Copyright	(C) 2011-2015 Edward Kmett,
License	BSD-style (see the file LICENSE)
Maintainer	Edward Kmett <ekmett@gmail.com>
Stability	experimental
Portability	non-portable
Safe Haskell	Trustworthy
Language	Haskell2010

Data.Constraint

Contents

The Kind of Constraints
Dictionary
Entailment
Dict is fully faithful
Reflection

Description

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.

Synopsis

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 #

Dictionary

data Dict :: Constraint -> * where Source #

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)) Source #
Methods ins :: a :- Monoid (Dict a) Source #
a :=> (Read (Dict a)) Source #
Methods ins :: a :- Read (Dict a) Source #
a :=> (Bounded (Dict a)) Source #
Methods ins :: a :- Bounded (Dict a) Source #
a :=> (Enum (Dict a)) Source #
Methods ins :: a :- Enum (Dict a) Source #
() :=> (Eq (Dict a)) Source #
Methods ins :: () :- Eq (Dict a) Source #
() :=> (Ord (Dict a)) Source #
Methods ins :: () :- Ord (Dict a) Source #
() :=> (Show (Dict a)) Source #
Methods ins :: () :- Show (Dict a) Source #
a => Bounded (Dict a) Source #
Methods minBound :: Dict a # maxBound :: Dict a #
a => Enum (Dict a) Source #
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) Source #
Methods (==) :: Dict a -> Dict a -> Bool # (/=) :: Dict a -> Dict a -> Bool #
(Typeable Constraint p, p) => Data (Dict p) Source #
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) Source #
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) Source #
Methods readsPrec :: Int -> ReadS (Dict a) # readList :: ReadS [Dict a] # readPrec :: ReadPrec (Dict a) # readListPrec :: ReadPrec [Dict a] #
Show (Dict a) Source #
Methods showsPrec :: Int -> Dict a -> ShowS # show :: Dict a -> String # showList :: [Dict a] -> ShowS #
a => Monoid (Dict a) Source #
Methods mempty :: Dict a # mappend :: Dict a -> Dict a -> Dict a # mconcat :: [Dict a] -> Dict a #

Entailment

newtype a :- b infixr 9 Source #

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 because 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 them 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 (:-) Source #	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)) Source #
Methods ins :: () :- Eq (a :- b) Source #
() :=> (Ord ((:-) a b)) Source #
Methods ins :: () :- Ord (a :- b) Source #
() :=> (Show ((:-) a b)) Source #
Methods ins :: () :- Show (a :- b) Source #
Eq ((:-) a b) Source #	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) Source #
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) Source #	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) Source #
Methods showsPrec :: Int -> (a :- b) -> ShowS # show :: (a :- b) -> String # showList :: [a :- b] -> ShowS #

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

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

weaken1 :: (a, b) :- a Source #

Weakening a constraint product

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

weaken2 :: (a, b) :- b Source #

Weakening a constraint product

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

contract :: a :- (a, a) Source #

Contracting a constraint / diagonal morphism

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

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

Constraint product

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

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

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 Source #

Transitivity of entailment

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

refl :: a :- a Source #

Reflexivity of entailment

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

class Any => Bottom Source #

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

Minimal complete definition

no

top :: a :- () Source #

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 Source #

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

Dict is fully faithful

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

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 Source #

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 Source #

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 Source #

Instances

Class () () Source #
Methods cls :: () :- () Source #
Class () (Bounded a) Source #
Methods cls :: Bounded a :- () Source #
Class () (Enum a) Source #
Methods cls :: Enum a :- () Source #
Class () (Eq a) Source #
Methods cls :: Eq a :- () Source #
Class () (Monad f) Source #
Methods cls :: Monad f :- () Source #
Class () (Functor f) Source #
Methods cls :: Functor f :- () Source #
Class () (Num a) Source #
Methods cls :: Num a :- () Source #
Class () (Read a) Source #
Methods cls :: Read a :- () Source #
Class () (Show a) Source #
Methods cls :: Show a :- () Source #
Class () (Monoid a) Source #
Methods cls :: Monoid a :- () Source #
Class b a => () :=> (Class b a) Source #
Methods ins :: () :- Class b a Source #
Class () ((:=>) b a) Source #
Methods cls :: (b :=> a) :- () Source #
Class () (Class b a) Source #
Methods cls :: Class b a :- () Source #
Class (Eq a) (Ord a) Source #
Methods cls :: Ord a :- Eq a Source #
Class (Fractional a) (Floating a) Source #
Methods cls :: Floating a :- Fractional a Source #
Class (Monad f) (MonadPlus f) Source #
Methods cls :: MonadPlus f :- Monad f Source #
Class (Functor f) (Applicative f) Source #
Methods cls :: Applicative f :- Functor f Source #
Class (Num a) (Fractional a) Source #
Methods cls :: Fractional a :- Num a Source #
Class (Applicative f) (Alternative f) Source #
Methods cls :: Alternative f :- Applicative f Source #
Class (Num a, Ord a) (Real a) Source #
Methods cls :: Real a :- (Num a, Ord a) Source #
Class (Real a, Fractional a) (RealFrac a) Source #
Methods cls :: RealFrac a :- (Real a, Fractional a) Source #
Class (Real a, Enum a) (Integral a) Source #
Methods cls :: Integral a :- (Real a, Enum a) Source #
Class (RealFrac a, Floating a) (RealFloat a) Source #
Methods cls :: RealFloat a :- (RealFrac a, Floating a) Source #

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

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 Source #

Instances

() :=> () Source #
Methods ins :: () :- () Source #
a :=> (Monoid (Dict a)) Source #
Methods ins :: a :- Monoid (Dict a) Source #
a :=> (Read (Dict a)) Source #
Methods ins :: a :- Read (Dict a) Source #
a :=> (Bounded (Dict a)) Source #
Methods ins :: a :- Bounded (Dict a) Source #
a :=> (Enum (Dict a)) Source #
Methods ins :: a :- Enum (Dict a) Source #
() :=> (Bounded Bool) Source #
Methods ins :: () :- Bounded Bool Source #
() :=> (Bounded Char) Source #
Methods ins :: () :- Bounded Char Source #
() :=> (Bounded Int) Source #
Methods ins :: () :- Bounded Int Source #
() :=> (Bounded Ordering) Source #
Methods ins :: () :- Bounded Ordering Source #
() :=> (Bounded ()) Source #
Methods ins :: () :- Bounded () Source #
() :=> (Enum Bool) Source #
Methods ins :: () :- Enum Bool Source #
() :=> (Enum Char) Source #
Methods ins :: () :- Enum Char Source #
() :=> (Enum Double) Source #
Methods ins :: () :- Enum Double Source #
() :=> (Enum Float) Source #
Methods ins :: () :- Enum Float Source #
() :=> (Enum Int) Source #
Methods ins :: () :- Enum Int Source #
() :=> (Enum Integer) Source #
Methods ins :: () :- Enum Integer Source #
() :=> (Enum Ordering) Source #
Methods ins :: () :- Enum Ordering Source #
() :=> (Enum ()) Source #
Methods ins :: () :- Enum () Source #
() :=> (Eq Bool) Source #
Methods ins :: () :- Eq Bool Source #
() :=> (Eq Double) Source #
Methods ins :: () :- Eq Double Source #
() :=> (Eq Float) Source #
Methods ins :: () :- Eq Float Source #
() :=> (Eq Int) Source #
Methods ins :: () :- Eq Int Source #
() :=> (Eq Integer) Source #
Methods ins :: () :- Eq Integer Source #
() :=> (Eq ()) Source #
Methods ins :: () :- Eq () Source #
() :=> (Eq ((:-) a b)) Source #
Methods ins :: () :- Eq (a :- b) Source #
() :=> (Eq (Dict a)) Source #
Methods ins :: () :- Eq (Dict a) Source #
() :=> (Floating Double) Source #
Methods ins :: () :- Floating Double Source #
() :=> (Floating Float) Source #
Methods ins :: () :- Floating Float Source #
() :=> (Fractional Double) Source #
Methods ins :: () :- Fractional Double Source #
() :=> (Fractional Float) Source #
Methods ins :: () :- Fractional Float Source #
() :=> (Integral Int) Source #
Methods ins :: () :- Integral Int Source #
() :=> (Integral Integer) Source #
Methods ins :: () :- Integral Integer Source #
() :=> (Monad ((->) a)) Source #
Methods ins :: () :- Monad ((->) a) Source #
() :=> (Monad []) Source #
Methods ins :: () :- Monad [] Source #
() :=> (Monad IO) Source #
Methods ins :: () :- Monad IO Source #
() :=> (Monad (Either a)) Source #
Methods ins :: () :- Monad (Either a) Source #
() :=> (Functor ((->) a)) Source #
Methods ins :: () :- Functor ((->) a) Source #
() :=> (Functor []) Source #
Methods ins :: () :- Functor [] Source #
() :=> (Functor Maybe) Source #
Methods ins :: () :- Functor Maybe Source #
() :=> (Functor IO) Source #
Methods ins :: () :- Functor IO Source #
() :=> (Functor (Either a)) Source #
Methods ins :: () :- Functor (Either a) Source #
() :=> (Functor ((,) a)) Source #
Methods ins :: () :- Functor ((,) a) Source #
() :=> (Num Double) Source #
Methods ins :: () :- Num Double Source #
() :=> (Num Float) Source #
Methods ins :: () :- Num Float Source #
() :=> (Num Int) Source #
Methods ins :: () :- Num Int Source #
() :=> (Num Integer) Source #
Methods ins :: () :- Num Integer Source #
() :=> (Ord Bool) Source #
Methods ins :: () :- Ord Bool Source #
() :=> (Ord Char) Source #
Methods ins :: () :- Ord Char Source #
() :=> (Ord Double) Source #
Methods ins :: () :- Ord Double Source #
() :=> (Ord Float) Source #
Methods ins :: () :- Ord Float Source #
() :=> (Ord Int) Source #
Methods ins :: () :- Ord Int Source #
() :=> (Ord Integer) Source #
Methods ins :: () :- Ord Integer Source #
() :=> (Ord ()) Source #
Methods ins :: () :- Ord () Source #
() :=> (Ord ((:-) a b)) Source #
Methods ins :: () :- Ord (a :- b) Source #
() :=> (Ord (Dict a)) Source #
Methods ins :: () :- Ord (Dict a) Source #
() :=> (Read Bool) Source #
Methods ins :: () :- Read Bool Source #
() :=> (Read Char) Source #
Methods ins :: () :- Read Char Source #
() :=> (Read Ordering) Source #
Methods ins :: () :- Read Ordering Source #
() :=> (Read ()) Source #
Methods ins :: () :- Read () Source #
() :=> (Real Double) Source #
Methods ins :: () :- Real Double Source #
() :=> (Real Float) Source #
Methods ins :: () :- Real Float Source #
() :=> (Real Int) Source #
Methods ins :: () :- Real Int Source #
() :=> (Real Integer) Source #
Methods ins :: () :- Real Integer Source #
() :=> (RealFloat Double) Source #
Methods ins :: () :- RealFloat Double Source #
() :=> (RealFloat Float) Source #
Methods ins :: () :- RealFloat Float Source #
() :=> (RealFrac Double) Source #
Methods ins :: () :- RealFrac Double Source #
() :=> (RealFrac Float) Source #
Methods ins :: () :- RealFrac Float Source #
() :=> (Show Bool) Source #
Methods ins :: () :- Show Bool Source #
() :=> (Show Char) Source #
Methods ins :: () :- Show Char Source #
() :=> (Show Ordering) Source #
Methods ins :: () :- Show Ordering Source #
() :=> (Show ()) Source #
Methods ins :: () :- Show () Source #
() :=> (Show ((:-) a b)) Source #
Methods ins :: () :- Show (a :- b) Source #
() :=> (Show (Dict a)) Source #
Methods ins :: () :- Show (Dict a) Source #
() :=> (Applicative ((->) a)) Source #
Methods ins :: () :- Applicative ((->) a) Source #
() :=> (Applicative []) Source #
Methods ins :: () :- Applicative [] Source #
() :=> (Applicative Maybe) Source #
Methods ins :: () :- Applicative Maybe Source #
() :=> (Applicative IO) Source #
Methods ins :: () :- Applicative IO Source #
() :=> (Applicative (Either a)) Source #
Methods ins :: () :- Applicative (Either a) Source #
() :=> (Monoid [a]) Source #
Methods ins :: () :- Monoid [a] Source #
() :=> (Monoid Ordering) Source #
Methods ins :: () :- Monoid Ordering Source #
() :=> (Monoid ()) Source #
Methods ins :: () :- Monoid () Source #
() :=> (Alternative []) Source #
Methods ins :: () :- Alternative [] Source #
() :=> (Alternative Maybe) Source #
Methods ins :: () :- Alternative Maybe Source #
() :=> (MonadPlus []) Source #
Methods ins :: () :- MonadPlus [] Source #
() :=> (MonadPlus Maybe) Source #
Methods ins :: () :- MonadPlus Maybe Source #
(:=>) b a => () :=> ((:=>) b a) Source #
Methods ins :: () :- (b :=> a) Source #
Class b a => () :=> (Class b a) Source #
Methods ins :: () :- Class b a Source #
Class () ((:=>) b a) Source #
Methods cls :: (b :=> a) :- () Source #
(Eq a) :=> (Eq (Ratio a)) Source #
Methods ins :: Eq a :- Eq (Ratio a) Source #
(Eq a) :=> (Eq (Complex a)) Source #
Methods ins :: Eq a :- Eq (Complex a) Source #
(Eq a) :=> (Eq (Maybe a)) Source #
Methods ins :: Eq a :- Eq (Maybe a) Source #
(Eq a) :=> (Eq [a]) Source #
Methods ins :: Eq a :- Eq [a] Source #
(Integral a) :=> (RealFrac (Ratio a)) Source #
Methods ins :: Integral a :- RealFrac (Ratio a) Source #
(Integral a) :=> (Fractional (Ratio a)) Source #
Methods ins :: Integral a :- Fractional (Ratio a) Source #
(Integral a) :=> (Real (Ratio a)) Source #
Methods ins :: Integral a :- Real (Ratio a) Source #
(Integral a) :=> (Num (Ratio a)) Source #
Methods ins :: Integral a :- Num (Ratio a) Source #
(Integral a) :=> (Enum (Ratio a)) Source #
Methods ins :: Integral a :- Enum (Ratio a) Source #
(Integral a) :=> (Ord (Ratio a)) Source #
Methods ins :: Integral a :- Ord (Ratio a) Source #
(Monad m) :=> (Applicative (WrappedMonad m)) Source #
Methods ins :: Monad m :- Applicative (WrappedMonad m) Source #
(Monad m) :=> (Functor (WrappedMonad m)) Source #
Methods ins :: Monad m :- Functor (WrappedMonad m) Source #
(Ord a) :=> (Ord [a]) Source #
Methods ins :: Ord a :- Ord [a] Source #
(Ord a) :=> (Ord (Maybe a)) Source #
Methods ins :: Ord a :- Ord (Maybe a) Source #
(Read a) :=> (Read (Maybe a)) Source #
Methods ins :: Read a :- Read (Maybe a) Source #
(Read a) :=> (Read [a]) Source #
Methods ins :: Read a :- Read [a] Source #
(Read a) :=> (Read (Complex a)) Source #
Methods ins :: Read a :- Read (Complex a) Source #
(RealFloat a) :=> (Floating (Complex a)) Source #
Methods ins :: RealFloat a :- Floating (Complex a) Source #
(RealFloat a) :=> (Fractional (Complex a)) Source #
Methods ins :: RealFloat a :- Fractional (Complex a) Source #
(RealFloat a) :=> (Num (Complex a)) Source #
Methods ins :: RealFloat a :- Num (Complex a) Source #
(Show a) :=> (Show (Maybe a)) Source #
Methods ins :: Show a :- Show (Maybe a) Source #
(Show a) :=> (Show [a]) Source #
Methods ins :: Show a :- Show [a] Source #
(Show a) :=> (Show (Complex a)) Source #
Methods ins :: Show a :- Show (Complex a) Source #
(Monoid a) :=> (Applicative ((,) a)) Source #
Methods ins :: Monoid a :- Applicative ((,) a) Source #
(Monoid a) :=> (Monoid (Maybe a)) Source #
Methods ins :: Monoid a :- Monoid (Maybe a) Source #
(MonadPlus m) :=> (Alternative (WrappedMonad m)) Source #
Methods ins :: MonadPlus m :- Alternative (WrappedMonad m) Source #
(Bounded a, Bounded b) :=> (Bounded (a, b)) Source #
Methods ins :: (Bounded a, Bounded b) :- Bounded (a, b) Source #
(Eq a, Eq b) :=> (Eq (Either a b)) Source #
Methods ins :: (Eq a, Eq b) :- Eq (Either a b) Source #
(Eq a, Eq b) :=> (Eq (a, b)) Source #
Methods ins :: (Eq a, Eq b) :- Eq (a, b) Source #
(Integral a, Read a) :=> (Read (Ratio a)) Source #
Methods ins :: (Integral a, Read a) :- Read (Ratio a) Source #
(Integral a, Show a) :=> (Show (Ratio a)) Source #
Methods ins :: (Integral a, Show a) :- Show (Ratio a) Source #
(Ord a, Ord b) :=> (Ord (Either a b)) Source #
Methods ins :: (Ord a, Ord b) :- Ord (Either a b) Source #
(Ord a, Ord b) :=> (Ord (a, b)) Source #
Methods ins :: (Ord a, Ord b) :- Ord (a, b) Source #
(Read a, Read b) :=> (Read (Either a b)) Source #
Methods ins :: (Read a, Read b) :- Read (Either a b) Source #
(Read a, Read b) :=> (Read (a, b)) Source #
Methods ins :: (Read a, Read b) :- Read (a, b) Source #
(Show a, Show b) :=> (Show (Either a b)) Source #
Methods ins :: (Show a, Show b) :- Show (Either a b) Source #
(Show a, Show b) :=> (Show (a, b)) Source #
Methods ins :: (Show a, Show b) :- Show (a, b) Source #
(Monoid a, Monoid b) :=> (Monoid (a, b)) Source #
Methods ins :: (Monoid a, Monoid b) :- Monoid (a, b) Source #