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\begin{document}

\title{Lucas's Theorem}
\author{Chelsea Edmonds}
\maketitle

\begin{abstract}
  This work presents a formalisation of a generating function proof for Lucas's theorem. We first outline extensions to the existing Formal Power Series (FPS) library, including an equivalence relation for coefficients modulo $n$, an alternate binomial theorem statement, and a formalised proof of the Freshman's dream (mod $p$) lemma.

  The second part of the work presents the formal proof of Lucas's Theorem. Working backwards, the formalisation first proves a well known corollary of the theorem which is easier to formalise and then applies induction to prove the original theorem statement. The proof of the corollary aims to provide a good example of a formalised generating function equivalence proof using the FPS library. The final theorem statement is intended to be integrated into the formalised proof of Hilbert's 10th Problem \cite{bayerDPRMTheoremIsabelle2019}.
\end{abstract}

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