Solution
= Solution
The <interval containment order> compares the lower endpoints in reverse and the upper endpoints in their ordinary order. The smallest containing <integer interval> uses the smaller lower endpoint and the larger upper endpoint.
``
class Interval(val l: Int, val u: Int) extends Ordered[Interval] {
require(l <= u)
def <=(that: Interval): Boolean = that.l <= l && u <= that.u
def lub(that: Interval): Interval =
new Interval(math.min(l, that.l), math.max(u, that.u))
}
``
Endpoint comparison immediately proves <reflexivity>, <transitivity>, and <antisymmetry> on interval values. If two separately allocated instances have the same endpoints, they denote the same abstract <interval>; object identity is not the mathematical equality used by this <partial order>.
The returned <interval> contains both operands. Every other containing <interval> has a lower endpoint at most both original lower endpoints and an upper endpoint at least both original upper endpoints, so it also contains the returned hull. Hence
$$
\boxed{[l,u]\vee[l',u']=[\min(l,l'),\max(u,u')].}
$$
This <least upper bound> need not equal the ordinary <union>: the hull of $[1,2]$ and $[5,6]$ is $[1,6]$, including the gap.