I've encountered two different generalizations of the Principle of Dependent Choices (DC) to initial ordinals $\kappa>\omega$. First, some notation. By $A\prec B$, I indicate that $A$ is injectable into, but not bijectable with, $B$. Given a function $f$ on a set $A$ and a subset $B$ of $A$, I denote the restriction of $f$ to $B$ by $f\restriction B$. Given an ordinal $\alpha$ and a set $X$, I denote the set of functions $\beta\to X$ where $\beta<\alpha$ by $X^{<\alpha}$.
Now, the generalizations are as follows:
$\text{DC}_\kappa(1)$: If $X$ is a non-empty set and $R\subseteq\mathcal{P}(X)\times X$ is a relation such that $\text{dom}(R)\supseteq\{Y\in\mathcal{P}(X):Y\prec\kappa\},$ then there exists some $f:\kappa\to X$ such that $\text{ran}(f\restriction\alpha)\:R\:f(\alpha)$ for all $\alpha<\kappa$.
$\text{DC}_\kappa(2)$: If $X$ is a non-empty set and $R\subseteq X^{<\kappa}\times X$ is a relation such that $\text{dom}(R)=X^{<\kappa},$ then there exists some $f:\kappa\to X$ such that $(f\restriction\alpha)\:R\:f(\alpha)$ for all $\alpha<\kappa$.
I can see that $\text{DC}_\omega(1)$ and $\text{DC}_\omega(2)$ are equivalent to DC.
I've been trying to show that $\text{DC}_\kappa(1)$ and $\text{DC}_\kappa(2)$ are equivalent. Showing that $\text{DC}_\kappa(2)$ implies $\text{DC}_\kappa(1)$ is fairly straightforward, but I have thus far been stymied on proving the other direction. The approach I've been trying to take is to suppose some relation $R$ satisfies the hypotheses of $\text{DC}_\kappa(2)$, constructed a related relation $S$ satisfying the hypotheses of $\text{DC}_\kappa(1)$, yielding a function $f$ satisfying the conclusion of the $\text{DC}_\kappa(1)$, and from that trying to show that $f$ satisfies the conclusions of $\text{DC}_\kappa(2)$, or using $f$ somehow to construct another function $g$ that does.
Can anyone give me any hints as to how I might accomplish this task?
Edit: Let me show you one of my abortive attempts to prove the trickier direction, to make it easier to advise me on this.
Suppose $\text{DC}_\kappa(1)$ holds for some initial ordinal $\kappa>\omega$, and suppose $R$ satisfies the hypotheses of $\text{DC}_\kappa(2)$.
Given $Y\subseteq X$, define $I(Y,\kappa)$ to be the set of all injections $Y\to\kappa$ if $Y\prec\kappa$, and otherwise define $I(Y,\kappa):=\emptyset$. Given $h\in I(Y,\kappa)$, and well-ordering $Y$ by proxy using $h$, there is a unique ordinal $\alpha_h$ isomorphic to $Y$ in this well-ordering, and a unique isomorphism. In this way, $h\in I(Y,\kappa)$ uniquely determines an ordinal $\alpha_h<\kappa$ and a bijection $F_h:\alpha_h\to Y$.
Now, given any $\alpha<\kappa$ and any $h:\alpha\to X$, the map $\text{ran}(h)\to\alpha$ given by $x\mapsto\min h^{-1}(x)$ lets us well-order $\text{ran}(h)$ by proxy, uniquely determining an ordinal $\beta_h<\kappa$ and an isomorphism $G_h:\text{ran}(h)\to\beta_h.$ Each $G_h$ is then an injection from a subset of $X$ into (but not onto) $\kappa$.
Define $S\subseteq\mathcal{P}(X)\times X$ by $Y\:S\:y$ iff there is some $\alpha<\kappa$ and some $h:\alpha\to X$ such that $Y=\text{dom}(G_h)$ and $h\:R\:y$.
Take any $Y\subseteq X$ with $Y\prec\kappa$ and any $h\in I(Y,\kappa)$. Then $F_h\in X^{<\kappa}$, so by assumption there is some $y\in X$ such that $F_h\:R\:y$. Moreover, $Y=\text{dom}(G_{F_h})$, so by definition, $Y\:S\:y$. Thus, by $\text{DC}_\kappa(1),$ there is some $f:\kappa\to X$ such that $\text{ran}(f\restriction\alpha)\:S\:f(\alpha)$ for all $\alpha<\kappa$.
Sticking Point: I don't believe we can conclude that $(f\restriction\alpha)\:R\:f(\alpha)$ for all $\alpha<\kappa$, but I'd like to be able to use $f$ to construct a function $\hat f:\kappa\to X$ such that $(\hat f\restriction\alpha)\:R\:\hat f(\alpha)$ for all $\alpha<\kappa$. I'm not sure how I might do this, though.