scalaz

Leibniz

object Leibniz extends AnyRef

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  1. type === [A, B] = Leibniz[$u22A5, $u22A4, A, B]

    (A === B) is a supertype of Leibniz[L,H,A,B]

  2. class LeibnizGroupoid [L_, H_ >: L_] extends GeneralizedGroupoid with Hom

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    sealed

Value Members

  1. def != (arg0: AnyRef): Boolean

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  2. def != (arg0: Any): Boolean

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  3. def ## (): Int

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  4. def == (arg0: AnyRef): Boolean

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  5. def == (arg0: Any): Boolean

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  6. def asInstanceOf [T0] : T0

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  7. def clone (): AnyRef

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  8. def eq (arg0: AnyRef): Boolean

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  9. def equals (arg0: Any): Boolean

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  10. def finalize (): Unit

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  11. def force [L, H >: L, A >: L <: H, B >: L <: H] : Leibniz[L, H, A, B]

    Unsafe coercion between types.

    Unsafe coercion between types. force abuses asInstanceOf to explicitly coerce types. It is unsafe, but needed where Leibnizian equality isn't sufficient

  12. def getClass (): java.lang.Class[_]

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  13. def hashCode (): Int

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  14. def isInstanceOf [T0] : Boolean

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  15. implicit def leibnizGroupoid [L, H >: L] : LeibnizGroupoid[L, H]

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  16. def lift [LA, LT, HA >: LA, HT >: LT, T[_ >: LA <: HA] >: LT <: HT, A >: LA <: HA, A2 >: LA <: HA] (a: Leibniz[LA, HA, A, A2]): Leibniz[LT, HT, T[A], T[A2]]

    We can lift equality into any type constructor

  17. def lift2 [LA, LB, LT, HA >: LA, HB >: LB, HT >: LT, T[_ >: LA <: HA, _ >: LB <: HB] >: LT <: HT, A >: LA <: HA, A2 >: LA <: HA, B >: LB <: HB, B2 >: LB <: HB] (a: Leibniz[LA, HA, A, A2], b: Leibniz[LB, HB, B, B2]): Leibniz[LT, HT, T[A, B], T[A2, B2]]

    We can lift equality into any type constructor

  18. def lift3 [LA, LB, LC, LT, HA >: LA, HB >: LB, HC >: LC, HT >: LT, T[_ >: LA <: HA, _ >: LB <: HB, _ >: LC <: HC] >: LT <: HT, A >: LA <: HA, A2 >: LA <: HA, B >: LB <: HB, B2 >: LB <: HB, C >: LC <: HC, C2 >: LC <: HC] (a: Leibniz[LA, HA, A, A2], b: Leibniz[LB, HB, B, B2], c: Leibniz[LC, HC, C, C2]): Leibniz[LT, HT, T[A, B, C], T[A2, B2, C2]]

    We can lift equality into any type constructor

  19. def lower [LA, HA >: LA, T[_ >: LA <: HA], A >: LA <: HA, A2 >: LA <: HA] (t: $eq$eq$eq[T[A], T[A2]]): Leibniz[LA, HA, A, A2]

    Emir Pasalic's PhD thesis mentions that it is unknown whether or not ((A,B) === (C,D)) => (A === C) is inhabited.

    Emir Pasalic's PhD thesis mentions that it is unknown whether or not ((A,B) === (C,D)) => (A === C) is inhabited.

    Haskell can work around this issue by abusing type families as noted in Leibniz equality can be injective (Oleg Kiselyov, Haskell Cafe Mailing List 2010) but we instead turn to force.

  20. def lower2 [LA, HA >: LA, LB, HB >: LB, T[_ >: LA <: HA, _ >: LB <: HB], A >: LA <: HA, A2 >: LA <: HA, B >: LB <: HB, B2 >: LB <: HB] (t: $eq$eq$eq[T[A, B], T[A2, B2]]): (Leibniz[LA, HA, A, A2], Leibniz[LB, HB, B, B2])

  21. def ne (arg0: AnyRef): Boolean

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  22. def notify (): Unit

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  23. def notifyAll (): Unit

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  24. implicit def refl [A] : Leibniz[A, A, A, A]

    Equality is reflexive -- we rely on subtyping to expand this type

    Equality is reflexive -- we rely on subtyping to expand this type

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  25. def symm [L, H >: L, A >: L <: H, B >: L <: H] (f: Leibniz[L, H, A, B]): Leibniz[L, H, B, A]

    Equality is symmetric

  26. def synchronized [T0] (arg0: ⇒ T0): T0

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  27. def toString (): String

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  28. def trans [L, H >: L, A >: L <: H, B >: L <: H, C >: L <: H] (f: Leibniz[L, H, B, C], g: Leibniz[L, H, A, B]): Leibniz[L, H, A, C]

    Equality is transitive

  29. def wait (): Unit

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  30. def wait (arg0: Long, arg1: Int): Unit

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  31. def wait (arg0: Long): Unit

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  32. implicit def witness [A, B] (f: $eq$eq$eq[A, B]): (A) ⇒ B

    We can witness equality by using it to convert between types We rely on subtyping to enable this to work for any Leibniz arrow

    We can witness equality by using it to convert between types We rely on subtyping to enable this to work for any Leibniz arrow

    Attributes
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