  The ring of Witt vectors $\mathbb{W} R$ over a base ring $R$ is an important
tool in algebraic number theory and lies at the foundations of modern $p$-adic
Hodge theory. $\mathbb{W} R$ has the interesting property that it constructs a
ring of characteristic $0$ out of a ring of characteristic $p > 1$, and it can
be used more specifically to construct from a finite field containing
$\mathbb{Z}/p\mathbb{Z}$ the corresponding unramified field extension of the
$p$-adic numbers $\mathbb{Q}_p$ (which is unique up to isomorphism).
  We formalize the notion of a Witt vector in the Lean proof assistant, along
with the corresponding ring operations and other algebraic structure. We prove
in Lean that, for prime $p$, the ring of Witt vectors over
$\mathbb{Z}/p\mathbb{Z}$ is isomorphic to the ring of $p$-adic integers
$\mathbb{Z}_p$. In the process we develop idioms to cleanly handle calculations
of identities between operations on the ring of Witt vectors. These
calculations are intractable with a naive approach, and require a proof
technique that is usually skimmed over in the informal literature. Our proofs
resemble the informal arguments while being fully rigorous.
