| /*@ Import Lean | |
| import Algorithms.catalan_numbers.lean.helper_lib | |
| open scoped SimpleC | |
| */ | |
| /*@ Extern Coq | |
| (StackSequenceCount : Z -> Z -> Prop) | |
| (StackCellBound : Z -> Z -> Z -> Prop) | |
| (StackCellCorrect : Z -> Z -> Z -> Z -> Prop) | |
| (StackRowsDone : Z -> list Z -> Z -> Prop) | |
| (StackRowProgress : Z -> list Z -> Z -> Z -> Prop) | |
| */ | |
| int id(int n, int x, int y) | |
| /*@ Require | |
| 0 <= n && n <= 7 && | |
| 0 <= x && x <= n && | |
| 0 <= y && y <= n + 1 | |
| Ensure | |
| __return == x * (n + 1) + y && | |
| 0 <= __return && __return <= (n + 1) * (n + 1) | |
| */ | |
| { | |
| return x * (n + 1) + y; | |
| } | |
| int solve(int n, int *f) | |
| /*@ Require | |
| 0 <= n && n <= 7 && | |
| IntArray::undef_full(f, (n + 1) * (n + 1)) | |
| Ensure | |
| exists table, | |
| StackSequenceCount(n, __return) && | |
| StackRowsDone(n, table, n + 1) && | |
| IntArray::full(f, (n + 1) * (n + 1), table) | |
| */ | |
| { | |
| /*@ Inv Assert | |
| exists table, | |
| n == n@pre && f == f@pre && | |
| 0 <= n@pre && n@pre <= 7 && | |
| 0 <= i && i <= n@pre + 1 && | |
| StackRowsDone(n@pre, table, i) && | |
| IntArray::seg(f, 0, i * (n@pre + 1), table) * | |
| IntArray::undef_seg(f, i * (n@pre + 1), | |
| (n@pre + 1) * (n@pre + 1)) | |
| */ | |
| for (int i = 0; i <= n; i++) { | |
| /*@ Inv Assert | |
| exists table, | |
| n == n@pre && f == f@pre && | |
| 0 <= n@pre && n@pre <= 7 && | |
| 0 <= i && i <= n@pre && | |
| 0 <= j && j <= n@pre + 1 && | |
| StackRowProgress(n@pre, table, i, j) && | |
| IntArray::seg(f, 0, i * (n@pre + 1) + j, table) * | |
| IntArray::undef_seg(f, i * (n@pre + 1) + j, | |
| (n@pre + 1) * (n@pre + 1)) | |
| */ | |
| for (int j = 0; j <= n; j++) { | |
| if (i == 0) { | |
| f[id(n, i, j)] = 1; | |
| } | |
| else if (j == 0) { | |
| /*@ Assert | |
| exists table, | |
| n == n@pre && f == f@pre && | |
| 0 <= n@pre && n@pre <= 7 && | |
| 1 <= i && i <= n@pre && | |
| j == 0 && | |
| 0 <= i * (n@pre + 1) + j && | |
| i * (n@pre + 1) + j < | |
| (n@pre + 1) * (n@pre + 1) && | |
| 0 <= (i - 1) * (n@pre + 1) + (j + 1) && | |
| (i - 1) * (n@pre + 1) + (j + 1) < | |
| i * (n@pre + 1) + j && | |
| StackRowProgress(n@pre, table, i, j) && | |
| IntArray::seg(f, 0, i * (n@pre + 1) + j, table) * | |
| IntArray::undef_seg(f, i * (n@pre + 1) + j, | |
| (n@pre + 1) * (n@pre + 1)) | |
| */ | |
| f[id(n, i, j)] = f[id(n, i-1, j+1)]; | |
| } | |
| else { | |
| /*@ Assert | |
| exists table, | |
| n == n@pre && f == f@pre && | |
| 0 <= n@pre && n@pre <= 7 && | |
| 1 <= i && i <= n@pre && | |
| 1 <= j && j <= n@pre && | |
| 0 <= i * (n@pre + 1) + j && | |
| i * (n@pre + 1) + j < | |
| (n@pre + 1) * (n@pre + 1) && | |
| 0 <= (i - 1) * (n@pre + 1) + (j + 1) && | |
| (i - 1) * (n@pre + 1) + (j + 1) < | |
| i * (n@pre + 1) + j && | |
| 0 <= i * (n@pre + 1) + (j - 1) && | |
| i * (n@pre + 1) + (j - 1) < | |
| i * (n@pre + 1) + j && | |
| StackRowProgress(n@pre, table, i, j) && | |
| IntArray::seg(f, 0, i * (n@pre + 1) + j, table) * | |
| IntArray::undef_seg(f, i * (n@pre + 1) + j, | |
| (n@pre + 1) * (n@pre + 1)) | |
| */ | |
| f[id(n, i, j)] = f[id(n, i-1, j+1)] + f[id(n, i, j-1)]; | |
| } | |
| /*@ Assert | |
| exists table, | |
| n == n@pre && f == f@pre && | |
| 0 <= n@pre && n@pre <= 7 && | |
| 0 <= i && i <= n@pre && | |
| 0 <= j && j <= n@pre && | |
| StackCellBound(i, j, table[i * (n@pre + 1) + j]) && | |
| StackCellCorrect(n@pre, i, j, | |
| table[i * (n@pre + 1) + j]) && | |
| StackRowProgress(n@pre, table, i, j + 1) && | |
| IntArray::seg(f, 0, i * (n@pre + 1) + j + 1, table) * | |
| IntArray::undef_seg(f, i * (n@pre + 1) + j + 1, | |
| (n@pre + 1) * (n@pre + 1)) | |
| */ | |
| } | |
| /*@ Assert | |
| exists table, | |
| n == n@pre && f == f@pre && | |
| 0 <= n@pre && n@pre <= 7 && | |
| 0 <= i && i <= n@pre && | |
| StackRowsDone(n@pre, table, i + 1) && | |
| IntArray::seg(f, 0, (i + 1) * (n@pre + 1), table) * | |
| IntArray::undef_seg(f, (i + 1) * (n@pre + 1), | |
| (n@pre + 1) * (n@pre + 1)) | |
| */ | |
| } | |
| /*@ Assert | |
| exists table, | |
| n == n@pre && f == f@pre && | |
| 0 <= n@pre && n@pre <= 7 && | |
| 0 <= n@pre * (n@pre + 1) && | |
| n@pre * (n@pre + 1) < | |
| (n@pre + 1) * (n@pre + 1) && | |
| StackRowsDone(n@pre, table, n@pre + 1) && | |
| StackSequenceCount(n@pre, | |
| table[n@pre * (n@pre + 1)]) && | |
| IntArray::full(f, (n@pre + 1) * (n@pre + 1), table) | |
| */ | |
| return f[id(n, n, 0)]; | |
| } | |