src/HOL/Tools/datatype_rep_proofs.ML
author wenzelm
Sun Feb 27 15:26:47 2000 +0100 (2000-02-27)
changeset 8305 93aa21ec5494
parent 8005 b64d86018785
child 8436 8a87fa482baf
permissions -rw-r--r--
HOLogic.dest_conj;
     1 (*  Title:      HOL/Tools/datatype_rep_proofs.ML
     2     ID:         $Id$
     3     Author:     Stefan Berghofer
     4     Copyright   1998  TU Muenchen
     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 ->
    21           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 (* figure out internal names *)
    33 
    34 val image_name = Sign.intern_const (Theory.sign_of Set.thy) "image";
    35 val UNIV_name = Sign.intern_const (Theory.sign_of Set.thy) "UNIV";
    36 val inj_on_name = Sign.intern_const (Theory.sign_of Fun.thy) "inj_on";
    37 val inv_name = Sign.intern_const (Theory.sign_of Fun.thy) "inv";
    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 thy =
    46   let
    47     val Datatype_thy = theory "Datatype";
    48     val node_name = Sign.intern_tycon (Theory.sign_of Datatype_thy) "node";
    49     val [In0_name, In1_name, Scons_name, Leaf_name, Numb_name, Lim_name,
    50       Funs_name, o_name, sum_case_name] =
    51       map (Sign.intern_const (Theory.sign_of Datatype_thy))
    52         ["In0", "In1", "Scons", "Leaf", "Numb", "Lim", "Funs", "op o", "sum_case"];
    53 
    54     val [In0_inject, In1_inject, Scons_inject, Leaf_inject, In0_eq, In1_eq,
    55          In0_not_In1, In1_not_In0, Funs_mono, FunsI, Lim_inject,
    56          Funs_inv, FunsD, Funs_rangeE, Funs_nonempty, sum_case_inject] = map (get_thm Datatype_thy)
    57         ["In0_inject", "In1_inject", "Scons_inject", "Leaf_inject", "In0_eq", "In1_eq",
    58          "In0_not_In1", "In1_not_In0", "Funs_mono", "FunsI", "Lim_inject",
    59          "Funs_inv", "FunsD", "Funs_rangeE", "Funs_nonempty", "sum_case_inject"];
    60 
    61     val Funs_IntE = (Int_lower2 RS Funs_mono RS
    62       (Int_lower1 RS Funs_mono RS Int_greatest) RS subsetD) RS IntE;
    63 
    64     val descr' = flat descr;
    65 
    66     val big_name = space_implode "_" new_type_names;
    67     val thy1 = add_path flat_names big_name thy;
    68     val big_rec_name = big_name ^ "_rep_set";
    69     val rep_set_names = map (Sign.full_name (Theory.sign_of thy1))
    70       (if length descr' = 1 then [big_rec_name] else
    71         (map ((curry (op ^) (big_rec_name ^ "_")) o string_of_int)
    72           (1 upto (length descr'))));
    73 
    74     val tyvars = map (fn (_, (_, Ts, _)) => map dest_DtTFree Ts) (hd descr);
    75     val leafTs' = get_nonrec_types descr' sorts;
    76     val branchTs = get_branching_types descr' sorts;
    77     val branchT = if null branchTs then HOLogic.unitT
    78       else fold_bal (fn (T, U) => Type ("+", [T, U])) branchTs;
    79     val unneeded_vars = hd tyvars \\ foldr add_typ_tfree_names (leafTs' @ branchTs, []);
    80     val leafTs = leafTs' @ (map (fn n => TFree (n, the (assoc (sorts, n)))) unneeded_vars);
    81     val recTs = get_rec_types descr' sorts;
    82     val newTs = take (length (hd descr), recTs);
    83     val oldTs = drop (length (hd descr), recTs);
    84     val sumT = if null leafTs then HOLogic.unitT
    85       else fold_bal (fn (T, U) => Type ("+", [T, U])) leafTs;
    86     val Univ_elT = HOLogic.mk_setT (Type (node_name, [sumT, branchT]));
    87     val UnivT = HOLogic.mk_setT Univ_elT;
    88 
    89     val In0 = Const (In0_name, Univ_elT --> Univ_elT);
    90     val In1 = Const (In1_name, Univ_elT --> Univ_elT);
    91     val Leaf = Const (Leaf_name, sumT --> Univ_elT);
    92     val Lim = Const (Lim_name, (branchT --> Univ_elT) --> Univ_elT);
    93 
    94     (* make injections needed for embedding types in leaves *)
    95 
    96     fun mk_inj T' x =
    97       let
    98         fun mk_inj' T n i =
    99           if n = 1 then x else
   100           let val n2 = n div 2;
   101               val Type (_, [T1, T2]) = T
   102           in
   103             if i <= n2 then
   104               Const ("Inl", T1 --> T) $ (mk_inj' T1 n2 i)
   105             else
   106               Const ("Inr", T2 --> T) $ (mk_inj' T2 (n - n2) (i - n2))
   107           end
   108       in mk_inj' sumT (length leafTs) (1 + find_index_eq T' leafTs)
   109       end;
   110 
   111     (* make injections for constructors *)
   112 
   113     fun mk_univ_inj ts = access_bal (fn t => In0 $ t, fn t => In1 $ t, if ts = [] then
   114         Const ("arbitrary", Univ_elT)
   115       else
   116         foldr1 (HOLogic.mk_binop Scons_name) ts);
   117 
   118     (* function spaces *)
   119 
   120     fun mk_fun_inj T' x =
   121       let
   122         fun mk_inj T n i =
   123           if n = 1 then x else
   124           let
   125             val n2 = n div 2;
   126             val Type (_, [T1, T2]) = T;
   127             val sum_case = Const (sum_case_name, [T1 --> Univ_elT, T2 --> Univ_elT, T] ---> Univ_elT)
   128           in
   129             if i <= n2 then
   130               sum_case $ (mk_inj T1 n2 i) $ Const ("arbitrary", T2 --> Univ_elT)
   131             else
   132               sum_case $ Const ("arbitrary", T1 --> Univ_elT) $ mk_inj T2 (n - n2) (i - n2)
   133           end
   134       in mk_inj branchT (length branchTs) (1 + find_index_eq T' branchTs)
   135       end;
   136 
   137     (************** generate introduction rules for representing set **********)
   138 
   139     val _ = message "Constructing representing sets ...";
   140 
   141     (* make introduction rule for a single constructor *)
   142 
   143     fun make_intr s n (i, (_, cargs)) =
   144       let
   145         fun mk_prem (DtRec k, (j, prems, ts)) =
   146               let val free_t = mk_Free "x" Univ_elT j
   147               in (j + 1, (HOLogic.mk_mem (free_t,
   148                 Const (nth_elem (k, rep_set_names), UnivT)))::prems, free_t::ts)
   149               end
   150           | mk_prem (DtType ("fun", [T, DtRec k]), (j, prems, ts)) =
   151               let val T' = typ_of_dtyp descr' sorts T;
   152                   val free_t = mk_Free "x" (T' --> Univ_elT) j
   153               in (j + 1, (HOLogic.mk_mem (free_t,
   154                 Const (Funs_name, UnivT --> HOLogic.mk_setT (T' --> Univ_elT)) $
   155                   Const (nth_elem (k, rep_set_names), UnivT)))::prems,
   156                     Lim $ mk_fun_inj T' free_t::ts)
   157               end
   158           | mk_prem (dt, (j, prems, ts)) =
   159               let val T = typ_of_dtyp descr' sorts dt
   160               in (j + 1, prems, (Leaf $ mk_inj T (mk_Free "x" T j))::ts)
   161               end;
   162 
   163         val (_, prems, ts) = foldr mk_prem (cargs, (1, [], []));
   164         val concl = HOLogic.mk_Trueprop (HOLogic.mk_mem
   165           (mk_univ_inj ts n i, Const (s, UnivT)))
   166       in Logic.list_implies (map HOLogic.mk_Trueprop prems, concl)
   167       end;
   168 
   169     val consts = map (fn s => Const (s, UnivT)) rep_set_names;
   170 
   171     val intr_ts = flat (map (fn ((_, (_, _, constrs)), rep_set_name) =>
   172       map (make_intr rep_set_name (length constrs))
   173         ((1 upto (length constrs)) ~~ constrs)) (descr' ~~ rep_set_names));
   174 
   175     val (thy2, {raw_induct = rep_induct, intrs = rep_intrs, ...}) =
   176       setmp InductivePackage.quiet_mode (!quiet_mode)
   177         (InductivePackage.add_inductive_i false true big_rec_name false true false
   178            consts [] (map (fn x => (("", x), [])) intr_ts) [Funs_mono] []) thy1;
   179 
   180     (********************************* typedef ********************************)
   181 
   182     val thy3 = add_path flat_names big_name (foldl (fn (thy, ((((name, mx), tvs), c), name')) =>
   183       setmp TypedefPackage.quiet_mode true
   184         (TypedefPackage.add_typedef_i_no_def name' (name, tvs, mx) c [] []
   185           (Some (QUIET_BREADTH_FIRST (has_fewer_prems 1)
   186             (resolve_tac (Funs_nonempty::rep_intrs) 1)))) thy)
   187               (parent_path flat_names thy2, types_syntax ~~ tyvars ~~
   188                 (take (length newTs, consts)) ~~ new_type_names));
   189 
   190     (*********************** definition of constructors ***********************)
   191 
   192     val big_rep_name = (space_implode "_" new_type_names) ^ "_Rep_";
   193     val rep_names = map (curry op ^ "Rep_") new_type_names;
   194     val rep_names' = map (fn i => big_rep_name ^ (string_of_int i))
   195       (1 upto (length (flat (tl descr))));
   196     val all_rep_names = map (Sign.intern_const (Theory.sign_of thy3)) rep_names @
   197       map (Sign.full_name (Theory.sign_of thy3)) rep_names';
   198 
   199     (* isomorphism declarations *)
   200 
   201     val iso_decls = map (fn (T, s) => (s, T --> Univ_elT, NoSyn))
   202       (oldTs ~~ rep_names');
   203 
   204     (* constructor definitions *)
   205 
   206     fun make_constr_def tname T n ((thy, defs, eqns, i), ((cname, cargs), (cname', mx))) =
   207       let
   208         fun constr_arg (dt, (j, l_args, r_args)) =
   209           let val T = typ_of_dtyp descr' sorts dt;
   210               val free_t = mk_Free "x" T j
   211           in (case dt of
   212               DtRec m => (j + 1, free_t::l_args, (Const (nth_elem (m, all_rep_names),
   213                 T --> Univ_elT) $ free_t)::r_args)
   214             | DtType ("fun", [T', DtRec m]) =>
   215                 let val ([T''], T''') = strip_type T
   216                 in (j + 1, free_t::l_args, (Lim $ mk_fun_inj T''
   217                   (Const (o_name, [T''' --> Univ_elT, T, T''] ---> Univ_elT) $
   218                     Const (nth_elem (m, all_rep_names), T''' --> Univ_elT) $ free_t))::r_args)
   219                 end
   220 
   221             | _ => (j + 1, free_t::l_args, (Leaf $ mk_inj T free_t)::r_args))
   222           end;
   223 
   224         val (_, l_args, r_args) = foldr constr_arg (cargs, (1, [], []));
   225         val constrT = (map (typ_of_dtyp descr' sorts) cargs) ---> T;
   226         val abs_name = Sign.intern_const (Theory.sign_of thy) ("Abs_" ^ tname);
   227         val rep_name = Sign.intern_const (Theory.sign_of thy) ("Rep_" ^ tname);
   228         val lhs = list_comb (Const (cname, constrT), l_args);
   229         val rhs = mk_univ_inj r_args n i;
   230         val def = equals T $ lhs $ (Const (abs_name, Univ_elT --> T) $ rhs);
   231         val def_name = (Sign.base_name cname) ^ "_def";
   232         val eqn = HOLogic.mk_Trueprop (HOLogic.mk_eq
   233           (Const (rep_name, T --> Univ_elT) $ lhs, rhs));
   234         val thy' = thy |>
   235           Theory.add_consts_i [(cname', constrT, mx)] |>
   236           (PureThy.add_defs_i o map Thm.no_attributes) [(def_name, def)];
   237 
   238       in (thy', defs @ [get_thm thy' def_name], eqns @ [eqn], i + 1)
   239       end;
   240 
   241     (* constructor definitions for datatype *)
   242 
   243     fun dt_constr_defs ((thy, defs, eqns, rep_congs, dist_lemmas),
   244         ((((_, (_, _, constrs)), tname), T), constr_syntax)) =
   245       let
   246         val _ $ (_ $ (cong_f $ _) $ _) = concl_of arg_cong;
   247         val sg = Theory.sign_of thy;
   248         val rep_const = cterm_of sg
   249           (Const (Sign.intern_const sg ("Rep_" ^ tname), T --> Univ_elT));
   250         val cong' = standard (cterm_instantiate [(cterm_of sg cong_f, rep_const)] arg_cong);
   251         val dist = standard (cterm_instantiate [(cterm_of sg distinct_f, rep_const)] distinct_lemma);
   252         val (thy', defs', eqns', _) = foldl ((make_constr_def tname T) (length constrs))
   253           ((add_path flat_names tname thy, defs, [], 1), constrs ~~ constr_syntax)
   254       in
   255         (parent_path flat_names thy', defs', eqns @ [eqns'],
   256           rep_congs @ [cong'], dist_lemmas @ [dist])
   257       end;
   258 
   259     val (thy4, constr_defs, constr_rep_eqns, rep_congs, dist_lemmas) = foldl dt_constr_defs
   260       ((thy3 |> Theory.add_consts_i iso_decls |> parent_path flat_names, [], [], [], []),
   261         hd descr ~~ new_type_names ~~ newTs ~~ constr_syntax);
   262 
   263     (*********** isomorphisms for new types (introduced by typedef) ***********)
   264 
   265     val _ = message "Proving isomorphism properties ...";
   266 
   267     (* get axioms from theory *)
   268 
   269     val newT_iso_axms = map (fn s =>
   270       (get_thm thy4 ("Abs_" ^ s ^ "_inverse"),
   271        get_thm thy4 ("Rep_" ^ s ^ "_inverse"),
   272        get_thm thy4 ("Rep_" ^ s))) new_type_names;
   273 
   274     (*------------------------------------------------*)
   275     (* prove additional theorems:                     *)
   276     (*  inj_on dt_Abs_i rep_set_i  and  inj dt_Rep_i  *)
   277     (*------------------------------------------------*)
   278 
   279     fun prove_newT_iso_inj_thm (((s, (thm1, thm2, _)), T), rep_set_name) =
   280       let
   281         val sg = Theory.sign_of thy4;
   282         val RepT = T --> Univ_elT;
   283         val Rep_name = Sign.intern_const sg ("Rep_" ^ s);
   284         val AbsT = Univ_elT --> T;
   285         val Abs_name = Sign.intern_const sg ("Abs_" ^ s);
   286 
   287         val inj_Abs_thm = 
   288 	    prove_goalw_cterm [] 
   289 	      (cterm_of sg
   290 	       (HOLogic.mk_Trueprop 
   291 		(Const (inj_on_name, [AbsT, UnivT] ---> HOLogic.boolT) $
   292 		 Const (Abs_name, AbsT) $ Const (rep_set_name, UnivT))))
   293               (fn _ => [rtac inj_on_inverseI 1, etac thm1 1]);
   294 
   295         val setT = HOLogic.mk_setT T
   296 
   297         val inj_Rep_thm =
   298 	    prove_goalw_cterm []
   299 	      (cterm_of sg
   300 	       (HOLogic.mk_Trueprop
   301 		(Const (inj_on_name, [RepT, setT] ---> HOLogic.boolT) $
   302 		 Const (Rep_name, RepT) $ Const (UNIV_name, setT))))
   303               (fn _ => [rtac inj_inverseI 1, rtac thm2 1])
   304 
   305       in (inj_Abs_thm, inj_Rep_thm) end;
   306 
   307     val newT_iso_inj_thms = map prove_newT_iso_inj_thm
   308       (new_type_names ~~ newT_iso_axms ~~ newTs ~~
   309         take (length newTs, rep_set_names));
   310 
   311     (********* isomorphisms between existing types and "unfolded" types *******)
   312 
   313     (*---------------------------------------------------------------------*)
   314     (* isomorphisms are defined using primrec-combinators:                 *)
   315     (* generate appropriate functions for instantiating primrec-combinator *)
   316     (*                                                                     *)
   317     (*   e.g.  dt_Rep_i = list_rec ... (%h t y. In1 ((Leaf h) $ y))        *)
   318     (*                                                                     *)
   319     (* also generate characteristic equations for isomorphisms             *)
   320     (*                                                                     *)
   321     (*   e.g.  dt_Rep_i (cons h t) = In1 ((dt_Rep_j h) $ (dt_Rep_i t))     *)
   322     (*---------------------------------------------------------------------*)
   323 
   324     fun make_iso_def k ks n ((fs, eqns, i), (cname, cargs)) =
   325       let
   326         val argTs = map (typ_of_dtyp descr' sorts) cargs;
   327         val T = nth_elem (k, recTs);
   328         val rep_name = nth_elem (k, all_rep_names);
   329         val rep_const = Const (rep_name, T --> Univ_elT);
   330         val constr = Const (cname, argTs ---> T);
   331 
   332         fun process_arg ks' ((i2, i2', ts, Ts), dt) =
   333           let val T' = typ_of_dtyp descr' sorts dt
   334           in (case dt of
   335               DtRec j => if j mem ks' then
   336                   (i2 + 1, i2' + 1, ts @ [mk_Free "y" Univ_elT i2'], Ts @ [Univ_elT])
   337                 else
   338                   (i2 + 1, i2', ts @ [Const (nth_elem (j, all_rep_names),
   339                     T' --> Univ_elT) $ mk_Free "x" T' i2], Ts)
   340             | (DtType ("fun", [_, DtRec j])) =>
   341                 let val ([T''], T''') = strip_type T'
   342                 in if j mem ks' then
   343                     (i2 + 1, i2' + 1, ts @ [Lim $ mk_fun_inj T''
   344                       (mk_Free "y" (T'' --> Univ_elT) i2')], Ts @ [T'' --> Univ_elT])
   345                   else
   346                     (i2 + 1, i2', ts @ [Lim $ mk_fun_inj T''
   347                       (Const (o_name, [T''' --> Univ_elT, T', T''] ---> Univ_elT) $
   348                         Const (nth_elem (j, all_rep_names), T''' --> Univ_elT) $
   349                           mk_Free "x" T' i2)], Ts)
   350                 end
   351             | _ => (i2 + 1, i2', ts @ [Leaf $ mk_inj T' (mk_Free "x" T' i2)], Ts))
   352           end;
   353 
   354         val (i2, i2', ts, Ts) = foldl (process_arg ks) ((1, 1, [], []), cargs);
   355         val xs = map (uncurry (mk_Free "x")) (argTs ~~ (1 upto (i2 - 1)));
   356         val ys = map (uncurry (mk_Free "y")) (Ts ~~ (1 upto (i2' - 1)));
   357         val f = list_abs_free (map dest_Free (xs @ ys), mk_univ_inj ts n i);
   358 
   359         val (_, _, ts', _) = foldl (process_arg []) ((1, 1, [], []), cargs);
   360         val eqn = HOLogic.mk_Trueprop (HOLogic.mk_eq
   361           (rep_const $ list_comb (constr, xs), mk_univ_inj ts' n i))
   362 
   363       in (fs @ [f], eqns @ [eqn], i + 1) end;
   364 
   365     (* define isomorphisms for all mutually recursive datatypes in list ds *)
   366 
   367     fun make_iso_defs (ds, (thy, char_thms)) =
   368       let
   369         val ks = map fst ds;
   370         val (_, (tname, _, _)) = hd ds;
   371         val {rec_rewrites, rec_names, ...} = the (Symtab.lookup (dt_info, tname));
   372 
   373         fun process_dt ((fs, eqns, isos), (k, (tname, _, constrs))) =
   374           let
   375             val (fs', eqns', _) = foldl (make_iso_def k ks (length constrs))
   376               ((fs, eqns, 1), constrs);
   377             val iso = (nth_elem (k, recTs), nth_elem (k, all_rep_names))
   378           in (fs', eqns', isos @ [iso]) end;
   379         
   380         val (fs, eqns, isos) = foldl process_dt (([], [], []), ds);
   381         val fTs = map fastype_of fs;
   382         val defs = map (fn (rec_name, (T, iso_name)) => ((Sign.base_name iso_name) ^ "_def",
   383           equals (T --> Univ_elT) $ Const (iso_name, T --> Univ_elT) $
   384             list_comb (Const (rec_name, fTs @ [T] ---> Univ_elT), fs))) (rec_names ~~ isos);
   385         val thy' = (PureThy.add_defs_i o map Thm.no_attributes) defs thy;
   386         val def_thms = map (get_thm thy') (map fst defs);
   387 
   388         (* prove characteristic equations *)
   389 
   390         val rewrites = def_thms @ (map mk_meta_eq rec_rewrites);
   391         val char_thms' = map (fn eqn => prove_goalw_cterm rewrites
   392           (cterm_of (Theory.sign_of thy') eqn) (fn _ => [rtac refl 1])) eqns;
   393 
   394       in (thy', char_thms' @ char_thms) end;
   395 
   396     val (thy5, iso_char_thms) = foldr make_iso_defs
   397       (tl descr, (add_path flat_names big_name thy4, []));
   398 
   399     (* prove isomorphism properties *)
   400 
   401     fun mk_funs_inv thm =
   402       let
   403         val [_, t] = prems_of Funs_inv;
   404         val [_ $ (_ $ _ $ R)] = Logic.strip_assums_hyp t;
   405         val _ $ (_ $ (r $ (a $ _)) $ _) = Logic.strip_assums_concl t;
   406         val [_ $ (_ $ _ $ R')] = prems_of thm;
   407         val _ $ (_ $ (r' $ (a' $ _)) $ _) = concl_of thm;
   408         val inv' = cterm_instantiate (map 
   409           ((pairself (cterm_of (sign_of_thm thm))) o
   410            (apsnd (map_term_types (incr_tvar 1))))
   411              [(R, R'), (r, r'), (a, a')]) Funs_inv
   412       in
   413         rule_by_tactic (atac 2) (thm RSN (2, inv'))
   414       end;
   415 
   416     (* prove  x : dt_rep_set_i --> x : range dt_Rep_i *)
   417 
   418     fun mk_iso_t (((set_name, iso_name), i), T) =
   419       let val isoT = T --> Univ_elT
   420       in HOLogic.imp $ 
   421         HOLogic.mk_mem (mk_Free "x" Univ_elT i, Const (set_name, UnivT)) $
   422           (if i < length newTs then Const ("True", HOLogic.boolT)
   423            else HOLogic.mk_mem (mk_Free "x" Univ_elT i,
   424              Const (image_name, [isoT, HOLogic.mk_setT T] ---> UnivT) $
   425                Const (iso_name, isoT) $ Const (UNIV_name, HOLogic.mk_setT T)))
   426       end;
   427 
   428     val iso_t = HOLogic.mk_Trueprop (mk_conj (map mk_iso_t
   429       (rep_set_names ~~ all_rep_names ~~ (0 upto (length descr' - 1)) ~~ recTs)));
   430 
   431     (* all the theorems are proved by one single simultaneous induction *)
   432 
   433     val iso_thms = if length descr = 1 then [] else
   434       drop (length newTs, split_conj_thm
   435         (prove_goalw_cterm [] (cterm_of (Theory.sign_of thy5) iso_t) (fn _ =>
   436            [indtac rep_induct 1,
   437             REPEAT (rtac TrueI 1),
   438             REPEAT (EVERY
   439               [rewrite_goals_tac [mk_meta_eq Collect_mem_eq],
   440                REPEAT (etac Funs_IntE 1),
   441                REPEAT (eresolve_tac [rangeE, Funs_rangeE] 1),
   442                REPEAT (eresolve_tac (map (fn (iso, _, _) => iso RS subst) newT_iso_axms @
   443                  map (fn (iso, _, _) => mk_funs_inv iso RS subst) newT_iso_axms) 1),
   444                TRY (hyp_subst_tac 1),
   445                rtac (sym RS range_eqI) 1,
   446                resolve_tac iso_char_thms 1])])));
   447 
   448     val Abs_inverse_thms' = (map #1 newT_iso_axms) @ map (fn r => r RS mp RS f_inv_f) iso_thms;
   449 
   450     val Abs_inverse_thms = map (fn r => r RS subst) (Abs_inverse_thms' @
   451       map mk_funs_inv Abs_inverse_thms');
   452 
   453     (* prove  inj dt_Rep_i  and  dt_Rep_i x : dt_rep_set_i *)
   454 
   455     fun prove_iso_thms (ds, (inj_thms, elem_thms)) =
   456       let
   457         val (_, (tname, _, _)) = hd ds;
   458         val {induction, ...} = the (Symtab.lookup (dt_info, tname));
   459 
   460         fun mk_ind_concl (i, _) =
   461           let
   462             val T = nth_elem (i, recTs);
   463             val Rep_t = Const (nth_elem (i, all_rep_names), T --> Univ_elT);
   464             val rep_set_name = nth_elem (i, rep_set_names)
   465           in (HOLogic.all_const T $ Abs ("y", T, HOLogic.imp $
   466                 HOLogic.mk_eq (Rep_t $ mk_Free "x" T i, Rep_t $ Bound 0) $
   467                   HOLogic.mk_eq (mk_Free "x" T i, Bound 0)),
   468               HOLogic.mk_mem (Rep_t $ mk_Free "x" T i, Const (rep_set_name, UnivT)))
   469           end;
   470 
   471         val (ind_concl1, ind_concl2) = ListPair.unzip (map mk_ind_concl ds);
   472 
   473         val rewrites = map mk_meta_eq iso_char_thms;
   474         val inj_thms' = flat (map (fn r => [r RS injD, r RS inj_o]) inj_thms);
   475 
   476         val inj_thm = prove_goalw_cterm [] (cterm_of (Theory.sign_of thy5)
   477           (HOLogic.mk_Trueprop (mk_conj ind_concl1))) (fn _ =>
   478             [indtac induction 1,
   479              REPEAT (EVERY
   480                [rtac allI 1, rtac impI 1,
   481                 exh_tac (exh_thm_of dt_info) 1,
   482                 REPEAT (EVERY
   483                   [hyp_subst_tac 1,
   484                    rewrite_goals_tac rewrites,
   485                    REPEAT (dresolve_tac [In0_inject, In1_inject] 1),
   486                    (eresolve_tac [In0_not_In1 RS notE, In1_not_In0 RS notE] 1)
   487                    ORELSE (EVERY
   488                      [REPEAT (etac Scons_inject 1),
   489                       REPEAT (dresolve_tac
   490                         (inj_thms' @ [Leaf_inject, Lim_inject, Inl_inject, Inr_inject]) 1),
   491                       REPEAT ((EVERY [etac allE 1, dtac mp 1, atac 1]) ORELSE
   492                               (dtac inj_fun_lemma 1 THEN atac 1)),
   493                       TRY (hyp_subst_tac 1),
   494                       rtac refl 1])])])]);
   495 
   496         val inj_thms'' = map (fn r => r RS datatype_injI)
   497                              (split_conj_thm inj_thm);
   498 
   499         val elem_thm = 
   500 	    prove_goalw_cterm []
   501 	      (cterm_of (Theory.sign_of thy5)
   502 	       (HOLogic.mk_Trueprop (mk_conj ind_concl2)))
   503 	      (fn _ =>
   504 	       [indtac induction 1,
   505 		rewrite_goals_tac (o_def :: rewrites),
   506 		REPEAT (EVERY
   507 			[resolve_tac rep_intrs 1,
   508 			 REPEAT (FIRST [atac 1, etac spec 1,
   509 				 resolve_tac (FunsI :: elem_thms) 1])])]);
   510 
   511       in (inj_thms @ inj_thms'', elem_thms @ (split_conj_thm elem_thm))
   512       end;
   513 
   514     val (iso_inj_thms, iso_elem_thms) = foldr prove_iso_thms
   515       (tl descr, (map snd newT_iso_inj_thms, map #3 newT_iso_axms));
   516 
   517     (******************* freeness theorems for constructors *******************)
   518 
   519     val _ = message "Proving freeness of constructors ...";
   520 
   521     (* prove theorem  Rep_i (Constr_j ...) = Inj_j ...  *)
   522     
   523     fun prove_constr_rep_thm eqn =
   524       let
   525         val inj_thms = map (fn (r, _) => r RS inj_onD) newT_iso_inj_thms;
   526         val rewrites = o_def :: constr_defs @ (map (mk_meta_eq o #2) newT_iso_axms)
   527       in prove_goalw_cterm [] (cterm_of (Theory.sign_of thy5) eqn) (fn _ =>
   528         [resolve_tac inj_thms 1,
   529          rewrite_goals_tac rewrites,
   530          rtac refl 1,
   531          resolve_tac rep_intrs 2,
   532          REPEAT (resolve_tac (FunsI :: iso_elem_thms) 1)])
   533       end;
   534 
   535     (*--------------------------------------------------------------*)
   536     (* constr_rep_thms and rep_congs are used to prove distinctness *)
   537     (* of constructors.                                             *)
   538     (*--------------------------------------------------------------*)
   539 
   540     val constr_rep_thms = map (map prove_constr_rep_thm) constr_rep_eqns;
   541 
   542     val dist_rewrites = map (fn (rep_thms, dist_lemma) =>
   543       dist_lemma::(rep_thms @ [In0_eq, In1_eq, In0_not_In1, In1_not_In0]))
   544         (constr_rep_thms ~~ dist_lemmas);
   545 
   546     fun prove_distinct_thms (_, []) = []
   547       | prove_distinct_thms (dist_rewrites', t::_::ts) =
   548           let
   549             val dist_thm = prove_goalw_cterm [] (cterm_of (Theory.sign_of thy5) t) (fn _ =>
   550               [simp_tac (HOL_ss addsimps dist_rewrites') 1])
   551           in dist_thm::(standard (dist_thm RS not_sym))::
   552             (prove_distinct_thms (dist_rewrites', ts))
   553           end;
   554 
   555     val distinct_thms = map prove_distinct_thms (dist_rewrites ~~
   556       DatatypeProp.make_distincts new_type_names descr sorts thy5);
   557 
   558     val simproc_dists = map (fn ((((_, (_, _, constrs)), rep_thms), congr), dists) =>
   559       if length constrs < !DatatypeProp.dtK then FewConstrs dists
   560       else ManyConstrs (congr, HOL_basic_ss addsimps rep_thms)) (hd descr ~~
   561         constr_rep_thms ~~ rep_congs ~~ distinct_thms);
   562 
   563     (* prove injectivity of constructors *)
   564 
   565     fun prove_constr_inj_thm rep_thms t =
   566       let val inj_thms = Scons_inject::sum_case_inject::(map make_elim
   567         ((map (fn r => r RS injD) iso_inj_thms) @
   568           [In0_inject, In1_inject, Leaf_inject, Inl_inject, Inr_inject, Lim_inject]))
   569       in prove_goalw_cterm [] (cterm_of (Theory.sign_of thy5) t) (fn _ =>
   570         [rtac iffI 1,
   571          REPEAT (etac conjE 2), hyp_subst_tac 2, rtac refl 2,
   572          dresolve_tac rep_congs 1, dtac box_equals 1,
   573          REPEAT (resolve_tac rep_thms 1), rewtac o_def,
   574          REPEAT (eresolve_tac inj_thms 1),
   575          REPEAT (ares_tac [conjI] 1 ORELSE (EVERY [rtac ext 1, dtac fun_cong 1,
   576                   eresolve_tac inj_thms 1, atac 1]))])
   577       end;
   578 
   579     val constr_inject = map (fn (ts, thms) => map (prove_constr_inj_thm thms) ts)
   580       ((DatatypeProp.make_injs descr sorts) ~~ constr_rep_thms);
   581 
   582     val thy6 = thy5 |> parent_path flat_names |>
   583       store_thmss "inject" new_type_names constr_inject |>
   584       store_thmss "distinct" new_type_names distinct_thms;
   585 
   586     (*************************** induction theorem ****************************)
   587 
   588     val _ = message "Proving induction rule for datatypes ...";
   589 
   590     val Rep_inverse_thms = (map (fn (_, iso, _) => iso RS subst) newT_iso_axms) @
   591       (map (fn r => r RS inv_f_f RS subst) (drop (length newTs, iso_inj_thms)));
   592     val Rep_inverse_thms' = map (fn r => r RS inv_f_f)
   593       (drop (length newTs, iso_inj_thms));
   594 
   595     fun mk_indrule_lemma ((prems, concls), ((i, _), T)) =
   596       let
   597         val Rep_t = Const (nth_elem (i, all_rep_names), T --> Univ_elT) $
   598           mk_Free "x" T i;
   599 
   600         val Abs_t = if i < length newTs then
   601             Const (Sign.intern_const (Theory.sign_of thy6)
   602               ("Abs_" ^ (nth_elem (i, new_type_names))), Univ_elT --> T)
   603           else Const (inv_name, [T --> Univ_elT, Univ_elT] ---> T) $
   604             Const (nth_elem (i, all_rep_names), T --> Univ_elT)
   605 
   606       in (prems @ [HOLogic.imp $ HOLogic.mk_mem (Rep_t,
   607             Const (nth_elem (i, rep_set_names), UnivT)) $
   608               (mk_Free "P" (T --> HOLogic.boolT) (i + 1) $ (Abs_t $ Rep_t))],
   609           concls @ [mk_Free "P" (T --> HOLogic.boolT) (i + 1) $ mk_Free "x" T i])
   610       end;
   611 
   612     val (indrule_lemma_prems, indrule_lemma_concls) =
   613       foldl mk_indrule_lemma (([], []), (descr' ~~ recTs));
   614 
   615     val cert = cterm_of (Theory.sign_of thy6);
   616 
   617     val indrule_lemma = prove_goalw_cterm [] (cert
   618       (Logic.mk_implies
   619         (HOLogic.mk_Trueprop (mk_conj indrule_lemma_prems),
   620          HOLogic.mk_Trueprop (mk_conj indrule_lemma_concls)))) (fn prems =>
   621            [cut_facts_tac prems 1, REPEAT (etac conjE 1),
   622             REPEAT (EVERY
   623               [TRY (rtac conjI 1), resolve_tac Rep_inverse_thms 1,
   624                etac mp 1, resolve_tac iso_elem_thms 1])]);
   625 
   626     val Ps = map head_of (HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of indrule_lemma)));
   627     val frees = if length Ps = 1 then [Free ("P", snd (dest_Var (hd Ps)))] else
   628       map (Free o apfst fst o dest_Var) Ps;
   629     val indrule_lemma' = cterm_instantiate (map cert Ps ~~ map cert frees) indrule_lemma;
   630 
   631     val dt_induct = prove_goalw_cterm [] (cert
   632       (DatatypeProp.make_ind descr sorts)) (fn prems =>
   633         [rtac indrule_lemma' 1, indtac rep_induct 1,
   634          EVERY (map (fn (prem, r) => (EVERY
   635            [REPEAT (eresolve_tac (Funs_IntE::Abs_inverse_thms) 1),
   636             simp_tac (HOL_basic_ss addsimps ((symmetric r)::Rep_inverse_thms')) 1,
   637             DEPTH_SOLVE_1 (ares_tac [prem] 1 ORELSE (EVERY [rewtac o_def,
   638               rtac allI 1, dtac FunsD 1, etac CollectD 1]))]))
   639                 (prems ~~ (constr_defs @ (map mk_meta_eq iso_char_thms))))]);
   640 
   641     val thy7 = thy6 |>
   642       Theory.add_path big_name |>
   643       PureThy.add_thms [(("induct", dt_induct), [])] |>
   644       Theory.parent_path;
   645 
   646   in (thy7, constr_inject, distinct_thms, dist_rewrites, simproc_dists, dt_induct)
   647   end;
   648 
   649 end;