// Part 1 about Regular Expression Matching+
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//==========================================+
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+
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abstract class Rexp+
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case object ZERO extends Rexp+
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case object ONE extends Rexp+
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case class CHAR(c: Char) extends Rexp+
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case class ALT(r1: Rexp, r2: Rexp) extends Rexp +
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case class SEQ(r1: Rexp, r2: Rexp) extends Rexp +
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case class STAR(r: Rexp) extends Rexp +
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+
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// some convenience for typing in regular expressions+
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+
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import scala.language.implicitConversions    +
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import scala.language.reflectiveCalls +
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+
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def charlist2rexp(s: List[Char]): Rexp = s match {+
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  case Nil => ONE+
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  case c::Nil => CHAR(c)+
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  case c::s => SEQ(CHAR(c), charlist2rexp(s))+
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}+
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implicit def string2rexp(s: String): Rexp = charlist2rexp(s.toList)+
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+
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implicit def RexpOps (r: Rexp) = new {+
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  def | (s: Rexp) = ALT(r, s)+
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  def % = STAR(r)+
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  def ~ (s: Rexp) = SEQ(r, s)+
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}+
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+
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implicit def stringOps (s: String) = new {+
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  def | (r: Rexp) = ALT(s, r)+
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  def | (r: String) = ALT(s, r)+
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  def % = STAR(s)+
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  def ~ (r: Rexp) = SEQ(s, r)+
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  def ~ (r: String) = SEQ(s, r)+
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}+
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+
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// (1a) Complete the function nullable according to+
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// the definition given in the coursework; this +
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// function checks whether a regular expression+
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// can match the empty string+
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+
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def nullable (r: Rexp) : Boolean = r match {+
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  case ZERO => false+
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  case ONE => true+
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  case CHAR(_) => false+
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  case ALT(r1, r2) => nullable(r1) || nullable(r2)+
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  case SEQ(r1, r2) => nullable(r1) && nullable(r2)+
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  case STAR(_) => true+
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}+
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+
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// (1b) Complete the function der according to+
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// the definition given in the coursework; this+
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// function calculates the derivative of a +
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// regular expression w.r.t. a character+
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+
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def der (c: Char, r: Rexp) : Rexp = r match {+
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  case ZERO => ZERO+
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  case ONE => ZERO+
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  case CHAR(d) => if (c == d) ONE else ZERO+
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  case ALT(r1, r2) => ALT(der(c, r1), der(c, r2))+
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  case SEQ(r1, r2) => +
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    if (nullable(r1)) ALT(SEQ(der(c, r1), r2), der(c, r2))+
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    else SEQ(der(c, r1), r2)+
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  case STAR(r1) => SEQ(der(c, r1), STAR(r1))+
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}+
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+
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// (1c) Complete the function der according to+
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// the specification given in the coursework; this+
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// function simplifies a regular expression;+
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// however it does not simplify inside STAR-regular+
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// expressions+
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+
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def simp(r: Rexp) : Rexp = r match {+
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  case ALT(r1, r2) => (simp(r1), simp(r2)) match {+
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    case (ZERO, r2s) => r2s+
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    case (r1s, ZERO) => r1s+
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    case (r1s, r2s) => if (r1s == r2s) r1s else ALT (r1s, r2s)+
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  }+
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  case SEQ(r1, r2) =>  (simp(r1), simp(r2)) match {+
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    case (ZERO, _) => ZERO+
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    case (_, ZERO) => ZERO+
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    case (ONE, r2s) => r2s+
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    case (r1s, ONE) => r1s+
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    case (r1s, r2s) => SEQ(r1s, r2s)+
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  }+
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  case r => r+
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}+
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+
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// (1d) Complete the two functions below; the first +
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// calculates the derivative w.r.t. a string; the second+
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// is the regular expression matcher taking a regular+
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// expression and a string and checks whether the+
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// string matches the regular expression+
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+
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def ders (s: List[Char], r: Rexp) : Rexp = s match {+
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  case Nil => r+
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  case c::s => ders(s, simp(der(c, r)))+
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}+
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+
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// main matcher function+
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def matcher(r: Rexp, s: String): Boolean = nullable(ders(s.toList, r))+
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+
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+
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// (1e) Complete the function below: it searches (from the left to +
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// right) in string s1 all the non-empty substrings that match the +
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// regular expression -- these substrings are assumed to be+
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// the longest substrings matched by the regular expression and+
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// assumed to be non-overlapping. All these substrings in s1 are replaced+
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// by s2.+
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+
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+
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+
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def splits(s: String): List[(String, String)] =+
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  (for (i <- (1 to s.length).toList) yield s.splitAt(i)).reverse+
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  +
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splits("abcde")+
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splits("")+
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+
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def first(r: Rexp, lst: List[(String, String)]): Option[String] = lst match {+
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  case Nil => None+
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  case (s1, s2)::xs => if (matcher(r, s1)) Some(s2) else first(r, xs)+
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}+
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 +
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"abcd".head+
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+
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def replace(r: Rexp, s1: String, s2: String): String = first(r, splits(s1)) match {+
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  case None if (s1 == "") => ""+
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  case None => s1.head.toString ++ replace(r, s1.tail, s2)+
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  case Some(s) => s2 ++ replace(r, s, s2) +
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}+
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 +
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val s1 =  "aabbbaaaaaaabaaaaabbaaaabb"+
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val r: Rexp = "aa".% | "bb"+
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splits(s1)+
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first(r, splits(s1))+
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+
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replace(r, s1, "c")+
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+
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splits("bb")+
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first(r, splits("bb"))+
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replace(r, "abb", "c")+
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+
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+
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// PART 2+
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//========+
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+
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+
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// (2a)+
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+
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import scala.annotation.tailrec+
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+
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@tailrec+
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def iterT[A](n: Int, f: A => A, x: A): A = +
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  if (n == 0) x else iterT(n - 1, f, f(x)) +
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+
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+
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//non-tail recursive iter+
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+
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//def iter[A](n: Int, f: A => A)(x: A): A = +
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//  if (n == 0) x else f(iter(n - 1,f, x)) +
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+
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+
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+
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// (2b)+
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+
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def size(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 + size(r1) + size (r2)+
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  case SEQ(r1, r2) => 1 + size(r1) + size (r2)+
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  case STAR(r1) => 1 + size(r1)+
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}+
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+
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+
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val EVIL = SEQ(STAR(STAR(CHAR('a'))), CHAR('b'))+
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size(iterT(20, (r: Rexp) => der('a', r), EVIL))        // should produce 7340068        +
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size(iterT(20, (r: Rexp) => simp(der('a', r)), EVIL))  // should produce 8+
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+
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+
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// (2c)+
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+
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@tailrec+
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def fixpT[A](f: A => A, x: A): A = {+
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  val fx = f(x)+
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  if (fx == x) x else fixpT(f, fx) +
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}+
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+
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def ctest(n: Long): Long =+
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  if (n == 1) 1 else+
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    if (n % 2 == 0) n / 2 else 3 * n + 1+
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+
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fixpT(ctest, 97L)+
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fixpT(ctest, 871L)+
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fixpT(ctest, 77031L)+
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fixpT(ctest, 837799L)+
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+
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def foo(s: String): String = {+
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  if (matcher("a", s)) "a" else+
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  if (matcher("aa" ~ STAR("aa"), s)) s.take(s.length / 2) +
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  else "a" ++ s * 3+
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}+
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+
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fixpT(foo, "a" * 97)+
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fixpT(foo, "a" * 871)+
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+
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+
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