Ordinals and recursively defined functions on the reals

Given a function f:ℝ→ℝ, call a decreasing sequence x_1>x_2>x_3>⋯ f-bad if f(x_1)>f(x_2)>f(x_3)>⋯, 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,…,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-⋯-g_k(-M(x-s(x)))⋯))))". Then M(x) halts and is ordinal decreasing for all x ∈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≥ 2, the function M(x) defined by the above algorithm satisfies o(M)≤φ_k-1(γ+o(s)+1), where γ is the smallest ordinal such that max{o(s),o(f),o(g_1), …, o(g_k)}<φ_k-1(γ).