Define Apply.product in term of ap and map

- Reverts many changes introduced by typelevel#555
- Changes `cats.syntax.all._` to `import cats.syntax.monoidal._` in doc

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

@@ -20,15 +20,15 @@ trait Apply[F[_]] extends Functor[F] with Monoidal[F] with ApplyArityFunctions[F
    * ap2 is a binary version of ap, defined in terms of ap.
    */
   def ap2[A, B, Z](fa: F[A], fb: F[B])(ff: F[(A, B) => Z]): F[Z] =
-    map(product(fa, product(fb, ff))) { case (a, (b, f)) => f(a, b) }
+    ap(fb)(ap(fa)(map(ff)(ff => (a: A) => (b: B) => ff(a, b))))
 
   /**
    * Applies the pure (binary) function f to the effectful values fa and fb.
    *
    * map2 can be seen as a binary version of [[cats.Functor]]#map.
    */
   def map2[A, B, Z](fa: F[A], fb: F[B])(f: (A, B) => Z): F[Z] =
-    map(product(fa, fb)) { case (a, b) => f(a, b) }
+    ap(fb)(map(fa)(a => (b: B) => f(a, b)))
 
   /**
    * Two sequentially dependent Applys can be composed.
@@ -46,6 +46,8 @@ trait Apply[F[_]] extends Functor[F] with Monoidal[F] with ApplyArityFunctions[F
       def G: Apply[G] = GG
     }
 
+  override def product[A, B](fa: F[A], fb: F[B]): F[(A, B)] =
+    ap(fa)(map(fb)(b => (a => (a, b))))
 }
 
 trait CompositeApply[F[_], G[_]]
@@ -55,8 +57,4 @@ trait CompositeApply[F[_], G[_]]
 
   def ap[A, B](fa: F[G[A]])(f: F[G[A => B]]): F[G[B]] =
     F.ap(fa)(F.map(f)(gab => G.ap(_)(gab)))
-
-  def product[A, B](fa: F[G[A]], fb: F[G[B]]): F[G[(A, B)]] =
-    F.map2(fa, fb)(G.product)
-
 }

core/src/main/scala/cats/FlatMap.scala

@@ -29,9 +29,6 @@ import simulacrum.typeclass
   override def ap[A, B](fa: F[A])(ff: F[A => B]): F[B] =
     flatMap(ff)(f => map(fa)(f))
 
-  override def product[A, B](fa: F[A], fb: F[B]): F[(A, B)] =
-    flatMap(fa)(a => map(fb)(b => (a, b)))
-
   /**
    *  Pair `A` with the result of function application.
    */

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

@@ -91,9 +91,6 @@ private[data] sealed abstract class ConstInstances0 extends ConstInstances1 {
 
     def map[A, B](fa: Const[C, A])(f: A => B): 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)]
   }
 }
 
@@ -107,9 +104,6 @@ private[data] sealed abstract class ConstInstances1 {
     def ap[A, B](fa: Const[C, A])(f: Const[C, A => B]): Const[C, B] =
       fa.retag[B] combine f.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)]
-
     def map[A, B](fa: Const[C, A])(f: A => B): Const[C, B] =
       fa.retag[B]
   }

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

@@ -62,8 +62,6 @@ sealed trait FuncApply[F[_], C] extends Apply[Lambda[X => Func[F, C, X]]] with F
   def F: Apply[F]
   def ap[A, B](fa: Func[F, C, A])(f: Func[F, C, A => B]): Func[F, C, B] =
     Func.func(c => F.ap(fa.run(c))(f.run(c)))
-  def product[A, B](fa: Func[F, C, A], fb: Func[F, C, B]): Func[F, C, (A, B)] =
-    Func.func(c => F.product(fa.run(c), fb.run(c)))
 }
 
 sealed trait FuncApplicative[F[_], C] extends Applicative[Lambda[X => Func[F, C, X]]] with FuncApply[F, C] {
@@ -125,8 +123,6 @@ private[data] sealed trait AppFuncApplicative[F[_], C] extends Applicative[Lambd
     fa.map(f)
   def ap[A, B](fa: AppFunc[F, C, A])(f: AppFunc[F, C, A => B]): AppFunc[F, C, B] =
     Func.appFunc[F, C, B](c => F.ap(fa.run(c))(f.run(c)))(F)
-  def product[A, B](fa: AppFunc[F, C, A], fb: AppFunc[F, C, B]): AppFunc[F, C, (A, B)] =
-    Func.appFunc[F, C, (A, B)](c => F.product(fa.run(c), fb.run(c)))(F)
   def pure[A](a: A): AppFunc[F, C, A] =
     Func.appFunc[F, C, A](c => F.pure(a))(F)
 }

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

@@ -148,9 +148,6 @@ private[data] sealed abstract class KleisliInstances1 extends KleisliInstances2
 
     def map[B, C](fb: Kleisli[F, A, B])(f: B => C): Kleisli[F, A, C] =
       fb.map(f)
-
-    def product[B, C](fb: Kleisli[F, A, B], fc: Kleisli[F, A, C]): Kleisli[F, A, (B, C)] =
-      Kleisli(a => Applicative[F].product(fb.run(a), fc.run(a)))
   }
 }
 
@@ -159,9 +156,6 @@ private[data] sealed abstract class KleisliInstances2 extends KleisliInstances3
     def ap[B, C](fa: Kleisli[F, A, B])(f: Kleisli[F, A, B => C]): Kleisli[F, A, C] =
       fa(f)
 
-    def product[B, C](fb: Kleisli[F, A, B], fc: Kleisli[F, A, C]): Kleisli[F, A, (B, C)] =
-      Kleisli(a => Apply[F].product(fb.run(a), fc.run(a)))
-
     def map[B, C](fa: Kleisli[F, A, B])(f: B => C): Kleisli[F, A, C] =
       fa.map(f)
   }

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

@@ -79,8 +79,6 @@ sealed trait ProdApply[F[_], G[_]] extends Apply[Lambda[X => Prod[F, G, X]]] wit
   def G: Apply[G]
   def ap[A, B](fa: Prod[F, G, A])(f: Prod[F, G, A => B]): Prod[F, G, B] =
     Prod(F.ap(fa.first)(f.first), G.ap(fa.second)(f.second))
-  def product[A, B](fa: Prod[F, G, A], fb: Prod[F, G, B]): Prod[F, G, (A, B)] =
-    Prod(F.product(fa.first, fb.first), G.product(fa.second, fb.second))
 }
 
 sealed trait ProdApplicative[F[_], G[_]] extends Applicative[Lambda[X => Prod[F, G, X]]] with ProdApply[F, G] {

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

@@ -117,17 +117,6 @@ sealed abstract class Validated[+E, +A] extends Product with Serializable {
       case (_, e@Invalid(_)) => e
     }
 
-  /**
-   * From Product
-   */
-  def product[EE >: E, B](fb: Validated[EE, B])(implicit EE: Semigroup[EE]): Validated[EE, (A, B)] =
-    (this, fb) match {
-      case (Valid(a), Valid(b)) => Valid((a, b))
-      case (Invalid(e1), Invalid(e2)) => Invalid(EE.combine(e1, e2))
-      case (e @ Invalid(_), _) => e
-      case (_, e @ Invalid(_)) => e
-    }
-
   /**
    * Apply a function to a Valid value, returning a new Valid value
    */
@@ -253,9 +242,6 @@ private[data] sealed abstract class ValidatedInstances extends ValidatedInstance
 
       def ap[A,B](fa: Validated[E,A])(f: Validated[E,A=>B]): Validated[E, B] =
         fa.ap(f)(E)
-
-      def product[A, B](fa: Validated[E, A], fb: Validated[E, B]): Validated[E, (A, B)] =
-        fa.product(fb)(E)
     }
 }
 

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

@@ -157,8 +157,6 @@ private[data] sealed trait WriterTApply[F[_], L] extends WriterTFunctor[F, L] wi
 
   def ap[A, B](fa: WriterT[F, L, A])(f: WriterT[F, L, A => B]): WriterT[F, L, B] =
     fa ap f
-  def product[A, B](fa: WriterT[F, L, A], fb: WriterT[F, L, B]): WriterT[F, L, (A, B)] =
-    WriterT(F0.map(F0.product(fa.run, fb.run)) { case ((l1, a), (l2, b)) => (L0.combine(l1, l2), (a, b)) })
 }
 
 private[data] sealed trait WriterTFlatMap[F[_], L] extends WriterTApply[F, L] with FlatMap[WriterT[F, L, ?]] {

core/src/main/scala/cats/free/FreeApplicative.scala

@@ -48,7 +48,7 @@ sealed abstract class FreeApplicative[F[_], A] extends Product with Serializable
     }
 
   /** Interpret this algebra into a Monoid */
-  final def analyze[M:Monoid](f: F ~> λ[α => M]): M =
+  final def analyze[M: Monoid](f: F ~> λ[α => M]): M =
     foldMap[Const[M, ?]](new (F ~> Const[M, ?]) {
       def apply[X](x: F[X]): Const[M,X] = Const(f(x))
     }).getConst
@@ -79,7 +79,6 @@ object FreeApplicative {
 
   implicit final def freeApplicative[S[_]]: Applicative[FA[S, ?]] = {
     new Applicative[FA[S, ?]] {
-      def product[A, B](fa: FA[S, A], fb: FA[S, B]): FA[S, (A, B)] = ap(fb)(fa.map(a => b => (a, b)))
       def map[A, B](fa: FA[S, A])(f: A => B): FA[S, B] = fa.map(f)
       override def ap[A, B](fa: FA[S, A])(f: FA[S, A => B]): FA[S, B] = fa.ap(f)
       def pure[A](a: A): FA[S, A] = Pure(a)

docs/src/main/tut/apply.md

@@ -26,19 +26,13 @@ implicit val optionApply: Apply[Option] = new Apply[Option] {
     fa.flatMap (a => f.map (ff => ff(a)))
 
   def map[A,B](fa: Option[A])(f: A => B): Option[B] = fa map f
-  
-  def product[A, B](fa: Option[A], fb: Option[B]): Option[(A, B)] =
-    fa.flatMap(a => fb.map(b => (a, b)))
 }
 
 implicit val listApply: Apply[List] = new Apply[List] {
   def ap[A, B](fa: List[A])(f: List[A => B]): List[B] =
     fa.flatMap (a => f.map (ff => ff(a)))
 
   def map[A,B](fa: List[A])(f: A => B): List[B] = fa map f
-  
-  def product[A, B](fa: List[A], fb: List[B]): List[(A, B)] =
-    fa.zip(fb)
 }
 ```
 
@@ -121,7 +115,7 @@ Apply[Option].tuple3(Some(1), Some(2), Some(3))
 
 The `|@|` operator offers an alternative syntax for the higher-arity `Apply`
 functions (`apN`, `mapN` and `tupleN`).
-In order to use it, first import `cats.syntax.all._` or `cats.syntax.apply._`.
+In order to use it, first import `import cats.syntax.monoidal._`.
 Here we see that the following two functions, `f1` and `f2`, are equivalent:
 
 ```tut

docs/src/main/tut/const.md

@@ -221,8 +221,6 @@ implicit def constApplicative[Z]: Applicative[Const[Z, ?]] =
     def ap[A, B](fa: Const[Z, A])(f: Const[Z, A => B]): Const[Z, B] = ???
 
     def map[A, B](fa: Const[Z, A])(f: A => B): Const[Z, B] = ???
-
-    def product[A, B](fa: Const[Z, A],fb: Const[Z, B]): Const[Z, (A, B)] = ???
   }
 ```
 
@@ -249,9 +247,6 @@ implicit def constApplicative[Z : Monoid]: Applicative[Const[Z, ?]] =
       
     def map[A, B](fa: Const[Z, A])(f: A => B): Const[Z, B] =
       Const(fa.getConst)
-
-    def product[A, B](fa: Const[Z, A],fb: Const[Z, B]): Const[Z, (A, B)] =
-      Const(Monoid[Z].combine(fa.getConst, fb.getConst))
   }
 ```
 

docs/src/main/tut/validated.md

@@ -200,7 +200,6 @@ implicit def validatedApplicative[E : Semigroup]: Applicative[Validated[E, ?]] =
 
     def pure[A](x: A): Validated[E, A] = Validated.valid(x)
     def map[A, B](fa: Validated[E, A])(f: A => B): Validated[E, B] = fa.map(f)
-    def product[A, B](fa: Validated[E, A], fb: Validated[E, B]): Validated[E, (A, B)] = ap(fb)(fa.map(a => b => (a, b)))
   }
 ```
 

laws/src/main/scala/cats/laws/TraverseLaws.scala

@@ -49,11 +49,6 @@ trait TraverseLaws[F[_]] extends FunctorLaws[F] with FoldableLaws[F] {
         val (mx, nx) = fx
         (M.map(mx)(f), N.map(nx)(f))
       }
-      def product[X, Y](fx: MN[X], fy: MN[Y]): MN[(X, Y)] = {
-        val (mx, nx) = fx
-        val (my, ny) = fy
-        (M.product(mx, my), N.product(nx, ny))
-      }
     }
     val lhs: MN[F[B]] = fa.traverse[MN, B](a => (f(a), g(a)))
     val rhs: MN[F[B]] = (fa.traverse(f), fa.traverse(g))