src/HOL/Tools/datatype_abs_proofs.ML
author wenzelm
Mon Nov 16 11:14:02 1998 +0100 (1998-11-16)
changeset 5891 92e0f5e6fd17
parent 5661 6ecb6ea25f19
child 6092 d9db67970c73
permissions -rw-r--r--
Attribute.tthms_of;
     1 (*  Title:      HOL/Tools/datatype_abs_proofs.ML
     2     ID:         $Id$
     3     Author:     Stefan Berghofer
     4     Copyright   1998  TU Muenchen
     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  - distinctness of constructors (external version)
    14  - equations for splitting "P (case ...)" expressions
    15  - datatype size function
    16  - "nchotomy" and "case_cong" theorems for TFL
    17 
    18 *)
    19 
    20 signature DATATYPE_ABS_PROOFS =
    21 sig
    22   val prove_casedist_thms : string list -> (int * (string * DatatypeAux.dtyp list *
    23     (string * DatatypeAux.dtyp list) list)) list list -> (string * sort) list ->
    24       thm -> theory -> theory * thm list
    25   val prove_primrec_thms : bool -> string list -> (int * (string * DatatypeAux.dtyp list *
    26     (string * DatatypeAux.dtyp list) list)) list list -> (string * sort) list ->
    27       DatatypeAux.datatype_info Symtab.table -> thm list list -> thm list list ->
    28         thm -> theory -> theory * string list * thm list
    29   val prove_case_thms : bool -> string list -> (int * (string * DatatypeAux.dtyp list *
    30     (string * DatatypeAux.dtyp list) list)) list list -> (string * sort) list ->
    31       string list -> thm list -> theory -> theory * string list * thm list list
    32   val prove_distinctness_thms : bool -> string list -> (int * (string * DatatypeAux.dtyp list *
    33     (string * DatatypeAux.dtyp list) list)) list list -> (string * sort) list ->
    34       thm list list -> thm list list -> theory -> theory * thm list list
    35   val prove_split_thms : string list -> (int * (string * DatatypeAux.dtyp list *
    36     (string * DatatypeAux.dtyp list) list)) list list -> (string * sort) list ->
    37       thm list list -> thm list list -> thm list -> thm list list -> theory ->
    38         theory * (thm * thm) list
    39   val prove_size_thms : bool -> string list -> (int * (string * DatatypeAux.dtyp list *
    40     (string * DatatypeAux.dtyp list) list)) list list -> (string * sort) list ->
    41       string list -> thm list -> theory -> theory * thm list
    42   val prove_nchotomys : string list -> (int * (string * DatatypeAux.dtyp list *
    43     (string * DatatypeAux.dtyp list) list)) list list -> (string * sort) list ->
    44       thm list -> theory -> theory * thm list
    45   val prove_case_congs : string list -> (int * (string * DatatypeAux.dtyp list *
    46     (string * DatatypeAux.dtyp list) list)) list list -> (string * sort) list ->
    47       thm list -> thm list list -> theory -> theory * thm list
    48 end;
    49 
    50 structure DatatypeAbsProofs : DATATYPE_ABS_PROOFS =
    51 struct
    52 
    53 open DatatypeAux;
    54 
    55 val thin = read_instantiate_sg (sign_of Set.thy) [("V", "?X : ?Y")] thin_rl;
    56 
    57 val (_ $ (_ $ (_ $ (distinct_f $ _) $ _))) = hd (prems_of distinct_lemma);
    58 
    59 (************************ case distinction theorems ***************************)
    60 
    61 fun prove_casedist_thms new_type_names descr sorts induct thy =
    62   let
    63     val _ = message "Proving case distinction theorems...";
    64 
    65     val descr' = flat descr;
    66     val recTs = get_rec_types descr' sorts;
    67     val newTs = take (length (hd descr), recTs);
    68 
    69     val induct_Ps = map head_of (dest_conj (HOLogic.dest_Trueprop (concl_of induct)));
    70 
    71     fun prove_casedist_thm ((i, t), T) =
    72       let
    73         val dummyPs = map (fn (Var (_, Type (_, [T', T'']))) =>
    74           Abs ("z", T', Const ("True", T''))) induct_Ps;
    75         val P = Abs ("z", T, HOLogic.imp $ HOLogic.mk_eq (Var (("a", 0), T), Bound 0) $
    76           Var (("P", 0), HOLogic.boolT))
    77         val insts = take (i, dummyPs) @ (P::(drop (i + 1, dummyPs)));
    78         val cert = cterm_of (sign_of thy);
    79         val insts' = (map cert induct_Ps) ~~ (map cert insts);
    80         val induct' = refl RS ((nth_elem (i,
    81           split_conj_thm (cterm_instantiate insts' induct))) RSN (2, rev_mp))
    82 
    83       in prove_goalw_cterm [] (cert t) (fn prems =>
    84         [rtac induct' 1,
    85          REPEAT (rtac TrueI 1),
    86          REPEAT ((rtac impI 1) THEN (eresolve_tac prems 1)),
    87          REPEAT (rtac TrueI 1)])
    88       end;
    89 
    90     val casedist_thms = map prove_casedist_thm ((0 upto (length newTs - 1)) ~~
    91       (DatatypeProp.make_casedists descr sorts) ~~ newTs)
    92 
    93   in
    94     (store_thms "exhaust" new_type_names casedist_thms thy, casedist_thms)
    95   end;
    96 
    97 (*************************** primrec combinators ******************************)
    98 
    99 fun prove_primrec_thms flat_names new_type_names descr sorts
   100     (dt_info : datatype_info Symtab.table) constr_inject dist_rewrites induct thy =
   101   let
   102     val _ = message "Constructing primrec combinators...";
   103 
   104     val big_name = space_implode "_" new_type_names;
   105     val thy0 = add_path flat_names big_name thy;
   106 
   107     val descr' = flat descr;
   108     val recTs = get_rec_types descr' sorts;
   109     val used = foldr add_typ_tfree_names (recTs, []);
   110     val newTs = take (length (hd descr), recTs);
   111 
   112     val induct_Ps = map head_of (dest_conj (HOLogic.dest_Trueprop (concl_of induct)));
   113 
   114     val big_rec_name' = big_name ^ "_rec_set";
   115     val rec_set_names = map (Sign.full_name (sign_of thy0))
   116       (if length descr' = 1 then [big_rec_name'] else
   117         (map ((curry (op ^) (big_rec_name' ^ "_")) o string_of_int)
   118           (1 upto (length descr'))));
   119 
   120     val rec_result_Ts = map TFree (variantlist (replicate (length descr') "'t", used) ~~
   121       replicate (length descr') HOLogic.termS);
   122 
   123     val reccomb_fn_Ts = flat (map (fn (i, (_, _, constrs)) =>
   124       map (fn (_, cargs) =>
   125         let
   126           val recs = filter is_rec_type cargs;
   127           val argTs = (map (typ_of_dtyp descr' sorts) cargs) @
   128             (map (fn r => nth_elem (dest_DtRec r, rec_result_Ts)) recs)
   129         in argTs ---> nth_elem (i, rec_result_Ts)
   130         end) constrs) descr');
   131 
   132     val rec_set_Ts = map (fn (T1, T2) => reccomb_fn_Ts ---> HOLogic.mk_setT
   133       (HOLogic.mk_prodT (T1, T2))) (recTs ~~ rec_result_Ts);
   134 
   135     val rec_fns = map (uncurry (mk_Free "f"))
   136       (reccomb_fn_Ts ~~ (1 upto (length reccomb_fn_Ts)));
   137     val rec_sets = map (fn c => list_comb (Const c, rec_fns))
   138       (rec_set_names ~~ rec_set_Ts);
   139 
   140     (* introduction rules for graph of primrec function *)
   141 
   142     fun make_rec_intr T set_name ((rec_intr_ts, l), (cname, cargs)) =
   143       let
   144         fun mk_prem (dt, (j, k, prems, t1s, t2s)) =
   145           let
   146             val T = typ_of_dtyp descr' sorts dt;
   147             val free1 = mk_Free "x" T j
   148           in (case dt of
   149              DtRec m =>
   150                let val free2 = mk_Free "y" (nth_elem (m, rec_result_Ts)) k
   151                in (j + 1, k + 1, (HOLogic.mk_Trueprop (HOLogic.mk_mem
   152                  (HOLogic.mk_prod (free1, free2), nth_elem (m, rec_sets))))::prems,
   153                    free1::t1s, free2::t2s)
   154                end
   155            | _ => (j + 1, k, prems, free1::t1s, t2s))
   156           end;
   157 
   158         val Ts = map (typ_of_dtyp descr' sorts) cargs;
   159         val (_, _, prems, t1s, t2s) = foldr mk_prem (cargs, (1, 1, [], [], []))
   160 
   161       in (rec_intr_ts @ [Logic.list_implies (prems, HOLogic.mk_Trueprop (HOLogic.mk_mem
   162         (HOLogic.mk_prod (list_comb (Const (cname, Ts ---> T), t1s),
   163           list_comb (nth_elem (l, rec_fns), t1s @ t2s)), set_name)))], l + 1)
   164       end;
   165 
   166     val (rec_intr_ts, _) = foldl (fn (x, ((d, T), set_name)) =>
   167       foldl (make_rec_intr T set_name) (x, #3 (snd d)))
   168         (([], 0), descr' ~~ recTs ~~ rec_sets);
   169 
   170     val (thy1, {intrs = rec_intrs, elims = rec_elims, ...}) =
   171       setmp InductivePackage.quiet_mode (!quiet_mode)
   172         (InductivePackage.add_inductive_i false true big_rec_name' false false true
   173            rec_sets rec_intr_ts [] []) thy0;
   174 
   175     (* prove uniqueness and termination of primrec combinators *)
   176 
   177     val _ = message "Proving termination and uniqueness of primrec functions...";
   178 
   179     fun mk_unique_tac ((tac, intrs), ((((i, (tname, _, constrs)), elim), T), T')) =
   180       let
   181         val distinct_tac = (etac Pair_inject 1) THEN
   182           (if i < length newTs then
   183              full_simp_tac (HOL_ss addsimps (nth_elem (i, dist_rewrites))) 1
   184            else full_simp_tac (HOL_ss addsimps
   185              ((#distinct (the (Symtab.lookup (dt_info, tname)))) @
   186                [Suc_Suc_eq, Suc_not_Zero, Zero_not_Suc])) 1);
   187 
   188         val inject = map (fn r => r RS iffD1)
   189           (if i < length newTs then nth_elem (i, constr_inject)
   190             else #inject (the (Symtab.lookup (dt_info, tname))));
   191 
   192         fun mk_unique_constr_tac n ((tac, intr::intrs, j), (cname, cargs)) =
   193           let
   194             val k = length (filter is_rec_type cargs)
   195 
   196           in (EVERY [DETERM tac,
   197                 REPEAT (etac ex1E 1), rtac ex1I 1,
   198                 DEPTH_SOLVE_1 (ares_tac [intr] 1),
   199                 REPEAT_DETERM_N k (etac thin 1),
   200                 etac elim 1,
   201                 REPEAT_DETERM_N j distinct_tac,
   202                 etac Pair_inject 1, TRY (dresolve_tac inject 1),
   203                 REPEAT (etac conjE 1), hyp_subst_tac 1,
   204                 REPEAT (etac allE 1),
   205                 REPEAT (dtac mp 1 THEN atac 1),
   206                 TRY (hyp_subst_tac 1),
   207                 rtac refl 1,
   208                 REPEAT_DETERM_N (n - j - 1) distinct_tac],
   209               intrs, j + 1)
   210           end;
   211 
   212         val (tac', intrs', _) = foldl (mk_unique_constr_tac (length constrs))
   213           ((tac, intrs, 0), constrs);
   214 
   215       in (tac', intrs') end;
   216 
   217     val rec_unique_thms =
   218       let
   219         val rec_unique_ts = map (fn (((set_t, T1), T2), i) =>
   220           Const ("Ex1", (T2 --> HOLogic.boolT) --> HOLogic.boolT) $
   221             absfree ("y", T2, HOLogic.mk_mem (HOLogic.mk_prod
   222               (mk_Free "x" T1 i, Free ("y", T2)), set_t)))
   223                 (rec_sets ~~ recTs ~~ rec_result_Ts ~~ (1 upto length recTs));
   224         val cert = cterm_of (sign_of thy1)
   225         val insts = map (fn ((i, T), t) => absfree ("x" ^ (string_of_int i), T, t))
   226           ((1 upto length recTs) ~~ recTs ~~ rec_unique_ts);
   227         val induct' = cterm_instantiate ((map cert induct_Ps) ~~
   228           (map cert insts)) induct;
   229         val (tac, _) = foldl mk_unique_tac
   230           ((rtac induct' 1, rec_intrs), descr' ~~ rec_elims ~~ recTs ~~ rec_result_Ts)
   231 
   232       in split_conj_thm (prove_goalw_cterm []
   233         (cert (HOLogic.mk_Trueprop (mk_conj rec_unique_ts))) (K [tac]))
   234       end;
   235 
   236     val rec_total_thms = map (fn r =>
   237       r RS ex1_implies_ex RS (select_eq_Ex RS iffD2)) rec_unique_thms;
   238 
   239     (* define primrec combinators *)
   240 
   241     val big_reccomb_name = (space_implode "_" new_type_names) ^ "_rec";
   242     val reccomb_names = map (Sign.full_name (sign_of thy1))
   243       (if length descr' = 1 then [big_reccomb_name] else
   244         (map ((curry (op ^) (big_reccomb_name ^ "_")) o string_of_int)
   245           (1 upto (length descr'))));
   246     val reccombs = map (fn ((name, T), T') => list_comb
   247       (Const (name, reccomb_fn_Ts @ [T] ---> T'), rec_fns))
   248         (reccomb_names ~~ recTs ~~ rec_result_Ts);
   249 
   250     val thy2 = thy1 |>
   251       Theory.add_consts_i (map (fn ((name, T), T') =>
   252         (Sign.base_name name, reccomb_fn_Ts @ [T] ---> T', NoSyn))
   253           (reccomb_names ~~ recTs ~~ rec_result_Ts)) |>
   254       Theory.add_defs_i (map (fn ((((name, comb), set), T), T') =>
   255         ((Sign.base_name name) ^ "_def", Logic.mk_equals
   256           (comb $ Free ("x", T),
   257            Const ("Eps", (T' --> HOLogic.boolT) --> T') $ absfree ("y", T',
   258              HOLogic.mk_mem (HOLogic.mk_prod (Free ("x", T), Free ("y", T')), set)))))
   259                (reccomb_names ~~ reccombs ~~ rec_sets ~~ recTs ~~ rec_result_Ts)) |>
   260       parent_path flat_names;
   261 
   262     val reccomb_defs = map ((get_def thy2) o Sign.base_name) reccomb_names;
   263 
   264     (* prove characteristic equations for primrec combinators *)
   265 
   266     val _ = message "Proving characteristic theorems for primrec combinators..."
   267 
   268     val rec_thms = map (fn t => prove_goalw_cterm reccomb_defs
   269       (cterm_of (sign_of thy2) t) (fn _ =>
   270         [rtac select1_equality 1,
   271          resolve_tac rec_unique_thms 1,
   272          resolve_tac rec_intrs 1,
   273          REPEAT (resolve_tac rec_total_thms 1)]))
   274            (DatatypeProp.make_primrecs new_type_names descr sorts thy2)
   275 
   276   in
   277     (thy2 |> Theory.add_path (space_implode "_" new_type_names) |>
   278              PureThy.add_tthmss [(("recs", Attribute.tthms_of rec_thms), [])] |>
   279              Theory.parent_path,
   280      reccomb_names, rec_thms)
   281   end;
   282 
   283 (***************************** case combinators *******************************)
   284 
   285 fun prove_case_thms flat_names new_type_names descr sorts reccomb_names primrec_thms thy =
   286   let
   287     val _ = message "Proving characteristic theorems for case combinators...";
   288 
   289     val thy1 = add_path flat_names (space_implode "_" new_type_names) thy;
   290 
   291     val descr' = flat descr;
   292     val recTs = get_rec_types descr' sorts;
   293     val used = foldr add_typ_tfree_names (recTs, []);
   294     val newTs = take (length (hd descr), recTs);
   295     val T' = TFree (variant used "'t", HOLogic.termS);
   296 
   297     val case_dummy_fns = map (fn (_, (_, _, constrs)) => map (fn (_, cargs) =>
   298       let
   299         val Ts = map (typ_of_dtyp descr' sorts) cargs;
   300         val Ts' = replicate (length (filter is_rec_type cargs)) T'
   301       in Const ("arbitrary", Ts @ Ts' ---> T')
   302       end) constrs) descr';
   303 
   304     val case_names = map (fn s =>
   305       Sign.full_name (sign_of thy1) (s ^ "_case")) new_type_names;
   306 
   307     (* define case combinators via primrec combinators *)
   308 
   309     val (case_defs, thy2) = foldl (fn ((defs, thy),
   310       ((((i, (_, _, constrs)), T), name), recname)) =>
   311         let
   312           val (fns1, fns2) = ListPair.unzip (map (fn ((_, cargs), j) =>
   313             let
   314               val Ts = map (typ_of_dtyp descr' sorts) cargs;
   315               val Ts' = Ts @ (replicate (length (filter is_rec_type cargs)) T');
   316               val frees' = map (uncurry (mk_Free "x")) (Ts' ~~ (1 upto length Ts'));
   317               val frees = take (length cargs, frees');
   318               val free = mk_Free "f" (Ts ---> T') j
   319             in
   320              (free, list_abs_free (map dest_Free frees',
   321                list_comb (free, frees)))
   322             end) (constrs ~~ (1 upto length constrs)));
   323 
   324           val caseT = (map (snd o dest_Free) fns1) @ [T] ---> T';
   325           val fns = (flat (take (i, case_dummy_fns))) @
   326             fns2 @ (flat (drop (i + 1, case_dummy_fns)));
   327           val reccomb = Const (recname, (map fastype_of fns) @ [T] ---> T');
   328           val decl = (Sign.base_name name, caseT, NoSyn);
   329           val def = ((Sign.base_name name) ^ "_def",
   330             Logic.mk_equals (list_comb (Const (name, caseT), fns1),
   331               list_comb (reccomb, (flat (take (i, case_dummy_fns))) @
   332                 fns2 @ (flat (drop (i + 1, case_dummy_fns))) )));
   333           val thy' = thy |>
   334             Theory.add_consts_i [decl] |> Theory.add_defs_i [def];
   335 
   336         in (defs @ [get_def thy' (Sign.base_name name)], thy')
   337         end) (([], thy1), (hd descr) ~~ newTs ~~ case_names ~~
   338           (take (length newTs, reccomb_names)));
   339 
   340     val case_thms = map (map (fn t => prove_goalw_cterm (case_defs @
   341       (map mk_meta_eq primrec_thms)) (cterm_of (sign_of thy2) t)
   342         (fn _ => [rtac refl 1])))
   343           (DatatypeProp.make_cases new_type_names descr sorts thy2);
   344 
   345     val thy3 = thy2 |> Theory.add_trrules_i
   346       (DatatypeProp.make_case_trrules new_type_names descr) |>
   347       parent_path flat_names;
   348 
   349   in
   350     (store_thmss "cases" new_type_names case_thms thy3, case_names, case_thms)
   351   end;
   352 
   353 (************************ distinctness of constructors ************************)
   354 
   355 fun prove_distinctness_thms flat_names new_type_names descr sorts dist_rewrites case_thms thy =
   356   let
   357     val thy1 = add_path flat_names (space_implode "_" new_type_names) thy;
   358 
   359     val descr' = flat descr;
   360     val recTs = get_rec_types descr' sorts;
   361     val newTs = take (length (hd descr), recTs);
   362 
   363     (*--------------------------------------------------------------------*)
   364     (* define t_ord - functions for proving distinctness of constructors: *)
   365     (*  t_ord C_i ... = i                                                 *)
   366     (*--------------------------------------------------------------------*)
   367 
   368     fun define_ord ((thy, ord_defs), (((_, (_, _, constrs)), T), tname)) =
   369       if length constrs < DatatypeProp.dtK then (thy, ord_defs)
   370       else
   371         let
   372           val Tss = map ((map (typ_of_dtyp descr' sorts)) o snd) constrs;
   373           val ts = map HOLogic.mk_nat (0 upto length constrs - 1);
   374           val mk_abs = foldr (fn (T, t') => Abs ("x", T, t'));
   375           val fs = map mk_abs (Tss ~~ ts);
   376           val fTs = map (fn Ts => Ts ---> HOLogic.natT) Tss;
   377           val ord_name = Sign.full_name (sign_of thy) (tname ^ "_ord");
   378           val case_name = Sign.intern_const (sign_of thy) (tname ^ "_case");
   379           val ordT = T --> HOLogic.natT;
   380           val caseT = fTs ---> ordT;
   381           val defpair = (tname ^ "_ord_def", Logic.mk_equals
   382             (Const (ord_name, ordT), list_comb (Const (case_name, caseT), fs)));
   383           val thy' = thy |>
   384             Theory.add_consts_i [(tname ^ "_ord", ordT, NoSyn)] |>
   385             Theory.add_defs_i [defpair];
   386           val def = get_def thy' (tname ^ "_ord")
   387 
   388         in (thy', ord_defs @ [def]) end;
   389 
   390     val (thy2, ord_defs) =
   391       foldl define_ord ((thy1, []), (hd descr) ~~ newTs ~~ new_type_names);
   392 
   393     (**** number of constructors < dtK ****)
   394 
   395     fun prove_distinct_thms _ [] = []
   396       | prove_distinct_thms dist_rewrites' (t::_::ts) =
   397           let
   398             val dist_thm = prove_goalw_cterm [] (cterm_of (sign_of thy2) t) (fn _ =>
   399               [simp_tac (HOL_ss addsimps dist_rewrites') 1])
   400           in dist_thm::(standard (dist_thm RS not_sym))::
   401             (prove_distinct_thms dist_rewrites' ts)
   402           end;
   403 
   404     val distinct_thms = map (fn ((((_, (_, _, constrs)), ts),
   405       dist_rewrites'), case_thms) =>
   406         if length constrs < DatatypeProp.dtK then
   407           prove_distinct_thms dist_rewrites' ts
   408         else 
   409           let
   410             val t::ts' = rev ts;
   411             val (_ $ (_ $ (_ $ (f $ _) $ _))) = hd (Logic.strip_imp_prems t);
   412             val cert = cterm_of (sign_of thy2);
   413             val distinct_lemma' = cterm_instantiate
   414               [(cert distinct_f, cert f)] distinct_lemma;
   415             val rewrites = ord_defs @ (map mk_meta_eq case_thms)
   416           in
   417             (map (fn t => prove_goalw_cterm rewrites (cert t)
   418               (fn _ => [rtac refl 1])) (rev ts')) @ [standard distinct_lemma']
   419           end) ((hd descr) ~~ (DatatypeProp.make_distincts new_type_names
   420             descr sorts thy2) ~~ dist_rewrites ~~ case_thms)
   421 
   422   in
   423     (thy2 |> parent_path flat_names |>
   424              store_thmss "distinct" new_type_names distinct_thms,
   425      distinct_thms)
   426   end;
   427 
   428 (******************************* case splitting *******************************)
   429 
   430 fun prove_split_thms new_type_names descr sorts constr_inject dist_rewrites
   431     casedist_thms case_thms thy =
   432   let
   433     val _ = message "Proving equations for case splitting...";
   434 
   435     val descr' = flat descr;
   436     val recTs = get_rec_types descr' sorts;
   437     val newTs = take (length (hd descr), recTs);
   438 
   439     fun prove_split_thms ((((((t1, t2), inject), dist_rewrites'),
   440         exhaustion), case_thms'), T) =
   441       let
   442         val cert = cterm_of (sign_of thy);
   443         val _ $ (_ $ lhs $ _) = hd (Logic.strip_assums_hyp (hd (prems_of exhaustion)));
   444         val exhaustion' = cterm_instantiate
   445           [(cert lhs, cert (Free ("x", T)))] exhaustion;
   446         val tacsf = K [rtac exhaustion' 1, ALLGOALS (asm_simp_tac
   447           (HOL_ss addsimps (dist_rewrites' @ inject @ case_thms')))]
   448       in
   449         (prove_goalw_cterm [] (cert t1) tacsf,
   450          prove_goalw_cterm [] (cert t2) tacsf)
   451       end;
   452 
   453     val split_thm_pairs = map prove_split_thms
   454       ((DatatypeProp.make_splits new_type_names descr sorts thy) ~~ constr_inject ~~
   455         dist_rewrites ~~ casedist_thms ~~ case_thms ~~ newTs);
   456 
   457     val (split_thms, split_asm_thms) = ListPair.unzip split_thm_pairs
   458 
   459   in
   460     (thy |> store_thms "split" new_type_names split_thms |>
   461             store_thms "split_asm" new_type_names split_asm_thms,
   462      split_thm_pairs)
   463   end;
   464 
   465 (******************************* size functions *******************************)
   466 
   467 fun prove_size_thms flat_names new_type_names descr sorts reccomb_names primrec_thms thy =
   468   let
   469     val _ = message "Proving equations for size function...";
   470 
   471     val big_name = space_implode "_" new_type_names;
   472     val thy1 = add_path flat_names big_name thy;
   473 
   474     val descr' = flat descr;
   475     val recTs = get_rec_types descr' sorts;
   476 
   477     val big_size_name = space_implode "_" new_type_names ^ "_size";
   478     val size_name = Sign.intern_const (sign_of (the (get_thy "Arith" thy1))) "size";
   479     val size_names = replicate (length (hd descr)) size_name @
   480       map (Sign.full_name (sign_of thy1))
   481         (if length (flat (tl descr)) = 1 then [big_size_name] else
   482           map (fn i => big_size_name ^ "_" ^ string_of_int i)
   483             (1 upto length (flat (tl descr))));
   484     val def_names = map (fn i => big_size_name ^ "_def_" ^ string_of_int i)
   485       (1 upto length recTs);
   486 
   487     val plus_t = Const ("op +", [HOLogic.natT, HOLogic.natT] ---> HOLogic.natT);
   488 
   489     fun make_sizefun (_, cargs) =
   490       let
   491         val Ts = map (typ_of_dtyp descr' sorts) cargs;
   492         val k = length (filter is_rec_type cargs);
   493         val t = if k = 0 then HOLogic.zero else
   494           foldl1 (app plus_t) (map Bound (k - 1 downto 0) @ [HOLogic.mk_nat 1])
   495       in
   496         foldr (fn (T, t') => Abs ("x", T, t')) (Ts @ replicate k HOLogic.natT, t)
   497       end;
   498 
   499     val fs = flat (map (fn (_, (_, _, constrs)) => map make_sizefun constrs) descr');
   500     val fTs = map fastype_of fs;
   501 
   502     val thy' = thy1 |>
   503       Theory.add_consts_i (map (fn (s, T) =>
   504         (Sign.base_name s, T --> HOLogic.natT, NoSyn))
   505           (drop (length (hd descr), size_names ~~ recTs))) |>
   506       Theory.add_defs_i (map (fn (((s, T), def_name), rec_name) =>
   507         (def_name, Logic.mk_equals (Const (s, T --> HOLogic.natT),
   508           list_comb (Const (rec_name, fTs @ [T] ---> HOLogic.natT), fs))))
   509             (size_names ~~ recTs ~~ def_names ~~ reccomb_names)) |>
   510       parent_path flat_names;
   511 
   512     val size_def_thms = map (get_axiom thy') def_names;
   513     val rewrites = size_def_thms @ map mk_meta_eq primrec_thms;
   514 
   515     val size_thms = map (fn t => prove_goalw_cterm rewrites
   516       (cterm_of (sign_of thy') t) (fn _ => [rtac refl 1]))
   517         (DatatypeProp.make_size new_type_names descr sorts thy')
   518 
   519   in
   520     (thy' |> Theory.add_path big_name |>
   521              PureThy.add_tthmss [(("size", Attribute.tthms_of size_thms), [])] |>
   522              Theory.parent_path,
   523      size_thms)
   524   end;
   525 
   526 (************************* additional theorems for TFL ************************)
   527 
   528 fun prove_nchotomys new_type_names descr sorts casedist_thms thy =
   529   let
   530     val _ = message "Proving additional theorems for TFL...";
   531 
   532     fun prove_nchotomy (t, exhaustion) =
   533       let
   534         (* For goal i, select the correct disjunct to attack, then prove it *)
   535         fun tac i 0 = EVERY [TRY (rtac disjI1 i),
   536               hyp_subst_tac i, REPEAT (rtac exI i), rtac refl i]
   537           | tac i n = rtac disjI2 i THEN tac i (n - 1)
   538       in 
   539         prove_goalw_cterm [] (cterm_of (sign_of thy) t) (fn _ =>
   540           [rtac allI 1,
   541            exh_tac (K exhaustion) 1,
   542            ALLGOALS (fn i => tac i (i-1))])
   543       end;
   544 
   545     val nchotomys =
   546       map prove_nchotomy (DatatypeProp.make_nchotomys descr sorts ~~ casedist_thms)
   547 
   548   in
   549     (store_thms "nchotomy" new_type_names nchotomys thy, nchotomys)
   550   end;
   551 
   552 fun prove_case_congs new_type_names descr sorts nchotomys case_thms thy =
   553   let
   554     fun prove_case_cong ((t, nchotomy), case_rewrites) =
   555       let
   556         val (Const ("==>", _) $ tm $ _) = t;
   557         val (Const ("Trueprop", _) $ (Const ("op =", _) $ _ $ Ma)) = tm;
   558         val cert = cterm_of (sign_of thy);
   559         val nchotomy' = nchotomy RS spec;
   560         val nchotomy'' = cterm_instantiate
   561           [(cert (hd (add_term_vars (concl_of nchotomy', []))), cert Ma)] nchotomy'
   562       in
   563         prove_goalw_cterm [] (cert t) (fn prems => 
   564           let val simplify = asm_simp_tac (HOL_ss addsimps (prems @ case_rewrites))
   565           in [simp_tac (HOL_ss addsimps [hd prems]) 1,
   566               cut_facts_tac [nchotomy''] 1,
   567               REPEAT (etac disjE 1 THEN REPEAT (etac exE 1) THEN simplify 1),
   568               REPEAT (etac exE 1) THEN simplify 1 (* Get last disjunct *)]
   569           end)
   570       end;
   571 
   572     val case_congs = map prove_case_cong (DatatypeProp.make_case_congs
   573       new_type_names descr sorts thy ~~ nchotomys ~~ case_thms)
   574 
   575   in
   576     (store_thms "case_cong" new_type_names case_congs thy, case_congs)
   577   end;
   578 
   579 end;