Kolmogorov complexity of string concatenation
Your conjecture is wrong. For some constants $C,D$, it holds that $K(sr) \leq 2K(s) + K(r) + C \leq 2K(s) + |r| + D$ (proof: use a universal Turing machine to generate $s$ and then $r$; you need somewhat more than $K(s)+K(r)$ to store both programs, though $2K(s)+K(r)$ is an overkill). Therefore if $2K(s) + D &lt; |s|$, your conjecture doesn't hold. Such easy strings $s$ certainly exist, for example $K(0^n) = O(\log n)$.