src/HOL/Library/Code_Target_Nat.thy
 author haftmann Fri Mar 22 19:18:08 2019 +0000 (2 months ago) changeset 69946 494934c30f38 parent 69593 3dda49e08b9d permissions -rw-r--r--
improved code equations taken over from AFP
```     1 (*  Title:      HOL/Library/Code_Target_Nat.thy
```
```     2     Author:     Florian Haftmann, TU Muenchen
```
```     3 *)
```
```     4
```
```     5 section \<open>Implementation of natural numbers by target-language integers\<close>
```
```     6
```
```     7 theory Code_Target_Nat
```
```     8 imports Code_Abstract_Nat
```
```     9 begin
```
```    10
```
```    11 subsection \<open>Implementation for \<^typ>\<open>nat\<close>\<close>
```
```    12
```
```    13 context
```
```    14 includes natural.lifting integer.lifting
```
```    15 begin
```
```    16
```
```    17 lift_definition Nat :: "integer \<Rightarrow> nat"
```
```    18   is nat
```
```    19   .
```
```    20
```
```    21 lemma [code_post]:
```
```    22   "Nat 0 = 0"
```
```    23   "Nat 1 = 1"
```
```    24   "Nat (numeral k) = numeral k"
```
```    25   by (transfer, simp)+
```
```    26
```
```    27 lemma [code_abbrev]:
```
```    28   "integer_of_nat = of_nat"
```
```    29   by transfer rule
```
```    30
```
```    31 lemma [code_unfold]:
```
```    32   "Int.nat (int_of_integer k) = nat_of_integer k"
```
```    33   by transfer rule
```
```    34
```
```    35 lemma [code abstype]:
```
```    36   "Code_Target_Nat.Nat (integer_of_nat n) = n"
```
```    37   by transfer simp
```
```    38
```
```    39 lemma [code abstract]:
```
```    40   "integer_of_nat (nat_of_integer k) = max 0 k"
```
```    41   by transfer auto
```
```    42
```
```    43 lemma [code_abbrev]:
```
```    44   "nat_of_integer (numeral k) = nat_of_num k"
```
```    45   by transfer (simp add: nat_of_num_numeral)
```
```    46
```
```    47 context
```
```    48 begin
```
```    49
```
```    50 qualified definition natural :: "num \<Rightarrow> nat"
```
```    51   where [simp]: "natural = nat_of_num"
```
```    52
```
```    53 lemma [code_computation_unfold]:
```
```    54   "numeral = natural"
```
```    55   "nat_of_num = natural"
```
```    56   by (simp_all add: nat_of_num_numeral)
```
```    57
```
```    58 end
```
```    59
```
```    60 lemma [code abstract]:
```
```    61   "integer_of_nat (nat_of_num n) = integer_of_num n"
```
```    62   by (simp add: nat_of_num_numeral integer_of_nat_numeral)
```
```    63
```
```    64 lemma [code abstract]:
```
```    65   "integer_of_nat 0 = 0"
```
```    66   by transfer simp
```
```    67
```
```    68 lemma [code abstract]:
```
```    69   "integer_of_nat 1 = 1"
```
```    70   by transfer simp
```
```    71
```
```    72 lemma [code]:
```
```    73   "Suc n = n + 1"
```
```    74   by simp
```
```    75
```
```    76 lemma [code abstract]:
```
```    77   "integer_of_nat (m + n) = of_nat m + of_nat n"
```
```    78   by transfer simp
```
```    79
```
```    80 lemma [code abstract]:
```
```    81   "integer_of_nat (m - n) = max 0 (of_nat m - of_nat n)"
```
```    82   by transfer simp
```
```    83
```
```    84 lemma [code abstract]:
```
```    85   "integer_of_nat (m * n) = of_nat m * of_nat n"
```
```    86   by transfer (simp add: of_nat_mult)
```
```    87
```
```    88 lemma [code abstract]:
```
```    89   "integer_of_nat (m div n) = of_nat m div of_nat n"
```
```    90   by transfer (simp add: zdiv_int)
```
```    91
```
```    92 lemma [code abstract]:
```
```    93   "integer_of_nat (m mod n) = of_nat m mod of_nat n"
```
```    94   by transfer (simp add: zmod_int)
```
```    95
```
```    96 context
```
```    97   includes integer.lifting
```
```    98 begin
```
```    99
```
```   100 lemma divmod_nat_code [code]: \<^marker>\<open>contributor \<open>RenĂ© Thiemann\<close>\<close> \<^marker>\<open>contributor \<open>Akihisa Yamada\<close>\<close>
```
```   101   "Divides.divmod_nat m n = (
```
```   102      let k = integer_of_nat m; l = integer_of_nat n
```
```   103      in map_prod nat_of_integer nat_of_integer
```
```   104        (if k = 0 then (0, 0)
```
```   105         else if l = 0 then (0, k) else
```
```   106           Code_Numeral.divmod_abs k l))"
```
```   107   by (simp add: prod_eq_iff Let_def; transfer)
```
```   108     (simp add: nat_div_distrib nat_mod_distrib)
```
```   109
```
```   110 end
```
```   111
```
```   112 lemma [code]:
```
```   113   "divmod m n = map_prod nat_of_integer nat_of_integer (divmod m n)"
```
```   114   by (simp only: prod_eq_iff divmod_def map_prod_def case_prod_beta fst_conv snd_conv; transfer)
```
```   115     (simp_all only: nat_div_distrib nat_mod_distrib
```
```   116         zero_le_numeral nat_numeral)
```
```   117
```
```   118 lemma [code]:
```
```   119   "HOL.equal m n = HOL.equal (of_nat m :: integer) (of_nat n)"
```
```   120   by transfer (simp add: equal)
```
```   121
```
```   122 lemma [code]:
```
```   123   "m \<le> n \<longleftrightarrow> (of_nat m :: integer) \<le> of_nat n"
```
```   124   by simp
```
```   125
```
```   126 lemma [code]:
```
```   127   "m < n \<longleftrightarrow> (of_nat m :: integer) < of_nat n"
```
```   128   by simp
```
```   129
```
```   130 lemma num_of_nat_code [code]:
```
```   131   "num_of_nat = num_of_integer \<circ> of_nat"
```
```   132   by transfer (simp add: fun_eq_iff)
```
```   133
```
```   134 end
```
```   135
```
```   136 lemma (in semiring_1) of_nat_code_if:
```
```   137   "of_nat n = (if n = 0 then 0
```
```   138      else let
```
```   139        (m, q) = Divides.divmod_nat n 2;
```
```   140        m' = 2 * of_nat m
```
```   141      in if q = 0 then m' else m' + 1)"
```
```   142 proof -
```
```   143   from div_mult_mod_eq have *: "of_nat n = of_nat (n div 2 * 2 + n mod 2)" by simp
```
```   144   show ?thesis
```
```   145     by (simp add: Let_def divmod_nat_def of_nat_add [symmetric])
```
```   146       (simp add: * mult.commute of_nat_mult add.commute)
```
```   147 qed
```
```   148
```
```   149 declare of_nat_code_if [code]
```
```   150
```
```   151 definition int_of_nat :: "nat \<Rightarrow> int" where
```
```   152   [code_abbrev]: "int_of_nat = of_nat"
```
```   153
```
```   154 lemma [code]:
```
```   155   "int_of_nat n = int_of_integer (of_nat n)"
```
```   156   by (simp add: int_of_nat_def)
```
```   157
```
```   158 lemma [code abstract]:
```
```   159   "integer_of_nat (nat k) = max 0 (integer_of_int k)"
```
```   160   including integer.lifting by transfer auto
```
```   161
```
```   162 definition char_of_nat :: "nat \<Rightarrow> char"
```
```   163   where [code_abbrev]: "char_of_nat = char_of"
```
```   164
```
```   165 definition nat_of_char :: "char \<Rightarrow> nat"
```
```   166   where [code_abbrev]: "nat_of_char = of_char"
```
```   167
```
```   168 lemma [code]:
```
```   169   "char_of_nat = char_of_integer \<circ> integer_of_nat"
```
```   170   including integer.lifting unfolding char_of_integer_def char_of_nat_def
```
```   171   by transfer (simp add: fun_eq_iff)
```
```   172
```
```   173 lemma [code abstract]:
```
```   174   "integer_of_nat (nat_of_char c) = integer_of_char c"
```
```   175   by (cases c) (simp add: nat_of_char_def integer_of_char_def integer_of_nat_eq_of_nat)
```
```   176
```
```   177 lemma term_of_nat_code [code]:
```
```   178   \<comment> \<open>Use \<^term>\<open>Code_Numeral.nat_of_integer\<close> in term reconstruction
```
```   179         instead of \<^term>\<open>Code_Target_Nat.Nat\<close> such that reconstructed
```
```   180         terms can be fed back to the code generator\<close>
```
```   181   "term_of_class.term_of n =
```
```   182    Code_Evaluation.App
```
```   183      (Code_Evaluation.Const (STR ''Code_Numeral.nat_of_integer'')
```
```   184         (typerep.Typerep (STR ''fun'')
```
```   185            [typerep.Typerep (STR ''Code_Numeral.integer'') [],
```
```   186          typerep.Typerep (STR ''Nat.nat'') []]))
```
```   187      (term_of_class.term_of (integer_of_nat n))"
```
```   188   by (simp add: term_of_anything)
```
```   189
```
```   190 lemma nat_of_integer_code_post [code_post]:
```
```   191   "nat_of_integer 0 = 0"
```
```   192   "nat_of_integer 1 = 1"
```
```   193   "nat_of_integer (numeral k) = numeral k"
```
```   194   including integer.lifting by (transfer, simp)+
```
```   195
```
```   196 code_identifier
```
```   197   code_module Code_Target_Nat \<rightharpoonup>
```
```   198     (SML) Arith and (OCaml) Arith and (Haskell) Arith
```
```   199
```
```   200 end
```