The following question is about a lower bound on the rank of a composition of functions given as a simple expression for the two terms of the sum involved in the inequality.
Consider finite-dimensional vector spaces $V_1,V_2, V_3,V_4$ and linear transformations of these spaces $f_1 : V_1 \rightarrow V_2$, $f_2: V_2 \rightarrow V_3$, $f_3: V_3 \rightarrow V_4$.
Is it true that $\def\rank{\operatorname{rank}}\rank(f_3 \circ f_2) + \rank(f_2 \circ f_1) \geq \rank(f_3 \circ f_2 \circ f_1) + \rank(f_2) $?