Add ContravariantMonoidal

build.sbt

@@ -198,6 +198,37 @@ def mimaSettings(moduleName: String) = Seq(
  exclude[MissingTypesProblem]("cats.data.OneAndLowPriority3"),
  exclude[MissingTypesProblem]("cats.data.OneAndLowPriority2"),
  exclude[MissingTypesProblem]("cats.data.OneAndLowPriority1"),
+ exclude[DirectMissingMethodProblem]("cats.implicits.catsContravariantSemigroupalForEq"),
+ exclude[DirectMissingMethodProblem]("cats.implicits.catsContravariantSemigroupalForOrder"),
+ exclude[DirectMissingMethodProblem]("cats.implicits.catsContravariantSemigroupalForOrdering"),
+ exclude[DirectMissingMethodProblem]("cats.implicits.catsContravariantSemigroupalEquiv"),
+ exclude[ReversedMissingMethodProblem]("cats.ContravariantSemigroupal.contramap2"),
+ exclude[ReversedMissingMethodProblem]("cats.ContravariantSemigroupal#Ops.contramap2"),
+ exclude[InheritedNewAbstractMethodProblem]("cats.ContravariantMonoidal#ToContravariantMonoidalOps.toContravariantMonoidalOps"),
+ exclude[InheritedNewAbstractMethodProblem]("cats.ContravariantSemigroupal#ToContravariantSemigroupalOps.toContravariantSemigroupalOps"),
+ exclude[DirectMissingMethodProblem]("cats.instances.package=uiv.catsContravariantSemigroupalEquiv"),
+ exclude[DirectMissingMethodProblem]("cats.instances.OrderingInstances.catsContravariantSemigroupalForOrdering"),
+ exclude[ReversedMissingMethodProblem]("cats.instances.OrderingInstances.cats$instances$OrderingInstances$_setter_$catsContravariantMonoidalForOrdering_="),
+ exclude[ReversedMissingMethodProblem]("cats.instances.OrderingInstances.catsContravariantMonoidalForOrdering"),
+ exclude[DirectMissingMethodProblem]("cats.instances.OrderInstances.catsContravariantSemigroupalForOrder"),
+ exclude[ReversedMissingMethodProblem]("cats.instances.OrderInstances.catsContravariantMonoidalForOrder"),
+ exclude[ReversedMissingMethodProblem]("cats.instances.OrderInstances.cats$instances$OrderInstances$_setter_$catsContravariantMonoidalForOrder_="),
+ exclude[DirectMissingMethodProblem]("cats.instances.package#ordering.catsContravariantSemigroupalForOrdering"),
+ exclude[DirectMissingMethodProblem]("cats.instances.package#all.catsContravariantSemigroupalForEq"),
+ exclude[DirectMissingMethodProblem]("cats.instances.package#all.catsContravariantSemigroupalForOrder"),
+ exclude[DirectMissingMethodProblem]("cats.instances.package#all.catsContravariantSemigroupalForOrdering"),
+ exclude[DirectMissingMethodProblem]("cats.instances.package#all.catsContravariantSemigroupalEquiv"),
+ exclude[DirectMissingMethodProblem]("cats.instances.EquivInstances.catsContravariantSemigroupalEquiv"),
+ exclude[ReversedMissingMethodProblem]("cats.instances.EquivInstances.catsContravariantMonoidalForEquiv"),
+ exclude[ReversedMissingMethodProblem]("cats.instances.EquivInstances.cats$instances$EquivInstances$_setter_$catsContravariantMonoidalForEquiv_="),
+ exclude[DirectMissingMethodProblem]("cats.instances.package=.catsContravariantSemigroupalForEq"),
+ exclude[DirectMissingMethodProblem]("cats.instances.package#order.catsContravariantSemigroupalForOrder"),
+ exclude[ReversedMissingMethodProblem]("cats.instances.Function1Instances.catsStdContravariantMonoidalForFunction1"),
+ exclude[DirectMissingMethodProblem]("cats.instances.EqInstances.catsContravariantSemigroupalForEq"),
+ exclude[ReversedMissingMethodProblem]("cats.instances.EqInstances.catsContravariantMonoidalForEq"),
+ exclude[ReversedMissingMethodProblem]("cats.instances.EqInstances.cats$instances$EqInstances$_setter_$catsContravariantMonoidalForEq_="),
+ exclude[InheritedNewAbstractMethodProblem]("cats.syntax.ContravariantSemigroupalSyntax.catsSyntaxContravariantSemigroupal"),
+ exclude[InheritedNewAbstractMethodProblem]("cats.syntax.ContravariantMonoidalSyntax.catsSyntaxContravariantMonoidal"),
  exclude[DirectMissingMethodProblem]("cats.data.OneAndLowPriority3.catsDataNonEmptyTraverseForOneAnd"),
  exclude[DirectMissingMethodProblem]("cats.data.OneAndLowPriority2.catsDataTraverseForOneAnd"),
  exclude[ReversedMissingMethodProblem]("cats.instances.ParallelInstances.catsStdNonEmptyParallelForZipVector"),

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

@@ -113,6 +113,19 @@ private[cats] trait ComposedContravariantCovariant[F[_], G[_]] extends Contravar
  F.contramap(fga)(gb => G.map(gb)(f))
 }
 
+private[cats] trait ComposedApplicativeContravariantMonoidal[F[_], G[_]] extends ContravariantMonoidal[λ[α => F[G[α]]]] { outer =>
+ def F: Applicative[F]
+ def G: ContravariantMonoidal[G]
+
+ override def unit[A]: F[G[A]] = 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))
+
+ override def product[A, B](fa: F[G[A]], fb: F[G[B]]): F[G[(A, B)]] =
+ F.map2(fa, fb)(G.product(_, _))
+}
+
 private[cats] trait ComposedSemigroupal[F[_], G[_]] extends ContravariantSemigroupal[λ[α => F[G[α]]]] with ComposedContravariantCovariant[F, G] { outer =>
  def F: ContravariantSemigroupal[F]
  def G: Functor[G]

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

@@ -0,0 +1,32 @@
+package cats
+
+import simulacrum.typeclass
+
+/**
+ * [[ContravariantMonoidal]] functors are functors that supply
+ * a unit along the diagonal map for the `contramap2` operation.
+ *
+ * Must obey the laws defined in cats.laws.ContravariantMonoidalLaws.
+ *
+ * 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 =>
+ /**
+ * `unit` 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 liftContravariant[A, B](f: A => B): F[B] => F[A] =
+ ContravariantMonoidal.contramap2(unit[B], _: F[B])(((b: B) => (b, b)) compose f)(self, self)
+
+ // Technically, this is not correct, as the Applicative is composed with the ContravariantMonoidal, not the other way around
+ def composeApplicative[G[_]: Applicative]: ContravariantMonoidal[λ[α => G[F[α]]]] =
+ new ComposedApplicativeContravariantMonoidal[G, F] {
+ val F = Applicative[G]
+ val G = self
+ }
+}
+object ContravariantMonoidal extends SemigroupalArityFunctions

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

@@ -13,3 +13,4 @@ import simulacrum.typeclass
  def G = Functor[G]
  }
 }
+object ContravariantSemigroupal extends SemigroupalArityFunctions

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

@@ -70,6 +70,14 @@ private[data] sealed abstract class ConstInstances extends ConstInstances0 {
  fa.retag[B]
  }
 
+ implicit def catsDataContravariantMonoidalForConst[D: Monoid]: ContravariantMonoidal[Const[D, ?]] = new ContravariantMonoidal[Const[D, ?]] {
+ override def unit[A] = Const.empty[D, A]
+ 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)] =
+ fa.retag[(A, B)] combine fb.retag[(A, B)]
+ }
+
  implicit def catsDataTraverseForConst[C]: Traverse[Const[C, ?]] = new Traverse[Const[C, ?]] {
  def foldLeft[A, B](fa: Const[C, A], b: B)(f: (B, A) => B): B = b
 

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

@@ -73,6 +73,18 @@ private[data] sealed trait IdTApplicative[F[_]] extends Applicative[IdT[F, ?]] w
  def pure[A](a: A): IdT[F, A] = IdT.pure(a)
 }
 
+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 contramap[A, B](fa: IdT[F, A])(f: B => A): IdT[F, B] =
+ IdT(F0.contramap(fa.value)(f))
+
+ override def product[A, B](fa: IdT[F, A], fb: IdT[F, B]): IdT[F, (A, B)] =
+ IdT(F0.product(fa.value, fb.value))
+}
+
 private[data] sealed trait IdTFlatMap[F[_]] extends FlatMap[IdT[F, ?]] with IdTApply[F] {
  implicit val F0: FlatMap[F]
 
@@ -123,7 +135,12 @@ private[data] sealed trait IdTNonEmptyTraverse[F[_]] extends IdTTraverse[F] with
  fa.reduceRightTo(f)(g)
 }
 
-private[data] sealed abstract class IdTInstances5 {
+private[data] sealed abstract class IdTInstances6 {
+ implicit def catsDataContravariantMonoidalForIdT[F[_]](implicit F: ContravariantMonoidal[F]): ContravariantMonoidal[IdT[F, ?]] =
+ new IdTContravariantMonoidal[F] { implicit val F0: ContravariantMonoidal[F] = F }
+}
+
+private[data] sealed abstract class IdTInstances5 extends IdTInstances6 {
  implicit def catsDataFunctorForIdT[F[_]](implicit F: Functor[F]): Functor[IdT[F, ?]] =
  new IdTFunctor[F] { implicit val F0: Functor[F] = F }
 }

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

@@ -230,6 +230,10 @@ 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 {
@@ -362,6 +366,28 @@ private[data] sealed abstract class IndexedStateTSemigroupK[F[_], SA, SB] extend
  IndexedStateT(s => G.combineK(x.run(s), y.run(s)))
 }
 
+private[data] sealed abstract class IndexedStateTContravariantMonoidal[F[_], S] extends ContravariantMonoidal[IndexedStateT[F, S, 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 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)
+
+ 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)
+
+ def contramap2[A, B, C](fb: IndexedStateT[F, S, S, B], fc: IndexedStateT[F, S, S, C])(f: A => (B, C)): IndexedStateT[F, S, S, A] =
+ IndexedStateT.applyF(
+ G.pure((s: S) =>
+ ContravariantMonoidal.contramap2(G.map(fb.runF)(_.apply(s)), G.map(fc.runF)(_.apply(s)))(
+ (tup: (S, A)) => f(tup._2) match {
+ case (b, c) => (G.pure((tup._1, b)), G.pure((tup._1, c)))
+ })(G, F)))
+}
+
 private[data] sealed abstract class IndexedStateTAlternative[F[_], S] extends IndexedStateTMonad[F, S] with Alternative[IndexedStateT[F, S, S, ?]] {
  def G: Alternative[F]
 

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

@@ -165,6 +165,9 @@ private[data] sealed abstract class KleisliInstances1 extends KleisliInstances2
 private[data] sealed abstract class KleisliInstances2 extends KleisliInstances3 {
  implicit def catsDataAlternativeForKleisli[F[_], A](implicit F0: Alternative[F]): Alternative[Kleisli[F, A, ?]] =
  new KleisliAlternative[F, A] { def F: Alternative[F] = F0 }
+
+ implicit def catsDataContravariantMonoidalForKleisli[F[_], A](implicit F0: ContravariantMonoidal[F]): ContravariantMonoidal[Kleisli[F, A, ?]] =
+ new KleisliContravariantMonoidal[F, A] { def F: ContravariantMonoidal[F] = F0 }
 }
 
 private[data] sealed abstract class KleisliInstances3 extends KleisliInstances4 {
@@ -294,6 +297,18 @@ private[data] trait KleisliAlternative[F[_], A] extends Alternative[Kleisli[F, A
  implicit def F: Alternative[F]
 }
 
+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 contramap[A, B](fa: Kleisli[F, D, A])(f: B => A): Kleisli[F, D, B] =
+ Kleisli(d => F.contramap(fa.run(d))(f))
+
+ override def product[A, B](fa: Kleisli[F, D, A], fb: Kleisli[F, D, B]): Kleisli[F, D, (A, B)] =
+ Kleisli(d => F.product(fa.run(d), fb.run(d)))
+}
+
 private[data] trait KleisliMonadError[F[_], A, E] extends MonadError[Kleisli[F, A, ?], E] with KleisliApplicativeError[F, A, E] with KleisliMonad[F, A] {
  def F: MonadError[F, E]
 }

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

@@ -62,6 +62,11 @@ private[data] sealed abstract class NestedInstances1 extends NestedInstances2 {
  new NestedFunctor[F, G] {
  val FG: Functor[λ[α => F[G[α]]]] = Contravariant[F].compose[G]
  }
+
+ implicit def catsDataContravariantMonoidalForApplicativeForNested[F[_]: Applicative, G[_]: ContravariantMonoidal]: ContravariantMonoidal[Nested[F, G, ?]] =
+ new NestedContravariantMonoidal[F, G] {
+ val FG: ContravariantMonoidal[λ[α => F[G[α]]]] = ContravariantMonoidal[G].composeApplicative[F]
+ }
 }
 
 private[data] sealed abstract class NestedInstances2 extends NestedInstances3 {
@@ -263,3 +268,15 @@ private[data] trait NestedContravariant[F[_], G[_]] extends Contravariant[Nested
  def contramap[A, B](fga: Nested[F, G, A])(f: B => A): Nested[F, G, B] =
  Nested(FG.contramap(fga.value)(f))
 }
+
+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 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))
+}

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

@@ -221,6 +221,9 @@ private[data] sealed abstract class OptionTInstances0 extends OptionTInstances1
  implicit def catsDataMonadErrorForOptionT[F[_], E](implicit F0: MonadError[F, E]): MonadError[OptionT[F, ?], E] =
  new OptionTMonadError[F, E] { implicit val F = F0 }
 
+ implicit def catsDataContravariantMonoidalForOptionT[F[_]](implicit F0: ContravariantMonoidal[F]): ContravariantMonoidal[OptionT[F, ?]] =
+ new OptionTContravariantMonoidal[F] { implicit val F = F0 }
+
  implicit def catsDataSemigroupKForOptionT[F[_]](implicit F0: Monad[F]): SemigroupK[OptionT[F, ?]] =
  new OptionTSemigroupK[F] { implicit val F = F0 }
 
@@ -281,6 +284,23 @@ private trait OptionTMonadError[F[_], E] extends MonadError[OptionT[F, ?], E] wi
  OptionT(F.handleErrorWith(fa.value)(f(_).value))
 }
 
+private trait OptionTContravariantMonoidal[F[_]] extends ContravariantMonoidal[OptionT[F, ?]] {
+ def F: ContravariantMonoidal[F]
+
+ override def unit[A]: OptionT[F, A] = OptionT (F.unit)
+
+ override def contramap[A, B](fa: OptionT[F, A])(f: B => A): OptionT[F, B] =
+ OptionT(F.contramap(fa.value)(_ map f))
+
+ override def product[A, B](fa: OptionT[F, A], fb: OptionT[F, B]): OptionT[F, (A, B)] =
+ OptionT(F.contramap(F.product(fa.value, fb.value))(
+ (t: Option[(A, B)]) => t match {
+ case Some((x, y)) => (Some(x), Some(y))
+ case None => (None, None)
+ }
+ ))
+}
+
 private[data] trait OptionTFoldable[F[_]] extends Foldable[OptionT[F, ?]] {
  implicit def F: Foldable[F]
 

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

@@ -33,6 +33,11 @@ private[data] sealed abstract class Tuple2KInstances extends Tuple2KInstances0 {
  def F: Contravariant[F] = FC
  def G: Contravariant[G] = GC
  }
+ implicit def catsDataContravariantMonoidalForTuple2k[F[_], G[_]](implicit FD: ContravariantMonoidal[F], GD: ContravariantMonoidal[G]): ContravariantMonoidal[λ[α => Tuple2K[F, G, α]]] =
+ new Tuple2KContravariantMonoidal[F, G] {
+ def F: ContravariantMonoidal[F] = FD
+ def G: ContravariantMonoidal[G] = GD
+ }
 }
 
 private[data] sealed abstract class Tuple2KInstances0 extends Tuple2KInstances1 {
@@ -123,6 +128,16 @@ private[data] sealed trait Tuple2KContravariant[F[_], G[_]] extends Contravarian
  def contramap[A, B](fa: Tuple2K[F, G, A])(f: B => A): Tuple2K[F, G, B] = Tuple2K(F.contramap(fa.first)(f), G.contramap(fa.second)(f))
 }
 
+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 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] =
+ Tuple2K(F.contramap(fa.first)(f), G.contramap(fa.second)(f))
+}
+
 private[data] sealed trait Tuple2KApply[F[_], G[_]] extends Apply[λ[α => Tuple2K[F, G, α]]] with Tuple2KFunctor[F, G] {
  def F: Apply[F]
  def G: Apply[G]

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

@@ -178,6 +178,11 @@ private[data] sealed abstract class WriterTInstances6 extends WriterTInstances7
  implicit val F0: Alternative[F] = F
  implicit val L0: Monoid[L] = L
  }
+
+ implicit def catsDataContravariantMonoidalForWriterT[F[_], L](implicit F: ContravariantMonoidal[F]): ContravariantMonoidal[WriterT[F, L, ?]] =
+ new WriterTContravariantMonoidal[F, L] {
+ implicit val F0: ContravariantMonoidal[F] = F
+ }
 }
 
 private[data] sealed abstract class WriterTInstances7 extends WriterTInstances8 {
@@ -353,6 +358,23 @@ private[data] sealed trait WriterTAlternative[F[_], L] extends Alternative[Write
  override implicit def F0: Alternative[F]
 }
 
+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 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))))
+
+ override def product[A, B](fa: WriterT[F, L, A], fb: WriterT[F, L, B]): WriterT[F, L, (A, B)] =
+ WriterT(
+ F0.contramap(
+ F0.product(fa.run, fb.run))(
+ (t: (L, (A, B))) => t match {
+ case (l, (a, b)) => ((l, a), (l, b))
+ }))
+}
+
 private[data] sealed trait WriterTSemigroup[F[_], L, A] extends Semigroup[WriterT[F, L, A]] {
  implicit def F0: Semigroup[F[(L, A)]]
 

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

@@ -2,11 +2,30 @@ package cats
 package instances
 
 trait EqInstances {
- implicit val catsContravariantSemigroupalForEq: ContravariantSemigroupal[Eq] =
- new ContravariantSemigroupal[Eq] {
- def contramap[A, B](fa: Eq[A])(fn: B => A): Eq[B] = Eq.by[B, A](fn)(fa)
+ implicit val catsContravariantMonoidalForEq: ContravariantMonoidal[Eq] =
+ new ContravariantMonoidal[Eq] {
+ /**
+ * Defaults to the trivial equivalence relation
+ * contracting the type to a point
+ */
+ def unit[A]: Eq[A] = Eq.allEqual
+
+ /** Derive an `Eq` for `B` given an `Eq[A]` and a function `B => A`.
+ *
+ * Note: resulting instances are law-abiding only when the functions used are injective (represent a one-to-one mapping)
+ */
+ def contramap[A, B](fa: Eq[A])(f: B => A): Eq[B] =
+ Eq.instance { (l: B, r: B) =>
+ fa.eqv(f(l), f(r))
+ }
 
  def product[A, B](fa: Eq[A], fb: Eq[B]): Eq[(A, B)] =
- Eq.instance { (left, right) => fa.eqv(left._1, right._1) && fb.eqv(left._2, right._2) }
+ Eq.instance { (l, r) =>
+ (l, r) match {
+ case ((aL, bL), (aR, bR)) =>
+ fa.eqv(aL, aR) &&
+ fb.eqv(bL, bR)
+ }
+ }
  }
 }