Equivalence rules for LTL - Getting stuck working with $ \square \lozenge $ & Until ($\textsf{ U}$ ) operators
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04-11-2019 - |
Question
Need to prove equivalence for (or disprove equivalence for):
$ \hspace{1cm}\square ϕ → \lozenge ψ ≡ ϕ\textsf{ U }(ψ ∨ ¬ϕ) \\ $
My current attempt using the LTL equivalnce rules to determine equivalence:
$ \square ϕ → \lozenge ψ \\ ≡\ ¬\square ϕ \lor \lozenge ψ \hspace{1cm}(\textsf{"Mutual implication"} ϕ →ψ≡¬ϕ\lor ψ )\\ ≡ \lozenge ¬ϕ \lor \lozenge ψ \hspace{1cm}(\textsf{"Duality"} ¬\square≡\lozenge ¬ )\\ ≡ \lozenge (¬ϕ \lor ψ) \hspace{1cm}(\textsf{"Distributive law on"}\lozenge )\\ ≡true \textsf{ U} (¬ϕ \lor ψ) \hspace{1cm}(\textsf{"Def of"}\lozenge )\\ ≡true \textsf{ U} (ψ \lor ¬ϕ) \hspace{1cm}(\textsf{"Symmetry"}\lor)\\ $
After this I got stuck. I am not sure how to proceed from here to get the right side. Are other propositional logic rules allowed to be used in LTL, like for example identity disjunction?
No correct solution