Towards Trans-Exponential O-minimal Expansion of $(\mathbb{R},+,\cdot ,0,1<)$

We add an analytic trans-exponential function $\phi$ to ${\mathbb{R}}_{an,\exp }$. We reduce the o-minimality of ${\mathbb{R}}_{an,\exp ,\phi }$ to the existence of “many” regular values for some definable systems of functions, which is a necessary condition for the o-minimality of ${\mathbb{R}}_{an,\exp ,\phi }$.