package scalaz

import scala.collection.Iterator
import Maybe.Just
import std.list.listMonoid
import std.option._
import std.stream.streamMonoid
import syntax.Ops
import syntax.semigroup._
import FingerTree._
import Tags.LastVal

/**
 * Finger trees with leaves of type A and Nodes that are annotated with type V.
 *
 * Finger Trees provide a base for implementations of various collection types,
 * as described in "Finger trees: a simple general-purpose data structure", by
 * Ralf Hinze and Ross Paterson.
 * A gentle introduction is presented in the blog post "Monoids and Finger Trees" by Heinrich Apfelmus.
 *
 * This is done by choosing a suitable type to annotate the nodes. For example,
 * a binary tree can be implemented by annotating each node with the size of its subtree,
 * while a priority queue can be implemented by labelling the nodes by the minimum priority of its children.
 *
 * The operations on FingerTree enforce the constraint measured (in the form of a Reducer instance).
 *
 * Finger Trees have excellent (amortized) asymptotic performance:
 *
 *  - Access to the first and last elements is `O(1)`
 *  - Appending/prepending a single value is `O(1)`
 *  - Concatenating two trees is `(O lg min(l1, l2))` where `l1` and `l2` are their sizes
 *  - Random access to an element at `n` is `O(lg min(n, l - n))`, where `l` is the size of the tree.
 *  - Constructing a tree with n copies of a value is O(lg n).
 *
 * @tparam V The type of the annotations of the nodes (the '''measure''')
 * @tparam A The type of the elements stored at the leaves
 *
 * @see [[https://www.staff.city.ac.uk/~ross/papers/FingerTree.pdf Finger trees: a simple general-purpose data structure]]
 * @see [[https://apfelmus.nfshost.com/articles/monoid-fingertree.html]]
 */
sealed abstract class FingerTree[V, A](implicit measurer: Reducer[A, V]) {
  import measurer.semigroup

  def measure: Maybe[V] =
    fold(Maybe.empty, (v, _) => Just(v), (v, _, _, _) => Just(v))

  def foldMap[B](f: A => B)(implicit s: Monoid[B]): B =
    fold(s.zero, (v, x) => f(x), (v, pr, m, sf) =>
      s.append(s.append(pr.foldMap(f), m.foldMap(x => x.foldMap(f))), sf.foldMap(f)))

  def foldRight[B](z: => B)(f: (A, => B) => B): B = {
    foldMap(a => Endo.endoByName[B](f(a, _))).apply(z)
  }

  def foldLeft[B](b: B)(f: (B, A) => B): B = {
    fold(b,
          (v, a) => f(b, a),
          (v, pr, m, sf) =>
              fingerFoldable[V].foldLeft(sf, m.foldLeft[B](fingerFoldable[V].foldLeft(pr, b)(f))((x, y) => nodeFoldable[V].foldLeft(y, x)(f)))(f))
  }

  /**
   * Fold over the structure of the tree. The given functions correspond to the three possible variations of the finger tree.
   *
   * @param empty value to return if the tree is empty
   * @param single if the tree contains a single element, convert the measure and this element to a `B`
   * @param deep otherwise, convert the measure, the two fingers, and the sub tree to a `B`.
   */
  def fold[B](empty: => B, single: (V, A) => B, deep: (V, Finger[V, A], => FingerTree[V, Node[V, A]], Finger[V, A]) => B): B

  /** Prepends an element to the left of the tree. O(1). */
  def +:(a: A): FingerTree[V, A] = {
    implicit val nm: Reducer[FingerTree.Node[V, A], V] = nodeMeasure[A, V]
    fold(
      single(measurer.unit(a), a),
      (v, b) => deep(measurer.cons(a, v), one(a), empty[V, Node[V, A]], one(b)),
      (v, pr, m, sf) => {
        val mz = m
        pr match {
          case Four(vf, b, c, d, e) => deep(measurer.cons(a, v), two(a, b), node3[V, A](c, d, e) +: mz, sf)
          case _ => deep(measurer.cons(a, v), a +: pr, mz, sf)
      }})
  }

  /** Appends an element to the right of the tree. O(1). */
  def :+(a: => A): FingerTree[V, A] = {
    implicit val nm: Reducer[FingerTree.Node[V, A], V] = nodeMeasure[A, V]
    val az = Need(a)
    fold(
      single(measurer.unit(az.value), az.value),
      (v, b) => deep(measurer.snoc(v, az.value), one(b), empty[V, Node[V, A]], one(az.value)),
      (v, pr, m, sf) => {
        val mz = m
        sf match {
          case Four(vf, b, c, d, e) => deep(measurer.snoc(v, az.value), pr, (mz :+ node3(b, c, d)), two(e, az.value))
          case _ => deep(measurer.snoc(v, az.value), pr, mz, sf :+ az.value)
      }})
  }

  /** Replace the first element of the tree with the given value. O(1) */
  def |-:(a: A): FingerTree[V, A] =
    fold(
      sys.error("Replacing first element of an empty FingerTree"),
      (v, b) => single(a),
      (v, pr, m, sf) => deep(a |-: pr, m, sf))

  /** Replace the last element of the tree with the given value. O(1) */
  def :-|(a: => A): FingerTree[V, A] = {
    val az = Need(a)
    fold(
      sys.error("Replacing last element of an empty FingerTree"),
      (v, b) => single(az.value),
      (v, pr, m, sf) => deep(pr, m, sf :-| az.value))
  }

  /** Appends the given finger tree to the right of this tree. */
  def <++>(right: => FingerTree[V, A]): FingerTree[V, A] = {
    val rightz = Need(right)
    fold(
      rightz.value,
      (v, x) => x +: rightz.value,
      (v1, pr1, m1, sf1) =>
        rightz.value.fold(
          this,
          (v, x) => this :+ x,
          (v2, pr2, m2, sf2) => deep(measurer.append(v1, v2), pr1, addDigits0(m1, sf1, pr2, m2), sf2)
        )
    )
  }

  private type ATree = FingerTree[V, A]
  private type AFinger = Finger[V, A]
  private type NodeTree = FingerTree[V, Node[V, A]]

  def add1(n: A, right: => ATree): ATree = {
    val rightz = Need(right)
    fold(
      n +: rightz.value,
      (v, x) => x +: n +: rightz.value,
      (v1, pr1, m1, sf1) =>
        rightz.value.fold(
          this :+ n,
          (v, x) => this :+ n :+ x,
          (v2, pr2, m2, sf2) =>
            deep(measurer.append((measurer.snoc(v1, n)), v2), pr1, addDigits1(m1, sf1, n, pr2, m2), sf2)
        )
    )
  }

  def add2(n1t: => A, n2t: => A, right: => ATree): ATree = {
    val rightz = Need(right)
    val n1 = Need(n1t)
    val n2 = Need(n2t)
    fold(
      n1.value +: n2.value +: rightz.value,
      (v, x) => x +: n1.value +: n2.value +: rightz.value,
      (v1, pr1, m1, sf1) =>
        rightz.value.fold(
          this :+ n1.value :+ n2.value,
          (v, x) => this :+ n1.value :+ n2.value :+ x,
          (v2, pr2, m2, sf2) =>
            deep(measurer.append(measurer.snoc(measurer.snoc(v1, n1.value), n2.value), v2),
              pr1, addDigits2(m1, sf1, n1.value, n2.value, pr2, m2), sf2)
        )
    )
  }

  def add3(n1t: => A, n2t: => A, n3t: => A, right: => ATree): ATree = {
    val rightz = Need(right)
    val n1 = Need(n1t)
    val n2 = Need(n2t)
    val n3 = Need(n3t)
    fold(
      n1.value +: n2.value +: n3.value +: rightz.value,
      (v, x) => x +: n1.value +: n2.value +: n3.value +: rightz.value,
      (v1, pr1, m1, sf1) =>
        rightz.value.fold(
           this :+ n1.value :+ n2.value :+ n3.value,
           (v, x) => this :+ n1.value :+ n2.value :+ n3.value :+ x,
           (v2, pr2, m2, sf2) =>
             deep(measurer.append(measurer.snoc(measurer.snoc(measurer.snoc(v1, n1.value), n2.value), n3.value), v2),
               pr1, addDigits3(m1, sf1, n1.value, n2.value, n3.value, pr2, m2), sf2)
        )
    )
  }

  def add4(n1t: => A, n2t: => A, n3t: => A, n4t: => A, right: => ATree): ATree = {
    val rightz = Need(right)
    val n1 = Need(n1t)
    val n2 = Need(n2t)
    val n3 = Need(n3t)
    val n4 = Need(n4t)
    fold(
      n1.value +: n2.value +: n3.value +: n4.value +: rightz.value,
      (v, x) => x +: n1.value +: n2.value +: n3.value +: n4.value +: rightz.value,
      (v1, pr1, m1, sf1) =>
        rightz.value.fold(
          this :+ n1.value :+ n2.value :+ n3.value :+ n4.value,
          (v, x) => this :+ n1.value :+ n2.value :+ n3.value :+ n4.value :+ x,
          (v2, pr2, m2, sf2) =>
            deep(measurer.append(measurer.snoc(measurer.snoc(measurer.snoc(measurer.snoc(v1, n1.value), n2.value), n3.value), n4.value), v2),
              pr1, addDigits4(m1, sf1, n1.value, n2.value, n3.value, n4.value, pr2, m2), sf2)
        )
    )
  }

  def addDigits0(m1: NodeTree, dig1: AFinger, dig2: AFinger, m2: => NodeTree): NodeTree = dig1 match {
    case One(_, a) => dig2 match {
      case One(_, b) => m1.add1(node2(a, b), m2)
      case Two(_, b,c) => m1.add1(node3(a,b,c), m2)
      case Three(_, b,c,d) => m1.add2(node2(a,b), node2(c,d),m2)
      case Four(_, b,c,d,e) => m1.add2(node3(a,b,c), node2(d,e), m2)
    }
    case Two(_, a,b) => dig2 match {
      case One(_, c) => m1.add1(node3(a,b,c), m2)
      case Two(_, c,d) => m1.add2(node2(a,b), node2(c,d), m2)
      case Three(_, c,d,e) => m1.add2(node3(a,b,c), node2(d,e), m2)
      case Four(_, c,d,e,f) => m1.add2(node3(a,b,c), node3(d,e,f), m2)
    }
    case Three(_, a,b,c) => dig2 match {
      case One(_, d) => m1.add2(node2(a,b), node2(c,d), m2)
      case Two(_, d,e) => m1.add2(node3(a,b,c), node2(d,e), m2)
      case Three(_, d,e,f) => m1.add2(node3(a,b,c), node3(d,e,f), m2)
      case Four(_, d,e,f,g) => m1.add3(node3(a,b,c), node2(d,e), node2(f,g), m2)
    }
    case Four(_, a,b,c,d) => dig2 match {
      case One(_, e) => m1.add2(node3(a,b,c), node2(d,e), m2)
      case Two(_, e,f) => m1.add2(node3(a,b,c), node3(d,e,f), m2)
      case Three(_, e,f,g) => m1.add3(node3(a,b,c), node2(d,e), node2(f,g), m2)
      case Four(_, e,f,g,h) => m1.add3(node3(a,b,c), node3(d,e,f), node2(g,h), m2)
    }
  }

  def addDigits1(m1: NodeTree, d1: AFinger, xt: => A, d2: AFinger, m2t: => NodeTree): NodeTree = {
    val x = Need(xt)
    val m2 = Need(m2t)
    d1 match {
      case One(_, a) => d2 match {
        case One(_, b) => m1.add1(node3(a,x.value,b), m2.value)
        case Two(_, b,c) => m1.add2(node2(a,x.value), node2(b,c), m2.value)
        case Three(_, b,c,d) => m1.add2(node3(a,x.value,b), node2(c,d), m2.value)
        case Four(_, b,c,d,e) => m1.add2(node3(a,x.value,b), node3(c,d,e), m2.value)
      }
      case Two(_, a,b) => d2 match {
        case One(_, c) => m1.add2(node2(a,b), node2(x.value,c), m2.value)
        case Two(_, c,d) => m1.add2(node3(a,b,x.value), node2(c,d), m2.value)
        case Three(_, c,d,e) => m1.add2(node3(a,b,x.value), node3(c,d,e), m2.value)
        case Four(_, c,d,e,f) => m1.add3(node3(a,b,x.value), node2(c,d), node2(e,f), m2.value)
      }
      case Three(_, a,b,c) => d2 match {
        case One(_, d) => m1.add2(node3(a,b,c), node2(x.value,d), m2.value)
        case Two(_, d,e) => m1.add2(node3(a,b,c), node3(x.value,d,e), m2.value)
        case Three(_, d,e,f) => m1.add3(node3(a,b,c), node2(x.value,d), node2(e,f), m2.value)
        case Four(_, d,e,f,g) => m1.add3(node3(a,b,c), node3(x.value,d,e), node2(f,g), m2.value)
      }
      case Four(_, a,b,c,d) => d2 match {
        case One(_, e) => m1.add2(node3(a,b,c), node3(d,x.value,e), m2.value)
        case Two(_, e,f) => m1.add3(node3(a,b,c), node2(d,x.value), node2(e,f), m2.value)
        case Three(_, e,f,g) => m1.add3(node3(a,b,c), node3(d,x.value,e), node2(f,g), m2.value)
        case Four(_, e,f,g,h) => m1.add3(node3(a,b,c), node3(d,x.value,e), node3(f,g,h), m2.value)
      }
    }
  }

  def addDigits2(m1: NodeTree, d1: AFinger, xt: => A, yt: => A, d2: AFinger, m2t: => NodeTree): NodeTree = {
    val x = Need(xt)
    val y = Need(yt)
    val m2 = Need(m2t)
    d1 match {
      case One(_, a) => d2 match {
        case One(_, b) => m1.add2(node2(a,x.value), node2(y.value,b), m2.value)
        case Two(_, b,c) => m1.add2(node3(a,x.value,y.value), node2(b,c), m2.value)
        case Three(_, b,c,d) => m1.add2(node3(a,x.value,y.value), node3(b,c,d), m2.value)
        case Four(_, b,c,d,e) => m1.add3(node3(a,x.value,y.value), node2(b,c), node2(d,e), m2.value)
      }
      case Two(_, a,b) => d2 match {
        case One(_, c) => m1.add2(node3(a,b,x.value), node2(y.value,c), m2.value)
        case Two(_, c,d) => m1.add2(node3(a,b,x.value), node3(y.value,c,d), m2.value)
        case Three(_, c,d,e) => m1.add3(node3(a,b,x.value), node2(y.value,c), node2(d,e), m2.value)
        case Four(_, c,d,e,f) => m1.add3(node3(a,b,x.value), node3(y.value,c,d), node2(e,f), m2.value)
      }
      case Three(_, a,b,c) => d2 match {
        case One(_, d) => m1.add2(node3(a,b,c), node3(x.value,y.value,d), m2.value)
        case Two(_, d,e) => m1.add3(node3(a,b,c), node2(x.value,y.value), node2(d,e), m2.value)
        case Three(_, d,e,f) => m1.add3(node3(a,b,c), node3(x.value,y.value,d), node2(e,f), m2.value)
        case Four(_, d,e,f,g) => m1.add3(node3(a,b,c), node3(x.value,y.value,d), node3(e,f,g), m2.value)
      }
      case Four(_, a,b,c,d) => d2 match {
        case One(_, e) => m1.add3(node3(a,b,c), node2(d,x.value), node2(y.value,e), m2.value)
        case Two(_, e,f) => m1.add3(node3(a,b,c), node3(d,x.value,y.value), node2(e,f), m2.value)
        case Three(_, e,f,g) => m1.add3(node3(a,b,c), node3(d,x.value,y.value), node3(e,f,g), m2.value)
        case Four(_, e,f,g,h) => m1.add4(node3(a,b,c), node3(d,x.value,y.value), node2(e,f), node2(g,h), m2.value)
      }
    }
  }

  def addDigits3(m1: NodeTree, d1: AFinger, xt: => A, yt: => A, zt: => A, d2: AFinger, m2t: => NodeTree): NodeTree = {
    val x = Need(xt)
    val y = Need(yt)
    val z = Need(zt)
    val m2 = Need(m2t)
    d1 match {
      case One(_, a) => d2 match {
        case One(_, b) => m1.add2(node3(a,x.value,y.value), node2(z.value,b), m2.value)
        case Two(_, b,c) => m1.add2(node3(a,x.value,y.value), node3(z.value,b,c), m2.value)
        case Three(_, b,c,d) => m1.add3(node3(a,x.value,y.value), node2(z.value,b), node2(c,d), m2.value)
        case Four(_, b,c,d,e) => m1.add3(node3(a,x.value,y.value), node3(z.value,b,c), node2(d,e), m2.value)
      }
      case Two(_, a,b) => d2 match {
        case One(_, c) => m1.add2(node3(a,b,x.value), node3(y.value,z.value,c), m2.value)
        case Two(_, c,d) => m1.add3(node3(a,b,x.value), node2(y.value,z.value), node2(c,d), m2.value)
        case Three(_, c,d,e) => m1.add3(node3(a,b,x.value), node3(y.value,z.value,c), node2(d,e), m2.value)
        case Four(_, c,d,e,f) => m1.add3(node3(a,b,x.value), node3(y.value,z.value,c), node3(d,e,f),m2.value)
      }
      case Three(_, a,b,c) => d2 match {
        case One(_, d) => m1.add3(node3(a,b,c), node2(x.value,y.value), node2(z.value,d), m2.value)
        case Two(_, d,e) => m1.add3(node3(a,b,c), node3(x.value,y.value,z.value), node2(d,e), m2.value)
        case Three(_, d,e,f) => m1.add3(node3(a,b,c), node3(x.value,y.value,z.value), node3(d,e,f), m2.value)
        case Four(_, d,e,f,g) => m1.add4(node3(a,b,c), node3(x.value,y.value,z.value), node2(d,e), node2(f,g), m2.value)
      }
      case Four(_, a,b,c,d) => d2 match {
        case One(_, e) => m1.add3(node3(a,b,c), node3(d,x.value,y.value), node2(z.value,e), m2.value)
        case Two(_, e,f) => m1.add3(node3(a,b,c), node3(d,x.value,y.value), node3(z.value,e,f), m2.value)
        case Three(_, e,f,g) => m1.add4(node3(a,b,c), node3(d,x.value,y.value), node2(z.value,e),node2(f,g), m2.value)
        case Four(_, e,f,g,h) => m1.add4(node3(a,b,c), node3(d,x.value,y.value), node3(z.value,e,f), node2(g,h), m2.value)
      }
    }
  }

  def addDigits4(m1: NodeTree, d1: AFinger, xt: => A, yt: => A, zt: => A, wt: => A, d2: AFinger, m2t: => NodeTree): NodeTree = {
      val x = Need(xt)
      val y = Need(yt)
      val z = Need(zt)
      val w = Need(wt)
      val m2 = Need(m2t)
      d1 match {
      case One(_, a) => d2 match {
        case One(_, b) => m1.add2(node3(a,x.value,y.value), node3(z.value,w.value,b), m2.value)
        case Two(_, b,c) => m1.add3(node3(a,x.value,y.value), node2(z.value,w.value), node2(b,c), m2.value)
        case Three(_, b,c,d) => m1.add3(node3(a,x.value,y.value), node3(z.value,w.value,b), node2(c,d), m2.value)
        case Four(_, b,c,d,e) => m1.add3(node3(a,x.value,y.value), node3(z.value,w.value,b), node3(c,d,e), m2.value)
      }
      case Two(_, a,b) => d2 match {
        case One(_, c) => m1.add3(node3(a,b,x.value), node2(y.value,z.value), node2(w.value,c), m2.value)
        case Two(_, c,d) => m1.add3(node3(a,b,x.value), node3(y.value,z.value,w.value), node2(c,d), m2.value)
        case Three(_, c,d,e) => m1.add3(node3(a,b,x.value), node3(y.value,z.value,w.value), node3(c,d,e), m2.value)
        case Four(_, c,d,e,f) => m1.add4(node3(a,b,x.value), node3(y.value,z.value,w.value), node2(c,d), node2(e,f),m2.value)
      }
      case Three(_, a,b,c) => d2 match {
        case One(_, d) => m1.add3(node3(a,b,c), node3(x.value,y.value,z.value), node2(w.value,d), m2.value)
        case Two(_, d,e) => m1.add3(node3(a,b,c), node3(x.value,y.value,z.value), node3(w.value,d,e), m2.value)
        case Three(_, d,e,f) => m1.add4(node3(a,b,c), node3(x.value,y.value,z.value), node2(w.value,d),node2(e,f), m2.value)
        case Four(_, d,e,f,g) => m1.add4(node3(a,b,c), node3(x.value,y.value,z.value), node3(w.value,d,e), node2(f,g), m2.value)
      }
      case Four(_, a,b,c,d) => d2 match {
        case One(_, e) => m1.add3(node3(a,b,c), node3(d,x.value,y.value), node3(z.value,w.value,e), m2.value)
        case Two(_, e,f) => m1.add4(node3(a,b,c), node3(d,x.value,y.value), node2(z.value,w.value), node2(e,f), m2.value)
        case Three(_, e,f,g) => m1.add4(node3(a,b,c), node3(d,x.value,y.value), node3(z.value,w.value,e),node2(f,g), m2.value)
        case Four(_, e,f,g,h) => m1.add4(node3(a,b,c), node3(d,x.value,y.value), node3(z.value,w.value,e), node3(f,g,h), m2.value)
      }
    }
  }

  /**
   * Splits this tree into a pair of subtrees at the point where the given predicate, based on the measure,
   * changes from `false` to `true`. O(log(min(i,n-i)))
   *
   * @param pred predicate on node measures. Must be a semigroup homomorphism from the semigroup `V` of
   *             node measures to the semigroup of `Boolean`s with `||` as the semigroup operation.
   *             Namely, it must hold that `pred(v1 |+| v2) = pred(v1) || pred(v2)`.
   * @return `(as, bs)`, where
   *          - `as`: the subtree containing elements before the point where `pred` first holds
   *          - `bs`: the subtree containing elements at and after the point where `pred` first holds. Empty if `pred` never holds.
   */
  def split(pred: V => Boolean): (FingerTree[V, A], FingerTree[V, A]) =
    measure match {
      case Just(v) if pred(v) =>
        val (l, x, r) = split1(pred)
        (l, x +: r)
      case _ =>
        (this, empty)
    }

  /**
   * Like `split`, but returns the element where `pred` first holds separately
   *
   * @throws if the tree is empty.
   */
  def split1(pred: V => Boolean): (FingerTree[V, A], A, FingerTree[V, A]) = split1(pred, Maybe.empty)

  private def split1(pred: V => Boolean, accV: Maybe[V]): (FingerTree[V, A], A, FingerTree[V, A]) = fold(
    sys.error("Splitting an empty FingerTree"), // we can never get here
    (v, x) => (empty, x, empty),
    (v, pr, m, sf) => {
      val accVpr = accV.map(measurer.append(_, pr.measure)).getOrElse(pr.measure)
      if (pred(accVpr)) {
        val (l, x, r) = pr.split1(pred, accV)
        (cata(l)(_.toTree, empty), x, deepL(r, m, sf))
      } else {
        val accVm = mappendVal(accVpr, m)
        if (pred(accVm)) {
          val (ml, xs, mr) = m.split1(pred, Just(accVpr))
          val (l, x, r) = xs.split1(pred, mappendVal(accVpr, ml))
          (deepR(pr, ml, l), x, deepL(r, mr, sf))
        } else {
          val (l, x, r) = sf.split1(pred, Just(accVm))
          (deepR(pr, m, l), x, cata(r)(_.toTree, empty))
        }
      }
    }
  )

  def isEmpty: Boolean = fold(true, (v, x) => false, (v, pr, m, sf) => false)

  def viewl: ViewL[FingerTree[V, *], A] =
    fold(
      EmptyL[FingerTree[V, *], A],
      (v, x) => OnL[FingerTree[V, *], A](x, empty[V, A]),
      (v, pr, m, sf) =>
        pr match {
          case One(v, x) => OnL[FingerTree[V, *], A](x, rotL(m, sf))
          case _ => OnL[FingerTree[V, *], A](pr.lhead, deep(pr.ltail, m, sf))
        })

  def viewr: ViewR[FingerTree[V, *], A] =
    fold(
      EmptyR[FingerTree[V, *], A],
      (v, x) => OnR[FingerTree[V, *], A](empty[V, A], x),
      (v, pr, m, sf) =>
        sf match {
          case One(v, x) => OnR[FingerTree[V, *], A](rotR(pr, m), x)
          case _ => OnR[FingerTree[V, *], A](deep(pr, m, sf.rtail), sf.rhead)
        })

  /**
   * Selects the first element in the tree.
   *
   * @throws if the tree is empty
   */
  def head: A = viewl.head

  /**
   * Selects the last element in the tree.
   *
   * @throws if the tree is empty
   */
  def last: A = viewr.last

  /**
   * Selects a subtree containing all elements except the first
   *
   * @throws if the tree is empty
   */
  def tail: FingerTree[V, A] = viewl.tail

  /**
   * Selects a subtree containing all elements except the last
   *
   * @throws if the tree is empty
   */
  def init: FingerTree[V, A] = viewr.init

  /** Maps the given function across the tree, annotating nodes in the resulting tree according to the provided `Reducer`. */
  def map[B, V2](f: A => B)(implicit r: Reducer[B, V2]): FingerTree[V2, B] = {
    import r.semigroup
    implicit val nm: Reducer[FingerTree.Node[V2, B], V2] = nodeMeasure[B, V2]
    fold(
      empty[V2, B],
      (v, x) => single(f(x)),
      (v, pr, mt, sf) => deep(pr map f, mt.map(x => x.map(f)), sf map f))
  }

  /**
   * Like traverse, but with a more constraint type: we need the additional measure to construct the new tree.
   */
  def traverseTree[F[_], V2, B](f: A => F[B])(implicit ms: Reducer[B, V2], F: Applicative[F]): F[FingerTree[V2, B]]
  = {
    import ms.semigroup
    def mkDeep(pr: Finger[V2, B])(m: FingerTree[V2, Node[V2, B]])(sf: Finger[V2, B]): FingerTree[V2, B] = deep(pr, m, sf)
    fold(F.pure(FingerTree.empty[V2, B]),
         (v, a) => F.map(f(a))(a => single(ms.unit(a), a)),
         (v, pr, m, sf) => {
           //F.ap(traverseFinger(sf)(f))(F.ap(m.traverseTree(n => traverseNode(n)(f)))(F.map(traverseFinger(pr)(f))(pr => mkDeep(pr)_)))
           //the implementation below seems most efficient. The straightforward implementation using F.map3 leads to an explosion of traverseTree calls
           val fmap2 = F.apply2(traverseFinger(pr)(f), m.traverseTree(n => traverseNode(n)(f)))((a,b) => mkDeep(a)(b)_)
           F.ap(traverseFinger(sf)(f))(fmap2)
        })
  }

  private def traverseNode[F[_], V2, B](node: Node[V, A])(f: A => F[B])(implicit ms: Reducer[B, V2], F: Applicative[F]): F[Node[V2, B]] = {
    def mkNode(x: B)(y: B)(z: B): Node[V2, B] = node3(x, y, z)
    node.fold((v, a, b) => F.apply2(f(a), f(b))((x, y) => node2(x, y)),
        (v, a, b, c) =>  {
          F.ap(f(c))(F.ap(f(b))(F.map(f(a))(x => mkNode(x)_)))
        }
    )
  }

  private def traverseFinger[F[_], A, B, V2](digit: Finger[V, A])(f: A => F[B])(implicit ms: Reducer[B, V2], F: Applicative[F]): F[Finger[V2, B]] = {
    def mkTwo(x: B)(y: B): Finger[V2, B] = two(x, y)
    def mkThree(x: B)(y: B)(z: B): Finger[V2, B] = three(x, y, z)
    def mkFour(w: B)(x: B)(y: B)(z: B): Finger[V2, B] = four(w, x, y, z)
    digit match {
      case One(v, a) => F.map(f(a))(x => one(x))
      case Two(v, a, b) => F.ap(f(b))(F.map(f(a))(x => mkTwo(x)_))
      case Three(v, a, b, c) => F.ap(f(c))(F.ap(f(b))(F.map(f(a))(x => mkThree(x)_)))
      case Four(v, a, b, c, d) => F.ap(f(d))(F.ap(f(c))(F.ap(f(b))(F.map(f(a))(x => mkFour(x)_))))
    }
  }

  /** Execute the provided side effect for each element in the tree. */
  def foreach(f: A => Unit): Unit = {
    fold(
      {},
      (_, x) => { f(x) },
      (_, pr, m, sf) => { pr.foreach(f); m.foreach(_.foreach(f)); sf.foreach(f) }
    )}

  /** An iterator that visits each element in the tree. */
  def iterator: Iterator[A] = fold(
    Iterator.empty,
    (_, x) => Iterator.single(x),
    (_, pr, m, sf) => pr.iterator ++ m.iterator.flatMap(_.iterator) ++ sf.iterator)

  /** An iterator that visits each element in the tree in reverse order. */
  def reverseIterator: Iterator[A] = fold(
    Iterator.empty,
    (_, x) => Iterator.single(x),
    (_, pr, m, sf) => sf.reverseIterator ++ m.reverseIterator.flatMap(_.reverseIterator) ++ pr.reverseIterator)

  /** Convert the leaves of the tree to a `scala.Stream` */
  def toStream: Stream[A] = to[Stream]

  /** Convert the leaves of the tree to a `scala.List` */
  def toList: List[A] = to[List]

  /** Convert the leaves of the tree to a `scalaz.IList` */
  def toIList: IList[A] = to[IList]

  /** Convert the leaves of the tree to an `M` */
  def reduceTo[M: Reducer[A, *]: Monoid]: M = map[A, M](x => x).measure.getOrElse(Monoid[M].zero)

  /** Convert the leaves of the tree to an `F[A]` */
  def to[F[_]](implicit R: Reducer[A, F[A]], M: Monoid[F[A]]): F[A] = reduceTo[F[A]]

  /** Convert the tree to a `String`. Unsafe: this uses `Any#toString` for types `V` and `A` */
  override def toString = {
    val showV = Show.showFromToString[V]
    val showA = Show.showFromToString[A]
    fingerTreeShow(showV, showA).shows(this)
  }
}

sealed abstract class FingerTreeInstances {
  import FingerTree._

  implicit def viewLFunctor[S[_]](implicit s: Functor[S]): Functor[ViewL[S, *]] =
    new Functor[ViewL[S, *]] {
      def map[A, B](t: ViewL[S, A])(f: A => B): ViewL[S, B] =
        t.fold(EmptyL[S, B], (x, xs) => OnL(f(x), s.map(xs)(f))) //TODO define syntax for &: and :&
    }

  implicit def viewRFunctor[S[_]](implicit s: Functor[S]): Functor[ViewR[S, *]] =
    new Functor[ViewR[S, *]] {
      def map[A, B](t: ViewR[S, A])(f: A => B): ViewR[S, B] =
        t.fold(EmptyR[S, B], (xs, x) => OnR(s.map(xs)(f), f(x)))
    }

  implicit def fingerFoldable[V]: Foldable[Finger[V, *]] =
    new Foldable[Finger[V, *]] with Foldable.FromFoldMap[Finger[V, *]] {
      override def foldMap[A, M: Monoid](v: Finger[V, A])(f: A => M) = v.foldMap(f)
    }

  implicit def fingerMeasure[A, V: Semigroup]: Reducer[Finger[V, A], V] =
    UnitReducer((a: Finger[V, A]) => a.measure)

  implicit def nodeMeasure[A, V: Semigroup]: Reducer[Node[V, A], V] =
    UnitReducer((a: Node[V, A]) => a fold (
            (v, _, _) => v,
            (v, _, _, _) => v))

  implicit def nodeFoldable[V]: Foldable[Node[V, *]] =
    new Foldable[Node[V, *]] {
      def foldMap[A, M: Monoid](t: Node[V, A])(f: A => M): M = t foldMap f
      def foldRight[A, B](v: Node[V, A], z: => B)(f: (A, => B) => B): B =
         foldMap(v)((a: A) => Endo.endoByName[B](f(a, _))) apply z
    }

  implicit def fingerTreeFoldable[V]: Foldable[FingerTree[V, *]] =
    new Foldable[FingerTree[V, *]] {
      override def foldLeft[A, B](t: FingerTree[V, A], b: B)(f: (B, A) => B) = t.foldLeft(b)(f)

      def foldMap[A, M: Monoid](t: FingerTree[V, A])(f: A => M): M = t foldMap(f)

      override def foldRight[A, B](t: FingerTree[V, A], z: => B)(f: (A, => B) => B) = t.foldRight(z)(f)
    }

  implicit def fingerTreeMonoid[V: Reducer[A, *], A]: Monoid[FingerTree[V, A]] =
    new Monoid[FingerTree[V, A]] {
      def append(f1: FingerTree[V, A], f2: => FingerTree[V, A]) = f1 <++> f2
      def zero = empty
    }

  implicit def fingerTreeShow[V, A](implicit V: Show[V], A: Show[A]): Show[FingerTree[V,A]] =
    new Show[FingerTree[V,A]] {
      import std.iterable._
      val AS = Show[List[A]]
      import syntax.show._
      override def show(t: FingerTree[V,A]) = t.fold(
        empty = cord"[]",
        single = (v, x) => cord"$v [$x]",
        deep = (v, pf, m, sf) => cord"$v [${AS.show(pf.toList)}, *, ${AS.show(sf.toList)}]"
      )
    }

  implicit def fingerTreeEqual[V, A : Equal]: Equal[FingerTree[V, A]] =
    new Equal[FingerTree[V, A]] {
      import std.stream._
      def equal(x: FingerTree[V, A], y: FingerTree[V, A]) =
        Equal[Stream[A]].equal(x.toStream, y.toStream)
    }
}

object FingerTree extends FingerTreeInstances {

  def Node2[V: Reducer[A, *], A](v: V, a1: => A, a2: => A): Node[V, A] =
    new Node[V, A] {
      def fold[B](two: (V, => A, => A) => B, three: (V, => A, => A, => A) => B) =
        two(v, a1, a2)
      val measure = v
    }

  def Node3[V: Reducer[A, *], A](v: V, a1: => A, a2: => A, a3: => A): Node[V, A] =
    new Node[V, A] {
      def fold[B](two: (V, => A, => A) => B, three: (V, => A, => A, => A) => B) =
        three(v, a1, a2, a3)
      val measure = v
    }

  def EmptyR[S[_], A]: ViewR[S, A] =
    new ViewR[S, A] {
      def fold[B](b: => B, f: (=> S[A], => A) => B) = b
    }

  def OnR[S[_], A](sa: => S[A], a: => A): ViewR[S, A] =
    new ViewR[S, A] {
      def fold[B](b: => B, f: (=> S[A], => A) => B) = f(sa, a)
    }

  def EmptyL[S[_], A]: ViewL[S, A] =
    new ViewL[S, A] {
      def fold[B](b: => B, f: (=> A, => S[A]) => B) = b
    }

  def OnL[S[_], A](a: => A, sa: => S[A]): ViewL[S, A] =
    new ViewL[S, A] {
      def fold[B](b: => B, f: (=> A, => S[A]) => B) = f(a, sa)
    }

  def one[V, A](a: A)(implicit measure: Reducer[A, V]): Finger[V, A] =
    One(measure.unit(a), a)

  def two[V, A](a1: A, a2: A)(implicit measure: Reducer[A, V]): Finger[V, A] =
    Two(measure.snoc(measure.unit(a1), a2), a1, a2)

  def three[V, A](a1: A, a2: A, a3: A)(implicit measure: Reducer[A, V]): Finger[V, A] =
    Three(measure.snoc(measure.snoc(measure.unit(a1), a2), a3), a1, a2, a3)

  def four[V, A](a1: A, a2: A, a3: A, a4: A)(implicit measure: Reducer[A, V]): Finger[V, A] =
    Four(measure.snoc(measure.snoc(measure.snoc(measure.unit(a1), a2), a3), a4), a1, a2, a3, a4)

  def node2[V, A](a: A, b: A)(implicit measure: Reducer[A, V]): Node[V, A] =
    Node2[V, A](measure.snoc(measure.unit(a), b), a, b)

  def node3[V, A](a: A, b: A, c: A)(implicit measure: Reducer[A, V]): Node[V, A] =
    Node3[V, A](measure.snoc(measure.snoc(measure.unit(a), b), c), a, b, c)

  def mappendVal[V: Semigroup, A](v: V, t: FingerTree[V, A]): V =
    t.fold(v, (vt, _) => v |+| vt, (vt, _, _, _) => v |+| vt)

  def empty[V, A](implicit ms: Reducer[A, V]): FingerTree[V, A] =
    new FingerTree[V, A] {
      def fold[B](b: => B, s: (V, A) => B, d: (V, Finger[V, A], => FingerTree[V, Node[V, A]], Finger[V, A]) => B): B = b
    }

  def single[V, A](a: A)(implicit ms: Reducer[A, V]): FingerTree[V, A] = single(ms.unit(a), a)

  def single[V: Reducer[A, *], A](v: V, a: => A): FingerTree[V, A] =
    new FingerTree[V, A] {
      def fold[B](b: => B, s: (V, A) => B, d: (V, Finger[V, A], => FingerTree[V, Node[V, A]], Finger[V, A]) => B): B = s(v, a)
    }

  def deep[V, A](pr: Finger[V, A], m: => FingerTree[V, Node[V, A]], sf: Finger[V, A])(implicit r: Reducer[A, V]): FingerTree[V, A] = {
    import r.semigroup
    val measure = fingerMeasure[A, V]
    deep(measure.snoc(mappendVal(measure.unit(pr), m), sf), pr, m, sf)
  }

  def deep[V: Reducer[A, *], A](v: V, pr: Finger[V, A], m: => FingerTree[V, Node[V, A]], sf: Finger[V, A]): FingerTree[V, A] =
    new FingerTree[V, A] {
      private[this] val mz = Need(m)
      def fold[B](b: => B, f: (V, A) => B, d: (V, Finger[V, A], => FingerTree[V, Node[V, A]], Finger[V, A]) => B): B =
        d(v, pr, mz.value, sf)
    }

  def deepL[V: Reducer[A, *], A](mpr: Option[Finger[V, A]], m: => FingerTree[V, Node[V, A]], sf: Finger[V, A]): FingerTree[V, A] =
    mpr match {
      case None => rotL(m, sf)
      case Some(pr) => deep(pr, m, sf)
    }

  def deepR[V: Reducer[A, *], A](pr: Finger[V, A], m: => FingerTree[V, Node[V, A]], msf: Option[Finger[V, A]]): FingerTree[V, A] =
    msf match {
      case None => rotR(pr, m)
      case Some(sf) => deep(pr, m, sf)
    }

  def rotL[V, A](m: FingerTree[V, Node[V, A]], sf: Finger[V, A])(implicit r: Reducer[A, V]): FingerTree[V, A] = {
    import r.semigroup
    m.viewl.fold(
      sf.toTree,
      (a, mm) => deep(m.measure.cata(_ |+| sf.measure, sf.measure), a.toDigit, mm, sf))
  }

  def rotR[V, A](pr: Finger[V, A], m: FingerTree[V, Node[V, A]])(implicit r: Reducer[A, V]): FingerTree[V, A] = {
    import r.semigroup
    m.viewr.fold(
      pr.toTree,
      (mm, a) => deep(mappendVal(pr.measure, m), pr, mm, a.toDigit))
  }

  /**View of the left end of a sequence.*/
  sealed abstract class ViewL[S[_], A] {
    def fold[B](b: => B, f: (=> A, => S[A]) => B): B
    def headOption: Option[A] = fold(None, (a, sa) => Some(a))
    def tailOption: Option[S[A]] = fold(None, (a, sa) => Some(sa))
    def head: A = headOption.getOrElse(sys.error("Head on empty view"))
    def tail: S[A] = tailOption.getOrElse(sys.error("Tail on empty view"))
  }

  /**View of the right end of a sequence.*/
  sealed abstract class ViewR[S[_], A] {
    def fold[B](b: => B, f: (=> S[A], => A) => B): B
    def lastOption: Option[A] = fold(None, (sa, a) => Some(a))
    def initOption: Option[S[A]] = fold(None, (sa, a) => Some(sa))
    def last: A = lastOption.getOrElse(sys.error("Last on empty view"))
    def init: S[A] = initOption.getOrElse(sys.error("Init on empty view"))
  }

  sealed abstract class Finger[V, A] {
    def foldMap[B](f: A => B)(implicit m: Semigroup[B]): B

    /**
      * Append the given element to the right
      *
      * @throws if the finger is `Four`.
      */
    def +:(a: A): Finger[V, A]

    /**
      * Prepends the given element to the left
      *
      * @throws if the finger is `Four`.
      */
    def :+(a: A): Finger[V, A]

    /** Replaces the first element of this finger with `a` */
    def |-:(a: A): Finger[V, A]

    /** Replaces the last element of this finger with `a` */
    def :-|(a: A): Finger[V, A]

    def lhead: A

    def ltail: Finger[V, A]

    def rhead: A

    def rtail: Finger[V, A]

    def toTree: FingerTree[V, A]

    def map[B, V2: Reducer[B, *]](f: A => B): Finger[V2, B]

    /** Apply the given side effect to each element. */
    def foreach(f: A => Unit): Unit

    /** An iterator that visits each element. */
    def iterator: Iterator[A]

    /** An iterator that visits each element in reverse order. */
    def reverseIterator: Iterator[A]

    def measure: V

    def toList: List[A] = map[A, List[A]](x => x).measure

    private[scalaz] def split1(pred: V => Boolean, accV: Maybe[V]): (Option[Finger[V, A]], A, Option[Finger[V, A]])
  }
  case class One[V, A](v: V, a1: A)(implicit r: Reducer[A, V]) extends Finger[V, A] {

    def foldMap[B](f: A => B)(implicit m: Semigroup[B]) = f(a1)

    def +:(a: A) = Two(r.cons(a, v), a, a1)

    def :+(a: A) = Two(r.snoc(v, a), a1, a)

    def |-:(a: A) = one(a)

    def :-|(a: A) = one(a)

    def lhead = a1

    def ltail = sys.error("Tail on the digit One")

    def rhead = a1

    def rtail = sys.error("Tail on the digit One")

    def toTree = single(a1)

    def map[B, V2: Reducer[B, *]](f: A => B) = one(f(a1))

    def foreach(f: A => Unit): Unit = {
      f(a1)
    }

    def iterator = Iterator.single(a1)

    def reverseIterator = Iterator.single(a1)

    val measure = v

    private[scalaz] def split1(pred: V => Boolean, accV: Maybe[V]) = (None, a1, None)
  }


  case class Two[V, A](v: V, a1: A, a2: A)(implicit r: Reducer[A, V]) extends Finger[V, A] {
    import r.semigroup

    def foldMap[B](f: A => B)(implicit m: Semigroup[B]) = m.append(f(a1), f(a2))

    def +:(a: A) = Three(r.cons(a, v), a, a1, a2)

    def :+(a: A) = Three(r.snoc(v, a), a1, a2, a)

    def |-:(a: A) = two(a, a2)

    def :-|(a: A) = two(a1, a)

    def lhead = a1

    def ltail = one(a2)

    def rhead = a2

    def rtail = one(a1)

    def toTree = {
      deep(v, one(a1), empty[V, Node[V, A]], one(a2))
    }

    def map[B, V2: Reducer[B, *]](f: A => B) = two(f(a1), f(a2))

    def foreach(f: A => Unit): Unit = {
      f(a1)
      f(a2)
    }

    def iterator = Iterator(a1, a2)

    def reverseIterator = Iterator(a2, a1)

    val measure = v

    private[scalaz] def split1(pred: V => Boolean, accV: Maybe[V]) = {
      val va1 = r.unit(a1)
      val accVa1 = accV.cata(_ |+| va1, va1)
      if (pred(accVa1))
        (None, a1, Some(one(a2)))
      else
        (Some(One(va1, a1)), a2, None)
    }
  }
  case class Three[V, A](v: V, a1: A, a2: A, a3: A)(implicit r: Reducer[A, V]) extends Finger[V, A] {
    import r.semigroup

    def foldMap[B](f: A => B)(implicit m: Semigroup[B]) = m.append(m.append(f(a1), f(a2)), f(a3))

    def +:(a: A) = Four(r.cons(a, v), a, a1, a2, a3)

    def :+(a: A) = Four(r.snoc(v, a), a1, a2, a3, a)

    def |-:(a: A) = three(a, a2, a3)

    def :-|(a: A) = three(a1, a2, a)

    def lhead = a1

    def ltail = two(a2, a3)

    def rhead = a3

    def rtail = two(a1, a2)

    def toTree = {
      deep(v, two(a1, a2), empty[V, Node[V, A]], one(a3))
    }

    def map[B, V2: Reducer[B, *]](f: A => B) = three(f(a1), f(a2), f(a3))

    def foreach(f: A => Unit): Unit = {
      f(a1)
      f(a2)
      f(a3)
    }

    def iterator = Iterator(a1, a2, a3)

    def reverseIterator = Iterator(a3, a2, a1)

    val measure = v

    private[scalaz] def split1(pred: V => Boolean, accV: Maybe[V]) = {
      val va1 = r.unit(a1)
      val accVa1 = accV.cata(_ |+| va1, va1)
      if (pred(accVa1))
        (None, a1, Some(two(a2, a3)))
      else {
        val accVa2 = r.snoc(accVa1, a2)
        if (pred(accVa2))
          (Some(One(va1, a1)), a2, Some(one(a3)))
        else
          (Some(two(a1, a2)), a3, None)
      }
    }
  }
  case class Four[V, A](v: V, a1: A, a2: A, a3: A, a4: A)(implicit r: Reducer[A, V]) extends Finger[V, A] {
    import r.semigroup

    def foldMap[B](f: A => B)(implicit m: Semigroup[B]) = m.append(m.append(f(a1), f(a2)), m.append(f(a3), f(a4)))

    def +:(a: A) = sys.error("Digit overflow")

    def :+(a: A) = sys.error("Digit overflow")

    def |-:(a: A) = four(a, a2, a3, a4)

    def :-|(a: A) = four(a1, a2, a3, a)

    def lhead = a1

    def ltail = three(a2, a3, a4)

    def rhead = a4

    def rtail = three(a1, a2, a3)

    def toTree = {
      deep(v, two(a1, a2), empty[V, Node[V, A]], two(a3, a4))
    }

    def map[B, V2: Reducer[B, *]](f: A => B) = four(f(a1), f(a2), f(a3), f(a4))

    def foreach(f: A => Unit): Unit = {
      f(a1)
      f(a2)
      f(a3)
      f(a4)
    }

    def iterator = Iterator(a1, a2, a3, a4)

    def reverseIterator = Iterator(a4, a3, a2, a1)

    val measure = v

    private[scalaz] def split1(pred: V => Boolean, accV: Maybe[V]) = {
      val va1 = r.unit(a1)
      val accVa1 = accV.cata(_ |+| va1, va1)
      if (pred(accVa1))
        (None, a1, Some(three(a2, a3, a4)))
      else {
        val accVa2 = r.snoc(accVa1, a2)
        if (pred(accVa2))
          (Some(One(va1, a1)), a2, Some(two(a3, a4)))
        else {
          val accVa3 = r.snoc(accVa2, a3)
          if (pred(accVa3))
            (Some(two(a1, a2)), a3, Some(one(a4)))
          else
            (Some(three(a1, a2, a3)), a4, None)
        }
      }
    }
  }

  sealed abstract class Node[V, A](implicit r: Reducer[A, V]) {
    def fold[B](two: (V, => A, => A) => B, three: (V, => A, => A, => A) => B): B

    def foldMap[B](f: A => B)(implicit m: Semigroup[B]): B = fold(
      (v, a1, a2) => m.append(f(a1), f(a2)),
      (v, a1, a2, a3) => m.append(m.append(f(a1), f(a2)), f(a3)))

    def toDigit: Finger[V, A] = fold(
      (v, a1, a2) => Two(v, a1, a2),
      (v, a1, a2, a3) => Three(v, a1, a2, a3))

    val measure: V

    def map[B, V2: Reducer[B, *]](f: A => B): Node[V2, B] = fold(
      (v, a1, a2) => node2(f(a1), f(a2)),
      (v, a1, a2, a3) => node3(f(a1), f(a2), f(a3)))

    def foreach(f: A => Unit): Unit = {
      fold(
        (_, a1, a2) => { f(a1); f(a2) },
        (_, a1, a2, a3) => { f(a1); f(a2); f(a3) }
      )}

    def iterator: Iterator[A] = fold(
      (_, a1, a2) => Iterator(a1, a2),
      (_, a1, a2, a3) => Iterator(a1, a2, a3))

    def reverseIterator: Iterator[A] = fold(
      (_, a1, a2) => Iterator(a2, a1),
      (_, a1, a2, a3) => Iterator(a3, a2, a1))

    private[scalaz] def split1(pred: V => Boolean, accV: V): (Option[Finger[V, A]], A, Option[Finger[V, A]]) = fold(
      (v, a1, a2) => {
        val va1 = r.unit(a1)
        val accVa1 = r.append(accV, va1)
        if (pred(accVa1))
          (None, a1, Some(one(a2)))
        else
          (Some(One(va1, a1)), a2, None)
      },
      (v, a1, a2, a3) => {
        val va1 = r.unit(a1)
        val accVa1 = r.append(accV, va1)
        if (pred(accVa1))
          (None, a1, Some(two(a2, a3)))
        else {
          val accVa2 = r.snoc(accVa1, a2)
          if (pred(accVa2))
            (Some(One(va1, a1)), a2, Some(one(a3)))
          else
            (Some(two(a1, a2)), a3, None)
        }
      })
  }

}

/** Indexed sequences, based on [[scalaz.FingerTree]]
 *
 * The measure is the count of the preceding elements, provided by `UnitReducer((e: Int) => 1)`.
 */
final class IndSeq[A](val self: FingerTree[Int, A]) {

  import IndSeq.indSeq
  import std.anyVal._

  private implicit def sizer[A]: Reducer[A, Int] = UnitReducer((_: A) => 1)
  def apply(i: Int): A =
    self.split(_ > i)._2.viewl.headOption.getOrElse(sys.error("Index " + i + " > " + self.measure))
  def replace(i: Int, a: => A): IndSeq[A] = {
    val (l, r) = self.split(_ > i)
    indSeq(l <++> (a |-: r))
  }
  def split(i: Int): (IndSeq[A], IndSeq[A]) = {
    val (l, r) = self.split(_ > i)
    (indSeq(l), indSeq(r))
  }
  def ++(xs: IndSeq[A]): IndSeq[A] = indSeq(self <++> xs.self)
  def :+(x: => A): IndSeq[A] = indSeq(self :+ x)
  def +:(x: A): IndSeq[A] = indSeq(x +: self)
  def length: Int = self.measure.getOrElse(0)
  def tail: IndSeq[A] = indSeq(self.tail)
  def init: IndSeq[A] = indSeq(self.init)
  def drop(n: Int): IndSeq[A] = split(n)._2
  def take(n: Int): IndSeq[A] = split(n)._1
  def map[B](f: A => B): IndSeq[B] = indSeq(self map f)

  import FingerTree.fingerTreeFoldable

  def flatMap[B](f: A => IndSeq[B]): IndSeq[B] = indSeq(fingerTreeFoldable.foldLeft(self, empty[Int, B])((ys, x) => ys <++> f(x).self))
}

object IndSeq extends IndSeqInstances {
  private def indSeq[A](v: FingerTree[Int, A]): IndSeq[A] = new IndSeq(v)

  import std.anyVal._

  def apply[A](as: A*): IndSeq[A] = fromSeq(as)
  def fromSeq[A](as: Seq[A]): IndSeq[A] = indSeq(as.foldLeft(empty[Int, A](UnitReducer(a => 1)))((x, y) => x :+ y))
}

sealed abstract class IndSeqInstances {

  implicit def indSeqEqual[A: Equal]: Equal[IndSeq[A]] =
    Equal.equalBy(_.self)

  implicit val indSeqInstance: MonadPlus[IndSeq] with Alt[IndSeq] with Traverse[IndSeq] with IsEmpty[IndSeq] =
    new MonadPlus[IndSeq] with Alt[IndSeq] with Traverse[IndSeq] with IsEmpty[IndSeq] with IsomorphismFoldable[IndSeq, FingerTree[Int, *]]{
      def G = implicitly
      override val naturalTrans = λ[IndSeq ~> FingerTree[Int, *]](_.self)
      def traverseImpl[G[_], A, B](fa: IndSeq[A])(f: A => G[B])(implicit G: Applicative[G]) = {
        import std.anyVal._
        implicit val r: Reducer[B, Int] = UnitReducer((_: B) => 1)
        G.map(fa.self.traverseTree(f))(new IndSeq(_))
      }
      override def length[A](fa: IndSeq[A]) =
        fa.length
      override def index[A](fa: IndSeq[A], i: Int) =
        if(0 <= i && i < fa.length) Some(fa(i)) else None
      override def isEmpty[A](fa: IndSeq[A]) =
        fa.self.isEmpty
      override def empty[A](fa: IndSeq[A]) =
        fa.self.isEmpty
      def point[A](a: => A) =
        IndSeq(a)
      def bind[A, B](fa: IndSeq[A])(f: A => IndSeq[B]) =
        fa flatMap f
      override def map[A, B](fa: IndSeq[A])(f: A => B) =
        fa map f
      def plus[A](a: IndSeq[A], b: => IndSeq[A]) =
        a ++ b
      def alt[A](a: => IndSeq[A], b: => IndSeq[A]) =
        plus(a, b)
      def empty[A] =
        IndSeq.apply()
    }
}

/** Ordered sequences, based on [[scalaz.FingerTree]]
 *
 *  `a` has a higher priority than `b` if `Order[A].greaterThan(a, b)`.
 *
 * `insert` and `++` maintains the ordering.
 *
 * The measure is calculated with a `Semigroup[A @@ LastVal]`, whose `append`
 * operation favours the first argument. Accordingly, the measure of a node is the
 * item with the highest priority contained recursively below that node.
 */
sealed abstract class OrdSeq[A] extends Ops[FingerTree[A @@ LastVal, A]] {
  import std.function._

  implicit val ord: Order[A]

  /**
   * @return (higher, lowerOrEqual) The sub-sequences that contain elements of higher and of lower-than-or-equal
   *                                priority than `a`, and of lower or equal priority respectively.
   */
  def partition(a: A): (OrdSeq[A], OrdSeq[A]) =
    function1Instance.product(OrdSeq.ordSeq[A](_: FingerTree[A @@ LastVal, A]))(self.split(a1 =>
      ord.greaterThanOrEqual(Tag.unwrap(a1), a)))

  /** Insert `a` at a the first point that all elements to the left are of higher priority */
  def insert(a: A): OrdSeq[A] = partition(a) match {
    case (l, r) => OrdSeq.ordSeq(l <++> (a +: r))
  }

  /** Append `xs` to this sequence, reordering elements to  */
  def ++(xs: OrdSeq[A]): OrdSeq[A] = xs.self.toList.foldLeft(this)(_ insert _)
}

object OrdSeq {
  private def ordSeq[A: Order](t: FingerTree[A @@ LastVal, A]): OrdSeq[A] = new OrdSeq[A] {
    val self = t
    val ord = Order[A]
  }

  implicit def unwrap[A](t: OrdSeq[A]): FingerTree[A @@ LastVal, A] = t.self

  def apply[A: Order](as: A*): OrdSeq[A] = {
    val z: OrdSeq[A] = {
      implicit val keyer: Reducer[A, A @@ LastVal] = UnitReducer((a: A) => LastVal(a))
      ordSeq(empty[A @@ LastVal, A])
    }
    as.foldLeft(z)((x, y) => x insert y)
  }
}