src/HOL/Library/Code_Target_Int.thy
 author haftmann Sun Oct 16 09:31:05 2016 +0200 (2016-10-16) changeset 64242 93c6f0da5c70 parent 63351 e5d08b1d8fea child 64997 067a6cca39f0 permissions -rw-r--r--
more standardized theorem names for facts involving the div and mod identity
```     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 "../GCD"
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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"
```
```    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"
```
```    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"
```
```    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"
```
```    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)"
```
```    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)"
```
```    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)"
```
```    79   by simp
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```    80
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```    81 lemma [code]:
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```    82   "divmod m n = map_prod int_of_integer int_of_integer (divmod m n)"
```
```    83   unfolding prod_eq_iff divmod_def map_prod_def case_prod_beta fst_conv snd_conv
```
```    84   by transfer simp
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```    85
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```    86 lemma [code]:
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```    87   "HOL.equal k l = HOL.equal (of_int k :: integer) (of_int l)"
```
```    88   by transfer (simp add: equal)
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```    89
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```    90 lemma [code]:
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```    91   "k \<le> l \<longleftrightarrow> (of_int k :: integer) \<le> of_int l"
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```    92   by transfer rule
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```    93
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```    94 lemma [code]:
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```    95   "k < l \<longleftrightarrow> (of_int k :: integer) < of_int l"
```
```    96   by transfer rule
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```    97
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```    98 declare [[code drop: "gcd :: int \<Rightarrow> _" "lcm :: int \<Rightarrow> _"]]
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```    99
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```   100 lemma gcd_int_of_integer [code]:
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```   101   "gcd (int_of_integer x) (int_of_integer y) = int_of_integer (gcd x y)"
```
```   102 by transfer rule
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```   103
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```   104 lemma lcm_int_of_integer [code]:
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```   105   "lcm (int_of_integer x) (int_of_integer y) = int_of_integer (lcm x y)"
```
```   106 by transfer rule
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```   107
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```   108 end
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```   109
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```   110 lemma (in ring_1) of_int_code_if:
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```   111   "of_int k = (if k = 0 then 0
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```   112      else if k < 0 then - of_int (- k)
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```   113      else let
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```   114        l = 2 * of_int (k div 2);
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```   115        j = k mod 2
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```   116      in if j = 0 then l else l + 1)"
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```   117 proof -
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```   118   from div_mult_mod_eq have *: "of_int k = of_int (k div 2 * 2 + k mod 2)" by simp
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```   119   show ?thesis
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```   120     by (simp add: Let_def of_int_add [symmetric]) (simp add: * mult.commute)
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```   121 qed
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```   122
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```   123 declare of_int_code_if [code]
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```   124
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```   125 lemma [code]:
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```   126   "nat = nat_of_integer \<circ> of_int"
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```   127   including integer.lifting by transfer (simp add: fun_eq_iff)
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```   128
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```   129 code_identifier
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```   130   code_module Code_Target_Int \<rightharpoonup>
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```   131     (SML) Arith and (OCaml) Arith and (Haskell) Arith
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```   132
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```   133 end
```