File size: 5,240 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 | /*@ 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)];
}
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