  We develop a general toolbox to study $W^{1,p}$ solutions of differential
inclusions $\nabla u \in K$ for unbounded sets $K$. A key notion is the concept
that a subset $K$ of the space $\mathbb{R}^{d \times m}$ of $d \times m$
matrices can be reduced to another set $K'$. We then use this framework to show
that the product rigidity for Sobolev maps fails for $p<2$, and also apply our
toolbox to simplify several examples from the literature.
