src/HOL/Library/Efficient_Nat.thy
author haftmann
Wed Sep 23 14:00:12 2009 +0200 (2009-09-23)
changeset 32657 5f13912245ff
parent 32348 36dbff4841ab
child 33343 2eb0b672ab40
permissions -rw-r--r--
Code_Eval(uation)
haftmann@23854
     1
(*  Title:      HOL/Library/Efficient_Nat.thy
haftmann@25931
     2
    Author:     Stefan Berghofer, Florian Haftmann, TU Muenchen
haftmann@23854
     3
*)
haftmann@23854
     4
haftmann@25931
     5
header {* Implementation of natural numbers by target-language integers *}
haftmann@23854
     6
haftmann@23854
     7
theory Efficient_Nat
haftmann@31203
     8
imports Code_Integer Main
haftmann@23854
     9
begin
haftmann@23854
    10
haftmann@23854
    11
text {*
haftmann@25931
    12
  When generating code for functions on natural numbers, the
haftmann@25931
    13
  canonical representation using @{term "0::nat"} and
haftmann@25931
    14
  @{term "Suc"} is unsuitable for computations involving large
haftmann@25931
    15
  numbers.  The efficiency of the generated code can be improved
haftmann@25931
    16
  drastically by implementing natural numbers by target-language
haftmann@25931
    17
  integers.  To do this, just include this theory.
haftmann@23854
    18
*}
haftmann@23854
    19
haftmann@25931
    20
subsection {* Basic arithmetic *}
haftmann@23854
    21
haftmann@23854
    22
text {*
haftmann@23854
    23
  Most standard arithmetic functions on natural numbers are implemented
haftmann@23854
    24
  using their counterparts on the integers:
haftmann@23854
    25
*}
haftmann@23854
    26
haftmann@25931
    27
code_datatype number_nat_inst.number_of_nat
haftmann@24715
    28
haftmann@32069
    29
lemma zero_nat_code [code, code_unfold_post]:
haftmann@25931
    30
  "0 = (Numeral0 :: nat)"
haftmann@25931
    31
  by simp
haftmann@24715
    32
haftmann@32069
    33
lemma one_nat_code [code, code_unfold_post]:
haftmann@25931
    34
  "1 = (Numeral1 :: nat)"
haftmann@25931
    35
  by simp
haftmann@24715
    36
haftmann@25931
    37
lemma Suc_code [code]:
haftmann@25931
    38
  "Suc n = n + 1"
haftmann@25931
    39
  by simp
haftmann@24715
    40
haftmann@25931
    41
lemma plus_nat_code [code]:
haftmann@25931
    42
  "n + m = nat (of_nat n + of_nat m)"
haftmann@25931
    43
  by simp
haftmann@24715
    44
haftmann@25931
    45
lemma minus_nat_code [code]:
haftmann@25931
    46
  "n - m = nat (of_nat n - of_nat m)"
haftmann@25931
    47
  by simp
haftmann@24715
    48
haftmann@25931
    49
lemma times_nat_code [code]:
haftmann@25931
    50
  "n * m = nat (of_nat n * of_nat m)"
haftmann@25931
    51
  unfolding of_nat_mult [symmetric] by simp
haftmann@24715
    52
haftmann@26009
    53
text {* Specialized @{term "op div \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat"} 
haftmann@26009
    54
  and @{term "op mod \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat"} operations. *}
haftmann@26009
    55
haftmann@28694
    56
definition divmod_aux ::  "nat \<Rightarrow> nat \<Rightarrow> nat \<times> nat" where
haftmann@29657
    57
  [code del]: "divmod_aux = Divides.divmod"
haftmann@24715
    58
haftmann@28522
    59
lemma [code]:
haftmann@29657
    60
  "Divides.divmod n m = (if m = 0 then (0, n) else divmod_aux n m)"
haftmann@26100
    61
  unfolding divmod_aux_def divmod_div_mod by simp
haftmann@26009
    62
haftmann@26100
    63
lemma divmod_aux_code [code]:
haftmann@26100
    64
  "divmod_aux n m = (nat (of_nat n div of_nat m), nat (of_nat n mod of_nat m))"
haftmann@26100
    65
  unfolding divmod_aux_def divmod_div_mod zdiv_int [symmetric] zmod_int [symmetric] by simp
haftmann@24715
    66
haftmann@25931
    67
lemma eq_nat_code [code]:
haftmann@28351
    68
  "eq_class.eq n m \<longleftrightarrow> eq_class.eq (of_nat n \<Colon> int) (of_nat m)"
haftmann@28351
    69
  by (simp add: eq)
haftmann@28351
    70
haftmann@28351
    71
lemma eq_nat_refl [code nbe]:
haftmann@28351
    72
  "eq_class.eq (n::nat) n \<longleftrightarrow> True"
haftmann@28351
    73
  by (rule HOL.eq_refl)
haftmann@24715
    74
haftmann@25931
    75
lemma less_eq_nat_code [code]:
haftmann@25931
    76
  "n \<le> m \<longleftrightarrow> (of_nat n \<Colon> int) \<le> of_nat m"
haftmann@25931
    77
  by simp
haftmann@23854
    78
haftmann@25931
    79
lemma less_nat_code [code]:
haftmann@25931
    80
  "n < m \<longleftrightarrow> (of_nat n \<Colon> int) < of_nat m"
haftmann@25931
    81
  by simp
haftmann@23854
    82
haftmann@25931
    83
subsection {* Case analysis *}
haftmann@23854
    84
haftmann@23854
    85
text {*
haftmann@25931
    86
  Case analysis on natural numbers is rephrased using a conditional
haftmann@25931
    87
  expression:
haftmann@23854
    88
*}
haftmann@23854
    89
haftmann@31998
    90
lemma [code, code_unfold]:
haftmann@25931
    91
  "nat_case = (\<lambda>f g n. if n = 0 then f else g (n - 1))"
haftmann@25931
    92
  by (auto simp add: expand_fun_eq dest!: gr0_implies_Suc)
haftmann@25615
    93
haftmann@23854
    94
haftmann@23854
    95
subsection {* Preprocessors *}
haftmann@23854
    96
haftmann@23854
    97
text {*
haftmann@23854
    98
  In contrast to @{term "Suc n"}, the term @{term "n + (1::nat)"} is no longer
haftmann@23854
    99
  a constructor term. Therefore, all occurrences of this term in a position
haftmann@23854
   100
  where a pattern is expected (i.e.\ on the left-hand side of a recursion
haftmann@23854
   101
  equation or in the arguments of an inductive relation in an introduction
haftmann@23854
   102
  rule) must be eliminated.
haftmann@23854
   103
  This can be accomplished by applying the following transformation rules:
haftmann@23854
   104
*}
haftmann@23854
   105
haftmann@29937
   106
lemma Suc_if_eq: "(\<And>n. f (Suc n) \<equiv> h n) \<Longrightarrow> f 0 \<equiv> g \<Longrightarrow>
haftmann@29937
   107
  f n \<equiv> if n = 0 then g else h (n - 1)"
haftmann@31954
   108
  by (rule eq_reflection) (cases n, simp_all)
haftmann@29937
   109
haftmann@25931
   110
lemma Suc_clause: "(\<And>n. P n (Suc n)) \<Longrightarrow> n \<noteq> 0 \<Longrightarrow> P (n - 1) n"
haftmann@29932
   111
  by (cases n) simp_all
haftmann@23854
   112
haftmann@23854
   113
text {*
haftmann@23854
   114
  The rules above are built into a preprocessor that is plugged into
haftmann@23854
   115
  the code generator. Since the preprocessor for introduction rules
haftmann@23854
   116
  does not know anything about modes, some of the modes that worked
haftmann@23854
   117
  for the canonical representation of natural numbers may no longer work.
haftmann@23854
   118
*}
haftmann@23854
   119
haftmann@23854
   120
(*<*)
haftmann@27609
   121
setup {*
haftmann@27609
   122
let
haftmann@23854
   123
haftmann@31954
   124
fun remove_suc thy thms =
haftmann@23854
   125
  let
haftmann@23854
   126
    val vname = Name.variant (map fst
haftmann@29937
   127
      (fold (Term.add_var_names o Thm.full_prop_of) thms [])) "n";
haftmann@23854
   128
    val cv = cterm_of thy (Var ((vname, 0), HOLogic.natT));
haftmann@23854
   129
    fun lhs_of th = snd (Thm.dest_comb
haftmann@31954
   130
      (fst (Thm.dest_comb (cprop_of th))));
haftmann@31954
   131
    fun rhs_of th = snd (Thm.dest_comb (cprop_of th));
haftmann@23854
   132
    fun find_vars ct = (case term_of ct of
haftmann@29932
   133
        (Const (@{const_name Suc}, _) $ Var _) => [(cv, snd (Thm.dest_comb ct))]
haftmann@23854
   134
      | _ $ _ =>
haftmann@23854
   135
        let val (ct1, ct2) = Thm.dest_comb ct
haftmann@23854
   136
        in 
haftmann@23854
   137
          map (apfst (fn ct => Thm.capply ct ct2)) (find_vars ct1) @
haftmann@23854
   138
          map (apfst (Thm.capply ct1)) (find_vars ct2)
haftmann@23854
   139
        end
haftmann@23854
   140
      | _ => []);
haftmann@23854
   141
    val eqs = maps
haftmann@23854
   142
      (fn th => map (pair th) (find_vars (lhs_of th))) thms;
haftmann@23854
   143
    fun mk_thms (th, (ct, cv')) =
haftmann@23854
   144
      let
haftmann@23854
   145
        val th' =
haftmann@23854
   146
          Thm.implies_elim
haftmann@23854
   147
           (Conv.fconv_rule (Thm.beta_conversion true)
haftmann@23854
   148
             (Drule.instantiate'
haftmann@23854
   149
               [SOME (ctyp_of_term ct)] [SOME (Thm.cabs cv ct),
haftmann@23854
   150
                 SOME (Thm.cabs cv' (rhs_of th)), NONE, SOME cv']
haftmann@31954
   151
               @{thm Suc_if_eq})) (Thm.forall_intr cv' th)
haftmann@23854
   152
      in
haftmann@23854
   153
        case map_filter (fn th'' =>
haftmann@23854
   154
            SOME (th'', singleton
haftmann@23854
   155
              (Variable.trade (K (fn [th'''] => [th''' RS th'])) (Variable.thm_context th'')) th'')
haftmann@23854
   156
          handle THM _ => NONE) thms of
haftmann@23854
   157
            [] => NONE
haftmann@23854
   158
          | thps =>
haftmann@23854
   159
              let val (ths1, ths2) = split_list thps
haftmann@23854
   160
              in SOME (subtract Thm.eq_thm (th :: ths1) thms @ ths2) end
haftmann@23854
   161
      end
haftmann@29937
   162
  in get_first mk_thms eqs end;
haftmann@29937
   163
haftmann@31954
   164
fun eqn_suc_preproc thy thms =
haftmann@29937
   165
  let
haftmann@31954
   166
    val dest = fst o Logic.dest_equals o prop_of;
haftmann@29937
   167
    val contains_suc = exists_Const (fn (c, _) => c = @{const_name Suc});
haftmann@29937
   168
  in
haftmann@29937
   169
    if forall (can dest) thms andalso exists (contains_suc o dest) thms
haftmann@32348
   170
      then thms |> perhaps_loop (remove_suc thy) |> (Option.map o map) Drule.zero_var_indexes
haftmann@29937
   171
       else NONE
haftmann@23854
   172
  end;
haftmann@23854
   173
haftmann@31954
   174
val eqn_suc_preproc1 = Code_Preproc.simple_functrans eqn_suc_preproc;
haftmann@29937
   175
haftmann@31954
   176
fun eqn_suc_preproc2 thy thms = eqn_suc_preproc thy thms
haftmann@29937
   177
  |> the_default thms;
haftmann@23854
   178
haftmann@23854
   179
fun remove_suc_clause thy thms =
haftmann@23854
   180
  let
haftmann@23854
   181
    val vname = Name.variant (map fst
wenzelm@29258
   182
      (fold (Term.add_var_names o Thm.full_prop_of) thms [])) "x";
haftmann@24222
   183
    fun find_var (t as Const (@{const_name Suc}, _) $ (v as Var _)) = SOME (t, v)
haftmann@23854
   184
      | find_var (t $ u) = (case find_var t of NONE => find_var u | x => x)
haftmann@23854
   185
      | find_var _ = NONE;
haftmann@23854
   186
    fun find_thm th =
haftmann@23854
   187
      let val th' = Conv.fconv_rule ObjectLogic.atomize th
haftmann@23854
   188
      in Option.map (pair (th, th')) (find_var (prop_of th')) end
haftmann@23854
   189
  in
haftmann@23854
   190
    case get_first find_thm thms of
haftmann@23854
   191
      NONE => thms
haftmann@23854
   192
    | SOME ((th, th'), (Sucv, v)) =>
haftmann@23854
   193
        let
haftmann@23854
   194
          val cert = cterm_of (Thm.theory_of_thm th);
haftmann@23854
   195
          val th'' = ObjectLogic.rulify (Thm.implies_elim
haftmann@23854
   196
            (Conv.fconv_rule (Thm.beta_conversion true)
haftmann@23854
   197
              (Drule.instantiate' []
haftmann@23854
   198
                [SOME (cert (lambda v (Abs ("x", HOLogic.natT,
haftmann@23854
   199
                   abstract_over (Sucv,
haftmann@23854
   200
                     HOLogic.dest_Trueprop (prop_of th')))))),
haftmann@24222
   201
                 SOME (cert v)] @{thm Suc_clause}))
haftmann@23854
   202
            (Thm.forall_intr (cert v) th'))
haftmann@23854
   203
        in
haftmann@23854
   204
          remove_suc_clause thy (map (fn th''' =>
haftmann@23854
   205
            if (op = o pairself prop_of) (th''', th) then th'' else th''') thms)
haftmann@23854
   206
        end
haftmann@23854
   207
  end;
haftmann@23854
   208
haftmann@23854
   209
fun clause_suc_preproc thy ths =
haftmann@23854
   210
  let
haftmann@23854
   211
    val dest = fst o HOLogic.dest_mem o HOLogic.dest_Trueprop
haftmann@23854
   212
  in
haftmann@23854
   213
    if forall (can (dest o concl_of)) ths andalso
wenzelm@29287
   214
      exists (fn th => exists (exists_Const (fn (c, _) => c = @{const_name Suc}))
wenzelm@29287
   215
        (map_filter (try dest) (concl_of th :: prems_of th))) ths
haftmann@23854
   216
    then remove_suc_clause thy ths else ths
haftmann@23854
   217
  end;
haftmann@27609
   218
in
haftmann@27609
   219
haftmann@31954
   220
  Codegen.add_preprocessor eqn_suc_preproc2
haftmann@23854
   221
  #> Codegen.add_preprocessor clause_suc_preproc
haftmann@31954
   222
  #> Code_Preproc.add_functrans ("eqn_Suc", eqn_suc_preproc1)
haftmann@27609
   223
haftmann@27609
   224
end;
haftmann@23854
   225
*}
haftmann@23854
   226
(*>*)
haftmann@23854
   227
haftmann@27609
   228
haftmann@25931
   229
subsection {* Target language setup *}
haftmann@25931
   230
haftmann@25931
   231
text {*
haftmann@25967
   232
  For ML, we map @{typ nat} to target language integers, where we
haftmann@25931
   233
  assert that values are always non-negative.
haftmann@25931
   234
*}
haftmann@25931
   235
haftmann@25931
   236
code_type nat
haftmann@27496
   237
  (SML "IntInf.int")
haftmann@25931
   238
  (OCaml "Big'_int.big'_int")
haftmann@25931
   239
haftmann@25931
   240
types_code
haftmann@25931
   241
  nat ("int")
haftmann@25931
   242
attach (term_of) {*
haftmann@25931
   243
val term_of_nat = HOLogic.mk_number HOLogic.natT;
haftmann@25931
   244
*}
haftmann@25931
   245
attach (test) {*
haftmann@25931
   246
fun gen_nat i =
haftmann@25931
   247
  let val n = random_range 0 i
haftmann@25931
   248
  in (n, fn () => term_of_nat n) end;
haftmann@25931
   249
*}
haftmann@25931
   250
haftmann@25931
   251
text {*
haftmann@25967
   252
  For Haskell we define our own @{typ nat} type.  The reason
haftmann@25967
   253
  is that we have to distinguish type class instances
haftmann@25967
   254
  for @{typ nat} and @{typ int}.
haftmann@25967
   255
*}
haftmann@25967
   256
haftmann@25967
   257
code_include Haskell "Nat" {*
haftmann@25967
   258
newtype Nat = Nat Integer deriving (Show, Eq);
haftmann@25967
   259
haftmann@25967
   260
instance Num Nat where {
haftmann@25967
   261
  fromInteger k = Nat (if k >= 0 then k else 0);
haftmann@25967
   262
  Nat n + Nat m = Nat (n + m);
haftmann@25967
   263
  Nat n - Nat m = fromInteger (n - m);
haftmann@25967
   264
  Nat n * Nat m = Nat (n * m);
haftmann@25967
   265
  abs n = n;
haftmann@25967
   266
  signum _ = 1;
haftmann@25967
   267
  negate n = error "negate Nat";
haftmann@25967
   268
};
haftmann@25967
   269
haftmann@25967
   270
instance Ord Nat where {
haftmann@25967
   271
  Nat n <= Nat m = n <= m;
haftmann@25967
   272
  Nat n < Nat m = n < m;
haftmann@25967
   273
};
haftmann@25967
   274
haftmann@25967
   275
instance Real Nat where {
haftmann@25967
   276
  toRational (Nat n) = toRational n;
haftmann@25967
   277
};
haftmann@25967
   278
haftmann@25967
   279
instance Enum Nat where {
haftmann@25967
   280
  toEnum k = fromInteger (toEnum k);
haftmann@25967
   281
  fromEnum (Nat n) = fromEnum n;
haftmann@25967
   282
};
haftmann@25967
   283
haftmann@25967
   284
instance Integral Nat where {
haftmann@25967
   285
  toInteger (Nat n) = n;
haftmann@25967
   286
  divMod n m = quotRem n m;
haftmann@25967
   287
  quotRem (Nat n) (Nat m) = (Nat k, Nat l) where (k, l) = quotRem n m;
haftmann@25967
   288
};
haftmann@25967
   289
*}
haftmann@25967
   290
haftmann@25967
   291
code_reserved Haskell Nat
haftmann@25967
   292
haftmann@25967
   293
code_type nat
haftmann@29793
   294
  (Haskell "Nat.Nat")
haftmann@25967
   295
haftmann@25967
   296
code_instance nat :: eq
haftmann@25967
   297
  (Haskell -)
haftmann@25967
   298
haftmann@25967
   299
text {*
haftmann@25931
   300
  Natural numerals.
haftmann@25931
   301
*}
haftmann@25931
   302
haftmann@32069
   303
lemma [code_unfold_post]:
haftmann@25931
   304
  "nat (number_of i) = number_nat_inst.number_of_nat i"
haftmann@25931
   305
  -- {* this interacts as desired with @{thm nat_number_of_def} *}
haftmann@25931
   306
  by (simp add: number_nat_inst.number_of_nat)
haftmann@25931
   307
haftmann@25931
   308
setup {*
haftmann@25931
   309
  fold (Numeral.add_code @{const_name number_nat_inst.number_of_nat}
haftmann@31295
   310
    false true) ["SML", "OCaml", "Haskell"]
haftmann@25931
   311
*}
haftmann@25931
   312
haftmann@25931
   313
text {*
haftmann@25931
   314
  Since natural numbers are implemented
haftmann@25967
   315
  using integers in ML, the coercion function @{const "of_nat"} of type
haftmann@25931
   316
  @{typ "nat \<Rightarrow> int"} is simply implemented by the identity function.
haftmann@25931
   317
  For the @{const "nat"} function for converting an integer to a natural
haftmann@25931
   318
  number, we give a specific implementation using an ML function that
haftmann@25931
   319
  returns its input value, provided that it is non-negative, and otherwise
haftmann@25931
   320
  returns @{text "0"}.
haftmann@25931
   321
*}
haftmann@25931
   322
haftmann@32073
   323
definition int :: "nat \<Rightarrow> int" where
haftmann@28562
   324
  [code del]: "int = of_nat"
haftmann@25931
   325
haftmann@28562
   326
lemma int_code' [code]:
haftmann@25931
   327
  "int (number_of l) = (if neg (number_of l \<Colon> int) then 0 else number_of l)"
haftmann@25931
   328
  unfolding int_nat_number_of [folded int_def] ..
haftmann@25931
   329
haftmann@28562
   330
lemma nat_code' [code]:
haftmann@25931
   331
  "nat (number_of l) = (if neg (number_of l \<Colon> int) then 0 else number_of l)"
huffman@28969
   332
  unfolding nat_number_of_def number_of_is_id neg_def by simp
haftmann@25931
   333
haftmann@32069
   334
lemma of_nat_int [code_unfold_post]:
haftmann@25931
   335
  "of_nat = int" by (simp add: int_def)
haftmann@25931
   336
haftmann@32073
   337
lemma of_nat_aux_int [code_unfold]:
haftmann@32073
   338
  "of_nat_aux (\<lambda>i. i + 1) k 0 = int k"
haftmann@32073
   339
  by (simp add: int_def Nat.of_nat_code)
haftmann@32073
   340
haftmann@25931
   341
code_const int
haftmann@25931
   342
  (SML "_")
haftmann@25931
   343
  (OCaml "_")
haftmann@25931
   344
haftmann@25931
   345
consts_code
haftmann@25931
   346
  int ("(_)")
haftmann@25931
   347
  nat ("\<module>nat")
haftmann@25931
   348
attach {*
haftmann@25931
   349
fun nat i = if i < 0 then 0 else i;
haftmann@25931
   350
*}
haftmann@25931
   351
haftmann@25967
   352
code_const nat
haftmann@25967
   353
  (SML "IntInf.max/ (/0,/ _)")
haftmann@25967
   354
  (OCaml "Big'_int.max'_big'_int/ Big'_int.zero'_big'_int")
haftmann@25967
   355
haftmann@25967
   356
text {* For Haskell, things are slightly different again. *}
haftmann@25967
   357
haftmann@25967
   358
code_const int and nat
haftmann@25967
   359
  (Haskell "toInteger" and "fromInteger")
haftmann@25931
   360
haftmann@25931
   361
text {* Conversion from and to indices. *}
haftmann@25931
   362
haftmann@31205
   363
code_const Code_Numeral.of_nat
haftmann@25967
   364
  (SML "IntInf.toInt")
haftmann@31377
   365
  (OCaml "_")
haftmann@27673
   366
  (Haskell "fromEnum")
haftmann@25967
   367
haftmann@31205
   368
code_const Code_Numeral.nat_of
haftmann@25931
   369
  (SML "IntInf.fromInt")
haftmann@31377
   370
  (OCaml "_")
haftmann@27673
   371
  (Haskell "toEnum")
haftmann@25931
   372
haftmann@25931
   373
text {* Using target language arithmetic operations whenever appropriate *}
haftmann@25931
   374
haftmann@25931
   375
code_const "op + \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat"
haftmann@25931
   376
  (SML "IntInf.+ ((_), (_))")
haftmann@25931
   377
  (OCaml "Big'_int.add'_big'_int")
haftmann@25931
   378
  (Haskell infixl 6 "+")
haftmann@25931
   379
haftmann@25931
   380
code_const "op * \<Colon> nat \<Rightarrow> nat \<Rightarrow> nat"
haftmann@25931
   381
  (SML "IntInf.* ((_), (_))")
haftmann@25931
   382
  (OCaml "Big'_int.mult'_big'_int")
haftmann@25931
   383
  (Haskell infixl 7 "*")
haftmann@25931
   384
haftmann@26100
   385
code_const divmod_aux
haftmann@26009
   386
  (SML "IntInf.divMod/ ((_),/ (_))")
haftmann@26009
   387
  (OCaml "Big'_int.quomod'_big'_int")
haftmann@26009
   388
  (Haskell "divMod")
haftmann@25931
   389
haftmann@28346
   390
code_const "eq_class.eq \<Colon> nat \<Rightarrow> nat \<Rightarrow> bool"
haftmann@25931
   391
  (SML "!((_ : IntInf.int) = _)")
haftmann@25931
   392
  (OCaml "Big'_int.eq'_big'_int")
haftmann@25931
   393
  (Haskell infixl 4 "==")
haftmann@25931
   394
haftmann@25931
   395
code_const "op \<le> \<Colon> nat \<Rightarrow> nat \<Rightarrow> bool"
haftmann@25931
   396
  (SML "IntInf.<= ((_), (_))")
haftmann@25931
   397
  (OCaml "Big'_int.le'_big'_int")
haftmann@25931
   398
  (Haskell infix 4 "<=")
haftmann@25931
   399
haftmann@25931
   400
code_const "op < \<Colon> nat \<Rightarrow> nat \<Rightarrow> bool"
haftmann@25931
   401
  (SML "IntInf.< ((_), (_))")
haftmann@25931
   402
  (OCaml "Big'_int.lt'_big'_int")
haftmann@25931
   403
  (Haskell infix 4 "<")
haftmann@25931
   404
haftmann@25931
   405
consts_code
haftmann@28522
   406
  "0::nat"                     ("0")
haftmann@28522
   407
  "1::nat"                     ("1")
haftmann@25931
   408
  Suc                          ("(_ +/ 1)")
haftmann@25931
   409
  "op + \<Colon>  nat \<Rightarrow> nat \<Rightarrow> nat"   ("(_ +/ _)")
haftmann@25931
   410
  "op * \<Colon>  nat \<Rightarrow> nat \<Rightarrow> nat"   ("(_ */ _)")
haftmann@25931
   411
  "op \<le> \<Colon>  nat \<Rightarrow> nat \<Rightarrow> bool"  ("(_ <=/ _)")
haftmann@25931
   412
  "op < \<Colon>  nat \<Rightarrow> nat \<Rightarrow> bool"  ("(_ </ _)")
haftmann@25931
   413
haftmann@25931
   414
haftmann@28228
   415
text {* Evaluation *}
haftmann@28228
   416
haftmann@28562
   417
lemma [code, code del]:
haftmann@32657
   418
  "(Code_Evaluation.term_of \<Colon> nat \<Rightarrow> term) = Code_Evaluation.term_of" ..
haftmann@28228
   419
haftmann@32657
   420
code_const "Code_Evaluation.term_of \<Colon> nat \<Rightarrow> term"
haftmann@28228
   421
  (SML "HOLogic.mk'_number/ HOLogic.natT")
haftmann@28228
   422
haftmann@28228
   423
haftmann@25931
   424
text {* Module names *}
haftmann@23854
   425
haftmann@23854
   426
code_modulename SML
haftmann@23854
   427
  Nat Integer
haftmann@23854
   428
  Divides Integer
haftmann@28683
   429
  Ring_and_Field Integer
haftmann@23854
   430
  Efficient_Nat Integer
haftmann@23854
   431
haftmann@23854
   432
code_modulename OCaml
haftmann@23854
   433
  Nat Integer
haftmann@23854
   434
  Divides Integer
haftmann@28683
   435
  Ring_and_Field Integer
haftmann@23854
   436
  Efficient_Nat Integer
haftmann@23854
   437
haftmann@23854
   438
code_modulename Haskell
haftmann@23854
   439
  Nat Integer
haftmann@24195
   440
  Divides Integer
haftmann@28683
   441
  Ring_and_Field Integer
haftmann@23854
   442
  Efficient_Nat Integer
haftmann@23854
   443
haftmann@25931
   444
hide const int
haftmann@23854
   445
haftmann@23854
   446
end