File size: 2,699 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 | /*@ Import Lean
import Data_structures.binary_indexed_tree.lean.spec_lib
open scoped SimpleC
*/
/*@ Extern Coq
(FenwickLowbit : Z -> Z)
(FenwickNodeLo : Z -> Z)
(FenwickNodeSum : list Z -> Z -> Z)
(FenwickPrefixSum : list Z -> Z -> Z)
(FenwickAddArray : list Z -> Z -> Z -> list Z)
(FenwickRep : list Z -> list Z -> Z -> Prop)
(FenwickIntervalsIntSafe : list Z -> Z -> Prop)
(FenwickAddProgress : list Z -> list Z -> Z -> Z -> Z -> Z -> Prop)
(FenwickQueryState : list Z -> Z -> Z -> Z -> Prop)
*/
int lowbit(int x)
/*@ Require 1 <= x && x <= INT_MAX
Ensure
__return == FenwickLowbit(x) &&
1 <= __return && __return <= x
*/
{
return x & (-x);
}
void add(int *bit, int n, int pos, int delta)
/*@ With (a : list Z) (bit_l : list Z)
Require
1 <= n && 2 * n <= INT_MAX &&
1 <= pos && pos <= n &&
FenwickRep(a, bit_l, n) &&
FenwickIntervalsIntSafe(a, n) &&
FenwickIntervalsIntSafe(FenwickAddArray(a, pos, delta), n) &&
IntArray::full(bit, n + 1, bit_l)
Ensure
exists bit_l1,
FenwickRep(FenwickAddArray(a, pos, delta), bit_l1, n) &&
Znth(0, bit_l1, 0) == Znth(0, bit_l, 0) &&
IntArray::full(bit, n + 1, bit_l1)
*/
{
/*@ Inv Assert
exists bit_cur,
bit == bit@pre && n == n@pre && delta == delta@pre &&
1 <= n@pre && 2 * n@pre <= INT_MAX &&
1 <= pos@pre && pos@pre <= n@pre &&
1 <= pos && pos <= 2 * n@pre &&
FenwickRep(a, bit_l, n@pre) &&
FenwickIntervalsIntSafe(a, n@pre) &&
FenwickIntervalsIntSafe(
FenwickAddArray(a, pos@pre, delta@pre), n@pre) &&
FenwickAddProgress(
bit_l, bit_cur, n@pre, pos@pre, pos, delta@pre) &&
IntArray::full(bit@pre, n@pre + 1, bit_cur)
*/
while (pos <= n) {
bit[pos] += delta;
pos += lowbit(pos);
}
}
int query(int *bit, int pos)
/*@ With (a : list Z) (bit_l : list Z) (n : Z)
Require
1 <= n && n <= INT_MAX &&
0 <= pos && pos <= n &&
FenwickRep(a, bit_l, n) &&
FenwickIntervalsIntSafe(a, n) &&
IntArray::full(bit, n + 1, bit_l)
Ensure
__return == FenwickPrefixSum(a, pos) &&
IntArray::full(bit, n + 1, bit_l)
*/
{
int sum = 0;
/*@ Inv Assert
bit == bit@pre &&
1 <= n && n <= INT_MAX &&
0 <= pos && pos <= pos@pre && pos@pre <= n &&
INT_MIN <= sum && sum <= INT_MAX &&
FenwickRep(a, bit_l, n) &&
FenwickIntervalsIntSafe(a, n) &&
FenwickQueryState(a, pos@pre, pos, sum) &&
IntArray::full(bit@pre, n + 1, bit_l)
*/
while (pos > 0) {
sum += bit[pos];
pos -= lowbit(pos);
}
return sum;
}
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