src/HOL/Tools/datatype_prop.ML
author wenzelm
Thu Mar 11 13:20:35 1999 +0100 (1999-03-11)
changeset 6349 f7750d816c21
parent 5695 898429dbb162
child 6394 3d9fd50fcc43
permissions -rw-r--r--
removed foo_build_completed -- now handled by session management (via usedir);
     1 (*  Title:      HOL/Tools/datatype_prop.ML
     2     ID:         $Id$
     3     Author:     Stefan Berghofer
     4     Copyright   1998  TU Muenchen
     5 
     6 Characteristic properties of datatypes
     7 *)
     8 
     9 signature DATATYPE_PROP =
    10 sig
    11   val dtK : int
    12   val make_injs : (int * (string * DatatypeAux.dtyp list *
    13     (string * DatatypeAux.dtyp list) list)) list list -> (string * sort) list ->
    14       term list list
    15   val make_ind : (int * (string * DatatypeAux.dtyp list *
    16     (string * DatatypeAux.dtyp list) list)) list list -> (string * sort) list -> term
    17   val make_casedists : (int * (string * DatatypeAux.dtyp list *
    18     (string * DatatypeAux.dtyp list) list)) list list -> (string * sort) list -> term list
    19   val make_primrecs : string list -> (int * (string * DatatypeAux.dtyp list *
    20     (string * DatatypeAux.dtyp list) list)) list list -> (string * sort) list ->
    21       theory -> term list
    22   val make_cases : string list -> (int * (string * DatatypeAux.dtyp list *
    23     (string * DatatypeAux.dtyp list) list)) list list -> (string * sort) list ->
    24       theory -> term list list
    25   val make_distincts : string list -> (int * (string * DatatypeAux.dtyp list *
    26     (string * DatatypeAux.dtyp list) list)) list list -> (string * sort) list ->
    27       theory -> term list list
    28   val make_splits : string list -> (int * (string * DatatypeAux.dtyp list *
    29     (string * DatatypeAux.dtyp list) list)) list list -> (string * sort) list ->
    30       theory -> (term * term) list
    31   val make_case_trrules : string list -> (int * (string * DatatypeAux.dtyp list *
    32     (string * DatatypeAux.dtyp list) list)) list list -> ast Syntax.trrule list
    33   val make_size : string list -> (int * (string * DatatypeAux.dtyp list *
    34     (string * DatatypeAux.dtyp list) list)) list list -> (string * sort) list ->
    35       theory -> term list
    36   val make_case_congs : string list -> (int * (string * DatatypeAux.dtyp list *
    37     (string * DatatypeAux.dtyp list) list)) list list -> (string * sort) list ->
    38       theory -> term list
    39   val make_nchotomys : (int * (string * DatatypeAux.dtyp list *
    40     (string * DatatypeAux.dtyp list) list)) list list -> (string * sort) list -> term list
    41 end;
    42 
    43 structure DatatypeProp : DATATYPE_PROP =
    44 struct
    45 
    46 open DatatypeAux;
    47 
    48 (*the kind of distinctiveness axioms depends on number of constructors*)
    49 val dtK = 7;
    50 
    51 fun make_tnames Ts =
    52   let
    53     fun type_name (TFree (name, _)) = implode (tl (explode name))
    54       | type_name (Type (name, _)) = 
    55           let val name' = Sign.base_name name
    56           in if Syntax.is_identifier name' then name' else "x"
    57           end;
    58 
    59     fun index_vnames (vn::vns) tab =
    60           (case assoc (tab, vn) of
    61              None => if vn mem vns
    62                      then (vn ^ "1") :: index_vnames vns ((vn, 2)::tab)
    63                      else vn :: index_vnames vns tab
    64            | Some i => (vn ^ (string_of_int i))::
    65                      index_vnames vns ((vn, i + 1)::tab))
    66       | index_vnames [] _ = []
    67 
    68   in index_vnames (map type_name Ts) []
    69   end;
    70 
    71 (** FIXME: move to hologic.ML ? **)
    72 val Not = Const ("Not", HOLogic.boolT --> HOLogic.boolT);
    73 
    74 (************************* injectivity of constructors ************************)
    75 
    76 fun make_injs descr sorts =
    77   let
    78     val descr' = flat descr;
    79 
    80     fun make_inj T ((cname, cargs), injs) =
    81       if null cargs then injs else
    82         let
    83           val Ts = map (typ_of_dtyp descr' sorts) cargs;
    84           val constr_t = Const (cname, Ts ---> T);
    85           val tnames = make_tnames Ts;
    86           val frees = map Free (tnames ~~ Ts);
    87           val frees' = map Free ((map ((op ^) o (rpair "'")) tnames) ~~ Ts);
    88         in (HOLogic.mk_Trueprop (HOLogic.mk_eq
    89           (HOLogic.mk_eq (list_comb (constr_t, frees), list_comb (constr_t, frees')),
    90            foldr1 (HOLogic.mk_binop "op &")
    91              (map HOLogic.mk_eq (frees ~~ frees')))))::injs
    92         end;
    93 
    94   in map (fn (d, T) => foldr (make_inj T) (#3 (snd d), []))
    95     ((hd descr) ~~ take (length (hd descr), get_rec_types descr' sorts))
    96   end;
    97 
    98 (********************************* induction **********************************)
    99 
   100 fun make_ind descr sorts =
   101   let
   102     val descr' = flat descr;
   103     val recTs = get_rec_types descr' sorts;
   104     val pnames = if length descr' = 1 then ["P"]
   105       else map (fn i => "P" ^ string_of_int i) (1 upto length descr');
   106 
   107     fun make_pred i T =
   108       let val T' = T --> HOLogic.boolT
   109       in Free (nth_elem (i, pnames), T') end;
   110 
   111     fun make_ind_prem k T (cname, cargs) =
   112       let
   113         val recs = filter is_rec_type cargs;
   114         val Ts = map (typ_of_dtyp descr' sorts) cargs;
   115         val recTs' = map (typ_of_dtyp descr' sorts) recs;
   116         val tnames = variantlist (make_tnames Ts, pnames);
   117         val rec_tnames = map fst (filter (is_rec_type o snd) (tnames ~~ cargs));
   118         val frees = tnames ~~ Ts;
   119         val prems = map (fn ((r, s), T) => HOLogic.mk_Trueprop
   120           (make_pred (dest_DtRec r) T $ Free (s, T))) (recs ~~ rec_tnames ~~ recTs');
   121 
   122       in list_all_free (frees, Logic.list_implies (prems,
   123         HOLogic.mk_Trueprop (make_pred k T $ 
   124           list_comb (Const (cname, Ts ---> T), map Free frees))))
   125       end;
   126 
   127     val prems = flat (map (fn ((i, (_, _, constrs)), T) =>
   128       map (make_ind_prem i T) constrs) (descr' ~~ recTs));
   129     val tnames = make_tnames recTs;
   130     val concl = HOLogic.mk_Trueprop (foldr1 (HOLogic.mk_binop "op &")
   131       (map (fn (((i, _), T), tname) => make_pred i T $ Free (tname, T))
   132         (descr' ~~ recTs ~~ tnames)))
   133 
   134   in Logic.list_implies (prems, concl) end;
   135 
   136 (******************************* case distinction *****************************)
   137 
   138 fun make_casedists descr sorts =
   139   let
   140     val descr' = flat descr;
   141 
   142     fun make_casedist_prem T (cname, cargs) =
   143       let
   144         val Ts = map (typ_of_dtyp descr' sorts) cargs;
   145         val frees = variantlist (make_tnames Ts, ["P", "y"]) ~~ Ts;
   146         val free_ts = map Free frees
   147       in list_all_free (frees, Logic.mk_implies (HOLogic.mk_Trueprop
   148         (HOLogic.mk_eq (Free ("y", T), list_comb (Const (cname, Ts ---> T), free_ts))),
   149           HOLogic.mk_Trueprop (Free ("P", HOLogic.boolT))))
   150       end;
   151 
   152     fun make_casedist ((_, (_, _, constrs)), T) =
   153       let val prems = map (make_casedist_prem T) constrs
   154       in Logic.list_implies (prems, HOLogic.mk_Trueprop (Free ("P", HOLogic.boolT)))
   155       end
   156 
   157   in map make_casedist
   158     ((hd descr) ~~ take (length (hd descr), get_rec_types descr' sorts))
   159   end;
   160 
   161 (*************** characteristic equations for primrec combinator **************)
   162 
   163 fun make_primrecs new_type_names descr sorts thy =
   164   let
   165     val sign = sign_of thy;
   166 
   167     val descr' = flat descr;
   168     val recTs = get_rec_types descr' sorts;
   169     val used = foldr add_typ_tfree_names (recTs, []);
   170 
   171     val rec_result_Ts = map TFree (variantlist (replicate (length descr') "'t", used) ~~
   172       replicate (length descr') HOLogic.termS);
   173 
   174     val reccomb_fn_Ts = flat (map (fn (i, (_, _, constrs)) =>
   175       map (fn (_, cargs) =>
   176         let
   177           val recs = filter is_rec_type cargs;
   178           val argTs = (map (typ_of_dtyp descr' sorts) cargs) @
   179             (map (fn r => nth_elem (dest_DtRec r, rec_result_Ts)) recs)
   180         in argTs ---> nth_elem (i, rec_result_Ts)
   181         end) constrs) descr');
   182 
   183     val rec_fns = map (uncurry (mk_Free "f"))
   184       (reccomb_fn_Ts ~~ (1 upto (length reccomb_fn_Ts)));
   185 
   186     val big_reccomb_name = (space_implode "_" new_type_names) ^ "_rec";
   187     val reccomb_names = map (Sign.intern_const sign)
   188       (if length descr' = 1 then [big_reccomb_name] else
   189         (map ((curry (op ^) (big_reccomb_name ^ "_")) o string_of_int)
   190           (1 upto (length descr'))));
   191     val reccombs = map (fn ((name, T), T') => list_comb
   192       (Const (name, reccomb_fn_Ts @ [T] ---> T'), rec_fns))
   193         (reccomb_names ~~ recTs ~~ rec_result_Ts);
   194 
   195     fun make_primrec T comb_t ((ts, f::fs), (cname, cargs)) =
   196       let
   197         val recs = filter is_rec_type cargs;
   198         val Ts = map (typ_of_dtyp descr' sorts) cargs;
   199         val recTs' = map (typ_of_dtyp descr' sorts) recs;
   200         val tnames = make_tnames Ts;
   201         val rec_tnames = map fst (filter (is_rec_type o snd) (tnames ~~ cargs));
   202         val frees = map Free (tnames ~~ Ts);
   203         val frees' = map Free (rec_tnames ~~ recTs');
   204         val reccombs' = map (fn (DtRec i) => nth_elem (i, reccombs)) recs
   205 
   206       in (ts @ [HOLogic.mk_Trueprop (HOLogic.mk_eq
   207         (comb_t $ list_comb (Const (cname, Ts ---> T), frees),
   208          list_comb (f, frees @ (map (uncurry ap) (reccombs' ~~ frees')))))], fs)
   209       end
   210 
   211   in fst (foldl (fn (x, ((dt, T), comb_t)) =>
   212     foldl (make_primrec T comb_t) (x, #3 (snd dt)))
   213       (([], rec_fns), descr' ~~ recTs ~~ reccombs))
   214   end;
   215 
   216 (****************** make terms of form  t_case f1 ... fn  *********************)
   217 
   218 fun make_case_combs new_type_names descr sorts thy fname =
   219   let
   220     val descr' = flat descr;
   221     val recTs = get_rec_types descr' sorts;
   222     val used = foldr add_typ_tfree_names (recTs, []);
   223     val newTs = take (length (hd descr), recTs);
   224     val T' = TFree (variant used "'t", HOLogic.termS);
   225 
   226     val case_fn_Ts = map (fn (i, (_, _, constrs)) =>
   227       map (fn (_, cargs) =>
   228         let val Ts = map (typ_of_dtyp descr' sorts) cargs
   229         in Ts ---> T' end) constrs) (hd descr);
   230 
   231     val case_names = map (fn s =>
   232       Sign.intern_const (sign_of thy) (s ^ "_case")) new_type_names
   233   in
   234     map (fn ((name, Ts), T) => list_comb
   235       (Const (name, Ts @ [T] ---> T'),
   236         map (uncurry (mk_Free fname)) (Ts ~~ (1 upto length Ts))))
   237           (case_names ~~ case_fn_Ts ~~ newTs)
   238   end;
   239 
   240 (**************** characteristic equations for case combinator ****************)
   241 
   242 fun make_cases new_type_names descr sorts thy =
   243   let
   244     val descr' = flat descr;
   245     val recTs = get_rec_types descr' sorts;
   246     val newTs = take (length (hd descr), recTs);
   247 
   248     fun make_case T comb_t ((cname, cargs), f) =
   249       let
   250         val Ts = map (typ_of_dtyp descr' sorts) cargs;
   251         val frees = map Free ((make_tnames Ts) ~~ Ts)
   252       in HOLogic.mk_Trueprop (HOLogic.mk_eq
   253         (comb_t $ list_comb (Const (cname, Ts ---> T), frees),
   254          list_comb (f, frees)))
   255       end
   256 
   257   in map (fn (((_, (_, _, constrs)), T), comb_t) =>
   258     map (make_case T comb_t) (constrs ~~ (snd (strip_comb comb_t))))
   259       ((hd descr) ~~ newTs ~~ (make_case_combs new_type_names descr sorts thy "f"))
   260   end;
   261 
   262 (************************* distinctness of constructors ***********************)
   263 
   264 fun make_distincts new_type_names descr sorts thy =
   265   let
   266     val descr' = flat descr;
   267     val recTs = get_rec_types descr' sorts;
   268     val newTs = take (length (hd descr), recTs);
   269 
   270     (**** number of constructors < dtK : C_i ... ~= C_j ... ****)
   271 
   272     fun make_distincts_1 _ [] = []
   273       | make_distincts_1 T ((cname, cargs)::constrs) =
   274           let
   275             val Ts = map (typ_of_dtyp descr' sorts) cargs;
   276             val frees = map Free ((make_tnames Ts) ~~ Ts);
   277             val t = list_comb (Const (cname, Ts ---> T), frees);
   278 
   279             fun make_distincts' [] = []
   280               | make_distincts' ((cname', cargs')::constrs') =
   281                   let
   282                     val Ts' = map (typ_of_dtyp descr' sorts) cargs';
   283                     val frees' = map Free ((map ((op ^) o (rpair "'"))
   284                       (make_tnames Ts')) ~~ Ts');
   285                     val t' = list_comb (Const (cname', Ts' ---> T), frees')
   286                   in
   287                     (HOLogic.mk_Trueprop (Not $ HOLogic.mk_eq (t, t')))::
   288                     (HOLogic.mk_Trueprop (Not $ HOLogic.mk_eq (t', t)))::
   289                       (make_distincts' constrs')
   290                   end
   291 
   292           in (make_distincts' constrs) @ (make_distincts_1 T constrs)
   293           end;
   294 
   295     (**** number of constructors >= dtK : t_ord C_i ... = i ****)
   296 
   297     fun make_distincts_2 T tname i constrs =
   298       let
   299         val ord_name = Sign.intern_const (sign_of thy) (tname ^ "_ord");
   300         val ord_t = Const (ord_name, T --> HOLogic.natT)
   301 
   302       in (case constrs of
   303           [] => [Logic.mk_implies (HOLogic.mk_Trueprop (Not $ HOLogic.mk_eq
   304              (ord_t $ Free ("x", T), ord_t $ Free ("y", T))),
   305                HOLogic.mk_Trueprop (Not $ HOLogic.mk_eq
   306                  (Free ("x", T), Free ("y", T))))]
   307         | ((cname, cargs)::constrs) =>
   308             let
   309               val Ts = map (typ_of_dtyp descr' sorts) cargs;
   310               val frees = map Free ((make_tnames Ts) ~~ Ts);
   311             in
   312               (HOLogic.mk_Trueprop (HOLogic.mk_eq (ord_t $
   313                 list_comb (Const (cname, Ts ---> T), frees), HOLogic.mk_nat i)))::
   314                   (make_distincts_2 T tname (i + 1) constrs)
   315             end)
   316       end;
   317 
   318   in map (fn (((_, (_, _, constrs)), T), tname) =>
   319       if length constrs < dtK then make_distincts_1 T constrs
   320       else make_distincts_2 T tname 0 constrs)
   321         ((hd descr) ~~ newTs ~~ new_type_names)
   322   end;
   323 
   324 (*************************** the "split" - equations **************************)
   325 
   326 fun make_splits new_type_names descr sorts thy =
   327   let
   328     val descr' = flat descr;
   329     val recTs = get_rec_types descr' sorts;
   330     val used' = foldr add_typ_tfree_names (recTs, []);
   331     val newTs = take (length (hd descr), recTs);
   332     val T' = TFree (variant used' "'t", HOLogic.termS);
   333     val P = Free ("P", T' --> HOLogic.boolT);
   334 
   335     fun make_split (((_, (_, _, constrs)), T), comb_t) =
   336       let
   337         val (_, fs) = strip_comb comb_t;
   338         val used = ["P", "x"] @ (map (fst o dest_Free) fs);
   339 
   340         fun process_constr (((cname, cargs), f), (t1s, t2s)) =
   341           let
   342             val Ts = map (typ_of_dtyp descr' sorts) cargs;
   343             val frees = map Free (variantlist (make_tnames Ts, used) ~~ Ts);
   344             val eqn = HOLogic.mk_eq (Free ("x", T),
   345               list_comb (Const (cname, Ts ---> T), frees));
   346             val P' = P $ list_comb (f, frees)
   347           in ((foldr (fn (Free (s, T), t) => HOLogic.mk_all (s, T, t))
   348                 (frees, HOLogic.imp $ eqn $ P'))::t1s,
   349               (foldr (fn (Free (s, T), t) => HOLogic.mk_exists (s, T, t))
   350                 (frees, HOLogic.conj $ eqn $ (Not $ P')))::t2s)
   351           end;
   352 
   353         val (t1s, t2s) = foldr process_constr (constrs ~~ fs, ([], []));
   354         val lhs = P $ (comb_t $ Free ("x", T))
   355       in
   356         (HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, mk_conj t1s)),
   357          HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs, Not $ mk_disj t2s)))
   358       end
   359 
   360   in map make_split ((hd descr) ~~ newTs ~~
   361     (make_case_combs new_type_names descr sorts thy "f"))
   362   end;
   363 
   364 (************************ translation rules for case **************************)
   365 
   366 fun make_case_trrules new_type_names descr =
   367   let
   368     fun mk_asts i j ((cname, cargs)::constrs) =
   369       let
   370         val k = length cargs;
   371         val xs = map (fn i => Variable ("x" ^ string_of_int i)) (i upto i + k - 1);
   372         val t = Variable ("t" ^ string_of_int j);
   373         val ast = Syntax.mk_appl (Constant "@case1")
   374           [Syntax.mk_appl (Constant (Sign.base_name cname)) xs, t];
   375         val ast' = foldr (fn (x, y) =>
   376           Syntax.mk_appl (Constant "_abs") [x, y]) (xs, t)
   377       in
   378         (case constrs of
   379             [] => (ast, [ast'])
   380           | cs => let val (ast'', asts) = mk_asts (i + k) (j + 1) cs
   381               in (Syntax.mk_appl (Constant "@case2") [ast, ast''],
   382                   ast'::asts)
   383               end)
   384       end;
   385 
   386     fun mk_trrule ((_, (_, _, constrs)), tname) =
   387       let val (ast, asts) = mk_asts 1 1 constrs
   388       in Syntax.ParsePrintRule
   389         (Syntax.mk_appl (Constant "@case") [Variable "t", ast],
   390          Syntax.mk_appl (Constant (tname ^ "_case"))
   391            (asts @ [Variable "t"]))
   392       end
   393 
   394   in
   395     map mk_trrule (hd descr ~~ new_type_names)
   396   end;
   397 
   398 (******************************* size functions *******************************)
   399 
   400 fun make_size new_type_names descr sorts thy =
   401   let
   402     val descr' = flat descr;
   403     val recTs = get_rec_types descr' sorts;
   404 
   405     val big_size_name = space_implode "_" new_type_names ^ "_size";
   406     val size_name = Sign.intern_const (sign_of (the (get_thy "Arith" thy))) "size";
   407     val size_names = replicate (length (hd descr)) size_name @
   408       map (Sign.intern_const (sign_of thy))
   409         (if length (flat (tl descr)) = 1 then [big_size_name] else
   410           map (fn i => big_size_name ^ "_" ^ string_of_int i)
   411             (1 upto length (flat (tl descr))));
   412     val size_consts = map (fn (s, T) =>
   413       Const (s, T --> HOLogic.natT)) (size_names ~~ recTs);
   414 
   415     val plus_t = Const ("op +", [HOLogic.natT, HOLogic.natT] ---> HOLogic.natT);
   416 
   417     fun make_size_eqn size_const T (cname, cargs) =
   418       let
   419         val recs = filter is_rec_type cargs;
   420         val Ts = map (typ_of_dtyp descr' sorts) cargs;
   421         val recTs = map (typ_of_dtyp descr' sorts) recs;
   422         val tnames = make_tnames Ts;
   423         val rec_tnames = map fst (filter (is_rec_type o snd) (tnames ~~ cargs));
   424         val ts = map (fn ((r, s), T) => nth_elem (dest_DtRec r, size_consts) $
   425           Free (s, T)) (recs ~~ rec_tnames ~~ recTs);
   426         val t = if ts = [] then HOLogic.zero else
   427           foldl1 (app plus_t) (ts @ [HOLogic.mk_nat 1])
   428       in
   429         HOLogic.mk_Trueprop (HOLogic.mk_eq (size_const $
   430           list_comb (Const (cname, Ts ---> T), map Free (tnames ~~ Ts)), t))
   431       end
   432 
   433   in
   434     flat (map (fn (((_, (_, _, constrs)), size_const), T) =>
   435       map (make_size_eqn size_const T) constrs) (descr' ~~ size_consts ~~ recTs))
   436   end;
   437 
   438 (************************* additional rules for TFL ***************************)
   439 
   440 (*---------------------------------------------------------------------------
   441  * Structure of case congruence theorem looks like this:
   442  *
   443  *    (M = M') 
   444  *    ==> (!!x1,...,xk. (M' = C1 x1..xk) ==> (f1 x1..xk = g1 x1..xk)) 
   445  *    ==> ... 
   446  *    ==> (!!x1,...,xj. (M' = Cn x1..xj) ==> (fn x1..xj = gn x1..xj)) 
   447  *    ==>
   448  *      (ty_case f1..fn M = ty_case g1..gn M')
   449  *---------------------------------------------------------------------------*)
   450 
   451 fun make_case_congs new_type_names descr sorts thy =
   452   let
   453     val case_combs = make_case_combs new_type_names descr sorts thy "f";
   454     val case_combs' = make_case_combs new_type_names descr sorts thy "g";
   455 
   456     fun mk_case_cong ((comb, comb'), (_, (_, _, constrs))) =
   457       let
   458         val Type ("fun", [T, _]) = fastype_of comb;
   459         val (_, fs) = strip_comb comb;
   460         val (_, gs) = strip_comb comb';
   461         val used = ["M", "M'"] @ map (fst o dest_Free) (fs @ gs);
   462         val M = Free ("M", T);
   463         val M' = Free ("M'", T);
   464 
   465         fun mk_clause ((f, g), (cname, _)) =
   466           let
   467             val (Ts, _) = strip_type (fastype_of f);
   468             val tnames = variantlist (make_tnames Ts, used);
   469             val frees = map Free (tnames ~~ Ts)
   470           in
   471             list_all_free (tnames ~~ Ts, Logic.mk_implies
   472               (HOLogic.mk_Trueprop
   473                 (HOLogic.mk_eq (M', list_comb (Const (cname, Ts ---> T), frees))),
   474                HOLogic.mk_Trueprop
   475                 (HOLogic.mk_eq (list_comb (f, frees), list_comb (g, frees)))))
   476           end
   477 
   478       in
   479         Logic.list_implies (HOLogic.mk_Trueprop (HOLogic.mk_eq (M, M')) ::
   480           map mk_clause (fs ~~ gs ~~ constrs),
   481             HOLogic.mk_Trueprop (HOLogic.mk_eq (comb $ M, comb' $ M')))
   482       end
   483 
   484   in
   485     map mk_case_cong (case_combs ~~ case_combs' ~~ hd descr)
   486   end;
   487 
   488 (*---------------------------------------------------------------------------
   489  * Structure of exhaustion theorem looks like this:
   490  *
   491  *    !v. (? y1..yi. v = C1 y1..yi) | ... | (? y1..yj. v = Cn y1..yj)
   492  *---------------------------------------------------------------------------*)
   493 
   494 fun make_nchotomys descr sorts =
   495   let
   496     val descr' = flat descr;
   497     val recTs = get_rec_types descr' sorts;
   498     val newTs = take (length (hd descr), recTs);
   499 
   500     fun mk_eqn T (cname, cargs) =
   501       let
   502         val Ts = map (typ_of_dtyp descr' sorts) cargs;
   503         val tnames = variantlist (make_tnames Ts, ["v"]);
   504         val frees = tnames ~~ Ts
   505       in
   506         foldr (fn ((s, T'), t) => HOLogic.mk_exists (s, T', t))
   507           (frees, HOLogic.mk_eq (Free ("v", T),
   508             list_comb (Const (cname, Ts ---> T), map Free frees)))
   509       end
   510 
   511   in map (fn ((_, (_, _, constrs)), T) =>
   512     HOLogic.mk_Trueprop (HOLogic.mk_all ("v", T, mk_disj (map (mk_eqn T) constrs))))
   513       (hd descr ~~ newTs)
   514   end;
   515 
   516 end;