src/HOL/Tools/Datatype/datatype_rep_proofs.ML
author wenzelm
Tue Sep 29 22:48:24 2009 +0200 (2009-09-29)
changeset 32765 3032c0308019
parent 32727 9072201cd69d
child 32874 5281cebb1a37
permissions -rw-r--r--
modernized Balanced_Tree;
     1 (*  Title:      HOL/Tools/datatype_rep_proofs.ML
     2     Author:     Stefan Berghofer, TU Muenchen
     3 
     4 Definitional introduction of datatypes
     5 Proof of characteristic theorems:
     6 
     7  - injectivity of constructors
     8  - distinctness of constructors
     9  - induction theorem
    10 *)
    11 
    12 signature DATATYPE_REP_PROOFS =
    13 sig
    14   include DATATYPE_COMMON
    15   val distinctness_limit : int Config.T
    16   val distinctness_limit_setup : theory -> theory
    17   val representation_proofs : config -> info Symtab.table ->
    18     string list -> descr list -> (string * sort) list ->
    19       (binding * mixfix) list -> (binding * mixfix) list list -> attribute
    20         -> theory -> (thm list list * thm list list * thm list list *
    21           DatatypeAux.simproc_dist list * thm) * theory
    22 end;
    23 
    24 structure DatatypeRepProofs : DATATYPE_REP_PROOFS =
    25 struct
    26 
    27 open DatatypeAux;
    28 
    29 (*the kind of distinctiveness axioms depends on number of constructors*)
    30 val (distinctness_limit, distinctness_limit_setup) =
    31   Attrib.config_int "datatype_distinctness_limit" 7;
    32 
    33 val (_ $ (_ $ (_ $ (distinct_f $ _) $ _))) = hd (prems_of distinct_lemma);
    34 
    35 val collect_simp = rewrite_rule [mk_meta_eq mem_Collect_eq];
    36 
    37 
    38 (** theory context references **)
    39 
    40 fun exh_thm_of (dt_info : info Symtab.table) tname =
    41   #exhaust (the (Symtab.lookup dt_info tname));
    42 
    43 (******************************************************************************)
    44 
    45 fun representation_proofs (config : config) (dt_info : info Symtab.table)
    46       new_type_names descr sorts types_syntax constr_syntax case_names_induct thy =
    47   let
    48     val Datatype_thy = ThyInfo.the_theory "Datatype" thy;
    49     val node_name = "Datatype.node";
    50     val In0_name = "Datatype.In0";
    51     val In1_name = "Datatype.In1";
    52     val Scons_name = "Datatype.Scons";
    53     val Leaf_name = "Datatype.Leaf";
    54     val Numb_name = "Datatype.Numb";
    55     val Lim_name = "Datatype.Lim";
    56     val Suml_name = "Datatype.Suml";
    57     val Sumr_name = "Datatype.Sumr";
    58 
    59     val [In0_inject, In1_inject, Scons_inject, Leaf_inject,
    60          In0_eq, In1_eq, In0_not_In1, In1_not_In0,
    61          Lim_inject, Suml_inject, Sumr_inject] = map (PureThy.get_thm Datatype_thy)
    62           ["In0_inject", "In1_inject", "Scons_inject", "Leaf_inject",
    63            "In0_eq", "In1_eq", "In0_not_In1", "In1_not_In0",
    64            "Lim_inject", "Suml_inject", "Sumr_inject"];
    65 
    66     val descr' = flat descr;
    67 
    68     val big_name = space_implode "_" new_type_names;
    69     val thy1 = Sign.add_path big_name thy;
    70     val big_rec_name = big_name ^ "_rep_set";
    71     val rep_set_names' =
    72       (if length descr' = 1 then [big_rec_name] else
    73         (map ((curry (op ^) (big_rec_name ^ "_")) o string_of_int)
    74           (1 upto (length descr'))));
    75     val rep_set_names = map (Sign.full_bname thy1) rep_set_names';
    76 
    77     val tyvars = map (fn (_, (_, Ts, _)) => map dest_DtTFree Ts) (hd descr);
    78     val leafTs' = get_nonrec_types descr' sorts;
    79     val branchTs = get_branching_types descr' sorts;
    80     val branchT = if null branchTs then HOLogic.unitT
    81       else Balanced_Tree.make (fn (T, U) => Type ("+", [T, U])) branchTs;
    82     val arities = get_arities descr' \ 0;
    83     val unneeded_vars = hd tyvars \\ List.foldr OldTerm.add_typ_tfree_names [] (leafTs' @ branchTs);
    84     val leafTs = leafTs' @ (map (fn n => TFree (n, (the o AList.lookup (op =) sorts) n)) unneeded_vars);
    85     val recTs = get_rec_types descr' sorts;
    86     val newTs = Library.take (length (hd descr), recTs);
    87     val oldTs = Library.drop (length (hd descr), recTs);
    88     val sumT = if null leafTs then HOLogic.unitT
    89       else Balanced_Tree.make (fn (T, U) => Type ("+", [T, U])) leafTs;
    90     val Univ_elT = HOLogic.mk_setT (Type (node_name, [sumT, branchT]));
    91     val UnivT = HOLogic.mk_setT Univ_elT;
    92     val UnivT' = Univ_elT --> HOLogic.boolT;
    93     val Collect = Const ("Collect", UnivT' --> UnivT);
    94 
    95     val In0 = Const (In0_name, Univ_elT --> Univ_elT);
    96     val In1 = Const (In1_name, Univ_elT --> Univ_elT);
    97     val Leaf = Const (Leaf_name, sumT --> Univ_elT);
    98     val Lim = Const (Lim_name, (branchT --> Univ_elT) --> Univ_elT);
    99 
   100     (* make injections needed for embedding types in leaves *)
   101 
   102     fun mk_inj T' x =
   103       let
   104         fun mk_inj' T n i =
   105           if n = 1 then x else
   106           let val n2 = n div 2;
   107               val Type (_, [T1, T2]) = T
   108           in
   109             if i <= n2 then
   110               Const ("Sum_Type.Inl", T1 --> T) $ (mk_inj' T1 n2 i)
   111             else
   112               Const ("Sum_Type.Inr", T2 --> T) $ (mk_inj' T2 (n - n2) (i - n2))
   113           end
   114       in mk_inj' sumT (length leafTs) (1 + find_index (fn T'' => T'' = T') leafTs)
   115       end;
   116 
   117     (* make injections for constructors *)
   118 
   119     fun mk_univ_inj ts = Balanced_Tree.access
   120       {left = fn t => In0 $ t,
   121         right = fn t => In1 $ t,
   122         init =
   123           if ts = [] then Const (@{const_name undefined}, Univ_elT)
   124           else foldr1 (HOLogic.mk_binop Scons_name) ts};
   125 
   126     (* function spaces *)
   127 
   128     fun mk_fun_inj T' x =
   129       let
   130         fun mk_inj T n i =
   131           if n = 1 then x else
   132           let
   133             val n2 = n div 2;
   134             val Type (_, [T1, T2]) = T;
   135             fun mkT U = (U --> Univ_elT) --> T --> Univ_elT
   136           in
   137             if i <= n2 then Const (Suml_name, mkT T1) $ mk_inj T1 n2 i
   138             else Const (Sumr_name, mkT T2) $ mk_inj T2 (n - n2) (i - n2)
   139           end
   140       in mk_inj branchT (length branchTs) (1 + find_index (fn T'' => T'' = T') branchTs)
   141       end;
   142 
   143     val mk_lim = List.foldr (fn (T, t) => Lim $ mk_fun_inj T (Abs ("x", T, t)));
   144 
   145     (************** generate introduction rules for representing set **********)
   146 
   147     val _ = message config "Constructing representing sets ...";
   148 
   149     (* make introduction rule for a single constructor *)
   150 
   151     fun make_intr s n (i, (_, cargs)) =
   152       let
   153         fun mk_prem (dt, (j, prems, ts)) = (case strip_dtyp dt of
   154             (dts, DtRec k) =>
   155               let
   156                 val Ts = map (typ_of_dtyp descr' sorts) dts;
   157                 val free_t =
   158                   app_bnds (mk_Free "x" (Ts ---> Univ_elT) j) (length Ts)
   159               in (j + 1, list_all (map (pair "x") Ts,
   160                   HOLogic.mk_Trueprop
   161                     (Free (nth rep_set_names' k, UnivT') $ free_t)) :: prems,
   162                 mk_lim free_t Ts :: ts)
   163               end
   164           | _ =>
   165               let val T = typ_of_dtyp descr' sorts dt
   166               in (j + 1, prems, (Leaf $ mk_inj T (mk_Free "x" T j))::ts)
   167               end);
   168 
   169         val (_, prems, ts) = List.foldr mk_prem (1, [], []) cargs;
   170         val concl = HOLogic.mk_Trueprop
   171           (Free (s, UnivT') $ mk_univ_inj ts n i)
   172       in Logic.list_implies (prems, concl)
   173       end;
   174 
   175     val intr_ts = maps (fn ((_, (_, _, constrs)), rep_set_name) =>
   176       map (make_intr rep_set_name (length constrs))
   177         ((1 upto (length constrs)) ~~ constrs)) (descr' ~~ rep_set_names');
   178 
   179     val ({raw_induct = rep_induct, intrs = rep_intrs, ...}, thy2) =
   180         Inductive.add_inductive_global (serial_string ())
   181           {quiet_mode = #quiet config, verbose = false, kind = Thm.internalK,
   182            alt_name = Binding.name big_rec_name, coind = false, no_elim = true, no_ind = false,
   183            skip_mono = true, fork_mono = false}
   184           (map (fn s => ((Binding.name s, UnivT'), NoSyn)) rep_set_names') []
   185           (map (fn x => (Attrib.empty_binding, x)) intr_ts) [] thy1;
   186 
   187     (********************************* typedef ********************************)
   188 
   189     val (typedefs, thy3) = thy2 |>
   190       Sign.parent_path |>
   191       fold_map (fn ((((name, mx), tvs), c), name') =>
   192           Typedef.add_typedef false (SOME (Binding.name name')) (name, tvs, mx)
   193             (Collect $ Const (c, UnivT')) NONE
   194             (rtac exI 1 THEN rtac CollectI 1 THEN
   195               QUIET_BREADTH_FIRST (has_fewer_prems 1)
   196               (resolve_tac rep_intrs 1)))
   197                 (types_syntax ~~ tyvars ~~
   198                   (Library.take (length newTs, rep_set_names)) ~~ new_type_names) ||>
   199       Sign.add_path big_name;
   200 
   201     (*********************** definition of constructors ***********************)
   202 
   203     val big_rep_name = (space_implode "_" new_type_names) ^ "_Rep_";
   204     val rep_names = map (curry op ^ "Rep_") new_type_names;
   205     val rep_names' = map (fn i => big_rep_name ^ (string_of_int i))
   206       (1 upto (length (flat (tl descr))));
   207     val all_rep_names = map (Sign.intern_const thy3) rep_names @
   208       map (Sign.full_bname thy3) rep_names';
   209 
   210     (* isomorphism declarations *)
   211 
   212     val iso_decls = map (fn (T, s) => (Binding.name s, T --> Univ_elT, NoSyn))
   213       (oldTs ~~ rep_names');
   214 
   215     (* constructor definitions *)
   216 
   217     fun make_constr_def tname T n ((thy, defs, eqns, i), ((cname, cargs), (cname', mx))) =
   218       let
   219         fun constr_arg (dt, (j, l_args, r_args)) =
   220           let val T = typ_of_dtyp descr' sorts dt;
   221               val free_t = mk_Free "x" T j
   222           in (case (strip_dtyp dt, strip_type T) of
   223               ((_, DtRec m), (Us, U)) => (j + 1, free_t :: l_args, mk_lim
   224                 (Const (nth all_rep_names m, U --> Univ_elT) $
   225                    app_bnds free_t (length Us)) Us :: r_args)
   226             | _ => (j + 1, free_t::l_args, (Leaf $ mk_inj T free_t)::r_args))
   227           end;
   228 
   229         val (_, l_args, r_args) = List.foldr constr_arg (1, [], []) cargs;
   230         val constrT = (map (typ_of_dtyp descr' sorts) cargs) ---> T;
   231         val abs_name = Sign.intern_const thy ("Abs_" ^ tname);
   232         val rep_name = Sign.intern_const thy ("Rep_" ^ tname);
   233         val lhs = list_comb (Const (cname, constrT), l_args);
   234         val rhs = mk_univ_inj r_args n i;
   235         val def = Logic.mk_equals (lhs, Const (abs_name, Univ_elT --> T) $ rhs);
   236         val def_name = Long_Name.base_name cname ^ "_def";
   237         val eqn = HOLogic.mk_Trueprop (HOLogic.mk_eq
   238           (Const (rep_name, T --> Univ_elT) $ lhs, rhs));
   239         val ([def_thm], thy') =
   240           thy
   241           |> Sign.add_consts_i [(cname', constrT, mx)]
   242           |> (PureThy.add_defs false o map Thm.no_attributes) [(Binding.name def_name, def)];
   243 
   244       in (thy', defs @ [def_thm], eqns @ [eqn], i + 1) end;
   245 
   246     (* constructor definitions for datatype *)
   247 
   248     fun dt_constr_defs ((thy, defs, eqns, rep_congs, dist_lemmas),
   249         ((((_, (_, _, constrs)), tname), T), constr_syntax)) =
   250       let
   251         val _ $ (_ $ (cong_f $ _) $ _) = concl_of arg_cong;
   252         val rep_const = cterm_of thy
   253           (Const (Sign.intern_const thy ("Rep_" ^ tname), T --> Univ_elT));
   254         val cong' = standard (cterm_instantiate [(cterm_of thy cong_f, rep_const)] arg_cong);
   255         val dist = standard (cterm_instantiate [(cterm_of thy distinct_f, rep_const)] distinct_lemma);
   256         val (thy', defs', eqns', _) = Library.foldl ((make_constr_def tname T) (length constrs))
   257           ((Sign.add_path tname thy, defs, [], 1), constrs ~~ constr_syntax)
   258       in
   259         (Sign.parent_path thy', defs', eqns @ [eqns'],
   260           rep_congs @ [cong'], dist_lemmas @ [dist])
   261       end;
   262 
   263     val (thy4, constr_defs, constr_rep_eqns, rep_congs, dist_lemmas) = Library.foldl dt_constr_defs
   264       ((thy3 |> Sign.add_consts_i iso_decls |> Sign.parent_path, [], [], [], []),
   265         hd descr ~~ new_type_names ~~ newTs ~~ constr_syntax);
   266 
   267     (*********** isomorphisms for new types (introduced by typedef) ***********)
   268 
   269     val _ = message config "Proving isomorphism properties ...";
   270 
   271     val newT_iso_axms = map (fn (_, td) =>
   272       (collect_simp (#Abs_inverse td), #Rep_inverse td,
   273        collect_simp (#Rep td))) typedefs;
   274 
   275     val newT_iso_inj_thms = map (fn (_, td) =>
   276       (collect_simp (#Abs_inject td) RS iffD1, #Rep_inject td RS iffD1)) typedefs;
   277 
   278     (********* isomorphisms between existing types and "unfolded" types *******)
   279 
   280     (*---------------------------------------------------------------------*)
   281     (* isomorphisms are defined using primrec-combinators:                 *)
   282     (* generate appropriate functions for instantiating primrec-combinator *)
   283     (*                                                                     *)
   284     (*   e.g.  dt_Rep_i = list_rec ... (%h t y. In1 (Scons (Leaf h) y))    *)
   285     (*                                                                     *)
   286     (* also generate characteristic equations for isomorphisms             *)
   287     (*                                                                     *)
   288     (*   e.g.  dt_Rep_i (cons h t) = In1 (Scons (dt_Rep_j h) (dt_Rep_i t)) *)
   289     (*---------------------------------------------------------------------*)
   290 
   291     fun make_iso_def k ks n ((fs, eqns, i), (cname, cargs)) =
   292       let
   293         val argTs = map (typ_of_dtyp descr' sorts) cargs;
   294         val T = nth recTs k;
   295         val rep_name = nth all_rep_names k;
   296         val rep_const = Const (rep_name, T --> Univ_elT);
   297         val constr = Const (cname, argTs ---> T);
   298 
   299         fun process_arg ks' ((i2, i2', ts, Ts), dt) =
   300           let
   301             val T' = typ_of_dtyp descr' sorts dt;
   302             val (Us, U) = strip_type T'
   303           in (case strip_dtyp dt of
   304               (_, DtRec j) => if j mem ks' then
   305                   (i2 + 1, i2' + 1, ts @ [mk_lim (app_bnds
   306                      (mk_Free "y" (Us ---> Univ_elT) i2') (length Us)) Us],
   307                    Ts @ [Us ---> Univ_elT])
   308                 else
   309                   (i2 + 1, i2', ts @ [mk_lim
   310                      (Const (nth all_rep_names j, U --> Univ_elT) $
   311                         app_bnds (mk_Free "x" T' i2) (length Us)) Us], Ts)
   312             | _ => (i2 + 1, i2', ts @ [Leaf $ mk_inj T' (mk_Free "x" T' i2)], Ts))
   313           end;
   314 
   315         val (i2, i2', ts, Ts) = Library.foldl (process_arg ks) ((1, 1, [], []), cargs);
   316         val xs = map (uncurry (mk_Free "x")) (argTs ~~ (1 upto (i2 - 1)));
   317         val ys = map (uncurry (mk_Free "y")) (Ts ~~ (1 upto (i2' - 1)));
   318         val f = list_abs_free (map dest_Free (xs @ ys), mk_univ_inj ts n i);
   319 
   320         val (_, _, ts', _) = Library.foldl (process_arg []) ((1, 1, [], []), cargs);
   321         val eqn = HOLogic.mk_Trueprop (HOLogic.mk_eq
   322           (rep_const $ list_comb (constr, xs), mk_univ_inj ts' n i))
   323 
   324       in (fs @ [f], eqns @ [eqn], i + 1) end;
   325 
   326     (* define isomorphisms for all mutually recursive datatypes in list ds *)
   327 
   328     fun make_iso_defs (ds, (thy, char_thms)) =
   329       let
   330         val ks = map fst ds;
   331         val (_, (tname, _, _)) = hd ds;
   332         val {rec_rewrites, rec_names, ...} = the (Symtab.lookup dt_info tname);
   333 
   334         fun process_dt ((fs, eqns, isos), (k, (tname, _, constrs))) =
   335           let
   336             val (fs', eqns', _) = Library.foldl (make_iso_def k ks (length constrs))
   337               ((fs, eqns, 1), constrs);
   338             val iso = (nth recTs k, nth all_rep_names k)
   339           in (fs', eqns', isos @ [iso]) end;
   340         
   341         val (fs, eqns, isos) = Library.foldl process_dt (([], [], []), ds);
   342         val fTs = map fastype_of fs;
   343         val defs = map (fn (rec_name, (T, iso_name)) => (Binding.name (Long_Name.base_name iso_name ^ "_def"),
   344           Logic.mk_equals (Const (iso_name, T --> Univ_elT),
   345             list_comb (Const (rec_name, fTs @ [T] ---> Univ_elT), fs)))) (rec_names ~~ isos);
   346         val (def_thms, thy') =
   347           apsnd Theory.checkpoint ((PureThy.add_defs false o map Thm.no_attributes) defs thy);
   348 
   349         (* prove characteristic equations *)
   350 
   351         val rewrites = def_thms @ (map mk_meta_eq rec_rewrites);
   352         val char_thms' = map (fn eqn => SkipProof.prove_global thy' [] [] eqn
   353           (fn _ => EVERY [rewrite_goals_tac rewrites, rtac refl 1])) eqns;
   354 
   355       in (thy', char_thms' @ char_thms) end;
   356 
   357     val (thy5, iso_char_thms) = apfst Theory.checkpoint (List.foldr make_iso_defs
   358       (Sign.add_path big_name thy4, []) (tl descr));
   359 
   360     (* prove isomorphism properties *)
   361 
   362     fun mk_funs_inv thy thm =
   363       let
   364         val prop = Thm.prop_of thm;
   365         val _ $ (_ $ ((S as Const (_, Type (_, [U, _]))) $ _ )) $
   366           (_ $ (_ $ (r $ (a $ _)) $ _)) = Type.freeze prop;
   367         val used = OldTerm.add_term_tfree_names (a, []);
   368 
   369         fun mk_thm i =
   370           let
   371             val Ts = map (TFree o rpair HOLogic.typeS)
   372               (Name.variant_list used (replicate i "'t"));
   373             val f = Free ("f", Ts ---> U)
   374           in SkipProof.prove_global thy [] [] (Logic.mk_implies
   375             (HOLogic.mk_Trueprop (HOLogic.list_all
   376                (map (pair "x") Ts, S $ app_bnds f i)),
   377              HOLogic.mk_Trueprop (HOLogic.mk_eq (list_abs (map (pair "x") Ts,
   378                r $ (a $ app_bnds f i)), f))))
   379             (fn _ => EVERY [REPEAT_DETERM_N i (rtac ext 1),
   380                REPEAT (etac allE 1), rtac thm 1, atac 1])
   381           end
   382       in map (fn r => r RS subst) (thm :: map mk_thm arities) end;
   383 
   384     (* prove  inj dt_Rep_i  and  dt_Rep_i x : dt_rep_set_i *)
   385 
   386     val fun_congs = map (fn T => make_elim (Drule.instantiate'
   387       [SOME (ctyp_of thy5 T)] [] fun_cong)) branchTs;
   388 
   389     fun prove_iso_thms (ds, (inj_thms, elem_thms)) =
   390       let
   391         val (_, (tname, _, _)) = hd ds;
   392         val induct = (#induct o the o Symtab.lookup dt_info) tname;
   393 
   394         fun mk_ind_concl (i, _) =
   395           let
   396             val T = nth recTs i;
   397             val Rep_t = Const (nth all_rep_names i, T --> Univ_elT);
   398             val rep_set_name = nth rep_set_names i
   399           in (HOLogic.all_const T $ Abs ("y", T, HOLogic.imp $
   400                 HOLogic.mk_eq (Rep_t $ mk_Free "x" T i, Rep_t $ Bound 0) $
   401                   HOLogic.mk_eq (mk_Free "x" T i, Bound 0)),
   402               Const (rep_set_name, UnivT') $ (Rep_t $ mk_Free "x" T i))
   403           end;
   404 
   405         val (ind_concl1, ind_concl2) = ListPair.unzip (map mk_ind_concl ds);
   406 
   407         val rewrites = map mk_meta_eq iso_char_thms;
   408         val inj_thms' = map snd newT_iso_inj_thms @
   409           map (fn r => r RS @{thm injD}) inj_thms;
   410 
   411         val inj_thm = SkipProof.prove_global thy5 [] []
   412           (HOLogic.mk_Trueprop (mk_conj ind_concl1)) (fn _ => EVERY
   413             [(indtac induct [] THEN_ALL_NEW ObjectLogic.atomize_prems_tac) 1,
   414              REPEAT (EVERY
   415                [rtac allI 1, rtac impI 1,
   416                 exh_tac (exh_thm_of dt_info) 1,
   417                 REPEAT (EVERY
   418                   [hyp_subst_tac 1,
   419                    rewrite_goals_tac rewrites,
   420                    REPEAT (dresolve_tac [In0_inject, In1_inject] 1),
   421                    (eresolve_tac [In0_not_In1 RS notE, In1_not_In0 RS notE] 1)
   422                    ORELSE (EVERY
   423                      [REPEAT (eresolve_tac (Scons_inject ::
   424                         map make_elim [Leaf_inject, Inl_inject, Inr_inject]) 1),
   425                       REPEAT (cong_tac 1), rtac refl 1,
   426                       REPEAT (atac 1 ORELSE (EVERY
   427                         [REPEAT (rtac ext 1),
   428                          REPEAT (eresolve_tac (mp :: allE ::
   429                            map make_elim (Suml_inject :: Sumr_inject ::
   430                              Lim_inject :: inj_thms') @ fun_congs) 1),
   431                          atac 1]))])])])]);
   432 
   433         val inj_thms'' = map (fn r => r RS @{thm datatype_injI})
   434                              (split_conj_thm inj_thm);
   435 
   436         val elem_thm = 
   437             SkipProof.prove_global thy5 [] [] (HOLogic.mk_Trueprop (mk_conj ind_concl2))
   438               (fn _ =>
   439                EVERY [(indtac induct [] THEN_ALL_NEW ObjectLogic.atomize_prems_tac) 1,
   440                 rewrite_goals_tac rewrites,
   441                 REPEAT ((resolve_tac rep_intrs THEN_ALL_NEW
   442                   ((REPEAT o etac allE) THEN' ares_tac elem_thms)) 1)]);
   443 
   444       in (inj_thms'' @ inj_thms, elem_thms @ (split_conj_thm elem_thm))
   445       end;
   446 
   447     val (iso_inj_thms_unfolded, iso_elem_thms) = List.foldr prove_iso_thms
   448       ([], map #3 newT_iso_axms) (tl descr);
   449     val iso_inj_thms = map snd newT_iso_inj_thms @
   450       map (fn r => r RS @{thm injD}) iso_inj_thms_unfolded;
   451 
   452     (* prove  dt_rep_set_i x --> x : range dt_Rep_i *)
   453 
   454     fun mk_iso_t (((set_name, iso_name), i), T) =
   455       let val isoT = T --> Univ_elT
   456       in HOLogic.imp $ 
   457         (Const (set_name, UnivT') $ mk_Free "x" Univ_elT i) $
   458           (if i < length newTs then HOLogic.true_const
   459            else HOLogic.mk_mem (mk_Free "x" Univ_elT i,
   460              Const (@{const_name image}, isoT --> HOLogic.mk_setT T --> UnivT) $
   461                Const (iso_name, isoT) $ Const (@{const_name UNIV}, HOLogic.mk_setT T)))
   462       end;
   463 
   464     val iso_t = HOLogic.mk_Trueprop (mk_conj (map mk_iso_t
   465       (rep_set_names ~~ all_rep_names ~~ (0 upto (length descr' - 1)) ~~ recTs)));
   466 
   467     (* all the theorems are proved by one single simultaneous induction *)
   468 
   469     val range_eqs = map (fn r => mk_meta_eq (r RS @{thm range_ex1_eq}))
   470       iso_inj_thms_unfolded;
   471 
   472     val iso_thms = if length descr = 1 then [] else
   473       Library.drop (length newTs, split_conj_thm
   474         (SkipProof.prove_global thy5 [] [] iso_t (fn _ => EVERY
   475            [(indtac rep_induct [] THEN_ALL_NEW ObjectLogic.atomize_prems_tac) 1,
   476             REPEAT (rtac TrueI 1),
   477             rewrite_goals_tac (mk_meta_eq choice_eq ::
   478               symmetric (mk_meta_eq @{thm expand_fun_eq}) :: range_eqs),
   479             rewrite_goals_tac (map symmetric range_eqs),
   480             REPEAT (EVERY
   481               [REPEAT (eresolve_tac ([rangeE, ex1_implies_ex RS exE] @
   482                  maps (mk_funs_inv thy5 o #1) newT_iso_axms) 1),
   483                TRY (hyp_subst_tac 1),
   484                rtac (sym RS range_eqI) 1,
   485                resolve_tac iso_char_thms 1])])));
   486 
   487     val Abs_inverse_thms' =
   488       map #1 newT_iso_axms @
   489       map2 (fn r_inj => fn r => @{thm f_inv_f} OF [r_inj, r RS mp])
   490         iso_inj_thms_unfolded iso_thms;
   491 
   492     val Abs_inverse_thms = maps (mk_funs_inv thy5) Abs_inverse_thms';
   493 
   494     (******************* freeness theorems for constructors *******************)
   495 
   496     val _ = message config "Proving freeness of constructors ...";
   497 
   498     (* prove theorem  Rep_i (Constr_j ...) = Inj_j ...  *)
   499     
   500     fun prove_constr_rep_thm eqn =
   501       let
   502         val inj_thms = map fst newT_iso_inj_thms;
   503         val rewrites = @{thm o_def} :: constr_defs @ (map (mk_meta_eq o #2) newT_iso_axms)
   504       in SkipProof.prove_global thy5 [] [] eqn (fn _ => EVERY
   505         [resolve_tac inj_thms 1,
   506          rewrite_goals_tac rewrites,
   507          rtac refl 3,
   508          resolve_tac rep_intrs 2,
   509          REPEAT (resolve_tac iso_elem_thms 1)])
   510       end;
   511 
   512     (*--------------------------------------------------------------*)
   513     (* constr_rep_thms and rep_congs are used to prove distinctness *)
   514     (* of constructors.                                             *)
   515     (*--------------------------------------------------------------*)
   516 
   517     val constr_rep_thms = map (map prove_constr_rep_thm) constr_rep_eqns;
   518 
   519     val dist_rewrites = map (fn (rep_thms, dist_lemma) =>
   520       dist_lemma::(rep_thms @ [In0_eq, In1_eq, In0_not_In1, In1_not_In0]))
   521         (constr_rep_thms ~~ dist_lemmas);
   522 
   523     fun prove_distinct_thms _ _ (_, []) = []
   524       | prove_distinct_thms lim dist_rewrites' (k, ts as _ :: _) =
   525           if k >= lim then [] else let
   526             (*number of constructors < distinctness_limit : C_i ... ~= C_j ...*)
   527             fun prove [] = []
   528               | prove (t :: ts) =
   529                   let
   530                     val dist_thm = SkipProof.prove_global thy5 [] [] t (fn _ =>
   531                       EVERY [simp_tac (HOL_ss addsimps dist_rewrites') 1])
   532                   in dist_thm :: standard (dist_thm RS not_sym) :: prove ts end;
   533           in prove ts end;
   534 
   535     val distinct_thms = DatatypeProp.make_distincts descr sorts
   536       |> map2 (prove_distinct_thms
   537            (Config.get_thy thy5 distinctness_limit)) dist_rewrites;
   538 
   539     val simproc_dists = map (fn ((((_, (_, _, constrs)), rep_thms), congr), dists) =>
   540       if length constrs < Config.get_thy thy5 distinctness_limit
   541       then FewConstrs dists
   542       else ManyConstrs (congr, HOL_basic_ss addsimps rep_thms)) (hd descr ~~
   543         constr_rep_thms ~~ rep_congs ~~ distinct_thms);
   544 
   545     (* prove injectivity of constructors *)
   546 
   547     fun prove_constr_inj_thm rep_thms t =
   548       let val inj_thms = Scons_inject :: (map make_elim
   549         (iso_inj_thms @
   550           [In0_inject, In1_inject, Leaf_inject, Inl_inject, Inr_inject,
   551            Lim_inject, Suml_inject, Sumr_inject]))
   552       in SkipProof.prove_global thy5 [] [] t (fn _ => EVERY
   553         [rtac iffI 1,
   554          REPEAT (etac conjE 2), hyp_subst_tac 2, rtac refl 2,
   555          dresolve_tac rep_congs 1, dtac box_equals 1,
   556          REPEAT (resolve_tac rep_thms 1),
   557          REPEAT (eresolve_tac inj_thms 1),
   558          REPEAT (ares_tac [conjI] 1 ORELSE (EVERY [REPEAT (rtac ext 1),
   559            REPEAT (eresolve_tac (make_elim fun_cong :: inj_thms) 1),
   560            atac 1]))])
   561       end;
   562 
   563     val constr_inject = map (fn (ts, thms) => map (prove_constr_inj_thm thms) ts)
   564       ((DatatypeProp.make_injs descr sorts) ~~ constr_rep_thms);
   565 
   566     val ((constr_inject', distinct_thms'), thy6) =
   567       thy5
   568       |> Sign.parent_path
   569       |> store_thmss "inject" new_type_names constr_inject
   570       ||>> store_thmss "distinct" new_type_names distinct_thms;
   571 
   572     (*************************** induction theorem ****************************)
   573 
   574     val _ = message config "Proving induction rule for datatypes ...";
   575 
   576     val Rep_inverse_thms = (map (fn (_, iso, _) => iso RS subst) newT_iso_axms) @
   577       (map (fn r => r RS @{thm inv_f_f} RS subst) iso_inj_thms_unfolded);
   578     val Rep_inverse_thms' = map (fn r => r RS @{thm inv_f_f}) iso_inj_thms_unfolded;
   579 
   580     fun mk_indrule_lemma ((prems, concls), ((i, _), T)) =
   581       let
   582         val Rep_t = Const (nth all_rep_names i, T --> Univ_elT) $
   583           mk_Free "x" T i;
   584 
   585         val Abs_t = if i < length newTs then
   586             Const (Sign.intern_const thy6
   587               ("Abs_" ^ (nth new_type_names i)), Univ_elT --> T)
   588           else Const (@{const_name Fun.inv}, [T --> Univ_elT, Univ_elT] ---> T) $
   589             Const (nth all_rep_names i, T --> Univ_elT)
   590 
   591       in (prems @ [HOLogic.imp $
   592             (Const (nth rep_set_names i, UnivT') $ Rep_t) $
   593               (mk_Free "P" (T --> HOLogic.boolT) (i + 1) $ (Abs_t $ Rep_t))],
   594           concls @ [mk_Free "P" (T --> HOLogic.boolT) (i + 1) $ mk_Free "x" T i])
   595       end;
   596 
   597     val (indrule_lemma_prems, indrule_lemma_concls) =
   598       Library.foldl mk_indrule_lemma (([], []), (descr' ~~ recTs));
   599 
   600     val cert = cterm_of thy6;
   601 
   602     val indrule_lemma = SkipProof.prove_global thy6 [] []
   603       (Logic.mk_implies
   604         (HOLogic.mk_Trueprop (mk_conj indrule_lemma_prems),
   605          HOLogic.mk_Trueprop (mk_conj indrule_lemma_concls))) (fn _ => EVERY
   606            [REPEAT (etac conjE 1),
   607             REPEAT (EVERY
   608               [TRY (rtac conjI 1), resolve_tac Rep_inverse_thms 1,
   609                etac mp 1, resolve_tac iso_elem_thms 1])]);
   610 
   611     val Ps = map head_of (HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of indrule_lemma)));
   612     val frees = if length Ps = 1 then [Free ("P", snd (dest_Var (hd Ps)))] else
   613       map (Free o apfst fst o dest_Var) Ps;
   614     val indrule_lemma' = cterm_instantiate (map cert Ps ~~ map cert frees) indrule_lemma;
   615 
   616     val dt_induct_prop = DatatypeProp.make_ind descr sorts;
   617     val dt_induct = SkipProof.prove_global thy6 []
   618       (Logic.strip_imp_prems dt_induct_prop) (Logic.strip_imp_concl dt_induct_prop)
   619       (fn {prems, ...} => EVERY
   620         [rtac indrule_lemma' 1,
   621          (indtac rep_induct [] THEN_ALL_NEW ObjectLogic.atomize_prems_tac) 1,
   622          EVERY (map (fn (prem, r) => (EVERY
   623            [REPEAT (eresolve_tac Abs_inverse_thms 1),
   624             simp_tac (HOL_basic_ss addsimps ((symmetric r)::Rep_inverse_thms')) 1,
   625             DEPTH_SOLVE_1 (ares_tac [prem] 1 ORELSE etac allE 1)]))
   626                 (prems ~~ (constr_defs @ (map mk_meta_eq iso_char_thms))))]);
   627 
   628     val ([dt_induct'], thy7) =
   629       thy6
   630       |> Sign.add_path big_name
   631       |> PureThy.add_thms [((Binding.name "induct", dt_induct), [case_names_induct])]
   632       ||> Sign.parent_path
   633       ||> Theory.checkpoint;
   634 
   635   in
   636     ((constr_inject', distinct_thms', dist_rewrites, simproc_dists, dt_induct'), thy7)
   637   end;
   638 
   639 end;