Is Regular compl
Cslib.ωLanguage.IsRegular.compl
Plain-language statement
The complementation of an ω-regular language is ω-regular.
Exact Lean statement
@[simp]
theorem IsRegular.compl {Symbol : Type} [Inhabited Symbol] {p : ωLanguage Symbol}
(h : p.IsRegular) : (pᶜ).IsRegularFormal artifact
Lean source
@[simp]theorem IsRegular.compl {Symbol : Type} [Inhabited Symbol] {p : ωLanguage Symbol} (h : p.IsRegular) : (pᶜ).IsRegular := by obtain ⟨State, h_fin, na, rfl⟩ := h have : Finite (Quotient na.BuchiCongruence.eq) := buchiCongruence_fin_index have h_sat := buchiFamily_saturation (na := na) have h_cov := buchiFamily_cover (na := na) apply IsRegular.fin_cover_saturates_compl h_sat h_cov have := Language.IsRegular.congr_fin_index (c := na.BuchiCongruence) grind [buchiFamily, IsRegular.hmul, IsRegular.omegaPow]- Project
- Lean Computer Science Library
- License
- Apache-2.0
- Commit
- f36649cff2c9
- Source
- Cslib/Computability/Languages/OmegaRegularLanguage.lean:248-257
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Related declarations
Unique minimal
Cslib.Automata.DA.FinAcc.unique_minimal
Plain-language statement
The minimal DFA M accepting the language l is unique up to unique isomorphism.
Source project: Lean Computer Science Library
Person-level attribution pending.
Buchi Family cover
Cslib.Automata.NA.Buchi.buchiFamily_cover
Project documentation
na.buchiFamily is a cover if na has only finitely many states. This theorem uses the Ramsey theorem for infinite graphs and does not depend on any details of na.BuchiCongruence other than that it is of finite index.
Source project: Lean Computer Science Library
Person-level attribution pending.
Buchi Family saturation
Cslib.Automata.NA.Buchi.buchiFamily_saturation
Plain-language statement
na.buchiFamily saturates the ω-language accepted by na.
Source project: Lean Computer Science Library
Person-level attribution pending.