{-# LANGUAGE TypeFamilies #-}

-- | `AtCoder.LazySegTree.SegAct` instance of one-dimensional affine transformation
-- \(f: x \rightarrow a \times x + b\).
--
-- @since 1.0.0
module AtCoder.Extra.Monoid.Affine1
  ( -- * Affine1
    Affine1 (..),
    Affine1Repr,

    -- * Constructor
    new,

    -- * Action
    act,
  )
where

import AtCoder.Extra.Math qualified as ACEM
import AtCoder.LazySegTree (SegAct (..))
import Data.Coerce (coerce)
import Data.Foldable (foldl')
import Data.List.NonEmpty (NonEmpty (..))
import Data.Semigroup (Dual (..), Semigroup (..), Sum (..))
import Data.Vector.Generic qualified as VG
import Data.Vector.Generic.Mutable qualified as VGM
import Data.Vector.Unboxed qualified as VU
import Data.Vector.Unboxed.Mutable qualified as VUM

-- Tuple is not the fastest representation, but it's easier to implement `Unbox`.

-- | `AtCoder.LazySegTree.SegAct` instance of one-dimensional affine transformation
-- \(f: x \rightarrow a \times x + b\).
--
-- ==== Composition and dual
-- `Semigroup` for `Affine1` is implemented like function composition, and rightmost affine
-- transformation is applied first: \((f_1 \circ f_2) v := f_1 (f_2(v))\). If you need 'foldr'
-- of \([f_l, f_{l+1}, .., f_r)\) on a segment tree, be sure to wrap `Affine1` in
-- `Data.Monoid.Dual`.
--
-- ==== Example
-- >>> import AtCoder.Extra.Monoid (SegAct(..), Affine1(..))
-- >>> import AtCoder.LazySegTree qualified as LST
-- >>> seg <- LST.build @_ @(Affine1 Int) @(Sum Int) $ VU.generate 3 Sum -- [0, 1, 2]
-- >>> LST.applyIn seg 0 3 $ Affine1 (2, 1) -- [1, 3, 5]
-- >>> getSum <$> LST.allProd seg
-- 9
--
-- @since 1.0.0
newtype Affine1 a = Affine1 (Affine1Repr a)
  deriving newtype
    ( -- | @since 1.0.0
      Affine1 a -> Affine1 a -> Bool
(Affine1 a -> Affine1 a -> Bool)
-> (Affine1 a -> Affine1 a -> Bool) -> Eq (Affine1 a)
forall a. Eq a => Affine1 a -> Affine1 a -> Bool
forall a. (a -> a -> Bool) -> (a -> a -> Bool) -> Eq a
$c== :: forall a. Eq a => Affine1 a -> Affine1 a -> Bool
== :: Affine1 a -> Affine1 a -> Bool
$c/= :: forall a. Eq a => Affine1 a -> Affine1 a -> Bool
/= :: Affine1 a -> Affine1 a -> Bool
Eq,
      -- | @since 1.0.0
      Eq (Affine1 a)
Eq (Affine1 a) =>
(Affine1 a -> Affine1 a -> Ordering)
-> (Affine1 a -> Affine1 a -> Bool)
-> (Affine1 a -> Affine1 a -> Bool)
-> (Affine1 a -> Affine1 a -> Bool)
-> (Affine1 a -> Affine1 a -> Bool)
-> (Affine1 a -> Affine1 a -> Affine1 a)
-> (Affine1 a -> Affine1 a -> Affine1 a)
-> Ord (Affine1 a)
Affine1 a -> Affine1 a -> Bool
Affine1 a -> Affine1 a -> Ordering
Affine1 a -> Affine1 a -> Affine1 a
forall a.
Eq a =>
(a -> a -> Ordering)
-> (a -> a -> Bool)
-> (a -> a -> Bool)
-> (a -> a -> Bool)
-> (a -> a -> Bool)
-> (a -> a -> a)
-> (a -> a -> a)
-> Ord a
forall a. Ord a => Eq (Affine1 a)
forall a. Ord a => Affine1 a -> Affine1 a -> Bool
forall a. Ord a => Affine1 a -> Affine1 a -> Ordering
forall a. Ord a => Affine1 a -> Affine1 a -> Affine1 a
$ccompare :: forall a. Ord a => Affine1 a -> Affine1 a -> Ordering
compare :: Affine1 a -> Affine1 a -> Ordering
$c< :: forall a. Ord a => Affine1 a -> Affine1 a -> Bool
< :: Affine1 a -> Affine1 a -> Bool
$c<= :: forall a. Ord a => Affine1 a -> Affine1 a -> Bool
<= :: Affine1 a -> Affine1 a -> Bool
$c> :: forall a. Ord a => Affine1 a -> Affine1 a -> Bool
> :: Affine1 a -> Affine1 a -> Bool
$c>= :: forall a. Ord a => Affine1 a -> Affine1 a -> Bool
>= :: Affine1 a -> Affine1 a -> Bool
$cmax :: forall a. Ord a => Affine1 a -> Affine1 a -> Affine1 a
max :: Affine1 a -> Affine1 a -> Affine1 a
$cmin :: forall a. Ord a => Affine1 a -> Affine1 a -> Affine1 a
min :: Affine1 a -> Affine1 a -> Affine1 a
Ord,
      -- | @since 1.0.0
      Int -> Affine1 a -> ShowS
[Affine1 a] -> ShowS
Affine1 a -> String
(Int -> Affine1 a -> ShowS)
-> (Affine1 a -> String)
-> ([Affine1 a] -> ShowS)
-> Show (Affine1 a)
forall a. Show a => Int -> Affine1 a -> ShowS
forall a. Show a => [Affine1 a] -> ShowS
forall a. Show a => Affine1 a -> String
forall a.
(Int -> a -> ShowS) -> (a -> String) -> ([a] -> ShowS) -> Show a
$cshowsPrec :: forall a. Show a => Int -> Affine1 a -> ShowS
showsPrec :: Int -> Affine1 a -> ShowS
$cshow :: forall a. Show a => Affine1 a -> String
show :: Affine1 a -> String
$cshowList :: forall a. Show a => [Affine1 a] -> ShowS
showList :: [Affine1 a] -> ShowS
Show
    )

-- | `Affine1` internal representation. Tuples are not the fastest representation, but it's easier
-- to implement `Data.Vector.Unboxed.Unbox`.
--
-- @since 1.0.0
type Affine1Repr a = (a, a)

-- | Creates `Affine1`.
--
-- @since 1.0.0
{-# INLINE new #-}
new :: a -> a -> Affine1 a
new :: forall a. a -> a -> Affine1 a
new !a
a !a
b = Affine1Repr a -> Affine1 a
forall a. Affine1Repr a -> Affine1 a
Affine1 (a
a, a
b)

-- | Applies \(f: x \rightarrow a \times x + b\).
--
-- @since 1.0.0
{-# INLINE act #-}
act :: (Num a) => Affine1 a -> a -> a
act :: forall a. Num a => Affine1 a -> a -> a
act (Affine1 (!a
a, !a
b)) a
x = a
a a -> a -> a
forall a. Num a => a -> a -> a
* a
x a -> a -> a
forall a. Num a => a -> a -> a
+ a
b

-- | Acts on @a@ with length in terms of `SegAct`. Works for `Sum a` only.
--
-- @since 1.0.0
{-# INLINE actWithLength #-}
actWithLength :: (Num a) => Int -> Affine1 a -> a -> a
actWithLength :: forall a. Num a => Int -> Affine1 a -> a -> a
actWithLength Int
len (Affine1 (!a
a, !a
b)) !a
x = a
a a -> a -> a
forall a. Num a => a -> a -> a
* a
x a -> a -> a
forall a. Num a => a -> a -> a
+ a
b a -> a -> a
forall a. Num a => a -> a -> a
* Int -> a
forall a b. (Integral a, Num b) => a -> b
fromIntegral Int
len

-- | @since 1.0.0
instance (Num a) => Semigroup (Affine1 a) where
  {-# INLINE (<>) #-}
  (Affine1 (!a
a1, !a
b1)) <> :: Affine1 a -> Affine1 a -> Affine1 a
<> (Affine1 (!a
a2, !a
b2)) = (a, a) -> Affine1 a
forall a. Affine1Repr a -> Affine1 a
Affine1 (a
a', a
b')
    where
      !a' :: a
a' = a
a1 a -> a -> a
forall a. Num a => a -> a -> a
* a
a2
      !b' :: a
b' = a
a1 a -> a -> a
forall a. Num a => a -> a -> a
* a
b2 a -> a -> a
forall a. Num a => a -> a -> a
+ a
b1
  {-# INLINE sconcat #-}
  sconcat :: NonEmpty (Affine1 a) -> Affine1 a
sconcat (Affine1 a
x :| [Affine1 a]
xs) = (Affine1 a -> Affine1 a -> Affine1 a)
-> Affine1 a -> [Affine1 a] -> Affine1 a
forall b a. (b -> a -> b) -> b -> [a] -> b
forall (t :: * -> *) b a.
Foldable t =>
(b -> a -> b) -> b -> t a -> b
foldl' Affine1 a -> Affine1 a -> Affine1 a
forall a. Semigroup a => a -> a -> a
(<>) Affine1 a
x [Affine1 a]
xs
  {-# INLINE stimes #-}
  stimes :: forall b. Integral b => b -> Affine1 a -> Affine1 a
stimes b
b = (Affine1 a -> Affine1 a -> Affine1 a)
-> Int -> Affine1 a -> Affine1 a
forall a. (a -> a -> a) -> Int -> a -> a
ACEM.power Affine1 a -> Affine1 a -> Affine1 a
forall a. Semigroup a => a -> a -> a
(<>) (b -> Int
forall a b. (Integral a, Num b) => a -> b
fromIntegral b
b)

-- | @since 1.0.0
instance (Num a) => Monoid (Affine1 a) where
  {-# INLINE mempty #-}
  mempty :: Affine1 a
mempty = Affine1Repr a -> Affine1 a
forall a. Affine1Repr a -> Affine1 a
Affine1 (a
1, a
0)
  {-# INLINE mconcat #-}
  mconcat :: [Affine1 a] -> Affine1 a
mconcat [] = Affine1 a
forall a. Monoid a => a
mempty
  mconcat (Affine1 a
x : [Affine1 a]
xs) = (Affine1 a -> Affine1 a -> Affine1 a)
-> Affine1 a -> [Affine1 a] -> Affine1 a
forall b a. (b -> a -> b) -> b -> [a] -> b
forall (t :: * -> *) b a.
Foldable t =>
(b -> a -> b) -> b -> t a -> b
foldl' Affine1 a -> Affine1 a -> Affine1 a
forall a. Semigroup a => a -> a -> a
(<>) Affine1 a
x [Affine1 a]
xs

-- | @since 1.0.0
instance (Num a) => SegAct (Affine1 a) (Sum a) where
  {-# INLINE segActWithLength #-}
  segActWithLength :: Int -> Affine1 a -> Sum a -> Sum a
segActWithLength !Int
len Affine1 a
f (Sum !a
x) = a -> Sum a
forall a. a -> Sum a
Sum (a -> Sum a) -> a -> Sum a
forall a b. (a -> b) -> a -> b
$! Int -> Affine1 a -> a -> a
forall a. Num a => Int -> Affine1 a -> a -> a
actWithLength Int
len Affine1 a
f a
x

-- | @since 1.0.0
instance (Num a) => SegAct (Affine1 (Sum a)) (Sum a) where
  {-# INLINE segActWithLength #-}
  segActWithLength :: Int -> Affine1 (Sum a) -> Sum a -> Sum a
segActWithLength = Int -> Affine1 (Sum a) -> Sum a -> Sum a
forall a. Num a => Int -> Affine1 a -> a -> a
actWithLength

-- | @since 1.0.0
instance (Num a) => SegAct (Dual (Affine1 a)) (Sum a) where
  {-# INLINE segActWithLength #-}
  segActWithLength :: Int -> Dual (Affine1 a) -> Sum a -> Sum a
segActWithLength !Int
len (Dual Affine1 a
f) (Sum !a
x) = a -> Sum a
forall a. a -> Sum a
Sum (a -> Sum a) -> a -> Sum a
forall a b. (a -> b) -> a -> b
$! Int -> Affine1 a -> a -> a
forall a. Num a => Int -> Affine1 a -> a -> a
actWithLength Int
len Affine1 a
f a
x

-- | @since 1.0.0
instance (Num a) => SegAct (Dual (Affine1 (Sum a))) (Sum a) where
  {-# INLINE segActWithLength #-}
  segActWithLength :: Int -> Dual (Affine1 (Sum a)) -> Sum a -> Sum a
segActWithLength !Int
len (Dual Affine1 (Sum a)
f) (Sum !a
x) = a -> Sum a
forall a. a -> Sum a
Sum (a -> Sum a) -> a -> Sum a
forall a b. (a -> b) -> a -> b
$! Int -> Affine1 a -> a -> a
forall a. Num a => Int -> Affine1 a -> a -> a
actWithLength Int
len (Affine1 (Sum a) -> Affine1 a
forall a b. Coercible a b => a -> b
coerce Affine1 (Sum a)
f) a
x

-- not works as SegAct for Product, Min, and Max.

-- | @since 1.0.0
newtype instance VU.MVector s (Affine1 a) = MV_Affine1 (VU.MVector s (Affine1Repr a))

-- | @since 1.0.0
newtype instance VU.Vector (Affine1 a) = V_Affine1 (VU.Vector (Affine1Repr a))

-- | @since 1.0.0
deriving instance (VU.Unbox a) => VGM.MVector VUM.MVector (Affine1 a)

-- | @since 1.0.0
deriving instance (VU.Unbox a) => VG.Vector VU.Vector (Affine1 a)

-- | @since 1.0.0
instance (VU.Unbox a) => VU.Unbox (Affine1 a)