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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 34 35 36 37 38 39 40 41 42 43 44 45 | import Mathlib
namespace OAI
namespace GeneralizedStarHeight
universe u
inductive Expression (Alphabet : Type u) : Type u
| zero : Expression Alphabet
| one : Expression Alphabet
| letter : Alphabet → Expression Alphabet
| union : Expression Alphabet → Expression Alphabet → Expression Alphabet
| concat : Expression Alphabet → Expression Alphabet → Expression Alphabet
| compl : Expression Alphabet → Expression Alphabet
| star : Expression Alphabet → Expression Alphabet
namespace Expression
open scoped Computability
def language {Alphabet : Type u} : Expression Alphabet → Language Alphabet
| .zero => 0
| .one => 1
| .letter a => {[a]}
| .union P Q => P.language + Q.language
| .concat P Q => P.language * Q.language
| .compl P => P.languageᶜ
| .star P => P.language∗
def height {Alphabet : Type u} : Expression Alphabet → ℕ
| .zero | .one | .letter _ => 0
| .union P Q | .concat P Q => max P.height Q.height
| .compl P => P.height
| .star P => 1 + P.height
end Expression
def HasHeightAtMost {Alphabet : Type u} (L : Language Alphabet) (n : ℕ) : Prop :=
∃ P : Expression Alphabet, P.language = L ∧ P.height ≤ n
end GeneralizedStarHeight
end OAI
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