diff --git a/compiler/src/dotty/tools/dotc/typer/Synthesizer.scala b/compiler/src/dotty/tools/dotc/typer/Synthesizer.scala
index 75e7c55ada28..2dd088d70671 100644
--- a/compiler/src/dotty/tools/dotc/typer/Synthesizer.scala
+++ b/compiler/src/dotty/tools/dotc/typer/Synthesizer.scala
@@ -102,7 +102,8 @@ class Synthesizer(typer: Typer)(using @constructorOnly c: Context):
     def canComparePredefinedClasses(cls1: ClassSymbol, cls2: ClassSymbol): Boolean =
 
       def cmpWithBoxed(cls1: ClassSymbol, cls2: ClassSymbol) =
-        cls2 == defn.boxedType(cls1.typeRef).symbol
+        cls2 == defn.NothingClass
+        || cls2 == defn.boxedType(cls1.typeRef).symbol
         || cls1.isNumericValueClass && cls2.derivesFrom(defn.BoxedNumberClass)
 
       if cls1.isPrimitiveValueClass then
@@ -129,7 +130,7 @@ class Synthesizer(typer: Typer)(using @constructorOnly c: Context):
       else if cls2 == defn.NullClass then
         cls1.derivesFrom(defn.ObjectClass)
       else
-        false
+        cls1 == defn.NothingClass || cls2 == defn.NothingClass
     end canComparePredefinedClasses
 
     /** Some simulated `CanEqual` instances for predefined types. It's more efficient
diff --git a/tests/pos/i13512.scala b/tests/pos/i13512.scala
new file mode 100644
index 000000000000..b400809d47b9
--- /dev/null
+++ b/tests/pos/i13512.scala
@@ -0,0 +1,7 @@
+import scala.language.strictEquality
+
+class NotEquatable
+
+def f = List(new NotEquatable) match
+  case Nil => ???
+  case _ =>
diff --git a/tests/pos/i13739.scala b/tests/pos/i13739.scala
new file mode 100644
index 000000000000..418e665f44e8
--- /dev/null
+++ b/tests/pos/i13739.scala
@@ -0,0 +1,46 @@
+import scala.language.strictEquality
+
+class Foo(i: Int) extends AnyVal
+
+val _ = summon[CanEqual[Nothing, Nothing]]
+
+val _ = summon[CanEqual[Int, Nothing]]
+val _ = summon[CanEqual[Nothing, Int]]
+val _ = summon[CanEqual[3, Nothing]]
+val _ = summon[CanEqual[Nothing, 3]]
+
+val _ = summon[CanEqual[Byte, Nothing]]
+val _ = summon[CanEqual[Nothing, Byte]]
+val _ = summon[CanEqual[Short, Nothing]]
+val _ = summon[CanEqual[Nothing, Short]]
+val _ = summon[CanEqual[Float, Nothing]]
+val _ = summon[CanEqual[Nothing, Float]]
+
+val _ = summon[CanEqual[Double, Nothing]]
+val _ = summon[CanEqual[Nothing, Double]]
+val _ = summon[CanEqual[3.0, Nothing]]
+val _ = summon[CanEqual[Nothing, 3.0]]
+
+val _ = summon[CanEqual[String, Nothing]]
+val _ = summon[CanEqual[Nothing, String]]
+val _ = summon[CanEqual["foo", Nothing]]
+val _ = summon[CanEqual[Nothing, "foo"]]
+
+val _ = summon[CanEqual[Char, Nothing]]
+val _ = summon[CanEqual[Nothing, Char]]
+val _ = summon[CanEqual['f', Nothing]]
+val _ = summon[CanEqual[Nothing, 'f']]
+
+val _ = summon[CanEqual[Boolean, Nothing]]
+val _ = summon[CanEqual[Nothing, Boolean]]
+val _ = summon[CanEqual[true, Nothing]]
+val _ = summon[CanEqual[Nothing, true]]
+
+val _ = summon[CanEqual[Foo, Nothing]]
+val _ = summon[CanEqual[Nothing, Foo]]
+
+val _ = summon[CanEqual[Option[Int], None.type]]
+val _ = summon[CanEqual[Option[Int], Option[Nothing]]]
+
+val _ = summon[CanEqual[Any & Nothing, Foo]]
+val _ = summon[CanEqual[Nothing & Any, Foo]]