Pair lang regular
Automata.pair_lang_regular
Plain-language statement
If M is finite-state, then M.PairLang s s' is regular for any pair of states s and s'.
Exact Lean statement
theorem pair_lang_regular [Inhabited A] [h_fin : Finite M.State] {s s' : M.State} :
RegLang (M.PairLang s s')Formal artifact
Lean source
theorem pair_lang_regular [Inhabited A] [h_fin : Finite M.State] {s s' : M.State} : RegLang (M.PairLang s s') := by use (M.SingleInit s), {s'} ; constructor · assumption ext al ; constructor · rintro ⟨n, as, ⟨ss, ⟨h_init, h_next⟩, rfl⟩, rfl⟩ use ss ; apply pair_path_fin_run.mpr simp [NA.FinRun, h_init, length_extract] intro k h_k have h1 : k < (as.extract 0 n).length := by simp [length_extract, h_k] simp (disch := omega) [padDefault_elt_left h1, get_extract', h_next] · rintro ⟨ss, h_path⟩ use al.length, al.padDefault ; simp [extract_padDefault] use ss ; exact pair_path_fin_run.mp h_path- Project
- Automata Theory
- License
- Apache-2.0
- Commit
- f196548710ce
- Source
- AutomataTheory/Automata/Pair.lean:260-273
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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.