Liskov

final def !=(arg0: Any): Boolean

final def ##(): Int

final def ==(arg0: Any): Boolean

final def asInstanceOf[T0]: T0

def clone(): AnyRef

def co[T[+_], A, A2](a: <~<[A, A2]): <~<[T[A], T[A2]]

We can lift subtyping into any invariant or covariant type constructor

def co2[T[+_, _], Z, A, B](a: <~<[A, Z]): <~<[T[A, B], T[Z, B]]

def co3[T[+_, _, _], Z, A, B, C](a: <~<[A, Z]): <~<[T[A, B, C], T[Z, B, C]]

def co4[T[+_, _, _, _], Z, A, B, C, D](a: <~<[A, Z]): <~<[T[A, B, C, D], T[Z, B, C, D]]

def contra[T[-_], A, A2](a: <~<[A, A2]): <~<[T[A2], T[A]]

We can lift subtyping into any contravariant type constructor

def contra1_2[T[-_, _], Z, A, B](a: <~<[A, Z]): <~<[T[Z, B], T[A, B]]

def contra1_3[T[-_, _, _], Z, A, B, C](a: <~<[A, Z]): <~<[T[Z, B, C], T[A, B, C]]

def contra1_4[T[-_, _, _, _], Z, A, B, C, D](a: <~<[A, Z]): <~<[T[Z, B, C, D], T[A, B, C, D]]

def contra2_2[T[_, -_], Z, A, B](a: <~<[B, Z]): <~<[T[A, Z], T[A, B]]

def contra2_3[T[_, -_, _], Z, A, B, C](a: <~<[B, Z]): <~<[T[A, Z, C], T[A, B, C]]

def contra2_4[T[_, -_, _, _], Z, A, B, C, D](a: <~<[B, Z]): <~<[T[A, Z, C, D], T[A, B, C, D]]

def contra3_3[T[_, _, -_], Z, A, B, C](a: <~<[C, Z]): <~<[T[A, B, Z], T[A, B, C]]

def contra3_4[T[_, _, -_, _], Z, A, B, C, D](a: <~<[C, Z]): <~<[T[A, B, Z, D], T[A, B, C, D]]

def contra4_4[T[_, _, _, -_], Z, A, B, C, D](a: <~<[D, Z]): <~<[T[A, B, C, Z], T[A, B, C, D]]

final def eq(arg0: AnyRef): Boolean

def equals(arg0: Any): Boolean

def finalize(): Unit

def force[A, B]: <~<[A, B]

Unsafely force a claim that A is a subtype of B

final def getClass(): Class[_]

def hashCode(): Int

final def isInstanceOf[T0]: Boolean

implicit def isa[A, B >: A]: <~<[A, B]

Lift Scala's subtyping relationship

def lift2[T[+_, +_], A, A2, B, B2](a: <~<[A, A2], b: <~<[B, B2]): <~<[T[A, B], T[A2, B2]]

lift2(a,b) = co1_2(a) compose co2_2(b)

def lift3[T[+_, +_, +_], A, A2, B, B2, C, C2](a: <~<[A, A2], b: <~<[B, B2], c: <~<[C, C2]): <~<[T[A, B, C], T[A2, B2, C2]]

lift3(a,b,c) = co1_3(a) compose co2_3(b) compose co3_3(c)

def lift4[T[+_, +_, +_, +_], A, A2, B, B2, C, C2, D, D2](a: <~<[A, A2], b: <~<[B, B2], c: <~<[C, C2], d: <~<[D, D2]): <~<[T[A, B, C, D], T[A2, B2, C2, D2]]

lift4(a,b,c,d) = co1_3(a) compose co2_3(b) compose co3_3(c) compose co4_4(d)

def liftF1[F[-_, +_], A, A2, R, R2](a: <~<[A, A2], r: <~<[R, R2]): <~<[F[A2, R], F[A, R2]]

Lift subtyping into a Function1-like type liftF1(a,r) = contra1_2(a) compose co2_2(b)

def liftF2[F[-_, -_, +_], A, A2, B, B2, R, R2](a: <~<[A, A2], b: <~<[B, B2], r: <~<[R, R2]): <~<[F[A2, B2, R], F[A, B, R2]]

Lift subtyping into a function liftF2(a,b,r) = contra1_3(a) compose contra2_3(b) compose co3_3(c)

def liftF3[F[-_, -_, -_, +_], A, A2, B, B2, C, C2, R, R2](a: <~<[A, A2], b: <~<[B, B2], c: <~<[C, C2], r: <~<[R, R2]): <~<[F[A2, B2, C2, R], F[A, B, C, R2]]

Lift subtyping into a function liftF3(a,b,c,r) = contra1_4(a) compose contra2_4(b) compose contra3_4(c) compose co3_4(d)

def liftF4[F[-_, -_, -_, -_, +_], A, A2, B, B2, C, C2, D, D2, R, R2](a: <~<[A, A2], b: <~<[B, B2], c: <~<[C, C2], d: <~<[D, D2], r: <~<[R, R2]): <~<[F[A2, B2, C2, D2, R], F[A, B, C, D, R2]]

Lift subtyping into a function

implicit def liskov: Category[<~<]

Subtyping forms a category

final def ne(arg0: AnyRef): Boolean

final def notify(): Unit

final def notifyAll(): Unit

implicit def refl[A]: <~<[A, A]

Subtyping is reflexive

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

def toString(): String

def trans[A, B, C](f: <~<[B, C], g: <~<[A, B]): <~<[A, C]

Subtyping is transitive

def unco[F[+_], Z, A](a: <~<[F[A], F[Z]])(implicit arg0: Injective[F]): <~<[A, Z]

def unco2_1[F[+_, _], Z, A, B](a: <~<[F[A, B], F[Z, B]])(implicit arg0: Injective2[F]): <~<[A, Z]

def unco2_2[F[_, +_], Z, A, B](a: <~<[F[A, B], F[A, Z]])(implicit arg0: Injective2[F]): <~<[B, Z]

def uncontra[F[-_], Z, A](a: <~<[F[A], F[Z]])(implicit arg0: Injective[F]): <~<[Z, A]

def uncontra2_1[F[-_, _], Z, A, B](a: <~<[F[A, B], F[Z, B]])(implicit arg0: Injective2[F]): <~<[Z, A]

def uncontra2_2[F[_, -_], Z, A, B](a: <~<[F[A, B], F[A, Z]])(implicit arg0: Injective2[F]): <~<[Z, B]

final def wait(): Unit

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

final def wait(arg0: Long): Unit

implicit def witness[A, B](lt: <~<[A, B]): (A) ⇒ B

We can witness equality by using it to convert between types

def inv[T[_], A, A2](a: <~<[A, A2]): <~<[T[A], T[A2]]