Rebase.Data.Type.Equality

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 :~~:.

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.

A type family to compute Boolean equality.

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.

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

Apply one equality to another, respectively

Extract equality of type constructors from an equality of applied types

Extract equality of the arguments from an equality of applied types

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.

Type-safe cast, using propositional equality

Generalized form of type-safe cast using propositional equality

Symmetry of equality

Transitivity of equality