// codesnip-guard: LazySegmentTreeMap
#[cfg_attr(any(),rustfmt::skip)]pub use self::lazy_segment_tree_map::LazySegmentTreeMap;#[cfg_attr(any(),rustfmt::skip)]mod lazy_segment_tree_map{use super::{FibHashMap,LazyMapAct,LazyMapAgg,LazyMapMonoid,RangeBoundsExt};use std::{fmt::{self,Debug,Formatter},mem::replace,ops::RangeBounds};pub struct LazySegmentTreeMap<M>where M:LazyMapMonoid{n:usize,seg:FibHashMap<usize,(LazyMapAgg<M>,LazyMapAct<M>)>}impl<M>Clone for LazySegmentTreeMap<M>where M:LazyMapMonoid{fn clone(&self)->Self{Self{n:self.n,seg:self.seg.clone()}}}impl<M>Debug for LazySegmentTreeMap<M>where M:LazyMapMonoid,LazyMapAgg<M>:Debug,LazyMapAct<M>:Debug{fn fmt(&self,f:&mut Formatter<'_>)->fmt::Result{f.debug_struct("LazySegmentTreeMap").field("n",&self.n).field("seg",&self.seg).finish()}}impl<M>LazySegmentTreeMap<M>where M:LazyMapMonoid{pub fn new(n:usize)->Self{Self{n,seg:Default::default()}}#[inline]fn get_mut(&mut self,k:usize)->&mut(LazyMapAgg<M>,LazyMapAct<M>){self.seg.entry(k).or_insert((M::agg_unit(),M::act_unit()))}#[inline]fn update_at(&mut self,k:usize,x:&LazyMapAct<M>){if M::is_act_unit(x){return;}let n=self.n;let a=self.get_mut(k);let nx=M::act_agg(&a.0,x);if k<n{a.1=M::act_operate(&a.1,x);}if let Some(nx)=nx{a.0=nx;}else if k<n{self.propagate_at(k);self.recalc_at(k);}else{panic!("act failed on leaf");}}#[inline]fn recalc_at(&mut self,k:usize){let x=match(self.seg.get(&(2*k)),self.seg.get(&(2*k+1))){(None,None)=>M::agg_unit(),(None,Some((y,_)))=>y.clone(),(Some((x,_)),None)=>x.clone(),(Some((x,_)),Some((y,_)))=>M::agg_operate(x,y),};self.get_mut(k).0=x;}#[inline]fn propagate_at(&mut self,k:usize){debug_assert!(k<self.n);let x=match self.seg.get_mut(&k){Some((_,x))=>replace(x,M::act_unit()),None=>M::act_unit(),};if M::is_act_unit(&x){return;}self.update_at(2*k,&x);self.update_at(2*k+1,&x);}#[inline]fn propagate(&mut self,k:usize,right:bool,nofilt:bool){let right=right as usize;for i in(1..(k+1-right).next_power_of_two().trailing_zeros()).rev(){if nofilt||(k>>i)<<i!=k{self.propagate_at((k-right)>>i);}}}#[inline]fn recalc(&mut self,k:usize,right:bool,nofilt:bool){let right=right as usize;for i in 1..(k+1-right).next_power_of_two().trailing_zeros(){if nofilt||(k>>i)<<i!=k{self.recalc_at((k-right)>>i);}}}pub fn update<R>(&mut self,range:R,x:LazyMapAct<M>)where R:RangeBounds<usize>{let range=range.to_range_bounded(0,self.n).expect("invalid range");if M::is_act_unit(&x){return;}let mut a=range.start+self.n;let mut b=range.end+self.n;self.propagate(a,false,false);self.propagate(b,true,false);while a<b{if a&1!=0{self.update_at(a,&x);a+=1;}if b&1!=0{b-=1;self.update_at(b,&x);}a/=2;b/=2;}self.recalc(range.start+self.n,false,false);self.recalc(range.end+self.n,true,false);}pub fn fold<R>(&mut self,range:R)->LazyMapAgg<M>where R:RangeBounds<usize>{let range=range.to_range_bounded(0,self.n).expect("invalid range");if range.is_empty(){return M::agg_unit();}let mut l=range.start+self.n;let mut r=range.end+self.n;self.propagate(l,false,true);self.propagate(r,true,true);let mut vl=M::agg_unit();let mut vr=M::agg_unit();while l<r{if l&1!=0{if let Some((x,_))=self.seg.get(&l){vl=M::agg_operate(&vl,x);}l+=1;}if r&1!=0{r-=1;if let Some((x,_))=self.seg.get(&r){vr=M::agg_operate(x,&vr);}}l/=2;r/=2;}M::agg_operate(&vl,&vr)}pub fn set(&mut self,k:usize,x:LazyMapAgg<M>){let k=k+self.n;self.propagate(k,false,true);*self.get_mut(k)=(x,M::act_unit());self.recalc(k,false,true);}pub fn get(&mut self,k:usize)->LazyMapAgg<M>{assert!(k<self.n);let k=k+self.n;self.propagate(k,false,true);self.seg.get(&k).map(|(x,_)|x.clone()).unwrap_or_else(M::agg_unit)}pub fn fold_all(&mut self)->LazyMapAgg<M>{self.fold(0..self.n)}fn partition_point_perfect<P>(&mut self,mut pos:usize,mut acc:LazyMapAgg<M>,mut pred:P)->(usize,LazyMapAgg<M>)where P:FnMut(&LazyMapAgg<M>)->bool{while pos<self.n{self.propagate_at(pos);pos<<=1;let nacc=match self.seg.get(&pos){Some((x,_))=>M::agg_operate(&acc,x),None=>acc.clone(),};if pred(&nacc){acc=nacc;pos+=1;}}(pos-self.n,acc)}fn rpartition_point_perfect<P>(&mut self,mut pos:usize,mut acc:LazyMapAgg<M>,mut pred:P)->(usize,LazyMapAgg<M>)where P:FnMut(&LazyMapAgg<M>)->bool{while pos<self.n{self.propagate_at(pos);pos=pos*2+1;let nacc=match self.seg.get(&pos){Some((x,_))=>M::agg_operate(x,&acc),None=>acc.clone(),};if pred(&nacc){acc=nacc;pos-=1;}}(pos-self.n,acc)}pub fn partition_point_acc<P>(&mut self,left:usize,mut pred:P)->usize where P:FnMut(&LazyMapAgg<M>)->bool{let mut acc=M::agg_unit();if left==self.n{return self.n;}let mut l=left+self.n;let r=2*self.n;self.propagate(l,false,true);self.propagate(r,true,true);let mut k=0usize;while l<r>>k{if l&1!=0{let nacc=match self.seg.get(&l){Some((x,_))=>M::agg_operate(&acc,x),None=>acc.clone(),};if!pred(&nacc){return self.partition_point_perfect(l,acc,pred).0;}acc=nacc;l+=1;}l>>=1;k+=1;}for k in(0..k).rev(){let r=r>>k;if r&1!=0{let nacc=match self.seg.get(&(r-1)){Some((x,_))=>M::agg_operate(&acc,x),None=>acc.clone(),};if!pred(&nacc){return self.partition_point_perfect(r-1,acc,pred).0;}acc=nacc;}}self.n}pub fn rpartition_point_acc<P>(&mut self,right:usize,mut pred:P)->usize where P:FnMut(&LazyMapAgg<M>)->bool{let mut acc=M::agg_unit();if right==0{return 0;}let mut l=self.n;let mut r=right+self.n;self.propagate(l,false,true);self.propagate(r,true,true);let mut c=0usize;let mut k=0usize;while l>>k<r{c<<=1;if l&(1<<k)!=0{l+=1<<k;c+=1;}if r&1!=0{r-=1;let nacc=match self.seg.get(&r){Some((x,_))=>M::agg_operate(x,&acc),None=>acc.clone(),};if!pred(&nacc){return self.rpartition_point_perfect(r,acc,pred).0+1;}acc=nacc;}r>>=1;k+=1;}for k in(0..k).rev(){if c&1!=0{l-=1<<k;let l=l>>k;let nacc=match self.seg.get(&l){Some((x,_))=>M::agg_operate(x,&acc),None=>acc.clone(),};if!pred(&nacc){return self.rpartition_point_perfect(l,acc,pred).0+1;}acc=nacc;}c>>=1;}0}}}
// codesnip-guard: AdditiveOperation
#[cfg_attr(any(),rustfmt::skip)]pub use self::additive_operation_impl::AdditiveOperation;#[cfg_attr(any(),rustfmt::skip)]mod additive_operation_impl{use super::*;use std::{marker::PhantomData,ops::{Add,Neg,Sub}};#[doc=" $+$"]pub struct AdditiveOperation<T>where T:Clone+Zero+Add<Output=T>{_marker:PhantomData<fn()->T>}impl<T>Magma for AdditiveOperation<T>where T:Clone+Zero+Add<Output=T>{type T=T;#[inline]fn operate(x:&Self::T,y:&Self::T)->Self::T{x.clone()+y.clone()}}impl<T>Unital for AdditiveOperation<T>where T:Clone+Zero+Add<Output=T>{#[inline]fn unit()->Self::T{Zero::zero()}}impl<T>Associative for AdditiveOperation<T>where T:Clone+Zero+Add<Output=T>{}impl<T>Commutative for AdditiveOperation<T>where T:Clone+Zero+Add<Output=T>{}impl<T>Invertible for AdditiveOperation<T>where T:Clone+Zero+Add<Output=T>+Sub<Output=T>+Neg<Output=T>{#[inline]fn inverse(x:&Self::T)->Self::T{-x.clone()}#[inline]fn rinv_operate(x:&Self::T,y:&Self::T)->Self::T{x.clone()-y.clone()}}}
// codesnip-guard: ArrayOperation
#[cfg_attr(any(),rustfmt::skip)]pub use self::array_operation_impl::ArrayOperation;#[cfg_attr(any(),rustfmt::skip)]mod array_operation_impl{use super::*;use crate::array;use std::marker::PhantomData;pub struct ArrayOperation<M,const N:usize>{_marker:PhantomData<fn()->M>}impl<M,const N:usize>Magma for ArrayOperation<M,N>where M:Magma{type T=[M::T;N];#[inline]fn operate(x:&Self::T,y:&Self::T)->Self::T{array!(|i|M::operate(&x[i],&y[i]);N)}}impl<M,const N:usize>Unital for ArrayOperation<M,N>where M:Unital{#[inline]fn unit()->Self::T{array!(||M::unit();N)}}impl<M,const N:usize>Associative for ArrayOperation<M,N>where M:Associative{}impl<M,const N:usize>Commutative for ArrayOperation<M,N>where M:Commutative{}impl<M,const N:usize>Idempotent for ArrayOperation<M,N>where M:Idempotent{}impl<M,const N:usize>Invertible for ArrayOperation<M,N>where M:Invertible{#[inline]fn inverse(x:&Self::T)->Self::T{array!(|i|M::inverse(&x[i]);N)}}}
// codesnip-guard: CountingOperation
#[cfg_attr(any(),rustfmt::skip)]pub use self::counting_operation_impl::CountingOperation;#[cfg_attr(any(),rustfmt::skip)]mod counting_operation_impl{use super::*;use std::marker::PhantomData;pub struct CountingOperation<M>{_marker:PhantomData<fn()->M>}impl<M>Magma for CountingOperation<M>where M:Magma<T:PartialEq>+Idempotent{type T=(M::T,usize);#[inline]fn operate(x:&Self::T,y:&Self::T)->Self::T{let z=M::operate(&x.0,&y.0);match(z==x.0,z==y.0){(true,true)=>(z,x.1+y.1),(true,false)=>(z,x.1),(false,true)=>(z,y.1),(false,false)=>(z,1),}}}impl<M>Unital for CountingOperation<M>where M:Unital<T:PartialEq>+Idempotent{#[inline]fn unit()->Self::T{(M::unit(),0)}}impl<M>Associative for CountingOperation<M>where M:Associative<T:PartialEq>+Idempotent{}impl<M>Commutative for CountingOperation<M>where M:Commutative<T:PartialEq>+Idempotent{}}
// codesnip-guard: FibonacciHash
#[cfg_attr(any(),rustfmt::skip)]pub use self::fibonacci_hash::{FibHashMap,FibHashSet,FibonacciHasher,FibonacciHasheru32,FibonacciHasheru64};#[cfg_attr(any(),rustfmt::skip)]mod fibonacci_hash{use std::{collections::{HashMap,HashSet},hash::{BuildHasherDefault,Hasher}};#[cfg(target_pointer_width="32")]pub type FibonacciHasher=FibonacciHasheru32;#[cfg(not(target_pointer_width="32"))]pub type FibonacciHasher=FibonacciHasheru64;pub type FibHashMap<K,V>=HashMap<K,V,BuildHasherDefault<FibonacciHasher>>;pub type FibHashSet<V>=HashSet<V,BuildHasherDefault<FibonacciHasher>>;#[derive(Debug,Default)]pub struct FibonacciHasheru32{hash:u32}impl FibonacciHasheru32{const A:u32=2654435769;const B:u32=5;fn push(&mut self,x:u32){self.hash=(self.hash.rotate_right(Self::B)^x).wrapping_mul(Self::A);}}impl Hasher for FibonacciHasheru32{fn finish(&self)->u64{self.hash as u64}fn write(&mut self,mut bytes:&[u8]){if bytes.len()%4>=2{let(chunk,rest)=bytes.split_first_chunk::<2>().unwrap();self.push(u16::from_ne_bytes(*chunk)as u32);bytes=rest;}if bytes.len()%2>=1{let(chunk,rest)=bytes.split_first_chunk::<1>().unwrap();self.push(u8::from_ne_bytes(*chunk)as u32);bytes=rest;}for chunk in bytes.as_chunks::<4>().0{self.push(u32::from_ne_bytes(*chunk));}}fn write_u8(&mut self,i:u8){self.push(i as u32)}fn write_u16(&mut self,i:u16){self.push(i as u32)}fn write_u32(&mut self,i:u32){self.push(i)}fn write_u64(&mut self,i:u64){self.push(i as u32);self.push((i>>32)as u32);}fn write_u128(&mut self,i:u128){self.push(i as u32);self.push((i>>32)as u32);self.push((i>>64)as u32);self.push((i>>96)as u32);}#[cfg(target_pointer_width="32")]fn write_usize(&mut self,i:usize){self.write_u32(i as u32)}#[cfg(target_pointer_width="64")]fn write_usize(&mut self,i:usize){self.write_u64(i as u64)}fn write_i8(&mut self,i:i8){self.write_u8(i as u8)}fn write_i16(&mut self,i:i16){self.write_u16(i as u16)}fn write_i32(&mut self,i:i32){self.write_u32(i as u32)}fn write_i64(&mut self,i:i64){self.write_u64(i as u64)}fn write_i128(&mut self,i:i128){self.write_u128(i as u128)}fn write_isize(&mut self,i:isize){self.write_usize(i as usize)}}#[derive(Debug,Default)]pub struct FibonacciHasheru64{hash:u64}impl FibonacciHasheru64{const A:u64=11400714819323198485;const B:u32=5;fn push(&mut self,x:u64){self.hash=(self.hash.rotate_right(Self::B)^x).wrapping_mul(Self::A);}}impl Hasher for FibonacciHasheru64{fn finish(&self)->u64{self.hash}fn write(&mut self,mut bytes:&[u8]){if bytes.len()%8>=4{let(chunk,rest)=bytes.split_first_chunk::<4>().unwrap();self.push(u32::from_ne_bytes(*chunk)as u64);bytes=rest;}if bytes.len()%4>=2{let(chunk,rest)=bytes.split_first_chunk::<2>().unwrap();self.push(u16::from_ne_bytes(*chunk)as u64);bytes=rest;}if bytes.len()%2>=1{let(chunk,rest)=bytes.split_first_chunk::<1>().unwrap();self.push(u8::from_ne_bytes(*chunk)as u64);bytes=rest;}for chunk in bytes.as_chunks::<8>().0{self.push(u64::from_ne_bytes(*chunk));}}fn write_u8(&mut self,i:u8){self.push(i as u64)}fn write_u16(&mut self,i:u16){self.push(i as u64)}fn write_u32(&mut self,i:u32){self.push(i as u64)}fn write_u64(&mut self,i:u64){self.push(i);}fn write_u128(&mut self,i:u128){self.push(i as u64);self.push((i>>64)as u64);}fn write_usize(&mut self,i:usize){self.write_u64(i as u64)}fn write_i8(&mut self,i:i8){self.write_u8(i as u8)}fn write_i16(&mut self,i:i16){self.write_u16(i as u16)}fn write_i32(&mut self,i:i32){self.write_u32(i as u32)}fn write_i64(&mut self,i:i64){self.write_u64(i as u64)}fn write_i128(&mut self,i:i128){self.write_u128(i as u128)}fn write_isize(&mut self,i:isize){self.write_usize(i as usize)}}}
// codesnip-guard: LastOperation
#[cfg_attr(any(),rustfmt::skip)]pub use self::last_operation_impl::LastOperation;#[cfg_attr(any(),rustfmt::skip)]mod last_operation_impl{use super::*;use std::marker::PhantomData;#[doc=" retain the last element"]pub struct LastOperation<T>where T:Clone{_marker:PhantomData<fn()->T>}impl<T>Magma for LastOperation<T>where T:Clone{type T=Option<T>;#[inline]fn operate(x:&Self::T,y:&Self::T)->Self::T{y.as_ref().or(x.as_ref()).cloned()}}impl<T>Unital for LastOperation<T>where T:Clone{#[inline]fn unit()->Self::T{None}}impl<T>Associative for LastOperation<T>where T:Clone{}impl<T>Idempotent for LastOperation<T>where T:Clone{}}
// codesnip-guard: LazyMapMonoid
#[cfg_attr(any(),rustfmt::skip)]pub use self::lazy_map::*;#[cfg_attr(any(),rustfmt::skip)]mod lazy_map{use super::*;use std::{cmp::Ordering,marker::PhantomData,ops::{Add,Mul,Sub}};pub type LazyMapAgg<L>=<<L as LazyMapMonoid>::AggMonoid as Magma>::T;pub type LazyMapAct<L>=<<L as LazyMapMonoid>::ActMonoid as Magma>::T;pub trait LazyMapMonoid{type Key;type AggMonoid:Monoid;type ActMonoid:Monoid<T:PartialEq>;fn single_agg(key:&Self::Key)->LazyMapAgg<Self>;fn toggle(_x:&mut LazyMapAgg<Self>){}#[inline]fn is_act_unit(act:&LazyMapAct<Self>)->bool{<Self::ActMonoid as Unital>::is_unit(act)}#[doc=" Returns `None` when descent is needed; must succeed on single-key aggregates."]fn act_agg(x:&LazyMapAgg<Self>,a:&LazyMapAct<Self>)->Option<LazyMapAgg<Self>>;#[inline]fn agg_unit()->LazyMapAgg<Self>{<Self::AggMonoid as Unital>::unit()}#[inline]fn act_unit()->LazyMapAct<Self>{<Self::ActMonoid as Unital>::unit()}#[inline]fn agg_operate(x:&LazyMapAgg<Self>,y:&LazyMapAgg<Self>)->LazyMapAgg<Self>{<Self::AggMonoid as Magma>::operate(x,y)}#[inline]fn act_operate(x:&LazyMapAct<Self>,y:&LazyMapAct<Self>)->LazyMapAct<Self>{<Self::ActMonoid as Magma>::operate(x,y)}#[inline]fn agg_operate_assign(x:&mut LazyMapAgg<Self>,y:&LazyMapAgg<Self>){*x=<Self::AggMonoid as Magma>::operate(x,y);}#[inline]fn act_operate_assign(x:&mut LazyMapAct<Self>,y:&LazyMapAct<Self>){*x=<Self::ActMonoid as Magma>::operate(x,y);}}pub trait LazyMapMonoidWithKeyAct:LazyMapMonoid{fn act_key(key:&Self::Key,act:&LazyMapAct<Self>)->Self::Key;}pub struct EmptyActLazy<M>{_marker:PhantomData<fn()->M>}impl<M>LazyMapMonoid for EmptyActLazy<M>where M:Monoid{type Key=M::T;type AggMonoid=M;type ActMonoid=();fn single_agg(key:&Self::Key)->LazyMapAgg<Self>{key.clone()}fn act_agg(x:&LazyMapAgg<Self>,_a:&LazyMapAct<Self>)->Option<LazyMapAgg<Self>>{Some(x.clone())}}impl<M>LazyMapMonoidWithKeyAct for EmptyActLazy<M>where M:Monoid{fn act_key(key:&Self::Key,act:&LazyMapAct<Self>)->Self::Key{EmptyAct::act(key,act)}}pub struct EmptyAggActLazy<T>{_marker:PhantomData<fn()->T>}impl<T>LazyMapMonoid for EmptyAggActLazy<T>{type Key=T;type AggMonoid=();type ActMonoid=();fn single_agg(_key:&Self::Key)->LazyMapAgg<Self>{}fn act_agg(_x:&LazyMapAgg<Self>,_a:&LazyMapAct<Self>)->Option<LazyMapAgg<Self>>{Some(())}}impl<T>LazyMapMonoidWithKeyAct for EmptyAggActLazy<T>where T:Clone{fn act_key(key:&Self::Key,act:&LazyMapAct<Self>)->Self::Key{EmptyAct::act(key,act)}}pub struct FlattenLazy<M>{_marker:PhantomData<fn()->M>}impl<M>LazyMapMonoid for FlattenLazy<M>where M:Monoid,M::T:PartialEq{type Key=M::T;type AggMonoid=M;type ActMonoid=M;fn single_agg(key:&Self::Key)->LazyMapAgg<Self>{key.clone()}fn act_agg(x:&LazyMapAgg<Self>,a:&LazyMapAct<Self>)->Option<LazyMapAgg<Self>>{Some(M::operate(x,a))}}impl<M>LazyMapMonoidWithKeyAct for FlattenLazy<M>where M:Monoid,M::T:PartialEq{fn act_key(key:&Self::Key,act:&LazyMapAct<Self>)->Self::Key{FlattenAct::<M>::act(key,act)}}pub struct RangeSumRangeAdd<T>{_marker:PhantomData<fn()->T>}impl<T>LazyMapMonoid for RangeSumRangeAdd<T>where T:Copy+Zero+One+Add<Output=T>+Mul<Output=T>+PartialEq{type Key=T;type AggMonoid=(AdditiveOperation<T>,AdditiveOperation<T>);type ActMonoid=AdditiveOperation<T>;fn single_agg(key:&Self::Key)->LazyMapAgg<Self>{(*key,T::one())}fn act_agg(&(x,y):&LazyMapAgg<Self>,a:&LazyMapAct<Self>)->Option<LazyMapAgg<Self>>{Some(if Self::is_act_unit(a){(x,y)}else{let a=*a;(x+a*y,y)})}}impl<T>LazyMapMonoidWithKeyAct for RangeSumRangeAdd<T>where T:Copy+Zero+One+Add<Output=T>+Mul<Output=T>+PartialEq{fn act_key(key:&Self::Key,act:&LazyMapAct<Self>)->Self::Key{FlattenAct::<Self::ActMonoid>::act(key,act)}}pub struct RangeSumRangeLinear<T>{_marker:PhantomData<fn()->T>}impl<T>LazyMapMonoid for RangeSumRangeLinear<T>where T:Copy+Zero+One+Add<Output=T>+Mul<Output=T>+PartialEq{type Key=T;type AggMonoid=(AdditiveOperation<T>,AdditiveOperation<T>);type ActMonoid=LinearOperation<T>;fn single_agg(key:&Self::Key)->LazyMapAgg<Self>{(*key,T::one())}fn act_agg(&(x,y):&LazyMapAgg<Self>,&(a,b):&LazyMapAct<Self>)->Option<LazyMapAgg<Self>>{Some((a*x+b*y,y))}}impl<T>LazyMapMonoidWithKeyAct for RangeSumRangeLinear<T>where T:Copy+Zero+One+Add<Output=T>+Mul<Output=T>+PartialEq{fn act_key(key:&Self::Key,act:&LazyMapAct<Self>)->Self::Key{LinearAct::act(key,act)}}pub struct RangePowerSumRangeLinear<T,const K:usize>{_marker:PhantomData<fn()->T>}impl<T,const K:usize>LazyMapMonoid for RangePowerSumRangeLinear<T,K>where T:Copy+Zero+One+Add<Output=T>+Mul<Output=T>+PartialEq{type Key=T;type AggMonoid=ArrayOperation<AdditiveOperation<T>,K>;type ActMonoid=LinearOperation<T>;fn single_agg(key:&Self::Key)->LazyMapAgg<Self>{let mut power=T::one();std::array::from_fn(|k|{if k!=0{power=power**key;}power})}fn act_agg(x:&LazyMapAgg<Self>,&(a,b):&LazyMapAct<Self>)->Option<LazyMapAgg<Self>>{let mut coefficients=[T::one();K];let mut result=*x;for k in 1..K{coefficients[k]=a*coefficients[k-1];for j in(1..k).rev(){coefficients[j]=b*coefficients[j]+a*coefficients[j-1];}coefficients[0]=b*coefficients[0];result[k]=coefficients[..=k].iter().zip(x).fold(T::zero(),|sum,(&c,&p)|sum+c*p);}Some(result)}}impl<T,const K:usize>LazyMapMonoidWithKeyAct for RangePowerSumRangeLinear<T,K>where T:Copy+Zero+One+Add<Output=T>+Mul<Output=T>+PartialEq{fn act_key(key:&Self::Key,act:&LazyMapAct<Self>)->Self::Key{LinearAct::act(key,act)}}pub struct RangeSumRangeUpdate<T>{_marker:PhantomData<fn()->T>}impl<T>LazyMapMonoid for RangeSumRangeUpdate<T>where T:Copy+Zero+One+Add<Output=T>+Mul<Output=T>+PartialEq{type Key=T;type AggMonoid=(AdditiveOperation<T>,AdditiveOperation<T>);type ActMonoid=LastOperation<T>;fn single_agg(key:&Self::Key)->LazyMapAgg<Self>{(*key,T::one())}fn act_agg(&(x,y):&LazyMapAgg<Self>,a:&LazyMapAct<Self>)->Option<LazyMapAgg<Self>>{Some((a.map(|a|a*y).unwrap_or(x),y))}}impl<T>LazyMapMonoidWithKeyAct for RangeSumRangeUpdate<T>where T:Copy+Zero+One+Add<Output=T>+Mul<Output=T>+PartialEq{fn act_key(key:&Self::Key,act:&LazyMapAct<Self>)->Self::Key{UpdateAct::act(key,act)}}pub struct RangeMaxRangeUpdate<T>{_marker:PhantomData<fn()->T>}impl<T>LazyMapMonoid for RangeMaxRangeUpdate<T>where T:Clone+PartialEq+Ord+Bounded{type Key=T;type AggMonoid=MaxOperation<T>;type ActMonoid=LastOperation<T>;fn single_agg(key:&Self::Key)->LazyMapAgg<Self>{key.clone()}fn act_agg(x:&LazyMapAgg<Self>,a:&LazyMapAct<Self>)->Option<LazyMapAgg<Self>>{Some(a.as_ref().unwrap_or(x).clone())}}impl<T>LazyMapMonoidWithKeyAct for RangeMaxRangeUpdate<T>where T:Clone+PartialEq+Ord+Bounded{fn act_key(key:&Self::Key,act:&LazyMapAct<Self>)->Self::Key{UpdateAct::act(key,act)}}pub struct RangeMinRangeUpdate<T>{_marker:PhantomData<fn()->T>}impl<T>LazyMapMonoid for RangeMinRangeUpdate<T>where T:Clone+PartialEq+Ord+Bounded{type Key=T;type AggMonoid=MinOperation<T>;type ActMonoid=LastOperation<T>;fn single_agg(key:&Self::Key)->LazyMapAgg<Self>{key.clone()}fn act_agg(x:&LazyMapAgg<Self>,a:&LazyMapAct<Self>)->Option<LazyMapAgg<Self>>{Some(a.as_ref().unwrap_or(x).clone())}}impl<T>LazyMapMonoidWithKeyAct for RangeMinRangeUpdate<T>where T:Clone+PartialEq+Ord+Bounded{fn act_key(key:&Self::Key,act:&LazyMapAct<Self>)->Self::Key{UpdateAct::act(key,act)}}pub struct RangeMaxRangeAdd<T>{_marker:PhantomData<fn()->T>}impl<T>LazyMapMonoid for RangeMaxRangeAdd<T>where T:Clone+Ord+Bounded+Zero+Add<Output=T>{type Key=T;type AggMonoid=MaxOperation<T>;type ActMonoid=AdditiveOperation<T>;fn single_agg(key:&Self::Key)->LazyMapAgg<Self>{key.clone()}fn act_agg(x:&LazyMapAgg<Self>,a:&LazyMapAct<Self>)->Option<LazyMapAgg<Self>>{Some(if Self::is_act_unit(a){x.clone()}else{x.clone()+a.clone()})}}impl<T>LazyMapMonoidWithKeyAct for RangeMaxRangeAdd<T>where T:Clone+Ord+Bounded+Zero+Add<Output=T>{fn act_key(key:&Self::Key,act:&LazyMapAct<Self>)->Self::Key{FlattenAct::<Self::ActMonoid>::act(key,act)}}pub struct RangeMinRangeAdd<T>{_marker:PhantomData<fn()->T>}impl<T>LazyMapMonoid for RangeMinRangeAdd<T>where T:Clone+Ord+Bounded+Zero+Add<Output=T>{type Key=T;type AggMonoid=MinOperation<T>;type ActMonoid=AdditiveOperation<T>;fn single_agg(key:&Self::Key)->LazyMapAgg<Self>{key.clone()}fn act_agg(x:&LazyMapAgg<Self>,a:&LazyMapAct<Self>)->Option<LazyMapAgg<Self>>{Some(if Self::is_act_unit(a){x.clone()}else{x.clone()+a.clone()})}}impl<T>LazyMapMonoidWithKeyAct for RangeMinRangeAdd<T>where T:Clone+Ord+Bounded+Zero+Add<Output=T>{fn act_key(key:&Self::Key,act:&LazyMapAct<Self>)->Self::Key{FlattenAct::<Self::ActMonoid>::act(key,act)}}pub struct RangeMinCountRangeAdd<T>{_marker:PhantomData<fn()->T>}impl<T>LazyMapMonoid for RangeMinCountRangeAdd<T>where T:Clone+Ord+Bounded+Zero+Add<Output=T>{type Key=T;type AggMonoid=CountingOperation<MinOperation<T>>;type ActMonoid=AdditiveOperation<T>;fn single_agg(key:&Self::Key)->LazyMapAgg<Self>{(key.clone(),1)}fn act_agg(x:&LazyMapAgg<Self>,a:&LazyMapAct<Self>)->Option<LazyMapAgg<Self>>{Some(if x.1==0{x.clone()}else{(x.0 .clone()+a.clone(),x.1)})}}impl<T>LazyMapMonoidWithKeyAct for RangeMinCountRangeAdd<T>where T:Clone+Ord+Bounded+Zero+Add<Output=T>{fn act_key(key:&Self::Key,act:&LazyMapAct<Self>)->Self::Key{FlattenAct::<Self::ActMonoid>::act(key,act)}}#[derive(Debug,Clone,Copy,PartialEq,Eq)]pub struct RangeChminChmaxAdd<T>{lb:T,ub:T,bias:T}impl<T>RangeChminChmaxAdd<T>where T:Zero+Bounded{pub fn chmin(x:T)->Self{Self{lb:T::minimum(),ub:x,bias:T::zero()}}pub fn chmax(x:T)->Self{Self{lb:x,ub:T::maximum(),bias:T::zero()}}pub fn add(x:T)->Self where T:Copy+Ord+Add<Output=T>{Self{lb:T::minimum()+x.max(T::zero()),ub:T::maximum()+x.min(T::zero()),bias:x}}}impl<T>Magma for RangeChminChmaxAdd<T>where T:Copy+Zero+Ord+Bounded+Add<Output=T>{type T=Self;fn operate(x:&Self::T,y:&Self::T)->Self::T{Self{lb:(x.lb+y.bias).max(y.lb).min(y.ub),ub:(x.ub+y.bias).max(y.lb).min(y.ub),bias:x.bias+y.bias}}}impl<T>Associative for RangeChminChmaxAdd<T>where T:Copy+Zero+Ord+Bounded+Add<Output=T>{}impl<T>Unital for RangeChminChmaxAdd<T>where T:Copy+Zero+Ord+Bounded+Add<Output=T>{fn unit()->Self::T{Self{lb:T::minimum(),ub:T::maximum(),bias:T::zero()}}}#[derive(Debug,Clone,PartialEq,Eq)]pub struct RangeSumRangeChminChmaxAdd<T>{min:T,max:T,min2:T,max2:T,pub sum:T,size:T,n_min:T,n_max:T}impl<T>RangeSumRangeChminChmaxAdd<T>where T:Copy+Zero+One+Ord+Bounded+Add<Output=T>+Sub<Output=T>+Mul<Output=T>+PartialEq{pub fn single(key:T,size:T)->Self{Self{min:key,max:key,min2:T::maximum(),max2:T::minimum(),sum:key*size,size,n_min:size,n_max:size}}}impl<T>Magma for RangeSumRangeChminChmaxAdd<T>where T:Copy+Zero+One+Ord+Bounded+Add<Output=T>+Sub<Output=T>+Mul<Output=T>+PartialEq{type T=Self;fn operate(x:&Self::T,y:&Self::T)->Self::T{Self{min:x.min.min(y.min),max:x.max.max(y.max),min2:if x.min==y.min{x.min2.min(y.min2)}else if x.min2<=y.min{x.min2}else if y.min2<=x.min{y.min2}else{x.min.max(y.min)},max2:if x.max==y.max{x.max2.max(y.max2)}else if x.max2>=y.max{x.max2}else if y.max2>=x.max{y.max2}else{x.max.min(y.max)},sum:x.sum+y.sum,size:x.size+y.size,n_min:match x.min.cmp(&y.min){Ordering::Less=>x.n_min,Ordering::Equal=>x.n_min+y.n_min,Ordering::Greater=>y.n_min,},n_max:match x.max.cmp(&y.max){Ordering::Less=>y.n_max,Ordering::Equal=>x.n_max+y.n_max,Ordering::Greater=>x.n_max,}}}}impl<T>Associative for RangeSumRangeChminChmaxAdd<T>where T:Copy+Zero+One+Ord+Bounded+Add<Output=T>+Sub<Output=T>+Mul<Output=T>+PartialEq{}impl<T>Unital for RangeSumRangeChminChmaxAdd<T>where T:Copy+Zero+One+Ord+Bounded+Add<Output=T>+Sub<Output=T>+Mul<Output=T>+PartialEq{fn unit()->Self::T{Self{min:T::maximum(),max:T::minimum(),min2:T::maximum(),max2:T::minimum(),sum:T::zero(),size:T::zero(),n_min:T::zero(),n_max:T::zero()}}}impl<T>MonoidAct for RangeChminChmaxAdd<T>where T:Copy+Zero+Ord+Bounded+Add<Output=T>{type Key=T;type Act=RangeChminChmaxAdd<T>;type ActMonoid=RangeChminChmaxAdd<T>;fn act(x:&Self::Key,a:&Self::Act)->Self::Key{(*x+a.bias).max(a.lb).min(a.ub)}}impl<T>LazyMapMonoid for RangeSumRangeChminChmaxAdd<T>where T:Copy+Zero+One+Ord+Bounded+Add<Output=T>+Sub<Output=T>+Mul<Output=T>+PartialEq{type Key=T;type AggMonoid=Self;type ActMonoid=RangeChminChmaxAdd<T>;fn single_agg(&key:&Self::Key)->LazyMapAgg<Self>{Self::single(key,T::one())}fn act_agg(x:&LazyMapAgg<Self>,a:&LazyMapAct<Self>)->Option<LazyMapAgg<Self>>{if Self::is_act_unit(a){return Some(x.clone());}if x.size.is_zero(){return Some(Self::unit());}let lb=a.lb-a.bias;let ub=a.ub-a.bias;let min=x.min.max(lb);let max=x.max.min(ub);Some(if min>=max{Self::single(RangeChminChmaxAdd::act(&x.min,a),x.size)}else if x.min2==x.max{let mut x=x.clone();let min=min+a.bias;let max=max+a.bias;x.min=min;x.max2=min;x.max=max;x.min2=max;x.sum=min*x.n_min+max*x.n_max;x}else if lb<x.min2&&x.max2<ub{let mut x=x.clone();x.sum=x.sum+(min-x.min)*x.n_min+(max-x.max)*x.n_max+a.bias*x.size;x.min=min+a.bias;x.max=max+a.bias;x.min2=x.min2+a.bias;x.max2=x.max2+a.bias;x}else{return None;})}}impl<T>LazyMapMonoidWithKeyAct for RangeSumRangeChminChmaxAdd<T>where T:Copy+Zero+One+Ord+Bounded+Add<Output=T>+Sub<Output=T>+Mul<Output=T>+PartialEq{fn act_key(key:&Self::Key,act:&LazyMapAct<Self>)->Self::Key{RangeChminChmaxAdd::act(key,act)}}}
// codesnip-guard: LinearOperation
#[cfg_attr(any(),rustfmt::skip)]pub use self::linear_operation_impl::LinearOperation;#[cfg_attr(any(),rustfmt::skip)]mod linear_operation_impl{use super::*;use std::{marker::PhantomData,ops::{Add,Div,Mul,Neg,Sub}};#[doc=" $(a, b) \\circ (c, d) = \\lambda x. c \\times (a \\times x + b) + d$"]pub struct LinearOperation<T>where T:Clone+Zero+Add<Output=T>+One+Mul<Output=T>{_marker:PhantomData<fn()->T>}impl<T>LinearOperation<T>where T:Clone+Zero+Add<Output=T>+One+Mul<Output=T>{pub fn apply(f:&(T,T),x:&T)->T{f.0 .clone()*x.clone()+f.1 .clone()}}impl<T>Magma for LinearOperation<T>where T:Clone+Zero+One+Add<Output=T>+Mul<Output=T>{type T=(T,T);#[inline]fn operate(x:&Self::T,y:&Self::T)->Self::T{(y.0 .clone()*x.0 .clone(),y.0 .clone()*x.1 .clone()+y.1 .clone())}}impl<T>Unital for LinearOperation<T>where T:Clone+Zero+One+Add<Output=T>+Mul<Output=T>{#[inline]fn unit()->Self::T{(One::one(),Zero::zero())}}impl<T>Associative for LinearOperation<T>where T:Clone+Zero+One+Add<Output=T>+Mul<Output=T>{}impl<T>Invertible for LinearOperation<T>where T:Clone+Zero+One+Add<Output=T>+Sub<Output=T>+Neg<Output=T>+Mul<Output=T>+Div<Output=T>{fn inverse(x:&Self::T)->Self::T{let y=<T as One>::one().div(x.0 .clone());(y.clone(),-y.mul(x.1 .clone()))}}}
// codesnip-guard: MaxOperation
#[cfg_attr(any(),rustfmt::skip)]pub use self::max_operation_impl::MaxOperation;#[cfg_attr(any(),rustfmt::skip)]mod max_operation_impl{use super::*;use std::marker::PhantomData;#[doc=" binary operation to select larger element"]pub struct MaxOperation<T>where T:Clone+Ord+Bounded{_marker:PhantomData<fn()->T>}impl<T>Magma for MaxOperation<T>where T:Clone+Ord+Bounded{type T=T;#[inline]fn operate(x:&Self::T,y:&Self::T)->Self::T{x.max(y).clone()}}impl<T>Unital for MaxOperation<T>where T:Clone+Ord+Bounded{#[inline]fn unit()->Self::T{<T as Bounded>::minimum()}}impl<T>Associative for MaxOperation<T>where T:Clone+Ord+Bounded{}impl<T>Commutative for MaxOperation<T>where T:Clone+Ord+Bounded{}impl<T>Idempotent for MaxOperation<T>where T:Clone+Ord+Bounded{}}
// codesnip-guard: MinOperation
#[cfg_attr(any(),rustfmt::skip)]pub use self::min_operation_impl::MinOperation;#[cfg_attr(any(),rustfmt::skip)]mod min_operation_impl{use super::*;use std::marker::PhantomData;#[doc=" binary operation to select smaller element"]pub struct MinOperation<T>where T:Clone+Ord+Bounded{_marker:PhantomData<fn()->T>}impl<T>Magma for MinOperation<T>where T:Clone+Ord+Bounded{type T=T;#[inline]fn operate(x:&Self::T,y:&Self::T)->Self::T{x.min(y).clone()}}impl<T>Unital for MinOperation<T>where T:Clone+Ord+Bounded{#[inline]fn unit()->Self::T{<T as Bounded>::maximum()}}impl<T>Associative for MinOperation<T>where T:Clone+Ord+Bounded{}impl<T>Commutative for MinOperation<T>where T:Clone+Ord+Bounded{}impl<T>Idempotent for MinOperation<T>where T:Clone+Ord+Bounded{}}
// codesnip-guard: MonoidAct
#[cfg_attr(any(),rustfmt::skip)]pub use self::action::*;#[cfg_attr(any(),rustfmt::skip)]mod action{use super::*;use std::{marker::PhantomData,ops::{Add,Mul}};pub trait MonoidAct{type Key;type Act:Clone;type ActMonoid:Monoid<T=Self::Act>;fn act(x:&Self::Key,a:&Self::Act)->Self::Key;fn act_assign(x:&mut Self::Key,a:&Self::Act){*x=Self::act(x,a);}fn unit()->Self::Act{<Self::ActMonoid as Unital>::unit()}fn operate(x:&Self::Act,y:&Self::Act)->Self::Act{<Self::ActMonoid as Magma>::operate(x,y)}fn operate_assign(x:&mut Self::Act,y:&Self::Act){*x=<Self::ActMonoid as Magma>::operate(x,y);}}pub struct EmptyAct<T>{_marker:PhantomData<fn()->T>}impl<T>MonoidAct for EmptyAct<T>where T:Clone{type Key=T;type Act=();type ActMonoid=();fn act(x:&Self::Key,_a:&Self::Act)->Self::Key{x.clone()}}pub struct FlattenAct<M>{_marker:PhantomData<fn()->M>}impl<M>MonoidAct for FlattenAct<M>where M:Monoid{type Key=M::T;type Act=M::T;type ActMonoid=M;fn act(x:&Self::Key,a:&Self::Act)->Self::Key{M::operate(x,a)}}pub struct LinearAct<T>{_marker:PhantomData<fn()->T>}impl<T>MonoidAct for LinearAct<T>where T:Clone+Zero+One+Add<Output=T>+Mul<Output=T>{type Key=T;type Act=(T,T);type ActMonoid=LinearOperation<T>;fn act(x:&Self::Key,(a,b):&Self::Act)->Self::Key{a.clone()*x.clone()+b.clone()}}pub struct UpdateAct<T>{_marker:PhantomData<fn()->T>}impl<T>MonoidAct for UpdateAct<T>where T:Clone{type Key=T;type Act=Option<T>;type ActMonoid=LastOperation<T>;fn act(x:&Self::Key,a:&Self::Act)->Self::Key{a.as_ref().unwrap_or(x).clone()}}}
// codesnip-guard: TupleOperation
#[cfg_attr(any(),rustfmt::skip)]mod tuple_operation_impl{use super::*;macro_rules!impl_tuple_operation{(@impl)=>{impl Magma for(){type T=();fn operate(_x:&Self::T,_y:&Self::T)->Self::T{}}impl Unital for(){fn unit()->Self::T{}}impl Associative for(){}impl Commutative for(){}impl Idempotent for(){}impl Invertible for(){fn inverse(_x:&Self::T)->Self::T{}}};(@impl$($T:ident$i:tt)*)=>{impl<$($T:Magma),*>Magma for($($T,)*){type T=($(<$T as Magma>::T,)*);fn operate(x:&Self::T,y:&Self::T)->Self::T{($(<$T as Magma>::operate(&x.$i,&y.$i),)*)}}impl<$($T:Unital),*>Unital for($($T,)*){fn unit()->Self::T{($(<$T as Unital>::unit(),)*)}}impl<$($T:Associative),*>Associative for($($T,)*){}impl<$($T:Commutative),*>Commutative for($($T,)*){}impl<$($T:Idempotent),*>Idempotent for($($T,)*){}impl<$($T:Invertible),*>Invertible for($($T,)*){fn inverse(x:&Self::T)->Self::T{($(<$T as Invertible>::inverse(&x.$i),)*)}}};(@inner$($T:ident$i:tt)*;$U:ident$j:tt$($t:tt)*)=>{impl_tuple_operation!(@impl$($T$i)*);impl_tuple_operation!(@inner$($T$i)*$U$j;$($t)*);};(@inner$($T:ident$i:tt)*;)=>{impl_tuple_operation!(@impl$($T$i)*);};($($t:tt)*)=>{impl_tuple_operation!(@inner;$($t)*);};}impl_tuple_operation!(A 0 B 1 C 2 D 3 E 4 F 5 G 6 H 7 I 8 J 9);}
// codesnip-guard: algebra
#[cfg_attr(any(),rustfmt::skip)]pub use self::magma::*;#[cfg_attr(any(),rustfmt::skip)]mod magma{#![doc=" algebraic traits"]#[doc=" binary operaion: $T \\circ T \\to T$"]pub trait Magma{#[doc=" type of operands: $T$"]type T:Clone;#[doc=" binary operaion: $\\circ$"]fn operate(x:&Self::T,y:&Self::T)->Self::T;fn reverse_operate(x:&Self::T,y:&Self::T)->Self::T{Self::operate(y,x)}fn operate_assign(x:&mut Self::T,y:&Self::T){*x=Self::operate(x,y);}}#[doc=" $\\forall a,\\forall b,\\forall c \\in T, (a \\circ b) \\circ c = a \\circ (b \\circ c)$"]pub trait Associative:Magma{}#[doc=" associative binary operation"]pub trait SemiGroup:Magma+Associative{}impl<S>SemiGroup for S where S:Magma+Associative{}#[doc=" $\\exists e \\in T, \\forall a \\in T, e \\circ a = a \\circ e = a$"]pub trait Unital:Magma{#[doc=" identity element: $e$"]fn unit()->Self::T;fn is_unit(x:&Self::T)->bool where Self::T:PartialEq{x==&Self::unit()}fn set_unit(x:&mut Self::T){*x=Self::unit();}}pub trait ExpBits{type Iter:Iterator<Item=bool>;fn bits(self)->Self::Iter;}pub trait SignedExpBits{type T:ExpBits;fn neg_and_bits(self)->(bool,Self::T);}pub struct Bits<T>{n:T}macro_rules!impl_exp_bits_for_uint{($($t:ty)*)=>{$(impl Iterator for Bits<$t>{type Item=bool;fn next(&mut self)->Option<bool>{if self.n==0{None}else{let bit=(self.n&1)==1;self.n>>=1;Some(bit)}}}impl ExpBits for$t{type Iter=Bits<$t>;fn bits(self)->Self::Iter{Bits{n:self}}}impl SignedExpBits for$t{type T=$t;fn neg_and_bits(self)->(bool,Self::T){(false,self)}})*};}impl_exp_bits_for_uint!(u8 u16 u32 u64 u128 usize);macro_rules!impl_signed_exp_bits_for_sint{($($s:ty,$u:ty;)*)=>{$(impl SignedExpBits for$s{type T=$u;fn neg_and_bits(self)->(bool,Self::T){(self<0,self.unsigned_abs())}})*};}impl_signed_exp_bits_for_sint!(i8,u8;i16,u16;i32,u32;i64,u64;i128,u128;isize,usize;);#[doc=" associative binary operation and an identity element"]pub trait Monoid:SemiGroup+Unital{#[doc=" binary exponentiation: $x^n = x\\circ\\ddots\\circ x$"]fn pow<E>(mut x:Self::T,exp:E)->Self::T where E:ExpBits{let mut bits=exp.bits();let Some(first)=bits.next()else{return Self::unit();};let mut res=if first{x.clone()}else{Self::unit()};for bit in bits{x=Self::operate(&x,&x);if bit{res=Self::operate(&res,&x);}}res}fn fold<I>(iter:I)->Self::T where I:IntoIterator<Item=Self::T>{let mut iter=iter.into_iter();if let Some(item)=iter.next(){iter.fold(item,|acc,x|Self::operate(&acc,&x))}else{Self::unit()}}}impl<M>Monoid for M where M:SemiGroup+Unital{}#[doc=" $\\exists e \\in T, \\forall a \\in T, \\exists b,c \\in T, b \\circ a = a \\circ c = e$"]pub trait Invertible:Magma+Unital{#[doc=" $a$ where $a \\circ x = e$"]fn inverse(x:&Self::T)->Self::T;fn rinv_operate(x:&Self::T,y:&Self::T)->Self::T{Self::operate(x,&Self::inverse(y))}fn rinv_operate_assign(x:&mut Self::T,y:&Self::T){*x=Self::rinv_operate(x,y);}}#[doc=" associative binary operation and an identity element and inverse elements"]pub trait Group:Monoid+Invertible{fn signed_pow<E>(x:Self::T,exp:E)->Self::T where E:SignedExpBits{let(neg,exp)=E::neg_and_bits(exp);let res=Self::pow(x,exp);if neg{Self::inverse(&res)}else{res}}}impl<G>Group for G where G:Monoid+Invertible{}#[doc=" $\\forall a,\\forall b \\in T, a \\circ b = b \\circ a$"]pub trait Commutative:Magma{}#[doc=" commutative monoid"]pub trait AbelianMonoid:Monoid+Commutative{}impl<M>AbelianMonoid for M where M:Monoid+Commutative{}#[doc=" commutative group"]pub trait AbelianGroup:Group+Commutative{}impl<G>AbelianGroup for G where G:Group+Commutative{}#[doc=" $\\forall a \\in T, a \\circ a = a$"]pub trait Idempotent:Magma{}#[doc=" idempotent monoid"]pub trait IdempotentMonoid:Monoid+Idempotent{}impl<M>IdempotentMonoid for M where M:Monoid+Idempotent{}#[macro_export]macro_rules!monoid_fold{($m:ty)=>{<$m as Unital>::unit()};($m:ty,)=>{<$m as Unital>::unit()};($m:ty,$f:expr)=>{$f};($m:ty,$f:expr,$($ff:expr),*)=>{<$m as Magma>::operate(&($f),&monoid_fold!($m,$($ff),*))};}#[macro_export]macro_rules!define_monoid{($Name:ident,$t:ty,|$x:ident,$y:ident|$op:expr,$unit:expr)=>{struct$Name;impl Magma for$Name{type T=$t;fn operate($x:&Self::T,$y:&Self::T)->Self::T{$op}}impl Unital for$Name{fn unit()->Self::T{$unit}}impl Associative for$Name{}};}}
// codesnip-guard: array
#[cfg_attr(any(),rustfmt::skip)]mod array{#[macro_export]macro_rules!array{[@inner$data:ident=[$init:expr;$len:expr]]=>{{use::std::mem::{ManuallyDrop,MaybeUninit};let mut$data:[MaybeUninit<_>;$len]=unsafe{MaybeUninit::uninit().assume_init()};$init;#[repr(C)]union __Transmuter<const N:usize,T:Clone>{src:ManuallyDrop<[MaybeUninit<T>;N]>,dst:ManuallyDrop<[T;N]>,}ManuallyDrop::into_inner(unsafe{__Transmuter{src:ManuallyDrop::new($data)}.dst})}};[||$e:expr;$len:expr]=>{$crate::array![@inner data=[data.iter_mut().for_each(|item|*item=MaybeUninit::new($e));$len]]};[|$i:pat_param|$e:expr;$len:expr]=>{$crate::array![@inner data=[data.iter_mut().enumerate().for_each(|($i,item)|*item=MaybeUninit::new($e));$len]]};[$e:expr;$len:expr]=>{{let e=$e;$crate::array![||Clone::clone(&e);$len]}};}}
// codesnip-guard: bounded
#[cfg_attr(any(),rustfmt::skip)]pub use self::bounded::Bounded;#[cfg_attr(any(),rustfmt::skip)]mod bounded{#[doc=" Trait for max/min bounds"]pub trait Bounded:Sized+PartialOrd{fn maximum()->Self;fn minimum()->Self;fn is_maximum(&self)->bool{self==&Self::maximum()}fn is_minimum(&self)->bool{self==&Self::minimum()}fn set_maximum(&mut self){*self=Self::maximum()}fn set_minimum(&mut self){*self=Self::minimum()}}macro_rules!impl_bounded_num{($($t:ident)*)=>{$(impl Bounded for$t{fn maximum()->Self{$t::MAX}fn minimum()->Self{$t::MIN}})*};}impl_bounded_num!(u8 u16 u32 u64 u128 usize i8 i16 i32 i64 i128 isize f32 f64);macro_rules!impl_bounded_tuple{(@impl$($T:ident)*)=>{impl<$($T:Bounded),*>Bounded for($($T,)*){fn maximum()->Self{($(<$T as Bounded>::maximum(),)*)}fn minimum()->Self{($(<$T as Bounded>::minimum(),)*)}}};(@inner$($T:ident)*,)=>{impl_bounded_tuple!(@impl$($T)*);};(@inner$($T:ident)*,$U:ident$($Rest:ident)*)=>{impl_bounded_tuple!(@impl$($T)*);impl_bounded_tuple!(@inner$($T)*$U,$($Rest)*);};($T:ident$($Rest:ident)*)=>{impl_bounded_tuple!(@inner$T,$($Rest)*);};}impl_bounded_tuple!(A B C D E F G H I J);impl Bounded for(){fn maximum()->Self{}fn minimum()->Self{}}impl Bounded for bool{fn maximum()->Self{true}fn minimum()->Self{false}}impl<T>Bounded for Option<T>where T:Bounded{fn maximum()->Self{Some(<T as Bounded>::maximum())}fn minimum()->Self{None}}impl<T>Bounded for std::cmp::Reverse<T>where T:Bounded{fn maximum()->Self{std::cmp::Reverse(<T as Bounded>::minimum())}fn minimum()->Self{std::cmp::Reverse(<T as Bounded>::maximum())}}}
// codesnip-guard: discrete_steps
#[cfg_attr(any(),rustfmt::skip)]pub use self::discrete_steps::{DiscreteSteps,RangeBoundsExt};#[cfg_attr(any(),rustfmt::skip)]mod discrete_steps{use super::Bounded;use std::{convert::TryFrom,ops::{Bound,Range,RangeBounds,RangeInclusive}};pub trait DiscreteSteps<Delta>:Clone{fn delta()->Delta;fn steps_between(start:&Self,end:&Self)->Option<Delta>;fn forward_checked(start:Self,delta:Delta)->Option<Self>;fn backward_checked(start:Self,delta:Delta)->Option<Self>;fn forward(start:Self,delta:Delta)->Self{Self::forward_checked(start,delta).expect("overflow in `DiscreteSteps::forward`")}fn backward(start:Self,delta:Delta)->Self{Self::backward_checked(start,delta).expect("overflow in `DiscreteSteps::backward`")}fn forward_delta_checked(start:Self)->Option<Self>{Self::forward_checked(start,Self::delta())}fn backward_delta_checked(start:Self)->Option<Self>{Self::backward_checked(start,Self::delta())}fn forward_delta(start:Self)->Self{Self::forward(start,Self::delta())}fn backward_delta(start:Self)->Self{Self::backward(start,Self::delta())}}macro_rules!impl_discrete_steps_integer{(@common$u_source:ident)=>{fn delta()->$u_source{1}fn forward(start:Self,delta:$u_source)->Self{assert!(Self::forward_checked(start,delta).is_some(),"attempt to add with overflow");start.wrapping_add(delta as Self)}fn backward(start:Self,delta:$u_source)->Self{assert!(Self::backward_checked(start,delta).is_some(),"attempt to subtract with overflow");start.wrapping_sub(delta as Self)}};($u_source:ident$i_source:ident;$($u_narrower:ident$i_narrower:ident),*;$($u_wider:ident$i_wider:ident),*)=>{$(impl DiscreteSteps<$u_source>for$u_narrower{impl_discrete_steps_integer!(@common$u_source);fn steps_between(start:&Self,end:&Self)->Option<$u_source>{if*start<=*end{$u_source::try_from(*end-*start).ok()}else{None}}fn forward_checked(start:Self,delta:$u_source)->Option<Self>{Self::try_from(delta).ok().and_then(|delta|start.checked_add(delta))}fn backward_checked(start:Self,delta:$u_source)->Option<Self>{Self::try_from(delta).ok().and_then(|delta|start.checked_sub(delta))}}impl DiscreteSteps<$u_source>for$i_narrower{impl_discrete_steps_integer!(@common$u_source);fn steps_between(start:&Self,end:&Self)->Option<$u_source>{if*start<=*end{$u_source::try_from(end.abs_diff(*start)).ok()}else{None}}fn forward_checked(start:Self,delta:$u_source)->Option<Self>{$u_narrower::try_from(delta).ok().and_then(|delta|{let wrapped=start.wrapping_add(delta as Self);if wrapped>=start{Some(wrapped)}else{None}})}fn backward_checked(start:Self,delta:$u_source)->Option<Self>{$u_narrower::try_from(delta).ok().and_then(|delta|{let wrapped=start.wrapping_sub(delta as Self);if wrapped<=start{Some(wrapped)}else{None}})}})*$(impl DiscreteSteps<$u_source>for$u_wider{impl_discrete_steps_integer!(@common$u_source);fn steps_between(start:&Self,end:&Self)->Option<$u_source>{if*start<=*end{$u_source::try_from(*end-*start).ok()}else{None}}fn forward_checked(start:Self,delta:$u_source)->Option<Self>{start.checked_add(delta as Self)}fn backward_checked(start:Self,delta:$u_source)->Option<Self>{start.checked_sub(delta as Self)}}impl DiscreteSteps<$u_source>for$i_wider{impl_discrete_steps_integer!(@common$u_source);fn steps_between(start:&Self,end:&Self)->Option<$u_source>{if*start<=*end{end.checked_sub(*start).and_then(|result|$u_source::try_from(result).ok())}else{None}}fn forward_checked(start:Self,delta:$u_source)->Option<Self>{start.checked_add(delta as Self)}fn backward_checked(start:Self,delta:$u_source)->Option<Self>{start.checked_sub(delta as Self)}})*};}impl_discrete_steps_integer!(u16 i16;u8 i8,u16 i16,usize isize;u32 i32,u64 i64,u128 i128);impl_discrete_steps_integer!(u32 i32;u8 i8,u16 i16,u32 i32,usize isize;u64 i64,u128 i128);impl_discrete_steps_integer!(u64 i64;u8 i8,u16 i16,u32 i32,u64 i64,usize isize;u128 i128);impl_discrete_steps_integer!(u128 i128;u8 i8,u16 i16,u32 i32,u64 i64,u128 i128,usize isize;);impl_discrete_steps_integer!(usize isize;u8 i8,u16 i16,u32 i32,u64 i64,usize isize;u128 i128);pub trait RangeBoundsExt<T>{fn start_bound_included_checked(&self)->Option<T>;fn start_bound_excluded_checked(&self)->Option<T>;fn end_bound_included_checked(&self)->Option<T>;fn end_bound_excluded_checked(&self)->Option<T>;fn start_bound_included(&self)->T;fn start_bound_excluded(&self)->T;fn end_bound_included(&self)->T;fn end_bound_excluded(&self)->T;fn start_bound_included_bounded(&self,lb:T)->Option<T>where T:Ord;fn start_bound_excluded_bounded(&self,lb:T)->Option<T>where T:Ord;fn end_bound_included_bounded(&self,ub:T)->Option<T>where T:Ord;fn end_bound_excluded_bounded(&self,ub:T)->Option<T>where T:Ord;fn to_range_checked(&self)->Option<Range<T>>{match(self.start_bound_included_checked(),self.end_bound_excluded_checked()){(Some(start),Some(end))=>Some(start..end),_=>None,}}fn to_range(&self)->Range<T>{self.start_bound_included()..self.end_bound_excluded()}fn to_range_bounded(&self,min:T,max:T)->Option<Range<T>>where T:Ord{Some(self.start_bound_included_bounded(min)?..self.end_bound_excluded_bounded(max)?)}fn to_range_inclusive_checked(&self)->Option<RangeInclusive<T>>{match(self.start_bound_included_checked(),self.end_bound_included_checked()){(Some(start),Some(end))=>Some(start..=end),_=>None,}}fn to_range_inclusive(&self)->RangeInclusive<T>{self.start_bound_included()..=self.end_bound_included()}fn to_range_inclusive_bounded(&self,min:T,max:T)->Option<RangeInclusive<T>>where T:Ord{Some(self.start_bound_included_bounded(min)?..=self.end_bound_included_bounded(max)?)}}macro_rules!impl_range_bounds_ext{($($source:ident=>$($target:ident)+);*$(;)?)=>{$($(impl<R>RangeBoundsExt<$target>for R where R:RangeBounds<$target>,{fn start_bound_included_checked(&self)->Option<$target>{match self.start_bound(){Bound::Included(x)=>Some(*x),Bound::Excluded(x)=>DiscreteSteps::<$source>::forward_delta_checked(*x),Bound::Unbounded=>Some(Bounded::minimum()),}}fn start_bound_excluded_checked(&self)->Option<$target>{match self.start_bound(){Bound::Included(x)=>DiscreteSteps::<$source>::backward_delta_checked(*x),Bound::Excluded(x)=>Some(*x),Bound::Unbounded=>None,}}fn end_bound_included_checked(&self)->Option<$target>{match self.end_bound(){Bound::Included(x)=>Some(*x),Bound::Excluded(x)=>DiscreteSteps::<$source>::backward_delta_checked(*x),Bound::Unbounded=>Some(Bounded::maximum()),}}fn end_bound_excluded_checked(&self)->Option<$target>{match self.end_bound(){Bound::Included(x)=>DiscreteSteps::<$source>::forward_delta_checked(*x),Bound::Excluded(x)=>Some(*x),Bound::Unbounded=>None,}}fn start_bound_included(&self)->$target{match self.start_bound(){Bound::Included(x)=>*x,Bound::Excluded(x)=>DiscreteSteps::<$source>::forward_delta(*x),Bound::Unbounded=>Bounded::minimum(),}}fn start_bound_excluded(&self)->$target{match self.start_bound(){Bound::Included(x)=>DiscreteSteps::<$source>::backward_delta(*x),Bound::Excluded(x)=>*x,Bound::Unbounded=>DiscreteSteps::<$source>::backward_delta(Bounded::minimum()),}}fn end_bound_included(&self)->$target{match self.end_bound(){Bound::Included(x)=>*x,Bound::Excluded(x)=>DiscreteSteps::<$source>::backward_delta(*x),Bound::Unbounded=>Bounded::maximum(),}}fn end_bound_excluded(&self)->$target{match self.end_bound(){Bound::Included(x)=>DiscreteSteps::<$source>::forward_delta(*x),Bound::Excluded(x)=>*x,Bound::Unbounded=>DiscreteSteps::<$source>::forward_delta(Bounded::maximum()),}}fn start_bound_included_bounded(&self,lb:$target)->Option<$target>where$target:Ord{match self.start_bound(){Bound::Included(x)=>Some(*x).filter(|&x|lb<=x),Bound::Excluded(x)=>DiscreteSteps::<$source>::forward_delta_checked(*x).filter(|&x|lb<=x),Bound::Unbounded=>Some(lb),}}fn start_bound_excluded_bounded(&self,lb:$target)->Option<$target>where$target:Ord{match self.start_bound(){Bound::Included(x)=>DiscreteSteps::<$source>::backward_delta_checked(*x).filter(|&x|lb<=x),Bound::Excluded(x)=>Some(*x).filter(|&x|lb<=x),Bound::Unbounded=>Some(lb),}}fn end_bound_included_bounded(&self,ub:$target)->Option<$target>where$target:Ord{match self.end_bound(){Bound::Included(x)=>Some(*x).filter(|&x|x<=ub),Bound::Excluded(x)=>DiscreteSteps::<$source>::backward_delta_checked(*x).filter(|&x|x<=ub),Bound::Unbounded=>Some(ub),}}fn end_bound_excluded_bounded(&self,ub:$target)->Option<$target>where$target:Ord{match self.end_bound(){Bound::Included(x)=>DiscreteSteps::<$source>::forward_delta_checked(*x).filter(|&x|x<=ub),Bound::Excluded(x)=>Some(*x).filter(|&x|x<=ub),Bound::Unbounded=>Some(ub),}}})+)*};}impl_range_bounds_ext!(u16=>u8 i8 u16 i16;u32=>u32 i32;u64=>u64 i64;u128=>u128 i128;usize=>isize usize;);}
// codesnip-guard: zero_one
#[cfg_attr(any(),rustfmt::skip)]pub use self::zero_one::{One,Zero};#[cfg_attr(any(),rustfmt::skip)]mod zero_one{pub trait Zero:Sized{fn zero()->Self;#[inline]fn is_zero(&self)->bool where Self:PartialEq{self==&Self::zero()}#[inline]fn set_zero(&mut self){*self=Self::zero();}}pub trait One:Sized{fn one()->Self;#[inline]fn is_one(&self)->bool where Self:PartialEq{self==&Self::one()}#[inline]fn set_one(&mut self){*self=Self::one();}}macro_rules!impl_zero_one{($({$(<$T:ident:$Bound:ident>)?$Trait:ident$method:ident$($t:ty)*,$e:expr})*)=>{$(impl_zero_one!(@impl[$(<$T:$Bound>)?]$Trait$method[$($t)*],$e);)*};(@impl[<$T:ident:$Bound:ident>]$Trait:ident$method:ident[$($t:ty)*],$e:expr)=>{$(impl<$T:$Bound>$Trait for$t{fn$method()->Self{$e}})*};(@impl[]$Trait:ident$method:ident[$($t:ty)*],$e:expr)=>{$(impl$Trait for$t{fn$method()->Self{$e}})*};}impl_zero_one!({Zero zero u8 u16 u32 u64 usize i8 i16 i32 i64 isize u128 i128,0}{Zero zero f32 f64,0.}{One one u8 u16 u32 u64 usize i8 i16 i32 i64 isize u128 i128,1}{One one f32 f64,1.}{<T:Zero>Zero zero std::num::Wrapping<T>,Self(T::zero())}{<T:One>One one std::num::Wrapping<T>,Self(T::one())});}