Linear Arithmetic - why calling it linear?

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This might sound like a stupid question, but I could not figure out, why Linear Arithmetic is called linear.

Linear Arithmetic often comes in play when talking about SMT Solving. For example, in this PDF linear arithmetic is defined as formulas using $=$, $\leq$, $<$ within the atoms. Somethink like 3*x_1 + 2*x_2 ≤ 5*x_3 is a linear arithmetic formula.

Because of which property is this arithmetic called linear?

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2 Answers

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It seems that linear in that document is related to linear programming, which is optimization with linear constraints.

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A function $F(x)$ is linear when $F(ax+by)=aF(x)+bF(y)$. This means that solutions to $F(x)=0$ are linear spaces and solutions to $F(x)\le 0$ are half spaces. The point, here, is that all your formulas are constructed to be linear.

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