    {-# LANGUAGE CPP #-}
    {-# LANGUAGE GADTs #-}
    {-# LANGUAGE Rank2Types #-}
    {-# LANGUAGE FlexibleContexts #-}
    {-# LANGUAGE FlexibleInstances #-}
    {-# LANGUAGE ScopedTypeVariables #-}
    {-# LANGUAGE MultiParamTypeClasses #-}
    {-# LANGUAGE KindSignatures #-}
    #if __GLASGOW_HASKELL__ >= 706
    {-# LANGUAGE PolyKinds #-}
    #endif
    -------------------------------------------------------------------------------
    -- |
    -- Module      :  Control.Lens.Type
    -- Copyright   :  (C) 2012-15 Edward Kmett
    -- License     :  BSD-style (see the file LICENSE)
    -- Maintainer  :  Edward Kmett <ekmett@gmail.com>
    -- Stability   :  provisional
    -- Portability :  Rank2Types
    --
    -- This module exports the majority of the types that need to appear in user
    -- signatures or in documentation when talking about lenses. The remaining types
    -- for consuming lenses are distributed across various modules in the hierarchy.
    -------------------------------------------------------------------------------
    module Control.Lens.Type
      (
      -- * Other
        Equality, Equality', As
      , Iso, Iso'
      , Prism , Prism'
      , Review , AReview
      -- * Lenses, Folds and Traversals
      , Lens, Lens'
      , Traversal, Traversal'
      , Traversal1, Traversal1'
      , Setter, Setter'
      , Getter, Fold
      , Fold1
      -- * Indexed
      , IndexedLens, IndexedLens'
      , IndexedTraversal, IndexedTraversal'
      , IndexedTraversal1, IndexedTraversal1'
      , IndexedSetter, IndexedSetter'
      , IndexedGetter, IndexedFold
      , IndexedFold1
      -- * Index-Preserving
      , IndexPreservingLens, IndexPreservingLens'
      , IndexPreservingTraversal, IndexPreservingTraversal'
      , IndexPreservingTraversal1, IndexPreservingTraversal1'
      , IndexPreservingSetter, IndexPreservingSetter'
      , IndexPreservingGetter, IndexPreservingFold
      , IndexPreservingFold1
      -- * Common
      , Simple
      , LensLike, LensLike'
      , Over, Over'
      , IndexedLensLike, IndexedLensLike'
      , Optical, Optical'
      , Optic, Optic'
      ) where

    import Control.Applicative
    import Control.Lens.Internal.Setter
    import Control.Lens.Internal.Indexed
    import Data.Bifunctor
    import Data.Functor.Identity
    import Data.Functor.Contravariant
    import Data.Functor.Apply
    import Data.Profunctor
    import Data.Tagged
    import Prelude ()

    -- $setup
    -- >>> :set -XNoOverloadedStrings
    -- >>> import Control.Lens
    -- >>> import Debug.SimpleReflect.Expr
    -- >>> import Debug.SimpleReflect.Vars as Vars hiding (f,g,h)
    -- >>> import Prelude
    -- >>> let f :: Expr -> Expr; f = Debug.SimpleReflect.Vars.f
    -- >>> let g :: Expr -> Expr; g = Debug.SimpleReflect.Vars.g
    -- >>> let h :: Expr -> Expr -> Expr; h = Debug.SimpleReflect.Vars.h
    -- >>> let getter :: Expr -> Expr; getter = fun "getter"
    -- >>> let setter :: Expr -> Expr -> Expr; setter = fun "setter"
    -- >>> import Numeric.Natural
    -- >>> let nat :: Prism' Integer Natural; nat = prism toInteger $ \i -> if i < 0 then Left i else Right (fromInteger i)

    -------------------------------------------------------------------------------
    -- Lenses
    -------------------------------------------------------------------------------

    -- | A 'Lens' is actually a lens family as described in
    -- <http://comonad.com/reader/2012/mirrored-lenses/>.
    --
    -- With great power comes great responsibility and a 'Lens' is subject to the
    -- three common sense 'Lens' laws:
    --
    -- 1) You get back what you put in:
    --
    -- @
    -- 'Control.Lens.Getter.view' l ('Control.Lens.Setter.set' l v s)  ≡ v
    -- @
    --
    -- 2) Putting back what you got doesn't change anything:
    --
    -- @
    -- 'Control.Lens.Setter.set' l ('Control.Lens.Getter.view' l s) s  ≡ s
    -- @
    --
    -- 3) Setting twice is the same as setting once:
    --
    -- @
    -- 'Control.Lens.Setter.set' l v' ('Control.Lens.Setter.set' l v s) ≡ 'Control.Lens.Setter.set' l v' s
    -- @
    --
    -- These laws are strong enough that the 4 type parameters of a 'Lens' cannot
    -- vary fully independently. For more on how they interact, read the \"Why is
    -- it a Lens Family?\" section of
    -- <http://comonad.com/reader/2012/mirrored-lenses/>.
    --
    -- There are some emergent properties of these laws:
    --
    -- 1) @'Control.Lens.Setter.set' l s@ must be injective for every @s@ This is a consequence of law #1
    --
    -- 2) @'Control.Lens.Setter.set' l@ must be surjective, because of law #2, which indicates that it is possible to obtain any 'v' from some 's' such that @'Control.Lens.Setter.set' s v = s@
    --
    -- 3) Given just the first two laws you can prove a weaker form of law #3 where the values @v@ that you are setting match:
    --
    -- @
    -- 'Control.Lens.Setter.set' l v ('Control.Lens.Setter.set' l v s) ≡ 'Control.Lens.Setter.set' l v s
    -- @
    --
    -- Every 'Lens' can be used directly as a 'Control.Lens.Setter.Setter' or 'Traversal'.
    --
    -- You can also use a 'Lens' for 'Control.Lens.Getter.Getting' as if it were a
    -- 'Fold' or 'Getter'.
    --
    -- Since every 'Lens' is a valid 'Traversal', the
    -- 'Traversal' laws are required of any 'Lens' you create:
    --
    -- @
    -- l 'pure' ≡ 'pure'
    -- 'fmap' (l f) '.' l g ≡ 'Data.Functor.Compose.getCompose' '.' l ('Data.Functor.Compose.Compose' '.' 'fmap' f '.' g)
    -- @
    --
    -- @
    -- type 'Lens' s t a b = forall f. 'Functor' f => 'LensLike' f s t a b
    -- @
    type Lens s t a b = forall f. Functor f => (a -> f b) -> s -> f t

    -- | @
    -- type 'Lens'' = 'Simple' 'Lens'
    -- @
    type Lens' s a = Lens s s a a

    -- | Every 'IndexedLens' is a valid 'Lens' and a valid 'Control.Lens.Traversal.IndexedTraversal'.
    type IndexedLens i s t a b = forall f p. (Indexable i p, Functor f) => p a (f b) -> s -> f t

    -- | @
    -- type 'IndexedLens'' i = 'Simple' ('IndexedLens' i)
    -- @
    type IndexedLens' i s a = IndexedLens i s s a a

    -- | An 'IndexPreservingLens' leaves any index it is composed with alone.
    type IndexPreservingLens s t a b = forall p f. (Conjoined p, Functor f) => p a (f b) -> p s (f t)

    -- | @
    -- type 'IndexPreservingLens'' = 'Simple' 'IndexPreservingLens'
    -- @
    type IndexPreservingLens' s a = IndexPreservingLens s s a a

    ------------------------------------------------------------------------------
    -- Traversals
    ------------------------------------------------------------------------------

    -- | A 'Traversal' can be used directly as a 'Control.Lens.Setter.Setter' or a 'Fold' (but not as a 'Lens') and provides
    -- the ability to both read and update multiple fields, subject to some relatively weak 'Traversal' laws.
    --
    -- These have also been known as multilenses, but they have the signature and spirit of
    --
    -- @
    -- 'Data.Traversable.traverse' :: 'Data.Traversable.Traversable' f => 'Traversal' (f a) (f b) a b
    -- @
    --
    -- and the more evocative name suggests their application.
    --
    -- Most of the time the 'Traversal' you will want to use is just 'Data.Traversable.traverse', but you can also pass any
    -- 'Lens' or 'Iso' as a 'Traversal', and composition of a 'Traversal' (or 'Lens' or 'Iso') with a 'Traversal' (or 'Lens' or 'Iso')
    -- using ('.') forms a valid 'Traversal'.
    --
    -- The laws for a 'Traversal' @t@ follow from the laws for 'Data.Traversable.Traversable' as stated in \"The Essence of the Iterator Pattern\".
    --
    -- @
    -- t 'pure' ≡ 'pure'
    -- 'fmap' (t f) '.' t g ≡ 'Data.Functor.Compose.getCompose' '.' t ('Data.Functor.Compose.Compose' '.' 'fmap' f '.' g)
    -- @
    --
    -- One consequence of this requirement is that a 'Traversal' needs to leave the same number of elements as a
    -- candidate for subsequent 'Traversal' that it started with. Another testament to the strength of these laws
    -- is that the caveat expressed in section 5.5 of the \"Essence of the Iterator Pattern\" about exotic
    -- 'Data.Traversable.Traversable' instances that 'Data.Traversable.traverse' the same entry multiple times was actually already ruled out by the
    -- second law in that same paper!
    type Traversal s t a b = forall f. Applicative f => (a -> f b) -> s -> f t

    -- | @
    -- type 'Traversal'' = 'Simple' 'Traversal'
    -- @
    type Traversal' s a = Traversal s s a a

    type Traversal1 s t a b = forall f. Apply f => (a -> f b) -> s -> f t
    type Traversal1' s a = Traversal1 s s a a

    -- | Every 'IndexedTraversal' is a valid 'Control.Lens.Traversal.Traversal' or
    -- 'Control.Lens.Fold.IndexedFold'.
    --
    -- The 'Indexed' constraint is used to allow an 'IndexedTraversal' to be used
    -- directly as a 'Control.Lens.Traversal.Traversal'.
    --
    -- The 'Control.Lens.Traversal.Traversal' laws are still required to hold.
    --
    -- In addition, the index @i@ should satisfy the requirement that it stays
    -- unchanged even when modifying the value @a@, otherwise traversals like
    -- 'indices' break the 'Traversal' laws.
    type IndexedTraversal i s t a b = forall p f. (Indexable i p, Applicative f) => p a (f b) -> s -> f t

    -- | @
    -- type 'IndexedTraversal'' i = 'Simple' ('IndexedTraversal' i)
    -- @
    type IndexedTraversal' i s a = IndexedTraversal i s s a a

    type IndexedTraversal1 i s t a b = forall p f. (Indexable i p, Apply f) => p a (f b) -> s -> f t
    type IndexedTraversal1' i s a = IndexedTraversal1 i s s a a

    -- | An 'IndexPreservingLens' leaves any index it is composed with alone.
    type IndexPreservingTraversal s t a b = forall p f. (Conjoined p, Applicative f) => p a (f b) -> p s (f t)

    -- | @
    -- type 'IndexPreservingTraversal'' = 'Simple' 'IndexPreservingTraversal'
    -- @
    type IndexPreservingTraversal' s a = IndexPreservingTraversal s s a a

    type IndexPreservingTraversal1 s t a b = forall p f. (Conjoined p, Apply f) => p a (f b) -> p s (f t)
    type IndexPreservingTraversal1' s a = IndexPreservingTraversal1 s s a a

    ------------------------------------------------------------------------------
    -- Setters
    ------------------------------------------------------------------------------

    -- | The only 'LensLike' law that can apply to a 'Setter' @l@ is that
    --
    -- @
    -- 'Control.Lens.Setter.set' l y ('Control.Lens.Setter.set' l x a) ≡ 'Control.Lens.Setter.set' l y a
    -- @
    --
    -- You can't 'Control.Lens.Getter.view' a 'Setter' in general, so the other two laws are irrelevant.
    --
    -- However, two 'Functor' laws apply to a 'Setter':
    --
    -- @
    -- 'Control.Lens.Setter.over' l 'id' ≡ 'id'
    -- 'Control.Lens.Setter.over' l f '.' 'Control.Lens.Setter.over' l g ≡ 'Control.Lens.Setter.over' l (f '.' g)
    -- @
    --
    -- These can be stated more directly:
    --
    -- @
    -- l 'pure' ≡ 'pure'
    -- l f '.' 'untainted' '.' l g ≡ l (f '.' 'untainted' '.' g)
    -- @
    --
    -- You can compose a 'Setter' with a 'Lens' or a 'Traversal' using ('.') from the @Prelude@
    -- and the result is always only a 'Setter' and nothing more.
    --
    -- >>> over traverse f [a,b,c,d]
    -- [f a,f b,f c,f d]
    --
    -- >>> over _1 f (a,b)
    -- (f a,b)
    --
    -- >>> over (traverse._1) f [(a,b),(c,d)]
    -- [(f a,b),(f c,d)]
    --
    -- >>> over both f (a,b)
    -- (f a,f b)
    --
    -- >>> over (traverse.both) f [(a,b),(c,d)]
    -- [(f a,f b),(f c,f d)]
    type Setter s t a b = forall f. Settable f => (a -> f b) -> s -> f t

    -- | A 'Setter'' is just a 'Setter' that doesn't change the types.
    --
    -- These are particularly common when talking about monomorphic containers. /e.g./
    --
    -- @
    -- 'sets' Data.Text.map :: 'Setter'' 'Data.Text.Internal.Text' 'Char'
    -- @
    --
    -- @
    -- type 'Setter'' = 'Setter''
    -- @
    type Setter' s a = Setter s s a a

    -- | Every 'IndexedSetter' is a valid 'Setter'.
    --
    -- The 'Setter' laws are still required to hold.
    type IndexedSetter i s t a b = forall f p.
      (Indexable i p, Settable f) => p a (f b) -> s -> f t

    -- | @
    -- type 'IndexedSetter'' i = 'Simple' ('IndexedSetter' i)
    -- @
    type IndexedSetter' i s a = IndexedSetter i s s a a

    -- | An 'IndexPreservingSetter' can be composed with a 'IndexedSetter', 'IndexedTraversal' or 'IndexedLens'
    -- and leaves the index intact, yielding an 'IndexedSetter'.
    type IndexPreservingSetter s t a b = forall p f. (Conjoined p, Settable f) => p a (f b) -> p s (f t)

    -- | @
    -- type 'IndexedPreservingSetter'' i = 'Simple' 'IndexedPreservingSetter'
    -- @
    type IndexPreservingSetter' s a = IndexPreservingSetter s s a a

    -----------------------------------------------------------------------------
    -- Isomorphisms
    -----------------------------------------------------------------------------

    -- | Isomorphism families can be composed with another 'Lens' using ('.') and 'id'.
    --
    -- Note: Composition with an 'Iso' is index- and measure- preserving.
    type Iso s t a b = forall p f. (Profunctor p, Functor f) => p a (f b) -> p s (f t)

    -- | @
    -- type 'Iso'' = 'Control.Lens.Type.Simple' 'Iso'
    -- @
    type Iso' s a = Iso s s a a

    ------------------------------------------------------------------------------
    -- Review Internals
    ------------------------------------------------------------------------------

    -- | This is a limited form of a 'Prism' that can only be used for 're' operations.
    --
    -- Like with a 'Getter', there are no laws to state for a 'Review'.
    --
    -- You can generate a 'Review' by using 'unto'. You can also use any 'Prism' or 'Iso'
    -- directly as a 'Review'.
    type Review t b = forall p f. (Choice p, Bifunctor p, Settable f) => Optic' p f t b

    -- | If you see this in a signature for a function, the function is expecting a 'Review'
    -- (in practice, this usually means a 'Prism').
    type AReview t b = Optic' Tagged Identity t b

    ------------------------------------------------------------------------------
    -- Prism Internals
    ------------------------------------------------------------------------------

    -- | A 'Prism' @l@ is a 'Traversal' that can also be turned
    -- around with 'Control.Lens.Review.re' to obtain a 'Getter' in the
    -- opposite direction.
    --
    -- There are two laws that a 'Prism' should satisfy:
    --
    -- First, if I 'Control.Lens.Review.re' or 'Control.Lens.Review.review' a value with a 'Prism' and then 'Control.Lens.Fold.preview' or use ('Control.Lens.Fold.^?'), I will get it back:
    --
    -- @
    -- 'Control.Lens.Fold.preview' l ('Control.Lens.Review.review' l b) ≡ 'Just' b
    -- @
    --
    -- Second, if you can extract a value @a@ using a 'Prism' @l@ from a value @s@, then the value @s@ is completely described by @l@ and @a@:
    --
    -- If @'Control.Lens.Fold.preview' l s ≡ 'Just' a@ then @'Control.Lens.Review.review' l a ≡ s@
    --
    -- These two laws imply that the 'Traversal' laws hold for every 'Prism' and that we 'Data.Traversable.traverse' at most 1 element:
    --
    -- @
    -- 'Control.Lens.Fold.lengthOf' l x '<=' 1
    -- @
    --
    -- It may help to think of this as a 'Iso' that can be partial in one direction.
    --
    -- Every 'Prism' is a valid 'Traversal'.
    --
    -- Every 'Iso' is a valid 'Prism'.
    --
    -- For example, you might have a @'Prism'' 'Integer' 'Numeric.Natural.Natural'@ allows you to always
    -- go from a 'Numeric.Natural.Natural' to an 'Integer', and provide you with tools to check if an 'Integer' is
    -- a 'Numeric.Natural.Natural' and/or to edit one if it is.
    --
    --
    -- @
    -- 'nat' :: 'Prism'' 'Integer' 'Numeric.Natural.Natural'
    -- 'nat' = 'Control.Lens.Prism.prism' 'toInteger' '$' \\ i ->
    --    if i '<' 0
    --    then 'Left' i
    --    else 'Right' ('fromInteger' i)
    -- @
    --
    -- Now we can ask if an 'Integer' is a 'Numeric.Natural.Natural'.
    --
    -- >>> 5^?nat
    -- Just 5
    --
    -- >>> (-5)^?nat
    -- Nothing
    --
    -- We can update the ones that are:
    --
    -- >>> (-3,4) & both.nat *~ 2
    -- (-3,8)
    --
    -- And we can then convert from a 'Numeric.Natural.Natural' to an 'Integer'.
    --
    -- >>> 5 ^. re nat -- :: Natural
    -- 5
    --
    -- Similarly we can use a 'Prism' to 'Data.Traversable.traverse' the 'Left' half of an 'Either':
    --
    -- >>> Left "hello" & _Left %~ length
    -- Left 5
    --
    -- or to construct an 'Either':
    --
    -- >>> 5^.re _Left
    -- Left 5
    --
    -- such that if you query it with the 'Prism', you will get your original input back.
    --
    -- >>> 5^.re _Left ^? _Left
    -- Just 5
    --
    -- Another interesting way to think of a 'Prism' is as the categorical dual of a 'Lens'
    -- -- a co-'Lens', so to speak. This is what permits the construction of 'Control.Lens.Prism.outside'.
    --
    -- Note: Composition with a 'Prism' is index-preserving.
    type Prism s t a b = forall p f. (Choice p, Applicative f) => p a (f b) -> p s (f t)

    -- | A 'Simple' 'Prism'.
    type Prism' s a = Prism s s a a

    -------------------------------------------------------------------------------
    -- Equality
    -------------------------------------------------------------------------------

    -- | A witness that @(a ~ s, b ~ t)@.
    --
    -- Note: Composition with an 'Equality' is index-preserving.
    type Equality s t a b = forall p (f :: * -> *). p a (f b) -> p s (f t)

    -- | A 'Simple' 'Equality'.
    type Equality' s a = Equality s s a a

    -- | Composable `asTypeOf`. Useful for constraining excess
    -- polymorphism, @foo . (id :: As Int) . bar@.
    type As a = Equality' a a

    -------------------------------------------------------------------------------
    -- Getters
    -------------------------------------------------------------------------------

    -- | A 'Getter' describes how to retrieve a single value in a way that can be
    -- composed with other 'LensLike' constructions.
    --
    -- Unlike a 'Lens' a 'Getter' is read-only. Since a 'Getter'
    -- cannot be used to write back there are no 'Lens' laws that can be applied to
    -- it. In fact, it is isomorphic to an arbitrary function from @(s -> a)@.
    --
    -- Moreover, a 'Getter' can be used directly as a 'Control.Lens.Fold.Fold',
    -- since it just ignores the 'Applicative'.
    type Getter s a = forall f. (Contravariant f, Functor f) => (a -> f a) -> s -> f s

    -- | Every 'IndexedGetter' is a valid 'Control.Lens.Fold.IndexedFold' and can be used for 'Control.Lens.Getter.Getting' like a 'Getter'.
    type IndexedGetter i s a = forall p f. (Indexable i p, Contravariant f, Functor f) => p a (f a) -> s -> f s

    -- | An 'IndexPreservingGetter' can be used as a 'Getter', but when composed with an 'IndexedTraversal',
    -- 'IndexedFold', or 'IndexedLens' yields an 'IndexedFold', 'IndexedFold' or 'IndexedGetter' respectively.
    type IndexPreservingGetter s a = forall p f. (Conjoined p, Contravariant f, Functor f) => p a (f a) -> p s (f s)

    --------------------------
    -- Folds
    --------------------------

    -- | A 'Fold' describes how to retrieve multiple values in a way that can be composed
    -- with other 'LensLike' constructions.
    --
    -- A @'Fold' s a@ provides a structure with operations very similar to those of the 'Data.Foldable.Foldable'
    -- typeclass, see 'Control.Lens.Fold.foldMapOf' and the other 'Fold' combinators.
    --
    -- By convention, if there exists a 'foo' method that expects a @'Data.Foldable.Foldable' (f a)@, then there should be a
    -- @fooOf@ method that takes a @'Fold' s a@ and a value of type @s@.
    --
    -- A 'Getter' is a legal 'Fold' that just ignores the supplied 'Data.Monoid.Monoid'.
    --
    -- Unlike a 'Control.Lens.Traversal.Traversal' a 'Fold' is read-only. Since a 'Fold' cannot be used to write back
    -- there are no 'Lens' laws that apply.
    type Fold s a = forall f. (Contravariant f, Applicative f) => (a -> f a) -> s -> f s

    -- | Every 'IndexedFold' is a valid 'Control.Lens.Fold.Fold' and can be used for 'Control.Lens.Getter.Getting'.
    type IndexedFold i s a = forall p f.  (Indexable i p, Contravariant f, Applicative f) => p a (f a) -> s -> f s

    -- | An 'IndexPreservingFold' can be used as a 'Fold', but when composed with an 'IndexedTraversal',
    -- 'IndexedFold', or 'IndexedLens' yields an 'IndexedFold' respectively.
    type IndexPreservingFold s a = forall p f. (Conjoined p, Contravariant f, Applicative f) => p a (f a) -> p s (f s)

    -- | A relevant Fold (aka 'Fold1') has one or more targets.
    type Fold1 s a = forall f. (Contravariant f, Apply f) => (a -> f a) -> s -> f s
    type IndexedFold1 i s a = forall p f.  (Indexable i p, Contravariant f, Apply f) => p a (f a) -> s -> f s
    type IndexPreservingFold1 s a = forall p f. (Conjoined p, Contravariant f, Apply f) => p a (f a) -> p s (f s)

    -------------------------------------------------------------------------------
    -- Simple Overloading
    -------------------------------------------------------------------------------

    -- | A 'Simple' 'Lens', 'Simple' 'Traversal', ... can
    -- be used instead of a 'Lens','Traversal', ...
    -- whenever the type variables don't change upon setting a value.
    --
    -- @
    -- 'Data.Complex.Lens._imagPart' :: 'Simple' 'Lens' ('Data.Complex.Complex' a) a
    -- 'Control.Lens.Traversal.traversed' :: 'Simple' ('IndexedTraversal' 'Int') [a] a
    -- @
    --
    -- Note: To use this alias in your own code with @'LensLike' f@ or
    -- 'Setter', you may have to turn on @LiberalTypeSynonyms@.
    --
    -- This is commonly abbreviated as a \"prime\" marker, /e.g./ 'Lens'' = 'Simple' 'Lens'.
    type Simple f s a = f s s a a

    -------------------------------------------------------------------------------
    -- Optics
    -------------------------------------------------------------------------------

    -- | A valid 'Optic' @l@ should satisfy the laws:
    --
    -- @
    -- l 'pure' ≡ 'pure'
    -- l ('Procompose' f g) = 'Procompose' (l f) (l g)
    -- @
    --
    -- This gives rise to the laws for 'Equality', 'Iso', 'Prism', 'Lens',
    -- 'Traversal', 'Traversal1', 'Setter', 'Fold', 'Fold1', and 'Getter' as well
    -- along with their index-preserving variants.
    --
    -- @
    -- type 'LensLike' f s t a b = 'Optic' (->) f s t a b
    -- @
    type Optic p f s t a b = p a (f b) -> p s (f t)

    -- | @
    -- type 'Optic'' p f s a = 'Simple' ('Optic' p f) s a
    -- @
    type Optic' p f s a = Optic p f s s a a

    -- | @
    -- type 'LensLike' f s t a b = 'Optical' (->) (->) f s t a b
    -- @
    --
    -- @
    -- type 'Over' p f s t a b = 'Optical' p (->) f s t a b
    -- @
    --
    -- @
    -- type 'Optic' p f s t a b = 'Optical' p p f s t a b
    -- @
    type Optical p q f s t a b = p a (f b) -> q s (f t)

    -- | @
    -- type 'Optical'' p q f s a = 'Simple' ('Optical' p q f) s a
    -- @
    type Optical' p q f s a = Optical p q f s s a a

    -- | Many combinators that accept a 'Lens' can also accept a
    -- 'Traversal' in limited situations.
    --
    -- They do so by specializing the type of 'Functor' that they require of the
    -- caller.
    --
    -- If a function accepts a @'LensLike' f s t a b@ for some 'Functor' @f@,
    -- then they may be passed a 'Lens'.
    --
    -- Further, if @f@ is an 'Applicative', they may also be passed a
    -- 'Traversal'.
    type LensLike f s t a b = (a -> f b) -> s -> f t

    -- | @
    -- type 'LensLike'' f = 'Simple' ('LensLike' f)
    -- @
    type LensLike' f s a = LensLike f s s a a

    -- | Convenient alias for constructing indexed lenses and their ilk.
    type IndexedLensLike i f s t a b = forall p. Indexable i p => p a (f b) -> s -> f t

    -- | Convenient alias for constructing simple indexed lenses and their ilk.
    type IndexedLensLike' i f s a = IndexedLensLike i f s s a a

    -- | This is a convenient alias for use when you need to consume either indexed or non-indexed lens-likes based on context.
    type Over p f s t a b = p a (f b) -> s -> f t

    -- | This is a convenient alias for use when you need to consume either indexed or non-indexed lens-likes based on context.
    --
    -- @
    -- type 'Over'' p f = 'Simple' ('Over' p f)
    -- @
    type Over' p f s a = Over p f s s a a