| 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 | |