  Brouwer (1927) claimed that every function from the Baire space to natural
numbers is induced by a neighbourhood function whose domain admits bar
induction. We show that Brouwer's claim is provable in Heyting arithmetic in
all finite types ($\mathrm{HA}^{\omega}$) for definable functions of the
system. The proof does not rely on elaborate proof theoretic methods such as
normalisation or ordinal analysis. Instead, we internalise in
$\mathrm{HA}^{\omega}$ the dialogue tree interpretation of G\"{o}del's system T
due to Escard\'{o} (2013). The interpretation determines a syntactic
translation of terms, which yields a neighbourhood function from a closed term
of $\mathrm{HA}^{\omega}$ with the required property. As applications of this
result, we prove some well-known properties of $\mathrm{HA}^{\omega}$: uniform
continuity of definable functions from $\mathbb{N}^{\mathbb{N}}$ to
$\mathbb{N}$ on the Cantor space; closure under the rule of bar induction; and
closure of bar recursion for the lowest type with a definable stopping
function.
