Constructing logical sentences that involve negative integers over the nonnegative integers

Question

Consider the following statement:

If $x$ and $y$ are integers and $z$ is a nonnegative integer and $x + z = y$, then $x$ is at most $y$.

I'd like to build a sentence for this statement in the model of $(\mathbb{N}, +)$, where $\mathbb{N}$ is the set of nonnegative integers and $+$ is just the relation (where here I define a relation as a mapping from tuples to a boolean value): $\{(a,b,c): a + b = c\}$. By "sentence", I mean a logical formula with no free variables.

How can I write the highlighted statement in terms of the given model if variables in this model can only take only take on nonnegative integers? It seems at best I can only build a sentence in this model for the following statement:

If $x$ and $y$ are nonnegative integers and $z$ is a nonnegative integer and $x + z = y$, then $x$ is at most $y$.

Or is it possible to define negative quantities in the model of $(\mathbb{N}, +)$?