src/HOL/Tools/datatype_rep_proofs.ML
author wenzelm
Wed Oct 17 20:25:19 2001 +0200 (2001-10-17)
changeset 11822 122834177ec1
parent 11806 e1fd22a657ae
child 11827 16ef206e6648
permissions -rw-r--r--
improved internal interface of typedef;
     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 = theory "Datatype";
    48     val node_name = Sign.intern_tycon (Theory.sign_of Datatype_thy) "node";
    49     val [In0_name, In1_name, Scons_name, Leaf_name, Numb_name, Lim_name,
    50       Funs_name, o_name, sum_case_name] =
    51       map (Sign.intern_const (Theory.sign_of Datatype_thy))
    52         ["In0", "In1", "Scons", "Leaf", "Numb", "Lim", "Funs", "op o", "sum_case"];
    53 
    54     val [In0_inject, In1_inject, Scons_inject, Leaf_inject, In0_eq, In1_eq,
    55          In0_not_In1, In1_not_In0, Funs_mono, FunsI, Lim_inject,
    56          Funs_inv, FunsD, Funs_rangeE, Funs_nonempty, sum_case_inject] = map (get_thm Datatype_thy)
    57         ["In0_inject", "In1_inject", "Scons_inject", "Leaf_inject", "In0_eq", "In1_eq",
    58          "In0_not_In1", "In1_not_In0", "Funs_mono", "FunsI", "Lim_inject",
    59          "Funs_inv", "FunsD", "Funs_rangeE", "Funs_nonempty", "sum_case_inject"];
    60 
    61     val Funs_IntE = (Int_lower2 RS Funs_mono RS
    62       (Int_lower1 RS Funs_mono RS Int_greatest) RS subsetD) RS IntE;
    63 
    64     val descr' = flat descr;
    65 
    66     val big_name = space_implode "_" new_type_names;
    67     val thy1 = add_path flat_names big_name thy;
    68     val big_rec_name = big_name ^ "_rep_set";
    69     val rep_set_names = map (Sign.full_name (Theory.sign_of thy1))
    70       (if length descr' = 1 then [big_rec_name] else
    71         (map ((curry (op ^) (big_rec_name ^ "_")) o string_of_int)
    72           (1 upto (length descr'))));
    73 
    74     val tyvars = map (fn (_, (_, Ts, _)) => map dest_DtTFree Ts) (hd descr);
    75     val leafTs' = get_nonrec_types descr' sorts;
    76     val branchTs = get_branching_types descr' sorts;
    77     val branchT = if null branchTs then HOLogic.unitT
    78       else fold_bal (fn (T, U) => Type ("+", [T, U])) branchTs;
    79     val unneeded_vars = hd tyvars \\ foldr add_typ_tfree_names (leafTs' @ branchTs, []);
    80     val leafTs = leafTs' @ (map (fn n => TFree (n, the (assoc (sorts, n)))) unneeded_vars);
    81     val recTs = get_rec_types descr' sorts;
    82     val newTs = take (length (hd descr), recTs);
    83     val oldTs = drop (length (hd descr), recTs);
    84     val sumT = if null leafTs then HOLogic.unitT
    85       else fold_bal (fn (T, U) => Type ("+", [T, U])) leafTs;
    86     val Univ_elT = HOLogic.mk_setT (Type (node_name, [sumT, branchT]));
    87     val UnivT = HOLogic.mk_setT Univ_elT;
    88 
    89     val In0 = Const (In0_name, Univ_elT --> Univ_elT);
    90     val In1 = Const (In1_name, Univ_elT --> Univ_elT);
    91     val Leaf = Const (Leaf_name, sumT --> Univ_elT);
    92     val Lim = Const (Lim_name, (branchT --> Univ_elT) --> Univ_elT);
    93 
    94     (* make injections needed for embedding types in leaves *)
    95 
    96     fun mk_inj T' x =
    97       let
    98         fun mk_inj' T n i =
    99           if n = 1 then x else
   100           let val n2 = n div 2;
   101               val Type (_, [T1, T2]) = T
   102           in
   103             if i <= n2 then
   104               Const ("Inl", T1 --> T) $ (mk_inj' T1 n2 i)
   105             else
   106               Const ("Inr", T2 --> T) $ (mk_inj' T2 (n - n2) (i - n2))
   107           end
   108       in mk_inj' sumT (length leafTs) (1 + find_index_eq T' leafTs)
   109       end;
   110 
   111     (* make injections for constructors *)
   112 
   113     fun mk_univ_inj ts = access_bal (fn t => In0 $ t, fn t => In1 $ t, if ts = [] then
   114         Const ("arbitrary", Univ_elT)
   115       else
   116         foldr1 (HOLogic.mk_binop Scons_name) ts);
   117 
   118     (* function spaces *)
   119 
   120     fun mk_fun_inj T' x =
   121       let
   122         fun mk_inj T n i =
   123           if n = 1 then x else
   124           let
   125             val n2 = n div 2;
   126             val Type (_, [T1, T2]) = T;
   127             val sum_case = Const (sum_case_name, [T1 --> Univ_elT, T2 --> Univ_elT, T] ---> Univ_elT)
   128           in
   129             if i <= n2 then
   130               sum_case $ (mk_inj T1 n2 i) $ Const ("arbitrary", T2 --> Univ_elT)
   131             else
   132               sum_case $ Const ("arbitrary", T1 --> Univ_elT) $ mk_inj T2 (n - n2) (i - n2)
   133           end
   134       in mk_inj branchT (length branchTs) (1 + find_index_eq T' branchTs)
   135       end;
   136 
   137     (************** generate introduction rules for representing set **********)
   138 
   139     val _ = message "Constructing representing sets ...";
   140 
   141     (* make introduction rule for a single constructor *)
   142 
   143     fun make_intr s n (i, (_, cargs)) =
   144       let
   145         fun mk_prem (DtRec k, (j, prems, ts)) =
   146               let val free_t = mk_Free "x" Univ_elT j
   147               in (j + 1, (HOLogic.mk_mem (free_t,
   148                 Const (nth_elem (k, rep_set_names), UnivT)))::prems, free_t::ts)
   149               end
   150           | mk_prem (DtType ("fun", [T, DtRec k]), (j, prems, ts)) =
   151               let val T' = typ_of_dtyp descr' sorts T;
   152                   val free_t = mk_Free "x" (T' --> Univ_elT) j
   153               in (j + 1, (HOLogic.mk_mem (free_t,
   154                 Const (Funs_name, UnivT --> HOLogic.mk_setT (T' --> Univ_elT)) $
   155                   Const (nth_elem (k, rep_set_names), UnivT)))::prems,
   156                     Lim $ mk_fun_inj T' free_t::ts)
   157               end
   158           | mk_prem (dt, (j, prems, ts)) =
   159               let val T = typ_of_dtyp descr' sorts dt
   160               in (j + 1, prems, (Leaf $ mk_inj T (mk_Free "x" T j))::ts)
   161               end;
   162 
   163         val (_, prems, ts) = foldr mk_prem (cargs, (1, [], []));
   164         val concl = HOLogic.mk_Trueprop (HOLogic.mk_mem
   165           (mk_univ_inj ts n i, Const (s, UnivT)))
   166       in Logic.list_implies (map HOLogic.mk_Trueprop prems, concl)
   167       end;
   168 
   169     val consts = map (fn s => Const (s, UnivT)) rep_set_names;
   170 
   171     val intr_ts = flat (map (fn ((_, (_, _, constrs)), rep_set_name) =>
   172       map (make_intr rep_set_name (length constrs))
   173         ((1 upto (length constrs)) ~~ constrs)) (descr' ~~ rep_set_names));
   174 
   175     val (thy2, {raw_induct = rep_induct, intrs = rep_intrs, ...}) =
   176       setmp InductivePackage.quiet_mode (!quiet_mode)
   177         (InductivePackage.add_inductive_i false true big_rec_name false true false
   178            consts (map (fn x => (("", x), [])) intr_ts) [Funs_mono] []) thy1;
   179 
   180     (********************************* typedef ********************************)
   181 
   182     val thy3 = add_path flat_names big_name (foldl (fn (thy, ((((name, mx), tvs), c), name')) =>
   183       setmp TypedefPackage.quiet_mode true
   184         (TypedefPackage.add_typedef_i false name' (name, tvs, mx) c None
   185           (QUIET_BREADTH_FIRST (has_fewer_prems 1)
   186             (resolve_tac (Funs_nonempty::rep_intrs) 1))) thy |> #1)
   187               (parent_path flat_names thy2, types_syntax ~~ tyvars ~~
   188                 (take (length newTs, consts)) ~~ new_type_names));
   189 
   190     (*********************** definition of constructors ***********************)
   191 
   192     val big_rep_name = (space_implode "_" new_type_names) ^ "_Rep_";
   193     val rep_names = map (curry op ^ "Rep_") new_type_names;
   194     val rep_names' = map (fn i => big_rep_name ^ (string_of_int i))
   195       (1 upto (length (flat (tl descr))));
   196     val all_rep_names = map (Sign.intern_const (Theory.sign_of thy3)) rep_names @
   197       map (Sign.full_name (Theory.sign_of thy3)) rep_names';
   198 
   199     (* isomorphism declarations *)
   200 
   201     val iso_decls = map (fn (T, s) => (s, T --> Univ_elT, NoSyn))
   202       (oldTs ~~ rep_names');
   203 
   204     (* constructor definitions *)
   205 
   206     fun make_constr_def tname T n ((thy, defs, eqns, i), ((cname, cargs), (cname', mx))) =
   207       let
   208         fun constr_arg (dt, (j, l_args, r_args)) =
   209           let val T = typ_of_dtyp descr' sorts dt;
   210               val free_t = mk_Free "x" T j
   211           in (case dt of
   212               DtRec m => (j + 1, free_t::l_args, (Const (nth_elem (m, all_rep_names),
   213                 T --> Univ_elT) $ free_t)::r_args)
   214             | DtType ("fun", [T', DtRec m]) =>
   215                 let val ([T''], T''') = strip_type T
   216                 in (j + 1, free_t::l_args, (Lim $ mk_fun_inj T''
   217                   (Const (o_name, [T''' --> Univ_elT, T, T''] ---> Univ_elT) $
   218                     Const (nth_elem (m, all_rep_names), T''' --> Univ_elT) $ free_t))::r_args)
   219                 end
   220 
   221             | _ => (j + 1, free_t::l_args, (Leaf $ mk_inj T free_t)::r_args))
   222           end;
   223 
   224         val (_, l_args, r_args) = foldr constr_arg (cargs, (1, [], []));
   225         val constrT = (map (typ_of_dtyp descr' sorts) cargs) ---> T;
   226         val abs_name = Sign.intern_const (Theory.sign_of thy) ("Abs_" ^ tname);
   227         val rep_name = Sign.intern_const (Theory.sign_of thy) ("Rep_" ^ tname);
   228         val lhs = list_comb (Const (cname, constrT), l_args);
   229         val rhs = mk_univ_inj r_args n i;
   230         val def = equals T $ lhs $ (Const (abs_name, Univ_elT --> T) $ rhs);
   231         val def_name = (Sign.base_name cname) ^ "_def";
   232         val eqn = HOLogic.mk_Trueprop (HOLogic.mk_eq
   233           (Const (rep_name, T --> Univ_elT) $ lhs, rhs));
   234         val (thy', [def_thm]) = thy |>
   235           Theory.add_consts_i [(cname', constrT, mx)] |>
   236           (PureThy.add_defs_i false o map Thm.no_attributes) [(def_name, def)];
   237 
   238       in (thy', defs @ [def_thm], eqns @ [eqn], i + 1) end;
   239 
   240     (* constructor definitions for datatype *)
   241 
   242     fun dt_constr_defs ((thy, defs, eqns, rep_congs, dist_lemmas),
   243         ((((_, (_, _, constrs)), tname), T), constr_syntax)) =
   244       let
   245         val _ $ (_ $ (cong_f $ _) $ _) = concl_of arg_cong;
   246         val sg = Theory.sign_of thy;
   247         val rep_const = cterm_of sg
   248           (Const (Sign.intern_const sg ("Rep_" ^ tname), T --> Univ_elT));
   249         val cong' = standard (cterm_instantiate [(cterm_of sg cong_f, rep_const)] arg_cong);
   250         val dist = standard (cterm_instantiate [(cterm_of sg distinct_f, rep_const)] distinct_lemma);
   251         val (thy', defs', eqns', _) = foldl ((make_constr_def tname T) (length constrs))
   252           ((add_path flat_names tname thy, defs, [], 1), constrs ~~ constr_syntax)
   253       in
   254         (parent_path flat_names thy', defs', eqns @ [eqns'],
   255           rep_congs @ [cong'], dist_lemmas @ [dist])
   256       end;
   257 
   258     val (thy4, constr_defs, constr_rep_eqns, rep_congs, dist_lemmas) = foldl dt_constr_defs
   259       ((thy3 |> Theory.add_consts_i iso_decls |> parent_path flat_names, [], [], [], []),
   260         hd descr ~~ new_type_names ~~ newTs ~~ constr_syntax);
   261 
   262     (*********** isomorphisms for new types (introduced by typedef) ***********)
   263 
   264     val _ = message "Proving isomorphism properties ...";
   265 
   266     (* get axioms from theory *)
   267 
   268     val newT_iso_axms = map (fn s =>
   269       (get_thm thy4 ("Abs_" ^ s ^ "_inverse"),
   270        get_thm thy4 ("Rep_" ^ s ^ "_inverse"),
   271        get_thm thy4 ("Rep_" ^ s))) new_type_names;
   272 
   273     (*------------------------------------------------*)
   274     (* prove additional theorems:                     *)
   275     (*  inj_on dt_Abs_i rep_set_i  and  inj dt_Rep_i  *)
   276     (*------------------------------------------------*)
   277 
   278     fun prove_newT_iso_inj_thm (((s, (thm1, thm2, _)), T), rep_set_name) =
   279       let
   280         val sg = Theory.sign_of thy4;
   281         val RepT = T --> Univ_elT;
   282         val Rep_name = Sign.intern_const sg ("Rep_" ^ s);
   283         val AbsT = Univ_elT --> T;
   284         val Abs_name = Sign.intern_const sg ("Abs_" ^ s);
   285 
   286         val inj_Abs_thm = 
   287 	    prove_goalw_cterm [] 
   288 	      (cterm_of sg
   289 	       (HOLogic.mk_Trueprop 
   290 		(Const ("Fun.inj_on", [AbsT, UnivT] ---> HOLogic.boolT) $
   291 		 Const (Abs_name, AbsT) $ Const (rep_set_name, UnivT))))
   292               (fn _ => [rtac inj_on_inverseI 1, etac thm1 1]);
   293 
   294         val setT = HOLogic.mk_setT T
   295 
   296         val inj_Rep_thm =
   297 	    prove_goalw_cterm []
   298 	      (cterm_of sg
   299 	       (HOLogic.mk_Trueprop
   300 		(Const ("Fun.inj_on", [RepT, setT] ---> HOLogic.boolT) $
   301 		 Const (Rep_name, RepT) $ Const ("UNIV", setT))))
   302               (fn _ => [rtac inj_inverseI 1, rtac thm2 1])
   303 
   304       in (inj_Abs_thm, inj_Rep_thm) end;
   305 
   306     val newT_iso_inj_thms = map prove_newT_iso_inj_thm
   307       (new_type_names ~~ newT_iso_axms ~~ newTs ~~
   308         take (length newTs, rep_set_names));
   309 
   310     (********* isomorphisms between existing types and "unfolded" types *******)
   311 
   312     (*---------------------------------------------------------------------*)
   313     (* isomorphisms are defined using primrec-combinators:                 *)
   314     (* generate appropriate functions for instantiating primrec-combinator *)
   315     (*                                                                     *)
   316     (*   e.g.  dt_Rep_i = list_rec ... (%h t y. In1 ((Leaf h) $ y))        *)
   317     (*                                                                     *)
   318     (* also generate characteristic equations for isomorphisms             *)
   319     (*                                                                     *)
   320     (*   e.g.  dt_Rep_i (cons h t) = In1 ((dt_Rep_j h) $ (dt_Rep_i t))     *)
   321     (*---------------------------------------------------------------------*)
   322 
   323     fun make_iso_def k ks n ((fs, eqns, i), (cname, cargs)) =
   324       let
   325         val argTs = map (typ_of_dtyp descr' sorts) cargs;
   326         val T = nth_elem (k, recTs);
   327         val rep_name = nth_elem (k, all_rep_names);
   328         val rep_const = Const (rep_name, T --> Univ_elT);
   329         val constr = Const (cname, argTs ---> T);
   330 
   331         fun process_arg ks' ((i2, i2', ts, Ts), dt) =
   332           let val T' = typ_of_dtyp descr' sorts dt
   333           in (case dt of
   334               DtRec j => if j mem ks' then
   335                   (i2 + 1, i2' + 1, ts @ [mk_Free "y" Univ_elT i2'], Ts @ [Univ_elT])
   336                 else
   337                   (i2 + 1, i2', ts @ [Const (nth_elem (j, all_rep_names),
   338                     T' --> Univ_elT) $ mk_Free "x" T' i2], Ts)
   339             | (DtType ("fun", [_, DtRec j])) =>
   340                 let val ([T''], T''') = strip_type T'
   341                 in if j mem ks' then
   342                     (i2 + 1, i2' + 1, ts @ [Lim $ mk_fun_inj T''
   343                       (mk_Free "y" (T'' --> Univ_elT) i2')], Ts @ [T'' --> Univ_elT])
   344                   else
   345                     (i2 + 1, i2', ts @ [Lim $ mk_fun_inj T''
   346                       (Const (o_name, [T''' --> Univ_elT, T', T''] ---> Univ_elT) $
   347                         Const (nth_elem (j, all_rep_names), T''' --> Univ_elT) $
   348                           mk_Free "x" T' i2)], Ts)
   349                 end
   350             | _ => (i2 + 1, i2', ts @ [Leaf $ mk_inj T' (mk_Free "x" T' i2)], Ts))
   351           end;
   352 
   353         val (i2, i2', ts, Ts) = foldl (process_arg ks) ((1, 1, [], []), cargs);
   354         val xs = map (uncurry (mk_Free "x")) (argTs ~~ (1 upto (i2 - 1)));
   355         val ys = map (uncurry (mk_Free "y")) (Ts ~~ (1 upto (i2' - 1)));
   356         val f = list_abs_free (map dest_Free (xs @ ys), mk_univ_inj ts n i);
   357 
   358         val (_, _, ts', _) = foldl (process_arg []) ((1, 1, [], []), cargs);
   359         val eqn = HOLogic.mk_Trueprop (HOLogic.mk_eq
   360           (rep_const $ list_comb (constr, xs), mk_univ_inj ts' n i))
   361 
   362       in (fs @ [f], eqns @ [eqn], i + 1) end;
   363 
   364     (* define isomorphisms for all mutually recursive datatypes in list ds *)
   365 
   366     fun make_iso_defs (ds, (thy, char_thms)) =
   367       let
   368         val ks = map fst ds;
   369         val (_, (tname, _, _)) = hd ds;
   370         val {rec_rewrites, rec_names, ...} = the (Symtab.lookup (dt_info, tname));
   371 
   372         fun process_dt ((fs, eqns, isos), (k, (tname, _, constrs))) =
   373           let
   374             val (fs', eqns', _) = foldl (make_iso_def k ks (length constrs))
   375               ((fs, eqns, 1), constrs);
   376             val iso = (nth_elem (k, recTs), nth_elem (k, all_rep_names))
   377           in (fs', eqns', isos @ [iso]) end;
   378         
   379         val (fs, eqns, isos) = foldl process_dt (([], [], []), ds);
   380         val fTs = map fastype_of fs;
   381         val defs = map (fn (rec_name, (T, iso_name)) => ((Sign.base_name iso_name) ^ "_def",
   382           equals (T --> Univ_elT) $ Const (iso_name, T --> Univ_elT) $
   383             list_comb (Const (rec_name, fTs @ [T] ---> Univ_elT), fs))) (rec_names ~~ isos);
   384         val (thy', def_thms) = (PureThy.add_defs_i false o map Thm.no_attributes) defs thy;
   385 
   386         (* prove characteristic equations *)
   387 
   388         val rewrites = def_thms @ (map mk_meta_eq rec_rewrites);
   389         val char_thms' = map (fn eqn => prove_goalw_cterm rewrites
   390           (cterm_of (Theory.sign_of thy') eqn) (fn _ => [rtac refl 1])) eqns;
   391 
   392       in (thy', char_thms' @ char_thms) end;
   393 
   394     val (thy5, iso_char_thms) = foldr make_iso_defs
   395       (tl descr, (add_path flat_names big_name thy4, []));
   396 
   397     (* prove isomorphism properties *)
   398 
   399     fun mk_funs_inv thm =
   400       let
   401         val [_, t] = prems_of Funs_inv;
   402         val [_ $ (_ $ _ $ R)] = Logic.strip_assums_hyp t;
   403         val _ $ (_ $ (r $ (a $ _)) $ _) = Logic.strip_assums_concl t;
   404         val [_ $ (_ $ _ $ R')] = prems_of thm;
   405         val _ $ (_ $ (r' $ (a' $ _)) $ _) = concl_of thm;
   406         val inv' = cterm_instantiate (map 
   407           ((pairself (cterm_of (sign_of_thm thm))) o
   408            (apsnd (map_term_types (incr_tvar 1))))
   409              [(R, R'), (r, r'), (a, a')]) Funs_inv
   410       in
   411         rule_by_tactic (atac 2) (thm RSN (2, inv'))
   412       end;
   413 
   414     (* prove  inj dt_Rep_i  and  dt_Rep_i x : dt_rep_set_i *)
   415 
   416     fun prove_iso_thms (ds, (inj_thms, elem_thms)) =
   417       let
   418         val (_, (tname, _, _)) = hd ds;
   419         val {induction, ...} = the (Symtab.lookup (dt_info, tname));
   420 
   421         fun mk_ind_concl (i, _) =
   422           let
   423             val T = nth_elem (i, recTs);
   424             val Rep_t = Const (nth_elem (i, all_rep_names), T --> Univ_elT);
   425             val rep_set_name = nth_elem (i, rep_set_names)
   426           in (HOLogic.all_const T $ Abs ("y", T, HOLogic.imp $
   427                 HOLogic.mk_eq (Rep_t $ mk_Free "x" T i, Rep_t $ Bound 0) $
   428                   HOLogic.mk_eq (mk_Free "x" T i, Bound 0)),
   429               HOLogic.mk_mem (Rep_t $ mk_Free "x" T i, Const (rep_set_name, UnivT)))
   430           end;
   431 
   432         val (ind_concl1, ind_concl2) = ListPair.unzip (map mk_ind_concl ds);
   433 
   434         val rewrites = map mk_meta_eq iso_char_thms;
   435         val inj_thms' = flat (map (fn r => [r RS injD, r RS inj_o])
   436           (map snd newT_iso_inj_thms @ inj_thms));
   437 
   438         val inj_thm = prove_goalw_cterm [] (cterm_of (Theory.sign_of thy5)
   439           (HOLogic.mk_Trueprop (mk_conj ind_concl1))) (fn _ =>
   440             [indtac induction 1,
   441              REPEAT (EVERY
   442                [rtac allI 1, rtac impI 1,
   443                 exh_tac (exh_thm_of dt_info) 1,
   444                 REPEAT (EVERY
   445                   [hyp_subst_tac 1,
   446                    rewrite_goals_tac rewrites,
   447                    REPEAT (dresolve_tac [In0_inject, In1_inject] 1),
   448                    (eresolve_tac [In0_not_In1 RS notE, In1_not_In0 RS notE] 1)
   449                    ORELSE (EVERY
   450                      [REPEAT (etac Scons_inject 1),
   451                       REPEAT (dresolve_tac
   452                         (inj_thms' @ [Leaf_inject, Lim_inject, Inl_inject, Inr_inject]) 1),
   453                       REPEAT ((EVERY [etac allE 1, dtac mp 1, atac 1]) ORELSE
   454                               (dtac inj_fun_lemma 1 THEN atac 1)),
   455                       TRY (hyp_subst_tac 1),
   456                       rtac refl 1])])])]);
   457 
   458         val inj_thms'' = map (fn r => r RS datatype_injI)
   459                              (split_conj_thm inj_thm);
   460 
   461         val elem_thm = 
   462 	    prove_goalw_cterm []
   463 	      (cterm_of (Theory.sign_of thy5)
   464 	       (HOLogic.mk_Trueprop (mk_conj ind_concl2)))
   465 	      (fn _ =>
   466 	       [indtac induction 1,
   467 		rewrite_goals_tac (o_def :: rewrites),
   468 		REPEAT (EVERY
   469 			[resolve_tac rep_intrs 1,
   470 			 REPEAT (FIRST [atac 1, etac spec 1,
   471 				 resolve_tac (FunsI :: elem_thms) 1])])]);
   472 
   473       in (inj_thms'' @ inj_thms, elem_thms @ (split_conj_thm elem_thm))
   474       end;
   475 
   476     val (iso_inj_thms_unfolded, iso_elem_thms) = foldr prove_iso_thms
   477       (tl descr, ([], map #3 newT_iso_axms));
   478     val iso_inj_thms = map snd newT_iso_inj_thms @ iso_inj_thms_unfolded;
   479 
   480     (* prove  x : dt_rep_set_i --> x : range dt_Rep_i *)
   481 
   482     fun mk_iso_t (((set_name, iso_name), i), T) =
   483       let val isoT = T --> Univ_elT
   484       in HOLogic.imp $ 
   485         HOLogic.mk_mem (mk_Free "x" Univ_elT i, Const (set_name, UnivT)) $
   486           (if i < length newTs then Const ("True", HOLogic.boolT)
   487            else HOLogic.mk_mem (mk_Free "x" Univ_elT i,
   488              Const ("image", [isoT, HOLogic.mk_setT T] ---> UnivT) $
   489                Const (iso_name, isoT) $ Const ("UNIV", HOLogic.mk_setT T)))
   490       end;
   491 
   492     val iso_t = HOLogic.mk_Trueprop (mk_conj (map mk_iso_t
   493       (rep_set_names ~~ all_rep_names ~~ (0 upto (length descr' - 1)) ~~ recTs)));
   494 
   495     (* all the theorems are proved by one single simultaneous induction *)
   496 
   497     val iso_thms = if length descr = 1 then [] else
   498       drop (length newTs, split_conj_thm
   499         (prove_goalw_cterm [] (cterm_of (Theory.sign_of thy5) iso_t) (fn _ =>
   500            [indtac rep_induct 1,
   501             REPEAT (rtac TrueI 1),
   502             REPEAT (EVERY
   503               [rewrite_goals_tac [mk_meta_eq Collect_mem_eq],
   504                REPEAT (etac Funs_IntE 1),
   505                REPEAT (eresolve_tac (rangeE ::
   506                  map (fn r => r RS Funs_rangeE) iso_inj_thms_unfolded) 1),
   507                REPEAT (eresolve_tac (map (fn (iso, _, _) => iso RS subst) newT_iso_axms @
   508                  map (fn (iso, _, _) => mk_funs_inv iso RS subst) newT_iso_axms) 1),
   509                TRY (hyp_subst_tac 1),
   510                rtac (sym RS range_eqI) 1,
   511                resolve_tac iso_char_thms 1])])));
   512 
   513     val Abs_inverse_thms' =
   514       map #1 newT_iso_axms @
   515       map2 (fn (r_inj, r) => f_myinv_f OF [r_inj, r RS mp])
   516         (iso_inj_thms_unfolded, iso_thms);
   517 
   518     val Abs_inverse_thms = map (fn r => r RS subst) (Abs_inverse_thms' @
   519       map mk_funs_inv Abs_inverse_thms');
   520 
   521     (******************* freeness theorems for constructors *******************)
   522 
   523     val _ = message "Proving freeness of constructors ...";
   524 
   525     (* prove theorem  Rep_i (Constr_j ...) = Inj_j ...  *)
   526     
   527     fun prove_constr_rep_thm eqn =
   528       let
   529         val inj_thms = map (fn (r, _) => r RS inj_onD) newT_iso_inj_thms;
   530         val rewrites = o_def :: constr_defs @ (map (mk_meta_eq o #2) newT_iso_axms)
   531       in prove_goalw_cterm [] (cterm_of (Theory.sign_of thy5) eqn) (fn _ =>
   532         [resolve_tac inj_thms 1,
   533          rewrite_goals_tac rewrites,
   534          rtac refl 1,
   535          resolve_tac rep_intrs 2,
   536          REPEAT (resolve_tac (FunsI :: iso_elem_thms) 1)])
   537       end;
   538 
   539     (*--------------------------------------------------------------*)
   540     (* constr_rep_thms and rep_congs are used to prove distinctness *)
   541     (* of constructors.                                             *)
   542     (*--------------------------------------------------------------*)
   543 
   544     val constr_rep_thms = map (map prove_constr_rep_thm) constr_rep_eqns;
   545 
   546     val dist_rewrites = map (fn (rep_thms, dist_lemma) =>
   547       dist_lemma::(rep_thms @ [In0_eq, In1_eq, In0_not_In1, In1_not_In0]))
   548         (constr_rep_thms ~~ dist_lemmas);
   549 
   550     fun prove_distinct_thms (_, []) = []
   551       | prove_distinct_thms (dist_rewrites', t::_::ts) =
   552           let
   553             val dist_thm = prove_goalw_cterm [] (cterm_of (Theory.sign_of thy5) t) (fn _ =>
   554               [simp_tac (HOL_ss addsimps dist_rewrites') 1])
   555           in dist_thm::(standard (dist_thm RS not_sym))::
   556             (prove_distinct_thms (dist_rewrites', ts))
   557           end;
   558 
   559     val distinct_thms = map prove_distinct_thms (dist_rewrites ~~
   560       DatatypeProp.make_distincts new_type_names descr sorts thy5);
   561 
   562     val simproc_dists = map (fn ((((_, (_, _, constrs)), rep_thms), congr), dists) =>
   563       if length constrs < !DatatypeProp.dtK then FewConstrs dists
   564       else ManyConstrs (congr, HOL_basic_ss addsimps rep_thms)) (hd descr ~~
   565         constr_rep_thms ~~ rep_congs ~~ distinct_thms);
   566 
   567     (* prove injectivity of constructors *)
   568 
   569     fun prove_constr_inj_thm rep_thms t =
   570       let val inj_thms = Scons_inject::sum_case_inject::(map make_elim
   571         ((map (fn r => r RS injD) iso_inj_thms) @
   572           [In0_inject, In1_inject, Leaf_inject, Inl_inject, Inr_inject, Lim_inject]))
   573       in prove_goalw_cterm [] (cterm_of (Theory.sign_of thy5) t) (fn _ =>
   574         [rtac iffI 1,
   575          REPEAT (etac conjE 2), hyp_subst_tac 2, rtac refl 2,
   576          dresolve_tac rep_congs 1, dtac box_equals 1,
   577          REPEAT (resolve_tac rep_thms 1), rewtac o_def,
   578          REPEAT (eresolve_tac inj_thms 1),
   579          REPEAT (ares_tac [conjI] 1 ORELSE (EVERY [rtac ext 1, dtac fun_cong 1,
   580                   eresolve_tac inj_thms 1, atac 1]))])
   581       end;
   582 
   583     val constr_inject = map (fn (ts, thms) => map (prove_constr_inj_thm thms) ts)
   584       ((DatatypeProp.make_injs descr sorts) ~~ constr_rep_thms);
   585 
   586     val (thy6, (constr_inject', distinct_thms'))= thy5 |> parent_path flat_names |>
   587       store_thmss "inject" new_type_names constr_inject |>>>
   588       store_thmss "distinct" new_type_names distinct_thms;
   589 
   590     (*************************** induction theorem ****************************)
   591 
   592     val _ = message "Proving induction rule for datatypes ...";
   593 
   594     val Rep_inverse_thms = (map (fn (_, iso, _) => iso RS subst) newT_iso_axms) @
   595       (map (fn r => r RS myinv_f_f RS subst) iso_inj_thms_unfolded);
   596     val Rep_inverse_thms' = map (fn r => r RS myinv_f_f) iso_inj_thms_unfolded;
   597 
   598     fun mk_indrule_lemma ((prems, concls), ((i, _), T)) =
   599       let
   600         val Rep_t = Const (nth_elem (i, all_rep_names), T --> Univ_elT) $
   601           mk_Free "x" T i;
   602 
   603         val Abs_t = if i < length newTs then
   604             Const (Sign.intern_const (Theory.sign_of thy6)
   605               ("Abs_" ^ (nth_elem (i, new_type_names))), Univ_elT --> T)
   606           else Const ("Inductive.myinv", [T --> Univ_elT, Univ_elT] ---> T) $
   607             Const (nth_elem (i, all_rep_names), T --> Univ_elT)
   608 
   609       in (prems @ [HOLogic.imp $ HOLogic.mk_mem (Rep_t,
   610             Const (nth_elem (i, rep_set_names), UnivT)) $
   611               (mk_Free "P" (T --> HOLogic.boolT) (i + 1) $ (Abs_t $ Rep_t))],
   612           concls @ [mk_Free "P" (T --> HOLogic.boolT) (i + 1) $ mk_Free "x" T i])
   613       end;
   614 
   615     val (indrule_lemma_prems, indrule_lemma_concls) =
   616       foldl mk_indrule_lemma (([], []), (descr' ~~ recTs));
   617 
   618     val cert = cterm_of (Theory.sign_of thy6);
   619 
   620     val indrule_lemma = prove_goalw_cterm [] (cert
   621       (Logic.mk_implies
   622         (HOLogic.mk_Trueprop (mk_conj indrule_lemma_prems),
   623          HOLogic.mk_Trueprop (mk_conj indrule_lemma_concls)))) (fn prems =>
   624            [cut_facts_tac prems 1, REPEAT (etac conjE 1),
   625             REPEAT (EVERY
   626               [TRY (rtac conjI 1), resolve_tac Rep_inverse_thms 1,
   627                etac mp 1, resolve_tac iso_elem_thms 1])]);
   628 
   629     val Ps = map head_of (HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of indrule_lemma)));
   630     val frees = if length Ps = 1 then [Free ("P", snd (dest_Var (hd Ps)))] else
   631       map (Free o apfst fst o dest_Var) Ps;
   632     val indrule_lemma' = cterm_instantiate (map cert Ps ~~ map cert frees) indrule_lemma;
   633 
   634     val dt_induct = prove_goalw_cterm [InductivePackage.inductive_forall_def] (cert
   635       (DatatypeProp.make_ind descr sorts)) (fn prems =>
   636         [rtac indrule_lemma' 1, indtac rep_induct 1,
   637          EVERY (map (fn (prem, r) => (EVERY
   638            [REPEAT (eresolve_tac (Funs_IntE::Abs_inverse_thms) 1),
   639             simp_tac (HOL_basic_ss addsimps ((symmetric r)::Rep_inverse_thms')) 1,
   640             DEPTH_SOLVE_1 (ares_tac [prem] 1 ORELSE (EVERY [rewtac o_def,
   641               rtac allI 1, dtac FunsD 1, etac CollectD 1]))]))
   642                 (prems ~~ (constr_defs @ (map mk_meta_eq iso_char_thms))))]);
   643 
   644     val (thy7, [dt_induct']) = thy6 |>
   645       Theory.add_path big_name |>
   646       PureThy.add_thms [(("induct", dt_induct),
   647         [Drule.rule_attribute (K InductivePackage.rulify), case_names_induct])] |>>
   648       Theory.parent_path;
   649 
   650   in (thy7, constr_inject', distinct_thms', dist_rewrites, simproc_dists, dt_induct')
   651   end;
   652 
   653 end;