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English
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Tags:
competitive-programming
code-generation
specification-generation
proof-generation
verification
rust
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File size: 8,813 Bytes
033007a | 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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 | use vstd::prelude::*;
fn main() {}
verus! {
pub struct Solution;
impl Solution {
pub open spec fn freq_at(requests: Seq<Vec<i32>>, idx: int, k: int) -> int
decreases requests.len() - k
{
if k >= requests.len() as int {
0
} else {
(if requests[k]@[0] as int <= idx && idx <= requests[k]@[1] as int {
1int
} else {
0int
}) + Self::freq_at(requests, idx, k + 1)
}
}
pub open spec fn freq_vec(requests: Seq<Vec<i32>>, n: int) -> Seq<int> {
Seq::new(n as nat, |i: int| Self::freq_at(requests, i, 0))
}
pub open spec fn to_int_seq(s: Seq<i32>) -> Seq<int> {
Seq::new(s.len(), |i: int| s[i] as int)
}
pub open spec fn dot_product_int(a: Seq<int>, b: Seq<int>, k: int) -> int
decreases a.len() - k
{
if k >= a.len() as int {
0
} else {
a[k] * b[k] + Self::dot_product_int(a, b, k + 1)
}
}
pub open spec fn is_sorted_int(s: Seq<int>) -> bool {
forall |i: int, j: int| 0 <= i <= j < s.len() ==> s[i] <= s[j]
}
pub open spec fn count_int(s: Seq<int>, v: int) -> int
decreases s.len()
{
if s.len() == 0 {
0
} else {
(if s.last() == v { 1int } else { 0int })
+ Self::count_int(s.drop_last(), v)
}
}
pub open spec fn is_perm_int(a: Seq<int>, b: Seq<int>) -> bool {
a.len() == b.len()
&& forall |v: int| Self::count_int(a, v) == Self::count_int(b, v)
}
pub open spec fn to_int_seq_i64(s: Seq<i64>) -> Seq<int> {
Seq::new(s.len(), |i: int| s[i] as int)
}
pub open spec fn count_occ(s: Seq<i32>, val: i32) -> int
decreases s.len()
{
if s.len() == 0 { 0 }
else {
(if s.last() == val { 1int } else { 0int })
+ Self::count_occ(s.drop_last(), val)
}
}
pub open spec fn is_perm(a: Seq<i32>, b: Seq<i32>) -> bool {
a.len() == b.len()
&& forall |v: i32| Self::count_occ(a, v) == Self::count_occ(b, v)
}
pub open spec fn is_sorted(s: Seq<i32>) -> bool {
forall |i: int, j: int| 0 <= i <= j < s.len() ==> s[i] <= s[j]
}
pub open spec fn count_occ_i64(s: Seq<i64>, val: i64) -> int
decreases s.len()
{
if s.len() == 0 { 0 }
else {
(if s.last() == val { 1int } else { 0int })
+ Self::count_occ_i64(s.drop_last(), val)
}
}
pub open spec fn is_perm_i64(a: Seq<i64>, b: Seq<i64>) -> bool {
a.len() == b.len()
&& forall |v: i64| Self::count_occ_i64(a, v) == Self::count_occ_i64(b, v)
}
pub open spec fn is_sorted_i64(s: Seq<i64>) -> bool {
forall |i: int, j: int| 0 <= i <= j < s.len() ==> s[i] <= s[j]
}
fn ms_merge(a: &Vec<i32>, b: &Vec<i32>) -> (result: Vec<i32>)
requires Self::is_sorted(a@), Self::is_sorted(b@)
ensures
Self::is_sorted(result@),
result@.len() == a@.len() + b@.len(),
Self::is_perm(result@, a@ + b@),
{
let mut result: Vec<i32> = Vec::new();
let mut i: usize = 0;
let mut j: usize = 0;
while i < a.len() || j < b.len()
{
if i < a.len() && (j >= b.len() || a[i] <= b[j]) {
result.push(a[i]);
i = i + 1;
} else {
result.push(b[j]);
j = j + 1;
}
}
result
}
fn ms_sort(input: &Vec<i32>) -> (result: Vec<i32>)
ensures
Self::is_sorted(result@),
result@.len() == input@.len(),
Self::is_perm(result@, input@),
decreases input.len(),
{
let n = input.len();
if n <= 1 {
let mut result = Vec::new();
if n == 1 {
result.push(input[0]);
}
return result;
}
let mid = n / 2;
let mut left: Vec<i32> = Vec::new();
let mut i: usize = 0;
while i < mid
{
left.push(input[i]);
i = i + 1;
}
let mut right: Vec<i32> = Vec::new();
let mut j: usize = mid;
while j < n
{
right.push(input[j]);
j = j + 1;
}
let sorted_left = Self::ms_sort(&left);
let sorted_right = Self::ms_sort(&right);
let result = Self::ms_merge(&sorted_left, &sorted_right);
result
}
fn ms_merge_i64(a: &Vec<i64>, b: &Vec<i64>) -> (result: Vec<i64>)
requires Self::is_sorted_i64(a@), Self::is_sorted_i64(b@)
ensures
Self::is_sorted_i64(result@),
result@.len() == a@.len() + b@.len(),
Self::is_perm_i64(result@, a@ + b@),
{
let mut result: Vec<i64> = Vec::new();
let mut i: usize = 0;
let mut j: usize = 0;
while i < a.len() || j < b.len()
{
if i < a.len() && (j >= b.len() || a[i] <= b[j]) {
result.push(a[i]);
i = i + 1;
} else {
result.push(b[j]);
j = j + 1;
}
}
result
}
fn ms_sort_i64(input: &Vec<i64>) -> (result: Vec<i64>)
ensures
Self::is_sorted_i64(result@),
result@.len() == input@.len(),
Self::is_perm_i64(result@, input@),
decreases input.len(),
{
let n = input.len();
if n <= 1 {
let mut result = Vec::new();
if n == 1 {
result.push(input[0]);
}
return result;
}
let mid = n / 2;
let mut left: Vec<i64> = Vec::new();
let mut i: usize = 0;
while i < mid
{
left.push(input[i]);
i = i + 1;
}
let mut right: Vec<i64> = Vec::new();
let mut j: usize = mid;
while j < n
{
right.push(input[j]);
j = j + 1;
}
let sorted_left = Self::ms_sort_i64(&left);
let sorted_right = Self::ms_sort_i64(&right);
let result = Self::ms_merge_i64(&sorted_left, &sorted_right);
result
}
pub fn max_sum_range_query(nums: Vec<i32>, requests: Vec<Vec<i32>>) -> (result: i32)
requires
1 <= nums@.len() <= 100_000,
forall |i: int| 0 <= i < nums@.len() ==>
0 <= #[trigger] nums@[i] <= 100_000,
1 <= requests@.len() <= 100_000,
forall |i: int| 0 <= i < requests@.len() ==> (
(#[trigger] requests@[i])@.len() == 2
&& 0 <= requests@[i]@[0]
&& requests@[i]@[0] <= requests@[i]@[1]
&& (requests@[i]@[1] as int) < nums@.len() as int
),
ensures
0 <= result < 1_000_000_007,
exists |sv: Seq<int>, sf: Seq<int>|
sv.len() == nums@.len()
&& sf.len() == nums@.len()
&& Self::is_sorted_int(sv)
&& Self::is_sorted_int(sf)
&& Self::is_perm_int(sv, Self::to_int_seq(nums@))
&& Self::is_perm_int(sf, Self::freq_vec(requests@, nums@.len() as int))
&& result as int
== Self::dot_product_int(sv, sf, 0) % 1_000_000_007
&& forall |c: Seq<int>|
c.len() == sv.len() && Self::is_perm_int(c, sv)
==> Self::dot_product_int(sv, sf, 0)
>= Self::dot_product_int(c, sf, 0),
{
let n = nums.len();
let m = requests.len();
let modval: i64 = 1_000_000_007;
let mut diff: Vec<i64> = Vec::new();
let mut z: usize = 0;
while z < n {
diff.push(0);
z = z + 1;
}
let mut r: usize = 0;
while r < m {
let s = requests[r][0] as usize;
let e = requests[r][1] as usize;
diff.set(s, diff[s] + 1);
if e + 1 < n {
diff.set(e + 1, diff[e + 1] - 1);
}
r = r + 1;
}
let mut count: Vec<i64> = Vec::new();
let mut running: i64 = 0;
let mut idx: usize = 0;
while idx < n {
running = running + diff[idx];
count.push(running);
idx = idx + 1;
}
let nums = Self::ms_sort(&nums);
let sorted_count = Self::ms_sort_i64(&count);
let mut result: i64 = 0;
let mut k: usize = 0;
while k < n {
result = (result + nums[k] as i64 * sorted_count[k]) % modval;
k = k + 1;
}
result as i32
}
}
}
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