The Sigmoid Function and the Universal Approximation Theorem

We present a machine-checked Isabelle/HOL development of the sigmoid function \[ \sigma(x)=\frac{e^{x}}{1+e^{x}}, \] together with its most important analytic properties. After proving positivity, strict monotonicity, \(C^{\infty}\) smoothness, and the limits at \(\pm\infty\), we derive a closed-form expression for the \(n\)-th derivative using Stirling numbers of the second kind, following the combinatorial argument of Minai and Williams. These results are packaged into a small reusable library of lemmas on \(\sigma\). Building on this analytic groundwork we mechanise a constructive version of the classical Universal Approximation Theorem: for every continuous function \(f\colon[a,b]\to\mathbb{R}\) and every \(\varepsilon>0\) there is a single‑hidden‑layer neural network with sigmoidal activations whose output is within \(\varepsilon\) of \(f\) everywhere on \([a,b]\). Our proof follows the method of Costarell and Spigler, giving the first fully verified end-to‑end proof of this theorem inside a higher‑order proof assistant.