src/HOL/Library/Target_Numeral.thy
 author Christian Sternagel Thu Aug 30 15:44:03 2012 +0900 (2012-08-30) changeset 49093 fdc301f592c4 parent 48075 ec5e62b868eb child 49834 b27bbb021df1 permissions -rw-r--r--
```     1 theory Target_Numeral
```
```     2 imports Main Code_Nat
```
```     3 begin
```
```     4
```
```     5 subsection {* Type of target language numerals *}
```
```     6
```
```     7 typedef (open) int = "UNIV \<Colon> int set"
```
```     8   morphisms int_of of_int ..
```
```     9
```
```    10 hide_type (open) int
```
```    11 hide_const (open) of_int
```
```    12
```
```    13 lemma int_eq_iff:
```
```    14   "k = l \<longleftrightarrow> int_of k = int_of l"
```
```    15   using int_of_inject [of k l] ..
```
```    16
```
```    17 lemma int_eqI:
```
```    18   "int_of k = int_of l \<Longrightarrow> k = l"
```
```    19   using int_eq_iff [of k l] by simp
```
```    20
```
```    21 lemma int_of_int [simp]:
```
```    22   "int_of (Target_Numeral.of_int k) = k"
```
```    23   using of_int_inverse [of k] by simp
```
```    24
```
```    25 lemma of_int_of [simp]:
```
```    26   "Target_Numeral.of_int (int_of k) = k"
```
```    27   using int_of_inverse [of k] by simp
```
```    28
```
```    29 hide_fact (open) int_eq_iff int_eqI
```
```    30
```
```    31 instantiation Target_Numeral.int :: ring_1
```
```    32 begin
```
```    33
```
```    34 definition
```
```    35   "0 = Target_Numeral.of_int 0"
```
```    36
```
```    37 lemma int_of_zero [simp]:
```
```    38   "int_of 0 = 0"
```
```    39   by (simp add: zero_int_def)
```
```    40
```
```    41 definition
```
```    42   "1 = Target_Numeral.of_int 1"
```
```    43
```
```    44 lemma int_of_one [simp]:
```
```    45   "int_of 1 = 1"
```
```    46   by (simp add: one_int_def)
```
```    47
```
```    48 definition
```
```    49   "k + l = Target_Numeral.of_int (int_of k + int_of l)"
```
```    50
```
```    51 lemma int_of_plus [simp]:
```
```    52   "int_of (k + l) = int_of k + int_of l"
```
```    53   by (simp add: plus_int_def)
```
```    54
```
```    55 definition
```
```    56   "- k = Target_Numeral.of_int (- int_of k)"
```
```    57
```
```    58 lemma int_of_uminus [simp]:
```
```    59   "int_of (- k) = - int_of k"
```
```    60   by (simp add: uminus_int_def)
```
```    61
```
```    62 definition
```
```    63   "k - l = Target_Numeral.of_int (int_of k - int_of l)"
```
```    64
```
```    65 lemma int_of_minus [simp]:
```
```    66   "int_of (k - l) = int_of k - int_of l"
```
```    67   by (simp add: minus_int_def)
```
```    68
```
```    69 definition
```
```    70   "k * l = Target_Numeral.of_int (int_of k * int_of l)"
```
```    71
```
```    72 lemma int_of_times [simp]:
```
```    73   "int_of (k * l) = int_of k * int_of l"
```
```    74   by (simp add: times_int_def)
```
```    75
```
```    76 instance proof
```
```    77 qed (auto simp add: Target_Numeral.int_eq_iff algebra_simps)
```
```    78
```
```    79 end
```
```    80
```
```    81 lemma int_of_of_nat [simp]:
```
```    82   "int_of (of_nat n) = of_nat n"
```
```    83   by (induct n) simp_all
```
```    84
```
```    85 definition nat_of :: "Target_Numeral.int \<Rightarrow> nat" where
```
```    86   "nat_of k = Int.nat (int_of k)"
```
```    87
```
```    88 lemma nat_of_of_nat [simp]:
```
```    89   "nat_of (of_nat n) = n"
```
```    90   by (simp add: nat_of_def)
```
```    91
```
```    92 lemma int_of_of_int [simp]:
```
```    93   "int_of (of_int k) = k"
```
```    94   by (induct k) (simp_all, simp only: neg_numeral_def numeral_One int_of_uminus int_of_one)
```
```    95
```
```    96 lemma of_int_of_int [simp, code_abbrev]:
```
```    97   "Target_Numeral.of_int = of_int"
```
```    98   by rule (simp add: Target_Numeral.int_eq_iff)
```
```    99
```
```   100 lemma int_of_numeral [simp]:
```
```   101   "int_of (numeral k) = numeral k"
```
```   102   using int_of_of_int [of "numeral k"] by simp
```
```   103
```
```   104 lemma int_of_neg_numeral [simp]:
```
```   105   "int_of (neg_numeral k) = neg_numeral k"
```
```   106   by (simp only: neg_numeral_def int_of_uminus) simp
```
```   107
```
```   108 lemma int_of_sub [simp]:
```
```   109   "int_of (Num.sub k l) = Num.sub k l"
```
```   110   by (simp only: Num.sub_def int_of_minus int_of_numeral)
```
```   111
```
```   112 instantiation Target_Numeral.int :: "{ring_div, equal, linordered_idom}"
```
```   113 begin
```
```   114
```
```   115 definition
```
```   116   "k div l = of_int (int_of k div int_of l)"
```
```   117
```
```   118 lemma int_of_div [simp]:
```
```   119   "int_of (k div l) = int_of k div int_of l"
```
```   120   by (simp add: div_int_def)
```
```   121
```
```   122 definition
```
```   123   "k mod l = of_int (int_of k mod int_of l)"
```
```   124
```
```   125 lemma int_of_mod [simp]:
```
```   126   "int_of (k mod l) = int_of k mod int_of l"
```
```   127   by (simp add: mod_int_def)
```
```   128
```
```   129 definition
```
```   130   "\<bar>k\<bar> = of_int \<bar>int_of k\<bar>"
```
```   131
```
```   132 lemma int_of_abs [simp]:
```
```   133   "int_of \<bar>k\<bar> = \<bar>int_of k\<bar>"
```
```   134   by (simp add: abs_int_def)
```
```   135
```
```   136 definition
```
```   137   "sgn k = of_int (sgn (int_of k))"
```
```   138
```
```   139 lemma int_of_sgn [simp]:
```
```   140   "int_of (sgn k) = sgn (int_of k)"
```
```   141   by (simp add: sgn_int_def)
```
```   142
```
```   143 definition
```
```   144   "k \<le> l \<longleftrightarrow> int_of k \<le> int_of l"
```
```   145
```
```   146 definition
```
```   147   "k < l \<longleftrightarrow> int_of k < int_of l"
```
```   148
```
```   149 definition
```
```   150   "HOL.equal k l \<longleftrightarrow> HOL.equal (int_of k) (int_of l)"
```
```   151
```
```   152 instance proof
```
```   153 qed (auto simp add: Target_Numeral.int_eq_iff algebra_simps
```
```   154   less_eq_int_def less_int_def equal_int_def equal)
```
```   155
```
```   156 end
```
```   157
```
```   158 lemma int_of_min [simp]:
```
```   159   "int_of (min k l) = min (int_of k) (int_of l)"
```
```   160   by (simp add: min_def less_eq_int_def)
```
```   161
```
```   162 lemma int_of_max [simp]:
```
```   163   "int_of (max k l) = max (int_of k) (int_of l)"
```
```   164   by (simp add: max_def less_eq_int_def)
```
```   165
```
```   166 lemma of_nat_nat_of [simp]:
```
```   167   "of_nat (nat_of k) = max 0 k"
```
```   168   by (simp add: nat_of_def Target_Numeral.int_eq_iff less_eq_int_def max_def)
```
```   169
```
```   170
```
```   171 subsection {* Code theorems for target language numerals *}
```
```   172
```
```   173 text {* Constructors *}
```
```   174
```
```   175 definition Pos :: "num \<Rightarrow> Target_Numeral.int" where
```
```   176   [simp, code_abbrev]: "Pos = numeral"
```
```   177
```
```   178 definition Neg :: "num \<Rightarrow> Target_Numeral.int" where
```
```   179   [simp, code_abbrev]: "Neg = neg_numeral"
```
```   180
```
```   181 code_datatype "0::Target_Numeral.int" Pos Neg
```
```   182
```
```   183
```
```   184 text {* Auxiliary operations *}
```
```   185
```
```   186 definition dup :: "Target_Numeral.int \<Rightarrow> Target_Numeral.int" where
```
```   187   [simp]: "dup k = k + k"
```
```   188
```
```   189 lemma dup_code [code]:
```
```   190   "dup 0 = 0"
```
```   191   "dup (Pos n) = Pos (Num.Bit0 n)"
```
```   192   "dup (Neg n) = Neg (Num.Bit0 n)"
```
```   193   unfolding Pos_def Neg_def neg_numeral_def
```
```   194   by (simp_all add: numeral_Bit0)
```
```   195
```
```   196 definition sub :: "num \<Rightarrow> num \<Rightarrow> Target_Numeral.int" where
```
```   197   [simp]: "sub m n = numeral m - numeral n"
```
```   198
```
```   199 lemma sub_code [code]:
```
```   200   "sub Num.One Num.One = 0"
```
```   201   "sub (Num.Bit0 m) Num.One = Pos (Num.BitM m)"
```
```   202   "sub (Num.Bit1 m) Num.One = Pos (Num.Bit0 m)"
```
```   203   "sub Num.One (Num.Bit0 n) = Neg (Num.BitM n)"
```
```   204   "sub Num.One (Num.Bit1 n) = Neg (Num.Bit0 n)"
```
```   205   "sub (Num.Bit0 m) (Num.Bit0 n) = dup (sub m n)"
```
```   206   "sub (Num.Bit1 m) (Num.Bit1 n) = dup (sub m n)"
```
```   207   "sub (Num.Bit1 m) (Num.Bit0 n) = dup (sub m n) + 1"
```
```   208   "sub (Num.Bit0 m) (Num.Bit1 n) = dup (sub m n) - 1"
```
```   209   unfolding sub_def dup_def numeral.simps Pos_def Neg_def
```
```   210     neg_numeral_def numeral_BitM
```
```   211   by (simp_all only: algebra_simps add.comm_neutral)
```
```   212
```
```   213
```
```   214 text {* Implementations *}
```
```   215
```
```   216 lemma one_int_code [code, code_unfold]:
```
```   217   "1 = Pos Num.One"
```
```   218   by simp
```
```   219
```
```   220 lemma plus_int_code [code]:
```
```   221   "k + 0 = (k::Target_Numeral.int)"
```
```   222   "0 + l = (l::Target_Numeral.int)"
```
```   223   "Pos m + Pos n = Pos (m + n)"
```
```   224   "Pos m + Neg n = sub m n"
```
```   225   "Neg m + Pos n = sub n m"
```
```   226   "Neg m + Neg n = Neg (m + n)"
```
```   227   by simp_all
```
```   228
```
```   229 lemma uminus_int_code [code]:
```
```   230   "uminus 0 = (0::Target_Numeral.int)"
```
```   231   "uminus (Pos m) = Neg m"
```
```   232   "uminus (Neg m) = Pos m"
```
```   233   by simp_all
```
```   234
```
```   235 lemma minus_int_code [code]:
```
```   236   "k - 0 = (k::Target_Numeral.int)"
```
```   237   "0 - l = uminus (l::Target_Numeral.int)"
```
```   238   "Pos m - Pos n = sub m n"
```
```   239   "Pos m - Neg n = Pos (m + n)"
```
```   240   "Neg m - Pos n = Neg (m + n)"
```
```   241   "Neg m - Neg n = sub n m"
```
```   242   by simp_all
```
```   243
```
```   244 lemma times_int_code [code]:
```
```   245   "k * 0 = (0::Target_Numeral.int)"
```
```   246   "0 * l = (0::Target_Numeral.int)"
```
```   247   "Pos m * Pos n = Pos (m * n)"
```
```   248   "Pos m * Neg n = Neg (m * n)"
```
```   249   "Neg m * Pos n = Neg (m * n)"
```
```   250   "Neg m * Neg n = Pos (m * n)"
```
```   251   by simp_all
```
```   252
```
```   253 definition divmod :: "Target_Numeral.int \<Rightarrow> Target_Numeral.int \<Rightarrow> Target_Numeral.int \<times> Target_Numeral.int" where
```
```   254   "divmod k l = (k div l, k mod l)"
```
```   255
```
```   256 lemma fst_divmod [simp]:
```
```   257   "fst (divmod k l) = k div l"
```
```   258   by (simp add: divmod_def)
```
```   259
```
```   260 lemma snd_divmod [simp]:
```
```   261   "snd (divmod k l) = k mod l"
```
```   262   by (simp add: divmod_def)
```
```   263
```
```   264 definition divmod_abs :: "Target_Numeral.int \<Rightarrow> Target_Numeral.int \<Rightarrow> Target_Numeral.int \<times> Target_Numeral.int" where
```
```   265   "divmod_abs k l = (\<bar>k\<bar> div \<bar>l\<bar>, \<bar>k\<bar> mod \<bar>l\<bar>)"
```
```   266
```
```   267 lemma fst_divmod_abs [simp]:
```
```   268   "fst (divmod_abs k l) = \<bar>k\<bar> div \<bar>l\<bar>"
```
```   269   by (simp add: divmod_abs_def)
```
```   270
```
```   271 lemma snd_divmod_abs [simp]:
```
```   272   "snd (divmod_abs k l) = \<bar>k\<bar> mod \<bar>l\<bar>"
```
```   273   by (simp add: divmod_abs_def)
```
```   274
```
```   275 lemma divmod_abs_terminate_code [code]:
```
```   276   "divmod_abs (Neg k) (Neg l) = divmod_abs (Pos k) (Pos l)"
```
```   277   "divmod_abs (Neg k) (Pos l) = divmod_abs (Pos k) (Pos l)"
```
```   278   "divmod_abs (Pos k) (Neg l) = divmod_abs (Pos k) (Pos l)"
```
```   279   "divmod_abs j 0 = (0, \<bar>j\<bar>)"
```
```   280   "divmod_abs 0 j = (0, 0)"
```
```   281   by (simp_all add: prod_eq_iff)
```
```   282
```
```   283 lemma divmod_abs_rec_code [code]:
```
```   284   "divmod_abs (Pos k) (Pos l) =
```
```   285     (let j = sub k l in
```
```   286        if j < 0 then (0, Pos k)
```
```   287        else let (q, r) = divmod_abs j (Pos l) in (q + 1, r))"
```
```   288   by (auto simp add: prod_eq_iff Target_Numeral.int_eq_iff Let_def prod_case_beta
```
```   289     sub_non_negative sub_negative div_pos_pos_trivial mod_pos_pos_trivial div_pos_geq mod_pos_geq)
```
```   290
```
```   291 lemma divmod_code [code]: "divmod k l =
```
```   292   (if k = 0 then (0, 0) else if l = 0 then (0, k) else
```
```   293   (apsnd \<circ> times \<circ> sgn) l (if sgn k = sgn l
```
```   294     then divmod_abs k l
```
```   295     else (let (r, s) = divmod_abs k l in
```
```   296       if s = 0 then (- r, 0) else (- r - 1, \<bar>l\<bar> - s))))"
```
```   297 proof -
```
```   298   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"
```
```   299     by (auto simp add: sgn_if)
```
```   300   have aux2: "\<And>q::int. - int_of k = int_of l * q \<longleftrightarrow> int_of k = int_of l * - q" by auto
```
```   301   show ?thesis
```
```   302     by (simp add: prod_eq_iff Target_Numeral.int_eq_iff prod_case_beta aux1)
```
```   303       (auto simp add: zdiv_zminus1_eq_if zmod_zminus1_eq_if div_minus_right mod_minus_right aux2)
```
```   304 qed
```
```   305
```
```   306 lemma div_int_code [code]:
```
```   307   "k div l = fst (divmod k l)"
```
```   308   by simp
```
```   309
```
```   310 lemma div_mod_code [code]:
```
```   311   "k mod l = snd (divmod k l)"
```
```   312   by simp
```
```   313
```
```   314 lemma equal_int_code [code]:
```
```   315   "HOL.equal 0 (0::Target_Numeral.int) \<longleftrightarrow> True"
```
```   316   "HOL.equal 0 (Pos l) \<longleftrightarrow> False"
```
```   317   "HOL.equal 0 (Neg l) \<longleftrightarrow> False"
```
```   318   "HOL.equal (Pos k) 0 \<longleftrightarrow> False"
```
```   319   "HOL.equal (Pos k) (Pos l) \<longleftrightarrow> HOL.equal k l"
```
```   320   "HOL.equal (Pos k) (Neg l) \<longleftrightarrow> False"
```
```   321   "HOL.equal (Neg k) 0 \<longleftrightarrow> False"
```
```   322   "HOL.equal (Neg k) (Pos l) \<longleftrightarrow> False"
```
```   323   "HOL.equal (Neg k) (Neg l) \<longleftrightarrow> HOL.equal k l"
```
```   324   by (simp_all add: equal Target_Numeral.int_eq_iff)
```
```   325
```
```   326 lemma equal_int_refl [code nbe]:
```
```   327   "HOL.equal (k::Target_Numeral.int) k \<longleftrightarrow> True"
```
```   328   by (fact equal_refl)
```
```   329
```
```   330 lemma less_eq_int_code [code]:
```
```   331   "0 \<le> (0::Target_Numeral.int) \<longleftrightarrow> True"
```
```   332   "0 \<le> Pos l \<longleftrightarrow> True"
```
```   333   "0 \<le> Neg l \<longleftrightarrow> False"
```
```   334   "Pos k \<le> 0 \<longleftrightarrow> False"
```
```   335   "Pos k \<le> Pos l \<longleftrightarrow> k \<le> l"
```
```   336   "Pos k \<le> Neg l \<longleftrightarrow> False"
```
```   337   "Neg k \<le> 0 \<longleftrightarrow> True"
```
```   338   "Neg k \<le> Pos l \<longleftrightarrow> True"
```
```   339   "Neg k \<le> Neg l \<longleftrightarrow> l \<le> k"
```
```   340   by (simp_all add: less_eq_int_def)
```
```   341
```
```   342 lemma less_int_code [code]:
```
```   343   "0 < (0::Target_Numeral.int) \<longleftrightarrow> False"
```
```   344   "0 < Pos l \<longleftrightarrow> True"
```
```   345   "0 < Neg l \<longleftrightarrow> False"
```
```   346   "Pos k < 0 \<longleftrightarrow> False"
```
```   347   "Pos k < Pos l \<longleftrightarrow> k < l"
```
```   348   "Pos k < Neg l \<longleftrightarrow> False"
```
```   349   "Neg k < 0 \<longleftrightarrow> True"
```
```   350   "Neg k < Pos l \<longleftrightarrow> True"
```
```   351   "Neg k < Neg l \<longleftrightarrow> l < k"
```
```   352   by (simp_all add: less_int_def)
```
```   353
```
```   354 lemma nat_of_code [code]:
```
```   355   "nat_of (Neg k) = 0"
```
```   356   "nat_of 0 = 0"
```
```   357   "nat_of (Pos k) = nat_of_num k"
```
```   358   by (simp_all add: nat_of_def nat_of_num_numeral)
```
```   359
```
```   360 lemma int_of_code [code]:
```
```   361   "int_of (Neg k) = neg_numeral k"
```
```   362   "int_of 0 = 0"
```
```   363   "int_of (Pos k) = numeral k"
```
```   364   by simp_all
```
```   365
```
```   366 lemma of_int_code [code]:
```
```   367   "Target_Numeral.of_int (Int.Neg k) = neg_numeral k"
```
```   368   "Target_Numeral.of_int 0 = 0"
```
```   369   "Target_Numeral.of_int (Int.Pos k) = numeral k"
```
```   370   by simp_all
```
```   371
```
```   372 definition num_of_int :: "Target_Numeral.int \<Rightarrow> num" where
```
```   373   "num_of_int = num_of_nat \<circ> nat_of"
```
```   374
```
```   375 lemma num_of_int_code [code]:
```
```   376   "num_of_int k = (if k \<le> 1 then Num.One
```
```   377      else let
```
```   378        (l, j) = divmod k 2;
```
```   379        l' = num_of_int l + num_of_int l
```
```   380      in if j = 0 then l' else l' + Num.One)"
```
```   381 proof -
```
```   382   {
```
```   383     assume "int_of k mod 2 = 1"
```
```   384     then have "nat (int_of k mod 2) = nat 1" by simp
```
```   385     moreover assume *: "1 < int_of k"
```
```   386     ultimately have **: "nat (int_of k) mod 2 = 1" by (simp add: nat_mod_distrib)
```
```   387     have "num_of_nat (nat (int_of k)) =
```
```   388       num_of_nat (2 * (nat (int_of k) div 2) + nat (int_of k) mod 2)"
```
```   389       by simp
```
```   390     then have "num_of_nat (nat (int_of k)) =
```
```   391       num_of_nat (nat (int_of k) div 2 + nat (int_of k) div 2 + nat (int_of k) mod 2)"
```
```   392       by (simp add: mult_2)
```
```   393     with ** have "num_of_nat (nat (int_of k)) =
```
```   394       num_of_nat (nat (int_of k) div 2 + nat (int_of k) div 2 + 1)"
```
```   395       by simp
```
```   396   }
```
```   397   note aux = this
```
```   398   show ?thesis
```
```   399     by (auto simp add: num_of_int_def nat_of_def Let_def prod_case_beta
```
```   400       not_le Target_Numeral.int_eq_iff less_eq_int_def
```
```   401       nat_mult_distrib nat_div_distrib num_of_nat_One num_of_nat_plus_distrib
```
```   402        mult_2 [where 'a=nat] aux add_One)
```
```   403 qed
```
```   404
```
```   405 hide_const (open) int_of nat_of Pos Neg sub dup divmod_abs num_of_int
```
```   406
```
```   407
```
```   408 subsection {* Serializer setup for target language numerals *}
```
```   409
```
```   410 code_type Target_Numeral.int
```
```   411   (SML "IntInf.int")
```
```   412   (OCaml "Big'_int.big'_int")
```
```   413   (Haskell "Integer")
```
```   414   (Scala "BigInt")
```
```   415   (Eval "int")
```
```   416
```
```   417 code_instance Target_Numeral.int :: equal
```
```   418   (Haskell -)
```
```   419
```
```   420 code_const "0::Target_Numeral.int"
```
```   421   (SML "0")
```
```   422   (OCaml "Big'_int.zero'_big'_int")
```
```   423   (Haskell "0")
```
```   424   (Scala "BigInt(0)")
```
```   425
```
```   426 setup {*
```
```   427   fold (Numeral.add_code @{const_name Target_Numeral.Pos}
```
```   428     false Code_Printer.literal_numeral) ["SML", "OCaml", "Haskell", "Scala"]
```
```   429 *}
```
```   430
```
```   431 setup {*
```
```   432   fold (Numeral.add_code @{const_name Target_Numeral.Neg}
```
```   433     true Code_Printer.literal_numeral) ["SML", "OCaml", "Haskell", "Scala"]
```
```   434 *}
```
```   435
```
```   436 code_const "plus :: Target_Numeral.int \<Rightarrow> _ \<Rightarrow> _"
```
```   437   (SML "IntInf.+ ((_), (_))")
```
```   438   (OCaml "Big'_int.add'_big'_int")
```
```   439   (Haskell infixl 6 "+")
```
```   440   (Scala infixl 7 "+")
```
```   441   (Eval infixl 8 "+")
```
```   442
```
```   443 code_const "uminus :: Target_Numeral.int \<Rightarrow> _"
```
```   444   (SML "IntInf.~")
```
```   445   (OCaml "Big'_int.minus'_big'_int")
```
```   446   (Haskell "negate")
```
```   447   (Scala "!(- _)")
```
```   448   (Eval "~/ _")
```
```   449
```
```   450 code_const "minus :: Target_Numeral.int \<Rightarrow> _"
```
```   451   (SML "IntInf.- ((_), (_))")
```
```   452   (OCaml "Big'_int.sub'_big'_int")
```
```   453   (Haskell infixl 6 "-")
```
```   454   (Scala infixl 7 "-")
```
```   455   (Eval infixl 8 "-")
```
```   456
```
```   457 code_const Target_Numeral.dup
```
```   458   (SML "IntInf.*/ (2,/ (_))")
```
```   459   (OCaml "Big'_int.mult'_big'_int/ 2")
```
```   460   (Haskell "!(2 * _)")
```
```   461   (Scala "!(2 * _)")
```
```   462   (Eval "!(2 * _)")
```
```   463
```
```   464 code_const Target_Numeral.sub
```
```   465   (SML "!(raise/ Fail/ \"sub\")")
```
```   466   (OCaml "failwith/ \"sub\"")
```
```   467   (Haskell "error/ \"sub\"")
```
```   468   (Scala "!sys.error(\"sub\")")
```
```   469
```
```   470 code_const "times :: Target_Numeral.int \<Rightarrow> _ \<Rightarrow> _"
```
```   471   (SML "IntInf.* ((_), (_))")
```
```   472   (OCaml "Big'_int.mult'_big'_int")
```
```   473   (Haskell infixl 7 "*")
```
```   474   (Scala infixl 8 "*")
```
```   475   (Eval infixl 9 "*")
```
```   476
```
```   477 code_const Target_Numeral.divmod_abs
```
```   478   (SML "IntInf.divMod/ (IntInf.abs _,/ IntInf.abs _)")
```
```   479   (OCaml "Big'_int.quomod'_big'_int/ (Big'_int.abs'_big'_int _)/ (Big'_int.abs'_big'_int _)")
```
```   480   (Haskell "divMod/ (abs _)/ (abs _)")
```
```   481   (Scala "!((k: BigInt) => (l: BigInt) =>/ if (l == 0)/ (BigInt(0), k) else/ (k.abs '/% l.abs))")
```
```   482   (Eval "Integer.div'_mod/ (abs _)/ (abs _)")
```
```   483
```
```   484 code_const "HOL.equal :: Target_Numeral.int \<Rightarrow> _ \<Rightarrow> bool"
```
```   485   (SML "!((_ : IntInf.int) = _)")
```
```   486   (OCaml "Big'_int.eq'_big'_int")
```
```   487   (Haskell infix 4 "==")
```
```   488   (Scala infixl 5 "==")
```
```   489   (Eval infixl 6 "=")
```
```   490
```
```   491 code_const "less_eq :: Target_Numeral.int \<Rightarrow> _ \<Rightarrow> bool"
```
```   492   (SML "IntInf.<= ((_), (_))")
```
```   493   (OCaml "Big'_int.le'_big'_int")
```
```   494   (Haskell infix 4 "<=")
```
```   495   (Scala infixl 4 "<=")
```
```   496   (Eval infixl 6 "<=")
```
```   497
```
```   498 code_const "less :: Target_Numeral.int \<Rightarrow> _ \<Rightarrow> bool"
```
```   499   (SML "IntInf.< ((_), (_))")
```
```   500   (OCaml "Big'_int.lt'_big'_int")
```
```   501   (Haskell infix 4 "<")
```
```   502   (Scala infixl 4 "<")
```
```   503   (Eval infixl 6 "<")
```
```   504
```
```   505 ML {*
```
```   506 structure Target_Numeral =
```
```   507 struct
```
```   508
```
```   509 val T = @{typ "Target_Numeral.int"};
```
```   510
```
```   511 end;
```
```   512 *}
```
```   513
```
```   514 code_reserved Eval Target_Numeral
```
```   515
```
```   516 code_const "Code_Evaluation.term_of \<Colon> Target_Numeral.int \<Rightarrow> term"
```
```   517   (Eval "HOLogic.mk'_number/ Target'_Numeral.T")
```
```   518
```
```   519 code_modulename SML
```
```   520   Target_Numeral Arith
```
```   521
```
```   522 code_modulename OCaml
```
```   523   Target_Numeral Arith
```
```   524
```
```   525 code_modulename Haskell
```
```   526   Target_Numeral Arith
```
```   527
```
```   528
```
```   529 subsection {* Implementation for @{typ int} *}
```
```   530
```
```   531 code_datatype Target_Numeral.int_of
```
```   532
```
```   533 lemma [code, code del]:
```
```   534   "Target_Numeral.of_int = Target_Numeral.of_int" ..
```
```   535
```
```   536 lemma [code]:
```
```   537   "Target_Numeral.of_int (Target_Numeral.int_of k) = k"
```
```   538   by (simp add: Target_Numeral.int_eq_iff)
```
```   539
```
```   540 declare Int.Pos_def [code]
```
```   541
```
```   542 lemma [code_abbrev]:
```
```   543   "Target_Numeral.int_of (Target_Numeral.Pos k) = Int.Pos k"
```
```   544   by simp
```
```   545
```
```   546 declare Int.Neg_def [code]
```
```   547
```
```   548 lemma [code_abbrev]:
```
```   549   "Target_Numeral.int_of (Target_Numeral.Neg k) = Int.Neg k"
```
```   550   by simp
```
```   551
```
```   552 lemma [code]:
```
```   553   "0 = Target_Numeral.int_of 0"
```
```   554   by simp
```
```   555
```
```   556 lemma [code]:
```
```   557   "1 = Target_Numeral.int_of 1"
```
```   558   by simp
```
```   559
```
```   560 lemma [code]:
```
```   561   "k + l = Target_Numeral.int_of (of_int k + of_int l)"
```
```   562   by simp
```
```   563
```
```   564 lemma [code]:
```
```   565   "- k = Target_Numeral.int_of (- of_int k)"
```
```   566   by simp
```
```   567
```
```   568 lemma [code]:
```
```   569   "k - l = Target_Numeral.int_of (of_int k - of_int l)"
```
```   570   by simp
```
```   571
```
```   572 lemma [code]:
```
```   573   "Int.dup k = Target_Numeral.int_of (Target_Numeral.dup (of_int k))"
```
```   574   by simp
```
```   575
```
```   576 lemma [code, code del]:
```
```   577   "Int.sub = Int.sub" ..
```
```   578
```
```   579 lemma [code]:
```
```   580   "k * l = Target_Numeral.int_of (of_int k * of_int l)"
```
```   581   by simp
```
```   582
```
```   583 lemma [code]:
```
```   584   "pdivmod k l = map_pair Target_Numeral.int_of Target_Numeral.int_of
```
```   585     (Target_Numeral.divmod_abs (of_int k) (of_int l))"
```
```   586   by (simp add: prod_eq_iff pdivmod_def)
```
```   587
```
```   588 lemma [code]:
```
```   589   "k div l = Target_Numeral.int_of (of_int k div of_int l)"
```
```   590   by simp
```
```   591
```
```   592 lemma [code]:
```
```   593   "k mod l = Target_Numeral.int_of (of_int k mod of_int l)"
```
```   594   by simp
```
```   595
```
```   596 lemma [code]:
```
```   597   "HOL.equal k l = HOL.equal (of_int k :: Target_Numeral.int) (of_int l)"
```
```   598   by (simp add: equal Target_Numeral.int_eq_iff)
```
```   599
```
```   600 lemma [code]:
```
```   601   "k \<le> l \<longleftrightarrow> (of_int k :: Target_Numeral.int) \<le> of_int l"
```
```   602   by (simp add: less_eq_int_def)
```
```   603
```
```   604 lemma [code]:
```
```   605   "k < l \<longleftrightarrow> (of_int k :: Target_Numeral.int) < of_int l"
```
```   606   by (simp add: less_int_def)
```
```   607
```
```   608 lemma (in ring_1) of_int_code:
```
```   609   "of_int k = (if k = 0 then 0
```
```   610      else if k < 0 then - of_int (- k)
```
```   611      else let
```
```   612        (l, j) = divmod_int k 2;
```
```   613        l' = 2 * of_int l
```
```   614      in if j = 0 then l' else l' + 1)"
```
```   615 proof -
```
```   616   from mod_div_equality have *: "of_int k = of_int (k div 2 * 2 + k mod 2)" by simp
```
```   617   show ?thesis
```
```   618     by (simp add: Let_def divmod_int_mod_div mod_2_not_eq_zero_eq_one_int
```
```   619       of_int_add [symmetric]) (simp add: * mult_commute)
```
```   620 qed
```
```   621
```
```   622 declare of_int_code [code]
```
```   623
```
```   624
```
```   625 subsection {* Implementation for @{typ nat} *}
```
```   626
```
```   627 definition Nat :: "Target_Numeral.int \<Rightarrow> nat" where
```
```   628   "Nat = Target_Numeral.nat_of"
```
```   629
```
```   630 definition of_nat :: "nat \<Rightarrow> Target_Numeral.int" where
```
```   631   [code_abbrev]: "of_nat = Nat.of_nat"
```
```   632
```
```   633 hide_const (open) of_nat Nat
```
```   634
```
```   635 lemma [code_unfold]:
```
```   636   "Int.nat (Target_Numeral.int_of k) = Target_Numeral.nat_of k"
```
```   637   by (simp add: nat_of_def)
```
```   638
```
```   639 lemma int_of_nat [simp]:
```
```   640   "Target_Numeral.int_of (Target_Numeral.of_nat n) = of_nat n"
```
```   641   by (simp add: of_nat_def)
```
```   642
```
```   643 lemma [code abstype]:
```
```   644   "Target_Numeral.Nat (Target_Numeral.of_nat n) = n"
```
```   645   by (simp add: Nat_def nat_of_def)
```
```   646
```
```   647 lemma [code abstract]:
```
```   648   "Target_Numeral.of_nat (Target_Numeral.nat_of k) = max 0 k"
```
```   649   by (simp add: of_nat_def)
```
```   650
```
```   651 lemma [code_abbrev]:
```
```   652   "nat (Int.Pos k) = nat_of_num k"
```
```   653   by (simp add: nat_of_num_numeral)
```
```   654
```
```   655 lemma [code abstract]:
```
```   656   "Target_Numeral.of_nat 0 = 0"
```
```   657   by (simp add: Target_Numeral.int_eq_iff)
```
```   658
```
```   659 lemma [code abstract]:
```
```   660   "Target_Numeral.of_nat 1 = 1"
```
```   661   by (simp add: Target_Numeral.int_eq_iff)
```
```   662
```
```   663 lemma [code abstract]:
```
```   664   "Target_Numeral.of_nat (m + n) = of_nat m + of_nat n"
```
```   665   by (simp add: Target_Numeral.int_eq_iff)
```
```   666
```
```   667 lemma [code abstract]:
```
```   668   "Target_Numeral.of_nat (Code_Nat.dup n) = Target_Numeral.dup (of_nat n)"
```
```   669   by (simp add: Target_Numeral.int_eq_iff Code_Nat.dup_def)
```
```   670
```
```   671 lemma [code, code del]:
```
```   672   "Code_Nat.sub = Code_Nat.sub" ..
```
```   673
```
```   674 lemma [code abstract]:
```
```   675   "Target_Numeral.of_nat (m - n) = max 0 (of_nat m - of_nat n)"
```
```   676   by (simp add: Target_Numeral.int_eq_iff)
```
```   677
```
```   678 lemma [code abstract]:
```
```   679   "Target_Numeral.of_nat (m * n) = of_nat m * of_nat n"
```
```   680   by (simp add: Target_Numeral.int_eq_iff of_nat_mult)
```
```   681
```
```   682 lemma [code abstract]:
```
```   683   "Target_Numeral.of_nat (m div n) = of_nat m div of_nat n"
```
```   684   by (simp add: Target_Numeral.int_eq_iff zdiv_int)
```
```   685
```
```   686 lemma [code abstract]:
```
```   687   "Target_Numeral.of_nat (m mod n) = of_nat m mod of_nat n"
```
```   688   by (simp add: Target_Numeral.int_eq_iff zmod_int)
```
```   689
```
```   690 lemma [code]:
```
```   691   "Divides.divmod_nat m n = (m div n, m mod n)"
```
```   692   by (simp add: prod_eq_iff)
```
```   693
```
```   694 lemma [code]:
```
```   695   "HOL.equal m n = HOL.equal (of_nat m :: Target_Numeral.int) (of_nat n)"
```
```   696   by (simp add: equal Target_Numeral.int_eq_iff)
```
```   697
```
```   698 lemma [code]:
```
```   699   "m \<le> n \<longleftrightarrow> (of_nat m :: Target_Numeral.int) \<le> of_nat n"
```
```   700   by (simp add: less_eq_int_def)
```
```   701
```
```   702 lemma [code]:
```
```   703   "m < n \<longleftrightarrow> (of_nat m :: Target_Numeral.int) < of_nat n"
```
```   704   by (simp add: less_int_def)
```
```   705
```
```   706 lemma num_of_nat_code [code]:
```
```   707   "num_of_nat = Target_Numeral.num_of_int \<circ> of_nat"
```
```   708   by (simp add: fun_eq_iff num_of_int_def of_nat_def)
```
```   709
```
```   710 lemma (in semiring_1) of_nat_code:
```
```   711   "of_nat n = (if n = 0 then 0
```
```   712      else let
```
```   713        (m, q) = divmod_nat n 2;
```
```   714        m' = 2 * of_nat m
```
```   715      in if q = 0 then m' else m' + 1)"
```
```   716 proof -
```
```   717   from mod_div_equality have *: "of_nat n = of_nat (n div 2 * 2 + n mod 2)" by simp
```
```   718   show ?thesis
```
```   719     by (simp add: Let_def divmod_nat_div_mod mod_2_not_eq_zero_eq_one_nat
```
```   720       of_nat_add [symmetric])
```
```   721       (simp add: * mult_commute of_nat_mult add_commute)
```
```   722 qed
```
```   723
```
```   724 declare of_nat_code [code]
```
```   725
```
```   726 text {* Conversions between @{typ nat} and @{typ int} *}
```
```   727
```
```   728 definition int :: "nat \<Rightarrow> int" where
```
```   729   [code_abbrev]: "int = of_nat"
```
```   730
```
```   731 hide_const (open) int
```
```   732
```
```   733 lemma [code]:
```
```   734   "Target_Numeral.int n = Target_Numeral.int_of (of_nat n)"
```
```   735   by (simp add: int_def)
```
```   736
```
```   737 lemma [code abstract]:
```
```   738   "Target_Numeral.of_nat (nat k) = max 0 (Target_Numeral.of_int k)"
```
```   739   by (simp add: of_nat_def of_int_of_nat max_def)
```
```   740
```
```   741 end
```
```   742
```