We show a deterministic simulation (or lifting) theorem for composed problems $f \circ \mathsf{Eq}_n$ where the inner function (the gadget) is Equality on $n$ bits. When $f$ is a total function on $p$ bits, it is easy to show via a rank argument that the communication complexity of $f\circ \mathsf{Eq}_n$ is $\Omega(deg(f) \cdot n)$. However, there is a surprising counter-example of a partial function $f$ on $p$ bits, such that any completion $f'$ of $f$ has $\deg(f') = \Omega(p)$, and yet $f \circ \mathsf{Eq}_n$ has communication complexity $O(n)$.
We develop a technique for proving lower bounds in the setting of asymmetric communication, a model that was introduced in the famous works of Miltersen (STOC'94) and Miltersen, Nisan, Safra and Wigderson (STOC'95). At the core of our technique is a novel simulation theorem. Alice gets a $p \times n$ matrix $x$ over $\mathbb F_2$ and Bob gets a vector $y \in \mathbb F_2^n$. Alice and Bob need to evaluate $f(x\cdot y)$ for a Boolean function $f$.