LeibnizFunctions

Value Members

1. final def !=(arg0: Any): Boolean

   Definition Classes

   AnyRef → Any

2. final def ##(): Int

   Definition Classes

   AnyRef → Any

3. final def ==(arg0: Any): Boolean

   Definition Classes

   AnyRef → Any

4. final def asInstanceOf[T0]: T0

   Definition Classes

   Any

5. def clone(): AnyRef

   Attributes

   protected[java.lang]

   Definition Classes

   AnyRef

   Annotations

   @throws( ... )

6. final def eq(arg0: AnyRef): Boolean

   Definition Classes

   AnyRef

7. def equals(arg0: Any): Boolean

   Definition Classes

   AnyRef → Any

8. def finalize(): Unit

   Attributes

   protected[java.lang]

   Definition Classes

   AnyRef

   Annotations

   @throws( classOf[java.lang.Throwable] )

9. 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

10. final def getClass(): Class[_]

    Definition Classes

    AnyRef → Any

11. def hashCode(): Int

    Definition Classes

    AnyRef → Any

12. final def isInstanceOf[T0]: Boolean

    Definition Classes

    Any

13. 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

14. 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

15. 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

16. def lower[LA, HA >: LA, T[_ >: LA <: HA], A >: LA <: HA, A2 >: LA <: HA](t: ===[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.

17. 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: ===[T[A, B], T[A2, B2]]): (Leibniz[LA, HA, A, A2], Leibniz[LB, HB, B, B2])

18. final def ne(arg0: AnyRef): Boolean

    Definition Classes

    AnyRef

19. final def notify(): Unit

    Definition Classes

    AnyRef

20. final def notifyAll(): Unit

    Definition Classes

    AnyRef

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

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

22. implicit def subst[A, B](a: A)(implicit f: ===[A, B]): B

23. 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

24. final def synchronized[T0](arg0: ⇒ T0): T0

    Definition Classes

    AnyRef

25. def toString(): String

    Definition Classes

    AnyRef → Any

26. 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

27. final def wait(): Unit

    Definition Classes

    AnyRef

    Annotations

    @throws( ... )

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

    Definition Classes

    AnyRef

    Annotations

    @throws( ... )

29. final def wait(arg0: Long): Unit

    Definition Classes

    AnyRef

    Annotations

    @throws( ... )

30. implicit def witness[A, B](f: ===[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