therefore
Sign up
Open navigation
All proofs
Project-declaredLean 4.33.0-rc1 · mathlib@169c26b52a38

Is Regular fin cover saturates

Cslib.ωLanguage.IsRegular.fin_cover_saturates

computer sciencecomputabilityprogram semanticsomega regular language

Plain-language statement

If an ω-language has a finite saturating cover made of ω-regular languages, then it is an ω-regular language.

Exact Lean statement

theorem IsRegular.fin_cover_saturates {I : Type*} [Finite I]
    {p : I → ωLanguage Symbol} {q : ωLanguage Symbol}
    (hs : Saturates (fun i ↦ (p i).toSet) q.toSet)
    (hc : ⨆ i, p i = ⊤) (hr : ∀ i, (p i).IsRegular) : q.IsRegular

Formal artifact

Lean source

Canonical source
Full Lean sourceLean 4
theorem IsRegular.fin_cover_saturates {I : Type*} [Finite I]    {p : I → ωLanguage Symbol} {q : ωLanguage Symbol}    (hs : Saturates (fun i ↦ (p i).toSet) q.toSet)    (hc : ⨆ i, p i = ⊤) (hr : ∀ i, (p i).IsRegular) : q.IsRegular := by  suffices hu : q = ⨆ i ∈ {i | ((p i) ⊓ q).toSet.Nonempty}, (p i) by    rw [hu]    apply IsRegular.iSup    grind  have hc' : ⋃ i, (p i).toSet = univ := by grind [iSup_def, top_def]  have hq := saturates_eq_biUnion hs hc'  ext : 1  rw [hq]  simp [iSup_def, inf_def]
Project
Lean Computer Science Library
License
Apache-2.0
Commit
f36649cff2c9
Source
Cslib/Computability/Languages/OmegaRegularLanguage.lean:224-236

Reuse this declaration

Bring the exact result into your workflow

The import identifies the source module. Your project still needs the pinned package dependency shown on this page.

What this badge means

This completion status comes from the project or community source. It has not yet been represented here as an independent rebuild and axiom audit.

Continue in this project

Related declarations

Project-declaredLean 4.33.0-rc1

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.

computer sciencecomputabilityprogram semantics

Source project: Lean Computer Science Library

Person-level attribution pending.

View proof record
Project-declaredLean 4.33.0-rc1

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.

computer sciencecomputabilityprogram semantics

Source project: Lean Computer Science Library

Person-level attribution pending.

View proof record