Unique Existence and Computability in Constructive Reverse Mathematics

Abstract

We introduce, and show the equivalences among, relativizedversions of Brouwer's fan theorem for detachable bars (FAN), weakKonig lemma with a uniqueness hypothesis (WKL!), and thelongest path lemma with a uniqueness hypothesis (LPL!) in thespirit of constructive reverse mathematics. We prove that acomputable version of minimum principle: if fis a realvalued computable uniformly continuous function with at most oneminimum on {0,1}N, then there exists a computableα in {0,1}Nsuch that $f(\alpha) = \inff(\{0,1\}^\mathbf{N})$, is equivalent to some computablyrelativized version of FAN, WKL! and LPL!.