  The minimization principle MIN($\prec$) studied in bounded arithmetic in
[Chiari, M. and Kraj\'i\v{c}ek, J. Witnessing Functions in Bounded Arithmetic
and Search Problems, Journal of Symbolic Logic, 63(3):1095-1115, 1998] says
that a strict linear ordering $\prec$ on any finite interval $[0,\dots,n)$ has
the minimal element. We shall prove that bounded arithmetic theory
$T^1_2(\prec)$ augmented by instances of the pigeonhole principle for all
$\Delta^b_1(\prec)$ formulas does not prove MIN($\prec$).
