lens-5.2.3: Lenses, Folds and Traversals

Copyright | (C) 2012-16 Edward Kmett
License | BSD-style (see the file LICENSE)
Maintainer | Edward Kmett <ekmett@gmail.com>
Stability | provisional
Portability | Rank2Types
Safe Haskell | Trustworthy
Language | Haskell2010

Control.Lens.Equality

Contents

- Type Equality
- The Trivial Equality
- `Iso`-like functions
- Implementation Details

Description

Synopsis

- 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)
- type Equality' (s :: k2) (a :: k2) = Equality s s a a
- type AnEquality (s :: k) (t :: k1) (a :: k) (b :: k2) = Identical a (Proxy b) a (Proxy b) -> Identical a (Proxy b) s (Proxy t)
- type AnEquality' (s :: k) (a :: k) = AnEquality s s a a
- data (a :: k) :~: (b :: k) where
  - Refl :: forall {k} (a :: k). a :~: a
- runEq :: forall {k1} {k2} (s :: k1) (t :: k2) (a :: k1) (b :: k2). AnEquality s t a b -> Identical s t a b
- substEq :: forall {k1} {k2} (s :: k1) (t :: k2) (a :: k1) (b :: k2) r. AnEquality s t a b -> ((s ~ a, t ~ b) => r) -> r
- mapEq :: forall k1 k2 (s :: k1) (t :: k2) (a :: k1) (b :: k2) f. AnEquality s t a b -> f s -> f a
- fromEq :: forall {k2} {k1} (s :: k2) (t :: k1) (a :: k2) (b :: k1). AnEquality s t a b -> Equality b a t s
- simply :: forall {k} {k1} p (f :: k -> k1) (s :: k) (a :: k) r. (Optic' p f s a -> r) -> Optic' p f s a -> r
- simple :: forall {k2} (a :: k2) k3 p (f :: k2 -> k3). p a (f a) -> p a (f a)
- equality :: forall {k1} {k2} (s :: k1) (a :: k1) (b :: k2) (t :: k2). (s :~: a) -> (b :~: t) -> Equality s t a b
- equality' :: forall {k2} (a :: k2) (b :: k2). (a :~: b) -> Equality' a b
- withEquality :: forall {k1} {k2} (s :: k1) (t :: k2) (a :: k1) (b :: k2) r. AnEquality s t a b -> ((s :~: a) -> (b :~: t) -> r) -> r
- underEquality :: forall {k1} {k2} (s :: k1) (t :: k2) (a :: k1) (b :: k2) p. AnEquality s t a b -> p t s -> p b a
- overEquality :: forall {k1} {k2} (s :: k1) (t :: k2) (a :: k1) (b :: k2) p. AnEquality s t a b -> p a b -> p s t
- fromLeibniz :: forall {k1} {k2} (a :: k1) (b :: k2) (s :: k1) (t :: k2). (Identical a b a b -> Identical a b s t) -> Equality s t a b
- fromLeibniz' :: forall {k2} (s :: k2) (a :: k2). ((s :~: s) -> s :~: a) -> Equality' s a
- cloneEquality :: forall {k1} {k2} (s :: k1) (t :: k2) (a :: k1) (b :: k2). AnEquality s t a b -> Equality s t a b
- data Identical (a :: k) (b :: k1) (s :: k) (t :: k1) where
  - Identical :: forall {k} {k1} (a :: k) (b :: k1). Identical a b a b

# Type Equality

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 witness that `(a ~ s, b ~ t)`.

Note: Composition with an `Equality` is index-preserving.

type Equality' (s :: k2) (a :: k2) = Equality s s a a #

A `Simple` `Equality`.

type AnEquality (s :: k) (t :: k1) (a :: k) (b :: k2) = Identical a (Proxy b) a (Proxy b) -> Identical a (Proxy b) s (Proxy t) #

When you see this as an argument to a function, it expects an `Equality`.

type AnEquality' (s :: k) (a :: k) = AnEquality s s a a #

A `Simple` `AnEquality`.

data (a :: k) :~: (b :: k) where infix 4 #

Propositional equality. If `a :~: b` is inhabited by some terminating value, then the type `a` is the same as the type `b`. To use this equality in practice, pattern-match on the `a :~: b` to get out the `Refl` constructor; in the body of the pattern-match, the compiler knows that `a ~ b`.

Since: base-4.7.0.0

Constructors

Refl :: forall {k} (a :: k). a :~: a |

#### Instances

Instances details

Category ((:~:) :: k -> k -> Type) | Since: base-4.7.0.0
Instance details Defined in Control.Category Methods id :: forall (a :: k). a :~: a # (.) :: forall (b :: k) (c :: k) (a :: k). (b :~: c) -> (a :~: b) -> a :~: c #
Groupoid ((:~:) :: k -> k -> Type) |
Instance details Defined in Data.Groupoid Methods inv :: forall (a :: k) (b :: k). (a :~: b) -> b :~: a #
Semigroupoid ((:~:) :: k -> k -> Type) |
Instance details Defined in Data.Semigroupoid Methods o :: forall (j :: k) (k1 :: k) (i :: k). (j :~: k1) -> (i :~: j) -> i :~: k1 #
TestCoercion ((:~:) a :: k -> Type) | Since: base-4.7.0.0
Instance details Defined in Data.Type.Coercion Methods testCoercion :: forall (a0 :: k) (b :: k). (a :~: a0) -> (a :~: b) -> Maybe (Coercion a0 b) #
TestEquality ((:~:) a :: k -> Type) | Since: base-4.7.0.0
Instance details Defined in Data.Type.Equality Methods testEquality :: forall (a0 :: k) (b :: k). (a :~: a0) -> (a :~: b) -> Maybe (a0 :~: b) #
NFData2 ((:~:) :: Type -> Type -> Type) | Since: deepseq-1.4.3.0
Instance details Defined in Control.DeepSeq Methods liftRnf2 :: (a -> ()) -> (b -> ()) -> (a :~: b) -> () #
NFData1 ((:~:) a) | Since: deepseq-1.4.3.0
Instance details Defined in Control.DeepSeq Methods liftRnf :: (a0 -> ()) -> (a :~: a0) -> () #
(a ~ b, Data a) => Data (a :~: b) | Since: base-4.7.0.0

Instance details

Defined in Data.Data

Methods

gfoldl :: (forall d b0. Data d => c (d -> b0) -> d -> c b0) -> (forall g. g -> c g) -> (a :~: b) -> c (a :~: b) #

gunfold :: (forall b0 r. Data b0 => c (b0 -> r) -> c r) -> (forall r. r -> c r) -> Constr -> c (a :~: b) #

toConstr :: (a :~: b) -> Constr #

dataTypeOf :: (a :~: b) -> DataType #

dataCast1 :: Typeable t => (forall d. Data d => c (t d)) -> Maybe (c (a :~: b)) #

dataCast2 :: Typeable t => (forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c (a :~: b)) #

gmapT :: (forall b0. Data b0 => b0 -> b0) -> (a :~: b) -> a :~: b #

gmapQl :: (r -> r' -> r) -> r -> (forall d. Data d => d -> r') -> (a :~: b) -> r #

gmapQr :: forall r r'. (r' -> r -> r) -> r -> (forall d. Data d => d -> r') -> (a :~: b) -> r #

gmapQ :: (forall d. Data d => d -> u) -> (a :~: b) -> [u] #

gmapQi :: Int -> (forall d. Data d => d -> u) -> (a :~: b) -> u #

gmapM :: Monad m => (forall d. Data d => d -> m d) -> (a :~: b) -> m (a :~: b) #

gmapMp :: MonadPlus m => (forall d. Data d => d -> m d) -> (a :~: b) -> m (a :~: b) #

gmapMo :: MonadPlus m => (forall d. Data d => d -> m d) -> (a :~: b) -> m (a :~: b) #

a ~ b => Bounded (a :~: b) | Since: base-4.7.0.0
Instance details Defined in Data.Type.Equality Methods minBound :: a :~: b # maxBound :: a :~: b #
a ~ b => Enum (a :~: b) | Since: base-4.7.0.0
Instance details Defined in Data.Type.Equality Methods succ :: (a :~: b) -> a :~: b # pred :: (a :~: b) -> a :~: b # toEnum :: Int -> a :~: b # fromEnum :: (a :~: b) -> Int # enumFrom :: (a :~: b) -> [a :~: b] # enumFromThen :: (a :~: b) -> (a :~: b) -> [a :~: b] # enumFromTo :: (a :~: b) -> (a :~: b) -> [a :~: b] # enumFromThenTo :: (a :~: b) -> (a :~: b) -> (a :~: b) -> [a :~: b] #
a ~ b => Read (a :~: b) | Since: base-4.7.0.0
Instance details Defined in Data.Type.Equality Methods readsPrec :: Int -> ReadS (a :~: b) # readList :: ReadS [a :~: b] # readPrec :: ReadPrec (a :~: b) # readListPrec :: ReadPrec [a :~: b] #
Show (a :~: b) | Since: base-4.7.0.0
Instance details Defined in Data.Type.Equality Methods showsPrec :: Int -> (a :~: b) -> ShowS # show :: (a :~: b) -> String # showList :: [a :~: b] -> ShowS #
NFData (a :~: b) | Since: deepseq-1.4.3.0
Instance details Defined in Control.DeepSeq Methods rnf :: (a :~: b) -> () #
Eq (a :~: b) | Since: base-4.7.0.0
Instance details Defined in Data.Type.Equality Methods (==) :: (a :~: b) -> (a :~: b) -> Bool # (/=) :: (a :~: b) -> (a :~: b) -> Bool #
Ord (a :~: b) | Since: base-4.7.0.0
Instance details Defined in Data.Type.Equality Methods compare :: (a :~: b) -> (a :~: b) -> Ordering # (<) :: (a :~: b) -> (a :~: b) -> Bool # (<=) :: (a :~: b) -> (a :~: b) -> Bool # (>) :: (a :~: b) -> (a :~: b) -> Bool # (>=) :: (a :~: b) -> (a :~: b) -> Bool # max :: (a :~: b) -> (a :~: b) -> a :~: b # min :: (a :~: b) -> (a :~: b) -> a :~: b #

runEq :: forall {k1} {k2} (s :: k1) (t :: k2) (a :: k1) (b :: k2). AnEquality s t a b -> Identical s t a b #

Extract a witness of type `Equality`.

substEq :: forall {k1} {k2} (s :: k1) (t :: k2) (a :: k1) (b :: k2) r. AnEquality s t a b -> ((s ~ a, t ~ b) => r) -> r #

Substituting types with `Equality`.

mapEq :: forall k1 k2 (s :: k1) (t :: k2) (a :: k1) (b :: k2) f. AnEquality s t a b -> f s -> f a #

We can use `Equality` to do substitution into anything.

fromEq :: forall {k2} {k1} (s :: k2) (t :: k1) (a :: k2) (b :: k1). AnEquality s t a b -> Equality b a t s #

`Equality` is symmetric.

simply :: forall {k} {k1} p (f :: k -> k1) (s :: k) (a :: k) r. (Optic' p f s a -> r) -> Optic' p f s a -> r #

This is an adverb that can be used to modify many other `Lens` combinators to make them require simple lenses, simple traversals, simple prisms or simple isos as input.

# The Trivial Equality

simple :: forall {k2} (a :: k2) k3 p (f :: k2 -> k3). p a (f a) -> p a (f a) #

Composition with this isomorphism is occasionally useful when your `Lens`, `Traversal` or `Iso` has a constraint on an unused argument to force that argument to agree with the type of a used argument and avoid `ScopedTypeVariables` or other ugliness.

# `Iso`-like functions

equality :: forall {k1} {k2} (s :: k1) (a :: k1) (b :: k2) (t :: k2). (s :~: a) -> (b :~: t) -> Equality s t a b #

Construct an `Equality` from explicit equality evidence.

equality' :: forall {k2} (a :: k2) (b :: k2). (a :~: b) -> Equality' a b #

A `Simple` version of `equality`

withEquality :: forall {k1} {k2} (s :: k1) (t :: k2) (a :: k1) (b :: k2) r. AnEquality s t a b -> ((s :~: a) -> (b :~: t) -> r) -> r #

A version of `substEq` that provides explicit, rather than implicit, equality evidence.

underEquality :: forall {k1} {k2} (s :: k1) (t :: k2) (a :: k1) (b :: k2) p. AnEquality s t a b -> p t s -> p b a #

The opposite of working `overEquality` is working `underEquality`.

overEquality :: forall {k1} {k2} (s :: k1) (t :: k2) (a :: k1) (b :: k2) p. AnEquality s t a b -> p a b -> p s t #

Recover a "profunctor lens" form of equality. Reverses `fromLeibniz`.

fromLeibniz :: forall {k1} {k2} (a :: k1) (b :: k2) (s :: k1) (t :: k2). (Identical a b a b -> Identical a b s t) -> Equality s t a b #

Convert a "profunctor lens" form of equality to an equality. Reverses `overEquality`.

The type should be understood as

    fromLeibniz :: (forall p. p a b -> p s t) -> Equality s t a b

fromLeibniz' :: forall {k2} (s :: k2) (a :: k2). ((s :~: s) -> s :~: a) -> Equality' s a #

Convert Leibniz equality to equality. Reverses `mapEq` in `Simple` cases.

The type should be understood as

    fromLeibniz' :: (forall f. f s -> f a) -> Equality' s a

cloneEquality :: forall {k1} {k2} (s :: k1) (t :: k2) (a :: k1) (b :: k2). AnEquality s t a b -> Equality s t a b #

# Implementation Details

data Identical (a :: k) (b :: k1) (s :: k) (t :: k1) where #

Provides witness that `(s ~ a, b ~ t)` holds.

Constructors

Identical :: forall {k} {k1} (a :: k) (b :: k1). Identical a b a b |