src/HOL/Tools/datatype_rep_proofs.ML
author wenzelm
Wed Nov 14 18:44:27 2001 +0100 (2001-11-14)
changeset 12180 91c9f661b183
parent 11957 f1657e0291ca
child 12922 ed70a600f0ea
permissions -rw-r--r--
inductive: removed con_defs;
     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 = "Datatype_Universe.node";
    49     val In0_name = "Datatype_Universe.In0";
    50     val In1_name = "Datatype_Universe.In1";
    51     val Scons_name = "Datatype_Universe.Scons";
    52     val Leaf_name = "Datatype_Universe.Leaf";
    53     val Numb_name = "Datatype_Universe.Numb";
    54     val Lim_name = "Datatype_Universe.Lim";
    55     val Funs_name = "Datatype_Universe.Funs";
    56     val o_name = "Fun.op o";
    57     val sum_case_name = "Datatype.sum.sum_case";
    58 
    59     val [In0_inject, In1_inject, Scons_inject, Leaf_inject, In0_eq, In1_eq,
    60          In0_not_In1, In1_not_In0, Funs_mono, FunsI, Lim_inject,
    61          Funs_inv, FunsD, Funs_rangeE, Funs_nonempty, sum_case_inject] = map (get_thm Datatype_thy)
    62         ["In0_inject", "In1_inject", "Scons_inject", "Leaf_inject", "In0_eq", "In1_eq",
    63          "In0_not_In1", "In1_not_In0", "Funs_mono", "FunsI", "Lim_inject",
    64          "Funs_inv", "FunsD", "Funs_rangeE", "Funs_nonempty", "sum_case_inject"];
    65 
    66     val Funs_IntE = (Int_lower2 RS Funs_mono RS
    67       (Int_lower1 RS Funs_mono RS Int_greatest) RS subsetD) RS IntE;
    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 = map (Sign.full_name (Theory.sign_of thy1))
    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 
    79     val tyvars = map (fn (_, (_, Ts, _)) => map dest_DtTFree Ts) (hd descr);
    80     val leafTs' = get_nonrec_types descr' sorts;
    81     val branchTs = get_branching_types descr' sorts;
    82     val branchT = if null branchTs then HOLogic.unitT
    83       else fold_bal (fn (T, U) => Type ("+", [T, U])) branchTs;
    84     val unneeded_vars = hd tyvars \\ foldr add_typ_tfree_names (leafTs' @ branchTs, []);
    85     val leafTs = leafTs' @ (map (fn n => TFree (n, the (assoc (sorts, n)))) unneeded_vars);
    86     val recTs = get_rec_types descr' sorts;
    87     val newTs = take (length (hd descr), recTs);
    88     val oldTs = drop (length (hd descr), recTs);
    89     val sumT = if null leafTs then HOLogic.unitT
    90       else fold_bal (fn (T, U) => Type ("+", [T, U])) leafTs;
    91     val Univ_elT = HOLogic.mk_setT (Type (node_name, [sumT, branchT]));
    92     val UnivT = HOLogic.mk_setT Univ_elT;
    93 
    94     val In0 = Const (In0_name, Univ_elT --> Univ_elT);
    95     val In1 = Const (In1_name, Univ_elT --> Univ_elT);
    96     val Leaf = Const (Leaf_name, sumT --> Univ_elT);
    97     val Lim = Const (Lim_name, (branchT --> Univ_elT) --> Univ_elT);
    98 
    99     (* make injections needed for embedding types in leaves *)
   100 
   101     fun mk_inj T' x =
   102       let
   103         fun mk_inj' T n i =
   104           if n = 1 then x else
   105           let val n2 = n div 2;
   106               val Type (_, [T1, T2]) = T
   107           in
   108             if i <= n2 then
   109               Const ("Inl", T1 --> T) $ (mk_inj' T1 n2 i)
   110             else
   111               Const ("Inr", T2 --> T) $ (mk_inj' T2 (n - n2) (i - n2))
   112           end
   113       in mk_inj' sumT (length leafTs) (1 + find_index_eq T' leafTs)
   114       end;
   115 
   116     (* make injections for constructors *)
   117 
   118     fun mk_univ_inj ts = access_bal (fn t => In0 $ t, fn t => In1 $ t, if ts = [] then
   119         Const ("arbitrary", Univ_elT)
   120       else
   121         foldr1 (HOLogic.mk_binop Scons_name) ts);
   122 
   123     (* function spaces *)
   124 
   125     fun mk_fun_inj T' x =
   126       let
   127         fun mk_inj T n i =
   128           if n = 1 then x else
   129           let
   130             val n2 = n div 2;
   131             val Type (_, [T1, T2]) = T;
   132             val sum_case = Const (sum_case_name, [T1 --> Univ_elT, T2 --> Univ_elT, T] ---> Univ_elT)
   133           in
   134             if i <= n2 then
   135               sum_case $ (mk_inj T1 n2 i) $ Const ("arbitrary", T2 --> Univ_elT)
   136             else
   137               sum_case $ Const ("arbitrary", T1 --> Univ_elT) $ mk_inj T2 (n - n2) (i - n2)
   138           end
   139       in mk_inj branchT (length branchTs) (1 + find_index_eq T' branchTs)
   140       end;
   141 
   142     (************** generate introduction rules for representing set **********)
   143 
   144     val _ = message "Constructing representing sets ...";
   145 
   146     (* make introduction rule for a single constructor *)
   147 
   148     fun make_intr s n (i, (_, cargs)) =
   149       let
   150         fun mk_prem (DtRec k, (j, prems, ts)) =
   151               let val free_t = mk_Free "x" Univ_elT j
   152               in (j + 1, (HOLogic.mk_mem (free_t,
   153                 Const (nth_elem (k, rep_set_names), UnivT)))::prems, free_t::ts)
   154               end
   155           | mk_prem (DtType ("fun", [T, DtRec k]), (j, prems, ts)) =
   156               let val T' = typ_of_dtyp descr' sorts T;
   157                   val free_t = mk_Free "x" (T' --> Univ_elT) j
   158               in (j + 1, (HOLogic.mk_mem (free_t,
   159                 Const (Funs_name, UnivT --> HOLogic.mk_setT (T' --> Univ_elT)) $
   160                   Const (nth_elem (k, rep_set_names), UnivT)))::prems,
   161                     Lim $ mk_fun_inj T' free_t::ts)
   162               end
   163           | mk_prem (dt, (j, prems, ts)) =
   164               let val T = typ_of_dtyp descr' sorts dt
   165               in (j + 1, prems, (Leaf $ mk_inj T (mk_Free "x" T j))::ts)
   166               end;
   167 
   168         val (_, prems, ts) = foldr mk_prem (cargs, (1, [], []));
   169         val concl = HOLogic.mk_Trueprop (HOLogic.mk_mem
   170           (mk_univ_inj ts n i, Const (s, UnivT)))
   171       in Logic.list_implies (map HOLogic.mk_Trueprop prems, concl)
   172       end;
   173 
   174     val consts = map (fn s => Const (s, UnivT)) rep_set_names;
   175 
   176     val intr_ts = flat (map (fn ((_, (_, _, constrs)), rep_set_name) =>
   177       map (make_intr rep_set_name (length constrs))
   178         ((1 upto (length constrs)) ~~ constrs)) (descr' ~~ rep_set_names));
   179 
   180     val (thy2, {raw_induct = rep_induct, intrs = rep_intrs, ...}) =
   181       setmp InductivePackage.quiet_mode (!quiet_mode)
   182         (InductivePackage.add_inductive_i false true big_rec_name false true false
   183            consts (map (fn x => (("", x), [])) intr_ts) [Funs_mono]) thy1;
   184 
   185     (********************************* typedef ********************************)
   186 
   187     val thy3 = add_path flat_names big_name (foldl (fn (thy, ((((name, mx), tvs), c), name')) =>
   188       setmp TypedefPackage.quiet_mode true
   189         (TypedefPackage.add_typedef_i false (Some name') (name, tvs, mx) c None
   190           (rtac exI 1 THEN
   191             QUIET_BREADTH_FIRST (has_fewer_prems 1)
   192             (resolve_tac (Funs_nonempty::rep_intrs) 1))) thy |> #1)
   193               (parent_path flat_names thy2, types_syntax ~~ tyvars ~~
   194                 (take (length newTs, consts)) ~~ new_type_names));
   195 
   196     (*********************** definition of constructors ***********************)
   197 
   198     val big_rep_name = (space_implode "_" new_type_names) ^ "_Rep_";
   199     val rep_names = map (curry op ^ "Rep_") new_type_names;
   200     val rep_names' = map (fn i => big_rep_name ^ (string_of_int i))
   201       (1 upto (length (flat (tl descr))));
   202     val all_rep_names = map (Sign.intern_const (Theory.sign_of thy3)) rep_names @
   203       map (Sign.full_name (Theory.sign_of thy3)) rep_names';
   204 
   205     (* isomorphism declarations *)
   206 
   207     val iso_decls = map (fn (T, s) => (s, T --> Univ_elT, NoSyn))
   208       (oldTs ~~ rep_names');
   209 
   210     (* constructor definitions *)
   211 
   212     fun make_constr_def tname T n ((thy, defs, eqns, i), ((cname, cargs), (cname', mx))) =
   213       let
   214         fun constr_arg (dt, (j, l_args, r_args)) =
   215           let val T = typ_of_dtyp descr' sorts dt;
   216               val free_t = mk_Free "x" T j
   217           in (case dt of
   218               DtRec m => (j + 1, free_t::l_args, (Const (nth_elem (m, all_rep_names),
   219                 T --> Univ_elT) $ free_t)::r_args)
   220             | DtType ("fun", [T', DtRec m]) =>
   221                 let val ([T''], T''') = strip_type T
   222                 in (j + 1, free_t::l_args, (Lim $ mk_fun_inj T''
   223                   (Const (o_name, [T''' --> Univ_elT, T, T''] ---> Univ_elT) $
   224                     Const (nth_elem (m, all_rep_names), T''' --> Univ_elT) $ free_t))::r_args)
   225                 end
   226 
   227             | _ => (j + 1, free_t::l_args, (Leaf $ mk_inj T free_t)::r_args))
   228           end;
   229 
   230         val (_, l_args, r_args) = foldr constr_arg (cargs, (1, [], []));
   231         val constrT = (map (typ_of_dtyp descr' sorts) cargs) ---> T;
   232         val abs_name = Sign.intern_const (Theory.sign_of thy) ("Abs_" ^ tname);
   233         val rep_name = Sign.intern_const (Theory.sign_of thy) ("Rep_" ^ tname);
   234         val lhs = list_comb (Const (cname, constrT), l_args);
   235         val rhs = mk_univ_inj r_args n i;
   236         val def = equals T $ lhs $ (Const (abs_name, Univ_elT --> T) $ rhs);
   237         val def_name = (Sign.base_name cname) ^ "_def";
   238         val eqn = HOLogic.mk_Trueprop (HOLogic.mk_eq
   239           (Const (rep_name, T --> Univ_elT) $ lhs, rhs));
   240         val (thy', [def_thm]) = thy |>
   241           Theory.add_consts_i [(cname', constrT, mx)] |>
   242           (PureThy.add_defs_i false o map Thm.no_attributes) [(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 sg = Theory.sign_of thy;
   253         val rep_const = cterm_of sg
   254           (Const (Sign.intern_const sg ("Rep_" ^ tname), T --> Univ_elT));
   255         val cong' = standard (cterm_instantiate [(cterm_of sg cong_f, rep_const)] arg_cong);
   256         val dist = standard (cterm_instantiate [(cterm_of sg distinct_f, rep_const)] distinct_lemma);
   257         val (thy', defs', eqns', _) = foldl ((make_constr_def tname T) (length constrs))
   258           ((add_path flat_names tname thy, defs, [], 1), constrs ~~ constr_syntax)
   259       in
   260         (parent_path flat_names thy', defs', eqns @ [eqns'],
   261           rep_congs @ [cong'], dist_lemmas @ [dist])
   262       end;
   263 
   264     val (thy4, constr_defs, constr_rep_eqns, rep_congs, dist_lemmas) = foldl dt_constr_defs
   265       ((thy3 |> Theory.add_consts_i iso_decls |> parent_path flat_names, [], [], [], []),
   266         hd descr ~~ new_type_names ~~ newTs ~~ constr_syntax);
   267 
   268     (*********** isomorphisms for new types (introduced by typedef) ***********)
   269 
   270     val _ = message "Proving isomorphism properties ...";
   271 
   272     (* get axioms from theory *)
   273 
   274     val newT_iso_axms = map (fn s =>
   275       (get_thm thy4 ("Abs_" ^ s ^ "_inverse"),
   276        get_thm thy4 ("Rep_" ^ s ^ "_inverse"),
   277        get_thm thy4 ("Rep_" ^ s))) new_type_names;
   278 
   279     (*------------------------------------------------*)
   280     (* prove additional theorems:                     *)
   281     (*  inj_on dt_Abs_i rep_set_i  and  inj dt_Rep_i  *)
   282     (*------------------------------------------------*)
   283 
   284     fun prove_newT_iso_inj_thm (((s, (thm1, thm2, _)), T), rep_set_name) =
   285       let
   286         val sg = Theory.sign_of thy4;
   287         val RepT = T --> Univ_elT;
   288         val Rep_name = Sign.intern_const sg ("Rep_" ^ s);
   289         val AbsT = Univ_elT --> T;
   290         val Abs_name = Sign.intern_const sg ("Abs_" ^ s);
   291 
   292         val inj_Abs_thm = 
   293 	    prove_goalw_cterm [] 
   294 	      (cterm_of sg
   295 	       (HOLogic.mk_Trueprop 
   296 		(Const ("Fun.inj_on", [AbsT, UnivT] ---> HOLogic.boolT) $
   297 		 Const (Abs_name, AbsT) $ Const (rep_set_name, UnivT))))
   298               (fn _ => [rtac inj_on_inverseI 1, etac thm1 1]);
   299 
   300         val setT = HOLogic.mk_setT T
   301 
   302         val inj_Rep_thm =
   303 	    prove_goalw_cterm []
   304 	      (cterm_of sg
   305 	       (HOLogic.mk_Trueprop
   306 		(Const ("Fun.inj_on", [RepT, setT] ---> HOLogic.boolT) $
   307 		 Const (Rep_name, RepT) $ Const ("UNIV", setT))))
   308               (fn _ => [rtac inj_inverseI 1, rtac thm2 1])
   309 
   310       in (inj_Abs_thm, inj_Rep_thm) end;
   311 
   312     val newT_iso_inj_thms = map prove_newT_iso_inj_thm
   313       (new_type_names ~~ newT_iso_axms ~~ newTs ~~
   314         take (length newTs, rep_set_names));
   315 
   316     (********* isomorphisms between existing types and "unfolded" types *******)
   317 
   318     (*---------------------------------------------------------------------*)
   319     (* isomorphisms are defined using primrec-combinators:                 *)
   320     (* generate appropriate functions for instantiating primrec-combinator *)
   321     (*                                                                     *)
   322     (*   e.g.  dt_Rep_i = list_rec ... (%h t y. In1 ((Leaf h) $ y))        *)
   323     (*                                                                     *)
   324     (* also generate characteristic equations for isomorphisms             *)
   325     (*                                                                     *)
   326     (*   e.g.  dt_Rep_i (cons h t) = In1 ((dt_Rep_j h) $ (dt_Rep_i t))     *)
   327     (*---------------------------------------------------------------------*)
   328 
   329     fun make_iso_def k ks n ((fs, eqns, i), (cname, cargs)) =
   330       let
   331         val argTs = map (typ_of_dtyp descr' sorts) cargs;
   332         val T = nth_elem (k, recTs);
   333         val rep_name = nth_elem (k, all_rep_names);
   334         val rep_const = Const (rep_name, T --> Univ_elT);
   335         val constr = Const (cname, argTs ---> T);
   336 
   337         fun process_arg ks' ((i2, i2', ts, Ts), dt) =
   338           let val T' = typ_of_dtyp descr' sorts dt
   339           in (case dt of
   340               DtRec j => if j mem ks' then
   341                   (i2 + 1, i2' + 1, ts @ [mk_Free "y" Univ_elT i2'], Ts @ [Univ_elT])
   342                 else
   343                   (i2 + 1, i2', ts @ [Const (nth_elem (j, all_rep_names),
   344                     T' --> Univ_elT) $ mk_Free "x" T' i2], Ts)
   345             | (DtType ("fun", [_, DtRec j])) =>
   346                 let val ([T''], T''') = strip_type T'
   347                 in if j mem ks' then
   348                     (i2 + 1, i2' + 1, ts @ [Lim $ mk_fun_inj T''
   349                       (mk_Free "y" (T'' --> Univ_elT) i2')], Ts @ [T'' --> Univ_elT])
   350                   else
   351                     (i2 + 1, i2', ts @ [Lim $ mk_fun_inj T''
   352                       (Const (o_name, [T''' --> Univ_elT, T', T''] ---> Univ_elT) $
   353                         Const (nth_elem (j, all_rep_names), T''' --> Univ_elT) $
   354                           mk_Free "x" T' i2)], Ts)
   355                 end
   356             | _ => (i2 + 1, i2', ts @ [Leaf $ mk_inj T' (mk_Free "x" T' i2)], Ts))
   357           end;
   358 
   359         val (i2, i2', ts, Ts) = foldl (process_arg ks) ((1, 1, [], []), cargs);
   360         val xs = map (uncurry (mk_Free "x")) (argTs ~~ (1 upto (i2 - 1)));
   361         val ys = map (uncurry (mk_Free "y")) (Ts ~~ (1 upto (i2' - 1)));
   362         val f = list_abs_free (map dest_Free (xs @ ys), mk_univ_inj ts n i);
   363 
   364         val (_, _, ts', _) = foldl (process_arg []) ((1, 1, [], []), cargs);
   365         val eqn = HOLogic.mk_Trueprop (HOLogic.mk_eq
   366           (rep_const $ list_comb (constr, xs), mk_univ_inj ts' n i))
   367 
   368       in (fs @ [f], eqns @ [eqn], i + 1) end;
   369 
   370     (* define isomorphisms for all mutually recursive datatypes in list ds *)
   371 
   372     fun make_iso_defs (ds, (thy, char_thms)) =
   373       let
   374         val ks = map fst ds;
   375         val (_, (tname, _, _)) = hd ds;
   376         val {rec_rewrites, rec_names, ...} = the (Symtab.lookup (dt_info, tname));
   377 
   378         fun process_dt ((fs, eqns, isos), (k, (tname, _, constrs))) =
   379           let
   380             val (fs', eqns', _) = foldl (make_iso_def k ks (length constrs))
   381               ((fs, eqns, 1), constrs);
   382             val iso = (nth_elem (k, recTs), nth_elem (k, all_rep_names))
   383           in (fs', eqns', isos @ [iso]) end;
   384         
   385         val (fs, eqns, isos) = foldl process_dt (([], [], []), ds);
   386         val fTs = map fastype_of fs;
   387         val defs = map (fn (rec_name, (T, iso_name)) => ((Sign.base_name iso_name) ^ "_def",
   388           equals (T --> Univ_elT) $ Const (iso_name, T --> Univ_elT) $
   389             list_comb (Const (rec_name, fTs @ [T] ---> Univ_elT), fs))) (rec_names ~~ isos);
   390         val (thy', def_thms) = (PureThy.add_defs_i false o map Thm.no_attributes) defs thy;
   391 
   392         (* prove characteristic equations *)
   393 
   394         val rewrites = def_thms @ (map mk_meta_eq rec_rewrites);
   395         val char_thms' = map (fn eqn => prove_goalw_cterm rewrites
   396           (cterm_of (Theory.sign_of thy') eqn) (fn _ => [rtac refl 1])) eqns;
   397 
   398       in (thy', char_thms' @ char_thms) end;
   399 
   400     val (thy5, iso_char_thms) = foldr make_iso_defs
   401       (tl descr, (add_path flat_names big_name thy4, []));
   402 
   403     (* prove isomorphism properties *)
   404 
   405     fun mk_funs_inv thm =
   406       let
   407         val [_, t] = prems_of Funs_inv;
   408         val [_ $ (_ $ _ $ R)] = Logic.strip_assums_hyp t;
   409         val _ $ (_ $ (r $ (a $ _)) $ _) = Logic.strip_assums_concl t;
   410         val [_ $ (_ $ _ $ R')] = prems_of thm;
   411         val _ $ (_ $ (r' $ (a' $ _)) $ _) = concl_of thm;
   412         val inv' = cterm_instantiate (map 
   413           ((pairself (cterm_of (sign_of_thm thm))) o
   414            (apsnd (map_term_types (incr_tvar 1))))
   415              [(R, R'), (r, r'), (a, a')]) Funs_inv
   416       in
   417         rule_by_tactic (atac 2) (thm RSN (2, inv'))
   418       end;
   419 
   420     (* prove  inj dt_Rep_i  and  dt_Rep_i x : dt_rep_set_i *)
   421 
   422     fun prove_iso_thms (ds, (inj_thms, elem_thms)) =
   423       let
   424         val (_, (tname, _, _)) = hd ds;
   425         val {induction, ...} = the (Symtab.lookup (dt_info, tname));
   426 
   427         fun mk_ind_concl (i, _) =
   428           let
   429             val T = nth_elem (i, recTs);
   430             val Rep_t = Const (nth_elem (i, all_rep_names), T --> Univ_elT);
   431             val rep_set_name = nth_elem (i, rep_set_names)
   432           in (HOLogic.all_const T $ Abs ("y", T, HOLogic.imp $
   433                 HOLogic.mk_eq (Rep_t $ mk_Free "x" T i, Rep_t $ Bound 0) $
   434                   HOLogic.mk_eq (mk_Free "x" T i, Bound 0)),
   435               HOLogic.mk_mem (Rep_t $ mk_Free "x" T i, Const (rep_set_name, UnivT)))
   436           end;
   437 
   438         val (ind_concl1, ind_concl2) = ListPair.unzip (map mk_ind_concl ds);
   439 
   440         val rewrites = map mk_meta_eq iso_char_thms;
   441         val inj_thms' = flat (map (fn r => [r RS injD, r RS inj_o])
   442           (map snd newT_iso_inj_thms @ inj_thms));
   443 
   444         val inj_thm = prove_goalw_cterm [] (cterm_of (Theory.sign_of thy5)
   445           (HOLogic.mk_Trueprop (mk_conj ind_concl1))) (fn _ =>
   446             [(indtac induction THEN_ALL_NEW ObjectLogic.atomize_tac) 1,
   447              REPEAT (EVERY
   448                [rtac allI 1, rtac impI 1,
   449                 exh_tac (exh_thm_of dt_info) 1,
   450                 REPEAT (EVERY
   451                   [hyp_subst_tac 1,
   452                    rewrite_goals_tac rewrites,
   453                    REPEAT (dresolve_tac [In0_inject, In1_inject] 1),
   454                    (eresolve_tac [In0_not_In1 RS notE, In1_not_In0 RS notE] 1)
   455                    ORELSE (EVERY
   456                      [REPEAT (eresolve_tac (Scons_inject :: sum_case_inject ::
   457                         map make_elim (inj_thms' @
   458                           [Leaf_inject, Lim_inject, Inl_inject, Inr_inject])) 1),
   459                       REPEAT ((EVERY [etac allE 1, dtac mp 1, atac 1]) ORELSE
   460                               (dtac inj_fun_lemma 1 THEN atac 1)),
   461                       REPEAT (hyp_subst_tac 1),
   462                       rtac refl 1])])])]);
   463 
   464         val inj_thms'' = map (fn r => r RS datatype_injI)
   465                              (split_conj_thm inj_thm);
   466 
   467         val elem_thm = 
   468 	    prove_goalw_cterm []
   469 	      (cterm_of (Theory.sign_of thy5)
   470 	       (HOLogic.mk_Trueprop (mk_conj ind_concl2)))
   471 	      (fn _ =>
   472 	       [(indtac induction THEN_ALL_NEW ObjectLogic.atomize_tac) 1,
   473 		rewrite_goals_tac (o_def :: rewrites),
   474 		REPEAT (EVERY
   475 			[resolve_tac rep_intrs 1,
   476 			 REPEAT (FIRST [atac 1, etac spec 1,
   477 				 resolve_tac (FunsI :: elem_thms) 1])])]);
   478 
   479       in (inj_thms'' @ inj_thms, elem_thms @ (split_conj_thm elem_thm))
   480       end;
   481 
   482     val (iso_inj_thms_unfolded, iso_elem_thms) = foldr prove_iso_thms
   483       (tl descr, ([], map #3 newT_iso_axms));
   484     val iso_inj_thms = map snd newT_iso_inj_thms @ iso_inj_thms_unfolded;
   485 
   486     (* prove  x : dt_rep_set_i --> x : range dt_Rep_i *)
   487 
   488     fun mk_iso_t (((set_name, iso_name), i), T) =
   489       let val isoT = T --> Univ_elT
   490       in HOLogic.imp $ 
   491         HOLogic.mk_mem (mk_Free "x" Univ_elT i, Const (set_name, UnivT)) $
   492           (if i < length newTs then Const ("True", HOLogic.boolT)
   493            else HOLogic.mk_mem (mk_Free "x" Univ_elT i,
   494              Const ("image", [isoT, HOLogic.mk_setT T] ---> UnivT) $
   495                Const (iso_name, isoT) $ Const ("UNIV", HOLogic.mk_setT T)))
   496       end;
   497 
   498     val iso_t = HOLogic.mk_Trueprop (mk_conj (map mk_iso_t
   499       (rep_set_names ~~ all_rep_names ~~ (0 upto (length descr' - 1)) ~~ recTs)));
   500 
   501     (* all the theorems are proved by one single simultaneous induction *)
   502 
   503     val iso_thms = if length descr = 1 then [] else
   504       drop (length newTs, split_conj_thm
   505         (prove_goalw_cterm [] (cterm_of (Theory.sign_of thy5) iso_t) (fn _ =>
   506            [indtac rep_induct 1,
   507             REPEAT (rtac TrueI 1),
   508             REPEAT (EVERY
   509               [rewrite_goals_tac [mk_meta_eq Collect_mem_eq],
   510                REPEAT (etac Funs_IntE 1),
   511                REPEAT (eresolve_tac (rangeE ::
   512                  map (fn r => r RS Funs_rangeE) iso_inj_thms_unfolded) 1),
   513                REPEAT (eresolve_tac (map (fn (iso, _, _) => iso RS subst) newT_iso_axms @
   514                  map (fn (iso, _, _) => mk_funs_inv iso RS subst) newT_iso_axms) 1),
   515                TRY (hyp_subst_tac 1),
   516                rtac (sym RS range_eqI) 1,
   517                resolve_tac iso_char_thms 1])])));
   518 
   519     val Abs_inverse_thms' =
   520       map #1 newT_iso_axms @
   521       map2 (fn (r_inj, r) => f_myinv_f OF [r_inj, r RS mp])
   522         (iso_inj_thms_unfolded, iso_thms);
   523 
   524     val Abs_inverse_thms = map (fn r => r RS subst) (Abs_inverse_thms' @
   525       map mk_funs_inv Abs_inverse_thms');
   526 
   527     (******************* freeness theorems for constructors *******************)
   528 
   529     val _ = message "Proving freeness of constructors ...";
   530 
   531     (* prove theorem  Rep_i (Constr_j ...) = Inj_j ...  *)
   532     
   533     fun prove_constr_rep_thm eqn =
   534       let
   535         val inj_thms = map (fn (r, _) => r RS inj_onD) newT_iso_inj_thms;
   536         val rewrites = o_def :: constr_defs @ (map (mk_meta_eq o #2) newT_iso_axms)
   537       in prove_goalw_cterm [] (cterm_of (Theory.sign_of thy5) eqn) (fn _ =>
   538         [resolve_tac inj_thms 1,
   539          rewrite_goals_tac rewrites,
   540          rtac refl 1,
   541          resolve_tac rep_intrs 2,
   542          REPEAT (resolve_tac (FunsI :: iso_elem_thms) 1)])
   543       end;
   544 
   545     (*--------------------------------------------------------------*)
   546     (* constr_rep_thms and rep_congs are used to prove distinctness *)
   547     (* of constructors.                                             *)
   548     (*--------------------------------------------------------------*)
   549 
   550     val constr_rep_thms = map (map prove_constr_rep_thm) constr_rep_eqns;
   551 
   552     val dist_rewrites = map (fn (rep_thms, dist_lemma) =>
   553       dist_lemma::(rep_thms @ [In0_eq, In1_eq, In0_not_In1, In1_not_In0]))
   554         (constr_rep_thms ~~ dist_lemmas);
   555 
   556     fun prove_distinct_thms (_, []) = []
   557       | prove_distinct_thms (dist_rewrites', t::_::ts) =
   558           let
   559             val dist_thm = prove_goalw_cterm [] (cterm_of (Theory.sign_of thy5) t) (fn _ =>
   560               [simp_tac (HOL_ss addsimps dist_rewrites') 1])
   561           in dist_thm::(standard (dist_thm RS not_sym))::
   562             (prove_distinct_thms (dist_rewrites', ts))
   563           end;
   564 
   565     val distinct_thms = map prove_distinct_thms (dist_rewrites ~~
   566       DatatypeProp.make_distincts new_type_names descr sorts thy5);
   567 
   568     val simproc_dists = map (fn ((((_, (_, _, constrs)), rep_thms), congr), dists) =>
   569       if length constrs < !DatatypeProp.dtK then FewConstrs dists
   570       else ManyConstrs (congr, HOL_basic_ss addsimps rep_thms)) (hd descr ~~
   571         constr_rep_thms ~~ rep_congs ~~ distinct_thms);
   572 
   573     (* prove injectivity of constructors *)
   574 
   575     fun prove_constr_inj_thm rep_thms t =
   576       let val inj_thms = Scons_inject::sum_case_inject::(map make_elim
   577         ((map (fn r => r RS injD) iso_inj_thms) @
   578           [In0_inject, In1_inject, Leaf_inject, Inl_inject, Inr_inject, Lim_inject]))
   579       in prove_goalw_cterm [] (cterm_of (Theory.sign_of thy5) t) (fn _ =>
   580         [rtac iffI 1,
   581          REPEAT (etac conjE 2), hyp_subst_tac 2, rtac refl 2,
   582          dresolve_tac rep_congs 1, dtac box_equals 1,
   583          REPEAT (resolve_tac rep_thms 1), rewtac o_def,
   584          REPEAT (eresolve_tac inj_thms 1),
   585          REPEAT (ares_tac [conjI] 1 ORELSE (EVERY [rtac ext 1, dtac fun_cong 1,
   586                   eresolve_tac inj_thms 1, atac 1]))])
   587       end;
   588 
   589     val constr_inject = map (fn (ts, thms) => map (prove_constr_inj_thm thms) ts)
   590       ((DatatypeProp.make_injs descr sorts) ~~ constr_rep_thms);
   591 
   592     val (thy6, (constr_inject', distinct_thms'))= thy5 |> parent_path flat_names |>
   593       store_thmss "inject" new_type_names constr_inject |>>>
   594       store_thmss "distinct" new_type_names distinct_thms;
   595 
   596     (*************************** induction theorem ****************************)
   597 
   598     val _ = message "Proving induction rule for datatypes ...";
   599 
   600     val Rep_inverse_thms = (map (fn (_, iso, _) => iso RS subst) newT_iso_axms) @
   601       (map (fn r => r RS myinv_f_f RS subst) iso_inj_thms_unfolded);
   602     val Rep_inverse_thms' = map (fn r => r RS myinv_f_f) iso_inj_thms_unfolded;
   603 
   604     fun mk_indrule_lemma ((prems, concls), ((i, _), T)) =
   605       let
   606         val Rep_t = Const (nth_elem (i, all_rep_names), T --> Univ_elT) $
   607           mk_Free "x" T i;
   608 
   609         val Abs_t = if i < length newTs then
   610             Const (Sign.intern_const (Theory.sign_of thy6)
   611               ("Abs_" ^ (nth_elem (i, new_type_names))), Univ_elT --> T)
   612           else Const ("Inductive.myinv", [T --> Univ_elT, Univ_elT] ---> T) $
   613             Const (nth_elem (i, all_rep_names), T --> Univ_elT)
   614 
   615       in (prems @ [HOLogic.imp $ HOLogic.mk_mem (Rep_t,
   616             Const (nth_elem (i, rep_set_names), UnivT)) $
   617               (mk_Free "P" (T --> HOLogic.boolT) (i + 1) $ (Abs_t $ Rep_t))],
   618           concls @ [mk_Free "P" (T --> HOLogic.boolT) (i + 1) $ mk_Free "x" T i])
   619       end;
   620 
   621     val (indrule_lemma_prems, indrule_lemma_concls) =
   622       foldl mk_indrule_lemma (([], []), (descr' ~~ recTs));
   623 
   624     val cert = cterm_of (Theory.sign_of thy6);
   625 
   626     val indrule_lemma = prove_goalw_cterm [] (cert
   627       (Logic.mk_implies
   628         (HOLogic.mk_Trueprop (mk_conj indrule_lemma_prems),
   629          HOLogic.mk_Trueprop (mk_conj indrule_lemma_concls)))) (fn prems =>
   630            [cut_facts_tac prems 1, REPEAT (etac conjE 1),
   631             REPEAT (EVERY
   632               [TRY (rtac conjI 1), resolve_tac Rep_inverse_thms 1,
   633                etac mp 1, resolve_tac iso_elem_thms 1])]);
   634 
   635     val Ps = map head_of (HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of indrule_lemma)));
   636     val frees = if length Ps = 1 then [Free ("P", snd (dest_Var (hd Ps)))] else
   637       map (Free o apfst fst o dest_Var) Ps;
   638     val indrule_lemma' = cterm_instantiate (map cert Ps ~~ map cert frees) indrule_lemma;
   639 
   640     val dt_induct = prove_goalw_cterm [InductivePackage.inductive_forall_def] (cert
   641       (DatatypeProp.make_ind descr sorts)) (fn prems =>
   642         [rtac indrule_lemma' 1, indtac rep_induct 1,
   643          EVERY (map (fn (prem, r) => (EVERY
   644            [REPEAT (eresolve_tac (Funs_IntE::Abs_inverse_thms) 1),
   645             simp_tac (HOL_basic_ss addsimps ((symmetric r)::Rep_inverse_thms')) 1,
   646             DEPTH_SOLVE_1 (ares_tac [prem] 1 ORELSE (EVERY [rewtac o_def,
   647               rtac allI 1, dtac FunsD 1, etac CollectD 1]))]))
   648                 (prems ~~ (constr_defs @ (map mk_meta_eq iso_char_thms))))]);
   649 
   650     val (thy7, [dt_induct']) = thy6 |>
   651       Theory.add_path big_name |>
   652       PureThy.add_thms [(("induct", dt_induct),
   653         [Drule.rule_attribute (K InductivePackage.rulify), case_names_induct])] |>>
   654       Theory.parent_path;
   655 
   656   in (thy7, constr_inject', distinct_thms', dist_rewrites, simproc_dists, dt_induct')
   657   end;
   658 
   659 end;