Loop run exists
Cslib.Automata.NA.loop_run_exists
Plain-language statement
For any infinite sequence xls of nonempty finite words from language na, there is an infinite run of na.loop corresponding to xls.flatten in which the state () marks the boundaries between the finite words in xls.
Exact Lean statement
theorem loop_run_exists [Inhabited Symbol] {xls : ωSequence (List Symbol)}
(h : ∀ k, (xls k) ∈ language na - 1) :
∃ ss, na.loop.Run xls.flatten ss ∧ ∀ k, ss (xls.cumLen k) = inl ()Formal artifact
Lean source
theorem loop_run_exists [Inhabited Symbol] {xls : ωSequence (List Symbol)} (h : ∀ k, (xls k) ∈ language na - 1) : ∃ ss, na.loop.Run xls.flatten ss ∧ ∀ k, ss (xls.cumLen k) = inl () := by let ts := ωSequence.const (inl () : Unit ⊕ State) have h_mtr (k : ℕ) : na.loop.MTr (ts k) (xls k) (ts (k + 1)) := by grind [loop_fin_run_mtr] have (k : ℕ) : xls k ≠ [] := by grind have h_pos (k : ℕ) : (xls k).length > 0 := List.length_pos_iff.mpr (this k) obtain ⟨ss, _, _⟩ := LTS.OmegaExecution.flatten_mTr h_mtr h_pos use ss grind [Run.mk, FinAcc.loop, cumLen_zero (ls := xls)]- Project
- Lean Computer Science Library
- License
- Apache-2.0
- Commit
- f36649cff2c9
- Source
- Cslib/Computability/Automata/NA/Loop.lean:126-135
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Related declarations
Unique minimal
Cslib.Automata.DA.FinAcc.unique_minimal
Plain-language statement
The minimal DFA M accepting the language l is unique up to unique isomorphism.
Source project: Lean Computer Science Library
Person-level attribution pending.
Buchi Family cover
Cslib.Automata.NA.Buchi.buchiFamily_cover
Project documentation
na.buchiFamily is a cover if na has only finitely many states. This theorem uses the Ramsey theorem for infinite graphs and does not depend on any details of na.BuchiCongruence other than that it is of finite index.
Source project: Lean Computer Science Library
Person-level attribution pending.
Buchi Family saturation
Cslib.Automata.NA.Buchi.buchiFamily_saturation
Plain-language statement
na.buchiFamily saturates the ω-language accepted by na.
Source project: Lean Computer Science Library
Person-level attribution pending.