I was going through the Dynamic Programming section of Introduction to Algorithms(2nd Edition) by Cormen et. al. where I came across the following recurrence relations in the assembly line scheduling portion.

$(1),(2),(3)$ are three relations as shown.

$$f_{1}(j) = begin{cases}

e_1+a_{1,1} &quadtext{if } j=1\

min(f_1(j-1)+a_{1,j},f_2(j-1)+t_{2,j-1}+a_{1,j})&quadtext{if} jgeq2\

end{cases}tag 1$$

Symmetrically,

$$f_{2}(j) = begin{cases}

e_2+a_{2,1} &quadtext{if } j=1\

min(f_2(j-1)+a_{2,j},f_1(j-1)+t_{1,j-1}+a_{2,j})&quadtext{if} jgeq2\

end{cases}tag 2$$

(where $e_i,a_{i,j},t_{2,j-1}$ are constants for $i=1,2$ and $j=1,2,3,…,n$)

$$f^star=min(f_1(n)+x_1,f_2(n)+x_2)tag 3$$

The text tries to find the recurrence relation of the number of times $f_i(j)$ ($i=1,2$ and $j=1,2,3,…,n$) is referenced if we write a mutual recursive code for $f_1(j)$ and $f_2(j)$. Let $r_i(j)$ denote the number of times $f_i(j)$ is referenced.

They say that,

From $(3)$,

$$r_1(n)=r_2(n)=1.tag4$$

From $(1)$ and $(2)$,

$$r_1(j)=r_2(j)=r_1(j+1)+r_2(j+1)tag 5$$

**I could not quite understand how the relations of $(4)$ and $(5)$ are obtained from the three corresponding relations.**

Thought I could make out intuitively that as there is only one place where $f_1(n)$ and $f_2(n)$ are called, which is in $f^star$, so probably in $(4)$ we get the required relation.

But as I had not encountered such concept before I do not quite know how to proceed. I would be grateful if someone guides me with the mathematical prove of the derivation as well as the intuition, however I would prefer an alternative to mathematical induction as it is a mechanical cookbook method without giving much insight into the problem though (but if in case there is no other way out, then I shall appreciate mathematical induction as well provided the intuition is explained to me properly).