object Leibniz extends LeibnizInstances
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- Leibniz.scala
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final
def
!=(arg0: Any): Boolean
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final
def
##(): Int
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final
def
==(arg0: Any): Boolean
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final
def
asInstanceOf[T0]: T0
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def
eq(arg0: AnyRef): Boolean
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def
equals(arg0: Any): Boolean
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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
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final
def
getClass(): Class[_]
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def
hashCode(): Int
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final
def
isInstanceOf[T0]: Boolean
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implicit
val
leibniz: Category[===]
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- LeibnizInstances
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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
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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
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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
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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.
- 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])
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final
def
ne(arg0: AnyRef): Boolean
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def
notify(): Unit
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final
def
notifyAll(): Unit
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implicit
def
refl[A]: Leibniz[A, A, A, A]
Equality is reflexive -- we rely on subtyping to expand this type
- implicit def subst[A, B](a: A)(implicit f: ===[A, B]): B
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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
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final
def
synchronized[T0](arg0: ⇒ T0): T0
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def
toString(): String
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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
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final
def
wait(): Unit
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def
wait(arg0: Long, arg1: Int): Unit
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final
def
wait(arg0: Long): Unit
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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