Challenge: Strictly Increasing

What if we need strictly increasing instead of non-decreasing? Transform differently. Set $b_i = a_i - i$. Then $a$ strictly increasing iff $b$ non-decreasing.

But we want non-decreasing. Alternative: $b_i = a_i - i \cdot \epsilon$ for tiny $\epsilon$. Conceptually, offset each position slightly. In practice: $b_i = a_i - i$ works for non-decreasing. For strictly increasing, use $b_i = a_i - i$ and require the transformed sequence to be strictly increasing (same cost structure).