src/HOL/Library/Code_Target_Nat.thy
 author blanchet Wed Sep 24 15:45:55 2014 +0200 (2014-09-24) changeset 58425 246985c6b20b parent 57512 cc97b347b301 child 58881 b9556a055632 permissions -rw-r--r--
simpler proof
```     1 (*  Title:      HOL/Library/Code_Target_Nat.thy
```
```     2     Author:     Florian Haftmann, TU Muenchen
```
```     3 *)
```
```     4
```
```     5 header {* Implementation of natural numbers by target-language integers *}
```
```     6
```
```     7 theory Code_Target_Nat
```
```     8 imports Code_Abstract_Nat
```
```     9 begin
```
```    10
```
```    11 subsection {* Implementation for @{typ nat} *}
```
```    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 lemma [code abstract]:
```
```    48   "integer_of_nat (nat_of_num n) = integer_of_num n"
```
```    49   by transfer (simp add: nat_of_num_numeral)
```
```    50
```
```    51 lemma [code abstract]:
```
```    52   "integer_of_nat 0 = 0"
```
```    53   by transfer simp
```
```    54
```
```    55 lemma [code abstract]:
```
```    56   "integer_of_nat 1 = 1"
```
```    57   by transfer simp
```
```    58
```
```    59 lemma [code]:
```
```    60   "Suc n = n + 1"
```
```    61   by simp
```
```    62
```
```    63 lemma [code abstract]:
```
```    64   "integer_of_nat (m + n) = of_nat m + of_nat n"
```
```    65   by transfer simp
```
```    66
```
```    67 lemma [code abstract]:
```
```    68   "integer_of_nat (m - n) = max 0 (of_nat m - of_nat n)"
```
```    69   by transfer simp
```
```    70
```
```    71 lemma [code abstract]:
```
```    72   "integer_of_nat (m * n) = of_nat m * of_nat n"
```
```    73   by transfer (simp add: of_nat_mult)
```
```    74
```
```    75 lemma [code abstract]:
```
```    76   "integer_of_nat (m div n) = of_nat m div of_nat n"
```
```    77   by transfer (simp add: zdiv_int)
```
```    78
```
```    79 lemma [code abstract]:
```
```    80   "integer_of_nat (m mod n) = of_nat m mod of_nat n"
```
```    81   by transfer (simp add: zmod_int)
```
```    82
```
```    83 lemma [code]:
```
```    84   "Divides.divmod_nat m n = (m div n, m mod n)"
```
```    85   by (fact divmod_nat_div_mod)
```
```    86
```
```    87 lemma [code]:
```
```    88   "HOL.equal m n = HOL.equal (of_nat m :: integer) (of_nat n)"
```
```    89   by transfer (simp add: equal)
```
```    90
```
```    91 lemma [code]:
```
```    92   "m \<le> n \<longleftrightarrow> (of_nat m :: integer) \<le> of_nat n"
```
```    93   by simp
```
```    94
```
```    95 lemma [code]:
```
```    96   "m < n \<longleftrightarrow> (of_nat m :: integer) < of_nat n"
```
```    97   by simp
```
```    98
```
```    99 lemma num_of_nat_code [code]:
```
```   100   "num_of_nat = num_of_integer \<circ> of_nat"
```
```   101   by transfer (simp add: fun_eq_iff)
```
```   102
```
```   103 end
```
```   104
```
```   105 lemma (in semiring_1) of_nat_code_if:
```
```   106   "of_nat n = (if n = 0 then 0
```
```   107      else let
```
```   108        (m, q) = divmod_nat n 2;
```
```   109        m' = 2 * of_nat m
```
```   110      in if q = 0 then m' else m' + 1)"
```
```   111 proof -
```
```   112   from mod_div_equality have *: "of_nat n = of_nat (n div 2 * 2 + n mod 2)" by simp
```
```   113   show ?thesis
```
```   114     by (simp add: Let_def divmod_nat_div_mod of_nat_add [symmetric])
```
```   115       (simp add: * mult.commute of_nat_mult add.commute)
```
```   116 qed
```
```   117
```
```   118 declare of_nat_code_if [code]
```
```   119
```
```   120 definition int_of_nat :: "nat \<Rightarrow> int" where
```
```   121   [code_abbrev]: "int_of_nat = of_nat"
```
```   122
```
```   123 lemma [code]:
```
```   124   "int_of_nat n = int_of_integer (of_nat n)"
```
```   125   by (simp add: int_of_nat_def)
```
```   126
```
```   127 lemma [code abstract]:
```
```   128   "integer_of_nat (nat k) = max 0 (integer_of_int k)"
```
```   129   including integer.lifting by transfer auto
```
```   130
```
```   131 lemma term_of_nat_code [code]:
```
```   132   -- {* Use @{term Code_Numeral.nat_of_integer} in term reconstruction
```
```   133         instead of @{term Code_Target_Nat.Nat} such that reconstructed
```
```   134         terms can be fed back to the code generator *}
```
```   135   "term_of_class.term_of n =
```
```   136    Code_Evaluation.App
```
```   137      (Code_Evaluation.Const (STR ''Code_Numeral.nat_of_integer'')
```
```   138         (typerep.Typerep (STR ''fun'')
```
```   139            [typerep.Typerep (STR ''Code_Numeral.integer'') [],
```
```   140          typerep.Typerep (STR ''Nat.nat'') []]))
```
```   141      (term_of_class.term_of (integer_of_nat n))"
```
```   142   by (simp add: term_of_anything)
```
```   143
```
```   144 lemma nat_of_integer_code_post [code_post]:
```
```   145   "nat_of_integer 0 = 0"
```
```   146   "nat_of_integer 1 = 1"
```
```   147   "nat_of_integer (numeral k) = numeral k"
```
```   148   including integer.lifting by (transfer, simp)+
```
```   149
```
```   150 code_identifier
```
```   151   code_module Code_Target_Nat \<rightharpoonup>
```
```   152     (SML) Arith and (OCaml) Arith and (Haskell) Arith
```
```   153
```
```   154 end
```