src/HOL/Library/Code_Target_Int.thy
 author haftmann Sat Aug 08 10:51:33 2015 +0200 (2015-08-08) changeset 60868 dd18c33c001e parent 60500 903bb1495239 child 61275 053ec04ea866 permissions -rw-r--r--
direct bootstrap of integer division from natural division
```     1 (*  Title:      HOL/Library/Code_Target_Int.thy
```
```     2     Author:     Florian Haftmann, TU Muenchen
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```     3 *)
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```     4
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```     5 section \<open>Implementation of integer numbers by target-language integers\<close>
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```     6
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```     7 theory Code_Target_Int
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```     8 imports Main
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```     9 begin
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```    10
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```    11 code_datatype int_of_integer
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```    12
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```    13 declare [[code drop: integer_of_int]]
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```    14
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```    15 context
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```    16 includes integer.lifting
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```    17 begin
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```    18
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```    19 lemma [code]:
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```    20   "integer_of_int (int_of_integer k) = k"
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```    21   by transfer rule
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```    22
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```    23 lemma [code]:
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```    24   "Int.Pos = int_of_integer \<circ> integer_of_num"
```
```    25   by transfer (simp add: fun_eq_iff)
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```    26
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```    27 lemma [code]:
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```    28   "Int.Neg = int_of_integer \<circ> uminus \<circ> integer_of_num"
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```    29   by transfer (simp add: fun_eq_iff)
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```    30
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```    31 lemma [code_abbrev]:
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```    32   "int_of_integer (numeral k) = Int.Pos k"
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```    33   by transfer simp
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```    34
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```    35 lemma [code_abbrev]:
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```    36   "int_of_integer (- numeral k) = Int.Neg k"
```
```    37   by transfer simp
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```    38
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```    39 lemma [code, symmetric, code_post]:
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```    40   "0 = int_of_integer 0"
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```    41   by transfer simp
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```    42
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```    43 lemma [code, symmetric, code_post]:
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```    44   "1 = int_of_integer 1"
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```    45   by transfer simp
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```    46
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```    47 lemma [code_post]:
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```    48   "int_of_integer (- 1) = - 1"
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```    49   by simp
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```    50
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```    51 lemma [code]:
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```    52   "k + l = int_of_integer (of_int k + of_int l)"
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```    53   by transfer simp
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```    54
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```    55 lemma [code]:
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```    56   "- k = int_of_integer (- of_int k)"
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```    57   by transfer simp
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```    58
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```    59 lemma [code]:
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```    60   "k - l = int_of_integer (of_int k - of_int l)"
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```    61   by transfer simp
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```    62
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```    63 lemma [code]:
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```    64   "Int.dup k = int_of_integer (Code_Numeral.dup (of_int k))"
```
```    65   by transfer simp
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```    66
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```    67 declare [[code drop: Int.sub]]
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```    68
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```    69 lemma [code]:
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```    70   "k * l = int_of_integer (of_int k * of_int l)"
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```    71   by simp
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```    72
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```    73 lemma [code]:
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```    74   "k div l = int_of_integer (of_int k div of_int l)"
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```    75   by simp
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```    76
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```    77 lemma [code]:
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```    78   "k mod l = int_of_integer (of_int k mod of_int l)"
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```    79   by simp
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```    80
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```    81 lemma [code]:
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```    82   "HOL.equal k l = HOL.equal (of_int k :: integer) (of_int l)"
```
```    83   by transfer (simp add: equal)
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```    84
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```    85 lemma [code]:
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```    86   "k \<le> l \<longleftrightarrow> (of_int k :: integer) \<le> of_int l"
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```    87   by transfer rule
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```    88
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```    89 lemma [code]:
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```    90   "k < l \<longleftrightarrow> (of_int k :: integer) < of_int l"
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```    91   by transfer rule
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```    92 end
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```    93
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```    94 lemma (in ring_1) of_int_code_if:
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```    95   "of_int k = (if k = 0 then 0
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```    96      else if k < 0 then - of_int (- k)
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```    97      else let
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```    98        l = 2 * of_int (k div 2);
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```    99        j = k mod 2
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```   100      in if j = 0 then l else l + 1)"
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```   101 proof -
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```   102   from mod_div_equality have *: "of_int k = of_int (k div 2 * 2 + k mod 2)" by simp
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```   103   show ?thesis
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```   104     by (simp add: Let_def of_int_add [symmetric]) (simp add: * mult.commute)
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```   105 qed
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```   106
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```   107 declare of_int_code_if [code]
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```   108
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```   109 lemma [code]:
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```   110   "nat = nat_of_integer \<circ> of_int"
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```   111   including integer.lifting by transfer (simp add: fun_eq_iff)
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```   112
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```   113 code_identifier
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```   114   code_module Code_Target_Int \<rightharpoonup>
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```   115     (SML) Arith and (OCaml) Arith and (Haskell) Arith
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```   116
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```   117 end
```