Inference of a measure for a decreasing chain
Yes.  This is possible.  $t=1$ suffices.  Take

$$\begin{align*}
f(m,n-1) &amp;= Cm + D\\
f(m,n-2) &amp;= f(m+v_{n-2},n-1) + 1 = Cm + C v_{n-2} + D + 1\\
f(m,n-3) &amp;= f(m+v_{n-3},n-2) + 1 = Cm + C v_{n-3} + C v_{n-2} + D + 2\\
&amp;\ldots\\
f(m,i) &amp;= Cm + C v_{i} + \dots + C v_{n-2} + D + n-1-i
\end{align*}$$

where we take the constant $C&gt;0$ to be at least $n$; we choose a constant $D&gt;0$ that's large enough that the output of $f$ is always positive; we take $t=1$; and we use the standard order on $\mathbb{N}$.  Then you can verify that this satisfies all the conditions.

How do you find this?  As always, start with small cases.  Start with $n=2$ and see what has to hold of $f$, and try to construct a $f$ that meets the conditions.  (One way to ensure that $x&gt;_t y$ is to set $x=y+1$; in the construction above, I used that to construct $f$ that satisfies the condition by definition.)  Then try $n=3$.  Then generalize.  The above definition follows as a straightforward consequence of your requirements; if you choose elements of $f$ in the right order, it becomes routine to fill out the truth table of $f$ so all the requirements are met.