Given a function $f:\mathbb R\to\mathbb R$, call a decreasing sequence $x_1>x_2>x_3>\cdots$ $f$-bad if $f(x_1)>f(x_2)>f(x_3)>\cdots$, and call the function $f$ "ordinal decreasing" if there exist no infinite $f$-bad sequences. We prove the following result, which generalizes results of Erickson et al. (2022) and Bufetov et al. (2024): Given ordinal decreasing functions $f,g_1,\ldots,g_k,s$ that are everywhere larger than $0$, define the recursive algorithm "$M(x)$: if $x<0$ return $f(x)$, else return $g_1(-M(x-g_2(-M(x-\cdots-g_k(-M(x-s(x)))\cdots))))$". Then $M(x)$ halts and is ordinal decreasing for all $x \in \mathbb{R}$. More specifically, given an ordinal decreasing function $f$, denote by $o(f)$ the ordinal height of the root of the tree of $f$-bad sequences. Then we prove that, for $k\ge 2$, the function $M(x)$ defined by the above algorithm satisfies $o(M)\le\varphi_{k-1}(\gamma+o(s)+1)$, where $\gamma$ is the smallest ordinal such that $\max{\{o(s),o(f),o(g_1), \ldots, o(g_k)\}}<\varphi_{k-1}(\gamma)$.

翻译：给定函数 $f:\mathbb R\to\mathbb R$，若存在递减序列 $x_1>x_2>x_3>\cdots$ 满足 $f(x_1)>f(x_2)>f(x_3)>\cdots$，则称该序列为 $f$ 坏序列；若不存在无限 $f$ 坏序列，则称函数 $f$ 为“序数递减”。我们证明了以下结果，该结果推广了 Erickson 等人（2022）及 Bufetov 等人（2024）的结论：给定处处大于 $0$ 的序数递减函数 $f,g_1,\ldots,g_k,s$，定义递归算法：“$M(x)$: 若 $x<0$ 则返回 $f(x)$，否则返回 $g_1(-M(x-g_2(-M(x-\cdots-g_k(-M(x-s(x)))\cdots))))$”。则对任意 $x \in \mathbb{R}$，$M(x)$ 停机且为序数递减。进一步地，给定序数递减函数 $f$，记 $o(f)$ 为 $f$ 坏序列树根节点的序数高度。我们证明，对于 $k\ge 2$，由上述算法定义的函数 $M(x)$ 满足 $o(M)\le\varphi_{k-1}(\gamma+o(s)+1)$，其中 $\gamma$ 是满足 $\max{\{o(s),o(f),o(g_1), \ldots, o(g_k)\}}<\varphi_{k-1}(\gamma)$ 的最小序数。