I am having difficulty understanding the recursive formula for the posterior distribution $p(x_{0:t+1}|y_{1:t+1})$ of the state space model in which we assume $x_t$ is Markovian and $y_t$ are conditionally independent given the process $x_t$.
The recursive formula for the posterior is given by $$p(x_{0:t+1}|y_{1+t+1}) = p(x_{0:t}|y_{1:t})\frac{p(y_{t+1}|x_{t+1})p(x_{t+1}|x_t)}{p(y_{t+1}|y_{1:t})}$$
and for the predictive distribution is given by $$p(x_t|y_{1:t-1})=\int p(x_t | x_{t-1})p(x_{t-1}|y_{1:t-1})dx_{t-1};$$ and the marginal distribution by
$$p(x_t|y_{1:t})=\frac{p(y_t|x_t)p(x_t|y_{1:t-1})}{\int p(y_t|x_t)p(x_t|y_{1:t-1})dx_t}.$$
However, I can't write down the equality based on the definition of conditional probabilities.
For the first one we have on the right side, $\frac{p(x_{0:t},y_{1:t})}{p(y_{1:t})}p(x_{t+1},y_{t+1})p(x_t, x_{t+1}) \frac{1}{p(x_t)} p(y_{1:t})\frac{1}{p(y_{1:t+1})}=\frac{p(x_{0:t},y_{1:t})}{p(y_{1:t})}p(x_{t+1},y_{t+1})p(x_t, x_{t+1})\frac{1}{p(y_{1:t+1})}.$ But how does this equal the left hand side?
For the second one, we get $\int \frac{p(x_{t-1},x_t)}{p(x_{t-1})} \frac{p(x_{t-1}, y_{1:t-1})}{p(y_{1:t-1})}d_{x_{t-1}}$. How does this equal $\frac{p(x_t, y_{1:t-1})}{p(y_{1:t-1})}$?
Similarly, I cannot derive the third one into the first.
I would greatly appreciate if anyone could show me how these equations hold.