src/HOL/Tools/datatype_rep_proofs.ML
author wenzelm
Sun Mar 01 23:36:12 2009 +0100 (2009-03-01)
changeset 30190 479806475f3c
parent 29579 cb520b766e00
child 30280 eb98b49ef835
child 30305 720226bedc4d
permissions -rw-r--r--
use long names for old-style fold combinators;
     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   val distinctness_limit : int Config.T
    15   val distinctness_limit_setup : theory -> theory
    16   val representation_proofs : bool -> DatatypeAux.datatype_info Symtab.table ->
    17     string list -> DatatypeAux.descr list -> (string * sort) list ->
    18       (string * mixfix) list -> (string * mixfix) list list -> attribute
    19         -> theory -> (thm list list * thm list list * thm list list *
    20           DatatypeAux.simproc_dist list * thm) * theory
    21 end;
    22 
    23 structure DatatypeRepProofs : DATATYPE_REP_PROOFS =
    24 struct
    25 
    26 open DatatypeAux;
    27 
    28 (*the kind of distinctiveness axioms depends on number of constructors*)
    29 val (distinctness_limit, distinctness_limit_setup) =
    30   Attrib.config_int "datatype_distinctness_limit" 7;
    31 
    32 val (_ $ (_ $ (_ $ (distinct_f $ _) $ _))) = hd (prems_of distinct_lemma);
    33 
    34 val collect_simp = rewrite_rule [mk_meta_eq mem_Collect_eq];
    35 
    36 
    37 (** theory context references **)
    38 
    39 val f_myinv_f = thm "f_myinv_f";
    40 val myinv_f_f = thm "myinv_f_f";
    41 
    42 
    43 fun exh_thm_of (dt_info : datatype_info Symtab.table) tname =
    44   #exhaustion (the (Symtab.lookup dt_info tname));
    45 
    46 (******************************************************************************)
    47 
    48 fun representation_proofs flat_names (dt_info : datatype_info Symtab.table)
    49       new_type_names descr sorts types_syntax constr_syntax case_names_induct thy =
    50   let
    51     val Datatype_thy = ThyInfo.the_theory "Datatype" thy;
    52     val node_name = "Datatype.node";
    53     val In0_name = "Datatype.In0";
    54     val In1_name = "Datatype.In1";
    55     val Scons_name = "Datatype.Scons";
    56     val Leaf_name = "Datatype.Leaf";
    57     val Numb_name = "Datatype.Numb";
    58     val Lim_name = "Datatype.Lim";
    59     val Suml_name = "Datatype.Suml";
    60     val Sumr_name = "Datatype.Sumr";
    61 
    62     val [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] = map (PureThy.get_thm Datatype_thy)
    65           ["In0_inject", "In1_inject", "Scons_inject", "Leaf_inject",
    66            "In0_eq", "In1_eq", "In0_not_In1", "In1_not_In0",
    67            "Lim_inject", "Suml_inject", "Sumr_inject"];
    68 
    69     val descr' = flat descr;
    70 
    71     val big_name = space_implode "_" new_type_names;
    72     val thy1 = add_path flat_names big_name thy;
    73     val big_rec_name = big_name ^ "_rep_set";
    74     val rep_set_names' =
    75       (if length descr' = 1 then [big_rec_name] else
    76         (map ((curry (op ^) (big_rec_name ^ "_")) o string_of_int)
    77           (1 upto (length descr'))));
    78     val rep_set_names = map (Sign.full_bname thy1) rep_set_names';
    79 
    80     val tyvars = map (fn (_, (_, Ts, _)) => map dest_DtTFree Ts) (hd descr);
    81     val leafTs' = get_nonrec_types descr' sorts;
    82     val branchTs = get_branching_types descr' sorts;
    83     val branchT = if null branchTs then HOLogic.unitT
    84       else BalancedTree.make (fn (T, U) => Type ("+", [T, U])) branchTs;
    85     val arities = get_arities descr' \ 0;
    86     val unneeded_vars = hd tyvars \\ List.foldr OldTerm.add_typ_tfree_names [] (leafTs' @ branchTs);
    87     val leafTs = leafTs' @ (map (fn n => TFree (n, (the o AList.lookup (op =) sorts) n)) unneeded_vars);
    88     val recTs = get_rec_types descr' sorts;
    89     val newTs = Library.take (length (hd descr), recTs);
    90     val oldTs = Library.drop (length (hd descr), recTs);
    91     val sumT = if null leafTs then HOLogic.unitT
    92       else BalancedTree.make (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     val UnivT' = Univ_elT --> HOLogic.boolT;
    96     val Collect = Const ("Collect", UnivT' --> UnivT);
    97 
    98     val In0 = Const (In0_name, Univ_elT --> Univ_elT);
    99     val In1 = Const (In1_name, Univ_elT --> Univ_elT);
   100     val Leaf = Const (Leaf_name, sumT --> Univ_elT);
   101     val Lim = Const (Lim_name, (branchT --> Univ_elT) --> Univ_elT);
   102 
   103     (* make injections needed for embedding types in leaves *)
   104 
   105     fun mk_inj T' x =
   106       let
   107         fun mk_inj' T n i =
   108           if n = 1 then x else
   109           let val n2 = n div 2;
   110               val Type (_, [T1, T2]) = T
   111           in
   112             if i <= n2 then
   113               Const ("Sum_Type.Inl", T1 --> T) $ (mk_inj' T1 n2 i)
   114             else
   115               Const ("Sum_Type.Inr", T2 --> T) $ (mk_inj' T2 (n - n2) (i - n2))
   116           end
   117       in mk_inj' sumT (length leafTs) (1 + find_index_eq T' leafTs)
   118       end;
   119 
   120     (* make injections for constructors *)
   121 
   122     fun mk_univ_inj ts = BalancedTree.access
   123       {left = fn t => In0 $ t,
   124         right = fn t => In1 $ t,
   125         init =
   126           if ts = [] then Const (@{const_name undefined}, Univ_elT)
   127           else foldr1 (HOLogic.mk_binop Scons_name) ts};
   128 
   129     (* function spaces *)
   130 
   131     fun mk_fun_inj T' x =
   132       let
   133         fun mk_inj T n i =
   134           if n = 1 then x else
   135           let
   136             val n2 = n div 2;
   137             val Type (_, [T1, T2]) = T;
   138             fun mkT U = (U --> Univ_elT) --> T --> Univ_elT
   139           in
   140             if i <= n2 then Const (Suml_name, mkT T1) $ mk_inj T1 n2 i
   141             else Const (Sumr_name, mkT T2) $ mk_inj T2 (n - n2) (i - n2)
   142           end
   143       in mk_inj branchT (length branchTs) (1 + find_index_eq T' branchTs)
   144       end;
   145 
   146     val mk_lim = List.foldr (fn (T, t) => Lim $ mk_fun_inj T (Abs ("x", T, t)));
   147 
   148     (************** generate introduction rules for representing set **********)
   149 
   150     val _ = message "Constructing representing sets ...";
   151 
   152     (* make introduction rule for a single constructor *)
   153 
   154     fun make_intr s n (i, (_, cargs)) =
   155       let
   156         fun mk_prem (dt, (j, prems, ts)) = (case strip_dtyp dt of
   157             (dts, DtRec k) =>
   158               let
   159                 val Ts = map (typ_of_dtyp descr' sorts) dts;
   160                 val free_t =
   161                   app_bnds (mk_Free "x" (Ts ---> Univ_elT) j) (length Ts)
   162               in (j + 1, list_all (map (pair "x") Ts,
   163                   HOLogic.mk_Trueprop
   164                     (Free (List.nth (rep_set_names', k), UnivT') $ free_t)) :: prems,
   165                 mk_lim free_t Ts :: ts)
   166               end
   167           | _ =>
   168               let val T = typ_of_dtyp descr' sorts dt
   169               in (j + 1, prems, (Leaf $ mk_inj T (mk_Free "x" T j))::ts)
   170               end);
   171 
   172         val (_, prems, ts) = List.foldr mk_prem (1, [], []) cargs;
   173         val concl = HOLogic.mk_Trueprop
   174           (Free (s, UnivT') $ mk_univ_inj ts n i)
   175       in Logic.list_implies (prems, concl)
   176       end;
   177 
   178     val intr_ts = maps (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 ({raw_induct = rep_induct, intrs = rep_intrs, ...}, thy2) =
   183         InductivePackage.add_inductive_global (serial_string ())
   184           {quiet_mode = ! quiet_mode, verbose = false, kind = Thm.internalK,
   185            alt_name = Binding.name big_rec_name, coind = false, no_elim = true, no_ind = false,
   186            skip_mono = true, fork_mono = false}
   187           (map (fn s => ((Binding.name s, UnivT'), NoSyn)) rep_set_names') []
   188           (map (fn x => (Attrib.empty_binding, x)) intr_ts) [] thy1;
   189 
   190     (********************************* typedef ********************************)
   191 
   192     val (typedefs, thy3) = thy2 |>
   193       parent_path flat_names |>
   194       fold_map (fn ((((name, mx), tvs), c), name') =>
   195           TypedefPackage.add_typedef false (SOME name') (name, tvs, mx)
   196             (Collect $ Const (c, UnivT')) NONE
   197             (rtac exI 1 THEN rtac CollectI 1 THEN
   198               QUIET_BREADTH_FIRST (has_fewer_prems 1)
   199               (resolve_tac rep_intrs 1)))
   200                 (types_syntax ~~ tyvars ~~
   201                   (Library.take (length newTs, rep_set_names)) ~~ new_type_names) ||>
   202       add_path flat_names big_name;
   203 
   204     (*********************** definition of constructors ***********************)
   205 
   206     val big_rep_name = (space_implode "_" new_type_names) ^ "_Rep_";
   207     val rep_names = map (curry op ^ "Rep_") new_type_names;
   208     val rep_names' = map (fn i => big_rep_name ^ (string_of_int i))
   209       (1 upto (length (flat (tl descr))));
   210     val all_rep_names = map (Sign.intern_const thy3) rep_names @
   211       map (Sign.full_bname thy3) rep_names';
   212 
   213     (* isomorphism declarations *)
   214 
   215     val iso_decls = map (fn (T, s) => (s, T --> Univ_elT, NoSyn))
   216       (oldTs ~~ rep_names');
   217 
   218     (* constructor definitions *)
   219 
   220     fun make_constr_def tname T n ((thy, defs, eqns, i), ((cname, cargs), (cname', mx))) =
   221       let
   222         fun constr_arg (dt, (j, l_args, r_args)) =
   223           let val T = typ_of_dtyp descr' sorts dt;
   224               val free_t = mk_Free "x" T j
   225           in (case (strip_dtyp dt, strip_type T) of
   226               ((_, DtRec m), (Us, U)) => (j + 1, free_t :: l_args, mk_lim
   227                 (Const (List.nth (all_rep_names, m), U --> Univ_elT) $
   228                    app_bnds free_t (length Us)) Us :: r_args)
   229             | _ => (j + 1, free_t::l_args, (Leaf $ mk_inj T free_t)::r_args))
   230           end;
   231 
   232         val (_, l_args, r_args) = List.foldr constr_arg (1, [], []) cargs;
   233         val constrT = (map (typ_of_dtyp descr' sorts) cargs) ---> T;
   234         val abs_name = Sign.intern_const thy ("Abs_" ^ tname);
   235         val rep_name = Sign.intern_const 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 = Logic.mk_equals (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 ([def_thm], thy') =
   243           thy
   244           |> Sign.add_consts_i [(cname', constrT, mx)]
   245           |> (PureThy.add_defs false o map Thm.no_attributes) [(Binding.name def_name, def)];
   246 
   247       in (thy', defs @ [def_thm], eqns @ [eqn], i + 1) end;
   248 
   249     (* constructor definitions for datatype *)
   250 
   251     fun dt_constr_defs ((thy, defs, eqns, rep_congs, dist_lemmas),
   252         ((((_, (_, _, constrs)), tname), T), constr_syntax)) =
   253       let
   254         val _ $ (_ $ (cong_f $ _) $ _) = concl_of arg_cong;
   255         val rep_const = cterm_of thy
   256           (Const (Sign.intern_const thy ("Rep_" ^ tname), T --> Univ_elT));
   257         val cong' = standard (cterm_instantiate [(cterm_of thy cong_f, rep_const)] arg_cong);
   258         val dist = standard (cterm_instantiate [(cterm_of thy distinct_f, rep_const)] distinct_lemma);
   259         val (thy', defs', eqns', _) = Library.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) = Library.foldl dt_constr_defs
   267       ((thy3 |> Sign.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     val newT_iso_axms = map (fn (_, td) =>
   275       (collect_simp (#Abs_inverse td), #Rep_inverse td,
   276        collect_simp (#Rep td))) typedefs;
   277 
   278     val newT_iso_inj_thms = map (fn (_, td) =>
   279       (collect_simp (#Abs_inject td) RS iffD1, #Rep_inject td RS iffD1)) typedefs;
   280 
   281     (********* isomorphisms between existing types and "unfolded" types *******)
   282 
   283     (*---------------------------------------------------------------------*)
   284     (* isomorphisms are defined using primrec-combinators:                 *)
   285     (* generate appropriate functions for instantiating primrec-combinator *)
   286     (*                                                                     *)
   287     (*   e.g.  dt_Rep_i = list_rec ... (%h t y. In1 (Scons (Leaf h) y))    *)
   288     (*                                                                     *)
   289     (* also generate characteristic equations for isomorphisms             *)
   290     (*                                                                     *)
   291     (*   e.g.  dt_Rep_i (cons h t) = In1 (Scons (dt_Rep_j h) (dt_Rep_i t)) *)
   292     (*---------------------------------------------------------------------*)
   293 
   294     fun make_iso_def k ks n ((fs, eqns, i), (cname, cargs)) =
   295       let
   296         val argTs = map (typ_of_dtyp descr' sorts) cargs;
   297         val T = List.nth (recTs, k);
   298         val rep_name = List.nth (all_rep_names, k);
   299         val rep_const = Const (rep_name, T --> Univ_elT);
   300         val constr = Const (cname, argTs ---> T);
   301 
   302         fun process_arg ks' ((i2, i2', ts, Ts), dt) =
   303           let
   304             val T' = typ_of_dtyp descr' sorts dt;
   305             val (Us, U) = strip_type T'
   306           in (case strip_dtyp dt of
   307               (_, DtRec j) => if j mem ks' then
   308                   (i2 + 1, i2' + 1, ts @ [mk_lim (app_bnds
   309                      (mk_Free "y" (Us ---> Univ_elT) i2') (length Us)) Us],
   310                    Ts @ [Us ---> Univ_elT])
   311                 else
   312                   (i2 + 1, i2', ts @ [mk_lim
   313                      (Const (List.nth (all_rep_names, j), U --> Univ_elT) $
   314                         app_bnds (mk_Free "x" T' i2) (length Us)) Us], Ts)
   315             | _ => (i2 + 1, i2', ts @ [Leaf $ mk_inj T' (mk_Free "x" T' i2)], Ts))
   316           end;
   317 
   318         val (i2, i2', ts, Ts) = Library.foldl (process_arg ks) ((1, 1, [], []), cargs);
   319         val xs = map (uncurry (mk_Free "x")) (argTs ~~ (1 upto (i2 - 1)));
   320         val ys = map (uncurry (mk_Free "y")) (Ts ~~ (1 upto (i2' - 1)));
   321         val f = list_abs_free (map dest_Free (xs @ ys), mk_univ_inj ts n i);
   322 
   323         val (_, _, ts', _) = Library.foldl (process_arg []) ((1, 1, [], []), cargs);
   324         val eqn = HOLogic.mk_Trueprop (HOLogic.mk_eq
   325           (rep_const $ list_comb (constr, xs), mk_univ_inj ts' n i))
   326 
   327       in (fs @ [f], eqns @ [eqn], i + 1) end;
   328 
   329     (* define isomorphisms for all mutually recursive datatypes in list ds *)
   330 
   331     fun make_iso_defs (ds, (thy, char_thms)) =
   332       let
   333         val ks = map fst ds;
   334         val (_, (tname, _, _)) = hd ds;
   335         val {rec_rewrites, rec_names, ...} = the (Symtab.lookup dt_info tname);
   336 
   337         fun process_dt ((fs, eqns, isos), (k, (tname, _, constrs))) =
   338           let
   339             val (fs', eqns', _) = Library.foldl (make_iso_def k ks (length constrs))
   340               ((fs, eqns, 1), constrs);
   341             val iso = (List.nth (recTs, k), List.nth (all_rep_names, k))
   342           in (fs', eqns', isos @ [iso]) end;
   343         
   344         val (fs, eqns, isos) = Library.foldl process_dt (([], [], []), ds);
   345         val fTs = map fastype_of fs;
   346         val defs = map (fn (rec_name, (T, iso_name)) => (Binding.name (Sign.base_name iso_name ^ "_def"),
   347           Logic.mk_equals (Const (iso_name, T --> Univ_elT),
   348             list_comb (Const (rec_name, fTs @ [T] ---> Univ_elT), fs)))) (rec_names ~~ isos);
   349         val (def_thms, thy') =
   350           apsnd Theory.checkpoint ((PureThy.add_defs false o map Thm.no_attributes) defs thy);
   351 
   352         (* prove characteristic equations *)
   353 
   354         val rewrites = def_thms @ (map mk_meta_eq rec_rewrites);
   355         val char_thms' = map (fn eqn => SkipProof.prove_global thy' [] [] eqn
   356           (fn _ => EVERY [rewrite_goals_tac rewrites, rtac refl 1])) eqns;
   357 
   358       in (thy', char_thms' @ char_thms) end;
   359 
   360     val (thy5, iso_char_thms) = apfst Theory.checkpoint (List.foldr make_iso_defs
   361       (add_path flat_names big_name thy4, []) (tl descr));
   362 
   363     (* prove isomorphism properties *)
   364 
   365     fun mk_funs_inv thy thm =
   366       let
   367         val prop = Thm.prop_of thm;
   368         val _ $ (_ $ ((S as Const (_, Type (_, [U, _]))) $ _ )) $
   369           (_ $ (_ $ (r $ (a $ _)) $ _)) = Type.freeze prop;
   370         val used = OldTerm.add_term_tfree_names (a, []);
   371 
   372         fun mk_thm i =
   373           let
   374             val Ts = map (TFree o rpair HOLogic.typeS)
   375               (Name.variant_list used (replicate i "'t"));
   376             val f = Free ("f", Ts ---> U)
   377           in SkipProof.prove_global thy [] [] (Logic.mk_implies
   378             (HOLogic.mk_Trueprop (HOLogic.list_all
   379                (map (pair "x") Ts, S $ app_bnds f i)),
   380              HOLogic.mk_Trueprop (HOLogic.mk_eq (list_abs (map (pair "x") Ts,
   381                r $ (a $ app_bnds f i)), f))))
   382             (fn _ => EVERY [REPEAT_DETERM_N i (rtac ext 1),
   383                REPEAT (etac allE 1), rtac thm 1, atac 1])
   384           end
   385       in map (fn r => r RS subst) (thm :: map mk_thm arities) end;
   386 
   387     (* prove  inj dt_Rep_i  and  dt_Rep_i x : dt_rep_set_i *)
   388 
   389     val fun_congs = map (fn T => make_elim (Drule.instantiate'
   390       [SOME (ctyp_of thy5 T)] [] fun_cong)) branchTs;
   391 
   392     fun prove_iso_thms (ds, (inj_thms, elem_thms)) =
   393       let
   394         val (_, (tname, _, _)) = hd ds;
   395         val {induction, ...} = the (Symtab.lookup dt_info tname);
   396 
   397         fun mk_ind_concl (i, _) =
   398           let
   399             val T = List.nth (recTs, i);
   400             val Rep_t = Const (List.nth (all_rep_names, i), T --> Univ_elT);
   401             val rep_set_name = List.nth (rep_set_names, i)
   402           in (HOLogic.all_const T $ Abs ("y", T, HOLogic.imp $
   403                 HOLogic.mk_eq (Rep_t $ mk_Free "x" T i, Rep_t $ Bound 0) $
   404                   HOLogic.mk_eq (mk_Free "x" T i, Bound 0)),
   405               Const (rep_set_name, UnivT') $ (Rep_t $ mk_Free "x" T i))
   406           end;
   407 
   408         val (ind_concl1, ind_concl2) = ListPair.unzip (map mk_ind_concl ds);
   409 
   410         val rewrites = map mk_meta_eq iso_char_thms;
   411         val inj_thms' = map snd newT_iso_inj_thms @
   412           map (fn r => r RS @{thm injD}) inj_thms;
   413 
   414         val inj_thm = SkipProof.prove_global thy5 [] []
   415           (HOLogic.mk_Trueprop (mk_conj ind_concl1)) (fn _ => EVERY
   416             [(indtac induction [] THEN_ALL_NEW ObjectLogic.atomize_prems_tac) 1,
   417              REPEAT (EVERY
   418                [rtac allI 1, rtac impI 1,
   419                 exh_tac (exh_thm_of dt_info) 1,
   420                 REPEAT (EVERY
   421                   [hyp_subst_tac 1,
   422                    rewrite_goals_tac rewrites,
   423                    REPEAT (dresolve_tac [In0_inject, In1_inject] 1),
   424                    (eresolve_tac [In0_not_In1 RS notE, In1_not_In0 RS notE] 1)
   425                    ORELSE (EVERY
   426                      [REPEAT (eresolve_tac (Scons_inject ::
   427                         map make_elim [Leaf_inject, Inl_inject, Inr_inject]) 1),
   428                       REPEAT (cong_tac 1), rtac refl 1,
   429                       REPEAT (atac 1 ORELSE (EVERY
   430                         [REPEAT (rtac ext 1),
   431                          REPEAT (eresolve_tac (mp :: allE ::
   432                            map make_elim (Suml_inject :: Sumr_inject ::
   433                              Lim_inject :: inj_thms') @ fun_congs) 1),
   434                          atac 1]))])])])]);
   435 
   436         val inj_thms'' = map (fn r => r RS @{thm datatype_injI})
   437                              (split_conj_thm inj_thm);
   438 
   439         val elem_thm = 
   440             SkipProof.prove_global thy5 [] [] (HOLogic.mk_Trueprop (mk_conj ind_concl2))
   441               (fn _ =>
   442                EVERY [(indtac induction [] THEN_ALL_NEW ObjectLogic.atomize_prems_tac) 1,
   443                 rewrite_goals_tac rewrites,
   444                 REPEAT ((resolve_tac rep_intrs THEN_ALL_NEW
   445                   ((REPEAT o etac allE) THEN' ares_tac elem_thms)) 1)]);
   446 
   447       in (inj_thms'' @ inj_thms, elem_thms @ (split_conj_thm elem_thm))
   448       end;
   449 
   450     val (iso_inj_thms_unfolded, iso_elem_thms) = List.foldr prove_iso_thms
   451       ([], map #3 newT_iso_axms) (tl descr);
   452     val iso_inj_thms = map snd newT_iso_inj_thms @
   453       map (fn r => r RS @{thm injD}) iso_inj_thms_unfolded;
   454 
   455     (* prove  dt_rep_set_i x --> x : range dt_Rep_i *)
   456 
   457     fun mk_iso_t (((set_name, iso_name), i), T) =
   458       let val isoT = T --> Univ_elT
   459       in HOLogic.imp $ 
   460         (Const (set_name, UnivT') $ mk_Free "x" Univ_elT i) $
   461           (if i < length newTs then Const ("True", HOLogic.boolT)
   462            else HOLogic.mk_mem (mk_Free "x" Univ_elT i,
   463              Const ("image", [isoT, HOLogic.mk_setT T] ---> UnivT) $
   464                Const (iso_name, isoT) $ Const ("UNIV", HOLogic.mk_setT T)))
   465       end;
   466 
   467     val iso_t = HOLogic.mk_Trueprop (mk_conj (map mk_iso_t
   468       (rep_set_names ~~ all_rep_names ~~ (0 upto (length descr' - 1)) ~~ recTs)));
   469 
   470     (* all the theorems are proved by one single simultaneous induction *)
   471 
   472     val range_eqs = map (fn r => mk_meta_eq (r RS @{thm range_ex1_eq}))
   473       iso_inj_thms_unfolded;
   474 
   475     val iso_thms = if length descr = 1 then [] else
   476       Library.drop (length newTs, split_conj_thm
   477         (SkipProof.prove_global thy5 [] [] iso_t (fn _ => EVERY
   478            [(indtac rep_induct [] THEN_ALL_NEW ObjectLogic.atomize_prems_tac) 1,
   479             REPEAT (rtac TrueI 1),
   480             rewrite_goals_tac (mk_meta_eq choice_eq ::
   481               symmetric (mk_meta_eq @{thm expand_fun_eq}) :: range_eqs),
   482             rewrite_goals_tac (map symmetric range_eqs),
   483             REPEAT (EVERY
   484               [REPEAT (eresolve_tac ([rangeE, ex1_implies_ex RS exE] @
   485                  maps (mk_funs_inv thy5 o #1) newT_iso_axms) 1),
   486                TRY (hyp_subst_tac 1),
   487                rtac (sym RS range_eqI) 1,
   488                resolve_tac iso_char_thms 1])])));
   489 
   490     val Abs_inverse_thms' =
   491       map #1 newT_iso_axms @
   492       map2 (fn r_inj => fn r => f_myinv_f OF [r_inj, r RS mp])
   493         iso_inj_thms_unfolded iso_thms;
   494 
   495     val Abs_inverse_thms = maps (mk_funs_inv thy5) Abs_inverse_thms';
   496 
   497     (******************* freeness theorems for constructors *******************)
   498 
   499     val _ = message "Proving freeness of constructors ...";
   500 
   501     (* prove theorem  Rep_i (Constr_j ...) = Inj_j ...  *)
   502     
   503     fun prove_constr_rep_thm eqn =
   504       let
   505         val inj_thms = map fst newT_iso_inj_thms;
   506         val rewrites = @{thm o_def} :: constr_defs @ (map (mk_meta_eq o #2) newT_iso_axms)
   507       in SkipProof.prove_global thy5 [] [] eqn (fn _ => EVERY
   508         [resolve_tac inj_thms 1,
   509          rewrite_goals_tac rewrites,
   510          rtac refl 3,
   511          resolve_tac rep_intrs 2,
   512          REPEAT (resolve_tac iso_elem_thms 1)])
   513       end;
   514 
   515     (*--------------------------------------------------------------*)
   516     (* constr_rep_thms and rep_congs are used to prove distinctness *)
   517     (* of constructors.                                             *)
   518     (*--------------------------------------------------------------*)
   519 
   520     val constr_rep_thms = map (map prove_constr_rep_thm) constr_rep_eqns;
   521 
   522     val dist_rewrites = map (fn (rep_thms, dist_lemma) =>
   523       dist_lemma::(rep_thms @ [In0_eq, In1_eq, In0_not_In1, In1_not_In0]))
   524         (constr_rep_thms ~~ dist_lemmas);
   525 
   526     fun prove_distinct_thms _ _ (_, []) = []
   527       | prove_distinct_thms lim dist_rewrites' (k, ts as _ :: _) =
   528           if k >= lim then [] else let
   529             (*number of constructors < distinctness_limit : C_i ... ~= C_j ...*)
   530             fun prove [] = []
   531               | prove (t :: ts) =
   532                   let
   533                     val dist_thm = SkipProof.prove_global thy5 [] [] t (fn _ =>
   534                       EVERY [simp_tac (HOL_ss addsimps dist_rewrites') 1])
   535                   in dist_thm :: standard (dist_thm RS not_sym) :: prove ts end;
   536           in prove ts end;
   537 
   538     val distinct_thms = DatatypeProp.make_distincts descr sorts
   539       |> map2 (prove_distinct_thms
   540            (Config.get_thy thy5 distinctness_limit)) dist_rewrites;
   541 
   542     val simproc_dists = map (fn ((((_, (_, _, constrs)), rep_thms), congr), dists) =>
   543       if length constrs < Config.get_thy thy5 distinctness_limit
   544       then FewConstrs dists
   545       else ManyConstrs (congr, HOL_basic_ss addsimps rep_thms)) (hd descr ~~
   546         constr_rep_thms ~~ rep_congs ~~ distinct_thms);
   547 
   548     (* prove injectivity of constructors *)
   549 
   550     fun prove_constr_inj_thm rep_thms t =
   551       let val inj_thms = Scons_inject :: (map make_elim
   552         (iso_inj_thms @
   553           [In0_inject, In1_inject, Leaf_inject, Inl_inject, Inr_inject,
   554            Lim_inject, Suml_inject, Sumr_inject]))
   555       in SkipProof.prove_global thy5 [] [] t (fn _ => EVERY
   556         [rtac iffI 1,
   557          REPEAT (etac conjE 2), hyp_subst_tac 2, rtac refl 2,
   558          dresolve_tac rep_congs 1, dtac box_equals 1,
   559          REPEAT (resolve_tac rep_thms 1),
   560          REPEAT (eresolve_tac inj_thms 1),
   561          REPEAT (ares_tac [conjI] 1 ORELSE (EVERY [REPEAT (rtac ext 1),
   562            REPEAT (eresolve_tac (make_elim fun_cong :: inj_thms) 1),
   563            atac 1]))])
   564       end;
   565 
   566     val constr_inject = map (fn (ts, thms) => map (prove_constr_inj_thm thms) ts)
   567       ((DatatypeProp.make_injs descr sorts) ~~ constr_rep_thms);
   568 
   569     val ((constr_inject', distinct_thms'), thy6) =
   570       thy5
   571       |> parent_path flat_names
   572       |> store_thmss "inject" new_type_names constr_inject
   573       ||>> store_thmss "distinct" new_type_names distinct_thms;
   574 
   575     (*************************** induction theorem ****************************)
   576 
   577     val _ = message "Proving induction rule for datatypes ...";
   578 
   579     val Rep_inverse_thms = (map (fn (_, iso, _) => iso RS subst) newT_iso_axms) @
   580       (map (fn r => r RS myinv_f_f RS subst) iso_inj_thms_unfolded);
   581     val Rep_inverse_thms' = map (fn r => r RS myinv_f_f) iso_inj_thms_unfolded;
   582 
   583     fun mk_indrule_lemma ((prems, concls), ((i, _), T)) =
   584       let
   585         val Rep_t = Const (List.nth (all_rep_names, i), T --> Univ_elT) $
   586           mk_Free "x" T i;
   587 
   588         val Abs_t = if i < length newTs then
   589             Const (Sign.intern_const thy6
   590               ("Abs_" ^ (List.nth (new_type_names, i))), Univ_elT --> T)
   591           else Const ("Inductive.myinv", [T --> Univ_elT, Univ_elT] ---> T) $
   592             Const (List.nth (all_rep_names, i), T --> Univ_elT)
   593 
   594       in (prems @ [HOLogic.imp $
   595             (Const (List.nth (rep_set_names, i), UnivT') $ Rep_t) $
   596               (mk_Free "P" (T --> HOLogic.boolT) (i + 1) $ (Abs_t $ Rep_t))],
   597           concls @ [mk_Free "P" (T --> HOLogic.boolT) (i + 1) $ mk_Free "x" T i])
   598       end;
   599 
   600     val (indrule_lemma_prems, indrule_lemma_concls) =
   601       Library.foldl mk_indrule_lemma (([], []), (descr' ~~ recTs));
   602 
   603     val cert = cterm_of thy6;
   604 
   605     val indrule_lemma = SkipProof.prove_global thy6 [] []
   606       (Logic.mk_implies
   607         (HOLogic.mk_Trueprop (mk_conj indrule_lemma_prems),
   608          HOLogic.mk_Trueprop (mk_conj indrule_lemma_concls))) (fn _ => EVERY
   609            [REPEAT (etac conjE 1),
   610             REPEAT (EVERY
   611               [TRY (rtac conjI 1), resolve_tac Rep_inverse_thms 1,
   612                etac mp 1, resolve_tac iso_elem_thms 1])]);
   613 
   614     val Ps = map head_of (HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of indrule_lemma)));
   615     val frees = if length Ps = 1 then [Free ("P", snd (dest_Var (hd Ps)))] else
   616       map (Free o apfst fst o dest_Var) Ps;
   617     val indrule_lemma' = cterm_instantiate (map cert Ps ~~ map cert frees) indrule_lemma;
   618 
   619     val dt_induct_prop = DatatypeProp.make_ind descr sorts;
   620     val dt_induct = SkipProof.prove_global thy6 []
   621       (Logic.strip_imp_prems dt_induct_prop) (Logic.strip_imp_concl dt_induct_prop)
   622       (fn {prems, ...} => EVERY
   623         [rtac indrule_lemma' 1,
   624          (indtac rep_induct [] THEN_ALL_NEW ObjectLogic.atomize_prems_tac) 1,
   625          EVERY (map (fn (prem, r) => (EVERY
   626            [REPEAT (eresolve_tac Abs_inverse_thms 1),
   627             simp_tac (HOL_basic_ss addsimps ((symmetric r)::Rep_inverse_thms')) 1,
   628             DEPTH_SOLVE_1 (ares_tac [prem] 1 ORELSE etac allE 1)]))
   629                 (prems ~~ (constr_defs @ (map mk_meta_eq iso_char_thms))))]);
   630 
   631     val ([dt_induct'], thy7) =
   632       thy6
   633       |> Sign.add_path big_name
   634       |> PureThy.add_thms [((Binding.name "induct", dt_induct), [case_names_induct])]
   635       ||> Sign.parent_path
   636       ||> Theory.checkpoint;
   637 
   638   in
   639     ((constr_inject', distinct_thms', dist_rewrites, simproc_dists, dt_induct'), thy7)
   640   end;
   641 
   642 end;