# A question regarding two Cauchy definitions of integrability

While doing research on the construction of the Riemann integral I've stumbled upon two different criterions for Riemann integrability. In this article (on page $$19$$) there is a theorem called the Cauchy Criterion for Henstock Integrals. This theorem states that $$f:[a,b]\rightarrow \mathbb{R}$$ is Henstock integrable on $$[a,b]$$ if and only if $$\forall \varepsilon > 0 \exists\delta:[a,b]\rightarrow \mathbb{R^+}$$ such that when $$P_1$$ and $$P_2$$ are any $$\delta$$-,fine tagged partitions of $$[a,b]$$ then $$|S(f,P_1,t_1)-S(f,P_2,t_2)|<\varepsilon$$, where $$t_1$$ and $$t_2$$ are tags. This theorem evidently applies for Riemann integrals as well if we let $$\delta$$ be constant. So with this slight tweak I suppose we could call the theorem the Cauchy criterion for Riemann integrability.

However, at other places on the internet I've seen this theorem be stated as $$\forall \varepsilon>0$$ there exists partitions $$P_1$$ and $$P_2$$ of $$[a,b]$$ such that $$U(f,P_1)-L(f,P_2)<\varepsilon$$.

I am aware that both the aforementioned theorems are equivalent. In terms of how these two theorems are stated I don't think it makes sense for them to be called the same. The second theorem should probably be called the Cauchy criterion for Darkboux integrability instead. However, I feel like it's a bit arrogant of me, as an undergraduate student of mathematics to name theorems that are not due to my own work. So if I published a paper on integration, and named the second theorem after Cauchy and Darboux, do you think anyone would object?