C++ Library for Competitive Programming
#include "emthrm/data_structure/lazy_segment_tree.hpp"
モノイドであるデータに対して高速に区間クエリを処理する(完全)二分木である。
$\langle O(N), O(\log{N}) \rangle$
template <typename T>
requires requires {
typename T::Monoid;
{T::id()} -> std::same_as<typename T::Monoid>;
{T::merge(std::declval<typename T::Monoid>(),
std::declval<typename T::Monoid>())}
-> std::same_as<typename T::Monoid>;
}
struct SegmentTree;
T
:モノイドを表す構造体であり、以下の型エイリアスと静的メンバ関数を必要とする。
Monoid
:要素型static constexpr Monoid id();
:単位元static Monoid merge(const Monoid&, const Monoid&);
:二項演算 $\circ$名前 | 効果・戻り値 |
---|---|
explicit SegmentTree(const int n); |
要素数 $N$ のオブジェクトを構築する。 |
explicit SegmentTree(const std::vector<Monoid>& a); |
$A$ に対してオブジェクトを構築する。 |
void set(int idx, const Monoid val); |
$A_{\mathrm{idx}} \gets \mathrm{val}$ |
Monoid get(int left, int right) const; |
$A_{\mathrm{left}} \circ \cdots \circ A_{\mathrm{right}}$ |
Monoid operator[](const int idx) const; |
$A_{\mathrm{idx}}$ |
template <typename G> int find_right(int left, const G g);
|
g(get(left, right + 1)) = false を満たす最小の $\mathrm{right}$。ただし存在しないときは $N$ を返す。 |
template <typename G> int find_left(int right, const G g);
|
g(get(left, right)) = false を満たす最大の $\mathrm{left}$。ただし存在しないときは $-1$ を返す。 |
名前 | 説明 |
---|---|
Monoid |
T::Monoid |
template <typename T>
requires requires {
typename T::Monoid;
typename T::OperatorMonoid;
{T::m_id()} -> std::same_as<typename T::Monoid>;
{T::o_id()} -> std::same_as<typename T::OperatorMonoid>;
{T::m_merge(std::declval<typename T::Monoid>(),
std::declval<typename T::Monoid>())}
-> std::same_as<typename T::Monoid>;
{T::o_merge(std::declval<typename T::OperatorMonoid>(),
std::declval<typename T::OperatorMonoid>())}
-> std::same_as<typename T::OperatorMonoid>;
{T::apply(std::declval<typename T::Monoid>(),
std::declval<typename T::OperatorMonoid>())}
-> std::same_as<typename T::Monoid>;
}
struct LazySegmentTree;
T
:モノイドを表す構造体であり、以下の型エイリアスと静的メンバ関数を必要とする。
Monoid
:要素型OperatorMonoid
:作用素モノイドの要素型static constexpr Monoid m_id();
:単位元static constexpr OperatorMonoid o_id();
:作用素モノイドの単位元static Monoid m_merge(const Monoid&, const Monoid&);
:二項演算 $\circ$static OperatorMonoid o_merge(const OperatorMonoid&, const OperatorMonoid&);
static Monoid apply(const Monoid&, const OperatorMonoid&);
名前 | 効果・戻り値 |
---|---|
explicit LazySegmentTree(const int n); |
要素数 $N$ のオブジェクトを構築する。 |
explicit LazySegmentTree(const std::vector<Monoid>& a); |
$A$ に対してオブジェクトを構築する。 |
void set(int idx, const Monoid val); |
$A_{\mathrm{idx}} \gets \mathrm{val}$ |
void apply(int idx, const OperatorMonoid val); |
$\mathrm{idx}$ における変更クエリ |
void apply(int left, int right, const OperatorMonoid val); |
$[\mathrm{left}, \mathrm{right})$ における変更クエリ |
Monoid get(int left, int right); |
$[\mathrm{left}, \mathrm{right})$ における解答クエリ |
Monoid operator[](const int idx); |
$A_{\mathrm{idx}}$ |
template <typename G> int find_right(int left, const G g);
|
g(get(left, right + 1)) = false を満たす最小の $\mathrm{right}$。ただし存在しない場合は $N$ を返す。 |
template <typename G> int find_left(int right, const G g);
|
g(get(left, right)) = false を満たす最大の $\mathrm{left}$。ただし存在しない場合は $-1$ を返す。 |
名前 | 説明 |
---|---|
Monoid |
T::Monoid |
OperatorMonoid |
T::OperatorMonoid |
セグメント木
遅延伝播セグメント木
get
#ifndef EMTHRM_DATA_STRUCTURE_LAZY_SEGMENT_TREE_HPP_
#define EMTHRM_DATA_STRUCTURE_LAZY_SEGMENT_TREE_HPP_
#include <algorithm>
#include <bit>
// #include <cassert>
#include <limits>
#include <type_traits>
#include <vector>
namespace emthrm {
template <typename T>
requires requires {
typename T::Monoid;
typename T::OperatorMonoid;
{T::m_id()} -> std::same_as<typename T::Monoid>;
{T::o_id()} -> std::same_as<typename T::OperatorMonoid>;
{T::m_merge(std::declval<typename T::Monoid>(),
std::declval<typename T::Monoid>())}
-> std::same_as<typename T::Monoid>;
{T::o_merge(std::declval<typename T::OperatorMonoid>(),
std::declval<typename T::OperatorMonoid>())}
-> std::same_as<typename T::OperatorMonoid>;
{T::apply(std::declval<typename T::Monoid>(),
std::declval<typename T::OperatorMonoid>())}
-> std::same_as<typename T::Monoid>;
}
struct LazySegmentTree {
using Monoid = typename T::Monoid;
using OperatorMonoid = typename T::OperatorMonoid;
explicit LazySegmentTree(const int n)
: LazySegmentTree(std::vector<Monoid>(n, T::m_id())) {}
explicit LazySegmentTree(const std::vector<Monoid>& a)
: n(a.size()), height(std::countr_zero(std::bit_ceil(a.size()))),
p2(1 << height) {
lazy.assign(p2, T::o_id());
data.assign(p2 << 1, T::m_id());
std::copy(a.begin(), a.end(), data.begin() + p2);
for (int i = p2 - 1; i > 0; --i) {
data[i] = T::m_merge(data[i << 1], data[(i << 1) + 1]);
}
}
void set(int idx, const Monoid val) {
idx += p2;
for (int i = height; i > 0; --i) {
propagate(idx >> i);
}
data[idx] = val;
for (int i = 1; i <= height; ++i) {
const int current_idx = idx >> i;
data[current_idx] =
T::m_merge(data[current_idx << 1], data[(current_idx << 1) + 1]);
}
}
void apply(int idx, const OperatorMonoid val) {
idx += p2;
for (int i = height; i > 0; --i) {
propagate(idx >> i);
}
data[idx] = T::apply(data[idx], val);
for (int i = 1; i <= height; ++i) {
const int current_idx = idx >> i;
data[current_idx] =
T::m_merge(data[current_idx << 1], data[(current_idx << 1) + 1]);
}
}
void apply(int left, int right, const OperatorMonoid val) {
if (right <= left) [[unlikely]] return;
left += p2;
right += p2;
const int ctz_left = std::countr_zero(static_cast<unsigned int>(left));
for (int i = height; i > ctz_left; --i) {
propagate(left >> i);
}
const int ctz_right = std::countr_zero(static_cast<unsigned int>(right));
for (int i = height; i > ctz_right; --i) {
propagate(right >> i);
}
for (int l = left, r = right; l < r; l >>= 1, r >>= 1) {
if (l & 1) apply_sub(l++, val);
if (r & 1) apply_sub(--r, val);
}
for (int i = left >> (ctz_left + 1); i > 0; i >>= 1) {
data[i] = T::m_merge(data[i << 1], data[(i << 1) + 1]);
}
for (int i = right >> (ctz_right + 1); i > 0; i >>= 1) {
data[i] = T::m_merge(data[i << 1], data[(i << 1) + 1]);
}
}
Monoid get(int left, int right) {
if (right <= left) [[unlikely]] return T::m_id();
left += p2;
right += p2;
const int ctz_left = std::countr_zero(static_cast<unsigned int>(left));
for (int i = height; i > ctz_left; --i) {
propagate(left >> i);
}
const int ctz_right = std::countr_zero(static_cast<unsigned int>(right));
for (int i = height; i > ctz_right; --i) {
propagate(right >> i);
}
Monoid res_l = T::m_id(), res_r = T::m_id();
for (; left < right; left >>= 1, right >>= 1) {
if (left & 1) res_l = T::m_merge(res_l, data[left++]);
if (right & 1) res_r = T::m_merge(data[--right], res_r);
}
return T::m_merge(res_l, res_r);
}
Monoid operator[](const int idx) {
const int node = idx + p2;
for (int i = height; i > 0; --i) {
propagate(node >> i);
}
return data[node];
}
template <typename G>
int find_right(int left, const G g) {
if (left >= n) [[unlikely]] return n;
left += p2;
for (int i = height; i > 0; --i) {
propagate(left >> i);
}
Monoid val = T::m_id();
do {
while (!(left & 1)) left >>= 1;
Monoid nxt = T::m_merge(val, data[left]);
if (!g(nxt)) {
while (left < p2) {
propagate(left);
left <<= 1;
nxt = T::m_merge(val, data[left]);
if (g(nxt)) {
val = nxt;
++left;
}
}
return left - p2;
}
val = nxt;
++left;
} while (!std::has_single_bit(static_cast<unsigned int>(left)));
return n;
}
template <typename G>
int find_left(int right, const G g) {
if (right <= 0) [[unlikely]] return -1;
right += p2;
for (int i = height; i > 0; --i) {
propagate((right - 1) >> i);
}
Monoid val = T::m_id();
do {
--right;
while (right > 1 && (right & 1)) right >>= 1;
Monoid nxt = T::m_merge(data[right], val);
if (!g(nxt)) {
while (right < p2) {
propagate(right);
right = (right << 1) + 1;
nxt = T::m_merge(data[right], val);
if (g(nxt)) {
val = nxt;
--right;
}
}
return right - p2;
}
val = nxt;
} while (!std::has_single_bit(static_cast<unsigned int>(right)));
return -1;
}
private:
const int n, height, p2;
std::vector<Monoid> data;
std::vector<OperatorMonoid> lazy;
void apply_sub(const int idx, const OperatorMonoid& val) {
data[idx] = T::apply(data[idx], val);
if (idx < p2) lazy[idx] = T::o_merge(lazy[idx], val);
}
void propagate(const int idx) {
// assert(1 <= idx && idx < p2);
apply_sub(idx << 1, lazy[idx]);
apply_sub((idx << 1) + 1, lazy[idx]);
lazy[idx] = T::o_id();
}
};
namespace monoid {
template <typename T>
struct RangeMinimumAndUpdateQuery {
using Monoid = T;
using OperatorMonoid = T;
static constexpr Monoid m_id() { return std::numeric_limits<Monoid>::max(); }
static constexpr OperatorMonoid o_id() {
return std::numeric_limits<OperatorMonoid>::max();
}
static Monoid m_merge(const Monoid& a, const Monoid& b) {
return std::min(a, b);
}
static OperatorMonoid o_merge(const OperatorMonoid& a,
const OperatorMonoid& b) {
return b == o_id() ? a : b;
}
static Monoid apply(const Monoid& a, const OperatorMonoid& b) {
return b == o_id() ? a : b;
}
};
template <typename T>
struct RangeMaximumAndUpdateQuery {
using Monoid = T;
using OperatorMonoid = T;
static constexpr Monoid m_id() {
return std::numeric_limits<Monoid>::lowest();
}
static constexpr OperatorMonoid o_id() {
return std::numeric_limits<OperatorMonoid>::lowest();
}
static Monoid m_merge(const Monoid& a, const Monoid& b) {
return std::max(a, b);
}
static OperatorMonoid o_merge(const OperatorMonoid& a,
const OperatorMonoid& b) {
return b == o_id() ? a : b;
}
static Monoid apply(const Monoid& a, const OperatorMonoid& b) {
return b == o_id()? a : b;
}
};
template <typename T, T Inf>
struct RangeMinimumAndAddQuery {
using Monoid = T;
using OperatorMonoid = T;
static constexpr Monoid m_id() { return Inf; }
static constexpr OperatorMonoid o_id() { return 0; }
static Monoid m_merge(const Monoid& a, const Monoid& b) {
return std::min(a, b);
}
static OperatorMonoid o_merge(const OperatorMonoid& a,
const OperatorMonoid& b) {
return a + b;
}
static Monoid apply(const Monoid& a, const OperatorMonoid& b) {
return a + b;
}
};
template <typename T, T Inf>
struct RangeMaximumAndAddQuery {
using Monoid = T;
using OperatorMonoid = T;
static constexpr Monoid m_id() { return -Inf; }
static constexpr OperatorMonoid o_id() { return 0; }
static Monoid m_merge(const Monoid& a, const Monoid& b) {
return std::max(a, b);
}
static OperatorMonoid o_merge(const OperatorMonoid& a,
const OperatorMonoid& b) {
return a + b;
}
static Monoid apply(const Monoid& a, const OperatorMonoid& b) {
return a + b;
}
};
template <typename T>
struct RangeSumAndUpdateQuery {
using Monoid = struct { T sum; int len; };
using OperatorMonoid = T;
static std::vector<Monoid> init(const int n) {
return std::vector<Monoid>(n, Monoid{0, 1});
}
static constexpr Monoid m_id() { return {0, 0}; }
static constexpr OperatorMonoid o_id() {
return std::numeric_limits<OperatorMonoid>::max();
}
static Monoid m_merge(const Monoid& a, const Monoid& b) {
return Monoid{a.sum + b.sum, a.len + b.len};
}
static OperatorMonoid o_merge(const OperatorMonoid& a,
const OperatorMonoid& b) {
return b == o_id() ? a : b;
}
static Monoid apply(const Monoid& a, const OperatorMonoid& b) {
return Monoid{b == o_id() ? a.sum : b * a.len, a.len};
}
};
template <typename T>
struct RangeSumAndAddQuery {
using Monoid = struct { T sum; int len; };
using OperatorMonoid = T;
static std::vector<Monoid> init(const int n) {
return std::vector<Monoid>(n, Monoid{0, 1});
}
static constexpr Monoid m_id() { return {0, 0}; }
static constexpr OperatorMonoid o_id() { return 0; }
static Monoid m_merge(const Monoid& a, const Monoid& b) {
return Monoid{a.sum + b.sum, a.len + b.len};
}
static OperatorMonoid o_merge(const OperatorMonoid& a,
const OperatorMonoid& b) {
return a + b;
}
static Monoid apply(const Monoid& a, const OperatorMonoid& b) {
return Monoid{a.sum + b * a.len, a.len};
}
};
} // namespace monoid
} // namespace emthrm
#endif // EMTHRM_DATA_STRUCTURE_LAZY_SEGMENT_TREE_HPP_
#line 1 "include/emthrm/data_structure/lazy_segment_tree.hpp"
#include <algorithm>
#include <bit>
// #include <cassert>
#include <limits>
#include <type_traits>
#include <vector>
namespace emthrm {
template <typename T>
requires requires {
typename T::Monoid;
typename T::OperatorMonoid;
{T::m_id()} -> std::same_as<typename T::Monoid>;
{T::o_id()} -> std::same_as<typename T::OperatorMonoid>;
{T::m_merge(std::declval<typename T::Monoid>(),
std::declval<typename T::Monoid>())}
-> std::same_as<typename T::Monoid>;
{T::o_merge(std::declval<typename T::OperatorMonoid>(),
std::declval<typename T::OperatorMonoid>())}
-> std::same_as<typename T::OperatorMonoid>;
{T::apply(std::declval<typename T::Monoid>(),
std::declval<typename T::OperatorMonoid>())}
-> std::same_as<typename T::Monoid>;
}
struct LazySegmentTree {
using Monoid = typename T::Monoid;
using OperatorMonoid = typename T::OperatorMonoid;
explicit LazySegmentTree(const int n)
: LazySegmentTree(std::vector<Monoid>(n, T::m_id())) {}
explicit LazySegmentTree(const std::vector<Monoid>& a)
: n(a.size()), height(std::countr_zero(std::bit_ceil(a.size()))),
p2(1 << height) {
lazy.assign(p2, T::o_id());
data.assign(p2 << 1, T::m_id());
std::copy(a.begin(), a.end(), data.begin() + p2);
for (int i = p2 - 1; i > 0; --i) {
data[i] = T::m_merge(data[i << 1], data[(i << 1) + 1]);
}
}
void set(int idx, const Monoid val) {
idx += p2;
for (int i = height; i > 0; --i) {
propagate(idx >> i);
}
data[idx] = val;
for (int i = 1; i <= height; ++i) {
const int current_idx = idx >> i;
data[current_idx] =
T::m_merge(data[current_idx << 1], data[(current_idx << 1) + 1]);
}
}
void apply(int idx, const OperatorMonoid val) {
idx += p2;
for (int i = height; i > 0; --i) {
propagate(idx >> i);
}
data[idx] = T::apply(data[idx], val);
for (int i = 1; i <= height; ++i) {
const int current_idx = idx >> i;
data[current_idx] =
T::m_merge(data[current_idx << 1], data[(current_idx << 1) + 1]);
}
}
void apply(int left, int right, const OperatorMonoid val) {
if (right <= left) [[unlikely]] return;
left += p2;
right += p2;
const int ctz_left = std::countr_zero(static_cast<unsigned int>(left));
for (int i = height; i > ctz_left; --i) {
propagate(left >> i);
}
const int ctz_right = std::countr_zero(static_cast<unsigned int>(right));
for (int i = height; i > ctz_right; --i) {
propagate(right >> i);
}
for (int l = left, r = right; l < r; l >>= 1, r >>= 1) {
if (l & 1) apply_sub(l++, val);
if (r & 1) apply_sub(--r, val);
}
for (int i = left >> (ctz_left + 1); i > 0; i >>= 1) {
data[i] = T::m_merge(data[i << 1], data[(i << 1) + 1]);
}
for (int i = right >> (ctz_right + 1); i > 0; i >>= 1) {
data[i] = T::m_merge(data[i << 1], data[(i << 1) + 1]);
}
}
Monoid get(int left, int right) {
if (right <= left) [[unlikely]] return T::m_id();
left += p2;
right += p2;
const int ctz_left = std::countr_zero(static_cast<unsigned int>(left));
for (int i = height; i > ctz_left; --i) {
propagate(left >> i);
}
const int ctz_right = std::countr_zero(static_cast<unsigned int>(right));
for (int i = height; i > ctz_right; --i) {
propagate(right >> i);
}
Monoid res_l = T::m_id(), res_r = T::m_id();
for (; left < right; left >>= 1, right >>= 1) {
if (left & 1) res_l = T::m_merge(res_l, data[left++]);
if (right & 1) res_r = T::m_merge(data[--right], res_r);
}
return T::m_merge(res_l, res_r);
}
Monoid operator[](const int idx) {
const int node = idx + p2;
for (int i = height; i > 0; --i) {
propagate(node >> i);
}
return data[node];
}
template <typename G>
int find_right(int left, const G g) {
if (left >= n) [[unlikely]] return n;
left += p2;
for (int i = height; i > 0; --i) {
propagate(left >> i);
}
Monoid val = T::m_id();
do {
while (!(left & 1)) left >>= 1;
Monoid nxt = T::m_merge(val, data[left]);
if (!g(nxt)) {
while (left < p2) {
propagate(left);
left <<= 1;
nxt = T::m_merge(val, data[left]);
if (g(nxt)) {
val = nxt;
++left;
}
}
return left - p2;
}
val = nxt;
++left;
} while (!std::has_single_bit(static_cast<unsigned int>(left)));
return n;
}
template <typename G>
int find_left(int right, const G g) {
if (right <= 0) [[unlikely]] return -1;
right += p2;
for (int i = height; i > 0; --i) {
propagate((right - 1) >> i);
}
Monoid val = T::m_id();
do {
--right;
while (right > 1 && (right & 1)) right >>= 1;
Monoid nxt = T::m_merge(data[right], val);
if (!g(nxt)) {
while (right < p2) {
propagate(right);
right = (right << 1) + 1;
nxt = T::m_merge(data[right], val);
if (g(nxt)) {
val = nxt;
--right;
}
}
return right - p2;
}
val = nxt;
} while (!std::has_single_bit(static_cast<unsigned int>(right)));
return -1;
}
private:
const int n, height, p2;
std::vector<Monoid> data;
std::vector<OperatorMonoid> lazy;
void apply_sub(const int idx, const OperatorMonoid& val) {
data[idx] = T::apply(data[idx], val);
if (idx < p2) lazy[idx] = T::o_merge(lazy[idx], val);
}
void propagate(const int idx) {
// assert(1 <= idx && idx < p2);
apply_sub(idx << 1, lazy[idx]);
apply_sub((idx << 1) + 1, lazy[idx]);
lazy[idx] = T::o_id();
}
};
namespace monoid {
template <typename T>
struct RangeMinimumAndUpdateQuery {
using Monoid = T;
using OperatorMonoid = T;
static constexpr Monoid m_id() { return std::numeric_limits<Monoid>::max(); }
static constexpr OperatorMonoid o_id() {
return std::numeric_limits<OperatorMonoid>::max();
}
static Monoid m_merge(const Monoid& a, const Monoid& b) {
return std::min(a, b);
}
static OperatorMonoid o_merge(const OperatorMonoid& a,
const OperatorMonoid& b) {
return b == o_id() ? a : b;
}
static Monoid apply(const Monoid& a, const OperatorMonoid& b) {
return b == o_id() ? a : b;
}
};
template <typename T>
struct RangeMaximumAndUpdateQuery {
using Monoid = T;
using OperatorMonoid = T;
static constexpr Monoid m_id() {
return std::numeric_limits<Monoid>::lowest();
}
static constexpr OperatorMonoid o_id() {
return std::numeric_limits<OperatorMonoid>::lowest();
}
static Monoid m_merge(const Monoid& a, const Monoid& b) {
return std::max(a, b);
}
static OperatorMonoid o_merge(const OperatorMonoid& a,
const OperatorMonoid& b) {
return b == o_id() ? a : b;
}
static Monoid apply(const Monoid& a, const OperatorMonoid& b) {
return b == o_id()? a : b;
}
};
template <typename T, T Inf>
struct RangeMinimumAndAddQuery {
using Monoid = T;
using OperatorMonoid = T;
static constexpr Monoid m_id() { return Inf; }
static constexpr OperatorMonoid o_id() { return 0; }
static Monoid m_merge(const Monoid& a, const Monoid& b) {
return std::min(a, b);
}
static OperatorMonoid o_merge(const OperatorMonoid& a,
const OperatorMonoid& b) {
return a + b;
}
static Monoid apply(const Monoid& a, const OperatorMonoid& b) {
return a + b;
}
};
template <typename T, T Inf>
struct RangeMaximumAndAddQuery {
using Monoid = T;
using OperatorMonoid = T;
static constexpr Monoid m_id() { return -Inf; }
static constexpr OperatorMonoid o_id() { return 0; }
static Monoid m_merge(const Monoid& a, const Monoid& b) {
return std::max(a, b);
}
static OperatorMonoid o_merge(const OperatorMonoid& a,
const OperatorMonoid& b) {
return a + b;
}
static Monoid apply(const Monoid& a, const OperatorMonoid& b) {
return a + b;
}
};
template <typename T>
struct RangeSumAndUpdateQuery {
using Monoid = struct { T sum; int len; };
using OperatorMonoid = T;
static std::vector<Monoid> init(const int n) {
return std::vector<Monoid>(n, Monoid{0, 1});
}
static constexpr Monoid m_id() { return {0, 0}; }
static constexpr OperatorMonoid o_id() {
return std::numeric_limits<OperatorMonoid>::max();
}
static Monoid m_merge(const Monoid& a, const Monoid& b) {
return Monoid{a.sum + b.sum, a.len + b.len};
}
static OperatorMonoid o_merge(const OperatorMonoid& a,
const OperatorMonoid& b) {
return b == o_id() ? a : b;
}
static Monoid apply(const Monoid& a, const OperatorMonoid& b) {
return Monoid{b == o_id() ? a.sum : b * a.len, a.len};
}
};
template <typename T>
struct RangeSumAndAddQuery {
using Monoid = struct { T sum; int len; };
using OperatorMonoid = T;
static std::vector<Monoid> init(const int n) {
return std::vector<Monoid>(n, Monoid{0, 1});
}
static constexpr Monoid m_id() { return {0, 0}; }
static constexpr OperatorMonoid o_id() { return 0; }
static Monoid m_merge(const Monoid& a, const Monoid& b) {
return Monoid{a.sum + b.sum, a.len + b.len};
}
static OperatorMonoid o_merge(const OperatorMonoid& a,
const OperatorMonoid& b) {
return a + b;
}
static Monoid apply(const Monoid& a, const OperatorMonoid& b) {
return Monoid{a.sum + b * a.len, a.len};
}
};
} // namespace monoid
} // namespace emthrm