Assume we are given inequalities $x \not\equiv a_i\text { (mod }b_i)$ for $i=1,\ldots,n$ where $1 \leq a_i \leq b_i, x \in \mathbb{Z}$.
Can we somehow reformulate the problem as $x \not\equiv a_i'\text { (mod }b), x \in r\mathbb{Z}$ for some $r \in \mathbb{Z}, i=1,\ldots,n'$, where we only use polynomially many new inequalities (in $n$)?
For example if we have the system $x \not \equiv 3\text { (mod }4),x \not \equiv 2\text { (mod }2)$ we could reformulate this as $x \not \equiv 2\text { (mod }4),x \not \equiv 3\text { (mod }4),x \not \equiv 4\text { (mod }4), x \in 1\mathbb{Z}$; but of course this approach will need exponentially many new inequalities in general.
Here reformulating means that the new system has a solution $x \in r\mathbb{Z}$ satisfying all inequalities if and only if the original system has a solution.