src/HOL/Tools/datatype_abs_proofs.ML
author berghofe
Fri May 21 21:48:03 2004 +0200 (2004-05-21)
changeset 14799 a405aadff16c
parent 13641 63d1790a43ed
child 14981 e73f8140af78
permissions -rw-r--r--
Added more flexible parse / print translations for case expressions.
     1 (*  Title:      HOL/Tools/datatype_abs_proofs.ML
     2     ID:         $Id$
     3     Author:     Stefan Berghofer, TU Muenchen
     4     License:    GPL (GNU GENERAL PUBLIC LICENSE)
     5 
     6 Proofs and defintions independent of concrete representation
     7 of datatypes  (i.e. requiring only abstract properties such as
     8 injectivity / distinctness of constructors and induction)
     9 
    10  - case distinction (exhaustion) theorems
    11  - characteristic equations for primrec combinators
    12  - characteristic equations for case combinators
    13  - equations for splitting "P (case ...)" expressions
    14  - datatype size function
    15  - "nchotomy" and "case_cong" theorems for TFL
    16 
    17 *)
    18 
    19 signature DATATYPE_ABS_PROOFS =
    20 sig
    21   val prove_casedist_thms : string list ->
    22     DatatypeAux.descr list -> (string * sort) list -> thm ->
    23     theory attribute list -> theory -> theory * thm list
    24   val prove_primrec_thms : bool -> string list ->
    25     DatatypeAux.descr list -> (string * sort) list ->
    26       DatatypeAux.datatype_info Symtab.table -> thm list list -> thm list list ->
    27         simpset -> thm -> theory -> theory * (string list * thm list)
    28   val prove_case_thms : bool -> string list ->
    29     DatatypeAux.descr list -> (string * sort) list ->
    30       string list -> thm list -> theory -> theory * (thm list list * string list)
    31   val prove_split_thms : string list ->
    32     DatatypeAux.descr list -> (string * sort) list ->
    33       thm list list -> thm list list -> thm list -> thm list list -> theory ->
    34         theory * (thm * thm) list
    35   val prove_size_thms : bool -> string list ->
    36     DatatypeAux.descr list -> (string * sort) list ->
    37       string list -> thm list -> theory -> theory * thm list
    38   val prove_nchotomys : string list -> DatatypeAux.descr list ->
    39     (string * sort) list -> thm list -> theory -> theory * thm list
    40   val prove_weak_case_congs : string list -> DatatypeAux.descr list ->
    41     (string * sort) list -> theory -> theory * thm list
    42   val prove_case_congs : string list ->
    43     DatatypeAux.descr list -> (string * sort) list ->
    44       thm list -> thm list list -> theory -> theory * thm list
    45 end;
    46 
    47 structure DatatypeAbsProofs: DATATYPE_ABS_PROOFS =
    48 struct
    49 
    50 open DatatypeAux;
    51 
    52 (************************ case distinction theorems ***************************)
    53 
    54 fun prove_casedist_thms new_type_names descr sorts induct case_names_exhausts thy =
    55   let
    56     val _ = message "Proving case distinction theorems ...";
    57 
    58     val descr' = flat descr;
    59     val recTs = get_rec_types descr' sorts;
    60     val newTs = take (length (hd descr), recTs);
    61 
    62     val {maxidx, ...} = rep_thm induct;
    63     val induct_Ps = map head_of (HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of induct)));
    64 
    65     fun prove_casedist_thm ((i, t), T) =
    66       let
    67         val dummyPs = map (fn (Var (_, Type (_, [T', T'']))) =>
    68           Abs ("z", T', Const ("True", T''))) induct_Ps;
    69         val P = Abs ("z", T, HOLogic.imp $ HOLogic.mk_eq (Var (("a", maxidx+1), T), Bound 0) $
    70           Var (("P", 0), HOLogic.boolT))
    71         val insts = take (i, dummyPs) @ (P::(drop (i + 1, dummyPs)));
    72         val cert = cterm_of (Theory.sign_of thy);
    73         val insts' = (map cert induct_Ps) ~~ (map cert insts);
    74         val induct' = refl RS ((nth_elem (i,
    75           split_conj_thm (cterm_instantiate insts' induct))) RSN (2, rev_mp))
    76 
    77       in prove_goalw_cterm [] (cert t) (fn prems =>
    78         [rtac induct' 1,
    79          REPEAT (rtac TrueI 1),
    80          REPEAT ((rtac impI 1) THEN (eresolve_tac prems 1)),
    81          REPEAT (rtac TrueI 1)])
    82       end;
    83 
    84     val casedist_thms = map prove_casedist_thm ((0 upto (length newTs - 1)) ~~
    85       (DatatypeProp.make_casedists descr sorts) ~~ newTs)
    86   in thy |> store_thms_atts "exhaust" new_type_names (map single case_names_exhausts) casedist_thms end;
    87 
    88 
    89 (*************************** primrec combinators ******************************)
    90 
    91 fun prove_primrec_thms flat_names new_type_names descr sorts
    92     (dt_info : datatype_info Symtab.table) constr_inject dist_rewrites dist_ss induct thy =
    93   let
    94     val _ = message "Constructing primrec combinators ...";
    95 
    96     val big_name = space_implode "_" new_type_names;
    97     val thy0 = add_path flat_names big_name thy;
    98 
    99     val descr' = flat descr;
   100     val recTs = get_rec_types descr' sorts;
   101     val used = foldr add_typ_tfree_names (recTs, []);
   102     val newTs = take (length (hd descr), recTs);
   103 
   104     val induct_Ps = map head_of (HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of induct)));
   105 
   106     val big_rec_name' = big_name ^ "_rec_set";
   107     val rec_set_names = map (Sign.full_name (Theory.sign_of thy0))
   108       (if length descr' = 1 then [big_rec_name'] else
   109         (map ((curry (op ^) (big_rec_name' ^ "_")) o string_of_int)
   110           (1 upto (length descr'))));
   111 
   112     val rec_result_Ts = map TFree (variantlist (replicate (length descr') "'t", used) ~~
   113       replicate (length descr') HOLogic.typeS);
   114 
   115     val reccomb_fn_Ts = flat (map (fn (i, (_, _, constrs)) =>
   116       map (fn (_, cargs) =>
   117         let
   118           val Ts = map (typ_of_dtyp descr' sorts) cargs;
   119           val recs = filter (is_rec_type o fst) (cargs ~~ Ts);
   120 
   121           fun mk_argT (dt, T) =
   122             binder_types T ---> nth_elem (body_index dt, rec_result_Ts);
   123 
   124           val argTs = Ts @ map mk_argT recs
   125         in argTs ---> nth_elem (i, rec_result_Ts)
   126         end) constrs) descr');
   127 
   128     val rec_set_Ts = map (fn (T1, T2) => reccomb_fn_Ts ---> HOLogic.mk_setT
   129       (HOLogic.mk_prodT (T1, T2))) (recTs ~~ rec_result_Ts);
   130 
   131     val rec_fns = map (uncurry (mk_Free "f"))
   132       (reccomb_fn_Ts ~~ (1 upto (length reccomb_fn_Ts)));
   133     val rec_sets = map (fn c => list_comb (Const c, rec_fns))
   134       (rec_set_names ~~ rec_set_Ts);
   135 
   136     (* introduction rules for graph of primrec function *)
   137 
   138     fun make_rec_intr T set_name ((rec_intr_ts, l), (cname, cargs)) =
   139       let
   140         fun mk_prem ((dt, U), (j, k, prems, t1s, t2s)) =
   141           let val free1 = mk_Free "x" U j
   142           in (case (strip_dtyp dt, strip_type U) of
   143              ((_, DtRec m), (Us, _)) =>
   144                let
   145                  val free2 = mk_Free "y" (Us ---> nth_elem (m, rec_result_Ts)) k;
   146                  val i = length Us
   147                in (j + 1, k + 1, HOLogic.mk_Trueprop (HOLogic.list_all
   148                      (map (pair "x") Us, HOLogic.mk_mem (HOLogic.mk_prod
   149                        (app_bnds free1 i, app_bnds free2 i),
   150                          nth_elem (m, rec_sets)))) :: prems,
   151                    free1::t1s, free2::t2s)
   152                end
   153            | _ => (j + 1, k, prems, free1::t1s, t2s))
   154           end;
   155 
   156         val Ts = map (typ_of_dtyp descr' sorts) cargs;
   157         val (_, _, prems, t1s, t2s) = foldr mk_prem (cargs ~~ Ts, (1, 1, [], [], []))
   158 
   159       in (rec_intr_ts @ [Logic.list_implies (prems, HOLogic.mk_Trueprop (HOLogic.mk_mem
   160         (HOLogic.mk_prod (list_comb (Const (cname, Ts ---> T), t1s),
   161           list_comb (nth_elem (l, rec_fns), t1s @ t2s)), set_name)))], l + 1)
   162       end;
   163 
   164     val (rec_intr_ts, _) = foldl (fn (x, ((d, T), set_name)) =>
   165       foldl (make_rec_intr T set_name) (x, #3 (snd d)))
   166         (([], 0), descr' ~~ recTs ~~ rec_sets);
   167 
   168     val (thy1, {intrs = rec_intrs, elims = rec_elims, ...}) =
   169       setmp InductivePackage.quiet_mode (!quiet_mode)
   170         (InductivePackage.add_inductive_i false true big_rec_name' false false true
   171            rec_sets (map (fn x => (("", x), [])) rec_intr_ts) []) thy0;
   172 
   173     (* prove uniqueness and termination of primrec combinators *)
   174 
   175     val _ = message "Proving termination and uniqueness of primrec functions ...";
   176 
   177     fun mk_unique_tac ((tac, intrs), ((((i, (tname, _, constrs)), elim), T), T')) =
   178       let
   179         val distinct_tac = (etac Pair_inject 1) THEN
   180           (if i < length newTs then
   181              full_simp_tac (HOL_ss addsimps (nth_elem (i, dist_rewrites))) 1
   182            else full_simp_tac dist_ss 1);
   183 
   184         val inject = map (fn r => r RS iffD1)
   185           (if i < length newTs then nth_elem (i, constr_inject)
   186             else #inject (the (Symtab.lookup (dt_info, tname))));
   187 
   188         fun mk_unique_constr_tac n ((tac, intr::intrs, j), (cname, cargs)) =
   189           let
   190             val k = length (filter is_rec_type cargs)
   191 
   192           in (EVERY [DETERM tac,
   193                 REPEAT (etac ex1E 1), rtac ex1I 1,
   194                 DEPTH_SOLVE_1 (ares_tac [intr] 1),
   195                 REPEAT_DETERM_N k (etac thin_rl 1 THEN rotate_tac 1 1),
   196                 etac elim 1,
   197                 REPEAT_DETERM_N j distinct_tac,
   198                 etac Pair_inject 1, TRY (dresolve_tac inject 1),
   199                 REPEAT (etac conjE 1), hyp_subst_tac 1,
   200                 REPEAT (EVERY [etac allE 1, dtac mp 1, atac 1]),
   201                 TRY (hyp_subst_tac 1),
   202                 rtac refl 1,
   203                 REPEAT_DETERM_N (n - j - 1) distinct_tac],
   204               intrs, j + 1)
   205           end;
   206 
   207         val (tac', intrs', _) = foldl (mk_unique_constr_tac (length constrs))
   208           ((tac, intrs, 0), constrs);
   209 
   210       in (tac', intrs') end;
   211 
   212     val rec_unique_thms =
   213       let
   214         val rec_unique_ts = map (fn (((set_t, T1), T2), i) =>
   215           Const ("Ex1", (T2 --> HOLogic.boolT) --> HOLogic.boolT) $
   216             absfree ("y", T2, HOLogic.mk_mem (HOLogic.mk_prod
   217               (mk_Free "x" T1 i, Free ("y", T2)), set_t)))
   218                 (rec_sets ~~ recTs ~~ rec_result_Ts ~~ (1 upto length recTs));
   219         val cert = cterm_of (Theory.sign_of thy1)
   220         val insts = map (fn ((i, T), t) => absfree ("x" ^ (string_of_int i), T, t))
   221           ((1 upto length recTs) ~~ recTs ~~ rec_unique_ts);
   222         val induct' = cterm_instantiate ((map cert induct_Ps) ~~
   223           (map cert insts)) induct;
   224         val (tac, _) = foldl mk_unique_tac
   225           (((rtac induct' THEN_ALL_NEW ObjectLogic.atomize_tac) 1
   226               THEN rewtac (mk_meta_eq choice_eq), rec_intrs),
   227             descr' ~~ rec_elims ~~ recTs ~~ rec_result_Ts);
   228 
   229       in split_conj_thm (prove_goalw_cterm []
   230         (cert (HOLogic.mk_Trueprop (mk_conj rec_unique_ts))) (K [tac]))
   231       end;
   232 
   233     val rec_total_thms = map (fn r => r RS theI') rec_unique_thms;
   234 
   235     (* define primrec combinators *)
   236 
   237     val big_reccomb_name = (space_implode "_" new_type_names) ^ "_rec";
   238     val reccomb_names = map (Sign.full_name (Theory.sign_of thy1))
   239       (if length descr' = 1 then [big_reccomb_name] else
   240         (map ((curry (op ^) (big_reccomb_name ^ "_")) o string_of_int)
   241           (1 upto (length descr'))));
   242     val reccombs = map (fn ((name, T), T') => list_comb
   243       (Const (name, reccomb_fn_Ts @ [T] ---> T'), rec_fns))
   244         (reccomb_names ~~ recTs ~~ rec_result_Ts);
   245 
   246     val (thy2, reccomb_defs) = thy1 |>
   247       Theory.add_consts_i (map (fn ((name, T), T') =>
   248         (Sign.base_name name, reccomb_fn_Ts @ [T] ---> T', NoSyn))
   249           (reccomb_names ~~ recTs ~~ rec_result_Ts)) |>
   250       (PureThy.add_defs_i false o map Thm.no_attributes) (map (fn ((((name, comb), set), T), T') =>
   251         ((Sign.base_name name) ^ "_def", Logic.mk_equals (comb, absfree ("x", T,
   252            Const ("The", (T' --> HOLogic.boolT) --> T') $ absfree ("y", T',
   253              HOLogic.mk_mem (HOLogic.mk_prod (Free ("x", T), Free ("y", T')), set))))))
   254                (reccomb_names ~~ reccombs ~~ rec_sets ~~ recTs ~~ rec_result_Ts)) |>>
   255       parent_path flat_names;
   256 
   257 
   258     (* prove characteristic equations for primrec combinators *)
   259 
   260     val _ = message "Proving characteristic theorems for primrec combinators ..."
   261 
   262     val rec_thms = map (fn t => prove_goalw_cterm reccomb_defs
   263       (cterm_of (Theory.sign_of thy2) t) (fn _ =>
   264         [rtac the1_equality 1,
   265          resolve_tac rec_unique_thms 1,
   266          resolve_tac rec_intrs 1,
   267          REPEAT (rtac allI 1 ORELSE resolve_tac rec_total_thms 1)]))
   268            (DatatypeProp.make_primrecs new_type_names descr sorts thy2)
   269 
   270   in
   271     thy2 |> Theory.add_path (space_implode "_" new_type_names) |>
   272     PureThy.add_thmss [(("recs", rec_thms), [])] |>>
   273     Theory.parent_path |> apsnd (pair reccomb_names o flat)
   274   end;
   275 
   276 
   277 (***************************** case combinators *******************************)
   278 
   279 fun prove_case_thms flat_names new_type_names descr sorts reccomb_names primrec_thms thy =
   280   let
   281     val _ = message "Proving characteristic theorems for case combinators ...";
   282 
   283     val thy1 = add_path flat_names (space_implode "_" new_type_names) thy;
   284 
   285     val descr' = flat descr;
   286     val recTs = get_rec_types descr' sorts;
   287     val used = foldr add_typ_tfree_names (recTs, []);
   288     val newTs = take (length (hd descr), recTs);
   289     val T' = TFree (variant used "'t", HOLogic.typeS);
   290 
   291     fun mk_dummyT dt = binder_types (typ_of_dtyp descr' sorts dt) ---> T';
   292 
   293     val case_dummy_fns = map (fn (_, (_, _, constrs)) => map (fn (_, cargs) =>
   294       let
   295         val Ts = map (typ_of_dtyp descr' sorts) cargs;
   296         val Ts' = map mk_dummyT (filter is_rec_type cargs)
   297       in Const ("arbitrary", Ts @ Ts' ---> T')
   298       end) constrs) descr';
   299 
   300     val case_names = map (fn s =>
   301       Sign.full_name (Theory.sign_of thy1) (s ^ "_case")) new_type_names;
   302 
   303     (* define case combinators via primrec combinators *)
   304 
   305     val (case_defs, thy2) = foldl (fn ((defs, thy),
   306       ((((i, (_, _, constrs)), T), name), recname)) =>
   307         let
   308           val (fns1, fns2) = ListPair.unzip (map (fn ((_, cargs), j) =>
   309             let
   310               val Ts = map (typ_of_dtyp descr' sorts) cargs;
   311               val Ts' = Ts @ map mk_dummyT (filter is_rec_type cargs);
   312               val frees' = map (uncurry (mk_Free "x")) (Ts' ~~ (1 upto length Ts'));
   313               val frees = take (length cargs, frees');
   314               val free = mk_Free "f" (Ts ---> T') j
   315             in
   316              (free, list_abs_free (map dest_Free frees',
   317                list_comb (free, frees)))
   318             end) (constrs ~~ (1 upto length constrs)));
   319 
   320           val caseT = (map (snd o dest_Free) fns1) @ [T] ---> T';
   321           val fns = (flat (take (i, case_dummy_fns))) @
   322             fns2 @ (flat (drop (i + 1, case_dummy_fns)));
   323           val reccomb = Const (recname, (map fastype_of fns) @ [T] ---> T');
   324           val decl = (Sign.base_name name, caseT, NoSyn);
   325           val def = ((Sign.base_name name) ^ "_def",
   326             Logic.mk_equals (list_comb (Const (name, caseT), fns1),
   327               list_comb (reccomb, (flat (take (i, case_dummy_fns))) @
   328                 fns2 @ (flat (drop (i + 1, case_dummy_fns))) )));
   329           val (thy', [def_thm]) = thy |>
   330             Theory.add_consts_i [decl] |> (PureThy.add_defs_i false o map Thm.no_attributes) [def];
   331 
   332         in (defs @ [def_thm], thy')
   333         end) (([], thy1), (hd descr) ~~ newTs ~~ case_names ~~
   334           (take (length newTs, reccomb_names)));
   335 
   336     val case_thms = map (map (fn t => prove_goalw_cterm (case_defs @
   337       (map mk_meta_eq primrec_thms)) (cterm_of (Theory.sign_of thy2) t)
   338         (fn _ => [rtac refl 1])))
   339           (DatatypeProp.make_cases new_type_names descr sorts thy2)
   340 
   341   in
   342     thy2 |>
   343     parent_path flat_names |>
   344     store_thmss "cases" new_type_names case_thms |>
   345     apsnd (rpair case_names)
   346   end;
   347 
   348 
   349 (******************************* case splitting *******************************)
   350 
   351 fun prove_split_thms new_type_names descr sorts constr_inject dist_rewrites
   352     casedist_thms case_thms thy =
   353   let
   354     val _ = message "Proving equations for case splitting ...";
   355 
   356     val descr' = flat descr;
   357     val recTs = get_rec_types descr' sorts;
   358     val newTs = take (length (hd descr), recTs);
   359 
   360     fun prove_split_thms ((((((t1, t2), inject), dist_rewrites'),
   361         exhaustion), case_thms'), T) =
   362       let
   363         val cert = cterm_of (Theory.sign_of thy);
   364         val _ $ (_ $ lhs $ _) = hd (Logic.strip_assums_hyp (hd (prems_of exhaustion)));
   365         val exhaustion' = cterm_instantiate
   366           [(cert lhs, cert (Free ("x", T)))] exhaustion;
   367         val tacsf = K [rtac exhaustion' 1, ALLGOALS (asm_simp_tac
   368           (HOL_ss addsimps (dist_rewrites' @ inject @ case_thms')))]
   369       in
   370         (prove_goalw_cterm [] (cert t1) tacsf,
   371          prove_goalw_cterm [] (cert t2) tacsf)
   372       end;
   373 
   374     val split_thm_pairs = map prove_split_thms
   375       ((DatatypeProp.make_splits new_type_names descr sorts thy) ~~ constr_inject ~~
   376         dist_rewrites ~~ casedist_thms ~~ case_thms ~~ newTs);
   377 
   378     val (split_thms, split_asm_thms) = ListPair.unzip split_thm_pairs
   379 
   380   in
   381     thy |> store_thms "split" new_type_names split_thms |>>>
   382       store_thms "split_asm" new_type_names split_asm_thms |> apsnd ListPair.zip
   383   end;
   384 
   385 (******************************* size functions *******************************)
   386 
   387 fun prove_size_thms flat_names new_type_names descr sorts reccomb_names primrec_thms thy =
   388   if exists (fn (_, (_, _, constrs)) => exists (fn (_, cargs) => exists (fn dt =>
   389     is_rec_type dt andalso not (null (fst (strip_dtyp dt)))) cargs) constrs)
   390       (flat descr)
   391   then
   392     (thy, [])
   393   else
   394   let
   395     val _ = message "Proving equations for size function ...";
   396 
   397     val big_name = space_implode "_" new_type_names;
   398     val thy1 = add_path flat_names big_name thy;
   399 
   400     val descr' = flat descr;
   401     val recTs = get_rec_types descr' sorts;
   402 
   403     val size_name = "Nat.size";
   404     val size_names = replicate (length (hd descr)) size_name @
   405       map (Sign.full_name (Theory.sign_of thy1)) (DatatypeProp.indexify_names
   406         (map (fn T => name_of_typ T ^ "_size") (drop (length (hd descr), recTs))));
   407     val def_names = map (fn s => s ^ "_def") (DatatypeProp.indexify_names
   408       (map (fn T => name_of_typ T ^ "_size") recTs));
   409 
   410     fun plus (t1, t2) = Const ("op +", [HOLogic.natT, HOLogic.natT] ---> HOLogic.natT) $ t1 $ t2;
   411 
   412     fun make_sizefun (_, cargs) =
   413       let
   414         val Ts = map (typ_of_dtyp descr' sorts) cargs;
   415         val k = length (filter is_rec_type cargs);
   416         val t = if k = 0 then HOLogic.zero else
   417           foldl1 plus (map Bound (k - 1 downto 0) @ [HOLogic.mk_nat 1])
   418       in
   419         foldr (fn (T, t') => Abs ("x", T, t')) (Ts @ replicate k HOLogic.natT, t)
   420       end;
   421 
   422     val fs = flat (map (fn (_, (_, _, constrs)) => map make_sizefun constrs) descr');
   423     val fTs = map fastype_of fs;
   424 
   425     val (thy', size_def_thms) = thy1 |>
   426       Theory.add_consts_i (map (fn (s, T) =>
   427         (Sign.base_name s, T --> HOLogic.natT, NoSyn))
   428           (drop (length (hd descr), size_names ~~ recTs))) |>
   429       (PureThy.add_defs_i true o map Thm.no_attributes) (map (fn (((s, T), def_name), rec_name) =>
   430         (def_name, Logic.mk_equals (Const (s, T --> HOLogic.natT),
   431           list_comb (Const (rec_name, fTs @ [T] ---> HOLogic.natT), fs))))
   432             (size_names ~~ recTs ~~ def_names ~~ reccomb_names)) |>>
   433       parent_path flat_names;
   434 
   435     val rewrites = size_def_thms @ map mk_meta_eq primrec_thms;
   436 
   437     val size_thms = map (fn t => prove_goalw_cterm rewrites
   438       (cterm_of (Theory.sign_of thy') t) (fn _ => [rtac refl 1]))
   439         (DatatypeProp.make_size descr sorts thy')
   440 
   441   in
   442     thy' |> Theory.add_path big_name |>
   443     PureThy.add_thmss [(("size", size_thms), [])] |>>
   444     Theory.parent_path |> apsnd flat
   445   end;
   446 
   447 fun prove_weak_case_congs new_type_names descr sorts thy =
   448   let
   449     fun prove_weak_case_cong t =
   450        prove_goalw_cterm [] (cterm_of (Theory.sign_of thy) t)
   451          (fn prems => [rtac ((hd prems) RS arg_cong) 1])
   452 
   453     val weak_case_congs = map prove_weak_case_cong (DatatypeProp.make_weak_case_congs
   454       new_type_names descr sorts thy)
   455 
   456   in thy |> store_thms "weak_case_cong" new_type_names weak_case_congs end;
   457 
   458 (************************* additional theorems for TFL ************************)
   459 
   460 fun prove_nchotomys new_type_names descr sorts casedist_thms thy =
   461   let
   462     val _ = message "Proving additional theorems for TFL ...";
   463 
   464     fun prove_nchotomy (t, exhaustion) =
   465       let
   466         (* For goal i, select the correct disjunct to attack, then prove it *)
   467         fun tac i 0 = EVERY [TRY (rtac disjI1 i),
   468               hyp_subst_tac i, REPEAT (rtac exI i), rtac refl i]
   469           | tac i n = rtac disjI2 i THEN tac i (n - 1)
   470       in 
   471         prove_goalw_cterm [] (cterm_of (Theory.sign_of thy) t) (fn _ =>
   472           [rtac allI 1,
   473            exh_tac (K exhaustion) 1,
   474            ALLGOALS (fn i => tac i (i-1))])
   475       end;
   476 
   477     val nchotomys =
   478       map prove_nchotomy (DatatypeProp.make_nchotomys descr sorts ~~ casedist_thms)
   479 
   480   in thy |> store_thms "nchotomy" new_type_names nchotomys end;
   481 
   482 fun prove_case_congs new_type_names descr sorts nchotomys case_thms thy =
   483   let
   484     fun prove_case_cong ((t, nchotomy), case_rewrites) =
   485       let
   486         val (Const ("==>", _) $ tm $ _) = t;
   487         val (Const ("Trueprop", _) $ (Const ("op =", _) $ _ $ Ma)) = tm;
   488         val cert = cterm_of (Theory.sign_of thy);
   489         val nchotomy' = nchotomy RS spec;
   490         val nchotomy'' = cterm_instantiate
   491           [(cert (hd (add_term_vars (concl_of nchotomy', []))), cert Ma)] nchotomy'
   492       in
   493         prove_goalw_cterm [] (cert t) (fn prems => 
   494           let val simplify = asm_simp_tac (HOL_ss addsimps (prems @ case_rewrites))
   495           in [simp_tac (HOL_ss addsimps [hd prems]) 1,
   496               cut_facts_tac [nchotomy''] 1,
   497               REPEAT (etac disjE 1 THEN REPEAT (etac exE 1) THEN simplify 1),
   498               REPEAT (etac exE 1) THEN simplify 1 (* Get last disjunct *)]
   499           end)
   500       end;
   501 
   502     val case_congs = map prove_case_cong (DatatypeProp.make_case_congs
   503       new_type_names descr sorts thy ~~ nchotomys ~~ case_thms)
   504 
   505   in thy |> store_thms "case_cong" new_type_names case_congs end;
   506 
   507 end;