File size: 3,671 Bytes
bff4ec7 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 | /*@ Import Lean
import Algorithms.maximum_subarray.lean.helper_lib
open scoped SimpleC
*/
/*@ Extern Coq
(max_Z : Z -> Z -> Z)
(MaxSuffixSumPrefix : list Z -> Z -> Z -> Prop)
(MaxSubarraySumPrefix : list Z -> Z -> Z -> Prop)
*/
int max(int a, int b)
/*@ Require emp
Ensure __return == max_Z(a, b) && emp
*/
{
return (a > b) ? a : b;
}
int max_sub_array(int *arr, int n)
/*@ With (l : list Z)
Require
1 <= n && n <= 100000 &&
Zlength(l) == n &&
IntArray::full(arr, n, l) &&
(forall (k : Z), (0 <= k && k < n) => (-10000 <= l[k] && l[k] <= 10000))
Ensure
MaxSubarraySumPrefix(l, n, __return) &&
IntArray::full(arr, n, l)
*/
{
if (n == 0) {
return 0;
}
int cur = arr[0];
int res = arr[0];
/*@ Assert
arr == arr@pre && n == n@pre &&
1 <= n@pre && n@pre <= 100000 &&
Zlength(l) == n@pre &&
cur == l[0] && res == l[0] &&
-10000 <= cur && cur <= 10000 &&
-10000 <= res && res <= 10000 &&
MaxSuffixSumPrefix(l, 1, cur) &&
MaxSubarraySumPrefix(l, 1, res) &&
IntArray::full(arr, n@pre, l) &&
(forall (k : Z), (0 <= k && k < n@pre) => (-10000 <= l[k] && l[k] <= 10000))
*/
/*@ Inv Assert
arr == arr@pre && n == n@pre &&
1 <= n@pre && n@pre <= 100000 &&
Zlength(l) == n@pre &&
1 <= i && i <= n@pre &&
-10000 <= cur && cur <= 1000000000 &&
-10000 <= res && res <= 1000000000 &&
MaxSuffixSumPrefix(l, i, cur) &&
MaxSubarraySumPrefix(l, i, res) &&
IntArray::full(arr, n@pre, l) &&
(forall (k : Z), (0 <= k && k < n@pre) => (-10000 <= l[k] && l[k] <= 10000))
*/
for (int i = 1; i < n; i++) {
/*@ Assert
arr == arr@pre && n == n@pre &&
1 <= n@pre && n@pre <= 100000 &&
Zlength(l) == n@pre &&
1 <= i && i < n@pre &&
-10000 <= cur && cur <= 1000000000 &&
-10000 <= res && res <= 1000000000 &&
INT_MIN <= cur + l[i] && cur + l[i] <= INT_MAX &&
MaxSuffixSumPrefix(l, i, cur) &&
MaxSubarraySumPrefix(l, i, res) &&
IntArray::full(arr, n@pre, l) &&
(forall (k : Z), (0 <= k && k < n@pre) => (-10000 <= l[k] && l[k] <= 10000))
*/
cur = max(arr[i], cur + arr[i]);
/*@ Assert
arr == arr@pre && n == n@pre &&
1 <= n@pre && n@pre <= 100000 &&
Zlength(l) == n@pre &&
1 <= i && i < n@pre &&
-10000 <= cur && cur <= 1000000000 &&
-10000 <= res && res <= 1000000000 &&
MaxSuffixSumPrefix(l, i + 1, cur) &&
MaxSubarraySumPrefix(l, i, res) &&
IntArray::full(arr, n@pre, l) &&
(forall (k : Z), (0 <= k && k < n@pre) => (-10000 <= l[k] && l[k] <= 10000))
*/
res = max(res, cur);
/*@ Assert
arr == arr@pre && n == n@pre &&
1 <= n@pre && n@pre <= 100000 &&
Zlength(l) == n@pre &&
1 <= i && i < n@pre &&
-10000 <= cur && cur <= 1000000000 &&
-10000 <= res && res <= 1000000000 &&
MaxSuffixSumPrefix(l, i + 1, cur) &&
MaxSubarraySumPrefix(l, i + 1, res) &&
IntArray::full(arr, n@pre, l) &&
(forall (k : Z), (0 <= k && k < n@pre) => (-10000 <= l[k] && l[k] <= 10000))
*/
}
/*@ Assert
arr == arr@pre && n == n@pre &&
1 <= n@pre && n@pre <= 100000 &&
Zlength(l) == n@pre &&
-10000 <= cur && cur <= 1000000000 &&
MaxSuffixSumPrefix(l, n@pre, cur) &&
MaxSubarraySumPrefix(l, n@pre, res) &&
IntArray::full(arr, n@pre, l)
*/
return res;
}
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