src/HOL/Code_Numeral.thy
 author haftmann Fri Mar 22 19:18:08 2019 +0000 (2 months ago) changeset 69946 494934c30f38 parent 69906 55534affe445 child 70009 435fb018e8ee permissions -rw-r--r--
improved code equations taken over from AFP
```     1 (*  Title:      HOL/Code_Numeral.thy
```
```     2     Author:     Florian Haftmann, TU Muenchen
```
```     3 *)
```
```     4
```
```     5 section \<open>Numeric types for code generation onto target language numerals only\<close>
```
```     6
```
```     7 theory Code_Numeral
```
```     8 imports Divides Lifting
```
```     9 begin
```
```    10
```
```    11 subsection \<open>Type of target language integers\<close>
```
```    12
```
```    13 typedef integer = "UNIV :: int set"
```
```    14   morphisms int_of_integer integer_of_int ..
```
```    15
```
```    16 setup_lifting type_definition_integer
```
```    17
```
```    18 lemma integer_eq_iff:
```
```    19   "k = l \<longleftrightarrow> int_of_integer k = int_of_integer l"
```
```    20   by transfer rule
```
```    21
```
```    22 lemma integer_eqI:
```
```    23   "int_of_integer k = int_of_integer l \<Longrightarrow> k = l"
```
```    24   using integer_eq_iff [of k l] by simp
```
```    25
```
```    26 lemma int_of_integer_integer_of_int [simp]:
```
```    27   "int_of_integer (integer_of_int k) = k"
```
```    28   by transfer rule
```
```    29
```
```    30 lemma integer_of_int_int_of_integer [simp]:
```
```    31   "integer_of_int (int_of_integer k) = k"
```
```    32   by transfer rule
```
```    33
```
```    34 instantiation integer :: ring_1
```
```    35 begin
```
```    36
```
```    37 lift_definition zero_integer :: integer
```
```    38   is "0 :: int"
```
```    39   .
```
```    40
```
```    41 declare zero_integer.rep_eq [simp]
```
```    42
```
```    43 lift_definition one_integer :: integer
```
```    44   is "1 :: int"
```
```    45   .
```
```    46
```
```    47 declare one_integer.rep_eq [simp]
```
```    48
```
```    49 lift_definition plus_integer :: "integer \<Rightarrow> integer \<Rightarrow> integer"
```
```    50   is "plus :: int \<Rightarrow> int \<Rightarrow> int"
```
```    51   .
```
```    52
```
```    53 declare plus_integer.rep_eq [simp]
```
```    54
```
```    55 lift_definition uminus_integer :: "integer \<Rightarrow> integer"
```
```    56   is "uminus :: int \<Rightarrow> int"
```
```    57   .
```
```    58
```
```    59 declare uminus_integer.rep_eq [simp]
```
```    60
```
```    61 lift_definition minus_integer :: "integer \<Rightarrow> integer \<Rightarrow> integer"
```
```    62   is "minus :: int \<Rightarrow> int \<Rightarrow> int"
```
```    63   .
```
```    64
```
```    65 declare minus_integer.rep_eq [simp]
```
```    66
```
```    67 lift_definition times_integer :: "integer \<Rightarrow> integer \<Rightarrow> integer"
```
```    68   is "times :: int \<Rightarrow> int \<Rightarrow> int"
```
```    69   .
```
```    70
```
```    71 declare times_integer.rep_eq [simp]
```
```    72
```
```    73 instance proof
```
```    74 qed (transfer, simp add: algebra_simps)+
```
```    75
```
```    76 end
```
```    77
```
```    78 instance integer :: Rings.dvd ..
```
```    79
```
```    80 lemma [transfer_rule]:
```
```    81   "rel_fun pcr_integer (rel_fun pcr_integer HOL.iff) Rings.dvd Rings.dvd"
```
```    82   unfolding dvd_def by transfer_prover
```
```    83
```
```    84 lemma [transfer_rule]:
```
```    85   "rel_fun (=) pcr_integer (of_bool :: bool \<Rightarrow> int) (of_bool :: bool \<Rightarrow> integer)"
```
```    86   by (unfold of_bool_def [abs_def]) transfer_prover
```
```    87
```
```    88 lemma [transfer_rule]:
```
```    89   "rel_fun (=) pcr_integer (of_nat :: nat \<Rightarrow> int) (of_nat :: nat \<Rightarrow> integer)"
```
```    90   by (rule transfer_rule_of_nat) transfer_prover+
```
```    91
```
```    92 lemma [transfer_rule]:
```
```    93   "rel_fun (=) pcr_integer (\<lambda>k :: int. k :: int) (of_int :: int \<Rightarrow> integer)"
```
```    94 proof -
```
```    95   have "rel_fun HOL.eq pcr_integer (of_int :: int \<Rightarrow> int) (of_int :: int \<Rightarrow> integer)"
```
```    96     by (rule transfer_rule_of_int) transfer_prover+
```
```    97   then show ?thesis by (simp add: id_def)
```
```    98 qed
```
```    99
```
```   100 lemma [transfer_rule]:
```
```   101   "rel_fun HOL.eq pcr_integer (numeral :: num \<Rightarrow> int) (numeral :: num \<Rightarrow> integer)"
```
```   102   by (rule transfer_rule_numeral) transfer_prover+
```
```   103
```
```   104 lemma [transfer_rule]:
```
```   105   "rel_fun HOL.eq (rel_fun HOL.eq pcr_integer) (Num.sub :: _ \<Rightarrow> _ \<Rightarrow> int) (Num.sub :: _ \<Rightarrow> _ \<Rightarrow> integer)"
```
```   106   by (unfold Num.sub_def [abs_def]) transfer_prover
```
```   107
```
```   108 lemma [transfer_rule]:
```
```   109   "rel_fun pcr_integer (rel_fun (=) pcr_integer) (power :: _ \<Rightarrow> _ \<Rightarrow> int) (power :: _ \<Rightarrow> _ \<Rightarrow> integer)"
```
```   110   by (unfold power_def [abs_def]) transfer_prover
```
```   111
```
```   112 lemma int_of_integer_of_nat [simp]:
```
```   113   "int_of_integer (of_nat n) = of_nat n"
```
```   114   by transfer rule
```
```   115
```
```   116 lift_definition integer_of_nat :: "nat \<Rightarrow> integer"
```
```   117   is "of_nat :: nat \<Rightarrow> int"
```
```   118   .
```
```   119
```
```   120 lemma integer_of_nat_eq_of_nat [code]:
```
```   121   "integer_of_nat = of_nat"
```
```   122   by transfer rule
```
```   123
```
```   124 lemma int_of_integer_integer_of_nat [simp]:
```
```   125   "int_of_integer (integer_of_nat n) = of_nat n"
```
```   126   by transfer rule
```
```   127
```
```   128 lift_definition nat_of_integer :: "integer \<Rightarrow> nat"
```
```   129   is Int.nat
```
```   130   .
```
```   131
```
```   132 lemma nat_of_integer_of_nat [simp]:
```
```   133   "nat_of_integer (of_nat n) = n"
```
```   134   by transfer simp
```
```   135
```
```   136 lemma int_of_integer_of_int [simp]:
```
```   137   "int_of_integer (of_int k) = k"
```
```   138   by transfer simp
```
```   139
```
```   140 lemma nat_of_integer_integer_of_nat [simp]:
```
```   141   "nat_of_integer (integer_of_nat n) = n"
```
```   142   by transfer simp
```
```   143
```
```   144 lemma integer_of_int_eq_of_int [simp, code_abbrev]:
```
```   145   "integer_of_int = of_int"
```
```   146   by transfer (simp add: fun_eq_iff)
```
```   147
```
```   148 lemma of_int_integer_of [simp]:
```
```   149   "of_int (int_of_integer k) = (k :: integer)"
```
```   150   by transfer rule
```
```   151
```
```   152 lemma int_of_integer_numeral [simp]:
```
```   153   "int_of_integer (numeral k) = numeral k"
```
```   154   by transfer rule
```
```   155
```
```   156 lemma int_of_integer_sub [simp]:
```
```   157   "int_of_integer (Num.sub k l) = Num.sub k l"
```
```   158   by transfer rule
```
```   159
```
```   160 definition integer_of_num :: "num \<Rightarrow> integer"
```
```   161   where [simp]: "integer_of_num = numeral"
```
```   162
```
```   163 lemma integer_of_num [code]:
```
```   164   "integer_of_num Num.One = 1"
```
```   165   "integer_of_num (Num.Bit0 n) = (let k = integer_of_num n in k + k)"
```
```   166   "integer_of_num (Num.Bit1 n) = (let k = integer_of_num n in k + k + 1)"
```
```   167   by (simp_all only: integer_of_num_def numeral.simps Let_def)
```
```   168
```
```   169 lemma integer_of_num_triv:
```
```   170   "integer_of_num Num.One = 1"
```
```   171   "integer_of_num (Num.Bit0 Num.One) = 2"
```
```   172   by simp_all
```
```   173
```
```   174 instantiation integer :: "{linordered_idom, equal}"
```
```   175 begin
```
```   176
```
```   177 lift_definition abs_integer :: "integer \<Rightarrow> integer"
```
```   178   is "abs :: int \<Rightarrow> int"
```
```   179   .
```
```   180
```
```   181 declare abs_integer.rep_eq [simp]
```
```   182
```
```   183 lift_definition sgn_integer :: "integer \<Rightarrow> integer"
```
```   184   is "sgn :: int \<Rightarrow> int"
```
```   185   .
```
```   186
```
```   187 declare sgn_integer.rep_eq [simp]
```
```   188
```
```   189 lift_definition less_eq_integer :: "integer \<Rightarrow> integer \<Rightarrow> bool"
```
```   190   is "less_eq :: int \<Rightarrow> int \<Rightarrow> bool"
```
```   191   .
```
```   192
```
```   193 lemma integer_less_eq_iff:
```
```   194   "k \<le> l \<longleftrightarrow> int_of_integer k \<le> int_of_integer l"
```
```   195   by (fact less_eq_integer.rep_eq)
```
```   196
```
```   197 lift_definition less_integer :: "integer \<Rightarrow> integer \<Rightarrow> bool"
```
```   198   is "less :: int \<Rightarrow> int \<Rightarrow> bool"
```
```   199   .
```
```   200
```
```   201 lemma integer_less_iff:
```
```   202   "k < l \<longleftrightarrow> int_of_integer k < int_of_integer l"
```
```   203   by (fact less_integer.rep_eq)
```
```   204
```
```   205 lift_definition equal_integer :: "integer \<Rightarrow> integer \<Rightarrow> bool"
```
```   206   is "HOL.equal :: int \<Rightarrow> int \<Rightarrow> bool"
```
```   207   .
```
```   208
```
```   209 instance
```
```   210   by standard (transfer, simp add: algebra_simps equal less_le_not_le [symmetric] mult_strict_right_mono linear)+
```
```   211
```
```   212 end
```
```   213
```
```   214 lemma [transfer_rule]:
```
```   215   "rel_fun pcr_integer (rel_fun pcr_integer pcr_integer) (min :: _ \<Rightarrow> _ \<Rightarrow> int) (min :: _ \<Rightarrow> _ \<Rightarrow> integer)"
```
```   216   by (unfold min_def [abs_def]) transfer_prover
```
```   217
```
```   218 lemma [transfer_rule]:
```
```   219   "rel_fun pcr_integer (rel_fun pcr_integer pcr_integer) (max :: _ \<Rightarrow> _ \<Rightarrow> int) (max :: _ \<Rightarrow> _ \<Rightarrow> integer)"
```
```   220   by (unfold max_def [abs_def]) transfer_prover
```
```   221
```
```   222 lemma int_of_integer_min [simp]:
```
```   223   "int_of_integer (min k l) = min (int_of_integer k) (int_of_integer l)"
```
```   224   by transfer rule
```
```   225
```
```   226 lemma int_of_integer_max [simp]:
```
```   227   "int_of_integer (max k l) = max (int_of_integer k) (int_of_integer l)"
```
```   228   by transfer rule
```
```   229
```
```   230 lemma nat_of_integer_non_positive [simp]:
```
```   231   "k \<le> 0 \<Longrightarrow> nat_of_integer k = 0"
```
```   232   by transfer simp
```
```   233
```
```   234 lemma of_nat_of_integer [simp]:
```
```   235   "of_nat (nat_of_integer k) = max 0 k"
```
```   236   by transfer auto
```
```   237
```
```   238 instantiation integer :: unique_euclidean_ring
```
```   239 begin
```
```   240
```
```   241 lift_definition divide_integer :: "integer \<Rightarrow> integer \<Rightarrow> integer"
```
```   242   is "divide :: int \<Rightarrow> int \<Rightarrow> int"
```
```   243   .
```
```   244
```
```   245 declare divide_integer.rep_eq [simp]
```
```   246
```
```   247 lift_definition modulo_integer :: "integer \<Rightarrow> integer \<Rightarrow> integer"
```
```   248   is "modulo :: int \<Rightarrow> int \<Rightarrow> int"
```
```   249   .
```
```   250
```
```   251 declare modulo_integer.rep_eq [simp]
```
```   252
```
```   253 lift_definition euclidean_size_integer :: "integer \<Rightarrow> nat"
```
```   254   is "euclidean_size :: int \<Rightarrow> nat"
```
```   255   .
```
```   256
```
```   257 declare euclidean_size_integer.rep_eq [simp]
```
```   258
```
```   259 lift_definition division_segment_integer :: "integer \<Rightarrow> integer"
```
```   260   is "division_segment :: int \<Rightarrow> int"
```
```   261   .
```
```   262
```
```   263 declare division_segment_integer.rep_eq [simp]
```
```   264
```
```   265 instance
```
```   266   by (standard; transfer)
```
```   267     (use mult_le_mono2 [of 1] in \<open>auto simp add: sgn_mult_abs abs_mult sgn_mult abs_mod_less sgn_mod nat_mult_distrib
```
```   268      division_segment_mult division_segment_mod intro: div_eqI\<close>)
```
```   269
```
```   270 end
```
```   271
```
```   272 lemma [code]:
```
```   273   "euclidean_size = nat_of_integer \<circ> abs"
```
```   274   by (simp add: fun_eq_iff nat_of_integer.rep_eq)
```
```   275
```
```   276 lemma [code]:
```
```   277   "division_segment (k :: integer) = (if k \<ge> 0 then 1 else - 1)"
```
```   278   by transfer (simp add: division_segment_int_def)
```
```   279
```
```   280 instance integer :: ring_parity
```
```   281   by (standard; transfer) (simp_all add: of_nat_div division_segment_int_def)
```
```   282
```
```   283 lemma [transfer_rule]:
```
```   284   "rel_fun (=) (rel_fun pcr_integer pcr_integer) (push_bit :: _ \<Rightarrow> _ \<Rightarrow> int) (push_bit :: _ \<Rightarrow> _ \<Rightarrow> integer)"
```
```   285   by (unfold push_bit_eq_mult [abs_def]) transfer_prover
```
```   286
```
```   287 lemma [transfer_rule]:
```
```   288   "rel_fun (=) (rel_fun pcr_integer pcr_integer) (take_bit :: _ \<Rightarrow> _ \<Rightarrow> int) (take_bit :: _ \<Rightarrow> _ \<Rightarrow> integer)"
```
```   289   by (unfold take_bit_eq_mod [abs_def]) transfer_prover
```
```   290
```
```   291 lemma [transfer_rule]:
```
```   292   "rel_fun (=) (rel_fun pcr_integer pcr_integer) (drop_bit :: _ \<Rightarrow> _ \<Rightarrow> int) (drop_bit :: _ \<Rightarrow> _ \<Rightarrow> integer)"
```
```   293   by (unfold drop_bit_eq_div [abs_def]) transfer_prover
```
```   294
```
```   295 instantiation integer :: unique_euclidean_semiring_numeral
```
```   296 begin
```
```   297
```
```   298 definition divmod_integer :: "num \<Rightarrow> num \<Rightarrow> integer \<times> integer"
```
```   299 where
```
```   300   divmod_integer'_def: "divmod_integer m n = (numeral m div numeral n, numeral m mod numeral n)"
```
```   301
```
```   302 definition divmod_step_integer :: "num \<Rightarrow> integer \<times> integer \<Rightarrow> integer \<times> integer"
```
```   303 where
```
```   304   "divmod_step_integer l qr = (let (q, r) = qr
```
```   305     in if r \<ge> numeral l then (2 * q + 1, r - numeral l)
```
```   306     else (2 * q, r))"
```
```   307
```
```   308 instance proof
```
```   309   show "divmod m n = (numeral m div numeral n :: integer, numeral m mod numeral n)"
```
```   310     for m n by (fact divmod_integer'_def)
```
```   311   show "divmod_step l qr = (let (q, r) = qr
```
```   312     in if r \<ge> numeral l then (2 * q + 1, r - numeral l)
```
```   313     else (2 * q, r))" for l and qr :: "integer \<times> integer"
```
```   314     by (fact divmod_step_integer_def)
```
```   315 qed (transfer,
```
```   316   fact le_add_diff_inverse2
```
```   317   unique_euclidean_semiring_numeral_class.div_less
```
```   318   unique_euclidean_semiring_numeral_class.mod_less
```
```   319   unique_euclidean_semiring_numeral_class.div_positive
```
```   320   unique_euclidean_semiring_numeral_class.mod_less_eq_dividend
```
```   321   unique_euclidean_semiring_numeral_class.pos_mod_bound
```
```   322   unique_euclidean_semiring_numeral_class.pos_mod_sign
```
```   323   unique_euclidean_semiring_numeral_class.mod_mult2_eq
```
```   324   unique_euclidean_semiring_numeral_class.div_mult2_eq
```
```   325   unique_euclidean_semiring_numeral_class.discrete)+
```
```   326
```
```   327 end
```
```   328
```
```   329 declare divmod_algorithm_code [where ?'a = integer,
```
```   330   folded integer_of_num_def, unfolded integer_of_num_triv,
```
```   331   code]
```
```   332
```
```   333 lemma integer_of_nat_0: "integer_of_nat 0 = 0"
```
```   334 by transfer simp
```
```   335
```
```   336 lemma integer_of_nat_1: "integer_of_nat 1 = 1"
```
```   337 by transfer simp
```
```   338
```
```   339 lemma integer_of_nat_numeral:
```
```   340   "integer_of_nat (numeral n) = numeral n"
```
```   341 by transfer simp
```
```   342
```
```   343
```
```   344 subsection \<open>Code theorems for target language integers\<close>
```
```   345
```
```   346 text \<open>Constructors\<close>
```
```   347
```
```   348 definition Pos :: "num \<Rightarrow> integer"
```
```   349 where
```
```   350   [simp, code_post]: "Pos = numeral"
```
```   351
```
```   352 lemma [transfer_rule]:
```
```   353   "rel_fun HOL.eq pcr_integer numeral Pos"
```
```   354   by simp transfer_prover
```
```   355
```
```   356 lemma Pos_fold [code_unfold]:
```
```   357   "numeral Num.One = Pos Num.One"
```
```   358   "numeral (Num.Bit0 k) = Pos (Num.Bit0 k)"
```
```   359   "numeral (Num.Bit1 k) = Pos (Num.Bit1 k)"
```
```   360   by simp_all
```
```   361
```
```   362 definition Neg :: "num \<Rightarrow> integer"
```
```   363 where
```
```   364   [simp, code_abbrev]: "Neg n = - Pos n"
```
```   365
```
```   366 lemma [transfer_rule]:
```
```   367   "rel_fun HOL.eq pcr_integer (\<lambda>n. - numeral n) Neg"
```
```   368   by (simp add: Neg_def [abs_def]) transfer_prover
```
```   369
```
```   370 code_datatype "0::integer" Pos Neg
```
```   371
```
```   372
```
```   373 text \<open>A further pair of constructors for generated computations\<close>
```
```   374
```
```   375 context
```
```   376 begin
```
```   377
```
```   378 qualified definition positive :: "num \<Rightarrow> integer"
```
```   379   where [simp]: "positive = numeral"
```
```   380
```
```   381 qualified definition negative :: "num \<Rightarrow> integer"
```
```   382   where [simp]: "negative = uminus \<circ> numeral"
```
```   383
```
```   384 lemma [code_computation_unfold]:
```
```   385   "numeral = positive"
```
```   386   "Pos = positive"
```
```   387   "Neg = negative"
```
```   388   by (simp_all add: fun_eq_iff)
```
```   389
```
```   390 end
```
```   391
```
```   392
```
```   393 text \<open>Auxiliary operations\<close>
```
```   394
```
```   395 lift_definition dup :: "integer \<Rightarrow> integer"
```
```   396   is "\<lambda>k::int. k + k"
```
```   397   .
```
```   398
```
```   399 lemma dup_code [code]:
```
```   400   "dup 0 = 0"
```
```   401   "dup (Pos n) = Pos (Num.Bit0 n)"
```
```   402   "dup (Neg n) = Neg (Num.Bit0 n)"
```
```   403   by (transfer, simp only: numeral_Bit0 minus_add_distrib)+
```
```   404
```
```   405 lift_definition sub :: "num \<Rightarrow> num \<Rightarrow> integer"
```
```   406   is "\<lambda>m n. numeral m - numeral n :: int"
```
```   407   .
```
```   408
```
```   409 lemma sub_code [code]:
```
```   410   "sub Num.One Num.One = 0"
```
```   411   "sub (Num.Bit0 m) Num.One = Pos (Num.BitM m)"
```
```   412   "sub (Num.Bit1 m) Num.One = Pos (Num.Bit0 m)"
```
```   413   "sub Num.One (Num.Bit0 n) = Neg (Num.BitM n)"
```
```   414   "sub Num.One (Num.Bit1 n) = Neg (Num.Bit0 n)"
```
```   415   "sub (Num.Bit0 m) (Num.Bit0 n) = dup (sub m n)"
```
```   416   "sub (Num.Bit1 m) (Num.Bit1 n) = dup (sub m n)"
```
```   417   "sub (Num.Bit1 m) (Num.Bit0 n) = dup (sub m n) + 1"
```
```   418   "sub (Num.Bit0 m) (Num.Bit1 n) = dup (sub m n) - 1"
```
```   419   by (transfer, simp add: dbl_def dbl_inc_def dbl_dec_def)+
```
```   420
```
```   421
```
```   422 text \<open>Implementations\<close>
```
```   423
```
```   424 lemma one_integer_code [code, code_unfold]:
```
```   425   "1 = Pos Num.One"
```
```   426   by simp
```
```   427
```
```   428 lemma plus_integer_code [code]:
```
```   429   "k + 0 = (k::integer)"
```
```   430   "0 + l = (l::integer)"
```
```   431   "Pos m + Pos n = Pos (m + n)"
```
```   432   "Pos m + Neg n = sub m n"
```
```   433   "Neg m + Pos n = sub n m"
```
```   434   "Neg m + Neg n = Neg (m + n)"
```
```   435   by (transfer, simp)+
```
```   436
```
```   437 lemma uminus_integer_code [code]:
```
```   438   "uminus 0 = (0::integer)"
```
```   439   "uminus (Pos m) = Neg m"
```
```   440   "uminus (Neg m) = Pos m"
```
```   441   by simp_all
```
```   442
```
```   443 lemma minus_integer_code [code]:
```
```   444   "k - 0 = (k::integer)"
```
```   445   "0 - l = uminus (l::integer)"
```
```   446   "Pos m - Pos n = sub m n"
```
```   447   "Pos m - Neg n = Pos (m + n)"
```
```   448   "Neg m - Pos n = Neg (m + n)"
```
```   449   "Neg m - Neg n = sub n m"
```
```   450   by (transfer, simp)+
```
```   451
```
```   452 lemma abs_integer_code [code]:
```
```   453   "\<bar>k\<bar> = (if (k::integer) < 0 then - k else k)"
```
```   454   by simp
```
```   455
```
```   456 lemma sgn_integer_code [code]:
```
```   457   "sgn k = (if k = 0 then 0 else if (k::integer) < 0 then - 1 else 1)"
```
```   458   by simp
```
```   459
```
```   460 lemma times_integer_code [code]:
```
```   461   "k * 0 = (0::integer)"
```
```   462   "0 * l = (0::integer)"
```
```   463   "Pos m * Pos n = Pos (m * n)"
```
```   464   "Pos m * Neg n = Neg (m * n)"
```
```   465   "Neg m * Pos n = Neg (m * n)"
```
```   466   "Neg m * Neg n = Pos (m * n)"
```
```   467   by simp_all
```
```   468
```
```   469 definition divmod_integer :: "integer \<Rightarrow> integer \<Rightarrow> integer \<times> integer"
```
```   470 where
```
```   471   "divmod_integer k l = (k div l, k mod l)"
```
```   472
```
```   473 lemma fst_divmod_integer [simp]:
```
```   474   "fst (divmod_integer k l) = k div l"
```
```   475   by (simp add: divmod_integer_def)
```
```   476
```
```   477 lemma snd_divmod_integer [simp]:
```
```   478   "snd (divmod_integer k l) = k mod l"
```
```   479   by (simp add: divmod_integer_def)
```
```   480
```
```   481 definition divmod_abs :: "integer \<Rightarrow> integer \<Rightarrow> integer \<times> integer"
```
```   482 where
```
```   483   "divmod_abs k l = (\<bar>k\<bar> div \<bar>l\<bar>, \<bar>k\<bar> mod \<bar>l\<bar>)"
```
```   484
```
```   485 lemma fst_divmod_abs [simp]:
```
```   486   "fst (divmod_abs k l) = \<bar>k\<bar> div \<bar>l\<bar>"
```
```   487   by (simp add: divmod_abs_def)
```
```   488
```
```   489 lemma snd_divmod_abs [simp]:
```
```   490   "snd (divmod_abs k l) = \<bar>k\<bar> mod \<bar>l\<bar>"
```
```   491   by (simp add: divmod_abs_def)
```
```   492
```
```   493 lemma divmod_abs_code [code]:
```
```   494   "divmod_abs (Pos k) (Pos l) = divmod k l"
```
```   495   "divmod_abs (Neg k) (Neg l) = divmod k l"
```
```   496   "divmod_abs (Neg k) (Pos l) = divmod k l"
```
```   497   "divmod_abs (Pos k) (Neg l) = divmod k l"
```
```   498   "divmod_abs j 0 = (0, \<bar>j\<bar>)"
```
```   499   "divmod_abs 0 j = (0, 0)"
```
```   500   by (simp_all add: prod_eq_iff)
```
```   501
```
```   502 lemma divmod_integer_eq_cases:
```
```   503   "divmod_integer k l =
```
```   504     (if k = 0 then (0, 0) else if l = 0 then (0, k) else
```
```   505     (apsnd \<circ> times \<circ> sgn) l (if sgn k = sgn l
```
```   506       then divmod_abs k l
```
```   507       else (let (r, s) = divmod_abs k l in
```
```   508         if s = 0 then (- r, 0) else (- r - 1, \<bar>l\<bar> - s))))"
```
```   509 proof -
```
```   510   have *: "sgn k = sgn l \<longleftrightarrow> k = 0 \<and> l = 0 \<or> 0 < l \<and> 0 < k \<or> l < 0 \<and> k < 0" for k l :: int
```
```   511     by (auto simp add: sgn_if)
```
```   512   have **: "- k = l * q \<longleftrightarrow> k = - (l * q)" for k l q :: int
```
```   513     by auto
```
```   514   show ?thesis
```
```   515     by (simp add: divmod_integer_def divmod_abs_def)
```
```   516       (transfer, auto simp add: * ** not_less zdiv_zminus1_eq_if zmod_zminus1_eq_if div_minus_right mod_minus_right)
```
```   517 qed
```
```   518
```
```   519 lemma divmod_integer_code [code]: \<^marker>\<open>contributor \<open>RenÃ© Thiemann\<close>\<close> \<^marker>\<open>contributor \<open>Akihisa Yamada\<close>\<close>
```
```   520   "divmod_integer k l =
```
```   521    (if k = 0 then (0, 0)
```
```   522     else if l > 0 then
```
```   523             (if k > 0 then Code_Numeral.divmod_abs k l
```
```   524              else case Code_Numeral.divmod_abs k l of (r, s) \<Rightarrow>
```
```   525                   if s = 0 then (- r, 0) else (- r - 1, l - s))
```
```   526     else if l = 0 then (0, k)
```
```   527     else apsnd uminus
```
```   528             (if k < 0 then Code_Numeral.divmod_abs k l
```
```   529              else case Code_Numeral.divmod_abs k l of (r, s) \<Rightarrow>
```
```   530                   if s = 0 then (- r, 0) else (- r - 1, - l - s)))"
```
```   531   by (cases l "0 :: integer" rule: linorder_cases)
```
```   532     (auto split: prod.splits simp add: divmod_integer_eq_cases)
```
```   533
```
```   534 lemma div_integer_code [code]:
```
```   535   "k div l = fst (divmod_integer k l)"
```
```   536   by simp
```
```   537
```
```   538 lemma mod_integer_code [code]:
```
```   539   "k mod l = snd (divmod_integer k l)"
```
```   540   by simp
```
```   541
```
```   542 definition bit_cut_integer :: "integer \<Rightarrow> integer \<times> bool"
```
```   543   where "bit_cut_integer k = (k div 2, odd k)"
```
```   544
```
```   545 lemma bit_cut_integer_code [code]:
```
```   546   "bit_cut_integer k = (if k = 0 then (0, False)
```
```   547      else let (r, s) = Code_Numeral.divmod_abs k 2
```
```   548        in (if k > 0 then r else - r - s, s = 1))"
```
```   549 proof -
```
```   550   have "bit_cut_integer k = (let (r, s) = divmod_integer k 2 in (r, s = 1))"
```
```   551     by (simp add: divmod_integer_def bit_cut_integer_def odd_iff_mod_2_eq_one)
```
```   552   then show ?thesis
```
```   553     by (simp add: divmod_integer_code) (auto simp add: split_def)
```
```   554 qed
```
```   555
```
```   556 lemma equal_integer_code [code]:
```
```   557   "HOL.equal 0 (0::integer) \<longleftrightarrow> True"
```
```   558   "HOL.equal 0 (Pos l) \<longleftrightarrow> False"
```
```   559   "HOL.equal 0 (Neg l) \<longleftrightarrow> False"
```
```   560   "HOL.equal (Pos k) 0 \<longleftrightarrow> False"
```
```   561   "HOL.equal (Pos k) (Pos l) \<longleftrightarrow> HOL.equal k l"
```
```   562   "HOL.equal (Pos k) (Neg l) \<longleftrightarrow> False"
```
```   563   "HOL.equal (Neg k) 0 \<longleftrightarrow> False"
```
```   564   "HOL.equal (Neg k) (Pos l) \<longleftrightarrow> False"
```
```   565   "HOL.equal (Neg k) (Neg l) \<longleftrightarrow> HOL.equal k l"
```
```   566   by (simp_all add: equal)
```
```   567
```
```   568 lemma equal_integer_refl [code nbe]:
```
```   569   "HOL.equal (k::integer) k \<longleftrightarrow> True"
```
```   570   by (fact equal_refl)
```
```   571
```
```   572 lemma less_eq_integer_code [code]:
```
```   573   "0 \<le> (0::integer) \<longleftrightarrow> True"
```
```   574   "0 \<le> Pos l \<longleftrightarrow> True"
```
```   575   "0 \<le> Neg l \<longleftrightarrow> False"
```
```   576   "Pos k \<le> 0 \<longleftrightarrow> False"
```
```   577   "Pos k \<le> Pos l \<longleftrightarrow> k \<le> l"
```
```   578   "Pos k \<le> Neg l \<longleftrightarrow> False"
```
```   579   "Neg k \<le> 0 \<longleftrightarrow> True"
```
```   580   "Neg k \<le> Pos l \<longleftrightarrow> True"
```
```   581   "Neg k \<le> Neg l \<longleftrightarrow> l \<le> k"
```
```   582   by simp_all
```
```   583
```
```   584 lemma less_integer_code [code]:
```
```   585   "0 < (0::integer) \<longleftrightarrow> False"
```
```   586   "0 < Pos l \<longleftrightarrow> True"
```
```   587   "0 < Neg l \<longleftrightarrow> False"
```
```   588   "Pos k < 0 \<longleftrightarrow> False"
```
```   589   "Pos k < Pos l \<longleftrightarrow> k < l"
```
```   590   "Pos k < Neg l \<longleftrightarrow> False"
```
```   591   "Neg k < 0 \<longleftrightarrow> True"
```
```   592   "Neg k < Pos l \<longleftrightarrow> True"
```
```   593   "Neg k < Neg l \<longleftrightarrow> l < k"
```
```   594   by simp_all
```
```   595
```
```   596 lift_definition num_of_integer :: "integer \<Rightarrow> num"
```
```   597   is "num_of_nat \<circ> nat"
```
```   598   .
```
```   599
```
```   600 lemma num_of_integer_code [code]:
```
```   601   "num_of_integer k = (if k \<le> 1 then Num.One
```
```   602      else let
```
```   603        (l, j) = divmod_integer k 2;
```
```   604        l' = num_of_integer l;
```
```   605        l'' = l' + l'
```
```   606      in if j = 0 then l'' else l'' + Num.One)"
```
```   607 proof -
```
```   608   {
```
```   609     assume "int_of_integer k mod 2 = 1"
```
```   610     then have "nat (int_of_integer k mod 2) = nat 1" by simp
```
```   611     moreover assume *: "1 < int_of_integer k"
```
```   612     ultimately have **: "nat (int_of_integer k) mod 2 = 1" by (simp add: nat_mod_distrib)
```
```   613     have "num_of_nat (nat (int_of_integer k)) =
```
```   614       num_of_nat (2 * (nat (int_of_integer k) div 2) + nat (int_of_integer k) mod 2)"
```
```   615       by simp
```
```   616     then have "num_of_nat (nat (int_of_integer k)) =
```
```   617       num_of_nat (nat (int_of_integer k) div 2 + nat (int_of_integer k) div 2 + nat (int_of_integer k) mod 2)"
```
```   618       by (simp add: mult_2)
```
```   619     with ** have "num_of_nat (nat (int_of_integer k)) =
```
```   620       num_of_nat (nat (int_of_integer k) div 2 + nat (int_of_integer k) div 2 + 1)"
```
```   621       by simp
```
```   622   }
```
```   623   note aux = this
```
```   624   show ?thesis
```
```   625     by (auto simp add: num_of_integer_def nat_of_integer_def Let_def case_prod_beta
```
```   626       not_le integer_eq_iff less_eq_integer_def
```
```   627       nat_mult_distrib nat_div_distrib num_of_nat_One num_of_nat_plus_distrib
```
```   628        mult_2 [where 'a=nat] aux add_One)
```
```   629 qed
```
```   630
```
```   631 lemma nat_of_integer_code [code]:
```
```   632   "nat_of_integer k = (if k \<le> 0 then 0
```
```   633      else let
```
```   634        (l, j) = divmod_integer k 2;
```
```   635        l' = nat_of_integer l;
```
```   636        l'' = l' + l'
```
```   637      in if j = 0 then l'' else l'' + 1)"
```
```   638 proof -
```
```   639   obtain j where k: "k = integer_of_int j"
```
```   640   proof
```
```   641     show "k = integer_of_int (int_of_integer k)" by simp
```
```   642   qed
```
```   643   have *: "nat j mod 2 = nat_of_integer (of_int j mod 2)" if "j \<ge> 0"
```
```   644     using that by transfer (simp add: nat_mod_distrib)
```
```   645   from k show ?thesis
```
```   646     by (auto simp add: split_def Let_def nat_of_integer_def nat_div_distrib mult_2 [symmetric]
```
```   647       minus_mod_eq_mult_div [symmetric] *)
```
```   648 qed
```
```   649
```
```   650 lemma int_of_integer_code [code]:
```
```   651   "int_of_integer k = (if k < 0 then - (int_of_integer (- k))
```
```   652      else if k = 0 then 0
```
```   653      else let
```
```   654        (l, j) = divmod_integer k 2;
```
```   655        l' = 2 * int_of_integer l
```
```   656      in if j = 0 then l' else l' + 1)"
```
```   657   by (auto simp add: split_def Let_def integer_eq_iff minus_mod_eq_mult_div [symmetric])
```
```   658
```
```   659 lemma integer_of_int_code [code]:
```
```   660   "integer_of_int k = (if k < 0 then - (integer_of_int (- k))
```
```   661      else if k = 0 then 0
```
```   662      else let
```
```   663        l = 2 * integer_of_int (k div 2);
```
```   664        j = k mod 2
```
```   665      in if j = 0 then l else l + 1)"
```
```   666   by (auto simp add: split_def Let_def integer_eq_iff minus_mod_eq_mult_div [symmetric])
```
```   667
```
```   668 hide_const (open) Pos Neg sub dup divmod_abs
```
```   669
```
```   670
```
```   671 subsection \<open>Serializer setup for target language integers\<close>
```
```   672
```
```   673 code_reserved Eval int Integer abs
```
```   674
```
```   675 code_printing
```
```   676   type_constructor integer \<rightharpoonup>
```
```   677     (SML) "IntInf.int"
```
```   678     and (OCaml) "Z.t"
```
```   679     and (Haskell) "Integer"
```
```   680     and (Scala) "BigInt"
```
```   681     and (Eval) "int"
```
```   682 | class_instance integer :: equal \<rightharpoonup>
```
```   683     (Haskell) -
```
```   684
```
```   685 code_printing
```
```   686   constant "0::integer" \<rightharpoonup>
```
```   687     (SML) "!(0/ :/ IntInf.int)"
```
```   688     and (OCaml) "Z.zero"
```
```   689     and (Haskell) "!(0/ ::/ Integer)"
```
```   690     and (Scala) "BigInt(0)"
```
```   691
```
```   692 setup \<open>
```
```   693   fold (fn target =>
```
```   694     Numeral.add_code \<^const_name>\<open>Code_Numeral.Pos\<close> I Code_Printer.literal_numeral target
```
```   695     #> Numeral.add_code \<^const_name>\<open>Code_Numeral.Neg\<close> (~) Code_Printer.literal_numeral target)
```
```   696     ["SML", "OCaml", "Haskell", "Scala"]
```
```   697 \<close>
```
```   698
```
```   699 code_printing
```
```   700   constant "plus :: integer \<Rightarrow> _ \<Rightarrow> _" \<rightharpoonup>
```
```   701     (SML) "IntInf.+ ((_), (_))"
```
```   702     and (OCaml) "Z.add"
```
```   703     and (Haskell) infixl 6 "+"
```
```   704     and (Scala) infixl 7 "+"
```
```   705     and (Eval) infixl 8 "+"
```
```   706 | constant "uminus :: integer \<Rightarrow> _" \<rightharpoonup>
```
```   707     (SML) "IntInf.~"
```
```   708     and (OCaml) "Z.neg"
```
```   709     and (Haskell) "negate"
```
```   710     and (Scala) "!(- _)"
```
```   711     and (Eval) "~/ _"
```
```   712 | constant "minus :: integer \<Rightarrow> _" \<rightharpoonup>
```
```   713     (SML) "IntInf.- ((_), (_))"
```
```   714     and (OCaml) "Z.sub"
```
```   715     and (Haskell) infixl 6 "-"
```
```   716     and (Scala) infixl 7 "-"
```
```   717     and (Eval) infixl 8 "-"
```
```   718 | constant Code_Numeral.dup \<rightharpoonup>
```
```   719     (SML) "IntInf.*/ (2,/ (_))"
```
```   720     and (OCaml) "Z.shift'_left/ _/ 1"
```
```   721     and (Haskell) "!(2 * _)"
```
```   722     and (Scala) "!(2 * _)"
```
```   723     and (Eval) "!(2 * _)"
```
```   724 | constant Code_Numeral.sub \<rightharpoonup>
```
```   725     (SML) "!(raise/ Fail/ \"sub\")"
```
```   726     and (OCaml) "failwith/ \"sub\""
```
```   727     and (Haskell) "error/ \"sub\""
```
```   728     and (Scala) "!sys.error(\"sub\")"
```
```   729 | constant "times :: integer \<Rightarrow> _ \<Rightarrow> _" \<rightharpoonup>
```
```   730     (SML) "IntInf.* ((_), (_))"
```
```   731     and (OCaml) "Z.mul"
```
```   732     and (Haskell) infixl 7 "*"
```
```   733     and (Scala) infixl 8 "*"
```
```   734     and (Eval) infixl 9 "*"
```
```   735 | constant Code_Numeral.divmod_abs \<rightharpoonup>
```
```   736     (SML) "IntInf.divMod/ (IntInf.abs _,/ IntInf.abs _)"
```
```   737     and (OCaml) "!(fun k l ->/ if Z.equal Z.zero l then/ (Z.zero, l) else/ Z.div'_rem/ (Z.abs k)/ (Z.abs l))"
```
```   738     and (Haskell) "divMod/ (abs _)/ (abs _)"
```
```   739     and (Scala) "!((k: BigInt) => (l: BigInt) =>/ if (l == 0)/ (BigInt(0), k) else/ (k.abs '/% l.abs))"
```
```   740     and (Eval) "Integer.div'_mod/ (abs _)/ (abs _)"
```
```   741 | constant "HOL.equal :: integer \<Rightarrow> _ \<Rightarrow> bool" \<rightharpoonup>
```
```   742     (SML) "!((_ : IntInf.int) = _)"
```
```   743     and (OCaml) "Z.equal"
```
```   744     and (Haskell) infix 4 "=="
```
```   745     and (Scala) infixl 5 "=="
```
```   746     and (Eval) infixl 6 "="
```
```   747 | constant "less_eq :: integer \<Rightarrow> _ \<Rightarrow> bool" \<rightharpoonup>
```
```   748     (SML) "IntInf.<= ((_), (_))"
```
```   749     and (OCaml) "Z.leq"
```
```   750     and (Haskell) infix 4 "<="
```
```   751     and (Scala) infixl 4 "<="
```
```   752     and (Eval) infixl 6 "<="
```
```   753 | constant "less :: integer \<Rightarrow> _ \<Rightarrow> bool" \<rightharpoonup>
```
```   754     (SML) "IntInf.< ((_), (_))"
```
```   755     and (OCaml) "Z.lt"
```
```   756     and (Haskell) infix 4 "<"
```
```   757     and (Scala) infixl 4 "<"
```
```   758     and (Eval) infixl 6 "<"
```
```   759 | constant "abs :: integer \<Rightarrow> _" \<rightharpoonup>
```
```   760     (SML) "IntInf.abs"
```
```   761     and (OCaml) "Z.abs"
```
```   762     and (Haskell) "Prelude.abs"
```
```   763     and (Scala) "_.abs"
```
```   764     and (Eval) "abs"
```
```   765
```
```   766 code_identifier
```
```   767   code_module Code_Numeral \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
```
```   768
```
```   769 export_code divmod_integer in Haskell
```
```   770
```
```   771
```
```   772 subsection \<open>Type of target language naturals\<close>
```
```   773
```
```   774 typedef natural = "UNIV :: nat set"
```
```   775   morphisms nat_of_natural natural_of_nat ..
```
```   776
```
```   777 setup_lifting type_definition_natural
```
```   778
```
```   779 lemma natural_eq_iff [termination_simp]:
```
```   780   "m = n \<longleftrightarrow> nat_of_natural m = nat_of_natural n"
```
```   781   by transfer rule
```
```   782
```
```   783 lemma natural_eqI:
```
```   784   "nat_of_natural m = nat_of_natural n \<Longrightarrow> m = n"
```
```   785   using natural_eq_iff [of m n] by simp
```
```   786
```
```   787 lemma nat_of_natural_of_nat_inverse [simp]:
```
```   788   "nat_of_natural (natural_of_nat n) = n"
```
```   789   by transfer rule
```
```   790
```
```   791 lemma natural_of_nat_of_natural_inverse [simp]:
```
```   792   "natural_of_nat (nat_of_natural n) = n"
```
```   793   by transfer rule
```
```   794
```
```   795 instantiation natural :: "{comm_monoid_diff, semiring_1}"
```
```   796 begin
```
```   797
```
```   798 lift_definition zero_natural :: natural
```
```   799   is "0 :: nat"
```
```   800   .
```
```   801
```
```   802 declare zero_natural.rep_eq [simp]
```
```   803
```
```   804 lift_definition one_natural :: natural
```
```   805   is "1 :: nat"
```
```   806   .
```
```   807
```
```   808 declare one_natural.rep_eq [simp]
```
```   809
```
```   810 lift_definition plus_natural :: "natural \<Rightarrow> natural \<Rightarrow> natural"
```
```   811   is "plus :: nat \<Rightarrow> nat \<Rightarrow> nat"
```
```   812   .
```
```   813
```
```   814 declare plus_natural.rep_eq [simp]
```
```   815
```
```   816 lift_definition minus_natural :: "natural \<Rightarrow> natural \<Rightarrow> natural"
```
```   817   is "minus :: nat \<Rightarrow> nat \<Rightarrow> nat"
```
```   818   .
```
```   819
```
```   820 declare minus_natural.rep_eq [simp]
```
```   821
```
```   822 lift_definition times_natural :: "natural \<Rightarrow> natural \<Rightarrow> natural"
```
```   823   is "times :: nat \<Rightarrow> nat \<Rightarrow> nat"
```
```   824   .
```
```   825
```
```   826 declare times_natural.rep_eq [simp]
```
```   827
```
```   828 instance proof
```
```   829 qed (transfer, simp add: algebra_simps)+
```
```   830
```
```   831 end
```
```   832
```
```   833 instance natural :: Rings.dvd ..
```
```   834
```
```   835 lemma [transfer_rule]:
```
```   836   "rel_fun pcr_natural (rel_fun pcr_natural HOL.iff) Rings.dvd Rings.dvd"
```
```   837   unfolding dvd_def by transfer_prover
```
```   838
```
```   839 lemma [transfer_rule]:
```
```   840   "rel_fun (=) pcr_natural (of_bool :: bool \<Rightarrow> nat) (of_bool :: bool \<Rightarrow> natural)"
```
```   841   by (unfold of_bool_def [abs_def]) transfer_prover
```
```   842
```
```   843 lemma [transfer_rule]:
```
```   844   "rel_fun HOL.eq pcr_natural (\<lambda>n::nat. n) (of_nat :: nat \<Rightarrow> natural)"
```
```   845 proof -
```
```   846   have "rel_fun HOL.eq pcr_natural (of_nat :: nat \<Rightarrow> nat) (of_nat :: nat \<Rightarrow> natural)"
```
```   847     by (unfold of_nat_def [abs_def]) transfer_prover
```
```   848   then show ?thesis by (simp add: id_def)
```
```   849 qed
```
```   850
```
```   851 lemma [transfer_rule]:
```
```   852   "rel_fun HOL.eq pcr_natural (numeral :: num \<Rightarrow> nat) (numeral :: num \<Rightarrow> natural)"
```
```   853 proof -
```
```   854   have "rel_fun HOL.eq pcr_natural (numeral :: num \<Rightarrow> nat) (\<lambda>n. of_nat (numeral n))"
```
```   855     by transfer_prover
```
```   856   then show ?thesis by simp
```
```   857 qed
```
```   858
```
```   859 lemma [transfer_rule]:
```
```   860   "rel_fun pcr_natural (rel_fun (=) pcr_natural) (power :: _ \<Rightarrow> _ \<Rightarrow> nat) (power :: _ \<Rightarrow> _ \<Rightarrow> natural)"
```
```   861   by (unfold power_def [abs_def]) transfer_prover
```
```   862
```
```   863 lemma nat_of_natural_of_nat [simp]:
```
```   864   "nat_of_natural (of_nat n) = n"
```
```   865   by transfer rule
```
```   866
```
```   867 lemma natural_of_nat_of_nat [simp, code_abbrev]:
```
```   868   "natural_of_nat = of_nat"
```
```   869   by transfer rule
```
```   870
```
```   871 lemma of_nat_of_natural [simp]:
```
```   872   "of_nat (nat_of_natural n) = n"
```
```   873   by transfer rule
```
```   874
```
```   875 lemma nat_of_natural_numeral [simp]:
```
```   876   "nat_of_natural (numeral k) = numeral k"
```
```   877   by transfer rule
```
```   878
```
```   879 instantiation natural :: "{linordered_semiring, equal}"
```
```   880 begin
```
```   881
```
```   882 lift_definition less_eq_natural :: "natural \<Rightarrow> natural \<Rightarrow> bool"
```
```   883   is "less_eq :: nat \<Rightarrow> nat \<Rightarrow> bool"
```
```   884   .
```
```   885
```
```   886 declare less_eq_natural.rep_eq [termination_simp]
```
```   887
```
```   888 lift_definition less_natural :: "natural \<Rightarrow> natural \<Rightarrow> bool"
```
```   889   is "less :: nat \<Rightarrow> nat \<Rightarrow> bool"
```
```   890   .
```
```   891
```
```   892 declare less_natural.rep_eq [termination_simp]
```
```   893
```
```   894 lift_definition equal_natural :: "natural \<Rightarrow> natural \<Rightarrow> bool"
```
```   895   is "HOL.equal :: nat \<Rightarrow> nat \<Rightarrow> bool"
```
```   896   .
```
```   897
```
```   898 instance proof
```
```   899 qed (transfer, simp add: algebra_simps equal less_le_not_le [symmetric] linear)+
```
```   900
```
```   901 end
```
```   902
```
```   903 lemma [transfer_rule]:
```
```   904   "rel_fun pcr_natural (rel_fun pcr_natural pcr_natural) (min :: _ \<Rightarrow> _ \<Rightarrow> nat) (min :: _ \<Rightarrow> _ \<Rightarrow> natural)"
```
```   905   by (unfold min_def [abs_def]) transfer_prover
```
```   906
```
```   907 lemma [transfer_rule]:
```
```   908   "rel_fun pcr_natural (rel_fun pcr_natural pcr_natural) (max :: _ \<Rightarrow> _ \<Rightarrow> nat) (max :: _ \<Rightarrow> _ \<Rightarrow> natural)"
```
```   909   by (unfold max_def [abs_def]) transfer_prover
```
```   910
```
```   911 lemma nat_of_natural_min [simp]:
```
```   912   "nat_of_natural (min k l) = min (nat_of_natural k) (nat_of_natural l)"
```
```   913   by transfer rule
```
```   914
```
```   915 lemma nat_of_natural_max [simp]:
```
```   916   "nat_of_natural (max k l) = max (nat_of_natural k) (nat_of_natural l)"
```
```   917   by transfer rule
```
```   918
```
```   919 instantiation natural :: unique_euclidean_semiring
```
```   920 begin
```
```   921
```
```   922 lift_definition divide_natural :: "natural \<Rightarrow> natural \<Rightarrow> natural"
```
```   923   is "divide :: nat \<Rightarrow> nat \<Rightarrow> nat"
```
```   924   .
```
```   925
```
```   926 declare divide_natural.rep_eq [simp]
```
```   927
```
```   928 lift_definition modulo_natural :: "natural \<Rightarrow> natural \<Rightarrow> natural"
```
```   929   is "modulo :: nat \<Rightarrow> nat \<Rightarrow> nat"
```
```   930   .
```
```   931
```
```   932 declare modulo_natural.rep_eq [simp]
```
```   933
```
```   934 lift_definition euclidean_size_natural :: "natural \<Rightarrow> nat"
```
```   935   is "euclidean_size :: nat \<Rightarrow> nat"
```
```   936   .
```
```   937
```
```   938 declare euclidean_size_natural.rep_eq [simp]
```
```   939
```
```   940 lift_definition division_segment_natural :: "natural \<Rightarrow> natural"
```
```   941   is "division_segment :: nat \<Rightarrow> nat"
```
```   942   .
```
```   943
```
```   944 declare division_segment_natural.rep_eq [simp]
```
```   945
```
```   946 instance
```
```   947   by (standard; transfer)
```
```   948     (auto simp add: algebra_simps unit_factor_nat_def gr0_conv_Suc)
```
```   949
```
```   950 end
```
```   951
```
```   952 lemma [code]:
```
```   953   "euclidean_size = nat_of_natural"
```
```   954   by (simp add: fun_eq_iff)
```
```   955
```
```   956 lemma [code]:
```
```   957   "division_segment (n::natural) = 1"
```
```   958   by (simp add: natural_eq_iff)
```
```   959
```
```   960 instance natural :: linordered_semidom
```
```   961   by (standard; transfer) simp_all
```
```   962
```
```   963 instance natural :: semiring_parity
```
```   964   by (standard; transfer) simp_all
```
```   965
```
```   966 lemma [transfer_rule]:
```
```   967   "rel_fun (=) (rel_fun pcr_natural pcr_natural) (push_bit :: _ \<Rightarrow> _ \<Rightarrow> nat) (push_bit :: _ \<Rightarrow> _ \<Rightarrow> natural)"
```
```   968   by (unfold push_bit_eq_mult [abs_def]) transfer_prover
```
```   969
```
```   970 lemma [transfer_rule]:
```
```   971   "rel_fun (=) (rel_fun pcr_natural pcr_natural) (take_bit :: _ \<Rightarrow> _ \<Rightarrow> nat) (take_bit :: _ \<Rightarrow> _ \<Rightarrow> natural)"
```
```   972   by (unfold take_bit_eq_mod [abs_def]) transfer_prover
```
```   973
```
```   974 lemma [transfer_rule]:
```
```   975   "rel_fun (=) (rel_fun pcr_natural pcr_natural) (drop_bit :: _ \<Rightarrow> _ \<Rightarrow> nat) (drop_bit :: _ \<Rightarrow> _ \<Rightarrow> natural)"
```
```   976   by (unfold drop_bit_eq_div [abs_def]) transfer_prover
```
```   977
```
```   978 lift_definition natural_of_integer :: "integer \<Rightarrow> natural"
```
```   979   is "nat :: int \<Rightarrow> nat"
```
```   980   .
```
```   981
```
```   982 lift_definition integer_of_natural :: "natural \<Rightarrow> integer"
```
```   983   is "of_nat :: nat \<Rightarrow> int"
```
```   984   .
```
```   985
```
```   986 lemma natural_of_integer_of_natural [simp]:
```
```   987   "natural_of_integer (integer_of_natural n) = n"
```
```   988   by transfer simp
```
```   989
```
```   990 lemma integer_of_natural_of_integer [simp]:
```
```   991   "integer_of_natural (natural_of_integer k) = max 0 k"
```
```   992   by transfer auto
```
```   993
```
```   994 lemma int_of_integer_of_natural [simp]:
```
```   995   "int_of_integer (integer_of_natural n) = of_nat (nat_of_natural n)"
```
```   996   by transfer rule
```
```   997
```
```   998 lemma integer_of_natural_of_nat [simp]:
```
```   999   "integer_of_natural (of_nat n) = of_nat n"
```
```  1000   by transfer rule
```
```  1001
```
```  1002 lemma [measure_function]:
```
```  1003   "is_measure nat_of_natural"
```
```  1004   by (rule is_measure_trivial)
```
```  1005
```
```  1006
```
```  1007 subsection \<open>Inductive representation of target language naturals\<close>
```
```  1008
```
```  1009 lift_definition Suc :: "natural \<Rightarrow> natural"
```
```  1010   is Nat.Suc
```
```  1011   .
```
```  1012
```
```  1013 declare Suc.rep_eq [simp]
```
```  1014
```
```  1015 old_rep_datatype "0::natural" Suc
```
```  1016   by (transfer, fact nat.induct nat.inject nat.distinct)+
```
```  1017
```
```  1018 lemma natural_cases [case_names nat, cases type: natural]:
```
```  1019   fixes m :: natural
```
```  1020   assumes "\<And>n. m = of_nat n \<Longrightarrow> P"
```
```  1021   shows P
```
```  1022   using assms by transfer blast
```
```  1023
```
```  1024 instantiation natural :: size
```
```  1025 begin
```
```  1026
```
```  1027 definition size_nat where [simp, code]: "size_nat = nat_of_natural"
```
```  1028
```
```  1029 instance ..
```
```  1030
```
```  1031 end
```
```  1032
```
```  1033 lemma natural_decr [termination_simp]:
```
```  1034   "n \<noteq> 0 \<Longrightarrow> nat_of_natural n - Nat.Suc 0 < nat_of_natural n"
```
```  1035   by transfer simp
```
```  1036
```
```  1037 lemma natural_zero_minus_one: "(0::natural) - 1 = 0"
```
```  1038   by (rule zero_diff)
```
```  1039
```
```  1040 lemma Suc_natural_minus_one: "Suc n - 1 = n"
```
```  1041   by transfer simp
```
```  1042
```
```  1043 hide_const (open) Suc
```
```  1044
```
```  1045
```
```  1046 subsection \<open>Code refinement for target language naturals\<close>
```
```  1047
```
```  1048 lift_definition Nat :: "integer \<Rightarrow> natural"
```
```  1049   is nat
```
```  1050   .
```
```  1051
```
```  1052 lemma [code_post]:
```
```  1053   "Nat 0 = 0"
```
```  1054   "Nat 1 = 1"
```
```  1055   "Nat (numeral k) = numeral k"
```
```  1056   by (transfer, simp)+
```
```  1057
```
```  1058 lemma [code abstype]:
```
```  1059   "Nat (integer_of_natural n) = n"
```
```  1060   by transfer simp
```
```  1061
```
```  1062 lemma [code]:
```
```  1063   "natural_of_nat n = natural_of_integer (integer_of_nat n)"
```
```  1064   by transfer simp
```
```  1065
```
```  1066 lemma [code abstract]:
```
```  1067   "integer_of_natural (natural_of_integer k) = max 0 k"
```
```  1068   by simp
```
```  1069
```
```  1070 lemma [code_abbrev]:
```
```  1071   "natural_of_integer (Code_Numeral.Pos k) = numeral k"
```
```  1072   by transfer simp
```
```  1073
```
```  1074 lemma [code abstract]:
```
```  1075   "integer_of_natural 0 = 0"
```
```  1076   by transfer simp
```
```  1077
```
```  1078 lemma [code abstract]:
```
```  1079   "integer_of_natural 1 = 1"
```
```  1080   by transfer simp
```
```  1081
```
```  1082 lemma [code abstract]:
```
```  1083   "integer_of_natural (Code_Numeral.Suc n) = integer_of_natural n + 1"
```
```  1084   by transfer simp
```
```  1085
```
```  1086 lemma [code]:
```
```  1087   "nat_of_natural = nat_of_integer \<circ> integer_of_natural"
```
```  1088   by transfer (simp add: fun_eq_iff)
```
```  1089
```
```  1090 lemma [code, code_unfold]:
```
```  1091   "case_natural f g n = (if n = 0 then f else g (n - 1))"
```
```  1092   by (cases n rule: natural.exhaust) (simp_all, simp add: Suc_def)
```
```  1093
```
```  1094 declare natural.rec [code del]
```
```  1095
```
```  1096 lemma [code abstract]:
```
```  1097   "integer_of_natural (m + n) = integer_of_natural m + integer_of_natural n"
```
```  1098   by transfer simp
```
```  1099
```
```  1100 lemma [code abstract]:
```
```  1101   "integer_of_natural (m - n) = max 0 (integer_of_natural m - integer_of_natural n)"
```
```  1102   by transfer simp
```
```  1103
```
```  1104 lemma [code abstract]:
```
```  1105   "integer_of_natural (m * n) = integer_of_natural m * integer_of_natural n"
```
```  1106   by transfer simp
```
```  1107
```
```  1108 lemma [code abstract]:
```
```  1109   "integer_of_natural (m div n) = integer_of_natural m div integer_of_natural n"
```
```  1110   by transfer (simp add: zdiv_int)
```
```  1111
```
```  1112 lemma [code abstract]:
```
```  1113   "integer_of_natural (m mod n) = integer_of_natural m mod integer_of_natural n"
```
```  1114   by transfer (simp add: zmod_int)
```
```  1115
```
```  1116 lemma [code]:
```
```  1117   "HOL.equal m n \<longleftrightarrow> HOL.equal (integer_of_natural m) (integer_of_natural n)"
```
```  1118   by transfer (simp add: equal)
```
```  1119
```
```  1120 lemma [code nbe]: "HOL.equal n (n::natural) \<longleftrightarrow> True"
```
```  1121   by (rule equal_class.equal_refl)
```
```  1122
```
```  1123 lemma [code]: "m \<le> n \<longleftrightarrow> integer_of_natural m \<le> integer_of_natural n"
```
```  1124   by transfer simp
```
```  1125
```
```  1126 lemma [code]: "m < n \<longleftrightarrow> integer_of_natural m < integer_of_natural n"
```
```  1127   by transfer simp
```
```  1128
```
```  1129 hide_const (open) Nat
```
```  1130
```
```  1131 lifting_update integer.lifting
```
```  1132 lifting_forget integer.lifting
```
```  1133
```
```  1134 lifting_update natural.lifting
```
```  1135 lifting_forget natural.lifting
```
```  1136
```
```  1137 code_reflect Code_Numeral
```
```  1138   datatypes natural
```
```  1139   functions "Code_Numeral.Suc" "0 :: natural" "1 :: natural"
```
```  1140     "plus :: natural \<Rightarrow> _" "minus :: natural \<Rightarrow> _"
```
```  1141     "times :: natural \<Rightarrow> _" "divide :: natural \<Rightarrow> _"
```
```  1142     "modulo :: natural \<Rightarrow> _"
```
```  1143     integer_of_natural natural_of_integer
```
```  1144
```
```  1145 end
```