# Can the two optimal subproblems of the recurence formula below be reduced to one subproblem given the assumtions stated in the description below

In CLRS (Intro to algorithms 3rd Edition) on page 362, it says eqn(1) : Lets Assume that you are given the cost of matrix multiplication for $$A_{i}..A_{j}$$ is $$C[i,j]$$ .
$$C[i,j]$$ is the Number of scalar multiplications of $$A_{i}..A_{j}$$.
$$C[i,j]$$ is not necessarily the optimal Number of scalar multiplications but obtained from this paranthesisation.
i.e $$C[i,j]$$ is derived from => $$(A_{i}(A_{i+1}(A_{i+2}\dots(A_{j-2}(A_{j-1}A_{j}))...)$$
Given that $$C[i,j]$$ has been provided to us. Can $$m[i,j]$$ be reduced the equation below.

$$m[i,j] = \begin{cases} 0 & \text{if } i = j, \\ \displaystyle\min_{i\le k\le j} \{C[i,k]+m[k+1,j] + p_{i-1}p_{k}p_{j}\} & \text{if } i \leq j. \end{cases}$$

Note $$C[i,j]$$ could be any parenthesizations so far as we are consistent.
I truly appreciate any help.