  The elementary arithmetic operations $+,\cdot,\le$ on integers are well-known
to be computable in the weak complexity class $\mathrm{TC}^0$, and it is a
basic question what properties of these operations can be proved using only
$\mathrm{TC}^0$-computable objects, i.e., in a theory of bounded arithmetic
corresponding to $\mathrm{TC}^0$. We will show that the theory $\mathit{VTC}^0$
extended with an axiom postulating the totality of iterated multiplication
(which is computable in $\mathrm{TC}^0$) proves induction for quantifier-free
formulas in the language $\langle +,\cdot,\le \rangle$ (IOpen), and more
generally, minimization for $\Sigma^b_0$ formulas in the language of Buss's
$S_2$.
