// Scala Lecture 3

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

// Pattern Matching

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

// A powerful tool which is supposed to come to Java in a few years

// time (https://www.youtube.com/watch?v=oGll155-vuQ)...Scala already

// has it for many years. Other functional languages have it already for

// decades. I think I would be really upset if a programming language 

// I have to use does not have pattern matching....its is just so 

// useful. ;o)

// The general schema:

//

//    expression match {

//       case pattern1 => expression1

//       case pattern2 => expression2

//       ...

//       case patternN => expressionN

//    }

// remember

val lst = List(None, Some(1), Some(2), None, Some(3)).flatten

def my_flatten(xs: List[Option[Int]]): List[Int] = {

  if (xs == Nil) Nil

  else if (xs.head == None) my_flatten(xs.tail)

  else xs.head.get :: my_flatten(xs.tail)

}

val lst = List(None, Some(1), Some(2), None, Some(3))

def my_flatten(lst: List[Option[Int]]): List[Int] = lst match {

  case Nil => Nil

  case None::xs => my_flatten(xs)

  case Some(n)::xs => n::my_flatten(xs)

}

my_flatten(lst)

Nil == List()

// another example including a catch-all pattern

def get_me_a_string(n: Int): String = n match {

  case 0 => "zero"

  case 1 => "one"

  case 2 => "two"

  case _ => "many"

}

get_me_a_string(10)

// you can also have cases combined

def season(month: String) = month match {

  case "March" | "April" | "May" => "It's spring"

  case "June" | "July" | "August" => "It's summer"

  case "September" | "October" | "November" => "It's autumn"

  case "December" | "January" | "February" => "It's winter"

}

println(season("November"))

// What happens if no case matches?

println(season("foobar"))

// we can also match more complicated pattern

//

// let's look at the Collatz function on binary strings

// adding two binary strings in a very, very lazy manner

def badd(s1: String, s2: String) : String = 

  (BigInt(s1, 2) + BigInt(s2, 2)).toString(2)

"111".dropRight(1)

"111".last

def bcollatz(s: String) : Long = (s.dropRight(1), s.last) match {

  case ("", '1') => 1                               // we reached 1

  case (rest, '0') => 1 + bcollatz(rest)            

                                  // even number => divide by two

  case (rest, '1') => 1 + bcollatz(badd(s + '1', s))

                                  // odd number => s + '1' is 2 * s + 1

                                  // add another s gives 3 * s + 1  

} 

bcollatz(6.toBinaryString)

bcollatz(837799.toBinaryString)

bcollatz(100000000000000000L.toBinaryString)

bcollatz(BigInt("1000000000000000000000000000000000000000000000000000000000000000000000000000").toString(2))

// User-defined Datatypes

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

abstract class Colour

case object Red extends Colour 

case object Green extends Colour 

case object Blue extends Colour

def fav_colour(c: Colour) : Boolean = c match {

  case Red   => false

  case Green => true

  case Blue  => false 

}

fav_colour(Green)

// actually colors can be written with "object",

// because they do not take any arguments

// ... a bit more useful: Roman Numerals

abstract class RomanDigit 

case object I extends RomanDigit 

case object V extends RomanDigit 

case object X extends RomanDigit 

case object L extends RomanDigit 

case object C extends RomanDigit 

case object D extends RomanDigit 

case object M extends RomanDigit 

type RomanNumeral = List[RomanDigit] 

def RomanNumeral2Int(rs: RomanNumeral): Int = rs match { 

  case Nil => 0

  case M::r    => 1000 + RomanNumeral2Int(r)  

  case C::M::r => 900 + RomanNumeral2Int(r)

  case D::r    => 500 + RomanNumeral2Int(r)

  case C::D::r => 400 + RomanNumeral2Int(r)

  case C::r    => 100 + RomanNumeral2Int(r)

  case X::C::r => 90 + RomanNumeral2Int(r)

  case L::r    => 50 + RomanNumeral2Int(r)

  case X::L::r => 40 + RomanNumeral2Int(r)

  case X::r    => 10 + RomanNumeral2Int(r)

  case I::X::r => 9 + RomanNumeral2Int(r)

  case V::r    => 5 + RomanNumeral2Int(r)

  case I::V::r => 4 + RomanNumeral2Int(r)

  case I::r    => 1 + RomanNumeral2Int(r)

}

RomanNumeral2Int(List(I,V))             // 4

RomanNumeral2Int(List(I,I,I,I))         // 4 (invalid Roman number)

RomanNumeral2Int(List(V,I))             // 6

RomanNumeral2Int(List(I,X))             // 9

RomanNumeral2Int(List(M,C,M,L,X,X,I,X)) // 1979

RomanNumeral2Int(List(M,M,X,V,I,I))     // 2017

// another example

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

// Once upon a time, in a complete fictional country there were Persons...

abstract class Person

case object King extends Person

case class Peer(deg: String, terr: String, succ: Int) extends Person

case class Knight(name: String) extends Person

case class Peasant(name: String) extends Person

case object Clown extends Person

def title(p: Person): String = p match {

  case King => "His Majesty the King"

  case Peer(deg, terr, _) => s"The ${deg} of ${terr}"

  case Knight(name) => s"Sir ${name}"

  case Peasant(name) => name

  case Clown => "My name is Boris Johnson"

}

title(Clown)

def superior(p1: Person, p2: Person): Boolean = (p1, p2) match {

  case (King, _) => true

  case (Peer(_,_,_), Knight(_)) => true

  case (Peer(_,_,_), Peasant(_)) => true

  case (Peer(_,_,_), Clown) => true

  case (Knight(_), Peasant(_)) => true

  case (Knight(_), Clown) => true

  case (Clown, Peasant(_)) => true

  case _ => false

}

val people = List(Knight("David"), 

                  Peer("Duke", "Norfolk", 84), 

                  Peasant("Christian"), 

                  King, 

                  Clown)

println(people.sortWith(superior(_, _)).mkString(", "))

// Tail recursion

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

def fact(n: Long): Long = 

  if (n == 0) 1 else n * fact(n - 1)

fact(10)              //ok

fact(10000)           // produces a stackoverflow

def factT(n: BigInt, acc: BigInt): BigInt =

  if (n == 0) acc else factT(n - 1, n * acc)

factT(10, 1)

factT(100000, 1)

// there is a flag for ensuring a function is tail recursive

import scala.annotation.tailrec

@tailrec

def factT(n: BigInt, acc: BigInt): BigInt =

  if (n == 0) acc else factT(n - 1, n * acc)

// for tail-recursive functions the Scala compiler

// generates loop-like code, which does not need

// to allocate stack-space in each recursive

// call; Scala can do this only for tail-recursive

// functions

// sudoku again

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

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) = 

  (empty(game) % MaxValue, empty(game) / 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))

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)))

}

// 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 pretty(game: String): String = 

  "\n" + (game sliding (MaxValue, MaxValue) mkString "\n")

/////////////////////

// not tail recursive 

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))).toList.flatten

  }

}

// tail recursive version that searches 

// for all solutions

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)

// 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)

// Moral: Whenever a recursive function is resource-critical

// (i.e. works with large recursion depth), then you need to

// write it in tail-recursive fashion.

// 

// Unfortuantely, Scala because of current limitations in 

// the JVM is not as clever as other functional languages. It can 

// only optimise "self-tail calls". This excludes the cases of 

// multiple functions making tail calls to each other. Well,

// nothing is perfect. 

// Polymorphic Types

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

// You do not want to write functions like contains, first 

// and so on for every type of lists.

def length_string_list(lst: List[String]): Int = lst match {

  case Nil => 0

  case x::xs => 1 + length_string_list(xs)

}

def length_int_list(lst: List[Int]): Int = lst match {

  case Nil => 0

  case x::xs => 1 + length_int_list(xs)

}

length_string_list(List("1", "2", "3", "4"))

length_int_list(List(1, 2, 3, 4))

//-----

def length[A](lst: List[A]): Int = lst match {

  case Nil => 0

  case x::xs => 1 + length(xs)

}

length(List("1", "2", "3", "4"))

length(List(King, Knight("foo"), Clown))

length(List(1, 2, 3, 4))

def map[A, B](lst: List[A], f: A => B): List[B] = lst match {

  case Nil => Nil

  case x::xs => f(x)::map_int_list(xs, f) 

}

map_int_list(List(1, 2, 3, 4), square)

// Remember?

def first[A, B](xs: List[A], f: A => Option[B]): Option[B] = ...

// Cool Stuff

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

// Implicits 

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

//

// 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.

implicit class MyString(s: String) {

  def increment = for (c <- s) yield (c + 1).toChar 

}

"HAL".increment

// 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 o r2  

case class STAR(r: Rexp) extends Rexp               // star         r*

// (ab)*

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(ALT("ab"))

val r3 = STAR(ALT("ab", "baa baa black sheep"))

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 = "+" | "-" | ""