License | BSD-style (see the LICENSE file in the distribution) |
---|---|

Maintainer | libraries@haskell.org |

Stability | experimental |

Portability | not portable |

Safe Haskell | Trustworthy |

Language | Haskell2010 |

Definition of propositional equality `(:~:)`

. Pattern-matching on a variable
of type `(a :~: b)`

produces a proof that `a ~ b`

.

*Since: 4.7.0.0*

- data a :~: b where
- class (~#) j k a b => (j ~~ k) a b
- sym :: (a :~: b) -> b :~: a
- trans :: (a :~: b) -> (b :~: c) -> a :~: c
- castWith :: (a :~: b) -> a -> b
- gcastWith :: (a :~: b) -> (a ~ b => r) -> r
- apply :: (f :~: g) -> (a :~: b) -> f a :~: g b
- inner :: (f a :~: g b) -> a :~: b
- outer :: (f a :~: g b) -> f :~: g
- class TestEquality f where
- type family (a :: k) == (b :: k) :: Bool

# 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: 4.7.0.0*

Category k ((:~:) k) Source # | |

TestEquality k ((:~:) k a) Source # | |

TestCoercion k ((:~:) k a) Source # | |

(~) k a b => Bounded ((:~:) k a b) Source # | |

(~) k a b => Enum ((:~:) k a b) Source # | |

Eq ((:~:) k a b) Source # | |

((~) * a b, Data a) => Data ((:~:) * a b) Source # | |

Ord ((:~:) k a b) Source # | |

(~) k a b => Read ((:~:) k a b) Source # | |

Show ((:~:) k a b) Source # | |

class (~#) j k a b => (j ~~ k) a b #

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.

# Working with equality

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

Generalized form of type-safe cast using propositional equality

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

Extract equality of the arguments from an equality of a 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.

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

Conditionally prove the equality of `a`

and `b`

.

TestEquality k ((:~:) k a) Source # | |

# Boolean type-level equality

type family (a :: k) == (b :: k) :: Bool infix 4 Source #

A type family to compute Boolean equality. Instances are provided
only for *open* kinds, such as `*`

and function kinds. Instances are
also provided for datatypes exported from base. A poly-kinded instance
is *not* provided, as a recursive definition for algebraic kinds is
generally more useful.

type (==) Bool a b Source # | |

type (==) Ordering a b Source # | |

type (==) * a b Source # | |

type (==) Nat a b Source # | |

type (==) Symbol a b Source # | |

type (==) () a b Source # | |

type (==) [k] a b Source # | |

type (==) (Maybe k) a b Source # | |

type (==) (k1 -> k2) a b Source # | |

type (==) (Either k k1) a b Source # | |

type (==) (k, k1) a b Source # | |

type (==) (k, k1, k2) a b Source # | |

type (==) (k, k1, k2, k3) a b Source # | |

type (==) (k, k1, k2, k3, k4) a b Source # | |

type (==) (k, k1, k2, k3, k4, k5) a b Source # | |

type (==) (k, k1, k2, k3, k4, k5, k6) a b Source # | |

type (==) (k, k1, k2, k3, k4, k5, k6, k7) a b Source # | |

type (==) (k, k1, k2, k3, k4, k5, k6, k7, k8) a b Source # | |

type (==) (k, k1, k2, k3, k4, k5, k6, k7, k8, k9) a b Source # | |

type (==) (k, k1, k2, k3, k4, k5, k6, k7, k8, k9, k10) a b Source # | |

type (==) (k, k1, k2, k3, k4, k5, k6, k7, k8, k9, k10, k11) a b Source # | |

type (==) (k, k1, k2, k3, k4, k5, k6, k7, k8, k9, k10, k11, k12) a b Source # | |

type (==) (k, k1, k2, k3, k4, k5, k6, k7, k8, k9, k10, k11, k12, k13) a b Source # | |

type (==) (k, k1, k2, k3, k4, k5, k6, k7, k8, k9, k10, k11, k12, k13, k14) a b Source # | |