An independence of the MIN principle from the PHP principle

Abstract

The minimization principle $\textsf{MIN}(\triangleleft)$ studied in bounded arithmetic says that a strict linear ordering $\triangleleft$ on any finite interval $[0,\dots,n)$ has the minimal element. We shall prove that bounded arithmetic theory $\textsf{T}^1_2(\triangleleft)$ augmented by instances of the pigeonhole principle for all $Δ^b_1(\triangleleft)$ formulas does not prove $\textsf{MIN}(\triangleleft)$.