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Thanks, that makes sense! Could you say a little about why the weak union axiom holds? I've been struggling to prove that from your definitions. I was hoping that  would hold, but I don't think that  satisfies the second condition in the definition of conditional history for .

I'm confused by the definition of conditional history, because it doesn't seem to be a generalisation of history. I would expect , but both of the conditions in the definition of  are vacuously true if . This is independent of what  is, so . Am I missing something?