src/HOL/Code_Numeral.thy
 author wenzelm Tue Sep 01 22:32:58 2015 +0200 (2015-09-01) changeset 61076 bdc1e2f0a86a parent 60868 dd18c33c001e child 61274 0261eec37233 permissions -rw-r--r--
eliminated \<Colon>;
```     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 Nat_Transfer 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 lemma [transfer_rule]:
```
```    79   "rel_fun HOL.eq pcr_integer (of_nat :: nat \<Rightarrow> int) (of_nat :: nat \<Rightarrow> integer)"
```
```    80   by (unfold of_nat_def [abs_def]) transfer_prover
```
```    81
```
```    82 lemma [transfer_rule]:
```
```    83   "rel_fun HOL.eq pcr_integer (\<lambda>k :: int. k :: int) (of_int :: int \<Rightarrow> integer)"
```
```    84 proof -
```
```    85   have "rel_fun HOL.eq pcr_integer (of_int :: int \<Rightarrow> int) (of_int :: int \<Rightarrow> integer)"
```
```    86     by (unfold of_int_of_nat [abs_def]) transfer_prover
```
```    87   then show ?thesis by (simp add: id_def)
```
```    88 qed
```
```    89
```
```    90 lemma [transfer_rule]:
```
```    91   "rel_fun HOL.eq pcr_integer (numeral :: num \<Rightarrow> int) (numeral :: num \<Rightarrow> integer)"
```
```    92 proof -
```
```    93   have "rel_fun HOL.eq pcr_integer (numeral :: num \<Rightarrow> int) (\<lambda>n. of_int (numeral n))"
```
```    94     by transfer_prover
```
```    95   then show ?thesis by simp
```
```    96 qed
```
```    97
```
```    98 lemma [transfer_rule]:
```
```    99   "rel_fun HOL.eq (rel_fun HOL.eq pcr_integer) (Num.sub :: _ \<Rightarrow> _ \<Rightarrow> int) (Num.sub :: _ \<Rightarrow> _ \<Rightarrow> integer)"
```
```   100   by (unfold Num.sub_def [abs_def]) transfer_prover
```
```   101
```
```   102 lemma int_of_integer_of_nat [simp]:
```
```   103   "int_of_integer (of_nat n) = of_nat n"
```
```   104   by transfer rule
```
```   105
```
```   106 lift_definition integer_of_nat :: "nat \<Rightarrow> integer"
```
```   107   is "of_nat :: nat \<Rightarrow> int"
```
```   108   .
```
```   109
```
```   110 lemma integer_of_nat_eq_of_nat [code]:
```
```   111   "integer_of_nat = of_nat"
```
```   112   by transfer rule
```
```   113
```
```   114 lemma int_of_integer_integer_of_nat [simp]:
```
```   115   "int_of_integer (integer_of_nat n) = of_nat n"
```
```   116   by transfer rule
```
```   117
```
```   118 lift_definition nat_of_integer :: "integer \<Rightarrow> nat"
```
```   119   is Int.nat
```
```   120   .
```
```   121
```
```   122 lemma nat_of_integer_of_nat [simp]:
```
```   123   "nat_of_integer (of_nat n) = n"
```
```   124   by transfer simp
```
```   125
```
```   126 lemma int_of_integer_of_int [simp]:
```
```   127   "int_of_integer (of_int k) = k"
```
```   128   by transfer simp
```
```   129
```
```   130 lemma nat_of_integer_integer_of_nat [simp]:
```
```   131   "nat_of_integer (integer_of_nat n) = n"
```
```   132   by transfer simp
```
```   133
```
```   134 lemma integer_of_int_eq_of_int [simp, code_abbrev]:
```
```   135   "integer_of_int = of_int"
```
```   136   by transfer (simp add: fun_eq_iff)
```
```   137
```
```   138 lemma of_int_integer_of [simp]:
```
```   139   "of_int (int_of_integer k) = (k :: integer)"
```
```   140   by transfer rule
```
```   141
```
```   142 lemma int_of_integer_numeral [simp]:
```
```   143   "int_of_integer (numeral k) = numeral k"
```
```   144   by transfer rule
```
```   145
```
```   146 lemma int_of_integer_sub [simp]:
```
```   147   "int_of_integer (Num.sub k l) = Num.sub k l"
```
```   148   by transfer rule
```
```   149
```
```   150 instantiation integer :: "{ring_div, equal, linordered_idom}"
```
```   151 begin
```
```   152
```
```   153 lift_definition divide_integer :: "integer \<Rightarrow> integer \<Rightarrow> integer"
```
```   154   is "divide :: int \<Rightarrow> int \<Rightarrow> int"
```
```   155   .
```
```   156
```
```   157 declare divide_integer.rep_eq [simp]
```
```   158
```
```   159 lift_definition mod_integer :: "integer \<Rightarrow> integer \<Rightarrow> integer"
```
```   160   is "Divides.mod :: int \<Rightarrow> int \<Rightarrow> int"
```
```   161   .
```
```   162
```
```   163 declare mod_integer.rep_eq [simp]
```
```   164
```
```   165 lift_definition abs_integer :: "integer \<Rightarrow> integer"
```
```   166   is "abs :: int \<Rightarrow> int"
```
```   167   .
```
```   168
```
```   169 declare abs_integer.rep_eq [simp]
```
```   170
```
```   171 lift_definition sgn_integer :: "integer \<Rightarrow> integer"
```
```   172   is "sgn :: int \<Rightarrow> int"
```
```   173   .
```
```   174
```
```   175 declare sgn_integer.rep_eq [simp]
```
```   176
```
```   177 lift_definition less_eq_integer :: "integer \<Rightarrow> integer \<Rightarrow> bool"
```
```   178   is "less_eq :: int \<Rightarrow> int \<Rightarrow> bool"
```
```   179   .
```
```   180
```
```   181 lift_definition less_integer :: "integer \<Rightarrow> integer \<Rightarrow> bool"
```
```   182   is "less :: int \<Rightarrow> int \<Rightarrow> bool"
```
```   183   .
```
```   184
```
```   185 lift_definition equal_integer :: "integer \<Rightarrow> integer \<Rightarrow> bool"
```
```   186   is "HOL.equal :: int \<Rightarrow> int \<Rightarrow> bool"
```
```   187   .
```
```   188
```
```   189 instance proof
```
```   190 qed (transfer, simp add: algebra_simps equal less_le_not_le [symmetric] mult_strict_right_mono linear)+
```
```   191
```
```   192 end
```
```   193
```
```   194 lemma [transfer_rule]:
```
```   195   "rel_fun pcr_integer (rel_fun pcr_integer pcr_integer) (min :: _ \<Rightarrow> _ \<Rightarrow> int) (min :: _ \<Rightarrow> _ \<Rightarrow> integer)"
```
```   196   by (unfold min_def [abs_def]) transfer_prover
```
```   197
```
```   198 lemma [transfer_rule]:
```
```   199   "rel_fun pcr_integer (rel_fun pcr_integer pcr_integer) (max :: _ \<Rightarrow> _ \<Rightarrow> int) (max :: _ \<Rightarrow> _ \<Rightarrow> integer)"
```
```   200   by (unfold max_def [abs_def]) transfer_prover
```
```   201
```
```   202 lemma int_of_integer_min [simp]:
```
```   203   "int_of_integer (min k l) = min (int_of_integer k) (int_of_integer l)"
```
```   204   by transfer rule
```
```   205
```
```   206 lemma int_of_integer_max [simp]:
```
```   207   "int_of_integer (max k l) = max (int_of_integer k) (int_of_integer l)"
```
```   208   by transfer rule
```
```   209
```
```   210 lemma nat_of_integer_non_positive [simp]:
```
```   211   "k \<le> 0 \<Longrightarrow> nat_of_integer k = 0"
```
```   212   by transfer simp
```
```   213
```
```   214 lemma of_nat_of_integer [simp]:
```
```   215   "of_nat (nat_of_integer k) = max 0 k"
```
```   216   by transfer auto
```
```   217
```
```   218 instance integer :: semiring_numeral_div
```
```   219   by intro_classes (transfer,
```
```   220     fact le_add_diff_inverse2
```
```   221     semiring_numeral_div_class.div_less
```
```   222     semiring_numeral_div_class.mod_less
```
```   223     semiring_numeral_div_class.div_positive
```
```   224     semiring_numeral_div_class.mod_less_eq_dividend
```
```   225     semiring_numeral_div_class.pos_mod_bound
```
```   226     semiring_numeral_div_class.pos_mod_sign
```
```   227     semiring_numeral_div_class.mod_mult2_eq
```
```   228     semiring_numeral_div_class.div_mult2_eq
```
```   229     semiring_numeral_div_class.discrete)+
```
```   230
```
```   231 lemma integer_of_nat_0: "integer_of_nat 0 = 0"
```
```   232 by transfer simp
```
```   233
```
```   234 lemma integer_of_nat_1: "integer_of_nat 1 = 1"
```
```   235 by transfer simp
```
```   236
```
```   237 lemma integer_of_nat_numeral:
```
```   238   "integer_of_nat (numeral n) = numeral n"
```
```   239 by transfer simp
```
```   240
```
```   241 subsection \<open>Code theorems for target language integers\<close>
```
```   242
```
```   243 text \<open>Constructors\<close>
```
```   244
```
```   245 definition Pos :: "num \<Rightarrow> integer"
```
```   246 where
```
```   247   [simp, code_abbrev]: "Pos = numeral"
```
```   248
```
```   249 lemma [transfer_rule]:
```
```   250   "rel_fun HOL.eq pcr_integer numeral Pos"
```
```   251   by simp transfer_prover
```
```   252
```
```   253 definition Neg :: "num \<Rightarrow> integer"
```
```   254 where
```
```   255   [simp, code_abbrev]: "Neg n = - Pos n"
```
```   256
```
```   257 lemma [transfer_rule]:
```
```   258   "rel_fun HOL.eq pcr_integer (\<lambda>n. - numeral n) Neg"
```
```   259   by (simp add: Neg_def [abs_def]) transfer_prover
```
```   260
```
```   261 code_datatype "0::integer" Pos Neg
```
```   262
```
```   263
```
```   264 text \<open>Auxiliary operations\<close>
```
```   265
```
```   266 lift_definition dup :: "integer \<Rightarrow> integer"
```
```   267   is "\<lambda>k::int. k + k"
```
```   268   .
```
```   269
```
```   270 lemma dup_code [code]:
```
```   271   "dup 0 = 0"
```
```   272   "dup (Pos n) = Pos (Num.Bit0 n)"
```
```   273   "dup (Neg n) = Neg (Num.Bit0 n)"
```
```   274   by (transfer, simp only: numeral_Bit0 minus_add_distrib)+
```
```   275
```
```   276 lift_definition sub :: "num \<Rightarrow> num \<Rightarrow> integer"
```
```   277   is "\<lambda>m n. numeral m - numeral n :: int"
```
```   278   .
```
```   279
```
```   280 lemma sub_code [code]:
```
```   281   "sub Num.One Num.One = 0"
```
```   282   "sub (Num.Bit0 m) Num.One = Pos (Num.BitM m)"
```
```   283   "sub (Num.Bit1 m) Num.One = Pos (Num.Bit0 m)"
```
```   284   "sub Num.One (Num.Bit0 n) = Neg (Num.BitM n)"
```
```   285   "sub Num.One (Num.Bit1 n) = Neg (Num.Bit0 n)"
```
```   286   "sub (Num.Bit0 m) (Num.Bit0 n) = dup (sub m n)"
```
```   287   "sub (Num.Bit1 m) (Num.Bit1 n) = dup (sub m n)"
```
```   288   "sub (Num.Bit1 m) (Num.Bit0 n) = dup (sub m n) + 1"
```
```   289   "sub (Num.Bit0 m) (Num.Bit1 n) = dup (sub m n) - 1"
```
```   290   by (transfer, simp add: dbl_def dbl_inc_def dbl_dec_def)+
```
```   291
```
```   292
```
```   293 text \<open>Implementations\<close>
```
```   294
```
```   295 lemma one_integer_code [code, code_unfold]:
```
```   296   "1 = Pos Num.One"
```
```   297   by simp
```
```   298
```
```   299 lemma plus_integer_code [code]:
```
```   300   "k + 0 = (k::integer)"
```
```   301   "0 + l = (l::integer)"
```
```   302   "Pos m + Pos n = Pos (m + n)"
```
```   303   "Pos m + Neg n = sub m n"
```
```   304   "Neg m + Pos n = sub n m"
```
```   305   "Neg m + Neg n = Neg (m + n)"
```
```   306   by (transfer, simp)+
```
```   307
```
```   308 lemma uminus_integer_code [code]:
```
```   309   "uminus 0 = (0::integer)"
```
```   310   "uminus (Pos m) = Neg m"
```
```   311   "uminus (Neg m) = Pos m"
```
```   312   by simp_all
```
```   313
```
```   314 lemma minus_integer_code [code]:
```
```   315   "k - 0 = (k::integer)"
```
```   316   "0 - l = uminus (l::integer)"
```
```   317   "Pos m - Pos n = sub m n"
```
```   318   "Pos m - Neg n = Pos (m + n)"
```
```   319   "Neg m - Pos n = Neg (m + n)"
```
```   320   "Neg m - Neg n = sub n m"
```
```   321   by (transfer, simp)+
```
```   322
```
```   323 lemma abs_integer_code [code]:
```
```   324   "\<bar>k\<bar> = (if (k::integer) < 0 then - k else k)"
```
```   325   by simp
```
```   326
```
```   327 lemma sgn_integer_code [code]:
```
```   328   "sgn k = (if k = 0 then 0 else if (k::integer) < 0 then - 1 else 1)"
```
```   329   by simp
```
```   330
```
```   331 lemma times_integer_code [code]:
```
```   332   "k * 0 = (0::integer)"
```
```   333   "0 * l = (0::integer)"
```
```   334   "Pos m * Pos n = Pos (m * n)"
```
```   335   "Pos m * Neg n = Neg (m * n)"
```
```   336   "Neg m * Pos n = Neg (m * n)"
```
```   337   "Neg m * Neg n = Pos (m * n)"
```
```   338   by simp_all
```
```   339
```
```   340 definition divmod_integer :: "integer \<Rightarrow> integer \<Rightarrow> integer \<times> integer"
```
```   341 where
```
```   342   "divmod_integer k l = (k div l, k mod l)"
```
```   343
```
```   344 lemma fst_divmod [simp]:
```
```   345   "fst (divmod_integer k l) = k div l"
```
```   346   by (simp add: divmod_integer_def)
```
```   347
```
```   348 lemma snd_divmod [simp]:
```
```   349   "snd (divmod_integer k l) = k mod l"
```
```   350   by (simp add: divmod_integer_def)
```
```   351
```
```   352 definition divmod_abs :: "integer \<Rightarrow> integer \<Rightarrow> integer \<times> integer"
```
```   353 where
```
```   354   "divmod_abs k l = (\<bar>k\<bar> div \<bar>l\<bar>, \<bar>k\<bar> mod \<bar>l\<bar>)"
```
```   355
```
```   356 lemma fst_divmod_abs [simp]:
```
```   357   "fst (divmod_abs k l) = \<bar>k\<bar> div \<bar>l\<bar>"
```
```   358   by (simp add: divmod_abs_def)
```
```   359
```
```   360 lemma snd_divmod_abs [simp]:
```
```   361   "snd (divmod_abs k l) = \<bar>k\<bar> mod \<bar>l\<bar>"
```
```   362   by (simp add: divmod_abs_def)
```
```   363
```
```   364 lemma divmod_abs_code [code]:
```
```   365   "divmod_abs (Pos k) (Pos l) = divmod k l"
```
```   366   "divmod_abs (Neg k) (Neg l) = divmod k l"
```
```   367   "divmod_abs (Neg k) (Pos l) = divmod k l"
```
```   368   "divmod_abs (Pos k) (Neg l) = divmod k l"
```
```   369   "divmod_abs j 0 = (0, \<bar>j\<bar>)"
```
```   370   "divmod_abs 0 j = (0, 0)"
```
```   371   by (simp_all add: prod_eq_iff)
```
```   372
```
```   373 lemma divmod_integer_code [code]:
```
```   374   "divmod_integer k l =
```
```   375     (if k = 0 then (0, 0) else if l = 0 then (0, k) else
```
```   376     (apsnd \<circ> times \<circ> sgn) l (if sgn k = sgn l
```
```   377       then divmod_abs k l
```
```   378       else (let (r, s) = divmod_abs k l in
```
```   379         if s = 0 then (- r, 0) else (- r - 1, \<bar>l\<bar> - s))))"
```
```   380 proof -
```
```   381   have aux1: "\<And>k l::int. sgn k = sgn l \<longleftrightarrow> k = 0 \<and> l = 0 \<or> 0 < l \<and> 0 < k \<or> l < 0 \<and> k < 0"
```
```   382     by (auto simp add: sgn_if)
```
```   383   have aux2: "\<And>q::int. - int_of_integer k = int_of_integer l * q \<longleftrightarrow> int_of_integer k = int_of_integer l * - q" by auto
```
```   384   show ?thesis
```
```   385     by (simp add: prod_eq_iff integer_eq_iff case_prod_beta aux1)
```
```   386       (auto simp add: zdiv_zminus1_eq_if zmod_zminus1_eq_if div_minus_right mod_minus_right aux2)
```
```   387 qed
```
```   388
```
```   389 lemma div_integer_code [code]:
```
```   390   "k div l = fst (divmod_integer k l)"
```
```   391   by simp
```
```   392
```
```   393 lemma mod_integer_code [code]:
```
```   394   "k mod l = snd (divmod_integer k l)"
```
```   395   by simp
```
```   396
```
```   397 lemma equal_integer_code [code]:
```
```   398   "HOL.equal 0 (0::integer) \<longleftrightarrow> True"
```
```   399   "HOL.equal 0 (Pos l) \<longleftrightarrow> False"
```
```   400   "HOL.equal 0 (Neg l) \<longleftrightarrow> False"
```
```   401   "HOL.equal (Pos k) 0 \<longleftrightarrow> False"
```
```   402   "HOL.equal (Pos k) (Pos l) \<longleftrightarrow> HOL.equal k l"
```
```   403   "HOL.equal (Pos k) (Neg l) \<longleftrightarrow> False"
```
```   404   "HOL.equal (Neg k) 0 \<longleftrightarrow> False"
```
```   405   "HOL.equal (Neg k) (Pos l) \<longleftrightarrow> False"
```
```   406   "HOL.equal (Neg k) (Neg l) \<longleftrightarrow> HOL.equal k l"
```
```   407   by (simp_all add: equal)
```
```   408
```
```   409 lemma equal_integer_refl [code nbe]:
```
```   410   "HOL.equal (k::integer) k \<longleftrightarrow> True"
```
```   411   by (fact equal_refl)
```
```   412
```
```   413 lemma less_eq_integer_code [code]:
```
```   414   "0 \<le> (0::integer) \<longleftrightarrow> True"
```
```   415   "0 \<le> Pos l \<longleftrightarrow> True"
```
```   416   "0 \<le> Neg l \<longleftrightarrow> False"
```
```   417   "Pos k \<le> 0 \<longleftrightarrow> False"
```
```   418   "Pos k \<le> Pos l \<longleftrightarrow> k \<le> l"
```
```   419   "Pos k \<le> Neg l \<longleftrightarrow> False"
```
```   420   "Neg k \<le> 0 \<longleftrightarrow> True"
```
```   421   "Neg k \<le> Pos l \<longleftrightarrow> True"
```
```   422   "Neg k \<le> Neg l \<longleftrightarrow> l \<le> k"
```
```   423   by simp_all
```
```   424
```
```   425 lemma less_integer_code [code]:
```
```   426   "0 < (0::integer) \<longleftrightarrow> False"
```
```   427   "0 < Pos l \<longleftrightarrow> True"
```
```   428   "0 < Neg l \<longleftrightarrow> False"
```
```   429   "Pos k < 0 \<longleftrightarrow> False"
```
```   430   "Pos k < Pos l \<longleftrightarrow> k < l"
```
```   431   "Pos k < Neg l \<longleftrightarrow> False"
```
```   432   "Neg k < 0 \<longleftrightarrow> True"
```
```   433   "Neg k < Pos l \<longleftrightarrow> True"
```
```   434   "Neg k < Neg l \<longleftrightarrow> l < k"
```
```   435   by simp_all
```
```   436
```
```   437 lift_definition integer_of_num :: "num \<Rightarrow> integer"
```
```   438   is "numeral :: num \<Rightarrow> int"
```
```   439   .
```
```   440
```
```   441 lemma integer_of_num [code]:
```
```   442   "integer_of_num num.One = 1"
```
```   443   "integer_of_num (num.Bit0 n) = (let k = integer_of_num n in k + k)"
```
```   444   "integer_of_num (num.Bit1 n) = (let k = integer_of_num n in k + k + 1)"
```
```   445   by (transfer, simp only: numeral.simps Let_def)+
```
```   446
```
```   447 lift_definition num_of_integer :: "integer \<Rightarrow> num"
```
```   448   is "num_of_nat \<circ> nat"
```
```   449   .
```
```   450
```
```   451 lemma num_of_integer_code [code]:
```
```   452   "num_of_integer k = (if k \<le> 1 then Num.One
```
```   453      else let
```
```   454        (l, j) = divmod_integer k 2;
```
```   455        l' = num_of_integer l;
```
```   456        l'' = l' + l'
```
```   457      in if j = 0 then l'' else l'' + Num.One)"
```
```   458 proof -
```
```   459   {
```
```   460     assume "int_of_integer k mod 2 = 1"
```
```   461     then have "nat (int_of_integer k mod 2) = nat 1" by simp
```
```   462     moreover assume *: "1 < int_of_integer k"
```
```   463     ultimately have **: "nat (int_of_integer k) mod 2 = 1" by (simp add: nat_mod_distrib)
```
```   464     have "num_of_nat (nat (int_of_integer k)) =
```
```   465       num_of_nat (2 * (nat (int_of_integer k) div 2) + nat (int_of_integer k) mod 2)"
```
```   466       by simp
```
```   467     then have "num_of_nat (nat (int_of_integer k)) =
```
```   468       num_of_nat (nat (int_of_integer k) div 2 + nat (int_of_integer k) div 2 + nat (int_of_integer k) mod 2)"
```
```   469       by (simp add: mult_2)
```
```   470     with ** have "num_of_nat (nat (int_of_integer k)) =
```
```   471       num_of_nat (nat (int_of_integer k) div 2 + nat (int_of_integer k) div 2 + 1)"
```
```   472       by simp
```
```   473   }
```
```   474   note aux = this
```
```   475   show ?thesis
```
```   476     by (auto simp add: num_of_integer_def nat_of_integer_def Let_def case_prod_beta
```
```   477       not_le integer_eq_iff less_eq_integer_def
```
```   478       nat_mult_distrib nat_div_distrib num_of_nat_One num_of_nat_plus_distrib
```
```   479        mult_2 [where 'a=nat] aux add_One)
```
```   480 qed
```
```   481
```
```   482 lemma nat_of_integer_code [code]:
```
```   483   "nat_of_integer k = (if k \<le> 0 then 0
```
```   484      else let
```
```   485        (l, j) = divmod_integer k 2;
```
```   486        l' = nat_of_integer l;
```
```   487        l'' = l' + l'
```
```   488      in if j = 0 then l'' else l'' + 1)"
```
```   489 proof -
```
```   490   obtain j where "k = integer_of_int j"
```
```   491   proof
```
```   492     show "k = integer_of_int (int_of_integer k)" by simp
```
```   493   qed
```
```   494   moreover have "2 * (j div 2) = j - j mod 2"
```
```   495     by (simp add: zmult_div_cancel mult.commute)
```
```   496   ultimately show ?thesis
```
```   497     by (auto simp add: split_def Let_def mod_integer_def nat_of_integer_def not_le
```
```   498       nat_add_distrib [symmetric] Suc_nat_eq_nat_zadd1)
```
```   499       (auto simp add: mult_2 [symmetric])
```
```   500 qed
```
```   501
```
```   502 lemma int_of_integer_code [code]:
```
```   503   "int_of_integer k = (if k < 0 then - (int_of_integer (- k))
```
```   504      else if k = 0 then 0
```
```   505      else let
```
```   506        (l, j) = divmod_integer k 2;
```
```   507        l' = 2 * int_of_integer l
```
```   508      in if j = 0 then l' else l' + 1)"
```
```   509   by (auto simp add: split_def Let_def integer_eq_iff zmult_div_cancel)
```
```   510
```
```   511 lemma integer_of_int_code [code]:
```
```   512   "integer_of_int k = (if k < 0 then - (integer_of_int (- k))
```
```   513      else if k = 0 then 0
```
```   514      else let
```
```   515        l = 2 * integer_of_int (k div 2);
```
```   516        j = k mod 2
```
```   517      in if j = 0 then l else l + 1)"
```
```   518   by (auto simp add: split_def Let_def integer_eq_iff zmult_div_cancel)
```
```   519
```
```   520 hide_const (open) Pos Neg sub dup divmod_abs
```
```   521
```
```   522
```
```   523 subsection \<open>Serializer setup for target language integers\<close>
```
```   524
```
```   525 code_reserved Eval int Integer abs
```
```   526
```
```   527 code_printing
```
```   528   type_constructor integer \<rightharpoonup>
```
```   529     (SML) "IntInf.int"
```
```   530     and (OCaml) "Big'_int.big'_int"
```
```   531     and (Haskell) "Integer"
```
```   532     and (Scala) "BigInt"
```
```   533     and (Eval) "int"
```
```   534 | class_instance integer :: equal \<rightharpoonup>
```
```   535     (Haskell) -
```
```   536
```
```   537 code_printing
```
```   538   constant "0::integer" \<rightharpoonup>
```
```   539     (SML) "!(0/ :/ IntInf.int)"
```
```   540     and (OCaml) "Big'_int.zero'_big'_int"
```
```   541     and (Haskell) "!(0/ ::/ Integer)"
```
```   542     and (Scala) "BigInt(0)"
```
```   543
```
```   544 setup \<open>
```
```   545   fold (fn target =>
```
```   546     Numeral.add_code @{const_name Code_Numeral.Pos} I Code_Printer.literal_numeral target
```
```   547     #> Numeral.add_code @{const_name Code_Numeral.Neg} (op ~) Code_Printer.literal_numeral target)
```
```   548     ["SML", "OCaml", "Haskell", "Scala"]
```
```   549 \<close>
```
```   550
```
```   551 code_printing
```
```   552   constant "plus :: integer \<Rightarrow> _ \<Rightarrow> _" \<rightharpoonup>
```
```   553     (SML) "IntInf.+ ((_), (_))"
```
```   554     and (OCaml) "Big'_int.add'_big'_int"
```
```   555     and (Haskell) infixl 6 "+"
```
```   556     and (Scala) infixl 7 "+"
```
```   557     and (Eval) infixl 8 "+"
```
```   558 | constant "uminus :: integer \<Rightarrow> _" \<rightharpoonup>
```
```   559     (SML) "IntInf.~"
```
```   560     and (OCaml) "Big'_int.minus'_big'_int"
```
```   561     and (Haskell) "negate"
```
```   562     and (Scala) "!(- _)"
```
```   563     and (Eval) "~/ _"
```
```   564 | constant "minus :: integer \<Rightarrow> _" \<rightharpoonup>
```
```   565     (SML) "IntInf.- ((_), (_))"
```
```   566     and (OCaml) "Big'_int.sub'_big'_int"
```
```   567     and (Haskell) infixl 6 "-"
```
```   568     and (Scala) infixl 7 "-"
```
```   569     and (Eval) infixl 8 "-"
```
```   570 | constant Code_Numeral.dup \<rightharpoonup>
```
```   571     (SML) "IntInf.*/ (2,/ (_))"
```
```   572     and (OCaml) "Big'_int.mult'_big'_int/ (Big'_int.big'_int'_of'_int/ 2)"
```
```   573     and (Haskell) "!(2 * _)"
```
```   574     and (Scala) "!(2 * _)"
```
```   575     and (Eval) "!(2 * _)"
```
```   576 | constant Code_Numeral.sub \<rightharpoonup>
```
```   577     (SML) "!(raise/ Fail/ \"sub\")"
```
```   578     and (OCaml) "failwith/ \"sub\""
```
```   579     and (Haskell) "error/ \"sub\""
```
```   580     and (Scala) "!sys.error(\"sub\")"
```
```   581 | constant "times :: integer \<Rightarrow> _ \<Rightarrow> _" \<rightharpoonup>
```
```   582     (SML) "IntInf.* ((_), (_))"
```
```   583     and (OCaml) "Big'_int.mult'_big'_int"
```
```   584     and (Haskell) infixl 7 "*"
```
```   585     and (Scala) infixl 8 "*"
```
```   586     and (Eval) infixl 9 "*"
```
```   587 | constant Code_Numeral.divmod_abs \<rightharpoonup>
```
```   588     (SML) "IntInf.divMod/ (IntInf.abs _,/ IntInf.abs _)"
```
```   589     and (OCaml) "Big'_int.quomod'_big'_int/ (Big'_int.abs'_big'_int _)/ (Big'_int.abs'_big'_int _)"
```
```   590     and (Haskell) "divMod/ (abs _)/ (abs _)"
```
```   591     and (Scala) "!((k: BigInt) => (l: BigInt) =>/ if (l == 0)/ (BigInt(0), k) else/ (k.abs '/% l.abs))"
```
```   592     and (Eval) "Integer.div'_mod/ (abs _)/ (abs _)"
```
```   593 | constant "HOL.equal :: integer \<Rightarrow> _ \<Rightarrow> bool" \<rightharpoonup>
```
```   594     (SML) "!((_ : IntInf.int) = _)"
```
```   595     and (OCaml) "Big'_int.eq'_big'_int"
```
```   596     and (Haskell) infix 4 "=="
```
```   597     and (Scala) infixl 5 "=="
```
```   598     and (Eval) infixl 6 "="
```
```   599 | constant "less_eq :: integer \<Rightarrow> _ \<Rightarrow> bool" \<rightharpoonup>
```
```   600     (SML) "IntInf.<= ((_), (_))"
```
```   601     and (OCaml) "Big'_int.le'_big'_int"
```
```   602     and (Haskell) infix 4 "<="
```
```   603     and (Scala) infixl 4 "<="
```
```   604     and (Eval) infixl 6 "<="
```
```   605 | constant "less :: integer \<Rightarrow> _ \<Rightarrow> bool" \<rightharpoonup>
```
```   606     (SML) "IntInf.< ((_), (_))"
```
```   607     and (OCaml) "Big'_int.lt'_big'_int"
```
```   608     and (Haskell) infix 4 "<"
```
```   609     and (Scala) infixl 4 "<"
```
```   610     and (Eval) infixl 6 "<"
```
```   611
```
```   612 code_identifier
```
```   613   code_module Code_Numeral \<rightharpoonup> (SML) Arith and (OCaml) Arith and (Haskell) Arith
```
```   614
```
```   615
```
```   616 subsection \<open>Type of target language naturals\<close>
```
```   617
```
```   618 typedef natural = "UNIV :: nat set"
```
```   619   morphisms nat_of_natural natural_of_nat ..
```
```   620
```
```   621 setup_lifting type_definition_natural
```
```   622
```
```   623 lemma natural_eq_iff [termination_simp]:
```
```   624   "m = n \<longleftrightarrow> nat_of_natural m = nat_of_natural n"
```
```   625   by transfer rule
```
```   626
```
```   627 lemma natural_eqI:
```
```   628   "nat_of_natural m = nat_of_natural n \<Longrightarrow> m = n"
```
```   629   using natural_eq_iff [of m n] by simp
```
```   630
```
```   631 lemma nat_of_natural_of_nat_inverse [simp]:
```
```   632   "nat_of_natural (natural_of_nat n) = n"
```
```   633   by transfer rule
```
```   634
```
```   635 lemma natural_of_nat_of_natural_inverse [simp]:
```
```   636   "natural_of_nat (nat_of_natural n) = n"
```
```   637   by transfer rule
```
```   638
```
```   639 instantiation natural :: "{comm_monoid_diff, semiring_1}"
```
```   640 begin
```
```   641
```
```   642 lift_definition zero_natural :: natural
```
```   643   is "0 :: nat"
```
```   644   .
```
```   645
```
```   646 declare zero_natural.rep_eq [simp]
```
```   647
```
```   648 lift_definition one_natural :: natural
```
```   649   is "1 :: nat"
```
```   650   .
```
```   651
```
```   652 declare one_natural.rep_eq [simp]
```
```   653
```
```   654 lift_definition plus_natural :: "natural \<Rightarrow> natural \<Rightarrow> natural"
```
```   655   is "plus :: nat \<Rightarrow> nat \<Rightarrow> nat"
```
```   656   .
```
```   657
```
```   658 declare plus_natural.rep_eq [simp]
```
```   659
```
```   660 lift_definition minus_natural :: "natural \<Rightarrow> natural \<Rightarrow> natural"
```
```   661   is "minus :: nat \<Rightarrow> nat \<Rightarrow> nat"
```
```   662   .
```
```   663
```
```   664 declare minus_natural.rep_eq [simp]
```
```   665
```
```   666 lift_definition times_natural :: "natural \<Rightarrow> natural \<Rightarrow> natural"
```
```   667   is "times :: nat \<Rightarrow> nat \<Rightarrow> nat"
```
```   668   .
```
```   669
```
```   670 declare times_natural.rep_eq [simp]
```
```   671
```
```   672 instance proof
```
```   673 qed (transfer, simp add: algebra_simps)+
```
```   674
```
```   675 end
```
```   676
```
```   677 lemma [transfer_rule]:
```
```   678   "rel_fun HOL.eq pcr_natural (\<lambda>n::nat. n) (of_nat :: nat \<Rightarrow> natural)"
```
```   679 proof -
```
```   680   have "rel_fun HOL.eq pcr_natural (of_nat :: nat \<Rightarrow> nat) (of_nat :: nat \<Rightarrow> natural)"
```
```   681     by (unfold of_nat_def [abs_def]) transfer_prover
```
```   682   then show ?thesis by (simp add: id_def)
```
```   683 qed
```
```   684
```
```   685 lemma [transfer_rule]:
```
```   686   "rel_fun HOL.eq pcr_natural (numeral :: num \<Rightarrow> nat) (numeral :: num \<Rightarrow> natural)"
```
```   687 proof -
```
```   688   have "rel_fun HOL.eq pcr_natural (numeral :: num \<Rightarrow> nat) (\<lambda>n. of_nat (numeral n))"
```
```   689     by transfer_prover
```
```   690   then show ?thesis by simp
```
```   691 qed
```
```   692
```
```   693 lemma nat_of_natural_of_nat [simp]:
```
```   694   "nat_of_natural (of_nat n) = n"
```
```   695   by transfer rule
```
```   696
```
```   697 lemma natural_of_nat_of_nat [simp, code_abbrev]:
```
```   698   "natural_of_nat = of_nat"
```
```   699   by transfer rule
```
```   700
```
```   701 lemma of_nat_of_natural [simp]:
```
```   702   "of_nat (nat_of_natural n) = n"
```
```   703   by transfer rule
```
```   704
```
```   705 lemma nat_of_natural_numeral [simp]:
```
```   706   "nat_of_natural (numeral k) = numeral k"
```
```   707   by transfer rule
```
```   708
```
```   709 instantiation natural :: "{semiring_div, equal, linordered_semiring}"
```
```   710 begin
```
```   711
```
```   712 lift_definition divide_natural :: "natural \<Rightarrow> natural \<Rightarrow> natural"
```
```   713   is "divide :: nat \<Rightarrow> nat \<Rightarrow> nat"
```
```   714   .
```
```   715
```
```   716 declare divide_natural.rep_eq [simp]
```
```   717
```
```   718 lift_definition mod_natural :: "natural \<Rightarrow> natural \<Rightarrow> natural"
```
```   719   is "Divides.mod :: nat \<Rightarrow> nat \<Rightarrow> nat"
```
```   720   .
```
```   721
```
```   722 declare mod_natural.rep_eq [simp]
```
```   723
```
```   724 lift_definition less_eq_natural :: "natural \<Rightarrow> natural \<Rightarrow> bool"
```
```   725   is "less_eq :: nat \<Rightarrow> nat \<Rightarrow> bool"
```
```   726   .
```
```   727
```
```   728 declare less_eq_natural.rep_eq [termination_simp]
```
```   729
```
```   730 lift_definition less_natural :: "natural \<Rightarrow> natural \<Rightarrow> bool"
```
```   731   is "less :: nat \<Rightarrow> nat \<Rightarrow> bool"
```
```   732   .
```
```   733
```
```   734 declare less_natural.rep_eq [termination_simp]
```
```   735
```
```   736 lift_definition equal_natural :: "natural \<Rightarrow> natural \<Rightarrow> bool"
```
```   737   is "HOL.equal :: nat \<Rightarrow> nat \<Rightarrow> bool"
```
```   738   .
```
```   739
```
```   740 instance proof
```
```   741 qed (transfer, simp add: algebra_simps equal less_le_not_le [symmetric] linear)+
```
```   742
```
```   743 end
```
```   744
```
```   745 lemma [transfer_rule]:
```
```   746   "rel_fun pcr_natural (rel_fun pcr_natural pcr_natural) (min :: _ \<Rightarrow> _ \<Rightarrow> nat) (min :: _ \<Rightarrow> _ \<Rightarrow> natural)"
```
```   747   by (unfold min_def [abs_def]) transfer_prover
```
```   748
```
```   749 lemma [transfer_rule]:
```
```   750   "rel_fun pcr_natural (rel_fun pcr_natural pcr_natural) (max :: _ \<Rightarrow> _ \<Rightarrow> nat) (max :: _ \<Rightarrow> _ \<Rightarrow> natural)"
```
```   751   by (unfold max_def [abs_def]) transfer_prover
```
```   752
```
```   753 lemma nat_of_natural_min [simp]:
```
```   754   "nat_of_natural (min k l) = min (nat_of_natural k) (nat_of_natural l)"
```
```   755   by transfer rule
```
```   756
```
```   757 lemma nat_of_natural_max [simp]:
```
```   758   "nat_of_natural (max k l) = max (nat_of_natural k) (nat_of_natural l)"
```
```   759   by transfer rule
```
```   760
```
```   761 lift_definition natural_of_integer :: "integer \<Rightarrow> natural"
```
```   762   is "nat :: int \<Rightarrow> nat"
```
```   763   .
```
```   764
```
```   765 lift_definition integer_of_natural :: "natural \<Rightarrow> integer"
```
```   766   is "of_nat :: nat \<Rightarrow> int"
```
```   767   .
```
```   768
```
```   769 lemma natural_of_integer_of_natural [simp]:
```
```   770   "natural_of_integer (integer_of_natural n) = n"
```
```   771   by transfer simp
```
```   772
```
```   773 lemma integer_of_natural_of_integer [simp]:
```
```   774   "integer_of_natural (natural_of_integer k) = max 0 k"
```
```   775   by transfer auto
```
```   776
```
```   777 lemma int_of_integer_of_natural [simp]:
```
```   778   "int_of_integer (integer_of_natural n) = of_nat (nat_of_natural n)"
```
```   779   by transfer rule
```
```   780
```
```   781 lemma integer_of_natural_of_nat [simp]:
```
```   782   "integer_of_natural (of_nat n) = of_nat n"
```
```   783   by transfer rule
```
```   784
```
```   785 lemma [measure_function]:
```
```   786   "is_measure nat_of_natural"
```
```   787   by (rule is_measure_trivial)
```
```   788
```
```   789
```
```   790 subsection \<open>Inductive representation of target language naturals\<close>
```
```   791
```
```   792 lift_definition Suc :: "natural \<Rightarrow> natural"
```
```   793   is Nat.Suc
```
```   794   .
```
```   795
```
```   796 declare Suc.rep_eq [simp]
```
```   797
```
```   798 old_rep_datatype "0::natural" Suc
```
```   799   by (transfer, fact nat.induct nat.inject nat.distinct)+
```
```   800
```
```   801 lemma natural_cases [case_names nat, cases type: natural]:
```
```   802   fixes m :: natural
```
```   803   assumes "\<And>n. m = of_nat n \<Longrightarrow> P"
```
```   804   shows P
```
```   805   using assms by transfer blast
```
```   806
```
```   807 lemma [simp, code]: "size_natural = nat_of_natural"
```
```   808 proof (rule ext)
```
```   809   fix n
```
```   810   show "size_natural n = nat_of_natural n"
```
```   811     by (induct n) simp_all
```
```   812 qed
```
```   813
```
```   814 lemma [simp, code]: "size = nat_of_natural"
```
```   815 proof (rule ext)
```
```   816   fix n
```
```   817   show "size n = nat_of_natural n"
```
```   818     by (induct n) simp_all
```
```   819 qed
```
```   820
```
```   821 lemma natural_decr [termination_simp]:
```
```   822   "n \<noteq> 0 \<Longrightarrow> nat_of_natural n - Nat.Suc 0 < nat_of_natural n"
```
```   823   by transfer simp
```
```   824
```
```   825 lemma natural_zero_minus_one: "(0::natural) - 1 = 0"
```
```   826   by (rule zero_diff)
```
```   827
```
```   828 lemma Suc_natural_minus_one: "Suc n - 1 = n"
```
```   829   by transfer simp
```
```   830
```
```   831 hide_const (open) Suc
```
```   832
```
```   833
```
```   834 subsection \<open>Code refinement for target language naturals\<close>
```
```   835
```
```   836 lift_definition Nat :: "integer \<Rightarrow> natural"
```
```   837   is nat
```
```   838   .
```
```   839
```
```   840 lemma [code_post]:
```
```   841   "Nat 0 = 0"
```
```   842   "Nat 1 = 1"
```
```   843   "Nat (numeral k) = numeral k"
```
```   844   by (transfer, simp)+
```
```   845
```
```   846 lemma [code abstype]:
```
```   847   "Nat (integer_of_natural n) = n"
```
```   848   by transfer simp
```
```   849
```
```   850 lemma [code abstract]:
```
```   851   "integer_of_natural (natural_of_nat n) = of_nat n"
```
```   852   by simp
```
```   853
```
```   854 lemma [code abstract]:
```
```   855   "integer_of_natural (natural_of_integer k) = max 0 k"
```
```   856   by simp
```
```   857
```
```   858 lemma [code_abbrev]:
```
```   859   "natural_of_integer (Code_Numeral.Pos k) = numeral k"
```
```   860   by transfer simp
```
```   861
```
```   862 lemma [code abstract]:
```
```   863   "integer_of_natural 0 = 0"
```
```   864   by transfer simp
```
```   865
```
```   866 lemma [code abstract]:
```
```   867   "integer_of_natural 1 = 1"
```
```   868   by transfer simp
```
```   869
```
```   870 lemma [code abstract]:
```
```   871   "integer_of_natural (Code_Numeral.Suc n) = integer_of_natural n + 1"
```
```   872   by transfer simp
```
```   873
```
```   874 lemma [code]:
```
```   875   "nat_of_natural = nat_of_integer \<circ> integer_of_natural"
```
```   876   by transfer (simp add: fun_eq_iff)
```
```   877
```
```   878 lemma [code, code_unfold]:
```
```   879   "case_natural f g n = (if n = 0 then f else g (n - 1))"
```
```   880   by (cases n rule: natural.exhaust) (simp_all, simp add: Suc_def)
```
```   881
```
```   882 declare natural.rec [code del]
```
```   883
```
```   884 lemma [code abstract]:
```
```   885   "integer_of_natural (m + n) = integer_of_natural m + integer_of_natural n"
```
```   886   by transfer simp
```
```   887
```
```   888 lemma [code abstract]:
```
```   889   "integer_of_natural (m - n) = max 0 (integer_of_natural m - integer_of_natural n)"
```
```   890   by transfer simp
```
```   891
```
```   892 lemma [code abstract]:
```
```   893   "integer_of_natural (m * n) = integer_of_natural m * integer_of_natural n"
```
```   894   by transfer (simp add: of_nat_mult)
```
```   895
```
```   896 lemma [code abstract]:
```
```   897   "integer_of_natural (m div n) = integer_of_natural m div integer_of_natural n"
```
```   898   by transfer (simp add: zdiv_int)
```
```   899
```
```   900 lemma [code abstract]:
```
```   901   "integer_of_natural (m mod n) = integer_of_natural m mod integer_of_natural n"
```
```   902   by transfer (simp add: zmod_int)
```
```   903
```
```   904 lemma [code]:
```
```   905   "HOL.equal m n \<longleftrightarrow> HOL.equal (integer_of_natural m) (integer_of_natural n)"
```
```   906   by transfer (simp add: equal)
```
```   907
```
```   908 lemma [code nbe]: "HOL.equal n (n::natural) \<longleftrightarrow> True"
```
```   909   by (rule equal_class.equal_refl)
```
```   910
```
```   911 lemma [code]: "m \<le> n \<longleftrightarrow> integer_of_natural m \<le> integer_of_natural n"
```
```   912   by transfer simp
```
```   913
```
```   914 lemma [code]: "m < n \<longleftrightarrow> integer_of_natural m < integer_of_natural n"
```
```   915   by transfer simp
```
```   916
```
```   917 hide_const (open) Nat
```
```   918
```
```   919 lifting_update integer.lifting
```
```   920 lifting_forget integer.lifting
```
```   921
```
```   922 lifting_update natural.lifting
```
```   923 lifting_forget natural.lifting
```
```   924
```
```   925 code_reflect Code_Numeral
```
```   926   datatypes natural = _
```
```   927   functions integer_of_natural natural_of_integer
```
```   928
```
```   929 end
```