Judging whether one convex polytope is inside of another when both are expressed as a system of linear inequalities seems not to be an easy problem. 
[This answer][1] on math.stackexchange.com claims the following proposition regarding a special case.

Using Farkas' lemma to show, a halfspace $\{x \mid a' x \leq b' \}$  contains the polytope $P= \{x |A x \leq b \}$ if and only if there exists $\lambda \in \mathbb{R}^m$ with  $\lambda \geq 0$ (understood component-wise) such that:
\begin{align*}
a' &= \lambda^T A \\
b' &\geq \lambda^T b
\end{align*}

It is easy to prove the sufficiency. I am able to show the necessity of $a' = \lambda^T A$ by Farkas' lemma, but not $b' \geq \lambda^T b$. I tried but failed to argue by contradiction. How does one prove that last inequality?

  [1]: https://math.stackexchange.com/a/198091/64809