Ramana Kumar comments on Eight Definitions of Observability

[Ramana Kumar]

Let $C\simeq {\text{Assume}}_{S_1}\left(C\right)\&\dots \&{\text{Assume}}_{S_n}\left(C\right)$. Thus, we also have that $C\simeq {\text{Assume}}_{S_1\cup S_2}\left(C\right)\&{\text{Assume}}_{S_3}\left(C\right)\&\dots \&{\text{Assume}}_{S_n}\left(C\right)$

I’m not seeing why this follows. I’ll look for a counterexample, but in the meantime maybe there’s a simple explanation for why we can combine the product of two assumes as an assume of the union? (I think the only relevant assumption in this context is that the $S_i$s partition the world; but I might be missing some other important assumption.)

EDIT: I can see how maybe this will follow from the definition of observability of a partition from subsets (which we are also assuming) and the fact that $Obs$ is closed under union… will try to figure that out. -- Yep I think this works out. Sorry for the confusion.