A Generalized Theorem of Katz and Motivic Integration

In what follows, we are interested in an extension of a theorem of Nicholas Katz, which will be useful in studying the cohomology of generalized arc spaces develop by Hans Schoutens. As is well known, one is typically interested in the motivic volume of a definable subset of $\mathcal{X} \times X \times \mathbb{Z}^n$ where $\mathcal{X}$ is a scheme over $k((t))$ and $X$ the special fiber of $\mathcal{X}$ . Schoutens has introduced the possibility of developing a motivic integration for limit points other than $k[[t]]$.