File size: 1,178 Bytes
a72760f | 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 | 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
|