intrs, elims = iter_elims, ...}) =

  1422       setmp InductivePackage.quiet_mode (!quiet_mode)

  1423         (InductivePackage.add_inductive_i false true big_iter_name false false false

  1424            iter_sets (map (fn x => (("", x), [])) iter_intr_ts) []) thy10;

  1426   in

  1427     thy11

  1428   end;

  1430 val add_nominal_datatype = gen_add_nominal_datatype read_typ true;

  1433 (* FIXME: The following stuff should be exported by DatatypePackage *)

  1435 local structure P = OuterParse and K = OuterKeyword in

  1437 val datatype_decl =

  1438   Scan.option (P.$$$ "(" |-- P.name --| P.$$$ ")") -- P.type_args -- P.name -- P.opt_infix --

  1439     (P.$$$ "=" |-- P.enum1 "|" (P.name -- Scan.repeat P.typ -- P.opt_mixfix));

  1441 fun mk_datatype args =

  1442   let

  1443     val names = map (fn ((((NONE, _), t), _), _) => t | ((((SOME t, _), _), _), _) => t) args;

  1444     val specs = map (fn ((((_, vs), t), mx), cons) =>

  1445       (vs, t, mx, map (fn ((x, y), z) => (x, y, z)) cons)) args;

  1446   in add_nominal_datatype false names specs end;

  1448 val nominal_datatypeP =

  1449   OuterSyntax.command "nominal_datatype" "define inductive datatypes" K.thy_decl

  1450     (P.and_list1 datatype_decl >> (Toplevel.theory o mk_datatype));

  1452 val _ = OuterSyntax.add_parsers [nominal_datatypeP];

  1454 end;

  1456 end

author	berghofe
	Wed Mar 15 17:59:33 2006 +0100 (2006-03-15)
changeset 19275	3d10de7a8ca7
parent 19251	6bc0dda66f32
child 19322	bf84bdf05f14
permissions	-rw-r--r--