λ Tony’s blog λ

object Main {
  def main(args: Array[String]) {
    import Monad._
    import MonadicFunctions._
 
    val plusOne = Inter(1+)
    val multTwo = Inter(2*)
    val squared = Inter(n => n*n)
    val plus = (_: Int) + (_: Int)
 
    val values = List(
// sequence
sequence(List(List(1, 2), List(3, 4)), ListMonad),
sequence(List(Some(7), Some(8), Some(9)), OptionMonad),
sequence(List(Some(7), None, Some(9)), OptionMonad),
sequence(List(plusOne, multTwo, squared), InterMonad) f 7,
sequence(List(Identity(7), Identity(4)), IdentityMonad),
// fmap
fmap(List(1, 2, 3), (x: Int) => x * 10, ListMonad),
fmap(Some(8), (x: Int) => x * 10, OptionMonad),
fmap(None: Option[Int], (x: Int) => x * 10, OptionMonad),
fmap(plusOne, (x: Int) => x * 10, InterMonad) f 7,
fmap(Identity(9), (x: Int) => x * 10, IdentityMonad),
// flatten
flatten(List(List(1, 2), List(3, 4)), ListMonad),
flatten(Some(Some(8)), OptionMonad),
flatten(Some(None: Option[Int]), OptionMonad),
flatten(None: Option[Option[Int]], OptionMonad),
flatten(Inter(a => Inter(a *)), InterMonad) f 7,
flatten(Identity(Identity(8)), IdentityMonad),
// apply
apply(List((a: Int) => a + 1,
           (a: Int) => a * 2,
           (a: Int) => a % 2), List(1, 2, 3), ListMonad),
apply(Some((a: Int) => a + 1), Some(8), OptionMonad),
apply(None: Option[Int => Int], Some(8), OptionMonad),
apply(Some((a: Int) => a + 1), None: Option[Int], OptionMonad),
apply(Inter(a => (b: Int) => a * b), Inter(1+), InterMonad) f 7,
apply(Identity((a: Int) => a + 1), Identity(7), IdentityMonad),
// filterM
filterM((a: Int) => List(a > 2, a % 2 == 0), List(1, 2, 3), ListMonad),
filterM((a: Int) => Some(a > 1), List(1, 2, 3), OptionMonad),
filterM((a: Int) => Inter(n => a * n % 2 == 0),
  List(1, 2, 3), InterMonad) f 7,
filterM((a: Int) => Identity(a > 1), List(1, 2, 3), IdentityMonad),
// replicateM
replicateM(2, List(7, 8), ListMonad),
replicateM(2, Some(7), OptionMonad),
replicateM(2, plusOne, InterMonad) f 7,
replicateM(2, Identity(6), IdentityMonad),
// lift2
lift2(plus, List(1, 2), List(3, 4), ListMonad),
lift2(plus, Some(7), Some(8), OptionMonad),
lift2(plus, Some(7), None: Option[Int], OptionMonad),
lift2(plus, None: Option[Int], Some(8), OptionMonad)
    )
 
    val verify = List(
// sequence
List(List(1, 3), List(1, 4), List(2, 3), List(2, 4)),
Some(List(7, 8, 9)),
None,
List(8, 14, 49),
Identity(List(7, 4)),
// fmap
List(10, 20, 30),
Some(80),
None,
80,
Identity(90),
// flatten
List(1, 2, 3, 4),
Some(8),
None,
None,
49,
Identity(8),
// apply
List(2, 3, 4, 2, 4, 6, 1, 0, 1),
Some(9),
None,
None,
56,
Identity(8),
// filterM
List(List(3), Nil, List(2, 3), List(2), List(3),
  Nil, List(2, 3), List(2)),
Some(List(2, 3)),
List(2),
Identity(List(2, 3)),
// replicateM
List(List(7, 7), List(7, 8), List(8, 7), List(8, 8)),
Some(List(7, 7)),
List(8, 8),
Identity(List(6, 6)),
// lift2
List(4, 5, 5, 6),
Some(15),
None,
None
)
 
    for((a, b) <- values zip verify)    
      println(if(a == b) "PASS"
              else "FAIL. Expected: " + b + " Actual: " + a)
  }
}

main :: IO ()
main =
  let plusOne = Inter (1+)
      multTwo = Inter (2*)
      squared = Inter (\n -> n*n)
      s x = show x
      (%) = f
      values =
        [
        -- sequence'
        s (sequence' [[1, 2], [3, 4]] listMonad),
        s (sequence' [Just 7, Just 8, Just 9] maybeMonad),
        s (sequence' [Just 7, Nothing, Nothing] maybeMonad),
        s (sequence' [plusOne, multTwo, squared] interMonad % 7),
        s (sequence' [Identity 7, Identity 4] identityMonad),
        -- fmap'
        s (fmap' [1..3] (*10) listMonad),
        s (fmap' (Just 8) (*10) maybeMonad),
        s (fmap' Nothing (*10) maybeMonad),
        s (fmap' plusOne (*10) interMonad % 7),
        s (fmap' (Identity 9) (*10) identityMonad),
        -- flatten
        s (flatten [[1, 2], [3, 4]] listMonad),
        s (flatten (Just (Just 8)) maybeMonad),
        s (flatten (Just (Nothing :: Maybe Int)) maybeMonad),
        s (flatten (Nothing :: Maybe (Maybe Int)) maybeMonad),
        s (flatten (Inter (Inter . (*))) interMonad % 7),
        s (flatten (Identity (Identity 8)) identityMonad),
        -- apply
        s (apply [(+1), (*2), (`mod` 2)] [1..3] listMonad),
        s (apply (Just (+1)) (Just 8) maybeMonad),
        s (apply (Nothing :: Maybe (Int -> Int)) (Just 8) maybeMonad),
        s (apply (Just (+1)) (Nothing :: Maybe Int) maybeMonad),
        s (apply (Inter (*)) (Inter (1+)) interMonad % 7),
        s (apply (Identity (+1)) (Identity 7) identityMonad),
        -- filterM'
        s (filterM' (\a -> [a > 2, a `mod` 2 == 0]) [1..3] listMonad),
        s (filterM' (\a -> Just (a > 1)) [1..3] maybeMonad),
        s (filterM' (\a -> Inter (\n -> a * n `mod` 2 == 0)) [1..3]
          interMonad % 7),
        s (filterM' (Identity . (>1)) [1..3] identityMonad),
        -- replicateM'
        s (replicateM' 2 [7, 8] listMonad),
        s (replicateM' 2 (Just 7) maybeMonad),
        s (replicateM' 2 plusOne interMonad % 7),
        s (replicateM' 2 (Identity 6) identityMonad),
        -- lift2
        s (lift2 (+) [1, 2] [3, 4] listMonad),
        s (lift2 (+) (Just 7) (Just 8) maybeMonad),
        s (lift2 (+) (Just 7) (Nothing :: Maybe Int) maybeMonad),
        s (lift2 (+) (Nothing :: Maybe Int) (Just 8) maybeMonad)
        ]
      verify =
        [
        -- sequence'
        s ([[1, 3], [1, 4], [2, 3], [2, 4]]),
        s (Just [7..9]),
        s (Nothing :: Maybe Int),
        s [8, 14, 49],
        s (Identity [7, 4]),
        -- fmap'
        s [10, 20, 30],
        s (Just 80),
        s (Nothing :: Maybe Int),
        s 80,
        s (Identity 90),
        -- flatten
        s [1..4],
        s (Just 8),
        s (Nothing :: Maybe Int),
        s (Nothing :: Maybe Int),
        s 49,
        s (Identity 8),
        -- apply
        s [2, 3, 4, 2, 4, 6, 1, 0, 1],
        s (Just 9),
        s (Nothing :: Maybe Int),
        s (Nothing :: Maybe Int),
        s 56,
        s (Identity 8),
        -- filterM'
        s [[3], [], [2, 3], [2], [3], [], [2, 3], [2]],
        s (Just [2, 3]),
        s [2],
        s (Identity [2, 3]),
        -- replicateM
        s [[7, 7], [7, 8], [8, 7], [8, 8]],
        s (Just [7, 7]),
        s [8, 8],
        s (Identity [6, 6]),
        -- lift2
        s [4, 5, 5, 6],
        s (Just 15),
        s (Nothing :: Maybe Int),
        s (Nothing :: Maybe Int)
        ]
  in mapM_
      (\(a, b) ->
        print(if a == b
                then "PASS"
                else "FAIL. Expected: " ++ b ++ " Actual: " ++ a))
      (values `zip` verify)

newtype Comparator a = C { compare :: a -> a -> Int }

class Comparator a where
  compare :: a -> a -> Int

// 1. Start here. Observe this trait
trait Monad[M[_]] {
  def flatMap[A, B](a: M[A], f: A => M[B]): M[B]
  def unital[A](a: A): M[A]
}
 
// A simple data type, which turns out to satisfy the above trait
case class Inter[A](f: Int => A)
 
// So does this.
case class Identity[A](a: A)
 
// Monad implementations
object Monad {
  // 2. Replace error("todo") with an implementation
  def ListMonad: Monad[List] = error("todo")
 
  // 3. Replace error("todo") with an implementation
  def OptionMonad: Monad[Option] = error("todo")
 
  // 4. Replace error("todo") with an implementation
  def InterMonad: Monad[Inter] = error("todo")
 
  // 5. Replace error("todo") with an implementation
  def IdentityMonad: Monad[Identity] = error("todo")
}
 
object MonadicFunctions {
  // 6. Replace error("todo") with an implementation
  def sequence[M[_], A](as: List[M[A]], m: Monad[M]): M[List[A]] =
    error("todo")
 
  // 7. Replace error("todo") with an implementation
  def fmap[M[_], A, B](a: M[A], f: A => B, m: Monad[M]): M[B] =
    error("todo")
 
  // 8. Replace error("todo") with an implementation
  def flatten[M[_], A](a: M[M[A]], m: Monad[M]): M[A] =
    error("todo")
 
  // 9. Replace error("todo") with an implementation
  def apply[M[_], A, B](f: M[A => B], a: M[A], m: Monad[M]): M[B] =
    error("todo")
 
  // 10. Replace error("todo") with an implementation
  def filterM[M[_], A](f: A => M[Boolean], as: List[A]
    , m: Monad[M]): M[List[A]] =
    error("todo")
 
  // 11. Replace error("todo") with an implementation
  def replicateM[M[_], A](n: Int, a: M[A], m: Monad[M]): M[List[A]] =
    error("todo: flatMap n times to produce a list")
 
  // 12. Replace error("todo") with an implementation
  def lift2[M[_], A, B, C](f: (A, B) => C, a: M[A], b: M[B]
    , m: Monad[M]): M[C] =
    error("todo")
 
  // lift3, lift4, etc. Interesting question: Can we have liftN?
}

{-# LANGUAGE RankNTypes #-}
 
-- 1. Start here. Observe this data type
data Monad' m = Monad' {
  unital :: forall a. a -> m a,
  flatMap :: forall a b. m a -> (a -> m b) -> m b
}
 
-- A simple data type, which turns out to satisfy the above trait
newtype Inter a = Inter { f :: Int -> a }
 
-- So does this.
newtype Identity a = Identity { a :: a }
  deriving Show
 
-- *** Monad implementations ***
 
-- 2. Replace error "todo" with an implementation
listMonad :: Monad' []
listMonad = error "todo"
 
-- 3. Replace error "todo" with an implementation
maybeMonad :: Monad' Maybe
maybeMonad = error "todo"
 
-- 4. Replace error "todo" with an implementation
interMonad :: Monad' Inter
interMonad = error "todo"
 
-- 5. Replace error "todo" with an implementation
identityMonad :: Monad' Identity
identityMonad = error "todo"
 
-- *** Monadic functions ***
 
-- 6. Replace error "todo" with an implementation
sequence' :: [m a] -> Monad' m -> m [a]
sequence' = error "todo"
 
-- 7. Replace error "todo" with an implementation
fmap' :: m a -> (a -> b) -> Monad' m -> m b
fmap' = error "todo"
 
-- 8. Replace error "todo" with an implementation
flatten :: m (m a) -> Monad' m -> m a
flatten = error "todo"
 
-- 9. Replace error "todo" with an implementation
apply :: m (a -> b) -> m a -> Monad' m -> m b
apply = error "todo"
 
-- 10. Replace error "todo" with an implementation
filterM' :: (a -> m Bool) -> [a] -> Monad' m -> m [a]
filterM' = error "todo"
 
-- 11. Replace error "todo" with an implementation
replicateM' :: Int -> m a -> Monad' m -> m [a]
replicateM' = error "todo: flatMap n times to produce a list"
 
-- 12. Replace error "todo" with an implementation
lift2 :: (a -> b -> c) -> m a -> m b -> Monad' m -> m c
lift2 = error "todo"
 
-- lift3, lift4, etc. Interesting question: Can we have liftN?

method(s1.charAt(i), s2.charAt(j));

sequence :: [m a] -> m [a]
sequence* :: [t -> a] -> t -> [a]

// A constructor accepting a Swizzle
SwizzleManager m = new SwizzleManager(t);
// One of the following usually follow:
//   A list constructed by the results of executing several methods on 'm'
List<R> = { m.a(x), m.b(y), m.c(z) }
//   A loop over a list of the results of executing several methods on 'm'
for(R r : { m.a(x), m.b(y), m.c(z) }) {
  use(r);
}
//   A list of effects to execute using 'm'
m.a(x);
m.b(y);
m.c(z);

public static <A> Semigroup<LinkedList<A>> LinkedListSemigroup() {
  return new Semigroup<LinkedList<A>>() {
    public LinkedList<A> append(final LinkedList<A> a1,
                                final LinkedList<A> a2) {
      final LinkedList<A> r = new LinkedList<A>(a1);
      r.addAll(a2);
      return r;
    }
  };
}

// int wrapper between 0 and 130
class Age {
  private final int i;
  private Age(final int i) { this.i = i; }
  public int value() { return i; }
  public static Age age(final int i) {
    if(i <= 0 || i >= 130)
      throw new Error("out of range");
    else
      return new Age(i);
  }
}

// String wrapper starts with upper-case
class Name {
  private final String s;
  private Name(final String s) { this.s = s; }
  public String value() { return s; }
  public static Name name(final String s) {
    if(s.isEmpty() || !Character.isUpperCase(s.charAt(0)))
      throw new java.lang.Error();
    else
      return new Name(s);
  }
}

F<String, F<String, <Validation<LinkedList<String>, Person>>>>

F<A, F<B, Validation<E, V>>

data Validation e a = Fail e | Success a deriving (Eq, Show)
 
-- Associative, binary operation (Monoid without identity)
class Semigroup a where
  append :: a -> a -> a
 
(<+>) :: Validation e a -> (a -> b) -> Validation e b
Fail e <+> _ = Fail e
Success a <+> f = Success (f a)
 
(<*>) :: (Semigroup e) => Validation e a
    -> Validation e (a -> b)
    -> Validation e b
Fail e1 <*> Fail e2 = Fail (e1 `append` e2)
Fail e1 <*> Success _ = Fail e1
Success _ <*> Fail e2 = Fail e2
Success a <*> Success f = Success (f a)

sealed trait Validation[E, A] {
  def <+>[B](f: A => B): Validation[E, B] = this match {
    case Fail(e) => Fail(e)
    case Success(a) => Success(f(a))
  }
 
  def <*>[B](f: Validation[E, A => B])(implicit s: Semigroup[E])
    : Validation[E, B] = this match {
    case Fail(e1) => f match {
      case Fail(e2) => Fail(s append (e1, e2))
      case Success(_) => Fail(e1)
    }
    case Success(a) => f match {
      case Fail(e2) => Fail(e2)
      case Success(f) => Success(f(a))
    }
  }
}
case class Fail[E, A](e: E) extends Validation[E, A]
case class Success[E, A](a: A) extends Validation[E, A]
 
// Associative binary operation (Monoid without identity)
case class Semigroup[A](append: (A, A) => A)

// Java pre-amble, sigh.
interface F<X, Y> {
  Y apply(X x);
}
 
// Associative binary operation (Monoid without identity)
interface Semigroup<A> {
  A append(A a1, A a2);
}
 
abstract class Validation<E, A> {
  // No algebraic data types in Java. Use a Church encoding (catamorphism).
  public abstract <X> X fold(F<E, X> fail, F<A, X> success);
 
  // Construction for fail
  public static <E, A> Validation<E, A> fail(final E e) {
    return new Validation<E, A>() {
      public <X> X fold(final F<E, X> fail, final F<A, X> success) {
        return fail.apply(e);
      }
    };
  }
 
  // Construction for success
  public static <E, A> Validation<E, A> success(final A a) {
    return new Validation<E, A>() {
      public <X> X fold(final F<E, X> fail, final F<A, X> success) {
        return success.apply(a);
      }
    };
  }
 
  // <+>
  public <B> Validation<E, B> map(final F<A, B> f) {
    return new Validation<E, B>() {
      public <X> X fold(final F<E, X> fail, final F<B, X> success) {
        return Validation.this.fold(fail, new F<A, X>() {
          public X apply(final A a) {
            return success.apply(f.apply(a));
          }
        });
      }
    };
  }
 
  // <*>
  public <B> Validation<E, B> applicativate(
      final Validation<E, F<A, B>> f,
      final Semigroup<E> s /* no implicits or type-classes in Java */) {
    return Validation.this.fold(new F<E, Validation<E, B>>() {
      public Validation<E, B> apply(final E e1) {
        return f.fold(new F<E, Validation<E, B>>() {
          public Validation<E, B> apply(final E e2) {
            // case (Fail(e1), Fail(e2))
            return Validation.fail(s.append(e1, e2)); 
          }
        }, new F<F<A, B>, Validation<E, B>>() {
          public Validation<E, B> apply(final F<A, B> f) {
            // case (Fail(e1), Success(f))
            return Validation.fail(e1); 
          }
        });
      }
    }, new F<A, Validation<E, B>>() {
      public Validation<E, B> apply(final A a) {
        return f.fold(new F<E, Validation<E, B>>() {
          public Validation<E, B> apply(final E e2) {
            // case (Success(a), Fail(e2))
            return Validation.fail(e2); 
          }
        }, new F<F<A, B>, Validation<E, B>>() {
          public Validation<E, B> apply(final F<A, B> f) {
            // case (Success(a), Success(f))
            return Validation.success(f.apply(a)); 
          }
        });
      }
    });
  }
}