Unique existence and computability in constructive reverse mathematics

Abstract:
We introduce, and show the equivalences among, relativized versions of
Brouwer's fan theorem for detachable bars (FAN), weak Koenig lemma
with a uniqueness hypothesis (WKL!), and the longest path lemma with
a uniqueness hypothesis (LPL!) in the spirit of constructive revrese
mathematics. We prove that a computable version of minimum principle:
if $f$ is a real valued computable uniformly continuous function with at
most one minimum on the Cantor space $\{0,1\}^\mathbf{N}$, then there
exists a computable $\alpha$ in $\{0,1\}^\mathbf{N}$ such that $f(\alpha)
= \inf f(\{0,1\}^\mathbf{N})$, is equivalent to some computably relativized
version of FAN, WKL!, and LPL!.