The second partial function accepts this one's outputs. Put this method inside the invariant class in part (b):
def compose[V](that: PartialFn[U, V]): PartialFn[T, V] = {
  val pairs: List[(T, V)] = data.flatMap {
    case (x, y) => that.get(y).map(z => (x, z)).toList
  }
  new PartialFn[T, V](pairs)
}
The represented mapping is , since the paper explicitly applies that second despite the method name. A pair is emitted exactly when this mapping supplies a y and that.get(y) supplies a z. Hence its domain of a partial function is and its value there is . This is strict composition of partial functions: either missing stage makes the composite undefined. Repeated equal input pairs still produce the same output, preserving the representation invariant. With and pairs, direct association-list lookup gives worst-case time complexity, not a hidden constant-cost dictionary lookup.

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