| /*@ 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; | |
| } | |