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copied!<p>This answer is a synthesis of the partial answers provided by both 0__ and Nicolas Rinaudo. </p> <p><strong>Summary:</strong></p> <p>There are many convenient (but also highly intertwined) assumptions being made by the Scala compiler.</p> <ol> <li>Scala treats <code>extends (A => B)</code> as synonymous with <code>extends Function1[A, B]</code> (<a href="http://www.scala-lang.org/api/current/index.html#scala.Function1" rel="nofollow">ScalaDoc for Function1[+T1, -R]</a>)</li> <li>A concrete implementation of Function1's inherited abstract method <code>apply(x: A): B</code> must be provided; <code>def apply(x: A): B = cache.getOrElseUpdate(x, f(x))</code></li> <li>Scala assumes an implied <code>match</code> for the code block starting with <code>= Memo {</code></li> <li>Scala passes the content between <code>{}</code> started in item 3 as a parameter to the Memo case class constructor</li> <li>Scala assumes an implied type between <code>{}</code> started in item 3 as <code>PartialFunction[Int, BigInt]</code> and the compiler uses the "match" code block as the override for the PartialFunction method's <code>apply()</code> and then provides an additional override for the PartialFunction's method <code>isDefinedAt()</code>. </li> </ol> <p><strong>Details:</strong></p> <p>The first code block defining the case class Memo can be written more verbosely as such: </p> <pre><code>case class Memo[A,B](f: A => B) extends Function1[A, B] { //replaced (A => B) with what it's translated to mean by the Scala compiler private val cache = mutable.Map.empty[A, B] def apply(x: A): B = cache.getOrElseUpdate(x, f(x)) //concrete implementation of unimplemented method defined in parent class, Function1 } </code></pre> <p>The second code block defining the val fibanocci can be written more verbosely as such:</p> <pre><code>lazy val fibonacci: Memo[Int, BigInt] = { Memo.apply( new PartialFunction[Int, BigInt] { override def apply(x: Int): BigInt = { x match { case 0 => 0 case 1 => 1 case n => fibonacci(n-1) + fibonacci(n-2) } } override def isDefinedAt(x: Int): Boolean = true } ) } </code></pre> <p>Had to add <code>lazy</code> to the second code block's val in order to deal with a self-referential problem in the line <code>case n => fibonacci(n-1) + fibonacci(n-2)</code>.</p> <p>And finally, an example usage of fibonacci is:</p> <pre><code>val x:BigInt = fibonacci(20) //returns 6765 (almost instantly) </code></pre>

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