feat: prove that an omega-language is regular iff it is a finite union of omega-languages of a special form #249
+521
−60
Conversation
This file contains hidden or bidirectional Unicode text that may be interpreted or compiled differently than what appears below. To review, open the file in an editor that reveals hidden Unicode characters.
Learn more about bidirectional Unicode characters
Contributor
Author
|
Rebase on #241. |
Contributor
Author
|
Rebase on #241. |
…n of omega-languages of a special form
Sign up for free
to join this conversation on GitHub.
Already have an account?
Sign in to comment
2 participants
Add this suggestion to a batch that can be applied as a single commit.
This suggestion is invalid because no changes were made to the code.
Suggestions cannot be applied while the pull request is closed.
Suggestions cannot be applied while viewing a subset of changes.
Only one suggestion per line can be applied in a batch.
Add this suggestion to a batch that can be applied as a single commit.
Applying suggestions on deleted lines is not supported.
You must change the existing code in this line in order to create a valid suggestion.
Outdated suggestions cannot be applied.
This suggestion has been applied or marked resolved.
Suggestions cannot be applied from pending reviews.
Suggestions cannot be applied on multi-line comments.
Suggestions cannot be applied while the pull request is queued to merge.
Suggestion cannot be applied right now. Please check back later.
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 PRs #239 and #241.