Project documentation
Helper lemma for reg_lang_concat below.
Exact Lean statement
lemma reg_lang_concat_e {L0 L1 : Set (List A)}
(h0 : RegLang L0) (h1 : RegLang L1) (h_e : [] ∈ L1) : RegLang (L0 * L1)Formal artifact
Lean source
lemma reg_lang_concat_e {L0 L1 : Set (List A)} (h0 : RegLang L0) (h1 : RegLang L1) (h_e : [] ∈ L1) : RegLang (L0 * L1) := by obtain ⟨M0, acc0, h_fin0, h_l0⟩ := h0 obtain ⟨M1, acc1, h_fin1, h_l1⟩ := h1 use (M0.Concat acc0 M1), (inl '' acc0 ∪ inr '' acc1) constructor · exact Finite.instSum · have h_l1' : L1 = (L1 \ {[]}) ∪ {[]} := by symm ; apply Set.diff_union_of_subset ; simp [h_e] have h_l1'' : [] ∉ L1 \ {[]} := by simp rw [← h_l1] at h_l1' h_l1'' rw [← h_l0, ← h_l1, h_l1', ConcatFin_union_distrib, ConcatFin_epsilon, Automata.acc_lang_acc_union, Automata.acc_lang_concat_e, Automata.acc_lang_concat_ne, union_comm]- Project
- Automata Theory
- License
- Apache-2.0
- Commit
- f196548710ce
- Source
- AutomataTheory/Languages/RegLang.lean:166-178
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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.