// Part 1 about Regular Expression Matching

//==========================================

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 

// 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)

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)

}

// (1a) Complete the function nullable according to

// the definition given in the coursework; this 

// function checks whether a regular expression

// can match 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

}

// (1b) Complete the function der according to

// the definition given in the coursework; this

// function calculates 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))

}

// (1c) Complete the function der according to

// the specification given in the coursework; this

// function simplifies a regular expression;

// however it does not simplify inside STAR-regular

// expressions

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

}

// (1d) Complete the two functions below; the first 

// calculates the derivative w.r.t. a string; the second

// is the regular expression matcher taking a regular

// expression and a string and checks whether the

// string matches the regular expression

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

  case Nil => r

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

}

// main matcher function

def matcher(r: Rexp, s: String): Boolean = nullable(ders(s.toList, r))

// (1e) Complete the function below: it searches (from the left to 

// right) in string s1 all the non-empty substrings that match the 

// regular expression -- these substrings are assumed to be

// the longest substrings matched by the regular expression and

// assumed to be non-overlapping. All these substrings in s1 are replaced

// by s2.

def splits(s: String): List[(String, String)] =

  (for (i <- (1 to s.length).toList) yield s.splitAt(i)).reverse

splits("abcde")

splits("")

def first(r: Rexp, lst: List[(String, String)]): Option[String] = lst match {

  case Nil => None

  case (s1, s2)::xs => if (matcher(r, s1)) Some(s2) else first(r, xs)

}

"abcd".head

def replace(r: Rexp, s1: String, s2: String): String = first(r, splits(s1)) match {

  case None if (s1 == "") => ""

  case None => s1.head.toString ++ replace(r, s1.tail, s2)

  case Some(s) => s2 ++ replace(r, s, s2) 

}

val s1 =  "aabbbaaaaaaabaaaaabbaaaabb"

val r: Rexp = "aa".% | "bb"

splits(s1)

first(r, splits(s1))

replace(r, s1, "c")

splits("bb")

first(r, splits("bb"))

replace(r, "abb", "c")

// PART 2

//========

// (2a)

import scala.annotation.tailrec

@tailrec

def iterT[A](n: Int, f: A => A, x: A): A = 

  if (n == 0) x else iterT(n - 1, f, f(x)) 

//non-tail recursive iter

//def iter[A](n: Int, f: A => A)(x: A): A = 

//  if (n == 0) x else f(iter(n - 1,f, x)) 

// (2b)

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(r1) => 1 + size(r1)

}

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

size(iterT(20, (r: Rexp) => der('a', r), EVIL))        // should produce 7340068        

size(iterT(20, (r: Rexp) => simp(der('a', r)), EVIL))  // should produce 8

// (2c)

@tailrec

def fixpT[A](f: A => A, x: A): A = {

  val fx = f(x)

  if (fx == x) x else fixpT(f, fx) 

}

def ctest(n: Long): Long =

  if (n == 1) 1 else

    if (n % 2 == 0) n / 2 else 3 * n + 1

fixpT(ctest, 97L)

fixpT(ctest, 871L)

fixpT(ctest, 77031L)

fixpT(ctest, 837799L)

def foo(s: String): String = {

  if (matcher("a", s)) "a" else

  if (matcher("aa" ~ STAR("aa"), s)) s.take(s.length / 2) 

  else "a" ++ s * 3

}

fixpT(foo, "a" * 97)

fixpT(foo, "a" * 871)