// A version which attempts to move whole strings, not

// just characters, under derivatives whenever possible

abstract class Rexp

case object ZERO extends Rexp

case object ONE extends Rexp

case class CHAR(c: Char) extends Rexp

case class ALT(r1: Rexp, r2: Rexp) extends Rexp 

case class SEQ(r1: Rexp, r2: Rexp) extends Rexp 

case class STAR(r: Rexp) extends Rexp 

case class NTIMES(r: Rexp, n: Int) extends Rexp 

// the nullable function: tests whether the regular 

// expression can recognise the empty string

def nullable (r: Rexp) : Boolean = r match {

  case ZERO => false

  case ONE => true

  case CHAR(_) => false

  case ALT(r1, r2) => nullable(r1) || nullable(r2)

  case SEQ(r1, r2) => nullable(r1) && nullable(r2)

  case STAR(_) => true

  case NTIMES(r, i) => if (i == 0) true else nullable(r)

}

// the derivative of a regular expression w.r.t. a character

def der (c: Char, r: Rexp) : Rexp = r match {

  case ZERO => ZERO

  case ONE => ZERO

  case CHAR(d) => if (c == d) ONE else ZERO

  case ALT(r1, r2) => ALT(der(c, r1), der(c, r2))

  case SEQ(r1, r2) => 

    if (nullable(r1)) ALT(SEQ(der(c, r1), r2), der(c, r2))

    else SEQ(der(c, r1), r2)

  case STAR(r1) => SEQ(der(c, r1), STAR(r1))

  case NTIMES(r1, i) => 

    if (i == 0) ZERO else SEQ(der(c, r1), NTIMES(r1, i - 1))

}

def simp(r: Rexp) : Rexp = r match {

  case ALT(r1, r2) => (simp(r1), simp(r2)) match {

    case (ZERO, r2s) => r2s

    case (r1s, ZERO) => r1s

    case (r1s, r2s) => if (r1s == r2s) r1s else ALT (r1s, r2s)

  }

  case SEQ(r1, r2) =>  (simp(r1), simp(r2)) match {

    case (ZERO, _) => ZERO

    case (_, ZERO) => ZERO

    case (ONE, r2s) => r2s

    case (r1s, ONE) => r1s

    case (r1s, r2s) => SEQ(r1s, r2s)

  }

  case r => r

}

// an example

val r = SEQ(SEQ(CHAR('x'), CHAR('y')), CHAR('z'))

der('x', r)

der('y', der('x', r))

der('z', der('y', der('x', r)))

simp(der('z', der('y', der('x', r))))

// *new*

// the derivative w.r.t. a string (iterates der)

def ders2(s: List[Char], r: Rexp) : Rexp = (s, r) match {

  case (Nil, r) => r

  case (s, ZERO) => ZERO

  case (s, ONE) => if (s == Nil) ONE else ZERO

  case (s, CHAR(c)) => if (s == List(c)) ONE else 

                       if (s == Nil) CHAR(c) else ZERO

  case (s, ALT(r1, r2)) => ALT(ders2(s, r1), ders2(s, r2))

  case (c::s, r) => ders2(s, simp(der(c, r)))

}

// the main matcher function

def matcher(r: Rexp, s: String) : Boolean = 

  nullable(ders2(s.toList, r))

// one or zero

def OPT(r: Rexp) = ALT(r, ONE)

// Test Cases

def EVIL1(n: Int) = SEQ(NTIMES(OPT(CHAR('a')), n), NTIMES(CHAR('a'), n))

val EVIL2 = SEQ(STAR(STAR(CHAR('a'))), CHAR('b'))

// 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()

  (end - start) / (i * 1.0e9)

}

// test: (a?{n}) (a{n})

for (i <- 0 to 11000 by 1000) {

  println(f"$i: ${time_needed(2, matcher(EVIL1(i), "a" * i))}%.5f")

}

// test: (a*)* b

for (i <- 0 to 7000000 by 500000) {

  println(f"$i: ${time_needed(2, matcher(EVIL2, "a" * i))}%.5f")

}

// the size of a regular expressions - for testing purposes 

def size(r: Rexp) : Int = r match {

  case ZERO => 1

  case ONE => 1

  case CHAR(_) => 1

  case ALT(r1, r2) => 1 + size(r1) + size(r2)

  case SEQ(r1, r2) => 1 + size(r1) + size(r2)

  case STAR(r) => 1 + size(r)

  case NTIMES(r, _) => 1 + size(r)

}

// string of a regular expressions - for testing purposes 

def string(r: Rexp) : String = r match {

  case ZERO => "0"

  case ONE => "1"

  case CHAR(c) => c.toString 

  case ALT(r1, r2) => s"(${string(r1)} + ${string(r2)})"

  case SEQ(CHAR(c), CHAR(d)) => s"${c}${d}"

  case SEQ(r1, r2) => s"(${string(r1)} ~ ${string(r2)})"

  case STAR(r) => s"(${string(r)})*"

  case NTIMES(r, n) =>  s"(${string(r)}){${n}}"

}

// test: ("a" | "aa")*

val EVIL3 = STAR(ALT(CHAR('a'), SEQ(CHAR('a'), CHAR('a'))))

// test: ("" | "a" | "aa")*

val EVIL4 = STAR(ALT(ONE, ALT(CHAR('a'), SEQ(CHAR('a'), CHAR('a')))))

val t1  = ders2("a".toList, EVIL3)

val t1s = simp(t1) 

val t2  = ders2("aa".toList, t1s)

val t2s = simp(t2)

val t3  = ders2("aaa".toList, t2s)

val t3s = simp(t3)

val t4  = ders2("aaaa".toList, t3s)

val t4s = simp(t4)

println(string(t1) + "    " + size(t1))

println("s " + string(t1s) + "    " + size(t1s))

println(string(t2) + "    " + size(t2))

println("s " + string(t2s) + "    " + size(t2s))

println(string(t3) + "    " + size(t3))

println("s " + string(t3s) + "    " + size(t3s))

println(string(t4) + "    " + size(t4))

println("s " + string(t4s) + "    " + size(t4s))