I Prod run iff
Cslib.Automata.NA.iProd_run_iff
Plain-language statement
Every run of the product automaton projects onto runs of its component automata, and vice versa.
Exact Lean statement
@[simp, scoped grind =]
theorem iProd_run_iff {na : (i : I) → NA (State i) Symbol}
{xs : ωSequence Symbol} {ss : ωSequence (Π i, State i)} :
(iProd na).Run xs ss ↔ ∀ i, (na i).Run xs (ss.map (· i))Formal artifact
Lean source
@[simp, scoped grind =]theorem iProd_run_iff {na : (i : I) → NA (State i) Symbol} {xs : ωSequence Symbol} {ss : ωSequence (Π i, State i)} : (iProd na).Run xs ss ↔ ∀ i, (na i).Run xs (ss.map (· i)) := by rw [iProd] constructor · rintro ⟨h_start, h_trans⟩ simp only [mem_iInter] at h_start grind [Run] · intro h constructor · simp only [mem_iInter] grind only [Run, = mem_preimage, Run.mk, = ωSequence.head_map] · intro n i exact (h i).trans n- Project
- Lean Computer Science Library
- License
- Apache-2.0
- Commit
- f36649cff2c9
- Source
- Cslib/Computability/Automata/NA/Prod.lean:31-45
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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.