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Project-declaredLean 4.33.0-rc1 · mathlib@169c26b52a38

Concat language eq

Cslib.Automata.NA.Buchi.concat_language_eq

Plain-language statement

The Buchi automaton formed from concat na1 na2 accepts the ω-language that is the concatenation of the language of na1 and the ω-language of na2.

Exact Lean statement

@[simp]
theorem concat_language_eq {acc2 : Set State2} :
    language (Buchi.mk (concat na1 na2) (inr '' acc2)) =
    language na1 * language (Buchi.mk na2 acc2)

Formal artifact

Lean source

Canonical source
Full Lean sourceLean 4
@[simp]theorem concat_language_eq {acc2 : Set State2} :    language (Buchi.mk (concat na1 na2) (inr '' acc2)) =    language na1 * language (Buchi.mk na2 acc2) := by  ext xs  constructor  · rintro ss, h_run, h_acc    obtain k, h_k :  k, (ss k).isRight := by grind [Frequently.exists h_acc]    obtain n, _, h_acc1, ss2, h_run2, h_map2 := concat_run_proj h_run h_k    use xs.take n, h_acc1, xs.drop n, ?_, by simp    use ss2, h_run2    grind [(drop_frequently_iff_frequently n).mpr h_acc]  · rintro xs1, h_xs1, xs2, ss2, h_run2, h_acc2, rfl    obtain ss, h_run, h_map := concat_run_exists h_xs1 h_run2    use ss, h_run    apply (drop_frequently_iff_frequently xs1.length).mp    grind
Project
Lean Computer Science Library
License
Apache-2.0
Commit
f36649cff2c9
Source
Cslib/Computability/Automata/NA/Concat.lean:124-140

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Related declarations

Project-declaredLean 4.33.0-rc1

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.

computer sciencecomputabilityprogram semantics

Source project: Lean Computer Science Library

Person-level attribution pending.

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Project-declaredLean 4.33.0-rc1

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.

computer sciencecomputabilityprogram semantics

Source project: Lean Computer Science Library

Person-level attribution pending.

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