Formalising the local compactness of the adele ring

The adele ring of a number field is a central object in modern number theory. Its status as a locally compact topological ring is one of the key reasons why. We describe a formal proof that the adele ring of a number field is locally compact implemented in the Lean 4 theorem prover. Our work includes the formalisations of new types, including the completion of a number field at an infinite place, the infinite adele ring and the finite $S$-adele ring, as well as formal proofs that completions of a number field are locally compact and that their rings of integers at finite places are compact.