License | BSD-style (see the LICENSE file in the distribution) |
---|---|
Maintainer | libraries@haskell.org |
Stability | stable |
Portability | not portable |
Safe Haskell | Safe |
Language | Haskell2010 |
Synopsis
- class a ~# b => (a :: k) ~ (b :: k)
- class a ~# b => (a :: k0) ~~ (b :: k1)
- data (a :: k) :~: (b :: k) where
- data (a :: k1) :~~: (b :: k2) where
- sym :: forall {k} (a :: k) (b :: k). (a :~: b) -> b :~: a
- trans :: forall {k} (a :: k) (b :: k) (c :: k). (a :~: b) -> (b :~: c) -> a :~: c
- castWith :: (a :~: b) -> a -> b
- gcastWith :: forall {k} (a :: k) (b :: k) r. (a :~: b) -> (a ~ b => r) -> r
- apply :: forall {k1} {k2} (f :: k1 -> k2) (g :: k1 -> k2) (a :: k1) (b :: k1). (f :~: g) -> (a :~: b) -> f a :~: g b
- inner :: forall {k1} {k2} (f :: k1 -> k2) (a :: k1) (g :: k1 -> k2) (b :: k1). (f a :~: g b) -> a :~: b
- outer :: forall {k1} {k2} (f :: k1 -> k2) (a :: k1) (g :: k1 -> k2) (b :: k1). (f a :~: g b) -> f :~: g
- class TestEquality (f :: k -> Type) where
- testEquality :: forall (a :: k) (b :: k). f a -> f b -> Maybe (a :~: b)
- type family (a :: k) == (b :: k) :: Bool where ...
The equality types
class a ~# b => (a :: k) ~ (b :: k) infix 4 Source #
Lifted, homogeneous equality. By lifted, we mean that it
can be bogus (deferred type error). By homogeneous, the two
types a
and b
must have the same kinds.
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 :: k) :~: (b :: k) 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
Instances
Category ((:~:) :: k -> k -> Type) Source # | Since: base-4.7.0.0 |
TestCoercion ((:~:) a :: k -> Type) Source # | Since: base-4.7.0.0 |
Defined in GHC.Internal.Data.Type.Coercion | |
TestEquality ((:~:) a :: k -> Type) Source # | Since: base-4.7.0.0 |
Defined in GHC.Internal.Data.Type.Equality | |
(a ~ b, Data a) => Data (a :~: b) Source # | Since: base-4.7.0.0 |
Defined in GHC.Internal.Data.Data gfoldl :: (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 |
a ~ b => Enum (a :~: b) Source # | Since: base-4.7.0.0 |
Defined in GHC.Internal.Data.Type.Equality succ :: (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 |
Show (a :~: b) Source # | Since: base-4.7.0.0 |
Eq (a :~: b) Source # | Since: base-4.7.0.0 |
Ord (a :~: b) Source # | Since: base-4.7.0.0 |
Defined in GHC.Internal.Data.Type.Equality |
data (a :: k1) :~~: (b :: k2) 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
Instances
Category ((:~~:) :: k -> k -> Type) Source # | Since: base-4.10.0.0 |
TestCoercion ((:~~:) a :: k -> Type) Source # | Since: base-4.10.0.0 |
Defined in GHC.Internal.Data.Type.Coercion | |
TestEquality ((:~~:) a :: k -> Type) Source # | Since: base-4.10.0.0 |
Defined in GHC.Internal.Data.Type.Equality | |
(Typeable i, Typeable j, Typeable a, Typeable b, a ~~ b) => Data (a :~~: b) Source # | Since: base-4.10.0.0 |
Defined in GHC.Internal.Data.Data gfoldl :: (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 |
a ~~ b => Enum (a :~~: b) Source # | Since: base-4.10.0.0 |
Defined in GHC.Internal.Data.Type.Equality succ :: (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 |
Show (a :~~: b) Source # | Since: base-4.10.0.0 |
Eq (a :~~: b) Source # | Since: base-4.10.0.0 |
Ord (a :~~: b) Source # | Since: base-4.10.0.0 |
Defined in GHC.Internal.Data.Type.Equality compare :: (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 # |
Working with equality
trans :: forall {k} (a :: k) (b :: k) (c :: k). (a :~: b) -> (b :~: c) -> a :~: c Source #
Transitivity of equality
gcastWith :: forall {k} (a :: k) (b :: k) r. (a :~: b) -> (a ~ b => r) -> r Source #
Generalized form of type-safe cast using propositional equality
apply :: forall {k1} {k2} (f :: k1 -> k2) (g :: k1 -> k2) (a :: k1) (b :: k1). (f :~: g) -> (a :~: b) -> f a :~: g b Source #
Apply one equality to another, respectively
inner :: forall {k1} {k2} (f :: k1 -> k2) (a :: k1) (g :: k1 -> k2) (b :: k1). (f a :~: g b) -> a :~: b Source #
Extract equality of the arguments from an equality of applied types
outer :: forall {k1} {k2} (f :: k1 -> k2) (a :: k1) (g :: k1 -> k2) (b :: k1). (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 :: k -> Type) where Source #
This class contains types where you can learn the equality of two types from information contained in terms.
The result should be Just Refl
if and only if the types applied to f
are
equal:
testEquality (x :: f a) (y :: f b) = Just Refl ⟺ a = b
Typically, only singleton types should inhabit this class. In that case type argument equality coincides with term equality:
testEquality (x :: f a) (y :: f b) = Just Refl ⟺ a = b ⟺ x = y
isJust (testEquality x y) = x == y
Singleton types are not required, however, and so the latter two would-be laws are not in fact valid in general.
testEquality :: forall (a :: k) (b :: k). f a -> f b -> Maybe (a :~: b) Source #
Conditionally prove the equality of a
and b
.
Instances
TestEquality SNat Source # | Since: base-4.18.0.0 |
Defined in GHC.Internal.TypeNats | |
TestEquality SChar Source # | Since: base-4.18.0.0 |
Defined in GHC.Internal.TypeLits | |
TestEquality SSymbol Source # | Since: base-4.18.0.0 |
Defined in GHC.Internal.TypeLits | |
TestEquality (TypeRep :: k -> Type) Source # | |
Defined in GHC.Internal.Data.Typeable.Internal | |
TestEquality ((:~:) a :: k -> Type) Source # | Since: base-4.7.0.0 |
Defined in GHC.Internal.Data.Type.Equality | |
TestEquality ((:~~:) a :: k -> Type) Source # | Since: base-4.10.0.0 |
Defined in GHC.Internal.Data.Type.Equality | |
TestEquality f => TestEquality (Compose f g :: k2 -> Type) Source # | The deduction (via generativity) that if Since: base-4.14.0.0 |
Defined in Data.Functor.Compose |