feat: prove that an omega-language is regular iff it is a finite union of omega-languages of a special form #249
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Rebase on #241. |
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Rebase on #241. |
…n of omega-languages of a special form
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Rebase on #241 . |
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This PR proves that an ω-language is regular iff it is the finite union of ω-languages of the form
L * M^ω, where allLs andMs are regular languages. In addition to being of independent interest, thiss result will also be used in proving that ω-regular languages are closed under complementation, where the complement of an ω-regular language will be given as a finite union of this form. As part of this PR, we prove the theoremLTS.ωTr.flatten, which says that an infinite sequence of finite executions of a LTS connecting an infinite sequence of intermediate states can be concatenated into an infinite execution of the LTS. We also use this theorem to simplify the proof ofloop_run_existsin Loop.lean.This PR depends on the PR #241.