src/HOL/Code_Numeral.thy
author haftmann
Mon Mar 23 19:05:14 2015 +0100 (2015-03-23)
changeset 59816 034b13f4efae
parent 59487 adaa430fc0f7
child 60352 d46de31a50c4
permissions -rw-r--r--
distributivity of partial minus establishes desired properties of dvd in semirings
     1 (*  Title:      HOL/Code_Numeral.thy
     2     Author:     Florian Haftmann, TU Muenchen
     3 *)
     4 
     5 section {* Numeric types for code generation onto target language numerals only *}
     6 
     7 theory Code_Numeral
     8 imports Nat_Transfer Divides Lifting
     9 begin
    10 
    11 subsection {* Type of target language integers *}
    12 
    13 typedef integer = "UNIV \<Colon> 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 div_integer :: "integer \<Rightarrow> integer \<Rightarrow> integer"
   154   is "Divides.div :: int \<Rightarrow> int \<Rightarrow> int"
   155   .
   156 
   157 declare div_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 semiring_numeral_div_class.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 {* Code theorems for target language integers *}
   242 
   243 text {* Constructors *}
   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 {* Auxiliary operations *}
   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 {* Implementations *}
   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, j) = divmod_int k 2;
   516        l' = 2 * integer_of_int l
   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 {* Serializer setup for target language integers *}
   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 {*
   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 *}
   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 {* Type of target language naturals *}
   617 
   618 typedef natural = "UNIV \<Colon> 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 div_natural :: "natural \<Rightarrow> natural \<Rightarrow> natural"
   713   is "Divides.div :: nat \<Rightarrow> nat \<Rightarrow> nat"
   714   .
   715 
   716 declare div_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 {* Inductive representation of target language naturals *}
   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 {* Code refinement for target language naturals *}
   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