src/HOL/Tools/datatype_rep_proofs.ML
author paulson
Thu Sep 26 10:51:29 2002 +0200 (2002-09-26)
changeset 13585 db4005b40cc6
parent 13340 9b0332344ae2
child 13641 63d1790a43ed
permissions -rw-r--r--
Converted Fun to Isar style.
Moved Pi, funcset, restrict from Fun.thy to Library/FuncSet.thy.
Renamed constant "Fun.op o" to "Fun.comp"
     1 (*  Title:      HOL/Tools/datatype_rep_proofs.ML
     2     ID:         $Id$
     3     Author:     Stefan Berghofer, TU Muenchen
     4     License:    GPL (GNU GENERAL PUBLIC LICENSE)
     5 
     6 Definitional introduction of datatypes
     7 Proof of characteristic theorems:
     8 
     9  - injectivity of constructors
    10  - distinctness of constructors
    11  - induction theorem
    12 
    13 *)
    14 
    15 signature DATATYPE_REP_PROOFS =
    16 sig
    17   val representation_proofs : bool -> DatatypeAux.datatype_info Symtab.table ->
    18     string list -> (int * (string * DatatypeAux.dtyp list *
    19       (string * DatatypeAux.dtyp list) list)) list list -> (string * sort) list ->
    20         (string * mixfix) list -> (string * mixfix) list list -> theory attribute
    21           -> theory -> theory * thm list list * thm list list * thm list list *
    22             DatatypeAux.simproc_dist list * thm
    23 end;
    24 
    25 structure DatatypeRepProofs : DATATYPE_REP_PROOFS =
    26 struct
    27 
    28 open DatatypeAux;
    29 
    30 val (_ $ (_ $ (_ $ (distinct_f $ _) $ _))) = hd (prems_of distinct_lemma);
    31 
    32 
    33 (** theory context references **)
    34 
    35 val f_myinv_f = thm "f_myinv_f";
    36 val myinv_f_f = thm "myinv_f_f";
    37 
    38 
    39 fun exh_thm_of (dt_info : datatype_info Symtab.table) tname =
    40   #exhaustion (the (Symtab.lookup (dt_info, tname)));
    41 
    42 (******************************************************************************)
    43 
    44 fun representation_proofs flat_names (dt_info : datatype_info Symtab.table)
    45       new_type_names descr sorts types_syntax constr_syntax case_names_induct thy =
    46   let
    47     val Datatype_thy =
    48       if PureThy.get_name thy = "Datatype" then thy
    49       else theory "Datatype";
    50     val node_name = "Datatype_Universe.node";
    51     val In0_name = "Datatype_Universe.In0";
    52     val In1_name = "Datatype_Universe.In1";
    53     val Scons_name = "Datatype_Universe.Scons";
    54     val Leaf_name = "Datatype_Universe.Leaf";
    55     val Numb_name = "Datatype_Universe.Numb";
    56     val Lim_name = "Datatype_Universe.Lim";
    57     val Funs_name = "Datatype_Universe.Funs";
    58     val o_name = "Fun.comp";
    59     val sum_case_name = "Datatype.sum.sum_case";
    60 
    61     val [In0_inject, In1_inject, Scons_inject, Leaf_inject, In0_eq, In1_eq,
    62          In0_not_In1, In1_not_In0, Funs_mono, FunsI, Lim_inject,
    63          Funs_inv, FunsD, Funs_rangeE, Funs_nonempty, sum_case_inject] = map (get_thm Datatype_thy)
    64         ["In0_inject", "In1_inject", "Scons_inject", "Leaf_inject", "In0_eq", "In1_eq",
    65          "In0_not_In1", "In1_not_In0", "Funs_mono", "FunsI", "Lim_inject",
    66          "Funs_inv", "FunsD", "Funs_rangeE", "Funs_nonempty", "sum_case_inject"];
    67 
    68     val Funs_IntE = (Int_lower2 RS Funs_mono RS
    69       (Int_lower1 RS Funs_mono RS Int_greatest) RS subsetD) RS IntE;
    70 
    71     val descr' = flat descr;
    72 
    73     val big_name = space_implode "_" new_type_names;
    74     val thy1 = add_path flat_names big_name thy;
    75     val big_rec_name = big_name ^ "_rep_set";
    76     val rep_set_names = map (Sign.full_name (Theory.sign_of thy1))
    77       (if length descr' = 1 then [big_rec_name] else
    78         (map ((curry (op ^) (big_rec_name ^ "_")) o string_of_int)
    79           (1 upto (length descr'))));
    80 
    81     val tyvars = map (fn (_, (_, Ts, _)) => map dest_DtTFree Ts) (hd descr);
    82     val leafTs' = get_nonrec_types descr' sorts;
    83     val branchTs = get_branching_types descr' sorts;
    84     val branchT = if null branchTs then HOLogic.unitT
    85       else fold_bal (fn (T, U) => Type ("+", [T, U])) branchTs;
    86     val unneeded_vars = hd tyvars \\ foldr add_typ_tfree_names (leafTs' @ branchTs, []);
    87     val leafTs = leafTs' @ (map (fn n => TFree (n, the (assoc (sorts, n)))) unneeded_vars);
    88     val recTs = get_rec_types descr' sorts;
    89     val newTs = take (length (hd descr), recTs);
    90     val oldTs = drop (length (hd descr), recTs);
    91     val sumT = if null leafTs then HOLogic.unitT
    92       else fold_bal (fn (T, U) => Type ("+", [T, U])) leafTs;
    93     val Univ_elT = HOLogic.mk_setT (Type (node_name, [sumT, branchT]));
    94     val UnivT = HOLogic.mk_setT Univ_elT;
    95 
    96     val In0 = Const (In0_name, Univ_elT --> Univ_elT);
    97     val In1 = Const (In1_name, Univ_elT --> Univ_elT);
    98     val Leaf = Const (Leaf_name, sumT --> Univ_elT);
    99     val Lim = Const (Lim_name, (branchT --> Univ_elT) --> Univ_elT);
   100 
   101     (* make injections needed for embedding types in leaves *)
   102 
   103     fun mk_inj T' x =
   104       let
   105         fun mk_inj' T n i =
   106           if n = 1 then x else
   107           let val n2 = n div 2;
   108               val Type (_, [T1, T2]) = T
   109           in
   110             if i <= n2 then
   111               Const ("Inl", T1 --> T) $ (mk_inj' T1 n2 i)
   112             else
   113               Const ("Inr", T2 --> T) $ (mk_inj' T2 (n - n2) (i - n2))
   114           end
   115       in mk_inj' sumT (length leafTs) (1 + find_index_eq T' leafTs)
   116       end;
   117 
   118     (* make injections for constructors *)
   119 
   120     fun mk_univ_inj ts = access_bal (fn t => In0 $ t, fn t => In1 $ t, if ts = [] then
   121         Const ("arbitrary", Univ_elT)
   122       else
   123         foldr1 (HOLogic.mk_binop Scons_name) ts);
   124 
   125     (* function spaces *)
   126 
   127     fun mk_fun_inj T' x =
   128       let
   129         fun mk_inj T n i =
   130           if n = 1 then x else
   131           let
   132             val n2 = n div 2;
   133             val Type (_, [T1, T2]) = T;
   134             val sum_case = Const (sum_case_name, [T1 --> Univ_elT, T2 --> Univ_elT, T] ---> Univ_elT)
   135           in
   136             if i <= n2 then
   137               sum_case $ (mk_inj T1 n2 i) $ Const ("arbitrary", T2 --> Univ_elT)
   138             else
   139               sum_case $ Const ("arbitrary", T1 --> Univ_elT) $ mk_inj T2 (n - n2) (i - n2)
   140           end
   141       in mk_inj branchT (length branchTs) (1 + find_index_eq T' branchTs)
   142       end;
   143 
   144     (************** generate introduction rules for representing set **********)
   145 
   146     val _ = message "Constructing representing sets ...";
   147 
   148     (* make introduction rule for a single constructor *)
   149 
   150     fun make_intr s n (i, (_, cargs)) =
   151       let
   152         fun mk_prem (DtRec k, (j, prems, ts)) =
   153               let val free_t = mk_Free "x" Univ_elT j
   154               in (j + 1, (HOLogic.mk_mem (free_t,
   155                 Const (nth_elem (k, rep_set_names), UnivT)))::prems, free_t::ts)
   156               end
   157           | mk_prem (DtType ("fun", [T, DtRec k]), (j, prems, ts)) =
   158               let val T' = typ_of_dtyp descr' sorts T;
   159                   val free_t = mk_Free "x" (T' --> Univ_elT) j
   160               in (j + 1, (HOLogic.mk_mem (free_t,
   161                 Const (Funs_name, UnivT --> HOLogic.mk_setT (T' --> Univ_elT)) $
   162                   Const (nth_elem (k, rep_set_names), UnivT)))::prems,
   163                     Lim $ mk_fun_inj T' free_t::ts)
   164               end
   165           | mk_prem (dt, (j, prems, ts)) =
   166               let val T = typ_of_dtyp descr' sorts dt
   167               in (j + 1, prems, (Leaf $ mk_inj T (mk_Free "x" T j))::ts)
   168               end;
   169 
   170         val (_, prems, ts) = foldr mk_prem (cargs, (1, [], []));
   171         val concl = HOLogic.mk_Trueprop (HOLogic.mk_mem
   172           (mk_univ_inj ts n i, Const (s, UnivT)))
   173       in Logic.list_implies (map HOLogic.mk_Trueprop prems, concl)
   174       end;
   175 
   176     val consts = map (fn s => Const (s, UnivT)) rep_set_names;
   177 
   178     val intr_ts = flat (map (fn ((_, (_, _, constrs)), rep_set_name) =>
   179       map (make_intr rep_set_name (length constrs))
   180         ((1 upto (length constrs)) ~~ constrs)) (descr' ~~ rep_set_names));
   181 
   182     val (thy2, {raw_induct = rep_induct, intrs = rep_intrs, ...}) =
   183       setmp InductivePackage.quiet_mode (!quiet_mode)
   184         (InductivePackage.add_inductive_i false true big_rec_name false true false
   185            consts (map (fn x => (("", x), [])) intr_ts) [Funs_mono]) thy1;
   186 
   187     (********************************* typedef ********************************)
   188 
   189     val thy3 = add_path flat_names big_name (foldl (fn (thy, ((((name, mx), tvs), c), name')) =>
   190       setmp TypedefPackage.quiet_mode true
   191         (TypedefPackage.add_typedef_i false (Some name') (name, tvs, mx) c None
   192           (rtac exI 1 THEN
   193             QUIET_BREADTH_FIRST (has_fewer_prems 1)
   194             (resolve_tac (Funs_nonempty::rep_intrs) 1))) thy |> #1)
   195               (parent_path flat_names thy2, types_syntax ~~ tyvars ~~
   196                 (take (length newTs, consts)) ~~ new_type_names));
   197 
   198     (*********************** definition of constructors ***********************)
   199 
   200     val big_rep_name = (space_implode "_" new_type_names) ^ "_Rep_";
   201     val rep_names = map (curry op ^ "Rep_") new_type_names;
   202     val rep_names' = map (fn i => big_rep_name ^ (string_of_int i))
   203       (1 upto (length (flat (tl descr))));
   204     val all_rep_names = map (Sign.intern_const (Theory.sign_of thy3)) rep_names @
   205       map (Sign.full_name (Theory.sign_of thy3)) rep_names';
   206 
   207     (* isomorphism declarations *)
   208 
   209     val iso_decls = map (fn (T, s) => (s, T --> Univ_elT, NoSyn))
   210       (oldTs ~~ rep_names');
   211 
   212     (* constructor definitions *)
   213 
   214     fun make_constr_def tname T n ((thy, defs, eqns, i), ((cname, cargs), (cname', mx))) =
   215       let
   216         fun constr_arg (dt, (j, l_args, r_args)) =
   217           let val T = typ_of_dtyp descr' sorts dt;
   218               val free_t = mk_Free "x" T j
   219           in (case dt of
   220               DtRec m => (j + 1, free_t::l_args, (Const (nth_elem (m, all_rep_names),
   221                 T --> Univ_elT) $ free_t)::r_args)
   222             | DtType ("fun", [T', DtRec m]) =>
   223                 let val ([T''], T''') = strip_type T
   224                 in (j + 1, free_t::l_args, (Lim $ mk_fun_inj T''
   225                   (Const (o_name, [T''' --> Univ_elT, T, T''] ---> Univ_elT) $
   226                     Const (nth_elem (m, all_rep_names), T''' --> Univ_elT) $ free_t))::r_args)
   227                 end
   228 
   229             | _ => (j + 1, free_t::l_args, (Leaf $ mk_inj T free_t)::r_args))
   230           end;
   231 
   232         val (_, l_args, r_args) = foldr constr_arg (cargs, (1, [], []));
   233         val constrT = (map (typ_of_dtyp descr' sorts) cargs) ---> T;
   234         val abs_name = Sign.intern_const (Theory.sign_of thy) ("Abs_" ^ tname);
   235         val rep_name = Sign.intern_const (Theory.sign_of thy) ("Rep_" ^ tname);
   236         val lhs = list_comb (Const (cname, constrT), l_args);
   237         val rhs = mk_univ_inj r_args n i;
   238         val def = equals T $ lhs $ (Const (abs_name, Univ_elT --> T) $ rhs);
   239         val def_name = (Sign.base_name cname) ^ "_def";
   240         val eqn = HOLogic.mk_Trueprop (HOLogic.mk_eq
   241           (Const (rep_name, T --> Univ_elT) $ lhs, rhs));
   242         val (thy', [def_thm]) = thy |>
   243           Theory.add_consts_i [(cname', constrT, mx)] |>
   244           (PureThy.add_defs_i false o map Thm.no_attributes) [(def_name, def)];
   245 
   246       in (thy', defs @ [def_thm], eqns @ [eqn], i + 1) end;
   247 
   248     (* constructor definitions for datatype *)
   249 
   250     fun dt_constr_defs ((thy, defs, eqns, rep_congs, dist_lemmas),
   251         ((((_, (_, _, constrs)), tname), T), constr_syntax)) =
   252       let
   253         val _ $ (_ $ (cong_f $ _) $ _) = concl_of arg_cong;
   254         val sg = Theory.sign_of thy;
   255         val rep_const = cterm_of sg
   256           (Const (Sign.intern_const sg ("Rep_" ^ tname), T --> Univ_elT));
   257         val cong' = standard (cterm_instantiate [(cterm_of sg cong_f, rep_const)] arg_cong);
   258         val dist = standard (cterm_instantiate [(cterm_of sg distinct_f, rep_const)] distinct_lemma);
   259         val (thy', defs', eqns', _) = foldl ((make_constr_def tname T) (length constrs))
   260           ((add_path flat_names tname thy, defs, [], 1), constrs ~~ constr_syntax)
   261       in
   262         (parent_path flat_names thy', defs', eqns @ [eqns'],
   263           rep_congs @ [cong'], dist_lemmas @ [dist])
   264       end;
   265 
   266     val (thy4, constr_defs, constr_rep_eqns, rep_congs, dist_lemmas) = foldl dt_constr_defs
   267       ((thy3 |> Theory.add_consts_i iso_decls |> parent_path flat_names, [], [], [], []),
   268         hd descr ~~ new_type_names ~~ newTs ~~ constr_syntax);
   269 
   270     (*********** isomorphisms for new types (introduced by typedef) ***********)
   271 
   272     val _ = message "Proving isomorphism properties ...";
   273 
   274     (* get axioms from theory *)
   275 
   276     val newT_iso_axms = map (fn s =>
   277       (get_thm thy4 ("Abs_" ^ s ^ "_inverse"),
   278        get_thm thy4 ("Rep_" ^ s ^ "_inverse"),
   279        get_thm thy4 ("Rep_" ^ s))) new_type_names;
   280 
   281     (*------------------------------------------------*)
   282     (* prove additional theorems:                     *)
   283     (*  inj_on dt_Abs_i rep_set_i  and  inj dt_Rep_i  *)
   284     (*------------------------------------------------*)
   285 
   286     fun prove_newT_iso_inj_thm (((s, (thm1, thm2, _)), T), rep_set_name) =
   287       let
   288         val sg = Theory.sign_of thy4;
   289         val RepT = T --> Univ_elT;
   290         val Rep_name = Sign.intern_const sg ("Rep_" ^ s);
   291         val AbsT = Univ_elT --> T;
   292         val Abs_name = Sign.intern_const sg ("Abs_" ^ s);
   293 
   294         val inj_Abs_thm = 
   295 	    prove_goalw_cterm [] 
   296 	      (cterm_of sg
   297 	       (HOLogic.mk_Trueprop 
   298 		(Const ("Fun.inj_on", [AbsT, UnivT] ---> HOLogic.boolT) $
   299 		 Const (Abs_name, AbsT) $ Const (rep_set_name, UnivT))))
   300               (fn _ => [rtac inj_on_inverseI 1, etac thm1 1]);
   301 
   302         val setT = HOLogic.mk_setT T
   303 
   304         val inj_Rep_thm =
   305 	    prove_goalw_cterm []
   306 	      (cterm_of sg
   307 	       (HOLogic.mk_Trueprop
   308 		(Const ("Fun.inj_on", [RepT, setT] ---> HOLogic.boolT) $
   309 		 Const (Rep_name, RepT) $ Const ("UNIV", setT))))
   310               (fn _ => [rtac inj_inverseI 1, rtac thm2 1])
   311 
   312       in (inj_Abs_thm, inj_Rep_thm) end;
   313 
   314     val newT_iso_inj_thms = map prove_newT_iso_inj_thm
   315       (new_type_names ~~ newT_iso_axms ~~ newTs ~~
   316         take (length newTs, rep_set_names));
   317 
   318     (********* isomorphisms between existing types and "unfolded" types *******)
   319 
   320     (*---------------------------------------------------------------------*)
   321     (* isomorphisms are defined using primrec-combinators:                 *)
   322     (* generate appropriate functions for instantiating primrec-combinator *)
   323     (*                                                                     *)
   324     (*   e.g.  dt_Rep_i = list_rec ... (%h t y. In1 ((Leaf h) $ y))        *)
   325     (*                                                                     *)
   326     (* also generate characteristic equations for isomorphisms             *)
   327     (*                                                                     *)
   328     (*   e.g.  dt_Rep_i (cons h t) = In1 ((dt_Rep_j h) $ (dt_Rep_i t))     *)
   329     (*---------------------------------------------------------------------*)
   330 
   331     fun make_iso_def k ks n ((fs, eqns, i), (cname, cargs)) =
   332       let
   333         val argTs = map (typ_of_dtyp descr' sorts) cargs;
   334         val T = nth_elem (k, recTs);
   335         val rep_name = nth_elem (k, all_rep_names);
   336         val rep_const = Const (rep_name, T --> Univ_elT);
   337         val constr = Const (cname, argTs ---> T);
   338 
   339         fun process_arg ks' ((i2, i2', ts, Ts), dt) =
   340           let val T' = typ_of_dtyp descr' sorts dt
   341           in (case dt of
   342               DtRec j => if j mem ks' then
   343                   (i2 + 1, i2' + 1, ts @ [mk_Free "y" Univ_elT i2'], Ts @ [Univ_elT])
   344                 else
   345                   (i2 + 1, i2', ts @ [Const (nth_elem (j, all_rep_names),
   346                     T' --> Univ_elT) $ mk_Free "x" T' i2], Ts)
   347             | (DtType ("fun", [_, DtRec j])) =>
   348                 let val ([T''], T''') = strip_type T'
   349                 in if j mem ks' then
   350                     (i2 + 1, i2' + 1, ts @ [Lim $ mk_fun_inj T''
   351                       (mk_Free "y" (T'' --> Univ_elT) i2')], Ts @ [T'' --> Univ_elT])
   352                   else
   353                     (i2 + 1, i2', ts @ [Lim $ mk_fun_inj T''
   354                       (Const (o_name, [T''' --> Univ_elT, T', T''] ---> Univ_elT) $
   355                         Const (nth_elem (j, all_rep_names), T''' --> Univ_elT) $
   356                           mk_Free "x" T' i2)], Ts)
   357                 end
   358             | _ => (i2 + 1, i2', ts @ [Leaf $ mk_inj T' (mk_Free "x" T' i2)], Ts))
   359           end;
   360 
   361         val (i2, i2', ts, Ts) = foldl (process_arg ks) ((1, 1, [], []), cargs);
   362         val xs = map (uncurry (mk_Free "x")) (argTs ~~ (1 upto (i2 - 1)));
   363         val ys = map (uncurry (mk_Free "y")) (Ts ~~ (1 upto (i2' - 1)));
   364         val f = list_abs_free (map dest_Free (xs @ ys), mk_univ_inj ts n i);
   365 
   366         val (_, _, ts', _) = foldl (process_arg []) ((1, 1, [], []), cargs);
   367         val eqn = HOLogic.mk_Trueprop (HOLogic.mk_eq
   368           (rep_const $ list_comb (constr, xs), mk_univ_inj ts' n i))
   369 
   370       in (fs @ [f], eqns @ [eqn], i + 1) end;
   371 
   372     (* define isomorphisms for all mutually recursive datatypes in list ds *)
   373 
   374     fun make_iso_defs (ds, (thy, char_thms)) =
   375       let
   376         val ks = map fst ds;
   377         val (_, (tname, _, _)) = hd ds;
   378         val {rec_rewrites, rec_names, ...} = the (Symtab.lookup (dt_info, tname));
   379 
   380         fun process_dt ((fs, eqns, isos), (k, (tname, _, constrs))) =
   381           let
   382             val (fs', eqns', _) = foldl (make_iso_def k ks (length constrs))
   383               ((fs, eqns, 1), constrs);
   384             val iso = (nth_elem (k, recTs), nth_elem (k, all_rep_names))
   385           in (fs', eqns', isos @ [iso]) end;
   386         
   387         val (fs, eqns, isos) = foldl process_dt (([], [], []), ds);
   388         val fTs = map fastype_of fs;
   389         val defs = map (fn (rec_name, (T, iso_name)) => ((Sign.base_name iso_name) ^ "_def",
   390           equals (T --> Univ_elT) $ Const (iso_name, T --> Univ_elT) $
   391             list_comb (Const (rec_name, fTs @ [T] ---> Univ_elT), fs))) (rec_names ~~ isos);
   392         val (thy', def_thms) = (PureThy.add_defs_i false o map Thm.no_attributes) defs thy;
   393 
   394         (* prove characteristic equations *)
   395 
   396         val rewrites = def_thms @ (map mk_meta_eq rec_rewrites);
   397         val char_thms' = map (fn eqn => prove_goalw_cterm rewrites
   398           (cterm_of (Theory.sign_of thy') eqn) (fn _ => [rtac refl 1])) eqns;
   399 
   400       in (thy', char_thms' @ char_thms) end;
   401 
   402     val (thy5, iso_char_thms) = foldr make_iso_defs
   403       (tl descr, (add_path flat_names big_name thy4, []));
   404 
   405     (* prove isomorphism properties *)
   406 
   407     fun mk_funs_inv thm =
   408       let
   409         val [_, t] = prems_of Funs_inv;
   410         val [_ $ (_ $ _ $ R)] = Logic.strip_assums_hyp t;
   411         val _ $ (_ $ (r $ (a $ _)) $ _) = Logic.strip_assums_concl t;
   412         val [_ $ (_ $ _ $ R')] = prems_of thm;
   413         val _ $ (_ $ (r' $ (a' $ _)) $ _) = concl_of thm;
   414         val inv' = cterm_instantiate (map 
   415           ((pairself (cterm_of (sign_of_thm thm))) o
   416            (apsnd (map_term_types (incr_tvar 1))))
   417              [(R, R'), (r, r'), (a, a')]) Funs_inv
   418       in
   419         rule_by_tactic (atac 2) (thm RSN (2, inv'))
   420       end;
   421 
   422     (* prove  inj dt_Rep_i  and  dt_Rep_i x : dt_rep_set_i *)
   423 
   424     fun prove_iso_thms (ds, (inj_thms, elem_thms)) =
   425       let
   426         val (_, (tname, _, _)) = hd ds;
   427         val {induction, ...} = the (Symtab.lookup (dt_info, tname));
   428 
   429         fun mk_ind_concl (i, _) =
   430           let
   431             val T = nth_elem (i, recTs);
   432             val Rep_t = Const (nth_elem (i, all_rep_names), T --> Univ_elT);
   433             val rep_set_name = nth_elem (i, rep_set_names)
   434           in (HOLogic.all_const T $ Abs ("y", T, HOLogic.imp $
   435                 HOLogic.mk_eq (Rep_t $ mk_Free "x" T i, Rep_t $ Bound 0) $
   436                   HOLogic.mk_eq (mk_Free "x" T i, Bound 0)),
   437               HOLogic.mk_mem (Rep_t $ mk_Free "x" T i, Const (rep_set_name, UnivT)))
   438           end;
   439 
   440         val (ind_concl1, ind_concl2) = ListPair.unzip (map mk_ind_concl ds);
   441 
   442         val rewrites = map mk_meta_eq iso_char_thms;
   443         val inj_thms' = flat (map (fn r => [r RS injD, r RS inj_o])
   444           (map snd newT_iso_inj_thms @ inj_thms));
   445 
   446         val inj_thm = prove_goalw_cterm [] (cterm_of (Theory.sign_of thy5)
   447           (HOLogic.mk_Trueprop (mk_conj ind_concl1))) (fn _ =>
   448             [(indtac induction THEN_ALL_NEW ObjectLogic.atomize_tac) 1,
   449              REPEAT (EVERY
   450                [rtac allI 1, rtac impI 1,
   451                 exh_tac (exh_thm_of dt_info) 1,
   452                 REPEAT (EVERY
   453                   [hyp_subst_tac 1,
   454                    rewrite_goals_tac rewrites,
   455                    REPEAT (dresolve_tac [In0_inject, In1_inject] 1),
   456                    (eresolve_tac [In0_not_In1 RS notE, In1_not_In0 RS notE] 1)
   457                    ORELSE (EVERY
   458                      [REPEAT (eresolve_tac (Scons_inject :: sum_case_inject ::
   459                         map make_elim (inj_thms' @
   460                           [Leaf_inject, Lim_inject, Inl_inject, Inr_inject])) 1),
   461                       REPEAT ((EVERY [etac allE 1, dtac mp 1, atac 1]) ORELSE
   462                               (dtac inj_fun_lemma 1 THEN atac 1)),
   463                       REPEAT (hyp_subst_tac 1),
   464                       rtac refl 1])])])]);
   465 
   466         val inj_thms'' = map (fn r => r RS datatype_injI)
   467                              (split_conj_thm inj_thm);
   468 
   469         val elem_thm = 
   470 	    prove_goalw_cterm []
   471 	      (cterm_of (Theory.sign_of thy5)
   472 	       (HOLogic.mk_Trueprop (mk_conj ind_concl2)))
   473 	      (fn _ =>
   474 	       [(indtac induction THEN_ALL_NEW ObjectLogic.atomize_tac) 1,
   475 		rewrite_goals_tac (o_def :: rewrites),
   476 		REPEAT (EVERY
   477 			[resolve_tac rep_intrs 1,
   478 			 REPEAT (FIRST [atac 1, etac spec 1,
   479 				 resolve_tac (FunsI :: elem_thms) 1])])]);
   480 
   481       in (inj_thms'' @ inj_thms, elem_thms @ (split_conj_thm elem_thm))
   482       end;
   483 
   484     val (iso_inj_thms_unfolded, iso_elem_thms) = foldr prove_iso_thms
   485       (tl descr, ([], map #3 newT_iso_axms));
   486     val iso_inj_thms = map snd newT_iso_inj_thms @ iso_inj_thms_unfolded;
   487 
   488     (* prove  x : dt_rep_set_i --> x : range dt_Rep_i *)
   489 
   490     fun mk_iso_t (((set_name, iso_name), i), T) =
   491       let val isoT = T --> Univ_elT
   492       in HOLogic.imp $ 
   493         HOLogic.mk_mem (mk_Free "x" Univ_elT i, Const (set_name, UnivT)) $
   494           (if i < length newTs then Const ("True", HOLogic.boolT)
   495            else HOLogic.mk_mem (mk_Free "x" Univ_elT i,
   496              Const ("image", [isoT, HOLogic.mk_setT T] ---> UnivT) $
   497                Const (iso_name, isoT) $ Const ("UNIV", HOLogic.mk_setT T)))
   498       end;
   499 
   500     val iso_t = HOLogic.mk_Trueprop (mk_conj (map mk_iso_t
   501       (rep_set_names ~~ all_rep_names ~~ (0 upto (length descr' - 1)) ~~ recTs)));
   502 
   503     (* all the theorems are proved by one single simultaneous induction *)
   504 
   505     val iso_thms = if length descr = 1 then [] else
   506       drop (length newTs, split_conj_thm
   507         (prove_goalw_cterm [] (cterm_of (Theory.sign_of thy5) iso_t) (fn _ =>
   508            [indtac rep_induct 1,
   509             REPEAT (rtac TrueI 1),
   510             REPEAT (EVERY
   511               [rewrite_goals_tac [mk_meta_eq Collect_mem_eq],
   512                REPEAT (etac Funs_IntE 1),
   513                REPEAT (eresolve_tac (rangeE ::
   514                  map (fn r => r RS Funs_rangeE) iso_inj_thms_unfolded) 1),
   515                REPEAT (eresolve_tac (map (fn (iso, _, _) => iso RS subst) newT_iso_axms @
   516                  map (fn (iso, _, _) => mk_funs_inv iso RS subst) newT_iso_axms) 1),
   517                TRY (hyp_subst_tac 1),
   518                rtac (sym RS range_eqI) 1,
   519                resolve_tac iso_char_thms 1])])));
   520 
   521     val Abs_inverse_thms' =
   522       map #1 newT_iso_axms @
   523       map2 (fn (r_inj, r) => f_myinv_f OF [r_inj, r RS mp])
   524         (iso_inj_thms_unfolded, iso_thms);
   525 
   526     val Abs_inverse_thms = map (fn r => r RS subst) (Abs_inverse_thms' @
   527       map mk_funs_inv Abs_inverse_thms');
   528 
   529     (******************* freeness theorems for constructors *******************)
   530 
   531     val _ = message "Proving freeness of constructors ...";
   532 
   533     (* prove theorem  Rep_i (Constr_j ...) = Inj_j ...  *)
   534     
   535     fun prove_constr_rep_thm eqn =
   536       let
   537         val inj_thms = map (fn (r, _) => r RS inj_onD) newT_iso_inj_thms;
   538         val rewrites = o_def :: constr_defs @ (map (mk_meta_eq o #2) newT_iso_axms)
   539       in prove_goalw_cterm [] (cterm_of (Theory.sign_of thy5) eqn) (fn _ =>
   540         [resolve_tac inj_thms 1,
   541          rewrite_goals_tac rewrites,
   542          rtac refl 1,
   543          resolve_tac rep_intrs 2,
   544          REPEAT (resolve_tac (FunsI :: iso_elem_thms) 1)])
   545       end;
   546 
   547     (*--------------------------------------------------------------*)
   548     (* constr_rep_thms and rep_congs are used to prove distinctness *)
   549     (* of constructors.                                             *)
   550     (*--------------------------------------------------------------*)
   551 
   552     val constr_rep_thms = map (map prove_constr_rep_thm) constr_rep_eqns;
   553 
   554     val dist_rewrites = map (fn (rep_thms, dist_lemma) =>
   555       dist_lemma::(rep_thms @ [In0_eq, In1_eq, In0_not_In1, In1_not_In0]))
   556         (constr_rep_thms ~~ dist_lemmas);
   557 
   558     fun prove_distinct_thms (_, []) = []
   559       | prove_distinct_thms (dist_rewrites', t::_::ts) =
   560           let
   561             val dist_thm = prove_goalw_cterm [] (cterm_of (Theory.sign_of thy5) t) (fn _ =>
   562               [simp_tac (HOL_ss addsimps dist_rewrites') 1])
   563           in dist_thm::(standard (dist_thm RS not_sym))::
   564             (prove_distinct_thms (dist_rewrites', ts))
   565           end;
   566 
   567     val distinct_thms = map prove_distinct_thms (dist_rewrites ~~
   568       DatatypeProp.make_distincts new_type_names descr sorts thy5);
   569 
   570     val simproc_dists = map (fn ((((_, (_, _, constrs)), rep_thms), congr), dists) =>
   571       if length constrs < !DatatypeProp.dtK then FewConstrs dists
   572       else ManyConstrs (congr, HOL_basic_ss addsimps rep_thms)) (hd descr ~~
   573         constr_rep_thms ~~ rep_congs ~~ distinct_thms);
   574 
   575     (* prove injectivity of constructors *)
   576 
   577     fun prove_constr_inj_thm rep_thms t =
   578       let val inj_thms = Scons_inject::sum_case_inject::(map make_elim
   579         ((map (fn r => r RS injD) iso_inj_thms) @
   580           [In0_inject, In1_inject, Leaf_inject, Inl_inject, Inr_inject, Lim_inject]))
   581       in prove_goalw_cterm [] (cterm_of (Theory.sign_of thy5) t) (fn _ =>
   582         [rtac iffI 1,
   583          REPEAT (etac conjE 2), hyp_subst_tac 2, rtac refl 2,
   584          dresolve_tac rep_congs 1, dtac box_equals 1,
   585          REPEAT (resolve_tac rep_thms 1), rewtac o_def,
   586          REPEAT (eresolve_tac inj_thms 1),
   587          REPEAT (ares_tac [conjI] 1 ORELSE (EVERY [rtac ext 1, dtac fun_cong 1,
   588                   eresolve_tac inj_thms 1, atac 1]))])
   589       end;
   590 
   591     val constr_inject = map (fn (ts, thms) => map (prove_constr_inj_thm thms) ts)
   592       ((DatatypeProp.make_injs descr sorts) ~~ constr_rep_thms);
   593 
   594     val (thy6, (constr_inject', distinct_thms'))= thy5 |> parent_path flat_names |>
   595       store_thmss "inject" new_type_names constr_inject |>>>
   596       store_thmss "distinct" new_type_names distinct_thms;
   597 
   598     (*************************** induction theorem ****************************)
   599 
   600     val _ = message "Proving induction rule for datatypes ...";
   601 
   602     val Rep_inverse_thms = (map (fn (_, iso, _) => iso RS subst) newT_iso_axms) @
   603       (map (fn r => r RS myinv_f_f RS subst) iso_inj_thms_unfolded);
   604     val Rep_inverse_thms' = map (fn r => r RS myinv_f_f) iso_inj_thms_unfolded;
   605 
   606     fun mk_indrule_lemma ((prems, concls), ((i, _), T)) =
   607       let
   608         val Rep_t = Const (nth_elem (i, all_rep_names), T --> Univ_elT) $
   609           mk_Free "x" T i;
   610 
   611         val Abs_t = if i < length newTs then
   612             Const (Sign.intern_const (Theory.sign_of thy6)
   613               ("Abs_" ^ (nth_elem (i, new_type_names))), Univ_elT --> T)
   614           else Const ("Inductive.myinv", [T --> Univ_elT, Univ_elT] ---> T) $
   615             Const (nth_elem (i, all_rep_names), T --> Univ_elT)
   616 
   617       in (prems @ [HOLogic.imp $ HOLogic.mk_mem (Rep_t,
   618             Const (nth_elem (i, rep_set_names), UnivT)) $
   619               (mk_Free "P" (T --> HOLogic.boolT) (i + 1) $ (Abs_t $ Rep_t))],
   620           concls @ [mk_Free "P" (T --> HOLogic.boolT) (i + 1) $ mk_Free "x" T i])
   621       end;
   622 
   623     val (indrule_lemma_prems, indrule_lemma_concls) =
   624       foldl mk_indrule_lemma (([], []), (descr' ~~ recTs));
   625 
   626     val cert = cterm_of (Theory.sign_of thy6);
   627 
   628     val indrule_lemma = prove_goalw_cterm [] (cert
   629       (Logic.mk_implies
   630         (HOLogic.mk_Trueprop (mk_conj indrule_lemma_prems),
   631          HOLogic.mk_Trueprop (mk_conj indrule_lemma_concls)))) (fn prems =>
   632            [cut_facts_tac prems 1, REPEAT (etac conjE 1),
   633             REPEAT (EVERY
   634               [TRY (rtac conjI 1), resolve_tac Rep_inverse_thms 1,
   635                etac mp 1, resolve_tac iso_elem_thms 1])]);
   636 
   637     val Ps = map head_of (HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of indrule_lemma)));
   638     val frees = if length Ps = 1 then [Free ("P", snd (dest_Var (hd Ps)))] else
   639       map (Free o apfst fst o dest_Var) Ps;
   640     val indrule_lemma' = cterm_instantiate (map cert Ps ~~ map cert frees) indrule_lemma;
   641 
   642     val dt_induct = prove_goalw_cterm [] (cert
   643       (DatatypeProp.make_ind descr sorts)) (fn prems =>
   644         [rtac indrule_lemma' 1, indtac rep_induct 1,
   645          EVERY (map (fn (prem, r) => (EVERY
   646            [REPEAT (eresolve_tac (Funs_IntE::Abs_inverse_thms) 1),
   647             simp_tac (HOL_basic_ss addsimps ((symmetric r)::Rep_inverse_thms')) 1,
   648             DEPTH_SOLVE_1 (ares_tac [prem] 1 ORELSE (EVERY [rewtac o_def,
   649               dtac FunsD 1, etac CollectD 1]))]))
   650                 (prems ~~ (constr_defs @ (map mk_meta_eq iso_char_thms))))]);
   651 
   652     val (thy7, [dt_induct']) = thy6 |>
   653       Theory.add_path big_name |>
   654       PureThy.add_thms [(("induct", dt_induct), [case_names_induct])] |>>
   655       Theory.parent_path;
   656 
   657   in (thy7, constr_inject', distinct_thms', dist_rewrites, simproc_dists, dt_induct')
   658   end;
   659 
   660 end;