Is Regular i Sup
Cslib.Language.IsRegular.iSup
Plain-language statement
The union of any finite number of regular languages is regular.
Exact Lean statement
@[simp]
theorem IsRegular.iSup {I : Type*} [Finite I] {s : Set I} {l : I → Language Symbol}
(h : ∀ i ∈ s, (l i).IsRegular) : (⨆ i ∈ s, l i).IsRegularFormal artifact
Lean source
@[simp]theorem IsRegular.iSup {I : Type*} [Finite I] {s : Set I} {l : I → Language Symbol} (h : ∀ i ∈ s, (l i).IsRegular) : (⨆ i ∈ s, l i).IsRegular := by generalize h_n : s.ncard = n induction n generalizing s case zero => obtain ⟨rfl⟩ := (ncard_eq_zero (s := s)).mp h_n simp only [mem_empty_iff_false, not_false_eq_true, iSup_neg, iSup_bot] exact IsRegular.zero case succ n h_ind => obtain ⟨i, t, h_i, rfl, rfl⟩ := (ncard_eq_succ (s := s)).mp h_n rw [iSup_insert] apply IsRegular.add <;> grind- Project
- Lean Computer Science Library
- License
- Apache-2.0
- Commit
- f36649cff2c9
- Source
- Cslib/Computability/Languages/RegularLanguage.lean:140-152
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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.