Inferring the Kolmogorov complexity of a string from its substrings' complexity I know that the Kolmogorov complexity of a substring $v$ of an incompressible string $x$ has $C(v)\geq |v|-O(\log{|x|})$ , but I'm wondering if it is also possible to infer the complexity of a string given its substrings' complexity:
Specifically, If $x$ is a string of length $n$ and every substring $v$ of $x$ having length $m$ has complexity $C(v)\geq k$, is it possible to infer any general lower bound for $C(x)$ in terms of $n$, $m$, and $k$? I'm thinking that $C(x)$ cannot be much smaller than $k$ or else the $C(v)$ would not have $k$ as a lower bound. On the other hand, since $n$ can be much larger than $m$ this might balance things out.
A simple bound is that we could use a program generating $x$, a starting index $1 \leq i \leq n$ and length $m$ to get a program generating any choice of $v$, and thus $C(v)$ is bounded as
$$\forall v: k \leq C(v) \leq C(x) + \log(n) + \log(m) + c,$$
where $c$ is some constant overhead independent from $x,v$ that represents the steps necessary to take a Turing machine, a starting index and length and return the requested substring.
This in turn gives us a lower bound for $C(x)$,
$$C(x)  \geq k - \log(n) - \log(m) - c.$$