openAI-math-dataset / lean /OAI /NumberTheory /PiExponent /Approximation /ApproximationSelection.lean
| import Mathlib | |
| namespace OAI | |
| namespace PiExponent | |
| theorem exists_large_log_approximation | |
| (nu X : ℝ) (hnu : 0 < nu) | |
| (hbad : ∀ Q : ℕ, ∃ p : ℤ, ∃ q : ℕ, | |
| Q ≤ q ∧ |Real.pi - (p : ℝ) / q| ≤ (q : ℝ) ^ (-nu)) : | |
| ∃ p : ℤ, ∃ q : ℕ, | |
| 2 ≤ q ∧ X < Real.log q ∧ p ≠ 0 ∧ | |
| |Real.pi - (p : ℝ) / q| ≤ (q : ℝ) ^ (-nu) := by | |
| obtain ⟨N, hN⟩ := exists_nat_gt (Real.exp X) | |
| obtain ⟨p, q, hq, happrox⟩ := hbad (max 2 N) | |
| have hq2 : 2 ≤ q := le_trans (le_max_left _ _) hq | |
| have hNq : N ≤ q := le_trans (le_max_right _ _) hq | |
| have hexp : Real.exp X < (q : ℝ) := lt_of_lt_of_le hN (by exact_mod_cast hNq) | |
| have hlog : X < Real.log q := by | |
| simpa using Real.log_lt_log (Real.exp_pos X) hexp | |
| have hp : p ≠ 0 := by | |
| intro hp0 | |
| have hpow : (q : ℝ) ^ (-nu) ≤ 1 := | |
| Real.rpow_le_one_of_one_le_of_nonpos (by exact_mod_cast (by omega : 1 ≤ q)) | |
| (by linarith) | |
| have hpi : Real.pi ≤ (q : ℝ) ^ (-nu) := by | |
| simpa [hp0, abs_of_pos Real.pi_pos] using happrox | |
| have hpi2 := Real.pi_gt_three | |
| linarith | |
| exact ⟨p, q, hq2, hlog, hp, happrox⟩ | |
| end PiExponent | |
| end OAI | |