subset Integer  < Number   { .is_int   }
subset Positive < Integer  { .is_pos   }
subset Negative < Integer  { .is_neg   }
subset Prime    < Positive { .is_prime }

func arithmetic_derivative((0)) { 0 }
func arithmetic_derivative((1)) { 0 }

func arithmetic_derivative(Prime _) { 1 }

func arithmetic_derivative(Negative n) {
    -arithmetic_derivative(-n)
}

func arithmetic_derivative(Positive n) is cached {

    var a = n.factor.rand
    var b = n/a

    arithmetic_derivative(a)*b + a*arithmetic_derivative(b)
}

func arithmetic_derivative(Number n) {
    var (a, b) = n.nude
    (arithmetic_derivative(a)*b - arithmetic_derivative(b)*a) / b**2
}