We show that $VTC^0$, the basic theory of bounded arithmetic corresponding to the complexity class $\mathrm{TC}^0$, proves the $IMUL$ axiom expressing the totality of iterated multiplication satisfying its recursive definition, by formalizing a suitable version of the $\mathrm{TC}^0$ iterated multiplication algorithm by Hesse, Allender, and Barrington. As a consequence, $VTC^0$ can also prove the integer division axiom, and (by results of Je\v{r}\'abek) the RSUV-translation of induction and minimization for sharply bounded formulas. Similar consequences hold for the related theories $\Delta^b_1$-$CR$ and $C^0_2$. As a side result, we also prove that there is a well-behaved $\Delta_0$ definition of modular powering in $I\Delta_0+WPHP(\Delta_0)$.

中文翻译：

---

$VTC^0$ 中的迭代乘法

我们证明了 $VTC^0$，对应于复杂性类 $\mathrm{TC}^0$ 的有界算术的基本理论，证明了 $IMUL$ 公理表示满足其递归定义的迭代乘法的整体，通过形式化一个Hesse、Allender 和 Barrington 的 $\mathrm{TC}^0$ 迭代乘法算法的合适版本。因此，$VTC^0$ 还可以证明整数除法公理，以及（根据 Je\v{r}\'abek 的结果）锐界公式的归纳和最小化的 RSUV 转换。类似的结果适用于相关理论 $\Delta^b_1$-$CR$ 和 $C^0_2$。作为附带的结果，我们还证明了 $I\Delta_0+WPHP(\Delta_0)$ 中模块化供电的良好的 $\Delta_0$ 定义。