Omega reg lang union
omega_reg_lang_union
Plain-language statement
ω-regular languages are closed under union.
Exact Lean statement
theorem omega_reg_lang_union {L0 L1 : Set (Stream' A)}
(h0 : OmegaRegLang L0) (h1 : OmegaRegLang L1) : OmegaRegLang (L0 ∪ L1)Formal artifact
Lean source
theorem omega_reg_lang_union {L0 L1 : Set (Stream' A)} (h0 : OmegaRegLang L0) (h1 : OmegaRegLang L1) : OmegaRegLang (L0 ∪ L1) := by obtain ⟨M0, acc0, h_fin0, h_l0⟩ := h0 obtain ⟨M1, acc1, h_fin1, h_l1⟩ := h1 let M_u : (i : Fin 2) → Automata.NA A | 0 => M0 | 1 => M1 let acc_u : (i : Fin 2) → Set (M_u i).State | 0 => acc0 | 1 => acc1 use (Automata.NA.Sum M_u), (Automata.NA.Sum_Acc M_u acc_u) constructor · have h_fin : ∀ i, Finite (M_u i).State := by simp [Fin.forall_fin_two, M_u, h_fin0, h_fin1] exact Finite.instSigma · ext as simp [h_l0, h_l1, Automata.acc_omega_lang_union M_u acc_u, Fin.exists_fin_two, M_u, acc_u]- Project
- Automata Theory
- License
- Apache-2.0
- Commit
- f196548710ce
- Source
- AutomataTheory/Languages/OmegaRegLang.lean:55-70
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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.