$E_{q(x_1,x_2,...,x_N|x_0)}[log(\frac{p(x_t│x_{t+1} )}{q(x_t│x_{t-1})})] = E_{q(x_{t-1},x_t,x_{t+1}) |x_0)}[log(\frac{p(x_t│x_{t+1} )}{q(x_t│x_{t-1})})]$
Knowing that the joint distribution $q$ can be factorized as: $ q(x_1,x_2,...,x_N|x_0)=q(x_1|x_0)*q(x_2|x_1)*...*q(x_N,x_{N-1}) $
That is, like a Markov chain where each variable depends only on the previous one.
Why does the equivalence hold? I understand that the expected value does not depend on variables $x_i$ with $i>t+1$, but why doesn't it depend on the entire sequence of variables before $x_{t-1}$? And moreover, since it does not depend on the previous variables, why does it still depend on $x_0$
Thanks in advance, please it's important for me to understand deeply why.