Lambda reduction template in Scala

reduction.scala

sealed trait Term

case class Var(name: String) extends Term
case class Lambda(param: Var, body: Term) extends Term
case class App(term1: Term, term2: Term) extends Term

object LambdaReduce {

  def substitute(term: Term, v: Var, body: Term): Term = body match {
    case Var(name) => ???
    case Lambda(param, body) => ???
    case App(term1, term2) => ???
  }

  val beta: Term => Option[Term] = {
    case App(Lambda(v, body), term) =>
      Some(substitute(term, v, body))
    case _ =>
      None
  }

  def toNormalForm(step: Term => Option[Term]): Term => Term =
    (term: Term) => step(term).fold(term)(toNormalForm(step))

  def mu(reduction: Term => Option[Term]): Term => Option[Term] = {
    case App(term, params) => reduction(term).map(App(_, params))
    case _ => None
  }

  def nu(reduction: Term => Option[Term]): Term => Option[Term] = {
    case App(term, param) => reduction(param).map(App(term, _))
    case _ => None
  }

  def deepMu(reduction: Term => Option[Term]): Term => Option[Term] = {
    case App(term, params) => mu(reduction)(term).map(App(_, params))
    case other => reduction(other)
  }

  def deepNu(reduction: Term => Option[Term]): Term => Option[Term] = {
    case App(term, param) => nu(reduction)(param).map(App(term, _))
    case other => reduction(other)
  }

}

import LambdaReduce._

val b = beta
val nf = toNormalForm(beta)
val munu = mu(nu(beta))
val mununf = toNormalForm(munu)