| 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
Description
ConstraintKinds made type classes into types of a new kind, Constraint.
Eq:: * ->ConstraintOrd:: * ->ConstraintMonad:: (* -> *) ->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
- data Constraint
- data Dict :: Constraint -> * where
- withDict :: Dict a -> (a => r) -> r
- newtype a :- b = Sub (a => Dict b)
- (\\) :: a => (b => r) -> (a :- b) -> r
- weaken1 :: (a, b) :- a
- weaken2 :: (a, b) :- b
- contract :: a :- (a, a)
- strengthen1 :: Dict b -> (a :- c) -> a :- (b, c)
- strengthen2 :: Dict b -> (a :- c) -> a :- (c, b)
- (&&&) :: (a :- b) -> (a :- c) -> a :- (b, c)
- (***) :: (a :- b) -> (c :- d) -> (a, c) :- (b, d)
- trans :: (b :- c) -> (a :- b) -> a :- c
- refl :: a :- a
- class Any => Bottom where
- no :: a
- top :: a :- ()
- bottom :: Bottom :- a
- mapDict :: (a :- b) -> Dict a -> Dict b
- unmapDict :: (Dict a -> Dict b) -> a :- b
- class Class b h | h -> b where
- class b :=> h | h -> b where
The Kind of Constraints
data Constraint #
The kind of constraints, like Show a
Dictionary
data Dict :: Constraint -> * where #
Values of type Dict pp.
e.g.
Dict::Dict(EqInt)
captures a dictionary that proves we have an:
instance Eq 'Int
Pattern matching on the Dict constructor will bring this instance into scope.
Instances
| a :=> (Monoid (Dict a)) # | |
| a :=> (Read (Dict a)) # | |
| a :=> (Bounded (Dict a)) # | |
| a :=> (Enum (Dict a)) # | |
| () :=> (Eq (Dict a)) # | |
| () :=> (Ord (Dict a)) # | |
| () :=> (Show (Dict a)) # | |
| () :=> (Semigroup (Dict a)) # | |
| a => Bounded (Dict a) # | |
| a => Enum (Dict a) # | |
Defined in Data.Constraint | |
| Eq (Dict a) # | |
| (Typeable p, p) => Data (Dict p) # | |
Defined in Data.Constraint 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) # | |
| a => Read (Dict a) # | |
| Show (Dict a) # | |
| Semigroup (Dict a) # | |
| a => Monoid (Dict a) # | |
| NFData (Dict c) # | |
Defined in Data.Constraint | |
Entailment
This is the type of entailment.
a is read as :- ba "entails" b.
With this we can actually build a category for Constraint resolution.
e.g.
Because Eq aOrd aOrd aEq a
Because instance exists, we can show that Ord a => Ord [a]Ord aOrd [a]
This relationship is captured in the :- entailment type here.
Since p and entailment composes, :- p:- 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]
What are the two ways?
Well, we can go from Ord a :- Eq aEq a :- Eq [a]Ord a :- Ord [a]Ord [a] :- Eq [a]
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.
Instances
| Category (:-) # | Possible since GHC 7.8, when |
| () :=> (Eq (a :- b)) # | |
| () :=> (Ord (a :- b)) # | |
| () :=> (Show (a :- b)) # | |
| Eq (a :- b) # | Assumes |
| (Typeable p, Typeable q, p, q) => Data (p :- q) # | |
Defined in Data.Constraint 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 |
Defined in Data.Constraint | |
| Show (a :- b) # | |
| a => NFData (a :- b) # | |
Defined in Data.Constraint | |
(\\) :: a => (b => r) -> (a :- b) -> r infixl 1 #
Given that a :- b, derive something that needs a context b, using the context a
Weakening a constraint product
The category of constraints is Cartesian. We can forget information.
Weakening a constraint product
The category of constraints is Cartesian. We can forget information.
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.
Any inhabits every kind, including Constraint but is uninhabited, making it impossible to define an instance.
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.
This demonstrates the law of classical logic "ex falso quodlibet"
Dict is fully faithful
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
instanceClass(Foo a) (Bar a) wherecls=SubDict
Now the user can use 'cls :: Bar a :- Foo a'
Instances
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] whereins=SubDict