base-4.16.1.0: Basic libraries

Data.Type.Equality

Description

Definition of propositional equality (:~:). Pattern-matching on a variable of type (a :~: b) produces a proof that a ~ b.

Since: base-4.7.0.0

Synopsis

# The equality types

data a :~: b where infix 4 Source #

Propositional equality. If a :~: b is inhabited by some terminating value, then the type a is the same as the type b. To use this equality in practice, pattern-match on the a :~: b to get out the Refl constructor; in the body of the pattern-match, the compiler knows that a ~ b.

Since: base-4.7.0.0

Constructors

 Refl :: a :~: a

#### Instances

Instances details
 Category ((:~:) :: k -> k -> Type) Source # Since: base-4.7.0.0 Instance detailsDefined in Control.Category Methodsid :: forall (a :: k0). a :~: a Source #(.) :: forall (b :: k0) (c :: k0) (a :: k0). (b :~: c) -> (a :~: b) -> a :~: c Source # TestCoercion ((:~:) a :: k -> Type) Source # Since: base-4.7.0.0 Instance detailsDefined in Data.Type.Coercion MethodstestCoercion :: forall (a0 :: k0) (b :: k0). (a :~: a0) -> (a :~: b) -> Maybe (Coercion a0 b) Source # TestEquality ((:~:) a :: k -> Type) Source # Since: base-4.7.0.0 Instance detailsDefined in Data.Type.Equality MethodstestEquality :: forall (a0 :: k0) (b :: k0). (a :~: a0) -> (a :~: b) -> Maybe (a0 :~: b) Source # (a ~ b, Data a) => Data (a :~: b) Source # Since: base-4.7.0.0 Instance detailsDefined in Data.Data Methodsgfoldl :: (forall d b0. Data d => c (d -> b0) -> d -> c b0) -> (forall g. g -> c g) -> (a :~: b) -> c (a :~: b) Source #gunfold :: (forall b0 r. Data b0 => c (b0 -> r) -> c r) -> (forall r. r -> c r) -> Constr -> c (a :~: b) Source #toConstr :: (a :~: b) -> Constr Source #dataTypeOf :: (a :~: b) -> DataType Source #dataCast1 :: Typeable t => (forall d. Data d => c (t d)) -> Maybe (c (a :~: b)) Source #dataCast2 :: Typeable t => (forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c (a :~: b)) Source #gmapT :: (forall b0. Data b0 => b0 -> b0) -> (a :~: b) -> a :~: b Source #gmapQl :: (r -> r' -> r) -> r -> (forall d. Data d => d -> r') -> (a :~: b) -> r Source #gmapQr :: forall r r'. (r' -> r -> r) -> r -> (forall d. Data d => d -> r') -> (a :~: b) -> r Source #gmapQ :: (forall d. Data d => d -> u) -> (a :~: b) -> [u] Source #gmapQi :: Int -> (forall d. Data d => d -> u) -> (a :~: b) -> u Source #gmapM :: Monad m => (forall d. Data d => d -> m d) -> (a :~: b) -> m (a :~: b) Source #gmapMp :: MonadPlus m => (forall d. Data d => d -> m d) -> (a :~: b) -> m (a :~: b) Source #gmapMo :: MonadPlus m => (forall d. Data d => d -> m d) -> (a :~: b) -> m (a :~: b) Source # a ~ b => Bounded (a :~: b) Source # Since: base-4.7.0.0 Instance detailsDefined in Data.Type.Equality MethodsminBound :: a :~: b Source #maxBound :: a :~: b Source # a ~ b => Enum (a :~: b) Source # Since: base-4.7.0.0 Instance detailsDefined in Data.Type.Equality Methodssucc :: (a :~: b) -> a :~: b Source #pred :: (a :~: b) -> a :~: b Source #toEnum :: Int -> a :~: b Source #fromEnum :: (a :~: b) -> Int Source #enumFrom :: (a :~: b) -> [a :~: b] Source #enumFromThen :: (a :~: b) -> (a :~: b) -> [a :~: b] Source #enumFromTo :: (a :~: b) -> (a :~: b) -> [a :~: b] Source #enumFromThenTo :: (a :~: b) -> (a :~: b) -> (a :~: b) -> [a :~: b] Source # a ~ b => Read (a :~: b) Source # Since: base-4.7.0.0 Instance detailsDefined in Data.Type.Equality MethodsreadsPrec :: Int -> ReadS (a :~: b) Source #readList :: ReadS [a :~: b] Source # Show (a :~: b) Source # Since: base-4.7.0.0 Instance detailsDefined in Data.Type.Equality MethodsshowsPrec :: Int -> (a :~: b) -> ShowS Source #show :: (a :~: b) -> String Source #showList :: [a :~: b] -> ShowS Source # Eq (a :~: b) Source # Since: base-4.7.0.0 Instance detailsDefined in Data.Type.Equality Methods(==) :: (a :~: b) -> (a :~: b) -> Bool Source #(/=) :: (a :~: b) -> (a :~: b) -> Bool Source # Ord (a :~: b) Source # Since: base-4.7.0.0 Instance detailsDefined in Data.Type.Equality Methodscompare :: (a :~: b) -> (a :~: b) -> Ordering Source #(<) :: (a :~: b) -> (a :~: b) -> Bool Source #(<=) :: (a :~: b) -> (a :~: b) -> Bool Source #(>) :: (a :~: b) -> (a :~: b) -> Bool Source #(>=) :: (a :~: b) -> (a :~: b) -> Bool Source #max :: (a :~: b) -> (a :~: b) -> a :~: b Source #min :: (a :~: b) -> (a :~: b) -> a :~: b Source #

class a ~# b => (a :: k0) ~~ (b :: k1) infix 4 Source #

Lifted, heterogeneous equality. By lifted, we mean that it can be bogus (deferred type error). By heterogeneous, the two types a and b might have different kinds. Because ~~ can appear unexpectedly in error messages to users who do not care about the difference between heterogeneous equality ~~ and homogeneous equality ~, this is printed as ~ unless -fprint-equality-relations is set.

In 0.7.0, the fixity was set to infix 4 to match the fixity of :~~:.

data a :~~: b where infix 4 Source #

Kind heterogeneous propositional equality. Like :~:, a :~~: b is inhabited by a terminating value if and only if a is the same type as b.

Since: base-4.10.0.0

Constructors

 HRefl :: a :~~: a

#### Instances

Instances details
 Category ((:~~:) :: k -> k -> Type) Source # Since: base-4.10.0.0 Instance detailsDefined in Control.Category Methodsid :: forall (a :: k0). a :~~: a Source #(.) :: forall (b :: k0) (c :: k0) (a :: k0). (b :~~: c) -> (a :~~: b) -> a :~~: c Source # TestCoercion ((:~~:) a :: k -> Type) Source # Since: base-4.10.0.0 Instance detailsDefined in Data.Type.Coercion MethodstestCoercion :: forall (a0 :: k0) (b :: k0). (a :~~: a0) -> (a :~~: b) -> Maybe (Coercion a0 b) Source # TestEquality ((:~~:) a :: k -> Type) Source # Since: base-4.10.0.0 Instance detailsDefined in Data.Type.Equality MethodstestEquality :: forall (a0 :: k0) (b :: k0). (a :~~: a0) -> (a :~~: b) -> Maybe (a0 :~: b) Source # (Typeable i, Typeable j, Typeable a, Typeable b, a ~~ b) => Data (a :~~: b) Source # Since: base-4.10.0.0 Instance detailsDefined in Data.Data Methodsgfoldl :: (forall d b0. Data d => c (d -> b0) -> d -> c b0) -> (forall g. g -> c g) -> (a :~~: b) -> c (a :~~: b) Source #gunfold :: (forall b0 r. Data b0 => c (b0 -> r) -> c r) -> (forall r. r -> c r) -> Constr -> c (a :~~: b) Source #toConstr :: (a :~~: b) -> Constr Source #dataTypeOf :: (a :~~: b) -> DataType Source #dataCast1 :: Typeable t => (forall d. Data d => c (t d)) -> Maybe (c (a :~~: b)) Source #dataCast2 :: Typeable t => (forall d e. (Data d, Data e) => c (t d e)) -> Maybe (c (a :~~: b)) Source #gmapT :: (forall b0. Data b0 => b0 -> b0) -> (a :~~: b) -> a :~~: b Source #gmapQl :: (r -> r' -> r) -> r -> (forall d. Data d => d -> r') -> (a :~~: b) -> r Source #gmapQr :: forall r r'. (r' -> r -> r) -> r -> (forall d. Data d => d -> r') -> (a :~~: b) -> r Source #gmapQ :: (forall d. Data d => d -> u) -> (a :~~: b) -> [u] Source #gmapQi :: Int -> (forall d. Data d => d -> u) -> (a :~~: b) -> u Source #gmapM :: Monad m => (forall d. Data d => d -> m d) -> (a :~~: b) -> m (a :~~: b) Source #gmapMp :: MonadPlus m => (forall d. Data d => d -> m d) -> (a :~~: b) -> m (a :~~: b) Source #gmapMo :: MonadPlus m => (forall d. Data d => d -> m d) -> (a :~~: b) -> m (a :~~: b) Source # a ~~ b => Bounded (a :~~: b) Source # Since: base-4.10.0.0 Instance detailsDefined in Data.Type.Equality MethodsminBound :: a :~~: b Source #maxBound :: a :~~: b Source # a ~~ b => Enum (a :~~: b) Source # Since: base-4.10.0.0 Instance detailsDefined in Data.Type.Equality Methodssucc :: (a :~~: b) -> a :~~: b Source #pred :: (a :~~: b) -> a :~~: b Source #toEnum :: Int -> a :~~: b Source #fromEnum :: (a :~~: b) -> Int Source #enumFrom :: (a :~~: b) -> [a :~~: b] Source #enumFromThen :: (a :~~: b) -> (a :~~: b) -> [a :~~: b] Source #enumFromTo :: (a :~~: b) -> (a :~~: b) -> [a :~~: b] Source #enumFromThenTo :: (a :~~: b) -> (a :~~: b) -> (a :~~: b) -> [a :~~: b] Source # a ~~ b => Read (a :~~: b) Source # Since: base-4.10.0.0 Instance detailsDefined in Data.Type.Equality MethodsreadsPrec :: Int -> ReadS (a :~~: b) Source #readList :: ReadS [a :~~: b] Source # Show (a :~~: b) Source # Since: base-4.10.0.0 Instance detailsDefined in Data.Type.Equality MethodsshowsPrec :: Int -> (a :~~: b) -> ShowS Source #show :: (a :~~: b) -> String Source #showList :: [a :~~: b] -> ShowS Source # Eq (a :~~: b) Source # Since: base-4.10.0.0 Instance detailsDefined in Data.Type.Equality Methods(==) :: (a :~~: b) -> (a :~~: b) -> Bool Source #(/=) :: (a :~~: b) -> (a :~~: b) -> Bool Source # Ord (a :~~: b) Source # Since: base-4.10.0.0 Instance detailsDefined in Data.Type.Equality Methodscompare :: (a :~~: b) -> (a :~~: b) -> Ordering Source #(<) :: (a :~~: b) -> (a :~~: b) -> Bool Source #(<=) :: (a :~~: b) -> (a :~~: b) -> Bool Source #(>) :: (a :~~: b) -> (a :~~: b) -> Bool Source #(>=) :: (a :~~: b) -> (a :~~: b) -> Bool Source #max :: (a :~~: b) -> (a :~~: b) -> a :~~: b Source #min :: (a :~~: b) -> (a :~~: b) -> a :~~: b Source #

# Working with equality

sym :: (a :~: b) -> b :~: a Source #

Symmetry of equality

trans :: (a :~: b) -> (b :~: c) -> a :~: c Source #

Transitivity of equality

castWith :: (a :~: b) -> a -> b Source #

Type-safe cast, using propositional equality

gcastWith :: (a :~: b) -> (a ~ b => r) -> r Source #

Generalized form of type-safe cast using propositional equality

apply :: (f :~: g) -> (a :~: b) -> f a :~: g b Source #

Apply one equality to another, respectively

inner :: (f a :~: g b) -> a :~: b Source #

Extract equality of the arguments from an equality of applied types

outer :: (f a :~: g b) -> f :~: g Source #

Extract equality of type constructors from an equality of applied types

# Inferring equality from other types

class TestEquality f where Source #

This class contains types where you can learn the equality of two types from information contained in terms. Typically, only singleton types should inhabit this class.

Methods

testEquality :: f a -> f b -> Maybe (a :~: b) Source #

Conditionally prove the equality of a and b.

#### Instances

Instances details
 TestEquality (TypeRep :: k -> Type) Source # Instance detailsDefined in Data.Typeable.Internal MethodstestEquality :: forall (a :: k0) (b :: k0). TypeRep a -> TypeRep b -> Maybe (a :~: b) Source # TestEquality ((:~:) a :: k -> Type) Source # Since: base-4.7.0.0 Instance detailsDefined in Data.Type.Equality MethodstestEquality :: forall (a0 :: k0) (b :: k0). (a :~: a0) -> (a :~: b) -> Maybe (a0 :~: b) Source # TestEquality ((:~~:) a :: k -> Type) Source # Since: base-4.10.0.0 Instance detailsDefined in Data.Type.Equality MethodstestEquality :: forall (a0 :: k0) (b :: k0). (a :~~: a0) -> (a :~~: b) -> Maybe (a0 :~: b) Source # TestEquality f => TestEquality (Compose f g :: k2 -> Type) Source # The deduction (via generativity) that if g x :~: g y then x :~: y.Since: base-4.14.0.0 Instance detailsDefined in Data.Functor.Compose MethodstestEquality :: forall (a :: k) (b :: k). Compose f g a -> Compose f g b -> Maybe (a :~: b) Source #

# Boolean type-level equality

type family a == b where ... infix 4 Source #

A type family to compute Boolean equality.

Equations

 (f a) == (g b) = (f == g) && (a == b) a == a = 'True _ == _ = 'False