Question about a TLA derivation

[Andrew Helwer]

On page 68 of A Science of Concurrent Programs we have the following derivation:

□◇□F ≡ ¬◇¬¬□¬¬◇¬F ≡ ¬◇□◇¬F ≡ ¬□◇¬F ≡ ¬¬◇¬¬□¬¬F ≡ ◇□F

I am having trouble understanding this step in particular:

¬◇□◇¬F ≡ ¬□◇¬F

How is this step done? I remember getting stuck on this exact transformation the last time this was discussed a few months ago. Thanks!

Andrew Helwer

[Stephan Merz]

As written in the text, the chain of equivalences shows that

(1) <>[]<>F <=> []<>F

implies the equivalence

(2) []<>[]F <=> <>[]F.

The validity of equivalence (1) – which is the step that you are having trouble with – is argued informally, but you can justify it formally using the semantic definitions of [] and <>. Alternatively, use a PTL tautology checker.

Cheers,

Stephan

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[Andrew Helwer]

Thanks Stephan! Ach of course, I thought it was some transformation involving the ¬ but in fact it was just an identity proven on the previous page.

Andrew