Move typelevel/algebra into cats repo

README.md

@@ -115,6 +115,7 @@ functionality, you can pick-and-choose from amongst these modules
  * `cats-kernel`: Small set of basic type classes (*required*).
  * `cats-core`: Most core type classes and functionality (*required*).
  * `cats-laws`: Laws for testing type class instances.
+ * `cats-algebra`: Type classes to represent algebraic structures.
  * `cats-free`: Free structures such as the free monad, and supporting type classes.
  * `cats-testkit`: lib for writing tests for type class instances using laws.
  * `alleycats-core`: Cats instances and classes which are not lawful. 
@@ -140,10 +141,11 @@ Links:
 2. ScalaDoc: [typelevel.org/cats/api/](https://typelevel.org/cats/api/)
 3. Type classes: [typelevel.org/cats/typeclasses](https://typelevel.org/cats/typeclasses.html)
 4. Data types: [typelevel.org/cats/datatypes.html](https://typelevel.org/cats/datatypes.html)
-5. Glossary: [typelevel.org/cats/nomenclature.html](https://typelevel.org/cats/nomenclature.html)
-6. Resources for Learners: [typelevel.org/cats/resources_for_learners.html](https://typelevel.org/cats/resources_for_learners.html)
-7. FAQ: [typelevel.org/cats/faq.html](https://typelevel.org/cats/faq.html)
-8. The Typelevel Ecosystem: [typelevel.org/cats/typelevelEcosystem.html](https://typelevel.org/cats/typelevelEcosystem.html)
+5. Algebra overview: [typelevel.org/cats/algebra.html](https://typelevel.org/cats/algebra.html)
+6. Glossary: [typelevel.org/cats/nomenclature.html](https://typelevel.org/cats/nomenclature.html)
+7. Resources for Learners: [typelevel.org/cats/resources_for_learners.html](https://typelevel.org/cats/resources_for_learners.html)
+8. FAQ: [typelevel.org/cats/faq.html](https://typelevel.org/cats/faq.html)
+9. The Typelevel Ecosystem: [typelevel.org/cats/typelevelEcosystem.html](https://typelevel.org/cats/typelevelEcosystem.html)
 
 ### Community
 

algebra-core/src/main/scala/cats/algebra/Priority.scala

@@ -0,0 +1,66 @@
+package cats
+package algebra
+
+/**
+ * Priority is a type class for prioritized implicit search.
+ *
+ * This type class will attempt to provide an implicit instance of `P`
+ * (the preferred type). If that type is not available it will
+ * fallback to `F` (the fallback type). If neither type is available
+ * then a `Priority[P, F]` instance will not be available.
+ *
+ * This type can be useful for problems where multiple algorithms can
+ * be used, depending on the type classes available.
+ */
+sealed trait Priority[+P, +F] {
+
+  import Priority.{Fallback, Preferred}
+
+  def fold[B](f1: P => B)(f2: F => B): B =
+    this match {
+      case Preferred(x) => f1(x)
+      case Fallback(y)  => f2(y)
+    }
+
+  def join[U >: P with F]: U =
+    fold(_.asInstanceOf[U])(_.asInstanceOf[U])
+
+  def bimap[P2, F2](f1: P => P2)(f2: F => F2): Priority[P2, F2] =
+    this match {
+      case Preferred(x) => Preferred(f1(x))
+      case Fallback(y)  => Fallback(f2(y))
+    }
+
+  def toEither: Either[P, F] =
+    fold[Either[P, F]](p => Left(p))(f => Right(f))
+
+  def isPreferred: Boolean =
+    fold(_ => true)(_ => false)
+
+  def isFallback: Boolean =
+    fold(_ => false)(_ => true)
+
+  def getPreferred: Option[P] =
+    fold[Option[P]](p => Some(p))(_ => None)
+
+  def getFallback: Option[F] =
+    fold[Option[F]](_ => None)(f => Some(f))
+}
+
+object Priority extends FindPreferred {
+
+  case class Preferred[P](get: P) extends Priority[P, Nothing]
+  case class Fallback[F](get: F) extends Priority[Nothing, F]
+
+  def apply[P, F](implicit ev: Priority[P, F]): Priority[P, F] = ev
+}
+
+private[algebra] trait FindPreferred extends FindFallback {
+  implicit def preferred[P](implicit ev: P): Priority[P, Nothing] =
+    Priority.Preferred(ev)
+}
+
+private[algebra] trait FindFallback {
+  implicit def fallback[F](implicit ev: F): Priority[Nothing, F] =
+    Priority.Fallback(ev)
+}

algebra-core/src/main/scala/cats/algebra/instances/StaticMethods.scala

@@ -0,0 +1,28 @@
+package cats
+package algebra.instances
+
+import scala.annotation.tailrec
+
+object StaticMethods {
+
+  /**
+   * Exponentiation function, e.g. x^y
+   *
+   * If base^ex doesn't fit in a Long, the result will overflow (unlike
+   * Math.pow which will return +/- Infinity).
+   */
+  final def pow(base: Long, exponent: Long): Long = {
+    @tailrec def loop(t: Long, b: Long, e: Long): Long =
+      if (e == 0L) t
+      else if ((e & 1) == 1) loop(t * b, b * b, e >>> 1L)
+      else loop(t, b * b, e >>> 1L)
+
+    if (exponent >= 0L) loop(1L, base, exponent)
+    else {
+      if (base == 0L) throw new ArithmeticException("zero can't be raised to negative power")
+      else if (base == 1L) 1L
+      else if (base == -1L) if ((exponent & 1L) == 0L) -1L else 1L
+      else 0L
+    }
+  }
+}

algebra-core/src/main/scala/cats/algebra/instances/all.scala

@@ -0,0 +1,26 @@
+package cats
+package algebra
+package instances
+
+package object all extends AllInstances
+
+trait AllInstances
+    extends ArrayInstances
+    with BigDecimalInstances
+    with BigIntInstances
+    with BitSetInstances
+    with BooleanInstances
+    with ByteInstances
+    with CharInstances
+    with DoubleInstances
+    with FloatInstances
+    with IntInstances
+    with ListInstances
+    with LongInstances
+    with MapInstances
+    with OptionInstances
+    with SetInstances
+    with ShortInstances
+    with StringInstances
+    with TupleInstances
+    with UnitInstances

algebra-core/src/main/scala/cats/algebra/instances/array.scala

@@ -0,0 +1,71 @@
+package cats
+package algebra
+package instances
+
+import scala.{specialized => sp}
+
+package object array extends ArrayInstances
+
+trait ArrayInstances {
+  implicit def arrayEq[@sp A: Eq]: Eq[Array[A]] =
+    new ArrayEq[A]
+  implicit def arrayOrder[@sp A: Order]: Order[Array[A]] =
+    new ArrayOrder[A]
+  implicit def arrayPartialOrder[@sp A: PartialOrder]: PartialOrder[Array[A]] =
+    new ArrayPartialOrder[A]
+}
+
+private object ArraySupport {
+
+  private def signum(x: Int): Int =
+    if (x < 0) -1
+    else if (x > 0) 1
+    else 0
+
+  def eqv[@sp A](x: Array[A], y: Array[A])(implicit ev: Eq[A]): Boolean = {
+    var i = 0
+    if (x.length != y.length) return false
+    while (i < x.length && i < y.length && ev.eqv(x(i), y(i))) i += 1
+    i == x.length
+  }
+
+  def compare[@sp A](x: Array[A], y: Array[A])(implicit ev: Order[A]): Int = {
+    var i = 0
+    while (i < x.length && i < y.length) {
+      val cmp = ev.compare(x(i), y(i))
+      if (cmp != 0) return cmp
+      i += 1
+    }
+    signum(x.length - y.length)
+  }
+
+  def partialCompare[@sp A](x: Array[A], y: Array[A])(implicit ev: PartialOrder[A]): Double = {
+    var i = 0
+    while (i < x.length && i < y.length) {
+      val cmp = ev.partialCompare(x(i), y(i))
+      // Double.NaN is also != 0.0
+      if (cmp != 0.0) return cmp
+      i += 1
+    }
+    signum(x.length - y.length).toDouble
+  }
+}
+
+final private class ArrayEq[@sp A: Eq] extends Eq[Array[A]] with Serializable {
+  def eqv(x: Array[A], y: Array[A]): Boolean =
+    ArraySupport.eqv(x, y)
+}
+
+final private class ArrayOrder[@sp A: Order] extends Order[Array[A]] with Serializable {
+  override def eqv(x: Array[A], y: Array[A]): Boolean =
+    ArraySupport.eqv(x, y)
+  def compare(x: Array[A], y: Array[A]): Int =
+    ArraySupport.compare(x, y)
+}
+
+final private class ArrayPartialOrder[@sp A: PartialOrder] extends PartialOrder[Array[A]] with Serializable {
+  override def eqv(x: Array[A], y: Array[A]): Boolean =
+    ArraySupport.eqv(x, y)
+  override def partialCompare(x: Array[A], y: Array[A]): Double =
+    ArraySupport.partialCompare(x, y)
+}

algebra-core/src/main/scala/cats/algebra/instances/bigDecimal.scala

@@ -0,0 +1,32 @@
+package cats
+package algebra
+package instances
+
+import java.math.MathContext
+
+import cats.algebra.ring._
+
+package object bigDecimal extends BigDecimalInstances
+
+trait BigDecimalInstances extends cats.kernel.instances.BigDecimalInstances {
+  implicit val bigDecimalAlgebra: BigDecimalAlgebra = new BigDecimalAlgebra()
+}
+
+class BigDecimalAlgebra(mc: MathContext) extends Field[BigDecimal] with Serializable {
+  def this() = this(MathContext.UNLIMITED)
+
+  val zero: BigDecimal = BigDecimal(0, mc)
+  val one: BigDecimal = BigDecimal(1, mc)
+
+  def plus(a: BigDecimal, b: BigDecimal): BigDecimal = a + b
+  def negate(a: BigDecimal): BigDecimal = -a
+  override def minus(a: BigDecimal, b: BigDecimal): BigDecimal = a - b
+
+  def times(a: BigDecimal, b: BigDecimal): BigDecimal = a * b
+  def div(a: BigDecimal, b: BigDecimal): BigDecimal = a / b
+
+  override def pow(a: BigDecimal, k: Int): BigDecimal = a.pow(k)
+
+  override def fromInt(n: Int): BigDecimal = BigDecimal(n, mc)
+  override def fromBigInt(n: BigInt): BigDecimal = BigDecimal(n, mc)
+}

algebra-core/src/main/scala/cats/algebra/instances/bigInt.scala

@@ -0,0 +1,57 @@
+package cats
+package algebra
+package instances
+
+import cats.algebra.ring._
+
+package object bigInt extends BigIntInstances
+
+trait BigIntInstances extends cats.kernel.instances.BigIntInstances {
+  implicit val bigIntAlgebra: BigIntAlgebra =
+    new BigIntAlgebra
+}
+
+class BigIntAlgebra extends EuclideanRing[BigInt] with Serializable {
+
+  val zero: BigInt = BigInt(0)
+  val one: BigInt = BigInt(1)
+
+  def plus(a: BigInt, b: BigInt): BigInt = a + b
+  def negate(a: BigInt): BigInt = -a
+  override def minus(a: BigInt, b: BigInt): BigInt = a - b
+
+  def times(a: BigInt, b: BigInt): BigInt = a * b
+
+  override def pow(a: BigInt, k: Int): BigInt = a.pow(k)
+
+  override def fromInt(n: Int): BigInt = BigInt(n)
+  override def fromBigInt(n: BigInt): BigInt = n
+
+  override def lcm(a: BigInt, b: BigInt)(implicit ev: Eq[BigInt]): BigInt =
+    if (a.signum == 0 || b.signum == 0) zero else (a / a.gcd(b)) * b
+  override def gcd(a: BigInt, b: BigInt)(implicit ev: Eq[BigInt]): BigInt = a.gcd(b)
+
+  def euclideanFunction(a: BigInt): BigInt = a.abs
+
+  override def equotmod(a: BigInt, b: BigInt): (BigInt, BigInt) = {
+    val (qt, rt) = a /% b // truncated quotient and remainder
+    if (rt.signum >= 0) (qt, rt)
+    else if (b.signum > 0) (qt - 1, rt + b)
+    else (qt + 1, rt - b)
+  }
+
+  def equot(a: BigInt, b: BigInt): BigInt = {
+    val (qt, rt) = a /% b // truncated quotient and remainder
+    if (rt.signum >= 0) qt
+    else if (b.signum > 0) qt - 1
+    else qt + 1
+  }
+
+  def emod(a: BigInt, b: BigInt): BigInt = {
+    val rt = a % b // truncated remainder
+    if (rt.signum >= 0) rt
+    else if (b > 0) rt + b
+    else rt - b
+  }
+
+}

algebra-core/src/main/scala/cats/algebra/instances/bitSet.scala

@@ -0,0 +1,22 @@
+package cats
+package algebra
+package instances
+
+import scala.collection.immutable.BitSet
+
+import cats.algebra.lattice._
+
+package object bitSet extends BitSetInstances
+
+trait BitSetInstances extends cats.kernel.instances.BitSetInstances {
+  implicit val bitSetAlgebra: BitSetAlgebra =
+    new BitSetAlgebra
+}
+
+class BitSetAlgebra extends GenBool[BitSet] with Serializable {
+  val zero: BitSet = BitSet.empty
+  def and(a: BitSet, b: BitSet): BitSet = a & b
+  def or(a: BitSet, b: BitSet): BitSet = a | b
+  def without(a: BitSet, b: BitSet): BitSet = a -- b
+  override def xor(a: BitSet, b: BitSet): BitSet = a ^ b
+}

algebra-core/src/main/scala/cats/algebra/instances/boolean.scala

@@ -0,0 +1,43 @@
+package cats
+package algebra
+package instances
+
+import cats.algebra.lattice.Bool
+import cats.algebra.ring.BoolRing
+import cats.algebra.ring.CommutativeRig
+
+package object boolean extends BooleanInstances
+
+trait BooleanInstances extends cats.kernel.instances.BooleanInstances {
+  implicit val booleanAlgebra: BooleanAlgebra =
+    new BooleanAlgebra
+
+  val booleanRing = new BoolRing[Boolean] {
+    def zero: Boolean = false
+    def one: Boolean = true
+    def plus(x: Boolean, y: Boolean): Boolean = x ^ y
+    def times(x: Boolean, y: Boolean): Boolean = x && y
+  }
+}
+
+/**
+ * This commutative rig is different than the one obtained from GF(2).
+ *
+ * It uses || for plus, and && for times.
+ */
+class BooleanAlgebra extends Bool[Boolean] with CommutativeRig[Boolean] {
+
+  def zero: Boolean = false
+  def one: Boolean = true
+
+  override def isZero(x: Boolean)(implicit ev: Eq[Boolean]): Boolean = !x
+  override def isOne(x: Boolean)(implicit ev: Eq[Boolean]): Boolean = x
+
+  def and(x: Boolean, y: Boolean): Boolean = x && y
+  def or(x: Boolean, y: Boolean): Boolean = x || y
+  def complement(x: Boolean): Boolean = !x
+
+  def plus(a: Boolean, b: Boolean): Boolean = a || b
+  override def pow(a: Boolean, b: Int): Boolean = a
+  override def times(a: Boolean, b: Boolean): Boolean = a && b
+}