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How to define a function that has these specific properties?

Suppose $x = (x_1,x_2,\dots,x_K) \in \mathbb{Z}^K_{\geq 0}$. For $x,y \in \mathbb{Z}^K_{\geq 0}$, we write $x \succ y$ or $y \prec x$ if $x \neq y$ and \begin{align*} x_{i(x,y)} > y_{i(x,y)}, …
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