Na Accepted Lang of Fin Accept
Automata.na_AcceptedLang_of_FinAccept'
Project documentation
The following theorem shows that under the assumption that the alphabet type A is inhabited, the definitions using infinite sequences and those using finite sequences actually define the same notion of the accepted language of an automaton.
Exact Lean statement
theorem na_AcceptedLang_of_FinAccept' [Inhabited A] :
M.AcceptedLang acc = { al | ∃ n as, M.FinAccept' acc n as ∧ List.ofFn as = al }Formal artifact
Lean source
theorem na_AcceptedLang_of_FinAccept' [Inhabited A] : M.AcceptedLang acc = { al | ∃ n as, M.FinAccept' acc n as ∧ List.ofFn as = al } := by rw [NA.AcceptedLang, Set.ext_iff] ; intro al ; constructor · rintro ⟨n, as, h_acc, rfl⟩ use n, (fun k : Fin n ↦ as k) constructor · exact na_FinAccept'_of_FinAccept h_acc · simp [extract_eq_ofFn] · rintro ⟨n, as, h_acc, h_al⟩ use n, (fun k ↦ if h : k < n then as ⟨k, h⟩ else default) constructor · exact na_FinAccept_of_FinAccept' h_acc · simpa [extract_eq_ofFn]- Project
- Automata Theory
- License
- Apache-2.0
- Commit
- f196548710ce
- Source
- AutomataTheory/Automata/Basic.lean:126-138
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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.