    {-# LANGUAGE CPP #-}
    {-# LANGUAGE GADTs #-}
    {-# LANGUAGE Rank2Types #-}
    {-# LANGUAGE FlexibleContexts #-}
    {-# LANGUAGE FlexibleInstances #-}
    {-# LANGUAGE DefaultSignatures #-}
    {-# LANGUAGE UndecidableInstances #-}
    {-# LANGUAGE MultiParamTypeClasses #-}
    {-# LANGUAGE FunctionalDependencies #-}

    #ifdef TRUSTWORTHY
    {-# LANGUAGE Trustworthy #-} -- vector, hashable
    #endif

    #if __GLASGOW_HASKELL__ >= 711
    {-# OPTIONS_GHC -fno-warn-redundant-constraints #-}
    #endif

    #ifndef MIN_VERSION_containers
    #define MIN_VERSION_containers(x,y,z) 1
    #endif
    -------------------------------------------------------------------------------
    -- |
    -- Module      :  Control.Lens.Indexed
    -- Copyright   :  (C) 2012-15 Edward Kmett
    -- License     :  BSD-style (see the file LICENSE)
    -- Maintainer  :  Edward Kmett <ekmett@gmail.com>
    -- Stability   :  provisional
    -- Portability :  Rank2Types
    --
    -- (The classes in here need to be defined together for @DefaultSignatures@ to work.)
    -------------------------------------------------------------------------------
    module Control.Lens.Indexed
      (
      -- * Indexing
        Indexable(..)
      , Conjoined(..)
      , Indexed(..)
      , (<.), (<.>), (.>)
      , selfIndex
      , reindexed
      , icompose
      , indexing
      , indexing64
      -- * Indexed Functors
      , FunctorWithIndex(..)
      -- * Indexed Foldables
      , FoldableWithIndex(..)
      -- ** Indexed Foldable Combinators
      , iany
      , iall
      , inone, none
      , itraverse_
      , ifor_
      , imapM_
      , iforM_
      , iconcatMap
      , ifind
      , ifoldrM
      , ifoldlM
      , itoList
      -- * Converting to Folds
      , withIndex
      , asIndex
      -- * Restricting by Index
      , indices
      , index
      -- * Indexed Traversables
      , TraversableWithIndex(..)
      -- * Indexed Traversable Combinators
      , ifor
      , imapM
      , iforM
      , imapAccumR
      , imapAccumL
      -- * Indexed Folds with Reified Monoid
      , ifoldMapBy
      , ifoldMapByOf
      -- * Indexed Traversals with Reified Applicative
      , itraverseBy
      , itraverseByOf
      ) where

    import Control.Applicative
    import Control.Applicative.Backwards
    import Control.Comonad.Cofree
    import Control.Comonad.Trans.Traced
    import Control.Monad (void, liftM)
    import Control.Monad.Trans.Identity
    import Control.Monad.Trans.Reader
    import Control.Monad.Trans.State.Lazy as Lazy
    import Control.Monad.Free
    import Control.Lens.Fold
    import Control.Lens.Getter
    import Control.Lens.Internal.Fold
    import Control.Lens.Internal.Indexed
    import Control.Lens.Internal.Level
    import Control.Lens.Internal.Magma
    import Control.Lens.Setter
    import Control.Lens.Traversal
    import Control.Lens.Type
    import Data.Array (Array)
    import qualified Data.Array as Array
    import Data.Foldable
    import Data.Functor.Compose
    import Data.Functor.Contravariant
    import Data.Functor.Product
    import Data.Functor.Reverse
    import Data.Hashable
    import Data.HashMap.Lazy as HashMap
    import Data.IntMap as IntMap
    import Data.Ix (Ix)
    import Data.List.NonEmpty as NonEmpty
    import Data.Map as Map
    import Data.Monoid hiding (Product)
    import Data.Profunctor.Unsafe
    import Data.Reflection
    import Data.Sequence hiding ((:<), index)
    #if !(MIN_VERSION_containers(0,5,0))
    import Data.Traversable (sequenceA)
    #endif
    import Data.Tree
    import Data.Tuple (swap)
    import Data.Vector (Vector)
    import qualified Data.Vector as V
    import Prelude

    infixr 9 <.>, <., .>

    -- $setup
    -- >>> :set -XNoOverloadedStrings
    -- >>> import Control.Lens

    -- | Compose an 'Indexed' function with a non-indexed function.
    --
    -- Mnemonically, the @<@ points to the indexing we want to preserve.
    --
    -- >>> let nestedMap = (fmap Map.fromList . Map.fromList) [(1, [(10, "one,ten"), (20, "one,twenty")]), (2, [(30, "two,thirty"), (40,"two,forty")])]
    -- >>> nestedMap^..(itraversed<.itraversed).withIndex
    -- [(1,"one,ten"),(1,"one,twenty"),(2,"two,thirty"),(2,"two,forty")]
    (<.) :: Indexable i p => (Indexed i s t -> r) -> ((a -> b) -> s -> t) -> p a b -> r
    (<.) f g h = f . Indexed $ g . indexed h
    {-# INLINE (<.) #-}

    -- | Compose a non-indexed function with an 'Indexed' function.
    --
    -- Mnemonically, the @>@ points to the indexing we want to preserve.
    --
    -- This is the same as @('.')@.
    --
    -- @f '.' g@ (and @f '.>' g@) gives you the index of @g@ unless @g@ is index-preserving, like a
    -- 'Prism', 'Iso' or 'Equality', in which case it'll pass through the index of @f@.
    --
    -- >>> let nestedMap = (fmap Map.fromList . Map.fromList) [(1, [(10, "one,ten"), (20, "one,twenty")]), (2, [(30, "two,thirty"), (40,"two,forty")])]
    -- >>> nestedMap^..(itraversed.>itraversed).withIndex
    -- [(10,"one,ten"),(20,"one,twenty"),(30,"two,thirty"),(40,"two,forty")]
    (.>) :: (st -> r) -> (kab -> st) -> kab -> r
    (.>) = (.)
    {-# INLINE (.>) #-}

    -- | Use a value itself as its own index. This is essentially an indexed version of 'id'.
    --
    -- Note: When used to modify the value, this can break the index requirements assumed by 'indices' and similar,
    -- so this is only properly an 'IndexedGetter', but it can be used as more.
    --
    -- @
    -- 'selfIndex' :: 'IndexedGetter' a a b
    -- @
    selfIndex :: Indexable a p => p a fb -> a -> fb
    selfIndex f a = indexed f a a
    {-# INLINE selfIndex #-}

    -- | Remap the index.
    reindexed :: Indexable j p => (i -> j) -> (Indexed i a b -> r) -> p a b -> r
    reindexed ij f g = f . Indexed $ indexed g . ij
    {-# INLINE reindexed #-}

    -- | Composition of 'Indexed' functions.
    --
    -- Mnemonically, the @\<@ and @\>@ points to the fact that we want to preserve the indices.
    --
    -- >>> let nestedMap = (fmap Map.fromList . Map.fromList) [(1, [(10, "one,ten"), (20, "one,twenty")]), (2, [(30, "two,thirty"), (40,"two,forty")])]
    -- >>> nestedMap^..(itraversed<.>itraversed).withIndex
    -- [((1,10),"one,ten"),((1,20),"one,twenty"),((2,30),"two,thirty"),((2,40),"two,forty")]
    (<.>) :: Indexable (i, j) p => (Indexed i s t -> r) -> (Indexed j a b -> s -> t) -> p a b -> r
    f <.> g = icompose (,) f g
    {-# INLINE (<.>) #-}

    -- | Composition of 'Indexed' functions with a user supplied function for combining indices.
    icompose :: Indexable p c => (i -> j -> p) -> (Indexed i s t -> r) -> (Indexed j a b -> s -> t) -> c a b -> r
    icompose ijk istr jabst cab = istr . Indexed $ \i -> jabst . Indexed $ \j -> indexed cab $ ijk i j
    {-# INLINE icompose #-}

    -------------------------------------------------------------------------------
    -- Restricting by index
    -------------------------------------------------------------------------------

    -- | This allows you to filter an 'IndexedFold', 'IndexedGetter', 'IndexedTraversal' or 'IndexedLens' based on a predicate
    -- on the indices.
    --
    -- >>> ["hello","the","world","!!!"]^..traversed.indices even
    -- ["hello","world"]
    --
    -- >>> over (traversed.indices (>0)) Prelude.reverse $ ["He","was","stressed","o_O"]
    -- ["He","saw","desserts","O_o"]
    indices :: (Indexable i p, Applicative f) => (i -> Bool) -> Optical' p (Indexed i) f a a
    indices p f = Indexed $ \i a -> if p i then indexed f i a else pure a
    {-# INLINE indices #-}

    -- | This allows you to filter an 'IndexedFold', 'IndexedGetter', 'IndexedTraversal' or 'IndexedLens' based on an index.
    --
    -- >>> ["hello","the","world","!!!"]^?traversed.index 2
    -- Just "world"
    index :: (Indexable i p, Eq i, Applicative f) => i -> Optical' p (Indexed i) f a a
    index j f = Indexed $ \i a -> if j == i then indexed f i a else pure a
    {-# INLINE index #-}

    -------------------------------------------------------------------------------
    -- FunctorWithIndex
    -------------------------------------------------------------------------------

    -- | A 'Functor' with an additional index.
    --
    -- Instances must satisfy a modified form of the 'Functor' laws:
    --
    -- @
    -- 'imap' f '.' 'imap' g ≡ 'imap' (\\i -> f i '.' g i)
    -- 'imap' (\\_ a -> a) ≡ 'id'
    -- @
    class Functor f => FunctorWithIndex i f | f -> i where
      -- | Map with access to the index.
      imap :: (i -> a -> b) -> f a -> f b
    #ifndef HLINT
      default imap :: TraversableWithIndex i f => (i -> a -> b) -> f a -> f b
      imap = iover itraversed
      {-# INLINE imap #-}
    #endif

      -- | The 'IndexedSetter' for a 'FunctorWithIndex'.
      --
      -- If you don't need access to the index, then 'mapped' is more flexible in what it accepts.
      imapped :: IndexedSetter i (f a) (f b) a b
      imapped = conjoined mapped (isets imap)
      {-# INLINE imapped #-}

    -------------------------------------------------------------------------------
    -- FoldableWithIndex
    -------------------------------------------------------------------------------

    -- | A container that supports folding with an additional index.
    class Foldable f => FoldableWithIndex i f | f -> i where
      --
      -- | Fold a container by mapping value to an arbitrary 'Monoid' with access to the index @i@.
      --
      -- When you don't need access to the index then 'foldMap' is more flexible in what it accepts.
      --
      -- @
      -- 'foldMap' ≡ 'ifoldMap' '.' 'const'
      -- @
      ifoldMap :: Monoid m => (i -> a -> m) -> f a -> m
    #ifndef HLINT
      default ifoldMap :: (TraversableWithIndex i f, Monoid m) => (i -> a -> m) -> f a -> m
      ifoldMap = ifoldMapOf itraversed
      {-# INLINE ifoldMap #-}
    #endif

      -- | The 'IndexedFold' of a 'FoldableWithIndex' container.
      --
      -- @'ifolded' '.' 'asIndex'@ is a fold over the keys of a 'FoldableWithIndex'.
      --
      -- >>> Data.Map.fromList [(2, "hello"), (1, "world")]^..ifolded.asIndex
      -- [1,2]
      ifolded :: IndexedFold i (f a) a
      ifolded = conjoined folded $ \f -> phantom . getFolding . ifoldMap (\i -> Folding #. indexed f i)
      {-# INLINE ifolded #-}

      -- | Right-associative fold of an indexed container with access to the index @i@.
      --
      -- When you don't need access to the index then 'Data.Foldable.foldr' is more flexible in what it accepts.
      --
      -- @
      -- 'Data.Foldable.foldr' ≡ 'ifoldr' '.' 'const'
      -- @
      ifoldr   :: (i -> a -> b -> b) -> b -> f a -> b
      ifoldr f z t = appEndo (ifoldMap (\i -> Endo #. f i) t) z
      {-# INLINE ifoldr #-}

      -- | Left-associative fold of an indexed container with access to the index @i@.
      --
      -- When you don't need access to the index then 'Data.Foldable.foldl' is more flexible in what it accepts.
      --
      -- @
      -- 'Data.Foldable.foldl' ≡ 'ifoldl' '.' 'const'
      -- @
      ifoldl :: (i -> b -> a -> b) -> b -> f a -> b
      ifoldl f z t = appEndo (getDual (ifoldMap (\i -> Dual #. Endo #. flip (f i)) t)) z
      {-# INLINE ifoldl #-}

      -- | /Strictly/ fold right over the elements of a structure with access to the index @i@.
      --
      -- When you don't need access to the index then 'foldr'' is more flexible in what it accepts.
      --
      -- @
      -- 'foldr'' ≡ 'ifoldr'' '.' 'const'
      -- @
      ifoldr' :: (i -> a -> b -> b) -> b -> f a -> b
      ifoldr' f z0 xs = ifoldl f' id xs z0
        where f' i k x z = k $! f i x z
      {-# INLINE ifoldr' #-}

      -- | Fold over the elements of a structure with an index, associating to the left, but /strictly/.
      --
      -- When you don't need access to the index then 'Control.Lens.Fold.foldlOf'' is more flexible in what it accepts.
      --
      -- @
      -- 'Control.Lens.Fold.foldlOf'' l ≡ 'ifoldlOf'' l '.' 'const'
      -- @
      ifoldl' :: (i -> b -> a -> b) -> b -> f a -> b
      ifoldl' f z0 xs = ifoldr f' id xs z0
        where f' i x k z = k $! f i z x
      {-# INLINE ifoldl' #-}

    -- | Return whether or not any element in a container satisfies a predicate, with access to the index @i@.
    --
    -- When you don't need access to the index then 'any' is more flexible in what it accepts.
    --
    -- @
    -- 'any' ≡ 'iany' '.' 'const'
    -- @
    iany :: FoldableWithIndex i f => (i -> a -> Bool) -> f a -> Bool
    iany f = getAny #. ifoldMap (\i -> Any #. f i)
    {-# INLINE iany #-}

    -- | Return whether or not all elements in a container satisfy a predicate, with access to the index @i@.
    --
    -- When you don't need access to the index then 'all' is more flexible in what it accepts.
    --
    -- @
    -- 'all' ≡ 'iall' '.' 'const'
    -- @
    iall :: FoldableWithIndex i f => (i -> a -> Bool) -> f a -> Bool
    iall f = getAll #. ifoldMap (\i -> All #. f i)
    {-# INLINE iall #-}

    -- | Return whether or not none of the elements in a container satisfy a predicate, with access to the index @i@.
    --
    -- When you don't need access to the index then 'none' is more flexible in what it accepts.
    --
    -- @
    -- 'none' ≡ 'inone' '.' 'const'
    -- 'inone' f ≡ 'not' '.' 'iany' f
    -- @
    inone :: FoldableWithIndex i f => (i -> a -> Bool) -> f a -> Bool
    inone f = not . iany f
    {-# INLINE inone #-}

    -- | Determines whether no elements of the structure satisfy the predicate.
    --
    -- @
    -- 'none' f ≡ 'not' '.' 'any' f
    -- @
    none :: Foldable f => (a -> Bool) -> f a -> Bool
    none f = not . Data.Foldable.any f
    {-# INLINE none #-}

    -- | Traverse elements with access to the index @i@, discarding the results.
    --
    -- When you don't need access to the index then 'traverse_' is more flexible in what it accepts.
    --
    -- @
    -- 'traverse_' l = 'itraverse' '.' 'const'
    -- @
    itraverse_ :: (FoldableWithIndex i t, Applicative f) => (i -> a -> f b) -> t a -> f ()
    itraverse_ f = getTraversed #. ifoldMap (\i -> Traversed #. void . f i)
    {-# INLINE itraverse_ #-}

    -- | Traverse elements with access to the index @i@, discarding the results (with the arguments flipped).
    --
    -- @
    -- 'ifor_' ≡ 'flip' 'itraverse_'
    -- @
    --
    -- When you don't need access to the index then 'for_' is more flexible in what it accepts.
    --
    -- @
    -- 'for_' a ≡ 'ifor_' a '.' 'const'
    -- @
    ifor_ :: (FoldableWithIndex i t, Applicative f) => t a -> (i -> a -> f b) -> f ()
    ifor_ = flip itraverse_
    {-# INLINE ifor_ #-}

    -- | Run monadic actions for each target of an 'IndexedFold' or 'Control.Lens.IndexedTraversal.IndexedTraversal' with access to the index,
    -- discarding the results.
    --
    -- When you don't need access to the index then 'Control.Lens.Fold.mapMOf_' is more flexible in what it accepts.
    --
    -- @
    -- 'mapM_' ≡ 'imapM' '.' 'const'
    -- @
    imapM_ :: (FoldableWithIndex i t, Monad m) => (i -> a -> m b) -> t a -> m ()
    imapM_ f = getSequenced #. ifoldMap (\i -> Sequenced #. liftM skip . f i)
    {-# INLINE imapM_ #-}

    -- | Run monadic actions for each target of an 'IndexedFold' or 'Control.Lens.IndexedTraversal.IndexedTraversal' with access to the index,
    -- discarding the results (with the arguments flipped).
    --
    -- @
    -- 'iforM_' ≡ 'flip' 'imapM_'
    -- @
    --
    -- When you don't need access to the index then 'Control.Lens.Fold.forMOf_' is more flexible in what it accepts.
    --
    -- @
    -- 'Control.Lens.Fold.forMOf_' l a ≡ 'iforMOf' l a '.' 'const'
    -- @
    iforM_ :: (FoldableWithIndex i t, Monad m) => t a -> (i -> a -> m b) -> m ()
    iforM_ = flip imapM_
    {-# INLINE iforM_ #-}

    -- | Concatenate the results of a function of the elements of an indexed container with access to the index.
    --
    -- When you don't need access to the index then 'concatMap' is more flexible in what it accepts.
    --
    -- @
    -- 'concatMap' ≡ 'iconcatMap' '.' 'const'
    -- 'iconcatMap' ≡ 'ifoldMap'
    -- @
    iconcatMap :: FoldableWithIndex i f => (i -> a -> [b]) -> f a -> [b]
    iconcatMap = ifoldMap
    {-# INLINE iconcatMap #-}

    -- | Searches a container with a predicate that is also supplied the index, returning the left-most element of the structure
    -- matching the predicate, or 'Nothing' if there is no such element.
    --
    -- When you don't need access to the index then 'find' is more flexible in what it accepts.
    --
    -- @
    -- 'find' ≡ 'ifind' '.' 'const'
    -- @
    ifind :: FoldableWithIndex i f => (i -> a -> Bool) -> f a -> Maybe (i, a)
    ifind p = ifoldr (\i a y -> if p i a then Just (i, a) else y) Nothing
    {-# INLINE ifind #-}

    -- | Monadic fold right over the elements of a structure with an index.
    --
    -- When you don't need access to the index then 'foldrM' is more flexible in what it accepts.
    --
    -- @
    -- 'foldrM' ≡ 'ifoldrM' '.' 'const'
    -- @
    ifoldrM :: (FoldableWithIndex i f, Monad m) => (i -> a -> b -> m b) -> b -> f a -> m b
    ifoldrM f z0 xs = ifoldl f' return xs z0
      where f' i k x z = f i x z >>= k
    {-# INLINE ifoldrM #-}

    -- | Monadic fold over the elements of a structure with an index, associating to the left.
    --
    -- When you don't need access to the index then 'foldlM' is more flexible in what it accepts.
    --
    -- @
    -- 'foldlM' ≡ 'ifoldlM' '.' 'const'
    -- @
    ifoldlM :: (FoldableWithIndex i f, Monad m) => (i -> b -> a -> m b) -> b -> f a -> m b
    ifoldlM f z0 xs = ifoldr f' return xs z0
      where f' i x k z = f i z x >>= k
    {-# INLINE ifoldlM #-}

    -- | Extract the key-value pairs from a structure.
    --
    -- When you don't need access to the indices in the result, then 'Data.Foldable.toList' is more flexible in what it accepts.
    --
    -- @
    -- 'Data.Foldable.toList' ≡ 'Data.List.map' 'snd' '.' 'itoList'
    -- @
    itoList :: FoldableWithIndex i f => f a -> [(i,a)]
    itoList = ifoldr (\i c -> ((i,c):)) []
    {-# INLINE itoList #-}

    -------------------------------------------------------------------------------
    -- TraversableWithIndex
    -------------------------------------------------------------------------------

    -- | A 'Traversable' with an additional index.
    --
    -- An instance must satisfy a (modified) form of the 'Traversable' laws:
    --
    -- @
    -- 'itraverse' ('const' 'Identity') ≡ 'Identity'
    -- 'fmap' ('itraverse' f) '.' 'itraverse' g ≡ 'Data.Functor.Compose.getCompose' '.' 'itraverse' (\\i -> 'Data.Functor.Compose.Compose' '.' 'fmap' (f i) '.' g i)
    -- @
    class (FunctorWithIndex i t, FoldableWithIndex i t, Traversable t) => TraversableWithIndex i t | t -> i where
      -- | Traverse an indexed container.
      --
      -- @
      -- 'itraverse' ≡ 'itraverseOf' 'itraversed'
      -- @
      itraverse :: Applicative f => (i -> a -> f b) -> t a -> f (t b)
    #ifndef HLINT
      default itraverse :: Applicative f => (Int -> a -> f b) -> t a -> f (t b)
      itraverse = traversed .# Indexed
      {-# INLINE itraverse #-}
    #endif

      -- | The 'IndexedTraversal' of a 'TraversableWithIndex' container.
      itraversed :: IndexedTraversal i (t a) (t b) a b
      itraversed = conjoined traverse (itraverse . indexed)
      {-# INLINE itraversed #-}

    -- | Traverse with an index (and the arguments flipped).
    --
    -- @
    -- 'for' a ≡ 'ifor' a '.' 'const'
    -- 'ifor' ≡ 'flip' 'itraverse'
    -- @
    ifor :: (TraversableWithIndex i t, Applicative f) => t a -> (i -> a -> f b) -> f (t b)
    ifor = flip itraverse
    {-# INLINE ifor #-}

    -- | Map each element of a structure to a monadic action,
    -- evaluate these actions from left to right, and collect the results, with access
    -- the index.
    --
    -- When you don't need access to the index 'mapM' is more liberal in what it can accept.
    --
    -- @
    -- 'mapM' ≡ 'imapM' '.' 'const'
    -- @
    imapM :: (TraversableWithIndex i t, Monad m) => (i -> a -> m b) -> t a -> m (t b)
    imapM f = unwrapMonad #. itraverse (\i -> WrapMonad #. f i)
    {-# INLINE imapM #-}

    -- | Map each element of a structure to a monadic action,
    -- evaluate these actions from left to right, and collect the results, with access
    -- its position (and the arguments flipped).
    --
    -- @
    -- 'forM' a ≡ 'iforM' a '.' 'const'
    -- 'iforM' ≡ 'flip' 'imapM'
    -- @
    iforM :: (TraversableWithIndex i t, Monad m) => t a -> (i -> a -> m b) -> m (t b)
    iforM = flip imapM
    {-# INLINE iforM #-}

    -- | Generalizes 'Data.Traversable.mapAccumR' to add access to the index.
    --
    -- 'imapAccumROf' accumulates state from right to left.
    --
    -- @
    -- 'Control.Lens.Traversal.mapAccumR' ≡ 'imapAccumR' '.' 'const'
    -- @
    imapAccumR :: TraversableWithIndex i t => (i -> s -> a -> (s, b)) -> s -> t a -> (s, t b)
    imapAccumR f s0 a = swap (Lazy.runState (forwards (itraverse (\i c -> Backwards (Lazy.state (\s -> swap (f i s c)))) a)) s0)
    {-# INLINE imapAccumR #-}

    -- | Generalizes 'Data.Traversable.mapAccumL' to add access to the index.
    --
    -- 'imapAccumLOf' accumulates state from left to right.
    --
    -- @
    -- 'Control.Lens.Traversal.mapAccumLOf' ≡ 'imapAccumL' '.' 'const'
    -- @
    imapAccumL :: TraversableWithIndex i t => (i -> s -> a -> (s, b)) -> s -> t a -> (s, t b)
    imapAccumL f s0 a = swap (Lazy.runState (itraverse (\i c -> Lazy.state (\s -> swap (f i s c))) a) s0)
    {-# INLINE imapAccumL #-}

    -------------------------------------------------------------------------------
    -- Instances
    -------------------------------------------------------------------------------

    instance FunctorWithIndex i f => FunctorWithIndex i (Backwards f) where
      imap f  = Backwards . imap f . forwards
      {-# INLINE imap #-}

    instance FoldableWithIndex i f => FoldableWithIndex i (Backwards f) where
      ifoldMap f = ifoldMap f . forwards
      {-# INLINE ifoldMap #-}

    instance TraversableWithIndex i f => TraversableWithIndex i (Backwards f) where
      itraverse f = fmap Backwards . itraverse f . forwards
      {-# INLINE itraverse #-}

    instance FunctorWithIndex i f => FunctorWithIndex i (Reverse f) where
      imap f = Reverse . imap f . getReverse
      {-# INLINE imap #-}

    instance FoldableWithIndex i f => FoldableWithIndex i (Reverse f) where
      ifoldMap f = getDual . ifoldMap (\i -> Dual #. f i) . getReverse
      {-# INLINE ifoldMap #-}

    instance TraversableWithIndex i f => TraversableWithIndex i (Reverse f) where
      itraverse f = fmap Reverse . forwards . itraverse (\i -> Backwards . f i) . getReverse
      {-# INLINE itraverse #-}

    instance FunctorWithIndex () Identity where
      imap f (Identity a) = Identity (f () a)
      {-# INLINE imap #-}

    instance FoldableWithIndex () Identity where
      ifoldMap f (Identity a) = f () a
      {-# INLINE ifoldMap #-}

    instance TraversableWithIndex () Identity where
      itraverse f (Identity a) = Identity <$> f () a
      {-# INLINE itraverse #-}

    instance FunctorWithIndex k ((,) k) where
      imap f (k,a) = (k, f k a)
      {-# INLINE imap #-}

    instance FoldableWithIndex k ((,) k) where
      ifoldMap = uncurry
      {-# INLINE ifoldMap #-}

    instance TraversableWithIndex k ((,) k) where
      itraverse f (k, a) = (,) k <$> f k a
      {-# INLINE itraverse #-}

    -- | The position in the list is available as the index.
    instance FunctorWithIndex Int []
    instance FoldableWithIndex Int []
    instance TraversableWithIndex Int [] where
      itraversed = traversed
      {-# INLINE itraversed #-}

    instance FunctorWithIndex Int NonEmpty
    instance FoldableWithIndex Int NonEmpty
    instance TraversableWithIndex Int NonEmpty where
      itraverse = itraverseOf traversed
      {-# INLINE itraverse #-}

    instance FunctorWithIndex () Maybe where
      imap f = fmap (f ())
      {-# INLINE imap #-}
    instance FoldableWithIndex () Maybe where
      ifoldMap f = foldMap (f ())
      {-# INLINE ifoldMap #-}
    instance TraversableWithIndex () Maybe where
      itraverse f = traverse (f ())
      {-# INLINE itraverse #-}

    -- | The position in the 'Seq' is available as the index.
    instance FunctorWithIndex Int Seq
    instance FoldableWithIndex Int Seq
    instance TraversableWithIndex Int Seq where
      itraversed = traversed
      {-# INLINE itraversed #-}

    instance FunctorWithIndex Int Vector where
      imap = V.imap
      {-# INLINE imap #-}
    instance FoldableWithIndex Int Vector where
      ifoldr = V.ifoldr
      {-# INLINE ifoldr #-}
      ifoldl = V.ifoldl . flip
      {-# INLINE ifoldl #-}
      ifoldr' = V.ifoldr'
      {-# INLINE ifoldr' #-}
      ifoldl' = V.ifoldl' . flip
      {-# INLINE ifoldl' #-}
    instance TraversableWithIndex Int Vector where
      itraversed = traversed
      {-# INLINE itraversed #-}

    instance FunctorWithIndex Int IntMap
    instance FoldableWithIndex Int IntMap
    instance TraversableWithIndex Int IntMap where
    #if MIN_VERSION_containers(0,5,0)
      itraverse = IntMap.traverseWithKey
    #else
      itraverse f = sequenceA . IntMap.mapWithKey f
    #endif
      {-# INLINE [0] itraverse #-}

    {-# RULES
    "itraversed -> mapIntMap"    itraversed = sets IntMap.map               :: ASetter (IntMap a) (IntMap b) a b;
    "itraversed -> imapIntMap"   itraversed = isets IntMap.mapWithKey       :: AnIndexedSetter Int (IntMap a) (IntMap b) a b;
    "itraversed -> foldrIntMap"  itraversed = foldring IntMap.foldr         :: Getting (Endo r) (IntMap a) a;
    "itraversed -> ifoldrIntMap" itraversed = ifoldring IntMap.foldrWithKey :: IndexedGetting Int (Endo r) (IntMap a) a;
     #-}

    instance FunctorWithIndex k (Map k)
    instance FoldableWithIndex k (Map k)
    instance TraversableWithIndex k (Map k) where
    #if MIN_VERSION_containers(0,5,0)
      itraverse = Map.traverseWithKey
    #else
      itraverse f = sequenceA . Map.mapWithKey f
    #endif
      {-# INLINE [0] itraverse #-}

    {-# RULES
    "itraversed -> mapMap"    itraversed = sets Map.map               :: ASetter (Map k a) (Map k b) a b;
    "itraversed -> imapMap"   itraversed = isets Map.mapWithKey       :: AnIndexedSetter k (Map k a) (Map k b) a b;
    "itraversed -> foldrMap"  itraversed = foldring Map.foldr         :: Getting (Endo r) (Map k a) a;
    "itraversed -> ifoldrMap" itraversed = ifoldring Map.foldrWithKey :: IndexedGetting k (Endo r) (Map k a) a;
     #-}

    instance (Eq k, Hashable k) => FunctorWithIndex k (HashMap k)
    instance (Eq k, Hashable k) => FoldableWithIndex k (HashMap k)
    instance (Eq k, Hashable k) => TraversableWithIndex k (HashMap k) where
      itraverse = HashMap.traverseWithKey
      {-# INLINE [0] itraverse #-}

    {-# RULES
    "itraversed -> mapHashMap"    itraversed = sets HashMap.map               :: ASetter (HashMap k a) (HashMap k b) a b;
    "itraversed -> imapHashMap"   itraversed = isets HashMap.mapWithKey       :: AnIndexedSetter k (HashMap k a) (HashMap k b) a b;
    "itraversed -> foldrHashMap"  itraversed = foldring HashMap.foldr         :: Getting (Endo r) (HashMap k a) a;
    "itraversed -> ifoldrHashMap" itraversed = ifoldring HashMap.foldrWithKey :: IndexedGetting k (Endo r) (HashMap k a) a;
     #-}

    instance FunctorWithIndex r ((->) r) where
      imap f g x = f x (g x)
      {-# INLINE imap #-}

    instance FunctorWithIndex i (Level i) where
      imap f = go where
        go (Two n l r) = Two n (go l) (go r)
        go (One i a)   = One i (f i a)
        go Zero        = Zero
      {-# INLINE imap #-}

    instance FoldableWithIndex i (Level i) where
      ifoldMap f = go where
        go (Two _ l r) = go l `mappend` go r
        go (One i a)   = f i a
        go Zero        = mempty
      {-# INLINE ifoldMap #-}

    instance TraversableWithIndex i (Level i) where
      itraverse f = go where
        go (Two n l r) = Two n <$> go l <*> go r
        go (One i a)   = One i <$> f i a
        go Zero        = pure Zero
      {-# INLINE itraverse #-}

    instance FunctorWithIndex i (Magma i t b) where
      imap f (MagmaAp x y)    = MagmaAp (imap f x) (imap f y)
      imap _ (MagmaPure x)    = MagmaPure x
      imap f (MagmaFmap xy x) = MagmaFmap xy (imap f x)
      imap f (Magma i a)      = Magma i (f i a)
      {-# INLINE imap #-}

    instance FoldableWithIndex i (Magma i t b) where
      ifoldMap f (MagmaAp x y)   = ifoldMap f x `mappend` ifoldMap f y
      ifoldMap _ MagmaPure{}     = mempty
      ifoldMap f (MagmaFmap _ x) = ifoldMap f x
      ifoldMap f (Magma i a)     = f i a
      {-# INLINE ifoldMap #-}

    instance TraversableWithIndex i (Magma i t b) where
      itraverse f (MagmaAp x y)    = MagmaAp <$> itraverse f x <*> itraverse f y
      itraverse _ (MagmaPure x)    = pure (MagmaPure x)
      itraverse f (MagmaFmap xy x) = MagmaFmap xy <$> itraverse f x
      itraverse f (Magma i a)      = Magma i <$> f i a
      {-# INLINE itraverse #-}

    instance FunctorWithIndex i f => FunctorWithIndex [i] (Free f) where
      imap f (Pure a) = Pure $ f [] a
      imap f (Free s) = Free $ imap (\i -> imap (f . (:) i)) s
      {-# INLINE imap #-}

    instance FoldableWithIndex i f => FoldableWithIndex [i] (Free f) where
      ifoldMap f (Pure a) = f [] a
      ifoldMap f (Free s) = ifoldMap (\i -> ifoldMap (f . (:) i)) s
      {-# INLINE ifoldMap #-}

    instance TraversableWithIndex i f => TraversableWithIndex [i] (Free f) where
      itraverse f (Pure a) = Pure <$> f [] a
      itraverse f (Free s) = Free <$> itraverse (\i -> itraverse (f . (:) i)) s
      {-# INLINE itraverse #-}

    instance Ix i => FunctorWithIndex i (Array i) where
      imap f arr = Array.listArray (Array.bounds arr) . fmap (uncurry f) $ Array.assocs arr
      {-# INLINE imap #-}

    instance Ix i => FoldableWithIndex i (Array i) where
      ifoldMap f = foldMap (uncurry f) . Array.assocs
      {-# INLINE ifoldMap #-}

    instance Ix i => TraversableWithIndex i (Array i) where
      itraverse f arr = Array.listArray (Array.bounds arr) <$> traverse (uncurry f) (Array.assocs arr)
      {-# INLINE itraverse #-}

    instance FunctorWithIndex i f => FunctorWithIndex [i] (Cofree f) where
      imap f (a :< as) = f [] a :< imap (\i -> imap (f . (:) i)) as
      {-# INLINE imap #-}

    instance FoldableWithIndex i f => FoldableWithIndex [i] (Cofree f) where
      ifoldMap f (a :< as) = f [] a `mappend` ifoldMap (\i -> ifoldMap (f . (:) i)) as
      {-# INLINE ifoldMap #-}

    instance TraversableWithIndex i f => TraversableWithIndex [i] (Cofree f) where
      itraverse f (a :< as) = (:<) <$> f [] a <*> itraverse (\i -> itraverse (f . (:) i)) as
      {-# INLINE itraverse #-}

    instance (FunctorWithIndex i f, FunctorWithIndex j g) => FunctorWithIndex (i, j) (Compose f g) where
      imap f (Compose fg) = Compose $ imap (\k -> imap (f . (,) k)) fg
      {-# INLINE imap #-}

    instance (FoldableWithIndex i f, FoldableWithIndex j g) => FoldableWithIndex (i, j) (Compose f g) where
      ifoldMap f (Compose fg) = ifoldMap (\k -> ifoldMap (f . (,) k)) fg
      {-# INLINE ifoldMap #-}

    instance (TraversableWithIndex i f, TraversableWithIndex j g) => TraversableWithIndex (i, j) (Compose f g) where
      itraverse f (Compose fg) = Compose <$> itraverse (\k -> itraverse (f . (,) k)) fg
      {-# INLINE itraverse #-}

    instance FunctorWithIndex i m => FunctorWithIndex i (IdentityT m) where
      imap f (IdentityT m) = IdentityT $ imap f m
      {-# INLINE imap #-}

    instance FoldableWithIndex i m => FoldableWithIndex i (IdentityT m) where
      ifoldMap f (IdentityT m) = ifoldMap f m
      {-# INLINE ifoldMap #-}

    instance TraversableWithIndex i m => TraversableWithIndex i (IdentityT m) where
      itraverse f (IdentityT m) = IdentityT <$> itraverse f m
      {-# INLINE itraverse #-}

    instance (FunctorWithIndex i f, FunctorWithIndex j g) => FunctorWithIndex (Either i j) (Product f g) where
      imap f (Pair a b) = Pair (imap (f . Left) a) (imap (f . Right) b)
      {-# INLINE imap #-}

    instance (FoldableWithIndex i f, FoldableWithIndex j g) => FoldableWithIndex (Either i j) (Product f g) where
      ifoldMap f (Pair a b) = ifoldMap (f . Left) a `mappend` ifoldMap (f . Right) b
      {-# INLINE ifoldMap #-}

    instance (TraversableWithIndex i f, TraversableWithIndex j g) => TraversableWithIndex (Either i j) (Product f g) where
      itraverse f (Pair a b) = Pair <$> itraverse (f . Left) a <*> itraverse (f . Right) b
      {-# INLINE itraverse #-}

    instance FunctorWithIndex i m => FunctorWithIndex (e, i) (ReaderT e m) where
      imap f (ReaderT m) = ReaderT $ \k -> imap (f . (,) k) (m k)
      {-# INLINE imap #-}

    instance FunctorWithIndex i w => FunctorWithIndex (s, i) (TracedT s w) where
      imap f (TracedT w) = TracedT $ imap (\k' g k -> f (k, k') (g k)) w
      {-# INLINE imap #-}

    instance FunctorWithIndex [Int] Tree where
      imap f (Node a as) = Node (f [] a) $ imap (\i -> imap (f . (:) i)) as
      {-# INLINE imap #-}

    instance FoldableWithIndex [Int] Tree where
      ifoldMap f (Node a as) = f [] a `mappend` ifoldMap (\i -> ifoldMap (f . (:) i)) as
      {-# INLINE ifoldMap #-}

    instance TraversableWithIndex [Int] Tree where
      itraverse f (Node a as) = Node <$> f [] a <*> itraverse (\i -> itraverse (f . (:) i)) as
      {-# INLINE itraverse #-}

    -------------------------------------------------------------------------------
    -- Misc.
    -------------------------------------------------------------------------------

    skip :: a -> ()
    skip _ = ()
    {-# INLINE skip #-}

    -------------------------------------------------------------------------------
    -- Indexed Folds with Reified Monoid
    -------------------------------------------------------------------------------

    ifoldMapBy :: FoldableWithIndex i t => (r -> r -> r) -> r -> (i -> a -> r) -> t a -> r
    ifoldMapBy f z g = reifyMonoid f z (ifoldMap (\i a -> ReflectedMonoid (g i a)))

    ifoldMapByOf :: IndexedFold i t a -> (r -> r -> r) -> r -> (i -> a -> r) -> t -> r
    ifoldMapByOf l f z g = reifyMonoid f z (ifoldMapOf l (\i a -> ReflectedMonoid (g i a)))

    itraverseBy :: TraversableWithIndex i t => (forall x. x -> f x) -> (forall x y. f (x -> y) -> f x -> f y) -> (i -> a -> f b) -> t a -> f (t b)
    itraverseBy pur app f = reifyApplicative pur app (itraverse (\i a -> ReflectedApplicative (f i a)))

    itraverseByOf :: IndexedTraversal i s t a b -> (forall x. x -> f x) -> (forall x y. f (x -> y) -> f x -> f y) -> (i -> a -> f b) -> s -> f t
    itraverseByOf l pur app f = reifyApplicative pur app (itraverseOf l (\i a -> ReflectedApplicative (f i a)))