The Complementary Filter, $$y=\alpha \times y+(1-\alpha) \times x$$ where $\alpha$ is the filter parameter, usually chosen to be ~0.98, is named as such, because effectively the filter highpasses $y$ and lowpasses $x$.
With this setup ($\alpha=0.98$), man claims that the filter can filter out the low-frequency part of $y$ and meanwhile the high-frequency part of $x$.
What is the mathematical proof here?
There is no rigorous proof so far available. All the sources simply take it as granted and validates it with experiment data, which is unacceptable in scientific validation.