// Main Part 3 about Regular Expression Matching+
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//=============================================+
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+
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object M3 {+
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+
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// Regular Expressions+
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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 ALTs(rs: List[Rexp]) extends Rexp      // alternatives +
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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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+
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//the usual binary choice can be defined in terms of ALTs+
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def ALT(r1: Rexp, r2: Rexp) = ALTs(List(r1, r2))+
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+
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// some convenience for typing in regular expressions+
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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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// (1) 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 and Returns a boolean+
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// accordingly.+
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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 ALTs(rs) => rs.exists(nullable)+
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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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// (2) 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 ALTs(rs) => ALTs(rs.map(der(c, _)))+
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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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+
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// (3) Implement the flatten function flts. It+
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// deletes 0s from a list of regular expressions+
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// and also 'spills out', or flattens, nested +
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// ALTernativeS.+
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+
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def flts(rs: List[Rexp]) : List[Rexp] = rs match {+
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  case Nil => Nil+
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  case ZERO::tl => flts(tl)+
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  case ALTs(rs1)::rs2 => rs1 ::: flts(rs2)  +
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  case r::rs => r :: flts(rs) +
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}+
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+
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// (4) Complete the simp function according to+
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// the specification given in the coursework; this+
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// function simplifies a regular expression from+
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// the inside out, like you would simplify arithmetic +
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// expressions; however it does not simplify inside +
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// STAR-regular expressions.+
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+
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+
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def simp(r: Rexp) : Rexp = r match {+
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  case ALTs(rs) => (flts(rs.map(simp)).distinct) match {+
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    case Nil => ZERO+
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    case r::Nil => r  +
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    case rs => ALTs(rs)+
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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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simp(ALT(ONE | CHAR('a'), CHAR('a') | ONE))+
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+
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// (5) 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) = nullable(ders(s.toList, r))+
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+
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// (6) Complete the size function for regular+
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// expressions according to the specification +
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// given in the coursework.+
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+
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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 ALTs(rs) => 1 + rs.map(size).sum+
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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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+
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// some testing data+
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+
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//matcher(("a" ~ "b") ~ "c", "abc")  // => true+
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//matcher(("a" ~ "b") ~ "c", "ab")   // => false+
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+
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// the supposedly 'evil' regular expression (a*)* b+
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// val EVIL = SEQ(STAR(STAR(CHAR('a'))), CHAR('b'))+
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+
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//println(matcher(EVIL, "a" * 1000 ++ "b"))   // => true+
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//println(matcher(EVIL, "a" * 1000))          // => false+
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+
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// size without simplifications+
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//println(size(der('a', der('a', EVIL))))             // => 28+
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//println(size(der('a', der('a', der('a', EVIL)))))   // => 58+
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+
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// size with simplification+
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//println(simp(der('a', der('a', EVIL))))          +
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//println(simp(der('a', der('a', der('a', EVIL)))))+
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+
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//println(size(simp(der('a', der('a', EVIL)))))           // => 8+
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//println(size(simp(der('a', der('a', der('a', EVIL)))))) // => 8+
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+
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// Python needs around 30 seconds for matching 28 a's with EVIL. +
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// Java 9 and later increase this to an "astonishing" 40000 a's in+
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// around 30 seconds.+
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//+
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// Lets see how long it takes to match strings with +
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// 5 Million a's...it should be in the range of a +
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// couple of seconds.+
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+
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def time_needed[T](i: Int, code: => T) = {+
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  val start = System.nanoTime()+
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  for (j <- 1 to i) code+
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  val end = System.nanoTime()+
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  "%.5f".format((end - start)/(i * 1.0e9))+
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}+
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+
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//for (i <- 0 to 5000000 by 500000) {+
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//  println(s"$i ${time_needed(2, matcher(EVIL, "a" * i))} secs.") +
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//}+
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+
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// another "power" test case +
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//simp(Iterator.iterate(ONE:Rexp)(r => SEQ(r, ONE | ONE)).drop(100).next) == ONE+
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+
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// the Iterator produces the rexp+
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//+
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//      SEQ(SEQ(SEQ(..., ONE | ONE) , ONE | ONE), ONE | ONE)+
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//+
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//    where SEQ is nested 50 times.+
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 +
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+
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+
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}+
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