Choueka lang omega power eq omega limit
Automata.choueka_lang_omega_power_eq_omega_limit
Project documentation
If the language accepted by M is of the form V∗, then V^ω = V∗ * (M.ChouekaLang acc)↗ω. Note that this theorem does need to assume that M is finite-state.
Exact Lean statement
theorem choueka_lang_omega_power_eq_omega_limit [Inhabited A]
{M : DA A} [Finite M.State] {acc : Set M.State}
{V : Set (List A)} (h_lang : M.toNA.AcceptedLang acc = V∗) :
V^ω = V∗ * (M.ChouekaLang acc)↗ωFormal artifact
Lean source
theorem choueka_lang_omega_power_eq_omega_limit [Inhabited A] {M : DA A} [Finite M.State] {acc : Set M.State} {V : Set (List A)} (h_lang : M.toNA.AcceptedLang acc = V∗) : V^ω = V∗ * (M.ChouekaLang acc)↗ω := by apply Subset.antisymm · apply choueka_lang_omega_power_subset_omega_limit h_lang · have h1 : (M.ChouekaLang acc)↗ω ⊆ (V∗)^ω := by rw [← h_lang] ; apply choueka_lang_omega_limit_subset_omega_power have h2 : V∗ * (M.ChouekaLang acc)↗ω ⊆ V∗ * (V∗)^ω := by exact ConcatInf_mono (by simp) h1 rw [IterOmega_IterStar, ConcatInf_IterStar_IterOmega] at h2 exact h2- Project
- Automata Theory
- License
- Apache-2.0
- Commit
- f196548710ce
- Source
- AutomataTheory/Languages/ChouekaLemma.lean:217-228
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Related declarations
Acc lang congr
acc_lang_congr
Plain-language statement
The language accepted by c.toDA with a unique accepting state s is exactly the equivalence class of s.
Source project: Automata Theory
Person-level attribution pending.
Acc lang concat e
Automata.acc_lang_concat_e
Plain-language statement
The language of the concatenation NA that is accepted by M0's accepting states is the language accepted by M0.
Source project: Automata Theory
Person-level attribution pending.
Acc lang concat ne
Automata.acc_lang_concat_ne
Plain-language statement
The language of the concatenation NA that is accepted by M1's accepting states is the language accepted by M0 concatenated with the language accepted by M1 minus the empty word.
Source project: Automata Theory
Person-level attribution pending.