Lucas's Theorem

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.