Length comparison is PolyTimeDecidable

The length comparison $V(x,w) = [|w| \le |x|]$ is polynomial-time decidable.

This predicate checks whether the certificate $w$ is no longer than the instance $x$. A multi-tape Turing machine can decide this by scanning both tapes simultaneously from left to right: if the head on the $w$-tape reaches a blank symbol before or at the same time as the head on the $x$-tape, accept; otherwise reject. This runs in $O(|x|+|w|)$ steps, which is clearly bounded by $c(|x|+|w|+1)^d$ for $c=1, d=1$.