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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 $(\mathbb{N}, +)$?

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Any integer $x \in \mathbb{Z}$ can be expressed in the form $x = x_+ - x_-$ where $x_+ \in \mathbb{N}$ and $x_- \in \mathbb{N}$ (i.e., every integer can be expressed as the difference of two natural numbers; and conversely, the difference of any two natural numbers is always an integer).

Try using that to formulate your statement. You'll need to work out how to define the $\le$ operator, but that's a separate issue.

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