Interval Overlap Check

Let me formalize the overlap check you just used. Two intervals $[a, b)$ and $[c, d)$ overlap if and only if: $a < d \text{ and } c < b$

Think about it: they do NOT overlap if one ends before the other starts. That means $b \leq c$ or $d \leq a$. The negation gives the overlap condition.

For closed intervals $[a, b]$ and $[c, d]$, use $a \leq d$ and $c \leq b$ instead. The boundary matters for touching intervals. Common bug: mixing half-open and closed interval semantics.