Data.Constraint

ConstraintKinds made type classes into types of a new kind, Constraint.

Eq :: * -> Constraint
Ord :: * -> Constraint
Monad :: (* -> *) -> Constraint

The need for this extension was first publicized in the paper

Scrap your boilerplate with class: extensible generic functions

by Ralf Lämmel and Simon Peyton Jones in 2005, which shoehorned all the things they needed into a custom Sat typeclass.

With ConstraintKinds we can put into code a lot of tools for manipulating these new types without such awkward workarounds.

The Kind of Constraints

data Constraint :: *

The kind of constraints, like Show a

Instances

Category Constraint (:-)

Possible since GHC 7.8, when Category was made polykinded.

Methods

id :: cat a a

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

(~) Constraint p c => InstV Constraint p c

Associated Types

type ForallV' p (c :: p) :: Constraint

Methods

instV :: ForallV p c :- c

(~) Constraint (p a) c => InstV (k -> Constraint) p c

Associated Types

type ForallV' p (c :: p) :: Constraint

Methods

instV :: ForallV p c :- c

Dictionary

data Dict :: Constraint -> * where

Values of type Dict p capture a dictionary for a constraint of type p.

e.g.

Dict :: Dict (Eq Int)

captures a dictionary that proves we have an:

instance Eq 'Int

Pattern matching on the Dict constructor will bring this instance into scope.

Constructors

Dict :: a => Dict a

Instances

a :=> (Monoid (Dict a))

Methods

ins :: a :- Monoid (Dict a)

a :=> (Read (Dict a))

Methods

ins :: a :- Read (Dict a)

a :=> (Bounded (Dict a))

Methods

ins :: a :- Bounded (Dict a)

a :=> (Enum (Dict a))

Methods

ins :: a :- Enum (Dict a)

() :=> (Eq (Dict a))

Methods

ins :: () :- Eq (Dict a)

() :=> (Ord (Dict a))

Methods

ins :: () :- Ord (Dict a)

() :=> (Show (Dict a))

Methods

ins :: () :- Show (Dict a)

a => Bounded (Dict a)

Methods

minBound :: Dict a

maxBound :: Dict a

a => Enum (Dict a)

Methods

succ :: Dict a -> Dict a

pred :: Dict a -> Dict a

toEnum :: Int -> Dict a

fromEnum :: Dict a -> Int

enumFrom :: Dict a -> [Dict a]

enumFromThen :: Dict a -> Dict a -> [Dict a]

enumFromTo :: Dict a -> Dict a -> [Dict a]

enumFromThenTo :: Dict a -> Dict a -> Dict a -> [Dict a]

Eq (Dict a)

Methods

(==) :: Dict a -> Dict a -> Bool

(/=) :: Dict a -> Dict a -> Bool

(Typeable Constraint p, p) => Data (Dict p)

Methods

gfoldl :: (forall d b. Data d => c (d -> b) -> d -> c b) -> (forall g. g -> c g) -> Dict p -> c (Dict p)

gunfold :: (forall b r. Data b => c (b -> r) -> c r) -> (forall r. r -> c r) -> Constr -> c (Dict p)

toConstr :: Dict p -> Constr

dataTypeOf :: Dict p -> DataType

dataCast1 :: Typeable (* -> *) t => (forall d. Data d => c (t d)) -> Maybe (c (Dict p))

dataCast2 :: Typeable (* -> * -> *) t => (forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c (Dict p))

gmapT :: (forall b. Data b => b -> b) -> Dict p -> Dict p

gmapQl :: (r -> r' -> r) -> r -> (forall d. Data d => d -> r') -> Dict p -> r

gmapQr :: (r' -> r -> r) -> r -> (forall d. Data d => d -> r') -> Dict p -> r

gmapQ :: (forall d. Data d => d -> u) -> Dict p -> [u]

gmapQi :: Int -> (forall d. Data d => d -> u) -> Dict p -> u

gmapM :: Monad m => (forall d. Data d => d -> m d) -> Dict p -> m (Dict p)

gmapMp :: MonadPlus m => (forall d. Data d => d -> m d) -> Dict p -> m (Dict p)

gmapMo :: MonadPlus m => (forall d. Data d => d -> m d) -> Dict p -> m (Dict p)

Ord (Dict a)

Methods

compare :: Dict a -> Dict a -> Ordering

(<) :: Dict a -> Dict a -> Bool

(<=) :: Dict a -> Dict a -> Bool

(>) :: Dict a -> Dict a -> Bool

(>=) :: Dict a -> Dict a -> Bool

max :: Dict a -> Dict a -> Dict a

min :: Dict a -> Dict a -> Dict a

a => Read (Dict a)

Methods

readsPrec :: Int -> ReadS (Dict a)

readList :: ReadS [Dict a]

readPrec :: ReadPrec (Dict a)

readListPrec :: ReadPrec [Dict a]

Show (Dict a)

Methods

showsPrec :: Int -> Dict a -> ShowS

show :: Dict a -> String

showList :: [Dict a] -> ShowS

a => Monoid (Dict a)

Methods

mempty :: Dict a

mappend :: Dict a -> Dict a -> Dict a

mconcat :: [Dict a] -> Dict a

withDict :: Dict a -> (a => r) -> r

From a Dict, takes a value in an environment where the instance witnessed by the Dict is in scope, and evaluates it.

Essentially a deconstruction of a Dict into its continuation-style form.

Entailment

newtype a :- b infixr 9

This is the type of entailment.

a :- b is read as a "entails" b.

With this we can actually build a category for Constraint resolution.

e.g.

Because Eq a is a superclass of Ord a, we can show that Ord a entails Eq a.

Because instance Ord a => Ord [a] exists, we can show that Ord a entails Ord [a] as well.

This relationship is captured in the :- entailment type here.

Since p :- p and entailment composes, :- forms the arrows of a Category of constraints. However, Category only became sufficiently general to support this instance in GHC 7.8, so prior to 7.8 this instance is unavailable.

But due to the coherence of instance resolution in Haskell, this Category has some very interesting properties. Notably, in the absence of IncoherentInstances, this category is "thin", which is to say that between any two objects (constraints) there is at most one distinguishable arrow.

This means that for instance, even though there are two ways to derive Ord a :- Eq [a], the answers from these two paths _must_ by construction be equal. This is a property that Haskell offers that is pretty much unique in the space of languages with things they call "type classes".

What are the two ways?

Well, we can go from Ord a :- Eq a via the superclass relationship, and then from Eq a :- Eq [a] via the instance, or we can go from Ord a :- Ord [a] via the instance then from Ord [a] :- Eq [a] through the superclass relationship and this diagram by definition must "commute".

Diagrammatically,

                   Ord a
               ins /     \ cls
                  v       v
            Ord [a]     Eq a
               cls \     / ins
                    v   v
                   Eq [a]

This safety net ensures that pretty much anything you can write with this library is sensible and can't break any assumptions on the behalf of library authors.

Constructors

Sub (a => Dict b)

Instances

Category Constraint (:-)

Possible since GHC 7.8, when Category was made polykinded.

Methods

id :: cat a a

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

() :=> (Eq ((:-) a b))

Methods

ins :: () :- Eq (a :- b)

() :=> (Ord ((:-) a b))

Methods

ins :: () :- Ord (a :- b)

() :=> (Show ((:-) a b))

Methods

ins :: () :- Show (a :- b)

Eq ((:-) a b)

Assumes IncoherentInstances doesn't exist.

Methods

(==) :: (a :- b) -> (a :- b) -> Bool

(/=) :: (a :- b) -> (a :- b) -> Bool

(Typeable Constraint p, Typeable Constraint q, p, q) => Data ((:-) p q)

Methods

gfoldl :: (forall d b. Data d => c (d -> b) -> d -> c b) -> (forall g. g -> c g) -> (p :- q) -> c (p :- q)

gunfold :: (forall b r. Data b => c (b -> r) -> c r) -> (forall r. r -> c r) -> Constr -> c (p :- q)

toConstr :: (p :- q) -> Constr

dataTypeOf :: (p :- q) -> DataType

dataCast1 :: Typeable (* -> *) t => (forall d. Data d => c (t d)) -> Maybe (c (p :- q))

dataCast2 :: Typeable (* -> * -> *) t => (forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c (p :- q))

gmapT :: (forall b. Data b => b -> b) -> (p :- q) -> p :- q

gmapQl :: (r -> r' -> r) -> r -> (forall d. Data d => d -> r') -> (p :- q) -> r

gmapQr :: (r' -> r -> r) -> r -> (forall d. Data d => d -> r') -> (p :- q) -> r

gmapQ :: (forall d. Data d => d -> u) -> (p :- q) -> [u]

gmapQi :: Int -> (forall d. Data d => d -> u) -> (p :- q) -> u

gmapM :: Monad m => (forall d. Data d => d -> m d) -> (p :- q) -> m (p :- q)

gmapMp :: MonadPlus m => (forall d. Data d => d -> m d) -> (p :- q) -> m (p :- q)

gmapMo :: MonadPlus m => (forall d. Data d => d -> m d) -> (p :- q) -> m (p :- q)

Ord ((:-) a b)

Assumes IncoherentInstances doesn't exist.

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

Show ((:-) a b)

Methods

showsPrec :: Int -> (a :- b) -> ShowS

show :: (a :- b) -> String

showList :: [a :- b] -> ShowS

(\\) :: a => (b => r) -> (a :- b) -> r infixl 1

Given that a :- b, derive something that needs a context b, using the context a

weaken1 :: (a, b) :- a

Weakening a constraint product

The category of constraints is Cartesian. We can forget information.

weaken2 :: (a, b) :- b

Weakening a constraint product

The category of constraints is Cartesian. We can forget information.

contract :: a :- (a, a)

Contracting a constraint / diagonal morphism

The category of constraints is Cartesian. We can reuse information.

strengthen1 :: Dict b -> (a :- c) -> a :- (b, c)

strengthen2 :: Dict b -> (a :- c) -> a :- (c, b)

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

Constraint product

trans weaken1 (f &&& g) = f
trans weaken2 (f &&& g) = g

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

due to the hack for the kind of (,) in the current version of GHC we can't actually make instances for (,) :: Constraint -> Constraint -> Constraint, but (,) is a bifunctor on the category of constraints. This lets us map over both sides.

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

Transitivity of entailment

If we view (:-) as a Constraint-indexed category, then this is (.)

refl :: a :- a

Reflexivity of entailment

If we view (:-) as a Constraint-indexed category, then this is id

class Any => Bottom

Any inhabits every kind, including Constraint but is uninhabited, making it impossible to define an instance.

Minimal complete definition

no

top :: a :- ()

Every constraint implies truth

These are the terminal arrows of the category, and () is the terminal object.

Given any constraint there is a unique entailment of the () constraint from that constraint.

bottom :: Bottom :- a

This demonstrates the law of classical logic "ex falso quodlibet"

Dict is fully faithful

mapDict :: (a :- b) -> Dict a -> Dict b

Apply an entailment to a dictionary.

From a category theoretic perspective Dict is a functor that maps from the category of constraints (with arrows in :-) to the category Hask of Haskell data types.

unmapDict :: (Dict a -> Dict b) -> a :- b

This functor is fully faithful, which is to say that given any function you can write Dict a -> Dict b there also exists an entailment a :- b in the category of constraints that you can build.

Reflection

class Class b h | h -> b where

Reify the relationship between a class and its superclass constraints as a class

Given a definition such as

class Foo a => Bar a

you can capture the relationship between 'Bar a' and its superclass 'Foo a' with

instance Class (Foo a) (Bar a) where cls = Sub Dict

Now the user can use 'cls :: Bar a :- Foo a'

Minimal complete definition

cls

Methods

cls :: h :- b

Instances

Class () ()

Methods

cls :: () :- ()

Class () (Bounded a)

Methods

cls :: Bounded a :- ()

Class () (Enum a)

Methods

cls :: Enum a :- ()

Class () (Eq a)

Methods

cls :: Eq a :- ()

Class () (Functor f)

Methods

cls :: Functor f :- ()

Class () (Num a)

Methods

cls :: Num a :- ()

Class () (Read a)

Methods

cls :: Read a :- ()

Class () (Show a)

Methods

cls :: Show a :- ()

Class () (Monoid a)

Methods

cls :: Monoid a :- ()

Class b a => () :=> (Class b a)

Methods

ins :: () :- Class b a

Class () ((:=>) b a)

Methods

cls :: (b :=> a) :- ()

Class () (Class b a)

Methods

cls :: Class b a :- ()

Class (Eq a) (Bits a)

Methods

cls :: Bits a :- Eq a

Class (Eq a) (Ord a)

Methods

cls :: Ord a :- Eq a

Class (Fractional a) (Floating a)

Methods

cls :: Floating a :- Fractional a

Class (Functor f) (Applicative f)

Methods

cls :: Applicative f :- Functor f

Class (Num a) (Fractional a)

Methods

cls :: Fractional a :- Num a

Class (Applicative f) (Monad f)

Methods

cls :: Monad f :- Applicative f

Class (Applicative f) (Alternative f)

Methods

cls :: Alternative f :- Applicative f

Class (Monad f, Alternative f) (MonadPlus f)

Methods

cls :: MonadPlus f :- (Monad f, Alternative f)

Class (Num a, Ord a) (Real a)

Methods

cls :: Real a :- (Num a, Ord a)

Class (Real a, Fractional a) (RealFrac a)

Methods

cls :: RealFrac a :- (Real a, Fractional a)

Class (Real a, Enum a) (Integral a)

Methods

cls :: Integral a :- (Real a, Enum a)

Class (RealFrac a, Floating a) (RealFloat a)

Methods

cls :: RealFloat a :- (RealFrac a, Floating a)

class b :=> h | h -> b where infixr 9

Reify the relationship between an instance head and its body as a class

Given a definition such as

instance Foo a => Foo [a]

you can capture the relationship between the instance head and its body with

instance Foo a :=> Foo [a] where ins = Sub Dict

Minimal complete definition

ins

Methods

ins :: b :- h

Instances

() :=> ()

Methods

ins :: () :- ()

a :=> (Monoid (Dict a))

Methods

ins :: a :- Monoid (Dict a)

a :=> (Read (Dict a))

Methods

ins :: a :- Read (Dict a)

a :=> (Bounded (Dict a))

Methods

ins :: a :- Bounded (Dict a)

a :=> (Enum (Dict a))

Methods

ins :: a :- Enum (Dict a)

() :=> (Bounded Bool)

Methods

ins :: () :- Bounded Bool

() :=> (Bounded Char)

Methods

ins :: () :- Bounded Char

() :=> (Bounded Int)

Methods

ins :: () :- Bounded Int

() :=> (Bounded Ordering)

Methods

ins :: () :- Bounded Ordering

() :=> (Bounded Word)

Methods

ins :: () :- Bounded Word

() :=> (Bounded ())

Methods

ins :: () :- Bounded ()

() :=> (Enum Bool)

Methods

ins :: () :- Enum Bool

() :=> (Enum Char)

Methods

ins :: () :- Enum Char

() :=> (Enum Double)

Methods

ins :: () :- Enum Double

() :=> (Enum Float)

Methods

ins :: () :- Enum Float

() :=> (Enum Int)

Methods

ins :: () :- Enum Int

() :=> (Enum Integer)

Methods

ins :: () :- Enum Integer

() :=> (Enum Ordering)

Methods

ins :: () :- Enum Ordering

() :=> (Enum Word)

Methods

ins :: () :- Enum Word

() :=> (Enum ())

Methods

ins :: () :- Enum ()

() :=> (Enum Natural)

Methods

ins :: () :- Enum Natural

() :=> (Eq Bool)

Methods

ins :: () :- Eq Bool

() :=> (Eq Double)

Methods

ins :: () :- Eq Double

() :=> (Eq Float)

Methods

ins :: () :- Eq Float

() :=> (Eq Int)

Methods

ins :: () :- Eq Int

() :=> (Eq Integer)

Methods

ins :: () :- Eq Integer

() :=> (Eq Word)

Methods

ins :: () :- Eq Word

() :=> (Eq ())

Methods

ins :: () :- Eq ()

() :=> (Eq Natural)

Methods

ins :: () :- Eq Natural

() :=> (Eq ((:-) a b))

Methods

ins :: () :- Eq (a :- b)

() :=> (Eq (Dict a))

Methods

ins :: () :- Eq (Dict a)

() :=> (Floating Double)

Methods

ins :: () :- Floating Double

() :=> (Floating Float)

Methods

ins :: () :- Floating Float

() :=> (Fractional Double)

Methods

ins :: () :- Fractional Double

() :=> (Fractional Float)

Methods

ins :: () :- Fractional Float

() :=> (Integral Int)

Methods

ins :: () :- Integral Int

() :=> (Integral Integer)

Methods

ins :: () :- Integral Integer

() :=> (Integral Word)

Methods

ins :: () :- Integral Word

() :=> (Integral Natural)

Methods

ins :: () :- Integral Natural

() :=> (Monad ((->) a))

Methods

ins :: () :- Monad ((->) a)

() :=> (Monad [])

Methods

ins :: () :- Monad []

() :=> (Monad IO)

Methods

ins :: () :- Monad IO

() :=> (Monad (Either a))

Methods

ins :: () :- Monad (Either a)

() :=> (Monad Identity)

Methods

ins :: () :- Monad Identity

() :=> (Functor ((->) a))

Methods

ins :: () :- Functor ((->) a)

() :=> (Functor [])

Methods

ins :: () :- Functor []

() :=> (Functor Maybe)

Methods

ins :: () :- Functor Maybe

() :=> (Functor IO)

Methods

ins :: () :- Functor IO

() :=> (Functor (Either a))

Methods

ins :: () :- Functor (Either a)

() :=> (Functor ((,) a))

Methods

ins :: () :- Functor ((,) a)

() :=> (Functor Identity)

Methods

ins :: () :- Functor Identity

() :=> (Functor (Const * a))

Methods

ins :: () :- Functor (Const * a)

() :=> (Num Double)

Methods

ins :: () :- Num Double

() :=> (Num Float)

Methods

ins :: () :- Num Float

() :=> (Num Int)

Methods

ins :: () :- Num Int

() :=> (Num Integer)

Methods

ins :: () :- Num Integer

() :=> (Num Word)

Methods

ins :: () :- Num Word

() :=> (Num Natural)

Methods

ins :: () :- Num Natural

() :=> (Ord Bool)

Methods

ins :: () :- Ord Bool

() :=> (Ord Char)

Methods

ins :: () :- Ord Char

() :=> (Ord Double)

Methods

ins :: () :- Ord Double

() :=> (Ord Float)

Methods

ins :: () :- Ord Float

() :=> (Ord Int)

Methods

ins :: () :- Ord Int

() :=> (Ord Integer)

Methods

ins :: () :- Ord Integer

() :=> (Ord Word)

Methods

ins :: () :- Ord Word

() :=> (Ord ())

Methods

ins :: () :- Ord ()

() :=> (Ord Natural)

Methods

ins :: () :- Ord Natural

() :=> (Ord ((:-) a b))

Methods

ins :: () :- Ord (a :- b)

() :=> (Ord (Dict a))

Methods

ins :: () :- Ord (Dict a)

() :=> (Read Bool)

Methods

ins :: () :- Read Bool

() :=> (Read Char)

Methods

ins :: () :- Read Char

() :=> (Read Ordering)

Methods

ins :: () :- Read Ordering

() :=> (Read Word)

Methods

ins :: () :- Read Word

() :=> (Read ())

Methods

ins :: () :- Read ()

() :=> (Read Natural)

Methods

ins :: () :- Read Natural

() :=> (Real Double)

Methods

ins :: () :- Real Double

() :=> (Real Float)

Methods

ins :: () :- Real Float

() :=> (Real Int)

Methods

ins :: () :- Real Int

() :=> (Real Integer)

Methods

ins :: () :- Real Integer

() :=> (Real Word)

Methods

ins :: () :- Real Word

() :=> (Real Natural)

Methods

ins :: () :- Real Natural

() :=> (RealFloat Double)

Methods

ins :: () :- RealFloat Double

() :=> (RealFloat Float)

Methods

ins :: () :- RealFloat Float

() :=> (RealFrac Double)

Methods

ins :: () :- RealFrac Double

() :=> (RealFrac Float)

Methods

ins :: () :- RealFrac Float

() :=> (Show Bool)

Methods

ins :: () :- Show Bool

() :=> (Show Char)

Methods

ins :: () :- Show Char

() :=> (Show Ordering)

Methods

ins :: () :- Show Ordering

() :=> (Show Word)

Methods

ins :: () :- Show Word

() :=> (Show ())

Methods

ins :: () :- Show ()

() :=> (Show Natural)

Methods

ins :: () :- Show Natural

() :=> (Show ((:-) a b))

Methods

ins :: () :- Show (a :- b)

() :=> (Show (Dict a))

Methods

ins :: () :- Show (Dict a)

() :=> (Applicative ((->) a))

Methods

ins :: () :- Applicative ((->) a)

() :=> (Applicative [])

Methods

ins :: () :- Applicative []

() :=> (Applicative Maybe)

Methods

ins :: () :- Applicative Maybe

() :=> (Applicative IO)

Methods

ins :: () :- Applicative IO

() :=> (Applicative (Either a))

Methods

ins :: () :- Applicative (Either a)

() :=> (Monoid [a])

Methods

ins :: () :- Monoid [a]

() :=> (Monoid Ordering)

Methods

ins :: () :- Monoid Ordering

() :=> (Monoid ())

Methods

ins :: () :- Monoid ()

() :=> (Bits Bool)

Methods

ins :: () :- Bits Bool

() :=> (Bits Int)

Methods

ins :: () :- Bits Int

() :=> (Bits Integer)

Methods

ins :: () :- Bits Integer

() :=> (Bits Word)

Methods

ins :: () :- Bits Word

() :=> (Bits Natural)

Methods

ins :: () :- Bits Natural

() :=> (Alternative [])

Methods

ins :: () :- Alternative []

() :=> (Alternative Maybe)

Methods

ins :: () :- Alternative Maybe

() :=> (MonadPlus [])

Methods

ins :: () :- MonadPlus []

() :=> (MonadPlus Maybe)

Methods

ins :: () :- MonadPlus Maybe

(:=>) b a => () :=> ((:=>) b a)

Methods

ins :: () :- (b :=> a)

Class b a => () :=> (Class b a)

Methods

ins :: () :- Class b a

Class () ((:=>) b a)

Methods

cls :: (b :=> a) :- ()

(Bounded a) :=> (Bounded (Const * a b))

Methods

ins :: Bounded a :- Bounded (Const * a b)

(Bounded a) :=> (Bounded (Identity a))

Methods

ins :: Bounded a :- Bounded (Identity a)

(Enum a) :=> (Enum (Const * a b))

Methods

ins :: Enum a :- Enum (Const * a b)

(Enum a) :=> (Enum (Identity a))

Methods

ins :: Enum a :- Enum (Identity a)

(Eq a) :=> (Eq (Const * a b))

Methods

ins :: Eq a :- Eq (Const * a b)

(Eq a) :=> (Eq (Identity a))

Methods

ins :: Eq a :- Eq (Identity a)

(Eq a) :=> (Eq (Ratio a))

Methods

ins :: Eq a :- Eq (Ratio a)

(Eq a) :=> (Eq (Complex a))

Methods

ins :: Eq a :- Eq (Complex a)

(Eq a) :=> (Eq (Maybe a))

Methods

ins :: Eq a :- Eq (Maybe a)

(Eq a) :=> (Eq [a])

Methods

ins :: Eq a :- Eq [a]

(Floating a) :=> (Floating (Const * a b))

Methods

ins :: Floating a :- Floating (Const * a b)

(Floating a) :=> (Floating (Identity a))

Methods

ins :: Floating a :- Floating (Identity a)

(Fractional a) :=> (Fractional (Const * a b))

Methods

ins :: Fractional a :- Fractional (Const * a b)

(Fractional a) :=> (Fractional (Identity a))

Methods

ins :: Fractional a :- Fractional (Identity a)

(Integral a) :=> (RealFrac (Ratio a))

Methods

ins :: Integral a :- RealFrac (Ratio a)

(Integral a) :=> (Fractional (Ratio a))

Methods

ins :: Integral a :- Fractional (Ratio a)

(Integral a) :=> (Integral (Const * a b))

Methods

ins :: Integral a :- Integral (Const * a b)

(Integral a) :=> (Integral (Identity a))

Methods

ins :: Integral a :- Integral (Identity a)

(Integral a) :=> (Real (Ratio a))

Methods

ins :: Integral a :- Real (Ratio a)

(Integral a) :=> (Num (Ratio a))

Methods

ins :: Integral a :- Num (Ratio a)

(Integral a) :=> (Enum (Ratio a))

Methods

ins :: Integral a :- Enum (Ratio a)

(Integral a) :=> (Ord (Ratio a))

Methods

ins :: Integral a :- Ord (Ratio a)

(Monad m) :=> (Applicative (WrappedMonad m))

Methods

ins :: Monad m :- Applicative (WrappedMonad m)

(Monad m) :=> (Functor (WrappedMonad m))

Methods

ins :: Monad m :- Functor (WrappedMonad m)

(Num a) :=> (Num (Const * a b))

Methods

ins :: Num a :- Num (Const * a b)

(Num a) :=> (Num (Identity a))

Methods

ins :: Num a :- Num (Identity a)

(Ord a) :=> (Ord (Const * a b))

Methods

ins :: Ord a :- Ord (Const * a b)

(Ord a) :=> (Ord (Identity a))

Methods

ins :: Ord a :- Ord (Identity a)

(Ord a) :=> (Ord [a])

Methods

ins :: Ord a :- Ord [a]

(Ord a) :=> (Ord (Maybe a))

Methods

ins :: Ord a :- Ord (Maybe a)

(Read a) :=> (Read (Const * a b))

Methods

ins :: Read a :- Read (Const * a b)

(Read a) :=> (Read (Identity a))

Methods

ins :: Read a :- Read (Identity a)

(Read a) :=> (Read (Maybe a))

Methods

ins :: Read a :- Read (Maybe a)

(Read a) :=> (Read [a])

Methods

ins :: Read a :- Read [a]

(Read a) :=> (Read (Complex a))

Methods

ins :: Read a :- Read (Complex a)

(Real a) :=> (Real (Const * a b))

Methods

ins :: Real a :- Real (Const * a b)

(Real a) :=> (Real (Identity a))

Methods

ins :: Real a :- Real (Identity a)

(RealFloat a) :=> (RealFloat (Const * a b))

Methods

ins :: RealFloat a :- RealFloat (Const * a b)

(RealFloat a) :=> (RealFloat (Identity a))

Methods

ins :: RealFloat a :- RealFloat (Identity a)

(RealFloat a) :=> (Floating (Complex a))

Methods

ins :: RealFloat a :- Floating (Complex a)

(RealFloat a) :=> (Fractional (Complex a))

Methods

ins :: RealFloat a :- Fractional (Complex a)

(RealFloat a) :=> (Num (Complex a))

Methods

ins :: RealFloat a :- Num (Complex a)

(RealFrac a) :=> (RealFrac (Const * a b))

Methods

ins :: RealFrac a :- RealFrac (Const * a b)

(RealFrac a) :=> (RealFrac (Identity a))

Methods

ins :: RealFrac a :- RealFrac (Identity a)

(Show a) :=> (Show (Const * a b))

Methods

ins :: Show a :- Show (Const * a b)

(Show a) :=> (Show (Identity a))

Methods

ins :: Show a :- Show (Identity a)

(Show a) :=> (Show (Maybe a))

Methods

ins :: Show a :- Show (Maybe a)

(Show a) :=> (Show [a])

Methods

ins :: Show a :- Show [a]

(Show a) :=> (Show (Complex a))

Methods

ins :: Show a :- Show (Complex a)

(Monoid a) :=> (Applicative (Const * a))

Methods

ins :: Monoid a :- Applicative (Const * a)

(Monoid a) :=> (Applicative ((,) a))

Methods

ins :: Monoid a :- Applicative ((,) a)

(Monoid a) :=> (Monoid (IO a))

Methods

ins :: Monoid a :- Monoid (IO a)

(Monoid a) :=> (Monoid (Identity a))

Methods

ins :: Monoid a :- Monoid (Identity a)

(Monoid a) :=> (Monoid (Const * a b))

Methods

ins :: Monoid a :- Monoid (Const * a b)

(Monoid a) :=> (Monoid (Maybe a))

Methods

ins :: Monoid a :- Monoid (Maybe a)

(Bits a) :=> (Bits (Const * a b))

Methods

ins :: Bits a :- Bits (Const * a b)

(Bits a) :=> (Bits (Identity a))

Methods

ins :: Bits a :- Bits (Identity a)

(MonadPlus m) :=> (Alternative (WrappedMonad m))

Methods

ins :: MonadPlus m :- Alternative (WrappedMonad m)

(Bounded a, Bounded b) :=> (Bounded (a, b))

Methods

ins :: (Bounded a, Bounded b) :- Bounded (a, b)

(Eq a, Eq b) :=> (Eq (Either a b))

Methods

ins :: (Eq a, Eq b) :- Eq (Either a b)

(Eq a, Eq b) :=> (Eq (a, b))

Methods

ins :: (Eq a, Eq b) :- Eq (a, b)

(Integral a, Read a) :=> (Read (Ratio a))

Methods

ins :: (Integral a, Read a) :- Read (Ratio a)

(Integral a, Show a) :=> (Show (Ratio a))

Methods

ins :: (Integral a, Show a) :- Show (Ratio a)

(Ord a, Ord b) :=> (Ord (Either a b))

Methods

ins :: (Ord a, Ord b) :- Ord (Either a b)

(Ord a, Ord b) :=> (Ord (a, b))

Methods

ins :: (Ord a, Ord b) :- Ord (a, b)

(Read a, Read b) :=> (Read (Either a b))

Methods

ins :: (Read a, Read b) :- Read (Either a b)

(Read a, Read b) :=> (Read (a, b))

Methods

ins :: (Read a, Read b) :- Read (a, b)

(Show a, Show b) :=> (Show (Either a b))

Methods

ins :: (Show a, Show b) :- Show (Either a b)

(Show a, Show b) :=> (Show (a, b))

Methods

ins :: (Show a, Show b) :- Show (a, b)

(Monoid a, Monoid b) :=> (Monoid (a, b))

Methods

ins :: (Monoid a, Monoid b) :- Monoid (a, b)