src/HOL/Tools/datatype_rep_proofs.ML
author wenzelm
Tue Jul 11 12:16:54 2006 +0200 (2006-07-11)
changeset 20071 8f3e1ddb50e6
parent 20046 9c8909fc5865
child 20820 58693343905f
permissions -rw-r--r--
replaced Term.variant(list) by Name.variant(_list);
     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 
    31 (** theory context references **)
    32 
    33 val f_myinv_f = thm "f_myinv_f";
    34 val myinv_f_f = thm "myinv_f_f";
    35 
    36 
    37 fun exh_thm_of (dt_info : datatype_info Symtab.table) tname =
    38   #exhaustion (the (Symtab.lookup dt_info tname));
    39 
    40 (******************************************************************************)
    41 
    42 fun representation_proofs flat_names (dt_info : datatype_info Symtab.table)
    43       new_type_names descr sorts types_syntax constr_syntax case_names_induct thy =
    44   let
    45     val Datatype_thy = ThyInfo.the_theory "Datatype" thy;
    46     val node_name = "Datatype_Universe.node";
    47     val In0_name = "Datatype_Universe.In0";
    48     val In1_name = "Datatype_Universe.In1";
    49     val Scons_name = "Datatype_Universe.Scons";
    50     val Leaf_name = "Datatype_Universe.Leaf";
    51     val Numb_name = "Datatype_Universe.Numb";
    52     val Lim_name = "Datatype_Universe.Lim";
    53     val Suml_name = "Datatype.Suml";
    54     val Sumr_name = "Datatype.Sumr";
    55 
    56     val [In0_inject, In1_inject, Scons_inject, Leaf_inject,
    57          In0_eq, In1_eq, In0_not_In1, In1_not_In0,
    58          Lim_inject, Suml_inject, Sumr_inject] = map (get_thm Datatype_thy o Name)
    59         ["In0_inject", "In1_inject", "Scons_inject", "Leaf_inject",
    60          "In0_eq", "In1_eq", "In0_not_In1", "In1_not_In0",
    61          "Lim_inject", "Suml_inject", "Sumr_inject"];
    62 
    63     val descr' = List.concat descr;
    64 
    65     val big_name = space_implode "_" new_type_names;
    66     val thy1 = add_path flat_names big_name thy;
    67     val big_rec_name = big_name ^ "_rep_set";
    68     val rep_set_names = map (Sign.full_name (Theory.sign_of thy1))
    69       (if length descr' = 1 then [big_rec_name] else
    70         (map ((curry (op ^) (big_rec_name ^ "_")) o string_of_int)
    71           (1 upto (length descr'))));
    72 
    73     val tyvars = map (fn (_, (_, Ts, _)) => map dest_DtTFree Ts) (hd descr);
    74     val leafTs' = get_nonrec_types descr' sorts;
    75     val branchTs = get_branching_types descr' sorts;
    76     val branchT = if null branchTs then HOLogic.unitT
    77       else fold_bal (fn (T, U) => Type ("+", [T, U])) branchTs;
    78     val arities = get_arities descr' \ 0;
    79     val unneeded_vars = hd tyvars \\ foldr add_typ_tfree_names [] (leafTs' @ branchTs);
    80     val leafTs = leafTs' @ (map (fn n => TFree (n, (the o AList.lookup (op =) sorts) n)) unneeded_vars);
    81     val recTs = get_rec_types descr' sorts;
    82     val newTs = Library.take (length (hd descr), recTs);
    83     val oldTs = Library.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 ("Sum_Type.Inl", T1 --> T) $ (mk_inj' T1 n2 i)
   105             else
   106               Const ("Sum_Type.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             fun mkT U = (U --> Univ_elT) --> T --> Univ_elT
   128           in
   129             if i <= n2 then Const (Suml_name, mkT T1) $ mk_inj T1 n2 i
   130             else Const (Sumr_name, mkT T2) $ mk_inj T2 (n - n2) (i - n2)
   131           end
   132       in mk_inj branchT (length branchTs) (1 + find_index_eq T' branchTs)
   133       end;
   134 
   135     val mk_lim = foldr (fn (T, t) => Lim $ mk_fun_inj T (Abs ("x", T, t)));
   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 (dt, (j, prems, ts)) = (case strip_dtyp dt of
   146             (dts, DtRec k) =>
   147               let
   148                 val Ts = map (typ_of_dtyp descr' sorts) dts;
   149                 val free_t =
   150                   app_bnds (mk_Free "x" (Ts ---> Univ_elT) j) (length Ts)
   151               in (j + 1, list_all (map (pair "x") Ts,
   152                   HOLogic.mk_Trueprop (HOLogic.mk_mem (free_t,
   153                     Const (List.nth (rep_set_names, k), UnivT)))) :: prems,
   154                 mk_lim free_t Ts :: ts)
   155               end
   156           | _ =>
   157               let val T = typ_of_dtyp descr' sorts dt
   158               in (j + 1, prems, (Leaf $ mk_inj T (mk_Free "x" T j))::ts)
   159               end);
   160 
   161         val (_, prems, ts) = foldr mk_prem (1, [], []) cargs;
   162         val concl = HOLogic.mk_Trueprop (HOLogic.mk_mem
   163           (mk_univ_inj ts n i, Const (s, UnivT)))
   164       in Logic.list_implies (prems, concl)
   165       end;
   166 
   167     val consts = map (fn s => Const (s, UnivT)) rep_set_names;
   168 
   169     val intr_ts = List.concat (map (fn ((_, (_, _, constrs)), rep_set_name) =>
   170       map (make_intr rep_set_name (length constrs))
   171         ((1 upto (length constrs)) ~~ constrs)) (descr' ~~ rep_set_names));
   172 
   173     val (thy2, {raw_induct = rep_induct, intrs = rep_intrs, ...}) =
   174       setmp InductivePackage.quiet_mode (!quiet_mode)
   175         (InductivePackage.add_inductive_i false true big_rec_name false true false
   176            consts (map (fn x => (("", x), [])) intr_ts) []) thy1;
   177 
   178     (********************************* typedef ********************************)
   179 
   180     val thy3 = add_path flat_names big_name (Library.foldl (fn (thy, ((((name, mx), tvs), c), name')) =>
   181       setmp TypedefPackage.quiet_mode true
   182         (TypedefPackage.add_typedef_i false (SOME name') (name, tvs, mx) c NONE
   183           (rtac exI 1 THEN
   184             QUIET_BREADTH_FIRST (has_fewer_prems 1)
   185             (resolve_tac rep_intrs 1))) thy |> snd)
   186               (parent_path flat_names thy2, types_syntax ~~ tyvars ~~
   187                 (Library.take (length newTs, consts)) ~~ new_type_names));
   188 
   189     (*********************** definition of constructors ***********************)
   190 
   191     val big_rep_name = (space_implode "_" new_type_names) ^ "_Rep_";
   192     val rep_names = map (curry op ^ "Rep_") new_type_names;
   193     val rep_names' = map (fn i => big_rep_name ^ (string_of_int i))
   194       (1 upto (length (List.concat (tl descr))));
   195     val all_rep_names = map (Sign.intern_const (Theory.sign_of thy3)) rep_names @
   196       map (Sign.full_name (Theory.sign_of thy3)) rep_names';
   197 
   198     (* isomorphism declarations *)
   199 
   200     val iso_decls = map (fn (T, s) => (s, T --> Univ_elT, NoSyn))
   201       (oldTs ~~ rep_names');
   202 
   203     (* constructor definitions *)
   204 
   205     fun make_constr_def tname T n ((thy, defs, eqns, i), ((cname, cargs), (cname', mx))) =
   206       let
   207         fun constr_arg (dt, (j, l_args, r_args)) =
   208           let val T = typ_of_dtyp descr' sorts dt;
   209               val free_t = mk_Free "x" T j
   210           in (case (strip_dtyp dt, strip_type T) of
   211               ((_, DtRec m), (Us, U)) => (j + 1, free_t :: l_args, mk_lim
   212                 (Const (List.nth (all_rep_names, m), U --> Univ_elT) $
   213                    app_bnds free_t (length Us)) Us :: r_args)
   214             | _ => (j + 1, free_t::l_args, (Leaf $ mk_inj T free_t)::r_args))
   215           end;
   216 
   217         val (_, l_args, r_args) = foldr constr_arg (1, [], []) cargs;
   218         val constrT = (map (typ_of_dtyp descr' sorts) cargs) ---> T;
   219         val abs_name = Sign.intern_const (Theory.sign_of thy) ("Abs_" ^ tname);
   220         val rep_name = Sign.intern_const (Theory.sign_of thy) ("Rep_" ^ tname);
   221         val lhs = list_comb (Const (cname, constrT), l_args);
   222         val rhs = mk_univ_inj r_args n i;
   223         val def = equals T $ lhs $ (Const (abs_name, Univ_elT --> T) $ rhs);
   224         val def_name = (Sign.base_name cname) ^ "_def";
   225         val eqn = HOLogic.mk_Trueprop (HOLogic.mk_eq
   226           (Const (rep_name, T --> Univ_elT) $ lhs, rhs));
   227         val ([def_thm], thy') =
   228           thy
   229           |> Theory.add_consts_i [(cname', constrT, mx)]
   230           |> (PureThy.add_defs_i false o map Thm.no_attributes) [(def_name, def)];
   231 
   232       in (thy', defs @ [def_thm], eqns @ [eqn], i + 1) end;
   233 
   234     (* constructor definitions for datatype *)
   235 
   236     fun dt_constr_defs ((thy, defs, eqns, rep_congs, dist_lemmas),
   237         ((((_, (_, _, constrs)), tname), T), constr_syntax)) =
   238       let
   239         val _ $ (_ $ (cong_f $ _) $ _) = concl_of arg_cong;
   240         val sg = Theory.sign_of thy;
   241         val rep_const = cterm_of sg
   242           (Const (Sign.intern_const sg ("Rep_" ^ tname), T --> Univ_elT));
   243         val cong' = standard (cterm_instantiate [(cterm_of sg cong_f, rep_const)] arg_cong);
   244         val dist = standard (cterm_instantiate [(cterm_of sg distinct_f, rep_const)] distinct_lemma);
   245         val (thy', defs', eqns', _) = Library.foldl ((make_constr_def tname T) (length constrs))
   246           ((add_path flat_names tname thy, defs, [], 1), constrs ~~ constr_syntax)
   247       in
   248         (parent_path flat_names thy', defs', eqns @ [eqns'],
   249           rep_congs @ [cong'], dist_lemmas @ [dist])
   250       end;
   251 
   252     val (thy4, constr_defs, constr_rep_eqns, rep_congs, dist_lemmas) = Library.foldl dt_constr_defs
   253       ((thy3 |> Theory.add_consts_i iso_decls |> parent_path flat_names, [], [], [], []),
   254         hd descr ~~ new_type_names ~~ newTs ~~ constr_syntax);
   255 
   256     (*********** isomorphisms for new types (introduced by typedef) ***********)
   257 
   258     val _ = message "Proving isomorphism properties ...";
   259 
   260     (* get axioms from theory *)
   261 
   262     val newT_iso_axms = map (fn s =>
   263       (get_thm thy4 (Name ("Abs_" ^ s ^ "_inverse")),
   264        get_thm thy4 (Name ("Rep_" ^ s ^ "_inverse")),
   265        get_thm thy4 (Name ("Rep_" ^ s)))) new_type_names;
   266 
   267     (*------------------------------------------------*)
   268     (* prove additional theorems:                     *)
   269     (*  inj_on dt_Abs_i rep_set_i  and  inj dt_Rep_i  *)
   270     (*------------------------------------------------*)
   271 
   272     fun prove_newT_iso_inj_thm (((s, (thm1, thm2, _)), T), rep_set_name) =
   273       let
   274         val sg = Theory.sign_of thy4;
   275         val RepT = T --> Univ_elT;
   276         val Rep_name = Sign.intern_const sg ("Rep_" ^ s);
   277         val AbsT = Univ_elT --> T;
   278         val Abs_name = Sign.intern_const sg ("Abs_" ^ s);
   279 
   280         val inj_Abs_thm = 
   281             Goal.prove_global sg [] []
   282               (HOLogic.mk_Trueprop 
   283                 (Const ("Fun.inj_on", [AbsT, UnivT] ---> HOLogic.boolT) $
   284                  Const (Abs_name, AbsT) $ Const (rep_set_name, UnivT)))
   285               (fn _ => EVERY [rtac inj_on_inverseI 1, etac thm1 1]);
   286 
   287         val setT = HOLogic.mk_setT T
   288 
   289         val inj_Rep_thm =
   290             Goal.prove_global sg [] []
   291               (HOLogic.mk_Trueprop
   292                 (Const ("Fun.inj_on", [RepT, setT] ---> HOLogic.boolT) $
   293                  Const (Rep_name, RepT) $ Const ("UNIV", setT)))
   294               (fn _ => EVERY [rtac inj_inverseI 1, rtac thm2 1]);
   295 
   296       in (inj_Abs_thm, inj_Rep_thm) end;
   297 
   298     val newT_iso_inj_thms = map prove_newT_iso_inj_thm
   299       (new_type_names ~~ newT_iso_axms ~~ newTs ~~
   300         Library.take (length newTs, rep_set_names));
   301 
   302     (********* isomorphisms between existing types and "unfolded" types *******)
   303 
   304     (*---------------------------------------------------------------------*)
   305     (* isomorphisms are defined using primrec-combinators:                 *)
   306     (* generate appropriate functions for instantiating primrec-combinator *)
   307     (*                                                                     *)
   308     (*   e.g.  dt_Rep_i = list_rec ... (%h t y. In1 (Scons (Leaf h) y))    *)
   309     (*                                                                     *)
   310     (* also generate characteristic equations for isomorphisms             *)
   311     (*                                                                     *)
   312     (*   e.g.  dt_Rep_i (cons h t) = In1 (Scons (dt_Rep_j h) (dt_Rep_i t)) *)
   313     (*---------------------------------------------------------------------*)
   314 
   315     fun make_iso_def k ks n ((fs, eqns, i), (cname, cargs)) =
   316       let
   317         val argTs = map (typ_of_dtyp descr' sorts) cargs;
   318         val T = List.nth (recTs, k);
   319         val rep_name = List.nth (all_rep_names, k);
   320         val rep_const = Const (rep_name, T --> Univ_elT);
   321         val constr = Const (cname, argTs ---> T);
   322 
   323         fun process_arg ks' ((i2, i2', ts, Ts), dt) =
   324           let
   325             val T' = typ_of_dtyp descr' sorts dt;
   326             val (Us, U) = strip_type T'
   327           in (case strip_dtyp dt of
   328               (_, DtRec j) => if j mem ks' then
   329                   (i2 + 1, i2' + 1, ts @ [mk_lim (app_bnds
   330                      (mk_Free "y" (Us ---> Univ_elT) i2') (length Us)) Us],
   331                    Ts @ [Us ---> Univ_elT])
   332                 else
   333                   (i2 + 1, i2', ts @ [mk_lim
   334                      (Const (List.nth (all_rep_names, j), U --> Univ_elT) $
   335                         app_bnds (mk_Free "x" T' i2) (length Us)) Us], Ts)
   336             | _ => (i2 + 1, i2', ts @ [Leaf $ mk_inj T' (mk_Free "x" T' i2)], Ts))
   337           end;
   338 
   339         val (i2, i2', ts, Ts) = Library.foldl (process_arg ks) ((1, 1, [], []), cargs);
   340         val xs = map (uncurry (mk_Free "x")) (argTs ~~ (1 upto (i2 - 1)));
   341         val ys = map (uncurry (mk_Free "y")) (Ts ~~ (1 upto (i2' - 1)));
   342         val f = list_abs_free (map dest_Free (xs @ ys), mk_univ_inj ts n i);
   343 
   344         val (_, _, ts', _) = Library.foldl (process_arg []) ((1, 1, [], []), cargs);
   345         val eqn = HOLogic.mk_Trueprop (HOLogic.mk_eq
   346           (rep_const $ list_comb (constr, xs), mk_univ_inj ts' n i))
   347 
   348       in (fs @ [f], eqns @ [eqn], i + 1) end;
   349 
   350     (* define isomorphisms for all mutually recursive datatypes in list ds *)
   351 
   352     fun make_iso_defs (ds, (thy, char_thms)) =
   353       let
   354         val ks = map fst ds;
   355         val (_, (tname, _, _)) = hd ds;
   356         val {rec_rewrites, rec_names, ...} = the (Symtab.lookup dt_info tname);
   357 
   358         fun process_dt ((fs, eqns, isos), (k, (tname, _, constrs))) =
   359           let
   360             val (fs', eqns', _) = Library.foldl (make_iso_def k ks (length constrs))
   361               ((fs, eqns, 1), constrs);
   362             val iso = (List.nth (recTs, k), List.nth (all_rep_names, k))
   363           in (fs', eqns', isos @ [iso]) end;
   364         
   365         val (fs, eqns, isos) = Library.foldl process_dt (([], [], []), ds);
   366         val fTs = map fastype_of fs;
   367         val defs = map (fn (rec_name, (T, iso_name)) => ((Sign.base_name iso_name) ^ "_def",
   368           equals (T --> Univ_elT) $ Const (iso_name, T --> Univ_elT) $
   369             list_comb (Const (rec_name, fTs @ [T] ---> Univ_elT), fs))) (rec_names ~~ isos);
   370         val (def_thms, thy') = (PureThy.add_defs_i false o map Thm.no_attributes) defs thy;
   371 
   372         (* prove characteristic equations *)
   373 
   374         val rewrites = def_thms @ (map mk_meta_eq rec_rewrites);
   375         val char_thms' = map (fn eqn => Goal.prove_global thy' [] [] eqn
   376           (fn _ => EVERY [rewrite_goals_tac rewrites, rtac refl 1])) eqns;
   377 
   378       in (thy', char_thms' @ char_thms) end;
   379 
   380     val (thy5, iso_char_thms) = foldr make_iso_defs
   381       (add_path flat_names big_name thy4, []) (tl descr);
   382 
   383     (* prove isomorphism properties *)
   384 
   385     fun mk_funs_inv thm =
   386       let
   387         val {sign, prop, ...} = rep_thm thm;
   388         val _ $ (_ $ (Const (_, Type (_, [U, _])) $ _ $ S)) $
   389           (_ $ (_ $ (r $ (a $ _)) $ _)) = Type.freeze prop;
   390         val used = add_term_tfree_names (a, []);
   391 
   392         fun mk_thm i =
   393           let
   394             val Ts = map (TFree o rpair HOLogic.typeS)
   395               (Name.variant_list used (replicate i "'t"));
   396             val f = Free ("f", Ts ---> U)
   397           in Goal.prove_global sign [] [] (Logic.mk_implies
   398             (HOLogic.mk_Trueprop (HOLogic.list_all
   399                (map (pair "x") Ts, HOLogic.mk_mem (app_bnds f i, S))),
   400              HOLogic.mk_Trueprop (HOLogic.mk_eq (list_abs (map (pair "x") Ts,
   401                r $ (a $ app_bnds f i)), f))))
   402             (fn _ => EVERY [REPEAT (rtac ext 1), REPEAT (etac allE 1), rtac thm 1, atac 1])
   403           end
   404       in map (fn r => r RS subst) (thm :: map mk_thm arities) end;
   405 
   406     (* prove  inj dt_Rep_i  and  dt_Rep_i x : dt_rep_set_i *)
   407 
   408     fun prove_iso_thms (ds, (inj_thms, elem_thms)) =
   409       let
   410         val (_, (tname, _, _)) = hd ds;
   411         val {induction, ...} = the (Symtab.lookup dt_info tname);
   412 
   413         fun mk_ind_concl (i, _) =
   414           let
   415             val T = List.nth (recTs, i);
   416             val Rep_t = Const (List.nth (all_rep_names, i), T --> Univ_elT);
   417             val rep_set_name = List.nth (rep_set_names, i)
   418           in (HOLogic.all_const T $ Abs ("y", T, HOLogic.imp $
   419                 HOLogic.mk_eq (Rep_t $ mk_Free "x" T i, Rep_t $ Bound 0) $
   420                   HOLogic.mk_eq (mk_Free "x" T i, Bound 0)),
   421               HOLogic.mk_mem (Rep_t $ mk_Free "x" T i, Const (rep_set_name, UnivT)))
   422           end;
   423 
   424         val (ind_concl1, ind_concl2) = ListPair.unzip (map mk_ind_concl ds);
   425 
   426         val rewrites = map mk_meta_eq iso_char_thms;
   427         val inj_thms' = map (fn r => r RS injD)
   428           (map snd newT_iso_inj_thms @ inj_thms);
   429 
   430         val inj_thm = Goal.prove_global thy5 [] []
   431           (HOLogic.mk_Trueprop (mk_conj ind_concl1)) (fn _ => EVERY
   432             [(indtac induction THEN_ALL_NEW ObjectLogic.atomize_tac) 1,
   433              REPEAT (EVERY
   434                [rtac allI 1, rtac impI 1,
   435                 exh_tac (exh_thm_of dt_info) 1,
   436                 REPEAT (EVERY
   437                   [hyp_subst_tac 1,
   438                    rewrite_goals_tac rewrites,
   439                    REPEAT (dresolve_tac [In0_inject, In1_inject] 1),
   440                    (eresolve_tac [In0_not_In1 RS notE, In1_not_In0 RS notE] 1)
   441                    ORELSE (EVERY
   442                      [REPEAT (eresolve_tac (Scons_inject ::
   443                         map make_elim [Leaf_inject, Inl_inject, Inr_inject]) 1),
   444                       REPEAT (cong_tac 1), rtac refl 1,
   445                       REPEAT (atac 1 ORELSE (EVERY
   446                         [REPEAT (rtac ext 1),
   447                          REPEAT (eresolve_tac (mp :: allE ::
   448                            map make_elim (Suml_inject :: Sumr_inject ::
   449                              Lim_inject :: fun_cong :: inj_thms')) 1),
   450                          atac 1]))])])])]);
   451 
   452         val inj_thms'' = map (fn r => r RS datatype_injI)
   453                              (split_conj_thm inj_thm);
   454 
   455         val elem_thm = 
   456             Goal.prove_global thy5 [] [] (HOLogic.mk_Trueprop (mk_conj ind_concl2))
   457               (fn _ =>
   458                EVERY [(indtac induction THEN_ALL_NEW ObjectLogic.atomize_tac) 1,
   459                 rewrite_goals_tac rewrites,
   460                 REPEAT ((resolve_tac rep_intrs THEN_ALL_NEW
   461                   ((REPEAT o etac allE) THEN' ares_tac elem_thms)) 1)]);
   462 
   463       in (inj_thms'' @ inj_thms, elem_thms @ (split_conj_thm elem_thm))
   464       end;
   465 
   466     val (iso_inj_thms_unfolded, iso_elem_thms) = foldr prove_iso_thms
   467       ([], map #3 newT_iso_axms) (tl descr);
   468     val iso_inj_thms = map snd newT_iso_inj_thms @ iso_inj_thms_unfolded;
   469 
   470     (* prove  x : dt_rep_set_i --> x : range dt_Rep_i *)
   471 
   472     fun mk_iso_t (((set_name, iso_name), i), T) =
   473       let val isoT = T --> Univ_elT
   474       in HOLogic.imp $ 
   475         HOLogic.mk_mem (mk_Free "x" Univ_elT i, Const (set_name, UnivT)) $
   476           (if i < length newTs then Const ("True", HOLogic.boolT)
   477            else HOLogic.mk_mem (mk_Free "x" Univ_elT i,
   478              Const ("image", [isoT, HOLogic.mk_setT T] ---> UnivT) $
   479                Const (iso_name, isoT) $ Const ("UNIV", HOLogic.mk_setT T)))
   480       end;
   481 
   482     val iso_t = HOLogic.mk_Trueprop (mk_conj (map mk_iso_t
   483       (rep_set_names ~~ all_rep_names ~~ (0 upto (length descr' - 1)) ~~ recTs)));
   484 
   485     (* all the theorems are proved by one single simultaneous induction *)
   486 
   487     val range_eqs = map (fn r => mk_meta_eq (r RS range_ex1_eq))
   488       iso_inj_thms_unfolded;
   489 
   490     val iso_thms = if length descr = 1 then [] else
   491       Library.drop (length newTs, split_conj_thm
   492         (Goal.prove_global thy5 [] [] iso_t (fn _ => EVERY
   493            [(indtac rep_induct THEN_ALL_NEW ObjectLogic.atomize_tac) 1,
   494             REPEAT (rtac TrueI 1),
   495             rewrite_goals_tac (mk_meta_eq choice_eq ::
   496               symmetric (mk_meta_eq expand_fun_eq) :: range_eqs),
   497             rewrite_goals_tac (map symmetric range_eqs),
   498             REPEAT (EVERY
   499               [REPEAT (eresolve_tac ([rangeE, ex1_implies_ex RS exE] @
   500                  List.concat (map (mk_funs_inv o #1) newT_iso_axms)) 1),
   501                TRY (hyp_subst_tac 1),
   502                rtac (sym RS range_eqI) 1,
   503                resolve_tac iso_char_thms 1])])));
   504 
   505     val Abs_inverse_thms' =
   506       map #1 newT_iso_axms @
   507       map2 (fn r_inj => fn r => f_myinv_f OF [r_inj, r RS mp])
   508         iso_inj_thms_unfolded iso_thms;
   509 
   510     val Abs_inverse_thms = List.concat (map mk_funs_inv Abs_inverse_thms');
   511 
   512     (******************* freeness theorems for constructors *******************)
   513 
   514     val _ = message "Proving freeness of constructors ...";
   515 
   516     (* prove theorem  Rep_i (Constr_j ...) = Inj_j ...  *)
   517     
   518     fun prove_constr_rep_thm eqn =
   519       let
   520         val inj_thms = map (fn (r, _) => r RS inj_onD) newT_iso_inj_thms;
   521         val rewrites = o_def :: constr_defs @ (map (mk_meta_eq o #2) newT_iso_axms)
   522       in Goal.prove_global thy5 [] [] eqn (fn _ => EVERY
   523         [resolve_tac inj_thms 1,
   524          rewrite_goals_tac rewrites,
   525          rtac refl 1,
   526          resolve_tac rep_intrs 2,
   527          REPEAT (resolve_tac iso_elem_thms 1)])
   528       end;
   529 
   530     (*--------------------------------------------------------------*)
   531     (* constr_rep_thms and rep_congs are used to prove distinctness *)
   532     (* of constructors.                                             *)
   533     (*--------------------------------------------------------------*)
   534 
   535     val constr_rep_thms = map (map prove_constr_rep_thm) constr_rep_eqns;
   536 
   537     val dist_rewrites = map (fn (rep_thms, dist_lemma) =>
   538       dist_lemma::(rep_thms @ [In0_eq, In1_eq, In0_not_In1, In1_not_In0]))
   539         (constr_rep_thms ~~ dist_lemmas);
   540 
   541     fun prove_distinct_thms (_, []) = []
   542       | prove_distinct_thms (dist_rewrites', t::_::ts) =
   543           let
   544             val dist_thm = Goal.prove_global thy5 [] [] t (fn _ =>
   545               EVERY [simp_tac (HOL_ss addsimps dist_rewrites') 1])
   546           in dist_thm::(standard (dist_thm RS not_sym))::
   547             (prove_distinct_thms (dist_rewrites', ts))
   548           end;
   549 
   550     val distinct_thms = map prove_distinct_thms (dist_rewrites ~~
   551       DatatypeProp.make_distincts new_type_names descr sorts thy5);
   552 
   553     val simproc_dists = map (fn ((((_, (_, _, constrs)), rep_thms), congr), dists) =>
   554       if length constrs < !DatatypeProp.dtK then FewConstrs dists
   555       else ManyConstrs (congr, HOL_basic_ss addsimps rep_thms)) (hd descr ~~
   556         constr_rep_thms ~~ rep_congs ~~ distinct_thms);
   557 
   558     (* prove injectivity of constructors *)
   559 
   560     fun prove_constr_inj_thm rep_thms t =
   561       let val inj_thms = Scons_inject :: (map make_elim
   562         ((map (fn r => r RS injD) iso_inj_thms) @
   563           [In0_inject, In1_inject, Leaf_inject, Inl_inject, Inr_inject,
   564            Lim_inject, Suml_inject, Sumr_inject]))
   565       in Goal.prove_global thy5 [] [] t (fn _ => EVERY
   566         [rtac iffI 1,
   567          REPEAT (etac conjE 2), hyp_subst_tac 2, rtac refl 2,
   568          dresolve_tac rep_congs 1, dtac box_equals 1,
   569          REPEAT (resolve_tac rep_thms 1),
   570          REPEAT (eresolve_tac inj_thms 1),
   571          REPEAT (ares_tac [conjI] 1 ORELSE (EVERY [REPEAT (rtac ext 1),
   572            REPEAT (eresolve_tac (make_elim fun_cong :: inj_thms) 1),
   573            atac 1]))])
   574       end;
   575 
   576     val constr_inject = map (fn (ts, thms) => map (prove_constr_inj_thm thms) ts)
   577       ((DatatypeProp.make_injs descr sorts) ~~ constr_rep_thms);
   578 
   579     val ((constr_inject', distinct_thms'), thy6) =
   580       thy5
   581       |> parent_path flat_names
   582       |> store_thmss "inject" new_type_names constr_inject
   583       ||>> store_thmss "distinct" new_type_names distinct_thms;
   584 
   585     (*************************** induction theorem ****************************)
   586 
   587     val _ = message "Proving induction rule for datatypes ...";
   588 
   589     val Rep_inverse_thms = (map (fn (_, iso, _) => iso RS subst) newT_iso_axms) @
   590       (map (fn r => r RS myinv_f_f RS subst) iso_inj_thms_unfolded);
   591     val Rep_inverse_thms' = map (fn r => r RS myinv_f_f) iso_inj_thms_unfolded;
   592 
   593     fun mk_indrule_lemma ((prems, concls), ((i, _), T)) =
   594       let
   595         val Rep_t = Const (List.nth (all_rep_names, i), T --> Univ_elT) $
   596           mk_Free "x" T i;
   597 
   598         val Abs_t = if i < length newTs then
   599             Const (Sign.intern_const (Theory.sign_of thy6)
   600               ("Abs_" ^ (List.nth (new_type_names, i))), Univ_elT --> T)
   601           else Const ("Inductive.myinv", [T --> Univ_elT, Univ_elT] ---> T) $
   602             Const (List.nth (all_rep_names, i), T --> Univ_elT)
   603 
   604       in (prems @ [HOLogic.imp $ HOLogic.mk_mem (Rep_t,
   605             Const (List.nth (rep_set_names, i), UnivT)) $
   606               (mk_Free "P" (T --> HOLogic.boolT) (i + 1) $ (Abs_t $ Rep_t))],
   607           concls @ [mk_Free "P" (T --> HOLogic.boolT) (i + 1) $ mk_Free "x" T i])
   608       end;
   609 
   610     val (indrule_lemma_prems, indrule_lemma_concls) =
   611       Library.foldl mk_indrule_lemma (([], []), (descr' ~~ recTs));
   612 
   613     val cert = cterm_of (Theory.sign_of thy6);
   614 
   615     val indrule_lemma = Goal.prove_global thy6 [] []
   616       (Logic.mk_implies
   617         (HOLogic.mk_Trueprop (mk_conj indrule_lemma_prems),
   618          HOLogic.mk_Trueprop (mk_conj indrule_lemma_concls))) (fn _ => EVERY
   619            [REPEAT (etac conjE 1),
   620             REPEAT (EVERY
   621               [TRY (rtac conjI 1), resolve_tac Rep_inverse_thms 1,
   622                etac mp 1, resolve_tac iso_elem_thms 1])]);
   623 
   624     val Ps = map head_of (HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of indrule_lemma)));
   625     val frees = if length Ps = 1 then [Free ("P", snd (dest_Var (hd Ps)))] else
   626       map (Free o apfst fst o dest_Var) Ps;
   627     val indrule_lemma' = cterm_instantiate (map cert Ps ~~ map cert frees) indrule_lemma;
   628 
   629     val dt_induct_prop = DatatypeProp.make_ind descr sorts;
   630     val dt_induct = Goal.prove_global thy6 []
   631       (Logic.strip_imp_prems dt_induct_prop) (Logic.strip_imp_concl dt_induct_prop)
   632       (fn prems => EVERY
   633         [rtac indrule_lemma' 1,
   634          (indtac rep_induct THEN_ALL_NEW ObjectLogic.atomize_tac) 1,
   635          EVERY (map (fn (prem, r) => (EVERY
   636            [REPEAT (eresolve_tac Abs_inverse_thms 1),
   637             simp_tac (HOL_basic_ss addsimps ((symmetric r)::Rep_inverse_thms')) 1,
   638             DEPTH_SOLVE_1 (ares_tac [prem] 1 ORELSE etac allE 1)]))
   639                 (prems ~~ (constr_defs @ (map mk_meta_eq iso_char_thms))))]);
   640 
   641     val ([dt_induct'], thy7) =
   642       thy6
   643       |> Theory.add_path big_name
   644       |> PureThy.add_thms [(("induct", dt_induct), [case_names_induct])]
   645       ||> Theory.parent_path;
   646 
   647   in
   648     ((constr_inject', distinct_thms', dist_rewrites, simproc_dists, dt_induct'), thy7)
   649   end;
   650 
   651 end;