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.

Instances should satisfy the following 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.

Minimal complete definition

arr, (first | (***))

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') infixr 3

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') infixr 3

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 (->)

Methods

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

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

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

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

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

Arrow Isomorphism

Methods

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

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

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

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

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

Arrow View

Methods

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

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

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

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

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

Arrow Matcher

Methods

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

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

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

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

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

Arrow Trans

Methods

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

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

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

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

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

Monad m => Arrow (Kleisli m)

Methods

arr :: (b -> c) -> Kleisli m b c

first :: Kleisli m b c -> Kleisli m (b, d) (c, d)

second :: Kleisli m b c -> Kleisli m (d, b) (d, c)

(***) :: Kleisli m b c -> Kleisli m b' c' -> Kleisli m (b, b') (c, c')

(&&&) :: Kleisli m b c -> Kleisli m b c' -> Kleisli m b (c, c')

Arrow (Encoder a)

Methods

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

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

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

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

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

class Arrow a => ArrowChoice a where

Choice, for arrows that support it. This class underlies the if and case constructs in arrow notation.

Instances should satisfy the following 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.

Minimal complete definition

left | (+++)

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') infixr 2

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 infixr 2

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 (->)

Methods

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

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

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

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

ArrowChoice Isomorphism

Methods

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

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

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

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

ArrowChoice View

Methods

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

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

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

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

ArrowChoice Matcher

Methods

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

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

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

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

ArrowChoice Trans

Methods

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

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

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

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

Monad m => ArrowChoice (Kleisli m)

Methods

left :: Kleisli m b c -> Kleisli m (Either b d) (Either c d)

right :: Kleisli m b c -> Kleisli m (Either d b) (Either d c)

(+++) :: Kleisli m b c -> Kleisli m b' c' -> Kleisli m (Either b b') (Either c c')

(|||) :: Kleisli m b d -> Kleisli m c d -> Kleisli m (Either b c) d

class Arrow a => ArrowZero a where

Minimal complete definition

zeroArrow

Methods

zeroArrow :: a b c

Instances

ArrowZero Matcher

Methods

zeroArrow :: Matcher b c

ArrowZero Trans

Methods

zeroArrow :: Trans b c

MonadPlus m => ArrowZero (Kleisli m)

Methods

zeroArrow :: Kleisli m b c

class ArrowZero a => ArrowPlus a where

A monoid on arrows.

Minimal complete definition

(<+>)

Methods

(<+>) :: a b c -> a b c -> a b c infixr 5

An associative operation with identity zeroArrow.

Instances

ArrowPlus Matcher

Methods

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

ArrowPlus Trans

Methods

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

MonadPlus m => ArrowPlus (Kleisli m)

Methods

(<+>) :: Kleisli m b c -> Kleisli m b c -> Kleisli m b c

(>>>) :: Category k cat => cat a b -> cat b c -> cat a c infixr 1

Left-to-right composition

(<<<) :: Category k cat => cat b c -> cat a b -> cat a c infixr 1

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

Methods

match :: Isomorphism a b -> a -> Maybe b

matcher :: Isomorphism a b -> Matcher a b

IsMatcher View

Methods

match :: View a b -> a -> Maybe b

matcher :: View a b -> Matcher a b

IsMatcher Matcher

Methods

match :: Matcher a b -> a -> Maybe b

matcher :: Matcher a b -> Matcher a b

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

Methods

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

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

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

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

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

ArrowZero Matcher

Methods

zeroArrow :: Matcher b c

ArrowPlus Matcher

Methods

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

ArrowChoice Matcher

Methods

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

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

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

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

IsMatcher Matcher

Methods

match :: Matcher a b -> a -> Maybe b

matcher :: Matcher a b -> Matcher a b

Category * Matcher

Methods

id :: cat a a

(.) :: cat b c -> cat a b -> cat a c

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

Methods

build :: Isomorphism a b -> b -> a

toView :: Isomorphism a b -> View a b

IsView View

Methods

build :: View a b -> b -> a

toView :: View a b -> View a b

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

Methods

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

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

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

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

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

ArrowChoice View

Methods

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

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

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

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

BiArrow View

Methods

(<->) :: (a -> b) -> (b -> a) -> View a b

(!->) :: (a -> b) -> View a b

(<-!) :: (b -> a) -> View a b

IsView View

Methods

build :: View a b -> b -> a

toView :: View a b -> View a b

IsMatcher View

Methods

match :: View a b -> a -> Maybe b

matcher :: View a b -> Matcher a b

Category * View

Methods

id :: cat a a

(.) :: cat b c -> cat a b -> cat a c

HasId (View a b)

Methods

getId :: View a b -> Id

changeId :: (Id -> Id) -> View a b -> View a b

Identify (View a b)

Methods

(@>) :: IsId n => n -> View a b -> 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

Methods

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

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

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

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

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

ArrowChoice Isomorphism

Methods

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

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

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

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

BiArrow Isomorphism

Methods

(<->) :: (a -> b) -> (b -> a) -> Isomorphism a b

(!->) :: (a -> b) -> Isomorphism a b

(<-!) :: (b -> a) -> Isomorphism a b

IsView Isomorphism

Methods

build :: Isomorphism a b -> b -> a

toView :: Isomorphism a b -> View a b

IsMatcher Isomorphism

Methods

match :: Isomorphism a b -> a -> Maybe b

matcher :: Isomorphism a b -> Matcher a b

Category * Isomorphism

Methods

id :: cat a a

(.) :: cat b c -> cat a b -> cat a c

HasId (Isomorphism a b)

Methods

getId :: Isomorphism a b -> Id

changeId :: (Id -> Id) -> Isomorphism a b -> Isomorphism a b

Identify (Isomorphism a b)

Methods

(@>) :: IsId n => n -> Isomorphism a b -> 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

Minimal complete definition

liftViewIn

Methods

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

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

Instances

LiftView Recognizer

Methods

liftView :: View a b -> Recognizer b -> Recognizer a

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

LiftView Rule

Methods

liftView :: View a b -> Rule b -> Rule a

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

LiftView Dynamic

Methods

liftView :: View a b -> Dynamic b -> Dynamic a

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

LiftView Leaf

Methods

liftView :: View a b -> Leaf b -> Leaf a

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

LiftView LabeledStrategy

Methods

liftView :: View a b -> LabeledStrategy b -> LabeledStrategy a

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

LiftView Strategy

Methods

liftView :: View a b -> Strategy b -> Strategy a

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

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

Methods

getId :: ViewPackage -> Id

changeId :: (Id -> Id) -> ViewPackage -> 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