src/HOL/Tools/datatype_rep_proofs.ML
author haftmann
Fri May 23 16:05:04 2008 +0200 (2008-05-23)
changeset 26969 cf3f998d0631
parent 26806 40b411ec05aa
child 27002 215d64dc971e
permissions -rw-r--r--
moved case distinction over number of constructors for distinctness rules from DatatypeProp to DatatypeRepProofs
berghofe@5177
     1
(*  Title:      HOL/Tools/datatype_rep_proofs.ML
berghofe@5177
     2
    ID:         $Id$
wenzelm@11539
     3
    Author:     Stefan Berghofer, TU Muenchen
berghofe@5177
     4
berghofe@5177
     5
Definitional introduction of datatypes
berghofe@5177
     6
Proof of characteristic theorems:
berghofe@5177
     7
berghofe@5177
     8
 - injectivity of constructors
berghofe@7228
     9
 - distinctness of constructors
berghofe@5177
    10
 - induction theorem
berghofe@5177
    11
berghofe@5177
    12
*)
berghofe@5177
    13
berghofe@5177
    14
signature DATATYPE_REP_PROOFS =
berghofe@5177
    15
sig
berghofe@5661
    16
  val representation_proofs : bool -> DatatypeAux.datatype_info Symtab.table ->
berghofe@13641
    17
    string list -> DatatypeAux.descr list -> (string * sort) list ->
wenzelm@18728
    18
      (string * mixfix) list -> (string * mixfix) list list -> attribute
haftmann@18314
    19
        -> theory -> (thm list list * thm list list * thm list list *
haftmann@18314
    20
          DatatypeAux.simproc_dist list * thm) * theory
berghofe@5177
    21
end;
berghofe@5177
    22
berghofe@5177
    23
structure DatatypeRepProofs : DATATYPE_REP_PROOFS =
berghofe@5177
    24
struct
berghofe@5177
    25
berghofe@5177
    26
open DatatypeAux;
berghofe@5177
    27
berghofe@5177
    28
val (_ $ (_ $ (_ $ (distinct_f $ _) $ _))) = hd (prems_of distinct_lemma);
berghofe@5177
    29
berghofe@21021
    30
val collect_simp = rewrite_rule [mk_meta_eq mem_Collect_eq];
berghofe@21021
    31
wenzelm@11435
    32
wenzelm@11435
    33
(** theory context references **)
berghofe@5177
    34
wenzelm@11435
    35
val f_myinv_f = thm "f_myinv_f";
wenzelm@11435
    36
val myinv_f_f = thm "myinv_f_f";
wenzelm@11435
    37
berghofe@5177
    38
berghofe@5177
    39
fun exh_thm_of (dt_info : datatype_info Symtab.table) tname =
wenzelm@17412
    40
  #exhaustion (the (Symtab.lookup dt_info tname));
berghofe@5177
    41
berghofe@5177
    42
(******************************************************************************)
berghofe@5177
    43
berghofe@5661
    44
fun representation_proofs flat_names (dt_info : datatype_info Symtab.table)
wenzelm@8436
    45
      new_type_names descr sorts types_syntax constr_syntax case_names_induct thy =
berghofe@5177
    46
  let
wenzelm@19540
    47
    val Datatype_thy = ThyInfo.the_theory "Datatype" thy;
wenzelm@20820
    48
    val node_name = "Datatype.node";
wenzelm@20820
    49
    val In0_name = "Datatype.In0";
wenzelm@20820
    50
    val In1_name = "Datatype.In1";
wenzelm@20820
    51
    val Scons_name = "Datatype.Scons";
wenzelm@20820
    52
    val Leaf_name = "Datatype.Leaf";
wenzelm@20820
    53
    val Numb_name = "Datatype.Numb";
wenzelm@20820
    54
    val Lim_name = "Datatype.Lim";
berghofe@13641
    55
    val Suml_name = "Datatype.Suml";
berghofe@13641
    56
    val Sumr_name = "Datatype.Sumr";
berghofe@7015
    57
berghofe@13641
    58
    val [In0_inject, In1_inject, Scons_inject, Leaf_inject,
berghofe@13641
    59
         In0_eq, In1_eq, In0_not_In1, In1_not_In0,
wenzelm@26343
    60
         Lim_inject, Suml_inject, Sumr_inject] = map (PureThy.get_thm Datatype_thy)
wenzelm@26343
    61
          ["In0_inject", "In1_inject", "Scons_inject", "Leaf_inject",
wenzelm@26343
    62
           "In0_eq", "In1_eq", "In0_not_In1", "In1_not_In0",
wenzelm@26343
    63
           "Lim_inject", "Suml_inject", "Sumr_inject"];
berghofe@5177
    64
skalberg@15570
    65
    val descr' = List.concat descr;
berghofe@5177
    66
berghofe@5661
    67
    val big_name = space_implode "_" new_type_names;
berghofe@5661
    68
    val thy1 = add_path flat_names big_name thy;
berghofe@5661
    69
    val big_rec_name = big_name ^ "_rep_set";
berghofe@21021
    70
    val rep_set_names' =
berghofe@5177
    71
      (if length descr' = 1 then [big_rec_name] else
berghofe@5177
    72
        (map ((curry (op ^) (big_rec_name ^ "_")) o string_of_int)
berghofe@5177
    73
          (1 upto (length descr'))));
wenzelm@22578
    74
    val rep_set_names = map (Sign.full_name thy1) rep_set_names';
berghofe@5177
    75
berghofe@5661
    76
    val tyvars = map (fn (_, (_, Ts, _)) => map dest_DtTFree Ts) (hd descr);
berghofe@5661
    77
    val leafTs' = get_nonrec_types descr' sorts;
berghofe@7015
    78
    val branchTs = get_branching_types descr' sorts;
berghofe@7015
    79
    val branchT = if null branchTs then HOLogic.unitT
wenzelm@23419
    80
      else BalancedTree.make (fn (T, U) => Type ("+", [T, U])) branchTs;
berghofe@13641
    81
    val arities = get_arities descr' \ 0;
skalberg@15574
    82
    val unneeded_vars = hd tyvars \\ foldr add_typ_tfree_names [] (leafTs' @ branchTs);
haftmann@17521
    83
    val leafTs = leafTs' @ (map (fn n => TFree (n, (the o AList.lookup (op =) sorts) n)) unneeded_vars);
berghofe@5177
    84
    val recTs = get_rec_types descr' sorts;
skalberg@15570
    85
    val newTs = Library.take (length (hd descr), recTs);
skalberg@15570
    86
    val oldTs = Library.drop (length (hd descr), recTs);
berghofe@5177
    87
    val sumT = if null leafTs then HOLogic.unitT
wenzelm@23419
    88
      else BalancedTree.make (fn (T, U) => Type ("+", [T, U])) leafTs;
berghofe@7015
    89
    val Univ_elT = HOLogic.mk_setT (Type (node_name, [sumT, branchT]));
berghofe@5177
    90
    val UnivT = HOLogic.mk_setT Univ_elT;
berghofe@21021
    91
    val UnivT' = Univ_elT --> HOLogic.boolT;
berghofe@21021
    92
    val Collect = Const ("Collect", UnivT' --> UnivT);
berghofe@5177
    93
berghofe@5177
    94
    val In0 = Const (In0_name, Univ_elT --> Univ_elT);
berghofe@5177
    95
    val In1 = Const (In1_name, Univ_elT --> Univ_elT);
berghofe@5177
    96
    val Leaf = Const (Leaf_name, sumT --> Univ_elT);
berghofe@7015
    97
    val Lim = Const (Lim_name, (branchT --> Univ_elT) --> Univ_elT);
berghofe@5177
    98
berghofe@5177
    99
    (* make injections needed for embedding types in leaves *)
berghofe@5177
   100
berghofe@5177
   101
    fun mk_inj T' x =
berghofe@5177
   102
      let
berghofe@5177
   103
        fun mk_inj' T n i =
berghofe@5177
   104
          if n = 1 then x else
berghofe@5177
   105
          let val n2 = n div 2;
berghofe@5177
   106
              val Type (_, [T1, T2]) = T
berghofe@5177
   107
          in
berghofe@5177
   108
            if i <= n2 then
paulson@15391
   109
              Const ("Sum_Type.Inl", T1 --> T) $ (mk_inj' T1 n2 i)
berghofe@5177
   110
            else
paulson@15391
   111
              Const ("Sum_Type.Inr", T2 --> T) $ (mk_inj' T2 (n - n2) (i - n2))
berghofe@5177
   112
          end
berghofe@5177
   113
      in mk_inj' sumT (length leafTs) (1 + find_index_eq T' leafTs)
berghofe@5177
   114
      end;
berghofe@5177
   115
berghofe@5177
   116
    (* make injections for constructors *)
berghofe@5177
   117
wenzelm@23419
   118
    fun mk_univ_inj ts = BalancedTree.access
wenzelm@23419
   119
      {left = fn t => In0 $ t,
wenzelm@23419
   120
        right = fn t => In1 $ t,
wenzelm@23419
   121
        init =
wenzelm@23419
   122
          if ts = [] then Const ("arbitrary", Univ_elT)
wenzelm@23419
   123
          else foldr1 (HOLogic.mk_binop Scons_name) ts};
berghofe@5177
   124
berghofe@7015
   125
    (* function spaces *)
berghofe@7015
   126
berghofe@7015
   127
    fun mk_fun_inj T' x =
berghofe@7015
   128
      let
berghofe@7015
   129
        fun mk_inj T n i =
berghofe@7015
   130
          if n = 1 then x else
berghofe@7015
   131
          let
berghofe@7015
   132
            val n2 = n div 2;
berghofe@7015
   133
            val Type (_, [T1, T2]) = T;
berghofe@13641
   134
            fun mkT U = (U --> Univ_elT) --> T --> Univ_elT
berghofe@7015
   135
          in
berghofe@13641
   136
            if i <= n2 then Const (Suml_name, mkT T1) $ mk_inj T1 n2 i
berghofe@13641
   137
            else Const (Sumr_name, mkT T2) $ mk_inj T2 (n - n2) (i - n2)
berghofe@7015
   138
          end
berghofe@7015
   139
      in mk_inj branchT (length branchTs) (1 + find_index_eq T' branchTs)
berghofe@7015
   140
      end;
berghofe@7015
   141
skalberg@15574
   142
    val mk_lim = foldr (fn (T, t) => Lim $ mk_fun_inj T (Abs ("x", T, t)));
berghofe@13641
   143
berghofe@5177
   144
    (************** generate introduction rules for representing set **********)
berghofe@5177
   145
wenzelm@6427
   146
    val _ = message "Constructing representing sets ...";
berghofe@5177
   147
berghofe@5177
   148
    (* make introduction rule for a single constructor *)
berghofe@5177
   149
berghofe@5177
   150
    fun make_intr s n (i, (_, cargs)) =
berghofe@5177
   151
      let
berghofe@13641
   152
        fun mk_prem (dt, (j, prems, ts)) = (case strip_dtyp dt of
berghofe@13641
   153
            (dts, DtRec k) =>
berghofe@13641
   154
              let
berghofe@13641
   155
                val Ts = map (typ_of_dtyp descr' sorts) dts;
berghofe@13641
   156
                val free_t =
berghofe@13641
   157
                  app_bnds (mk_Free "x" (Ts ---> Univ_elT) j) (length Ts)
berghofe@13641
   158
              in (j + 1, list_all (map (pair "x") Ts,
berghofe@21021
   159
                  HOLogic.mk_Trueprop
berghofe@21021
   160
                    (Free (List.nth (rep_set_names', k), UnivT') $ free_t)) :: prems,
skalberg@15574
   161
                mk_lim free_t Ts :: ts)
berghofe@5177
   162
              end
berghofe@13641
   163
          | _ =>
berghofe@5177
   164
              let val T = typ_of_dtyp descr' sorts dt
berghofe@5177
   165
              in (j + 1, prems, (Leaf $ mk_inj T (mk_Free "x" T j))::ts)
berghofe@13641
   166
              end);
berghofe@5177
   167
skalberg@15574
   168
        val (_, prems, ts) = foldr mk_prem (1, [], []) cargs;
berghofe@21021
   169
        val concl = HOLogic.mk_Trueprop
berghofe@21021
   170
          (Free (s, UnivT') $ mk_univ_inj ts n i)
berghofe@13641
   171
      in Logic.list_implies (prems, concl)
berghofe@5177
   172
      end;
berghofe@5177
   173
skalberg@15570
   174
    val intr_ts = List.concat (map (fn ((_, (_, _, constrs)), rep_set_name) =>
berghofe@5177
   175
      map (make_intr rep_set_name (length constrs))
berghofe@21021
   176
        ((1 upto (length constrs)) ~~ constrs)) (descr' ~~ rep_set_names'));
berghofe@5177
   177
wenzelm@21365
   178
    val ({raw_induct = rep_induct, intrs = rep_intrs, ...}, thy2) =
wenzelm@26475
   179
        InductivePackage.add_inductive_global (serial_string ())
berghofe@26531
   180
          {quiet_mode = ! quiet_mode, verbose = false, kind = Thm.internalK,
berghofe@26531
   181
           alt_name = big_rec_name, coind = false, no_elim = true, no_ind = false,
berghofe@26531
   182
           skip_mono = true}
wenzelm@26128
   183
          (map (fn s => ((s, UnivT'), NoSyn)) rep_set_names') []
wenzelm@26475
   184
          (map (fn x => (("", []), x)) intr_ts) [] thy1;
berghofe@5177
   185
berghofe@5177
   186
    (********************************* typedef ********************************)
berghofe@5177
   187
berghofe@21021
   188
    val (typedefs, thy3) = thy2 |>
berghofe@21021
   189
      parent_path flat_names |>
berghofe@21021
   190
      fold_map (fn ((((name, mx), tvs), c), name') =>
wenzelm@26475
   191
          TypedefPackage.add_typedef_i false (SOME name') (name, tvs, mx)
berghofe@21021
   192
            (Collect $ Const (c, UnivT')) NONE
berghofe@21021
   193
            (rtac exI 1 THEN rtac CollectI 1 THEN
berghofe@21021
   194
              QUIET_BREADTH_FIRST (has_fewer_prems 1)
wenzelm@26475
   195
              (resolve_tac rep_intrs 1)))
berghofe@21021
   196
                (types_syntax ~~ tyvars ~~
berghofe@21021
   197
                  (Library.take (length newTs, rep_set_names)) ~~ new_type_names) ||>
berghofe@21021
   198
      add_path flat_names big_name;
berghofe@5177
   199
berghofe@5177
   200
    (*********************** definition of constructors ***********************)
berghofe@5177
   201
berghofe@5177
   202
    val big_rep_name = (space_implode "_" new_type_names) ^ "_Rep_";
berghofe@5177
   203
    val rep_names = map (curry op ^ "Rep_") new_type_names;
berghofe@5177
   204
    val rep_names' = map (fn i => big_rep_name ^ (string_of_int i))
skalberg@15570
   205
      (1 upto (length (List.concat (tl descr))));
wenzelm@22578
   206
    val all_rep_names = map (Sign.intern_const thy3) rep_names @
wenzelm@22578
   207
      map (Sign.full_name thy3) rep_names';
berghofe@5177
   208
berghofe@5177
   209
    (* isomorphism declarations *)
berghofe@5177
   210
berghofe@5177
   211
    val iso_decls = map (fn (T, s) => (s, T --> Univ_elT, NoSyn))
berghofe@5177
   212
      (oldTs ~~ rep_names');
berghofe@5177
   213
berghofe@5177
   214
    (* constructor definitions *)
berghofe@5177
   215
berghofe@5177
   216
    fun make_constr_def tname T n ((thy, defs, eqns, i), ((cname, cargs), (cname', mx))) =
berghofe@5177
   217
      let
berghofe@5177
   218
        fun constr_arg (dt, (j, l_args, r_args)) =
berghofe@5177
   219
          let val T = typ_of_dtyp descr' sorts dt;
berghofe@5177
   220
              val free_t = mk_Free "x" T j
berghofe@13641
   221
          in (case (strip_dtyp dt, strip_type T) of
skalberg@15574
   222
              ((_, DtRec m), (Us, U)) => (j + 1, free_t :: l_args, mk_lim
skalberg@15574
   223
                (Const (List.nth (all_rep_names, m), U --> Univ_elT) $
skalberg@15574
   224
                   app_bnds free_t (length Us)) Us :: r_args)
berghofe@5177
   225
            | _ => (j + 1, free_t::l_args, (Leaf $ mk_inj T free_t)::r_args))
berghofe@5177
   226
          end;
berghofe@5177
   227
skalberg@15574
   228
        val (_, l_args, r_args) = foldr constr_arg (1, [], []) cargs;
berghofe@5177
   229
        val constrT = (map (typ_of_dtyp descr' sorts) cargs) ---> T;
wenzelm@22578
   230
        val abs_name = Sign.intern_const thy ("Abs_" ^ tname);
wenzelm@22578
   231
        val rep_name = Sign.intern_const thy ("Rep_" ^ tname);
berghofe@5177
   232
        val lhs = list_comb (Const (cname, constrT), l_args);
berghofe@5177
   233
        val rhs = mk_univ_inj r_args n i;
berghofe@5177
   234
        val def = equals T $ lhs $ (Const (abs_name, Univ_elT --> T) $ rhs);
berghofe@5177
   235
        val def_name = (Sign.base_name cname) ^ "_def";
berghofe@5177
   236
        val eqn = HOLogic.mk_Trueprop (HOLogic.mk_eq
berghofe@5177
   237
          (Const (rep_name, T --> Univ_elT) $ lhs, rhs));
haftmann@18377
   238
        val ([def_thm], thy') =
haftmann@18377
   239
          thy
wenzelm@24712
   240
          |> Sign.add_consts_i [(cname', constrT, mx)]
haftmann@18377
   241
          |> (PureThy.add_defs_i false o map Thm.no_attributes) [(def_name, def)];
berghofe@5177
   242
wenzelm@8436
   243
      in (thy', defs @ [def_thm], eqns @ [eqn], i + 1) end;
berghofe@5177
   244
berghofe@5177
   245
    (* constructor definitions for datatype *)
berghofe@5177
   246
berghofe@5177
   247
    fun dt_constr_defs ((thy, defs, eqns, rep_congs, dist_lemmas),
berghofe@5177
   248
        ((((_, (_, _, constrs)), tname), T), constr_syntax)) =
berghofe@5177
   249
      let
berghofe@5177
   250
        val _ $ (_ $ (cong_f $ _) $ _) = concl_of arg_cong;
wenzelm@22578
   251
        val rep_const = cterm_of thy
wenzelm@22578
   252
          (Const (Sign.intern_const thy ("Rep_" ^ tname), T --> Univ_elT));
wenzelm@22578
   253
        val cong' = standard (cterm_instantiate [(cterm_of thy cong_f, rep_const)] arg_cong);
wenzelm@22578
   254
        val dist = standard (cterm_instantiate [(cterm_of thy distinct_f, rep_const)] distinct_lemma);
skalberg@15570
   255
        val (thy', defs', eqns', _) = Library.foldl ((make_constr_def tname T) (length constrs))
berghofe@5661
   256
          ((add_path flat_names tname thy, defs, [], 1), constrs ~~ constr_syntax)
berghofe@5177
   257
      in
berghofe@5661
   258
        (parent_path flat_names thy', defs', eqns @ [eqns'],
berghofe@5177
   259
          rep_congs @ [cong'], dist_lemmas @ [dist])
berghofe@5177
   260
      end;
berghofe@5177
   261
skalberg@15570
   262
    val (thy4, constr_defs, constr_rep_eqns, rep_congs, dist_lemmas) = Library.foldl dt_constr_defs
wenzelm@24712
   263
      ((thy3 |> Sign.add_consts_i iso_decls |> parent_path flat_names, [], [], [], []),
berghofe@5177
   264
        hd descr ~~ new_type_names ~~ newTs ~~ constr_syntax);
berghofe@5177
   265
berghofe@5177
   266
    (*********** isomorphisms for new types (introduced by typedef) ***********)
berghofe@5177
   267
wenzelm@6427
   268
    val _ = message "Proving isomorphism properties ...";
berghofe@5177
   269
berghofe@21021
   270
    val newT_iso_axms = map (fn (_, td) =>
berghofe@21021
   271
      (collect_simp (#Abs_inverse td), #Rep_inverse td,
berghofe@21021
   272
       collect_simp (#Rep td))) typedefs;
berghofe@5177
   273
berghofe@21021
   274
    val newT_iso_inj_thms = map (fn (_, td) =>
berghofe@21021
   275
      (collect_simp (#Abs_inject td) RS iffD1, #Rep_inject td RS iffD1)) typedefs;
berghofe@5177
   276
berghofe@5177
   277
    (********* isomorphisms between existing types and "unfolded" types *******)
berghofe@5177
   278
berghofe@5177
   279
    (*---------------------------------------------------------------------*)
berghofe@5177
   280
    (* isomorphisms are defined using primrec-combinators:                 *)
berghofe@5177
   281
    (* generate appropriate functions for instantiating primrec-combinator *)
berghofe@5177
   282
    (*                                                                     *)
berghofe@13641
   283
    (*   e.g.  dt_Rep_i = list_rec ... (%h t y. In1 (Scons (Leaf h) y))    *)
berghofe@5177
   284
    (*                                                                     *)
berghofe@5177
   285
    (* also generate characteristic equations for isomorphisms             *)
berghofe@5177
   286
    (*                                                                     *)
berghofe@13641
   287
    (*   e.g.  dt_Rep_i (cons h t) = In1 (Scons (dt_Rep_j h) (dt_Rep_i t)) *)
berghofe@5177
   288
    (*---------------------------------------------------------------------*)
berghofe@5177
   289
berghofe@5177
   290
    fun make_iso_def k ks n ((fs, eqns, i), (cname, cargs)) =
berghofe@5177
   291
      let
berghofe@5177
   292
        val argTs = map (typ_of_dtyp descr' sorts) cargs;
skalberg@15570
   293
        val T = List.nth (recTs, k);
skalberg@15570
   294
        val rep_name = List.nth (all_rep_names, k);
berghofe@5177
   295
        val rep_const = Const (rep_name, T --> Univ_elT);
berghofe@5177
   296
        val constr = Const (cname, argTs ---> T);
berghofe@5177
   297
berghofe@7015
   298
        fun process_arg ks' ((i2, i2', ts, Ts), dt) =
berghofe@13641
   299
          let
berghofe@13641
   300
            val T' = typ_of_dtyp descr' sorts dt;
berghofe@13641
   301
            val (Us, U) = strip_type T'
berghofe@13641
   302
          in (case strip_dtyp dt of
berghofe@13641
   303
              (_, DtRec j) => if j mem ks' then
skalberg@15574
   304
                  (i2 + 1, i2' + 1, ts @ [mk_lim (app_bnds
skalberg@15574
   305
                     (mk_Free "y" (Us ---> Univ_elT) i2') (length Us)) Us],
berghofe@13641
   306
                   Ts @ [Us ---> Univ_elT])
berghofe@5177
   307
                else
skalberg@15574
   308
                  (i2 + 1, i2', ts @ [mk_lim
skalberg@15574
   309
                     (Const (List.nth (all_rep_names, j), U --> Univ_elT) $
skalberg@15574
   310
                        app_bnds (mk_Free "x" T' i2) (length Us)) Us], Ts)
berghofe@7015
   311
            | _ => (i2 + 1, i2', ts @ [Leaf $ mk_inj T' (mk_Free "x" T' i2)], Ts))
berghofe@5177
   312
          end;
berghofe@5177
   313
skalberg@15570
   314
        val (i2, i2', ts, Ts) = Library.foldl (process_arg ks) ((1, 1, [], []), cargs);
berghofe@5177
   315
        val xs = map (uncurry (mk_Free "x")) (argTs ~~ (1 upto (i2 - 1)));
berghofe@7015
   316
        val ys = map (uncurry (mk_Free "y")) (Ts ~~ (1 upto (i2' - 1)));
berghofe@5177
   317
        val f = list_abs_free (map dest_Free (xs @ ys), mk_univ_inj ts n i);
berghofe@5177
   318
skalberg@15570
   319
        val (_, _, ts', _) = Library.foldl (process_arg []) ((1, 1, [], []), cargs);
berghofe@5177
   320
        val eqn = HOLogic.mk_Trueprop (HOLogic.mk_eq
berghofe@5177
   321
          (rep_const $ list_comb (constr, xs), mk_univ_inj ts' n i))
berghofe@5177
   322
berghofe@5177
   323
      in (fs @ [f], eqns @ [eqn], i + 1) end;
berghofe@5177
   324
berghofe@5177
   325
    (* define isomorphisms for all mutually recursive datatypes in list ds *)
berghofe@5177
   326
berghofe@5177
   327
    fun make_iso_defs (ds, (thy, char_thms)) =
berghofe@5177
   328
      let
berghofe@5177
   329
        val ks = map fst ds;
berghofe@5177
   330
        val (_, (tname, _, _)) = hd ds;
wenzelm@17412
   331
        val {rec_rewrites, rec_names, ...} = the (Symtab.lookup dt_info tname);
berghofe@5177
   332
berghofe@5177
   333
        fun process_dt ((fs, eqns, isos), (k, (tname, _, constrs))) =
berghofe@5177
   334
          let
skalberg@15570
   335
            val (fs', eqns', _) = Library.foldl (make_iso_def k ks (length constrs))
berghofe@5177
   336
              ((fs, eqns, 1), constrs);
skalberg@15570
   337
            val iso = (List.nth (recTs, k), List.nth (all_rep_names, k))
berghofe@5177
   338
          in (fs', eqns', isos @ [iso]) end;
berghofe@5177
   339
        
skalberg@15570
   340
        val (fs, eqns, isos) = Library.foldl process_dt (([], [], []), ds);
berghofe@5177
   341
        val fTs = map fastype_of fs;
berghofe@5177
   342
        val defs = map (fn (rec_name, (T, iso_name)) => ((Sign.base_name iso_name) ^ "_def",
berghofe@5177
   343
          equals (T --> Univ_elT) $ Const (iso_name, T --> Univ_elT) $
berghofe@5177
   344
            list_comb (Const (rec_name, fTs @ [T] ---> Univ_elT), fs))) (rec_names ~~ isos);
haftmann@18358
   345
        val (def_thms, thy') = (PureThy.add_defs_i false o map Thm.no_attributes) defs thy;
berghofe@5177
   346
berghofe@5177
   347
        (* prove characteristic equations *)
berghofe@5177
   348
oheimb@5553
   349
        val rewrites = def_thms @ (map mk_meta_eq rec_rewrites);
berghofe@26531
   350
        val char_thms' = map (fn eqn => SkipProof.prove_global thy' [] [] eqn
wenzelm@20046
   351
          (fn _ => EVERY [rewrite_goals_tac rewrites, rtac refl 1])) eqns;
berghofe@5177
   352
berghofe@5177
   353
      in (thy', char_thms' @ char_thms) end;
berghofe@5177
   354
skalberg@15574
   355
    val (thy5, iso_char_thms) = foldr make_iso_defs
skalberg@15574
   356
      (add_path flat_names big_name thy4, []) (tl descr);
berghofe@5177
   357
berghofe@5177
   358
    (* prove isomorphism properties *)
berghofe@5177
   359
berghofe@7015
   360
    fun mk_funs_inv thm =
berghofe@7015
   361
      let
wenzelm@26626
   362
        val thy = Thm.theory_of_thm thm;
wenzelm@26626
   363
        val prop = Thm.prop_of thm;
berghofe@21021
   364
        val _ $ (_ $ ((S as Const (_, Type (_, [U, _]))) $ _ )) $
wenzelm@16287
   365
          (_ $ (_ $ (r $ (a $ _)) $ _)) = Type.freeze prop;
berghofe@13641
   366
        val used = add_term_tfree_names (a, []);
berghofe@13641
   367
berghofe@13641
   368
        fun mk_thm i =
berghofe@13641
   369
          let
berghofe@13641
   370
            val Ts = map (TFree o rpair HOLogic.typeS)
wenzelm@20071
   371
              (Name.variant_list used (replicate i "'t"));
berghofe@13641
   372
            val f = Free ("f", Ts ---> U)
berghofe@26531
   373
          in SkipProof.prove_global thy [] [] (Logic.mk_implies
berghofe@13641
   374
            (HOLogic.mk_Trueprop (HOLogic.list_all
berghofe@21021
   375
               (map (pair "x") Ts, S $ app_bnds f i)),
berghofe@13641
   376
             HOLogic.mk_Trueprop (HOLogic.mk_eq (list_abs (map (pair "x") Ts,
wenzelm@17985
   377
               r $ (a $ app_bnds f i)), f))))
berghofe@26806
   378
            (fn _ => EVERY [REPEAT_DETERM_N i (rtac ext 1),
berghofe@26806
   379
               REPEAT (etac allE 1), rtac thm 1, atac 1])
berghofe@13641
   380
          end
berghofe@13641
   381
      in map (fn r => r RS subst) (thm :: map mk_thm arities) end;
berghofe@7015
   382
berghofe@5177
   383
    (* prove  inj dt_Rep_i  and  dt_Rep_i x : dt_rep_set_i *)
berghofe@5177
   384
berghofe@26806
   385
    val fun_congs = map (fn T => make_elim (Drule.instantiate'
berghofe@26806
   386
      [SOME (ctyp_of thy5 T)] [] fun_cong)) branchTs;
berghofe@26806
   387
berghofe@5177
   388
    fun prove_iso_thms (ds, (inj_thms, elem_thms)) =
berghofe@5177
   389
      let
berghofe@5177
   390
        val (_, (tname, _, _)) = hd ds;
wenzelm@17412
   391
        val {induction, ...} = the (Symtab.lookup dt_info tname);
berghofe@5177
   392
berghofe@5177
   393
        fun mk_ind_concl (i, _) =
berghofe@5177
   394
          let
skalberg@15570
   395
            val T = List.nth (recTs, i);
skalberg@15570
   396
            val Rep_t = Const (List.nth (all_rep_names, i), T --> Univ_elT);
skalberg@15570
   397
            val rep_set_name = List.nth (rep_set_names, i)
berghofe@5177
   398
          in (HOLogic.all_const T $ Abs ("y", T, HOLogic.imp $
berghofe@5177
   399
                HOLogic.mk_eq (Rep_t $ mk_Free "x" T i, Rep_t $ Bound 0) $
berghofe@5177
   400
                  HOLogic.mk_eq (mk_Free "x" T i, Bound 0)),
berghofe@21021
   401
              Const (rep_set_name, UnivT') $ (Rep_t $ mk_Free "x" T i))
berghofe@5177
   402
          end;
berghofe@5177
   403
berghofe@5177
   404
        val (ind_concl1, ind_concl2) = ListPair.unzip (map mk_ind_concl ds);
berghofe@5177
   405
oheimb@5553
   406
        val rewrites = map mk_meta_eq iso_char_thms;
berghofe@21021
   407
        val inj_thms' = map snd newT_iso_inj_thms @
haftmann@26359
   408
          map (fn r => r RS @{thm injD}) inj_thms;
berghofe@5177
   409
berghofe@26531
   410
        val inj_thm = SkipProof.prove_global thy5 [] []
wenzelm@17985
   411
          (HOLogic.mk_Trueprop (mk_conj ind_concl1)) (fn _ => EVERY
berghofe@25678
   412
            [(indtac induction [] THEN_ALL_NEW ObjectLogic.atomize_prems_tac) 1,
berghofe@5177
   413
             REPEAT (EVERY
berghofe@5177
   414
               [rtac allI 1, rtac impI 1,
berghofe@5177
   415
                exh_tac (exh_thm_of dt_info) 1,
berghofe@5177
   416
                REPEAT (EVERY
berghofe@5177
   417
                  [hyp_subst_tac 1,
berghofe@5177
   418
                   rewrite_goals_tac rewrites,
berghofe@5177
   419
                   REPEAT (dresolve_tac [In0_inject, In1_inject] 1),
berghofe@5177
   420
                   (eresolve_tac [In0_not_In1 RS notE, In1_not_In0 RS notE] 1)
berghofe@5177
   421
                   ORELSE (EVERY
berghofe@13641
   422
                     [REPEAT (eresolve_tac (Scons_inject ::
berghofe@13641
   423
                        map make_elim [Leaf_inject, Inl_inject, Inr_inject]) 1),
berghofe@13641
   424
                      REPEAT (cong_tac 1), rtac refl 1,
berghofe@13641
   425
                      REPEAT (atac 1 ORELSE (EVERY
berghofe@13641
   426
                        [REPEAT (rtac ext 1),
berghofe@13641
   427
                         REPEAT (eresolve_tac (mp :: allE ::
berghofe@13641
   428
                           map make_elim (Suml_inject :: Sumr_inject ::
berghofe@26806
   429
                             Lim_inject :: inj_thms') @ fun_congs) 1),
wenzelm@20046
   430
                         atac 1]))])])])]);
berghofe@5177
   431
haftmann@26359
   432
        val inj_thms'' = map (fn r => r RS @{thm datatype_injI})
paulson@6171
   433
                             (split_conj_thm inj_thm);
berghofe@5177
   434
paulson@6171
   435
        val elem_thm = 
berghofe@26531
   436
            SkipProof.prove_global thy5 [] [] (HOLogic.mk_Trueprop (mk_conj ind_concl2))
wenzelm@20046
   437
              (fn _ =>
berghofe@25678
   438
               EVERY [(indtac induction [] THEN_ALL_NEW ObjectLogic.atomize_prems_tac) 1,
wenzelm@20046
   439
                rewrite_goals_tac rewrites,
wenzelm@20046
   440
                REPEAT ((resolve_tac rep_intrs THEN_ALL_NEW
wenzelm@20046
   441
                  ((REPEAT o etac allE) THEN' ares_tac elem_thms)) 1)]);
berghofe@5177
   442
berghofe@11471
   443
      in (inj_thms'' @ inj_thms, elem_thms @ (split_conj_thm elem_thm))
berghofe@11471
   444
      end;
berghofe@11471
   445
skalberg@15574
   446
    val (iso_inj_thms_unfolded, iso_elem_thms) = foldr prove_iso_thms
skalberg@15574
   447
      ([], map #3 newT_iso_axms) (tl descr);
berghofe@21021
   448
    val iso_inj_thms = map snd newT_iso_inj_thms @
haftmann@26359
   449
      map (fn r => r RS @{thm injD}) iso_inj_thms_unfolded;
berghofe@11471
   450
berghofe@21021
   451
    (* prove  dt_rep_set_i x --> x : range dt_Rep_i *)
berghofe@11471
   452
berghofe@11471
   453
    fun mk_iso_t (((set_name, iso_name), i), T) =
berghofe@11471
   454
      let val isoT = T --> Univ_elT
berghofe@11471
   455
      in HOLogic.imp $ 
berghofe@21021
   456
        (Const (set_name, UnivT') $ mk_Free "x" Univ_elT i) $
berghofe@11471
   457
          (if i < length newTs then Const ("True", HOLogic.boolT)
berghofe@11471
   458
           else HOLogic.mk_mem (mk_Free "x" Univ_elT i,
berghofe@11471
   459
             Const ("image", [isoT, HOLogic.mk_setT T] ---> UnivT) $
berghofe@11471
   460
               Const (iso_name, isoT) $ Const ("UNIV", HOLogic.mk_setT T)))
berghofe@5177
   461
      end;
berghofe@5177
   462
berghofe@11471
   463
    val iso_t = HOLogic.mk_Trueprop (mk_conj (map mk_iso_t
berghofe@11471
   464
      (rep_set_names ~~ all_rep_names ~~ (0 upto (length descr' - 1)) ~~ recTs)));
berghofe@11471
   465
berghofe@11471
   466
    (* all the theorems are proved by one single simultaneous induction *)
berghofe@11471
   467
haftmann@26359
   468
    val range_eqs = map (fn r => mk_meta_eq (r RS @{thm range_ex1_eq}))
berghofe@13641
   469
      iso_inj_thms_unfolded;
berghofe@13641
   470
berghofe@11471
   471
    val iso_thms = if length descr = 1 then [] else
skalberg@15570
   472
      Library.drop (length newTs, split_conj_thm
berghofe@26531
   473
        (SkipProof.prove_global thy5 [] [] iso_t (fn _ => EVERY
berghofe@25678
   474
           [(indtac rep_induct [] THEN_ALL_NEW ObjectLogic.atomize_prems_tac) 1,
berghofe@11471
   475
            REPEAT (rtac TrueI 1),
berghofe@13641
   476
            rewrite_goals_tac (mk_meta_eq choice_eq ::
haftmann@26359
   477
              symmetric (mk_meta_eq @{thm expand_fun_eq}) :: range_eqs),
berghofe@13641
   478
            rewrite_goals_tac (map symmetric range_eqs),
berghofe@11471
   479
            REPEAT (EVERY
berghofe@13641
   480
              [REPEAT (eresolve_tac ([rangeE, ex1_implies_ex RS exE] @
skalberg@15570
   481
                 List.concat (map (mk_funs_inv o #1) newT_iso_axms)) 1),
berghofe@11471
   482
               TRY (hyp_subst_tac 1),
berghofe@11471
   483
               rtac (sym RS range_eqI) 1,
wenzelm@20046
   484
               resolve_tac iso_char_thms 1])])));
wenzelm@11435
   485
wenzelm@11435
   486
    val Abs_inverse_thms' =
wenzelm@11435
   487
      map #1 newT_iso_axms @
haftmann@18330
   488
      map2 (fn r_inj => fn r => f_myinv_f OF [r_inj, r RS mp])
haftmann@18330
   489
        iso_inj_thms_unfolded iso_thms;
wenzelm@11435
   490
skalberg@15570
   491
    val Abs_inverse_thms = List.concat (map mk_funs_inv Abs_inverse_thms');
berghofe@5177
   492
berghofe@5177
   493
    (******************* freeness theorems for constructors *******************)
berghofe@5177
   494
wenzelm@6427
   495
    val _ = message "Proving freeness of constructors ...";
berghofe@5177
   496
berghofe@5177
   497
    (* prove theorem  Rep_i (Constr_j ...) = Inj_j ...  *)
berghofe@5177
   498
    
berghofe@5177
   499
    fun prove_constr_rep_thm eqn =
berghofe@5177
   500
      let
berghofe@21021
   501
        val inj_thms = map fst newT_iso_inj_thms;
haftmann@26359
   502
        val rewrites = @{thm o_def} :: constr_defs @ (map (mk_meta_eq o #2) newT_iso_axms)
berghofe@26531
   503
      in SkipProof.prove_global thy5 [] [] eqn (fn _ => EVERY
berghofe@5177
   504
        [resolve_tac inj_thms 1,
berghofe@5177
   505
         rewrite_goals_tac rewrites,
berghofe@21021
   506
         rtac refl 3,
berghofe@5177
   507
         resolve_tac rep_intrs 2,
wenzelm@20046
   508
         REPEAT (resolve_tac iso_elem_thms 1)])
berghofe@5177
   509
      end;
berghofe@5177
   510
berghofe@5177
   511
    (*--------------------------------------------------------------*)
berghofe@5177
   512
    (* constr_rep_thms and rep_congs are used to prove distinctness *)
berghofe@7015
   513
    (* of constructors.                                             *)
berghofe@5177
   514
    (*--------------------------------------------------------------*)
berghofe@5177
   515
berghofe@5177
   516
    val constr_rep_thms = map (map prove_constr_rep_thm) constr_rep_eqns;
berghofe@5177
   517
berghofe@5177
   518
    val dist_rewrites = map (fn (rep_thms, dist_lemma) =>
berghofe@5177
   519
      dist_lemma::(rep_thms @ [In0_eq, In1_eq, In0_not_In1, In1_not_In0]))
berghofe@5177
   520
        (constr_rep_thms ~~ dist_lemmas);
berghofe@5177
   521
haftmann@26969
   522
    fun prove_distinct_thms _ _ (_, []) = []
haftmann@26969
   523
      | prove_distinct_thms lim dist_rewrites' (k, ts as _ :: _) =
haftmann@26969
   524
          if k >= lim then [] else let
haftmann@26969
   525
            fun prove [] = []
haftmann@26969
   526
              | prove (t :: _ :: ts) =
haftmann@26969
   527
                  let
haftmann@26969
   528
                    val dist_thm = SkipProof.prove_global thy5 [] [] t (fn _ =>
haftmann@26969
   529
                      EVERY [simp_tac (HOL_ss addsimps dist_rewrites') 1])
haftmann@26969
   530
                  in dist_thm :: standard (dist_thm RS not_sym) :: prove ts end;
haftmann@26969
   531
          in prove ts end;
berghofe@7015
   532
haftmann@26969
   533
    val distincts = DatatypeProp.make_distincts descr sorts;
haftmann@26969
   534
    val distinct_thms = map2 (prove_distinct_thms
haftmann@26969
   535
      (Config.get_thy thy5 DatatypeProp.distinctness_limit))
haftmann@26969
   536
      dist_rewrites distincts;
berghofe@7015
   537
berghofe@7015
   538
    val simproc_dists = map (fn ((((_, (_, _, constrs)), rep_thms), congr), dists) =>
wenzelm@24112
   539
      if length constrs < Config.get_thy thy5 DatatypeProp.distinctness_limit
wenzelm@24098
   540
      then FewConstrs dists
berghofe@7015
   541
      else ManyConstrs (congr, HOL_basic_ss addsimps rep_thms)) (hd descr ~~
berghofe@7015
   542
        constr_rep_thms ~~ rep_congs ~~ distinct_thms);
berghofe@7015
   543
berghofe@5177
   544
    (* prove injectivity of constructors *)
berghofe@5177
   545
berghofe@5177
   546
    fun prove_constr_inj_thm rep_thms t =
berghofe@13641
   547
      let val inj_thms = Scons_inject :: (map make_elim
berghofe@21021
   548
        (iso_inj_thms @
berghofe@13641
   549
          [In0_inject, In1_inject, Leaf_inject, Inl_inject, Inr_inject,
berghofe@13641
   550
           Lim_inject, Suml_inject, Sumr_inject]))
berghofe@26531
   551
      in SkipProof.prove_global thy5 [] [] t (fn _ => EVERY
berghofe@5177
   552
        [rtac iffI 1,
berghofe@5177
   553
         REPEAT (etac conjE 2), hyp_subst_tac 2, rtac refl 2,
berghofe@5177
   554
         dresolve_tac rep_congs 1, dtac box_equals 1,
berghofe@13641
   555
         REPEAT (resolve_tac rep_thms 1),
berghofe@5177
   556
         REPEAT (eresolve_tac inj_thms 1),
berghofe@13641
   557
         REPEAT (ares_tac [conjI] 1 ORELSE (EVERY [REPEAT (rtac ext 1),
berghofe@13641
   558
           REPEAT (eresolve_tac (make_elim fun_cong :: inj_thms) 1),
wenzelm@20046
   559
           atac 1]))])
berghofe@5177
   560
      end;
berghofe@5177
   561
berghofe@5177
   562
    val constr_inject = map (fn (ts, thms) => map (prove_constr_inj_thm thms) ts)
berghofe@5177
   563
      ((DatatypeProp.make_injs descr sorts) ~~ constr_rep_thms);
berghofe@5177
   564
haftmann@18314
   565
    val ((constr_inject', distinct_thms'), thy6) =
haftmann@18314
   566
      thy5
haftmann@18314
   567
      |> parent_path flat_names
haftmann@18314
   568
      |> store_thmss "inject" new_type_names constr_inject
haftmann@18314
   569
      ||>> store_thmss "distinct" new_type_names distinct_thms;
berghofe@5177
   570
berghofe@5177
   571
    (*************************** induction theorem ****************************)
berghofe@5177
   572
wenzelm@6427
   573
    val _ = message "Proving induction rule for datatypes ...";
berghofe@5177
   574
berghofe@5177
   575
    val Rep_inverse_thms = (map (fn (_, iso, _) => iso RS subst) newT_iso_axms) @
berghofe@11471
   576
      (map (fn r => r RS myinv_f_f RS subst) iso_inj_thms_unfolded);
berghofe@11471
   577
    val Rep_inverse_thms' = map (fn r => r RS myinv_f_f) iso_inj_thms_unfolded;
berghofe@5177
   578
berghofe@5177
   579
    fun mk_indrule_lemma ((prems, concls), ((i, _), T)) =
berghofe@5177
   580
      let
skalberg@15570
   581
        val Rep_t = Const (List.nth (all_rep_names, i), T --> Univ_elT) $
berghofe@5177
   582
          mk_Free "x" T i;
berghofe@5177
   583
berghofe@5177
   584
        val Abs_t = if i < length newTs then
wenzelm@22578
   585
            Const (Sign.intern_const thy6
skalberg@15570
   586
              ("Abs_" ^ (List.nth (new_type_names, i))), Univ_elT --> T)
wenzelm@11435
   587
          else Const ("Inductive.myinv", [T --> Univ_elT, Univ_elT] ---> T) $
skalberg@15570
   588
            Const (List.nth (all_rep_names, i), T --> Univ_elT)
berghofe@5177
   589
berghofe@21021
   590
      in (prems @ [HOLogic.imp $
berghofe@21021
   591
            (Const (List.nth (rep_set_names, i), UnivT') $ Rep_t) $
berghofe@5177
   592
              (mk_Free "P" (T --> HOLogic.boolT) (i + 1) $ (Abs_t $ Rep_t))],
berghofe@5177
   593
          concls @ [mk_Free "P" (T --> HOLogic.boolT) (i + 1) $ mk_Free "x" T i])
berghofe@5177
   594
      end;
berghofe@5177
   595
berghofe@5177
   596
    val (indrule_lemma_prems, indrule_lemma_concls) =
skalberg@15570
   597
      Library.foldl mk_indrule_lemma (([], []), (descr' ~~ recTs));
berghofe@5177
   598
wenzelm@22578
   599
    val cert = cterm_of thy6;
berghofe@5177
   600
berghofe@26531
   601
    val indrule_lemma = SkipProof.prove_global thy6 [] []
berghofe@5177
   602
      (Logic.mk_implies
berghofe@5177
   603
        (HOLogic.mk_Trueprop (mk_conj indrule_lemma_prems),
wenzelm@17985
   604
         HOLogic.mk_Trueprop (mk_conj indrule_lemma_concls))) (fn _ => EVERY
wenzelm@17985
   605
           [REPEAT (etac conjE 1),
berghofe@5177
   606
            REPEAT (EVERY
berghofe@5177
   607
              [TRY (rtac conjI 1), resolve_tac Rep_inverse_thms 1,
wenzelm@20046
   608
               etac mp 1, resolve_tac iso_elem_thms 1])]);
berghofe@5177
   609
wenzelm@8305
   610
    val Ps = map head_of (HOLogic.dest_conj (HOLogic.dest_Trueprop (concl_of indrule_lemma)));
berghofe@5177
   611
    val frees = if length Ps = 1 then [Free ("P", snd (dest_Var (hd Ps)))] else
berghofe@5177
   612
      map (Free o apfst fst o dest_Var) Ps;
berghofe@5177
   613
    val indrule_lemma' = cterm_instantiate (map cert Ps ~~ map cert frees) indrule_lemma;
berghofe@5177
   614
wenzelm@17985
   615
    val dt_induct_prop = DatatypeProp.make_ind descr sorts;
berghofe@26531
   616
    val dt_induct = SkipProof.prove_global thy6 []
wenzelm@17985
   617
      (Logic.strip_imp_prems dt_induct_prop) (Logic.strip_imp_concl dt_induct_prop)
wenzelm@26711
   618
      (fn {prems, ...} => EVERY
berghofe@13641
   619
        [rtac indrule_lemma' 1,
berghofe@25678
   620
         (indtac rep_induct [] THEN_ALL_NEW ObjectLogic.atomize_prems_tac) 1,
berghofe@5177
   621
         EVERY (map (fn (prem, r) => (EVERY
berghofe@13641
   622
           [REPEAT (eresolve_tac Abs_inverse_thms 1),
berghofe@5177
   623
            simp_tac (HOL_basic_ss addsimps ((symmetric r)::Rep_inverse_thms')) 1,
berghofe@13641
   624
            DEPTH_SOLVE_1 (ares_tac [prem] 1 ORELSE etac allE 1)]))
wenzelm@20046
   625
                (prems ~~ (constr_defs @ (map mk_meta_eq iso_char_thms))))]);
berghofe@5177
   626
haftmann@18377
   627
    val ([dt_induct'], thy7) =
haftmann@18377
   628
      thy6
wenzelm@24712
   629
      |> Sign.add_path big_name
haftmann@18377
   630
      |> PureThy.add_thms [(("induct", dt_induct), [case_names_induct])]
wenzelm@24712
   631
      ||> Sign.parent_path;
berghofe@5177
   632
haftmann@18314
   633
  in
haftmann@18314
   634
    ((constr_inject', distinct_thms', dist_rewrites, simproc_dists, dt_induct'), thy7)
berghofe@5177
   635
  end;
berghofe@5177
   636
berghofe@5177
   637
end;