Inputs:
- L, the list of x-coordinates, each a NonZeroScalar.
- x_i, an x-coordinate contained in L, a NonZeroScalar.

Outputs:
- value, a Scalar.

Errors:
- "invalid parameters", if 1) x_i is not in L, or if 2) any
  x-coordinate is represented more than once in L.

def derive_interpolating_value(L, x_i):
  if x_i not in L:
    raise "invalid parameters"
  for x_j in L:
    if count(x_j, L) > 1:
      raise "invalid parameters"

  numerator = Scalar(1)
  denominator = Scalar(1)
  for x_j in L:
    if x_j == x_i: continue
    numerator *= x_j
    denominator *= x_j - x_i

  value = numerator / denominator
  return value