src/HOL/Tools/datatype_rep_proofs.ML
author wenzelm
Sun Nov 26 18:07:19 2006 +0100 (2006-11-26)
changeset 21525 1b18b5892dc4
parent 21365 4ee8e2702241
child 22578 b0eb5652f210
permissions -rw-r--r--
InductivePackage.add_inductive_global;
     1 (*  Title:      HOL/Tools/datatype_rep_proofs.ML
     2     ID:         $Id$
     3     Author:     Stefan Berghofer, TU Muenchen
     4 
     5 Definitional introduction of datatypes
     6 Proof of characteristic theorems:
     7 
     8  - injectivity of constructors
     9  - distinctness of constructors
    10  - induction theorem
    11 
    12 *)
    13 
    14 signature DATATYPE_REP_PROOFS =
    15 sig
    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 val (_ $ (_ $ (_ $ (distinct_f $ _) $ _))) = hd (prems_of distinct_lemma);
    29 
    30 val collect_simp = rewrite_rule [mk_meta_eq mem_Collect_eq];
    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 = ThyInfo.the_theory "Datatype" thy;
    48     val node_name = "Datatype.node";
    49     val In0_name = "Datatype.In0";
    50     val In1_name = "Datatype.In1";
    51     val Scons_name = "Datatype.Scons";
    52     val Leaf_name = "Datatype.Leaf";
    53     val Numb_name = "Datatype.Numb";
    54     val Lim_name = "Datatype.Lim";
    55     val Suml_name = "Datatype.Suml";
    56     val Sumr_name = "Datatype.Sumr";
    57 
    58     val [In0_inject, In1_inject, Scons_inject, Leaf_inject,
    59          In0_eq, In1_eq, In0_not_In1, In1_not_In0,
    60          Lim_inject, Suml_inject, Sumr_inject] = map (get_thm Datatype_thy o Name)
    61         ["In0_inject", "In1_inject", "Scons_inject", "Leaf_inject",
    62          "In0_eq", "In1_eq", "In0_not_In1", "In1_not_In0",
    63          "Lim_inject", "Suml_inject", "Sumr_inject"];
    64 
    65     val descr' = List.concat descr;
    66 
    67     val big_name = space_implode "_" new_type_names;
    68     val thy1 = add_path flat_names big_name thy;
    69     val big_rec_name = big_name ^ "_rep_set";
    70     val rep_set_names' =
    71       (if length descr' = 1 then [big_rec_name] else
    72         (map ((curry (op ^) (big_rec_name ^ "_")) o string_of_int)
    73           (1 upto (length descr'))));
    74     val rep_set_names = map (Sign.full_name (Theory.sign_of thy1)) rep_set_names';
    75 
    76     val tyvars = map (fn (_, (_, Ts, _)) => map dest_DtTFree Ts) (hd descr);
    77     val leafTs' = get_nonrec_types descr' sorts;
    78     val branchTs = get_branching_types descr' sorts;
    79     val branchT = if null branchTs then HOLogic.unitT
    80       else fold_bal (fn (T, U) => Type ("+", [T, U])) branchTs;
    81     val arities = get_arities descr' \ 0;
    82     val unneeded_vars = hd tyvars \\ foldr add_typ_tfree_names [] (leafTs' @ branchTs);
    83     val leafTs = leafTs' @ (map (fn n => TFree (n, (the o AList.lookup (op =) sorts) n)) unneeded_vars);
    84     val recTs = get_rec_types descr' sorts;
    85     val newTs = Library.take (length (hd descr), recTs);
    86     val oldTs = Library.drop (length (hd descr), recTs);
    87     val sumT = if null leafTs then HOLogic.unitT
    88       else fold_bal (fn (T, U) => Type ("+", [T, U])) leafTs;
    89     val Univ_elT = HOLogic.mk_setT (Type (node_name, [sumT, branchT]));
    90     val UnivT = HOLogic.mk_setT Univ_elT;
    91     val UnivT' = Univ_elT --> HOLogic.boolT;
    92     val Collect = Const ("Collect", UnivT' --> UnivT);
    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 ("Sum_Type.Inl", T1 --> T) $ (mk_inj' T1 n2 i)
   110             else
   111               Const ("Sum_Type.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             fun mkT U = (U --> Univ_elT) --> T --> Univ_elT
   133           in
   134             if i <= n2 then Const (Suml_name, mkT T1) $ mk_inj T1 n2 i
   135             else Const (Sumr_name, mkT T2) $ mk_inj T2 (n - n2) (i - n2)
   136           end
   137       in mk_inj branchT (length branchTs) (1 + find_index_eq T' branchTs)
   138       end;
   139 
   140     val mk_lim = foldr (fn (T, t) => Lim $ mk_fun_inj T (Abs ("x", T, t)));
   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 (dt, (j, prems, ts)) = (case strip_dtyp dt of
   151             (dts, DtRec k) =>
   152               let
   153                 val Ts = map (typ_of_dtyp descr' sorts) dts;
   154                 val free_t =
   155                   app_bnds (mk_Free "x" (Ts ---> Univ_elT) j) (length Ts)
   156               in (j + 1, list_all (map (pair "x") Ts,
   157                   HOLogic.mk_Trueprop
   158                     (Free (List.nth (rep_set_names', k), UnivT') $ free_t)) :: prems,
   159                 mk_lim free_t Ts :: ts)
   160               end
   161           | _ =>
   162               let val T = typ_of_dtyp descr' sorts dt
   163               in (j + 1, prems, (Leaf $ mk_inj T (mk_Free "x" T j))::ts)
   164               end);
   165 
   166         val (_, prems, ts) = foldr mk_prem (1, [], []) cargs;
   167         val concl = HOLogic.mk_Trueprop
   168           (Free (s, UnivT') $ mk_univ_inj ts n i)
   169       in Logic.list_implies (prems, concl)
   170       end;
   171 
   172     val intr_ts = List.concat (map (fn ((_, (_, _, constrs)), rep_set_name) =>
   173       map (make_intr rep_set_name (length constrs))
   174         ((1 upto (length constrs)) ~~ constrs)) (descr' ~~ rep_set_names'));
   175 
   176     val ({raw_induct = rep_induct, intrs = rep_intrs, ...}, thy2) =
   177       setmp InductivePackage.quiet_mode (! quiet_mode)
   178         (InductivePackage.add_inductive_global false big_rec_name false true false
   179            (map (fn s => (s, SOME UnivT', NoSyn)) rep_set_names') []
   180            (map (fn x => (("", []), x)) intr_ts) []) thy1;
   181 
   182     (********************************* typedef ********************************)
   183 
   184     val (typedefs, thy3) = thy2 |>
   185       parent_path flat_names |>
   186       fold_map (fn ((((name, mx), tvs), c), name') =>
   187         setmp TypedefPackage.quiet_mode true
   188           (TypedefPackage.add_typedef_i false (SOME name') (name, tvs, mx)
   189             (Collect $ Const (c, UnivT')) NONE
   190             (rtac exI 1 THEN rtac CollectI 1 THEN
   191               QUIET_BREADTH_FIRST (has_fewer_prems 1)
   192               (resolve_tac rep_intrs 1))))
   193                 (types_syntax ~~ tyvars ~~
   194                   (Library.take (length newTs, rep_set_names)) ~~ new_type_names) ||>
   195       add_path flat_names big_name;
   196 
   197     (*********************** definition of constructors ***********************)
   198 
   199     val big_rep_name = (space_implode "_" new_type_names) ^ "_Rep_";
   200     val rep_names = map (curry op ^ "Rep_") new_type_names;
   201     val rep_names' = map (fn i => big_rep_name ^ (string_of_int i))
   202       (1 upto (length (List.concat (tl descr))));
   203     val all_rep_names = map (Sign.intern_const (Theory.sign_of thy3)) rep_names @
   204       map (Sign.full_name (Theory.sign_of thy3)) rep_names';
   205 
   206     (* isomorphism declarations *)
   207 
   208     val iso_decls = map (fn (T, s) => (s, T --> Univ_elT, NoSyn))
   209       (oldTs ~~ rep_names');
   210 
   211     (* constructor definitions *)
   212 
   213     fun make_constr_def tname T n ((thy, defs, eqns, i), ((cname, cargs), (cname', mx))) =
   214       let
   215         fun constr_arg (dt, (j, l_args, r_args)) =
   216           let val T = typ_of_dtyp descr' sorts dt;
   217               val free_t = mk_Free "x" T j
   218           in (case (strip_dtyp dt, strip_type T) of
   219               ((_, DtRec m), (Us, U)) => (j + 1, free_t :: l_args, mk_lim
   220                 (Const (List.nth (all_rep_names, m), U --> Univ_elT) $
   221                    app_bnds free_t (length Us)) Us :: r_args)
   222             | _ => (j + 1, free_t::l_args, (Leaf $ mk_inj T free_t)::r_args))
   223           end;
   224 
   225         val (_, l_args, r_args) = foldr constr_arg (1, [], []) cargs;
   226         val constrT = (map (typ_of_dtyp descr' sorts) cargs) ---> T;
   227         val abs_name = Sign.intern_const (Theory.sign_of thy) ("Abs_" ^ tname);
   228         val rep_name = Sign.intern_const (Theory.sign_of thy) ("Rep_" ^ tname);
   229         val lhs = list_comb (Const (cname, constrT), l_args);
   230         val rhs = mk_univ_inj r_args n i;
   231         val def = equals T $ lhs $ (Const (abs_name, Univ_elT --> T) $ rhs);
   232         val def_name = (Sign.base_name cname) ^ "_def";
   233         val eqn = HOLogic.mk_Trueprop (HOLogic.mk_eq
   234           (Const (rep_name, T --> Univ_elT) $ lhs, rhs));
   235         val ([def_thm], thy') =
   236           thy
   237           |> Theory.add_consts_i [(cname', constrT, mx)]
   238           |> (PureThy.add_defs_i false o map Thm.no_attributes) [(def_name, def)];
   239 
   240       in (thy', defs @ [def_thm], eqns @ [eqn], i + 1) end;
   241 
   242     (* constructor definitions for datatype *)
   243 
   244     fun dt_constr_defs ((thy, defs, eqns, rep_congs, dist_lemmas),
   245         ((((_, (_, _, constrs)), tname), T), constr_syntax)) =
   246       let
   247         val _ $ (_ $ (cong_f $ _) $ _) = concl_of arg_cong;
   248         val sg = Theory.sign_of thy;
   249         val rep_const = cterm_of sg
   250           (Const (Sign.intern_const sg ("Rep_" ^ tname), T --> Univ_elT));
   251         val cong' = standard (cterm_instantiate [(cterm_of sg cong_f, rep_const)] arg_cong);
   252         val dist = standard (cterm_instantiate [(cterm_of sg distinct_f, rep_const)] distinct_lemma);
   253         val (thy', defs', eqns', _) = Library.foldl ((make_constr_def tname T) (length constrs))
   254           ((add_path flat_names tname thy, defs, [], 1), constrs ~~ constr_syntax)
   255       in
   256         (parent_path flat_names thy', defs', eqns @ [eqns'],
   257           rep_congs @ [cong'], dist_lemmas @ [dist])
   258       end;
   259 
   260     val (thy4, constr_defs, constr_rep_eqns, rep_congs, dist_lemmas) = Library.foldl dt_constr_defs
   261       ((thy3 |> Theory.add_consts_i iso_decls |> parent_path flat_names, [], [], [], []),
   262         hd descr ~~ new_type_names ~~ newTs ~~ constr_syntax);
   263 
   264     (*********** isomorphisms for new types (introduced by typedef) ***********)
   265 
   266     val _ = message "Proving isomorphism properties ...";
   267 
   268     val newT_iso_axms = map (fn (_, td) =>
   269       (collect_simp (#Abs_inverse td), #Rep_inverse td,
   270        collect_simp (#Rep td))) typedefs;
   271 
   272     val newT_iso_inj_thms = map (fn (_, td) =>
   273       (collect_simp (#Abs_inject td) RS iffD1, #Rep_inject td RS iffD1)) typedefs;
   274 
   275     (********* isomorphisms between existing types and "unfolded" types *******)
   276 
   277     (*---------------------------------------------------------------------*)
   278     (* isomorphisms are defined using primrec-combinators:                 *)
   279     (* generate appropriate functions for instantiating primrec-combinator *)
   280     (*                                                                     *)
   281     (*   e.g.  dt_Rep_i = list_rec ... (%h t y. In1 (Scons (Leaf h) y))    *)
   282     (*                                                                     *)
   283     (* also generate characteristic equations for isomorphisms             *)
   284     (*                                                                     *)
   285     (*   e.g.  dt_Rep_i (cons h t) = In1 (Scons (dt_Rep_j h) (dt_Rep_i t)) *)
   286     (*---------------------------------------------------------------------*)
   287 
   288     fun make_iso_def k ks n ((fs, eqns, i), (cname, cargs)) =
   289       let
   290         val argTs = map (typ_of_dtyp descr' sorts) cargs;
   291         val T = List.nth (recTs, k);
   292         val rep_name = List.nth (all_rep_names, k);
   293         val rep_const = Const (rep_name, T --> Univ_elT);
   294         val constr = Const (cname, argTs ---> T);
   295 
   296         fun process_arg ks' ((i2, i2', ts, Ts), dt) =
   297           let
   298             val T' = typ_of_dtyp descr' sorts dt;
   299             val (Us, U) = strip_type T'
   300           in (case strip_dtyp dt of
   301               (_, DtRec j) => if j mem ks' then
   302                   (i2 + 1, i2' + 1, ts @ [mk_lim (app_bnds
   303                      (mk_Free "y" (Us ---> Univ_elT) i2') (length Us)) Us],
   304                    Ts @ [Us ---> Univ_elT])
   305                 else
   306                   (i2 + 1, i2', ts @ [mk_lim
   307                      (Const (List.nth (all_rep_names, j), U --> Univ_elT) $
   308                         app_bnds (mk_Free "x" T' i2) (length Us)) Us], Ts)
   309             | _ => (i2 + 1, i2', ts @ [Leaf $ mk_inj T' (mk_Free "x" T' i2)], Ts))
   310           end;
   311 
   312         val (i2, i2', ts, Ts) = Library.foldl (process_arg ks) ((1, 1, [], []), cargs);
   313         val xs = map (uncurry (mk_Free "x")) (argTs ~~ (1 upto (i2 - 1)));
   314         val ys = map (uncurry (mk_Free "y")) (Ts ~~ (1 upto (i2' - 1)));
   315         val f = list_abs_free (map dest_Free (xs @ ys), mk_univ_inj ts n i);
   316 
   317         val (_, _, ts', _) = Library.foldl (process_arg []) ((1, 1, [], []), cargs);
   318         val eqn = HOLogic.mk_Trueprop (HOLogic.mk_eq
   319           (rep_const $ list_comb (constr, xs), mk_univ_inj ts' n i))
   320 
   321       in (fs @ [f], eqns @ [eqn], i + 1) end;
   322 
   323     (* define isomorphisms for all mutually recursive datatypes in list ds *)
   324 
   325     fun make_iso_defs (ds, (thy, char_thms)) =
   326       let
   327         val ks = map fst ds;
   328         val (_, (tname, _, _)) = hd ds;
   329         val {rec_rewrites, rec_names, ...} = the (Symtab.lookup dt_info tname);
   330 
   331         fun process_dt ((fs, eqns, isos), (k, (tname, _, constrs))) =
   332           let
   333             val (fs', eqns', _) = Library.foldl (make_iso_def k ks (length constrs))
   334               ((fs, eqns, 1), constrs);
   335             val iso = (List.nth (recTs, k), List.nth (all_rep_names, k))
   336           in (fs', eqns', isos @ [iso]) end;
   337         
   338         val (fs, eqns, isos) = Library.foldl process_dt (([], [], []), ds);
   339         val fTs = map fastype_of fs;
   340         val defs = map (fn (rec_name, (T, iso_name)) => ((Sign.base_name iso_name) ^ "_def",
   341           equals (T --> Univ_elT) $ Const (iso_name, T --> Univ_elT) $
   342             list_comb (Const (rec_name, fTs @ [T] ---> Univ_elT), fs))) (rec_names ~~ isos);
   343         val (def_thms, thy') = (PureThy.add_defs_i false o map Thm.no_attributes) defs thy;
   344 
   345         (* prove characteristic equations *)
   346 
   347         val rewrites = def_thms @ (map mk_meta_eq rec_rewrites);
   348         val char_thms' = map (fn eqn => Goal.prove_global thy' [] [] eqn
   349           (fn _ => EVERY [rewrite_goals_tac rewrites, rtac refl 1])) eqns;
   350 
   351       in (thy', char_thms' @ char_thms) end;
   352 
   353     val (thy5, iso_char_thms) = foldr make_iso_defs
   354       (add_path flat_names big_name thy4, []) (tl descr);
   355 
   356     (* prove isomorphism properties *)
   357 
   358     fun mk_funs_inv thm =
   359       let
   360         val {sign, prop, ...} = rep_thm thm;
   361         val _ $ (_ $ ((S as Const (_, Type (_, [U, _]))) $ _ )) $
   362           (_ $ (_ $ (r $ (a $ _)) $ _)) = Type.freeze prop;
   363         val used = add_term_tfree_names (a, []);
   364 
   365         fun mk_thm i =
   366           let
   367             val Ts = map (TFree o rpair HOLogic.typeS)
   368               (Name.variant_list used (replicate i "'t"));
   369             val f = Free ("f", Ts ---> U)
   370           in Goal.prove_global sign [] [] (Logic.mk_implies
   371             (HOLogic.mk_Trueprop (HOLogic.list_all
   372                (map (pair "x") Ts, S $ app_bnds f i)),
   373              HOLogic.mk_Trueprop (HOLogic.mk_eq (list_abs (map (pair "x") Ts,
   374                r $ (a $ app_bnds f i)), f))))
   375             (fn _ => EVERY [REPEAT (rtac ext 1), REPEAT (etac allE 1), rtac thm 1, atac 1])
   376           end
   377       in map (fn r => r RS subst) (thm :: map mk_thm arities) end;
   378 
   379     (* prove  inj dt_Rep_i  and  dt_Rep_i x : dt_rep_set_i *)
   380 
   381     fun prove_iso_thms (ds, (inj_thms, elem_thms)) =
   382       let
   383         val (_, (tname, _, _)) = hd ds;
   384         val {induction, ...} = the (Symtab.lookup dt_info tname);
   385 
   386         fun mk_ind_concl (i, _) =
   387           let
   388             val T = List.nth (recTs, i);
   389             val Rep_t = Const (List.nth (all_rep_names, i), T --> Univ_elT);
   390             val rep_set_name = List.nth (rep_set_names, i)
   391           in (HOLogic.all_const T $ Abs ("y", T, HOLogic.imp $
   392                 HOLogic.mk_eq (Rep_t $ mk_Free "x" T i, Rep_t $ Bound 0) $
   393                   HOLogic.mk_eq (mk_Free "x" T i, Bound 0)),
   394               Const (rep_set_name, UnivT') $ (Rep_t $ mk_Free "x" T i))
   395           end;
   396 
   397         val (ind_concl1, ind_concl2) = ListPair.unzip (map mk_ind_concl ds);
   398 
   399         val rewrites = map mk_meta_eq iso_char_thms;
   400         val inj_thms' = map snd newT_iso_inj_thms @
   401           map (fn r => r RS injD) inj_thms;
   402 
   403         val inj_thm = Goal.prove_global thy5 [] []
   404           (HOLogic.mk_Trueprop (mk_conj ind_concl1)) (fn _ => EVERY
   405             [(indtac induction THEN_ALL_NEW ObjectLogic.atomize_tac) 1,
   406              REPEAT (EVERY
   407                [rtac allI 1, rtac impI 1,
   408                 exh_tac (exh_thm_of dt_info) 1,
   409                 REPEAT (EVERY
   410                   [hyp_subst_tac 1,
   411                    rewrite_goals_tac rewrites,
   412                    REPEAT (dresolve_tac [In0_inject, In1_inject] 1),
   413                    (eresolve_tac [In0_not_In1 RS notE, In1_not_In0 RS notE] 1)
   414                    ORELSE (EVERY
   415                      [REPEAT (eresolve_tac (Scons_inject ::
   416                         map make_elim [Leaf_inject, Inl_inject, Inr_inject]) 1),
   417                       REPEAT (cong_tac 1), rtac refl 1,
   418                       REPEAT (atac 1 ORELSE (EVERY
   419                         [REPEAT (rtac ext 1),
   420                          REPEAT (eresolve_tac (mp :: allE ::
   421                            map make_elim (Suml_inject :: Sumr_inject ::
   422                              Lim_inject :: fun_cong :: inj_thms')) 1),
   423                          atac 1]))])])])]);
   424 
   425         val inj_thms'' = map (fn r => r RS datatype_injI)
   426                              (split_conj_thm inj_thm);
   427 
   428         val elem_thm = 
   429             Goal.prove_global thy5 [] [] (HOLogic.mk_Trueprop (mk_conj ind_concl2))
   430               (fn _ =>
   431                EVERY [(indtac induction THEN_ALL_NEW ObjectLogic.atomize_tac) 1,
   432                 rewrite_goals_tac rewrites,
   433                 REPEAT ((resolve_tac rep_intrs THEN_ALL_NEW
   434                   ((REPEAT o etac allE) THEN' ares_tac elem_thms)) 1)]);
   435 
   436       in (inj_thms'' @ inj_thms, elem_thms @ (split_conj_thm elem_thm))
   437       end;
   438 
   439     val (iso_inj_thms_unfolded, iso_elem_thms) = foldr prove_iso_thms
   440       ([], map #3 newT_iso_axms) (tl descr);
   441     val iso_inj_thms = map snd newT_iso_inj_thms @
   442       map (fn r => r RS injD) iso_inj_thms_unfolded;
   443 
   444     (* prove  dt_rep_set_i x --> x : range dt_Rep_i *)
   445 
   446     fun mk_iso_t (((set_name, iso_name), i), T) =
   447       let val isoT = T --> Univ_elT
   448       in HOLogic.imp $ 
   449         (Const (set_name, UnivT') $ mk_Free "x" Univ_elT i) $
   450           (if i < length newTs then Const ("True", HOLogic.boolT)
   451            else HOLogic.mk_mem (mk_Free "x" Univ_elT i,
   452              Const ("image", [isoT, HOLogic.mk_setT T] ---> UnivT) $
   453                Const (iso_name, isoT) $ Const ("UNIV", HOLogic.mk_setT T)))
   454       end;
   455 
   456     val iso_t = HOLogic.mk_Trueprop (mk_conj (map mk_iso_t
   457       (rep_set_names ~~ all_rep_names ~~ (0 upto (length descr' - 1)) ~~ recTs)));
   458 
   459     (* all the theorems are proved by one single simultaneous induction *)
   460 
   461     val range_eqs = map (fn r => mk_meta_eq (r RS range_ex1_eq))
   462       iso_inj_thms_unfolded;
   463 
   464     val iso_thms = if length descr = 1 then [] else
   465       Library.drop (length newTs, split_conj_thm
   466         (Goal.prove_global thy5 [] [] iso_t (fn _ => EVERY
   467            [(indtac rep_induct THEN_ALL_NEW ObjectLogic.atomize_tac) 1,
   468             REPEAT (rtac TrueI 1),
   469             rewrite_goals_tac (mk_meta_eq choice_eq ::
   470               symmetric (mk_meta_eq expand_fun_eq) :: range_eqs),
   471             rewrite_goals_tac (map symmetric range_eqs),
   472             REPEAT (EVERY
   473               [REPEAT (eresolve_tac ([rangeE, ex1_implies_ex RS exE] @
   474                  List.concat (map (mk_funs_inv o #1) newT_iso_axms)) 1),
   475                TRY (hyp_subst_tac 1),
   476                rtac (sym RS range_eqI) 1,
   477                resolve_tac iso_char_thms 1])])));
   478 
   479     val Abs_inverse_thms' =
   480       map #1 newT_iso_axms @
   481       map2 (fn r_inj => fn r => f_myinv_f OF [r_inj, r RS mp])
   482         iso_inj_thms_unfolded iso_thms;
   483 
   484     val Abs_inverse_thms = List.concat (map mk_funs_inv Abs_inverse_thms');
   485 
   486     (******************* freeness theorems for constructors *******************)
   487 
   488     val _ = message "Proving freeness of constructors ...";
   489 
   490     (* prove theorem  Rep_i (Constr_j ...) = Inj_j ...  *)
   491     
   492     fun prove_constr_rep_thm eqn =
   493       let
   494         val inj_thms = map fst newT_iso_inj_thms;
   495         val rewrites = o_def :: constr_defs @ (map (mk_meta_eq o #2) newT_iso_axms)
   496       in Goal.prove_global thy5 [] [] eqn (fn _ => EVERY
   497         [resolve_tac inj_thms 1,
   498          rewrite_goals_tac rewrites,
   499          rtac refl 3,
   500          resolve_tac rep_intrs 2,
   501          REPEAT (resolve_tac iso_elem_thms 1)])
   502       end;
   503 
   504     (*--------------------------------------------------------------*)
   505     (* constr_rep_thms and rep_congs are used to prove distinctness *)
   506     (* of constructors.                                             *)
   507     (*--------------------------------------------------------------*)
   508 
   509     val constr_rep_thms = map (map prove_constr_rep_thm) constr_rep_eqns;
   510 
   511     val dist_rewrites = map (fn (rep_thms, dist_lemma) =>
   512       dist_lemma::(rep_thms @ [In0_eq, In1_eq, In0_not_In1, In1_not_In0]))
   513         (constr_rep_thms ~~ dist_lemmas);
   514 
   515     fun prove_distinct_thms (_, []) = []
   516       | prove_distinct_thms (dist_rewrites', t::_::ts) =
   517           let
   518             val dist_thm = Goal.prove_global thy5 [] [] t (fn _ =>
   519               EVERY [simp_tac (HOL_ss addsimps dist_rewrites') 1])
   520           in dist_thm::(standard (dist_thm RS not_sym))::
   521             (prove_distinct_thms (dist_rewrites', ts))
   522           end;
   523 
   524     val distinct_thms = map prove_distinct_thms (dist_rewrites ~~
   525       DatatypeProp.make_distincts new_type_names descr sorts thy5);
   526 
   527     val simproc_dists = map (fn ((((_, (_, _, constrs)), rep_thms), congr), dists) =>
   528       if length constrs < !DatatypeProp.dtK then FewConstrs dists
   529       else ManyConstrs (congr, HOL_basic_ss addsimps rep_thms)) (hd descr ~~
   530         constr_rep_thms ~~ rep_congs ~~ distinct_thms);
   531 
   532     (* prove injectivity of constructors *)
   533 
   534     fun prove_constr_inj_thm rep_thms t =
   535       let val inj_thms = Scons_inject :: (map make_elim
   536         (iso_inj_thms @
   537           [In0_inject, In1_inject, Leaf_inject, Inl_inject, Inr_inject,
   538            Lim_inject, Suml_inject, Sumr_inject]))
   539       in Goal.prove_global thy5 [] [] t (fn _ => EVERY
   540         [rtac iffI 1,
   541          REPEAT (etac conjE 2), hyp_subst_tac 2, rtac refl 2,
   542          dresolve_tac rep_congs 1, dtac box_equals 1,
   543          REPEAT (resolve_tac rep_thms 1),
   544          REPEAT (eresolve_tac inj_thms 1),
   545          REPEAT (ares_tac [conjI] 1 ORELSE (EVERY [REPEAT (rtac ext 1),
   546            REPEAT (eresolve_tac (make_elim fun_cong :: inj_thms) 1),
   547            atac 1]))])
   548       end;
   549 
   550     val constr_inject = map (fn (ts, thms) => map (prove_constr_inj_thm thms) ts)
   551       ((DatatypeProp.make_injs descr sorts) ~~ constr_rep_thms);
   552 
   553     val ((constr_inject', distinct_thms'), thy6) =
   554       thy5
   555       |> parent_path flat_names
   556       |> store_thmss "inject" new_type_names constr_inject
   557       ||>> store_thmss "distinct" new_type_names distinct_thms;
   558 
   559     (*************************** induction theorem ****************************)
   560 
   561     val _ = message "Proving induction rule for datatypes ...";
   562 
   563     val Rep_inverse_thms = (map (fn (_, iso, _) => iso RS subst) newT_iso_axms) @
   564       (map (fn r => r RS myinv_f_f RS subst) iso_inj_thms_unfolded);
   565     val Rep_inverse_thms' = map (fn r => r RS myinv_f_f) iso_inj_thms_unfolded;
   566 
   567     fun mk_indrule_lemma ((prems, concls), ((i, _), T)) =
   568       let
   569         val Rep_t = Const (List.nth (all_rep_names, i), T --> Univ_elT) $
   570           mk_Free "x" T i;
   571 
   572         val Abs_t = if i < length newTs then
   573             Const (Sign.intern_const (Theory.sign_of thy6)
   574               ("Abs_" ^ (List.nth (new_type_names, i))), Univ_elT --> T)
   575           else Const ("Inductive.myinv", [T --> Univ_elT, Univ_elT] ---> T) $
   576             Const (List.nth (all_rep_names, i), T --> Univ_elT)
   577 
   578       in (prems @ [HOLogic.imp $
   579             (Const (List.nth (rep_set_names, i), UnivT') $ Rep_t) $
   580               (mk_Free "P" (T --> HOLogic.boolT) (i + 1) $ (Abs_t $ Rep_t))],
   581           concls @ [mk_Free "P" (T --> HOLogic.boolT) (i + 1) $ mk_Free "x" T i])
   582       end;
   583 
   584     val (indrule_lemma_prems, indrule_lemma_concls) =
   585       Library.foldl mk_indrule_lemma (([], []), (descr' ~~ recTs));
   586 
   587     val cert = cterm_of (Theory.sign_of thy6);
   588 
   589     val indrule_lemma = Goal.prove_global thy6 [] []
   590       (Logic.mk_implies
   591         (HOLogic.mk_Trueprop (mk_conj indrule_lemma_prems),
   592          HOLogic.mk_Trueprop (mk_conj indrule_lemma_concls))) (fn _ => EVERY
   593            [REPEAT (etac conjE 1),
   594             REPEAT (EVERY
   595               [TRY (rtac conjI 1), resolve_tac Rep_inverse_thms 1,
   596                etac mp 1, resolve_tac iso_elem_thms 1])]);
   597 
   598     val Ps = map head_of (HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of indrule_lemma)));
   599     val frees = if length Ps = 1 then [Free ("P", snd (dest_Var (hd Ps)))] else
   600       map (Free o apfst fst o dest_Var) Ps;
   601     val indrule_lemma' = cterm_instantiate (map cert Ps ~~ map cert frees) indrule_lemma;
   602 
   603     val dt_induct_prop = DatatypeProp.make_ind descr sorts;
   604     val dt_induct = Goal.prove_global thy6 []
   605       (Logic.strip_imp_prems dt_induct_prop) (Logic.strip_imp_concl dt_induct_prop)
   606       (fn prems => EVERY
   607         [rtac indrule_lemma' 1,
   608          (indtac rep_induct THEN_ALL_NEW ObjectLogic.atomize_tac) 1,
   609          EVERY (map (fn (prem, r) => (EVERY
   610            [REPEAT (eresolve_tac Abs_inverse_thms 1),
   611             simp_tac (HOL_basic_ss addsimps ((symmetric r)::Rep_inverse_thms')) 1,
   612             DEPTH_SOLVE_1 (ares_tac [prem] 1 ORELSE etac allE 1)]))
   613                 (prems ~~ (constr_defs @ (map mk_meta_eq iso_char_thms))))]);
   614 
   615     val ([dt_induct'], thy7) =
   616       thy6
   617       |> Theory.add_path big_name
   618       |> PureThy.add_thms [(("induct", dt_induct), [case_names_induct])]
   619       ||> Theory.parent_path;
   620 
   621   in
   622     ((constr_inject', distinct_thms', dist_rewrites, simproc_dists, dt_induct'), thy7)
   623   end;
   624 
   625 end;