# Unifying constraint matrices in sparse situations

$$\DeclareMathOperator\Set{Set}$$

Let $$Set=\{x\in\mathbb Z^{n}:\exists y\in\mathbb Z^m\text{ satisfying } A[x,y]'\leq b\}$$ where $$A$$ has $$r=km$$ rows and $$k=O(1)$$.

I am trying to write $$Set=\{x\in\mathbb Z^n:C[x]'\leq d\}\quad\text{where C has r' rows.}$$

Is the number of rows $$r'\leq r^{m}$$?

In my situation in addition to $$A$$ having $$r=O(m)$$ rows and every row has $$1$$, $$2$$ or $$3$$ entries. We further divide $$n+m$$ columns to first $$m_0=n$$ columns, next $$m_1$$ columns until final $$m_d$$ columns ($$m=\sum_{i=1}^d m_i$$) and so $$A$$ is block matrix of $$d(d+1)$$ blocks (rows are $$k$$ times number of columns). A block is non-zero iff it is $$(i,i)$$ and $$(i,i+1)$$ block (diagonal and one above).

$$A=\begin{bmatrix} A_{0,0}&A_{0,1}&0&\dots&0&0\\ 0&A_{1,1}&A_{1,2}&\dots&0&0\\ \vdots\\ 0&0&0&\dots&A_{d-1,d-1}&A_{d-1,d} \end{bmatrix}$$ where $$A_{j-1,j}=\mathbb I_{m_j,m_j}\otimes\mathbb 1_{k}$$ where $$\mathbb I_{m_j,m_j}$$ is $$m_j\times m_j$$ identity and $$\mathbb 1_k$$ is $$k\times 1$$ vectors of every entry being $$1$$ and $$A_{j,j}$$ has $$km_{j+1}$$ rows and $$m_j$$ columns. In addition at every block matrix $$A_{j,j+1}$$ every column is non-zero and every row has $$1$$ non-zero entry. Every row of $$A$$ has $$1$$, $$2$$ or $$3$$ non-zero entries.

In the situation involved is it possible that $$r'=\operatorname{poly}(2^d{m}r)=\operatorname{poly}(m2^d)$$ holds?