Ideas.Common.View

This module defines views on data-types, as described in Canonical Forms in Interactive Exercise Assistants

Documentation

class Category a => Arrow a where

The basic arrow class.

Minimal complete definition: arr and first, satisfying the laws

• arr id = id

• arr (f >>> g) = arr f >>> arr g

• first (arr f) = arr (first f)

• first (f >>> g) = first f >>> first g

• first f >>> arr fst = arr fst >>> f

• first f >>> arr (id *** g) = arr (id *** g) >>> first f

• first (first f) >>> arr assoc = arr assoc >>> first f

where

 assoc ((a,b),c) = (a,(b,c))

The other combinators have sensible default definitions, which may be overridden for efficiency.

Methods

arr :: (b -> c) -> a b c

Lift a function to an arrow.

first :: a b c -> a (b, d) (c, d)

Send the first component of the input through the argument arrow, and copy the rest unchanged to the output.

second :: a b c -> a (d, b) (d, c)

A mirror image of first.

The default definition may be overridden with a more efficient version if desired.

(***) :: a b c -> a b' c' -> a (b, b') (c, c')

Split the input between the two argument arrows and combine their output. Note that this is in general not a functor.

The default definition may be overridden with a more efficient version if desired.

(&&&) :: a b c -> a b c' -> a b (c, c')

Fanout: send the input to both argument arrows and combine their output.

The default definition may be overridden with a more efficient version if desired.

Instances

Arrow (->)

Arrow Isomorphism

Arrow View

Arrow Matcher

Arrow Trans

Monad m => Arrow (Kleisli m)

Arrow (EncoderState st)

class Arrow a => ArrowChoice a where

Choice, for arrows that support it. This class underlies the if and case constructs in arrow notation. Minimal complete definition: left, satisfying the laws

• left (arr f) = arr (left f)

• left (f >>> g) = left f >>> left g

• f >>> arr Left = arr Left >>> left f

• left f >>> arr (id +++ g) = arr (id +++ g) >>> left f

• left (left f) >>> arr assocsum = arr assocsum >>> left f

where

 assocsum (Left (Left x)) = Left x
 assocsum (Left (Right y)) = Right (Left y)
 assocsum (Right z) = Right (Right z)

The other combinators have sensible default definitions, which may be overridden for efficiency.

Methods

left :: a b c -> a (Either b d) (Either c d)

Feed marked inputs through the argument arrow, passing the rest through unchanged to the output.

right :: a b c -> a (Either d b) (Either d c)

A mirror image of left.

The default definition may be overridden with a more efficient version if desired.

(+++) :: a b c -> a b' c' -> a (Either b b') (Either c c')

Split the input between the two argument arrows, retagging and merging their outputs. Note that this is in general not a functor.

The default definition may be overridden with a more efficient version if desired.

(|||) :: a b d -> a c d -> a (Either b c) d

Fanin: Split the input between the two argument arrows and merge their outputs.

The default definition may be overridden with a more efficient version if desired.

Instances

ArrowChoice (->)

ArrowChoice Isomorphism

ArrowChoice View

ArrowChoice Matcher

ArrowChoice Trans

Monad m => ArrowChoice (Kleisli m)

ArrowChoice (EncoderState st)

class Arrow a => ArrowZero a where

Methods

zeroArrow :: a b c

Instances

ArrowZero Matcher

ArrowZero Trans

MonadPlus m => ArrowZero (Kleisli m)

ArrowZero (EncoderState st)

class ArrowZero a => ArrowPlus a where

A monoid on arrows.

Methods

(<+>) :: a b c -> a b c -> a b c

An associative operation with identity zeroArrow.

Instances

ArrowPlus Matcher

ArrowPlus Trans

MonadPlus m => ArrowPlus (Kleisli m)

ArrowPlus (EncoderState st)

(>>>) :: Category cat => cat a b -> cat b c -> cat a c

Left-to-right composition

(<<<) :: Category cat => cat b c -> cat a b -> cat a c

Right-to-left composition

IsMatch type class

class IsMatcher f where

Methods

match :: f a b -> a -> Maybe b

matcher :: f a b -> Matcher a b

Instances

IsMatcher Isomorphism

IsMatcher View

IsMatcher Matcher

matchM :: (Monad m, IsMatcher f) => f a b -> a -> m b

generalized monadic variant of match

belongsTo :: IsMatcher f => a -> f a b -> Bool

viewEquivalent :: (IsMatcher f, Eq b) => f a b -> a -> a -> Bool

viewEquivalentWith :: IsMatcher f => (b -> b -> Bool) -> f a b -> a -> a -> Bool

data Matcher a b

Instances

Arrow Matcher

ArrowZero Matcher

ArrowPlus Matcher

ArrowChoice Matcher

Category Matcher

IsMatcher Matcher

makeMatcher :: (a -> Maybe b) -> Matcher a b

IsView type class

class IsMatcher f => IsView f where

Minimal complete definition: toView or both match and build.

Methods

build :: f a b -> b -> a

toView :: f a b -> View a b

Instances

IsView Isomorphism

IsView View

simplify :: IsView f => f a b -> a -> a

simplifyWith :: IsView f => (b -> b) -> f a b -> a -> a

simplifyWithM :: IsView f => (b -> Maybe b) -> f a b -> a -> a

canonical :: IsView f => f a b -> a -> Maybe a

canonicalWith :: IsView f => (b -> b) -> f a b -> a -> Maybe a

canonicalWithM :: IsView f => (b -> Maybe b) -> f a b -> a -> Maybe a

isCanonical :: (IsView f, Eq a) => f a b -> a -> Bool

isCanonicalWith :: IsView f => (a -> a -> Bool) -> f a b -> a -> Bool

Views

data View a b

Instances

Arrow View

ArrowChoice View

Category View

BiArrow View

IsView View

IsMatcher View

Identify (View a b)

HasId (View a b)

identity :: Category f => f a a

makeView :: (a -> Maybe b) -> (b -> a) -> View a b

matcherView :: Matcher a b -> (b -> a) -> View a b

Isomorphisms

data Isomorphism a b

Instances

Arrow Isomorphism

ArrowChoice Isomorphism

Category Isomorphism

BiArrow Isomorphism

IsView Isomorphism

IsMatcher Isomorphism

Identify (Isomorphism a b)

HasId (Isomorphism a b)

from :: Isomorphism a b -> a -> b

to :: Isomorphism a b -> b -> a

inverse :: Isomorphism a b -> Isomorphism b a

Lifting with views

class LiftView f where

Methods

liftView :: View a b -> f b -> f a

liftViewIn :: View a (b, c) -> f b -> f a

Instances

LiftView Recognizer

LiftView Rule

LiftView LabeledStrategy

LiftView Strategy

Some combinators

swapView :: Isomorphism (a, b) (b, a)

listView :: View a b -> View [a] [b]

Specialized version of traverseView

traverseView :: Traversable f => View a b -> View (f a) (f b)

($<) :: Traversable f => View a (f b) -> View b c -> View a (f c)

Packaging a view

data ViewPackage where

Constructors

ViewPackage :: (Show a, Show b, Eq a) => (String -> Maybe a) -> View a b -> ViewPackage

Instances

HasId ViewPackage

Properties on views

propIdempotence :: (Show a, Eq a) => Gen a -> View a b -> Property

propSoundness :: Show a => (a -> a -> Bool) -> Gen a -> View a c -> Property

propNormalForm :: (Show a, Eq a) => Gen a -> View a b -> Property