Add tests for Signed/TruncatedDivision

Finishes merging typelevel/algebra#248

Author: armanbilge

algebra-laws/shared/src/main/scala/algebra/laws/OrderLaws.scala

@@ -8,6 +8,11 @@ import org.scalacheck.{Arbitrary, Cogen, Prop}
 import org.scalacheck.Prop._
 
 import cats.kernel.instances.all._
+import algebra.ring.Signed
+import algebra.ring.CommutativeRing
+import algebra.ring.TruncatedDivision
+import algebra.ring.AdditiveCommutativeGroup
+import algebra.ring.GCDRing
 
 @deprecated("Provided by cats.kernel.laws", since = "2.7.0")
 object OrderLaws {
@@ -114,6 +119,105 @@ trait OrderLaws[A] extends Laws {
     }
   )
 
+  def signed(implicit A: Signed[A]) = new OrderProperties(
+    name = "signed",
+    parent = Some(order(A.order)),
+    "abs non-negative" -> forAll((x: A) => A.sign(A.abs(x)) != Signed.Negative),
+    "signum returns -1/0/1" -> forAll((x: A) => A.signum(A.abs(x)) <= 1),
+    "signum is sign.toInt" -> forAll((x: A) => A.signum(x) == A.sign(x).toInt),
+    "ordered group" -> forAll { (x: A, y: A, z: A) =>
+      A.order.lteqv(x, y) ==> A.order.lteqv(A.additiveCommutativeMonoid.plus(x, z),
+                                            A.additiveCommutativeMonoid.plus(y, z)
+      )
+    },
+    "triangle inequality" -> forAll { (x: A, y: A) =>
+      A.order.lteqv(A.abs(A.additiveCommutativeMonoid.plus(x, y)), A.additiveCommutativeMonoid.plus(A.abs(x), A.abs(y)))
+    }
+  )
+
+  def signedAdditiveCommutativeGroup(implicit signedA: Signed[A], A: AdditiveCommutativeGroup[A]) = new DefaultRuleSet(
+    name = "signedAdditiveAbGroup",
+    parent = Some(signed),
+    "abs(x) equals abs(-x)" -> forAll { (x: A) =>
+      signedA.abs(x) ?== signedA.abs(A.negate(x))
+    }
+  )
+
+  // more a convention: as GCD is defined up to a unit, so up to a sign,
+  // on an ordered GCD ring we require gcd(x, y) >= 0, which is the common
+  // behavior of computer algebra systems
+  def signedGCDRing(implicit signedA: Signed[A], A: GCDRing[A]) = new DefaultRuleSet(
+    name = "signedGCDRing",
+    parent = Some(signedAdditiveCommutativeGroup),
+    "gcd(x, y) >= 0" -> forAll { (x: A, y: A) =>
+      signedA.isSignNonNegative(A.gcd(x, y))
+    },
+    "gcd(x, 0) === abs(x)" -> forAll { (x: A) =>
+      A.gcd(x, A.zero) ?== signedA.abs(x)
+    }
+  )
+
+  def truncatedDivision(implicit ring: CommutativeRing[A], A: TruncatedDivision[A]) = new DefaultRuleSet(
+    name = "truncatedDivision",
+    parent = Some(signed),
+    "division rule (tquotmod)" -> forAll { (x: A, y: A) =>
+      A.isSignNonZero(y) ==> {
+        val (q, r) = A.tquotmod(x, y)
+        x ?== ring.plus(ring.times(y, q), r)
+      }
+    },
+    "division rule (fquotmod)" -> forAll { (x: A, y: A) =>
+      A.isSignNonZero(y) ==> {
+        val (q, r) = A.fquotmod(x, y)
+        x ?== ring.plus(ring.times(y, q), r)
+      }
+    },
+    "|r| < |y| (tmod)" -> forAll { (x: A, y: A) =>
+      A.isSignNonZero(y) ==> {
+        val r = A.tmod(x, y)
+        A.order.lt(A.abs(r), A.abs(y))
+      }
+    },
+    "|r| < |y| (fmod)" -> forAll { (x: A, y: A) =>
+      A.isSignNonZero(y) ==> {
+        val r = A.fmod(x, y)
+        A.order.lt(A.abs(r), A.abs(y))
+      }
+    },
+    "r = 0 or sign(r) = sign(x) (tmod)" -> forAll { (x: A, y: A) =>
+      A.isSignNonZero(y) ==> {
+        val r = A.tmod(x, y)
+        A.isSignZero(r) || (A.sign(r) ?== A.sign(x))
+      }
+    },
+    "r = 0 or sign(r) = sign(y) (fmod)" -> forAll { (x: A, y: A) =>
+      A.isSignNonZero(y) ==> {
+        val r = A.fmod(x, y)
+        A.isSignZero(r) || (A.sign(r) ?== A.sign(y))
+      }
+    },
+    "tquot" -> forAll { (x: A, y: A) =>
+      A.isSignNonZero(y) ==> {
+        A.tquotmod(x, y)._1 ?== A.tquot(x, y)
+      }
+    },
+    "tmod" -> forAll { (x: A, y: A) =>
+      A.isSignNonZero(y) ==> {
+        A.tquotmod(x, y)._2 ?== A.tmod(x, y)
+      }
+    },
+    "fquot" -> forAll { (x: A, y: A) =>
+      A.isSignNonZero(y) ==> {
+        A.fquotmod(x, y)._1 ?== A.fquot(x, y)
+      }
+    },
+    "fmod" -> forAll { (x: A, y: A) =>
+      A.isSignNonZero(y) ==> {
+        A.fquotmod(x, y)._2 ?== A.fmod(x, y)
+      }
+    }
+  )
+
   class OrderProperties(
     name: String,
     parent: Option[RuleSet],

algebra-laws/shared/src/test/scala/algebra/laws/FPApprox.scala

@@ -66,7 +66,7 @@ object FPApprox {
   def exact[A: Rng: Order](a: A): FPApprox[A] = FPApprox(a, abs(a), 0)
   def approx[A: Rng: Order](a: A): FPApprox[A] = FPApprox(a, abs(a), 1)
 
-  trait Epsilon[A] {
+  trait Epsilon[A] extends Serializable {
     def minValue: A
     def epsilon: A
     def isFinite(a: A): Boolean

algebra-laws/shared/src/test/scala/algebra/laws/LawTests.scala

@@ -123,7 +123,9 @@ class LawTests extends munit.DisciplineSuite {
   checkAll("Long", RingLaws[Long].commutativeRing)
   checkAll("Long", LatticeLaws[Long].boundedDistributiveLattice)
 
+  checkAll("BigInt", OrderLaws[BigInt].truncatedDivision)
   checkAll("BigInt", RingLaws[BigInt].euclideanRing)
+  checkAll("BigInt", OrderLaws[BigInt].signedGCDRing)
 
   checkAll("FPApprox[Float]", RingLaws[FPApprox[Float]].approxField)
   checkAll("FPApprox[Double]", RingLaws[FPApprox[Double]].approxField)