In the OpenStax Calculus III book ( page 287-286) , an example is given of arc length parametrization.
The original function is : $ r(t)= 4\cos(t)$i$+4\sin(t)$j.
The arc length function is : $s(t)= 4t$, implying that $t=s/4$.
Then comes the substitution : $r(s)= 4\cos(s/4)$i$+4\sin(s/4)$j.
It looks as though the substitution were not uniform. I cannot understand why on the LHS the parameter $t$ is replaced by $s$, while on the RHS, it is $s/4$ that is substituted for $t$.
Sure I'm missing something, but I cannot see what.
You're not missing anything. If that's what they wrote, it is just an abuse of notation. We are defining a new function $r_0(s):=r\left(\frac{s}{4}\right)$. They are abusing notation by calling $r_0$ and $r$ (which are completely different functions) by the same letter $r$.