type Variable = String
  type Env[A] = Map[Variable,A]

  // Arithmetic expressions

  abstract class Expr
  case class Num(n: Integer) extends Expr
  case class Plus(e1: Expr, e2: Expr) extends Expr
  case class Minus(e1: Expr, e2: Expr) extends Expr
  case class Times(e1: Expr, e2: Expr) extends Expr

  // Booleans
  case class Bool(n: Boolean) extends Expr
  case class Eq(e1: Expr, e2:Expr) extends Expr
  case class IfThenElse(e: Expr, e1: Expr, e2: Expr) extends Expr
  
   // Strings
  case class Str(s: String) extends Expr
  case class Length(e: Expr) extends Expr
  case class Index(e1: Expr, e2: Expr) extends Expr
  case class Concat(e1: Expr, e2: Expr) extends Expr

  // Variables and let-binding
  case class Var(x: Variable) extends Expr
  case class Let(x: Variable, e1: Expr, e2: Expr) extends Expr
  case class LetFun(f: Variable, arg: Variable, ty: Type, e1:Expr, e2:Expr)
      extends Expr
  case class LetRec(f: Variable, arg: Variable, xty: Type, ty: Type, e1:Expr, e2:Expr)
      extends Expr
  case class LetPair(x: Variable,y: Variable, e1:Expr, e2:Expr) extends Expr

  // Pairing
  case class Pair(e1: Expr, e2: Expr) extends Expr
  case class First(e: Expr) extends Expr
  case class Second(e: Expr) extends Expr

  // Functions
  case class Lambda(x: Variable, ty: Type, e: Expr) extends Expr
  case class Apply(e1: Expr, e2: Expr) extends Expr
  case class Rec(f: Variable, x: Variable, tyx:Type, ty: Type, e: Expr) extends Expr

  // Values
  abstract class Value
  case class NumV(n: Integer) extends Value
  case class BoolV(n: Boolean) extends Value
  case class StringV(s: String) extends Value
  case class PairV(v1: Value, v2: Value) extends Value
  case class ClosureV(env: Env[Value], x: Variable, e: Expr) extends Value
  case class RecV(env: Env[Value], f:Variable, x: Variable, e: Expr) extends Value

  // Types
  abstract class Type
  case object IntTy extends Type
  case object BoolTy extends Type
  case object StringTy extends Type
  case class PairTy(ty1: Type, ty2: Type) extends Type
  case class FunTy(ty1: Type, ty2: Type) extends Type