I'm reading the wikipedia page on Markov property. It mentioned that the Markov property
$$P(X_t\in A|F_s)=P(X_t\in A| X_s)$$ for $s,t\in I$ with $s<t$, where $(F_s,s\in I)$ was a filtration
, given $S$ and $I$ being discrete(it mentioned $I=\mathbb{N}$ but I assumed it can be relaxed to just discrete), could be reformulated to
$$P(X_n=x_n|X_{n-1}=x_{n-1},...,X_0=x_0)=P(X_n=x_n|X_{n-1}=x_{n-1})$$
I don't get why the $s<t$ condition could be replaced by $X_n=x_n|X_{n-1}=x_{n-1}$. For example, the "growth model" evaluated by markov matrix seemed to satisfy the first equation but may dependent on $x_{t-1}, x_{t-2},...,x_{t-N}$. The regression analysis similarly also dependent on $x_{t-1}, x_{t-2},...,x_{t-N}$. They both satisfy the first definition but obviously more extensive than the reformulated version.
In continuous case(when consider this case I already had a hint of the reasoning but not quite sure), the first definition translated to $$\frac{d}{dt}P(X_t=x_t)\in \mathbb{C}$$, i.e. the probability being a $\mathbb{C}^1$ function?
How did the Markov property reformulated from $X_t|F_s$ to $x_t|x_{t-1}$?