In the course of a proof, I used the following principle, which seems so intuitive that it should have a name:

Suppose one has a stochastic process $X_t$, for $t \in \omega$, on a (possibly infinite) state space $S$. Suppose one has $n$ Boolean functions $A_i, B_i: S \rightarrow \{ 0, 1 \}$. Suppose that, for all $s \in S$ and all $t \in \omega$, one has the condition that $P(B_i(X_{t+1}) = 1 \mid X_t = s) \leq p_i$.

Now consider the following event $E$:

For all $i = 1, ..., n$, there is a unique time $t_i$ such that $A(X_{t_i}) = 1$;

For all $i \neq i'$ we have $t_i \neq t_{i'}$

For all $i = 1, ..., n$ we have $B(X_{t_i+1}) = 1$

Then the probability of the event $E$ is at most $p_1 p_2 \dots p_n$.

Is the principle already known? Would one consider it "obvious" ?