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760f90e | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 | import OAI.Computability.StarHeight.Recognition
namespace OAI
namespace GeneralizedStarHeight
universe u
theorem main {Alphabet : Type u} [Finite Alphabet] (L : Language Alphabet) (hL : L.IsRegular) :
HasHeightAtMost L 3 := by
classical
rcases isEmpty_or_nonempty Alphabet with h|h
· let : IsEmpty Alphabet := h
apply HasHeightAtMost.finite L _ 3
apply (Set.finite_singleton ([] : List Alphabet)).subset
intro w _
cases w with
| nil => rfl
| cons a w => exact isEmptyElim a
· let : Nonempty Alphabet := h
obtain ⟨M,hM,hfin,T,F,hTF⟩ := FiniteRecognition.of_regular hL
let : Monoid M := hM
let : Fintype M := hfin
have heq : L=({w | T (FreeMonoid.ofList w)∈F} : Language Alphabet) := by
ext w
exact hTF w
rw [heq]
exact FiniteRecognition.recognized_height_three T F
end GeneralizedStarHeight
end OAI
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