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b797a5b | 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 | import Mathlib
namespace OAI
namespace PiExponent
def GoodRationalApproximations (x ν : ℝ) : Set ℚ :=
{r | 2 ≤ r.den ∧ 0 < |x - (r : ℝ)| ∧ |x - (r : ℝ)| < (r.den : ℝ) ^ (-ν)}
def ApproximationExponents (x : ℝ) : Set ℝ :=
{ν | 0 < ν ∧ (GoodRationalApproximations x ν).Infinite}
noncomputable def irrationalityExponent (x : ℝ) : ℝ := sSup (ApproximationExponents x)
def EventualLowerBound (x : ℝ) : Prop :=
∀ ν : ℝ, 2 < ν → ∃ Q : ℕ, 2 ≤ Q ∧
∀ (p : ℤ) (q : ℕ), Q ≤ q → (q : ℝ) ^ (-ν) ≤ |x - (p : ℝ) / (q : ℝ)|
def PiEventualLowerBound : Prop := EventualLowerBound Real.pi
def IntegerEventualLowerBound (x : ℝ) : Prop :=
∀ ν : ℝ, 2 < ν → ∃ Q : ℤ, 2 ≤ Q ∧
∀ (p q : ℤ), Q ≤ q → (q : ℝ) ^ (-ν) ≤ |x - (p : ℝ) / (q : ℝ)|
end PiExponent
end OAI
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