We prove the following result, which generalizes results of Erickson et al. (2022) and Bufetov et al. (2024): Call a function $f:\mathbb{R}\to \mathbb{R}$ ordinal decreasing if for every infinite decreasing sequence $x_0>x_1>x_2>\cdots$ there exist $i<j$ such that $f(x_j) \geq f(x_i)$. Given ordinal decreasing functions $f,g_0,\ldots,g_k,s$ that are larger than $0$, define the recursive algorithm "$M(x)$: if $x<0$ return $f(x)$, else return $g_0(-M(x-g_1(-M(x-\cdots-g_k(-M(x-s(x)))\cdots))))$". Then $M(x)$ halts and is ordinal decreasing for all $x \in \mathbb{R}$.

翻译：我们证明以下结果，该结果推广了Erickson等人（2022）和Bufetov等人（2024）的结论：称函数$f:\mathbb{R}\to \mathbb{R}$为序数递减，若对于任意无穷递减序列$x_0>x_1>x_2>\cdots$，存在$i<j$使得$f(x_j) \geq f(x_i)$。给定大于$0$的序数递减函数$f,g_0,\ldots,g_k,s$，定义递归算法"$M(x)$：若$x<0$则返回$f(x)$，否则返回$g_0(-M(x-g_1(-M(x-\cdots-g_k(-M(x-s(x)))\cdots))))$"。则$M(x)$对所有$x \in \mathbb{R}$停机且为序数递减。