Add InvariantSemigroupal and ability to turn Monoidals to Monoids

build.sbt

@@ -340,6 +340,26 @@ def mimaSettings(moduleName: String) = Seq(
       exclude[DirectMissingMethodProblem]("cats.data.RWSTAlternative.sequence"),
       exclude[ReversedMissingMethodProblem]("cats.MonadError.rethrow"),
       exclude[ReversedMissingMethodProblem]("cats.syntax.MonadErrorSyntax.catsSyntaxMonadErrorRethrow"),
+      exclude[InheritedNewAbstractMethodProblem]("cats.InvariantSemigroupal.composeApply"),
+      exclude[DirectMissingMethodProblem]("cats.Semigroupal.catsSemigroupalForMonoid"),
+      exclude[ReversedMissingMethodProblem]("cats.instances.InvariantMonoidalInstances.catsSemigroupalForMonoid"),
+      exclude[DirectMissingMethodProblem]("cats.implicits.catsContravariantSemigroupalForPartialOrder"),
+      exclude[DirectMissingMethodProblem]("cats.instances.PartialOrderingInstances.catsContravariantSemigroupalForPartialOrdering"),
+      exclude[ReversedMissingMethodProblem]("cats.instances.PartialOrderingInstances.catsContravariantMonoidalForPartialOrdering"),
+      exclude[DirectMissingMethodProblem]("cats.instances.package#partialOrdering.catsContravariantSemigroupalForPartialOrdering"),
+      exclude[DirectMissingMethodProblem]("cats.instances.package#all.catsContravariantSemigroupalForPartialOrder"),
+      exclude[DirectMissingMethodProblem]("cats.instances.package#all.catsContravariantSemigroupalForPartialOrdering"),
+      exclude[DirectMissingMethodProblem]("cats.data.ConstInstances1.catsConstInvariantMonoidal"),
+      exclude[ReversedMissingMethodProblem]("cats.instances.PartialOrderInstances.cats$instances$PartialOrderInstances$_setter_$catsContravariantMonoidalForPartialOrder_="),
+      exclude[ReversedMissingMethodProblem]("cats.instances.PartialOrderingInstances.cats$instances$PartialOrderingInstances$_setter_$catsContravariantMonoidalForPartialOrdering_="),
+      exclude[DirectMissingMethodProblem]("cats.instances.package#partialOrder.catsContravariantSemigroupalForPartialOrder"),
+      exclude[DirectMissingMethodProblem]("cats.instances.PartialOrderInstances.catsContravariantSemigroupalForPartialOrder"),
+      exclude[ReversedMissingMethodProblem]("cats.instances.PartialOrderInstances.catsContravariantMonoidalForPartialOrder"),
+      exclude[DirectMissingMethodProblem]("cats.implicits.catsContravariantSemigroupalForPartialOrdering"),
+      exclude[InheritedNewAbstractMethodProblem]("cats.InvariantMonoidal.point"),
+      exclude[DirectMissingMethodProblem]("cats.InvariantMonoidal.pure"),
+      exclude[ReversedMissingMethodProblem]("cats.InvariantMonoidal.point"),
+      exclude[ReversedMissingMethodProblem]("cats.InvariantMonoidal.unit"),
       exclude[DirectMissingMethodProblem]("cats.data.CokleisliArrow.id"),
       exclude[IncompatibleResultTypeProblem]("cats.data.CokleisliArrow.id"),  
       exclude[DirectMissingMethodProblem]("cats.Apply#Ops.followedBy"),

core/src/main/scala/cats/Applicative.scala

@@ -14,7 +14,9 @@ import simulacrum.typeclass
  *
  * Must obey the laws defined in cats.laws.ApplicativeLaws.
  */
-@typeclass trait Applicative[F[_]] extends Apply[F] { self =>
+@typeclass trait Applicative[F[_]] extends Apply[F] with InvariantMonoidal[F] { self =>
+
+
 
   /**
    * `pure` lifts any value into the Applicative Functor.
@@ -184,6 +186,7 @@ import simulacrum.typeclass
    */
   def whenA[A](cond: Boolean)(f: => F[A]): F[Unit] =
     if (cond) void(f) else pure(())
+
 }
 
 object Applicative {

core/src/main/scala/cats/Apply.scala

@@ -9,7 +9,7 @@ import simulacrum.noop
  * Must obey the laws defined in cats.laws.ApplyLaws.
  */
 @typeclass(excludeParents = List("ApplyArityFunctions"))
-trait Apply[F[_]] extends Functor[F] with Semigroupal[F] with ApplyArityFunctions[F] { self =>
+trait Apply[F[_]] extends Functor[F] with InvariantSemigroupal[F] with ApplyArityFunctions[F] { self =>
 
   /**
    * Given a value and a function in the Apply context, applies the
@@ -94,6 +94,7 @@ trait Apply[F[_]] extends Functor[F] with Semigroupal[F] with ApplyArityFunction
       val F = self
       val G = Apply[G]
     }
+
 }
 
 object Apply {

core/src/main/scala/cats/Composed.scala

@@ -125,7 +125,7 @@ private[cats] trait ComposedApplicativeContravariantMonoidal[F[_], G[_]] extends
   def F: Applicative[F]
   def G: ContravariantMonoidal[G]
 
-  override def unit[A]: F[G[A]] = F.pure(G.unit)
+  override def unit: F[G[Unit]] = F.pure(G.unit)
 
   override def contramap[A, B](fa: F[G[A]])(f: B => A): F[G[B]] =
     F.map(fa)(G.contramap(_)(f))
@@ -144,6 +144,18 @@ private[cats] trait ComposedSemigroupal[F[_], G[_]] extends ContravariantSemigro
     }
 }
 
+private[cats] trait ComposedInvariantApplySemigroupal[F[_], G[_]] extends InvariantSemigroupal[λ[α => F[G[α]]]] with ComposedInvariantCovariant[F, G] { outer =>
+  def F: InvariantSemigroupal[F]
+  def G: Apply[G]
+
+  def product[A, B](fa: F[G[A]], fb: F[G[B]]): F[G[(A, B)]] =
+    F.imap(F.product(fa, fb)) { case (ga, gb) =>
+      G.map2(ga, gb)(_ -> _)
+    } { g: G[(A, B)] =>
+      (G.map(g)(_._1), G.map(g)(_._2))
+    }
+}
+
 private[cats] trait ComposedCovariantContravariant[F[_], G[_]] extends Contravariant[λ[α => F[G[α]]]] { outer =>
   def F: Functor[F]
   def G: Contravariant[G]

core/src/main/scala/cats/ContravariantMonoidal.scala

@@ -11,12 +11,20 @@ import simulacrum.typeclass
  * Based on ekmett's contravariant library:
  * https://hackage.haskell.org/package/contravariant-1.4/docs/Data-Functor-Contravariant-Divisible.html
  */
-@typeclass trait ContravariantMonoidal[F[_]] extends ContravariantSemigroupal[F] { self =>
+@typeclass trait ContravariantMonoidal[F[_]] extends ContravariantSemigroupal[F] with InvariantMonoidal[F] {
   /**
-   * `unit` produces an instance of `F` for any type `A`
+   * `trivial` produces an instance of `F` for any type `A`
    * that is trivial with respect to `contramap2` along
    * the diagonal
    */
-  def unit[A]: F[A]
+  def trivial[A]: F[A] = contramap(unit)(_ => ())
+
+}
+object ContravariantMonoidal extends SemigroupalArityFunctions {
+  def monoid[F[_], A](implicit f: ContravariantMonoidal[F]): Monoid[F[A]] =
+    new ContravariantMonoidalMonoid[F, A](f)
+}
+
+private[cats] class ContravariantMonoidalMonoid[F[_], A](f: ContravariantMonoidal[F]) extends ContravariantSemigroupalSemigroup[F, A](f) with Monoid[F[A]] {
+  def empty: F[A] = f.trivial
 }
-object ContravariantMonoidal extends SemigroupalArityFunctions

core/src/main/scala/cats/ContravariantSemigroupal.scala

@@ -6,11 +6,21 @@ import simulacrum.typeclass
  * [[ContravariantSemigroupal]] is nothing more than something both contravariant
  * and Semigroupal. It comes up enough to be useful, and composes well
  */
-@typeclass trait ContravariantSemigroupal[F[_]] extends Semigroupal[F] with Contravariant[F] { self =>
+@typeclass trait ContravariantSemigroupal[F[_]] extends InvariantSemigroupal[F] with Contravariant[F] { self =>
   override def composeFunctor[G[_]: Functor]: ContravariantSemigroupal[λ[α => F[G[α]]]] =
     new ComposedSemigroupal[F, G] {
       def F = self
       def G = Functor[G]
     }
+
+}
+
+object ContravariantSemigroupal extends SemigroupalArityFunctions {
+  def semigroup[F[_], A](implicit f: ContravariantSemigroupal[F]): Semigroup[F[A]] =
+    new ContravariantSemigroupalSemigroup[F, A](f)
+}
+
+private[cats] class ContravariantSemigroupalSemigroup[F[_], A](f: ContravariantSemigroupal[F]) extends Semigroup[F[A]] {
+  def combine(a: F[A], b: F[A]): F[A] =
+    ContravariantSemigroupal.contramap2(a, b)((a: A) => (a, a))(f, f)
 }
-object ContravariantSemigroupal extends SemigroupalArityFunctions

core/src/main/scala/cats/InvariantMonoidal.scala

@@ -7,6 +7,34 @@ import simulacrum.typeclass
  *
  * Must obey the laws defined in cats.laws.InvariantMonoidalLaws.
  */
-@typeclass trait InvariantMonoidal[F[_]] extends Invariant[F] with Semigroupal[F] {
-  def pure[A](a: A): F[A]
+@typeclass trait InvariantMonoidal[F[_]] extends InvariantSemigroupal[F] {
+  /**
+    * `point` lifts any value into a Monoidal Functor.
+    *
+    * Example:
+    * {{{
+    * scala> import cats.implicits._
+    *
+    * scala> InvariantMonoidal[Option].point(10)
+    * res0: Option[Int] = Some(10)
+    * }}}
+    */
+  def point[A](a: A): F[A] = imap(unit)(_ => a)(_ => ())
+
+  def unit: F[Unit]
+
+
+}
+
+object InvariantMonoidal {
+  /**
+    * Gives a `Monoid` instance if A itself has a `Monoid` instance.
+    */
+  def monoid[F[_], A](implicit F: InvariantMonoidal[F], A: Monoid[A]): Monoid[F[A]] =
+    new InvariantMonoidalMonoid[F, A](F, A)
+}
+
+
+private[cats] class InvariantMonoidalMonoid[F[_], A](f: InvariantMonoidal[F], monoid: Monoid[A]) extends InvariantSemigroupalSemigroup(f, monoid) with Monoid[F[A]] {
+  def empty: F[A] = f.point(monoid.empty)
 }

core/src/main/scala/cats/InvariantSemigroupal.scala

@@ -0,0 +1,30 @@
+package cats
+
+import simulacrum.typeclass
+
+/**
+  * [[InvariantSemigroupal]] is nothing more than something both invariant
+  * and Semigroupal. It comes up enough to be useful, and composes well
+  */
+@typeclass trait InvariantSemigroupal[F[_]] extends Semigroupal[F] with Invariant[F] { self =>
+
+   def composeApply[G[_]: Apply]: InvariantSemigroupal[λ[α => F[G[α]]]] =
+     new ComposedInvariantApplySemigroupal[F, G] {
+       def F = self
+       def G = Apply[G]
+     }
+
+}
+
+object InvariantSemigroupal extends SemigroupalArityFunctions {
+  /**
+    * Gives a `Semigroup` instance if A itself has a `Semigroup` instance.
+    */
+  def semigroup[F[_], A](implicit F: InvariantSemigroupal[F], A: Semigroup[A]): Semigroup[F[A]] =
+    new InvariantSemigroupalSemigroup[F, A](F, A)
+}
+
+private[cats] class InvariantSemigroupalSemigroup[F[_], A](f: InvariantSemigroupal[F], sg: Semigroup[A]) extends Semigroup[F[A]] {
+  def combine(a: F[A], b: F[A]): F[A] =
+    InvariantSemigroupal.imap2(a, b)(sg.combine)(a => (a, a))(f, f)
+}

core/src/main/scala/cats/Semigroupal.scala

@@ -16,11 +16,4 @@ import simulacrum.typeclass
   def product[A, B](fa: F[A], fb: F[B]): F[(A, B)]
 }
 
-object Semigroupal extends SemigroupalArityFunctions {
-  implicit def catsSemigroupalForMonoid: Semigroupal[Monoid] = new Semigroupal[Monoid] {
-    def product[A, B](fa: Monoid[A], fb: Monoid[B]): Monoid[(A, B)] = new Monoid[(A, B)] {
-      val empty = fa.empty -> fb.empty
-      def combine(x: (A, B), y: (A, B)): (A, B) = fa.combine(x._1, y._1) -> fb.combine(x._2, y._2)
-    }
-  }
-}
+object Semigroupal extends SemigroupalArityFunctions

core/src/main/scala/cats/data/Const.scala

@@ -66,7 +66,7 @@ private[data] sealed abstract class ConstInstances extends ConstInstances0 {
   }
 
   implicit def catsDataContravariantMonoidalForConst[D: Monoid]: ContravariantMonoidal[Const[D, ?]] = new ContravariantMonoidal[Const[D, ?]] {
-    override def unit[A] = Const.empty[D, A]
+    override def unit = Const.empty[D, Unit]
     override def contramap[A, B](fa: Const[D, A])(f: B => A): Const[D, B] =
       fa.retag[B]
     override def product[A, B](fa: Const[D, A], fb: Const[D, B]): Const[D, (A, B)] =
@@ -135,16 +135,6 @@ private[data] sealed abstract class ConstInstances0 extends ConstInstances1 {
 }
 
 private[data] sealed abstract class ConstInstances1 {
-  implicit def catsConstInvariantMonoidal[C: Monoid]: InvariantMonoidal[Const[C, ?]] = new InvariantMonoidal[Const[C, ?]] {
-    def pure[A](a: A): Const[C, A] =
-      Const.empty
-
-    def imap[A, B](fa: Const[C, A])(f: A => B)(g: B => A): Const[C, B] =
-      fa.retag[B]
-
-    def product[A, B](fa: Const[C, A], fb: Const[C, B]): Const[C, (A, B)] =
-      fa.retag[(A, B)] combine fb.retag[(A, B)]
-  }
 
   implicit def catsDataEqForConst[A: Eq, B]: Eq[Const[A, B]] = new Eq[Const[A, B]] {
     def eqv(x: Const[A, B], y: Const[A, B]): Boolean =

core/src/main/scala/cats/data/IdT.scala

@@ -76,7 +76,7 @@ private[data] sealed trait IdTApplicative[F[_]] extends Applicative[IdT[F, ?]] w
 private[data] sealed trait IdTContravariantMonoidal[F[_]] extends ContravariantMonoidal[IdT[F, ?]] {
   implicit val F0: ContravariantMonoidal[F]
 
-  override def unit[A]: IdT[F, A] = IdT(F0.unit[A])
+  override def unit: IdT[F, Unit] = IdT(F0.unit)
 
   override def contramap[A, B](fa: IdT[F, A])(f: B => A): IdT[F, B] =
     IdT(F0.contramap(fa.value)(f))

core/src/main/scala/cats/data/IndexedStateT.scala

@@ -247,10 +247,6 @@ private[data] sealed abstract class IndexedStateTInstances extends IndexedStateT
   implicit def catsDataAlternativeForIndexedStateT[F[_], S](implicit FM: Monad[F],
     FA: Alternative[F]): Alternative[IndexedStateT[F, S, S, ?]] with Monad[IndexedStateT[F, S, S, ?]] =
     new IndexedStateTAlternative[F, S] { implicit def F = FM; implicit def G = FA }
-
-  implicit def catsDataContravariantMonoidalForIndexedStateT[F[_], S](implicit FD: ContravariantMonoidal[F],
-    FA: Applicative[F]): ContravariantMonoidal[IndexedStateT[F, S, S, ?]] =
-    new IndexedStateTContravariantMonoidal[F, S] { implicit def F = FD; implicit def G = FA }
 }
 
 private[data] sealed abstract class IndexedStateTInstances1 extends IndexedStateTInstances2 {
@@ -387,11 +383,11 @@ private[data] sealed abstract class IndexedStateTContravariantMonoidal[F[_], S]
   implicit def F: ContravariantMonoidal[F]
   implicit def G: Applicative[F]
 
-  override def unit[A]: IndexedStateT[F, S, S, A]  =
-    IndexedStateT.applyF(G.pure((s: S) => F.unit[(S, A)]))
+  override def unit: IndexedStateT[F, S, S, Unit]  =
+    IndexedStateT.applyF(G.pure((s: S) => F.trivial[(S, Unit)]))
 
   override def contramap[A, B](fa: IndexedStateT[F, S, S, A])(f: B => A): IndexedStateT[F, S, S, B] =
-    contramap2(fa, unit)(((a: A) => (a, a)) compose f)
+    contramap2(fa, trivial)(((a: A) => (a, a)) compose f)
 
   override def product[A, B](fa: IndexedStateT[F, S, S, A], fb: IndexedStateT[F, S, S, B]): IndexedStateT[F, S, S, (A, B)] =
     contramap2(fa, fb)(identity)

core/src/main/scala/cats/data/Kleisli.scala

@@ -338,7 +338,7 @@ private[data] trait KleisliAlternative[F[_], A] extends Alternative[Kleisli[F, A
 private[data] sealed trait KleisliContravariantMonoidal[F[_], D] extends ContravariantMonoidal[Kleisli[F, D, ?]] {
   implicit def F: ContravariantMonoidal[F]
 
-  override def unit[A]: Kleisli[F, D, A] = Kleisli(Function.const(F.unit[A]))
+  override def unit: Kleisli[F, D, Unit] = Kleisli(Function.const(F.unit))
 
   override def contramap[A, B](fa: Kleisli[F, D, A])(f: B => A): Kleisli[F, D, B] =
     Kleisli(d => F.contramap(fa.run(d))(f))

core/src/main/scala/cats/data/Nested.scala

@@ -147,27 +147,34 @@ private[data] sealed abstract class NestedInstances8 extends NestedInstances9 {
 }
 
 private[data] sealed abstract class NestedInstances9 extends NestedInstances10 {
+  implicit def catsDataInvariantSemigroupalApplyForNested[F[_]: InvariantSemigroupal, G[_]: Apply]: InvariantSemigroupal[Nested[F, G, ?]] =
+    new NestedInvariantSemigroupalApply[F, G] {
+      val FG: InvariantSemigroupal[λ[α => F[G[α]]]] = InvariantSemigroupal[F].composeApply[G]
+    }
+}
+
+private[data] sealed abstract class NestedInstances10 extends NestedInstances11 {
   implicit def catsDataFunctorForNested[F[_]: Functor, G[_]: Functor]: Functor[Nested[F, G, ?]] =
     new NestedFunctor[F, G] {
       val FG: Functor[λ[α => F[G[α]]]] = Functor[F].compose[G]
     }
 }
 
-private[data] sealed abstract class NestedInstances10 extends NestedInstances11 {
+private[data] sealed abstract class NestedInstances11 extends NestedInstances12 {
   implicit def catsDataInvariantForNested[F[_]: Invariant, G[_]: Invariant]: Invariant[Nested[F, G, ?]] =
     new NestedInvariant[F, G] {
       val FG: Invariant[λ[α => F[G[α]]]] = Invariant[F].compose[G]
     }
 }
 
-private[data] sealed abstract class NestedInstances11 extends NestedInstances12 {
+private[data] sealed abstract class NestedInstances12 extends NestedInstances13 {
   implicit def catsDataInvariantForCovariantNested[F[_]: Invariant, G[_]: Functor]: Invariant[Nested[F, G, ?]] =
     new NestedInvariant[F, G] {
       val FG: Invariant[λ[α => F[G[α]]]] = Invariant[F].composeFunctor[G]
     }
 }
 
-private[data] sealed abstract class NestedInstances12 {
+private[data] sealed abstract class NestedInstances13 {
   implicit def catsDataInvariantForNestedContravariant[F[_]: Invariant, G[_]: Contravariant]: Invariant[Nested[F, G, ?]] =
     new NestedInvariant[F, G] {
       val FG: Invariant[λ[α => F[G[α]]]] = Invariant[F].composeContravariant[G]
@@ -284,11 +291,21 @@ private[data] trait NestedContravariant[F[_], G[_]] extends Contravariant[Nested
 private[data] trait NestedContravariantMonoidal[F[_], G[_]] extends ContravariantMonoidal[Nested[F, G, ?]] {
   def FG: ContravariantMonoidal[λ[α => F[G[α]]]]
 
-  def unit[A]: Nested[F, G, A] = Nested(FG.unit)
+  def unit: Nested[F, G, Unit] = Nested(FG.unit)
 
   def contramap[A, B](fa: Nested[F, G, A])(f: B => A): Nested[F, G, B] =
     Nested(FG.contramap(fa.value)(f))
 
   def product[A, B](fa: Nested[F, G, A], fb: Nested[F, G, B]): Nested[F, G, (A, B)] =
     Nested(FG.product(fa.value, fb.value))
 }
+
+private[data] trait NestedInvariantSemigroupalApply[F[_], G[_]] extends InvariantSemigroupal[Nested[F, G, ?]] {
+  def FG: InvariantSemigroupal[λ[α => F[G[α]]]]
+
+  def imap[A, B](fa: Nested[F, G, A])(f: A => B)(g: B => A): Nested[F, G, B] =
+    Nested(FG.imap(fa.value)(f)(g))
+
+  def product[A, B](fa: Nested[F, G, A], fb: Nested[F, G, B]): Nested[F, G, (A, B)] =
+    Nested(FG.product(fa.value, fb.value))
+}

core/src/main/scala/cats/data/OptionT.scala

@@ -300,7 +300,7 @@ private trait OptionTMonadError[F[_], E] extends MonadError[OptionT[F, ?], E] wi
 private trait OptionTContravariantMonoidal[F[_]] extends ContravariantMonoidal[OptionT[F, ?]] {
   def F: ContravariantMonoidal[F]
 
-  override def unit[A]: OptionT[F, A] = OptionT (F.unit)
+  override def unit: OptionT[F, Unit] = OptionT(F.trivial)
 
   override def contramap[A, B](fa: OptionT[F, A])(f: B => A): OptionT[F, B] =
     OptionT(F.contramap(fa.value)(_ map f))

core/src/main/scala/cats/data/Tuple2K.scala

@@ -145,7 +145,7 @@ private[data] sealed trait Tuple2KContravariant[F[_], G[_]] extends Contravarian
 private[data] sealed trait Tuple2KContravariantMonoidal[F[_], G[_]] extends ContravariantMonoidal[λ[α => Tuple2K[F, G, α]]] {
   def F: ContravariantMonoidal[F]
   def G: ContravariantMonoidal[G]
-  def unit[A]: Tuple2K[F, G, A] = Tuple2K(F.unit, G.unit)
+  def unit: Tuple2K[F, G, Unit] = Tuple2K(F.unit, G.unit)
   def product[A, B](fa: Tuple2K[F, G, A], fb: Tuple2K[F, G, B]): Tuple2K[F, G, (A, B)] =
     Tuple2K(F.product(fa.first, fb.first), G.product(fa.second, fb.second))
   def contramap[A, B](fa: Tuple2K[F, G, A])(f: B => A): Tuple2K[F, G, B] =

core/src/main/scala/cats/data/WriterT.scala

@@ -378,7 +378,7 @@ private[data] sealed trait WriterTAlternative[F[_], L] extends Alternative[Write
 private[data] sealed trait WriterTContravariantMonoidal[F[_], L] extends ContravariantMonoidal[WriterT[F, L, ?]] {
   implicit def F0: ContravariantMonoidal[F]
 
-  override def unit[A]: WriterT[F, L, A] = WriterT(F0.unit[(L, A)])
+  override def unit: WriterT[F, L, Unit] = WriterT(F0.trivial[(L, Unit)])
 
   override def contramap[A, B](fa: WriterT[F, L, A])(f: B => A): WriterT[F, L, B] =
     WriterT(F0.contramap(fa.run)((d: (L, B)) => (d._1, f(d._2))))

core/src/main/scala/cats/instances/eq.scala

@@ -8,7 +8,7 @@ trait EqInstances {
        * Defaults to the trivial equivalence relation
        * contracting the type to a point
        */
-      def unit[A]: Eq[A] = Eq.allEqual
+      def unit: Eq[Unit] = Eq.allEqual
 
       /** Derive an `Eq` for `B` given an `Eq[A]` and a function `B => A`.
        *

core/src/main/scala/cats/instances/equiv.scala

@@ -8,8 +8,8 @@ trait EquivInstances {
        * Defaults to trivially contracting the type
        * to a point
        */
-      def unit[A]: Equiv[A] = new Equiv[A] {
-        def equiv(x: A, y: A): Boolean = true
+      def unit: Equiv[Unit] = new Equiv[Unit] {
+        def equiv(x: Unit, y: Unit): Boolean = true
       }
 
       /** Derive an `Equiv` for `B` given an `Equiv[A]` and a function `B => A`.