// Scala Lecture 4+
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//=================+
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+
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// pattern-matching+
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// tail-recursion+
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// polymorphic types+
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+
−
+
−
+
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// Pattern Matching+
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//==================+
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+
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// A powerful tool which has even landed in Java during +
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// the last few years (https://inside.java/2021/06/13/podcast-017/).+
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// ...Scala already has it for many years and the concept is+
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// older than your friendly lecturer, that is stone old  ;o)+
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+
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// The general schema:+
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//+
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//    expression match {+
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//       case pattern1 => expression1+
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//       case pattern2 => expression2+
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//       ...+
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//       case patternN => expressionN+
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//    }+
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+
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+
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// recall+
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def len(xs: List[Int]) : Int = {+
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    if (xs == Nil) 0+
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    else 1 + len(xs.tail)+
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}    +
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+
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def len(xs: List[Int]) : Int = xs match {+
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    case Nil => 0+
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    case _::xs => 1 + len(xs)+
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}  +
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+
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len(Nil)+
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len(List(1,2,3,4))+
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+
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+
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List(1,2,3,4).map(x => x * x)+
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+
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def my_map_int(lst: List[Int], f: Int => Int) : List[Int] = +
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  lst match {+
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    case Nil => Nil+
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    case foo::xs => f(foo) :: my_map_int(xs, f)+
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  }+
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+
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def my_map_option(opt: Option[Int], f: Int => Int) : Option[Int] = +
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  opt match {+
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    case None => None+
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    case Some(x) => {+
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      Some(f(x))+
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    }+
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  }+
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+
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my_map_option(None, x => x * x)+
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my_map_option(Some(8), x => x * x)+
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+
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+
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// you can also have cases combined+
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def season(month: String) : String = month match {+
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  case "March" | "April" | "May" => "It's spring"+
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  case "June" | "July" | "August" => "It's summer"+
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  case "September" | "October" | "November" => "It's autumn"+
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  case "December" => "It's winter"+
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  case "January" | "February" => "It's unfortunately winter"+
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  case _ => "Wrong month"+
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}+
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+
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// pattern-match on integers+
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+
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def fib(n: Int) : Int = n match { +
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  case 0 | 1 => 1+
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  case _ => fib(n - 1) + fib(n - 2)+
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}+
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+
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fib(10)+
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+
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// pattern-match on results+
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+
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// Silly: fizz buzz+
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def fizz_buzz(n: Int) : String = (n % 3, n % 5) match {+
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  case (0, 0) => "fizz buzz"+
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  case (0, _) => "fizz"+
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  case (_, 0) => "buzz"+
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  case _ => n.toString  +
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}+
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+
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for (n <- 1 to 20) +
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 println(fizz_buzz(n))+
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+
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// guards in pattern-matching+
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+
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def foo(xs: List[Int]) : String = xs match {+
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  case Nil => s"this list is empty"+
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  case x :: xs if x % 2 == 0 +
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     => s"the first elemnt is even"+
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  case x if len(x) ==+
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     => s"this list has exactly two elements"   +
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  case x :: y :: rest if x == y+
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     => s"this has two elemnts that are the same"+
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  case hd :: tl => s"this list is standard $hd::$tl"+
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}+
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+
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foo(Nil)+
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foo(List(1,2,3))+
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foo(List(1,1))+
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foo(List(1,1,2,3))+
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foo(List(2,2,2,3))+
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+
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+
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+
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+
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abstract class Colour+
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case object Red extends Colour +
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case object Green extends Colour +
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case object Blue extends Colour+
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case object Yellow extends Colour+
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+
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+
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def fav_colour(c: Colour) : Boolean = c match {+
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  case Green => true+
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  case Red => true+
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  case _  => false +
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}+
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+
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fav_colour(Blue)+
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+
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+
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// ... a tiny bit more useful: Roman Numerals+
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+
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sealed abstract class RomanDigit +
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case object I extends RomanDigit +
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case object V extends RomanDigit +
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case object X extends RomanDigit +
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case object L extends RomanDigit +
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case object C extends RomanDigit +
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case object D extends RomanDigit +
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case object M extends RomanDigit +
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+
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type RomanNumeral = List[RomanDigit] +
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+
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List(I, M,C,D,X,X,V,I,I, A)+
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+
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/*+
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I    -> 1+
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II   -> 2+
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III  -> 3+
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IV   -> 4+
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V    -> 5+
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VI   -> 6+
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VII  -> 7+
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VIII -> 8+
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IX   -> 9+
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X    -> 10+
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*/+
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+
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def RomanNumeral2Int(rs: RomanNumeral): Int = rs match { +
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  case Nil => 0+
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  case M::r    => 1000 + RomanNumeral2Int(r)  +
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  case C::M::r => 900 + RomanNumeral2Int(r)+
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  case D::r    => 500 + RomanNumeral2Int(r)+
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  case C::D::r => 400 + RomanNumeral2Int(r)+
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  case C::r    => 100 + RomanNumeral2Int(r)+
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  case X::C::r => 90 + RomanNumeral2Int(r)+
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  case L::r    => 50 + RomanNumeral2Int(r)+
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  case X::L::r => 40 + RomanNumeral2Int(r)+
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  case X::r    => 10 + RomanNumeral2Int(r)+
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  case I::X::r => 9 + RomanNumeral2Int(r)+
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  case V::r    => 5 + RomanNumeral2Int(r)+
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  case I::V::r => 4 + RomanNumeral2Int(r)+
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  case I::r    => 1 + RomanNumeral2Int(r)+
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}+
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+
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RomanNumeral2Int(List(I,V))             // 4+
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RomanNumeral2Int(List(I,I,I,I))         // 4 (invalid Roman number)+
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RomanNumeral2Int(List(V,I))             // 6+
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RomanNumeral2Int(List(I,X))             // 9+
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RomanNumeral2Int(List(M,C,M,L,X,X,I,X)) // 1979+
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RomanNumeral2Int(List(M,M,X,V,I,I))     // 2017+
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+
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abstract class Tree+
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case class Leaf(x: Int)+
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case class Branch(tl: Tree, tr: Tree)+
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+
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+
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abstract class Rexp+
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case object ZERO extends Rexp                      // matches nothing+
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case object ONE extends Rexp                       // matches the empty string+
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case class CHAR(c: Char) extends Rexp              // matches a character c+
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case class ALT(r1: Rexp, r2: Rexp) extends Rexp    // alternative+
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case class SEQ(r1: Rexp, r2: Rexp) extends Rexp    // sequence+
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case class STAR(r: Rexp) extends Rexp              // star+
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+
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def depth(r: Rexp) : Int = r match {+
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  case ZERO => 1+
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  case ONE => 1+
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  case CHAR(_) => 1+
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  case ALT(r1, r2) => 1 + List(depth(r1), depth(r2)).max+
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  case SEQ(r1, r2) => 1 + List(depth(r1), depth(r2)).max+
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  case STAR(r1) => 1 + depth(r1)+
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}+
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+
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+
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// Trees (example of an Algebraic Datatype)+
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+
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+
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abstract class Tree+
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case class Leaf(x: Int) extends Tree+
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case class Node(s: String, left: Tree, right: Tree) extends Tree +
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+
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val lf = Leaf(20)+
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val tr = Node("foo", Leaf(10), Leaf(23))+
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+
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val lst : List[Tree] = List(lf, tr)+
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+
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+
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+
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// expressions (essentially trees)+
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+
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sealed abstract class Exp+
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case class N(n: Int) extends Exp                  // for numbers+
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case class Plus(e1: Exp, e2: Exp) extends Exp+
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case class Times(e1: Exp, e2: Exp) extends Exp+
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+
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def string(e: Exp) : String = e match {+
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  case N(n) => s"$n"+
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  case Plus(e1, e2) => s"(${string(e1)} + ${string(e2)})" +
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  case Times(e1, e2) => s"(${string(e1)} * ${string(e2)})"+
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}+
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+
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val e = Plus(N(9), Times(N(3), N(4)))+
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println(e.toString)+
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println(string(e))+
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+
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def eval(e: Exp) : Int = e match {+
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  case N(n) => n+
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  case Plus(e1, e2) => eval(e1) + eval(e2) +
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  case Times(e1, e2) => eval(e1) * eval(e2) +
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}+
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+
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println(eval(e))+
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+
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// simplification rules:+
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// e + 0, 0 + e => e +
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// e * 0, 0 * e => 0+
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// e * 1, 1 * e => e+
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//+
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// (....9 ....)+
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+
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def simp(e: Exp) : Exp = e match {+
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  case N(n) => N(n)+
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  case Plus(e1, e2) => (simp(e1), simp(e2)) match {+
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    case (N(0), e2s) => e2s+
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    case (e1s, N(0)) => e1s+
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    case (e1s, e2s) => Plus(e1s, e2s)+
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  }  +
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  case Times(e1, e2) => (simp(e1), simp(e2)) match {+
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    case (N(0), _) => N(0)+
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    case (_, N(0)) => N(0)+
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    case (N(1), e2s) => e2s+
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    case (e1s, N(1)) => e1s+
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    case (e1s, e2s) => Times(e1s, e2s)+
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  }  +
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}+
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+
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+
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val e2 = Times(Plus(N(0), N(1)), Plus(N(0), N(9)))+
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println(string(e2))+
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println(string(simp(e2)))+
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+
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+
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+
−
+
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+
−
+
−
+
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// Tokens and Reverse Polish Notation+
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abstract class Token+
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case class T(n: Int) extends Token+
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case object PL extends Token+
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case object TI extends Token+
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+
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// transfroming an Exp into a list of tokens+
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def rp(e: Exp) : List[Token] = e match {+
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  case N(n) => List(T(n))+
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  case Plus(e1, e2) => rp(e1) ::: rp(e2) ::: List(PL) +
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  case Times(e1, e2) => rp(e1) ::: rp(e2) ::: List(TI) +
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}+
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println(string(e2))+
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println(rp(e2))+
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+
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def comp(ls: List[Token], st: List[Int] = Nil) : Int = (ls, st) match {+
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  case (Nil, st) => st.head +
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  case (T(n)::rest, st) => comp(rest, n::st)+
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  case (PL::rest, n1::n2::st) => comp(rest, n1 + n2::st)+
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  case (TI::rest, n1::n2::st) => comp(rest, n1 * n2::st)+
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}+
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+
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comp(rp(e))+
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+
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def proc(s: String) : Token = s match {+
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  case  "+" => PL+
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  case  "*" => TI+
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  case  _ => T(s.toInt) +
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}+
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+
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comp("1 2 + 4 * 5 + 3 +".split(" ").toList.map(proc), Nil)+
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+
−
+
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// Tail recursion+
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//================+
−
+
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def fact(n: BigInt): BigInt = +
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  if (n == 0) 1 else n * fact(n - 1)+
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+
−
+
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fact(10)          +
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fact(1000)        +
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fact(100000)       +
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+
−
+
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def factT(n: BigInt, acc: BigInt): BigInt =+
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  if (n == 0) acc else factT(n - 1, n * acc)+
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+
−
+
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factT(10, 1)+
−
println(factT(100000, 1))+
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+
−
+
−
// there is a flag for ensuring a function is tail recursive+
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import scala.annotation.tailrec+
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+
−
@tailrec+
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def factT(n: BigInt, acc: BigInt): BigInt =+
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  if (n == 0) acc else factT(n - 1, n * acc)+
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+
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factT(100000, 1)+
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+
−
// for tail-recursive functions the Scala compiler+
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// generates loop-like code, which does not need+
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// to allocate stack-space in each recursive+
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// call; Scala can do this only for tail-recursive+
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// functions+
−
+
−
// Moral: Whenever a recursive function is resource-critical+
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// (i.e. works with a large recursion depth), then you need to+
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// write it in tail-recursive fashion.+
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// +
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// Unfortuantely, Scala because of current limitations in +
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// the JVM is not as clever as other functional languages. It can +
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// only optimise "self-tail calls". This excludes the cases of +
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// multiple functions making tail calls to each other. Well,+
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// nothing is perfect. +
−
+
−
+
−
+
−
// Polymorphic Types+
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//===================+
−
+
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// You do not want to write functions like contains, first, +
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// length and so on for every type of lists.+
−
+
−
def length_int_list(lst: List[Int]): Int = lst match {+
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  case Nil => 0+
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  case _::xs => 1 + length_int_list(xs)+
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}+
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+
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length_int_list(List(1, 2, 3, 4))+
−
+
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def length_string_list(lst: List[String]): Int = lst match {+
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  case Nil => 0+
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  case _::xs => 1 + length_string_list(xs)+
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}+
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+
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length_string_list(List("1", "2", "3", "4"))+
−
+
−
+
−
// you can make the function parametric in type(s)+
−
+
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def length[A](lst: List[A]): Int = lst match {+
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  case Nil => 0+
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  case x::xs => 1 + length(xs)+
−
}+
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length(List("1", "2", "3", "4"))+
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length(List(1, 2, 3, 4))+
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+
−
+
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length[String](List(1, 2, 3, 4))+
−
+
−
+
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def map[A, B](lst: List[A], f: A => B): List[B] = lst match {+
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  case Nil => Nil+
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  case x::xs => f(x)::map(xs, f) +
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}+
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+
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map(List(1, 2, 3, 4), (x: Int) => x.toString)+
−
+
−
+
−
// should be+
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def first[A, B](xs: List[A], f: A => Option[B]) : Option[B] = ???+
−
+
−
// Type inference is local in Scala+
−
+
−
def id[T](x: T) : T = x+
−
+
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val x = id(322)          // Int+
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val y = id("hey")        // String+
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val z = id(Set(1,2,3,4)) // Set[Int]+
−
+
−
+
−
// The type variable concept in Scala can get really complicated.+
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//+
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// - variance (OO)+
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// - bounds (subtyping)+
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// - quantification+
−
+
−
// Java has issues with this too: Java allows+
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// to write the following incorrect code, and+
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// only recovers by raising an exception+
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// at runtime.+
−
+
−
// Object[] arr = new Integer[10];+
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// arr[0] = "Hello World";+
−
+
−
+
−
// Scala gives you a compile-time error, which+
−
// is much better.+
−
+
−
var arr = Array[Int]()+
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arr(0) = "Hello World"+
−
+
−
+
−
+
−
+
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// Function definitions again+
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//============================+
−
+
−
// variable arguments+
−
+
−
def printAll(strings: String*) = {+
−
  strings.foreach(println)+
−
}+
−
+
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printAll()+
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printAll("foo")+
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printAll("foo", "bar")+
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printAll("foo", "bar", "baz")+
−
+
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// pass a list to the varargs field+
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val fruits = List("apple", "banana", "cherry")+
−
+
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printAll(fruits: _*)+
−
+
−
+
−
// you can also implement your own string interpolations+
−
import scala.language.implicitConversions+
−
import scala.language.reflectiveCalls+
−
+
−
implicit def sring_inters(sc: StringContext) = new {+
−
    def i(args: Any*): String = s"${sc.s(args:_*)}\n"+
−
}+
−
+
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i"add ${3+2} ${3 * 3}" +
−
+
−
+
−
// default arguments+
−
+
−
def length[A](xs: List[A]) : Int = xs match {+
−
  case Nil => 0+
−
  case _ :: tail => 1 + length(tail)+
−
}+
−
+
−
def lengthT[A](xs: List[A], acc : Int = 0) : Int = xs match {+
−
  case Nil => acc+
−
  case _ :: tail => lengthT(tail, 1 + acc)+
−
}+
−
+
−
lengthT(List.fill(100000)(1))+
−
+
−
+
−
def fact(n: BigInt, acc: BigInt = 1): BigInt =+
−
  if (n == 0) acc else fact(n - 1, n * acc)+
−
+
−
fact(10)+
−
+
−
+
−
+
−
// currying    (Haskell Curry)+
−
+
−
def add(x: Int, y: Int) = x + y+
−
+
−
List(1,2,3,4,5).map(x => add(3, x))+
−
+
−
def add2(x: Int)(y: Int) = x + y+
−
+
−
List(1,2,3,4,5).map(add2(3))+
−
+
−
val a3 : Int => Int = add2(3)+
−
+
−
// currying helps sometimes with type inference+
−
+
−
def find[A](xs: List[A])(pred: A => Boolean): Option[A] = {+
−
  xs match {+
−
    case Nil => None+
−
    case hd :: tl =>+
−
      if (pred(hd)) Some(hd) else find(tl)(pred)+
−
  }+
−
}+
−
+
−
find(List(1, 2, 3))(x => x % 2 == 0)+
−
+
−
// Source.fromURL(url)(encoding)+
−
// Source.fromFile(name)(encoding)+
−
+
−
+
−
+
−
+
−
+
−
+
−
+
−
// Sudoku +
−
//========+
−
+
−
// THE POINT OF THIS CODE IS NOT TO BE SUPER+
−
// EFFICIENT AND FAST, just explaining exhaustive+
−
// depth-first search+
−
+
−
+
−
val game0 = """.14.6.3..+
−
              |62...4..9+
−
              |.8..5.6..+
−
              |.6.2....3+
−
              |.7..1..5.+
−
              |5....9.6.+
−
              |..6.2..3.+
−
              |1..5...92+
−
              |..7.9.41.""".stripMargin.replaceAll("\\n", "")+
−
+
−
+
−
+
−
type Pos = (Int, Int)+
−
val EmptyValue = '.'+
−
val MaxValue = 9+
−
+
−
def pretty(game: String): String = +
−
  "\n" + (game.grouped(MaxValue).mkString("\n"))+
−
+
−
pretty(game0)+
−
+
−
+
−
val allValues = "123456789".toList+
−
val indexes = (0 to 8).toList+
−
+
−
def empty(game: String) = game.indexOf(EmptyValue)+
−
def isDone(game: String) = empty(game) == -1 +
−
def emptyPosition(game: String) = {+
−
  val e = empty(game)+
−
  (e % MaxValue, e / MaxValue)+
−
}+
−
+
−
def get_row(game: String, y: Int) = +
−
  indexes.map(col => game(y * MaxValue + col))+
−
def get_col(game: String, x: Int) = +
−
  indexes.map(row => game(x + row * MaxValue))+
−
+
−
//get_row(game0, 0)+
−
//get_row(game0, 1)+
−
//get_col(game0, 0)+
−
+
−
def get_box(game: String, pos: Pos): List[Char] = {+
−
    def base(p: Int): Int = (p / 3) * 3+
−
    val x0 = base(pos._1)+
−
    val y0 = base(pos._2)+
−
    val ys = (y0 until y0 + 3).toList+
−
    (x0 until x0 + 3).toList+
−
      .flatMap(x => ys.map(y => game(x + y * MaxValue)))+
−
}+
−
+
−
+
−
//get_box(game0, (3, 1))+
−
+
−
+
−
// this is not mutable!!+
−
def update(game: String, pos: Int, value: Char): String = +
−
  game.updated(pos, value)+
−
+
−
def toAvoid(game: String, pos: Pos): List[Char] = +
−
  (get_col(game, pos._1) ++ +
−
   get_row(game, pos._2) ++ +
−
   get_box(game, pos))+
−
+
−
def candidates(game: String, pos: Pos): List[Char] = +
−
  allValues.diff(toAvoid(game, pos))+
−
+
−
//candidates(game0, (0,0))+
−
+
−
+
−
def search(game: String): List[String] = {+
−
  if (isDone(game)) List(game)+
−
  else {+
−
    val cs = candidates(game, emptyPosition(game))+
−
    cs.map(c => search(update(game, empty(game), c))).flatten+
−
  }+
−
}+
−
+
−
pretty(game0)+
−
search(game0).map(pretty)+
−
+
−
val game1 = """23.915...+
−
              |...2..54.+
−
              |6.7......+
−
              |..1.....9+
−
              |89.5.3.17+
−
              |5.....6..+
−
              |......9.5+
−
              |.16..7...+
−
              |...329..1""".stripMargin.replaceAll("\\n", "")+
−
+
−
search(game1).map(pretty)+
−
+
−
// a game that is in the hard category+
−
val game2 = """8........+
−
              |..36.....+
−
              |.7..9.2..+
−
              |.5...7...+
−
              |....457..+
−
              |...1...3.+
−
              |..1....68+
−
              |..85...1.+
−
              |.9....4..""".stripMargin.replaceAll("\\n", "")+
−
+
−
search(game2).map(pretty)+
−
+
−
// game with multiple solutions+
−
val game3 = """.8...9743+
−
              |.5...8.1.+
−
              |.1.......+
−
              |8....5...+
−
              |...8.4...+
−
              |...3....6+
−
              |.......7.+
−
              |.3.5...8.+
−
              |9724...5.""".stripMargin.replaceAll("\\n", "")+
−
+
−
search(game3).map(pretty).foreach(println)+
−
+
−
// for measuring time+
−
def time_needed[T](i: Int, code: => T) = {+
−
  val start = System.nanoTime()+
−
  for (j <- 1 to i) code+
−
  val end = System.nanoTime()+
−
  s"${(end - start) / 1.0e9} secs"+
−
}+
−
+
−
time_needed(1, search(game2))+
−
+
−
+
−
+
−
// tail recursive version that searches +
−
// for all Sudoku solutions+
−
import scala.annotation.tailrec+
−
+
−
@tailrec+
−
def searchT(games: List[String], sols: List[String]): List[String] = +
−
 games match {+
−
    case Nil => sols+
−
    case game::rest => {+
−
      if (isDone(game)) searchT(rest, game::sols)+
−
      else {+
−
        val cs = candidates(game, emptyPosition(game))+
−
        searchT(cs.map(c => update(game, empty(game), c)) ::: rest, sols)+
−
     }+
−
   }+
−
 }+
−
+
−
searchT(List(game3), List()).map(pretty)+
−
+
−
+
−
// tail recursive version that searches +
−
// for a single solution+
−
+
−
def search1T(games: List[String]): Option[String] = games match {+
−
  case Nil => None+
−
  case game::rest => {+
−
    if (isDone(game)) Some(game)+
−
    else {+
−
      val cs = candidates(game, emptyPosition(game))+
−
      search1T(cs.map(c => update(game, empty(game), c)) ::: rest)+
−
    }+
−
  }+
−
}+
−
+
−
search1T(List(game3)).map(pretty)+
−
time_needed(1, search1T(List(game3)))+
−
time_needed(1, search1T(List(game2)))+
−
+
−
// game with multiple solutions+
−
val game3 = """.8...9743+
−
              |.5...8.1.+
−
              |.1.......+
−
              |8....5...+
−
              |...8.4...+
−
              |...3....6+
−
              |.......7.+
−
              |.3.5...8.+
−
              |9724...5.""".stripMargin.replaceAll("\\n", "")+
−
+
−
searchT(List(game3), Nil).map(pretty)+
−
search1T(List(game3)).map(pretty)+
−
+
−
+
−
+
−
+
−
+
−
+
−
// Cool Stuff in Scala+
−
//=====================+
−
+
−
+
−
// Implicits or How to Pimp your Library+
−
//======================================+
−
//+
−
// For example adding your own methods to Strings:+
−
// Imagine you want to increment strings, like+
−
//+
−
//     "HAL".increment+
−
//+
−
// you can avoid ugly fudges, like a MyString, by+
−
// using implicit conversions.+
−
+
−
print("\n")+
−
print("""\n""")+
−
+
−
implicit class MyString(s: String) {+
−
  def increment = s.map(c => (c + 1).toChar) +
−
}+
−
+
−
"HAL".increment+
−
+
−
+
−
// Abstract idea:+
−
// In that version implicit conversions were used to solve the +
−
// late extension problem; namely, given a class C and a class T, +
−
// how to have C extend T without touching or recompiling C. +
−
// Conversions add a wrapper when a member of T is requested +
−
// from an instance of C.+
−
+
−
+
−
+
−
import scala.concurrent.duration.{TimeUnit,SECONDS,MINUTES}+
−
+
−
case class Duration(time: Long, unit: TimeUnit) {+
−
  def +(o: Duration) = +
−
    Duration(time + unit.convert(o.time, o.unit), unit)+
−
}+
−
+
−
implicit class Int2Duration(that: Int) {+
−
  def seconds = Duration(that, SECONDS)+
−
  def minutes = Duration(that, MINUTES)+
−
}+
−
+
−
5.seconds + 2.minutes   //Duration(125L, SECONDS )+
−
2.minutes + 60.seconds+
−
+
−
+
−
+
−
+
−
// Regular expressions - the power of DSLs in Scala+
−
//==================================================+
−
+
−
abstract class Rexp+
−
case object ZERO extends Rexp                     // nothing+
−
case object ONE extends Rexp                      // the empty string+
−
case class CHAR(c: Char) extends Rexp             // a character c+
−
case class ALT(r1: Rexp, r2: Rexp) extends Rexp   // alternative  r1 + r2+
−
case class SEQ(r1: Rexp, r2: Rexp) extends Rexp   // sequence     r1 . r2  +
−
case class STAR(r: Rexp) extends Rexp             // star         r*+
−
+
−
+
−
+
−
// writing (ab)* in the format above is +
−
// tedious+
−
val r0 = STAR(SEQ(CHAR('a'), CHAR('b')))+
−
+
−
+
−
// some convenience for typing in regular expressions+
−
import scala.language.implicitConversions    +
−
import scala.language.reflectiveCalls +
−
+
−
def charlist2rexp(s: List[Char]): Rexp = s match {+
−
  case Nil => ONE+
−
  case c::Nil => CHAR(c)+
−
  case c::s => SEQ(CHAR(c), charlist2rexp(s))+
−
}+
−
+
−
implicit def string2rexp(s: String): Rexp = +
−
  charlist2rexp(s.toList)+
−
+
−
val r1 = STAR("ab")+
−
val r2 = STAR("hello") | STAR("world")+
−
+
−
+
−
implicit def RexpOps (r: Rexp) = new {+
−
  def | (s: Rexp) = ALT(r, s)+
−
  def % = STAR(r)+
−
  def ~ (s: Rexp) = SEQ(r, s)+
−
}+
−
+
−
implicit def stringOps (s: String) = new {+
−
  def | (r: Rexp) = ALT(s, r)+
−
  def | (r: String) = ALT(s, r)+
−
  def % = STAR(s)+
−
  def ~ (r: Rexp) = SEQ(s, r)+
−
  def ~ (r: String) = SEQ(s, r)+
−
}+
−
+
−
//example regular expressions+
−
+
−
+
−
val digit = ("0" | "1" | "2" | "3" | "4" | +
−
              "5" | "6" | "7" | "8" | "9")+
−
val sign = "+" | "-" | ""+
−
val number = sign ~ digit ~ digit.% +
−
+
−
+
−
+
−
+
−
// In mandelbrot.scala I used complex (imaginary) numbers +
−
// and implemented the usual arithmetic operations for complex +
−
// numbers.+
−
+
−
case class Complex(re: Double, im: Double) { +
−
  // represents the complex number re + im * i+
−
  def +(that: Complex) = Complex(this.re + that.re, this.im + that.im)+
−
  def -(that: Complex) = Complex(this.re - that.re, this.im - that.im)+
−
  def *(that: Complex) = Complex(this.re * that.re - this.im * that.im,+
−
                                 this.re * that.im + that.re * this.im)+
−
  def *(that: Double) = Complex(this.re * that, this.im * that)+
−
  def abs = Math.sqrt(this.re * this.re + this.im * this.im)+
−
}+
−
+
−
val test = Complex(1, 2) + Complex (3, 4)+
−
+
−
+
−
// ...to allow the notation n + m * i+
−
import scala.language.implicitConversions   +
−
+
−
val i = Complex(0, 1)+
−
implicit def double2complex(re: Double) = Complex(re, 0)+
−
+
−
val inum1 = -2.0 + -1.5 * i+
−
val inum2 =  1.0 +  1.5 * i+
−