  This paper shows how techniques for linear dynamical systems can be used to
reason about the behavior of general loops. We present two main results. First,
we show that every loop that can be expressed as a transition formula in linear
integer arithmetic has a best model as a deterministic affine transition
system. Second, we show that for any linear dynamical system $f$ with integer
eigenvalues and any integer arithmetic formula $G$, there is a linear integer
arithmetic formula that holds exactly for the states of $f$ for which $G$ is
eventually invariant. Combining the two, we develop a monotone conditional
termination analysis for general loops.
