content: {-# LANGUAGE CPP #-}
  {-# LANGUAGE GADTs #-}
  {-# LANGUAGE Rank2Types #-}
  {-# LANGUAGE FlexibleContexts #-}
  {-# LANGUAGE FlexibleInstances #-}
  {-# LANGUAGE ScopedTypeVariables #-}
  {-# LANGUAGE MultiParamTypeClasses #-}
  {-# LANGUAGE Trustworthy #-}
  
  #if __GLASGOW_HASKELL__ >= 806
  {-# LANGUAGE PolyKinds #-}
  #else
  {-# LANGUAGE TypeInType #-}
  #endif
  -------------------------------------------------------------------------------
  -- |
  -- Module      :  Control.Lens.Type
  -- Copyright   :  (C) 2012-16 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 Prelude ()
  
  import Control.Lens.Internal.Prelude
  import Control.Lens.Internal.Setter
  import Control.Lens.Internal.Indexed
  import Data.Bifunctor
  import Data.Functor.Apply
  import Data.Kind
  
  -- $setup
  -- >>> :set -XNoOverloadedStrings
  -- >>> import Control.Lens
  -- >>> import Debug.SimpleReflect.Expr
  -- >>> import Debug.SimpleReflect.Vars as Vars hiding (f,g,h)
  -- >>> 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
  
  -- | A 'Traversal' which targets at least one element.
  --
  -- Note that since 'Apply' is not a superclass of 'Applicative', a 'Traversal1'
  -- cannot always be used in place of a 'Traversal'. In such circumstances
  -- 'Control.Lens.Traversal.cloneTraversal' will convert a 'Traversal1' into a 'Traversal'.
  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 'IndexPreservingTraversal' 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'' = 'Simple' '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'.
  --
  -- Since every 'Iso' is both a valid 'Lens' and a valid 'Prism', the laws for those types
  -- imply the following laws for an 'Iso' 'f':
  --
  -- @
  -- f '.' 'Control.Lens.Iso.from' f ≡ 'id'
  -- 'Control.Lens.Iso.from' f '.' f ≡ '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 three 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@:
  --
  -- @
  -- 'Control.Lens.Fold.preview' l s ≡ 'Just' a ⟹ 'Control.Lens.Review.review' l a ≡ s
  -- @
  --
  -- Third, if you get non-match @t@, you can convert it result back to @s@:
  --
  -- @
  -- 'Control.Lens.Combinators.matching' l s ≡ 'Left' t ⟹ 'Control.Lens.Combinators.matching' l t ≡ 'Left' s
  -- @
  --
  -- The first 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 an '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 :: k1) (t :: k2) (a :: k1) (b :: k2) = forall k3 (p :: k1 -> k3 -> Type) (f :: k2 -> k3) .
      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
  