Let $(M_t)$ be a martingale w.r.t. $(\mathcal F_t)_{t\geq 0}$ and $\tau$ is a stopping time. I must prove that $(M_{t\wedge \tau})$ is a martingale. What should I prove ? That for $s<t$ $$\mathbb E[M_{t\wedge \tau}\mid \mathcal F_s]=M_{s\wedge \tau}\quad \text{or}\quad \mathbb E[M_{t\wedge \tau}\mid \mathcal F_{s\wedge \tau}]=M_{s\wedge \tau} \ \ \ ?$$
I guess it's the second one, but in several post on MSE I saw that they tried to prove that $\mathbb E[M_{t\wedge \tau}\mid \mathcal F_s]$, so I'm a bit confused. What do you think ?
So, what I tried : Let $s<t$ and $F\in \mathcal F_{s\wedge \tau}$. Then \begin{align*} \mathbb E[\mathbb E[M_{t\wedge \tau}\mid \mathcal F_{s\wedge \tau}]\boldsymbol 1_F]&=\mathbb E[M_{\tau}\boldsymbol 1_{F\cap \{\tau\leq t\}}]+\mathbb E[M_t\boldsymbol 1_{F\cap \{\tau>t\}}].\\ \end{align*} How can I continue ?